HYDRAULICS 
 
 BY 
 
 F. C. LEA, 
 
 B.Sc. (LONDON ENGINEERING), 
 
 SENIOR WHITWORTH SCHOLAR; ASSOC.R. COL. SO. ; A.M.INST. C. E.; 
 
 TELFORD PRIZEMAN ; LECTURER IN APPLIED MECHANICS AND ENGINEERING DESIGN 
 IN THE CITY AND GUILDS OF LONDON CENTRAL TECHNICAL COLLEQ2. 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 NEW YORK: 
 
 LONGMANS, GREEN & CO 
 LONDON: EDWARD ARNOLD 
 
 [All Rights reserved] 
 
'\ A** 1 ^ 
 
 OF THE 
 
 f UNIVERSITY ] 
 
 OF 
 C*L!F> 
 
 PREFACE. 
 
 WHEN the author undertook some time ago to write this 
 work, it was under the impression, which impression was 
 shared by many teachers, that a book was required by Engineering 
 students dealing with the subject of Hydraulics in a wider sense 
 than that covered by existing text books. In addition the author 
 felt that though several excellent text books were in existence, 
 the large amount of experimental research carried out during the 
 last 10 or 15 years, very little of which has been done in this 
 country, on the subject of the flow of water, had not received the 
 attention it deserved. The great developments in turbines and 
 centrifugal pumps also merited some notice. 
 
 An attempt has been made to embody the results of the latest 
 researches in the book, and to give sufficient details to indicate 
 the methods used in obtaining these results, especially in those 
 cases where such information and the references thereto, are 
 likely to prove of value to those desirous of carrying out ex- 
 periments on the flow of water. 
 
 Perhaps in no branch of Applied Science is it more difficult 
 to co-ordinate results and express them by general formulae than 
 in Hydraulics. Practical Engineers engaged in the design of 
 water channels frequently complain of the large differences they 
 obtain in the calculated dimensions of such channels by using 
 the formulae put forward by different authorities. Before any 
 formula can be used with assurance it is necessary to have some 
 knowledge of the data used in determining the empirical con- 
 stants in the formula. For this reason a little attention has been 
 given to the historical development of the formulae for determining 
 the flow in pipes and channels, and some particulars of the data 
 from which the constants were determined are given. In this 
 respect the logarithmic analysis of experimental data, especially 
 in Chapter YI, together with the plottings of Fig. 114 and the 
 references to experiments, will it is hoped be of assistance to 
 
IV PREFACE 
 
 engineers in enabling them to choose the coefficients suitable to 
 given circumstances, and it is further hoped that the methods of 
 analysis given will be educational and useful to students, and 
 helpful in the interpretation of experiments. 
 
 The chapter on the flow of water in pipes is arranged so that a 
 student who reads as far as section 93 should be able to solve 
 a large number of problems on flow of water in pipes, without 
 further reading. At the end of the chapter the formulae derived 
 in the chapter are summarised, and various kinds of practical 
 problems solved, and arithmetical examples worked out. In the 
 chapter on flow in channels the student who reads to section 119, 
 and then sections 124 and 129 should be able to follow the 
 problems at the end of the chapter, and to work the examples. 
 Chapter YIII enables the student who is desirous of studying 
 the elementary theory of the impact of water on vanes, and of 
 turbines, to do so apart from the details of turbines, and the more 
 practical problems that arise in connection with their design. 
 
 The principles of construction of the various types of turbines 
 are illustrated in Chapter IX by diagrams of the simpler and 
 older types, as well as by drawings of the more complicated 
 modern turbines. The drawings have been made to scale, and 
 in particular cases sufficient dimensions are given to enable the 
 student acquainted with the principles of machine design to 
 design a turbine. The author believes the analysis given of the 
 form of the vanes for mixed flow turbines and also for parallel 
 flow turbines is new. 
 
 The subject of centrifugal pumps is treated somewhat fully, 
 because of the complaint the author has often heard of the 
 difficulty engineers and students have in determining what the 
 performance of a centrifugal pump is likely to be under varying 
 conditions. The method of analysis of the losses at entrance and 
 exit as given in the text, the author believes, is due to Professor 
 Unwin, and he willingly acknowledges his obligation to him. 
 The general formula given in article 237 is believed to be new, 
 and the examples given of its application in sections 235, etc., 
 show that by such an equation, which may be called the character- 
 istic equation for the pump, the performance of the pump under 
 varying conditions can be approximately determined. 
 
 The effects of inertia forces in plunger pumps and the effect of 
 air vessels in diminishing these forces are only imperfectly treated, 
 as no attempt is made to deal with the variations of pressure in 
 the air vessel. Sufficient attention is however given to the 
 subject to emphasise the importance of it, and it is probably 
 treated as fully as is desirable, considered from a practical 
 
PREFACE v 
 
 engineering standpoint. The analysis of section 260, although 
 too refined for practical purposes, is of value to the student in 
 that, neglecting losses which cannot very well be determined, it 
 enables him to realise how the energy given as velocity head to 
 the water both in the cylinder and in the suction pipe is recovered 
 before the end of the stroke is reached. The examples given of 
 " Hydraulic Machines " have been chosen as types, and no attempt 
 has been made to introduce very special kinds of machines. The 
 author has had a wide experience of this class of machinery, and 
 he thinks the examples . illustrate sufficiently the principles and 
 practice of the design of such machines. 
 
 The last two chapters have been introduced in the hope that 
 they will be of assistance to University students, and to candidates 
 for the Institution of Civil Engineers examinations. 
 
 Mr Froude's experiments, on the frictional resistance of boards 
 moving through water, are considered in Chapter XII simply in 
 their relationship to the resistance of ships, and no attempt has 
 been made, as is frequently done, to use them to determine so- 
 called laws of fluid friction for water flowing in pipes and 
 channels. 
 
 The author hardly dares to hope that in the large amount of 
 arithmetical work involved in the exercises given, mistakes will 
 not have crept in, and he will be grateful if those discovering 
 mistakes will kindly point them out. 
 
 The author wishes to express his sincerest thanks to his 
 friend, Mr W. A. Taylor, Wh.Sc., A.R.C.S., for his kindness in 
 reading proofs, and for many valuable suggestions, and also to 
 Mr W. Hewson, B.Sc., who has kindly read through some of the 
 proofs. 
 
 To the following firms the author is under great obligation for 
 the ready way in which they acceded to his request for information : 
 
 Messrs Escher, Wyss and Co. of Zurich for drawings of 
 turbines and for loan of block of turbine filter. 
 
 Messrs Piccard, Pictet and Co. of Geneva for drawings of 
 turbines. 
 
 Messrs Worthington and Co. for drawings of centrifugal 
 pumps and for loan of block. 
 
 Messrs Fielding and Platt of Gloucester for drawings of 
 accumulator. 
 
 Messrs Tangye of Birmingham for drawings of pumps. 
 
 Messrs Glenfield and Kennedy of Kilmarnock for drawings of 
 meter and for loan of blocks. 
 
 Messrs G. W. Kent of London for description and loan of 
 blocks of Venturi meter recording gear. 
 
V] PREFACE 
 
 Messrs W. and L. E. Gurley of Troy, N.Y., U.S.A. for loan 
 of block of current meter. 
 
 Messrs Holden and Brooke of Manchester for drawing of 
 Leinert meter. 
 
 Messrs W. H. Bailey and Co. of Manchester for drawing of 
 hydraulic ram. 
 
 Messrs Armstrong, Whitworth and Co. for drawings of crane 
 valves. 
 
 Messrs Davy of Sheffield for loan of block of forging press. 
 
 F. C. LEA. 
 
 CENTRAL TECHNICAL COLLEGE, 
 November, 1907. 
 
CONTENTS. 
 CHAPTER I. 
 
 FLUIDS AT REST. 
 
 Introduction. Fluids and their properties. Compressible and incom- 
 pressible fluids. Density and specific gravity. Hydrostatics. Intensity 
 of pressure. The pressure at a point in a fluid is the same in all directions. 
 The pressure on any horizontal plane in a fluid must be constant. Fluids 
 at rest with free surface horizontal. Pressure measured in feet of water. 
 Pressure head. Piezometer tubes. The barometer. The differential gauge. 
 Transmission of fluid pressure. Total or whole pressure. Centre of 
 pressure. Diagram of pressure on a plane area. Examples . Page 1 
 
 CHAPTER II. 
 
 FLOATING BODIES. 
 
 Conditions of equilibrium. Principle of Archimedes. Centre of 
 buoyancy. Condition of stability of equilibrium. Small displacements. 
 Metacentre. Stability of rectangular pontoon. Stability of floating vessel 
 containing water. Stability of floating body wholly immersed in water. 
 Floating docks. Stability of floating dock. Examples . . Page 21 
 
 CHAPTER III. 
 
 FLUIDS IN MOTION. 
 
 Steady motion. Stream line motion. Definitions relating to flow of 
 water. Energy per pound of water passing any section in a stream line. 
 Bernouilli's theorem. Venturi meter. Steering of canal boats. Extension 
 of Bernouilli's theorem. Examples Page 37 
 
viii CONTENTS 
 
 CHAPTER IV. 
 
 FLOW OF WATER THROUGH ORIFICES AND OVER WEIRS. 
 
 Velocity of discharge from an orifice. Coefficient of contraction for 
 sharp-edged orifice. Coefficient of velocity for sharp-edged orifice. Bazin's 
 experiments on a sharp-edged orifice. Distribution of velocity in the plane 
 of the orifice. Pressure in the plane of the orifice. Coefficient of discharge. 
 Effect of suppressed contraction on the coefficient of discharge. The form 
 of the jet from sharp-edged orifices. Large orifices. Drowned orifices. 
 Partially drowned orifice. Velocity of approach. Coefficient of resistance. 
 Sudden enlargement of a current of water. Sudden contraction of a 
 current of water. Loss of head due to sharp-edged entrance into a pipe or 
 mouthpiece. Mouthpieces. Borda's mouthpiece. Conical mouthpieces 
 and nozzles. Flow through orifices and mouthpieces under constant 
 pressure. Time of emptying a tank or reservoir. Notches and weirs. 
 Rectangular sharp-edged weir. Derivation of the weir formula from that 
 of a large orifice. Thomson's principle of similarity. Discharge through 
 a trianglar notch by the principle of similarity. Discharge through a 
 rectangular weir by the principle of similarity. Rectangular weir with 
 end contractions. Bazin's formula for the discharge of a weir. Bazin's 
 and the Cornell experiments on weirs. Velocity of approach. Influence of 
 the height of the weir sill above the bed of the stream on the contraction. 
 Discharge of a weir when the air is not freely admitted beneath the nappe. 
 Form of the nappe. Depressed nappe. Adhering nappes. Drowned or 
 wetted nappes. Instability of the form of the nappe. Drowned weirs with 
 sharp crests. Vertical weirs of small thickness. Depressed and wetted 
 nappes for flat-crested weirs. Drowned nappes for flat-crested weirs. Wide 
 flat-crested weirs. Flow over dams. Form of weir for accurate gauging. 
 Boussinesq's theory of the discharge over a weir. Determining by ap- 
 proximation the discharge of a weir, when the velocity of approach is 
 unknown. Time required to lower the water in a reservoir a given distance 
 by means of a weir. Examples ....... Page 50 
 
 CHAPTER V. 
 
 FLOW THROUGH PIPES. 
 
 Resistances to the motion of a fluid in a pipe. Loss of head by friction. 
 Head lost at the entrance to the pipe. Hydraulic gradient and virtual 
 slope. Determination of the loss of head due to friction. Reynold's 
 apparatus. Equation of flow in a pipe of uniform diameter and determi- 
 nation of the head lost due to friction. Hydraulic mean depth. Empirical 
 
CONTENTS i x 
 
 formulae for loss of head due to friction. Formula of Darcy. Variation 
 of C in the formula v = G\ f mi with service. Ganguillet and Kutter's 
 formula. Reynold's experiments and the logarithmic formula. Critical 
 velocity. Critical velocity by the method of colour bands. Law of 
 frictional resistance for velocities above the critical velocity. The de- 
 termination of the values of C given in Table XII. Variation of fc, in the 
 formula i = kv n , with the diameter. Criticism of experiments. Piezometer 
 fittings. Effect of temperature on the velocity of flow. Loss of head due 
 to bends and elbows. Variations of the velocity at the cross section of a 
 cylindrical pipe. Head necessary to give the mean velocity v m to the 
 water in the pipe. Practical problems. Velocity of flow in pipes. Trans- 
 mission of power along pipes by hydraulic pressure. The limiting diameter 
 of cast iron pipes. Pressures on pipe bends. Pressure on a plate in a pipe 
 filled with flowing water. Pressure on a cylinder. Examples . Page 112 
 
 CHAPTER VI. 
 
 FLOW IN OPEN CHANNELS. 
 
 Variety of the forms of channels. Steady motion in uniform channels* 
 Formula for the flow when the motion is uniform in a channel of uniform 
 section and slope. Formula of Chezy. Formulae of Prony and Eytelwein. 
 Formula of Darcy and Bazin. Ganguillet and Kutter's formula. Bazin's 
 formula. Variations of the coefficient C. Logarithmic formula for flow in 
 channels. Approximate formula for the flow in earth channels. Distribu- 
 tion of velocity in the cross section of open channels. Form of the curve 
 of velocities on a vertical section. The slopes of channels and the velocities 
 allowed in them. Sections of aqueducts and sewers. Siphons forming 
 part of aqueducts. The best form of channel. Depth of flow in a circular 
 channel for maximum velocity and maximum discharge. Curves of velocity 
 and discharge for a channel. Applications of the formulae. Problems. 
 Examples Page 178 
 
 CHAPTER VII. 
 
 GAUGING THE FLOW OF WATER. 
 
 Measuring the flow of water by weighing. Meters. Measuring the flow 
 by means of an orifice. Measuring the flow in open channels. Surface 
 floats. Double floats. Rod floats. The current meter. Pitot tube. Cali- 
 bration of Pitot tubes. Gauging by a weir. The hook gauge. Gauging 
 the flow in pipes ; Venturi meter. Deacon's waste-water meter. Kennedy's 
 meter. Gauging the flow of streams by chemical means. Examples 
 
 Page 234 
 
X CONTENTS 
 
 CHAPTER VIII. 
 
 IMPACT OF WATER ON VANES. 
 
 Definition of vector. Sum of two vectors. Resultant of two velocities. 
 Difference of two vectors. Impulse of water on vanes. Relative velocity. 
 Definition of relative velocity as a vector. To find the pressure on a 
 moving vane, and the rate of doing work. Impact of water on a vane 
 when the directions of motion of the vane and jet are not parallel. 
 Conditions which the vanes of hydraulic machines should satisfy. 
 Definition of angular momentum. Change of angular momentum. Two 
 important principles. Work done on a series of vanes fixed to a wheel 
 expressed in terms of the velocities of whirl of the water entering and 
 leaving the wheel. Curved vanes. Pelton wheel. Force tending to move 
 a vessel from which water is issuing through an orifice. The propulsion 
 of ships by water jets. Examples Page 261 
 
 CHAPTER IX. 
 
 WATER WHEELS AND TURBINES. 
 
 Overshot water wheels. Breast wheel. Sagebien wheels. Impulse 
 wheels. Poncelet wheel. Turbines. Reaction turbines. Outward flow 
 turbines. Losses of head due to frictional and other resistances in outward 
 flow turbines. Some actual outward flow turbines. Inward flow turbines. 
 Some actual inward flow turbines. The best peripheral velocity for 
 inward and outward flow turbines. Experimental determination of the 
 best peripheral velocity for inward and outward flow turbines. Value of e 
 
 to be used in the formula = eR. The ratio of the velocity of whirl V to 
 
 \J 
 
 the velocity of the inlet periphery v. The velocity with which water 
 leaves a turbine. Bernouilli's equations for inward and outward flow 
 turbines neglecting friction. Bernouilli's equations for the inward and 
 outward flow turbines including friction. Turbine to develope a given 
 horse-power. Parallel or axial flow turbines. Regulation of the flow to 
 parallel flow turbines. Bernouilli's equations for axial flow turbines. 
 Mixed flow turbines. Cone turbine. Effect of changing the direction of 
 the guide blade, when altering the flow of inward flow and mixed flow 
 turbines. Effect of diminishing the flow through turbines on the velocity 
 of exit. Regulation of the flow by means of cylindrical gates. The Swain 
 gate. The form of the wheel vanes between the inlet and outlet of 
 turbines. The limiting head for a single stage reaction turbine. Series 
 or multiple stage reaction turbines. Impulse turbines. The form of the 
 vanes for impulse turbines, neglecting friction. Triangles of velocity for 
 an axial flow impulse turbine considering friction. Impulse turbine for 
 high head. Pelton wheel. Oil pressure governor or regulator. Water 
 pressure regulators for impulse turbines. Hammer blow in a long turbine 
 supply pipe. Examples p age 283 
 
CONTENTS XI 
 
 CHAPTER X. 
 
 PUMPS. 
 
 Centrifugal and turbine pumps. Starting centrifugal or turbine pumps. 
 Form of the vanes of centrifugal pumps. Work done on the water by the 
 wheel. Ratio of velocity of whirl to peripheral velocity. The kinetic energy 
 of the water at exit from the wheel. Gross lift of a centrifugal pump. 
 Efficiencies of a centrifugal pump. Experimental determination of the 
 efficiency of a centrifugal pump. Design of pump to give a discharge Q. 
 The centrifugal head impressed on the water by the wheel. Head-velocity 
 curve of a centrifugal pump at zero discharge. Variation of the discharge 
 of a centrifugal pump with the head when the speed is kept constant. 
 Bernouilli's equations applied to centrifugal pumps. Losses in centrifugal 
 pumps. Variation of the head with discharge and with the speed of a 
 centrifugal pump. The effect of the variation of the centrifugal head and 
 the loss by friction on the discharge of a pump. The effect of the diminu- 
 tion of the centrifugal head and the increase of the friction head as the 
 flow increases, on the velocity. Discharge curve at constant head. Special 
 
 U 2 
 arrangements for converting the velocity head -^- , with which the water 
 
 *9 
 
 leaves the wheel, into pressure head. Turbine pumps. Losses in the 
 spiral casings of centrifugal pumps. General equation for a centrifugal 
 pump. The limiting height to which a single wheel centrifugal pump can 
 be used to raise water. The suction of a centrifugal pump. Series or 
 multi-stage turbine pumps. Advantages of centrifugal pumps. Pump 
 delivering into a long pipe line. Parallel flow turbine pump. Inward flow 
 turbine pump. Reciprocating pumps. Coefficient of discharge of the 
 pump. Slip. Diagram of work done by the pump. The accelerations 
 of the pump plunger and the water in the suction pipe. The effect of 
 acceleration of the plunger on the pressure in the cylinder during the 
 suction stroke. Accelerating forces in the delivery pipe. Variation of 
 pressure in the cylinder due to friction. Air vessel on the suction pipe. 
 Air vessel on the delivery pipe. Separation during the suction stroke. 
 Negative slip. Separation in the delivery pipe. Diagram of work done 
 considering the variable quantity of water in the cylinder. Head lost at 
 the suction valve. Variation of the pressure in hydraulic motors due to 
 inertia forces. Worked examples. High pressure plunger pump. Tangye 
 Duplex pump. The hydraulic ram. Lifting water by compressed air. 
 Examples Page 392 
 
 CHAPTER XL 
 
 HYDRAULIC MACHINES. 
 
 Joints and packings used in hydraulic work. The accumulator. Dif- 
 ferential accumulator. Air accumulator. Intensifies. Steam intensifiers. 
 Hydraulic forging press. Hydraulic cranes. Double power cranes. 
 Hydraulic crane valves. Hydraulic press. Hydraulic riveter. Brother- 
 hood and Rigg hydraulic engines. Examples .... Page 485 
 
Xll CONTENTS 
 
 CHAPTER XII. 
 
 RESISTANCE TO THE MOTION OF BODIES IN WATER. 
 
 Froude's experiments on the resistance of thin boards. Stream line 
 theory of the resistance offered to motion of bodies in water. Determination 
 of the resistance of a ship from that of the model. Examples . Page 507 
 
 CHAPTER XIII. 
 
 STREAM LINE MOTION. 
 
 Hele Shaw's experiments. Curved stream line motion. Scouring of 
 river banks at bends Page 517 
 
 ANSWERS TO EXAMPLES . . . . . . . . Page 521 
 
 INDEX Page 525 
 
UNIVERSITY 
 
 OF 
 
 HYDRAULICS. 
 
 CHAPTER I. 
 
 FLUIDS AT BEST. 
 
 ERRATA. 
 
 Page 25, formula 3. Insert 6 after . 
 
 82, for (L-O'lN) substitute (L-O'INH). 
 Pages 122 and 148, for 192 substitute 197. 
 Page 309, line 21, substitute cos for cot/3. 
 
 444, 29, OC OD. 
 
 > > OK CK. 
 
 4L lines 19 and 20, interchange " full " and " dotted." 
 
 orifices in the sides of tanks and through short pipes, probably 
 * The Aqueducts of Rome. Frootinus, translated by Herschel. 
 
 L - B - l 
 
Xli CONTENTS 
 
 CHAPTER XII. 
 
 RESISTANCE TO THE MOTION OF BODIES IN WATER. 
 
 Fronde's experiments on the resistance of thin boards. Stream line 
 theory of the resistance offered to motion of bodies in water. Determination 
 of the resistance of a ship from that of the model. Examples . Page 507 
 
 CHAPTER XIII. 
 
 STREAM LINE MOTION. 
 
XV^^JN, 
 
 / OF THE A 
 
 f UNIVERSITY } 
 
 OF / 
 
 \^IFC*^/ 
 
 HYDRAULICS. 
 
 CHAPTER I. 
 
 FLUIDS AT BEST. 
 
 1. Introduction. 
 
 The science of Hydraulics in its limited sense as originally 
 understood, had for its object the consideration of the laws 
 regulating the flow of water in channels, but it has come to 
 have a wider significance, and it now embraces, in addition, the 
 study of the principles involved in the pumping of water and other 
 fluids and their application to the working of different kinds of 
 machines. 
 
 The practice of conveying water along artificially constructed 
 channels for irrigation and domestic purposes dates back into 
 great antiquity. The Egyptians constructed transit canals for 
 warlike purposes, as early as 3000 B.C., and works for the better 
 utilisation of the waters of the Nile were carried out at an even 
 earlier date. According to Josephus, the gardens of Solomon 
 were made beautiful by fountains and other water works. The 
 aqueducts of Rome*, some of which were constructed more than 
 2000 years ago, were among the " wonders of the world," and 
 to-day the city of Athens is partially supplied with water by 
 means of an aqueduct constructed probably some centuries before 
 the Christian era. 
 
 The science of Hydraulics, however, may be said to have only 
 come into existence at the end of the seventeenth century when 
 the attention of philosophers was drawn to the problems involved 
 in the design of the fountains, which came into considerable use 
 in Italian landscape gardens, and which, according to Bacon, 
 were of "great beauty and refreshment." The founders were 
 principally Torricelli and Marriott from the experimental, and 
 Bernouilli from the theoretical, side. The experiments of Torri- 
 celli and of Marriott to determine the discharge of water through 
 orifices in the sides of tanks and through short pipes, probably 
 
 * The Aqueducts of Rome. Frontinus, translated by Herschel. 
 L. H. 1 
 
2 HYDRAULICS 
 
 mark the first attempts to determine the laws regulating the 
 flow of water, and Torricelli's famous theorem may be said to 
 be the foundation of modern Hydraulics. But, as shown in the 
 chapter on the flow of water in pipes, it was not until a century 
 later that any serious attempt was made to give expression to the 
 laws regulating the flow in long pipes and channels, and practi- 
 cally the whole of the knowledge we now possess has been 
 acquired during the last century. Simple machines for the 
 utilisation of the power of natural streams have been made for 
 many centuries, examples of which are to be found in an interest- 
 ing work Hydrostatics and Hydraulics written in English by 
 Stephen Swetzer in 1729, but it has been reserved to the workers 
 of the nineteenth century to develope all kinds of hydraulic 
 machinery, and to discover the principles involved in their correct 
 design. Poncelet's enunciation of the correct principles which 
 should regulate the design of the "floats" or buckets of water 
 wheels, and Fourneyron's application of the triangle of velocities 
 to the design of turbines, marked a distinct advance,- but it must 
 be admitted that the enormous development of this class of 
 machinery, and the very high standard of efficiency obtained, is 
 the outcome, not of theoretical deductions, but of experience, 
 and the careful, scientific interpretation of the results of 
 experiments. 
 
 2. Fluids and their properties. 
 
 The name fluid is given, in general, to a body which offers 
 very small resistance to deformation, and which takes the shape 
 of the body with which it is in contact. 
 
 If a solid body rests upon a horizontal plane, a force is required 
 to move the body over the plane, or to overcome the friction 
 between the body and the plane. If the plane is very smooth 
 the force may be very small, and if we conceive the plane to be 
 perfectly smooth the smallest imaginable force would move the 
 body. 
 
 If in a fluid, a horizontal plane be imagined separating the 
 fluid into two parts, the force necessary to cause the upper 
 part to slide over the lower will be very small indeed, and 
 any force, however small, applied to the fluid above the plane 
 and parallel to it, will cause motion, or in other words will cause 
 a deformation of the fluid. 
 
 Similarly, if a very thin plate be immersed in the fluid in any 
 direction, the plate can be made to separate the fluid into two 
 parts by the application to the plate of an infinitesimal force, 
 and in the imaginary perfect fluid this force would be zero. 
 
FLUIDS AT REST 3 
 
 Viscosity. Fluids found in nature are not perfect and are 
 said to have viscosity; but when they are at rest the conditions 
 of equilibrium can be obtained, with sufficient accuracy, on 
 the assumption that they are perfect fluids, and that therefore 
 no tangential stresses can exist along any plane in a fluid. 
 This branch of the study of fluids is called Hydrostatics ; when 
 the laws of movement of fluids are considered, as in Hydraulics, 
 these tangential, or frictional forces have to be taken into 
 consideration. 
 
 3. Compressible and incompressible fluids. 
 
 There are two kinds of fluids, gases and liquids, or those which 
 are easily compressed, and those which are compressed with 
 difficulty. The amount by which the volumes of the latter are 
 altered for a very large variation in the pressure is so small that 
 in practical problems this variation is entirely neglected, and 
 they are therefore considered as incompressible fluids. 
 
 In this volume only incompressible fluids are considered, and 
 attention is confined, almost entirely, to the one fluid, water. 
 
 4. Density and specific gravity. 
 
 The density of any substance is the weight of unit volume at 
 the standard temperature and pressure. 
 
 The specific gravity of any substance at any temperature and 
 pressure is the ratio of the weight of unit volume to the weight 
 of unit volume of pure water at the standard temperature and 
 pressure. 
 
 The variation of the volume of liquid fluids, with the pressure, 
 as stated above, is negligible, and the variation due to changes of 
 temperature, such as are ordinarily met with, is so small, that in 
 practical problems it is unnecessary to take it into account. 
 
 In the case of water, the presence of salts in solution is of 
 greater importance in determining the density than variations 
 of temperature, as will be seen by comparing the densities of sea 
 water and pure water given in the following table. 
 
 TABLE I. 
 
 Useful data. 
 
 One cubic foot of water at 39-1 F. weighs 62-425 Ibs. 
 60 F. 62-36 
 
 One cubic foot of average sea water at 60 F. weighs 64 Ibs. 
 One gallon of pure water at 60 F. weighs 10 Ibs. 
 One gallon of pure water has a volume of 277'25 cubic inches. 
 One ton of pure water at 60 F. has a volume of 35-9 cubic feet. 
 
 12 
 
4 HYDRAULICS 
 
 Table of densities of pure water. 
 
 Temperature 
 
 Degrees Fahrenheit Density 
 32 -99987 
 
 39-1 1-000000 
 
 50 0-99973 
 
 60 0-99905 
 
 80 0-99664 
 
 104 0-99233 
 
 From the above it will be seen that in practical problems it 
 will be sufficiently near to take the weight of one cubic foot of 
 fresh water as 62*4 Ibs., one gallon as 10 pounds, 6'24 gallons in a 
 cubic foot, and one cubic foot of sea water as 64 pounds. 
 
 5. Hydrostatics. 
 
 A knowledge of the principles of hydrostatics is very helpful 
 in approaching the subject of hydraulics, and in the wider sense 
 in which the latter word is now used it may be said to include the 
 former. It is, therefore, advisable to consider the laws of fluids 
 at rest. 
 
 There are two cases to consider. First, fluids at rest under the 
 action of gravity, and second, those cases in which the fluids are 
 at rest, or are moving very slowly, and are contained in closed 
 vessels in which pressures of any magnitude act upon the fluid, 
 as, for instance, in hydraulic lifts and presses. 
 
 6. Intensity of pressure. 
 
 The intensity of pressure at any point in a fluid is the pressure 
 exerted upon unit area, if the pressure on the unit area is uniform 
 and is exerted at the same rate as at the point. 
 
 Consider any little element of area a, about a point in the fluid,, 
 and upon which the pressure is uniform. 
 
 If P is the total pressure on a, the Intensity of Pressure p, is then 
 
 P 
 
 P= a> 
 or when P and a are indefinitely diminished, 
 
 3P 
 P= "da' 
 
 7. The pressure at any point in a fluid is the same in all 
 directions. 
 
 It has been stated above that when a fluid is at rest its resist- 
 ance to lateral deformation is practically zero and that on any 
 plane in the fluid tangential stresses cannot exist. From this 
 experimental fact it follows that the pressure at any point in the 
 fluid is the same in all directions. 
 
FLUIDS AT REST 5 
 
 Consider a small wedge ABC, Fig. 1, floating immersed in a 
 fluid at rest. 
 
 Since there cannot be a tangential i^ 
 
 stress on any of the planes AB, BC, or AC, Av. A ^- .B 
 
 the pressures on them must be normal. 
 
 Let p, P! and p 2 be the intensities of -p^ 
 
 pressures on these planes respectively. 
 
 The weight of the wedge will be very Fig. 1. 
 
 small and may be neglected. 
 
 As the wedge is in equilibrium under the forces acting on 
 its three faces, the resolved components of the force acting on 
 AC in the directions of p and p l must balance the forces acting 
 on AB and BC respectively. 
 
 Therefore p. 2 . AC cos = p . AB, 
 
 and puACsinfl^^BC. 
 
 But AB = ACcos0, 
 
 and BC = ACsin0. 
 
 Therefore p = p l = p^ . 
 
 8. The pressure on any horizontal plane in a fluid must 
 be constant. 
 
 Consider a small cylinder of a fluid joining any two points A 
 and B on the same horizontal plane in the fluid. 
 
 Since there can be no tangential forces acting on the cylinder 
 parallel to the axis, the cylinder must be in equilibrium under the 
 pressures on the ends A and B of the cylinder, and since these 
 are of equal area, the pressure must be the same at each end of 
 the cylinder. 
 
 9. Fluids at rest, with the free surface horizontal. 
 
 The pressure per unit area at any depth h below the free 
 surface of a fluid due to the weight of the fluid is equal to the 
 weight of a column of fluid of height h and of unit sectional area. 
 
 Let the pressure per unit area acting on the surface of the 
 fluid be p Ibs. If the fluid is in a closed vessel, the pressure p may 
 have any assigned value, but if the free surface is exposed to the 
 atmosphere, p will be the atmospheric pressure. 
 
 If a small open tube AB, of length h, and cross sectional area a, 
 be placed in the fluid, the weight per unit volume of which is 
 w Ibs., with its axis vertical, and its upper end A coincident with 
 the surface of the fluid, the weight of fluid in the cylinder must be 
 w . a . h Ibs. The pressure acting on the end A of the column 
 is pa Ibs. 
 
6 
 
 HYDRAULICS 
 
 Since there cannot be any force acting on the column parallel 
 to the sides of the tube, the force of wah Ibs. + pa Ibs. must be 
 kept in equilibrium by the pressure of the external fluid acting on 
 the fluid in the cylinder at the end B. 
 
 The pressure per unit area at B, therefore, 
 
 wah + pa , , 
 = = (wfi + p) Ibs. 
 
 The pressure per unit area, therefore, due to the weight of the 
 fluid only is wh Ibs. 
 
 In the case of water, w may be taken as 62*40 Ibs. per cubic 
 foot and the pressure per sq. foot at a depth of h feet is, therefore, 
 62'407i Ibs., and per sq. inch '433/1 Ibs. 
 
 It should be noted that the pressure is independent of the form 
 of the vessel, and simply depends upon the vertical depth of the 
 point considered below the surface of the fluid. This can be 
 illustrated by the different vessels shown in Fig. 2. If these 
 were all connected together by means of a pipe, the fluid when 
 at rest would stand at the same level in all of them, and on any 
 horizontal plane AB the pressure would be the same. 
 
 D 
 
 B 
 
 Pressure an, the Plaite A B= w-h Ibs per sq Foot. 
 Fig. 2. 
 
 If now the various vessels were sealed from each other 
 by closing suitable valves, and the pipe taken away without 
 disturbing the level CD in any case, the intensity of pressure on 
 AB would remain unaltered, and would be, in all cases, equal 
 to wh. 
 
 Example. In a condenser containing air and water, the pressure of the air is 
 2 Ibs. per sq. inch absolute. Find the pressure per sq. foot at a point 3 feet below 
 the free surface of the water. 
 
 ^ = 2x144 + 3x62-4 
 = 475-2 Ibs. per sq. foot. 
 
FLUIDS AT REST 7 
 
 10. Pressures measured in feet of water. Pressure head. 
 It is convenient in hydrostatics and hydraulics to express the 
 
 pressure at any point in a fluid in feet of the fluid instead of pounds 
 per sq. foot or sq. inch. It follows from the previous section that 
 if the pressure per sq. foot is p Ibs. the equivalent pressure in feet 
 
 of water, or the pressure head, is h = ^ ft. and for any other fluid 
 having a specific gravity p, the pressure per sq. foot for a head 
 
 h of the fluid is - = w . p . h, or h = . 
 
 wp 
 
 11. Piezometer tubes. 
 
 The pressure in a pipe or other vessel can conveniently be 
 measured by fixing a tube in the pipe and noting the height to 
 which the water rises in the tube. 
 
 Such a tube is called a pressure, or piezometer, tube. 
 
 The tube need not be made straight but may be bent into any 
 form and carried, within reasonable limits, any distance horizon- 
 tally. 
 
 The vertical rise h of the water will be always 
 
 h=p, 
 
 w' 
 where p is the pressure per sq. foot in the pipe. 
 
 If instead of water, a liquid of specific gravity p is used the 
 height h to which the liquid will rise in the tube is 
 
 w .p 
 
 Example. A tube having one end open to the atmosphere is fitted into a pipe 
 containing water at a pressure of 10 Ibs. per sq. inch above the atmosphere. Find 
 the height to which the water will rise in the tube. 
 
 The water will rise to such a height that the pressure at the end of the tube in 
 the pipe due to the column of water will be 10 Ibs. per sq. inch. 
 
 Therefore h= 10 X 144 = 23 -08 feet. 
 
 w 
 
 12. The barometer. 
 
 The method of determining the atmospheric 
 pressure by means of the barometer can now be 
 understood. 
 
 If a tube about 3 feet long closed at one end be 
 completely filled with mercury, Fig. 3, and then 
 turned into a vertical position with its open end 
 in a vessel containing mercury, the liquid in the 
 tube falls until the length h of the column is about 
 30 inches above the surface of the mercury in the 
 vessel. 
 
HYDRAULICS 
 
 Since the pressure p on the top of the mercury is now zero, the 
 pressure per unit area acting on the section of the tube, level with 
 the surface of the mercury in the vessel, must be equal to the 
 weight of a column of mercury of height h. 
 
 The specific gravity of the mercury is 13'596 at the standard 
 temperature and pressure, and therefore the atmospheric pressure 
 per sq. inch, p a , is, 
 
 30" x 13'596 x 62'4 
 Pa = - 10.. tAA - - 14 7 Ibs. per sq. inch. 
 
 12 x 144 
 Expressed in feet of water, 
 
 _ 14-7 x 144 
 62-4 
 
 - 33-92 feet. 
 
 This is so near to 34 feet that for the standard atmospheric 
 pressure this value will be taken throughout this book. 
 
 A similar tube can be conveniently used for measuring low 
 pressures, lighter liquids being used when a more sensitive gauge 
 is required. 
 
 13. The differential gauge. 
 
 A more convenient arrangement for measuring pressures, and 
 one of considerable utility in many hydraulic experiments, is 
 known as the differential gauge. 
 
 Let ABCD, Fig. 4, be a simple U tube 
 containing in the lower part some fluid of 
 known density. 
 
 If the two limbs of the tube are open to 
 the atmosphere the two surfaces of the fluid 
 will be in the same horizontal plane. 
 
 If, however, into the limbs of the tube a 
 lighter fluid, which does not mix with the 
 lower fluid, be poured until it rises to C in 
 one tube and to D in the other, the two 
 surfaces of the lower fluid will., now be at 
 different levels. 
 
 Let B and E be the common surfaces of 
 the two fluids, h being their difference of 
 level, and hi and h 2 the heights of the free 
 surfaces of the lighter fluid above E and B respectively. 
 
 Let p be the pressure of the atmosphere per unit area, and d 
 and di the densities of the lower and upper fluids respectively. 
 Then, since upon the horizontal plane AB the fluid pressure must 
 be constant, 
 
 p + dih 2 = p + dihi + dh, 
 or di (h 2 - hi) = dh. 
 
 
 Q 
 
 ^ 
 c 
 
 
 V 
 
 D 
 
 
 T~ 
 
 Cs 
 
 < 
 
 r*J 
 
 F 
 
 
 f 
 
 L A 
 
 ll 
 
 Fig. 4. 
 
FLUIDS AT REST 
 
 9 
 
 If now, instead of the two limbs of the U tube being open to 
 the atmosphere, they are connected by tubes to closed vessels in 
 which the pressures are p l and p. 2 pounds per sq. 'foot -respectively, 
 and /ij and /^ are the vertical lengths of the columns of fluid above 
 E and B respectively, then 
 
 . h, 
 
 or 
 
 An application of such a tube to determine the difference of 
 pressure at two points in a pipe containing flowing water is shown 
 in Fig. 88, page 116. 
 
 Fluids generally used in such U tubes. In hydraulic experiments 
 the upper part of the tube is filled with water, and therefore the 
 fluid in the lower part must have a greater density than water. 
 When the difference of pressure is fairly large, mercury is generally 
 used, the specific gravity of which is 13*596. When the difference 
 of pressure is small, the height h is difficult to measure with 
 precision, so that, if this form of gauge is to be used, it is desirable 
 to replace the mercury by a lighter liquid. Carbon bisulphide 
 has been used but its action is sluggish and the meniscus between 
 it and the water is not always well defined. 
 Nitro-benzine gives good results, its prin- 
 cipal fault being that the falling meniscus 
 does not very quickly assume a definite 
 shape. 
 
 The inverted air gauge. A more sen- 
 sitive gauge can be made by inverting a 
 U tube and enclosing in the upper part 
 a certain quantity of air as in the tube 
 BHC, Fig. 5. 
 
 Let the pressure at D in the limb DF 
 be pi pounds peM square foot, equivalent 
 to a head hi of the fluid in the lower part 
 of the gauge, and at A in the limb AE let 
 the pressure be p 2 , equivalent to a head 7i 2 . 
 Let h be the difference of level of G and C. 
 
 Then if CGH contains air, and the weight of the air be 
 neglected, being very small, the pressure at C must equal the 
 pressure at G ; and since in a fluid the pressure on any horizontal 
 plane is constant the pressure at C is equal to the pressure at D, 
 and the pressure at A equal to the pressure at B. Again the 
 pressure at G- is equal to the pressure at K. 
 
 Therefore h 2 -h = hi, 
 
 h. 
 
 g= 
 
 H 
 
 ^ 
 
 F 
 
 fr 
 
 ~f 
 
 -C 
 
 ^. 
 
 E 
 
 i 
 
 G 
 
 1 T" 
 
 f 
 
 F 
 
 A 
 
 L 
 
 Bf C 
 
 L 
 
 D 
 
 
 
 
 Fig. 5. 
 
 or 
 
 - Pi = 
 
10 
 
 HYDRAULICS 
 
 If the fluid is water p may then be taken as unity ; for a given 
 difference of pressure the value of h will clearly be much greater 
 than for the mercury gauge, and it has the further advantage that 
 In gives directly the difference of pressure in feet of water. The 
 temperature of the air in the tube does not affect the readings, as 
 any rise in temperature will simply depress the two columns 
 without affecting the value of h. 
 
 The inverted oil gauge. A still more sensitive gauge can 
 however be obtained by using, in the 
 upper part of the tube, an oil lighter 
 than water instead of air, as shown 
 in Fig. 6. 
 
 Let pi and p 2 be the pressures in 
 the two limbs of the tube on a given 
 horizontal plane AB, hi and h 2 being 
 the equivalent heads of water. The 
 oil in the bent tube will then take up 
 some such position as shown, the 
 plane AD being supposed to coincide 
 with the lower surface C. 
 
 Then, since upon any horizontal 
 plane in a homogeneous fluid the 
 pressure must be constant, the pres- 
 sures at Gr and H are equal and also 
 those at D and C. 
 
 Let PI be the specific gravity of 
 the water, and p of the oil. 
 
 Then Pihi-ph = pi (fe h). 
 
 Therefore h (PI - p) = pi (h 2 - hi) 
 
 / 
 
 E 
 
 
 
 \ 
 
 
 ~T" 
 
 
 
 
 if- 
 
 j 
 
 G 
 
 
 H 
 
 ,* 
 
 k 
 
 
 B |C 
 
 
 D 
 
 
 \ ^ 
 
 
 v^ 
 
 
 Fig. 6. 
 
 and 
 
 (p.- 
 
 (1) 
 
 Substituting for hi and h 2 the values 
 
 and 
 
 (2), 
 
 to . Oh - P) 
 
 or p 2 PJ = <# . (pj p) /j, (3) . 
 
 From (2) it is evident that, if the density of the oil is not very 
 different from that of the water, h may be large for very small 
 differences of pressure. Williams, Hubbell and Fenkell* found 
 * Proceedings Am.S.C.E., Vol. xxvii. p. 384. 
 
FLUIDS AT REST 
 
 11 
 
 that either kerosene, gasoline, or sperm oil gave excellent results, 
 but sperm oil was too sluggish in its action for rapid work. 
 Kerosene gave the best results. 
 
 Temperature coefficient of the inverted oil gauge. Unlike the 
 inverted air gauge the oil gauge has a considerable temperature 
 coefficient, as will be seen from the table of specific gravities at 
 various temperatures of water and the kerosene and gasoline used 
 by Williams, Hubbell and Fenkell. 
 
 In this table the specific gravity of water is taken as unity 
 at 60 F. 
 
 Temperature F. 
 Specific gravity 
 
 Water 
 
 Kerosene 
 
 Gasoline 
 
 40 
 1-00092 
 
 60 100 
 1-0000 '9941 
 
 40 60 100 
 7955 -7879 '7725 
 
 40 60 80 
 72147 -71587 '70547 
 
 The calibration of the inverted oil gauge. Messrs Williams, 
 Hubbell and Fenkell have adopted an ingenious method of 
 calibrating the oil gauge. This will readily be understood on 
 reference to Fig. 6. 
 
 The difference of level of E and F clearly gives the difference 
 of head acting on the plane AD in feet of water, and this from 
 
 equation (1) equals . 
 
 Water is put into AE and FD so that the surfaces E and F 
 are on the same level, the common surfaces of the oil and the 
 water also being on the same level, this level being zero for the 
 oil. Water is then run out of FD until the surface F is 
 exactly 1 inch below E and a reading for h taken. The surface F 
 is again lowered 1 inch and a reading of h taken. This process 
 is continued until F is lowered as far as convenient, and then 
 the water in EA is drawn out in a similar manner. When E 
 and F are again level the oil in the gauge should read zero. 
 
 14. Transmission of fluid pressure. 
 
 If an external pressure be applied at any point in a fluid, it is 
 transmitted equally in all direc- 
 tions through the whole mass. 
 This is proved experimentally 
 by means of a simple apparatus 
 such as shown in Fig. 7. 
 
 If a pressure P is exerted upon 
 a small piston Q of a sq. inches 
 
 
 
 Fig. 7. 
 
12 HYDRAULICS 
 
 p 
 
 area, the pressure per unit area p = , and the piston at R on the 
 
 a 
 
 same level as Q, which has an area A, can be made to lift a load W 
 
 p 
 equal to A ; or the pressure per sq. inch at R is equal to the 
 
 CL 
 
 pressure at Q. The piston at R is assumed to be on the same level 
 as Q so as to eliminate the consideration of the small differences of 
 pressure due to the weight of the fluid. 
 
 If a pressure gauge is fitted on the connecting pipe at any 
 point, and p is so large that the pressure due to the weight of the 
 fluid may be neglected, it will be found that the intensity of 
 pressure is p. This result could have been anticipated from that 
 of section 8. 
 
 Upon this simple principle depends the fact that enormous 
 forces can be exerted by means of hydraulic pressure. 
 
 If the piston at Q is of small area, while that at R is large, 
 then, since the pressure per sq. inch is constant throughout the 
 fluid, 
 
 W A 
 P a' 
 
 or a very large force W can be overcome by the application of 
 a small force P. A very large mechanical advantage is thus 
 obtained. 
 
 It should be clearly understood that the rate of doing work 
 at W, neglecting any losses, is equal to that at P, the distance 
 moved through by W being to that moved through by P in 
 the ratio of P to W, or in the ratio of a to A. 
 
 Example. A pump ram has a stroke of 3 inches and a diameter of 1 inch. The 
 pump supplies water to a lift which has a ram of 5 inches diameter. The force 
 driving the pump ram is 1500 Ibs. Neglecting all losses due to friction etc., 
 determine the weight lifted, the work done in raising it 5 feet, and the number 
 of strokes made by the pump while raising the weight. 
 
 Area of the pump ram = -7854 sq. inch. 
 
 Area of the lift ram= 19'6 sq. inches. 
 
 Therefore W = ,= 37,500 Ibs. 
 
 78o4 
 
 Work done = 37,500 x 5 = 187,500 ft. Ibs. 
 
 Let N equal the number of strokes of the pump ram. 
 Then N x T ^ x 1500 Ibs. = 187,500 ft. Ibs. 
 
 and N = 500 strokes. 
 
 15. Total or whole pressure. 
 
 The whole pressure acting on a surface is the sum of all the 
 normal pressures acting on the surface. If the surface is plane all 
 the forces are parallel, and the whole pressure is the sum of these 
 parallel forces. 
 
FLUIDS AT REST 
 
 13 
 
 Let any surface, which need not be a plane, be immersed 
 in a fluid. Let A be the area of the wetted surface, and h the 
 pressure head at the centre of gravity of the area. If the area 
 is immersed in a fluid the pressure on the surface of which is zero, 
 the free surface of the fluid will be at a height h above the centre 
 of gravity of the area. In the case of the area being immersed in 
 a fluid, the surface of which is exposed to a pressure p, and below 
 which the depth of the centre of gravity of the area is 7t , then 
 
 w 
 
 If the area exposed to the fluid pressure is one face of a body, 
 the opposite face of which is exposed to the atmospheric pressure, 
 as in the case of the side of a tank containing water, or the 
 masonry dam of Fig. 14, or a valve closing the end of a pipe as 
 in Fig. 8, the pressure due to the 
 atmosphere is the same on the two 
 faces and therefore may be neglected. 
 
 Let w be the weight of a cubic 
 foot of the fluid. Then, the whole 
 pressure on the area is 
 
 If the surface is in a horizontal 
 plane the theorem is obviously true, 
 since the intensity of pressure is con- 
 stant and equals w . h. 
 
 In general, imagine the surface, 
 
 B 
 
 Fig. 8. 
 
 Fig. 9, divided into a large number of small areas a, Oi, a^ ... . 
 
 Let x be the depth below the free surface FS, of any element 
 of area a ; the pressure on this element = w . x . a. 
 
 The whole pressure P = ^w .x.a. 
 
 But w is constant, and the sum of the moments of the elements 
 of the area about any axis equals the moment of the whole area* 
 about the same axis, therefore 
 
 2# . a = A . h, 
 and P = w.A.h. 
 
 16. Centre of pressure. 
 
 The centre of pressure of any plane 
 surface acted upon by a fluid is the 
 point of action of the resultant pressure 
 acting upon the surface. 
 
 Depth of the centre of pressure. Let 
 DBG, Fig. 9, be any plane surface 
 exposed to fluid pressure. 
 
 * See text-books on Mechanics. 
 
14 HYDRAULICS 
 
 Let A be the area, and h the pressure head at the centre of 
 gravity of the surface, or if FS is the free surface of the fluid, h is 
 the depth below FS of the centre of gravity. 
 Then, the whole pressure 
 
 P = w.A.h. 
 
 Let X be the depth of the centre of pressure. 
 Imagine the surface, as before, divided into a number of small 
 areas a, 0,1, 0$, ... etc. 
 
 The pressure on any element a 
 
 = w.a.x, 
 
 and P = %wax. 
 
 Taking moments about FS, 
 
 P.X= (wax 2 
 
 or X- 
 
 wAh 
 
 ~ Ah ' 
 
 When the area is in a vertical plane, which intersects the 
 surface of the water in FS, 3ax 2 is the "second moment" of the 
 area about the axis FS, or what is sometimes called the moment 
 of inertia of the area about this axis. 
 
 Therefore, the depth of the centre of pressure of a vertical 
 area below the free surface of the fluid 
 
 moment of inertia of the area about an axis in its own plane 
 _ _ and in the free surface _ 
 
 area x the depth of the centre of gravity 
 or, if I is the moment of inertia, 
 
 Moment of Inertia about any axis. Calling I the Moment 
 of Inertia about an axis through the centre of gravity, and I the 
 Moment of Inertia about any axis parallel to the axis through the 
 centre of gravity and at a distance h from it, 
 
 I = I + A/> 2 . 
 
 Examples. (1) Area is a rectangle breadth b and depth d. 
 
 P = w.b.d.h, 
 
FLUIDS AT REST 
 
 15 
 
 If the free surface of the water is level with the upper edge of the rectangle, 
 ^, and X = |.d. 
 
 (2) Area is a circle of radius E. 
 
 X: 
 
 R 2 
 
 If the top of the circle is just in the free surface or fc= 
 
 TABLE II. 
 
 Table of Moments of Inertia of areas. 
 
 
 Form of area 
 
 Moment of inertia about 
 an axis AB through the 
 C. of G. of the section 
 
 Rectangle 
 
 & 
 
 ft 
 
 Triangle 
 
 S^f- 
 
 r*-6-t 
 
 m bd3 
 
 Circle 
 
 J7~^B 
 
 jfcz 
 
 Trd 4 
 64 
 
 Semicircle 
 
 V 
 
 About the axis AB 
 
 Trr* 
 8 
 
 Parabola 
 
 i b-+* 
 '*AT7*hjB 
 .W 
 
 1* 
 
16 
 
 HYDRAULICS 
 
 17. Diagram of pressure on a plane area. 
 
 If a diagram be drawn showing the intensity of pressure on 
 a plane area at any depth, the whole pressure is equal to the volume 
 of the solid thus formed, and the centre of pressure of the area is 
 found by drawing a line through the centre 
 of gravity of this solid perpendicular to the 
 area. 
 
 For a rectangular area ABCD, having the 
 side AB in the surface of the water, the 
 diagram of pressure is AEFCB, Fig. 10. The 
 volume of AEFCB is the whole pressure and 
 equals \bd?w, b being the width and d the 
 depth of the area. 
 
 Since the rectangle is of constant width, 
 the diagram of pressure may be represented 
 by the triangle BCF, Fig. 11, the resultant pressure acting 
 through its centre of gravity, and therefore at f d from the surface. 
 
 h g 
 
 & b -Intensify of pressure 
 
 OJV 
 
 Fig. 12. 
 
 - For a vertical circle the diagram of pressure is as shown in 
 Figs. 12 and 13. The intensity of pressure ab on any strip at a 
 depth h is wh^ . The whole pressure is the volume of the truncated 
 cylinder efkh and the centre of pressure is found by drawing a 
 line perpendicular to the circle, through the centre of gravity 
 of this truncated cylinder. 
 
 
FLUIDS AT REST 
 
 17 
 
 Another, and frequently a very convenient method of deter- 
 mining the depth of the centre of pressure, when the whole of the 
 area is at some distance below the surface of the water, is to 
 consider the pressure on the area as made up of a uniform pressure 
 over the whole surface, and a pressure of variable intensity. 
 
 Take again, as an example, the vertical circle the diagrams of 
 pressure for which are shown in Figs. 12 and 13. 
 
 At any depth h the intensity of pressure on the strip ad is 
 
 The pressure on any strip ad is, therefore, made up of a 
 constant pressure per unit area wh^ and a variable pressure whi - f 
 and the whole pressure is equal to the volume of the cylinder efgh, 
 Fig. 12, together with the circular wedge fkg. 
 
 The wedge fkg is equal to the whole pressure on a vertical 
 circle, the tangent to which is in the free surface of the water and 
 
 equals w . A . ^ , and the centre of gravity of this wedge will be at 
 
 the same vertical distance from the centre of the circle as the 
 centre of pressure when the circle touches the surface. The whole 
 pressure P may be supposed therefore to be the resultant of two 
 parallel forces PI and P 2 acting through the centres of gravity of 
 the cylinder efgh, and of the circular wedge fkg respectively, the 
 magnitudes of PI and P 2 being the volumes of the cylinder and 
 the wedge respectively. 
 
 To find the centre of pressure on the circle AB it is only 
 necessary to find the resultant of two parallel forces 
 
 of which Pj acts at the centre c, and P 2 at a point Ci which is at 
 a distance from A of f r. 
 
 Example. A masonry dam, Fig. 14, 
 has a height of 80 feet from the founda- 
 tions and the water face is inclined at 
 10 degrees to the vertical ; find the whole 
 pressure on the face due to the water per 
 foot width of the dam, and the centre of 
 pressure, when the water surface is level 
 with the top of the dam. The atmo- 
 spheric pressure may be neglected. 
 
 The whole pressure will be the force 
 tending to overturn the dam, since the 
 horizontal component of the pressure 
 on AB due to the atmosphere will be 
 counterbalanced by the horizontal com- 
 ponents of the atmospheric pressure on 
 the back of the dam. Since the pressure 
 on the face is normal, and the intensity 
 of pressure is proportional to the depth, 
 
 L. H. 
 
 thrust 
 
 bcLS&D Bands acts 
 cut th& points . 
 
 Fig. 14. 
 
18 
 
 HYDRAULICS 
 
 the diagram of pressure on the face AB will be the triangle ABC, BC being equal 
 to wd and perpendicular to AB. 
 
 The centre of pressure is at the centre of gravity of the pressure diagram and is, 
 therefore, at ^ the height of the dam from the base. 
 
 The whole pressure acts perpendicular to AB, and is equal to the area ABC 
 
 = ^wd 2 x sec 10 per foot width 
 
 = . 62-4 x 6400 x 1-054 = 20540 Ibs. 
 
 Combining P with W, the weight of the dam, the resultant thrust E on the base 
 and its point of intersection E with the base is determined. 
 
 Example. A vertical flap valve closes the end of a pipe 2 feet diameter ; the 
 pressure at the centre of the pipe is equal to a head of 8 feet of water. To determine 
 the whole pressure on the valve and the centre of pressure. The atmospheric 
 pressure may be neglected. 
 
 The whole pressure P = ivirR* . 8' 
 
 = 62-4. TT. 8 = 1570 Ibs. 
 
 Depth of the centre of pressure. 
 
 The moment of inertia about the free surface, which is 8 feet above the centre 
 of the valve, is 
 
 Therefore 
 
 That is, | inch below the centre of the valve. 
 
 The diagram of pressure is a truncated cylinder efkh, Figs. 12 and 13, ef and hk 
 being the intensities of pressure at the top and bottom of the valve respectively. 
 
 Example. The end of a pontoon which floats in sea water is as shown in Fig. 15. 
 The level "WL of the water is also shown. Find the whole pressure on the end of 
 the pontoon and the centre of pressure. 
 
 The whole pressure on BE 
 
 = 64 Ibs. x 10' x 4-5' x 2-25' = 6480 Ibs. 
 The depth of the centre of pressure of BE is 
 
 f .4-5 = 3'. 
 The whole pressure on each of the rectangles above the quadrants 
 
 w. 5 = 320 Ibs., 
 and the depth of the centre of pressure is % feet. 
 
 The two quadrants, since they are symmetrically placed about the vertical 
 centre line, may be taken together to form a semicircle. Let d be the distance 
 below the centre of the semicircle of any element of area, the distance of the 
 element below the surface being h . 
 
FLUIDS AT REST 19 
 
 Then the intensity of pressure at depth h 
 
 w . 2 + w . d. 
 And the whole pressure on the semicircle is P=w .^-^- . 2' +the whole pressure 
 
 on the semicircle when the diameter is in the surface of the water. 
 
 The distance of the centre of gravity of a semicircle from the centre of the 
 circle is 
 
 4R 
 
 Therefore, P = wirW + 
 
 2 Sir 
 
 = 201R 2 + 42 -66 R 3 = 1256 + 666 Ibs. 
 
 The depth of the centre of pressure of the semicircle when the surface of the 
 water is at the centre of the circle, is 
 
 TrR 4 
 8 3.7T.R 
 
 2 ' Sir 
 
 And, therefore, the whole pressure on the semicircle is the sum of two forces, 
 oue of which, 1256 Ibs., acts at the centre of gravity, or at a distance of 3-06' from 
 AD, and the other of 666 Ibs. acts at a distance of 3-47' from AD. 
 
 Then taking moments about AD the product of the pressure on the whole area 
 into the depth of the centre of pressure is equal to the moments of all the forces 
 acting on the area, about AD. The depth of the centre of pressure is, therefore, 
 
 6480 Ibs. x 3' + 320 Ibs. x 2 x $ ' + 1256 Ibs. x 3-06 + 666 Ibs. x 3-47' 
 
 6480 + 640 + 1256 + 666 
 = 2-93 feet. 
 
 EXAMPLES. 
 
 (1) A rectangular tank 12 feet long, 5 feet wide, and 5 feet deep is 
 filled with water. 
 
 Find the total pressure on an end and side of the tank. 
 
 (2) Find the total pressure and the centre of pressure, on a vertical 
 sluice, circular in form, 2 feet in diameter, the centre of which is 4 feet 
 below the surface of the water. [M. S. T. Cambridge, 1901.] 
 
 (3) A masonry dam vertical on the water side supports water of 
 120 feet depth. Find the pressure per square foot at depths of 20 feet and 
 70 feet from the surface; also the total pressure on 1 foot length of the dam. 
 
 (4) A dock gate is hinged horizontally at the bottom and supported in 
 a vertical position by horizontal chains at the top. 
 
 Height of gate 45 feet, width 30 ft. Depth of water at one side of the 
 gate 32 feet and 20 feet on the other side. Find the tension in the chains. 
 Sea-water weighs 64 pounds per cubic foot. 
 
 (5) If mercury is 13 times as heavy as water, find the height of a 
 column corresponding to a pressure of 100 Ibs. per square inch. 
 
 (6) A straight pipe 6 inches diameter has a right-angled bend connected 
 to it by bolts, the end of the bend being closed by a flange. 
 
 The pipe contains water at a pressure of 700 Ibs. per sq. inch. Determine 
 the total pull in the bolts at both ends of the elbow. 
 
 22 
 
20 
 
 HYDRAULICS 
 
 (7) The end of a dock caisson is as shown in Fig. 16 and the water 
 level is AB. 
 
 Determine the whole pressure and the centre of pressure. 
 
 ----------- 48'.C-" 
 
 4.C -> 
 
 Fig. 16. 
 
 (8) An U tube contains oil having a specific gravity of I'l in the lower 
 part of the tube. Above the oil in one limb is one foot of water, and above 
 the other 2 feet. Find the difference of level of the oil in the two limbs. 
 
 (9) A pressure gauge, for use in a stokehold, is made of a glass U tube 
 with enlarged ends, one of which is exposed to the pressure in the stokehold 
 and the other connected to the outside air. The gauge is filled with water 
 on one side, and oil having a specific gravity of 0*95 on the other the 
 surface of separation being in the tube below the enlarged ends. If the 
 area of the enlarged end is fifty times that of the tube, how many inches of 
 water pressure in the stokehold correspond to a displacement of one inch 
 in the surface of separation ? [Lond. Un. 1906.] 
 
 (10) An inverted oil gauge has its upper U filled with oil having a 
 specific gravity of 0'7955 and the lower part of the gauge is filled with 
 water. The two limbs are then connected to two different points on a pipe 
 in which there is flowing water. 
 
 Find the difference of the pressure at the two points in the pipe when 
 the difference of level of the oil surfaces in the limbs of the U is 
 15 inches. 
 
 (11) An opening in a reservoir dam is closed by a plate 3 feet square, 
 which is hinged at the upper horizontal edge ; the plate is inclined at an 
 angle of 60 to the horizontal, and its top edge is 12 feet below the surface 
 of the water. If this plate is opened by means of a chain attached to the 
 centre of the lower edge, find the necessary pull in the chain if its line of 
 action makes an angle of 45 with the plate. The weight of the plate is 
 400 pounds. [Lond. Un. 1905.] 
 
 (12) The width of a lock is 20 feet and it is closed by two gates at each 
 end, each gate being 12' long. 
 
 If the gates are closed and the water stands 16' above the bottom on one 
 side and 4' on the other side, find the magnitude and position of the resultant 
 pressure on each gate, and the pressure between the gates. Show also that 
 the reaction at the hinges is equal to the pressure between the gates. One 
 cubic foot of water =62-5 Ibs. [Lond. Un. 1905.] 
 
CHAPTER II. 
 
 FLOATING BODIES. 
 
 18. Conditions of equilibrium. 
 
 When a body floats in a fluid the surface of the body in 
 contact with the fluid is subject to hydrostatic pressures, the 
 intensity of pressure on any element of the surface depend- 
 ing upon its depth below the surface. The resultant of the 
 vertical components of these hydrostatic forces is called the 
 buoyancy, and its magnitude must be exactly equal to the weight 
 of the body, for if not the body will either rise or sink. Again 
 the horizontal components of these hydrostatic forces must 
 be in equilibrium amongst themselves, otherwise the body will 
 have a lateral movement. 
 
 The position of equilibrium for a floating body is obtained 
 when (a) the buoyancy is exactly equal to the weight of the 
 body, and (6) the vertical forces the weight and the buoyancy- 
 act in the same vertical line, or in other words, in such a way as 
 to produce no couple tending to make the body rotate. 
 
 Let Gr, Fig. 17, be the centre of gravity of a floating ship and 
 BK, which does not pass through Gr, the line of action of the 
 resultant of the vertical buoyancy forces. Since the buoyancy 
 
 Fig. 17. 
 
 Fig. 18. 
 
 must equal the weight of the ship, there are two parallel forces 
 each equal to W acting through Gr and along BK respectively, 
 and these form a couple of magnitude Wa?, which tends to bring 
 the ship into the position shown in Fig. 18, that is, so that BK 
 
22 
 
 HYDRAULICS 
 
 passes through Gr. The above condition (6) can therefore only be 
 realised, when the resultant of the buoyancy forces passes through 
 the centre of gravity of the body. If, however, the body is 
 displaced from this position of equilibrium, as for example a ship 
 at sea would be when made to roll by wave motions, there will 
 generally be a couple, as in Fig. 17, acting upon the body, which 
 should in all cases tend to restore the body to its position of 
 equilibrium. Consequently the floating body will oscillate about 
 its equilibrium position and it is then said to be in stable equi- 
 librium. On the other hand, if when the body is given a small 
 displacement from the position of equilibrium, the vertical forces 
 act in such a way as to cause a couple tending to increase the 
 displacement, the equilibrium is said to be unstable. 
 
 The problems connected with floating bodies acted upon by 
 forces due to gravity and the hydrostatic pressures only, 
 resolve themselves therefore into two, 
 
 (a) To find the position of equilibrium of the body. 
 
 (6) To find whether the equilibrium is stable. 
 
 19. Principle of Archimedes. 
 
 When a body floats freely in a fluid the weight of the body is 
 equal to the weight of the fluid displaced. 
 
 Since the weight of the body is equal to the resultant of the 
 vertical hydrostatic pressures, or to the buoyancy, this principle 
 will be proved, if the weight of the water displaced is shown to be 
 equal to the buoyancy. 
 
 Let ABC, Fig. 19, be a body floating in equilibrium, AC being 
 in the surface of the fluid. 
 
 Fig. 19. 
 
 Consider any small element ab of the surface, of area a and 
 depth h, the plane of the element being inclined at any angle to 
 the horizontal. Then, if w is the weight of unit volume of the 
 fluid, the whole pressure on the area a is wha, and the vertical 
 component of this pressure is seen to be wha cos 0. 
 
FLOATING BODIES 23 
 
 Imagine now a vertical cylinder standing on this area, the top 
 of which is in the surface AC. 
 
 The horizontal sectional area of this cylinder is a cos 0, the 
 volume is ha cos and the weight of the water filling this volume 
 is whacosO, and is, therefore, equal to the buoyancy on the 
 area ab. 
 
 If similar cylinders be imagined on all the little elements 
 of area which make up the whole immersed surface, the total 
 volume of these cylinders is the volume of the water displaced, 
 and the total buoyancy is, therefore, the weight of this displaced 
 water. 
 
 If the body is wholly immersed as in Fig. 20 and the 
 body is supposed to be made up of small . _ 
 
 vertical cylinders intersecting the surface of 
 the body in the elements of area ab and ab', 
 which are inclined to the horizontal at angles 
 and < and having areas a and ai respectively, 
 the vertical component of the pressure on ab 
 will be wha cos and on ab' will be whidi cos <. 
 But acos# must equal o^cos^, each being 
 equal to the horizontal section of the small cylinder. The whole 
 buoyancy is therefore 
 
 2>wha cos ^whiai cos <, 
 and is again equal to the weight of the water displaced. 
 
 In this case if the fluid be assumed to be of constant density 
 and the weight of the body as equal to the weight of the fluid 
 of the same volume, the body will float at any depth. The 
 slightest increase in the weight of the body would cause it to 
 sink until it reached the bottom of the vessel containing the fluid, 
 while a very small diminution of its weight or increase in its 
 volume would cause it to rise immediately to the surface. It 
 would clearly be practically impossible to maintain such a body 
 in equilibrium, by endeavouring to adjust the weight of the body, 
 by pumping out, or letting in, water, as has been attempted in a 
 certain type of submarine boat. In recent submarines the lowering 
 and raising of the boat are controlled by vertical screw propellers. 
 
 20. Centre of buoyancy. 
 
 Since the buoyancy on any element of area is the weight of 
 the vertical cylinder of the fluid above this area, and that the 
 whole buoyancy is the sum of the weights of all these cylinders, it 
 at once follows, that the resultant of the buoyancy forces must 
 pass through the centre of gravity of the water displaced, and this 
 point is, therefore, called the Centre of Buoyancy. 
 
24 
 
 HYDRAULICS 
 
 21. Condition of stability of equilibrium. 
 
 Let AND, Fig. 21, be the section made by a vertical plane 
 containing Gr the centre of gravity and B the centre of buoyancy 
 of a floating vessel, AD being the surface of the fluid when the 
 centre of gravity and centre of buoyancy are in the same vertical 
 line. 
 
 B 
 
 Fig. 21. 
 
 M, 
 
 B, 
 
 Fig. 22. 
 
 Let the vessel be heeled over about a horizonal axis, FE being 
 now the fluid surface, and let B x be the new centre of buoyancy, 
 the above vertical sectional plane being taken to contain Gr, B, 
 and BI. Draw BiM, the vertical through B 1} intersecting the line 
 GrB in M. Then, if M is above Gr the couple W . x will tend to 
 restore the ship to its original position of equilibrium, but if M is 
 below Gr, as in Fig. 22, the couple will tend to cause a further 
 displacement, and the ship will either topple over, or will heel over 
 into a new position of equilibrium. 
 
 In designing ships it is necessary that, for even large displace- 
 ments such as may be caused by the rolling of the vessel, the 
 point M shall be above Gr. To determine M, it is necessary to 
 determine Gr and the centres of buoyancy for the two positions 
 of the floating body. This in many cases is a long and somewhat 
 tedious operation. 
 
 22. Small displacements. Metacentre. 
 
 When the angular displacement is small the point M is called 
 the Metacentre, and the distance of M from Gr can be calculated. 
 
 Assume the angular displacement in Fig. 21 to be small and 
 equal to 0. 
 
 Then, since the volume displacement is constant the volume of 
 the wedge ODE must equal CAF, or in Fig. 23, dC 2 DE must equal 
 dC 2 AF. 
 
FLOATING BODIES 
 
 25 
 
 Let Gri and Gr 2 be the centres of gravity of the wedges CiC 2 AF 
 and CiC 2 DE respectively. 
 
 d-f D 
 
 Fig. 23. 
 
 The heeling of the ship has the effect of moving a mass of 
 water equal to either of these wedges from GK to Gr 2 , and this 
 movement causes the centre of gravity of the whole water 
 displaced to move from B to B a . 
 
 Let Z be the horizontal distance between (TI and G 2 , when FE 
 is horizontal, and S the perpendicular distance from B to BiM. 
 
 Let V be the total volume displacement, v the volume of the 
 wedge and w the weight of unit volume of the fluid. 
 
 Then w . v . Z = w . Y . S 
 
 = .V.BM.sin0. 
 
 Or, since is small, = w.V.BM.0 (1). 
 
 The restoring couple is 
 
 = w.V.G-M.0 
 
 rr ~\T ~DC* A /O^ 
 
 = IV V J 10 . V . JjljT . " \) 
 
 But w . v . Z = twice the sum of the moments about the axis 
 of all the elements such as acdb which make up the wedge 
 CADE. 
 
 Taking ab as x, bf is x& 9 and if ac is dl, the volume of the 
 element is Ja? 2 . dl. 
 
 The centre of gravity of the element is at fa? from CiC 2 . 
 
 Therefore to . t> . Z = 2iflfl -^- (3). 
 
 , i s the Second Moment or Moment of Inertia of the 
 
 o 
 
 element of area aceb about C 2 Ci, and 2 J -g- is, therefore, the 
 Moment of Inertia I of the water-plane area ACiDC a about CiC 2 . 
 Therefore w.v.Z = w.I.O (4). 
 
26 HYDRAULICS 
 
 The restoring couple is then 
 
 If this is positive, the equilibrium is stable, but if negative it is 
 unstable. 
 
 Again since from (1) 
 
 wv . Z - w . Y . BM . 0, 
 therefore w . Y . BM . 6 = wI0, 
 
 and 
 
 If BM is greater than BGr the equilibrium is stable, if less than 
 BGr it is unstable, and the body will heel over until a new position 
 of equilibrium is reached. If BGr is equal to BM the equilibrium 
 is said to be neutral. 
 
 The distance G-M is called the Metacentric Height, and varies 
 in various classes of ships from a small negative value to a positive 
 value of 4 or 5 feet. 
 
 When the metacentric height is negative the ship heels until 
 it finds a position of stable equilibrium. This heeling can be 
 corrected by ballasting. 
 
 Example. A ship has a displacement of 15,400 tons, and a draught of 27*5 feet. 
 The height of the centre of buoyancy from the bottom of the keel is 15 feet. 
 
 The moment of inertia of the horizontal section of the ship at the water line 
 is 9,400,000 feet 4 units. 
 
 Determine the position of the centre of gravity that the metacentric height shall 
 not be less than 4 feet in sea water. 
 
 9,400,000x64 
 "15,400x2240 
 = 17-1 feet. 
 
 Height of metacentre from the bottom of the keel is, therefore, 32*1 feet. 
 As long as the centre of gravity is not higher than 0'6 feet above the surface of 
 the water, the metacentric height is more than 4 feet. 
 
 23. Stability of a rectangular pontoon. 
 
 Let EFJS, Fig. 24, be the section of the pontoon and Gr its 
 centre of gravity. 
 
 Let YE be the surface of the water when the sides of the 
 pontoon are vertical, and AL the surface of the water when the 
 pontoon is given an angle of heel 0. 
 
 Then, since the weight of water displaced equals the weight of 
 the pontoon, the area AFJL is equal to the area YFJE. 
 
 Let B be the centre of buoyancy for the vertical position, 
 B being the centre of area of VF JE, and BI the centre of buoyancy 
 for the new position, BI* being the centre of area of AFJL. Then 
 the line joining BGr must be perpendicular to the surface YE and 
 
 * In the Fig., B t is not the centre of area of APJL, as, for the sake of clearness, 
 it is further removed from B than it actually should be. 
 
FLOATING BODIES 
 
 27 
 
 is the direction in which the buoyancy force acts when the sides 
 of the pontoon are vertical, and BjM perpendicular to AL is the 
 direction in which the buoyancy force acts when the pontoon is 
 heeled over through the angle 0. M is the metacentre. 
 
 F 
 
 Fig. 24. 
 
 The forces acting on the pontoon in its new position are, W the 
 weight of the pontoon acting vertically through G and an equal and 
 parallel buoyancy force W through BI . 
 
 There is, therefore, a couple, W.HGr, tending to restore the 
 pontoon to its vertical position. 
 
 If the line BiH were to the right of the vertical through Gr, or 
 in other words the point M was below Gr, the pontoon would be in 
 unstable equilibrium. 
 
 The new centre of buoyancy B a can be found in several ways. 
 The following is probably the simplest. 
 
 The figure AFJL is formed by moving the triangle, or really 
 the wedge-shaped piece GEL to CYA, and therefore it may be 
 imagined that a volume of water equal to the volume of this wedge 
 is moved from Gr 2 to GK . This will cause the centre of buoyancy 
 to move parallel to GriGr 2 to a new position BI, such that 
 
 BBi x weight of pontoon = GiGa x weight of water in GEL. 
 
 Let b be half the breadth of the pontoon, 
 Z the length, 
 
 D the depth of displacement for the upright position, 
 d the length LE, or AY, 
 and w the weight of a cubic foot of water. 
 
 Then, the weight of the pontoon 
 
 bd 
 and the weight of the wedge CLE = -^ x I w. 
 
28 HYDRAULICS 
 
 Therefore BBi . 26 . D = G ^ b ' d , 
 
 d rr 
 and -D>i = ^g Irilr 2 . 
 
 Resolving BBi and GriGr 2 , which are parallel to each other, along 
 and perpendicular to BM respectively, 
 
 _d_ J_/2 \ Id ^VtanO 
 
 
 To find the distance of the point M from G and the value of the 
 restoring couple. Since BiM is perpendicular to AL and BM to 
 YE, the angle BMBi equals 6. 
 
 Therefore QM = B X Q cot 6 = |g cot = j^ . 
 
 Let z be the distance of the centre of gravity Gr from C. 
 Then Qa = QC-z = BC-BQ-z 
 
 D & 2 tan 2 
 
 
 ~ 2 6D 
 Therefore 
 
 & ^ fr 2 tan 2 
 
 And since HGr = GrM sin 0, 
 
 the righting couple, 
 
 The distance of the metacentre from the point B, is 
 QM + QB^Qcottf + ^j^ 
 
 V_ 6 2 tan 2 
 ~ 3D + 6D 
 When is small, the term containing tan 2 6 is negligible, and 
 
 This result can be obtained from formula (4) given in 
 section 22. 
 
 I for the rectangle is T V (26) 3 = f Z6 3 , and V = 26DZ. 
 
 Therefore BM=Ji. 
 
 <L>U 
 
 If BGr is known, the metacentric height can now be found. 
 
FLOATING BODIES 
 
 29 
 
 Example. A pontoon has a displacement of 200 tons. Its length is 50 feet. 
 The centre of gravity is 1 foot above the centre of area of the cross section. Find 
 the breadth and depth of the pontoon so that for an angular displacement of 10 degrees 
 the metacentre shall not be less than 3 feet from the centre of gravity, and the free- 
 board shall not be less than 2 feet. 
 
 Kef erring to Fig. 24, G is the centre of gravity of the pontoon and is the 
 centre of the cross section KJ. 
 
 Then, GO = 1 foot, 
 
 F = 2feet, 
 
 GM = 3 feet. 
 
 Let D be the depth of displacement. Then 
 
 D x 26 x 62-4 x 50 Ibs. = 200 tons x 2240 Ibs. 
 Therefore D6 = 71'5 
 
 (1). 
 
 The height of the centre of buoyancy B above the bottom of pontoon is 
 
 Since the free-board is to be 2 feet, 
 
 Then 
 
 Therefore 
 
 But 
 
 = l' and BG = 2'. 
 BM = 5'. 
 
 tan 2 
 
 6D 
 
 (2). 
 
 Multiplying numerator and denominator by b, and substituting from equation (1) 
 
 214-5 
 
 from which 
 
 therefore 
 
 and 
 
 429 
 
 -^ 6 = 10-1 ft., 
 
 D = 7'lft., 
 
 The breadth B = 20-2 ft.l 
 ,, depth 
 
 ! = 20-2 ft.) 
 = 7-1 ft.) 
 
 Ans. 
 
 24. Stability of a floating vessel containing water. 
 
 If a vessel contains water with a free surface, as for instance 
 the compartments of a floating dock, such as is described on page 
 31, the surface of the water in these compartments will remain 
 horizontal as the vessel heels over, and the centre of gravity of 
 the water in any compartment will change its position in such 
 a way as to increase the angular displacement of the vessel. 
 
 In considering the stability 
 of such vessels, therefore, the 
 turning moments due to the 
 water in the vessel must be 
 taken into account. 
 
 As a simple case consider 
 the rectangular vessel, Fig. 25, 
 which, when its axis is vertical, 
 floats with the plane AB in the Fig - 25 ' 
 
 A 
 
 _---~ 
 E 
 H 
 
 _____ 
 
 _.,- '-"" 
 
 . 
 11 
 
 K 
 J) 
 
 __^ 
 
 ^' 
 
 >^-A 
 
30 
 
 HYDRAULICS 
 
 surface of the fluid, DE being the surface of the fluid in the 
 vessel. 
 
 When the vessel is heeled through an angle 0, the surface of 
 fluid in the vessel is KH. 
 
 The effect has been, therefore, to move the wedge of fluid OEH 
 to ODK, and the turning couple due to this movement is w . v . Z, 
 v being the volume of either wedge and Z the distance between 
 the centre of gravity of the wedges. 
 
 6 2 
 If 26 is the width of the vessel and I its length, v is -~ I tan 0, 
 
 Z is ^b tan 0, and the turning couple is w f 6 3 1 tan 2 0. 
 
 If is small wvZ is equal to wW, I being the moment of inertia 
 of the water surface KH about an axis through 0, as shown in 
 section 22. 
 
 For the same width and length of water surface in the 
 compartment, the turning couple is the same wherever the 
 compartment is situated, for the centre of gravity of the wedge 
 OHE, Fig. 26, is moved by the same amount in all cases. 
 
 If, therefore, there are free fluid surfaces in the floating vessel, 
 for any small angle of heel 0, the tippling-moment due to these 
 surfaces is 2wI0, I being in all cases the moment of inertia of the 
 fluid surface about its own axis of oscillation, or the axis through 
 the centre of gravity of the surface. 
 
 H 
 
 
 
 Fig. 26. 
 
 Fig. 27. 
 
 25. Stability of a floating body wholly immersed. 
 
 It has already been shown that a floating body wholly im- 
 mersed in a fluid, as far as vertical motions are concerned, can 
 only with great difficulty be maintained in equilibrium. 
 
 If further the body is made to roll through a small angle, the 
 equilibrium will be unstable unless the centre of gravity of the 
 body is below the centre of buoyancy. This will be seen at once 
 on reference to Fig. 27. Since the body is wholly immersed the 
 centre of buoyancy cannot change its position on the body itself, 
 as however it rolls the centre of buoyancy must be the centre of 
 gravity of the displaced water, and this is not altered in form by 
 
FLOATING BODIES 
 
 31 
 
 any movement of the body. If, therefore, Gr is above B and the 
 body be given a small angular displacement to the right say, Gr 
 will move to the right relative to B and the couple will not restore 
 the body to its position of equilibrium. 
 
 On the other hand, if Gr is below B, the couple will act so as to 
 bring the body to its position of equilibrium. 
 
 26. Floating docks. 
 
 Figs. 28 and 29 show a diagrammatic outline of the pontoons 
 forming a floating dock, and in the section is shown the outline of 
 a ship on the dock. 
 
 w 
 
 L. 
 
 Fig. 29. 
 
 To dock a ship, the dock is sunk to a sufficient depth by 
 admitting water into compartments formed in the pontoons, and the 
 ship is brought into position over the centre of the dock. 
 
 Water is then pumped from the pontoon chambers, and the 
 dock in consequence rises until the ship just rests on the keel 
 blocks of the dock. As more water is pumped from the pontoons 
 the dock rises with the ship, which may thus be lifted clear of 
 the water. 
 
 Let Gri be the centre of gravity of the ship, Gr 2 of the dock and i 
 water ballast and Gr the centre of gravity of the dock and the 
 
 ship. 
 
 The position of the centre of gravity of the dock will vary 
 
32 HYDRAULICS 
 
 relative to the bottom of the dock, as water is pumped from the 
 pontoons. 
 
 As the dock is raised care must be taken that the metacentre 
 is above Gr or the dock will " list." 
 
 Suppose the ship and dock are rising and that WL is the 
 water line. 
 
 Let B 2 be the centre of buoyancy of the dock and B : of the 
 portion of the ship still below the water line. 
 
 Then if Vi and V 2 are the volume displacements below 
 the water line of the ship and dock respectively, the centre of 
 buoyancy B of the whole water displaced divides B 2 B 3 , so that 
 
 BB 2 ~Y/ 
 
 The centre of gravity Gr of the dock and the ship divides GriGr 2 
 in the inverse ratios of their weights. 
 
 As the dock rises the centre of gravity Gr of the dock and the 
 ship must be on the vertical through B, and water must be 
 pumped from the pontoons so as to fulfil this condition and as 
 nearly as possible to keep the deck of the dock horizontal. 
 
 The centre of gravity Gri of the ship is fixed, while the centre of 
 buoyancy of the ship BI changes its position as the ship is raised. 
 
 The centre of buoyancy B 2 of the dock will also be changing, 
 but as the submerged part of the dock is symmetrical about its 
 centre lines, B 2 will only move vertically. As stated above, B 
 must always lie on the line joining BI and B 2 , and as Gr is to be 
 vertically above B, the centre of gravity Gr 2 and the weight of 
 the pontoon must be altered by taking water from the various 
 compartments in such a way as to fulfil this condition. 
 
 Quantity of water to be pumped from the pontoons in raising the 
 dock. Let V be the volume displacement of the dock in its lowest 
 position, Yo the volume displacement in its highest position. To 
 raise the dock without a ship in it the volume of the water to be 
 pumped from the pontoons is Y - Y . 
 
 If, when the dock is in its highest position, a weight W is put 
 on to the dock, the dock will sink, and a further volume of water 
 
 W 
 
 - cubic feet will be required to be taken from the pontoons to 
 
 raise the dock again to its highest position. 
 
 To raise the dock, therefore, and the ship, a total quantity of 
 water 
 
 cubic feet will have to be taken from the pontoons. 
 
FLOATING BODIES 33 
 
 Example. A floating dock as shown dimensioned in Fig. 28 is made up of a 
 bottom pontoon 540 feet long x 96 feet wide x 14-75 feet deep, two side pontoons 
 380 feet long x 13 feet wide x 4-8 feet deep, the bottom of these pontoons being 
 2 feet above the bottom of the dock, and two side chambers on the top of the 
 bottom pontoon 447 feet long by 8 feet deep and 2 feet wide at the top and 8 feet at 
 the bottom. The keel blocks may be taken as 4 feet deep. 
 
 The dock is to lift a ship of 15,400 tons displacement and 27' 6" draught. 
 
 Determine the amount of water that must be pumped from the dock, to raise 
 the ship so that the deck of the lowest pontoon is in the water surface. 
 
 When the ship just takes to the keel blocks on the dock, the bottom of the 
 dock is 27-5' + 14-75' + 4' = 46 -25 feet below the water line. 
 
 The volume displacement of the dock is then 
 
 14-75 x 540 x 96 + 2 x 44-25 x 13 x 380 + 447 x 8 x 5' = 1,237, 600 cubic feet. 
 The volume of dock displacement when the deck is just awash is 
 
 540 x 96 x 14-75 + 2 x 380 x 13' x (14-75 - 2) = 890,000 cubic feet. 
 The volume displacement of the ship is 
 15,400 x 2240 
 
 64 
 
 = 540,000 cubic feet, 
 
 and this equals the weight of the ship in cubic feet. 
 
 Of the 891,000 cubic feet displacement when the ship is clear of the water, 
 351,000 cubic feet is therefore required to support the dock alone. 
 
 Simply to raise the dock through 31 '5 feet the amount of water to be pumped is 
 the difference of the displacements, and is, therefore, 347,600 cubic feet. 
 
 To raise the ship with the dock an additional 540,000 cubic feet must be 
 extracted from the pontoons. 
 
 The total quantity, therefore, to be taken from the pontoons from the time the 
 ship takes to the keel blocks to when the pontoon deck is in the surface of the 
 water is 
 
 887,600 cubic feet = 25,380 tons. 
 
 27. Stability of the floating dock. 
 
 As some of the compartments of the dock are partially filled 
 with water, it is necessary, in considering the stability, to take 
 account of the tippling-moments caused by the movement of the 
 free surface of the water in these compartments. 
 
 If Gr is the centre of gravity of the dock and ship on the 
 dock, B the centre of buoyancy, I the moment of inertia of the 
 section of the ship and dock by the water-plane about the axis of 
 oscillation, and I 1? I 2 etc. the moments of inertia of the water 
 surfaces in the compartments about their axes of oscillation, the 
 righting moment when the dock receives a small angle of 
 heel 0, is 
 
 wIO - (Va + Y 2 ) RGO-wO (Ix + 1 2 +...). 
 
 The moment of inertia of the water-plane section varies 
 considerably as the dock is raised, and the stability varies 
 accordingly. 
 
 When the ship is immersed in the water, I is equal to the 
 moment of inertia of the horizontal section of the ship at the 
 water surface, together with the moment of inertia of the 
 horizontal section of the side pontoons, about the axis of 
 oscillation 0. 
 
 L.H. 3 
 
34 HYDRAULICS 
 
 When the tops of the keel blocks are just above the surface 
 of the water, the water-plane is only that of the side pontoons, 
 and I has its minimum value. If the dock is L-shaped as in 
 Fig. 30, which is a very convenient form 
 for some purposes, the stability when 
 the tops of the keel blocks come to the 
 surface simply depends upon the moment 
 of inertia of the area AB about an axis : 
 through the centre of AB. This critical 
 point can, however, be eliminated by 
 
 
 fitting an air box, shown dotted, on the Fi 30 
 
 outer end of the bottom pontoon, the 
 
 top of which is slightly higher than the top of the keel blocks. 
 
 Example. To find the height of the metacentre above the centre of buoyancy of 
 the dock of Fig. 28 when 
 
 (a) the ship just takes to the keel blocks, 
 (6) the keel is just clear of the water, 
 (c) the pontoon deck is just above the water. 
 
 Take the moment of inertia of the horizontal section of the ship at the 
 water line as 9,400,000 ft. 4 units, and assume that the ship is symmetrically 
 placed on the dock, and that the dock deck is horizontal. The horizontal distance 
 between the centres of the side tanks is 111 ft. 
 
 (a) Total moment of inertia of the horizontal section is 
 
 9, 400,000 + 2 (380 x 13' x 55 "5 2 + T \ x 380 x 13 3 ) = 9,400,000 + 30,430,000 + 139,000. 
 The volume of displacement 
 
 = 540,000 + 1,237,600 cubic feet. 
 The height of the metacentre above the centre of buoyancy is therefore 
 
 (b) When the keel is just clear of the water the moment of inertia is 
 30,569,000. 
 
 The volume displacement is 
 
 540 x 96 x 14-75 + 380 x 2 x 13 x (14-75 + 4-2) 
 
 = 930,000 cubic feet. 
 Therefore BM = 32-8 feet. 
 
 (c) When the pontoon deck is just above the surface of the water, 
 
 I = 30,569,000 + ~ T V x 510' x 96 3 
 
 = 70,269,000. 
 
 The volume displacement is 890,000 cubic feet. 
 Therefore BM = 79'8 feet. 
 
 The height of the centre of buoyancy above the bottom of the dock can be 
 determined by finding the centre of buoyancy of each of the parts of the dock, and 
 of the ship if it is in the water, and then taking moments about any axis. 
 
 For example. To find the height h of the centre of buoyancy of the dock and 
 the ship, when the ship just comes on the keel blocks. 
 
 The centre of buoyancy for the ship is at 15 feet above the bottom of the keel. 
 The centre of buoyancy of the bottom pontoon is at 7 -375' from the bottom. 
 5 ,, side pontoons ,, 24-125' ,, ,, 
 
 > ,, chambers ,, 17'94' 
 
FLOATING BODIES 35 
 
 Taking moments about the bottom of the dock 
 
 h (540,000 + 437,000 + 765,000 + 35,760) 
 = 540,000 x 33-75 + 765,000 x 7-375 
 + 437,000 x 24-125 + 35, 760 x 17 "95, 
 therefore h = 19 '1 feet. 
 
 For case (a) the metacentre is, therefore, 40-3' above the bottom of the dock. If 
 now the centre of gravity of the dock and ship is known the metacentric height 
 
 can be found. 
 
 EXAMPLES. 
 
 (1) A ship when fully loaded has a total burden of 10,000 tons. Find 
 the volume displacement in sea water. 
 
 (2) The sides of a ship are vertical near the water line and the area of 
 the horizontal section at the water line is 22,000 sq. feet. The total weight 
 of the ship is 10,000 tons when it leaves the river dock. 
 
 Find the difference in draught in the dock and at sea after the weight 
 of the ship has been reduced by consumption of coal, etc., by 1500 tons. 
 Let 9 be the difference in draught. 
 Then d x 22,000 = the difference in volume displacement 
 _ 10,000 x 2240 8500 x 2240 
 
 62-43 64 
 
 =6130 cubic feet. 
 Therefore 8 = '278 feet 
 
 =3'34 inches. 
 
 (3) The moment of inertia of the section at the water line of a boat 
 is 1200 foot 4 units; the weight of the boat is 11-5 tons. 
 
 Determine the height of the metacentre above the centre of buoyancy. 
 
 (4) A ship has a total displacement of 15.000 tons and a draught of 
 27 feet. 
 
 When the ship is lifted by a floating dock so that the depth of the bottom 
 of the keel is 16'5 feet, the centre of buoyancy is 10 feet from the bottom of 
 the keel and the displacement is 9000 tons. 
 
 The moment of inertia of the water-plane is 7,600,000 foot 4 units. 
 
 The horizontal section of the dock, at the plane 16'5 feet above the 
 bottom of the keel, consists of two rectangles 380 feet x 11 feet, the distance 
 apart of the centre lines of the rectangles being 114 feet. 
 
 The volume displacement of the dock at this level is 1,244,000 cubic feet. 
 
 The centre of buoyancy for the dock alone is 24'75 feet below the surface 
 of the water. 
 
 Determine (a) The centre of buoyancy for the whole ship and the dock. 
 
 (b) The height of the metacentre above the centre of buoyancy. 
 
 (5) A rectangular pontoon 60 feet long is to have a displacement of 
 220 tons, a free-board of not less than 3 feet, and the metacentre is not to 
 be less than 3 feet above the centre of gravity when the angle of heel 
 is 15 degrees. The centre of gravity coincides with the centre of figure. 
 
 Find the width and depth of the pontoon. 
 
 32 
 
36 HYDRAULICS 
 
 (6) A rectangular pontoon 24 feet wide, 50 feet long and 14 feet deep, 
 has a displacement of 180 tons. 
 
 A vertical diaphragm divides the pontoon longitudinally into two 
 compartments each 12 feet wide and 50 feet long. In the lower part 
 of each of these compartments there is water ballast, the surface of the 
 water being free to move. 
 
 Determine the position of the centre of gravity of the pontoon that it 
 may be stable for small displacements. 
 
 (7) Define "metacentric height" and show how to obtain it graphically 
 or otherwise. A ship of 16,000 tons displacement is 600 feet long, 60 feet 
 beam, and 26 feet draught. A coefficient of ^ may be taken in the moment 
 of inertia term instead of ^ to allow for the water-line section not being 
 a rectangle. The depth of the centre of buoyancy from the water line is 
 10 feet. Find the height of the metacentre above the water line and 
 determine the position of the centre of gravity to give a metacentric height 
 of 18 inches. [Lond. Un. 1906.] 
 
 (8) The total weight of a fully loaded ship is 5000 tons, the water line 
 encloses an area of 9000 square feet, and the sides of the ship are vertical 
 at the water line. The ship was loaded in fresh water. Find the change 
 in the depth of immersion after the ship has been sufficiently long at sea to 
 burn 500 tons of coal. 
 
 Weight of 1 cubic foot of fresh water 62 Ibs. 
 Weight of 1 cubic foot of salt water 64 Ibs. 
 
CHAPTER III. 
 
 FLUIDS IN MOTION. 
 
 28. Steady motion. 
 
 The motion of a fluid is said to be steady or permanent, when 
 the particles which succeed each other at any point whatever 
 have the same density and velocity, and are subjected to the same 
 pressure. 
 
 In practice it is probably very seldom that such a condition of 
 flow is absolutely realised, as even in the case of the water flowing 
 steadily along a pipe or channel, except at very low velocities, the 
 velocities of succeeding particles of water which arrive at any 
 point in the channel, are, as will be shown later, not the same 
 either in magnitude or direction. 
 
 For practical purposes, however, it is convenient to assume 
 that if the rate at which a fluid is passing through any finite area 
 is constant, then at all points in the area the motion is steady. 
 
 For example, if a section of a stream be taken at right angles 
 to the direction of flow of the stream, and the mean rate at which 
 water flows through this section' is constant, it is convenient 
 to assume that at any point in the section, the velocity always 
 remains constant both in magnitude and direction, although the 
 velocity at different points may not be the same. 
 
 Mean velocity. The mean velocity through the section, or the 
 mean velocity of the stream, is equal to the quantity of flow per 
 unit time divided by the area of the section. 
 
 29. Stream line motion. 
 
 The particles of a fluid are generally regarded as flowing along 
 definite paths, or, in other words, the fluid may be supposed to 
 flow in thread-like filaments, and when the motion is steady these 
 filaments may be supposed to be fixed in position. 
 
 In a pipe or channel of constant section, the filaments are 
 generally supposed to be parallel to the sides of the channel. 
 
38 
 
 HYDRAULICS 
 
 30. Definitions relating to flow of water. 
 
 Pressure head. The pressure head at a point in a fluid at rest 
 has been denned as the vertical distance of the point from the free 
 
 surface of the fluid, and is equal to , where p is the pressure per 
 
 sq. foot and w is weight per cubic foot of 
 the fluid. Similarly, the pressure head at 
 any point in a moving fluid at which the 
 
 /Y\ 
 
 pressure is p Ibs. per sq. foot, is feet, 
 
 and if a vertical tube, called a piezometer 
 tube, Fig. 31, be inserted in the fluid, it 
 will rise in the tube to a height h, which 
 equals the pressure head above the atmo- 
 spheric pressure. If p is the pressure per 
 sq. foot, above the atmospheric pressure, 
 
 h = , but if p is the absolute pressure per - 
 sq. foot, and p A the atmospheric pressure, 
 
 Datum,], Level 
 
 Fig. 31. 
 
 w w 
 
 Velocity head. If through a small area around the point B, 
 the velocity of the fluid is v feet per second, the velocity head is 
 
 5- , g being the acceleration due to gravity in feet per second per 
 
 second. 
 
 Position head. If the point B is at a height z feet above any 
 convenient datum level, the position head of the fluid at B above 
 the given datum is said to be z feet. 
 
 31. Energy per pound of water passing any section in 
 a stream line. 
 
 The total amount of work that can be obtained from every 
 pound of water passing the point B, Fig. 31, assuming it can fall to 
 the datum level and that no energy is lost, is 
 
 + I" + z ft. Ibs. 
 w 20 
 
 Proof. Work available due to pressure head. That the work 
 which can be done by the pressure head per pound is - foot 
 
 pounds can be shown as follows. 
 
 Imagine a piston fitting into the end of a small tube of cross 
 sectional area a, in which the pressure is h feet of water as in 
 
UNIVERSITY 
 
 FLUIDS IN MOTION 
 
 Fig. 32, and let a small quantity <3Q cubic feet of water enter the 
 tube and move the piston through a small dis- 
 tance dx. 
 
 Then dQ=a.dx. 
 
 The work done on the piston as it enters 
 will be 
 
 w . h . a . dx = w . 
 
 1 
 
 But the weight of dQ cubic feet is w . 8Q pounds, Fi 8- 32 - 
 
 and the work done per pound is, therefore, h, or foot pounds. 
 
 w 
 
 A pressure head h is therefore equivalent to h foot pounds of 
 energy per pound of water. 
 
 Work available due to velocity. When a body falls through 
 a height h feet, the work done on the body by gravity is h foot 
 pounds per pound. It is shown in books on mechanics that if the 
 body is allowed to fall freely, that is without resistance, the 
 velocity the body acquires in feet per second is 
 
 v= \/2gh, 
 
 v* z. * 
 
 Yg = h ' 
 
 And ,since no resistance is offered to the motion, the whole of 
 the work done on the body has been utilised in giving kinetic 
 
 energy to it, and therefore the kinetic energy per pound is ~- . 
 
 In the case of the fluid moving with velocity v, an amount of 
 
 v 2 
 energy equal to ~- foot pounds per pound is therefore available 
 
 before the velocity is destroyed. 
 
 Work available due to position. If a weight of one pound 
 falls through the height z the work done on it by gravity will be 
 z foot pounds, and, therefore, if the fluid is at ^height z feet above 
 any datum, as for example, water at a given height above the 
 sea level, the available energy on allowing the fluid to fall to 
 the datum level is z foot pounds per jpund. 
 
 32. Bernouilli's theorem. 
 
 In a steady moving stream of an incompressible fluid in which 
 the particles of fluid are moving in stream lines, and there is no 
 loss by friction or other causes 
 
 p v 2 
 
 + o~ + z 
 w 2g 
 
 is constant for all sections of the stream. This is a most important 
 theorem and should be carefully studied by the reader. 
 
40 
 
 HYDRAULICS 
 
 It has been shown in the last paragraph that this expression 
 represents the total amount of energy per pound of water flowing 
 through any section of a stream, and since, between any two 
 points in the stream no energy is lost, by the principle of the 
 conservation of energy it can at once be inferred that this 
 expression must be constant for all sections of a steady flowing 
 stream. A more general proof is as follows. 
 
 Let DB, Fig. 33, be the path of a particle of the fluid. 
 
 Fig. 33. 
 
 Imagine a small tube to be surrounding DE, and let the flow 
 in this be steady, and let the sectional area of the tube be so small 
 that the velocity through any section normal to DE is uniform. 
 
 Then the amount of fluid that flows in at D through the area 
 AB equals the amount that flows out at E through the area OF. 
 
 Let pv and 17 D , and p E and V E be the pressures and velocities at 
 D and E respectively, and A and a the corresponding areas of the 
 tube. 
 
 Let z be the height of D above some datum and z l the height 
 of E. 
 
 Then, if a quantity of fluid ABAiBi equal to 3Q enters at D, 
 and a similar quantity CFCiFi leaves at E, in a time 9, the 
 velocity at D is 
 
 and the velocity at E is 
 
 Aut 
 
 . 
 aot 
 
 The kinetic energy of the quantity of fluid uQ, entering at D 
 
FLUIDS IN MOTION 41 
 
 and that of the liquid leaving at E 
 
 Since the flow in the tube is steady, the kinetic energy o%the 
 portion ABCF does not alter, and therefore the increase of the 
 kinetic energy of the quantity dQ 
 
 w .c 
 
 The work done by gravity is the same as if ABBiAi fell to 
 i and therefore equals 
 
 The total pressure on the area AB is p D . A, and the work done 
 at D in time dt 
 
 and the work done by the pressure at E in time t 
 
 But the gain of kinetic energy must equal the work done, and 
 therefore 
 
 . E 2 - V) - wdQ, (z - Zi) + p D 9Q - p E dQ. 
 
 From which 
 
 V_V_ ,PD_^E 
 
 2g 2g ~ Zl+ w w 1 
 
 VE PE ^D 2 PD 
 
 or ^ + r*+ Zl = 7 ?-+^ + z = constant. 
 
 Zg w 2g w 
 
 From this theorem it is seen that, if at points in a steady 
 moving stream, a vertical ordinate equal to the velocity head plus 
 the pressure head is erected, the upper extremities of these 
 ordinates will be in the same horizontal plane, at a height H 
 
 T) V^ 
 
 equal to + ~- + z above the datum level. 
 w 2g 
 
 Mr Froude* has given some very beautiful experimental illus- 
 trations of this theorem. 
 
 In Fig. 34 water is taken from a tank or reservoir in which 
 the water is maintained at a constant level by an inflowing 
 stream, through a pipe of variable diameter fitted with tubes 
 at various points. Since the pipe is short it may be supposed to 
 be frictionless. If the end of the pipe is closed the water will rise 
 in all the tubes to the same level as the water in the reservoir, but 
 if the end C is opened, water will flow through the pipe and the 
 water surfaces in the tubes will be found to be at different levels. 
 * British Assoc. Report 1875. 
 
42 
 
 HYDRAULICS 
 
 The quantity of water flowing per second through the pipe can be 
 measured, and the velocities at A, B, and C can be found by 
 dividing this quantity by the cross-sectional areas of the pipe at 
 these points. 
 
 Fig. 34. 
 
 If to the' head of water in the tubes at A and B the ordinates 
 - and - be added respectively, the upper extremities of these 
 
 ordinates will be practically on the same level and nearly level 
 with the surface of the water in the reservoir, the small difference 
 being due to frictional and other losses of energy. 
 
 At C the pressure is equal to the atmospheric pressure, and 
 neglecting friction in the pipe, the whole of the work done by 
 gravity on any water leaving the pipe while it falls from the 
 surface of the water in the reservoir through the height H, which 
 is H ft. Ibs. per pound, is utilised in giving velocity of motion to 
 the water, and, as will be seen later, in setting up internal motions. 
 
 Neglecting these resistances, 
 
 Due to the neglected losses, the actual velocity measured will be 
 less than v c as calculated from this equation. 
 
 If at any point D in the pipe, the sectional area is less than the 
 area at C, the velocity will be greater than v c , and the pressure 
 will be less than the atmospheric pressure. 
 
 If v is the velocity at any section of the pipe, which is supposed 
 to be horizontal, the absolute pressure head at that section is 
 
 . = ^+H- = 2^ + ^- 
 w w 2g w 2g 2g' 
 
 p a being the atmospheric pressure at the surface of the water in 
 the reservoir. 
 
 At D the velocity V-Q is greater than v c and therefore p& is less 
 
FLUIDS IN MOTION 
 
 43 
 
 than p a . If coloured water be put into the vessel E, it will rise in 
 the tube DE to a height 
 
 w 
 
 w 
 
 2<7 2<7* 
 
 If the area at the section is so small, that p becomes negative, the 
 fluid will be in tension, and discontinuity of flow will take place. 
 
 If the fluid is water which has been exposed to the atmosphere 
 and which consequently contains gases in solution, these gases 
 will escape from the water if the pressure becomes less than the 
 tension of the dissolved gases, and there will be discontinuity even 
 before the pressure becomes zero. 
 
 Figs. 35 and 36 show two of Froude's illustrations of the 
 theorem. 
 
 Fig. 35. 
 
 Fig. 36. 
 
 At the section B, Fig. 36, the pressure head is h E and the 
 velocity head is 
 
 If a is the section of the pipe at A, and a\ at B, since there 
 is continuity of flow, 
 
 2 2 
 
 and 2^ + Ji A = ^- + h E = h. 
 
 If now a is made so that 
 
 the pressure head h A becomes equal to the atmospheric pressure, 
 and the pipe can be divided at A, as shown in the figure. 
 
 Professor Osborne Reynolds devised an interesting experiment, 
 to show that when the velocity is high, the pressure is small. 
 
 He allowed water to flow through a tube f inch diameter 
 under a high pressure, the tube being diminished at one section to 
 0'05 inch diameter. 
 
44 HYDRAULICS 
 
 At this diminished section, the velocity was very high and the 
 pressure fell so low that the water boiled and made a hissing 
 noise. 
 
 33. Venturi meter. 
 
 An application of Bernoulli's theorem is found in the Yenturi 
 meter, as invented by Mr Clemens Herschel*. The meter takes 
 its name from an Italian philosopher who in the last decade of the 
 lth century made experiments upon the flow of water through 
 conical pipes. In its usual form the Yenturi meter consists of two 
 truncated conical pipes connected together by a short cylindrical 
 pipe called the throat, as shown in Figs. 37 and 38. The meter is 
 inserted horizontally in a line of piping, the diameter of the large 
 ends of the frustra being equal to that of the pipe. 
 
 Piezometer tubes or other pressure gauges are connected to 
 the throat and to one or both of the large ends of the cones. 
 
 Let a be the area of the throat. 
 
 Let di be the area of the pipe or the large end of the cone 
 at A. 
 
 Let a 2 be the area of the pipe or the large end of the cone 
 atC. 
 
 Let p be the pressure head at the throat. 
 
 Let pi be the pressure head at the up-stream gauge A. 
 
 Let > 2 be the pressure head at the down-stream gauge C. 
 
 Let H and HI be the differences of pressure head at the throat 
 and large ends A and C of the cone respectively, or 
 
 H = 2-'-, 
 
 w w 
 
 and H, = -. 
 
 W W 
 
 Let Q be the flow through the meter in cubic feet per sec. 
 Let v be the velocity through the throat. 
 Let Vi be the velocity at the up-stream large end of cone A. 
 Let v z be the velocity at the down-stream large end of cone C. 
 Then, assuming Bernouilli's theorem, and neglecting friction, 
 
 and 
 
 . = = 
 
 w 2g w 2g w 
 
 TT v 9 v^ 
 
 If v 2 is equal to Vi, p 2 is theoretically equal to pi, but there is 
 always in practice a slight loss of head in the meter, the difference 
 Pi - p* being equal to this loss of head. 
 
 * Transactions Am.S.C.E., 1887. 
 
FLUIDS IN MOTION 
 
 45 
 
 ' 1 
 
46 
 
 HYDRAULICS 
 
 The velocity v is , and Vi is . 
 a, a\ 
 
 Therefore Q 2 (~ 2 ~ ^) = 20 . H, 
 
 \tl QJ\ / 
 
 and 
 
 Due to friction, and eddy motions that may be set up in the 
 meter, the discharge is slightly less than this theoretical value, or 
 
 Jc being a coefficient which has to be determined by experiment. 
 
 For a meter having a diameter of 25'5 inches at the throat and 
 54 inches at the large end of the cone, Herschel found the 
 following values for &, given in Table III, so that the coefficient 
 varies but little for a large variation of H. 
 
 TABLE III. 
 
 Herschel 
 
 Coker 
 
 Hfeet 
 
 k 
 
 Discharge 
 
 k 
 
 
 
 in cu. ft. 
 
 
 1 
 
 995 
 
 0418 
 
 9494 
 
 2 
 
 992 
 
 0319 
 
 9587 
 
 6 
 
 985 
 
 0254 
 
 9572 
 
 12 
 
 9785 
 
 0185 
 
 9920 
 
 18 
 
 977 
 
 0096 
 
 1-2021 
 
 23 
 
 970 
 
 0084 
 
 1-3583 
 
 Professor Coker*, from careful experiments on an exceedingly 
 well designed small Yenturi meter, Fig. 38, the area of the throat 
 of which was '014411 sq. feet, found that for small flows the 
 coefficient was very variable as shown in Table III. 
 
 These results show, as pointed out by Professor Coker from an 
 analysis of his own and Herschel's experiments on meters of 
 various sizes, that in large Venturi meters, the discharge is very 
 approximately proportional to the square root of the head, but for 
 small meters it only follows this law for high heads, and for low 
 heads they require special calibration. 
 
 Example. A Venturi meter having a diameter at the throat of 36 inches is 
 inserted in a 9 foot diameter pipe. 
 
 The pressure head at the throat gauge is 20 feet of water and at the pipe gauge 
 is 26 feet. 
 
 * Canadian Society of Civil Engineers, 1902. 
 
FLUIDS IN MOTION 
 
 Find the discharge, and the velocity of flow through the throat. 
 The area of the pipe is 63-5 sq. feet. 
 
 throat 7-05 
 The difference in pressure head at the two gauges is 6 feet. 
 
 63-5x7-05 
 Therefore Q = 
 
 47 
 
 = 137 c. ft. per second. 
 The velocity of flow in the pipe is 2-15 ft. per sec. 
 
 ,, ,, through the throat is 19-4 ft. per sec. 
 
 34. Steering of canal boats. 
 
 An interesting application of Bernoulli's theorem is to show 
 the effect of speed and position on the steering of a canal boat. 
 
 When a boat is moved at a high velocity along a narrow 
 and shallow canal, the boat tends to leave behind it a hollow 
 which is filled by the water rushing past the boat as shown 
 in Figs. 39 and 40, while immediately in front of the boat the 
 impact of the bow on the still water causes an increase in the 
 pressure and the water is " piled up " or is at a higher level than 
 the still water, and what is called a bow wave is formed. 
 
 Fig. 39. Fig. 41. 
 
 A 
 
 Fig. 40. 
 
 Let it be assumed that the water moves past the boat in 
 stream lines. 
 
 If vertical sections are taken at E and F, and the points E and 
 F are on the same horizontal line, by Bernoulli's theorem 
 
 w 2g w 2g ' 
 
 At E the water is practically at rest, and therefore V E is 
 zero, and 
 
 w ~ w 2g' 
 The surface at E will therefore be higher than at F. 
 
48 HYDRAULICS 
 
 When the boat is at the centre of the canal the stream lines on 
 both sides of the boat will have the same velocity, but if the boat 
 is nearer to one bank than the other, as shown in the figures, the 
 velocity v?> of the stream lines between the boat and the nearer 
 bank, Fig. 41, will be higher than the velocity V F on the other 
 side. But for each side of the boat 
 
 PV = PV + ^ = PP' + ^F 2 
 
 w w 2g w 2g ' 
 
 And since v$> is greater than VF, the pressure head p v is 
 greater than p F ', or in other words the surface of the water at 
 the right side D of the boat will be higher than on the left side B. 
 
 The greater pressure 011 the right side D tends to push the 
 boat towards the left bank A, and at high speeds considerably 
 increases the difficulty of steering. 
 
 This difficulty is diminished if the canal is made sufficiently 
 deep, so that flow can readily take place underneath the boat. 
 
 35. Extension of Bernouilli's theorem. 
 
 In deducing this theorem it has been assumed that the fluid 
 is a perfect fluid moving with steady motion and that there are no 
 losses of energy, by friction of the surfaces with which the fluid 
 may be in contact, or by the relative motion of consecutive ele- 
 ments of the fluid, or due to internal motions of the fluid. 
 
 In actual cases the value of 
 
 73 v 2 
 
 ~ + ^r + z 
 w 2g 
 
 diminishes as the motion proceeds. 
 
 If h f is the loss of head, or loss of energy per pound of fluid, 
 between any two given points A and B in the stream, then more 
 generally 
 
 ZA = ls f ............... . 
 
 w 2g w 2g 
 
 EXAMPLES. 
 
 (1) The diameter of the throat of a Venturi meter is f inch, and of 
 the pipe to which it is connected If inches. The discharge through the 
 meter in 20 minutes was found to be 314 gallons. 
 
 The difference in pressure head at the two gauges was 49 feet. 
 Determine the coefficient of discharge. 
 
 (2) A Venturi meter has a diameter of 4 ft. in the large part and 
 1-25 ft. in the throat. With water flowing through it, the pressure head is 
 100 ft. in the large part and 97 ft. at the throat. Find the velocity in the 
 small part and the discharge through the meter. Coefficient of meter 
 taken as unity. 
 
FLUIDS IN MOTION 49 
 
 (3) A pipe AB, 100 ft. long, has an inclination of 1 in 5. The head due 
 to the pressure at A is 45 ft., the velocity is 3 ft. per second, and the section 
 of the pipe is 3 sq. ft. Find the head due to the pressure at B, where the 
 section is 1| sq. ft. Take A as the lower end of the pipe. 
 
 (4) The suction pipe of a pump is laid at an inclination of 1 hi 5, and 
 water is pumped through it at 6 ft. per second. Suppose the air hi the 
 water is disengaged if the pressure falls to more than 10 Ibs. below 
 atmospheric pressure. Then deduce the greatest practicable length of 
 suction pipe. Friction neglected. 
 
 (5) Water is delivered to an inward-flow turbine under a head of 100 feet 
 (see Chapter IX). The pressure just outside the wheel is 25 Ibs. per 
 sq. inch by gauge. 
 
 Find the velocity with which the water approaches the wheel. Friction 
 neglected. 
 
 (6) A short conical pipe varying in diameter from 4' 6" at the large end 
 to 2 feet at the small end forms part of a horizontal water main. The 
 pressure head at the large end is found to be 100 feet, and at the small end 
 96-5 feet. 
 
 Find the discharge through the pipe. Coefficient of discharge unity. 
 
 (7) Three cubic feet of water per second flow along a pipe which as it 
 falls varies in diameter from 6 inches to 12 inches. In 50 feet the pipe 
 falls 12 feet. Due to various causes there is a loss of head of 4 feet. 
 
 Find (a) the loss of energy in foot pounds per minute, and in horse- 
 power, and the difference in pressure head at the two points 50 feet apart.. 
 (Use equation 1, section 35.) 
 
 (8) A horizontal pipe in which the sections vary gradually has sections 
 of 10 square feet, 1 square foot, and 10 square feet at sections A, B, and C. 
 The pressure head at A is 100 feet, and the velocity 3 feet per second. 
 Find the pressure head and velocity at B. 
 
 Given that in another case the difference of the pressure heads at A 
 and B is 2 feet. Find the velocity at A. 
 
 (9) A Venturi meter in a water main consists of a pipe converging to 
 the throat and enlarging again gradually. The section of main is 9 sq. ft. 
 and the area of throat 1 sq. ft. The difference of pressure in the main and 
 at the throat is 12 feet of water. Find the discharge of the main per hour. 
 
 (10) If the inlet area of a Venturi meter is n times the throat area, and 
 v and p are the velocity and pressure at the throat, and the inlet pressure 
 is mp, show that 
 
 and show that if p and mp are observed, v can be found. 
 
 L. H. 
 
CHAPTER IV. 
 
 FLOW OF WATER THROUGH ORIFICES AND 
 OVER WEIRS. 
 
 36. Plow of fluids through orifices. 
 
 The general theory of the discharge of fluids through orifices, 
 as for example the flow of steam and air, presents considerable 
 difficulties, and is somewhat outside the scope of this treatise. 
 Attention is, therefore, confined to the problem of determining the 
 quantity of water which flows through a given orifice in a given 
 time, and some of the phenomena connected therewith. 
 
 In what follows, it is assumed that the density of the fluid is 
 constant, the effect of small changes of temperature and pressure 
 in altering the density being thus neglected. 
 
 Consider a vessel, Fig. 42, filled with water, the free surface of 
 which is maintained at a constant level ; in the lower part of the 
 vessel there is an orifice AB. 
 
 Fig. 42. 
 
 Let it be assumed that although water flows into the vessel so 
 as to maintain a constant head, the vessel is so large that at some 
 surface CD, the velocity of flow is zero. 
 
 Imagine the water in the vessel to be divided into a number of 
 stream lines, and consider any stream line EF. 
 
 Let the velocities at E and F be V E and V F , the pressure heads 
 h E and h F , and the position heads above some datum, Z E and Z F , 
 respectively. 
 
FLOW THROUGH ORIFICES 51 
 
 Then, applying Bernoulli's theorem to the stream line EF, 
 
 If v? is zero, then 
 
 2 
 
 But from the figure it is seen that 
 is equal to h, and therefore 
 
 or VE 
 
 Since HE is the pressure head at E, the water would rise in 
 a tube having its end open at E, a height 7i E , and h may thus 
 be called following Thomson the fall of "free level for the 
 point E." 
 
 At some section GK near to the orifice the stream lines are all 
 practically normal to the section, and the pressure head will be 
 equal to the atmospheric pressure ; and if the orifice is small the fall 
 of free level for all the stream lines is H, the distance of the centre 
 of the section GK below the free surface of the water. If the 
 orifice is circular and sharp-edged, as in Figs. 44 and 45, the section 
 GK is at a distance, from the plane of the orifice, about equal to 
 its radius. For vertical orifices, and small horizontal orifices, 
 H may be taken as equal to the distance of the centre of the 
 orifice below the free surface. 
 
 The theoretical velocity of flow through the small section GK 
 is, therefore, the same for all the stream lines, and equal to the 
 velocity which a body will acquire, in falling, in a vacuum, 
 through a height, equal to the depth of the centre of the orifice 
 below the free surface of the water in the vessel. 
 
 The above is Thomson's proof of Torricelli's theorem, which 
 was discovered experimentally, by him, about 
 the middle of the 17th century. 
 
 The theorem is proved experimentally as 
 follows. 
 
 If the aperture is turned upwards, as in 
 Fig. 43, it is found that the water rises 
 nearly to the level of the water in the vessel, 
 and it is inferred, that if the resistance of the 
 air and of the orifice could be eliminated, the 
 jet would rise exactly to the level of the 
 surface of the water in the vessel. 
 
52 
 
 HYDRAULICS 
 
 Other experiments described on pages 5456, also show that, 
 with carefully constructed orifices, the mean velocity through the 
 orifice differs from \/2^H by. a very small quantity. 
 
 37. Coefficient of contraction for sharp-edged orifice. 
 
 If an orifice is cut in the flat side, or in the bottom of a vessel, 
 and has a sharp edge, as shown in Figs. 44 and 45, the stream lines 
 set up in the water approach the orifice in all directions, as shown 
 in the figure, and the directions of flow of the particles of water, 
 except very near the centre, are not normal to the plane of the 
 orifice, but they converge, producing a contraction of the jet. 
 
 i 
 
 Fig. 44. Fig. 45. 
 
 At a small distance from the orifice the stream lines become 
 practically parallel, but the cross sectional area of the jet is 
 considerably less than the area of the orifice. 
 
 If w is the area of the jet at this section and a the area of the 
 
 orifice the ratio - is called the coefficient of contraction and may 
 a 
 
 be denoted by c. Weisbach states, that for a circular orifice, the 
 jet has a minimum area at a distance from the orifice slightly less 
 than the radius of the orifice, and defines the coefficient of 
 contraction as this area divided by the area of the orifice. For a 
 circular orifice he gives to c the value 0*64. Recent careful 
 measurements of the sections of jets from horizontal and vertical 
 sharp-edged circular and rectangular orifices, by Bazin, the 
 results of some of which are shown in Table IY, show, however, 
 that the section of the jet diminishes continuously and in fact has 
 no minimum value. Whether a minimum occurs for square orifices 
 is doubtful. 
 
 The diminution in section for a greater distance than that 
 given by Weisbach is to be expected, for, as the jet moves away 
 from the orifice the centre of the jet falls, and the theoretical 
 velocity becomes */2g (H + y), y being the vertical distance between 
 the centre of the orifice and the centre of the jet. 
 
FLOW THROUGH ORIFICES 53 
 
 At a small distance away from the orifice, however, the stream 
 lines are practically parallel, and very little error is introduced in 
 the coefficient of contraction by measuring the stream near the 
 orifice. 
 
 Poncelet and Lesbros in 1828 found, for an orifice "20 m. square, 
 a minimum section of the jet at a distance of '3 m. from the orifice 
 and at this section c was '563. M. Bazin, in discussing these 
 results, remarks that at distances greater than 0*3 m. the section 
 becomes very difficult to measure, and although the vein appears 
 to expand, the sides become hollow, and it is uncertain whether 
 the area is really diminished. 
 
 Complete contraction. The maximum contraction of the jet 
 takes place when the orifice is sharp edged and is well removed 
 from the sides and bottom of the vessel. In this case the contrac- 
 tion is said to be complete. Experiments show, that for complete 
 contraction the distance from the orifice to the sides or bottom of 
 the vessel should not be less than one and a half to twice the least 
 diameter of the orifice. 
 
 Incomplete or suppressed contraction. An example of incom- 
 plete contraction is shown in Fig. 46, the lower edge of the 
 rectangular orifice being made level with the bottom of the vessel. 
 The same effect is produced by placing a horizontal plate in 
 the vessel level with the bottom of the orifice. The stream 
 lines at the lower part of the orifice are normal to its plane 
 and the contraction at the lower edge is consequently suppressed. 
 
 Fig. 46. Fig. 47. 
 
 Similarly, if the width of a rectangular orifice is made equal 
 to that of the vessel, or the orifice abed is provided with side walls 
 as in Fig. 47, the side or lateral contraction is suppressed. In any 
 case of suppressed contraction the discharge is increased, but, as 
 will be seen later, the discharge coefficient may vary more than 
 when the contraction is complete. To suppress the contraction 
 completely, the orifice must be made of such a form that the 
 stream lines become parallel at the orifice and normal to its plane. 
 
54 
 
 HYDRAULICS 
 
 Experimental determination of c. The section of the stream 
 from a circular orifice can be obtained with considerable accu- 
 racy by the apparatus shown in Fig. 49, which consists of a 
 ring having four radial set 
 screws of fine pitch. The 
 screws are adjusted until the 
 points thereof touch the jet. 
 M. Bazin has recently used an 
 octagonal frame with twenty- 
 four set screws, all radiating 
 to a common centre, to deter- 
 mine the form of the section 
 of jets from various kinds of 
 orifices. Fi g . 48> Fi g . 49. 
 
