LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class THE FLOW OF WATER A NEW THEOBY OF THE MOTION OF WATER UNDER PRESSURE AND IN OPEN CONDUITS AND ITS PRACTICAL APPLICATION BY LOUIS ^CHMEER CIVIL AND IRRIGATION ENGINEER NEW YORK: D. VAN NOSTRAND COMPANY 23 MURRAY AND 27 WARREN STS. 1909 COPYRIGHT, 1909, BY D. VAN NOSTRAND COMPANY NEW YORK Stanhope ipreaa F. H. GILSON COMPANT BOSTON. U.S.A. PREFACE. THE present work is the outcome of a series of investigations begun several years ago with the object of finding a simple expression for the phenomenon of flow in irrigation channels. The author hopes that his work will prove of interest and value to the student and useful to the practical engineer. He also hopes that it will stimulate further research and thus tend to widen the field of hydraulic knowledge. LOUIS SCHMEER. Los GATOS, CALIFORNIA, October, 1909 I iff 194610 NOTATION. 8,8 = LJ Velocity in feet per second. The mean hydraulic radius of a conduit. Area of cross section. Wet Perimeter. Diameter of circular or semicircular conduit. f Diameter of a circular conduit. < Depth of Semi square or semi circle. [ Depth of Water in a Channel. Slope of Water Surface. Head in feet. Length of conduit in feet. Fall of surface in feet. Distance in feet. The variable coefficient in the formula v = c \/r . s. The coefficient of friction, loss of head per unit area of surface at unit velocity. A coefficient indicating the resistance of an impediment to flow. Coefficients indicating the degree of roughness of the wet peri- meter. A coefficient indicating the variation of the coefficient c with the velocity of flow. A vertical distance, a head of water. Length of a conduit, a horizontal distance. Width of surface of water. TABLE OF CONTENTS. PAGE INTRODUCTION 1 PRIMARY LAWS OF PRESSURE AND FALL 4 PRIMARY LAWS OF FLUID FRICTION 8 DISTRIBUTION OF HEAD 12 DISTRIBUTION OF ENERGY 14 THE COEFFICIENT c IN THE FORMULA v = c Vrs 15 PRIMARY DETERMINATION OF THE COEFFICIENT c 17 VARIATION OF THE COEFFICIENT c (a) with the roughness of the wet perimeter 21 (6) with the velocity of flow 24 MATHEMATICAL EXPRESSIONS FOR THE VARIATION OF THE COEFFICIENT c WITH THE VELOCITY: (a) for conduits under pressure 32 (6) for open conduits 43 (c) for channels in earth 47 THE RESISTANCE DUE TO CURVES 55 THE RESISTANCES DUE TO ENTRANCES, ELBOWS, ETC 57 RIVETED CONDUITS 59 PRACTICAL APPLICATION OF THE FORMULA 63 VALUES OF a, THE COEFFICIENT OF VARIATION OF c 70 VALUES OF THE COEFFICIENTS c AND /FOR CONDUITS UNDER PRESSURE . . 71 Loss OF HEAD IN WELDED CONDUITS 72 DIAMETERS, INTERNAL AREAS, RADII AND THEIR ROOTS 73 ROOTS OF MEAN HYDRAULIC RADII 74 VALUES OF m AND K, THE COEFFICIENTS INDICATING THE DEGREE OF ROUGHNESS 76 ALPHABETICAL LIST OF AUTHORITIES 77 EXPERIMENTAL DATA 82 FORMS OF SECTIONS OF CONDUITS 113 SEWERS 118 EXPONENTIAL EQUATIONS 121 (a) for conduits under pressure 124 (6) for sewers 126 (c) for open conduits 129 EXPLANATION OF THE USE OF THE TABLES OF VELOCITIES AND QUANTITIES 136 SINES OF SLOPES AND THEIR ROOTS 143 v vi TABLE OF CONTENTS PAGE POWERS OF DIAMETERS OF CIRCULAR CONDUITS 145 POWERS OF ME AN. HYDRAULIC RADII OR OF DEPTHS OF WATER IN THE ' FORM OF SECTION MOST FAVORABLE TO FLOW 147, 151 QUANTITIES OF DISCHARGE IN CUBIC FEET PER SECOND OF A CONDUIT ONE FOOT IN DIAMETER. 155 VELOCITY OF FLOW IN A SEMI SQUARE 1 FOOT DEEP 159 DISCHARGE OF A SEMI SQUARE 1 FOOT DEEP 163 WEIR DISCHARGES: (a) Francis' formula 167 (6) Bazin's formula 168 WEIR FORMULAE 173 METHODS OF MEASUREMENT: (a) loss of head , 183 (6) discharges 184 SURFACE, MEAN AND BOTTOM VELOCITIES 193 VARIATION OF THE COEFFICIENT c WITH THE SLOPE 196 THE FORMULA IN METRIC MEASURE 205 ENGLISH AND METRIC EQUIVALENTS 209 GREATEST EFFICIENCY OF A CONDUIT OF A GIVEN DIAMETER AS A TRANS- MITTER OF ENERGY 211 MOST ECONOMICAL DIAMETER OF A CONDUIT UNDER PRESSURE 212 THE FLOW OF WATER. INTRODUCTION. THERE is no branch of the science of physics on which more has been written than on hydraulics. The master minds of the last four centuries have wrestled with the problem and thread by thread they have torn away the veil of mystery that enveloped the phenomenon of flow, The universal mind of Leonardo da Vinci (14521519), painter, sculptor, scientist and engineer, was the first to pierce the darkness and although he did not give his thoughts on the flow of water mathematical expression, we are to-day, with all the knowledge and experience gained since his time, astounded at his clear and comprehensive reasoning. The great Galileo (1564-1642) admitted that he had less trouble in finding the law of motion of the planets millions of miles away than in discerning any law in the motion of water in the stream flowing at his feet. Torricelli (1608-1644), inventor of the barometer, investi- gated the laws of falling bodies and found that the velocities of bodies falling free vary with the square roots of the heights fallen through, or with VTT. Huygens (1629-1695) first found the numerical value of g, the acceleration due to gravity; and following him Bernoulli was (in 1738) able to write the fundamental formula for the velocities of bodies falling free, On this general theoretical foundation our present system of hydraulics has gradually been built. Brahms (Dyke and 2 THE FLOW OF WATER other Hydraulic Constructions 1753) made the first step towards a practical application of the then existing theories of motion to the motion of water flowing in a channel. He found that the motion of water flowing in a channel is not like the motion of water falling free, or that of a body rolling down an inclined plane continually accelerated in speed, but moves with a uniform velocity, and that the resistance due to the friction of a fluid against the walls of the conduit depends on the relation of the wet perimeter to the area of the cross-section or on the mean hydraulic depth. Chezy (in 1776), gave the ideas of Brahms an elegant mathe- matical expression by writing for the velocity of flow v = c \/r . s in which c is a coefficient, which Chezy assumed to be constant, and r the mean hydraulic depth. This simple formula found general application in practice and is still in use. Subsequent writers occupied themselves chiefly with the definition of variations of the coefficient c in the formula pro- posed by Chezy. Owing to the researches of Coulomb (1736-1806) on the resistance of fluids to slow motions, the variation of the coefficient c with the velocity of flow was the first to be recognized and Weisbach and others found expressions for this variation. If Darcy was not the first to perceive the influence of the degree of roughness of the walls of a conduit on the velocity of flow, he at any rate was the first who thoroughly investigated the subject. (Mouvement de 1'eau dans les tuyaux, Paris, 1851.) Beginning his investigations on flow in conduits under pressure he extended them to flow in open conduits and under the auspices of the government of France constructed a special test channel 596.5 meters (1956.5 feet) long and 2 meters wide. This channel was successively lined with materials possessing characteristic degrees of roughness, the cross-section was given various forms and the bottom various slopes. To regulate the discharge two reservoirs were constructed at the head of the INTRODUCTION 3 channel and the water admitted through carefully tested sharp- edged orifices 20 centimeters square. The experiments were extended also to flow in channels lined with masonry and to flow in channels in earth. Darcy's work was after his death completed by Bazin, his successor in the office of Chief Engineer of Bridges and Roads in France. Darcy-Bazin's experiments were made with the utmost care and precision and the tabulated data (Darcy- Bazin, Recherches Hydrauliques, Paris, 1856) bear the stamp of scientific exactness and truth; they are mines of reliable information on all matters relating to flow. Darcy's experiments on flow in pipes have since his time been supplemented by many others. Hamilton Smith in California carefully gauged the discharge of sheet-iron riveted pipes under great pressures, and his data rank in reliability with those of Darcy. Clemens Herschel gauged the discharge of N large steel-riveted pipes; Iben that of pipes coated with tar; Adams and Noble the discharge of circular pipes of planed V boards. Kutter, a Swiss engineer, extended the researches of Darcy- Bazin on flow in open conduits to channels of greater slopes and greater dimensions and published (in 1869) the results of his investigations under the title " Versuch zur Aufstellung einer allgemeinen Formel," etc. Kutter and Ganguillet elaborated a general formula intended to define the variation of the coefficient c in the formula of Chezy with the mean hydraulic radius, the degree of roughness of the walls of the channel and also with the slope. Despite its cumbrousness this formula found universal appli- cation. It has, however, many defects and is no longer regarded as embodying any true law of flow. Bazin, in his memoir, " Etudes sur les mouvements des eaux dans les canaux decouverts " (Annales des Fonts et Chaussees, Faris, 1898), reviews the accumulated experimental data and proposes a formula of great simplicity. It does not, however, express the variation of c with the velocity or with the slope. 4 THE FLOW OF WATER PRIMARY LAWS OF PRESSURE AND FALL. A. The physical laws relating to fluids at rest, which are of interest in their relation to fluid motion, are briefly as follows : 1. The pressure of water on a surface is proportional to the depth below the free surface. Let H be the vertical distance of a horizontal plane below the free surface, G the weight of one cubic foot of water = 62.37 pounds. P the pressure in pounds per square foot, then P = GH = 62.37 H and the pressure per square inch p = H = 0<43 3 H 144 2. The pressure of water is the same at all points in a hori- zontal plane irrespective of the horizontal distance of any point in the plane from the free surface. No matter what the shape of the vessel or the length of the conduit may be the pressure at any point is always proportional to the vertical distance below the free surface. At the bottom of a stand pipe 80 feet below the free surface of the water the pressure on the area of a circle 4 inches in diameter will be 0.433 80.0 4 2 0.7854 = 435.2 pounds. Let a 4-inch pipe 5 miles long be connected with the standpipe at any point below the free surface, and the end of the pipe be placed in the same horizontal plane as the bottom of the stand- pipe, then, no matter how many curves or elbows there may be in the length of the conduit, the pressure will be as before, equal to 435.2 pound. 3. If a pressure be applied to the free surface of the water, this pressure is transmitted equally and undiminished in all directions, and to any distance, horizontal or vertical. Into the upper end of a pipe 1 foot in diameter and filled PRIMARY LAWS OF PRESSURE AND FALL 5 with water let a piston be inserted and a pressure of 100 pounds 100 applied. Then a pressure equal to p-^rr = 129.5 pounds per 0.7 854 square foot will be exerted on any square foot of the inner surface of the pipe, no matter how great the distance. Let the depth of the water below the surface be 20 feet. Then the total pressure per square foot will be 129.5 + (20 62.37) = 1403.9 pounds. If Pj is the external pressure in pounds per square foot, the total pressure will be, for any distance H, The external pressure due to the atmosphere is equal to 14.7 pounds per square inch. It is consequently equal to that of a 14 7 column of water ^ = 33.9 feet in height. B Torricelli's fundamental theorem for the velocity of bodies falling free is expressed by the equation : 1. v = gt 2. v 2 = 2gh 3. h = }$ 2 Or: 1. The speed of fall is proportional to the time of fall. 2. The square of the speed is proportional to the distance fallen through. 3. The distance fallen through is proportional to the square of the time of fall. The velocity of fall in feet per second is consequently: At the end of the first second of fall equal to g = 32.2 ft. At the end of the second second of fall equal to 2g = 64.4 ft. At the end of the tenth second of fall equal to 10 g = 322.0 ft. THE FLOW OF WATER The velocity of fall in feet per second is equal. At the end of the first foot of space fallen through to VW = 8-025. At the end of the second foot of space fallen through to Vg = n.34. At the end of the tenth foot of space fallen through to = 25.35. The distance fallen through is equal: At the end of the first second of the time of fall to J g = 16.1 ft. At the end of the second second of the time of fall to J g 2 2 = 64.4 ft. At the end of the tenth second of the time of fall to J g 10 2 = 1610.0 ft. C. The laws of fall thus stated apply to any body, solid or liquid falling free in vacuo. For bodies falling in the atmosphere, the resistance of the air has to be considered. This resistance is proportionally the greater, the less the density of the body. Disregarding the resistance of the air, a jet of water issuing from a well-formed orifice has a velocity proportional to the square root of the height of the column of water above the centre of gravity of the orifice. Let h be the head of water above the centre of gravity of the orifice. b a coefficient of velocity differing with the nature of the orifice, and the velocity of the jet will be v = b V2 If the discharge is into free space the speed of the motion will continue to increase with the distance fallen through, and if h l be the vertical distance fallen through in the atmosphere, the water will have, at the end of its journey, acquired a velocity equal to v = b \/2g(h~+~h^j nearly. PRIMARY LAWS OF PRESSURE AND FALL 7 * D. The motion of a rigid body descending in an inclined plane infinitely smooth is continually accelerated; the law of fall still holds, only with this difference, that in the equation g is replaced by g sin d, d being the angle which the inclined plane makes with the horizon. The kinetic energy or living force aquired by a body descending in a plane infinitely smooth is equal to Wh or \ m v 2 W in which m = = the mass of the body. The weight of the \j body W, divides into two components; one, equal to W sin d acts parallel to the plane and produces motion ; the other, equal to W cos d, acts at right angles to the plane. When the frictional resistance between the plane and the descending body is considered, the force that produces the motion or W sin d reduces to W sin d, zW cos d, z being a coefficient of friction. The acceleration of motion continues as long as W sin d is greater than zW cos d. If they are equal, or if - -,, or tangent d is equal to z, the coefficient of friction, the motion will cease. Following the laws of motion of a rigid body, the motion of a perfect fluid flowing down an inclined plane infinitely smooth would be continually accelerated. Owing, however, to internal friction, to its adhesive qualities, and the friction of the fluid against the surface of the channel in which it flows, water soon spends its accelerating force and the motion arrives at a state of steadiness more or less approaching uniformity. The motion of water is said to be steady, when at a given point of the cross-section the fluid arrives with the same velocity and in the same direction. The motion is said to be uniform, if in following a given course the mass of water has a constant velocity. 8 THE FLOW OF WATER The motion is said to be varying, if in following a given course the velocity varies from point to point. In our subsequent discussions of flow we always assume the motion to be uniform, or conditions to be such that there is no acceleration of velocity with increase of the distance fallen through, that the accelerating forces are equalized by frictional resistances and that the velocity of flow at any point in a given course remains constant as long as the slope remains constant. PRIMARY LAWS OF FLUID FRICTION. A plane surface moving in a still body of water is retarded in its motion by a resistance due to the friction of the fluid against the surface. The subject of fluid friction was investigated by Coulomb by rotating disks of greater or lesser diameters and having surfaces of a greater or lesser degree of roughness with more or less speed in a still body of water, at greater or lesser depths, and ascertaining the work done under the various conditions. The researches of Coulomb were extended by Froude in his investigations on the resistance of the surfaces of ships (1870- 1874). For the rotating disks of Coulomb, Froude substituted sharp-edged planks or metal plates of greater or lesser length and coated with various substances. These he impelled to move in a still body of water and ascertained the resistance by a suit- able device. The laws deduced from experiments made by these investi- gators may be summed up as follows: 1. The pressure existing in any horizontal plane below the free surface or in any part of a conduit under pressure has no influence on the friction of the fluid against a solid surface. Though the pressure in pounds per unit area may be much greater in one part of a conduit than in another, the frictional resistance of the area is not thereby increased. This is demon- strated as follows: A plank of suitable shape is immersed in a still body of water just below the surface, impelled to move at a certain constant PRIMARY LAWS OF FLUID FRICTION 9 speed, and the resistance to motion ascertained. If the plank is subsequently placed at a greater depth and impelled to move at the same constant speed, it is found that the resistance to motion has not been increased. If a pipe of constant dimensions is resting on an inclined plane, it can also be shown that the loss of head due to the frictional resistance is for equal lengths of the conduit the same in the lower part of the conduit where the pressure is greatest, as in the upper part, where it is least. 2. The resistance to motion, due to the friction of a fluid against a solid surface, is proportional to the area of the surface. This is demonstrated as follows: A plank of a certain length and width is impelled to move at a certain constant speed in a still body of water and the work done in foot pounds noted. If the width of the plank is subsequently doubled, thus doubling the area of its surface, and it is impelled to move at the same constant speed, it is found that the work done in foot pounds is also doubled. If water flows in a pipe running full it is found that the amount of head consumed in overcoming the resistance of the walls is proportional to the length of the pipe. Let AQ be the area of a surface in square feet : W the weight in pounds required to move a plank in a still body of water at a velocity of one foot per second; / the frictional resistance in pounds per square foot of surface then / = y , ^o and the total resistance to motion in pounds at any velocity W = fA v*, x being the variable exponent of the power of v, to which the resistance is proportional. As the frictional resistance in pounds per square foot for a velocity of one foot per second corresponds to an equal pressure per square foot, the head corresponding to the resistance is equal to h = -77 (JT 10 THE FLOW OF WATER The head equal to the resistance or --, multiplied by 20, the accel- Cr eration due to gravity or G is termed the coefficient of friction and denoted by z. As / z G = - the total resistance of a surface in pounds is equal to t7 W = z GA V The velocity of flow remaining after the frictional resistance is equalized acts through a distance equal to v. The total work done in foot pounds in overcoming the frictional resistance of a surface is consequently: 3. The resistance to motion due to the friction of a fluid against a solid surface is for equal areas of the surface greater for a short than for a long surface. This is demonstrated by impelling two planks of equal areas but different lengths to move at equal constant speeds in a still body of water. It will be found that more power is consumed in moving the shorter plank. There is a resistance due to the cutting edge of the plank, this resistance is proportionally more apparent the shorter the plank, because the total surface is proportionally less. At the entrance of any kind of a conduit head is consumed by a resistance due to shock. For short conduits this head is an appreciable part of the total head consumed. With increasing length of the conduit the head thus consumed becomes pro- portionally less and less in comparison with the total loss of head and becomes insignificant for very long conduits. 4. The resistance to motion due to the friction of a fluid against a solid surface is increased by elbows, curves, etc. Joessel, experimenting on the resistance of ships, found the resistance of oblique planes to be equal to = J ^' ** J A v 1 0.39 + 0.61 sin. a 2g PRIMARY LAWS OF FLUID FRICTION 11 in which / is a coefficient indicating the degree of roughness of the surface, varying between 1.1 and 1.7, d the density of the fluid, A the area of the surface, a the angle the plane makes with the line of motion. The resistance to motion in conduits is proportional to the angle of deflection, the radius of a curve and its length. 5. The resistance to motion due to the friction of a fluid against a solid surface varies with the degree of roughness of the surface. It increases rapidly as the roughness of the surface increases. By impelling surfaces coated with different materials to move in a still body of water Coulomb found the following values of 2, the coefficient of friction and /, the resistance in pounds per square ft. Description of Surface. z / For a varnished surface 00258 00250 For a planed and painted plank . ... 00350 00339 For the surface of iron ships 00362 00351 For a new painted iron plate . 00489 00443 For a surface coated with fine sand 00418 00405 For a surface coated with coarse sand 00503 00488 6. The power of the velocity to which the frictional resistance is proportional is not constant. It varies with the degree of roughness of the surface; with the length of the surface in the direction of motion: it is also influenced by angles, curves, etc., in the surface. By impelling surfaces coated with various materials and of various lengths in the direction of motion to move in a still body of water Froude found the following values of #, the exponent of the power of v to which the resistance is proportional : 2 8 20 50 Varnished surface . . 2 1 85 1 85 1 83 Surface coated with paraffin 1 94 1 93 Surface coated with tinfoil 2 16 1 99 1 90 1 83 Surface coated with sand 2 2 2 2 Length of Surface in Feet. 12 THE FLOW OF WATER DISTRIBUTION OF HEAD. Water issuing from a well-formed orifice flows with a velocity directly proportional to the square root of the vertical distance between the centre of gravity of the orifice and the free surface, and the velocity will continue to increase if the discharge is into free space. A stream of water entering a conduit encounters various frictional resistance tending to equalize the accelerating forces and uniform motion ensues. The total head consumed in producing this uniform motion may be resolved into several components : 1. Head consumed in producing the velocity. This is always equal to and usually but a small fraction of the total head. 2. Head consumed in overcoming the frictional resistance due to the entrance of the conduit. Let 2 be a coefficient indicating the resistance due to the entrance and the head con- sumed will be 3. Head consumed in overcoming the frictional resistance of the wet perimeter, or of the walls of the conduit. We have previously seen that the energy expended in over- coming the resistance of a surface is v 3 E = z^ GA Q foot pounds. *9 Replacing A , the area of the surface by its equivalent P, the wet perimeter multiplied by L, the length of the conduit, this is ~s and since Q, the discharge, is equal to AI, the area of the cross- section multiplied by v, the velocity, DISTRIBUTION OF HEAD 13 P L and as "7~ = "#" A i li E L v 2 we have - ^ =z. -= A G 1 R 2g As #, the total force in foot pounds, is the product of height of fall, quantity and weight we have E ~Q~G~ l and consequently h - z- 4. Head consumed in overcoming the frictional resistances due to curves, elbows, changes of section, etc. If z n is a coefficient indicating the resistances due to these impediments to flow, the head consumed will be equal to v 2 * n W Summing up all the components we have H = h + h Q + h l + h n jj v 2 v n Lv 2 v n - + * ; From this we have for the velocity 2gH This is on the assumption that the resistance of a surface is proportional to the square of the speed. We have already observed, however, that this is not always the case; it is in fact an exception. But we are not yet in a position to give the true indexes of the powers of v to which the resistance is pro- portional. 14 THE FLOW OF WATER DISTRIBUTION OF ENERGY. A quantity of water, GQ, impounded at a vertical distance,//", above a horizontal plane, possesses with reference to that plane, a stored up or potential energy equal to QGH. If by means of a conduit of greater or lesser length the water is transported to the horizontal plane at the vertical distance H, below the free surface the stored-up energy is transformed into work. The total stored-up energy resolves into several components. Let the difference of level between the free surface and the horizontal plane be 80 feet, the length of the asphalt-coated cast-iron conduit transporting the water 10,000 feet, and its diameter one foot. Assuming for Zi the average value 0.00489 we have for the velocity of flow from the data given f 64.4 .80 1* 1 + 0.505 + 0.00489 OK *- O.Zo - or v = 5.11 feet per second. The discharge in cubic feet per second will be Q = 5.11 d* 0.7854 = 4.013 cubic feet. The total energy expended in transporting this quantity is equal to E = 4.013 . 62.4 . 80 = 20,033 foot pounds. This total energy of 20,033 foot pounds is consumed as follows: 1. A quantity of work is done in producing the velocity of flow. This is equal to QG ^- =4.013 . 62.4?^ = 101.6 foot pounds. 2g 64.4 2. Another quantity of work is done in overcoming the resistance at the entrance. This is equal to QGz ^ = 4.013 . 