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THE 
 
 STEAM ENGINE 
 
 BY 
 
 > 
 
 GEORGE C. V. HOLMES * 
 
 VVHITWORTH SCHOLAR; SECRETARY OF THE INSTITUTION OF NAVAL ARCHITECTS 
 
 CORRESPONDING MEMBER OF THE SOCIETY OF NATURAL SCIENCES AT 
 
 CHERBOURG ; MEMBER OF THE JURIES OF THE INTERNATIONAL 
 
 EXHIBITION OF VIENNA, 1873, ANr> OF THE INTERNATIONAL 
 
 INVENTIONS EXHIBITION, LONDON, 1885 ; KNIGHT OF 
 
 THE ORDER OF FRANCIS JOSEPH OF AUSTRIA 
 
 SECOND EDITION 
 
 LONDON 
 LONGMANS, GREEN, AND CO. 
 
 AND NEW YORK : 15 EAST i6th STREET 
 1888 
 
PRINTED BY 
 
 SPOTTISWOODE AND CO., NEW-STREET SQUARE 
 LONDON 
 

 PREFACE. 
 
 THE progress of technical education in this country during 
 the last few years has rendered necessary the production of 
 an elementary text book on the Steam Engine, containing 
 information upon branches of the subject which have 
 hitherto received but scant notice in works of this nature. 
 I have endeavoured, as far. as .the limits of space in this 
 small volume permitted, to make good these deficiencies, 
 which were for the most part brought under my notice by 
 engineering students. 
 
 There are four important points in which I venture to 
 hope this book will be found to contain information, put 
 in a form suitable for beginners, which has hitherto only 
 been accessible in works of a more advanced character or 
 in those which only profess to treat special branches of the 
 subject. 
 
 They are as follows : 
 
 1. The modern science of thermodynamics, which is the 
 foundation of all knowledge of the steam engine considered 
 as an apparatus for converting heat into mechanical work. 
 
 2. The very important effects exercised on the motion 
 of quick running engines by the inertia of their reciprocat- 
 ing parts. 40848 
 
vi Preface. 
 
 3. The geometrical methods of fixing the dimensions 
 and the setting of slide valves. 
 
 4. The investigation of the methods in use for diminish- 
 ing the losses of efficiency in expansive engines, due to the 
 cooling of the cylinders by the expanding steam, the prin- 
 cipal of which methods are, superheating, steam jacketing, 
 and compounding. 
 
 The space required for even an elementary treatment 
 of the above subjects could not be gained without a certain 
 sacrifice, and after full consideration I came to the con- 
 clusion to sacrifice altogether the historical part of the 
 subject, partly because there are already in existence many 
 elementary works full of historical information, and partly 
 because I doubted whether a history of the steam engine 
 has any legitimate place in a text book for students. I 
 have endeavoured throughout this work to make the de- 
 scriptions as simple as possible, and their sequence as 
 systematic as the nature of the work allowed. I believe 
 that fully one half of the difficulties experienced by students 
 in mastering new subjects is due to the want of system which 
 characterises too much of our older technical literature. It 
 is the rule rather than the exception in many books to 
 present to the student ready-made formulae without indicat- 
 ing the steps by which they are reached. I have carefully 
 avoided this source of difficulty to beginners, for I conceive 
 it to be the duty of all who attempt to teach even the most 
 elementary subjects to husband the powers of their readers 
 by saving them all unnecessary trouble. 
 
 I cannot claim anything original in the 'cook, but I do 
 claim that I have endeavoured to render the information 
 which it contains very easy to understand, so that it can be 
 
Preface. vii 
 
 followed from first to last by any student who possesses a 
 slight acquaintance with elementary mathematics. Wherever 
 it has been advantageous to do so I have used geometrical 
 instead of analytical methods of demonstration. I have not 
 assumed the slightest acquaintance on the part of the reader 
 with the sciences of heat or of motion, and have consequently 
 devoted many pages to the explanation of such parts of 
 these sciences as are necessary for the proper understanding 
 of the working of engines. In this I have followed the 
 precedent set in many excellent works included in this 
 series. 
 
 If I were to acknowledge in detail all the sources of 
 information from which I have freely drawn, I fear this Pre- 
 face would run to an inordinate length ; but I cannot forbear 
 to express my deep obligations to my old friend and private 
 tutor at Cambridge, Professor James Stuart, M.P., who has 
 kindly revised the proofs of the entire work, and to the 
 Editors and Proprietors of * Engineering ' and the ' Engineer,' 
 who have allowed me free use of many of the illustrations 
 and of the inexhaustible stores of information which have 
 appeared in their journals. Students of thermodynamics 
 would be in a bad way without the writings of the late 
 Professor Rankine, F.R.S., and of Professor Cotteril, F.R.S., 
 and I have availed myself freely of the information contained 
 in their invaluable books. I have also found much that 
 was valuable in the published papers of the Institutions of 
 Civil Engineers, Mechanical Engineers and Naval Archi- 
 tects, and am greatly indebted to the Councils of these 
 Societies for permission to make use of many drawings 
 which are reproduced in these pages. Among the other 
 authors whom I have consulted, I may mention Mr. Arthur 
 
viii Preface. 
 
 Rigg, whose very ingenious system of circular diagrams 
 of twisting moments on crank-shafts I have adopted in 
 Chapter V. ; Professor Zeuner, whose invaluable system of 
 valve diagrams is explained in Chapter VII. ; Mr. Cowling 
 Welch, Mr. Porter, Professors Galbraith and Haughton, Clerk 
 Maxwell and Cawthorn Unwin, Mr. A. E. Seaton, and lastly 
 Mr. R. Sennett, to whom I am indebted for several illustra- 
 tions and for much valuable information on the subject of 
 the distribution of the steam in Compound Engines. 
 
 GEORGE C. V. HOLMES. 
 
 5 ADELPHI TERRACE: 
 December, 1886. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 PAGE 
 
 The elementary conception of a steam engine The essential ele- 
 ments of steam engines Description of a simple form of 
 modern steam engine and boiler Distribution of steam by an 
 ordinary slide valve -The use of the fly wheel Various pur- 
 poses for which steam engines are employed Importance of 
 the accurate study of the engine in all its bearings The natu- 
 ral subdivisions of the subject ...... I 
 
 CHAPTER II. 
 
 NATURE OF HEAT. THE MODE OF MEASURING IT. ITS 
 EFFECTS ON GASES AND WATER. 
 
 General ideas of nature of heat Old notions regarding it Mate- 
 rial theory and its refutation by Davy Modern theory that 
 heat is a form of energy Definitions and examples of energy 
 and work Example of conversion of heat into work 
 Measurement of heat Temperature Thermometers, their 
 graduation and defects Quantity of heat Specific heat 
 British thermal unit Capacity of substances for heat Rela- 
 tion between heat and work -The mechanical equivalent of 
 heat Joule's experiments Effect of application of heat to 
 gases Nature of gas -Boyle's law connecting the pressure 
 and volume of gas Graphic representation of Boyle's law 
 Definition of an Isothermal Charles's law connecting the 
 volume and temperature of gas Dalton's law connecting the 
 volume and temperature of gas The air thermometer Abso- 
 
Contents. 
 
 PAGE 
 
 lute temperature Combination of Boyle's and Charles's laws 
 The specific heat of gases Difference in the specific heats 
 according as the gas is heated at constant volume or at con- 
 stant pressure External and internal work done when a gas 
 is heated at constant pressure Effect of application of heat to 
 water and ice Heat absorbed in liquefying ice Heat ab- 
 sorbed in evaporating water at various pressures External 
 and internal work of evaporation Law connecting the pres- 
 sure and temperature of steam Total heat of steam analysed 
 ^-Specific volume and relative volumes of steam Law con- 
 necting the pressure, volume, and density of steam Graphic 
 representation of the expenditure of heat in evaporating water 
 Expansion of gas and steam Isothermal expansion of gas 
 Isothermal expansion of steam Adiabatic expansion of gas 
 Adiabatic expansion of steam ...... 20 
 
 CHAPTER III. 
 
 THEORETICALLY PERFECT HEAT ENGINES. 
 
 Application of Boyle's and Charles's laws to gases Specific heat 
 of gases at constant pressure and at constant volume Cycle 
 of operations Ratio of heat expended to work done Graphic 
 representation of external work done during the expansion of 
 a gas Nature of the curves of expansion of gases as influenced 
 by the supply of heat Heat supplied : (i) when the curve of 
 expansion is a rectangular hyperbola, the equation for which 
 \<$, pv-c ; (2) when the equation of the curve takes the form 
 pv n = c, where n has any value except unity Nature of the 
 curve when no heat is supplied or abstracted The ideally 
 perfect heat engine Calculation of the efficiency of such 
 engines The reversed action of the ideal heat engine Proof 
 that no other engine has a greater efficiency than the ideal 
 heat engine Practical limits of efficiency in the ideal heat 
 engine Laws connecting the pressure, temperature, and 
 volume of dry saturated steam Specific heats of M ater and 
 steam Law connecting the pressure, volume, and tempera- 
 ture of superheated steam Total heat expended in converting 
 water into steam Proportion of total heat expended in doing 
 external and internal work Expenditure of heat in a steam 
 engine when the steam is not used expansively Method of 
 
Contents. xi 
 
 PAGE 
 
 representing heat by an 'equivalent pressure' Expenditure 
 of heat in a steam engine when the steam is used expansively, 
 1st, when the curve of expansion is a rectangular hyperbola; 
 2nd, when the steam remains dry and saturated throughout 
 whole stroke To realise latter condition steam jacket is 
 necessary Rankine's formulae for the expenditure of heat in 
 a steam engine Theory of the ideally perfect heat engine 
 applied to steam How actual steam engines differ from the 
 ideal heat engine Summary of laws and formulae . . 73 
 
 CHAPTER IV. 
 
 CONNECTION BETWEEN THE SIZE OF AN ENGINE, THE EVA- 
 PORATIVE POWER OF THE BOILER, AND THE EXTERNAL 
 WORK WHICH IT CAN DO. 
 
 Navier's modification of Boyle's law applied to steam Work 
 done during expansion of steam calculated by Navier's formula 
 - - De Pambour's theory of the double-acting steam engine 
 Its two fundamental principles and their mathematical ex- 
 pression-Analysis of the resistance to the motion of the 
 piston of a steam engine Back pressure Engine friction 
 Load Horse power Examples of the connection between 
 the size of cylinder, the piston-speed, the rate of expansion, 
 the evaporation in the boiler, and the power developed by 
 the engine Locomotive engines Analysis of the resistance 
 to be overcome by the pistons of locomotives --Back pressure 
 Engine friction Resistance to uniform motion of engine, 
 tender, and train Resistance due to gradient Resistance 
 due to inertia of weights to be moved Atmospheric resist- 
 ance Application of De Pambour's theory to locomotives . 126 
 
 CHAPTER V. 
 
 THE MECHANICS OF THE STEAM ENGINE. 
 
 Elementary principles of dynamics Definitions of mass, weight, 
 velocity, motion, force Units employed in their measure- 
 ment The laws of motion and examples of their application 
 Work and energyMotions of bodies in circles Applica- 
 tion to fly-wheels Centrifugal force Conversion of work 
 
xii Contents. 
 
 PAGE 
 
 done in the cylinder into work done on the crank ist case, 
 when pressure of steam is uniform, connecting rod supposed 
 to be of infinite length and moving parts without weight 
 2nd case, when steam is allowed to expand, the other con- 
 ditions remaining unchanged Curve of effort on crank pin 
 3rd case, when the length of the connecting rod is taken into 
 account 4th case, when the weights and velocities of the 
 reciprocating parts are taken into account Power absorbed in 
 accelerating these parts Power restored by their retardation 
 The consequent modification of the indicator diagrams neces- 
 sary for calculating effort of crank pin Effect of steam distri- 
 bution on the action of the moving parts Means of equalising 
 the tangential effort of crank-pin Fly-wheels Theory of 
 their action Graphic diagrams illustrating their action . . 145 
 
 CHAPTER VI. 
 
 THE MECHANISM AND DETAILS OF STEAM ENGINES. 
 
 Cylinders with their fittings -Lubricators Pistons Piston pack- 
 ings Piston rods Cross heads and slide bars Connecting 
 rods Cranks and eccentrics Eccentric rods Shaft and 
 plummer blocks Axle boxes Governors Fly wheels . . 201 
 
 CHAPTER VII. 
 
 VALVES AND VALVE GEARS. 
 
 Action of the simplest form of D slide valve driven by single 
 eccentric Definitions of ' lap ' and ' lead ' .-- Position of 
 eccentric as affected by the lap and lead of the valve Effect 
 on the steam distribution of the lap and lead of the valve - 
 Effect of ratio of length of connecting rod to length of crank 
 in modifying steam distribution Means of varying the rate 
 of expansion and of reversing Stephenson's link motion - 
 Effect of diminishing the throw of the eccentric Reversing 
 lever Ramsbottom's reversing screw Variations in the 
 details of Stephenson's link motion Other systems of link 
 motion Other means of varying the rate of expansion 
 Meyer's separate expansion valve Corliss's valve gear 
 Varieties of valves -Valve gears in which eccentrics are dis- 
 
Contents xiii 
 
 PAGE 
 
 pensed with Joy's gear Geometrical representations of the 
 action of slide valves Zeuner's valve diagrams Case of 
 valve without lap or lead Case of valve with lap and lead 
 Problems on simple valve setting Zeuner's diagrams applied 
 to valves driven by link motions Analytical method of fixing 
 centres of valve circles Graphical method Problems in link 
 motion The method of suspending link motions Zeuner's 
 diagrams applied to Meyer's valve gear Reversing by Meyer's 
 gear Problems on valve setting with Meyer's gear . . 249 
 
 CHAPTER VIII. 
 
 INDICATORS AND INDICATOR DIAGRAMS. 
 
 Uses of indicator diagrams Richards' indicator General cha- 
 racter of indicator diagrams The lines of admission, expan- 
 sion, release, exhaust, and compression How to measure 
 the power exerted during a stroke of the piston How to 
 ascertain from diagram the horse power exerted by the 
 engine How peculiarities and defects are revealed by the 
 diagram Loss of pressure during admission Slow cut-off of 
 steam Effects of clearance on expansion curve Effects of 
 condensation and re-evaporation in cylinder on expansion curve 
 Usual form of the expansion curve Late and early release 
 Effect of wet steam on the exhaust line Causes affecting the 
 back pressure Cushioning or compression of exhaust steam 
 Principal causes affecting forms of diagrams Examples of 
 diagrams from defective engines How to draw the hyperbolic 
 curve of expansion Initial condensation and re-evaporation 
 shown by diagram Gross and net indicated power Cause 
 which limits the economical rate of expansion How to deduce 
 from indicator diagrams the effective pressure on piston How 
 to ascertain the expenditure of heat and steam accounted for 
 by the diagram . . . . . . . . -3 1 ? 
 
 CHAPTER IX. 
 
 FUEL COMBUSTION THE GENERATION OF STEAM 
 BOILERS AND THEIR FITTINGS. 
 
 Combustion Combination of oxygen with carbon Combination 
 of oxygen with compounds of carbon and hydrogen 
 
xiv Contents. 
 
 PACK 
 
 Chemical symbols and atomic weights of constituents of fuel 
 --Principal compounds of carbon, hydrogen, and oxygen 
 Constituents of fuel Heat of combustion of carbon and 
 hydrogen with oxygen Description of fuels in common use 
 Table of the chemical constituents and evaporative power of 
 various fuels Weight and temperature of the products of 
 combustion Waste of fuel by splintering, distillation, in- 
 sufficient air supply, smoke forming, draught creation, radia- 
 tion, and conduction Conduction of heat through the plates 
 of furnaces Importance of preventing an over-supply of air to 
 fuel The separate parts of boilers Cylindrical boiler with 
 external firing Cornish boiler Lancashire boiler Galloway 
 tubes The stiffening of internal flues Tubulous boilers 
 Locomotive boilers Marine boilers for low-pressure steam 
 Marine boilers for high-pressure steam Proportions of parts of 
 boilers Firegrate area Evaporative power of fuel in various 
 types of boilers Consumption of fuel per square foot of fire- 
 grate area Efficiency of heating surface Cubic capacities of 
 boilers of different types Steam room Strength of boilers 
 Hollow cylinder pressed from within : 1st, longitudinal 
 strain ; 2nd, transverse strain Strength of riveted joints 
 Hollow cylinder pressed from without Flat stayed surfaces 
 Effects of unequal heating in straining boilers Materials of 
 construction Boiler fittings Safety valves Pressure gauges 
 Feed pumps Injectors Water gauges Draught creation 
 by chimnies Forced draught Closed stoke-holds Closed 
 ashpits .......... 346 
 
 CHAPTER X. 
 
 CONDENSATION AND CONDENSERS. 
 
 The object and advantages of condensing steam General descrip- 
 tion of condensers Quantity of water required to condense 
 steam Object of surface condensation for marine engines 
 Description of a jet condenser for a stationary engine De- 
 scription of a marine surface condenser Air pumps Method 
 of indicating the vacuum Ejector condensers . . . 4 2 3 
 
Contents. xv 
 
 CHAPTER XI. 
 
 ON SOME OF THE PRINCIPAL CAUSES OF LOSS OF EFFICIENCY 
 IN STEAM ENGINES, AND THE METHODS EMPLOYED FOR 
 REDUCING THE LOSS SUPERHEATING STEAM JACKETING 
 COMPOUNDING. 
 
 PAGE 
 
 Early improvement in the steam engine consisted in the separa- 
 tion of the functions of steam generating, steam using, and 
 condensing The cylinder, even in modern engines, still acts 
 as a generator and condenser Experimental confirmation of 
 the foregoing Cause of condensation and re- evaporation in 
 engine cylinders Hypothetical example showing the succes- 
 sive stages of condensation and re-evaporation Injurious 
 effects of presence of water in cylinders Four principal 
 causes of the presence of water in cylinders : I, priming ; 2, 
 excess of condensation over re-evaporation ; 3, liquefaction 
 due to work done ; 4, loss of heat by radiation Influence of 
 dimensions of cylinder, pressure of steam, and rate of expan- 
 sion on the initial condensat.on Experiments on steam con- 
 sumption in conducting and non-conducting cylinders Means 
 employed to diminish the loss due to liquefaction: I, super- 
 heating the steam ; 2, steam jacketing The benefits derived 
 from the use of steam jackets Cases in which jackets are 
 useless Precautions to be observed in jacketing Experi- 
 ments showing the use of jackets in simple and compound 
 engines 3, cushioning, or compressing the exhaust steam ; 
 4, compounding Variations in temperature of cylinders due 
 to working high-pressure steam expansively Compounding 
 reduces the variation of temperature in each cylinder Tan- 
 dem compound engines Two-cylinder receiver compounds 
 Three-cylinder ordinary compounds Triple compound, or 
 triple expansive engines Experimental demonstration of the 
 saving in fuel to be effected by compounding The distribu- 
 tion of the steam in the various types of compound engines 
 Actual indicator diagrams of the various types of compound 
 engines The manner of reducing the diagrams of compound 
 engines The mechanical advantages of compound as com- 
 pared with simple expansive engines Example of the curve of 
 twisting moments on the crank of a triple compound engine 
 The relative sizes of the cylinders of compound engines 
 
xvi Contents. 
 
 PAGE 
 
 Means of equalising the power developed in the separate 
 cylinders of compound engines Table of cylinder ratios for 
 various types of compound engines working with different 
 pressures of steam ........ 438 
 
 APPENDIX : 
 
 TABLE 1 489 
 
 TABLE II 49$ 
 
 EXAMPLES 55 
 
 INDEX . 523 
 
THE 
 
 STEAM ENGINE. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 The elementary conception of a steam engine The essential elements of 
 steam engines Description of a simple form of modern steam engine 
 and boiler Distribution of steam by an ordinary slide valve The 
 use of the fly wheel Various purposes for which steam engines are 
 employed Importance of the accurate study of the engine in all its 
 bearings The natural subdivisions of the subject. 
 
 THE complete study of the steam engine is, in its nature, 
 somewhat complex, involving as it does an acquaintance 
 with the sciences of heat, of chemistry, and of pure and 
 applied mechanics, as well as a knowledge of the theory of 
 mechanism and the strength of materials. It is proposed, 
 therefore, to begin this work by showing, in a very simple 
 case, how steam can be used to do work, and then to pro- 
 ceed to describe an actual steam engine of the most modern 
 construction, but one which at the same time is remarkably 
 free from complexity. When studying this description, the 
 student will soon find out how it is that the perfect know- 
 ledge of the steam engine involves an acquaintance with so 
 many branches of science ; and the order in which these 
 subjects must be studied, so far as they bear on the matter 
 in hand, will naturally be suggested by the description. 
 
 B 
 
The Steam Engine. 
 
 Take a hollow cylinder (fig. i) of indefinite height, the 
 bottom of which is closed while the top remains open, and 
 fill this cylinder to the height of a few inches with water. 
 Next cover in the water by means of a flat 
 plate, or piston, which fits perfectly the in- 
 terior of the cylinder, and then apply heat 
 to the water ; we shall witness the following 
 phenomena. After the lapse of some minutes 
 the water will begin to boil, and steam will 
 accumulate at its upper surface between it 
 and the piston, which latter will be raised 
 slightly in order to make room for the steam. 
 As the boiling process continues, more and 
 more steam will be formed, and the piston 
 will be raised higher and higher, till the 
 whole of the water is boiled away, and 
 nothing but steam is contained in the 
 cylinder. Now this apparatus, consisting of 
 cylinder, piston, water, and fire, is an elementary form of 
 steam engine of the simplest kind. For a steam engine 
 may be defined as an apparatus for doing work by means of 
 heat applied to water ; and it is manifest that the appliance 
 just described, inconvenient and clumsy though it may be, 
 perfectly answers to the definition ; for the piston is a 
 weight, and this weight has been raised to a certain height by 
 the formation of steam from the water. Now the raising 
 of a weight through a height is a particular form of doing 
 work, and consequently this combination is an apparatus 
 capable of doing work by means of heat applied to water. 
 
 If, instead of a simple piston, we had taken one loaded 
 with weights, and applied heat as before, the result would 
 have been similar but not precisely the same. The water 
 would not have begun to boil so soon, and when it was all 
 boiled away the loaded piston would not have risen to the 
 same height as did the simple one. The reason of this will 
 be amply explained in the chapters on heat. Supposing 
 
Introduction. 3 
 
 that, having raised the weight to the utmost height it would 
 go, we then removed it from the piston, and wished to 
 employ the apparatus in order to raise a similar weight to 
 the same height, we should have to bring back the steam to 
 its original condition of water. This we could do by remov- 
 ing the fire and by surrounding the cylinder instead with cold 
 water. The result would be that the steam would all con- 
 dense into water, and fall back to its original place, the 
 piston following it, and everything would be ready for a 
 fresh start. Now, though this apparatus answers the defini- 
 tion of a steam engine, it is, nevertheless, a very bad one, 
 for the following reasons. The only kind of work it can do 
 is the raising of weights through certain heights. When we 
 want to repeat the operation we have to remove the fire and 
 surround the cylinder with cold water, and then replace the 
 fire, which is a most cumbrous process. While condensing 
 the steam we made the cylinder cold, and a large quantity 
 of heat is wasted in warming it again. Moreover, when, at 
 the cost of a considerable amount of fuel, we have heated 
 the water and turned it into steam, we allow the whole of 
 the heat in the steam to escape into the cold water, and thus 
 become wasted, though it is capable of doing much more 
 work if properly used. Thus we see that our elementary 
 engine is limited in its scope, clumsy in use, and extremely 
 wasteful of fuel. It is in obviating these disadvantages that 
 actual engines differ from the one we have described. 
 
 It will have been observed that this engine consists of 
 four principal elements, viz. : the fire, or source of heat ; the 
 water, or medium to which the heat is applied, and by the 
 conversion of which into steam the work is done; the cylin- 
 der with movable piston, which contains the water and 
 steam, and which prevents the latter from escaping into the 
 air when formed and becoming lost ; and, lastly, the source 
 of cold, or the water by means of which the steam was con- 
 densed and brought back to its original condition. The 
 great majority of actual engines consist of precisely the same 
 
 B 2 
 
4 The Steam Engine. 
 
 elements, more advantageously arranged, with the addition 
 of certain mechanism for changing the straight line move- 
 ment of the piston into circular, or any other kind of motion. 
 This mechanism has also to effect other subsidiary objects 
 which will be fully described hereafter. It should also here 
 be mentioned that if, instead of condensing the steam by 
 means of cold water, we had opened a temporary communi- 
 cation between the steam space inside the cylinder and the 
 open air, we should have equally well succeeded in bring- 
 ing the piston back to its original position, when, by intro- 
 ducing into the cylinder a fresh quantity of water, we could 
 have again raised the weights. 
 
 In practice the arrangement adopted is as follows : 
 
 1. The source of heat, and the vessel containing the 
 water to be boiled, are kept quite separate and distinct 
 from the cylinder. These parts of the apparatus are called 
 respectively the furnace and boiler. The steam is supplied 
 from the boiler, where it is generated, to the cylinder where 
 it is used, as it is wanted, by means of a pipe, called the 
 steam pipe. 
 
 2. The steam, after doing its work in the cylinder, is led 
 away through a second pipe, called the exhaust pipe, into 
 the open air, or else to be condensed in a separate vessel 
 kept quite apart from the cylinder, and which is called the 
 condenser. 
 
 3. The cylinder, instead of being open at one end, and 
 of indefinite length, is closed at both ends, and in length 
 seldom exceeds twice the diameter of the piston. 
 
 4. The steam, instead of being used only on one side of 
 the piston, is admitted alternately to and exhausted from 
 each side in succession, so that when the engine is in use, 
 the piston is constantly travelling backwards and forwards 
 from one end to the other of the cylinder. 
 
 5. Suitable openings are made at each end of the 
 cylinder, to allow the steam alternately to enter and escape, 
 and valves driven by suitable mechanism are provided in 
 
Introduction. 5 
 
 order to ensure that the admission and escape of the steam 
 shall take place at the proper moments. 
 
 6. Instead of placing the weights to be lifted directly 
 upon the piston, a cylindrical bar or rod called the piston 
 rod is attached firmly to the centre of the piston, and is 
 continued through one end of the cylinder to the open air, 
 so that the outside end of the rod moves backwards and 
 forwards in a straight line, exactly as the piston does. By 
 means of suitable mechanism, which will be fully described 
 hereafter, this straight line motion of the piston rod end is 
 changed into rotatory or circular motion, so that the engine 
 can be used, not only for lifting weights up in a vertical 
 direction, but for doing any kind of work which may be 
 required of it. 
 
 The manner in which all this may be accomplished in 
 practice will be shown in the following description and 
 drawings of an engine and boiler, which are here selected 
 for description on account of their simplicity of construction. 
 We will commence with the source of heat, and apparatus 
 for turning the water into steam ; then go on to the engine 
 proper, i.e. the cylinder with the mechanism belonging to it. 
 The abstracter of heat, or condenser, will be considered in 
 a separate chapter. Fig. 2 is an elevation of the boiler, fig. 3 
 a vertical section through its axis, and fig. 4 a horizontal 
 section through the furnace bars. 
 
 The type of steam generator here exhibited is what is 
 known as a vertical tubular boiler. The outside casing or 
 shell is cylindrical in shape, and is composed of wrought 
 iron or steel plates riveted together as shown in fig. 2. 
 The top, which is likewise composed of the same material, 
 is slightly dome-shaped, except at the centre, which is cut 
 away in order to receive the chimney, a, which is cylindrical 
 in shape and formed of thin wrought-iron plates. The 
 interior is shown in vertical section in fig. 3. It consists of 
 a furnace chamber, , which contains the fire. The furnace 
 is formed like the shell of the boiler of wrought iron or steel 
 
The Steam Engine. 
 
 plates in the form of a cylinder, the top of which is covered 
 by a flat circular plate, a, firmly attached to the cylindrical 
 
 Fig. 2. 
 
 Fig. 3- 
 
 portion by flanging and riveting. The bottom is occupied 
 by the grating, on which rests the incandescent fuel. The 
 grating consists of a number of cast-iron bars, d (fig. 3), and 
 shown in plan in fig. 4, placed so as to 
 have interstices between them like the 
 grate of an ordinary fireplace. The 
 bottom of the furnace is firmly secured 
 to the outside shell of the boiler in the 
 manner shown in fig. 3. The top cover- 
 ing plate, cc, is perforated with a number 
 of circular holes of from one and a half 
 to three inches diameter, according to the size of the boiler. 
 Into each of these holes is fixed a vertical tube made of 
 
 Fig. 4. 
 
C 
 Introduction. 
 
 brass, wrought iron, or steel, shown at fff (fig. 3). These 
 tubes pass through similar holes, at their top ends in the 
 plate gg, which latter is firmly riveted to the outside shell 
 of the boiler. The tubes are also firmly attached to 
 the two plates, cc, gg. They serve to convey the flame, 
 smoke, and hot air from the fire to the smoke box, /*, and 
 the chimney, a, and at the same time their sides provide 
 ample heating surface to allow the heat contained in the 
 products of combustion to escape into the water. The 
 fresh fuel is thrown on to the grating when required 
 through the fire door, A (fig. 2). The ashes, cinders, &c., 
 fall between the fire bars into the ash pit, B (fig. 3). The 
 water is contained in the space between the shell of the 
 boiler, the furnace chamber, and the tubes. It is kept at 
 or about the level, ww (fig. 3), the space above this part being 
 reserved for the steam as it rises. The heat, of course, 
 escapes into the water, through the sides and top plate of 
 the furnace, and through the sides of the tubes. The steam 
 which, as it rises from the boiling water, ascends into the 
 space above ww t is thence led away by the steam pipe to the 
 engine. Unless consumed quickly enough by the engine, 
 the steam would accumulate too much within the boiler, and 
 its pressure would rise to a dangerous point. To provide 
 against this contingency, the steam is enabled to escape 
 when it rises above a certain pressure through the safety 
 valve, which is shown in sketch on the top of the boiler in 
 fig. 2. The details of the construction of safety valves will 
 be found fully described in Chapter IX., which is devoted 
 exclusively to the consideration of boilers and their fittings. 
 In the same chapter will be found full descriptions of the 
 various fittings and accessories of boilers, which it would be 
 out of place here to describe in detail, such as the water 
 and pressure gauges, the apparatus for feeding the boiler 
 with water, for producing the requisite draught of air to 
 maintain the combustion, and also the particulars of the 
 construction of the boilers themselves and their furnaces, 
 
The Steam Engine. 
 
Introduction. 9 
 
 and the principles on which their strength is determined, 
 and their various parts proportioned, so as to fully realise 
 the effects intended. 
 
 We now come to the description of the engine, and the 
 type selected for illustration is that usually called horizontal 
 single cylinder, direct acting. 
 
 Fig. 5 is an elevation of the exterior. Fig. 6 is a hori 
 zontal section of the cylinder, piston, and valve box. Fig. 7 
 is a plan. The cylinder is shown at A, figs. 5, 6, 7 ; its 
 construction is best seen from the section, fig. 6. It is 
 formed of cast iron, the ends being flanged to allow of the 
 cylinder cover or end plate, aa^ and the frame, PP, being 
 
 Fig. 6. 
 
 bolted to it. The piston is shown at B ; it is a circular cast- 
 iron disc, made to fit the cylinder in a steam-tight manner. 
 Into the piston is fixed the piston rod, C, which passes 
 through the front cylinder cover, the place where it passes 
 through being made steam-tight by the stuffing box, D. The 
 front end of the piston rod is fastened to the crosshead, E 
 (fig. 5), which is a joint used for connecting the piston rod to 
 the connecting rod, F, in such a manner as to allow the 
 latter to swing in a vertical plane as the piston travels back- 
 wards and forwards. The crosshead is also provided with 
 two slides, ^(fig. 5), which move between the guide bars, ff 
 (figs. 5 and 6), and which prevent the piston rod from being 
 bent, and from moving otherwise than in a straight line. 
 The connecting rod, F (figs. 5 and 7), joins the end of the 
 piston rod to the crank pin, G. The crank axle in which 
 
10 
 
 The Steam Engine, 
 
Introduction. 1 1 
 
 the crank is formed is shown in section at H (fig. 5), but is 
 seen more clearly in the plan, fig. 7, where it is shown pass- 
 ing through the two bearings, LL. The distance between 
 the centre of the crank pin, G, and the centre of the crank 
 axle, H (fig. 5), is called the length of the crank arm, and is 
 exactly equal to half the distance which the piston moves 
 from one end to the other of the cylinder. 
 
 Supposing now that steam were allowed to flow from the 
 boiler into the cylinder in such a manner as to obtain ad- 
 mission behind the piston, B ; this latter would commence 
 to travel towards the front cover of the cylinder, and in 
 doing so would push forward the piston rod and the cross- 
 head. The end of the connecting rod next the crosshead 
 would also be pushed forward, but the other end of the con- 
 necting rod which encircles the crank pin, not being free to 
 move simply forward, would describe an arc of a circle round 
 the centre of the crank axle, H, and in so doing the direction 
 of the rod would become inclined so as to form an angle 
 with the axis of the cylinder. By the time the piston has 
 travelled to the front end of the cylinder, the crank pin 
 will have been turned round into the position G' (fig. 5), 
 diametrically opposite to its initial position. Suppose that, 
 just before this takes place, the steam is shut off from the 
 back of the piston, and the steam already in the cylinder is 
 allowed to escape, while at the same time fresh steam from 
 the boiler is allowed to enter the cylinder at the front side 
 of the piston, this latter will commence to travel back to its 
 original position, 1 and in doing so will cause the crank 
 pin to revolve from the position G' (fig. 5), through a semi- 
 circle, till it reaches its original position, it having thus 
 described a complete revolution round the centre of the 
 crank axle, while the piston was making a double stroke 
 backwards and forwards. This operation may be repeated 
 as often as we like provided we have a suitable apparatus 
 
 1 For the sake of simplifying the description, no account is here 
 taken of the action at the dead centres. See p. 14. 
 
12 The Steam Engine. 
 
 for admitting the steam alternately on each side of the piston, 
 and then allowing it to escape either into the open air or 
 a condenser. 
 
 The manner in which the steam admission is regulated 
 is as follows. By referring to the section (fig. 6), it will be 
 seen that a box-like casing, MM, is cast in one piece with 
 the cylinder and on one side of it. This box contains the 
 valve, V, which controls the flow of the steam. It will be 
 noticed that the side of the cylinder next the valve box 
 contains two passages, ss ; these are called the steam ports 
 because the steam by means of them gains access to and 
 escapes from either end of the cylinder. For the sake of 
 clearness the following diagram, fig. 8, is given, showing the 
 
 valve and side of a cylinder to a larger scale. The cast-iron 
 box containing the valve is always filled, when the engine 
 is at work, with steam from the boiler. If the valve occu- 
 pies the position shown in fig. 8, the steam cannot enter the 
 cylinder at all, because both ports are covered up by the 
 valve. If the latter, however, be moved a little to the right 
 so as to uncover the steam port s, two things will happen. 
 The steam will be enabled to pass through the port s into 
 the cylinder, and push the piston forward from left to right, 
 while at the same time the port s' will be uncovered by the 
 inner edge of the valve, and any steam which may be con- 
 tained in the cylinder on the right-hand side of the piston 
 
Introduction. ' 13 
 
 will be enabled to escape through the port s' into the interior 
 hollow of the valve, and thence into the exhaust passage e, 
 whence it can escape to the air of the condenser. This 
 condition of things is shown by fig. 9. 
 
 Fig. 9. 
 
 If when the piston has reached the end of its forward 
 stroke the valve be moved backwards into the corresponding 
 position on the other side, the steam port s f will then be 
 uncovered and will allow the boiler steam to enter the cylin- 
 der, and force the piston back from right to left, while the 
 steam on the left-hand side of the piston will be enabled to 
 escape into the exhaust passage. 
 
 The foregoing remarks must be looked upon as merely 
 an elementary sketch of the working of this particular sort 
 of valve (which is commonly called the D slide valve). The 
 proper way of proportioning the parts of the valve, the widths 
 of the steam ports, and the methods of driving the valve so 
 as to admit and cut off the fresh steam and release the ex- 
 haust steam precisely at the right moments during the stroke 
 of the piston, are points of the greatest nicety and require the 
 most careful study, and are fully described in Chapter VII. ; 
 but enough has been now said to illustrate the method of 
 working in a general way without going into complexities. 
 
 It will be noticed that the valve is connected by a rod 
 (see fig. 7) with a cam, C, fixed to the crank axle of the engine. 
 This cam, which is called an eccentric, drives the valve 
 
14 The Steam Engine. 
 
 backwards and forwards ; its manner of working will be 
 found described in the chapter already referred to. 
 
 When the centre of the crank-pin occupies either the 
 point G', fig. 5, or the diametrically opposite position, the 
 centre line of the crank is in the prolongation of the axis of 
 the cylinder and connecting rod, and it is evident that when 
 in either of these positions, which are called the dead centres, 
 the steam would only tend to press the crank axle against 
 its bearings, LL, fig. 7, and would exercise no rotating effect 
 whatever. Consequently unless some means can be devised 
 for getting the crank over the dead centres the engine will 
 stick fast. 
 
 The plan invariably adopted with a single cylinder engine 
 is to provide a heavy fly-wheel, shown in elevation in fig. 5, 
 and in plan in fig. 7. The momentum acquired by this fly- 
 wheel during the stroke carries the crank over the dead centre. 
 In addition to the above the fly-wheel exercises other useful 
 functions which are explained in Chapter V., but which need 
 not be dwelt upon at present. 
 
 The engine which has been described above is mounted 
 on the heavy combined bed plate and frame PPP, shown in 
 elevation fig. 5, and in plan fig. 7. The bed plate is bolted 
 down to a solid mass of masonry as shown in fig. 5. 
 
 For our present purposes it is not necessary to examine 
 into the other details of the mechanism, such as the governor 
 and feed pump shown on fig. 7. 
 
 The engine which we have just described belongs to a 
 type which is very much employed to drive machinery on 
 land. It must not be supposed, however, that the steam 
 engine, as originally invented, was anything like so simple 
 a machine. On the contrary, it has taken two centuries 
 of time to attain its present degree of perfection. We have 
 no intention of entering into the history of the steam 
 engine ; indeed, the limits of this volume would pre- 
 clude any such idea ; moreover, the historical part of the 
 subject has been dealt with over and over again in special 
 
Introduction. 1 5 
 
 books, and in the biographies of the great engineers. At 
 present we are only concerned with the engine as we actually 
 find it, and with its possible future. The past will only be 
 referred to for the purpose of showing what increase in effi- 
 ciency has been attained in more modern times. 
 
 The importance of the accurate study of the steam 
 engine will not be disputed when it is remembered to what 
 purposes the engine is applied now-a-days, and to what an 
 extent this manufacturing and sea-trading country is depend- 
 ent upon its efficiency. Foremost among these purposes 
 are : 
 
 1. Locomotion on railways. The steam engine is em- 
 ployed in effecting nearly the whole of the internal goods 
 and passenger traffic of the country. At the present moment 
 there are in this country over 18,000 miles of railway opened 
 for traffic, and the various railway companies employ between 
 them many thousands of locomotives. 
 
 2. Marine locomotion. For this, purpose the engine is 
 employed in propelling the numerous steam vessels, which 
 effect the greater part of the ocean-carrying trade of the 
 world. Another important use of marine engines is the 
 propulsion of those ships of war on which we depend for 
 the protection of our coasts and our mercantile navy from 
 foreign enemies. 
 
 3. The driving of machinery in our factories. The im- 
 portance of the engine for this purpose can hardly be over- 
 estimated, when it is remembered that we depend on our 
 factories, and on the export of our manufactures, for the 
 means of maintaining our present population, which is far 
 too large to be supported by the produce of the country. 
 
 4. The winding of coal and other minerals, and the 
 pumping of water out of mines. 
 
 5. The tillage of the soil, and the preparation of its pro- 
 duce for the use of mankind. This is a comparatively novel 
 purpose for which the steam engine is employed, but one 
 which is daily increasing in importance. 
 
1 6 The Steam Engine. 
 
 Each of these purposes requires a different type of engine 
 for itself. In a small volume like this, it would be out of 
 the question to describe every variety of engine at present 
 in use. It will only be possible, at best, to explain the 
 principles on which they should all alike be designed. The 
 great importance of an accurate study of the subject is this : 
 that without this study we cannot make our engines econo- 
 mical in the use of fuel. This economy should be one of 
 the first objects of every constructor of a steam engine ; for 
 even if our supply of fuel at present prices were inexhaustible, 
 nevertheless in many cases economy is of paramount im- 
 portance. Take only the case of steam vessels which have 
 to make long voyages. Up to a comparatively recent period 
 it was not found commercially practicable to run merchant 
 steamers on the longest trade routes, such as to China ; for 
 the mere coal required to develop the power necessary for 
 propulsion would have occupied so much of the carrying 
 capacity of the vessel as to leave insufficient room for a re- 
 munerative cargo. Thanks, however, to the fuel economies 
 introduced during the last twenty years, steamers can now 
 be employed with advantage on the longest voyages. Simi- 
 larly the magnificent passenger steamers which now cross 
 the Atlantic owe their high speed mainly to the modern im- 
 provements which have ^enabled great power to be attained 
 with a comparatively moderate weight of machinery and fuel. 
 
 The ultimate object of all study of the steam engine is 
 this : to enable us to attain the maximum economy in the 
 use of fuel with the greatest efficiency of the machinery. 
 Hence the theoretical portion of the subject naturally divides 
 itself into two principal parts. First, the study of the engine 
 as a heat engine ; that is, as an apparatus for the conversion 
 of the heat supplied to it into mechanical work. Second, 
 the study of the engine as a piece of machinery. 
 
 The study of the heat engine involves a knowledge of the 
 nature of heat, and the laws of its conversion into mechanical 
 work ; hence we shall have the following divisions : 
 
Introduction. 1 7 
 
 A chapter (II.) in which is explained the nature of heat, 
 and the mode of measuring it. This chapter will only deal 
 with the subject so far as it bears directly upon the heat 
 engine, and all reference to other branches of this science 
 will be avoided. 
 
 A chapter (III.) which deals with the conversion of heat 
 into mechanical work, by its application to gas and water. 
 This chapter will give an exact account of the physical pro- 
 perties of these bodies, and an explanation as to how the 
 heat supplied to them under given circumstances is actually 
 spent. It will also contain a description of the theoretically 
 perfect heat engine, and show what proportion of the total 
 heat supplied to it can, under the most favourable circum- 
 stances, be in theory turned into work, and also the con- 
 ditions to be observed in order that this ratio of work done, 
 to heat supplied, may be realised. It will, lastly, show how 
 to apply the principle of the theoretically perfect heat engine 
 to actual steam engines, and will explain why these latter 
 are comparatively wasteful of heat. 
 
 We now come to the consideration of the engine, as a 
 piece of machinery, and the student will require to study 
 in detail, both theoretically and practically, the nature 
 of the mechanical means, or mechanism, by which the 
 pressure of the steam is transformed into work. The study 
 of this part of the subject is contained in the following 
 divisions. 
 
 Chapter IV. shows the connection between the size of 
 the cylinder, the pressure of the steam, the velocity of the 
 piston, the useful work to be done, and the incidental 
 resistances which have to be overcome ; and will show prac- 
 tically how to proportion the size of the cylinder to the work 
 it has to do. 
 
 Chapter V., on the laws of motion, as applied to the 
 separate moving parts of an engine, so that the effects of 
 their weights, velocities, and directions of motion, on the 
 working of the whole may be understood. 
 
 c 
 
1 8 The Steam Engine. 
 
 The practical part of the book contains descriptions of 
 the various organs of which different types of engines and 
 boilers are made up, and the rules for proportioning them 
 to their several purposes. 
 
 Chapter VI. is on the practical details of the mechanism 
 employed. This chapter will contain illustrations of the 
 working parts of various sorts of engines. 
 
 Chapter VII. , on a part of the mechanism viz., the 
 valves and valve gear which, on account of its importance 
 and complexity, requires a separate detailed description. 
 
 Intimately connected with the subject of valve-gearing is 
 the instrument which is used in practice in order to ascertain 
 if the valves are effecting a proper distribution of the steam. 
 This instrument is called the indicator, and it is used not only 
 for the above-mentioned purpose, but also to measure the 
 power which is being exerted by the engine. The indicator 
 records the performance of the engine by inscribing a geo- 
 metrical figure called an indicator diagram on a piece of 
 paper. 
 
 Chapter VIII. is devoted to the consideration of indi- 
 cators and the interpretation of their diagrams, and is illus- 
 trated by numerous diagrams taken from actual engines, 
 each of them being remarkable for some peculiarity. 
 
 Chapter IX. deals with the means of generating steam 
 in practice, and contains an account of the nature of com- 
 bustion, the constituents of fuel, and the various descriptions 
 of furnaces, boilers, and their fittings. 
 
 The subject of the condensation of steam, and the 
 various forms of condensers, air and circulating pumps, are 
 dealt with in Chapter X. 
 
 In the chapters containing descriptions of the mechanism 
 of steam engines, several arrangements, which may be looked 
 upon in the light of refinements, have been omitted. Most 
 of these contrivances have been designed with the object 
 of minimising the losses of efficiency of actual engines, as 
 compared with those which are theoretically perfect. These 
 
Introduction. 19 
 
 sources of loss are enumerated at the end of Chapter III., 
 and a special chapter (XL) is devoted to the various remedies, 
 and contains an examination into the merits of steam jackets, 
 super-heating, and the compounding of engines. 
 
 Students who approach this subject for the first time, or 
 those who wish only to acquire a general knowledge of the 
 construction of engines and boilers, are recommended to 
 omit Chapters III., IV., V., and the latter part of Chap- 
 ter VII. 
 
2O The Steam Engine. 
 
 CHAPTER II. 
 
 Nature of heat The mode of measuring it Its effects on gases and water 
 General ideas of nature of heat Old notions regarding it Material 
 theory and its refutation by Davy Modern theory that heat is a form of 
 energy Definitions and examples of energy and work Example of 
 conversion of heat into work Measurement of heat Temperature 
 Thermometers, their graduation and defects Quantity of heat 
 Specific heat British thermal unit Capacity of substances for heat 
 Relation between heat and work The mechanical equivalent of heat 
 joule's experiments Effect of application of heat to gases Nature of 
 g as Boyle's law connecting the pressure and volume of gas Graphic 
 representation of Boyle's law Definition of an Isothermal Charles's 
 law connecting the volume and temperature of gas Dalton's law con- 
 necting the volume and temperature of gas The air thermometer 
 Absolute temperature Combination of Boyle's and Charles's laws 
 The specific heat of gases Difference in the specific heats according 
 as the gas is heated at constant volume or at constant pressure 
 External and internal work done when a gas is heated at constant 
 pressure Effect of application of heat to water and ice Heat absorbed 
 in liquefying ice Heat absorbed in evaporating water at various pres- 
 sures External and internal work of evaporation Law connecting the 
 pressure and temperature of steam Total heat of steam analysed 
 Specific volume and relative volumes of steam Law connecting the 
 pressure, volume, and density of steam Graphic representation of the 
 expenditure of heat in evaporating water Expansion of gas and steam 
 Isothermal expansion of gas Isothermal expansion of-steam Adiabatic 
 expansion of gas Adiabatic expansion of steam. 
 
 IN this and the following chapter it is not by any means 
 proposed to go into the study of heat, otherwise than as it 
 bears directly upon the heat engine. Consequently no refer- 
 ence will be made to theories and phenomena of heat, 
 other than those which affect gases and water : nor will any 
 attempt be made to describe the numerous experiments 
 which are usually dwelt upon in treatises devoted exclu- 
 sively to this branch of science. On the contrary, these 
 chapters will be found to be mere summaries of certain 
 
Nature of Heat. ITY: 
 
 parts of the subject, inserted here because they are absolutely 
 necessary to the correct knowledge of the heat engine. 
 
 Everyone is familiar with the sensations produced by 
 heat on the human body, as, for instance, when the hand is 
 exposed to the action of a fire, or plunged into boiling 
 water. The agency which produces this sensation is called 
 Heat. The nature of the agency has, ever since the physical 
 sciences were first studied, been the subject of speculation 
 with natural philosophers. In the last and the beginning of 
 the present century, heat was supposed to be a kind of 
 matter which differed from all other forms of matter with 
 which we are acquainted, in that it had no weight. It was, 
 in fact, supposed to be a subtle and imponderable fluid, 
 which was capable of spreading and insinuating itself between 
 all the elementary particles which constitute matter, and of 
 flying from hot bodies to colder ones, no matter at what 
 distance apart these bodies might be. This theory did good 
 service in its time, in helping philosophers to account for 
 many of the effects of heat ; it had, however, ultimately to 
 be discarded, because it failed altogether to account for the 
 fact that heat, in apparently illimitable quantities, could be 
 evolved from cold bodies, by rubbing them together ; that 
 is to say, by the process of friction, cold bodies could be 
 made hot and could be made to communicate heat to any 
 quantity of other cold bodies. This phenomenon was 
 accounted for, by the. believers in the material theory of 
 heat, in the following manner : The bodies to be rubbed 
 together possessed in their state of heat, or thermal condi- 
 tion, before the commencement of the experiment, a certain 
 quantity of the fluid called heat, which caused them to be as 
 hot as, say, for example, the human body. This was expressed 
 by saying that the bodies when as warm as the human body 
 had a certain capacity for heat, i.e. they required a certain 
 quantity of the imponderable fluid to be absorbed between 
 their particles, in order that they might become as warm as 
 aforesaid. Now, when the bodies were rubbed together, 
 
22 The Steam Engine. 
 
 and became eventually hotter than the human body, this 
 was accounted for by saying that their capacity for heat 
 became diminished by the action of friction ; that is to say, 
 they could not, when rubbed, retain the same amount of the 
 imponderable fluid as before, without becoming hotter. If 
 any experiment could be devised which should prove that 
 the capacity of bodies for heat is not diminished by friction, 
 then the material theory of heat would fail to account for 
 the fact that bodies become hotter when rubbed. 
 
 The first absolutely conclusive experiment, which estab- 
 lished the fact that friction makes bodies hot, while it does 
 not diminish their capacities for heat, was made by Davy 
 in 1799. His experiment consisted in rubbing together two 
 pieces of ice till they melted into water, due care having 
 been taken to prevent heat from entering the ice by any 
 other means than friction alone. Now, according to the 
 old theory, the resulting water ought to have a less capacity 
 for heat than the original ice ; but it has been proved over 
 and over again by experiment that the capacity of water for 
 heat is not only not less than, but about double that of ice ; 
 consequently the material theory failed completely to account 
 for the facts, and Davy, after reasoning on his experiments 
 for some years, came to the following conclusion, which we 
 repeat in his own words : 
 
 ' Heat, then, or that power which prevents the actual 
 contact of the corpuscles of bodies, and which is the cause 
 of our own sensations of heat and cold, may be defined as a 
 peculiar motion, probably a vibration of the corpuscles of 
 bodies tending to separate them.' Again, in 1812, Davy 
 thus states his theory : ' The immediate cause of the phe- 
 nomenon of heat, then, is motion, and the laws of its 
 communication are precisely the same as the laws of the 
 communication of motion.' 
 
 Another way of stating the above is that heat is a form 
 of energy. To make this point clear before going further 
 into the nature of heat, we must first define what is under- 
 
Energy and Work. 23 
 
 stood by the term energy and the involved term work, and 
 illustrate the definitions by examples. 
 
 Energy is the power of doing work. 
 
 Work is the overcoming of a resistance through a certain 
 space, and is measured by the amount of the resistance 
 multiplied by the length of space through which it is over- 
 come. 
 
 The simplest possible example of doing work is to raise 
 a weight through a space against the resistance of the earth's 
 attraction, that is to say, against the force of gravity. For 
 instance, if a hundred pounds be raised vertically upwards, 
 through a space of three feet, work is done, and, according 
 to the above, the amount of work done is measured by the 
 resistance due to the attraction of the earth or gravity, i.e. 
 one hundred pounds, multiplied by the space of three feet, 
 through which it is lifted. The product formed by multi- 
 plying a pound by a foot is called a foot-pound. Thus, in 
 the above instance, the amount of work done is 300 foot- 
 pounds. Had the weight been only three pounds, but the 
 height to which it was raised been 100 feet, the quantity of 
 work done would have been precisely the same, i.e. 300 
 foot-pounds. 
 
 In Great Britain, the unit of work is a resistance, equal 
 to the attraction of the earth upon a pound of matter, over- 
 come through a space of one foot j or, in other words, one 
 foot-pound. 
 
 RATE OF DOING WORK. HORSE-POWER. 
 
 The rate of work of any agent means the quantity of 
 work which it performs in a given time, and is measured by 
 the number of foot-pounds done in an hour, or a minute, 
 or a second. 
 
 A quantity of work equivalent to the raising of 33,000 
 pounds through one foot, in one minute, is called a horse-power. 
 This is the unit generally employed to represent the rate 
 
24 The Steam Engine. 
 
 of work of a steam engine, and is adopted to avoid the use 
 of the very high numbers which would result if foot-pounds 
 per minute were chosen. Thus an engine which can over- 
 come resistances equivalent to raising 10,000 pounds verti- 
 cally upwards through 33 feet every minute is said to be an 
 engine of 10 horse-power or 10 H.P. If the engine raised 
 the same weight through the same height once every second, 
 instead of every minute, then by the definition the work 
 done would be equal to sixty times ten horse-power, or 600 
 H.P. Hence if r = the resistance expressed in pounds, 
 h = the height in feet through which r is overcome, and 
 / = the time in minutes which it takes to do the work, then 
 
 The horse-power exerted = l Xr . 
 
 33,000 x / 
 
 The lifting of weights is only one special form of doing 
 work, but there are also many other ways of doing it. For 
 example : if a carriage be pulled along a level road, it is 
 well known that its progress is resisted by the friction of its 
 wheels against the surface of the road and against their own 
 axles. Hence the pulling of such a carriage answers per- 
 fectly to the definition of doing work, for resistance is thereby 
 overcome through a space. 
 
 Again, it is well understood by those who have studied 
 the laws of motion that if a mass as, for example, a stone 
 be projected upwards it will rise to a certain height, depend- 
 ing on the velocity with which it left the hand. The exact 
 height to which it will rise is precisely equal to the height 
 through which it must fall, under the action of gravity, in 
 order that, at the end of its fall, it may have acquired a 
 velocity equal to that with which it was projected upwards. 
 Now the imparting of this velocity to the mass is evidently 
 a way of enabling work to be done, for the mass is thereby 
 caused to rise to a certain height, against the attraction of 
 the earth, and the amount of the work done is measured by 
 the weight of the mass multiplied by the height to which it 
 rises. 
 
Horse-power. 25 
 
 It is not necessary to impart velocity to the mass in a 
 vertical direction only, in order to do work. Whenever 
 motion is given to a body in any direction the resistance due 
 to the inertia of the body is overcome through a space, and 
 consequently, by the definition, work is done. If, for in- 
 stance, a train were capable of moving without friction on a 
 level railway, in order to start it from a state of rest and 
 give it a speed of, say, forty miles an hour, work would have 
 to be done in order to overcome the mere inertia of the train. 
 When once the given speed had been imparted to the train, 
 it would, of course, move on for ever on a level railroad, 
 provided it met with no frictional resistances. If in its 
 course it came to an inclined plane, it would run up the 
 plane till it had attained a vertical height above the level 
 equal to the height through which the train must fall down- 
 wards, in order to attain the given speed of forty miles an 
 hour. The measure of the work done in giving motion to 
 the train is equal to the weight of the train multiplied by 
 this height. 
 
 On actual railroads the work done by the engine partakes 
 of the character of each of these examples. When starting 
 the train from a station and giving it a certain speed, the 
 resistance due to the inertia of the whole moving mass is 
 overcome. The going up an incline corresponds to lifting 
 a weight up a height ; and throughout the entire run the 
 friction of the wheels and axles and the resistance of the 
 air are being overcome. 
 
 ENERGY. 
 
 It has been necessary to dwell thus at considerable length 
 on the nature of Work, in order that the term Energy, i.e. 
 the power of doing Work, might be thoroughly understood. 
 This power of doing work exists in many different ways. 
 For instance, a coiled spring is capable of doing work in 
 driving a clock, and therefore possesses energy. Similarly a 
 
26 The Steam Engine. 
 
 weight raised to a height, and attached to a string passing 
 over a pulley, is capable, during its fall, of raising another 
 weight, or of driving machinery, and consequently it also 
 possesses energy. Again, a body in motion, such, for in- 
 stance, as a railway train, is capable of overcoming the fric- 
 tion of the brakes for a certain time till it is brought to a 
 standstill, and therefore possesses energy. Similarly a pro- 
 jectile from a modern rifled gun possesses very great energy 
 owing to the high speed at which it moves, so much so that 
 before it is forcibly brought to a rest it can do work repre- 
 sented by piercing many inches of iron armour. It will be 
 noticed that there is a great difference between the kind of 
 energy of which the first two cases are examples and the last 
 two. The first two are instances of bodies which, though 
 themselves at rest, are capable at any moment of doing work. 
 In the case of the coiled spring its energy was due to the 
 relative position of its parts with regard to each other and 
 to the mutual forces acting between them. In the case of 
 the raised weight, its energy was solely due to its position 
 with regard to the earth and to the forces acting between 
 the earth and it. This energy, due to position, is called 
 potential energy, a term- which signifies that the energy is 
 capable of being exerted. The last two instances on the 
 contrary are cases of bodies possessing energy by virtue of 
 their motion. This kind has been called actual or kinetic 
 energy. The last term, which is derived from a Greek 
 word signifying motion, is, perhaps, the most appropriate of 
 the two. 
 
 Bodies may be possessed of both descriptions of energy 
 at one and the same time. For instance, when the raised 
 weight of the former example begins to fall, it possesses 
 kinetic energy by virtue of the motion which it has acquired, 
 while it still possesses potential energy, for it is capable of 
 falling further still. For every foot which it descends its 
 kinetic energy increases, while the potential diminishes. 
 Just as it touches the earth its kinetic energy is a maximum. 
 
Modern Theory of Energy. 27 
 
 while the potential has vanished altogether. Thus during 
 the fall the energy has changed from being all potential into 
 being all kinetic. Moreover, the kinetic energy acquired at 
 the end of the fall is exactly equal in amount to the poten- 
 tial energy possessed at the commencement. For, before its 
 fall the mass was capable of pulling up another mass of nearly 
 equal weight with itself, to the same height above the ground 
 which it occupied ; while at the end of the fall it has 
 acquired a velocity, sufficient, if reversed, to send itself back 
 to whence it came. Moreover, it is clear that at any time 
 during the fall, the sum of the potential energy left, and the 
 kinetic energy acquired, are equal to the original energy, for 
 what the one has lost in amount the other has gained. 
 
 This is an example of what is called the transmutation of 
 energy, by which is meant that the energy is changed from 
 one form into another, and also of the conservation of 
 energy, by which is meant that the total energy of the two 
 bodies viz. the earth and the weight is not altered in 
 amount, but only in kind. It is one of the cardinal doctrines 
 of modern science, and one which has done more to extend 
 our knowledge of heat than any other, that energy, like 
 matter, can neither be created nor destroyed by material 
 agency, but can only be transmuted from one form to an- 
 other. This doctrine is called the Principle of the Conser- 
 vation of Energy. In books on Dynamics, the principle is 
 proved by mathematical reasoning to be true for certain 
 cases, and it has, moreover, been proved by experiment to be 
 true in all cases which can be tested by experiment. Hence 
 it is believed to be universally true. The following is a 
 general statement of the principle. 
 
 The energy of any system of bodies cannot be altered in 
 quantity by the mutual action of the bodies ; it can only be 
 transmuted in kind into one or more of the forms which 
 energy takes. 
 
 We are now in a position to return to the subject of 
 Heat, and to understand how it is that heat is a form of 
 
28 The Steam Engine. 
 
 energy i.e. a form of the capability of doing work. For 
 Davy's statement is, that heat may be defined to be a pecu- 
 liar motion of the corpuscles of bodies ; now, we have seen 
 that matter in motion is capable of doing work, and is 
 therefore possessed of energy, and consequently if heat be 
 motion, or the cause of the motion of the ultimate corpuscles 
 of matter, heat is also a form of energy. 
 
 HEAT A FORM OF ENERGY. 
 
 The reasons for believing that Davy's definition of heat 
 is a true one are the following : 
 
 1. It seems impossible to believe that heat is a substance; 
 for if it be such, then no theory has yet been advanced 
 which can account for certain phenomena, such for instance 
 as the production of heat from bodies in boundless quanti- 
 ties by means of friction or other mechanical action. 
 
 2. Heat can always be generated by doing work upon 
 bodies. For example, we have seen how Davy melted ice 
 by friction. Again, let the student attempt to file a piece 
 of metal, and after a very few strokes of the file, he will 
 find that both it and the metal have become perceptibly 
 warmer, and if he continues the action smartly for some 
 time on a small piece of metal, he will not be able to touch 
 it without burning his hand. As an example of another 
 kind of mechanical action producing heat, it is well known 
 that a smith can hammer a small piece of iron to a red heat. 
 Again, if water be allowed to fall several times from a height 
 into a nonconducting receptacle, and care be taken to pre- 
 vent the escape of heat, it will be found that after its fall 
 the water will be warmer than at the commencement of the 
 experiment. Another and most important example is the 
 effect of compression upon gases. If, for instance, a portion 
 of air be inclosed together with a piece of easily inflammable 
 tinder in a cylinder provided with a movable piston, and 
 the piston be driven down suddenly, it will be found that 
 
Heat a Form of Energy. 29 
 
 the contained air has become so warm that it can cause the 
 ignition of the tinder. Now in all the above instances, unless 
 we are prepared to admit that energy is destructible, that is, 
 that it can be put out of existence altogether, we are forced 
 to confess that it is merely transmuted into heat, for heat is 
 apparently the only thing we have to show for the energy 
 expended in the majority of these examples. 
 
 3. The converse of the above is also true, viz. heat can 
 be made to produce work, and for every unit of work which 
 it does, a certain amount of heat disappears, and there is 
 nothing to show for the disappearance of the heat but the 
 work done. As an example of this, we need only refer to 
 the elementary steam engine described in the first chapter, 
 where we saw that the heat of the fire, communicated to the 
 water contained in the cylinder, was partially converted into 
 work done ; for by the agency of heat alone the piston with 
 its weights was raised to a certain height. The conditions of 
 this experiment were not such as to enable us to ascertain 
 what heat, if any, had disappeared in consequence of the 
 work done ; but the following modification of the experiment 
 will render this fact also demonstrable. 
 
 Instead of holding water, let the cylinder contain a 
 portion of air of a certain warmth, and let the piston, instead 
 of being loaded with common weights, have a vessel con- 
 taining a quantity of water placed on it. Then, so long as 
 the weight on the piston remains constant, and so long as no 
 heat is communicated to the gas from outside, nothing will 
 happen. 
 
 If, however, a little of the water be removed from the 
 can, the pressure of the inclosed air will cause the piston 
 with its load to rise through a small space, and again come 
 to rest. For, the air occupying a larger space in the cylinder, 
 its pressure becomes diminished by a well-known law, which 
 will be explained hereafter (see page 43), and as soon as 
 this diminished pressure on the bottom of the piston is 
 equal to the diminished pressure of the piston on the in- 
 
3O The Steam Engine. 
 
 closed air (caused by some of the water having been taken 
 away) then the whole must come to rest. Let now a little 
 more water be abstracted ; the piston will rise a little higher, 
 and so on, till the whole of the water has been removed, 
 when the piston will have risen higher still. A convenient 
 way of abstracting the water, as fast or as slowly as we like, 
 is by means of a syphon. If now we have any means for 
 ascertaining the warmth of the air at the beginning and at 
 the end of the experiment, it will be found to have lost heat 
 at the end, after having done work, measured by lifting the 
 piston, together with the pressure of the external atmosphere 
 and the empty can through the whole height, and different 
 portions of the water through different heights. Now, unless 
 this heat has been spent in effecting internal changes in the 
 constitution of the gas itself, it must have been spent in 
 doing the above work, for no other effects have been pro- 
 duced. It is of course assumed that in the experiment no 
 heat has been allowed to escape from the air in the cylinder 
 to external bodies, or, vice versa^ to reach the air from 
 external bodies. 
 
 Most elaborate experiments have been made on steam- 
 engines when at work, in which the following quantities have 
 been measured: i. All the heat which enters the engine in 
 the shape of steam ; 2. All the heat which leaves the con- 
 denser in the shape of warm water; and 3. All the heat which 
 escapes during the working of the engine in various ways ; 
 and it has been found that the quantities comprised under 
 the 2nd and 3rd headings are not equal to but less than 
 the heat which enters the engine ; so that a certain quantity 
 remains to be accounted for, the disappearance of which can 
 only be explained on the supposition that it has been turned 
 into the mechanical work done by the engine. 
 
 From all the above considerations we conclude that heat 
 is a form of energy. It is further supposed that the special 
 form which this energy takes is that of a motion of the 
 
Heat a Form of Energy. 31 
 
 molecules which constitute matter. Into the nature of this 
 motion, however, it is not proposed to enter here. 
 
 The next thing which we shall want to know is this : What 
 is the exact relation between heat and work ; that is to say, 
 What quantity of heat can be produced by the doing of a 
 certain quantity of work, and, vice versa, How much work 
 is a given quantity of heat capable of doing ? Before it is 
 possible to answer these questions, it must first be explained 
 what is meant by a quantity of heat, and how heat is measured 
 at all. 
 
 MEASUREMENT OF HEAT. TEMPERATURE. 
 
 Everybody is familiar with the sensations caused by diffe- 
 rent intensities of heat. For instance, the sensation produced 
 by plunging the hand into boiling water is very different from 
 that caused by contact with cold water taken direct from a 
 well. The quality of heat which causes these sensations is 
 called temperature. In the first case the immediate cause of 
 the sensation experienced was the heat leaving the boiling 
 water and entering the comparatively cold hand ; while in 
 the second instance exactly the reverse took place, heat 
 entering the cold water from the comparatively warm hand. 
 This communication of heat from one body to another 
 depends on the differences between their temperatures ; so 
 much so that temperature has been defined as follows: 'The 
 temperature of a body is its thermal state considered with 
 reference to its power of communicating heat to other 
 bodies.' 1 
 
 If now two bodies be so placed that they can freely com- 
 municate heat to one another, and are isolated from the in- 
 fluence of all other bodies, then if neither of them loses heat 
 they are said each to have the same temperature, but if one 
 of them loses and the other gains heat, then the body which 
 loses is said to have the highest temperature. 
 
 See Clerk Maxwell's Theory of Heat, p. 32. 
 
32 The Steam Engine. 
 
 Temperatures are measured and compared by noting 
 the effects which heat has upon bodies. One of the most 
 remarkable effects of heat is that it expands most substances 
 to which it is communicated, so that the higher the tempera- 
 ture the greater the expansion. If then we want to compare 
 the temperatures of two bodies, we have only to bring each 
 of them in turn into thermal communication with some 
 third substance which expands readily under the action of 
 heat. 
 
 If care be now taken that each of the bodies in turn 
 remains sufficiently long in contact with the third body, so 
 that the latter may acquire, first, the exact temperature of one 
 of them, and afterwards that of the other, and if its expansion 
 in each case be carefully measured, then that body which 
 causes the greatest expansion has the highest temperature. 
 An instrument designed to serve the purpose of this third 
 body is called a Thermometer. 
 
 THERMOMETERS. 
 
 A thermometer for practical use should be portable, 
 readily acted upon by slight differences of temperature, and 
 difficult to put out of order ; it should be furnished with an 
 index, or scale, for reading off differences of temperature, 
 and should always give the same reading on the scale, for the 
 same temperature, under the same circumstances. Thermo- 
 meters are made of various substances, but we propose at 
 present to describe only the one which is in most common 
 use, viz. the ordinary mercurial thermometer. This instru- 
 ment (see fig. 10) is made by taking a tube of glass, a few 
 inches in length, having a capillary bore, that is to say a bore 
 of very small calibre. A bulb is blown at one end of the 
 tube, and while the bulb is warm, so that most of the air it 
 contains is expelled, the tube is plunged into mercury. The 
 effect of this is to cool the tube, and, as we shall see after- 
 wards, to reduce the pressure of the air which remains in the 
 
Thermometers. 33 
 
 bore and bulb. Some of the mercury then enters the bore, 
 and partly fills the bulb. By boiling this mercury while in 
 the bulb, the remainder of the air is expelled, its place being 
 taken by the vapour of mercury. If now the open end of the 
 tube be again plunged into mercury, 
 both tube and bulb will be completely fi*AnAei| 
 filled, and while still warm the open 2l2 j-100 w l r E N B G 
 
 end is closed hermetically. As soon 
 as the tube and its contents have 
 cooled down, the mercury will be 
 found to have contracted, leaving part 
 of the bore quite empty. The instru- 
 ment is now ready for graduation. 
 This is done by first marking on the 
 tube the position at which the mercury 
 stands for two different temperatures, 
 and then dividing the intermediate 
 space into an arbitrary number of 
 equal spaces, each of which is said to 
 represent one degree of temperature. 
 The two temperatures always chosen 
 are those of melting ice and boiling 
 water. The temperature of melting 
 ice is always the same, at the varying 
 pressures of our atmosphere. The 
 thermometer is plunged into a mix- 
 ture of melting ice and water, and, 
 after remaining immersed for some 
 time, the point at which the mercury 
 stands is marked on the tube. We 
 may be certain that we have thus 
 marked the exact temperature of melt- Fig - I0 - 
 
 ing ice ; for if, during the process of immersion, heat enters 
 the mixture of ice and water, its effect will be simply to 
 melt some more of the ice, and not to raise the tempe- 
 rature of the water. This action of heat will be explained 
 
 D 
 
34 The Steam Engine. 
 
 hereafter. It is more difficult to mark the point at which 
 the mercury stands for boiling water, for it is known that 
 water does not always boil at the same temperature. In 
 fact, the greater the pressure under which the water boils, 
 the higher will be its temperature. Consequently, on a 
 day when the barometer stands high, boiling water is hotter 
 than when the opposite is the case. It is therefore neces- 
 sary to fix upon some one special atmospheric pressure in 
 order to settle a standard boiling-point, and the pressure 
 always adopted in this country is that marked by the 
 barometer when the mercury stands at a height of 29-905 
 inches, the temperature of the mercury being that of melting 
 ice. If then, on a day when the barometer indicates the 
 above pressure, the thermometer be immersed in the steam 
 of boiling water, it will be found that the contained mercury 
 will rise to a certain point, and remain there ; and by mark- 
 ing the tube at this place we obtain a point which always 
 shows the temperature of boiling water at the standard pres- 
 sure of the air. This temperature is always fixed and invari- 
 able, so long as the pressure under which the water boils 
 remains fixed ; for the effect of adding more heat to the 
 water is only to turn more of it into steam, but not to raise 
 its temperature. The reason of this will be explained here- 
 after. 
 
 Having now got two fixed points on the tube of the 
 thermometer, we are at liberty to call them by any numbers 
 we please, and to divide the space between them into, any 
 convenient number of divisions, or degrees, and to carry 
 these divisions above the boiling and below the freezing 
 points, as far as the length of the tube permits. There are 
 three modes of numbering in common use in various 
 countries: 
 
 i. The Centigrade scale, in which the temperature of ice 
 
 is called zero, or o, and the temperature of boiling water 100. 
 
 The space between these two is divided into one hundred 
 
 . equal parts, each of which, if the bore of the tube be per- 
 
Graduation of Thermometers. 35 
 
 fectly even, is assumed to represent an equal increment of 
 temperature, and the divisions are carried up above boiling- 
 point, and down below freezing-point as far as the tube per- 
 mits. Those degrees below freezing-point are called negative. 
 This scale of temperature is in common use in nearly all the 
 countries of the continent excepting Russia. 
 
 2. The Fahrenheit scale, used in the British Empire and 
 the United States. On this scale the freezing-point is called 
 32, the boiling-point 212, and the intermediate space is 
 divided into one hundred and eighty equal parts. The 
 zero of this scale is 32 below freezing-point, and below this 
 zero the numbers are negative. 
 
 3. The Reaumur scale is used chiefly in Russia. This 
 scale differs from the Centigrade, in that the boiling-point is 
 called 80, and the space between it and zero or melting ice 
 is divided into eighty equal parts. This scale is less in use 
 than either of the others. Throughout this book the Fahren- 
 heit scale is the one generally referred to. Whenever the 
 Centigrade scale is made use of, it will be specially indicated 
 by writing C. after the numeral showing the number of 
 degrees. Fig. 10 shows the scales of the Fahrenheit and 
 Centigrade thermometers side by side. 
 
 To compare degrees on the Fahrenheit and Centigrade 
 scales it is only necessary to remember that the freezing- 
 point on the Fahrenheit scale is 32, and on the Centigrade 
 o, while the number of degrees between this and boiling- 
 point is in the former case 180, and in the latter 100. 
 Consequently the length of one degree F. is f of one degree 
 C. Now the actual number of degrees F. above freezing- 
 point is equal to the number on the scale minus 32. Let 
 T stand for the number of degrees on either scale, then 
 Tc=f(T F -32 ) and conversely T F =fT c + 32. 
 
 It is not proposed to enter here into the refinements of 
 thermometer-making, but it will be necessary now to point 
 out how far the mercurial thermometer may and may not 
 be trusted, as a measurer of temperature, and what errors 
 
 D 2 
 
36 T/ie Steam Engine. 
 
 are inseparably connected with its use, no matter how 
 perfectly it may be made. 
 
 For the mere purpose of ascertaining whether two or 
 more bodies are precisely of the same temperature, or for 
 stating generally in which of them the temperature is highest, 
 the instrument is trustworthy enough. It is only when it is 
 wanted to measure the thermal condition of bodies quanti- 
 tatively that its indications can no longer be accepted. For 
 instance we cannot be certain that a difference of tempera- 
 ture of one degree between say 32 and 33 in any body 
 measured on a mercurial thermometer represents the same 
 difference in its thermal condition as does a difference of 
 one degree between, say, 200 and 201. In other words, if 
 we heat a certain quantity of water from 32 to 33, and a 
 similar quantity of water from 200 to 201, we cannot by 
 any means state that we have in each operation altered the 
 thermal condition of the water by the same amount. 
 
 There are two reasons for this. The first has to do with 
 the thermometer, and the second with the substance of 
 which the differences of temperature have to be measured. 
 It will be remembered that the way in which the length of 
 degrees was arrived at when making the thermometer was 
 by dividing the space between freezing and boiling point 
 into 1 80 equal divisions, each of which was called one 
 degree of temperature. Now in order that each of these 
 degrees should represent an equal increase of heat of the 
 mercury we should have to prove that if we add successive 
 equal quantities of heat to the mercury, we thereby expand 
 it by each operation by an absolutely equal quantity. Now 
 we have no right to assume that this is the case, for the 
 action of heat in causing some bodies to expand is known 
 to be most irregular. If, for instance, the thermometer had 
 contained water instead of mercury, then, commencing at 
 freezing-point, it is known that the first effect of increasing 
 the temperature is to cause the water to contract. This 
 contraction would go on till the water had reached the tern- 
 
Defects of Thermometers. 37 
 
 perature of about 39, after which further additions of heat 
 would cause the water to expand. In the same way, careful 
 experiments made with mercury have proved that its rate of 
 expansion at high temperatures is considerably greater than 
 at low ones, for equal increments of heat ; consequently the 
 errors in the high part of the scale become considerable. 
 
 The second reason has only to do with the substance 
 the temperature of which has to be measured. Even as- 
 suming that our thermometer were quite perfect, we should 
 still be unable to use it by itself alone to determine quanti- 
 tatively the thermal condition of bodies ; for the thermo- 
 meter in the first instance shows only the temperature of its 
 own mercury, and though its degrees might be so marked 
 that each successive one would correspond with succes- 
 sive equal additions of heat to the mercury, still it does 
 not follow that this would also be true of the substance 
 whose temperature had to be ascertained. On the contrary, 
 experiments on some substances, such as water, show that 
 'it takes more heat to raise their temperature by one degree 
 at the high part of the scale than at the low part. 
 
 This last remark leads us at once to the object of all 
 experiments on thermometry, viz. the measurement of 
 quantities of heat. It might at first be supposed that the 
 measurement of temperature was the same thing as the 
 measurement of quantity of heat, but an easy experiment will 
 prove that this is not the case. Take two vessels, one con- 
 taining a pound of water and the other a pound of olive oil, 
 each liquid having the temperature of the air of the room, 
 say 55. Take also two pieces of copper, each weighing a 
 pound. For the purpose of the experiment sheet copper 
 about one-sixteenth of an inch thick is best ; and for conveni- 
 ence sake it should be bent round nearly to the form of a 
 cylinder. Bring each of these pieces to a certain high tem- 
 perature. This is best accomplished by boiling them for a 
 short time in water, so that their temperature becomes 212. 
 Next, plunge one of the pounds of copper into the pound of 
 
38 77/ Steam Engine. 
 
 water, and the other into the pound of oil. The two liquids 
 will of course receive heat, each from its own piece of 
 copper, and they will therefore rise in temperature. Let 
 the rise in temperature be carefully noted by means of two 
 identical thermometers, one immersed in the water and the 
 other in the oil. As soon as the mercury in the two ther- 
 mometers has ceased to rise, we may assume that the pieces 
 of copper have parted with their surplus heat, but it will be 
 found that the temperature of the water is 68^, while that 
 of the oil is nearly 92. Here then we have a pound of 
 copper at 212, which is only capable of heating a pound of 
 water, having the original temperature of 55, up to 68^. 
 In other words, while the copper has lost 212 -68^= 143^ 
 the water has gained only 68^-55=i3| . While in the 
 case of the oil, the copper has lost 212 - 92 = 120, and the 
 oil has gained 92 -55 = 37. Now the amount of heat 
 lost by the copper in each case is of course exactly equal to 
 that gained by the water in the one instance, and by the oil 
 in the other ; therefore it is evident that it takes less heat to 
 raise the temperature of the oil by 37 than it does to raise 
 that of the water by 13^, while the same quantity which 
 suffices to produce this latter effect upon the water is compe- 
 tent to raise the temperature of the copper 143 \. 
 
 These figures show conclusively how very differently the 
 temperatures of different bodies are affected by different 
 quantities of heat. 
 
 Specific Heat. The amount of heat which a body of 
 unit mass absorbs in order that its temperature may be raised 
 by one degree ; or, vice versa, the amount of heat which the 
 body parts with while its temperature is lowered one degree, 
 is called its Capacity for Heat. 
 
 To compare this quantity for different bodies we must 
 first fix upon some unit of quantity of heat. The unit 
 generally adopted in Great Britain is the quantity of heat 
 required to raise one pound of pure water from the tempe- 
 rature of 39 to 40 ; or, vice versd, the quantity of heat 
 
Specific Heat. 39 
 
 parted with by the water in cooling from 40 to 39. It is 
 necessary thus to specify the temperature, because water, 
 and indeed most bodies, have different capacities for heat 
 at different temperatures. This quantity of heat is called the 
 British Thermal Unit. The capacities for heat of other 
 bodies are designated numerically, by comparing the 
 quantities of heat necessary to raise their temperatures by 
 one degree with unity. 
 
 The ratio of the quantity of heat required to raise the 
 temperature of a body one degree, to the quantity re- 
 quired to raise an equal weight of water one degree, is 
 called the Specific Heat of the body. Thus, if it take half 
 the quantity of heat to raise one pound of ice from 20 to 
 21 that it does to raise a like quantity of water from 39 to 
 40, then the specific heat of ice is said to be \ or -5. 
 
 It is often necessary in questions connected with the steam 
 engine to know how much matter at one temperature it will 
 take, in order to raise a certain quantity of matter of another 
 temperature to some third temperature. These questions 
 are easily solved in the following manner : 
 Let M be the mass of one of the bodies. 
 
 ,, M' be the mass of the other body. 
 
 T be the temperature of the body of mass M. 
 
 ,, T' be the temperature of the body of mass M'. 
 
 ,, S be the specific heat of the body of mass M. 
 