 The screws were adjusted 
 
 until they just touched the jet. The frame was then placed upon 
 a sheet of paper and the positions of the ends of the screws 
 marked upon the paper. The forms of the sections could then 
 be obtained, and the areas measured with considerable accuracy. 
 Some of the results obtained are shown in Table IV and also in 
 the section on the form of the liquid vein. 
 
 38. Coefficient of velocity for sharp-edged orifice. 
 
 The theoretical velocity through the contracted section is, as 
 shown in section 36, equal to \/20H, but the actual velocity 
 v l is slightly less than this due to friction at the orifice. The 
 
 ratio - l = Jc is called the coefficient of velocity. 
 
 Experimental determination of k. There are two methods 
 adopted for determining k experimentally. 
 
 First method. The velocity is determined by measuring the 
 discharge in a given time under a given head, and the cross 
 sectional area w of the jet, as explained in the last paragraph, is 
 also obtained. Then, if Vi is the actual velocity, and Q the 
 discharge per second, 
 
 and 
 
 
 
 
 Second method. An orifice, Fig. 50, is formed in the side of a 
 vessel and water allowed to flow from it. The water after leaving 
 the orifice flows in a parabolic curve. Above the orifice is fixed 
 a horizontal scale on which is a slider carrying a vertical scale, 
 to the bottom of which is clamped a bent piece of wire, with a sharp 
 
FLOW THROUGH ORIFICES 
 
 55 
 
 point. The vertical scale can be adjusted so that the point touches 
 the upper or lower surface of the jet, and the horizontal and vertical 
 distances of any point in the axis of the jet from the centre of the 
 orifice can thus be obtained. 
 
 Fig. 50. 
 
 Assume the orifice is vertical, and let Vi be the horizontal 
 velocity of flow. At a time t seconds after a particle has passed 
 the orifice, the distance it has moved horizontally is 
 
 X = Vit (1). 
 
 The vertical distance is 
 
 y = \qt\. (2). 
 
 Therefore 
 
 
 and 
 
 The theoretical velocity of flow is 
 
 Therefore 
 
 It is better to take two values of x and y so as to make 
 allowance for the plane of the orifice not being exactly perpen- 
 dicular. 
 
 If the orifice has its plane inclined at an angle 6 to the 
 vertical, the horizontal component of the velocity is v l cos and 
 the vertical component v\ sin 0. 
 
 At a time t seconds after a particle has passed the orifice, the 
 horizontal movement from the orifice is, 
 
 and the vertical movement is, 
 
 y = v l smet + 
 After a time h seconds x 1 = v 1 cosOt lL 
 
 (2). 
 
56 HYDRAULICS 
 
 Substituting the value of t from (1) in (2) and ti from (3) 
 in (4), 
 
 and, ^ ]tan , + _ ..................... (6) . 
 
 From (5), W = jt**fO 
 
 g y-x tan 
 
 Substituting for Vi in (6), 
 
 =^-^' 3 . ........................ (8). 
 
 Having calculated tan 0, sec can be found from mathematical 
 tables, and from (7) v\ can be calculated. Then 
 
 ft- 
 
 39. Bazin's experiments on a sharp-edged orifice. 
 
 In Table IV are given values of k as obtained by Bazin from 
 experiments on vertical and horizontal sharp-edged orifices, for 
 various values of the head. 
 
 The section of the jet at various distances from the orifice was 
 carefully measured by the apparatus described above, and the 
 actual discharge per second was determined by noting the time 
 taken to fill a vessel of known capacity. 
 
 The mean velocity through any section was then 
 
 Q being the discharge per second and A the area of the section. 
 
 The fall of free level for the various sections was different, and 
 allowance is made for this in calculating the coefficient k in the 
 fourth column. 
 
 Let y be the vertical distance of the centre of any section 
 below the centre of the orifice ; then the fall of free level for that 
 section is H + y and the theoretical velocity is 
 
 V2^(HT7). 
 
 The coefficients given in column 3 were determined by dividing 
 the actual mean velocity through different sections of the jet by 
 v 2gH, the theoretical velocity at the centre of the orifice. 
 
 Those in column 4 were found by dividing the actual mean 
 velocity through the section by \/2# (H + y), the theoretical 
 velocity at any section of the jet. 
 
 The coefficient of column 3 increases as the section is taken 
 further from the jet, and in nearly all cases is greater than unity. 
 
FLOW THROUGH ORIFICES 
 
 TABLE IV. 
 
 Sharp-edged Orifices Contraction Complete. 
 
 Table showing the ratio of the area of the jet to the area of 
 the orifice at definite distances from the orifice, and the ratio of 
 the mean velocity in the section to \/2#H and to \l2g . (H +y), 
 H being the head at the centre of the orifice and y the vertical 
 distance of the centre of the section of the jet from the centre of 
 the orifice. 
 
 Vertical circular orifice 0'20 m. ('656 feet) diameter, H = '990 m. 
 (3-248 feet). 
 
 Coefficient of discharge m, by actual measurement of the flow is 
 
 m = '5977*. 
 
 Distance of the section Area of Jet Mean Velocity Mean Velocity 
 
 from the plane of the 
 
 orifice in metres 
 
 0-08 
 
 0-13 
 
 0-17 
 
 0-235 
 
 0-335 
 
 0-515 
 
 Area of Orince 
 
 = c 
 
 6079 
 5971 
 5951 
 5904 
 5830 
 5690 
 
 0-983 
 1-001 
 1-004 
 1-012 
 1-025 
 1-050 
 
 998 
 
 999 
 
 1-003 
 
 1-007 
 
 1-010 
 
 Horizontal circular orifice 0'20 m. ('656 feet) diameter, 
 = '975m. (3-198 feet). 
 
 m = 0'6035. 
 
 0-075 
 0-093 
 0-110 
 0-128 
 0-145 
 0-163 
 
 0-6003 
 0-5939 
 0-5824 
 0-5734 
 0-5658 
 0-5597 
 
 1-005 
 1-016 
 1-036 
 1-053 
 1-067 
 1-078 
 
 0-968 
 0-971 
 0-982 
 0-990 
 0-996 
 0-998 
 
 Vertical orifice '20 m. ('656 feet) square, H = '953 m. (3126 feet). 
 m = 0-6066. 
 
 0-151 
 0-175 
 0-210 
 0-248 
 0-302 
 0-350 
 
 0-6052 
 0-6029 
 0-5970 
 0-5930 
 0-5798 
 0-5783 
 
 1-002 
 1-006 
 1-016 
 1-023 
 1-046 
 1-049 
 
 997 
 1-000 
 1-007 
 1-010 
 1-027 
 1-024 
 
 The real value of the coefficient for the various sections is 
 however that given in column 4. 
 
 For the horizontal orifice, for every section, it is less than 
 unity, but for the vertical orifice it is greater than unity. 
 
 Bazin's results confirm those of Lesbros and Poncelet, who m 
 
 * See section 42. 
 
58 HYDRAULICS 
 
 1828 found that the actual velocity through the contracted section 
 of the jet, even when account was taken of the centre of the 
 section of the jet being below the centre of the orifice, was 
 -gV greater than the theoretical value. 
 
 This result appears at first to contradict the principle of the 
 conservation of energy, and Bernoulli's theorem. 
 
 It should however be noted that the vertical dimensions of the 
 orifice are not small compared with the head, and the explanation 
 of the apparent anomaly is no doubt principally to be found in the 
 fact that the initial velocities in the different horizontal filaments 
 of the jet are different. 
 
 Theoretically the velocity in the lower part of the jet is greater 
 than \/2# (H + y\ and in the upper part less than v2<7 (H + y). 
 
 Suppose for instance a section of a jet, the centre of which is 
 1 metre below the free surface, and assume that all the filaments 
 have a velocity corresponding to the depth below the free surface, 
 and normal to the section. This is equivalent to assuming that 
 the pressure in the section of the jet is constant, which is probably 
 not true. 
 
 Let the jet be issuing from a square orifice of '2 m. ('656 feet) 
 side, and assume the coefficient of contraction is '6, and for 
 simplicity that the section of the jet is square. 
 
 Then the side of the jet is '1549 metres. 
 
 The theoretical velocity at the centre is v 20, and the discharge 
 assuming this velocity for the whole section is 
 
 "6 x -04 x v/2# - '024 J2g cubic metres. 
 
 The actual discharge, on the above assumption, through any 
 horizontal filament of thickness dh, and depth h, is 
 
 oQ = G'1549 xdfcx J2gh t 
 
 and the total discharge is 
 
 r 1-0775 
 Q - 01549 x/20 h*dh 
 
 J '9225 
 
 - '0241 J2g. 
 
 The theoretical discharge, taking account of the varying heads 
 is, therefore, 1'004 times the discharge calculated on the assumption 
 that the head is constant. 
 
 As the head is increased this difference diminishes, and when 
 the head is greater than 5 times the depth of the orifice, is very 
 small indeed. 
 
 The assumed data agrees very approximately with that given 
 in Table I for a square orifice, where the value of ~k is given as 
 1-006. 
 
FLOW THROUGH ORIFICES 59 
 
 This partly then, explains the anomalous values of k, but it 
 cannot be looked upon as a complete explanation. 
 
 The conditions in the actual jet are not exactly those assumed, 
 and the variation of velocity normal to the plane of the section is 
 probably much more complicated than here assumed. 
 
 As Bazin further points out, it is probable that, in jets like 
 those from the square orifice, which, as will be seen later when the 
 form of the jet is considered, are subject to considerable deformation, 
 the divergence of some of the filaments gives rise to pressures less 
 than that of the atmosphere. 
 
 Bazin has attempted to demonstrate this experimentally, and 
 his instrument, Fig. 150, registered pressures less than that of the 
 atmosphere; but he doubts the reliability of the results, and 
 points out the extreme difficulty of satisfactorily determining the 
 pressure in the jet. 
 
 That the inequality of the velocity of the filaments is the 
 primary cause, receives support from the fact that for the 
 horizontal orifice, discharging downwards, the coefficient fc is 
 always slightly less than unity. In this case, in any horizontal 
 section below the orifice, the head is the same for all the stream 
 lines, and the velocity of the filaments is practically constant. 
 The coefficient of velocity is never less than '96, so that the loss 
 due to the internal friction of the liquid is very small. 
 
 40. Distribution of velocity in the plane of the orifice. 
 Bazin has examined the distribution of the velocity in the 
 
 various sections of the jet by means of a fine Pitot tube (see 
 page 245). In the plane of the orifice a minimum velocity 
 occurs, which for vertical orifices is just above the centre, but at a 
 little distance from the orifice the minimum velocity is at the top 
 of the jet. 
 
 For orifices having complete contraction Bazin found the 
 minimum velocity to be '62 to '64 \/2#H, and for the rectangular 
 orifice, with lateral contraction suppressed, 0*69 \/2pH. 
 
 As the distance from the plane of the orifice increases, the 
 velocities in the transverse section of the jets from horizontal 
 orifices, rapidly become uniform throughout the transverse section. 
 
 For vertical orifices, the velocities below the centre of the jet 
 are greater than those in the upper part. 
 
 41. Pressure in the plane of the orifice. 
 
 M. Lagerjelm stated in 1826 that if a vertical tube open at 
 both ends was placed with its lower end near the centre, and not 
 perceptibly below the plane of the inner edge of a horizontal 
 
60 HYDRAULICS 
 
 orifice made in the bottom of a large reservoir, the water rose in 
 the tube to a height equal to that of the water in the reservoir, 
 that is the pressure at the centre of the orifice is equal to the head 
 over the orifice even when flow is taking place. 
 
 M. Bazin has recently repeated this experiment and found, 
 that the water in the tube did not rise to the level of the water in 
 the reservoir. 
 
 If Lager j elm's statement were correct it would follow that the 
 velocity at the centre of the orifice must be zero, which again does 
 not agree with the results of Bazin's experiments quoted above. 
 
 42. Coefficient of discharge. 
 
 The discharge per second from an orifice, is clearly the area 
 of the jet at the contracted section GrK multiplied by the mean 
 velocity through this section, and is therefore, 
 
 Q = c . k . a \/2#H. 
 Or, calling m the coefficient of discharge, 
 
 This coefficient m is equal to the product c.Jc. It is the only 
 coefficient required in practical problems and fortunately it can 
 be more easily determined than the other two coefficients c and Jc. 
 
 Experimental determination of the coefficient of discharge. 
 The most satisfactory method of determining the coefficient of 
 discharge of orifices is to measure the volume, or the weight of 
 water, discharged under a given head in a known time. 
 
 The coefficients quoted in the Tables from M. Bazin*, were 
 determined by finding accurately the time required to fill a vessel 
 of known capacity. 
 
 The coefficient of discharge m, has been determined with 
 a great degree of accuracy for sharp-edged orifices, by Poncelet 
 and Lesbrost, WeisbachJ, Bazin and others . In Table IY 
 Bazin's values for m are given. 
 
 The values as given in Tables Y and YI may be taken as 
 representative of the best experiments. 
 
 For vertical, circular and square orifices, and for a head of 
 about 3 feet above the centre of the orifice, Mr Hamilton Smith, 
 junr. ||, deduces the values of m given in Table YI. 
 
 * Annales des Fonts et Chaussees, October, 1888. 
 
 f Floiv through Vertical Orifices. 
 
 J Mechanics of Engineering. 
 
 Experiments upon the Contraction of the Liquid Vein. Bazin translated by 
 Trautwine. 
 
 || The Flow of Water through Orifices and over Weirs and through open Conduits 
 and Pipe*, Hamilton Smith, junr., 1886. 
 
FLOW THROUGH ORIFICES 
 
 TABLE Y. 
 
 61 
 
 Experimenter 
 
 Particulars of orifice 
 
 Coefficient of 
 discharge m 
 
 Bazin 
 
 Poncelet and 
 
 Vertical square orifice side of square 0*6562 ft. 
 
 0-606 
 
 Lesbros 
 
 i 
 
 0-605 
 
 Bazin 
 
 Vertical Rectangular orifice -656 ft. high x 2'624 
 ft. wide with side contraction suppressed 
 Vertical circular orifice 0*6562 ft. diameter 
 
 0-627 
 0-598 
 
 
 
 Horizontal } 
 
 0-6035 
 
 " 
 
 0-3281 
 
 0-6063 
 
 TABLE VI. 
 
 Circular orifices. 
 
 Diameter of 
 orifice in ft. 
 
 m 
 
 0-0197 
 0-627 
 
 0-0295 
 0-617 
 
 0-039 
 0-611 
 
 0-0492 
 0-606 
 
 0-0984 
 0-603 
 
 0-164 
 0-600 
 
 0-328 
 0-599 
 
 0-6562 
 0-598 
 
 0-9843 
 0-597 
 
 Square orifices. 
 
 Side of square 
 in feet 
 
 m 
 
 0-0197 
 0-631 
 
 0-0492 
 0-612 
 
 0-0984 
 0-607 
 
 0-197 
 0-605 
 
 0-5906 
 0-604 
 
 0-9843 
 0-603 
 
 TABLE VII. 
 
 Table showing coefficients of discharge for square and rect- 
 angular orifices as determined by Poncelet and Lesbros. 
 
 
 Width of orifice -6562 feet 
 
 Width of orifice 
 1 -968 feet 
 
 Head of water 
 
 
 
 above the top 
 
 
 of the orifice 
 
 Depth of orifice in feet 
 
 in feet 
 
 
 
 0328 
 
 0656 
 
 0984 
 
 1640 
 
 3287 
 
 6562 
 
 0656 
 
 6562 
 
 0328 
 
 701 
 
 660 
 
 630 
 
 607 
 
 
 
 
 
 0656 
 
 694 
 
 659 
 
 634 
 
 615 
 
 596 
 
 572 
 
 643 
 
 
 1312 
 
 683 
 
 658 
 
 640 
 
 623 
 
 603 
 
 582 
 
 642 
 
 595 
 
 2624 
 
 670 
 
 656 
 
 638 
 
 629 
 
 610 
 
 589 
 
 640 
 
 601 
 
 3937 
 
 663 
 
 653 
 
 636 
 
 630 
 
 612 
 
 593 
 
 638 
 
 603 
 
 6562 
 
 655 
 
 648 
 
 633 
 
 630 
 
 615 
 
 598 
 
 635 
 
 605 
 
 1-640 
 
 642 
 
 638 
 
 630 
 
 627 
 
 617 
 
 604 
 
 630 
 
 607 
 
 3-281 
 
 632 
 
 633 
 
 628 
 
 626 i -615 
 
 605 
 
 626 
 
 605 
 
 4-921 
 
 615 
 
 619 
 
 620 
 
 620 
 
 611 
 
 602 
 
 623 
 
 602 
 
 6-562 
 
 611 
 
 612 
 
 612 
 
 613 
 
 607 
 
 601 
 
 620 
 
 602 
 
 9-843 
 
 609 
 
 610 
 
 608 
 
 606 
 
 603 
 
 601 
 
 615 
 
 601 
 
62 HYDRAULICS 
 
 The heads for which Bazin determined the coefficients in 
 Tables IY and V varied only from 2'6 to 3*3 feet, but, as will be 
 seen from Table VII, deduced from results given by Poncelet and 
 Lesbros* in their classical work, when the variation of head is not 
 small, the coefficients for rectangular and square orifices vary 
 considerably with the head. 
 
 43. Effect of suppressed contraction on the coefficient 
 of discharge. 
 
 Sharp-edged orifice. When some part of the contraction of a 
 transverse section of a jet issuing from an orifice is suppressed, 
 the cross sectional area of the jet can only be obtained with 
 difficulty. 
 
 The coefficient of discharge can, however, be easily obtained, 
 as before, by determining the discharge in a given time. The 
 most complete and accurate experiments on the effect of contrac- 
 tion are those of Lesbros, some of the results of which are quoted 
 in Table VIII. The coefficient is most constant for square or 
 rectangular orifices when the lateral contraction is suppressed. The 
 reason being, that whatever the head, the variation in the section 
 of the jet is confined to the top and bottom of the orifice, the 
 width of the stream remaining constant, and therefore in a greater 
 part of the transverse section the stream lines are normal to the 
 plane of the orifice. 
 
 According to Bidone, if x is the fraction of the periphery of a 
 sharp-edged orifice upon which the contraction is suppressed, and 
 m the coefficient of discharge when the contraction is complete, 
 then the coefficient for incomplete contraction is, 
 
 for rectangular orifices, and 
 
 mi = m (1 
 for circular orifices. 
 
 Bidone's formulae give results agreeing fairly well with 
 Lesbros' experiments. 
 
 His formulae are, however, unsatisfactory when x approaches 
 unity, as in that case mi should be nearly unity. 
 
 If the form of the formula is preserved, and m taken as '606, 
 for mi to be unity it would require to have the value, 
 
 For accurate measurements, either orifices with perfect con- 
 traction or, if possible, rectangular or square orifices with the 
 lateral contraction completely suppressed, should be used. It will 
 
 * Experiences hydrauliques sur les lois de Vecoulement de Veau a travers les 
 orifices, etc., 1832. Poncelet and Lesbros. 
 
FLOW THROUGH ORIFICES 
 
 63 
 
 generally be necessary to calibrate the orifice for various heads, 
 but as shown above the coefficient for the latter kind is more 
 likely to be constant. 
 
 TABLE VIII. 
 
 Table showing the effect of suppressing the contraction on the 
 coefficient of discharge. Lesbros*. 
 
 Square vertical orifice 0'656 feet square. 
 
 Head of water 
 above the upper 
 edge of the orifice 
 
 Sharp-edged 
 
 Side con- 
 traction 
 suppressed 
 
 Contraction 
 suppressed at 
 the lower edge 
 
 Contraction 
 suppressed at 
 the lower and 
 side edges 
 
 0-6562 
 
 0-572 
 
 
 0-599 
 
 
 0-1640 
 
 0-585 
 
 0-631 
 
 0-608 
 
 
 0-3281 
 
 0-592 
 
 0-631 
 
 0-615 
 
 
 0-6562 
 
 0-598 
 
 0-632 
 
 0-621 
 
 0-708 
 
 1-640 
 
 0-603 
 
 0-631 
 
 0-623 
 
 0-680 
 
 3-281 
 
 0-605 
 
 0-628 
 
 0-624 
 
 0-676 
 
 4-921 
 
 0-602 
 
 0-627 
 
 0-624 
 
 0-672 
 
 6-562 
 
 0-601 
 
 0-626 
 
 0-619 
 
 0-668 
 
 9-843 
 
 0-601 
 
 0-624 
 
 0-614 
 
 0-665 
 
 circular orifice. 
 * .4 
 
 44. The form of the jet from sharp-edged orifices. 
 
 From a circular orifice the jet emerges like a cylindrical rod 
 and retains a form nearly cylindrical for some distance from the 
 orifice. 
 
 Fig. 51 shows three sections of a jet from a vertical circular 
 orifice at varying distances from the Fig. 51. Section of jet from 
 orifice, as given by M. Bazin. 
 
 The flow from square orifices is 
 accompanied by an interesting and 
 curious phenomenon called the in- 
 version of the jet. 
 
 At a very small distance from 
 the orifice the section becomes as 
 shown in Fig. 52. The sides of the 
 jet are concave and the corners are 
 cut off by concave sections. The Figs. 5254. Section of jet from 
 section then becomes octagonal as in 8 4 uare orifice " 
 
 Fig. 53 and afterwards takes the form of a square with concave 
 sides and rounded corners, the diagonals of the square being 
 perpendicular to the sides of the orifice, Fig. 54. 
 
 * Experiments hydrauliques sur les lois de Vecoulement de Veau. 
 
64 
 
 HYDRAULICS 
 
 45. Large orifices. 
 
 Table YII shows very clearly that if the depth of a vertical orifice 
 is not small compared with the head, the coefficient of discharge 
 varies very considerably with the head, and in the discussion of 
 the coefficient of velocity k, it has already been shown that the 
 distribution of velocity in jets issuing from such orifices is not 
 uniform. As the jet moves through a large orifice the stream 
 lines are not normal to its plane, but at some section of the stream 
 very near to the orifice they are practically normal. 
 
 If now it is assumed that the pressure is constant and equal to 
 the atmospheric pressure and that the shape of this section is 
 known, the discharge through it can be calculated. 
 
 Rectangular orifice. Let efgh, Fig. 55, be the section by a 
 vertical plane EF of the stream issuing from a vertical rectangular 
 orifice. Let the crest E of the stream be at a depth h below 
 the free surface of the water in the vessel and the under edge 
 F at a depth h 2 . 
 
 
 
 ^ 
 
 
 e K 
 
 
 
 
 
 f ff 
 
 
 \ J 
 
 ^x 
 
 
 i 
 
 r B 
 
 Fig. 55. 
 
 At any depth h, since the pressure is assumed constant in the 
 section, the fall of free level is h, and the velocity of flow through 
 the strip of width dh is therefore, kv2gh, and the discharge is 
 
 If Jc be assumed constant for all the filaments the total discharge 
 in cubic feet per second is 
 
 Q - 
 
 Here at once a difficulty is met with. The dimensions h , hi 
 and b cannot easily be determined, and experiment shows that 
 they vary with the head of water over the orifice, and that they 
 cannot therefore be written as fractions of H , Hi, and B. 
 
FLOW THROUGH ORIFICES 
 
 65 
 
 By replacing h , h^ and b by H , Hj and B an empirical 
 formula of the same form is obtained which, by introducing a 
 coefficient c, can be made to agree with experiments. Then 
 
 or replacing f c by n, 
 
 (1). 
 
 The coefficient n varies with the head H , and for any orifice 
 the simpler formula 
 
 Q=m.a.N/2<7H .............................. (2), 
 
 a being the area of the orifice and H the head at the centre, 
 can be used with equal confidence, for if n is known for the 
 particular orifice for various values of H , ra will also be known. 
 
 From Table VII probable values of m for any large sharp- 
 edged rectangular orifices can be interpolated. 
 
 Rectangular sluices. If the lower edge of a sluice opening is 
 some distance above the bottom of the channel the discharge 
 through it will be practically the same as through a sharp-edged 
 orifice, but if it is flush with the bottom of the channel, the 
 contraction at this edge is suppressed and the coefficient of 
 discharge will be slightly greater as shown in Table VIII. 
 
 46. Drowned orifices. 
 
 When an orifice is submerged as in Fig. 56 and the water in 
 the up-stream tank or reservoir is moving so slowly that its velocity 
 may be neglected, the head causing velocity of flow through any 
 filament is equal to the difference of the up- and down-stream 
 levels. Let H be the difference of level of the water on the twa 
 sides of the orifice. 
 
 Fig. 56. 
 
 L. H. 
 
66 
 
 HYDRAULICS 
 
 Consider any stream line FE which passes through the orifice 
 at E. The pressure head at E is equal to h 2 , the depth of E below 
 the down-stream level. If then at F the velocity is zero, 
 
 or 
 
 or taking a coefficient of velocity 
 
 which, since H is constant, is the same for all filaments of the 
 orifice. 
 
 If the coefficient of contraction is c the whole discharge through 
 the orifice is then 
 
 Q = 
 
 47. Partially drowned orifice. 
 
 If the orifice is partially drowned, as in 
 Fig. 57, the discharge may be considered in 
 two parts. Through the upper part AC the 
 discharge, using (2) section 45, is 
 
 and through the lower part BC 
 
 Hx. B 
 
 48. Velocity of approach. 
 
 It is of interest to consider the effect of the 
 water approaching an orifice having what is Fl &- 57 - 
 
 called a velocity of approach, which will be equal to the velocity 
 of the water in the stream above the orifice. 
 
 In Fig. 56 let the water at F approaching the drowned orifice 
 have a velocity VL 
 
 Bernoulli's equation for the stream line drawn is then 
 
 V . * V 
 
 ^ ' "H fl<> li ~l~ ^ 
 
 and 
 
 which is again constant for all filaments of the orifice. 
 Then Q = m.c 
 
SUDDEN ENLARGEMENT OF A STREAM 67 
 
 49. Effect of velocity of approach on the discharge 
 through a large rectangular orifice. 
 
 If the water approaching the large orifice, Fig. 55, has 
 a velocity of approach v 1} Bernoulli's equation for the stream line 
 passing through the strip at depth h, will be 
 
 fe + *-fe +*+*', 
 
 W 2ff . ID 2g' 
 
 p a being the atmospheric pressure, or putting in a coefficient of 
 velocity, 
 
 The discharge through the orifice is now, 
 
 50. Coefficient of resistance. 
 
 In connection with the flow through orifices, and hydraulic 
 plant generally, the term " coefficient of resistance " is frequently 
 used. Two meanings have been attached to the term. Some- 
 times it is defined as the ratio of the head lost in a hydraulic 
 system to the effective head, and sometimes as the ratio of the 
 head lost to the total head available. According to the latter 
 method, if H is the total head available and h f the head lost, 
 the coefficient of resistance is 
 
 51. Sudden enlargement of a current of water. 
 
 It seems reasonable to proceed from the consideration of flow 
 through orifices to that of the flow through mouthpieces, but 
 before doing so it is desirable that the effect of a sudden 
 enlargement of a stream should be considered. 
 
 Suppose for simplicity that a pipe as 
 in Fig. 58 is suddenly enlarged, and that 
 there is a continuous sinuous flow along 
 the pipe. (See section 284.) 
 
 At the enlargement of the pipe, the 
 stream suddenly enlarges, and, as shown 
 in the figure, in the corners of the large 
 pipe it may be assumed that eddy motions 
 are set up which cause a loss of energy. 
 
 52 
 
68 HYDRAULICS 
 
 Consider two sections aa and dd at such a distance from 66 
 that the flow is steady. 
 
 Then, the total head at dd equals the total head at aa minus 
 the loss of head between aa and dd, or if h is the loss of head due 
 to shock, then 
 
 & + v/. = p + vji +h . 
 
 w 2g w 2g 
 
 Let A and A^ be the area at aa and dd respectively. 
 Since the flow past aa equals that past dd, 
 
 Then, assuming that each filament of fluid at aa has the 
 velocity v a , and v d at dd, the momentum of the quantity of water 
 
 nt\ 
 
 which passes aa in unit time is equal to A^Va, and the momentum 
 
 y 
 of the water that passes dd is 
 
 the momentum of a mass of M pounds moving with a velocity 
 v feet per second being M.V pounds feet. 
 The change of momentum is therefore, 
 
 The forces acting on the water between aa and dd to produce 
 this change of momentum, are 
 
 p a A a acting on aa, p d ^. d acting on dd, 
 
 and, if p is the mean pressure per unit area on the annular ring 
 66, an additional force p(A. d A a ). 
 
 There is considerable doubt as to what is the magnitude of the 
 pressure p, but it is generally assumed that it is equal to p a , for 
 the following reason. 
 
 The water in the enlarged portion of the pipe may be looked 
 upon as divided into two parts, the one part having a motion of 
 translation, while the other part, which is in contact with the 
 annular ring, is practically at rest. (See section 284.) 
 
 If this assumption is correct, then it is to be expected that the 
 pressure throughout this still water will be practically equal at all 
 points and in all directions, and must be equal to the pressure in 
 the stream at the section 66, or the pressure p is equal to p a . 
 
 Therefore 
 
 v 
 
 - A a ) - p a A^ = W 
 
 y 
 from which (p d - p a ) A d = w (v a - v d ) ; 
 
SUDDEN ENLARGEMENT OF A STREAM 
 
 69 
 
 and since 
 therefore 
 
 w w g g 
 
 v 
 
 Adding -?j- to both sides of the equation and separating 
 
 . 2 
 
 - into two parts, 
 
 w 2g w 2g "" 20 ' 
 or /& the loss of head due to shock is equal to 
 
 According to St Venant this quantity should be increased by 
 an amount equal to ~ ~- , but this correction is so small that as 
 a rule it can be neglected. 
 
 52. Sudden contraction of a current of water. 
 
 Suppose a pipe partially closed by means of a diaphragm as in 
 Fig. 59. 
 
 As the stream approaches the diaphragm 
 which is supposed to be sharp-edged 
 it contracts in a similar way to the stream 
 passing through an orifice on the side of 
 a vessel, so that the minimum cross sec- 
 tional area of the flow will be less than the 
 area of the orifice. 
 
 The loss of head due to this contraction, or due to passing 
 through the orifice is small, as seen in section 39, but due to 
 the sudden enlargement of the stream to fill the pipe again, there 
 is a considerable loss of head. 
 
 Let A be the area of the pipe and a of the orifice, and let c be 
 the coefficient of contraction at the orifice. 
 
 Then the area of the stream at the contracted section is ca, and, 
 therefore, the loss of head due to shock 
 
 Fig. 59. 
 
 \ ca 
 
70 HYDRAULICS 
 
 If the pipe simply diminishes in diameter as in Fig. 58, the 
 section of the stream enlarges from the contracted area ca to fill 
 the pipe of area a, therefore the loss of head in this case is 
 
 Or making St Venant correction 
 
 ' ..................... * 
 
 Value of the coefficient c. The mean value of c for a sharp-edged 
 circular orifice is, as seen in Table IY, about 0'6, and this may be 
 taken as the coefficient of contraction in this formula. 
 
 Substituting this value in equation (1) the loss of head is 
 
 found to be -- , and in equation (2), - - , v being the velocity in 
 *9 ' ^9 
 
 C\* K. 2 
 
 the small pipe. It may be taken therefore as -~ . Further 
 experiments are required before a correct value can be assigned. 
 
 53. Loss of head due to sharp-edged entrance into a pipe 
 or mouthpiece. 
 
 When water enters a pipe or mouthpiece from a vessel through 
 a sharp-edged entrance, as in Fig. 61, there is first a contraction, and 
 then an enlargement, as in the second case considered in section 52. 
 
 The loss of head may be, therefore, taken as approximately ~ 
 
 O'SOSv 2 
 and this agrees with the experimental value of ~ -- given by 
 
 Weisbach. 
 
 This value is probably too high for small pipes and too low for 
 large pipes*. 
 
 54, Mouthpieces. 
 
 If an orifice is provided with a short pipe or mouthpiece, through 
 which the liquid can flow, the discharge may be very different 
 from that of a sharp-edged orifice, the difference depending upon 
 the length and form of the mouthpiece. If the orifice is cylindrical 
 as shown in Fig. 60, being sharp at the inner edge, and so short 
 that the stream after converging at the inner edge clears the 
 outer edge, it behaves as a sharp-edged orifice. 
 
 Short external cylindrical mouthpieces. If the mouthpiece is 
 cylindrical as ABFE, Fig. 61, having a sharp edge at AB and 
 a length of from one and a half to twice its diameter, the jet 
 
 * See M. Bazin, Experiences nouvelles sur la distribution des vitesses dans 
 les tuyaux. 
 
FLOW THROUGH MOUTHPIECES 
 
 71 
 
 contracts to CD, and then expands to fill the pipe, so that at EF 
 it discharges full bore, and the coefficient of contraction is then 
 unity. Experiment shows, that the coefficient of discharge is 
 
 Fig. 60. 
 
 Fig. 61. 
 
 from 0'80 to 0*85, the coefficient diminishing with the diameter 
 of the tube. The coefficient of contraction being unity, the 
 coefficients of velocity and discharge are equal. Good mean 
 values, according to Weisbach, are 0*815 for cylindrical tubes, 
 and 0*819 for tubes of prismatic form. 
 
 These coefficients agree with those determined on the assump- 
 tion that the only head lost in the mouthpiece is that due to 
 sudden enlargement, and is 
 
 0*5 
 
 v being the velocity of discharge at EF. 
 
 Applying Bernoulli's theorem to the sections CD and EF, and 
 
 '5v 2 
 
 taking into account the loss of head of -~ , and p a as the atmo- 
 spheric pressure, 
 
 PCD , ^CD 2 Pa , ^ , 'V* _ TT Pa 
 r p: = r p: P ~p: -CL T , 
 
 w 2g w 2g 2g w 
 
 or 
 
 Therefore 
 
 -66 x 
 
 and v 
 
 The area of the jet at EF is a, and therefore, the discharge 
 per second is 
 
 Or m, the coefficient of discharge, is 0*812. 
 The pressure head at the section CD. Taking the area at CD 
 as 0*606 the area at EF, 
 
72 HYDRAULICS 
 
 Therefore C->.fc + "^ , 
 
 w w Zg Zg w Zg 
 
 or the pressure at C is less than the atmospheric pressure. 
 
 If a pipe be attached to the mouthpiece, as in Fig. 61, and the 
 lower end dipped in water, the water should rise to a height of about 
 
 1'22-y 2 
 
 2 feet above the water in the vessel. 
 
 55. Borda's mouthpiece. 
 
 A short cylindrical mouthpiece projecting into the vessel, as in 
 Fig. 62, is called a Borda's mouthpiece, and is of interest, as the 
 coefficient of discharge upon certain assumptions can be readily 
 calculated. Let the mouthpiece be so short 
 that the jet issuing at EF falls clear of GrH. 
 The orifice projecting into the liquid has 
 the effect of keeping the liquid in contact 
 with the face AD practically at rest, and 
 at all points on it except the area EF the 
 hydrostatic pressure will, therefore, simply 
 depend upon the depth below the free 
 
 surface AB. Imagine the mouthpiece produced to meet the 
 face BC in the area IK. Then the hydrostatic pressure on AD, 
 neglecting EF, will be equal to the hydrostatic pressure on BC, 
 neglecting IK. 
 
 Again, BC is far enough away from EF to assume that the 
 pressure upon it follows the hydrostatic law. 
 
 The hydrostatic pressure on IK, therefore, is the force which 
 gives momentum to the water escaping through the orifice, over- 
 comes the pressure on EF, and the resistance of the mouthpiece. 
 
 Let H be the depth of the centre of the orifice below the free 
 surface and p the atmospheric pressure. Neglecting frictional 
 resistances, the velocity of flow v, through the orifice, is X/20H. 
 
 Let a be the area of the orifice and <o the area of the transverse 
 section of the jet. The discharge per second will be w . w \/2gH Ibs. 
 
 The hydrostatic pressure on IK is 
 
 pa + waH. Ibs. 
 
 The hydrostatic pressure on EF is pa Ibs. 
 
 The momentum given to the issuing water per second, is 
 
 Therefore pa + w 2gR = pa + waK, 
 
 and w = a. 
 
FLOW THROUGH MOUTHPIECES 73 
 
 The coefficient of contraction is then, in this case, equal to 
 one half. 
 
 Experiments by Borda and others, show that this result is 
 justified, the experimental coefficient being slightly greater 
 than J. 
 
 56. Conical mouthpieces and nozzles. 
 
 These are either convergent as in Fig. 63, or divergent as in 
 Fig. 64. 
 
 Fig. 63. 
 
 Fig. 64. 
 
 Calling the diameter of the mouthpiece the diameter at the 
 outlet, a divergent tube gives a less, and a convergent 
 tube a greater discharge than a cylindrical tube of the 
 same diameter. 
 
 Experiments show that the maximum discharge for a 
 convergent tube is obtained when the angle of the cone 
 is from 12 to 13J degrees, and it is then 0'94 . a . J2gh. 
 If, instead of making the convergent mouthpiece conical, 
 its sides are curved as in Fig. 65, so that it follows as 
 near as possible the natural form of the stream lines, the 
 coefficient of discharge may, with high heads, approxi- 
 mate very nearly to unity. 
 
 Weisbach*, using the method described on page 55 
 to determine the velocity of flow, obtained, for this 
 mouthpiece, the following values of k. Since the mouth- Fig< 65 
 piece discharges full the coefficients of velocity k and 
 discharge m are practically equal. 
 
 Head in feet 
 k and m 
 
 0-66 
 959 
 
 1-64 
 967 
 
 11-48 
 975 
 
 55-8 
 994 
 
 338 
 994 
 
 According to Freeman t, the fire-hose nozzle shown in Fig. 66 
 has a coefficient of velocity of "977. 
 
 * Mechanics of Engineering. 
 
 f Transactions Am. Soc. C.E., Vol. xxi. 
 
74 HYDRAULICS 
 
 If the mouthpiece is first made convergent, and then divergent, 
 
 Fig. 66. 
 
 as in Fig. 67, the divergence being sufficiently gradual for the 
 stream lines to remain in contact with the tube, the coefficient of 
 contraction is unity and there is but a 
 small loss of head. The velocity of efflux 
 from EF is then nearly equal to \/2grH 
 and the discharge is m . a . v/2011, a being 
 the area of EF, and the coefficient m 
 approximates to unity. 
 
 It would appear, that the discharge 
 could be increased indefinitely by length- 
 ening the divergent part of the tube and 
 
 Fig. 67. 
 
 thus increasing a, but as the length increases, the velocity 
 decreases due to the friction of the sides of the tube, and further, 
 as the discharge increases, the velocity through the contracted 
 section CD increases, and the pressure head at CD consequently 
 falls. 
 
 Calling p a the atmospheric pressure, pi the pressure at CD, 
 and Vi the velocity at CD, then 
 
 w 2g 
 
 w 
 
 and 
 
 Fi_j + /^a_ "1 
 
 w w 2g ' 
 
 Pa 
 
 If - is greater than H + , p x becomes negative. 
 
 
 As pointed out, however, in connection with Froude's apparatus, 
 page 43, if continuity is to be maintained, the pressure cannot be 
 negative, and in reality, if water is the fluid, it cannot be less 
 than 3- the atmospheric pressure, due to the separation of the air 
 from the water. The velocity Vi cannot, therefore, be increased 
 indefinitely. 
 
FLOW THROUGH MOUTHPIECES 
 
 75 
 
 Assuming the pressure can just become zero, and taking the 
 atmospheric pressure as equivalent to a head of 34 ft. of water, the 
 maximum possible velocity, is 
 
 and the maximum ratio of the area of EF to CD is 
 
 34ft. 
 H ' 
 
 Practically, the maximum value of Vi may be taken as 
 
 and the maximum ratio of EF to CD as 
 
 24 ft. 
 
 H ' 
 
 The maximum discharge is 
 
 a J2g (H + 24) 
 = m. y v ' 
 
 The ratio given of EF to CD may be taken as the maximum 
 ratio between the area of a pipe and the throat of a Yenturi meter 
 to be used in the pipe. 
 
 5 7. Flow through orifices and mouthpieces under constant 
 pressure. 
 
 The head of water causing flow through an orifice may be 
 produced by a pump or other mechanical means, and the discharge 
 may take place into a vessel, such as the condenser of a steam 
 engine, in which the pressure is less than that of the atmosphere. 
 
 For example, suppose water to be discharged from a cylinder 
 A, into a vessel B, Fig. 68, through 
 an orifice or mouthpiece by means 
 of a piston loaded with P Ibs., and 
 let the pressure per sq. foot in B 
 be po Ibs. 
 
 Let the area of the piston be 
 A square feet. Let h be the height 
 of the water in the cylinder above 
 the centre of the orifice and h of 
 the water in the vessel B. The 
 
 Fig. 68. 
 
 theoretical effective head forcing water through the orifice may 
 be written 
 
 H= r- 
 
 A.W 
 
 w 
 
76 HYDRAULICS 
 
 If P is large h and h will generally be negligible. 
 At the orifice the pressure head is h + , and therefore for 
 any stream line through the orifice, if there is no friction, 
 
 ^ +h+ p- p +h 
 
 d ~ /t/O T . Til/ 
 
 2g w A.W 
 
 tf P , , p, 
 
 or fr- = -7 + h-ho . 
 
 2g Aw w 
 
 The actual velocity will be less than v t due to friction, and if Jc 
 is a coefficient of velocity, the velocity is then 
 
 and the discharge is Q = m . a 
 
 In practical examples the cylinder and the vessel will generally 
 be connected by a short pipe, for which the coefficient of velocity 
 will depend upon the length. 
 
 If it is only a few feet long the principal loss of head will be 
 at the entrance to the pipe, and the coefficient of discharge will 
 probably vary between 0'65 and 0*85. 
 
 The effect of lengthening the pipe will be understood after the 
 chapter on flow through pipes has been read. 
 
 Example. Water is discharged from a pump into a condenser in which the 
 pressure is 3 Ibs. per sq. inch through a short pipe 3 inches diameter. 
 The pressure in the pump is 20 Ibs. per sq. inch. 
 
 Find the discharge into the condenser, taking the coefficient of discharge 0*75. 
 The effective head is 
 
 20x144 3x144 
 : 62-4 62-4 
 
 = 39-2 feet. 
 
 Therefore, Q = -75 x -7854 x ^ x ^64-4 x 39-2 cubic feet per sec. 
 = 1-84 cubic ft. per sec. 
 
 58. Time of emptying a tank or reservoir. 
 
 Suppose a reservoir to have a sharp-edged horizontal orifice 
 as in Fig. 44. It is required to find the time taken to empty 
 the reservoir. 
 
 Let the area of the horizontal section of the reservoir at any 
 height h above the orifice be A sq. feet, and the area of the 
 
 orifice a sq. feet, and let the ratio be sufficiently large that the 
 
 CL 
 
 velocity of the water in the reservoir may be neglected. 
 
 When the surface of the water is at any height h above the 
 orifice, the volume which flows through the orifice in any time dt 
 will be ma \/2gh . dt. 
 
FLOW THROUGH MOUTHPIECES 77 
 
 The amount dh by which the surface of water in the reservoir 
 falls in the time dtis 
 
 or d = 
 
 ma 
 
 The time for the water to fall from a height H to Hj is 
 (H Adh 1 /= Adh 
 
 I/ I ~ ' .____ y_ I 
 
 / HJ ma \/2gh a *J2g J H, m h^ ' 
 
 If A is constant, and m is assumed constant, the time required 
 for the surface to fall from a height H to H! above the orifice is 
 
 [ H Adh 
 
 ma 
 
 and the time to empty the vessel is 
 
 2A 
 
 IcT ' 
 
 ma v Zg 
 
 or is equal to twice the time required for the same volume of 
 water to leave the vessel under a constant head H. 
 
 Time of emptying a lock with vertical drowned sluice. Let the 
 water in the lock when the sluice is closed be at a height H, 
 Fig. 56, above the down-stream level. 
 
 Then the time required is that necessary to reduce the level in 
 the lock by an amount H. 
 
 When the flow is taking place, let x be the height of the water 
 in the lock at any instant above the down-stream water. 
 
 Let A be the sectional area of the lock, at the level of the 
 water in the lock, a the area of the sluice, and m its coefficient of 
 discharge. 
 
 The discharge through the sluice in time dt is 
 
 c)Q = m . a \l2gx . dt. 
 
 If dx is the distance the surface falls in the lock in time dt, then 
 Adx = ma *j2gxdt, 
 
 Adx 
 
 or dt = - . . . 
 
 ma v2<7#52 
 
 To reduce the level by an amount H, 
 
 ' H Adx 
 
 o ma 
 
78 HYDRAULICS 
 
 If m and A are constant, 
 
 ma \/2g 
 
 Example. A reservoir, 200 yards long and 150 yards wide at the bottom, and 
 having side slopes of 1 to 1, has a depth of water in it of 25 feet. A short pipe 
 3 feet diameter is used to draw off water from the reservoir. 
 
 Find the time taken for the water in the reservoir to fall 10 feet. The 
 coefficient of discharge for the pipe is 0-7. 
 
 When the water has a depth h the area of the water surface is 
 
 A = (600 + 27i)(450 + 27i). 
 The area of the pipe is a = 7'068 sq. feet. 
 
 Therefore t= L- /* (600 + 2*) (460 + 8*) M 
 
 0-70 J2g . 7-068 J 15 h b 
 
 = -* a f 25 2 x 27000071* + 1 x 21007** + f 7*~| 
 
 o9"O l_15 _l 
 
 = ^ig (610200 + 93800 + 3606) 
 
 = 17,850 sees. 
 = 4-95 hours. 
 
 Example: A horizontal boiler 6 feet diameter and 30 feet long is half full of 
 water. 
 
 Find the time of emptying the boiler through a short vertical pipe 3 inches 
 diameter attached to the bottom of the boiler. 
 
 The pipe may be taken as a mouthpiece discharging full, the coefficient of 
 velocity for which is 0*8. 
 
 Let r be the radius of the boiler. 
 
 When the water has any depth h above the bottom of the boiler the area A is 
 
 The area of the pipe is 0-049 sq. feet. 
 
 , 
 
 Therefore t = 
 
 8 x 0-049 *j2g J o ,Jk 
 
 = 95-5 / (2r-h$dh 
 o 
 
 = 63-8x9-5 
 = 606 sees. 
 
 EXAMPLES. 
 
 (1) Find the velocity due to a head of 100 ft. 
 
 (2) Find the head due to a velocity of 500 ft. per sec. 
 
 (3) Water issues vertically from an orifice under a head of 40 ft. To 
 what height will the jet rise, if the coefficient of velocity is 0'97 ? 
 
 (4) What must be the size of a conoidal orifice to discharge 10 c. ft. 
 per second under a head of 100 ft. ? m = -925. 
 
FLOW THKOUGH ORIFICES AND MOUTHPIECES 79 
 
 (5) A jet 3 in. diameter at the orifice rises vertically 50 ft. Find its 
 diameter at 25 ft. above the orifice. 
 
 (6) An orifice 1 sq. ft. in area discharges 18 c. ft. per second under a 
 head of 9 ft. Assuming coefficient of velocity =0'98, find coefficient of 
 contraction. 
 
 (7) The pressure in the pump cylinder of a fire-engine is 14,400 Ibs. 
 per sq. ft. ; assuming the resistance of the valves, hose, and nozzle is such 
 that the coefficient of resistance is 0'5, find the velocity of discharge, and 
 the height to which the jet will rise. 
 
 (8) The pressure in the hose of a fire-engine is 100 Ibs. per sq. inch; 
 the jet rises to a height of 150 ft. Find the coefficient of velocity. 
 
 (9) A horizontal jet issues under a head of 9 ft. At 6 ft. from the 
 orifice it has fallen vertically 15 ins. Find the coefficient of velocity. 
 
 (10) Required the coefficient of resistance corresponding to a coefficient 
 of velocity =0-97. 
 
 (11) A fluid of one quarter the density of water is discharged from a 
 vessel in which the pressure is 50 Ibs. per sq. in. (absolute) into the 
 atmosphere where the pressure is 15 Ibs. per sq. in. Find the velocity of 
 discharge. 
 
 (12) Find the diameter of a circular orifice to discharge 2000 c. ft. per 
 hour, under a head of 6 ft. Coefficient of discharge 0'60. 
 
 (13) A cylindrical cistern contains water 16 ft. deep, and is 1 sq. ft. in 
 cross section. On opening an orifice of 1 sq. in. in the bottom, the water 
 level fell 7 ft. in one minute. Find the coefficient of discharge. 
 
 (14) A miner's inch is defined to be the discharge through an orifice in 
 a vertical plane of 1 sq. in. area, under an average head of 6| ins. Find 
 the supply of water per hour in gallons. Coefficient of discharge 0'62. 
 
 (15) A vessel fitted with a piston of 12 sq. ft. area discharges water 
 under a head of 10 ft. What weight placed on the piston would double the 
 rate of discharge ? 
 
 (16) An orifice 12 inches square discharges under a head of 100 feet 
 1*338 cubic feet per second. Taking the coefficient of velocity at 0'97, find 
 the coefficient of contraction. 
 
 (17) Find the discharge per minute from a circular orifice 1 inch 
 diameter, under a constant pressure of 34 Ibs. per sq. inch, taking 0'60 as 
 the coefficient of discharge. 
 
 (18) The plunger of a fire-engine pump of one quarter of a sq. ft. in 
 area is driven by a force of 9542 Ibs. and the jet is observed to rise to a 
 height of 150 feet. Find the coefficient of resistance of the apparatus. 
 
 (19) An orifice 3 feet wide and 2 feet deep has 12 feet head of water 
 above its centre on the up-stream side, and the backwater on the other 
 side is at the level of the centre of the orifice. Find the discharge if 
 
80 HYDRAULICS 
 
 (20) Ten c. ft. of water per second flow through a pipe of 1 sq. ft. area, 
 which suddenly enlarges to 4 sq. ft. area. Taking the pressure at 100 Ibs. 
 per sq. ft. in the smaller part of the pipe, find (1) the head lost in shock, 
 (2) the pressure in the larger part, (3) the work expended in forcing the 
 water through the enlargement. 
 
 (21) A pipe of 3" diameter is suddenly enlarged to 5" diameter. A U 
 tube containing mercury is connected to two points, one on each side of the 
 enlargement, at points where the flow is steady. Find the difference in 
 level in the two limbs of the U when water flows at the rate of 2 c. ft. per 
 second from the small to the large section and vice versd. The specific 
 gravity of mercury is 13'6. Lond. Un. 
 
 (22) A pipe is suddenly enlarged from 2 inches in diameter to 3 
 inches in diameter. Water flows through these two pipes from the smaller 
 to the larger, and the discharge from the end of the bigger pipe is two 
 gallons per second. Find: 
 
 (a) The loss of head, and gain of pressure head, at the enlarge- 
 ment. 
 
 (b) The ratio of head lost to velocity head in small pipe. 
 
 (23) The head and tail water of a vertical- sided lock differ in level 
 12 ft. The area of the lock basin is 700 sq. ft. Find the time of emptying 
 the lock, through a sluice of 5 sq. ft. area, with a coefficient 0'5. The 
 sluice discharges below tail water level. 
 
 (24) A tank 1200 sq. ft. in area discharges through an orifice 1 sq. ft. 
 in area. Calculate the time required to lower the level in the tank from 
 50 ft. to 25 ft. above the orifice. Coefficient of discharge 0'6. 
 
 (25) A vertical-sided lock is 65 ft. long and 18 ft. wide. Lift 15 ft. 
 Find the area of a sluice below tail water to empty the lock in 5 minutes. 
 Coefficient 0'6. 
 
 (26) A reservoir has a bottom width of 100 feet and a length of 250 
 feet. 
 
 The sides of the reservoir are vertical. 
 
 The reservoir is connected to a second reservoir of the same dimensions 
 by means of a pipe 2 feet diameter. The surface of the water in the first 
 reservoir is 17 feet above that in the other. The pipe is below the surface 
 of the water in both reservoirs. Find the time taken for the water in the 
 two reservoirs to become level. Coefficient of discharge 0'8. 
 
 59. Notches and Weirs. 
 
 When the sides of an orifice are 
 produced, so that' they extend be- 
 yond the free surface of the water, 
 as in Figs. 69 and 70, it is called a 
 notch. 
 
 Notches are generally made tri- 
 angular or rectangular as shown 
 in the figures and are largely used 
 for gauging the flow of water. Fig. 69. Triangular Notch. 
 
FLOW OVER WEIRS 81 
 
 For example, if the flow of a small stream is required, a dam is 
 constructed across the stream and the water allowed to pass 
 through a notch cut in a board or metal plate. 
 
 - -L---H 
 
 mimr 
 
 Fig. 70. Rectangular Notch. 
 
 They can conveniently be used for measuring the compensation 
 water to be supplied from collecting reservoirs, and also to gauge 
 the supply of water to water wheels and turbines. 
 
 The term weir is a name given to a structure used to dam up 
 a stream and over which the water flows. 
 
 The conditions of flow are practically the same as through 
 a rectangular notch, and hence such notches are generally called 
 weirs, and in what follows the latter term only is used. The top 
 of the weir corresponds to the horizontal edge of the notch and is 
 called the sill of the weir. 
 
 The sheet of water flowing over a weir or through a notch is 
 generally called the vein, sheet, or nappe. 
 
 The shape of the nappe depends upon the form of the sill and 
 sides of the weir, the height of the sill above the bottom of the 
 up-stream channel, the width of the up-stream channel, and the 
 construction of the channel into which the nappe falls. 
 
 The effect of the form of the sill and of the down-stream 
 channel will be considered later, but, for the present, attention 
 will be confined to weirs with sharp edges, and to those in which 
 the air has free access under the nappe so that it detaches itself 
 entirely from the weir as shown in Fig. 70. 
 
 60. Rectangular sharp-edged weir. 
 
 If the crest and sides of the weir are made sharp-edged, as 
 shown in Fig. 70, and the weir is narrower than the approaching 
 channel, and the sill some distance above the bed of the stream, 
 there is at the sill and at the sides, contraction similar to that at 
 a sharp-edged orifice. 
 
 The surface of the water as it approaches the weir falls, taking 
 a curved form, so that the thickness h s , Fig. 70, of the vein over 
 the weir, is less than H, the height, above the sill, of the water at 
 L. H. 6 
 
82 HYDRAULICS 
 
 some distance from the weir. The height H, which is called the 
 head over the weir, should be carefully measured at such a distance 
 from- it, that the water surface has not commenced to curve. 
 Fteley -and Stearns state, that this distance should be equal to 
 2^ times the height of the weir above the bed of the stream. 
 
 For the present, let it be assumed that at the point where H is 
 measured the water is at rest. In actual cases the water will 
 always have some velocity, and the effect of this velocity will have 
 to be considered later. H may be called the still water head over 
 the weir, and in all the formulae following it has this meaning. 
 
 Side contraction. According to Fteley and Stearns the amount 
 by which the stream is contracted when the weir is sharp-edged 
 is from 0'06 to 0'12H at each side, and Francis obtained a mean of 
 O'lH. A wide weir may be divided into several bays by parti- 
 tions, and there may then be more than two contractions, at each 
 of which the effective width of the weir will be diminished, if 
 Francis' value be taken, by O'lH. 
 
 If L is the total width of a rectangular weir and N the number 
 of contractions, the effective width I, Fig. 70, is then, 
 
 (L-O'INH) 
 
 When L is very long the lateral contraction may be neglected. 
 
 Suppression of the contraction. The side contraction can be 
 completely suppressed by making the approaching channel with 
 vertical sides and of the same width as the weir, as was done for 
 the orifice shown in Fig. 47. The width of the stream is then 
 equal to the width of the sill. 
 
 61. Derivation of the weir formula from that of a large 
 orifice. 
 
 If in the formula for large orifices, p. 64, h Q is made equal to 
 zero and for the effective width of the stream the length I is 
 substituted for 6, and Jc is unity, the formula becomes 
 
 Q=t>/2^.Z.fci* ........................ (1). 
 
 If instead of hi the head H, Fig. 70, is substituted, and 
 a coefficient H introduced, 
 
 The actual width I is retained instead of L, to make allowance 
 for the end contraction which as explained above is equal to O'lH 
 for each contraction. 
 
 If the width of the approaching channel is made equal to the 
 width of the weir I is equal to L. 
 
 With N contractions I = (L - Q'1NJ$ 
 
 and Q = f C >/2cj . (L - O'lN) H*. 
 
 If C is given a mean value of 0*625, 
 
 Q = 3'33 (L- O'lN) H f ..................... (2). 
 
FLOW OVER WEIRS 83 
 
 This is the well-known formula deduced by Francis* from 
 a careful series of experiments on sharp-edged weirs. 
 
 The formula, as an empirical one, is approximately correct and 
 gives reliable values for the discharge. 
 
 The method of obtaining it from that for large orifices is, 
 however, open to very serious objection, as the velocity at F on 
 the section EF, Fig. 70, is clearly not equal to zero, neither is the 
 direction of flow at the surface perpendicular to the section EF, 
 and the pressure on EF, as will be understood later (section 83) 
 is not likely to be constant. 
 
 That the directions and the velocities of the stream lines are 
 different from those through a section taken near a sharp-edged 
 orifice is seen by comparing the thickness of the jet in the two 
 cases with the coefficient of discharge. 
 
 For the sharp-edged orifice with side contractions suppressed, 
 the ratio of the thickness of the jet to the depth of the orifice is not 
 very different from the coefficient of discharge, being about 0'625, 
 but the thickness EF of the nappe of the weir is very nearly 0'78H, 
 whereas the coefficient of discharge is practically 0*625, and the 
 thickness is therefore 1'24 times the coefficient of discharge. 
 
 It appears therefore, that although the assumptions made in 
 calculating the flow through an orifice may be justifiable, providing 
 the head above the top of the orifice is not very small, yet when 
 it approaches zero, the assumptions are not approximately true. 
 
 The angles which the stream lines make with the plane of EF 
 cannot be very different from 90 degrees, so that it would appear, 
 that the error principally arises from the assumption that the 
 pressure throughout the section is uniform. 
 
 Bazin for special cases has carefully measured the fall of the 
 point F and the thickness EF, and if the assumptions of constant 
 pressure and stream lines perpendicular to EF are made, the 
 discharge through EF can be calculated. 
 
 For example, the height of the point E above the sill of the 
 weir for one of Bazin's experiments was 0'112H and the thickness 
 EF was 0'78H. The fall of the point F is, therefore, O'lOSH. 
 Assuming constant pressure in the section, the discharge per foot 
 width of the weir is, then, 
 
 0108H 
 
 = >/2g . H* {('888)* - (-108)*} 
 
 Lowell, Hydraulic Experiments, New York, 1858. 
 
 62 
 
84 HYDRAULICS 
 
 The actual discharge per foot width, by experiment, was 
 
 g = 0'433N/20.H^, 
 
 so that the calculation gives the discharge 1'228 greater than the 
 actual, which is approximately the ratio of the thickness EF to 
 the thickness of the stream from a sharp-edged orifice having 
 a depth H. The assumption of constant pressure is, therefore, 
 quite erroneous. 
 
 62. Thomson's principle of similarity. 
 
 " When a frictionless liquid flows out of similar and similarly 
 placed orifices in similar vessels in which the same kind of liquid 
 is at similar heights, the stream lines in the different flows are 
 similar in form, the velocities at similar points are proportional to 
 the square roots of the linear dimensions, and since the areas of 
 the stream lines are proportional to the squares of the linear 
 dimensions, the discharges are proportional to the linear dimensions 
 raised to the power of f *." 
 
 Let A and B, Figs. 71 and 72, be exactly similar vessels with 
 similar orifices, and let all the dimensions of A be n times those 
 of B. Let c and Ci be similarly situated areas on similar stream 
 lines. 
 
 Fig. 71. 
 
 Fig. 72. 
 
 Then, since the dimensions of A are n times those of B, the 
 fall of free level at c is n times that at Ci . Let v be the velocity 
 at c and Vi at Ci. 
 
 Then, since it has been shown (page 36) that the velocity in 
 any stream line is proportional to the square root of the fall of 
 free level, 
 
 '. v : Vi :: *Jn : 1. 
 
 Again the area at c is n 2 times the area at Ci and, therefore, 
 the discharge through c o ,- 5 
 the discharge through c, = n " ^ n = ^^ 
 which proves the principle. 
 
 * British Association Eeports 1858 and 1876. 
 
FLOW OVER WEIRS 85 
 
 63. Discharge through a triangular notch by the 
 principle of similarity. 
 
 Let ADC, Figs. 73 and 74, be a triangular notch. 
 A C A 
 
 X r -^- r ^- 
 
 D D 
 
 Fig. 73. Fig. 74. 
 
 Let the depth of the flow through the notch at one time be H 
 and at another n . H. 
 
 Suppose the area of the stream in the two cases to be divided 
 into the same number of horizontal elements, such as ab and aj>i. 
 
 Then clearly the thickness of ab will be n times the thickness 
 of a-J}\. 
 
 Let Oi&i be at a distance x from the apex B, and ab at a 
 distance rac; then the width of ab is clearly n times the width 
 of Oibi. 
 
 The area of ab will therefore be ri* times the area of a^ . 
 
 Again, the head above ab is n times the head above a$i and 
 therefore the velocity through ab will be \ln times the velocity 
 through aibi and the discharge through ab will be r$ times 
 that through 0-161. 
 
 More generally Thomson expresses this as follows : 
 
 " If two triangular notches, similar in form, have water flowing 
 through them at different depths, but with similar passages of 
 approach, the cross sections of the jets at the notches may be 
 similarly divided into the same number of elements of area, and 
 the areas of corresponding elements will be proportional to the 
 squares of the lineal dimensions of the cross sections, or pro- 
 portional to the squares of the heads." 
 
 As the depth h of each element can be expressed as a fraction 
 of the head H, the velocities through these elements are propor- 
 tional to the square root of the head, and, therefore, the discharge 
 is proportional to H*. 
 
 Therefore Q oo H*, 
 
 or Q = C . H", 
 
 C being a coefficient which has to be determined by experiment. 
 
 From experiments with a sharp-edged notch having an angle 
 at the vertex of 90 degrees, he found C to be practically constant 
 for all heads and equal to 2'635. Then 
 
 a* .. ...(3). 
 
HYDRAULICS 
 
 64. Flow through a triangular notch. 
 
 The flow through a triangular notch is frequently given as 
 
 in which B is the top width of the notch and n an experimental coefficient. 
 
 It is deduced as follows : 
 
 Let ADC, Fig. 74, be the triangular notch, H being the still water head over 
 the apex, and B the width at a height H above the apex. At any depth h the 
 
 T> /TT I- \ 
 
 width b of the strip a-Jb-, is - - - . 
 ti 
 
 If the velocity through this strip is assumed to be v = k*j2gh, the width of the 
 
 stream through a^, , and the thickness dh, the discharge through it is 
 
 H 
 
 The section of the jet just outside the orifice is really less than the area EFD. 
 The width of the stream through any strip a^ is less than a^, the surface is lower 
 than EF, and the apex of the jet is some distance above B. 
 
 The diminution of the width of a^ has been allowed for by the coefficient c, and 
 the diminution of depth might approximately be allowed for by integrating between 
 h = and h = H, and introducing a third coefficient Cj. 
 
 Then = fccc, B ^ T ~ h) 
 
 ti 
 
 Keplacing cc-Jt by n 
 
 Q = T VW27.BH ....................................... (4). 
 
 Calling the angle ADC, 0, 
 
 B=2Htan|, 
 
 When B is 90 degrees, B is equal to 2H, and 
 
 Taking a mean value for n of 0'617 
 Q = 2-635. H*, 
 
 which agrees with Thomson's formula for a right-angled notch. 
 
 The result is the same as obtained by the method of similarity, but the method 
 of reasoning is open to very serious objection, as at no section of the jet are all the 
 stream lines normal to the section, and k cannot therefore be constant. The 
 assumption that the velocity through any strip is proportional to >Jh is also open 
 to objection, as the pressure throughout the section can hardly be uniform. 
 
 65. Discharge through a rectangular weir by the 
 principle of similarity. 
 
 The discharge through a rectangular weir can also be obtained 
 by the principle of similarity. 
 
FLOW OVER WEIRS 
 
 87 
 
 Consider two rectangular weirs each of length L, Figs. 75 
 and 76, and let the head over the sill be H in the one case and 
 Hi, or nTL, in the other. Assume the approaching channel to be 
 of such a form that it does not materially alter the flow in either 
 case. 
 
 H 
 
 H 
 
 J) 
 
 Fig. 75. 
 
 B 
 
 Fig. 76. 
 
 To simplify the problem let the weirs be fitted with sides 
 projecting up stream so that there is no side contraction. 
 
 Then, if each of the weirs be divided into any number of equal 
 parts the flow through each of these parts in any one of the weirs 
 will be the same. 
 
 Suppose the first weir to be divided into N equal parts. If 
 
 then, the second weir is divided into ' equal parts, the parts 
 
 in the second weir will be exactly similar to those of the first. 
 
 By the principle of similarity, the discharge through each of 
 the parts in the first weir will be to the discharge in the second 
 
 as j , and the total discharge through the first weir is to the 
 HI* 
 
 discharge through the second as 
 
 N.H^ 5 _Hi_! 
 AT TT TT,i TT,I *ii' 
 
 H! 
 
 Instead of two separate weirs the two cases may refer to the 
 same weir, and the discharge for any head H is, therefore, pro- 
 portional to H^ ; and since the flow is proportional to L 
 
 Q = C.L.H*, 
 
 in which C is a coefficient which should be constant. 
 
 66. Rectangular weir with end contractions. 
 
 If the width of the channel as it approaches the weir is greater 
 than the width of the weir, contraction takes place at each side, 
 and the effectual width of the stream or nappe is diminished ; the 
 amount by which the stream is contracted is practically inde- 
 pendent of the width and is a constant fraction of H, as explained 
 above, or is equal to &H, k being about O'l 
 
88 HYDRAULICS 
 
 Let the total width of each weir be now divided into three 
 parts, the width of each end part being equal to n . ~k . H. The 
 width of the end parts of the transverse section of the stream will 
 each be (n- 1) ~k . H, and the width of central part L - 2nkTL. 
 
 The flow through the central part of the weir will be equal to 
 
 Now, whatever the head on the weir, the end pieces of the 
 stream, since the width is (n - 1) &H and k is a constant, will be 
 similar figures, and, therefore, the flow through them can be 
 expressed as 
 
 The total flow is, therefore, 
 
 Q = C (L - 2wfcH) Ht + 20, (n - 1) fcHH*. 
 If now Ci is assumed equal to C 
 
 Q=C(L-2&H)Hi 
 
 If instead of two there are N contractions, due to the weir 
 being divided into several bays by posts or partitions, the formula 
 becomes 
 
 Q0(L-N01.H)'H*. 
 
 This is Francis 7 formula, and by Thomson's theory it is thus 
 shown to be rational. 
 
 67. Bazin's* formula for the discharge of a weir. 
 
 The discharge through a weir with no side contraction may be 
 written 
 
 or 
 
 the coefficient m being equal to 
 
 Taking Francis' value for C as 3*33, m is then 0'415. 
 From experiments on sharp-crested weirs with no side con- 
 traction Bazin deduced for mt the value 
 
 A/mK '00984 
 m = 0'405 + TT . 
 1 
 
 In Table IX, and Fig. 77, are shown Bazin's values for m for 
 different heads, and also those obtained by Eafter at Cornell upon 
 a weir similar to that used by Bazin, the maximum head in the 
 Cornell experiments being much greater than that in Bazin's 
 experiments. In Fig. 77 are also shown several values of m, as 
 calculated by the author, from Francis' experimental data. 
 
 * Annales des Fonts et Chaussees, 1888 1898. 
 
 t "Experiments on flow over Weirs," Am.S.C.E. Vol. xxvii. 
 
FLOW OVER WEIRS 89 
 
 TABLE IX. 
 
 Values of the coefficient m in the formula Q - mL \/2gr H*. 
 
 Weir, sharp-crested, 6'56 feet wide with free overfall and lateral 
 contraction suppressed, H being the still water head over the weir, 
 or the measured head h* corrected for velocity of approach. 
 
 Bazin. 
 
 Head in feet 0-164 0*328 O656 0'984 1-312 1-64 1-968 
 m 0-448 0-432 0-421 0-417 0-414 0-412 0-409 
 
 m 
 
 11 . 
 
 Rafter. 
 Head in feet m C 
 
 0-1 0-4286 3-437 
 
 0-5 0-4230 3-392 
 
 1-0 0-4174 3-348 
 
 1'5 0-4136 3-317 
 
 2-0 0-4106 3-293 
 
 2-5 0-4094 3-283 
 
 3-0 0-4094 3-283 
 
 3-5 0-4099 3-288 
 
 4-0 0-4112 3-298 
 
 4-5 0-4125 3-308 
 
 5-0 0-4133 3-315 
 
 5-5 0-4135 3-316 
 
 6-0 0-4136 3-317 
 
 68. Bazin's and the Cornell experiments on weirs. 
 
 Bazin's experiments were made on a weirt 6*56 feet long 
 having the approaching channel the same width as the weir, so 
 that the lateral contractions were suppressed, and the discharge 
 was measured by noting the time taken to fill a concrete trench of 
 known capacity. 
 
 The head over the weir was measured by means of the hook 
 gauge, page 249. Side chambers were constructed and connected 
 to the channel by means of circular pipes O'l m. diameter. 
 
 The water in the chambers was very steady, and its level 
 could therefore be accurately gauged. The gauges were placed 
 5 metres from the weir. The maximum head over the weir in 
 Bazin's experiments was however only 2 feet. 
 
 The experiments for higher heads at Cornell University were 
 made on a weir of practically the same width as Bazin's, 6'53 feet, 
 the other conditions being made as nearly the same as possible ; 
 the maximum head on the weir was 6 feet. 
 
 * See page 90. 
 
 t Annales des Fonts et Chaussees, p. 445, Vol. n. 1891. 
 
90 
 
 HYDRAULICS 
 
 The results of these experiments, Fig. 77, show that the 
 coefficient m diminishes and then increases, having a minimum 
 value when H is between 2'5 feet and 3 feet. 
 
 44 
 
 -43 
 
 V3 
 
 
 Head in, Feet. 
 
 Mean, coefficient, carves for Sharp -edged. Weirs 
 *! BOL^LTLS E.ccperi?Tiefit$ 
 
 n (Deduced, by the author) 
 
 Fig. 77. 
 
 o CorneLL 
 A Fronds 
 
 It is doubtful, however, although the experiments were made 
 with great care and skill, whether at high heads the deduced 
 coefficients are absolutely reliable. 
 
 To measure the head over the weir a 1 inch galvanised pipe 
 with holes Jinch diameter and opening downwards, 6 inches 
 apart, was laid across the channel. To this pipe were connected 
 f inch pipes passing through the weir to a convenient point below 
 the weir where they could be connected to the gauges by rubber 
 tubing. The gauges were glass tubes f inch diameter mounted 
 on a frame, the height of the water being read on a scale 
 graduated to 2mm. spaces. 
 
 69. Velocity of approach. 
 
 It should be clearly understood that in the formula given, it 
 has been assumed, in giving values to the coefficient m that H is 
 the height, above the sill of the weir, of the still water surface. 
 
FLOW OVER WEIRS 91 
 
 In actual cases the water where the head is measured will have 
 some velocity, and due to this, the discharge over the weir will be 
 increased. 
 
 If Q is the actual discharge over a weir, and A is the area of 
 the up-stream channel approaching the weir, the mean velocity in 
 
 the channel is v = -^ . 
 A 
 
 There have been a number of methods suggested to take into 
 account this velocity of approach, the best perhaps being that 
 adopted by Hamilton Smith, and Bazin. 
 
 This consists in considering the equivalent still water head H, 
 over the weir, as equal to 
 
 a being a coefficient determined by experiment, and h the 
 measured head. 
 
 The discharge is then 
 
 (5), 
 
 
 2 
 
 Expanding (5), and remembering that ~ r is generally a small 
 
 quantity, 
 
 The velocity v depends upon the discharge Q to be determined 
 and is equal to - . 
 
 Therefore Q = mL hj2gh ( 
 
 1 + 
 
 From five sets of experiments, the height of the weir above the 
 bottom of the channel being different for each set, Bazin found 
 the mean value of a to be 1*66. 
 
 This form of the formula, however, is not convenient for use, 
 since the unknown Q appears upon both sides of the equation. 
 
 If, however, the discharge Q is expressed as 
 
 Q - nL ^2gh. h, 
 the coefficient n for any weir can be found by measuring Q and h. 
 
 It will clearly be different from the coefficient m, since for m 
 to be used h has to be corrected. 
 
 From his experimental results Bazin calculated n for various 
 heads, some of which are shown in Table X. 
 
92 HYDRAULICS 
 
 Substituting this value of Q in the above formula, 
 
 3 a - 
 
 /( _, 
 (7). 
 
 Let fan 2 be called fe. 
 Then Q - 
 
 1 + 
 
 Or, when the width of channel of approach is equal to "the 
 width of the weir, and the height of the sill, Fig. 78, is p feet above 
 the bed of the channel, and h the measured head, 
 
 and 
 
 Fig. 78. 
 
 The mean value given to the coefficient Jc by Bazin is 0'55, 
 so that 
 
 This may be written 
 
 (9). 
 
 Q = 
 
 \r2gh, 
 
 in which 
 
 Substituting for m the value given on page 88, 
 
 \ 
 
 mi may be called the absolute coefficient of discharge. 
 
 The coefficient given in the Tables. 
 
 It should be clearly understood that in determining the values 
 of m as given in the Tables and in Fig. 77 the measured head h 
 was corrected for velocity of approach, and in using this 
 
FLOW OVER WEIRS 93 
 
 coefficient to determine Q, h must first be corrected, or Q 
 calculated from formula 9. 
 
 Rafter in determining the values of ra from the Cornell ex- 
 
 periments, increased the observed head h by ~ only, instead of 
 by 1-66 g. 
 
 Fteley and Stearns*, from their researches on the flow over 
 weirs, found the correction necessary for velocity of approach to 
 be from 
 
 1-45 to 1'5 . 
 20 
 
 Hamilton Smith t adopts for weirs with end contractions 
 suppressed the values 
 
 T33 to 1-40 1^, 
 
 and for a weir with two end contractions, 
 11 to 1-25 |^. 
 
 TABLE X. 
 Coefficients n and m as calculated by Bazin from the formulae 
 
 and Q=mW20H*, 
 
 h being the head actually measured and H the head corrected for 
 
 velocity of approach. 
 
 Coefficient 
 
 0-448 
 0-417 
 0-4118 
 
 An example is now taken illustrating the method of deducing 
 the coefficients n and m from the result of an experiment, and the 
 difference between them for a special case. 
 
 Example. In one of Bazin's experiments the width of the weir and the 
 approaching channel were both 6 -56 feet. The depth of the channel approaching 
 the weir measured at a point 2 metres up stream from the weir was 7'544 feet and 
 the head measured over the weir, which may be denoted by h, was 0*984 feet. The 
 measured discharge was 21-8 cubic ft. per second. 
 
 * Transactions Am.S.C.E., Vol. xn. 
 f Hydraulics. 
 
 Head 
 h in feet 
 
 Height of sill 
 p in feet 
 
 Coefficient 
 n 
 
 0-164 
 
 0-656 
 
 0-458 
 
 
 6-560 
 
 0-448 
 
 0-984 
 
 0-656 
 
 0-500 
 
 
 6-560 
 
 0-421 
 
 1-640 
 
 0-656 
 
 0-500 
 
 
 6-560 
 
 0-421 
 
94 HYDRAULICS 
 
 The velocity at the section where h was measured, and which may be called the 
 velocity of approach was, therefore, 
 
 7-544x6-56' 7-544x6-56 
 = 0-44 feet per second. 
 If now the formula for discharge be written 
 
 and n is calculated from this formula by substituting the known values of 
 Q, L and h 
 
 7i=0-421. 
 Correcting h for velocity of approach, 
 
 Then 
 from which 
 
 . -9888 
 
 It will seem from Table X that when the height p of the sill of the weir above 
 the stream bed is small compared with the head, the difference may be much 
 larger than for this example. 
 
 When the head is 1-64 feet and larger than p, the coefficient n is eighteen 
 per cent, greater than m. In such cases failure to correct the coefficient will lead 
 to considerable inaccuracy. 
 
 70. Influence of the height of the weir sill above the bed 
 of the stream on the contraction. 
 
 The nearer the sill is to the bottom of the stream, the less the 
 contraction at the sill, and if the depth is small compared with H, 
 the diminution on the contraction may considerably affect the 
 flow. 
 
 When the sill was 1*15 feet above the bottom of a channel, 
 
 of the same width as the weir, Bazin found the ratio ^ (Fig. 85) 
 
 JH 
 
 to be 0*097, and when it was 3*70 feet, to be 0112. For greater 
 
 p 
 
 heights than these the mean value of ^ was 0'13. 
 
 1 
 
 71. Discharge of a weir when the air is not freely 
 admitted beneath the nappe. Form of the nappe. 
 
 Francis in the Lowell experiments, found that, by making the 
 width of the channel below the weir equal to the width of the 
 weir, and thus preventing free access of air to the underside of the 
 nappe, the discharge was increased. Bazin*, in the experiments 
 already referred to, has investigated very fully the effect upon 
 the discharge and upon the form of the nappe, of restricting the 
 free passage of the air below the nappe. He finds, that when the 
 flow is sufficient to prevent the air getting under the nappe, it may 
 assume one of three distinct forms, and that the discharge for 
 * Annales des Fonts et Chaussees, 1891 and 1898. 
 
FLOW OVER WEIRS 
 
 95 
 
 one of them may be 28 per cent, greater than when the air is 
 freely admitted, or the nappe is "free." Which of these three 
 forms the nappe assumes and the amount by which the discharge 
 is greater than for the "free nappe," depends largely upon the 
 head over the weir, and also upon the height of the weir above 
 the water in the down-stream channel. 
 
 The phenomenon is, however, very complex, the form of the 
 nappe for any head depending to a very large extent upon 
 whether the head has been decreasing, or increasing, and for a 
 given head may possibly have any one of the three forms, so that 
 the discharge is very uncertain. M. Bazin distinguishes the forms 
 of nappe as follows : 
 
 (1) Free nappe. Air under nappe at atmospheric pressure, 
 Figs. 70 and 78. 
 
 (2) Depressed nappe enclosing a limited volume of air at a 
 pressure less than that of the atmosphere, Fig. 79. 
 
 (3) Adhering nappe. No air enclosed and the nappe adher- 
 ing to the down-stream face of the weir, Fig. 80. The nappe in this 
 case may take any one of several forms. 
 
 Top of (haravd\ 
 
 Fig. 80. 
 
 No air enclosed but 
 the nappe encloses a mass of turbulent water which does not move 
 with the nappe, and which is said to wet the nappe. 
 
 Fig. 79. 
 (4) Drowned or wetted nappe, Fig. 81. 
 
 Fig. 81. 
 
96 HYDRAULICS 
 
 72. Depressed nappe. 
 
 The air below the nappe being at less than the atmospheric 
 pressure the excess pressure on the top of the nappe causes it to 
 be depressed. There is also a rise of water in the down-stream 
 channel under the nappe. 
 
 The discharge is slightly greater than for a free nappe. On a 
 weir 2*46 feet above the bottom of the up-stream' channel, the 
 nappe was depressed for heads below 0*77 feet, and at this head 
 the coefficient of discharge was 1'08 m ly m l being the absolute 
 coefficient for the free nappe. 
 
 73. Adhering nappes. 
 
 As the head for this weir approached 0*77 feet the air was 
 rapidly expelled, and the nappe became vertical as in Fig. 80, its 
 surface having a corrugated appearance. The coefficient of dis- 
 charge changed from 1*08 m : to 1*28 m^ This large change in 
 the coefficient of discharge caused the head over the weir to fall 
 to 0'69 feet, but the nappe still adhered to the weir. 
 
 74. Drowned or wetted nappes. 
 
 As the head was further increased, and approached 0'97 feet, 
 the nappe came away from the weir face, assuming the drowned 
 form, and the coefficient suddenly fell to 1*19 mi. As the head 
 was further increased the coefficient diminished, becoming 1'12 
 when the head was above 1'3 feet. 
 
 The drowned nappes are more stable than the other two, but 
 whereas for the depressed and adhering nappes the discharge is 
 not affected by the depth of water in the down-stream channel, 
 the height of the water may influence the flow of the drowned 
 nappe. If when the drowned nappe falls into the down stream 
 the rise of the water takes place at a distance from the foot of the 
 nappe, Fig. 81, the height of the down-stream water does not affect 
 the flow. On the other hand if the rise encloses the foot of the 
 nappe, Fig. 82, the discharge is affected. Let h 2 be the difference 
 
 Fig. 82. 
 
FLOW OVER WEIRS 97 
 
 of level of the sill of the weir and the water below the weir. The 
 coefficient of discharge in the first case is independent of /i 2 but is 
 dependent upon p the height of the sill above the head of the up- 
 stream channel, and is 
 
 ...(11). 
 
 Bazin found that the drowned nappe could not be formed if h 
 is less than 0*4 p and, therefore, j? cannot be greater than 2*5. 
 Substituting for m a its value 
 
 from (10) page 92 
 
 mo = 0-470 + 0-0075 1* (12). 
 
 In the second case the coefficient depends upon h^, and is, 
 
 mo = wi(l'06 + 0-16 > )('--0-05 > )? . ...(13), 
 
 / \p / h 
 
 for which, with a sufficient degree of approximation, may be 
 substituted the simpler formula, 
 
 7710^/1-05 + 1-15^ ...(14). 
 
 The limiting value of m is l'2wi, for if /^ becomes greater 
 than h the nappe is no longer drowned. 
 
 Further, the rise can only enclose the foot of the nappe when 
 h 2 is less than (f p - h} . As h 2 passes this value the rise is pushed 
 down stream away from the foot of the nappe and the coefficient 
 changes to that of the preceding case. 
 
 75. Instability of the form of the nappe. 
 
 The head at which the form of nappe changes depends upon 
 whether the head is increasing or diminishing, and the depressed 
 and adhering nappes are very unstable, an accidental admission 
 of air or other interference causing rapid change in their form. 
 Further, the adhering nappe is only formed under special circum- 
 stances, and as the air is expelled the depressed nappe generally 
 passes directly to the drowned form. 
 
 If, therefore, the air is not freely admitted below the nappe 
 the form for any given head is very uncertain and the discharge 
 cannot be obtained with any great degree of assurance. 
 
 With the weir 2*46 feet above the bed of the channel and 6*56 
 feet long Bazin obtained for the same head of 0*656 feet, the four 
 kinds of nappe, the coefficients of discharge being as follows : 
 L. H. 7 
 
98 HYDRAULICS 
 
 Free nappe, 0*433 
 
 Depressed nappe, 0*460 
 Drowned nappe, level of water down stream 
 
 0*41 feet below the crest of the weir, 0*497 
 
 Nappe adhering to down-stream face, 0*554 
 The discharge for this weir while the head was kept constant, 
 thus varied 26 per cent. 
 
 76. Drowned weirs with sharp crests*. 
 
 When the surface of the water down stream is higher than the 
 sill of the weir, as in Fig. 83, the weir is said to be drowned. 
 
 Fig. 83. 
 
 Bazin gives a formula for deducing the coefficients for such a 
 weir from those for the sharp-edged weirs with a free nappe, which 
 in its simplest form is, 
 
 .-.(15), 
 
 hz being the height of the down-stream water above the sill of 
 the weir, h the head actually measured above the weir, p the 
 height of the sill above the up-stream channel, and ra x the 
 coefficient ((10), p. 92) for a sharp-edged weir. This expression 
 gives the same value within 1 or 2 per cent, as the formulae (13) 
 and (14). 
 
 Example. The head over a weir is 1 foot, and the height of the sill above the 
 up-stream channel is 5 feet. The length is 8 feet and the surface of the water 
 in the down-stream channel is 6 inches above the sill. Find the discharge. 
 
 From formula (10), page 92, the coefficient m x for a sharp-edged weir with free 
 nappe is 
 
 = 4215. 
 
 * Attempts have been made to express the discharge over a drowned weir as 
 equivalent to that through a drowned orifice of an area equal to Lft 2 , under a head 
 h - 7i 2 together with a discharge over a weir of length L when the head ish-h^. 
 
 The discharge is then 
 
 n and m being coefficients. 
 
FLOW OVER WEIRS 99 
 
 Therefore m = -4215 [1 -05 (1 + -021) 0-761] 
 
 = 3440. 
 
 Then Q = -344 x 
 
 = 22-08 cubic ft. per second. 
 
 77. Vertical weirs of small thickness. 
 
 Instead of making the sill of a weir sharp-edged, it may 
 have a flat sill of thickness c. This will frequently be the case in 
 practice, the weir being constructed of timbers of uniform width 
 placed one upon the other. The conditions of flow for these weirs 
 may be very different from those of a sharp-edged weir. 
 
 The nappes of such weirs present two distinct forms, according 
 as the water is in contact with the crest of the weir, or becomes 
 detached at the up-stream edge and leaps over the crest without 
 touching the down-stream edge. In the second case the discharge 
 is the same as if the weir were sharp-edged. When the head h 
 over the weir is more than 2c this condition is realised, and may 
 obtain when h passes f c. Between these two values the nappe is 
 in a condition of unstable equilibrium ; when h is less than f c the 
 nappe adheres to the sill, and the coefficient of discharge is 
 
 m, (0'70 + 0185^), 
 
 any external perturbation such as the entrance of air or the 
 passage of a floating body causing the detachment. 
 
 If the nappe adheres between f c and 2c the coefficient m varies 
 from '98wi to l'07mi, but if it is free the coefficient m = mi. 
 When H = |c, m is *79rai. If therefore the coefficients for a 
 sharp-edged weir are used it is clear the error may be con- 
 siderable. 
 
 The formula for ra gives approximately correct results when 
 the width of the sill is great, from 3 to 7 feet for example. 
 
 If the up-stream edge of the weir is rounded the discharge is 
 increased. The discharge* for a weir having a crest 6*56 feet 
 wide, when the up-stream edge was rounded to a radius of 4 inches, 
 was increased by 14 per cent., and that of a weir 2*624 feet wide 
 by 12 per cent. 
 
 The rounding of the corners, due to wear, of timber weirs of 
 ordinary dimensions, to a radius of 1 inch or less, will, therefore, 
 affect the flow considerably. 
 
 78. Depressed and wetted nappes for flat-crested weirs. 
 
 The nappes of weirs having flat sills may be depressed, and 
 may become drowned as for sharp-edged weirs. 
 
 * Annales des Fonts et Chaiisstes, Vol. n. 1896. 
 
 72 
 
100 
 
 HYDRAULICS 
 
 The coefficient of discharge for the depressed nappes, whether 
 the nappe leaps over the crest or adheres to it, is practically the 
 same as for the free nappes, being slightly less for low heads and 
 becomes greater as the head increases. In this respect they differ 
 from the sharp-crested weirs, the coefficients for which are always 
 greater for the depressed nappes than for the free nappes. 
 
 79. Drowned nappes for flat-crested weirs. 
 
 As long as the nappe adheres to the sill the coefficient m may 
 be taken the same as when the nappe is free, or 
 
 'OTO+^. 
 
 When the nappe is free from the sill and becomes drowned, 
 the same formula 
 
 m = w 1 (0'878+ 0128 1) , 
 
 as for sharp-crested weirs with drowned nappes, may be used. 
 For a given limiting value of the head h these two formulae give 
 the same value of m . When the head is less than this limiting 
 value, the former formula should be used. It gives values of m 
 slightly too small, but the error is never more than 3 to 4 per cent. 
 When the head is greater than the limiting value, the second 
 formula should be used. The error in this case may be as 
 great as 8 per cent. 
 
 80. Wide flat-crested weirs. 
 
 When the sill is very wide the surface of the water falls 
 towards the weir, but the stream lines, as they pass over the weir, 
 are practically parallel to the top of the weir. 
 
 Let H be the height of the still water surface, and h the depth 
 of the water over the weir, Fig. 84. 
 
 Fig. 84. 
 
 Then, assuming that the pressure throughout the section of the 
 nappe is atmospheric, the velocity of any stream line is 
 
 (16). 
 
 and if L is the length of the weir, the discharge is 
 
FLOW OVER WEIRS 101 
 
 For the flow to be permanent (see page 106) Q must be a 
 maximum for a given value of h, or -^ must equal zero. 
 Therefore 
 
 From which 2 (H - h) - h = 0, 
 
 and h = f H. 
 
 Substituting for h in (16) 
 
 - 0-385L \20H . H = 3'08L VS . H. 
 
 The actual discharge will be a little less than this due to 
 friction on the sill, etc. 
 
 Bazin found for a flat-crested weir 6'56 feet wide the coefficient 
 ra was G'373, or C =2'991. 
 
 Lesbros' experiments on weirs sufficiently wide to approximate 
 to the conditions assumed, gave *35 for the value of the co- 
 efficient m. 
 
 In Table XI the coefficient C for such weirs varies from 2'66 
 to 3-10. 
 
 81. Flow over dams. 
 
 Weirs of various forms. M. Bazin has experimentally investi- 
 gated the flow over weirs having (a) sharp crests and (b) flat 
 crests, the up- and down-stream faces, instead of both being vertical, 
 being 
 
 (1) vertical on the down-stream face and inclined on the 
 up-stream face, 
 
 (2) vertical on the up-stream face and inclined on the down- 
 stream face, 
 
 (3) inclined on both the up- and down-stream faces, 
 and (c) weirs of special sections. 
 
 The coefficients vary very considerably from those for sharp- 
 crested vertical weirs, and also for the various kinds of weirs. 
 Coefficients are given in Table XI for a few cases, to show the 
 necessity of the care to be exercised in choosing the coefficient for 
 any weir, and the errors that may ensue by careless evaluation of 
 the coefficient of discharge. 
 
 For a full account of these experiments and the coefficients 
 obtained, the reader is referred to Bazin's* original papers, or to 
 Rafter's! paper, in which also will be found the results of experi- 
 
 * Annale* des Fonts et Cliauxes, 1898. 
 
 t Transactions of tlic Am.S.C.E., Vol. XLIV., 1900. 
 
102 
 
 HYDRAULICS 
 
 TABLE XL 
 
 Values of the coefficient C in the formula Q = CL . h*, for weirs 
 of the sections shown, for various values of the observed head h. 
 
 Bazin. 
 
 Section of 
 weir 
 
 Head in feet 
 
 0-3 0-5 1-0 1-3 2-0 3-0 4-0 5-0 6-0 
 
 1-31$ 
 
 2-66 
 
 2-66 
 
 2-90 
 
 3-10 
 
 3-61 
 
 3-80 
 
 4-01 
 
 3-91 
 
 4-02 
 
 4-15 
 
 4-18 
 
 4-15 
 
 3-46 
 
 3-57 
 
 3-86 
 
 3-80 
 
 3-46 
 
 3-08 
 
 3-49 
 
 3-08 
 
 3-59 
 
 3-19 
 
 3-63 
 
 3-22 
 
FLOW OVER WEIRS 
 
 103 
 
 TABLE XI (continued). 
 Bazin. 
 
 Section of 
 weir 
 
 -66 
 
 Head in feet 
 
 3-10 
 
 2-75 
 
 0-5 
 
 3-27 
 
 3-05 
 
 1-0 1-3 
 
 3-73 
 
 3-52 
 
 3-90 
 
 3-73 
 
 2-0 3-0 4-0 5-0 6-0 
 
 Rafter. 
 
 Section of 
 weir 
 
 Head in feet 
 
 0-3 0-5 1-0 1-3 2-0 3-0 4-0 5-0 6-0 
 
 3-35 
 
 3-14 
 
 2-95 
 
 3-68 
 
 3-42 
 
 3-16 
 
 3-83 
 
 3-52 
 
 3-27 
 
 3-77 
 
 3-61 
 
 3-45 
 
 3-68 
 
 3-66 
 
 3'56 
 
 3-70 
 
 3'66 
 
 3-61 
 
 3-71 
 
 3-64 
 
 3-65 
 
 3-71 
 
 3-63 
 
 3-67 
 
104 HYDRAULICS 
 
 ments made at Cornell University on the discharge of weirs, similar 
 to those used by Bazin and for heads higher than he used, and 
 also weirs of sections approximating more closely to those of 
 existing masonry dams, used as weirs. From Bazin's and Rafter's 
 experiments, curves of discharge for varying heads for some of 
 these actual weirs have been drawn up. 
 
 82. Form of weir for accurate gauging. 
 
 The uncertainty attaching itself to the correction to be applied 
 to the measured head for velocity of approach, and the difficulty 
 of making proper allowance for the imperfect contraction at the 
 sides and at the sill, when the sill is near the bed of the channel 
 and is not sharp-edged, and the instability of the nappe and 
 uncertainty of the form for any given head when the admission of 
 air below the nappe is imperfect, make it desirable that as far as 
 possible, when accurate gaugings are required, the weir should 
 comply with the following four conditions, as laid down by 
 Bazin. 
 
 (1) The sill of the weir must be made as high as possible 
 above the bed of the stream. 
 
 (2) Unless the weir is long compared with the head, the 
 lateral contraction should be suppressed by making the channel 
 approaching the weir with vertical sides and of the same width as 
 the weir. 
 
 (3) The sill of the weir must be made sharp-crested. 
 
 (4) Free access of air to the sides and under the nappe of 
 the weir must be ensured. 
 
 83. Boussinesq's * theory of the discharge over a weir. 
 
 As stated above, if air is freely admitted below the nappe of 
 a weir there is a contraction of the stream at the sharp edge of the 
 sill, and also due to the falling curved surface. 
 
 If the top of the sill is well removed from the bottom of the 
 channel, the amount by which the arched under side of the nappe 
 is raised above the sill of the weir is assumed by Boussinesq and 
 this assumption has been verified by Bazin's experiments to be 
 some fraction of the head H on the weir. 
 
 Let CD, Fig. 85, be the section of the vein at which the 
 maximum rise of the bottom of the vein occurs above the sill, and 
 let e be the height of D above S. 
 
 Let it be assumed that through the section CD the stream 
 lines are moving in curved paths normal to the section, and that 
 they have a common centre of curvature 0. 
 
 * Cowptes Rendus, 1887 and 1889. 
 
FLOW OVER WEIRS 
 
 105 
 
 Let H be the height of the surface of the water up stream 
 above the sill. Let R be the radius of the stream line at any 
 point E in CD at a height x above S, and RI and R 2 the radii of 
 curvature at D and C respectively. Let Y, Vi and V 2 be the 
 velocities at E, D, and C respectively. 
 
 
 
 Fig. 85. 
 
 Consider the equilibrium of any element of fluid at the point 
 E, the thickness of which is SR and the horizontal area is a. If w 
 is the weight of unit volume, the weight of the element is w . a8R. 
 
 Since the element is moving in a circle of radius R the centri- 
 
 fugal force acting on the element is wa 
 
 Y 2 8R 
 
 Ibs. 
 
 The force acting on the element due to gravity is wa SR Ibs. 
 Let p be the pressure per unit area on the lower face of the 
 element and p + &p on the upper face. 
 
 Then, equating the upward and downward forces, 
 
 (p + ftp) a + waSR = pa + 
 
 Fromwhioh - =-! + -% (D- 
 
 w cZR #R 
 
 Assuming now that Bernoulli's theorem is applicable to the 
 stream line at EF, 
 
 + p + V! = H 
 
 w 2g 
 Differentiating, and remembering H is constant, 
 
 = 0, 
 
 9 
 
 or 
 
 w dx 
 
 g .dx ' 
 
106 * HYDRAULICS 
 
 dp dp 
 And since fo = dR> 
 
 V 2 VdV 
 tnereiore -^- = ,T> > 
 
 or RdV + VdR = 0. 
 
 Integrating, VR = constant. 
 
 Therefore VR - VA = V 2 R 2 . 
 
 At the upper and lower surfaces of the vein the pressure is 
 atmospheric, and therefore, 
 
 Since VR = ViRi, and R from the figure is (R a + x e), therefore, 
 
 The total flow over the weir is 
 
 (3). 
 
 Now if the flow over the weir is permanent, the thickness h of 
 the nappe must adjust itself, so that for the given head H the 
 discharge is the maximum possible. 
 
 The maximum flow however can only take place if each 
 filament at the section GrF has the maximum velocity possible to 
 the conditions, otherwise the filaments will be accelerated ; and 
 for a given discharge the thickness h is therefore a minimum, or 
 for a given value of h the discharge is a maximum. That is, when 
 
 Q is a maximum, -55* = 0. 
 (trio 
 
 If therefore RI can be written as a function of h Q) the value of 
 ho, which makes Q a maximum, can be determined by differ- 
 
 entiating (3) and equating -~ to zero. 
 
 Then, since Ra = RI + ho, 
 
 V 2 2 H - e - i 
 and -____ 
 
FLOW OVER WEIRS 107 
 
 Therefore, h = (H - e) (1 - 2 ), 
 
 and KI = n (1 + n) (H e). 
 
 Substituting this value of Ba in the expression for Q, 
 
 which, since Q is a maximum when -^ = 0, and h is a function 
 
 Ctrl 
 
 of n. is a maximum when -~ = 0. 
 
 dn 
 
 Differentiating and equating to zero, 
 
 the solution of which gives 
 
 n = G'4685, 
 
 and therefore, Q = 0'5216 *J%g (H - e) 
 
 = 0-5216 J2g(l- 4 
 
 = 0'5216 (l - 4: 
 
 the coefficient m being equal to 
 
 0*5216 (l - g 
 M. Bazin has found by actual measurement, that the mean 
 
 p 
 
 value for TT, when the height of the weir is at considerable 
 distance from the bottom of the channel, is 0*13. 
 Then, (l-^= Q-812, 
 
 and m = 0*423. 
 
 It will be seen on reference to Fig. 77, that this value is very 
 near to the mean value of m as given by Francis and Bazin, and 
 the Cornell experiments. Giving to g the value 32'2, 
 
 Q = 3'39 H^ per foot length of the weir. 
 
 If the length of the weir is L feet and there are no end con- 
 tractions the total discharge is 
 
 and if there are N contractions 
 
 Q = 3-39(L-NO'lH)Hi 
 
108 HYDRAULICS 
 
 The coefficient 3'39 agrees remarkably well with the mean 
 value of C obtained from experiment. 
 
 The value of a theory must be measured by the closeness of 
 the results of experience with those given by the theory, and in 
 this respect Boussinesq's theory is the most satisfactory, as it not 
 only, in common with the other theories, shows that the flow is 
 proportional to H^, but also determines the value of the 
 constant C. 
 
 84. Solving for Q, by approximation, when the velocity 
 of approach is unknown. 
 
 A simple method of determining the discharge over a weir 
 when the velocity of approach is unknown, is, by approximation, 
 as follows. 
 
 Let A be the cross-sectional area of the channel. 
 
 First find an approximation to Q, without correcting for 
 velocity of approach, from the formula 
 
 Q = mLh */2gh. 
 The approximate velocity of approach is, then, 
 
 and H is approximately 
 
 A nearer approximation to Q can then be obtained by sub- 
 stituting H for h, and if necessary a second value for v can be 
 found and a still nearer approximation to H. 
 
 In practical problems this is, however, hardly necessary. 
 
 Example. A weir without end contractions has a length of 16 feet. The head 
 as measured on the weir is 2 feet and the depth of the channel of approach below 
 the sill of the weir is 10 feet. Find the discharge. 
 
 Therefore C = 3-28. 
 
 Approximately, Q = 3-28 2^.16 
 
 = 148 cubic feet per second. 
 
 1 4.ft^i 
 The velocity v = -rz T = * 77 '* P er sec -> 
 
 -Llj X JLu 
 
 and 1^ 2 = . 0147 feet. 
 
 A second approximation to Q is, therefore, 
 
 Q = 3-28 (2-0147)*. 16 
 
 = 150 cubic feet per second. 
 
 A third value for Q can be obtained, but the approximation is sufficiently near 
 for all practical purposes. 
 
 In this case the error in neglecting the velocity of approach altogether, is 
 probably less than the error involved in taking m as 0*4099. 
 
FLOW OVER WEIRS 109 
 
 85. Time required to lower the water in a reservoir a 
 given distance by means of a weir. 
 
 A reservoir has a weir of length L feet made in one of its sides, 
 and having its sill H feet below the original level of the water in 
 the reservoir. 
 
 It is required to find the time necessary for the water to fall to 
 a level H feet above the sill of the weir. It is assumed that the 
 area of the reservoir is so large that the velocity of the water as 
 it approaches the weir may be neglected. 
 
 When the surface of the water is at any height h above the sill 
 the flow in a time dt is 
 
 Let A be the area of the water surface at this level and dh the 
 distance the surface falls in time dt. 
 
 Then, CU$dt = A.dh, 
 
 Adh 
 and ot = -- . 
 
 CL&* 
 
 The time required for the surface to fall (H-H ) feet is, 
 therefore, 
 
 I H Adh 
 
 The coefficient C may be supposed constant and equal to 3'34. 
 If then A is constant 
 
 . dh 
 
 ~ 
 
 = 2A/J_ J_ 
 CLWHo N/H 
 
 To lower the level to the sill of the weir, H must be made 
 equal to and t is then infinite. 
 
 That is, on the assumptions made, the surface of the water 
 never could be reduced to the level of the sill of the weir. The 
 time taken is not actually infinite as the water in the reservoir is 
 not really at rest, but has a small velocity in the direction of the 
 weir, which causes the time of emptying to be less than that 
 given by the above formula. But although the actual time is 
 not infinite, it is nevertheless very great. 
 
 9 A 
 
 When Ho is iH, t=7^FF- 
 
 OLvH 
 
 When Ho is T VH, t = ^jj=. 
 
 So that it takes three times as long for the water to fall from 
 iH to T VH as from H to iH. 
 
110 HYDRAULICS 
 
 Example 1. A reservoir has an area of 60,000 sq. yards. A weir 10 feet long 
 has its sill 2 feet below the surface. Find the time required to reduce the level of 
 the water 1' 11". 
 
 H = T y, H = 2'. 
 
 Therefore t= *' (3-46-0-708), 
 
 = 89,000 sees. 
 = 24-7 hours. 
 
 So that, neglecting velocity of approach, there will be only one inch of water on 
 the weir after 24 hours. 
 
 Example 2. To find in the last example the discharge from the reservoir in 
 15 hours. 
 
 O A /I 1 \ 
 
 Therefore 54,000 = -=- ( = -= \ . 
 
 From which ^/H^ 0-421, 
 
 H = 0-176 feet. 
 The discharge is, therefore, 
 
 (2 - 0-176) 540,000 cubic feet 
 = 984,960 cubic feet. 
 
 EXAMPLES. 
 
 (1) A weir is 100 feet long and the head is 9 inches. Find the discharge 
 in c. ft. per minute. C = 3'34. 
 
 (2) The discharge through a sharp-edged rectangular weir is 500 
 gallons per minute, and the still water head is 2| inches. Find the effective 
 length of the weir, m = '43. 
 
 (3) A weir is 15 feet long and the head over the crest is 15 inches. 
 Find the discharge. If the velocity of approach to this weir were 5 feet 
 per second, what would be the discharge ? 
 
 (4) Deduce an expression for the discharge through a right-angled 
 triangular notch. If the head over apex of notch is 12 ins., find the 
 discharge in c. ft. per sec. 
 
 (5) A rectangular weir is to discharge 10,000,000 gallons per day 
 (1 gallon = 10 Ibs.), with a normal head of 15 ins. Find the length of the 
 weir. Choose a coefficient, stating for what kind of weir it is applicable, 
 or take the coefficient C as 3'33. 
 
 (6) What is the advantage in gauging, of using a weir without end 
 contractions ? 
 
 (7) Deduce Francis' formula by means of the Thomson principle of 
 similarity. 
 
 Apply the formula to calculate the discharge over a weir 10 feet wide 
 under a head of 1-2 feet, assuming one end contraction, and neglecting the 
 effect of the velocity of approach. 
 
FLOW OVER WEIRS 111 
 
 (8) A rainfall of ^ inch per hour is discharged from a catchment area 
 of 5 square miles. Find the still water head when this volume flows over 
 a weir with free overfall 30 feet in length, constructed in six bays, each 
 5 feet wide, taking 0'415 as Bazin's coefficient. 
 
 (9) A district of 6500 acres (1 acre =43,560 sq. ft.) drains into a large 
 storage reservoir. The maximum rate at which rain falls in the district is 
 2 ins. in 24 hours. When rain falls after the reservoir is full, the water 
 requires to be discharged over a weir or bye-wash which has its crest at 
 the ordinary top-water level of the reservoir. Find the length of such a 
 weir for the above reservoir, under the condition that the water in the 
 reservoir shall never rise more than 18 ins. above its top-water level. 
 
 The top of the weir may be supposed flat and about 18 inches wide 
 (see Table XI). 
 
 (10) Compare rectangular and V notches in regard to accuracy and 
 convenience when there is considerable variation in the flow. 
 
 In a rectangular notch 50" wide the still water surface level is 15" above 
 the sill. 
 
 If the same quantity of water flowed over a right-angled V notch, what 
 would be the height of the still water surface above the apex ? 
 
 If the channels are narrow how would you correct for velocity of 
 approach in each case? Lon. Un. 1906. 
 
 (11) The heaviest daily record of rainfall for a catchment area was 
 found to be 42*0 million gallons. Assuming two-thirds of the rain to reach 
 the storage reservoir and to pass over the waste weir, find the length of 
 the sill of the waste weir, so that the water shall never rise more than two 
 feet above the sill. 
 
 (12) A weir is 300 yards long. What is the discharge when the head 
 is 4 feet ? Take Bazin's coefficient 
 
 AnK '00984 
 m = '405 + -= . 
 h 
 
 (13) Suppose the water approaches the weir in the last question in a 
 channel 8' 6" deep and 500 yards wide. Find by approximation the dis- 
 charge, taking into account the velocity of approach. 
 
 (14) The area of the water surface of a reservoir is 20,000 square 
 yards. Find the time required for the surface to fall one foot, when the 
 water discharges over a sharp-edged weir 5 feet long and the original head 
 over the weir is 2 feet. 
 
 (15) Find, from the following data, the horse-power available in a given 
 waterfall : 
 
 Available height of fall 120 feet. 
 
 A rectangular notch above the fall, 10 feet long, is used to measure 
 the quantity of water, and the mean head over the notch is found to be 
 15 inches, when the velocity of approach at the point where the head 
 is measured is 100 feet per minute. Lon. Un. 1905. 
 
CHAPTER V. 
 
 FLOW THROUGH PIPES. 
 
 86. Resistances to the motion of a fluid in a pipe. 
 
 When a fluid is made to flow through a pipe, certain resistances 
 are set up which oppose the motion, and energy is consequently 
 dissipated. Energy is lost, by friction, due to the relative motion 
 of the water and the pipe, by sudden enlargements or contractions 
 of the pipe, by sudden changes of direction, as at bends, and by 
 obstacles, such as valves which interfere with the free flow of the 
 fluid. 
 
 It will be necessary to consider these causes of the loss of 
 energy in detail. 
 
 Loss of head. Before proceeding to do so, however, the student 
 should be reminded that instead of loss of energy it is convenient 
 to speak of the loss of head. 
 
 It has been shown on page 39 that the work that can be 
 obtained from a pound of water, at a height z above datum, 
 moving with a velocity v feet per second, and at a pressure head 
 
 79 / D V^ 
 
 , is + + z foot pounds. 
 w* w 2g 
 
 If now water flows along a pipe and, due to any cause, h foot 
 pounds of work are lost per pound, the available head is clearly 
 diminished by an amount h. 
 
 In Fig. 86 water is supposed to be flowing from a tank through 
 a pipe of uniform diameter and of considerable length, the end B 
 being open to the atmosphere. 
 
 D 
 
 Fig. 86. Loss of head by friction in a pipe. 
 
FLOW THROUGH PIPES 113 
 
 Let be the head due to the atmospheric pressure. 
 
 Then if there were no resistances and assuming stream line 
 flow, Bernoulli's equation for the point B is 
 
 , Pa , v* p a 
 
 ZB + + 77- = Zip + , 
 w 2g w ' 
 
 from which ^ - Z P - Z B - H, 
 
 or V E = \/2#H. 
 
 The whole head H above the point B has therefore been 
 utilised to give the kinetic energy to the water leaving the pipe at 
 B. Experiment would show, however, that the mean velocity of 
 the water would have some value v less than V B , and the kinetic 
 
 energy would be ~- . 
 
 B 
 
 A head h = -= -- -r- = H - 
 
 Y B 2 
 -= 
 
 Zg 2g 
 
 has therefore been lost in the pipe. 
 
 By carefully measuring H, the diameter of the pipe d, and the 
 discharge Q in a given time, the loss of head h can be determined. 
 
 For v = -- , 
 
 (H 
 
 Q 2 
 
 and therefore h = H -- 
 
 The head h clearly includes all causes of loss of head, which, 
 in this case, are loss at the entrance of the pipe and loss by 
 friction. 
 
 87. Loss of head by friction. 
 
 Suppose tubes 1, 2, 3 are fitted into the pipe AB, Fig. 86, at 
 equal distance apart, and with their lower ends flush with the inside 
 of the pipe, and the direction of the tube perpendicular to the 
 direction of flow. If flow is prevented by closing the end B of the- 
 pipe, the water would rise in all the tubes to the level of the water- 
 in the reservoir. 
 
 Further, if the flow is regulated at B by a valve so that thfr 
 mean velocity through the pipe is v feet per second, a permanent 
 regime being established, and the pipe is entirely full, the mean 
 velocity at all points along the pipe will be the same ; and there- 
 fore, if between the tank and the point B there were no resistances 
 offered to the motion, and it be assumed that all the particles 
 L. H. 8 
 
114 HYDRAULICS 
 
 have a velocity equal to the mean velocity, the water would again 
 
 rise in all the tubes to the same height, but now lower than the 
 
 0i 
 surface of the water in the tank by an amount equal to ^- . 
 
 It is found by experiment, however, that the water does not 
 rise to the same height in the three tubes, but is lower in 2 than 
 in 1 and in 3 than in 2 as shown in the figure. As the fluid moves 
 along the pipe there is, therefore, a loss of head. 
 
 The difference of level h 2 of the water in the tubes 1 and 2 is 
 called the head lost by friction in the length of pipe 1 2. In any 
 length I of the pipe the loss of head is h. 
 
 This head is not wholly lost simply by the relative movement 
 of the water and the surface of the pipe, as if the water were 
 a solid body sliding along the pipe, but is really the sum of the 
 losses of energy, by friction along the surface, and due to relative 
 motions in the mass of water. 
 
 It will be shown later that, as the water flows along the pipe, 
 there is relative motion between consecutive filaments in the pipe, 
 and that, when the velocity is above a certain amount, the water 
 has a sinuous motion along the pipe. Some portion of this head h 2 
 is therefore lost, by the relative motion of the filaments of water, 
 and by the eddy motions which take place in the mass of the 
 water. 
 
 When the pipe is uniform the loss of head is proportional 
 to the length of the pipe, and the line CB, drawn through the tops 
 of the columns of water in the tubes and called the hydraulic 
 gradient, is a straight line. 
 
 It should be noted that along CB the pressure is equal to that 
 of the atmosphere. 
 
 88. Head lost at the entrance to the pipe. 
 
 For a point E just inside the pipe, Bernoulli's equation is 
 
 2 
 
 + Q- + head lost at entrance to the pipe = h& + , 
 
 being the absolute pressure head at E. 
 
 The head lost at entrance has been shown on page 70 to be 
 
 , , 
 
 about -^ , and therefore, 
 
 p E p a -, 
 
 
 . 
 w w 2(7 
 
 That is, the point C on the hydraulic gradient vertically above 
 
 1*50* 
 
 E, is - below the surface FD. 
 
FLOW THROUGH PIPES 
 
 115 
 
 If the pipe is bell-mouthed, there will be no head lost at entrance, 
 and the point C is a distance equal to below the surface. 
 
 89. Hydraulic gradient and virtual slope. 
 
 The line CB joining the tops of the columns of water in the 
 tube, is called the hydraulic gradient, and the angle i which it 
 makes with the horizontal is called the slope of the hydraulic 
 gradient, or the virtual slope. The angle i is generally small, and 
 
 sin i may be taken therefore equal to i, so that j = i. 
 In what follows the virtual slope -y is denoted by i. 
 
 i 
 
 More generally the hydraulic gradient may be defined as the 
 line, the vertical distance between which and the centre of the 
 pipe gives the pressure head at that point in the pipe. This line 
 will only be a straight line between any two points of the pipe, 
 when the head is lost uniformly along the pipe. 
 
 If the pressure head is measured above the atmospheric 
 pressure, the hydraulic gradient in Fig. 87 is AD, but if above 
 zero, AiDi is the hydraulic gradient, the vertical distance between 
 
 r> 144 
 AD and AiDi being equal to - , p a being the atmospheric 
 
 pressure per sq. inch. 
 
 Fig. 87. Pipe rising above the Hydraulic Gradient. 
 
 If the pipe rises above the hydraulic gradient AD, as in Fig. 87, 
 the pressure in the pipe at C will be less than that of the atmosphere 
 by a head equal to CE. If the pipe is perfectly air-tight it will 
 act as a siphon and the discharge for a given length of pipe will 
 not be altered. But if a tube open to the atmosphere be fitted at 
 
 82 
 
116 
 
 HYDRAULICS 
 
 the highest point, the pressure at C is equal to the atmospheric 
 pressure, and the hydraulic gradient will be now AC, and the flow 
 will be diminished, as the available head to overcome the resist- 
 ances between B and C, and to give velocity to the water, will only 
 be CF, and the part of the pipe CD will not be kept full. 
 
 In practice, although the pipe is closed to the atmosphere, yet 
 air will tend to accumulate and spoil the siphon action. 
 
 As long as the .point C is below the level of the water in the 
 reservoir, water will flow along the pipe, but any accumulation of 
 air at C tends to diminish the flow. In an ordinary pipe line it is 
 desirable, therefore, that no point in the pipe should be allowed to 
 rise above the hydraulic gradient. 
 
 90. Determination of the loss of head due to friction. 
 Reynolds' apparatus. 
 
 Fig. 88 shows the apparatus as used by Professor Reynolds* for 
 determining the loss of head by friction in a pipe. , 
 
 Fig. 88. Reynolds' apparatus for determining loss of head by friction in a pipe. 
 
 A horizontal pipe AB, 16 feet long, was connected to the water 
 main, a suitable regulating device being inserted between the 
 main and the pipe. 
 
 At two points 5 feet apart near the end B, and thus at a distance 
 sufficiently removed from the point at which the water entered 
 the pipe, that any initial eddy motions might be destroyed and a 
 steady regime established, two holes of about 1 mm. diameter were 
 pierced into the pipe for the purpose of gauging the pressure, at 
 these points of the pipe. 
 
 Short tubes were soldered to the pipe, so that the holes 
 communicated with these tubes, and these were connected by 
 
 * Phil. Trans. 1883, or Vol. n. Scientific Papers, Reynolds. 
 
FLOW THROUGH PIPES 117 
 
 indiarubber pipes to the limbs of a siphon gauge G-, made of glass, 
 and which contained mercury or bisulphide of carbon. Scales 
 were fixed behind the tubes so that the height of the columns 
 in each limb of the gauge could be read. 
 
 For very small differences of level a cathetometer was used *. 
 When water was made to flow through the pipe, the difference in 
 the heights of the columns in the two limbs of the siphon measured 
 the difference of pressure at the two points A and B of the pipe, 
 and thus measured the loss of head due to friction. 
 
 If s is the specific gravity of the liquid, and H the difference 
 in height of the columns, the loss of head due to friction in feet of 
 water is h = H (s - 1). 
 
 The quantity of water flowing in a time t was obtained by 
 actual measurement in a graduated flask. 
 
 Calling v the mean velocity in the pipe in feet per second, Q 
 the discharge in cubic feet per second, and d the diameter of the 
 pipe in feet, 
 
 The loss of head at different velocities was carefully measured, 
 and the law connecting head lost in a given length of pipe, with 
 the velocity, determined. 
 
 The results obtained by Keynolds, and others, using this 
 method of experimenting, will be referred to later. 
 
 91. Equation of flow in a pipe of uniform diameter 
 and determination of the head lost due to friction. 
 
 .Let 8Z be the length of a small element of pipe of uniform 
 diameter, Fig. 89. 
 
 A 
 
 C 
 
 Fig. 89. 
 
 Let the area of the transverse section be w, P the length of 
 the line of contact of the water and the surface on this section, or 
 the wetted perimeter, a the inclination of the pipe, p the pressure 
 per unit area on AB, and p dp the pressure on CD. 
 * p. 258, Vol. i. Scientific Papers, Keynolds. 
 
118 HYDRAULICS 
 
 Let v be the mean velocity of the fluid, Q the flow in cubic 
 feet per second, and w the weight of one cubic foot of the fluid. 
 The work done by gravity as the fluid flows from AB to CD 
 
 = Qw . dz <*> . v . w . dz. 
 
 The work done on ABCD by the pressure acting upon the area 
 AB 
 
 = p . w . v ft. Ibs. per sec. 
 
 The work done by the pressure acting upon CD against the 
 flow 
 
 = (p dp) . o> . v ft. Ibs. per sec. 
 
 The frictional force opposing the motion is proportional to the 
 area of the wetted surface and is equal to F . P . ol, where F is some 
 coefficient which must be determined by experiment and is the 
 frictional force per unit area. The work done by friction per sec. 
 is, therefore, F . P . bl . v. 
 
 The velocity being constant, the velocity head is the same at 
 both sections, and therefore, applying the principle of the con- 
 servation of energy, 
 
 p.<o.v + <a.v .w .dz = (p- dp) to . v + F . P . 9Z . v. 
 
 Therefore w . w . dz = - dp . w + F . P . 3Z, 
 
 , dp F.P.dl 
 
 or dz = - + - - . 
 
 w w . to 
 
 Integrating this equation between the limits of z and Zi, p and 
 Pi being the corresponding pressures, and Z the length of the pipe, 
 
 p 1 p F . P I 
 
 z Zi = + - . 
 
 W W W to 
 
 Therefore, + * = & + z 1+ ^ 1 . 
 
 W W W to 
 
 The quantity - - is equal to h of equation (1), page 52, and is 
 
 the loss of head due to friction. The head lost by friction is 
 therefore proportional to the area of the wetted surface of the pipe 
 PZ, and inversely proportional to the cross sectional area of the 
 pipe and to the density of the fluid. 
 
 92. Hydraulic mean depth. 
 
 The quantity p is called the hydraulic radius, or the hydraulic 
 
 mean depth. 
 
 If then this quantity is denoted by m, the head h lost by 
 friction, is 
 
 FZ 
 
 h = 
 
 w .m 
 
FLOW THROUGH PIPES 119 
 
 The quantity F, which has been called above the friction per 
 unit area, is found by experiments to vary with the density, 
 viscosity, and velocity of the fluid, and with the diameter and 
 roughness of the internal surface of the pipe. 
 
 In Hydraulics, the fluid considered is water, and any variations 
 in density or viscosity, due to changes of temperature, are generally 
 negligible. F, therefore, may be taken as proportional to the 
 density, or to the weight w per cubic foot, to the roughness of the 
 pipe, and as some function, f(v) of the mean velocity, and f(d) of 
 the diameter of the pipe. 
 
 Then, h 
 
 m 
 
 in which expression p. may be called the coefficient of friction. 
 
 It will be seen later, that the mean velocity v is different from 
 the relative velocity u of the water and the surface of the pipe, 
 and it probably would be better to express F as a function of u, 
 but as u itself probably varies with the roughness of the pipe and 
 with other circumstances, and cannot directly be determined, it 
 simplifies matters to express F, and thus h, as a function of v. 
 
 93. Empirical formulae for loss of head due to friction. 
 
 The difficulty of correctly determining the exact value of 
 fW fWy nas led to the use of empirical formulae, which have 
 proved of great practical service, to express the head h in terms of 
 the velocity and the dimensions of the pipe. 
 
 The simplest formula assumes that the friction simply varies as 
 the square of the velocity, and is independent of the diameter of 
 the pipe, or f(v) f(d) = av 2 . 
 
 mi, -, av 2 l /-.N 
 
 Then, h = - .. ............................. (1), 
 
 or writing for a, 
 
 from which is deduced the -well-known Chezy formula, 
 
 n /* 
 
 t> = Cy m. j, 
 
 or v = C \/mi. 
 
 Another form in which formula (1) is often found is 
 
 h -f^L 
 
 1 1 ^r , 
 
 2g m' 
 
120 
 
 HYDRAULICS 
 
 or since m = -7 for a circular pipe full of water, 
 
 h 
 
 .(3), 
 
 in wliicli for a of (1) is substituted - . 
 
 The quantity Zg was introduced by Weisbach so that h is 
 expressed in terms of the velocity head. 
 
 Adopting either of these forms, the values of the coefficients C 
 and / are determined from experiments on various classes of pipes. 
 
 It should be noticed that C = 
 
 _ /^ 
 
 -V ?' 
 
 Values of these constants are shown in Tables XII to XIY for 
 different kinds and diameters of pipes and different velocities. 
 
 TABLE XII. 
 
 Values of C in the formula v = C *Jmi for new and old cast-iron 
 pipes. 
 