62.4 . 0.505 ?~^ = 51.3 This is on the assumption that Z Q = 0.505. DISTRIBUTION OF ENERGY 15 3. The principal part of the work is done in overcoming the frictional resistance of the interior surface of the conduit. This is equal to QG Z L*L = 4.013 . 62.4 . 0.00489 ^2 ^i* = 1 9? 880 R 2g 0.25 64.4 foot pounds. The sum of the several quantities of work done in trans- porting 4.013 cubic feet of water a vertical distance of 80 and a horizontal distance of 10 ; 000 feet is equal to 101.6 + 51.3 + 19,880 = 20,033 foot pounds, or Dividing both sides of the equation by QG we have as before v 2 v 2 Lv 2 The C oefficient C in the Formula v = C Vr~s. Neglecting the loss of head due to the velocity, the loss of head due to the frictional resistance of the entrance, and the loss of head due to the resistance of other obstructions to flow, which severally or combined, form but a small part of the total head lost if the conduit is of a length of 4,000 times the mean hydraulic depth or the velocity not great, we have L v 2 H = z 1 as the loss of head due to the frictional resistance R 2g of the walls of the conduit. From this we have v 2 HR and y is equal The term is eual to the coefficient c first introduced into C = 16 THE FLOW OF WATER hydraulic calculations by Chezy, a French engineer (in 1776). On account of its brevity, this term is almost exclusively used to indicate the frictional resistance of long conduits of all descriptions. AS *, = ?| : . in which / = the frictional resistance in pounds per square foot of surface, G = the weight of one cubic foot of water = 62.4 we may write 2l! 2gj .G and as -^ = head lost per unit area of surface at unit velocity, we have finally c~\f^ i * head lost per unit area at unit velocity. Chezy and many of his followers up to the middle of the last century considered the coefficient c to be a constant. The researches of Coulomb, the investigations of Prony, Eytelwein, Weisbach and others, however, revealed the fact, that it varies with the velocity of flow. Later researches by Darcy and Darcy-Bazin brought to light the astounding influence of the degree of roughness of the walls of a channel and of the value of the mean hydraulic radius on the value of c. The manifold variations of c render the problem of its exact valuation one of great difficulty. A mathematical expression embodying all variations will necessarily be very complex; to be of practical value, however, it should be as simple as possible. It is some- what difficult to harmonize great exactness and great simplicity without making sacrifices at one end or the other. On this account two expressions are often found embodying the same idea and rendering it with great exactitude or great sim- plicity. DISTRIBUTION OF ENERGY 17 We will now proceed to investigate the laws on which the variation of c depends and to find suitable mathematical expres- sions embodying these laws. I. Primary Determination of the Coefficient c. Going back to first principles we may ask the question : To what power of R, the mean hydraulic radius, is the velocity of flow proportional? Using the exponential equation gives x - ; g % - ; g * log R, - log R Q we find that the value of x is, in the case of channels in earth, such as rivers and canals and with R varying between 1 and 50 feet in the majority of cases equal to 1 2 or 3 1.333 2.666 4 For this class of conduits we may consequently write: in which y is variable, differing with the degree of roughness and with the slope of the conduit. As v = c \/rTsTand R* = t/r Vr~ we have C-vW, hence c increases directly with Vr. c Column 5, Table I, gives values of y = ir=for conduits of several degrees of roughness. It will be observed that the formula gives fairly constant values of y only for large conduits, such as rivers and canals, For small conduits however y increases with increase of R if the wet perimeter be smooth, but decreases with increase of R if the contrary is the case. Applying the exponential equation to other classes of conduits, the following values of x, the power of R } to which the velocity is proportional were found. 18 THE FLOW OF WATER For a semi-circular channel of fine cement x = 0.67 For a semi-circular channel of concrete x = 0.68 For a rectangular channel of rough boards x = 0.69 For a rectangular channel of rough masonry x = 0.75 For a channel carrying coarse detritus x = 1.00 The conclusions to be drawn from these data may be summed up as follows: 1. For rivers and canals the power of R, to which the velocity is proportional, is approximately equal to |. 2 For small channels the power varies with the degree of roughness of the perimeter a/id the form of the cross-section of the conduit. 3. For small channels the power of R increases with increase of roughness. 4. For the smoothest class of conduits the velocity is pro- portional to R' Q7 for the very roughest to R 1 ' . Hence the rougher the wet perimeter, the more conditions are approached resembling those pertaining to flow in permeable strata, in which instance the velocity is proportional to the square of the diameter of the channel. 5. No formula, based on anyone sjjigle power of R can give satisfactory results when applied to all classes of conduits. UNIVERSITY OF VARIATION OF THE COEFFICIENT C TABLE I. 19 _ c Description of Conduit. R 1000 S V t/r Sudbury Conduit. Smooth hard brick well 0.5 0.189 1 .134 138 pointed. 0.6 0.189 1.371 135 0.8 0.189 1.515 131 1.0 0.189 1.754 127 1.2 0.189 1 .948 124 1 .4 0.189 2.148 121 1.6 0.189 2.382 119 1.8 0.189 2.514 119 : . 2.0 0.189 2.683 116 2.2 0.189 2.843 114 2.33 0.189 2.929 113 Semicircular channel lined with pebbles f 0.454 1.5 2.17 95.1 to inch diameter. 0.546 1.5 2.50 95.3 0.619 1.5 2.69 92.5 0.681 1.5 2.93 92.3 0.731 1.5 3.05 91.3 0.784 1.5 3.22 90.4 0.826 1 .5 3.33 88.4 0.900 1 .5 3.54 87.6 0.968 1.5 3.73 85.8 1.012 1.5 3 .95 87.9 Solani Embankment. 6.32 0.140 2.63 55.9 Jaoli Site. 6.53 0.144 2.70 55.0 Sides of brick set in mud, bottoms very 6.79 0.145 2.80 54.9 rough. 7.05 0.146 2.81 53.7 7.46 0.160 2.94 51 .5 Linth Canal, channel in earth, fairly 5.14 0.29 3.414 58.6 regular. 5.93 0.30 3.830 58.2 6.48 0.31 4.152 58.2 7.12 0.32 4.418 56.5 7.52 0.33 4.753 57.4 8.09 0.34 4.920 55.8 8.28 0.34 5.058 56.1 8.62 0.35 5.225 57.7 8.87 0.36 5.392 55.5 9.18 0.37 5.530 54.5 River Seine at Paris. 9.48 0.14 3.37 53.1 10.92 0.14 3.74 52.8 12.19 0.14 3.81 49.6 14.50 0.14 4.23 49.4 15.02 0.14 5.11 49.8 15.93 0.14 4.68 49.5 16.85 0.131 4.80 48.6 18.39 0.103 4.69 51 .8 Mill race at Pricbam, Hungary. 0.316 2.2 0.389 20.0 0.336 2.2 0.588 28.4 Irregular channel lined with rubble 0.442 2.2 0.953 35.7 masonry. 0.548 2.2 1.135 37.7 0.560 2.2 1.190 39.1 0.566 2.2 1.270 41 .3 20 THE FLOW OF WATER TABLE II. Description of Conduit. V c a I New straight asphalt-coated wrought-iron riveted pipe with screw joints. 0.328 1.171 3.117 76.7 99.9 108.4 0.80 1 .04 1 .13 6.148 117.1 1 .219 m = 0.94 R = 0.0677 10 .535 12 .786 124.0 124.3 1 .289 1.291 II 2.78 139.1 1.122 Test pipe of clear cement. 3.65 4.20 139.2 139.5 1.139 1 .140 m = 0.95 4.72 140.4 1 .141 R = 0.658 4.79 141 .2 .155 4.92 141 .4 .157 5.81 141 .4 .157 6.58 142.5 .166 III 1.0 101.2 .0 New cylinder joint asphalt-coated steel-riveted pipe with many curves. 2.0 3.0 3.5 108.3 112.8 113.4 .09 .113 .119 m = 0.53 4.0 113.2 .118 R = 1.0 5.0 112.0 .105 6.0 111 .6 .091 IV 1.007 73.6 .0 Old cast-iron pipe. 2.32 75.5 .023 m = 0.45 5.075 75.1 .02 R = 0.1995 6.801 75.2 .02 12 .576 75.3 .02 V Heavily in crusted cast-iron pipe. Twenty- five years in use. 1.60 2.70 3.60 64.0 60.0 59.0 0.948 0.900 0.874 m = 0.30 R = 0.416 4.50 58.0 0.858 VI Channel of dry rubble masonry of large stones, bed somewhat damaged. Six years old. 8.442 8.905 9.181 9.427 57.4 54.3 50.7 49.6 0.890 0.842 0.784 0.769 m = 0.30 10 .145 47.2 0.731 R = 0.19 VII 0.50 126.9 1.089 Short conduit. 1.0 116.6 1 .00 Wrought-iron riveted pipe, somewhat rusty. 1.5 111 .9 0.959 2.0 109.4 0.938 2.5 109.0 0.934 Length 152.9 feet. 3.0 108.2 0.928 Diameter 8.58 feet. 3.5 107.0 0.917 m = 0.54 4.0 106.2 0.910 R = 2.145 4.5 105 .6 0.905 VARIATION OF THE COEFFICIENT C 21 Variation of the Coefficient C with the Roughness of the Wet Perimeter of a Conduit. Although the primary formula v = y R Vs does not give satisfactory results when applied to all classes of conduits it may be made the basis of formulae of general application. Regarding y Vr as an approximate value of c expressions may be found defining the variation of c with the roughness of the wet perimeter as depending on Vr. The primary value of c from which its variations with the slope or the velocity of flow must be derived is that value which corresponds to a velocity of one foot per second. In order to retain if possible a straight line formula we may choose the expression c = (y Vr) 1 + m, m indicating the condition of the wet perimeter of the conduit. For a primary determination of y and m Darcy's values of c for clean iron pipes were selected. These data give c = 112.0 for R = 1.0 and c = 80.4 for R = 0.0208 (or a one-inch pipe). These values of c are merely average values found by Darcy from a great number of experiments on clean pipes, which, how- ever, did not include pipes of great diameters. Taking 50 as a trial value for y we find 112.0 = = 2.24, hence 1 + m = 1 + 1.24 4 if = 4.21, hence 1 + m = 1 + 3.21. 50 Vr Dividing 1.24 by 3.21 that the quotient is 0.386. This is almost equal to 0.38, the fourth root of 0.0208, the value of R for the one-inch pipe. We have consequently in both instances 4,- 1.24 c = (50 Vr) 1 f -47=- /YY) or in general c = (50 Vr) 1 + -57=- vr Testing this formula by experimental data pertaining to flow in conduits differing widely in their degree of roughness it did 22 THE FLOW OF WATER not prove entirely satisfactory. As already stated, Darcy's experiments were made on conduits of comparatively small diameters and his coefficients for the larger conduits do not quite agree with those found by recent experiments. For the final determination of the value of y we choose the graphical method. If then = (If ^r) 1 + -TF- Vr m y This is the equation of a straight line (Fig. 1) having for 1 C abscissae values of -j - , for ordinates values of j and having Vr yvr Values of FIG. 1. 1.0 as the common distance from the axis of abscissae where all the lines intersect the axis of ordinates; the tangent _10 y ^Jr~ of the angle a b c will give the value of m. Identical values will be obtained by putting 7 selecting data in which v = 1.0 foot per second. VARIATION OF THE COEFFICIENT C 23 Experimental data giving values of c corresponding to a veloc- ity of one foot per second are not numerous while those giving values of c corresponding to a velocity of one metre per second are quite abundant, this coming nearer to being an average velocity. On this account data given in metric measure were chosen, taken chiefly from the writings of Darcy-Bazin. After numerous trials, and using all the reliable material available, a constant value of y and corresponding values of m were found, producing a straight line in every instance. As our subsequent work depends much on the reliability of this constant, great pains were taken to find its exact value. In metric measure its value is equal to 50.0 for which in English measure we substitute 66.0. We have consequently for the value of c corresponding to a velocity of one foot per second c = 66 1 + -> or, reducing to a straight line c = 66 (fa + m). As 66 (t/r + m) = c = y ?-2 and (66 (^ r + m ))* = c 2 = -j- we have z = 7^- or z = (66 (Vr + m)) 2 0.01478 (Vr + m) 2 ' As primary expressions for the velocity, in most instances true only when the velocity is equal to one foot per second we have now the formulae v = 66 (-Vr + m) VrTs. .... (1) 24 THE FLOW OF WATER In the formula c = 66 (Vr + m), when applied to calculations of flow in channels in earth of a great degree of roughness of the bed, the coefficient m, which indicates the degree of roughness will have a negative value and c will in consequence vanish for very small values of Vr. To avoid this defect the formula may be written, when applied to channels in earth, so that it reads 66 (Vr + 1) _ 66 (Vr + Vr) = ~ Vr in which K is a coefficient increasing in value with increasing roughness of the wet perimeter. The relation between m and K is given by K--M--1.0 1 + m Variation of the Coefficient C with the Velocity of Flow. A. The characteristics which distinguish water from a perfect fluid are its adhesive qualities, its viscosity. All fluids, includ- ing gases, have these qualities in a greater or lesser degree. It is even asserted that solids like ice become viscous under great pressures. The adhesive qualities of tar or crude oil are apparent to the eye, those of other fluids can only be inferred from their effects. To its viscosity is due the fact, that water flowing in a channel perfectly smooth, is not, in accord with the law of falling bodies, continually increasing in speed. The retarding forces due to viscosity equalize the accelerating forces due to gravity and distance fallen through, the speed of the water shows no increase from point to point, in other words, the motion is uniform. The layer of water immediately in contact with the walls of the channel in which it flows does not change except by diffusion ; VARIATION OF THE COEFFICIENT C 25 it is held fast by surface adhesion. If the wall is perfectly smooth there is consequently no friction between it and the fluid directly in contact; the resistances to flow are entirely due to shearing stresses between the infinitely fine film coating the wall and the moving body of water. Frictional resistances are always proportional to the areas of \ the surfaces in contact; surface areas near the periphery of a conduit are always greater than near the centre and the retarda- tion will in consequence be greater and the velocity less. This decrease of speed from the centre towards the periphery is in a measure counteracted by difference of pressure. Greater velocities are always accompanied by a corresponding fall of pressure and the pressure in the centre is in consequence less than near the wall. This difference of pressure continually tends to draw the water towards the centre and thus to equalize the speeds. When this equalizing tendency is for a moment interrupted, we suddenly perceive a wave or flash-like motion, clearly indicating the speed the water would acquire were it not for the resistances near the periphery. In conduits having smooth walls the equalization of velocities is performed so rapidly that a difference of speed between the centre and the periphery is scarcely perceptible. A wave-like rotation is set up and the water glides through the conduit very much like a bullet through a rifled channel. Let R be the force required to keep up the flow of a liquid in two parallel planes past each other, let the surface area of each plane be A, let the respective distances of the two planes from a common plane of reference be D t and Z) , let the velocities be v t and v Q and e a coefficient indicating the degree of viscosity of the liquid and we have: _ eA (v l - v ) or: the resistance is proportional to the degree of viscosity into the area and the relative velocity v 1 - V Q , the whole divided by the difference in the distance of the two layers from a common plane of reference. 26 THE FLOW OF WATER For a circular conduit the total force required to set up motion in a stream line is given by 4el (v, - v ) 2flv 1 = - ; - ; ~r - Tj 2 - r 2 r t in which r l is the semi-diameter of the conduit, r a distance from the axis of the conduit, I its length, and / the coefficient indicating the degree of roughness of the surface. This indicates, that the resistance due to viscosity is least in the centre or the axis of the conduit where r 2 = and greatest at the periphery where r t 2 r 2 = 0. The last expression gives for the velocity of flow in a circular conduit / v = / , 2gh + from which, the coefficients e and / being known, the value of v and the discharge may be computed. The coefficient e depends for its value on the temperature of the liquid ; its value diminishes rapidly with increase of temperature and is five times less for water at the boiling point than for water at the freezing point. According to Mayer its values are (in c. g. s. units) : at 0.6 Celsius e = 0.0173 at 10 Celsius e = 0.0131 at 20 Celsius e = 0.010 at 45 Celsius e = 0.005833 at 90 Celsius e = 0.00339. The influence of the temperature on the flow of water through capillary tubes has been minutely studied by Poisseule. Slichter has demonstrated the immense influence of the temperature on the movement of water through permeable strata and Saph and Shoder have shown its influence on the dis- charge of pipes. A tube having a diameter of 0.5 millimetre (0.02 inch) or less is considered to be a capillary tube. VARIATION OF THE COEFFICIENT C 27 Poisseule's experiments demonstrated, that the velocity of flow in such tubes is equal to and the discharge to This shows, that in capillary tubes the velocity is proportional to the head and not to the square root of the head, to the square of the radius and not to its square root. In investigations on the movement of water through porous strata it has been found, that the velocity of flow is proportional to the square of the diameter of the soil grains through which the water percolates; from which it follows that it is also pro- portional to the square of the voids between the soil grains. The general equation for the movement of water in a per- meable stratum may be written (v and Q per minute) - 0.0189d 2 (0.7 4- 0.03 t'c) 2L Q = mv hb. In these equations h is the elevation of the water table at the point of efflux, h + z its elevation at the distance L, d the FIG. 2. diameter of the soil grains in millimetres, t 0, the temperature of the water in degrees centigrade, m the percentage of the voids in the material, b the breadth of the stratum. 28 THE FLOW OF WATER The elevation of the water table at the distance x from the point of efflux is equal to y 2Qx hm 0.0198 d 2 b The discharge of a well is given by Q T log L - log R 0.0189 d* (0.7 + 0.03 in which R is the semi-diameter of the well, and the logarithms the Naperian FIG. 3. The elevation of the water table at a distance x from the well is given by The surface is consequently a logarithmic curve. These equations serve to illustrate that between the movement in capillary tubes and in a porous stratum there is only this dif- Ji 2 ference, that h is displaced by - , the velocity is not proportional to the head but to the square of the head. On account of internal motions the phenomenon of flow in VARIATION OF THE COEFFICIENT C 29 pipes and other channels is much more complex than in capillary tubes or porous strata. The equation for the velocity of flow in a cylindrical conduit we have given above may be transformed so it will read - - % + r- JM! T7-v M f 2 ~, 2 I O 7. '1 'n l^~7't which shows that one of the terms above the line, denoting the internal resistance, is directly proportional to the velocity, the p other to its square. This is also true of the two terms 2-^ denoting the friction of the fluid against the walls of the conduit. Moreover, /, the coefficient denoting the surface friction, depends for its value on e, the coefficient denoting the internal friction; its value is consequently modified by temperature. Even in conduits having the smoothest walls there are always rotary and wave-like motions tending to equalize pressures to speeds. Wherever there are cross-currents there naturally is impact, one stream impinging on the other. To this impact and the attend- ing shearing stresses between a streamline and its surroundings are due the increasing powers of the velocity to which the resistances are proportional. Furthermore, if the walls of the conduit are not perfectly smooth there are streamlines constantly impinging on projections, however small they may be. Conditions existing at the entrance, curves, elbows, changes of section, etc., also affect the power of the speed to which the total resistance is proportional. It was formerly assumed that the resistances due to these impediments were proportional to the square of the velocity. From experiments made by Hubbel and Fenkell (Detroit) to determine the resistances due to curves, the writer found, neglecting curves the radius R of which is less than 2.5 diameters of the conduit, that the resistance of a curve is equal to 30 THE FLOW OF WATER times the resistance of a tangent of equal length, and the excess of frictional resistance in a curve equal to times the resistance in a tangent of equal length. The length of tangent equal in frictional resistance to the resistance in a curve of 90 is equal to This evidently vanishes when = (4.9)^ d = 539.3 and is a a maximum when = (4.9) * d. d Hubbel and Fenkell's experiments were made on 30", 16" and 12" conduits, and comparison showed that the influence of the diameter on the resistance depends on the value of d. The value of z, or /, the coefficient of friction is therefore for any curve. . - 1.0 360 in which n is equal to the number of degrees in the curve and d' 45 substituted for d for diameter less than a foot. Hubbel and Fenkell's experiments were supplemented by those of Saph and Schoder on 2-inch brass tubes and more recently by those of Alexander on a IJ-inch wooden tube. Although we cannot accept the formulae the latter deduced from his own ex- periments and those of Hubbel and Fenkell, Saph and Schoder, his experiments are valuable in indicating the powers of the velocity to which resistances in curves are proportional. While Alexander's experiments show that resistances in a curve are proportional to the same powers of the speed as resistances in a tangent, provided there is no shock, the experiments of Saph and Schoder indicate that the power of the speed increases rapidly with increasing values of Their data indicate that ti VARIATION OF THE COEFFICIENT C 31 7? 7? for = 10 the resistance is proportional to V 1 ' 8 , for =4 to a a 7 2 ' 87 . How far this holds good for diameters greater than 2 inches we are not prepared to say. It is probable, however, that with increasing diameter the force of the shock decreases and the powers of v with it. It is probable that the resistances due to right-angled entrances right-angled elbows are also proportional to powers of v higher than 2.0. The effect of the temperature on the variation of the power of v has so far not been determined with precision. Saph and Schoder, experimenting with a 2-inch brass pipe, found for a rise of 10 F. an increase in the discharge of 4%. That the resistances to flow are not proportional to the square of the speed was recognized long before Darcy and Bazin demonstrated the great influence of the degree of roughness of the walls of a channel on its discharge. The laws of fluid friction were first investigated by Coulomb. He states, that the total resistance to motion is a compound of two factors, one being proportional to v, the other to v 2 . Dubois's experiments on flow confirmed this view and from his data Prony found for the resistance the expression (in metric measure) Rv = 0.000044 v + 0.000309 v 2 , this corresponds to ff -?TT' Weisbach put Prony's formula into the form H = 0.00741 v I r which in our day is still used. The relation of the power of the velocity, to which the resist- ance is proportional, to the variation of the coefficient c with the velocity is such, that c remains constant for all velocities if the resistance is proportional to v 2 ; it increases with increase of velocity if the resistance is proportional to v*~*, and decreases if the resistance is proportional to v** x 32 THE FLOW OF WATEK B. If the value of the coefficient c corresponding to any velocity is divided by its value corresponding to a velocity of one foot per second; the quotient is a variable which we will call the coefficient of variation of c and denote by (a). Hence c a 66 (Vr + m) While the term i [66 (Vr + m)] 2 represents the frictional resistance per unit area of surface at unit velocity, the term c 66 (t/r + m) indicates the power of the velocity to which the resistance is proportional. We shall presently see, that under normal con- ditions, that is if resistances proportional to different powers of v do not enter, the coefficient a is merely a root of v. An analysis of the values of a found in Column 4, Table II, shows that its value does not entirely depend on the velocity, but is affected by the degree of roughness of the walls of the conduit, by its length and alignment, by conditions existing at the entrance, by changes of section, etc. According to the manner in which the coefficient a is affected we may classify conduits as follows: 1. Long straight conduits without internal obstructions and a great degree of smoothness of the wet perimeter. 2. Long conduits of a great degree of smoothness of the wet perimeter but with some easy curves or other impediments, also long straight conduits of a fair degree of smoothness of the wet perimeter. 3. Long conduits of a great or fair degree of smoothness of the wet perimeter but with sharp curves, angles or other impedi- ments to flow. 4. Conduits whose walls are coated with rust, slimy or sticky substances. VARIATION OF THE COEFFICIENT C 33 5. Conduits of a great degree of roughness of the wet peri- meter; badly tuberculated pipes, damaged masonry, channels in earth with sharp bends, bars or other obstructions. 