 S' be the specific heat of the body of mass M'. 
 When the bodies are mixed together the hotter of them 
 will lose heat to the colder, till at last they attain some 
 common temperature ; and the quantity of heat lost by the 
 one substance must be exactly equal to the quantity gained 
 by the other, since the total quantity of heat remains un- 
 changed. Let the body of mass M be the hotter of the 
 two, and let the common temperature which both attain 
 when mixed be called 0. Then, the quantity of heat. lost 
 by one pound of the hotter body in cooling from T to 
 !* T ~~ ^)'" and consec l uentl y tne quantity lost by M 
 
40 Tlie Steam Engine. 
 
 pounds, is M.S (T -6). Similarly, the quantity gained by 
 the other body is M'.S' (# T'), and since these two quan- 
 tities are equal, we obtain the equation 
 
 M.S. (T-0; = M'.S'. (0 T /0 ). 
 
 This equation is only true provided the whole effect of heat 
 upon bodies is the changing of their temperatures ; but it is 
 known that this is not the only effect. We shall afterwards 
 see that a large quantity of heat, may be added to bodies 
 without changing their temperatures in the least, and that its 
 effect is in these cases to change the state of constitution of 
 the body ; as, for instance, when ice of 32 is changed into 
 water of 32, or water of 212 changed into steam of 212. 
 For such cases, therefore, as the mixing of ice and water 
 together, the above equation does not hold good. The 
 equation is also only true on the supposition that the specific 
 heat of bodies is the same at all temperatures. This also is 
 not, strictly speaking, true, but for ordinary purposes the 
 error thus introduced may be neglected, 
 
 MECHANICAL EQUIVALENT OF HEAT. 
 
 Having thus examined the question of the measurement 
 of heat, we are now in a position to revert to the subject of 
 the equivalence of heat and energy. What we want to know 
 is, firstly, how much heat can be got by the doing of a cer- 
 tain quantity of work. The converse question, viz., how 
 much work can be got out of a certain quantity of heat, is 
 of a more complicated character, and its discussion must be 
 postponed till the following chapter. 
 
 The first question was accurately settled experimentally 
 by Dr. Joule, of Manchester. He conducted an immense 
 number of experiments on the friction of various solids and 
 liquids, and on the compression of gases. His experiments 
 on the friction of fluids were carried out in the following 
 way. The work was done by causing a known weight to 
 
Mechanical Equivalent of Heat. 41 
 
 descend through a given distance ; the weight during its 
 descent caused, by means of a suitable mechanism, a paddle 
 to revolve inside a closed vessel filled with the liquid to be 
 experimented upon. This paddle, by agitating the contents 
 of the vessel, produced friction between the particles of the 
 liquid, the walls of the vessel and the paddle, which friction 
 would of course be converted into heat, and would raise 
 the temperature of the vessel and its contents. Careful 
 allowance was made for the resistance caused by the friction 
 of all mechanism exterior to the vessel, and also for all the 
 heat which escaped into the sides of the vessel or into the 
 air. The temperature of the liquid was carefully noted, first 
 before the experiment commenced, and next after the weight 
 had descended through a given distance. The rise of tem- 
 perature multiplied by the mass of liquid, multiplied by its 
 specific heat, gave, after making all allowances, the quantity 
 of heat generated by the descent of the weight. 
 
 The result was that a quantity of work represented by 
 772 foot-pounds is capable, when all converted into heat, of 
 raising the temperature of a pound of pure water, weighed 
 in vacuo, and having the temperature of 50, through one 
 degree Fahrenheit. 
 
 In other words : The British Thermal Unit is equivalent 
 to 772 foot-pounds ofivork. This number is called the me- 
 chanical equivalent of heat. 
 
 This result has been fully confirmed by numerous other 
 experiments, made on various substances and in various 
 ways, and it constitutes by far the most important practical 
 discovery which has yet been made in the science of heat. 
 Another way of putting the above result is this. If a pound of 
 water be allowed to fall in vacuo down a height of 772 feet, 
 and if all the heat generated by its impact at the end of its 
 descent be collected into the pound of water, its temperature 
 will be raised one degree. 
 
 The equivalent of the units of mechanical work in 
 thermal units can now be readily expressed. For example, 
 
42 The Steam Engine. 
 
 one foot-pound of work is equivalent to yj^nd P art of a 
 thermal unit. One horse-power exerted for a minute, or 
 33,000 foot-pounds, is equivalent to 4274 thermal units. 
 
 It might at first be supposed that if by doing 772 foot- 
 pounds of work on a pound of water, we thereby raise its 
 temperature one degree, the converse of this must also be 
 true, viz. that by cooling a pound of water by one degree we 
 should thereby be enabled with a suitable apparatus to do 
 772 foot-pounds of w r ork. It will be seen hereafter that 
 this is not possible ; but before this question can be 
 thoroughly understood w r e must examine into the effect of 
 heat upon gases and water, as it is in general through the 
 medium of these substances that heat is converted into 
 mechanical effect. 
 
 THE EFFECT OF THE APPLICATION OF HEAT TO GAS. 
 
 The effect of heat upon \vater has more to do with the 
 subject-matter of this book than has its effect upon gases; 
 but as steam is an imperfect gas, and as the laws which 
 govern the behaviour of gases are much simpler than those 
 for water and steam, we will commence with the subject of 
 gases. 
 
 Gases differ from solids and liquids in that they have 
 no tendency to keep to any fixed form and volume. A small 
 portion of gas if introduced into a closed empty vessel will 
 at once expand, so as to fill the whole of it, and will press 
 with a certain equal pressure against every equal portion of 
 the containing surface of the vessel. If by any means the 
 vessel be enlarged, the gas will expand still further, so as to 
 fill it completely as before, but its density or weight per unit 
 of volume will be less, and the pressure which it exerts 
 against the sides of the vessel will also be less. 
 
 Take for instance a cylinder closed permanently at one 
 end, and containing a movable piston, by means of which 
 the volume of the portion below the piston can be changed 
 
Boyle's Law. 43 
 
 at pleasure. Let the area of the horizontal section of the 
 cylinder be one square foot, and let the piston, which is sup- 
 posed to be without weight, be placed at a height of one 
 foot above the bottom of the cylinder, so that the space 
 beneath it is filled with one cubic foot of gas or air at the 
 ordinary temperature and pressure (say 147 Ibs. per square 
 inch) of the atmosphere. The volume of the air is now one 
 cubic foot \ and its pressure is 147 Ibs. per square inch, or 
 2116*8 Ibs. per square foot. Next, let the piston be raised 
 by hand to a new position, two feet from the bottom of 
 the cylinder, the temperature of the inclosed air being 
 maintained constant. The volume of the air is now doubled 
 or two cubic feet. If we had a proper instrument for 
 measuring pressures, we should find that the pressure 
 per square foot is only half what it was before, or 10584 
 per square inch, and as the same weight of air now occupies 
 twice the original bulk, its weight per cubic foot is halved 
 If instead of raising the piston we had weighted it, or 
 pushed it down to within half a foot of the bottom of the 
 cylinder, and if we had taken care to keep the temperature 
 of the inclosed air constantly the same, we should thus 
 have halved the volume, which would now be half a cubic 
 foot ; and doubled the pressure to 4233-6 Ibs. per square 
 foot, and also doubled the density or weight per cubic foot. 
 
 These facts are expressed generally by Boyle's law of the 
 pressure and volume of gases, which is as follows : 
 
 The volume of a portion of gas varies inversely as the 
 pressure, so long as the temperature is constant. 
 
 This law may be expressed in other words as follows: 
 
 The product of the pressure multiplied by the volume of a 
 portion of gas is a constant quantity, so long as the tempe- 
 rature is constant. 
 
 If we remember, that in the above experiment the weight 
 of the gas per cubic foot diminished in proportion as the 
 volume increased, we may express the law otherwise thus : 
 
 The pressure of a gas is proportional to its density. 
 
44 TJie Steam Engine. 
 
 If P represent the pressure in any units, say in pounds, 
 per square foot, and V represent the volume in cubic feet, 
 and C is a constant quantity, to be determined experimentally 
 for one particular case, then the algebraical expression for 
 Boyle's law is 
 
 PxV=C or P= 
 
 GRAPHIC REPRESENTATION OF BOYLE'S LAW. 
 
 The law may be exhibited graphically in the following 
 manner. Take a line OV, fig. n, along which volumes are 
 
 measured to any scale ; and 
 a line OP at right-angles to 
 OV, along which pressures 
 are measured, likewise to 
 scale. Now, reverting to the 
 former experiment, measure 
 off along OV a distance Qa to 
 scale, representing the original 
 volume of the air beneath the 
 piston, viz. one cubic foot, 
 and draw aA at right-angles 
 to Qa so as to represent to 
 scale the original pressure, 
 viz. 2116-8 Ibs. per square foot, then the product of this vo- 
 lume and pressure is represented by the area of the rectangle 
 OaAfi, which is the constant quantity in the above equation. 
 Next draw Ob to represent the volume in the second stage 
 of the experiment, viz. two cubic feet, and B to represent 
 the corresponding pressure, viz. 1058*4 Ibs. per square foot. 
 Then the product of these two quantities is represented by the 
 rectangle O^Ba which is equal to the original rectangle OaA(3 
 because its base O = 2Oa, while its height B = \aA. 
 Next, for the third stage, measure Og equal to half a cubic 
 foot, and^G equal to twice the original pressure, or 4233-6 
 
Isothermal Lines. 45 
 
 Ibs. per square foot. Then the product of this pressure and 
 volume is represented by the rectangle O^Gy, which like- 
 wise is equal to the original rectangle. In a similar way we 
 can obtain the point D such that the rectangle QdxdT) equals 
 the original rectangle, and similarly any other number of 
 points. Now, a curve drawn through the angles G A B D 
 &c. of any number of rectangles of equal area, arranged as 
 in the figure, is called a rectangular hyperbola ; and not 
 merely for these points, but also for every other point along 
 the curve, the condition holds good, viz. that the rectangle 
 formed by drawing perpendiculars to the lines OV and OP 
 is equal to the original rectangle. 
 
 If we designate all lines measured parallel to OV by the 
 symbol 7', and all corresponding lines measured parallel to 
 OP by the symbol/, then it is evident that the curve GABD 
 is expressed by the equation pvc, in which c represents the 
 area of the original rectangle. This equation corresponds 
 exactly, in form, with the equation used to express Boyle's 
 law. The lines OV and OP are called respectively the axes 
 of volume and of pressure ; they are also what are called the 
 asymptotes of the hyperbola GABD. The lines drawn from 
 any point on the curve, perpendicular to the axes, are called 
 the ordinates of the point. 
 
 ISOTHERMAL LINES. 
 
 We see, therefore, that the volume and pressure of a 
 portion of gas, when expanding or being compressed, the 
 temperature remaining always the same, may be represented 
 graphically by the ordinates of an hyperbola. Such an hyper- 
 bola as has just been drawn is called an isothermal line 
 of expansion or compression of a gas. or briefly an isothermal. 
 ( This term, which is derived from two Greek words signifying 
 equal and heat, signifies that the temperature is constant 
 throughout the expansion or compression represented by the 
 line. 
 
46 77ie Steam Engine. 
 
 The system of representing graphically the varying 
 pressures and volumes of a portion of gas, by means of a 
 line should be carefully marked, for it is, as we shall after- 
 wards see, the basis of the graphic representation of the 
 working of all gas or steam engines, called indicator dia- 
 grams. The method may also be used in many theoretical 
 investigations connected with the expansion of gases and 
 steam under different circumstances, in place of complicated 
 algebraical expressions, or also to supplement and explain 
 these latter. 
 
 Effect of varying the temperature of a gas. Let us now 
 revert to the cylinder of the previous experiment, the 
 piston being, as before, one foot above the bottom, and 
 the air beneath it being at the pressure of the atmosphere, 
 but at the temperature of melting ice. Let us now heat 
 the air by any means so as to bring it up to the tempe- 
 rature of boiling water. During the rise of temperature 
 the piston will gradually ascend, and when the tempera- 
 ture of 212 has been attained, it will have risen to "366 feet 
 or 4-39 inches above its original position, and will remain 
 there so long as the temperature is maintained at 212, and 
 the other circumstances continue unchanged. 
 
 If the piston had not been "allowed to rise, but the air 
 beneath it had been heated as before up to 212, its pressure 
 would then have increased from i to i'3654 atmospheres 
 i.e. from 147 to 20*08 Ibs. per square inch. During the 
 first of these operations the air is said to be heated at con- 
 stant pressure, and during the second at constant volume. 
 
 The fact that air, and indeed all gases, increase in bulk 
 when heated from freezing to boiling point by a fixed frac- 
 tion (which is nearly the same for every gas) of their original 
 volumes at freezing-point, was first discovered in France by 
 Charles. Hence the numerical statement of the amount of 
 the expansion is usually called Charles' law. The dilatation 
 of gases was afterwards investigated more completely by 
 Dalton in this country, and by Gay Lussac in France ; hence 
 
Gay Lussac s Lctiv. 47 
 
 the statement is also often called the law of Dalton and of 
 Gay Lussac. Though both these philosophers agreed as to 
 the total increase in the bulk of a gas when raised in tem- 
 perature from 32 to 212, there was, nevertheless, an im- 
 portant difference between them in investigating the increase 
 for each individual degree between these two temperatures. 
 Gay Lussac's view was that the increase for each degree 
 is a certain fixed quantity, which quantity is a definite frac- 
 tion of the volume at any temperature. Thus, if V be the 
 volume of the gas at o, and V the volume at any other 
 temperature tf, and if a be the fraction of V by which the 
 volume is increased for each degree of rise of temperature 
 (commonly called the coefficient of expansion), then, 
 according to Gay Lussac, 
 
 V = V + /.*.V = V (f + fa). 
 
 The total increase in volume between 32 and 212 is, as 
 stated above, in the ratio of i to 1*3654, and the increase per 
 degree is therefore '3654 ~ 180 = '00203. If tne original 
 volume be taken at o, the corresponding increase per 
 degree = '00217. 
 
 In a similar manner, if the volume of a gas be kept con- 
 stant and it be heated from 32 to 212, the pressure will 
 be increased in the ratio of i to 1*3654, and the increase 
 per degree of temperature will be as before, '00203. This 
 experimental result can 'also be deduced by the aid of 
 Boyle's law from the known increase of volume when the 
 temperature is raised and the pressure kept constant. For, 
 let the pressure be P and the original volume at 32 be V. 
 When heated to 212 at constant pressure the volume 
 becomes V x i '3654. Let the gas now be compressed at 
 the constant temperature 212 back to its original volume, 
 
 and the pressure by Boyle's law will become P \7 
 
 = P xi -3654. 
 
 According to Dalton, however, if a be the coefficient of 
 
48 The Steam Engine. 
 
 expansion for a rise of one degree from 32 to 33, so that 
 the volume at 33=V + V; then at 34 the volume 
 becomes (V + 0V) + (V + aV) = V (i+tf) 2 , and at 35 it 
 is V (i+a) 3 and so on. In other words, a gas at a given 
 temperature, say 50, for a rise of one degree, say to 51, 
 increases in bulk by a fixed fraction of its volume at 50 
 and not of its volume at 32 as stated by Gay Lussac 
 and so on for any other temperature. It will be seen that 
 there is a most important difference between the two laws, 
 and many of the theoretical calculations relating to steam 
 would be largely modified by the adoption of Dalton's 
 instead of Gay Lussac's statement. It might have been 
 thought that the question of accuracy between the two 
 could be easily settled by experimental investigation, and 
 the subject has naturally received much attention at the 
 hands of Regnault, Stewart, and others. It is, however, not 
 possible absolutely to prove the truth of either law experi- 
 mentally, because, as we have seen, no accurate experimental 
 method exists of measuring temperature quantitatively. The 
 experiments above referred to are, however, generally 
 accepted as proving that Gay Lussac's statement is the 
 nearer to truth of the two, and, in conformity with this 
 generally received opinion, his law will be made use of 
 throughout the remainder of this book. 
 
 It should here be stated that both Boyle's and Gay 
 Lussac's laws are only true for perfect gases. For actual 
 gases Regnault has found : 
 
 1. That the product P x V is not quite constant, especi- 
 ally for those gases which can be liquefied. 
 
 2. That the coefficients of expansion of air and other 
 simple and compound gases are not quite identical. 
 
 3. That the coefficients of expansion of all gases, with 
 the exception of hydrogen, increase somewhat as their 
 density increases. 
 
 4. The coefficients of expansion of all gases become 
 more nearly equal to each other as their pressures diminish. 
 
A ir Thermometers. 
 
 49 
 
 THE AIR THERMOMETER. 
 
 One of the most important practical 
 deductions from the above laws is the con- 
 struction of the air thermometer. There 
 are several reasons why air, or any other 
 gas, is a better substance for measuring 
 temperature than any liquid or solid. 
 Foremost among these is the fact, that no 
 two different liquids or solids will agree 
 when used for measuring temperatures 
 other than the two originally selected 
 temperatures, which are invariably those 
 of freezing and boiling water. On the 
 other hand, air and all gases, when at the 
 same pressures, will expand by almost 
 exactly identical amounts when subjected 
 to the same temperatures. Another reason 
 for preferring gases to other substances 
 for thermometrical purposes is that the 
 specific heats of gases remain the same 
 at all temperatures. 
 
 Without going into the question of air 
 thermometers in actual use, a very simple 
 theoretically possible form of such an in- 
 strument will now be described for the 
 purpose of illustrating what is meant by 
 absolute temperature. 
 
 Take a tube with a uniform bore, and 
 inclose in it a portion of air or other gas, 
 in such a way that, at the temperature of 
 melting ice, the air will occupy a space 
 of one foot measured from the bottom of 
 the tube and will be separated from the 
 external atmosphere by an air-tight piston, which, 
 
 212 
 
 MELTING 
 ICE 
 
 32 
 
 ABSOLUTE 
 ZERO 
 
 Fig. 12. 
 
 however, is 
 
50 TJie Steam Engine. 
 
 free to move up and down the bore of the tube (see fig. 12). 
 The pressure on the inclosed gas is to be maintained 
 constant. If, now, the tube be exposed to the steam of 
 boiling water, the inclosed air will expand, causing the 
 piston to rise to a point 1*3654 feet above the end of 
 the tube. A mark at this point will indicate the tempe- 
 rature of 212. If we divide the space between this and 
 freezing-point, viz. '3654 feet, into 180 equal divisions, 
 each of these will, if Gay Lussac's law be true, represent 
 one degree Fahrenheit. Hence we see that for each degree 
 of rise of temperature the volume of the gas inclosed in 
 the thermometer increases by a fraction of its original 
 
 volume at 32 ==-3_54 =-002 03. 
 ioo 
 
 ABSOLUTE TEMPERATURE. 
 
 If we choose we can now extend these subdivisions below 
 freezing and above boiling point as far as we like. As the space 
 between freezing and boiling points, viz. '3654 feet, contains 
 1 80 divisions, the space below freezing-point to the bottom 
 of the tube, viz. one foot, will contain 492^6 divisions or 
 degrees. In other words, the bottom of the tube is 
 492'6 32 460*6, or in round numbers, 461 below 
 zero. As we shall see hereafter, great use is made in thermo- 
 dynamical calculations of the above fact. 
 
 This point has been called the absolute zero of tempe- 
 rature, and the first conclusion which would be deduced 
 from the foregoing would be, that by depriving a portion of 
 gas of all its heat we should thereby reduce its bulk to 
 nothing. It is needless to say that all reasoning as to the 
 condition of a gas when deprived of all its heat is mere 
 speculation, as we have no experimental knowledge of the 
 subject whatever. Dalton's law, it will be observed, would 
 lead to quite other conclusions. It is, however, very con- 
 venient in calculations respecting gases to measure tempera- 
 
Absolute Temperature. 51 
 
 ture from the absolute zero, instead of from the zeros of any 
 of the scales in common use. Temperature thus measured 
 is called absolute temperature. To convert temperature 
 measured on Fahrenheit's scale to absolute temperature, we 
 have only to add 461 to the reading. Thus the absolute 
 temperature of boiling water would be 46i + 2i2 =673. 
 
 Combination of Boyle's and Gay Lussac's laws. Let V 
 be the volume of a portion of gas at the pressure/ and the 
 temperature o. 
 
 Let V be the volume at any other temperature /, the 
 pressure remaining p. 
 
 Then by Gay Lussac's law V = V (i + at) :. V = -^ 
 
 i + at' 
 
 Now by Boyle's law the volume V'^ at any pressure /', and 
 temperature t', multiplied by its pressure/', equals the 
 volume at pressure/, and same temperature /', multiplied 
 by its pressure/, 
 
 or 
 
 at 
 
 Now by reference to the description of the air thermometer 
 and the definition of the absolute zero, it will be seen that 
 i -+- at' and i + at bear the same relations to t' and / that 
 the absolute temperatures do to the temperatures on the 
 Fahrenheit scale ; therefore writing T' and r for the absolute 
 temperatures instead of i+at' and !-}-#/, we get the very 
 simple equation 
 
 or in other words, the product of the volume and pressure of 
 a portion of gas is proportional to the absolute temperature. 
 
 The above equation will be found of the greatest use in 
 solving all questions as to the varying volumes, pressures 
 and temperatures of a portion of gas when its condition at 
 any one temperature is given. 
 
 The same result might have been arrived at in a simpler 
 
 E 2 
 
52 The Steam Engine. 
 
 way by the mere inspection of the air thermometer, for this 
 latter is nothing more nor less than the record of a series of 
 experiments on the varying volume of a portion of gas when 
 the pressure remains constant and the temperature changes. 
 The varying volume is in fact the exact measure of the 
 varying temperature. Hence the product of the pressure 
 and volume is exactly proportional to the length of the 
 column of gas i.e. to the absolute temperature. Now there 
 is no reason why any one pressure should be chosen rather 
 than any other, hence the above statement is perfectly, 
 general. 
 
 The specific heat of gases. The specific heat of a gas, is, 
 in accordance with the definition on page 39, the ratio of 
 the amount of heat required to raise a given weight of it one 
 degree in temperature, to the amount required to raise the 
 same weight of water one degree. In heating a gas, it is 
 possible to do so when the volume is kept constant, or the 
 pressure constant, or partly in the one way and partly in the 
 other. The first two cases are the most important. It will 
 be seen that, if the mechanical theory of heat be true, it will 
 take more heat to raise the gas one degree in temperature 
 when the pressure is kept constant while the volume is 
 variable than in the reverse case. For, take a cylinder closed 
 at one end, having a section of one square foot, and confine 
 a cubic foot of gas in it, by means of a piston free to move. 
 The cubic foot of gas has then to sustain the pressure due 
 to the weight of the atmosphere plus the weight of the 
 piston. When, therefore, the inclosed gas is heated and 
 expands, the whole weight of the atmosphere and piston is 
 raised through a certain space, and work is done. Conse- 
 quently, the heat supplied to the gas is partly expended in 
 raising its temperature, and partly in doing external work. 
 Now, in the case of heating at constant volume, the weight 
 of the atmosphere is not raised at all, and no external work 
 is done, and therefore less heat is required in this than in the 
 former case. 
 
Specific Heat of Gases, 5 3 
 
 The ratio which these two specific heats of a gas bear 
 to each other can be easily ascertained. Reverting to the 
 cylinder, let the cubic foot of gas be at the temperature 
 of 32, and let it be heated till its volume is doubled. To 
 do this the temperature must be raised, according to Gay 
 Lussac's law, by 492-6. 
 
 Now, according to Regnault's experiments the specific 
 heat of gas say air is, when heated at constant pressure, 
 0-2 3 7, therefore the quantity of heat in thermal units re- 
 quired to effect the above operation is equal to the original 
 weight of the cubic foot of air at the atmospheric pressure of 
 147 Ibs. per square inch, and temperature of 32, mul- 
 tiplied by the rise in temperature, multiplied by the specific 
 heat. Now the weight of the cubic foot of air, undei 
 the above circumstances, is '0807 Ib. The rise of tempe- 
 rature is 492-6 and the specific heat is 0*2375 ; therefore, 
 the quantity of heat required is, 
 
 0807 x 492-6 x 0-2375=9-422 thermal units. 
 
 Now the external work done during the process is equal 
 to the weight of the atmosphere, viz. 147 Ibs. per square 
 inch, resting on the surface of the piston, viz. 144 square 
 inches, multiplied by the height through which it is raised 
 viz. one foot. 
 
 = 14-7 x 144 xi = 2 1 1 6-8 foot-pounds. 
 Now 2116-8 foot-pounds is equivalent to - - =274 
 
 thermal units. Therefore, of the above quantity of thermal 
 units 2-74 have been expended in doing external work, and 
 
 9-422 2-74=6-682 thermal units 
 
 represent the quantity of heat expended in raising the 
 temperature of the air, provided that no other effect has been 
 produced by the heat. 
 
 Now, before we can establish a ratio from the above data 
 between the two specific heats, we must ascertain whether 
 
54 The Steam Engine. 
 
 heat has been expended in any other way than in doing 
 external work, and in raising the temperature of the mass of 
 gas. It will be noticed, that the gas, at the end of the above 
 experiment, is in a different molecular condition to what it 
 was at the commencement, the particles composing the gas 
 being much further apart. It might be thought that part of 
 the heat was expended in effecting this molecular change. 
 Whatever maybe the case with other substances, it has been 
 experimentally proved by Joule that no heat is expended in 
 this way in the case of the permanent gases. The experiment 
 consisted in warming a mass of gas in a closed vessel, and 
 then, by opening a cock, allowing the confined and heated 
 gas to expand into another vessel, in communication with 
 the first, and which was in a condition of vacuum. In this 
 case the gas evidently expanded without doing any work ; 
 therefore, no heat was consumed in this way. At the 
 end of the experiment the temperature was found to be 
 unchanged, showing that no heat had been expended in 
 altering the distance apart of the particles. 
 
 Reverting now to the original experiment, we see that 
 the only effects produced when the gas was heated at con- 
 stant pressure were : i. The doing of external work; 2. The 
 raising the temperature of the mass of the gas ; and 3. The 
 further separation of the particles of the gas. We have 
 seen that the last effect required no expenditure of heat 
 at all, and consequently we may be sure that the above- 
 named quantity of 6*682 thermal units represents the 
 amount of heat expended in raising the temperature of the 
 gas. 
 
 If, now, the mass of gas be heated at constant volume 
 instead of at constant pressure, no external work is done, 
 and no separation of the particles takes place, and the whole 
 heat is expended in raising the temperature of the gas. The 
 quantity of heat required for this purpose is, as we have seen, 
 only 6*682 thermal units. The specific heat of the gas at con- 
 stant volume is, therefore, less than the specific heat at con- 
 
Specific Heat of Gases. 5 5 
 
 stant prebsure in the ratio of 6*682 to 9-422, and consequently 
 the numerical value of the specific heat at constant volume is 
 
 9^22 
 
 Our knowledge of the specific heats of various gases is 
 derived chiefly from Regnault's experiments, which were all 
 made upon gases at constant pressure. The experimental 
 determination of the specific heat at constant volume is a 
 very difficult operation, hence it is usually derived theoreti- 
 cally from the specific heat at constant pressure in the manner 
 explained above. Though the true value of the ratio of the 
 two specific heats has not been confirmed by direct experi- 
 ment, still it has been assumed in calculations, such as the 
 theoretical determination of the velocity of sound in air ; 
 and as the calculated velocity agrees practically with the result 
 of experiment, we may assume that the value for the ratio has 
 been indirectly confirmed by experiment, 
 
 THE EFFECT OF THE APPLICATION OF HEAT TO WATER. 
 
 Having thus briefly considered the effect of heat when 
 applied to gases, we must now consider its effect on water. 
 This latter is the more complicated subject of the two, but 
 is at the same time of the greater importance in the 
 study of the steam engine. 
 
 It will be convenient to commence with the solid form of 
 water or ice at the temperature of o F. In this condition 
 the specific heat is about 0-5, that is, it takes half as much 
 heat to raise a pound of ice one degree in temperature that 
 it does to raise the same weight of water by the same 
 amount. To raise the temperature of the ice up to 32 
 requires then 32 x 0-5 = 16 units of heat. At this tempera- 
 ture its volume compared with water at its maximum den- 
 sity is as 1-0908 to i. 
 
 If we continue to heat the pound of ice at 32 it will 
 begin to melt, but the temperature will remain stationary 
 
56 The Steam Engine. 
 
 till the whole of the ice is turned into water. To effect 
 this transformation 144 units of heat must be supplied, 
 equivalent to 144x772 = 111,168 foot-pounds of work. 
 In other words, it would require as much heat to raise 
 a pound of ice at 32 through a height of about 21 
 miles, as it does to convert it into water of 32. As 
 the temperature remains stationary during the melting 
 process, the question arises what becomes of the heat 
 which has been expended. The early discoverers of this 
 phenomenon, being unable to account for the heat thus 
 apparently lost, invented the theory that it had become 
 latent.., or concealed in the water, and, in accordance with 
 this theory, it was said that the latent heat of water was 144. 
 In accordance with the mechanical theory, it is recognised 
 that the heat thus expended is spent in doing internal work 
 on the particles of the ice, which results in their cohesion 
 being overcome so that the condition of the ice is changed 
 from the solid to the liquid state. We should say, there- 
 fore, that 144 units of internal work, or of latent work, 
 are done upon the ice in order to transform it into water. 
 The term, internal or latent work, is used in contradistinc- 
 tion to the external work, which, as we have seen before, 
 a body can perform when increasing in volume under the 
 influence of heat. 
 
 It should here be noticed how enormous are the internal 
 forces which have to be overcome in changing the molecular 
 condition of the ice from the solid to the liquid state. The 
 1 1 1, 1 68 foot-pounds of work necessary to effect the trans- 
 formation in a pound of the substance being equivalent to 
 about 595 tons raised one inch high. At the same time, 
 the volume of the water is slightly less than that of the ice 
 from which it was formed, being in the ratio of 1-0908 to 
 1-000127, thus showing that there is no great change in the 
 distance apart of the molecules. 
 
 If we continue to apply heat to the water, its effect is to 
 raise the temperature till the boiling-point is reached. This 
 
Application of Heat to Water. 57 
 
 point being 180 above the temperature of melting ice, the 
 number of units of heat required to accomplish this rise in 
 temperature would, if the specific heat were unity through- 
 out the whole process, be 180. As, however, the specific 
 heat is only unity at 39*1, and after that increases slightly 
 with the temperature, the actual number of heat units re- 
 quired has been found to be iSo'Q. During this rise in tem- 
 perature the volume of the water decreases slightly to 39 *i, 
 the point at which it is reckoned unity, and after that in- 
 creases with the temperature till it reaches 1*04315 at 212. 
 
 The further application of heat to the water will not 
 increase its temperature, so long as the pressure is that of 
 the atmosphere, but will only result in the formation of 
 steam of atmospheric pressure. In other words, the water 
 will boil and will continue to do so till it is all turned into 
 steam. To effect this change from water of 212 into steam 
 of the same temperature, 965 7 units of heat are required, 
 equal to 745,520 foot-pounds. That is to say, to turn a 
 pound of water having the temperature of 212 into steam of 
 atmospheric pressure, heat has to be supplied to it, equi- 
 valent to the work involved in raising the weight of water up 
 to a height of about 146 miles, or, in raising about 346 tons 
 a foot high. When the whole of the water is turned into 
 steam its volume is about 1650 times that of the water from 
 which it was formed, and in this state it is called dry satu- 
 rated steam, and in many of its qualities it resembles a gas. 
 Its temperature is the same as that of the water from which 
 it was formed, viz. 212. 
 
 If more heat be added, the pressure remaining that of 
 the atmosphere, the temperature of the steam will rise, and it 
 will become what is called superheated, which means that it 
 is of higher temperature than the water from which it was 
 formed. The specific heat of steam is only 0-4805, so that 
 for every unit of heat now supplied to it the temperature will 
 rise 2 -08, and the volume will also increase directly as the 
 absolute temperature. 
 
58 The Steam Engine. 
 
 Just as in the case of the liquefaction of ice, so with the 
 vaporisation of boiling water, the 9657 units of heat which 
 have been supplied for this purpose, and which produce no 
 effect on the thermometer, have all been expended in doing 
 work. Part of the work so. done is internal or latent work 
 expended in overcoming the molecular resistances of the 
 water, and part is expended in doing external work against 
 the pressure of the atmosphere. To make this point clear, 
 and to show how much heat is spent in each of these sorts 
 of work, suppose a cubic foot of water at the temperature of 
 212 to be inclosed in a cylinder of indefinite length and of 
 one square foot in section. Suppose the water to be covered in 
 by a piston without weight, and free to move. The pressure 
 of the air on this piston will be 147 Ibs. x 144=21 16*8. 
 The cubic foot of water weighs 62-42 Ibs. To turn it into 
 steam requires, therefore, 62*42 x 9657=60279 units of heat, 
 equivalent to 46,535,388 foot-pounds. At the end of the 
 process the piston is lifted 1,650 feet from the bottom of the 
 cylinder, or 1,649 ^ eet from its original position. The 
 external work done is, therefore, this height multiplied by the 
 pressure on the piston, or 1649x2116*8 = 3,490,603 foot- 
 pounds ; while the internal or latent work is equal to the 
 total minus the. external work = 46,535,388 3,490,603 
 =43,044,785 foot-pounds. The external is to the internal 
 work, therefore, in the ratio of i to 12*33 n early. 
 
 Suppose now the piston in the above experiment to be 
 loaded with some other weight in addition to that of the 
 atmosphere, say with another 147 Ibs. to the square inch, 
 and that heat be applied as before. The result would 
 be that the water would rise in temperature to nearly 249" 
 before steam began to form. When it did form, the steam 
 would have the same temperature as the water, viz., 249, 
 and the same pressure as the piston sustains, viz. 29*4 Ibs. 
 per square inch. When the water was all turned into steam, 
 the piston would have risen to a height of 858 feet above 
 the cylinder bottom, i.e. to 857 feet above its original po- 
 
Heat of Steam. 59 
 
 sition. The quantity of heat requisite to effect this trans- 
 formation is found by experiments to be 1157*85 units for 
 each pound of water, whereas in the last example only 
 965-7 + 180-9=1146-6 units were required, thus showing 
 an increase of 11*25 units. Also the way in which the heat 
 is expended is different. For instance, in the second example 
 the final temperature of the water is 249 instead of 212* 
 consequently, the heat expended in raising it from 32 
 is about 218*4 units against 180-9. The heat expended in 
 merely vaporising the water is 1157-85 218-4=939*45 in 
 the second example against 965-7 in the first, showing, there- 
 fore, a decrease of 26-45 units. Now, of these 939*45 units, 
 a certain quantity is spent in doing external work, against 
 the load on the piston. The whole heat thus expended on 
 the cubic foot of water =i 44 x 29-4 x 857 = 3,628,195 foot- 
 pounds = 4699- 7 thermal units. Consequently, the internal 
 
 or latent work done per pound of water = 939-45^-^ 
 
 = 864*15 units; and the ratio of the external to the inter- 
 nal work is i to n'47 instead of i to 12*33, as m tne fi rst 
 example. 
 
 We see, therefore, that by increasing the load on the 
 piston we have changed everything, viz. the temperature at 
 which the water boils ; the temperature, pressure, volume, 
 and consequently density of the steam ; the total heat 
 necessary to effect the change, and also the proportions of 
 the heat which are expended in raising the temperature of the 
 water, in vaporising it and in doing external and internal work. 
 
 The laws which regulate many of these changes are 
 not yet perfectly understood, and consequently at present 
 only empirical formulae are available to express them. The 
 formulae are founded upon the results of experiments which 
 have been carried out in the most exhaustive manner. The 
 results of these experiments are recorded in tables, so that 
 the student is, except for the purpose of analytical calculation, 
 rendered independent of the formulae. 
 
60 The Steam Engine. 
 
 Connection between pressure and temperature of steam. 
 The connection between the pressure and temperature of 
 steam was determined by Regnault, and the numerical 
 results are given in the Table, page 489, transformed into 
 English measure for every degree between 100 and 401 
 Fahrenheit. Regnault's experiments were made at pressures 
 varying from 3 Ibs. to 200 Ibs. per square inch. It will be 
 seen by studying the table that the pressure increases with 
 the temperature, but not in a uniform manner as in the case 
 of gas. For instance, starting at 212, the pressure is 147 
 Ibs. per square inch and the increment of pressure per degree 
 of rise of temperature is 0-29 Ib. At 300, however, the 
 pressure is 67-22 Ibs. and the increment of pressure per 
 degree is i Ib. ; while at 408 the pressure is 270-99 Ibs., 
 and the increment of pressure per degree is 3 Ibs. 
 
 Total heat of steam. The connection between the tem- 
 perature of the steam and the total quantity of heat required 
 to raise the water from 32 and vaporise it was also deter- 
 mined by Regnault, and the numerical results are given 
 both in thermal units and in foot-pounds in col. 4 of the 
 Table. It will be seen that the total heat increases with the 
 temperature, the rate of increase being about 0-305 of a 
 thermal unit for each degree above 212, so that if 1146-6 
 units is the total heat of one pound of steam at 212, and 
 if we want to know the total heat at any other temperature 
 /, it will be given by the expression 
 
 Total heat=ii46-6-p3o5 (/ 212). 
 
 Heat of vaporisation of steam. The heat of vaporisation, 
 as distinguished from the total heat, is easily calculated, if 
 we know the total heat, by subtracting from this latter the 
 number of units of heat required to raise the water from 32 
 to the boiling-point ; see col. 3 of the Table. The Table 
 shows that the heat of vaporisation diminishes as the tem- 
 perature of the steam increases, but not by a constant rate. 
 The rate of diminution increases with the temperature. 
 
Heat of Vaporisation. 61 
 
 When a table is not at hand, very approximate results can 
 be obtained by assuming that the rate of diminution is 071 
 of one thermal unit for every degree above 212. Thus for 
 steam of / temperature, 
 
 Heat of vaporisation = 9657 71 (/ 2 1 2 ), 
 
 9657 units being the heat of vaporisation of steam of 212. 
 
 Volume of steam. The heat spent in external work de- 
 pends of course on the volume which 'the steam occupies 
 when formed. Experiments have also been made upon this 
 point. The volume in cubic feet occupied by a pound of 
 water when turned into steam is called the specific volume of 
 the steam. The term relative volume is used to denote the 
 comparison between the volume occupied by the steam, and 
 that occupied by the water from which it is formed. 
 
 Connection between pressure and volume of steam. The 
 weight in pounds of a cubic foot of steam is called its density. 
 In the case of a gas the connection between the pressure, 
 the volume, and the density is, as we have seen, extremely 
 simple. The equation pv = constant, giving the connection 
 between pressure and volume, while the density is exactly 
 proportional to the pressure. In the case of steam, the re- 
 lationship is not so simple. No rational formula has ever 
 been devised to express the relationship, but experiments 
 have been made for each separate case, the results of which 
 are given in coL 5 of the Table. An empirical formula has 
 been given by Rankine, which very nearly gives the results 
 of the experiments on pressure and volume, and is of the 
 same form as pv = constant. Rankine's formula, connecting 
 the pressure and volume of steam, is as follows, 
 
 17 
 
 y 16 " = constant 
 
 where/ is the pressure in pounds per square inch, and v is 
 the volume in cubic feet, the value of the constant being 
 475- 
 
62 The Steam Engine. 
 
 External work done during vaporisation of water. This 
 formula enables the external work done during the vaporisa- 
 tion of water to be calculated, but except where it is neces- 
 sary to use a formula in analytical investigations, the figures 
 are best taken from the Table by multiplying the volume 
 as given in col. 5 by the pressure per square foot. It will 
 be noticed on studying the results that the external work 
 done increases slowly with the temperature, but not by a 
 uniform rate of increase. The rate diminishes as the tem- 
 perature rises. 
 
 Internal work done during vaporisation. The heat ex- 
 pended in doing internal or latent work during vaporisation, in 
 altering the molecular constitution of the water, is the differ- 
 ence between the heat of vaporisation and the heat expended 
 in doing external work. It may be deduced from the Table 
 by subtracting the external work, plus the heat expended 
 in raising the temperature of the water as given in col. 3 
 from the total heat as given in col. 4. It diminishes with 
 the temperature by about 0792 unit for every degree. 
 
 The heat necessary for turning water of 32 into steam, 
 at constant pressure, is expended in the three following ways, 
 which must be kept distinct from one another. 
 
 1. In raising the temperature of the water from 32 to 
 the temperature of the boiling-point, which last depends 
 upon the pressure. 
 
 2. In changing the physical constitution of the water 
 from the liquid state to the condition of steam. This is 
 what has been called above internal or latent work. 
 
 3. In doing external work, by overcoming the resistance 
 of the atmosphere, or other external resistance through a 
 certain space, corresponding to the volume which the steam 
 occupies at the particular pressure. 
 
 It should here be noticed that when steam is formed 
 in a boiler, in connection with a non-expansive engine at 
 work, it is generated under the condition of nearly constant 
 pressure ; the piston which is constantly moving backwards 
 
Heat expended in making Steam. 
 
 and forwards in a cylinder which is in communication with 
 the boiler, corresponding to the piston, in the example given 
 above, while the forces which in a steam engine oppose the 
 motion of the piston correspond to the weights placed upon 
 the piston in the example. The 
 case of steam formed in a close 
 vessel is different, for here no 
 heat has to be expended in doing 
 external work, for by the nature 
 of the case none can be done. 
 
 The relative proportions of 
 the three separate quantities of 
 heat necessary to raise a pound 
 of water from 32 to boiling tem- 
 perature, and then to evaporate 
 it, may perhaps best be exhibited 
 by a graphical diagram. Draw 
 a line OX (fig. 13) along which 
 to measure the volume of one 
 pound of water when turned into 
 steam. Suppose the water as 
 before contained in a cylinder 
 having a section of one square 
 foot. 
 
 Draw a line OY along which 
 to measure the total pressure 
 in pounds on the piston. Let 
 us take for the first illustration 
 steam of 30 Ibs. pressure to the 
 square inch, the temperature of 
 which, according to the Table, 
 is about 250, and the specific 
 volume 13*49 feet. Now the original volume occupied by 
 the pound of water is o'oi6 cubic foot, therefore the space 
 through which the steam lifts the piston when doing work 
 is 13-49 016=13-474 feet. Measure off a length OA to 
 
 Fig. 13- 
 
64 TJie Steam Engine. 
 
 scale along OX to represent 13*474 feet, and a length OB 
 along OY to represent to scale the pressure on the piston 
 = 144 x 30=4320 Ibs. Then the exterior work done by the 
 steam when formed is =4320x13*474=58207-68 foot- 
 pounds, and this is represented on the diagram by completing 
 the rectangle AB, the area of which is of course OA x OB, and 
 which therefore represents the above number of foot-pounds. 
 Similarly the heat spent in internal work and in raising the 
 temperature of the water may be represented by the areas of 
 rectangles. For the sake of comparison these rectangles 
 should have the same base OA as the original rectangle, and 
 should be drawn below OX. Now the heat spent in internal 
 work may be calculated from the Table to be 680,021 foot- 
 pounds, which is about 1 1 *6 times as much as the external 
 work. Draw therefore OC downwards at right angles to 
 O A and in length n -6 times as much as OB. The area of 
 the rectangle AC will then represent the heat calculated in 
 foot-pounds expended in internal work. 
 
 Similarly the heat expended in raising the temperature 
 of the water from 32 to 250 can be represented. This 
 heat is 219-5 thermal units=i69,454 foot-pounds, which is 
 about 2-91 times as much as the heat spent in external work. 
 From C therefore draw CD downwards, in length equal to 
 2-91 times OB, and complete the rectangle. Its area will then 
 represent the amount of heat calculated in foot-pounds re- 
 quired to raise the temperature of the water from 32 to 250. 
 
 An inspection of this diagram will show what a very 
 wasteful kind of steam engine such a cylinder and piston 
 would constitute ; for the whole of the work done by the 
 heat expended is represented by the rectangle AB, while the 
 whole heat supplied is represented by the big rectangle BE, 
 which is 15*51 times the area of AB. Therefore for every 
 15-51 units of heat supplied to such an engine only one 
 unit can possibly be expended in doing useful work. 
 
 By constructing, with the aid of the tables, similar 
 diagrams for every pressure of steam, we should be able to 
 
Isothermal Lines. 65 
 
 see at a glance the proportion between heat supplied and 
 useful work done. 
 
 EXPANSION OF GAS AND STEAM. 
 
 Isothermal lines. Boyle's law, connecting the volume 
 and pressure of gas, viz. pv=- constant, assumes that during 
 the variation of pressure and volume the temperature re- 
 mains constant. Suppose a portion of air or gas inclosed 
 in a cylinder, provided as before with a movable weighted 
 piston. The inclosed gas would attain a certain definite 
 volume, pressure, and temperature, the pressure being, of 
 course, in equilibrium with the weight of the loaded piston. 
 If now the load on the piston be diminished exactly as in 
 the example on page 29, the gas will expand, raising the 
 remaining weights through a certain space, and conse- 
 quently doing external work. This work is done at the 
 expense of the heat contained in the gas below the piston. 
 The result will be that the temperature of the gas will fall 
 by an amount which can be easily calculated when we 
 know the quantity of external work done and the specific 
 heat and the weight of the gas. Such expansion, then, 
 does not fulfil the condition laid down in Boyle's law, that 
 the temperature should remain constant. 
 
 In order that this latter condition should hold, heat 
 must be supplied to the gas from some external source. It 
 was shown before, that when the pressure and volume vary 
 according to Boyle's law, the different states of the substance 
 as regards pressure and volume for any given temperature 
 may be represented graphically by the ordinates of a rect- 
 angular hyperbola. Such a line is said to be an isothermal 
 curve of expansion, or, shortly, an isothermal ; so called from 
 two Greek words which signify equal and temperature, 
 because the temperature is supposed to remain the same 
 throughout all the changes in pressure and volume indicated 
 by the co-ordinates of the curve. There is, of course, a 
 separate isothermal for every temperature ; for, with a given 
 
 F 
 
66 
 
 The Steam Engine. 
 
 mass of gas, the variations in pressure and volume are 
 different for different temperatures, though following the 
 same law. For instance, if, at any given volume, the tempe- 
 rature is in one case 32, and in another 100, the pressure 
 will be greater in the second than in the first instance, in 
 accordance with Gay Lussac's law. 
 
 If we have a series of isothermal lines drawn to scale, as 
 in fig. 14, for a portion of any gas, such as air, we can 
 
 Fig. 14. 
 
 immediately find out by simple measurement either the 
 temperature, the pressure, or the volume, when any two of 
 these puantities are given. Each isothermal should be 
 marked with the degree of temperature for which it is drawn. 
 Suppose in the figure that there is a separate line drawn for 
 each degree, and suppose that lines measured parallel to O/> 
 represent pressures, and those parallel to Qv volumes. Let 
 1 Figs. 14 and 15 are taken from Clerk Maxwell's Theory of Heat. 
 
Isothermals of Steam. 67 
 
 the temperature 4, and volume equal to OL, be given, and 
 let it be required to find the pressure, we have simply to 
 draw an ordinate LP vertically upward, till it meets the iso- 
 thermal for 4, then LP will be the required pressure. 
 
 The isothermals of dry saturated steam are very different 
 to those of a gas. Suppose that a pound of water in a 
 cylinder closed by a piston be turned into steam of 
 atmospheric pressure, and that the piston be then pressed 
 down, while the temperature is maintained at 212, the 
 pressure will not rise at all, while the volume will diminish, 
 
 and, to permit of this taking place, part of the steam will be 
 converted back into water. The reason of this is that dry 
 saturated steam at a given temperature, say 212, can exist at 
 no higher pressure than the natural pressure of formation at 
 that temperature, as shown by Regnault's tables. Take, for 
 instance, steam of 341 This is the temperature at which 
 the steam forms when the pressure is 120 pounds on the 
 
 1-2 
 
68 The Steam Engine. 
 
 square inch, and it can exist at no higher pressure so long 
 as the temperature remains the same. If, however, instead 
 of pressing the piston down, it were raised up, the volume 
 of the steam would increase, and if the temperature were 
 maintained constant the pressure would diminish as the 
 volume increased, but not strictly in accordance with Boyle's 
 law; that is, as has been before explained, the product/^ 
 would not be quite constant, though nearly so ; and the 
 isothermal curve would consequently not be a perfect 
 hyperbola. Fig. 15 illustrates the isothermal lines of steam 
 and water. Take for instance the full line cba which is the 
 isothermal for the temperature 212; we see that as the 
 volume is enlarged the pressure diminishes as the ordinates 
 of the curve ba, which is not a hyperbola. When, however, 
 the mass of steam is compressed from the point b, the tem- 
 perature remaining constant the isothermal is a straight line 
 parallel to the base <??', showing that the pressure remains 
 unaltered as the volume is diminished. The isothermal for 
 the temperature 302 is shown by the full \\\\efed. The 
 dotted line shows the pressures and volumes at which con- 
 densation commences for the temperatures marked on the 
 left hand side of the diagram. 
 
 Adiabatic lines. In what has been said above regarding 
 variations in the pressure and volume of gas and steam, 
 the only condition observed when determining the shape of 
 the curves was that the temperature was to be maintained 
 constant throughout the changes. If, however, a portion of 
 gas be inclosed in a cylinder as before and the cylinder and 
 piston be supposed to be absolute non-conductors of heat, 
 and to be also incapable of communicating any heat of their 
 own to the gas, and if the pressure be then made to vary, 
 the temperature will fall when the volume increases ; for 
 external work will be done, and, as this work can only be 
 done at the expense of the heat contained in the gas, its tem- 
 perature must fall. Now the pressure of the gas, other 
 things being equal, depends on the temperature ; conse- 
 
Adiabatic Lines. 69 
 
 quently, when a gas expands in a non-conducting cylinder 
 doing external work, its pressure will be less for a given 
 volume than when it expands isothermally to the same 
 volume. 
 
 Let us now, instead of allowing the gas to expand in the 
 cylinder, compress it by adding weights to the piston. The 
 piston and weights, when closing in upon the gas, do work 
 upon it, and consequently raise its temperature. As the 
 cylinder is non-conducting, none of the heat represented by 
 this increased temperature can escape, and its effect conse- 
 quently is to increase the pressure of the gas to a greater 
 degree than would be the case if the compression took place 
 isothermally. 
 
 The effect of this difference in the conditions can easily 
 be exhibited graphically. Take two cylinders of equal size 
 and inclose in each of them an equal portion of gas, so that 
 in their initial state the volumes, pressures, and temperatures 
 shall be the same. Let the ordinates of the point of inter- 
 section of the two curves on 
 fig. 1 6 represent the initial 
 pressure and volume of each 
 portion of gas. Let one of 
 the cylinders be a non-con- 
 ductor and in the other let the "*** isothermal 
 gas change its volume iso- 
 thermallv. In the latter case 
 
 .' . . Fig. 16. 
 
 the variations in its pressure 
 
 and volume will be represented by the isothermal curve or 
 hyperbola. If, however, the gas be prevented from re- 
 ceiving or parting with heat 'by the non-conducting cylinder, 
 as it expands from the point of intersection, the fall in 
 the temperature will cause the pressure ordinates at each 
 point to be less than the corresponding ordinates of the 
 isothermal, and the new curve will consequently fall below 
 the hyperbola. When, on the other hand, the volume is 
 diminished, this can only be accomplished by doing work 
 
The Steam Engine. 
 
 upon the gas, and, as no heat can escape, the temperature 
 rises, and the pressure ordinates will consequently be greater 
 than in the first case, and the curve will lie above the 
 hyperbola. The second curve is represented on the diagram 
 by the second dotted line. It is called an adiabatic line in 
 contradistinction to the isothermal. The term adiabatic is 
 derived from two Greek words signifying not passing through, 
 and the line is so called because, in the operation above 
 described, no heat passes through the cylinder either in 
 or out. 
 
 Adiabatic lines of steam. The adiabatic lines of dry 
 saturated steam differ essentially from the corresponding 
 lines for gases. Take a given mass of steam at the pres- 
 
 Fig. 17. 
 
 sure of the atmosphere, and the corresponding temperature 
 Then, as we have seen, if the volume of the steam 
 
 2I2 V 
 
 be diminished and the temperature preserved at 212, the 
 pressure will remain constant, and part of the steam will 
 be changed back into water, because steam at the tempe- 
 rature 212 can exist at no higher pressure than that of 
 the atmosphere. The isothermal will be then, as before 
 explained, a horizontal straight line A#, fig. 17, while the 
 volume is being diminished from the point A; and in order 
 
Adiabatics of Steam. 71 
 
 to effect such a diminution and keep the temperature 
 constant, heat must be abstracted from the steam. If, 
 however, the compression be effected adiabatically, so that 
 no heat can escape, the work done upon the steam by the 
 act of compressing it will raise its temperature above 212, 
 and consequently enable it to exist at a higher pressure ; and 
 the more it is compressed, the greater will be the increase in 
 the temperature, and consequently in the pressure. The 
 adiabatic line therefore, during the diminution of volume of 
 dry saturated steam, will be a curved line Ac, resembling 
 very closely the corresponding line for a gas. It has been 
 shown by Rankine and Clausius that if the cylinder, in addi- 
 tion to containing the steam, held also some water at the 
 same temperature, the heat generated by the compression 
 is sufficient to cause some of this water to become steam. 
 
 If we now return to the point A, and allow the volume to 
 increase, so that the steam does external work, without 
 gaining or losing any additional heat by conduction from 
 outside bodies, the temperature will fall, for the external 
 work is done at the expense of the heat contained in the 
 steam. The heat will diminish so much that when the 
 pressure is reduced to any given extent there will no longer 
 be the quantity of heat present in the mass of steam, neces- 
 sary, according to Regnault's experiments, to maintain it in the 
 dry saturated condition (see p. 60 and the Table, p. 489), 
 and consequently part of the steam will condense back into 
 water, and in so doing will part with that portion of its heat 
 which effected its transformation from water into steam. 
 The heat thus liberated will maintain the remainder of the 
 steam in the dry and saturated condition. Consequently, 
 during expansion from the point A, the volume for a given 
 pressure will be less being reduced by the condensation 
 of so much steam than if the whole of the steam were 
 maintained throughout in the dry and saturated condition, 
 and less still than if the temperature were maintained 
 uniform throughout. 
 
/ 2 The Steam Engine. 
 
 The above example is applied to steam of the atmo- 
 spheric pressure and corresponding temperature, but any 
 other temperature might have been chosen, and the same 
 reasoning would have applied. Just as in the case of the 
 isothermals, so with the adiabatic lines, a separate one can 
 be drawn for every separate degree. 
 
 We have now examined into the nature of the expansion 
 lines of gases and steam for two separate cases, viz. first, the 
 case of the temperature being maintained constant through- 
 out the change, and second, the case of no heat being 
 allowed to escape from or reach the gas and steam by 
 conduction, radiation, &c., from or to other matter. 
 
 It is evident, however, that these are not the only 
 possible cases, for we might, had we wished, have supplied 
 or abstracted any quantity of heat we chose, to or from the 
 gas, during the process of alteration of volume and pressure, 
 and thus have made the shapes of the curves ot expansion 
 anything we pleased. The two cases above described are, 
 however, the most important. 
 
73 
 
 CHAPTER III. 
 
 THEORETICALLY PERFECT HEAT ENGINES. 
 
 Application of Boyle's and Charles' laws to gases Specific heat of gases' at 
 constant pressure and at constant volume Cycle of operations Ratio 
 of heat expended to work done Graphic representation of external 
 work done during the expansion of a gas Nature of the curves of 
 expansion of gases as influenced by the supply of heat Heat supplies : 
 (i) when the curve of expansion is a rectangular hyperbola, the equa- 
 tion for which is pv c\ (2) when the equation of the curve takes the 
 form />" = , where n has any value except unity Nature of the curve 
 when no heat is supplied or abstracted The ideally perfect heat engine 
 Calculation of the efficiency of such engines The reversed action 
 of the ideal heat engine Proof that no other engine has a greater 
 efficiency than the ideal heat engine Practical limits of efficiency in 
 the ideal heat engine Laws connecting the pressure, temperature, arid 
 volume of dry saturated steam Specific heats of water and steam 
 Law connecting the pressure, volume, and temperature of superheated 
 steam Total heat expended in converting water into steam Propor- 
 tion of total heat expended in doing external and internal work- 
 Expenditure of heat in a steam engine when the steam is not used 
 expansively Method of representing heat by an ' equivalent pressure ' 
 Expenditure of heat in a steam engine when the steam is used ex- 
 pansively, ist, when the curve of expansion is a rectangular hyperbola, 
 2nd, when the steam remains dry and saturated throughout whole 
 stroke To realise latter condition steam jacket is necessary Rankine's 
 formulas for the expenditure of heat in a steam engine Theory of the 
 ideally perfect heat engine applied to steam How actual steam en- 
 gines differ from the ideal heat engine Summary of laws and formula?. 
 
 THE last chapter contained a sketch of the principles of the 
 science of heat and an account of the effects of heat upon 
 gases and water. The present chapter will deal with the 
 conversion of heat into mechanical work through the instru- 
 mentality of heat engines, and will contain an account of an 
 ideal heat engine which is perfect in theory ; that is to say, 
 no other conceivable engine can get more work out of the 
 heat supplied to it than the one about to be described. 
 Practical difficulties render the realisation of such an engine 
 
74 The Steam Engine. 
 
 impossible, but the study of it is nevertheless of the greatest 
 importance, as enabling us to find out the deficiencies of 
 existing engines, and to ascribe to each of these deficiencies 
 its due share in causing waste of heat. 
 
 On account of the greater simplicity of gas, it will be 
 found convenient, first to describe the mode of operation of 
 the ideal engine when worked with gas or air, and afterwards 
 to apply the results obtained to the case of steam. Before 
 doing so, however, it will be necessary to recapitulate the 
 laws affecting gases which were explained in the last chapter, 
 but with greater numerical exactitude, and then, from these 
 laws, to make certain deductions, which, as they refer to the 
 power of doing work through the medium of gases, are 
 commonly classed under the head of the Thermodynamics 
 of Gases. 
 
 Numerical application of Boyle's law to gases The first 
 of the laws referred to is Boyle's law, connecting the pressure 
 and the volume of the gas when the temperature is main- 
 tained constant. The algebraical expression for this law was 
 shown to be /?>=<:. 
 
 If one pound's weight of air be taken, at the pressure / 
 of the atmosphere, equal to 2116-8 Ibs. on the square foot, 
 at the temperature 32, then the volume of this pound of air, 
 or ^ , multiplied by the pressure on the square foot has been 
 proved by Regnault's experiments to be 
 
 26,2i4 foot-pounds. 
 
 This quantity, 26,214 foot-pounds, is therefore the value of 
 the constant c, so long as the temperature remains 32. 
 
 If the temperature be changed, the value of the constant 
 is changed also. This leads us to Gay Lussac's law (see 
 page 46) connecting the pressure and volume with the 
 temperature. This law states that the product pv is in- 
 creased when the temperature is raised from 32 to 212, 
 in the ratio of i to 1-3654 ; and that for each of the 180 
 degrees intermediate between 32 and 212 the increase is 
 
Boyle's Law applied to Gases. 7 5 
 
 T-J-oth part of the increase at 212. If then p Q v Q be the 
 pressure and volume at 32, and pv be the pressure and 
 volume at any other temperature f, then if /=2i2, 
 
 and if / be any other temperature then 
 
 This rate of increase of course applies also when the tem- 
 perature is raised above 212. 
 
 It was also shown (see page 51) that if the tempera- 
 ture be reckoned from the bottom of the tube of the air 
 thermometer, which was shown to be 492 below 32 
 Fahrenheit, the above law could be greatly simplified. 
 
 For, the product of the pressure and volume of a portion 
 of gas is proportional to the absolute temperature, so that if 
 r be the absolute temperature corresponding to f* then, 
 remembering that 492-6 absolute, corresponds to 32 on 
 the ordinary scale, and attaching the same values as before 
 to all the other symbols, we have 
 
 pv \ T '.' P^VO '. 492*6 
 
 A p v r A)? ; o 
 492-6. 
 o?' as stated above =26,2 14. 
 
 493 
 
 Hence we get /z / =53' 2 - r > 
 
 which is a very simple expression, connecting the pressure, 
 the volume, and the absolute temperature. 
 
 Specific heat of gases at constant volume^ and at constant 
 pressure. The next law, which is now mentioned for the first 
 time, relates to the specific heat of gases, and asserts that, if 
 a gas be heated at constant pressure, it requires the same 
 quantity of heat to raise its temperature from any point, say 
 212 to 213, as it does from any other point, say 32 to 
 33. In other words, the specific heat of a gas at constant 
 
76 The Steam Engine. 
 
 pressure does not change with the temperature, as is the case 
 with water. 
 
 The capacity of air for heat, that is, the amount of heat 
 required to raise one pound of it through i of temperature, 
 the pressure being maintained constant, is, according to 
 Regnault's experiments, 0-2375 thermal units, equal to 
 J^S'SS foot-pounds. This quantity of heat is, as has been 
 shown before, not all expended in merely raising the tem- 
 perature of the air ; for, the heating having been accom- 
 plished at constant pressure, part of the heat has been 
 spent in doing external work. 
 
 Let #i,"i, ?'2> T 2 be the original and final volumes and 
 absolute temperatures of a pound of air ; and let p be the 
 pressure which remains constant. Then the external work 
 is measured by the increase in the volume, viz. ?' 2 v\ 
 multiplied by the pressure p ; therefore 
 
 External work=(?; 2 v^p ; 
 and, as we have seen, ^=53-2.7 ; therefore 
 
 The external work 53*2 (r 2 T } ). 
 
 Also the total heat expended equals the specific heat multi- 
 plied by the number of foot-pounds in one thermal unit 
 multiplied by the number of degrees of rise of temperature. 
 The usual symbol for the specific heat at constant pressure 
 multiplied by the number of foot-pounds in one thermal 
 unit is K p ; l and as the rise in temperature is r 2 r l5 we have 
 
 Total heat expended =Kp(r 2 TJ). 
 
 Hence the heat expended in doing internal work that is, in 
 merely raising the temperature of the air is the difference 
 between the total heat expended and that part which is spent 
 in doing external work 
 
 1 K p and K v are spoken of hereafter for the sake of brevity and in 
 accordance with a usual custom as specific heats ; but in reality a 
 specific heat is only a ratio, whereas K p and K v are absolute quantities. 
 
Specific Heat of Gases. 77 
 
 Therefore the internal work = (r 2 r,) (K p 53*2). 
 
 Now this latter quantity within the right-hand bracket is 
 also the value of the specific heat of air when heated at 
 constant volume ; because, as we know by Joule's experiment 
 (see page 54), the mere separation of the particles of air 
 requires no heat to effect it when no external work is done, 
 and as the heat is only expended in doing external and 
 internal work, and as, moreover, when the air is heated at 
 constant volume no external work is done, therefore the 
 specific heat of air heated at constant volume is the same 
 as the internal specific heat at constant pressure, and, calling 
 the specific heat at constant volume K v , we have 
 
 K V =K P 53'2 = i3o-25 foot-pounds. 
 
 Consequently the heat required to raise the temperature of 
 one pound of air at constant volume from rj to r 2 is 
 
 From the equation K V = K P 53*2 we get, by simply 
 transposing, K p K v = 53'2. That is to say, the difference in 
 the two specific heats of air is equal to the constant quantity 
 53-2, which, as we have seen before, when multiplied by the 
 absolute temperature, equals the product pv. 
 
 From the result given above for the value of the heat 
 expended in internal work, when the air was heated at con 
 stant pressure, viz. (K p 5 3 *2 ) (r 2 r j ) = K v (r 2 r \ ), we see 
 that the internal work is proportional to the change of 
 temperature, and is equal to the change of temperature 
 multiplied by the specific heat at constant volume. 
 
 This result is true whether the air be heated at constant 
 pressure, or at constant volume, or partly in the one way 
 and partly in the other, or in fact in any way we can con- 
 ceive of. For, as an example, first change the air from 
 volume 7 1 ,, and temperature r,, to volume v.>, keeping the 
 pressure constant at/ t ; let the new temperature be r ; then 
 by the above the heat expended in internal work is K v (r TJ). 
 
78 The Steam Engine. 
 
 Next change the pressure from p to / 2 , the volume being 
 kept constant at v z . To do this we must add heat to 
 the gas, raising its temperature to r 2 ; the heat spent is 
 K v (r 2 r), which is all internal work ; adding to this the 
 heat spent in doing internal work during the first part of 
 the operation, we get 
 
 Total heat spent in internal work = K v (r r { + r 2 -) 
 = K v ('- 2 -r 1 ). 
 
 This result might be proved to be true for any other case 
 which might arise, by similar reasoning to the above, but it 
 may also be shown to be generally true from the following 
 considerations. 
 
 Cycle of Operations. If a substance such as gas or water 
 be subjected to the action of heat, and be thus brought 
 through a series of changes of state, and eventually brought 
 back to its original condition, it is said to have undergone a 
 Cycle of Operations. During these changes of state heat has 
 been expended in doing two things only, viz. external work, 
 and internal work of various sorts, such as altering the 
 temperature or the molecular condition of the substance. 
 When, however, the body is brought back to its original 
 condition, the sum of all the quantities of heat expended in 
 doing internal work must be nil, because if during one part 
 of the operation heat has been thus expended, when the 
 substance is brought back to its original condition this heat 
 is again liberated or rejected. 
 
 Now when the state of gas or air is changed by the 
 action of heat in any way whatever we can analyse the ope- 
 ration into three distinct sets of processes, viz., 
 
 i st. Heating at constant pressure, the volume being 
 changed ; 
 
 2nd. Heating at constant volume, the pressure being 
 changed ; and 
 
 3rd. One or more cyclical processes. 
 
 Now during the latter processes no heat is spent in 
 
Cycle of Operations. 79 
 
 internal work, and during the two former the heat thus spent 
 is as we have seen = K v (r 2 T } ). Hence the proposition is 
 universally true that when the state of a gas is changed by 
 the action of heat, the quantity of heat spent in doing internal 
 work depends only on the difference of temperatures of the 
 two states, and is equal to the specific heat at constant 
 volume multiplied by this difference of temperatures. 
 
 From the above fundamental laws we are enabled to 
 reason on all the questions which may arise regarding the 
 thermodynamics of gases. All that we require to know is, 
 how much heat is expended in doing external, and how 
 much in doing internal work. The total heat expended is 
 equal to the sum of these two quantities. When we possess 
 this information we can deduce all that it is requisite to 
 know regarding the pressure and temperature at every stage 
 of the process. Conversely if we know the pressure and 
 temperature we can calculate the external and internal work 
 done, and the expenditure of heat. The internal work is, 
 as has been proved above, always given by the expression 
 K v (r 2 T } ). The external work is different in different cases. 
 For instance, if during the changes of volume and pressure 
 sufficient heat be supplied to keep the temperature uniform, 
 we get a certain quantity of external work. If on the 
 contrary no heat be supplied we get quite another quantity, 
 and if more than enough or less than enough heat be sup- 
 plied to keep the temperature uniform, we get still different 
 quantities of external work done in each case. 
 
 The quantity of external work done is perhaps best 
 calculated and exhibited by means of diagrams. We have 
 seen (see page 44) how the varying pressure and volume of 
 a portion of gas can be represented by the ordinates of a 
 line GD, fig. n. We also saw (see page 63) how work done 
 could be represented by the area of a rectangle. An exten- 
 sion of these methods will now be explained. 
 
 Let ac, cO, fig. 18, represent the initial, and &/, dQ the 
 final pressures and volumes of a portion of gas. Let the 
 
8o 
 
 The Steam Engine. 
 
 intermediate co-ordinates of the curve ab represent the inter- 
 mediate pressures and volumes while the gas is expanding. 
 Draw a line ef, indefinitely near and parallel to the line ac. 
 While the volume of the gas has been increasing from O<r to 
 Of, the pressure has been falling from ado ef. Now the ex- 
 ternal work done is represented by the increase in volume cf, 
 multiplied by the pressure. The pressure in this case is not 
 uniform, but- decreases as the ordinates of the curved line 
 ae. We must therefore multiply the increase of volume by 
 the average pressure. It is difficult to find the average 
 pressure when the line ae is curved ; but if we take ef as 
 being very near indeed to ac t we may regard ae as being to 
 
 o. 
 
 Fig. 18 
 
 all intents and purposes a straight line, and in this case the 
 average pressure will be represented by the mean between 
 
 the two lines ac and ef, viz. - -5^ The external work 
 
 done is therefore represented by the expression cfx 
 
 But this expression also represents the area of the strip aefc, 
 therefore the external work done while the volume of the 
 gas is increasing from Oc to Of is represented by the area 
 aefc. We can divide the whole figure abdc up into a series 
 of such strips, and the above reasoning would hold good for 
 
GrapJtic Representation of Work. Si 
 
 each of them. Now the sum of tiie areas of these strips 
 equals the area of the figure abdc ; therefore when the 
 volume of a portion of gas increases from Qc to O*/, the 
 pressure at the same time varying as the vertical ordinates 
 of the line ab, the external work done during the process is 
 represented by the area inclosed by the base line cd repre- 
 senting the increase of volume, the vertical lines ac and bd 
 representing the initial and final pressures, and the line ab. 
 which represents the way in which the pressure varies. 
 
 The line ab may be anything that we please. For 
 instance, if during the expansion of the gas enough heat 
 were supplied to it to keep the pressure uniform throughout, 
 it would be a straight line drawn through a, parallel to ov. 
 If enough heat were supplied to keep the temperature uni- 
 form, the line would, as has been proved, be an isothermal, 
 which for gases is a common rectangular hyperbola denoted 
 by the equation pv = constant. 
 
 If no heat were supplied and none allowed to escape, the 
 line would be an adiabatic, the equation for which will be 
 
 shown to be piV = constant. The symbol y- which is of very 
 
 ^ 
 constant occurrence denotes the ratio /. 
 
 K* 
 
 Most of the lines occurring in the theory of the heat 
 engine are denoted by the equation pv n = constant, where 
 the index n varies according to the supply of heat. For 
 instance, the two preceding cases are special instances of this 
 equation in the first of which =i, and in the second =y. 
 
 The area of the figure abdc depends, of course, upon the 
 special form of the line ab, and can be readily calculated for 
 each case by those who are familiar with the processes of 
 analytical geometry. 
 
 We can now examine into the most important cases 
 that arise. 
 
 i. Heat expended in changing the state of a gas when 
 the temperature remains constant throughout the change. 
 The total heat expended = the internal work done + the 
 
82 The Steam Engine. 
 
 external work done. The internal work in this case is nil, 
 because the temperature does not change, and consequently 
 the expression K v (r 2 T l ) vanishes. 
 
 The external work is obtained by calculating the area of 
 the diagram abdc (see fig. 18). We assume that the initial 
 pressure and volume are represented respectively by the line 
 ac, drawn to scale so as to represent pounds on the square 
 foot, and Oc drawn to represent cubic feet. Similarly the 
 lines bd and Od represent the final pressure and volume to 
 the same scale. As the temperature is uniform the line ab 
 is a rectangular hyperbola, having for its asymptots OV and 
 OP; 
 
 and the area 1 of the figure abdc=acxQcx\o^ . 
 
 Also, as by the principle of the hyperbola ac xO = 
 bdxOd 
 
 .'. the area is also = bdxQd x log e . 
 
 It will be noted that the logarithms used are hyperbolic. 
 A table of the hyperbolic logarithms of the most useful 
 numbers will be found at the end of the book (see page 498). 
 
 1 The area is calculated in the following manner : 
 Let ab, fig. 18, be a curve of the equation. 
 
 pv = constant. 
 
 The area abdc, is the sum of a number of small stiips such as af. 
 By making these strips indefinitely narrow they may each be repre- 
 sented by the expression pxdv, where p represents the height, and 
 ' dv the indefinitely small width. 
 
 Let a c=pi\ Oe = v l and bd=p^; Od = v. i . 
 
 Then the area= \ V J p.dv. 
 Now as p v -p l v } 
 
 and substituting 
 
 v 
 
 =/i Vi (log* g - 
 
Expenditure of Heat during Expansion. 83 
 
 The ratio of the final volume Od to the initial volume Of is 
 called the ratio of expansion, and is generally denoted by the 
 symbol r. Calling the initial and final volumes v l and z> 2 , 
 and the initial and final pressures^ and/ 2 respectively, the 
 expression for the area becomes p^v^ log e r^ or / 2 z/ 2 lge r 
 foot-pounds. Also if T be the absolute temperature of the 
 gas, then, as we have seen, ^ 1 ^ 1 =^ 2 z; 2 =^r, 
 
 .*. the area=<rr log e r foot-pounds. 
 
 This quantity is, therefore, the expenditure of heat in 
 foot-pounds when a gas expands isothermally from volume 
 v\ to z> 2 . 
 
 2. Let the curve ab, instead of being an hyperbola of the 
 equation pv=. constant, be a curve of the form 
 
 /# n = constant, 
 
 where n may have any value we like to assign to it except 
 unity. In this case the area ! of the figure abdc 
 
 n i 
 
 1 The area is calculated in the following manner : 
 Let a I), fig. 1 8, be a curve of the equation 
 
 pv* = constant. 
 To find the arc abdc 
 
 Leta=/j; Oc = v l and bd=p.j,', Od=v 2 ; 
 also let / be any pressure ordinate, and v the corresponding volume. 
 
 Then the area= f^ 2 p.dv. 
 Now as p v n =/! vf 
 
 and substituting 
 
 Area= ; 
 
 ni 
 
 v\-piVi m 
 n-i 
 
 G 2 
 
84 The Steam Engine 
 
 Let r, be the initial, and r 2 the final absolute tempera- 
 tures. The expression for the area becomes then 
 
 fr, <rr. 2 _ 
 
 n i ^ i 
 
 As before, the heat expended=the internal work + the 
 external work 
 
 Also, as has been proved before, <r=K / K w ; 
 
 / \ 
 
 (r 2 -r,) ; 
 
 /. substituting this value of ( r i Te i) i n the above 
 
 n i 
 
 equation, we get heat expended 
 
 3. Let no heat be communicated to or taken from the gas 
 during the expansion. In other words, let the line ab be 
 an adiabatic curve. 
 
 The last expression for heat expended, viz. (r 2 T-J) 
 
 2-1 must, when no heat is expended, equal zero. 
 n i / 
 
 Hence one or other of the terms within the brackets must 
 equal zero. We know, however, that (r 2 r,) cannot equal 
 zero ; because during the expansion the temperature falls, 
 and therefore r 2 is less than T } . If, therefore, we make 
 
 the other term, viz. n 7; ~ f =o, we get 
 
 n i 
 
 The equation /z/ n = constant, becomes therefore pi-= 
 constant for the case of adiabatic expansion. 
 
 As no heat is supplied to the gas, the external work 
 
Expenditure of Heat during Expansion. 85 
 
 must be done at the expense of the heat already existing in 
 the gas, consequently the temperature falls, and the internal 
 work done is of a negative character 
 
 The temperature at the end of the expansion may be 
 found in the following manner, using the same symbols as 
 before. We have CT l =-p ] v } , and 
 
 Now 
 
 which, as 7=1-408 becomes T^=T I (-) 4 
 
 an expression which, when the initial pressure and volume 
 and the ratio of expansion are given, enables us to find the 
 final temperature. 
 
 Supposing in all the above examples that the gas were 
 compressed back to its original condition, the varying pres- 
 sures and volumes could be represented graphically, just as 
 in the case of expansion. If the conditions of compression 
 were the same as those of expansion, the same curve would 
 represent each operation. For instance, if the temperature 
 were maintained constant the curve would be an hyperbola. 
 If no heat were added or abstracted, the compression curve 
 would be an adiabatic line, and the temperature would ris^ 
 as the compression continued. If the conditions of the 
 compression were different to those of the expansion, the 
 curve would also be different. 
 
 THE IDEALLY PERFECT HEAT ENGINE. 
 
 We are now in a position to examine into the theory and 
 conditions of working of the ideal heat engine referred to at 
 the beginning of the 'chapter. This engine requires to be 
 made of materials which do not exist in practice ; the only 
 
86 
 
 The Steam Engine. 
 
 object in discussing it is therefore to separate the action of 
 the heat on the gas from the accidents of its surroundings, 
 so that we may be enabled to ascribe to the surroundings in 
 actual engines their proper influence. The efficiency of the 
 action of this engine does not depend in any way upon the 
 mechanism by which its motion may be converted, but only 
 on the manner in which it receives and rejects heat. We will 
 therefore, for the sake of simplicity, suppose it to consist of 
 
 Fig. 19. 
 
 a working cylinder, connected by means of a piston and 
 connecting rods with a crank. 
 
 Let AP, fig. 19, be the working cylinder which contains 
 between the end AP and the piston when at its initial 
 position i a certain quantity, say one pound, of gas at the 
 temperature r v The space in front of the piston is sup- 
 posed to be a perfect vacuum, so that the engine is single 
 acting. The sides of the cylinder are supposed to be made 
 of an ideal substance which can neither give heat to nor 
 
Perfect Heat Engines. 87 
 
 receive it from the gas. The bottom of the cylinder is, on 
 the contrary, supposed to be made of a substance which 
 though it has no capacity for heat itself, is nevertheless a 
 perfect conductor of heat. C is a source of heat having the 
 absolute temperature T } . D is similarly a body having a 
 temperature r 2 less than T I which is used for the reception 
 of the heat rejected by the gas. B is a cover made of 
 the same material as the sides of the cylinder which, when 
 applied to the bottom AP, renders it perfectly non-con- 
 ducting like the rest of the cylinder. 
 
 Let the piston commence to move forwards from the 
 position i. To prevent the temperature of the gas from 
 falling, apply the source of heat C to the end of the cylinder. 
 As this end is a perfect conductor, heat will flow into the 
 gas and maintain the temperature constant. Let the co- 
 ordinates of the point i' with reference to the lines OV, OP 
 denote the initial volume and pressure of the gas. So long 
 as the body r , is kept in contact with the cylinder end, the 
 gas will expand isothermally, and the variations in its pres- 
 sure and volume will be denoted by the co-ordinates of the 
 hyperbolic curve iV. When the piston has reached the 
 point 2, the exact position of which will be presently deter- 
 mined, withdraw the body r l and apply the non-conducting 
 cover B. The gas will now continue to expand adiabatically, 
 as represented by the curve 2 '3'. During this part of the 
 expansion the temperature will fall, and the point 2' must 
 be so chosen that the temperature will, when the piston 
 has reached the end of its stroke, have fallen to that of the 
 cold body, viz. - 2 . The piston must now be caused to 
 return in the opposite direction. As the engine is single- 
 acting, this can only be done by the application of forces 
 external to the engine. The piston when returning will 
 compress the gas and do work upon it. To prevent the 
 temperature from rising apply the cold body D to the end 
 of the cylinder. The compression will then take place 
 isothermally, at the temperature r 2 , along the hyperbolic 
 
88 The Steam Engine. 
 
 curve 3 / 4 / . When the piston has reached the point 4 the 
 position of which will be presently determined the cold 
 body must be withdrawn, and the non-conducting cover 
 reapplied to the end of the cylinder. The compression of 
 the gas must then be continued, and as no heat can escape 
 it will take place adiabatically along the line 4 / i / . During 
 this part of the compression the temperature will rise, and, if 
 the point 4 has been rightly chosen, it will reach r l when 
 the piston has returned to its original position. 
 
 During the first operation the gas has been receiving 
 heat, and doing external work, which is represented by the 
 area of the figure i' 2' v. 2 v v During the second operation 
 the gas has received no heat from without, but, at the 
 expense of the heat which it already possessed, it has done 
 external work, represented by the area 2' 3' v% ?' 2 . During 
 the third operation, work has been done upon the gas, 
 measured by the area 3' 4' v ?; 3 , and the gas has rejected 
 heat into the body D ; and during the fourth and last 
 operation work has been done upon the gas represented by 
 the area 4' i' v l v 4 , with the result of restoring it to the 
 original pressure, volume, and temperature. We see there- 
 fore that the work done by the gas exceeds the work done upon 
 the gas, by the difference between the sum of the two first, 
 and the sum of the two last-mentioned areas. This difference 
 is equal to the area i' 2' 3' 4', which therefore represents 
 the effective work done by the engine. 
 
 Calculation of the efficiency of perfect heat engines. 
 We must now calculate the heat expended, and the work 
 done. During the first operation, the heat supplied to the 
 gas is all expended in doing external work : for no internal 
 work is done, as the temperature of the gas is not raised. 
 The heat supplied therefore in foot-pounds is equal to the 
 area i' 2' ?' 2 7^ = ^' v' log e r = cr l log c r, where ; is as 
 
 before the ratio . During the second operation no heat 
 is supplied to the gas. During the third operation, the gas 
 
Efficiency of Perfect Heat Engines. 89 
 
 rejects heat equal in amount to the work done upon the gas 
 =c-^ log e ?; where r is the ratio -? which will be presently 
 
 ?; 4 
 
 proved. During the fourth operation the gas rejects 
 
 v\ 
 
 no heat. The total heat supplied therefore =cr^ log e ^, and 
 the total heat rejected - CT^ log e r. 
 