 
 New cast-iron pipes 
 
 Old cast-iron pipes 
 
 Velocities in ft. per second 
 
 1 
 
 3 
 
 6 
 
 10 
 
 1 
 
 3 
 
 6 
 
 10 
 
 Diameter of pipe 
 
 
 
 
 
 
 
 
 
 3" 
 
 95 
 
 98 
 
 100 
 
 102 
 
 63 
 
 68 
 
 71 
 
 73 
 
 6" 
 
 96 
 
 101 
 
 104 
 
 106 
 
 69 
 
 74 
 
 77 
 
 79 
 
 9" 
 
 98 
 
 105 
 
 109 
 
 112 
 
 73 
 
 78 
 
 80 
 
 84 
 
 12" 
 
 100 
 
 108 
 
 112 
 
 117 
 
 77 
 
 82 
 
 85 
 
 88 
 
 15" 
 
 102 
 
 110 
 
 117 
 
 122 
 
 81 
 
 86 
 
 89 
 
 91 
 
 18" 
 
 105 
 
 112 
 
 119 
 
 125 
 
 86 
 
 91 
 
 94 
 
 97 
 
 24" 
 
 111 
 
 120 
 
 126 
 
 131 
 
 92 
 
 98 
 
 101 
 
 104 
 
 30" 
 
 118 
 
 126 
 
 131 
 
 136 
 
 98 
 
 103 
 
 106 
 
 109 
 
 36" 
 
 124 
 
 131 
 
 136 
 
 140 
 
 103 
 
 108 
 
 111 
 
 114 
 
 42" 
 
 130 
 
 136 
 
 140 
 
 144 
 
 105 
 
 111 
 
 114 
 
 117 
 
 48" 
 
 135 
 
 141 
 
 145 
 
 148 
 
 106 
 
 112 
 
 115 
 
 118 
 
 60" 
 
 142 
 
 147 
 
 150 
 
 152 
 
 
 
 
 
 For method of determining the values of C given in the tables, 
 see page 102. 
 
 On reference to these tables, it will be seen, that C and / are 
 by no means constant, but vary very considerably for different 
 kinds of pipes, and for different values of the velocity in any 
 given pipe. 
 
FLOW THROUGH PIPES 
 
 121 
 
 The fact that C varies with the velocity, and the diameter of 
 the pipe, suggests that the coefficient C is itself some function of 
 the velocity of flow, and of the diameter of the pipe, and that 
 jjf(v) f(d) does not, therefore, equal CLV*. 
 
 TABLE XIII. 
 
 Values of / in the formula 
 
 2gd 
 
 
 New cast-iron pipes 
 
 Old cast-iron pipes 
 
 Velocities in 
 ft. per second 
 
 1 
 
 3 
 
 6 
 
 10 
 
 1 
 
 3 
 
 6 
 
 10 
 
 Diam. of pipe 
 
 
 
 
 
 
 
 
 
 3" 
 
 0071 
 
 0067 
 
 0064 
 
 0062 
 
 0152 
 
 0139 
 
 0128 
 
 0122 
 
 6" 
 
 007 
 
 0063 
 
 006 
 
 0057 
 
 0135 
 
 0117 
 
 0108 
 
 0103 
 
 9" 
 
 0067 
 
 0058 
 
 0055 
 
 0051 
 
 0122 
 
 0105 
 
 010 
 
 0092 
 
 12" 
 
 0064 
 
 0056 
 
 0051 
 
 0048 
 
 0108 
 
 0096 
 
 0089 
 
 0084 
 
 15" 
 
 0062 
 
 0053 
 
 0048 
 
 0043 
 
 0099 
 
 0087 
 
 0081 
 
 0078 
 
 18" 
 
 0058 
 
 0051 
 
 0045 
 
 0041 
 
 0087 
 
 0078 
 
 0073 
 
 0069 
 
 24" 
 
 0053 
 
 0045 
 
 0040 
 
 0037 
 
 0076 
 
 0067 
 
 0063 
 
 0060 
 
 30" 
 
 0046 
 
 0040 
 
 0037 
 
 0035 
 
 0067 
 
 0061 
 
 0057 
 
 0055 
 
 36" 
 
 0042 
 
 0037 
 
 0035 
 
 0033 
 
 0061 
 
 0056 
 
 0052 
 
 0050 
 
 42" 
 
 0038 
 
 0035 
 
 0033 
 
 0031 
 
 0058 
 
 0052 
 
 005 
 
 0048 
 
 48" 
 
 0036 
 
 0032 
 
 0031 
 
 0029 
 
 0057 
 
 0051 
 
 0049 
 
 0046 
 
 60" 
 
 0032 
 
 0030 
 
 0029 
 
 0028 
 
 
 
 
 
 TABLE XIV. 
 
 Values of C in the formula v = C *Jmi for steel riveted pipes. 
 
 Velocities in ft. per second 
 
 1 
 
 3 
 
 5 
 
 10 
 
 Diameter of pipe 
 
 
 
 
 
 3" 
 
 81 
 
 86 
 
 89 
 
 92 
 
 11" 
 
 92 
 
 102 
 
 107 
 
 115 
 
 llf 
 15" 
 
 93 
 
 109 
 
 99 
 112 
 
 102 
 114 
 
 105 
 117 
 
 38" 
 
 113 
 
 113 
 
 113 
 
 113 
 
 42" 
 
 102 
 
 106 
 
 108 
 
 111 
 
 48" 
 
 105 
 
 105 
 
 105 
 
 105 
 
 72"* 
 
 110 
 
 110 
 
 111 
 
 111 
 
 72" 
 
 93 
 
 101 
 
 105 
 
 110 
 
 103" 
 
 114 
 
 109 
 
 106 
 
 104 
 
 * See pages 124 and 137. 
 
122 HYDRAULICS 
 
 94. Formula of Darcy. 
 
 In 1857 Darcy* published an account of a series of experiments 
 on flow of water in pipes, previous to the publication of which, it 
 had been assumed by most writers that the friction and consequently 
 the constant C was independent of the nature of the wetted surface 
 of the pipe (see page 232). He, however, showed by experiments 
 upon pipes of various diameters and of different materials, 
 including wrought iron, sheet iron covered with bitumen, lead, 
 glass, and new and old cast-iron, that the condition of the internal 
 surface was of considerable importance and that the resistance was 
 by no means independent of it. 
 
 He also investigated the influence of the diameter of the pipe 
 upon the resistance. The results of his experiments he expressed 
 by assuming the coefficient a in the formula 
 
 7 Q>1 2 
 
 h = . tr 
 
 m 
 
 was of the form a = a + - 
 
 T 
 
 r being the radius of the pipe. 
 
 For new cast-iron, and wrought-iron pipes of the same 
 roughness, Darcy's values of a and ft when transferred to English 
 units are, 
 
 a = 0-000077, 
 ft = 0-000003235. 
 
 For old cast-iron pipes Darcy proposed to double these values. 
 Substituting the diameter d for the radius r, and doubling /?, for 
 new pipes, 
 
 = 0-000077 
 
 12d+l 
 
 Substituting for m its value -^ , and multiplying and dividing 
 by 20, 
 
 For old cast-iron pipes, 
 
 h = 0-00001294 
 
 d / m 
 
 .J ........................ (7). 
 
 Eecherches Experlmentales. 
 
FLOW THROUGH PIPES 123 
 
 As the student cannot possibly retain, without unnecessary 
 labour, values of / and C for different diameters it is convenient 
 to remember the simple forms, 
 
 f=- 5 ( 1 + 
 
 for new pipes, and 
 
 for old pipes. 
 
 According to Darcy, therefore, the coefficient C in the Chezy 
 formula varies only with the diameter and roughness of the pipe. 
 
 The values of C as calculated from his experimental results, for 
 some of the pipes, were practically constant for all velocities, and 
 notably for those pipes which had a comparatively rough internal 
 surface, but for smooth pipes, the value of C varied from 10 to 
 20 per cent, for the same pipe as the velocity changed. The 
 experiments of other workers show the same results. 
 
 The assumption that pf(v) f(d) - at; 2 in which a is made to 
 vary only with the diameter and roughness, or in other words, the 
 assumption that h is proportional to v 2 is therefore not in general 
 justified by experiments. 
 
 95. As stated above, the formulae given must be taken as 
 purely empirical, and though by the introduction of suitable 
 constants they can be made to agree with any particular experi- 
 ment, or even set of experiments, yet none of them probably 
 expresses truly the laws of fluid friction. 
 
 The formula of Chezy by its simplicity has found favour, and 
 it is likely, that for some time to come, it will continue to be used, 
 either in the form v = C vW, or in its modified form 
 
 ~ - ^ 
 
 2gd 
 
 In making calculations, values of C or /, which most nearly suit 
 any given case, can be taken from the tables. 
 
 96. Variation of C in the formula v = C */m~i with service. 
 
 It should be clearly borne in mind, however, that the dis- 
 charging capacity of a pipe may be considerably diminished after 
 a few years' service. 
 
 Darcy's results show that the loss of head in an old pipe may 
 be double that in a new one, or since the velocity v is taken as 
 
124 HYDRAULICS 
 
 proportional to the square root of Ji, the discharge of the old pipe 
 for the same head will be ^ times that of the new pipe, or about 
 
 30 per cent. less. 
 
 An experiment by Sherman* on a 36-inch cast-iron main showed 
 that after one year's service the discharge was diminished by 
 23 per cent., but a second year's service did not make any further 
 alteration. 
 
 Experiments by Kuichlingt on a 36-inch cast-iron main showed 
 that the discharge during four years diminished 36 per cent., while 
 experiments by Fitzgerald J on a cast-iron main, coated with tar, 
 which had been in use for 16 years, showed that cleaning increased 
 the discharge by nearly 40 per cent. Fitzgerald also found that 
 the discharge of the Sudbury aqueduct diminished 10 per cent, in 
 one year due to accumulation of slime. 
 
 The experiments of Marx, Wing, and Hoskins on a 72-inch steel 
 main, when new, and after two years' service, showed that there 
 had been a change in the condition of the internal surface of the 
 pipe, and that the discharge had diminished by 10 per cent, at low 
 velocities and about 5 per cent, at the higher velocities. 
 
 If, therefore, in calculations for pipes, values of C or / are used 
 for new pipes, it will in most cases be advisable to make the pipe 
 of such a size that it will discharge under the given head at least 
 from 10 to 30 per cent, more than the calculated value. 
 
 97. Ganguillet and Kutter's formula. 
 
 Granguillet and Kutter endeavoured to determine a form for 
 the coefficient C in the Chezy formula v = C \/W', applicable 
 to all forms of channels, and in which C is made a function of the 
 virtual slope i, and also of the diameter of the pipe. 
 
 They gave C the value, 
 
 11-6 i 1 ' 811 i ' 00281 
 
 to n 
 
 = ............... 
 
 This formula is very cumbersome to use, and the value of the 
 coefficient of roughness n for different cases is uncertain. Tables 
 have however been prepared which considerably facilitate the use 
 of the formula. 
 
 * Trans. Am.S.C.E. Vol. XLIV. p. 85. 
 t Trans. Am.S. C.E. Vol. XLIV. p. 56. 
 $ Trans. Am.S.C.E. Vol. XLIV. p. 87. 
 See Table No. XIV. 
 
FLOW THROUGH PIPES 125 
 
 Values of n in Ganguillet and Kutter's formula. 
 Wood pipes = '01, may be as high as '015. 
 
 Cast-iron and steel pipes = 'Oil, *02. 
 
 Grlazed earthenware = '013. 
 
 98. Reynolds' experiments and the logarithmic formula. 
 
 The formulae for loss of head due to friction previously given 
 have all been founded upon a probable law of variation of h 
 with v, but no rational basis for the assumptions has been adduced. 
 
 It has been stated in section 93, that on the assumption that h 
 varies with v 2 , the coefficient C in the formula 
 
 ' m^ 
 
 is itself a function of the velocity. 
 
 The experiments and deductions of Reynolds, and of later 
 workers, throw considerable light upon this subject, and show that 
 h is proportional to v n , where n is an index which for very small 
 velocities* as previously shown by Poiseuille by experiments on 
 capillary tubes is equal to unity, and for higher velocities may 
 have a variable value, which in many cases approximates to 2. 
 
 As Darcy's experiments marked a decided advance, in showing 
 experimentally that the roughness of the wetted surface has an 
 effect upon the loss due to friction, so Reynolds' work marked 
 a further step in showing that the index n depends upon the state 
 of the internal surface, being generally greater the rougher the 
 surface. 
 
 The student will be better able to follow Reynolds, by a brief 
 consideration of one of his experiments. 
 
 In Table XV are shown the results of an experiment made 
 by Reynolds with apparatus as illustrated in Fig. 88. 
 
 In columns 1 and 5 are shown the experimental values of 
 
 i = j, and v respectively. 
 
 The curves, Fig. 90, were obtained by plotting v as abscissae 
 and i as ordinates. 
 
 For velocities up to 1*347 feet per second, the points lie very close 
 to a straight line and i is simply proportional to the velocity, or 
 
 i = kiv (11), 
 
 Jci being a coefficient for this particular pipe. 
 
 Above 2 feet per second, the points lie very near to a continuous 
 curve, the equation to which is 
 
 < = * (12). 
 
 * Phil. Tram. 1883. 
 
126 
 
 HYDRAULICS 
 
 Taking logarithms, 
 
 log i = log k + n log v. 
 
 Curve N?2 is the part AB of 
 
 Carve N?l drawn to larger 
 Scale. y 
 
 VeLodty. 
 
 Fig. 90. 
 
 The curve, Fig. 90 a, was determined by plotting log i as 
 ordinate and logv as abscissae. Eeynolds calls the lines of this 
 figure the logarithmic homologues. 
 
 Calling log ij y, and log v, x, the equation has the form 
 
 nx. 
 
 which is an equation to a straight line, the inclination of which to 
 the axis of x is 
 
 = tan" 1 n. 
 
 or 
 
 n = tan 0. 
 
 Further, when x = 0, y = &, so that the value of ~k can readily be 
 found as the ordinate of the line when x or \ogv = Q, that is, 
 when v = 1. 
 
 Up to a velocity of 1'37 feet per second, the points lie near to 
 a line inclined at 45 degrees to the axis of v, and therefore, n is 
 unity, or as stated above, i = kv. 
 
 The ordinate when 'v is equal to unity is 0'038, so that for the 
 first part of the curve k = '038, and i = '03&v. 
 
FLOW THROUGH PIPES 
 
 127 
 
 Above the velocity of 2 feet per second the points lie about 
 a second straight line, the inclination of which to the axis of v is 
 
 6 = tan- 1 170. 
 
 Therefore log i = 170 log v + k. 
 
 The ordinate when v equals 1 is 0*042, so that 
 
 Je = 0-042, 
 and i = 0*042v r7 . 
 
 -4-0 
 
 -3-0 
 
 -2-0 
 
 -1-0 
 
 -9 
 
 --S 
 
 _-7 
 6 
 
 -5 
 
 veLocUy 
 
 -02 
 
 01 
 
 \ Log v 
 4 -5 -7-8-91\0 
 
 3 4 5 6 789W 
 
 20 
 
 Fig. 90 a. Logarithmic plottings of i and v to determine the index n in 
 the formula for pipes, i=kv n . 
 
 In the table are given values of i as determined experimentally 
 and as calculated from the equation i = Jc . v n . 
 
 The quantities in the two columns agree within 3 per cent. 
 
123 
 
 HYDRAULICS 
 
 TABLE XV. 
 
 Experiment on Resistance in Pipes. 
 Lead Pipe. Diameter 0'242". Water from Manchester Main. 
 
 Slope t=* 
 
 k 
 
 n 
 
 Velocity ft. per 
 second 
 
 Experimental value 
 
 Calculated from 
 
 
 
 
 0086 
 
 0092 
 
 038 
 
 1 
 
 239 
 
 0172 
 
 0172 
 
 038 
 
 1 
 
 451 
 
 0258 
 
 0261 
 
 038 
 
 1 
 
 690 
 
 0345 
 
 0347 
 
 038 
 
 1 
 
 914 
 
 0430 
 
 0421 
 
 038 
 
 1 
 
 1-109 
 
 0516 
 
 0512 
 
 038 
 
 1 
 
 1-349 
 
 0602 
 
 
 ... 
 
 ... 
 
 1-482 
 
 0682 
 
 
 ... 
 
 ... 
 
 1-573 
 
 0861 
 
 
 
 
 1-671 
 
 1033 
 
 ... 
 
 
 
 1-775 
 
 1206 
 
 ... 
 
 
 
 1-857 
 
 1378 
 
 1352 
 
 042 
 
 1-70 
 
 1-987 
 
 1714 
 
 1610 
 
 042 
 
 1-70 
 
 2-203 
 
 3014 
 
 2944 
 
 042 
 
 1-70 
 
 3-141 
 
 4306 
 
 4207 
 
 042 
 
 1-70 
 
 3-93 
 
 8185 
 
 8017 
 
 042 
 
 1-70 
 
 5-66 
 
 1-021 
 
 1-033 
 
 042 
 
 1-70 
 
 6-57 
 
 1-433 
 
 1-476 
 
 042 
 
 1-70 
 
 8-11 
 
 2-455 
 
 2-404 
 
 042 
 
 1-70 
 
 10-79 
 
 3-274 
 
 3-206 
 
 042 
 
 1-70 
 
 12-79 
 
 3-873 
 
 3-899 
 
 042 
 
 1-70 
 
 14-29 
 
 NOTE. To make the columns shorter, only part of Reynolds' results are given. 
 
 99. Critical velocity. 
 
 It appears, from Reynolds' experiment, that up to a certain 
 velocity, which is called the Critical Velocity, the loss of head is 
 proportional to v, but above this velocity there is a definite change 
 in the law connecting i and v. 
 
 By experiments upon pipes of different diameters and the 
 water at variable temperatures, Reynolds found that the critical 
 velocity, which was taken as the point of intersection of the two 
 straight lines, was 
 
 0388P 
 
 the value of P being 
 P = 
 
 (13), 
 
 1+ 0-0336 T + -0000221T 2 
 T being the temperature in degrees centigrade and D the diameter 
 of the pipe. 
 
OF 
 < FLOW THROUGH PIPES 129 
 
 100. Critical velocity by the method of colour bands. 
 
 The existence of the critical velocity has been beautifully 
 shown by Reynolds, by the method of colour bands, and his 
 experiments also explain why there is a sudden change in the law 
 connecting i and v. 
 
 " Water was drawn through tubes (Figs. 91 and 92), out of 
 a large glass tank in which the tubes were immersed, and in 
 which the water had been allowed to come to rest, arrangements 
 being made as shown in the figure so that a streak or streaks of 
 highly coloured water entered the tubes with the clear water." 
 
 Fig. 91. 
 
 Fig. 92. 
 
 The results were as follows : 
 
 " (1) When the velocities were sufficiently low, the streak 
 of colour extended in a beautiful straight line through the tube " 
 (Fig. 91). 
 
 " (2) As the velocity was increased by small stages, at 
 some point in the tube, always at a considerable distance from the 
 trumpet-shaped intake, the colour band would all at once mix up 
 with the surrounding water, and fill the rest of the tube with 
 a mass of coloured water" (Fig. 92). 
 
 This sudden change takes place at the critical velocity. 
 
 That such a change takes place is also shown by the apparatus 
 illustrated in Fig. 88; when the critical velocity is reached there is 
 a violent disturbance of the mercury in the U tube. 
 
 There is, therefore, a definite and sudden change in the con- 
 dition of flow. For velocities below the critical velocity, the flow 
 is parallel to the tubes, or is " Stream Line " flow, but after the 
 critical velocity has been passed, the motion parallel to the tube is 
 accompanied by eddy motions, which cause a definite change to 
 take place in the law of resistance. 
 
 Barnes and Coker* have determined the critical velocity by 
 noting the sudden change of temperature of the water when its 
 motion changes. They have also found that the critical velocity, 
 as determined by noting the velocity at which stream-line flow 
 
 * Proceedings of the Royal Society, Vol. LXXIV. 1904; Phil. Transactions, 
 Royal Society, Vol. xx. pp. 4561. 
 
 9 
 
130 HYDRAULICS 
 
 breaks up into eddies, is a much more variable quantity than 
 that determined from the points of intersection of the two lines 
 as in Fig. 90. In the former case the critical velocity depends 
 upon the condition of the water in the tank, and when it is 
 perfectly at rest the stream lines may be maintained at much 
 higher velocities than those given by the formula of Reynolds. 
 If the water is not perfectly at rest, the results obtained by both 
 methods agree with the formula. 
 
 Barnes and Coker have called the critical velocity obtained by 
 the method of colour bands the upper limit, and that obtained by 
 the intersection of the logarithmic homologues the lower critical 
 velocity. The first gives the velocity at which water flowing from 
 rest in stream-line motion breaks up into eddy motion, while the 
 second gives the velocity at which water that is initially disturbed 
 persists in flowing with eddy motions throughout a long pipe, or 
 in other words the velocity is too high to allow stream lines to be 
 formed. 
 
 That the motion of the water in large conduits is in a similar 
 condition of motion is shown by the experiment of Mr Gr. H. 
 Benzenberg* on the discharge through a sewer 12 feet in diameter, 
 2534 ft. long. 
 
 In order to measure the velocity of water in the sewer, red 
 eosine dissolved in water was suddenly injected into the sewer, 
 and the time for the coloured water to reach the outlet half a 
 mile away was noted. The colour was readily perceived and it 
 was found that it was never distributed over a length of more than 
 9 feet. As will be seen by reference to section 130, the velocities 
 of translation of the particles on any cross section at any instant 
 are very different, and if the motion were stream line the colour 
 must have been spread out over a much greater length. 
 
 101. Law of frictional resistance for velocities above the 
 critical velocity. 
 
 As seen from Reynolds' formula, the critical velocity except 
 for very small pipes is so very low that it is only necessary in 
 practical hydraulics to consider the law of frictional resistance for 
 velocities above the critical velocity. 
 
 For any particular pipe, 
 
 i = Jcv n , 
 and it remains to determine Jc and n. 
 
 From the plottings of the results of his own and Darcy's 
 
 p. 1173. 
 
 Transactions Am.S.C.E. 1893; and also Proceedings Am.S.C.E., Vol. xxvn. 
 
FLOW THROUGH PIPES 
 
 131 
 
 experiments, Reynolds found that the law of resistance " for all 
 pipes and all velocities " could be expressed as 
 
 AD 3 . /BD Y 
 
 ~p~* = V~p~) ( 14 )- 
 
 Transposing, 
 
 and 
 
 AP*.D 3 
 B 71 2 - M 
 
 = T- 
 
 (15), 
 
 A D 3 -' 
 
 D is diameter of pipe, A and B are constants, and P is obtained 
 from formula (13). 
 
 Taking the temperature in degrees centigrade and the metre 
 as unit length, 
 
 A - 67,700,000, 
 
 B - 396, 
 
 -r-*, J- 
 
 or 
 
 in which 
 
 -0036T + -000221T 2 ' 
 " . v n . P 2 -" _ y . v n 
 
 67,700,000 D 3 -' ] 
 
 7 = 67,700,000 ' 
 Values of y when the temperature is 10 C. 
 
 (16), 
 
 71 
 
 7 
 
 1-75 
 
 0-000265 
 
 1-85 
 
 0-000388 
 
 1-95 
 
 0-000587 
 
 2-00 
 
 0-000704 
 
 The values for A and B, as given by Reynolds, are, however, 
 only applicable to clean pipes, and later experiments show that 
 although 
 
 it is doubtful whether 
 
 p = 3 - n, 
 
 as given by Reynolds, is correct. 
 
 Value of n. For smooth pipes n appears to be nearly 175. 
 Reynolds found the mean value of n for lead pipes was 1723. 
 
 Saph and Schoder*, in an elaborate series of experiments 
 carried out at Cornell University, have determined for smooth 
 
 * Transactions of the American Society of Civil Engineers, May, 1903. See 
 exercise 31, page 172. 
 
 92 
 
132 HYDRAULICS 
 
 brass pipes a mean value for n of 1'75. Coker and Clements 
 found that n for a brass pipe '3779 inches diameter was 1*731. In 
 column 5 of Table XVI are given values of n, some taken from 
 Saph and Schoder's paper, and others as determined by the 
 author by logarithmic plotting of a large number of experiments. 
 
 It will be seen that n varies very considerably for pipes of 
 different materials, and depends upon the condition of the surface 
 of a given material, as is seen very clearly from Nos. 3 and 4. 
 The value for n in No. 3 is 1'72, while for No. 4, which is the 
 same pipe after two years' service, the value of n is 1'93. The 
 internal surface had no doubt become coated with a deposit of 
 some kind. 
 
 Even very small differences in the condition of the surface, 
 such as cannot be seen by the unaided eye, make a considerable 
 difference in the value of n, as is seen by reference to the values 
 for galvanised pipes, as given by Saph and Schoder. For large 
 pipes of riveted steel, riveted wrought iron, and cast iron, the 
 value of n approximates to 2. 
 
 The method, of plotting the logarithms of i and v determined 
 by experiment, allows of experimental errors being corrected 
 without difficulty and with considerable assurance. 
 
 102. The determination of the values of C given in 
 Table XII. 
 
 The method of logarithmic plotting has been employed for 
 determining the values of C given in Table XII. 
 
 If values of C are calculated by the substitution of the 
 experimental values of v and i in the formula 
 
 many of the results are apparently inconsistent with each other 
 due to experimental errors. 
 
 The values of C in the table were, therefore, determined as 
 follows. 
 
 Since i = kv n 
 
 and in the Chezy formula 
 
 v = C v if, 
 
 <--* 
 
 therefore -^ = kv n 
 
 wO 
 
 and 2 log C = 21og v - (log m +log k + n log v) ...... (17). 
 
 The index n and the coefficient k were determined for a 
 number of cast-iron pipes. 
 
FLOW THROUGH PIPES 133 
 
 Values of C for velocities from 1 to 10 were calculated. Curves 
 were then plotted, for different velocities, having C as ordinates 
 and diameters as abscissae, and the values given in the table were 
 deduced from the curves. 
 
 The values of C so interpolated differ very considerably, in 
 some cases, from the experimental values. The difficulties 
 attending the accurate determination of i and v are very great, 
 and the values of C, for any given pipe, as calculated by substi- 
 tuting in the Chezy formula the losses of head in friction and the 
 velocities as determined in the experiments, were frequently 
 inconsistent with each other. 
 
 As, for example, in the pipe of 3*22 ins. diameter given in 
 Table XVI which was one of Darcy's pipes, the variation of C as 
 calculated from In and v given by Darcy is from 78'8 to 100. 
 
 On plotting log h and log v and correcting the readings so 
 that they all lie on one line and recalculating C the variation was 
 found to be only from 95'9 to 101. 
 
 Similar corrections have been made in other cases. 
 
 The author thinks this procedure is justified by the fact that 
 many of the best experiments do not show any such inconsistencies. 
 
 An attempt to draw up an interpolated table for riveted pipes 
 was not satisfactory. The author has therefore in Table XIV 
 given the values of C as calculated by formula (17), for various 
 velocities, and the diameters of the pipes actually experimented 
 upon. If curves are plotted from the values of C given in 
 Table XIV, it will be seen that, except for low velocities, the 
 curves are not continuous, and, until further experimental evidence 
 is forthcoming for riveted pipes, the engineer must be content, 
 with choosing values of C, which most nearly coincide, as far as 
 he can judge, with the case he is considering. 
 
 103. Variation of k, in the formula i = kv M , with the 
 diameter. 
 
 It has been shown in section 98 how the value of k, for a 
 given pipe, can be obtained by the logarithmic plotting of i and v. 
 
 In Table XVI, are given values of &, as determined by the 
 author, by plotting the results of different experiments. Saph 
 and Schoder found that for smooth hard-drawn brass pipes 
 of various sizes n varied between 173 and 1*77, the mean value 
 being 175. 
 
 By plotting log d as abscissae and log A; as ordinates, as in 
 Fig. 93, for these brass pipes the points lie nearly in a straight line 
 which has an inclination V with the axis of d, such that 
 
134 
 
 HYDRAULICS' 
 
 and the equation to the line is, therefore, 
 
 log ~k = log y - p log d, 
 where p = 1'25, 
 
 and log y = log Jc 
 
 when d = I. 
 
 From the figure 
 
 y = G'000296 per foot length of pipe. 
 
 i 
 
 05. 
 04 
 03 
 
 02 
 
 01 
 
 003 
 
 Equation tu line 
 Log.lo-Log m,-1-25Lcg d 
 
 126 
 
 02 -03 -04- -06 -OS W -2O 
 Log d 
 
 Fig. 93. Logarithmic plottings of fe and d to determine the index p in the formula 
 
 . _ 7 . v n 
 
 On the same figure are plotted logd and log&, as deduced 
 from experiments on lead and glass pipes by various workers. It 
 will be seen that all the points lie very close to the same line. 
 
 For smooth pipes, therefore, and for velocities above the 
 critical velocity, the loss of head due to friction is given by 
 
 yv n 
 
 the mean value for y being 0'000296, for ra, 175, and for p T25. 
 
 From which, v = 104f m d, 
 
 or log v = 2-017 + 0-572 log i + 0715 log d. 
 
FLOW THROUGH PIPES 135 
 
 The value of p in this formula agrees with that given by 
 Reynolds in his formula 
 
 Professor Unwin* in 1886, by an examination of experiments 
 on cast-iron pipes, deduced the formula, for smooth cast-iron 
 pipes, 
 
 and for rough pipes, 
 
 d 
 
 1-1 
 
 M. Flamantt in 1892 examined carefully the experiments 
 available on flow in pipes and proposed the formula, 
 
 yv^ 
 
 1 ^1-25 ) 
 
 for all classes of pipes, and suggested for y the following values : 
 Lead pipes \ 
 
 G-lass \ -000236 to '00028, 
 
 Wrought-iron (smooth) J 
 Cast-iron new '000336, 
 
 in service '000417. 
 
 If the student plots from Table XVI, log d as ordinates, and 
 log k as abscissae, it will be found, that the points all lie between 
 two straight lines the equations to which are 
 
 log k = log '00069 - T25 log d, 
 and log Jc = log '00028 - 1'25 log d. 
 
 Further, the points for any class of pipes not only lie between 
 these two lines, but also lie about some line nearly parallel to 
 these lines. So that p is not very different from 1'25. 
 From the table, n is seen to vary from 1*70 to 2'08. 
 A general formula is thus obtained, 
 
 7 -00028 to -00069t; r70to2 - 08 Z 
 II = -jj^: . 
 
 The variations in y, n t and p are, however, too great to admit 
 of the formula being useful for practical purposes. 
 For new cast-iron pipes, 
 
 fe = -000296 to -0004 W 84 * 01 ' 9 ^ 
 
 If the pipes are lined with bitumen the smaller values of y and 
 n may be taken. 
 
 * Industries, 1886. 
 
 t Annales des Fonts et Chaussees, 1892, Vol. n. 
 
1.36 
 
 HYDRAULICS 
 
 For new, steel, riveted pipes, 
 
 0004to'00054^ 1 ' 93to2 ' 08 Z 
 
 Fig. 94 shows the result ' of plotting log k and log d for all 
 the pipes in Table XVI having a value of n between 1'92 and 1'94. 
 They are seen to lie very close to a line having a slope of 1'25, 
 and the ordinate of which, when d is 1 foot, is '000364. 
 000364^ r9:3 Z 
 
 Therefore h = 
 
 d 1 
 
 or v = 
 
 very approximately expresses the law of resistance for particular 
 pipes of wood, new cast iron, cleaned cast iron, and galvanised 
 iron. 
 
 and lo d rronv Table 16, 
 
 Loyk. 
 
 plotting s of loq h 
 og c 
 
 to deter\nin 
 formula^ i, = 
 
 he indea> p in- the. 
 
 is atxwbl'93 
 
 Fig. 94. 
 
 Taking a pipe 1 foot diameter and the velocity as 3 feet per 
 second, the value of i obtained by this formula agrees with that 
 from Darcy's formula for clear cast-iron pipes within 1 per cent. 
 
 Use of the logarithmic formula for practical calculations. A 
 very serious difficulty arises in the use of the logarithmic 
 formula, as to what value to give to n for any given case, and 
 consequently it has for practical purposes very little advantage 
 over the older and simpler formula of Chezy. 
 
TABLE XVI. 
 
 
 Experimenter 
 
 Kind of pipe 
 
 Diameter 
 (in ins.) 
 
 Velocity in 
 ft. per sec. 
 from to 
 
 Value of n 
 in formula 
 
 i = kv n 
 
 Value of k 
 in formula 
 
 / = kv 
 
 Noble 
 
 Wood 
 
 44 
 
 3-46 _ 4-415 
 
 1-73 
 
 0001254 
 
 J} 
 
 55 
 
 54 
 
 2-28 4-68 
 
 1-75 
 
 000083 
 
 Marx, Wing ) 
 
 55 
 
 72-5 
 
 1 - 4 
 
 1-72 
 
 000061 
 
 and Hoskins } 
 
 55 
 
 72-5 
 
 1 5-5 
 
 1-93 
 
 000048 
 
 raltner Kitcham 
 
 Riveted 
 
 3 
 
 
 1-88 
 
 00245 
 
 H. Smith 
 
 Wrought 
 
 11 
 
 
 1-81 
 
 000515 
 
 55 
 
 iron or steel 
 
 11| 
 
 
 1-90 
 
 000470 
 
 H 
 
 55 
 
 15 
 
 
 1-94 
 
 000270 
 
 Kinchling 
 
 55 
 
 38 
 
 505 1-254 
 
 2-0 
 
 000099 
 
 Herschel 
 
 
 42 
 
 2-10 4-99 
 
 1-93 
 
 00011 
 
 55 
 
 55 
 
 48 
 
 2 - 5 (?) 
 
 2-0 
 
 000090 
 
 Marx, Wing ) 
 
 55 
 
 72 
 
 1 4 
 
 1-99 
 
 000055 
 
 and Hoskins ( 
 
 55 
 
 72 
 
 1 5-5 
 
 1-85 
 
 000077 
 
 Herschel 
 
 55 
 
 103 
 
 1 - 4-5 
 
 2-08 
 
 000036 
 
 Darcy 
 
 Cast iron 
 
 3-22 
 
 28910-71 
 
 1-97 
 
 00156 
 
 59 
 
 new 
 
 5-39 
 
 48 15-3 
 
 1-97 
 
 00079 
 
 ,9 
 
 55 
 
 7-44 
 
 67316-17 
 
 1-956 
 
 00062 
 
 99 
 
 55 
 
 12 
 
 
 1-779 
 
 000323 
 
 Williams 
 
 55 
 
 16-25 
 
 
 1-858 
 
 000214 
 
 Lampe 
 
 55 
 
 16-5 
 
 2-48 3-09 
 
 1-80 
 
 000267 
 
 57 
 
 55 
 
 19-68 
 
 1-38 3-7 
 
 1-84 
 
 00022 
 
 Sherman 
 
 55 
 
 36 
 
 4 - 7 
 
 2* 
 
 000062 
 
 Stearns 
 
 55 
 
 48 
 
 1-243 3-23 
 
 1-92 
 
 0000567 
 
 (tubbell&Fenkell 
 
 55 
 
 30 
 
 
 2 
 
 00003 
 
 Darcy 
 
 Cast iron 
 
 1-4136 
 
 167 2-077 
 
 1-99 
 
 0098 
 
 
 old and 
 
 3-1296 
 
 403 3-747 
 
 1-94 
 
 0035 
 
 55 
 
 tuberculated 
 
 9-575 
 
 1-00712-58 
 
 1-98 
 
 0009 
 
 Sherman 
 
 
 20 
 
 2-71 5-11 
 
 
 
 
 ii 
 
 36 
 
 1-1 4-5 
 
 2 
 
 000105 
 
 Fitzgerald 
 
 55 
 
 48 
 
 1-176 3-533 
 
 2-04 
 
 000083 
 
 n 
 
 55 
 
 48 
 
 1-135 3-412 
 
 2-00 
 
 000085 
 
 Darcy 
 
 Cast-iron 
 
 1-4328 
 
 371 3-69 
 
 1-85 
 
 0041 
 
 
 old pipes 
 
 3-1536 
 
 633 5-0 
 
 1-97 
 
 00185 
 
 
 cleaned 
 
 11-68 
 
 8 10-368 
 
 2-0 
 
 000375 
 
 Fitzgerald 
 
 55 
 
 55 
 55 
 
 48 
 48 
 
 3-67 5-6 
 395 7-245 
 
 2-02 
 1-94 
 
 000082 
 000059 
 
 Darcy 
 
 Sheet -iron 
 
 1-055 
 
 -098 8-225 
 
 1-76 
 
 0074 
 
 
 j5 
 
 3-24 
 
 32812-78 
 
 1-81 
 
 00154 
 
 
 
 7-72 
 
 59119-72 
 
 1-78 
 
 00059 
 
 >5 
 
 H 
 
 11-2 
 
 1-29610-52 
 
 1-81 
 
 00039 
 
 
 Gas 
 
 48 
 
 113 3-92 
 
 1-83 
 
 0278 
 
 
 pi 
 
 1-55 
 
 205 8-521 
 
 1-86 
 
 00418 
 
 
 
 55 
 
 
 
 1-91 
 
 0072 
 
 aph and Schoder 
 
 Galvanised 
 
 364 
 
 
 1-96 
 
 0352 
 
 ? 
 
 . 
 
 494 
 
 
 1-91 
 
 0181 
 
 
 
 623 
 
 
 1-86 
 
 0132 
 
 ,9 
 
 9, 
 
 824 
 
 
 1-80 
 
 0095 
 
 55 
 
 55 
 
 1-048 
 
 
 1-93 
 
 0082 
 
 tt 
 
 Hard-drawn 
 brass 
 
 15 pipes 
 up to 1-84 
 
 
 1-75 
 
 00025 to 
 00035 
 
 Reynolds 
 
 Lead 
 
 
 
 1-732 
 
 
 Darcy 
 
 
 55 
 
 
 1-761 
 
 0126 
 
 / 
 
 55 
 
 1-61 
 
 
 1-783 
 
 00425 
 
1.38 
 
 HYDRAULICS 
 
 TABLE XVII. 
 
 Showing reasonable values of y, and n, for pipes of various 
 kinds, in the formula, 
 
 L YV*I 
 
 
 Reasonable 
 
 
 values for 
 
 
 7 
 
 n 
 
 7 
 
 n 
 
 Clean cast-iron pipes 
 
 00029 to -000418 
 
 1-80 to 1-97 
 
 00036 
 
 1-93 
 
 Old cast-iron pipes 
 
 00047 to -00069 
 
 1-94 to 2-04 
 
 00060 
 
 2 
 
 Riveted pipes 
 Galvanised pipes 
 
 00040 to -00054 
 00035 to -00045 
 
 1-93 to 2-08 
 1-80 to 1-96 
 
 00050 
 00040 
 
 2 
 
 1-88 
 
 Sheet-iron pipes cover- 
 ed with bitumen 
 
 00030 to -00038 
 
 1-76 to 1-81 
 
 00034 
 
 1-78 
 
 Clean wood pipes 
 Brass and lead pipes 
 
 00056 to -00063 
 
 1-72 to 1-75 
 
 00060 
 00030 
 
 1-75 
 1-75 
 
 When further experiments have been performed on pipes, of 
 which the state of the internal surfaces is accurately known, and 
 special care taken to ensure that all the loss of head in a given 
 length of pipe is due to friction only, more definiteness may be 
 given to the values of y, ?i, and p. 
 
 Until such evidence is forthcoming the simple Chezy formula 
 may be used with almost as much confidence as the more 
 complicated logarithmic formula, the values of C or / being taken 
 from Tables XII XIY. Or the formula h = kv n may be used, 
 values of k and n being taken from Table XYI, which most nearly 
 fits the case for which the calculations are to be made. 
 
 104. Criticism of experiments. 
 
 The difficulty of differentiating the loss of head due to friction 
 from other sources of loss, such as loss due to changes in direction, 
 change in the diameter of the pipe and other causes, as well as the 
 possibilities of error in experiments on long pipes of large diameter, 
 makes many experiments that have been performed of very little 
 value, and considerably increases the difficulty of arriving at 
 correct formulae. 
 
 The author has found in many cases, when log i and log d were 
 plotted, from the records of experiments, that, although the results 
 seemed consistent amongst themselves, yet compared with other 
 experiments, they seemed of little value. 
 
FLOW THROUGH PIPES 
 
 139 
 
 The value of n for one of Couplet's* experiments on a lead and 
 earthenware pipe being as low as 1*56, while the results of an 
 experiment by Simpson t on a cast-iron pipe gave n as 2'5. In the 
 latter case there were a number of bends in the pipe. 
 
 In making experiments for loss of head due to friction, it is 
 desirable that the pipe should be of uniform diameter and as 
 straight as possible between the points at which the pressure head 
 is measured. Further, special care should be taken to ensure the 
 removal of all air, and that a perfectly steady flow is established 
 at the point where the pressure is taken. 
 
 105. Piezometer fittings. 
 
 It is of supreme importance that the 
 piezometer connections shall be made 
 so that the difference in the pressures 
 registered at any two points shall be 
 that lost by friction, and friction only, 
 between the points. 
 
 This necessitates that there shall 
 be no obstructions to interfere with the 
 free flow of the water, and it is, there- 
 fore, very essential that all burrs shall 
 be removed from the inside of the pipe. 
 
 In experiments on small pipes in 
 the laboratory the best results are no 
 doubt obtained by cutting the pipe 
 completely through at the connection 
 as shown in Fig. 95, which illustrates 
 the form of connection used by Dr 
 Coker in the experiments cited on 
 
 page 129. The two ends of the pipe are not more than 
 of an inch apart. 
 
 Fig. 96 shows the method adopted by Marx, Wing and Hoskins 
 in their experiments on a 72-inch wooden pipe to ensure a correct 
 reading of the pressure. 
 
 The gauge X was connected to the top of the pipe only while 
 Y was connected at four points as shown. 
 
 Small differences were observed in the readings of the two 
 gauges, which they thought were due to some accidental circum- 
 stance affecting the gauge X only, as no change was observed 
 in the reading of Y when the points of communication to Y were 
 changed by means of the cocks. 
 
 * Hydraulics, Hamilton Smith, Junr. 
 
 t Proceedings of the Institute of Civil Engineers, 1855. 
 
 Fig. 95. 
 
140 
 
 HYDRAULICS 
 
 106. Effect of temperature on the velocity of flow. 
 
 Poiseuille found that by raising the temperature of the water 
 from 50 C. to 100 C. the discharge of capillary tubes was 
 doubled. 
 
 Fig. 96. Piezometer connections to a wooden pipe. 
 
 Reynolds* showed that for pipes of larger diameter, the effect 
 of changes of the temperature was very marked for velocities 
 below the critical velocity, but for velocities above the critical 
 velocity the effect is comparatively small. 
 
 The reason for this is seen, at once, from an examination of 
 Reynolds'* formula. Above the critical velocity n does not differ 
 very much from 2, so that P 2 ~ u is a small quantity compared with 
 its value when n is 1. 
 
 Saph and Schodert, for velocities above the critical velocity, 
 found that, as the temperature rises, the loss of head due to 
 friction decreases, but only in a small degree. For brass pipes of 
 small diameter, the correction at 60 F. was about 4 per cent, per 
 
 * Scientific Papers, Vol. n. 
 
 t See also Barnes and Coker, Proceedings of the Eoyal Society, Vol. LXX. 1904 ; 
 Coker and Clements, Transactions of the Royal Society, Vol. cci. Proceedings 
 Am.S.G.E. Vol. xxix. 
 
FLOW THROUGH PIPES 141 
 
 10 degrees F. With galvanised pipes the correction appears to 
 be from 1 per cent, to 5 per cent, per 10 degrees F. 
 
 Since the head lost increases, as the temperature falls, the 
 discharge for any given head diminishes with the temperature, 
 but for practical purposes the correction is generally negligible. 
 
 107. Loss of head due to bends and elbows. 
 
 The loss of head due to bends and elbows in a long pipe is 
 generally so small compared with the loss of head due to friction 
 in the straight part of the pipe, that it can be neglected, and 
 consequently the experimental determination of this quantity has 
 not received much attention. 
 
 Weisbach*, from experiments on a pipe 1J inches diameter, 
 with bends of various radii, expressed the loss of head as 
 
 r being the radius of the pipe, ~R the radius of the bend on the 
 centre line of the pipe and v the velocity of the water in feet per 
 second. If the formula be written in the form 
 
 ' 2 
 
 at' 
 
 /\+ 
 
 the table shows the values of a for different values of ^ . 
 
 XV 
 
 r 
 B 
 
 1 -157 
 
 2 -250 
 
 5 -526 
 
 St Venantt has given as the loss of head h^ at a bend, 
 A B = '00152 ^ y^M)-l|Ay| nearly, 
 
 I being the length of the bend measured on the centre line of the 
 bend and d the diameter of the pipe. 
 When the bend is a right angle 
 
 l_ /~d_v /d 
 R V R ~ 2 V R ' 
 
 When 5-1, '5, '2, 
 
 n 
 
 - 1-57, Ml, 702 
 
 and 7* B = "157 -, 'Ills-, '07^-. 
 
 * Mechanics of Engineering. 
 t Comptes Rendus, 1862. 
 
142 
 
 HYDRAULICS 
 
 Recent experiments by Williams, Hubbell and Fenkell * on cast- 
 iron pipes asphalted, by Saph and Schoder on brass pipes, and 
 others by Alexander t on wooden pipes, show that the loss of head 
 in bends, as in a straight pipe, can be expressed as 
 
 /IB = kv n , 
 n being a variable for different kinds of pipes, while 
 
 d* 
 
 y being a constant coefficient for any pipe. 
 
 For the cast-iron pipes of Hubbell and Fenkell, y, n, m, and p 
 have approximately the following values. 
 
 
 
 
 
 Diameter of pipe 
 
 7 
 
 m 
 
 n 
 
 f 
 
 12" 
 
 0040 
 
 0-83 
 
 1-78 
 
 1-09 
 
 16" 
 
 ?5 
 
 5? 
 
 1-86 
 
 ?? 
 
 30" 
 
 5? 
 
 ?5 
 
 2-0 
 
 J5 
 
 
 
 When v is 3 feet per second and ^- is ;}, the bend being a right 
 
 angle, the loss of head as calculated by this formula for the 
 
 , . . -206&; 2 
 12-inch pipe is ~ - 
 
 , ,, Q n -, 
 
 * or the ^0- lncn P 1 P& ~^~ 
 
 For the brass pipes of Saph and Schoder, 2 inches diameter, 
 Alexander found 
 
 7i B = '00858 5 
 
 /T 
 
 and for varnished wood pipes when - is less than 0'2, 
 7i B = '008268 i 
 
 and when ^ is between 0*2 and 0'5, 
 
 He further found for varnished wood pipes that, a bend of 
 radius equal to 5 times the radius of the pipe gives the minimum 
 loss of head, and that its resistance is equal to a straight pipe 3'38 
 times the length of the bend. 
 
 Messrs Williams, Hubbell and Fenkell also state at the end of 
 their elaborate paper, that a bend having a radius equal to 2J 
 
 * Proc. Amer. Soc. Civil Engineers, Vol. xxvu. 
 t Proc. Inst> Civil Engineers, Vol. CLIX. 
 
FLOW THROUGH PIPES 143 
 
 diameters, offers less resistance to the flow of water than those of 
 longer radius. It should not be overlooked, however, that although 
 the loss of head in a bend of radius equal to 2J diameters of the 
 pipe is less than for any other, it does not follow that the loss of 
 head per unit length of the pipe measured along its centre line 
 has its minimum value for bends of this radius. 
 
 108. Variations of the velocity at the cross section of a 
 cylindrical pipe. 
 
 Experiments show that when water flows through conduits of 
 any form, the velocities are not the same at all points of any 
 transverse section, but decrease from the centre towards the 
 circumference. 
 
 The first experiments to determine the law of the variation of 
 the velocity in cylindrical pipes were those of Darcy, the pipes 
 varying in diameter from 7*8 inches to 19 inches. A complete 
 account of the experiments is to be found in his Recherches 
 Experimentales dans les tuyaux. 
 
 The velocity was measured by means of a Pitot tube at five 
 points on a vertical diameter, and KVV , xxxxxxxxxxxxxx ^^^^^ 
 the results plotted as shown in 
 Fig. 97. 
 
 Calling V the velocity at the 
 centre of a pipe of radius R, u the 
 velocity at the circumference, v m 
 the mean velocity, v the velocity 
 at any distance r from the centre, 
 
 and i the loss of head per unit 
 
 length of the pipe, Darcy deduced the formulae 
 
 & ^ T 4 7 /^r 
 
 and v m = - = - =\-~k v R^. 
 
 When the unit is the metre the value of k is 11*3, and 20*4 when 
 the unit is the English foot. 
 
 Later experiments commenced by Darcy and continued by 
 Bazin, on the distribution of velocity in a semicircular channel, 
 the surface of the water being maintained at the horizontal 
 diameter, and in which it was assumed the conditions were similar 
 to those in a cylindrical pipe, showed that the velocity near the 
 surface of the pipe diminished much more rapidly than indicated 
 by the formula of Darcy. 
 
144 HYDRAULICS 
 
 Bazin substituted therefore a new formula, 
 
 CD, 
 
 /"i 
 
 or since v m = C *Jmi = ^ J^i 
 
 V-t; = 53^ /_r y (2) 
 
 It was open to question, however, whether the conditions of flow 
 in a semicircular pipe are similar to those in a pipe discharging 
 full bore, and Bazin consequently carried out at Dijon*, experi- 
 ments on the distribution of velocity in a cement pipe, 2*73 feet 
 diameter, the discharge through which was measured by means 
 of a weir, and the velocities at different points in the transverse 
 section by means of a Pitot tubet. 
 
 From these experiments Bazin concluded that both formulae (1) 
 and (2) were incorrect and deduced the three formulae 
 
 5{l- x /l--95() 2 } (5), 
 
 V-tJ = Vl*53'5|l 
 
 the constants in these formulae being obtained from Bazin's by 
 changing the unit from 1 metre to the English foot. 
 
 Equation (5) is the equation to an ellipse to which the sides of 
 the pipes are not tangents but are nearly so, and this formula 
 gives values of v near to the surface of the pipe, which agree much 
 more nearly with the experimental values, than those given by 
 any of the other formulae. 
 
 Experiments of Williams, HuHbdl and Fenkell+. An elaborate 
 series of experiments by these three workers have been carried out 
 to determine the distribution of velocity in pipes of various 
 diameters, Pitot tubes being used to determine the velocities. 
 
 The pipes at Detroit were of cast iron and had diameters of 12, 
 16, 30 and 42 inches respectively. 
 
 The Pitot tubes were calibrated by preliminary experiments 
 on the flow through brass tubes 2 inches diameter, the total 
 
 * " Memoire de 1'Academie des Sciences de Paris, Kecueil des Savants Etrangeres," 
 Vol. xxxn. 1897. Proc. Am.S.C.E. Vol. xxvu. p. 1042. 
 
 t See page 241. 
 
 "Experiments at Detroit, Mich., on the effect of curvature on the flow of 
 water in pipes," Proc. Am.S.C.E. Vol. xxvn. p. 313. 
 
 See page 246. 
 
FLOW THROUGH PIPES 145 
 
 discharge being determined by weighing, and the mean velocity 
 thus determined. From the results of their experiments they 
 came to the conclusion that the curve of velocities should be an 
 ellipse to which the sides of the pipe are tangents, and that the 
 velocity at the centre of the pipe V is l'19v m , v m being the mean 
 velocity. 
 
 These results are consistent with those of Bazin. His experi- 
 
 y 
 mental value for for the cement pipe was 1'1675, and if the 
 
 v m 
 
 constant '95, in formula (5), be made equal to 1, the velocity curve 
 becomes an ellipse to which the walls of the pipe are tangents. 
 
 The ratio can be determined from any of Bazin's formulae. 
 
 Substituting p for \R in (1), (3), (4) or (5), the value of 
 v at radius r can be expressed by any one of them as 
 
 i 
 
 c 
 
 Then, since the flow past any section in unit time is -Um^R 2 , and 
 that the flow is also equal to 
 
 f E 
 
 
 therefore ^ m 7rR 2 = 2?r I |Y ~7y^/(~|? }\ T dr (6). 
 
 Substituting for /(-5J , its value -^3- from equation (1), and 
 integrating, 
 
 ^ = 1 + ^ ...(7), 
 
 (T \ 
 ^} from equation (4), 
 
 V ,23 
 
 ~ = H 
 
 so that the ratio is not very different when deduced from the 
 
 v m 
 
 simple formula (2) or the more complicated formula (4). 
 When C has the values 
 
 = 80, 100, 120, 
 
 from (8) = 1-287, T23, T19. 
 
 Vm 
 
 The value of C, in the 30-inch pipe referred to above, varied 
 between 109'6 and 123'4 for different lengths of the pipe, and 
 L. H. 10 
 
<: V 
 
 < V- 
 
 -* _JB 
 
 146 HYDRAULICS 
 
 the mean value was 116, so that there is a remarkable agreement 
 between the results of Bazin, and Williams, Hubbell and Fenkell. 
 
 The velocity at the surface of a pipe. Assuming that the 
 velocity curve is an ellipse to which 
 the sides of the pipe are tangents, as _ 
 in Fig. 98, and that Y = l'19v m , the 
 velocity at the surface of the pipe 
 can readily be determined. 
 
 Let u = the velocity at the surface 
 of the pipe and v the velocity at any 
 radius r. 
 
 Let the equation to the ellipse be Fl S- 98< 
 
 ^2 ~2 
 
 + --1 
 
 B* P 
 
 in which x = v-u, 
 
 and b = V - u. 
 
 Then, if the semi-ellipse be revolved about its horizontal axis, 
 the volume swept out by it will be firR 2 6, and the volume of 
 discharge per second will be 
 
 Zirrdr .v = TrW.u + f 7rR 2 , 
 
 ' v m = u + |(Y - u) = lu + x l'19v m , 
 and u = '621v m . 
 
 Using Bazin' s elliptical formula, the values of for 
 C = 80, 100, 120, 
 
 are --'552, '642, '702. 
 
 v m 
 
 The velocities, as above determined, give the velocity of 
 translation in a direction parallel to the pipe, but as shown by 
 Reynolds' experiments the particles of water may have a much 
 more complicated motion than here assumed. 
 
 109. Head necessary to give the mean velocity v m to 
 the water in the pipe. 
 
 It is generally assumed that the head necessary to give a .mean 
 
 Vm 
 
 velocity v m to the water flowing in a pipe is 7^-, which would be 
 
 correct if all the particles of water had a common velocity v m . 
 
 If, however, the form of the velocity curve is known, and on the 
 assumption that the water is moving in stream lines with definite 
 velocities parallel to the axis of the pipe, the actual head can 
 be determined by calculating the mean kinetic energy per Ib. of 
 
 water flowing in the pipe, and this is slightly greater than -- . 
 
FLOW THROUGH PIPES 147 
 
 As before, let v be the velocity at radius r. 
 The kinetic energy of the quantity of water which flows past 
 any section per second 
 
 w . Z-n-rdr . v . ^- , 
 o 2gr' 
 
 w being the weight of 1 c. ft. of water. 
 The kinetic energy per lb., therefore, 
 
 = / 
 
 J 
 
 _ J o 
 
 E 
 
 w . 
 o 
 
 v/2 
 
 I'm 
 
 f(S>] 
 
 rdr 
 
 .(9). 
 
 The simplest value for / (T>) is that of Bazin's formula (1) 
 above, from which 
 
 v=#, 
 
 and f ( ) = 38 
 
 Substituting these values and integrating, the kinetic energy 
 per lb. is ^ , and when 
 
 C is 80, 100, 
 a is 112, 1-076. 
 
 On the assumption that the velocity curve is an ellipse to which 
 the walls of the pipe are tangents the integration is easy, and the 
 value of a is 1*047. 
 
 Using the other formulae of Bazin the calculations are tedious 
 and the values obtained differ but slightly from those given. 
 
 The head necessary to give a mean velocity v m to the water in 
 
 the pipe may therefore be taken to be ~- , the value of a being 
 
 about 1*12. This value* agrees with the value of 1*12 for a, 
 obtained by M. Boussinesq, and with that of M. J. Delemer who 
 finds for a the value 1*1346. 
 
 110. Practical problems. 
 
 Before proceeding to show how the formulae relating to the 
 loss of head in pipes may be used for the solution of various 
 problems, it will be convenient to tabulate them. 
 * Flamant's Hydraulique. 
 
 102 
 
148 HYDRAULICS 
 
 NOTATION. 
 h = loss of head due to friction in a length I of a straight pipe. 
 
 i = the virtual slope = y . 
 
 v = the mean velocity of flow in the pipe. 
 d = the diameter. 
 m = the hydraulic mean depth 
 
 = ^ ea . = = T when the pipe is cylindrical and full. 
 
 Wetted Perimeter P 4 
 
 , vH 4v 2 l 
 
 Formula 1. h = 7^ = 7^-, . 
 
 Cm C d 
 
 7i v 2 
 This may be written y = , 
 
 or v C vmi. 
 
 The values of C for cast-iron and steel pipes are shown in 
 Tables XII and XIV. 
 
 4fZ^ 2 
 Formula 2. h = ^- -3 , 
 
 *- in this formula being equal to 7^2 of formula (1). 
 "9 
 
 Values of f are shown in Table XIII. 
 
 Either of these formulae can conveniently be used for 
 calculating h, v, or d when /, and Z, and any two of three 
 quantities /&, v, and d, are known. 
 
 Formula 3. As values of C and / cannot be remembered for 
 variable velocities and diameters, the formulae of Darcy are 
 convenient as giving results, in many cases, with sufficient 
 accuracy. For smooth clean cast-iron pipes 
 
 For rough and dirty pipes 
 
 
 mi. 
 
FLOW THROUGH PIPES 149 
 
 If d is the unknown, Darcy's formulae can only be used to solve 
 
 for d by approximation. The coefficient ( 1 + ^75-7 ) is first neglected 
 
 \ LZa/ 
 
 and an approximate value of d determined. The coefficient can 
 then be obtained from this approximate value of d with a greater 
 degree of accuracy, and a new value of d can then be found, and 
 so on. (See examples.) 
 
 Formula 4. Known as the logarithmic formula. 
 
 r_* 
 
 or 
 
 d* ' 
 
 h . y . v n 
 - 
 
 Values of y, n, and p are given on page 138. 
 By taking logarithms 
 
 log h = log y + n log v + log I p log d, 
 from which h can be found if I, v, and d are known. 
 If h, I, and d are known, by writing the formula as 
 
 n log v = log Tz, log I - log y + p log d, 
 -u can be found. 
 
 If h, I, and v are known, d can be obtained from 
 p log d = log y + n log v + log I - log /i. 
 
 This formula is a little more cumbersome to use than either (1) or 
 (2) but it has the advantage that y is constant for all velocities. 
 Formula 5. The head necessary to give a mean velocity v to 
 
 l"12i; 2 
 the water flowing along the pipe is about ^ , but it is generally 
 
 v 2 
 convenient and sufficiently accurate to take this head as g-, as 
 
 was done in Fig. 87. Unless the pipe is short this quantity is 
 negligible compared with the friction head. 
 
 Formula 6. The loss of head at the sharp-edged entrance to a 
 
 K-Jl 
 
 pipe is about -~- and is generally negligible. 
 
 Formula 7. The loss of head due to a sudden enlargement in 
 a pipe where the velocity changes from Vi to v 2 is ^ 2 
 
 Formula 8. The loss of head at bends and elbows is a very 
 variable quantity. It can be expressed as equal to -^- in which 
 a varies from a very small quantity to unity. 
 
 Problem 1. The difference in level of the water in two reservoirs is h feet, 
 Fig. 99, and they are connected by means of a straight pipe of length I and 
 diameter d; to find the discharge through the pipe. 
 
150 
 
 HYDRAULICS 
 
 Let Q be the number of cubic feet discharged per second. The head h is utilised 
 in giving velocity to the water and in overcoming resistance at the entrance to the 
 pipe and the frictional resistances. 
 
 Fig. 99. Pipe connecting two reservoirs. 
 
 Let v be the mean velocity of the water. The head necessary to give the water 
 this mean velocity may be taken as , and to overcome the resistance at the 
 
 entrances 
 Then 
 
 l-12t; 2 -5t? 2 
 ~~ ^ 
 
 2g. d 
 Or using in the expression for friction, the coefficient C, 
 
 h = -0174tr + -0078v 2 + ~ 
 
 If - is greater than 300 the head lost due to friction is generally great compared 
 
 a 
 with the other quantities, and these may be neglected. 
 
 Then h= 
 
 and 
 
 2gd _C 2 
 C /dh 
 
 =2VT- 
 
 As the velocity is not known, the coefficient C cannot be obtained from the 
 table, but an approximate value can be assumed, or Darcy's value 
 
 = 394 
 
 J 
 
 = 278 
 
 I2d + l 
 
 I2d + l 
 
 for clean pipes, 
 if the pipe is dirty, 
 
 and 
 
 can be taken. 
 
 An approximation to v which in many cases will be sufficiently near or will be 
 as near probably as the coefficient can be known is thus obtained. From the 
 table a value of C for this velocity can be taken and a nearer approximation to 
 v determined. 
 
 Then Q=^d*.v. 
 
 yv n l 
 The velocity can be deduced directly from the logarithmic formula h= -=^ > 
 
 provided 7 and n are known for the pipe. 
 
FLOW THROUGH PIPES 151 
 
 The hydraulic gradient is EF. 
 
 At any point C distant x from A the pressure head is equal to the distance 
 between the centre of the pipe and the hydraulic gradient. The pressure head 
 just inside the end A of the pipe is ft A -- 5 , and at the end B the pressure head 
 must be equal to h#. The head lost due to friction is h, which, neglecting the 
 small quantity ~ , is equal to the difference of level of the water in the two 
 
 tanks. 
 
 Example 1. A pipe 3 inches diameter 200 ft. long connects two tanks, the 
 difference of level of the water in which is 10 feet, and the pressure is atmospheric. 
 Find the discharge assuming the pipe dirty. 
 
 Using Darcy's coefficient 
 
 = 3-88 ft. per sec. 
 
 For a pipe 3 inches diameter, and this velocity, C from the table is about 69, so 
 that the approximation is sufficiently near. 
 
 00064t?i*. I 
 Taking h = -^ , 
 
 i> = 3-88 ft. per sec., 
 
 gives v = 3 -85 ft. per sec. 
 
 Example 2. A pipe 18 inches diameter brings water from a reservoir 100 feet 
 above datum. The total length of the pipe is 15,000 feet and the last 5000 feet 
 are at the datum level. For this 5000 feet the water is drawn off by service pipes at 
 the uniform rate of 20 cubic feet per minute, per 500 feet length. Find the pressure 
 at the end of the pipe. 
 
 The total quantity of flow per minute is 
 
 5000 x 20 4 
 
 Q = = 200 cubic feet per minute. 
 
 ouo 
 
 Area of the pipe is 1-767 sq. feet. 
 
 The velocity in the first 10,000 feet is, therefore, 
 
 =_??!!_.= 1-888 ft. per sec. 
 
 The head lost due to friction in this length, is 
 
 k _*-f- 10,000.1-888 2 
 
 In the last 5000 feet of the pipe the velocity varies uniformly. At a distance 
 x feet from the end of the pipe the velocity is . 
 
 In a length dx the head lost due to friction is 
 
 4./ t l-888 a .a? a ds 
 : 20.1-5.5000* ' 
 and the total loss by friction is 
 
 4/. 1-888 2 pooo _4/. (1-888) 2 5000 
 "o-20.1-5.5000 2 J 20.1-5 ' 3 
 
 The total head lost due to friction in the whole pipe is, therefore, 
 
 H= 4 { ,. 
 
152 HYDRAULICS 
 
 Taking / as -0082, H = 14-3 feet. 
 
 Neglecting the velocity head and the loss of head at entrance, the pressure head 
 at the end of the pipe is (100 - H) feet = 85-7 feet. 
 
 Problem 2. Diameter of pipe to give a given discharge. 
 
 Eequired the diameter of a pipe of length I feet which will discharge Q cubic feet 
 per second between the two reservoirs of the last problem. 
 Let v be the mean velocity and d the diameter of the pipe. 
 
 Then v=-Q- (1), 
 
 and 
 
 Therefore, 
 
 Squaring and transposing, 
 c 
 If I is long compared with d, 
 
 and 
 
 0-0406. Q 2 d_ 6-5ZQ 2 
 
 h &h { >' 
 
 Q /dh 
 
 r- =c v 47' 
 
 Since v and d are unknown C is unknown, and a value for C must be pro- 
 visionally assumed. 
 
 Assume C is 100 for a new pipe and 80 for an old pipe, and solve equation (3) 
 ford. 
 
 From (1) find v, and from the tables find the value of C corresponding to the 
 values of d and v thus determined. 
 
 If C differs much from the assumed value, recalculate d and v using this second 
 value of C, and from the tables find a third value for C. This will generally be 
 found to be sufficiently near to the second value to make it unnecessary to calculate 
 d and v a third time. 
 
 The approximation, assuming the values of C in the tables are correct, can be 
 taken to any degree of accuracy, but as the values of C are uncertain it will not as 
 a rule be necessary to calculate more than two values of d. 
 
 yv n l 
 Logarithmic formula. If the formula li=! be used, d can be found direct, 
 
 from 
 
 p log d = nlog v + logy + log l-log h. 
 
 Example 3. Find the diameter of a steel riveted pipe, which will discharge 
 14 cubic feet per second, the loss of head by friction being 2 feet per mile. It is 
 assumed that the pipe has become dirty and that provisionally C = 110. 
 
 From equation (3) 
 
 2-55 . 14 /5280 
 
 =-110 V2' 
 
 or f log d = log 16 -63, 
 
 therefore d = 3-08 feet. 
 
 For a thirty-eight inch pipe Kuichling found C to be 113. 
 
 The assumption that C is 110 is nearly correct and the diameter may be taken 
 as 37 inches. 
 
 Using the logarithmic formula 
 
 00045V 1 - 95 ; 
 h= 
 
FLOW THROUGH PIPES 
 
 153 
 
 and substituting for v the value -2- 
 
 /,r\ 
 
 ) 
 
 from which 
 
 5 -15 log d = log -00045 - 1 -95 log 0-7854 + 1-95 log 14 + log 2640, 
 and d = 3-07 feet. 
 
 Short pipe. If the pipe is short so that the velocity head and the head lost at 
 entrance are not negligible compared with the loss due to friction, the equation 
 
 ' -0406Q 2 d 6-5/Q 2 
 
 "T" : ~c^T' 
 
 when a value is given to C, can be solved graphically by plotting two curves 
 
 Vi- 
 
 0406Q 2 
 
 .d + 
 
 6-5ZQ 2 
 
 ~mr 
 
 The point of intersection of the two curves will give the 
 diameter d. 
 
 It is however easier to solve by approximation in the 
 following manner. 
 
 Neglect the term in d and solve as for a long pipe. 
 
 Choose a new value for C corresponding to this ap- 
 proximate diameter, and the velocity corresponding to it, 
 and then plot three points on the curve y = d 5 , choosing 
 values of d which are nearly equal to the calculated value 
 of d, and two points of the straight line 
 _-Q406Q 2 d 6-5/Q 2 
 2/1 ~ ~~h~ "~C*}T' T 
 
 The curve y = d 5 between the three points can easily * '5 
 
 be drawn, as in Fig. 100, and where the straight line cuts -pig. 100. 
 
 the curve, gives the required diameter.' 
 
 Example 4. One hundred and twenty cubic feet of water are to be taken 
 per minute from a tank through a cast-iron pipe 100 feet long, having a square- 
 edged entrance. The total head is 10 feet. Find the diameter of the pipe. 
 
 Neglecting the term in d and assuming C to be 100, 
 
 and 
 
 Therefore 
 
 100 . 100 . 10 
 d= -4819 feet. 
 2 
 
 |(-4819) 2 
 
 = 10-9 ft. per sec. 
 
 From Table XII, the value of C is seen to be about 106 for these values of 
 d and v. 
 
 A second value for d 5 is 
 
 6-5. 100. 4 
 " 1062.10 " 
 from which d=-476'. 
 
 The schedule shows the values of d 5 and y for values of d not very different 
 from the calculated value, and taking C as 106. 
 
 d -4 -5 -6 
 
 d 5 -01024 -03125 -0776 
 
 y 1 -0297 -0329 
 
 The line and curve plotted in Fig. 100, from this schedule, intersect at 4? for which 
 
 d = -498 feet. 
 
154 HYDRAULICS 
 
 It is seen therefore that taking 106 as the value of C, neglecting the term in d, 
 makes an error of -022' or -264". 
 
 This problem shows that when the ratio is about 200, and the virtual slope is 
 even as great as j 1 ^, for all practical purposes, the friction head only need be con- 
 sidered. For smaller values of the ratio the quantity -025^ may become im- 
 
 portant, but to what extent will depend upon the slope of the hydraulic gradient. 
 
 The logarithmic formula may be used for short pipes but it is a little more 
 cumbersome. 
 
 Using the logarithmic formula to express the loss of head for short pipes with 
 square-edged entrance, 
 
 or dan+i-26 _ .0406Q-d 2n - 2 ' 75 = . 
 
 w 
 
 When suitable values are given to 7 and ?i, this can be solved by plotting the 
 two curves 
 
 and y l = - 
 
 the intersection of the two curves giving the required value of d. 
 
 Problem 3. To find wbat the discharge between the reservoirs of problem (1) 
 would be, if for a given distance Zj the pipe , 
 
 of diameter d is divided into two branches i I 
 
 laid side by side having diameters d l and d 2 , [< --- >V --- , --- ^ 
 Fi g- 101. i ! 
 
 Assume all the head is lost in friction. A[ . ty^j d* 1^ 
 
 Let Qj be the discharge in cubic feet. -CJ | - ^ -- K f *~ 
 
 Then, since both the branches BC and BD | - 1 -- ^ V 
 are connected at B and to the same reservoir, v^ ok | L) 
 
 the head lost in friction must be the same in 
 
 BC as in BD, and if there were any number - L - 
 
 of branches connected at B the head lost in p- -^Q-^ 
 
 them all would be the same. 
 
 The case is analogous to that of a conductor joining two points between which 
 a definite difference of potential is maintained, the conductor being divided between 
 the points into several circuits in parallel. 
 
 The total head lost between the reservoirs is, therefore, the head lost in AB 
 together with the head lost in any one of the branches. 
 
 Let v be the velocity in AB, v l in BC and v z in BD. 
 
 Then vd' 2 = v l d l - + v 2 d^ .................................... (1), 
 
 and the difference of level between the reservoirs 
 
 
 And since the head lost in BC is the same as in BD, therefore, 
 
 4W-4W 
 
 qtt -< 
 
 If provisionally Cj be taken as equal to C 2 , 
 
FLOW THROUGH PIPES 
 
 155 
 
 Therefore, 
 
 and 
 
 v.d? 
 
 ___ 
 
 A 
 Vd, 
 
 From (2), v can be found by substituting for v l from (4), and thus Q can 
 be determined. 
 
 If AB, BC, and CD are of the same diameter and Z x is equal to Z 2 , then 
 
 and h= 
 
 or Qi 
 
 Problem 4. Pipes connecting three reservoirs. As in Fig. 102, let three pipes 
 AB, BC, and BD, connect three reservoirs A, C, D, the level of the water in each 
 of which remains constant. 
 
 Let v l , v 2 , and v 3 be the velocities in AB, BC, .and BD respectively, Q 15 Q 2 , 
 and Q 3 the quantities flowing along these pipes in cubic feet per sec., l lt Z 2 , and 1 3 
 the lengths of the pipes, and d l , d 2 and d 3 their diameters. 
 
 Fig. 102. 
 
 Let z lt z 2 , and z 3 be the heights of the surfaces of the water in the reservoirs, 
 and z the height of the junction B above some datum. 
 Let h Q be the pressure head at B. 
 
 Assume all losses, other than those due to friction in the pipes, to be negligible. 
 The head lost due to friction for the pipe AB is 
 
 A 1 _ O 
 
 (1), 
 
 (2), 
 
 andfor the pipe BC, 
 
 the upper or lower signs being taken, according as to whether the flow is from, or 
 towards, the reservoir C. 
 
 For the pipe BD the head lost is 
 
 z + h -z s (3). 
 
 ^3"3 
 
 Since the flow from A and C must equal the flow into D, or else the flow 
 from A must equal the quantity entering C and D, therefore, 
 
 or 
 
 . 
 
 There are four equations, from which four unknowns may be found, if it is 
 further known which sign to take in equations (2) and (4). There are two cases to 
 consider. 
 
156 HYDRAULICS 
 
 Case (a). Given the levels of the surfaces of the water in the reservoirs and 
 of the junction B, and the lengths and diameters of the pipes, to find the quantity 
 flowing along each of the pipes. 
 
 To solve this problem, it is first necessary to obtain by trial, whether water flows 
 to, or from, the reservoir C. 
 
 First assume there is no flow along the pipe BC, that is, the pressure head h Q at 
 B is equal to z 2 -z . 
 
 Then from (1), substituting for v t its value , 
 
 _, z 
 
 .. ! 2 ' 
 
 = 
 
 from which an approximate value for Q 1 can be found. By solving (3) in the same 
 way, an approximate value for Q 3 , is, 
 
 CTT 
 
 k 
 
 If Q 3 is found to be equal to Q 1? the problem is solved ; but if Q 3 is greater than 
 Q! , the assumed value for 7z is too large, and if less, h is too small, for a diminu- 
 tion in the pressure head at B will clearly diminish Q 3 and increase Q I} and will 
 also cause flow to take place from the reservoir C along CB. Increasing the 
 pressure head at B will decrease Q lf increase Q 3 , and cause flow from B to C. 
 
 This preliminary trial will settle the question of sign in equations (2) and (4) 
 and the four equations may be solved for the four unknowns, v l , v 2 , v 3 and h . It 
 is better, however, to proceed by "trial and error." 
 
 The first trial shows whether it is necessary to increase or diminish 7? and new 
 values are, therefore, given to h until the calculated values of v 1 , v 2 and v 3 satisfy 
 equation (4). 
 
 Case (b). Given Q 15 Q 2 , Q 3 , and the levels of the surfaces of the water in 
 the reservoirs and of the junction B, to find the diameters of the pipes. 
 
 In this case, equation (4) must be satisfied by the given data, and, therefore, 
 only three equations are given from which to calculate the four unknowns d lt 
 d 2 , d s and /z . For a definite solution a fourth equation must consequently be 
 found, from some other condition. The further condition that may be taken is 
 that the cost of the pipe lines shall be a minimum. 
 
 The cost of pipes is very nearly proportional to the product of the length and 
 diameter, and if, therefore, 1^ + 1^ + 1 s d s is made a minimum, the cost of the 
 pipes will be as small as possible. 
 
 Differentiating, with respect to 7z , the condition for a minimum is, that 
 
 dd^ 7 dd% . dd% n . 
 
 l-i -TT + to -r: \~ I") ~rp =0 ( 7 ). 
 
 dh dh dh 
 Substituting in (1), (2) and (3) the values for v lt v 2 and v 3 , 
 
 differentiating and substituting in (7) 
 
FLOW THROUGH PIPES 157 
 
 Putting the values of Q lf Q 2 , and Q 3 in (1), (2), (3), and (8), there are four 
 equations as before for four unknown quantities. 
 
 It will be better however to solve by approximation. 
 
 Give some arbitrary value to say d 2 , and calculate h from equation (2). 
 
 Then calculate d l and d 3 by putting 7i in (1) and (3), and substitute in 
 equation (8). 
 
 If this equation is satisfied the problem is solved, but if not, assume a second 
 value for d 2 and try again, and so on until such values of d lt d 2 , d 3 are obtained 
 that (8) is satisfied. 
 
 In this, as in simpler systems, the pressure at any point in the pipes ought not 
 to fall below the atmospheric pressure. 
 
 Flow through a pipe of constant diameter when the flow is diminishing at a 
 uniform rate. Let I be the length of the pipe and d its diameter. 
 
 Let h be the total loss of head in the pipe, the whole loss being assumed to be 
 by friction. 
 
 Let Q be the number of cubic feet per second that enters the pipe at a section A, 
 and Q, the number of cubic feet that passes the section B, I feet from A, the 
 quantity Q - Q x being taken from the pipe, by branches, at a uniform rate of 
 
 ~ l cubic feet per foot. 
 
 Then, if the pipe is assumed to be continued on, it is seen from Fig. 103, that 
 if the rate of discharge per foot length of the 
 pipe is kept constant, the whole of Q will be 
 discharged in a length of pipe, 
 
 ZQ 
 
 '"(Q-Qi)' 
 
 The discharge past any section, x feet from 
 C, will be 
 
 L Fig. 103. 
 
 The velocity at the section is 
 
 Assuming that in an element of length dx the loss of head due to friction is 
 
 7^/9* 
 
 d m 
 and substituting for v x its value 
 
 Q x 
 
 the loss of head due to friction in the length I is 
 
 d 1 " 25 
 7 / 4Q 
 
 If Q x is zero, I is equal to L, and 
 
 7 (4 
 
 ~?l + l \7I 
 
 The result is simplified by taking for dh the value 
 
 and assuming C constant. 
 
158 HYDRAULICS 
 
 Problem 5. Pumping water through long pipes. Required the diameter of a 
 long pipe to deliver a given quantity of water, against a given effective head, in 
 order that the charges on capital outlay and working expenses shall be a minimum. 
 
 Let I be the length of the pipe, d its diameter, and h feet the head against which 
 Q cubic feet of water per second is to be pumped. 
 
 Let the cost per horse-power of the pumping plant and its accommodation 
 be N, and the cost of a pipe of unit diameter n per foot length. 
 
 Let the cost of generating power be m per cent, of the capital outlay in the 
 pumping station, and the interest, depreciation, and cost of upkeep of the pumping 
 plant, taken together, be r per cent, of the capital outlay, and that of the pipe line 
 r per cent. ; r : will be less than r. The horse-power required to lift the water 
 against a head h and to overcome the frictional resistance of the pipe is 
 
 Let e be the ratio of the average effective horse-power to the total horse-power, 
 including the stand-by plant. The total horse-power of the plant is then 
 
 _> 0-1136Q /, 6-48Q 2 r 
 I ~^~ r<2 
 
 The cost of the pumping plant is N times this quantity. 
 The total cost per year, P, of the station, is 
 
 Assuming that the cost of the pipe line is proportional to the diameter and to 
 the length, the capital outlay for the pipe is, nld, and the cost of upkeep and 
 
 . r,nld 
 interest is ' . 
 
 nrJd 
 
 . B ., 
 
 Therefore 0-1136 ..-k + -- + 
 
 is to be a minimum. 
 
 Differentiating with respect to d and equating to zero, 
 
 and d6= 3*68(m + r)N.Q3 
 
 That is, d is independent of the length I and the head against which the water 
 is pumped. 
 
 Taking C as 80, e as 0-6 and as 50, then 
 
 68 x 50 /Q 
 
 80 x 80 x -6 
 
 = 0-603 */QT 
 If (m r)N 
 
 and if is 25, 
 
 w, 
 
 d =-535^/0^ 
 
 Problem 6. Pipe with a nozzle at the end. Suppose a pipe of length I and 
 diameter D has at one end a nozzle of diameter d, through which water is dis- 
 charged from a reservoir, the level of the water in which is h feet above the centre 
 of the nozzle. 
 
 Required the diameter of the nozzle so that the kinetic energy of the jet is 
 a maximum. 
 
FLOW THROUGH PIPES 159 
 
 Let V be the velocity of the water in the pipe. 
 
 Then, since there is continuity of flow, v the velocity with which the water 
 
 leaves the nozzle is ^ . 
 d 2 
 
 The head lost by friction in the pipe is 
 
 4/ V-l _ fvn . d 4 
 
 The kinetic energy of the jet per Ib. of flow as it leaves the nozzle is 
 
 Therefore * = | ( 1 + < ) 
 
 from which by transposing and taking the square root, 
 
 The weight of water which flows per second d 2 . v . w where M? = the weight of 
 
 a cubic foot of water. 
 
 Therefore, the kinetic energy of the jet, is 
 
 This is a maximum when -^ = 0. 
 ad 
 
 Therefore 
 
 .(3). 
 
 4 
 
 from which D 5 + 4/W 4 = 12/W 4 , 
 
 and 
 
 or 
 
 v "/ " 
 
 If the nozzle is not circular but has an area a, then since in the circular nozzle 
 of the same area 
 
 16a 2 
 from which d 4 = 
 
 Therefore D 
 
 and a = 0-278 /y- 
 
 By substituting the value of D 5 from (5) in (1) it is at once seen that, for 
 maximum kinetic energy, the head lost in friction is 
 
 iv 2 
 
 V or ** 
 
 Problem 1. Taking the same data as in problem 6, to find the area of the 
 nozzle that the momentum of the issuing jet is a maximum. 
 
 The momentum of the quantity of water Q which flows per second, as it leaves 
 
 the nozzle, is W ' ^ V Ibs. feet. The momentum M is, therefore, 
 
 9 
 
 Substituting for ^ from equation (1), problem 6, 
 
 M= 
 
160 HYDRAULICS 
 
 Differentiating, and equating to zero, 
 
 and 
 
 d= \f ^' 
 64 
 
 If the nozzle has an area a, D 5 = 
 
 and a=-3 
 
 Substituting for D 5 in equation (1) it is seen that when the momentum is a 
 maximum half the head h is lost in friction. 
 
 Problem 6 has an important application, in determining the ratio of the size 
 of the supply pipe to the orifice supplying water to a Pelton Wheel, while problem 7 
 gives the ratio, in order that the pressure exerted by the jet on a fixed plane 
 perpendicular to the jet should be a maximum. 
 
 Problem 8. Loss of head due to friction in a pipe, the diameter of which varies 
 uniformly. Let the pipe be of length I and its diameter vary uniformly from d 
 to dj_. 
 
 Suppose the sides of the pipe produced until they meet in P, Fig. 104. 
 
 Then ^=4 and S = ^- . ...(1). 
 
 The diameter of the pipe at any distance x from the small end is 
 
 The loss of head in a small element of length dx is 2 , v being the velocity 
 when the diameter is d. 
 T 
 
 Fig. 104. 
 If Q is the flow in cubic ft. per second 
 
 i* 
 
 The total loss of head h in a length I is 
 
 dx 
 
 4 Q 
 
 TTd 2 
 
 16Q 2 . S 5 
 
 S 5 /_!_ __ 1 \ 
 
 f \s 4 (s + ov 
 
 Substituting the value of S from equation (1) the loss of head due to friction 
 can be determined. 
 
 Problem 9. Pipe line consisting of a number of pipes of different diameters. In 
 practice only short conical pipes are used, as for instance in the limbs of a Venturi 
 meter. 
 
 If it is desirable to diminish the diameter of a long pipe line, instead of using 
 a pipe the diameter of which varies uniformly with the length, the line is made up 
 of a number of parallel pipes of different diameters and lengths. 
 
FLOW THROUGH PIPES 161 
 
 Let Zj, Z 2 , 1 3 ... be the lengths and d^, d 2 , d s ... the diameters respectively, of 
 the sections of the pipe. 
 
 The total loss of head due to friction, if C be assumed constant, is 
 
 The diameter d of the pipe, which, for the same total length, would give the 
 same discharge for the same loss of head due to friction, can be found from the 
 equation 
 
 -_ h . ** ** . 
 
 The length L of a pipe, of constant diameter D, which will give the same 
 discharge for the same loss of head by friction, is 
 
 Problem 10. Pipe acting as a siphon. It is sometimes necessary to take a 
 pipe line over some obstruction, such as a hill, which necessitates the pipe rising, 
 not only above the hydraulic gradient as in Fig. 87, but even above the original 
 level of the water in the reservoir from which the supply is derived. 
 
 Let it be supposed, as in Fig. 105, that water is to be delivered from the reservoir 
 B to the reservoir C through the pipe BAG, which at the point A rises ^ feet above 
 the level of the surface of the water in the upper reservoir. 
 
 f 
 
 Fig. 105. 
 
 Let the difference in level of the surfaces of the water in the reservoirs 
 be fe 2 feet. 
 
 Let h a be the pressure head equivalent to the atmospheric pressure. 
 
 To start the flow in the pipe, it will be necessary to fill it by a pump or other 
 artificial means. 
 
 Let it be assumed that the flow is allowed to take place and is regulated so that 
 it is continuous, and the velocity v is as large as possible. 
 
 Then neglecting the velocity head and resistances other than that due to friction, 
 
 / 
 
 L and d being the length and diameter of the pipe respectively. 
 
 The hydraulic gradient is practically the straight line DE. 
 
 Theoretically if AF is made greater than h a , which is about 34 feet, the pressure 
 at A becomes negative and the flow will cease. 
 
 Practically AF cannot be made much greater than 25 feet. 
 
 To find the maximum velocity possible in the rising limb AB, so that the pressure 
 head at A shall just be zero. 
 
 Let v m be this velocity. Let the datum level be the surface of the water in C. 
 
 L. H. 
 
 11 
 
162 HYDRAULICS 
 
 But 
 Therefore 
 
 If the pressure head is not to be less than 10 feet of water, 
 
 limiting velocity, and it will be necessary to throttle the pipe at C by means of a 
 valve, so as to keep the limb AC full and to keep the " siphon " from being broken. 
 
 In designing such a siphon it is, therefore, necessary to determine whether the 
 flow through the pipe as a whole under a head h 2 is greater, or less than, the flow 
 in the rising limb under a head h a - h l . 
 
 If AB is short, or 
 friction in AB will be 
 
 2gd 
 
 If the end C of the pipe is open to the atmosphere instead of being connected to 
 a reservoir, the total head available will be h 3 instead of h^. 
 
 111. Velocity of flow in pipes. 
 
 The mean velocity of flow in pipes is generally about 3 feet 
 per second, but in pipes supplying water to hydraulic machines, 
 and in short pipes, it may be as high as 10 feet per second. 
 
 If the velocity is high, the loss of head due to friction in long 
 pipes becomes excessive, and the risk of broken pipes and valves 
 through attempts to rapidly check the flow, by the sudden closing 
 of valves, or other causes, is considerably increased. 
 
 On the other hand, if the velocity is too small, unless the water 
 is very free from suspended matter, sediment* tends to collect at 
 the lower parts of the pipe, and further, at low velocities it is 
 probable that fresh water sponges and polyzoa will make their 
 abode on the surface of the pipe, and thus diminish its carrying 
 capacity. 
 
 112. Transmission of power along pipes by hydraulic 
 pressure. 
 
 Power can be transmitted hydraulically through a considerable 
 distance, with very great efficiency, as at high pressures the per 
 centage loss due to friction is small. 
 
 Let water be delivered into a pipe of diameter d feet under a 
 head of H feet, or pressure of p Ibs. per sq. foot, for which the 
 
 equivalent head is H = feet. 
 w 
 
 * An interesting example of this is quoted on p. 82 Trans. Am.S.G.E. 
 Vol. XLIV. 
 
FLOW THROUGH PIPES 163 
 
 Let the velocity of flow be v feet per second, and the length of 
 the pipe L feet. 
 
 The head lost due to friction is 
 
 -f ........................... >, 
 
 and the energy per pound available at the end of the pipe is, 
 therefore, 
 
 ' 
 
 vl. C\ t 
 
 w 2gd 
 The efficiency is 
 
 H-fr h 
 
 H H 
 
 The fraction of the given energy lost is 
 
 h 
 
 m = H' 
 
 For a given pipe the efficiency increases as the velocity 
 diminishes. 
 
 If / and L are supposed to remain constant, the efficiency is 
 
 i? 2 
 constant if -^FT is constant, and since v is generally fixed from 
 
 other conditions it may be supposed constant, and the efficiency 
 then increases as the product dH. increases. 
 
 If W is the weight of water per second passing through the 
 pipe, the work put into the pipe is W . H foot Ibs. per second, the 
 available work per second at the end of the pipe is W (H - h), and 
 the horse-power transmitted is 
 
 Hp _W.(H-/Q WH 
 
 ~ =1 ~ 
 
 Since 
 
 
 the horse-power = -- H - 
 
 ,-, /, dmS. 
 
 therefore, v = 4*01 
 
 and the horse-power = 0'357 * - d?~H.% (1 - m). 
 
 112 
 
164 HYDRAULICS 
 
 If p is the pressure per sq. inch 
 
 and the horse-power = 1'24 */ -^ d*p^ (1 - m) . 
 
 From this equation if m is given and L is known the diameter d 
 to transmit a given horse-power can be found, and if d is known the 
 longest length L that the loss shall not be greater than the given 
 fraction m can be found. 
 
 The cost of the pipe line before laying is proportional to its 
 weight, and the cost of laying approximately proportional to its 
 diameter. 
 
 If t is the thickness of the pipe in inches the weight per foot 
 length is 8l'5irdt Ibs., approximately. 
 
 Assuming the thickness of the pipe to be proportional to the 
 pressure, i.e. to the head H, 
 
 t = kp = &H, 
 
 and the weight per foot may therefore be written 
 
 w = Jcid . H. 
 
 The initial cost of the pipe per foot will then be 
 
 C=& 2 MH = K.d.H, 
 
 and since the cost of laying is approximately proportional to d, 
 the total cost per foot is 
 
 P-K.d.H + K^. 
 And since the horse-power transmitted is 
 
 HP = '357 d*H* (1 - m), 
 
 for a given horse-power and efficiency, the initial cost per horse- 
 power including laying will be a minimum when 
 
 0^7./i-dH*(l-) 
 
 is a maximum. 
 
 In large works, docks, and goods yards, the hydraulic trans- 
 mission of power to cranes, capstans, riveters and other machines 
 is largely used. 
 
 A common pressure at which water is supplied from the pumps 
 is 700 to 750 Ibs. per sq. inch, but for special purposes, it is 
 sometimes as high as 3000 Ibs. per sq. inch. These high pressures 
 are, however, frequently obtained by using an intensifier (Ch. XI) 
 to raise the ordinary pressure of 700 Ibs. to the pressure required. 
 
FLOW THROUGH PIPES 165 
 
 The demand for hydraulic power for the working of lifts, etc. 
 has led to the laying down of a network of mains in several of the 
 large cities of Great Britain. In London a mean velocity of 4 feet 
 per second is allowed in the mains and the pressure is 750 Ibs. 
 per sq. inch. In later installations, pressures of 1100 Ibs. per 
 sq. inch are used. 
 
 113. The limiting diameter of cast-iron pipes. 
 
 The diameter d for a cast-iron pipe cannot be made very large 
 if the pressure is high. 
 
 If p is the safe internal pressure per sq. inch, and s the safe 
 stress per sq. inch of the metal, and r x and r 2 the internal and 
 external radii of the pipe, 
 
 - 
 
 rf+rf 
 
 For a pressure p = 1000 Ibs. per sq. inch, and a stress s of 
 3000 Ibs. per sq. inch, r 2 is 5*65 inches when r\ is 4 inches, or the 
 pipe requires to be 1'65 inches thick. 
 
 If, therefore, the internal diameter is greater than 8 inches, the 
 pipe becomes very thick indeed. 
 
 The largest cast-iron pipe used for this pressure is between 
 7" and 8" internal diameter. 
 
 Using a maximum velocity of 5 feet per second, and a pipe 
 7 J inches diameter, the maximum horse-power, neglecting friction, 
 that can be transmitted at 1000 Ibs. per sq. inch by one pipe is 
 T^_ 4418x1000x5 
 
 550 
 = 400. 
 
 The following example shows that, if the pipe is 13,300 feet 
 long, 15 per cent, of the power is lost and the maximum power 
 that can be transmitted with this length of pipe is, therefore, 
 320 horse-power. 
 
 Steel mains are much more suitable for high pressures, as the 
 working stress may be as high as 7 tons per sq. inch. The greater 
 plasticity of the metal enables them to resist shock more readily 
 than cast-iron pipes and slightly higher velocities can be used. 
 
 A pipe 15 inches diameter and J inch thick in which the 
 pressure is 1000 Ibs. per sq. inch, and the velocity 5 ft. per second, 
 is able to transmit 1600 horse-power. 
 
 Example. Power is transmitted along a cast-iron main 7 inches diameter at 
 a pressure of 1000 Ibs. per sq. inch. The velocity of the water is 5 feet per second. 
 
 Find the maximum distance the power can be transmitted so that the efficiency 
 is uot less than 85/ . 
 
 * Ewing's Strength of Materials. 
 
166 
 
 therefore 
 
 Then 
 from which 
 
 HYDRAULICS 
 d = 0-625 feet, 
 
 345' = 
 
 = 0-15x2300 
 = 345 feet. 
 4 x 0-0104 x 25 . L 
 
 L = 
 
 2.9 x 0-625 
 345 x 64-4 x 0-625 
 
 0-0104 x 100 
 = 13,300 feet. 
 
 D 
 
 114. Pressures on pipe bends. 
 
 If a bent pipe contain a fluid at rest, the intensity of pressure 
 being the same in all directions, 
 the resultant force tending to move 
 the pipe in any direction will be 
 the pressure per unit area multiplied 
 by the projected area of the pipe 
 on a plane perpendicular to that 
 direction. 
 
 If one end of a right-angled 
 elbow, as in Fig. 106, be bolted to 
 a pipe full of water at a pressure p 
 
 Fig. 106. 
 
 Fig. 107. 
 
 pounds per sq. inch by gauge, and on the other end of the elbow 
 is bolted a flat cover, the tension in the bolts at A will be the 
 same as in the bolts at B. The pressure on the cover B is clearly 
 '7854p^ 2 , d being the diameter of the pipe in inches. If the elbow 
 be projected on to a vertical plane the projection of ACB is daefc, 
 the projection of DEF is abcfe. The resultant pressure on the 
 elbow in the direction of the arrow is, therefore, p . abed = *7854p<i 2 . 
 
 If the cover B is removed, and water flows through the pipe 
 with a velocity v feet per second, the horizontal momentum of the 
 water is destroyed and there is an additional force in the direction 
 of the arrow equal to '7854w^V. 
 
 When flow is taking place the vertical force tending to lift the 
 elbow or to shear the bolts at A is 
 
 If the elbow is less than a right 
 angle, as in Fig. 108, the total 
 tension in the bolts at A is 
 T = p (daehgc - aefgc) 
 
 e 
 
 A 
 
 9? 
 
 Fig. 108. 
 
 and since the area aehgcb is common to the two projected areas, 
 T = '7854d 2 (p - p cos + wv* cos 0) . 
 
FLOW THROUGH PIPES 
 
 167 
 
 Consider now a pipe bent as shown in Fig. 109, the limbs AA 
 and FF being parallel, and the water being supposed at rest. 
 
 The total force acting in the direction A A is 
 
 P = p {dcghea - aefgcb + d'cg'h'e'a - a'ef'g'cb'}, 
 which clearly is equal to 0. 
 
 If now instead of the fluid being at rest it has a uniform 
 velocity, the pressure must remain constant, and since there is no 
 change of velocity there is no change of momentum, and the re- 
 sultant pressure in the direction parallel to AA is still zero. 
 
 There is however a couple acting upon the bend tending to 
 rotate it in a clockwise direction. 
 
 Let p and q be the centres of gravity of the two areas daehgc 
 and aefgcb respectively, and m and n the centres of gravity of 
 d'a'e'lig'c and aefgcb'. 
 
 Through these points there are parallel forces acting as shown 
 by the arrows, and the couple is 
 
 M = P' . mn - P . pq. 
 
 The couple P . pq is also equal to the pressure on the semicircle 
 adc multiplied by the distance between the centres of gravity of 
 adc and efg, and the couple P' . mn is equal to the pressure on a'd'c 
 multiplied by the distance between the centres of gravity of a'd'c 
 and e'f'g'. 
 
 Then the resultant couple is the pressure on the semicircle efg 
 multiplied by the distance between the centres of gravity of efg 
 and e'f'g'. 
 
 If the axes of FF and AA are on the same straight line the 
 couple, as well as the force, becomes zero. 
 
 It can also be shown, by similar reasoning, that, as long as the 
 diameters at F and A are equal, the velocities at these sections 
 being therefore equal, and the two ends A and F are in the same 
 straight line, the force and the couple are both zero, whatever the 
 form of the pipe. If, therefore, as stated by Mr Froude, "the 
 
168 HYDRAULICS 
 
 two ends of a tortuous pipe are in the same straight line, there is 
 no tendency for the pipe to move." 
 
 115. Pressure on a plate in a pipe filled with flowing water. 
 
 The pressure on a plate in a pipe filled with flowing water, with 
 its plane perpendicular to the direction of flow, on certain assump- 
 tions, can be determined. 
 
 Let PQ, Fig. 110, be a thin plate of area a and let the sectional 
 area of the pipe be A. 
 
 The stream as it passes the edge of fL gj & 
 
 the plate will be contracted, and the 
 section of the stream on a plane gd will 
 be c(A-a), c being some coefficient of 
 contraction. 
 
 It has been shown on page 52 that 
 for a sharp-edged orifice the coefficient 
 of contraction is about 0*625, and when 
 
 part of the orifice is fitted with sides so that the contraction is 
 incomplete and the stream lines are in part directed perpendi- 
 cular to the orifice, the coefficient of contraction is larger. 
 
 If a coefficient in this case of 0'66 is assumed, it will probably 
 be not far from the truth. 
 
 Let YI be the velocity through the section gd and V the mean 
 velocity in the pipe. 
 
 The loss of head due to sudden enlargement from gd to ef is 
 
 (Vi-V) 2 
 
 2<7 
 
 Let the pressures at the sections ab, gd, ef be p, pi and p 2 pounds 
 per square foot respectively. 
 
 Bernouilli's equations for the three sections are then, 
 
 w 2g w 2g 
 
 and 5*^5*5*^?- (2) - 
 
 Adding (1) and (2) 
 
 (2 _ P*\ = (Vi- y ) 2 
 \w w/ 2g 
 
 The whole pressure on the plate in the direction of motion is then 
 
 Y 2 
 = w .a 
 
 '2g V66(A-a] 
 
FLOW THROUGH PIPES 
 
 169 
 
 If a = JA, 
 
 v a 
 
 ~- nearly. 
 
 0-46.g.Y 2 
 
 116. Pressure on a cylinder. 
 
 If instead of a thin plate a cylinder be placed in the pipe, 
 with its axis coincident with the axis of the pipe, Fig. Ill, there 
 are two enlargements of the section of the water. 
 
 As the stream passes the up-stream edge of the cylinder, it 
 contracts to the section at cd, and then enlarges to the section 
 ef. It again enlarges at the down-stream end of the cylinder 
 from the section ef to the section gh. 
 
 cu 
 
 g^ 
 
 Fig. 111. 
 
 Let Vi, v 2j v 3) v* be the velocities at ab, cd, ef and gh re- 
 spectively, v 4 and t;i being equal. 
 
 Between cd and ef there is a loss of head 
 
 and between ef and gh there is a loss of 
 
 2<7 
 
 The Bernouilli's equations for the sections are 
 
 Pi + v = P? + '? 
 w Zg w 2g' 
 
 Adding (2) and (3), 
 
 + = . = 
 
 w 2g w 2g w 2g 2g 
 
 P3^ = P4 + ^L 
 w 2g w 2g 
 
 2g 
 
 (1), 
 (2), 
 (3). 
 
 2g 
 
170 HYDRAULICS 
 
 If the coefficient of contraction at cd is c, the area at cd 
 
 A a 
 
 A 
 
 Then 
 
 c.(A-a) ' ~A-a' 
 
 Therefore 
 
 A 
 
 and the pressure on the cylinder is 
 
 P = (Pi - pO . 
 
 EXAMPLES. 
 
 (1) A new cast-iron pipe is 2000 ft. long and 6 ins. diameter. It is to 
 discharge 50 c. ft. of water per minute. Find the loss of head in friction 
 and the virtual slope. 
 
 (2) What is the head lost per mile in a pipe 2 ft. diameter, discharging 
 6,000,000 gallons in 24 hours ? /= '007. 
 
 (3) A pipe is to supply 40,000 gallons in 24 hours. Head of water 
 above point of discharge = 36 ft. Length of pipe = 2^ miles. Find its 
 diameter. Take C from Table XII. 
 
 (4) A pipe is 12 ins. in diameter and 3 miles in length. It connects 
 two reservoirs with a difference of level of 20 ft. Find the discharge per 
 minute in c. ft. Use Darcy's coefficient for corroded pipes. 
 
 (5) A water main has a virtual slope of 1 in 900 and discharges 35 c. ft. 
 per second. Find the diameter of the main. Coefficient / is 0'007. 
 
 (6) A pipe 12 ins. diameter is suddenly enlarged to 18 ins., and then to 
 24 ins. diameter. Each section of pipe is 100 feet long. Find the loss of 
 head in friction in each length, and the loss due to shock at each enlarge- 
 ment. The discharge is 10 c. ft. per second, and the coefficient of friction 
 /= '0106. Draw, to scale, the hydraulic gradient of the pipe. 
 
 (7) Find an expression for the relative discharge of a square, and a 
 circular pipe of the same section and slope. 
 
 (8) A pipe is 6 ins. diameter, and is laid for a quarter mile at a slope 
 of 1 in 50; for another quarter mile at a slope of 1 in 100; and for a third 
 quarter mile is level. The level of the water is 20 ft. above the inlet end, 
 and 9 ft. above the outlet end. Find the discharge (neglecting all losses 
 except skin friction) and draw the hydraulic gradient. Mark the pressure 
 in the pipe at each quarter mile. 
 
 (9) A pipe 2000 ft. long discharges Q c. ft. per second. Find by how 
 much the discharge would be increased if to the last 1000 ft. a second pipe 
 of the same size were laid alongside the first and the water allowed to flow 
 equally well along either pipe. 
 
FLOW THROUGH PIPES 171 
 
 (10) A reservoir, the level of which is 50 ft. above datum, discharges 
 into a second reservoir 30 ft. above datum, through a 12 in. pipe, 5000 ft. 
 in length ; find the discharge. Also, taking the levels of the pipe at the 
 upper reservoir, and at each successive 1000 ft., to be 40, 25, 12, 12, 10, 15, 
 ft. above datum, write down the pressure at each of these points, and 
 sketch the position of the line of hydraulic gradient. 
 
 (11) It is required to draw off the water of a reservoir through a 
 pipe placed horizontally. Diameter of pipe 6 ins. Length 40 ft. Ef- 
 fective head 20 ft. Find the discharge per second. 
 
 (12) Given the data of Ex. 11 find the discharge, taking into account 
 the loss of head if the pipe is not bell-mouthed at either end. 
 
 (13) A pipe 4 ins. diameter and 100 ft. long discharges \ c. ft. per 
 second. Find the head expended in giving velocity of entry, in overcoming 
 mouthpiece resistance, and in friction. 
 
 (14) Required the diameter of a pipe having a fall of 10 ft. per mile, 
 and capable of delivering water at a velocity of 3 ft. per second when dirty. 
 
 (15) Taking the coefficient / as O'Ol (l + ?oj) nn ^ h w much water 
 
 would be discharged through a 12 -inch pipe a mile long, connecting two 
 reservoirs with a difference of level of 20 feet. 
 
 (16) Water flows through a 12-inch pipe having a virtual slope of 3 feet 
 per 1000 feet at a velocity of 3 feet per second. 
 
 Find the friction per sq. ft. of surface of pipe in Ibs. 
 
 Also the value of / in the ordinary formula for flow in pipes. 
 
 (17) Find the relative discharge of a 6-inch main with a slope of 
 1 in 400, and a 4 -inch main with a slope of 1 in 50. 
 
 (18) A 6-inch main 7 miles in length with a virtual slope of 1 in 100 
 is replaced by 4 miles of 6-inch main, and 3 miles of 4-inch main. Would 
 the discharge be altered, and, if so, by how much ? 
 
 (19) Find the velocity of flow in a water main 10 miles long, con- 
 necting two reservoirs with a difference of level of 200 feet. Diameter of 
 main 15 inches. Coefficient /=0'009. 
 
 (20) Find the discharge, if the pipe of the last question is replaced for 
 the first 5 miles by a pipe 20 inches diameter and the remainder by a pipe 
 12 inches diameter. 
 
 (21) Calculate the loss of head per mile in a 10-inch pipe (area of cross 
 section 0'54 sq. ft.) when the discharge is 2 c. ft. per second. 
 
 (22) A pipe consists of a mile of 10 inch, and a mile of 5 -inch pipe, 
 and conveys 2 c. ft. per second. State from the answer to the previous 
 question the loss of head in each section and sketch a hydraulic gradient. 
 The head at the outlet is 5 ft. 
 
 (23) What is the head lost in friction in a pipe 3 feet diameter 
 discharging 6,000,000 gallons in 12 hours? 
 
 (24) A pipe 2000 feet long and 8 inches diameter is to discharge 85 c. ft. 
 per minute. What must be the head of water ? 
 
172 HYDRAULICS 
 
 (25) A pipe 6 inches diameter, 50 feet long, is connected to the bottom 
 of a tank 50 feet long by 40 feet wide. The original head over the open 
 end of the pipe is 15 feet. Find the time of emptying the tank, assuming 
 the entrance to the pipe is sharp-edged. 
 
 If ft = the head over the exit of the pipe at any moment, 
 h _v^ 'JM 4/t? 2 50' 
 2gr 2g 2g x 0*5' 
 
 from which, v =l^mf 
 
 In time dt, the discharge is 
 
 144 
 In time dt the surface falls an amount dh. 
 
 Therefore 
 
 Integrating, 
 
 2000 (1-5 +400/) / 79000 (1-5 +400/) 
 
 C = - ; - - A V lO = - ; - - 
 
 0-196 \/20 \/20 
 
 (26) The internal diameter of the tubes of a condenser is 0'654 inches. 
 The tubes are 7 feet long and the number of tubes is 400. The number of 
 gallons per minute flowing through the condenser is 400. Find the loss of 
 head due to friction as the water flows through the tubes. /=0'006. 
 
 (27) Assuming fluid friction to vary as the square of the velocity, find 
 an expression for the work done in rotating a disc of diameter d at an 
 angular velocity a in water. 
 
 (28) What horse-power can be conveyed through a 6-in. main if the 
 working pressure of the water supplied from the hydraulic power station is 
 700 Ibs. per sq. in.? Assume that the velocity of the water is limited to 
 3 ft. per second. 
 
 (29) Ten horse-power is to be transmitted by hydraulic pressure a 
 distance of a mile. Find the diameter of pipe and pressure required for an 
 efficiency of '9 when the velocity is 5 ft. per sec. 
 
 The frictional loss is given by equation 
 
 v* 4Z 
 h = '0l 7r . -T . 
 2g d 
 
 (30) Find the inclination necessary to produce a velocity of 4^ feet per 
 second in a steel water main 31 inches diameter, when running full and 
 discharging with free outlet, using the formula 
 
 . 
 
 (31) The following values of the slope i and the velocity v were 
 determined from an experiment on flow in a pipe '1296 ft. diam. 
 i -00022 -00182 -00650 '02389 -04348 -12315 -22408 
 v -205 -606 1-252 2'585 3'593 6'310 8-521 
 
FLOW THROUGH PIPES 173 
 
 Determine k and n in the formula 
 
 i=kv n . 
 
 Also determine values of C for this pipe for velocities of -5, 1, 3, 5 and 
 7 feet per sec. 
 
 (32) The total length of the Coolgardie steel aqueduct is 307 miles 
 and the diameter 30 inches. The discharge per day may be 5,600,000 
 gallons. The water is lifted a total height of 1499 feet. 
 
 Find (a) the head lost due to friction, 
 
 (6) the total work done per minute in raising the water. 
 
 (33) A pipe 2 feet diameter and 500 feet long without bends furnishes 
 water to a turbine. The turbine works under a head of 25 feet and uses 
 10 c. ft. of water per second. What percentage of work of the fall is lost 
 
 n friction in the pipe ? 
 
 Coefficient /= '007 ( 1 + \ . 
 
 (34) Eight thousand gallons an hour have to be discharged through 
 each of six nozzles, and the jet has to reach a height of 80 ft. 
 
 If the water supply is l miles away, at what elevation above the 
 nozzles would you place the required reservoir, and what would you 
 make the diameter of the supply main ? 
 
 Give the dimensions of the reservoir you would provide to keep a 
 constant supply for six hours. Lond. Un. 1903. 
 
 (35) The pipes laid to connect the Vyrnwy dam with Liverpool are 
 42 inches diameter. How much water will such a pipe supply in gallons 
 per diem if the slope of the pipe is 4^ feet per mile ? 
 
 At one point on the line of pipes the gradient is 6| feet per mile, and the 
 pipe diameter is reduced to 39 inches ; is this a reasonable reduction in the 
 dimension of the cross section ? Lond. Un. 1905. 
 
 (36) Water under a head of 60 feet is discharged through a pipe 
 6 inches diameter and 150 feet long, and then through a nozzle the area of 
 which is one-tenth the area of the pipe. Neglecting all losses except friction, 
 find the velocity with which the water leaves the nozzle. 
 
 (37) Two rectangular tanks each 50 feet long and 50 feet broad are 
 connected by a horizontal pipe 4 inches diameter, 1000 feet long. The 
 head over the centre of the pipe at one tank is 12 feet, and over the other 
 4 feet when flow commences. 
 
 Determine the time taken for the water in the two tanks to come to the 
 same level. Assume the coefficient C to be constant and equal to 90. 
 
 (38) Two reservoirs are connected by a pipe 1 mile long and 10" 
 diameter; the difference in the water surface levels being 25 ft. 
 
 Determine the flow through the pipe in gallons per hour and find by 
 how much the discharge would be increased if for the last 2000 ft. a second 
 pipe of 10" diameter is laid alongside the first. Lond. Un. 1905. 
 
 (39) A pipe 18" diameter leads from a reservoir, 300 ft. above the 
 datum, and is continued for a length of 5000 ft. at the datum, the length 
 being 15,000 ft. For the last 5000 ft. of its length water is drawn off by 
 
174 HYDRAULICS 
 
 service pipes at the rate of 10 c. ft. per min. per 500 ft. uniformly. Find 
 the pressure at the end of the pipe. Lond. Un. 1906. 
 
 (40) 350 horse-power is to be transmitted by hydraulic pressure a 
 distance of 1^ miles. 
 
 Find the number of 6 ins. diameter pipes and the pressure required for 
 an efficiency of 92 per cent. /='01. Take v as 3 ft. per sec. 
 
 (41) Find the loss of head due to friction in a water main L feet long, 
 which receives Q cubic feet per second at the inlet end and delivers 
 
 j- cubic feet to branch mains for each foot of its length. 
 What is the form of the hydraulic gradient ? 
 
 (42) A reservoir A supplies water to two other reservoirs B and C. 
 The difference of level between the surfaces of A and B is 75 feet, and 
 between A and C 97'5 feet. A common 8-inch cast-iron main supplies for 
 the first 850 feet to a point D. A 6-inch main of length 1400 feet is then 
 carried on in the same straight line to B, and a 5 -inch main of length 
 630 feet goes to C. The entrance to the 8-inch main is bell-mouthed, and 
 losses at pipe exits to the reservoirs and at the junction may be neglected. 
 Find the quantity discharged per minute into the reservoirs B and C. 
 Take the coefficient of friction (/) as '01. Lond. Un. 1907. 
 
 (43) Describe a method of finding the "loss of head" in a pipe due to 
 the hydraulic resistances, and state how you would proceed to find the 
 loss as a function of the velocity. 
 
 (44) A pipe, Z feet long and D feet in diameter, leads water from a 
 tank to a nozzle whose diameter is d, and whose centre is h feet below 
 the level of water in the tank. The jet impinges on a fixed plane 
 surface. Assuming that the loss of head due to hydraulic resistance is 
 given by 
 
 show that the pressure on the surface is a maximum when 
 
 *-* 
 
 (45) Find the flow through a sewer consisting of a cast-iron pipe 
 12 inches diameter, and having a fall of 3 feet per mile, when discharging 
 full bore. c = 100. 
 
 (46) A pipe 9 inches diameter and one mile long slopes for the first 
 half mile at 1 in 200 and for the second half mile at 1 in 100. The pressure 
 head at the higher end is found to be 40 feet of water and at the lower 
 20 feet. 
 
 Find the velocity and flow through the pipe. 
 
 Draw the hydraulic gradient and find the pressure in feet at 500 feet 
 and 1000 feet from the higher end. 
 
 (47) A town of 250,000 inhabitants is to be supplied with water. Half 
 the daily supply of 32 gallons per head is to be delivered in 8 hours. 
 
 The service reservoir is two miles from the town, and a fall of 10 feet 
 per mile can be allowed in the pipe. 
 
 What must be the size of the pipe? C = 90. 
 
FLOW THROUGH PIPES 175 
 
 (48) A water pipe is to be laid in a street 800 yards long with houses 
 on both sides of the street of 24 feet frontage. The average number of 
 inhabitants of each house is 6, and the average consumption of water for 
 each person is 30 gallons in 8 hrs. On the assumption that the pipe is laid 
 in four equal lengths of 200 yards and has a uniform slope of j^y, and that 
 the whole of the water flows through the first length, three-fourths through 
 the second, one half through the third and one quarter through the fourth, 
 and that the value of C is 90 for the whole pipe, find the diameters of the 
 four parts of the pipe. 
 
 (49) A pipe 3 miles long has a uniform slope of 20 feet per mile, and is 
 18 inches diameter for the first mile, 30 inches for the second and 21 
 inches for the third. The pressure heads at the higher and lower ends of 
 the pipe are 100 feet and 40 feet respectively. Find the discharge through 
 the pipe and determine the pressure heads at the commencement of the 
 30 inches diameter pipe, and also of the 21 inches diameter pipe. 
 
 (50) The difference of level of two reservoirs ten miles apart is 80 feet. 
 A pipe is required to connect them and to convey 45,000 gallons of water 
 per hour from the higher to the lower reservoir. 
 
 Find the necessary diameter of the pipe, and sketch the hydraulic 
 gradient, assuming /=0'01. 
 
 The middle part of the pipe is 120 feet below the surface of the upper 
 reservoir. Calculate the pressure head in the pipe at a point midway 
 between the two reservoirs. 
 
 (51) Some hydraulic machines are served with water under pressure 
 by a pipe 1000 feet long, the pressure at the machines being 600 Ibs. per 
 square inch. The horse-power developed by the machine is 300 and the 
 friction horse-power in the pipes 120. Find the necessary diameter of the 
 
 I v 2 
 
 pipe, taking the loss of head in feet as 0'03 -3 x =- and '43 Ib. per square 
 
 a 2g 
 
 inch as equivalent to 1 foot head. Also determine the pressure at which 
 the water is delivered by the pump. 
 
 What is the maximum horse-power at which it would be possible to 
 work the machines, the pump pressure remaining the same ? Lond. Un. 
 1906. 
 
 (52) Discuss Reynolds' work on the critical velocity and on a general 
 law of resistance, describing the experimental apparatus, and showing the 
 connection with the experiments of Poiseuille and D'Arcy. Lond. Un. 
 1906. 
 
 (53) In a condenser, the water enters through a pipe (section A) at the 
 bottom of the lower water head, passes through the lower nest of tubes, 
 then through the upper nest of tubes into the upper water head (section B). 
 The sectional areas at sections A and B are 0'196 and 0'95 sq. ft. respec- 
 tively ; the total sectional area of flow of the tubes forming the lower nest 
 is 0*814 sq. ft., and of the upper nest 0'75 sq. ft., the number of tubes being 
 respectively 353 and 326. The length of all the tubes is 6 feet 2 inches. 
 When the volume of the circulating water was 1*21 c. ft. per sec., the 
 observed difference of pressure head (by gauges) at A and B was 6'5 feet. 
 Find the total actual head necessary to overcome frictional resistance, and 
 
176 HYDRAULICS 
 
 the coefficient of hydraulic resistance referred to A. If the coefficient of 
 friction (4/) for the tubes is taken to be '015, find the coefficient of hydraulic 
 resistance for the tubes alone, and compare with the actual experiment. 
 Lond. Un. 1906. (C r = head lost divided by vel. head at A.) 
 
 (54) An open stream, which is discharging 20 c. ft. of water per 
 second is passed under a road by a siphon of smooth stoneware pipe, the 
 section of the siphon being cylindrical, and 2 feet in diameter. When the 
 stream enters this siphon, the siphon descends vertically 12 feet, it 
 then has a horizontal length of 100 feet, and again rises 12 feet. If all the 
 bends are sharp right-angled bends, what is the total loss of head in the 
 tunnel due to the bends and to the friction? C = 117. Lond. Un. 1907. 
 
 (55) It has been shown on page 159 that when the kinetic energy of a 
 jet issuing from a nozzle on a long pipe line is a maximum, 
 
 Hence find the minimum diameter of a pipe that will supply a Pelton 
 Wheel of 70 per cent, efficiency and 500 brake horse-power, the available 
 head being 600 feet and the length of pipe 3 miles. 
 
 (56) A fire engine supplies water at a pressure of 40 Ibs. per square 
 inch by gauge, and at a velocity of 6 feet per second into a pipe 3 inches 
 diameter. The pipe is led a distance of 100 feet to a nozzle 25 feet above 
 the pump. If the coefficient/ (of friction) in the pipe be "01, and the actual 
 lift of the jet is f of that due to the velocity of efflux, find the actual height 
 to which the jet will rise, and the diameter of the nozzle to satisfy the 
 conditions of the problem. 
 
 (57) Obtain expressions (a) for the head lost by friction (expressed in 
 feet of gas) in a main of given diameter, when the main is horizontal, and 
 when the variations of pressure are not great enough to cause any important 
 change of volume, and (6) for the discharge in cubic feet per second. 
 
 Apply your results to the following example : 
 
 The main is 16 inches diameter, the length of the main is 300 yards, 
 the density of the gas is 0'56 (that of air = l), and the difference of pressure 
 at the two ends of the pipe is ^ inch of water ; find : 
 (a) The head lost in feet of gas. 
 (&) The discharge of gas per hour in cubic feet. 
 
 Weight of 1 cubic foot of air = 0'08 lb.; weight of 1 cubic foot of water 
 62-4 Ibs. ; coefficient / (of friction) for the gas against the walls of the pipe 
 0-005. Lond. Un. 1905. 
 
 (See page 118 ; substitute for w the weight of cubic foot of gas.) 
 
 (58) Three reservoirs A, B and C are connected by a pipe leading 
 from each to a junction box P situated 450' above datum. 
 
 The lengths of the pipes are respectively 10,000', 5000' and 6000' and the 
 levels of the still water surface in A, B and C are 800', 600' and 200' above 
 datum. 
 
 Calculate the magnitude and indicate the direction of mean velocity in 
 each pipe, taking v = 100 VW, the pipes being all the same diameter, 
 namely 15". Lond. Un. 1905. 
 
FLOW THROUGH PIPES 177 
 
 (59) A pipe 3' 6" diameter bends through 45 degrees on a radius of 
 25 feet. Determine the displacing force in the direction of the radial line 
 bisecting the angle between the two limbs of the pipe, when the head of 
 water in the pipe is 250 feet. 
 
 Show also that, if a uniformly distributed pressure be applied in the 
 plane of the centre lines of the pipe, normally to the pipe on its outer 
 surface, and of intensity 
 
 49M 2 
 
 per unit length, the bend is in equilibrium. 
 R = radius of bend in feet. 
 d= diameter of pipe. 
 7& = head of water in the pipe. 
 
 L. n. 
 
 12 
 
CHAPTER VI. 
 
 FLOW IN OPEN CHANNELS. 
 
 117. Variety of the forms of channels. 
 
 The study of the flow of water in open channels is much more 
 complicated than in the case of closed pipes, because of the 
 infinite variety of the forms of the channels and of the different 
 degrees of roughness of the wetted surfaces, varying, as they do, 
 from channels lined with smooth boards or cement, to the irregular 
 beds of rivers and the rough, pebble or rock strewn, mountain 
 stream. 
 
 Attempts have been made to find formulae which are applicable 
 to any one of these very variable conditions, but as in the case of 
 pipes, the logarithmic formulae vary with the roughness of the 
 pipe, so in this case the formulae for smooth regular shaped channels 
 cannot with any degree of assurance be applied to the calculation 
 of the flow in the irregular natural streams. 
 
 118. Steady motion in uniform channels. 
 
 The experimental study of the distribution of velocities of 
 water flowing in open channels reveals the fact that, as in the 
 case of pipes, the particles of water at different points in a cross 
 section of the stream may have very different velocities, and the 
 direction of flow is not always actually in the direction of the flow 
 of the stream. 
 
 The particles of water have a sinuous motion, and at any point 
 it is probable that the condition of flow is continually changing. 
 In a channel of uniform section and slope, and in which the total 
 flow remains constant for an appreciable time, since the same 
 quantity of water passes each section, the mean velocity v in the 
 direction of the stream is constant, and is the same for all the 
 sections, and is simply equal to the discharge divided by the area 
 of the cross section. This mean velocity is purely an artificial 
 quantity, and does not represent, either in direction or magnitude, 
 the velocity of the particles of water as they pass the section. 
 
FLOW IN OPEN CHANNELS 
 
 179 
 
 Experiments with, current meters, to determine the distribution 
 of velocity in channels, show, however, that at any point in the 
 cross section, the component of velocity in a direction parallel to 
 the direction of flow remains practically constant. The considera- 
 tion of the motion is consequently simplified by assuming that 
 the water moves in parallel fillets or stream lines, the velocities in 
 which are different, but the velocity in each stream line remains 
 constant. This is the assumption that is made in investigating 
 so-called rational formulae for the velocity of flow in channels, 
 but it should not be overlooked that the actual motion may be 
 much, more complicated. 
 
 119. Formula for the flow when the motion is uniform 
 in a channel of uniform section and slope. 
 
 On this assumption, the conditions of flow at similarly situated 
 points C and D in any two cross sections AA and BB, Figs. 112 
 and 113, of a channel of uniform slope and section are exactly the 
 same ; the velocities are equal, and since C and D are at the same 
 distance below the free surface the pressures are also equal. For 
 the filament CD, therefore, 
 
 PC + vJ = PD + V 
 w 2g w 2g' 
 
 and therefore, since the same is true for any other filament, 
 
 is constant for the two sections. 
 
 Fig. 112. 
 
 Let v be the mean velocity of the stream, i the fall per foot 
 length of the surface of the water, or the slope, dZ the length 
 between AA and BB, ^ the cross sectional area EFGH of the 
 stream, P the wetted perimeter, i.e. the length EF + FG + GrH, 
 and w the weight of a cubic foot of water. 
 
 Let ^ = m be called the hydraulic mean depth. 
 
 Let dz be the fall of the surface between AA and BB. Since 
 the slope is small dz = i.dl. 
 
 122 
 
180 HYDRAULICS 
 
 If Q cubic feet per second fall from AA to BB, the work done 
 upon it by gravity will be : 
 
 wQ.dz = w . <*> . v . i . dZ. 
 