6. Short conduits. For classes 1 and 2 the resistances are proportional to powers of v less than 2.0 and the coefficients c and a continue to increase in value with increasing velocity. For the third class some resistances proportional to a power higher than 2.0 enter, a increases with increase of velocity and then decreases. For class 4 the resistance is proportional to v 2 or nearly so and a is constant. For classes 5 and 6 the resist- ances are proportional to powers of v higher than 2.0 and a continually decreases with increasing velocity. C. We have so far only found expressions for the value of c corresponding to a velocity of one foot per second. These give for the velocity v = 66 (t/r + m) Vrs (1) 2 H .... (2) 0.01478 L (Sir + my R We will now proceed to find in what relation the 'value of v as found from the formula v = 66 (tfr + m) Vr . s stands to the true mean velocity in all cases where v is more or less than unity or the value of a, the coefficient of variation, is affected by the conditions we have enumerated. Using the exponential equation (66 (^r + m) _[ L (66 (r + m) which gives log y t _ _ log v x = log (66 (#r + m) Vr.s), - log (66 (Vf + m) Vrs), OF THE UNIVFRRITY 34 THE FLOW OF WATER we find from the experimental data given by Darcy and Hamilton Smith for straight or nearly straight clean cast-iron, wrought- iron and sheet-iron riveted pipes of all diameters and for velocities up to 20 feet per second *-|i in other words, from the data given by Darcy and Hamilton Smith we find, that the true mean velocity is equal to v = (66 (vV + m)^r.s)* ..... (3> which may be written v = 66 (-r + m) ree (^r + m) hence the coefficient a, indicating the variation of the coefficient c with the velocity is equal to a = 466 (\V + m) \/r~7s. From Formula 3 we have also yt = (66 tJr + m) consequently a = V* and y* = iV(66 A/r + m) VrTsT Table III contains a number of experimental data relating to flow in conduits under pressure. They are purposely selected in order to show the variation of the coefficient c as affected by various conditions of flow. The values of the coefficient a found in columns 3, 6 and 9 show that for 1-inch pipes of tin and wrought iron, for sheet-iron riveted pipes up to 2.43 feet in diameter, for new cast-iron pipes up to 1.393 feet in diameter, for pipes of planed shares up to 4.5 feet in diameter the coefficient a is equal to y^ or nearly so. The fact that a = F holds good for a tin or wrought-iron pipe 1 inch in diameter, and also for a pipe of planed staves 54 inches in diameter allows us to conclude, that it holds good also, be- tween these limits for other conduits having walls of a similar degree of roughness such as asphalt-coated, cast and wrought- iron or cement-lined pipes. VARIATION OF THE COEFFICIENT C 35 This, however, holds good only when the value of , the ratio ct between the length of a pipe and its diameter, is at least 1,000. For lesser values of the value of (a) decreases with - The a a experimental values given by Stearns and Fitzgerald relating to flow in four-foot cast-iron pipes indicate this plainly. In the case of the four-foot Sudbury conduit (Stearns) the ratio is equal to 439. If the formula v = (66 (tfr + m) Vr . s)* is put into the form V = the term -Vr + m) 2 R includes all the resistances, those due to the velocity itself, those due to the entrance, and those due to the friction of the fluid against the walls of the conduit. The loss of head due to the velocity itself is proportional to the square of the speed; resistances due to the entrance are pro- portional, according to the nature of the entrance all the way from the square of the speed up to its cube. An average value is probably 2.5. The value of the coefficient of resistance due to the velocity itself is equal to 1.0; the value of z , the coefficient representing the resistance due to the entrance is, according to Weisbach, for a well rounded entrance, equal to 0.505; hence the value of z 1} the coefficient representing the resistance due to the walls of the conduit, is equal to ' 01478 L - 1.505. + m) 2 R 36 THE FLOW OF WATER If the conduit is long, above 1,000 diameters in length, 1.505 is a quantity small in comparison with 0.01478 L Vr + m) 2 R and does in consequence not appreciably affect the variation of c. With decreasing length of the conduit, however, the ratio between the two quantities changes at an increasing rate and more and more affects the variation of c. In the case of the four foot Sudbury conduit (Stearns), we have the following data, taking m = 0.97 : v = 3.738, H = 1.2421 ft., L 1747 ft. 0.01478 L = 6.656 (1 + 0.97) 2 R 6.656 - 1.505 = 5.151 = z r ( 3 jJ 38 ) 2 X 1.0 = 0.217 = h = loss of head due to velocity. 7 (3 ' 738)2>5 X 0.505 = 0.2119 = h = loss of head due to entrance. 0.217 + 0.2119 = 0.4289 = h + h . 1.2421 - 0.4289 = 0.8182 = ^ = loss of head due to friction in the pipe itself. Using the formula - = v x and inserting values we have 64.4 X 0.8142 = = 5.151 Dividing log 10.17 = 1.0073209 by log 3.738 = 0.5736293 the quotient = 1.76 very near. Consequently the frictional resistance in the pipe itself is proportional to v 1 ' 76 , corresponding closely to FT", the value we have found for long pipes. The data relating to flow in a riveted flume 8.58 feet in diameter and 152.9 feet long (Herschel, Holyoke Testing Flume) show the great influence of the length of the conduit on the variation of c most plainly. VARIATION OF THE COEFFICIENT C 37 TABLE III. EXPERIMENTAL DATA SHOWING EXTENT OF VARIATION OF c WITH THE VELOCITY OF FLOW. Tin pipe, straight. Dubuat. Wrought-iron pipe. Darcy. Asphalt- coated riveted pipe. Darcy. d = 0.0888 ft. d = 0.1296 ft. d = 0.271 ft. Z/not given. L = 372 ft. L = 365 ft. m = 0.98 m = 0.83 m = 0.94 a =F* a = H a = V V c a V c a V c a 0.141 67.6 0.751 0.205 76.9 0.932 0.328 76.7 0.80 0.772 82.8 0.92 0.858 82.3 0.995 1.171 99.9 1 .043 1.183 91.4 1.019 2.585 92.9 1 .112 3.117 108.4 1.132 2.546 98.9 1.099 6.3 99.8 1.21 6.148 117.4 1 .223 2.606 100.4 1.115 8.521 100.0 1.212 10 .535 124.0 1.274 5.223 111.4 1 .237 12 .786 124.3 1.298 Asphalt-coated riveted pipe. Darcy. Asphalt-coated riveted pipe. H. Smith. Asphalt-coated riveted pipe. H. Smith. d = 0.643 ft. d = 0.911 ft. d = 1.229 ft. L = 365 ft. L = 700 ft. L = 700 ft. m = 0.92 m = 0.68 m = 0.69 a = V? a = 7? a = 7^ V c a V c a V c a 0.591 104.1 1.013 4.712 107.1 1.19 4.283 111 .6 1.181 1.529 106.2 1 .035 6.094 110.6 1 .229 6.841 117.8 1 .246 1.53 115.6 1 .125 6.927 111 .5 1 .24 7.314 119.1 1 .261 5.509 125.4 1.22 8.659 113.4 1 .26 8.462 119.1 1.26 9.0 130.2 1 .267 10 .021 115.5 1.283 10 .593 121.6 1 .285 19.72 141.0 1.372 12.09 121.3 1.280 38 THE ELOW OF WATER TABLE III. Continued. Asphalt-coated cast- Asphalt-coated cast- Asphalt-coated cast- iron pipe. Darcy. iron pipe. Hubbel & Fenkell. iron pipe. Lampe. d = 0.4495 ft. d = 1.0ft. d = 1.373 ft. L = 366 ft. L not given. L = 26,000 ft. m = 0.90 m = 0.83 m = 0.83 a -F* a = V? a = V? V c a V c a V c a 0.489 94.1 0.96 1 .0 101 .5 .0 1 .577 110.5 1.072 2 .503 108.4 1.107 2.0 109.6 .06 2.489 114.1 1.107 5.625 112.5 1.15 3.0 114.6 .13 2.709 114.6 1.112 11 .942 113.5 1.16 4.0 118.3 .166 3.090 119.4 1.162 15 .397 112.2 1.15 5.0 121 .3 .196 Redwood Stave Pipes. A. L. Adams. Cedar Stave Pipe. Th. A. Noble Cedar Stave Pipe. Th. A. Noble. d = 1.166 ft. d == 3.667 ft. d = 4.5 ft. L = 80,006 ft. Lnot given. Lnot given. m = 0.93 m = 0.50 m = 58 a = V? a = V* a = V^ V c a V c a V c a 0.698 97 0.908 3.468 110.1 .134 2.282 116.8 1 .095 0.698 101 0.926 3.522 108.6 .12 2.276 115.5 1 .086 0.751 104 0.953 3 .685 110.9 .144 2.650 119.9 1 .12 0.691 105 0.963 3 .853 112.6 .163 3 067 122.1 1 .151 1.167 109 1.0 3.964 112.9 .164 3 .045 121 .4 1 .138 1 .531 112 1 .027 3 .972 113.1 .164 3.408 123.7 1 .164 1.181 113 1 .036 4.415 113.7 .164 3.724 125.2 1.176 4.635 114.9 .183 3.929 126.2 1.179 4.831 115.5 .190 4.688 129.0 1.205 VARIATION OF THE COEFFICIENT C 39 TABLE III. Continued. New steel-riveted pipe. New steel-riveted pipe. New steel-riveted pipe. Herschel. Herschel. Marx-Wing. d = 3.5 ft. d = 4.0 ft. d = 6.0 ft. L = 81,339 ft. L = 24,648 ft. L = 4,367 ft. m = 0.56 m = 0.47 m = 0.50 a = V& a = V& a = V^ V c a V c a V c a 1.0 101.0 1.0 1 97.1 1.0 1.07 103.5 1.0 2.0 104.3 1.032 2 101.3 1 .043 1.67 108.0 1 .024 3.0 106.4 1 .053 3 102.2 1 .052 2.14 113.0 1 .091 4.0 107.8 1 .067 4 104.2 1.073 2.50 108.0 1 .024 5.0 108.4 1.073 5 105.1 1.083 3.0 112.0 1.082 6.0 108.5 1.074 6 105.2 1.084 3.84 113.0 1.091 Asphalt-coated cast- Cleaned cast-iron pipe. Cedar stave pipe iron pipe. Stearns. Fitzgerald. Marx-Wing. d = 4.0 ft. d = 4.0 ft. d = 6.0 ft. L = 1,747 ft. L not given. L = 4,000 ft. m = 0.97 m = 0.98 m = 0.66 a, = FIT a = yr a = V&i V c a V c a V c a 3.738 140.1 1.077 2.472 137.5 1.051 1.0 116.0 1.0 4.965 142.1 1 .093 3.723 139.1 1.064 1.5 118.7 1.023 6.193 144.1 1.109 4.796 141 .1 1.085 2.0 119.9 1.032 6.141 143.6 1.100 3.0 121.4 1.046 4.0 122.0 1.051 5.0 122.4 1 .055 6.0 122.5 1 .056 40 THE FLOW OF WATER The experimental data relating to flow in riveted conduits show great diversities both in the values of m and a. The coefficient m is equal to 0.94 for a riveted pipe 0.270 feet in diameter (Darcy) and equal to 0.51 for a butt-jointed riveted pipe six feet in diameter (Marx -Wing). This great difference in the values of m is mainly due to the size of the rivet heads. In pipes of small diameters the rivet heads, especially when coated with asphalt, do not offer an appreciable impediment to flow. In large conduits, however, their size is such, that they not only produce constriction of the section, but also vortex motions, thus reducing the discharge in a twofold manner. From data relating to flow in steel-riveted pipes exceeding three feet in diameter, given by Herschel, and by him considered the most reliable (see " Herschel "115 Experiments), we find that the coefficient of variation of c for these conduits is fairly, though not precisely, equal to (Vr + m) Vf~7. Consequently (66 (Vr Hh m) Vr . s) i? . . . . (5) f 2gH 1* (6"\ 0.01478 L .(^r + m) 2 ^J or v = This value of a = V we find to hold good also for flow in rectangular pipes (Darcy). The experimental data relating to flow in old iron pipes, those not heavily incrusted or tuberculated show that the coefficient c does not vary to any extent with variations in the value of v. For this class of conduits we have consequently a = 1.0. The data relating to flow in badly incrusted or heavily tuber- culated pipes indicate a decrease in the value of c with increasing velocity. Using as before the equation log v, log^p x = log (66 (Vr + m) Vrs), - (66 (Vr + m) Vrs) ( VARIATION OF THE COEFFICIENT C 41 we find for incrusted pipes: x = if, hence a = 19/ _ V V 66 (^r + m) Vr.s and for very badly tuberculated pipes: x = y 9 0, hence a = - - = ^66 (>/r + m) Vr . a The experimental data relating to flow in a 12-foot brick sewer at Milwaukee, a 7.5-foot brick sewer at Dorchester Bay, in a siphon aqueduct of 119 feet cross-section at the river Elvo all show a slight decrease in the value of c with increasing velocity. This decrease is due, in the first two cases, partly to the fact that these conduits are discharging under water against a hydraulic counter pressure, partly it is due to the greater viscosity of the sewerage and partly also to the relative shortness of these conduits. In the case of the siphon aqueduct, its length is so short comparatively, that it can only be considered as a short pipe, conditions being much the same as in the case of the Holyoke Testing Flume. D. The variation of the coefficient c as deduced from experi- mental data relating to flow in conduits under pressure may be summarized as follows: 1. For long, straight conduits fairly clean, such as pipes of glass, tin, lead, galvanized iron, cast and wrought iron, planed staves, cement, riveted pipes up to 3 feet in diameter, the coeffi- cient of variation of c is equal to a = 7* and the frictional resistance is proportional to 7"*". 2. For pipes rectangular in section, for riveted pipes exceeding 3 feet in diameter, for those enumerated under (1) between 300 and 1,000 diameters in length, the coefficient of variation of c is equal to a = 7 fs and the frictional resistance is proportional to 7~*~. 42 THE FLOW OF WATER 3. For the classes of pipes enumerated under (1) and (2) discharging against a hydraulic counterpressure, or between 100 and 300 diameters in length, for old pipes not incrusted or tuberculated the coefficient c does not appreciably vary with the velocity, and consequently a = 1.0 and the frictional resistance is proportional to F 2 . 4. For incrusted pipes and those enumerated under (1) and (2) less than 100 diameters in length the coefficient of variation of c is equal to 1 a = and the frictional resistance is proportional to 3. For very heavily tuberculated pipes the coefficient of vari- ation of c is equal to 1 a = i v* and the frictional resistance is proportional to V s . In our collection of experimental data we find many instances relating to flow in one and the same conduit which do not fit any of the values of (a) enumerated above and which indicate : 1. First an increase in the value of c with increasing velocity up to a certain critical velocity. 2. Then a decrease in the value of c with increasing velocity. As instances of this kind we mention: Two new steel riveted pipes at East Jersey, 3.5 and 4 feet in diameter (Herschel). A cement lined pipe with elbows (Fanning). This peculiar variation of c indicates the presence of resistances which are proportional to powers of the velocity greater than 2.0, that is resistances which produce shocks. In case of the steel pipes the shocks are no doubt due to the rivet heads, in the second to the elbows in the line of the conduit. This peculiar variation of the coefficient c is also very plainly indicated in the data relating to flow in channels of rough boards with cleats nailed crosswise to bottom and sides of the channel OPEN CONDUITS 43 (Darcy-Bazin, series 12-17). These cleats or laths were 1 centi- metre thick and 2.5 centimetres wide. In one channel they were spaced apart 1 centimetre, in the other 5.0. The data relating to flow in the channel with the cleats spaced 1 centi- metre indicate the highest values of both the coefficients m and a, plainly showing the effect of the shock due to the wider spacing of the cleats. In the first case the coefficients are m = 0.41, a = V, in the second m = 0.03 a = , indicating that the frictional resistance was proportional in the first case to 7 1 ' 94 , in the second to F 2 ' 25 . Open Conduits. E. An analysis of experimental data relating to flow in open conduits of permanent cross-section, such as aqueducts, flumes, etc., indicates, that the coefficient c is affected in its variation with the velocity by the shape of the cross-section, or by the depth of the water in the channel. For semicircular or well rounded channels, for the semi-square when flowing full, for all sections for which the mean hydraulic radius is equal to half the depth, for the triangle with sides inclined 45 the variation of the coefficient c with the velocity does not seem to be affected by slight variations in the value of r. For rectangular channels, however, and others having very steep side walls (excluding those mentioned above) the varia- tion of c is affected by the depth of water in the conduit. The coefficient a seems to have its normal value in all instances when the depth of water is equal to one-half the mean width of the channel, it increases in value as the depth of water decreases, and decreases in value as the depth of water increases. This peculiar influence of the steepness of the walls of a conduit on the frictional resistance has been revealed by nu- merous current metre observations in rectangular flumes and aqueducts and other channels with steep side-walls. It has been 44 THE FLOW OF WATER found that in such channels the position of the thread of max- imum velocity is situated at a greater distance from the surface than in channels having side walls more inclined: thus clearly indicating the retarding influence of the steepness of the walls. Experimental data relating to flow in rectangular flumes fre- quently indicate values of the coefficient (a) as high as vf for small depths, its value is generally equal to v\ when the mean hydraulic radius is equal to one-fourth the width of the channel. Its value is less than the normal when the depth exceeds one -half the mean width of the channel. Applying the exponential equa- tion = log V t - log ^o ~ log (66 (Sir + m) Vrs\ - log (66 (Sir + m) Vr s) to data relating to flow in a semi-circular channel lined with neat cement (Darcy-Bazin, series 24) we find x = iJ ver y near - Applying the same equation to data relating to flow in channels lined with rough boards, semicircular in section, we find (Darcy-Bazin, series 26) x ^ it thus indicating a slight decrease in the value of a with increasing roughness of the conduit's wet perimeter. As a mean between these two values and differing but slightly from either we may take *- which corresponds to a = V or a =\/66 (Vr + m) Vrs and the frictional resistance is proportional to V"*~ = V ' This value of the power of the velocity we observe is the identical value Froude found for smooth plain surfaces in his investiga- tions on the resistance of ships. Besides the two series mentioned, given by Darcy-Bazin, we find that x = if holds also good for the following : OPEN CONDUITS 45 Darcy-Bazin, series 25, semicircular channel lined with smooth concrete. McDougall, Provo Canal Flume, semicircular channel of planed staves. Th. Horton, Conduit of North Metropolitan Sewage System of Massachusetts. Brickwork washed with cement. Diameter 9 feet. Values of R up to 2.31 feet. F. Applying the experimental equation as indicated above to data relating to flow in channels not semicircular in section and lined with cement or concrete, planed or rough boards, brickwork and good ashlar masonry we find x= If a = y 66 (\'r + m)V rs and the frictional resistance is proportional to F . This we find to hold good for the following : Darcy-Bazin, series 2, neat cement, section rectangular. Darcy-Bazin, series 6, 7, 8, 9, 10, 11, 18, 19, 21, 22, and 23, sawed boards, section rectangular, triangular or trapezoidal. Darcy-Bazin series 32, 33, 39, channels lined with good ashlar masonry, section trapezoidal. Darcy-Bazin, series 3, rough brick work, section rectangular. Darcy-Bazin, series 4, channel lined with pebbles up to J inch in diameter, section rectangular. Fteley and Stearns, Sudbury conduit, very good brickwork, sides of channel nearly vertical, bottom flat arch. Fairlie Bruce, Aqueduct of Glasgow, smooth concrete, sides of channel nearly vertical, bottom flat arch. Th. Horton, Conduit of North Metropolitan Sewage System of Massachusetts, brickwork washed with cement, covered with sewer slime, sides of conduit vertical, bottom flat arch. 46 THE FLOW OF WATER Lippincott, San Bernardino Canal Trapezoidal channels in earth, lined with concrete. Kutter, Gontenbachschale, new and well built channel of dry rubble masonry. Passim and Gioppi, Aqueduct of the Cervo, Canal Cavour. Floor of concrete, sides of brick, section rectangular. Values of R up to 7.2 feet. G. Applying the exponential equation as indicated to data relating to flow in channels having walls possessing a greater degree of roughness than those enumerated above we find x = 1.0 a = 1.0 and the frictional resistance is proportional to v 2 . This, amongst others, holds good for the following : Darcy-Bazin, series 1, 34, 35, channels lined with roughly hammered stone masonry. Darcy-Bazin, series 5, channel lined with pebbles 1J inch to 1J inch in diameter. Kutter, numerous channels lined with dry rubble masonry. Perrone, Torlonia drain tunnel, channel in rockwork, partly lined with rubble masonry. We mention here also: Cunningham, Aqueduct of the Solani, Ganges Canal. Floor of brick, laid flat, sides of masonry, length 920 feet. In this case the fact that c does not vary with the velocity of flow is due to the shortness of the conduit. It has no independent slope and the movement of the water is influenced by the greater resistance in the rough channel in earth downstream. This is plainly indicated by the low value of the coefficient m. Of open conduits, not channels in earth, there are few possess- ing a degree of roughness still greater than those enumerated, exceptional cases of old and damaged rubble masonry. CHANNELS IN EARTH 47 H. The variation of the coefficient c with the velocity of flow as deduced from experimental data relating to flow in open conduits not channels in earth may be briefly summarized as follows: 1. For semicircular channels lined with cement, concrete, good brickwork, planed or rough boards, the value of the coeffi- cient a is equal to F n . 2. For rectangular, triangular or trapezoidal channels of the same description, for channels lined with rough brickwork, ashlar and very good rubble masonry, for channels lined with pebbles up to J inch diameter the value of the coefficient a is equal to F n . 3. For channels lined with roughly hammered stone or common rubble masonry, for channels lined with pebbles up to 1^ inch in diameter, for channels in rockwork, for aqueducts of any description discharging into channels in earth and having no independent slopes, the value of the coefficient a is equal to 1.0. 4. For channels with obstructions producing shocks, such as channels with cleats nailed crosswise to retard the flow, for channels lined with old and damaged masonry the value of the coefficient a is equal to j _ 1 Channels in Earth. I. When we scrutinize the data relating to flow in rivers and other channels in earth we perceive that these data contain many irregularities and contradictions which make them appear doubt- ful and untrustworthy. Even those given by the best authori- ties are not entirely free from anomalies. These irregularities and contradictions are occasionally the result of inaccurate measurements; more often, however, they must be attributed to the unstable character of the beds of these channels. This instability of the bed of the channels makes the phenomenon of flow a problem of great complexity. An exact valuation of all the facts entering is as yet, with the incomplete data at present available, out of the question. We here leave the path 48 THE FLOW OF WATER of exactitude and enter a labyrinth, satisfied if we come out with the gain of an increment of knowledge which may prove useful. Natural and artificial channels in rock work or earth may be divided according to the stability of their beds, into three classes : 1. Channels having beds in a regime of stability at velocities exceeding the ordinary. Channels in rockwork, cemented gravel, channels in earth protected by riprap or masonry side walls. 2. Channels in a regime of stability at ordinary velocities. Channels in gravel, stiff clay, clayey loam, sandy soils with over 50 per cent clay. 3. Channels in a regime of instability at ordinary velocities. Channels in sand, sand with fine gravel, sandy loam with less than 50 per cent clay. The beds of the second and third class are in a regime of stability until the velocity becomes sufficiently great to erode the bed. The velocity at which erosion begins varies with the cohesion of the material. In channels in sand, sandy gravel, sandy soils with small percentages of clay, erosion begins at very low velocities; these channels are consequently very unstable. Omitting channels in firm rock or cemented gravel, the stability of the bed depends mainly on the percentages of clay in the material. According to W. A. Burr pure clay resists erosion up to a velocity of 7.35 feet per second. The following table, based chiefly on Burr's experiments, gives the mean velocities at which erosion begins: Nature of Material Forming the Bed. Mean Velocity. Fine sand 72 Coarse sand, sand with pebbles up to pea size 1 10 Sandy soil 15 per cent clay 1 20 Fine gravel up to inch in diameter 1 50 Sandy loam 45 per cent clay 1 80 Common loam 65 per cent clay 3 00 Gravel or pebbles from ^ to 1 inch in diameter . 3 15 Coarse gravel ... .... 