 The work done can be calculated in two ways. We may 
 either compute the area i' 2' 3' 4', or we may make use of 
 the principle of the cycle of operations. For since the gas 
 returns to its original condition no heat is spent in doing in- 
 ternal work upon the gas and the heat expended must there- 
 fore equal the external work done, plus the heat rejected. 
 
 Consequently the external work 
 
 The efficiency of the engine is the ratio of the work 
 done to the heat expended 
 
 That is to say the efficiency of the engine is the ratio of 
 the difference of temperatures of the sources of heat and of 
 cold to the temperature of the source of heat ; the tempera- 
 tures being reckoned in absolute measure. 
 
 The efficiency of the engine can only become equal to 
 unity, i.e. the engine can only turn the whole of the heat 
 supplied to it into mechanical work, when the temperature 
 r 2 =o ; that is to say, when the cold body has the absolute 
 zero of temperature ; a result which is of course unattain- 
 able. 
 
 On the other hand, the nearer to unity the fraction 
 
 TI ~~ T ' 2 becomes, the greater is the efficiency of the engine. 
 r i 
 
 This result can only be attained by making r l r 2 as 
 nearly as possible equal to - } . To do this we must make 
 T! as large and r 2 as small as possible. 
 
po The Steam Engine. 
 
 In practical engines, the limits of temperature which we 
 can make use of are soon reached, and consequently the 
 efficiency of such engines is necessarily low. 
 
 In order to fix the points 2 and 4 with precision, we 
 must remember that the isothermal expansion and com- 
 pression must be stopped soon enough to allow the tempera- 
 ture to fall to r 2 and rise to r l respectively during the 
 subsequent adiabatic expansion and compression. We have 
 seen (see page 85) that when a gas expands adiabatically, 
 the temperatures and the ratio of expansion are con- 
 
 nected together by the equation r a = 
 
 i 
 Consequently r y *='-L and r =(- 1 ) y ~ J 
 
 r 2 \r 2 / 
 
 I 
 
 That is to say, in fig. 19, Oz' 3 -T-O^ 2 =( ) **, which 
 
 V2/ 
 
 fixes the point 2. 
 
 Similarly during the adiabatic compression from r 2 to 
 
 i 
 
 r l we have Ov^Ov l (\ y ~\ which fixes the point 4. 
 
 Hence we see that ^L=_^ 2 or the ratio of adiabatic 
 O 4 Ov 3 
 
 compression equals the ratio of adiabatic expansion. It may 
 likewise easily be proved that the ratio of the isothermal 
 compression equals the ratio of isothermal expansion. 
 
 We can now prove that no other engine can have a 
 greater efficiency than the one which has just been described. 
 We must, however, first state what is meant by the reversed 
 action of the engine. Let the piston, starting from the 
 position i, move forwards, the 7 " Conducting cover B being 
 applied to the cylinder end. ''Yhe gas will expand adia- 
 batically along the curve i'4 f . As soon as the temperature 
 has fallen to r 2 apply the cold body D so as to keep the 
 temperature constant, and allow the gas to expand still 
 
Efficiency of Perfect Heat Engines. - 91 
 
 further along the isothermal curve 4' 3'. When the point 3 
 is reached reverse the piston, by means of external forces, 
 and compress the gas adiabatically along the line 3' 2'. As 
 soon as the temperature has risen to T-J apply the hot body 
 C and continue the compression isothermally till the piston 
 reaches its original position, and the gas its original state. 
 The result will be, that we shall have done the exact reverse 
 of all that was done when the engine was worked in the 
 usual way. Instead of the engine having done any work, 
 external work has been done upon the gas represented by 
 the area i^Y/j.'- Instead of heat having been taken from 
 the hot and rejected into the cold body, it has been taken 
 from the cold and rejected into the hot body. The quantity 
 of heat taken from the cold body is <r.- 2 log e r and the 
 quantity rejected into the hot body is cr\ log e r, the work 
 done upon the gas being the difference of these two quan- 
 tities. 
 
 If now any engine can be devised having a greater 
 efficiency than the one described, let it be employed to 
 drive this latter in the reversed manner. Then engine 
 ,/o No. 2 takes heat from the hot body and rejects heat into 
 the cold body, while engine No. i does exactly the 
 reverse, taking heat from the cold body and rejecting it into 
 the hot one. Let the power of each engine be the same. 
 In the case of engine No. i this work is done upon the gas, 
 and in the case of No. 2 it is done by the gas, so that upon 
 the whole the engine does no external work, and the com- 
 bination is self-acting, and can, if we neglect friction, go on 
 running for ever. 
 
 Let Rj be the heat which No. i takes from the cold 
 body, and H! be the heat which it rejects into the hot 
 body. Let H 2 and R 2 be the quantities which engine 
 No. 2 takes from the hot body and rejects into the cold 
 body. Now, the heat taken from the cold body by No i, 
 plus the work done on the gas, equals the heat given to the 
 hot body ; therefore H 4 R t is the work done on the gas in 
 
92 The Steaui Engine. 
 
 the case of No. i, and H 2 R 2 is the work done by No. 2. 
 But, since the power of each engine is the same, 
 
 Now, the efficiency is the ratio of the work done to th 
 
 heat supplied=^ 1 -~--i for engine No. i, and H2 ~ R%2 for 
 HI H 2 
 
 No. 2. According to the hypothesis the efficiency of No. 2 
 
 TT _ -p TT -p 
 
 is the greater, therefore 2 r > ] ~- I. In order that 
 H 2 Hj 
 
 this may be so, the denominator of the first fraction must 
 be less than that of the second, for the numerators are equal, 
 therefore H 2 is less than H lt 
 
 Now, H 2 is the quantity of heat which No. 2 takes from 
 the hot body; and H, is the quantity which No. i rejects 
 into it, and as H 2 is less than Hj upon the whole the hot 
 body receives heat ; and similarly upon the whole the cold 
 body parts with heat, and if the engine were kept at work 
 long enough the whole of the heat in the cold body could 
 be taken out of it and conveyed to the hot body, that is to 
 say, heat could be transferred from a cold to a hot body by 
 a self-acting contrivance, which is the exact opposite of all 
 experience ; consequently we must conclude that engine 
 No. 2 has not got a greater efficiency than No. i, and that 
 no arrangement we can imagine can have a greater efficiency 
 than No. i. 
 
 The Second Law of Thermodynamics. The statement of 
 the principle from which the above conclusion is drawn is 
 called the Second Law of Thermodynamics. It may be ex- 
 pressed as follows : 
 
 Heat cannot pass from a cold to a hot body by a self-acting 
 process, unaided by external agency. 
 
 The peculiarity of engine No. i is its reversibility, and 
 this is due to its receiving heat always at the temperature of 
 the hot body, and rejecting it at that of the cold body. If 
 this condition did not obtain, we could not work the engine 
 
Conditions of Maximum Efficiency. 93 
 
 in the reverse sense. Hence we conclude, that for a heat 
 engine to develope a maximum of work out of the heat 
 supplied to the gas or other working substance it must receive 
 all its heat at the temperature of the hot body, and reject it 
 all at the temperature of the cold body. If these conditions 
 are complied with, the maximum of work obtainable is got by 
 multiplying the heat supplied to the engine, measured in 
 
 foot-pounds by the ratio TI ~ ' 2 . 
 
 T \ 
 
 It may at first appear anomalous to the student that the 
 whole of the heat supplied to the gas cannot be converted 
 into work. Perhaps the best way to overcome this difficulty 
 is, if he can conceive of any arrangement by which he hopes 
 to get more work out of an engine, to calculate out the 
 various steps in the process, and compare the heat expended 
 with the work done. 
 
 Take, for instance, a pound of air inclosed in a cylinder 
 as before, and at the pressure and temperature r of the 
 atmosphere. Heat the air, keeping the volume constant till 
 it attains any desired pressure /,, and corresponding tempe- 
 rature r l ; next expand it adiabatically, till the original 
 temperature of the atmosphere is reached, the corresponding 
 pressure, / 2 , being attained. During this part of the process 
 the exact equivalent of the heat expended on the gas is 
 converted into mechanical work. For the heat so ex- 
 pended is K^rj r), and the work done is (see page 84) 
 
 - (r l r) = K^T-J r). But in order to bring the gas 
 
 back to its original pressure, volume, and temperature, so as 
 to be able to make another revolution of the engine, we 
 must either compress the air at the constant temperature 
 r, ; or else we must temporarily open the end of the 
 cylinder to the outer air, and force back the piston against 
 the constant pressure of the atmosphere to its original 
 position. In the first case, to compress the air at constant 
 temperature r h we must do work upon it represented by 
 
94 
 
 The Steam Engine. 
 
 CT log e l -j (see page 83). In the second case we do work 
 
 equal to the pressure of the atmosphere=2ii6 Ibs. per 
 square foot, multiplied by the space through which it has to 
 be overcome, viz. v 2 v l feet. In either case the work so 
 done upon the air must be subtracted from the work done 
 by the air during the first half of the stroke, so that in 
 neither instance can we realise the full equivalent of the 
 heat expended. It may in fact be proved by a calculation 
 similar to that on pages 88, 89 that in this case the efficiency 
 
 is less than the maximum, viz. TI ~~- T . 
 
 TI 
 
 To take another case. If a portion of gas be expanded 
 at constant temperature, it is known that the heat absorbed 
 equals the external work done. It might therefore be sup- 
 posed that a ready means was hereby presented of converting 
 
 Fig. 20. 
 
 the whole of the heat supplied to an engine into work. Let 
 the point i, fig. 20, represent the initial condition of the gas 
 as regards pressure and volume. Let it expand at constant 
 temperature till the pressure and volume are indicated by 
 the point 2. During this part of the operation external 
 
Efficiency of Heat Engines. HIT'S 405 
 
 work would be done represented by : ^be~a^flt^*2Z' 2 z; 1 
 = CT^ log e 2 =the heat supplied to the gas during expansion. 
 
 If we were now to restore the gas to its original pressure, 
 volume, and temperature, we could do so by compressing 
 it isothermally ; but in this case the compression curve 
 would be identical with the expansion curve, and con- 
 sequently in compressing the gas we should have to do the 
 same work upon it that we got out of it during the expansion, 
 and the .resulting work done would be nil. In order to get 
 work out of the gas, we must therefore, when the point 2 is 
 reached, make it reject heat. Let this be done at constant 
 volume. The pressure will fall, and this part of the opera- 
 tion is represented on the diagram by the vertical straight 
 line 2 3. When the pressure has fallen to the point 3, the 
 temperature has also fallen to r 2 , and the heat rejected during 
 the fall is K,, (T\ r 2 ). The gas may now be compressed 
 isothermally at the temperature r. 2 , till the end of the stroke 
 
 is reached, and the work done upon it equals <rr 2 log e ^?. We 
 
 v \ 
 
 thus arrive at the point 4, where we see that the pressure 
 and temperature are less than at the initial position. In 
 order, therefore, to complete the cycle of operations, we 
 must now heat the gas at constant volume till its original 
 temperature and pressure are restored, and in so doing heat 
 equal to K v (T l T 2 ) will have been expended. We shall, 
 therefore, have the following quantities : ' 
 
 Heat expended = fr l logg + K^Ti r 2 ). 
 
 Heat rejected =<rr 2 log e -f K^(T! r 2 ) 
 
 v \ 
 
 Work done = difference of two above quantities 
 
96 The Steam Engine 
 
 Efficiency = ratio of work done to heat expended 
 
 Now the first factor in the above result, viz. ^LIT_ T2 } re- 
 
 Tj 
 
 presents the maximum efficiency attainable under any circum- 
 stances, and the second factor is a fraction, the denominator 
 of which is greater than unity ; therefore the efficiency in 
 this case is less than the maximum efficiency. 
 
 It has been already pointed out (see page 89) that 
 in order to approximate to perfect efficiency' the fraction 
 
 Tl ~" T2 must be made as large as possible, unity being its 
 
 T i 
 
 limit. This can only be done by increasing r l and 
 diminishing TO as far as practicable. In other words the 
 efficiency of a perfect engine will become greater in propor- 
 tion as the temperature of the source of heat is increased, and 
 that of the source of cold diminished. In practice we are 
 extremely limited in the power of choosing both temperatures 
 for two reasons : firstly, because the range of temperatures 
 in nature is limited, for commercial purposes, by those of the 
 products of combustion of coal on the one hand and ice on 
 the other ; and, secondly, because the highest temperatures 
 thus available are far more than sufficient to destroy the 
 substances of which engines are constructed. 
 
 If, by special materials and appliances, a heat engine 
 could be made to use air having the higher limit of 1000, 
 and the lower limit of 493 absolute, equal respectively to 
 539 and 32 Fahrenheit, the theoretical efficiency of such 
 
 an engine would be given by the fraction z23= 57_ 
 
 1000 1000 
 
 or nearly \ ; that is to say, for every unit of heat supplied to 
 'such an engine, only half a unit could possibly be converted 
 into mechanical work. 
 
Pressure, Volume and Temperature of Steam. 97 
 
 LAWS CONNECTING THE PRESSURE, VOLUME, AND 
 TEMPERATURE OF DRY STEAM. 
 
 It will now be possible to apply the principles of the 
 foregoing pages to the case of steam. As was explained in 
 the last portions of Chapter II., dry saturated steam is a very 
 imperfect gas, and the laws connecting its pressure, tempera- 
 ture, and volume cannot be expressed with the same sim- 
 plicity as those relating to a perfect gas. In fact, the formulae 
 hitherto devised are purely empirical. The general results 
 which were stated in Chapter II. will now, for convenience 
 in calculation, be expressed in algebraical language. 
 
 Rankings and Zeuner's laws connecting the pressure and 
 volume of dry steam. The product of the pressure and volume 
 of dry saturated steam, instead of being expressed by the very 
 simple law, pv=> constant, is given according to Rankine by 
 the equation 
 
 17 
 
 pv 1^= constant. 
 
 According to Zeuner the index -}= i -0625 should be i -0646. 
 If the pressure be expressed in pounds per square inch and 
 the volume in cubic feet, the constant has the value 475, 
 and adopting Zeuner's index the equation becomes 
 
 The above formula relates to one pound weight of steam. 
 The volume occupied by one pound of steam is called 
 the specific volume. The density of the steam of course 
 diminishes as the specific volume increases, and can be 
 calculated by solving the above equation for z', which can 
 be done with the help of a table of common logarithms, 
 thus, log ^ + 1-0646 log z'=log 475 = 2-6766936 
 
 .-. log -,= 
 
 Connection between the pressure and temperature of dry 
 steam. The connection between the pressure and tempera- 
 
 H 
 
98 The Steam Engine. 
 
 ture of steam is also much more complicated than the corre- 
 sponding relation for gases. It will be remembered (see 
 page 75) that for common air the relation was expressed by 
 the formula/z>=53'2. T. In the case of steam there is no 
 formula of general application, and the temperature must be 
 taken for each pressure from the Table, page 489 et seqq. 
 
 Specific heat of ivater and steam. The specific heat for 
 water and steam is equally complicated. In the case of air, 
 we have seen (see page 76) that the specific heat at constant 
 pressure is a fixed quantity, K p =i83'35 foot-pounds, and 
 tire specific heat at constant volume K v is also a constant 
 quantity = 1 30 '2 5 foot-pounds. Also the heat expended in 
 effecting any change of state in the air can be easily calcu- 
 lated, when we know these specific heats, and the quantity 
 of external work done. In the case of water, however, the 
 specific heat is constantly increasing from 772 foot-pounds 
 at the point of maximum density, 39F., and is about 802 '88 
 foot-pounds when the temperature is 400. While the water 
 is being changed into steam, large quantities of heat are 
 supplied to it which produce no effect upon the thermo- 
 meter, and which consequently cannot be measured by any 
 reference to the specific heat of the substance at that stage. 
 When the water has become dry saturated steam, any further 
 heat supplied to it does certainly raise the temperature, but it 
 also changes the state from the saturated to the superheated 
 condition. 
 
 When once thoroughly superheated, the properties of 
 steam ^resemble those of a perfect gas, and may be reasoned 
 on accordingly. The equation connecting its pressure, 
 volume, and absolute temperature in this state is flv=8$ -5 T, 
 and differs from the corresponding equation for air only 
 in the value of the numerical constant, which for air was 
 proved to be 53-2. The heat required to raise one pound 
 of superheated steam at constant pressure through one 
 degree is given as 370*56 foot-pounds, and at constant 
 volume as 285-03 foot-pounds, while the ratio of the first 
 of these quantities to the last is y= i '3. 
 
Total Heat of Steam. 99 
 
 Total heat required to convert water into steam.- In con- 
 sequence of all these complications, we cannot deal with 
 the quantities of heat required to effect changes of state in 
 water and steam with the same ease and simplicity as in the 
 case of gases and air. The total heat required to change 
 water of a given temperature into steam at a given constant 
 pressure is (see page 62) divisible into 3 parts, viz. 
 
 i st The quantity required to heat the water from the 
 given temperature to the natural temperature of formation 
 of the steam ; 
 
 2nd. The quantity required to change the substance 
 from the liquid to the gaseous state ; 
 
 3rd. The quantity required to do external work, that 
 is, to overcome the external pressure through the space 
 represented by the difference between the volumes of the 
 steam and the water from which it was formed. 
 
 The two latter quantities taken together are usually 
 called the latent heat of evaporation. It is, however, 
 necessary to bear in mind that this latent heat consists of 
 two elements. 
 
 The total heat required for any particular case may be 
 extracted from the tables, the original temperature of the 
 water being supposed to be 32 ; or it may be calculated 
 very approximately by the following empirical rule : 
 
 The total heat required to change one pound of water of 
 32 into steam .of atmospheric pressure is known by experi- 
 ment to be 885,200 foot-pounds, and for every degree of 
 increase of temperature of the steam about 235*46 foot- 
 pounds have to be added. Thus if T be the temperature of 
 the steam, the total heat required is 
 
 885,200 -|- 235 -46 (T 212). 
 
 If the water be warmer than 32 to start with, less heat 
 will be required. If the specific heat of water were constantly 
 772 foot-pounds, we should have no trouble in calculating the 
 quantity to be subtracted, and we may without sensible 
 
ioo The Steam Engine. 
 
 error regard the specific heat -as constant for the usual 
 temperatures of water. Thus if / be the temperature of 
 the water, it would take 772 (/ 32) foot-pounds to heat 
 the water from 32 to t ; this quantity must therefore be 
 subtracted in the above formula, so that we get total heat 
 required 
 
 = 885,200 + 235-46 (T-2I2)-772(/-32). 
 
 The quantity of heat required to do external work in 
 the above is not difficult to calculate. Let v be the volume 
 occupied by the steam when formed in cubic feet, v' the 
 space occupied originally by the water. Then the external 
 work done is equal to the increased space occupied by the 
 steam above that occupied by the water, viz. (v v'\ 
 multiplied by the constant pressure of formation of the 
 steam, viz. p pounds per square foot ; or, external work 
 done=/(z> v') foot-pounds. Now v' is the space occupied 
 by a pound of water, viz. -016 cubic foot, and is so small that 
 it may in general be neglected. Hence we may write : 
 
 External work=/z; foot-pounds. 
 
 The product pv may be calculated from the equation 
 given above (see page 97), viz. pv 1 - 646 =475. 
 
 Another equation has been devised by Zeuner, and is 
 commonly used, on account of its convenience, to express 
 the external work empirically. Let h be the quantity of 
 heat in foot-pounds required to heat the water from 32 to 
 T the temperature of the steam ; then, 
 
 External work i5,45o + 846T h foot-pounds. 
 
 If we know the total heat required to raise the water 
 from 32 to the temperature of the steam, and then evaporate 
 it, and if we also know by calculation the external work 
 done, we can deduce the quantity cf heat spent in doing 
 internal or latent work during evaporation, for, 
 
 The total heat =/* + internal work + external work. 
 
 /. 885,200 + 235-46 (T- 212) 
 
 =/fc + internal work + (15, 450 + 846!" li) 
 .*. internal work=8i9,832 611 T foot-pounds, 
 
Heat Expended in Steam Engines. 101 
 
 EXPENDITURE OF HEAT IN A STEAM ENGINE. 
 
 We can now examine a question of great practical impor- 
 tance, viz. what quantity of heat must be supplied to a steam 
 engine, in order to get a certain amount of work out of it. 
 Steam engines regarded as heat engines may be divided into 
 two principal classes, viz. i. Those which work with full pres- 
 sure of steam throughout the entire stroke, and 2. Those 
 which use the full pressure of the steam during a portion only 
 of the stroke, and then cut off all connection between the 
 cylinder and the boiler, allowing the steam which is isolated 
 in the cylinder to expand to the end of the stroke. The 
 former are called non-expansive, the latter expansive engines. 
 We will first consider the case of a non-expansive engine, and 
 will suppose the steam supplied to it to be in the dry 
 saturated condition, and free from moisture. It is necessary 
 to mention the subject of moisture, because in the generality 
 of cases the steam which enters a cylinder is not pure, but 
 carries over with it from the boiler a large per-centage of 
 suspended moisture. 
 
 How to represent heat by an equivalent pressure on the 
 piston. We have seen, in the case of the external work done 
 by an engine, that the heat expended in doing this work 
 can be represented by the pressure in pounds on the piston 
 multiplied by the space in feet through which the piston 
 moves. 
 
 The heat expended in raising the temperature of the 
 water and in doing any internal work can be represented in 
 a similar manner, by a pressure on the piston multiplied by 
 the space through which it moves. Thus referring to fig. 13, 
 page 63, it will be remembered that the total heat expended 
 in converting a pound of water into steam was represented 
 by the area of the three rectangles BA, AC, and CE, of 
 which CE represented the heat spent in raising the temper- 
 ature of the water, AC the heat spent in doing internal 
 work while changing the water into steam, and BA the heat 
 
IO2 The Steam Engine. 
 
 spent in doing external work while the volume of the 
 steam was increasing at constant pressure. 
 
 Now, we may take the line OA as representing the stroke 
 of a steam engine, working with full steam throughout. The 
 pressure, due to the resistance to the motion of the piston 
 (see pages 62 and 63), is represented by the vertical line OB, 
 and the external work done during one stroke of the piston 
 is represented by the area of the rectangle BA. Now, the 
 internal work done when changing the water into steam is 
 represented by the area of the rectangle AC, and as both 
 these rectangles have the same base, OA, their areas are to 
 each other in the same ratios as the vertical lines, OB and 
 OC ; and just as OB represents the pressure due to external 
 work, so OC may be said to represent an ideal pressure 
 due to internal work, and CD an ideal pressure due to the 
 heating of the water, and the sum of the three lines, viz. 
 BD, represents a pressure due to the total expenditure of 
 heat. 
 
 Now, if the area of the piston be one square foot, then 
 the volume occupied by the steam at the end of the stroke 
 or z>=OA, and if H is the total heat expended expressed in 
 foot-pounds, viz. the area BE, then the line BD, represent- 
 ing the pressure due to the heat expended, multiplied by #, 
 equals the area BE, or denoting the line BD by P, we have 
 
 Pxz/ = H, or P = ?. 1 
 V 
 
 That is to say, the ideal pressure, due to the heat expended, 
 equals the heat expended expressed in foot-pounds, divided 
 by the volume in cubic feet occupied by the steam. 
 
 If v equals unity, then the work done by the piston equals 
 the pressure on it, equals / ; that is to say, when the piston 
 travels through one cubic foot, the external work done equals 
 
 1 Properly speaking v should be diminished by the volume of the 
 original pound of water, but as this is very small the correction has 
 been omitted in this and the succeeding calculations. 
 
Heat Expended in Steam Engines. 103 
 
 /, and the total heat expended equals P ; or expressed 
 generally, the work done per cubic foot swept through by 
 the piston =/, and the heat expended = P. 
 
 Let us take the case of a condensing engine, working 
 with dry saturated steam of 60 pounds pressure on the 
 square inch, and find out how much useful work it can do 
 per pound weight of steam used, and how much heat has to 
 be expended in order to do the work. 
 
 At first sight it might be thought that the useful work 
 done equals the pressure on the piston multiplied by the 
 space through which it moves, and this would be the case if 
 there were a perfect vacuum at the back of the piston. It 
 is, however, impossible to realise a perfect vacuum with a 
 condenser, and consequently the back of the piston experi- 
 ences a pressure varying according to the perfection of the 
 vacuum, and acting in the contrary direction to the steam 
 pressure. This back pressure in a condensing engine usually 
 varies from 2 to 4 pounds per square inch. In the case of 
 a non- condensing engine, the back of the piston is pressed 
 upon by the whole weight of the atmosphere, or 147 pounds 
 per square inch, as well as by the residual pressure of the 
 exhaust steam, which cannot escape quickly enough through 
 the narrow exhaust passages, so that with these engines the 
 back pressure varies from 16 to 19 pounds per square inch. 
 
 In the present example the back pressure is taken as 
 three pounds, and the effective forward pressure of the 
 steam is 60 3'= 5 7 pounds per square inch. Now the 
 specific volume of steam of 60 Ibs. is (see page 493) 7*037 
 cubic feet, and the effective external work or pv consequently 
 equals 7*037 X57 x 144=57758-4 foot-pounds. This then 
 is the useful work which one pound of steam can realise 
 when worked non- expansively. Also as one horse-power 
 per hour is 33000 x 60 foot-pounds, an engine working 
 
 under the above conditions would require 33 000> < := .^. 2 g 
 
 5775 8 
 pounds weight of steam per horse-power per hour. 
 
IO4 The Steam Engine. 
 
 Now, the heat expended in order to attain this result is 
 the whole heat of formation of the pound of steam at the 
 temperature corresponding to 60 pounds pressure, viz. 293. 
 If the water were taken originally at 32, this quantity of 
 heat would be 904,106 foot-pounds (see Table I.), but if 
 the water were taken originally at the temperature of the 
 condenser, which would be about 104, we should have to 
 subtract from the above quantity the heat requisite in order 
 to raise the water from 32 to 104. This latter quantity 
 expressed in foot-pounds would be approximately (104 
 3 2) 7 7 2 =5 5, 5 84, and consequently the total heat expended 
 would be 904,106 55,584=848,522 foot-pounds. 
 
 Thus we see that, in order to obtain 57,758 foot-pounds 
 of external work we have to expend 848,522 foot-pounds of 
 heat, and the consequent efficiency of the engine would 
 
 only be -ZZSaE'o68 ; a result which shows how wasteful 
 848522 
 
 such a form of steam engine is. The above result, bad 
 though it be, is far more favourable than anything that 
 would take place in practice ; for it must be remembered 
 that we have not taken account of any losses due to radia- 
 tion, conduction, leakage, &c., and we have supposed the 
 steam to be formed without any waste of fuel ; whereas we 
 know that even in the best boilers this waste is very con- 
 siderable. 
 
 At the end of the stroke the whole of the heat in the 
 steam, which is equal to the heat of formation minus 
 the external work done, is rejected into the condenser, 
 part of it heating the injection water and the remainder is 
 wasted. 
 
 If we desire to express the above quantities, per cubic 
 foot swept through by the piston, instead of per pound of 
 steam used, we can do so very simply. For the heat ex- 
 pended equals the work done plus the heat rejected. Now, 
 if H be the total heat of formation of the steam from the 
 temperature of the water, and v its specific volume in cubic 
 
Expansion of Steam. 105 
 
 TT 
 
 feet, then equals the heat of formation of one cubic foot 
 
 of the steam. Also the external work done per cubic foot 
 equals the pressure of the steam per square foot, equals /, 
 and the heat rejected is the heat left in the cubic foot of 
 
 TT 
 
 steam after it has done its work = /. Now, part of the 
 
 work done is spent in overcoming the resistance of the back 
 pressure. If J> b be the back pressure per square foot, the 
 work so spent per cubic foot swept through by the piston is 
 also equal to / b , and as this work is done upon the con- 
 densing steam it reappears as heat in the condenser, and 
 must consequently be reckoned as so much heat rejected. 
 
 TT 
 
 Therefore the heat rejected, instead of being /, will be 
 
 EXPANSION OF STEAM. 
 
 It is evident from the foregoing that the heat rejected is 
 very nearly equal to the total heat supplied. The only way 
 of increasing the efficiency of a steam engine is to utilise 
 some of this wasted heat. This object can be attained by 
 cutting off all communication between the cylinder and the 
 boiler when the former is partly filled, and then allowing 
 the steam to expand during the remainder of the stroke. 
 The amount of the expansion may be varied, according as 
 the steam is cut off earlier or later during the stroke. There 
 is theoretically no reason why the expansion should not be 
 carried on till the final pressure of the steam equals the 
 back pressure ; but there are practical reasons, which will 
 be explained hereafter, which render such high degrees of 
 expansion inexpedient. 
 
 The great economical advantage of using steam expan- 
 sively will be seen at once from the diagram, fig. 21. Let 
 the steam be, as before, of 60 pounds pressure per square inch 
 above zero, and the original temperature of the water 104. 
 
io6 
 F 
 
 The Steam Engine. 
 
 * 7-037 cub feet v = $rw** 
 
 f \ cubic feet 
 
 Measure off Op to represent 60 pounds, 
 and Oz to represent 7*037 cubic feet, 
 viz. the specific volume of one pound of 
 steam of 60 pounds pressure. Then, 
 neglecting the small initial volume oc- 
 cupied by the pound of water, the rect- 
 angle pQv represents the external work 
 = 60 x 144 x 7*037 = 60,800 foot-pounds. 
 Also the total heat of formation of the 
 steam, the initial temperature of the 
 water being 104, is 848,61 6 foot-pounds, 
 represented as before by the rectangle /B, 
 the area of which is 13*95 times that of 
 the rectangle pQv. 
 
 Let, now, the steam be expanded 
 till its final volume Oz> f is four times the 
 original volume Ov. The work done 
 during this expansion will depend upon 
 the curve p'p & which may be anything 
 we please, according as heat is or is not 
 supplied to the steam during its expan- 
 sion. Let us suppose, for the sake of 
 simplicity, that the curve is a rectangular 
 Fig. 21. hyperbola, and that the heat supply 
 
 necessary to make it one may be neglected and it may 
 here be noted that this is the curve to which the ex- 
 pansion line of steam most nearly approximates, under the 
 ordinary circumstances of a well- constructed steam engine. 
 
 B 
 
Heat Expended in Steam Engines, 107 
 
 The area of the portion p'w { p { of the diagram is then 
 pv loggf or pfVf log e r ; for since the curve is a rect- 
 angular hyperbola p x v = p f x v { . Also the total area 
 pQv\pfp'i representing the total external work done, is 
 ^7'+/f'log e r=/7'(i+log e r). In the present instance ?'=4, 
 and log e 7-= i -3863, and /z;=6o,8oo foot-pounds, c.pv 
 (i + log e r) = 60,800 x 2-3863 = 145,087 foot-pounds. In 
 order to obtain the useful work done, we must subtract 
 from the above amount the work expended in overcoming the 
 back pressure, say of 3 Ibs. per square inch. As the back 
 pressure is overcome through a space of 4 x 7-03 7 feet, the 
 work done=3 x 144X4x7-037 = 12,160 foot-pounds, and 
 the useful work therefore = 132,92 7 foot-pounds, against 
 57758-4 in the previous example, when there was no 
 expansion. Now the expenditure of heat was shown to be 
 
 848,616 foot-pounds, and the efficiency is therefore * 3 ^;? 2 ? 
 
 040,616 
 
 = 156, as against -068 in the case of the non-expansive 
 engine. The ratio of the external work done to heat 
 expended is represented graphically by the ratio of the area 
 of the figure p/p f v) to the area of the rectangle /B. 
 
 To find expenditure of heat when condition of steam at end 
 of stroke is given. The above result is only true if the expan- 
 sion curve be a rectangular hyperbola. If, however, some 
 other condition had been given, such as, that throughout, and 
 at the end of the expansion, the steam should be dry and 
 saturated, we should have had a different result \ for we know 
 that the expansion curve of dry saturated steam is not/z>= 
 constant but pv 1*0646= constant. 
 
 In practical experiments it is found easier to ascertain the 
 quantity and state of the steam at the end of the stroke, rather 
 than at the point of cut off ; we shall therefore next show how 
 to find the expenditure of heat, when the condition of the 
 steam at the end of the stroke is given, and the work done is 
 known. 
 
 In the first instance, suppose that the original pressure of 
 
io8 The Steam Engine. 
 
 the pound of steam is /, the final pressure A and that the 
 steam is dry and saturated at the end of the stroke. Now, 
 the heat rejected is the amount of heat in the steam at the 
 end of the expansion, together with the work done upon the 
 exhausting steam, by the piston overcoming the back pres- 
 sure / b . The heat in the steam at the end of the expansion 
 is the total heat of formation of dry saturated steam of the 
 pressure A minus the amount due to doing external work= 
 pfVf ; for, of course, the heat spent in doing external work 
 disappears from the steam, having been transmuted into the 
 work done. 
 
 Consequently heat rejected =H f 
 
 where A 7; f is the work done in overcoming the back 
 pressure A through the space v ft and H f is the total heat 
 of formation of dry saturated steam of the pressure A 
 
 Now, the heat expended equals the heat rejected, plus 
 the external work done by the steam during admission and 
 expansion. If p m be the mean or average pressure through- 
 out the stroke, then (p m A) Z V is the external work done, 
 and consequently 
 
 Heat expended=H f -A?'f 
 
 If we wish to express these results per cubic foot swept 
 through by the piston, we have only to divide by ?' f , the 
 number of feet occupied by the steam at the end of the 
 stroke, and we get 
 
 "FT 
 Heat expended = ^- f +A #. 
 
 Steam Jackets. We must now ascertain whether the heat 
 contained in the steam, as supplied by the boiler, is as 
 
 TT 
 
 much as the above quantity f +/ m A for if not, one of 
 
 Vf 
 
 two things must happen, viz. either more heat must be 
 supplied to the steam while it is in the cylinder from some 
 
Steam Jackets. 109 
 
 external source, or else at the end of the stroke it will not be 
 dry and saturated, but a certain proportion will be condensed 
 into water. 
 
 Let Hj be the total heat contained in a pound of the 
 steam, in its initial condition, as supplied by the boiler, 
 then, as i\ equals the contents of the cylinder, or the 
 number of cubic feet swept through by the piston in one 
 
 TT 
 
 stroke, therefore ' is the expenditure of heat per cubic foot 
 
 Z'f 
 
 swept through, provided that no heat is obtained from any 
 other source than from the boiler. But the heat actually 
 
 TT TT 
 
 expended per cubic foot swept through is not J but f + 
 
 Vf V ( 
 
 An -pf Subtracting, therefore, the first from the last of 
 
 TT TT 
 
 these quantities, we get a difference =/ m / f +- 
 
 Now, the numerical value of the two quantities Hj and H 
 may be taken from Table L, and as Hj is always greater 
 
 TT __ TT 
 
 than H f , the quantity will always be negative, and for 
 
 ?l 
 
 TT TT 
 
 every particular case the difference p n / f + - - 'will be 
 
 Vf 
 
 found to be a positive quantity ; therefore, the heat actually 
 wanted for the steam in order that it may remain dry and 
 saturated is greater than the quantity present in the steam 
 as it is supplied by the boiler. 
 
 The difference must therefore be supplied to the steam 
 while in the cylinder, and this is usually effected by 
 surrounding the latter with a casing always kept full of 
 boiler steam or hot air. The temperature of the steam in 
 the jacket is evidently greater than the mean temperature of 
 the steam in the cylinder, and consequently heat will flow 
 from the former to the latter, and will either check or wholly 
 prevent condensation, according to the quantity of heat 
 thus supplied. 
 
 The question, whether or no it is desirable to prevent 
 
1 10 The Steam Engine. 
 
 condensation during expansion, is a rather complicated one,, 
 and will be discussed in Chapter XI. 
 
 In the foregoing we were only concerned with proving 
 that if the condition be given that the steam must be dry 
 and saturated at the end of the expansion, in order to 
 fulfil this condition, heat must be supplied to the steam 
 from a hot casing, which is generally called a steam jacket. 
 
 Rankings empirical formula for the expenditure of heat 
 in a steam engine. From the above formula for the ex- 
 
 TT 
 
 penditure of heat, viz. : - f +/ m pi it would be easy to 
 v t 
 
 construct a numerical formula involving only the mean 
 and final pressures, and the temperature of the steam and 
 feed water and certain constants. It has, however, been 
 found by Rankine that the results are equally well given 
 by a very simple empirical formula which for condensing 
 engines is : 
 
 Heat expended =/ m -f i5/ f ; 
 and for non-condensing engines : 
 
 Heat expended =/ m + 1 4/ f : 
 
 the results being expressed in foot-pounds per cubic foot: 
 swept through by the piston. 
 
 THEORY OF THE PERFECT HEAT ENGINE APPLIED TO 
 THE CASE OF STEAM. 
 
 We can now proceed to apply the principles laid down 1 
 with regard to perfect heat engines to the case where steam 
 is employed instead of a gas. The amount of steam and 
 fuel necessary for a perfect steam engine under given 
 circumstances will first be considered. The nature of the 
 diagram indicating the varying states of the steam in such 
 an engine will then be examined, and finally the question 
 will be discussed how far actual steam engines of the best 
 
Theory of Perfect Engine applied to Steam, in 
 
 construction comply with, and how far they depart from, the 
 conditions of maximum efficiency. 
 
 The efficiency of a perfect heat engine has been shown 
 
 (see page 89) to be . Tl ~ T2 , where T I and r 2 are the 
 
 absolute temperatures of the sources of heat and cold. 
 Hence, in such an engine, if H be the quantity of heat 
 supplied, and W the exterior work done, we obtain the 
 relation 
 
 H. JJHl2_ = W or H=W. _ Il_. 
 
 Hence if we require to know the least amount of heat 
 necessary in order to obtain one horse-power per hour 
 when the limits of temperature within which the engine 
 works are known, we have W=33,ooo x 60= 1,980,000 foot- 
 pounds, 
 
 and H= i, 980,000 x - foot-pounds. 
 TI r 2 
 
 In a steam engine the limits of temperature ought to be 
 the temperature of the hot gases in the furnace of the 
 boiler on the one hand, and the temperature of the con- 
 denser on the other, or in the case of non-condensing 
 engines, the lower limit is the temperature due to the pres- 
 sure of the atmosphere, i.e. 2 12 + 461 absolute. We pos- 
 sess at present, however, no mear/s of utilising the tempera- 
 ture of the furnace gases, and consequently the higher limit 
 in a steam engine must be taken to be the temperature of 
 the steam in the boiler. 
 
 Let us consider the case of a perfect engine working 
 with steam of 60 pounds pressure, as before; the temperature 
 of the condenser being 104 + 461 absolute. The absolute 
 temperature of steam of 60 pounds pressure is 293 + 461 
 = 754. The quantity of work obtained per pound of 
 steam is the total heat contained in the steam, multiplied 
 
 by the fraction Z^ 5 5=J very nearly. Now, a perfect 
 
1 1 2 The Steam Engine. 
 
 engine, as will be seen presently, always uses the same 
 water over and over again, and always evaporates it from 
 the temperature of the steam ; consequently the heat 
 supplied to the water in order to turn it into steam of 
 293 is less than the quantity given in Table I., by the 
 amount necessary to heat the water from 32 to 293 ; that 
 is to say, the quantity of heat in question is 904, 1 06 202,444 
 = 701,662 foot-pounds. Therefore the work obtainable 
 per pound of steam is 701,622 xj=i75,4i3 foot-pounds. 
 In order to obtain from this engine one horse-power 
 
 per hour we should require to expend therefore Ii9 c ^ 00 
 
 I754i3 
 
 = 1 1 '3 pounds of steam. If we wish to find out at what ex- 
 penditure of fuel this power is attained, we must know what 
 heat can be developed by the combustion of a given weight 
 of fuel. This subject will be fully dealt with in the chapter 
 on boilers ; but at present it may be stated that one pound of 
 good average coal properly burnt should give 12,000,000 foot- 
 pounds of heat. Now as one pound of steam of 60 pounds 
 pressure requires for its formation, from water of 293, 
 701,662 foot-pounds, the pound of coal should theoreti- 
 cally be able to evaporate '- ' 000 = 17 pounds of water, 
 
 701,662 
 
 and consequently we ought to require --^=.-66 pound 
 
 of coal per horse-power per hour. 
 
 The actual amount of water which a pound of fuel can 
 evaporate in a good boiler is, however, much less than the 
 above ; in fact, as will be seen hereafter, it seldom exceeds 
 eleven, and is more often from seven to eight pounds. If, 
 for the sake of simplicity, we suppose that 11*3 pounds of 
 water are evaporated by a pound of coal, then, in the case 
 of the engine under discussion, we should require to burn 
 one pound of fuel per horse-power per hour. In the best 
 constructed modern strain engines working with steam of 
 the pressure under discussion, viz. sixty pounds absolute, 
 
Losses of Efficiency in Steam Engines. 1 1 3 
 
 or about forty-five pounds above the atmosphere, the amount 
 of fuel burnt per horse-power per hour is far greater than one 
 pound. It is, in fact, never less than two pounds, while in 
 engines of inferior construction, from eight to ten pounds, 
 and even larger quantities, are consumed. We see, there- 
 fore, plainly, that, in addition to the loss of heat which takes 
 place in the boiler, there are other sources of waste. It be- 
 comes, then, necessary to compare the working of an actual 
 engine with that of the theoretical engine, step by step, in 
 order to ascertain all the causes of inefficiency. 
 
 Causes of loss of efficiency in steam engines. In accordance 
 with the principles laid down, the water should receive all 
 its heat at the fixed higher temperature ; in other words, it 
 should be turned into steam at constant pressure. The 
 steam should then be allowed to expand adiabatically, 
 till the temperature falls to that of the condenser. It 
 should next reject heat into the condenser at this fixed 
 temperature, though 
 none of the steam it- 
 self is supposed to enter 
 the condenser. Finally, 
 at a given point, the 
 cooled steam, or mix- 
 ture of steam and water 
 as it would probably be, 
 should be compressed 
 till it returns to its first 
 condition of water, of 
 the original tempera- 
 ture of the steam. 
 
 These changes are indicated by fig. 2-2. 
 Let the point i represent the volume and pressure of one 
 pound of water, the pressure being that at which the steam 
 is formed. In strict theory, the steam ought to be formed 
 at the temperature of the furnace gases. This, however, as 
 has been before stated, is practically impossible, and we will 
 
 i 
 
 Fig. 22. 
 
H4 The Steam Engine. 
 
 suppose for the moment that the water receives its heat at 
 the constant temperature, due to the pressure at which the 
 steam is formed. During evaporation the pressure remains 
 constant and the volume increases, till the whole of the 
 water is converted into steam. This state of things is 
 indicated by the point 2. The steam is now allowed to 
 expand adiabatically along the curve 2 3 till the temperature 
 has fallen to that of the source of cold. The volume is then 
 reduced at the constant pressure, corresponding to the 
 temperature of the source of cold, till the point 4 is reached, 
 when the mixture of steam and water is compressed adia- 
 batically along the line 4 i, till the whole is re-converted 
 into water of the original temperature, pressure, and volume. 
 The area of this diagram, as before, represents the external 
 work done during the cycle of operations, and it may either 
 be computed analytically, if we know the equations of the 
 adiabatic curves, or, calculated more simply on the principle 
 that the work done equals the heat supplied, multiplied by 
 the efficiency of the engine ; in other words it equals the 
 heat necessary to turn a pound of water of the temperature 
 
 T I into steam of T b multiplied by Tl ~_ 2 . 
 
 r \ 
 
 Now, in an actual steam-engine, the series of operations 
 which takes place differs more or less at every step from 
 that which has been just described. In the first place, the 
 water, instead of receiving all its heat at the higher tempera- 
 ture TJ, is introduced into the boiler as feed water at a 
 much lower temperature. During the process of evaporation, 
 the condition of receiving heat at constant pressure is ful- 
 filled, so long as the pressure in the boiler is kept constantly 
 the same, which it never can be when the steam is worked 
 expansively. 
 
 During the expansion, the condition that heat shall not 
 be supplied to, or abstracted from, the steam is not fulfilled, 
 because the metals of which cylinders are constructed 
 render the fulfilment of this condition impossible. Cylinders 
 
Losses of Efficiency in Steam Engines. 115 
 
 are of three sorts ; the first sort is made of metal directly 
 exposed to the outer air. In this case, the metal being a 
 good conductor is rapidly heated by the steam, and parts 
 with its heat to surrounding bodies by radiation and con- 
 duction, thus causing the steam to be cooled during its 
 passage through the cylinder, so that the expansion line 
 falls below the proper adiabatic curve. 
 
 The second class of cylinder is clothed with some non- 
 conducting substance, so as to prevent the escape of heat 
 to outside bodies. For the sake of simplicity, we will 
 suppose the substance to be a perfect non-conductor. 
 When first the steam enters such a cylinder it finds the 
 metal cool, and parts with some of its heat ; after a few 
 strokes, however, the cylinder gets warmed, and if the 
 temperature of the steam remained uniform throughout the 
 entire stroke, no further loss would ensue from this cause. 
 But the temperature of the steam is only uniform while the 
 line i, 2 is being described ; after that point, and during ex- 
 pansion, the temperature drops from T, to r 2 ; consequently, 
 when the steam enters, it finds the sides and end of the 
 cylinder cooled down, they having just been in contact with 
 steam of the temperature r 2 . Part of the heat of the steam, 
 therefore, is spent in re-heating the metal of the cylinder. 
 This causes part of the steam to condense, if it be originally 
 in the dry and saturated condition. When, however, the 
 expansion begins, the temperature of the steam rapidly drops 
 below the temperature of the walls of the cylinder. These 
 latter consequently give up part of their heat again to the steam, 
 and partially re-evaporate the condensed portions. This re- 
 evaporation is facilitated by the circumstance that a portion of 
 the condensed steam has the temperature r b and conse- 
 quently, when the expansion commences and the pressure 
 falls, it is too hot to remain any longer in the condition 
 of water, and its surplus heat helps in re-evaporating 
 it. Thus, though no heat is lost to external objects 
 during the stroke, when the cylinder is perfectly clothed, 
 
 I 2 
 
1 1 6 The Steam Engine. 
 
 still, during one period, heat is taken from, and during 
 another period given back to the steam, by the cylinder, 
 and consequently, the curve of expansion is not, strictly 
 speaking, adiabatic. The exact effect of this peculiar action 
 on the shape of the expansion curve is difficult to ascertain, 
 because the rapidity with which the metal of the cylinder 
 can take up and give off heat is not accurately known. The 
 subject will, however, be further examined in Chapter XI. 
 
 The third description of cylinder is surrounded by a 
 jacket or casing filled with steam from the boiler, and which 
 is itself covered with non-conducting substances so as to 
 prevent the escape of heat to the outside. It is evident 
 that in this case the temperature in the jacket is higher than 
 the average temperature in the cylinder, and consequently 
 heat will flow from the former to the latter throughout a great 
 part of the stroke, and will thus tend to raise the curve of 
 expansion above the adiabatic line. The effect of the 
 action of the jacket upon the working of the steam engine 
 will be more particularly considered in Chapter XI. 
 
 During the third operation in an actual steam engine, 
 viz. the rejection of heat, the condition of maximum efficiency 
 is not fulfilled, for the heat is not all rejected at the tem- 
 perature of the condenser. If the expansion were carried 
 so far that the temperature of the steam were reduced to the 
 temperature of the condenser, this condition could be ful- 
 filled, but in practice it is not found possible to carry the ex- 
 pansion so far, and consequently, when condensation com- 
 mences, there is a sudden drop from the temperature due 
 to the terminal pressure of the steam, to that of the conden- 
 ser. This is illustrated by the diagram fig. 23, where the 
 point 3 represents the pressure of the steam at the end 
 of the expansion, and the vertical ordinate of the point 4 
 represents the back pressure due to the temperature of the 
 condenser. When the steam commences to reject its heat, 
 the temperature suddenly falls from that due to the pressure 
 of the point 3 to that due to point 4. In strict theory the 
 
Losses of Efficiency in Steam Engines. 1 1 7 
 
 expansion should have been continued to the point 3', 
 where the curve intersects the horizontal line of back, or 
 condenser pressure. 
 
 During the remainder of the period of heat-rejection the 
 condition of maximum efficiency is fulfilled very approxi- 
 mately. At this part of the process, however, another evil 
 arises, for the metal of the cylinder was heated up to a 
 certain temperature during the admission and expansion of 
 the steam; when, however, the steam is being condensed, 
 its temperature is much lower, and consequently the 
 cylinder parts with some of its heat to the condensing steam, 
 thus retarding condensation, and cooling the cylinder down ; 
 
 Fig. 23. 
 
 so that, as has been before stated, some of the fresh steam 
 on entering is condensed. If the cylinder be provided with 
 a steam jacket the first of these evils may be increased, for, 
 during half the time such an engine is running, the steam 
 jacket is employed in wasting heat on the condensing steam. 
 The fourth condition of maximum efficiency, viz. that at 
 a certain point the rejection of heat should be stopped, and 
 the mixture of steam and water in the cylinder should be 
 compressed into water of the original pressure and tempera- 
 ture, which water should be used over again in the boiler, is 
 not fulfilled at all in the ordinary steam engine. If the 
 engine be of the condensing type, the condensation is 
 
Ii8 The Steam Engine. 
 
 carried out completely, and all the heat in the steam is spent 
 in warming up from 20 to 30 times its own weight of water 
 employed in the condensation to a temperature about one 
 third of that of the water in the boiler. Of this water, only 
 one twentieth to one thirtieth can be used over again to feed 
 the boiler, and must, when in the boiler, be suddenly raised 
 from the temperature of the condenser to that of the steam, 
 thus infringing the first condition of efficiency. 
 
 If, on the other hand, the engine be a non-condenser, 
 the steam all escapes into the open air, and is there con- 
 densed, and the boiler is fed with cold water, unless some 
 special provision is made for heating the latter with waste 
 steam, or furnace gases, which arrangement has, of course, 
 nothing to do with the engine, properly so called. 
 
 We thus see that the actual engine differs at every stage 
 of its working from the theoretically perfect heat engine, and 
 these differences are multiplied and rendered more compli- 
 cated by numerous other circumstances which will presently 
 be referred to. For instance, it has been taken for granted, 
 in all that has gone before, that the engine receives dry 
 saturated steam from the boiler. Now, as a matter of fact, 
 boilers do not usually deliver dry steam, but send over 
 large quantities of hot water with the steam into the cylin- 
 ders. When this takes place, the calculations for heat 
 expended and work done have to be materially modified ; 
 for it is evident that a large quantity of heat has been spent 
 in warming this water up, from the temperature of the feed 
 to that of the steam ; which heat is wholly or in greater part 
 wasted, as no work can be done by this water unless it 
 evaporates in the cylinder. Under the most favourable 
 circumstances the water can only be partially evaporated, 
 viz. when a jacket supplies heat to the steam in the cylinder, 
 and when by expansion the pressure of the steam is so far 
 lowered that the water is too hot to remain water at the 
 lower pressure, and consequently expends its surplus heat in 
 partially evaporating itself. This subject of wet steam is 
 
Losses of Efficiency in Steam Engines. 119 
 
 chiefly of interest in so far as it affects the subject of 
 jacketing ; and will consequently be referred to again in 
 Chapter XI. 
 
 Again, in all that has gone before, it has been assumed 
 that the action of the valves which admitted the steam from 
 the boiler to the cylinder, and from the cylinder to the 
 condenser, was perfect ; that is to say, that they opened and 
 closed fully and quickly, precisely -at the proper moment, 
 and in no way by their slowness of motion or imperfect 
 design strangled the steam on its passage to or from the 
 cylinder. In actual steam engines the valves not infrequently 
 fair short of this ideal perfection. 
 
 The results usually attained in practice fall short of what 
 they should do owing to the three following sets of causes : 
 
 1. The boiler is imperfect, inasmuch as it wastes heat, 
 and delivers water along with the steam to the cylinder. 
 
 2. The engine, considered merely as a heat engine, is 
 imperfect for the following reasons : 
 
 a. The limits of temperature within which it is possible 
 to work it in practice are narrow, so that the numerical value 
 
 of the fraction Tl ~" T2 is very small. 
 
 b. The series of operations does not comply with the 
 conditions of maximum efficiency, for the heat is neither 
 received nor rejected at constant temperatures, and the 
 metal of which the cylinder is made is capable of absorbing, 
 transmitting, and radiating heat, so that adiabatic expansion 
 is impossible. 
 
 3. The engine, considered as a piece of mechanism, is 
 defective for the following reasons : 
 
 a. Work is lost in friction of the different moving parts. 
 
 b. The passages conveying the steam from the boiler to 
 the cylinder, and from the latter to the open air or condenser, 
 always impede somewhat the motion of the steam, thus 
 diminishing the useful pressure on the piston and increasing 
 the back pressure. 
 
I2O The Steam Engine. 
 
 c. It is impossible to avoid leaving, and is even necessary 
 to allow a certain space between the piston, when at its 
 extreme positions and the face of the cylinder cover, which 
 space, together with the cubic contents of -the steam port, is 
 called clearance. Now, it is evident that this clearance space 
 has to be filled with fresh steam at every stroke, which does 
 no work except when expanding, and consequently causes a 
 loss of efficiency. 
 
 The losses due to the imperfections of the boiler and of 
 the mechanism will be duly considered in the chapters 
 devoted to these subjects. The losses due to the defects of 
 the engine proper considered as a heat engine have been 
 considered so far as space and the scope of this work will 
 allow in the present chapter. Some of them will, however, 
 be referred to again in Chapter XL, which deals principally 
 with the refinements of the engine, contrived to neutralise 
 the deficiencies. 
 
 SUMMARY OF THE LAWS AND FORMULAE OF 
 THERMODYNAMICS. 
 
 It may be useful and convenient to sum up here the 
 principal laws and formulas of the science of heat, as ex- 
 plained in this and the preceding chapter. 
 
 First Law of Thermodynamics. Heat and mechanical 
 energy are mutually convertible. A unit of heat requires for 
 its production, and produces by its disappearance, a fixed 
 amount of mechanical energy. 
 
 British unit of heat. The unit of heat used in this 
 country is the quantity of heat required to raise one pound 
 of water of the temperature 39*3 through one degree 
 Fahrenheit. 
 
 Mechanical equivalent of heat. One British thermal 
 unit is equivalent to 772 foot-pounds of mechanical work. 
 
 Second Law of Thermodynamics. Heat cannot pass from 
 a cold to a hot body by a self-acting process unaided by 
 external agency. 
 
Summary of Formulcz. 121 
 
 Boyle's law applied to gases. The product of the pressure 
 and volume of a portion of gas is a constant quantity so long 
 as the temperature remains constant. 
 
 For air at 32- the constant quantity is 26,214 foot-pounds. 
 Hence the expression of the law for air is : 
 
 ^=26,214 foot-pounds. 
 
 Law of Charles and Gay Lussac applied to gases. When 
 the pressure is constant all gases expand alike for the same 
 increase of temperature. The amount of the expansion 
 between 32 and 212 is '3654 of the original volume ; and 
 
 for each degree between 32 and 212 it equals 3 54 == 
 
 00203. 
 
 Similarly, when the volume remains constant the pressure 
 varies in the above proportion. 
 
 Combination of the two foregoing laws. The product of 
 the pressure and volume of a portion of gas is proportional 
 to the absolute temperature. Thus : 
 
 T l T 
 
 N.B. The absolute temperature is equal to the ordinary 
 temperature on Fahrenheit's scale plus 461. 
 
 Hence, remembering that the absolute temperature of 
 32 is 493, and that the value of pv for 32 is 26,214, we 
 get the very important law 
 
 /^=53' 2 r. 
 
 The specific heat of gas at constant pressure is the same at 
 all temperatures. 
 
 The mechanical equivalent of the heat required to raise 
 one pound of air, one degree, at constant pressure is : 
 
 Kf = '2375 thermal unit = 183-35 foot-pounds. 
 
 If a gas expand without doing external work, its tempera* 
 ture is unchanged. 
 
122 The Steam Engine. 
 
 The mechanical equivalent of the heat required to raise 
 one pound of air, one degree, at constant volume is : 
 
 K B = -1686 thermal unit = 130-2 foot-pounds. 
 The ratio of the two above numbers is : 
 
 LS3-25 o8 . 
 130.2 
 
 Expenditure of heat during isothermal expansion : 
 For isothermal expansion Boyle's law applies. 
 
 /. pv = constant, 
 
 and the external work done during the expansion 
 = pv log e r = cr log e r foot-pounds. 
 
 As the temperature of the gas does not alter during the 
 expansion, there is no internal work done, and, consequently, 
 the above expression represents the total heat supply. 
 
 Expenditure of heat when expansion takes place according 
 to the formula : 
 
 pv n = constant. 
 
 The external work done during expansion 
 
 = A^-A^ 2 = ____ (ri-T,) foot-pounds. 
 n i n i 
 
 The internal work done in changing the temperature from 
 
 TI tO T 2 
 
 = K, (T S -T,) 
 
 Therefore, the total heat expended is the sum of the two 
 above quantities : 
 
 Expenditure of heat during adiabatic expansion : 
 
 The results in this case are got by substituting y = i -408 
 
 for n in the above formula. Hence 
 
 heat expended in doing external work during expansion 
 
 = (T! r 2 ) foot-pounds ; 
 400 
 
Summary of Formula. 123 
 
 the internal work 
 
 = K z; (T 2 -T 1 ) 
 
 and the total heat supplied 
 
 The final temperature in adiabatic expansion is 
 
 -408 
 
 T 2 = 
 
 The efficiency of a perfect heat engine is the ratio of the 
 difference of the absolute temperatures of the sources of heat 
 and cold, to the absolute temperature of the source of heat 
 
 Law connecting the pressure and volume of dry saturated 
 steam. 
 
 where the pressure is expressed in pounds per square inch 
 and the volume in cubic feet. 
 
 /. log. v = 2-516 -939 log./. 
 
 The specific heat of water is constantly varying. It has 
 the value unity = 772 foot-pounds only at the temperature 
 39-3. At 400 the specific heat = 802*88 foot-pounds. 
 
 Superheated steam. 
 
 The law connecting the pressure, volume, and absolute 
 temperature of superheated steam is 
 
 /*=-85'5r 
 
 as against/ v = 53-2 - in the case of air. 
 
 The mechanical equivalent of the heat required to raise 
 one pound of superheated steam, one degree, is 
 
 At constant pressure, 370-56 foot-pounds. 
 At constant volume, 285-03 foot-pounds. 
 
 The ratio of the two numbers, or y = i -3. 
 
124 The Steam Engine. 
 
 Total heat required to change one pound of water of 32 
 into steam 0/V =885,2oo + 235*46 (t 2 12) approximately. 
 
 If the water were originally hotter than 32, say / 1 . 
 
 Total heat required = 885, 200 + 23546 (/ 212) 772 
 (/! 32) approximately. 
 
 Of the above the quantity required to do external work 
 =.pv foot-pounds. 
 
 where pv is calculated from the equation/ v I 2 $ 475. 
 
 Zeuner's empirical equation for the heat expended in doing 
 external work. 
 
 External work= 15,450 + 846 th foot-pounds. 
 where h is the quantity of heat required to raise the water 
 from 32 to the temperature of the steam. 
 
 Expenditure of heat per pound of steam expressed by an 
 equivalent pressure. 
 
 where P is the equivalent pressure required. H is the heat 
 of formation of one pound of steam in foot pounds, and v 
 the corresponding volume of the steam. 
 
 Expenditure of heat per cubic foot swept through by the 
 piston. 
 
 TT 
 
 Heat expended = , 
 
 v 
 
 TT 
 
 and heat rejected = p-\-p b 
 
 where / = pressure of the steam per square foot, and p b 
 the back pressure per square foot. 
 
 Expansive working of steam. 
 
 To find expenditure of heat when steam at end of stroke 
 is dry and saturated. 
 
 Let H r be the total heat of formation of one pound of 
 steam having the pressure p f per square foot at some point 
 just before the end of the stroke and the corresponding 
 
Summary of Formula. 125 
 
 volume Vf, Let p m be the mean pressure, and let the other 
 symbols have the same meanings as before. 
 
 Heat rejected H/ p f v f -\-p b Vj. 
 
 Heat expended = tt f p f ? ; /+A ^/+(A A) zy= HH- 
 A ^/ A ^V- 
 
 Heat expended per cubic foot swept through by the 
 piston 
 
 Rankings empirical formula for the expenditure of heat 
 in a steam engine. 
 
 Heat expended =fl m + 15 p f for condensing engines. 
 
 = p m + 14/7 for non-condensing engines. 
 
1 26 The Steam Engine. 
 
 CHAPTER IV. 
 
 CONNECTION BETWEEN THE SIZE OF AN ENGINE, THE 
 EVAPORATIVE POWER OF THE BOILER, AND THE 
 EXTERNAL WORK WHICH CAN BE DEVELOPED. 
 
 Navier's modification of Boyle's law applied to steam Work done during 
 expansion of steam calculated by Navier's formula De Pambour's 
 theory of the double-acting steam engine Its two fundamental princi- 
 ples and their mathematical expression Analysis of the resistance to 
 the motion of the piston of a steam engine Back pressure Engine 
 friction Load ; horse power Examples of the connection between the 
 size of cylinder, the piston-speed, the rate of expansion, the evapora- 
 tion in the boiler, and the power developed by the engine Locomotive 
 engines Analysis of the resistance to be overcome by the pistons of 
 locomotives Back pressure Engine friction Resistance to uniform 
 motion of engine, tender, and train Resistance due to gradient 
 Resistance due to inertia of weights to be moved Atmospheric resis- 
 tance Application of De Pambour's theory to locomotives Examples. 
 
 IN the preceding chapters the physical properties of steam 
 and gases, and the laws which regulate their changes of state, 
 have been briefly considered. The purpose of the present 
 chapter is more practical, it being proposed to show what 
 mechanical work can be accomplished by an engine of 
 a given size working at a given rate of expansion, when 
 attached to a boiler capable of yielding a given quantity 
 steam. Or,- vice versa, what quantity of steam must be 
 evaporated in order that the engine may under the given 
 circumstances perform the given quantity of work. 
 
 It would be very easy to devise suitable formulae to solve 
 these questions if the law of expansion of dry saturated steam 
 in a cylinder could be expressed with the same simplicity as 
 Boyle's law for the expansion of gases. To facilitate 'the 
 calculation of the questions we will in this chapter make 
 
Boyle's Law applied to Steam. 127 
 
 use of an empirical modification of Boyle's law, discovered 
 by Navier, and which is more suitable for the purposes of 
 calculation than Rankine's law. 
 
 If we were to make use of Boyle's law, and to start with 
 the constant obtained by multiplying the pressure by the 
 corresponding volume of a pound of steam of atmospheric 
 pressure, the calculated volumes would all be too small for 
 pressures above the atmosphere, when compared with the 
 volumes as given by actual experiment. To avoid this 
 error, Navier increased the value of the constant in Boyle's 
 law, and added a small constant to the number showing 
 the pressure. Thus, instead of pv = c, Navier adopted a 
 formula of the form 
 
 C 
 
 where C is a larger number than c, and k is the constant 
 added to the pressure. 
 
 The numbers which give the best results are, for steam 
 between the pressures of 20 Ibs. and 180 Ibs. absolute C = 
 28,200 and / = 4 ; and for steam below 20 Ibs. = 31,000 
 and k = 4. Hence the formula becomes in the two cases 
 
 =- and* = . 
 
 The relative volume of steam is its volume compared with 
 that of the water from which it is formed. The Tables give 
 the absolute volumes qf steam formed from one pound of 
 
 water. Now as the volume of one pound of water is - 
 
 62-42 
 
 cubic feet, we have only got to multiply the volumes as given 
 in the Tables by 62-42 in order to get the relative volumes. 
 According to this formula, the ratio of the volumes V P1 to V p 
 at two different pressures Pj and P would no longer be equal 
 to the ratio of the pressures P to Pj but to the ratio 
 *>?!+& Thus, 
 
 Vpi 
 
 V P 
 
128 TJie Steam Engine., 
 
 Calculation of the work done during the expansion of steam 
 by Navier's formula. 
 
 Let P be the pressure of the steam on entering the 
 cylinder in pounds per square inch. After it has raised the 
 piston through the height /, let it be cut off and allowed to 
 expand to the end of the stroke L. At any point x between 
 I and L the pressure p may be got from the proportion 
 p+k : P +/::/ : x 
 
 and multiplying each side of the equation by the elementary 
 space dx and integrating between the limits x=l and x=~L, 
 we have 
 
 = (P + )/log e ^-/(L-/). 
 / / 
 
 The quantity on the right-hand side of the equation is 
 the work done per square inch of piston during the 
 expansion of the steam. If the area of the piston be A 
 square feet, then the work done by the steam up to the 
 time it was cut off is equal to the total pressure on the 
 piston, or I44AP multiplied by the space through which it 
 moves = I44AP/. Add to this the total work done during 
 expansion 
 
 = i44A|~(P + /)/log e - - k (L /)"], 
 L I J 
 
 we get 
 
 Let the resistance which can be just overcome by the 
 steam be at the rate of R pounds per square inch, then we 
 must have, 
 
Work done by Expanding Steam. 129 
 
 In the above the fraction -= the rate of expansion 
 
 T-> . 7 
 
 usually denoted by the letter E and the fraction is, 
 
 according toNavier's formula, the ratio of the relative volumes 
 of the steam at the pressures P and R respectively. If we 
 represent these relative volumes by the symbols V p and V R 
 we get the equation : 
 
 V R 
 
 by means of which all questions relating to the work done 
 by steam when used expansively in a cylinder can be 
 calculated. 
 
 For example, take an engine having a cylinder 30 inches 
 diameter, and 60 inches stroke, working with steam having 
 a pressure of 60 Ibs. per square inch, cutting off at a quarter 
 stroke. Find the work done per stroke. 
 
 Y 
 
 ^ Ve have ( i + log e 4) = 4 x ^. 6 -. 
 
 R 
 
 Now, log e 4=1-3862, and V 60 =44o, from table of 
 
 volumes ; 
 
 > 1760 
 
 2-3862 2-3862 
 
 = 737*5 
 
 which number corresponds, according to the table, to the 
 relative volume for the pressure, 34*5 Ibs. per square inch. 
 
 This, then, is the value of all the resistances to the motion 
 of the piston, of whatever nature, reduced to pounds per 
 square inch ; and the work accomplished in overcoming 
 them, per stroke of the piston, 
 
 =34-5 xarea of piston x 5 feet 
 
 = 35x707x5 
 = 12 1, 95 7 '5 foot-pounds. 
 
130 The Steam Engine. 
 
 Similarly, we might have used the formula in order to 
 determine at what pressure steam should enter the cylinder, 
 in order that it might overcome a resistance of 34/5 Ibs. per 
 square inch, if cut off at a quarter-stroke. 
 
 DE PAMBOUR'S THEORY OF THE DOUBLE-ACTING 
 STEAM ENGINE. 1 
 
 De Pambour was the first to form a theory of the 
 mechanical action of the steam engine, connecting the size 
 of the cylinder, the evaporative power of the boiler, the rate 
 of expansion, and the work done. His theory depends on 
 the two following principles: 
 
 1. When the engine is running at uniform speed, the work 
 done by the steam on the piston is equal to the work due 
 to overcoming the resistance to the motion of the piston. 
 
 2. The steam which is evaporated in the boiler is equal 
 to that used in the cylinder. 
 
 The first of these propositions is evident to anyone 
 acquainted with the elements of mechanics. If the average 
 pressure on the piston were greater than the average resist- 
 ance, the motion of the piston would be accelerated, and the 
 engine would, consequently, not be running at a uniform 
 rate of speed. If, on the contrary, the resistance predomi- 
 nated, the motion of the piston would be retarded. 
 
 As to the second principle, it is now known not to be 
 strictly true. At first sight it would appear that, with the 
 exception of preventible leakages through safety valves and 
 imperfect joints, the steam generated in the boiler must all 
 be used in the cylinder, but we know from Chapter III. 
 that when expansion takes place in a cylinder, part of the 
 
 1 The term double acting, which has not been previously explained, 
 is applied to those engines in which steam acts alternately on either 
 side of the piston instead of on one side only, as is the case with a few 
 engines in use in Cornwall and elsewhere, for pumping water from mines. 
 
De Pamboufs Theory of the Steam Engine. 1 3 1 
 
 steam is condensed back into water, and, as we shall after- 
 wards see, if this condensation is prevented in the cylinder 
 by the use of a steam-jacket, the condensation takes place 
 in the jacket, instead of in the cylinder, and in neither case 
 can the steam be said to be used in the cylinder, in the 
 manner contemplated by De Pambour, who, in his theory, 
 supposed that the whole of the steam was used in producing 
 mechanical effects. In spite of this defect, the theory may 
 be accepted as sufficiently accurate for many practical 
 purposes. 
 
 Mathematical expression of De Pambour's first principle. 
 
 Let P be the pressure of the steam when admitted to the 
 
 cylinder. 
 /be the pressure of the steam when the piston has 
 
 moved over the space x, after the steam has 
 
 been cut off. 
 
 L be the length of the stroke in feet. 
 /be the distance traversed before the steam is cut off. 
 Cbe the clearance, i.e. the space between the initial 
 
 position of the piston and the bottom of the 
 
 cylinder. 1 
 
 Before expansion commences, the work done per square 
 inch of area of piston is equal to the pressure P multiplied 
 by/, the space traversed=P/. After expansion commences, 
 the pressure must be calculated by Navier's formula. Thus, 
 at the moment expansion commences, the space occupied 
 by the steam is /+C. At any point x, the space occupied 
 is x + c, and we have, as before, the following proportion: 
 
 : x + C\:p+k : 
 from which we find the pressure p at the position x is 
 
 1 The clearance includes also the contents of the admission passage 
 for the steam, reduced to a corresponding length of cylinder. 
 
 K 2 
 
132 The Steam Engine. 
 
 Multiplying by dx to obtain the work done during the pas- 
 sage of the piston over the small space dx, and integrating 
 between the limits / and L, we obtain for the total work 
 done during expansion the expression 
 
 log e *1C (p+j) (/+Q-* (L-/). 
 
 If to this we add the work done before the steam was cut 
 off, viz. P/, we obtain the total work done during the 
 stroke 
 
 Now, according to De Pambour's first principle, this 
 quantity of work equals the total resistance which the 
 piston encounters, multiplied by the length of the stroke. 
 The resistance is composed of different elements, some of 
 which vary in amount at different parts of the stroke ; but, 
 calling the average resistance R pounds per square inch of 
 piston area, and RL=work done per stroke in overcoming 
 the resistance ; we have, 
 
 10K L + C / _ (R + *) L 
 
 > e 7Tc + 7+C~"(P7I) (7+C) 
 
 Now, by Navier's law - is the ratio of the relative 
 
 i T~ K 
 
 volumes of the steam at the pressures P and R, or -. 
 Also - - is a ratio which it is 
 
 I ~r Vx 
 
 the separate symbol E. Hence 
 
 R 
 
 Also - - is a ratio which it is convenient to represent by 
 
 I ~r Vx 
 
 is the mathematical expression of the first principle. 
 
De Pambour's Theory, 133 
 
 The mathematical expression of the second principle is 
 as follows. Let F be the number of cubic feet of water 
 evaporated per minute in the boiler, then the steam formed 
 from this water at the pressure P=FV P cubic feet. This 
 quantity, according to De Pambour, is equal to the quantity 
 consumed in the cylinder. Now the amount of steam used 
 in the cylinder per minute depends on the piston speed, 
 i.e. on the number of strokes made per minute and on the 
 point of cut-off, viz. /+C. 
 
 If v be the piston-speed per minute, 
 
 then ~ = the number of strokes per minute, 
 J_j 
 
 and T 7 ' x a (1+ C) 
 
 the number of cubic feet of steam used per minute, where 
 t7= area of piston. 
 
 111 
 
 but we have already seen that 
 
 Therefore, va x =FV P , or f^=E.FV p , 
 
 and multiplying both sides by F we have, 
 
 Therefore, =E.F.V p =(log e 
 
 The resistance, R, to the motion of the piston is made up 
 of the back pressure, the friction of the engine, and the 
 load. The back pressure in non-condensing engines is 
 made up of the pressure of the atmosphere plus an additional 
 quantity due to the resistance of the steam ports to the 
 exhaust steam, and to the compression of the steam remain- 
 ing in the cylinder after the exhaust is closed by the valve. 
 
134 The Steam Engine. 
 
 The pressure of the atmosphere may be taken on the average 
 as 147 Ibs. to the square inch ; the remainder of the back 
 pressure varies in amount according to the size of the 
 exhaust passages, and the point of the return stroke where 
 the exhaust is closed. It may be roughly taken at 3 Ibs. 
 per square inch; making the total back pressure 17 7 Ibs. 
 In the case of condensing engines the back pressure depends 
 on the perfection of the vacuum maintained in the condenser,, 
 and also on the resistance of the exhaust passages, and the: 
 point where these latter are closed by the valve. Its amount,, 
 as a rule, in engines in fairly good condition, does not. 
 exceed 4 Ibs. per square inch. 
 
 The friction of the engine is made up of two parts, viz- 
 one due to the friction of the unloaded engine, and the: 
 other due to the load. The friction of the unloaded engine 
 depends on a variety of conditions, such as the diameter 
 of cylinder relative to the power given out by the engine, the 
 length of stroke, the relative length of the connecting rod 
 to the stroke, the nature of the valves whether balanced or 
 not, and, lastly, the general condition of the engine as to 
 workmanship, repair, and lubrication. It is consequently 
 impossible to assign a general value to this quantity. De 
 Pambour took it as being i Ib. per square inch for engines 
 of average size and in fair condition, and 0-5 Ib. per 
 square inch for large engines in good Condition. The ad- 
 ditional friction due to the load on the engine is extremely 
 difficult to calculate. As a general rule it may be said to 
 increase directly with the load, but this statement is by no 
 means universally true, for the load may in many cases be 
 driven from the main axle in such a way as to diminish the 
 friction on the main bearings, and experiments with the 
 friction brake have been made which show that in some 
 cases the loss due to friction remains nearly constant, while 
 the load on the brake is increased. 
 
 The addition to the friction assumed by De Pambour in 
 his calculations was one seventh of the load 
 
Resistances to Motion of Piston. 135 
 
 The load multiplied by the distance moved by the 
 piston in a given time is the useful work done by the 
 engine during that time. The load here referred to is not 
 the actual weight lifted, or pressure overcome by the engine 
 at the point of application of the weight or pressure, but is 
 supposed to be ' reduced ' to the motion of the piston in 
 the ratio of the length of the stroke to the actual distance 
 the real load is overcome during a stroke. Thus, supposing 
 an engine having a cylinder of 10 inches diameter, or 
 7 5 '5 square inches area, and 2 feet length of stroke to be 
 employed in turning a winding drum by its direct action, the 
 drum having a diameter of 3 feet ; and suppose further that 
 the drum is used to lift a weight of 10 cwt. vertically up- 
 wards; we have in this case the actual load of 1,120 Ibs., 
 which for each stroke of the engine is lifted up a height 
 equal to half the circumference of the drum or 471 feet, but 
 the load ' reduced ' to the piston is 
 
 iiQX4- 7 i = a6 lbs . 
 
 2 
 
 and the pressure due to the load per square inch of piston 
 area is 
 
 If we call the pressure per square inch due to the load=?/, 
 and assign to the other elements of the resistance their 
 values as given above, we have for non-condensing engines 
 
 and for condensing engines 
 
 o 
 
 R= 
 
 The horse-power of the engine is found by calculating the 
 number of foot-pounds of work done in the cylinder per 
 minute, and dividing the result by 33,000, which latter 
 number is the number of foot-pounds per minute equal to 
 
136 T/te Steam Engine. 
 
 one horse-power. In calculating the power of the engine, the 
 work done in overcoming the back pressure, which is for 
 the most part pure waste, is left out of the account. The 
 useful power of the engine is simply the power required to 
 overcome the resistance due to the load, and is equivalent 
 to the power found as above, minus that required to over- 
 come the friction of the engine. Thus, if w be the pressure 
 per square inch due to the load, then i^a.w total pressure 
 on piston due to load, and if v be the velocity of the piston 
 in feet per minute, then 1 440.2;. w work done per minute in 
 
 overcoming resistance of load, and I A^ a ' v ' w '= useful horse- 
 
 33,000 
 
 power exerted. 
 
 EXAMPLE. 
 
 The diameter of cylinder of a non-condensing engine is 15 inches, 
 the stroke is 30 inches ; the number of revolutions per minute = 70 ; 
 the steam is cut off at one third of the stroke, and the boiler is able to 
 evaporate -4 cubic ft. of water per hour: find out the useful power 
 given out by the engine. 
 
 We have area of cylinder in square feet = I -227. Piston speed in 
 
 30 x 2 x 70 
 feet per minute = 2. ! = 350. 
 
 We have already established the relation, 
 
 pg=nog e .* + v---pj'F.V R) we require to find R. 
 
 The quantity within the brackets has for a cut-off -i stroke, and 
 clearance = 5 per cent, of stroke, the value I -87. Also F = -4 cubic 
 feet. Hence we have 
 
 which number, by the table of Volumes, corresponds with the pressure 
 of 45 -5 Ibs. per square inch. 
 
 This pressure is the total resistance to the piston per square inch, 
 
 which, as the engine is non-condensing = + 187 Ibs. per square 
 inch. 
 
 .-.R- +187 -45-5 
 
Examples on De Pambour's T/ieory. 137 
 
 And the useful power 
 
 = 23-4 x 144.*.* = 43 . 6HP . 
 
 33,000 
 
 Find the initial pressure of steam in the cylinder. 
 We have, 
 
 which number corresponds very nearly with the relative volume for the 
 pressure 65 Ibs. per square inch. 
 
 Find the power of the engine with the above evaporation and cut- 
 off provided a condenser were used. 
 
 We should obtain as before V R = 573, and R = 45'5 Ibs. per square 
 
 o 
 
 inch, but in this case - iv + 5 = R = 45 -5, 
 
 .". '('- - ^ - =35 '5 Ibs. P. er square inch nearly, 
 
 and the useful horse-power exerted 
 
 = i_44..px35. 5c66 
 
 33,000 
 
 Find the power, and the initial pressure in the cylinder, if the cut- 
 off were at half the stroke, all the other data as above. 
 Using the same equation as before, we have 
 
 The quantity within the bracket has in this case the value 1-55 
 
 = 37'5 Ibs. P er square inch- /+ 187, 
 '.iv= 1 6 '4 Ibs. 
 
 and the useful power exerted 
 
138 The Steam Engine. 
 
 This last example shows the advantage gained by working expan- 
 sively, when the evaporation remains constant, the powers exerted at 
 cuts-off and \ being respectively 43-6 and 30-5. 
 
 Let everything remain the same as in the last example except the 
 piston speed, which is reduced to 300, we have 
 
 3 68=-62V R , 
 
 .*. VR = 593 .*. R = 43'5 Ibs. per square inch, 
 .*. TV = 217 Ibs. per square inch, 
 
 and the useful horse-power 
 
 33 5 ooo 
 
 This example shows that by reducing the piston-speed, 
 all the other conditions remaining constant, we can increase 
 the power exerted, because the slower speed, with a fixed 
 evaporation, enables a higher pressure to be maintained in 
 the boiler, and consequently a higher average pressure in the 
 cylinder. Hence the conclusion might be drawn that, the 
 slower the piston-speed with a fixed rate of evaporation, the 
 more advantageously the engine could be worked, and this 
 conclusion is true in so far as slow piston-speed favours the 
 attainment of high pressure, though, as will be explained 
 subsequently, slow speeds cause a loss of power in pro- 
 moting the condensation of the steam in the cylinder, 
 a circumstance of which De Pambour's theory takes no 
 account. The problem how to work the engine to most 
 advantage with a given boiler-power is not the one which 
 most frequently presents itself to the engineer ; on the 
 contrary, it is more often his business to proportion the boiler 
 power to the engine, rather than to adjust the rate of 
 expansion of the engine so as best to suit the boiler. 
 