 Then, since 2 ^ + ~ 
 
 \w lg> 
 
 is constant for the two sections, the work done by gravity must 
 be equal to the work done by the frictional and other resistances 
 opposing the motion of the water. 
 
 As remarked above, all the filaments have not the same velocity, 
 so that there is relative motion between consecutive filaments, 
 and since water is not a perfect fluid some portion of the work 
 done by gravity is utilised in overcoming the friction due to this 
 relative motion. Energy is also lost, due to the cross currents or 
 eddy motions, which are neglected in assuming stream line flow, 
 and some resistance is also offered to the flow by the air on the 
 surface of the water. 
 
 The principal cause of loss is, however, the frictional resistance 
 of the sides of the channel, and it is assumed that the whole of 
 work done by gravity is utilised in overcoming this resistance. 
 
 Let F . v be the work done per unit area of the sides of the 
 channel, v being the mean velocity of flow. F is often called the 
 frictional resistance per unit area, but this assumes that the relative 
 velocity of the water and the sides of the channel is equal to the 
 mean velocity, which is not correct. 
 
 The area of the surface of the channel between AA and BB 
 
 Then, 
 
 o> . F 
 
 thereiore ^r i = 
 
 P w 
 
 . F 
 
 or mi = . 
 
 w 
 
 F is found by experiment to be a function of the velocity and 
 also of the hydraulic mean depth, and may be written 
 
 6 being a numerical coefficient. 
 
 Since for water w is constant may be replaced by k and 
 
 therefore, mi If . f (v) f (m) . 
 
 The form of f(v) f(m) must be determined by experiment. 
 
 120. Formula of Chezy. 
 
 The first attempts to determine the flow of water in channels 
 
FLOW IN OPEN CHANNELS 181 
 
 with precision were probably those of Chezy made on an earthen 
 canal, at Coupalet in 1775, from which he concluded that 
 *= an 
 
 and therefore mi = Jcv 2 .............................. (1). 
 
 Writing C for 4= 
 
 which is known as the Chezy formula, and has already been given 
 in the chapter on pipes. 
 
 121. Formulae of Prony and Eytelwein. 
 Prony adopted the same formula for channels and for pipes, and 
 assumed that F was a function of v and also of v 2 , and therefore, 
 
 mi = av + bv 2 . 
 
 By an examination of the experiments of Chezy and those of 
 Du Buat* made in 1782 on wooden channels, 20 inches wide and 
 less than 1 foot deep, and others on the Jard canal and the river 
 Hayne, Prony gave to a and b the values 
 
 a = '000044, 
 b = '000094. 
 This formula may be written 
 
 or 
 
 The coefficient C of the Chezy formula is then, according to Prony, 
 a function of the velocity v. 
 
 If the first term containing v be neglected, the formula is the 
 same as that of Chezy, or 
 
 v = 103 Jmi. 
 
 Eytelwein by a re-examination of the same experiments 
 together with others on the flow in the rivers Rhinet and Wesert, 
 gave values to a and b of 
 
 a = '000024, 
 
 b = -0001114. 
 
 Neglecting the term containing a, 
 
 v = 95 vmJ. 
 
 * Principes d'hydraulique. 
 
 t Experiments by Funk, 1803-6. 
 
 J Experiments by Brauings, 1790-92 
 
182 HYDRAULICS 
 
 As in the case of pipes, Prony and Eytelwein incorrectly 
 assumed that the constants a and b were independent of the 
 nature of the bed of the channel. 
 
 122. Formula of Darcy and Bazin. 
 
 After completing his classical experiments on flow in pipes 
 M. Darcy commenced a series of experiments upon open channels 
 afterwards completed by M. Bazin to determine, how the 
 frictional resistances varied with the material with which the 
 channels were lined and also with the form of the channel. 
 
 Experimental channels of semicircular and rectangular section 
 were constructed at Dijon, and lined with different materials. 
 Experiments were also made upon the flow in small earthen 
 channels (branches of the Burgoyne canal), earthen channels lined 
 with stones, and similar channels the beds of which were covered 
 with mud and aquatic herbs. The results of these experiments, 
 published in 1858 in the monumental work, Recherches Hydrau- 
 liques, very clearly demonstrated the inaccuracy of the assump- 
 tions of the old writers, that the frictional resistances were 
 independent of the nature of the wetted surface. 
 
 From the results of these experiments M. Bazin proposed for 
 the coefficient fc, section 120, the form used by Darcy for pipes, 
 
 a and ft being coefficients both of which depend upon the nature 
 of the lining of the channel. 
 
 Thus, mi = ( a. + }v 2 
 
 \ mj 
 
 or v = , = \mi. 
 
 The coefficient C in the Chezy formula is thus made to vary 
 with the hydraulic mean depth m, as well as with the roughness 
 of the surface. 
 
 It is convenient to write the coefficient k as 
 
 Taking the unit as 1 foot, Bazin's values for a and /?, and 
 values of Jc are shown in Table XVIII. 
 
 It will be seen that the influence of the second term increases 
 very considerably with the roughness of the surface. 
 
 123. Granguillet and Kutter, from an examination of Bazin' s 
 
FLOW IN OPEN CHANNELS 
 
 183 
 
 experiments, together with some of their own, found that the 
 coefficient C in the Chezy formula could be written in the form 
 
 c= A b 
 
 in which a is a constant for all channels, and b is a coefficient of 
 roughness. 
 
 TABLE XVIII. 
 
 Showing the values of a, ft, and k in Bazin's formula for 
 channels. 
 
 
 a 
 
 ft 
 
 k 
 
 Planed boards and smooth 
 cement 
 
 0000457 
 
 0000045 
 
 0000457 I 1 4- "A 
 
 
 Rough boards, bricks and 
 concrete 
 
 0000580 
 
 0000133 
 
 000058 (lH -) 
 V m J 
 
 Ashlar masonry 
 
 0000730 
 
 00006 
 
 (QO\ 
 1 + ) 
 m / 
 
 Earth 
 
 0000854 
 
 00035 
 
 / 4-1 \ 
 0000854(1 + ) 
 
 
 
 
 \ m / 
 
 Gravel (Ganguillet and 
 Kutter) 
 
 0001219 
 
 00070 
 
 0001219(^1 + ^ 
 V m J 
 
 The results of experiments by Humphreys and Abbott upon 
 the flow in the Mississippi* were, however, found to give results 
 inconsistent with this formula and also that of Bazin. 
 
 They then proposed to make the coefficient depend upon the 
 slope of the channel as well as upon the hydraulic mean depth. 
 
 From experiments which they conducted in Switzerland, upon 
 the flow in rough channels of considerable slope, and from an 
 examination of the experiments of Humphreys and Abbott on the 
 flow in the Mississippi, in which the slope is very small, and 
 a large number of experiments on channels of intermediate slopes, 
 they gave to the coefficient C, the unit being 1 foot, the value 
 
 0-00281 1-811 
 
 41'6-t- 
 
 c = 
 
 n 
 
 in which n is a coefficient of roughness of the channel and has the 
 values given in Tables XIX and XIX A. 
 
 * Report on the Hydraulics of the Mississippi River, 1861 ; Flow of water in 
 rivers and canals, Trautwine and Bering, 1893. 
 
184 HYDKAULICS 
 
 TABLE XIX. 
 
 Showing values of n in the formula of Granguillet and Kutter. 
 
 Channel n 
 
 Yery smooth, cement and planed boards '009 to '01 
 
 Smooth, boards, bricks, concrete '012 to '013 
 
 Smooth, covered with slime or tuberculated '015 
 
 Rough ashlar or rubble masonry '017 to '019 
 
 Very firm gravel or pitched with stones ... ... ... '02 
 
 Earth, in ordinary condition free from stones and weeds ... '025 
 
 Earth, not free from stones and weeds -030 
 
 Gravel in bad condition '035 to '040 
 
 Torrential streams with rough stony beds -05 
 
 TABLE XIX A. 
 
 ^howing values of n in the formula of Granguillet and Kutter, 
 
 determined from recent experiments. 
 
 n 
 
 Rectangular wooden flume, very smooth '0098 
 
 Wood pipe 6 ft. diameter -0132 
 
 Brick, washed with cement, basket shaped sewer, 6' x 6' 8", nearly 
 
 new -0130 
 
 Brick, washed with cement, basket shaped sewer, 6'x6'8", one 
 
 year old " -0148 
 
 Brick, washed with cement, basket shaped sewer, 6'x6'8", four 
 
 years old '0152 
 
 Brick, washed with cement, circular sewer, 9 ft. diameter, nearly 
 
 new -0116 
 
 Brick, washed with cement, circular sewer, 9 ft. diameter, four 
 
 years old -0133 
 
 Old Croton aqueduct, lined with brick '015 
 
 New Croton aqueduct* -012 
 
 Sudbury aqueduct ... ... ... ... ... ... ... '01 
 
 Glasgow aqueduct, lined with cement ... ... ... ... '0124 
 
 Steel pipe, wetted, clean, 1897 (mean) -0144 
 
 Steel pipe, 1899 (mean) -0155 
 
 This formula has found favour with English, American and 
 German engineers, but French writers favour the simpler formula 
 of Bazin. 
 
 It is a peculiarity of the formula, that when m equals unity 
 
 then C = - and is independent of the slope ; and also when m is 
 
 large, C increases as the slope decreases. 
 
 It is also of importance to notice that later experiments upon 
 the Mississippi by a special commission, and others on the flow of 
 the Irrawaddi and various European rivers, are inconsistent with 
 
 * Report New York Aqueduct Commission. 
 
FLOW IN OPEN CHANNELS 185 
 
 the early experiments of Humphreys and Abbott, to which 
 G-anguillet and Kutter attached very considerable importance in 
 framing their formula, and the later experiments show, as described 
 later, that the experimental determination of the flow in, and the 
 slope of, large natural streams is beset with such great difficulties, 
 that any formula deduced for channels of uniform section and 
 slope cannot with confidence be applied to natural streams, and 
 vice versa. 
 
 The application of this formula to the calculation of uniform 
 channels gives, however, excellent results, and providing the value 
 of n is known, it can be used with confidence. 
 
 It is, however, very cumbersome, and does not appear to give 
 results more accurate than a new and simpler formula suggested 
 recently by Bazin and which is given in the next section. 
 
 124. M. Bazin's later formula for the flow in channels. 
 
 M. Bazin has recently (Annales des Ponts et Chaussees, 1897, 
 Vol. IV. p. 20), made a careful examination of practically all the 
 available experiments upon channels, and has proposed for the 
 coefficient C in the Chezy formula a form originally proposed by 
 G-anguillet and Kutter, which he writes 
 
 1 
 
 or = 
 
 in which a is constant for all channels and ft is a coefficient of 
 roughness of the channel. 
 
 O 
 
 Taking 1 metre as the unit a = "0115, and writing y for -, 
 
 (1), 
 
 i , y_ 
 
 *Jm 
 
 or when the unit is one foot, 
 
 1+4= 
 
 (2), 
 
 the value of y in (2) being l'811y, in formula (1). 
 
 The values of y as found by Bazin for various kinds of channels 
 are shown in Table XX, and in Table XXI are shown values of 
 
186 HYDRAULICS 
 
 C, to the nearest whole number, as deduced from Bazin's 
 coefficients for values of m from '2 to 50. 
 
 For the channels in the first four columns only a very few 
 experimental values for C have been obtained for values of m 
 greater than 3, and none for m greater than 7'3. For the earth 
 channels, experimental values for C are wanting for small values 
 of m, so that the values as given in the table when m is greater 
 than 7 '3 for the first four columns, and those for the first three 
 columns for m less than 1, are obtained on the assumption, that 
 Bazin's formula is true for all values of m within the limits of the 
 table. 
 
 TABLE XX. 
 
 Values of y in the formula, 
 
 o= 157 - 5 
 
 unit metre unit foot 
 
 Very smooth surfaces of cement and planed boards ... '06 -1085 
 
 Smooth surfaces of boards, bricks, concrete '16 '29 
 
 Ashlar or rubble masonry "46 '83 
 
 Earthen channels, very regular or pitched with stones, 
 
 tunnels and canals in rock '85 1'54 
 
 Earthen channels in ordinary condition T30 2'35 
 
 Earthern channels presenting an exceptional resistance, 
 the wetted surface being covered with detritus, 
 
 stones or weed, or very irregular rocky surface 1*7 3'17 
 
 125. Glazed earthenware pipes. 
 
 Vellut* from experiments on the flow in earthenware pipes has 
 given to C the value 
 
 in which 
 or 
 
 This gives values of C, not very different from those given by 
 Bazin's formula when y is 0'29. 
 
 In Table XXI, column 2, glazed earthenware pipes have been 
 included with the linings given by Bazin. 
 
 * Proc. I. C. E., Vol. CLI. p. 482. 
 
FLOW IN OPEN CHANNELS 
 
 187 
 
 TABLE XXI. 
 
 Values of C in the formula v = C\/wn calculated from Bazin's 
 formula, the unit of length being 1 foot, 
 
 c= 157 ' 5 
 
 
 Channels 
 
 Hydraulic 
 mean 
 depth 
 
 Very smooth 
 cement and 
 planed 
 boards 
 
 Smooth 
 boards, brick, 
 concrete, 
 glazed 
 earthenware 
 
 Smooth 
 but dirty 
 brick, 
 concrete 
 
 Ashlar 
 masonry 
 
 Earth canals 
 in very good 
 condition, 
 and canals 
 pitched with 
 
 Earth canals 
 in ordinary 
 condition 
 
 Earth canals 
 exceptionally 
 rough 
 
 m. 
 
 
 pipes 
 
 
 
 stones 
 
 
 
 
 y = '1085 
 
 y = '29 
 
 7 = -50 
 
 y = '83 
 
 7 = 1-54 
 
 7 = 2-35 
 
 y = 3-17 
 
 2 
 
 127 
 
 96 
 
 74 
 
 55 
 
 35 
 
 25 
 
 19 
 
 3 
 
 131 
 
 103 
 
 82 
 
 63 
 
 41 
 
 30 
 
 23 
 
 4 
 
 135 
 
 108 
 
 88 
 
 68 
 
 46 
 
 32 
 
 26 
 
 5 
 
 137 
 
 112 
 
 92 
 
 72 
 
 50 
 
 37 
 
 29 
 
 6 
 
 139 
 
 116 
 
 96 
 
 76 
 
 53 
 
 39 
 
 31 
 
 8 
 
 141 
 
 119 
 
 101 
 
 82 
 
 58 
 
 43 
 
 35 
 
 1-0 
 
 142 
 
 122 
 
 105 
 
 86 
 
 62 
 
 47 
 
 38 
 
 1-3 
 
 144 
 
 126 
 
 109 
 
 91 
 
 67 
 
 51 
 
 42 
 
 1-5 
 
 145 
 
 128 
 
 112 
 
 94 
 
 70 
 
 54 
 
 44 
 
 1-75 
 
 146 
 
 130 
 
 114 
 
 97 
 
 73 
 
 57 
 
 46 
 
 2-0 
 
 147 
 
 132 
 
 116 
 
 99 
 
 76 
 
 59 
 
 49 
 
 2-5 
 
 148 
 
 134 
 
 119 
 
 103 
 
 80 
 
 64 
 
 53 
 
 3-0 
 
 149 
 
 136 
 
 122 
 
 107 
 
 84 
 
 67 
 
 56 
 
 4-0 
 
 150 
 
 138 
 
 126 
 
 111 
 
 89 
 
 72 
 
 61 
 
 5-0 
 
 151 
 
 140 
 
 129 
 
 115 
 
 94 
 
 77 
 
 65 
 
 6-0 
 
 151 
 
 142 
 
 131 
 
 118 
 
 98 
 
 80 
 
 69 
 
 8-0 
 
 152 
 
 144 
 
 134 
 
 122 
 
 102 
 
 86 
 
 74 
 
 10-0 
 
 153 
 
 145 
 
 136 
 
 125 
 
 106 
 
 90 
 
 79 
 
 12-0 
 
 
 
 
 
 109 
 
 94 
 
 82 
 
 15-0 
 
 
 
 
 
 113 
 
 98 
 
 87 
 
 20-0 
 
 
 
 
 
 117 
 
 103 
 
 92 
 
 30-0 
 
 
 
 
 
 123 
 
 110 
 
 100 
 
 50-0 
 
 
 
 
 
 129 
 
 119 
 
 108 
 
 126. Bazin's method of determining a and ft. 
 The method used by Bazin to determine the values of a and ft 
 is of sufficient interest and importance to be considered in detail. 
 
 He first calculated values of -7= and 
 
 Vm 
 
 from experimental 
 
 
 data, and plotted these values as shown in Fig. 114, -j= 
 
 abscissae, and - '- as ordinates. 
 v 
 
 as 
 
188 
 
 HYDRAULICS 
 
 As will be seen on reference to the figure, points have been 
 plotted for four classes of channels, and the points lie close to four 
 straight lines passing through a common point P on the axis 
 of y. 
 
 The equation to each of these lines is 
 
FLOW IN OPEN CHANNELS 189 
 
 'Jmi 8 
 
 or - = a + - - 
 
 ^ vm 
 
 a being the intercept on the axis of y, or the ordinate when -f= is 
 
 v/m 
 zero, and /?, which is variable, is the inclination of any one of 
 
 these lines to the axis o 
 transposing the equation, 
 
 these lines to the axis of x ; for when = is zero, ^- = a, and 
 
 vm 
 
 \l 
 
 which is clearly the tangent of the angle of inclination of the line 
 to the axis of x. 
 
 It should be noted, that since - = ~ , the ordinates give 
 
 actual experimental values of g , or by inverting the scale, values 
 
 of C. Two scales for ordinates are thus shown. 
 
 In addition to the points shown on the diagram, Fig. 114, 
 Bazin plotted the results of some hundreds of experiments for all 
 kinds of channels, and found that the points lay about a series of 
 lines, all passing through the point P, Fig. 114, for which a is '00635, 
 
 and the values of - , i.e. y, are as shown in Table XX. 
 
 CL 
 
 Bazin therefore concluded, that for all channels 
 
 ^'= -00635 +*, 
 
 v Vm 
 
 the value of /? depending upon the roughness of the channel. 
 
 For very smooth channels in cement and planed boards, Bazin 
 plotted a large number of points, not shown in Fig. 114, and the 
 line for which 7 = '109 passes very nearly through the centre of 
 the zone occupied by these points. 
 
 The line for which y is 0'29 coincides well with the mean of 
 the plotted points for smooth channels, but for some of the points 
 y may be as high as 0'4. 
 
 It is further of interest to notice, that where the surfaces and 
 sections of the channels are as nearly as possible of the same 
 character, as for instance in the Boston and New York aqueducts, 
 the values of the coefficient C differ by about 6 per cent., the 
 difference being probably due to the pointing of the sides and 
 arch of the New York aqueduct not being so carefully executed 
 as for the Boston aqueduct. By simply washing the walls of the 
 latter with cement, Fteley found that its discharge was increased 
 20 per cent. 
 
190 HYDRAULICS 
 
 y is also greater for rectangular-shaped channels, or those 
 which approximate to the rectangular form, than for those of 
 circular form, as is seen by comparing the two channels in wood 
 W and P, and also the circular and basket-shaped sewers. 
 
 M. Bazin also found that y was slightly greater for a very 
 smooth rectangular channel lined with cement than for one of 
 semicircular section. 
 
 In the figure the author has also plotted the results of some 
 recent experiments, which show clearly the effect of slime and 
 tuberculations, in increasing the resistance of very smooth channels. 
 The value of y for the basket-shaped sewer lined with brick, 
 washed with cement, rising from *4 to '642 during 4 years' service. 
 
 127. Variations in the coefficient C. 
 
 For channels lined with rubble, or similar materials, some of 
 the experimental points give values of C differing very considerably 
 from those given by points on the line for which y is 0*83, Fig. 114, 
 but the values of C deduced from experiments on particular 
 channels show similar discrepancies among themselves. 
 
 On reference to Bazin's original paper it will be seen that, for 
 channels in earth, there is a still greater variation between the 
 experimental values of C, and those given by the formula, but the 
 experimental results in these cases, for any given channel, are 
 even more inconsistent amongst themselves. 
 
 An apparently more serious difficulty arises with respect to 
 Bazin's formula in that C cannot be greater than 157'5. The 
 maximum value of the hydraulic mean depth m recorded in 
 any series of experiments is 74*3, obtained by Humphreys and 
 Abbott from measurements of the Mississippi at Carrollton in 1851. 
 Taking y as 2'35 the maximum value for C would then be 124. 
 Humphreys and Abbott deduced from their experiments values 
 of C as large as 254. If, therefore, the experiments are reliable 
 the formula of Bazin evidently gives inaccurate results for excep- 
 tional values of m. 
 
 The values of C obtained at Carrollton are, however, incon- 
 sistent with those obtained by the same workers at Vicksburg, 
 and they are not confirmed by later experiments carried out at 
 Carrollton by the Mississippi commission. Further the velocities 
 at Carrollton were obtained by double floats, and, according to 
 G-ordon*, the apparent velocities determined by such floats should 
 be at least increased, when the depth of the water is large, by ten 
 per cent. 
 
 Bazin has applied this correction to the velocities obtained by 
 
 * Gordon, Proceedings Inst. Civil Eng., 1893. 
 
FLOW IN OPEN CHANNELS 191 
 
 Humphreys and Abbott at Vicksburg and also to those obtained 
 by the Mississippi Commission at Carrollton, and shows, that the 
 maximum value for C is then, probably, only 122. 
 
 That the values of C as deduced from the early experiments on 
 the Mississippi are unreliable, is more than probable, since the 
 smallest slope, as measured, was only '0000034, which is less than 
 ^ inch per mile. It is almost impossible to believe that such small 
 differences of level could be measured with certainty, as the 
 smallest ripple would mean a very large percentage error, and 
 it is further probable that the local variations in level would be 
 greater than this measured difference for a mile length. Further, 
 assuming the slope is correct, it seems probable that the velocity 
 under such a fall would be less than some critical velocity similar 
 to that obtained in pipes, and that the velocity instead of being 
 proportional to the square root of the slope i, is proportional 
 to i. That either the measured slope was unreliable, or that the 
 velocity was less than the critical velocity, seems certain from the 
 fact, that experiments at other parts of the Mississippi, upon the 
 Irrawaddi by Gordon, and upon the large rivers of Europe, in no 
 case give values of C greater than 124. 
 
 The experimental evidence for these natural streams tends, 
 however, clearly to show, that the formulae, which can with 
 confidence be applied to the calculation of flow in channels of 
 definite form, cannot with assurance be used to determine the 
 discharge of rivers. The reason for this is not far to seek, as 
 the conditions obtaining in a river bed are generally very far 
 removed from those assumed, in obtaining the formula. The 
 assumption that the motion is uniform over a length sufficiently 
 great to be able to measure with precision the fall of the surface, 
 must be far from the truth in the case of rivers, as the irregu- 
 larities in the cross section must cause a corresponding variation 
 in the mean velocities in those sections. 
 
 In the derivation of the formula, frictional resistances only 
 are taken into account, whereas a considerable amount of the 
 work done on the falling water by gravity is probably dissipated 
 by eddy motions, set up as the stream encounters obstructions in 
 the bed of the river. These eddy motions must depend very 
 much on local circumstances and will be much more serious in 
 irregular channels and those strewn with weeds, stones or other 
 obstructions, than in the regular channels. Another and probably 
 more serious difficulty is the assumption that the slope is uniform 
 throughout the whole length over which it is measured, whereas 
 the slope between two cross sections may vary considerably 
 between bank and bank. It is also doubtful whether locally 
 
192 HYDRAULICS 
 
 there is always equilibrium between the resisting and accelerating 
 forces. In those cases, therefore, in which the beds are rocky or 
 covered with weeds, or in which the stream has a very irregular 
 shape, the hypotheses of uniform motion, slope, and section, will 
 not even be approximately realised. 
 
 128. Logarithmic formula for the flow in channels. 
 
 In the formulae discussed, it has been assumed that the f rictional 
 resistance of the channel varies as the square of the velocity, and 
 in order to make the formulae fit the experiments, the coefficient C 
 has been made to vary with the velocity. 
 
 As early as 1816, Du Buat* pointed out, that the slope i 
 increased at a less rate than the square of the velocity, and 
 half a century later St Tenant proposed the formula 
 
 m*= -000401 tt . 
 
 To determine the discharge of brick-lined sewers, Mr Santo 
 Crimp has suggested the formula 
 
 v = 124m' 6 V 5 
 
 and experiments show that for sewers that have been in use some 
 time it gives good results. The formula may be written 
 
 An examination of the results of experiments, by logarithmic 
 plotting, shows that in any uniform channel the slope 
 
 . -w; 
 
 ~ m p > 
 
 ~k being a numerical coefficient which depends upon the roughness 
 of the surface of the channel, and n and p also vary with the 
 nature of the surface. 
 
 Therefore, in the formula, 
 
 From what follows it will be seen that n varies between 1'75 
 and 2'1, while p varies between 1 and 1'5. 
 
 kv n 
 Since m is constant, the formula i=j, may be written i = bv n , 
 
 b being equal to - p . 
 
 Therefore log i = log b + n log v. 
 
 * Principes d'Hydraulique, Vol. i. p. 29, 1816. 
 
FLOW IN OPEN CHANNELS 
 
 193 
 
 In Fig. 115 are shown plotted the logarithms of i and v 
 obtained from an experiment by Bazin on the flow in a semi- 
 circular cement-lined pipe. The points lie about a straight line, 
 the tangent of the inclination of which to the axis of v is 1'96 
 and the intercept on the axis of i through v = 1, or log v = 0, is 
 0000808. 
 
 Fig. 115. Logarithmic plottings of i and v to determine the index n in 
 
 kv n 
 
 the formula for channels, t= . 
 m*> 
 
 For this experimental channel, therefore, 
 
 In the same figure are shown the plottings of log i and log v for 
 the siphon-aqueduct* of St Elvo lined with brick and for which 
 m is 278 feet. In this case n is 2 and b is '000283. Therefore 
 
 i= '000283^. 
 
 If, therefore, values of v and i are determined for a channel, 
 while m is kept constant, n can be found. 
 
 * Annales des Fonts et Cliaussees, Vol. iv. 1897. 
 
 L. H. 
 
 13 
 
194 HYDRAULICS 
 
 To determine the ratio - . The formula, 
 P 
 
 m 
 may be written in the form, 
 
 i 
 
 v, n 
 
 or om = o- +- 
 
 By determining experimentally m and v y while the slope i is 
 kept constant, and plotting log m as ordinates and log v as 
 abscissae, the plottings lie about a straight line, the tangent of the 
 
 *Y1 
 
 inclination of which to the axis of v is equal to -, and the 
 intercept on the axis of m is equal to 
 
 In Fig. 116 are shown the logarithmic plottings of m and v for 
 a number of channels, of varying degrees of roughness. 
 
 m 
 
 The ratio - varies considerably, and for very regular channels 
 
 increases with the roughness of the channel, being about 1*40 for 
 very smooth channels, lined with pure cement, planed wood or 
 cement mixed with very fine sand, 1*54 for channels in unplaned 
 wood, and 1*635 for channels lined with hard brick, smooth 
 concrete, or brick washed with cement. For channels of greater 
 
 roughness, - is very variable and appears to become nearly equal 
 to or even less than its value for smooth channels. Only in one 
 case does the ratio - become equal to 2, and the values of m and 
 
 v for that case are of very doubtful accuracy. 
 
 As shown above, from experiments in which m is kept constant, 
 
 n can be determined, and since by keeping i constant - can be 
 
 found, n and p can be deduced from two sets of experiments. 
 
 Unfortunately, there are wanting experiments in which m is 
 kept constant, so that, except for a very few cases, n cannot 
 directly be determined. 
 
 There is, however, a considerable amount of experimental data 
 for channels similarly lined, and of different slopes, but here 
 
FLOW IN OPEN CHANNELS 
 
 195 
 
 Fig. 116. Logarithmic plottings of m and v to determine the 
 
 n . , kv 
 
 ratio - in the formula i = - . 
 
 TABLE XXII. 
 Particulars of channels, plottings for which are shown in Fig. 116. 
 
 1. Semicircular channel, very smooth, lined with wood 
 
 2. ,, ,, ,, ,, ,, ,, cement mixed with 
 very fine sand 
 
 3. Eectangular channel, very smooth, lined with cement 
 
 4. , ,, ,, , wood, 1' 1" wide 
 
 5. , smooth , slope -00208 ... 
 
 6. , , '0043 ... 
 
 7. , , -0049 ... 
 
 8. , , -00824 ... 
 
 9. New Croton aqueduct, smooth, lined with bricks (Keport New York 
 
 Water Supply) 
 
 10. Glasgow aqueduct, smooth, lined with concrete (Proc. I. C. E. 1896) 
 
 11. Sudbury ,, ,, ,, brick well pointed (Tr. Am. 
 
 S.C.E. 1883) 
 
 12. Boston sewer, circular, smooth, lined with brick washed with cement 
 
 (Tr. Am.S. G. E. 1901) 
 
 13. Eectangular channel, smooth, lined with brick 
 
 14. ,, wood 
 
 15. ,, ,, ,, ,, ,, small pebbles 
 15a. Kectangular sluice channel lined with hammered ashlar 
 
 16. ,, channel lined with large pebbles ... 
 
 17. Torlonia tunnel, rock, partly lined 
 
 18. Ordinary channel lined with stones covered with mud and weeds 
 
 19- 
 
 20. Eiver Weser 
 
 21. 
 
 22. 
 
 23. Earth channel. Gros bois 
 
 24. Cavour canal 
 
 25. Eiver Seine ... 
 
 n 
 
 P 
 
 1-45 
 
 1-36 
 1-44 
 1-38 
 1-54 
 1-54 
 1-54 
 1-54 
 
 1-74 
 1-635 
 
 1-635 
 
 1-635 
 1-635 
 1-655 
 1-49 
 1-36 
 1-36 
 1-29 
 1-49 
 1-18 
 94 
 1-615 
 1-65 
 2-1 
 1-49 
 1-5 
 1-37 
 
 132 
 
196 
 
 HYDRAULICS 
 
 again, as will appear in the context, a difficulty is encountered, as 
 even with similarly lined channels, the roughness is in no two 
 cases exactly the same, and as shown by the plottings in Fig. 116, 
 no two channels of any class give exactly the same values 
 
 n 
 
 for - , but for certain classes the ratio is fairly constant. 
 
 Taking, for example, the wooden channels of the group (Nos. 4 
 
 n 
 
 to 8), the values of are all nearly equal to 1*54. 
 
 The plottings for these channels are again shown in Fig. 117. 
 The intercepts on the axis of m vary from 0'043 to 0'14. 
 
 09 
 -08 
 
 07 
 
 -06 
 -OS 
 
 e 
 
 4. Leg v 
 
 Fig. 117. Logarithmic plottings to determine the ratio - for smooth channels. 
 
 Let the intercepts on the axis of m be denoted by y, then, 
 
 i 
 
 y = 
 
FLOW IN OPEN CHANNELS 
 
 197 
 
 and 
 
 logy=-logfc--log*. 
 
 If k and p are constant for these channels, and logi and 
 log y are plotted as abscissae and ordinates, the plottings should lie 
 about a straight line, the tangent of the inclination of which to the 
 
 axis of i is - , and when log y = 0, or y is unity, the abscissa i = k, 
 
 i.e. the intercept on the axis of i is k. 
 
 In Fig. 118 are shown the plottings of log i and log y for these 
 channels, from which p=l'14> approximately, and k = '00023. 
 
 Therefore, n is approximately 1'76, and taking - as 1'54 
 
 . 00023i; 176 
 *~ m 1 ' 14 ' 
 
 01 
 
 OU5 
 
 
 
 t 
 
 OOO2 
 
 '-OOO23 
 
 -0005. 
 
 \ 
 
 N; 
 
 -OO1 
 
 -O02 
 
 -005 
 
 Log. is 
 
 Fig. 118. Logarithmic plottings to determine the value of p for smooth 
 channels, in the formula i -. 
 
 
 Since the ratio - is not exactly 1'54 for all these channels, the 
 
 P 
 
 values of n and p cannot be exactly correct for the four channels, 
 but, as will be seen on reference to Table XXIII, in which are 
 shown values of v as observed and as calculated by the formula, 
 the calculated and observed values of v agree very nearly. 
 
198 
 
 HYDRAULICS 
 
 TABLE XXIII. 
 
 Values of v, for rectangular channels lined with wood, as 
 determined experimentally, and as calculated from the formula 
 
 ii 1 ' 76 
 i = -00028 . 
 
 Slope -00208 
 
 Slope -0049 
 
 Slope -00824 
 
 
 v ob- 
 
 v calcu- 
 
 
 v ob- 
 
 v calcu- 
 
 
 v ob- 
 
 v calcu- 
 
 m in 
 
 served 
 
 lated 
 
 m in 
 
 served 
 
 lated 
 
 m in 
 
 served 
 
 lated 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 metres 
 
 
 per sec. 
 
 per sec. 
 
 
 per sec. 
 
 per sec. 
 
 
 per sec. 
 
 per sec. 
 
 0-1381 
 
 0-962 
 
 0-972 
 
 0-1042 
 
 1-325 
 
 1-314 
 
 0882 
 
 1-594 
 
 1-589 
 
 1609 
 
 1-076 
 
 1-07 
 
 1224 
 
 1-479 
 
 1-459 
 
 1041 
 
 1-776 
 
 1-764 
 
 1832 
 
 1-152 
 
 1-165 
 
 1382 
 
 1-612 
 
 1-58 
 
 1197 
 
 1-902 
 
 1-932 
 
 1976 
 
 1-259 ' 
 
 1-223 
 
 1535 
 
 1-711 
 
 1-690 
 
 1313 
 
 2-053 
 
 2-051 
 
 2146 
 
 1-324 
 
 1-290 
 
 1668 
 
 1-818 
 
 1-782 
 
 1420 
 
 2-186 
 
 2-158 
 
 2313 
 
 1-374 
 
 1-354 
 
 1789 
 
 1-898 
 
 1-858 
 
 1543 
 
 2-268 
 
 2-275 
 
 2441 
 
 1-440 
 
 1-402 
 
 1913 
 
 1-967 
 
 1-947 
 
 1649 
 
 2-357 
 
 2-377 
 
 2578 
 
 1-487 
 
 1-452 
 
 2018 
 
 2-045 
 
 2-014 
 
 1744 
 
 2-447 
 
 2-460 
 
 2681 
 
 1-552 
 
 1-49 
 
 2129 
 
 2-102 
 
 2-089 
 
 1842 
 
 2-518 
 
 2-553 
 
 2809 
 
 1-587 
 
 1-552 
 
 2215 
 
 2-179 
 
 2-143 
 
 1919 
 
 2-612 
 
 2-618 
 
 As a further example, which also shows how n and p increase 
 with the roughness of the channel, consider two channels built in 
 hammered ashlar, for which the logarithmic plottings of m and v 
 
 are shown in Fig. 116, Nos. 15 a and 156, and is 1'36. 
 
 The slopes of these channels are '101 and '037. By plotting 
 log^ and logy, p is found to be 1'43 and k '000149. So that for 
 these two channels 
 
 000149t? r95 
 
 The calculated and observed velocities are shown in Table XXXI 
 and agree remarkably well. 
 
 Very smooth channels. 
 
 
 The ratio - for the four very smooth 
 
 channels, shown in Fig. 116, varies between 1'36 and 1'45, the 
 average value being about 1'4. On plotting logy and logi the 
 points did not appear to lie about any particular line, so that p 
 could not be determined, and indicates that k is different for the 
 four channels. Trial values of n = 175 and p = 1*25 were taken, or 
 
 and values of k calculated for each channel. 
 
FLOW IN OPEN CHANNELS 199 
 
 Velocities as determined experimentally and as calculated for 
 three of the channels are shown in Table XXIII from which it will 
 be seen that k varies from '00006516 for the channel lined with 
 pure cement, to '0001072 for the rectangular shaped section lined 
 with carefully planed boards. 
 
 It will be seen, that although the range of velocities is con- 
 siderable, there is a remarkable agreement between the calculated 
 and observed values of v, so that for very smooth channels the 
 values of n and p taken, can be used with considerable confidence. 
 
 Channels moderately smooth. The plottings of log m and log v 
 for channels lined with brick, concrete, and brick washed with 
 cement are shown in Fig. 116, Nos. 9 to 13. 
 
 It will be seen that the value of - is not so constant as for the 
 
 P 
 two classes previously considered, but the mean value is about 
 
 nj 
 
 1*635, which is exactly the value of - for the Sudbury aqueduct. 
 
 *! 
 
 For the New Croton aqueduct - is as high as 1'74, and, as shown 
 in Fig. 114, this aqueduct is a little rougher than the Sudbury. 
 
 The variable values of -- show that for any two of these 
 
 P 
 
 channels either n, or p, or both, are different. On plotting logi 
 and log?; as was done in Fig. 115, the points, as in the last case, 
 could not be said to lie about any particular straight line, and the 
 value of p is therefore uncertain. It was assumed to be 1'15, and 
 
 *l 
 
 therefore, taking - as 1'635, n is 1*88. 
 
 Since no two channels have the same value for - , it is to be 
 
 P 
 expected that the coefficient Jc will not be constant. 
 
 In the Tables XXIV to XXXIII the values of v as observed 
 and as calculated from the formula 
 
 . kv 
 
 and also the value of k are given. 
 
 It will be seen that Jc varies very considerably, but, for the 
 three large aqueducts which were built with care, it is fairly 
 constant. 
 
 The effect of the sides of the channel becoming dirty with 
 time, is very well seen in the case of the circular and basket- 
 shaped sewers. In the one case the value of &, during four years' 
 service, varied from '00006124 to '00007998 and in the other from 
 '00008405 to '0001096. It is further of interest to note, that when 
 
200 HYDRAULICS 
 
 m and v are both unity and k is equal to "000067, the value of i is 
 the same as given by Bazin's formula, when y is '29, and when Jc is 
 "0001096, as in the case of the dirty basket-shaped sewer, the value 
 of y is '642, which agrees with that shown for this sewer on 
 Fig. 114. 
 
 Channels in masonry. Hammered ashlar and rubble. Attention 
 has already been called, page 198, to the results given in 
 Table XXXI for the two channels lined with hammered ashlar. 
 
 The values of n and p for these two channels were determined 
 directly from the logarithmic plottings, but the data is insufficient 
 to give definite values, in general, to n, p, and k. 
 
 In addition to these two channels, the results for one of 
 Bazin's channels lined with small pebbles, and for other channels 
 lined with rubble masonry and large pebbles are given. The 
 
 ratio - is quoted at the head of the tables where possible. In the 
 p 
 
 other cases n and p were determined by trial. 
 
 The value of n, for these rough channels, approximates to 2, 
 and appears to have a mean value of about 1'96, while p varies 
 from 1'36 to 1'5. 
 
 Earthen channels. A very large number of experiments have 
 been made on the flow in canals and rivers, but as it is generally 
 
 impracticable to keep either i or m constant, the ratio - can only 
 
 be determined in a very few cases, and in these, as will be seen 
 from the plottings in Fig. 116, the results are not satisfactory, and 
 
 'I? 
 
 appear to be unreliable, as - varies between "94 and 2'18. It seems 
 
 probable that p is between 1 and 1*5 and n from 1*96 to 2'15. 
 Logarithmic formulae for various classes of channels. 
 Very smooth channels, lined with cement, or planed boards, 
 
 i= ('000065 to '00011) . 
 lined with brick we 
 * = '000065 to '00011 
 
 Smooth channels, lined with brick well pointed, or concrete, 
 
 m x " 
 Channels lined with ashlar masonry, or small pebbles, 
 
 ,1-96 
 
 * -00015^. 
 
 m 14 
 
 Channels lined with rubble masonry, large pebbles, rock, and 
 exceptionally smooth earth channels free from deposits, 
 
 i; 1 ' 96 
 t = '00028 -^n-,. 
 
 Ill 
 
FLOW IN OPEN CHANNELS 201 
 
 Earth channels, 
 
 w 1 *' 
 
 k varies from '00033 to '00050 for channels in ordinary condition 
 and from '00050 to "00085 for channels of exceptional resistance. 
 
 129. Approximate formula for the flow in earth 
 channels. 
 
 The author has by trial found n and p for a number of 
 channels, and except for very rough channels, n is not very 
 different from 2, and p is nearly 1'5. The approximate formula 
 
 v = C v m% i, 
 
 may, therefore, be taken for earth channels, in which C is about 
 50 for channels in ordinary condition. 
 
 In Table XXXIII are shown values of v as observed and 
 calculated from this formula. 
 
 The hydraulic mean depth varies from '958 to 14*1 and for all 
 values between these external limits, the calculated velocities 
 agree with the observed, within 10 per cent., whereas the variation 
 of C in the ordinary Chezy formula is from 40 to 103, and 
 according to Bazin's formula, C would vary from about 60 to 115. 
 With this formula velocities can be readily calculated with the 
 ordinary slide rule. 
 
 TABLE XXIY. 
 Yery smooth channels. 
 Planed wood, rectangular, 1'575 wide. 
 
 = -0001072^, 
 
 II V 
 
 log k = 4'0300. 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 2372 3-55 3'57 
 
 2811 4-00 4-03 
 
 3044 4-20 4-26 
 
 3468 4-67 4'68 
 
 3717 4-94 4-94 
 
 3930 5-11 5-12 
 
 4124 5-26 5-30 
 
 4311 5-49 5-47 
 
202 HYDRAULICS 
 
 TABLE XXIV (continued). 
 Pure cement, semicircular. 
 
 ra 
 
 " 
 
 00006516 ^ 5 , 
 
 log fc = 5-8141. 
 
 m t; observed v calculated 
 
 503 3-72 3-66 
 
 682 4-59 4-55 
 
 750 4-87 4-87 
 
 915 5-57 5-62 
 
 1-034 6-14 6-14 
 
 Cement and very fine sand, semicircular. 
 i = '0000759^, 
 log fc = 5-8802. 
 
 m feet 
 
 379 
 529 
 636 
 706 
 787 
 839 
 900 
 941 
 983 
 
 1-006 
 
 1-02 
 
 1-04 
 
 TABLE XXV. 
 
 Boston circular sewer, 9 ft. diameter. 
 
 Brick, washed with cement, t = srnn7 (Horton). 
 
 i = -00006124^, 
 m 115 
 
 log t? = -61 18 log m + '5319 logt + 2*2401, 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 928 2-21 2-34 
 
 1-208 2-70 2-76 
 
 1-408 3-03 3-03 
 
 1-830 3-48 3-56 
 
 1-999 3-73 3-75 
 
 2-309 4-18 4-10 
 
 v ft. per sec. 
 
 v ft. per sec. 
 
 observed 
 
 calculated 
 
 2-87 
 
 2-74 
 
 3-44 
 
 3-49 
 
 3-87 
 
 3-98 
 
 4-30 
 
 4-30 
 
 4-51 
 
 4-59 
 
 4-80 
 
 4-84 
 
 4-94 
 
 5-10 
 
 5-20 
 
 5-26 
 
 5-38 
 
 5-43 
 
 5-48 
 
 5-53 
 
 5-55 
 
 5-58 
 
 5-66 
 
 5-66 
 
FLOW IN OPEN CHANNELS 203 
 
 TABLE XXV (continued). 
 The same sewer after 4 years' service. 
 
 i = '00007998^, 
 m 115 
 
 log v = '6118 log m+ -5319 log i + 21795. 
 
 m v observed v calculated 
 1-120 2-38 2-29 
 
 1-606 2-82 2-78 
 
 1-952 3-16 3-22 
 
 2-130 3-30 3-39 
 
 TABLE XXYI. 
 
 New Croton aqueduct. Lined with concrete. 
 
 i = -000073^, 
 logv = '61 18 log m + '5319 log *+ 2'200. 
 
 
 v ft. per sec. 
 
 v ft. per sec. 
 
 m feet 
 
 observed 
 
 calculated 
 
 1-000 
 
 1-37 
 
 1-37 
 
 1-250 
 
 1-59 
 
 1-57 
 
 1-499 
 
 1-79 
 
 1-76 
 
 1-748 
 
 1-95 
 
 1-93 
 
 2-001 
 
 2-11 
 
 2-10 
 
 2-250 
 
 2-27 
 
 2-26 
 
 2-500 
 
 2-41 
 
 2-40 
 
 2-749 
 
 2-52 
 
 2-55 
 
 2-998 
 
 2-65 
 
 2-68 
 
 3-251 
 
 2-78 
 
 2-82 
 
 3-508 
 
 2-89 
 
 2-96 
 
 3-750 
 
 3-00 
 
 3-08 
 
 3-838 
 
 3-02 
 
 3-12 
 
 TABLE XXVII. 
 
 Sudbury aqueduct. Lined with well pointed brick. 
 
 i = -00006427 ^ps, 
 m 115 
 
 log & = 5-808 log v = -6118 log m + '5319 log i + 2'23. 
 
 v ft. per sec. v ft. per sec. 
 
 m feet observed calculated 
 
 4987 1-135 1-142 
 
 6004 1-269 1-279 
 
 8005 1-515 1-525 
 
 1-000 1-755 1-752 
 
 1-200 1-948 1-954 
 
 1-400 2-149 2-147 
 
 1-601 2-332 2-331 
 
 1-801 2-513 2-511 
 
 2-001 2-651 2-672 
 
 2-201 2-844 2-832 
 
 2-336 2-929 2-937 
 
204 HYDRAULICS 
 
 TABLE XXVIII. 
 Rectangular channel lined with, brick (Bazin). 
 
 4,1-88 
 
 i= '000107^. 
 m 11 " 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 1922 2-75 2-90 
 
 2838 3-67 3'68 
 
 3654 4-18 4-30 
 
 4235 4-72 4-71 
 
 4812 5-10 5-09 
 
 540 5-34 5-46 
 
 5823 5-68 5'77 
 
 6197 6-01 5-94 
 
 6682 6-15 6-22 
 
 6968 6-47 6'39 
 
 7388 6-60 6-62 
 
 7788 6-72 6'83 
 
 G-lasgow aqueduct. Lined with concrete. 
 
 i = '0000696^, 
 
 m 115 
 
 log v = "6118 log m + -5319 log i + 2'2113. 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 1-227 1-87 1'89 
 
 1-473 2-07 2-11 
 
 1-473 2-106 2-11 
 
 1-489 2-214 2-13 
 
 1-499 2-13 2-14 
 
 1-499 2-15 2-14 
 
 1-548 2-18 2-22 
 
 1-597 2-21 2-23 
 
 1-607 2-23 2-23 
 
 1-610 2-22 2-24 
 
 1-620 2-24 2-24 
 
 1-627 2-25 2-27 
 
 1-738 2-26 2-33 
 
 1-811 2-47 2-40 
 
 TABLE XXIX. 
 
 Charlestown basket-shaped sewer 6' x 6' 8". 
 Brick, washed with cement, *' = 21^577 (Horton). 
 
 i= -00008405^, 
 m 115 
 
 log v = '6118 log m + "5319 log i + 2'1678. 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 688 1-99 2-05 
 
 958 2-46 2-52 
 
 1-187 2-82 2-87 
 
 1-539 3-44 3-36 
 
FLOW IN OPEN CHANNELS 
 
 205 
 
 TABLE XXIX (continued). 
 
 The same sewer after 4 years' service, 
 
 -.1-88 
 
 ; = -0001096 ~- 5 , 
 log v = -6118 log m + -5319 log i + 2'1065. 
 
 m feet 
 1-342 
 1-508 
 1-645 
 
 v ft. per sec. v ft. per sec. 
 observed calculated 
 
 2-66 
 2-86 
 3-04 
 
 2-68 
 2-88 
 3-04 
 
 TABLE XXX. 
 
 Left aqueduct of the Solani canal, rectangular in section, lined 
 with rubble masonry (Cunningham), 
 
 i 
 
 000225 
 000206 
 000222 
 000207 
 000189 ? 
 
 V96 
 
 i= -00026-^. 
 m 14 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 6-43 
 
 6-81 
 
 7-21 
 
 7-643 
 
 7-94 
 
 3-46 
 3-49 
 3-70 
 
 3-87 
 4-06 
 
 Right aqueduct, 
 
 = -0002213 ~, 
 m 14 
 
 000195 
 000225 
 000205 
 000193 
 000193 
 000190 
 
 m 
 
 3-42 
 5-86 
 6-76 
 7-43 
 7-77 
 7-96 
 
 v observed 
 2-43 
 3-61 
 3-73 
 3-87 
 3-93 
 4-06 
 
 3-50 
 3-47 
 3-84 
 3-83 
 3-83 
 
 v calculated 
 2-26 
 3-58 
 3-76 
 3-89 
 4-04 
 4-06 
 
 Torlonia tunnel, partly in hammered ashlar, partly in solid 
 rock, 
 
 i= '00104, 
 
 ,1 "95 
 
 i = '00022 
 
 TO J 
 
 1-932 
 2-172 
 2-552 
 2-696 
 3-251 
 3-438 
 3-531 
 3-718 
 
 v observed 
 
 v calculated 
 
 3-382 
 
 3-45 
 
 3-625 
 
 3-73 
 
 4-232 
 
 4-16 
 
 4-324 
 
 4-32 
 
 5-046 
 
 4-90 
 
 4-965 
 
 5-08 
 
 4-908 
 
 5-18 
 
 5-358 
 
 5-37 
 
206 
 
 HYDRAULICS 
 
 TABLE XXXI. 
 
 Channel lined with hammered ashlar, 
 
 P~ 
 
 1-96 
 
 t =-000149 ^~ 
 
 log fc = 41740. 
 
 -" -> 
 
 v ft. per sec. 
 
 v ft. per sec. 
 
 m feet 
 
 observed 
 
 calculated 
 
 324 
 
 12-30 
 
 12-30 
 
 467 
 
 16-18 
 
 16-18 
 
 580 
 
 18-68 
 
 18-97 
 
 562 
 
 21-09 
 
 20-8 
 
 m feet 
 
 424 
 620 
 
 745 
 852 
 
 ft. per sec. v ft. per sec. 
 observed calculated 
 
 9-04 
 11-46 
 13-55 
 15-08 
 
 9-02 
 11-86 
 13-52 
 14-93 
 
 Channel lined with small pebbles, ^ = '0049 (n=l'96, 
 will give equally good results). 
 
 %l-4, 
 
 p 
 
 1*95 
 
 t -000152^= 
 
 log fc = 41913. 
 
 m feet 
 
 250 
 357 
 450 
 520 
 588 
 644 
 700 
 746 
 785 
 832 
 871 
 910 
 
 v ft. per sec. 
 observed 
 
 2-16 
 2-95 
 3-40 
 3-84 
 4-14 
 4-43 
 4-64 
 4-88 
 5-12 
 5-26 
 5-43 
 5-57 
 
 v ft. per sec. 
 calculated 
 
 2-34 
 2-97 
 3-47 
 3-82 
 4-15 
 4-43 
 4-66 
 4-88 
 5-05 
 5-25 
 5-43 
 5-58 
 
FLOW IN OPEN CHANNELS 
 
 207 
 
 TABLE XXXII. 
 
 Channel lined with large pebbles (Bazin), 
 
 -.1-96 
 
 i = '000229^, 
 log Je -4-3605. 
 
 v ft. per sec. v ft. per sec. 
 m feet observed calculated 
 
 291 
 417 
 510 
 587 
 656 
 712 
 772 
 823 
 867 
 909 
 946 
 987 
 
 1-79 
 2-43 
 2-90 
 3-27 
 3-56 
 3-85 
 4-03 
 4-23 
 4-43 
 4-60 
 4-78 
 4-90 
 
 1-84 
 2-44 
 2-90 
 3-18 
 3-45 
 3-67 
 3-91 
 4-33 
 4-53 
 4-69 
 4-84 
 5-00 
 
 TABLE XXXIII. 
 
 Velocities as observed, and as calculated by the formula 
 v=cJ^$~i. = 50. 
 
 Ganges Canal. 
 
 000155 
 000229 
 000174 
 000227 
 000291 
 
 m feet 
 
 5-40 
 8-69 
 7-82 
 9-34 
 4-50 
 
 v ft. per sec. v ft. per sec. 
 observed calculated 
 
 2-4 
 3-71 
 2-96 
 4-02 
 
 2-82 
 
 2-34 
 3-80 
 3-08 
 4-00 
 2-63 
 
 0005503 
 0005503 
 0002494 
 0002494 
 
 River Weser. 
 m v observed 
 
 8-93 
 13-35 
 14-1 
 10'5 
 
 6-29 
 7'90 
 5-69 
 4-75 
 
 v calculated 
 
 6-0 
 
 8-18 
 
 5-70 
 
 4-78 
 
 0001183 
 0001782 
 0001714 
 0002130 
 
 Missouri, 
 m v observed v calculated 
 
 10-7 
 12-3 
 15-4 
 17-7 
 
 3-6 
 4-38 
 5-03 
 6-19 
 
 3-23 
 4-37 
 4-80 
 6-26 
 
208 
 
 00029 
 00029 
 00033 
 00033 
 
 HYDRAULICS 
 
 Cavour 
 
 Canal. 
 
 m 
 
 v observed 
 
 7-32 
 5-15 
 5-63 
 4-74 
 
 3-70 
 3-10 
 3-40 
 3-04 
 
 v calculated 
 
 3-80 
 2-92 
 3-14 
 2-91 
 
 Earth channel (branch of Burgoyne canal). 
 Some stones and a few herbs upon the surface. 
 
 = 48. 
 
 000957 
 000929 
 000993 
 000986 
 000792 
 000808 
 000858 
 000842 
 
 m feet 
 
 958 
 1-181 
 1-405 
 1-538 
 
 958 
 1-210 
 1-436 
 1-558 
 
 v ft. per sec. v ft. per sec. 
 observed calculated 
 
 1-243 
 1-702 
 1-797 
 1-958 
 1-233 
 1-666 
 1-814 
 1-998 
 
 1-30 
 1-66 
 1-94 
 2-06 
 1-25 
 1-56 
 1-79 
 2-08 
 
 130. Distribution of the velocity in the cross section 
 of open channels. 
 
 The mean velocity of flow in channels and pipes of small cross 
 sectional area can be determined by actually measuring the weight 
 or the volume of the water discharged, as shown in Chapter VII, 
 and dividing the volume discharged per second by the cross 
 section of the pipe. For large channels this is impossible, and 
 the mean velocity has to be determined by other means, usually 
 by observing the velocity at a large number of points in the same 
 transverse section by means of floats, current meters*, or Pi tot 
 tubes t. If the bed of the stream is carefully sounded, the cross 
 section can be plotted and divided into small areas, at the centres 
 of which the velocities have been observed. If then, the observed 
 velocity be assumed equal to the mean velocity over the small 
 area, the discharge is found by adding the products of the areas 
 and velocities. 
 
 Or Q - 2a . v. 
 
 M. Bazin+, with a thoroughness that has characterised his 
 experiments in other branches of hydraulics, has investigated the 
 distribution of velocities in experimental channels and also in 
 natural streams. 
 
 In Figs. 119 and 120 respectively are shown the cross sections 
 of an open and closed rectangular channel with curves of equal 
 
 See page 238. 
 
 Bazin, Recherches Hydraulique, 
 
 t See page 241. 
 
FLOW IN OPEN CHANNELS 
 
 209 
 
 velocity drawn on the section. Curves showing the distribution 
 of velocities at different depths on vertical and horizontal sections 
 are also shown. 
 
 r elo cities onJfcrizontaLSectio 
 
 cu- 
 
 Curves ofeqiLoL Velocity 
 
 Fig. 119. 
 
 Velocities on 
 Vertical* Sections. 
 
 
 /" 
 
 * 
 
 '' 
 
 
 V \ 
 
 \ 
 
 ^ 
 
 
 / 
 
 ' 
 
 ,' 
 
 -r* --K, 
 
 X 
 
 \ 
 
 v^ 
 
 
 / 
 
 
 
 Velocities on, 
 ttori^oTital Sections. 
 
 
 \ 
 
 1 
 
 ab c d e 
 
 a, 
 
 b 
 
 
 C A ^ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 ^- 
 
 4 /? X X / ^ 
 
 
 / 
 
 
 
 
 ^ 
 
 
 el f/ / / 
 
 
 
 ( 
 
 
 N 
 
 
 
 
 
 
 
 (^ "N 
 
 
 
 
 "g"! j- * & 
 
 
 
 
 7-763 1-176 
 
 - 
 
 
 
 ^'1 < v \ 
 
 
 
 
 
 
 
 
 ^>fe \\\ x \ i. 
 
 \ 
 
 \ 
 
 
 ^ J 
 
 _ 
 
 
 1 
 
 ^ \\'\ N \ 
 
 (L 
 
 *> 
 
 v 
 
 C tt' & 
 
 J 
 
 
 
 \\ \ \y 
 
 \ 
 
 ^. 
 
 
 
 
 y 
 
 / 
 
 v \i^ Xx -^ ^ v ^- 
 
 
 
 
 
 
 
 
 Fig. 120. 
 
 It will be seen that the maximum velocity does not occur in 
 the free surface of the water, but on the central vertical section 
 at some distance from the surface, and that the surface velocity 
 may be very different from the mean velocity. As the maximum 
 velocity does not occur at the surface, it would appear that in 
 L. H. H 
 
210 HYDRAULICS 
 
 assuming the wetted perimeter to be only the wetted surface of 
 the channel, some error is introduced. That the air has not the 
 same influence as if the water were in contact with a surface 
 similar to that of the sides of the channel, is very clearly 
 shown by comparing the curves of equal velocity for the closed 
 rectangular channel as shown in Fig. 119 with those of Fig. 120. 
 The air resistance, no doubt, accounts in some measure for the 
 surface velocity not being the maximum velocity, but that it does 
 not wholly account for it is shown by the fact that, whether the 
 wind is blowing up or down stream, the maximum velocity is still 
 below the surface. M. Flamant* suggests as the principal reason 
 why the maximum velocity does not occur at the surface, that 
 the water is less constrained at the surface, and that irregular 
 movements of all kinds are set up, and energy is therefore 
 utilised in giving motions to the water not in the direction of 
 translation. 
 
 Depth on any vertical at which the velocity is equal to the mean 
 velocity. Later is discussed, in detail, the distribution of velocity 
 on the verticals of any cross section, and it will be seen, that if u 
 is the mean velocity on any vertical section of the channel, the 
 depth at which the velocity is equal to the mean velocity is about 
 0'6 of the total depth. This depth varies with the roughness of 
 the stream, and is deeper the greater the ratio of the depth to 
 the width of the stream. It varies between '5 and '55 of the depth 
 for rivers of small depth, having beds of fine sand, and from '55 
 to '66 in large rivers from 1 to 3J feet deep and having strong 
 bedst. 
 
 As the banks of the stream are approached, the point at which 
 the mean velocity occurs falls nearer still to the bed of the stream, 
 but if it falls very low there is generally a second point near the 
 surface at which the velocity is also equal to the mean velocity. 
 
 When the river is covered with ice the maximum velocity of 
 the current is at a depth of "35 to '45 of the total depth, and the 
 mean velocity at two points at depths of '08 to '13 and '68 to '74 
 of the total depth {. 
 
 If, therefore, on various verticals of the cross section of a stream 
 the velocity is determined, by means of a current meter, or Pitot 
 tube, at a depth of about '6 of the total depth from the surface, 
 the velocity obtained may be taken as the mean velocity upon the 
 vertical. 
 
 * Hydraulique. 
 
 t Le Genie Civil, April, 1906, "Analysis of a communication by Murphy to 
 the Hydrological section of the Institute of Geology of the United States." 
 J Cunningham, Experiments on the Ganges Canal. 
 
FLOW IN OPEN CHANNELS 
 
 211 
 
 The total discharge can then be found, approximately, by 
 dividing the cross section into a number of rectangles, such as 
 abed, Fig. 120 a, and multiplying the area of the rectangle by the 
 velocity measured on the median line at 0*6 of its depth. 
 
 cu d 
 
 Fig. 120a. 
 
 The flow of the Upper Nile has recently been determined in 
 this way. 
 
 Captain Cunningham has given several formulae, for the mean 
 velocity u upon a vertical section, of which two are here quoted. 
 
 (1), 
 (2), 
 
 "V being the velocity at the surface, va the velocity at f of the depth, 
 v at one quarter of the depth, and so on. 
 
 131. Form of the curve of velocities on a vertical 
 section. 
 
 M. Bazin* and Cunningham have both taken the curve of 
 velocities upon a vertical section as a parabola, the maximum 
 velocity being at some distance h m below the free surface of the 
 water. 
 
 Let V be the velocity measured at the centre of a current and 
 as near the surface as possible. This point will really be at 1 inch 
 or more below the surface, but it is supposed to be at the surface. 
 
 Let v be the velocity on the same vertical section at any depth 
 h, and H the depth of the stream. 
 
 Bazin found that, if the stream is wide compared to its depth, 
 the relationship between v, V, h, and i the slope, is expressed by 
 the formula, 
 
 " 
 
 - I- 
 
 ?ffi~*VH 
 
 or 
 
 k being a numerical coefficient, which has a nearly constant value 
 of 36'2 when the unit of length is one foot. 
 
 * Recherches Hydraulique, p. 228 ; Annales des Fonts et Cliaussees, 2nd Vol., 
 1875. 
 
 142 
 
212 HYDRAULICS 
 
 To determine the depth on any vertical at which the velocity is 
 equal to the mean velocity. Let u be the mean velocity on any 
 vertical section, and h u the depth at which the velocity is equal to 
 the mean velocity. 
 
 The discharge through a vertical strip of width 9Z is 
 
 /H 
 
 u~Kdl=dll v.dh. 
 Jo 
 
 /H / 1.7, 2 _ . \ 
 
 Therefore ^ H = J (V - j </H3 )fik> 
 
 and u = V-|VHJ .............................. (2). 
 
 Substituting u and h u in (1) and equating to (2), 
 
 H 
 and fc = '577H. 
 
 This depth, at which the velocity is equal to the mean velocity, 
 is determined on the assumption that fc is constant, which is only 
 true for sections very near to the centre of streams which are 
 wide compared with their depth. 
 
 It will be seen from the curves of Fig. 120 that the depth at 
 which the maximum velocity occurs becomes greater as the sides. 
 of the channel are approached, and the law of variation of velocity 
 also becomes more complicated. M. Bazin also found that the 
 depth at the centre of the stream, at which the maximum velocity 
 occurs, depends upon the ratio of the width to the depth, the 
 reason apparently being that, in a stream which is wide compared 
 to its depth, the flow at the centre is but slightly affected by the 
 resistance of the sides, but if the depth is large compared with the 
 width, the effect of the sides is felt even at the centre of the 
 stream. The farther the vertical section considered is removed 
 from the centre, the effect of the resistance of the sides is 
 increased, and the distribution of velocity is influenced to a 
 greater degree. This effect of the sides, Bazin expressed by 
 making the coefficient k to vary with the depth h m at which 
 the maximum velocity occurs. 
 
 The coefficient is then, 
 
 , 36'2 
 
 FIT 
 
 Further, the equation to the parabola can be written in terms 
 of v m , the maximum velocity, instead of V. 
 
FLOW IN OPEN CHANNELS 213 
 
 THUS, . - M* (*-*->' .................. ( 3 ). 
 
 The mean velocity u, upon the vertical section, is then, 
 
 u = ^pp / vdh 
 H; o 
 
 36-2 
 = Vm 
 
 H 
 
 Therefore 
 
 H; u V A H 
 
 When v u, h = h u , 
 
 and therefore, "* " m . 
 
 o 1 1 H 
 
 The depth 7i m at which the velocity is a maximum is generally 
 less than *2H, except very near the sides, and Ti u is, therefore, not 
 very different from *6H, as stated above. 
 
 Ratio of maximum, velocity to the mean velocity. From 
 equation (4), 
 
 hnt, hn?\ 
 
 H, 
 
 In a wide stream in which the depth of a cross section is fairly 
 constant the hydraulic mean depth m does not differ very much 
 from H, and since the mean velocity of flow through the section is 
 C *Jmi and is approximately equal to u t therefore, 
 
 36-2 /I h m 
 
 Assuming 7t m to vary from to "2 and C to be 100, varies 
 
 from T12 to 1'09. The ratio of maximum velocity to mean 
 velocity is, therefore, probably not very different from I'l. 
 
 132. The slopes of channels and the velocities allowed 
 in them. 
 
 The discharge of a channel being the product of the area and 
 the velocity, a given discharge can be obtained by making the 
 area small and the velocity great, or vice versa. And since the 
 velocity is equal to CVmi, a given velocity can be obtained by 
 
214 HYDRAULICS 
 
 varying either m or i. Since m will in general increase with the 
 area, the area will be a minim urn when i is as large as possible. 
 But, as the cost of a channel, including land, excavation and 
 construction, will, in many cases, be almost proportional to its 
 cross sectional area, for the first cost to be small it is desirable 
 that i should be large. It should be noted, however, that the 
 discharge is generally increased in a greater proportion, by an 
 increase in A, than for the same proportional increase in i. 
 
 Assume, for instance, the channel to be semicircular. 
 
 The area is proportional to d 2 , and the velocity v to *Jd. i. 
 
 Therefore Q oc d? 4di. 
 
 If d is kept constant and i doubled, the discharge is increased 
 to s/2Q, but if d is doubled, i being kept constant, the discharge 
 will be increased to 5'6Q. The maximum slope that can be given 
 will in many cases be determined by the difference in level of the 
 two points connected by the channel. 
 
 When water is to be conveyed long distances, it is often 
 necessary to have several pumping stations en route, as sufficient 
 fall cannot be obtained to admit of the aqueduct or pipe line being 
 laid in one continuous length. 
 
 The mean velocity in large aqueducts is about 3 feet per 
 second, while the slopes vary from 1 in 2000 to 1 in 10,000. The 
 slope may be as high as 1 in 1000, but should not, only iri excep- 
 tional circumstances, be less than 1 in 10,000. 
 
 In Table XXXIV are given the slopes and the maximum 
 velocities in them, of a number of brick and masonry lined 
 aqueducts and earthen channels, from which it will be seen that 
 the maximum velocities are between 2 and 5| feet per second, 
 and the slopes vary from 1 in 2000 to 1 in 7700 for the brick and 
 masonry lined aqueducts, and from 1 in 300 to 1 in 20,000 for the 
 earth channels. The slopes of large natural streams are in some 
 cases even less than 1 in 100,000. If the velocity is too small 
 suspended matter is deposited and slimy growths adhere to the sides. 
 
 It is desirable that the smallest velocity in the channel shall be 
 such, that the channel is " self-cleansing," and as far as possible 
 the growth of low forms of plant life prevented. 
 
 In sewers, or channels conveying unfiltered waters, it is 
 especially desirable that the velocity shall not be too small, and 
 should, if possible, not be less than 2 ft. per second. 
 
 TABLE XXXIY. 
 
 Showing the slopes of, and maximum velocities, as determined 
 experimentally, in some existing channels. 
 
FLOW IN OPEN CHANNELS 
 
 215 
 
 Smooth aqueducts. 
 
 New Croton aqueduct 
 Sudbury aqueduct 
 Glasgow aqueduct 
 Paris Dhuis 
 Avre, 1st part 
 2nd part 
 Manchester Thirlmere 
 Naples 
 Boston Sewer 
 
 Slope 
 
 0001326 
 
 000189 
 
 000182 
 
 000130 
 
 0004 
 
 00033 
 
 000315 
 
 00050 
 
 0005 
 
 000333 
 
 Maximum velocity 
 
 3 ft. per second 
 
 2-94 
 
 2-25 
 
 4-08 
 3-44 
 
 4-18 
 
 Earth channels. 
 Slope Maximum velocity 
 
 Ganges canal 
 
 Escher 
 
 Linth 
 
 Cavour 
 
 Simmen ,, 
 
 Chazilly cut 
 
 Marseilles canal 
 
 Chicago drainage canal 
 
 (of the bottom of the canal) 
 
 Lining 
 
 000306 
 003 
 00037 
 00033 
 0070 
 00085 
 00043 
 
 4-16 ft. p 
 4-08 
 5-53 
 3-42 
 3-74 
 1-70 
 1-70 
 
 er sec 
 
 ond earth 
 
 V 
 
 ( gravel and 
 } some stones 
 earth 
 ( earth, stony, 
 ( few weeds 
 
 00005 
 
 TABLE XXXV. 
 
 Showing for varying values of the hydraulic mean depth m, the 
 minimum slopes, which brick channels and glazed earthenware 
 pipes should have, that the velocity may not be less than 2 ft. 
 per second. 
 
 in feet 
 
 1 
 2 
 3 
 4 
 5 
 6 
 8 
 1-0 
 
 slope 
 
 1 in 93 
 275 
 510 
 
 775 
 1058 
 1380 
 2040 
 
 m feet 
 
 1-25 
 
 1-5 
 
 1-75 
 
 2-0 
 
 2-5 
 
 3-0 
 
 4-0 
 
 slope 
 
 1 in 3700 
 4700 
 5710 
 6675 
 9000 
 11200 
 15850 
 
 2760 
 
 The slopes are calculated from the formula 
 
 The value of y is taken as 0*5 to allow for the channel becoming 
 dirty. For the minimum slope for any other velocity v, multiply 
 
 (2\ 2 
 -J . For example, the minimum slope 
 
 for a velocity of 3 feet per second when m is 1, is 1 in 1227. 
 
216 HYDRAULICS 
 
 Velocity of flow in, and slope of earth channels. If the velocity 
 is high in earth channels, the sides and bed of the channel are 
 eroded, while on the other hand if it is too small, the capacity of 
 the channel will be rapidly diminished by the deposition of sand 
 and other suspended matter, and the growth of aquatic plants. 
 Du Buat gives '5 foot per second as the minimum velocity that 
 mud shall not be deposited, while Belgrand allows a minimum 
 of *8 foot per second. 
 
 TABLE XXXVI. 
 
 Showing the velocities above which, according to Du Buat, 
 and as quoted by Rankine, erosion of channels of various materials 
 takes place. 
 
 Soft clay 0'25 ft. per second 
 
 Fine sand 0-50 
 
 Coarse sand and gravel as large as peas 0'70 
 
 Gravel 1 inch diameter 2*25 
 
 Pebbles 1| inches diameter 3 '33 
 
 Heavy shingle 4-00 
 
 Soft rock, brick, earthenware 4'50 
 
 Rock, various kinds 6'00 and upwards 
 
 133. Sections of aqueducts and sewers. 
 
 The forms of sections given to some aqueducts and sewers are 
 shown in Figs. 121 to 131. In designing such aqueducts and 
 sewers, consideration has to be given to problems other than the 
 comparatively simple one of determining the size and slope to 
 be given to the channel to convey a certain quantity of water. 
 The nature of the strata through which the aqueduct is to be 
 cut, and whether the excavation can best be accomplished by 
 tunnelling, or by cut and cover, and also, whether the aqueduct 
 is to be lined, or cut in solid rock, must be considered. In many 
 cases it is desirable that the aqueduct or sewer should have such 
 a form that a man can conveniently walk along it, although its 
 sectional area is not required to be exceptionally large. In 
 such cases the section of the channel is made deep and narrow. 
 For sewers, the oval section, Figs. 126 and 127, is largely 
 adopted because of the facilities it gives in this respect, and it has 
 the further advantage that, as the flow diminishes, the cross 
 section also diminishes, and the velocity remains nearly constant 
 for all, except very small, discharges. This is important, as at 
 small velocities sediment tends to collect at the bottom of the 
 sewer. 
 
 134. Siphons forming part of aqueducts. 
 
 It is frequently necessary for some part of an aqueduct to be 
 constructed as a siphon, as when a valley has to be crossed or the 
 
FLOW IN OPEN CHANNELS 
 
 217 
 
 aqueduct taken under a stream or other obstruction, and the 
 aqueduct must, therefore, be made capable of resisting con- 
 siderable pressure. As an example the New Croton aqueduct 
 from Croton Lake to Jerome Park reservoir, which is 33*1 miles 
 
 Fig. 121. 
 
 Fig. 122. 
 
 Fig. 123. 
 
 Fig. 124. 
 
 Fig. 125. 
 
 Metal 
 
 Fig. 127. 
 
 Fig. 128. 
 
 Fig. 129. 
 
 Fig. 130. 
 
 Fig. 131. 
 
218 HYDRAULICS 
 
 long, is made up of two parts. The first is a masonry conduit of 
 the section shown in Fig. 121, 23'9 miles long and having a slope 
 of '0001326, the second consists almost entirely of a brick lined 
 siphon 6'83 miles long, 12' 3" diameter, the maximum head in 
 which is 126 feet, and the difference in level of the two ends is 
 6'19 feet. In such cases, however, the siphon is frequently made 
 of steel, or cast-iron pipes, as in the case of the new Edinburgh 
 aqueduct (see Fig. 131) which, where it crosses the valleys, is 
 made of cast-iron pipes 33 inches diameter. 
 
 135. The best form of channel. 
 
 The best form of channel, or channel of least resistance, is 
 that which, for a given slope and area, will give the maximum 
 discharge. 
 
 Since the mean velocity in a channel of given slope is proper- 
 
 j^ 
 tional to p , and the discharge is A . v, the best form of channel for 
 
 a given area, is that for which P is a minimum. 
 
 The form of the channel which has the minimum wetted peri- 
 meter for a given area is a semicircle, for which, if r is the radius > 
 
 the hydraulic mean depth is ~. 
 
 More convenient forms, for channels to be excavated in rock 
 or earth, are those of the rectangular or trapezoidal section > 
 Fig. 133. For a given discharge, the best forms for these 
 channels, will be those for which both A and P are a minimum ; 
 that is, when the differentials d A and dP are respectively equal to 
 zero. 
 
 Rectangular channel. Let L be the width and h the depth, 
 Fig. 132, of a rectangular channel ; it is required to find the ratio 
 
 j- that the area A and the wetted perimeter P may both be a 
 minimum, for a given discharge. 
 A = Lfc, 
 therefore 3A = h.o~L+ ~Ldh = ..................... (1), 
 
 therefore 3P = dL + 2;)7i = ........................... (2). 
 
 Substituting the value of c*L from (2) in (1), 
 
 L = 27i. 
 
 Therefore m== l^ = |- 
 
 Since L = 27t, the sides and bottom of the channel touch a circle 
 having h as radius and the centre of which is in the free surface 
 of the water. 
 
FLOW IN OPEN CHANNELS 
 
 219 
 
 Earth channels of trapezoidal form. In Fig. 133 let 
 
 I be the bottom width, 
 
 h the depth, 
 
 A the cross sectional area FBCD, 
 
 P the length of FBCD or the wetted perimeter, 
 
 i the slope, 
 
 and let the slopes of the sides be t horizontal to one vertical ; CG 
 is then equal to th and tan CDGr = t. 
 
 IT* fiC 
 
 K lt" th\ 
 
 k-i-4 
 
 Fig. 132. 
 
 Fig. 133. 
 
 Let Q be the discharge in cubic feet per second. 
 
 Then A=hl + t~h? (3), 
 
 ~1 . ...(4), 
 
 and 
 
 " 
 
 .(5). 
 
 For the channel to be of the best form dP and dA both equal 
 zero. 
 
 From (3) A = hl+ th\ 
 and therefore dA = hdl + ldh + 2thdh = (6). 
 
 From (4) P = I 
 
 and 
 
 Substituting the value of dl from (7) in (6) 
 Therefore, n 
 
 (7). 
 .(8). 
 
 1-2M 
 
 h 
 2* 
 
 Let O be the centre of the water surface AD, then since from (8) 
 
 therefore, in Fig. 133, CD = EG - OD. 
 
220 HYDRAULICS 
 
 Draw OF and OE perpendicular to CD and BC respectively. 
 
 Then, because the angle OFD is a right angle, the angles CDGr 
 and FOD are equal ; and since OF - OD cos FOD, and DG- - OE, 
 and Da = CDcosCDG, therefore, OE-OF; and since OEC and 
 OFC are right angles, a circle with as centre will touch the sides 
 of the channel, as in the case of the rectangular channel. 
 
 136. Depth of flow in a channel of given form that, 
 (a) the velocity may be a maximum, (b) the discharge may 
 be a maximum. 
 
 Taking the general formula 
 
 . k.v n 
 
 and transposing, 
 
 1 p 
 i n m n 
 
 For a given slope and roughness of the channel v is, therefore, 
 proportional to the hydraulic mean depth and will be a maximum 
 when m is a maximum. 
 
 That is, when the differential of p is zero, or 
 
 For maximum discharge, Aw is a maximum, and therefore, 
 
 P 
 A\*. 
 
 A f A 
 
 MP) 
 
 pi is a maximum. 
 Differentiating and equating to zero, 
 
 PdA. A.dP = 0... ...(2). 
 
 n n 
 
 Affixing values to n and p this differential equation can be 
 solved for special cases. It will generally be sufficiently accurate 
 to assume n is 2 and p - 1, as in the Chezy formula, then 
 
 n + p _ 3 
 and the equation becomes 
 
 137. Depth of flow in a circular channel of given 
 radius and slope, when the velocity is a maximum. 
 
 Let r be the radius of the channel, and 2^ the angle subtended 
 by the surface of the water at the centre of the channel, Fig. 134. 
 
FLOW IN OPEN CHANNELS 221 
 
 Then the wetted perimeter 
 
 and 
 
 / sm 2(\ 
 The area A = r 2 </> - r 2 sin <f> cos < = r 2 {$ ^ y > 
 
 and dA = r*d<}>-r 2 co$2<f>d<}>. 
 
 Substituting these values of dP and dA in equation (3), 
 section 136, 
 
 The solution in this case is obtained 
 directly as follows, 
 
 sin i 
 
 m = ^FT T^ 
 
 This will be a maximum when sin 
 is negative, and 
 
 is a maximum, or when Fi g- 134 - 
 
 d /sinf ' 
 
 and tan 2</> - 2<. 
 
 The solution to this equation, for which 2< is less than 360, is 
 
 2< = 25727'. 
 Then A = 2'738r 2 , 
 
 m = '608r, 
 and the depth of flow d = l'626r. 
 
 138. Depth of flow in a circular channel for maximum 
 discharge. 
 
 Substituting for dP and dA in equation (3), section 136, 
 
 6r 3 <f>d<f> - er 3 ^ cos 2<i>d<}> - 2r 3 <W + r 3 sin 2<W - 0, 
 from which 4< - 6< cos 2^> + sin 2</> = 0, 
 
 and therefore <t> = 154. 
 
 Then A = 3'044r 2 , 
 
 P = 5'30r, 
 m = '573r, 
 and the depth of flow d = l'899r. 
 
 Similar solutions can be obtained for other forms of channels, 
 and may be taken by the student as useful mathematical exercises 
 but they are not of much practical utility. 
 
222 
 
 HYDRAULICS 
 
 139. Curves of velocity and discharge for a given 
 channel. 
 
 The depth of flow for maximum velocity, or discharge, can be 
 determined very readily by drawing curves of velocity and dis- 
 charge for different depths of flow in the channel. This method 
 is useful and instructive, especially to those students who are not 
 familiar with the differential calculus. 
 
 As an example, velocities and discharges, for different depths 
 of flow, have been calculated for a large aqueduct, the profile of 
 which is shown in Fig. 135, and the slope i of which is 0*0001326. 
 The velocities and discharges are shown by the curves drawn in 
 the figure. 
 
 Fig. 135. 
 
 Values of A and P for different depths of flow were first deter- 
 mined and m calculated from them. 
 
 The velocities were calculated by the formula 
 
 v = C >Jmi, 
 
 using values of C from column 3, Table XXI. 
 
 It will be seen that the velocity does not vary very much for 
 all depths of flow greater than 3 feet, and that neither the velocity 
 nor the discharge is a maximum when the aqueduct is full; the 
 reason being that, as in the circular channel, as the surface of the 
 water approaches the top of the aqueduct the wetted perimeter 
 increases much more rapidly than the area. 
 
 The maximum velocity is obtained when m is a maximum 
 and equal to 3'87, but the maximum discharge is given, when the 
 depth of flow is greater than that which gives the greatest 
 
FLOW IN OPEN CHANNELS 223 
 
 Telocity. A circle is shown on the figure which gives the same 
 maximum discharge. 
 
 The student should draw similar curves for the egg-shaped 
 sewer or other form of channel. 
 
 140. Applications of the formula. 
 
 Problem 1. To find the flow in a channel of given section and slope. 
 
 This is the simplest problem and can be solved by the application of either the 
 logarithmic formula or by Bazin's formula. 
 
 The only difficulty that presents itself, is to affix values to k, n, and p in the 
 logarithmic formula or to y in Bazin's formula. 
 
 (1) By the logarithmic formula. 
 
 First assign some value to k, n, and p by comparing the lining of the channel 
 with those given in Tables XXIV to XXXIII. Let w be the cross sectional area of 
 the water. 
 
 . v 
 
 Then since i = 
 
 m'' 
 
 log v = - log i + log m log k, 
 
 and Q u.v, 
 
 or log Q = log o> + - log i + - log m log k. 
 
 (2) By the C he zy formula, using Bazin's coefficient. 
 
 The coefficient for a given value of m must be first calculated from the formula 
 
 or taken from Table XXI. 
 Then 
 
 and 
 
 Example. Determine the flow in a circular culvert 9 ft. diameter, lined with 
 smooth brick, the slope being 1 in 2000, and the channel half full. 
 
 Area = !?=2-25'. 
 
 T . r ^ , 
 
 Wetted perimeter 4 
 
 (1) By the logarithmic formula 
 
 ml*88 
 
 i = -00073 -^T.. 
 m 115 
 
 Therefore, log V =-L log -0005 + ^ log 2 -25 - -L i og -00007, 
 v = 4'55 ft. per sec., 
 w = Tj! = 31-8 sq.ft., 
 
 Q = 145 cubic feet per sec. 
 
 (2) By the Chezy formula, using Bazin's coefficient, 
 157 \5 
 
 x/2-25 
 
 ' = 132 Jfr25~^fa = 4-43 ft. per sec. 
 = 31-8 x 3-35 = 141 cubic ft. per sec. 
 
224 HYDRAULICS 
 
 Problem 2. To find the diameter of a circular channel of given slope, for which 
 the maximum discharge is Q cubic feet per second. 
 
 The hydraulic mean depth m for maximum discharge is -573r '(section 138) and 
 A = 3-044r 2 . 
 
 Then the velocity is v = -757C JrT, 
 
 and Q = 2-370*^ 
 
 and the diameter 
 
 The coefficient C is unknown, but by assuming a value for it, an approximation 
 to D can be obtained ; a new value for C can then be taken and a nearer approxi- 
 mation to D determined ; a third value for C will give a still nearer approximation 
 to D. 
 
 Example. A circular aqueduct lined with concrete has a diameter of 5' 9" and 
 a slope of 1 foot per mile. 
 
 To find the diameter of two cast-iron siphon pipes 5 miles long, to be put in 
 series with the aqueduct, and which shall have the same discharge ; the difference 
 of level between the two ends of the siphon being 12-5 feet. 
 
 The value of m for the brick lined aqueduct of circular -section when the 
 discharge is a maximum is '573?' =-64 feet. 
 
 The area A-3'044r 2 = 25 sq. ft. 
 
 Taking C as 130 from Table XXI for the brick culvert and 110 for the cast-iron 
 pipe from Table XII, then 
 
 Therefore 
 
 = 4-00 feet. 
 
 Problem 3. Having given the bottom width I, the slope i, and the side slopes t 
 of a trapezoidal earth channel, to calculate the discharge for a given depth. 
 
 First calculate m from equation (5), section 135. 
 
 From Table XXI determine the corresponding value of C, or calculate C from 
 Bazin's formula, 
 
 then v = C >/mi, 
 
 and Q = A.v. 
 
 A convenient formula to remember is the approximate formula for ordinary 
 earth channels 
 
 For values of m greater than 2, v as calculated from this formula is very nearly 
 equal to v obtained by using Bazin's formula. 
 
 The formula 
 
 m 1 ' 5 
 may also be used. 
 
FLOW IN OPEN CHANNELS 225 
 
 Example. An ordinary earth channel has a width 1= 10 feet, a depth, d = feet, 
 and a slope i = - s -^-^. Side slopes 1 to 1. To find Q 
 
 A = 46 sq. ft., 
 P = 21-212ft., 
 m = 2-16 ft., 
 
 35- 60 - 5 ' 
 N /2 7 16 
 .. v = 1-625 ft. per sec., 
 
 Q = 74-7 cubic ft. per sec. 
 From the formula 
 
 v = 1-63 ft. per sec., 
 Q = 75 cubic ft. per sec., 
 From the logarithmic formula 
 
 i = ^j^> 
 
 v = 1-649 ft. per sec., 
 
 Q = 75-8 cubic feet per sec. 
 
 Problem 4. Having given the flow in a canal, the slope, and the side slopes, to 
 find the dimensions of the profile and the mean velocity of flow, 
 
 (a) When the canal is of the best form. 
 
 (b) When the depth is given. 
 
 In the first case m = - , and from equations (8) and (4) respectively, section 135- 
 2 
 
 Therefore 
 Substituting - for m 
 
 and A 2 
 
 But v = ? = CJmi. 
 
 Therefore C 2 - 
 
 
 A value for C should be chosen, say C = 70, and h calculated, from which a mean 
 value for m = ^ can be obtained. 
 
 A nearer approximation to h can then be determined by choosing a new value of C, 
 from Table XXI corresponding to this approximate value of m, and recalculating 
 h from equation (1). 
 
 Example. An earthen channel to be kept in very good condition, having a slope 
 of 1 in 10,000, and side slopes 2 to 1, is required to discharge 100 cubic feet 
 per second ; to find the dimensions of the channel ; take C = 70. 
 
 L. H. 15 
 
226 HYDRAULICS 
 
 Then M 20 ' 00 
 
 ^900 
 10,000 ( ' 
 20,000 
 
 49 x 6-1 
 = 6700, 
 
 and h= 5-4 feet. 
 
 Therefore m = 2 -7. 
 
 From Table XXI, C = 82 for this value of m, therefore a nearer approximation 
 to h is now found from 
 
 M _ 20,000 _ 20,000 
 
 82' - -67 x 6-1 ' 
 
 x6'l 
 
 10,000 
 from which /t = 5'22 ft. and m = 2'61. 
 
 The approximation is now sufficiently near for all practical purposes and may 
 be taken as 5 feet. 
 
 Problem 5. Having given the depth d of a trapezoidal channel, the slope i, and 
 the side slopes t, to find the bottom width I for a given discharge. 
 First using the Chezy formula, 
 
 and 
 
 The mean velocity 
 
 Therefore . . . 
 
 In this equation the coefficient C is unknown, since it depends upon the value 
 of m which is unknown, and even if a value for C be assumed the equation cannot 
 very readily be solved. It is desirable, therefore, to solve by approximation. 
 
 * Assume any value for m, and find from column 4, Table XXI, the corresponding 
 value for C, and use these values of m and C. 
 
 Then, calculate v from the formula 
 
 Since T 
 
 A 
 
 and k 
 
 Therefore dl + td* = ., ...(1). 
 
 v 
 
 From this equation a value of I can be obtained, which will probably not be the 
 correct value. 
 
 With this value of I calculate a new value for m, from the formula 
 
 For this value of m obtain a new value of C from the table, recalculate v, and 
 by substitution in formula (1) obtain a second value for I. 
 
 On now again calculating m by substituting for d in formula (2), it will generally 
 be found that m differs but little from m previously calculated ; if so, the approxi- 
 mation has proceeded sufficiently far, and d as determined by using this value of m 
 will agree with the correct value sufficiently nearly for all practical purposes. 
 
 The problem can be solved in a similar way by the logarithmic formula 
 
 The indices n andp may be taken as 2-1, and 1'5 respectively, and k as -00037. 
 
FLOW IN OPEN CHANNELS 227 
 
 Example. The depth of an ordinary earth channel is 4 feet, the side slopes 
 1 to 1, the slope 1 in 6000 and the discharge is to be 7000 cubic feet per minute. 
 Find the bottom width of the channel. 
 Assume a value for ra, say 2 feet. 
 From the logarithmic formula 
 
 2 -1 log v = log i + 1-5 log m- 4-5682 ........................... (3), 
 
 v = 1-122 feet per sec. 
 7000 
 
 TheQ A =dk6o= 104s <- feet - 
 
 But A 
 
 Substituting this value for I in equation (2) 
 
 Kecalculating v from formula (3) 
 
 Then A = 75*fee 
 
 1= 14-75 feet, 
 and m = 2-88 feet. 
 
 The first value of I is, therefore, too large, and this second value is too small. 
 Third values were found to be v = 1'455, 
 
 A = 80-2, 
 J = 16-05, 
 m= 2-935. 
 This value of I is again too large. 
 
 A fourth calculation gave v = 1-475, 
 
 A =79-2, 
 
 m = 2-92. 
 
 The approximation has been carried sufficiently far, and even further than is 
 necessary, as for such channels the coefficient of roughness k cannot be trusted to 
 an accuracy corresponding to the small difference between the third and fourth 
 values of I. 
 
 Problem 6. Having given the bottom width I, the slope i and the side slopes of 
 a trapezoidal channel, to find the depth d for a given discharge. 
 
 This problem is solved exactly as the last, by first assumingji_value for m, and 
 calculating an approximate value for v from the formula v = C*Jmi. 
 
 Then, by substitution in equation (1) of the last problem and solving the 
 quadratic, 
 
 by substituting this value for d in equation (2), a new value for m can be found, 
 and hence, a second approximation to d, and so on. 
 
 Using the logarithmic formula the procedure is exactly the same as for 
 problem 5. 
 
 Problem 7 *. Having a natural stream BC, Fig. 135 a, of given slope, it is required 
 to determine the point C, at which a canal, of trapezoidal section, which is to 
 deliver a definite quantity of water to a given point A at a given level, shall be 
 made to join the stream so that the cost of the canal is a minimum. 
 
 * The solution here given is practically the same as that given by M. Flamant 
 in his excellent treatise Hydraulique. 
 
 152 
 
228 HYDRAULICS 
 
 Let I be the slope of the stream, i of the canal, h the height above some datum 
 of the surface of the water at A, and J^ of the 
 water in the stream at B, at some distance L 
 from C. 
 
 Let L be also the length and A the |T> \r* 
 
 sectional area of the canal, and let it be j J\ ^ 
 
 assumed that the section of the canal is of the A j y 
 
 most economical form, or m=-. 
 
 2 Fig. 135 a. 
 
 The side slopes of the canal will be fixed 
 
 according to the nature of the strata through which the canal is cut, and may be 
 supposed to be known. 
 
 Then the level of the water at C is 
 
 Therefore L = TZ- 
 
 Let I be the bottom width of the canal, and t the slope of the sides. The cross 
 section is then dl + td?, and 
 
 _A_ dl + tcP 
 
 ~P~ 
 
 Substituting 2m for d, 
 
 Z=4w N / 2 +l-4tra, 
 
 and therefore m= == =- , 
 
 4m tjf + 1 - 4tm + 4w Jf 2 + 1 
 
 from which m 2 =- 
 
 8 
 
 The coefficient C in the formula v = G,Jmi may be assumed constant. 
 Then v 2 = C 2 wi, 
 
 and v 4 =C 4 w 2 i 2 . 
 
 For v substituting ^ , and for ra 2 the above value, 
 O 4 C 4 Ai 2 
 
 and A.H 2 = (2 Jtf + l - 1). 
 
 Therefore A = 
 
 The cost of the canal will be approximately proportional to the product of the 
 length L and the cross sectional area, or to the cubical content of the excavation. 
 Let k be the price per cubic yard including buying of land, excavation etc. Let x 
 be the total cost. 
 
 Then c = fc.L.A 
 
 = k.(h-h l ) 
 
 This will be a minimum when -TT = O. 
 
 di 
 
 Differentiating therefore, and equating to zero, 
 
 and i = fl. 
 
 The most economical slope is therefore f of the slope of the natural stream. 
 
 If instead of taking the channel of the best form the depth is fixed, the 
 slope i=.I. 
 
FLOW IN OPEN CHANNELS 229 
 
 There have been two assumptions made in the calculation, neither of which is 
 rigidly true, the first being that the coefficient C is constant, and the second that 
 the price of the canal is proportional to its cross sectional area. 
 
 It will not always be possible to adopt the slope thus found, as the mean 
 velocity must be maintained within the limits given on page 216, and it is not 
 advisable that the slope should be less than 1 in 10,000. 
 
 EXAMPLES. 
 
 (1) The area of flow in a sewer was found to be 0'28 sq. feet; the 
 wetted perimeter T60 feet ; the inclination 1 in 38*7. The mean velocity 
 of flow was 6*12 feet per second. Find the value of C in the formula 
 
 (2) The drainage area of a certain district was 19*32 acres, the whole 
 area being impermeable to rain water. The maximum intensity of the 
 rainfall was 0'360 ins. per hour and the maximum rate of discharge regis- 
 tered in the sewer was 96% of the total rainfall. 
 
 Find the size of a circular glazed earthenware culvert having a slope of 
 
 1 in 50 suitable for carrying the storm water. 
 
 (3) Draw a curve of mean velocities and a curve of discharge for an 
 egg-shaped brick sewer, using Bazin's coefficient. Sewer, 6 feet high by 
 4 feet greatest width ; slope 1 in 1200. 
 
 (4) The sewer of the previous question is required to join into a main 
 outfall sewer. To cheapen the junction with the main outfall it is thought 
 advisable to make the last 100 feet of the sewer of a circular steel pipe 
 3 feet diameter, the junction between the oval sewer and the pipe being 
 carefully shaped so that there is no impediment to the flow. 
 
 Find what fall the circular pipe should have so that its maximum 
 discharge shall be equal to the maximum discharge of the sewer. Having 
 found the slope, draw out a curve of velocity and discharge. 
 
 (5) A canal in earth has a slope of 1 foot in 20,000, side slopes of 
 
 2 horizontal to 1 vertical, a depth of 22 feet, and a bottom width of 
 200 feet; find the volume of discharge. 
 
 Bazin's coefficient -y=2'35. 
 
 (6) Give the diameter of a circular brick sewer to run half -full for a 
 population of 80,000, the diurnal volume of sewage being 75 gallons per 
 head, the period of maximum flow 6 hours, and the available fall 1 in 1000. 
 
 Inst. C. E. 1906. 
 
 (7) A channel is to be cut with side slopes of 1| to 1 ; depth of water, 
 
 3 feet; slope, 9 inches per mile: discharge, 6,000 cubic feet per minute. 
 Find by approximation dimensions of channel. 
 
 (8) An area of irrigated land requires 2 cubic yards of water per hour 
 per acre. Find dimensions of a channel 3 feet deep and with a side slope 
 of 1 to 1. Fall, 1 feet per mile. Area to be irrigated, 6000 acres. (Solve 
 by approximation.) y=2'35. 
 
 (9) A trapezoidal channel in earth of the most economical form has a 
 depth of 10 feet and side slopes of 1 to 1. Find the discharge when the 
 slope is 18 inches per mile. y=2'35. 
 
230 HYDRAULICS 
 
 (10) A river has the following section : top width, 800 feet ; depth of 
 water, 20 feet ; side slopes 1 to 1 ; fall, 1 foot per mile. Find the discharge, 
 using Bazin's coefficient for earth channels. 
 
 (11) A channel is to be constructed for a discharge of 2000 cubic feet 
 per second ; the fall is 1^ feet per mile ; side slopes, 1 to 1 ; bottom width, 
 10 times the depth. Find dimensions of channel. Use the approximate 
 
 formula, v- 
 
 (12) Find the dimensions of a trapezoidal earth channel, of the most 
 economical form, to convey 800 cubic feet per second, with a fall of 2 feet 
 per mile, and side slopes, 1^ to 1. (Approximate formula.) 
 
 (13) An irrigation channel, with side slopes of l to 1, receives 600 
 cubic feet per second. Design a suitable channel of 3 feet depth and 
 determine its dimensions and slope. The mean velocity is not to exceed 
 2 feet per second. y = 2'35. 
 
 (14) A canal, excavated in rock, has vertical sides, a bottom width of 
 160 feet, a depth of 22 feet, and the slope is 1 foot in 20,000 feet. Find the 
 discharge, y = 1 '54. 
 
 (15) A length of the canal referred to in question (14) is in earth. It 
 has side slopes of 2 horizontal to 1 vertical; its width at the water line 
 is 290 feet and its depth 22 feet. 
 
 Find the slope this portion of the canal should have, taking y as 2-35. 
 
 (16) An aqueduct 95| miles long is made up of a culvert 50 1 miles 
 long and two steel pipes 3 feet diameter and 45 miles long laid side by side. 
 The gradient of the culvert is 20 inches to the mile, and of the pipes 2 feet 
 to the mile. Find the dimensions of a rectangular culvert lined with well 
 pointed brick, so that the depth of flow shall be equal to the width of the 
 culvert, when the pipes are giving their maximum discharge. 
 
 Take for the culvert the formula 
 
 000061 v 1 ' 88 
 
 1-15 
 
 m 
 and for the pipes the formula 
 
 ._ -00050. v 2 
 d 1 ' 26 ' 
 
 (17) The Ganges canal at Taoli was found to have a slope of 0*000146 
 and its hydraulic mean depth m was 7'0 feet ; the velocity as determined 
 by vertical floats was 2-80 feet per second; find the value of C and the 
 value of y in Bazin's equation. 
 
 (18) The following data were obtained from an aqueduct lined with 
 brick carefully pointed : 
 
 m i v 
 
 in metres in metres per sec. 
 
 229 0*0001326 '336 
 
 381 -484 
 
 533 -596 
 
 686 -691 
 
 838 -769 
 
 991 -848 
 
 1-143 -913 
 
 1-170 -922 
 
FLOW IN OPEN CHANNELS 231 
 
 Plot -= as ordinates, -- as abscissae ; find values of a and 8 in Bazin's 
 
 Vra v 
 
 formula, and thus deduce a value of y for this aqueduct. 
 
 (19) An aqueduct 107^ miles long consists of 13J miles of siphon, and 
 the remainder of a masonry culvert 6 feet 10^ inches diameter with a gradient 
 of 1 in 8000. The siphons consist of two lines of cast-iron pipes 43 inches 
 diameter having a slope of 1 in 500. Determine the maximum discharge. 
 
 (20) An aqueduct consists partly of the section shown in Fig. 131, 
 page 217, and partly (i.e. when crossing valleys) of 33 inches diameter cast- 
 iron pipe siphons. 
 
 Determine the minimum slope of the siphons, so that the aqueduct 
 may discharge 15,000,000 gallons per day, and the slope of the masonry 
 aqueduct so that the water shall not be more than 4 feet 6 inches deep in 
 the aqueduct. 
 
 (21) Calculate the quantity delivered by the water main in question (30) , 
 page 172, per day of 24 hours. 
 
 This amount, representing the water supply of a city, is discharged into 
 the sewers at the rate of one-half the total daily volume in 6 hours, and is 
 then trebled by rainfall. Find the diameter of the circular brick outfall 
 sewer which will carry off the combined flow when running half full, the 
 available fall being 1 in 1500. Use Bazin's coefficient for brick channels. 
 
 (22) Determine for a smooth cylindrical cast-iron pipe the angle 
 subtended at the centre by the wetted perimeter, when the velocity of flow 
 is a maximum. Determine the hydraulic mean depth of the pipe under 
 these conditions. Lond. Un. 1905. 
 
 (23) A 9 -inch drain pipe is laid at a slope of 1 in 150, and the value of 
 c is 107 (v = c>Jmi). Find a general expression for the angle subtended at 
 the centre by the water line, and the velocity of flow; and indicate how the 
 general equations may be solved when the discharge is given. Lond. Un. 
 1906. 
 
 141. Short account of the historical development of the pipe and channel formulae. 
 It seems remarkable that, although the practice of conducting water along pipes 
 and channels for domestic and other purposes has been carried on for many 
 centuries, no serious attempt to discover the laws regulating the flow seems 
 to have been attempted until the eighteenth century. It seems difficult to realise 
 how the gigantic schemes of water distribution of the ancient cities could have been 
 executed without such knowledge, but certain it is, that whatever information they 
 possessed, it was lost during the middle ages. 
 
 It is of peculiar interest to note the trouble taken by the Roman engineers in 
 the construction of their aqueducts. In order to keep the slope constant they 
 tunnelled through hills and carried their aqueducts on magnificent arches. The 
 Claudian aqueduct was 38 miles long and had a constant slope of five feet per mile. 
 Apparently they were unaware of the simple fact that it is not necessary for a pipe 
 or aqueduct connecting two reservoirs to be laid perfectly straight, or else they 
 wished the water at all parts of the aqueducts to be at atmospheric pressure. 
 
 Stephen Schwetzer in his interesting treatise on hydrostatics and hydraulics 
 published in 1729 quotes experiments by Marriott showing that, a pipe 1400 yards 
 long, 1| inches diameter, only gave of the discharge which a hole If inches diameter 
 in the side of a tank would give under the same head, and also explains that the 
 motion of the liquid in the pipes is diminished by friction, but he is entirely 
 ignorant of the laws regulating the flow of fluids through pipes. Even as late as 
 
232 HYDRAULICS 
 
 1786 Du Buat* wrote, "We are yet in absolute ignorance of the laws to which the 
 movement of water is subjected." 
 
 The earliest recorded experiments of any value on long pipes are those of 
 Couplet, in which he measured the flow through the pipes which supplied the 
 famous fountains of Versailles in 1732. In 1771 Abbe" Bossut made experiments on 
 flow in pipes and channels, these being followed by the experiments of Du Buat, who 
 erroneously argued that the loss of head due to friction in a pipe was independent 
 of the internal surface of the pipe, and gave a complicated formula for the velocity 
 of flow when the head and the length of the pipe were known. 
 
 In 1775 M. Chezy from experiments upon the flow in an open canal, came to 
 the conclusion that the fluid friction was proportional to the velocity squared, and 
 that the slope of the channel multiplied by the cross sectional area of the stream, 
 was equal to the product of the length of the wetted surface measured on the cross 
 section, the velocity squared, and some constant, or 
 
 iK = Pav z (1), 
 
 i being the slope of the bed of the channel, A the cross sectional area of the stream, 
 P the wetted perimeter, and a a coefficient. 
 
 From this is deduced the well-known Chezy formula 
 
 Pronyf, applying to the flow of water in pipes the results of the classical experi- 
 ments of Coulomb on fluid friction, from which Coulomb had deduced the law that 
 fluid friction was proportional to av + bv 2 , arrived at the formula 
 
 = f- 
 
 This is similar to the Chezy formula, I - + j being equal to ^ . 
 
 By an examination of the experiments of Couplet, Bossut, and Du Buat, Prony 
 gave values to a and /3 which when transformed into British units are, 
 
 a 00001733, 
 p= -00010614. 
 
 For velocities, above 2 feet per second, Prony neglected the term containing the 
 first power of the velocity and deduced the formula 
 
 v = 48-6^/^77. 
 
 He continued the mistake of Du Buat and assumed that the friction was in- 
 dependent of the condition of the internal surface of the pipe and gave the following 
 explanation : " When the fluid flow? in a pipe or upon a wetted surface a film of 
 fluid adheres to the surface, and this film may be regarded as enclosing the mass 
 of fluid in motion J." That such a film encloses the moving water receives support 
 from the experiments of Professor Hele Shaw. The experiments were made upon 
 such a small scale that it is difficult to say how far the results obtained are indica- 
 tive of the conditions of flow in large pipes, and if the film exists it does not seem 
 to act in the way argued by Prony. 
 
 TT 
 
 The value of i in Prony's formula was equal to , H including, not only the 
 
 loss of head due to friction but, as measured by Couplet, Bossut and Du Buat, 
 it also included the head necessary to give velocity to the water and to overcome 
 resistances at the entrance to the pipe. 
 
 Eytelwein and also Aubisson, both made allowances for these losses, by sub- 
 tracting from H a quantity , and then determined new values for a and b in the 
 
 &9 
 formula 
 
 * Le Discours preliminaire de ses Principes d'hydr antique. 
 
 t See also Girard's Movement des fluids dans les tubes capillaires, 1817. 
 
 J Traite d'hydraulique. Engineer, Aug. 1897 and May 1898. 
 
FLOW IN OPEN CHANNELS 233 
 
 They gave to a and b the following values. 
 
 Eytelwein a = -000023584, 
 
 6 =-000085434. 
 
 Aubisson* a = -000018837, 
 
 6 =-000104392. 
 
 By neglecting the term containing v to the first power, and transforming the 
 terms, Aubisson's formula reduces to 
 
 I + 35-5d ' 
 
 Young, in the Encyclopaedia Britannica, gave a complicated formula for v when 
 H and d were known, but gave the simplified formula, for velocities such as 
 are generally met with in practice, 
 
 St 
 
 ^ 
 
 Venant made a decided departure by making - proportional to v~?~ instead of 
 
 to v 2 as in the Chezy formula. 
 
 When expressed in English feet as units, his formula becomes 
 
 Weisbach by an examination of the early experiments together with ten others by 
 himself and one by M. Gueynard gave to the coefficient a in the formula h = 
 
 771 
 
 the value 
 
 that is, he made it to vary with the velocity. 
 Then, mi 
 
 the values of a and ft being a = 0-0144, 
 
 j3 = 0-01716. 
 
 From this formula tables were drawn up by Weisbach, and in England by 
 Hawkesley, which were considerably used for calculations relating to flow of 
 water in pipes. 
 
 Darcy, as explained in Chapter V, made the coefficient a to vary with the 
 diameter, and Hagen proposed to make it vary with both the velocity and the 
 diameter. 
 
 His formula then became 
 
 The formulae of Ganguillet and Kutter and of Bazin have been given in 
 Chapters V and VI. 
 
 Dr Lampe from experiments on the Dantzig mains and other pipes proposed 
 the formula 
 
 thus modifying St Venant's formula and anticipating the formulae of Eeynolds, 
 Flamant and Unwin, in which, 
 
 n and p being variable coefficients. 
 
 * Traite d'hydraulique. 
 
CHAPTER VII. 
 
 GAUGING THE FLOW OF WATER. 
 
 142. Measuring the flow of water by weighing. 
 
 In the laboratory or workshop a flow of water can generally 
 be measured by collecting the water in tanks, and either by 
 direct weighing, or by measuring the volume from the known 
 capacity of the tank, the discharge in a given time can be 
 determined. This is the most accurate method of measuring 
 water and should be adopted where possible in experimental 
 work. 
 
 In pump trials or in measuring the supply of water to boilers, 
 determining the quantity by direct weighing has the distinct 
 advantage that the results are not materially affected by 
 changes of temperature. It is generally necessary to have two 
 tanks, one of which is filling while the other is being weighed 
 and emptied. For facility in weighing the tanks should stand 
 on the tables of weighing machines. 
 
 143. Meters. 
 
 Liner t meter. An ingenious direct weighing meter suitable for 
 gauging practically any kind of liquid, is constructed as shown in 
 Figs. 136 and 137. 
 
 It consists of two tanks A 1 and A 2 , each of which can swing 
 on knife edges BB. The liquid is allowed to fall into a shoot F, 
 which swivels about the centre J, and from which it falls into 
 either A 1 or A 2 according to the position of the shoot. The tanks 
 have weights D at one end, which are so adjusted that when a 
 certain weight of water has run into a tank, it swings over into 
 the dotted position, Fig. 136, and flow commences through a 
 siphon pipe C. When the level of the liquid in the tank has 
 fallen sufficiently, the weights D cause the tank to come back to 
 its original position, but the siphon continues in action until the 
 tank is empty. As the tank turns into the dotted position 
 
GAUGING THE FLOW OF WATER 
 
 235 
 
 it suddenly tilts over the shoot F, and the liquid is discharged 
 into the other tank. An indicator H registers the number of 
 times the tanks are filled, and as at each tippling a definite weight 
 of fluid is emptied from the tank, the indicator can be marked 
 off in pounds or in any other unit. 
 
 Fig. 136. Fig. 137. 
 
 Linert direct weighing meter. 
 
 144. Measuring the flow by means of an orifice. 
 
 The coefficient of discharge of sharp-edged orifices can be 
 obtained, with considerable precision, from the tables of Chapter IY, 
 or the coefficient for any given orifice can be determined for 
 various heads by direct measurement of the flow in a given time, 
 as described above. Then, knowing the coefficient of discharge at 
 various heads a curve of rate of discharge for the orifice, as in 
 Fig. 138, may be drawn, and the orifice can then be used to 
 measure a continuous flow of water. 
 
 The orifice should be made in the side or bottom of a tank. If 
 in the side of the tank the lower edge should be at least one and 
 a half to twice its depth above the bottom of the tank, and the 
 sides of the orifice whether horizontal or vertical should be at 
 least one and a half to twice the width from the sides of the tank. 
 The tank should be provided with baffle plates, or some other 
 arrangement, for destroying the velocity of the incoming water 
 and ensuring quiet water in the neighbourhood of the orifice. The 
 coefficient of discharge is otherwise indefinite. The head over the 
 orifice should be observed at stated intervals. A head-time curve 
 having head as ordinates and time as abscissae can then be plotted 
 as in Fig. 139. 
 
 From the head-discharge curve of Fig. 138 the rate of discharge 
 can be found for any head h, and the curve of Fig. 139 plotted. 
 The area of this curve between any two ordinates AB and CD, 
 
236 
 
 HYDRAULICS 
 
 which is the mean ordinate between AB and CD multiplied by the 
 time t, gives the discharge from the orifice in time t. 
 
 The head h can be measured by fixing a scale, having its zero 
 coinciding with the centre of the orifice, behind a tube on the side 
 of the tank. 
 
 Fig. 138. 
 
 B TUne D 
 
 Fig. 139. 
 
 
 B 
 
 A 
 
 
 E 
 
 D 
 
 
 
 ^xf 
 
 
 
 / x \ 
 
 A 
 
 Fig. 140. 
 
 145. Measuring the flow in open channels. 
 
 Large open channels : floats. The oldest and simplest method 
 of determining approximately the discharge in an open channel is 
 by means of floats. 
 
 A part of the channel as straight as possible is selected, and in 
 which the flow may be considered as uniform. 
 
 The readings should be taken on a calm day as a down-stream 
 wind will accelerate the floats and an up-stream wind retard them. 
 
 Two cords are stretched across the channel, as near to the 
 surface as possible, and perpendicular to the direction of flow. The 
 distance apart of the cords should be as great as possible consistent 
 with uniform flow, and should not be less than 150 feet. From a 
 boat, anchored at a point not less than 50 to 70 feet above stream, 
 so that the float shall acquire before reaching the first line a 
 uniform velocity, the float is allowed to fall into the stream and 
 
GAUGING THE FLOW OF WATER 237 
 
 the time carefully noted by means of a chronometer at which it 
 passes both the first and second line. If the velocity is slow, the 
 observer may walk along the bank while the float is moving from 
 one cord to the other, but if it is greater than 200 feet per minute 
 two observers will generally be required, one at each line. 
 
 A better method, and one which enables any deviation of the 
 float from a path perpendicular to the lines to be determined, is, 
 for two observers provided with box sextants, or theodolites, to be 
 stationed at the points A and B, which are in the planes of the 
 two lines. As the float passes the line AA at D, the observer 
 at A signals, and the observer at B measures the angle ABD 
 and, if both are provided with watches, each notes the time. 
 When the float passes the line BB at E, the observer at B signals, 
 and the observer at A measures the angle BAE, and both 
 observers again note the time. The distance DE can then be 
 accurately determined by calculation or by a scale drawing, and 
 the mean velocity of the float obtained, by dividing by the time. 
 
 To ensure the mean velocities of the floats being nearly equal 
 to the mean velocity of the particles of water in contact with 
 them, their horizontal dimensions should be as small as possible, 
 so as to reduce friction, and the portion of the float above the 
 surface of the water should be very small to diminish the effect of 
 the wind. 
 
 As pointed out in section 130, the distribution of velocity in 
 any transverse section is not by any means uniform and it is 
 necessary, therefore, to obtain the mean velocity on a number of 
 vertical planes, by finding not only the surface velocity, but also 
 the velocity at various depths on each vertical. 
 
 146. Surface floats. 
 
 Surface floats may consist of washers of cork, or wood, or 
 other small floating bodies, weighted so as to just project above 
 the water surface. The surface velocity is, however, so likely to 
 be aifected by wind, that it is better to obtain the velocity a 
 short distance below the surface. 
 
 147. Double floats. 
 
 To measure the velocity at points below the surface double 
 floats are employed. They consist of two bodies connected by 
 means of a fine wire or cord, the upper one being made as small 
 as possible so as to reduce its resistance. 
 
 Gordon*, on the Irrawaddi, used two wooden floats connected 
 by a fine fishing line, the lower float being a cylinder 1 foot long, 
 
 * Proc.Inst. C. E., 1893. 
 
238 HYDRAULICS 
 
 and 6 inches diameter, hollow underneath and loaded with clay to 
 sink it to any required depth ; the upper float, which swam on the 
 surface, was of light wood 1 inch thick, and carried a small flag. 
 
 The surface velocity was obtained by sinking the lower float 
 to a depth of 3J feet, the velocity at this depth being not very 
 different from the surface velocity and the motion of the float more 
 independent of the effect of the wind. 
 
 Fig. 141. Gurley's current meter. 
 
 Subsurface velocities were measured by increasing the depths 
 of the lower float by lengths of 3j feet until the bottom was 
 reached. 
 
GAUGING THE FLOW OF WATER 239 
 
 Gordon has compared the results obtained by floats with those 
 obtained by means of a current meter (see section 149). For 
 small depths and low velocities the results obtained by double 
 floats are fairly accurate, but at high velocities and great depths, 
 the velocities obtained are too high. The error is from to 10 
 per cent. 
 
 Double floats are sometimes made with two similar floats, of 
 the same dimensions, one of which is ballasted so as to float at any 
 required depth and the other floats just below the surface. The 
 velocity of the float is then the mean of the surface velocity 
 and the velocity at the depth of the lower float. 
 
 148. Rod floats. 
 
 The mean velocity, on any vertical, may be obtained ap- 
 proximately by means of a rod float, which consists of a long rod 
 having at the lower end a small hollow cylinder, which may be 
 filled with lead or other ballast so as to keep the rod nearly 
 vertical. 
 
 The rod is made sufficiently long, and the ballast adjusted, so 
 that its lower end is near to the bed of the stream, and its upper 
 end projects slightly above the water. Its velocity is approximately 
 the mean velocity in the vertical plane in which it floats. 
 
 149. The current meter. 
 
 The discharge of large channels or rivers can be obtained most 
 conveniently and accurately by determining the velocity of flow 
 at a number of points in a transverse section by means of a current 
 meter. 
 
 The arrangement shown in Fig. 141 is a meter of the anemo- 
 meter type. A wheel is mounted on a vertical spindle and has 
 five conical buckets. The spindle revolves in bearings, from 
 which all water is excluded, and which are carefully made so 
 that the friction shall remain constant. The upper end of the 
 spindle extends above its bearing, into an air-tight chamber, and 
 is shaped to form an eccentric. A light spring presses against 
 the eccentric, and successively makes and breaks an electric 
 circuit as the wheel revolves. The number of revolutions of the 
 wheel is recorded by an electric register, which can be arranged 
 at any convenient distance from the wheel. When the circuit is 
 made, an electro-magnet in the register moves a lever, at the end 
 of which is a pawl carrying forward a ratchet wheel one tooth 
 for each revolution of the spindle. The frame of the meter, which 
 is made of bronze, is pivoted to a hollow cylinder which can be 
 clamped in any desired position to a vertical rod. At the right- 
 
240 HYDRAULICS 
 
 hand side is a rudder having four light metal wings, which 
 balances the wheel and its frame. When the meter is being used 
 in deep waters it is suspended by means of a fine cable, and to 
 the lower end of the rod is fixed a lead weight. The electric 
 circuit wires are passed through the trunnion and so have no 
 tendency to pull the meter out of the line of current. When 
 placed in a current the meter is free to move about the horizontal 
 axis, and also about a vertical axis, so that it adjusts itself to 
 the direction of the current. 
 
 The meters are rated by experiment and the makers recommend 
 the following method. The meter should be attached to the bow 
 of a boat, as shown in Fig. 142, and immersed in still water not 
 less than two feet deep. A thin rope should be attached to the 
 boat, and passed round a pulley in line with the course in which 
 the boat is to move. Two parallel lines about 200 feet apart 
 should be staked on shore and at right angles to the course of the 
 boat. The boat should be without a rudder, but in the boat with 
 the observer should be a boatman to keep the boat from running 
 
 Fig. 142. 
 
 into the shore. The boat should then be hauled between the two 
 ranging lines at varying speeds, which during each passage should 
 be as uniform as possible. With each meter a reduction table is 
 supplied from which the velocity of the stream in feet per second 
 can be at once determined from the number of revolutions recorded 
 per second of the wheel. 
 
 The Haskell meter has a wheel of the screw propeller type 
 revolving upon a horizontal axis. Its mode of action is very 
 similar to the one described. 
 
 Comparative tests of the discharges along a rectangular canal 
 as measured by these two meters and by a sharp-edged weir which 
 had been carefully calibrated, in no case differed by more than 
 5 per cent, and the agreement was generally much closer*. 
 
 * Murphy on current Meter and Weir discharges, Proceedings Am.S.C.E., 
 Vol. xxvii., p. 779. 
 
GAUGING THE FLOW OF WATER 
 
 241 
 
 150. Pitot tube. 
 
 Another apparatus which can be used for determining the 
 velocity at a point in a flowing stream, even when the stream is of 
 small dimensions, as for example a small pipe, is called a Pitot 
 tube. 
 
 In its simplest form, as originally proposed by Pitot in 1732, 
 it consists of a glass tube, with a 
 small orifice at one end which may 
 be turned to receive the impact of 
 the stream as shown in Fig. 143. 
 The water in the tube rises to a 
 height h above the free surface of 
 the water, the value of h depending 
 upon the velocity v at the orifice of 
 the tube. If a second tube is placed 
 
 h 
 
 1^0 
 
 Fig. 143. Pitot tube. 
 
 beside the first with an orifice parallel to the direction of flow, 
 the water will rise in this tube nearly to the level of the free 
 surface, the fall hi being due to a slight diminution in pressure 
 at the mouth of the tube, caused probably by the stream lines, 
 having their directions changed at the mouth of the tube. A. 
 further depression of the free surface in the tube takes place,, 
 if the tube, as EF, is turned so that the orifice faces down stream.. 
 
 Theory of the Pitot tube. Let v be the velocity of the stream 
 at the orifice of the tube in ft. per sec. and a the area of the 
 orifice in sq. ft. 
 
 The quantity of water striking the orifice per second is wav 
 pounds. 
 
 w 
 
 The momentum is therefore - . a . v* pounds feet. 
 
 If the momentum of this water is entirely destroyed, the 
 pressure on the orifice which, according to Newton's second law of 
 motion is equal to the rate of change of momentum, is 
 
 -p _ 
 
 wav 
 
 and the pressure per unit area is 
 
 The equivalent head 
 
 , _ WIT _ V' 
 il/ . 
 WO Cf 
 
 According to this theory, the head of water in the tube, due to 
 
 v* 
 the impact, is therefore twice ~- > the head due to the velocity v, and 
 
 L. H. 
 
 16 
 
242 
 
 HYDRAULICS 
 
 the water should rise in the tube to a height above the surface 
 equal to h. 
 
 Experiment shows that the actual height the water rises in the 
 
 v 2 v 2 
 
 tube is more nearly equal to the velocity head ^- than to , 
 
 s & 
 
 and the head h is thus generally taken as 
 
 c being a coefficient for any given tube, which experiment shows 
 is fairly constant. 
 
 Similarly for given tubes 
 
 and 
 
 , 2 
 
 2 = ~ 
 
 The coefficients are determined by placing the tubes in streams 
 the velocities of which are known, or by attaching them to some 
 body which moves through still water with a known velocity, and 
 carefully measuring h for different velocities. 
 
 B 
 
 D 
 
 Fig. 144. 
 
 Fig. 145. 
 
 Darcy* was the first to use the Pitot tube as an instrument of 
 precision. His improved apparatus as ^used in open channels con- 
 sisted of two tubes placed side by side as in Fig. 144, the orifices 
 in the tubes facing up-stream and down-stream respectively. The 
 
 Eecherches Hydrauliques, etc., 1857. 
 
GAUGING THE FLOW OF WATER 243 
 
 two tubes were connected at the top, a cock C 1 being placed in the 
 common tube to allow the tubes to be opened or closed to the 
 atmosphere. At the lower end both tubes could be closed at the 
 same time by means of cock C. When the apparatus is put into 
 flowing water, the cocks C and C 1 being open, the free surface 
 rises in the tube B a height hi and is depressed in D an amount 
 Tia. The cock C 1 is then closed, and the apparatus can be taken 
 from the water and the difference in the level of the two columns, 
 
 h = hi + h%, 
 measured with considerable accuracy. 
 
 If desired, air can be aspirated from the tubes and the columns 
 made to rise to convenient levels for observation, without moving 
 the apparatus. The difference of level will be the same, whatever 
 the pressure in the upper part of the tubes. 
 
 Fig. 145 shows one of the forms of Pitot tubes, as experimented 
 upon by Professor Gardner Williams*, and used to determine 
 the distribution of velocities of the water flowing in circular pipes. 
 
 The arrangement shown in Fig. 146, is a modified form of the 
 apparatus used by Freeman t to determine the distribution of 
 velocities in a jet of water issuing from a fire hose under con- 
 siderable pressure. As shown in the sketch, the small orifice 
 receives the impact of the stream and two small holes Q are drilled 
 in the tube T in a direction perpendicular to the flow. The lower 
 part of the apparatus OY, as shown in the sectional plan, is made 
 boat-shaped so as to prevent the formation of eddies in the 
 neighbourhood of the orifices. The pressure at the orifice O is 
 transmitted through the tube OS, and the pressure at Q through 
 the tube QR. To measure the difference of pressure, or head, 
 in the two tubes, OS and QE. were connected to a differential 
 gauge, similar to that described in section 13 and very small 
 differences of head could thus be obtained with great accuracy. 
 