4 00 Clayey loam, 85 per cent clay 4 80 Clay soil, 95 per cent clay, loose rock 6 .20 Stratified rock slaty rock 7 45 Hard rock 12 00 CHANNELS IN EARTH 49 In the process of erosion energy is consumed which varies with the specific gravity and the cohesion of the material. The erosive power of a current is proportional to the square of its speed. Its transporting power, however, varies (according to Le Conte) : When the surface is constant with v 2 . When the velocity is constant with the surface of the object or with d\ When both vary the assistance is equal to v 2 d 2 . But the weight of the object is proportional to'd 3 . Hence, when the forces are in equilibrium or the weight equal to the energy d 3 = v 2 d 2 . Dividing by the surface or d 2 we have d = v*. Consequently when the forces are in equilibrium the resistance is proportional to v 6 . In other words, the transporting power of a current is proportional to the sixth power of the speed. This indicates that powers of r ranging between 2 and 6 enter the problem of flow when erosion begins. With the beginning of erosion the destruction of the bed will be the greater ; the less the cohesion of material the greater the velocity. Changes and alterations in course and section generally continue till a channel is formed which, owing to its greater length, its deflections, curves and bars offers such resistances that the power of the current is reduced and course and section again become stable when force and resistance are in equi- librium. A stream will pick up material in a narrow, deep section of its course where the force of the current is great, and deposit it in a wide and shallow section where the current is feeble. At high water, the greater depth of the water in the shallow section will result in greater velocities, the material previously deposited will again be put in motion and carried to a place where the current is feeble. The work done during these processes of building and rebuild- ing cannot be accurately measured, and on this account slope formulae, when applied to flow in channels where erosion is going on, are always more or less deficient. They cannot be depended on in computing discharges; this falls into the province of the 60 THE FLOW OF WATER current metre and the rod float. They are useful, however, as a guide to the engineer in the design of new conduits, alterations in courses or sections, etc., etc. The banks of channels having unstable beds are frequently protected by riprap or masonry walls. Frequently the bottoms of such channels are also protected by artificial bars made of boulders or masonry. Rittinger, Borneman, Epper, Cunningham, and others, have given us data relating to flow in such channels. An analysis of these data gives surprising results. Using the exponential equation x = log v l - log y log r t - log r we find the following values of x, the power of the mean hydraulic radius to which the velocity is proportional: Rittinger, millrace of dry rubble side walls, bed very rough, depth of water 0.40 to 0.90, x = 3.0. Rittinger, mill race, bed sand and gravel, side walls of masonry, depth 0.28 to 0.90 ft., x = 1.77. Rittinger, Aqueduct in earth lined with dry rubble side walls, depth 0.61 to 1.27 feet, x = 1.19. Cunningham, Solani Embankment, sides of masonry built in steps, bed of clay and boulders, with frequent artificial bars to prevent erosion. Main site, width, 150 to 170 ft.; depth of water, 1.7 to 4.1 ft., x = 1.49; depth of water, 5.6 to 9.34 ft., x = 0.9; Jaoli site, depth, 6.8 to 8.1 ft., x = 0.93. Excluding extremes, the powers of R, to which the velocity is proportional as expressed in these data, may be given by the equation x = 1.8 - O.IR so that for R = 1.0 x = 1.7 R = 2.0 x = 1.6 R = 9.0 x = 0.9. A high value of x indicates a low value of the coefficient c, but a rapid increase in its value with increasing value of R] a CHANNELS IN EARTH 51 low value of x indicates a high value of c and a slow increase in its value with increasing values of R. The influence of the roughness of the bed is necessarily much greater when the water in the channel is shallow than when it is high; the diminishing values of x indicate a rapid decrease in the relative influence of the character of the bed. But, on the other hand, while the powers of R are abnormally high for shallow water in rough channels, the powers of the sine of the slope. to which the velocity is proportional are abnormally low. This may be illustrated by data deduced from experimental values relating to flow in rough channels in earth. Amongst others we find: Wampfler, Simme Canal, coarse gravel and detritus, #1.104 0.2 3> La Nicca, Rhine in the Forest, coarse gravel and detritus, depth 0.42 to 0.9 feet. R?' g S ' 4 . La Nicca, Plessur River, coarse gravel to detritus, depth 1.25 to 4.58 feet. #' 64 S ' 4 . Darcy-Bazin, Grosbois Canal, Chazilly Canal, channels in earth, with stones and vegetation, depth 1.5 to 3.0 feet. #0.87 0.43 to #1.59 0.4^ Reich, River Salzach, gravel and detritus, depth 3.53 to 7.39 ft. #0.8 0.333^ Funk, Weser River, depth 4.5 to 11 ft. ft ' 79 S ' 5 . Villevert, River Seine, depth 5.66 to 18.39 ft. R ' 63 S' 443 . In general therefore, for shallow water in rough channels the power of the sine of the slope to which the velocity is proportional is equal to 0.4 and equal to 0.473 for depths exceeding 4 feet. The variations in the powers of both r and S with the depth of the water in the channel are chiefly due to the fact, that the bottoms of such channels are in most cases much rougher than the sides. In shallow water, the resistance due to the bottom preponderates, with increasing depth the influence of the less rough sides more and more reduces the mean resistance per unit area of surface. The powers of r vary not only with the degree of roughness 52 THE FLOW OF WATER in general and with the depth of the water, but also with the value of a, the coefficient of variation of c. % For the same degree of roughness, the powers of r have their highest value for the highest value of a. For m = 0.33 or K = 2.0 for instance, and a = 1.0 R* = R ' 795 . But for a = V" R- = R ' 835 for a = A R- = # 9 ' 745 yrs for a = A- R* = R ' 66 . V* This shows the great influence of bends, bars, or other impedi- ments on the powers of R. Our general equation expresses the variation of the powers of r with the depth with a fair degree of accuracy. Greater accuracy is obtained if the formula is put into the form c = 66 f \/r + ("2" 1 + ^r JJ and giving m a negative value, as for instance : for K = 1.20 m = - 0.10 for K = 1.50 m = - 0.20 for K = 2.0m = - 0.33. For values of R less than 1.0 foot the formula 66 (\/r + Vf) Vr+K gives slightly excessive results. Amongst the mass of experimental data accumulated during recent years those given by Fortier for irrigation channels are, considered from the practical standpoint, the most valuable. They relate to flow in channels possessing all possible degrees of roughness and a minute description of the nature of the bed is always given. Gaugings were, however, taken only for a single depth and a single slope at each section and on this account no deductions can be made in regard to the variation of the coefficient c with the velocity. CHANNELS IN EARTH 53 Besides these Dubuat, Darcy-Bazin, Legler, Cunningham, Rittinger and others have given valuable data relating to flow in canals to ditches; Funk, Villevert, Revy, Gordon and the U. S. Engineers, interesting data relating to flow in rivers. After a careful analysis of all the material available we come to the following conclusions in regard to the variation of the coefficient c with the velocity: 1. For channels of fairly regular cross-sections and courses having tolerably smooth beds, such as channels in firm clay, clayey loam, sandy soil with over 50 per cent clay, fine cemented gravel, the coefficient c increases at ordinary velocities with the velocity of flow. Under ordinary velocities in this sense we understand velocities which do not cause erosion. The increase in the value of c with increasing velocities is equal to a = V h for the smoothest down to a = V for the roughest ohannels of this class. Examples : S. Fortier, Bear River Canal Branch. S. Fortier, Providence Canal. S. Fortier, Solveron and Logan City Canals, Utah. Darcy-Bazin, rectangular channel lined with pebbles up to J inch diameter. Epper, millrace, channel in earth, bottom covered with fine gravel. Dubuat, Canal du Jard. Channel in earth. Reich, River Salzach, reach very regular. 2. At velocities exceeding the ordinary, or when erosion begins, the coefficient c decreases in value for the classes of channels enumerated above. The decrease is usually such that 1 a = jr m 54 THE FLOW OF WATER Examples : Legler, Linth Canal. The coefficient c increases until v is equal to 4.72 ft. per second, then decreases. Gordon, Irrawaddi River. The coefficient c increases until v is equal to 2.62 ft. per second, then decreases. In the first case the bed is firm earth, in the second sand. 3. For channels of fairly regular cross-section and course in rock work, firm gravel up to 2 inches diameter, for channels in firm earth or sand, or sand with gravel, with stones or vegeta- tion, the coefficient c does not appreciably vary with the velocity of flow. Consequently a = 1.0. Examples: Perrone, Torlonia Drain tunnel, channel in rock work. Darcy-Bazin, series 5, rectangular channel lined with pebbles up to 1^-inches diameter. Darcy-Bazin, series 36, 37, 38, 41, 43, 47, 48, 50, Grosbois and Chazilly Canals. Channels in earth of regular cross-section but with stones or weeds. La Nicca, Moesa River, coarse gravel. La Nicca, Plessur River, coarse gravel. Funk, Weser River. Passini and Gioppi, Canal Cavour, below the Syphon of the Sesia. 4. For the class of channels enumerated under (3) the co- efficient c decreases in value whenever the velocity becomes sufficient to cause erosion. The decrease usually corresponds to 1 a = r ' V* 5. For channels with very rough beds, channels with boulders, loose cobblestones, loose coarse gravel or detritus, for channels with artificial bars to prevent scour, the coefficient c decreases rapidly in value with increasing velocities. The decrease is equal to 1 a = ? V* CHANNELS IN EARTH 55 Example : Cunningham, Solani Embankment, bed in clay and boulders with artificial bars to prevent erosion, sides of masonry. Omitting the extremes, we may briefly sum up the variation of the coefficient c with the velocity as follows: 1. For channels of very regular cross- sections and courses in clay, clayey loam, sandy soils with large percentages of clay, cemented gravel up to one inch in diameter, the coefficient of variation of c is equal up to the eroding limit to a = 2. For channels in rock work or cemented gravel exceeding one inch in diameter, for ordinary channels in earth, channels with some stones or vegetation, the coefficient a is equal up to the eroding limit to a = 1.0. 3. For channels in sand at any velocity and for all others at velocities exceeding the eroding limit, the coefficient c decreases in value with increasing velocities and the coefficient of variation is fairly equal to 1 a = j- vt* K. In a preceding chapter we have mentioned the experiments made by Hubbel and Fenkell, Saph and Schoder to determine the loss of head due to the resistance in curves. From data r> given by them we computed, that, omitting values of -j less (jL than 2.5, the friction per unit length of curve, in terms of the friction per unit length of tangent is equal to 56 THE FLOW OF WATER and the excess of friction per unit length of curve in terms of tangent friction is equal to and the length of tangent equal in the amount of frictional resistance to the frictional resistance in a curve of 90 equal to 0.5 xR U.Qd* 3 ^M - i.o. ~R 7? This vanishes when -y = 4.9 3 d, it is a maximum when -r d a = 4.9 3 d and the total excess of friction is greatest. loss of head due to any curve is consequently The ) 2 r 2g TABLE IV. FRICTION IN CURVES. 2.5 4 5 6 10 15 20 25 50 100 2.5 4 5 6 10 15 20 25 50 100 Values of |4.9d* f-1 ) 1.0. Diameters 1 to 72 Inches. I" 2" 4" 6" 12" 18" 24" 30" 36" 48" 60" 72" 0.375 0.777 1.422 1.907 2.971 3.360 3.657 3.903 4.113 4.462 4.750 4.996 0.271 0.595 1.174 1.609 2.564 2.913 3.080 3.400 3.588 3.903 4.160 4.382 0.225 0.515 1.065 1.478 2.386 2.717 2.971 3.180 3.359 3.657 3.903 4.113 0.188 0.453 0.980 1.377 2.247 2.565 2.808 3.009 3.180 3.466 3.701 3.903 0.091 0.292 0.761 1.113 1.887 2.170 2.388 2.564 2.717 2.971 3.186 3.369 0.020 0.177 0.604 0.925 1.630 1.887 2.084 2.247 2.386 2.617 2.808 2.971 0.101 0.501 0.802 1.461 1.703 1.887 2.039 2.168 2.386 2.564 2.717 0.046 0.426 0.712 1.337 1.567 1.742 1.887 2.010 2.216 2.386 2.531 0.216 0.460 0.994 1.189 1.338 1.462 1.567 1.743 1.887 2.010 0.037 0.244 0.700 0.867 0.994 1.099 1.189 1.348 1.461 1.565 Values of z. Curve of 90 degrees, m = 0.95. 0.049 0.091 0.151 0.184 0.249 0.258 0.263 0.266 0.268 0.270 0.271 0.272 0.057 0.112 0.210 0.248 0.344 0.358 0.366 0.371 0.375 0.378 0.380 0.382 0.059 0.121 0.227 0.285 0.401 0.416 0.427 0.432 0.438 0.443 0.446 0.448 0.059 0.128 0.251 0.319 0.451 0.472 0.484 0.492 0.478 0.504 0.507 0.510 0.048 0.137 0.325 0.429 0.634 0.666 0.686 0.699 0.710 0.720 0.727 0.734 0.015 0.147 0.386 0.535 0.821 0.869 0.899 0.918 0.935 0.951 0.963 0.971 0.095 0.427 0.619 0.982 1.046 1.085 1.111 1.133 1.157 1.173 1.184 0.054 0.454 0.687 1.123 1.203 1.252 1.296 1.313 1.343 1.364 1.379 0.460 0.888 1.670 1.825 1.924 2.001 2.047 2.113 2.157 2.190 0.158 0.931 2.352 2.836 2.858 2.995 3.107 3.268 3.340 3.410 CHANNELS IN EARTH 57 TABLE IV. A. WEISBACH'S COEFFICIENTS FOR RESISTANCES DUE TO ENTRANCES, ELBOWS, CURVES, CHANGES OF SECTION, ETC., ETC. Values of z. Description of Resistance. 0.054 Funnel-shaped or bell-mouthed entrance not pro- truding into the reservoir. 0.505 Well rounded entrance not protruding into the reservoir. 0.505 Funnel-shaped or bell-mouthed entrance protruding into the reservoir. 1.957 Ordinary pipe protruding into the reservoir. 0.9457 sine 2 1 + 2.047 sine 4 ^ Elbows d = angle of deflection. o.i3i + uW^y Curves. Section circular, d = diameter, R = radius of curve. 0.124 + 3.104^^ Curves. Section rectangular, d = Width of side parallel to R, the radius of the bend. a-'J n . . Section contracted Section not contracted a = 1.225 + 1.45 w 2 - 1.675m. (*-') Enlargements or Contractions. A l Section not contracted, A 2 = Section contracted. & Bends of Rivers, n = Number of degrees in arc of bend. i - B7i (^?- 1 ) Obstructions in Rivers, m = Percentage not ob- structed. v x being equal to V~* , V^~, V~, etc., according to the degree of roughness of the conduit. The coefficient of frictional resistance is given by + m) 2 r 58 THE FLOW OF WATER in these equations n = number of degrees in curve. TT = 3.1416. d = diameter of conduit in feet. R = radius of curve in feet. x = -$ for diameters greater than 1 foot. x = 0.45 for diameters less than 1 foot. y = | for a diameter of 1 inch. y = A f r an y ther diameter. From the foregoing we draw the conclusion, that the value of z depends: r> 1. On the value of -7- and the value of d. a n 2. On the value of - 3. On the value of m. For any arc, multiply the values of 2, found in the table, by the number of degrees and divide by 90. For any degree of roughness multiply the values of z by the following: m = 0.95, multiply by 1.0. m = 0.83, multiply by 1.166. m = 0.68, multiply by 1.436. m = 0.53, multiply by 1.802. m = 0.45, multiply by 2.060. m = 0.30, multiply by 2.717. If in the formula for the loss of head due to a curve we substitute 2 grs , .. . , 0.01478 for its equivalent -r-= - V 2 ( Vr + m) 2 and L for the length of the curve the formula will read, after reduction, RIVETED CONDUITS 59 which simply expresses the theory outlined at the beginning of this chapter that the excess loss of head due to a curve is / 4 .9 f fCA - 1.0 times the loss due to an equal length of straight pipe; S being the sine of the slope to which velocities in the tangent are due. Riveted Conduits. L. Riveted conduits form a class apart in so far as the degree of roughness varies with the diameter. Up to date the coefficients for such conduits have been fairly well determined for diameters up to 8.5 feet (Holyoke Testing Flumes); for larger sections they are as yet problematical. Fairly reliable values of the coefficients for riveted conduits may be found by computing the losses of head due to the resistance of rivet heads, or to enlargements and contractions of the section as follows : If in an 18-foot steel-riveted pipe we allow an internal pressure of 140 pounds per square inch, in the steel a tension of 20,000 pounds per square inch; and if we assume the efficiency of the riveted joints to be 70 per cent of the metal, we have for the thickness of the metal in inches 140 X diameter in inches, 0.7 X 40,000 which gives t = 1.08 inches. It is usual to take for the diameter of the rivet in inches d = 0.15 + 1.5 t, and for the pitch of the rivets in a single row s l = 0.375 + 2 d, and s 2 = 0.75 + 3 d for the pitch in a double row. 60 THE FLOW OF WATER Hence in our case d = 1.75, s, = 3.875, s 2 = 6.0. The usual diameter of the rivet head is 1.8 d and its depth 0.6 d. This gives for the sectional area of the rivet at right angles to the line of flow 3.15 X 1.05 = 3.3075 square inches nearly. As the circumference of the conduit is 12 X 18 X 3.14 = 678.25 inches and the spacing 3.875 inches, there will be 175 rivets in the single circumferential row. The open space between the rivets will only be 3.875 - 3.25 = 0.725 inches. The dis- turbance in the motion in this narrow space will be such, that it will be safe to consider the row of rivet heads as an unbroken line of a depth 0.6 d = 1.05 inches. Weisbach gives for the loss of head due to constrictions -> 2g in which A l = section not constricted, A 2 = section constricted, a = 1.225 + * - 1.695 - \4J A i In our case A, = 18 2 X 0.7854 = 254.34, A 2 = (17.825) 2 X 0.7854 = 249.5. Inserting these values in Weisbach's formula we find h = 00187489 ^- - Assuming the metal sheets to be 10 feet each way there will be six sheets in the circumference, and as the pipe is double riveted longitudinally there will be twelve longitudinal rows of rivets, and allowing 1.6 d for the outside rim on each side there RIVETED CONDUITS 61 will be twenty circumferential rows, the pitch being six inches. The twelve rivets in each row will cause a constriction of 12 x 3.3075 = 39.69 square inches = 0.275 / 2 . According to Weis- bach's formula this constriction causes a loss of head equal to h = 00005936^-, and the twenty rows a loss equal to h = 0001 1907 ~ Adding the resistances due to all the circumferential rows in a section of 9.5 feet we have Z, = 00187489 + 00011907 = 00199396. Assuming the conduit to be 20,000 feet long the total resistance due to the rivet heads will be 20000 Z.= - = 2105 X 00199396 = 4.196985. To this must be added the resistance due to the enlargement or contraction caused by the circumferential lap of the sheets. As the thickness of the metal is 1.08 inches the diameter is enlarged or contracted 2.16 inches at each lap. The loss of head due to enlargements or contractions is, according to Weisbach, r / 216" \ 2 V V 2 V 2 hence in our ease [ ( m ^J - ij ^ = 00041209 ^ The total resistance due to all the enlargements or contractions is consequently Z 2 = 2105 X 00041209 = 0.86755. If the conduit had no rivet heads or enlargements and con- tractions to increase the resistance, the value of the coefficient m 62 THE FLOW OF WATER would be the same as for a cast-iron pipe, or equal to 0.83, and the frictional resistance per unit area of surface would be 0.01478 /=! (1.456 + 0.83) 2 ' and the total resistance of the wet perimeter Z 3 = 002829 , .. = 12.473. 4.5 Adding, we have for the sum of all the resistances Z l + z 2 + z 3 = 17.5375. This gives for the total frictional resistance per unit area of surface 17 . 5325 x ^ or / = 00394594; hence the coefficient c is equal to y ' = 127.7, and m is ( M ) "iVy^-rOr/ / 127 7 equal to -^ 1.456 = 0.48. OD Practical Applications of the Formulae M. 1. From the formula v = (66 (Vr + m) we have and We have also 1 V -i' v _ ^66(^r+ra) Vr ] Vr + m)Vr = Rl 4- m Vr. - 66 Vs PRACTICAL APPLICATIONS OF THE FORMULAE 63 Putting 'Vr = x and transposing we have X 3 + mX 2 + 5-^-=- = 0, " *- 66 Vs from which the value of x = *vV is found by Horner's method. We have also m = 66 If the coefficient of variation of c is equal to 7^, V^* j- etc., these values are substituted in the given equations. Values of a =V% 7*, 7* y^ y& yrV > are found in Table V. Example: Let it be required to find the slope for a rectangular aqueduct of common brickwork or concrete 100 feet wide, 12.5 feet deep, the velocity to be 4 feet per second. The cross- sec- tion is 1,250 / 2 , the wet perimeter 125 /, hence R = 10.0. In the table of roots of mean radii we find \/10 = 3.163 VlO = 1.78. The value of m for common brickwork or concrete is 0.57. The value of a = 7 T * for v = 4.0 is, according to Table V, equal to 1.08. Inserting these values into our formula we have for the slope r _ 4 _ 7 Ll.08 X 66 X (1.78 + 0.57) X 3.163J V530 = 0.0000569. Example: Let it be required to find the diameter of a semi- circular channel lined with common ashlar or very good rubble masonry, the slope being 1 in 1,000, and the permissible velocity 10 feet per second. In this case m = 0.30 V7 = 0.0316 a = ^To = 1.137. 64 THE FLOW OF WATER Solving by Homer's method and inserting values we have X 3 + - - 1.187 XMX0.0316 0.3 X 2 + 0.0 - 4.217 = 0.0 \c = 1.521 1.0 + 1.3 + 1.300 - " 1.3 1.0 + 1.3 - 2.917 + 2.3 2.3 1.0 + 3.6 3.3 0.5 + 3.6 + 1.9 - 2.917 + 2.750 3.8 0.5 + 5.5 + 2.15 - 0.167 4.3 0.5 + 7.65 4.8 0.02 + 7.65 + 0.096 - 0.167 + 0.1544 4.82 + 7.746 - 0.0121 0.001 + 0.005 + 0.0077 4.821 + 7.751 - 0.0044. This gives x = 1.521; hence the mean hydraulic radius r (1.521) 4 = 5.352, and the diameter = 21.408 ft. 2. From the formula 2gh 1ft 0.01478 L tlr + m) 2 ~R v = we have 2gh 0.01478 L _ 0.01478 L r 4/ \~2 D ~O L V + m) 2 R = 0.01478 -=7 ^ /i z g m )Vr = R* + mVr - V 0.01478 PRACTICAL APPLICATIONS OF THE FORMULAE 65 and putting x = t/r we have ho-yo.r 1 "*" L X 3 + m X 2 + 0.0 - V 0.01478 ^ = 0.0, a. 2 g which may be solved by Homer's method. To facilitate calculations it is well to remember that y&=7*x 7*. and 7V-/4Y V7V Values of 7 * and 7^ are found in Table V. Resistances due to entrance and the velocity itself are included in the term ^= and need not be further considered (Vr 4- m) 2 R unless the length of the conduit is less than 1,000 diameters. For pipes between 300 and 1,000 diameters in length (as also for riveted pipes exceeding 3 feet in diameter), the coefficient of variation is equal to a = 7^, and A and V are substituted in the given equations for A and V. If the pipe is between 100 and 300 diameters in length (or an old pipe not very clean) the coefficient a is equal to 1.0, and % and 2.0 are substituted in the given equations for A and V 6 - In case the conduit is less than 100 diameters in length the coefficient a is equal to j- and A and V are substituted for T 9 = 5.0. In Table IV we find the coefficient z 2 for - = 5.0 and a curve of 90 to be equal to 0.466. As there are 20 curves of 10 degrees we have Z 2 = 2Q X X ' 448 - 0.995. For m = 0.53 yu 66 THE FLOW OF WATER this is to be multiplied by 1.802, which gives for the total resist- ance due to curves Z 2 = 1.782. Inserting values into our formula we have 64.4 X 5.0 v = 0.01478 X 10000 J1.107 + 0.53) 2 X 1.5 = (8.312)* Remembering that V* 7 = 7* X V& we first draw the square root out of the quotient and multiply this by the seventeenth root of the square root. The quotient is 8.312, V8312 = 2.884. In the table of roots we find ^3 = 1.065, X/2.75 = 1.059. Interpolating we have for *!J 2^884 = 1.062. Consequently!; = 2.884 X 1.062 = 3.0628 feet per second. 3. From the formula v = (66 (r + m) we have for the discharge of a circular conduit in cubic feet per second Q = (66 (j/r + m) V77) * d 2 0.7854. From this we have for the head in feet _y_ -i Ti. 0.7854/ 66 ($r + m)J R and for the diameter in feet ).7854/ [66(-Vr + m)] 2 H If a = 7 T * the index if is substituted for |, if for f , V for V 6 and A for 4 9 T- From this equation the value of d can only be found by trial, assuming a value of $lr in the term ^Ir + m. For a first trial a value of $r = 1.0 will give good results. From the formula 2gH 0.01478 L f 4/ \ o -W-*L .vr + m) 2 R PRACTICAL APPLICATIONS OF THE FORMULAE 67 we have Q 2gH 0.01478 L ( Vr + m) 2 R d 2 0.7854 0.01478 L A. i I H _( _ __\ r + \ d 2 0.7854/ 2g 0-01478 , 7 p, 7 p" ' * From this equation also d can only be found by a second or third trial, assuming a value of Vr and R. For a = FT* the index T 9 7 is substituted for tV, V for V and A for 4 9 T- For a = 1.0 the index is substituted for y 9 s, 2 for V and A = t for A. Example: What will be the loss of head corresponding to a discharge of 5 cubic feet per second, the conduit being a 2-foot riveted pipe 20,000 feet long and having 30 curves of 15 each and a radius of 20 feet. In this case m = 0.68; R = 0.5; Vr = 0.84, consequently 0.01478 0.01478 (Vr+TO) 2 (0.84 + 0.68) 2 In Table IV we find for the resistance of a 90 curve for the 7? OC\ relation = - = 10 Z 2 = 0.686, consequently for 30 curves d of 15 each Z 2 - 30 X 15 X ' 686 - 3.430. For m = 0.68 this yo is to be multiplied by 1.436 which gives z 2 = 4.925. Inserting these values we have 5 [0.00638 X -- + 4.925 L4 X 0.7854J 64.4 or H = 9.24 feet. 68 THE FLOW OF WATER 5. The Kinetic energy or living force acquired by a body falling free or descending in a plane infinitely smooth is equal to E = \rntf = Q.W.H., Weight m = the mass of a body = -= ^ , Gravity Q = the discharge in cubic feet per second, W = the weight of one cubic foot, H = the total fall in feet. Expressed in horsepowers the energy is equal to or, in kilowatts, to K.W. If a body of water is not falling free, the total head is reduced by an amount which depends on the velocity, the length of the conduit, its diameter and its degree of roughness. The loss of head is equal to V 2 0.01478 2g 2g 2 g (t/r +m) 2 R as the case may be. For conduits of equal length the loss is evidently the least for the greatest diameter and for the lesser speed of flow. For a given diameter the efficiency of a conduit as a trans- mitter of energy is greatest when the speed of flow is such, that one-third of the total available head is consumed in overcoming frictional resistances (see " Adams and Gummel," Eng. News, May 4, 1893) . Example: A new four foot steel riveted conduit 2,000 feet long, under a head of 300 feet is to deliver water to the gener- ator at such a velocity that its efficiency will be a maximum. What will be the discharge and the horsepower transmitted? PRACTICAL APPLICATION OF THE FORMULAE. 