 The problem of finding the evaporation, i.e. the necessary 
 supply of steam, when the size of cylinder, rate of expansion, 
 speed of piston and power of the engine are given, can also 
 be easily solved by means of the equation made use of in 
 solving the above examples. Also, by the help of this same 
 formula, and the empirical formula for the relative volume 
 
Examples on De Pambour^s Theory. 139 
 
 of steam (see page 127), the diameter of the cylinder may be 
 calculated when the power of the engine, the piston-speed, 
 the cut-off, and the evaporation are known. 
 
 EXAMPLE. 
 
 The useful power exerted by an engine is 43 '6 H. P. ; the rate of 
 evaporation is "4 cubic foot per minute ; the piston-speed, 350 per 
 minute, the cut-off takes place at one third the stroke : find the diameter 
 of the cylinder. We have as before, 
 
 /. 350 a= I -87 x -4 xV R ; 
 
 7 
 
 and, according to Navier, V R = ----- ; 
 
 R + 4 
 
 substituting the value of VR, and reducing, and remembering that 
 i^.a.v.w _ tlie use f ul horse-power = 43 -6, 
 
 we obtain a - \ -20 square feet, 
 or d--^ 14*98 inches, 
 
 a result which agrees very well with the previous example when the 
 approximate nature of the formula for V R is taken into account. 
 
 LOCOMOTIVE ENGINES. 
 
 In the case of locomotives the equations hitherto used 
 apply in principle, but the expression for R, the resist- 
 ance, requires modification. To understand the nature 
 of the resistance a short account must be given of the 
 method of action of locomotives. Fig. 24 is an outline 
 diagram of such an engine. A is the boiler, a descrip- 
 tion of the details of which will be found commencing 
 at page 371 ; B is one of the cylinders. It will be noticed 
 that the connecting rod is attached to a crank formed on the 
 axle of the main or driving wheels, one of which is seen at 
 c ; the wheels are keyed to their axles. Two of the other 
 
140 
 
 The Steam Engine. 
 
 wheels, which support the weight of the engine and boiler, 
 are seen at c',c'. The frame to which the cylinders are 
 bolted, and on which the boiler rests, and which also carries 
 the bearings of the three axles, is shown at e, e. With the 
 details of the construction of the locomotive we have now 
 nothing to do ; we merely wish to arrive at the relationship 
 between the resistance to be overcome by the pistons, the 
 
 Fig. 24. 
 
 size of the cylinders, and the evaporative power of the boiler. 
 The locomotive engine has not merely to propel itself along 
 the rails, but also to pull after it a tender containing coals 
 and water, as well as the train. When steam is admitted to 
 the cylinders, and the pistons move backwards and forwards, 
 the driving wheels, r, revolve. When this takes place, one 
 of two things must happen either the wheels will revolve, 
 the whole engine remaining stationary, the wheels merely 
 slipping on the rails ; or else the wheels will revolve and 
 at the same time roll on the rails, thus propelling the whole 
 engine, together with the tender and train attached to it, 
 through space. The former of these two effects would take 
 place if the surface of the rails and the outside rim of the 
 wheels were perfectly smooth and hard, so that there was no 
 friction between them ; or, even if friction subsisted between 
 the wheel and rails, the former might merely slip on the 
 latter, if the resistance to motion of the train were very great. 
 The latter of the two effects would take place whenever the 
 
Resistances of Trains and Locomotives. 141 
 
 friction between wheel and rails is sufficient to allow of the 
 resistance to motion of the train being overcome. 
 
 The friction between the driving wheels and rail is 
 called the adhesion, and is equal to the weight on the 
 wheels multiplied by a coefficient which depends on the 
 condition of the surface of the rail. In ordinary dry weather 
 the coefficient equals 0-15, but in damp weather or when the 
 rails are greasy it falls to 0*07. 
 
 The resistance which has to be overcome by the pistons 
 is made up of several elements. 
 
 i st. There is the back pressure which, as a locomotive 
 is a non-condensing engine, is equal to the atmospheric 
 pressure plus an extra quantity due to the forcing of the 
 exhaust steam through the blast pipe (see page 376), and to 
 the compression caused by the closing of the exhaust port 
 before the end of the stroke when the engine is working ex- 
 pansively (see page 338). 
 
 The excess of pressure due to the blast pipe when all the 
 other conditions remain constant varies 
 
 As the square of the speed of piston ; 
 
 As the pressure of the steam at the moment the exhaust 
 begins, i.e. at the point of release, which point in the case 
 of locomotives is implicitly connected with the rate of 
 expansion (see page 338) ; 
 
 Inversely as the square of the area of the nozzle of the 
 blast pipe. 
 
 The above is only applicable to the case of dry 
 steam. When much water is present in the cylinder at the 
 point of release, the back pressure may be increased up to 
 as much as seventy per cent, in excess of what it would 
 be with dry steam. 
 
 2nd. The resistance due to friction of the mechanism of 
 the engine. This quantity, as in the case of stationary 
 engines, in part depends upon the load, but it is usual in 
 calculations affecting locomotives to make use of a formula 
 which includes not only the friction of the mechanism but 
 
142 The Steam Engine. 
 
 also the resistance to rolling motion at a uniform speed of 
 the whole weight of the engine and tender. This formula 
 is given below. 
 
 3rd. The frictional resistance to uniform motion of the 
 whole train including the engine and tender. This is 
 usually expressed by giving the direct pull in pounds neces- 
 sary in order to propel each ton's weight of the train along a 
 level line at a given speed. The pull varies with the condition 
 of the line, the state of the surface of the rails, the state of 
 the rolling stock and the speed. For instance, with line and 
 rolling stock in good condition, dry rails, and a speed of ten 
 miles an hour, it would be necessary to exert a pull of about 
 8^ Ibs. per ton of weight of train, including engine and 
 tender. 
 
 If M be the speed in miles per hour, and T the weight 
 of the train in tons, exclusive of engine and tender, the 
 resistance to uniform motion may be expressed by the 
 formula 
 
 {6 + ' 3 (M-io)}T. 
 
 If T 1 be the weight of the engine and tender, the corre- 
 sponding resistance is 
 
 which expression includes the frictional resistance of the 
 mechanism of the engine referred to in the preceding para- 
 graph. 
 
 4th. Resistance due to gradient. If the train be moving 
 up an incline, its whole weight has to be raised up a vertical 
 height equal to the difference in levels between the foot and 
 the summit of the incline. If the gradient be represented 
 
 by the fraction -, then, by the laws of statics, the force in 
 a 
 
 pounds necessary to lift a weight of T tons up such an 
 incline, neglecting friction, is 
 
Resistances of Trains and Locomotives. 143 
 
 This quantity has to be added to the frictional resis- 
 tances, as set forth in the last paragraph, if the incline is 
 ascending, and subtracted if descending. 
 
 5th. Resistance due to the passage of the train through the 
 air. This, of course, depends largely upon the force and 
 direction of the wind. In calm air, it is proportional to the 
 square of the velocity of the train. A strong side wind, by 
 pressing the tires of the wheels against the rails, may 
 increase the frictional resistance of the train by as much as 
 twenty per cent. No formula has yet been devised which 
 satisfactorily takes account of the resistance due to the ever- 
 varying force and direction of the wind and speed of the 
 train. 
 
 6th. When a train is being started from a state of rest, 
 in addition to the frictional resistances to motion, the whole 
 mass of the train has to be put in motion that is to say, the 
 inertia of the train has to be overcome. The resistance due 
 to this cause is the principal one which has to be considered 
 in the case of urban railways. It does not, however, enter 
 into the computations affecting trains which have attained a 
 uniform rate of motion, and it is these latter only which are 
 dealt with in De Pambour's theory. 
 
 It will be noticed that many of these elements of resis- 
 tance, such, for instance, as the back pressure and the force 
 of the wind, depend, for their numerical value, on so many 
 variable circumstances that it is impossible to express them 
 accurately with simplicity. By substituting their values, 
 when obtained, for R in the general equation for double- 
 acting engines, viz. 
 
 va= 
 
 it would, of course, be possible to obtain an equation which 
 would enable all problems, connecting the speed of the 
 engine, the rate of evaporation, the dimensions of the 
 
144 The Steam Engine. 
 
 cylinders, the rate of expansion, the weight of engine, 
 tender, and train, and the varying resistances, to be solved ; 
 but such an equation would be too complicated, and, when 
 used for finding the speed, of too high dimensions for 
 ordinary use. The analysis of the resistances given above 
 will, however, be useful to students, as it will often facilitate 
 the solution of individual problems. 
 
HS 
 
 CHAPTER V. 
 
 THE MECHANICS OF THE STEAM ENGINE. 
 
 F.lerheniary principles of dynamics Definitions of mass, weight, velocity, 
 motion, force Units employed in their measurement The laws of 
 motion and examples of their application Work and energy Motion 
 of bodies in circles App'ication 10 fly-wheels Centrifugal force Con- 
 version of work done in the cylinder into work done on the crank 
 ist case, when pressure of steam is uniform, connecting rod supposed 
 to be of infinite length and moving parts without weight 2nd case, 
 when steam is allowed to expand, the other conditions remaining 
 unchanged Curve of effort on crank pin 3rd case, when the length 
 of the connecting rod is taken into account 4th case, when the 
 weights and velocities of the reciprocating parts are taken into account 
 Power absorbed in accelerating these parts Power restored by their 
 retardation The consequent modification of the pressures shown by 
 indicator diagrams necessary for calculating effort on crank pin Effect 
 of steam distribution on the action of the moving parts Means of 
 equalising the tangential effoit on crank-pin Fly-wheels Theory of 
 their action Graphic diagrams il.ustrating their action. 
 
 BEFORE discussing the questions of applied mechanics 
 which arise in the study of the moving parts of the steam 
 engine, it will be useful for the sake of accuracy, to re- 
 capitulate briefly the elementary principles of dynamics, a 
 previous acquaintance with the principles of the composition 
 of resolution of forces and velocities, on the part of the 
 reader being, however, assumed. We will start with the 
 following definitions : 
 
 1. Mass. This word denotes the quantity of matter 
 contained in a body. 
 
 2. Weight is the attraction which the earth exercises on 
 a mass. 
 
 3. Velocity is the speed at which a body moves, i.e. the 
 space which it traverses in a given time. 
 
 4. Motion. This word is employed in dynamics, not 
 
 L 
 
146 The Steam Engine. 
 
 merely to denote movement on the part of a body, but also 
 takes account of the mass of the body moved. Thus, if two 
 bodies have each the same velocity, but the mass of one 
 be double that of the other, then the motion, or quantity of 
 motion, or momentum, as it is variously termed, of the body 
 having the larger mass is double the motion of the smaller 
 one. if the bodies had each the same mass, but the 
 velocity of one were double that of the other, then the 
 motion of the body having the greater velocity would be 
 likewise double that of the other. When the velocity 
 remains constant, the motion varies as the mass moved. 
 When the mass is constant, the motion varies as the velocity. 
 Therefore, generally, the motion varies as the mass x the 
 velocity. 
 
 5. Force is any cause which produces, or tends to 
 produce, motion in a body, or which changes, or tends to 
 change, the motion of a body. 
 
 UNITS ADOPTED IN MEASURING MASS, WEIGHT, 
 VELOCITY, AND FORCE. 
 
 The only means we have of measuring the masses of 
 different bodies, i.e. the quantities of matter in them, is by 
 weighing them; that is to say, by comparing the attractions 
 of the earth on them relatively to some standard substance. 
 Consequently, the measure of mass is dependent upon the 
 unit chosen to measure weight. 
 
 The unit of weight adopted in this country is the weight 
 in London of a certain piece of platinum, kept in the office 
 of the Exchequer, and called a pound avoirdupois. The 
 weight of this piece of platinum varies in different parts of 
 the globe. Its weight depends on the attraction exercised 
 by the earth upon the matter contained in it. This force 
 of attraction, called gravitation, was discovered by Newton 
 to depend on the distance between the centre of the earth 
 and the object attracted. In consequence of the flattening 
 
Units of Mass, Weight, Velocity and Force. 147 
 
 of the earth towards the poles, and its bulging out towards 
 the equator, the surface of the earth is nearer to the centre 
 in London than in more southern places, and consequently 
 the weight of the standard piece of platinum is greater in 
 London than it is at the equator. 
 
 The mass of a body is then measured by its weight at 
 a given place. There are two units of mass made use of in 
 dynamics. The so-called gravitation unit of mass is the 
 quantity of matter contained in a body weighing 32-2 pounds. 
 
 The so-called absolute unit of mass is the quantity of 
 matter in a body weighing one pound. 
 
 Let M denote the mass of a body measured by the gravita- 
 tion unit, then its weight by the definition is W=M X32'2 
 Ibs. The symbol is used to denote the number 32-2. 
 Hence we have 
 
 The velocity of a body is measured in different ways 
 according as it is a linear velocity, i.e. due to a motion of 
 translation of the body from one point to another ; or an 
 angular velocity, i.e. due to the rotation of the body round 
 an axis. 
 
 Velocity is uniform when the body traverses equal spaces 
 in equal times. When uniform, linear velocity is always 
 measured by the number of feet of linear space, traversed 
 in one second of time. Thus, for instance, we speak of a 
 body having a velocity of two thousand feet a second. 
 
 Let v denote the velocity of a body moving uniformly. Let 
 s denote total the number of feet which it passes over, and / 
 denote the number of seconds occupied in describing the s 
 feet. 
 
 Then v= s or s=t.v. . 
 
 Further, the centrifugal force (seepage 159) which in all 
 cases tends to diminish the weight is greater at the equator 
 
 than at the poles. 
 
 L 2 
 
148 The Steam Engine. 
 
 Force is also, like mass, measured in two ways. Accord- 
 ing to the gravitation system, a force is measured by the 
 weight which it can support. Thus a string is said to 
 exercise a force of ten pounds when the tension in the string 
 is sufficient to prevent a force of ten pounds from falling to 
 the earth. The unit of force in gravitation measure is the 
 force which can support a weight of one pound. 
 
 The second or absolute system of measuring force is 
 more in harmony with the definition. By this system a 
 force is measured by the velocity which it can impart to a 
 given mass in a given time, when acting continuously on the 
 mass for that time. Thus, for instance, the force of gravity 
 is measured by the velocity which it can generate in a given 
 mass when acting on it for a second of time. The unit of 
 force in absolute measure is the force which can generate a 
 velocity of one foot per second, when acting on a mass 
 weighing a pound, during one second of time. 
 
 THE LAWS OF MOTION. 
 
 First Law. Every body continues in a state ot rest or 
 of uniform motion in a straight line, unless compelled by 
 impressed forces to change that state. 
 
 This law lays down that matter has of itself no power to 
 change its own condition of rest or of uniform motion. In 
 other words, it possesses what is called inertia. Consequently, 
 when we note that a body is not moving with a uniform velo- 
 city, we know that it is being acted upon by external force. 
 
 Second Law. Change of motion is proportional to the 
 impressed force, and takes place in the straight line in which 
 the force is impressed. 
 
 In the above statement the word motion has the meaning 
 already explained, viz. Mass multiplied by velocity. 
 
 Thus if two equal forces act on two unequal masses, the 
 quantity of motion generated in each case will be the same, 
 but the greater mass will have the least velocity, and the 
 
Laws of Motion. 149 
 
 product of the mass multiplied by the velocity will be the 
 same in each case. If the velocities of the two masses are 
 to be equal, then the force acting on the greater mass must 
 be greater than the other in the same ratio that the mass 
 itself is greater. 
 
 This law enables us to compare the relative magnitudes 
 of forces, for we have only to observe the velocities generated 
 by the various forces in the same mass, when acting for the 
 same time. The standard for comparison is the velocity 
 generated in a mass by the force of gravity, i.e. by its own 
 weight when acting for a second of time. This velocity is g 
 or 32-2 feet per second ; that is to say, if the force of gravity 
 act on a free body for one second, it will at the end of the 
 second have imparted to the body a velocity of 32-2 feet per 
 second. Thus a force F acting on a body weighing 20 Ibs. 
 for a second generates a velocity of 50 feet per second ; 
 what is the magnitude of the force in gravitation measure ? 
 
 The force of gravity, i.e. the weight of the body, or 
 20 Ibs., would generate a velocity in the body of 32-2 feet 
 per second. Therefore we have 
 
 F : 20:: 50 : 32'2. 
 
 ^ 
 
 In the general case, if the velocity v be generated in one 
 second by a force F in a body weighing W Ibs., then 
 
 F : W::z; : . 
 
 . 
 
 S 
 
 The statement in the above law that the change of 
 motion takes place in the direction in which the impressed 
 force acts may be illustrated by the following example : 
 
 A ball is projected from a rifle in a perfectly horizontal 
 direction, AB (fig. 2 5), from a height above the ground, AH. 
 Directly it leaves the muzzle of the gun it is acted on by two 
 
ISO 
 
 The Steam Engine. 
 
 forces, viz. the momentum acquired while in the barrel, and 
 which would in a given number of seconds carry it to, say, B ; 
 and the force of gravity which acting alone would in the same 
 time carry it to, say, H. According to the law, each force gene- 
 rates motion in the ball in the direction in which each acts, and 
 the consequence is, that at the end of the given time, the ball 
 
 H 
 
 Fig. 25. 
 
 will have travelled as far forward as B, and as far downwards 
 as H, and hence its resultant position will be C. The law 
 in effect states that dynamical forces may be compounded 
 and resolved in the same manner as statical forces. 
 
 The motion of bodies moving under the action of a 
 constant force. Let us take, for example, a body falling 
 freely from a state of rest under the action of gravity. At 
 the end of the first second, its velocity, having been zero to 
 start with, will have increased up to 32-2 or g feet per second. 
 As the force continues to act uniformly it will have produced 
 precisely the same effect by the end of another second ; that 
 is to say, the velocity will have been gradually increased by 
 another 32*2 feet per second. Hence the velocity at the 
 end of the second second will be 64/4 or 2g feet per second. 
 At the end of 3 seconds it will be 3^ and so on, and generally 
 at the end of / seconds vgt. 
 
 Next, as to the space traversed at the end of / seconds. 
 If it were travelling all the time with its final velocity gt 
 we should, using the formula s=^vt, have for the space 
 
Laws of Motion. 1 5 1 
 
 s = gt.t gf 2 . As, however, the velocity is constantly 
 increasing, we can only take the average, which, in conse- 
 quence of the perfectly uniform nature of the rate of increase, 
 or acceleration as it is called, is very easy to calculate, and 
 is in fact half the sum of the initial and final velocities, and 
 
 in this particular case = . * =^-. 
 2 2 
 
 Hence, instead of s=gt 2 we have 
 
 where v is the final velocity. These formulae connect the 
 space, time, and velocity. 
 
 If we wish to connect the space traversed with the final 
 velocity, without taking account of the time we can eliminate 
 / by combining the formulae, 
 
 s = ^gP, and v=gt. 
 Hence ,= k .|=J 
 
 Therefore v*-=2g.s. 
 
 v* 
 or s= . 
 
 *g 
 
 Third Law. To every action there is always an equal 
 and contrary reaction. 
 
 EXAMPLES OF THE APPLICATION OF THE LAWS OF 
 MOTION. 
 
 EXAMPLE (i). 
 
 A body falls from rest under the influence of gravity ; what will be 
 its velocity at the end of 10 seconds, what will be the total space tra- 
 versed, and what the space traversed during the tenth second ? 
 
 1. v=gt=32'2 x 10 = 322 feet per second. 
 
 2. s = $gt 2 = 16-1 x 100=1,610 feet. 
 
 3. Space described in nine seconds. 
 s = gt*=i(>'i x 81 = 1304-1 feet. 
 
 /. Space described in tenth second^ 1610 1304 'I 
 = 3S '9 f eet 
 
152 77/6' Steam Engine. 
 
 EXAMPLE (2). 
 
 A rifle bullet is shot vertically upwards with a velocity of 1,000 
 feet per second ; find the maximum height to which it would reach if 
 it experienced no resistance from the air. How long a time will it 
 take to reach this height ? The height to which the bullet will reach 
 is equal to the height down which a body must fall from rest in order 
 to acquire a velocity of 1,000 feet per second. For, from the moment 
 it commences to rise, its velocity is being diminished at the rate of 
 32*2 feet per second, till the initial velocity is all expended and the 
 bullet come to rest at the top of its flight. If from this point it com- 
 menced to fall it would attain a velocity of 1,000 feet a second on 
 reaching the starting point. Using, therefore, the formula 
 
 s _ v- _ i, 000,000 
 ~2'~ "2x32-2 ' 
 we have s= 15,527 feet. 
 
 To find the time occupied. As a velocity of 32*2 feet is generated 
 in each second, a velocity of 1,000 feet per second will be generated 
 
 32-2 
 
 = 31-05 seconds. 
 
 EXAMPLE (3). 
 
 A weight of 12 Ibs. rests on a perfectly smooth surface and is con- 
 nected by a string passing over a smooth 
 pulley with a weight of 6 Ibs. hanging 
 vertically downwards. What velocity per 
 second will the weights have at the end of 
 the first second from rest ? 
 
 The tension, T, in the string is uni- 
 form throughout the string. Also, the 
 L 6 j weight 12 is caused to move by the 
 Fig. 26. tension of the string, while the weight 
 
 6 is caused to move by its own weight 
 acting downwards, minus the tension in the string acting upwards. 
 
 Let v be the velocity generated in the mass of 1 2 Ibs. by T ; then 
 T : 1 2 : mass x v : mass x g 
 
 T * 
 
I 
 Examples on Laws of Motion. 153 
 
 As the velocity of the mass of 6 Ibs. is the same = v, and as this 
 velocity is generated by the force 6--T, we have 
 6 T I 6 ' mass x v '. mass x g 
 
 .. 
 
 12 6 
 
 .'. iST-72 /.T = 4lbs. 
 
 To find the velocity generated at the end of one second. 
 
 Gravity, or the weight 12, acting on the mass of 12 Ibs. for one 
 second, would generate a velocity of 32-2 feet per second, therefore the 
 tension T = 4 Ibs. will generate a velocity 
 
 = - - = 1073 feet per second. 
 
 EXAMPLE (4). 
 
 A mass of 200 Ibs. is moved from rest by a constant force F, and 
 passes over a space of 60 feet in the first second ; what is the measure 
 of the force ? 
 
 As the space passed over from rest is 60 feet, the velocity at the 
 end of the second is 2 x 60= 120 feet per second. The force of gravity, 
 or 200 Ibs., would in the same time generate a velocity of 32-2. 
 
 .*. F : 2oo::i2o : 32-2. 
 
 j2 2 
 
 The following example illustrates the application of the 
 laws of motion to the moving parts of a steam engine. 
 
 EXAMPLE (5). 
 
 The piston, piston rod, and connecting rod of an engine weigh 
 400 Ibs., the stroke if ft., and the number of revolutions 200 per 
 minute. During each revolution this mass starts from a state of rest ; 
 its velocity is gradually increased up to a maximum, and from this 
 point it diminishes till it comes to rest again at the end of the stroke. 
 During the first part of the stroke a certain proportion of the total 
 steam pressure is required to generate this velocity in the moving parts, 
 and whatever pressure remains over is all that is available for trans- 
 mission to the crank. During the latter part of the stroke, after the 
 
154 Tfo Steam Engine. 
 
 point of maximum velocity is reached, the speed of the moving parts 
 has to be reduced to nothing by the end of the stroke, and while this 
 reduction is taking place the moving parts press upon the crank-pin 
 with more or less severity depending on their weight, their maximum 
 velocity, and the space in which the velocity is reduced from the 
 maximum to zero. Consequently, while during the first part of the 
 stroke the crank-pin has only a portion of the steam pressure trans- 
 mitted to it, during the latter portion, on the contrary, it is subjected 
 not only to the full steam pressure acting on the piston, but also to the 
 extra pressure due to the pulling up of the moving parts. 
 
 In the present case the length of stroke and speed of rotation of the 
 crank are such, that at the commencement of the stroke the piston 
 moves through -000183 ft- in the thousandth part of a second ; what 
 is the total pressure, P, required to move the mass of 400 Ibs. from 
 rest over this space in a given time ? 
 
 The velocity at the end of the time = 2 x -000183 ft- P er n?oo" 
 .*. the velocity per second at the end of one second 
 
 = 2 x -000183 x iooo 2 = 365-4. 
 
 Now, gravity, or the weight of the moving parts, i.e. 400 Ibs., is 
 capable of generating in them a velocity of 32 -2 feet per second. 
 
 .-.400 : 32-2 :: P : 365-4. 
 
 The diameter of the cylinder is 10 inches, its area /. = 78'5sq. inches^ 
 and the pressure of steam at the commencement of the stroke necessary 
 to impart the required velocity to the moving parts 
 
 = = 57*8 Ibs. per square inch. 
 78'5 
 
 At the extreme end of the stroke the motion of the moving parts 
 is arrested at the same rate as it is imparted at the commencement, 1 
 and consequently a pressure at the rate of 57 -8 Ibs. per square inch 
 of piston area is transmitted to the crank in addition to whatever 
 pressure of steam may happen to be acting on the piston at that 
 moment. 
 
 Energy. The term energy, or capacity for doing work, 
 has been already explained (see page 25). The matter is 
 
 1 For the sake of simplicity the influence of the length of the con- 
 necting rod relatively to the length of the crank in retarding the motion 
 of the piston, c., is here neglected. 
 
Energy and Work. 155 
 
 now referred to again for the purpose of showing the effect 
 on the working of steam engines, of the energy stored up in 
 masses in motion. A body in motion possesses energy ; for, 
 if the motion be, for instance, vertically upwards, it will carry 
 the body up to a certain height before it is brought to rest, 
 i.e. it will overcome the attraction of the earth through a 
 certain space. The height to which the body will rise is, as 
 explained in Ex. (2), p. 152, equal to the height down which 
 the body must fall in order to acquire the same velocity. 
 
 In questions concerning the steam engine we are chiefly 
 concerned with the energy of bodies in motion. Very fre- 
 quently work is said to be stored up in a body in motion, 
 or in a raised weight What is really meant is that energy 
 or the capacity of doing work is stored. 
 
 Take as an example the case of a cannon-ball weighing 
 100 Ibs. and having a velocity of 2,000 feet a second ; what 
 is its capacity for doing work ? The velocity of 2,000 feet a 
 second would be acquired by falling down a height S, 
 calculated by the formula 
 
 8==,,= ft 
 
 2g 64-4 
 
 Therefore the velocity of 2,000 ft. per second is capable of 
 raising the body to a height of 62,111 feet, and the work 
 which would be done = the height multiplied by the weight 
 
 o 
 
 = 7 '_ xw=62,in x zoo foot-pounds. 
 
 2g 
 
 Vice versa, in order to impart this velocity to the can- 
 non-ball, 6,211,100 foot-pounds of work would have to be 
 done upon it before it left the bore of the gun. If the bore 
 of the latter were six inches in diameter, and ten feet long 
 from the front of the powder- cartridge to the muzzle, what 
 would be the average pressure of the powder gases per 
 square inch? As 6,211,100 foot-pounds of work have 
 to be done on the shot while it traverses the space of 
 ten feet, the total average pressure on the shot must be 
 
156 The Steam Engine. 
 
 6,21^200 _ 6 IIO lbs> Also as the di ameter o f the bore 
 
 10 
 is six inches, its area is 28-27 square inches, and the average 
 
 pressure per square inch= -? - = 21,970 Ibs., or a little 
 
 less than ten tons. 
 
 Similarly take the case of the steam engine given in Ex. 
 (5) p. 153. The moving parts which weigh 400 Ibs. attain a 
 maximum velocity towards the middle of the stroke, which 
 is reduced to nothing at the end of the stroke. Required 
 to find the work which the moving parts are capable of doing 
 after having attained their maximum velocity, the length of 
 stroke being if feet and the number of revolutions 200 per 
 minute. The path described by the crank-pin in each 
 rev. = i7T ft. = 5 '236 ft. and the velocity of the crank-pin 
 per second 
 
 ^ i . ft _ 
 
 The energy stored up in the moving mass at this velocity 
 
 is obtained from the formula 
 
 
 = 400 x 17-5 xi 7 -5 = I902 foot . poundSt 
 64-4 
 
 This energy is given out while the piston is traversing 
 half the stroke, 1 i.e. ten inches, and is consequently equiva- 
 
 lent to a pressure of I 9 02 x I2 2282*4 Ibs., acting through 
 10 
 
 this space. As the area of the piston is 78-54 square inches 
 the energy stored up in the moving parts is equivalent to 
 
 an average pressure of 22 2 4 =29'o6 Ibs. per square inch 
 
 7 8 '54 
 during the latter half of the stroke. 
 
 1 This statement is only true when the connecting rod is infinitely 
 long. It is also true for finite connecting rods if taken to apply to the 
 mean of the forward and back strokes. 
 
Motion of Bodies in Circles. 
 
 157 
 
 Motion of bodies in circles. In all the cases hitherto 
 considered, the motion has been in a straight line, but in 
 dealing with the mechanics of the steam engine cases of 
 great importance occur in which the motion takes place in 
 a circular path. Such for instance is the motion of the fly- 
 wheel, which is a wheel having a heavy rim. It is generally 
 keyed to the crank axle of the engine, and is used for 
 modifying the effects of any irregularity either in the driving 
 power or in the resistance to be overcome. When, for 
 instance, the driving power is in excess of the resistance to 
 be overcome, the surplus is expended in increasing the 
 velocity of the fly-wheel ; and, vice versa, when the resistance 
 is in excess of the driving power, the energy stored up in 
 the fly-wheel is expended in helping to overcome the 
 resistance, during which operation its velocity is lowered. 
 
 The consideration of the motion of bodies in circles is 
 somewhat complicated by the fact that different parts of the 
 bodies may be at diffe- 
 rent distances from the 
 centres of the circles in 
 which they are moving, 
 and as the velocities 
 necessarily vary directly 
 with the distance from 
 the centre, so also do 
 the quantities of motion. 
 Take, for instance, such 
 a body as a fly-wheel re- 
 presented by fig. 27. It 
 is composed of a rim, a 
 set of arms, and a central 
 boss. The velocity of 
 the rim is many times greater than that of the boss, and 
 again the velocity of the exterior portion, a, of the rim is 
 greater than that of the interior portion, b ; consequently 
 it is usual in calculations respecting fly-wheels, to consider 
 
 Fig 27 
 
158 T/ie Steam Engine. 
 
 the whole of the weight as concentrated at a certain distance 
 from the centre, where its effect will be the same as the sum 
 of the effects of the various portions of the real wheel, each 
 acting at its own distance from the centre. It is very often 
 a complicated calculation to determine this distance with 
 accuracy, but for all practical purposes we shall be suffi- 
 ciently correct if we take the mean radius of the rim as the 
 distance at which the whole of the weight is supposed to be 
 concentrated. 
 
 The laws of motion, as already stated and illustrated, 
 apply equally when the direction of the motion is in a 
 circle. Thus, for instance, if a weight w move round a 
 
 centre with a velocity ^, the energy stored up in it= -- For 
 
 a given number, N, of revolutions per second the velocity 
 v varies with the length of the radius r, and equals 2vrrN. 
 Substituting this expression for v in the above equation we 
 
 and consequently the energy varies as the square of the 
 radius, that is of distance of the weight moved from the 
 centre. A fly-wheel, therefore, of a given weight, the mean 
 radius of the rim of which is five feet in length, is rather 
 more than twice as efficient as a reservoir of energy as if 
 its mean radius were 3-5 feet. 
 
 EXAMPLE (6). 
 
 How much energy is stored in a fly-wheel of 5,000 Ibs. weight, the 
 mean radius of the rim of which is 4 feet, and the number of revolutions 
 60 per minute ? N.B. The whole of the weight is, for simplicity, sup- 
 posed to be concentrated at the end of the mean radius. 
 
 The mean velocity per second, v= 2<7r ' 4 ' =25-13 feet. 
 
 The energy = ^ = 5,000x631 "5 
 
 2g 64-4 
 
 = 49,029 foot-pounds. 
 
Motion of Bodies in Circles. 1 59 
 
 EXAMPLE (7). 
 
 A fly-wheel weighs 5,000 Ibs. and the mean rim moves with a maxi- 
 mum velocity of 35 feet per second. On account of the inequality of the 
 force transmitted to the crank, the fly-wheel has, during a portion of the 
 stroke, to expend 9,000 foot-pounds of energy ; what will its velocity be 
 after having done so ? 
 
 The maximum energy = 5 ' OO x 35 x 35 ^ g^ Io g foot-pounds. 
 64-4 
 
 After expending 9,000 foot-pounds, the energy remaining 
 = 95, 108 - 9,000 = 86, 108 foot-pounds. 
 
 /.^:= 86,108. 
 
 2 ^ 
 
 ,, 86,108 xy 
 5000 
 
 .*. = 33'3 feet per second, 
 
 being a loss of 1 7 foot per second from the maximum velocity, which is 
 equivalent to a variation of 4-8 per cent, from the maximum or of 2-4 
 per cent, from the mean velocity. 
 
 Centrifugal force. By the first law of motion a body will 
 continue to move in a straight line unless compelled to do 
 otherwise by impressed forces. When a body moves in a 
 circle it is, however, changing its direction from instant to 
 instant, and consequently must be continuously under the 
 influence of some force. Suppose this force were removed, 
 the body would no longer move in the circle, but would fly 
 off in a straight line at a tangent to the circle from the point 
 at which the force was removed. This is true of any and 
 every point on the circumference, from which it is evident 
 that the direction of the force which compels the body to 
 move in the circle is always at right angles to the tangent at 
 any point, and consequently always points to the centre. This 
 force, which keeps a body moving in a circle, is, on account 
 of its direction, always called the centripetal force. The 
 resistance which the mass of the body opposes to being moved 
 towards the centre, and which by the third law of motion is 
 
i6o 
 
 The Steam Engine. 
 
 equal to the centripetal force, is called centrifugal force. This 
 
 force may be measured as follows. 
 
 Let a body be supposed to start from the point a, fig. 28, 
 
 and move in the circle represented, with the uniform velocity 
 
 v feet per second. If the centripetal force F were removed, 
 
 the body would during a very 
 short time / move in a straight 
 line over the space ab. By the 
 second law of motion the effect 
 of the centripetal force would 
 therefore be to cause the body 
 to move over the space be during 
 the time t. 
 
 By a well-known proposition 
 in Euclid bcxbd= ab 2 . Calling 
 bc=x we have x (2r+x)=a& 2 . 
 
 As ab is supposed to be very small, and consequently also 
 
 be, we may neglect x 2 and put ab=ac. 
 
 Also, since the motion in the circle is uniform, and since ac 
 is the space moved in the time /, we have 
 
 Calling the weight of the body w, and / the velocity which 
 the centripetal force F can generate in w in one second, we 
 have 
 
 g 
 
 . We have next to express / in terms of v and r. Now x 
 is the space be which the body would move over from rest 
 under the influence of the centripetal force in time / sees. 
 
Centrifugal Force. 161 
 
 Therefore the velocity at the end of time t" zx per t" 
 = per sec. 
 
 Therefore the velocity which would be acquired at the 
 end of one second is 
 
 f 2X / 2X 
 
 Jf-^ 1 = -p 
 
 and substituting the value of x as given above, we have 
 
 This important expression which is constantly made use 
 of gives the centripetal force in terms of the weight of the 
 body, its velocity, and the radius of the circle in which it 
 
 moves. 
 
 If the velocity is given in revolutions per second, , we 
 have 
 
 V = 27T77Z, 
 
 and the above formula becomes 
 
 F== x 4 
 
 g ' r 
 1*226. 
 
 If the revolutions are given as so many per minute, N, we 
 
 have _ N 
 
 n 
 60 
 
 (]SJ\ 2 
 60) rxi ' 
 
 = ze/NVx 0-00034. 
 
 M 
 
162 The Steam Engine. 
 
 CONVERSION OF THE PRESSURE OF STEAM ON THE PISTON 
 INTO ROTATIVE EFFECT ON THE CRANK AXLE. 
 
 One of the most important applications of mechanical 
 science to questions relating to the steam engine is, to 
 ascertain the exact effect which the pressure of the steam on 
 the piston has in causing the crank to rotate. In dealing 
 with this question there are several points to consider : 
 
 First of all, in the great majority of cases the pressure 
 of the steam varies considerably at different parts of the 
 stroke. 
 
 Secondly, this variable pressure is transmitted to the 
 crank-pin through a connecting rod, which is constantly 
 changing its angle of inclination to the axis of the 
 cylinder, as it swings between its extreme positions on either 
 side of this axis. 
 
 Thirdly, the varying pressure transmitted through the 
 connecting rod meets the crank at an angle which is 
 constantly changing. The pressure may be resolved at the 
 crank-pin into two components, one in the direction of the 
 crank, and the other at right angles to it, i.e. tangential to 
 the circle described by the crank-pin. Of these the latter 
 alone produces any turning effect on the crank, the former 
 producing merely pressure on the bearing. The tangential 
 component, or turning effort on the crank, as it may be 
 called, varies in value continually, for it depends not only on 
 the net pressure of the steam on the piston, but also on the 
 varying angles of inclination of the connecting rod, and the 
 crank. 
 
 Fourthly, the effective turning effort on the crank 
 depends not only on the above-mentioned variables, but 
 also on the weights and velocities of the reciprocating parts, 
 viz. the piston, and piston and connecting rods ; for, as we 
 have seen before, p. 153, Ex. 5, a considerable proportion of 
 the steam pressure may, during a portion of the stroke, be 
 
Twisting Moment on Crank Shafts. 163 
 
 absorbed in merely imparting motion to the reciprocating 
 parts, and may consequently never reach the crank-pin at 
 all ; while on the other hand these parts as they come to rest 
 may impart a considerable pressure to the crank-pin quite 
 independently of the pressure due to the steam on the 
 piston. 
 
 The problem will be investigated in the first instance 
 freed from all possible complications. The pressure of the 
 steam will be supposed to be uniform throughout the 
 stroke. The connecting rod will be taken to be of infinite 
 length, in other words it will be supposed to act always 
 parallel to the axis of the cylinder. Lastly, the moving 
 parts will be imagined to be without weight, or their velocity 
 may be supposed to be so small that no appreciable part of 
 the steam pressure is absorbed in imparting motion to them. 
 
 In the next instance the pressure of the steam will be 
 supposed to vary during the stroke ; then the angular vibra- 
 tion of the connecting rod will be taken into account, and 
 finally the effects of the weights and velocities of the reci- 
 procating parts will be considered, In every case graphical 
 methods will be employed, in preference to analytical, to 
 investigate the problems. 
 
 In the diagram, fig. 29, let the circle ABC represent 
 the path of the crank-pin. Let AC represent the direction 
 of the axis of the cylinder. Let the pressure of the steam 
 on the piston throughout the stroke be P Ibs. per square 
 inch, and let the scale of the diagram be such that the 
 length of the radius OA represents P Ibs. The reason for 
 so doing will soon become apparent. First assume that the 
 crank lies in the position AO. The pressure transmitted 
 through the crank at this moment acts radially through the 
 centre O, and has no effect whatever in turning the crank. 
 The same is true when the crank occupies the position OC 
 hence the two positions OA, OC are called the dead centres. 
 Next suppose the crank to occupy the position OB, at 
 right angles to the dead centres. As the connecting rod is 
 
 M 2 
 
1 64 
 
 The Steam Engine. 
 
 supposed to be of infinite length it acts in the direction 
 B'B parallel to AC, and consequently the whole of the force 
 
 B'^e 
 
 Fig. 29. 
 
 transmitted has the effect of turning the crank. The arm of 
 the lever with which the force acts is BO, viz. the radius of 
 the crank, and the turning moment per square inch area of 
 piston=P xBO. The same is true for the position D dia- 
 metrically opposite to B. Hence we see that while the 
 crank is at A and C the steam pressure has no effect whatever 
 in turning it, at B and D, on the contrary, its whole effect is in 
 turning. At any other point in any of the four quadrants the 
 force of the steam is partly expended in turning the crank, 
 and is partly transmitted through the crank as mere pressure 
 on the main bearing at O. 
 
 Take, for instance, the point E. At this position, the 
 force acts with a leverage measured by the length of the 
 perpendicular let fall from the point O on the direction 
 of EE', viz. E // O=EF=EO sin a, and the turning moment 
 consequently 
 
 =PxEOsina. 
 
 The tangential force which acts at the end of the crank, 
 and tends to turn it round, as distinguished from the 
 
Twisting Moment on Crank Shafts. 165 
 
 turning moment is got by dividing the above quantity by the 
 length of the crank arm. Calling this force P T , we have 
 
 P x EO sin a T. 
 P T = -- EO =Pxsma. 
 
 This expression is equally true for any point on the 
 circumference of the circle. Hence, we see that the tangen- 
 tial pressure on the crank, when the connecting rod is 
 infinitely long, is equal to the pressure on the piston 
 multiplied by the sine of the angle of inclination of the 
 crank to the axis of the cylinder. 
 
 The same result may be got by resolving the force P at 
 the point E, into two components, viz. one acting radially, 
 ER, and the other tangentially, ET. Of these, ER merely 
 produces pressure on the main bearing, while ET alone 
 tends to turn the crank. 
 
 Now, ET=EE' sin EET. 
 
 Also, EE'=P=EO, the scale of the figure being such 
 that EO represents P. 
 
 And the angle EE'T=o, because E'T is parallel to EO 
 and the angles ETE' and EFO are both right angles. There- 
 fore the two triangles are equal, and ET=EF=P sin a. 
 
 Thus we see that, though the pressure on the piston may- 
 be perfectly uniform throughout the stroke, the turning effort 
 on the crank is very variable, and begins by being zero at 
 the dead centre, increases to a maximum when the crank 
 is at right angles to the axis of the cylinder, again decreases 
 to zero by the time the other dead centre is reached, and so 
 on during the return stroke, or second half of the revolution. 
 
 It may be here noted that it was this fact, that the 
 tangential pressure on the crank is always less than the 
 pressure on the piston, except for two positions of the crank, 
 which led old writers on the steam-engine into the blunder 
 of asserting that there is a loss in the employment of the 
 crank as a means for converting reciprocating into circular 
 motion. We now know, by the definition of work, that 
 there is no such loss ; for, although the average tangential 
 
1 66 
 
 The Steam Engine 
 
 pressure on the crank is much less than the pressure on the 
 piston, on the other hand, the path traversed by the crank 
 in a revolution is greater than that traversed by the piston in 
 a double stroke, in the ratio of the circumference of a circle 
 to its double diameter, i.e. 2 : TT- i : 1-57079. 
 
 By the principle of work, the lesser average pressure on 
 the crank, multiplied by the path described by the crank- 
 pin, must equal the greater pressure on the* piston multi- 
 plied by the space traversed by the latter. 
 
 Graphic representation of the tangential effort on the crank- 
 pin. The variable tangential pressure on the crank-pin 
 
 
 
 Fig. 30. 
 
 throughout a revolution can be very well shown, graphically, 
 by means of a diagram. Let the semicircle ABC (fig. 30) 
 represent the path described by the crank-pin during half a 
 revolution. Draw Qa to represent the uniform pressure on the 
 piston to scale, and with centre O and radius Oa, draw the 
 inner semicircle abc. Divide the circumference of this semi- 
 circle into 10 equal divisions, for the sake of convenience, and 
 draw radial lines through each point of division, intersecting 
 the semicircle ABC. Then, at each position of the crank 
 
Graphic Representation of Twisting Moments. 167 
 
 represented by the points of division of the outer semicircle 
 the tangential force on the crank is equal to the pressure on 
 the piston multiplied by the sine of the angle of the crank. 
 As aO represents the pressure on the piston, the tangential 
 forces are represented in magnitude by the perpendiculars 
 i, 2, 3, 4, 5, &c. let fall from the points of division of the 
 inner circle on the line aO. On the prolongations of the 
 radial lines beyond the outer circle set off the lines i', 2', 3', 
 4', 5', &c., equal, respectively, to i, 2, 3, 4, 5. Join the 
 extremities of these lines by the curved line ADC. Hence, 
 ADC represents, graphically, the tangential pressure at every 
 position of the crank ; since, for any position, we have only 
 to draw a radial line through the point in question, and the 
 piece intersected between the outer circle and the curved 
 line will represent the tangential force. If the tangential 
 pressure were uniform all round the circle the curved line 
 ADC would be a circle concentric with the path of the 
 crank-pin. Its deviation from concentricity is the measure 
 of its want of uniformity. The average tangential pressure 
 on the crank-pin may be represented by drawing the circle 
 EFG from the centre O, the line EA which represents this 
 average pressure being obtained by the following proportion 
 
 EA : aO or P.* .'2 : TT. 
 
 When the engine is running at a uniform rate of speed, 
 this average tangential pressure on the crank, is, of course, 
 exactly equal to the resistance which the work to be done 
 offers to the motion of the crank-pin ; for, if the resistance 
 were greater, the speed would be reduced, and if the resis- 
 tance were less, the speed would be increased, and in 
 neither case would the engine be running uniformly. Con- 
 sequently, this average tangential pressure circle may 
 equally well be called the Resistance Circle. 
 
 The diagram (fig. 30) only shows the tangential pres- 
 sures for one half of the revolution, but the other half is, of 
 course, a precisely similar figure, and need not, therefore, be 
 
i68 
 
 The Steam Engine. 
 
 shown. By inspection of the diagram, we see that there are 
 four points during a complete revolution where the actual 
 tangential pressure exactly equals the average, viz. the 
 points ee\ &c. ; where the resistance circle intersects the 
 curves ADC, AD'C. Between the points e e' and the corre- 
 sponding points e"e'" the pressure is in excess of the resistance, 
 while between e'"e and e'e", the resistance is in excess of the 
 pressure ; consequently, during the two first intervals, the 
 surplus work is poured into the fly-wheel, and during the 
 last two intervals the deficiency in work is supplied by the 
 energy of the fly-wheel being diminished. As the fly-wheel 
 can only receive or restore energy by having its velocity 
 increased or diminished, we see that the velocity of the 
 crank-pin is not, strictly speaking, uniform, but it can be 
 kept within any assigned limits of deviation from uniformity, 
 by altering the weight of the fly-wheel. 
 
 The diagram shown on fig. 30 can also be drawn on a 
 straight base instead of on the circumference ABC. This 
 form of the diagram is more generally used in practice, 
 because it is easier to test the accuracy of the work; but 
 it is not so graphic to the eye as the circular form of 
 fig. 30. To construct the diagram on a straight base, draw 
 a straight line AC equal in length to the semicircumference 
 of the circle described by the crank pin. Divide AC into 
 
 4 5 
 
 Fig. 30 A. 
 
 ten equal parts corresponding with the divisions of the semi- 
 circle ABC. From each of the points of division, i, 2, 3, &c., 
 erect a perpendicular, and mark off the lengths la, i/;, ic, 
 &c., equal to the lines i', 2', 3', &c., in fig. 30. 
 
Graphic Representation of Twisting Moments, 1 69 
 
 Through the points a b, c., draw the curve 
 then the ordinates of this curve will give the tangential pres- 
 sures on the crank for any position of the latter. 
 
 It is evident that the area bounded between the straight 
 line AC and the curve measures the work done upon the 
 crank ; for the ordinates represent the effective pressures on 
 the crank pin, and the abscissae the spaces through which 
 they are exerted. Now the amount of work done upon the 
 crank is, as has been shown above, equal to 'the work done 
 upon the piston. Hence the area of fig. 30A should be 
 exactly equal to the area of the indicator diagram. By 
 measuring the area and comparing it with that of the indi- 
 cator diagram, we have a ready check of the accuracy of the 
 work. 
 
 If we wish to show the diagram of tangential pressure 
 for the whole revolution, we have only to prolong AC to 
 double its original length and construct on the prolonged 
 portion another curve precisely similar to Aa&C. 
 
 We will now take the case of an expansion diagram and 
 show the effect which the want of uniformity in the steam 
 pressure acting on the piston has upon the form of the 
 diagram which shows the tangential effort on the crank-pin. 
 We will further suppose as before that the connecting rod is 
 infinitely long, and that the moving parts possess no weight, 
 or are moving at a very slow velocity. 
 
 The steam diagram is shown at the upper part of fig. 31. 
 The cut-off is supposed to take place at f\ of the stroke. 
 The engine is non-condensing. The first thing to ascertain 
 is the net pressure of the steam which urges the piston 
 forward. In order to find this it will in most cases be 
 necessary to construct from the ordinary indicator diagram 
 a new diagram showing the actual pressures after deducting 
 the corresponding back pressures (see page 340). In the 
 present instance this step will not be necessary, because to 
 avoid complication an indicator diagram has been chosen 
 in which the back pressure is uniform throughout the stroke 
 
The Steam Engine. 
 
 and the compression curve at the end of the return stroke 
 is exactly similar to the exhaust curve at the commencement; 
 consequently to obtain the net pressures on the piston we 
 have only to measure the length of vertical ordinate bounded 
 between the upper and lower boundary lines of the diagram. 1 
 
 Fig. 31. 
 
 Divide the length of the diagram into ten equal parts, and 
 from each point of division draw a vertical ordinate to the 
 upper boundary of the diagram. Draw a line AB of the 
 
 1 This is a case which would probably never occur in practice. In 
 all ordinary cases two diagrams are required : viz. one from the top 
 and the second from the bottom cover of the piston. The net pres- 
 sures are then taken by deducting from the gross pressure as shown by 
 one diagram the simultaneous back pressure as shown by the other 
 (see page 340). This precaution is very frequently neglected, and has 
 led to serious errors in the calculation of curves of twisting moments. 
 
Tivisting Moment derived from Indicator Diagram. 171 
 
 same length as the diagram to represent the diameter of the 
 crank-pin circle. Divide the line AB into ten equal parts, 
 and from each point 123 &c. erect a perpendicular 1,1' 
 2,2' 3,3' &c., intersecting the circumference at the points 
 i' 2' 3' &c. These lines are not actually drawn, so as to 
 avoid complicating the diagram. Then at each position of 
 the crank-pin i' 2' 3' &c., the direct pressure on the pin is 
 represented by the corresponding ordinate taken from the 
 indicator diagram. From the centre O draw radial lines 
 01', 02', 03', &c. intersecting the circumference. It is only 
 necessary to show the portions of these lines which are 
 prolonged beyond the circumference ACB. On these lines 
 measure off the parts Otf, O, CV, Qd, &c., equal respec- 
 tively to the ordinates of the steam diagram i, 2, 3, 4, &c. 
 Then, as in the first case, the actual tangential pressures on 
 the crank-pin will be equal to these lines O#, O, Or, O*/, 
 &c., multiplied by the sines of the angles which the crank 
 makes with AB. In other words, the tangential pressures 
 will be equal to the perpendiculars let fall from the points 
 a, b, c, d, &c., on AB, i.e. aa' , bb', cc 1 , dd , &c. Produce the 
 radial lines Oa, Ob, &c., and from the points i', 2', 3', 4', 
 &c. on the circumference set off the parts I'a", 2'b", 3V' 7 , 4 f d", 
 equal in length respectively to aa', bb', a ', dd', &c., then the 
 curve drawn from A to B through the points a", b", c", a 7 ", 
 &c., will be the diagram of tangential effort, or twisting 
 moment on the crank-pin. 
 
 To ascertain the diameter of the circle of average 
 tangential pressure, i.e. the resistance circle, we have only 
 to compute the average steam pressure as shown by the 
 
 indicator diagram and multiply the same by the fraction - 
 
 7T 
 
 the product will give the radius of the resistance circle. The 
 tangential effort on the return stroke is, of course, exactly 
 similar to the curve Aa"fr"B, and is obtained in the same 
 way. 
 
 The above method is purely graphical. In actual 
 
1/2 
 
 The Steam Engine. 
 
 practice it will probably be found more expeditious to con- 
 struct a curve of twisting moments, the radial ordinates of 
 which are found by multiplying the steam pressure for any 
 given position of the crank by the leverage at which it works, 
 and then setting off the moment thus obtained to scale. Thus, 
 for the position of the crank 2', fig. 31 A, we have a pressure 
 of iSlbs. to the square inch, according to the scale on which 
 
 the diagram is drawn 
 Scale i"= soils when the piston oc- 
 cupies the position 
 2 corresponding with 
 the position 2' of 
 the crank. Also the 
 leverage at which it 
 acts is z'b. If the 
 radius of the crank 
 be one foot, then 
 to the same scale 
 2^=799 ft., and the 
 product 18x799= 
 14-3 = the twisting 
 moment. Draw a 
 radial line through 
 2' and set off z'b'=- 
 14-3 to any conveni- 
 ent scale. Proceed 
 in a similar manner 
 for all other positions 
 of the crank, and the curve A'B drawn through the 
 extremities of the radial lines is the curve of twisting 
 moments. 
 
 Influence of the connecting rod in modifying the curve of 
 tangential effort on the crank-pin. In actual steam engines 
 the connecting rod is of course always of finite length, and 
 consequently is always acting at an angle to the axis of the 
 cylinder, except when the crank-pin is on the dead centres. 
 
 Scale & /6 - 1 
 Fig. 31 A. 
 
 ioot 
 
Influence of Connecting Rod on Twisting Moment. 173 
 
 The size of the angle which the connecting-rod makes with 
 the axis for any position of the crank-pin depends on the 
 length of the rod compared to the length of the crank. The 
 shorter the relative length of the rod, the greater the angle 
 for any given position of the pin. To find the inclination 
 of the connecting rod for any given position of the crank 
 we have only to take the length of the rod as a radius, and 
 
 C D 
 
 Fig. 32. 
 
 from the centre of the crank-pin to describe an arc inter- 
 secting the axis of the cylinder prolonged. The line joining 
 the point of intersection with the centre of the crank-pin gives 
 the angular position of the connecting rod and also the position 
 of the piston for the given position of the crank-pin. Thus, let 
 the length of the connecting rod be four times that of the 
 crank, so that when the crank-pin is at A the piston rod end 
 of the connecting rod will be at A', and AA'=4AO. When 
 the crank has moved round a quarter of a revolution to D, 
 with centre D and radius =AA' describe an arc intersecting 
 the line AO. The point of intersection will be D'. the 
 distance AD' being greater than the half-stroke of the piston, 
 which latter equals A'C'. When the piston is at half- stroke, 
 the crank will occupy the position OC. 
 
 The tangential effort on the crank due to the piston pres- 
 sure may be calculated by estimating the effect of this pressure 
 in the direction of the connecting rod, and then resolving 
 this tangentially and radially to the crank circle ; or, it may 
 be more conveniently computed geometrically by finding 
 
The Steam Engine. 
 
 the leverage with which the connecting rod acts on the 
 crank-pin. The amount of the tangential pressure will, as 
 before, be proportional to the length of the arm of the lever. 
 Take, for instance, the position E of the crank-pin, the 
 connecting rod assumes the position EE', and the leverage 
 with which it acts, instead of being EP, as would be the case 
 were the rod of infinite length, is ^O, found by producing 
 EE and letting fall a perpendicular upon it from O. The 
 product of the line eO into the pressure or tension in the 
 connecting rod gives the twisting moment. Now the pres- 
 sure or tension in the connecting rod is to the pressure on 
 the piston as the line EE' is to the line E'P. Also the tri- 
 angle Qee' is similar to the triangle EE'P, and 
 
 Oe : CV::E'P : E'E 
 .:Oe'xET=OexE'E. 
 
 That is to say, the pressure on the piston multiplied by Oe' 
 equals the force in the connecting rod multiplied by Oe, 
 which latter product is the twisting moment. 
 
 Hence to obtain the twisting moment we have only to 
 prolong the line of the connecting rod till it intersects the 
 position DD' of the crank, and the product of this line into 
 the pressure on the piston gives the moment required. By 
 proceeding in this way fig. 31 A may be modified so as to take 
 account of the influence of the connecting rod. By an in- 
 spection of fig. 32 it is evident that the new arms of the 
 moments are greater than those obtained with infinite con- 
 necting rods until the axis of the connecting rod intersects 
 the point D, after which they are less till the end of the 
 stroke is reached. During the return stroke the opposite 
 eifect takes place, the arms of the levers being shorter during 
 the first portion, and longer during the latter part of the 
 stroke, than for the corresponding positions when the con- 
 necting rod is of infinite length. The arm of the moment 
 is a maximum when the axis of the connecting rod makes 
 a tangent with the circle described by the crank-pin. 
 
Influence of Connecting Rod on Twisting Moment. 1 7 5 
 
 An inspection of the diagrams, figs. 30 to 31 A, shows how 
 far from uniform is the tangential effort on the crank-pin, 
 and consequently how irregular is the driving power, in the 
 case of single cylinder engines, even when, as in the first 
 case illustrated, the steam pressure is uniform throughout 
 the stroke, and the angularity of the connecting rod is 
 neglected. There are various methods of diminishing this 
 irregularity of driving power. One plan is to fit on to the 
 crank-shaft a fly-wheel of adequate weight and dimensions 
 to overcome the irregularity. The principle of the action 
 of fly-wheels has already been explained (page 157). Another 
 and very usual plan is to use two or more cylinders with the 
 cranks forming angles with each other. These are usually 
 so arranged that the tangential effort on one crank is a 
 maximum when it is a minimum on the other crank. This 
 subject will be again referred to, and examples will be 
 illustrated, when all the disturbing causes which influence 
 the forms of twisting moment diagrams have been ex- 
 plained, but in the meantime it will be a useful exercise 
 for the student to prepare such diagrams for the two 
 cases illustrated in figs. 30, 31 (pp. 166 and 170), assuming 
 that in each case a second cylinder of equal size with the 
 first, and working under precisely similar conditions, is 
 added, and that the cranks are set at right angles to each 
 other. 
 
 The method of proceeding is as follows: A precisely 
 
 similar diagram to the curve Ka"b"c" B, fig. 31 (p. 170), 
 
 must be constructed on the diameter COD which is at right 
 angles to AOB. When complete, there will be four of these 
 curves round the circle ACBD, viz. one for each stroke of 
 each cylinder. The radial lines intercepted between the cir- 
 cumference and the curves for each position or', 02', &c. 
 of the crank, &c., must then be added together, and a 
 resultant curve drawn through their extremities. This curve 
 will be a diagram of the tangential effort for the two 
 cylinders combined. 
 
176 The Steam Engine. 
 
 Influence of the weights and velocities of the reciprocating 
 parts. We must next consider the effects of the weights 
 and velocities of the reciprocating parts on the form of the 
 tangential effort diagram. In the first part of this chapter 
 (p. 153) it was shown that a large portion or even the whole 
 of the steam pressure during the first part of the stroke 
 might be absorbed in generating the velocity in these parts ; 
 and consequently, that only a portion, or in some case not 
 any of the steam pressure on the piston was available for 
 transmission to the crank-pin ; while, on the other hand, 
 the effect of the velocities of the parts being reduced during 
 the latter portion of the stroke would be to cause a greater 
 pressure to appear on the crank-pin than is due to the steam 
 pressure in the cylinder. It is evident, therefore, that to 
 construct a tangential effort diagram for such cases we must 
 first deduce from the indicator diagram a new diagram, by 
 taking away from the steam pressure, during the first part 
 of the stroke, such a portion of it as goes to accelerate the 
 velocity of the moving parts, and vice versa, adding to the 
 steam pressure during the remainder of the stroke such a 
 portion as represents the effect of their retardation. 
 
 If the steam pressure were free to act on the recipro- 
 cating masses in the same way that gravity acts on a falling 
 body, or in the way that the pressure of powder-gases in a 
 gun acts on a projectile, it would be easy to calculate the 
 effects produced ; but when the motion is controlled by a 
 crank revolving uniformly, it is a little more difficult to 
 calculate the increments of velocity imparted to the piston 
 in successive intervals of time. The pressures required to 
 impart these accelerations, as they are called, are of course 
 proportional to the amounts of the increments. 
 
 We must again suppose for the sake of simplicity that 
 the connecting rod is of infinite length. In such a case 
 while the crank travels through successive equal angles 
 corresponding to the positions A, A', A," (fig. 33) the piston 
 moves through successive spaces AA^ A^, A 2 A 3 &c. The 
 
Influence of Inertia of Reciprocating Parts. 177 
 
 velocity of the piston at each point A! A 2 A 3 &c. may be 
 calculated graphically as follows : 
 
 Let the radius of the circle ABC represent to scale 
 the linear velocity of the crank-pin in feet per second, At 
 any point A" for example, 
 corresponding to the posi- 
 tion A 2 of the piston, the 
 velocity of the crank-pin 
 may be resolved into two 
 components, one horizontal 
 and the other vertical. The 
 horizontal component will 
 be the velocity of the 
 piston. From the point 
 A" draw a tangent A"T 
 
 c 
 
 As O 
 
 Fig. 33- 
 
 to the circle. Make A"T=the radius of the circle, then 
 A"T represents the velocity of the crank-pin. Draw A"H 
 horizontal and TH vertical, then will A ;/ H and TH repre- 
 sent respectively the horizontal and vertical components 
 of the velocity at the point A", and A"H represents also 
 the velocity of the piston when at a point in the stroke 
 corresponding to the position A 2 in the line AC. In the 
 same way the piston velocity may be obtained for any 
 other point in the stroke. It may, however, be ascertained 
 more simply as follows. The two triangles A"TH and 
 A"A 2 O are in every respect equal to each other and A"H 
 equals A"A 2 . Now A"A 2 is the sine of the angle of the 
 crank x A"O, therefore the velocity of the piston at any 
 point A 2 is proportional to the sine of the angle of the crank 
 for that position, and is in fact equal to the length of the 
 perpendicular drawn from the given point, such as A 2 , to 
 meet the circumference of the circle, when the velocity of 
 the crank-pin is represented by the length of the radius of 
 the crank circle. Hence we see that the velocity of the 
 piston at a series of successive positions A l A 2 A 3 &c. is 
 represented by the vertical ordinates A'A^ A"A 2 A'"A 3 &c. 
 
 N 
 
The Steam Engine. 
 
 As the crank-pin is supposed to travel at a uniform 
 velocity, the crank-pin circle represents time, just as does the 
 hour circle of a watch, and equal divisions of this circle, such 
 as AA', A' A", A" A'", c., represent equal divisions of time. 
 Now, the difference between the velocity of the piston at the 
 beginning and end of any such interval is the increment of 
 velocity, or acceleration imparted to the piston during the 
 interval. Thus the difference between the velocities at A' 
 and A" equals A" a', and similarly between A'" and A" 
 equals A!" a". Now, the force or pressure required to im- 
 part a velocity to a given mass in a given interval of time is 
 proportional to the velocity imparted, and when this latter 
 and the mass are known the force may be calculated (see 
 p. 149). If we suppose the divisions AA' &c. to be taken so 
 small that the force acting throughout the interval may be 
 considered as uniform and the acceleration imparted uni- 
 formly, then in this case any division such as A" A'" may 
 be considered a straight line, and the two triangles A" A'" a" 
 and A // A 2 O are similar, because the angle QA"A'" may be 
 considered a right angle and a"A"A 2 is a right angle, and 
 taking away the common angle the remainder a"A'A'"= 
 the remainder OA"A 2 . Also the angles at a" and A 2 are 
 right angles, therefore the third angles in each triangle are 
 
 A'" a" 
 
 equal and the two triangles are similar, therefore = 
 
 .A. A. 
 
 A O 
 
 ?__ =cosine of angle of crank. Consequently the accelera- 
 A' O 
 
 tion at any position of the crank equals the velocity of the 
 crank-pin multiplied by the cosine of angle of crank, and 
 the forces required to produce the accelerations are pro- 
 portional to them. 
 
 The magnitudes of the forces may be found in the follow- 
 ing manner. Suppose that the weight of the reciprocating 
 parts is all concentrated round the crank-pin, the connecting 
 rod being, as before, infinite. The weight is kept moving in 
 the circular path by the action of centripetal force (see p. 159). 
 
Influence of Inertia of Reciprocating Parts. 179 
 
 The centripetal force always acts radially, and at any point 
 D, fig. 34, may be resolved horizontally and vertically. The 
 horizontal component is the measure of the force which 
 imparts motion to the reciprocating parts. The vertical com- 
 ponent merely produces an 
 upward or downward pres- 
 sure on the bearings. At 
 the dead centre A the 
 force has no vertical com- 
 ponent, and therefore the 
 entire centripetal force 
 produces acceleration of 
 the reciprocating parts. 
 At B there is no horizontal component, and therefore there 
 is at this point no acceleration, at any other point D, the 
 horizontal component =DE - FO=DO cos a. If the radius 
 of the crank circle represent the centripetal force, then the 
 horizontal component at any point = the centripetal force 
 multiplied by the cosine 
 of the angle of the crank. 
 The expression for 
 the centripetal force in 
 terms of the weight, the 
 revolutions per minute, 
 and the radius of the 
 crank, is given on p. 
 161, and is F=wN 2 . rx 
 00034. This may be 
 expressed in pounds per 
 square inch of piston 
 area by dividing by the 
 area of the piston. Draw 
 a circle A B C D (fig. 
 35) of which the length of the radius represents the centri- 
 petal pressure per square inch of piston area. Then the 
 pressures per square inch of piston area required to accele- 
 
 N 2 
 
i8o 
 
 The Steam Engine, 
 
 rate the reciprocating parts at any points corresponding 
 to the positions A' A" &c. of the crank will be represented 
 by the lengths of the horizontal lines AV, A" a" &c. While 
 the crank is moving from A to B the pressures go to accele- 
 rate the reciprocating parts, and must therefore be subtracted 
 from the indicator diagram pressures for the corresponding 
 points, if we wish to arrive at the true turning effort on the 
 crank. On the other hand, as the crank moves from B to 
 C the opposite effect takes place, the motion of the recipro- 
 cating parts is being retarded, and imparts pressures to the 
 crank-pin, which for any given positions A 2 , A 1 &c are 
 
 Fig. 36- 
 
 measured by the horizontal lines AV, AV &c. These 
 pressures must therefore be added to the pressures shown 
 by the indicator diagram for the corresponding points. 
 
 A simpler diagram for use with the indicator diagrams is 
 made as follows : Let AC, fig. 36, represent the stroke of 
 the piston or diameter of crank-pin circle, to the same scale 
 as the indicator diagram is drawn. Calculate the centrifugal 
 force per square inch of piston area from the known weights 
 of the reciprocating parts, radius of the crank circle, and 
 number of revolutions per minute, by the formula given, 
 p. 1 6 1, and draw a perpendicular Aa to the same scale as 
 the pressure scale of the indicator diagram, to represent this 
 
Influence of Inertia of Reciprocating Parts. 1 8 1 
 
 force. Erect an equal perpendicular from C. Join ac. 
 Divide the line AC into ten equal parts corresponding to the 
 ten divisions of an indicator diagram, then will the ordinates 
 Aa, A.' a', A" a", &c. between A and B represent the pressures 
 to be subtracted from those given by the indicator diagram, 
 while the corresponding ordinates between B and ^represent 
 pressures to be added to those shown by the diagram. 
 
 Influence of the weights of the reciprocating parts in verti- 
 cal engines. If the engine were of the vertical type, we should 
 also have to take account of the pressures required merely 
 to overcome the force of gravity acting on the weights of the 
 piston, &c. During the up stroke the weights act against, 
 and during the down stroke in the same direction, as the 
 motion of the piston. Hence we should have to modify 
 fig. 36 as follows. Add to Aa a portion aa', fig. 37, represent- 
 ing the weight of the reciprocating parts per square inch of 
 piston, to the same scale as Aa represents the pressure re- 
 quired to accelerate the parts. At the other end of the 
 stroke the action of the weights is to reduce the pressure 
 restored by the retardation of the reciprocating parts to the 
 crank pin. Set off therefore cc'-=ad and join a'c 1 ; then the 
 ordinates of the line a'E' represent the pressures to be sub- 
 tracted from those given by the indicator diagram, while the 
 ordinates of BV represent pressures to be added to those 
 shown by the diagram. 
 
 In the reverse stroke the action of the weights is in the 
 same direction as the steam pressure, and aids the accelera- 
 tion of the reciprocating parts at the commencement of the 
 stroke, and increases the pressures on the crank-pin at the 
 end. This is clearly shown by the ordinates of the line ad 
 in fig. 38. The influence of the direct weights of the 
 reciprocating parts becomes of great practical importance in 
 the case of the large low pressure cylinders of quick running 
 compound engines in which the average steam pressures are 
 low, and the weights often reach as much as 3*5 Ibs. per 
 square inch of piston. 
 
1 82 
 
 The Steam Engine. 
 
 It is impossible to exaggerate the benefit to be derived by 
 testing the proposed indicator diagrams of any engine under 
 design, in the manner described, before finally settling the 
 weights of the moving parts, the pressure and distribu- 
 
 Ar 
 
 Fig. 37- 
 
 Fig. 38 
 
 tion of the steam in a quick-running engine. This will be 
 clearly shown from an example taken from actual practice, 
 after we have considered the last remaining complicating 
 circumstance, viz. the effect of the length of the connecting 
 rod on the pressures required to accelerate and retard the 
 reciprocating parts. 
 
Effect of Connecting Rod on Reciprocating Parts. 183 
 
 Effect of the connecting rod in modifying the influence of 
 the reciprocating parts. A finite connecting rod, instead of 
 moving always parallel to the axis of the engine, vibrates 
 from side to side, and is always inclined at an angle to the 
 axis of the engine, except when the crank is on the dead 
 centres. The result of this is that during the first quarter 
 of a revolution the piston moves through more than half the 
 stroke, and its average velocity is therefore greater than 
 when the connecting rod is infinite, and vice versa during 
 the second quadrant the piston has to move through less 
 than half stroke and its average velocity is therefore less. 
 These changes are illustrated by fig. 39, in which the length 
 of the connecting rod ab is three times the length of the 
 
 Fig. 39- 
 
 crank. While the crank has been moving from A to b' the 
 piston has moved through a distance A/=AO + (V, where 
 Qc'=a'b' versin Qa'b'. While the crank moves from b' to B 
 the piston moves through a distance =r'B= OB a'b' versin 
 Qa'b'. During the return stroke the opposite effect takes 
 place, while the crank moves from B to D, the piston moves 
 through B<:', and while the crank moves from D to A, the 
 piston moves through t'A. Generally speaking, if n be the 
 ratio of the length of the connecting rod to that of the crank, 
 then for any position of the crank, say ^, in the first and second 
 quadrants, the distance moved through by the piston=A^ + 
 &r=versin angle of crank + n versin angle of connecting rod. 
 On the other hand for any position, say #", in the second 
 
184 The Steam Engine. 
 
 or third quadrants the distance traversed by the piston = 
 versin angle of crank n versin angle of connecting rod. 
 
 The consequence of the increased velocity during the 
 first part of the forward stroke is that more of the steam 
 pressure will be required to accelerate the moving parts than 
 in the case of an infinite connecting rod. The amount of 
 this pressure varies from point to point, and is proportional 
 to the amount of the acceleration at each point. During 
 the latter part of the stroke, the retardation is less sudden, 
 and consequently the moving parts never exert as great a 
 pressure on the crank pin in coming to rest as they would 
 do were the connecting rod infinite. During the return 
 stroke the converse takes place. During the first portion 
 of this stroke the steam pressure required to produce 
 acceleration is less, and during th'e latter portion the 
 pressure exerted by the moving parts on the crank pin is 
 greater, than in the case of an infinite connecting rod. 
 As, however, the length of the connecting rod cannot alter 
 the area of the diagram, fig. 36, but only its shape, we shall 
 find that the period during which high pressure is being ex- 
 erted is less, and that during which low pressure is being 
 exerted is greater than in the case of the infinite rod. 
 
 To calculate the exact amount of these pressures for 
 every point of the stroke would be rather a tedious process. 
 It is, however, easy to ascertain the pressures for three 
 positions of the piston, and a curve drawn through these 
 points will enable us to measure the pressures expended 
 in accelerating the reciprocating parts, or given out during 
 their retardation. The relative velocities of the crank-pin 
 and the piston may easily be ascertained geometrically in 
 the following manner. 
 
 Let AB, fig. 40, be a portion of the path of the crank pin, 
 and Cb any position of the crank, and ab the corresponding 
 position of the connecting rod. The velocities of the two 
 ends of the connecting rod are in different directions, the 
 cross-head end always moving in the fixed direction aC 
 
Effect of Connecting Rod on Reciprocating Parts. 185 
 
 with variable velocity, while the crank pin end always moves 
 with fixed velocity at right angles to the momentary position 
 of the crank arm. As the velocity of the crank pin is 
 known, that of the cross-head, which is the same as that of 
 the piston, can be ascertained. Produce ab till it cuts the 
 perpendicular CB in e. From b draw bf perpendicular to 
 C. Let the velocity of the crank pin be represented by 
 the radius Cb. Make bf=.Cb. Then bf represents the 
 
 Fig. 40. 
 
 velocity of b both in magnitude and direction. From / let 
 fall a perpendicular, //, to the direction of the connecting 
 rod, and produce the line fi. From b draw bd parallel to 
 the direction of velocity of the point a. Then, since the 
 connecting rod is of invariable length, the components of 
 the velocities of each end resolved along the rod are equal. 
 Now bi is this component for the velocity of , therefore it 
 is also the corresponding component for 0, and therefore 
 bd represents the magnitude as well as the direction of the 
 velocity of a. Now, comparing the triangles Cfo, bfd^ their 
 angles are equal, because the sides of one triangle are per- 
 pendicular to those of the other; also the side /^equals the 
 corresponding side C#, therefore the two triangles are equal, 
 and therefore the line O, cut off from CB by the prolonga- 
 tion of the line ab, represents the variable velocity of the 
 point <z, while Cb represents the constant velocity of b. 
 
 We can now construct a very simple curve of piston 
 velocities. For any positions of the crank, C, C', &c. 
 (fig. 41), set off O, Cz/, &c.=O, (>', &c. Through all the 
 
1 86 
 
 The Steam Engine. 
 
 points ?;, ?/, &c., draw the closed curve Cvv' . . e"C, then the 
 portion of the radius intercepted, at any position, between the 
 centre C and the circumference of this curve represents the 
 velocity of the piston for that position. It will be noticed 
 that a portion of this curve travels outside the crank pin 
 circle, showing that the piston velocity during a part of the 
 revolution is greater than that of the crank pin. The 
 maximum velocity and corresponding position of the crank 
 can be obtained from the diagram. In practice, when the 
 connecting rod is three or more times as long as the 
 crank, the position fox maximum velocity corresponds very 
 nearly with the position when the connecting rod makes a 
 tangent with the crank pin circle. Now for this position, 
 
 length of connecting rod , ,. , 
 
 s- ------ . - _ =tan angle of crank. 
 
 length of crank 
 
 For instance, when the connecting rod is 3, 4, 5, or 6 
 times the length of the crank we have 
 
 tan </>=3, 4, 5, and 6 respectively, 
 
 which, by means of a table of natural tangents we find 
 correspond with values of of 71 34', 75 58', 78 42', and 
 80 32' respectively. 
 
 Take the case of a piston rod four times the length of 
 
 c 
 
 Fig. 42. 
 
 the crank. It is required to find three points on the curve 
 a'c' , fig. 42 which corresponds with the straight line ac> 
 
Effect of Connecting Rod on Reciprocating Parts. 1 87 
 
 fig. 36, and the ordinates of which measure the pressures which 
 have to be added to or subtracted from those of the steam 
 diagram. The point D corresponds with the position which 
 the piston occupies when the connecting rod makes a 
 tangent with the crank circle that is to say, when the 
 crank is at the angle 75 58' ; for, at this point, as has been 
 stated above, there is neither acceleration nor retardation. 
 The points a' and c' of the curve may be found from the 
 following considerations. 
 
 When the crank-pin gets into line with the axis of the 
 cylinder i.e. at the two dead centres, the weights of the re- 
 ciprocating parts act with a full centrifugal effect on the 
 crank. At the near dead centre they tend to pull the crank 
 towards the cylinder, while at the far centre their action on 
 the crank is reversed. The centripetal force in the crank 
 at the near centre gives the measure and direction of the 
 force absorbed in accelerating the reciprocating parts when 
 the connecting rod is supposed infinite. When, however, 
 the rod is of finite length another effect is produced ; for on 
 passing the near centre the crank-pin end of the connecting 
 rod describes an infinitely small arc of a circle round the 
 piston rod end as a centre, and a centrifugal effect is pro- 
 duced which increases that in the crank. On the other hand, 
 when the far dead centre is being passed the centrifugal 
 tendency in the connecting rod end diminishes that in the 
 crank. The amount of the centrifugal force in the con- 
 necting rod circle may be derived from that in the crank, 
 very simply, because the weights are common to both, 
 also the velocities are the same, and nothing differs but 
 the length of the radii. Now the radius of the connecting 
 rod is four times that of the crank, therefore, as the centri- 
 fugal forces vary inversely as the radii when other things are 
 equal, the centrifugal force in the connecting rod is one 
 fourth of that in the crank. Add therefore to Aa, a piece 
 
 aa'= ~ a , and subtract from O a piece c = c then will the 
 4 4 
 
1 88 The Steam Engine. 
 
 points a' d be the initial and terminal points on the curve. 
 The point D has been already found. It may be proved 
 that the curve to be drawn through the three points a , D, 
 d is a parabola, but for all practical purposes a circular arc 
 is sufficiently accurate. The ordinates of this curve between 
 a! and D represent pressures which are expended in accele- 
 rating the moving parts, wh:ch pressures must therefore be 
 subtracted from the pressures as shown by the indicator 
 diagram ; on the other hand, the ordinates between D and 
 c 1 represent pressures which are exerted by the moving parts 
 on the crank-pin when they are being brought to rest, and 
 which must therefore be added to the pressures as shown by 
 an indicator diagram. For the return stroke the same 
 diagram may be used, starting from C as the commence- 
 ment of the stroke. The ordinates of c'D represent the 
 pressures absorbed in accelerating, and those of Da' the 
 pressures restored during retardation. 
 
 If the engine be of the vertical type a correction must 
 be applied to fig. 42, similar to that already explained in 
 the case of fig. 37. 
 
 An inspection of fig. 42 shows what a powerful influence 
 on the working of the engine may be exerted by the action 
 of the reciprocating parts. This influence may, according 
 to circumstances, be either good or bad. Thus, take the 
 case of a quick running single cylinder expansion engine. 
 The steam pressure in such an engine would be high at the 
 commencement of the stroke and low at the end, but the 
 power required to impart motion to the reciprocating parts 
 absorbs pressure at the beginning of the stroke, and thus 
 relieves the pressure that would otherwise come on the 
 crank ; while at the end of the stroke the opposite effect 
 takes place, and thus the inertia of the moving parts may 
 tend to equalise the pressure throughout the stroke, and 
 may consequently promote steady running. In this respect 
 the action is similar to that of a fly-wheel. On the other 
 hand, it may happen, if the speed at which the engine runs 
 
Example of Action of Reciprocating Parts. 189 
 
 is very high, or if the reciprocating parts are very heavy, that 
 the whole of the steam pressure in the cylinder is insufficient 
 to impart the requisite motion at the commencement of the 
 stroke, and consequently the deficiency of force must be sup- 
 plied by the fly-wheel, the result being that at the commence- 
 ment of the forward stroke the connecting rod is actually 
 being dragged by the crank-pin instead of turning the latter. 
 As soon, however, as a certain amount of motion is 
 imparted, the pressure of the steam begins to be felt on 
 the crank-pin, and the strain on the rod changes from 
 tension into compression, and a knock or jar is experienced 
 at the joints which greatly tends to wear out the parts. 
 
 The following example, taken from the well-known Allen 
 engine, has occasionally been adduced to illustrate this 
 point. The essential particulars of this engine are given 
 below : 
 
 Diameter of cylinder = i foot. 
 
 Stroke =24 inches. 
 
 Revolutions per minute =200 
 
 Weight of reciprocating parts =470 Ibs. 
 
 Steam pressure =60 Ibs. per sq. inch. 
 
 Ratio of length of connecting 1 = ^. I ^ 
 
 rod to crank J 
 
 ABC, fig. 43, shows an indicator diagram when steam is 
 cut off at one twentieth of the stroke. 
 
 If the connecting rod were infinite, the pressure per 
 square inch of piston area required to accelerate the moving 
 parts at the commencement and end of the stroke would 
 be obtained by the formula given on page 161, viz.: 
 
 -p_ ff/NVooo34_47o * 40000 x i x '00034 
 ^ H3' 1 
 
 7T 
 
 4 
 = 56*5 Ibs. per square inch. 
 
 Measure from A to C, and from D to C, 56-5 Ibs. to 
 scale, and join CC'. Then the ordinates of the triangle 
 
The Steam Engine. 
 