 The tube shown in Fig. 145 has a cigar-shaped bulb, the 
 impact orifice being at one end and communicating with the 
 tube OS. There are four small openings in the side of the bulb, 
 so that any variations of pressure outside are equalised in the 
 bulb. The pressures are transmitted through the tubes OS and 
 TR to a differential gauge as in the case above. 
 
 In Fig. 147 is shown a special stuffing-box used by Professor 
 Williams, to allow the tube to be moved to the various positions in 
 
 * For other forms of Pitot tubes as used by Professor Williams, E. S. Cole and 
 others, see Proceedings of the Am.S.C.E., Vol. xxvu. 
 t Transactions of the Am.S.C.E., Vol. xxi. 
 
 162 
 
244 
 
 HYDRAULICS 
 
 the cross section of a pipe, at which it was desired to determine 
 the velocity of translation of the water*. 
 
 Mr E. S. Colet has used the Pitot tube as a continuous meter, 
 the arrangement being shown in Fig. 148. The tubes were con- 
 nected to a U tube containing a mixture of carbon tetrachloride 
 and gasoline of specific gravity 1*25. The difference of level of 
 the two columns was registered continuously by photography. 
 
 To Gauge 
 
 Fig. 147. 
 
 Fig. 146. Fig. 148. 
 
 The tubes shown in Figs. 149 150, were used by Bazin to 
 determine the distribution of velocity in the interior of jets issuing 
 
 * See page 144. 
 
 t Proc. A.M.S.C.E., Vol. xxvu. See also experiments by Murphy and Torrance 
 in same volume. 
 
GAUGING THE FLOW OF WATER 
 
 245 
 
 from orifices, and in the interior of the nappes of weirs. Each 
 tube consisted of a copper plate 1'89 inches wide, by "1181 inch 
 thick, sharpened on the upper edge and having two brass tubes 
 "0787 inch diameter, soldered along the other edge, and having 
 orifices '059 inch diameter, 0'394 inch apart. The opening in tube 
 A was arranged perpendicular to the stream, and in B on the face 
 of the plate parallel to the stream. 
 
 Fig. 149. 
 
 151. Calibration of Pitot tubes. 
 
 Whatever the form of the Pitot tube, the head h can be 
 expressed as 
 
 or 
 
 k being called the coefficient of the tube. 
 
 This coefficient k must be determined by experiment under 
 conditions as near as possible like those under which the tube will 
 be used to determine velocities. 
 
 To calibrate the tubes used in the determination of the distri- 
 bution of velocities in open channels, Darcy * and Bazin used three 
 distinct methods. 
 
 (a) The tube was placed in front of a boat which was drawn 
 through still water at different velocities. The coefficient was 
 1'034. This was considered too large as the bow of the boat 
 probably tilted a little, as it moved through the water, thus tilting 
 the tube so that the orifice was not exactly vertical. 
 
 (6) The tube was placed in a stream, the velocity of which 
 was determined by floats. The coefficient was 1*006. 
 
 (c) Readings were taken at different points in the cross 
 section of a channel, the total flow Q through which was carefully 
 measured by means of a weir. The water section was divided 
 
 * Eecherches Hydrauliques. 
 
246 HYDRAULICS 
 
 into areas, and about the centre of each a reading of the tube 
 was taken. Calling a the area of one of these sections, and h the 
 reading of the tube, the coefficient 
 
 and was found to be '993. 
 
 Darcy* and Bazin also found that by changing the position of 
 the orifice in the pressure tube the coefficients changed con- 
 siderably. 
 
 Williams, Hubbell and Fenkell used two methods of calibration 
 which gave very different results. 
 
 The first method was to move the tubes through still water at 
 known velocities. For this purpose a circumferential trough, 
 rectangular in section, 9 inches wide and 8 inches deep was built of 
 galvanised iron. The diameter of its centre line, which was made 
 the path of the tube, was 11 feet 10 inches. The tube to be rated 
 was supported upon an arm attached to a central shaft which was 
 free to revolve in bearings on the floor and ceiling, and which also 
 supported the gauge and a seat for the observer. The gauge was 
 connected with the tube by rubber hose. The arm carrying the 
 tube was revolved by a man walking behind it, at as uniform a 
 rate as possible, the time of the revolution being taken by means 
 of a watch reading to -5- of a second. The velocity was main- 
 tained as nearly constant as possible for at least a period of 
 5 minutes. The value of k as determined by this method was '926 
 for the tube shown in Fig. 145. 
 
 In the second method adopted by these workers, the tube was 
 inserted into a brass pipe 2 inches in diameter, the discharge 
 through which was obtained by weighing. Readings were taken 
 at various positions on a diameter of the pipe, while the flow in the 
 pipe was kept constant. The values of */2gh, which may be called 
 the tube velocities, could then be calculated, and the mean value 
 V, of them obtained. It was found that, in the cases in which the 
 form of the tube was such that the volume occupied by it in the pipe 
 was not sufficient to modify the flow, the velocity was a maximum 
 at, or near, the centre of the pipe. Calling this maximum velocity 
 
 V c , the ratio ^p for a given set of readings was found to be '81. 
 V c 
 
 Previous experiments on a cast-iron pipe line at Detroit having 
 
 y 
 shown that the ratio ^ was practically constant for all velocities, 
 
 a similar condition was assumed to obtain in the case of the brass 
 
 * Recherches Hydrauliques. 
 
GAUGING THE FLOW OF WATER 247 
 
 pipe. The tube was then fixed at the centre of the pipe, and 
 readings taken for various rates of discharge, the mean velocity 
 U, as determined by weight, varying from to 6 feet per second. 
 
 For the values of Ji thus determined, it was found that , 
 
 was practically constant. This ratio was '729 for the tube shown 
 in Fig. 145. 
 
 Then since for any reading h of the tube, the velocity v is 
 
 the actual mean velocity U = kV m , 
 
 *= 
 T~T- 
 
 V m V c V m 
 
 Therefore 
 
 j ratio of U to Y c _ '729 _ ft 
 ratioofV m toV c ~~'814~ 
 
 For the tube shown in Fig. 146, some of the values of k as 
 determined by the two methods differed very considerably. 
 
 Comparison of the values of k by the two methods. It will be 
 seen that the value of k as determined by moving the tube through 
 still water differs very considerably from that obtained in 
 running water. In the latter case the pressure was considerably 
 higher than in the former, and it appears therefore, that k depends 
 not only upon the form of the tube but upon the pressure under 
 which it is working. It is, clearly, of considerable importance 
 that the value of k shall be determined for conditions similar 
 to those under which the tube is to be finally used. This 
 uncertainty of the value of the coefficient under varying con- 
 ditions of pressure, and the difficulty in any case of accurately 
 determining it, and the danger of its alteration by objects floating 
 in the stream, makes the use of the Pitot tube as a velocity 
 measurer somewhat uncertain, and it should be used with con- 
 siderable care. In the hands of Darcy and Bazin it proved an 
 excellent instrument in the measurement of small velocities in 
 open canals, but for the determination of velocities in closed 
 channels in which the pressure is greater, it does not seem so 
 reliable. 
 
 152. Gauging by a weir. 
 
 When a stream is so small that a barrier or dam can be easily 
 constructed across it, or when a large quantity of water is required 
 to be gauged in the laboratory, the flow can be determined by 
 means of a notch or weir. 
 
248 
 
 HYDRAULICS 
 
 The channel as it approaches the weir should be as far as 
 possible uniform in section, and it is desirable for accurate 
 gauging*, that the sides of the channel be made vertical, and the 
 width equal to the width of the weir. The sill should be sharp- 
 edged, and perfectly horizontal, and as high as possible above the 
 
 bed of the stream, and the down-stream channel 
 
 should be wider than the weir to ensure atmospheric 
 pressure under the nappe. The difference in level 
 of the sill and the surface of the water, before it 
 begins to slope towards the weir, should be ac- 
 curately measured. This is best done by a Boyden 
 hook gauge. 
 
 153. The hook gauge. 
 
 A simple form of hook gauge as made by Grurley 
 is shown in Fig. 151. In a rectangular groove formed 
 in a frame of wood, three or four feet long, slides 
 another piece of wood S to which is attached a scale 
 graduated in feet and hundredths, similar to a level 
 staff. To the lower end of the scale is connected a 
 hook H, which has a sharp point. At the upper end 
 of the scale is a screw T which passes through a lug, 
 connected to a second sliding piece L. This sliding 
 piece can be clamped to the frame in any position 
 by means of a nut, not shown. The scale can then 
 be moved, either up or down, by means of the milled 
 nut. A vernier Y is fixed to the frame by two small 
 screws passing through slot holes, which allow for a 
 slight adjustment of the zero. At some point a few 
 feet up-stream from the weir*, the frame can be 
 fixed to a post, or better still to the side of a box 
 from which a pipe runs into the stream. The level 
 of the water in the box will thus be the same as the 
 level in the stream. The exact level of the crest of 
 the weir must be obtained by means of a level and a 
 line marked on the box at the same height as the 
 crest. The slider L can be moved, so that the hook 
 point is nearly coincident with the mark, and the 
 final adjustment made by means of the screw T. 
 The vernier can be adjusted so that its zero is 
 coincident with the zero of the scale, and the slider 
 again raised until the hook approaches the surface of the water. 
 By means of the screw, the hook is raised slowly, until, by piercing 
 
 * See section 82. 
 
GAUGING THE FLOW OF WATER 
 
 c 
 
 Fig. 152. Bazin's Hook Gauge. 
 
250 
 
 HYDRAULICS 
 
 the surface of the water, it causes a distortion of the light reflected 
 from the surface. On moving the hook downwards again very 
 slightly, the exact surface will be indicated when the distortion 
 disappears. 
 
 A more elaborate hook gauge, as used by Bazin for his experi- 
 mental work, is shown in Fig. 152. 
 
 For rough gaugings a post can be driven into the bed of the 
 channel, a few feet above the weir, until the top of the post is 
 level with the sill of the weir. The height of the water surface 
 
 Fig. 154. .Recording Apparatus Kent Venturi Meter. 
 
GAUGING THE FLOW OF WATER 
 
 251 
 
 above the top of the post can then be measured by any convenient 
 scale. 
 
 154. Gauging the flow in pipes; Venturi meter. 
 
 Such methods as already described are inapplicable to the 
 measurement of the flow in pipes, in which it is necessary that 
 there shall be no discontinuity in the flow, and special meters have 
 accordingly been devised. 
 
 For large pipes, the Yenturi meter, Fig. 153, is largely used in 
 America, and is coming into favour in this country. 
 
 The theory of the meter has already been discussed (p. 44), 
 and it was shown that the discharge is proportional to th square 
 root of the difference H of the head at the throat and the bend in 
 the pipe, or 
 
 Q= *^^ 
 
 V Oj\ Q," 
 
 fc* being a coefficient. 
 
 For measuring the pressure heads at the two ends of the cone, 
 Mr W. Gr. Kent uses the arrangement shown in Fig. 154. 
 
 Fig. 155. Recording drum of the Kent Venturi Meter. 
 * See page 46. 
 
252 
 
 HYDRAULICS 
 
 The two pressure tubes from the meter are connected to a U tube 
 consisting of two iron cylinders containing mercury. Upon the 
 surface of the mercury in each cylinder is a float made of iron and 
 vulcanite; these floats rise or fall with the surfaces of the mercury. 
 
 Fig. 156. Integrating drum of the Kent Venturi Meter. 
 
 When no water is passing through the meter, the mercury in the 
 two cylinders stands at the same level. When flow takes place 
 the mercury in the left cylinder rises, and that in the right 
 cylinder is depressed until the difference of level of the surfaces 
 
GAUGING THE 1 FLOW OF WATER 
 
 253 
 
 TT 
 
 of the mercury is equal to , s being the specific gravity of the 
 
 s 
 
 mercury and H the difference of pressure head in the two 
 cylinders. The two tubes are equal in diameter, so that the rise 
 in the one is exactly equal to the fall in the other, and the move- 
 ment of either rack is proportional to H. The discharge is 
 proportional to \/H, and arrangements are made in the recording 
 apparatus to make the revolutions of the counter proportional to 
 \/H. To the floats, inside the cylinders, are connected racks, as 
 shown in Fig. 154, gearing with small pinions. Outside the 
 mercury cylinders are two other racks, to each of which vertical 
 motion is given by a pinion fixed to the same spindle as the pinion 
 gearing with the rack in the cylinder. The rack outside the left 
 cylinder has connected to it a light pen carriage, the pen of which 
 
 Ci 
 
 CE-nv/W 
 
 Fig. 157. Kent Venturi Meter. Development of Integrating drum. 
 
 makes a continuous record on the diagram drum shown in 
 Fig. 155. This drum is rotated at a uniform rate by clockwork, 
 and on suitably prepared paper a curve showing the rate of 
 discharge at any instant is thus recorded. The rack outside the 
 right cylinder is connected to a carriage, the function of which is 
 to regulate the rotations of the counter which records the total 
 flow. Concentric with the diagram drum shown in Fig. 155, and 
 within it, is a second drum, shown in Fig. 156, which also rotates 
 at a uniform rate. Fig. 157 shows this internal drum developed. 
 The surface of the drum below the parabolic curve FEG- is recessed. 
 If the right-hand carriage is touching the drum on the recessed 
 
254 
 
 HYDRAULICS 
 
 portion, the counter gearing is in action, but is put out of action 
 when the carriage touches the cylinder on the raised portion 
 above FGL Suppose the mercury in the right cylinder to fall a 
 height proportional to H, then the carriage will be in contact 
 with the drum, as the drum rotates, along the line CD, but the 
 recorder will only be in operation while the carriage is in 
 contact along the length CE. Since FGr is a parabolic curve the 
 fraction of the circumference CE = ra . \/H, m being a constant, 
 and therefore for any displacement H of the floats the counter for 
 each revolution of the drum will be in action for a period propor- 
 tional to \/H. When the float is at the top of the right cylinder, 
 the carriage is at the top of the drum, and in contact with the 
 raised portion for the whole of a revolution and no flow is 
 registered. When the right float is in its lowest position the 
 carriage is at the bottom of the drum, and flow is registered 
 during the whole of a revolution. The recording apparatus can 
 be placed at any convenient distance less than 1000 feet from 
 the meter, the connecting tubes being made larger as the distance 
 is increased. 
 
 155. Deacon's waste-water meter. 
 
 An ingenious and very simple meter designed by Mr Gr. F. 
 Deacon principally for detecting the leakage of water from pipes 
 is as shown in Fig. 158. 
 
 Fig. 158. Deacon waste-water meter. 
 
 The body of the meter which is made of cast-iron, has fitted 
 into it a hollow cone C made of brass. A disc D of the same diameter 
 as the upper end of the cone is suspended in this cone by means of 
 a fine wire, which passes over a pulley not shown ; the other end 
 of the wire carries a balance weight. 
 
GAUGING THE FLOW OF WATER 255 
 
 When no water passes through the meter the disc is drawn to 
 the top of the cone, but when water is drawn through, the disc is 
 pressed downwards to a position depending upon the quantity of 
 water passing. A pencil is attached to the wire, and the motion 
 of the disc can then be recorded upon a drum made to revolve by 
 clockwork. The position of the pencil indicates the rate of flow 
 passing through the meter at any instant. 
 
 When used as a waste-water meter, it is placed in a by-pass 
 leading from the main, as shown diagrammatically in Fig. 159. 
 
 B 
 
 'S.V 
 
 A 
 
 xS.Y D 
 
 Fig. 159. 
 
 The valves A and B are closed and the valve C opened. The 
 rate of consumption in the pipe AD at those hours of the night 
 when the actual consumption is very small, can thus be deter- 
 mined, and an estimate made as to the probable amount wasted. 
 
 If waste is taking place, a careful inspection of the district 
 supplied by the main AD may then be made to detect where the 
 waste is occurring. 
 
 156. Kennedy's meter. 
 
 This is a positive meter in which the volume of water passing 
 through the meter is measured by the displacement of a piston 
 working in the measuring cylinder. 
 
 The long hollow piston P, Fig. 157, fits loosely in the cylinder 
 Co, but is made water-tight by means of a cylindrical ring of 
 rubber which rolls between the piston and the inside of the 
 cylinder, the friction being thus reduced to a minimum. At each 
 end of the cylinder is a rubber ring, which makes a water-tight 
 joint when the piston is forced to either end of the cylinder, so 
 that the rubber roller has only to make a joint while the piston is 
 free to move. 
 
 The water enters the meter at A, Fig. 1616, and for the 
 position shown of the regulating cock, it flows down the passage 
 D and under the piston. The piston rises, and as it does so the 
 rack R turns the pinion S, and thus the pinion p which is keyed 
 to the same spindle as S. This spindle also carries loosely 
 a weighted lever W, which is moved as the spindle revolves by 
 either of two projecting fingers. As the piston continues to 
 ascend, the weighted lever is moved by one of the fingers until its 
 
256 
 
 HYDRAULICS 
 
 centre of gravity passes the vertical position, when it suddenly 
 falls on to a buffer, and in its motion moves the lever L, which 
 turns the cock, Fig. 161 b, into a position at right angles to that 
 
 Rubber Seating 
 
 -Rubber Rolling 
 Packing 
 
 Rubber Seating 
 
 Fig. 160. 
 
GAUGING THE FLOW OF WATER 
 
 257 
 
 shown. The water now passes from A through the passage C, 
 and thus to the top of the cylinder, and as the piston descends, 
 
 Fig. 161 a. 
 
 Fig. 161 &. 
 
 L. H. 
 
 17 
 
258 
 
 HYDRAULICS 
 
 the water that is below it passes to the outlet B. The motion of 
 the pinion S is now reversed, and the weight W lifted until it 
 again reaches the vertical position, when it falls, bringing the 
 cock C into the position shown in the figure, and another up 
 
 S 
 
 Fig. 161 c. 
 
 stroke is commenced. The oscillations of the pinion p are trans- 
 ferred to the counter mechanism through the pinions pi and p 2 , 
 Fig. 161 a, in each of which is a ratchet and pawl. The counter 
 is thus rotated in the same direction whichever way the piston 
 moves. 
 
 157. Gauging the flow of streams by chemical means. 
 
 Mr Stromeyer* has very successfully gauged the quantity of 
 water supplied to boilers, and also 
 the flow of streams by mixing 
 with the stream during a definite 
 time and at a uniform rate, a 
 known quantity of a concentrated 
 solution of some chemical, the 
 presence of which in water, even 
 in very small quantities, can be 
 easily detected by some sensitive 
 reagent. Suppose for instance 
 water is flowing along a small 
 stream. Two stations at a known 
 distance apart are taken, and the 
 time determined which it takes 
 the water to traverse the dis- 
 tance between them. At a stated 
 time, by means of a special ap- 
 paratus Mr Stromeyer uses the 
 arrangement shown in Fig. 162 
 sulphuric acid, say, of known 
 strength, is run into the stream at a known rate, at the upper 
 
 Jv 
 
 i 
 
 Fig. 162. 
 
 * Transactions of Naval Architects, 1896 ; Proceedings Inst. C.E., Vol. CLX. 
 
GAUGING THE FLOW OF WATER 259 
 
 station. While the acid is being put into the stream, a small 
 distance up-stream from where the acid is introduced samples of 
 water are taken at definite intervals. At the lower station 
 sampling is commenced, at a time, after the insertion of the 
 acid at the upper station is started, equal to that required by the 
 water to traverse the distance between the stations, and samples 
 are then taken, at the same intervals, as at the upper station. 
 The quantity of acid in a known volume of the samples taken 
 at the upper and lower station is then determined by analysis. 
 In a volume V of the samples, let the difference in the amount of 
 sulphuric acid be equivalent to a volume v of pure sulphuric 
 acid. If in a time t, a volume Y of water, has flowed down the 
 stream, and there has been mixed with this a volume v of pure 
 sulphuric acid, then, if the acid has mixed uniformly with the 
 water, the ratio of the quantity of water flowing down the stream 
 to the quantity of acid put into the stream, is the same as the 
 ratio of the volume of the sample tested to the difference of the 
 volume of the acid in the samples at the two stations, or 
 
 V = V 
 
 V V ' 
 
 Mr Stromeyer considers that the flow in the largest rivers can 
 be determined by this method within one per cent, of its true value. 
 
 In large streams special precautions have to be taken in 
 putting the chemical solution into the water, to ensure a uniform 
 mixture, and also special precautions must be adopted in taking 
 samples. 
 
 For other important information upon this interesting method 
 of measuring the flow of water the reader is referred to the two 
 papers cited above. 
 
 An apparatus for accurately gauging the flow of the solution 
 is shown in Fig. 162. The chemical solution is delivered into 
 a cylindrical tank by means of a pipe I. On the surface of the 
 solution floats a cork which carries a siphon pipe SS, and a balance 
 weight to keep the cork horizontal. After the flow has been 
 commenced, the head h above the orifice is clearly maintained 
 constant, whatever the level of the surface of the solution in the 
 tank. 
 
 172 
 
260 
 
 HYDRAULICS 
 
 EXAMPLES. 
 
 (1) Some observations are made by towing a current meter, with the 
 following results : 
 
 Speed in ft. per sec. Bevs. of meter per min. 
 1 80 
 
 5 560 
 
 Find an equation for the meter. 
 
 (2) Describe two methods of gauging a large river, from observations 
 in vertical and horizontal planes; and state the nature of the results 
 obtained. 
 
 If the cross section of a river is known, explain how the approximate 
 discharge may be estimated by observation of the mid-surface velocity 
 alone. 
 
 (3) The following observations of head and the corresponding discharge 
 were made in connection with a weir 6'53 feet wide. 
 
 Head in feet O'l 
 
 Discharge in cubic feet per 
 
 sec. per foot width 
 
 0-17 
 
 0-5 
 1-2 
 
 1-0 
 
 3-35 
 
 1-5 
 
 6-1 
 
 2-0 
 9-32 
 
 2-5 
 
 13-03 
 
 3-0 
 17-03 
 
 3-5 
 
 21-54 
 
 4-0 
 
 26-4 
 
 Assuming the law connecting the head h with the discharge Q as 
 
 find m and n. (Plot logarithms of Q and h.) 
 
 (4) The following values of Q and h were obtained for a sharp -edge 
 weir 6'53 feet long, without lateral contraction. Find the coefficient of 
 discharge at various heads. 
 
 Head ft ... -1 1-4 
 Q per foot- 
 length ... -17 | -87 
 
 6 | -8 
 1-56 2-37 
 
 1-0 
 3-35 
 
 2-0 
 9-32 
 
 2-5 
 13-03 
 
 3-0 
 17-03 
 
 3-5 
 21-54 
 
 4-0 
 26-4 
 
 4-5 
 31-62 
 
 5-0 
 37-09 
 
 5-5 
 42-81 
 
 (5) The following values of the head over a weir 10 feet long were 
 obtained at 5 minutes intervals. 
 
 Head in feet '35 -36 '37 '37 '38 -39 -40 -41 -42 -40 -39 -41 
 Taking the coefficient of discharge C as 3'36, find the discharge in 
 one hour. 
 
 (6) A Pitot tube was calibrated by moving it through still water in a 
 tank, the tube being fixed to an arm which was made to revolve at 
 constant speed about a fixed centre. The following were the velocities of 
 the tube and the heads measured in inches of water. 
 
 Velocities ft. per sec. 1-432 
 Head in inches 
 
 of water -448 
 
 1-738 
 
 663 
 
 2-275 
 1-02 
 
 2-713 
 
 1-69 
 
 3-235 
 2-07 
 
 3-873 
 
 2-88 
 
 4-983 
 5-40 
 
 5-584 
 6-97 
 
 6-142 
 8-51 
 
 Determine the coefficient of the tube. 
 
 For examples on Venturi meters see Chapter II. 
 
CHAPTER VIII. 
 
 IMPACT OF WATER ON VANES. 
 
 158. Definition of a vector. A right line AB, considered as 
 having not only length, but also direction, and sense, is said to be 
 a vector*. The initial point A is said to be the origin. 
 
 It is important that the difference between sense and direction 
 should be clearly recognised. 
 
 Suppose for example, from any point A, a line AB of 
 definite length is drawn in a northerly direction, then the 
 direction of the line is either from south to north or north to 
 south, but the sense of the vector is definite, and is from A to B, 
 that is from south to north. 
 
 The vector AB is equal in magnitude to the vector BA, but 
 they are of opposite sign or, 
 
 AB = -BA. 
 
 The sense of the vector is indicated by an arrow, as on AB, 
 Fig. 163. 
 
 Any quantity which has magnitude, direction, and sense, may 
 be represented by a vector. 
 
 B 
 
 Fig. 163. 
 
 For example, a body is moving with a given velocity in a 
 given direction, sense being now implied. Then a line AB drawn 
 parallel to the direction of motion, and on some scale equal in 
 
 * Sir W. Hamilton, Quaternions. 
 
262 
 
 HYDRAULICS 
 
 length to the velocity of the body is the velocity vector ; the sense 
 is from A to B. 
 
 159. * Sum of two vectors. 
 
 If a and ft Fig. 163, are two vectors the sum of these vectors 
 is found, by drawing the vectors, so that the beginning of ft is at 
 the end of a, and joining the beginning of a to the end of ft 
 Thus 7 is the vector sum of a and ft. 
 
 160. Resultant of two velocities. 
 
 When a body has impressed upon it at any instant two 
 velocities, the resultant velocity of the body in magnitude and 
 direction is the vector sum of the two impressed velocities. This 
 may be stated in a way that is more definitely applicable to the 
 problems to be hereafter dealt with, as follows. If a body is 
 moving with a given velocity in a given direction, and a second 
 velocity is impressed upon the body, the resultant velocity is the 
 vector sum of the initial and impressed velocities. 
 
 Example. Suppose a particle of water to be moving along a vane DA, Fig. 164, 
 with a velocity V r , relative to the vane. 
 
 If the vane is at rest, the particle will leave it at A with this velocity. 
 
 If the vane is made to move in the direction EF with a velocity v, and the 
 particle has still a velocity V r relative to the vane, and remains in contact with the 
 vane until the point A is reached, the velocity of the water as it leaves the vane at 
 A, will be the vector sum y of a and /3, i.e. of V r and v, or is equal to u. 
 
 161. Difference of two vectors. 
 
 The difference of two vectors a and ft is found by drawing both 
 vectors from a common origin A, and joining the end of ft to the 
 end of a. Thus, CB, Fig. 165, is the difference of the two vectors 
 a and ft or y = a ft and BC is equal to ft - a, or ft - a = - y. 
 
 162. Absolute velocity. 
 
 By the terms " absolute velocity " or " velocity " without the 
 adjective, as used in this chapter, it should be clearly understood, 
 is meant the velocity of the moving water relative to the earth, or 
 to the fixed part of any machine in which the water is moving. 
 
 * Henrici and Turner, Vectors and Rotors. 
 
IMPACT OF WATER ON VANES 263 
 
 To avoid repetition of the word absolute, the adjective is 
 frequently dropped and " velocity " only is used. 
 
 163. When a body is moving with a velocity U, Fig. 166, in 
 any direction, and has its velocity changed to U' in any other 
 direction, by an impressed force, the change in velocity, or the 
 velocity that is impressed on the body, is the vector difference of 
 the final and the initial velocities. If AB is U, and AC, U', the 
 impressed velocity is BC. 
 
 By Newton's second law of motion, the resultant impressed 
 force is in the direction of the change of velocity, and if W is the 
 weight of the body in pounds and t is the time taken to change 
 the velocity, the magnitude of the impressed force is 
 
 W 
 
 P = (change of velocity) Ibs. 
 Qt 
 
 This may be stated more generally as follows. 
 The rate of change of momentum, in any direction, is equal to 
 the impressed force in that direction, or 
 
 W dt?,, 
 P = . JT Ibs. 
 g at 
 
 In hydraulic machine problems, it is generally only necessary 
 to consider the change of momentum of the mass of water that 
 acts upon the machine per second. W in the above equation then 
 becomes the weight of water per second, and t being one second, 
 
 W 
 
 P = - - (change of velocity). 
 
 164. Impulse of water on vanes. 
 
 It follows that when water strikes a vane which is either 
 moving or at rest, and has its velocity changed, either in magni- 
 tude or direction, pressure is exerted on the vane. 
 
 As an example, suppose in one second a mass of water, weighing 
 W Ibs. and moving with a velocity U feet per second, strikes a 
 fixed vane AD, and let it glide upon the vane at A, Fig. 167, and 
 leave at D in a direction at right angles to its original direction 
 of motion. The velocity of the water is altered in direction but 
 not in magnitude, the original velocity being changed to a velocity 
 at right angles to it by the impressed force the vane exerts upon 
 the water. 
 
 The change of velocity in the direction AC is, therefore, 
 
 W 
 
 equal to U, and the change of momentum per second is . U 
 
 foot Ibs. 
 
264 
 
 HYDRAULICS 
 
 Since W Ibs. of water strike the vane per second, the pressure 
 P, acting in the direction CA, required to hold the vane in position 
 is, therefore, 
 
 Fig. 167. 
 
 Again, the vane has impressed upon the water a velocity U in 
 the direction DF which it originally did not possess. 
 The pressure PI in the direction DF is, therefore, 
 
 W 
 
 P 1 = P = -11.TJ. 
 9 
 
 The resultant reaction of the vane in magnitude and direction 
 is, therefore, R the resultant of P and P a . 
 
 This resultant force could have been 
 found at once by finding the resultant c 
 change in velocity. Set out ac, Fig. 168, 
 equal to the initial velocity in magnitude 
 and direction, and ad equal to the final x 
 velocity. The change in velocity is the 
 vector difference cd, or cd is the velocity 
 that must be impressed on a particle of 
 water to change its velocity from ac to 
 ad. 
 
 Fig. 168. 
 
 The impressed velocity cd is Y = \/U 2 + U 2 , and the total 
 impressed force is 
 
IMPACT OF WATER ON VANES 265 
 
 It at once follows, that if a jet of water strikes a fixed plane 
 perpendicularly, with a velocity U, and glides along the plane, the 
 
 W 
 
 normal pressure on the plane is U. 
 
 y 
 
 Example. A stream of water 1 sq. foot in section and having a velocity of 
 10 feet per second glides on to a fixed vane in a direction making an angle of 
 30 degrees with a given direction AB. 
 
 The vane turns the jet through an angle of 90 degrees. 
 
 Find the pressure on the vane in the direction parallel to AB and the resultant 
 pressure on the vane. 
 
 In Fig. 167, AC is the original direction of the jet and DF the final direction. 
 The vane simply changes the direction of the water, the final velocity being equal 
 to the initial velocity. 
 
 The vector triangle is acd, Fig. 168, ac and ad being equal. 
 
 The change of velocity in magnitude and direction is cd, the vector difference of 
 ad and ac ; resolving cd parallel to, and perpendicular to AB, ce is the change of 
 velocity parallel to AB. 
 
 Scaling off ce and calling it v a , the force to be applied along B A to keep the 
 vane at rest is, 
 
 But cd=*j2.10 
 
 and ce = cdcos!5 
 
 = J% . 10 .0-9659 ; 
 
 10 x 62-4 . 
 
 therefore, PBA = . x 13> 
 
 oa'a 
 
 = 2641bs. 
 The pressure normal to AB is 
 
 9 
 = V2.10sinl5=721bs. 
 
 The resultant is 
 
 165. Relative velocity. 
 
 Before going on to the consideration of moving vanes it is 
 important that the student should have clear ideas as to what is 
 meant by relative velocity. 
 
 A train is said to have a velocity of sixty miles an hour when, 
 if it continued in a straight line at a constant velocity for one 
 hour, it would travel sixty miles. What is meant is that the train 
 is moving at sixty miles an hour relative to the earth. 
 
 If two trains run on parallel lines in the same direction, one 
 at sixty and the other at forty miles an hour, they have a 
 relative velocity to each other of 20 miles an hour. If they move 
 in opposite directions, they have a relative velocity of 100 miles 
 an hour. If one of the trains T is travelling in the direction AB, 
 Fig. 169, and the other T a in the direction AC, and it be supposed 
 that the lines on which they are travelling cross each other at A, 
 
266 
 
 HYDRAULICS 
 
 and the trains are at any instant over each other at A, at the end 
 
 of one minute the two trains will be at B and C respectively, at 
 
 distances of one mile and two-thirds of a 
 
 mile from A. Relatively to the train T 
 
 moving along AB, the train TI moving 
 
 along AC has, therefore, a velocity equal 
 
 to BC, in magnitude and direction, and 
 
 relatively to the train T a the train T has 
 
 a velocity equal to CB. But AB and AC 
 
 may be taken as the vectors of the two 
 
 velocities, and BC is the vector difference 
 
 of AC and AB, that is, the velocity of T 
 
 vector difference of AC and AB. 
 
 T, 
 
 Fig. 169. 
 relative to T is the 
 
 166. Definition of relative velocity as a vector. 
 
 If two bodies A and B are moving with given velocities v and 
 Vi in given directions, the relative velocity of A to B is the vector 
 difference of the velocities v and Vi. 
 
 Thus when a stream of water strikes a moving vane the 
 magnitude and direction of the relative velocity of the water and 
 the vane is the vector difference of the velocity of the water and 
 the edge of the vane where the water meets it. 
 
 167. To find the pressure on a moving vane, and the 
 rate of doing work. 
 
 A jet of water having a velocity U strikes a flat vane, the 
 plane of which is perpendicular to the direction of the jet, and 
 which is moving in the same direction as the jet with a velocity v. 
 
 ..*__ 
 Fig. 170 
 
 Fig. 171. 
 
 The relative velocity of the water and the vane is \J v, the 
 vector difference of U and v, Fig. 170. If the water as it strikes 
 the vane is supposed to glide along it as in Fig. 171, it will do 
 
IMPACT OF WATER ON VANES 267 
 
 so with a velocity equal to (U 1?), and as it moves with the vane 
 it will still have a velocity v in the direction of motion of the 
 vane. Instead of the water gliding along the vane, the velocity 
 ~U-v may be destroyed by eddy motions, but the water will still 
 have a velocity v in the direction of the vane. The change in 
 velocity in the direction of motion is, therefore, the relative 
 velocity U-v, Fig. 170. 
 
 For every pound of water striking the vane, the horizontal 
 
 U 1 
 
 change in momentum is - , and this equals the normal pressure 
 
 y 
 
 P on the vane, per pound of water striking the vane. 
 The work done per second per pound is 
 
 g 
 
 The original kinetic energy of the jet per pound of water 
 
 IT 2 
 
 striking the vane is ~ , and the efficiency of the vane is, therefore, 
 
 _ < 2v(U-v) 
 
 U 2 
 
 which is a maximum when v is JU, and e = \. An application of 
 such vanes is illustrated in Fig. 185, page 292. 
 
 Nozzle and single vane. Let the water striking a vane issue 
 from a nozzle of area a, and suppose that there is only one vane. 
 
 Let the vane at a given instant be supposed at A, Fig. 172. At 
 the end of one second the front of the jet, if perfectly free to 
 move, would have arrived at B and the vane at C. Of the water 
 that has issued from the jet, therefore, only the quantity BC will 
 have hit the vane. 
 
 Fig. 172. 
 The discharge from the nozzle is 
 
 and the weight that hits the vane per second is 
 
 U 
 
 The change of momentum per second is 
 
 U 
 
268 HYDKAULICS 
 
 and the work done is, therefore, 
 
 Or the work done per Ib. of water issuing from the nozzle is 
 
 This is purely a hypothetical case and has no practical 
 importance. 
 
 Nozzle and a number of vanes. If there are a number of 
 vanes closely following each other, the whole of the water issuing 
 from the nozzle hits the vanes, and the work done is 
 
 W(U-v)v 
 
 The efficiency is 
 
 9 
 
 2v (U - v) 
 U 2 
 
 and the maximum efficiency is -|. 
 
 It follows that an impulse water wheel, with radial blades, as 
 in Fig. 185, cannot have an efficiency of more than 50 per cent. 
 
 168. Impact of water on a vane when the directions of 
 motion of the vane and jet are not parallel. 
 
 Let U be the velocity of a jet of water and AB its direction, 
 Fig. 173. 
 
 A, 
 
 B 
 
 Fig. 173. 
 
 Let the edge A of the vane AC be moving with a velocity v ; 
 the relative velocity Y r of the water and the vane at A is DB. 
 From the triangle DAB it is seen that, the vector sum of the 
 velocity of the vane and the relative velocity of the jet and the 
 vane is equal to the velocity of the jet ; for clearly U is the vector 
 sum of v and V r . 
 
 If the direction of the tip of the vane at A is made parallel to 
 DB the water will glide on to the vane in exactly the same way 
 
IMPACT OF WATER ON VANES 2C9 
 
 as if it were at rest, and the water were moving in the direction 
 DB. This is the condition that no energy shall be lost by shock. 
 
 When the water leaves the vane, the relative velocity of the 
 water and the vane must be parallel to the direction of the 
 tangent to the vane at the point where it leaves, and it is equal to 
 the vector difference of the absolute velocity of the water, and 
 the vane. Or the absolute velocity with which the water leaves 
 the vane is the vector sum of the velocity of the tip of the vane 
 and the relative velocity of the water to the vane. 
 
 Let CG- be the direction of the tangent to the vane at C. Let 
 CE be Vi, the velocity of C in magnitude and direction, and let OF 
 be the absolute velocity Ui with which the water leaves the vane. 
 
 Draw EF parallel to CGr to meet the direction OF in F, then 
 the relative velocity of the water and the vane is EF, and the 
 velocity with which the water leaves the vane is equal to CF. 
 
 If Vi and the direction CGr are given, and the direction in which 
 the water leaves the vane is given, the triangle CEF can be 
 drawn, and CF determined. 
 
 If on the other hand v l is given, and the relative velocity v r is 
 given in magnitude and direction, CF can be found by measuring 
 off along EF the known relative velocity v r and joining CF. 
 
 If Vi and Ui are given, the direction of the tangent to the vane 
 is then, as at inlet, the vector difference of Ui and Vi . 
 
 It will be seen that when the water either strikes or leaves the 
 vane, the relative velocity of the water and the vane is the vector 
 difference of the velocity of the water and the vane, and the actual 
 velocity of the water as it leaves the vane is the vector sum of the 
 velocity of the vane and the relative velocity of the water and 
 the vane. 
 
 Example. The direction of the tip of the vane at the outer circumference of a 
 wheel fitted with vanes, makes an angle of 165 degrees with the direction of motion 
 of the tip of the vane. 
 
 The velocity of the tip at the outer circumference is 82 feet per second. 
 
 The water leaves the wheel in such a direction and with such a velocity that the 
 radial component is 13 feet per second. 
 
 Find the absolute velocity of the water in direction and magnitude and the 
 relative velocity of the water and the wheel. 
 
 To draw the triangle of velocities, set out AB equal to 82 feet, and make the 
 angle ABC equal to 15 degrees. BC is then parallel to the tip of the vane. 
 
 Draw EC parallel to AB, and at a distance from it equal to 13 feet and 
 intersecting BC in C. 
 
 Then AC is the vector sum of AB and BC, and is the absolute velocity of the 
 water in direction and magnitude. 
 
 Expressed trigonometrically 
 
 AC 2 =(82-13cotl5) 2 + 13 2 
 
 = 33-5 8 + 13 2 and AC = 36-7 ft. per sec. 
 
 sinBAC=i?=-354. 
 AU 
 
 Therefore BAG = 20 45'. 
 
270 
 
 HYDRAULICS 
 
 169. Conditions which the vanes of hydraulic machines 
 should satisfy. 
 
 In all properly designed hydraulic machines, such as turbines, 
 water wheels, and centrifugal pumps, in which water flowing in 
 a definite direction impinges on moving vanes, the relative velocity 
 of the water and the vanes should be parallel to the direction of 
 the vanes at the point of contact. If not, the water breaks into 
 eddies as it moves on to the vanes and energy is lost. 
 
 Again, if in such machines the water is required to leave the 
 vanes with a given velocity in magnitude and direction, it is only 
 necessary to make the tip of the vane parallel to the vector 
 difference of the given velocity with which the water is to leave 
 the vane and the velocity of the tip of the vane. 
 
 Example (1). A jet of water, Fig. 174, moves in a direction AB making an angle 
 of 30 degrees with the direction of motion AC of a vane moving in the atmosphere. 
 The jet has a velocity of 30 ft. per second and the vane of 15 ft. per second. To find 
 (a) the direction of the vane at A so that the water may enter without shock; (&) the 
 direction of the tangent to the vane where the water leaves it, so that the absolute 
 velocity of the water when it leaves the vane is in a direction perpendicular to AC ; 
 (c) the pressure on the vane and the work done per second per pound of water 
 striking the vane. Friction is neglected. 
 
 u, 
 
 CFiange> oCVeLodty in/ the 
 ofmotiori. 
 
 Fig. 174. 
 
 The relative velocity V r of the water and the vane at A is CB, and for no shock 
 the vane at A must be parallel to CB. 
 
 Since there is no friction, the relative velocity V r of the water and the vane 
 cannot alter, and therefore, the triangle of velocities at exit is ACD or FAjCj . 
 
 The point D is found, by taking C as centre and CB as radius and striking the 
 arc BD to cut the known direction AD in D. 
 
 The total change of velocity of the jet is the vector difference DB of the initial 
 and final velocities, and the change of velocity in the direction of motion is BE. 
 Calling this velocity V, the pressure exerted upon the vane in the direction of 
 motion is 
 
 Y 
 
 Ibs. per Ib. of water striking the vane. 
 
 y 
 
 The work done per Ib. is, therefore, ft. Ibs. and the efficiency, since there is 
 
 no loss by friction, or shock, is 
 
 Vv 2Vt; 
 
 T5*-1F' 
 
 9?r, 
 
IMPACT OF WATER ON VANES 271 
 
 The change in the kinetic energy of the jet is equal to the work done by the jet. 
 
 U 2 
 
 The kinetic energy per Ib. of the original jet is and the final kinetic energy is 
 
 *9 
 
 Hi! 
 
 2<7 ' 
 
 U 2 U, 2 
 The work done is, therefore, ^ -- -J- ft. Ibs. and the efficieny is 
 
 It can at once be seen from the geometry of the figure that 
 Vv = U 2 _ iy 
 
 g ~ 20 20 ' 
 
 For AB 2 =AC 2 + CB 2 + 2AC.CG, 
 
 and since CD = CB and CD-' = AC 2 + AD 2 , 
 
 therefore, AB 2 - AD 2 = 2 AC (AC + CG) 
 
 = 2uV. 
 
 But AB 2 -AD 2 = U 2 -U 1 2 , 
 
 IP-T vV 
 
 .u , - 
 
 therefore, 
 
 2< 
 
 If the water instead of leaving the vane in a direction perpendicular to v, leaves 
 it with a velocity U l having a component V l parallel to v, the work done on the 
 vane per pound of water is 
 
 9 
 
 If U l be drawn on the figure it will be seen that the change of velocity in the 
 
 direction of motion is now (V - VJ, the impressed force per pound is 1 , and 
 
 / V V \ 
 the work done is, therefore, ( l \ v l ft. Ibs. per pound. 
 
 As before, the work done on the vane is the loss of kinetic energy of the jet, and 
 therefore, 
 
 9 20 
 
 The work done on the vane per pound of water for any given value of Uj , is, 
 therefore, independent of the direction of Uj . 
 
 Example (2). A series of vanes such as AB, Fig. 175, are fixed to a (turbine) 
 wheel which revolves about a fixed centre C, with an angular velocity u. 
 
 The radius of B is K and of A, r. Within the wheel are a number of guide 
 passages, through which water is directed with a velocity U, at a definite inclination 
 B with the tangent to the wheel. The air is supposed to have free access to the 
 wheel. 
 
 To draw the triangles of velocity, at inlet and outlet, and to find the directions 
 of the tips of the vanes, so that the water moves on to the vanes without shock and 
 leaves the wheel with a given velocity Uj. Friction neglected. 
 
 As in the last example the velocity relative to the vane must remain constant, 
 and therefore, V r and v r are equal, but v and v l are unequal. 
 
 The tangent AH to the vane at A makes an angle < with the tangent AD to the 
 wheel, so that CD makes an angle <f> with AD. The triangle of velocities ACD at 
 inlet is, therefore, as shown in the figure and does not need explanation. 
 
 To draw the triangle of velocities at exit, set out BG equal to v l and perpen- 
 dicular to the radius BC, and with B and G as centres, describe circles with U l and 
 v r which is equal to V r as radii respectively, intersecting in E. Then GE is 
 parallel to the tangent to the vane at B. 
 
272 
 
 HYDRAULICS 
 
 If there is a loss of head, hf, by friction, as the water moves over the vane then 
 v r is less than V r , if hf is known, it can be found from 
 
 (See Impulse turbines.) 
 
 Work done on the wheel. Neglecting friction etc. the work done per pound of 
 water passing through the wheel, since the pressure is constant, being equal to the 
 atmospheric pressure, is the loss of kinetic energy of the water, and is 
 
 2 2 1 " ft * ^ S ' 
 
 The work done on the wheel can also be found from the consideration of the 
 change of the angular momentum of the water passing through the wheel. Before 
 going on however to determine the work per pound by this method, the notation 
 that has been used is summarised and several important principles considered. 
 
 Notation used in connection with vanes, turbines and centrifugal 
 pumps. Let U be the velocity with which the water approaches 
 the vane, Fig. 175, and v the velocity, perpendicular to the radius 
 AC, of the edge A of the vane at which water enters the wheel. 
 
 Let Y be the component of U in the direction of v, 
 
 u the component of U perpendicular to v, 
 
 V r the relative velocity of the water and vane at A, 
 
 Ui the velocity, perpendicular to BC, of the edge B of the vane 
 at which water leaves the wheel, 
 
 Ui the velocity with which the water leaves the wheel, 
 
 Yi the component of Ui in the direction of Vi, 
 
UNIVERSITY 
 
 OF 
 
 IMPACT OF WATER ON VANES 
 
 273 
 
 Ui the component of Ui perpendicular to Vj, or along BC, 
 v r the relative velocity of the water and the vane at B. 
 Velocities of whirl. The component velocities V and Vi are 
 
 called the velocities of whirl at inlet and outlet respectively. 
 
 This term will frequently be used in the following chapters. 
 
 170. Definition of angular momentum. 
 
 If a weight of W pounds is moving with a velocity U, Figs. 175 
 and 176, in a given direction, the perpendicular distance of which 
 is S feet from a fixed centre C, the angular momentum of W is 
 
 W 
 
 . U . S pounds feet. 
 
 171. Change of angular momentum. 
 
 If after a small time t the mass is moving with a velocity TJi in 
 a direction, which is at a perpendicular distance Si from C, the 
 
 W 
 
 angular momentum is now UiSij the change of angular 
 
 momentum in time t is 
 
 W 
 
 -(US-U.SO; 
 y 
 
 and the rate of change of angular momentum is 
 
 Fig. 176. Fig. 177. 
 
 172. Two important principles. 
 
 (1) Work done by a couple, or turning moment. When a 
 body is turned through an angle a measured in radians, under the 
 action of a constant turning moment, or couple, of T pounds feet, 
 the work done is Ta foot pounds. 
 
 If the body is rotating with an angular velocity o> radians 
 per second, the rate of doing work is T<o foot pounds per second, 
 
 and the horse-power is ^=7: . 
 
 L. II. 
 
 18 
 
274 HYDRAULICS 
 
 Suppose a body rotates about a fixed centre C, Fig. 177, and 
 a force P Ibs. acts on the body, the perpendicular distance from 
 C to the direction of P being S. 
 
 The moment of P about C is 
 
 T=P.S. 
 
 If the body turns through an angle w in one second, the 
 distance moved through by the force P is w . S, and the work 
 done by P in foot pounds is 
 
 P<oS=To>. 
 
 And since one horse-power is equivalent to 33,000 foot pounds 
 per minute or 550 foot pounds per second the horse-power is 
 
 HP- Ta> 
 -550- 
 
 (2) The rate of change of angular momentum of a body 
 rotating about a fixed centre is equal to the couple acting upon 
 the body. Suppose a weight of W pounds is moving at any instant 
 with a velocity U, Fig. 176, the perpendicular distance of which 
 from a fixed centre C is S, and that a couple is exerted upon W 
 so as to change its velocity from U to Ui in magnitude and 
 direction. 
 
 The reader may be helped by assuming the velocity U is 
 changed to Ui by a wheel such as that shown in Fig. 175. 
 
 Suppose now at the point A the velocity Ui is destroyed in a 
 time dt, then a force will be exerted at the point A equal to 
 
 P = W 5 
 
 ~ g 'ct> 
 and the moment of this force about C is P . S. 
 
 At the end of the time dt, let the weight W leave the wheel 
 with a velocity Ui. During this time dt the velocity Ui might 
 have been given to the moving body by a force 
 
 acting at the radius Si. 
 
 The moment of PI is PI Si ; and therefore if the body has been 
 acting on a wheel, Fig. 175, the reaction of the wheel thus exerting 
 the couple upon the body, or a couple has been exerted upon it in 
 any other way, the couple required to change the velocity of W 
 from U to Ui is 
 
 a). 
 
 Let the wheel of Fig. 175, or the couple which is acting upon 
 the body, have an angular velocity w. 
 
IMPACT OF WATER ON VANES 275 
 
 In a time dt the angle moved through by the couple is wd, 
 and therefore the work done in time dt is 
 
 W 
 
 UiS,) .................. (2). 
 
 Suppose now W is the weight of water in pounds per second 
 which strikes the vanes of a moving wheel of any form, and this 
 water has its velocity changed from U to Ui, then by making dt 
 in either equation (1) or (2) equal to unity, the work done per 
 second is 
 
 W 
 
 and the work done per second per pound of water entering the 
 wheel is 
 
 -(US-ILSO. 
 
 I/ 
 
 This result, as will be seen later (page 337), is entirely inde- 
 pendent of the change of pressure as the water passes through the 
 wheel, or of the direction in which the water passes. 
 
 173. Work done on a series of vanes fixed to a wheel 
 expressed in terms of the velocities of whirl of the water 
 entering and leaving the wheel. 
 
 Outward flow turbine. If water enters a wheel at the inner 
 circumference, as in Fig. 175, the flow is said to be outward. 
 On reference to the figure it is seen that since r is perpendicular 
 to Y, and S to U, therefore 
 
 r _ U 
 
 S~ V 
 
 and for a similar reason 
 
 R Ui 
 
 srvi- 
 
 Again the angular velocity of the wheel 
 
 V Vi 
 
 - = r = S' 
 
 therefore the work done per second is 
 
 W 
 
 -^(y-Vio, 
 
 i/ 
 
 and the work done per pound of flow is 
 
 9 9 
 
 Inward flow turbine. If the water enters at the outer cir- 
 cumference of a wheel with a velocity of whirl V, and leaves at 
 the inner circumference with a velocity of whirl Vi, the velocities 
 
 182 
 
276 
 
 HYDRAULICS 
 
 of the inlet and outlet tips of the vanes being v and 
 the work done on the wheel is still 
 
 V0 
 
 respectively 
 
 9 9 
 The flow in this case is said to be inward. 
 
 Parallel flow or axial flow turbine. If vanes, such as those 
 shown in Fig. 174, are fixed to a wheel, the flow is parallel to the 
 axis of the wheel, and is said to be axial. 
 
 For any given radius of the wheel, Vi is equal to v, and the 
 work done per pound is 
 
 which agrees with the result already found on page 271. 
 
 174. Curved vanes. Pelton wheel. 
 
 Let a series of cups, similar to Figs. 178 and 179, be moving 
 with a velocity v } and a stream with a greater velocity U in the 
 same direction. 
 
 The relative velocity is 
 
 Neglecting friction, the relative velocity Y r will remain con- 
 stant, and the water will, therefore, leave the cup at the point B 
 with a velocity, Y r , relative to the cup. 
 
 Fig. 178. 
 
 If the tip of the cup at B, Fig. 178, makes an angle with the 
 direction of v, the absolute velocity with which the water leaves 
 the cup will be the vector sum of v and Y r , and is therefore Ui . 
 The work done on the cups is then 
 
IMPACT OF WATER ON VANES 277 
 
 per Ib. of water, and the efficiency is 
 
 23 
 
 For Ui, the value 
 
 Uj - J{v - (U - v) cos 0} a + (U - v)* sin 6 2 
 
 can be substituted, and the efficiency thus determined in terms of 
 v, U and 0. 
 
 Pelton wheel cups. If is zero, as in Fig. 178, and U v is 
 equal to v, or U is twice v, Ui clearly becomes zero, and the water 
 drops away from the cup, under the action of gravity, without 
 possessing velocity in the direction of motion. 
 
 The whole of the kinetic energy of the jet is thus absorbed 
 and the theoretical efficiency of the cups is unity. 
 
 The work done determined from consideration of the change of 
 momentum. The component of Ui, Fig. 178, in the direction of 
 
 motion, is 
 
 v (U - v) cos 0, 
 
 and the change of momentum per pound of water striking the 
 vanes is, therefore, 
 
 U - v + (U - v) cos 
 
 9 
 
 The work done per Ib. is 
 
 9 
 and the efficiency is 
 
 -v+ (U -v) cos0} 
 
 
 U 2 
 9 -29 
 
 U 2 
 When 6 is 0, cos 6 is unity, and 
 
 _4a)(U-v) 
 U 
 
 which is a maximum, and equal to unity, when v is ^- . 
 
 ft 
 
 175. Force tending to move a vessel from which water 
 is issuing through an orifice. 
 
 When water issues from a vertical orifice of area a sq. feet, 
 in the side of a vessel at rest, in which the surface of the water is 
 maintained at a height h feet above the centre of the orifice, the 
 
278 HYDRAULICS 
 
 pressure on the orifice, or the force tending to move the vessel 
 in the opposite direction to the movement of the water, is 
 
 = 2w.a.h Ibs., 
 w being the weight of a cubic foot of water in pounds. 
 
 The vessel being at rest, the velocity with which the water 
 leaves the orifice, neglecting friction, is 
 
 and the quantity discharged per second in cubic feet is 
 
 Q = av. 
 The momentum given to the water per second is 
 
 , f w .a.v* 
 M = 
 
 9 
 
 = 2w .a.h. 
 
 But the momentum given to the water per second is equal to 
 the impressed force, and therefore the force tending to move the 
 vessel is 
 
 Y**2w.a.h, 
 
 or is equal to twice the pressure that would be exerted upon a 
 plate covering the orifice. When a fireman holds the nozzle of a 
 hose-pipe through which water is issuing with a velocity -u, there 
 is, therefore, a pressure on his hand equal to 
 
 2wav 2 _ wav 2 
 
 ~W~ ~9~ 
 
 If the vessel has a velocity V backwards, the velocity U of the 
 water relative to the earth is 
 
 U=t>-V, 
 and the pressure exerted upon the vessel is 
 
 \J 
 
 The work done per second is 
 
 = 
 
 g 
 
 per Ib. of flow from the nozzle. 
 
 The efficiency is e = Y (v ~ Y) 
 
 gfi 
 
 = 2V(i7-Y) 
 
 v 2 
 which is a maximum, when 
 
 v = 2V 
 and e = i. 
 
IMPACT OF WATER ON VANES 279 
 
 176. The propulsion of ships by water jets. 
 
 A method of propelling ships by means of jets of water issuing 
 from orifices at the back of the ship, has been used with some 
 success, and is still employed to a very limited extent, for the 
 propulsion of lifeboats. 
 
 Water is taken by pumps carried by the ship from that 
 surrounding the vessel, and is forced through the orifices. Let 
 v be the velocity of the water issuing from the orifice relative 
 
 to the ship, and Y the velocity of the ship. Then ^- is the 
 
 head h forcing water from the ship, and the available energy 
 per pound of water leaving the ship is h foot pounds. 
 
 The whole of this energy need not, however, be given to the 
 water by the pumps. 
 
 Imagine the ship to be moving through the water and having 
 a pipe with an open end at the front of the ship. The water in 
 front of the ship being at rest, water will enter the pipe with a 
 
 Y 2 
 
 velocity Y relative to the ship, and having a kinetic energy -~- 
 
 per pound. If friction and other losses are neglected, the work 
 that the pumps will have to do upon each pound of water to eject 
 it at the back with a velocity v is, clearly, 
 
 <tf Y 2 
 
 As in the previous example, the velocity of the water issuing 
 from the nozzles relative to the water behind the ship is v - Y, 
 
 V 
 
 and the change of momentum per pound is, therefore, - . If & 
 
 \y 
 
 is the area of the nozzles the propelling force on the ship is 
 
 and the work done is 
 
 
 
 The efficiency is the work done on the ship divided by the 
 
 work done by the engines, which equals wav ( ~- - ) and, 
 
 \^9 ^9/ 
 therefore, 
 
 _2YO-Y) 
 
 v 2 -Y 2 
 
 2Y 
 
280 HYDRAULICS 
 
 which can be made as near unity as is desired by making v and 
 Y approximate to equality. 
 
 But for a given area a of the orifices, and velocity v, the nearer 
 v approximates to Y the less the propelling force F becomes, and 
 the size of ship that can be driven at a given velocity Y for the 
 given area a of the orifices diminishes. 
 
 Et?is2V, e = f. 
 
 EXAMPLES. 
 
 (1) Ten cubic feet of water per second are discharged from a stationary 
 jet, the sectional area of which is 1 square foot. The water impinges nor- 
 mally on a flat surface, moving in the direction of the jet with a velocity 
 of 2 feet per second. Find the pressure on the plane in Ibs., and the work 
 done on the plane in horse-power. 
 
 (2) A jet of water delivering 100 gallons per second with a velocity of 
 20 feet per second impinges perpendicularly on a wall. Find the pressure 
 on the wall. 
 
 (3) A jet delivers 160 cubic feet of water per minute at a velocity of 
 20 feet per second and strikes a plane perpendicularly. Find the pressure 
 on the plane (1) when it is at rest ; (2) when it is moving at 5 feet per 
 second in the direction of the jet. In the latter case find the work done 
 per second in driving the plane. 
 
 (4) A fire-engine hose, 3 inches bore, discharges water at a velocity of 
 100 feet per second. Supposing the jet directed normally to the side of a 
 building, find the pressure. 
 
 (5) Water issues horizontally from a fixed thin-edged orifice, 6 inches 
 square, under a head of 25 feet. The jet impinges normally on a plane 
 moving in the same direction at 10 feet per second. Find the pressure on 
 the plane in Ibs., and the work done in horse-power. Take the coefficient 
 of discharge as f 64 and the coefficient of velocity as '97. 
 
 (6) A jet and a plane surface move in directions inclined at 30, with 
 velocities of 30 feet and 10 feet per second respectively. What is the 
 relative velocity of the jet and surface ? 
 
 (7) Let AB and BC be two lines inclined at 30. A jet of water moves 
 in the direction AB, with a velocity of 25 feet per second, and a series of 
 vanes move in the direction CB with a velocity of 15 feet per second. Find 
 the form of the vane so that the water may come on to it tangentially, and 
 leave it in the direction BD, perpendicular to CB. 
 
 Supposing that the jet is 1 foot wide and 1 inch thick before impinging, 
 find the effort of the jet on the vanes. 
 
 (8) A curved plate is mounted on a slide so that the plate is free to 
 move along the slide. It receives a jet of water at an angle of 30 with a 
 normal to the direction of sliding, and the jet leaves the plate at an angle 
 
IMPACT OF WATER ON VANES 281 
 
 of 120 with the same normal. Find the force which must be applied to 
 the plate in the direction of sliding to hold it at rest, and also the normal 
 pressure on the slide. Quantity of water flowing is 500 Ibs. per minute 
 with a velocity of 35 feet per second. 
 
 (9) A fixed vane receives a jet of water at an angle of 120 with a 
 direction AB. Find what angle the jet must be turned through in order 
 that the pressure on the vane in the direction AB may be 40 Ibs., when the 
 flow of water is 45 Ibs. per second at a velocity of 30 feet per second. 
 
 (10) Water under a head of 60 feet is discharged through a pipe 6 inches 
 diameter and 150 feet long, and then through a nozzle, the area of which 
 is one-tenth the area of the pipe. 
 
 Neglecting all losses but the friction of the pipe, determine the pressure 
 on a fixed plate placed in front of the nozzle. 
 
 (11) A jet of water 4 inches diameter impinges on a fixed cone, the 
 axis coinciding with that of the jet, and the apex angle being 30 degrees, 
 at a velocity of 10 feet per second. Find the pressure tending to move the 
 cone in the direction of its axis. 
 
 (12) A vessel containing water and having in one of its vertical sides 
 a circular orifice 1 inch diameter, which at first is plugged up, is 
 suspended in such a way that any displacing force can be accurately 
 measured. On the removal of the plug, the horizontal force required to 
 keep the vessel in place, applied opposite to the orifice, is 3'6 Ibs. By the 
 use of a measuring tank the discharge is found to be 31 gallons per minute, 
 the level of the water in the vessel being maintained at a constant height 
 of 9 feet above the orifice. Determine the coefficients of velocity, con- 
 traction and discharge. 
 
 (13) A train carrying a Ramsbottom's scoop for taking water into the 
 tender is running at 24 miles an hour. What is the greatest height at 
 which the scoop will deliver the water ? 1 f 0^ 
 
 (14) A locomotive going at 40 miles an hour scoops up water from a 
 trough. The tank is 8 feet above the mouth of the scoop, and the delivery 
 pipe has an area of 50 square inches. If half the available head is wasted 
 at entrance, find the velocity at which the water is delivered into the tank, 
 and the number of tons lifted in a trench 500 yards long. What, under 
 these conditions, is the increased resistance; and what is the minimum 
 speed of train at which the tank can be filled ? Lond. Un. 1906. 
 
 (15) A stream delivering 3000 gallons of water per minute with a 
 velocity of 40 feet per second, by impinging on vanes is caused freely to 
 deviate through an angle of 10, the velocity being diminished to 35 feet 
 per second. Determine the pressure on the vanes due to impact. If the 
 vanes be moving in the direction of that pressure, find their velocity and 
 deduce the useful horse-power. Lond. Un. 1906. 
 
 (16) Water flows from a 2-inch pipe, without contraction, at 45 feet per 
 second. 
 
 Determine the maximum work done on a machine carrying moving 
 plates in the following cases and the respective efficiencies : 
 
282 HYDRAULICS 
 
 (a) When the water impinges on a single flat plate at right angles and 
 leaves tangentially. 
 
 (b) Similar to (a) but a large number of equidistant flat plates are 
 interposed in the path of the jet. 
 
 (c) When the water glides on and off a single semi -cylindrical cup. 
 
 (d) When a large number of cups are used as in a Pelton wheel. 
 
 (17) In hydraulic mining, a jet 6 inches in diameter, discharged under 
 a head of 400 feet, is delivered horizontally against a vertical cliff face. 
 Find the pressure on the face. What is the horse-power delivered by the 
 jet? 
 
 (18) If the action on a Pelton wheel is equivalent to that of a jet on a 
 series of hemispherical cups, find the efficiency when the speed of the 
 wheel is five-eighths of the speed of the jet. 
 
 (19) If in the last question the jet velocity is 50 feet per second, 
 and the jet area 0'15 square foot, find the horse -power of the wheel. 
 
 (20) A ship has jet orifices 3 square feet in aggregate area, and dis- 
 charges through the jets 100 cubic feet of water per second. The speed of 
 the ship is 15 feet per second. Find the propelling force of the jets, the 
 efficiency of the propeller, and, neglecting friction, the horse-power of the 
 engines. 
 
CHAPTER IX. 
 
 WATER WHEELS AND TURBINES. 
 
 Water wheels can be divided into two classes as follows. 
 
 (a) Wheels upon which the water does work partly by 
 impulse but almost entirely by weight, the velocity of the water 
 when it strikes the wheel being small. There are two types of 
 this class of wheel, Overshot Wheels, Figs. 180 and 181, and 
 Breast Wheels, Figs. 182 and 184. 
 
 (b) Wheels on which the water acts by impulse as when 
 the wheel utilises the kinetic energy of a stream, or if a head h is 
 available the whole of the head is converted into velocity before 
 the water comes in contact with the wheel. In most impulse 
 wheels the water is made to flow under the wheel and hence 
 they are called Undershot Wheels. 
 
 It will be seen that in principle, there is no line of demarcation 
 between impulse water wheels and impulse turbines, the latter 
 only differing from the former in constructional detail. 
 
 177. Overshot water wheels. 
 
 This type of wheel is not suitable for very low or very high 
 heads as the diameter of the wheel cannot be made greater than 
 the head, neither can it conveniently be made much less. 
 
 Figs. 180 and 181 show two arrangements of the wheel, the 
 only difference in the two cases being that in Fig. 181, the top of 
 the wheel is some distance below the surface of the water in the 
 up-stream channel or penstock, so that the velocity v with which 
 the water reaches the wheel is larger than in Fig. 180. This has 
 the advantage of allowing the periphery of the wheel to have a 
 higher velocity, and the size and weight of the wheel is conse- 
 quently diminished. 
 
 The buckets, which are generally of the form shown in the 
 figures, or are curved similar to those of Fig. 182, are con- 
 nected to a rim M coupled to the central hub of the wheel by 
 
284 
 
 HYDRAULICS 
 
 suitable spokes or framework. This class of wheel has been 
 considerably used for heads varying from 6 to 70 feet, but is now 
 becoming obsolete, being replaced by the modern turbine, which 
 for the same head and power can be made much more compact, 
 and can be run at a much greater number of revolutions per unit 
 time. 
 
 E D K 
 
 Fig. 180. Overshot Water Wheel. 
 
 Fig. 181. Overshot Water Wheel. 
 
 The direction of the tangent to the blade at inlet for no shock 
 can be found by drawing the triangle of velocities as in Figs. 180 
 and 181. The velocity of the periphery of the wheel is v and the 
 velocity of the water U. The tip of the blade should be parallel 
 to V r . The mean velocity U, of the water, as it enters the wheel 
 
WATER WHEELS 285 
 
 in Fig. 181, will be v + k v/2^H, v being the velocity of approach 
 of the water in the channel, H the fall of the free surface and k 
 a coefficient of velocity. The water is generally brought to the 
 wheel along a wooden flume, and thus the velocity U and the 
 supply to the wheel can be maintained fairly constant by a simple 
 sluice placed in the flume. 
 
 The best velocity v for the periphery is, as shown below, 
 equal to JU cos 0, but in practice the velocity v is frequently 
 much greater than this. 
 
 In order that U may be about 2v the water must enter the 
 wheel at a depth not less than 
 
 U 2 _2i; 2 
 = ^ = T 
 below the water in the penstock. When 
 
 v = 4'5 feet, H = 0'63 feet, 
 and when v = 8 feet, H = 1 foot. 
 
 If the total fall to the level of the water in the tail race is h, 
 the diameter of the wheel may, therefore, be between h and 
 
 w 
 
 ri . 
 
 9 
 
 Since U is equal to j2gK, for given values of U and of h, the 
 larger the wheel is made the greater must be the angular distance 
 from the top of the wheel at which the water enters. 
 
 With the type of wheel and penstock shown in Fig. 181, the 
 head H is likely to vary and the velocity U will not, therefore, be 
 constant. 
 
 If, however, the wheel is designed for the required power at 
 minimum flow, when the head increases, and there is a greater 
 quantity of water available, a loss in efficiency will not be 
 important. 
 
 The horse-power of the wheel. Let D be the diameter of the 
 wheel in feet which in actual wheels is from 10 to 70 feet. 
 
 Let N be the number of buckets, which in actual wheels is 
 generally from 2 to 3D. 
 
 Let Q be the volume of water in cubic feet of water supplied 
 per second. 
 
 Let w be the angular velocity of the wheel in radians, and n 
 the number of revolutions per sec. 
 
 Let 6 be the width of the wheel. 
 
 Let d, which equals r 2 r } , be the depth of the shroud, which 
 on actual wheels is from 10" to 20". 
 
286 HYDRAULICS 
 
 Whatever the form of the buckets the capacity of each bucket is 
 
 bd . -^- , nearly. 
 The number of buckets which pass the stream per second is 
 
 N<* AT 
 
 ~2^ = ^' n ' 
 
 If a fraction k of each bucket is filled with water 
 
 . 
 
 or 
 
 The fraction ~k in actual wheels is from J to ^ . 
 If h is the fall of the water to the level of the tail race and e 
 the efficiency of the wheel, the horse-power is 
 
 62-4 . e . h 
 
 550 
 
 and the width b for a given horse-power, HP, is 
 1100HP ? HP 
 
 ^ />r\. j 1 -iTHk ~i J. / w - - 
 
 Effect of centrifugal forces. As the wheel revolves, the surface 
 of the water in the buckets, due to centrifugal forces, takes up a 
 parabolic form. 
 
 It is shown on page 335 that when a mass of water having an 
 inner radius n and outer radius r. 2 revolves about a fixed centre 
 with angular velocity <>, the pressure head, due to centrifugal 
 forces, at any radius r, is 
 
 p _ o>V - Q)Vi 2 
 
 w 2g 
 
 To balance this pressure head the surface of the water in any 
 bucket, at the point C, of radius r, must be raised above the 
 horizontal through A a distance 
 
 This is the equation to a parabola, and the surface of the water, 
 therefore, assumes the form of a parabolic curve. 
 
 Let r be the radius at the centre of the surface of the water in 
 any cup and <f> the inclination of the radius r to the horizontal. 
 
 Then since n is nearly equal to r z , TI O r = r nearly. 
 
WATER WHEELS 287 
 
 Then y = j- (n + r) (r-rj 
 
 = r (r-ri) nearly. 
 y 
 
 Therefore, y is approximately proportional to r-r 1? and the 
 surface AB is approximately a straight line inclined at an angle 
 0, the tangent of which is 
 
 tan = - cos <. 
 
 Losses of energy in overshot wheels. 
 
 (a) The whole of the velocity head -^- is lost in eddies in the 
 
 buckets. 
 
 In addition, as the water falls in the bucket through the 
 vertical distance EM, its velocity will be increased by gravity, 
 and the velocity thus given will be practically all lost by eddies. 
 
 Again, if the direction of the tip of the bucket is not parallel to 
 V r the water will enter with shock, and a further head will be 
 lost. The total loss by eddies and shock may, therefore, be 
 written 
 
 or 7i! + &!-, 
 
 k and &i being coefficients and hi the vertical distance EM. 
 
 (b) The water begins to leave the buckets before the level of 
 the tail race is reached. This is increased by the centrifugal 
 forces, as clearly, due to these forces, the water will leave the 
 buckets earlier than it otherwise would do. If h m is the mean 
 height above the tail level at which the water leaves the buckets, 
 a head equal to h m is lost. By fitting an apron GrH in front of the 
 wheel the water can be prevented from leaving the wheel until it 
 is very near the tail race. 
 
 (c) The water leaves the buckets with a velocity of whirl 
 equal to the velocity of the periphery of the wheel and a further 
 
 2 
 
 head - is lost. 
 
 (d) If the level of the tajl water rises above the bottom of 
 the wheel there will be a further loss due to, (1) the head h equal to 
 the height of the water above the bottom of the wheel, (2) the 
 impact of the tail water stream on the buckets, and (3) the 
 tendency for the buckets to lift the water on the ascending side of 
 the wheel. 
 
288 HYDRAULICS 
 
 In times of flood there may be a considerable rise of the 
 down-stream, and h may then be a large fraction of In. If on 
 the other hand the wheel is raised to such a height above the tail 
 water that the bottom of the wheel may be always clear, the 
 head h m will be considerable during dry weather flow, and the 
 greatest possible amount of energy will not be obtained from the 
 water, just when it is desirable that no energy shall be wasted. 
 
 If h is the difference in level between the up and down-stream 
 surfaces, the maximum hydraulic efficiency possible is 
 
 
 h 
 
 and the actual hydraulic efficiency will be 
 
 / y } 2 v *\ 
 
 h - (fefc, + Wo + h m + k^- + ^ 
 
 e = -TJ- 
 
 &, ki and A? being coefficients. 
 
 The efficiency as calculated from equation (1), for any given 
 value of h m , is a maximum when 
 
 is a minimum. 
 
 From the triangles EKF and KDF, Fig. 180, 
 
 (U cos B - vY + (U sin 0) 2 - V r 2 . 
 Therefore, adding v 2 to both sides of the equation, 
 Y r 2 + v* = U 2 cos 2 B - 2Uv cos 6 + Zv* + U 2 sin 2 0, 
 
 which is a minimum for a given value of U, when Wv cos W 
 is a maximum. Differentiating and equating to zero this, and 
 therefore the efficiency, is seen to be a maximum, when 
 
 v = ~n cos 0- 
 
 The actual efficiencies obtained from overshot wheels vary 
 from 60 to 80 per cent. 
 
 178. Breast wheel. 
 
 This type of wheel, like the overshot wheel, is becoming 
 obsolete. Fig. 182 shows the form of the wheel, as designed by 
 Fairbairn. 
 
 The water is admitted to the wheel through a number of 
 passages, which may be opened or closed by a sluice as shown in 
 the figure. The directions of these passages may be made so that 
 the water enters the wheel without shock. The water is retained 
 
WATER WHEELS 
 
 289 
 
 in the bucket, by the breast, until the bucket reaches the tail race, 
 and a greater fraction of the head is therefore utilised than in 
 the overshot wheel. In order that the air may enter and leave 
 the buckets freely, they are partly open at the inner rim. Since 
 the water in the tail race runs in the direction of the motion of 
 the bottom of the wheel there is no serious objection to the tail 
 race level being 6 inches above the bottom of the wheel. 
 
 The losses of head will be the same as for the overshot wheel 
 except that h m will be practically zero, and in addition, there will 
 be loss by friction in the guide passages, by friction of the water 
 as it moves over the breast, and further loss due to leakage 
 between the breast and the wheel. 
 
 Fig. 182. Breast Wheel. 
 
 According to Rankine the velocity of the rim for overshot and 
 breast wheels, should be from 4J to 8 feet per second, and the 
 velocity U should be about 2v. 
 
 The depth of the shroud which is equal to r 2 - n is from 1 to 
 If feet. Let it be denoted by d. Let H be the total fall and let 
 it be assumed that the efficiency of the wheel is 65 per cent. Then, 
 L. H. 19 
 
290 HYDRAULICS 
 
 the quantity of water required per second in cubic feet for a 
 given horse-power N is 
 
 0= N.550 
 ^ 62-4 x H x 0'65 
 13-5N 
 
 H 
 
 From | to of the volume of each bucket, or from \ to of the 
 total volume of the buckets on the 
 loaded part of the wheel is filled with 
 water. 
 
 Let b be the breadth of the buckets. 
 If now v is the velocity of the rim, and 
 an arc AB, Fig. 183, is set off on the 
 outer rim equal to v, and each bucket 
 is half full, the quantity of water 
 carried down per second is 
 
 JABCD.6. 
 
 Therefore 
 
 Q = if 
 
 vdb. 
 
 2r 2 
 Equating this value of Q to the above value, the width b is 
 
 27ND 
 
 "(n + rOwSH' 
 D being the outer diameter of the wheel. 
 
 Breast wheels are used for falls of from 5 to 15 feet and the 
 diameter should be from 12 to 25 feet. The width may be as 
 great as 10 feet. 
 
 Example. A breast wheel 20 feet diameter and 6 feet wide, working on a fall 
 of 14 feet and having a depth of shroud of 1' 3", has its buckets f full. The mean 
 velocity of the buckets is 5 feet per second. Find the horse-power of the wheel, 
 assuming the efficiency 70 per cent. 
 
 TTT> K , ~ ,5 62-4x0-70x14' 
 
 HP = 5 x l-2o x 6 x - x - 
 
 o 550 
 
 = 26-1. 
 
 The dimensions of this wheel should be compared with those calculated for an 
 inward flow turbine working under the same head and developing the same horse- 
 power. See page 339. 
 
 179. Sagebien wheels. 
 
 These wheels, Fig. 184, have straight buckets inclined to the 
 radius at an angle of from 30 to 45 degrees. 
 
 The velocity of the periphery of the wheel is very small, never 
 exceeding 2| to 3 feet per second, so that the loss due to the water 
 leaving the wheel with this velocity and due to leakage between 
 the wheel and breast is small. 
 
WATER WHEELS 
 
 291 
 
 An efficiency of over 80 per cent, has been obtained with 
 these wheels. 
 
 The water enters the wheel in a horizontal direction with 
 a velocity U equal to that in the penstock, and the triangle of 
 velocities is therefore ABC. 
 
 If the bucket is made parallel to V r the water enters without 
 shock, while at the same time there is no loss of head due to 
 friction of guide passages, or to contraction as the water enters or 
 leaves them ; moreover the direction of the stream has not to be 
 changed. 
 
 Fig. 184. Sagebien Wheel. 
 
 The inclined straight bucket has one disadvantage ; when the 
 lower part of the wheel is drowned, the buckets as they ascend are 
 more nearly perpendicular to the surface of the tail water than 
 when the blades are radial, but as the peripheral speed is very 
 low the resistance due to this cause is not considerable. 
 
 180. Impulse wheels. 
 
 In Overshot and Breast wheels the work is done principally 
 by the weight of the water. In the wheels now to be considered 
 the whole of the head available is converted into velocity before 
 the water strikes the wheel, and the work is done on the wheel 
 by changing the momentum of the mass of moving water, or in 
 other words, by changing the kinetic energy of the water. 
 
 192 
 
292 
 
 HYDRAULICS 
 
 Undershot wheel with flat blades. The simplest case is when 
 a wheel with radial blades, similar to that shown in Fig. 185, is 
 put into a running stream. 
 
 If 6 is the width of the wheel, d the depth of the stream under 
 the wheel, and U the velocity in feet per second, the weight of 
 water that will strike the wheel per second is b . d . w . U Ibs., and 
 the energy available per second is 
 
 U 3 
 b . d . w - foot Ibs. 
 
 Let v. be the mean velocity of the blades. 
 
 The radius of the wheel being large the blades are similar to 
 a series of flat blades moving parallel to the stream and the water 
 leaves them with a velocity v in the direction of motion. 
 
 As shown on page 268, the best theoretical value for the 
 velocity v of such blades is JU and the maximum possible 
 efficiency of the wheel is 0*5. 
 
 Fig. 185. Impulse Wheel. 
 
 By placing a gate across the channel and making the bed near 
 the wheel circular as in Fig. 185, and the width of the wheel 
 equal to that of the channel, the supply is more under control, and 
 loss by leakage is reduced to a minimum. 
 
 The conditions are now somewhat different to those assumed 
 for the large number of flat vanes, and the maximum possible 
 efficiency is determined as follows. 
 
 Let Q be the number of cubic feet of water passing through 
 the wheel per second. The mean velocity with which the water 
 leaves the penstock at ab is U = fc \/2gh. Let the depth of the 
 
WATER WHEELS 293 
 
 stream at ab be t. The velocity with which the water leaves the 
 wheel at the section cd is v, the velocity of the blades. If the 
 width of the stream at cd is the same as at ab and the depth 
 is h , then, 
 
 h x v = t x U, 
 
 , *U 
 or h = . 
 
 v 
 
 Since U is greater than v, h is greater than t, as shown in 
 the figure. 
 
 The hydrostatic pressure on the section cd is ?h *bw and on 
 the section ab it is $Fbw. 
 
 The change in momentum per second is 
 
 and this must be equal to the impressed forces acting on the mass 
 of water flowing per second through ab or cd. 
 
 These impressed forces are P the driving pressure on the wheel 
 blades, and the difference between the hydrostatic pressures acting 
 on cd and ab. 
 
 If, therefore, the driving force acting on the wheel is P Ibs., 
 then, 
 
 P + h<?bw - tfbw = ^ (U - v). 
 Substituting for h , , the work done per second is 
 
 Or, since Q = b . t . U, 
 
 The efficiency is then, 
 
 v (U - v) _ t_ /U _ ^ 
 g 2\v I 
 
 
 U 
 
 which is a maximum when 
 
 2t; 2 U 2 - 4v 3 U + grrtJ* + gtv* = 0. 
 
 The best velocity, v, for the mean velocity of the blades, has 
 been found in practice to be about 0'4U, the actual efficiency is 
 from 30 to 35 per cent., and the diameters of the wheel are 
 generally from 10 to 23 feet. 
 
 Floating wheels. To adapt the wheel to the rising and 
 lowering of the waters of a stream, the wheel may be mounted on 
 
294 
 
 HYDRAULICS 
 
 a frame which may be raised or lowered as the stream rises, or the 
 axle carried upon pontoons so that the wheel rises automatically 
 with the stream. 
 
 181. Poncelet wheel. 
 
 The efficiency of the straight blade impulse wheels is very 
 small, due to the large amount of energy lost by shock, and to the 
 velocity with which the water leaves the wheel in the direction of 
 motion. 
 
 The efficiency of the wheel is doubled, if the blades are of such 
 a form, that the direction of the blade at entrance is parallel to 
 the relative velocity of the water and the blade, as first suggested 
 by Poncelet, and the water is made to leave the wheel with no 
 component in the direction of motion of the periphery of the 
 wheel. 
 
 Fig. 186 shows a Poncelet wheel. 
 
 Fig. 186. Undershot Wheel. 
 
 Suppose the water to approach the edge A of a blade with a 
 velocity U making an angle with the tangent to the wheel at A. 
 
 Then if the direction of motion of the water is in the direction 
 AC, the triangle of velocities for entrance is ABC. 
 
 The relative velocity of the water and the wheel is Y r , and if 
 the blade is made sufficiently deep that the water does not overflow 
 the upper edge and there is no loss by shock and by friction, a 
 particle of water will rise up the blade a vertical height 
 
WATER WHEELS 295 
 
 It then begins to fall and arrives at the tip of the blade with the 
 velocity V r relative to the blade in the inverse direction BE. 
 
 The triangle of velocities for exit is, therefore, ABE, BE being 
 equal to BC. 
 
 The velocity with which the water leaves the wheel is then 
 
 It has been assumed that no energy is lost by friction or by 
 shock, and therefore the work done on the wheel is 
 
 and the theoretical hydraulic efficiency* is 
 
 E- 
 
 2. 
 
 20 
 
 .il.w a) 
 
 IP" 
 
 This will be a maximum when Ui is a minimum. 
 
 Now since BE = BC, the perpendiculars EF and CD, on to 
 AB and AB produced, from the points E and C respectively, are 
 equal. And since AC and the angle 6 are constant, CD is constant 
 for all values of v, and therefore FE is constant. But AE, that is 
 Ui, is always greater than FE except when AE is perpendicular 
 to AD. The velocity Ui will have its minimum value, therefore, 
 when AE is equal to FE or Ui is perpendicular to v. 
 
 The triangles of velocities are then as in Fig. 187, the point B 
 bisects AD, and 
 
 For maximum efficiency, therefore, 
 v = ill cos 6. 
 
 * In what follows, the terms theoretical hydraulic efficiency and hydraulic 
 efficiency will be frequently used. The maximum work per Ib. that can be utilised 
 by any hydraulic machine supplied with water under a head H, and from which 
 
 the water exhausts with a velocity u is H - 1- . The ratio 
 
 H- 
 
 is the theoretical hydraulic efficiency. If there are other hydraulic losses in the 
 machine equivalent to a head hf per Ib. of flow, the hydraulic efficiency is 
 
 The actual efficiency of the machine is the ratio of the external work done per Ib. 
 of water by the machine to H. 
 
296 HYDRAULICS 
 
 The efficiency can also be found by considering the change of 
 momentum. 
 
 The total change of velocity impressed on the water is CE, and 
 the change in the direction of motion is 
 therefore FD, Fig. 186. 
 
 And since BE is equal to BC, FB is 
 equal to BD, and therefore, 
 
 FD = 2 (U cos 0-v). Fig. 187. 
 
 The work done per Ib. is, then, 
 
 2(Ucos0-?;) 
 
 and the efficiency is 
 
 
 ,, 
 
 _ 4 CUv cos 6 - v 2 ) , v 
 
 ~~ 
 
 Differentiating with respect to v and equating to zero, 
 
 Ucos0-20=0, 
 or v = ^U cos 6. 
 
 The velocity Ui with which the water leaves the wheel, is then 
 perpendicular to v and is 
 
 Ui=Usm0. 
 
 Substituting for v its value JU cos 6 in (2), the maximum efficiency 
 is cos 2 6. 
 
 The same result is obtained from equation (1), by substituting 
 
 The maximum efficiency is then 
 
 ,, 1 IP sin 2 
 
 .L = 1 -- Yya = cos 0. 
 
 A common value for & is 15 degrees, and the theoretical 
 hydraulic efficiency is then 0*933. 
 
 This increases as diminishes, and would become unity if & 
 could be made zero. 
 
 If, however, 6 is zero, U and v are parallel and the tip of the 
 blade will be perpendicular to the radius of the wheel. 
 
 This is clearly the limiting case, which practically is not 
 realisable, without modifying the construction of the wheel. The 
 necessary modification is shown in the Pelt-on wheel described on 
 page 377. 
 
 The actual efficiency of Poncelet wheels is from 55 to 65 per 
 cent. 
 
WATER WHEELS 297 
 
 Form of the bed. Water enters the wheel at all points between 
 Q and R, and for no shock the bed of the channel PQ should be 
 made of such a form that the direction of the stream, where it 
 enters the wheel at any point A between R and Q, should make 
 a constant angle 6 with the radius of the wheel at A. 
 
 With as centre, draw a circle touching the line AS which 
 makes the given angle with the radius AO. Take several 
 other points on the circumference of the wheel between R and 
 Q, and draw tangents to the circle STV. If then a curve 
 PQ is drawn normal to these several tangents, and the stream 
 lines are parallel to PQ, the water entering any part of the 
 wheel between R and Q, will make a constant angle with the 
 radius, and if it enters without shock at A, it will do so at all 
 points. The actual velocity of the water U, as it moves along the 
 race PQ, will be less than \/2#H, due to friction, etc. The 
 coefficient of velocity k v in most cases will probably be between 
 0'90 and 0'95, so that taking a mean value for k v of 0*925, 
 
 U = 0-925 
 
 The best value for the velocity v taking friction into account. 
 In determining the best velocity for the periphery of the wheel no 
 allowance has been made for the loss of energy due to friction in 
 the wheel. 
 
 If V r is the relative velocity of the water and wheel at entrance, 
 it is to be expected that the velocity relative to the wheel at exit 
 will be less than V r , due to friction and interference of the rising 
 and falling particles of water. 
 
 The case is somewhat analogous to that of a stone thrown 
 vertically up in the atmosphere with a velocity v. If there were 
 no resistance to its motion, it would rise to a certain height, 
 
 and then descend, and when it again reached the earth it would 
 have a velocity equal to its initial velocity v. Due to resistances, 
 the height to which it rises will be less than hi, and the velocity 
 with which it reaches the ground will be even less than that due 
 to falling freely through this diminished height. 
 
 Let the velocity relative to the wheel at exit be riV r , n being 
 a fraction less than unity. 
 
 The triangle of velocities at exit will then be ABE, Fig. 188. 
 The change of velocity in the direction of motion is GH, which 
 equals 
 
298 HYDRAULICS 
 
 If the velocity at exit relative to the wheel is only nV r , there 
 must have been lost by friction etc., a head equal to 
 
 The work done on the wheel per Ib. of water is, therefore, 
 
 -v)}v V, 2 
 
 9 
 
 H 
 
 Fig. 188. 
 
 Let (1 - n. 2 ) be denoted by /, then since 
 
 Y r 2 - BH 2 + CH 2 = (U cos - vy + U 2 sin 2 0, 
 the efficiency 
 
 Differentiating with respect to v and equating to zero, 
 2 (1 + n) U cos - 4 (1 + w) v + 2U/ cos 6 - 2t?/= 0, 
 from which 
 
 = 
 
 /+ 2(1+71) 
 
 If / is now supposed to be 0'5, i.e. the head lost by friction, etc. 
 
 . 0'5V r 2 . Af71 
 
 is s > w 1S vl an( i 
 
 t; = '56U cos 6. 
 If /is taken as 075, 
 
 v - 0'6U cos 0. 
 
 Dimensions of Poncelet wheels. The diameter of the wheel 
 should not be less than 10 feet when the bed is curved, and not 
 less than 15 feet for a straight bed, otherwise there will be con- 
 siderable loss by shock at entrance, due to the variation of the 
 angle 6 which the stream lines make with the blades between R 
 and Q, Fig. 186. The water will rise on the buckets to a height 
 
WATER WHEELS 299 
 
 V 2 
 nearly equal to -- , and since the water first enters at a point R, 
 
 the blade depth d must, therefore, be greater than this, or the 
 water will overflow at the upper edge. The clearance between 
 the bed and the bottom of the wheel should not be less than f ". 
 The peripheral distance between the consecutive blades is taken 
 from 8 inches to 18 inches. 
 
 Horse-power of Poncelet wheels. If H is the height of the 
 surface of water in the penstock above the bottom of the wheel, 
 the velocity U will be about 
 
 0' 
 and v may be taken as 
 
 0'55 x 0'92 \2H - 0'5 
 
 Let D be the diameter of the wheel, and b the breadth, and let 
 t be the depth of the orifice RP. Then the number of revolutions 
 per minute is 
 
 0'5 
 
 . 
 
 7T.D 
 
 The coefficient of contraction c for the orifice may be from 0*6, 
 if it is sharp-edged, to 1 if it is carefully rounded, and may be 
 taken as 0'8 if the orifice is formed by a flat-edged sluice. 
 
 The quantity of water striking the wheel per second is, then, 
 
 If the efficiency is taken as 60 per cent., the work done per 
 second is 0'6 x 62'4QH ft. Ibs. 
 The horse-power N is then 
 
 34'5.c.. W2J7H.H 
 550 
 
 182. Turbines. 
 
 Although the water wheel has been developed to a considerable 
 degree of perfection, efficiencies of over 80 per cent, having been 
 obtained, it is being almost entirely superseded by the turbine. 
 
 The old water wheels were required to drive slow moving 
 machinery, and the great disadvantage attaching to them of 
 having a small angular velocity was not felt. Such slow moving 
 wheels are however entirely unsuited to the driving of modern 
 machinery, and especially for the driving of dynamos, and they 
 are further quite unsuited for the high heads which are now 
 utilised for the generation of power. 
 
 Turbine wheels on the other hand can be made to run at either 
 low or very high speeds, and to work under any head varying 
 
300 HYDRAULICS 
 
 from . 1 foot to 2000 feet, and the speed can be regulated with 
 much greater precision. 
 
 Due to the slow speeds, the old water wheels could not develope 
 large power, the maximum being about 100 horse-power, whereas 
 at Niagara Falls, turbines of 10,000 horse-power have recently 
 been installed. 
 
 Types of Turbines. 
 
 Turbines are generally divided into two classes; impulse, or 
 free deviation turbines, and reaction or pressure turbines. 
 
 In both kinds of turbines an attempt is made to shape the 
 vanes so that the water enters the wheel without shock ; that is 
 the direction of the relative velocity of the water and the vane is 
 parallel to the tip of the vane, and the direction of the leaving 
 edge of the vane is made so that the water leaves in a specified 
 direction. 
 
 In the first class, the whole of the available head is converted 
 into velocity before the water strikes the turbine wheel, and the 
 pressure in the driving fluid as it moves over the vanes remains 
 constant, and equal to the atmospheric pressure. The wheel and 
 vanes, therefore, must be so formed that the air has free access 
 between the vanes, and the space between two consecutive vanes 
 must not be full of water. Work is done upon the vanes, or in 
 other words, upon the turbine wheel to which they are fixed, in 
 virtue of the change of momentum or kinetic energy of the 
 moving water, as in examples on pages 270 2. 
 
 Suppose water supplied to a turbine, as in Fig. 258, under an 
 effective head H, which may be supposed equal to the total head 
 minus losses of head in the supply pipe and at the nozzle. The 
 water issues from the nozzle with a velocity U = v/2grH, and the 
 available energy per pound is 
 
 Work is done on the wheel by the absorption of the whole, or 
 part, of this kinetic energy. 
 
 If Ui is the velocity with which the water leaves the wheel, 
 the energy lost by the water per pound is 
 
 U 2 _U L 2 
 2<7 20' 
 
 and this is equal to the work done on the wheel together with 
 energy lost by friction etc.. in the wheel. 
 
 In the second class, only part of the available head is con- 
 verted into velocity before the water enters the wheel, and the 
 
TURBINES 301 
 
 velocity and pressure both vary as the water passes through the 
 wheel. It is therefore essential, that the wheel shall always be 
 kept full of water. Work is done upon the wheel, as will be seen 
 in the sequence, partly by changing the kinetic energy the water 
 possesses when it enters the wheel, and partly by changing its 
 pressure or potential energy. 
 
 Suppose water is supplied to the turbine of Fig. 191, under 
 the effective head H ; the velocity U with which the water enters 
 the wheel, is only some fraction of v/2#H, and the pressure head 
 at the inlet to the wheel will depend upon the magnitude of U 
 and upon the position of the wheel relative to the head and tail 
 water surfaces. The turbine wheel always being full of water, 
 there is continuity of flow through the wheel, and if the head 
 impressed upon the water by centrifugal action is determined, as 
 on page 335, the equations of Bemouilli * can be used to determine 
 in any given case the difference of pressure head at the inlet and 
 outlet of the wheel. 
 
 If the pressure head at inlet is and at outlet , and the 
 
 velocity with which the water leaves the wheel is Ui, the work 
 done on the wheel (see page 338) is 
 
 - + o 7p- per pound of water, 
 
 w w 2g 2g 
 
 or work is done on the wheel, partly by changing the velocity 
 head and partly by changing the pressure head. Such a turbine 
 is called a reaction turbine, and the amount of reaction is measured 
 by the ratio 
 
 p_pi 
 
 w w 
 
 ~TT~ 
 
 Clearly, if p is made equal to p lf the limiting case is reached, 
 and the turbine becomes an impulse, or free-deviation turbine. 
 
 It should be clearly understood that in a reaction turbine no 
 work is done on the wheel merely by hydrostatic pressure, in the 
 sense in which work is done by the pressure on the piston of a 
 steam engine or the ram of a hydraulic lift. 
 
 183. Reaction turbines. 
 
 The oldest form of turbine is the simple reaction, or Scotch 
 turbine, which in its simplest form is illustrated in Fig. 189. 
 
 A vertical tube T has two horizontal tubes connected to it, the 
 outer ends of which are bent round at right angles to the direction 
 
 * See page 334. 
 
302 
 
 HYDKAULICS 
 
 of length of the tube, or two holes and Oi are drilled as in the 
 figure. 
 
 Water is supplied to the central tube at such a rate as to keep 
 the level of the water in the tube 
 constant, and at a height h above 
 the horizontal tubes. Water escapes 
 through the orifices and Oi and 
 the wheel rotates in a direction 
 opposite to the direction of flow of 
 the water from the orifices. Tur- 
 bines of this class are frequently 
 used to act as sprinklers for distri- 
 buting liquids, as for example for 
 distributing sewage on to bacteria Fig. 189. Scotch Turbine, 
 beds. 
 
 A better practical form, known as the Whitelaw turbine, is 
 shown in Fig. 190. 
 
 i 
 
 Fig. 190. Whitelaw Turbine. 
 
 To understand the action of the turbine it is first necessary to 
 consider the effect of the whirling of the water in the arm upon 
 
TURBINES 303 
 
 the discharge from the wheel. Let v be the velocity of rotation 
 of the orifices, and h the head of water above the orifices. 
 
 Imagine the wheel to be held at rest and the orifices opened ; 
 then the head causing velocity of flow relative to the arm is 
 simply 7i, and neglecting friction the water will leave the nozzle 
 with a velocity 
 
 v = \f2gh. 
 
 Now suppose the wheel is filled with water and made to rotate 
 at an angular velocity w, the orifices being closed. There will 
 now be a pressure head at the orifice equal to h plus the head 
 impressed on the water due to the whirling of each particle of 
 water in the arm. 
 
 Assume the arm to be a straight tube, Fig. 189, having a cross 
 sectional area a. At any radius r take an element of thickness dr. 
 
 The centrifugal force due to this element is 
 
 -, ,. w . a . wVdr 
 of = -- - . 
 9 
 
 The pressure per unit area at the outer periphery is, therefore, 
 
 R 
 
 a Jo 
 
 and the head impressed on the water is 
 
 p eoV 2 
 
 w~ 2g ' 
 Let v be the velocity of the orifice, then v = wr, and therefore 
 
 p _ ir 
 w~2g' 
 
 If now the wheel be assumed frictionless and the orifices are 
 opened, and the wheel rotates with the angular velocity w, the 
 head causing velocity of flow relative to the wheel is 
 
 H_ = h + 2-=h+ ...(1). 
 
 w 2g 
 
 Let V r be the velocity relative to the wheel with which the 
 water leaves the orifice. 
 
 The velocity relative to the ground, with which the water 
 leaves the wheel, is V r v, the vector sum of V r and v. 
 
304 HYDRAULICS 
 
 The water leaves the wheel, therefore, with a velocity relative 
 to the ground of p = V r - v, and the kinetic energy lost is 
 
 ^ per pound of water. 
 
 The theoretical hydraulic efficiency is then, 
 
 _ 2V (V r ~ V} 
 
 V r 2 -t? 2 
 2v 
 
 = Vr + V* 
 
 Since from (2), V r becomes more nearly equal to v as v 
 increases, the energy lost per pound diminishes as v increases, 
 and the efficiency E, therefore, increases with v. 
 
 The efficiency of the reaction wheel when friction is considered. 
 As before, 
 
 JcV 2 
 Assuming the head lost by friction to be ^- , the total head 
 
 must be equal to 
 
 H=^ + |-= ^- (1 + fc) (4). 
 
 The work done on the wheel, per pound, is now 
 
 Z.~U" 2 .,2 
 
 and the hydraulic efficiency is 
 
 1,~\T 2 
 
 Substituting for h from (4) and for ft, V r - v, 
 
 2v(V r -v) 
 (l+k)V r *-V 2 ' 
 
 Let V r = nv, 
 
 then ^ 2( - 
 
 Differentiating and equating to zero, 
 
From which 
 
 TURBINES 
 
 Tfc 
 
 305 
 
 + k' 
 
 Or the efficiency is a maximum when 
 
 and 
 
 v. 
 
 Fig. 191. Outward Flow Turbine. 
 
 L. H. 
 
306 HYDRAULICS 
 
 184. Outward flow turbines. 
 
 The outward flow turbine was invented in 1828 by Four- 
 neyron. A cylindrical wheel W, Figs. 191, 192, and 201, having 
 a number of suitably shaped vanes, is fixed to a vertical axis. 
 The water enters a cylindrical chamber at the centre of the 
 turbine, and is directed to the wheel by suitable fixed guide 
 blades Gr, and flows through the wheel in a radial direction 
 outwards. Between the guide blades and the wheel is a cylindri- 
 cal sluice E which is used to control the flow of water through 
 the wheel. 
 
 Fig. 191 a. 
 
 This method of regulating the flow is very imperfect, as when 
 the gate partially closes the passages, there must be a sudden 
 enlargement as the water enters the wheel, and a loss of head 
 ensues. The efficiency at "part gate" is consequently very 
 much less than when the flow is unchecked. This difficulty is 
 partly overcome by dividing the wheel into several distinct 
 compartments by horizontal diaphragms, as shown in Fig. 192, 
 so that when working at part load, only the efficiency of one 
 compartment is affected. 
 
 The wheels of outward flow turbines may have their axes, 
 either horizontal or vertical, and may be put either above, or 
 below, the tail water level. 
 
 The "suction tube" If placed above the tail water, the 
 exhaust must take place down a "suction pipe," as in Fig. 201, 
 page 317, the end of which must be kept drowned, and the pipe 
 air-tight, so that at the outlet of the wheel a pressure less than 
 the atmospheric pressure may be maintained. If hi is the height 
 of the centre of the discharge periphery of the wheel above the 
 tail water level, and p a is the atmospheric pressure in pounds per 
 square foot, the pressure head at the discharge circumference is 
 
 ^-^ = 34-^. 
 w 
 
TURBINES 
 
 307 
 
 The wheel cannot be more than 34 feet above the level of the tail 
 water, or the pressure at the outlet of the wheel will be negative, 
 and practically, it cannot be greater than 25 feet. 
 
 It is shown later that the effective head, under which the 
 turbine works, whether it is drowned, or placed in a suction tube, 
 is H, the total fall of the water to the level of the tail race. 
 
 Fig. 192. Fourneyron Outward Flow Turbine. 
 
 The use of the suction tube has the advantage of allowing the 
 turbine wheel to be placed at some distance above the tail water 
 level, so that the bearings can be readily got at, and repairs can 
 be more easily executed. 
 
 By making the suction tube to enlarge as it descends, the 
 velocity of exit can be diminished very gradually, and its final 
 
 202 
 
308 HYDRAULICS 
 
 value kept small. If the exhaust takes place direct from the 
 wheel, as in Fig. 192, into the air, the mean head available is the 
 head of water above the centre of the wheel. 
 
 Triangles of velocities at inlet and outlet. For the water to 
 enter the wheel without shock, the relative velocity of the water 
 and the wheel at inlet must be parallel to the inner tips of the 
 vanes. The triangles of velocities at inlet and outlet are shown 
 in Figs. 193 and 194. 
 
 Fig. 194. 
 
 Let AC, Fig. 193, be the velocity U in direction and magnitude 
 of the water as it flows out of the guide passages, and let AD be 
 the velocity v of the receiving edge of the wheel. Then DC is Y r 
 the relative velocity of the water and vane, and the receiving 
 edge of the vane must be parallel to DC. The radial component 
 GrC, of AC, determines the quantity of water entering the wheel 
 per unit area of the inlet circumference. Let this radial velocity 
 be denoted by u. Then if A is the peripheral area of the inlet 
 face of the wheel, the number of cubic feet Q per second entering 
 the wheel is 
 
 Q=A.th 
 
 or, if d is the diameter and b the depth of the wheel at inlet, and 
 t is the thickness of the vanes, and n the number of vanes, 
 
 Q = (ird - n . t) . b . u. 
 
 Let D be the diameter, and AI the area of the discharge peri- 
 phery of the wheel. 
 
 The peripheral velocity Vi at the outlet circumference is 
 
TURBINES 309 
 
 Let HI be the radial component of velocity of exit, then what- 
 ever the direction with which the water leaves the wheel the 
 radial component of velocity for a given discharge is constant. 
 
 The triangle of velocity can now be drawn as follows : 
 
 Set off BE equal to Vi, Fig. 194, and BK radial and equal 
 to UL 
 
 Let it now be supposed that the direction EF of the tip of the 
 vane at discharge is known. Draw EF parallel to the tip of the 
 vane at D, and through K draw KF parallel to BE to meet EF 
 in F. 
 
 Then BF is the velocity in direction and magnitude with which 
 the water leaves the wheel, relative to the ground, or to the fixed 
 casing of the turbine. Let this velocity be denoted by Ui. If, 
 instead of the direction EF being given, the velocity Ui is given 
 in direction and magnitude, the triangle of velocity at exit can be 
 drawn by setting out BE and BF equal to v l and Ui respectively, 
 and joining EF. Then the tip of the blade must be made parallel 
 toEF. 
 
 For any given value of Ui the quantity of water flowing 
 through the wheel is 
 
 Work done on the wheel neglecting friction, etc. The kinetic 
 energy of the water as it leaves the turbine wheel is 
 
 Uj 2 
 
 2~ per pound, 
 
 and if the discharge is into the air or into the tail water this 
 energy is of necessity lost. Neglecting friction and other losses, 
 the available energy per pound of water is then 
 
 H-^footlbs., 
 and the theoretical hydraulic efficiency is 
 
 and is constant for any given value of Ui, and independent of the 
 direction of Ui. This efficiency must not be confused with the 
 actual efficiency, which is much less than E. 
 
 The smaller Ui, the -greater the theoretical hydraulic efficiency, 
 and since for a given flow through the wheel, Ui will be least 
 when it is radial and equal to Ui, the greatest amount of work 
 will be obtained for the given flow, or the efficiency will be a 
 maximum, when the water leaves the wheel radially. If the 
 
310 HYDRAULICS 
 
 water leaves with a velocity Ui in any other direction, the 
 efficiency will be the same, but the power of the wheel will be 
 diminished. If the discharge takes place down a suction tube, 
 and there is no loss between the wheel and the outlet from the 
 tube, the velocity head lost then depends upon the velocity Ui 
 with which the water leaves the tube, and is independent of the 
 velocity or direction with which the water leaves the wheel. 
 
 The velocity of whirl at inlet and outlet. The component of 
 U, Fig. 193, in the direction of v is the velocity of whirl at inlet, 
 and the component of Ui, Fig. 194, in the direction of t?i, is the 
 velocity of whirl at exit. 
 
 Let Y and Vi be the velocities of whirl at inlet and outlet 
 respectively, then 
 
 and Vi = Ui sin ft = Ui tan /?. 
 
 Work done on the wheel. It has already been shown, 
 section 173, page 275, that when water enters a wheel, rotating 
 about a fixed centre, with a velocity U, and leaves it with velocity 
 Ui, the component Vi of which is in the same direction as v iy the 
 work done on the wheel is 
 
 Vv !! 
 
 - per pound, 
 9 9 
 
 and therefore, neglecting friction, 
 
 5p.5a.H-g ..................... (i). 
 
 99 %9 
 
 This is a general formula for all classes of turbines and should 
 be carefully considered by the student. 
 Expressed trigonometrically, 
 
 Vi HI tan /3 _ ^ Ui 2 , . 
 
 - 
 
 ............... < 
 
 9 9 %9 
 
 If F is to the left of BK, V is negative. 
 
 Again, since the radial flow at inlet must equal the radial flow 
 at outlet, therefore 
 
 AU8in0 = AiUiCosj8 ..................... (3). 
 
 When Ui is radial, YI is zero, and u\ equals v\ tan a. 
 
 i-i TT i /r , 
 
 from which - ^H-- 2 -^ - ..................... (5), 
 
 y ^y 
 
 and from (3) AU sin = A^ tan a . . . . (6) . 
 
TURBINES 311 
 
 If the tip of the vane is radial at inlet, i.e. Y r is radial, 
 
 and 
 
 tan 2 a, 
 
 2(7 
 
 .(7) 
 (8). 
 
 . 
 
 In actual turbines - is from *02H to '07H. 
 
 Example. An outward flow turbine wheel, Fig. 195, has an internal diameter of 
 5*249 feet, and an external diameter of 6-25 feet, and it makes 250 revolutions per 
 minute. The wheel has 32 vanes, which may be taken as f inch thick at inlet and 
 1 inches thick at outlet. The head is 141-5 feet above the centre of the wheel and 
 the exhaust takes place into the atmosphere. The effective width of the wheel face 
 at inlet and outlet is 10 inches. The quantity of water supplied per second is 
 215 cubic feet. 
 
 Neglecting all frictional losses, determine the angles of the tips of the vanes at 
 inlet and outlet so that the water shall leave radially. 
 
 The peripheral velocity at inlet is 
 
 v = ir x 5-249 x -Vu69 ft. per sec., 
 and at outlet ^ = w x 6-25 x a#p = 82 ft. 
 
 Fig. 195. 
 
 The radial velocity of flow at inlet is 
 
 215 
 
 ~ir x 5-249 x^-f 
 = 18- 35 ft. per sec. 
 The radial velocity of flow at exit is 
 
 215 
 
 Therefore, 
 
 = 16-5 ft. per sec. 
 ^-" = 4-23 ft. 
 
312 
 
 HYDRAULICS 
 
 Then 
 
 = 141-5-4-23 
 
 and 
 
 V= 
 
 : 137-27 ft. 
 137-27 x 32-2 
 69 
 
 = 64 ft. per sec. 
 
 To draw the triangle of velocities at inlet set out v and u at right angles. 
 
 Then since V is 64, and is the tangential component of U, and u is the radial 
 component of U, the direction and magnitude of U is determined. 
 
 By joining B and C the relative velocity V r is obtained, and BC is parallel to the 
 tip of the vane. 
 
 The triangle of velocities at exit is DEF, and the tip of the vane must be parallel 
 toEF. 
 
 Fig. 196. 
 
 Fig. 197. 
 
 Outlet 
 
 The angles 0, <f>, and a can be calculated ; for 
 
 and 
 
 and, therefore, 
 
 6-4 
 18-35 
 
 = -3-670 
 
 tan a= 
 
 = 0-1994, 
 
 0*16, 
 
 = 105 14', 
 a = 11 17'. 
 
 It will be seen later how these angles are modified when friction is considered. 
 
 Fig. 198 shows the form the guide blades and vanes of the wheel would 
 probably take. 
 
 The path of the water through the wheel. The average radial velocity through 
 the wheel may be taken as 17-35 feet. 
 
 The time taken for a particle of water to get through the wheel is, therefore, 
 
 The angle turned through by the wheel in this time is 0-39 radians. 
 
 Set off the arc AB, Fig. 198, equal to -39 radian, and divide it into four equal 
 parts, and draw the radii ea t fb, gc and Bd. 
 
 Divide AD also into four equal parts, and draw circles through A x , A 2 , and A 3 . 
 
 Suppose a particle of water to enter the wheel at A in contact with a vane and 
 suppose it to remain in contact with the vane during its passage through the wheel. 
 Then, assuming the radial velocity is constant, while the wheel turns through the 
 arc Ae the water will move radially a distance AAj and a particle that came on to 
 
TURBINES 
 
 313 
 
 the vane at A will, therefore, be in contact with the vane on the arc through A x . 
 The vane initially passing through A will be now in the position el, al being 
 equal to hJ and the particle will therefore be at 1. When the particle arrives on 
 the arc through A 2 the vane will pass through /, and the particle will consequently 
 be at 2, 62 being equal to mn. The curve A4 drawn through Al 2 etc. gives the 
 path of the water relative to the fixed casing. 
 
 Fig. 198. 
 
 185. Losses of head due to frictional and other resistances 
 in outward flow turbines. 
 
 The losses of head may be enumerated as follows : 
 
 (a) Loss by friction at the sluice and in the penstock or 
 supply pipe. 
 
 If v is the velocity, and h a the head lost by friction in 
 the pipe, 
 
 (6) As the water enters and moves through the guide 
 passages there will be a loss due to friction and by sudden changes 
 in the velocity of flow. 
 
 This head may be expressed as 
 
 Jc being a coefficient. 
 
 * See page 119. 
 
314 
 
 HYDRAULICS 
 
 (c) There is a loss of head at entrance due to shock as 
 the direction of the vane at entrance cannot be determined 
 with precision. 
 
 This may be written 
 
 Vr 2 
 Iu(* n/i "7^ * 
 
 2# 
 
 that is, it is made to depend upon V r the relative velocity of the 
 water, and the tip of the vane. 
 
 (d) In the wheel there is a loss of head h d , due to friction, 
 which depends upon the relative velocity of the water and the 
 wheel. This relative velocity may be changing, and on any small 
 element of surface of the wheel the head lost will diminish, as the 
 relative velocity diminishes. 
 
 It will be seen on reference to Figs. 193 and 194, that as the 
 velocity of whirl Vi is diminished the relative velocity of flow v r at 
 exit increases, but the relative velocity Y r at inlet passes through 
 a minimum when V is equal to v, or the tip of the vane is radial. 
 If V is the relative velocity of the water and the vane at any 
 radius, and b is the width of the vane, and dl an element of 
 length, then, 
 
 & 2 being a third coefficient. 
 
 If there is any sudden change of velocity as the water passes 
 through the wheel there will be a further loss, and if the turbine 
 has a suction tube there may be also a small loss as the water 
 enters the tube from the wheel. 
 
 The whole loss of head in the penstock and guide passages may 
 be called H/ and the loss in the wheel h f . Then if U is the 
 
 Rotor 
 
 Boyden Dtffuser 
 
 Fiaed 
 
 Fig. 199. 
 
TURBINES 315 
 
 velocity with which the water leaves the turbine the effective 
 head is 
 
 In well designed inward and outward flow turbines 
 
 varies from O'lOH to *22H and the hydraulic efficiency is, therefore, 
 from 90 to 78 per cent. 
 
 The efficiency of inward and outward flow turbines including 
 mechanical losses is from 75 to 88 per cent. 
 
 Calling the hydraulic efficiency e } the general formula (1), 
 section 184, may now be written 
 
 9 9 
 
 = '78to'9H. 
 
 Outward flow turbines were made by Boy den* about 1848 for 
 which he claimed an efficiency of 88 per cent. The workmanship 
 was of the highest quality and great care was taken to reduce 
 all losses by friction and shock. The section of the crowns of the 
 wheel of the Boyden turbine is shown in Fig. 199. Outside of 
 the turbine wheel was fitted a "diffuser" through which, after 
 leaving the wheel, the water moved radially with a continuously 
 diminishing velocity, and finally entered the tail race with a 
 velocity much less, than if it had done so direct from the wheel. 
 The loss by velocity head was thus diminished, and Boyden 
 claimed that the diffuser increased the efficiency by 3 per cent. 
 
 186. Some actual outward flow turbines. 
 
 Double outward flow turbines. The general arrangement of an 
 outward flow turbine as installed at Chevres is shown in Fig. 200. 
 There are four wheels fixed to a vertical shaft, two of which 
 receive the water from below, and two from above. The fall 
 varies from 27 feet in dry weather to 14 feet in time of flood. 
 
 The upper wheels only work in time of flood, while at other 
 times the full power is developed by the lower wheels alone, the 
 cylindrical sluices which surround the upper wheels being set in 
 such a position as to cover completely the exit to the wheel. 
 
 The water after leaving the wheels, diminishes gradually in 
 velocity, in the concrete passages leading to the tail race, and the 
 loss of head due to the velocity with which the water enters the 
 
 * Lowell Hydraulic Experiments, J. B. Francis, 1855. 
 
316 
 
 HYDRAULICS 
 
 tail race is consequently small. These passages serve the same 
 purpose as Boyden's diffuser, and as the enlarging suction tube, 
 in that they allow the velocity of exit to diminish gradually. 
 
 Fig. 200. Double Outward Flow Turbine. (Escher Wyss and Co.) 
 
 Outward flow turbine with horizontal axis. Fig. 201 shows a 
 section through the wheel, and the supply and exhaust pipes, of an 
 outward flow turbine, having a horizontal axis and exhausting 
 down a " suction pipe." The water after leaving the wheel enters 
 a large chamber, and then passes down the exhaust pipe, the 
 lower end of which is below the tail race. 
 
 The supply of water to the wheel is regulated by a horizontal 
 cylindrical gate S, between the guide blades Gr and the wheel. The 
 gate is connected to the ring R, which slides on guides, outside 
 the supply pipe P, and is under the control of the governor. 
 
 The pressure of the water in the supply pipe is prevented from 
 causing end thrust on the shaft by the partition T, and between 
 T and the wheel the exhaust water has free access. 
 
 Outward flow turbines at Niagara Falls. The first turbines 
 installed at Niagara Falls for the generation of electric power, 
 
TURBINES 
 
 317 
 
 were outward flow turbines of the type shown in Figs. 202 and 
 203. 
 
 There are two wheels on the same vertical shaft, the water 
 being brought to the chamber between the wheels by a vertical 
 penstock 7 6" diameter. The water passes upwards to one wheel 
 and downwards to the other. 
 
 Fig. 201. Outward Flow Turbine with Suction Tube. 
 
 As shown in Fig. 202 the water pressure in the chamber is 
 prevented from acting on the lower wheel by the partition MN, 
 but is allowed to act on the lower side of the upper wheel, the 
 upper partition HK having holes in it to allow the water free access 
 underneath the wheel. The weight of the vertical shaft, and of 
 the wheels, is thus balanced, by the water pressure itself. 
 
 The lower wheel is fixed to a solid shaft, which passes through 
 the centre of the upper wheel, and is connected to the hollow 
 shaft of the upper wheel as shown diagrammatically in Fig. 202. 
 Above this connection, the vertical shaft is formed of a hollow 
 
318 
 
 HYDRAULICS 
 
 tube 38 inches diameter, except where it passes through the 
 bearings, where it is solid, and 11 inches diameter. 
 
 A thrust block is also provided to carry the unbalanced 
 weight. 
 
 The regulating sluice is external to the wheel. To maintain a 
 high efficiency at part gate, the wheel is divided into three separate 
 compartments as in Fourneyron's wheel. 
 
 HoLUovr and 
 Solid Shaft 
 
 Water cwLrrvilted, 
 urui&r the/ upper 
 wh&e/L to support 
 the, weight 
 of the, shaft 
 
 Fig. 202. Diagrammatic section of Outward Flow Turbine at Niagara Falls. 
 
 A vertical section through the lower wheel is shown in Fig. 
 203, and a part sectional plan of the wheel and guide blades in 
 Fig. 195. 
 
 (Further particulars of these turbines and a description of the 
 governor will be found in Cassier's Magazine, Yol. III., and in 
 Turbines Actuelle) Buchetti, Paris 1901. 
 
 187. Inward flow turbines. 
 
 In an inward flow turbine the water is directed to the wheel 
 through guide passages external to the wheel, and after flowing 
 radially finally leaves the wheel in a direction parallel to the axis. 
 
 Like the outward flow turbine it may work drowned or with a 
 suction tube. 
 
 The water only acts upon the blades during the radial 
 movement. 
 
TUKBINES 
 
 319 
 
 As improved by Francis*, in 1849, the wheel was of the form 
 shown in Fig. 204 and was called by its inventor a "central vent 
 wheel." 
 
 The wheel is carried on a vertical shaft, resting on a footstep, 
 and supported by a collar bearing placed above the staging S. 
 
 * Lowell Hydraulic Experiments, F. B. Francis, 1855. 
 
320 
 
 HYDRAULICS 
 
 Above the wheel is a heavy casting C, supported by bolts 
 from the staging S, which acts as a guide for the cylindrical 
 sluice F, and carries the bearing B for the shaft. There are 
 40 vanes in the wheel shown, and 40 fixed guide blades, the former 
 being made of iron one quarter of an inch thick and the latter 
 three-sixteenths of an inch. 
 
 Fig. 204. Francis' Inward flow or Central vent Turbine. 
 
 The triangles of velocities at inlet and outlet, Fig. 205, are 
 drawn, exactly as for the outward flow turbine, the only difference 
 being that the velocities v t U, Y, V r and u refer to the outer 
 
TURBINES 
 
 321 
 
 periphery, and Vi, Ui, Vi, v r and u^ to the inner periphery of the 
 wheel. 
 
 The work done on the wheel is 
 
 ft. Ibs. per lb., 
 9 9 
 
 and neglecting friction, 
 
 VlJ "Vl^l TT Ul 2 
 
 ~g~ ~g~ ~~20' 
 
 For maximum efficiency, for a given flow through the wheel, 
 Ui should be radial exactly as for the outward flow turbine. 
 
 Fig. 205. 
 
 The student should work the following example. 
 
 The outer diameter of an inward flow turbine wheel is 7'70 feet, and the inner 
 diameter 6-3 feet, the wheel makes 55 revolutions per minute. The head is 
 14-8 feet, the velocity at inlet is 25 feet per sec., and the radial velocity may be 
 assumed constant and equal to 7'5 feet. Neglecting friction, draw the triangles of 
 velocities at inlet and outlet, and find the directions of the tips of the vanes at 
 inlet and outlet so that there may be no shock and the water may leave radially. 
 
 Loss of head by friction. The losses of head by friction are 
 similar to those for an outward flow turbine (see page 313) and 
 the general formula becomes 
 
 9 9 
 When the flow is radial at exit, 
 
 = eH. 
 9 
 
 The value of e varying as before between 0'78 and 0'90. 
 
 Example (I). An inward flow turbine working under a head of 80 feet has 
 radial blades at inlet, and discharges radially. The angle the tip of the guide 
 blade makes with the tangent at the inlet is 30 degrees and the radial velocity is 
 constant. The ratio of the radii at inlet and outlet is 1-75. Find the velocity of 
 the inlet circumference of the wheel. Neglect friction. 
 
 L. H. 
 
 21 
 
322 
 
 HYDRAULICS 
 
 Since the discharge is radial, the velocity at exit is 
 'U l = v l tan 30 
 
 v 
 
 1-75 
 
 tan 30. 
 
 Then 
 
 Vt?_ v z tan 2 30 
 
 ~ '~ 
 
 1-75 2 2g 
 and since the blades are radial at inlet V is equal to v, 
 
 v* tan 2 30 
 v2 = 5-- 8 -TT^2 o 
 
 ^ e 
 therefore 
 
 from 4 which 
 
 /3 
 V = Yr 
 
 32 x 80 
 
 = 49-3 ft. per sec. 
 
 Fig. 206. 
 
 Example (2). The outer diameter of the wheel of an inward flow turbine of 
 200 horse-power is 2-46 feet, the inner diameter is 1-968 feet. The effective width 
 of the wheel at inlet 1 -15 feet. The head is 39*5 feet and 59 cubic feet of 
 water per second are supplied. The radial velocity with which the water leaves 
 the wheel may be taken as 10 feet per second. 
 
 Determine the theoretical hydraulic efficiency E and the actual efficiency e l of 
 the turbine, and design suitable vanes. 
 
 200 x 550 
 x 59x62-5 
 
 " 5 /0> 
 
 Theoretical hydraulic efficiency 
 
 The radial velocity of flow at inlet, 
 
 W =S-TT: 7-^ = 6-7 feet per sec. 
 
 2-46 XTTX 1-15 
 
TUKBINES 323 
 
 The peripheral velocity 
 
 i? = 2-46 . TT x 3 ^j- = 38-6 feet. 
 
 The velocity of whirl V. Assuming a hydraulic efficiency of 85/ , from 
 the formula 
 
 
 9 
 39-5 x 32-2 x -85 
 
 38-6 
 
 = 28-0 feet per sec. 
 The angle 6. Since w = 6'7 ft. per sec. and V = 28'0 ft. per sec. 
 
 tan = = 
 
 = 13 27'. 
 The angle 0. Since V is less than v, is greater than 90. 
 
 tan ^=-^=-^1= -0-531, 
 
 and = 152. 
 
 For the water to discharge radially with a velocity of 10 feet per sec. 
 
 and a =18 nearly. 
 
 The theoretical vanes are shown in Fig. 206. 
 Example (3). Find the values of and a on the assumption that e is 0-80. 
 