69 Allowing one-third of the total head to be spent in overcoming 100 frictional resistances we have v = = 0.05. For this value zooo of v the velocity will be v = (66 (1 + 0.53)* Vl X 0.05) & = 2.369 feet per second; the discharge, Q = 2.369 X 16 2 X |j =30.4 cubic feet per second; r> 30.4 X 62.4 X 200 _ ftQ n the horsepower, H P = = 708.0. o5u TABLE V. Table V contains roots of velocities or values of (a), the coefficient of variation of c. To find the value of c corresponding to any velocity multiply the value of 66 (t/r + m) by the value of (a) = 7*, 7 T \ r-.as the case may be. 1 To find the velocity multiply the value of 66 (^r 4- m) Vr.s by the value of (66 ( Vr + m) VTJS) * which in Table V is given as 7*, (66 ( V r + rn) Vr.s) " which in Table V is given as (66 ( Vr + m) Vr.s) T7 which in Table V is given as T-= . which in Table V is given as (66 ( Vr + m) Vr.s) A as the case may be. Also , to find the velocity, multiply the value of which in Table V is given as f- ' R 70 THE FLOW OF WATER TABLE V. ROOTS OF VELOCITIES OR VALUES OF (a), THE COEFFICIENT OF VARIATION OF c. V V* V \ V* yfs yk yh jrA 1 F A 1 yr& 0.25 0.841 0.857 0.882 0.891 0.917 0.925 0.925 1.081 1.075 0.50 0.918 0.925 0.939 0.944 .958 0.959 0.961 1 .040 1.037 0.75 0.964 0.964 0.974 0.976 0.982 0.981 0.982 1.018 1.015 1.0 1.0 1.0 1.0 1.0 1.0 1 .0 1.0 1.0 1.0 1.25 1 .026 1.025 1 .0205 .019 1.013 .013 1.013 0.987 0.989 1.50 1 .052 1.046 1.038 .034 1.025 .024 .022 0.978 0.979 1 .75 1 .072 1.064 1 .052 .048 1.035 .033 .031 0.97 0.971 2.0 1 .090 1.080 1.065 .0594 1 .044 .042 .039 0.962 0.964 2.25 1 .107 1 .095 1.077 .070 1 .052 .049 1 .046 0.956 0.959 2.50 1.121 1.107 1 .087 1 .080 1.059 .056 1.052 0.95 0.953 2.75 1 .135 1.118 1.096 1.086 .065 .061 1.058 0.945 0.948 3.0 1 .147 1.130 1.105 1.096 .071 .067 1.063 0.940 0.943 3.25 .159 1.140 1.113 1.103 .076 .072 .068 0.936 0.940 3.50 .170 1.150 1 .121 1 .110 .081 .077 .072 0.932 0.936 3.75 .18 1.158 1.128 1.116 .086 1.081 .076 0.929 0.933 4.0 .189 1 .166 1 .134 1.123 .090 1.085 .080 0.926 0.930 4.25 .198 1 .174 1.141 1 .128 .094 1.088 .083 0.923 0.927 4.50 .207 1 .182 1 .146 1 .133 1 .098 1 .093 .087 0.919 0.924 4.75 1 .215 1 .189 1 .152 1 .139 1.102 1 .096 .090 0.917 0.921 5.0 1 .223 1 .196 1.158 1.144 1.106 1 .10 .093 0.914 0.919 5.25 1 .231 1.203 1.163 1 .149 1.109 1 .103 .098 0.912 0.917 5.50 1 .237 1.209 1.168 1.153 1 .112 1.106 .099 0.910 0.914 5.75 .244 1.215 1.173 1.159 1 .115 1.108 .102 0.907 0.912 6.0 .251 1 .220 1.177 1.161 1 .118 1.110 .104 0.905 0.910 6.25 .258 1 .226 1.181 1 .165 1 .121 1 .114 .107 0.903 0.908 6.50 .262 .231 1.186 1 .169 1.123 1 .116 .109 0.901 0.906 6.75 .269 .237 I .190 1 .173 1.126 .119 .112 0.899 0.904 7.0 .275 .241 1 .194 1 .176 .129 .121 .114 0.897 0.902 7.25 .281 .246 1.197 1.18 .131 .124 .116 0.895 0.901 7.50 .286 1.251 1.201 1.183 .134 .126 1.118 0.894 0.899 7.75 .292 1 .256 1.205 1 .186 .137 .128 1.12 0.893 0.898 8.0 1 .297 1 .260 1 .208 1.189 .139 .130 1 .122 0.891 0.896 8.25 1.302 1 .264 1.212 1 .192 .141 .132 1 .124 0.889 0.895 8.50 1.307 1 .268 1.215 1 .195 .143 .134 1 .126 0.888 0.893 8.75 1 .311 1 .272 1.218 1.198 .145 .136 1.128 0.886 0.892 9.0 1 .316 1.277 1 .221 1.201 .147 .138 1 .130 0.885 0.891 9.25 1 .320 1.280 1 .224 1 .204 .149 .140 1.131 0.884 0.889 9.50 1.325 1 .284 1 .227 .207 .151 .142 1.133 0.882 0.888 9.75 1.329 1.288 1 .230 .209 .153 .143 1 .135 0.881 0.887 10.0 1.333 1 .292 1 .233 .212 1.155 .145 1 .137 0.880 0.886 10.5 1 .341 1 .298 1.238 .216 1.158 .149 1 .139 0.877 0.884 11 .0 1 .348 1 .305 1.244 .221 1 .161 1 .152 1 .142 0.875 0.881 11 .5 1 .357 1 .310 1 .249 .226 1 .165 1 .155 1 .145 0.873 0.879 12.0 1 .364 1.318 1 .254 .230 1 .169 1 .158 .148 0.871 0.877 14.0 1.39 1.340 1 .271 .246 1 .179 1.168 .158 0.863 0.870 16.0 1 .414 1.365 1 .286 .260 1 .189 1 .178 .169 0.855 0.864 18.0 .435 1 .38 1 .301 .272 1 .199 1.185 .175 0.851 0.859 20.0 1.450 1.395 1 .311 .288 1.204 1 .193 .182 0.846 0.854 PRACTICAL APPLICATIONS OF THE FORMULA 71 TABLE VI. VALUES OF 66 (^r + m) AND CORRESPONDING VALUES OF /, THE COEFFICIENT OF FRICTION. CONDUITS UNDER PRESSURE. sj 3 m = 0.95 m = 0.83 m = 0.68 m = 0.53 TO = 0.45 TO = 0.30 S-S S c / c / c / c / c / c / i 87.3 00839 79.9 01007 70.0 01313 60.1 01780 54.8 02142 44 .9 03190 2 92.4 00750 84.5 00901 74.6 01156 64.7 01534 59.4 01823 49.5 02625 3 95.7 00699 87.8 00834 77.9 01060 68.0 01391 62.7 01636 52.8 02307 4 97.1 00679 90.2 00790 80.3 00984 70.4 01298 65.1 01528 55.2 02111 6 102.0 00615 94.1 00726 84.2 00907 74.3 01166 69.0 01351 59.1 01841 8 104.9 00581 97.0 00684 87.1 00848 77.2 01080 71.9 01244 62.0 01673 10 106.3 00566 99.4 00651 89.5 00803 79.6 01015 74.3 01165 64 .4 01551 12 109.4 00535 101 .5 00624 91 .6 00767 81.7 00964 76.4 01102 66.5 01454 14 111 .2 00517 103.3 00603 93.4 00737 83.5 00922 78.2 01052 68.3 01379 16 112.9 00502 105.0 00583 95.1 00711 85.2 00886 79.9 01007 70.0 01313 18 114.4 00489 106.5 00567 96.6 00689 86.7 00856 81.4 00971 71.5 01258 20 115.9 00498 107.8 00554 97.9 00671 88.0 00829 82.7 0094 72.8 01214 22 117.0 00467 109.1 00540 99.2 00654 89.3 00807 84.0 00912 74.1 01171 24 118.2 00458 110.3 00528 100.4 00638 90.5 00785 85.2 00886 75.3 01134 26 119.4 00449 111 .5 00517 101 .6 00623 91.7 00765 86.4 00862 76.5 0110 28 120.4 00441 112.5 00507 102.4 00611 92.8 00747 87.5 00840 77.6 01068 30 121 .4 00434 113.5 00499 103.6 00599 93.7 00733 88.4 00823 78.5 01044 32 122.2 00428 114.3 00492 104.4 00590 94.5 00720 89.2 00808 79.3 01023 34 123.1 00422 115.2 00485 105.3 00580 95.4 00707 90.1 00792 80.2 00100 36 124.0 00416 116.1 00477 106.2 00570 96.3 00693 91.0 00777 81 .1 00978 38 124.9 00410 117.0 00470 107.1 00560 97.2 0068 91.9 00762 82.0 00957 40 125.7 00405 117.8 00464 107.9 00552 98.0 0067 92.7 00748 82.8 00938 42 126.4 0040 118.5 00458 108.6 00545 98.7 0066 93 .4 00737 83.5 00923 44 127.0 00396 119.1 00453 109.2 00539 99.3 00651 94.2 00728 84.1 00909 46 127.9 00391 120.0 00446 110.1 00531 100.2 00641 94.9 00714 85.0 00890 48 128.4 00386 120.8 00441 110.9 00523 101.0 00631 95.7 00702 85.8 00874 50 129.4 00382 121 .5 00436 111 .6 00516 101 .7 00622 96.4 00692 86.5 0086 52 130.0 00379 122.1 00431 112.2 00511 102.3 00614 97.0 00684 87.1 00848 54 130.7 00375 122.8 00426 112.9 00504 103.0 00606 97.7 00674 87.8 00834 56 131 .2 00372 123.3 00423 113.4 00500 103.5 00600 98.2 00667 88.3 00826 58 131 .9 00368 124.0 00418 114.1 00494 104.2 00592 98.9 00657 89.0 00812 60 132.5 00364 124.6 00414 114.7 00489 104.8 00586 99.5 00650 89.6 00801 62 133.1 00361 125.2 00410 115.3 00484 105.4 00579 100.1 00642 90.2 00791 64 133.6 00358 125.7 00407 115.8 00480 105.9 00573 100.6 00636 90.7 00782 66 134.2 00355 126.3 00403 116.4 00475 106.5 00567 101 .2 00628 91 .3 00772 68 134.7 00352 126.8 00400 116.9 00471 107.0 00562 101 .7 00622 91 .8 00763 70 135.2 00350 127.3 00397 117.4 00467 107.5 00557 102.2 00616 92.3 00755 72 135.8 00347 127.9 00393 118.0 00461 108.1 00551 102.8 00609 92.9 00745 ' 78 137.2 00340 129.3 00385 119.4 00451 109.5 00536 104.2 00592 94.3 00723 84 138.5 00333 130.6 00374 120.7 00441 110.8 00524 105.5 00574 95.6 00704 90 139.9 00327 132.0 00369 122.1 00431 112.2 00511 106.9 00563 97.0 00684 96 141 .2 00321 133.3 00362 123.4 00423 113.5 00499 108.2 00549 98.3 00666 102 142.2 00316 134.3 00356 124.4 00416 114.5 00490 109.2 00539 99.3 00652 108 143.2 00312 135.3 00351 125.4 00409 115.5 00482 110.2 00530 100 .3 00639 114 144.5 00306 136.6 00344 126.7 00401 116.8 00472 111 .5 00518 101 .6 00624 120 145.7 00301 137.8 00339 127.9 00393 118.0 00462 112.7 00506 102.8 00609 126 146.7 00297 138.8 00334 128.9 00387 119.0 00454 113.7 00497 103.8 00597 132 147.7 00293 139.8 00329 129.9 00381 120.0 00446 114.7 00489 104.8 00590 138 148.6 00290 140.7 00325 130.8 00376 120.9 00440 115.6 00481 105.7 00576 144 149.5 00286 141 .6 00321 131 .7 00371 121 .8 00434 116.5 00474 106.6 00566 156 151.4 00279 143.5 00313 133 .6 00360 123.7 0042 118.4 00459 108.5 00557 168 153 .0 00273 145.1 00305 135.2 00352 125.3 00410 120.0 00447 110.1 00546 180 154.6 00268 146.7 00299 136 .8 00343 126.9 00400 121 .6 00435 111 .7 00538 72 THE FLOW OF WATER TABLE VI. A. WELDED PIPES. TUBES OF BRASS, GALVANIZED IRON, SHEET IRON, STEEL, ETC. 1 Actual Values of 66 ( ^r + ra) Vr / Loss of Head in Feet per Unit Length of Conduit ss So Diam- Area = = c Vr at Unit Velocity. 1-1 S Is eter in d 2 0.7854 feet. m = m = m = m = m = m = m = m = & 0.95 0.83 0.68 0.45 0.95 0.83 0.68 0.45 t .0225 .00038 6.058 5.467 4.722 3.584 1 .714 2.154 2.888 5.025 .0303 .00072 7.111 6.423 5.56 4.239 1 .274 1 .561 2.083 3 .585 .| 0.0411 .00133 8.483 7.665 6.677 5 .139 .8949 1 .096 * 1 .444 2.439 I .0516 .00209 9.646 8.745 7.623 5.398 0.692 0.842 1 .108 1 .850 I .0686 .00370 11.34 10.31 9.006 7.018 .5008 .6063 0.794 1 .308 i .0873 .00599 13.02 11.85 10.38 8.142 .3801 .4587 0.597 .9496 U 0.1150 .01039 15.22 13.90 12.22 9.646 .2781 .3331 0.431 0.692 li .1341 0.01412 16.65 15.20 13.39 10.61 .2268 .2790 0.359 0.572 2 .1722 .02339 19.24 17.59 15.54 12.40 0.1740 .2081 0.267 .4155 2^ .2056 .03320 21 .34 18.23 17.30 13.85 0.1414 .1936 0.215 .3354 3 .2556 .05130 24.24 22.25 19.75 15.90 .1095 .1302 0.165 .2548 3| .2956 .06863 26.61 24.24 21 .56 17.42 .0910 .1095 0.139 .2122 4 .3356 .08840 28.44 26.15 23.29 18.89 .0796 .0941 0.119 0.1806 4* .3756 .1168 30.40 27.98 24.94 20.39 .0696 .0823 0.103 .1565 5 .4204 .1388 32.48 29.93 26.72 21 .80 .0610 .0703 0.090 0.1355 6 .5056 .2008 36.26 33.45 29.94 24.54 .0489 .0575 .0718 0.1069 7 .5857 .2694 39.61 36 .60 33.05 27.00 .0410 .0481 .0598 .0883 8 .6651 .3474 42.73 39.53 35.49 29.30 .0352 .0412 0.0511 .0750 9 .7449 .4356 45.74 42.25 38.08 31.52 .0308 .0351 .0444 .0648 10 .8348 .5473 48.97 45.38 40.85 33.91 .0268 .0303 .0377 0.056 11 .9166 .6599 51.87 48.06 43.35 36.08 .0239 .0279 .0343 .0495 12 1.0 .7854 54.70 50.70 45.77 38.18 0.0215 .0250' .0304 .0442 13 1 .1641 .9531 57.99 53.84 49.76 40.66 .0191 .0222 .0272 0.039 14 1 .1875 1 .1075 60.70 56.4 50.21 42.72 .0174 .0202 .0248 .0353 15 1 .2708 1 .2675 63.29 58.78 53.22 44.66 .0160 .0186 .0227 .0323 PRACTICAL APPLICATIONS OF THE FORMULA 73 TABLE VII. CIRCULAR CONDUITS. DIAMETERS, INTERNAL AREAS, MEAN HYDRAULIC RADII AND THEIR ROOTS. -, 3 a! .s ti J3 fe ll R Y^ V? If ll R Vr 4^ "3 c3 _, 11 "o> ^ ** a II | | | |GC <& "fiCQ H- 1 333 1 .0408 1 OfiO 1 .0202 ^HJ 366 1 .UOU 1 .0801 '.0393 ERR A TA )83 1 .0993 .0485 5 1 .118 .057 Page 72, Table VI A, for Loss of Head in Feet per Unit Length of 533 1 .137 1 .415 .173 .066 .0684 .083 Conduit at Unit Velocity L66 .1903 1.091 )83 .2076 1.099 substitute 55 .226 .275 1 .107 1.129 Z , or Loss of Head in Feet in a Length of '5 .323 .369 1.150 1 .170 R 414 .189 2 g Feet at Unit Velocity. '5 .457 .208 .5 .224 5 .541 .244 " .581 .257 _ e 620 273 16 1 .396 .3333 .5771 0.759 132 95 03 275 ^658 l'288 18 1 .767 .3750 .6124 0.782 138 103 '.87 2.875 .696 1 .302 20 2.234 .4166 0.645 .8031 144 113.10 3.0 1 .732 1 .316 22 2.640 .4583 .6770 .8220 150 122 .72 3.125 1 .768 1 .330 24 3.142 0.5 .7071 0.841 156 132 .76 3.25 1 .803 1.343 26 3.687 .5416 .7360 0.859 162 143 .16 3 .375 1 .837 1.355 28 4.275 .5833 .7637 0.874 168 153 .96 3.5 1 .871 .368 30 4.909 0.625 .7906 0.889 174 165.17 3 .625 1 .904 .380 32 5.585 .6666 .8165 0.904 180 176 .70 3.75 1 .937 .392 34 6.305 .7084 .8416 0.917 186 188 .70 3.875 1 .968 .403 36 7.069 0.75 0.866 0.931 192 201 .03 4.0 2.0 .414 38 7.876 .7916 .8898 .9435 198 213.8 4.125 2.031 .425 40 8.927 .8333 .9129 .9555 204 227.0 4.25 2.062 .437 42 9.621 0.875 .9355 0.967 210 240.5 4.375 2.091 .446 44 10 .559 .9166 .9575 .9784 216 254.5 4.5 2.121 .456 46 11 .509 .9584 .9983 .9991 222 268.8 4.625 2.151 .466 48 12.566 1 .0 1.0 1 .0 228 283.5 4.75 2.180 .476 50 13 .635 1 .0416 1 .0206 1 .0103 240 314.2 5.0 2.236 .496 72 THE FLOW OF WATER TABLE VI. A. WELDED PIPES. TUBES OF BRASS, GALVANIZED IRON, SHEET IRON, STEEL, ETC. 1 Actual Values of 66 ( \/r + ^ /- w)VV Loss of Head in Feet per Unit Length of Conduit S| Diam- t>t r Area = eter in (P0.7854 1 feet. n i .0225 .00038 6 .0303 .00072 7 1 0.0411 .00133 g 'l .0516 .00209 c 1 .0686 .00370 11 1 .0873 .00599 Ic a 0.1150 .01039 1* a .1341 0.01412 1( 2 0.1722 .02339 J( 2^ .2056 .03320 23 3 .2556 .05130 2' 3^ .2956 .06863 2( 4 .3356 .08840 ft 1 1 R8 2< 5 .4204 U .1 IDo .1388 32 .48 29.93 tfX. .7- 26.72 21 .80 u .uuyu .0610 .0703 XT" . J.T7O 0.090 .1355 6 .5056 .2008 36.26 33.45 29.94 24.54 .0489 .0575 .0718 0.1069 7 .5857 .2694 39.61 36 .60 33.05 27.00 .0410 .0481 .0598 .0883 8 .6651 .3474 42.73 39.53 35.49 29.30 .0352 .0412 0.0511 .0750 9 .7449 .4356 45.74 42.25 38.08 31 .52 .0308 .0351 .0444 .0648 10 .8348 .5473 48.97 45.38 40.85 33.91 .0268 .0303 .0377 0.056 11 .9166 .6599 51.87 48.06 43.35 36.08 .0239 .0279 .0343 .0495 12 1.0 .7854 54.70 50.70 45.77 38.18 .0215 .0250" .0304 .0442 13 1 .1641 .9531 57.99 53.84 49.76 40.66 .0191 .0222 .0272 0.039 14 1 .1875 1 .1075 60.70 56.4 50.21 42.72 .0174 .0202 .0248 .0353 15 1 .2708 1 .2675 63.29 58.78 53.22 44.66 .0160 .0186 .0227 .0323 PEACTICAL APPLICATIONS OF THE FORMULAE 73 TABLE VII. CIRCULAR CONDUITS. DIAMETERS, INTERNAL AREAS, MEAN HYDRAULIC RADII AND THEIR ROOTS. < - .9 I 'cS .S fi S-. ~^ CD CD < R v7 V? fl ( 1 |.s 13 R Vr V' *c3 g g c8 1 1 c3 g.s |I i.i Jt i .001364 .010417 .10206 .3195 52 14 .750 1 .0833 1 .0408 1 .0202 1 .005454 .02083 .1444 0.380 54 15 .904 1.125 1 .060 1 .030 H .012272 .03125 .1768 .4302 56 17 .106 1 .1666 1 .0801 1 .0393 2 .02182 .04166 .2039 .4516 58 18 .347 1 .2083 1 .0993 1 .0485 2J .03163 .0502 .2240 .4733 60 19 .635 1 .25 1 .118 1.057 3 .04909 .0625 0.25 0.5 62 20 .964 1 .2966 1 .137 1 .066 3i .06681 .07292 0.270 .5196 64 22 .340 1 .3333 1 .415 1 .0684 4 .08726 .08333 0.291 .5370 66 23 .758 1 .375 1 .173 1.083 4^ 0.11045 .09375 .3062 .5533 68 25 .220 .4166 1 .1903 1 .091 5 .1364 .10416 .3227 .5681 70 26 .725 .4583 1 .2076 1.099 6 .1963 0.125 .3535 0.594 72 28.27 .5 .226 1.107 7 .2672 .1458 .3819 .6180 78 33.18 .625 .275 1.129 8 .3490 .1666 .4082 0.639 84 38.48 .75 .323 1.150 9 .4418 .1875 0.433 0.658 90 44.18 .875 .369 1.170 10 .5585 .2083 .4564 0.675 96 50.27 2.0 .414 1.189 11 .6599 .2297 .4787 0.692 102 56.75 2.125 1 .457 1.208 12 .7854 0.25 0.5 .7071 108 63 .62 2.25 1 .5 1 .224 13 .9217 .2708 .5204 0.721 114 70.88 2.375 1 .541 1 .244 14 1 .0689 .2916 0.540 0.735 120 78.54 2.5 1 .581 1 .257 15 1 .2272 .3125 .5339 0.748 126 86.59 2.625 1 .620 1 .273 16 1.396 .3333 .5771 0.759 132 95.03 2.75 .658 1.288 18 1.767 .3750 .6124 0.782 138 103 .87 2.875 .696 1 .302 20 2.234 .4166 0.645 .8031 144 113.10 3.0 .732 1 .316 22 2.640 .4583 .6770 .8220 150 122 .72 3.125 .768 .330 24 3.142 0.5 .7071 0.841 156 132 .76 3.25 .803 .343 26 3.687 .5416 .7360 0.859 162 143 .16 3.375 .837 .355 28 4.275 .5833 .7637 0.874 168 153 .96 3.5 .871 .368 30 4.909 0.625 .7906 0.889 174 165.17 3.625 .904 .380 32 5.585 .6666 .8165 0.904 180 176 .70 3.75 .937 1 .392 34 6.305 .7084 .8416 0.917 186 188 .70 3.875 .968 1 .403 36 7.069 0.75 0.866 0.931 192 201 .03 4.0 2.0 1 .414 38 7.876 .7916 .8898 .9435 198 213.8 4.125 2.031 1.425 40 8.927 .8333 .9129 .9555 204 227.0 4.25 2.062 1.437 42 9.621 0.875 .9355 0.967 210 240.5 4.375 2.091 1 .446 44 10 .559 .9166 .9575 .9784 216 254.5 4.5 2.121 1 .456 46 11 .509 .9584 .9983 .9991 222 268.8 4.625 2.151 1 .466 48 12.566 1.0 1.0 1 .0 228 283 .5 4.75 2.180 1 .476 50 13 .635 1 .0416 1 .0206 1 .0103 240 314.2 5.0 2.236 1 .496 74 THE FLOW OF WATER TABLE VII. A. ROOTS OF MEAN HYDRAULIC RADII. R v7 V? R Vr ^ R Vr v^ 0.05 0.224 0.473 2.75 1.658 1.287 5.9 2.429 1.558 0.1 0.316 0.562 2.80 1.673 1.293 6.0 2.449 1.565 0.15 0.387 0.622 2.85 1 .688 1.299 6.1 2.470 1.571 0.20 0.447 0.668 2.90 1.703 1.305 6.2 2.490 1 .578 0.25 0.5 0.707 2.95 1 .718 1 .311 6.3 2.510 1 .584 0.30 0.548 0.740 3.0 1 .732 .316 6.4 2.530 1 .590 0.35 0.592 0.769 3.05 .746 .322 6.5 2.550 1 .597 0.40 .632 0.803 3.10 .761 .327 6.7 2.588 1.609 0.45 0.671 0.819 3.15 .775 .332 6.8 2.608 1.615 0.5 0.707 0.841 3.20 .789 .338 6.9 2.627 1.621 0.55 0.742 0.861 3.25 .803 1.343 7.0 2.644 1 .624 0.60 0.775 0.881 3.30 .817 1.347 7.1 2.665 1 .630 0.65 0.806 0.898 3.35 .830 1.352 7.2 2.683 1 .637 0.70 0.837 0.914 3.40 .844 1.358 7.3 2.702 1 .643 0.75 0.866 0.930 3.45 1.857 1.363 7.4 2.720 1.649 0.80 0.894 0.946 3.50 1.871 1.368 7.5 2.739 1.655 0.85 0.922 0.960 3.55 1.884 1.373 7.6 2.757 1.661 0.90 0.949 0.974 3.60 1 .898 1.378 7.7 2.775 1.664 0.95 0.975 0.987 3.65 1 .910 1 .382 7.8 2.793 1.670 1.0 1.0 1.0 3.70 1.924 1 .387 7.9 2.811 1 .676 1.05 1.025 1.012 3.75 1.936 1 .392 8.0 2.828 1 .682 1.10 1 .049 1 .024 3.80 1.949 .396 8.1 2.846 1 .688 1.15 1.072 1 .036 3.85 1.962 .401 8.2 2.868 1 .692 1.20 .095 1.047 3.90 1.975 .405 8.3 2.881 1.697 1.25 .118 1 .057 3.95 1 .987 .410 8.4 2.898 1.702 1 .30 .140 1.068 4.0 2.0 .414 8.5 2.915 1 .707 .35 .162 1 .079 4.05 2.012 .419 8.6 2.933 1.712 .40 .183 1.088 4.10 2.025 .423 8.7 2.950 1.717 .45 .204 1.097 4.15 2.037 .427 8.8 2.966 1.722 .50 .225 .107 4.20 2.049 1 .432 8.9 2.983 1.727 .55 .245 .115 4.25 2.062 1 .436 9.0 3.0 1.732 .60 .265 .125 4.30 2.074 1 .440 9.1 3.017 1.737 .65 1 .285 .133 4.35 2.086 1 .444 9.2 3 .043 1.741 .70 1 .304 .142 4.40 2.098 1.448 9.3 3.056 1.746 .75 1 .323 .150 4.45 2.111 1 .453 9.4 3.066 1 .750 .80 1 .342 .158 4.50 2.121 1 .457 9.5 3.082 1.755 .85 1 .360 .166 4.55 2.133 1 .461 9.6 3.098 1.760 .90 1 .378 .174 4.60 2.145 1 .466 9.7 3.114 1.764 .95 1 .396 .182 4.65 2.156 .469 9.8 3.130 .769 2.0 1 .414 .189 4.70 2.168 .473 9.9 3.146 .773 2.05 1.432 .196 4.75 2.179 .476 10.0 3.162 .777 2.10 1 .449 1.204 4.80 2.191 .480 10.5 3.240 .800 2.15 1 .466 1.211 4.85 2.202 .484 11.0 3.317 .820 2.20 1 .483 1.218 4.90 2.214 .488 11.5 3.391 .845 2.25 1 .5 1.225 4.95 2.225 .492 12.0 3.464 .860 2.30 1 .517 1 .232 5.0 2.236 .495 12.5 3.536 .880 2.35 1 .533 1 .238 5.1 2.258 .503 13.0 3.606 .899 2.40 .549 .245 5.2 2.280 .511 13.5 3.674 .918 2.45 .565 .251 5.3 2.302 .518 14.0 3.742 1 .934 2.50 .581 .257 5.4 2.324 .526 14.5 3.808 1 .951 2.55 .597 .263 5.5 2.346 .532 15.0 3.873 1 .968 2.60 .612 .270 5.6 2.366 .539 15.5 3.937 1 .984 2.65 .628 .276 5.7 2.387 .548 16.0 4.0 2.0 2.70 1 .643 .282 5.8 2.408 .552 PRACTICAL APPLICATIONS OF THE FORMULAE 75 TABLE VIII. Table VIII contains the practically most useful coefficients indicating the degree of roughness of a conduit. In the design of a new conduit it is well to remember, that the degree of roughness of a conduit is not a permanent quantity. Conduits lined with cement, smooth concrete, good brickwork, planed boards, metals, etc., gradually deteriorate and assume a degree of roughness which closely resembles that of a sawed board (m = 0.68) , in case of sewers that of common brick work (m = 0.57) , in case of large riveted pipes that of very rough brick work (m = 0.45) . If the velocity is feeble, or the flow often interrupted, crypto- gamic plants sooner or later appear on the walls of open conduits and rust or calcareous matter coates the walls of pipes. In such a condition the degree of roughness corresponds to that of very rough brick work (m = 0.45) . If left to themselves, channels in earth of all descriptions likewise deteriorate and gradually assume a degree of roughness corresponding to that of a natural channel (k = 1.93 in most cases) . For artificial channels in earth Table VIII gives values of both m and k. Owing to the abnormally rapid decreases in the value of c with the decrease of the depth of the water in rough channels in earth a negative value of m gives better results than k. The k formula, however, gives good results in all cases where R is greater than one foot. The relation between K and m and the coefficient n of the for- mula of Kutter is given by 1 + K 0.02 100 = 1 + m ' 76 THE FLOW OF WATER TABLE VIII. VALUES OF m AND k, THE COEFFICIENTS INDICATING THE DEGREE OF ROUGH- NESS. A. CONDUITS UNDER PRESSURE. 1.0 0.95 0.83 0.68 0.57 0.53 0.45 0.30 0.20 Description of Conduit. New, straight tin or plated pipes. Pipes of planed boards or clean cement, new. Very smooth new asphalt-coated cast and wrought-iron pipes. New asphalt-coated riveted pipes not exceeding 6 inches in diameter. Ordinary new asphalt-coated cast and wrought-iron pipes. Wrought- iron pipes not coated, new. Glass and lead pipes. Pipes lined with smooth concrete or cement plaster. Pipes lined with cement or smooth concrete, pipes of planed or rough boards, cast and wrought-iron pipes, coated or not coated, steel and wrought-iron riveted pipes not exceeding 3 feet in diameter (all some time in use but fairly clean). Sewer pipe. Conduits lined with common brickwork or rough concrete. New asphalt-coated steel-riveted pipe exceeding 3 feet in diameter. Conduits lined with very rough brickwork or very rough concrete. Steel-riveted pipe exceeding 3 feet in diameter, some years in use. Old cast and wrought-iron pipes of all descriptions, not very clean. Old steel-riveted pipe exceeding 3 feet in diameter. Drain tile. B. OPEN CONDUITS. Description of Conduit. 1 .0 0.95 0.83 0.80 0.70 0.57 0.45 0.30 0.15 0.0 Conduits lined with neat cement exceptionally smooth. New conduits lined with neat cement or planed boards. New brick conduits washed with cement, conduits smoothly dressed with cement mortar. New conduits lined with smooth concrete or very good brick work. Conduits lined with sawed boards or fairly good brick work. Aqueducts lined with neat cement, cement plaster, smooth concrete very good brickwork, planed boards (all some time in use). Channels lined with common brickwork, rough concrete or smoothly dressed ashlar masonry. Sewers lined with neat cement, smooth concrete, brickwork washed with cement or plastered with cement mortar, fairly good and very good brickwork (all some time in use.) Channels lined with very rough brickwork or concrete, fairly good ashlar masonry. Channels lined with common ashlar or very good rubble masonry. Channels lined with roughly hammered stone masonry. Channels lined with common rubble masonry. Channels in rockwork. PRACTICAL APPLICATIONS OF THE FORMULAE 77 TABLE VIII. Continued. C. CHANNELS IN EARTH. m k Description of Conduit. 0.57 0.15 0.0 -0.10 -0.20 -0.32 0.27 0.74 1.0 1.2 1 .5 1.93 Channels of very regular cross-section in stiff clay or clayey loam. Channels of fairly regular cross-section in fine cemented gravel. Channels of fairly regular cross-section in coarse cemented gravel. Channels in rock work. Fairly regular channels in sand or sand with gravel imbedded. Fairly regular channels in earth, tolerably free from stones and plants. Ordinary channels in earth or gravel. Channels with stones vegetation or other impediments to flow. Natural channels, creeks, rivers. TABLE IX. ALPHABETICAL LIST OF AUTHORITIES WHOSE EXPERIMENTAL DATA ARE GIVEN IN TABLE X, METHODS OF GAUGING AND PUBLICATION CONTAINING ORIGINAL RECORD OF EXPERIMENTS. Author. Description of Channel Gauged. Method of Gauging. Where Recorded. Adams, A. L. Wooden Stave pipes. Discharge meas- ured by rise in reservoir surface, loss of head by open standpipes. Engineering News. Sept. 1898. Baumgarten . . Aqueduct. Bot- tom of cement, sides of brick. Piezometer. Darcy-Bazin. Recherches hy- drauliques. Benzenberg . . Brick sewer Floats probably Trans A S C E Bossut .... Tin and Lead pipes. Discharge meas- ured in tanks. Hamilton Smith, Hydraulics, 1886. Bruce Aqueduct. Con- crete. Discharge meas- ured by rise in reservoir surface. Proceedings of Institute of C. E. London, 1896. Brush Cast-iron pipes. Quantities meas- ured at pumps. Quoted by Kutter. "The Flow of Water." Clarke Brick sewer. Discharge meas- ured by rise in reservoir surface. H. Smith. Hy- draulics, 1886. 78 THE FLOW OF WATER TABLE IX. Continued. Author. Description of Channel Gauged. Method of Gauging. Where Recorded. Cunningham . . Aqueduct of masonry. Ganges Canal. Velocities meas- ured by one-inch tin rod floats. Roorkee Hydr. Ex- periments, 1880. Darcy Pipes. Discharge meas- ured in tanks, loss of head by piezo- meter or mercury column. Experiments sur le mouvement de Peau dans les tuyaux. Darcy-Bazin . . Cement and con- crete conduits. Discharges meas- ured by orifices previously tested. Recherches hydr. Paris 1865. Darcy-Bazin . . Conduits of planed and rough boards, or lined with brick. Discharges meas- ured by orifices previously tested, 20 centimeter square. Recherches hydr. Paris, 1865. Darcy-Bazin . . Canal lined with ashlar masonry. Piezometer. Recherches hydr. Paris, 1865. Darcy-Bazin . . Tailrace lined with ashlar masonry. Discharge meas- ured by orifice 50 centimeter square. Recherches hydr. Paris, 1865. Darcy-Bazin . . Tunnel lined with ashlar masonry. Current meter and reservoir. Recherches hydr. Paris, 1865. Darcy-Bazin . . Section of Gros- bois canal lined with masonry. Current meter and reservoir. Recherches hydr. Paris, 1865. Darcy-Bazin Chazilly Canal. Piezometer, cur- rent meter and reservoir. Recherches hydr. Paris, 1865. Darcy-Bazin Grosbois Canal. Piezometer, cur- rent meter and reservoir. Elecherches hydr. Paris, 1865. Dubuat Canal du Jard. Surface floats. Principes hydr. Paris, 1786. Ehman .... Galvanized and cast-iron pipes. Discharges meas- ured by volumes. Iben Druckho- henverlust. Fanning .... Cement lined pipe. Weir measure- ment probably. Water Supply Engineering. PRACTICAL APPLICATIONS OF THE FORMULAE 79 TABLE IX. Continued. Author. Description of Channel Gauged. Method of Gauging. Where Recorded. Fteley & Stearns Sudbury Conduit. Brick coated with cement and not coated. Weir measure- ment. H. Smith. Hydraulics, 1886. Fortier .... Irrigation Chan- nels. Current meter. U. S. Geol. Sur- vey. Irr. Papers, 1901. Hawks .... Steel-riveted pipe. Weir measure- ment. Tr. A. Soc. C. E., 1899. Herschel . . . Steel-riveted pipes. Discharges meas- ured by Ventury meter, loss of head by Bourdon gauges. 115 Experiments. Horton .... Brick sewers washed with ce- ment. Weir measure- ment probably. Eng. News. Hubbel & Fenkell Cast-iron ppes. Tr. A. Soc C E 1898. Iben Oast-iron pipes. Pipes coated with tar. Discharges meas- ured by volumes, loss of head by pressure gauges. Druckhohenver- lust. Kuichling . . . Riveted pipes. Cast-iron pipes. Quantities meas- ured by rise in reservoir surface, loss of head by mercury gauges. Marx-Wing & Hoskins, Tr. A. S. C. E., 1898-1899. Kutter .... Channels lined with rubble ma- sonry. Surface floats. )ie neue Theorie. La Nicca . . . Alpine Streams. Surface floats. Kutter, Die neue Theorie. Lampe .... 3ast-iron pipes. Discharges meas- ured by reservoir contents, loss of head by pressure gauges. ben. )ruckhohenver- ' lust. Legler . . . Canals. Rodfloats. lydrotechnische Mittheilungen. 80 THE FLOW OF WATER TABLE IX. Continued. Author. Description of Channel Gauged. Method of Gauging. Where Recorded. McDougall . . Irrigation chan- nels. Current meter. U. S. Geolog. Sur- vey Irr. Papers. Marx-Wing & Hoskins . . . Riveted pipe. Stave pipe. Discharges meas- ured by Venturi meters, loss of head by mercury gauges. Trans. A. S. C. E., 1898. Noble . . Stave pipes. Trans. A S C E , 1902. Passini & Gioppi Aqueduct. Bot- tom of concrete, sides of brick. Current meter. Giornale del. Genio Civile. Ro- ma, 1893. Passini & Gioppi Syphon aqueduct of brick. " tt Passini & Gioppi Canal Cavour. " 11 Perrone .... Aqueduct coated with clean cement. " Zoppi: Sul Vol- turno, Carte hy- Perrone .... Tunnel in rock- work. " Italic. Rafter Riveted pipes Discharge meas- Tr. A. S. C. Eng , ured by rise in reservoir surface, loss of head by piezo-meter. XXVI. Revy . . . . . La Plata and Pa- rana Rivers. Current meter. Hydraulics of great rivers, Lon- don, 1881. Rittinger . . . Channels lined with rubble masonry. Discharges meas- ured in tanks. Bornemann : Der Civil Ingenieur. Roff Saalach River. Piezometer. Grebenau, Theorie der Bewegung des Wassers. Rowland . . . Wrought-iron pipes. Discharge meas- ured by volumes. Brush. Smith, H . . . Riveted pipes Velocities meas- Tr A S C E ured by weirs and Standard orifices. "Hydraulics," N. Y., 1886. PRACTICAL APPLICATIONS OF THE FORMULAE 81 TABLE IX. Concluded. Author. Description of Channel Gauged. Method of Gauging. Where Recorded. Smith, J. W. . . Riveted pipes. Discharge meas- ured by weir, loss of head by piezometers. Tr. A. S. C. E Vol. XXVI. Stearns .... Brick aqueduct. Current meter. Report to the New Croton Aque- duct Com., 1895. Schwartz . . . Weser River. Current meter. Funk: Beit rag zur allgemeinen Wasserbaukunst Lemgo, 1808. Wampfler . . . 