 ACE represent pressures to be subtracted from the cor- 
 responding steam pressures as deduced from the diagram, 
 
 Fig. 43- 
 
 and the ordinates of the triangle DC'E represent pressures 
 to be added to those of the diagram in order to arrive at 
 
Example of Action of Reciprocating Parts. 191 
 
 the true pressures transmitted through the connecting 
 rod to the crank-pin, on the assumption that the length of 
 the connecting rod is infinite. The effect of the connecting 
 
 rod being 6'i6 times the crank in length is to add -5 sL 
 
 6*i6 
 
 = 9*17 to AC and to subtract the same quantity from DC'. 
 Mark off therefore <:C and SC' each = 9*17 Ibs. Ascertain the 
 point G in the manner already described, and through cGc' 
 describe a parabolic curve. The ordinates of this curve 
 above and below the line AD represent pressures to be 
 respectively added to and subtracted from those deduced 
 from the indicator diagram. We must next find from the 
 indicator diagram the true net pressures of steam on the 
 piston. This is done by deducting from the pressures, as 
 shown by the diagram, the simultaneous back pressures as 
 shown by another diagram, taken from the other cylinder 
 cover. In the absence of such a diagram, we may make 
 use of the back pressure line of the diagram on fig. 43, 
 remembering, however, that from the steam pressure at A 
 must be deducted the back pressure at the exhaust end. 
 We thus obtain the true curve of pressures shown by the 
 line GF beneath the expansion line of the diagram, and 
 terminating below BB at the point F. Having now deducted 
 from the true steam pressures the amounts given by the 
 ordinates of the curve c&c', we obtain the curve defg^ 
 the ordinates of which represent the true pressures trans- 
 mitted through the connecting rod. The pressures above 
 the line BB' are positive, while those below are negative, 
 and show that the steam pressure on the piston, wherever 
 the curve falls below the line, are insufficient even to 
 accelerate the moving parts. Thus, for instance, we have 
 seen that at the commencement of the stroke 56-5 + 9-1 7 = 
 65-67 Ibs. per square inch are required for this purpose, 
 whereas only 60 Ibs. are available, without even deducting 
 for the back pressure. The deficiency has to be supplied by 
 the fly wheel, and the consequence is that at the commence- 
 
192 The Steam Engine. 
 
 ment of the stroke the piston is actually being dragged 
 forward by the crank-pin, instead of pushing the latter 
 round, as it should do. When the curve defg rises above 
 the line there is a small amount of pressure transmitted to 
 the crank-pin, and consequently the strains in the piston and 
 connecting rods are reversed from tension to compression, 
 and a knock must occur if there is the least wear in the 
 brasses. Between e and / the steam pressure is again 
 insufficient, and the piston is again drawn round by the 
 crank-pin, and at each of these points a knock will occur. 
 
 It will be noticed that the general effect of the action 
 of the reciprocating parts is to completely reverse the 
 pressures as deduced from the indicator diagram ; for whereas 
 these are greatest at the commencement of the stroke and 
 dwindle down to nearly nothing at the end, when the re- 
 ciprocating parts are taken into account, the pressures are 
 greatest at the end of the stroke and are actually negative at 
 the beginning. 
 
 During the return stroke the state of things at the com- 
 mencement will not be so bad because the pressure required 
 is only 56-5 9-17 = 47^33 Ibs., and there is sufficient steam 
 power available for this purpose and to leave a balance over 
 to transmit to the crank pin. On the other hand, at the end of 
 the stroke the state of things will be very unfavourable for easy 
 running because of the enormous accumulation of pressure. 
 
 In order to improve this engine, one of three plans 
 might be adopted. The initial steam pressure might be 
 increased : the piston speed might be diminished ; or, lastly, 
 the weights of the moving parts might be reduced, A com- 
 bination of these methods would, of course, also be effectual. 
 
 Effect of distribution of steam on the action of the recipro- 
 cating parts. 1 Great care must be taken when designing the 
 valve gear of high-speed engines to see that the distribution 
 of the steam is properly effected, otherwise the ill effects of 
 
 1 Students who have no knowledge of indicator diagrams should 
 read this section after studying Chapter VIII. 
 
Steam Distribution and Reciprocating Parts. 193 
 
 a bad distribution may be greatly aggravated. Take, for 
 example, the very common fault of a late exhaust, as repre- 
 sented by fig. 44 Assuming that the diagram of the return 
 stroke is equally bad, the result will be that there will be a 
 back pressure a'b' ab 
 at the commencement of 
 the stroke, and the effec- 
 tive steam pressure will 
 only be A0', which, if the 
 piston speed is high, and 
 the weight of the recipro- 
 cating parts considerable, 
 may not be sufficient to ~ 
 
 impart the desired velo- 
 city to these latter. A late admission of steam will, of 
 course, produce a similar result to a still greater degree. 
 
 On the other hand a good compression curve, which, as 
 we shall hereafter see, is an excellent feature in an indicator 
 diagram, is productive of great good from the point of view 
 of steady running, when its effect on the action of the 
 reciprocating parts is taken into account. We have seen 
 that the tendency with high-speed expansive engines is that 
 the effective pressure on the crank-pin is transferred to the 
 end of the stroke. Now it is very undesirable that the 
 pressure on the pin should be very high at the extreme 
 end of the stroke, as it causes heavy strains on many 
 parts of the machinery; but the effect of a marked curve of 
 compression is to cause a considerable back pressure at the 
 end of the stroke, which pressure must of course be de- 
 ducted from the pressure on the crank-pin, due to the 
 combined effort of the steam and the retardation of the 
 reciprocating parts. This action is of course much assisted 
 if the exhaust opens early. In fact, in such cases it often 
 happens that when a strong compression curve exists the 
 back pressure on the piston is much in excess of the direct 
 Dressure. 
 
194 
 
 The Steam Engine. 
 
 L 
 
 This effect is clearly shown in fig. 43, and is also illus- 
 trated by fig. 45. The indicator diagram is clearly recog- 
 nisable ; FGH, the curve showing the effect of the recipro- 
 cating parts, is indicated by the full line. The irregular 
 
 dotted line AB is the resultant 
 curve, giving the true nett 
 pressures on the piston, and 
 showing a negative pressure at 
 the end of the stroke, on the 
 assumption that a diagram 
 taken from the other end of 
 the cylinder would give a 
 similar line of back pressure 
 to that shown on fig. 45. 
 
 The student is now in 
 possession of the means of 
 constructing a curve of twist- 
 ing moment, or tangential 
 effort on the crank-pin, taking 
 into account all the principal modifying circumstances. It 
 will have been observed that the effect of the connecting 
 rod is twofold. Firstly, by its varying inclination it alters 
 the length of the tangential component of the pressure 
 transmitted to the crank-pin (see p. 173) ; and secondly, it 
 modifies the action of the reciprocating parts in the manner 
 just explained. 
 
 The various steps to be taken to produce a complete 
 curve of effort on the crank-pin are as follows : 
 
 ist. To obtain a pair of indicator diagrams, viz. one 
 from each end of the cylinder. 
 
 2nd. By taking account of the back pressure at each 
 point in the stroke to deduce a pair of resultant diagrams 
 showing the true pressures on the driving sides of the piston, 
 as explained pages 190 and 340. 
 
 3rd. To find the effect of the weights and velocities of 
 the reciprocating parts as modified by the length of the 
 
 Fig. 45- 
 
Drawing true Curve of Twisting Moments. 195 
 
 connecting rod, and to correct the last found diagrams 
 accordingly, as explained pages 183 to 192. 
 
 4th. To deduce from the last diagrams the circular 
 diagrams of effort on the crank-pin, taking into account the 
 obliquity of the connecting rod, as explained pages 172 to 175, 
 
 These various steps have been explained at such length 
 that it will not be necessary to give a final example illustrat- 
 ing them. The student will have no difficulty in applying 
 the principles to any example whatever. 
 
 In the case of two or more cylinders being coupled on 
 to one crank shaft, the diagram of effort will have to be 
 made for each cylinder separately, unless they are identical 
 in dimensions and steam distribution, and a resultant 
 diagram formed by adding their separate effects together ; 
 see page 175. 
 
 How to approximate to uniformity of effort on the crank- 
 pin. In all of the examples given the engines have had 
 single cylinders, and though it is possible in such engines to 
 attain to great uniformity of driving power throughout the 
 greater portion of the revolution, even when very high rates 
 of expansions are made use of, by running them at high 
 speeds so as to utilise the action of the reciprocating parts, 
 nevertheless, the difficulty exists that at the beginning of 
 each stroke there is no rotative effort whatever on the crank. 
 There are two methods of overcoming this defect. 
 
 First. Two or more cylinders may be made use of 
 coupled on to the same crank shaft, but with the cranks set 
 at angles to each other, in such a manner that one engine is 
 producing its maximum rotative effort when the other is at 
 the dead centre. Each of these cylinders may be identical 
 in dimensions and in steam distribution, as is the case with 
 locomotives and many types of land engines ; or the engine 
 may be compound, one cylinder being much larger than the 
 other, and the steam which has been partially expanded in 
 the small cylinder being used over again in the larger. This 
 type of engine, which presents many advantages from the 
 
 o 2 
 
196 
 
 The Steam Engine. 
 
 point of view of fuel economy, will be more particularly de- 
 scribed in Chapter XI. It has, on account of its economical 
 properties, come into very general use for marine purposes. 
 
 Second. A fly wheel may be used which, by absorbing 
 energy when the turning power is in excess of the resistance, 
 gives it out again when this condition is reversed, and thus 
 enables the engine to pass the dead centre. 
 
 The double cylinder type of engine is invariably used 
 where there is much stopping and starting and reversing to 
 be done, because it may be easily started from any position 
 of the cranks. The single cylinder engine with a suitable 
 fly wheel is used with advantage whenever there is much 
 steady running to be done in one direction. This is the 
 case with the majority of factory engines. The single 
 cylinder type with a fly wheel, and a high rate of expansion, 
 has even been used successfully for marine purposes. 
 
 Graphic representation of the action of a fly wheel. In 
 cases where the resistance is uniform the power applied 
 tangentially to the crank-pin in single cylinder engines is in 
 
Graphic Representation of Action of Fly Wheel. 197 
 
 excess of the resistance during two portions of the revolu- 
 tion and is less than the resistance during the other two 
 portions. This is illustrated in fig. 46, where AB is the 
 crank circle, ab the circle of uniform resistance. The curve 
 showing the twisting moment on the crank is denoted by 
 ACB, BCA. The tangential pressure is in excess of the 
 resistance from e to e' and from e, to e n and during the other 
 two arcs the resistance is in excess. The average value of 
 the excess of twisting moment is represented by ed t and of 
 the resistance by eg. 
 
 The same thing may be illustrated also by a diagram 
 constructed on a straight base similar to fig. 3QA. The line 
 ABA' is equal in length to the circumference of the crank- 
 pin circle. ACB and BC'A' are the curves of twisting 
 
 Fig. 47- 
 
 moment during the forward and back strokes respectively. 
 The uniform resistance is shown by the line aa', the ordinate 
 aA being the mean of all the ordinates of the curves ACB, 
 BC'A'. The excess of tangential pressure between ee* and 
 e { e n above the resistance is shown by the ordinates of the 
 curves eCe' and fiC f e l} , and the excess of resistance above 
 tangential pressure during the remainder of the stroke is 
 shown by the ordinates of Ae, e'B, Ee }) and e l} A'. At the 
 four points e,e',e^ e n , the resistance exactly equals the tan- 
 gential pressure. The average values of the excess of tan- 
 gential pressure is given by the lines ed and e^d^ the areas 
 edd'e' and e v d } d n e n being equal respectively to the areas eCe* 
 and^CVn- Similarly the average values of the excess of 
 resistance are given by the ordinates eg and e'g 1 . Also the 
 
198 The Steam Engine. 
 
 sum of the areas edd'e' and e\d l d ll e ll \?> exactly equal to the 
 sum of the areas ag, e'g^ and g u a'. 
 
 The weight of fly wheel necessary in any given case de- 
 pends on the following conditions : 
 
 1. The mean diameter of the fly wheel rim. 
 
 2. The velocity at which this rim moves. 
 
 3. The variation from uniform speed which is to be 
 permitted. 
 
 4. The fractions of the entire revolution during which 
 the tangential force is above or below the uniform resist- 
 ance. 
 
 5. The average amount of the excess of power or 
 resistance. 
 
 When these conditions are known we can compute the 
 weight of the fly wheel. 
 
 The value of elements 4 and 5 may be computed for 
 any given case by constructing a diagram similar to fig. 47. 
 
 Let W=weight of the fly wheel in Ibs. 
 
 V=-the mean velocity of its rim in feet per second. 
 
 YI and V 2 =the maximum and minimum velocities. 
 
 -=the fraction of the mean velocity allowed for varia- 
 
 K, 
 
 tion, that is to say the difference between V l and V 2 - 
 
 Then the energy stored up in the fly wheel when at its 
 maximum velocity 
 
 = wv i 2 
 
 zg 
 
 and when at its least velocity the energy remaining 
 
 WV 2 2 
 
 *g 
 
 Therefore the work given out by the fly wheel while its 
 velocity falls from V, to V 2 
 
Calculation of Weight of Fly Wheels. 199 
 
 Now this work is measured by the product cdxce' x A 
 the area of the piston (see fig. 47), ed being the average excess 
 of tangential pressure in Ibs. per square inch of piston. 
 
 V 
 
 X 2V X XT 
 
 w _ x k X ed x ed X A 
 
 ~vr~ 
 
 Now V, the mean velocity of the fly wheel rim in feet per 
 second, may be expressed as follows in terms of the diameter 
 D, and the number of revolutions per minute R 
 
 v _ T^D.R 
 60 
 
 Substituting this value of V in the above equation and 
 reducing to tons, we have very nearly 
 
 It will generally be found that the value of ed differs in 
 the two strokes ; we must therefore take the value which 
 shows the greatest inequality in making the above calcu- 
 lation for the weight of the fly wheel, otherwise it will 
 permit a greater fluctuation than is desired. The value of 
 k varies according to the kind of work the engine has to 
 do. When employed in driving dynamo machines or 
 spinning machinery, or in any kind of work where very 
 uniform motion is required, the variation from the mean 
 velocity should not exceed from i to i per cent., that is 
 to say the value of k would vary from 100 to 80. In other 
 cases the variation may be as much as one twentieth, or 5 
 per cent, or =20. It will be seen from the foregoing 
 formula that the weight of the fly wheel varies directly as k, 
 that is to say the less the permissible variation from the mean 
 
2oo The Steau? Engine. 
 
 speed the greater the weight of the wheel. It also varies 
 inversely as the square of the diameter and inversely as the 
 square of the number of revolutions per minute. 
 
 The foregoing investigation applies to the case where 
 the resistance is practically uniform. If the resistance fluc- 
 tuates during each revolution, as for instance when the engine 
 is driving a two-bladed screw propeller, a proper curve of 
 resistance would have to be drawn in lieu of the line a a', 
 fig. 47. When the resistance is liable to sudden fluctuations, 
 as for instance in the case of factories where a large number 
 of machines are occasionally thrown on or off, the fly wheel 
 by itself is powerless to produce even an approximation 
 to steady running. To effect this in such cases is the duty 
 of the governor, which controls the actual power developed 
 by the engine, either by throttling the steam on the way 
 from the boiler, so that its pressure is reduced considerably 
 by the time it enters the cylinder ; or else, by varying the 
 rate of expansion so that the engine develops more or less 
 power in proportion to the work it has to do. For descrip- 
 tion of various governors see p. 239. 
 
201 
 
 CHAPTER VI. 
 
 THE MECHANISM AND DETAILS OF STEAM ENGINES. 
 
 Cylinders with their fittings Clearance Steam passages Valve boxes 
 Jacketing Lubricators Pistons Piston packings Piston rods- 
 Cross heads and slide bars Connecting rods Cranks and eccentrics 
 Eccentric rods The strains in crank shafts Journals Shaft bear- 
 ings and pedestals Axle boxes Governors Fly wheels. 
 
 IT is intended in this chapter to give an account of the 
 separate parts which constitute the mechanism of the steam 
 engine, excepting only the valves and valve-gear, which re- 
 quire separate treatment. The variety of form and arrange- 
 ment- of the parts of steam engines is so great, that it will 
 only be possible to give a few representative examples. A 
 more extended knowledge of the mechanism can only be 
 gained by close observation of numerous engines, or working 
 drawings. 
 
 The Cylinder. The cylinder of a steam engine is the 
 closed vessel in which the piston works backwards and 
 forwards. It is so called because the interior is cylindrical 
 in shape, though the form of the exterior is complicated by 
 sundry additions. Examples of stationary engine cylinders 
 are given in figs. 5 to 7, and of marine engine cylinders in 
 figs. 194, 195, 197 to 200. Fig. 48 is a longitudinal section 
 of a steam cylinder of a locomotive engine. It is made of 
 cast iron, the interior being carefully bored so as to form a 
 smooth and cylindrical surface for the passage of the piston. 
 It consists of the following principal parts. The cylin- 
 drical body AA, which is cast in one piece ; the valve 
 box BB, in the thickness of which are formed the two 
 
2O2 
 
 TJie Steam Engine, 
 
 steam passages SS and the exhaust passage E ; the two 
 covers CC, which are flanged, and which are attached to the 
 body of the cylinder by means of studs and nuts. The 
 cover through which the piston rod works is provided 
 with a stuffing box D and gland e, to prevent the steam 
 escaping round the rod. This object is accomplished in 
 the following manner. The space aa between the rod and 
 the inner cylindrical surface of the stuffing-box is filled with 
 plaited hemp saturated with tallow, or with one of the 
 numerous patent packings now procurable. The gun-metal 
 gland ^, through which the piston rod passes, is forced up 
 
 Fig. 48. 
 
 against the packing by means of the two nuts and screwed 
 studs shown in the drawing. The result is that the packing 
 can be squeezed with any desired degree of tightness round 
 the piston rod, and can thus prevent the escape of steam. 
 The opening by which the piston rod passes through the 
 substance of the cover is lined with a gun-metal bush, c. 
 In many engines, especially those of foreign manufacture, 
 there is a stuffing box on, the other cylinder cover, through 
 which a prolongation of the piston rod works. The object 
 of this arrangement is to prevent the piston bearing unequally 
 on the lower side of the cylinder. It is also adopted in the 
 
Stecun Cylinders. 203 
 
 case of condensing engines, when the plunger of the air- 
 pump is driven direct from the piston. It will be noticed 
 that the interior faces of the covers are cast so as to fit into 
 the corresponding faces of the piston. The reason of this 
 arrangement is that the piston has in many cases to be 
 formed in the shape shown in fig. 48, viz. with a broad rim 
 or flange, and a thin disc ; now if the cover faces were not 
 shaped correspondingly, there would be at each end of the 
 stroke a large space to be filled with steam before the piston 
 began to move, which steam would do no work till expan- 
 sion began. 
 
 Clearance. The interior length of the cylinder bore, from 
 cover to cover, is always a little longer than the stroke, plus 
 the thickness of the piston, so that a small vacant space called 
 the clearance is invariably left at the end of each stroke. If 
 it were not for this precaution the covers might be knocked 
 off whenever water accumulated in the cylinder. The 
 clearance spaces and also the contents of the steam passages 
 SS, between the valve face and the inner surface of the bore 
 of the cylinder must at each stroke be filled with steam 
 before the piston can be moved. This steam of course does 
 no work till expansion begins, but a great portion of the loss 
 due to this cause may be recovered by compressing the 
 exhaust steam before the end of the stroke. This operation 
 is called cushioning the piston, and is most essential for 
 many reasons. It helps to bring the piston gradually to 
 rest, and partially restores the temperature of the sides of 
 the cylinder which become cooled during the exhaust. It 
 further tends to produce uniformity of tangential effort on 
 crank-pin in the case of quick running engines ; see p. 193. 
 
 The joints of the covers are made steam tight by placing 
 between the flanges a layer of red lead cement, or soft 
 copper wire, or one of the numerous patent packings, and 
 then tightening up the bolts. 
 
 Steam passages. The design of the steam ports and pas- 
 sages is a matter of the greatest importance. It is desirable 
 
204 The Steam Engine, 
 
 to make the length of the passage as short as possible, so 
 that its cubic contents may not add unduly to the contents 
 of the clearance spaces, and on the other hand it is essential 
 that the area of the passages should be ample, so that the 
 fresh steam may not be throttled on its way into the cylinder, 
 nor the exhaust steam on its way out, the result of which 
 would be a considerable loss of power by reducing the 
 pressure of the incoming steam and increasing the t back 
 pressure. It is not easy to reconcile these desiderata with 
 the Vise ol a single slide valve ; for, if the passages were made 
 as short as possible, the ports would necessarily be situated 
 near the ends of the cylinder, and a valve that would cover 
 both of them would be of unwieldy dimensions. Again, if 
 the ports were made very wide so as to give a very free 
 passage to the steam, the distance which the valve would 
 have to move over commonly called the travel of the 
 valve in order to fully uncover the ports would be so great 
 that the work of moving the valve would absorb no incon- 
 siderable portion of the power of the engine. Consequently 
 in engines which are worked with a short slide valve, and 
 these constitute the great majority of engines in actual use, 
 a compromise has to be effected between conflicting evils 
 The area of the steam ports is obviously connected with the 
 piston speed of the engine ; for, the greater the speed the 
 greater the quantity of steam which has to be admitted in a 
 given time ; and consequently a port which would be found 
 ample for a slow running engine might be totally inadequate 
 if the speed were considerably increased. The following 
 empirical rule has been found to give satisfactory practical 
 results in the case of engines having long steam passages. 
 
 The area of steam port : area of piston : : speed of piston 
 in feet per sec. : 100. 
 
 or area of port = gl_ ea of P iston * s P eed of P ! 
 
 iston 
 
 100. 
 
 In some types of modern engines, which are designed 
 
Area of Steam Ports. 205 
 
 with the object of economising fuel, the steam ports are 
 placed quite close to the cylinder ends, and separate passages 
 are provided for the escape of the exhaust steam. In such 
 case the simple slide valve is, of course, dispensed with, 
 and each passage is worked by a separate valve (see page 264 
 and fig. 99). The object of providing separate exhaust passages 
 is, that the fresh steam on entering the cylinder may not 
 be cooled down by coming in contact with the sides of 
 passages through which the -cold exhaust steam has just 
 escaped. 
 
 Valve boxes. The valve box is generally flanged and pro- 
 vided with a cover, which is bolted to the face of the box, just 
 as are the covers to the cylinder ends. The valve box is pro- 
 vided with a stuffing box, through which works the rod which 
 actuates the valve, and also with an opening to which is 
 attached the steam pipe from the boiler. The exhaust steam, 
 after it has done its work, escapes through each steam passage 
 alternately, and passes through the hollow portion of the 
 valve V into the exhaust port E, whence it is led through 
 an exhaust passage cast on the body of the cylinder, and is 
 suffered to escape through a pipe into the open air, or else 
 in the case of condensing engines, is conducted by a pipe to 
 the condenser. 
 
 The face on which the slide valve works is, in large 
 engines, often cast separately from the body of the cylinder, 
 to which it is attached by bolts. In this manner it is 
 possible to secure a sound face for the valves to work on, 
 and to renew it when it becomes worn. 
 
 Jacketing. In many engines of the better class, the cylinder 
 is surrounded by an outer casing, sometimes cast in one piece 
 with it, and sometimes attached to it, in such a manner as 
 to leave an annular chamber between casing and cylinder. 
 In this case the cylinder is said to be jacketed. The 
 annular chamber is filled with fresh steam direct from the 
 boiler. The object of this arrangement is to keep the body 
 of the cylinder proper at, as nearly as possible, a uniform 
 
206 
 
 The Steam Engine. 
 
 temperature, and thus to avoid the injurious effects due to 
 the cooling of the cylinder sides by the expanding and 
 exhaust steam. Fig. 49 illustrates, in a longitudinal section, 
 a jacketed cylinder. Where jacketing is carried out very 
 thoroughly, the covers are jacketed as well as the body of the 
 cylinder. For further examples of jacketed cylinders see 
 pages 456 to 463. 
 
 In many modern marine engines, the inner surface or 
 barrel of the cylinder is cast as a separate piece and fitted 
 into the outer body. One end is attached by a flange to 
 the outer body, while the other end is left free to expand 
 and contract with the variations in temperature that take 
 
 Fig. 49- 
 
 place. The free end must be packed steam tight. This 
 inner portion is called the liner. There is usually a space 
 between the liner and the body which forms the steam- 
 jacket. Cylinders thus constructed possess many advantages. 
 They are easier to make. When the inner surface wears 
 down, it can be easily taken out, and a new one inserted. 
 The cylinder is not unequally strained by variations of 
 temperature in its different parts. Lastly, the liner can be 
 made of cast steel, which is the strongest and most durable 
 material for the purpose. For further on the subject of 
 jackets, see page 450. The exterior surface of the cylinder, 
 or of the jacket casing, should always be covered with a layer 
 
Lubricators. 207 
 
 of non-conducting material, such as wood, felt, or asbestos 
 cloth, to prevent losses by radiation. 
 
 In order to discharge condensed water, or water which 
 may have lodged in the cylinder on account of priming in 
 the boiler, a small opening is made in the bottom of each 
 end of the cylinder, and is provided with a cock called the 
 'pet-cock.' Sometimes, instead of the pet-cocks, small 
 relief valves are provided, which open outwards, whenever 
 the pressure within the cylinder exceeds the pressure which 
 holds the valve down. 
 
 In order to lubricate the rubbing surfaces of the piston 
 
 Fig. 50. Fig. 51. 
 
 and slide valve, small openings have to be provided, into 
 which are screwed lubricators containing oil or tallow. Fig. 
 50 represents a form of lubricator in common use. The 
 bulb A contains the oil or tallow which is introduced 
 through the funnel B. The lubricator is screwed into the 
 cylinder or valve box by means of the screw C. The cocks, 
 shown in the sketch, make or close communication between 
 the bulb and cylinder, or bulb and funnel. 
 
 Fig. 51 represents Mr. Ramsbottom's continuous feed 
 lubricators. The bulb A is filled with oil introduced by the 
 plug B. The pipe C is screwed into the cylinder. The 
 
208 The Steam Engine. 
 
 steam, entering this pipe, condenses by degrees on the 
 surface of the oil, and the water thus formed sinks to the 
 bottom of the bulb, displacing a small quantity of oil. This 
 process continues till the bulb becomes, entirely filled with 
 water, which can then be drawn off through the lower plug. 
 Great care should be devoted to the choice of a good 
 material for lubrication. Tallow, even of the best quality, 
 and animal oils generally, are unsuited for use with high- 
 pressure steam, as they are decomposed by the high tempe- 
 rature into stearic and other acids, which readily attack the 
 iron of the cylinders. If the waste steam from the cylinders 
 is mixed with the feed water for the boilers, the effects of 
 the decomposed tallow on the boiler are fatal and rapid. 
 Sometimes the iron plates are eaten away, and sometimes a 
 soapy deposit, of a very non-conducting nature, forms on 
 the crowns of the furnaces, which permits the crown plates 
 to become overheated, and to collapse. Whenever high- 
 pressure steam is employed, some preparation of mineral oil 
 should be used as the lubricating material. 
 
 \b 
 
 Fig. 52. 
 
 Pistons. The piston is the metallic disc or moveable 
 diaphragm which accurately fits the bore of the cylinder, 
 and which receives and transmits the pressure of the steam 
 to the other moving parts of the engine. Fig. 52 illustrates 
 
Pistons. 209 
 
 in section a piston which is used for locomotives on the 
 London & North-Western Railway. It consists of three 
 principal parts, viz. the central disc #, a, which forms the 
 body of the piston, the circumferential portion $, , which 
 contains the packing that enables the piston to move steam- 
 tight along the cylinder, and which is so shaped as to form 
 a large surface of contact with the sides of the latter, and the 
 central boss <r, c, which receives the coned end of the piston rod. 
 
 The forms of pistons are innumerable, and depend alto- 
 gether upon the purpose for which the engine is intended, 
 and the size of the cylinder, which in different classes of 
 engines varies from a few inches to over 9 feet in diameter. 
 The chief points to attend to in the design of a piston are 
 the following : it should be strong enough to withstand the 
 pressure of the steam, and to hold the end of the piston rod 
 immovably; the packing round the circumference should 
 be steam-tight, without causing undue friction, and not 
 liable to get out of order; the width of the circumferential 
 portion should be such that the pressure per square inch 
 due, in the case of horizontal engines, to the weight of the 
 piston be not sufficient to cause undue wear of the inner 
 surface of the cylinder. In many engines the weight of tne 
 piston is taken off the surface of the cylinder by prolonging 
 the piston rod backwards through the back cover of the 
 cylinder ; in such cases the weight is partly transferred to 
 the stuffing-boxes and slide-bars, which are easily got at for 
 purposes of examination and adjustment. 
 
 The importance of keeping the surface of the cylinder 
 true, and of keeping the piston in steam-tight contact with 
 it, will be readily recognised when it is borne in mind that 
 a leakage of steam past the piston means that during the 
 whole time the engine is at work there is an open passage 
 from the boiler to the condenser or outer air, through which 
 steam is continuously escaping without doing any work. 
 
 Methods of packing pistons. In the eaily days of steam 
 engines, when the pressure used was low, a good packing was 
 
 p 
 
2io 
 
 The Steam Enine. 
 
 obtained by winding plaits of hemp, soaked in tallow, round 
 the flange of the piston. These plaits were secured in place 
 and packed tightly together by a ring, called the junk ring, 
 which was bolted down to one face of the piston. Such a 
 system would be quite inapplicable to modern high-pressure 
 engines. One of the simplest and best modes of packing 
 pistons of moderate dimensions was devised by Mr. Rams- 
 bottom ; it is illustrated in fig. 53. The flange of the piston 
 
 b (fig. 52) is broad enough to 
 allow of three rectangular 
 channels being turned in it. 
 Into each of these channels 
 or grooves is placed a steel 
 ring of rectangular section, as 
 shown in the figure. These 
 rings are not bent into an 
 exact circle of the same dia- 
 meter as the piston, but are 
 so formed that they have a 
 tendency to spring outwards, 
 and consequently, when the 
 piston is confined in the 
 cylinder, they press against the working surface of the latter, 
 and thus effect a steam-tight joint. Fig. 53 shows one of the 
 rings, before it has been sprung into its place. The inner 
 complete circle represents the diameter of the cylinder. 
 The three rings are so arranged that the joints are not in a 
 straight line. It has been found that if the spring on the 
 rings is sufficient to cause them to press against the cylinder 
 with a pressure of 3 Ibs. to the square inch, the joint will be 
 steam-tight. 
 
 It is not necessary to make the rings of steel. In many 
 cases they are made of cast iron, the ring when cast being 
 slotted across. The outer face of the ring is turned, and 
 the inner face left as it comes from the mould. This insures 
 a good amount of spring. 
 
Packing of Pistons. 
 
 211 
 
 There are many varieties of metallic packing rings in use. 
 In some cases they are made without any initial outward 
 spring of their own, but are pressed outwards by the action 
 of the steam, which is admitted behind the ring by small 
 holes in the body of the piston. In other cases the metallic 
 packing, instead of being a simple ring, is made in the form 
 of a spiral which makes rather more than two complete 
 turns round the circumference of the piston, the groove in 
 the latter being of course formed to correspond. In this 
 manner the butt joint of the simple ring is avoided, and the 
 leakage of steam through this joint is prevented. 
 
 In the case of marine engines, the packing ring is usually 
 pressed against the barrel of the cylinder by means of a 
 
 Fig- 54- 
 
 series of independent adjustable springs contained within 
 the body of the piston. Figs. 54 and 55 l show the details 
 of such an arrangement. The spring ring, which is of con- 
 siderable depth, is held up against the sides of the cylinder 
 by a series of steel springs, aaa, fig. 55. The joint in the 
 ring is formed as in fig. 54 to prevent leakage. An oblique 
 slot is taken out of the ring. A plate, fitted with a tongue 
 piece, is fastened behind the slot, as shown in the section, 
 fig. 54, and the tongue piece, which slides in a groove, 
 allows the ring to expand and contract, and at the same 
 
 1 Taken by permission from Mr. R. Sennett's work, The Marine 
 Steam Engine. 
 
 P 2 
 
212 
 
 The Steam 
 
 Engine. 
 
 time makes a steam-tight joint. The ring, with its springs, 
 is covered by a flat circular piece of iron called the junk 
 ring, which is shown in plan on one half of fig. 55. This 
 enables the springs to be got at easily for examination and 
 repair. The junk ring is attached to the body of the piston 
 by bolts which work into brass nuts embedded in the metal 
 of the piston. When the threads work loose, the nuts can 
 be easily replaced. In the case of horizontal marine engines 
 the part of the spring ring which is in contact with the lower 
 
 Fig. 55- 
 
 side of the cylinder is not backed by springs, but by solid 
 blocks. These are necessary in order to support the weight 
 of the piston, as otherwise the springs at the bottom would 
 be abnormally compressed and those at the upper part of 
 the circumference correspondingly slack. Within the last 
 two or three years it has been possible, owing to the im- 
 provements introduced into the methods of casting steel, to 
 manufacture pistons of that metal instead of cast iron, which 
 is the material which has hitherto been almost invariably 
 used. In marine engine pistons, it has been found possible 
 
Piston Rods. 213 
 
 by the use of cast steel to effect a saving in weight of between 
 thirty and forty per cent, as compared with cast iron. 
 
 Piston rods. The piston rod is the member which trans- 
 mits the motion imparted to the piston to the mechanism 
 outside the cylinder. It consists of a truly cylindrical bar of 
 wrought iron or steel, one end of which is fastened securely 
 into the piston. The rod passes through the cylinder cover by 
 means of a steam-tight stuffing-box, as shown in fig. 48, and 
 the outer end terminates in the cross-head which will be de- 
 scribed presently. There are various methods in vogue of 
 fastening the rod into the body of the piston. Sometimes the 
 end of the rod is turned cylindrical and a hole bored in the 
 piston slightly less in diameter than the rod. The piston is 
 then heated, which causes it to expand, when the rod can be 
 inserted. After cooling, the piston contracts, and holds the 
 rod firmly in its place. In the majority of cases the end of 
 the rod is turned conical as in fig. 52, with a screw thread on 
 the extreme end, by means of which, together with a nut, the 
 rod is firmly embedded in a conical recess bored in the piston. 
 Sometimes instead of a nut the rod is fastened by means of 
 a cotter, and occasionally it is screwed and pinned into the 
 boss of the piston. 
 
 The strength of piston rods has to be fixed with special re- 
 ference to the fact that they are subject to alternating strains. 
 When the piston is making the stroke towards the crank shaft 
 the rod is in compression, and when making the return stroke 
 the rod is in tension. The maximum stress per square inch 
 of cross section at any part of the stroke is equal to the total 
 pressure of the steam on the piston divided by the area of 
 the rod. It is usual in designing pieces of machinery which 
 have to bear alternating strains of tension and compression to 
 make them much stronger than would be necessary were the 
 strain always of one sort. 
 
 Cross-heads and slide-bars. The outer end of the piston 
 rod is attached to the cross-head, or motion block, which 
 serves the double purpose of forming the means of connection 
 
214 T/ie Steam Engine. 
 
 between the piston rod and the connecting rod, and of 
 guiding the piston rod so as to keep it straight and in the 
 line of the axis of the cylinder, in spite of the bending moment 
 due to the angular position of the connecting rod. 
 
 The cross-head generally consists of three principal parts, 
 viz. (i) the body which often contains a conical hole into 
 which the coned end of the piston rod is fastened by means 
 of a cotter ; (2) the part by which the joint with the con- 
 necting rod is effected ; and (3) the guides or motion blocks 
 which travel between fixed bars parallel to the axis of the 
 cylinder, called slide-bars, and which prevent the end of 
 the piston rod from being deflected as the connecting rod 
 assumes an angular position. Fig. 56 shows a piston rod, 
 cross-head, slide-bars, and connecting rod in position. 
 
 Fig. 56. 
 
 Pressure upon slide-bars. Before giving examples of 
 cross-heads, it is necessary to explain the nature of the 
 strains which they transmit to the slide-bars. As the crank- 
 pin revolves, the connecting rod assumes an angular position, 
 the amount of the angularity increasing till the crank is at 
 right angles to the axis of the cylinder and then again 
 diminishing to nothing. 
 
 In the accompanying diagram (fig. 5 7) suppose the piston 
 to be making a stroke towards the crank axle, the crank-pin 
 revolving in the direction of the arrow. In this case the 
 piston and connecting rods are both in a state of compression, 
 and consequently there is a downward resulting force acting 
 at the cross-head, which is opposed by the resistance of the 
 slide-bar. Next take the return stroke, the connecting rod 
 being in the position shown by the dotted line. In this case 
 
Pressure on Slide Bars. 215 
 
 both piston and connecting rods are in a state of tension, 
 and consequently we have again a downward resultant force 
 acting at the cross-head. Therefore so long as the engine 
 runs in the direction shown by the circular arrow the pressure 
 is always downwards. Similarly if the engine were reversed 
 so as to run in the opposite direction, there would always be 
 an upward resultant pressure acting at the cross-head and we 
 should require a top slide-bar in order to prevent this pressure 
 from deflecting the end of the piston rod. If the steam ceased 
 to act on the piston during a portion of the stroke, and if at 
 the same time the valves were so set that there should be 
 considerable compression or back pressure on the other face 
 of the piston, the piston rod might cease to be in a state of 
 
 \ 
 
 I 
 
 / 
 
 / 
 
 Fig- 57- 
 
 compression (or tension as the case might be) at a certain 
 point in the stroke, and the above reasoning would not apply. 
 In such a case we might have a downward pressure on the 
 slide-bars during one portion of the stroke and an upward 
 pressure during the remainder (or vice versa), and thus two 
 slide -bars are often necessary even in factory engines which 
 are usually not reversed. Of course in the case of loco- 
 motives, marine and winding engines, which are constantly 
 reversing, two slide-bars or some corresponding arrangement 
 are an absolute necessity ; but in factory engines if the valves 
 are so set that there is no fear of a reversal of the direction of 
 stress on the piston^ one slide-bar is often sufficient. Of 
 course in such cases the engine is arranged to run so that 
 the pressure may act downwards, as it is much easier to keep 
 
216 T/ie Steam Engine. 
 
 the working face of the bottom bar lubricated than the 
 corresponding top face. 
 
 The amount of the pressure on the slide-bar depends 
 upon the angle () made by the axis of the connecting rod 
 with the axis of the piston rod. The greater the angle 
 (other things being equal) the greater the pressure. The 
 size of the angle depends on two things : first, the position 
 of the crank, the angle constantly increasing till the crank 
 is at right angles to the axis of the piston rod, and then 
 again diminishing ; and second, the relative lengths of the 
 crank arm and the connecting rod ; the shorter the connect- 
 ing rod, as compared with the crank arm, the greater will 
 be the value of a for a given position of the crank. In the 
 great majority of engines the connecting rod lies between 
 6 times and 3*5 times the length of the crank. The amount 
 of the pressure on the slide- bar equals the vertical com- 
 ponent of the pressure or tension in the connecting rod. 
 When the crank is at right angles to the axis of the piston 
 rod this vertical component equals the horizontal pressure 
 or tension in the pieton rod multiplied by the tangent of the 
 angle between the piston and connecting rods. With the 
 proportions of length of connecting rod to length of crank 
 arm usually adopted, this angle is so small that there is no 
 material difference between its sine and its tangent. Hence 
 the usual rule is : Maximum pressure on slide-bar=pressure 
 on piston x sine of angle between piston and connecting 
 rods = pressure on piston x crank radius -7- length of con- 
 necting rod. 
 
 We are only concerned to know the maximum pressure 
 because the slide bar must be made wide enough to bear 
 this pressure with safety. The safe amount of working 
 pressure per square inch of slide-bar surface varies with the 
 material of which the bar is constructed. In locomotives 
 with steel or case-hardened wrought-iron bars the pressure 
 is sometimes as high as 120 Ibs. per square inch. This is, 
 however, too high a pressure, and occasions very rapid wear 
 
Pressure on Slide 
 
 of both bars and motion blocks. In the est moern prac 
 tice the width of bars and areas of motion blocks are so 
 chosen that the pressure shall not exceed 40 Ibs. per square 
 inch when the bearing surfaces are of cast iron, and half as 
 much again when they are of steel. The actual pressure 
 acting on the piston rod when the crank arm is at right 
 angles to the axis of the cylinder must be calculated from 
 diagrams constructed as explained in Chapter V. page 194 ; 
 it can only be taken from the indicator diagrams when the 
 piston speed is very moderate. 
 
 The cross-head should be so designed that the point at 
 which the vertical pressure is transmitted by the connecting 
 rod shall be exactly over the centre of the guide or motion 
 block, otherwise an undue proportion of pressure will be 
 transmitted to the edge of the block nearest to the point 
 
 Fig. 58. 
 
 where the pressure is applied. Fig. 58 is an illustration of 
 a cross-head in which this consideration has not been 
 attended to. 
 
 Fig. 59- 
 
 Fig. 59 shows a longitudinal and transverse section of a 
 type of cross-head commonly in use in factory engines 
 which require only one slide-bar. 
 
218 
 
 Steam Engine. 
 
 Fig. 60 is an example of a cross-head provided with but 
 one slide -bar. The motion block is so arranged as to 
 envelope the bar, and the pressure comes on the upper or 
 lower surface of this latter, according to the direction in 
 which the engine is running. 
 
 Fig. 60. 
 
 Fig. 61 is an illustration of the cross-head of the largest 
 type of marine engine showing also the ends of the piston 
 and connecting rods, and sections of the large slides and 
 massive slide bars, the latter being attached to the main 
 frames of the engines. Specimens of cross-head and slide 
 bars are also given in pages 458 to 461. 
 
 In the older types of engines slide-bars were not used. 
 The end of the piston rod was kept straight by means of a 
 
Cross Heads. 219 
 
 parallel motion, which is an arrangement of link-work so 
 contrived that a pressure is always brought to bear on the 
 end of the rod equal in amount, but opposite in direction 
 to the pressure due to the angular position of the connecting 
 rod. Parallel motions are now seldom used for this purpose, 
 and need not therefore be described in this volume. A 
 
 Fig. 61. 
 
 simple form of parallel motion, as used in the construction 
 of indicators, is illustrated on page 319. 
 
 Connecting rods and cranks. The connecting rod is 
 the link which enables the reciprocating rectilinear motion 
 of the piston to be converted into the circular motion of the 
 crank pin. It is a link or rod of metal so formed at the 
 two ends that it can be jointed to both the cross-head and 
 
22O The Steam Engine. 
 
 the crank-pin. Suppose the piston to be at the commence- 
 ment or end of the stroke as shown in the upper diagram 
 fig. 62. Corresponding to these positions, the crank lies 
 in the direction of the axis of the piston rod prolonged. If 
 pressure be applied to the piston when at rest in either 
 of these positions, that pressure will be transmitted direct 
 to the bearings of the crank axle, there will be no turning 
 force applied to the crank, and consequently the latter will 
 have no tendency to revolve. Hence these two positions of 
 the crank are called its dead centres. Let, however, the 
 crank occupy the position a'c, fig. 62. The connecting rod 
 will now be inclined to the directions of both piston rod and 
 crank. If when in this position pressure be applied to the 
 
 Fig. 62. 
 
 piston^ the pressure will be transmitted through the con- 
 necting rod to the crank-pin, and can there be resolved in 
 two directions, viz. along the crank and at right angles to it, 
 i.e. tangential to the crank circle. The former component 
 merely exerts pressure on the bearings of the crank axle, 
 but the latter tends to cause the crank to revolve. The two 
 components are constantly varying in size and in relation 
 to each other, as is fully explained in Chapter V. When 
 the crank is on either dead centre the tangential component 
 vanishes, while the radial one is a maximum ; and vice versd y 
 when the crank occupies such a position that the axis of the 
 connecting rod forms a tangent with the crank circle, the 
 radial component vanishes, while the tangential is a maximum. 
 
Connecting Rods and Cranks. 
 
 221 
 
 This position, which occurs twice in each revolution, depends 
 upon the length of the connecting rod relatively to that of 
 the crank. If the connecting rod were of infinite length it 
 would form a tangent with the crank circle when the crank 
 was at right angles to the direction of the axis of the 
 piston rod. The shorter the connecting rod relatively to 
 the crank, the nearer to the dead centre will be the posi- 
 tion of the crank when the tangent is formed. This is 
 
 Fig. 63. 
 
 illustrated by the three diagrams, fig. 63, which represent 
 the tangential positions for connecting rods having respec- 
 tively lengths equal to infinity, and to six times and three 
 times the length of the crank. 
 
 The force transmitted by the connecting rod is estimated 
 as follows. Let P be the force acting on the piston rod, and 
 a the angle which this latter forms with the connecting rod, 
 then P sec a is the force in the connecting rod. If P were 
 
222 The Steam Engine. 
 
 constant the value of this expression would increase with a. 
 The maximum value attained by a is when the crank is at 
 right angles with direction of the line PC. Let the con- 
 necting rod be r times the length of the crank arm. In 
 this position ABC is a right-angled triangle and sec a 
 
 r 
 
 = AB -=- AC = _ ===. Suppose that r=6, then the force 
 
 in the connecting rod = P - =. =P x 1-0142. If r~ 3, then 
 
 ^35 
 
 the force = P A. =P x ro6. 
 V8 
 
 It will thus be seen that the connecting rod transmits a 
 constantly varying force to the crank-pin, and the component 
 of the force which causes the crank to rotate is also 
 constantly varying from zero, which value it attains at the 
 
 dead centres, to Psec a=P ^ r "*~ \ , which it attains at the 
 
 r 
 
 two positions when the connecting rod is tangential to the 
 crank circle. The proper method of exhibiting graphically the 
 tangential effort transmitted by the connecting rod to the 
 crank-pin is explained in detail in Chapter V., which also 
 shows how to take account of the varying steam pressure on 
 the piston, the effect of the inertia of the reciprocating parts, 
 and the length of the connecting rod relatively to the crank 
 arm. 
 
 It will be noticed that while the piston is moving once 
 backwards and forwards in the cylinder, the crank-pin is 
 describing a circle, having a diameter equal to the length of 
 stroke. Consequently the relative lengths of the paths 
 travelled by piston and crank-pin are as the diameter to the 
 semicircumference of a circle, or as ir : irr=2 : 3'i4i59. 
 Now, by the principle of work, the pressure on the piston 
 multiplied by the length of the path it travels equals the 
 tangential pressure on the crank pin multiplied by the 
 length of its path ; hence the average pressure on the piston 
 
Stresses on Connecting Rods. 223 
 
 is to the average tangential pressure on the crank pin as 
 3-14159 : 2. The fact that the average tangential pressure 
 on the crank-pin was less than that on the piston led old 
 writers on the steam engine into the fallacy of asserting that 
 the oblique action of the connecting rod caused a loss of 
 power in the steam engine. They forgot, however, that 
 though the full pressure on the piston does not in general 
 appear tangentially to the crank, the path travelled by the 
 latter is correspondingly greater than that of the piston. 
 
 In addition to the stress due to the pressure of steam on 
 the piston, which is alternately one of compression and ten- 
 sion, the connecting rod is also subject to stresses, due to 
 the inertia of the piston and piston rods and also due to its 
 own inertia. These stresses, which are dealt with in 
 Chapter V., may become very considerable in high-speed 
 engines. The rapid vibration of the rod from side to side, 
 somewhat after the manner of a pendulum, also tends to 
 produce a peculiar motion of the rod called whipping. This 
 arises from a bending stress due to the resistance of the rod 
 to acceleration in a direction at right angles to the motion 
 of the piston. The tendency to whip is greatest in rods of 
 great relative length. It may be guarded against by sparing 
 unnecessary weight in the rod, by making it strongest where 
 the tendency to whip is greatest, and by so distributing the 
 material in the cross-section of the rod that its resistance to 
 bending stresses may be a maximum, For instance the rod 
 is usually tapered from the piston rod end to the crank-pin 
 end ; and instead of being made round, it is usually given a 
 rectangular section, the side of the rectangle which lies in 
 the plane of the motion being the greatest. In this manner 
 a given weight of material is distributed so as best to resist 
 a bending stress. Sometimes the rod is given a section 
 resembling that of a flanged girder, the greater part of the 
 material being distributed in the two flanges. 1 
 
 1 The rules for estimating the stresses on and fixing the proportions 
 of the various moving parts of engines are given with great clearness in 
 Professor Umvin's Elements of Machine Design, published in this series. 
 
22 4 
 
 The Steam Engine. 
 
 Fig. 64 represents a connecting rod, and its various 
 details as used in a stationary engine. The parts surrounding 
 
Examples of Connecting Rods. 
 
 225 
 
 the crank and cross-head pins are made of gun metal, brass, 
 or white metal so as to diminish friction. They are made 
 in separate pieces, called steps, and are held in place by the 
 straps aa, which are fastened to the rod by means of the 
 gibs bb t and the cotters cc. When the brasses wear they can 
 be tightened up by driving in or screwing up the cotter, 
 which draws up the strap, and 
 thus tends to shorten the rod. 
 
 Fig. 65 shows a solid ended 
 rod. In this case the brass 
 steps are placed in the rect- 
 angular opening formed in the 
 end of the rod, their flanges 
 being partly cut away so as to 
 allow of their being inserted. 
 They are held in position by 
 a cotter, which in its turn is 
 prevented from moving by one 
 or more set screws. When 
 the brasses wear they can be tightened up by driving in the 
 cotter, which has the effect of lengthening the rod. 
 
 Fig. 66 shows a fork- . 
 
 rod 
 with 
 
 ended connecting 
 which is necessary 
 certain sorts of cross-heads. 
 Fig. 67 is the ordinary type 
 of marine connecting rod. 
 In this rod the brass steps 
 are not held by gibs and 
 cotters, but by bolts, which 
 are clearly shown in the 
 sketch. 
 
 The lubrication of the 
 surfaces of the brass steps 
 where they rub on the crank 
 and cross-head pins must be very carefully attended to. 
 
 Q 
 
 Fig. 65. 
 
 LP" 
 
 Fig. 66. 
 
226 
 
 The Steam Engine* 
 
 The ends are for this purpose provided with lubricators, 
 which are sometimes of brass, and can be removed, but are 
 more commonly forged solid on to the straps or ends. If 
 
 Fig. 67. 
 
 the lubrication is well attended to, and if the diameters and 
 lengths of the pins are made sufficiently large so as to keep 
 the pressure per square inch within moderate limits, the 
 wear of the brass steps should give very little trouble. 
 
 In old-fashioned engines the connecting rod was often 
 made of cast iron, but nowadays it is invariably of case- 
 hardened wrought iron, or of mild steel. 
 
 Cranks and eccentrics. The crank is simply a lever of 
 the first order, either attached to, or forged in one piece with 
 the main shaft of the engine. By means of it, the recipro- 
 cating motion of the piston is finally converted into circular 
 motion, as has been explained in the previous section. The 
 manner in which the force, transmitted by the connecting 
 rod to the crank, is resolved into a radial component 
 which has no effect but to exert pressure on the main 
 bearing, and a tangential component which alone tends to 
 revolve the shaft, is also explained in that section, and more 
 fully in Chapter V. 
 
 Fig. 68 shows two views of one of the simplest forms of 
 crank ; a is the crank shaft, c the crank-pin. The distance 
 from the centre of a to the centre of c is the length of the 
 
Cranks. 
 
 227 
 
 crank arm, which is, of course, equal to half the stroke of the 
 piston, b is the web of the crank, </, d' the bosses. Cranks of 
 this form are generally of cast iron, and are attached to the 
 
 Fig. 68. 
 
 main shaft by means of a key e. The fastening of a move- 
 able crank on a shaft requires the greatest care, because all 
 the stresses thrown on the crank are liable to reversion during 
 each stroke, especially in the case of a slow-running engine 
 working with a considerable cushion of steam. Very severe 
 reversals of pressure'also occur if water is allowed to accumulate 
 in the cylinder. In such cases the piston is brought up dead 
 before the end of the stroke is reached, while the crank 
 endeavours to pull it on, thus throwing a heavy strain on all 
 the connections, and amongst others on the key. Cranks 
 of this type in addition to being keyed are generally shrunk 
 on to the shaft, or else are forced on by hydraulic pressure. 
 
 Fig. 69 shows an end elevation and cross-section of 
 another form of cast-iron crank, called a disc crank. It is, as 
 its name implies, formed of a disc of cast iron, attached to 
 the shaft by the methods just described, and provided with a 
 wrought-iron or steel crank-pin. The portion of the disc 
 opposite to the pin is usually much thicker and heavier 
 than the remainder of the disc, this extra weight being used 
 as a balance to the weights of the reciprocating parts. 
 
 Very often it is impossible to arrange that the crank 
 shall overhang the end of the shaft, and in such cases it has 
 
 Q3 
 
228 
 
 The Steam Engine. 
 
 to be forged in one piece with the latter. A great number 
 of engines also work with two or more cylinders, and 
 a corresponding number of cranks. In the case of two- 
 cylindered engines, the cranks are, as a rule, set at right 
 angles to each other, so that when one of them is on its 
 
 Fig. 69. 
 
 dead centre, the other is in the most favourable position, 
 and thus, the resultant force tending to turn the shaft is 
 made more uniform. Fig. 70 shows two views of the crank 
 
 Fig. 70. 
 
 shaft of a locomotive engine, with two cranks forged in one 
 piece with the shaft, the whole being made of steel, which 
 metal, on account of its great strength, is especially useful 
 in cases where, from want of space, the thickness of the 
 webs or side pieces of the crank is limited. It will be noted 
 
Cranks and Eccentrics. 229 
 
 that in the case illustrated in fig. 70 the want of thickness 
 in the webs is made up for by increasing their width, which 
 is nearly double the diameter of the shaft. 
 
 The crank-pin is the portion of the engine which receives 
 the greatest stress, and special care must therefore be given 
 to its design and lubrication. The pressure which comes 
 upon it varies in practice, according to the type and speed of 
 the engine, from 500 Ibs. to 2,000 Ibs. per square inch. This 
 latter figure is far too high for safety. The diameter and 
 length of the pin should be so chosen that the pressure 
 may not exceed 1,000 Ibs. per square inch. It is desirable 
 to make the pin as long as is consistent with strength and 
 other considerations, because bearings retain their lubrication 
 better when long than short. In high speed-engines, there 
 is a great tendency to expel the lubricating oil from the 
 crank pin by centrifugal force, and this tendency must be 
 carefully guarded against, otherwise excessive wear of the 
 connecting rod brasses will be the result. 
 
 In the crank proper, the pin a, fig. 7 1 (A), is of such a 
 diameter that it does not come in contact with the shaft. 
 There is, however, a species of crank called the eccentric, in 
 which the pin is so large that it completely envelopes the 
 shaft. Such a crank is shown at fig. 7 1 (B). The distance 
 
230 The Steam Engine. 
 
 ac is the same as at A, but the pin has assumed the diameter 
 of the outer circle de. In such a case, of course, the web is 
 impossible and unnecessary. A crank is used for converting 
 the rectilinear motion of the piston into circular motion ; 
 the eccentric, on the other hand, is usually employed for 
 converting the circular motion of the main shaft back into 
 rectilinear motion. The ordinary crank is, of course, equally 
 capable of effecting this conversion, but it would in many 
 cases be impossible to apply it to such a purpose. A crank 
 always weakens a shaft except when overhung ; hence, the 
 fewer the number of cranks made use of, the better. 
 Moreover, when circular motion is converted into rectilinear, 
 as a rule, the rectilinear path travelled is short, relatively to 
 the diameter of the crank shaft ; hence a crank would be 
 an impossibility, because the crank-pin would fall either 
 wholly or partly within the section of the shaft. With an 
 eccentric, however, the case is different, the distance ac, 
 fig. 71 (B), corresponding to the length of the crank arm, 
 may be as small as we please. The most frequent uses to 
 which eccentrics are put are to drive slide valves and pumps, 
 the travels of which are very much less than that of the 
 piston. 
 
 The distance ac, from the centre of the shaft to the 
 centre of the eccentric, is called the eccentric radius, the 
 eccentricity, or the half-throw of the eccentric, and is equal 
 in length to the half-travel of the part to be driven, such as 
 the pump plunger, or slide-valve. 
 
 Fig. 7 2 represents side elevation and a longitudinal sec- 
 tion of an eccentric and rod as used for driving an ordinary 
 slide-valve. The circular portion a, which corresponds 
 to the crank pin, is called the sheave of the eccentric. It is 
 usually made of cast iron in two halves, which are bolted 
 together round the shaft, and keyed on in the proper 
 position. The piece bb is called the strap, and corresponds 
 with the big end of a connecting rod. The strap is made of 
 cast iron, wrought iron, or steel, according to circumstances, 
 
Eccentrics and Rods. 
 
 231 
 
 and is lined with a brass or white metal ring, where it comes 
 in contact with the sheave. This ring is grooved, as shown 
 in the section at cc, so as to prevent it from getting off the 
 
 sheave. The strap is made in two halves bolted together 
 so that it can be readily put on, or taken off the sheaf. It 
 should be made sufficiently rigid not to spring when the 
 
232 The Steam Engine. 
 
 engine is running, as the effect of this would be to cause a 
 great deal of local friction between the sheave and strap. 
 An oil cup d is usually forged solid on one half of the strap, 
 and particular care must be given to the lubrication, the 
 friction of eccentrics being much greater than that of cranks. 
 The eccentric rod, which corresponds with the connecting 
 rod of a crank, is sometimes forged solid with one half of 
 the strap ; it is, however, often made separate, and attached 
 by bolts to the strap. The length of the eccentric rod, 
 relatively to the half throw of the eccentric, is always much 
 greater than that of the connecting rod, relatively to the 
 crank arm. Hence, the disturbing influence on the point 
 driven, due to the finite length of the rod, is usually very 
 slight. 
 
 The manner in which the eccentric, or combinations of 
 eccentrics, are used to actuate the valves, will be explained 
 in the next chapter. 
 
 Crank Shafts. The shaft of the engine is the part 
 which receives circular motion from the crank and the 
 reciprocating pieces. By means of the shaft, the power 
 generated in the cylinder is transmitted to the machinery 
 intended to be driven. Thus, in the case of factory engines, 
 a pulley is usually keyed on to the shaft, and by means of a 
 leather belt passing over this pulley, the various lines of 
 shafting throughout .the building are driven. In locomotive 
 engines, the driving wheels are keyed direct on to the shaft, 
 and rotate with it, and in the case of marine engines, the 
 paddles or screw are also attached direct on to the shaft or 
 its prolongation. 
 
 Strains in Crank Shafts. Shafts are subjected to a 
 variety of strains. In the first place, they undergo bending 
 stresses from any weights which may be attached to them, 
 the most considerable of which is that of the fly-wheel, acting 
 vertically downwards. Also the pull of the driving belt 
 causes a bending stress, which acts in the line joining the 
 driving and the driven shaft The most important stresses. 
 
Strains in Crank Shafts. 233 
 
 however, are due to the direct thrust and pull of the con- 
 necting rod, or rods, which, when at their maximum, act in 
 the line of the axes of the cylinders. These various bending 
 stresses may act in such directions as either to partly neu- 
 tralise or to reinforce each other. In any case a resultant 
 stress can always be found which represents their combined 
 action. A shaft under the action of these stresses behaves 
 in exactly the same manner as a loaded girder. 
 
 The foregoing strains act in the planes of the axis of the 
 shaft. Shafts are, however, subjected to a totally different 
 class of strains, which act in planes at right angles to the 
 axle, viz. those which result from the twisting or torsional 
 effects of the power and resistance acting at the ends of 
 levers represented by the length of the crank arm, and the 
 radius of the pulley, over which the main belt passes in the 
 case of factory engines ; or the radius of the driving wheels, 
 or of the centres of effort of propellers, in the cases of 
 locomotive and marine engines respectively. The tendency 
 of these torsional strains is to shear the metal, composing 
 the shaft, in a plane at right angles to the axis. They are 
 consequently opposed, by the resistance of the metal to 
 
 Fig. 73- 
 
 shearing. In fig. 73 let A represent a section of a shaft. Let 
 bc=l be the length of the crank arm, and P be the maxi- 
 mum force in Ibs. transmitted to the crank pin by the con- 
 necting rod, estimated in the direction of the arrow. Then, 
 P x / is the twisting moment applied to the shaft. If / is 
 expressed in feet, then the moment is in foot-pounds, and if 
 in inches, then the moment is in inch-pounds. The forces 
 
234 The Steam Engine. 
 
 at work are prevented from shearing the shaft by the resis- 
 tance to shearing of all the metal in the section A. If the 
 shaft were just upon the point of shearing, the moment got 
 by multiplying the resistance of the metal by the distance of 
 its centre of resistance from c, would be exactly equal to the 
 twisting moment P/. To calculate the value of this moment 
 of resistance, we must proceed in the following manner : 
 Conceive the section of the shaft to be divided up into a 
 number of triangles (one of which is shown), and the triangle 
 to be divided up by lines parallel to its base into a series of 
 bands (one of which is also shown shaded); then, the power 
 of any one of these bands to oppose the shearing effect of 
 the twisting moment P/, relatively to the corresponding 
 power of the exterior band, depends upon the area of the 
 given band, and its distance from the centre, relatively to the 
 exterior radius of the shaft. Call the variable distance of 
 the bands from the centre p, and the angle at the apex c of 
 the triangle, dd, then the breadth of the band, measured 
 along the radius, is dp, and its length p.dd consequently, 
 its area is p.dp.dd, and if the band were carried right round the 
 circle, its area would be 2-rr.p.dp. The distance of this annular 
 band from the centre, relatively to that of the exterior cir- 
 cumference of the shaft =- ; hence, the power of the strip 
 
 to resist shearing, relatively to that of the outside, = 2 rr.p.dp.^- 
 
 2 3 
 
 = - 7r P-dp and its moment about the centre c= - d/>. Simi- 
 r r 
 
 larly with every other circular band into which the section of 
 the shaft can be divided. Hence, the moment of resistance 
 of the whole shaft to resist shearing about the centre c, is 
 proportional to the above expression integrated between the 
 limits p=o and p=r, or 
 
Strains in Crank Shafts. 235 
 
 If, instead of the radius r, we take the half diameter - 
 the above expression becomes - =o'i$6d 3 . 
 
 In order to make this expression practically useful we 
 require to know first, the ultimate resistance to shearing of the 
 metal of which the shaft is made ; and second, the factor 
 of safety to be employed, that is to say, the number of times 
 which the moment of resistance of the shaft should exceed 
 the twisting moment, in order that it may be safe in practice. 
 The following are usually taken as the ultimate shearing 
 strengths of the metals usually employed in making shafts : 
 
 Cast iron. . . . 28,000 j lbg 
 
 Wrought iron . . 54,000^.^ 
 
 Steel ..... 80,000 ) 
 
 As a rule the factor of safety employed for shafts is 6, 
 therefore in designing a shaft for a given purpose we must 
 only take one sixth of the above figures. Thus to find the 
 proper diameter for a wrought-iron shaft subject to a given 
 twisting moment P/ in inch Ibs., we have 
 
 o 
 /. d (in inches) = -082 75 x^/P/. 
 
 Sometimes the horse-power (HP) transmitted by the shaft 
 and the number of revolutions (N) are given. One horse- 
 power = 33,000 foot-pounds per minute. The total horse- 
 power equals the average pressure (P) applied to the end of 
 the crank arm (/) x by the path of the crank pin in feet 
 
 [ -- J x by the number of revolutions (N) per minute, 
 ,. 33,o x HP = P -N 
 
 = 63,024 x- 
 
236 The Steam Engine. 
 
 and, equating this latter expression to ' 
 
 3 /ZJp 
 
 we obtain d (in inches) = 3 -2 95 A/- . 
 
 Journals. The part of the shaft which is supported by 
 the bearing is called the journal. The usual form of the 
 journal of an engine crank is shown in fig. 74. The part 
 
 Fig. 74- 
 
 which runs in the bearings is turned so as to be truly cylin- 
 drical. The end play of the shaft is limited by the two 
 raised collars. The length of the journal, or the distance 
 between the inner faces of the collars, relatively to the 
 diameter depends principally upon the number of revolu- 
 tions which the shaft has to make per minute. For slow- 
 running engines the length is sometimes equal to the dia- 
 meter, whereas in cases of high speed it may be as much 
 as from two to three times the diameter of the journal. The 
 following figures give the proportions usually adopted * for 
 wrought-iron journals. 
 
 No. of revolutions per minute 50, 100, 150, 200, 250, 500. 
 
 Ratio of length to diameter 1*2, 1*4, 1*6, 1*8, 2*0, 3 - o. 
 
 Great care must be taken in designing journals not to 
 pass abruptly from one section of the metal to another. All 
 such differences should be gradually rounded off as shown 
 in fig. 74. 
 
 The strains to which the journals of crank shafts are 
 subjected are due to the combined action of the twisting 
 forces and the transverse loads. 
 
 Shaft Bearings and Pedestals. The bearing usually con- 
 sists of brass steps supported by a cast-iron pedestal or 
 
 1 Elements of Machine Design. By Professor W. Cawthorne Umvin. 
 
Bearings and Pedestals. 
 
 237 
 
 plummer block. Fig. 75 shows three views in half elevation 
 and half section of a common form of pedestal which is 
 used with a masonry foundation. It consists of a wall plate 
 
 Fig. 75. 
 
 which is bolted to the foundation and on which is fixed the 
 pedestal proper. The nature of the arrangement and the 
 means by which the steps are adjusted and secured are 
 sufficiently explained by the drawing. 
 
 In most stationary engines one or both of the pedestals 
 are attached to the cast- 
 iron framework as shown 
 in fig. 76, which represents 
 the principal pedestal of 
 a horizontal engine. In 
 this case the steps are not 
 divided horizontally, but in 
 an oblique plane, so that 
 the direction of the re- 
 sultant force of the pull or 
 thrust in the connecting 
 rod and of the other forces 
 
 Fig. 76. 
 
 which act on the shaft, may pass through the solid metal of 
 the step and not through the junction between the steps. 
 In the case of locomotives the bearings are not fixed, 
 
238 
 
 The Steam Engine. 
 
 but are free to slide up and down in a vertical plane, 
 within the limits allowed by the springs. These bearings 
 are called axle boxes. The whole weight of the engine is 
 transmitted through them to the journals by means of the 
 springs. Fig. 77 explains the structure of an axle box, 
 which consists of an outer case, arranged so as to be capable 
 of sliding up and down, between guides called horn plates 
 which are bolted to the frame of the engine. The casing 
 
 contains one brass step, the whole of the pressure being of 
 course on the upper half of the journal. The lower part of 
 the casing contains a receptacle for the oil which escapes 
 after lubricating the bearings. The upper portion contains 
 the oil box, and has also a socket formed in it which receives 
 the foot of a spindle by means of which the pressure from 
 the springs is transmitted. 
 
 Governors. If, during the working of a steam engine, 
 the load were wholly or partially removed while the supply 
 
Governors. 
 
 239 
 
 of steam to the cylinder remained undiminished, the engine 
 would commence to race. If, on the contrary, the load 
 were increased, the speed of the engine would be reduced 
 below the proper rate. To prevent such variations in the 
 speed, a contrivance called a governor is made use of which 
 acts upon the steam supply in one of two ways viz. either 
 by partially closing or opening the throttle valve which 
 regulates the flow of steam from the boiler; or else, by 
 acting directly on the valve gear in such a way as to vary 
 the point in the stroke where the steam is cut off, and thus 
 alter the rate of expansion. 
 
 The most common form of governor was invented by 
 Watt. It consists (see fig. 78) 
 of two heavy metal balls A,D, 
 attached to two inclined arms, 
 which latter are jointed at 
 the point E, to the central 
 vertical spindle. The latter is 
 connected by gearing with the 
 main shaft of the engine so as 
 to revolve at a rate strictly pro- 
 portional to that of the shaft. 
 The effect of rotation is that 
 the balls tend to fly away from 
 the vertical spindle and, being 
 controlled by the arms, they 
 can only rise and fall in arcs 
 of circles about the centre E. 
 Supposing that the velocity of rotation were increased be- 
 yond the normal rate, the balls would fly out and occupy 
 some new position D', at the same time lifting the collar H 
 which slides on the central spindle and which is attached 
 by the links L and K and to the ball arms M and N. Into 
 the collar H gears the forked end of a bell crank lever 
 which is connected by a link with the throttle valve. When 
 H is lifted the link acts upon the throttle valve, partly closing 
 
 Fig. 78- 
 
240 
 
 Tlie Steam Engine. 
 
 arm 
 
 it, and reducing the supply of steam. Conversely when the 
 balls fall, H falls also and the throttle valve is opened. 
 
 The theory of the conical pendulum governor is as follows. 
 Let us suppose that the weight of the arms may be neglected. 
 When the balls occupy any position as in fig. 79, each of 
 
 them is maintained in posi- 
 .A. tion by the three following 
 
 forces. The weight of the 
 ball W acting downwards ; 
 the tension T in the inclined 
 ; the centrifugal force 
 
 (where v is the velocity 
 
 gr 
 
 of rotation of the ball), acting 
 radially and horizontally out- 
 wards. Since the ball is in 
 equilibrium, the three forces 
 may be represented in magnitude and direction by the three 
 sides of the triangle ABC. Let the radius BC be denoted 
 by r, and the height of the cone of revolution AC by h feet, 
 W 
 
 then *=W=: /. ^J 1 -, :. r -=j,l h . 
 
 r v V* p v 'v & 
 
 g r 
 
 Also since the ball is supposed to move in a circle in a 
 horizontal plane with a uniform velocity, let / = the time in 
 seconds occupied in making one revolution, 
 2irr Ih 
 
 then 
 
 S 
 
 Consequently the time of a revolution is proportional to the 
 square root of the height of the cone of revolution. 
 
 If we are given the number of revolutions N per minute, 
 
 then N/=6o, .'.^= ; /h 
 
 3. /-_5- 
 
Governors. 241 
 
 If h be given in inches instead of feet, the above formula 
 becomes ^ 
 
 N= ==-. 
 N/ h 
 
 As the speed of rotation of the governor and consequently 
 of the engine is inversely proportional to the square root of 
 the height of the cone of revolution, it is clear that the 
 possible variations in the height of the cone have a very 
 direct influence upon the sensitiveness of the governor. 
 
 For instance, if the governor were so contrived that the 
 height of the cone were a constant quantity, the speed of the 
 engine would remain constant. The object aimed at in 
 the practical design of governors is to keep the variations in 
 the height of the cone of revolution within convenient limits. 
 It will "be noted that in fig. 78 the ball arms are jointed on 
 the axis of the vertical spindle, while in other cases the 
 joints are at some distance from the axis ; in other instances 
 the arms are crossed as in fig. 81, so that the joints are 
 on the sides of the spindle opposite to the corresponding 
 balls. 
 
 Each of these arrangements affects the height of the cone 
 of revolution for a given position of the balls. 
 
 It is evident, from a mere inspection of the figures, for a 
 given deviation of the balls from the axis of revolution, that 
 the variation in the height is greatest in fig. 78, and least 
 in fig. 8 1. 
 
 It is quite possible to make a governor so inconveniently 
 sensitive, that it is never still for a moment, but is affected 
 by even the small periodic changes of velocity, which occui 
 in each revolution of the engine. As the governor can, by 
 its nature, never act until after the change of velocity actually 
 occurs, which it is designed to control, and as, moreover, 
 the periodic changes of velocity above referred to are only 
 momentary in their duration, it may easily happen that the 
 effect of a hypersensitive governor is only felt by the 
 engine when it has resumed its normal speed of rotation, or 
 
 R 
 
242 
 
 The Steam Engine. 
 
 even attained a rate of speed which has fluctuated in the 
 opposite sense to that which had affected the governor. In 
 such cases, only harm can be done by the sensitiveness of 
 the governor: for, supposing that it is affected at a period of 
 the revolution when the crank-pin velocity is slightly above 
 the average of the revolution, the supply of steam will be 
 diminished when the crank-pin has re-attained its average 
 speed, or even sunk below it, in which case the effect of the 
 governor will be to aggravate the evil it was designed to 
 cure. 
 
 Many advantages are found to attend the use of high- 
 speed governors. They are more sensitive to alterations in 
 speed, the parts may be made lighter and move with less 
 friction. In order, however, to prevent the balls from flying 
 out too far, in consequence of the increased speed of rotation, 
 
 Fig. 80. 
 
 a weight, or else a spring, is so arranged as to act on the 
 ball arms in such a manner as to develope a radial force in 
 the contrary direction to the line of action of the centrifugal 
 force. Fig. 80 shows a loaded high-speed governor. Each 
 ball is attached to two sets of links. The weight is arranged 
 
Governors. 243 
 
 to slide on the central spindle, and presses directly upon the 
 lower pair of ball links. To find the height of the cone h, 
 corresponding to a given speed of rotation, we reason as 
 follows : Each ball is at rest under the action of its weight 
 acting downwards, the centrifugal force acting radially 
 outwards, and the tensions in the two ball arms due to the 
 weights which they support. 
 
 Calling the weight of each ball, W, that of the load, W, 
 the radius BC=^, and the height of the cone AC=/J, the 
 tension in BA=T, in BD=T, the angle BAC=a, and BDC 
 =ft we have : 
 
 The centrifugal force in BC is balanced by the compo- 
 nents of the tensions in the two arms, estimated in the direc- 
 tion BC 
 
 gr 
 
 Also, the vertical component of the tension in BA balances 
 the weight of the ball W, and the vertical component of 
 the tension in BD 
 
 /. Tcosa=W + T'cos/3. 
 
 The tension in BD is due to half the weight W 
 , /. W'= 2 T / cosft 
 
 and, finally, r=BAsina=BDsin/3, from which equations it 
 is possible to eliminate the values of the tensions in the ball 
 arms, and also the angles a and /3, so that /&, for any velocity z>, 
 may be expressed in terms of the weights of the balls and 
 load, and of the radius r. The simplest case is when a=/3, 
 whence 
 
 gr 
 W 
 W / = 2T / COSa. 
 
 Also, h=.rcoia. 
 
 R 2 
 
244 T/te Steam Engine. 
 
 Eliminating T, T, and coto, we obtain 
 
 r w 
 
 It has been already shown how the governor can be 
 arranged to act on the throttle valve. In many modern 
 
 Fig. 81. 
 
 engines the throttle valve is, however, not interfered with 
 during the working, and the governor is arranged to act 
 directly on the expansion gear of the slide valves. Fig. 81 
 shows a simple method of effecting this object. The collar 
 
Governors. 245 
 
 #, on the vertical spindle of the governor, works a lever b, 
 which is connected by a link c with the end of the sliding 
 block d which works in a rocking link as shown. The block 
 d is attached to the end of the eccentric rod g'. On the 
 position of d depends the amount of swing given to the 
 rocking link. An expansion valve, working on the back of 
 the main valve (see p. 263) is driven from another point of 
 the rocking link, and on the travel of this expansion valve 
 depends the point at which the steam is cut off. The 
 main valve is driven by the eccentric rod g in the ordinary 
 manner. 1 
 
 The forms of governors are so numerous, that it has been 
 impossible here to do more than explain the principles upon 
 which they act. 
 
 Locomotive engines are never fitted with governors, but 
 in marine engines they are very necessary, as racing may 
 ensue whenever the propeller is partially out of water, or 
 whenever the propeller or crank shaft may give way. On 
 account of the motion on board ship, the forms of governors 
 used on land engines could not be employed for marine 
 purposes. Marine governors are of two principal sorts, viz. 
 those that are actuated by variations in the water pressure at 
 the stern of the ship, and those which depend for their 
 motion on variations in the velocity of the engine. The 
 former class only provide for cases due to the incomplete 
 immersion of the propeller, but the latter will guard against 
 every contingency. In consequence of the great size of the 
 throttle valves and expansion gear of marine engines, an 
 ordinary governor cannot conveniently be employed to act 
 directly on the controlling parts ; hence, in this class of 
 engines, what are called steam governors are now generally 
 employed. The governor proper is arranged to move the 
 slide valve of a small steam cylinder, which, in its turn, 
 actuates the throttle valve. 
 
 1 This will be better understood after reading the succeeding chapter 
 on valve gearing. 
 
246 The Steam Engine. 
 
 Fly-wheels. The functions of fly wheels have been ex- 
 plained in Chapter V., pp. 157 and 196. It is only necessary 
 now to consider the principles involved in their construction. 
 The greater portion of the mass of a fly wheel is con- 
 centrated in its rim, and when revolving, every particle of 
 the rim is under the action of centrifugal force, and tends to 
 fly away radially from the centre ; hence the rim, when in a 
 state of revolution, resembles the condition of a ring put in 
 a state of tension by a force from within acting outwards. 
 The tension developed in the rim is opposed by the tensile 
 strength of the metal of which it is formed, and should the 
 
 former exceed the latter 
 the rim will inevitably 
 burst asunder, just as a' 
 boiler would burst if the 
 steam pressure were too 
 great for the strength of 
 the shell plates. 
 
 Suppose the rim in fig. 
 82 to be divided up into 
 a number of segments. 
 The centrifugal force on 
 FI 8a any one of them, such 
 
 as A, acts radially along 
 
 the line AC, and may be resolved into two components, 
 one along the diameter BD, and the other at right angles 
 to it, and similarly for all the other segments. The sum 
 of all the components at right angles to BD is the force 
 which tends to tear the ring asunder at the sections B 
 and D, 
 
 It is well known that if a force press uniformly outwards 
 all along the semicircumference of a ring, the components 
 at right angles to a given diameter equal the total radial 
 force multiplied by the ratio of the diameter to the circum- 
 ference. In the case of a fly wheel, the radial pressure 
 acting along any given semicircumference is half the 
 
Fly Wheels. 
 
 247 
 
 centrifugal force of the entire wheel, and the sum of the 
 components at right angles to the diameter BD 
 
 2T 
 
 = ^ centrifugal force of wheel x 
 
 Hence, the tension at either B, or at D, is half the above 
 quantity = half centrifugal force of wheel X . 
 
 27T 
 
 If W=the weight of the wheel in Ibs., r its mean radius 
 in feet, and N the number of revolutions per minute, we 
 have (see p. 161) 
 
 Tension at B, or at D= WxN'xrx 100034 
 
 2 XTT 
 
 EXAMPLE. 
 
 A fly wheel, the mean radius of which is 8 feet, weighs 15,000 Ibs., 
 the whole of which weight is supposed to act at the mean radius ; what 
 is the tension on the metal of the rim at either end of any diameter, the 
 wheel making 60 revolutions per minute ? 
 
 Answer. Tension = J5ooox_6o_x 60 x 8 x-ooo 34 = 2 ^ 6 lbs . 
 2x3-14159 
 
 Taking the tensile strength of cast iron at 15, 680 lbs. per square 
 inch and allowing a factor of safety of 5, it is evident that the section 
 of the rim of the wheel must not be less than about 8 inches. In 
 order to attain the given weight, the section of the rim would have 
 to be far greater than this quantity, hence the rim would possess ample 
 strength. 
 
 Fig. 83. 
 
 Small fly wheels are usually cast in one piece, but when 
 so large that the weight would be unwieldy, the rim is cast in 
 pieces which are afterwards put together by bars and cottars, 
 or by bars and bolts, as shown in fig. 83. 
 
248 TJie Steam Engine. 
 
 The rim in very large wheels is generally fastened to the 
 arms, as shown in fig. 84. The arms are fastened to the 
 
 Fig. 84. 
 
 boss in a similar manner, and the latter, in the case of large 
 wheels, is often cast in two halves, which are either bolted 
 together or attached by wrought-iron rings shrunk on. 
 
249 
 
 CHAPTER VII. 
 
 VALVES AND VALVE GEARS. 
 
 Action of the simplest form of D slide valve driven by single eccentric 
 Definitions of 'lap' and 'lead' Position of eccentric as affected by 
 the lap and lead of the valve Effect on the steam distribution of the 
 lap and lead of the valve Effect of ratio of length of connecting rod 
 to length of crank in modifying steam distribution Means of varying 
 the rate of expansion and of reversing Stephenson's link motion 
 Effect of diminishing the throw of the eccentric Reversing lever 
 Ramsbottom's reversing screw Variations in the details of Stephen- 
 son's link motion Other systems of link motion Other means of 
 varying the rate of expansion Meyer's separate expansion valve 
 Corliss's valve gear Varieties of valves Valve gears in which 
 eccentrics are dispensed with Joy's gear Geometrical representa- 
 tions of the action of slide valves Zeuner's valve diagrams Case of 
 valve without lap or lead Case of valve with lap and lead Problems 
 on simple valve setting Zeuner's diagrams applied to valves driven by 
 link motions Analytical method of fixing centres of valve circles 
 Graphical method Problems in link motion The method of suspend- 
 ing link motions Zeuner's diagrams applied to Meyer's valve gear 
 Reversing by Meyer's gear Problems on valve setting with Meyer's 
 gear. 
 