 Thomson's inward flow turbine. In 1851 Professor James 
 Thomson invented an inward flow turbine, the wheel of which 
 was surrounded by a large chamber set eccentrically to the wheel, 
 as shown in Figs. 207 to 210. 
 
 Between the wheel and the chamber is a parallel passage, in 
 which are four guide blades Gr, pivoted on fixed centres C and 
 which can be moved about the centres C by bell crank levers, 
 external to the casing, and connected together by levers as shown 
 in Fig. 207. The water is distributed to the wheel by these guide 
 blades, and by turning the worm quadrant Q by means of the 
 worm, the supply of water to the wheel, and thus the power of 
 the turbine, can be varied. The advantage of this method of 
 regulating the flow, is that there is no sudden enlargement from 
 the guide passages to the wheel, and the efficiency at part load 
 is not much less than at full load. 
 
 Figs. 209 and 210 show an enlarged section and part sectional 
 elevation of the turbine wheel, and one of the guide blades Gr. 
 The details of the wheel and casing are made slightly different 
 from those shown in Figs. 207 and 208 to illustrate alternative 
 methods. 
 
 The sides or crowns of the wheel are tapered, so that the 
 peripheral area of the wheel at the discharge is equal to the 
 peripheral area at inlet. The radial velocities of flow at inlet 
 and outlet are, therefore, equal. 
 
 212 
 
324 
 
 HYDRAULICS 
 
 The inner radius r in Thomson's turbine, and generally in 
 turbines of this class made by English makers, is equal to one-half 
 the external radius R. 
 
 Fig. 207. Guide blades and casing of Thomson Inward Flow Turbine. 
 
 The exhaust for the turbine shown takes place down two 
 suction tubes, but the turbine can easily be adapted to work below 
 the tail water level. 
 
 As will be seen from the drawing the vanes of the wheel are 
 made alternately long and short, every other one only continuing 
 from the outer to the inner periphery. 
 
TURBINES 
 
 325 
 
 The triangles of velocities for the inlet and outlet are shown in 
 Fig. 211, the water leaving the wheel radially. 
 
 The path of the water through the wheel, relative to the fixed 
 casing, is also shown and was obtained by the method described 
 on page 312. 
 
 Inward flow turbines with adjustable guide blades, as made by 
 the continental makers, have a much greater number of guide 
 blades (see Fig. 233, page 352). 
 
 Fig. 208. Section through wheel and casing of Thomson Inward Flow Turbine. 
 
 188. Some actual inward flow turbines. 
 
 A later form of the Francis inward flow turbine as designed by 
 Pictet and Co., and having a horizontal shaft, is shown in Fig. 212. 
 
 The wheel is double and is surrounded by a large chamber 
 from which water flows through the guides Gr to the wheel W. 
 After leaving the wheel, exhaust takes place down the two suction 
 tubes S, thus allowing the turbine to be placed well above the 
 tail water while utilising the full head. 
 
 The regulating sluice F consists of a steel cylinder, which 
 slides in a direction parallel to the axis between the wheel and 
 guides. 
 
HYDRAULICS 
 
 Gwicie 
 
 Fig. 209. Fig. 210. 
 
 Detail of wheel and guide blade of Thomson Inward Flow Turbine. 
 
 Fig. 211. 
 
TURBINES 
 
 327 
 
 The wheel is divided into five separate compartments, so that 
 at any time only one can be partially closed, and loss of head by 
 contraction and sudden enlargement of the stream, only takes 
 place in this one compartment. 
 
328 
 
 HYDRAULICS 
 
 The sluice F is moved by two screws T, which slide through 
 stuffing boxes B, and which can be controlled by hand or by the 
 governor B,. 
 
 Inward flow turbine for low falls and variable head. The 
 turbine shown in Fig. 213 is an example of an inward flow turbine 
 suitable to low falls and variable head. It has a vertical axis and 
 works drowned. The wheel and the distributor surrounding the 
 wheel are divided into five stages, the two upper stages being 
 shallower than the three lower ones, and all of which stages can 
 
 ,,.C 1Z34-S67S 9FET 
 
 Fig. 213. Inward Flow Turbine for a low and variable fall. (Pictet and Co. 
 
TURBINES 329 
 
 be opened or closed as required by the steel cylindrical sluice CC 
 surrounding the distributor. 
 
 When one of the stages is only partially closed by the sluice, 
 a loss of efficiency must take place, but the efficiency of this one 
 stage only is diminished, the stages that are still open working 
 with their full efficiency. With this construction a high efficiency 
 of the turbine is maintained for partial flow. With normal flows, 
 and a head of about 6'25 feet, the three lower stages only are 
 necessary to give full power, and the efficiency is then a 
 maximum. In times of flood there is a large volume of water 
 available, but the tail water rises so that the head is only about 
 4'9 feet, the two upper stages can then be brought into operation 
 to accommodate a larger flow, and thus the same power may be 
 obtained under a less head. The efficiency is less than when the 
 three stages only are working, but as there is plenty of water 
 available, the loss of efficiency is not serious. 
 
 The cylinder C is carried by four vertical spindles S, having 
 racks R fixed to their upper ends. G-earing with these racks, are 
 pinions p, Fig. 213, all of which are worked simultaneously by the 
 regulator, or by hand. A bevel wheel fixed to the vertical shaft 
 gears with a second bevel wheel on a horizontal shaft, the velocity 
 ratio being 3 to 1. 
 
 189. The best peripheral velocity for inward and outward 
 flow turbines. 
 
 When the discharge is radial, the general formula, as shown on 
 page 315, is 
 
 = eH = 0-78toO'90H.. ...(1). 
 
 9 
 
 If the blades are radial at inlet, for no shock, v should be equal 
 to Y, and 
 
 or v = V - 0'624 to 0'67 \20H. 
 
 This is sometimes called the best velocity for v, but it should be 
 clearly understood that it is only so when the blades are radial at 
 inlet. 
 
 190. Experimental determination of the best peripheral 
 velocity for inward and outward flow turbines. 
 
 For an outward flow turbine, working under a head of 14 feet, 
 with blades radial at inlet, Francis * found that when v was 
 
 0-626 
 
 * Lowell, Hydraulic Experiments. 
 
330 HYDRAULICS 
 
 the efficiency was a maximum and equal to 79'37 per cent. The 
 efficiency however was over 78 per cent, for all values of v 
 between 0*545 \/2^H and "671 \/20H. If 3 per cent, be allowed 
 for the mechanical losses the hydraulic efficiency may be taken 
 as 82*4 per cent. 
 
 Yv 
 
 From the formula = *824H, and taking V equal to v, 
 
 v = '64 
 so that the result of the experiment agrees well with the formula. 
 
 For an inward flow turbine having vanes as shown in Fig. 205, 
 the total efficiency was over 79 per cent, for values of v between 
 0*624 \/2^H and 0*708 *J2gK, the greatest efficiency being 797 
 per cent, when v was 0'708 \/20H and again when v was 
 637 s/2^H. 
 
 It will be seen from Fig. 205 that although the tip of the vane 
 at the convex side is nearly radial, the general direction of the 
 vane at inlet is inclined at an angle greater than 90 degrees to 
 the direction of motion, and therefore for no shock V should be 
 less than v. 
 
 When v was '708 \/2#H, V, Fig. 205, was less than v. The 
 value of Y was deduced from the following data, which is also 
 useful as being taken from a turbine of very high efficiency. 
 
 Diameter of wheel 9*338 feet. 
 
 Width between the crowns at inlet 0*999 foot. 
 
 There were 40 vanes in the wheel and an equal number of 
 fixed guides external to the wheel. 
 
 The minimum width of each guide passage was 0*1467 foot and 
 .the depth 1*0066 feet. 
 
 The quantity of water supplied to the wheel per second was 
 112*525 cubic feet, and the total fall of the water was 13*4 feet. 
 The radial velocity of flow u was, therefore, 3*86 feet per second. 
 
 The velocity through the minimum section of the guide passage 
 was 19 feet per second. 
 
 When the efficiency was a maximum, v was 20*8 feet per sec. 
 Then the radial velocity of flow at inlet to the wheel being 
 3*86 feet, and U being taken as 19 feet per second, the triangle 
 of velocities at inlet is ABC, Fig. 205, and Y is 18*4 feet per sec. 
 
 If it is assumed that the water leaves the wheel radially, then 
 
 eH = = 11*85 feet. 
 9 
 
 1 1 'Q\ 
 
 The efficiency e should be ..~, = 88*5 per cent., which is 9 per 
 cent, higher than the actual efficiency. 
 
TURBINES 331 
 
 The actual efficiency however includes not only the fluid losses 
 but also the mechanical losses, and these would probably be from 
 2 to 8 per cent., and the actual work done by the turbine on the 
 shaft is probably between 80 and 86*5 per cent, of the work done 
 by the water. 
 
 Vv 
 191. Value of e to be used in the formula = eH. 
 
 g 
 
 ~Vv 
 
 In general, it may be said that, in using the formula = eH, 
 
 y 
 
 the value of e to be used in any given case is doubtful, as even 
 though the efficiency of the class of turbines may be known, it is 
 difficult to say exactly how much of the energy is lost mechanically 
 and how much hydraulically. 
 
 A trial of a turbine without load, would be useless to deter- 
 mine the mechanical efficiency, as the hydraulic losses in such a 
 trial would be very much larger than when the turbine is working 
 at full load. By revolving the turbine without load by means of 
 an electric motor, or through the medium of a dynamometer, the 
 work to overcome friction of bearings and other mechanical losses 
 could be found. At all loads, from no load to full load, the 
 frictional resistances of machines are fairly constant, and the 
 mechanical losses for a given class of turbines, at the normal load 
 for which the vane angles are calculated, could thus approximately 
 be obtained. If, however, in making calculations the difference 
 between the actual and the hydraulic efficiency be taken as, say, 
 5 per cent., the error cannot be very great, as a variation of 5 per 
 cent, in the value assumed for the hydraulic efficiency e, will only 
 make a difference of a few degrees in the calculated value of 
 the angle <. 
 
 The best value for e, for inward flow turbines, is probably 0'80, 
 and experience shows that this value may be used with confidence. 
 
 Example. Taking the data as given in the example of section 184, and assuming 
 an efficiency for the turbine of 75 per cent., the horse-power is 
 ._ 215 x 62-4 x 141-5 x -75 x 60 
 
 33,000 
 
 = 2600 horse-power. 
 
 If the hydraulic efficiency is supposed to be 80 per cent., the velocity of 
 whirl V should be 
 
 .H 0-8.32.141-5 
 
 y 
 
 Then 
 
 v 69 
 
 = 52 feet per sec. 
 18-35 -18-35 
 
 and = 132 47'. 
 
 Now suppose the turbine to be still generating 2600 horse-power, and to have 
 an efficiency of 80 per cent., and a hydraulic efficiency of 85 per cent. 
 
332 
 
 HYDRAULICS 
 
 Then the quantity of water required per second, is 
 215x0-75 
 
 Q=- 
 
 0-8 
 
 = 200 cubic feet per sec. 
 
 and the radial velocity of flow at inlet will be 
 18-35x200 
 
 215 
 85.32. 141-5 
 
 = 17*1 ft. per sec. 
 
 Then 
 
 tan <f> = 
 
 69 
 17-1 
 
 :55'4 ft. per sec. 
 
 -17-1 
 
 55-4-69" 13-6 
 
 = 128. 24'. 
 
 192. The ratio of the velocity of whirl V to the velocity 
 of the inlet periphery v. 
 
 Experience shows that, consistent with. Vv satisfying the general 
 
 formula, the ratio ^ may vary between very wide limits without 
 considerably altering the efficiency of the turbine. 
 
 Table XXXYII shows actual values of the ratio 
 
 taken 
 
 from a number of existing turbines, and also corresponding values 
 
 Fig. 214. 
 
TURBINES 
 
 333 
 
 y v 
 , V being calculated from = 0*8H. The corresponding 
 
 variation in the angle <, Fig. 214, is from 20 to 150 degrees. 
 
 For a given head, v may therefore vary within wide limits, 
 which allows a very large variation in the angular velocity of the 
 wheel to suit particular circumstances. 
 
 TABLE XXXVII. 
 
 Showing the heads, and the velocity of the receiving circum- 
 ference v for some existing inward and outward, and mixed flow 
 turbines. 
 
 
 
 
 
 Ratio 
 
 
 v 
 
 Ratio 
 
 
 
 Hfeet 
 
 v feet 
 per sec. 
 
 v/2. 9 H 
 
 v 
 
 H.P. 
 
 V being calculated 
 Vv 
 
 /9^H 
 
 
 
 
 
 v &gti 
 
 
 from = -8H 
 
 
 
 
 
 
 
 9 
 
 Inward flow : 
 
 
 
 
 
 
 
 Niagara Falls* 
 
 146 
 
 70 
 
 96-8 
 
 0-72 
 
 5000 
 
 0-555 
 
 Rheinfelden 
 
 14-8 
 
 22 
 
 30-7 
 
 0-71 
 
 840 
 
 0-565 
 
 By Theodor ) 
 Bell and Co. J" 
 
 28-4 
 
 39 
 
 42-6 
 
 0-91 
 
 
 0-44 
 
 
 60-4 
 
 32-2 
 
 62-3 
 
 0-52 
 
 
 0-77 
 
 Pictet and Co. 
 
 183-7 
 
 51-1 
 
 76-8 
 
 0-47 
 
 300 
 
 0-85 
 
 M 
 
 134-5 
 
 46-6 
 
 65-6 
 
 0-505 
 
 300 
 
 0-79 
 
 
 6-25 
 
 16-6 
 
 20 
 
 0-83 
 
 
 0-48 
 
 11 
 
 30 
 
 25-75 
 
 44 
 
 0-58 
 
 700 
 
 0-69 
 
 11 
 
 
 38-5 
 
 50-3 
 
 0-77 
 
 200 
 
 0-52 
 
 Ganz and Co. 
 
 112 
 
 64-3 
 
 84-6 
 
 0-54 
 
 
 0-74 
 
 
 225 
 
 64-7 
 
 120 
 
 0-54 
 
 682 
 
 0-58 
 
 Rioter and Co. 
 
 10-66 
 
 15-2 
 
 26 
 
 0-585 
 
 30 
 
 0-69 
 
 Outward flow : 
 
 
 
 
 
 
 
 Niagara Falls 
 
 141-5 
 
 69 
 
 95-2 
 
 0-725) 
 
 rnnn 
 
 0-55 
 
 Pictet and Co. 
 
 130-5 
 
 69 
 
 91-6 
 
 0-750) 
 
 O\J\J\J 
 
 0-53 
 
 Ganz and Co. 
 
 95-1 
 
 38-7 
 
 78-0 
 
 0-495 
 
 290 
 
 0-81 
 
 
 
 223 
 
 55-6 
 
 120-0 
 
 0-46 
 
 1200 
 
 0-87 
 
 * Escher Wyss and Co. 
 
 For example, if a turbine is required to drive alternators 
 direct, the number of revolutions will probably be fixed by the 
 alternators, while, as shown later, the diameter of the wheel is 
 practically fixed by the quantity of water, which it is required to 
 pass through the wheel, consistent with the peripheral velocity of 
 the wheel, not being greater than 100 feet per second, unless, as 
 in the turbine described on page 373, special precautions are 
 taken. This latter condition may necessitate the placing of two 
 or more wheels on one shaft. 
 
334 HYDRAULICS 
 
 Suppose then, the number of revolutions of the wheel to be 
 given and d is fixed, then v has a definite value, and V must be 
 made to satisfy the equation 
 
 V* R 
 = ei. 
 
 9 
 
 Fig. 214 is drawn to illustrate three cases for which Yy is 
 constant. The angles of the vanes at outlet are the same for all 
 three, but the guide angle and the vane angle < at inlet vary 
 considerably. 
 
 193. The velocity with which water leaves a turbine. 
 In a well-designed turbine the velocity with which the water 
 
 leaves the turbine should be as small as possible, consistent with 
 keeping the turbine wheel and the down-take within reasonable 
 dimensions. 
 
 In actual turbines the head lost due to this velocity head 
 varies from 2 to 8 per cent. If a turbine is fitted with a 
 suction pipe the water may be allowed to leave the wheel itself 
 with a fairly high velocity and the discharge pipe can be made 
 conical so as to allow the actual discharge velocity to be as small 
 as desired. It should however be noted that if the water leaves 
 the wheel with a high velocity it is more than probable that there 
 will be some loss of head due to shock, as it is difficult to ensure 
 that water so discharged shall have its velocity changed gradually. 
 
 194. Bernouilli's equations applied to inward and out- 
 ward flow turbines neglecting friction. 
 
 Centrifugal head impressed on the water by the wheel. The 
 theory of the reaction turbines is best considered from the point 
 of view of Bernoulli's equations ; but before proceeding to discuss 
 them in detail, it is necessary to consider the " centrifugal head " 
 impressed on the water by the wheel. 
 
 This head has already been considered in connection with the 
 Scotch turbine, page 303. 
 
 Let r, Fig. 216, be the internal radius of a wheel, and E the 
 external radius. 
 
 At the internal circumference let the wheel be covered with a 
 cylinder c so that there can be no flow through the wheel, and let 
 it be supposed that the wheel is made to revolve at the angular 
 velocity u> which it has as a turbine, the wheel being full of water 
 and surrounded by water at rest, the pressure outside the wheel 
 being sufficient to prevent the water being whirled out of the 
 wheel. Let d be the depth of the wheel between the crowns. 
 Consider any element of a ring of radius r and thickness dr, and 
 subtending a small angle at the centre C, Fig. 216. 
 
TURBINES 
 The weight of the element is 
 
 335 
 
 .dr.d, 
 and the centrifugal force acting on the element is 
 
 .dr.d. w 
 
 - Ibs. 
 
 Let p be the pressure per unit area on the inner force of the 
 element and p + dp on the outer. 
 
 wr 6 .dr.d. wV 
 
 Then 
 
 . d 
 
 PC 
 
 m 
 
 PC 
 
 Fig. 215. 
 
 Fig. 216. 
 
 The increase in the pressure, due to centrifugal forces, between 
 r and R is, therefore, 
 
 ( R w<o 2 w 
 
 tvZr.= 
 
 and 
 
 For equilibrium, therefore, the pressure in the water surround- 
 ing the wheel must be p c . 
 
 If now the cylinder c be removed and water is allowed to flow 
 through the wheel, either inwards or outwards, this centrifugal 
 head will always be impressed upon the water, whether the wheel 
 is driven by the water as a turbine, or by some external agency, 
 and acts as a pump. 
 
 Bernoulli? s equations. The student on first reading these 
 equations will do well to confine his attention to the inward flow 
 turbine, Fig. 217, and then read them through again, confining his 
 .attention to the outward flow turbine, Fig. 191. 
 
.336 
 
 HYDRAULICS 
 
 Let p be the pressure at A, the inlet to the wheel, or in the 
 clearance between the wheel and the guides, pi the pressure at 
 the outlet B, Fig. 217, and p a the atmospheric pressure, in pounds 
 per square foot. Let H be the total head, and H the statical 
 head at the centre of the wheel. The triangles of velocities are 
 as shown in Figs. 218 and 219. 
 
 Then at A 
 
 w 
 
 w 
 
 CD. 
 
 Between B and A the wheel impresses upon the water the 
 centrifugal head 
 
 v being greater than 
 outward flow. 
 
 _ 
 
 2<7 20' 
 for an inward flow turbine and less for the 
 
 Fig. 217. 
 
 Consider now the total head relative to the wheel at A and B. 
 
 V 2 p 
 
 The velocity head at A is -- and the pressure head is , and 
 
 
 at B the velocity and pressure heads are ^j- and respectively. 
 
 If no head were impressed on the water as it flows through 
 the wheel, the pressure head plus the velocity head at A and B 
 would be equal to each other. But between A and B there is 
 impressed on the water the centrifugal head, and therefore, 
 
 w 20 ' 20 20 w 20 
 
TURBINES 337 
 
 This equation can be used to deduce the fundamental equation, 
 
 * V 1^1 _ 7 ,^, 
 
 y~ ~g~~ h (3) * 
 
 From the triangles ODE and ADE, Fig. 218, 
 
 Y r 2 - (Y - v) 2 + u 2 and Y 2 + w 2 = U 2 , 
 and from the triangle BFGr, Fig 219, 
 
 v r 2 = (vi - Yi) 2 + u 2 and Yi 2 + u? = Ui 2 . 
 Therefore by substitution in (2), 
 
 Pi fa-YQ 2 v*__vl UL = P_ (V-vY u? 
 
 w 2g 2g 2g + 2g w + 2g + 2g - (4!) ' 
 
 From which 
 
 UI' = P , u 2 ^v 
 
 2g w 2g g ' 
 
 and 
 
 v~V v l "V 1= p P! | U 2 W 
 g g w w 2g 2g 
 
 Substituting for P + 5? f rom (1) 
 
 _ - TT u. i l 
 
 -do "" 
 
 g g w w 2g 
 
 (5). 
 
 .(6). 
 
 Fig. 218. 
 
 Fig. 219. 
 
 Wheel in suction tube. If the centre of the wheel is 7i feet 
 above the surface of the tail water, and U is the velocity with 
 which the water leaves the down-pipe, then 
 
 w 
 
 Substituting for + ~ in (6), 
 
 ^-^=H 0+ ^-&^ -ni 2 
 
 g g w w 2g 
 
 = H-?^. 
 
 L. H. 
 
 22 
 
338 HYDRAULICS 
 
 IfVisO, ? = H -1K 
 
 The wheel can therefore take full advantage of the head H 
 even though it is placed at some distance above the level of the 
 tail water. 
 
 Drowned wheel. If the level of the tail water is CD, Fig. 217, 
 or the wheel is drowned, and hi is the depth of the centre of the 
 wheel below the tail race level, 
 
 fe-fc+fc, 
 
 w w 
 
 and the work done on the wheel per pound of water is again 
 vV 
 
 g g g 
 
 Y 
 
 IfViisO, = &. 
 
 From equation (5), 
 
 VV _ Pi Y! = p _ Pi + IP _ Ui 2 
 
 g g w w 2g 2g ' 
 
 so that the work done on the wheel per pound is the difference 
 between the pressure head plus the velocity head at entrance and 
 the pressure head plus velocity head at exit. 
 
 In an impulse turbine p and p l are equal, and the work done 
 is then the change in the kinetic energy of the jet when it strikes 
 and when it leaves the wheel. 
 
 A special ase arises when pi is equal to p. In this case a 
 considerable clearance may be allowed between the wheel and the 
 fixed guide without danger of leakage. 
 
 Equation (2), for this case, becomes 
 
 and if at exit v r is made equal to v 1} or the triangle BFGr, 
 Fig. 219, is isosceles, 
 
 and the triangle of velocities at entrance is also isosceles. 
 The pressure head at entrance is 
 
 and at exit is either + hi , or h . 
 w w 
 
TURBINES 339 
 
 Therefore, since the pressures at entrance and exit are equal, 
 
 U2 TT I TT 
 2~ - Mo - fii = J, 
 
 or else H + ho = H. 
 
 The water then enters the wheel with a velocity equal to that 
 due to the total head H, and the turbine becomes a free-deviation 
 or impulse turbine. 
 
 195. Bernouilli's equations for the inward and outward 
 flow turbines including friction. 
 
 If H/ is the loss of head in the penstock and guide passages, 
 h/ the loss of head in the wheel, h e the loss at exit from the wheel 
 and in the suction pipe, and Ui the velocity of exhaust, 
 
 P + ^! = __ 
 
 w 2 w 
 
 Pi + v l + ^L _ ?L = P + Zn 2 _ T, (9\ 
 
 w 2g 2g 2g w 2g f 
 
 and 
 
 from which = H-(^ + fc,+ Hr+fc) .................. (4). 
 
 If the losses can be expressed as a fraction of H, or equal to KH, 
 then 
 
 V 
 
 = (l-K)H = eH 
 
 -078H to 0'90H*. 
 
 196. Turbine to develop a given horse-power. 
 
 Let H be the total head in feet under which the turbine works. 
 
 Let n be the number of revolutions of the wheel per minute. 
 
 Let Q be the number of cubic feet of water per second required 
 by the turbine. 
 
 Let E be the theoretical hydraulic efficiency. 
 
 Let e be the hydraulic efficiency. 
 
 Let e m be the mechanical efficiency. 
 
 Let ei be the actual efficiency including mechanical losses. 
 
 Let Ui be the radial velocity with which the water leaves the 
 wheel. 
 
 Let D be the diameter of the wheel in feet at the inlet circum- 
 ference and d the diameter at the outlet circumference. 
 
 Let B be the width of the wheel in feet between the crowns 
 at the inlet circumference, and b be the width between the crowns 
 at the outlet circumference. 
 
 Let N be the horse-power of the turbine. 
 * See page 315. 
 
 222 
 
340 HYDRAULICS 
 
 The number of cubic feet per second required is 
 
 N. 33,000 
 
 
 ei H.62'4.60 ' 
 A reasonable value for ei is 75 per cent. 
 The velocity U with which the water leaves the turbine, since 
 
 is U =>/20(l-E)Hft. per sec (2). 
 
 If it be assumed that this is equal to Ui, which would of 
 necessity be the case when the turbine works drowned, or 
 exhausts into the air, then, if t is the peripheral thickness of the 
 vanes at outlet and m the number of vanes, 
 
 If U is not equal to u 1} then 
 
 (ird - mt) u\ b = Q (3) . 
 
 The number of vanes m and the thickness t are somewhat 
 arbitrary, but in well-designed turbines t is made as small as 
 possible. 
 
 As a first approximation mt may be taken as zero and (3) 
 becomes 
 
 vdbu^Q (4). 
 
 For an inward flow turbine the diameter d is fixed from 
 consideration of the velocity with which the water leaves the 
 wheel in an axial direction. 
 
 If the water leaves at both sides of the wheel as in Fig. 208, 
 and the diameter of the shaft is d , the axial velocity is 
 
 U Q = ft. per sec. 
 
 2 (^ 2 - d 2 ) 
 
 The diameter d can generally be given an arbitrary value, or 
 for a first approximation to d it may be neglected, and UQ may be 
 taken as equal to %. Then 
 
 d = 
 
 From (4) and (5) b and d can now be determined. 
 
 A ratio for -T having been decided upon, D can be calculated, 
 
 and if the radial velocity at inlet is to be the same as at outlet, 
 and t is the thickness of the vanes at inlet, 
 
 (7rD-mo)B=SL(7rd-mO& (6). 
 
TURBINES 341 
 
 For rolled brass or wrought steel blades, t may be very small, 
 and for blades cast with the wheel, by shaping them as in Fig. 227, 
 t is practically zero. Then 
 
 T* _ Q 
 
 ~^D' 
 
 If now the number of revolutions is fixed by any special 
 condition, such as having to drive an alternator direct, at some 
 definite speed, the peripheral velocity is 
 
 TrDfi p, m , 
 
 V = -QQ- ft- per sec ......................... (7). 
 
 V 
 
 Then ^T = eH > 
 
 y 
 
 and if e is given a value, say 80 per cent., 
 
 V = '^5 ft. per sec. ...(8). 
 
 v 
 
 Since u, V, and v are known, the triangle of velocities at inlet 
 can be drawn and the direction of flow and of the tip of vanes 
 at inlet determined. Or and </>, Fig. 214, can be calculated from 
 
 tan^ .............................. (9) 
 
 and tan<f>-~ v ........................ (10). 
 
 Then U, the velocity of flow at inlet, is 
 
 ., 
 
 At exit Vi = -QQ ft. per sec., 
 
 and taking u\ as radial and equal to u, the triangle of velocities 
 can be drawn, or a calculated from 
 
 u 
 tan a = . 
 
 Vi 
 
 If Ho is the head of water at the centre of the wheel and H/ the 
 head lost by friction in the supply pipe and guide passages, the 
 pressure head at the inlet is 
 
 P TT U ' 2 TT 
 = o o -- **/ 
 
 w 2g 
 
 Example. An inward flow turbine is required to develop 300 horse-power under 
 a head 60 feet, and to run at 250 revolutions per minute. 
 To determine the leading dimensions of the turbine. 
 Assuming e l to be 75 per cent., 
 
 300 x 33,000 
 ^ ~ -75 x 60 x 62-4 x 60 
 = 58-7 cubic feet per sec. 
 
342 HYDRAULICS 
 
 Assuming E is 95 per cent., or five per cent, of the head is lost by velocity 
 of exit and u^=u, 
 
 I = -05. 60 
 
 and u = 13-8 feet per sec. 
 
 Then from (5), page 340, 
 
 W2.1-36 
 
 = 1-65 feet, 
 say 20 inches to make allowance for shaft and to keep even dimension. 
 
 Then from (4), 6 = ^ = -82 foot 
 
 = 9 inches say. 
 Taking ? as 1-8, D = 3-0 feet, and 
 
 v = TT . 3 . *#$ = 39-3 feet per sec., 
 and B = 5 inches say. 
 
 Assuming e to be 80 per cent., 
 
 8a =8(H)fl 
 
 13-8 
 
 and 0=19 30', 
 
 13-8 
 
 and 0=91 15'. 
 
 13-8x1-8 
 
 and a = 32 18'. 
 
 The velocity U at inlet is 
 
 U = N /39-0 2 +(13-8) 2 
 = 41-3 ft. per sec. 
 The absolute pressure head at the inlet to the wheel is 
 
 - ^ -- - 
 
 - -- f 
 
 h f , the head lost by friction in the down pipe 
 
 The pressure head at the outlet of the wheel will depend upon the height of the 
 wheel above or below the tail water. 
 
 197. Parallel or axial flow turbines. 
 
 Fig. 220 shows a double compartment axial flow turbine, the 
 guide blades bein g placed above the wheel and the flow through 
 the wheel being parallel to the axis. The circumferential section 
 of the vanes at any radius when turned into the plane of the 
 paper is as shown in Fig. 221. A plan of the wheel is also shown. 
 
 The triangles of velocities at inlet and outlet for any radius 
 are similar to those for inward and outward flow turbines, the 
 velocities v and vi, Figs. 222 and 223, being equal. 
 
TUKBINES 
 
 The general formula now becomes 
 
 .343 
 
 For maximum efficiency for a given flow, the water should 
 leave the wheel in a direction parallel to the axis, so that it has 
 no momentum in the direction of v. 
 
 Fig. 220. Double Compartment Parallel Flow Turbine. 
 
 Figs. 221, 222, 22?. 
 
 Then, taking friction and other losses into account, 
 
 Vv 
 - =eH. 
 9 
 
344 
 
 HYDRAULICS 
 
 The velocity v will be proportional to the radius, so that if the 
 water is to enter and leave the wheel without shock, the angles 0, 
 <f>, and a must vary with the radius. 
 
 The variation in the form of the vane with the radius is shown 
 by an example. 
 
 A Jonval wheel has an internal diameter of 5 feet and an 
 external diameter of 8' 6". The depth of the wheel is 7 inches. 
 The head is 15 feet and the wheel makes 55 revolutions per 
 minute. The flow is 300 cubic feet per second. 
 
 To find the horse-power of the wheel, and to design the wheel 
 vanes. 
 
 Let TI be the mean radius, and r and r 2 the radii of the wheel 
 at the inner and outer circumference respectively. Then 
 
 r = 2"5 feet and v = 2irr = 14*4 feet per sec., 
 TI = 3"375 feet and Vi = 2nri J- = 21*5 feet per sec., 
 r 2 = 4'25 feet and v 2 = 27rr 2 f-J = 24" 5 feet per sec. 
 The mean axial velocity is 
 
 300 
 
 u= 7~2 1\ =0 15 ft. per sec. 
 TT (r, 1 - r 2 ) 
 
 y I 
 
 B D 
 
 Fig. 224. Triangles of velocities at inlet and outlet at three different 
 radii of a Parallel Flow Turbine. 
 
 Taking e as 0'80 at each radius, 
 0-8.32-2.15 385 
 
 ft. per Bee, 
 
 Vi= = 17-9 ft. per sec., 
 
 Y 2 = o7:H = 15*7 ft. per sec. 
 
 Inclination of the vanes at inlet. The triangles of velocities 
 for the three radii r, r a , r 2 are shown in Fig. 224. For example, 
 at radius r, ADC is the triangle of velocities at inlet and ABC the 
 
TURBINES 345 
 
 triangle of velocities at outlet. The inclinations of the vanes at 
 inlet are found from 
 
 0.1 ~ 
 
 tan <f> = og-7 _ 14-4 f rom which < = 33 30', 
 tan * = 17-9 8 -21'5 and * = 113 50/ ' 
 
 Q.1 * 
 
 tan 02 = TE^J A - 5 from which < 2 = 137 6'. 
 JLO / Z4 O 
 
 The inclination of the guide blade at each of the three radii, 
 fi 8-15 
 = 2fr7> 
 from which 6 17, 
 
 tan O-i = -j ,-Q and 1 = 24 30', 
 
 8'15 
 tan,= j^an 
 
 T/ie inclination of the vanes at exit. 
 
 ten 0!= f|t =20 48', 
 
 Zl o 
 
 ten ,= |^=18' 22'. 
 
 If now the lower tips of the guide blades and the upper tips 
 of the wheel vanes are made radial as in the plan, Fig. 221, the 
 inclination of the guide blade will have to vary from 17 to 
 27?, degrees or else there will be loss by shock. To get over this 
 difficulty the upper edge only of each guide blade may be made 
 radial, the lower edge of the guide blade and the upper edge of 
 each vane, instead of being radial, being made parallel to the 
 upper edge of the guide. In Fig. 225 let r and E be the radii 
 of the inner and outer crowns of the wheel and also of the guide 
 blades. Let MN be the plan of the upper edge of a guide blade 
 and let DGr be the plan of the lower edge, DGr being parallel to 
 MN. Then as the water runs along the guide at D, it will leave 
 the guide in a direction perpendicular to OD. At Gr it will leave 
 in a direction HGr perpendicular to OGr. Now suppose the guide 
 at the edge DGr to have an inclination ft to the plane of the paper. 
 If then a section of the guide is taken by a vertical plane XX 
 perpendicular to DGr, the elevation of the tip of the vane on this 
 plane will be AL, inclined at ft to the horizontal line AB, and AC 
 
846 
 
 HYDRAULICS 
 
 will be the intersection of the plane XX with the plane tangent 
 to the tip of the vane. 
 
 Now suppose DB and GrH to be the projections on the plane 
 of the paper of two lines lying on the tangent plane AC and 
 perpendicular to CD and OGr respectively. Draw EF and HK 
 perpendicular to DE and GrH respectively, and make each of 
 them equal to BC. Then the angle EDF is the inclination of the 
 stream line at D to the plane of the paper, and the angle HGrK is 
 the inclination of the stream line at Gr to the plane of the paper. 
 These should be equal to and 2 
 
 k 
 
 p t* 
 
 Flew, of lower edge of guide, 
 bfadesl of upper &dg& of rone, 
 
 Fig. 225. Plan of guide blades and vanes of Parallel Flow Turbines. 
 
 Let y be the perpendicular distance between MN and DG. 
 Let the angles GOD and GOH be denoted by < an da respectively. 
 
 Since EF, BC and HK are equal, 
 
 EDtan0 = #tan/3 (1), 
 
 and GH tan 2 = y tan j3 (2). 
 
 But 
 
 and 
 
 Therefore 
 and 
 
 Again, 
 
 - cos (a + </>), 
 
 tan = cos (a + <) tan ft (3), 
 
 tan 2 = cos a tan ft (4). 
 
 since- ^ ...(5). 
 
 There are thus three equations from which a, <f> and ft can be 
 determined. 
 
 Let x and y be the coordinates of the point D, being the 
 intersection of the axes. 
 
TURBINES 
 
 347 
 
 Then 
 
 and from (5) 
 
 cos (a + <) = - , 
 
 cosa = 
 
 Substituting for cos (a + <) and cos a and the known values of 
 tan and tan 2 in the three equations (3 5), three equations are 
 obtained with a?, y, and ft as the unknowns. 
 
 Solving simultaneously 
 
 x = 1'14 feet, 
 
 y = 2'23 feet, 
 
 and tan ft = 0'67, 
 
 from which ft = 34. 
 
 Fig. 226. 
 
 Fig. 228. 
 
 The length of the guide blade is thus found, and the constant 
 slope at the edge DGr so that the stream lines at D and Gr shall 
 have the correct inclination. 
 
 If now the upper edge of the vane is just below DGr, and the 
 tips of the vane at D and Gr are made as in Figs. 226 228, < and 
 
348 
 
 HYDRAULICS 
 
 < 2 being 33 30' and 137 6' respectively, the water will move on to 
 the vane without shock. 
 
 The plane of the lower edge of the vane may now be taken as 
 D'Gr', Fig. 225, and the circular sections DD', PQ, and GrGT at the 
 three radii, r, ri, and r 2 are then as in Figs. 226 228. 
 
 198. Regulation of the flow to parallel flow turbines. 
 
 To regulate the flow through a parallel flow turbine, Fontaine 
 placed sluices in the guide passages, as in Fig. 229, connected to 
 a ring which could be raised or lowered by three vertical rods 
 having nuts at the upper ends fixed to toothed pinions. When 
 
 Fig. 229. Fontaine's Sluices. 
 
 Fig. 230. Adjustable guide blades for Parallel Flow Turbine. 
 
 the sluices required adjustment, the nuts were revolved together 
 by a central toothed wheel gearing with the toothed pinions 
 carrying the nuts. Fontaine fixed the turbine wheel to a hollow 
 shaft which was carried on a footstep above the turbine. In some 
 modern parallel flow turbines the guide blades are pivoted, as in 
 Fig. 230, so that the flow can be regulated. The wheel may be 
 made with the crowns opening outwards, in section, similar to 
 the Grirard turbine shown in Fig. 254, so that the axial velocity 
 with which the water leaves the wheel may be small. 
 
 The axial flow turbine is well adapted to low falls with variable 
 head, and may be made in several compartments as in Fig. 220. 
 In this example, only the inner ring is provided with gates. In 
 dry weather flow the head is about 3 feet and the gates of the 
 inner ring can be almost closed as the outer ring will give the full 
 
TURBINES 349 
 
 power. During times of flood, and when there is plenty of water, 
 the head falls to 2 feet, and the sluices of the inner ring are 
 opened. A larger supply of water at less head can thus be 
 allowed to pass through the wheel, and although, due to the shock 
 in the guide passages of the inner ring, the wheel is not so efficient, 
 the abundance of water renders this unimportant. 
 
 Example. A double compartment Jonval turbine has an outer diameter of 
 12' 6" and an inner diameter of 6 feet. 
 
 The radial width of the inner compartment is V 9" and of the outer compart- 
 ment 1' 6". Allowing a velocity of flow of 3-25 ft. per second and supposing the 
 minimum fall is 1' 8", and the number of revolutions per minute 14, find the horse- 
 power of the wheel when all the guide passages are open, and find what portion of 
 the inner compartment must be shut off so that the horse-power shall be the same 
 under a head of 3 feet. Efficiency 70 per cent. 
 
 Neglecting the thickness of the blades, 
 
 the area of the outer compartment = ^ (12-5 2 -9'5 2 ) = 52'6 sq. feet. 
 
 inner =^ (9'5 2 -6 2 ) = 42-8 sq. feet. 
 
 Total area = 95 -4 sq. feet. 
 
 The weight of water passing through the wheel is 
 
 W = 95-4 x 62-4 x 3-25 Ibs. per sec. 
 
 = 19,300 Ibs. per sec. 
 and the horse-power is 
 
 19,300x1-66x0-7 
 
 550 
 
 Assuming the velocity of flow constant the area required when the head 
 is 3 feet is 
 
 40-8x33,000 
 
 '~60x62-5x3x-7 
 = 55-6 sq. feet, 
 or the outer wheel will nearly develop the horse-power required. 
 
 199. Bernouilli's equations for axial flow turbines. 
 
 The Bernouilli's equations for an axial flow turbine can be 
 written down in exactly the same way as for the inward and 
 outward flow turbines, page 335, except that for the axial flow 
 turbine there is no centrifugal head impressed on the water 
 between inlet and outlet. 
 
 w 2g w 2g 
 from which, since v is equal to Vi , 
 
 p + 
 
 w 2g 2g w 2g 
 
 p V 2 Vv <u? Pl YJ* u? 
 therefore - + s ---- + cT = + ~cT + t 
 
 w 2g g 2g w 2g 2g g 
 
 Vv y lV p u 2 IL 2 Pl , 
 
 and ~ =+ o -- - ft/. 
 
 g g w 2g 2g w 
 
350 HYDRAULICS 
 
 But in Fig. 220, ^ + ^ = H + ^ - H P , 
 
 and -^ = -^ + ^i 
 
 Therefore, = H -^ - H/ h/. 
 
 99 ^9 
 
 If Ui is axial and equal to u, as in Fig. 223, 
 
 200. Mixed flow turbines. 
 
 By a modification of the shape of the vanes of an inward flow 
 turbine, the mixed flow turbine is obtained. In the inward and 
 outward flow turbine the water only acts upon the wheel while it 
 is moving in a radial direction, but in the mixed flow turbine the 
 vanes are so formed that the water acts upon them also, while 
 flowing axially. 
 
 Fig. 231. Mixed Flow Turbine. 
 
 Fig. 231 shows a diagrammatic section through the wheel of 
 a mixed flow turbine, the axis of which is vertical. The water 
 
TURBINES 
 
 351 
 
 enters the wheel in a horizontal direction and leaves it vertically, 
 but it leaves the discharging edge of the vanes in different 
 directions. At the upper part B it leaves the vanes nearly 
 radially, and at the lower part A, axially. The vanes are spoon- 
 shaped, as shown in Fig. 232, and should be so formed, or in other 
 words, the inclination of the discharging edge should so vary, 
 that wherever the water leaves the vanes it should do so with no 
 component in a direction perpendicular to the axis of the turbine, 
 i.e. with no velocity of whirl. The regulation of the supply to 
 the wheel in the turbine of Fig. 231 is effected by a cylindrical 
 sluice or speed gate between the fixed guide blades and the wheel. 
 
 Fig. 232. Wheel of Mixed Flow Turbine. 
 
 Fig. 233 shows a section through the wheel and casing of a 
 double mixed flow turbine having adjustable guide blades to 
 regulate the flow. Fig. 234 shows a half longitudinal section of 
 the turbine, and Fig. 235 an outside elevation of the guide blade 
 regulating gear. The guide blades are surrounded by a large 
 
352 
 
 HYDRAULICS 
 
 vortex chamber, and the outer tips of the guide blades are of 
 variable shapes, Fig. 233, so as to diminish shock at the entrance 
 to the guide passages. Bach guide blade is really made in two 
 parts, one of which is made to revolve about the centre C, while 
 the outer tip is fixed. The moveable parts are made so that the 
 flow can be varied from zero to its maximum value. It will be 
 
 Fig. 233. Section through wheel and guide blades of Mixed Flow Turbine. 
 
 noticed that the mechanism for moving the guide blades is 
 entirely external to the turbine, and is consequently out of the 
 water. A further special feature is that between the ring R 
 and each of the guide blade cranks is interposed a spiral spring. 
 In the event of a solid body becoming wedged between two of 
 the guide blades, and thus locking one of them, the adjustment of 
 the other guide blades is not interfered with, as the spring con- 
 nected to the locked blade by its elongation will allow the ring 
 to rotate. 
 
 As with the inward and outward flow turbine, the mixed 
 flow turbine wheel may either work drowned, or exhaust into a 
 "suction tube." 
 
TURBINES 
 
 353 
 
 For a given flow, and width of wheel, the axial velocity 
 with which the water finally flows away from the wheel being the 
 same for the two cases, the diameter of a mixed flow turbine can 
 be made less than an inward flow turbine. As shown on page 340, 
 the diameter of the inward flow turbine is in large measure fixed 
 
 Fig. 234. Half-longitudinal section of Mixed Flow Turbine. 
 
 by the diameter of the exhaust openings of the wheel. For the 
 same axial velocity, and the same total flow, whether the turbine 
 is an inward or mixed flow turbine, the diameter d of the exhaust 
 openings must be about equal. The external diameter, therefore, 
 of the latter will be much smaller than for the former, and the 
 L. H. 23 
 
354 
 
 HYDRAULICS 
 
 general dimensions of the turbine will be also diminished. For 
 a given head H, the velocity v of the inlet edge being the same in 
 the two cases, the mixed flow turbine can be run at a higher 
 angular velocity, which is sometimes an advantage in driving 
 dynamos. 
 
 s 3 
 
TURBINES 355 
 
 Form of the vanes. At the receiving edge, the direction of the 
 blade is found in the same way as for an inward flow turbine. 
 
 ABC, Fig. 236, is the triangle of velocities, and BC is parallel 
 to the tip of the blade. This triangle has been drawn for the data 
 of the turbine shown in Figs. 233 235 ; v is 46'5 feet per second, 
 and from 
 
 Y = 33'5 feet per second. 
 The angle < is 139 degrees. 
 
 Triangle of Velocities 
 at receiving edge 
 
 Fig. 236. 
 
 The best form for the vane at the discharge is somewhat 
 difficult to determine, as the exact direction of flow at any point 
 on the discharging edge of the vane is not easily found. The 
 condition to be satisfied is that the water must leave the wheel 
 without any component in the direction of motion. 
 
 The following construction gives approximately the form of 
 the vane. 
 
 Make a section through the wheel as in Fig. 237. The outline 
 of the discharge edge FGrH is shown. This edge of the vane is 
 supposed to be on a radial plane, and the plan of it is, therefore, 
 a radius of the wheel, and upon this radius the section is taken. 
 
 It is now necessary to draw the form of the stream lines, as 
 they would be approximately, if the water entered the wheel 
 radially and flowed out axially, the vanes being removed. 
 
 Divide 04, Fig. 237, at the inlet, into any number of equal 
 parts, say four, and subdivide by the points a, 6, d, e. 
 
 Take any point A, not far from c, as centre, and describe 
 a circle MMi touching the crowns of the wheel at M and MI. 
 Join AM and AMi. 
 
 Draw a flat curve Mi MI touching the lines AM and AMi in M 
 and MI respectively, and as near as can be estimated, perpendicular 
 
 232 
 
356 
 
 HYDRAULICS 
 
 to the probable stream lines through a, 6, d, e, which can be 
 sketched in approximately for a short distance from 04. 
 
 Taking this curve MMi as approximately perpendicular to the 
 stream lines, two points/ and g near the centres of AM and 
 are taken. 
 
 Fig. 237. 
 
 Let the radius of the points g and / be r and r L respectively. 
 If any point Ci on MM! is now taken not far from A, the 
 peripheral area of Mci is nearly 27rrMci, and the .peripheral area 
 of MiCi is nearly 27rriM 1 Ci. 
 
 On the assumption that the mean velocity through MiM is 
 constant, the flow through Mci will be equal to that through 
 Mid, when, 
 
TURBINES 357 
 
 If, therefore, MM X is divided at the point c l so that 
 MI Ci _ r 
 McT~ri' 
 the point Ci will approximately be on the stream line through c. 
 
 If now when the stream line cc l is carefully drawn in, it is 
 perpendicular to MMi, the point Ci cannot be much in error. 
 
 A nearer approximation to c,\ can be found by taking new values 
 for r and n, obtained by moving the points / and g so that they 
 more nearly coincide with the centres of CiM and CiMi. If the 
 two curves are not perpendicular, the curve MMi and the point c l 
 are not quite correct, and new values of r and n will have to be 
 obtained by moving the points / and g. By approximation Ci can 
 be thus found with considerable accuracy. 
 
 By drawing other circles to touch the crown of the wheels, the 
 curves M 2 M 3 , M 4 M 5 etc. normal to the stream lines, and the points 
 c-2, c 3 , etc. on the centre stream line, can be obtained. 
 
 The curve 22, therefore, divides the stream lines into equal 
 parts. 
 
 Proceeding in a similar manner, the curves 11 and 33 can be 
 obtained, dividing the stream lines into four equal parts, and 
 these again subdivided by the curves aa, 66, dd, and ee, which 
 intersect the outlet edge of the vane at the points F, Gr, H and e 
 respectively. 
 
 To determine the direction of the tip of the vane at points on the 
 discharging edge. At the points F, Gr, H, the directions of the 
 stream lines are known, and the velocities U F , u&, Uu can be found, 
 since the flows through 01, 12, etc. are equal, and therefore 
 
 = ~ . 
 
 O7T 
 
 Draw a tangent FK to the stream line at F. This is the inter- 
 section, with the plane of the paper, of a plane perpendicular to 
 the paper and tangent to the stream line at F. 
 
 The point F in the plane of FK is moving perpendicular to the 
 plane of the paper with a velocity equal to w.R , <> being the 
 angular velocity of the wheel, and R the radius of the point F. 
 
 If a circle be struck on this plane with K as centre, this circle 
 may be taken as an imaginary discharge circumference of an 
 inward flow turbine, the velocity v of which is wR 0j and the tip of 
 the blade is to have such an inclination, that the water shall 
 discharge radially, i.e. along FK, with a velocity U F . Turning this 
 circle into the plane of the paper and drawing the triangle of 
 velocities FST, the inclination F of the tip of the blade at F in 
 the plane FK is obtained. 
 
 OF THE 
 
 UNIVERSITY 
 
358 
 
 HYDRAULICS 
 
 At Gr the stream line is nearly vertical, but o>R 2 can be set out 
 in the plane of the paper, as before, perpendicular to U G and the 
 inclination G , on this plane, is found. 
 
 At H, a H is fo und in tne same wa F> and tne direction o the 
 vane, in definite planes, at other points on its outlet edge, can be 
 
 similarly found. 
 
 Fig. 238. 
 
 Fig. 239. 
 
 Sections of the vane by planes 0Gb, and OiHd. These are 
 shown in Figs. 238 and 239, and are determined as follows. 
 
 Imagine a vertical plane tangent to the tip of the vane at 
 inlet. The angle this plane makes with the tangent to the wheel 
 at b is the angle </>, Fig. 236. Let BC of the same figure be the 
 
TURBINES 359 
 
 plan of a horizontal line lying in this plane, and BD the plan of 
 the radius of the wheel at b. The angle between these lines is y. 
 Let ft be the inclination of the plane OG6 to the horizontal. 
 From D, Fig. 236, set out DE, inclined to BD at an angle /?, 
 and intersecting AB produced in E, and draw BF perpendicular 
 toCB. 
 
 Make BF equal to BE and join OF. 
 
 Now set out a triangle BGrDi having BG- equal to OF, D X G 
 equal to DE, and the angle BGrDi a right angle. In the figure 
 Dj and D happen to coincide. 
 
 The angle BGD is the angle y 1? which the line of intersection of 
 the plane 0Gb, Fig. 237, with the plane tangent to the inlet tip of 
 the vane, makes with the radius 06. 
 
 In Fig. 238 the inclination of the inlet tip of the blade is y x as 
 shown. 
 
 To determine the angle a at the outlet edge, resolve U G , Fig. 
 237, along and perpendicular to OGr, U QG being the component 
 along OG-. 
 
 Draw the triangle of velocities DEF, Fig. 238. 
 The tangent to the vane at D is parallel to FE. 
 In the same way, the section on the plane lid, Fig. 237, may be 
 determined; the inclination at the inlet is y 2 , Fig. 239. 
 
 Mixed flow turbine working in open stream. A double turbine 
 working in open stream and discharging through a suction tube 
 is shown in Fig. 243. This is a convenient arrangement for 
 moderately low falls. Turbines, of this class, of 1500 horse- 
 power, having four wheels on the same shaft and working under 
 a head of 25 feet, and making 150 revolutions per minute, have 
 recently been installed by Messrs Escher Wyss at Wangen an der 
 Aare in Switzerland. 
 
 201. Cone turbine. 
 
 Another type of inward flow turbine, which is partly axial and 
 partly radial, is shown in Fig. 241, and is known as the cone 
 turbine. It has been designed by Messrs Escher Wyss to meet 
 the demand for a turbine that can be adapted to variable flows. 
 
 The example shown has been erected at Gusset near Lyons and 
 makes 120 revolutions per minute. 
 
 The wheel is divided into three distinct compartments, the 
 supply of water being regulated by three cylindrical sluices S, Si 
 and S 2 . The sluices S and Si are each moved by three vertical 
 spindles such as A and AI which carry racks at their upper ends. 
 These two sluices move in opposite directions and thus balance 
 each other. The sluice S 2 is normally out of action, the upper 
 
360 
 
 HYDRAULICS 
 
 compartment being closed. At low heads this upper compartment 
 is allowed to come into operation. The sluice S 2 carries a rack 
 which engages with a pinion P, connected to the vertical shaft T. 
 
 Feet,5 
 
 ZOFeeb 
 
 Fig. 240. 
 
 The shaft T is turned by hand by means of a worm and 
 wheel W. When it is desired to raise the sluice S 2 , it is revolved 
 by means of the pinion P until the arms F come between collars 
 D and E on the spindles carrying the sluice Si, and the sluice S 2 
 then rises and falls with Si . The pinion, gearing with racks on A 
 and AI, is fixed to the shaft M, which is rotated by the rack R 
 gearing with the bevel pinion Q. The rack R, is rotated by two 
 connecting rods, one of which C is shown, and which are under 
 the control of the hydraulic governor as described on page 378. 
 The wheel shaft can be adjusted by nuts working on the 
 square-threaded screw shown, and is carried on a special collar 
 bearing supported by the bracket B. The weight of the shaft is 
 partly balanced by the water-pressure piston which has acting 
 underneath it a pressure per unit area equal to that in the supply 
 chamber. The dimensions shown are in millimetres. 
 
TURBINES 
 
 361 
 
 Fig. 241. Cone Turbine. 
 
362 HYDRAULICS 
 
 202. Effect of changing the direction of the guide blade, 
 when altering the flow of inward flow and mixed flow 
 turbines. 
 
 As long as the velocity of a wheel remains constant, the 
 backward head impressed on the water by the wheel is the same, 
 and the pressure head, at the inlet to the wheel, will remain 
 practically constant as the guides are moved. The velocity of 
 flow U, through the guides, will, therefore, remain constant; 
 but as the angle 0, which the guide makes with the tangent to the 
 wheel, diminishes the radial component u, of U, diminishes. 
 
 Fig. 242. 
 
 Let ABC, Fig. 242, be the triangle of velocities for full opening, 
 and suppose the inclination of the tip of the blade is made parallel 
 to BC. On turning the guides into the dotted position, the incli- 
 nation being <f>\ , the triangle of velocities is ABCi , and the relative 
 velocity of the water and the periphery of the wheel is now BCj 
 which is inclined to the vane, and there is, consequently, loss due 
 to shock. 
 
 It will be seen that in the dotted position the tips of the guide 
 blades are some distance from the periphery of the wheel and it is 
 probable that the stream lines on leaving the guide blades follow 
 the dotted curves SS, and if so, the inclination of these stream 
 lines to the tangent to the wheel will be actually greater than <f>\, 
 and BCi will then be more nearly parallel to BC. The loss may 
 be approximated to as follows : 
 
 As the water enters the wheel its radial component will remain 
 unaltered, but its direction will be suddenly changed from BCi to 
 BC, and its magnitude to BC 2 ; CiC 2 is drawn parallel to AB. 
 A velocity equal to dC 2 has therefore to be suddenly impressed on 
 the water. 
 
 On page 68 it has been shown that on certain assumptions the 
 
TURBINES 
 
 363 
 
 head lost when the velocity of a stream is suddenly changed 
 from Vi to v 2 is 
 
 that is, it is equal to the head due to the relative velocity of 
 Vi and v 2 . 
 
 But CiCa is the relative velocity of Bd and BC 2 , and therefore 
 the head lost at inlet may be taken as 
 
 2? 
 
 k being a coefficient which may be taken as approximately unity. 
 
 203. Effect of diminishing the flow through turbines on 
 the velocity of exit. 
 
 If water leaves a wheel radially when the flow is a maximum, 
 it will not do so for any other flow. 
 
 The angle of the tip of the blade at exit is unalterable, and if 
 u and UQ are the radial velocities of flow, at full and part load 
 respectively, the triangles of velocity are DBF and DEFi, Fig. 243. 
 
 For part flow, the velocity with which the water leaves the 
 wheel is Ui. If this is greater than u, and the wheel is drowned, 
 or the exhaust takes place into the air, the theoretical hydraulic 
 efficiency is less than for full load, but if the discharge is down a 
 suction tube the velocity with which the water leaves the tube is 
 less than for full flow and the theoretical hydraulic efficiency is 
 greater for the part flow. The loss of head, by friction in the 
 wheel due to the relative velocity of the water and the vane, 
 which is less than at full load, should also be diminished, as also, 
 the loss of head by friction in the supply and exhaust pipes. 
 The mechanical losses remain practically constant at all loads. 
 
 Fig. 243. 
 
 Fig. 244. 
 
 The fact that the efficiency of turbines diminishes at part loads 
 must, therefore, in large measure be due to the losses by shock 
 being increased more than the friction losses are diminished. 
 
 By suitably designing the vanes, the greatest efficiency of 
 inward flow and mixed flow turbines can be obtained at some 
 fraction of full load. 
 
364 
 
 HYDRAULICS 
 
 204. Regulation of the flow by cylindrical gates. 
 
 When the speed of the turbine is adjusted by a gate between 
 the guides and the wheel, and the flow is less than the normal, the 
 velocity U with which the water leaves the guide is altered in 
 magnitude but not in direction. 
 
 Let ABC be the triangle of velocities, Fig. 244, when the flow is 
 normal. 
 
 Let the flow be diminished until the velocity with which the 
 water leaves the guides is U ,. equal to AD. 
 
 Then BD is the relative velocity of U and v, and U Q is the 
 radial velocity of flow into the wheel. 
 
 Draw DK parallel to AB. Then for the water to move along 
 the vane a sudden velocity equal to KD must be impressed on 
 
 & (KD) 2 
 the water, and there is a head lost equal to ^ . 
 
 To keep the velocity U more nearly constant Mr Swain has 
 introduced the gate shown in Fig. 245. The gate g is rigidly 
 connected to the guide blades, and to adjust the flow the guide 
 blades as well as the gate are moved. The effective width of the 
 guides is thereby made approximately proportional to the quantity 
 of flow, and the velocity U remains more nearly constant. If the 
 gate is raised, the width b of the wheel opening will be greater 
 than bi the width of the gate opening, and the radial velocity u 
 
 Fig. 245. Swain Gate. 
 
 Fig. 246. 
 
TURBINES 
 
 365 
 
 into the wheel will consequently be less than the radial velocity u 
 from the guides. If U is assumed constant the relative velocity of 
 the water and the vane will suddenly change from BC to BCi, 
 Fig. 248. Or it may be supposed that in the space between the 
 guide and the wheel the velocity U changes from AC to ACi. 
 
 The loss of head will now be ~ - . 
 
 205. The form of the wheel vanes between the inlet and 
 outlet of turbines. 
 
 The form of the vanes between inlet and outlet of turbines 
 should be such, that there is no sudden change in the relative 
 velocity of the water and the wheel. 
 
 Consider the case of an inward flow turbine. Having given 
 a form to the vane and fixed the width between the crowns of the 
 wheel the velocity relative to the wheel at any radius r can be 
 found as follows. 
 
 Take any circumferential section ef at radius r, Fig. 247. Let 
 b be the effective width between the crowns, and d the effective 
 width ef between the vanes, and let q be the flow in cubic feet 
 per second between the vanes Ae and B/. 
 
 Fig. 247. Relative velocity of the water and the vanes. 
 
 Fig. 248. 
 
366 HYDRAULICS 
 
 The radial velocity through e/is 
 
 Ur= bd' 
 
 Find by trial a point near the centre of ef such that a circle 
 drawn with as centre touches the vanes at M and MI. 
 
 Suppose the vanes near e and / to be struck with arcs of circles. 
 Join to the centres of these circles and draw a curve MCMi 
 touching the radii OM and OMi at M and Mi respectively. 
 
 Then MCMi will be practically normal to the stream lines 
 through the wheel. The centre of MCMi may not exactly 
 coincide with the centre of ef, but a second trial will probably 
 make it do so. 
 
 If then, b is the effective width between the crowns at C, 
 
 can be scaled off the drawing and v r calculated. 
 The curve of relative velocities for varying radii can then be 
 plotted as shown in the figure. 
 
 B 
 
 Fig. 249. 
 
 It will be seen that in this case the curve of relative velocities 
 changes fairly suddenly between c and h. By trial, the vanes 
 should be made so that the variation of velocity is as uniform 
 as possible. 
 
 If the vanes could be made involutes of a circle of 'radius K , 
 
TURBINES 367 
 
 as in Fig. 249, and the crowns of the wheel parallel, the relative 
 velocity of the wheel and the water would remain constant. 
 This form of vane is however entirely unsuitable for inward 
 flow turbines and could only be used in very special cases for 
 outward flow turbines, as the angles <f> and which the involute 
 makes with the circumferences at A and B are not independent, 
 for from the figure it is seen that, 
 
 
 
 i . O 
 
 and sin 9 ^7 
 
 XV 
 
 sin(9 R 
 
 or . - = . 
 
 sin 9 r 
 
 The angle must clearly always be greater than 9. 
 
 206. The limiting head for a single stage reaction 
 turbine. 
 
 Reaction turbines have not yet been made to work under heads 
 higher than 430 feet, impulse turbines of the types to be presently 
 described being used for heads greater than this value. 
 
 From the triangle of velocities at inlet of a reaction turbine, 
 e.g. Fig. 226, it is seen that the whirling velocity V cannot be 
 greater than 
 
 v + ucot<j>. 
 
 Assuming the smallest value for 9 to be 30 degrees, and the 
 maximum value for u to be 0'25 \/2grH, the general formula 
 
 V=eH 
 
 g 
 
 becomes, for the limiting case, 
 
 If v is assumed to have a limiting value of 100 feet per second, 
 which is higher than generally allowed in practice, and e to 
 be 0*8, then the maximum head H which can be utilised in a one 
 stage reaction turbine, is given by the equation 
 
 25'6H- 346 >/H = 10,000, 
 from which H = 530 feet. 
 
 207. Series or multiple stage reaction turbines. 
 
 Professor Osborne Reynolds has suggested the use of two 
 or more turbines in series, the same water passing through them 
 successively, and a portion of the head being utilised in each. 
 
 For parallel flow turbines, Reynolds proposed that the wheels 
 
368 
 
 HYDRAULICS 
 
 and fixed blades be arranged alternately as shown in Fig. 250*. 
 This arrangement, although not used in water turbines, is very 
 largely used in reaction steam turbines. 
 
 Fig. 250. 
 
 Figs. 251, 252. Axial Flow Impulse Turbine. 
 Taken from Prof. Reynolds' Scientific Papers, Vol. i. 
 
TURBINES 369 
 
 208. Impulse turbines. 
 
 Girard turbine. To overcome the difficulty of diminution of 
 efficiency with diminution of flow, 
 Girard introduced, about 1850, the 
 free deviation or partial admission 
 turbine. 
 
 Instead of the water being 
 admitted to the wheel throughout 
 the whole circumference as in the 
 reaction turbines, in the Grirard 
 turbine it is only allowed to enter 
 the wheel through guide passages 
 in two diametrically opposite 
 quadrants as shown in Figs. 252 
 254. In the first two, the flow is 
 axial, and in the last radial. 
 
 In Fig. 252 above the guide crown are two quadrant-shaped 
 plates or gates 2 and 4, which are made to rotate about a vertical 
 axis by means of a toothed wheel. When the gates are over the 
 quadrants 2 and 4, all the guide passages are open, and by turning 
 the gates in the direction of the arrow, any desired number of the 
 passages can be closed. In Fig. 254 the variation of flow is- 
 effected by means of a cylindrical quadrant-shaped sluice, which, 
 as in the previous case, can be made to close any desired number 
 of the guide passages. Several other types of regulators for 
 impulse turbines were introduced by Girard and others. 
 
 Fig. 253 shows a regulator employed by Fontaine. Above the 
 guide blades, and fixed at the opposite ends of a diameter DD, 
 are two indiarubber bands, the other ends of the bands being 
 connected to two conical rollers. The conical rollers can rotate 
 on journals, formed on the end of the arms which are connected 
 to the toothed wheel TW. A pinion P gears with TW, and by 
 rotating the spindle carrying the pinion P, the rollers can be made 
 to unwrap, or wrap up, the indiarubber band, thus opening or 
 closing the guide passages. 
 
 As the Girard turbine is not kept full of water, the whole of 
 the available head is converted into velocity before the water 
 enters the wheel, and the turbine is a pure impulse turbine. 
 
 To prevent loss of head by broken water in the wheel, the air 
 should be freely admitted to the buckets as shown in Figs. 252 
 and 254. 
 
 For small heads the wheel must be horizontal but for large 
 heads it may be vertical. 
 
 This class of turbine has the disadvantage that it cannot 
 L. H. 24 
 
370 
 
 HYDRAULICS 
 
 run drowned, and hence must always be placed above the tail 
 water. For low and variable heads the full head cannot therefore 
 be utilised, for if the wheel is to be clear of the tail water, an 
 amount of head equal to half the width of the wheel must of 
 necessity be lost. 
 
 Fig. 254. Girard Radial flow Impulse Turbine. 
 
 To overcome this difficulty Grirard placed the wheel in an air- 
 tight tube, Fig. 254, the lower end of which is below the tail water 
 level, and into which air is pumped by a small auxiliary air-pump, 
 the pressure being maintained at the necessary value to keep the 
 surface of the water in the tube below the wheel. 
 
TURBINES 371 
 
 Let H be the total head above the tail water level of the supply 
 water, the pressure head due to the atmospheric pressure, H 
 
 the distance of the centre of the wheel below the surface of the 
 supply water, and Ji the distance of the surface of the water in 
 the tube below the tail water level. Then the air-pressure in 
 the tube must be 
 
 &+*., 
 
 w 
 and the head causing velocity of flow into the wheel is, therefore, 
 
 W \W 
 
 So that wherever the wheel is placed in the tube below the tail 
 water the full fall H is utilised. 
 
 This system, however, has not found favour in practice, owing 
 to the difficulty of preserving the pressure in the tube. 
 
 209. The form of the vanes for impulse turbines, neg- 
 lecting friction. 
 
 The receiving tip of the vane should be parallel to the relative 
 velocity V r of the water and the edge of the vane, Fig. 255. 
 
 At exit the relative velocity v r , Fig. 256, neglecting friction, 
 must be equal to the relative velocity V r at inlet. 
 
 If the angle a which the tip of the vane at exit makes with 
 the direction of v l is known the triangle of velocities can be drawn, 
 by setting out DB equal to v : and EF at an angle a with it and 
 equal to V r . Then DF is the velocity with which the water leaves 
 the wheel. 
 
 For the axial flow turbine Vi equals v, and the triangle of 
 velocities at exit is AGB, Fig. 255. 
 
 If the velocity with which the water leaves the wheel is Ui, 
 the theoretical hydraulic efficiency is 
 
 H-SL 
 
 E ^ 1 U * 2 
 
 H IP 
 
 and is independent of the direction of Ui . 
 
 It should be observed, however, that in the radial flow turbine 
 the area of the section of the stream by the circumference of the 
 wheel, for a given flow, will depend upon the radial component of 
 Ui, and in the axial flow turbine the area of the section of the 
 stream by a plane perpendicular to the axis will depend upon the 
 axial component of Ui . That is, in each case the area will depend 
 upon the component of Ui perpendicular to Vi . 
 
 242 
 
372 
 
 HYDRAULICS 
 
 Now the section of the stream must not fill the outlet area r of 
 the wheel, and the minimum area of this outlet so that it is just 
 not filled will clearly be obtained for a given value of Ui when Ui 
 is perpendicular to Vi*, or is radial in the outward flow and axial in 
 the parallel flow turbine. 
 
 For the parallel flow turbine since BC and BGr, Fig. 255, are 
 equal, Ui is clearly perpendicular to v l when 
 
 and the inclinations a and <f> of the tips of the vanes are equal. 
 
 Figs. 255, 256. 
 
 D 
 
 Fig. 257. 
 
 If R and r are the outer and inner radii of the radial flow 
 turbine respectively, 
 
 R 
 
 It is often stated that this is the condition for maximum efficiency but it only 
 is so, as stated above, for maximum flow for the given machine. The efficiency 
 >nly depends upon the magnitude of U x and not upon its direction. 
 
TURBINES 373 
 
 For Ui to be radial 
 
 V r = Vi sec a 
 
 v.K 
 -~- sec a, 
 
 y y 
 
 and if v is made equal to -~ , V r from Fig. 255 is equal to 77 sec <, 
 
 Z Z 
 
 and therefore, 
 
 210. Triangles of velocity for an axial flow impulse tur- 
 bine considering friction. 
 
 The velocity with which the water leaves the guide passages 
 may be taken as from 0'94 to 0*97 \/2#H, and the hydraulic losses 
 in the wheel are from 5 to 10 per cent. 
 
 If the angle between the jet and the direction of motion of the 
 vane is taken as 30 degrees, and U is assumed as 0'95 \/20H, and v 
 as 0'45\/2#H, the triangle of velocities is ABC, Fig. 257. 
 
 Taking 10 per cent, of the head as being lost in the wheel, the 
 relative velocity v r at exit can be obtained from the expression 
 
 If now the velocity of exit Ui be taken as 0*22 V20H, and 
 circles with A and B as centres, and Ui and v r as radii be 
 described, intersecting in D, ABD the triangle of velocities at exit 
 is obtained, and Ui is practically axial as shown in the figure. 
 On these assumptions the best velocity for the rim of the wheel is 
 therefore '45 v2<?H instead of " 
 
 The head lost due to the water leaving the wheel with velocity 
 u is '048H, and the theoretical hydraulic efficiency is therefore 
 95'2 per cent. 
 
 The velocity head at entrance is 0'9025H and, therefore, '097H 
 has been lost when the water enters the wheel. 
 The efficiency, neglecting axle friction, will be 
 H - O'lH - 0-048H - 0-097H 
 
 -TT 
 = 76 per cent, nearly. 
 
 211. Impulse turbine for high heads. 
 
 For high heads Girard introduced a form of impulse turbine, 
 of which the turbine shown in Figs. 258 and 259, is the modern 
 development. 
 
 The water instead of being delivered through guides over an 
 arc of a circle, is delivered through one or more adjustable nozzles. 
 
TURBINES 375 
 
 In the example shown, the wheel has a mean diameter of 6'9 feet 
 and makes 500 revolutions per minute; it develops 1600 horse- 
 power under a head of 1935 feet. 
 
 The supply pipe is of steel and is 1'312 feet diameter. 
 
 The form of the orifices has been developed by experience, and 
 is such that there is no sudden change in the form of the liquid 
 vein, and consequently no loss due to shock. 
 
 The supply of water to the wheel is regulated by the sluices 
 shown in Fig. 258, which, as also the axles carrying the same, 
 are external to the orifices, and can consequently be lubricated 
 while the turbine is at work. The sluices are under the control 
 of a sensitive governor and special form of regulator. 
 
 As the speed of the turbine tends to increase the regulator 
 moves over a bell crank lever and partially closes both the orifices, 
 Any decrease in speed of the turbine causes the reverse action to 
 take place. 
 
 The very high peripheral speed of the wheel, 205 feet per 
 second, produces a high stress in the wheel due to centrifugal 
 forces. Assuming the weight of a bar of the metal of which the 
 rim is made one square inch in section and one foot long as 
 3'36 Ibs., the stress per sq. inch in the hoop surrounding the 
 wheel is 
 
 3'36. i> 2 
 /= 9 
 = 4400 Ibs. per sq. inch. 
 
 To avoid danger of fracture, steel laminated hoops are shrunk 
 on to the periphery of the wheel. 
 
 The crown carrying the blades is made independent of the disc 
 of the wheel, so that it may be replaced when the blades become 
 worn, without an entirely new wheel being provided. 
 
 The velocity of the vanes at the inner periphery is 171 feet per 
 second, and is, therefore, 0'484 \/2gH. 
 
 If the velocity U with which the water leaves the orifice is 
 taken as 0'97 \/2gH, and the angle the jet makes with the tangent 
 to the wheel is 30 degrees, the triangle of velocities at entrance is 
 ABC, Fig. 260, and the angle <f> is 53'5 degrees. 
 
 The velocity v l of the outer edges of the vanes is 205 feet per 
 second, and assuming there is a loss of head in the wheel, equal to 
 6 per cent, of H, 
 
 2g 2g 
 and v r = 123'5 ft. per second. 
 
376 
 
 HYDRAULICS 
 
 If then the angle a is 30 degrees the triangle of velocities at 
 exit is DEF, Fig. 261. 
 
 The velocity with which the water leaves the wheel is then 
 Ui = 95 feet per sec., and the head lost by this velocity is 140 feet 
 or -073H. 
 A v=m J}___ 
 
 5*T-8> 
 
 A 
 
 D 
 
 Fig. 260. 
 
 Fig. 261. 
 
 The head lost in the pipe and nozzle is, on the assumption 
 made above, 
 
 H-(0'97) 2 H=0'06H, 
 
 and the total percentage loss of head is, therefore, 
 
 and the hydraulic efficiency is 80*7 per cent. 
 
 \\ >> \1 
 
 \\ \ V 
 
 \ \ 1 
 
 \ \ \ 
 
 
 // I 
 
 
 ^^"^ 
 
 m 
 
 Fig. 262. Pelton Wheel. 
 
TURBINES 
 
 377 
 
 The actual efficiency of a similar turbine at full load was found 
 by experiment to be 78 per cent., which allows a mechanical loss 
 of 2' 7 per cent. 
 
 212. Pelton wheel. 
 
 A form of impulse turbine now very largely used for high heads 
 is known as the Pelton wheel. 
 
 A number of cups, as shown in Figs. 262 and 266, is fixed to a 
 wheel which is generally mounted on a horizontal axis. The 
 water is delivered to the wheel through a rectangular shaped 
 nozzle, the opening of which is generally made adjustable, either 
 by means of a hand wheel as in Fig. 262, or automatically by a 
 regulator as in Fig. 266. 
 
 As shown on page 276, the theoretical efficiency of the wheel is 
 unity and the best velocity for the cups is one-half the velocity of 
 the jet. This is also the velocity generally given to the cups 
 in actual examples. The width of the cups is from 2J to 
 4 times the thickness of the jet, and the width of the jet is about 
 twice its thickness. 
 
 The actual efficiency is between 70 and 82 per cent. 
 
 Table XXXVIII gives the numbers of revolutions per minute, 
 the diameters of the wheels and the nett head at the nozzle in 
 a number of examples. 
 
 TABLE XXXVIII. 
 Particulars of some actual Pelton wheels. 
 
 Head 
 
 in feet 
 
 Diameter 
 of wheel 
 (two wheels) 
 
 Eevolutions 
 per minute 
 
 V 
 
 U 
 
 H. P. 
 
 262 
 
 39-4" 
 
 375 
 
 64-5 
 
 129 
 
 500 
 
 *233 
 
 7" 
 
 2100 
 
 64 
 
 125 
 
 5 
 
 *197 
 
 20" 
 
 650 
 
 56-5 
 
 112 
 
 10 
 
 722 
 
 39" 
 
 650 
 
 111 
 
 215 
 
 167 
 
 382 
 
 60" 
 
 300 
 
 79 
 
 156 
 
 144 
 
 *289 
 
 54" 
 
 310 
 
 73 
 
 136 
 
 400 
 
 508 
 
 90" 
 
 200 
 
 79 
 
 180 
 
 300 
 
 * Picard Pictet and Co., the remainder by Escher Wyss and Co. 
 
 213. Oil pressure governor or regulator. 
 
 The modern applications of turbines to the driving of electrical 
 machinery, has made it necessary for particular attention to be 
 paid to the regulation of the speed of the turbines. 
 
 The methods of regulating the flow by cylindrical speed gates 
 and moveable guide blades have been described in connection with 
 
378 
 
 HYDRAULICS 
 
 various turbines but the means adopted for moving the gates and 
 guides have not been discussed. 
 
 Until recent years some form of differential governor was 
 almost entirely used, but these have been almost completely 
 superseded by hydraulic and oil governors. 
 
 Figs. 263 and 264 show an oil governor, as constructed by 
 Messrs Escher Wyss of Zurich. 
 
 Figs. 263, 264. Oil Pressure Regulator for Turbines. 
 
 A piston P having a larger diameter at one end than at the 
 other, and fitted with leathers I and Zi, fits into a double cylinder 
 Ci. Oil under pressure is continuously supplied through a pipe S 
 into the annulus A between the pistons, while at the back of the 
 large piston the pressure of the oil is determined by the regulator. 
 
TURBINES 
 
 379 
 
 Suppose the regulator to be in a definite position, the space 
 behind the large piston being full of oil, and the 
 turbine running at its normal speed. The valve V 
 (an enlarged diagrammatic section is shown in 
 Fig. 265) will be in such a position that oil cannot 
 enter or escape from the large cylinder, and the 
 pressure in the annular ring between the pistons 
 will keep the regulator mechanism locked. 
 
 If the wheel increases in speed, due to a 
 diminution of load, the balls of the spring loaded 
 governor Gr move outwards and the sleeve M 
 rises. For the moment, the point D on the lever 
 MD is fixed, and the lever turns about D as a 
 fulcrum, and thus raises the valve rod NV. This 
 allows oil under pressure to enter the large 
 cylinder and the piston in consequence moves to 
 
 Fig. 265. 
 
 the right, and moves the turbine gates in the manner described later. 
 As the piston moves to the right, the rod R, which rests on the 
 wedge W connected to the piston, falls, and the point D of the 
 lever MD consequently falls and brings the valve Y back to its 
 original position. The piston P thus takes up a new position 
 corresponding to the required gate opening. The speed of the 
 turbine and of the governor is a little higher than before, the 
 increase in speed depending upon the sensitiveness of the governor. 
 On the other hand, if the speed of the wheel diminishes, the 
 sleeve M and also the valve V falls and the oil from behind the 
 large piston escapes through the exhaust E, the piston moving 
 to the left. The wedge W then lifts the fulcrum D, the valve V 
 is automatically brought to its central position, and the piston P 
 takes up a new position, consistent with the gate opening being 
 sufficient to supply the necessary water required by the wheel. 
 
 A hand wheel and screw, Fig. 264, are also provided, so that 
 the gates can be moved by hand when necessary. 
 
 The piston P is connected by the connecting rod BE to a crank 
 EF, which rotates the vertical shaft T. A double crank KK is 
 connected by the two coupling rods shown to a rotating toothed 
 wheel R, Fig. 241, turning about the vertical shaft of the turbine, 
 and the movement, as described on page 360, causes the adjust- 
 ment of the speed gates. 
 
 214. Water pressure regulators for impulse turbines. 
 Fig. 266 shows a water pressure regulator as applied to regulate 
 the flow to a Pel ton wheel. 
 
 The area of the supply nozzle is adjusted by a beak B which 
 
380 
 
 HYDRAULICS 
 
 Figs. 266, 267. Pelton Wheel and Water Pressure Eegulator. 
 
TURBINES 
 
 381 
 
 rotates about the centre 0. The pressure of the water in the 
 supply pipe acting on this beak tends to lift it and thus to open 
 the orifice. The piston P, working in a cylinder C, is also acted 
 upon, on its under side, by the pressure of the water in the supply 
 pipe and is connected to the beak by the connecting rod DE. 
 The area of the piston is made sufficiently large so that when the 
 top of the piston is relieved of pressure the pull on the connecting 
 rod is sufficient to close the orifice. 
 
 The pipe p conveys water under the same pressure, to the 
 valve Y, which may be similar to that described in connection with 
 the oil pressure governor, Fig. 265. 
 
 A piston rod passes through the top of the cylinder, and carries 
 a nut, which screws on to the square thread cut on the rod. A 
 lever eg, Fig. 268, which is carried on the fixed fulcrum e, is made 
 to move with the piston. A link /A connects ef with the lever 
 MN, one end M of which moves with the governor sleeve and the 
 other end N is connected to the valve rod NV. The valve V is 
 shown in the neutral position. 
 
 Fig. 268. 
 
 Suppose now the speed of the turbine to increase. The 
 governor sleeve rises, and the lever MN turns about the fulcrum 
 A which is momentarily at rest. The valve V falls and opens the 
 top of the cylinder to the exhaust. The pressure on the piston 
 P now causes it to rise, and closes the nozzle, thus diminishing 
 the supply to the turbine. As the piston rises it lifts again the 
 lever MN by means of the link A/, and closes the valve V. A 
 new position of equilibrium is thus reached. If the speed of the 
 
382 
 
 HYDRAULICS 
 
 governor decreases the governor sleeve falls, the valve Y rises, 
 and water pressure is admitted to the top of the piston, which is 
 then in equilibrium, and the pressure on the beak B causes it to 
 move upwards and thus open the nozzle. 
 
 Hydraulic valve for water regulator. Instead of the simple 
 piston valve controlled mechanically, Messrs Bscher Wyss use, for 
 high heads, a hydraulic double-piston valve Pp, Fig. 269. 
 
 This piston valve has a small bore through its centre by means 
 of which high pressure water which is admitted below the valve 
 can pass to the top of the large piston P. Above the piston is a 
 small plug valve Y which is opened and closed by the governor. 
 
 Fig. 269. Hydraulic valve for automatic regulation. 
 
 If the speed of the governor decreases, the valve Y is opened, 
 thus allowing water to escape from above the piston valve, and the 
 pressure on the lower piston p raises the valve. Pressure water is 
 thus admitted above the regulator piston, and the pressure on the 
 beak opens the nozzle. As the governor falls the valve Y closes, 
 the exhaust is throttled, and the pressure above the piston P rises. 
 When the exhaust through Y is throttled to such a degree that 
 the pressure on P balances the pressure on the under face of the 
 piston p, the valve is in equilibrium and the regulator piston is 
 locked. 
 
TURBINES 383 
 
 If the speed of the governor increases, the valve V is closed, 
 and the excess pressure on the upper face of the piston valve 
 causes it to descend, thus connecting the regulator cylinder to 
 exhaust. The pressure on the under face of the regulator piston 
 then closes the nozzle. 
 
 Filter. Between the conduit pipe and the governor valve V, 
 is placed a filter, Figs. 270 and 271, to remove any sand or grit 
 contained in the water. 
 
 Within the cylinder, on a hexagonal frame, is stretched a 
 piece of canvas. The water enters the cylinder by the pipe E, and 
 after passing through the canvas, enters the central perforated 
 pipe and leaves by the pipe S. 
 
 Figs. 270, 271. Water Filter for Impulse Turbine Regulator. 
 
 To clean the filter while at work, the canvas frame is revolved 
 by means of the handle shown, and the cock R is opened. Each 
 side of the hexagonal frame is brought in turn opposite the 
 chamber A, and water flows outwards through the canvas and 
 through the cock R, carrying away any dirt that may have 
 collected outside the canvas. 
 
 Auxiliary valve to prevent hammer action. When the pipe line 
 is long an auxiliary valve is frequently fitted on the pipe near to 
 the nozzle, which is automatically opened by means of a cataract 
 motion* as the nozzle closes, and when the movement of the nozzle 
 beak is finished, the valve" slowly closes again. 
 
 If no such provision is made a rapid closing of the nozzle 
 means that a large mass of water must have its momentum 
 quickly changed and very large pressures may be set up, or in 
 other words hammer action is produced, which may cause fracture 
 of the pipe. 
 
 When there is an abundant supply of water, the auxiliary 
 valve is connected to the piston rod of the regulator and opened 
 and closed as the piston rod moves, the valve being adjusted so 
 that the opening increases by the same amount that the area of 
 the orifice diminishes. 
 
 * See Engineer, Vol. xc., p. 255. 
 
384 HYDRAULICS 
 
 If the load on the wheel does not vary through a large range 
 the quantity of water wasted is not large. 
 
 215. Hammer blow in a long turbine supply pipe. 
 
 Let L be the length of the pipe and d its diameter. 
 The weight of water in the pipe is 
 
 Let the velocity change by an amount dt? in time dt. Then the 
 
 Wdv 
 
 rate of change of momentum is --TJ-, and on a cross section of 
 
 got 
 
 the lower end of the column of water in the pipe a force P must 
 be applied equal to this. 
 
 -o w w^d 2 dv 
 
 Therefore P = 7 -- . . 
 
 4 g ct 
 
 Referring to Fig. 266, let b be the depth of the orifice and di its 
 width. 
 
 Then, if r is the distance of D from the centre about which the 
 beak turns, and n is the distance of the closing edge of the beak 
 from this centre, and if at any moment the velocity of the piston 
 is v feet per second, the velocity of closing of the beak will be 
 
 In any small element of time dt the amount by which the 
 nozzle will close is 
 
 96= a*. 
 
 r 
 
 Let it be assumed that U, the velocity of flow through the 
 nozzle, remains constant. It will actually vary, due to the 
 resistances varying with the velocity, but unless the pipe is very 
 long the error is not great in neglecting the variation. If then v 
 is the velocity in the pipe at the commencement of this element of 
 time and v - Cv at the end of it, and A the area of the pipe, 
 
 v.A=b.d l .U .............................. (1) 
 
 and (^-8^)A = 6-a.^.U ............... (2). 
 
 Subtracting (2) from (1), 
 
 or 
 
 Ot 
 
TURBINES 385 
 
 If W is the weight of water in the pipe, the force P in pounds 
 that will have to be applied to change the velocity of this water 
 by dv in time dt is 
 
 mi -D ri Vo 
 
 Therefore ~r~ , 
 
 g T j\. 
 
 and the pressure per sq. inch produced in the pipe near the 
 nozzle is 
 
 W n dJJvo 
 W -- 7-9- . 
 g r A 2 
 
 Suppose the nozzle to be completely closed in a time t seconds, 
 and during the closing the piston P moves with simple harmonic 
 motion. 
 
 Then the distance moved by the piston to close the nozzle is 
 
 br 
 
 n' 
 
 and the time taken to move this distance is t seconds. 
 The maximum velocity of the piston is then 
 
 and substituting in (3), the maximum value of is, therefore, 
 
 c/c 
 
 dv _ 
 ot~ 
 and the maximum pressure per square inch is 
 
 i.U ir.W.Q * Wv 
 Pm ~ 2gtA* ~2g.t.A*~2t' gA> 
 
 where Q is the flow in cubic feet per second before the orifice 
 began to close, and v is the velocity in the pipe. 
 
 Example. A 500 horse-power Pelton Wheel of 75 per cent, efficiency, and working 
 under a head of 260 feet, is supplied with water by a pipe 1000 feet long and 
 2' 3" diameter. The load is suddenly taken off, and the time taken by the 
 regulator to close the nozzle completely is 5 seconds. 
 
 On the assumption that the nozzle is completely closed (1) at a uniform rate, 
 and (2) with simple harmonic motion, and that no relief valve is provided, 
 determine the pressure produced at the nozzle. 
 
 The quantity of water delivered to the wheel per second when working at full 
 power is 
 
 500 x 33,000 
 
 The weight of water in the pipe is 
 
 W = 62-4xJ. (2-25) 2 xlOOO 
 
 = 250,000 Ibs. 
 L. H. 25 
 
386 HYDRAULICS 
 
 21'7 
 The velocity is gTg^ 5 ' 25 ft - P er sec - 
 
 In case (1) the total pressure acting on the lower end of the column of water in 
 pipe is 
 
 the pipe is 
 
 p _ 250, 000x5-25 
 
 9 x 5 
 
 = 8200 Ibs. 
 
 The pressure per sq. inch is 
 ft 900 
 
 p = ZlL = 14-5 Ibs. per sq. inch. 
 
 -27 2 
 
 In case (2) p m =^ -^^ =22-8 Ibs. per sq. inch. 
 
 & i . g . A 
 
 EXAMPLES. 
 
 (1) Find the theoretical horse-power of an overshot water-wheel 22 feet 
 diameter, using 20,000,000 gallons of water per 24 hours under a total head 
 of 25 feet. 
 
 (2) An overshot water-wheel has a diameter of 24 feet, and makes 3'5 
 revolutions per minute. The velocity of the water as it enters the buckets 
 is to be twice that of the wheel's periphery. 
 
 If the angle which the water makes with the periphery is to be 15 
 degrees, find the direction of the tip of the bucket, and the relative velocity 
 of the water and the bucket. 
 
 (3) The sluice of an overshot water-wheel 12 feet diameter is vertically 
 above the centre of the wheel. The surface of the water in the sluice 
 channel is 2 feet 6 inches above the top of the wheel and the centre of the 
 sluice opening is 8 inches above the top of the wheel. The velocity of the 
 wheel periphery is to be one-half that of the water as it enters the buckets. 
 Determine the number of rotations of the wheel, the point at which the 
 water enters the buckets, and the direction of the edge of the bucket. 
 
 (4) An overshot wheel 25 feet diameter having a width of 5 feet, and 
 depth of crowns 12 inches, receives 450 cubic feet of water per minute, and 
 makes 6 revolutions per minute. There are 64 buckets. 
 
 The water enters the wheel at 15 degrees from the crown of the wheel 
 with a velocity equal to twice that of the periphery, and at an angle of 20 
 degrees with the tangent to the wheel. 
 
 Assuming the buckets to be of the form shown in Fig. 180, the length 
 of the radial portion being one -half the length of the outer face of the 
 bucket, find how much water enters each bucket, and, allowing for centri- 
 fugal forces, the point at which the water begins to leave the buckets. 
 
 (5) An overshot wheel 32 feet diameter has shrouds 14 inches deep, 
 and is required to give 9 horse -power when making 5 revolutions per minute. 
 
 Assuming the buckets to be one-third filled with water and of the same 
 form as in the last question, find the width of the wheel, when the total 
 fall is 32 feet and the efficiency 60 per cent. 
 
TURBINES 387 
 
 Assuming the velocity of the water in the penstock to be If times that 
 of the wheel's periphery, and the bottom of the penstock level with the top 
 of the wheel, find the point at which the water enters the wheel. Find also 
 where water begins to discharge from the buckets. 
 
 (6) A radial blade impulse wheel of the same width as the channel in 
 which it runs, is 15 feet diameter. The depth of the sluice opening is 
 12 inches and the head above the centre of the sluice is 3 feet. Assuming 
 a coefficient of velocity of 0'8 and that the edge of the sluice is rounded so 
 that there is no contraction, and the velocity of the rim of the wheel is 0*4 
 the velocity of flow through the sluice, find the theoretical efficiency of 
 the wheel. 
 
 (7) An overshot wheel has a supply of 30 cubic feet per second on a fall 
 of 24 feet. 
 
 Determine the probable horse-power of the wheel, and a suitable 
 width for the wheel. 
 
 (8) The water impinges on a Poncelet float at 15 with the tangent to 
 the wheel, and the velocity of the water is double that of the wheel. Find, 
 by construction, the proper inclination of the tip of the float. 
 
 (9) In a Poncelet wheel, the direction of the jet impinging on the floats 
 makes an angle of 15 with the tangent to the circumference and the tip of 
 the floats makes an angle of 30 with the same tangent. Supposing the 
 velocity of the jet to be 20 feet per second, find, graphically or otherwise, 
 (1) the proper velocity of the edge of the wheel, (2) the height to which the 
 water will rise on the float above the point of admission, (3) the velocity 
 and direction of motion of the water leaving the float. 
 
 (10) Show that the efficiency of a simple reaction wheel increases 
 with the speed when frictional resistances are neglected, but is greatest 
 at a finite speed when they are taken into account. 
 
 If the speed of the orifices be that due to the head (1) find the efficiency, 
 neglecting friction ; (2) assuming it to be the speed of maximum efficiency, 
 show that f of the head is lost by friction, and ^ by final velocity of water. 
 
 (11) Explain why, in a vortex turbine, the inner ends of the vanes are 
 inclined backwards instead of being radial. 
 
 (12) An inward flow turbine wheel has radial blades at the outer 
 
 periphery, and at the inner periphery the blade makes an angle of 30 with 
 
 -p 
 
 the tangent. The total head is 70 feet and r=- . Find the velocity of the 
 rim of the wheel if the water discharges radially. Friction neglected. 
 
 (13) The inner and outer diameters of an inward flow turbine wheel 
 are 1 foot and 2 feet respectively. The water enters the outer circumference 
 at 12 with the tangent, and leaves the inner circumference radially. The 
 radial velocity of flow is 6 feet at both circumferences. The wheel makes 
 3*6 revolutions per second. Determine the angles of the vanes at both 
 circumferences, and the theoretical hydraulic efficiency of the turbine. 
 
 (14) Water is supplied to an inward flow turbine at 44 feet per second, 
 and at 10 degrees to the tangent to the wheel. The wheel makes 200 
 
 252 
 
388 HYDRAULICS 
 
 revolutions per minute. The inlet radius is 1 foot and the outer radius 
 2 feet. The radial velocity of flow through the wheel is constant. 
 
 Find the inclination of the vanes at inlet and outlet of the wheel. 
 
 Determine the ratio of the kinetic energy of the water entering the 
 wheel per pound to the work done on the wheel per pound. 
 
 (15) The supply of water for an inward flow reaction turbine is 500 
 cubic feet per minute and the available head is 40 feet. The vanes are 
 radial at the inlet, the outer radius is twice the inner, the constant 
 velocity of flow is 4 feet per second, and the revolutions are 350 per 
 minute. Find the velocity of the wheel, the guide and vane angles, 
 the inner and outer diameters, and the width of the bucket at inlet and 
 outlet. Lond. Un. 1906. 
 
 (16) An inward flow turbine on 15 feet fall has an inlet radius of 1 foot 
 and an outlet radius of 6 inches. Water enters at 15 with the tangent to 
 the circumference and is discharged radially with a velocity of 3 feet per 
 second. The actual velocity of water at inlet is 22 feet per second. The 
 circumferential velocity of the inlet surface of the wheel is 19^ feet per 
 second. 
 
 Construct the inlet and outlet angles of the turbine vanes. 
 Determine the theoretical hydraulic efficiency of the turbine. 
 If the hydraulic efficiency of the turbine is assumed 80 per cent, find the 
 vane angles. 
 
 (17) A quantity of water Q cubic feet per second flows through a 
 turbine, and the initial and final directions and velocities are known. 
 Apply the principle of equality of angular impulse and moment of 
 momentum to find the couple exerted on the turbine. 
 
 (18) The wheel of an inward flow turbine has a peripheral velocity of 
 50 feet per second. The velocity of whirl of the incoming water is 40 feet 
 per second, and the radial velocity of flow 5 feet per second. Determine 
 the vane angle at inlet. 
 
 Taking the flow as 20 cubic feet per second and the total losses as 
 20 per cent, of the available energy, determine the horse-power of the 
 turbine, and the head H. 
 
 If 5 per cent, of the head is lost in friction in the supply pipe, and the 
 centre of the turbine is 15 feet above the tail race level, find the pressure 
 head at the inlet circumference of the wheel. 
 
 (19) An inward flow turbine is required to give 200 horse-power under 
 a head of 100 feet when running at 500 revolutions per minute. The 
 velocity with which the water leaves the wheel axially may be taken as 
 10 feet per second, and the wheel is to have a double outlet. The diameter 
 of the outer circumference may be taken as If times the inner. Determine 
 the dimensions of the turbine and the angles of the guide blades and 
 vanes of the turbine wheel. The actual efficiency is to be taken as 75 per 
 cent, and the hydraulic efficiency as 80 per cent. 
 
 (20) An outward flow turbine wheel has an internal diameter of 5'249 
 feet and an external diameter of 6'25 feet. The head above the turbine is 
 141-5 feet. The width of the wheel at inlet is 10 inches, and the quantity 
 
TURBINES 389 
 
 of water supplied per second is 215 cubic feet. Assuming the hydraulic 
 losses are 20 per cent., determine the angles of tips of the vanes so that 
 the water shall leave the wheel radially. Determine the horse-power of 
 the turbine and verify the work done per pound from the triangles of 
 velocities. 
 
 (21) The total head available for an inward-flow turbine is 100 feet. 
 
 The turbine wheel is placed 15 feet above the tail water level. 
 
 When the flow is normal, there is a loss of head in the supply pipe of 
 3 per cent, of the head ; in the guide passages a loss of 5 per cent. ; in the 
 wheel 9 per cent. ; in the down pipe 1 per cent. ; and the velocity of flow 
 from the wheel and in the supply pipe, and also from the down pipe is 
 8 feet per second. 
 
 The diameter of the inner circumference of the wheel is 9 inches and 
 of the outer 19 inches, and the water leaves the wheel vanes radially. 
 The wheel has radial vanes at inlet. 
 
 Determine the number of revolutions of the wheel, the pressure head in 
 the eye of the wheel, the pressure head at the circumference to the wheel, 
 the pressure head at the entrance to the guide chamber, and the velocity 
 which the water has when it enters the wheel. From the data given 
 
 9 
 
 (22) A horizontal inward flow turbine has an internal diameter of 
 5 feet 4 inches and an external diameter of 7 feet. The crowns of the 
 wheel are parallel and are 8 inches apart. The difference in level of the 
 head and tail water is 6 feet, and the upper crown of the wheel is just below 
 the tail water level. Find the angle the guide blade makes with the tangent 
 to the wheel, when the wheel makes 32 revolutions per minute, and the 
 flow is 45 cubic feet per second. Neglecting friction, determine the vane 
 angles, the horse-power of the wheel and the theoretical hydraulic efficiency. 
 
 (23) A parallel flow turbine has a mean diameter of 11 feet. 
 
 The number of revolutions per minute is 15, and the axial velocity of 
 flow is 3'5 feet per second. The velocity of the water along the tips of the 
 guides is 15 feet per second. 
 
 Determine the inclination of the guide blades and the vane angles that 
 the water shall enter without shock and leave the wheel axially. 
 
 Determine the work done per pound of water passing through the wheel. 
 
 (24) The diameter of the inner crown of a parallel flow pressure turbine 
 is 5 feet and the diameter of the outer crown is 8 feet. The head over the 
 wheel is 12 feet. The number of revolutions per minute is 52. The radial 
 velocity of flow through the wheel is 4 feet per second. 
 
 Assuming a hydraulic efficiency of 0'8, determine the guide blade angles 
 and vane angles at inlet for the three radii 2 feet 6 inches, 3 feet 3 inches 
 and 4 feet. 
 
 Assuming the depth of the wheel is 8 inches, draw suitable sections of 
 the vanes at the three radii. 
 
 Find also the width of the guide blade in plan, if the upper and lower 
 edges are parallel, and the lower edge makes a constant angle with the 
 
390 HYDRAULICS 
 
 plane of the wheel, so that the stream lines at the inner and the outer 
 crown may have the correct inclinations. 
 
 (25) A parallel flow impulse turbine works under a head of 64 feet. 
 The water is discharged from the wheel in an axial direction with a 
 velocity due to a head of 4 feet. The circumferential speed of the wheel 
 at its mean diameter is 40 feet per second. 
 
 Neglecting all frictional losses, determine the mean vane and guide 
 angles. Lond. Un. 1905. 
 
 (26) An outward flow impulse turbine has an inner diameter of 5 feet, 
 an external diameter of 6 feet 3 inches, and makes 450 revolutions per 
 minute. 
 
 The velocity of the water as it leaves the nozzles is double the velocity 
 of the periphery of the wheel, and the direction of the water makes an 
 angle of 30 degrees with the circumference of the wheel. 
 
 Determine the vane angle at inlet, and the angle of the vane at outlet so 
 that the water shall leave the wheel radially. 
 
 Find the theoretical hydraulic efficiency. If 8 per cent, of the head 
 available at the nozzle is lost in the wheel, find the vane angle at exit that 
 the water shall leave radially. 
 
 What is now the hydraulic efficiency of the turbine ? 
 
 (27) In an axial flow Girard turbine, let V be the velocity due to the 
 effective head. Suppose the water issues from the guide blades with the 
 velocity (V95V, and is discharged axially with a velocity '12V. Let the 
 velocity of the receiving and discharging edges be 0*55 V. 
 
 Find the angle of the guide blades, receiving and discharging angles of 
 wheel vanes and hydraulic efficiency of the turbine. 
 
 (28) Water is supplied to an axial flow impulse turbine, having a mean 
 diameter of 6 feet, and making 144 revolutions per minute, under a head of 
 100 feet. The angle of the guide blade at entrance is 30, and the angle the 
 vane makes with the direction of motion at exit is 30. Eight per cent, of 
 the head is lost in the supply pipe and guide. Determine the relative 
 velocity of water and wheel at entrance, and on the assumption that 10 per 
 cent, of the total head is lost in friction and shock in the wheel, determine 
 the velocity with which the water leaves the wheel. Find the hydraulic 
 efficiency of the turbine. 
 
 (29) The guide blades of an inward flow turbine are inclined at 30 
 degrees, and the velocity U along the tip of the blade is 60 feet per second. 
 The velocity of the wheel periphery is 55 feet per second. The guide blades 
 are turned so that they are inclined at an angle of 15 degrees, the velocity 
 U remaining constant. Find the loss of head due to shock at entrance. 
 
 If the radius of the inner periphery is one-half that of the outer and the 
 radial velocity through the wheel is constant for any flow, and the water 
 left the wheel radially in the first case, find the direction in which it leaves 
 in the second case. The inlet radius is twice the outlet radius. 
 
 (30) The supply of water to a turbine is controlled by a speed gate 
 between the guides and the wheel. If when the gate is fully open the 
 velocity with which the water approaches the wheel is 70 feet per second 
 
TURBINES 391 
 
 and it makes an angle of 15 degrees with the tangent to the wheel, find 
 the loss of head by shock when the gate is half closed. The velocity of 
 the inlet periphery of the wheel is 75 feet per second. 
 
 (31) A Pelton wheel, which may be assumed to have semi-cylindrical 
 buckets, is 2 feet diameter. The available pressure at the nozzle when it 
 is closed is 200 Ibs. per square inch, and the supply when the nozzle is 
 open is 100 cubic feet per minute. If the revolutions are 600 per minute, 
 estimate the horse-power of the wheel and its efficiency. 
 
 (32) Show that the efficiency of a Pelton wheel is a maximum 
 neglecting frictional and other losses when the velocity of the cups equals 
 half the velocity of the jet. 
 
 25 cubic feet of water are supplied per second to a Pelton wheel through 
 a nozzle, the area of which is 44 square inches. The velocity of the cups 
 is 41 feet per second. Determine the horse-power of the wheel assuming 
 an efficiency of 75 per cent. 
 
CHAPTER X. 
 
 PUMPS. 
 
 Pumps are machines driven by some prime mover, and used 
 for raising fluids from a lower to a higher level, or for imparting 
 energy to fluids. For example, when a mine has to be drained 
 the water may be simply raised from the mine to the surface, and 
 work done upon it against gravity. Instead of simply raising the 
 water through a height h, the same pumps might be used to 
 deliver water into pipes, the pressure in which is wh pounds per 
 square foot. 
 
 A pump can either be a suction pump, a pressure pump, or 
 both. If the pump is placed above the surface of the water in 
 the well or sump, the water has to be first raised by suction; 
 the maximum height through which a pump can draw water, 
 or in other words the maximum vertical distance the pump can 
 be placed above the water in the well, is theoretically 34 feet, but 
 practically the maximum is from 25 to 30 feet. If the pump 
 delivers the water to a height h above the pump, or against a 
 pressure-head h, it is called a force pump. 
 
 216. Centrifugal and turbine pumps. 
 
 Theoretically any reaction turbine could be made to work as 
 a pump by rotating the wheel in the opposite direction to that in 
 which it rotates as a turbine, and supplying it with water at the 
 circumference, with the same velocity, but in the inverse direction 
 to that at which it was discharged when acting as a turbine. Up 
 to the present, only outward flow pumps have been constructed, 
 and, as will be shown later, difficulty would be experienced in 
 starting parallel flow or inward flow pumps. 
 
 Several types of centrifugal pumps (outward flow) are shown 
 in Figs. 272 to 276.. 
 
 The principal difference between the several types is in the 
 form of the casing surrounding the wheel, and this form has con- 
 siderable influence upon the efficiency of the pump. The reason 
 
PUMPS 
 
 393 
 
 for this can be easily seen in a general way from the following 
 consideration. The water approaches a turbine wheel with a 
 high velocity and in a direction making a small angle with the 
 direction of motion of the inlet circumference of the wheel, and 
 
 Fig. 272. Diagram of Centrifugal Pump. 
 
 thus it has a large velocity of whirl. When the water leaves the 
 wheel its velocity is small and the velocity of whirl should be zero. 
 In the centrifugal pump these conditions are entirely reversed; 
 the water enters the wheel with a small velocity, and leaves 
 
394 
 
 HYDRAULICS 
 
 it with a high velocity. If the case surrounding the wheel 
 admits of this velocity being diminished gradually, the kinetic 
 energy of the water is converted into useful work, but if not, it is 
 destroyed by eddy motions in the casing, and the efficiency of the 
 pump is accordingly low. 
 
 In Fig. 272 a circular casing surrounds the wheel, and prac- 
 tically the whole of the kinetic energy of the water when it leaves 
 the wheel is destroyed ; the efficiency of such pumps is generally 
 much less than 50 per cent. 
 
 Fig. 273. Centrifugal Pump with spiral casing. 
 
 The casing of Fig. 273 is made of spiral form, the sectional 
 area increasing uniformly towards the discharge pipe, and thus 
 being proportional to the quantity of water flowing through the 
 section. It may therefore be supposed that the mean velocity of 
 flow through any section is nearly constant, and that the stream 
 lines are continuous. 
 
 The wheel of Fig. 274 is surrounded by a large whirlpool 
 chamber in which, as shown later, the velocity with which the 
 water rotates round the wheel gradually diminishes, and the 
 velocity head with which the water leaves the wheel is partly 
 converted into pressure head. 
 
 The same result is achieved in the pump of Figs. 275 and 276 
 
PUMPS 395 
 
 by allowing the water as it leaves the wheel to enter guide 
 passages, similar to those used in a turbine to direct the water 
 to the wheel. The area of these passages gradually increases 
 and a considerable portion of the velocity head is thus converted 
 into pressure head and is available for lifting water.' 
 
 This class of centrifugal pump is known as the turbine pump. 
 
 Fig. 274. Diagram of Centrifugal Pump with Whirlpool Chamber. 
 
 217. Starting centrifugal or turbine pumps. 
 
 A centrifugal pump cannot commence delivery unless the wheel, 
 casing, and suction pipe are full of water. 
 
 If the pump is below the water in the well there is no difficulty 
 in starting as the casing will be maintained full of water. 
 
 When the pump is above the water in the well, as in Fig. 272, 
 a non-return valve V must be fitted in the suction pipe, to prevent 
 the pump when stopped from being drained. If the pump becomes 
 empty, or when the pump is first set to work, special means have 
 to be provided for filling the pump case. In large pumps the air 
 may be expelled by means of steam, which becomes condensed and 
 the water rises from the well, or they should be provided with 
 
396 
 
 HYDRAULICS 
 
 an air-pump or ejector as an auxiliary to the pump. Small pumps 
 can generally be easily filled by hand through a pipe such as 
 shown at P, Fig. 276. 
 
 With some classes of pumps, if the pump has to commence 
 delivery against full head, a stop valve on the rising main, 
 Fig. 296, is closed until the pump has attained the speed necessary 
 to commence delivery*, after which the stop valve is slowly 
 opened. 
 
 Fig. 275. 
 
 Turbine Pump. 
 
 Fig. 276. 
 
 It will be seen later that, under special circumstances, other 
 provisions will have to be made to enable the pump to commence 
 delivery. 
 
 218. Form of the vanes of centrifugal pumps. 
 
 The conditions to be satisfied by the vanes of a centrifugal 
 pump are exactly the same as for a turbine. At inlet the direction 
 of the vane should be parallel to the direction of the relative 
 velocity of the water and the tip of the vane, and the velocity 
 with which the water leaves the wheel, relative to the pump case, 
 is the vector sum of the velocity of the tip of the vane and the 
 velocity relative to the vane. 
 
 * See page 409. 
 
PUMPS 397 
 
 Suppose the wheel and casing of Fig. 272 is full of water, and 
 the wheel is rotated in the direction of the arrow with such a 
 velocity that water enters the wheel in a known direction with a 
 velocity U, Fig. 277, not of necessity radial. 
 
 Let v be the velocity of the receiving edge of the vane or inlet 
 circumference of the wheel; Vi the velocity of the discharging 
 circumference of the wheel ; Ui the absolute velocity of the water 
 as it leaves the wheel ; V and Yi the velocities of whirl at inlet 
 and outlet respectively; V P and v r the relative velocities of the 
 water and the vane at inlet and outlet respectively ; u and HI the 
 radial velocities at inlet and outlet respectively. 
 
 The triangle of velocities at inlet is ACD, Fig. 277, and if the 
 vane at A, Fig. 272, is made parallel to CD the water will enter 
 the wheel without shock. 
 
 C B 
 
 velocities 
 
 idngle oC velocities Triangle of velocities 
 
 cub inlet. at e^cct. 
 
 Fig. 277. Fig. 278. 
 
 The wheel being full of water, there is continuity of flow, and 
 if A and Ax are the circumferential areas of the inner and outer 
 circumferences, the radial component of the velocity of exit at the 
 outer circumference is 
 
 AM 
 
 If the direction of the tip of the vane at the outer circum- 
 ference is known the triangle of velocities at exit, Fig. 278, can be 
 drawn as follows. 
 
 Set out BGr radially and equally to Ui, and BE equal to VL 
 
 Draw GrF parallel to BE at a distance from BE equal to Ui, 
 and EF parallel to the tip of the vane to meet GrF in F. 
 
 Then BF is the vector sum of BE and EF and is the velocity 
 with which the water leaves the wheel relative to the fixed casing. 
 
 219. Work done on the water by the wheel. 
 
 Let R and r be the radii of the discharging and receiving 
 circumferences respectively. 
 
 The change in angular momentum of the water as it passes 
 through the wheel is ViR + ~Vr/g per pound of flow, the plus sign 
 being used when V is in the opposite direction to V 1? as in 
 Figs. 277 and 278. 
 
398 HYDRAULICS 
 
 Neglecting frictional and other losses, the work done by the 
 wheel on the water per pound (see page 275) is 
 
 , 
 
 9 ' 9 
 
 If U is radial, as in Fig. 272, Y is zero, and the work done on 
 the water by the wheel is 
 
 ^ foot Ibs. per Ib. flow. 
 y 
 
 If then Ho, Fig. 272, is the total height through which the water 
 is lifted from the sump or well, and u d is the velocity with which 
 the water is delivered from the delivery pipe, the work done on 
 each pound of water is 
 
 Ho+ | 2 ' 
 
 and therefore, 
 
 = H + 7; = H. 
 
 g 2g 
 
 Let (180 - <) be the angle which the direction of the vane at 
 exit makes with the direction of motion, and (180 - 0) the angle 
 which the vane makes with the direction of motion at inlet. Then 
 ACD is and BEF is <. 
 
 In the triangle HEF, HE = HF cot <, and therefore, 
 
 YI Vi Ui COt <j>. 
 
 The theoretical lift, therefore, is 
 
 Ua Vi (vi - Uj. cot </>) 
 JtiL -tio ~*~ n 
 
 If Q is the discharge and AI the peripheral area of the dis- 
 charging circumference, 
 
 V? Vi ~ COt < 
 
 and H> ^ (1). 
 
 9 
 If, therefore, the water enters the wheel without shock and all 
 
 T> 
 
 resistances are neglected, the lift is independent of the ratio , and 
 
 depends only on the velocity and inclination of the vane at the 
 discharging circumference. 
 
 220. Ratio of V x to v,. 
 
 As in the case of the turbine, for any given head H, Yi and v^ 
 can theoretically have any values consistent with the product 
 
PUMPS 
 
 399 
 
 Vi^i being equal to gH, the ratio of Vi to Vi simply depending upon 
 the magnitude of the angle <. 
 
 The greater the angle </> is made the less the velocity Vi of the 
 periphery must be for a given lift. 
 
 \ 
 
 if 
 
 */ 
 
 Fig. 279. 
 
 This is shown at once by equation (1), section 219, and is 
 illustrated in Fig. 279. The angle <f> is given three values, 
 30 degrees, 90 degrees and 150 degrees, and the product Yt; and 
 also the radial velocity of flow tti are kept constant. The theo- 
 retical head and also the discharge for the three cases are there- 
 fore the same. The diagrams are drawn to a common scale, and it 
 can therefore be seen that as <j> increases v l diminishes, and Ui 
 the velocity with which the water leaves the wheel increases. 
 
 221. The kinetic energy of the water at exit from the 
 wheel. 
 
 Part of the head H impressed upon the water by the wheel 
 increases the pressure head between the inlet and outlet, and the 
 remainder appears as the kinetic energy of the water as it leaves 
 
400 HYDEAULICS 
 
 U 2 
 the wheel. This kinetic energy is equal to -^- , and can only be 
 
 utilised to lift the water if the velocity can be gradually diminished 
 so as to convert velocity head into pressure head. This however 
 is not very easily accomplished, without being accompanied by a 
 considerable loss by eddy motions. If it be assumed that the same 
 
 TJ 2 
 
 proportion of the head 7^- in all cases is converted into useful 
 
 work, it is clear that the greater Ui, the greater the loss by eddy 
 motions, and the less efficient will be the pump. It is to be ex- 
 pected, therefore, that the less the angle <f>, the greater will be 
 the efficiency, and experiment shows that for a given form of 
 casing, the efficiency does increase as < is diminished. 
 
 222. Gross lift of a centrifugal pump. 
 
 Let Ji a be the actual height through which water is lifted; 
 Ji s the head lost in the suction pipe ; ha the head lost in the delivery 
 pipe ; and u& the velocity of flow along the delivery pipe. 
 
 Any other losses of head in the wheel and casing are incident 
 
 Ud 
 
 to the pump, but h s , h d , and the head ^- should be considered as 
 
 external losses. 
 
 The gross lift of a pump is then 
 
 h = h a + h s + h d + ~^- , 
 and this is always less than H. 
 
 223. Efficiencies of a centrifugal pump. 
 Manometric efficiency. The ratio fr > or 
 
 g .h 
 
 e = 
 
 Q 
 
 is the manometric efficiency of the pump at normal discharge. 
 
 The reason for specifically defining e as the manometric 
 efficiency at normal discharge is simply that the theoretical lift H 
 has been deduced from consideration of a definite discharge Q, 
 and only for this one discharge can the conditions at the inlet edge 
 be as assumed. 
 
 A more general definition is, however, generally given to e, and 
 for any discharge Q, therefore, the manometric efficiency may 
 be taken as the ratio of the gross lift at that discharge to the 
 theoretical head 
 
 ^i 2 ^i A c t * 
 
 AI 
 
 9 
 
CENTRIFUGAL PUMPS 401 
 
 This manometric efficiency of the pump must not be confused 
 with the efficiency obtained by dividing the work done by the 
 pump, by the energy required to do that work, as the latter in 
 many pumps is zero, when the former has its maximum value. 
 
 Hydraulic efficiency. The hydraulic efficiency of a pump is 
 the ratio of the work done on the pump wheel to the gross work 
 done by the pump. 
 
 Let W = the weight of water lifted per second. 
 
 Let h = the gross head 
 
 - h a + h 8 + h d + ~- . 
 
 *g 
 
 Let E = the work done on the pump wheel in foot pounds 
 per second. 
 
 Let e h = the hydraulic efficiency. Then 
 
 _W.h 
 E ' 
 
 The work done on the pump wheel is less than the work done 
 on the pump shaft by the belt or motor which drives the pump, 
 by an amount equal to the energy -lost by friction at the bearings 
 of the machine. This generally, in actual machines, can be 
 approximately determined by running the machine without load. 
 
 Actual efficiency. From a commercial point of view, what is 
 generally required is the ratio of the useful work .done by the 
 pump, taking it as a whole, to the work done on the pump shaft. 
 
 Let E be the energy given to the pump shaft per sec. and. 
 e m the mechanical efficiency of the pump, then 
 
 and the actual efficiency 
 
 
 E. 
 
 Gross efficiency of the pump. The gross efficiency of the pump 
 itself, including mechanical as well as fluid losses, is 
 
 W.h 
 
 e a = 
 
 E. 
 
 224. Experimental determination of the efficiency of a 
 centrifugal pump. 
 
 The actual and gross efficiencies of a pump can be determined 
 directly by experiment, but the hydraulic efficiency can only be 
 determined when at all loads the mechanical efficiency of the 
 pump is known. 
 
 To find the actual efficiency, it is only necessary to measure 
 the height through which water is lifted, the quantity of water 
 
 L. H. 26 
 
402 HYDRAULICS 
 
 discharged, and the energy E 8 given to the pump shaft in unit 
 
 time. 
 
 A. very convenient method of determining E 8 with a fair 
 degree of accuracy is to drive the pump shaft direct by an electric 
 motor, the efficiency curve * for which at varying loads is known. 
 A better method is to use some form of transmission dynamo- 
 meter t. 
 
 225. Design of pump to give a discharge Q. 
 
 If a pump is required to give a discharge Q under a gross 
 lift h, and from previous experience the probable manometric 
 efficiency e at this discharge is known, the problem of determining 
 suitable dimensions for the wheel of the pump is not difficult. 
 The difficulty really arises in giving a correct value to e and in 
 making proper allowance for leakage. 
 
 This difficulty will be better appreciated after the losses in 
 various kinds of pumps have been considered. It will then be 
 seen that e depends upon the angle <, the velocity of the wheel, 
 the dimensions of the wheel, the form of the vanes of the wheel, 
 the discharge through the wheel, and upon the form of the casing 
 surrounding the wheel; the form of the casing being just as 
 important, or more important, than the form of the wheel in 
 determining the probable value of e. 
 
 Design of the wheel of a pump for a given discharge under a 
 given head. If a pump is required to give a discharge Q under an 
 effective head h a , the gross head h can only be determined if h S) 
 
 h d , and =^- , are known. 
 
 Any suitable value can be given to the velocity Ud. If the 
 pipes are long it should not be much greater than 5 feet per second 
 for reasons explained in the chapter on pipes, and the velocity u s 
 in the suction pipe should be equal to or less than u d . The 
 velocities u s and u d having been settled, the losses h s and h d can be 
 approximated to and the gross head h found. In the suction pipe, 
 as explained on page 395, a foot valve is generally fitted, at which, 
 at high velocities, a loss of head of several feet may occur. 
 The angle <j> is generally made from 10 to 90 degrees. Theoreti- 
 cally, as already stated, it can be made much greater than 
 90 degrees, but the efficiency of ordinary centrifugal pumps might 
 be very considerably diminished as < is increased. 
 
 The manometric efficiency e varies very considerably; with 
 radial blades and a circular casing, the efficiency is not generally 
 
 * See Electrical Engineering, Thomalen-Howe, p. 195. 
 t See paper by Stanton, Proc. Inst. Mech. Engs., 1903. 
 
CENTRIFUGAL PUMPS 403 
 
 more than 0'3 to 0'4. With a vortex chamber, or a spiral casing, 
 and the vanes at inlet inclined so that the tip is parallel to the 
 relative velocity of the water and the vane, and < not greater than 
 90 degrees, the manometric efficiency e is from 0'5 to 0*75, being 
 greater the less the angle <, and with properly designed guide 
 blades external to the wheel, e is from 0'6 to '85. 
 
 The ratio of the diameter of the discharging circumference to 
 the inlet circumference is somewhat arbitrary and is generally 
 made from 2 to 3. Except for the difficulty of starting (see 
 section 226), the ratio might with advantage be made much 
 smaller, as by so doing the f rictional losses might be considerably 
 reduced. The radial velocity HI may be taken from 2 to 10 feet 
 per second. 
 
 Having given suitable values to u, and to any two of the three 
 quantities, e, v, and <, the third can be found from the equation 
 , _ e (vi - ViUi cot <fr) 
 
 9 
 
 The internal diameter d of the wheel will generally be settled from 
 consideration of the velocity of flow u 2 into the wheel. This may 
 be taken as equal to or about equal to u } but in special cases 
 it may be larger than u. 
 
 Then if the water is admitted to the wheel at both sides, as in 
 Fig. 273, 
 
 from which d can be calculated when u^ and Q are known. 
 
 Let b be the width of the vane at inlet and B at outlet, and D 
 the diameter of the outlet circumference. 
 
 Then l> > 
 
 -rrdu 
 
 and B = -- 
 
 TTL)Ui 
 
 If the water moves toward the vanes at inlet radially, the 
 inclination of the vane that there shall be no shock is such that 
 
 and if guide blades are to be provided external to the wheel, as in 
 Fig. 275, the inclination a of the tip of the guide blade with the 
 direction of Vi is found from 
 
 Ui 
 
 tan a = . 
 
 The guide passages should be so proportioned that the velocity 
 Ui is gradually diminished to the velocity in the delivery pipe. 
 
 262 
 
404 HYDRAULICS 
 
 Limiting velocity of the rim of the wheel. Quite apart from 
 head lost by friction in the wheel due to the relative motion of 
 the water and the wheel, there is also considerable loss of energy 
 external to the wheel due to the relative motion of the water and 
 the wheel. Between the wheel and the casing there is in most 
 pumps a film of water, and between this film and the wheel, 
 frictional forces are set up which are practically proportional to 
 the square of the velocity of the wheel periphery and to the area 
 of the wheel crowns. An attempt is frequently made to diminish 
 this loss by fixing the vanes to a central diaphragm only, the 
 wheel thus being without crowns, the outer casing being so 
 formed that there is but a small clearance between it and the 
 outer edges of the vanes. At high velocities these frictional resist- 
 ances may be considerable. To keep them small the surface of 
 the wheel crowns and vanes must be made smooth, and to this 
 end many high speed wheels are carefully finished. 
 
 Until a few years ago the peripheral velocity of pump wheels 
 was generally less than 50 feet per second, and the best velocity 
 was supposed to be about 30 feet per second. They are now, how- 
 ever, run at much higher speeds, and the limiting velocities are 
 fixed from consideration of the stresses in the wheel due to centri- 
 fugal forces. Peripheral velocities of nearly 200 feet per second 
 are now frequently used, and Rateau has constructed small pumps 
 with a peripheral velocity of 250 feet per second *. 
 
 Example. To find the proportions of a pump with radial blades at outlet 
 (i.e. = 90) to lift 10 cubic feet of water per second against a head of 50 feet. 
 