3anal. Surface floats. Kutter: Die neue Theorie. TABLE X. Table X contains the most reliable experimental data from which the general formula is deduced. For conduits under pressure the numerical values of (a), the coefficient of variation of c is generally given. This is done in order to show the details of the variation of c. For open conduits the variation is generally indicated by giving the velocity roots. These roots are found from the formula = log. v t log. % log (66 %/r + m) Vrs)^ log. (66 ( t/r + m)Vrs~) Q The value of x being thus found the value of m is found by putting -\Jr = m or 66 a = m For artificial channels in earth the values of m have been given in addition to the values of K. Owing to the abnormally rapid decrease in the values of c with the decrease of the depth of the water in very rough channels a negative value of m gives better results than K. The K formula, however, gives good results in all cases where R is greater than one foot. 82 THE FLOW OF WATER TABLE X. EXPERIMENTAL DATA. I. RIVETED PIPES. Description of Conduit. m L d R 1000s V c a 0.80 .04 .130 .219 .287 .291 New straight asphalt- coated wrought-iron riveted pipe with screw joints. Darcy. 0.94 365 0.271 .0677 0.27 2.028 12.20 40.70 106 .54 156 .05 0.328 1.171 3.117 6.148 10 .535 12 .786 76.7 99.9 108.4 117.1 124.0 124.3 Do. 0.92 365 0.643 .1607 0.20 1 .29 5.80 12.0 29.7 121 .56 0.591 1 .529 3.53 5.509 9.00 19.72 104.1 106.2 115.6 125.4 130.2 141 .0 1.013 1 .035 1 .125 1 .220 1.267 1.372 ... ... Do. 0.82 365 0.935 0.234 0.70 4.33 11 .90 28.07 1 .296 3.868 6.673 10 .522 101.3 121.6 126.5 129.9 1.013 1 .216 1 .265 1 .299 Sheet-iron riveted pipe with funnel mouthpiece 7.8 ft. long. Hamilton Smith. 0.68 700 0.911 0.228 8.50 13.34 16.95 25.59 33.09 4.712 6.094 6.927 8.659 10 .021 107.1 110.6 111 .5 113.4 115.5 1.19 1.229 1.240 1 .260 1 .283 Do. Coated with asphalt. Funnel mouthpiece 12 ft. long. 0.68 700 1.056 0.264 6.68 14.28 22.19 33.18 4.595 6.962 8.646 10 .706 109.4 113.4 113.0 114.4 .200 .242 .237 .253 Do. Funnel mouthpiece 14.8 ft. long. 0.69 700 1.229 0.307 5.02 10.97 12.27 16.46 24.70 32.31 4.383 6.841 7.314 8.462 10 .593 12.09 111 .6 119.8 119.1 119.2 121 .6 121.3 .181 .246 .261 .260 .286 .285 Do. Double riveted pipe with some easy curves. 0.65 4440 1.416 0.354 66.72 20 .143 131 .1 1.395 Do. 0.69 1200 2.154 0.538 16.41 12 .605 134.1 1 .325 Do. Inverted syphon with 887 ft. depression. 0.63 12800 2.43 0.607 11.72 10.78 127.8 1.30 DESCRIPTION OF CONDUIT, ETC. 83 TABLE X. Continued. Description of Conduit. m L d R 1000s V c a Wrought-iron riveted pipe with lap joints. Paint coating worn off, somewhat rusty. Clemens-Herschel. 0.54 152.9 8.58 2.145 .0079 0.032 0.0837 .1557 .2453 0.354 .4991 .6619 .8470 0.50 1.0 1 .5 2.0 2.5 3.0 3.5 4.0 4.5 126.9 116.6 111.9 109.4 109.0 108.2 107.0 106.2 105.6 1 .089 1.00 0.959 0.938 0.934 0.928 0.917 0.910 0.905 Asphalt- coated steel- riveted pipe. A. McL. Hawks. 0.55 1.166 .2915 .4550 0.584 0.932 1 .136 82.2 86.0 0.98 1.026 Asphalt coated cylin- der joint steel pipe. A.L.Adams. 0.65 16416 1.33 0.333 5.0 4.58 110.0 1.183 Asphalt- coated cylin- der joint steel- riveted pipe with curves. -E. Kuichling. 0.57 91641 3.166 .7915 1.01 0.99 3.23 3.27 114.0 116.6 1.14 1.166 Asphalt-coated taper joint steel- riveted pipe. New. Clem- ens-Herschel. 0.56 81139 3.5 0.875 0.112 1 .0 2.0 3.0 4.0 5.0 6.0 101 .0 104.3 106.4 107.8 108.4 108.5 1.0 1.032 1.053 1.067 1.073 1.074 Do. 0.54 5574 3.5 0.875 0.13 1.0 2.0 3.0 3.5 4.0 5.0 6.0 96.0 107.9 112.6 113.0 112.8 110.8 110.0 0.96 1.08 1.128 1.132 1.130 1.111 1.102 Do. Cylinder joint, many curves. 0.53 24000 4.0 1 .0 .0976 1.0 2.0 3.0 3.5 4.0 5.0 6.0 101 .2 108.3 112.8 113.4 113.2 112.0 111.6 1.0 1 07 1 113 1 .119 1 .118 1.105 1.091 Asphalt-coated cylin- der joint steel- riveted pipe. J. W. Smith. 0.80 0.74 39809 34176 2.916 2.75 0.584 0.55 1.31 1.31 3.52 3.96 126.8 123.2 1.15 1.166 84 THE FLOW OF WATER TABLE X. Continued. Description of Conduit. m L d R 1000 s V c a Asphalt-coated butt- jointed steel-riveted pipe with many curves. Marx- Wing. 0.50 4367 6.00 1.5 0.07 0.16 0.24 0.559 0.495 0.776 1 .08 1.57 2.14 2.59 3.02 3.84 108.0 114.0 113.0 110.0 112.0 113.0 i.o 1.055 1 .046 1 .018 1 .037 1 .046 II. OLD RIVETED PIPES. Cylinder joint asphalt- coated steel- riveted pipe. Fourteen years in use. George W. Rafter. 0.31 45400 2.0 0.50 3.83 3.58 3.46 3.32 3.32 3.35 76.0 78.0 80.5 1 .0 1.030 1.06 Do. 0.29 45400 3.0 0.75 0.45 0.43 1.47 1.49 80.4 83.0 1.0 Do. One year in use. E. Kuichling 0.59 45400 3.17 .7915 1 .59 1.61 1 .62 3.88 3.91 3.90 109.3 109.3 109.1 1 .08 1 .08 1.08 Taper joint, steel riveted. Four years old . Clemens-Her- schel. 0.26 24648 4.0 1.0 ... 1 .0 1.5 2.0 2.5 3.0 3.5 4.0 5.0 6.0 78.0 84.6 89.6 92.4 93.0 93.2 94.2 94.4 94.9 0.94 1 .019 1 .080 1 .113 1.120 1.121 1.135 1 .137 1 .143 ... Cylinder joint steel- riveted pipe, four years in use. C. Herschel. 0.47 24600 4 1 .0 1.0 1 .5 2.0 2.5 3.0 3.5 4.0 5.0 6.0 97.2 100.8 103 .3 104 .9 105.3 104.8 104.0 103.9 103.7 1.0 1.024 1.062 1 .079 1 .083 1 .079 1.069 1.066 1 .066 III. NEW WROUGHT-IRON PIPES, NOT COATED. Straight pipe. Darcy. 0.83 372 .0873 .0218 0.33 10.15 43.48 105 .71 309 .52 0.19 1.207 2.612 4.203 7.166 70.7 81.1 84.8 87.5 87.2 0.877 1 .006 1 .052 1.098 1 .094 ... DESCRIPTION OF CONDUIT, ETC. 85 TABLE X. Continued. Description of Conduit. m L d R 1000 s V c a Straight pipe. Darcy. 0.83 372 .1296 .0324 0.22 3.36 23.89 123 .15 224 .08 0.205 0.858 2.585 6.300 8.521 76.9 82.3 92.9 99.8 100.0 0.824 0.989 1 .128 1 .212 1 .215 Do. Rowland. 0.83 31 .0 31.0 31.0 97.0 .0833 .0833 .0208 .0208 6258 .6 8935 .5 10741 .9 2000 .0 2855 .6 3432 .9 36.1 43.4 48.1 19.9 24.5 27.2 100.0 100.6 101.7 97.5 100.5 101 .7 1 .240 1.248 1 .261 1 .209 1.247 1 .261 IV. PIPES COATED WITH TAR. New cast-iron pipe. Iben 1876 0.66 415 0.335 0.084 1.98 4.11 1.0 1 .70 79.0 92.0 1.0 1 164 Quoted by Kutter 6.56 2.10 90 1 139 7.83 2.30 90 .0 1 139 11 .07 2.80 91 .0 1 .152 Do. 0.56 1093 0.50 0.125 4.59 11 .62 2.00 3 30 82.0 87 1 .080 1 144 16.21 3.90 88 1 156 22.32 4.80 92 1 210 30.27 5.30 87.0 1 .144 Do. 0.65 1795 1 .001 0.25 1.46 1 830 1.60 2 10 85.0 97 0.944 1 080 2.19 2.60 112 1 244 3 .84 3 .80 121 1 .344 6.03 4.80 125.0 1 .388 Do. 0.69 3514 1.667 0.417 0.12 0.48 0.70 1 .60 105.0 110.0 1.077 1 .125 0.76 1 .90 109 1 115 1.21 2.50 109.0 1 .115 V. ASPHALT-COATED, WROUGHT AND CAST-IRON PlPE. NEW. Asphalt-coated wrought-iron pipe with funnel mouth- piece. Hamilton Smith. 0.89 60 .0875 .0218 26.93 52.19 103 .38 130 .64 2.22 3.224 4.761 5.443 91.6 95.5 100.2 101 .9 1.095 1 .140 1.189 1.205 ... Asphalt-coated cast- iron pipe. Darcy. 0.90 366 .4495 0.1124 0.24 4.25 22.25 98.52 167 .56 0.489 2.503 5.623 11 .942 15 .397 94.1 108.4 112.5 113.5 112.2 0.960 1 .107 1.150 1.160 1.150 86 THE FLOW OF WATER TABLE X. Continued. Description of Conduit. m L d R 1000s V c a Asphalt-coated pipe five years old, in good condition. Lampe. 0.83 26000 1.373 0.343 0.594 1.376 1.63 1.95 1.577 2.479 2.709 3.090 110.5 114.1 114.6 119.4 1.072 1 .107 1 .112 1.162 Cast-iron pipe. Darcy. 0.81 365 .6168 .1542 3.68 22.50 109 .80 145 .91 2.487 6.342 14 .183 16.168 104.4 107.7 109.0 107.8 1.107 1.142 1.155 1 .142 Asphalt-coated pipe, four years in use. Ehmann. 0.80 810 0.662 0.166 0.367 0.850 1 .332 1.883 0.73 1.12 1.45 1 .69 92.7 94.7 97.9 96.0 0.984 1.01 1 .039 1.017 Asphalt-coated cast- iron pipe. Hu li- bel and Fenkel. 0.83 ... 1.0 0.25 ... 1.0 2.0 3.0 4.0 5.0 101 .5 109.6 114.6 118.3 121 .5 1 .0 1 .08 1.13 1.166 1 .196 Cast-iron pipe. Darcy. 0.77 365 1 .6404 0.41 0.45 1 .20 2.10 2.60 1 .472 2.602 3.416 3 .674 108.4 117.3 116.4 112.5 1.047 1 .135 1.126 1.090 Cast-Iron Force Main. Large num- ber of summits, angles and curves, amongst which there are four right angles and ten quadrants of 30 ft. radius. Brush. 0.80 75000 1.667 0.417 0.733 0.880 .026 1.187 .333 .493 1 .64 1.800 2.0 2.24 2.36 2.52 2.68 2.76 2.92 3.0 114.4 117.0 .114 .1 113.3 113.7 110.6 111 .7 109.5 1 .08 1 .105 1 .080 1 .071 1.071 1.045 1 .055 1 .035 . .. Asphalt- coated cast- iron pipe. Some easy vertical cur- ves. Stearns. 0.97 1747 4.0 1.0 0.318 0.711 1 .221 1 .849 2.616 3.738 4.965 6.195 146.7 140.1 142.1 144.1 F A 1.077 1.093 1 .109 ... VI. OLD CAST AND WROUGHT-!RON PIPES. Old cast-iron pipe. Darcy. 0.52 366 .2628 .0657 0.84 7.25 16.10 45.35 0.458 1.463 2.224 3.777 62.0 67.3 68.7 68.9 0.94 1.04 1.062 1.065 DESCRIPTION OF CONDUIT, ETC. 87 TABLE X. Continued. Description of Conduit. m L d R 1000s V c a Old cast-iron pipe, cleaned. Darcy. 0.85 0.84 7.23 15.57 44.73 0.633 2.014 2.835 5.007 85.2 92.4 88.6 93.4 1 .00 1 .08 1 .026 1 .092 Old cast-iron pipe. Darcy. 0.45 365 0.798 .1995 0.94 4.73 22.90 41 .05 139 .81 1 .007 2.32 5.095 6.801 12 .576 73.6 75.5 75.1 75.2 75.3 1 .0 1.023 1.02 1 .02 1 .02 Old cast-iron pipe, twelve years in use. Slightly tubercu- lated. I ben. 0.45 541 1.0 0.25 2.24 2.84 1.79 2.03 75.7 76.2 1 .0 1.01 Do., two years in use, slightly incrusted. 0.56 2149 1.0 0.25 0.26 0.41 0.81 1.28 2.99 0.60 0.80 1.20 1.60 2.40 74.5 81.0 85.0 92.0 86.0 0.878 0.962 1 .01 1 .092 1 .021 Do., fourteen years in use, slightly in- crusted. 0.39 7179 1.00 0.25 0.42 1 .65 4.44 9.43 0.70 1 .60 2.70 3.90 71.0 78.0 80.0 80.0 0.979 1 .075 1 .133 1.133 Do., fifteen years in use. Heavily in- crustated. 0.30 1808 1.00 0.25 0.65 3.76 6.12 7.13 0.90 1 .80 2.30 2.60 67 58 59 58 0.991 0.860 0.875 0.860 Do., twenty -two years in use. Very heav- ily incrustated. 0.05 1736 1.00 0.25 1.08 4.29 10.91 23.86 0.80 1.50 2.40 3.50 50 44 45 46 1 .041 0.179 0.937 0.958 New asphalt-coated cast-iron pipe. Rochester, N.Y. E.Kuichling,1895. 0.83 3 0.75 1.38 1 .50 1 .50 4.204 4.234 4.234 129.4 125 .25 125 .25 FA 1.0 Do. 1897. 0.45 2.27 4.82 4.85 4.128 4.045 4.022 91 .22 66.84 66.24 Do. 1898. 0.13 ... ... 4.34 3.76 4.034 4.026 70.23 75.32 1 .0 Do. 1899. 0.28 ... ... 3.44 3.25 4.084 4.079 79.79 81 .93 1.0 88 THE FLOW OF WATER TABLE X. Continued. Description of Conduit. m 0.95 L d i R 1000s V c a 7?* Old cast-iron pipe 16 years old, tuber- cles removed. Fitzgerald. ... 4 1.0 .4167 1 .241 1 .8283 3.723 4.973 6.141 139.1 141 .1 143.6 Cast iron intake pipe at Erie, Pa., 8 years in use. 0.48 8215 5 1.25 ... 0.178 1.088 99.8 102.1 1.0 Old cast-iron pipe in good condition, some easy bends. Jas. M. Gale. 0.70 19600 4 1 0.947 3.458 112.4 1 .0 VII. GALVANIZED PIPES, GLASS, TIN AND LEAD PIPES. New wrought-iron galvanized pipe, straight. Eh- mann. 0.92 301 .8 .0842 0.021 7.61 29.35 113 .04 225.0 239 .13 1 .11 2.13 3.71 5.80 5.90 87.1 85.9 79.2 84.5 83.2 1.016 1.00 1 .035 0.984 0.970 Glass pipe with fun- nel-mouthpiece. Hamilton Smith. 0.87 63.9 .0764 .0191 25.01 50.77 75.30 102.6 129.18 1 .955 2.945 3.685 4.383 5.009 89.5 94.6 92.2 99.3 100.8 1.078 1.140 1.171 1.199 1.219 Do., no funnel. 0.84 63.9 .0764 .0191 17.97 132 .51 1 .398 4.373 83.6 96.3 1 .030 1 .185 Glass pipe, straight. Darcy. 0.84 147.0 0.163 .0407 0.96 7.71 57.62 111 .91 0.502 1 .591 4.849 6.916 80.3 89.8 100.1 102.4 0.930 1.044 1.164 1.191 New lead pipe, straight. Darcy. 0.84 172 .0886 .0221 0.44 8.14 54.36 146 .32 0.213 1 .089 3.35 5.509 68.3 81 .1 96.5 96.8 0.843 1 .00 1 .191 1 .195 New lead pipe, straight. Darcy. 0.84 172 .1345 .0336 0.82 7.48 56.00 158 .82 0.394 1.404 4.318 7.562 75.0 86.8 99.5 103.5 0.90 1.038 1 .178 1 .238 Tin pipe, straight. Dubuat. 0.98 .0888 .0222 0.196 0.641 3.91 5.39 7.54 9.91 13.7 29.82 30.31 99.01 0.141 0.322 0.772 0.927 1 .183 1.342 1 .476 2.546 2.606 5 .223 67.6 85.3 82.8 84.8 91 .4 90.5 92.9 98.9 100.4 111 .4 0.746 0.942 0.914 6.937 1.009 1.00 1 .015 1 .092 1 .109 1 .120 DESCRIPTION OF CONDUIT, ETC. 89 TABLE X. Continued. Description of Conduit. m L d R 1000 s V t a Tin pipe. Straight. 0.95 192 .1184 1 .0296 5.40 1.116 88.2 1 .00 Bossut. 192 u " 10.76 1.678 94.0 1 .068 64 ({ " 15.08 2.075 98.2 1 .116 32 (C tt 26.94 2.946 104.3 1 .185 32 ft (t 52.98 4.31 108.8 1.234 Do. 0.94 63 0.1184 .0296 113.4 6.143 106.0 1 .220 126 n 113.5 6.15 106.1 1 .220 189 tt u 113.4 6.157 106.2 1 .220 VIII. PIPES AND OPEN CONDUITS OF PLANED OR ROUGH BOARDS. Redwood stave pipe. Los Angles, Cal. A. L.Adams,1898. 0.93 4-8000 1.166 0.292 0.17 0.161 0.178 0.145 0.391 0.638 1.355 0.698 0.698 0.751 0.691 1.167 1.531 1.181 99 101 104 105 109 112 113 0.908 0.926 0.953 0.963 1.01 1.027 1.043 ... ... Do. At Astoria, Ore, A. L. Adams. 0.96 4188 1 .5 0.375 2.07 3.605 132.9 jri Wooden stave pipe, at Cedar River, Wash. Long but easy curves. Sev- eral years in use. Some slight de- posits. Theron A. Noble. 0.50 3.67 0.917 1.067 1.134 1 .191 1 .262 1 .33 1 .331 1 .401 1 .627 1 .757 1.757 1.888 3.468 3.522 3.685 3.853 3.964 3.972 4.072 4.415 4.595 4.635 4.831 110.1 108.6 110.9 112.6 112.9 113.1 112.9 113.7 113.8 114.8 115.6 V* ... ... Do. 0.58 4.5 1.125 0.342 0.342 0.436 0.558 0.557 0.672 0.783 0.856 0.983 1.076 1 .162 2.282 2.276 2.65 3.07 3.05 3.41 3.724 3.924 4.215 4.42 4.69 116.8 115.8 119.4 122.1 121 .4 123.7 125 .2 126.2 126.5 126 .7 129.2 V* 90 THE FLOW OF WATER TABLE X. Continued. Description of Conduit. m 0.51 L d R 1000 s V c a Wooden stave pipe at Ogden, Utah. Many easy curves two years in use Marx-Wing anc Hoskins. 4.000 6.0 1.5 1 .40 1.68 2.14 2.43 2.96 3.59 3.63 110. 112. 115. 119. 122. 126. 124. y\ Rectangular pipes of unplaned board. Darcy. 0.68 145.7 0.319 0.523 1.067 1.933 2.733 3.867 6.267 7.267 8.80 1.23 1.778 2.267 2.939 3.529 4.349 4.625 5.307 94. 96. 96. 99. 100. 97. 96. 100. yA ... ... ... Rectanglar pipe of unplaned boards. Darcy. 0.73 230 0.505 0.475 1.076 1.90 2.91 4.27 5.06 5.76 6.61 1.67 2.52 3.37 4.23 5.07 5.52 5.91 6.37 107. 108. 108. 110. 109. 109. 109.7 110.3 F rV Provo Canal Flume, Utah. Semicircu- lar conduit of planed staves, several years in use. W. B. Mc- Dougall. w^, P 8.21 8.72 .ZUo 0.198 0.200 .(Jl 3.01 3.12 to .4 74.7 74.7 F rV 188.4 9.6 9.02 0.191 3.17 76.4 XVII. NATURAL CHANNELS IN EARTH. Description of K W d R 1000s r c a Channel. 1 La Plata River, 1 .03 16.22 0.007 1.391 128.3 i Catalina channel. F T8 Width of channel many miles. Bed fine sand. Slopes measured with great accuracy for a distance of 85 miles. J.J. Re vy. 1 Parana de las Pal- 1.12 1222 50.3 0.007 3.07 160.0 r~ mas. 71* Do. . 49.7 .0068 2.95 160.3 ... 49.5 " 2.87 156.6 1 Parana. Rosario 1.18 2460 44.6 .0058 2.63 152.9 _ Section. v^ Do. 1 Seine River at Paris. Section between 1 .45 9.48 0.14 3.37 92.^ vs the bridges of Jena 10.92 K 3.74 95.6 and "The Inva- 12.19 " 3.80 92.4 lides." Ville- . 14.50 (( 4.23 94.0 '. . '. vert. 15.02 (( 4.51 98.3 15.93 0.173 4.68 89.5 16.85 0.131 4.80 102.1 ... ... ... 18.39 0.103 4.69 107.6 1 Qfi 9C 01 1 Q 9 RO 00 C I JtxlVCr JL O lt/ Jj OSS3. d'Albero. Com- i .yo ... .0 .iy .DU oo .O v & mission of Italian 10.1 0.12 2.92 88.1 Engineers, 1878. 11.0 0.12 3.15 112.0 9.8 0.103 3.25 96.0 11 .8 0.104 3.28 100.0 Do. At Porto Mo- . . 12.4 0.093 3.37 95.0 . '. '. rone. . . . . . . . . . 8.5 0.165 3.04 81 .5 DESCRIPTION OF CONDUIT, ETC. Ill TABLE X. Continued. Description of Channel. K W d R 1000s X c a Weser River near 1 .92 9.4 .2499 4.07 83.8 1.0 Minden. 9.82 tt 4.37 87.7 Schwartz, 1808. 10.57 tt 4.75 92.8 11 .18 " 4.94 93.2 12.07 tt 5.24 95.0 12.64 " 5.26 94.2 12.66 tt 5.35 97.0 ... ... 14.13 ec 5.67 96.0 0^98 Do. near Minden. 1 .98 4.50 .5032 3.38 71.0 1 .0 5.33 tt 4.0 78.6 6.14 tt 4.88 88.0 6.66 " 5.24 90.5 7.59 " 5.75 93.2 8.14 " 5.95 93.2 8.61 " 6.15 93.2 10.23 tt 6.66 93.2 10.45 tt 6.56 90.6 10.71 " 6.56 94.0 ... 11.28 ei 6.94 92.0 I'oi T")n At Vlrvtnw I K1 A Off OKKfJO Ol f\ 1 UO. jt\.\j VlOtOW. 1 .Ol 7.41 .oouo tt 5.32 ol .U 84.0 F A 8.92 " 6.30 90.0 9.3 tt 6.52 91.2 9.72 tt 6.65 91.2 11 .70 If 7.53 93.8 13.0 tt 7.92 93.9 13.35 tt 7.90 92.1 Pless^ir River. Some 1 .55 1 .25 9.65 6.6 54.7 1.0 stones and gravel. 2.33 it 9.99 66.4 La Nicca, 1839. . 3.48 " 10.19 66.4 3 .58 " 13.58 72.4 ... 3.59 M 13.94 74.8 Saalach River. Some 1.61 1.54 0.875 2.073 56.5 1 .0 stones. Roff, 1.31 1.100 2.246 58.8 1854. 1.91 1 .242 3.077 63.0 1 .98 1.240 3.385 68.2 ... 2.16 3.660 5.474 64.3 River Rhine in Dom- 1.71 0.25 5.74 1.25 32.8 1 leschger Valley. Gravel and detri- 1.32 7.73 4.75 47.0 F " tus. La Nicca. 2.95 7.95 7.42 48.3 . . . 112 THE FLOW OF WATER TABLE X. Concluded. Description of K W d R 1000s r c a Channel. 1 Salzach River. From 1.94 3.53 0.94 3.48 60.3 yh Bergheim to 4.20 0.94 4.03 63.9 Wildshut. Gravel 7.39 1.12 5.786 63.4 and detritus. 3.51 1.55 4.10 55.4 Reich. 4.64 1.55 4.67 67.5 3.87 1.79 4.45 53.4 4.26 1.79 5.15 58.8 ... Zihl River near Gott- 1.98 3.52 0.4 2.296 61.0 1 statt. Bed very "j/rir irregular, covered 5.02 " 3.706 77.1 with mud and de- 5.53 " 4.625 69.1 tritus. Trechsel, 1825. Mississippi River at 414 Q100 64 Q Orjo 91 9 1 Columbus, Ky., at ^t .-LTT O L44 vT . t7 .Uo ' y& high water, 1895. Rep. of Miss. River Com. of 1896. Do. At Helena, Ark. 2.91 5100 40.5 0.07 5.207 97.8 Do. At Arkansas City Do. At Wilson Point. 4.31 2.68 3453 3944 ... 65.2 56.4 0.064 0.054 5.807 6.145 89.7 111 .4 (C La. Do. At Natchez, 1 .13 2173 69.5 .0459 9.512 161 .2 (t Miss. Do. At Red River 1 .59 4044 57.2 .0284 5.636 140.9 tl Landing, La. Do. At Carrollton, 1.22 2338 71 .0 .0219 6.494 162.3 it La., at high water. 72.7 .0254 6.254 161.8 Bed sand. Do. At low water. 1 .51 2338 65.5 .0021 1.842 158.0 " 1 Irrawaddi River at 2.66 3395 35 16.28 .00861 1 .007 85 .1 1 .0 Saiktha, Burmah. 39 18.49 .0172 1 .783 99.9 Bed sand with 43 19.99 .0218 2.360 103.9 stones, right bank 47 21.13 .0344 2.857 105.9 1 17 rocky in places. 53 26.42 .0474 3.548 100.3 Gordon, 1873. . 57 29.80 .0559 3.993 97.8 . . . 63 35.44 .0688 4.052 94.2 . . . . 69 41 .01 .0817 5.382 92.9 73 44.47 .0904 6.147 97.0 .84 FOKMS OF SECTIONS OF CONDUITS 113 Forms of Sections of Conduits. RELATION OF MEAN HYDRAULIC RADIUS TO WET PERIMETER. In the design of the form of cross-section of an artificial con- duit two factors enter into consideration: 1. The material composing the walls of the channel. 2. The special purpose for which it is intended. Conduits under pressure, whether constructed of metal, wood, earthenware, concrete or masonry are nearly always circular in section, because this form can best be given the strength to resist internal and external pressures. The thickness of the material forming the walls of a circular conduit is found from the formula: PD , < = -jr + c in which P is the pressure in pounds per square inch; D the internal diameter in inches; T the safe tensile strength of the material; c a constant added to guard against defects in the casting or the welding. For such conduits as are subject to water ram a pressure of 100 pounds per square inch is allowed in addition to the pressure due to the head which is equal to P = 0.434 h. The stresses allowed in the material and the constants added are : For cast iron T = 4,000, c = 0.33 For wrought iron T = 17,000, c = 0.06 For steel . T = 20,000 For lead T = 450, c = 0.3 For concrete, 2 per cent steel T = 480, c 1.0 Since the advent of reinforced concrete, conduits constructed of this material are coming more and more into favor. Steel- concrete water pipes resisting pressures of heads exceeding 100 feet are now in use. The two aqueduct-syphons of Sosa have internal diameters of 12.46 feet, and resist the pressure of a head of 92 feet. 114 THE FLOW OF WATER FIG. 4. Forms of Sections of Masonry Conduits. The Numbers are Proportional. FORMS OF SECTIONS OF CONDUITS 115 Steel concrete sewer pipe is now made in diameters from 15 to 120 inches. Open conduits lined with concrete are most frequently made semicircular in section. Wooden flumes which are acting simply as aqueducts are generally made semi-square in section, if they are intended to carry lumber or wood the triangular section is used. For aqueducts lined with masonry a section is generally preferred whose sides are vertical or nearly so, whose bottom is a flat segmental arch, and whose top (if covered) is a semi- circle. Very large aqueducts, those crossing valleys, streams or other depressions are given a rectangular section. In designing channels in earth the velocity enters into the problem. In those of some dimensions the bottoms are well rounded and the sides given slopes ranging from J to 1 for cemented gravel, to 3 to 1 for loose sand. In a preceding chapter we have observed, that the form of the cross-section of a conduit has an appreciable influence on the power of the velocity to which the frictional resistance is pro- portional and that the circular or semicircular form is the one most favorable to flow. For rectangular conduits lined with boards, for instance, we have found the value of the coefficient a to be equal to V& and equal to W* for semicircular conduits lined with the same material. The circular form has the additional advantage of having a wet perimeter less in propor- tion to the area of the section than any other form. >A F E D Let AD (Fig. 5) be the top width of a trap- ezoidal channel, BC its bottom width, and FB the depth. The area of the cross- section will then be : y FIG. 5. 116 THE FLOW OF WATER and the wet circumference P = AB + BC + C D. Let BC = 6, BF = d, AD = t, AF_ BF Then the area A = db + Id 2 = d (b + Id); the wet perimeter P = b + 2 d \/l + Z 2 ; the bottom width b = -=- Id; d p and the relation , the reciprocal of R, P 1 d -T = 2 + -T ( 2 Vl * + ! - ' -il Cc- ./I Let the angle which the side of the conduit CD makes with the horizontal be denoted by a, and we have for the conditions P A most favorable to flow or T- a miminum, and a maximum, A sr the depth d = if _ A sin a 2 - cos a' the top width t = b +.2ld = + d cotangent a; A the bottom width b = -r - d cotangent a; a , . , , . Wet Perimeter the relation Area of Section jP __6_ 2rf ^4. ^ sin a = l + /2-cosa)\ d d \ A sin a I Consequently, for a given value of i? and given side slopes the area of section is least if R is equal to one-half the actual depth of water. For the semicircular section R is equal to one- half the radius, hence this forms fulfils the conditions best and other forms of SEWERS 117 section fulfil it the better the nearer they approach the semi- circle. Table A contains values of R and areas of sections in terms of the radius or semi-diameter *for semicircular conduits. TABLE A. Depth of Water in Terms of Radius. Value of R in Terms of Radius. Wetted Sec- tion in Terms of Radius. Depth of Water in Terms of Radius. Value of R in Terms of Radius. Wetted Sec- tion in Terms of Radius. 0.05 0.0321 0.0211 0.55 0.320 0.709 0.10 0.0524 0.0598 0.60 0.343 0.795 0.15 0.0963 0.1067 0.65 0.365 0.885 0.20 0.1278 0.1651 0.70 0.387 0.979 0.25 0.1574 0.228 0.75 0.408 1.075 0.30 0.1852 0.298 0.80 0.429 1.175 0.35 0.2142 0.370 0.85 0.439 1.276 0.40 0.2424 0.450 0.90 0.446 1.371 0.45 0.2690 0.530 0.95 0.484 1.470 0.50 0.2930 0.614 1.00 0.500 1.571 Table B contains proportions of channels of maximum values of R, the mean hydraulic radius for a given area and given side slopes. Half the top width is the length of each side slope and the wet perimeter is the sum of the top and bottom widths. The mean hydraulic radius is equal to one-half the depth of water. TABLE B. Description of Form of Section. Inclination of Sides to Horizon. Ratio of Side Slopes. Area of Section in Terms of Depth. Bottom Width in Terms of Depth of Water. Top Width in Terms of Depth of Water. Semi Circle 571 d 2 Semi Hexagon . . . Semi Square .... Trapezoid Do Do 60 90 78 58' 63 26' 53 8' 3 5 1 1 4 1 2 3 4 .732 d 2 2d 2 .812d 2 .736d 2 .750d 2 1.155d 2d 1 . 562 d 1 . 236 d d 2.31d 2d 2.062d 2.236d 2.50d Do 45.0 1 1.828d 2 0.828d 2.828d Do 38 40' U 1 952 d 2 702 d 3 022 d Do Do 33 42' 29 44' 1* if 2.106d 2 2 282 d 2 0.606d 532 d 3.606d 4 032 d Do Do ... 26 34' 23 58' 2 2i 2.472d 2 2 674 d 2 0.472d 424 d 4.472d 4 924 d Do. 21 48' 2i 1 2 885 d 2 385 d 5 385 d Do. . . . 19 58' 2| 1 3 104 d 2 354 d 5 854 d Do 18 26' 3 1 3 325 d 2 325 d 6 325 d 118 THE FLOW OF WATER Sewers. The forms of the cross-section most frequently adopted for sewers are the circular and the oval or egg-shaped. Only sewers of very great dimensions are given a rectangular section, the roof being a flat segmental arch. For sewers less than two feet in diameter glazed earthenware pipe is mostly used, less frequently concrete pipe. Sewers con- structed of brick, masonry or concrete are, however, found with diameters down to two feet. When the discharge of a sewer is estimated to be fairly con- stant the circular section is preferred, when it varies con- siderably, however, some form of an oval sewer is used. Two forms of egg-shaped sewers are in general use. The one most frequently adopted has the proportional parts as given in the annexed figure. In the other form the lower circle has a diameter of one-fourth the diameter of the upper circle only. This form is used for sewers of small dimensions and greatly varying in discharges. The vertical diameter in both forms is always equal to 1J diameters of the upper circle. SEWERS 119 In the figure Tangent = = 0.75. Hence the angle BCG = 36 53', and the angle EGG - 180 - (90 + 36 53') = 53 7', hence, the angle FGD = 2 X 53 7' = 106 14'. Using trigonometry, these data enable us to compute the different parts of the area and the circumference with precision. For the proportional parts given in the figure we find by this method : The area = 18.35 which is equal to 1.147 d 2 , the circumference = 15.8488 which is equal to 3.9622 d, the mean hydraulic radius = 1.1584, which is equal to 0.2896 d, d being the horizontal diameter. By the same method we may compute the value of the mean hydraulic radius or the area for any depth of water in the sewer. The mean hydraulic radius has its greatest value when the con- duit is about 0.85 full. It is equal, being 0.85 full to 0.345 the horizontal diameter; | full to 0.33 the horizontal diameter; i full to 0.28 the horizontal diameter; J full to 0.20 the horizontal diameter. The speed of flow necessary to prevent a deposit of sewage is given, for all forms of the cross-section by the equation or, in exceptional cases, when the sewer is very well constructed by 0.0625 v = 2 H , r being the mean hydraulic radius. Hence for the very greatest section the least permissible velocity is two feet per second. In order that the velocity should not fall below the permissible limit the value of 66 ( Vr + ) Vr, 120 THE FLOW OF WATER on which for equal slopes the velocity depends should not, for any quantity of discharge vary greatly. If we assume the horizontal diameter to be equal to 4 feet, the value of r is, for the sewer running full, equal to 1.1584 feet, for the sewer running \ full to 0.80 feet. Taking m = 0.57 (corresponding to common brickwork) the value of 66 ( Vr + m) A/r is in the first case equal to 113.6, in the second to 89.48. For r = 0.8 the velocity necessary to prevent a deposit is equal to 2 + ^^ = 2.156 ft. per second. 