 THE successful and economical working of a steam engine 
 depends in a very large degree upon the design and adjust- 
 ment of the valve or valves which regulate the distribution 
 of the steam in the cylinders. The subject is, perhaps, 
 more complicated in its nature than any other question 
 affecting the design of the engine. In order to treat it simply 
 and, at the same time, systematically, it is intended in 
 this chapter, first to explain the simplest examples, and then 
 to proceed to the description of the cases which more 
 frequently occur in practice. 
 
 Action of the simplest form of slide valve driven by a single 
 eccentric. As the motion of a slide valve is modified by the 
 length of the connecting and eccentric rods relatively to the 
 
250 
 
 The Steam Engine. 
 
 length of the crank arm and to the throw of the eccentric 
 respectively, we will suppose, in the first case, that these rods 
 are infinite in length. Fig. 85 represents a portion of a 
 cylinder with steam chest, slide valve, and passages, in which 
 the above condition as to length of rods is supposed to 
 obtain. The slide valve D is exactly the length contained 
 between the outer edges of the steam ports. This point is 
 important to notice, as will appear presently. Also the faces 
 d d' of the valve are just sufficient to cover the width of the 
 steam ports and no more. 
 
 The general arrangement of the slide valve, steam chests, 
 steam and exhaust passages, in relation to the piston and 
 
 Fig. 85. 
 
 cylinder having been already explained (see page 202), the 
 reader will have no difficulty in understanding what follows. 
 In practice the eccentric is always mounted on the crank 
 shaft, and revolves in a plane parallel with that containing 
 the crank arm, while the slide valve works on the side of the 
 cylinder. In figs. 85 to 93 the valve is shown on the top of 
 the cylinder, and the eccentric E revolves in the same plane 
 as the crank K, and above the latter, as in this way alone 
 can the relative motions of valve, piston, crank, and eccentric 
 be shown simultaneously. The arrangement of the eccentric 
 and crank circles is not one that could be carried out in an 
 actual steam engine. 
 
 The piston P is represented at one end of the cylinder. 
 
Simplest Form of Slide Valve. 
 
 251 
 
 and the valve is shown in its middle position exactly cover- 
 ing the two steam passages, so that no steam can pass 
 from the steam chest C to either side of the piston ; while, 
 on the other hand, none can escape from the cylinder through 
 either of the passages into the exhaust. As the piston is at 
 the end of its stroke the crank is on the dead centre, and as 
 the valve is at mid stroke the arm of the eccentric must also 
 be in the position midway between the two dead centres ; 
 that is to say, it is at right angles to the crank. Let the 
 piston, however, be moved in the slightest degree to the 
 right-hand side, turning the crank K and the eccentric radius 
 E in the direction of the arrows, and two things will in- 
 
 Fig. 86. 
 
 stantly happen. The eccentric, whose action has been 
 already explained (see page 229), will pull the valve D 
 slightly to the right, so that the outside edge of the face d 
 will open the left-hand steam port, and thus admit steam to 
 the left side of the piston, while the inner edge of the face 
 d' will uncover the right-hand steam port, and thus permit 
 whatever steam may be in the cylinder on the right side of 
 the piston to escape along the passage to the under or hollow 
 side of the valve, whence it finds its way to the exhaust 
 passage H. The result will be that the entering steam will 
 propel the piston forward, and the crank and eccentric will 
 continue to rotate in the direction of the arrows. 
 
 When the crank reaches the position shown in fig. 86, 
 
252 
 
 The Steam Engine. 
 
 which is at right angles to its first position, the piston will 
 be at half-stroke, and the eccentric radius will also be at right 
 angles to its first position. From an inspection of the diagram 
 it is evident that the eccentric has now pulled the slide valve 
 as far to the right as it will go, and that, as the crank continues 
 to revolve, the eccentric will commence to travel back from 
 right to left. Also it is evident that the total space through 
 which the eccentric has moved the valve from its original or 
 central position is exactly equal to the half-diameter of the 
 circle described by the eccentric radius. As will be seen from 
 the figure, the left-hand steam port is now fully open to the 
 entering steam, while the other is fully open to the exhaust. 
 
 Fig. 87. 
 
 At the end of another quarter-revolution the piston will 
 have reached the end of its stroke. The crank and eccentric 
 will occupy the positions shown in fig. 87, while the valve has 
 now regained its central position, closing both ports. The 
 slightest motion to the left will now admit the steam to 
 the right side of the piston, and cause it to commence to 
 move backwards from right to left, and will at the same 
 time open the left-hand port to the exhaust. At the end of 
 the next quarter-revolution the crank and eccentric will 
 occupy the positions shown in fig. 88 ; the right-hand port is 
 now fully open to the entering steam, while the other is fully 
 open to the exhaust, and the valve has reached the farthest 
 limit of its travel to the left. From the above we see that 
 
Simplest Form of Slide Valve. 
 
 253 
 
 the total distance moved by the valve, called the travel of the 
 valve, is exactly equal to the diameter of the circle described 
 by the radius of the eccentric. 
 
 The next quarter-revolution will bring everything to the 
 positions occupied at starting, and the whole series of opera- 
 tions may be repeated over and over again so long as the 
 steam supply lasts. We thus see that by means of a slide 
 valve and a single eccentric, the operations of opening and 
 closing the steam and exhaust inlets can be satisfactorily 
 accomplished. 
 
 It will be noted, however, that the valve just described, 
 which only just covers the steam ports when in its central 
 
 Fig. 88. 
 
 position, when driven by an eccentric set at right angles to 
 the crank, keeps the admission steam port open during the 
 whole length of the stroke, and thus does not permit of the 
 expansive working of the steam. Similarly it keeps the 
 exhaust open during the whole length of the stroke, thus 
 rendering compression or cushioning of the exhaust steam 
 impossible. Moreover, it only opens the admission and ex- 
 haust ports just after the stroke has commenced. In practice 
 these features are inadmissible. In order to effect economy 
 in working the supply of steam must be cut off compara- 
 tively early, and expanded during the remainder of the 
 stroke. Similarly the exhaust should be closed before 
 the end of the stroke, so that the steam left in may be 
 
254 The Steam Engine. 
 
 compressed before the advancing piston, and aid in bringing 
 the reciprocating parts to rest. Moreover, the steam must 
 be admitted just before, instead of just after, the commence- 
 ment of the stroke. 
 
 These objects may all be accomplished, within certain 
 limits, by adding to the length of the slide valve so that- 
 it overlaps the outer edges of the steam ports, by diminishing 
 the width of the hollow portion D so that the faces of the 
 valve overlap the inner edges of the ports, and lastly by 
 altering the position of the eccentric on the shaft relatively 
 to the crank. 
 
 Before investigating this question the following defini- 
 tions must be stated. 
 
 Outside lap. Any portion added to the length of a 
 valve more than is absolutely necessary in order to cover 
 
 Fig. 89. 
 
 the outside edges of the steam ports, is called the out- 
 side lap of the valve. In fig. 89 the portions c c are the 
 outside lap. 
 
 Inside lap. Any portion added to the hollow portion D 
 of the valve more than is necessary in order to cover the 
 inner edges of the steam ports, is called the inside lap of the 
 valve. In fig. 89 the portions //are the inside lap. 
 
 Lead. The amount by which the admission steam port 
 is open when the piston is at the commencement of the stroke 
 is called the lead of the valve. Thus in fig. 89 the space b 
 is the lead. 
 
 Action of a slide valve provided with lap and set with lead. 
 Referring back to fig. 85, it is obvious that if the valve were 
 provided with outside lap, and had a certain lead, it would 
 have to be moved out of its central position by an amount 
 
Lap and Lead. 255 
 
 equal to the lap and the lead together when the piston was 
 at the commencement of its stroke Consequently the 
 position of the radius of the eccentric can no longer be central, 
 that is to say, at right angles to that of the crank, but must 
 be inclined forward at such an angle, DCE, fig. 90, that the 
 space CL intercepted between the perpendicular EL and 
 the centre C shall be equal to the lap and lead added 
 together. It is also obvious that the travel of the valve will 
 have to be increased, for it has to move sufficiently to 
 uncover fully the steam port, and in order to do so it must 
 travel over a space equal to the width of the port phis the 
 lap. The result of altering the position of the eccentric 
 is that all the operations effected by the valve will be 
 completed earlier than in the previous examples. Con- 
 sequently, not only will the admission steam port be partly 
 open at the commencement of the stroke, but it will also 
 be entirely closed to the steam before the end of the stroke, 
 and the steam will consequently expand during the interval. 
 Similarly the exhaust will be closed before the end of the 
 stroke, and the exhaust steam remaining in the cylinder will 
 undergo compression. This is shown in the following four 
 diagrams, which show the points of the stroke at which the 
 steam and exhaust ports are opened and closed. 
 
 The relative positions of valve and piston during the 
 stroke will of course be very materially modified if we take 
 into account the ratios of the lengths of the connecting and 
 eccentric rods to the length of the crank and the throw of 
 the eccentric respectively. As a rule the length of the con- 
 necting rod is from three to six times the length of the 
 crank, according to the class of engine employed. The 
 travel of the valve, and consequently the throw of the 
 eccentric, is, however, always kept as small as possible, so as 
 to diminish to the utmost the waste work expended in 
 overcoming the friction of the valve on its seating 1 ; hence 
 the ratio of the length of the rod to the throw of the 
 eccentric is usually from \} to -\ ; and it will be easily seen 
 
25 6 
 
 The Steam Engine. 
 
 Fig. 90. Left-hand port just opened to steam by the amount of lead. 
 
 Fig. 91. Left-hand port just closed to steam ; expansion commencing. 
 
 Fig. 02. Right-hand port just closed to exhaust ; compression commencing. 
 
 93. Left-hand port just about to open to exhaust : piston not yet 
 at end of stroke. 
 
Action of Slide Valve with Lap and Lead. 257 
 
 that the position of the valve is much less affected by the 
 obliquity of the eccentric rod than is the piston by that of 
 the connecting rod ; also the effect on the position of the 
 valve is most apparent at those points in its travel which 
 cause the least possible disturbance to the steam distribution. 
 The effect of the obliquity of the connecting rod upon 
 the position of the piston has already been explained 
 (see page 183). The general effect of a connecting rod of 
 finite length is that it causes the piston in its advance 
 towards the crank to be always in advance of the position 
 which it would occupy were the xod of infinite length ; and 
 vice versa, in the return stroke the piston lags behind. 
 Hence, as the movements of the valve are practically the 
 same, but those of the piston quite different relatively to the 
 positions of the crank, so also will the steam distribution 
 be different in the two strokes. 
 
 Means of reversing the engine and of varying the rate of 
 expansion. The arrangement above described of a single 
 eccentric driving a properly proportioned slide valve will 
 answer very well for engines which have always to work in 
 one direction at a uniform rate of expansion. In many 
 engines, however, the rate of expansion has to be constantly 
 varied, and in some types, such as locomotives, marine, wind- 
 ing, and rolling-mill engines, the direction of working has to 
 be constantly changed. In order to provide for these require- 
 ments other arrangements have to be adopted. Referring to 
 fig. 86 it will be readily understood, that if there were a second 
 eccentric keyed on the shaft exactly opposite to the original 
 one, and if the valve were by any means connected with 
 this second, and disengaged from the first eccentric, the 
 valve would be moved to the opposite end of its travel, and 
 the steam port, which in the figure is shown as open to the 
 exhaust, would be opened to the fresh steam, and vice versa ; 
 the result of which would be that the engine would commence 
 running in the reverse direction. 
 
 A convenient arrangement for effecting this reversal and 
 
 s 
 
258 The Steam Engine. 
 
 for regulating the distribution of the steam is the link 
 motion invented by Stephenson, which is illustrated in fig. 94. 
 The centre of the crank shaft is at C, and the centres of 
 the two eccentrics at o o'. It will be noticed that the two 
 latter are not exactly opposite to each other, as they would 
 be if the valve had no lead, but are brought nearer to each 
 other by twice the amount of the angular advance. The 
 ends of the two eccentrics are connected to a slotted link, L, 
 at the two points P P'. The link is curved, the radius of 
 curvature being the length of the eccentric rod, and is 
 suspended from the point P by means of a system of levers 
 
 Fig. 94. 
 
 and link work clearly shown in fig. 94. In the slotted portion 
 of the link is a block, U, which fits the slot exactly ; and 
 when the link is raised or lowered in position by means of 
 the hand lever H, acting through the bell-crank lever K, 
 and the two rods / /', the block U slides in the slot, and is 
 capable of taking up an intermediate position between P' 
 and P, as is shown in the two diagrams, fig. 95, which represent 
 the link raised to such positions that the block occupies 
 first the centre of the link, and then a point opposite the end 
 of the lower eccentric. 
 
 The block U, fig. 94, is connected directly to the spindle 
 S, which drives the valve V. Now when the block occupies 
 
Link Motions. 
 
 259 
 
 the position nearest to P' it is almost wholly under the 
 influence of the eccentric E', and the engine will run in one 
 direction. Let, however, the link be raised so that U comes 
 into the position nearest P, the block is then under the influ- 
 ence of the eccentric E, and the position of the valve will be 
 so shifted that the engine will run in the reverse direction. 
 
 When U occupies a position intermediate between P' and 
 P it is under the control of both eccentrics, but most under 
 
 Fig. 95- 
 
 the control of the one to which it is nearest. When it 
 occupies the exact centre of the link it is equally under the 
 influence of both, and the consequence will be that the 
 engine will not run in either direction. When, however, it 
 occupies intermediate positions the effect is very curious 
 and most important. The block, being actuated by two 
 eccentrics working to a great extent against each other, will 
 not travel the full horizontal distance due to the throw of 
 either eccentric, but a distance which gets gradually less as 
 
 S2 
 
260 The Steam Engine. 
 
 the block approaches the central point of the link. The 
 effect on the valve is the same as if it were driven by a new 
 eccentric of lesser throw than the original one. Now if 
 we refer back to figs. 90 to 93, and keep everything, with the 
 exception of the throw of the eccentric, the same as in these 
 figures, it can easily be proved, by drawing the positions of the 
 valve corresponding to the positions of the crank, that the 
 distribution of the steam is entirely altered, the cut-off being 
 effected at a much earlier period of the stroke, and the rate 
 of expansion consequently increased. It will thus be seen 
 in a general way, that the result of raising the link so as to 
 bring the block nearer to its central position is to effect the 
 cut-off of the steam at an earlier period of the stroke. It 
 will hereafter be shown that not only is the point of cut-off 
 affected, but also the angle of advance of the ideal eccentric 
 which represents the combined action of the two actual 
 eccentrics, as well as the lead and the points of compression 
 and release of the exhaust steam. The combination of a 
 pair of eccentrics with a link and sliding block is, therefore, 
 capable not only of effecting the reversal of the direction of 
 running of an engine, but also of entirely altering the dis- 
 tribution of the steam, and for this reason we find it very 
 generally employed in locomotives, marine, winding, and 
 rolling-mill engines. 
 
 In order to fix the position of the block U in the slide, 
 the hand lever H, fig. 94, is provided with a catch and a 
 notched quadrant Q. Each notch corresponds with a 
 separate rate of expansion in forward or back running. The 
 central notch is the position of no motion of the engine. By 
 dropping the catch on the hand lever into any given notch 
 the link is kept in its new position relatively to the 
 block. Thus a quadrant with three notches on either 
 side of the centre is capable of running an engine at as 
 many different rates of expansion. In order to provide for 
 any possible rate of expansion the arrangement shown in 
 fig. 96 was invented by Mr. Ramsbottom. In this plan a 
 
Reversing Gears. 
 
 261 
 
 screwed spindle works the lever, which also makes it easier 
 to reverse the engine when running. 
 
 The above is a general description of the action of a 
 particular sort of link motion. The actual motion of the link 
 
 under the influence of two eccentrics is extremely difficult to 
 follow accurately, and is further complicated by the method 
 of suspension. So complicated, indeed, is the motion, that 
 the travel of the slide valve can only be approximately ex- 
 pressed by mathematical calculation or by geometrical con- 
 struction. The approximate geometrical method of deter- 
 mining the motion of a slide valve driven by link motion 
 will be given hereafter. 
 
 There are many different types of link motion in use, 
 and even considerable variations in the details of Stephenson's 
 motion. For instance, in some cases the shape of the link 
 is that shown by figs. 94 and 97A, in which the points of/'/are 
 respectively above and below the extreme positions of the 
 block U, the result of which arrangement is that the valve 
 is never exclusively under the control of either eccentric, 
 and never receives its full motion. Consequently with this 
 arrangement the throw of the eccentric requires to be greater 
 than the full travel of the valve. Fig. 97 represents a link 
 
262 
 
 The Steam Engine. 
 
 Fig. 97. 
 
 Fig. 97 A. 
 
 in which the block in its extreme positions is exactly oppo- 
 site the ends of the eccentric rods. Sometimes the link 
 
 is suspended from the 
 middle, and sometimes 
 from one or other of the 
 ends. Sometimes the ec- 
 centric rods are worked 
 in the open position as 
 shown in fig. 94, and 
 sometimes the rods are 
 crossed. All of these 
 details effect variations 
 in the distribution of the 
 steam. The other link 
 motions best known in this country are Gooch's, in which 
 the link is not shifted up or down by the reversing gear, 
 but the block U slides within a link suspended to a fixed 
 point ; and Allen's, in which both link and block are made 
 to shift in opposite directions. 
 
 It will be readily perceived from the description that the 
 link motion is a very efficient method of reversing an engine, 
 and a convenient means of regulating the rate of expansion. 
 Its performance of the latter function is, however, attended 
 with certain drawbacks. In addition to altering the point 
 at which the valve cuts off the steam, each new position 
 of the link also alters the lead, and the periods of release 
 and compression of the exhaust steam. As, moreover, the 
 travel of the valve is altered while the lap remains con- 
 stant, the extent to which the steam port is opened is very 
 seriously curtailed as the sliding block approaches its 
 central position. These peculiarities, coupled with the fact 
 that all simple slide valves open and close the ports with 
 a comparatively slow motion, thus causing the corners of 
 the indicator diagrams to assume a rounded instead of a 
 sharp appearance, have led to the adoption of modifica- 
 tions of the above ^ or other methods of effecting the dis- 
 
Meyer's Valve Gear. 263 
 
 tribution of the steam when economy of fuel is much sought 
 after. 
 
 Meyer's valve gear. The method in most common use 
 to obviate the disadvantages belonging to the common D 
 slide valve and link motion is the adoption of a second valve 
 to control the cut-off, while the ordinary valve, or a modifica- 
 tion of it, regulates all the other points connected with the 
 steam distribution. There are several varieties of these 
 double valve motions in use, but the one selected for descrip- 
 tion, on account of its simplicity and efficiency, is that known 
 as Meyer's valve gear, illustrated in fig. 98. 
 
 In fig. 98, A, A', E are the two steam and the exhaust ports, 
 which are worked by a valve B B, differing from an ordinary 
 slide valve only in the fact that it is much longer, and that 
 two steam passages a a', 
 for enabling the fresh 
 steam to get to the ports, 
 are made through the 
 substance of the pro- 
 longed portions. This 
 valve is driven in the 
 usual manner either by Fig. 98. 
 
 one eccentric or, when the engine is required to reverse, 
 by two eccentrics set opposite to each other in the ordinary 
 way and connected by a link, which latter, however, is not 
 used for altering the rate of expansion. The valve B B is 
 called the distribution valve, and effects the admission, the 
 release, and the closing of the exhaust precisely in the same 
 way as an ordinary slide valve. The cut-off is effected by 
 closing the passages a a' at the proper point in the stroke by 
 means of the expansion valve C C', which slides on the back 
 of B B. The expansion valve is formed in two halves, which, 
 by means of the threaded valve spindle D can be caused to 
 approach or recede from each other, thus varying the point 
 at which they close the passages a a' and cut off the steam. 
 The expansion valve is usually driven by a fixed eccentric, 
 
264 The Steam Engine. 
 
 and means are provided exterior to the valve box for turning 
 the spindle D by hand, so as to put the point of cut-off under 
 the control of the driver, or, in some cases, of the governor. 
 
 It is evident that, with this gear, the lead and the points 
 of admission, release, and compression are quite independ- 
 ent of the rate of expansion, and, when once fixed, are in- 
 variable. Moreover, as the expansion valve, when closing 
 the passages a a', is always moving in the opposite direction 
 to the distribution valve, the cut-off can be effected much 
 more rapidly than with the ordinary D slide. The proper 
 method of proportioning the two valves, and of setting the 
 eccentrics so as to provide for any desired range of expansion, 
 will be explained in the theoretical portion of the chapter 
 (see page 302). 
 
 Corliss valve motion. Meyer's valve gear, though it 
 possesses many advantages, does not remedy the evil of 
 admitting the fresh steam to the cylinder through the passages 
 which have been cooled down by the low temperature of the 
 exhaust steam. Moreover, the extra power required to drive 
 it, in consequence of the friction of two sliding surfaces is 
 so considerable, that many eminent authorities doubt the 
 advantage of using a comparatively complicated gear, the 
 chief object of which is to prevent the compression curve in- 
 creasing too rapidly with the rate of expansion. In order to 
 provide against the evil of admitting the fresh steam through 
 comparatively cold passages, and at the same time to dimin- 
 ish friction and retain the good points of the Meyer class of 
 gearing, the Corliss valve motion was brought out, and is 
 now very often employed, in some of its modifications, in 
 non-compound expansive engines when fuel economy is a 
 point of primary importance. 
 
 The leading peculiarities of the Corliss gear are that 
 there are four steam ports and four valves, the two of the 
 latter which control the steam admission and cut-off being 
 driven by a special mechanism which ensures a very sharp 
 closing of the steam ports when the cut-off takes place. The 
 
Corliss Valve Gear. 
 
 26 5 
 
 two upper ports are intended solely for the admission of the 
 steam, the remaining two serving for the exhaust. Thanks 
 to this arrangement, the fresh steam on entering the cylinder 
 does not come in contact with surfaces which have just been 
 cooled down by the passage of the comparatively cold exhaust 
 steam. The valves do not belong to the class of D slides, 
 but are formed of portions of cylinders, A A, B B, fig. 99, 
 
 Fig. 99. 
 
 which oscillate on a cylindrical face. The valves are so ar- 
 ranged that the pressure of the steam forces them against 
 their seats only when the port is closed. In all other posi- 
 tions there is no friction due to steam pressure, consequently 
 the operation of these valves absorbs very little of the power 
 of the engines. Fig. 99 is a section of a cylinder pro- 
 vided with Corliss gear. It shows very clearly the four steam 
 
266 
 
 The Steam Engine. 
 
 passages, and the cylindrical valves with the peculiarities of 
 seatings just explained. 
 
 Fig. 100 gives a general view of the mechanism by which 
 the valves are worked. The disc L is caused to oscillate in 
 an arc of a circle round its own centre by means of the 
 eccentric and the eccentric rod F. To the disc are jointed 
 four valve rods, G G and H H, which are also attached to 
 the four valves. The two belonging to the lower valves 
 B B are perfectly simple ; all they have to do is to open the 
 
 \ 
 
 Fig. 100. 
 
 exhaust ports at the proper moment, and keep them open 
 during the required fixed period of the stroke. The two 
 upper rods are more complicated. Their function is, not 
 only to open the steam ports at the proper moment, but also 
 to keep them open during whatever period of the stroke is 
 required by the rate of expansion, and then to liberate the 
 valve so as to permit of its being closed sharply by an inde- 
 pendent piece of mechanism which will be described here- 
 after. The upper rods are made in two independent pieces, 
 one of which slides within the other. One of these pieces 
 
Corliss Valve Gear. 267 
 
 is attached to the disc L, and the other to the rocking arm 
 of the steam valves A A. By means of two side-clip springs 
 D D, fig. 1 01, which are permanently fastened to the disc end 
 of the rod, and which engage with the projecting shoulders e e t 
 on the valve end of the rod, the two halves can at will be 
 united into one rigid rod, or be disunited, so that the motion 
 of the disc no longer controls that of the valve. At a certain 
 
 D 
 
 Fig. loi. 
 
 period in each stroke, dependent on the action of the go- 
 vernor, the clip springs D D are prised apart ; the valve rod 
 ceases to act as a whole, the valve is consequently liberated 
 from the control of the disc L, and is instantly closed by 
 the action of the mechanism contained in C. The mechan- 
 ism by which the clip springs D D are prised apart is shown 
 on an enlarged scale in fig. 101 ; it consists of a toe lever 
 centred on *, and having an arm E, the inclination of which 
 is capable of being altered by the rod /, which in its turn is 
 under the immediate control of the governor rod R, fig. TOO. 
 The toe lever rocks during each stroke of the valve rod, and 
 the period that it reaches its extreme position, and prises open 
 the clip springs, is determined by the inclination of the arm 
 E, and consequently by the governor. Hence we see that 
 in this type of valve gear the rate of expansion is controlled 
 automatically. When the clip springs are opened the valve is 
 closed in the following manner. The valve spindle A, fig. 100, 
 is provided with a second arm which is attached to a piston 
 or plunger contained in C. The back of this piston is always 
 in communication with the condenser, while the front receives 
 the atmospheric pressure ; and, consequently, whenever the 
 valve is released from the action of the valve rod, the piston 
 
268 
 
 The Steam Engine. 
 
 is sharply forced inwards by the atmospheric pressure, and 
 actuates the arm which closes the valve. There is also an 
 air buffer, or dash pot, contained in C, which prevents the 
 concussion, due to the very sharp action of the piston, which 
 would otherwise ensue when the valve is closed. 
 
 Varieties of valves. There are many varieties of valves 
 besides the short D slides which have been described above. 
 These latter, though in common use in the smaller types of 
 land engines and in locomotives, would be quite inapplicable 
 to the large cylinders and high pressures of modern marine 
 
 engines. It will be 
 readily understood that 
 the friction to be over- 
 come in driving the 
 ordinary type of D slide 
 valve is very consider- 
 able when high pres- 
 sures are used, as is 
 the case in locomotives. 
 In fact, it frequently 
 happens that each of 
 the valves of a large 
 locomotive is pressed 
 against the surface on 
 which it slides with a 
 total net pressure of nine 
 or ten tons, so that the 
 mere driving of the 
 valves probably absorbs 
 some five or six per 
 cent, of the total power 
 exerted by the engine. 
 F - I02 This defect would be 
 
 very much exaggerated 
 
 if the D type of slide valve were applied without modifica- 
 tion to marine engines. 
 
Piston and Double Ported Valves. 269 
 
 The types of valve generally made use of nowadays in 
 marine engines are either the double ported slide valve or 
 else what is known as the piston valve, an example of 
 which is shown in fig. 102. Here it will be seen the valve 
 consist of two pistons, one to each port, connected together 
 by a common spindle. The pistons are provided with the 
 ordinary spring ring packings to keep them steam-tight ; 
 they move up and down in the two cylindrical spaces 
 shown, in which are cast the openings to the steam ports. 
 The latter are not made continuously open as is the case 
 with ordinary slide valves, but are cast with bars of metal 
 as shown in the section, to prevent the packing rings of the 
 pistons from springing out into the ports, and also in order 
 to afford a continuous guide to the pistons. It will be seen 
 that this type of valve is perfectly balanced, as far as the 
 steam pressure is concerned. The only friction to be over- 
 come in its motion is that due to the pressure of the spring 
 packing rings. In the piston valve shown in fig. 102 the steam 
 is admitted between the two pistons, the inner edges of which 
 cut off the steam. 
 
 The double ported slide valve is an ingenious modifica- 
 tion of the ordinary D slide already described. Examples 
 are shown in section in the valves of the low-pressure cylin- 
 der, pages 456 and 458, and also in fig. 103. It will be 
 noticed that the steam passages of these cylinders have each 
 two ports or openings on the valve face. The steam is 
 admitted to the outside ports in the usual way, over the 
 outer edges of the valves, but the two inner ports get their 
 steam from the passages cast in the body of the valve, and 
 which are shown in section in the figures above referred to. 
 The advantage of this arrangement is, that for a given move- 
 ment of the valve twice the area of steam port is uncovered 
 as with the ordinary single ported valve, and consequently 
 for a given area of opening the travel of the valve may be 
 greatly reduced. 
 
 In order to relieve the faces of the slide valves of marine 
 
2/0 
 
 The Steam Engine. 
 
 engines of a portion of the pressure brought to bear on them 
 by the action of the steam on their backs, it is usual to fit 
 relief rings on to these latter, in order to cut off a portion of 
 their area from the pressure of the steam. It will be readily 
 understood that if the back of the slide valve were planed 
 perfectly flat, and if it could be so arranged as to work in 
 perfect contact with the inner face of the valve box cover, 
 so that no steam could get between the two faces, then the 
 valve would only be held against its seating by the unbal- 
 anced portion of the pressure acting on the lips of the valve 
 
 Fig. 103. 
 
 c c, fig. 89. Such an arrangement could not however be 
 made to work satisfactorily in practice, because, in conse- 
 quence of the different expansions under heat of the metals 
 of the valve and the box, the former would either jamb or 
 work loose. The relief ring is intended to get over this 
 difficulty. It consists of a flat ring on the back of the slide 
 valve, which is pressed outwards against the face of the 
 valve box cover by means of a spring. The ring fits steam- 
 tight into the back of the valve and works steam-tight against 
 the face of the valve box, and thus excludes a large portion 
 of the back of the valve from the direct pressure of the 
 
Joy's Valve Gear. 
 
 271 
 
 steam. In case any steam should leak into the hollow space 
 within the ring, the latter is generally placed in communica- 
 tion with the condenser. Fig. 103 is a section of a double 
 ported valve fitted with a relief ring, which is shown in 
 section in black. 
 
 Joy's valve gear. There are other methods in common 
 use for driving valves besides the eccentric system which has 
 been described. It has constantly been an object with in- 
 ventors to get rid of the complications of the two eccentrics, 
 link, &c., required for an expansion gear. Several success- 
 ful gears have been brought out, both in this country and 
 the Continent, in which the valve is driven from some 
 moving part of the engine. One of the best known of these 
 
 Fig. 104. 
 
 is the Joy valve gear, which has been largely used both "for 
 locomotives and marine engines. Fig. 104 illustrates a simple 
 form of this gear applied to a horizontal stationary engine. 
 A vibrating rod or link, B, is attached at one end to a point 
 A, near the middle of the connecting rod ; while the lower 
 end is jointed to the radius rod C, which compels B to 
 move in a vertical plane. To a point D in the link B is 
 jointed the end of the long arm of a lever E F, of which the 
 end of the small arm works the valve rod G, and the fulcrum 
 F is attached to a block which slides in the curved slot J. 
 This slot is formed in a disc, the centre of which is the 
 position of the fulcrum F when the piston is at either 
 end of its stroke. The radius of the slot is equal to the 
 length of the valve rod G. The disc can be made to 
 
272 The Steam Engine. 
 
 rotate through an arc by means of the worm and wheel 
 shown. Thus the slot can be inclined to either side of the 
 vertical. The slot allows the fulcrum of the lever to 
 move up and down with the motion of the point A of the 
 connecting rod. The forward or backward motion of the 
 engine, and the rate of expansion, are controlled by inclin- 
 ing the slot to one or other side of the vertical, the central 
 position corresponding with mid-gear. If the end D of the 
 lever were attached direct to the connecting rod, the motion 
 of the fulcrum F about the centre of the slot would not be 
 symmetrical, and the result would be that the cut-off would 
 
 Fig. 105. 
 
 be unequal in the two strokes. This error is corrected by 
 attaching the end of the lever to the point D of the 
 vibrating link. For while the point A on the connecting 
 rod describes a nearly true ellipse, as shown in fig. 105, the 
 point D describes a bulged figure, and the amount of the 
 bulge is so regulated as to correct the unequal motion 
 of the fulcrum above and below its central position. It is 
 obvious that by shifting the point D the amount of the 
 bulge may be altered, and thus the error may be corrected 
 too little, or too much, and by taking advantage of this cir- 
 cumstance a later cut-off may be given to either end of the 
 cylinder if found desirable. 
 
Zeuner's Valve Diagrams. 273 
 
 Several advantages are claimed for this type of gear over 
 the ordinary link motion driven by eccentrics. Foremost 
 among these is the fact that it gives an almost mathe- 
 matically correct motion to the valve, which the older gear 
 does not. It is also considerably cheaper ; and from the 
 peculiarity that the valve boxes are on the top of horizontal 
 cylinders, and in front of vertical marine engine cylinders, 
 instead of being at the sides, as is the case when the valves 
 are driven by ordinary eccentrics, a considerable saving of 
 space is effected. 
 
 GEOMETRICAL REPRESENTATION OF ACTION OF 
 SLIDE VALVE. 
 
 There are few points connected with the successful work- 
 ing of steam engines of greater importance than the design 
 of the valve gearing. Mathematical calculations intended to 
 show the connection between the dimensions of the valve, 
 the position and throw of the eccentric, and the various 
 points connected with the distribution of the steam such as 
 the lead, the period of admission, the cut-off, release, 
 duration of exhaust and compression are too complicated 
 and cumbersome to be of general use. Many geome- 
 trical diagrams have been designed to meet the drawbacks 
 of mathematical calculation, and of these the most simple 
 and comprehensive are those designed by Dr. G. Zeuner, 
 which we will now proceed to describe and illustrate. 
 
 Zeuner's valve diagrams. It will be assumed for the 
 sake of simplicity that the obliquity of the eccentric rod has 
 no appreciable effect upon the position of the valve. The 
 diagram will be made to show the particular angles of 
 the crank at which the various critical points of the steam 
 distribution take place, from which the corresponding posi- 
 tions of the piston can be deduced for the forward and 
 backward strokes when we know the ratio of the length of 
 the connecting rod to the arm of the crank. 
 
 T 
 
274 
 
 The Steam Engine. 
 
 Referring to fig. 85, which represents a valve with- 
 out lap or lead and an eccentric set at right angles to the 
 crank, let AO, fig. 106, represent the position of the crank 
 
 at one of its dead 
 centres, and OE 
 at right angles to 
 A O, the corre- 
 sponding position 
 of the eccentric, 
 the valve being 
 
 then in its central 
 position ; also let 
 the amount of ec- 
 If 
 
 O 
 
 Fig. 106. 
 
 centricity or half throw of the eccentric equal OE. 
 the crank now occupies successively the positions OB, 
 OC, &c., the eccentric will take up the corresponding 
 positions OF, OG, &c. ; and if from the points F, G, &c. 
 we let fall perpendiculars F/j G/', &c., upon the dia- 
 meter AOH, the lengths Of, O/', &c., intercepted between 
 the centre of the circle and the feet of the perpendicu- 
 lars, will represent the distances by which the valve has 
 been moved from its central position when the eccentric 
 takes up successively the positions OF, OG, &c. 
 
 Now, however, 
 suppose the ec- 
 centric to be fixed 
 in position inde- 
 pendently of the 
 crankshaft and 
 suppose the latter 
 to revolve with the 
 
 Fig. 107. ~ engine cylinder 
 
 and all the mov- 
 ing parts attached round the centre O. It is evident that the 
 fixed eccentric will in this case impart exactly the same 
 motion to the valve as was done in the former case, only 
 
Z diner's Valve Diagrams. 275 
 
 that in this instance the revolution will have to start from 
 the dead centre H, fig. 107, and proceed in the reverse 
 direction, so that 'when the crank occupies successively the 
 positions OB, OC, &c., it will be represented in the diagram 
 as occupying the positions OB', OC', &c. From the centre 
 of the eccentric E let fall perpendiculars E, E^', &c. upon 
 the lines OB', OC', &c. ; then the lines Oe, Oe' represent 
 the distances travelled by the valve from its central position 
 when the crank occupies the respective positions OB', OC', 
 &c. For in the diagram, fig. 106, we saw that the distances 
 moved by the valve were represented by the lines Of, Of ; 
 and it can easily be proved that Of Of are respectively 
 equal to Oe, Oe'. In the triangles OEe and OF/ we have 
 OF = OE, both being radii of the same circle ; also the right 
 angle OeE = the right angle OfF. Also by construction the 
 angle EOF = the angle E'Of Therefore, adding to each of 
 these the common angle FO^, we have the whole angle 
 EOe = the whole angle Of, and consequently the two 
 triangles are equal, and the side Oe = the side Of, and 
 similarly Oe' = Of, and so on for every other position of 
 the crank ; therefore Oe, Oe' t &c. represent the distances 
 travelled by the valve from its central position when the 
 crank occupies positions opposite OB', OC', &c. Hence 
 we see that the distances moved by the valve for any 
 positions of the crank OB, OC may be found graphically 
 by dropping perpendiculars from the centre of the eccentric 
 E upon the opposite positions OB', OC', and measuring the 
 lines intercepted between the feet of these perpendiculars 
 and the centre O. 
 
 Now it will be noticed that in this way a series of right- 
 angled triangles, O^E, Oe E, &c. are constructed upon a 
 common base OE ; and it is a well-known fact (depending 
 upon Prop. 31, Euclid, Bk. iii.) that when such a series of 
 right-angled triangles is constructed, their apices, e, e', e", &c., 
 all lie upon the circumference of a circle of which the base- 
 line is the diameter. 
 
 T 2 
 
276 
 
 The Steam Engine. 
 
 The above suggests a very simple method of ascertaining 
 the position of the valve for every angle occupied by the 
 crank, and this method is the basis of all Zeuner's valve 
 diagrams. We have only to draw the line OE, connect- 
 ing the cen-tre of 
 the eccentric with 
 the centre of the 
 crank-shaft, when 
 the crank is at 
 either of the dead 
 centres, and upon 
 this line as a dia- 
 meter to describe 
 a circle ; then the 
 chords of this 
 circle, Qe, Oe f , 
 Oe", &c., will re- 
 present the spaces 
 traversed by the 
 valve from its cen- 
 tral position when the crank occupies successively the posi- 
 tions opposite to OeB', Oe'C', Oe"D f , &c. During the return 
 stroke the motion of the valve will be indicated by the cor- 
 responding chords of the circle described on the line OK. 
 
 The application of this diagram to the valve shown in 
 fig. 85 is very easy, it being remembered that the valve 
 has no lap, and that it occupies its central position when 
 the piston is at the commencement of the stroke. Com- 
 mencing with the edge of the valve d which works the left- 
 hand steam port, we see that when the crank occupies the 
 position opposite OH, fig. 108, in consequence of the position 
 of the circle on OE, there is no chord intercepted by any part 
 of the line OH, and consequently when the crank is on the 
 dead centre the port is not open at all ; but directly the crank 
 moves through any arc, no matter how small, there will be a 
 chord intercepted by the periphery of the circle, and the valve 
 
 Fig. 108. 
 
Zeuner's Valve Diagrams. 277 
 
 will be opened by an amount equal to the length of the chord. 
 The port will continue to open till the crank-pin reaches the 
 position E, at which point the valve will have travelled to 
 one end of its beat, for no arc of a circle can be drawn 
 longer than the diameter. From this point the valve com- 
 mences to close the port, but does not completely close it 
 till the end of the stroke. Similarly the motion of the valve 
 during the return stroke can be ascertained by means of the 
 circle on OK. 
 
 The diagram is equally useful in tracing the exhaust 
 side of the valve. While the crank is travelling in the 
 direction of the arrow, fig. 85 (represented of course in the 
 diagram by the opposite direction), the outer edge d of 
 the valve is keeping the steam port open, but as soon as the 
 piston reaches the end of its forward stroke the valve has 
 returned to its central position, and during the first half of 
 the return stroke continues its motion towards the left. 
 Directly the crank is over the dead centre the inner edge of 
 the valve opens the left-hand port to the exhaust, and the 
 arcs intercepted between O and the periphery of the circle 
 OK, fig. 1 08, measure the extent to which the port is opened. 
 Following the motion as before, we see that the exhaust is 
 fully opened when the crank is at OK, and that it remains 
 open till the end of the stroke (compare fig. 88). 
 
 How to indicate lap and lead on the valve diagram. 
 When a valve is provided with outside lap, and when the 
 port has to be opened by the amount of the lead at the 
 commencement of the stroke, the valve can no longer be in 
 its central position when the crank is on the dead centre. 
 This has been shown in fig. 90, which also illustrates the 
 manner of setting the eccentric. Describe a circle, fig. 109, 
 with radius OA equal to the half-throw of the eccentric. 
 From O measure off OB equal to the outside lap, and BC 
 equal to the lead. When the crank-pin occupies the dead 
 centre A, the valve has already moved to the right of its 
 central position by the space OB + BC. From C erect the 
 
278 
 
 The Steam Engine. 
 
 perpendicular CE, and join OE. Then will OE be the posi- 
 tion occupied by the line joining the centre of the eccentric 
 with the centre of the crank-shaft at the commencement of the 
 stroke. On the line OE as diameter describe the circle OCE ; 
 then, as before, the chords Oe, OE, Oe f , will represent the 
 spaces travelled by the valve from its central position when 
 the crank-pin occupies respectively the positions opposite 
 
 cteacL\ 
 cenft-e. 
 
 Fig. 109. 
 
 to D, E, and F. But these chords will no longer represent 
 the extent to which the outer edge of the valve has opened 
 the steam port, because before the port is opened at all 
 the valve must have moved from its central position by an 
 amount equal to the lap OB. Hence to obtain the space 
 by which the port is opened we must subtract from each of 
 the arcs Oe, OE, &c., a length equal to OB. This is re- 
 
Z diner's Valve Diagrams. 279 
 
 presented graphically by describing from centre O a circle 
 with radius equal to the lap OB ; then the spaces fe, gE, 
 &c., intercepted between the circumferences of the lap 
 circle ~Bfe' and the valve circle OCE, will give the extent to 
 which the steam port is opened. Tracing the motion of the 
 valve as before, and remembering that, when we speak of 
 the crank occupying the position, say, OD, it really occupies 
 the position symmetrically opposite on the other side of the 
 diameter OG, we shall see at once how different is the 
 distribution of the steam to that illustrated in the last case. 
 
 To begin with, take the point k, at which the chord O/ 
 is common to both valve and lap circles. At this point it is 
 evident that the valve has moved to the right by the amount 
 of the lap, and is consequently just on the point of opening 
 the steam port. Hence the steam is admitted before the 
 commencement of the stroke, when the crank occupies the 
 position OH, and while the portion H A of the revolution 
 still remains to be accomplished. When the crank-pin 
 reaches the position A, that is to say at the commencement 
 of the stroke, the port is already opened by the space OC 
 OB = BC, called the lead. From this point forward till the 
 crank occupies the position OE the port continues to open, 
 but when the crank is at OE the valve has reached the 
 furthest limit of its travel to the right, and then commences 
 to return, till when in the position OF the edge of the valve 
 just covers the steam port, as is shown by the chord O^, 
 being again common to both lap and valve circles. Hence 
 when the crank occupies the position OF the cut-off takes 
 place and the steam commences to expand, and continues 
 to do so till the exhaust opens. For the return stroke the 
 steam port opens again at H' and closes at F'. 
 
 So far we have traced the action of the valve in admitting 
 and cutting off the steam. There remains only the exhaust 
 to be considered. When the line joining the centres of the 
 eccentric and crank-shaft occupies the position opposite to 
 OG at right angles to the line of dead centres, the crank is 
 
280 The Steam Engine. 
 
 in the line OP, at right angles to OE ; and as OP does 
 not intersect either valve circle the valve occupies its central 
 position, and consequently closes the port S, fig. 89, by the 
 amount of the inside lap /. The crank must, therefore, move 
 through such an angular distance that its line of direction 
 OQ must intercept a chord on the valve circle OK, equal in 
 length to the inside lap, before the port can be opened to 
 the exhaust. This point is ascertained precisely in the same 
 manner as for the outside lap, namely, by drawing a circle 
 from centre O, with a radius equal to the inside lap ; this is 
 the small inner circle in fig. 109. Where this circle intersects 
 the two valve circles we get four points which show the 
 positions of the crank when the exhaust opens and closes 
 during each revolution. Thus at Q the valve opens the 
 exhaust on the side of the piston which we have been con- 
 sidering, while at R the exhaust closes and compression 
 commences, and continues till the fresh steam is readmitted 
 at H. 
 
 Thus we see the diagram enables us to ascertain the 
 exact position of the crank when each critical operation of 
 the valve takes place. Making a resume of these operations 
 for one side of the piston, we have 
 
 Steam admitted before the commencement of the stroke 
 atH. 
 
 At the dead centre A the valve is already opened by the 
 amount BC. 
 
 At E the port is fully opened, and valve has reached 
 one end of its travel. 
 
 At F steam cut off, consequently admission lasted from 
 H to F. 
 
 At P valve occupies central position, and ports are closed 
 both to steam and exhaust. 
 
 At Q exhaust opened, consequently expansion lasted 
 from F to Q. 
 
 At K exhaust opened to maximum extent, and valve 
 reached the end of its travel to the left. 
 
Problems in Valve Setting. 281 
 
 At R exhaust closed and compression begins, and con- 
 tinues till the fresh steam is admitted at H. 
 
 Solution of problems relating to simple valve gearing by 
 Zeuner's diagrams. All problems bearing on valve gearing 
 involve relations between the following variables : 
 
 The inside and outside laps of the valve. 
 
 The angle of advance and the throw of the eccentric. 
 
 The angles of the crank or points of the stroke at which 
 take place the admission and cut-off of the steam, and the 
 opening and closing of the exhaust. 
 
 Occasionally, and more especially when the valves of an 
 old engine have to be altered, we have also to take account 
 of the width of the steam ports, and the extent to which 
 they have to be opened. 
 
 PROBLEM I. The simplest problem which occurs is the 
 following. Given the length of throw, the angle of ad- 
 vance of the eccentric, and the laps of the valve, find the 
 angles of the crank at which the steam is admitted and cut 
 off and the exhaust opened and closed. 
 
 This problem is solved in the manner shown in fig. no. 
 Draw the line OE, representing the half-throw of the eccentric 
 at the given angle of advance with the perpendicular OG. 
 Produce OE to K. On OE and OK as diameters describe 
 the two valve circles. With centre and radii equal to the 
 given laps, describe the outside and inside lap circles. Then 
 the intersection of these circles with the two valve circles 
 give points through which the lines OH, OF, OQ, and 
 O R can be drawn. These lines give the required positions 
 of the crank. 
 
 PROBLEM II. Given the points at which the steam is to 
 be admitted and cut off, and the exhaust to be opened, also 
 the throw of the eccentric, find the proper angle of 
 advance and the laps of the valve, also the point at which 
 the exhaust closes. 
 
 Describe a circle AG A', fig. no, with radius equal to the 
 half-throw of the eccentric, AA' being the dead centres. Let 
 
282 
 
 The Steam Engine. 
 
 OF be the crank angle when steam is cut off, OQ when the 
 exhaust opens, OH when the steam is admitted. Now the 
 valve is in exactly the same position when the steam is ad- 
 mitted and cut off ; consequently it reaches the end of its 
 travel midway between these positions. Bisect the angle 
 HOF by the line OE, then OE is the direction of the 
 crank when the valve is at the end of its stroke ; that is to 
 say, the chord of the valve circle made by the line of direction 
 of the crank will be a maximum at E, or in other words, 
 OE is the diameter of the valve circle. Produce EO to 
 
 K ; on OE and 
 OK describe the 
 valve circles. Now 
 the circle on OE 
 intersects the line 
 OF at the point e. 
 Therefore Od=the 
 outside lap. Simi- 
 larly OQ intersects 
 the circle on OK 
 at the point /', there- 
 fore O/ is the in- 
 side lap. Describe 
 the two lap circles 
 with radii Oe md 
 Ot. The intersec- 
 tion of the smaller lap circle with the valve circle OK gives 
 the direction of the crank OR when the exhaust closes. 
 The line cd gives the lead, that is, the extent to which the 
 steam port is admitted when the stroke commences. 
 
 If the amount of the lead cd had been given instead of 
 the angle of lead we should have had to proceed in a differ- 
 ent manner. 
 
 First assume that there is no lead, but that the port opens 
 when the crank is on the dead centre. Bisect the arc AOF 
 at the point E', fig in. From E' let fall the perpendicular 
 
 Fi 
 
Problems in Valve Setting. 
 
 283 
 
 EV on OA. From c' mark off c'd equal to half the required 
 lead. From d erect a perpendicular cutting the circumfer 
 ence in E. Mark off the arc EH equal to EF. Then OH is 
 the required angle 
 of lead. The re- 
 mainder of the con- 
 struction will be as 
 before. 
 
 Another, and a 
 very simple method 
 of finding the posi- 
 tion of the crank 
 when steam is ad- 
 mitted, the amount 
 of the lead being 
 
 Fig. in. 
 
 given, is to describe a small circle with centre A and radius 
 equal to the amount of the lead. Call this the lead circle. 
 From F draw a straight line tangential to the lead circle, 
 and prolong it to meet the circumference of the circle AFA' 
 in the point H. Join OH, then OH is the required posi- 
 tion of the crank. 
 
 PROBLEM III. Given the throw of the eccentric, the ex- 
 ternal lap, and the 
 lead, find the point 
 where the steam 
 is cut off, and the 
 angle of advance. 
 Let OA=the half- 
 throw. With this 
 radius describe a 
 circle. Let Qc ~ 
 the external lap. 
 With this radius 
 describe another circle. Let cd = the lead. From d erect 
 the perpendicular ^E, cutting the circumference of the 
 outer circle in E. Join OE, and on OE as diameter 
 
 Fig. 112. 
 
284 The Steam Engine. 
 
 describe the primary valve circle. The angle GOE is the 
 angle of advance, and the steam is cut off at the point F, 
 obtained by joining O with the point of intersection of the 
 primary valve and the lap circles. 
 
 PROBLEM IV. Given the outside lap, the lead, and the 
 point where the steam is cut off, find the throw of the 
 eccentric and the angle of advance. 
 
 Let Of, fig. 112= the external lap, cd = the lead, and OF 
 the position of the crank when the steam is cut off. The 
 problem then resolves itself into describing a circle eEO which 
 shall pass through the three points *?, O, d. The diameter 
 of this circle OE gives the length and position of the half- 
 throw of the eccentric, and the line OH gives the position of 
 the crank when the steam is admitted so as to secure a lead 
 = cd. 
 
 PROBLEM V. 1 Given the position of the crank when 
 the steam is cut off, the lead, and the amount the valve is 
 to be open for any particular position of the crank, find the 
 throw of the eccentric, the angle of advance, and the lap. 
 
 Let cE, fig. 113, represent the required lead, and ca the 
 extent to which the port is to be opened when the crank 
 occupies a position parallel to EG'. Also let EF' be parallel 
 to the position of the crank when steam is cut off. 
 
 Draw cc' and act! at right angles to ca ; Eg at right angles 
 to EF' ; Eb at right angles to EG', intersecting act' in the 
 point k. Bisect the angles gtS, Eka'. The point O where 
 the bisecting lines meet will be the centre of the lap circle. 
 
 Draw OA parallel to ca. O A will represent the direction 
 of the cranks when on their dead centres. With centre O 
 describe a circle to touch the two lines ic' , tg. The radius 
 of this circle represents the lap. Join OE, and on OE as 
 diameter describe the valve circle EnO. Then OE represents 
 the half-throw and position of the eccentric. From O draw 
 through the point of contact e of the lap circle with the line 
 
 1 The very beautiful geometrical solutions of this and the two follow- 
 ing problems are due to Mr. Cowling Welch. 
 
Problems in Valve Setting. 
 
 285 
 
 Eg the line OF. Then OF is evidently parallel to the line 
 EF', because it cuts ^E at right angles. Also the point e is 
 on the circumference of the primary valve circle because 
 the angle OeE is a right angle, and as e is also on the cir- 
 cumference of the lap circle, therefore the steam is cut off 
 when the crank is at OF parallel to the given direction EF.' 
 Also the lead is evidently equal to the given lead cE ; 
 for, joining Ed, we have the angle QdE in a semicircle equal 
 to a right angle, therefore Ed is parallel to ^', and therefore 
 cE equals the lead as shown by the part of OA intercepted 
 
 Fig 113. 
 
 between the circumferences of the lap and the primary valve 
 circles. 
 
 Also through O draw OG parallel to EG'. Then n'n 
 equals c'a', because the two triangles kOn, kQa' are equal, 
 and as c'a' equals ca by construction : therefore n'n equals 
 ca as required, and all the conditions are fulfilled. 
 
 PROBLEM VI. Given the lead, the angle at which the 
 steam is cut off, and also the angles at which the release 
 takes place and compression begins, find the angle of advance 
 and thow of the eccentric, and the outside and inside laps 
 of the valve. 
 
286 
 
 The Steam Engine. 
 
 This problem is useful in solving questions connected 
 with Meyer's valve gear, in which the angles of release and 
 compression, as well as the lead, are regulated by the main 
 or distribution valve. 
 
 Take any point E in a horizontal line. Mark off EC 
 equal to the lead, and draw EF', EQ', and ER', parallel 
 respectively to the positions of the crank when the steam is 
 
 cut off, the release 
 takes place, and 
 the compression 
 commences. 
 
 At c erect a 
 perpendicular cc'. 
 At E erect a per- 
 pendicular to EF'. 
 Then the direc- 
 tion of the radius 
 of the eccentric 
 must lie halfway 
 between the points 
 of lead and cut- 
 off, or, what is the 
 same thing, be- 
 tween the points 
 of release and of 
 compression. Bi- 
 sect the angle Q'ER' by the line EE'. Bisect also the 
 angle c'ie by the line ii'. Through the point O where 
 these two lines intersect, draw the horizontal line OA. 
 From O draw the lines OF, OQ, and OR parallel to EF', 
 EQ', and ER' respectively. Upon OE as diameter de- 
 scribe the primary valve circle. This will of necessity 
 pass through d, because the angle OdE, is a right angle. 
 With centre O and radius O^ describe the outer lap circle. 
 This will of necessity pass through the point of intersection 
 e of the valve circle with the line OF. Hence with the 
 
 Fig. 114. 
 
Problems in Valve Setting. 
 
 dimensions arrived at the lead will be Sd = cE, and the 
 steam will be cut off at F. In order to provide for the 
 release taking place at Q and for the compression com- 
 mencing at R we have only to draw the other valve circle on 
 OE', and with centre O to draw the inner lap circle through 
 the intersections of OQ and OR with the circumference of 
 the circle OE', 
 
 The following problem is of great practical utility in the 
 design of steam engines . 
 
 PROBLEM VII. Given the position of the crank when 
 the steam is cut off, the lead, and the maximum opening of 
 the steam port, find the throw of the eccentric, the angle of 
 advance, and the outside lap. 
 
 Fig. 115. 
 
 Let CE represent the given lead, and Ca the maximum 
 opening of the port Let EF' be drawn parallel to the 
 direction of the crank when steam is cut off. Through C 
 
288 The Steam Engine. 
 
 and a draw O' and GA at right angles to Ca, and from E draw 
 E/ at right angles to EF'. Bisect the angle etc' by the line 
 O, which produce till it intersects AaG in G. Join GE. 
 With centre / and radius = Ca describe an arc intersecting 
 GE in k. Join ki, and from E draw EO parallel to ki, and 
 intersecting /O in the point O. Then O is the centre of 
 the lap circle. Through O draw OA parallel to Ca. With 
 centre O and radius OA describe a circle. Join OE, and on 
 OE as diameter describe the primary valve circle, and with 
 O as centre describe the lap circle touching the line ie in e, 
 and C^' in c*. Join Oe and produce it to F. Then it can 
 be proved, as in the last problem, that the steam is cut off 
 when the crank occupies the position OF, which is parallel 
 to EF'. Also that c'd = CE, the given lap. 
 
 It remains only to prove that E, which is evidently the 
 greatest opening of the port = fa. Through / drawn ii' 
 parallel to ca. Then, because ik is parallel to the side OE 
 of the triangle GOE, we have 
 
 GO : G/:: OE : ik. 
 
 Also, because ii' is parallel to the side OA of the triangle 
 GO A, we have 
 
 GO : Gi :: OA : ii'. 
 
 But ii' = ik, both being radii of the same circle ; therefore 
 also OA = OE ; and subtracting from each the equal radii 
 Of' and On, we have the remainder c'K = ^E. But c'A is 
 by construction equal to Ca, the maximum port opening ; 
 therefore also nE = Ca. 
 
 Before passing on to the consideration of the more compli- 
 cated diagrams used to explain the action of expansion gears it 
 may be useful to show how, from any diagram such as fig. no, 
 to set out the dimensions of the valve and ports. The valve 
 travels to and fro on a plane surface, which is in general 
 made of such a length that when the valve reaches the end 
 of its beat, its edges do not overhang the end of the plane. 
 Take any straight line AB, and suppose the valve to occupy 
 
Problems in Valve Setting. 
 
 289 
 
 its central position. It will travel from this position in either 
 direction by an amount equal to the radius of the eccentric. 
 From A mark off AC equal to the radius OE, fig. no. 
 From C set off CD, equal to Qc, the radius of the lap circle. 
 Now the greatest amount of opening which can be given to 
 the steam port equals the diameter OE, minus the radius of 
 the lap circle. From D, therefore, set off DE equal to this 
 difference ; this will be the width of the steam port. Next, 
 to determine the width EF of the bridge between the steam 
 and exhaust ports. It is evident that this width must be 
 great enough to render it impossible for the outer edge of the 
 valve, when at the right-hand end of its travel, to open the 
 exhaust port to the fresh steam. Therefore the length CF 
 
 Fig. 116. 
 
 must in any case be greater than AC. The only other 
 condition to observe in proportioning EF is that the bridge 
 must be strong enough to withstand the pressure of steam 
 upon it. If the width of the steam port has been properly 
 set off as described, the above-mentioned contingency can 
 never arrive, for the lap of the valve plus the width of 
 the port, should together equal the travel. In the present 
 instance the width is three quarters of that of the steam port. 
 From E set off EG equal to the radius Oz of the inner lap 
 circle, fig. no. In proportioning the width of the exhaust 
 port the principal point to remember is that it must never 
 be throttled, when the valve is at the end of its travel, to 
 such an extent as to affect the back pressure. It is conse- 
 quently usual to make it of such a width, that when the 
 
 u 
 
290 
 
 The Steam Engine. 
 
 valve is at the end of its beat the port is still open to the 
 same extent as the steam port. From G, therefore, set off 
 the length GH equal to the half-travel of the valve plus the 
 width of the steam port, then will FH represent the width 
 of the exhaust port. The remaining dimensions are of 
 course the same as those of the corresponding parts which 
 have just been found. 
 
 Zeuner^s Diagrams applied to Link Motion. When a slide 
 valve is driven by two eccentrics connected to a link, as 
 
 Fig. 117. 
 
 shown in fig. 94, the distance moved by the valve for any 
 angle turned through by the crank depends upon the 
 position of the block U in the link, and is also influenced by 
 the manner in which the link is suspended. Neglecting for 
 the moment the latter consideration, it is capable of analy- 
 tical proof 1 that the distances moved by the valve from its 
 central position may be represented by the chords of a pair 
 of circles touching each other, precisely as in the case of a 
 
 1 The proof is too long and too complicated for insertion in an 
 elementary work. Those who wish to study it should refer to the 
 English translation of Zeuner's work. 
 
Zeuner's Diagrams applied to Link Motion. 291 
 
 simple slide-gear. Moreover, to each new position of the 
 block in the link corresponds a separate pair of valve circles, 
 which differ both in their diameter and angle of lead from 
 the circles corresponding to any other position of the block. 
 It has also been found that the centres of these valve circles, 
 figs. 117, 1 1 8, all lie upon a curve, which in the case of 
 Stephenson's link motion with open arms is a parabola, 
 C 4 . . . .C , concave to the centre of the crank circle, fig. 1 1 7 ; 
 while for the same motion with crossed arms the parabola is 
 convex to the centre (see fig. 118). 
 
 Y 
 
 Fig. 118. 
 
 Calling the angle of advance a, the throw of the eccen- 
 tric /, the length of the eccentric rod /, and the half-length 
 
 // cos a 
 
 of link = , the parameter of this parabola = ' 
 
 2k 
 
 while 
 
 the distance between the vertex of the parabola and the 
 centre O 
 
 = (sin a cos a), 
 
 2\ / ' 
 
 the plus or minus sign being used respectively according as 
 the arms are crossed or open. 
 
 U2 
 
292 The Steam Engine. 
 
 Analytical method of finding the centres of the valve circles 
 in link motions. One of the main problems in connection 
 with the diagrams for link motion is to fix the positions of 
 the centres of the primary valve circles, fig. 117, correspond- 
 ing to the given positions of the sliding block in the link. 
 These circles and the inside and outside lap circles being 
 drawn in place, it is perfectly easy to trace the variations in 
 the lead, release, compression, &c., corresponding to the 
 varying points of cut-off. These points may be fixed either 
 by analytical means, or with very approximate accuracy by 
 a graphical method. 
 
 Let x and y be the co-ordinates of the centres, and u 
 the distance which the slide has been moved in the link 
 for the given degree of expansion \ and using all the other 
 letters in the same sense as above, we have the following 
 formulae, which are obtained from the investigation above 
 referred to ; viz. 
 
 x =-( si 
 
 2\ 
 
 smcH cos a 
 
 tu COS a 
 
 by means of which we can describe the primary valve circles 
 when the angle of advance, the throw of the eccentric, the 
 lengths of link and eccentric rods, and the position of the 
 block in the link are given. 
 
 Let, for instance, the link be of the sort illustrated in 
 fig. 97, so that the block when fully raised or lowered in the 
 link comes exactly opposite the ends of the eccentric rods. 
 
 Let the half-length k of the link be divided into say four 
 divisions, called grades of expansion. Then u may be ex- 
 pressed as a fraction of k. Thus, at the third grade, u = ^ 
 
 k 4 
 
 At the fourth grade u = = k ; and so on. 
 
 4 
 Substituting these values of u in the formulae given 
 
Link Motions. 293 
 
 above, we have, when the fourth grade of expansion is 
 used, 
 
 x - sin a ; 
 2 
 
 y = - cos a. 
 
 At the third grade, when u = 3-, we have 
 
 4 
 
 = - (sin a + i COS a) ; 
 
 y = cos a. 
 For the middle or dead point u = , and 
 
 X = - (sin a + y COS a) ; 
 
 which shows that for this position of the slide block the 
 centre of the valve circle lies in the straight line OX. 
 
 Let, for example, the angle of advance be 30, the throw 
 of the eccentric ij inches, the half-length of link 3 inches, 
 and the length of the eccentric rod 30 inches. Substituting 
 these numerical values in the above formulae, we obtain the 
 centres of the primary valve circles as shown in fig. 117. 
 Thus, for the largest circle, 
 
 The ordinates of the centres r 4 , ^, &c., in fig. 117, are 
 obtained in this way. In order to avoid unnecessary com- 
 plications of the diagram, the valve circles for the second 
 and third grades of expansion are omitted. 
 
 We can see at a glance from fig. 117 how completely all 
 the critical points connected with the distribution of the 
 steam are altered by the position of the sliding block. 
 For instance, with the laps as given in the fig., and the link 
 
294 Tfo Steam Engine. 
 
 in full forward gear, the steam is cut off at F 4 , and the lead 
 equals el while at the first grade of expansion the cut-off 
 is earlier, viz. at F 1} while the lead is increased to e/ } , and 
 similarly the alteration in the points of compression and re- 
 lease may be ascertained by tracing the intersections of the 
 primary valve circles with the inner lap circle. 
 
 The alteration of the lead with the rate of expansion is 
 one of the peculiarities of the Stephenson link motion. In 
 the example first given, in which the eccentric rods are open 
 the lead increases with the rate of expansion ; but if the rods 
 are crossed, the contrary takes place, the lead decreasing. 
 It will also be observed that the travel of the valve, which is 
 represented by twice the diameter of the primary valve 
 circles, varies with each rate of expansion, continually 
 diminishing till it reaches a minimum, when the block 
 occupies the middle of the link. Consequently, if in full 
 forward gear, the valve completely uncovers the steam port 
 and no more, for each succeeding rate of expansion the 
 maximum opening of the port is reduced, till, at the central 
 position, the maximum opening only equals the lead. As a 
 consequence of this peculiarity it is necessary to make the 
 ports of engines provided with link motion unusually broad, 
 as otherwise the steam would be dangerously throttled at 
 the higher rates of expansion. 
 
 The distribution of the steam when the block is in mid- 
 gear is very peculiar. By reference to fig. 117 it will be clear 
 that the steam is cut off at about a quarter-stroke, while it is 
 released shortly after half-stroke, and admitted to the other 
 side of the piston at about three-quarter stroke ; also com- 
 pression commences on one side of the piston very soon 
 after expansion begins on the other. The consequence is 
 that with the block at mid-gear it is impossible for the 
 piston to make a stroke. 
 
 Geometrical method of finding the centres of valve circles 
 in link motions. It is found practically more convenient to 
 fix the positions of the centres of the primary valve circles 
 
Geometry of Link Motions 
 
 295 
 
 by a graphic method of 
 construction rather than by 
 calculation. The method 
 in common use, which will 
 now be explained, though 
 not theoretically exact, is 
 quite accurate enough for 
 all practical purposes. 
 
 As before, the throw 
 and position of the eccen- 
 tric and the lengths of the 
 link and eccentric rods are 
 supposed to be given. One 
 of the results obtained by 
 the analytical investigation 
 of the subject is that the 
 radius of the curvature of 
 the link should always be 
 equal to the length of the 
 eccentric rod. This fact is 
 made use of in the follow- 
 ing construction. Let LL' 
 (fig. 119) represent the 
 length of the link. Bisect 
 LL' in O, and through O 
 draw OA at right angles to 
 LL'. With centre L, or L', 
 and radius equal to the 
 length of the eccentric rod, 
 describe an arc intersecting 
 OA in C. With centre 
 C, and radius equal to the 
 length of the eccentric rod, 
 describe the arc LL', which 
 represents the centre line 
 of the link. With centre C 
 
296 
 
 The Steam Engine. 
 
 and radius CE equal to the 
 half throw of the eccentrics 
 describe a circle EE' A, and 
 draw CE, CE', having the 
 given angle of advance. 
 Upon CE as diameter de- 
 scribe a primary valve 
 circle. Through the point 
 E draw the line EDB, 
 passing through the pri- 
 mary valve circle at the 
 point D where the circum- 
 ference is intersected by 
 the line OL. Describe an 
 arc of a circle through the 
 points EBE'. Divide the 
 arc EB into four equal 
 parts at the points *', e", 
 e"'. Join these points with 
 the centre C ; then the 
 lines O', Ce", G?'", CE, 
 will represent the lengths 
 and positions of the dia- 
 meters of the primary valve 
 circles, which will show the 
 distribution of the steam 
 when the slide block occu- 
 pies the corresponding po- 
 sitions in the link. Gene- 
 rally speaking, if we wish 
 to know the effect on the 
 steam distribution due to 
 the slide block occupying 
 any position b in the link, 
 we have only to divide trie 
 arc EE' into two portions, 
 
Geometry of Link Motions. 
 
 297 
 
 at e bearing to each other the same proportion which IJb bears 
 to L'. By connecting the point e with the centre C we 
 obtain the length and position of the primary valve circle, 
 which will illustrate the distribution of the steam. 
 
 If the eccentric 
 rods had been 
 crossed instead of 
 open we should 
 have proceeded as 
 above, except that 
 to find the position 
 of the point B we 
 should have joined 
 C with L ; then the 
 line joining E' and 
 the intersection of 
 CL with the cir- 
 cumference of the 
 primary valve cir- 
 cle will, at its in- 
 tersection with the line CO give a point B, such that the curve 
 drawn through EBE' is convex to C, instead of being con- 
 cave as in the former case. This is illustrated in fig. 120, 
 which is merely a portion of fig. 119, the rods being supposed 
 to be crossed. 
 
 Supposing that, instead of having to find the distribution 
 of the steam when the position of the block in the link 'is 
 given, the problem be reversed, and we are given the positions 
 of the crank at which the steam is to be cui off, and are 
 required to find the proper method of dividing the quadrant 
 of the reversing lever, so as to provide for the given rates of 
 expansion, the other data being as above, we should pro- 
 ceed as follows. The link is supposed to be of the type 
 illustrated in fig. 97, which admits of the block being put 
 into full gear. 
 
 Let CE, fig. 121, be the radius and position of the eccen- 
 
 Fig. 121. 
 
298 The Steam Engine. 
 
 trie. Let the arc EE' be drawn in the manner already de- 
 scribed from the known dimensions of link and eccentric rods. 
 Let CF be the position of the crank when the steam is to 
 be cut off, when the block is in full forward gear, and let 
 f f" be the points where the cut-off is required to be 
 effected when the engine is worked at higher grades of ex- 
 pansion. The length Ci gives the radius of the lap circle. 
 The points where the lap circle intersects the two radii cf ' 
 cf give points on the circumferences of the two primary 
 valve circles corresponding to the points of cut-off/ 7 ',/ 7 . 
 One end of the diameters of these valve circles is in O, the 
 centre of the circle EAE', and the other ends are situated 
 in the arc EE'. From the points of intersection of Cf" and 
 Cf, with the lap circle draw tangents to this circle inter- 
 secting the arc EE' in e" and e'. Join Ce" and O'. These 
 lines are the diameters of the valve circles required. It now 
 only remains to divide the half-link LO, in the same ratio 
 as the arc EB is divided at e" and e'. These points of divi- 
 sion will give the positions of the slide block necessary in 
 order to effect the required expansion ; and the half of the 
 expansion lever quadrant, fig. 94, requires to be divided 
 in the same ratio, in order to obtain the positions of the 
 notches necessary to bring the slide block into the required 
 positions. 
 
 If we had not originally been given both the length and 
 position of CE, but had been furnished instead with some of 
 the data given in Problems II. to VI., we could have pro- 
 ceeded to find the throw and angle of advance in the manner 
 explained in those problems by means of a separate diagram, 
 afterwards proceeding as above. 
 
 We have hitherto considered cases in which the data 
 relate either to the position and throw of the eccentric, or 
 else to the lead, cut-off, and opening of the valve when the 
 link is in full gear. We might, however, have to solve a 
 question, the data referring to, say, the lead, point of cut-off, 
 and maximum opening of the valve when the slide block 
 
Geometry of Link Motions, 
 
 occupied some position 
 in the link intermediate 
 between full and central 
 gear, the lengths of link 
 and eccentric rods being 
 given as before. Find the 
 centre C, fig. 122, and de- 
 scribe the curve of the link 
 as explained. Next, with 
 the aid of a separate dia- 
 gram, find the position 
 and throw of an eccentric 
 which, if connected direct 
 to the valve spindle, will 
 give the required lead, 
 cut-off, and maximum 
 opening of the valve (see 
 Problem VI.). Transfer 
 the result to fig. 122, O 
 being the position and 
 length of the throw as 
 found. On Ce as dia- 
 meter describe a primary 
 valve circle. Let b be the 
 given intermediate point 
 in the link which has to 
 drive the valve in the 
 manner required. Join 
 Z>C, intersecting the cir- 
 cumference of the pri- 
 mary valve circle in D. 
 Join eD, and prolong it 
 to intersect the line CO 
 in B. Describe an arc of 
 a circle which shall pass 
 through e and B, and be 
 
30O The Steam Engine. 
 
 symmetrical about the line CO. Prolong this arc, and 
 find in it a point E such that Ee is to eft as VL is to O. 
 Join EC, then EC and the corresponding line E'C will give 
 the positions and throws of the eccentrics, which with the 
 given length of link and eccentric rods will produce the 
 required effects when the slide block is at b. This con- 
 struction is useful in solving problems connected with the 
 working of links which are joined to the eccentric rods 
 in such a way that the slide block can never be brought 
 opposite to the connecting-rod ends. Such a link is shown 
 in fig. 9 7 A. Once the position and length of the line CE are 
 fixed, other problems can be solved in the same manner as 
 in the preceding example. 
 
 In all the diagrams which have been given to illustrate 
 the action of link gearing it has been assumed that the 
 motion of the link was in no way affected by the manner in 
 which it was suspended. This assumption is, however, far 
 from being justified by practice, for, the link being held up 
 by a rod /', of finite length (see fig. 94), which oscillates about 
 a point M, the point of suspension P consequently moves 
 in an arc of which M is the centre, and must therefore 
 move up and down a little during each stroke. The point 
 of the link which drives the valve spindle must therefore 
 also slide a little up and down in the link, instead of 
 keeping to the position intended. It is very easy to aggra- 
 vate this irregularity by adopting a wrong method of sus- 
 pension. The end of the rod /' moves in an arc of a circle 
 of which K is the centre. The object aimed at will be best 
 attained when the point of suspension is made to oscillate 
 for every position of the block in the link in arcs, the 
 chords of which are parallel to the line of the valve spindle. 
 In practice the point of suspension is either the lower end 
 or else the central point of the link. Theoretical investi- 
 gation proves that in the former case the point M, fig. 94, 
 should move in a parabolic curve, the highest position of M 
 (when the block is at the bottom of the link) being the 
 
Suspension of Link Motions. 
 
 301 
 
 vertex of the parabola, and the parameter being equal to 
 twice the length of the eccentric rods ; the co-ordinates of the 
 vertex referred to C, fig. 
 123, as origin and to CX 
 and C Y as axes, being Cx 
 = the length of the ec- 
 centric rod, and Cy = the 
 length of the lifting link /. 
 In practice, instead of 
 the parabolic curve we 
 may make use of a circu- 
 lar arc, the radius of 
 
 Fig. 123. 
 
 which is equal to the length of the eccentric rod, and 
 the centre of which is vertically above the point C at a 
 distance = /. 
 
 When the link is suspended from its central point it is 
 found that the point M, fig. 124, should also move in a 
 parabola, the middle posi- 
 tion of M corresponding 
 with the vertex, and the 
 parameter being twice the 
 length of the eccentric rod. 
 
 The axis of the para- 
 bola is parallel to the line 
 CX, and the co-ordinates 
 of the vertex are Coc = 
 
 2 
 
 the length of the eccentric rod -. (where k = the half- 
 length of the link, and / = the length of the eccentric rod), 
 and Cy = the length of the lifting link /. In practice we 
 may substitute for the parabola a circular arc of which the 
 centre lies in a line KM parallel to CX, and at a distance 
 above it equal to /, the central point K lying to the left of Y 
 
 at a distance = . 
 
 2/ 
 
 In actual practice it is never possible to give the arm 
 
 Fig. 124. 
 
302 The Steam Engine. 
 
 KM a length equal to the eccentric rod, but it is always 
 desirable to make it as long as possible. 
 
 Zeuner's Diagrams applied to Meyer's Valve Motion. 
 These diagrams are peculiarly applicable to the investigation 
 of those gears which work with an expansion valve on the 
 back of the ordinary distribution valve. It is true that in 
 consequence of the large number of circles employed the dia- 
 grams look somewhat complex, but the principles on which 
 they are constructed are very easy to understand and to 
 apply. 
 
 It is perfectly obvious, from what has gone before, that a 
 separate primary valve circle may be employed to show the 
 distance which each valve circle has travelled from its 
 central position, for every position of the crank. Conse- 
 quently the difference between the lengths of the chords of 
 the two circles got by drawing the direction of the crank 
 in any position, gives the distance apart of the centres of the 
 two valves for that position of the crank. Thus in fig. 125 
 let CE denote the length and position of the radius of the 
 eccentric which drives the main or distribution valve, and 
 CK the corresponding throw and position of the eccentric 
 for the expansion valve. On each of these lines as diameter 
 describe a circle, and describe the circle AEA', to represent 
 the path of the crank-pin. Then the circle described on 
 CE shows in the usual way, in conjunction with the lap 
 circle, the lead, cut-off, &c., as provided for by the distribution 
 valve. Also for any position of the crank such as CD, 
 the chord Ce shows the distance moved by the main valve 
 from its central position, while the chord Ce' shows the 
 corresponding distance travelled by the expansion valve ; 
 therefore Ce Ce' = e'e shows the distance apart of the 
 central lines of the two valves for the position CD of the 
 crank. 
 
 We will next prove that it is possible to draw a third 
 circle, viz. CG, the chords of which, such as Ce, will represent 
 the differences Ce Ce', and which chords will consequently 
 
Zeuner's Diagrams applied to Meyer's Gear. 303 
 
 show at a glance the distances apart of the centres of the 
 two valves for any position of the crank. 
 
 Join EK, and from C draw CG parallel to EK, and from 
 E draw EG parallel to CK ; then CG will represent the 
 length and position of the diameter of the third or resultant 
 circle. The problem is to prove that Cc = e'e. Join Ge and 
 Ke'. From E let fall a perpendicular EB on K<?' produced. 
 Then in the two triangles GCe and KEB we have the side 
 GC equal to the side KE, being the opposite sides of a 
 
 Fig. 125. 
 
 parallelogram ; also the angle GCc equals the angle BEK 
 since each of them is the sum of two alternate and conse- 
 quently equal angles. Also the angle GeC is a right angle, 
 being the angle in a semicircle ; and EBK is a right angle 
 by construction ; therefore the two triangles are equal, and 
 Ce equals BE. But BE can easily be proved to be equal 
 to e'e. For join Ee ; then in the quadrilateral ~Be we have 
 the angle at B a right angle by construction ; also the angle 
 Be'e is equal to the vertically opposite angle O'K, which 
 being the angle in a semicircle is also a right angle. For 
 
304 The Steam Engine. 
 
 the same reason the angle OE is a right angle, therefore 
 the remaining angle BE^ is a right angle, and the quadrilateral 
 is a parallelogram, and consequently BE equals e'e ; and 
 therefore Ce = e'e ; that is to say, the chord intercepted be- 
 tween C and the circumference of the circle CG in the direc- 
 tion CD is equal to the differences of the chords intercepted 
 in the same direction between the point C and the circum- 
 ferences of the circles CE and CK. 
 
 This proof is perfectly general for all cases where the 
 
 Fig. 126. 
 
 direction of the crank intercepts the circumferences of all 
 three circles. Take, however, the case when it passes through 
 the point of intersection , fig. 126, of two of the circles. 
 In this case the difference between the chords of the two 
 circles on CE and CK equals Ce, which is a chord to both 
 circles CE and CG ; therefore the line Ce should not form 
 any chord with the circumference of the circle on CK. Join 
 eG and eE. Then the angle G?G, being in a semicircle, is 
 a right angle ; and for the same reason C^E is a right angle ; 
 
Zeuner's Diagrams applied to Meyer's Gear. 305 
 
 therefore GE is a straight line, and as CG is drawn equal and 
 parallel to EK, therefore GE is also parallel to CK, and 
 therefore the alternate angle G^C is equal to the alternate 
 angle ^CK ; therefore ^CK is a right angle, and consequently 
 the line eC is a tangent to the circle CK at the point C, and 
 therefore the direction of the crank Ce does not intercept 
 any chord from the circle CK. A similar proof would hold 
 good for the point e'. 
 
 It must be borne in mind that the chords of the circles 
 CK and CE represent the movements of their respective 
 valves to the right of their central positions, while the corre- 
 sponding circles on CE produced to E' and CK produced to 
 K' indicate movements to the left. If, therefore, any line of 
 direction of the crank CD should intersect simultaneously 
 say the expansion valve circle CK' and the distribution 
 valve circle CE, then the distances apart of the centres of 
 these two valves will not be represented by the difference of 
 the chords G/and Cd f , but by their sum ; and it may readily 
 be proved that the chord CD of the circle CG equals this 
 sum. Hence we see that for every possible position of the 
 crank the chords of the resultant circles CG and CG' repre- 
 sent the distances apart of the central lines of the valves. 
 
 We must next show the connection between the distances 
 apart of the centres of the valves, and the opening which 
 the expansion valve EE', fig. 127, permits in the port of the 
 distribution valve DD'. 
 
 First let the centres of the two valves coincide as in the 
 upper diagram, fig. 127, and next let both valves be moved 
 to the right, but of course to different extents, so that their 
 centres no longer coincide as shown in the lower diagram. 
 Let k be the distance from the centre of the distribution 
 valve to the outer edge -of the port. Let / be the length of 
 the half of the expansion valve. Let u be the distance 
 which the inner edge of the expansion valve is moved by 
 the screw spindle (see fig. 98) from the centre line. The 
 distance u is variable, so as to allow of the rate of expansion 
 
 x 
 
306 
 
 The Steam Engine. 
 
 being altered. Let r be the distance from the edge E of 
 the expansion valve to the outer edge of the port in the 
 distribution valve. The value of r is of course variable, and 
 depends upon z/, so that 
 
 r = k I u. 
 