 Assume there are two suction pipes and that the water enters the wheel from 
 both sides, as in Fig. 273, also that the velocity in the suction and delivery pipes 
 and the radial velocity through the wheel are 6 feet per second, and the manometric 
 efficiency is 75 per cent. 
 
 First to find v lt 
 
 Since the blades are radial, -75 = 50, 
 
 y 
 
 from which 1^ = 46 feet per sec. 
 
 To find the diameter of the suction pipes. 
 The discharge is 10 cubic feet per second, therefore 
 
 2.d 2 . 6 = 10, 
 from which d = l'03' = 12 3 ". 
 
 If the radius E of the external circumference be taken as 2r and r is taken equal 
 to the radius of the suction pipes, then R = 12f", and the number of revolutions 
 per second^will be 
 
 The velocity of the inner edge of the vane is 
 u = 23 feet per sec. 
 
 * Engineer, 1902. 
 
CENTRIFUGAL PUMPS 405 
 
 The inclination of the vane at inlet that the water may move on to the vane 
 without shock is 
 
 and the water when it leaves the wheel makes an angle a with v l such that 
 
 If there are guide blades surrounding the wheel, a gives the inclination of these 
 blades. 
 
 The width of the wheel at discharge is 
 
 = 3 inches about. 
 The width of the wheel at inlet = 6^ inches. 
 
 226. The centrifugal head impressed on the water by 
 the wheel. 
 
 Head against which a pump will commence to discharge. As 
 shown on page 335, the centrifugal head impressed on the water as 
 it passes through the wheel is 
 
 , _vl_rf 
 ~2<7 2g> 
 
 but this is not the lift of the pump. Theoretically it is the head 
 which will be impressed on the water when there is no flow 
 through the wheel, and is accordingly the difference between the 
 pressure at inlet and outlet when the pump is first set in motion ; 
 or it is the statical head which the pump will maintain when 
 
 running at its normal speed. If this is less than - - , the pump 
 
 y 
 
 theoretically cannot start lifting against its full head without 
 being speeded up above its normal velocity. 
 
 The centrifugal head is, however, always greater than 
 
 as the water in the eye of the wheel and in the casing surrounding 
 the wheel is made to rotate by friction. 
 
 For a pump having a wheel seven inches diameter surrounded 
 by a circular casing 20 inches diameter, Stanton* found that, when 
 the discharge was zero and the vanes were radial at exit, h c was 
 
 I'OW , , . T12t; 2 
 
 ~ , and with curved vanes, < being 30 degrees, h c was ~ . 
 
 ^9 ^9 
 
 For a pump with a spiral case surrounding the wheel, the 
 centrifugal head h c when there is no discharge, cannot be much 
 
 2 
 
 greater than ~ , as the water surrounding the wheel is prevented 
 
 *9 
 
 from rotating by the casing being brought near to the wheel at 
 one point. 
 
 * Proceedings Inst. M. E., 1903. 
 
406 HYDRAULICS 
 
 Parsons found for a pump having a wheel 14 inches diameter 
 with radial vanes at outlet, and running at 300 revolutions per 
 
 minute, that the head maintained without discharge was ~ , 
 and with an Appold* wheel running at 320 revolutions per minute 
 the statical head was ^ . For a pump, with spiral casing, 
 
 having a rotor 1*54 feet diameter, the least velocity at which 
 it commenced to discharge against a head of 14*67 feet was 
 
 392 revolutions per minute, and thus h c was ~ * > an( i the least 
 
 velocity against a head of 17*4 feet was 424 revolutions per 
 
 0*951) 2 
 minute or h c was again ~ ~ For a pump with circular casing 
 
 larger than the wheel, h c was x - . For a pump having guide 
 
 passages surrounding the wheel, and outside the guide passages 
 a circular chamber as in Fig. 275, the centrifugal head may also 
 
 2 
 
 be larger than <-; the mean actual value for this pump was 
 
 9 2 
 found to be 1*087 ^-. 
 
 Stanton found, when the seven inches diameter wheels mentioned 
 above discharged into guide passages surrounded by a circular 
 
 chamber 20 inches diameter, that h c was ^ - when the vanes of 
 
 1*39?; 2 
 the wheel were radial, and ^ - when <f> was 30 degrees. 
 
 That the centrifugal head when the wheel has radial vanes is 
 likely to be greater than when the vanes of the wheel are set back 
 is to be seen by a consideration of the manner in which the water 
 in the chamber outside the guide passages is probably set in 
 motion, Fig. 280. Since there is no discharge, this rotation cannot 
 be caused by the water passing through the pump, but must be 
 due to internal motions set up in the wheel and casing. The 
 water in the guide chamber cannot obviously rotate about the 
 axis 0, but there is a tendency for it to do so, and consequently 
 stream line motions, as shown in the figure, are probably set 
 up. The layer of water nearest the outer circumference of the 
 wheel will no doubt be dragged along by friction in the direction 
 shown by the arrow, and water will flow from the outer casing to 
 take its place ; the stream lines will give motion to the water in 
 the outer casing. 
 
 * See page 415. 
 
CENTRIFUGAL PUMPS 
 
 407 
 
 When the vanes in the wheel are radial and as long as a vane is 
 moving between any two guide vanes, the straight vane prevents 
 the friction between the water outside the wheel and that inside, 
 from dragging the water backwards along the vane, but when the 
 vane is set back and the angle < is greater than 90 degrees, there 
 will be a tendency for the water in the wheel to move backwards 
 while that in the guide chamber moves forward, and consequently 
 the velocity of the stream lines in the casing will be less in the 
 latter case than in the former. In either case, the general 
 direction of flow of the stream lines, in the guide chamber, will 
 be in the direction of rotation of the wheel, but due to friction 
 and eddy motions, even with radial vanes, the velocity of the stream 
 
 Fig. 280. 
 
 lines will be less than the velocity t\ of the periphery of the wheel. 
 Just outside the guide chambers the velocity of rotation will be 
 less than Vi. In the outer chamber it is to be expected that the 
 water will rotate as in a free vortex, or its velocity of whirl will 
 be inversely proportional to the distance from the centre of the 
 rotor, or will rotate in some manner approximating to this. 
 
 The head which a pump, with a vortex chamber, will theoreti- 
 cally maintain when the discharge is zero. In this case it is 
 probable that as the discharge approaches zero, in addition to the 
 water in the wheel rotating, the water in the vortex chamber will 
 also rotate because of friction. 
 
408 HYDRAULICS 
 
 The centrifugal head due to the water in the wheel is 
 
 2g~2g' 
 
 q ,.. 2 
 
 If R = 2r, this becomes 7 <r- 
 
 The centrifugal head due to the water in the chamber is, 
 Fig. 281, 
 
 TO and VQ being the radius and tangential velocity respectively of 
 any ring of water of thickness dr. 
 
 Fig. 281. 
 
 If it be assumed that v r is a constant, the centrifugal head 
 due to the vortex chamber is 
 
 
 The total centrifugal head is then 
 
 z, v * 
 c = 2 
 
 If r w is 2r and E, w is 2r w , 
 
 _vf_v^ v^/J. 1 
 
 IVr: r^. /-. ~T~ ^-v I ~~" -w^ 
 
 The conditions here assumed, however, give h e too high. In 
 Stanton's experiments h c was only ^ Vl . Decouer from experi- 
 
CENTRIFUGAL PUMPS 409 
 
 ments on a small pump with a vortex chamber, the diameter being 
 about twice the diameter of the wheel, found h c to be ~ l . 
 
 Let it be assumed that h c is -~- in any pump, and that the lift 
 
 of the pump when working normally is 
 
 e (i?i 2 - ViUi cot 
 
 Then if h is greater than -^ l , the pump will not commence to 
 
 t/ 
 
 discharge unless speeded up to some velocity v z such that 
 
 e (vi 2 Vi HI cot <) 
 
 After the discharge has been commenced, however, the speed 
 may be diminished, and the pump will continue to deliver against 
 the given head*. 
 
 For any given values of m and e the velocity v 2 at which delivery 
 commences decreases with the angle <. If < is 90 or greater than 
 90 degrees, and m is unity, the pump will only commence to 
 discharge against the normal head when the velocity is v^ , if the 
 manometric efficiency e is less than 0'5. If < is 30 degrees and m 
 is unity, v% is equal to Vi when e is 0'6, but if < is 150 degrees v z 
 is equal to v l when e is 0'428. 
 
 Nearly all actual pumps are run at such a speed that the 
 centrifugal head at that speed is greater than the actual head 
 against which the pump works, so that there is never any 
 difficulty in starting the pump. This is accounted for (1) by the 
 low manometric efficiencies of actual pumps, (2) by the angle <j> 
 never being greater than 90 degrees, and (3) by the wheels being 
 surrounded by casings which allow the centrifugal head to be 
 
 greater than ^- . 
 
 It should be observed that it does not follow, because in many 
 cases the manometric efficiency is small, the actual efficiency of 
 the pump is of necessity low. (See Fig. 286.) 
 
 227. Head-velocity curve of a centrifugal pump at zero 
 discharge. 
 
 For any centrifugal pump a curve showing the head against 
 which it will start pumping at any given speed can easily be 
 determined as follows. 
 
 On the delivery pipe fit a pressure gauge, and at the top 
 
 * See pages 411 and 419. 
 
410 
 
 HYDRAULICS 
 
 of the suction pipe a vacuum gauge. Start the pump with 
 the delivery valve closed, and observe the pressure on the two 
 gauges for various speeds of the pump. Let p be the absolute 
 pressure per sq. foot in the delivery pipe and pi the absolute 
 
 pressure per sq. foot at the top of the suction pipe, then ~-~ 
 is the total centrifugal head h c . 
 
 1000 1800 200Q 2200 
 
 per Minute. 
 
 Fig. 282. 
 
 2400 
 
 A curve may now be plotted similar to that shown in Fig.[282 
 which has been drawn from data obtained from the pump, shown 
 in Fig. 275. 
 
 When the head is 44 feet, the speed at which delivery would 
 just start is 2000 revolutions per minute. 
 
 On reference to Fig. 293, which shows the discharge under 
 different heads at various speeds, the discharge at 2000 revolutions 
 per minute when the head is 44 feet is seen to be 12 cubic feet 
 per minute. This means, that if the pump is to discharge against 
 this head at this speed it cannot deliver less than 12 cubic feet 
 per minute. 
 
 228. Variation of the discharge of a centrifugal pump 
 with the head when the speed is kept constant*. 
 
 Head-discharge curve at constant velocity. If the speed of a 
 centrifugal pump is kept constant and the head varied, the dis- 
 charge varies as shown in Figs. 283, 285, 289, and 292. 
 
 * See also page 418. 
 
CENTRIFUGAL PUMPS 
 
 411 
 
 The curve No. 2, of Fig. 283, shows the variation of the head 
 with discharge for the pump shown in Fig. 275 when running at 
 1950 revolutions per minute; and that of Fig. 285 was plotted 
 from experimental data obtained by M. Eateau on a pump having 
 a wheel 11*8 inches diameter. 
 
 The data for plotting the curve shown in Fig. 289* was 
 obtained from a large centrifugal pump having a spiral chamber. 
 In the case of the dotted curve the head is always less than the 
 centrifugal head when the flow is zero, and the discharge against 
 a given head has only one value. 
 10 
 
 Fig. 283. 
 
 1 2 3 4 
 
 Radii Velocity of flow from, Wheel. 
 Head-discharge curve for Centrifugal Pump. Velocity Constant. 
 
 Fig. 284. Velocity-discharge curve for Centrifugal Pump. Head Constant. 
 
 In Fig. 285 the discharge when the head is 80 feet may be 
 either '9 or 3'5 cubic feet per minute. The work required to drive 
 the pump will be however very different at the two discharges, 
 and, as shown by the curves of efficiency, the actual efficiencies 
 for the two discharges are very different. At the given velocity 
 therefore and at 80 feet head, the flow is ambiguous and is 
 unstable, and may suddenly change from one value to the other, 
 or it may actually cease, in which case the pump would not start 
 again without the velocity v l being increased to 70'7 feet per 
 second. This value is calculated from the equation 
 
 " =80', 
 
 * Proceedings Inst. Mech. Engs., 1903. 
 
412 
 
 HYDRAULICS 
 
 the coefficient m for this pump being 1'02. For the flow to be 
 stable when delivering against a head of 80 feet, the pump should 
 be run with a rim velocity greater than 70' 7 feet per second, in 
 which case the discharge cannot be less than 4J cubic feet per 
 minute, as shown by the velocity-discharge curve of Fig. 287. 
 The method of determining this curve is discussed later. 
 
 Pomp Wheel 
 
 90 
 80 
 
 
 
 ^ 
 
 
 ^ 
 
 ^^^ 
 
 
 
 ^ 
 
 ^' l 
 $60 
 
 $* 
 '* 4V 
 
 ^30 
 ^20 
 tn\ 
 
 ^1-02 V? jr 
 ?,n 1 Hl 
 
 ad- Disci 
 
 large Car 
 
 ye 
 
 ^ 
 
 
 
 
 
 v, = 6G\ 
 
 oersec. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 J 
 
 1 3 4 
 
 Fig. 285. 
 
 . 
 
 Discharge in, c.ft. per mini. 
 
 Fig. 286. 
 
 75 
 
 60 
 
 
 
 
 
 / 
 
 \^ 
 
 Velocity-] 
 
 discharge 
 Constant - 
 
 Curve ^ 
 
 / 
 
 
 ^~"^~~~-. . 
 
 
 
 
 
 1 Z 3 4 
 
 Fig. 287. 
 
 Example. A centrifugal pump, when discharging normally, has a peripheral 
 velocity of 50 feet per second. 
 
 The angle at exit is 30 degrees and the manometric efficiency is 60 per cent. 
 The radial velocity of flow at exit is 2^/T. 
 
 Determine the lift h and the velocity of the wheel at which it will start delivery 
 under full head. 
 
 V = 50 -(2jh) cos 130 
 = 50-1-73^. 
 
CENTRIFUGAL PUMPS 413 
 
 , (50 - 1-73 Jh) 50 
 Therefore ft = 0-6. - , 
 
 from which ft = 37 feet. 
 
 Let v 2 be the velocity of the rim of the wheel at which pumping commences. 
 Then assuming the centrifugal head, when there is no discharge, is 
 v '* 
 
 l=*> 
 
 i? 2 =:48-6 ft. per sec. 
 
 229. Bernouilli's equations applied to centrifugal pumps. 
 
 Consider the motion of the water in any passage between two 
 consecutive vanes of a wheel. Let p be the pressure head at 
 inlet, pi at outlet and p a the atmospheric pressure per sq. foot. 
 
 If the wheel is at rest and the water passes through it in 
 the same way as it does when the wheel is in motion, and all 
 losses are neglected, and the wheel is supposed to be horizontal, by 
 Bernouilli's equations (see Figs. 277 and 278), 
 
 Pi + ^ - P + In 2 m 
 
 w 2g w 2g 
 
 But since, due to the rotation, a centrifugal head 
 
 is impressed on the water between inlet and outlet, therefore, 
 
 - + - 
 
 w 2g w 2g 2g 2g ' 
 
 P! p V* V* V r 2 V? 
 
 -S = 2T2^-2 .................. (4) ' 
 
 From (3) by substitution as on page 337, 
 
 a + g,.* + g + 2ifl S ......(5), 
 
 w 20 w 2g g g 
 and when U is radial and therefore equal to u, 
 
 _ 
 
 w 2g w 2g g 
 
 If now the velocity Ui is diminished gradually and without 
 shock, so that the water leaves the delivery pipe with a velocity 
 u dy and if frictional losses be neglected, the height to which the 
 water can be lifted above the centre of the pump is, by Bernouilli's 
 equation, 
 
 -5*g-57j{ ..................... 0). 
 
 If the centre of the wheel is h feet above the level of the water 
 in the sump or well, and the water in the well is at rest, 
 
 *= = *. + *+ ...(8). 
 
 w w 2 
 
414 HYDRAULICS 
 
 Substituting from (7) and (8) in (6) 
 
 ll 7^7 ^ d 
 
 - fi + ho + TJ- 
 9 2 2<7 
 
 = H +g=H ..................... (9). 
 
 This result verifies the fundamental equation given on page 398. 
 Further from equation (6) 
 
 Example. The centre of a centrifugal pump is 15 feet above the level of the 
 water in the sump. The total lift is 60 feet and the velocity of discharge from the 
 delivery pipe is 5 feet per second. The angle < at discharge is 135 degrees, and 
 the radial velocity of flow through the wheel is 5 feet per second. Assuming there 
 are no losses, find the pressure head at the inlet and outlet circumferences. 
 
 At inlet = 34'-15'-! 
 
 = 18-6 feet. 
 The radial velocity at outlet is 
 
 w x = 5 feet per second, 
 
 and ^ 
 
 and therefore, ^2 + 5^ = 1940 ....................................... (1), 
 
 from which 1 = 41-6 feet per second, 
 
 and Vj 
 
 The pressure head at outlet is then 
 
 ft^ + tt'-HL 1 
 
 w w 2g 
 
 = 45 feet. 
 To find the velocity v l when <j> is made 30 degrees. 
 
 therefore (1) becomes v-f - 5 */3 . v x = 1940, 
 
 from which ^ = 48-6 ft. per sec. 
 
 and V 1 = 40 
 
 Then Hi. = 25-4 feet, and ^ = 53-6 feet. 
 
 2g w 
 
 230. Losses in centrifugal pumps. 
 
 The losses of head in a centrifugal pump are due to the same 
 causes as the losses in a turbine. 
 
 Loss of head at exit. The velocity Ui with which the water 
 leaves the wheel is, however, usually much larger than in the 
 case of the turbine, and as it is not an easy matter to diminish 
 this velocity gradually, there is generally a much larger loss of 
 velocity head at exit from the wheel in the pump than in the 
 turbine. 
 
CENTRIFUGAL PUMPS 415 
 
 In many of the earlier pumps, which had radial vanes at exit, 
 
 IT 2 
 the whole of the velocity head ~ was lost, no special precautions 
 
 being taken to diminish it gradually and the efficiency was 
 constantly very low, being less than 40 per cent. 
 
 The effect of the angle <f> on the efficiency of the pump. To 
 increase the efficiency Appold suggested that the blade should be 
 set back, the angle <f> being thus less than 90 degrees, Fig. 272. 
 
 Theoretically, the effect on the efficiency can be seen by 
 considering the three cases considered in section 220 and illustrated 
 
 U 2 
 
 in Fig. 279. When <f> is 90 degrees ~~ is '54H, and when < is 
 
 U 2 
 30 degrees ~ is '36H. If, therefore, in these two cases this head 
 
 *j 
 
 is lost, while the other losses remain constant, the efficiency in 
 the second case is 18 per cent, greater than in the first, and the 
 efficiencies cannot be greater than 46 per cent, and 64 per cent. 
 respectively. 
 
 In general when there is no precaution taken to utilise the 
 energy of motion at the outlet of the wheel, the theoretical lift is 
 
 and the maximum possible manometric efficiency is 
 
 Substituting for Yi, Vi- Ui cot <, and for Ui 2 , Yi 2 + ^i 2 , 
 
 Vi U? 
 
 = 
 
 and 
 
 2 (vi v l u l cot 
 u^ cosec 2 <f> 
 
 vi vi Ui cot < 
 
 When v l is 30 feet per second, u^ 5 feet per second and < 
 150 degrees, e is 56 per cent, and when <f> is 90 degrees e is 
 48'5 per cent. 
 
 Experiments also show that in ordinary pumps for a given lift 
 and discharge the efficiency is greater the smaller the angle <. 
 
 Parsons* found that when < was 90 degrees the efficiency of a 
 pump in which the wheel was surrounded by a circular casing 
 was nearly 10 per cent, less than when the angle <f> was made 
 about 165 degrees. 
 
 * Proceedings List. C. E., Vol. XLVII. p. 272. 
 
416 HYDRAULICS 
 
 Stanton found that a pump 7 inches diameter having radial 
 vanes at discharge had an efficiency of 8 per cent, less than when 
 the angle <f> at delivery was 150 degrees. In the first case the 
 maximum actual efficiency was only 39'6 per cent., and in the 
 second case 50 per cent. 
 
 It has been suggested by Dr Stanton that a second reason for 
 the greater efficiency of the pump having vanes curved back at 
 outlet is to be found in the fact that with these vanes the variation 
 of the relative velocity of the water and the wheel is less than 
 when the vanes are radial at outlet. It has been shown experi- 
 mentally that when the section of a stream is diverging, that is 
 the velocity is diminishing and the pressure increasing, there is 
 a tendency for the stream lines to flow backwards towards the 
 sections of least pressure. These return stream lines cause a loss 
 of energy by eddy motions. Now in a pump, when the vanes are 
 radial, there is a greater difference between the relative velocity 
 of the water and the vane at inlet and outlet than when the angle 
 < is less than 90 degrees (see Fig. 279), and it is probable there- 
 fore that there is more loss by eddy motions in the wheel in the 
 former case. 
 
 Loss of head at entry. To avoid loss of head at entry the vane 
 must be parallel to the relative velocity of the water and the 
 vane. 
 
 Unless guide blades are provided the exact direction in which 
 the water approaches the edge of the vane is not known. If there 
 were no friction between the water and the eye of the wheel it 
 would be expected that the stream lines, which in the suction pipe 
 are parallel to the sides of the pipe, would be simply turned to 
 approach the vanes radially. 
 
 It has already been seen that when there is no flow the water 
 in the eye of the wheel is made to rotate by friction, and it is 
 probable that at all flows the water has some rotation in the eye 
 of the wheel, but as the delivery increases the velocity of rotation 
 probably diminishes. If the water has rotation in the same 
 direction as the wheel, the angle of the vane at inlet will clearly 
 have to be larger for no shock than if the flow is radial. That 
 the water has rotation before it strikes the vanes seems to be 
 indicated by the experiments of Mr Livens on a pump, the vanes 
 of which were nearly radial at the inlet edge. (See section 236.) 
 The efficiencies claimed for this pump are so high, that there 
 could have been very little loss at inlet. 
 
 If the pump has to work under variable conditions and the 
 water be assumed to enter the wheel at all discharges in the same 
 direction, the relative velocity of the water and the edge of the 
 
CENTRIFUGAL PUMPS 417 
 
 vane can only be parallel to the tip of the vane for one discharge, 
 and at other discharges in order to make the water move along 
 the vane a sudden velocity must be impressed upon it, which 
 causes a loss of energy. 
 
 Let 2^2, Fig. 288, be the velocity with which the water enters a 
 wheel, and and v the inclination 
 and velocity of the tip of the vane \e--u* ->j 
 at inlet respectively. 
 
 The relative velocity of u% and v 
 is V/, the vector difference of u% 
 and v. 
 
 The radial component of flow 
 through the opening of the wheel 
 must be equal to the radial com- 
 ponent of u 2 , and therefore the 
 relative velocity of the water along the tip of the vane is V r . 
 
 If u* is assumed to be radial, a sudden velocity 
 
 u 8 - v u?, cot 
 has thus to be given to the water. 
 
 If Uz has a component in the direction of rotation u s will be 
 diminished. 
 
 It has been shown (page 67), on certain assumptions, that if 
 a body of water changes its velocity from v a to v d suddenly, the 
 
 head lost is ^-^ , or is the head due to the change of velocity. 
 In this case the change of velocity is u 8 , and the head lost may 
 
 JfU 2 
 
 reasonably be taken as -n~~ If & is assumed to be unity, the 
 effective work done on the water by the wheel is diminished by 
 
 If now this loss takes place in addition to the velocity head 
 being lost outside the wheel, and friction losses are neglected, 
 then 
 
 g 20 20 
 
 
 Zg 
 
 V* 
 
 L. H. 27 
 
418 
 
 HYDRAULICS 
 
 Example. The radial velocity of flow through a pump is 5 feet per second. 
 The angle is 30 degrees and the angle d is 15 degrees. The velocity of the 
 outer circumference is 50 feet per sec. and the radius is twice that of the inner 
 circumference. 
 
 Find the theoretical lift on the assumption that the whole of the kinetic energy 
 is lost at exit. 
 
 
 
 2g 
 
 = 37-5 feet. 
 
 The theoretical lift neglecting all losses is 64-2 feet, and the manometric 
 efficiency is therefore 58 per cent. 
 
 231. Variation of the head with discharge and with the 
 speed of a centrifugal pump. 
 
 It is of interest to study by means of equation (1), section 230, 
 the variation of the discharge Q with the velocity of the pump 
 when h is constant, and the variation of the head with the 
 discharge when the velocity of the pump is constant, and to 
 compare the results with the actual results obtained from 
 experiment. 
 
 The full curve of Fig. 289 shows the variations of the head 
 with the discharge when the velocity of a wheel is kept constant. 
 The data for which the curve has been plotted is indicated in 
 the figure. 
 
 JO 
 
 n 
 
 13 
 
 I. 
 1" 
 
 $10 
 
 
 
 
 
 
 
 
 
 
 
 >< 
 
 **">. 
 
 *""* 
 
 ^ 
 
 \ 
 
 
 
 / 
 
 
 
 \ 
 
 
 \ 
 
 
 / 
 
 Z% 
 
 (f=9t 
 
 e-15 
 
 orma 
 
 i 
 
 )Ft.p< 
 ) 
 
 
 
 zrSet 
 
 >. 
 
 \ 
 
 
 \ 
 
 / 
 
 
 
 
 \ 
 
 N 
 o 
 
 1 rcudbiaJL velocity =1i 
 
 2 I* l Is'*" 
 
 rir 
 
 velocity of FLow= Q 
 
 A, 
 
 Fig. 289. Head-discharge curve at constant velocity. 
 When the discharge is zero 
 
 The velocity of flow -- at outlet has been assumed equal to 
 
 AI 
 
 9- at inlet. 
 
 Values of 1, 2, etc. were given to ~ and the corresponding 
 
 .XX 
 
 values of h found from equation (1). 
 
CENTRIFUGAL PUMPS 419 
 
 When the discharge is normal, that is, the water enters the 
 wheel without shock, -^ is 4 feet and h is 14 feet. The theoretical 
 head assuming no losses is then 28 feet and the manometric 
 efficiency is thus 50 per cent. For less or greater values of - 
 
 the head diminishes and also the efficiency. 
 
 The curve of Fig. 290 shows how the flow varies with the 
 velocity for a constant value of h, which is taken as 12 feet. 
 
 2 fee 
 
 RadiaL Velocity througk Wheel. 
 
 Fig. 290. Velocity-discharge curve at constant head for Centrifugal Pump. 
 
 It will be seen that when the velocity Vi is 31'9 feet per second 
 the velocity of discharge may be either zero or 8*2 feet per second. 
 This means that if the head is 12 feet, the pump, theoretically, 
 will only start when the velocity is 31 '9 feet per second and the 
 velocity of discharge will suddenly become 8*2 feet per second. 
 If now the velocity Vi is diminished the pump still continues to 
 discharge, and will do so as long as v l is greater than 26'4 feet per 
 second. The flow is however unstable, as at any velocity v c it may 
 suddenly change from CE to CD, or it may suddenly cease, and it 
 will not start again until Vi is increased to 31*9 feet per second. 
 
 232. The effect of the variation of the centrifugal head 
 and the loss by friction on the discharge of a pump. 
 
 If then the losses at inlet and outlet were as above and were 
 the only losses, and the centrifugal head in an actual pump was 
 equal to the theoretical centrifugal head, the pump could not be 
 made to deliver water against the normal head at a small velocity 
 of discharge. In the case of the pump considered in section 231, 
 it could not safely be run with a rim velocity less than 31*9 ft. 
 per sec., and at any greater velocity the radial velocity of flow 
 could not be less than 8 feet per second. 
 
 272 
 
420 HYDRAULICS 
 
 In actual pumps, however, it has been seen that the centrifugal 
 head at commencement is greater than 
 
 There is also loss of head, which at high velocities and in small 
 pumps is considerable, due to friction. These two causes consider- 
 ably modify the head-discharge curve at constant velocity and the 
 velocity-discharge curve at constant head, and the centrifugal 
 head at the normal speed of the pump when the discharge is zero, 
 is generally greater than any head under which the pump works, 
 and many actual pumps can deliver variable quantities of water 
 against the head for which they are designed. 
 
 The centrifugal head when the flow is zero is 
 
 2? ' 
 
 m being generally equal to, or greater than unity. As the flow 
 increases, the velocity of whirl in the eye of the wheel and in 
 the casing will diminish and the centrifugal head will therefore 
 diminish. 
 
 Let it be assumed that when the velocity of flow is u (supposed 
 constant) the centrifugal head is 
 
 (lev nu) 2 
 
 Jc and n being constants which must be determined by experiment. 
 When u is zero 
 
 V W 
 
 2g 2g 2g ~ 2g > 
 
 and if m is known k can at once be found. 
 
 Let it further be assumed that the loss by friction* and eddy 
 
 motions, apart from the loss at inlet and outlet is ^ . 
 
 * The loss of head by friction will no doubt depend not only upon u but also 
 upon the velocity ^ of the wheel, and should be written as 
 
 Cw 2 C 
 
 .... 
 
 If it be supposed it can be expressed by the latter, then the correction 
 Wv* _ 2nku lVl k^ 
 2<7 20 ~~27' 
 
 if proper values are given to fc, n x and k lt takes into account the variation of the 
 centritugal head and also the friction head v, . 
 
CENTRIFUGAL PUMPS 421 
 
 The gross head h is then, 
 
 , v? v 2 u* , 2vu cot 
 
 cV 
 
 If now the head h and flow Q be determined experimentally, 
 the difference between h as determined from equation (1), page 4J 7, 
 and the experimental value of h, must be equal to 
 
 it 2 , 2 2 , 
 
 ~ (n ~ 
 
 ki being equal to (c 2 - n 2 ). 
 
 The coefficient k being known from an experiment when u is 
 zero, two other experiments giving corresponding values of h and 
 u will determine the coefficients n and ki . 
 
 The head-discharge curve at constant velocity, for a pump such 
 as the one already considered, would approximate to the dotted 
 curve of Fig. 289. This curve has been plotted from equation (2), 
 by taking k as 0*5, n as 7'64 and ki as - 38. 
 
 Substituting values for k, n, k l} cosec < and cot <, equation (2) 
 becomes 
 
 , mv? Cu . v 
 
 h= ~~ 
 
 C and Ci being new coefficients ; or it may be written 
 
 h = l + ^ + C 3 Q> ..................... (4), 
 
 Q being the flow in any desired units, the coefficients C 2 and C 3 
 varying with the units. If equation (4) is of the correct form, 
 three experiments will determine the constants m, C 2 and C 3 
 directly, and having given values to any two of the three 
 variables h, v, and Q the third can be found. 
 
 233. The effect of the diminution of the centrifugal head 
 and the increase of the friction head as the flow increases, on 
 the velocity-discharge curve at constant head. 
 
 Using the corrected equation (2), section 232, and the given 
 values of k, n t and ki the dotted curve of Fig. 290 has been plotted. 
 
 From the dotted curve of Fig. 289 it is seen that u cannot 
 be greater than 5 feet when the head is 12 feet, and therefore the 
 new curve of Fig. 290 is only drawn to the point where u is 5. 
 
 The pump starts delivering when v is 27*7 feet per second and 
 the discharge increases gradually as the velocity increases. 
 
422 HYDRAULICS 
 
 The pump will deliver, therefore, water under a head of 
 12 feet at any velocity of flow from zero to 5 feet per second. 
 
 In such a pump the manometric efficiency must have its 
 maximum value when the discharge is zero and it cannot be 
 
 greater than 
 
 my* 
 
 Vi a - ViUi CQt ' 
 
 9 
 
 This is the case with many existing pumps and it explains why, 
 when running at constant speed, they can be made to give any 
 discharge varying from zero to a maximum, as the head is 
 diminished. 
 
 234. Special arrangements for converting the velocity 
 
 U 2 
 head =- with which the water leaves the wheel into pressure 
 
 head. 
 
 The methods for converting the velocity head with which the 
 water leaves the wheel into pressure head have been indicated on 
 page 394. They are now discussed in greater detail. 
 
 Thomson's vortex or whirlpool chamber. Professor James 
 Thomson first suggested that the wheel should be surrounded by 
 a chamber in which the velocity of the water should gradually 
 change from Ui to u d the velocity of flow in the pipe. Such a 
 chamber is shown in Fig. 274. In this chamber the water forms 
 a free vortex, so called because no impulse is given to the water 
 while moving in the chamber. 
 
 Any fluid particle ab, Fig. 281, may be considered as moving 
 in a circle of radius r with a velocity v and to have also a 
 radial velocity u outwards. 
 
 Let it be supposed the chamber is horizontal. 
 
 If W is the weight of the element in pounds, its momentum 
 
 perpendicular to the radius is - and the moment of mo- 
 
 9 
 
 mentum or angular momentum about the centre C is . 
 
 For the momentum of a body to change, a force must act upon 
 it, and for the moment of momentum to change, a couple must act 
 upon the body. 
 
 But since no turning effort, or couple, acts upon the element 
 after leaving the wheel its moment of momentum must be 
 constant. 
 
CENTRIFUGAL PUMPS 423 
 
 Therefore, 
 
 is constant or v r = constant. 
 
 If the sides of the chamber are parallel the peripheral area of 
 the concentric rings is proportional to r , and the radial velocity of 
 flow u for any ring will be inversely proportional to r , and there- 
 fore, the ratio -- is constant, or the direction of motion of any 
 
 element with its radius r is constant, and the stream lines are 
 equiangular spirals. 
 
 If no energy is lost, by friction and eddies, Bernoulli's theorem 
 will hold, and, therefore, when the chamber is horizontal 
 
 U 2 VQ PQ 
 
 2g + 2g + w 
 
 is constant for the stream lines. 
 
 This is a general property of the free vortex. 
 If u is constant 
 
 V V 
 
 ~- + = constant. 
 2g w 
 
 Let the outer radius of the whirlpool chamber be R, and 
 the inner radius r ll} . Let v r , w and v Rw be the whirling velocities 
 at the inner and outer radii respectively. 
 
 Then since v r is a constant, 
 
 and + ^- = constant, 
 
 w g 
 
 w ~ w 2g 2g 
 
 w 
 When R* = 2r v 
 
 w w 4* 2g 
 
 If the velocity head which the water possesses when it leaves 
 the vortex chamber is supposed to be lost, and hi is the head of 
 water above the pump and p a the atmospheric pressure, then 
 neglecting friction 
 
 P*v_ , U<? Pa 
 
 fl/l ~r ~, ~ T . 
 
 w 2g w 
 
 or 
 
 w 2g w' 
 
424 HYDRAULICS 
 
 If then h Q is the height of the pump above the well, the total 
 lift h* is hi + ho. 
 Therefore, 
 
 also Prw = PI? fw R, and v rw = 
 
 Therefore 
 
 But from equation (6) page 413, 
 
 w w 2g g 2g ' 
 Therefore 
 
 R 2 
 
 This might have been written down at once from equation (1), 
 
 section 230. For clearly if there is a gain of pressure head 
 
 y 2 e j^2 \ 
 
 in the vortex chamber of -^ ( 1 -5-= ) , the velocity head to 
 
 g \ n, w / 
 
 be lost will be less by this amount than when there is no vortex 
 chamber. 
 
 Substituting for Vi and Ui the theoretical lift h is now 
 
 , _ i - ii cot <^> u^ fa - HI cot <^>) 2 E 2 x 1 x 
 
 ~%- ~W "'B? 
 When the discharge or rim velocity is not normal, there is a 
 further loss of head at entrance equal to 
 
 - 
 
 and 
 
 h = 
 
 (2). 
 
 When there is no discharge v rw is equal to t?i and 
 
 V 
 
CENTRIFUGAL PUMPS 425 
 
 If R = JR, and v = \v\ , 
 
 O 2 
 
 Correcting equation (1) in order to allow for the variation of 
 the centrifugal head with the discharge, and the friction losses, 
 
 7 _ Vi 2 V\Ui COt < U\ (v\ Ksi COt <) 2 R 2 
 
 ~g~ ~%9~ 
 
 _ (v u cot O) 2 fcV _ 2nkuvi _ kiu z 
 
 ~W~ "~W~ ~W ""20"' 
 
 2 /~1 /^\ /*^ /^\2 
 
 1 r i 7 WUi (jz^JV U.sU 
 
 which reduces to h c + ~ + ^ . 
 
 2g 2g 2g 
 
 The experimental data on the value of the vortex chamber 
 per 86 t in increasing the efficiency is very limited. 
 
 Stanton* showed that for a pump having a rotor 7 inches 
 diameter surrounded by a parallel sided vortex chamber 18 inches 
 diameter, the efficiency of the chamber in converting velocity head 
 to pressure head was about 40 per cent. It is however questionable 
 whether the design of the pump was such as to give the best results 
 possible. 
 
 So far as the author is aware, centrifugal pumps with vortex 
 chambers are not now being manufactured, but it seems very 
 probable that by the addition of a well-designed chamber small 
 centrifugal pumps might have their efficiencies considerably in- 
 creased. 
 
 235. Turbine pumps. 
 
 Another method, first suggested by Professor Reynolds, and 
 now largely used, for diminishing the velocity of discharge Ui 
 gradually, is to discharge the water from the wheel into guide 
 passages the sectional area of which should gradually increase 
 from the wheel outwards, Figs. 275 and 276, and the tangents to the 
 tips of the guide blades should be made parallel to the direction 
 of Ux. 
 
 The number of guide passages in small pumps is generally four 
 or five. 
 
 If the guide blades are fixed as in Fig. 275, the direction of 
 the tips can only be correct for one discharge of the pump, 
 but except for large pumps, the very large increase in initial cost 
 of the pump, if adjustable guide blades were used, as well as 
 the mechanical difficulties, would militate against their adoption. 
 
 Single wheel pumps of this type can be used up to a head of 
 100 feet with excellent results, efficiencies as high as 85 per cent. 
 * Proceedings Imt. C. E., 1903. 
 
426 HYDRAULICS 
 
 having been claimed. They are now being used to deliver water 
 against heads of over 350 feet, and M. Rateau has used a single 
 wheel 3*16 inches diameter running at 18,000 revolutions per 
 minute to deliver against a head of 936 feet. 
 
 Loss of head at the entrance to the guide passages. If the 
 guide blades are fixed, the direction of the tips can only be correct 
 for one discharge of the pump. For any other discharge than the 
 normal, the direction of the water as it leaves the wheel is not 
 parallel to the fixed guide and there is a loss of head due to 
 shock. 
 
 Let cc be the inclination of the guide blade and </> the vane 
 angle at exit. 
 
 Let 1*1 be the radial velocity of 
 flow. Then BE, Fig. 291, is the 
 velocity with which the water leaves 
 the wheel. 
 
 The radial velocity with which 
 the water enters the guide passages must be Ui and the velocity 
 along the guide is, therefore, BF. 
 
 There is a sudden change of velocity from BE to BF, and on 
 the assumption that the loss of head is equal to the head due to the 
 relative velocity FE, the head lost is 
 
 fa - Ui cot <j> ui cot ) 2 
 
 ~W~ 
 At inlet the loss of head is 
 
 (v - u cot 0) 2 
 
 and the theoretical lift is 
 
 -, _ Vi - ViUi cot <fr _ (v-u cot 6>) 2 _ (^ Ui cot < - Ui cot a) 
 ~~ ~ 
 
 __ l cot a 2vu cot 6 
 
 2g 2g + 2g 2g 
 
 u? (cot < + cot a) 2 u 2 cot 2 
 
 2g 2g ...... (1) ' 
 
 To correct for the diminution of the centrifugal head and to 
 allow for friction, 
 
 feV 2^71 . ^ u^ 
 20 ~ 2<f~ I 2g> 
 must be added, and the lift is then 
 
 h = ^ - ^- + ??MM5ota 2vu cot 6 ui* (cot <j> + cot a) 2 
 2 2 20 2g ~^g~ 
 
 2knv lUl 
 
 _ _ 
 
 2g 2g 2g ~ 2g 
 
CENTRIFUGAL PUMPS 
 
 427 
 
 which, since u can always be written as a multiple of t&i, reduces 
 to the form 
 
 2gh = mv l * + Cu l v 1 + C l u 1 * ..................... (2). 
 
 Equations for the turbine pump shown in Fig. 275. Character- 
 istic curves. Taking the data 
 
 0= 5 degrees, cot<9 = ll'43 
 </> = 30 cot< = 1*732 
 
 a= 3 
 
 cota=19'6 
 
 equation (2) above becomes 
 
 2gh = ' 
 
 eo- 
 
 (3). 
 
 irv Cubic Feet per Itfinjuute. 
 
 01 34-5 
 
 JLculiaJj Velocity cub Eacib from, fheWheeU. Teet per Sec onaU. 
 
 Fig. 292. Head-discharge curves at constant speed for Turbine Pump. 
 
 From equation (3) taking Vi as 50 feet per second, the head- 
 discharge curve No. 1, of Fig. 283, has been drawn, and taking h 
 as 35 feet, the velocity-discharge curve No. 1, of Fig. 284, has been 
 plotted. 
 
 In Figs. 292 4 are shown a sanies of head-discharge curves at 
 
428 
 
 HYDRAULICS 
 
 constant speed, velocity-discharge curves at constant head, and 
 head-velocity curves at constant discharge, respectively. 
 
 The points shown near to the curves were determined experi- 
 mentally, and the curves, it will be seen, are practically the mean 
 curves drawn through the experimental points. They were how- 
 ever plotted in all cases from the equation 
 
 2gh = 1-087^ + 2'26tM?i - 62' W, 
 
 obtained by substituting for m, C and d in equation (2) the values 
 1*087, 2'26 and - 62'1 respectively. The value of m was obtained 
 by determining the head h, when the stop valve was closed, for 
 speeds between 1500 and 2500 revolutions per minute, Fig. 282. 
 The values of C and Ci were first obtained, approximately, by 
 taking two values of u^ and v t respectively from one of the 
 actual velocity-discharge curves near the middle of the series, for 
 which h was known, and from the two quadratic equations thus 
 obtained C and Ci were calculated. By trial C and Ci were then 
 corrected to make the equation more nearly fit the remaining 
 curves. 
 
 mo 1900 . 
 
 Speed ReYoluALon& per Mirvuube,. 
 
 Fig. 293. Velocity-Discharge curves at Constant Head. 
 
 No attempt has been made to draw the actual mean curves in 
 the figures, as in most cases the difference between them and the 
 calculated curves drawn, could hardly be distinguished. The 
 reader can observe for himself what discrepancies there are between 
 the mean curves through the points and the calculated curves. It 
 
CENTRIFUGAL PUMPS 
 
 429 
 
 will be seen that for a very wide range of speed, head, and 
 discharge, the agreement between the curves and the observed 
 points is very close, and the equation can therefore be used with 
 confidence for this particular pump to determine its performance 
 under stated conditions. 
 
 It is interesting to note, that the experiments clearly indicated 
 the unstable condition of the discharge when the head was kept 
 constant and the velocity was diminished below that at which the 
 discharge commenced. 
 
 20 
 
 -2300- 
 
 ieoo 
 
 1900 
 
 2000 
 
 S per 
 
 2409 
 
 Fig. 294. Head-velocity curves at Constant Discharge. 
 
 236. Losses in the spiral casings of centrifugal pumps. 
 
 The spiral case allows the mean velocity of flow toward the 
 discharge pipe to be fairly constant and the results of experiment 
 seem to show that a large percentage of the velocity of the water 
 at the outlet of the wheel is converted into pressure head. 
 Mr Livens* obtained, for a pump having a wheel 19 \ inches 
 diameter running at 550 revolutions per minute, an efficiency of 
 71 per cent, when delivering 1600 gallons per minute against a 
 head of 25 feet. The angle < was about 13 degrees and the mean 
 of the angle for the two sides of the vane 81 degrees. 
 
 For a similar pump 21f inches diameter an efficiency of 82 per 
 cent, was claimed. 
 
 Proceedings Inst. Mech. Engs., 1903. 
 
430 HYDRAULICS 
 
 The author finds the equation to the head-discharge curve for 
 the 19 J inches diameter pump from Mr Livens' data to be 
 
 and for the 21| inches diameter pump 
 
 Tl&Y 5 - 4-5^ = 2^ ..................... (2). 
 
 The velocity of rotation of the water round the wheel will be 
 less than the velocity with which the water leaves the wheel and 
 there will be a loss of head due to the sudden change in velocity. 
 
 k U 2 
 Let this loss of head be written -^- L . The head, when ^ is the 
 
 radial velocity of flow at exit and assuming the water enters the 
 wheel radially, is then 
 
 _Vi z - ViUi cot <f> _ fc 3 Ui 2 _ (v-ucotO) 2 
 
 9 ~W %9 
 
 Taking friction and the diminution of centrifugal head into 
 account, 
 
 , _v*ViUiCot<l> & 3 Ui 2 (v ucotOy 
 
 ~~i~ "W ~W~ 
 
 which again may be written 
 , _ 
 = 
 
 The values of m } C and Cj. are given for two pumps in equations 
 (1) and (2). 
 
 237. General equation for a centrifugal pump. 
 
 The equations for the gross head h at discharge Q as determined 
 for the several classes of pumps have been shown to be of the form 
 
 C 3 Q 2 
 
 or, if u is the velocity of flow from the wheel, 
 , _ mv* Cuv du* 
 ~-W"^9~~ 2? ' 
 
 in which ra varies between 1 and 1*5. The coefficients C 2 and C 3 
 for any pump will depend upon the unit of discharge. 
 
 As a further example and illustrating the case in which at 
 certain speeds the flow may be unstable, the curves of Figs. 
 285 287 may be now considered. When Vi is 66 feet per second 
 the equation to the head discharge curve is 
 , = 1-02^ 15'5Q^ 
 
 20 2g 
 
 Q being in cubic feet per minute. 
 
CENTRIFUGAL PUMPS 431 
 
 The velocity-discharge curve for a constant head of 80 feet as 
 calculated from this equation is shown in Fig. 287. 
 
 To start the pump against a head of 80 feet the peripheral 
 velocity has to be 70*7 feet per second, at which velocity the 
 discharge Q suddenly rises to 4*3 cubic feet per minute. 
 
 The curves of actual and manometric efficiency are shown in 
 Fig. 286, the maximum for the two cases occurring at different 
 discharges. 
 
 238. The limiting height to which a single wheel centri- 
 fugal pump can be used to raise water. 
 
 The maximum height to which a centrifugal pump can raise 
 water, depends theoretically upon the maximum velocity at which 
 the rim of the wheel can be run. 
 
 It has already been stated that rim velocities up to 250 feet 
 per second have been used. Assuming radial vanes and a mano- 
 metric efficiency of 50 per cent., a pump running at this velocity 
 would lift against a head of 980 feet. 
 
 At these very high velocities, however, the wheel must be of 
 some material such as bronze or cast steel, having considerable 
 resistance to tensile stresses, and special precautions must be 
 taken to balance the wheel. The hydraulic losses are also 
 considerable, and manometric efficiencies greater than 50 per 
 cent, are hardly to be expected. 
 
 According to M. Eateau *, the limiting head against which it is 
 advisable to raise water by means of a single wheel is about 
 100 feet, and the maximum desirable velocity of the rim of the 
 wheel is about 100 feet per second. 
 
 Single wheel pumps to lift up to 350 feet are however being 
 used. At this velocity the stress in a hoop due to centrifugal forces 
 is about 7250 Ibs. per sq. incht. 
 
 239. The suction of a centrifugal pump. 
 
 The greatest height through which a centrifugal or other class 
 of pump will draw water is about 27 feet. Special precaution has 
 to be taken to ensure that all joints on the suction pipe are perfectly 
 air-tight, and especially is this so when the suction head is greater 
 than 15 feet; only under special circumstances is it therefore de- 
 sirable for the suction head to be greater than this amount, and it 
 is always advisable to keep the suction head as small as possible. 
 
 * "Pompes Centrifuges," etc.. Bulletin de la Societe de VIndustrie minerale, 
 1902 ; Engineer, p. 236, March, 1902. 
 
 f See Ewing's Strength of Materials ; Wood's Strength of Structural Members ; 
 The Steam Turbine, Stodola. 
 
432 
 
 HYDRAULICS 
 
CENTRIFUGAL PUMPS 
 
 433 
 
 240. Series or multi-stage turbine pumps. 
 
 It has been stated that the limiting economical head for a single 
 wheel pump is about 100 feet, and for high heads series pumps 
 are now generally used. 
 
 Fig. 296. General Arrangement of Worthington Multi-stage Turbine Pump. 
 
 By putting several wheels or rotors in series on one shaft, each 
 rotor giving a head varying from 100 to 200 feet, water can be 
 lifted to practically any height, and such pumps have been 
 L. H. 28 
 
434 
 
 HYDRAULICS 
 
 constructed to work against a head of 2000 feet. The number 
 of rotors, on one shaft, may be from one to twelve according 
 to the total head. For a given head, the greater the number of 
 rotors used, the less the peripheral velocity, and within certain 
 limits the greater the efficiency. 
 
 Figs. 295 and 296 show a longitudinal section and general 
 arrangement, respectively, of a series, or multi-stage pump, as 
 constructed by the Worthington Pump Company. On the motor 
 shaft are fixed three phosphor-bronze rotors, alternating with fixed 
 guides, which are rigidly connected to the outer casing, and to 
 the bearings. The water is drawn in through the pipe at the left 
 of the pump and enters the first wheel axially. The water leaves 
 the first wheel at the outer circumference and passes along an 
 expanding passage in which the velocity is gradually diminished 
 and enters the second wheel axially. The vanes in the passage 
 are of hard phosphor-bronze made very smooth to reduce friction 
 losses to a minimum. The water passes through the remaining 
 rotors and guides in a similar manner and is finally discharged 
 into the casing and thence into the delivery pipe. 
 
 Fig. 297. Sulzer Multi-stage Turbine Pump. 
 
 The difference in pressure head at the entrances to any two 
 consecutive wheels is the head impressed on the water by one 
 wheel. If the head is h feet, and there are n wheels the total 
 lift is nearly nh feet. The vanes of each wheel and the directions 
 of the guide vanes are determined as explained for the single 
 wheel so that losses by shock are reduced to a minimum, and 
 the wheels and guide passages are made smooth so as to reduce 
 friction. 
 
 Through the back of each wheel, just above the boss, are 
 a number of holes which allow water to get behind part of the 
 wheel, under the pressure at which it enters the wheel, to balance 
 the end thrust which would otherwise be set up. 
 
CENTRIFUGAL PUMPS 435 
 
 The pumps can be arranged to work either vertically or 
 horizontally, and to be driven by belt, or directly by any form 
 of motor. 
 
 Fig. 297 shows a multi-stage pump as made by Messrs Sulzer. 
 The rotors are arranged so that the water enters alternately 
 from the left and right and the end thrust is thus balanced. 
 Efficiencies as high as 84 per cent, have been claimed for multi- 
 stage pumps lifting against heads of 1200 feet and upwards. 
 
 The Worthington Pump Company state that the efficiency 
 diminishes as the ratio of the head to the quantity increases, the 
 best results being obtained when the number of gallons raised 
 per minute is about equal to the total head. 
 
 Example. A pump is to be driven by a motor at 1450 revolutions per minute, and 
 is required to lift 45 cubic feet of water per minute against a head of 320 feet. 
 Kequired the diameter of the suction, and delivery pipes, and the diameter and 
 number of the rotors, assuming a velocity of 5 '5 feet per second in the suction and 
 delivery pipes, and a manometric efficiency at the given delivery of 50 per cent. 
 
 Assume provisionally that the diameter of the boss of the wheel is 3 inches. 
 
 Let d be the external diameter of the annular opening, Fig. 295. 
 
 Then, l(#-V} 4 . 
 
 144 = 60x5-5' 
 from which d = Q inches nearly. 
 
 Taking the external diameter D of the wheel as 2d, D is 1 foot. 
 
 Then, v l = ^ x w = 76 feet per sec. 
 
 uo 
 
 Assuming radial blades at outlet the head lifted by each wheel is 
 
 h = 0-5. ^ feet 
 
 = 90 feet. 
 Four wheels would therefore be required. 
 
 241. Advantages of centrifugal pumps. 
 
 There are several advantages possessed by centrifugal pumps. 
 
 In the first place, as there are no sliding parts, such as occur in 
 reciprocating pumps, dirty water and even water containing com- 
 paratively large floating bodies can be pumped without greatly 
 endangering the pump. 
 
 Another advantage is that as delivery from the wheel is 
 constant, there is no fluctuation of speed of the water in the 
 suction or delivery pipes, and consequently there is no necessity 
 for air vessels such as are required on the suction and delivery 
 pipes of reciprocating pumps. There is also considerably less 
 danger of large stress being engendered in the pipe lines by 
 "water hammer*." 
 
 Another advantage is the impossibility of the pressure in the 
 * See page 384. 
 
 282 
 
436 HYDRAULICS 
 
 pump casing rising above that of the maximum head which the 
 rotor is capable of impressing upon the water. If the delivery 
 is closed the wheel will rotate without any danger of the pressure 
 in the casing becoming greater than the centrifugal head (page 
 335). This may be of use in those cases where a pump is de- 
 livering into a reservoir or pumping from a reservoir. In the first 
 case a float valve may be fitted, which, when the water rises to 
 a particular height in the reservoir, closes the delivery. The 
 pump wheel will continue to rotate but without delivering water, 
 and if the wheel is running at such a velocity that the centri- 
 fugal head is greater than the head in the pipe line it will start 
 delivery when the valve is opened. In the second case a similar 
 valve may be used to stop the flow when the water falls below a 
 certain level. This arrangement although convenient is uneco- 
 nomical, as although the pump is doing no effective work, the 
 power required to drive the pump may be more than 50 per cent, 
 of that required when the pump is giving maximum discharge. 
 
 It follows that a centrifugal pump may be made to deliver 
 water into a closed pipe system from which water may be taken 
 regularly, or at intervals, while the pump continues to rotate at a 
 constant velocity. 
 
 Pump delivering into a long pipe line. When a centrifugal 
 pump or air fan is delivering into a long pipe line the resistances 
 will vary approximately as the square of the quantity of water 
 delivered by the pump. 
 
 Let p> 2 be the absolute pressure per square inch which has 
 to be maintained at the end of the pipe line, and let the 
 resistances vary as the square of the velocity v along the pipe. 
 Then if the resistances are equivalent to a head hf = kv 2 , the 
 
 pressure head at the pump end of the delivery pipe must be 
 
 w w 
 
 = P* frQ 2 
 
 w + A 2 ' 
 A being the sectional area of the pipe. 
 
 /v\ 
 
 Let be the pressure head at the top of the suction pipe, then 
 the gross lift of the pump is 
 
 r _Pl __P _Pz &Q 2 P 
 
 w w w A 2 w ' 
 If, therefore, a curve, Fig. 298, be plotted having 
 
 fo-p) , feQ 2 
 
 w ~ A 2 
 
CENTRIFUGAL PUMPS 
 
 437 
 
 as ordinates, and Q as abscissae, it will be a parabola. If on 
 the same figure a curve having h as ordinates and Q as abscissae 
 be drawn for any given speed, the intersection of these two 
 curves at the point P will give the maximum discharge the pump 
 will deliver along the pipe at the given speed. 
 
 iris C. Ft. per Second/, 
 Fig. 298. 
 
 242. Parallel flow turbine pump. 
 
 By reversing the parallel flow turbine a pump is obtained 
 which is similar in some respects to the centrifugal pump, but 
 differs from it in an essential feature, that no head is impressed on 
 the water by centrifugal forces between inlet and outlet. It 
 therefore cannot be called a centrifugal pump. 
 
 The vanes of such a pump might be arranged as in Fig. 299, 
 the triangles of velocities for inlet and outlet being as shown. 
 
 The discharge may be allowed to take place into guide 
 passages above or below the wheel, where the velocity can be 
 gradually reduced. 
 
 Since there is no centrifugal head impressed on the water 
 between inlet and outlet, Bernoulli's equation is 
 
 .+*'= +^'. 
 
 w 2g w 2gr 
 
 From which, as in the centrifugal pump, 
 
 - _ 
 
 g w w 2g 2g' 
 
 If the wheel has parallel sides as in Fig. 299, the axial velocity 
 of flow will be constant and if the angles < and are properly 
 chosen, V r and v r may be equal, in which case the pressure at 
 inlet and outlet of the wheel will be equal. This would have 
 the advantage of stopping the tendency for leakage through the 
 clearance between the wheel and casing. 
 
438 
 
 HYDRAULICS 
 
 Such a pump is similar to a reversed impulse turbine, the 
 guide passages of which are kept full. The velocity with which 
 the water leaves the wheel would however be great and the lift 
 above the pump would depend upon the percentage of the velocity 
 head that could be converted into pressure head. 
 
 Fig. 299. 
 
 Since there is no centrifugal head impressed upon the water, 
 the parallel-flow pump cannot commence discharging unless the 
 water in the pump is first set in motion by some external means, 
 but as soon as the flow is commenced through the wheel, the full 
 discharge under full head can be obtained. 
 
 Fig. 300. 
 
 Fig. 301. 
 
 To commence the discharge, the pump would generally have to 
 be placed below the level of the water to be lifted, an auxiliary 
 lischarge pipe being fitted with a discharging valve, and a non- 
 return valve in the discharge pipe, arranged as in Fig. 300. 
 
CENTRIFUGAL PUMPS 439 
 
 The pump could be started when placed at a height h above 
 the water in the sump, by using an ejector or air pump to exhaust 
 the air from the discharge chamber, and thus start the flow 
 through the wheel. 
 
 243. Inward flow turbine pump. 
 
 Like the parallel flow pump, an inward flow pump if constructed 
 could not start pumping unless the water in the wheel were first 
 set in motion. If the wheel is started with the water at rest 
 the centrifugal head will tend to cause the flow to take place 
 outwards, but if flow can be commenced and the vanes are 
 properly designed, the wheel can be made to deliver water at its 
 inner periphery. As in the centrifugal and parallel flow pumps, 
 if the water enters the wheel radially, the total lift is 
 
 TT _ ii =2 i_ ,_ m 
 
 g w w + 2g 2g .................. U; " 
 
 From the equation 
 
 ^V L 2 = pitv^_i^ 
 w + 2g w*2g + 2g 2g> 
 it will be seen that unless V r 2 is greater than 
 
 vl tf__v? 
 2g + 2g 2g> 
 
 IP 
 
 P! is less than p, and ~- will then be greater than the total 
 
 lift H. 
 
 Very special precautions must therefore be made to diminish 
 the velocity U gradually, or otherwise the efficiency of the pump 
 will be very low. 
 
 The centrifugal head can be made small by making the 
 difference of the inner and outer radii small. 
 
 ^ + ^.^i 
 
 2g + 2g 2g 
 
 V 2 
 is made equal to -~ , the pressure at inlet and outlet will be the 
 
 same, and if the wheel passages are carefully designed, the 
 pressure throughout the wheel may be kept constant, and the 
 pump becomes practically an impulse pump. 
 
 There seems no advantage to be obtained by using either 
 a parallel flow pump or inward flow pump in place of the centri- 
 fugal pump, and as already suggested there are distinct dis- 
 advantages. 
 
 244. Reciprocating pumps. 
 
 A simple form of reciprocating force pump is shown dia- 
 grammatically in Fig. 301. It consists of a plunger P working in 
 
440 
 
 HYDRAULICS 
 
 Fig. 301 a. Vertical Single-acting Reciprocating Pump. 
 
RECIPROCATING PUMPS 441 
 
 a cylinder C and has two valves Y s and Y D , known as the suction 
 and delivery valves respectively. A section of an actual pump 
 is shown in Fig. 301 a. 
 
 Assume for simplicity the pump to be horizontal, with the 
 centre of the barrel at a distance h from the level of the water 
 in the well; h may be negative or positive according as the 
 pump is above or below the surface of the water in the well. 
 
 Let B be the height of the barometer in inches of mercury. 
 The equivalent head H, in feet of water, is 
 
 13-596. B V1 
 12 = l lddB > 
 
 which may be called the barometric height in feet of water. 
 
 When B is 30 inches H is 34 feet. 
 
 When the plunger is at rest, the valve Y D is closed by the head 
 of water above it, and the water in the suction pipe is sustained by 
 the atmospheric pressure. 
 
 Let h be the pressure head in the cylinder, then 
 
 h = H-h, 
 or the pressure in pounds per square inch in the cylinder is 
 
 P--43KH-W, 
 
 p cannot become less than the vapour tension of the water. At 
 ordinary temperatures this is nearly zero, and h n cannot be greater 
 than 34 feet. 
 
 If now the plunger is moved outwards, very slowly, and there 
 is no air leakage the valve Y s opens, and the atmospheric pressure 
 causes water to rise up the suction pipe and into the cylinder, 
 h remaining practically constant. 
 
 On the motion of the plunger being reversed, the valve Y s 
 closes, and the water is forced through Y D into the delivery 
 pipe. 
 
 In actual pumps if h is less than from 4 to 9 feet the 
 dissolved gases that are in the water are liberated, and it is there- 
 fore practically impossible to raise water more than from 25 to 
 30 feet. 
 
 Let A be the area of the plunger in square inches and L the 
 stroke in feet. The pressure on the end of the plunger outside the 
 cylinder is equal to the atmospheric pressure, and neglecting 
 the friction between the plunger and the cylinder, the force neces- 
 sary to move the plunger is 
 
 P - '43 {H - (H - h)} A = -43/i . A Ibs., 
 and the work done by the plunger per stroke is 
 E = -437i . A . L ft. Ibs. 
 
442 HYDRAULICS 
 
 If Y is the volume displacement per stroke of the plunger 
 
 in cubic feet 
 
 E = 62'4/t . V ft. Ibs. 
 
 The weight of water lifted per stroke is '43AL Ibs., and the 
 work done per pound is, therefore, h foot pounds. 
 
 Let Z be the head in the delivery pipe above the centre of the 
 pump, and u d the velocity with which the water leaves the delivery 
 pipe. 
 
 Neglecting friction, the work done by the plunger during the 
 
 2 
 
 delivery stroke is Z + ^ foot pounds per pound, and the total work 
 *9 
 
 in the two strokes is therefore h + Z + -- foot pounds per pound. 
 
 The actual work done on the plunger will be greater than this 
 due to mechanical friction in the pump, and the frictional and 
 other hydraulic losses in the suction and delivery pipes, and at the 
 valves; and the volume of water lifted per suction stroke will 
 generally be slightly less than the volume moved through by the 
 plunger. 
 
 Let W be the weight of water lifted per minute, and h t the 
 total height through which the water is lifted. 
 
 The effective work done by the pump is W . h t foot pounds per 
 minute, and the effective horse-power is 
 
 HP = 
 
 33,000 ' 
 
 245. Coefficient of discharge of the pump. Slip. 
 
 The theoretical discharge of a plunger pump is the volume 
 displaced by the plunger per stroke multiplied by the number of 
 delivery strokes per minute. 
 
 The actual discharge may be greater or less than this amount. 
 The ratio of the discharge per stroke to the volume displaced by 
 the plunger per stroke is the Coefficient of discharge, and the 
 difference between these quantities is called the Slip. 
 
 If the actual discharge is less than the theoretical the slip is 
 said to be positive, and if greater, negative. 
 
 Positive slip is due to leakage past the valves and plunger, 
 and in a steady working pump, with valves in proper condition, 
 should be less than five per cent. 
 
 The causes of negative slip and the conditions under which it 
 takes place will be discussed later*. 
 
 * See page 461. 
 
RECIPROCATING PUMPS 
 
 443 
 
 246. Diagram of work done by the pump. 
 
 Theoretical Diagram. Let a diagram be drawn, Fig. 302, the 
 ordinates representing the pressure in the cylinder and the abscissae 
 the corresponding volume displacements of the plunger. The 
 volumes will clearly be proportional to the displacement of the 
 plunger from the end of its stroke. During the suction stroke, 
 on the assumption made above that the plunger moves very 
 slowly and that therefore all frictional resistances, and also the 
 inertia forces, may be neglected, the absolute pressure behind the 
 plunger is constant and equal to H - h feet of water, or 62'4 (H - h) 
 pounds per square foot, and on the delivery stroke the pressure is 
 
 / u 2x 
 
 62*4 ( Z + H + ~ J pounds per square foot. 
 
 The effective work done per suction stroke is ABCD which equals 
 62'4 . h . V, and during the delivery stroke is EADF which equals 
 
 62v 
 
 and EBCF is the work done per cycle, that is, during one suction 
 and one delivery stroke. 
 
 E F 
 
 so 
 
 T 
 
 
 40 
 
 ^JC) 
 
 if|* 
 
 
 20 
 
 z 
 
 
 . 20- 
 Atrrv 
 
 A ] D 
 
 Pressure, 
 
 B 
 
 T r^ 
 
 c 
 
 ' T<-p 
 
 
 
 
 
 Fig. 302. Theoretical diagram of pressure in a Reciprocating Pump. 
 
 59 Strokes per nvuvuuU, 
 
 Fig. 303. 
 
 Actual diagram. Fig. 303 shows an actual diagram taken by 
 means of an indicator from a single acting pump, when running 
 at a slow speed. 
 
 The diagram approximates to the rectangular form and only 
 
444 HYDRAULICS 
 
 differs from the above in that at any point p in the suction stroke, 
 pq in feet of water is equal to h plus the losses in the suction 
 pipe, including loss at the valve, plus the head required to 
 accelerate the water in the suction pipe, and qr is the head 
 required to lift the water and overcome all losses, and to accelerate 
 the water in the delivery pipe. The velocity of the plunger being 
 small, these correcting quantities are practically inappreciable. 
 
 The area of this diagram represents the actual work done on 
 the water per cycle, and is equal to W (Z + h), together with the 
 head due to velocity of discharge and all losses of energy in the 
 suction and delivery pipes. 
 
 It will be seen later that although at any instant the pressure 
 in the cylinder is effected by the inertia forces, the total work 
 done in accelerating the water is zero. 
 
 247. The accelerations of the pump plunger and of the 
 water in the suction pipe. 
 
 The theoretical diagram, Fig. 302, has been drawn on the 
 assumption that the velocity of the plunger is very small and 
 without reference to the variation of the velocity and of the 
 acceleration of the plunger, but it is now necessary to consider 
 this variation and its effect on the motion of the water in the suction 
 and delivery pipes. To realise how the velocity and acceleration 
 of the plunger vary, suppose it to be driven by a crank and 
 connecting rod, as in Fig. 304, and suppose the crank rotates with 
 a uniform angular velocity of w radians per second. 
 
 Fig. 304. 
 
 If r is the radius of the crank in feet, the velocity of the crank 
 pin is V = o>r feet per second. For any crank position OC, it is 
 proved in books on mechanism, that the velocity of the point B is 
 
 01TC' By making BD equal to^K a diagram of velocities 
 EDF is found. 
 
 When CB is very long compared with CO, OK is equal to 
 OC sin 6, and the velocity v of the plunger is then V sin 0, and 
 
RECIPROCATING PUMPS 
 
 445 
 
 EDF is a semicircle. The plunger then moves with simple 
 harmonic motion. 
 
 If now the suction pipe is as in Fig. 300, and there is to be 
 continuity in the column of water in the pipe and cylinder, the 
 velocity of the water in the pipe must vary with the velocity of 
 the plunger. 
 
 Let v be the velocity of the plunger at any instant, A and 
 a the cross-sectional areas of the plunger and of the pipe respect- 
 ively. Then the velocity in the pipe must be - - . 
 
 As the velocity of the plunger is continuously changing, it is 
 continuously being accelerated, either positively or negatively. 
 
 Let Z be the length of the connecting rod in feet. The 
 acceleration* F of the point B in Fig. 305, for any crank angle 
 0, is approximately 
 
 Plotting F as BGr, Fig. 305, a curve of accelerations MNQ is 
 obtained. 
 
 When the connecting rod is very long compared with the 
 length of the crank, the motion is simple harmonic, and the 
 acceleration becomes 
 
 F = wV cos 0, 
 
 and the diagram of accelerations is then a straight line. 
 
 Velocity and acceleration of the water in the suction pipe. The 
 velocity and acceleration of the plunger being v and F respectively, 
 for continuity, the velocity of the water in the pipe must be 
 
 v and the acceleration 
 a 
 
 F.A 
 
 * See Balancing of Engines, W. E. Dalby. 
 
446 HYDRAULICS 
 
 248. The effect of acceleration of the plunger on the 
 pressure in the cylinder during the suction stroke. 
 
 When the velocity of the plunger is increasing, F is positive, 
 and to accelerate the water in the suction pipe a force P is 
 required. The atmospheric pressure has, therefore, not only to 
 lift the water and overcome the resistance in the suction pipe, 
 but it has also to provide the necessary force to accelerate the 
 water, and the pressure in the cylinder is consequently diminished. 
 
 On the other hand, as the velocity of the plunger decreases, 
 F is negative, and the piston has to exert a reaction upon the 
 water to diminish its velocity, or the pressure on the plunger is 
 increased. 
 
 Let L be the length of the suction pipe in feet, a its cross- 
 sectional area in square feet, f a the acceleration of the water in 
 the pipe at any instant in feet per second per second, and w the 
 weight of a cubic foot of water. 
 
 Then the mass of water in the pipe to be accelerated is w . a . L 
 pounds, and since by Newton's second law of motion 
 accelerating force = mass x acceleration, 
 the accelerating force required is 
 
 -PJ w . a . L ~ ,, 
 - ./lbs. 
 
 The pressure per unit area is 
 
 P to.L - ,, 
 
 - = _./ a lbs., 
 
 and the equivalent head of water is 
 
 7, L f 
 ll <* = - ?> 
 
 \j 
 
 A 
 
 or since / = 
 
 . 
 g.a 
 
 This may be large if any one of the three quantities, L, , or 
 
 . ^ 
 
 F is large. 
 
 Neglecting friction and other losses the pressure in the 
 cylinder is now 
 
 H-&-&., 
 and the head resisting the motion of the piston is h + h a . 
 
 249. Pressure in the cylinder during the suction stroke 
 when the plunger moves with simple harmonic motion. 
 
 If the plunger be supposed driven by a crank and very long 
 
RECIPROCATING PUMPS 
 
 447 
 
 connecting rod, the crank rotating uniformly with angular velocity 
 o> radians per second, for any crank displacement #, 
 
 and 
 
 , 
 h n = 
 
 , 
 . cos 0. 
 
 g . a 
 The pressure in the cylinder is 
 
 TT 7 L Ao>V cos 
 Jti fi . 
 
 ga 
 
 When is zero, cos 6 is unity, and when 6 is 90 degrees, cos 
 is zero. For values of G between 90 and 180 degrees, cos# is 
 negative. 
 
 The variation of the pressure in the cylinder is seen in 
 Fig. 306, which has been drawn for the following data. 
 
 Fig. 306. 
 
 Diameter of suction pipe 3J inches, length 12 feet 6 inches. 
 Diameter of plunger 4 inches, length of stroke 7| inches. 
 
 Number of strokes per minute 136. Height of the centre of 
 the pump above the water in the sump, 8 feet. The plunger is 
 assumed to have simple harmonic motion. 
 
 The plunger, since its motion is simple harmonic, may be 
 supposed to be driven by a crank 3| inches long, making 68 revo- 
 lutions per minute, and a very long connecting rod. 
 
 The angular velocity of the crank is 
 
 27T.68 ,_- 
 to = =71 radians per second. 
 
 The acceleration at the ends of the stroke is 
 
 F2 n, *7*12 ., A'QIO 
 = (y.7* t L X U OL& 
 
 = 15*7 feet per sec. per sec., 
 
 t-y)'~ 116 * 
 
 , 12-5 . 157 . 1-63 1AJ 
 
 and fi a = ^5 = 10 teet. 
 
 oZ 
 
448 HYDRAULICS 
 
 The pressure in the cylinder neglecting the water in the 
 cylinder at the beginning of the stroke is, therefore, 
 
 34 _ (10 + 8) -16 feet, 
 
 and at the end it is 34 - 8 + 10 = 36 feet. That is, it is greater 
 than the atmospheric pressure. 
 
 When is 90 degrees, cos# is zero, and h a is therefore zero, 
 and when is greater than 90 degrees, cos is negative. 
 
 The area AEDF is clearly equal to GrADH, and the work done 
 per suction stroke is, therefore, not altered by the accelerating 
 forces; but the rate at which the plunger is working at various 
 points in the stroke is affected by them, and the force required to 
 move the plunger may be very much increased. 
 
 In the above example, for instance, the force necessary to 
 move the piston at the commencement of the stroke has been 
 more than doubled by the accelerating force, and instead of 
 remaining constant and equal to '43 . 8 . A during the stroke, it 
 
 varies from 
 
 P = '43(8 + 10)A 
 
 to P = '43 (8 -10) A. 
 
 Air vessels. In quick running pumps, or when the length 
 of the pipe is long, the effects of these accelerating forces tend to 
 become serious, not only in causing a very large increase in the 
 stresses in the parts of the pump, but as will be shown later, under 
 certain circumstances they may cause separation of the water in 
 the pipe, and violent hammer actions may be set up. To reduce 
 the effects of the accelerating forces, air vessels are put on the 
 suction and delivery pipes, Figs. 310 and 311. 
 
 250. Accelerating forces in the delivery pipe of a plunger 
 pump when there is no air vessel. 
 
 When the plunger commences its return stroke it has not only 
 to lift the water against the head in the delivery pipe, but, if no 
 air vessel is provided, it has also to accelerate the water in the 
 cylinder and the delivery pipe. Let D be the diameter, a* the area, 
 and Li the length of the pipe. Neglecting the water in the 
 cylinder, the acceleration head when the acceleration of the piston 
 is F, is 
 
 , Lj.A.F 
 
 h a = , 
 
 00i 
 
 and neglecting head lost by friction etc., and the water in the 
 cylinder, the head resisting motion is 
 
 z + *.+. 
 
 If F is negative, h a is also negative. 
 
RECIPROCATING PUMPS 
 
 449 
 
 When the plunger moves with simple harmonic motion the 
 diagram is as shown in Fig. 307, which is drawn for the same 
 data as for Fig. 306, taking Z as 20 feet, Ln as 30 feet, and the 
 diameter D as 3| inches. 
 
 Fig. 307. 
 
 The total work done on the water in the cylinder is NJKM, 
 which is clearly equal to HJKL. If the atmospheric pressure is 
 acting on the outer end of the plunger, as in Fig. 301, the nett 
 work done on the plunger will be SNRMT, which equals HSTL. 
 
 251. Variation of pressure in the cylinder due to friction 
 when there is no air vessel. 
 
 Head lost by friction in the suction and delivery pipes. If v is 
 the velocity of the plunger at any instant during the suction 
 stroke, d the diameter, and a the area of the suction pipe, the 
 velocity of the water in the pipe, when there is no air vessel, is 
 
 0A 
 
 , and the head lost by friction at that velocity is 
 
 = WAL 
 
 f * 
 
 Similarly, if ai, D, and LI are the area, diameter and length 
 respectively of the delivery pipe, the head lost by friction, when 
 the plunger is making the delivery stroke and has a velocity v, is 
 
 When the plunger moves with simple harmonic motion, 
 
 v = o>r sin 0, 
 
 and 
 
 h,= 
 
 4/AVr 2 sin 2 0L 
 
 2gda* 
 
 L. H. 
 
 29 
 
450 
 
 HYDRAULICS 
 
 If the pump makes n strokes per second, or the number of 
 revolutions of the crank is ~ per second, and 1 8 is the length of 
 the stroke, 
 
 and 
 
 = 2r. 
 
 Substituting for w and r, 
 
 ~ 2gda? 
 
 Plotting values of h f at various points along the stroke, the 
 parabolic curve EMF, Fig. 308, is obtained. 
 
 When & is 90 degrees, sin# is unity, and h f is a maximum. 
 The mean ordinate of the parabola, which is the mean frictional 
 head, is then 
 
 2 /AWU 2 
 
 3 2gda? ' 
 
 M 
 
 Fig. 308. 
 
 and since the mean frictional head is equal to the energy lost per 
 pound of water, the work done per stroke by friction is 
 
 foot Ibs., 
 
 all dimensions being in feet. 
 
 Fig. 309. 
 Let Do be the diameter of the plunger in feet. Then 
 
 and 
 
 d* 
 
RECIPROCATING PUMPS 451 
 
 Therefore, work done by friction per suction stroke, when 
 there is no air vessel on the suction pipe, is 
 
 d 5 
 
 The pressure in the cylinder for any position of the plunger 
 during the suction stroke is now, Fig. 309, 
 h/Q = H h fi a hf. 
 
 At the ends of the stroke h/ is zero, and for simple harmonic 
 motion h a is zero at the middle of the stroke. 
 
 The work done per suction stroke is equal to the area 
 AEMFD, which equals 
 
 ARSD + EMF = ' 
 
 Similarly, during the delivery stroke the work done is 
 
 > 
 
 The friction diagram is HKGr, Fig. 309, and the resultant 
 diagram of total work done during the two strokes is EMFGKH. 
 
 252. Air vessel on the suction pipe. 
 
 As remarked above, in quick running pumps, or when the 
 lengths of the pipes are long, the effects of the accelerating forces 
 become serious, and air vessels are put on the suction and delivery 
 pipes, as shown in Figs. 310 and 311. By this means the velocity 
 in the part of the suction pipe between the well and the air 
 vessel is practically kept constant, the water, which has its 
 velocity continually changing as the velocity of the piston 
 changes, being practically confined to the water in the pipe 
 between the air vessel and the cylinder. The head required to 
 accelerate the water at any instant is consequently diminished, 
 and the friction head also remains nearly constant. 
 
 Let /! be the length of the pipe between the air vessel and 
 the cylinder, I the length from the well to the air vessel, a the 
 cross-sectional area of each of the pipes and d the diameter of the 
 pipe. 
 
 Let h v be the pressure head in the air vessel and let the air 
 vessel be of such a size that the variation of the pressure may for 
 simplicity be assumed negligible. 
 
 Suppose now that water flows from the well up the pipe AB 
 continuously and at a uniform velocity. The pump being single 
 acting, while the crank makes one revolution, the quantity of 
 water which flows along AB must be equal to the volume the 
 plunger displaces per stroke. 
 
 292 
 
452 
 
 HYDRAULICS 
 
 The time for the crank to make one revolution is 
 
 t = sees., 
 
 U) 
 
 therefore, the mean velocity of flow is 
 
 ^A^rw^ Ao>r 
 
 / 2Awr \ 
 
 (For a double acting pump v m = . J 
 
 During the delivery stroke, all the water is entering the air 
 vessel, the water in the pipe BC being at rest. 
 
 Fig. 310. 
 
 Then by Bernoulli's theorem, including friction and the velocity 
 head, other losses being neglected, the atmospheric head 
 
 A 2 o>V 4/AWZ n , 
 
 K = x + h v + n 2 r + ^ , a 2 ............... (1). 
 
 2ga 7T 2 2 
 
 The third and fourth quantities of the right-hand part of the 
 equation will generally be very small and h v is practically equal 
 to H-. 
 
 When the suction stroke is taking place, the water in the pipe 
 BC has to be accelerated. 
 
 Let H B be the pressure head at the point B, when the velocity 
 of the plunger is v feet per second, and the acceleration F feet per 
 second per second. 
 
RECIPROCATING PUMPS 453 
 
 Let hf be the loss of head by friction in AB, and In,} the loss in 
 
 BC. The velocity of flow along BC is , and the velocity of 
 
 a 
 
 flow from the air vessel is, therefore, 
 
 v.A 
 
 a ira 
 
 Then considering the pipe AB, 
 
 AW 
 
 HB ~ ~Wtf~ " 
 
 and from consideration of the pressures above B, 
 
 HR 
 
 Neglecting losses at the valve, the pressure in the cylinder is 
 then approximately 
 
 % TT 1' AZlF 
 
 ft,=H,-*,- 
 
 , AW , ,, AZ.F 
 - h - 
 
 Neglecting the small quantity ^ ^ z , 
 
 /to = H - h - (h f + h f ') - A ^ lF . 
 
 For a plunger moving with simple harmonic motion 
 
 TT i. 4/a>V 2 AV^ 7 2 /j\ A^<o 2 rcos<9 
 h = H - h - ^ o~ ~2 + Zi sm 2 ^ - - . 
 
 2ga?d \7r 2 Jag 
 
 By putting the air vessel near to the cylinder, thus making 
 li small, the acceleration head becomes very small and 
 
 ho = H-h-hf nearly, 
 and for simple harmonic motion 
 
 ^ = 
 
 The mean velocity in the suction pipe can very readily be 
 determined as follows. 
 
 Let Q be the quantity of water lifted per second in cubic feet. 
 Then since the velocity along the suction pipe is practically 
 
 constant v m = and the friction head is 
 
454 
 
 HYDKAULICS 
 
 When the pump is single acting and there are n strokes per 
 second, 
 
 K.ls.n 
 and therefore, 
 
 and 
 
 If the pump is double acting, 
 
 h = 2fAWl 
 
 For the same length of suction pipe the mean friction head, 
 when there is no air vessel and the pump is single acting, is \T? 
 times the friction head when there is an air vessel. 
 
 253. Air vessel on the delivery pipe. 
 
 An air vessel on the delivery pipe serves the same purpose 
 as on the suction pipe, in diminishing the mass of water which 
 changes its velocity as the piston velocity changes. 
 
 Fig. 311. 
 
 As the delivery pipe is generally much longer than the suction 
 pipe, the changes in pressure due to acceleration may be much 
 greater, and it accordingly becomes increasingly desirable to 
 provide an air vessel. 
 
 Assume the air vessel so large that the pressure head in it 
 remains practically constant. 
 
RECIPROCATING PUMPS 455 
 
 Let Z 2 , Fig. 311, be the length of the pipe between the pump and 
 the air vessel, l d be the length of the whole pipe, and &i and D the 
 area and diameter respectively of the pipe. 
 
 Let 7^2 be the height of the surface of the water in the air vessel 
 above the centre of the pipe at B, and let H r be the pressure head 
 in the air vessel. On the assumption that H r remains constant, 
 the velocity in the part BC of the pipe is practically constant. 
 
 Let Q be the quantity of water delivered per second. 
 
 The mean velocity in the part BC of the delivery pipe will be 
 
 . . . . 
 
 The friction head in this part of the pipe is constant and equal to 
 
 Considering then the part BC of the delivery pipe, the total 
 head at B required to force the water along the pipe will be 
 
 But the head at B must be equal to H v + h 2 nearly, therefore, 
 
 -W + H ............... (1). 
 
 In the part AB of the pipe the velocity of the water will vary 
 with the velocity of the plunger. 
 
 Let v and F be the velocity and acceleration of the plunger 
 respectively. 
 
 Neglecting the water in the cylinder, the head H r resisting the 
 motion of the plunger will be the head at B, plus the head 
 necessary to overcome friction in AB, and to accelerate the water 
 in AB. 
 
 * TT T, 4/Z 2 ^ 2 A 2 F.A.Z 2 
 
 Therefore, H r = H, + h> 
 
 For the same total length of the delivery pipe the acceleration 
 head is clearly much smaller than when there is no air vessel. 
 Substituting for H r + h from (1), 
 
 r.A.fc 
 
 If the pump is single acting and the plunger moves with simple 
 harmonic motion and makes n strokes per second, 
 
 Am 
 and u = -- 
 
456 HYDRAULICS 
 
 Therefore. 
 
 Neglecting the friction head in Z 2 and assuming 1 2 small com 
 pared with l d , 
 
 254. Separation during the suction stroke. 
 
 In reciprocating pumps it is of considerable importance that 
 during the stroke no discontinuity of flow shall take place, or 
 in other words, no part of the water in the pipe shall separate 
 from the remainder, or from the water in the cylinder of the pump. 
 Such separation causes excessive shocks in the working parts of 
 the pump and tends to broken joints and pipes, due to the hammer 
 action caused by the sudden change of momentum of a large mass 
 of moving water overtaking the part from which it has become 
 separated. 
 
 Consider a section AB of the pipe, Fig. 301, near to the inlet 
 valve. For simplicity, neglect the acceleration of the water in the 
 cylinder or suppose it to move with the plunger, and let the 
 acceleration of the plunger be F feet per second per second. 
 
 If now the water in the pipe is not to be separated from that in 
 the cylinder, the acceleration f a of the water in the pipe must not 
 
 FA 
 
 be less than feet per second per second, or separation will not 
 
 a 
 
 FA 
 take place as long as - - </ a . 
 
 -HI A 
 
 If f a at any instant becomes equal to , and f a is not to be- 
 
 CL 
 
 FA 
 
 come less than , the diminution d/of / a , when F is diminished 
 
 CL 
 
 by a small amount dF, must not be less than oF, or in general 
 
 CL 
 
 the differential of f a must not be less than times the differential 
 
 a 
 of F. 
 
 The general condition for no separation is, therefore, 
 
 Perhaps a simpler way to look at the question is as follows. 
 
 Let it be supposed that for given data the curve of pressures 
 in the cylinder during the suction stroke has been drawn as in 
 Fig. 309. In this figure the pressure in the cylinder always remains 
 positive, but suppose some part of the curve of pressures EF to 
 
RECIPROCATING PUMPS 
 
 457 
 
 come below the zero line BC as in Fig. 312 *. The pressure in the 
 cylinder then becomes negative; but it is impossible for a fluid 
 to be in tension and therefore discontinuity in the flow must 
 occur t. 
 
 In actual pumps the discontinuity will occur, if the curve EFGr 
 falls below the pressure at which the dissolved gases are liberated, 
 or the pressure head becomes less than from 4 to 10 feet. 
 
 B 
 
 D 
 
 Fig. 312. 
 
 At the dead centre the pressure in the cylinder just becomes 
 zero when h + h a = H, and will become negative when h + h a > H. 
 Theoretically for no separation at the dead centre, therefore, 
 
 - or 
 
 ga 
 
 If separation takes place when the pressure head is less than 
 some head h m , for no separation, 
 
 h a = H h m h, 
 
 ci 
 
 and 
 
 a ' I 
 
 Neglecting the water in the cylinder, at any other point in the 
 stroke, the pressure is negative when 
 
 ^ A *>H 
 2atf >H ' 
 
 ~P A T 
 
 That is, when In + - - + 
 a g 
 
 And the condition for no separation, therefore, is 
 
 FA 
 
 a 
 
 (2). 
 
 * See also Fig. 315, page 459. 
 
 t Surface tension of fluids at rest is not alluded to. 
 
458 HYDRAULICS 
 
 255. Separation during the suction stroke when the 
 plunger moves with simple harmonic motion. 
 
 When the plunger is driven by a crank and very long con- 
 necting rod, the acceleration for any crank angle is 
 
 F = a, 2 r cos 0, 
 or if the pump makes n single strokes per second, 
 
 (o= Trn, 
 
 o o 
 
 and F = "rf-tf . r cos 0= ~ - . l s cos 0, 
 
 l s being the length of the stroke. 
 
 F is a maximum when is zero, and separation will not take 
 place at the end of the stroke if 
 
 A 2 ^g(H.-h m -h) 
 a a L 
 
 and will just not take place when 
 
 A 7T 2 A 27 /H-fr m - 
 
 a wV r 2-a wZ ' = n L 
 The minimum area of the suction pipe for no separation is, 
 therefore, 
 
 AwVL x Q v 
 
 a= Tff 7 -- y-r- ........................ (o) 
 
 gCBL-h m -h) 
 
 and the maximum number of single strokes per second is 
 
 _ 
 
 ~ 
 
 A.Z..L 
 
 Separation actually takes place at the dead centre at a less 
 number of strokes than given by formula (4), due to causes 
 which could not very well be considered in deducing the formula. 
 
 Example. A single acting pump has a stroke of 1\ inches and the plunger is 
 4 inches diameter. The diameter of the suction pipe is 3^ inches, the length 
 12-5 feet, and the height of the centre of the pump above the water in the well is 
 8 feet. 
 
 To find the number of strokes per second at which separation will take place, 
 assuming it to do so when the pressure head falls below 10 feet. 
 
 H-/z = 26feet, 
 
 -=1-63, 
 a 
 
 and, therefore, = 1 / 64x26 
 
 TT V 1-63 x 7-5 
 
 x!2 
 5 x 12-5 
 
 7T 
 
 = 210 strokes per minute. 
 
 Nearly all actual diagrams taken from pumps, Figs. 313315, 
 have the corner at the commencement of the suction stroke 
 
RECIPROCATING PUMPS 
 
 459 
 
 rounded off, so that even at very slow speeds slight separation 
 occurs. The two principal causes of this are probably to be found 
 first, in the failure of the valves to open instantaneously, and 
 second, in the elastic yielding of the air compressed in the water 
 at the end of the delivery stroke. 
 
 136 Strokes per Truuvuutt> 
 
 Line 
 
 Fig. 315. 
 
 The diagrams Figs. 303 and 313 315, taken from a single-acting 
 pump, having a stroke of 7J inches, and a ram 4 inches diameter, 
 illustrate the effect of the rounding of the corner in producing 
 separation at a less speed than that given by equation (4). 
 
 Even at 59 strokes per minute, Fig. 303, at the dead centre a 
 momentary separation appears to have taken place, and the water 
 has then overtaken the plunger, the hammer action producing 
 vibration of the indicator. In Figs. 313 315, the ordinates to the 
 line rs give the theoretical pressures during the suction stroke. 
 The actual pressures are shown by the diagram. At 136 strokes 
 
460 HYDRAULICS 
 
 per minute at the point e in the stroke the available pressure is 
 clearly less than ef the head required to lift the water and to 
 produce acceleration, and the water lags behind the plunger. 
 This condition obtains until the point a is passed, after which 
 the water is accelerated at a quicker rate than the piston, and 
 finally overtakes it at the point b, when it strikes the plunger and 
 the indicator spring receives an impulse which makes the wave 
 form on the diagram. At 230 strokes per minute, the speed being 
 greater than that given by the formula when h m is assumed to 
 be 10 feet, the separation is very pronounced, and the water does 
 not overtake the piston until '7 of the stroke has taken place. It 
 is interesting to endeavour to show by calculation that the water 
 should overtake the plunger at b. 
 
 While the piston moves from a to b the crank turns through 
 70 degrees, in T e T T . ; seconds = '101 seconds. Between these two 
 points the pressure in the cylinder is 2 Ibs. per sq. inch, and 
 therefore the head available to lift the water, to overcome all 
 resistances and to accelerate the water in the pipe is 29'3 feet. 
 
 The height of the centre of the pump is 6' 3" above the water 
 in the sump. The total length of the suction pipe is about 
 12*5 feet, and its diameter is 3J inches. 
 
 Assuming the loss of head at the valve and due to friction etc., 
 to have a mean value of 2*5 feet, the mean effective head accele- 
 rating the water in the pipe is 20'5 feet. The mean acceleration 
 is, therefore, 
 
 , 20-5 x 32 _ , 
 j a = ~~<vc~~ = 52 5 feet per sec. per sec. 
 
 When the piston is at g the water will be at some distance 
 behind the piston. Let this distance be z inches and let the 
 velocity of the water be u feet per sec. Then in the time it 
 takes the crank to turn through 70 degrees the water will move 
 through a distance 
 
 S = ut + %f a t 2 
 
 = O'lOlu + J52'5 x -0102 feet 
 - l'2u + 3'2 inches. 
 
 The horizontal distance ab is 4'2 inches, so that z + 4*2 inches 
 should be equal to l'2u + 3'2 inches. 
 
 The distance of the point g from the end of the stroke is 
 '84 inch and the time taken by the piston to move from rest to g, 
 is 0*058 second. The mean pressure accelerating the water during 
 this time is the mean ordinate of akm when plotted on a time 
 base; this is about 5 Ibs. per sq. inch, and the equivalent head is 
 12-8 feet. 
 
RECIPROCATING PUMPS 461 
 
 The frictional resistances, which vary with the velocity, will be 
 small. Assuming the mean frictional head to be '25 foot, the head 
 causing acceleration is 12'55 feet and the mean acceleration of the 
 water in the pipe while the piston moves from rest to g is, 
 therefore, 
 
 , 12-55 x 32 
 
 fa = 
 
 The velocity in the pipe at the end of 0'058 second, should 
 therefore be 
 
 v = 32 x -058 - 1'86 feet per sec. 
 
 and the velocity in the cylinder 
 
 T86 110 - 
 u = ^7^5 = 1 153 ieet per sec. 
 
 1 Oo 
 
 Since the water in the pipe starts from rest the distance it 
 should move in 0'058 second is 
 
 12.J32.('058) 2 =-65in., 
 
 and the distance it should advance in the cylinder is 
 
 0-65 . 
 
 so that z is 0'4 in. 
 
 Then z + 4'2 ins. - 4'6, 
 
 and \'2u + 3'2 ins. = 4*57 ins. 
 
 The agreement is, therefore, very close, and the assumptions 
 made are apparently justified. 
 
 256. Negative slip in a plunger pump. 
 
 Fig. 315 shows very clearly the momentary increase in the 
 pressure due to the blow, when the water overtakes the plunger, 
 the pressure rising above the delivery pressure, and causing 
 discharge before the end of the stroke is reached. If no separa- 
 tion had taken place, the suction pressure diagram would have 
 approximated to the line rs and the delivery valve would still 
 have opened before the end of the stroke was reached. 
 
 The coefficient of discharge is T025, whereas at 59 strokes 
 per minute it is only 0'975. 
 
 257. Separation at points in the suction stroke other than 
 at the end of the stroke. 
 
 The acceleration of the plunger for a crank displacement 6 
 
 is cos 6, and therefore for no separation at any crank 
 
 a 
 
 angle 
 
462 HYDRAULICS 
 
 Putting in the value of h/ 9 and differentiating both sides of the 
 equation, and using the result of equation (1), page 456, 
 
 - o>V sin 0^^-Jl + sin 6 cos 6, 
 
 a L or \ d J 
 
 ( 4f L\ 
 from which aL A f 1 + ~j- 1 ^ cos 0. 
 
 Separation will just not take place if 
 
 aL = A.r(l + 4k) cos 6, 
 
 or when cos0 = - a . ..................... (2). 
 
 Since cos cannot be greater than unity, there is no real 
 solution to this equation, unless Ar (l + ^j-J is equal to or 
 greater than aL 
 
 If, therefore, 3*r is supposed equal to zero, and aL the volume 
 
 of the suction pipe is greater than half the volume of the cylinder, 
 separation cannot take place if it does not take place at the dead 
 centre. 
 
 In actual pumps, aL is not likely to be less than Ar, and 
 consequently it is only necessary to consider the condition for no 
 separation at the dead centre. 
 
 258. Separation with a large air vessel on the suction pipe. 
 
 To find whether separation will take place with a large air 
 vessel on the suction pipe, it is only necessary to substitute in 
 equations (2), section 255, and (3), (4), section 256, h v of Fig. 310 
 for H, li for L, and hi for h. In Fig. 310, hi is negative. 
 
 For no separation when the plunger is at the end of the stroke 
 the minimum area of the pipe between the air vessel and the 
 cylinder is 
 
 a = 
 
 (h v -h m -hi) ' 
 
 Substituting for h v its value from equation (1), section 253, and 
 h torn -h, Fig. 310, 
 
 a = 
 
 If the velocity and friction heads, in the denominator, be 
 neglected as being small compared with (H - h), then, 
 
 g(H-h m -h)' 
 
RECIPROCATING PUMPS 
 The maximum number of strokes is 
 
 463 
 
 A. pump can therefore be run at a much greater speed, without 
 fear of separation, with an air vessel on the suction pipe, than 
 without one. 
 
 259. Separation in the delivery pipe. 
 
 Consider a pipe as shown in Fig. 316, the centre of CD being at 
 a height Z above the centre of AB. 
 
 Let the pressure head at D be H , which, when the pipe 
 discharges into the atmosphere, becomes H. 
 
 Let I, li and Z 2 be the lengths of AB, BC and CD respectively, 
 hf, h fl and h fz the losses of head by friction in these pipes when the 
 plunger has a velocity v, and h m the pressure at which separation 
 actually takes place. 
 
 I ~H 
 
 Fig. 316. 
 
 Suppose now the velocity of the plunger is diminishing, and its 
 retardation is F feet per second per second. If there is to be 
 
 F A 
 
 continuity, the water in the pipe must be also retarded by - 
 
 ft 
 
 feet per second per second, and the pressure must always be 
 positive and greater than h m . 
 
 Let H c be the pressure at C ; then the head due to acceleration 
 in the pipe DC is 
 
 and if the pipe CD is full of water 
 
 w FAk 
 
 -tic -tJ-o 
 
 9 
 which becomes negative when 
 
4G4 HYDRAULICS 
 
 The condition for no separation at C is, therefore, 
 H.-JL.-A 
 
 -H-O ~ U m /I/ r 
 
 or separation takes place when 
 
 At the point B separation will take place if 
 
 a g 
 and at the point A if 
 
 At the dead centre i? is zero, and the friction head vanishes. 
 For no separation at the point C it is then necessary that 
 
 for no separation at B 
 
 , 
 
 + ZJ Al m 
 
 ga 
 
 and for no separation at A 
 
 For given values of H , F and Z, the greater Z 2 , the more likely 
 is separation to take place at C, and it is therefore better, for 
 a given total length of the discharge pipe, to let the pipe rise near 
 the delivery end, as shown by fesS" lines, rather than as shown 
 by the <rf lines. 
 
 If separation does not take place at A it clearly will not take 
 place at B. 
 
 Example. The retardation of the plunger of a pump at the end of its stroke 
 is 8 feet per second per second. The ratio of the area of the delivery pipe to the 
 plunger is 2, and the total length of the delivery pipe is 152 feet. The pipe is 
 horizontal for a length of 45 feet, then vertical for 40 feet, then rises 5 feet on 
 a slope of 1 vertical to 3 horizontal and is then horizontal, and discharges into 
 the atmosphere. Will separation take place on the assumption that the pressure 
 head cannot be less than 7 feet ? 
 
 Ans. At the bottom of the sloping pipe the pressure is 
 
 The pressure head is therefore less than 7 feet and separation will take place. 
 The student should also find whether there is separation at any other point. 
 
RECIPROCATING PUMPS 4G5 
 
 260. Diagram of pressure in the cylinder and work done 
 during the suction stroke, considering the variable quantity of 
 water in the cylinder. 
 
 It is instructive to consider the suction stroke a little more in 
 detail. 
 
 Let v and F be the velocity and acceleration respectively of 
 the piston at any point in the stroke. 
 
 As the piston moves forward, water will enter the pipe from the 
 well and its velocity will therefore be increased from zero to 
 
 A 
 
 v . -, the head required to give this velocity is 
 
 & 
 
 On the other hand water that enters the cylinder from the pipe 
 is diminished in velocity from - - to v, and neglecting any loss due 
 
 CL 
 
 to shock or due to contraction at the valve there is a gain of 
 pressure head in the cylinder equal to 
 
 -~ 
 
 *~2ga? 2g 
 The friction head in the pipe is 
 
 The head required to accelerate the water in the pipe is 
 
 FAL 
 
 The mass of water to be accelerated in the cylinder is a 
 variable quantity and will depend upon the plunger displacement. 
 Let the displacement be x feet from the end of the stroke. 
 
 /7M A SY> 
 
 The mass of water in the cylinder is - - Ibs. and the force 
 
 g 
 
 required to accelerate it is 
 
 and the equivalent head is 
 
 P 
 
 g 
 
 wA. g 
 
 The total acceleration head is therefore 
 F/ LA N 
 
 / LA\ 
 
 (SB + ). 
 
 \ a / 
 
 L. H. 30 
 
6 HYDRAULICS 
 
 Now let Ho be the pressure head in the cylinder, then 
 
 _ 7, _ j^! A! ^ A 2 _ ^ _ 4/LAV _ F / LAN 
 H = ~ k ~2g^ + 2g a 2 2g 2g.da 2 g V ' a j 
 
 g \ a 
 
 When the plunger moves with simple harmonic motion, and is 
 driven by a crank of radius r rotating uniformly with angular 
 velocity w, the displacement of the plunger from the end of the 
 stroke is r (1 - cos 0), the velocity wr sin and its acceleration is 
 o>V cos 0. 
 Therefore 
 
 oiV.Bm'0 4/LAV 
 
 H - 2g 2gda> 
 
 L A , oVcosfl eoVcos 2 ^ ,-, 
 -- w-r cos0 -- -+- - ...(6). 
 
 9 a 9 9 
 
 Work done, during the suction stroke. Assuming atmospheric 
 pressure on the face of the plunger, the pressure per square foot 
 resisting its motion is 
 
 For any small plunger displacement dx, the work done is, 
 therefore, 
 
 A (H - Ho) w . dx, 
 
 and the total work done during the stroke is 
 
 o 
 The displacement from the end of the stroke is 
 
 x = r (I -cos0), 
 ;and therefore dx = r sin QdO, 
 
 and E = fw . A (H - H ) r sin OdO. 
 
 Jo 
 
 Substituting for H its value from equation (6) 
 
 -^ f 4/LAW 2 sin 2 6 o>Vsin 2 
 
 E = w . Ar h + -* j 2 -- + - 
 
 7o 2gda 2 2g 
 
 wV cos <oV cos 2 L A /!/ JA 
 
 + - - + -- w 2 r cos 6} sm OdO. 
 
 9 9 9 a 
 
 The sum of the integration of the last four quantities of this 
 expression is equal to zero, so that the work done by the 
 accelerating forces is zero, and 
 
 Jo 
 
 4/LA 2 coV 2 
 
RECIPROCATING PUMPS 
 
 467 
 
 Or the work done is that required to lift the water through 
 a height h together with the work done in overcoming the 
 resistance in the pipe. 
 
 Diagrams of pressure in the cylinder and of work done per 
 stroke. The resultant pressure in the cylinder, and the head 
 resisting the motion of the piston can be represented diagram- 
 matically, by plotting curves the ordinates of which are equal to 
 Ho and H-H as calculated from equations (2) and (3). For 
 clearness the diagrams corresponding to each of the parts of 
 equation (2) are drawn in Figs. 318 321 and in Fig. 317 is shown 
 the combined diagram, any ordinate of which equals 
 
 -^ 
 
 Figs. 318, 319, 320. Figs. 321, 322. 
 
 In Fig. 318 the ordinate cd is equal to 
 
 and the curve HJK is a parabola, the area of which is 
 
 302 
 
468 HYDRAULICS 
 
 In Fig. 319, the ordinate e/is 
 
 and the ordinate gh of Fig. 320 is 
 
 + ^cos 2 *. 
 9 
 
 The areas of the curves are respectively 
 
 2 toV 7 , 1 eoV 7 
 
 3 W * 3^~ ' 
 
 and are therefore equal; and since the ordinates are always of 
 opposite sign the sum of the two areas is zero. 
 In Fig. 322, ~km is equal to 
 
 o>V 2 cos 
 
 ~T~ 
 
 and Id to 
 
 o>V a / L . A\ 
 
 - cos 1 x + - - ) . 
 a / 
 
 Since cos is negative between 90 and 180 the area WXY is 
 equal to YZU. 
 
 Fig. 321 has for its ordinate at any point of the stroke, the 
 head H-H resisting the motion of the piston. 
 
 This equals h + Id + cd + ef gh, 
 
 and the curve NPS is clearly the curve GrFE, inverted. 
 
 The area VNST measured on the proper scale, is the work done 
 per stroke, and is equal to VMRT + HJK. 
 
 The scale of the diagram can be determined as follows. 
 
 Since h feet of water = 62'4/z, Ibs. per square foot, the pressure 
 in pounds resisting the motion of the piston at any point in the 
 stroke is 
 
 62'4 . A . h Ibs. 
 
 If therefore, VNST be measured in square feet the work done 
 per stroke in ft.-lbs. 
 
 = 62-4 A. VNST. 
 
 261. Head lost at the suction valve. 
 
 In determining the pressure head H in the cylinder, no account 
 has been taken of the head lost due to the sudden enlargement 
 from the pipe into the cylinder, or of the more serious loss of head 
 due to the water passing through the valve. It is probable that the 
 
 whole of the velocity head, ~ 5 > ^ ^ ne water entering the cylinder 
 
 from the pipe is lost at the valve, in which case the available head 
 H will not only have to give this velocity to the water, but will 
 
RECIPROCATING PUMPS 469 
 
 v 2 
 also have to give a velocity head ~- to any water entering the 
 
 cylinder from the pipe. 
 
 The pressure head H in the cylinder then becomes 
 
 v 2 A" tf 4/L^A 2 F/ IA 
 - h ---~ 
 
 262. Variation of the pressure in hydraulic motors due 
 to inertia forces. 
 
 The description of hydraulic motors is reserved for the next 
 chapter, but as these motors are similar to reversed reciprocating 
 pumps, it is convenient here to refer to the effect of the inertia 
 forces in varying the effective pressure on the motor piston. 
 
 If L is the length of the supply pipe of a hydraulic motor, a 
 the cross-sectional area of the supply, A the cross-sectional area 
 of the piston of the motor, and F the acceleration, the acceleration 
 
 F.A 
 
 of the water in the pipe is and the head required to accelerate 
 
 a 
 
 the water in the pipe is 
 
 ga 
 
 If p is the pressure per square foot at the inlet end of the 
 supply pipe, and h f is equal to the losses of head by friction in the 
 pipe, and at the valve etc., when the velocity of the piston is v, the 
 pressure on the piston per square foot is 
 
 When the velocity of the piston is diminishing, F is negative, 
 and the inertia of the water in the pipe increases the pressure on 
 the piston. 
 
 Example (1). The stroke of a double acting pump is 15 inches and the number of 
 strokes per minute is 80. The diameter of the plunger is 12 inches and it moves 
 with simple harmonic motion. The centre of the pump is 13 feet above the water 
 in the well and the length of the suction pipe is 25 feet. 
 
 To find the diameter of the suction pipe that no separation shall take place, 
 assuming it to take place when the pressure head becomes less than 7 feet. 
 
 As the plunger moves with simple harmonic motion, it may be supposed driven 
 by a crank of 7 inches radius and a very long connecting rod, the angular 
 velocity of the crank being 2?r40 radians per minute. 
 
 The acceleration at the end of the stroke is then 
 
 47r 2 . 40 2 . r 
 
 Theref ore, 
 
 from which = 1'64. 
 
 a 
 
470 HYDRAULICS 
 
 Therefore d" = 1 28 
 
 and d = 9-4". 
 
 Ar is clearly less than al, therefore separation cannot take place at any other 
 point in the stroke. 
 
 Example (2). The pump of example (1) delivers water into a rising main 
 1225 feet long and 5 inches diameter, which is fitted with an air vessel. 
 
 The water is lifted through a total height of 220 feet. 
 
 Neglecting all losses except friction in the delivery pipe, determine the horse- 
 power required to work the pump. /=-Ol05. 
 
 Since there is an air vessel in the delivery pipe the velocity of flow u will be 
 practically uniform. 
 
 Let A and a be the cross-sectional areas of the pump cylinder and pipe respect- 
 
 A . 2r . 80 D 2 2r . 80 
 Then > 60^=^~60- 
 
 12 2 10 80 
 = 25--60 
 The head h lost due to friction is 
 
 042 x 9'6 2 x 1225 
 
 *p-A 
 
 = 176-4 feet. 
 The total lift is therefore 
 
 220 + 176-4 = 396-4 feet. 
 The weight of water lifted per minute is 
 
 . . 80 x 62-5 Ibs. = 4900 Ibs. 
 
 Example (3). If in example (2) the air vessel is near the pump and the mean 
 level of the water in the vessel is to be kept at 2 feet above the centre of the 
 pump, find the pressure per sq. inch in the air vessel. 
 
 The head at the junction of the air vessel and the supply pipe is the head 
 necessary to lift the water 207 feet and overcome the friction of the pipe. 
 Therefore, H w + 2' = 207 + 176-4, 
 
 H w = 381-4 feet, 
 
 381-4 x 62-5 
 ** : -144- 
 = 165 Ibs. per sq. inch. 
 
 Example (4). A single acting hydraulic motor making 50 strokes per minute 
 has a cylinder 8 inches diameter and the length of the stroke is 12 inches. The 
 diameter of the supply pipe is 3 inches and it is 500 feet long. The motor is 
 supplied with water from an accumulator, see Fig. 339, at a constant pressure of 
 300 Ibs. per sq. inch. 
 
 Neglecting the mass of water in the cylinder, and assuming the piston moves 
 with simple harmonic motion, find the pressure on the piston at the beginning and 
 the centre of its stroke. The student should draw a diagram of pressure for one 
 stroke. 
 
 There are 25 useful strokes per minute and the volume of water supplied 
 per minute is, therefore, 
 
 25 . ~ d 2 = 8-725 cubic feet. 
 4 
 
 K/VJ 
 
 At the commencement of the stroke the acceleration is ?r 2 ^- r, and the velocity 
 in the supply pipe is zero. 
 
RECIPROCATING PUMPS 471 
 
 The head required to accelerate the water in the pipe is, therefore, 
 
 = 7T 2 . 50 2 . 1 . 8 2 . 500 
 a ~ 60 a .2.3 2 .32 
 
 = 380 feet, 
 which is equivalent to 165 Ibs. per sq. inch. 
 
 The effective pressure on the piston is therefore 135 Ibs. per sq. inch. 
 At the end of the stroke the effective pressure on the piston is 465 Ibs. 
 per sq. inch. 
 
 At the middle of the stroke the acceleration is zero and the velocity of the 
 piston is 
 
 |$ TIT =1*31 feet per second. 
 The friction head is then 
 
 04.1-31 2 .8 2 .500' 
 
 2i7BTT~ 
 
 = 15-2 feet. 
 The pressure on the plunger at the middle of the stroke is 
 
 300 Ibs. - 15 ' 2 * 62 ' 5 = 293-4 Ibs. per sq. inch. 
 
 J.44 
 
 The mean friction head during the stroke is f . 15-2 = 10*1 feet, and the mean 
 loss of pressure is 4-4 Ibs. per sq. inch. 
 
 The work lost by friction in the supply pipe per stroke is 4 '4 . - . 8 2 . 1 K 
 
 = 222 ft. Ibs. 
 
 The work lost per minute = 5500 ft. Ibs. 
 The net work done per minute neglecting other losses is 
 
 (300 Ibs. -4-4).^.Z 8 .8 2 .25 
 
 = 370,317 ft. Ibs., 
 
 and therefore the work lost by friction is comparatively small, being less than 
 2 per cent. 
 
 Other causes of loss in this case, are the loss of head due to shock where the 
 water enters the cylinder, and losses due to bends and contraction at the valves. 
 
 It can safely be asserted, that at any instant a head equal to the velocity head, 
 of the water in the pipe, will be lost by shock at the valves, and a similar quantity 
 at the entrance to the cylinder. These quantities are however always small, and 
 even if there are bends along the pipe, which cause a further loss of head equal to 
 the velocity head, or even some multiple of it, the percentage loss of head will still 
 be small, and the total hydraulic efficiency will be high. 
 
 This example shows clearly that power can be transmitted hydraulically very 
 efficiently over comparatively long distances. 
 
 263. High pressure plunger pump. 
 
 Fig. 323 shows a section through a high pressure pump 
 suitable for pressures of 700 or 800 Ibs. per sq. inch. 
 
 Suction takes place on the outward stroke of the plunger, and 
 delivery on both strokes. 
 
 A brass liner is fitted in the cylinder and the plunger which, 
 as shown, is larger in diameter at the right end than at the left, 
 is also made of brass ; the piston rod is of steel. Hemp packing 
 is used to prevent leakage past the piston and also in the gland 
 box. 
 
 The plunger may have leather packing as in Fig. 324. 
 
 On the outward stroke neglecting slip the volume of water 
 
472 
 
 HYDRAULICS 
 
 ffTl fTll 
 
RECIPROCATING PUMPS 
 
 473 
 
 drawn into the cylinder is 7 D 2 . L cubic feet, D being the dia- 
 meter of the piston and L the length of the stroke. The quantity 
 of water forced into the delivery pipe through the valve V D is 
 
 (D 2 d 2 ) L cubic feet, 
 
 \ 4< 
 
 d being the diameter of the small part of the 
 plunger. 
 
 On the in-stroke, the suction valve is 
 closed and water is forced through the 
 delivery valve; part of this water enters 
 the delivery pipe and part flows behind the 
 piston through the port P. Yi %' 324 - 
 
 The amount that flows into the delivery pipe is 
 
 If, therefore, (D 2 - d 2 ) is made equal to d 2 , or D is \/2d, the 
 delivery, during each stroke, is | D 2 L cubic feet, and if there are 
 n strokes per minute, the delivery is 42*45 D 2 Im gallons per 
 
 minute. 
 
 Fig. 325. Tangye Duplex Pump. 
 
 264. Duplex feed pump. 
 
 Fig. 325 shows a section through one pump and steam cylinder 
 of a Tangye double-acting pump. 
 
474 
 
 HYDRAULICS 
 
 There are two steam cylinders side by side, one of which only 
 is shown, and two pump cylinders in line with the steam cylinders. 
 
 In the pump the two lower valves are suction valves and the 
 two upper delivery valves. As the pump piston P moves to the 
 right, the left-hand lower valve opens and water is drawn into the 
 pump from the suction chamber C. During this stroke the right 
 upper valve is open, and water is delivered into the delivery Ci. 
 When the piston moves to the left, the water is drawn in through 
 the lower right valve and delivered through the upper left valve. 
 
 The steam engine has double ports at each end. As the piston 
 approaches the end of its stroke the steam valve, Fig. 326, is at rest 
 and covers the steam port 1 while the "inner steam port 2 is open 
 to exhaust. When the piston passes the steam port 2, the steam 
 enclosed in the cylinder acts as a cushion and brings the piston 
 and plunger gradually to rest. 
 
 Fig. 326. 
 
 Fig. 327. 
 
 Let the one engine and pump shown in section be called A and 
 the other engine and pump, not shown, be called B. 
 
 As the piston of A moves from right to left, the lever L, Figs. 
 325 and 327, rotates a spindle to the other end of which is fixed a 
 crank M, which moves the valve of the cylinder B from left to 
 right and opens the left port of the cylinder B. Just before the 
 piston of A reaches the left end of its stroke, the piston of B, 
 therefore, commences its stroke from left to right, and by a lever 
 L! and crank MI moves the valve of cylinder A also from left to 
 right, and the piston of A can then commence its return stroke. 
 .t should be noted that while the piston of A is moving, that of 
 B is practically at rest, and vice versa. 
 
 265. The hydraulic ram. 
 
 The hydraulic ram is a machine which utilises the momentum 
 of a stream of water falling a small height to raise a part of the 
 water to a greater height. 
 
 In the arrangement shown in Fig. 328 water is supplied from a 
 tank, or stream, through a pipe A into a chamber B, which has two 
 
PUMPS 
 
 475 
 
 valves Y and Yi . When no flow is taking place the valve V falls 
 off its seating and the valve Vi rests on its seating. If water is 
 allowed to flow along the pipe B it will escape through the open 
 valve Y. The contraction of the jet through the valve opening, 
 exactly as in the case of the plate obstructing the flow in a pipe, 
 page 168, causes the pressure to be greater on the under face of 
 the valve, and when the pressure is sufficiently large the valve 
 will commence to close. As it closes the pressure will increase 
 and the rate of closing will be continually accelerated. The rapid 
 closing of the valve arrests the motion of the water in the pipe, 
 and there is a sudden rise in pressure in B, which causes the 
 valve YI to open, and a portion of the water passes into the air 
 vessel C. The water in the supply pipe and in the vessel B, after 
 being brought to rest, recoils, like a ball thrown against a wall, 
 and the pressure in the vessel is again diminished, allowing the 
 water to once more escape through the valve Y. The cycle of 
 operations is then repeated, more, water being forced into the air 
 chamber C, in which the air is compressed, and water is forced up 
 the delivery pipe to any desired height. 
 
 Fig. 328. 
 
 Let h be the height the water falls to the ram, H the height to 
 which the water is lifted. 
 
 If W Ibs. of water descend the pipe per second, the work 
 available per second is Wh foot Ibs., and if e is the efficiency of the 
 ram, the weight of water lifted through a height H will be 
 
 W.h 
 
 w = 
 
 e.H 
 
 The efficiency e diminishes as H increases and may be taken as 
 60 per cent, at high heads. 
 
 Fig. 329 shows a section through the De Cours hydraulic 
 ram, the valves of which are controlled by springs. The springs 
 
476 
 
 HYDRAULICS 
 
 can be regulated so that the number of beats per minute is com- 
 pletely under control, and can be readily adjusted to suit varying 
 
 heads. 
 
 With this type of ram Messrs Bailey claim to have obtained at 
 low heads, an efficiency of more than 90 per cent., and with H 
 equal to 8h an efficiency of 80 per cent. 
 
 Fig. 329. De Cours Hydraulic Earn. 
 
 As the water escapes through the valve Vi into the air vessel C, 
 a little air should be taken with it to maintain the air pressure in 
 C constant. 
 
 This is effected in the De Cours ram by allowing the end of the 
 exhaust pipe F to be under water. At each closing of the valve 
 
PUMPS 
 
 477 
 
 V, the siphon action of the water escaping from the discharge 
 causes air to be drawn in past the spindle of the valve. A cushion 
 of air is thus formed in the box B every stroke, and some of this 
 air is carried into C when the valve Vi opens. 
 
 The extreme simplicity of the hydraulic ram, together with 
 the ease with which* it can be adjusted to work with varying 
 quantities of water, render it particularly suitable for pumping 
 in out-of-the-way places, and for supplying water, for fountains 
 and domestic purposes, to country houses situated near a stream. 
 
 266. Lifting water by compressed air. 
 
 A very simple method of raising water from deep wells is by 
 means of compressed air. A delivery pipe is sunk into a well, 
 the open end of the pipe being placed at a considerable distance 
 below the surface of the water in the well. 
 
 AirTuube 
 
 Fig. 330. 
 
 Fig. 331. 
 
 In the arrangement shown in Fig. 330, there is surrounding the 
 delivery tube a pipe of larger diameter into which air is pumped 
 by a compressor. 
 
 The air rises up the delivery pipe carrying with it a quantity of 
 water. An alternative arrangement is shown in Fig. 331. 
 
 Whether the air acts as a piston and pushes the water in front 
 of it, or forms a mixture with the water, according to Kelly*, 
 depends very largely upon the rate at which air is supplied to the 
 pump. 
 
 In the pump experimented upon by Kelly, at certain rates of 
 
 * Proc. List. C. E. Vol. CLXIII. 
 
478 HYDRAULICS 
 
 working the discharge was continuous, the air and the water being 
 mixed together, while at low discharges the action was intermittent 
 and the pump worked in a definite cycle; the discharge commenced 
 slowly; the velocity then gradually increased until the pipe 
 discharged full bore; this was followed by a rush of air, after 
 which the flow gradually diminished and finally stopped ; after a 
 period of no flow the cycle commenced again. When the rate at 
 which air was supplied was further diminished, the water rose 
 up the delivery tube, but not sufficiently high to overflow, and the 
 air escaped without doing useful work. 
 
 The efficiency of these pumps is very low and only in exceptional 
 cases does it reach 50 per cent. The volume v of air, in cubic feet, 
 at atmospheric pressure, required to lift one cubic foot of water 
 through a height h depends upon the efficiency. With an ef- 
 ficiency of 30 per cent, it is approximately v = ^) an( i with an 
 
 efficiency of 40 per cent, v = ^ approximately. 
 
 It is necessary that the lower end of the delivery be at a greater 
 distance below the surface of the water in the well, than the height 
 of the lift above the free surface, and the well has consequently to 
 be made very deep. 
 
 On the other hand the well is much smaller in diameter than 
 would be required for reciprocating or centrifugal pumps, and the 
 initial cost of constructing the well per foot length is considerably 
 less. 
 
 EXAMPLES. 
 
 (1) Find the horse-power required to raise 100 cubic feet of water per 
 minute to a height of 125 feet, by a pump whose efficiency is ^. 
 
 (2) A centrifugal pump has an inner radius of 4 inches and an outer 
 radius of 12 inches. The angle the blade makes with the direction of 
 motion at exit is 153 degrees. The wheel makes 545 revolutions per minute. 
 
 The discharge of the pump is 3 cubic feet per second. The sides of the 
 wheel are parallel and 2 inches apart. 
 
 Determine the inclination of the tip of the blades at inlet so that there 
 shall be no shock, the velocity with which the water leaves the wheel, and 
 the theoretical lift. If the head due to the velocity with which the water 
 leaves the wheel is lost, find the theoretical lift. 
 
 (3) A centrifugal pump wheel has a diameter of 7 inches and makes 
 1358 revolutions per minute. 
 
 The blades are formed so that the water enters and leaves the wheel 
 without shock and the blades are radial at exit. The water is lifted by the 
 pump 29-4 feet. Find the manometric efficiency of the pump. 
 
PUMPS 479 
 
 (4) A centrifugal pump wheel 11 inches diameter which runs at 1203 
 revolutions per minute is surrounded by a vortex chamber 22 inches 
 diameter, and has radial blades at exit. The pressure head at the circum- 
 ference of the wheel is 23 feet. The water is lifted to a height of 43'5 
 feet above the centre of the pump. Find the efficiency of the whirlpool 
 chamber. 
 
 (5) The radial velocity of flow through a pump is 5 feet per second, and 
 the velocity of the outer periphery is 60 feet per second. 
 
 The angle the tangent to the blade at outlet makes with the direction 
 of motion is 120 degrees. Determine the pressure head and velocity head 
 where the water leaves the wheel, assuming the pressure head in the eye 
 of the wheel is atmospheric, and thus determine the theoretical lift. 
 
 (6) A centrifugal pump with vanes curved back has an outer radius of 
 10 inches and an inlet radius of 4 inches, the tangents to the vanes at outlet 
 being inclined at 40 to the tangent at the outer periphery. The section of 
 the wheel is such that the radial velocity of flow is constant, 5 feet per 
 second ; and it runs at 700 revolutions per minute. 
 
 Determine : 
 
 (1) the angle of the vane at inlet so that there shall be no shock, 
 
 (2) the theoretical lift of the pump, 
 
 (3) the velocity head of the water as it leaves the wheel. Lond. 
 Un. 1906. 
 