0.8 s = 7~v=T T~7 = l = R^ 1 I = 00053 L66 ( Vr + m} V r J L 8 r For this 1 velocity the slope will be .156m 2 ( Vr + m) Vr J ~ L 89.48 J For the same slope and the sewer running full the velocity will be v = (66 ( Vr + m) Vrs)& = (113.6 X 0.02304)if = 2.575 feet per second. The cross-section is equal to 1.147cP = 18.864 / 2 , hence the discharge, for the sewer running full Q = 18.864 X 2.575 = 48.57 / 3 and Q = x 2.516 = 9.09 / 3 for the sewer running \ full. Thus, while the actual discharge in cubic feet per second for the sewer running full is 5.34 times the discharge of the sewer running J full, the difference in the speed of flow is only 2.575 2.156 = 0.419 feet per second. EXPONENTIAL EQUATIONS 121 EXPONENTIAL EQUATIONS. General Relations between Diameters and Velocities or Quan- tities. General Relations between Slopes and Velocities or Quantities. Long Circular Conduits Running Full. A. If in our general equation for the velocity of flow we substitute d, the diameter of a conduit in feet, for r, its mean hydraulic radius, the formula thus transformed will read: v = 23.34 ( V5 + 1.414 ra) Vds and for a = vv v = (23.34 (V5 + 1.414m) V~ds)$ which may be written v = 34.607 ( V5 + 1.414 m)* d& s*. For the term ( *V3 + 1.414 m) and its variation with the velocity we have no adequate substitute, containing as it does two variables in an everchanging relation. This fact makes the problem of finding an exponential equation, giving values as exact as those found from the general formula an impossibility. The powers of the diameter (or the mean hydraulic radius) to which velocities and quantities are proportional are not con- stant, even for the same degree of roughness, but vary with the diameter (or the mean hydraulic radius) itself. On this account exponential equations with constant values of the powers of d or r are only approximations, fairly true between certain limits, but the more incorrect the farther outside of these limits. Such equations should only be considered as brief empirical expressions, valuable only on account of their brevity; they should never be substituted for the general formula when a great degree of accuracy is desired. Computing the velocities and discharges of two long straight circular conduits of different diameters, but of the same degree of roughness and having the same slope from the general equation, we may by means of the data thus obtained find an expression for the relation between the diameter and the velocity or the discharge which holds good between the limits of the two values of d. 122 THE FLOW OF WATER To find the exponents of the powers of d to which velocities and quantities are proportional we may put x _ log V, - log V Q log d l --log d Q y _ log Q t - log Q By means of these equations and for values of d between 1 and 50 inches for a = v\, and between 1 foot and 20 feet for other values of a we find the following values of x and y. a = v\ m = 1.0 x = 0.67 y = 2.67 a = v l m = 0.95 x = 0.67 y = 2.67 a = v \ m = 0.83 x = 0.68 y = 2.68 a = v? m = 0.68 x = 0.695 y = 2.695 a = V& m = 0.57 x = 0.70 y = 2.70 a = m \ m = 0.53 x = 0.70 y = 2.70 a = i.o m = 0.57 x = 0.66 y = 2.66 a = 1.0 m = 0.45 x = 0.67 y = 2 - 67 a = i.o m = 0.30 z = 0.68 y = 2.68. Consequently, for m = 0.68 (pipes of planed staves, cast and wrought iron, etc., all some time in use), velocities are pro- portional to d ' 695 , quantities to d 2 ' 695 and we have between V and d the relation /7 - 695 (1) J 0.695 1 ^ 0-695 '1=^03^695- () a 1.453 ,,- - < and between Q and d Q, _ ^ r2) f} ~~ J 2-695 W ^o a o rf i\ 2 ' 695 / ^ (a) (6) EXPONENTIAL EQUATIONS 123 Equations (a) 1 and 2 enable us to find velocities and quantities for a diameter d lf provided velocities and quantities for a diameter d are known. From equations (6)1 and 2 we may find the diameter d t for a velocity V l or a quantity Q v provided the diameter d for the Velocity V Q or the discharge Q is known. In the same manner we find for the relation between the slope and velocities and quantities: (3) Co * (4) ( } Combining equation (1) and (3) we have: 124 THE FLOW OF WATER Combining (2) and (4) we have: (6) (a) = <* (c) By means of these equations we may find : The value of V 1 or Q 1 for any value of d l or $ r The value of d^ for any value of 7 l; Q l; or $ r The value of S t for any value of V lf Q or d 1 provided values of F 07 Q , d , S are known. In these equations S& is substituted for S^ 5 when a = v& and stituted for its equivalent 1 m. In particular we have for the semicircle: m = 1.0 Semicircular channels lined with clean cement, v = 129 d' 69 S&. m = 0.83 Semicircular channels of brickwork washed with cement, v - 116 d' 7 S&. m = 0.70 Semicircular channels lined with rough boards, v = 104. 5 d' 71 S&. For the semisquare we have: a = 7*. m = 0.95 Channels lined with clean cement, planed boards, v = 108.43 d' n S&, Q = 216.86d 2 ' 67 T 7 . m = 0.80 Channels lined with smooth concrete, very good brickwork, v = 98.87 d' M S^ t Q = 197.74 d 2 ' 68 S&. m = . 70 Channels lined with sawed boards or good brickwork, v = 92.48d' 69 /S T97 , Q = 184.96 d S&. m = 0.57 Channels lined with common brickwork, rough con- crete or very good ashlar, v = 84.23 d' n Q = 168.46 d 2 ' 70 m = 0.45 Channels lined with rough brickwork, common ashlar or very rough concrete, *- 76.67 d' Q = 153.34 d 2 ' 715 m 0.30 Channels lined with good rubble masonry, v = 67.27 d*- m S& Q = 134.54 d*' OPEN CONDUITS 133 m = 0.15 Channels lined with roughly hammered masonry, channels in cemented gravel up to one inch in diameter, v = 57.8 d' 75 S&, Q = 114.6 d 2 ' 75 S&, a = 1.0. m = . Channels lined with common rubble masonry, tunnels K = 1.0 in rockwork, channels in cemented gravel exceeding one inch in diameter, v = 39.24 d' 75 4, Q = 78.48 d 2 ' 75 S*. m = 0.1 Fairly regular channels in loose sand, or sand with K = 1.2 gravel imbedded v = 35.39 d*' m S*, Q = 70.78 d 2 - 785 ^. m = - 0.2 Fairly regular channels in earth, free from debris or K = 1.5 vegetation, v. = 30.86 d' 775 S*, Q = 61.72 d 2 - 775 i m = 0.32 Channels in earth with debris or vegetation, K= 1.93 v = 26.1 d' S*, Q = 52.2 d 2 ' 795 Si If the cross-section of the conduit is a trapezoid, the dis- charges are multiplied by the proportional areas found in the table given at the beginning of this chapter. For a trapezoid having side slopes of 1: 1, for instance, the discharges found from any of the equations given above are mul- tiplied by . 914, for side slopes of 2 : 1 by 1 . 236 etc. The depth being the same, the velocity is not affected by the side slope. Values of the powers of d relating to Velocities are found in Table E ; those relating to Quantities of discharge in Table F. Values of the sines of the slope and their roots are found in Table C. 134 THE FLOW OF WATER For channels in earth, in case the velocity exceeds the limit where erosion begins, the following equations may be used: m = - 0.1 v = 28.68 d' K = 1.20 Q = 57.36 d 2 ' m = - 0.20 v = 25.14 d' 735 K= 1.50 = 50.28 d 2 ' m = - 0.32 v = 20.90 d' 755 K= 1.93 Q-41.80 d 2 ' m It is, however, more convenient to use the equations previ- ously given, for which the powers of d and S are found in the tables, and multiply the Velocities or Quantities found from the formulas by the coefficient of variation of C, which in these cases is equal to a = - - Values of a = - are found in yh column 10, Table V. II. GENERAL EQUATIONS. In the design of cross-sections of channels it is not always possible to use the form of section most favorable to flow. Other forms are frequently required for special purposes, are constructed at less cost, or offer other advantages. For wooden flumes, for instance, the triangular section is fre- quently adopted. If the sideslopes of a triangle are 1 : 1, or the sides inclined 45 (which is the usual sideslope for a flume), the area of its cross- section, its mean hydraulic radius and consequently its velocity and discharge are equal to those of a semisquare when (1) the depth = Varea of semisquare. (2) the top width = \/4 X area of semisquare. In the design of channels in earth it is frequently necessary, in order to keep the velocity below the eroding limit, to make the sections wide and shallow, so that the frictional resistance may be increased and the flow retarded. In many cases a shallow section is also more easily constructed and at less cost. OPEN CONDUITS 135 The general exponential equations, derived as previously indicated, are as follows: a = 7*, v = 243 - 131 .6 (1 - m) r* S A , a v = 205.8 - 112 (1 - m) r* S a = 7* v = 176.3 - 93.5 (1 - m) r* V2gH - and the discharge through OP q 2 = cf 2 b V therefore the whole discharges through NP Q = cb V2g (f 1 VH -0.5/! + / 2 Vh). m FIG. 16. For a given discharge Q, a given effective head H(MO) and a given height / 2 of the sill below the surface of the run-off water, the height f iy or the distance of the lower edge of the sluice board above the surface of the runoff water may be found by putting Q LOSS OF HEAD 183 METHODS OF MEASUREMENT. Loss of Head. A. When a conduit discharges into an open tank or reservoir, or into the open air, the loss of head is ascertained by levelling between the surface of the source of supply, and the surface of the discharge tank, reservoir or outflowing stream. When this is not the case, or when the loss in part of the conduit only is to be .found, other methods must be employed. Where the pressures are not great, open stand pipes or piezometers are most convenient, otherwise the pressures are measured by means of manometers. A mercury manometer of the form generally used has the following essentials: A cast-iron mercury reservoir into one side of which a glass plate is fitted through which the height of the mercury within may be observed. A metal tube with a gate valve connects the top of the reservoir with the main pipe at the point at which the pressure is to be measured. At its highest point, this tube has an air valve. Into the mercury reservoir, which is about half filled with mercury, a vertical tube is placed, nearly reaching to the bottom. This tube, usually one quarter of an inch in diameter, is of brass or wrought iron in its lower part and of glass in its upper part. To the glass tube a graduated scale is attached. As mercury is very sensitive to changes of temperature the tube is surrounded by a water- jacket, in its upper parts also of glass. When the gauge is to be used the air valve in the connecting tube is opened and also, by degrees, the gate valve. When the air is wholly removed the air valve is closed and the gate valve fully opened. The pressure of the water in the reservoir depresses the surface of the mercury and causes it to rise in the tube. The height of the mercury column above the surface of the mercury in the reservoir is read on the graduated scale, both at times of discharge and times of no discharge. If a is the difference of the heights of the mercury columns at two sections at times of no discharge, and A the difference at 184 THE FLOW OF WATER times of discharge, the loss of head between the two sections whose pressures are measured is equal to H = 13.6 A - a, 13.6 being the specific gravity of mercury. When the conduit is of great length and the difference between the pressures at two sections considerable, a form of the manom- eter known as the Bourdon gauge, is used with good results. The essential parts of this instrument, universally used as a steam- gauge, consist of a hollow curved metal spring, one end of which is free to curve, while the other is fastened to the case of the instrument. A pipe connects the interior of the tube, which is oval in cross-section, with the main pipe at the point where the pressure is to be measured. The pressure of the liquid expands the spring, the free end moves and by a lever the move- ment is transmitted to a toothed bar lever, which again transmits the motion to a toothed wheel. The movement of the spring, thus converted into rotary motion, is, by a pointer, indicated upon a graduated circular scale. The pressure is indicated in pounds per square inch. This is converted into feet of pressure by dividing it by 0.434. If A is the difference between the indicated pressures at two sections at times of discharge and a the difference at times of no discharge, the loss of head between the two sections is equal to A a H = 0.434 Discharge of Conduits under Pressure. B. Discharges are measured by means of vessels, tanks, by the rise in the surface of a reservoir, or the overflowing stream is measured by a weir, an orifice or the current meter. When these methods are not feasible, some form of water meter is used. The best known devise of this kind is the Venturi meter, invented by Herschel and named for a celebrated Italian hydraulician. DISCHARGE OF CONDUITS UNDER PRESSURE 185 The theory of the Venturi meter is based on the principles enunciated by Bernouilli: "The fall of the free surface level between two sections of a conduit is equal to the difference of the heights due to the velocities at the sections." If p is the pressure at one section of a conduit and v l the velocity and p 2 and v 2 the pressure and velocity at another sec- tion and y and y : elevations above datum, then GG +y ~ In Fig. 17 the line p v p 2 , p 3 , shows the theoretical variation of the free surface level due to the contraction and subsequent enlargement of a conduit. The line p v p v p 5 , shows the actual variation, the difference being due to the pressure expended in overcoming the frictional resistance of the walls of the con- duit. It will be observed that this difference increases with the distance. FIG. 17. Differences of pressure in sections of conduits not far apart are most conveniently measured by mercury difference gauges. In Fig. 18 is shown a gauge of this kind connected to sections, the "pressures at which are to be compared. 186 THE FLOW OF WATER The bottom of the gauge is filled with mercury. When the gate valves are opened, the pressure of the water causes the mercury to rise or fall to heights which indicate the pressures at the points of the main to which the gauge is attached. A graduated scale allows a comparison of the pressures. PRESSURE FROM LOWER END OF. VENTURI METER PRESSURE FROM THROAT OF VENTURI METER FIG. 18. The difference between the pressures at the full section and the section most contracted indicates the difference between the velocities at the two points; the difference between the pressures at the full sections above and below the contraction corresponds to the loss of head between the two points. Denoting the area of the section not contracted by A, the area of the section most contracted by a, and the difference between the pressures converted into feet of head of water by H, the theoretical quantity passing through the section most contracted per second is given by the equation A a For the actual discharge this is multiplied by a coefficient, which, however, differs little from unity. In the Venturi meter, DISCHARGE OF CONDUITS UNDER PRESSURE 187 as usually constructed, the area of the throat is contracted to one ninth the area of the full section of the main. Its length is from eight to sixteen times the diameter of the full section. It has a registering device which mechanically converts differences of pressure into corresponding velocities and these, for a given diameter and a certain interval of time (10 minutes), into gallons of discharge. The meter is made in sizes from 2 up to 100 inches in diameter. The loss of head is insignificant and the condition of the water does not affect its working. The discharge from vertical tubes was recently determined by Lawrence and Braunworth and formulae deduced, which not only will prove to be of great value in computing the discharge of artesian wells, but furnish another method to determine the discharge of any conduit under pressure with a fair degree of accuracy. To do this, it will simply be necessary to give the end of the conduit a vertical direction and observe the elevation of the crest of the outflow above the rim of the conduit. The investigators mentioned experimented with tubes rang- ing between 2 and 12 inches in diameter and 15 feet long and three conditions of out-flow were observed, depending on the pressure head. Under a feeble head the water flows simply over the rim of the conduit as it does over a sharp edged weir and the discharge is equal to Q - 8.8 d 1 ' 25 h lM . When the issuing water forms a jet the discharge is equal to Q = 5.57^'" ft ' 53 , in which Q = cub. ft. per sec. d = actual internal diameter in feet. h = elevation of crest of water above the rim of the conduit, in feet, determined by sighting rod. For the condition intermediate between the weirflow and the jetflow no formula was deduced. 188 THE FLOW OF WATER Discharge of Open Conduits. c. When the discharge of an open conduit cannot be measured by a weir or an orifice, it is necessary to find the mean velocity of flow. The mean velocity in a vertical section is ascertained directly by means of rod-floats or by making measurements at the point where the thread of mean velocity is found, either with a current meter or with a double float. Indirectly the mean velocity is found by means of surface floats or by current meter observa- tions at different points in the vertical section. If the channel is narrow, measurements in one vertical section are generally sufficient, especially if a rod-float is used. With increasing width of the channel observations in two or more vertical sections are necessary. When the mean velocity of flow in a river is to be ascertained, the channel is divided, at right angles to the line of flow, into sections 5, 10, 20 or more feet wide, the distance depending on the degree of accuracy desired. The mean velocity at each section is found by means of rod- floats, by observations at the surface, at mid-depth, at the position of the thread of mean velocity or at points of propor- tional depth. The mean velocity for the whole channel is found by taking the mean of the mean velocities of all the sections. For the discharge of the whole channel the mean velocity of each section is multiplied by its area and the discharges of all the sections summed up. If floats are used, the stretch over which the float is to pass should be carefully measured and staked off. If possible ropes or wires should be stretched across the stream, at right angles to the line of flow. The float should be started some distance above the rope and the time of its passage carefully observed. The distance measured out may be 250 to 500 feet for swift streams; 50 feet will suffice if the current is feeble. The longer the stretch the more reliable the time observation. On the other hand, if the stretch is long it is often exceedingly difficult DISCHARGE OF OPEN CONDUITS 189 to keep the float in a position parallel to the axis of the stream. This is especially so near the banks. On this account it may be necessary to measure stretches as short as 20 feet. A surface float may be a ball of wood or some other light material, or else a watertight metal cylinder, so loaded as to float flush with the surface of the stream. A small flag will render the float more visible. Double floats are used to find the velocities at different depths below the surface. They consist of light surface floats con- nected by a fine strong cord, to a large sub-surface float. A ball of wood or a flat watertight metal box makes a good surface float, a watertight metal cylinder, heavy enough to keep the cord in tension, but not to drag it below the surface is an excellent sub-surface float. The speed of the surface float is identical with that of the larger float and observations of its passage will give the speed of the latter. Usually the subsurface float is placed at the point where the thread of mean velocity is found. The use of double floats generally leads to trouble of one kind or another; they are rarely used, except to measure velocities in very deep channels. A cylindrical wooden pole two inches in diameter and loaded at the bottom, so that it will float vertically, makes an excellent rod-float. It may be made in sections and screwed together. A brass cylinder screwed to the bottom makes an excellent weight. Into it shot may be placed to suit the weight to all requirements. Watertight tin tubes also make good rod- FIG. 19. Channel of River Divided into Vertical Sections. floats. Rod floats should be loaded so that they nearly reach to the bottom of the channel, but never touch it. On the other hand they must not be too short, or else they will travel with a speed exceeding the mean velocity. 190 THE FLOW OF WATER The rod-float is the ideal instrument to measure the velocity in a flume or aqueduct. The fact that it integrates the velocity of the whole section and thus indicates the mean velocity directly is an advantage not possessed by any other measuring device. If properly used it gives results whose accuracy cannot be ques- tioned. However, if the bottom of the channel is very rough, covered with plants or else very deep, its use is not indicated. FIG. 20. Velocity measurements are made in the centre of each sec- tion. Depths are taken by soundings. Line (1) indicates the position of the thread of maximum velocity in each section, line (2) the position of the thread of mean velocity in each section, and line (3) the position of the thread of mean velocity for the whole section. The current meter, like so many other hydraulic measuring devices, originated centuries ago in the Valley of the Po, Italy, the cradle of hydraulics. The earliest form consisted of a small paddle wheel mounted in a floating frame. It could only be used at the surface. When Woltman, in 1790, added a recording device the instru- ment could be used at any depth. The recording mechanism consisted of an endless screw fitted to the horizontal axis; and a series of toothed wheels which transmitted the motion of the axis to a register. The recording mechanism was thrown in and out of gear by a string, attached to a lever. The instrument was fitted and clamped to a one-inch pole on which it could be DISCHARGE OF OPEN CONDUITS 191 slid up and down. To read the number of revolutions recorded the current meter had to be taken out of the water. The instru- ment was generally known as "Wolt man's Tachometer." Many modifications of this instrument appeared, mostly of the windmill pattern, with propellers and vanes. Some have the axis of the propeller horizontal, others vertical, and the shape of the propellers is variable. The general form of the instrument is, however, always the same. The present day current meter has an electrical signalling or registering device. The best known patterns are those of Harlacher in Europe, and those of Price and Ritchie-Haskell in the United States. The Harlacher meter is of the windmill pattern ; its propeller has four blades. A vane about 12 inches long and 5 inches wide is fitted to a prolongation of the axis of the wheel. This direct- ing device keeps the face of the wheel at right angles to the line of flow. To the axis of the propeller is fitted an endless screw, operating a toothed wheel. A pin in the side of the wheel strikes an electric wire at each revolution, thus completing an electrical circuit. The battery with the registering or sounding device is kept at the surface. The meter slides up or down on a vertical rod. To move the meter up and down with a uniform speed an apparatus consisting of ropes, pulleys and weights is often used. The propeller of the Price current meter has four cup-shaped wings; its axis is vertical and its revolutions are indicated by an electrical buzzer. The instrument is generally used without a rod; it is kept vertical by a weight attached to the frame and moved up and down by a cord. Its vane consists of two blades, one horizontal, the other vertical, intersecting in the middle at right angles. It is made in two sizes. The small meter measures velocities as low as 0.2 feet per second with a fair degree of accuracy; the large meter gives good results down to velocities of 0.5 foot per second. The latest design in the line of meters is the Ritchie-Haskell so-called "direction current meter." Like the Harlacher and the Price this instrument has a device recording the number of revolutions of the propeller electrically. It has also a device 192 THE FLOW OF WATER indicating the direction of the current. The body of the instru- ment is a compass with a magnetic needle. An electrical circuit measures the angle between the direction of the needle and the direction in which the vane points and indicates the angle on a graduated dial. Current meters must be rated; that is, the relation between the velocities and the number of revolutions of the propeller must be ascertained. This is done by pulling the meter at various constant speeds through a still body of water, and deter- mining the relation between speeds and revolutions. Current meters as furnished by the makers are always rated, but they must subsequently be rerated at frequent intervals, if good results are desired. As with floats, measurements with the current meter are made in various ways. The best method is no doubt the one adopted by Harlacher of sliding the instru- ment by means of a mechanism at a uniform speed up and down on a pole. By this process the velocity of the whole section is integrated and a very good mean value found. If no pole is used the instrument is most conveniently moved up or down by means of a cord thrown over a small pulley. A good current meter, properly rated and carefully handled, surpasses any other instrument in the facility and extent of its application; it gives results nearly as trustworthy as the rod- float, and for average velocities nearly as accurate as a weir. The Darcy gauge, an instrument formerly in great favor, is at present, owing to the great perfection of the current meter, but rarely used. The instrument consists of a combination of two Pitot tubes, fastened to a supporting frame. A Pitot tube is a vertical glass tube with a right-angled bend. If such a tube is placed into a stream, with its mouth facing up- stream and at right angles to the line of motion, the water will ascend in the tube to a height which is equal to v 2 b=*, nearly. If the mouth of the tube faces the bank of the stream, and is in line with the line of motion, there will be no difference of level between the surface of the water in the tube and the surface of the stream. SURFACE MEAN AND BOTTOM VELOCITIES 193 If the mouth of the tube faces downstream and is at right angles to the line of motion, the surface of the water in the tube will be below the surface of the stream, the difference being equal to In this case the velocity is somewhat modified by the retard- ing influence of the tube. Darcy combined two tubes having their mouths at right angles, and provided their lower parts with stopcocks, which can be operated, when the instrument is in the water, by means of a string. If the cocks are open and the mouth of one of the tubes faces upstream at right angles to the line of motion the water will ascend in it while it will not ascend in the other tube. If the corks are then closed, the instrument may be lifted out of the water and the difference of level in the two tubes read off on a graduated scale. Surface Mean and Bottom Velocities. Position of Thread of Mean Velocity. From 82 observations of flow in small channels Bazin deduces the following: Mean Velocity = Maximum Velocity 25.4 Bottom Velocity = Maximum Velocity 36.3 Vr.s Bottom Velocity = Mean Velocity 10 . 