 Now let both valves be moved to the right by their 
 respective eccentrics; let the distance moved by the distribu. 
 
 Fig. 127. 
 
 tion valve be x, and that by the expansion valve x' also let 
 the opening in the port in the distribution valve be e ; then 
 it is evident from the figure that 
 
 r+x } = x + e. 
 /. e = r(x x\) = k - lu (x x { \ 
 
 which expression gives the opening of the port in the valve 
 in terms of the two fixed dimensions k and /, the distance 
 u which is under control, and the distances x and x', traversed 
 by the valves. 
 
 It is very easy to show this opening of the port in the 
 valve graphically. As has already been proved, the chords 
 
Zeuner's Diagrams applied to Meyers Gear. 307 
 
 of the two resultant circles CG, CG', fig. 128, give the values of 
 % aj for any position of the crank. If with centre C and 
 radius CH =.k I u we describe a circle, then the differ- 
 ence between any radius of this circle, such as Ce, and the 
 corresponding chord of the resultant valve circle, viz. C<?', 
 gives the opening of the port in the distribution valve. 
 
 We have now the means of tracing the distribution of 
 the steam throughout the stroke, and also of ascertaining 
 the effect on the ex- 
 pansion due to the 
 alteration of u, the 
 half-distance apart 
 of the two portions 
 of the expansion 
 valve. 
 
 We see that when 
 the crank lies in the 
 direction Ci the port 
 in the distribution 
 valve is just closed, 
 for at the point / 
 where the resultant 
 valve circle intersects 
 the circle described 
 with, k-l-u as ra- 
 dius we have kl 
 u=x x l , and con- 
 sequently the value of e in the above equation is zero. 
 Between the points i and i the port in the distribution 
 valve remains closed, but at /' it is reopened, and the steam 
 would be readmitted to the cylinder were it not for the 
 fact that by this time the cylinder port is closed by the 
 distribution valve, as is proved by drawing the line CF 
 through the intersection of the primary valve circle CE and 
 the lap circle CL. 
 
 The tracing out of the steam distribution by means of 
 
 X 2 
 
 Fig. 128. 
 
308 The Steain Engine. 
 
 this diagram will be greatly facilitated if we bear in mind 
 the following points. 
 
 i st. The opening and closing of the exhaust are determined 
 solely by the distribution valve, are not affected in the 
 slightest degree by the existence of the expansion valve, and 
 may consequently be traced by noting the intersections of 
 the valve circle CE with the outside and inside lap circles 
 in the manner already explained. 
 
 2nd. No matter what the action of the expansion valve, 
 the distribution valve always cuts off the steam on its own 
 account at the, fixed position of the crank determined by 
 the intersection of the outside lap circle with the valve 
 circle CE. In fig. 128 this position is denoted by the line 
 CF. 
 
 3rd. Provided the plates of the expansion valve are not 
 screwed so far apart as to cover the ports in the distribution 
 valve at the commencement of the stroke (which would of 
 course be absurd, as it would prevent all admission of steam), 
 the lead is determined solely by the valve circle CE and 
 the outside lap circle. 
 
 4th. The amount by which the port in the distribution 
 valve is opened, and consequently the point at which the 
 steam is cut off by the expansion valve, is determined by 
 the resultant valve circle CG and the circle described from 
 centre C and radius CH = / u. That is to say, it is 
 so determined for all positions of the crank between the 
 dead point and the position CF ; for once the position CF 
 is passed the steam is cut off by the action of the distribution 
 valve, and the expansion valve can have no further effect. 
 The amount by which the ports in the distribution valve 
 are opened are measured for the port farthest from the 
 crank shaft by the differences between the radii of the circle 
 described with radius CH and the corresponding chords of 
 the resultant valve circle CG, but for the other port the 
 chords must be taken of the circle CG'. Thus we see that the 
 far port in the distribution valve is, in the case of fig. 128, 
 
Zeuner's Diagrams applied to Meyer's Gear. 309 
 
 opened during the whole of the stroke except while the arc 
 //' is being described, and similarly the near port is closed 
 only during the arc nn'. 
 
 5th. The details of the valves and eccentrics having 
 been fixed, the rate of expansion is altered solely by varying 
 the distance apart of the two halves of the expansion valve, 
 that is to say, by varying the radius CH. 
 
 We must now examine more closely the results obtained 
 by varying this radius. Fig. 129 is similar to a portion of 
 fig. 128 to a larger scale, but with the eccentrics set at differ- 
 ent angles, and with all unnecessary complications left out. 
 
 Fig. 129. 
 
 The length CA corresponds with the value- kl. The 
 length Ch corresponds to the value k-lu shown in fig. 
 127. In this case we see that the steam is cut off at the 
 position CH of the crank, the port in the distribution valve 
 is closed while the arc HH, is described, and at the point 
 H, the port in the valve is reopened, and steam would be 
 readmitted into the cylinder were it not for the fact that the 
 port in the cylinder itself is closed by the distribution valve 
 when the crank is at CF. If now we diminish the value 
 of #, so that/ / u = C^j, the expansion valve is so set that 
 it cuts off the steam at the position CH 1 of the crank and 
 
3io The Steam Engine. 
 
 reopens the valve port at the position CH^ ; that is to say, 
 at the same moment that the distribution valve covers the 
 cylinder port. The position CH 1 is therefore the latest cut- 
 off which it is possible to effect by means of the valve gear 
 represented by the above diagram ; for it is obvious that if 
 u is still further diminished, then the valve port will be re- 
 opened before the cylinder port is closed by the distribu- 
 tion valve, and consequently steam will be readmitted to 
 the cylinder during a portion of the stroke. If the value of 
 u be still further diminished, say to A/2 4 , the circle described 
 with C/* 4 as radius will not intersect the circle CG at all, 
 but will merely touch it at the point G, showing that the 
 expansion valve then only closes the valve port for an 
 instant, and consequently does not affect the expansion. 
 While if u be still further diminished, the only effect of the 
 expansion valve will be to reduce the available width of the 
 valve port during a short portion of each stroke. Thus we 
 see that with the dimensions actually chosen this valve gear 
 ceases to have any useful effect when we wish to cut off 
 steam later than at the angle CH 1 . 
 
 On the other hand, its action between the positions CH 1 
 and CA is perfect. For instance, if the value of u be in- 
 creased to AJi 6 , then the steam is cut off at the position 
 CH 6 . If u be still further increased, so as to be greater 
 than kl t say to A/* 3 , then the cut-off will be effected at a 
 position found by prolonging the line H 3 C backwards ; 
 while if u be still further increased to A/r 5 the steam can be 
 cut off at the commencement of the stroke, while in each of 
 the latter cases the reopening of the valve port does not take 
 place till after the position CF has been passed. 
 
 Hence we see that the valve gear represented above is 
 perfectly efficient for any cut-off between the dead point and 
 the position CH 1 . If we wish to be able to cut off at later 
 positions than CH 1 , we must alter the positions and throws 
 of the eccentrics. An inspection of fig. 129 will prove that 
 the nearer the diameter CG of the resultant circle approaches 
 
Zeuners Diagrams applied to Meyer's Gear. 3 1 1 
 
 the position CF, where the distribution valve cuts off the 
 steam, the later will be the position when the expansion 
 valve can cut off the steam. Fig. 130 will show that it is 
 
 Fig. 130. 
 
 perfectly possible so to arrange the eccentrics that the 
 steam may be cut off at any point that may be desired 
 between the dead point and the position CF where the dis- 
 tribution valve closes the cylinder port. In the case of 
 fig. 130 this result is attained by keeping the throw and angle 
 of advance of the eccentric represented by CE as before, 
 and by altering the angle and throw of the o.ther eccentric to 
 such an extent that the diameter CG of the resultant circle 
 coincides with the direction of the line CF. It may here be 
 noticed that in designing a Meyer valve gear it is unnecessary 
 to give the length AC =//, fig. 129, a greater value than 
 CG, the diameter of the resultant valve circle, for by so doing 
 we merely admit of the possibility of the expansion valve 
 being absolutely useless under certain conditions. This 
 would be the case in fig. 129, for instance, whenever u had a 
 less value than A/* 4 . 
 
 Reversing by Meyer's valve gear. If the engine to which 
 Meyer's valve gear is applied is expected to be able to run 
 backwards as well as forwards, the choice of the angle of 
 advance of the expansion valve eccentric will require a good 
 
The Steam Engine. 
 
 deal of consideration, for it is evident that unless this latter 
 eccentric is situated exactly midway between the two distri- 
 bution valve eccentrics, the diameter of the resultant circle 
 will be quite different according as the back or forward 
 eccentric of the distribution valve is in gear. This is illus- 
 trated in fig. 131, where CE and CE' represent respectively 
 the positions and length of the radii of the forward and back 
 gear eccentrics working the distribution valve, and CK that 
 
 Fig. 131. 
 
 of the expansion valve. We see that owing to the angle 
 of advance of the expansion valve eccentric DCK not being 
 90, the length and position of the diameter of the resultant 
 circle CG is quite different from that of the resultant circle 
 CG', and consequently the limits within which we can vary 
 the expansion are quite different according as the engine is 
 running backwards or forwards. For instance, when the 
 eccentric CE is in gear, the expansion can be controlled by 
 
Problems on Meyer's Valve Gear. 313 
 
 the expansion valve as far as the position CH of the crank ; 
 but when the eccentric CE' is in gear, then the cut-off can 
 only be effected by the expansion valve between the limits 
 CA and CH', which is a much smaller per-centage of the 
 stroke. 
 
 It is easy to see that the more perfect the action of this 
 gear when the eccentric CE is in gear, the less perfect will 
 it be for the eccentric CE'. Hence, when an engine has to 
 do the greater part of its work running in one direction, it 
 is desirable so to arrange the eccentrics that the action of 
 the valve gear will be most perfect for this direction. 
 
 As in the case of simple valve gears and link motions, 
 Zeuner's diagrams may be applied not only to analyse the 
 steam distribution when the dimensions, &c., of valves and 
 eccentrics are given, but also to solve problems that is to 
 say, to find out some of the dimensions when other dimen- 
 sions or conditions are given. 
 
 PROBLEM. Given the lead, the angles of the crank when 
 steam is cut off, the release takes place, and the compression 
 commences, find the position and throw of the eccentric 
 of the expansion valve which will permit of all degrees of 
 expansion between the dead point and the point where 
 the distribution valve closes the cylinder port. Also find 
 the length of the plates of the expansion valve, and the 
 distances apart of these plates for any given degree of ex- 
 pansion. 
 
 The data all refer to the distribution valve. From them 
 we proceed to deduce the position and throw of the eccentric 
 belonging to this valve in the manner explained in Pro- 
 blem VI., page 285. Having thus found the angle of advance 
 and the length of the throw CE, fig. 132, the external and 
 internal lap circles are found in the usual manner. 
 
 We have next to find the angle of advance and throw of 
 the expansion valve eccentric. If the main valve is to close 
 the cylinder port at the same instant that the expansion 
 valve covers the port in the distribution valve, the diameter 
 
3*4 
 
 The Steam Engine. 
 
 CG of the resultant circle should coincide in direction with 
 the line CF, which indicates the direction of the crank when 
 steam is cut off by the distribution valve. The length of 
 the diameter CG may be anything we like so as to suit the 
 remainder of the design. The longer we make CG, the 
 greater will be the angle ECK between the eccentrics of 
 the two valves ; and for engines which have to reverse, it is 
 desirable that the angle of advance of CK should be as 
 nearly as possible a right angle. Having fixed the length 
 
 Fig. 132. 
 
 CG, join GE, and draw CK parallel to GE, and EK parallel 
 to CG ; then CK is the position and half length of throw of 
 the expansion valve eccentric. 
 
 The minimum length of k I equals CG (see page 311). 
 With centre C and radius CG describe an arc of a circle 
 intersecting CA in H ; then CH corresponds with the line 
 CA, fig. 129, and equals k /. We have now to fix the distance 
 apart of the plates of the expansion valve for any degree of 
 expansion. Let AL to A A 7 be the required ratio of expan- 
 sion ; that is to say, supposing AA' to represent the stroke, 
 the steam is supposed to be cut off at L. Erect a perpen- 
 
Problems on Meyer's Valve Gear. 315 
 
 dicular from L intersecting the circumference of the circle 
 described on AA' in b. Join bC ; then, neglecting the 
 influence of the length of the connecting rod, Cb is the angle 
 of the crank when steam is cut off. Cb intersects the arc 
 GH in P' and the resultant valve circle in H'. With centre 
 C and radius CH' describe an arc of a circle intersecting 
 the line CA in h. Then it will be evident, from the descrip- 
 tion accompanying fig. 129, that H/J is the half-distance 
 apart of the plates of the expansion valve for the given ratio 
 of expansion. But ^H = P'H', hence we can measure the 
 distance required directly, as the part of the line Cb inter- 
 cepted between the arc GH and the circumference of the 
 resultant valve circle. Similarly, if steam were cut off when 
 the crank stood at the angle CE, the half-distance apart of 
 the plates would be represented by P^H' 7 . If steam were 
 cut off at the commencement of the stroke the half- distance 
 would be HH'". 
 
 To find the length of the plate / of the expansion valve 
 we must first fix the highest grade of expansion. Having 
 done so, we can find the half-distance apart of the two valves 
 in the manner above described. This gives the value u in 
 the formula r=k lu. And as we know that k I CH, 
 we can deduce r, that is the distance between the outer edge 
 E of the expansion valve, fig. 127, and the outer edge D of 
 the port in the distribution valve when both valves simulta- 
 neously occupy their central position. If the grade of 
 expansion is very high, say one-eighth of the stroke, we 
 shall find that u is greater than k - /, which means that E 
 would stand to the left of D when the valves were simul- 
 taneously central. Having thus fixed the position of E 
 when the two valves are supposed to be in their central 
 position, we must find its position when the valves are at 
 their greatest distance apart. Now the greatest distance 
 apart is when the crank lies in the direction of the diameter 
 CG of the resultant valve circle, and the distance apart 
 equals CG. Hence it will readily be understood by refer- 
 
316 The Steam Engine. 
 
 ence to fig. 127 that when the crank occupies the position 
 CG, fig. 128, the edge E of the expansion valve must lie still 
 further to the left of the point D by the distance CG, con- 
 sequently the edge E will be distant from D by the total 
 length r+CG. But when the crank, is in this position the 
 plate of the expansion valve must fully cover the port in the 
 distribution valve, otherwise steam would be momentarily 
 readmitted into the cylinder. Hence the length of the 
 plate must be equal to T'+CG + the width of the port in the 
 distribution valve -f a small overlap. Now the width of the 
 port is supposed to be known, and the small overlap may 
 be arbitrarily chosen, hence we know the length of the plate 
 /; and as r+l+u = k (see page 306), we can deduce /, the 
 half-length of the distribution valve. 
 
 It would be possible to multiply examples of the appli- 
 cation of Zeuner's valve gear to the solution of problems 
 connected with Meyer's valve gear, but the explanations 
 given are quite sufficient to enable the student to solve by 
 himself the great majority of the questions which may arise. 
 
317 
 
 CHAPTER VIII. 
 
 INDICATORS AND INDICATOR DIAGRAMS. 
 
 Uses of indicator diagrams Richards' indicator General character of 
 indicator diagrams The lines of admission, expansion, release, 
 exhaust, and compression How to measure the power exerted during 
 a stroke of the piston How to ascertain from diagram the horse- 
 power exerted by the engine How peculiarities and defects are re- 
 vealed by the diagram Loss of pressure during admission Slow cut- 
 off of steam Effects of clearance on expansion curve Effects of con- 
 densation and re-evaporation in cylinder on expansion curve Usual 
 form of the expansion curve Late and early release Effect of wet 
 steam on the exhaust line Causes affecting the back pressure 
 Cushioning or compression of exhaust steam Principal causes affect- 
 ing forms of diagrams Examples of diagrams from defective engines 
 How to draw the hyperbolic curve of expansion Initial condensa- 
 tion and re-evaporation shown by diagram Leaky pistons and slide 
 valves Gross and net indicated power Cause which limits the 
 economical rate of- expansion How to deduce from indicator dia- 
 grams the effective pressure on piston How to ascertain the expendi- 
 ture of steam accounted for by the diagram. 
 
 IT has been already explained (p. 79) how the work done 
 during the expansion of a gas can be represented graphically 
 by an area, while the exact way in which the work is done is 
 shown by the nature of the lines bounding the area. It is 
 of the greatest importance to be able to take diagrams of the 
 work done in the cylinder of a steam-engine, for by com- 
 puting the area of the diagram we can ascertain exactly the 
 power exerted by the engine, and by examining closely the 
 bounding lines we are enabled to see how the work is being 
 done at each successive instant, and can tell, for instance, 
 when the steam is cut off, whether expansion proceeds in 
 the proper manner, when the cylinder is open to the exhaust, 
 what is the amount of the back pressure, when the valves 
 are closed to the exhaust, when the readmission of steam 
 
318 The Steam Engine. 
 
 takes place, as well as numerous other details connected with 
 the working of the engine a knowledge of which enables 
 us to ascertain whether the latter is working correctly, and, 
 if not, where the fault lies. 
 
 Without the help of these diagrams it would be impos- 
 sible to tell in the majority of instances what are the faults 
 and shortcomings of engines at work ; and it is no exaggera- 
 tion to state that all the improvements which have been 
 effected in the economical working of the steam-engine, con- 
 sidered as a means of turning heat into mechanical work, 
 have been the result of the knowledge obtained from them. 
 
 Steam- Engine Indicator. An instrument for automatic- 
 ally drawing these diagrams is called a Steam-Engine Indi- 
 cator, and the figures which it draws are called indicator 
 diagrams. The indicator which is at present in most common 
 use is illustrated in fig. 133. It is called, after its inventor, 
 Richards' indicator. The part A is a steam cylinder con- 
 taining a piston, B. The part of the cylinder below the 
 piston can be placed in connection with either end of the 
 engine cylinder. Above the piston is a spiral spring, C. 
 Whenever steam is admitted below the piston it presses 
 against the spring, which is so constructed that the distance 
 through which it is compressed is proportional to the pres- 
 sure on the piston. Thus the height to which the piston 
 ascends under the action of the steam is a measure of the 
 pressure of the steam. When the steam is allowed to escape, 
 or its pressure is diminished, the piston is immediately 
 forced downwards by the spring. D is a piston rod, to the 
 head of which, E, it would be easy to fasten a pencil, the 
 point of which could be made to press against a piece of 
 paper, and by its upward and downward motion register 
 the steam pressure beneath the piston. This was, in fact, 
 the mode of action of the older forms of indicators. It is, 
 however, very desirable to limit the range of motion of the 
 piston to something like three-quarters of an inch ; while, 
 on the other hand, the indications traced by the pencil may, 
 
Richards' Indicator. 
 
 319 
 
 with advantage, be three or four inches long. The reason 
 why the range of the piston is limited in practice is that its 
 velocity, and consequently its momentum, are thus reduced. 
 In the older indicators the momentum was often so con- 
 
 Fig. 133. 
 
 siderable that the work stored up in the moving parts enabled 
 the resistance of the spring to be overcome beyond the point 
 which represented the true pressure of the steam, and thus 
 wavy lines were generated and the indications were un- 
 trustworthy. In order to obtain a long range of the pencil, 
 
320 The Steam Engine. 
 
 combined with a small motion of the piston, the piston rod 
 is attached, not directly to the pencil, but to the arm of a 
 lever, F, the fulcrum of wh ; ch is at G, while the pencil is 
 fixed to the end, H. This lever is so proportioned that the 
 total length, GH, is three or four times as long as the portion 
 between the fulcrum and the head of the piston rod, and 
 consequently the pencil sweeps through three to four times 
 the range of the piston. The pencil would now, however, 
 describe circular arcs instead of straight lines up and down. 
 To prevent this a parallel motion is made use of, which 
 consists of another lever arm, KL, equal in length to the 
 lever GH. The ends H and K are connected together by 
 a link, in the centre of which the pencil is fixed. What- 
 ever effect the lever GH may have in deflecting the part 
 which holds the pencil from the vertical straight line, it is 
 clear that the other lever exercises the opposite effect, and 
 consequently the line of motion of the pencil keeps straight. 
 The function of the indicator, however, is not merely to 
 register pressures, but to register them in relation to the 
 position of the piston of the steam-engine. In order to 
 effect this object, a drum, M, is provided, capable of revolu- 
 tion on its axis. Round the lower end of the drum is coiled 
 a piece of string, the free end of which is attached to some 
 one of the moving parts of the engine, the motion of which 
 is proportional to the motion of the piston. Round the 
 drum is fastened, by means of the clip N, the piece of paper 
 on which the diagram is to be drawn. The pencil is brought 
 to bear lightly against the paper by moving round the frame- 
 work which carries the parallel motion and lever. When 
 the piston of the engine begins to move forward, the string 
 on the barrel M is uncoiled, causing the barrel to revolve ; 
 the pencil at the same time moving up or down under the 
 varying pressure of the steam traces a line on the paper, 
 which shows the pressure at each successive position of 
 the piston. When the latter has reached the end of its 
 stroke the barrel revolves in the opposite direction by means 
 
Steam Engine Indicators. 321 
 
 of an internal spring, as fast as the tension of the string will 
 permit, the pencil meantime continuing its indications. If 
 the movement of the part of the mechanism chosen for 
 attaching the string is greater than the circumference of the 
 barrel, and if no part can be found having the requisite 
 movement, then some special means must be made use of 
 for reducing the motion of the piston to that of the circum- 
 ference of the barrel. 
 
 General Character of Indicator Diagrams, Suppose that 
 the end, O, of the indicator js screwed into one end of the 
 engine cylinder, or into a pipe which can be connected at 
 will with either end of it. The cock, P, is furnished with a 
 three-way plug, formed thus H, and is also provided with a 
 small hole communicating with the outside air. By means 
 of the three-way plug the under side of the indicator piston 
 can be placed either in communication with the atmosphere, 
 or with the engine cylinder, or can be shut off from either. 
 First*^ let it be placed in communication with the atmo- 
 sphere ; the spring above the indicator piston is then neither 
 in tension nor compression. Let the pencil now be pressed 
 against the paper on the barrel, and draw the string by hand 
 so as to make the barrel revolve ; the pencil will then trace 
 a horizontal straight line (aa r , fig. 134), which is the line of 
 atmospheric pressure. Next draw the line ^parallel to aa' 
 at a distance below it to represent 147 Ibs. pressure to the 
 scale corresponding to the spring above the piston. This 
 will be the line of perfect vacuum when the barometer stands 
 at its usual height. The atmospheric pressure is frequently 
 spoken of as 15 Ibs. to the square inch. 
 
 Let the cock now be turned so that the indicator piston 
 shall be placed in communication with the steam cylinder at 
 the commencement of a stroke. The piston will immediately 
 rise, compressing the spiral spring above it, through a space 
 proportional to the steam pressure. Suppose the pressure 
 in the engine cylinder, when the steam is first admitted, to 
 be 65 Ibs. by the gauge = 65 + 15 = 80 Ibs. above the line 
 
 Y 
 
3 22 
 
 The Steam Engine. 
 
 of perfect vacuum or zero of pressure. The pencil will, when 
 steam is first admitted to the engine cylinder, rise vertically 
 upwards, tracing the line to represent 80 Ibs. on the same 
 scale that/tf represents 15 Ibs. 
 
 As the piston of the engine moves forward, let the steam 
 continue to enter at undiminished pressure, say for one-third 
 
 of the stroke, and 
 let it then be 
 sharply cut off. 
 During this period 
 the pencil will re- 
 main at the same 
 height and will, 
 consequently, as 
 the drum of the 
 indicator revolves 
 beneath it, trace 
 out the horizontal 
 straight line AB 
 parallel to/g, and 
 proportional in length to one-third of the stroke of the piston 
 of the engine. 
 
 When the steam is cut off it will commence to expand, 
 and the pressure will consequently drop. We will suppose, 
 for the sake of simplicity, that the expansion takes place 
 according to Boyle's law, and at the end of the stroke it will 
 be % = 26 Ibs. absolute, and at intermediate points will 
 (see p. 45) be represented by the ordinates of a rectangular 
 hyperbola. The point of the pencil drops as the pressure 
 diminishes, and as at the same time the drum revolves, a 
 curve, BC, will be traced, which will be a portion of a rec- 
 tangular hyperbola. Erect, therefore, at the point g, which 
 represents the end of the stroke, the perpendicular ^C, in 
 length equal to one-third of A/j and through the points 
 BC draw the curve BC, such that any ordinate, b'b, multi- 
 plied by its corresponding abscissa /<', shall be equal to the 
 product BB'x/B'. 
 
Nature of Indicator Diagrams. 323 
 
 When the piston has reached the end of its stroke, let 
 the exhaust valve open fully and sharply ; then, if there be 
 no condenser, the pressure will fall to that of the atmosphere, 
 and the pencil will descend, describing the straight line Ca'. 
 The piston will now commence its return stroke, during 
 which there is no pressure of steam, but only the pressure 
 of the atmosphere opposing its return, and the pencil will 
 consequently trace the horizontal straight line a'a, till the 
 piston reaches its original position, when the same series of 
 operations recommences. If, however, the engine is pro- 
 vided with a condenser, when communication with the 
 exhaust is opened the pressure falls below the atmospheric 
 line to a point D, the position of which depends upon the 
 amount of vacuum maintained in the condenser. Suppose, 
 for instance, that twelve pounds was the amount of vacuum, 
 then the pencil will continue moving in the direction Ca' till 
 the point D is reached, such that 0'D=-}-f-=J- of af. If, 
 during the return stroke, the vacuum is maintained at this 
 figure, the pencil will trace the horizontal straight line DE, 
 parallel to aa' ; and when the piston has regained its origi- 
 nal position and steam is readmitted, the pencil will rise 
 vertically upwards to A, and the above series of operations 
 will be repeated. 
 
 How to measure the Power exerted during the Stroke of the 
 Piston. It must be remembered that all distances measured 
 along aa! represent spaces traversed by the piston, while all 
 distances measured at right angles to this direction represent 
 pressures of steam in pounds per square inch. And conse- 
 quently the work done, or pressure multiplied by space 
 traversed under that pressure, is represented by an area. If 
 the pressure were uniform, as it is, for instance, along the 
 line AB, it would be easy to calculate the area, which would 
 in this case, if the engine were of the condensing tfpe 
 = AB x AE. As, however, the pressure varies at different 
 parts of the stroke, it will be necessary to find the length of 
 line representing the mean or average pressure, which, when 
 
 Y 2 
 
324 The Steam Engine. 
 
 multiplied into the length of the stroke, will give an area 
 equal to the irregular figure ABCDE. The usual method 
 of finding the mean pressure is as follows. The line ED, 
 representing the length of the stroke in feet to scale, is 
 divided into ten equal parts, and from each point of division 
 a perpendicular is erected intersecting the bounding lines of 
 the diagram. The mean pressure in each of the " ten divi- 
 sions of the diagram must next be computed. This is easy 
 to do by eye if the diagram is not very irregular. In the 
 latter event it will, for the sake of accuracy, be better to 
 divide the diagram into twenty instead of ten equal divisions. 
 The ten mean pressures must next be added together, and 
 their sum divided by ten, the quotient being the mean pres- 
 sure for the entire diagram, which, being multiplied by the 
 length of stroke in feet, gives the work in foot-pounds exerted 
 during each stroke for every square inch of area of the 
 piston. 
 
 To ascertain from the Diagram the Horse-power at which 
 the Engine is working. A horse-power is, as has been before 
 explained, an expression used to denote the amount of work 
 done in raising 33,000 pounds one foot high in a minute. 
 Having ascertained the mean pressure from the diagram, we 
 multiply it by the number of square inches of area of the 
 piston ; the product gives the total average pressure on the 
 piston. This number, multiplied by the length of the stroke 
 in feet, gives the number of foot-pounds of work per stroke, 
 which, multiplied by the number of strokes per minute and 
 divided by 33,000, gives the indicated horse-power. 
 
 EXAMPLE. 
 
 The cylinder of an engine is 14*- inches diameter, the stroke is 3 
 feet, and the number of strokes per minute 88 ; the mean pressure in 
 the cylinder is ascertained from an indicator diagram to be 40 Ibs. per 
 square inch : what horse-power is being exerted ? 
 
 The area of a cylinder of 14^ inches diameter is 165 square inches. 
 
Defects revealed by Indicator Diagrams. 325 
 
 How peculiarities and defects are revealed by the indicator 
 diagram. The indicator diagrams of most engines differ in 
 many important respects from the ideal diagram of fig. 1 34. 
 In a fairly well-constructed engine the following modifications 
 of form will probably take place. When the cylinder is fitted 
 with an ordinary slide valve, the pressure during the portion 
 of the stroke represented by AB will not be uniform, for two 
 principal reasons. In the first place, the steam ports are, 
 even when fully open, necessarily of limited area. At the 
 beginning of the stroke the piston is moving comparatively 
 slowly, while the valve, on the contrary, is at the quickest 
 portion of its stroke, and is rapidly uncovering the port, and 
 the steam pressure is consequently at first fairly well main- 
 tained. As the point of cut-off is approached, the piston is, 
 however, moving nearly at its maximum velocity, while the 
 port is being gradually closed by the valve, and the area 
 available for the admission of the steam is gradually dimin- 
 ished. It results that the steam cannot enter fast enough 
 to follow up the piston. It is, in fact, throttled at its en- 
 trance into the cylinder, and the pressure falls in conse- 
 quence, so that the line AB is no longer horizontal, but 
 droops, as represented by the full line. 
 
 Another cause of loss of pressure at the commencement 
 of the stroke when the steam is worked expansively is the 
 partial condensation of the entering steam, which takes 
 place in consequence of its coming in contact with the 
 sides of the port and walls of the cylinder, which have 
 been previously cooled down by contact with the exhaust 
 steam of the preceding stroke. This condensation of the 
 fresh steam causes a very serious loss of efficiency in the 
 steam engine, and will be referred to again in Chap. XI. It 
 is sought to minimise the loss by keeping the cylinder hot by 
 surrounding it with a jacket or outer cylinder, which ought 
 always to be filled with fresh steam from the boiler. 
 
 The point of cut-off, B, instead of being sharply defined, 
 as in fig. 134, is usually a rounded corner. This results from 
 
326 The Steam Engine. 
 
 the very gradual manner in which a simple slide valve cuts 
 off the steam, causing excessive wire drawing and fall of 
 pressure at this point. 
 
 The line of expansion. The expansion curve BC will not, 
 as a rule, follow the line shown in fig. 1 34. It may either fall 
 above or below it. If the valves and piston are properly steam- 
 tight, it will probably rise above BC, for the following reasons. 
 In the first place, the line BC is calculated on the supposition 
 that only the portion of the cylinder represented by the line 
 AB is filled with steam. In reality this is not the case, because 
 the piston never quite reaches the two covers of the cylinder ; 
 there is always a vacant space between them, which, at the 
 commencement of each stroke, is filled with steam, which 
 expands with the remainder when the cut off takes place. 
 Moreover, the steam port, which is often of very considerable 
 cubic contents, is also filled with steam, which expands in 
 the same manner. These extra steam spaces are called the 
 Clearance. The steam contained in them at each stroke 
 would, under ordinary circumstances, but for expansion, be 
 utterly wasted. The influence of clearance on the curve of 
 expansion is, of course, to raise it ; for the clearance is vir- 
 tually equivalent to an increase of the length of the cylinder, 
 often of about five per cent. Consequently, instead of cal- 
 culating the curve on the supposition that AB represents the 
 volume of steam, we must also allow for the portion A' A, 
 which in this case is represented as one-twentieth, or five 
 per cent, of the stroke aa'. 
 
 The second principal cause of the raising of the expan- 
 sion curve is the re-evaporation of some of the water which 
 was condensed at the commencement of the stroke ; for this 
 water, having the temperature of the incoming steam, is pos- 
 sessed of too much heat to remain in the aqueous condition 
 when the pressure is lowered. A portion of it is consequently 
 re-evaporated towards the end of the stroke. This effect is 
 increased when the incoming steam, instead of being per- 
 fectly dry, contains a certain quantity of water. The waste 
 
The Expansion Curve. 327 
 
 due to condensation and subsequent re-evaporation is some- 
 times very considerable, for though work is done to a certain 
 extent during the re-evaporation, it is only done at the end 
 of the stroke ; whereas, had the steam not been condensed, 
 it would have done work throughout the whole stroke. The 
 only way to mitigate the evil is to jacket the cylinder with 
 boiler steam, and thus, as far as possible, to prevent conden- 
 sation. 
 
 A third cause of the raising of the expansion curve may, 
 in the case of jacketed cylinders, be traced to the action 
 of the jacket. The flow of heat from this latter to the 
 contents of the cylinder is proportional to the difference of 
 temperatures between these contents and the steam in the 
 jacket. The temperature of the contents of the cylinder 
 falls rapidly during the expansion, and consequently takes 
 up most heat from the jacket towards the end of the stroke, 
 the effect of the added heat being to raise the pressure of 
 the steam above what it would otherwise be. 
 
 The true curve of expansion of dry saturated steam is 
 not a real hyperbola like BC (fig. 134), but is a curve which 
 falls slightly below BC. As the steam expands it does work. 
 At the same time its pressure, temperature, and total heat 
 diminish. The difference of heat between the states of higher 
 and lower pressure is all converted into the mechanical 
 work done during the expansion. The heat so supplied 
 is, however, not nearly sufficient to account for the whole 
 work done. In order to supply the total heat required 
 some of the steam must be condensed, and yield up its in- 
 ternal or latent heat for conversion into mechanical work. 
 As the expansion continues this condensation progresses, 
 while, at the same time, some of the water condensed from 
 the steam during the earlier period of the expansion is re- 
 evaporated, because its temperature is too high to permit of 
 its all remaining in the aqueous condition when the pressure 
 is lowered. Thus we see that the actual curve of expansion 
 of steam in the ordinary conditions of working of an engine 
 
328 The Steam Engine. 
 
 is most complex, being, in fact, the resultant of a series of 
 opposing condensations and re-evaporations. The final 
 form of the curve, for a given ratio of expansion, is deter- 
 mined by the amount of the clearance, by the original con- 
 dition of the steam as to dryness, and the efficiency of the 
 jacketing, The general result with steam of the usual degree 
 of moisture is that the curve of expansion is often raised up 
 to, and even beyond, the true hyperbolic curve, as shown 
 by the full curve in fig. 134. 
 
 The release and back-pressure lines. In cylinders provided 
 with an ordinary slide valve the expansion is rarely carried 
 to the end of the stroke. The steam port is generally opened 
 to the exhaust a little before this latter point is reached, in 
 order to allow the exhaust steam plenty of time to get 
 away before the new stroke commences, and in order to 
 allow of the port being pretty well opened at the end of 
 the stroke. The result of this arrangement is that the 
 expansion curve terminates at a point c (fig. 134) near 
 the end of the stroke, and is succeeded by a rounded 
 corner at <r, which is caused by the exhaust being -throttled 
 at first by the smallness of the aperture of the port when it 
 commences to open. After this point the curve falls rapidly 
 to the end of the stroke. As the whole of the exhaust 
 steam will probably not have escaped by the end of the 
 stroke, instead of having a vertical line, a'D (fig. 134), bound- 
 ing the end of the diagram, we shall have a curved line, a'S, 
 showing that the pressure gradually falls till the point of 
 minimum pressure, c ', is reached. The character of the 
 exhaust curve, ca'c ', cannot be determined beforehand by 
 theoretical considerations. It depends on the size of the 
 exhaust opening, relatively to the size of the cylinder, on the 
 efficiency of the condenser, and on the pressure- and con- 
 dition of the steam as to moisture at the point where 
 the exhaust commences. If the steam contains much sus- 
 pended moisture, it is more difficult to expel it from the 
 cylinder than when in the dry condition ; and. this is one of 
 
Back-pressure Lines. 
 
 329 
 
 those indirect causes of loss due to condensation in the 
 cylinder and to priming, which is to a great extent corrected 
 by the use of a jacket. 
 
 The height cf the point d above the line of perfect 
 vacuum fg measures the minimum pressure which opposes 
 the return motion of the piston. This opposing pressure is 
 called the back pressure and, of course, the greater its 
 amount, the greater the loss of efficiency of the engine. No 
 matter how perfect the vacuum in the condenser, it is im- 
 possible to attain an equally good vacuum in the cylinder, 
 on account of the 
 resistance opposed 
 to the passage of 
 the exhaust steam 
 by the ports, and 
 the pipe connect- 
 ing the cylinder 
 with the condenser. 
 
 If the engine 
 be non-condensing. - 
 
 the back pressure ; 
 
 will at the least be 
 
 equal to the pressure of the atmosphere, and in addition 
 there will be an increment of pressure due to the difficulty 
 of expelling the steam from the cylinder through the ports 
 and exhaust-pipe. In the case of locomotives, where the 
 exhaust passes through a blast-pipe, with a narrow orifice, 
 up the chimney, for the purpose of promoting a draught, 
 the increase of back pressure due to this cause may under 
 certain circumstances (see p. 141) be very considerable. 
 Fig. 135 shows the usual character line of exhaust and of 
 back pressure for a non-condensing engine, the same letters 
 being used as in fig. 134. 
 
 Reverting again to fig. 134, after the point c' of minimum 
 back pressure is reached, the pencil will, if the vacuum be 
 maintained uniformly, trace the horizontal line c'd' till a 
 
330 The Steam Engine. 
 
 point d' is reached near the end of the stroke. At this 
 point, if the engine be fitted with a slide valve, worked 
 by an eccentric the exhaust will be closed ; for just as the 
 exhaust was made to open before the termination of the 
 forward stroke, so it must close before the termination of the 
 back stroke. The exact position of the point d' relatively 
 to <ris explained in the chapter on valve setting (see fig. 109). 
 All the steam remaining in the cylinder and in the clearance 
 spaces when the exhaust opening is closed, will be com- 
 pressed as the piston advances, and consequently the pressure 
 will rise, and the indicator pencil will describe a curve, 
 which at some point or other, depending on the position of 
 d' and on the pressure of the steam at </, will coalesce with 
 the vertical line EA. 
 
 Cushioning^ or compression of exhaust steam. This com- 
 pression of the steam at the end of the stroke or cushion- 
 ing, as it is usually called by raising the mean value of 
 the back pressure, detracts from the effective power of the 
 engine, and would at first sight appear to be a source of 
 waste. It must, however, be remembered that the steam 
 which is compressed by the piston is, at the end of the 
 stroke, in a condition to exert its pressure usefully on the 
 piston when making the next stroke. It, moreover, fills the 
 clearance spaces, which would otherwise have to be filled at 
 each stroke with fresh steam from the boiler, the whole 
 of which, excepting in so far as it would raise the curve of 
 expansion, would be wasted. Consequently, cushioning, 
 though it diminishes the effective power exerted, increases 
 the efficiency, or the ratio of work done to steam expended. 
 Some engineers recommend that the exhaust be closed at such 
 a point, that the compressed steam, by the time the end of the 
 stroke is reached, shall attain the boiler pressure, in which case 
 the curve of compression would terminate at the point A, 
 fig. 134. Cushioning is also very useful in engines having 
 a high speed of piston, inasmuch as it assists in bringing 
 the reciprocating masses, viz. the piston and the piston and 
 
Indications of Diagrams. 331 
 
 connecting rods, quietly to a state of rest. The lower diagram, 
 fig. 146, is intended to illustrate a very marked curve of com- 
 pression. It is taken from a locomotive engine working at a 
 high grade of expansion. 
 
 We have now seen how the actual diagram in an ordi- 
 nary steam engine may be expected to differ from the 
 theoretical diagram of fig. 134. We have also seen how the 
 diagram records the following facts and operations, viz. 
 
 1. The exact point of the stroke at which steam is 
 admitted. 
 
 2. The initial pressure of the steam in the cylinders, 
 which being compared to the boiler pressure, shows us 
 whether the steam pipes and passages are of the necessary 
 dimensions. 
 
 3. The way in which the initial pressure is maintained 
 or otherwise during the period of admission. 
 
 4. The point of cut-off. 
 
 5. The pressure during the whole period of expansion. 
 
 6. The point of release, i.e. when the exhaust is opened. 
 
 7. The rapidity with which the exhaust takes place, as 
 shown by the nature of the exhaust curve. 
 
 8. The minimum back pressure, which in a condensing 
 engine is also the test of the perfection of the vacuum, and 
 in a non-condensing engine shows what the effect of the 
 friction of the exhaust pipes and passages is in addition to 
 the unavoidable pressure due to the atmosphere. 
 
 9. The period when the exhaust is closed. 
 
 10. The nature of the curve of compression. 
 
 11. The power which is being given off by the engine. 
 Principal causes affecting the forms of diagrams. We have 
 
 also seen that the principal causes which disturb the shape 
 of the theoretical diagram are 
 
 1. The friction of the steam pipes and ports. 
 
 2. The variable size of the opening of the steam ports 
 as caused by the gradual motion of the slide valve. 
 
 3. The action of the sides of the cylinder in causing 
 
33 2 The Steam Engine. 
 
 condensation and partial re-evaporation of some of the 
 entering steam. 
 
 4. The steam contained in the clearance spaces which 
 affects the curve of expansion. 
 
 5. The gradual opening of the exhaust port, which 
 makes it necessary to release the steam too early in the 
 stroke. 
 
 6. The friction of the exhaust passages, which in the 
 case of condensing engines prevents the attainment in the 
 cylinder of the same degree of vacuum as in the condenser, 
 and in the case of non-condensing engines adds to the back 
 pressure. 
 
 7. The momentum of the moving parts, which, combined 
 with cause 4, and also with the unavoidable nature of the 
 simple slide valve driven by an eccentric, renders a curve of 
 compression necessary. 
 
 Examples of diagrams from actual engines. We shall 
 afterwards see that the indicator possesses other important 
 uses in addition to those named above, but it is first pro- 
 posed to give a few examples of good and bad diagrams, 
 
 taken from various 
 
 a a> engines, and to point 
 
 out some of the 
 peculiarities which 
 they reveal. 
 
 The diagram re- 
 presented by fig. 136 
 shows a very late 
 FIg I3g> admission of steam, 
 
 the maximum pres- 
 sure hot being attained till the piston has traversed a portion 
 of the stroke represented by ad. The proper position of 
 the line ba' is the dotted line ba. The fault is evidently 
 due to the valve having been badly set, either through be- 
 coming displaced, or else through the eccentric not having 
 been given sufficient advance (see page 255). 
 
Examples of Indicator Diagrams. 3^3 
 
 Fig. 137 shows that the steam pressure during admissioj 
 is injuriously affected by throttling, owing to insufficient 
 opening or area of the 
 port. 
 
 Fig. 138 repre- 
 sents a diagram 
 which in addition 
 to numerous other 
 defects shows a very 
 high back pressure. 
 This was due to the 
 fact that the steam, 
 instead of being al- 
 lowed to escape di- 
 
 Fig. 137- 
 
 rectly into the atmosphere, was passed first into a feed 
 
 water heater. With some classes of feed heaters it happens 
 
 that much more 
 
 power is lost by the 
 
 increase in the back 
 
 pressure than is 
 
 gained by raising the 
 
 temperature of the 
 
 feed. 
 
 How to draw the 
 hyperbolic curve of 
 expansion. The dia- 
 grams (figs. 139 to 142) are given to show the effects of 
 condensation and re-evaporation on the curves of expansion. 
 In order to be able to mark this effect more accurately it 
 is advisable in all cases to lay down on the diagram 
 the hyperbolic curve of expansion, which is the graphic 
 representation of Boyle's law ; for though this curve repre- 
 sents neither the curve of expansion of steam nor the 
 curve of its relative volumes, it is found, nevertheless 
 that it is the line to which the expansion of steam in 
 the best types of engines most closely approximates, and 
 
 Fig. 138. 
 
334 T/ie Steam Engine. 
 
 is for this reason the best curve to use as a standard of 
 comparison. 
 
 In order to draw the curve of expansion for a given 
 diagram, such as fig. 139, erect a perpendicular ab^ to the line 
 of perfect vacuum ac, the distance aa' representing the 
 clearance reduced to an equivalent fraction of the stroke. 
 The lines ab, ac, will be then the asymptotes of the hyper- 
 bola ; and ad, drawn at an angle of 45 with ab and ac, will 
 be the axis of the curve. We must now select some point 
 in the expansion curve of the diagram from which to com- 
 mence the hyperbolic curve. This latter will in general 
 vary for every point which we may choose, for if there be 
 condensation at the commencement and re-evaporation to- 
 wards the end of 
 the stroke, it is 
 evident that there 
 may be a higher 
 pressure of steam 
 in the cylinder at 
 the end of the 
 
 a' 9 y 9" 9" c stro ke than there 
 
 should be if the 
 true curve of saturated steam expanding and doing work 
 were followed. It is usual to choose a point either at the 
 commencement of the curve, such as e (fig. 139 ), when the 
 steam port has been completely closed, or else a point /, 
 just before the exhaust is opened. The co-ordinates, eg 
 and ek, of the point e (or of /if we select the latter point) 
 must then be measured and multiplied together. The points 
 e' t e") e'", &c., corresponding to the positions of the piston, 
 g'-> g"-> '"> are sucn tnat tne products of their co-ordinates 
 equal the product of the co-ordinates of the original point, e. 
 
 Thus e'g' xg'a = eg x ga. .-. e'g 1 =MI. 
 
 Initial condensation and re-evaporation shown by 
 diagrams. In the diagram, fig. 139, we see that the actual 
 
Condensation and Re-evaporation. 
 
 335 
 
 Fig. 140. 
 
 expansion curve, ^ of the steam lies throughout its whole 
 length above the hyperbola line, showing a considerable re- 
 evaporation of water, which has either been formed by con- 
 densation at the commencement of the stroke or carried 
 over in the form of spray from the boiler. 
 
 Fig. 140 shows the same effect in a still more marked 
 degree. In this case it was ascertained that a large quantity 
 of water was carried 
 over into the cylin- 
 der from the boiler, 
 which was partially 
 re- evaporated by the 
 end of the stroke. 
 That the water was 
 not wholly re- eva- 
 porated, before the 
 return stroke com- 
 menced, is shown by 
 the bad vacuum line. The effect of water in the cylinder 
 in increasing the back pressure is most marked, and is, 
 no doubt, in part due to the re -evaporation which goes on 
 during the exhaust, when the diminished pressure must 
 enable considerable 
 quantities of the 
 highly heated water 
 to burst into steam. 
 
 The diagram on 
 fig. 141 is given to 
 illustrate the case of 
 condensation at the 
 commencement and 
 re-evaporation at the 
 end of the stroke. Here, it will be noticed, the expan- 
 sion curve falls below the hyperbola at the commencement, 
 then crosses it, and for the remainder of the stroke lies 
 above it. 
 
 Fig. 
 
336 The Steam Engine. 
 
 Leaky pistons and slide valves. In engines which have 
 been long at work, the expansion curve may be injuriously 
 affected by leakage of steam through the valve or piston. 
 The best method of ascertaining whether this is going on is 
 to block the fly-wheel at any point in the stroke while the 
 admission port is open, then to admit steam, and to open 
 the lubricating cock at the other end of the cylinder. If 
 steam continues to pour steadily forth from the cock, it shows 
 that there is a leak. By blocking the fly-wheel when the 
 valve is at mid-stroke, and consequently covering both ports, 
 and then opening the lubricating cock, or looking at the 
 
 mouth of the exhaust 
 pipe, we can ascertain 
 whether the leakage is 
 through the valve. 
 
 Fig. 142 is the dia- 
 gram of an engine in 
 which a very consider- 
 able leak took place 
 \ past the valve. The 
 ^ initial pressure of 
 
 Fig I42 steam in the cylinder 
 
 was about 67 Ibs. abso- 
 lute, and the cut-off was supposed to take place at about 
 one-tenth of the stroke. Hence the pressure at release 
 should have been about seven pounds absolute, whereas 
 it is shown by the indicator to have been nearly twenty 
 pounds. 
 
 The diagrams on fig. 143 are intended to show differences 
 of the exhaust line. In No. i the exhaust port is opened 
 at a before the end of the stroke, and by the end of the 
 stroke the steam pressure has fallen very low. In No. 2,- 
 which was taken from the same engine as No. i, but with 
 the eccentric badly set, so as to cause a late admission of 
 the steam, the exhaust is not opened till just before the very 
 end of the stroke, and the terminal pressure, is much higher 
 
The ExJiaust L 
 
 tne. 
 
 337 
 
 than in the case of No. i, although the initial pressure is 
 less, and the rate of expansion the same. The vacuum line 
 is inferior to that of No. i 
 at the commencement, as 
 the steam has not had 
 time to escape before the 
 return stroke begins. It 
 very often happens that 
 what is gained by post- 
 poning the release till the 
 stroke is finished is lost 
 again through the increase 
 in the back pressure. The 
 terminal pressures are, 
 however, very different in 
 the two cases, and conse- 
 quently, also, the twisting moments on the crank towards 
 the end of the stroke. 
 
 Fig. 144 is taken from a cylinder provided with exhaust 
 valves and ports quite inde- 
 pendent of the steam valves 
 and ports. As will be seen from 
 the sharp corner at a and the 
 sudden fall of the pressure, 
 the exhaust port opens sharply 
 and fully, thus allowing the 
 steam to escape very readily. 
 
 The diagram, fig. 145, shows as bad a distribution of the 
 
 Fig 
 
 Fig. 144. 
 
 Fig. 145- 
 
 steam as it is possible to conceive. The valve is deficient 
 both in lap and lead, consequently the admission of the 
 
 z 
 
338 
 
 The Steam Engine. 
 
 steam is late ; the engine works with full steam during the 
 whole stroke ; the port is opened so gradually that the full 
 
 pressure is not attained 
 
 till the end of the stroke. 
 The exhaust opens so 
 late and so gradually 
 
 Cut off 75 % that the pressure of the 
 
 exhaust steam had not 
 fallen to its proper point 
 till nearly the end of the 
 return stroke. 
 
 The diagrams on fig. 
 146 have reference to 
 the line of compres- 
 sion. In an engine with 
 a single slide valve to 
 regulate the admission 
 and exhaust of the 
 steam from each end of 
 the cylinder, the point 
 of the stroke where the 
 exhaust closes is de- 
 pendent on the point 
 where it opens t.e. 
 the point of release ; 
 and in proportion as 
 the latter occurs early 
 in the forward stroke, 
 so will the former take 
 place early during the 
 return stroke. This 
 Fi ~ 6 ' effect is more particu- 
 larly apparent in the 
 
 diagrams of locomotives which are driven by a single slide, 
 the rate of expansion being varied by means of the link 
 motion. Nos. i, 2, 3, and 4 are taken from the same loco- 
 
 Culvf/19 % 
 
 tilt off 15 % 
 
Line of Compression. 
 
 339 
 
 motive engine, working at different rates of expansion, and 
 with varying points of release and compression. In Nos. 3 
 and 4, the exhaust is closed so early that the curve of com- 
 pression rises to a great height before the end of the return 
 stroke, so much so that the back pressure at the end of 
 the return stroke rises nearly to the initial pressure at the 
 commencement of the forward stroke. The advantage of 
 compressing the exhaust steam has already been explained. 
 
 It would be impossible to give in this chapter examples 
 of all the peculiarities which may arise in diagrams from 
 defective valve setting, or leaky valves and pistons, priming 
 boilers, and unjacketed cylinders ; but, as the nature of the 
 best attainable diagram has been explained, and also the 
 principal faults and peculiarities which occur in ordinary 
 engines, enough has been said to enable the student to in- 
 vestigate the condition of most engines from an inspection 
 of their indicator diagrams. 
 
 Gross and net indicated power. The indicator diagram 
 gives us, as we have seen, an exact account of the working 
 of the engine and of 
 the power which is 
 being exerted and 
 which is available for 
 transmission, either 
 to the working parts 
 of the engine proper 
 or to some external 
 train of mechanism. 
 It also gives the total 
 power which is being 
 exerted by the engine 
 which includes the power which is being thrown away 
 in overcoming back pressure. Take, for instance, the dia- 
 gram, fig. 147, which is taken from a high-pressure expan- 
 sive engine, the line AE being the line of absolute vacuum. 
 It is evident that during the forward stroke the work done by 
 
 z 2 
 
 Fig. 147- 
 
340 The Steam Engine. 
 
 the steam on the piston is equal to the area ABCDE. 
 During the back stroke the return of the piston is opposed 
 by the pressure of the atmosphere and of the exhaust steam, 
 on which it does work, measured by the area DEAF, so that 
 the useful work available for transmission to the parts of the 
 engine and external objects is represented by the difference 
 between these two figures i.e. BCDF. 
 
 The work represented by ABCDE is called the gross in- 
 dicated power, and BCt)F is called the net indicated power. 
 This latter, again, is divisible into two parts, one being the 
 work necessary to overcome the friction of the moving parts 
 of the engine, and the other being the remainder, which is 
 called the useful power, and which is all that is available for 
 doing external work. 
 
 The above explanation will render clear the reason why 
 the economy due to expansion in non-condensing engines is 
 so very limited. For instance, taking fig. 148, it is evident 
 that the power thrown away in overcoming the back pressure 
 is about forty per cent, of the total power exerted. If a still 
 greater degree of expansion were used, the gross power 
 would be diminished, while the power wasted would remain 
 the same, and the comparison between the useful and the 
 gross power would be still more disadvantageous. Of 
 course, condensation removes this evil to a great extent, but 
 in condensing engines the theoretical gain due to high rates 
 of expansion is also limited by causes which are explained 
 in Chapter XL 
 
 How to deduce from indicator diagrams the effective 
 pressure on the piston. The indicator diagram, though it 
 shows the pressure at every point of the stroke, does not 
 show the pressure which is available for transmission through 
 the piston and connecting rods to the crank of the engine. 
 This pressure is the difference between the total pressure, 
 as shown by the diagram, on the driving side of the piston, 
 and the simultaneous back pressure on the other side of the 
 piston, which latter can only be obtained by taking a sepa- 
 
Effective Pressure on Piston. 
 
 341 
 
 rate diagram from the other end of the cylinder. In well- 
 designed horizontal engines the diagrams from the two 
 cylinder ends should in most 'cases be as nearly as possible 
 alike, but in many engines, especially those in which the 
 valve setting is defective, or in which the connecting rod is 
 short compared to the length of the crank, the diagrams 
 differ from each other very considerably, and consequently a 
 pair of diagrams is often absolutely essential to enable us to 
 compute the net pressure transmitted by the piston. 
 
 In order to obtain the pressure available for transmission 
 externally at any point of the stroke we must construct a 
 new diagram, which shall show the differences between the 
 simultaneous forward and back pressures on the two sides 
 of the piston. 
 
 Let ABCD, A'B'C'D', fig. 148, represent diagrams taken 
 from the opposite ends of a cylinder. The diagrams are 
 arranged so as to 
 overlap, the point 
 D' marking the 
 end of the for- 
 ward stroke, being 
 in the same verti- 
 cal straight line 
 as the point A, 
 which marks the 
 commencement of 
 the back stroke. 
 This arrangement 
 is convenient, as 
 it enables the 
 
 simultaneous forward and back pressures to be seen at 
 a glance. Thus at the commencement of the forward 
 stroke the pressure on the working face of the piston is 
 measured by the distance of the point A from the line of 
 absolute vacuum GG'. At the same moment the other side 
 of the piston is acted on by a back pressure measured by 
 
342 
 
 The Steam Engine. 
 
 the distance of the point D' from gg f , and therefore the 
 effective pressure is the difference between these two, and 
 is measured by the line AD'. When the piston has reached 
 the position K, being the place where the exhaust line of 
 one diagram crosses the line of compression of the other 
 diagram, the two pressures are equal, and consequently the 
 piston is urged neither forwards nor backwards by the 
 steam, and continues its forward motion only by reason 
 of the energy stored up in the moving parts of the engine 
 and the fly-wheel. From the point K to the end of the 
 stroke D, the pressure urging the piston forward is actually 
 less than the back pressure, which latter consequently tends 
 to bring the piston to a state of rest. 
 
 The true diagram of resultant pressure on the piston 
 is formed by setting off the differences between the forward 
 
 and back pres- 
 
 A/ N 8 sures as ordinates, 
 
 measuring from 
 the line of abso- 
 lute vacuum as a 
 base and drawing 
 a curve through 
 the ends of the 
 ordinates. When- 
 ever the forward 
 pressure is in ex- 
 cess, the ordinate 
 is drawn above 
 the line GG', and 
 whenever the back pressure is in excess it is drawn below. 
 We thus get the dotted curve of fig. 148 which is reproduced 
 for the sake of clearness as a separate diagram in the curve 
 D'AB^DA', fig. 149, which shows the resultant pressure on 
 the piston at any point of the stroke. 
 
 We are enabled by this method to see what important 
 modifications of the resultant diagram may be caused by 
 
 ID' 
 
 Fig. 149. 
 
Resultant Diagrams. 343 
 
 alterations in the positions of the points of release C and 
 compression E, fig. 148. If the release occurred at the end 
 of the stroke instead of at the point C, the pressure between 
 the points C and D would be greater than that shown in 
 the figure. On the other hand, the vacuum at the com- 
 mencement of the return stroke would not be so good, and 
 consequently the back pressure between the points D and E' 
 would be greater than is represented Moreover, if the steam 
 distribution were controlled by an ordinary slide valve the 
 points of release and of compression would be interdependent, 
 and in the case under consideration there would be no ap- 
 preciable compression. The general result would be that the 
 positive pressure at the commencement and the negative 
 pressure at the end of the stroke would each be diminished, 
 and consequently the pressure on the piston would fluctuate 
 between less wide limits. 
 
 The resultant diagram will be found of great use when 
 we require to calculate the twisting moment on the crank- 
 pin throughout a revolution (see Chapter V.). 
 
 How to ascertain the expenditure of steam accounted for by 
 the diagram. Another and most important use of the indi- 
 cator diagram is to 
 enable us to ac- 
 count for the ex- 
 penditure of the 
 steam and the 
 
 heat supplied from 
 the boiler to the 
 engine. In order 
 to measure, from 
 the diagram, the 
 steam consumed, 
 
 Fig. 150. 
 
 we must ascertain 
 
 the pressure at a point #, fig. 150, just before the release 
 takes place. By reference to Col. 5 of Table I. we can deduce 
 the weight of a cubic foot of steam of this pressure. We 
 
344 The Steam Engine. 
 
 must ascertain the cubic contents of the cylinder, includ- 
 ing the clearance, in cubic feet, up to the point a, and 
 multiplying this number by the weight of the cubic foot 
 of steam, we obtain the weight of steam present in the 
 cylinder immediately before the release. When there is 
 any compression, we must deduct the quantity of steam 
 saved by the early closing of the exhaust. To do this we 
 have only to measure the pressure at any point fr, after the 
 exhaust has closed, and to ascertain the weight of the con- 
 tents of the cylinder up to the point b. The difference 
 between the steam spent and saved is the quantity accounted 
 for by the indicator. By comparing the quantity thus 
 accounted for in an hour with the weight of water which 
 leaves the boiler in the same time we can ascertain how 
 much of it is lost by the combined effects of priming, con- 
 densation in the pipes, and condensation in the cylinder 
 itself. 
 
 If we ascertain the pressure and weight of the steam 
 contained in the cylinder at a point immediately after the 
 admission is closed, we can, by the help of Table I., as- 
 certain the number of thermal units contained in the steam 
 at that point of the stroke. This number of thermal units 
 will always be less than the total heat which has been sup- 
 plied to the engine up to that point, for a certain number 
 will have been expended in restoring the temperature of the 
 ends of the cylinder and piston. Also during the expan- 
 sion the steam loses the heat which has been converted into 
 mechanical work, and consequently, if we were to measure the 
 heat contained in the steam at any point, say a, fig. 150, before 
 the release takes place, we should expect to find that the 
 number of thermal units was less than that contained in the 
 steam at the point of cut-off by the number converted into 
 mechanical work As a matter of fact, however, the number 
 is never less, and is often considerably greater, thus showing 
 that a great deal of the heat which passes from the steam to 
 
Heat Expenditure shown by Diagram. 34$ 
 
 the sides of the cylinder during the early part of the stroke 
 is, during the remainder of the stroke, re-transmitted from 
 the cylinder to the steam, and passes out with the exhaust 
 and is partly wasted. 
 
 The diagrams of compound engines will be considered 
 in Chapter XI. 
 
346 The Steam Engine. 
 
 CHAPTER IX. 
 
 FUEL COMBUSTION THE GENERATION OF STEAM BOILERS 
 AND THEIR FITTINGS. 
 
 Combustion Combination of oxygen with carbon Combination of oxygen 
 with compounds of carbon and hydrogen Chemical, symbols and 
 atomic weights of constituents of fuel Principal compounds of carbon, 
 hydrogen, and oxygen Constituents of fuel Heat of combustion of 
 carbon and hydrogen with oxygen Description of fuels in common 
 use Table of the chemical constituents and evaporative power of 
 various fuels Weight and temperature of the products of combustion 
 Waste of fuel by splintering, distillation, insufficient air supply 
 Smoke forming Draught creation, radiation, and conduction Con- 
 duction of heat through the plates of furnaces Importance of pre- 
 venting an over-supply of air to fuel The various types of boilers 
 Cylindrical boiler with external firing Cornish boiler Tubulous 
 boilers Lancashire boiler Galloway tubes The stiffening of internal 
 flues Locomotive boilers Marine boilers for low-pressure steam 
 Marine boilers for high-pressure steam Proportions of parts of boilers 
 Firegrate area Evaporative power of fuel in various types of boilers 
 Consumption of fuel per square foot of firegrate area Efficiency of 
 heating surface Cubic capacities of boilers of different types Steam 
 room Strength of boilers Hollow cylinder pressed from within : ist, 
 longitudinal strain ; 2nd, transverse strain Strength of riveted joints 
 Hollow cylinder pressed from without Flat stayed surfaces Effects 
 of unequal heating in straining boilers Materials of construction 
 Boiler fittings Safety valves Pressure gauges Feed pumps In- 
 jectors Water gauges. 
 
 WE have hitherto considered chiefly the nature and the laws 
 of heat, and the details of the engine which is employed to 
 convert the heat into mechanical work ; but of the source of 
 heat viz. the fuel, and the medium by means of which it is 
 conveyed to the engine, viz. water, and the apparatus by 
 which the heat of the fuel is transferred to the water, viz. 
 the boiler we have up till now said but little. 
 
 The source of heat which is always employed is fuel, 
 such as coal, wood, peat, or mineral oil, the principal calorific 
 
Chemical Combination. 347 
 
 constituents of which are carbon and hydrogen. The 
 chemical combination of these elements with oxygen pro- 
 duces intense heat, which, for the purposes of the steam- 
 engine, is transmitted to the water contained in the boiler in 
 a manner to be hereafter described. 
 
 The amount of heat which can be generated by the 
 chemical combination of fuel and oxygen, commonly called 
 combustion, depends upon the relative proportions of car- 
 bon ana hydrogen which the fuel contains, as well as on the 
 amount of oxygen which is supplied to it. It is well known 
 that chemical elements combine with each other in certain 
 definite proportions only ; that is to say, for instance, a 
 certain definite weight of carbon viz. twelve units will 
 combine with a certain other definite weight of oxygen viz. 
 sixteen units to form the compound called carbonic oxide, 
 and these two elements will only combine in these propor- 
 tions or in certain simple multiples of them. Thus, for 
 example, no chemical combinations can be formed out of 
 seven parts by weight of carbon to five parts by weight of 
 oxygen ; but, on the other hand, twelve parts of carbon will 
 combine with twice sixteen parts of oxygen, forming the 
 compound called carbonic acid or carbonic anhydride, 
 which is the term applied to the product of the complete 
 combustion of carbon in oxygen. 
 
 The numbers 12 and 16 applied to carbon and oxygen 
 are called the atomic weights of these two substances, and it 
 is supposed that the ultimate atoms of which they are com- 
 posed have to each other the relative weights of 12 to 16. 
 These numbers are also called the chemical equivalents of 
 the substances. 
 
 If, instead of pure elementary substances, such as carbon 
 and oxygen, we had to deal with the combination of a com- 
 pound with a simple element, we should find that the 
 chemical equivalent of the compound would be the sum of 
 the equivalents of its constituents ; thus, for example, a 
 pound of olefiant gas, which is composed of carbon and 
 
343 
 
 The Steam Engine. 
 
 hydrogen in the proportion of six parts by weight of carbon 
 to one of hydrogen, would, in order to effect its combustion, 
 require a quantity of oxygen computed as follows. The 
 carbon weighs f of a pound, and would require |f x 2 of 
 its own weight of oxygen in order to form carbonic acid. 
 The hydrogen weighs -f of a pound ; its chemical equiva- 
 lent or atomic weight is i, and it combines with oxygen in 
 the proportion of two parts by weight to sixteen of the 
 oxygen. That is to say, the weight of oxygen required is 
 \ X V 6 of a pound. Consequently for the pound of olefiant 
 gas we shall want (y x if x 2 ) + (y x V 6 ) = 3y I DS - of oxygen. 
 The following are the chemical symbols and atomic 
 weights of the principal elementary constituents of fuel. 
 Carbon . . C . .12 
 
 Hydrogen . . H . i 
 
 Oxygen . . . O . .16 
 
 The following are the principal chemical combinations 
 of the foregoing. 
 
 Name 
 
 Chemical composition 
 
 Chemical 
 symbol 
 
 Carbonic oxide 
 
 Carbonic acid 
 or carbonic 
 anhydride. 
 
 Formed of carbon and oxygen, in 
 the proportion of twelve parts by 
 weight of the former to sixteen of 
 the latter, or C I2 + O 16 . 
 Formed of carbon and oxygen, in 
 the proportion of twelve parts by 
 weight of the former to thirty-two 
 of the latter, or C 12 + O 32 . 
 
 CO 
 CO, 
 
 The above are the products of the combustion of carbon 
 in oxygen, the former being the result of imperfect, the 
 latter of perfect combustion. 
 
 Name 
 
 Chemical composition 
 
 Chemical 
 symbol 
 
 Water 
 
 P'ormed of hydrogen and oxygen, in 
 the proportion of two parts by 
 weight of the former to sixteen of 
 the latter, or H 2 -i-O 16 . 
 
 HP 
 
Constituents of Fuel. 
 
 349 
 
 The above is the product of the combustion of hydrogen 
 in oxygen. 
 
 Name 
 
 Chemical composition 
 
 Chemical 
 symbol 
 
 Olefiant gas 
 Marsh gas 
 
 Formed of carbon and hydrogen, in 
 the proportion of twelve parts by 
 weight of carbon two of hydrogen, 
 or C 12 + H 2 . 
 Formed of carbon and hydrogen, in 
 the proportion of twelve parts by 
 weight of carbon to four of hydro- 
 gen, or C 12 + H 4 . 
 
 CH, 
 CH 4 
 
 The two above are gaseous forms of the large family of 
 hydrocarbons, which exist largely in fuel in the solid and 
 liquid states also. To this category belong the vegetable and 
 mineral oils, animal fats, and the bituminous portions of coal. 
 
 Fuel. Ordinary fuel is composed chiefly of carbon, 
 hydrogen, oxygen, and mineral matters, or of chemical com- 
 binations of these three elements. Its heating power, as 
 before mentioned, depends on the relative proportions of 
 the two first elements, and on the manner in which they 
 are supplied with oxygen. 
 
 The heat evolved by a pound of hydrogen when burnt 
 with oxygen so as to form water is 62,032 thermal units, 
 which is sufficient to evaporate 64-2 Ibs. of water from and 
 at 212. It requires for its combustion 8 Ibs. of oxygen. 
 
 The heat evolved by a pound of carbon when burnt 
 completely is 14,500 thermal units, which is sufficient to 
 evaporate 15 Ibs. of water from and at 212. The amount 
 of oxygen required to effect the combustion is 2 Ibs. 
 
 When imperfectly burned so as to form carbonic oxide 
 the quantity of heat evolved is only 4,400 units, equivalent 
 to an evaporative power of 4-55 Ibs. of water from and at 
 212. The amount of oxygen required is ij Ibs. 
 
 In every case the oxygen is obtained from the air, which 
 is a mechanical mixture of nitrogen and oxygen in. the 
 
350 The Steam Engine. 
 
 proportion of 77 parts by weight of the former to 23 of the 
 latter. The nitrogen plays no part in the combustion, 
 except that it mingles with the products of combustion, and 
 reduces their temperature. 
 
 It will be noticed that the heat of combustion of hydro 
 gen is about four times as great as that of carbon, and 
 consequently those fuels which contain a relatively large 
 quantity of hydrogen, such as the hydrocarbons, possess 
 the greatest evaporative power ; for the heating power of a 
 compound of these two elements is in most cases nearly 
 equal to the sum of the heating powers of the constituents. 
 
 In making an exact estimate of the calorific value of 
 fuel, it is necessary to take account of the heat lost by 
 breaking up any existing chemical compounds ; for, just as 
 the chemical union of carbon and oxygen, or of carbon and 
 hydrogen, produces heat, so the separation of, say carbon 
 from hydrogen, requires the expenditure of heat. If, then, 
 a fuel contains hydrocarbons, which, during the combus- 
 tton, are broken up into their constituent proportions of 
 carbon and hydrogen, and each of these latter then com- 
 bined with oxygen so as to form carbonic acid and water, 
 we must not calculate the heat produced by the combina- 
 tion as being the same as if equal quantities of free carbon 
 and free hydrogen were so combined. The proper way to 
 proceed is to calculate first the heat that would be pro- 
 duced supposing the substances were all originally in the 
 free or uncombined state, and then to subtract from this 
 quantity the heat required in order to dissociate the 
 hydrogen from the carbon. The latter quantity is always 
 equal to the heat which would be produced by the com- 
 bination of the equivalent quantities of free carbon and 
 hydrogen. Thus, for example, one pound of marsh gas 
 consists of three-quarters of a pound of carbon and one- 
 quarter of a pound of hydrogen. If these constituents 
 were in the uncombined state their combustion with oxygen 
 would yield the following quantities of heat : 
 
Heat of Combustion of Fuel. 351 
 
 Carbon, Ib. . 14,500 x \ = 10,875 thermal units. 
 
 Hydrogen, j Ib. . . 62,032x^=15,508 
 
 Total . . . =26,383 
 
 But experiments prove that the heat developed by the 
 combustion of one pound of marsh gas in oxygen is only 
 23,582 thermal units, thus leaving a deficiency of 2,801 
 units due to the heat absorbed in splitting up the chemical 
 compound of carbon and hydrogen. In those compounds 
 of carbon and hydrogen in which only two equivalents of 
 hydrogen are combined with one of carbon, the heat of 
 combination is so little that it is not necessary to take 
 account of it in computing theoretically the calorific value 
 of fuel. When a fuel contains oxygen, in addition to carbon 
 and hydrogen, it is found that so much of the hydrogen as 
 would be required to combine with the oxygen present in 
 order to form water, must be left out of account in calculat- 
 ing the calorific effect. Any excess of hydrogen above this 
 quantity must, however, be taken into consideration. 
 
 The fuels in most common use are coal, coke, peat, 
 wood, and in some countries, such as South Russia, mineral 
 oils. Of these coal is by far the most important ; it is 
 therefore the only kind of fuel which will be considered in 
 detail in this chapter. 
 
 There are numerous varieties of coal found in this 
 country, which differ from each other in appearance, in 
 chemical constitution, and in their behaviour when under- 
 going combustion. Of these the principal varieties are 
 anthracite, dry bituminous, and caking bituminous coals. 
 Anthracite is found chiefly in South Wales. Chemically it 
 consists almost entirely of pure carbon. It burns without 
 flame or smoke. It is a very difficult coal to ignite, and 
 unless gradually heated it splits up, when thrown on the fire, 
 into small pieces. 
 
 Dry bituminous coal is the most useful fuel for steam 
 generation. It consists chemically of carbon, hydrogen, 
 
352 
 
 The Steam Engine. 
 
 oxygen, and mineral matter which forms ash. It is lighter 
 than anthracite, and burns easily with very little smoke. 
 
 Caking bituminous coal contains less carbon than the 
 foregoing, and more hydrogen and oxygen. It is called 
 caking because it softens when exposed to heat. It burns 
 easily with a good deal of smoke. The following table * 
 gives the chemical composition and theoretical heating 
 power of various kinds of coal. The theoretical heating 
 power is calculated in the manner already explained. The 
 practical heating power differs very considerably from the 
 
 Name of fuel 
 
 Chemical constituents 
 
 Heat of 
 
 combustion 
 of fuel in 
 thermal 
 units 
 
 Evaporative 
 power of one 
 pound of the 
 fuel in pounds 
 of water from 
 and at 212 
 
 C 
 
 H 
 
 
 
 I. Charcoal from wood 
 
 0-93 
 
 
 
 13,500 
 
 14 
 
 peat. 
 
 
 
 
 
 
 
 1 1, 600 
 
 12 
 
 II. Coke, good . 
 
 0-94 
 
 
 
 
 
 I3,62O 
 
 14 
 
 ,, middling 
 
 0-88 
 
 
 
 
 
 12,760 13-2 
 
 bad . 
 
 0-82 
 
 
 
 
 
 11,890 
 
 I2'3 
 
 III. Coal: 
 
 
 
 
 
 
 I. Anthracite . 
 
 0-915 
 
 0-035 
 
 0-026 
 
 I5>225 
 
 1575 
 
 2. Dry bituminous 
 
 0-90 
 
 0-04 
 
 O'O2 
 
 15,370 : IS'Q 
 
 3- 
 
 0-87 
 
 0*04 
 
 0-03 
 
 14,860 
 
 I5-4 
 
 4- 
 
 0-80 
 
 0-054 
 
 0-016 
 
 14,790 
 
 i5'3 
 
 5- 
 
 077 
 
 0-05 
 
 0-06 
 
 13,775 
 
 14-25 
 
 6. Caking 
 
 0-88 
 
 0-052 
 
 0-054 
 
 15,837 
 
 16-0 
 
 7. . 
 
 0-8 1 
 
 0-052 
 
 0-04 
 
 14,645 
 
 I5-I5 
 
 8. Cannel. 
 
 0-84 
 
 0-056 
 
 0-08 
 
 15,080 
 
 15-6 
 
 9. Dry long flaming . 
 
 077 
 
 0-052 
 
 0-15 
 
 M,I95 
 
 I3-65 
 
 10. Lignite 
 
 070 
 
 0-05 
 
 O'2O 
 
 n,745 
 
 12-15 
 
 IV. Peat, dry 
 
 0-58 
 
 0-06 
 
 0-3I 
 
 9,660 
 
 10-0 
 
 Peat containing 25 
 
 
 
 
 
 
 per cent, moisture 
 
 ~ 
 
 
 
 
 
 7,000 
 
 7-25 
 
 V. Wood, dry . 
 
 0-50 
 
 
 
 
 
 7,245 
 
 7-5 
 
 Wood containing 20 
 
 
 
 
 
 
 per cent, moisture 
 
 
 
 
 
 
 
 5,600 
 
 5-8 
 
 VI. Mineral oil from . 
 
 0-84 
 
 0-16 
 
 o- 
 
 21,930 
 
 22-7 
 
 to . 
 
 0-85 
 
 0-15 
 
 o- 
 
 2i,735 
 
 22-5 
 
 1 Taken from the Journal of the Royal United Service Institution. 
 Vol. XL 
 
Evaporating Power of Fuel. 353 
 
 theoretical, and depends chiefly on the stoking, and on the 
 furnace being suitably made to develope complete combus- 
 tion. The proper method of designing furnaces will be 
 explained later on, but it may here be mentioned that with 
 the best of fuel and the most suitable of furnaces it is 
 possible by bad stoking to obtain the most indifferent 
 results. For instance, if the fuel is laid on in such a 
 manner that air in sufficient quantities cannot reach it, the 
 coal will be partly distilled instead of being burnt ; the more 
 volatile constituents, such as the hydrocarbons, will be 
 driven off in the shape of unburnt gas, and a large propor- 
 tion of the carbon proper will burn incompletely, forming 
 carbonic oxide instead of carbonic acid, its heating power 
 being thus reduced by more than 70 per cent. 
 
 The heating powers in the above table are calculated 
 on the supposition that one pound of pure carbon is capable 
 of evaporating fifteen pounds of water from 212. This is 
 an experimental result arrived at by chemists, and is greatly 
 in excess of anything that has yet been realised in steam 
 boilers. 
 
 Weight and temperature of the products of combustion. 
 The temperature of the products of combustion of fuel 
 depends upon their weight and specific heat. The weight 
 of the products of combustion depends upon the quantity 
 of air which is supplied to the fuel. As stated above, one 
 pound of carbon requires for its perfect combustion two 
 and two-thirds pounds of oxygen, or about twelve pounds 
 of air. When imperfectly burned it requires one and one- 
 third pounds of oxygen, or six pounds of air. One pound 
 of hydrogen gas consumes eight pounds of oxygen, or 
 thirty-six pounds of air. 
 
 It is found, however, that in practice more air than the 
 above quantities must be supplied to the fuel in order to 
 effect complete combustion. The extra quantity required 
 depends upon the nature of the draught. Thus when the 
 draught is produced bv a chimney it is usual to estimate 
 
 A A 
 
354 Tfo Steam Engine. 
 
 that twice the theoretical quantity is required, i.e. twenty- 
 four pounds of air per pound of carbon. When the draught 
 is artificial, such as that produced by a blower, or by a fan, 
 or by the blast-pipe, one and a half times the theoretical 
 quantity, or eighteen pounds of air per pound of carbon, 
 is usually required. Although common coal is a compli- 
 cated mixture of carbon, hydrogen, and oxygen, no serious 
 error will be committed by estimating the quantity of air 
 required for its combustion on the supposition that it is pure 
 carbon. 
 
 From the above it will be evident that, even with the 
 same fuel, the temperature of the products of combustion 
 will vary according to the nature of the draught. Thus 
 taking, again, the case of pure carbon, burnt under an 
 artificial blast, and, therefore, requiring eighteen pounds of 
 air per pound of fuel, we have the total weight of the pro- 
 ducts of combustion = 18 + 1 = 19 pounds. The total heat 
 of combustion is, as stated above, 14,500 units. The mean 
 specific heat of the products is, according to Rankine, 
 237 for constant pressure, and the temperature is found 
 by dividing the total heat of combustion by Hhe weight 
 multiplied by the specific heat. Thus, in the present in- 
 
 stance, the temperature =- 
 19 x -237 
 
 If the draught were produced by means of a chimney, 
 so that twenty- four pounds of air would be required, instead 
 
 of eighteen, the temperature would only be = 2,440. 
 
 2 5 * ' 2 37 
 
 On the other hand, if it were possible to burn the fuel 
 completely with only the theoretical quantity of air neces- 
 sary, viz. twelve pounds, the weight of the products of 
 combustion would be only thirteen pounds, and the tem- 
 
 perature - ^= - =4,580, or very nearly double the tem- 
 
 perature which is usually obtained in practice. It will be 
 Been presently, when considering the waste of fuel in the 
 
Air required for Combustion. 355 
 
 gaseous state, that this question of initial temperature and 
 weight of the products of combustion assumes an important 
 aspect. 
 
 In practice it is found that a pound of coal falls very 
 far short of the evaporative power stated in the table. The 
 reasons for this are twofold. First, the fuel is wasted in 
 various ways, which will presently be enumerated ; and 
 second, the boiler is unable to abstract from the fuel all 
 the heat which actually is generated and to convey it to the 
 water, so the residue passes up the chimney unutilised. 
 
 Waste of fuel. The ways in which fuel is wasted are 
 various. Many kinds of coal, such as anthracite and dry 
 steam coal, are extremely brittle when exposed suddenly to 
 great heat, and small splinters are broken off which fall 
 through between the bars of the grate into the ashpit. In 
 the majority of cases, however, the great waste takes place 
 not so much in the solid as in the gaseous state. In an 
 ordinary coal fire, kindled from below, the upper layers of 
 fuel are heated through long before they become incandes- 
 cent. When thus warmed, the coal is partially distilled, 
 instead of being burnt, and many of its most valuable 
 constituents are driven off in a gaseous state, and escape up 
 the chimney unburnt, unless special provision is made to 
 mingle fresh air with the gases as they arise, and to burn 
 them, as it were, above the fuel. 
 