 (7) A centrifugal pump 4 feet diameter running at 200 revolutions per 
 minute, pumps 5000 tons of water from a dock in 45 minutes, the mean 
 lift being 20 feet. The area through the wheel periphery is 1200 square 
 inches and the angle of the vanes at outlet is 26. Determine the hydraulic 
 efficiency and estimate the average horse -power. Find also the lowest 
 speed to start pumping against the head of 20 feet, the inner radius being 
 half the outer. Lond. Un. 1906. 
 
 (8) A centrifugal pump, delivery 1500 gallons per minute with a lift of 
 25 feet, has an outer diameter of 16 inches, and the vane angle is 30. All 
 the kinetic energy at discharge is lost, and is equivalent to 50 per cent, of 
 the actual lift. Find the revolutions per minute and the breadth at the 
 inlet, the velocity of whirl being half the velocity of the wheel. Lond. 
 Un. 1906. 
 
 (9) A centrifugal pump has a rotor 19| inches diameter ; the width of 
 the outer periphery is 3 T 7 ^ inches. Using formula (1), section 236, deter- 
 mine the discharge of the pump when the head is 30 feet and Vi is 50. 
 
 (10) The angle $ at the outlet of the pump of question (9) is 13. 
 Find the velocity with which the water leaves the wheel, and the 
 
 minimum proportion of the velocity head that must be converted into work, 
 if the other losses are 15 per cent, and the total efficiency 70 per cent. 
 
 (11) The inner diameter of a centrifugal pump is 12 inches, the outer 
 diameter 21| inches. The width of the wheel at outlet is 3| inches. Using 
 equation (2), section 236, find the discharge of the pump when the head is 
 21'5 feet, and the number of revolutions per minute is 440. 
 
480 HYDRAULICS 
 
 (12) The efficiency of a centrifugal pump when running at 550 revolu- 
 tions per minute is 70 per cent. The mean angle the tip of the vane makes 
 with the direction of motion of the inlet edge of the vane is 99 degrees. 
 The angle the tip of the vane makes with the direction of motion of the 
 edge of the vane at exit is 167 degrees. The radial velocity of flow is 3'6 
 feet per second. The internal diameter of the wheel is 11| inches and the 
 external diameter 19 inches. 
 
 Find the kinetic energy of the water when it leaves the wheel. 
 
 Assuming that 5 per cent, of the energy is lost by friction, and that one- 
 half of the kinetic energy at exit is lost, find the head lost at inlet when the 
 lift is 30 feet. Hence find the probable velocity impressed on the water as 
 it enters the wheel. 
 
 (13) Describe a forced vortex, and sketch the form of the free surface 
 when the angular velocity is constant. 
 
 In a centrifugal pump revolving horizontally under water, the diameter 
 of the inside of the paddles is 1 foot, and of the outside 2 feet, and the 
 pump revolves at 400 revolutions per minute. Find approximately how 
 high the water would be lifted above the tail water level. 
 
 (14) Explain the action of a centrifugal pump, and deduce an expression 
 for its efficiency. If such a pump were required to deliver 1000 gallons an 
 hour to a height of 20 feet, how would you design it ? Lond. Un. 1903. 
 
 (15) Find the speed of rotation of a wheel of a centrifugal pump which 
 is required to lift 200 tons of water 5 feet high in one minute ; having given 
 the efficiency is 0'6. The velocity of flow through the wheel is 4*5 feet per 
 second, and the vanes are curved backward so that the angle between their 
 directions and a tangent to the circumference is 20 degrees. Lond. Un. 
 1905. 
 
 (16) A centrifugal pump is required to lift 2000 gallons of water per 
 minute through 20 feet. The velocity of flow through the wheel is 7 feet 
 per second and the efficiency 0'6. The angle the tip of the vane at outlet 
 makes with the direction of motion is 150 degrees. The outer radius of the 
 wheel is twice the inner. Determine the dimensions of the wheel. 
 
 (17) A double-acting plunger pump has a piston 6 inches diameter 
 and the length of the strokes is 12 inches. The gross head is 500 feet, 
 and the pump makes 80 strokas per minute. Assuming no slip, find the 
 discharge and horse-power of the pump. Find also the necessary diameter 
 for the steam cylinder of an engine driving the pump direct, assuming the 
 steam pressure is 100 Ibs. per square inch, and the mechanical efficiency 
 of the combination is 85 per cent. 
 
 (18) A plunger pump is placed above a tank containing water at a 
 temperature of 200 F. The weight of the suction valve is 2 Ibs. and its 
 diameter l inches. Find the maximum height above the tank at which 
 the pump may be placed so that it will draw water, the barometer standing 
 at 30 inches and the pump being assumed perfect and without clearance. 
 (The vapour tension of water at 200 F. is about 11-6 Ibs. per sq. inch.) 
 
 (19) A pump cylinder is 8 inches diameter and the stroke of the plunger 
 is one foot. Calculate the maximum velocity, and the acceleration of the 
 
PUMPS 481 
 
 water in the suction and delivery pipes, assuming their respective diameters 
 to be 7 inches and 5 inches, the motion of the piston to be simple harmonic, 
 and the piston to make 36 strokes per minute. 
 
 (20) Taking the data of question (19) calculate the work done on the 
 suction stroke of the pump, 
 
 (1) neglecting the friction in the suction pipe, 
 
 (2) including the friction in the suction pipe and assuming that the 
 
 suction pipe is 25 feet long and that/=0'01. 
 
 The height of the centre of the pump above the water in the sump is 
 18 feet. 
 
 (21) If the pump in question (20) delivers into a rising main against 
 a head of 120 feet, and if the length of the main itself is 250 feet, 
 find the total work done per revolution. Assuming the pump to be double 
 acting, find the i. H. p. required to drive the pump, the efficiency being *72 
 and no slip in the pump. Find the delivery of the pump, assuming a slip 
 of 5 per cent. 
 
 (22) The piston of a pump moves with simple harmonic motion, and it 
 is driven at 40 strokes per minute. The stroke is one foot. The suction 
 pipe is 25 feet long, and the suction valve is 19 feet above the surface of the 
 water in the sump. Find the ratio between the diameter of the suction 
 pipe and the pump cylinder, so that no separation may take place at the. 
 dead points. Water barometer 34 feet. 
 
 (23) Two double-acting pumps deliver water into a main without an 
 air vessel. Each is driven by an engine with a fly-wheel heavy enough to 
 keep the speed of rotation uniform, and the connecting rods are very long. 
 
 Let Q be the mean delivery of the pumps per second, Q L the quantity of 
 water in the main. Find the pressure due to acceleration (a) at the begin- 
 ning of a stroke when one pump is delivering water, (b) at the beginning 
 of the stroke of one of two double-acting pumps driven by cranks at right 
 angles when both are delivering. When is the acceleration zero ? 
 
 (24) A double-acting horizontal pump has a piston 6 inches diameter 
 (the diameter of the piston rod is neglected) and the stroke is one foot. 
 The water is pumped to a height of 250 feet along a delivery pipe 450 feet 
 long and 4^ inches diameter. An air vessel is put on the delivery pipe 
 10 feet from the delivery valve. 
 
 Find the pressure on the pump piston at the two ends of the stroke 
 when the pump is making 40 strokes per minute, assuming the piston 
 moves with simple harmonic motion and compare these pressures with the 
 pressures when there is no air vessel. /='0075. 
 
 (25) A single acting hydraulic motor makes 160 strokes per minute and 
 moves with simple harmonic motion. 
 
 The motor is supplied with water from an accumulator in which the 
 pressure is maintained at 200 Ibs. per square inch. 
 
 The cylinder is 8 inches diameter and 12 inches stroke. The delivery 
 pipe is 200 feet long, and the coefficient, which includes loss at bends, etc. 
 may be taken as/=0'2. 
 
 L. H. 31 
 
482 HYDRAULICS 
 
 Neglecting the mass of the reciprocating parts and of the variable 
 quantity of water in the cylinder, draw a curve of effective pressure on the 
 piston. 
 
 (26) The suction pipe of a plunger pump is 35 feet long and 4 inches 
 diameter, the diameter of the plunger is 6 inches and the stroke 1 foot. 
 
 The delivery pipe is 2| inches diameter, 90 feet long, and the head at 
 the delivery valve is 40 feet. There is no air vessel on the pump. The 
 centre of the pump is 12 feet 6 inches above the level of the water in the 
 sump. 
 
 Assuming the plunger moves with simple harmonic motion and makes 
 50 strokes per minute, draw the theoretical diagram for the pump. 
 
 Neglect the effect of the variable quantity of water in the cylinder and 
 the loss of head at the valves. 
 
 (27) Will separation take place anywhere in the delivery pipe of the 
 pump, the data of which is given in question (26), if the pipe first runs 
 horizontally for 50 feet and then vertically for 40, or rises 40 feet im- 
 mediately from the pump and then runs horizontally for 50 feet, and 
 separation takes place when the pressure head falls below 5 feet ? 
 
 (28) A pump has three single-acting plungers 29^ inches diameter 
 driven by cranks at 120 degrees with each other. The stroke is 5 feet and 
 the number of strokes per minute 40. The suction is 16 feet and the length 
 of the suction pipe is 22 feet. The delivery pipe is 3 feet diameter and 
 350 feet long. The head at the delivery valve is 214 feet. 
 
 Find (a) the minimum diameter of the suction pipe so that there is no 
 separation, assuming no air vessel and that separation takes place when 
 the pressure becomes zero. 
 
 (b) The horse-power of the pump when there is an air vessel on the 
 delivery very near to the pump. /= '007. 
 
 [The student should draw out three cosine curves differing in phase by 
 120 degrees. Then remembering that the pump is single acting, the 
 resultant curve of accelerations will be found to have maximum positive 
 
 o)V A 
 and also negative values of every 60 degrees. The maximum 
 
 acceleration head is then h a = +^^ - . 
 
 For no separation, therefore, a = n * - . 
 
 lo(/ (o4 ID) _J 
 
 (29) The piston of a double-acting pump is 5 inches in diameter and 
 the stroke is 1 foot. The delivery pipe is 4 inches diameter and 400 feet 
 long and it is fitted with an air vessel 8 feet from the pump cylinder. The 
 water is pumped to a height of 150 feet. Assuming that the motion of the 
 piston is simple harmonic, find the pressure per square inch on the piston 
 at the beginning and middle of its stroke and the horse-power of the pump 
 when it makes 80 strokes per minute. Neglect the effect of the variable 
 quantity of water in the cylinder. Lond. Un. 1906. 
 
PUMPS 483 
 
 (30) The plunger of a pump moves with simple harmonic motion. 
 Find the condition that separation shall not take place on the suction 
 stroke and show why the speed of the pump may be increased if an air 
 vessel is put in the suction pipe. Sketch an indicator diagram showing 
 separation. Explain " negative slip." Lond. Un. 1906. 
 
 (31) In a single-acting force pump, the diameter of the plunger is 
 4 inches, stroke 6 inches, length of suction pipe 63 feet, diameter of suction 
 pipe 2 1 inches, suction head 0'07 ft. When going at 10 revolutions per 
 minute, it is found that the average loss of head per stroke between the 
 suction tank and plunger cylinder is 0'23 ft. Assuming that the frictional 
 losses vary as the square of the speed, find the absolute head on the suction 
 side of the plunger at the two ends and at the middle of the stroke, the 
 revolutions being 50 per minute, and the barometric head 34 feet. Draw a 
 diagram of pressures on the plunger simple harmonic motion being 
 assumed. Lond. Un. 1906. 
 
 (32) A single-acting pump without an air vessel has a stroke of 
 7 inches. The diameter of the plunger is 4 inches and of the suction 
 pipe 3 inches. The length of the suction pipe is 12 feet, and the centre 
 of the pump is 9 feet above the level in the sump. 
 
 Determine the number of single strokes per second at which theoreti- 
 cally separation will take place, and explain why separation will actually 
 take place when the number of strokes is less than the calculated value. 
 
 (33) Explain carefully the use of an air vessel in the delivery pipe of a 
 pump. The pump of question (32) makes 100 single strokes per minute, 
 and delivers water to a height of 100 feet above the water in the well 
 through a delivery pipe 1000 feet long and 2 inches diameter. Large air 
 vessels being put on the suction and delivery pipes near to the pump. 
 
 On the assumption that all losses of head other than by friction in 
 the delivery pipe are neglected, determine the horse-power of the pump. 
 There is no slip. 
 
 (34) A pump plunger has an acceleration of 8 feet per second per 
 second when at the end of the stroke, and the sectional area of the plunger 
 is twice the sectional area of the delivery pipe. The delivery pipe is 152 
 feet long. It runs from the pump horizontally for a length of 45 feet, then 
 vertically for 40 feet, then rises 5 feet, on a slope of 1 vertical to 3 hori- 
 zontal, and finally runs in a horizontal direction. 
 
 Find whether separation will take place, and if so at which section 
 of the pipe, if it be assumed that separation takes place when the pressure 
 head in the pipe becomes 7 feet. 
 
 (35) A pump of the duplex kind, Fig. 325, in which the steam piston is 
 connected directly to the pump piston, works against a head of h feet of 
 water, the head being supplied by a column of water in the delivery pipe. 
 The piston area is AQ, the plunger area A, the delivery pipe area a, the 
 length of the delivery pipe I and the constant steam pressure on the piston 
 Po Ibs. per square foot. The hydraulic resistance may be represented by 
 
 Fv 2 
 
 - , v being the velocity of the plunger and F a coefficient. 
 
 312 
 
484 HYDRAULICS 
 
 Show that when the plunger has moved a distance x from the beginning 
 of the stroke 
 
 _ e -^. Lend. Un. 1906. 
 
 (36) A pump valve of brass has a specific gravity of 8| with a lift of 
 ^ foot, the stroke of the piston being 4 feet, the head of water 40 feet and 
 the ratio of the full valve area to the piston area one-fifth. 
 
 If the valve is neither assisted nor meets with any resistance to closing, 
 find the time it will take to close and the "slip" due to this gradual closing. 
 
 Time to close is given by formula, S=^ 2 . f=L^ X 32-2. Lond. Un. 1906. 
 
 ' 
 
CHAPTER XI. 
 
 HYDRAULIC MACHINES. 
 
 267. Joints and packings used in hydraulic work. 
 
 The high pressures used in hydraulic machinery make it 
 necessary to use special precautions in making joints. 
 
 Figs. 332 and 333 show methods of connecting two lengths of 
 pipe. The arrangement shown in Fig. 332 is used for small 
 
 Left hcuricL 
 thread 
 
 Right 
 thread 
 
 Fig. 332. 
 
 Fig. 333. 
 
 Fig. 334. 
 
486 
 
 HYDRAULICS 
 
 wrought-iron pipes, no packing being required. In Fig. 333 the 
 packing material is a gutta-percha ring. Fig. 336 shows an 
 ordinary socket joint for a cast-iron hydraulic main. To make 
 the joint, a few cords of hemp or tarred rope are driven into 
 the socket. Clay is then put round the outside of the socket and 
 molten lead run in it. The lead is then jammed into the socket 
 with a caulking tool. Fig. 335 shows various forms of packing 
 leathers, the applications of which will be seen in the examples 
 given of hydraulic machines. 
 
 Neck leather 
 
 Cup leather 
 Fig. 335. 
 
 Fig. 336. 
 
 Hemp twine, carefully plaited, and dipped in hot tallow, 
 makes a good packing, when used in suitably designed glands 
 (see Fig. 339) and is also very suitable for pump buckets, 
 Fig. 323. Metallic packings are also used as shown in Figs. 337 
 and 338. 
 
 Fig. 337. 
 
 Fig. 338. 
 
 268. The accumulator. 
 
 The accumulator is a device used in connection with hydraulic 
 machinery for storing energy. 
 
 In the form generally adopted in practice it consists of a long 
 cylinder C, Fig. 339, in which slides a ram B, and into which water 
 is delivered from pumps. At the top of the ram is fixed a rigid 
 cross head which carries, by means of the bolts, a large cylinder 
 which can be filled with slag or other heavy material, or it may 
 be loaded with cast-iron weights as in Fig. 340. The water is 
 
HYDRAULIC MACHINES 
 
 487 
 
 Fig. 339. Hydraulic Accumulator. 
 
488 HYDRAULICS 
 
 admitted to the cylinder at any desired pressures through a pipe 
 connected to the cylinder by the flange shown dotted, and the 
 weight is so adjusted that when the pressure per sq. inch in 
 the cylinder is a given amount the ram rises. 
 
 If d is the diameter of the ram in inches, p the pressure 
 in Ibs. per sq. inch, and h the height in feet through which the 
 ram can be lifted, the weight of the ram and its load is 
 
 and the energy that can be stored in the accumulator is 
 
 E = p.7#.fe foot Ibs. 
 4 
 
 The principal object of the accumulator is to allow hydraulic 
 machines, or lifts, which are being supplied with hydraulic power 
 from the pumps, to work for a short time at a much greater rate 
 than the pumps can supply energy. If the pumps are connected 
 directly to the machines the rate at which the pumps can supply 
 energy must be equal to the rate at which the machines are 
 working, together with the rate at which energy is being lost by 
 friction, etc., and the pump must be of such a capacity as to supply 
 energy at the greatest rate required by the machines, and the 
 frictional resistances. If the pump supplies water to an accumu- 
 lator, it can be kept working at a steady rate, and during the time 
 when the demand is less than the pump supply, energy can be 
 stored in the accumulator. 
 
 In addition to acting as a storer of energy, the accumulator 
 acts as a pressure regulator and as an automatic arrangement for 
 starting and stopping the pumps. 
 
 When the pumps are delivering into a long main, the demand 
 upon which is varying, the sudden cutting off of the whole or 
 a part of the demand may cause such a sudden rise in the pressure 
 as to cause breakage of the pipe line, or damage to the pump. 
 With an accumulator on the pipe line, unless the ram is 
 descending and is suddenly brought to rest, the pressure cannot 
 rise very much higher than the pressure p which will lift the ram. 
 
 To start and stop the pump automatically, the ram as it 
 approaches the top of its stroke moves a lever connected to 
 a chain which is led to a throttle valve on the steam pipe of the 
 pumping engine, and thus shuts off steam. On the ram again 
 falling below a certain level, it again moves the lever and opens 
 the throttle valve. The engine is set in motion, pumping re- 
 commences, and the accumulator rises. 
 
HYDRAULIC MACHINES 489 
 
 Example. A hydraulic crane working at a pressure of 700 Ibs. per sq. inch has 
 to lift 30 cwts. at a rate of 200 feet per minute through a height of 50 feet, once 
 every 1 minutes. The efficiency of the crane is 70 per cent, and an accumulator 
 is provided. 
 
 Find the volume of the cylinder of the crane, the minimum horse-power for the 
 pump, and the minimum capacity of the accumulator. 
 
 Let A be the sectional area of the ram of the crane cylinder in sq. feet and L 
 the length of the stroke in feet. 
 
 Then, p . 144. A . L x 0'70 = 30 x 112 x 50', 
 
 30 x 112 x 50 
 AL = V = 0-70x 144x700 
 
 = 2-38 cubic feet. 
 The rate of doing work in the lift cylinder is 
 
 per minute, 
 
 and the work done in lifting 50 feet is 240,000 ft. Ibs. Since this has to be done 
 once every one and half minutes, the work the pump must supply in one and half 
 minutes is at least 240,000 ft. Ibs. , and the minimum horse-power is 
 
 240,000 
 
 = 33,000 x 1-5 = ' 
 The work done by the pump while the crane is lifting is 
 
 I'D 
 
 The energy stored in the accumulator must be, therefore, at least 200,000 ft. Ibs. 
 Therefore, if V a is its minimum capacity in cubic feet, 
 
 V a x 700 x 144 = 200,000, 
 or V a = 2 cubic feet nearly. 
 
 269. Differential accumulator*. 
 
 Tweddell's differential accumulator, shown in Fig. 340, has a 
 fixed ram, the lower part of which is made slightly larger than 
 the upper by forcing a brass liner upon it. A cylinder loaded 
 with heavy cast-iron weights slides upon the ram, water-tight 
 joints being made by means of the cup leathers shown. Water 
 is pumped into the cylinder through a pipe, and a passage drilled 
 axially along the lower part of the ram. 
 
 Let p be the pressure in Ibs. per sq. inch, d and di the dia- 
 meters of the upper and lower parts of the ram respectively. 
 The weight lifted (neglecting friction) is then 
 
 and if h is the lift in feet, the energy stored is 
 E - p. ^W-d 2 ) h. foot Ibs. 
 
 The difference of the diameters d^ and d being small, the pres- 
 sure p can be very great for a comparatively small weight W. 
 
 The capacity of the accumulator is, however, very small. 
 This is of advantage when being used in connection with 
 
 * Proceedings Inst. Mech. Engs., 1874. 
 
490 
 
 HYDRAULICS 
 
 Fig. 340. 
 
 Fig. 341. Hydraulic Intensifies 
 
HYDRAULIC MACHINES 491 
 
 hydraulic riveters, as when a demand is made upon the ac- 
 cumulator, the ram falls quickly, but is suddenly arrested when 
 the ram of the riveter comes to rest, and there is a consequent 
 increase in the pressure in the cylinder of the riveter which 
 clinches the rivet. Mr Tweddell estimates that when the ac- 
 cumulator is allowed to fall suddenly through a distance of from 
 18 to 24 inches, the pressure is increased by 50 per cent. 
 
 270. Air accumulator. 
 
 The air accumulator is simply a vessel partly filled with air and 
 into which the pumps, which are supplying power to machinery, 
 deliver water while the machinery is not at work. 
 
 Such an air vessel has already been considered in connection 
 with reciprocating pumps and an application is shown in connection 
 with a forging press, Fig. 343. 
 
 If V is the volume of air in the vessel when the pressure is 
 p pounds per sq. inch and a volume v of water is pumped into 
 the vessel, the volume of air is (V v). 
 
 Assuming the temperature remains constant, the pressure pi in 
 the vessel will now be 
 
 p.V 
 
 *"V^' 
 
 If V is the volume of air, and a volume of water v is taken out 
 of the vessel, 
 
 p.V 
 P' = VTV 
 
 271. Intensifies. 
 
 It is frequently desirable that special machines shall work at 
 a higher pressure than is available from the hydraulic mains. To 
 increase the pressure to the desired amount the intensifier is used. 
 
 One form is shown in Fig. 341. A large hollow ram works in 
 a fixed cylinder C, the ram being made water-tight by means of a 
 stuffing-box. Connected to the cylinder by strong bolts is a cross 
 head which has a smaller hollow ram projecting from it, and 
 entering the larger ram, in the upper part of which is made a 
 stuffing-box. Water from the mains is admitted into the large 
 cylinder and also into the hollow ram through the pipe and 
 the lower valve respectively shown in Fig. 342. 
 
 If p Ibs. per sq. inch is the pressure in the main, then on 
 the underside of the large ram there is a total force acting 
 
 of p 7 D 2 pounds, and the pressure inside the hollow ram rises to 
 
 D 2 
 
 p-r 2 pounds per sq. inch, D and d being the external diameters 
 cL 
 
 of the large ram and the small ram respectively. 
 
492 
 
 HYDRAULICS 
 
 The form of intensifier here shown is used in connection with 
 a large flanging press. The cylinder of the press and the upper 
 part of the intensifier are filled with water at 700 Ibs. per sq. inch 
 and the die brought to the work. Water at the same pressure is 
 admitted below the large ram of the intensifier and the pressure 
 in the upper part of the intensifier, and thus in the press cylinder, 
 rises to 2000 Ibs. per sq. inch, at which pressure the flanging 
 is finished. 
 
 ToSjnaJL 
 
 Cylinder 
 
 of Intensifier 
 
 To press. 
 atWOOlbs. 
 per sq. inch, 
 
 Tb Large Cylinder erf Intensifier 
 
 From Pumps 
 
 NOTL Return Valves for 
 Intensifier. 
 
 Fig. 342. 
 
 atlOOWs. 
 
 persq. inch. 
 
 272. Steam intensifiers. 
 
 The large cylinder of an intensifier may be supplied with 
 steam, instead of water, as in Fig. 343, which shows a steam in- 
 tensifier used in conjunction with a hydraulic forging press. These 
 intensifiers have also been used on board ship* in connection with 
 hydraulic steering gears. 
 
 273. Hydraulic forging press, with steam intensifier and 
 air accumulator. 
 
 The application of hydraulic power to forging presses is illus- 
 trated in Fig. 343. This press is worked in conjunction with a 
 steam intensifier and air accumulator to allow of rapid working. 
 The whole is controlled by a single lever K, and the press is 
 capable of making 80 working strokes per minute. 
 
 When the lever K is in the mid position everything is at rest ; 
 
 on moving the lever partly to the right, steam is admitted into the 
 
 cylinders D of the press through a valve. On moving the lever to 
 
 its extreme position, a finger moves the valve M and admits water 
 
 * Proceedings Inst. Mech. Engs., 1874. 
 
HYDRAULIC MACHINES 
 
 493 
 
 under a relay piston shown at the top of the figure, which opens 
 a valve E at the top of the air vessel. In small presses the valve 
 E is opened by levers. The ram B now ascends at the rate of 
 
 about 1 foot per second, the water in the cylinder c being forced 
 into the accumulator. On moving the lever K to the left, as soon 
 as it has passed the central position the valve L is opened to 
 
494 
 
 HYDRAULICS 
 
 exhaust, and water from the air vessel, assisted by gravity, forces 
 down the ram B, the velocity acquired being about 2 feet per 
 second, until the press head A touches the work. The movement 
 of the lever K being continued, a valve situated above the valve 
 J is opened, and steam is admitted to the intensifier cylinder H ; 
 the valve E closes automatically, and a large pressure is exerted 
 on the work under the press head. 
 
 If only a very short stroke is required, the bye-pass valve L is 
 temporarily disconnected, so that steam is supplied continuously 
 to the lifting cylinders D. The lever K is then simply used to 
 admit and exhaust steam from the intensifier H, and no water 
 enters or leaves the accumulator. An automatic controlling gear 
 is also fitted, which opens the valve J sufficiently early to prevent 
 the intensifier from overrunning its proper stroke. 
 
 Fig. 346. 
 
 Fig. 344. Fig. 345. 
 
 274. Hydraulic cranes. 
 
 Fig. 344 shows a section through, and Fig. 345 an elevation 
 of, a hydraulic crane cylinder. 
 
HYDRAULIC MACHINES 495 
 
 One end of a wire rope, or chain, is fixed to a lug L on the 
 cylinder, and the rope is then passed alternately round the upper 
 and lower pulleys, and finally over the pulley on the jib of the 
 crane, Fig. 346. In the crane shown there are three pulleys on 
 the ram, and neglecting friction, the pressure on the ram is equally 
 divided among the six ropes. The weight lifted is therefore one- 
 sixth of the pressure on the ram, but the weight is lifted a distance 
 equal to six times the movement of the ram. 
 
 Let the number of pulleys on the end of the ram of any crane 
 
 be -a , arranged as in Fig. 347. 
 
 The movement of the weight will then be n times that of 
 the ram. 
 
 Let p be the pressure in Ibs. per sq. inch in the cylinder and 
 d the diameter of the ram in inches. 
 
 The pressure on the ram is 
 
 and the energy supplied to the crane per foot travel of the ram is 
 therefore P foot pounds. 
 
 The energy supplied per unit volume displacement is 144 . p. 
 
 The actual weight lifted is 
 
 W = e ^- pd 2 Ibs., 
 4m, 
 
 e being the efficiency. 
 
 When full load is being lifted e is between 0*7 and 0*8. 
 
 For a given lift of the weight, the number of cubic feet of water 
 used, and consequently the energy supplied, is the same whatever 
 the load lifted, and at light loads the efficiency is very small. 
 
 275. Double power cranes. 
 
 To enable a crane designed for heavy work to lift light loads 
 with reasonable efficiency, two lifting rams of different diameters 
 are employed, the smaller of which can be used at light loads. 
 
 A convenient arrangement is as shown in Figs. 348 and 349, 
 the smaller ram R' working inside the large ram R. 
 
 When light loads are to be lifted, the large ram is prevented 
 from moving by strong catches C, and the volume of water used 
 is only equal to the diameter of the small ram into the length of 
 the stroke. For large loads, the catches are released and the 
 two rams move together. 
 
 Another arrangement is shown in Fig. 350, water being ad- 
 mitted to both faces of the piston when light loads are to be 
 lifted, and to the face A only when heavy loads are to be raised. 
 
496 
 
 HYDRAULICS 
 
 .SP 
 
HYDRAULIC MACHINES 
 
 497 
 
 For a given stroke s of the ram, the energy supplied in the 
 first case is 
 
 ;. Ibs., 
 
 and in the second case 
 
 . Ibs. 
 
 Fig. 350. Armstrong Double-power Hydraulic Crane Cylinder. 
 
 276. Hydraulic crane valves. 
 
 Figs. 351 and 352 show two forms of lifting and lowering 
 valves used by Armstrong, Whitworth and Co. for hydraulic 
 cranes. 
 
 In the arrangement shown in Fig. 351 there are two inde- 
 pendent valves, the one on the left being the pressure, and that 
 on the right the exhaust valve. 
 
 Fig. 351. Armstrong-Whitworth 
 Hydraulic Crane Valve. 
 
 L. H. 
 
 Fig. 352. Armstrong-Whitworth 
 Hydraulic Crane Slide Valve. 
 
 32 
 
498 HYDRAULICS 
 
 In the arrangement shown in Fig. 352 a single D slide valve s 
 used. Water enters the valve chest through the pressure passage 
 P. The valve is shown in the neutral position. If the valve 
 is lowered, the water enters the cylinder, but if it is right, 
 water escapes from the cylinder through the port of the slide 
 valve. 
 
 277. Small hydraulic press. Fig. 353 is a section through 
 the cylinder of a small hydraulic press, used for testing springs. 
 
 The cast-iron cylinder is fitted with a brass liner, and axially 
 with the cylinder a rod P r , having a piston P at the free end, 
 is screwed into the liner. The steel ram is hollow, the inner 
 cylinder being lined with a brass liner. 
 
 Water is admitted and exhausted from the large cylinder 
 through a Luthe valve, fixed to the top of the cylinder and 
 operated by the lever A. The small cylinder inside the ram is 
 connected directly to the pressure pipe by a hole drilled along the 
 rod P r , so that the full pressure of the water is continuously 
 exerted upon the small piston P and the annular ring RR. 
 
 Leakage to the main cylinder is prevented by means of a 
 gutta-percha ring Gr and a ring leather c, and leakage past the 
 steel' ram and piston P by cup leathers L and LI . 
 
 When the valve spindle is moved to the right, the port p is 
 connected with the exhaust, and the ram is forced up by the 
 pressure of the water on the annular ring RR. On moving the 
 valve spindle over to the left, pressure water is admitted into the 
 cylinder and the ram is forced down. Immediately the pressure 
 is released, the ram comes back again. 
 
 Let D be the diameter of the ram, d the diameter of the 
 rod P r , di the diameter of the piston P, and p the water pressure 
 in pounds per sq. inch. 
 
 The resultant force acting on the ram is 
 
 and the force lifting the ram when pressure is released from the 
 main cylinder is, 
 
 The cylindrical valve spindle S, has a chamber C cast in it, 
 and two rings of six holes in each ring are drilled through 
 the external shell of the chamber. These rings of holes are at 
 such a distance apart that, when the spindle is moved to the 
 right, one ring is opposite to the exhaust and the other opposite 
 to the port p, and when the spindle is moved to the left, the holes 
 
HYDRAULIC MACHINES 
 
 499 
 
 are respectively opposite to the port p and the pressure water 
 inlet. 
 
 Leakage past the spindle is prevented by the four ring leathers 
 shown. 
 
 Fig. 353. Hydraulic Press with Luthe Valve. 
 
 278. Hydraulic riveter. 
 
 A section through the cylinder and ram of a hydraulic riveter 
 is shown in Fig. 354. 
 
 322 
 
500 
 
 HYDRAULICS 
 
 Fig. 354. Hydraulic Riveter. 
 
 Spring 
 forclosuLq 
 " 
 
 Fig. 355. Valves for Hydraulic Eiveter. 
 
HYDRAULIC MACHINES 
 
 501 
 
 The mode of working is exactly the same as that of the small 
 press described in section 277. 
 
 An enlarged section of the valves is shown in Fig. 355. On 
 pulling the lever L to the right, the inlet valve V is opened, and 
 pressure water is admitted to the large cylinder, forcing out 
 the ram. When the lever is in mid position, both valves are 
 closed by the springs S, and on moving the lever to the left, the 
 exhaust valve Vi is opened, allowing the water to escape from the 
 cylinder. The pressure acting on the annular ring inside the 
 large ram then brings back the ram. The methods of preventing 
 leakage are clearly shown in the figures. 
 
 279. Hydraulic engines. 
 
 Hydraulic power is admirably adapted for machines having a 
 reciprocating motion only, especially in those cases where the load 
 is practically constant. 
 
 Fig. 356. Hydraulic Capstan. 
 
502 
 
 HYDRAULICS 
 
 It has moreover been successfully applied to the driving of 
 machines such as capstans and winches in which a reciprocating 
 motion is converted into a rotary motion. 
 
 The hydraulic-engine shown in Figs. 356 and 357, has three 
 cylinders in one casting, the axes of which meet on the axis of the 
 crank shaft S. The motion of the piston P is transmitted to the 
 crank pin by short connecting rods R. Water is admitted and 
 exhausted through a valve Y, and ports p. 
 
 Fig. 357. 
 
 The face of the valve is as shown in Fig. 358, E being the 
 exhaust port connected through the centre of the valve to the 
 exhaust pipe, and KM the pressure port, connected to the supply 
 chamber H by a small port through the side of the valve. The 
 valve seating is generally made of lignum-vitae, and has three 
 circular ports as shown dotted in Fig. 358. The valve receives its 
 motion from a small auxiliary crank T, revolved by a projection 
 from the crank pin Gr. When the piston 1 is at the end of its 
 stroke, Fig. 359, the port pi should be just opening to the pressure 
 port, and just closing to the exhaust port E. The port p s should 
 be fully open to pressure and port p 2 fully open to exhaust. 
 When the crank has turned through 60 degrees, piston 3 will 
 
HYDRAULIC MACHINES 
 
 503 
 
 be at the inner end of its stroke, and the edge M of the pressure 
 port should be just closing to the port p 3 . At the same instant the 
 edge N of the exhaust port should be coincident with the lower 
 edge of the port p s . The angles QOM, and LON, therefore, 
 should each be 60 degrees. A little lead may be given to the 
 valve ports, i.e. they may be made a little longer than shown in 
 the Fig. 358, so as to ensure full pressure on the piston when 
 commencing its stroke. There is no dead centre, as in whatever 
 position the crank stops one or more of the pistons can exert a 
 turning moment on the shaft, and the engine will, therefore, start 
 in any position. 
 
 Fig. 358. 
 
 Fig. 359. 
 
 The crank* effort, or turning moment diagram, is shown in 
 Fig. 359, the turning moment for any crank position OK being 
 OM. The turning moment can never be less than ON, which is 
 the magnitude of the moment when any one of the pistons is at 
 the end of its stroke. 
 
 This type of hydraulic engine has been largely used for the 
 driving of hauling capstans, and other machinery which works 
 intermittently. It has the disadvantage, already pointed out in 
 connection with hydraulic lifts and cranes, that the amount of 
 water supplied is independent of the effective work done by the 
 machine, and at light loads it is consequently very inefficient. 
 There have been many attempts to overcome this difficulty, 
 notably as in the Hastie engine t, and Rigg engine. 
 
 * See text book on Steam Engine. 
 
 t Proceedings Inst. Mech. Engs. , 1874. 
 
504 
 
 HYDRAULICS 
 
 280. Rigg hydraulic engine. 
 
 To adapt the quantity of water used to the work done, Rigg* 
 has modified the three cylinder engine by fixing the crank pin, and 
 allowing the cylinders to revolve about it as centre. 
 
 The three pistons Pi, P 2 and P 3 are connected to a disc, 
 Fig. 360, by three pins. This disc revolves about a fixed centre A. 
 The three cylinders rotate about a centre Gr, which is capable of 
 being moved nearer or further away from the point A as desired. 
 The stroke of the pistons is twice AGr, whether the crank or the 
 cylinders revolve, and since the cylinders, for each stroke, have to 
 be filled with high pressure water, the quantity of water supplied 
 per revolution is clearly proportional to the length AGr. 
 
 Fig. 360. Kigg Hydraulic Engine. 
 
 The alteration of the length of the stroke is effected by means 
 of the subsidiary hydraulic engine, shown in Fig. 361. There are 
 two cylinders C and Ci, in which slide a hollow double ended 
 ram PP l which carries the pin Gr, Fig. 360. Cast in one piece with 
 the ram is a valve box B. R is a fixed ram, and through it water 
 enters the cylinder Ci, in which the pressure is continuously 
 maintained. The difference between the effective areas of P and 
 PI when water is in the two cylinders, is clearly equal to the area 
 of the ram head RL 
 
 * See also Engineer, Vol. LXXXV, 1898. 
 
HYDRAULIC MACHINES 
 
 505 
 
 From the cylinder Ci the water is led along the passages 
 shown to the valve V. On opening this valve high-pressure 
 water is admitted to the cylinder C. A second valve similar to 
 V, but not shown, is used to regulate the exhaust from the 
 cylinder C. When this valve is opened, the ram PPj moves to 
 the left and carries with it the pin Gr, Fig. 360. On the exhaust 
 being closed and the valve V opened, the full pressure acts upon 
 both ends of the ram, and since the effective area of P is greater 
 than PI it is moved to the right carrying the pin Gr. If both 
 valves are closed, water cannot escape from the cylinder C and 
 the ram is locked in position by the pressure on the two ends. 
 
 To Governor 
 
 Water 
 inlets 
 
 Fig. 361. 
 
 EXAMPLES. 
 
 (1) The ram of a hydraulic crane is 7 inches diameter. Water is 
 supplied to the crane at 700 Ibs. per square inch. By suitable gearing the 
 load is lifted 6 times as quickly as the ram. Assuming the total efficiency 
 of the crane is 70 per cent., find the weight lifted. 
 
 (2) An accumulator has a stroke of 23 feet ; the diameter of the ram is 
 23 inches; the working pressure is 700 Ibs. per square inch. Find the 
 capacity of the accumulator in horse-power hours. 
 
 (3) The total weight on the cage of an ammunition hoist is 3250 Ibs. 
 The velocity ratio between the cage and the ram is six, and the extra load 
 on the cage due to friction may be taken as 30 per cent, of the load on the 
 cage. The steady speed of the ram is 6 inches per second and the available 
 pressure at the working valve is 700 Ibs. per square inch. 
 
 Estimate the loss of head at the entrance to the ram cylinder, and 
 assuming this was to be due to a sudden enlargement in passing through 
 the port to the cylinder, estimate, on the usual assumption, the area of the 
 port, the ram cylinder being 9| inches diameter. Lond. Un. 1906. 
 
 3250x1-3x6 
 The effective pressure p = 
 
506 HYDRAULICS 
 
 Loss of head 
 
 w 2g 
 
 v= velocity through the valve. 
 
 Area of port 
 
 (4) Describe, with sketches, some form of hydraulic accumulator suit- 
 able for use in connection with riveting. Explain by the aid of diagrams, 
 if possible, the general nature of the curve of pressure on the riveter ram 
 during the stroke ; and point out the reasons of the variations. Lond. Un. 
 1905. (See sections 262 and 269.) 
 
 (5) Describe with sketches a hydraulic intensifier. 
 
 An intensifier is required to increase the pressure of 700 Ibs. per square 
 inch on the mains to 3000 Ibs. per square inch. The stroke of the intensi- 
 fier is to be 4 feet and its capacity three gallons. Determine the diameters 
 of the rams. Inst. C. E. 1905. 
 
 (6) Sketch in good proportion a section through a differential hydraulic 
 accumulator. What load would be necessary to produce a pressure of 1 ton 
 per square inch, if the diameters of the two rams are 4 inches and 4^ inches 
 respectively ? Neglect the friction of the packing. Give an instance of the 
 use of such a machine and state why accumulators are used. 
 
 (7) A Tweddell's differential accumulator is supplying water to riveting 
 machines. The diameters of the two rams are 4 inches and 4^ inches 
 respectively, and the pressure in the accumulator is 1 ton per square inch. 
 Suppose when the valve is closed the accumulator is falling at a velocity 
 of 5 feet per second, and the time taken to bring it to rest is 2 seconds, find 
 the increase in pressure in the pipe. 
 
 (8) A lift weighing 12 tons is worked by water pressure, the pressure 
 in the main at the accumulator being 1200 Ibs. per square inch ; the length 
 of the supply pipe which is 3| inches in diameter is 900 yards. What is 
 the approximate speed of ascent of this lift, on the assumption that the 
 friction of the stuffing-box, guides, etc. is equal to 6 per cent, of the gross 
 load lifted and the ram is 8 inches diameter ? 
 
 (9) Explain what is meant by the " coefficient of hydraulic resistance " 
 as applied to a whole system, and what assumption is usually made regard- 
 ing it ? A direct acting lift having a ram 10 inches diameter is supplied 
 from an accumulator working under a pressure of 750 Ibs. per square inch. 
 When carrying no load the ram moves through a distance of 60 feet, at a 
 uniform speed, in one minute, the valves being fully open. Estimate the 
 coefficient of hydraulic resistance referred to the velocity of the ram, and 
 also how long it would take to move the same distance when the ram 
 carries a load of 20 tons. Lond. Un. 1905. 
 
 ( g^ =head lost = . Assumption is made that resistance varies as v 2 . \ 
 
CHAPTER XII. 
 
 RESISTANCE TO THE MOTION OF BODIES IN WATER. 
 
 281. Froude's* experiments to determine frictional re- 
 sistances of thin boards when propelled in water. 
 
 It has been shown that the frictional resistance to the flow of 
 water along pipes is proportional to the velocity raised to some 
 power n, which approximates to two, and Mr Froude's classical 
 experiments, in connection with the resistance of ships, show that 
 the resistance to motion of plane vertical boards when propelled 
 in water, follows a similar law. 
 
 Fig. 362. 
 
 The experiments were carried out near Torquay in a parallel 
 sided tank 278 feet long, 36 feet broad and 10 feet deep. A light 
 railway on "which ran a stout framed truck, suspended from the 
 axles of two pairs of wheels," traversed the whole length of the 
 tank, about 20 inches above the water level. The truck was pro- 
 pelled by an endless wire rope wound on to a barrel, which could 
 be made to revolve at varying speeds, so that the truck could 
 traverse the length of the tank at any desired velocity between 
 100 and 1000 feet per minute. 
 
 * Brit. Ass. Reports, 1872-4. 
 
508 HYDRAULICS 
 
 Planes of wood, about T 3 F inch thick, the surfaces of which were 
 covered with various materials as set out in Table XXXIX, were 
 made of a uniform depth of 19 inches, and when under experi- 
 ment were placed on edge in the water, the upper edge being 
 about 1| inches below the surface. The lengths were varied from 
 2 to 50 feet. 
 
 The apparatus as used by Froude is illustrated and described 
 in the British Association Reports for 1872. 
 
 A later adaptation of the apparatus as used at Haslar for 
 determining the resistance of ships' models is shown in Fig. 362. 
 An arm L is connected to the model and to a frame beam, which 
 is carried on a double knife edge at H. A spring S is attached to 
 a knife edge on the beam and to a fixed knife edge N on the 
 frame of the truck. A link J connects the upper end of the beam 
 to a multiplying lever which moves a pen D over a recording 
 cylinder. This cylinder is made to revolve by means of a worm 
 and wheel, the worm being driven by an endless belt from the axle 
 of the truck. The extension of the spring S and thus the move- 
 ment of the pen D is proportional to the resistance of the model, 
 and the rotation of the drum is proportional to the distance moved. 
 A pen A actuated by clockwork registers time on the cylinder. 
 The time taken by the truck to move through a given distance 
 can thus be determined. 
 
 To calibrate the spring S, weights W are hung from a knife 
 edge, which is exactly at the same distance from H as the points 
 of attachment of L and the spring S. 
 
 From the results of these experiments, Mr Froude made the 
 following deductions. 
 
 (1) The frictional resistance varies very nearly with the 
 square of the velocity. 
 
 (2) The mean resistance per square foot of surface for lengths 
 up to 50 feet diminishes as the length is increased, but is prac- 
 tically constant for lengths greater than 50 feet. 
 
 (3) The frictional resistance varies very considerably with 
 the roughness of the surface. 
 
 Expressed algebraically the frictional resistance to the motion 
 of a plane surface of area A when moving with a velocity v feet 
 per second is 
 
 _/Q.Ai? 
 
 ^ 
 
 / being equal to 
 
RESISTANCE TO THE MOTION OF BODIES IN WATER 
 
 509 
 
 TABLE XXXIX. 
 
 Showing the result of Mr Fronde's experiments on the frictional 
 resistance to the motion of thin vertical boards towed through 
 water in a direction parallel to its plane. 
 
 Width of boards 19 inches, thickness f\ inch. 
 
 n = power or index of speed to which resistance is approxi- 
 mately proportional. 
 
 / = the mean resistance in pounds per square foot of a surface, 
 the length of which is that specified in the heading, when the 
 velocity is 10 feet per second. 
 
 /i = the resistance per square foot, at a distance from the 
 leading edge of the board, equal to that specified in the heading, 
 at a velocity of 10 feet per second. 
 
 As an example, the resistance of the tinfoil surface per square 
 foot at 8 feet from the leading edge of the board, at 10 feet per 
 second, is estimated at 0*263 pound per square foot; the mean 
 resistance is 0'278 pound per square foot. 
 
 
 Length of planes 
 
 
 2 feet 
 
 8 feet 
 
 20 feet 
 
 50 feet 
 
 Surface 
 covered with 
 
 n 
 
 f. 
 
 /i 
 
 n 
 
 /o 
 
 /i 
 
 n 
 
 /o 
 
 /, 
 
 n 
 
 /o 
 
 /i 
 
 Varnish 
 
 2-0 
 
 0-41 
 
 0-390 
 
 1-85 
 
 0-325 
 
 0-264 
 
 1-85 
 
 0-278 
 
 0-240 
 
 1-83 
 
 0-250 
 
 0-226 
 
 Tinfoil 
 
 2-16 
 
 0-30 
 
 0-295 
 
 1-99 
 
 0-278 
 
 0-263 
 
 1-90 
 
 0-262 
 
 0-244 
 
 1-83 
 
 0-246 
 
 0-232 
 
 Calico 
 
 1-93 
 
 0-87 
 
 0-725 
 
 1-92 
 
 0-626 
 
 0-504 
 
 1-89 
 
 0-531 
 
 0-447 
 
 1-87 
 
 0-474 
 
 0-423 
 
 Fine sand 
 
 2-0 
 
 0-81 
 
 0-690 
 
 2-0 
 
 0-583 
 
 0-450 
 
 2-0 
 
 0-480 
 
 0-384 
 
 2-06 
 
 0-405 
 
 0-337 
 
 Medium sand 
 
 2-0 
 
 0-90 
 
 0-730 
 
 2-0 
 
 0-625 
 
 0-488 
 
 2-0 
 
 0-534 
 
 0-465 
 
 2-00 
 
 0-488 
 
 0-456 
 
 Coarse sand 
 
 2-0 
 
 1-10 
 
 0-880 
 
 2-0 
 
 0-714 
 
 0-520 
 
 2-0 
 
 0-588 
 
 0-490 
 
 
 
 
 The diminution of the resistance per unit area, with the length, 
 is principally due to the relative velocity of the water and the 
 board not being constant throughout the whole length. 
 
 As the board moves through the water the frictional resistance 
 of the first foot length, say, of the board, imparts momentum to 
 the water in contact with it, and the water is given a velocity in 
 the direction of motion of the board. The second foot length will 
 therefore be rubbing against water having a velocity in its own 
 direction, and the frictional resistance will be less than for the 
 first foot. The momentum imparted to the water up to a certain 
 point, is accumulative, and the total resistance does not therefore 
 increase proportionally with the length of the board. 
 
510 
 
 HYDRAULICS 
 
 282. Stream line theory of the resistance offered to the 
 motion of bodies in water. 
 
 Resistance of ships. In considering the motion of water along 
 pipes and channels of uniform section, the water has been assumed 
 to move in " stream lines," which have a relative motion to the 
 sides of the pipe or channel and to each other, and the resistance 
 to the motion of the water has been considered as due to the 
 friction between the consecutive stream lines, and between the 
 water and the surface of the channel, these frictional resistances 
 above certain speeds being such as to cause rotational motions in 
 the mass of the water. 
 
 
 Fig. 363. 
 
 Fig. 364. 
 
 It has also been shown that at any sudden enlargement of a 
 stream, energy is lost due to eddy motions, and if bodies, such 
 as are shown in Figs. 110 and 111, be placed in the pipe, there is 
 a pressure acting on the body in the direction of motion of the 
 water. The origin of the resistance of ships is best realised by 
 the "stream line" theory, in which it is assumed that relative to 
 the ship the water is moving in stream lines as shown in Figs. 
 363, 364, consecutive stream lines also having relative motion. 
 
RESISTANCE TO THE MOTION OF BODIES IN WATER 511 
 
 According to this theory the resistance is divided into three 
 parts. 
 
 (1) Frictional resistance due to the relative motions of con- 
 secutive stream lines, and of the stream lines and the surface 
 of the ship. 
 
 (2) Eddy motion resistances due to the dissipation of the 
 energy of the stream lines, all of which are not gradually brought 
 to rest. 
 
 (3) Wave making resistances due to wave motions set up at 
 the surface of the water by the ship, the energy of the waves 
 being dissipated in the surrounding water. 
 
 According to the late Mr Froude, the greater proportion of 
 the resistance is due to friction, and especially is this so in long 
 ships, with fine lines, that is the cross section varies very gradually 
 from the bow towards midships, and again from the midships 
 towards the stern. At speeds less than 8 knots, Mr Froude has 
 shown that the frictional resistance of ships, the full speed of 
 which is about 13 knots, is nearly 90 per cent, of the whole 
 resistance, and at full speed it is not much less than 60 per cent. 
 He has further shown that it is practically the same as that 
 resisting the motion of a thin rectangle, the length and area of 
 the two sides of which are equal to the length and immersed 
 area respectively of the ship, and the surface of which has the 
 same degree of roughness as that of the ship. 
 
 If A is the area of the immersed surface, / the coefficient of 
 friction, which depends not only upon the roughness but also 
 upon the length, V the velocity of the ship in feet per second, the 
 resistance due to friction is 
 
 r,=/.A.V", 
 
 the value of the index n approximating to 2. 
 
 The eddy resistance depends upon the bluntness of the stern of 
 the boat, and can be reduced to a minimum by diminishing the 
 section of the ship gradually, as the stern is approached, and by 
 avoiding a thick stern and stern post. 
 
 As an extreme case consider a ship of the section shown in 
 Fig. 364, and suppose the stream lines to be as shown in the 
 figure. At the stern of the boat a sudden enlargement of the 
 stream lines takes place, and the kinetic energy, which has been 
 given to the stream lines by the ship, is dissipated. The case is 
 analogous to that of the cylinder, Fig. Ill, p. 169. Due to the 
 loss of energy, or head, there is a resultant pressure acting upon 
 the ship in the direction of flow of the stream lines, and con- 
 sequently opposing its motion. 
 
512 HYDRAULICS 
 
 If the ship has fine lines towards the stern, as in Fig. 363, 
 the velocities of the stream lines are diminished gradually and the 
 loss of energy by eddy motions becomes very small. In actual 
 ships it is probably not more than 8 per cent, of the whole 
 resistance. 
 
 The wave making resistance depends upon the length and the 
 form of the ship, and especially upon the length of the "entrance" 
 and " run." By the " entrance " is meant the front part of the 
 ship, which gradually increases in section* until the middle body, 
 which is of uniform section, is reached, and by the "run," the 
 hinder part of the ship, which diminishes in section from the 
 middle body to the stern post. 
 
 Beyond a certain speed, called the critical speed, the rate of 
 increase in wave making resistance is very much greater than 
 the rate of increase of speed. Mr Froude found that for the 
 S.S. "Merkara" the wave making resistance at 13 knots, the 
 normal speed of the ship, was 17 per cent, of the whole, but at 19 
 knots it was 60 per cent. The critical speed was about 18 knots. 
 
 An approximate formula for the critical speed Y in knots is 
 
 L being the length of entrance, and LI the length of the run in 
 feet. 
 
 The mode of the formation by the ship of waves can be partly 
 realised as follows. 
 
 Suppose the ship to be moving in smooth water, and the stream 
 lines to be passing the ship as in Fig. 363. As the bow of the 
 boat strikes the dead water in front there is an increase in 
 pressure, and in the horizontal plane SS the pressure will be 
 greater at the bow than at some distance in front of it, and 
 consequently the water at the bow is elevated above the normal 
 surface. 
 
 Now let AA, BB, and CO be three sections of the ship and the 
 stream lines. 
 
 Near the midship section CO the stream lines will be more 
 closely packed together, and the velocity of flow will be greater, 
 therefore, than at AA or BB. Assuming there is no loss of energy 
 in a stream line between AA and BB and applying Bernoulli's 
 theorem to any stream line, 
 
 PA + ^A 2 = PC + ^ = B + V 
 w 2g w 2g w 2g> 
 
 * See Sir W. White's Naval Architecture, Transactions of Naval Architects, 
 1877 and 1881. 
 
RESISTANCE TO THE MOTION OF BODIES IN WATER 513 
 and since V A and v^ are less than V G , 
 
 ^ and ? are greater than ^. 
 w w w 
 
 The surface of the water at A A and BB is therefore higher 
 than at CC and it takes the form shown in Fig. 363. 
 
 Two sets of waves are thus formed, one by the advance of the 
 bow and the other by the stream lines at the stern, and these 
 wave motions are transmitted to the surrounding water, where 
 their energy is dissipated. This energy, as well as that lost in 
 eddy motions, must of necessity have been given to the water by 
 the ship, and a corresponding amount of work has to be done by 
 the ship's propeller. The propelling force required to do work 
 equal to the loss of energy by eddy motions is the eddy resist- 
 ance, and the force required to do work equal to the energy of 
 the waves set up by the ship is the wave resistance. 
 
 To reduce the wave resistance to a minimum the ship should 
 be made very long, and should have no parallel body, or the 
 entire length of the ship should be devoted to the entrance and 
 run. On the other hand for the frictional resistance to be small, 
 the area of immersion must be small, so that in any attempt 
 to design a ship the resistance of which shall be as small as 
 possible, two conflicting conditions have to be met, and, neglecting 
 the eddy resistances, the problem resolves itself into making the 
 sum of the frictional and wave resistances a minimum. 
 
 Total resistance. If K, is the total resistance in pounds, r f the 
 frictional resistance, r e the eddy resistance, and r w the wave 
 resistance, 
 
 The frictional resistance r f can easily be determined when the 
 nature of the surface is known. For painted steel ships / is 
 practically the same as for the varnished boards, and at 10 feet 
 per second the frictional resistance is therefore about Ib. per 
 square foot, and at 20 feet per second 1 Ib. per square foot. The 
 only satisfactory way to determine r e and r w for any ship is to 
 make experiments upon a model, from which, by the principle of 
 similarity, the corresponding resistances of the ship are deduced. 
 The horse-power required to drive the ship at a velocity of V feet 
 per second is 
 
 TTP- RV 
 
 = ' 
 
 To determine the total resistance of the model the apparatus 
 shown in Fig. 362 is used in the same way as in determining the 
 frictional resistance of thin boards. 
 
 L. H. 33 
 
514 HYDRAULICS 
 
 283. Determination of the resistance of a ship from the 
 resistance of a model of the ship. 
 
 To obtain the resistance of the ship from the experimental 
 resistance of the model the principle of similarity, as stated by 
 Mr Froude, is used. Let the linear dimensions of the ship be D 
 times those of the model. 
 
 Corresponding speeds. According to Mr Fronde's theory, for 
 any speed V of the model, the speed of the ship at which its 
 resistance must be compared with that of the model, or the 
 corresponding speed Y s of the ship, is 
 
 Corresponding resistances. If R m is the resistance of the model 
 at the velocity Y m , and it be assumed that the coefficients of 
 friction for the ship and the model are the same, the resistance R s 
 of the ship at the corresponding speed V s is 
 
 As an example, suppose a model one-sixteenth of the size 
 of the ship ; the corresponding speed of the ship will be four times 
 the speed of the model, and the resistance of the ship at corre- 
 sponding speeds will be 16 3 or 4096 times the resistance of the 
 model. 
 
 Correction for the difference of the coefficients of friction for the 
 model and ship. The material of which the immersed surface of 
 the model is made is not generally the same as that of the ship, 
 and consequently R 8 must be corrected to make allowance for the 
 difference of roughness of the surfaces. In addition the ship is 
 very much longer than the model, and the coefficient of friction, 
 even if the surfaces were of the same degree of roughness, would 
 therefore be less than for the model. 
 
 Let A m be the immersed surface of the model and A s of 
 the ship. 
 
 Let f m be the coefficient of friction for the model and f s for the 
 ship, the values being made to depend not only upon the roughness 
 but also upon the length. If the resistance is assumed to vary as 
 V 2 , the frictional resistance of the model at the velocity V m is 
 
 r f A V 2 
 
 ' m J m-E*-m, " m j 
 
 and for the ship at the corresponding speed V s the frictional 
 resistance is 
 
 But A s -A m D 2 
 
RESISTANCE TO THE MOTION OF BODIES IN WATER 
 
 and, therefore, r = /A m V m 2 D 3 
 
 = rJD 3 . 
 
 Jm 
 
 Then the resistance of the ship is 
 
 515 
 
 Determination of the curve of resistance of the ship from the 
 curve of resistance of the model. From the experiments on the 
 model a curve having resistances as ordinates and velocities as 
 abscissae is drawn as in Fig. 365. If now the coefficients of 
 friction for the ship and the model are the same, this curve, by 
 an alteration of the scales, becomes a curve of resistance for the 
 ship. 
 
 For example, in the figure the dimensions of the ship are 
 supposed to be sixteen times those of the model. The scale of 
 velocities for the ship is shown on CD, corresponding velocities 
 being four times as great as the velocity of the model, and the 
 scale of resistances for the ship is shown at EH, corresponding 
 resistances being 4096 times the resistance of the model. 
 
 H 
 
 JE 
 
 C D 
 
 Fig. 365. 
 
 Mr Froude's method of correcting the curve for the difference of 
 the coefficients of friction for the ship and the model. From the 
 formula 
 
 r m = f m A.mVm n , 
 
 332 
 
516 HYDRAULICS 
 
 the frictional resistance of .the model for several values of V m 
 is calculated, and the curve FF plotted on the same scale as used 
 for the curve RE,. The wave and eddy making resistance at any 
 velocity is the ordinate between FF and RE. At velocities of 
 200 feet per second for the model and 800 feet per second for 
 the ship, for example, the wave and eddy making resistance is fee, 
 measured on the scale BGr for the model and on the scale EH for 
 the ship. 
 
 The frictional resistance of the ship is now calculated from the 
 formula r s =f 8 AsV 8 n , and ordinates are set down from the curve 
 FF, equal to r s , to the scale for ship resistance. A third curve is 
 thus obtained, and at any velocity the ordinate between this curve 
 and EE is the resistance of the ship at that velocity. For example, 
 when the ship has a velocity of 800 feet per second the resistance 
 is ac, measured on the scale EH. 
 
 EXAMPLES. 
 
 (1) Taking skin friction to be 0'4 Ib. per square foot at 10 feet per 
 second, find the skin resistance of a ship of 12,000 square feet immersed 
 surface at 15 knots (1 knot = T69 feet per second). Also find the horse-power 
 to drive the ship against this resistance. 
 
 (2) If the skin friction of a ship is 0*5 of a pound per square foot of 
 immersed surface at a speed of 6 knots, what horse-power will probably 
 be required to obtain a speed of 14 knots, if the immersed surface is 18,000 
 square feet ? You may assume the maximum speed for which the ship is 
 designed is 17 knots. 
 
 (3) The resistance of a vessel is deduced from that of a model j^th the 
 linear size. The wetted surface of the model is 29*4 square feet, the skin 
 friction per square foot, in fresh water, at 10 feet per second is 0'3 Ib., and 
 the index of velocity is 1'94. The skin friction of the vessel in salt water 
 is 60 Ibs. per 100 square feet at 10 knots, and the index of velocity is T83. 
 The total resistance of the model in fresh water at 200 feet per minute is 
 1-46 Ibs. Estimate the total resistance of the vessel in salt water at the 
 speed corresponding to 200 feet per minute in the model. Lond. Un. 1906. 
 
 (4) How from model experiments may the resistance of a ship be 
 inferred? Point out what corrections have to be made. At a speed of 
 300 feet per minute in fresh water, a model 10 feet in length with a wet 
 skin of 24 square feet has a total resistance of 2'39 Ibs., 2 Ibs. being due to 
 skin resistance, and -39 Ib. to wave-making. What will be the total resist- 
 ance at the corresponding speed in salt water of a ship 25 times the linear 
 dimensions of the model, having given that the surface friction per square 
 foot of the ship at that speed is 1-3 Ibs. ? Lond. Un. 1906. 
 
CHAPTER XIII. 
 
 STREAM LINE MOTION. 
 
 284. Hele Shaw's experiments on the flow of thin 
 sheets of water. 
 
 Professor Hele Shaw* has very beautifully shown, on a small 
 scale, the form of the stream lines in moving masses of water 
 under varying circumstances, and has exhibited the change from 
 stream line to sinuous, or rotational flow, by experiments on the 
 flow of water at varying velocities between two parallel glass 
 plates. In some of the experiments obstacles of various forms 
 were placed between the plates, past which the water had to flow, 
 and in others, channels of various sections were formed through 
 which the water was made to flow. The condition of the water 
 as it flowed between the plates was made visible by mixing with 
 it a certain quantity of air, or else by allowing thin streams of 
 coloured water to flow between the plates along with the other 
 water. When the velocity of flow was kept sufficiently low, 
 whatever the form of the obstacle in the path of the water, or 
 the form of the channel along which it flowed, the water persisted 
 in stream line flow. When the channel between the plates was 
 made to enlarge suddenly, as in Fig. 58, or to pass through an 
 orifice, as in Fig. 59, and as long as the flow was in stream lines, 
 no eddy motions were produced and there were no indications 
 of loss of head. When the velocity was sufficiently high for the 
 flow to become sinuous, the eddy motions were very marked. 
 When the motion was sinuous and the water was made to flow 
 past obstacles similar to those indicated in Figs. 110 and 111, the 
 water immediately in contact with the down-stream face was 
 shown to be at rest. Similarly the water in contact with the 
 annular ring surrounding a sudden enlargement appeared to be 
 at rest and the assumption made in section 51 was thus justified. 
 
 * Proceedings of Naval Architects, 1897 and 1898. Engineer, Aug. 1897 and 
 May 1898. 
 
518 HYDRAULICS 
 
 When the flow was along channels and sinuous, the sinuously 
 moving water appeared to be separated from the sides of the 
 channel by a thin film of water, which Professor Hele Shaw 
 suggested was moving in stream lines, the velocity of which in 
 the film diminish as the surface of the channel is approached. 
 The experiments also indicated that a similar film surrounded 
 obstacles of ship-like and other forms placed in flowing water, 
 and it was inferred by Professor Hele Shaw that, surrounding 
 a ship as it moves through still water, there is a thin film moving 
 in stream lines relatively to the ship, the shearing forces between 
 which and the surrounding water set up eddy motions which 
 account for the skin friction of the ship. 
 
 285. Curved stream line motion. 
 
 Let a mass of fluid be moving in curved stream lines, and let 
 AB, Fig. 366, be any one of the stream lines. 
 
 At any point c let the radius of curvature of the stream line 
 be r and let be the centre of curvature. 
 
 Consider the equilibrium of an element dbde surrounding the 
 point c. 
 
 Let W be the weight of this element. 
 
 p be the pressure per unit area on the face bd. 
 
 p + dploe the pressure per unit area on the face ae. 
 
 be the inclination of the tangent to the stream line at c 
 
 to the horizontal. 
 
 a be the area of each of the faces bd and ae. 
 v be the velocity of the stream line at c. 
 dr be the thickness ab of the stream line. 
 
 If then the stream line is in a vertical plane the forces acting 
 on the element are 
 
 (1) W due to gravity, 
 
 -TTTT- 2 
 
 (2) the centrifugal force acting along the radius away 
 from the centre, and 
 
 (3) the pressure adp acting along the radius towards the 
 centre of curvature 0. 
 
 Resolving along the radius through c, 
 
 adp + W cos = 0, 
 
 gr 
 
 or since W = wadr, 
 
 dp wv 2 
 dr = ^~ 
 If the stream line is horizontal, as in the case of water flowing 
 
STREAM LINE MOTION 
 
 519 
 
 round the bend of a river, Oc is horizontal and the component of 
 W along Oc is zero. 
 
 Then f= V - ...(2). 
 
 dr g r 
 
 Integrating between the limits R and R a the difference of 
 pressure on any horizontal plane at the radii R and R! is 
 
 W [ R ' V* , 
 
 Pi p = dr 
 
 g J R r 
 
 (3), 
 
 which can be integrated when v can be written as a function of r. 
 Now for any horizontal stream line, applying Bernoulli's 
 equation, 
 
 
 . 
 - is constant, 
 
 or 
 
 Differentiating 
 
 !_ dp vdv _ dK 
 
 w dr gdr dr 
 
 Fig. 366. 
 
 Fig. 367. 
 
 Free vortex. An important case arises when H is constant for 
 all the stream lines, as when water flows round a river bend, or as 
 in Thomson's vortex chamber. 
 
 Then 
 
 1 dp_ -vdv 
 w dr~ gdr 
 dp 
 
 .(5). 
 
 Substituting the value of -gf from (5) in (2) 
 
 wv dv _w tf 
 g dr ~ g ' r ' 
 
 from which rdv + vdr = 0, 
 
 and therefore by integration 
 
 vr = constant = C 
 
520 HYDRAULICS 
 
 Equation (3) now becomes 
 
 gh-p = C 2 /*' dr 
 w g JR r s 
 
 _ 
 
 -20VK 2 
 
 Forced vortex. If, as in the turbine wheel and centrifugal 
 pump, the angular velocities of all the stream lines are the same, 
 then in equation (3) 
 
 and 
 
 w 2 
 
 = v Rl ~ 
 
 Scouring of the banks of a river at the bends. When water 
 runs round a bend in a river the stream lines are practically 
 concentric circles, and since at a little distance from the bend the 
 surface of the water is horizontal, the head H on any horizontal 
 in the bend must be constant, and the stream lines form a free 
 vortex. The velocity of the outer stream lines is therefore less 
 than the inner, while the pressure head increases as the outer 
 bank is approached, and the water is consequently heaped up 
 towards the outer bank. The velocity being greater at the inner 
 bank it might be expected that it will be scoured to a greater 
 extent than the outer. Experience shows that the opposite effect 
 takes place. Near the bed of the river the stream lines have a 
 less velocity (see page 209) than in the mass of the fluid, and, as 
 Lord Kelvin has pointed out, the rate of increase of pressure near 
 the bed of the stream, due to the centrifugal forces, will be less 
 than near the surface. The pressure head near the bed of the 
 stream, due to the centrifugal forces, is thus less than near the 
 surface, and this pressure head is consequently unable to balance 
 the pressure head due to the heaping of the surface water, and 
 cross-currents are set up, as indicated in Fig. 367, which cause 
 scouring of the outer bank and deposition at the inner bank. 
 
ANSWERS TO EXAMPLES. 
 
 Chapter I. 
 
 (1) 3900 Ibs. 9372 Ibs. (2) 784 Ibs. (3) 78'6 tons. 
 
 (4) 5880 Ibs. (5) 17'1 feet. (6) 19800 Ibs. 
 
 (7) P = 665,600 Ibs. X = 12'5 ft. (8) '91 foot. (9) '089 in. 
 
 (10) 15-95 Ibs. per sq. ft. (11) 5400 Ibs. (12) 87040 Ibs. 
 
 Chapter II. 
 
 (1) 35,000 c. ft. (3) 2-98 ft. 
 
 (4) Depth of C. of B. = 21-95 ft. BM = 14-48 ft. (5) 19'1 ft. 6'9 ft. 
 
 (6) Less than 13'8 ft. from the bottom. (7) 1'57 ft. (8) 2'8 ins. 
 
 Chapter III. 
 
 (1) -945. (2) 14-6 ft. per sec. 18'3 c. ft. per sec. (3) 25 '01 ft. 
 
 (4) 115 ft. (5) 53-3 ft. per sec. (6) 63 c. ft. per sec. 
 
 (7) 44928 ft. Ibs. T36 H.P. 8'84 ft. (8) 86'2 ft. 11-4 ft. per sec. 
 (9) 1048 gallons. 
 
 Chapter IV, page 78. 
 
 (1) 80-25. (2) 3906. (3) 37'636. (4) 5 ins. diam. 
 
 (5) 3-567 ins. (6) -763. (7) 86 ft. per sec. 115ft. 
 
 (8) -806. (9) -895. (10) -058. (11) 144-3 ft. per sec. 
 
 (12) 2-94 ins. (13) -60. (14) 572 gallons. (15) 22464 Ibs. 
 (16) -6206. (17) 5-53 c. ft. (18) '755. (19) 102 c. ft. 
 
 (20) -875 ft. 136 Ibs. per sq. foot. 545 ft. Ibs. 
 
 (21) 10-5 ins. 29-85 ins. (22) -683 ft. (23) 4-52 minutes. 
 (24) 17-25 minutes. (25) -629 sq. ft. (26) 1'42 hours. 
 
 Chapter IV, page 110. 
 
 (1) 13,170 c. ft. (2) 4-15 ft. 
 
 (8) 69-9 c. ft. per sec. 129-8 c. ft. per sec. (4) 2-635. 
 (5) 13-28. (7) 43-3 c. ft. per see. (8) 1-675 ft. 
 
 (9) 89-2 ft. (10) 2-22 ft. (11) 5 -52 ft. (12) 23,500 c. ft. 
 
 (13) 24,250 c. ft. (14) 105 minutes. (15) 284 H.P. 
 
522 ANSWERS TO EXAMPLES 
 
 Chapter V. 
 
 (1) 27'8 ft. (2) 142 ft. (4) '65. (5) 2'388 ft. 
 
 (6) 10-75. 1-4 ft. '33ft. -782ft. -0961ft. 
 
 (8) -61 c. ft. 28-54 ft. 25-8 ft. 9 ft. (9) 26 per cent. 
 
 (10) 1-97. 21ft. 30ft. 26ft. 24ft. 15ft. (11) 3'64 c. ft. 
 
 (12) 3-08 c. ft. (13) -574 ft. -267 ft. 7'72 ft. (14) 2'1 ft. 
 
 (15) 1-86 c. ft. per sec. (16) F = '0318lbs. /= -005368. 
 
 (17) 1-023. (18) -704. (19) 2-9 ft. per sec. 
 
 (20) 4-4 c. ft. per sec. (21) If pipe is clean 46 ft. 
 
 (22) 23 ft. 736 ft. (23) Dirty cast-iron 6'1 feet per mile. 
 
 (24) 8-18 feet. (26) 1 foot. 
 
 (27) ~TrT~ ' F = friction per 'unit area at unit velocity. 
 
 (28) 108 H. P. (29) 1480 Ibs. 1 -03 ins. (30) -002825. 
 
 (31) k= -004286. w=l-84. (32) (a) 940ft. (b) 2871 H. P. (33) -0458, 
 
 (34) If d=9", v=5 ft. per sec., and /= -0056, 7i = 102 and H = 182. 
 
 (35) 1487 xlO 4 . Yes. (36) 58-15 ft. (37) 1 hour 48 min. 
 (38) 46,250 gallons. Increase 17 per cent. (39) 295'7 feet. 
 (40) 6 pipes. 480 Ibs. per sq. inch. 
 
 (42) Velocities 6'18, 5-08, 8-15 ft. per sec. Quantity to B=60 c. ft. per min. 
 Quantity to C=66'6 c. ft. per min. (45) -468 c. ft. per sec. 
 
 (46) Using formula for old cast-iron pipes from page 138, ?; = 3-62 ft. per sec. 
 
 (47) 2-91 ft. (48) d= 3'8 ins. ^ = 3-4 ins. d 2 = 2'9 ins. d 3 = 2'2 ins. 
 (49) Taking C as 120, first approximation to Q is 14-4 c. ft. per sec. 
 
 (51) d=4-13 ins. v = 20'55 ft. per sec. _p = 840 Ibs. per sq. inch. 
 
 (53) 7-069 ft. 3-01 ft. C r =ll-9. C r for tubes = 5 -06. 
 
 (54) Loss of head by friction = -73 ft. 
 
 v 2 
 A head equal to ^- will probably be lost at each bend. 
 
 (56) 43-9 ft. -936 in. 
 
 (57) h = 58'. Taking -005 to be / in formula h = ^? , v = 16'6 ft. per sec. 
 
 (58) ^ = 8-8 ft. per sec. from A to P. v 2 = 4-95 ft. per sec. from B to P. 
 
 V 3 = 13-75 ft. per sec. from P to C. 
 
 Chapter VI. 
 
 (!) 88-5. (2) 1-1 ft. diam. 
 
 (3) Value of ra when discharge is a maximum is 1-357. o> = 17*62. C = 127, 
 
 Q = 75 c. ft. per sec. 
 
 (4) -0136. (5) 16,250 c. ft. per sec. (6) 3ft. 
 
 (7) Bottom width 15 ft. nearly. (8) Bottom width 10 ft. nearly. 
 
 (9) 630 c. ft. per sec. (10) 96,000 c. ft. per sec. 
 
 (11) Depth 7-35 ft. (12) Depth 10'7 ft. 
 
 (13) Bottom width 75 ft. Slope -00052. (17) C = 87'5. 
 
ANSWERS TO EXAMPLES 52S 
 
 Chapter VIII. 
 
 (1) 124-8 Ibs. -456 H. P. (2) 623 Ibs. 
 
 (3) 104 Ibs. 58-7 Ibs. 294 ft. Ibs. (4) 960 Ibs. 
 
 (5) 261 Ibs. 4-7 H. P. (6) 21-8. (7) 57 Ibs. (8) 194 Ibs. 
 
 (9) Impressed velocity = 28-5 ft. per sec. Angle = 57. (10) 131 Ibs. 
 
 (12) -93. -678. -63. (13) 19'2. 
 
 (14) Vel. into tank = 34-8 ft. per sec. Vel. through the orifice = 41-5 ft. per 
 
 sec. Wt. lifted = 10-3 tons. Increased resistance =2330 Ibs. 
 
 (15) 125 Ibs. 8-4 ft. per sec. 1-91 H.P. 
 
 (16) Work done, 575, 970, 1150, 1940 ft. Ibs. Efficiencies fy, '50, f , 1. 
 
 (17) 1420 H.P. (18) -9375. (19) 32 H.P. 
 (20) 3666 Ibs. 161 H.P. 62 per cent. 
 
 Chapter IX. 
 
 (1) 105 H.P. (2) 0=29. V r = 7 ft. per sec. 
 
 (3) 14-3 per min. 11 from the top of wheel. $ = 70. 
 
 (4) 1-17 c. ft. (5) 4-14 ft. (8) 32 12'. 
 (9) 10-25 ft. per sec. 1-7 ft. 5'3 ft. per sec. 11 to radius. 
 
 (12) u = 24-7 ft. per sec. (13) = 47 30'. a = 27 20'. 
 
 (14) 79 15'. 19 26'. -53. 
 
 (15) 35-6 ft. per sec. 6 24'. 23| ins. llf ins. 12 39'. 16|ins. 32|ins. 
 
 (16) 99 per cent. 0=73, a = 18. $ = 120, a = 18. 
 
 (18) = 153 23'. H = 77-64 ft. H.P. = 141-16. Pressure head = 48'53 ft. 
 
 (19) d = 1-22 ft. D = 2-14 ft. Angles 12 45', 125 22', 16 4'. 
 
 (20) = 134 53', 6 = 16 25', a = 9 10'. H. p. = 2760. 
 
 (21) 616. Heads by gauge, - 14, 35'6, 81. U = 51'5 ft. per sec. 
 
 (22) = 153 53', a = 25. H. p. = 29'3. Eff . = "957. 
 
 (23) Blade angle 13 30'. Vane angle 30 25'. 3'92 ft. Ibs. per Ib. 
 
 (24) At 2' 6" radius, 6 = 10, = 23 45', a = 16 24'. At 3' 3" radius, 6 = 12 11' 
 
 = 78 47', a = 12 45'. At 4' radius, 6 = 15 46', = 152 11', a = 10 21 . 
 
 (25) 79 30'. 21 40'. 41 30'. 
 
 (26) 53 40'. 36. 24. 86'8 per cent. 87 per cent. 
 
 (27) 12 45'. 62 15'. 31 45'. 
 
 (28) v = 45-35. U=77. V r =44. v r = 36. Ux = 23. e = 73'75 per cent. 
 
 (29) -36ft. 40 to radius. (30) About 22 ft. 
 (31) H.P. =80-8. Eff. = 92-5 per cent. 
 
 Chapter X. 
 
 (1) 47-4 H.P. (2) 25. 53-1 ft. per sec. 94ft. 50ft. 
 
 (3) 55 per cent. (4) 52*5 per cent. 
 
 (5) ^ = 106 ft. ^L 2 =51ft. ^1 = 55 ft. 
 
 g 2g w 
 
 (6) 11 36'. 105ft. 47-4 ft. 
 
 (7) 60 per cent. 151 H.P. 197 revs, per min. 
 
 (8) 700 revs, per min. -81 in. Radial velocity 14'2 ft. per sec. 
 (12) 15-6 ft. Ibs. per Ib. 3 -05 ft. 14 ft. per sec. 
 
524 ANSWERS TO EXAMPLES 
 
 (15) v = 23-64 ft. per sec. V = 11'3. 
 
 (16) d=9ins. D = 19 ins. Kevs. per min. 472 or higher. 
 
 (17) 15 H. P. 9-6 ins. diam. (18) 5'5 ft. 
 
 (19) Vels. 1-23 and 2-41 ft. per sec. Max. accel. 2-32 and 4-55 ft. per sec. 
 
 per sec. 
 
 (20) 393 ft. Ibs. Mean friction head = '0268, therefore work due to friction 
 
 is very small. 
 
 (21) 4-61 H. P. 11-91 c. ft. per min. (22) -338. 
 
 (23) p ^ 1 . Acceleration is zero when 0=j(ra-{-2), m being any 
 
 integer. 
 
 (27) Separation in second case. 
 
 (29) 67'6 and 66-1 Ibs. per sq. inch respectively. H.p. = 3'14. 
 (31) 7'93 ft. 25'3 ft. 41 '93 ft. (32) '643. (33) '6. 
 
 (34) Separation in the sloping pipe. 
 
 Chapter XI. 
 
 (1) 3150 Ibs. (2) 3-38 H.P. hours. (5) 4'7 ins. and 9'7 ins. 
 
 (6) 3-338 tons. (7) 175 Ibs. per sq. inch. 
 
 (8) 2-8 ft. per sec. (9) 4-2 minutes. 
 
 Chapter XII. 
 
 (1) 30,890 Ibs. 1425 H. P. (2) 3500 H. P. 
 (3) 4575 Ibs. (4) 25,650 Ibs. 
 
INDEX. 
 
 [All numbers refer to pages.] 
 
 Absolute velocity 262 
 
 Acceleration in pumps, effect of (see 
 
 Reciprocating pump) 
 Accumulators 
 
 air 491 
 
 differential 489 
 
 hydraulic 486 
 Air gauge, inverted 9 
 Air vessels on pumps 451, 455 
 Angular momentum 273 
 Angular momentum, rate of change of 
 
 equal to a couple 274 
 Appold centrifugal pump 415 
 Aqueducts 1, 189, 195 
 
 sections of 216 
 Archimedes, principle of 22 
 Armstrong double power hydraulic crane 
 
 497 
 Atmospheric pressure 8 
 
 Bacon 1 
 
 Barnes and Coker 129 
 
 Barometer 7 
 
 Bazin's experiments on 
 
 calibration of Pitot tube 245 
 distribution of pressure in the plane 
 
 of an orifice 59 
 distribution of velocity in the cross 
 
 section of a channel 208 
 distribution of velocity in the cross 
 
 section of a pipe 144 
 distribution of velocity in the plane of 
 
 an orifice 59, 244 
 flow in channels 182 
 flow over dams 102 
 flow over weirs 89 
 flow through orifices 56 
 form of the jet from orifices 63 
 Bazin's formulae for 
 channels 182, 185 
 orifices, sharp-edged 57, 61 
 velocity at any depth in a vertical 
 
 section of a channel 212 
 velocity at any point in the cross 
 
 section of a pipe 144 
 weir, flat crested 99 
 weir, sharp-crested 97-99 
 weir, sill of small thickness 99 
 
 Bends, loss of head due to 140 
 Bernouilli's theorem 39 
 
 applied to centrifugal pumps 413, 
 423, 437, 439 
 
 applied to turbines 334, 349 
 
 examples on 48 
 
 experimental illustrations of 41 
 
 extension of 48 
 Borda's mouthpiece 72 
 Boussinesq's theory for discharge of a 
 
 weir 104 
 
 Boyden diffuser 314 
 Brotherhood hydraulic engine 501 
 Buoyancy of floating bodies 21 
 
 centre of 23 
 
 Canal boats, steering of 47 
 
 Capstan, hydraulic 501 
 
 Centre of buoyancy 23 
 
 Centre of pressure 13 
 
 Centrifugal force, effect of in discharge 
 
 from water-wheel 286 
 Centrifugal head 
 
 in centrifugal pumps 405, 408, 409, 
 
 419, 421 
 
 in reaction turbines 303, 334 
 Centrifugal pumps, see Pumps 
 Channels 
 
 circular, depth of flow for maximum 
 
 discharge 221 
 circular, depth of flow for maximum 
 
 velocity 220 
 
 coefficients for, in formulae of 
 Bazin 186, 187 
 Darcy and Bazin 183 
 Ganguillet and Kutter 184 
 coefficients for, in logarithmic for- 
 mulae 200-208 
 coefficients, variation of 190 
 curves of velocity and discharge for 222 
 dimensions of, for given flow deter- 
 mined by approximation 225-227 
 diameter of, for given maximum dis- 
 charge 224 
 
 distribution of velocity in cross sec- 
 tion of 208 
 
 earth, of trapezoidal form 219 
 erosion of earth 216 
 
526 
 
 INDEX 
 
 Channels (cont.) 
 
 examples on 223-231 
 
 flow in 178 
 
 flow in, of given section and slope 
 
 223 
 
 forms of 
 best 218 
 variety of 178 
 formula for flow in 
 applications of 223 
 approximate for earth 201, 207 
 Aubisson's 233 
 Bazin's 185 
 Bazin's method of determining the 
 
 constants in 187 
 Chezy 180 
 
 Darcy and Bazin's 182 
 Eytelwein's 181, 232 
 Ganguillet and Kutter s 182, 184 
 historical development of 231 
 logarithmic 192, 198-200 
 Prony 181, 232 
 hydraulic mean depth of 179 
 lined with 
 
 ashlar 183, 184, 186, 187, 200, 206 
 boards 183, 184, 187, 195, 201 
 brick 183, 184, 187, 193, 195, 197, 
 
 203 
 cement 183, 184, 186, 187, 193, 
 
 202 
 
 earth 183, 184, 186, 187, 201, 207 
 gravel 183, 184 
 pebbles 184, 186, 187, 206 
 rubble masonry 184, 186, 187, 205 
 logarithmic plottings for 193-198 
 minimum slopes of, for given velocity 
 
 215 
 
 particulars of 195 
 problems 223 (see Problems) 
 sections of 216 
 siphons forming part of 216 
 slope of for minimum cost 227 
 slopes of 213, 215 
 steady motion in 178 
 variation of the coefficient for 190 
 Coefficients 
 for mouthpieces 71, 73, 76 (see 
 
 Mouthpieces) 
 
 for rectangular notches (see Weirs) 
 for triangular notches 85 
 for Venturi meter 46 
 for weirs, 88, 89, 93 (see Weirs) 
 of resistance 67 
 Condenser 6 
 Condition of stability of floating bodies 
 
 24 
 
 Contraction of jet from orifice 53 
 Convergent mouthpiece 73 
 Couple, work done by 274 
 Cranes, hydraulic 494 
 Crank effort diagram 'for three cylinder 
 
 engine 503 
 Critical velocity 129 
 
 Current meters 239 
 
 calibration of 240 
 
 Gurley 238 
 
 Haskell 240 
 
 Curved stream line motion 518 
 Cylindrical mouthpiece 73 
 
 Dams, flow over 101 
 
 Darcy 
 
 experiments on flow in channels 182 
 experiments on flow in pipes 122 
 formula for flow in channels 182 
 formula for flow in pipes 122 
 
 Deacon's waste-water meter 254 
 
 Density 3 
 
 of gasoline 11 
 of kerosine 11 
 of mercury 8 
 of pure water 4, 11 
 
 Depth of centre of pressure 13 
 
 Diagram of pressure on a plane area 
 16 
 
 Diagram of pressure on a vertical circle 
 16 
 
 Diagram of work done in a reciprocating 
 pump 443, 459, 467 
 
 Differential accumulator 489 
 
 Differential gauge 8 
 
 Discharge 
 
 coefficient of, for orifices 60 (see 
 
 Orifices) 
 
 coefficient of, for Venturi meter 46 
 of a channel 178 (see Channels) 
 over weirs 82 (see Weirs) 
 through notches 85 (see Notches) 
 through orifices 50 (see Orifices) 
 through pipes 112 (see Pipes) 
 
 Distribution of velocity on cross section 
 of a channel 208 
 
 Distribution of velocity on cross section 
 of a pipe 143 
 
 Divergent mouthpieces 73 
 
 Dock caisson 181, 192, 216 
 
 Docks, floating 31 
 
 Drowned nappes of weirs 96, 100 
 
 Drowned orifices 65 
 
 Drowned weirs 98 
 
 Earth channels 
 
 approximate formula for 201, 207 
 coefficients for in Bazin's formula 
 
 187 
 coefficients for in Darcy and Bazin's 
 
 formula 183 
 coefficients for in Ganguillet and 
 
 Kutter's formula 184 
 erosion of 216 
 
 Elbows, loss of head due to 140 
 Engines, hydraulic 501 
 Brotherhood 501 
 Hastie 503 
 Bigg 504 
 Erosion of earth channels 216 
 
INDEX 
 
 527 
 
 Examples, solutions to which are given 
 
 in the text 
 Boiler, time of emptying through a 
 
 mouthpiece 78 
 Centrifugal pumps, determination of 
 
 pressure head at inlet and outlet 
 
 410 
 Centrifugal pumps, dimensions for a 
 
 given discharge 404 
 Centrifugal pumps, series, number 
 
 of wheels for a given lift 435 
 Centrifugal pumps, velocity at which 
 
 delivery starts 412 
 Channels, circular diameter, for a 
 
 given maximum discharge 224 
 Channels, diameter of siphon pipes 
 
 to given same discharge as an 
 
 aqueduct 224 
 Channels, dimensions of a canal for 
 
 a given flow and slope 225, 226, 227 
 Channels, discharge of an earth 
 
 channel 225 
 Channels, flow in, for given section 
 
 and slope 223 
 Cranes 12, 489 
 Floating docks, height of metacentre 
 
 of 34 
 Floating docks, water to be pumped 
 
 from 33 
 
 Head of water 7 
 Hydraulic machinery, capacity of 
 
 accumulator for working a hy- 
 draulic crane 489 
 Hydraulic motor, variation of the 
 
 pressure on the plunger 470 
 Impact on vanes, form of vane for 
 
 water to enter without shock and 
 
 leave in a given direction 271 
 Impact on vanes, pressure on a vane 
 
 when a jet in contact with is turned 
 
 through a given angle 267 
 Impact on vanes, turbine wheel, 
 
 form of vanes on 272 
 Impact on vanes, turbine wheel, 
 
 water leaving the vanes of 269 
 Impact on vanes, work done on a 
 
 vane 271 
 Metacentre, height of, for a floating 
 
 dock 34 
 
 Metacentre. height of, for a ship 26 
 Mouthpiece, discharge through, into 
 
 a condenser 76 
 Mouthpiece, time of emptying a 
 
 boiler by means of 78 
 Mouthpiece, time of emptying a 
 
 reservoir by means of 78 
 Pipes, diameter of, for a given dis- 
 charge 152, 153 
 Pipes, discharge along pipe connecting 
 
 two reservoirs 151, 154 
 Pipes in parallel 154 
 Pipes, pressure at end of a service 
 
 pipe 151 
 
 Examples (cont.) 
 
 Pontoon, dimensions for given dis- 
 placement 29 
 
 Pressure on a flap valve 13 
 
 Pressure on a masonry dam 13 
 
 Pressure on the end of a pontoon 13 
 
 Reciprocating pump fitted with an 
 air vessel 470 
 
 Reciprocating pump, horse-power of, 
 with long delivery pipe 470 
 
 Reciprocating pump, pressure in an 
 air vessel 470 
 
 Reciprocating pump, separation in, 
 diameter of suction pipe for no 469 
 
 Reciprocating pump, separation in 
 the delivery pipe 464 
 
 Reciprocating pump, separation in, 
 number of strokes at which sepa- 
 ration takes place 458 
 
 Reciprocating pump, variation of 
 pressure in, due to inertia forces 
 470 
 
 Reservoirs, time of emptying by weir 
 108 
 
 Reservoirs, time of emptying through 
 orifice 78 
 
 Ship, height of metacentre of 26 
 
 Transmission of fluid pressure 12 
 
 Turbine, design of vanes and de- 
 termination of efficiency of, con- 
 sidering friction 331 
 
 Turbine, design of vanes and de- 
 termination of efficiency of, fric- 
 tion neglected 322 
 
 Turbine, dimensions and form of 
 vanes for given horse-power 341 
 
 Turbine, double compartment parallel 
 flow 349 
 
 Turbine, form of vanes for an out- 
 ward flow 311 
 
 Turbine, hammer blow in a supply 
 pipe 385 
 
 Turbine, velocity of the wheel for a 
 given head 321 
 
 Venturi meter 46 
 
 Water wheel, diameter of breast 
 wheel for given horse-power 290 
 
 Weir, correction of coefficient for 
 velocity of approach 94 
 
 Weir, discharge of 94 
 
 Weir, discharge of by approximation 
 108 
 
 Weir, time of emptying reservoir by 
 means of 110 
 
 Fall of free level 51 
 Fire hose nozzle 73 
 Flap valve, pressure on 18 
 
 centre of 18 
 Floating bodies 
 
 Archimedes, principle of 22 
 
 buoyancy of 21 
 
 centre of buoyancy of 23 
 
528 
 
 INDEX 
 
 Floating bodies (cont.) 
 
 conditions of equilibrium of 21 
 
 containing water, stability of 29 
 
 examples on 34, 516 
 
 metacentre of 24 
 
 resistance to the motion of 507 
 
 small displacements of 24 
 
 stability of equilibrium, condition of 
 
 24 
 
 stability of floating dock 33 
 stability of rectangular pontoon 26 
 stability of vessel containing water 29 
 stability of vessel wholly immersed 
 
 30 
 
 weight of fluid displaced 22 
 Floating docks 31 
 
 stability of 33 
 Floats, double 237 
 rod 239 
 surface 237 
 Flow of water 
 
 definitions relating to 38 
 energy per pound of flowing water 38 
 in open channels 178 (see Channels) 
 over dams 101 (see Dams) 
 over weirs 81 (see Weirs) 
 through notches 80 (see Notches) 
 through orifices 50 (see Orifices) 
 through pipes 112 (see Pipes) 
 Fluids (liquids) 
 at rest 3-19 
 examples on 19 
 compressible 3 
 density of 3 
 
 flow of, through orifices 50 
 incompressible 3 
 in motion 37 
 
 pressure in, is the same in all direc- 
 tions 4 
 
 pressure on an area in 12 
 pressure on a horizontal plane in, is 
 
 constant 5 
 specific gravity of 3 
 steady motion of 37 
 stream line motion in 37, 517 
 transmission of pressure by 11 
 used in U tubes 9 
 viscosity of 2 
 
 Forging press, hydraulic 492 
 Fourneyron turbine 307 
 Friction 
 
 coefficients of, for ships' surfaces 509, 
 
 515 
 effect of, on discharge of centrifugal 
 
 pump 421 
 
 effect of, on velocity of exit from Im- 
 pulse Turbine 373 
 effect of, on velocity of exit from 
 
 Poncelet Wheel 297 
 Froude's experiments on fluid 507 
 in centrifugal pumps 400 
 in channels 180 
 in pipes 113, 118 
 
 Friction (cont.) 
 
 in reciprocating pumps 449 
 in turbines 313, 321, 339, 373 
 
 Ganguillet and Kutter 
 
 coefficients in formula of 125, 184 
 
 experiments of 183 
 
 formula for channels 184 
 
 formula for pipes 124 
 Gasoline, specific gravity of 11 
 Gauges, pressure 
 
 differential 8 
 
 inverted air 9 
 
 inverted oil 10 
 Gauging the flow of water 234 
 
 by an orifice 235 
 
 by a weir 247 
 
 by chemical means 258 
 
 by floats 239 (see Floats) 
 
 by meters 234, 251 (see Meters) 
 
 by Pitot tubes 241 
 
 by weighing 234 
 
 examples on 260 
 
 in open channels 236 
 
 in pipes 251 
 
 Glazed earthenware pipes 186 
 Gurley's current meter 238 
 
 Hammer blow in a long pipe 384 
 Haskell's current meter 240 
 Hastie's engine 503 
 Head 
 
 position 39 
 
 pressure 7, 39 
 
 velocity 39 
 
 High pressure pump 471 
 Historical development of pipe and 
 
 channel formulae 231 
 Hook gauge 248 
 Hydraulic accumulator 486 
 Hydraulic capstan 501 
 Hydraulic crane 494 
 
 double power 495 
 
 valves 497 
 
 Hydraulic differential accumulator 490 
 Hydraulic engines 501 
 
 crank effort diagram for 503 
 Hydraulic forging press 492 
 Hydraulic gradient 115 
 Hydraulic intensifier 491 
 Hydraulic machines 485 
 
 conditions which vanes of, must 
 satisfy 270 
 
 examples on 489, 505 
 
 joints for 485 
 
 maximum efficiency of 295 
 
 packings for 485 
 Hydraulic mean depth 119 
 Hydraulic motors, variations of pressure 
 
 in, due to inertia forces 469 
 Hydraulic ram 474 
 Hydraulic riveter 499 
 Hydraulics, definition of 1 
 
INDEX 
 
 529 
 
 Hydrostatics 4-19 
 
 Impact of water on vanes 261 (see Vanes) 
 Inertia forces in hydraulic motors 469 
 Inertia forces in reciprocating pumps 
 
 44o 
 
 Inertia, moment of 14 
 Inverted air gauge 9 
 Inverted oil gauge 9 
 Intensifies, hydraulic 491 
 
 non-return valves for 492 
 Intensitiers, steam 493 
 Inward flow turbines 275, 318 (see 
 Turbines) 
 
 Joints used in hydraulic work 485 
 
 Kennedy meter 255 
 Kent Venturi meter 253 
 Kerosene, specific gravity of 11 
 
 Leathers for hydraulic packings 486 
 Logarithmic formulae for flow 
 
 in channels 192 
 
 in pipes 125 
 Logarithmic plottings 
 
 for channels 195 
 
 for pipes 127, 133 
 Luthe valve 499 
 
 Masonry dam 17 
 Mercury 
 
 specific gravity of 8 
 
 use of, in barometer 7 
 
 use of, in U tubes 8 
 Metacentre, height of 24 
 Meters 
 
 current 239 
 
 Deacon's waste water 254 
 
 Kennedy 255 
 
 Leinert 234 
 
 Venturi 44, 75, 251 
 Moment of inertia 14 
 
 of water plane of floating body 25 
 
 table of 15 
 
 Motion, second law of 263 
 Mouthpieces 54 
 
 Borda's 72 
 
 coefficients of discharge for 
 Borda's 73 
 conical 73 
 cylindrical 71, 76 
 fire nozzle 73 
 
 coefficients of velocity for 71, 73 
 
 conical 73 
 
 convergent 73 
 
 cylindrical 73 
 
 divergent 73 
 
 examples on 78 
 
 flow through, under constant pressure 
 75 
 
 loss of head at entrance to 70 
 
 time of emptying boiler through 78 
 
 L. H. 
 
 Mouthpieces (cont.) 
 
 time of emptying reservoir through 
 
 78 
 
 Nappe of a weir 81 
 
 adhering 95 
 
 depressed 95 
 
 drowned or wetted 95 
 
 free 95 
 
 instability of the form of 97 
 Newton's second law of motion 263 
 Notation used in connection with vanes, 
 turbines and centrifugal pumps 272 
 Notches 
 
 coefficients for rectangular (see Weirs) 
 
 coefficients for triangular 85 
 
 rectangular 80 (see Weirs) 
 
 triangular 80 
 
 Nozzle at end of a pipe 159 
 Nozzle, fire 74 
 
 Oil pressure gauge, inverted 10 
 
 calibration of 11 
 
 Oil pressure regulator for turbines 377 
 Orifices 
 
 Bazin's coefficients for 57, 61 
 Bazin's experiments on 56 
 coefficients of contraction 52, 56 
 coefficients of discharge 57, 60, 61, 
 
 63 
 
 coefficients of velocity 54, 57 
 contraction complete 53, 57 
 contraction incomplete or suppressed 
 
 53, 63 
 distribution of pressure in plane of 
 
 59 
 
 distribution of velocity in plane of 59 
 drowned 65 
 drowned partially 66 
 examples on 78 
 flow of fluids through 50 
 flow of fluids through, under constant 
 
 pressure 75 
 force acting on a vessel when water 
 
 issues from 277 
 form of jet from 63 
 large rectangular 64 
 partially drowned 66 
 pressure in the plane of 59 
 sharp-edged 52 
 time of emptying a lock or tank by 
 
 76, 77 
 
 Torricelli's theorem 51 
 velocity of approach to 66 
 velocity of approach to, effect on dis- 
 charge from 67 
 
 Packings for hydraulic machines 485 
 Parallel flow turbine 276, 342, 368 
 Parallel flow turbine pump 437 
 Pelton wheel 276, 377, 380 
 Piezometer fittings 139 
 Piezometer tubes 7 
 
 34 
 
530 
 
 INDEX 
 
 Pipes 
 
 bends, loss of head due to 141 
 coefficients 
 
 C in formula v = G*Jmi, 
 
 and 
 
 /-HB 
 
 for cast iron, new and old 120, 
 
 121, 122, 123, 124 
 for steel riveted 121 
 for Darcy's formula 122 
 for logarithmic formulae 
 brass pipes 133, 138 
 cast iron, new and old 125, 137, 
 
 138 
 
 glass 135 
 riveted 137, 138 
 wood 135, 138 
 wrought iron 122, 135, 138 
 n in Ganguillet and Kutter's formula 
 cast iron, new and old 125 
 for glazed earthenware 125 
 for steel riveted 184 
 for wood pipes 125, 184 
 variation of, with service 123 
 connecting three reservoirs 155 
 connecting two reservoirs 149 
 connecting two reservoirs, diameter of 
 
 for given discharge 152 
 critical velocity in 128 
 Darcy's formula for 122 
 determination of the coefficient C, 
 as given in tables by logarithmic 
 plotting 132 
 diameter of, for given discharge 
 
 152 
 
 diameter for minimum cost 158 
 diameter varying 160 
 divided into two branches 154 
 elbows for 141 
 
 empirical formula for head lost in 119 
 empirical formula for velocity of flow 
 
 in 119 
 
 equation of flow in 117 
 examples on flow in 149-162, 170 
 experimental determination of loss of 
 
 head by friction in 116 
 experiments on distribution of velocity 
 
 experiments on flow in, criticism of 
 
 138 
 experiments on loss of head at bends 
 
 142 
 experiments on loss of head in 122 
 
 125, 129, 131, 132, 136 
 experiments on loss of head in, 
 
 criticism of 138 
 flow through 112 
 flow diminishing at uniform rate in 
 
 157 
 formula for 
 
 Chezy 119 
 
 Darcy 122 
 
 Pipes (cont.) 
 
 formula for (cont.) 
 
 logarithmic 125, 131, 133, 137-138 
 
 Reynolds 131 
 
 summary of 148 
 
 velocity at any point in a cross 
 
 section of 143 
 friction in, loss of head by 113 
 
 determination of 116 
 Ganguillet and Kutter's formula for 
 
 124 
 
 gauging the flow in 251 
 hammer blow in 384 
 head lost at entrance of 70, 114 
 head lost by friction in 113 
 head lost by friction in, empirical 
 
 formula for 119 
 head lost by friction in, examples on 
 
 150-162, 170 
 head lost by friction in, logarithmic 
 
 formula for 125, 133 
 head required to give velocity to 
 
 water in the pipe 146 
 head required to give velocity to water 
 
 in the pipe, approximate value 113 
 hydraulic gradient for 115 
 hydraulic mean depth of 118 
 joints for 485 
 law of frictional resistance for, above 
 
 the critical velocity 130 
 law of frictional resistance for, below 
 
 the critical velocity 125 
 limiting diameter of 165 
 logarithmic formula for 125 
 logarithmic formula for, coefficients 
 
 in 138 
 
 logarithmic formula, use of, for prac- 
 tical calculations 136 
 logarithmic plottings for 126 
 nozzle at discharge end of, area of 
 
 when energy of jet is a maximum 
 159 
 
 when momentum of jet is a maxi- 
 mum 159 
 
 piezometer fittings for 139 
 pressure on bends of 166 
 pressure on a cylinder in 169 
 pressure on a plate in 168 
 problems 147 (see Problems) 
 pumping water through long pipe, 
 
 diameter of for minimum cost 158 
 resistance to motion of fluid in 112 
 rising above hydraulic gradient 115 
 short 153 
 siphon 161 
 temperature, effect of, on velocity of 
 
 flow in 131, 140 
 
 transmission of power along, by hy- 
 draulic pressure 162 
 values of C in the formula v = C \lrni 
 
 for 120, 121 
 variation of C in the formula v = 
 
 for 123 
 
INDEX 
 
 531 
 
 Pipes (cont.) 
 
 variation of the discharge of, with 
 
 service 123 
 
 velocity of flow allowable in 162 
 velocity, head required to give velocity 
 
 to water in 146 
 velocity, variation of, in a cross section 
 
 of a pipe 143 
 virtual slope of 115 
 Pitot tube 241 
 
 calibration of 245 
 Poncelet water wheel 294 
 Pontoon, pressure on end of 18 
 Position head 29 
 Press, forging 493 
 Press, hydraulic 493, 498 
 Pressure 
 
 at any point in a fluid 4 
 atmospheric, in feet of water 8 
 gauges 8 
 head 7 
 
 measured in feet of water 7 
 on a horizontal plane in a fluid 5 
 on a plate in a pipe 168 
 on pipe bends 166 
 Principle of Archimedes 19 
 Principle of similarity 84 
 Problems, solutions of which are given 
 
 in the text 
 channels 
 
 diameter of, for a given maximum 
 
 discharge 224 
 dimensions of, for a given flow 
 
 225-227 
 
 earth discharge along, of given di- 
 mensions and slope 224 
 flow in, of given section and slope 
 
 223 
 
 slope of, for minimum cost 227 
 solutions of, by approximation 
 
 225-227 
 pipes 
 
 acting as a siphon 161 
 connecting three reservoirs 155 
 connecting two reservoirs 149 
 diameter of, for a given discharge 
 
 152 
 
 divided into two branches 154 
 head lost in, when flow diminishes 
 
 at uniform rate 157 
 loss of head in, of varying diameter, 
 
 160, 161 
 pumping water along, diameter of, 
 
 for minimum cost 158 
 with nozzle at the end 158, 159 
 Propulsion of ships by water jets 279 
 Pumping water through long pipes 158 
 Pumps 
 
 centrifugal 392 
 advantages of 435 
 Appold 415 
 
 Bernoulli's equation applied to 
 413 
 
 Pumps (cont.) 
 centrifugal (cont.) 
 centrifugal head, effect of variation 
 
 of on discharge 421 
 centrifugal head, impressed on the 
 
 water by the wheel 405 
 design of, for given discharge 402 
 discharge, effect of the variation 
 
 of tne centrifugal head and loss 
 
 by friction on 419 
 discharge, head-velocity curve at 
 
 zero 409 
 discharge, variation of with the 
 
 head at constant speed 410 
 discharge, variation of with speed 
 
 at constant head 410 
 efficiencies of 401 
 efficiencies of, experimental de- 
 termination of 401 
 examples on 404, 412, 414, 418, 
 
 435, 478 
 
 form of vanes 396 
 friction, effect of on discharge 419, 
 
 421 
 general equation for 421, 425, 428, 
 
 430 
 
 gross lift of 400 
 head-discharge curve at constant 
 
 velocity 410, 412, 427 
 head lost in 414 
 head, variation of with discharge 
 
 and speed 418 
 head-velocity curve at constant 
 
 discharge 429 
 
 head-velocity curve at zero dis- 
 charge 409 
 kinetic energy of water at exit 
 
 399 
 limiting height to which single 
 
 wheel pump will raise water 431 
 limiting velocity of wheel 404 
 losses of head in 414 
 multi-stage 433 
 series 433 
 
 spiral casing for 394, 429 
 starting of 395 
 suction of 431 
 Sulzer series 434 
 Thomson's vortex chamber 397, 407, 
 
 422 
 triangles of velocities at inlet and 
 
 exit 397 
 vane angle at exit, effect of variation 
 
 of on the efficiency 415 
 velocity-discharge curve at constant 
 
 head 411, 412, 421, 428 
 velocity, head-discharge curve for at 
 
 constant 410 
 velocity head, special arrangement 
 
 for converting into pressure head 
 
 422 
 velocity, limiting, of rim of wheel 
 
 404 
 
532 
 
 INDEX 
 
 Pumps (cont.) 
 centrifugal (cont.) 
 velocity of whirl, ratio of, to velocity 
 
 of outlet edge of vane 398 
 vortex chamber of 397, 407, 422 
 with whirlpool or vortex chamber 
 
 397, 407, 422 
 
 work done on water by 397 
 compressed air 477 
 duplex 473 
 
 examples on 458, 464, 469, 478 
 force 392 
 high pressure 472 
 hydraulic ram 476 
 packings for plungers of 472, 486 
 reciprocating 439 
 
 acceleration, effect of on pressure 
 
 in cylinder of a 446, 448 
 acceleration of the plunger of 444 
 acceleration of the water in delivery 
 
 pipe of 448 
 acceleration of the water in suction 
 
 pipe of 445 
 
 air vessel on delivery pipe of 454 
 air vessel on suction pipe of 451 
 air vessel on suction pipe, effect of 
 
 on separation 462 
 coefficient of discharge of 442 
 diagram of work done by 443, 450, 
 
 459, 467 
 
 discharge, coefficient of 443 
 duplex 473 
 examples on 458, 464, 469, 470, 
 
 480 
 friction, variation of pressure in the 
 
 cylinder due to 449 
 head lost at suction, valve of 468 
 head lost by friction in the suction 
 
 and delivery pipes 449 
 high pressure plunger 471 
 pressure in cylinder of when the 
 
 plunger moves with simple har- 
 monic motion 446 
 pressure in the cylinder, variation 
 
 of due to friction 449 
 separation in delivery pipe 463 
 separation during suction stroke 
 
 456 
 separation during suction stroke 
 
 when plunger moves with simple 
 
 harmonic motion 458, 461 
 slip of 442, 461 
 suction stroke of 441 
 suction stroke, separation in 456, 
 
 461, 462 
 
 Tangye duplex 473 
 vertical single acting 440 
 work done by 441 
 work done by, diagram of 443, 459, 
 
 turbine 396, 425 
 
 head-discharge curves at constant 
 speed 427 
 
 Pumps (cont.) 
 turbine (cont.) 
 head-velocity curves at constant 
 
 discharge 429 
 inward flow 439 
 multi-stage 433 
 parallel flow 437 
 velocity-discharge curves at constant 
 
 head 428 
 Worthington 432 
 work done by 443 
 work done by, diagram of (see Re- 
 
 ciprocating pumps) 
 work done by, series 433 
 
 Eeaction turbines 301 
 
 limiting head for 367 
 
 series 367 
 Reaction wheels 301 
 
 efficiency of 304 
 
 Reciprocating pumps 439 (see Pumps) 
 Rectangular pontoon, stability of 26 
 Rectangular sharp-edged weir 81 
 Rectangular sluices 65 
 Rectangular weir with end contrac- 
 tions 88 
 Regulation of turbines 306, 317, 318, 
 
 323, 348 
 Regulators 
 
 oil pressure, for impulse turbine 377 
 
 water pressure, for impulse turbine 
 
 379 
 Relative velocity 265 
 
 as a vector 266 
 Reservoirs, time of emptying through 
 
 orifice 76 
 Reservoirs, time of emptying over weir 
 
 109 
 
 Resistance of ship 510 
 Rigg hydraulic engine 503 
 Rivers, flow of 191, 207, 211 
 Rivers, scouring banks of 520 
 Riveter, hydraulic 500 
 
 Scotch turbine 301 
 
 Second law of motion 263 
 
 Separation (see Pumps) 
 
 Sharp-edged orifices 
 Bazin's experiments on 56 
 distribution of velocity in the plane of 59 
 pressure in the plane of 59 
 table of coefficients for, when con- 
 traction is complete 57, 61 
 table of coefficients for, when con- 
 traction is suppressed 63 
 
 Sharp-edged weir 81 (see Weirs) 
 
 Ships 
 
 propulsion of by water jets 279 
 resistance of 510 
 resistance of, from model 515 
 stream line theory of the resistance 
 of 510 
 
 Similarity, principle of 84 
 
INDEX 
 
 533 
 
 "Siphon, forming part of aqueduct 216 
 
 pipe 161 
 
 Slip of pumps 442, 461 
 Sluices 65 
 
 for regulating turbines (see Turbines) 
 Specific gravity 3 
 
 of gasoline 11 
 
 of kerosene 11 
 
 of mercury 8 
 
 of oils, variation of, with temperature 
 11 
 
 of pure water 4 
 
 variation of, with temperature 11 
 "Stability of 
 
 floating body 24, 25 
 
 floating dock 31 
 
 floating vessel containing water 29 
 
 rectangular pontoon 26 
 Steady motion of fluids 37 
 Steam intensifier 493 
 Stream line motion 37, 129, 517 
 
 curved 518 
 
 Hele Shaw's experiments on 284 
 Stream line theory of resistance of 
 
 ships 510 
 
 Suction in centrifugal pump 431 
 Suction in reciprocating pump 441 
 Suction tube of turbine 306 
 Sudden contraction of a current of 
 
 water 69 
 Sudden enlargement of a current of 
 
 water 67 
 
 Sulzer, multi-stage pump 434 
 Suppressed contraction 53 
 
 effect of, on discharge from orifice 
 62 
 
 effect of, on discharge of a weir 82 
 
 Tables 
 
 channels, sewers and aqueducts, par- 
 
 ticulars of, and values of - in 
 
 kvn P 
 
 formula i= 195 
 mP 
 
 channels 
 
 slopes and maximum velocities of 
 
 flow in 215 
 values of a and ft in Bazin's formula 
 
 183 
 
 values of v and i as determined 
 experimentally and as calculated 
 from logarithmic formulae 198, 
 201-208 
 
 coefficients for dams 102 
 coefficients for sharp-edged orifice, 
 
 contraction complete 57, 61 
 coefficients for sharp-edged orifice, 
 
 contraction suppressed 63 
 coefficients for sharp-edged weirs 89, 
 
 93 
 
 coefficients for Venturi meters 46 
 earth channels, velocities above which 
 erosion takes place 216 
 
 Tables (cow*.) 
 
 minimum slopes for varying values 
 of the hydraulic mean depth of 
 brick channels that the velocity 
 may not be less than 2 ft. per 
 second 215 
 
 moments of Inertia 15 
 
 Pelton wheels, particulars of 377 
 
 pipes 
 
 lead, slope of and velocity of flow 
 
 in 128 
 reasonable values of 7 and n in 
 
 the formula h = ?S- 138 
 
 values of_ C in the formula 
 
 v = CJmi 120, 121 
 values of / in the formula 
 
 values of n in Ganguillet and 
 
 Kutter's formula 125, 184 
 values of n and k in the formula 
 
 i = kv n 137 
 resistance to motion of boards in 
 
 fluids 509 
 
 turbines, peripheral velocities and 
 heads of inward and outward flow 
 333 
 
 useful data 3 
 Thomson, centrifugal pump, vortex 
 
 chamber for 397, 407, 422 
 principle of similarity 62 
 turbine 323 
 Time of emptying tank or reservoir by 
 
 an orifice 76 
 Time of emptying a tank or reservoir 
 
 by a weir 109 
 Torricelli's theorem 1 
 
 proof of 51 
 Total pressure 12 
 Triangular notches 80 
 
 discharge through 85 
 Turbines 
 
 axial flow 276, 342 
 
 axial flow, impulse 368 
 
 axial flow, pressure or reaction 342 
 
 axial flow, section of the vane with 
 
 the variation of the radius 344 
 Bernouilli's equations for 334 
 best peripheral velocity for 329 
 central vent 320 
 centrifugal head impressed on water 
 
 by wheel of 334 
 cone 359 
 
 design of vanes for 346 
 efficiency of 315, 331 
 examples on 311, 321, 323, 331, 341, 
 
 349, 385, 387 
 
 flow through, effect of diminishing, 
 by means of moveable guide blades 
 362 
 
 flow through, effect of diminishing 
 by means of sluices 364 
 
534 
 
 INDEX 
 
 Turbines (cont.) 
 flow through, effect of diminishing 
 
 on velocity of exit 363 
 Fontaine, regulating sluices 348 
 form of vanes for 308, 347, 365 
 Fourneyron 306 
 general formula for 31 
 general formula, including friction 
 
 315 
 guide blades for 320, 326, 348, 352, 
 
 362 
 guide blades, effect of changing the 
 
 direction of 362 
 guide blades, variation of the angle 
 
 of, for parallel flow turbines 344 
 horse power, to develop a given 
 
 339 
 impulse 300, 369-384 
 
 axial flow 368 
 
 examples 387 
 
 for high heads 373 
 
 form of vanes for 371 
 
 Girard 369, 370, 373 
 
 hydraulic efficiency of 371, 373 
 
 in airtight chamber 370 
 
 oil pressure regulator for 377 
 
 radial flow 370 
 
 triangles of velocities for 372 
 
 triangles of velocities for considering 
 friction 373, 376 
 
 water pressure regulator for 379 
 
 water pressure regulator, hydraulic 
 valve for 382 
 
 water pressure regulator, water filter 
 for 383 
 
 work done on wheel per Ib. of water 
 
 272, 277, 323 
 inclination of vanes at inlet of wheel 
 
 308, 321, 344 
 inclination of vanes at outlet of wheel 
 
 308, 321, 345 
 in open stream 360 
 inward flow 275, 318 
 
 Bernouilli's equations for 334, 339 
 
 best peripheral velocity for, at 
 inlet 329 
 
 central vent 320 
 
 examples on 321, 331, 341, 387 
 
 experimental determination of the 
 best velocity for 329 
 
 for low and variable falls 328 
 
 Francis 320 
 
 horizontal axis 327 
 
 losses in 321 
 
 Thomson 324 
 
 to develop a given horse- power 
 339 
 
 triangles of velocities for 322, 326, 
 332 
 
 work done on the wheel per Ib. of 
 
 water 321 
 limiting head for reaction turbine 
 
 367 
 
 Turbines (cont.) 
 
 loss of head in 313, 321 
 mixed flow 350 
 
 form of vanes of 355 
 
 guide blade regulating gear for 
 352-354 
 
 in open stream 360 
 
 Swain gate for 374 
 
 triangles of velocities for 355- 
 356 
 
 wheel of 351 
 Niagara falls 318 
 oil pressure regulator for 377 
 outward flow, 275, 306 
 
 Bernouilli's equations for 334 r 
 339 
 
 best peripheral velocity for, at inlet 
 329 
 
 Boyden 314 
 diffuser for 314 
 
 double 316 
 
 examples on 311, 387 
 
 experimental determination of the 
 best velocity for 329 
 
 Fourneyron 307 
 
 losses of head in 313 
 
 Niagara falls 318 
 
 suction tube of 308, 317 
 
 triangles of velocities for 308 
 
 work done on the wheel per Ib. of 
 
 water 310, 315 
 parallel flow 276, 342 
 
 adjustable guide blades for 348 
 
 Bernouilli's equations for 348 
 
 design of vanes for 344 
 
 double compartment 343 
 
 examples on 349, 387 
 
 regulation of the flow to 348 
 
 triangles of velocities for 344 
 reaction 301 
 
 axial flow 276-342 
 
 cone 359 
 
 inward flow 275, 318 
 
 mixed flow 350 
 
 outward flow 306 
 
 parallel flow 276-342 
 
 Scotch 302 
 
 series 368 
 regulation of 306, 317, 318, 323, 348, 
 
 350, 352, 360, 362, 364 
 Scotch 301 
 sluices for 305, 307, 316, 317, 319, 
 
 327, 328, 348, 350, 361, 364 
 suction tube of 306 
 Swain gate for 364 
 Thomson's inward flow 323 
 to develop given horse- power 339 
 triangles of velocities at inlet and 
 
 outlet of impulse 372, 376 
 triangles of velocities at inlet and 
 
 outlet of inward flow 308 
 triangles of velocities at inlet and 
 
 outlet of mixed flow 356 
 
INDEX 
 
 535 
 
 Turbines (cont.) 
 
 triangles of velocities at inlet and 
 
 outlet of outward flow 344 
 triangles of velocities at inlet and 
 
 outlet of parallel flow 344 
 types of 300 
 vanes, form of 
 
 between inlet and outlet 365 
 
 for inward flow 321 
 
 for mixed flow 351, 356 
 
 for outward flow 311 
 
 for parallel flow 344 
 velocity of whirl 273, 310 
 
 ratio of, to velocity of inlet edge 
 
 of vane 332 
 
 velocity with which water leaves 334 
 wheels, path of water through 312 
 wheels, peripheral velocity of 333 
 Whitelaw 302 
 work done on per Ib. of flow, 275, 
 
 304, 315 
 
 Turning moment, work done by 273 
 Tweddell's differential accumulator 489 
 
 U tubes, fluids used in 9 
 Undershot water wheels 292 
 
 Valves 
 
 crane 497 
 
 hydraulic ram 476 
 
 intensiner 492 
 
 Luthe 499 
 
 pump 470-472 
 Vanes 
 
 conditions which vanes of hydraulic 
 machines should satisfy 270 
 
 examples on impact on 269, 272, 280 
 
 impulse of water on 263 
 
 notation used in connection with 
 272 
 
 Pelton wheel 276 
 
 pressure on moving 266 
 
 work done 266, 271, 272, 275 
 Vectors 
 
 definition of 261 
 
 difference of two 262 
 
 relative velocity defined as vector 
 266 
 
 sum of two 262 
 Velocities, resultant of two 26 
 Velocity 
 
 coefficient of, for orifices 54 
 
 head 39 
 
 of approach to orifices 66 
 
 of approach to weirs 90 
 
 relative 265 
 
 Venturi meter 44, 75, 251 
 Virtual slope 115 
 Viscosity 2 
 
 Water 
 
 definitions relating to flow of 38 
 
 Water (cant.) 
 density of 3 
 specific gravity of 3 
 viscosity of 2 
 Water wheels 
 Breast 288 
 effect of centrifugal forces on water 
 
 286 
 
 examples on 290, 386 
 Impulse 291 
 Overshot 283 
 Poncelet 294 
 Sagebien 290 
 
 Undershot, with flat blades 292 
 Weirs 
 
 Bazin's experiments on 89 
 Boussinesq's theory of 104 
 coefficients 
 
 Bazin's formula for 
 adhering nappe 98 
 depressed nappe 98 
 drowned nappe 97 
 flat crested 99, 100 
 free nappe 88, 98 
 
 Bazin's tables of 89, 93 
 
 for flat-crested 99, 100 
 
 for sharp-crested 88, 89, 93, 97, 
 98 
 
 for sharp-crested, curve of 90 
 
 Rafter's table of 89 
 Cornell experiments on 89 
 dams acting as, flow over 101 
 discharge of, by principle of simi- 
 larity 86 
 
 discharge of, when air is not ad- 
 mitted below the nappe 94 
 drowned, with sharp crests 98 
 examples on 93, 98, 108, 110 
 experiments at Cornell 89 
 experiments of Bazin 89 
 flat-crested 100 
 
 form of, for accurate gauging 104 
 formula for, derived from that of a 
 
 large orifice 82 
 Francis' formula for 83 
 gauging flow of water by 247 
 nappe of 
 
 adhering 9j, 96 
 
 depressed 95, 98, 99 
 
 drowned 95, 96, 98 
 
 free 88, 95, 98 
 
 instability of 97 
 
 wetted 95, 96, 99 
 of various forms 101 
 principle of similarity applied to 86 
 rectangular sharp-edged 81 
 rectangular, with end contractions 
 
 82 
 
 side contraction, suppression of 82 
 sill, influence of the height of, on 
 
 discharge 94 
 sill of small thickness 99 
 
536 
 
 INDEX 
 
 Weirs (cont.) 
 
 time required to lower water in 
 
 reservoir by means of 109 
 various forms of 101 
 velocity of approach, correction of 
 
 coefficient for 92 
 velocity of approach, correction of 
 
 coefficient for, examples on 94 
 
 Weirs (cont.) 
 
 velocity of approach, effect of on 
 discharge 90 
 
 wide flat-crested 100 
 Whitelaw turbine 302 
 Whole pressure 12 
 Worthington multi-stage pump 433 
 
 CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. 
 
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THE UNIVERSITY OF CALIFORNIA LIBRARY