87 \/r\s From this we have Fmean + 25.4 Vi\s V max. V mean 1 1.0, V max. 1 + 25.4 Vr.a and as = c, r.s , 1 V mean 1 we have also = ., . V max. - 25.4 c i vi y bottom 1 and likewise V max. 1 , 36.3 1 H C 194 THE FLOW OF WATER Comparison of values of __ mean = with values of V max. 1 25.4 c mean f ounc j ^ observations of flow in a great variety of V max. channels shows that Bazin's formula is not of general application. It fails because the influence of the value of the total depth of the channel is not considered. V mean The following values of are given by the most reliable authorities : V mean V surface Revy, Parana de las Palmas, La Plata 0.835 Harlacher, Bohemian Rivers, 28 observations 0.838 Swiss Engineers, Swiss Rivers, 200 observations 0.835 Lippincott, Sacramento River, Cal., Depth, 3-5 feet ... 0.88 Lippincott, Tuolumne River, Cal., Depth, 1.12-1.84 feet . 0.88 Lippincott, San Gabriel and Santa Anna, Rough channels, 10-20 feet wide. Depth, 0.25-1.0 feet 0.92 Pressey, Catskill Creek, partial section 0.82 Pressey, Fishkill Creek, partial section 0.93 Pressey, Mean of 28 observations of flow in rivers with rough bottoms, Average depth, 5.05 feet 0.80 Prony, Small wooden channels 0.8164 Prony and Destrem, Neva River, Russia 0.78 Boileau, Canals 0.82 Baumgartner, Garrone River, France 0.80 Cunningham, Solani Aqueduct 0.823 Humphreys & Abbot, Mississippi 0.79-0.82 From these and other data given by Murphy (Cornell testing flume) and others, the writer found that the relation between the surface velocity and the mean velocity may be expressed by the equation Mean velocity = j-=. surface velocity (1) !+- Vc in which n is a coefficient ranging in value between 0.25 for the roughest and 0.35 for the smoothest classes of conduits. Its value is n = 0.32 for K = 1.25 n = 0.30 for K = 1.75 n = 0.27 for K = 2.25 For the velocity at any point x, depth d, in the vertical section SURFACE MEAN AND BOTTOM VELOCITIES 195 we found from data relating to flow in channels with rough bottoms, such as rivers with detritus or coarse gravel, 1 1 + )' (2) in which D is the total depth. This is on the assumption that the bottom velocity is equal to one half the surface velocity, a relation which holds good only for channels with rough bottoms. Bazin found from observations of flow in small artificial channels that the difference between the surface and the bottom velocity ranges between 0.25 and 0.5 of the surface velocity, the differ- ence increasing in value with 'the roughness of the walls. In canals and rivers with comparatively smooth bottoms the difference ranges between 0.3 and 0.4, the average difference being 0.35 of the surface velocity. Combining the two equations (1) and (2), we have for the position of the thread of mean velocity in the vertical section of rivers and canals with somewhat rough bottoms and whose width is several times the depth 41) (3) as the depth below the surface at which the thread of mean velocity is found. The formula does not apply to flumes and other narrow, deep channels. From equations (1) and (3) we find the following values of mean the relation re lative position of the thread V surface of mean velocity in a vertical section, assuming K = 1 . and v = 3 feet. V mean Relative E> V mean Relative V surface Depth. V surface Depth. 1.0 0.898 0.538 10.0 0.854 0.604 2.5 0.881 0.563 15.0 0.842 0.616 5.0 0.369 0.583 25.0 0.832 0.631 7.5 0.859 0.596 30.0 0.813 0.656 APPENDIX I. Variation of the Coefficient c with the Slope. IN the preceding chapters we have defined the variation of the coefficient c with the mean hydraulic radius, with the degree of roughness of the wet perimeter and with the velocity of flow. We will now proceed to investigate if it is possible to find a true expression for the variation of the coefficient c with the slope by the graphical method. From Formula III we have 66 ( *f? + m) V* = c, 66 ( + m) I ^ \ 9 v = [ - 1 \66 (S/r + m)/' or substituting for v its equivalent (66 ($fr + m) Vr7s)* = ( - -4 - Y; V66 ( Vr + m)/ hence 66 ( tfr + m) Vr . s = ( - T= - ) , \66 ( -N/F + m)) _ and (66 ( Vr + m)) 9 Vr . s = c 8 ; consequently (66(Vr+m))*(r.a)*- c; or 66 ( Vr +m) (66 ( Vr + m))* (r . s}& = c. This goes to show that c increases with (rs)^-, consequently the variation of the coefficient c with the slope depends on the value of R. The variation of the coefficient c follows the law of the para- bola. If values of the coefficients a = V% and V& are plotted as ordinates to values of v as abscissae, the points so found lie in curves which are parabolas of the ninth or eighteenth order. A curve somewhat resembling a parabola is the equilateral hyper- 196 APPENDIX I 197 bola, and it is possible to draw a curve of this kind which nearly coincides with the parabola. The equation of the equilateral hyperbola concave towards the axis of abscissae may be put into the simple form The curve in Fig. 1 represents the hyperbola of this equation. In the figure ZO is the vertical asymptote, Zd the horizontal asymptote, YK the axis of ordinates, KX the axis of abscissae, Zg the axis of the hyperbola, X the distance between the vertical asymptote ZO and the axis of ordinates YK, c the ordinate of any point in the curve. The area of the rectangle ZOKY is the constant which deter- mines the hyperbola. It is equal to the square zfgh or the area of any rectangle comprised between the asymptotes and per- pendiculars drawn to them from any point in the curve. 198 THE FLOW OF WATER Consequently, if lines are drawn from the center z to points R on the axis of abscissae, these lines will intersect the axis of ordinates in points which give the values of c corresponding to the values of R. In this way the hyperbola may be easily constructed. Bazin in his paper, " Etude d'une nouvelle formule," etc., put the equation for the coefficient c into the form in which y is constant and equal to 157.5 in English measure, and 0, a variable, indicating the degree of roughness. Dividing by y we have c = substituting x for g. y yVr Transposing we have 11 xj_ c y y \/r This is the equation of a straight line having values of _ as Vr abscissae, values of - as ordinates. If this equation would hold good, points of values of - pertaining to one slope would lie in Of straight lines intersecting the axis of ordinates in a point. If, y however, values of - and are plotted as indicated it appears c vr that only those points - pertaining to data of flow in old pipes C or fairly regular channels in earth lie in straight lines, while those pertaining to data of flow in very smooth conduits lie in APPENDIX I 199 curved lines convex towards the axis of abscissae, and those pertaining to data of flow in very irregular channels lie in curved lines concave towards the axis of abscissae. If straight lines are drawn averaging between the points as much as possible, these lines will intersect the axis of ordinates FIG. 2. in points giving values of - for the greatest value of v and the y greatest value of R included in the series plotted. These lines will also intersect the axis of abscissae in points which give the value of - pertaining to each value of - . In Fig. 2 we thus plotted the experimental data of Darcy-Bazin, series 7, 8, 9 and one series given by Rittinger (s = 0.0343), all pertaining to flow in testing channels of rough boards. It will be observed that the lines pertaining to the steeper slopes intersect each other in a point whose abscissa for is Vr 1.0. This is due to the fact that for the greater slopes s = 0.0049, 0.00824, and 0.0343, the velocity is so high that c varies but very little, while it varies much for the feebler slope s = 0.0015. 200 THE FLOW OF WATER The highest value of - corresponding to - = 1.0 is 0.0084, the c Vr lowest 0.0080, average 0.0082. Denoting the abscissa of the point of intersection by a and the average ordinate by K we have y y a and x = Kay a, TZ and considering = Ka as a tangent and denoting it by I, a we have x = ly a. Consequently in our case x = 0.0082 y - 1.0, which gives values of x very nearly equal to those found graph- ically. This formula will, however, only hold good for the values of R included in the series, the highest of which is 1.0. In Fig. 3 the values of y found graphically from Fig. 2 are plotted as ordinates to values of - as abscissae. The points y s are seen to lie in a curved line, intersecting the axis of ordinates at a point B = 131.0 nearly. If the line CD is produced, it will intersect the axis of ordinates in y = 157.5, which is the constant in the formula of Bazin mentioned above. The tan- gent of the angle CEF (in this instance 0.29) corresponds to Bazin's coefficient, g, indicating the degree of roughness. The value of m obtained from the given data is . 70 ; hence the value of ^r + m, for the highest value of R is 1 .70. Dividing 131 .0 by 1 . 70 we have y' = 77 ( 'N/r + m), nearly,. as the value of y corresponding to the highest velocity included in the series plotted. If from the point B = 131 . = 77 ( Vr + m) a line is drawn parallel to the axis of abscissae, any increase in the value of APPENDIX I 201 77 ( 'N/r + m) due to any slope less than 0.0343 will appear as an ordinate above this line BG. It will be observed that values of y" t y'" y s/ v , etc., increase with the decrease of the slope or increasing values of - We nay therefore put y" ', y'", etc., = 77 ( Vr + m) + - in which o j is a coefficient still to be determined. C ^-^^^ C F B^^^ . "* G " yll - , Values of IT FIG. 3. The line RDC is evidently a parabola. If a line is drawn from B to C the tangent of the angle CBG will be equal to z, for s = . 0015. For this slope we have from the figure y = 198.0 2/' = 131.0; therefore - = 198 - 131 = 67 and z = 0.0015 X 67, s or z = 0.1005. Hence y = 77 ( N/r + m) + - 1 s From experimental data pertaining to flow in small channels in earth, R ranging between 1 and 1 . 75 (Darcy-Bazin, Grosbois 202 THE FLOW OF WATER canal) which are, however, somewhat doubtful, we found z = 0. 0936, while from data pertaining to flow in the La Plata and its tributaries we found = 0.00293. From this it is evident that z is a variable and that its value depends on the value of R. Having found an expression for y, the value of x may be found from experimental data without resorting to graphical methods. No. 12, series 6, Darcy-Bazin gives R = 0.922 s = 0.00208 c = 118.9. Hence y = 77 ( ^922 + 0.68) + or y = 127.87 + 47.6 = 175.47. Dividing 175.47 by c = 118.9 we find x = 1.475, = 1 +0.475. But 0.475 is equal to 0.01, **'* or 0.01, 47.6. U . Denoting the term 0.01, which is variable, by I, we have from the given data for the variation of the coefficient c with the slope the expression 77 (tyr + m)+ L s c = :r~: ; ' which, within certain limits corresponds to c = 66 ( S/r + m) V&. From data relating to flow in a semicircular channel of rough boards (Darcy-Bazin, series 26) we find 01 * y + = 210, hence y = 210 - 67, s which is equal to 0.1 84 ( Vr + m) + APPENDIX I 203 and which corresponds within certain limits to c = 66 ( Vr + m) F^. Dividing 84 by 66 the quotient 1.272 is the value of the coefficient of variation of c for v = 18.0. Hence v = 18.0 is the limit up to which the formula holds good. The formula apparently gives good values of c up to the limit indicated. By trial we find, however, that it does not hold for values of R greater than 1.0; unless rs is substituted in the equation for s. Consequently the variation of c with the slope is dependent on the value of R, a fact we demonstrated at the beginning of this chapter. The facts related plainly show that a formula derived in the manner indicated can only be of limited application. It holds good only within the range of values of R, s and m included in the series of data from which it is derived. In other words: We cannot get out of a formula what we do not put into it. A general formula, like that of Ganguillet and Kutter, derived by the methods we have indicated, cannot embody true laws of flow, it naturally must be deficient in one respect or the other. The more so, if the data on which the formula is based are erroneous. The experimental data derived from observations of flow in the lower Mississippi by Humphreys and Abbot and embodied by Ganguillet and Kutter in their formula have been found to be incorrect, greatly at variance with those time and again found by the United States engineers. The contention of Ganguillet and Kutter, that, if values of - are plotted as ordinates c to values of - as abscissae and lines drawn through all the points Vr - these lines will intersect each other in a point = 1 meter, c Vr and that therefore c will increase with increasing values of s if R is less than 1 meter, and decrease if R is greater than 1 meter, is also plainly a fallacy. If values of and - derived from the numerous series given 204 THE FLOW OF WATER by Darcy-Bazin are plotted as indicated, it will be observed that for many of the series the lines intersect at = 1 foot. Vr It would be absurd, however, to draw the conclusion there- from that c will increase or decrease with increase of the slope if R is less or greater than 1 foot. The intersection of the lines at = 1 foot is due to the fact, that for the greater slopes values of c are nearly constant for values of R equal for 1 foot or more, because the value of F 1 * increases slowly at high velocities. APPENDIX II. The Formula in Metric Measure. THE general equation for the velocity of flow reads, for Metric measure, V = 50(-N/r>m) VrTs 2gH 0.007844 L ^Jr + m) 2 R_ The coefficients of variation of c are equal, as for English measure, to a = V^ holds good also for semicircular open conduits. Values of the coefficient m, indicating the degree of rough- ness, are found in the following table, mE signifying the English and mM the Metric values. Exponential Equations. The constants of the exponential equations which we have found for English measure are converted into Metric equivalents 'by putting log constant Metric measure = log constant English measure + " 3.281s - " 3.281. x being the variable power of R or D. The equations for conduits under pressure are as follows, diameters being in meters, velocities in meters per second and quantities in cubic meters (1000 liters) per second : 205 206 THE FLOW OF WATER VALUES OF mE WHICH APPLY IN THE ENGLISH AND VALUES OF mM WHICH APPLY IN THE METRIC SYSTEM. mE mM Description of Conduits. 1.0 0.95 0.85 0.83 0.80 0.70 0.68 0.57 0.53 0.50 0.45 0.30 0.20 0.10 0.20 0.27 0.32 0.85 0.80 0.70 0.75 0.65 0.62 0.60 0.48 0.45 0.47 0.42 0.25 0.20 -0.1 -0.2 -0.27 -0.32 Semicircular and circular conduits lined with pure cement. Long straight brass, tin, nickel and glass pipes. Rectangular conduits lined with pure cement. New pipes of planed boards and very smooth asphalt- coated cast iron. Semicircular conduits lined with cement plaster, 1 part cement, 2 parts sand. Ordinary new straight asphalt-coated cast, wrought iron welded and wrought iron riveted pipes with screw joints, common lead, tin, glass, brass and galvanized pipes. Rectangular conduits lined with cement plaster, smooth concrete or very good brickwork. Semicircular channels lined with rough boards . Chan- nels lined with fairly good brickwork or fairly smooth concrete. Rectangular channels lined with rough boards. Sewer pipe very well laid. Pipes of planed boards, asphalt-coated cast and wrought iron, riveted wrought iron pipes of small diameters or with screw joints, pipes coated with tar or lined with cement or smooth concrete, all some time in use. Common brickwork or concrete. Very good ashlar masonry. Ordinary sewer pipe. Asphalt-coated riveted pipe above 3 feet in diameter. Channels in earth roughly lined with cement mortar. Old pipes of all descriptions, fairly clean. Channels lined with rough brickwork or rough concrete. Old riveted pipes over 3 feet in diameter. Ordinary ashlar and very good rubble masonry. Channels of regular cross-section in fine cemented gravel. Tile drains. Channels of regular cross-section in coarse cemented gravel or rockwork. Channels of fairly regular cross-section in firm sand or sand with pebbles, no vegetation. Channels in earth somewhat above the average in regularity and condition, no stones or vegetation. Ordinary channels in earth, with stones or vegetation here and there. Channels of irregular cross-sections or channels of fairly regular cross-sections but with stones or plants. The values of K corresponding to in = 0.1, 0.2, -0.27, -0.32 are 1.2, 1.5, 1.75, 1.93. APPENDIX II 207 mE mM V in Meters per Second. Q in Cubic Meters per Second. .95 83 60. 92 0.67 S* 47 85 2-67 S* .83 75 56. 54 0.68 44 41 2.68 u .68 60 51. 28 0-69 II 40 28 2.69 11 .57 48 36. 76 0.7 s* 28 87 2.7 at, .57 48 40. 21 0.7 Egg-shaped 47 40 2.7 Egg-shaped .53 45 35. 55 0.7 * 27 92 2.7 stV .45 26 25. 48 0-66 S^ 20 2.66 s^ .30 22 22. 45 0.67 u 17 64 2.67 11 Values of D -' 67 , etc., are found in Table E, values of D 2 ' 67 , etc., in Table F. These tables give the values of the powers of diameters for diameters of 0.05, 0.10, 0.15, 0.20, 0.25 meters, etc. These correspond closely to 2, 4, 6, 8, 10 inches, one foot being 0.3048 meters, one meter 39.4 inches. In order that the powers of the diameters found in Tables E and F may apply to a greater range of diameters we shall find equations in which the unit is 1 deci- meter = 0.1 meter, so that diameters must be taken in deci- meters and fractions thereof. The results will be velocities in decimeters per second and quantities in cubic decimeters or liters per second. The diameters found in the tables as 0.05, 0.10, 0.15, 0.20, 0.25, 0.50, 0.75 when taken as fractions of a decimeter corre- spond to 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 3.0 inches respectively. The discharge of a new wrought iron pipe (m = 0.83) of one inch diameter (0.025 meter or 0.25 decimeter) for a slope of 1 : 100 is for instance : Q = 92.78 (0.25) 2 ' 68 (0.01) & = 92.78 . 0.02435 . 0.075 = 0.1694 liters per second or 10.164 liters per minute. mE mM Velocity in Decimeters per Second. Discharge in Liters per Second. 0.95 0.83 0.68 0.57 0.57 0.53 0.83 0.75 0.60 0.48 0.48 0.45 130.2 Z) - 67 S T * 118.1 Z) - 68 " 104.7 Z) - 69 " 73.34 Z) - 7 S& 80.23 Z) - 7 Egg-shaped 70.93 Z) - 7 S& 102.3 Z) 2 ' 67 S& 92.78 Z) 2 ' 68 " 82.23 Z) 2 - 69 " 57.60 D 2 ' 7 S l9>7 ' 94.58 Z) 2 - 7 Egg-shaped 55.71 Z) 2 ' 7 S & 0.45 0.30 0.36 0.22 55.74 D- 66 S 2 48.0 D- 67 " 43.78 Z) 2 - 66 S* 37.7 D 2 - 67 " 208 THE FLOW OF WATER Exponential Equations Relating to Flow in Open Conduits. Of the following sets of equations the first three relate to flow in the semicircle, the rest to flow in the semisquare, the depth being in meters, the velocities in meters per second, the dis- charges in cubic meters per second : mE mM V in Meters per Second. Q in Cubic Meters per Second. 1.0 0.85 89.25 D' 68 S& 70.12 Z) 2 - 68 " 0.85 0.70 83.0 D - 69 65.1 D 2 - 69 0.70 0.62. 74.0 D- 70 58.2 D 2 - 70 0.95 0.80 73.3 D- 67 S r 9 r 146.6 D 2 - 67 S V 0.80 0.65 67.6 D - 68 135.2 D 2 - 68 0.70 0.62 64.0 D - 69 128.0 D 2 - 89 0.57 0.48 59.0 Z>- 7 118.0 D 2 ' 7 0.50 0.49 57.0 D - 715 114.0 D 2 ' 715 0.45 0.42 54.7 D - 715 109.4 D 2 ' 715 0.30 0.25 49.1 D- 735 98.2 D 2 ' 735 0.20 0.20 44.6 Z>- 735 89.2 D 2 - 735 29.2 D' 75 S* 58.4 D 2 - 75 S* K K 1.2 1.2 26.75 Z>- 765 " 53.5 JD 2 - 765 " 1.5 1.5 23 . 6 D' 775 " 47.2 D 2 ' 775 " 1.75 1.25 21 6 D' 785 " 43.2 D 2 - 785 " 1.93 1.93 20.5 Z>- 795 " 41.0 D 2 - 795 " Of the following equations the first three apply to any depth of water in the semicircular section, the rest to any depth of water in any other form of section, R being in meters, velocities in meters per second. mE mM Velocities in Meters per Second. mE mM Velocities in Meters per Second. 1.0 0.85 142.4 E ' 68 S& 0.30 0.25 81.0 #- 735 S* 7 0.85 0.70 134.0 fl ' 69 0.20 0.20 74.0 .R - 735 " 0.70 0.62 122.0 tf . 70 1 49 ft ' 75 S* 0.95 0.80 116 #- 67 S T7 K K 0.80 0.65 108 R - 68 1.2 1.2 45.4 - 785 " 0.70 0.62 102.5 J?' 69 1.5 1.5 40.4 .R ' 775 " 0.57 0.48 95.3 JR' 70 1.75 1.75 37.2 #- 785 " 0.50 0.47 92.3 .R - 715 1.93 1.93 35.3 .R - 795 " 0.45 0.42 89.0 #- 715 APPENDIX II 209 English and Metric Equivalents. The following relations between the units of the English and the Metric Systems of Measurements are of interest in their relation to the flow of water. 1 meter = 10 decimeters = 100 centimeters = 1000 millimeters. 1 sq. meter = 100 sq. decimeters = 10,000 sq. centimeters. 1 cu. meter = 10 hectoliters = 1000 liters. 1 liter of water at 4 degrees centigrade weighs 1 kilogram. 1 kilogram = 1000 grams. 1 meter = 3.280899 feet = 39.37079 inches. 1 foot = 0.304794 meter = 30.4794 centimeters. 1 inch = 25.3995 millimeters = 2.53995 centimeters. = 0.253995 decimeter = 0.0253995 meter. 1 sq. meter = 10.7643 sq. feet = 1550 sq. inches. 1 sq. foot = 0.0928997 sq. meter = 928.997 sq. centimeters. 1 sq. inch = 6.451368 sq. centimeters. 1 cu. meter = 35.316585 cu. feet = 264.1863 gallons. 1 liter = 0.035316585 cu. feet; = 0.2641863 gallons. 1 cu. foot = 0.0283153 cu. meters, = 28.3153 liters. 1 cu. inch = 0.0163861 liters, = 16.38618 cu. centimeters. 1 gallon = 3.7852 liters. 1 liter weighs 2.204672 English pounds. 1 cu. foot weighs 62.425 English pounds. 1 gallon weighs 8.3448 English pounds. 1 gallon = 231 cubic inches. The pressure of water in kilograms is equal per square meter to 1000 h (h in meters) " " decimeter " 10 h centimeter " 0.1 h " " millimeter " 0.001 h. A pressure of one pound per square inch is equal to a pressure of 0.07031 kilo per square centimeter " 0.0007031 " " " millimeter. The tensile, shearing, or compressive strength of any material in pounds per square inch multiplied by 0.0007031 gives the value in kilos per square millimeter and multiplied by 0.07031, the value in kilos per square centimeter. A pressure of 1 atmosphere = 14.7 pounds per square inch corresponds to a pressure of 1.03296 kilos per square centimeter or a head of 10.3296 meters. 2 g = 19.61. 210 THE FLOW OF WATER Thickness of walls of conduits : PD t m c, t, D, C and m in millimeters. P in kilos per square millimeter = 0.001 h. Material. m C Cast iron 2 8 7 6 Wrought iron 12 1 5 Steel 14 Lead , 3 7 6 APPENDIX III. Greatest Efficiency of a Conduit of a Given Diameter as a Transmitter of Energy. Most Economical Diameter of a Conduit Transmitting Energy under Pressure. I. IN a preceding chapter the ratio between the total head and the head lost in overcoming frictional resistances, which for a conduit of a given diameter under a given head corresponds to a maximum of efficiency, has been mentioned. The potential energy of Qf 3 of water delivered per second at a vertical distance H above the generator is equal to Q 62.4 H foot-pounds, or 0.1134 QH horsepowers. The discharge of a steel-riveted conduit in / 3 per second is equal to Q = 40 26 = 40 d 2 - 1 0.57 S. Hence the efficiency of the conduit is greatest when the velocity and the discharge are 0.57 times the velocity and discharge corresponding to the total head H. II. Of much greater importance is the quest after the most eco- nomical diameter of a conduit for a given discharge and under a given head, a subject recently investigated by A. L. Adams. The function of a pressure pipe is the transmission of energy with a minimum of loss; the usefulness of a power plant as a whole depends on several factors, chief amongst which is the amount of revenue derived from its operation. In comparison with the power transmitted the cost of a con- duit transmitting all or nearly all the energy would be exces- sive. The conduit having a diameter just sufficient to carry the given quantity of water under the given head delivers but a small percentage of the gross energy and its cost per horsepower transmitted is equally excessive as the cost of the conduit deliver- ing all the energy. The diameter of a conduit just sufficient to carry a given quantity under a given head is equal to APPENDIX III 213 The diameter necessary to carry the same quantity with a loss of TsW f the gross energy is equal to = (1000)** = 3.875 times the diameter, just sufficient to carry the given quantity. A quantity of 100 f 3 of water delivered at an elevation of 1000 feet above the generator possesses a potential energy of 100 X 1000 X 0.1134 = 11,340 H.P. The diameter of a vertical steel-riveted conduit just sufficient to carry the given quantity, d = /I09\ 27 = 1.404 feet. \40/ The velocity corresponding to this diameter is equal to V = 50.8 X (1.404) - 7 = 64.38 feet per second. The energy transmitted is ( 64 - 38 ) 2 X 0.1134 X 100 - 729.9 H.P. 2# This is 6.43 per cent of the gross energy. The percentage transmitted by the conduit just sufficient is not constant but decreases with decreasing quantities and slopes. For Q = 10, H = 100, L = 1000, for instance, the gross energy is 113.4 H.P. and the energy transmitted 3.635 H.P., which is 3.21 per cent. The diameter corresponding to a loss of T oV