 An insufficient supply of air to the fuel itself is often a 
 source of very great waste. It has been stated before that 
 if only enough oxygen be present to burn the carbon into 
 carbonic oxide, instead of into carbonic acid, the units of 
 heat so generated will be only 4,400 per pound of carbon, 
 instead of 14,500. Carbonic oxide is a perfectly colourless 
 gas, and its formation in very large quantities may easily 
 escape detection. If mingled, however, with a sufficient 
 supply of fresh air, and suitably ignited, it will burn into 
 carbonic acid, and in so doing will give out the missing 
 iQjToo units of heat. 
 
 A A 2 
 
356 The Steam Engine. 
 
 The formation of srnoke is also a most fruitful, as it is 
 one of the most common sources of waste. Smoke is pure 
 unburnt carbon in a finely divided state which floats about 
 in the hot gases and air proceeding from the fire, so that 
 whenever we see dense volumes of smoke escaping from 
 a chimney, we know that it represents so much valuable 
 fuel absolutely thrown away, beyond the reach of recovery. 
 Smoke when once formed is extremely difficult to ignite, 
 and the greatest art of good firing consists in its prevention. 
 Coals which are rich in hydrocarbons are also the most 
 fruitful smoke producers. It is supposed that these volatile 
 hydrocarbons when^driven off at a considerable temperature, 
 in the manner described, evolve free carbon before they are 
 mingled with the air above the fuel, and becoming cooled 
 down by contact with the air, the suspended particles of 
 carbon show themselves in the form of smoke. Many 
 arrangements have been contrived for mingling fresh air 
 with the gases arising from the fuel in order to effect their 
 combustion. Some of these will be referred to hereafter, when 
 the practical details of boilers come under consideration. 
 
 Fuel is often largely wasted in forming the draught 
 which feeds the furnaces with air. The draught is pro- 
 duced either by means of a chimney or by some more 
 artificial blower, such as the steam blast-pipe or the 
 revolving fan. In the case of a chimney it is found that 
 the best temperature for the ascending gases is about 600, 
 whereas the temperature of the fire is about 2,440 above 
 that of the outside air. Consequently about one- fourth of 
 the total heat of combustion is wasted in forming the 
 draught. Hence it appears that a chimney is a most waste- 
 ful expedient, for it involves the necessity of supplying the 
 fuel with a double allowance of air, and in order to maintain 
 the draught efficiently it carries off this air at a high tem- 
 perature. With a blast-pipe or fan it is not necessary, as 
 far as the draught is concerned, that the escaping gases 
 should have any higher temperature than that of the 
 
Waste of Fuel. 357 
 
 atmosphere, and, moreover, the quantity of air which must 
 be supplied to the fuel is one fourth less than when a 
 chimney is employed, 
 
 Radiation and conduction are usually set down among 
 the causes of waste of heat, but when the fire is properly 
 enclosed, and the boiler surrounded with non-conducting 
 materials, losses from this cause may be rendered extremely 
 small. 
 
 The inability of the boiler to abstract all the heat which 
 the fuel gives out is a consequence of the nature of the 
 conduction of heat through the plates which separate the 
 water in the boiler from the fire. The rate at which con- 
 duction takes place between the plates depends, first, upon 
 the difference in temperature between the two sides of the 
 plate ; the greater the difference the quicker being the rate of 
 conduction ; second, upon the conductivity of the metal which 
 forms the plate ; and third, upon the thickness of the plate. 
 
 As regards the difference in temperature between the 
 sides of the plate, it is evident that when the temperature 
 on each side is the same no conduction of heat can take 
 place. If, for example, a boiler be used to form steam of 
 100 Ibs. pressure to the square inch, the temperature of this 
 steam and of the water from which it is formed is 337*5 ; 
 consequently the hot gases coming from the fire cannot be 
 cooled down below this point, and must at the least escape up 
 the chimney at this temperature ; and therefore all the heat 
 due to the difference between this temperature and that of 
 the atmosphere is of necessity wasted. As a matter of fact, 
 it is impossible to retain the hot gases long enough in 
 contact with the plates to allow of their temperature drop- 
 ping to ,that of the steam, and consequently the waste from 
 this source is considerably greater than what has been stated 
 above. This evil may be reduced, to a certain extent, by 
 introducing the comparatively cold feed water at that part 
 of the boiler where the gases are coldest, an arrangement 
 which is always carried out in carefully designed boilers. 
 
358 The Steam Engine. 
 
 From the above it will be readily understood how 
 important it is to reduce the supply of air to the fuel to 
 the minimum which is consistent with perfect combustion. 
 Any excess quantity of air, in the first place, reduces the 
 temperature within the furnace, and thus diminishes the rate 
 of conduction through the heating surface ; and, in the next 
 place, it augments the bulk of the gaseous products of 
 combustion, and thus makes it more difficult than it other- 
 wise would be for the heating surface to reduce the tempe- 
 rature of the products to that of the steam and water within 
 the boiler ; for it is self-evident that a given area of heating 
 surface is more efficient in abstracting the heat from a small 
 than from a large bulk of gases. 
 
 DESCRIPTION OF VARIOUS TYPES OF BOILERS. 
 
 The essential parts of all boilers are as follows : 
 
 A furnace, which contains the fuel to be burnt ; a water 
 receptacle, which contains the water to be evaporated ; a 
 steam space to hold the steam when generated ; heating 
 surface to transmit the heat from the burning fuel to the 
 water ; a chimney, or other apparatus to cause a draught 
 to the furnace and to carry away the products of com- 
 bustion ; and various fittings for supplying the boiler with 
 water, for carrying away the steam, when formed, to the 
 engine in which it is used ; for allowing steam to escape 
 into the open air when it forms faster than it can be used ; 
 for ascertaining the quantity of water in the boiler ; for 
 ascertaining the pressure of the steam, &c. 
 
 The forms of steam generators are most numerous, and 
 depend chiefly upon the purposes for which they are re- 
 quired. They may all be divided into three principal cate- 
 gories ; viz. stationary, locomotive, and marine boilers. 
 
 In stationary boilers the size and weight are of secondary 
 consideration, whereas for locomotive purposes, they are 
 paramount. 
 
Land Boilers. 
 
 359 
 
 The great majority of stationary boliers are cylindrical in 
 shape, because the cylinder is the best practical form for 
 strength as against internal pressure ; the ends are either 
 flat or else segments of spheres. It would be impossible 
 within the limits of this short work to describe all the 
 varieties of stationary boilers which have been contrived in 
 various countries, but a. few representative types will be con- 
 sidered. 
 
 STATIONARY LAND BOILERS. 
 
 Cylindrical boiler with external firing. The simplest of 
 all steam generators, and one which is now but seldom used 
 
 Fig. 151. 
 
 in this country, is illustrated 
 in longitudinal and transverse 
 section in figs. 151, 152. 
 It consists of a cylinder A, 
 formed of iron plate with 
 hemispherical ends BB, set 
 horizontally in brickwork C. 
 The lower part of this cylinder 
 contains the water, the upper 
 part the steam. The furnace 
 D is external to the cylinder, 
 being underneath one end. It consists simply of a series 
 
 Fig. 152. 
 
360 The Steam Engine. 
 
 of grate bars ee set in the brickwork at a convenient dis- 
 tance below the bottom of the boiler. 
 
 The sides and front of the furnace are walls of brick- 
 work, which, being continued upwards, support the end of 
 the cylinder. The fuel is thrown on to the bars through 
 the furnace door /, which is set in the front brickwork. The 
 air enters between the grate bars from below. The portion 
 beneath the bars is called the ashpit. The flame and hot 
 gases, when formed, first impinge on the bottom of the 
 boiler, and are then carried forward by the draught to the 
 so-called bridge 0, which is a projecting piece of brickwork, 
 which contracts the area of the flue ^, and forces all the 
 products of combustion to keep close to the bottom of the 
 boiler. Thence the gases pass along the flue ;z, and return 
 past one side of the cylinder in the flue m (fig. 152), and back 
 again by the other side flue m' to the far end of the boiler, 
 whence they escape up the chimney. This latter is provided 
 with a door or damper/, which can be closed or opened at 
 will, so as to regulate the draught. 
 
 This boiler has two great defects. The first is that the 
 area of heating surface, which is represented by those 
 portions of the flues which are bounded by the surface of 
 the cylindrical shell, is too small in proportion to the bulk 
 of the boiler. The second is that if the water contains 
 solid matter in solution, as nearly all water does to a 
 greater or less extent, this matter becomes deposited on the 
 bottom of the boiler, just where the greatest evaporation 
 takes place. The deposit, being a non-conductor, prevents 
 the heat of the fuel from reaching the water in sufficient 
 quantities, thus rendering the heating surface inefficient ; 
 and further, by preventing the heat from escaping to the 
 water, it causes the plates to become unduly heated, and 
 quickly burnt out. This defect is sometimes obviated by 
 placing within the shell of the cylinder a segment shaped 
 trough, shown in transverse section in fig. 153. The 
 trough a is fixed a few inches from the bottom of the boiler. 
 
Cylindrical Boiler. 
 
 361 
 
 In the water space below it the ebullition is most violent, 
 and the circulation of the water so rapid, that no deposit 
 can take place ; while within the trough the water is corn- 
 
 Fig. 153- 
 
 paratively quiet, and the mud consequently deposits itself 
 here and does no harm, and can easily be removed 
 periodically. 
 
 There is another defect belonging to this system of 
 boiler to which many engineers attach great importance, 
 viz. that the temperature in each of the three flues n, m, m' 
 is very different, and consequently the metal of which the 
 shell of the boiler is composed expands very unequally in 
 each of the flues, and cracks are very likely to take place 
 where the effects of the changes of temperature are most 
 felt. 
 
 Cornish boiler. The Cornish boiler obviates most of the 
 defects of the system just described. It consists also of a 
 cylindrical shell A (figs. 154, 155), with flat or semi -circular 
 ends. The furnace, however, instead of being situated 
 underneath the front end of the shell, is enclosed within it 
 in a second cylinder B, having usually a diameter rather 
 
362 
 
 The Steam Engine. 
 
 greater than half that of the boiler shell. The arrangement 
 of the grate and bridge is evident from the diagram. After 
 passing the bridge the flame and gases travel along through 
 the internal cylinder B, till they reach the back end of the 
 boiler ; they then return to the front again by the two side 
 flues ;//, ;;/, and thence back again to the chimney by the 
 bottom flue n. 
 
 Fig. 154- 
 
 In this form of boiler the heating 
 surface exceeds that of the last de- 
 scribed by an amount equal to the 
 area of the internal flue, while the 
 internal capacity is diminished by its 
 cubic contents ; hence for boilers of 
 equal external dimensions the ratio 
 of heating surface to mass of water 
 to be heated is greatly increased. 
 Boilers of this sort can, however, 
 never be made of as small diameters as the plain cylindrical 
 sort on account of the necessity of finding room inside, 
 below the water level, for the furnace and flue. The dis- 
 advantage attending the deposits in the plain cylindrical type 
 is, to a great extent, got over in the Cornish boiler ; for the 
 bottom, where the deposit chiefly takes place, is the coolest, 
 instead of being the hottest, part of the heating surface. 
 
 Fig- 155. 
 
Cornish Boiler. 363 
 
 The internal flue in the Cornish system is the hottest 
 portion of the boiler, and consequently undergoes a greater 
 linear expansion than the outside shell. The result is a 
 tendency to bulge out the ends, and when the boiler is out 
 of use the flue returns to its normal size, and thus has a 
 tendency to work loose from the ends to which it is riveted. 
 If the ends are too rigid to move, a very serious strain 
 comes on the joints of the flue. To remedy this the latter 
 is often made up of a number of short cylindrical lengths 
 jointed together by being riveted to a ring, the section of 
 which is shown in fig. 156. 
 This ring is intended to serve 
 as a spring, and to allow the 
 ends a, a' to approach or re- 
 cede from each other, when 
 undergoing change of tempe- 
 rature. It also serves to stiffen 
 the flue against the pressure from the outside, which tends 
 to collapse it. Another way of effecting the same end is 
 shown in the section of the Lancashire boiler, fig 159. 
 
 Even while in use, the flue of a Cornish boiler is liable 
 to undergo great changes in temperature, according to 
 the state of the fire. When this latter is very low, or when 
 fresh fuel has been thrown on, the temperature is a mini- 
 mum, and reaches a maximum again when the fresh fuel 
 commences to burn fiercely. 
 
 Lancashire boiler. To remedy this inconvenience, and 
 also in order to attain a more perfect combustion, Fair- 
 bairn contrived the double-flued, or Lancashire boiler, the 
 arrangement of the furnaces of which is shown in trans- 
 verse section in fig. 160. It will be observed that there 
 are two internal furnaces instead of one, as in the Cornish 
 type. These furnaces are sometimes each continued as a 
 separate flue to the other end of the boiler, as shown in 
 fig. 1 60 ; but as a rule they merge into one internal flue. 
 They are supposed to be fired alternately, and the smoke 
 
3^4 
 
 The Steam Engine. 
 
 and unburnt gases issuing from the fresh fuel are ignited 
 in the flue by the hot air proceeding from the other furnace, 
 the fuel in which is in a state of incandescence. Thus all 
 violent changes of temperature in the flue are avoided, and 
 the waste of fuel due to unburnt smoke is avoided, if the 
 firing is properly conducted. 
 
 The disadvantage of the Lancashire boiler is the dif- 
 ficulty of finding adequate room for the two furnaces 
 without unduly increasing the diameter of the shell. Low 
 furnaces are extremely unfavourable to complete com- 
 bustion, the comparatively cold crown plates, where they 
 are in contact with the water of the boiler, extinguishing 
 the flames from the fuel when they are just formed, while 
 the narrow space between the fuel and the crown does 
 not permit of the proper quantity of air being supplied 
 above the fuel to complete the combustion of the gases as 
 they arise. On the other hand, though this type of boiler 
 favours the distillation of the fuel and the formation of 
 
 j smoke, it supplies the 
 means of completing the 
 combustion afterwards by 
 means of the hot air from 
 the second furnace. 
 
 Another disadvantage 
 which the Lancashire 
 boiler has in common 
 with all steam generators 
 having circular internal 
 furnaces, is the danger 
 to which the flues are 
 exposed of collapsing, 
 because of the pressure 
 which they have to sustain from without. There are many 
 ways of getting over this evil. 
 
 In the Galloway form of boiler the flue is sustained and 
 stiffened by the introduction of numerous conical tubes, 
 
 Fig. 157- 
 
Lancashire Boiler. 
 
 365 
 
 Fig. 158. 
 
 flanged at the two ends and riveted across the flue. These 
 tubes, a sketch of two of which is given in fig. 157, are in 
 free communication with the water of the boiler, and, 
 besides acting as stirTeners, they also serve to increase the 
 heating surface and to pro- 
 mote circulation. Other 
 methods of strengthening 
 the flues have already been 
 described under the head 
 of Cornish boilers. 
 
 Mr. Fox of Leeds cor- 
 rugates the flues in the 
 manner shown in fig. 158. 
 This plan increases the re- 
 sisting power of the flues enormously, and, moreover, in- 
 creases the heating surface and provides for the contraction 
 and expansion of the flue. 
 
 The annexed illustrations give all the principal details 
 of a Lancashire boiler fitted with Galloway tubes. Fig. 159 
 represents a longitudinal section, and fig. 160 shows to a larger 
 scale, an end view of the front of the boiler with its fittings, 
 and also a transverse section. The arrangement of the 
 furnaces, flues, and the Galloway tubes a a a is sufficiently 
 obvious from the drawings. The usual length of these 
 boilers is 27 feet, though they are occasionally made as short 
 as e 2 1 feet. 
 
 The minimum diameter of the furnaces is 33 inches, 
 and in order to contain these comfortably the diameter of 
 the boiler should not be less than 7 feet. The ends of the 
 boiler are flat, and are prevented from bulging outwards by 
 being held in place by the furnaces and flues, which stay the 
 two ends together, and also by the so-called gusset stays, e , 
 fig. 159, which are explained in greater detail on page 397. 
 Great care should be taken to keep the lower ends of the 
 gusset stays 8 or 9 inches away from the nearest points of 
 furnaces and flues, otherwise when the latter expand under 
 
366 
 
 The Steam Engine. 
 
Lancashire Boiler. 
 
 367 
 
368 The Steam Engine. 
 
 the influence of the heat the boiler ends will be so stiff, that 
 instead of slightly bulging out to accommodate themselves 
 to the increased length of the flues, they will be very severely 
 strained in the neighbourhood of the angle irons by which 
 they are fastened to the flues, The result of this local strain- 
 ing will be the opening of the grain of the iron, and its 
 subsequent rapid corrosion. For the same reason the 
 thickness of the end plates should not exceed a half inch 
 for pressures up to 75 Ibs. per square inch. In addition to 
 the gusset stays, the flat ends of the boilers frequently have 
 longitudinal rods to tie them together. One of these is 
 shown at AA, fig. 159. 
 
 The steam is collected in the pipe S, which is perforated 
 all along the top so as to admit the steam, and exclude the 
 water spray which may rise from the surface during ebul- 
 lition. The steam thence passes to the stop-valve T, out- 
 side the boiler, and thence by the steam pipes to the engines. 
 
 There are two safety valves on the top of the boiler, one, 
 B (fig. 159), being of the dead weight type explained on page 
 404, and the other, C, being a so-called low-water safety valve. 
 It is attached by means of a lever and rod to the float F, 
 which ordinarily rests on the surface of the water. When, 
 through any neglect, the water sinks below its proper level, 
 the float sinks also, and causes the valve to open, thus 
 allowing steam to escape and giving an alarm. 
 
 M is the man-hole, with its covering plate, which admits 
 of access to the interior of the boiler, and H is the mud-hole 
 by which the sediment which accumulates all along the 
 bottom is raked out. Below the front end and underneath 
 are shown the pipe and stop-valve by which the boiler can 
 be emptied or blown off. 
 
 On the front of the boiler (fig. 160) are shown the pres- 
 fure gauge, the water gauges, and the furnace doors, which 
 are described in detail on pages 401, 407, 413. K is the 
 seed-pipe ; RR, a pipe and cock for blowing off scum. In the 
 front of the setting, fig. 1 60, are shown two iron doors by which 
 
Lancashire and Tubulous Boilers. 369 
 
 access may be gained to the lower external flues for clearing 
 purposes. 
 
 In the Lancashire boiler it is considered advisable to 
 take the products of combustion, after they leave the 
 internal flues, along the bottom of the boiler, and then back 
 to the chimney by the sides. When this plan is adopted 
 the bottom is kept hotter than would otherwise be the case, 
 and circulation is promoted, which prevents the coldest 
 water from accumulating at the bottom. If this precaution 
 be neglected, the boiler is very apt to strain locally, from 
 the fact of the top and sides being hotter, and consequently 
 expanding more than the bottom. The result is that the 
 lower portions of the transverse seams of rivets give way. 
 
 The principal dimensions and other particulars of the 
 boiler shown in figs. 159 and 160 are as follows : 
 
 Steam pressure . . . -75 Ibs. per sq. inch 
 
 Length 27 feet 
 
 Diameter . . . . . 7 ,, 
 
 Weight (total) . . . . 15^ tons 
 
 Shell plates . . . . . v inch 
 
 Furnace diameter . . 33 inches 
 
 Furnace plates . . . inch 
 
 Grate area . . . . 33 sq. feet 
 Heating surface : 
 
 In furnace and flues . . . 450 ,, 
 
 In Galloway pipes . . 30 ,, 
 
 In external flues . . . 370 ,, 
 
 Total. . . . 850 ,, 
 
 Fuel consumption, 17 to 23 Ibs. of coal per sq. ft. of grate per hour. 
 Water evaporated per Ib. of coal, at and from 212, from io to 
 1 1 Ibs. with the help of a feed -heater. 
 
 TUBULOUS BOILERS. 
 
 The above term is applied to a class of boilers in which 
 the water is contained in a series of tubes, of comparatively 
 small diameter, which communicate with one another and 
 with a common steam-chamber. The flame and hot gases 
 
 B B 
 
370 
 
 The Steam Engine. 
 
 from the furnace circulate between the tubes, and are 
 usually guided by baffle plates or partitions, so as to act 
 equally on all portions of the tubes. There are many 
 varieties of this type of boiler. Fig. 161 illustrates Root's 
 patent boiler. Each tube is screwed at either end into a 
 square cast-iron head, and each of these heads has two 
 
 Fig. 161. 
 
 openings, one communicating with the tube below, and the 
 other with the tube above. The communication is effected 
 by means of hollow cast-iron caps shown at the ends of the 
 tubes. The caps have openings in them corresponding with 
 the openings in the tube heads to which they are bolted, the 
 joints being made by india-rubber washers. 
 
Locomotive Boilers. 371 
 
 LOCOMOTIVE BOILERS. 
 
 The essential features of locomotive boilers are dictated 
 by the duties which they have to perform under peculiar 
 conditions. The size and weight are limited by the fact 
 that the boiler has to be transported rapidly from place to 
 place, and also that it has to fit in between the frames of the 
 locomotive ; while, at the same time, the pressure of the 
 steam has to be very great, in order that with comparatively 
 small cylinders the engine may develope great power ; 
 moreover, the quantity of water which has to be evaporated 
 in a given time is very considerable. To fulfil these latter 
 conditions a large quantity of coal must be burned on a fire- 
 grate of limited area ; hence intense combustion is necessary 
 under a forced blast. To utilise advantageously the heat 
 thus generated, a large heating surface must be provided, 
 and this can only be obtained by passing the products of 
 combustion through a great number of tubes of small 
 diameter. The manner in which these conditions are 
 carried out in practice will be best understood by reference 
 to the accompanying illustrations (figs. 162 to 165 ). Fig. 162 
 is a longitudinal vertical section of a boiler of a modern 
 locomotive. Fig. 163 is a vertical transverse section, half 
 through the fire-box, and half through the cylindrical body 
 of the boiler showing the tubes in section. Fig. 164 is half 
 a section through the smoke box and funnel, and half an 
 elevation of the front end of the locomotive, showing the 
 smoke-box door. Fig. 165 is a horizontal section through the 
 fire-box, showing some of the fire-bars in plan. The furnace 
 (figs. 162, 163, 165), or fire-box as it is called, is a box-shaped 
 casing of copper. In horizontal plan the fire-box is also rect- 
 angular, the width in this example being 40 inches, and the 
 length from front to back 60 inches, and the height from top 
 of fire-bars to crown 64^ inches. The fire-bars B usually form 
 the bottom of the box, though in some boilers there is a water 
 
 B B 2 
 
372 
 
 The Steam Engine. 
 
 space provided with two openings for the admission of air 
 below the bars. The lower edge of the furnace door is about 
 
Locomotive Boilers. 
 
 373 
 
 30 inches from the grate. Within the fire-box and below the 
 tubes a bridge or arch of fire-brick is often built, which serves 
 
 Fig. 163. 
 
 Fig. 164. 
 
 Fig. 165. 
 
374 The Steam Engine. 
 
 to deflect the products of combustion, which would otherwise 
 rush direct into the tubes, and causes them to impinge on the 
 sides and crown. The fire-box is enclosed completely within 
 the body of the boiler, and consequently the four sides, and 
 also the top or crown, are available as heating surface. The 
 sides and top, being flat, would quickly collapse under the 
 pressure of the steam unless special provision were made to 
 stiffen them. The plan invariably adopted is to connect the 
 sides of the fire-box with the outside shell of the boiler by a 
 number of short bolts or stays CCC, screwed and riveted into 
 each, as shown in fig. 163. The shell of the boiler exterior to 
 the fire-box is also in plan a rectangular box made of wrought 
 iron plates. The tendency of the steam being to bulge the 
 shell outwards, and the sides of the fire-box inwards, the two 
 pressures neutralise each other in the stays, which latter are of 
 course put into a state of tension. They are clearly shown in 
 the drawing, and are spaced about four inches apart all over 
 the flat surfaces. The portion of the shell immediately over 
 the crown of the fire-box is not flat, but semi-circular. It 
 would consequently be often inconvenient to stay these two 
 surfaces together in the manner described. The crown of 
 the fire-box is therefore stiffened in a different manner, and 
 usually by means of stout girders, though in the case of the 
 boiler under description, the crown is supported by stays EE 
 hanging from the shell plate. 
 
 The remainder of the shell of the boiler consists of a 
 cylindrical barrel united to the rectangular portion sur- 
 rounding the fire-box. This barrel is 4 feet 2-^ inches in 
 diameter and 10 feet n inches long. The products of com- 
 bustion from the furnace are conveyed through the barrel 
 to the smoke-box E, figs. 162 and 164, by means of 195 thin 
 tubes, made of brass, if inch in external diameter, and 10 feet 
 1 1 1 inches long. In this manner a very large heating surface 
 is obtained. Care must be taken in designing these tubes 
 to make their diameter sufficiently large to carry off the 
 products of combustion with ease. In some of the earlier 
 
Locomotive Boilers. 375 
 
 locomotive boilers, the diameter was made too small, with 
 the object of getting as many tubes as possible into the 
 available space, and thus increasing the heating surface ; 
 but it was found that the narrow tubes offered a most serious 
 impediment to the escape of the smoke and gases, and they 
 were moreover, on account of their small diameter, very 
 liable to become choked by soot, so that this plan had to be 
 avoided. It was formerly the custom to make the tubes 
 much longer than shown in fig. 162, with the object of 
 gaining heating surface; but modern experience has shown 
 that the last three or four feet next the smoke-box were of 
 little or no use, because, by the time the products of com- 
 bustion reached this part of the heating surface, their tem- 
 perature was so reduced that but little additional heat could 
 be abstracted from them, The tubes, in addition to acting 
 as flues and heating surface, fulfil also the function of stays 
 to the flat end of the barrel of the boiler, and the portion 
 of the fire-box opposite to it. They always tend to work loose 
 and consequently to leak at the tube plates (T,T, fig. 162), 
 because they expand and contract more than the outside 
 shell. They must therefore be very securely fastened. This 
 is accomplished either by riveting the ends over the tube- 
 plates, and driving in ring ferrules, or else by expanding the 
 tube immediately behind the tube plates by an instrument 
 specially made for this purpose. In addition to the staying 
 power derived from the tubes, the smoke-box tube plate and 
 the front shell plate F are stayed together by several long 
 rods, or else the ends are strengthened by gusset stays CCC. 
 In the boiler under consideration the heating surface given 
 by the tubes is 964 square feet in area, while the sides and 
 crown of the fire-box, or the direct heating surface, as it is 
 called, is 101 square feet. The grate area is 16 square 
 feet. 
 
 The forced draught in a locomotive boiler is obtained 
 by causing the steam from the cylinders after it has done its 
 work to be discharged into the chimney by means of a pipe 
 
376 The Steam Engine. 
 
 R (figs. 162, 164) called the blast-pipe. The lower portion of 
 the blast-pipe consists of two branches, one in communica- 
 tion with the exhaust port of each cylinder. One of these 
 branches and one cylinder L are shown in section, fig. 164. 
 The most advantageous position for the mouth of the blast- 
 pipe is some few inches below the base of the funnel. As 
 each puff of steam from the blast-pipe escapes up the 
 chimney, it forces the air out in front of it, causing a partial 
 vacuum which can only be supplied by the air rushing 
 through the furnace and tubes. The greater the body of 
 steam escaping at each puff, and the more rapid the succes- 
 sion of puffs, the more violent is the action of the blast-pipe 
 in producing a draught, and consequently this contrivance 
 regulates the consumption of fuel and the evaporation of 
 water, to a certain extent automatically, because when the 
 engine is working its hardest, and using most steam, the 
 blast is at the same time most efficacious. The blast- 
 pipe is perhaps the most distinctive feature of the loco- 
 motive boiler, and the one which alone has rendered it 
 possible to obtain large quantities of steam from so small a 
 generator. 
 
 The chamber E (figs. 162, 164) into which the smoke and 
 other products of combustion issue on leaving the tubes is 
 called the smoke-box. It is provided with a door in front, 
 for giving access to the interior and to the tubes. The 
 chimney is placed on the top of the smoke-box. The steam is 
 either collected in a dome, on the top of the barrel, and 
 which contains the mouth of the steam pipe S, leading to the 
 cylinders ; or else, as in the case of fig. 162, a perforated 
 pipe, S, is used, which runs along the top of the steam space 
 in the barrel of the boiler. Unless these precautions were 
 taken, the steam would carry over quantities of spray into the 
 cylinders ; in other words, the boiler would prime. Priming, 
 besides being a great inconvenience, is also a source of waste 
 of heat, for the hot water carried over into the cylinders 
 is incapable of doing work itself, and, moreover, lowers 
 
Locomotive and Marine Boilers. 377 
 
 the temperature of the steam in contact with it, and in 
 this way may indirectly become a most prolific source of 
 waste. 
 
 On account of the oscillations to which the boilers of 
 locomotive engines are subject, weighted safety valves are 
 inadmissible, and springs are used instead to hold the valves 
 in place. A spring safety valve is described on p. 406. 
 
 MARINE BOILERS. 
 
 The boilers used on board steamships are of two 
 principal types. The older sort used for steam of com- 
 paratively low pressure, viz. up to 35 Ibs. per square inch, 
 is usually made of flat plates stayed together, after the 
 manner of the exterior and interior fire-boxes of a locomotive 
 boiler. Modern high-pressure marine boilers, constructed 
 for steam of 60 to 150 Ibs. per square inch, are circular or 
 oval in cross section, and are fitted with cylindrical interior 
 furnaces and flues like land boilers. 
 
 Figs. 1 66, 167 represent the general arrangement of the 
 older type of marine boiler, in longitudinal and transverse 
 sections. A A, fig. 167, are the grate bars ; B, the furnace 
 door ; D, the ashpit. After passing the bridge the hot air 
 and flame enter a large chamber E, called the back up-take, 
 thence they return through the tubes eee, to the front up- 
 take F and the chimney. The heating surface consists of 
 the sides and crown of the furnace, the sides of the back up- 
 take and the tubes. The front up-take F is provided with 
 doors G for giving access to the tubes and chimney for clean- 
 ing and repairs. The outside shell of this type of boiler is 
 rectangular box-shaped. Some of the stays are represented 
 at a a a in both views. 
 
 The general arrangements and construction of this type 
 of boiler are rendered clear by the illustrations. As the form 
 of the boiler contributes nothing to its strength, the latter is 
 maintained by staying all the opposite surfaces together in 
 
378 
 
 The Steam Engine. 
 
 the manner already described for the flat fire-boxes of loco- 
 motive boilers. 
 
 The use of high-pressure steam of from 60 to 150 Ibs. 
 
 per square inch in modern marine engines has necessitated 
 the abandonment of the type of boiler just described, as it 
 would not be safe when dealing with such great pressures 
 
 1 Figs. 1 66 and 167 are taken by permission from Mr. R. Sennett's 
 work on the Marine Steam Engine. 
 
Marine Boilers. 
 
 379 
 
 and large surfaces to depend solely on the strength of the 
 stays. Accordingly we find that modern marine boilers are 
 circular, or nearly so, in cross section, with flat ends. 
 
 Fig. 167. 
 
 Fig. 1 68 shows a front elevation and partial sections of a 
 pair of such boilers, together with their up-takes, steam-chest, 
 and other fittings ; and fig. 169 shows one of them in longi- 
 tudinal vertical section. It will be seen from these drawings 
 that there are three internal cylindrical furnaces in each end 
 of these boilers, making in all six furnaces per boiler. The 
 firing takes place at both ends. The flame and hot gases 
 from each furnace, after passing over the bridge, enter a flat- 
 sided rectangular combustion chamber, and thence travel 
 through tubes to the front up-take, and so on to the 
 chimney. The sides of the combustion chambers are stayed 
 to each other and to the shell plate of the boiler. The tops 
 
380 
 
 The Steam Engine. 
 
 are strengthened in the same manner as the crowns of loco- 
 motive fire-boxes already described. The flat-end plates of 
 the boiler shell are stayed together by means of long bolts, 
 
 which can be tightened up by means of nuts at their ends. 
 Access is gained to the up- takes for purposes of cleaning, 
 repair of tubes, &c. by means of doors on their fronts, just 
 above the furnace doors. The steam is collected in the 
 
Marine Boilers. 381 
 
 large cylindrical receivers shown above each boiler. The 
 material of construction is mild steel. 
 
 Fig. 169. 
 
 The following are the principal dimensions and other 
 particulars of one of these boilers : 
 
 Length from front to back, 20 ft. 
 Diameter of shell, 15 ft. 6 in. 
 Length of furnace, 6 ft. 10 in. 
 Diameter of furnace, 3 ft. 10 in. 
 Length of tubes, 6 ft. 9 in. 
 Diameter, 3^ in. 
 
 No. of tubes 516. 
 Thickness of shell plates, ~. 
 Thickness of tube plates, f . 
 Grate area, 126*5 SC L- fi- 
 ll eating surface, 4015 sq. ft. 
 Steam pressure, Solbs. per. sq.in. 
 
382 
 
 The Steam Engine. 
 
 There are many varieties of marine boilers, adapted to 
 suit special circumstances. Fig. 170 for instance is a sketch 
 of a modern boiler, which is only fired from one end, and is 
 in consequence much shorter in proportion to its diameter 
 than the type illustrated in fig. 1 68. The cross section is often 
 not circular. The sides are sometimes flat, and are prevented 
 from bulging by being stayed to each other. The top and 
 bottom are semicircular in shape. This form of section has 
 been adopted in order to save some of the space which is 
 wasted when the true circular shape is adopted, and which 
 can be ill spared on board ship. It will have been noticed 
 that the boiler illustrated in fig. 168 has a separate com- 
 bustion chamber for each of the six furnaces. This 
 
 arrangement is very good, 
 because if a tube gives way 
 in one chamber the fires in 
 the other furnaces are not 
 affected by it, but it is never- 
 theless not always adopted. 
 Sometimes, even in double- 
 ended boilers, all the fur- 
 naces have only one chamber 
 in common. The disad- 
 vantages of this plan are so 
 serious that it is now but 
 seldom adopted. Very often 
 the two opposite furnaces of 
 a double-ended boiler have 
 a chamber in common, and 
 in single-ended boilers with two furnaces we frequently 
 find the same arrangement. The number of furnaces de- 
 pends upon the diameter of the boiler shell, and the very 
 confined natural limits, which are set to the diameter of fur- 
 naces. If the latter are less than 36 inches, the crown of 
 the furnace is so low that a large proportion of the heating 
 value of the fuel is lost by the process of distillation. On 
 
 Fig. 170. 
 
Marine Boilers. 383 
 
 the other hand, if over 48 inches, the thickness of plates 
 necessary to give sufficient strength to the structure is so 
 great, that the metal would be liable to be burnt, and its 
 heat-transmitting powers would be greatly diminished. 
 Boilers over nine feet in diameter have generally two fur- 
 naces, those over thirteen to fourteen feet three, while the 
 very largest boilers used on first-class mail steamers, and 
 which often exceed fifteen feet in diameter, have four furnaces. 
 
 THE PROPORTIONS OF THE PARTS OF BOILERS. 
 
 In designing a boiler of a given type to furnish a certain 
 amount of steam in a given time, it is requisite to know the 
 following things : 
 
 The size of fire-grate necessary to burn the requisite 
 quantity of fuel in a given time with various kinds of blast. 
 
 The capacity of different sorts of heating surface to trans- 
 mit heat. 
 
 The area of heating surface required to evaporate the 
 given quantity of water in the given time. 
 
 The relative proportions of the cubic contents of the 
 boiler, which should be occupied by steam and water 
 respectively. 
 
 The quantity of water which a pound of fuel will convert 
 into steam of a given pressure depends upon the pressure 
 of the steam, the nature of the coal, and the efficiency of 
 the type of boiler. For the purposes of comparison, all 
 rates of evaporation at various pressures and various tempe- 
 ratures of feed are reduced to the corresponding rates at 
 and from 212. When the above data are known, it is easy 
 to fix the size of fire-grate necessary in order to effect the 
 combustion of the fuel. 
 
 Evaporative power of fuel in different types of boilers. 
 As a general rule, with fair average coal, it may be stated 
 that the following rates of evaporation are obtained with 
 the different types of boilers named : 
 
384 The Steam Engine. 
 
 Per Ib. of coal 
 Lancashire boiler using feed heater 9 to 10-5 Ibs. of water at and 
 
 from 212 
 Marine boiler of old type . -87 Ibs. of water at and from 
 
 212 
 Marine boiler of new type . .8-1 Ibs. of water at and from 
 
 212 
 
 Locomotive boiler . . . 9 to 12 Ibs. of water at and 
 
 from 212 
 
 With bad fuel, such as steamers have often to take in at 
 foreign ports, these figures will have to be reduced about 
 twenty per cent. ; on the other hand, with picked fuel they 
 may all be increased about fifteen per cent. 
 
 Fire grate area. A given grate area will burn very 
 different weights of fuel in a given time according to 
 the nature of the draught. Where the size of the boiler is 
 a matter of no importance, as in most land boilers, a slow 
 rate of combustion is maintained, with a natural draught, on 
 account of the saving in wear and tear of the furnaces. In 
 such cases a common rate is from ten to twenty pounds of 
 fuel per square foot of grate area per hour. On the other 
 hand, when the size of the boiler is limited by the circum- 
 stances, as in the case of locomotives and torpedo boats, a 
 rate of from forty to one hundred and twenty pounds per 
 square foot can be maintained by using a forced draught. 
 Most of the following figures are given in Rankine's ' Manual 
 of the Steam Engine.' 
 
 Per sq. ft. per hour 
 Slowest rate of combustion in Cornish boilers . 4 Ibs. 
 
 Ordinary rate of combustion in Cornish boilers . 10 Ibs. 
 
 ,, ,, in factory boilers . 12 to 1 6 Ibs. 
 
 ,, ,, in marine boilers . 15 to 24 Ibs. 
 
 Quickest rate of complete combustion of dry coal, 
 
 air coming through grate alone . . 20 to 23 Ibs. 
 
 Rate in locomotive boilers with blast-pipe . . 40 to 120 Ibs. 
 
 Ordinary rate of locomotive boilers with blast-pipe 65 Ibs. 
 
 The length of fire-grate is limited by the distance to 
 which a stoker can throw the coals back with accuracy. It 
 
Proportions of Boilers. 385 
 
 is usual to fix six feet as the utmost limit. The breadth of 
 the grate depends chiefly on the breadth or diameter of 
 the boiler, and on the arrangement of the furnaces. In 
 Lancashire boilers with two internal flues the breadth is 
 extremely limited. Narrow furnaces have the great disad- 
 vantage that they allow the fire, which is necessarily of small 
 bulk, to be chilled by the proximity of the cold sides of the 
 furnace, and allow but little room above the fuel for the 
 introduction of air to complete the combustion. 
 
 Grates of large area are difficult to cover evenly with the 
 fuel. As a consequence the fire is apt to burn through too 
 quickly in the thin places, and the air rushing in most where 
 it finds the easiest entrance, causes the imperfect and slow 
 combustion of the fuel wherever it is piled on too thickly. 
 
 Efficiency of heating surface. The capacity of the heating 
 surface to transmit heat to the water depends on the con- 
 ductivity and the thickness of the metal, also on the position 
 of the surface, and the difference in temperature between 
 the water in the boiler and the hot gases in the furnace. 
 The metals most commonly used to separate the water from 
 the fuel are wrought iron and steel. These materials are, 
 however, inferior to copper in conducting power, in the ratio 
 of about one to three, and for this reason the latter metal 
 is used to form the sides of furnaces in all cases where it 
 is necessary to obtain a high rate of evaporation from a 
 boiler of limited size. The inner fire-boxes in locomotives 
 form a case in point. For the tubes of locomotive and 
 many marine boilers brass is very generally used, both on 
 account of its high conducting power, and also because of 
 the facility with which tubes can be drawn from this 
 material ; but tubes of steel and wrought iron are also used, 
 especially in the boilers of the mercantile marine. The 
 most effective portions of the heating surface are the sides, 
 and especially the crown of the furnace and combustion 
 chamber, and the first foot or two of the flues or tubes. The 
 reason of this is that the products of combustion are much 
 
 c c 
 
386 The Steam Engine. 
 
 hotter at these parts than elsewhere, and the effects of radia- 
 tion are also most strongly felt in these portions of the 
 boiler. 
 
 In a boiler with horizontal flues and tubes the lower 
 portions of these latter are considered of no value as heat- 
 ing surface, because of the difficulty with which the steam 
 escapes from them. For this reason the effective value of 
 tube-heating surface is usually estimated to be only three- 
 fourths of the total area of the tubes. In estimating the 
 amount of heating surface in a boiler the surfaces of the 
 furnace below the fire-bars, and of the combustion chamber 
 below the bridge, and also of the tube-plate, farthest from 
 the flame are altogether omitted. 
 
 It is impossible to lay down general rules for the evapo- 
 rating power of a given area of heating surface ; for, as has 
 been stated above, so much depends on the temperature 
 which is maintained in the furnace, and also on the position 
 of the surface relatively to the hottest part of the fire. For 
 these reasons the effects of different portions of the heating 
 surface in evaporating the water are widely different, and 
 nothing but an average of effect can be taken. In designing 
 boilers it will, therefore, be safest to follow the proportions of 
 heating surface to grate area, in the various types, which 
 experience has proved to give the best results. The follow- 
 ing are the proportions in a few representative cases : 
 
 Lancashire boiler, ratio of grate area to heating surface I : 26 
 Lancashire boiler, including surface of feed-water 
 
 heater, ratio of grate area to heating surface . I I 44 
 
 Marine boiler, ratio of grate area to heating surface . I : 22 to 35 
 Locomotive boiler, ratio of grate area to heating 
 
 surface . . . . . . . i : 60 to 90 
 
 Roughly speaking, it may be said that for every foot of 
 heating surface in a Lancashire boiler 6'8 to 9 Ibs. of water 
 can be evaporated per hour, excluding the surface of the 
 feed-heater ; in marine boilers from 6 to 8 Ibs. ; and in loco- 
 motives from 10 to 15, for good average coal and ordinary 
 
Proportions of Boilers. 387 
 
 conditions. The higher figures apply to the case of the 
 boilers being forced. With the locomotive type of boiler 
 applied to torpedo boats 18 Ibs. of water have been evapo- 
 rated per square foot of heating surface, with a forced 
 draught equivalent to six inches of water. The ratio of 
 grate area to heating surface was, however, only i : 34, and 
 the water evaporated per pound of coal was consequently 
 very low, having been only about 6 Ibs. 
 
 As a general rule it may be stated that the proportion 
 of heating surface to fuel burnt is as follows : 
 
 Per Ib. of coal burnt per hour 
 
 For Lancashire boilers . . . I'l to 1-5 sq. ft. of surface 
 For modern marine boilers . . I to I -5 sq. ft. of surface 
 For modern locomotive boilers . -9 to 1-5 sq. ft. of surface 
 
 In order to obtain good evaporative results the higher 
 figure should be chosen, but little is gained by any further 
 increase beyond the allowance of i -5 square ft. per Ib. of 
 coal per hour. When the allowance is less than 7 square 
 ft. per Ib. of coal the results are distinctly uneconomical. 
 
 Cubic capacities of boilers of different types. The absolute 
 cubic capacity and the relative capacities of the water and 
 steam rooms in a boiler are determined very much by the 
 nature of the work which is expected to be done. Of course, 
 in the case of boilers of a portable nature the circumstances 
 limit the absolute capacity, but in the case of land boilers, 
 where the bulk is of no particular account, the cubic con- 
 tents are determined solely by the nature of the work to be 
 done. Thus in cases where steam is only required occa- 
 sionally, and where, when wanted, it must be raised with 
 great rapidity, the capacity of the boiler is necessarily small, 
 and the heating surface and grate area large, relatively to 
 the cubic contents ; but where steady continuous work is 
 required the capacity is always large. 
 
 In boilers of small capacity the greatest care must neces- 
 sarily be bestowed on the feed ; otherwise it would be 
 impossible to maintain a uniform steam pressure, and more- 
 
 c cs 
 
388 The Stecun Engine. 
 
 over, on account of the rapidity of evaporation, the upper 
 portions of the heating surface are liable to be denuded of 
 water, in which case serious damage is likely to ensue. In 
 boilers of small capacity and great evaporative power it is 
 usual to put a lead plug into the crown of the furnace, in 
 order that, if the water has been allowed to sink below the 
 level of this portion, the plug may melt, and allow the steam 
 to enter the furnace and extinguish the fire. 
 
 The absolute capacity of steam and water room in 
 cubic feet, per pound of water evaporated per hour in the 
 boiler, varies greatly in different types of boilers. The 
 following figures give the proportions adopted in a few cases 
 of the best examples. 
 
 Lancashire boilers . I cubic foot capacity for every 3 Ibs, of 
 
 water to be evaporated per hour 
 
 Marine boilers . . I cubic foot capacity for every 7 to 10 Ibs. 
 of water to be evaporated per hour 
 
 In the case of locomotive boilers the rate of evaporation 
 varies within such wide limits in the same boiler according 
 to the work that has to be done, that it is impossible to give 
 any general rule. 
 
 The relative cubic capacities of the steam and water 
 rooms also vary very considerably in the different types of 
 boilers. For instance, where high pressure is used, and 
 small quantities of steam are very frequently withdrawn 
 from the boiler, as in the case of locomotives, the steam 
 room need not be relatively so great as when large volumes 
 of steam are slowly withdrawn, in the case, for example, 
 of paddle engines. The effect of withdrawing a large vol- 
 ume of steam from a confined space is to lower the pressure 
 considerably ; at the same time the water in the boiler has 
 the temperature due to the higher pressure ; consequently, 
 when the pressure falls, the surplus heat in the water at once 
 generates immense volumes of steam, which rushing to the 
 surface carry large quantities of spray with them, and thus 
 give rise to serious priming. 
 
Proportions of Boilers. 389 
 
 In ordinary practice \ve find that in Lancashire boilers the 
 water occupies about three -fourths of the diameter of the 
 boiler. 
 
 In marine boilers the proportion varies from j? to J 
 of the total shell capacity, according as they supply quick- 
 running short-stroke screw engines, or slow and long stroke 
 paddle engines. It should be borne in mind that when 
 very steady running is required, and comparatively unskilled 
 stoking is all that can be had, it is imperative that there 
 should be not only a large steam room, but also a large water 
 room to back it up with. The importance of a large cubic 
 capacity of water as an equaliser of pressure lies in the fact 
 that the specific heat of water is so high, that whenever the 
 pressure tends to drop through the neglect of the firing, the 
 immense store of heat in the water, at the temperature due 
 to the higher pressure, is at once available for the immediate 
 generation of steam. 
 
 THE STRENGTH OF BOILERS. 
 
 Hollow cylinder pressed from within. In considering the 
 strength of boilers the first point to be examined is the 
 case of a hollow cylinder pressed 
 from within. Let the circle (fig. 
 171) represent the transverse 
 section of such a cylinder, which 
 is supposed to be filled with 
 steam of a pressure of P Ibs. to 
 the square inch. It is required 
 to find the stress on the shell of 
 the cylinder at any two points Fig> 1?Ie 
 
 AA, diametrically opposite. It is 
 
 evident that every inch of the circumference of the shell is 
 subjected to a pressure acting radially outwards from the 
 centre C, so that the total pressure acting on the semi- 
 circumference of a ring one inch wide=7r . r . P, where r= 
 
390 The Steam Engine. 
 
 the radius in inches. The force, however, which tends to 
 separate the metal at AA is not the whole radial pressure 
 TT . r . P, but the sum of the components of this force re- 
 solved in a direction at right angles to the diameter AA. 
 The sum of these components on a ring one inch in width 
 may be proved to be equal to the pressure on an area equal 
 to the diameter of the circle (2^) multiplied by the width 
 (one inch). Hence the sum = 2rP. The cross section of 
 the metal at AA has, therefore, to sustain a tension = 2rP ; 
 or at either of these points separately, the tension = rP. 
 
 Let the thickness of the shell at A be / inches, and let 
 the tensile strength of the metal be W Ibs. per square inch ; 
 then the total strength of the ring at A per inch run of 
 length of cylinder = W/ Ibs. ; and when the boiler is on the 
 
 W/ 
 
 point of bursting we must have W/= rP, or P =- . In 
 
 other words, the pressure P, which a cylindrical boiler will 
 support, is directly proportional to the strength and thick- 
 ness of the metal, and inversely proportional to the 
 diameter. This reasoning applies only to the case of the 
 thickness of the shell being small compared with the dia- 
 meter ; had the thickness at AA been considerable, as in 
 the case of hydraulic presses and heavy guns, we should 
 have had to argue in a different manner. 
 
 As an example, take the case of a cylindrical boiler of 
 wrought iron of 6 feet diameter, the plates being \ an inch 
 thick, and the steam pressure 80 Ibs. per square inch, the 
 tensile strength of the iron being 48,000 Ibs. per square inch. 
 We have rP = the tension at any point in the circumference 
 of the shell = 3 x 12 x 80 = 2,880 Ibs. per inch of length. 
 On the other hand, the strength of a ring one inch in length 
 
 and half an inch thick is i_> = 24,000 Ibs. ; that is to 
 
 say, the strength of the boiler in this case is about eight and 
 
 a half times greater than the stress brought to bear upon it. 
 
 In actual practice, however, we could not take the full 
 
Strength of Boilers. 391 
 
 strength of the metal, because the riveted joints are much 
 weaker than the solid plate. The tensile strength of a 
 double riveted joint, for such a boiler, would be about 34,000 
 Ibs. per square inch, and consequently the strength of the 
 
 boiler at the weakest part would be 34'_ j^ooolbs. per 
 
 inch of length ; that is to say, the strength would be about 
 six times greater than the stress. 
 
 Factors of safety. The number which expresses the ratio 
 of the strength of a boiler to the working strain is called the 
 factor of safety. Thus in the above example the factor of 
 safety is six. The proper factor of safety is a point not yet 
 fully settled. The number adopted by the Board of Trade 
 for the shells of marine boilers, subject to their inspection, 
 is five for the most favourable cases ; that is to say, when 
 materials, construction, and workmanship are all of the best. 
 In Lancashire boilers the factor four is considered sufficient 
 for the weakest strip in the boiler. The French Govern- 
 ment has fixed three as the factor in land boilers, and this low 
 number has been found to give perfect security. 
 
 In addition to the strength of the longitudinal section of 
 a boiler we must also consider the case of the transverse or 
 ring-shaped section. The stress in this instance is brought 
 to bear by means of the pressure of the steam on the two 
 ends, and tends to pull the shell out like a telescope. 
 
 No matter what the shape of the end, the pressure on it 
 tending to pull the boiler in two is exactly the same as if 
 the ends were flat. Taking, therefore, flat ends, and using 
 the same symbols as before, 
 
 we have -n-r 2 = the area of the end, 
 
 7r;- 2 P = the total pressure on the area, 
 
 2-n-r = the circumference of any transverse section, 
 
 and 27rr/= the area of such section. 
 The area 2-xrt has, therefore, to sustain the pressure 7rr 2 P. 
 
3Q2 The Steam Engine. 
 
 When the boiler is on the point of bursting in this manner 
 we must have, therefore, 
 
 ?/w P*- 
 
 -7r/ 2 P = 27rr/W. /. P = _ , or W =-1, whereas in the 
 r 2t 
 
 -p r 
 former case we had W = ; that is to say, W, or the 
 
 tensile strength of the metal, requires to be only half as 
 great to resist the transverse stress as the longitudinal. 
 
 Taking the same example as before, we have the area of 
 the end = 3-14159 X (3 X 12)2 = 407 1-5 square inches ; while 
 the pressure on the end = 4071-5 x 80 = 325,720 Ibs. Now 
 the area of the transverse section of the boiler to resist this 
 stress=3'i4i59 x 6x 12 x^= 113*1 square inches, the tensile 
 strength of which is 5,428,800 Ibs.; that is to say, the strength 
 is about sixteen and three quarter times greater than the 
 stress brought to bear on it. As before, however, we can- 
 not in practice consider the full strength of the plate, 
 because all boilers are made up of two or more rings riveted 
 together. On account, however, of the comparatively light 
 load which the joints have to bear, it is not considered 
 necessary to double rivet the joints. The strength of a 
 single riveted joint in the above example would be only 
 about 26,000 Ibs. per square inch, and consequently the 
 factor of safety would be between eight and nine. The 
 transverse strength of a cylindrical boiler with internal fur- 
 naces is of course much greater than in the above example, 
 for while the area of the ends is diminished by the transverse 
 area of the furnace or flue, the section of metal which resists 
 the stress is increased by the area of metal contained in a 
 transverse section of the flue. The way in which the strength 
 is calculated is so apparent from what has gone before that it 
 is unnecessary to give another example. See also pp. 398-9. 
 
 Strength of riveted joints. The principles of the con- 
 struction of riveted joints are fully explained in the treatise 
 on the ' Elements of Machine Design,' published in this 
 series of text-books. 1 It is here only necessary to state that, 
 
 1 Elements of Machine Design. By W. C. Unwin. 
 
Riveted Joints. 
 
 393 
 
 according to Fairbairn's experiments, the strength of a single 
 riveted joint when properly proportioned is 56 per cent, of 
 the strength of the plate, and that of a double riveted joint 
 70 per cent. Fairbairn's experiments were made on plates ^ 
 inch in thickness, and it seems highly probable that with 
 the much thicker plates now in use, and the consequent 
 alteration in the pitch and proportions of the rivets, his 
 figures can no longer be accepted as correct. They appear 
 to err in representing the strength of the joint as being 
 greater than it really is. 
 
 Single Riveted Joints. 
 
 Iron Plates, iron Rivets 
 
 Steel Plates, iron Rivets 
 
 Thick- ! 
 ness of 
 Plates 
 
 Diameter of 
 Rivets 
 
 Pitch of 
 Rivets 
 
 Effici- 
 ency of 
 Joints 
 
 Thick- 
 ness of 
 Plates 
 
 Dia- 
 meter of 
 Rivets 
 
 Pitch of 
 Rivets 
 
 Effici- 
 ency of 
 Joints 
 
 8 
 10 
 
 670= li- 
 
 i -82= ill 
 
 621 
 
 I 5 e 
 
 fi 
 
 1-54=14 
 
 552 
 
 f 
 
 735= 2 
 
 i-fr-ii 
 
 606 
 
 1 
 
 1 
 
 I-S8=l4 
 
 538 
 
 & 790= I* 
 
 i '94 -ill 
 
 59 8 
 
 
 
 u 
 
 i '66= iH 
 
 512 
 
 i 
 
 849= | 
 
 1-98 = 2 
 
 571 
 
 
 
 1 
 
 170= ii| 
 
 501 
 
 1 
 
 949= 11 
 
 2-08 = 2^ 
 
 '543 
 
 1 
 
 IS 
 
 i -80 ilf 
 
 472 
 
 * 
 
 i'4 = ITS 
 
 2-17 = 2& 
 
 521 
 
 3 
 4 
 
 14 
 
 1-89-iJ 
 
 450 
 
 1 
 
 I-I2 -!j. 
 
 2-25 = 2^ 
 
 500 
 
 1 
 
 ii 
 
 I-97--2 
 
 431 
 
 1 
 
 1-20 =l 
 
 2-33 = 2^ 
 
 485 
 
 1 
 
 ii 
 
 2-05 = 2^ 
 
 415 
 
 Double Riveted Joints. 
 
 1 ' J ! 3 
 
 75 
 
 1 
 
 f 
 
 *4 
 
 6 9 
 
 I 7 
 
 13 
 
 To 
 
 3^ 
 
 73 
 
 1 7 B 
 
 -ii 
 
 2-i 
 
 6 7 
 
 k 
 
 1 
 
 38 
 
 72 
 
 i 
 
 1 
 
 4 
 
 66 
 
 9 
 16 
 
 1 
 
 3^ 
 
 72 
 
 9 
 16 
 
 1 
 
 2^ 
 
 66 
 
 * 
 
 if 
 
 3^ 
 
 71 
 
 1 
 
 H 
 
 2f 
 
 64 
 
 1 
 
 i 
 
 3^ 
 
 69 
 
 3 
 4 
 
 ii 1 . 
 
 2| 
 
 6 1 
 
 1 
 
 4 
 
 31 
 
 66 
 
 1 
 
 4 
 
 *H 
 
 59 
 
 I 
 
 H 
 
 3^ 
 
 64 
 
 I 
 
 H 
 
 2^ 
 
 57 
 
394 
 
 The Steam Engine. 
 
 The above tables, and figs. 172 to 174, extracted from 
 Mr. Unwin's work, give the proportions, together with the 
 
 efficiency of riveted joints 
 that is to say, the ratio 
 of their strength to that of 
 the solid plate, for iron 
 and steel plates, and for 
 single and double riveting. 
 Fig. 172 illustrates a single 
 riveted lap joint; fig. 173 
 a similar butt joint, in both 
 single and double shear 
 
 Fig. 172. 
 
 Fig. 174. 
 
 o 
 
 o 
 
 Fig. 173- 
 
 that is to say, with single and double cover plates. Fig. 
 174 shows a double riveted lap joint of the usual dimen- 
 
Strength of Hollow Cylinders. 39 $ 
 
 sions. All dimensions are given in terms of the diameter 
 of the rivet as unit. See also App., Examples 140 et seqq. 
 
 Holloiv cylinder pressed from without. The strength of 
 cylinders pressed from without is much more difficult to 
 determine than when they are pressed from within. Theo- 
 retically the metal would for similar pressures be in a state 
 of compression equal to the tension as determined above. 
 There is, however, a great practical difference between the 
 two cases. When a cylinder is pressed from without, unless 
 it is of a mathematically perfect shape, and perfectly homo- 
 geneous in strength, the pressure tends to change its shape, 
 so that it may yield by deformation long before the limit 
 of the crushing strength of the metal is approached. Thus 
 it is frequently found that internal furnace flues give way 
 by collapsing. On the other hand, when pressed from with- 
 in, the tendency of the pressure is to keep the cylinder in 
 shape, so that it can only give way when the metal yields. 
 The standard experiments on the strength of cylinders to 
 resist external pressure were those made by Fairbairn thirty- 
 three years ago, under conditions widely different from those 
 now common. The results arrived at by Fairbairn were as 
 follows : The strength varies inversely as the length of the 
 cylinder, inversely also as the diameter, and directly as the 
 square of the thickness of the metal. The following formula 
 is given by Rankine to determine the collapsing pressure 
 of such cylinders : 
 
 / 2 
 
 P 806000 -. 
 La 
 
 when P is the pressure in pounds per square inch, / the 
 thickness of the sides in inches, d the diameter in inches, 
 and L the length in feet. 
 
 In order to obviate the weakness of long flues, rings of 
 angle or tee iron are riveted round them at fixed intervals, 
 as shown in fig. 175, or else the joints are made as repre- 
 sented in fig. 156, or as shown in the furnace of the boiler 
 in fig. 159. The strength in these cases, according to Fair- 
 
396 
 
 The Steam Engine. 
 
 Fig. 175- 
 
 bairn, is to be calculated on the supposition that the length 
 is equal to the distance between two consecutive rings. 
 Those portions of cylindrical flues 
 which do not contain the furnace 
 are very successfully strengthened 
 by means of Galloway's tubes, de- 
 scribed on p. 364. The method 
 of strengthening the furnaces them- 
 selves which appears to be most 
 
 successful is the plan of corrugating the plates introduced 
 by Mr. Fox (fig. 158), and now much used in the furnaces 
 of. high-pressure marine boilers. This system has been 
 already described, see p. 365. 
 
 Staying of flat surfaces. When boilers are formed prin- 
 cipally of flat plates, like low-pressure marine boilers, or the 
 fire-boxes of locomotive boilers, the form contributes nothing 
 to the strength, which must, therefore, be provided for by 
 staying the opposite surfaces together. Fig. 176 shows the 
 arrangement of the stays in a locomotive fire-box. They 
 are usually pitched about 4 inches from centre to centre, and 
 are fastened into the opposite plates by screwing, as shown, 
 
 the heads being riveted over. 
 Each stay has to bear the 
 pressure of steam on a square 
 aa, and the sectional area of 
 the stay must be so chosen 
 that the tensile strength will 
 be sufficient to bear this strain 
 with the proper factor of 
 safety. Thus if a be the area 
 of the section of the stay in 
 inches, and a! that of the square of plate which it supports, 
 P the pressure of the steam per square inch, and / the 
 tensile strength of the stay, we must have 
 
 a'P 
 t ' 
 
 Fig. 176. 
 
 a'P =. af, or a = 
 
Staying of Boilers. 
 
 397 
 
 It is usual to allow a factor of safety of eight for loco- 
 motive boilers, while in marine boilers the factor is from 
 nine to ten, a large margin of strength being necessary on 
 account of the liability of the stays to corrosion. 
 
 If the spaces between the stays are too great, or the plate 
 too thin, there is a danger of the structure yielding through 
 the plate bulging outwards between the points of attach- 
 ment of' the stays, thus allowing the latter to draw through 
 the screwed holes made in the plates. Rankine recommends 
 that if the material of the plate is equal in strength to that 
 of the stay, the thickness of the plate should equal half the 
 diameter of the stay; and that if the material of the plate be 
 weaker, its thickness should be proportionately increased. 
 
 The flat ends of cylindrical boilers are usually stayed to 
 the cylindrical portions by triangular plates of iron, called 
 gusset stays (see figs. 159, 162, 177). Gusset stays should 
 never be brought too 
 close to any internal 
 flues riveted to the flat 
 ends for the reasons 
 explained on p. 365. 
 The two opposite ends 
 are also stayed together 
 by long bar stays, run 
 ning the whole length 
 of the boiler. It is dangerous, however, to trust too much 
 to the latter class of stays ; for, in consequence of the 
 alternate expansion and contraction which takes place every 
 time the boiler is heated and cooled, they have a tendency 
 to work loose at the joints ; and if the portion of the boiler 
 in which they are situated should happen to be hotter 
 than the outside shell, they have a tendency to droop, and 
 are then perfectly useless. 
 
 In designing boilers with stayed surfaces care should be 
 taken that the opposite plates connected by any system of 
 stays should, as far as possible, be of equal area, otherwise 
 
 Fig. 177. 
 
398 The Steam Engine. 
 
 there is sure to be an unequal distribution of load in the 
 stays, some receiving more than their proper share, and, 
 moreover, the least supported plate is exposed to the danger 
 of buckling. 
 
 THE EFFECTS OF UNEQUAL EXPANSION AND CONTRACTION 
 IN STRAINING A BOILER. 
 
 It every portion of a boiler were, when heated, raised 
 to exactly the same temperature, and if the same description 
 of metal were used throughout the entire structure, there 
 would of course be no strains set up by the change of 
 temperature, because all the parts would expand and 
 contract proportionately to their dimensions. In the 
 majority of boilers, however, the various portions are at very 
 different temperatures, and the more highly heated parts 
 expand to a greater extent than the remainder, thus distort- 
 ing the shape of the boiler, and inducing sometimes very 
 serious strains. Take, for instance, the internal furnace and 
 flue of a Cornish boiler : this portion, containing, as it does, 
 the fire, is considerably hotter than the outside shell, and 
 consequently expands more. One of two things must then 
 happen : either the flat ends of the boiler must bulge out, or 
 if these are too rigid to yield, or are stayed too stiffly, the 
 whole of the metal of the flue will be put into a state of com- 
 pression, the effects of which are sometimes most con- 
 spicuously seen in the joints. 1 Again, the flue, though hotter 
 on the whole than the rest of the boiler, is not itself uniformly 
 heated, the upper portions above the fire-bars being at a 
 higher temperature than the lower. The result of this is to 
 twist the flue out of shape, provided the ends of the boiler 
 can yield, the flue cambering up towards the top of the shell. 
 If the ends were quite rigid the top of the flue would be 
 put into compression, and the bottom in tension. 
 
 The outside shells are also subject to considerable dif- 
 1 See also page 363. 
 
Straining of Boilers by unequal Heating. 399 
 
 ferences of temperature, caused by the top of the boiler 
 being filled with hot steam, while the bottom contains water, 
 often not much warmer than the feed. In the case of 
 Cornish and Lancashire boilers with external return flues, 
 this difference in temperature is compensated for, and some- 
 times more than compensated for, by the high temperature 
 of the hot gases circulating underneath, but in the case of 
 boilers having no external heating surface there may be a 
 considerable difference in temperature, unless means are 
 taken to circulate the water. 
 
 In estimating the intensity of the strains due to tempera- 
 ture it should be borne in mind that one degree of rise in 
 temperature elongates a bar of ordinary boiler iron by the 
 same amount as would a tensile stress of the intensity of 
 about 190 Ibs. per square inch. Hence such a bar, if held 
 rigidly at the ends, so that these could not move, and then 
 heated ten degrees, would be subjected to a force of 
 compression equal to 1900 Ibs. per square inch. Similarly, 
 if cooled ten degrees below the normal temperature, it would 
 be subjected to a tensile stress of the same amount. 
 
 MATERIALS OF CONSTRUCTION. 
 
 The metals principally used in the construction of boilers 
 are wrought iron, mild steel, copper, and brass. Copper is 
 used almost exclusively for the inner fire-boxes of loco- 
 motive furnaces, on account of its great conductivity, and the 
 property which it possesses of resisting the intense tempera- 
 ture of combustion usual in this class of boiler. The use 
 of brass is limited to the tubes, but even these are now often 
 made of steel or iron. 
 
 Wrought iron has till lately been the principal metal 
 used in the structure of boilers, but it is now rapidly being 
 superseded by mild steel. The advantages of mild steel are 
 very great. Its strength to resist strains of tension and 
 compression is considerably greater than that of iron, thus 
 
4OO 
 
 T/te Steam Engine. 
 
 enabling lesser scantlings to do the work. Its ductility is 
 greater, its structure more homogeneous, and its quality 
 more uniform ; while its power to resist corrosion, when 
 proper precautions are taken, is reported on most favourably 
 by those who have had the best practical opportunities of 
 watching its behaviour in use. The only drawback which 
 retarded its general introduction was a certain difficulty 
 which the boiler-makers experienced in working the new 
 metal safely into shape, especially when at a black heat ; 
 but this difficulty has to a great extent been got over with 
 increased experience. 
 
 The following table gives the approximate numerical 
 value of the tensile strength of the three metals. It must 
 be understood that samples of the same metal vary so much 
 that nothing but approximate or rough average values can be 
 given. 
 
 
 Tensile Strength, Ibs. per 
 
 Name of Metal 
 
 sq. inch 
 
 
 With grain 
 
 Across grain 
 
 Best Lowmoor plate . 
 
 58,487 55,033 
 
 Ordinary wrought-iron boiler plate . ! 50,000 
 Mild Siemen's steel plates, average . 64,600 
 
 46,500 
 64,500 
 
 Brass tubes .... . 80,000 
 
 
 
 Copperplates. ... . 30,000 
 
 - 
 
 Copper bolts .... . j 36,000 
 
 
 
 Lloyd's rules for marine boilers require that when the 
 material of the shell plates is mild steel it shall have a tensile 
 strength of not less than 26 tons and not more than 30 tons 
 per square inch of section, and the ultimate elongation of a 
 test piece 8 inches in length after fracture, must be not less 
 than 20 per cent, of the original length. 
 
 The Board of Trade rules for steel marine boilers require 
 that the tensile strength of plates not exposed to flame should 
 be about 28 tons and should not exceed 32 tons per square 
 inch of section. The tensile strength of furnace, flanging 
 
Boiler Fittings, 
 
 40! 
 
 and combustion box plates should range from 26 tons to 30 
 tons per square inch. 
 
 FITTINGS OF BO!LERS 
 
 The principal parts of boilers which flow remain to be 
 considered are furnace doors and grates, safety valves, 
 pressure and water gauges, feeding apparatus, and feed 
 heaters. 
 
 Furnace Doors. The chief points to be considered in 
 the design of furnace doors are to prevent the radiation of 
 heat through them, and to provide for the admission of air 
 above the burning fuel in order to aid in the consumption 
 of smoke and unburnt gases. In all cases where the doors 
 are exposed to very rough usage such, for instance, as in 
 locomotive and marine boilers the means for admitting air 
 must be of the simplest, and consist generally of simple 
 perforations, as shown in fig. 178, which represents a front 
 view and section 
 of the furnace 
 door of a. locomo- 
 tive boiler. The 
 heat from the 
 burning fuel is 
 prevented from 
 radiating through 
 the perforations 
 in the outer door 
 by attaching to it 
 a second or baffle 
 plate a, at a distance of about i^ inch, the holes in which 
 do not coincide in direction with those of the door proper. 
 By the constant entry of cold air from the outside the greater 
 part of any heat which may be communicated to the door 
 by radiation or conduction is returned to the furnace. 
 
 Doors similar to the above provide for the constant 
 
 D D 
 
 Fig. 178. 
 
4O2 The Steam Engine. 
 
 admission of limited quantities of fresh air above the fuel. 
 In actual practice, however, air is only needed above the 
 fire for a few minutes after fresh fuel has been thrown on 
 the grate, and is then required in considerable quantities. 
 In the case of land boilers, the furnace doors of which 
 undergo comparatively mild treatment, it is possible to 
 introduce the necessary complications for effecting the 
 above objects. Fig. 160 show's an arrangement in common 
 use in Cornish and Lancashire boilers, and consists of a 
 number of radial slits in the outer door plate, which can be 
 closed or opened at will in the same manner as an ordinary 
 window ventilator. Other and more complicated arrange- 
 ments have been frequently devised which work admirably 
 so long as they remain in order, but the frequent banging 
 to which furnace doors are subjected, even in factory boilers, 
 soon deranges delicate mechanism. 
 
 Furnace doors should be kept as small as te compatible 
 with the proper distribution of the fuel over the grate area, 
 as otherwise the great rush of cold air, when the door is 
 opened, rapidly cools down the flues, and does considerable 
 injury to tube plates, crowns of furnaces, &c. For this 
 reason it is desirable, when grates are over forty inches in 
 width, to have two doors to each furnace, which can be 
 fired alternately. 
 
 Dead-plate and Fire-bars. The dead- plate is a flat plate 
 of iron immediately inside the furnace door, and which is 
 used in many boilers in order to insure the combustion of 
 the volatile portions of bituminous coal. When the fresh 
 fuel is laid on it is placed on the dead-plate instead of on 
 the grate. In this position the coal is coked, the volatile 
 hydrocarbons being driven off by the radiated heat from the 
 incandescent fuel, and ignited as they pass over the latter by 
 the surplus air coming through the grate, or by a special 
 admission through the furnace door. As soon as the cok- 
 ing process is complete the fuel is pushed forward from the 
 dead-plate over the fire-bars. Dead-plates are also frequently 
 
Fire-bars. 403 
 
 used where anthracite coal is burned, as this fuel is apt to 
 crack and splinter into small pieces if thrown fresh on to the 
 grate without having been previously warmed through. 
 
 The grate consists of a number of cast-iron bars, called 
 fire-bars, which are supported on wrought-iron bearers. In- 
 numerable forms of fire-bars have been contrived to meet 
 the cases of special kinds of fuel. The type in common 
 use is represented in fig. 179, which shows a side view and a 
 section of a single bar, and a plan of three bars in position 
 Each bar is, in fact, a small girder, the top surface of which 
 is wider than the bottom. On each bar are cast lugs, the 
 width of which determines the size of the interstices for the 
 
 Fig. 179. 
 
 passage of air. In marine boilers the usual width of the 
 bar on the top surface is \\ inch, tapering down to one- 
 third of this size at the bottom. The interstice varies in 
 width according to the character of the fuel. For anthracite 
 \ inch is a maximum, while for caking coals f inch is often 
 used. For long furnaces the bars are usually made in two 
 lengths, with a bearer in the middle of the grate. In the 
 Lancashire boiler, illustrated in fig. 159, the bars are in three 
 lengths of two feet each. They are f inch wide on the top, 
 and spaced inch apart. In locomotive boilers the bars 
 are generally in one length. As a rule long grates are set 
 with a considerable slope towards the bridge, in order to 
 facilitate the distribution of the fuel. A slope of an inch 
 
 D D 2 
 
404 
 
 TJie Steam Engine. 
 
 to the foot is the rule. The grates of locomotive engines 
 are nearly always flat. 
 
 Safety valves. The safety valve is a circular valve seated 
 on the outside of the boiler, and weighted to such an extent 
 that when the pressure of the steam exceeds a certain 
 point, the valve is lifted from its seating and allows the 
 steam to escape. Safety valves can be loaded directly with 
 
 weights, in which case they 
 are called dead-weight valves, 
 or the load can be trans- 
 mitted to the valve by a lever. 
 Again, the end of the lever 
 is sometimes held down by a 
 simple weight attached to it, 
 a plan commonly adopted in 
 land boilers ; while some- 
 times, as in the case of loco- 
 motive and marine boilers, 
 the lever is weighted by means 
 of a spring, the tension of 
 which can be adjusted. 
 
 Fig. 1 80 shows a form 
 of dead-weight safety valve, 
 where a is the valve which 
 rests on the seating b. 
 
 The valve is attached to 
 the circular casting AAA, so 
 that both rise and fall to- 
 gether. The weights WW, &c., are disposed on the casting 
 in rings, which can be adjusted to the desired blow-off 
 pressure. Owing to the centre of gravity of the casting 
 and weights being below the valve, the latter requires no 
 guides to keep it in position. This is a great advantage, 
 as guides frequently stick, and prevent the valve from 
 acting. Another advantage of this form of valve is that 
 it is difficult to tamper with. For instance, a four-inch 
 
 FIG. 1 80. 
 
Safety Valves. 405 
 
 valve, intended to blow off at 100 Ibs. per square inch, 
 would require weights of n cwt, which occupy a consider- 
 able bulk. An unauthorised addition of a few pounds to 
 such a mass would make no appreciable addition to the 
 blowing-off pressure, while any effectual increment of weight 
 would be immediately noticed. It is quite different with 
 the lever safety valve, about to be described. A small addi- 
 tion to the weight at the end of the lever is multiplied 
 several times at the valve. 
 
 Fig. 181. 
 
 The second form of safety valve is shown in fig. 181 
 Here the load is attached to the end C of the lever ABC, 
 the fulcrum of which is at A. 
 
 Calling the weight W, 
 
 the weight of the lever w, 
 
 the weight of the valve, a/, 
 
 the distance of the centre of gravity of the lever from A, /, 
 
 w ' + w m + ^ 
 
 is the pressure brought to bear on the seat bb of the valve a. 
 The effective pressure on the valve, and consequently the 
 blowing-off pressure in the boiler, can be regulated, within 
 certain limits, by sliding the weight W along the arm of the 
 lever. In locomotive engines the weight would, on account 
 of the oscillations, be inadmissible, and a spring is used to 
 hold down the end of the lever. The pressure on the valve 
 can be regulated by altering the tension of the spring. 
 
406 
 
 TJie Steam Engine. 
 
 A valve much used in locomotives is shown in fig. 182. 
 It is called, after the name of its inventor, Ramsbottom's 
 patent safety valve. It consists really of two valves AA, placed 
 side by side, at a little distance apart. A cross-piece B bears 
 upon each valve, and to the cross-piece is attached a power- 
 ful spiral spring D, the lower end of which is so fixed at C 
 
 that its tension 
 can be adjusted 
 by means of a set 
 screw at E which 
 is out of reach of the engine 
 driver. Before the valves can 
 rise they have to overcome the 
 resistance of the spring, to which 
 the pressure is communicated by 
 means of the cross-piece B. The 
 spring is attached to the cross- 
 piece below the bearing points 
 of the cross-piece on the valves. 
 If one of the valves should rise 
 from its seating before the other, 
 the spring leans a little towards 
 this latter, easing the pressure on it, and allowing it to open. 
 The rise of the valves from the seating is much greater 
 with these directly loaded valves than when the pressure is 
 transmitted through a lever, and thus the steam escapes with 
 much greater rapidity. 
 
 Every boiler should be provided with two safety 
 valves, one of which should be put beyond the control 
 of the attendant. The size of the opening depends of 
 course upon the steam-producing power of the boiler, 
 the object to be attained being to reduce the pressure 
 within the boiler to its normal point as quickly as pos- 
 sible. The following rule is given by Rankine for valves 
 having a lift of one-twentieth of their own diameter. Let 
 a = area of valve ; A = area of heating surface in square 
 
 Fig. 182. 
 
Pressure Gauges. 
 
 407 
 
 a = -=, 
 
 feet ; P = pressure of steam in pounds per square inch. 
 Then 
 
 A 
 
 3P" 
 
 The Board of Trade rule for marine boilers is to allow half 
 a square inch of safety valve for every square foot of fire- 
 grate area. 
 
 Pressure Gauges. These instruments are used for show- 
 ing the pressure at which the steam happens to be within the 
 boiler. The one in most common use is Bourdon's, and is 
 illustrated in fig. 183. It con- 
 sists of a bent metal tube aa, 
 which is put in connection with 
 the interior of the boiler by 
 means of the pipe b, which is 
 provided with a stopcock. The 
 tube aa is elliptical in cross 
 section, as shown at A. The 
 effect of internal pressure on 
 the tube is to tend to transform 
 the elliptical into a circular 
 cross section. This, however, 
 cannot be done without partially 
 unbending or straightening the 
 tube aa ; that is to say, the 
 effect of internal pressure is 
 ultimately to straighten the 
 tube, and the greater the pres- 
 sure the more the tube is unbent, and consequently the 
 more the free end c is moved from its normal position. 
 The free end is connected by means of a link with an index 
 like the hand of a watch, either directly, or else through the 
 medium of a small rack and pinion, which multiplies the 
 motion of the index ; and when the free end of the tube 
 moves under the influence of pressure, the end of the index 
 describes an arc of a circle. By placing a dial behind the 
 
 Fig 183. 
 
408 The Steam Engine. 
 
 index, and graduating the former experimentally, so that a 
 given position of the needle corresponds with a given pres- 
 sure in the tube, we obtain an exact pressure gauge. The 
 experimental division of the circumference of the dial is 
 made by connecting the Bourdon gauge with a mercurial 
 syphon gauge and a force-pump. The force-pump is then 
 worked, so that the syphon gauge registers successive incre- 
 ments of pressure of one pound per square inch, and at 
 each of these a mark is made on the dial of the Bourdon 
 gauge opposite the position of the index finger. These 
 gauges should be tested from time to time by a mercurial 
 gauge, as they are apt to get out of order, in consequence of 
 water lodging in the end of the bent tube and corroding 
 the latter. It may easily be known when they are out of 
 order by raising the pressure of the steam in the boiler, and 
 watching till it commences to blow off at the safety valve, 
 and then noting the position of the index finger. The pres- 
 sure registered by the finger should of course then corre- 
 spond with the known blow-off pressure of the valves ; if it 
 does not, one or other or both of these instruments must 
 be out of order : but the safety valve is usually kept in 
 order ; therefore when this is the case, and a disagreement 
 occurs, the Bourdon gauge may be presumed to need 
 correction. 
 
 Feeding Apparatus. The water of a boiler is replenished 
 by means of force-pumps or injectors, or by both. For 
 safety's sake every boiler ought to have two feeds, in order 
 to avoid accidents when one of them gets out of order. 
 Pumps for feeding are of two principal kinds, viz. those 
 driven by a crank or eccentric on the main axle of the 
 engine, and those which are connected direct to a separate 
 small engine, which is only employed for pumping purposes : 
 these latter are called donkey engines. The feed-pumps of 
 land boilers are usually made large enough to supply, if 
 kept continuously at work, from two to two and a half times 
 the quantity of water actually consumed by the engine, 
 
Injectors. 409 
 
 In old-fashioned marine boilers, where the engine is not pro- 
 vided with a surface condenser, the pumps had to be made 
 still larger, in order to allow for the waste occasioned by the 
 discharge of brine from the boilers. The pumps themselves, 
 being ordinary force-pumps, require no special description. 
 
 Injectors. The injector, which was invented by Giffard, is 
 in many respects the most peculiar and interesting apparatus 
 connected with the steam engine. It is an instrument which 
 converts the energy of the heat in the steam into mechanical 
 work without the aid of any moving mechanism whatever. 
 Before describing it, it is necessary to notice the difference 
 between the velocity of steam escaping from a boiler, and 
 water escaping from the same vessel under the same pressure 
 of steam. The velocity of the water is, in accordance with a 
 well-known law of hydrodynamics, and neglecting the effect 
 of friction, the same as it would acquire by falling down a 
 height equal to the length of the column of water which 
 would produce the same pressure as the steam. Thus, let 
 the pressure of the steam in the boiler be five atmospheres 
 above that of the external air ; the pressure of one atmo- 
 sphere will balance the weight of a column of water. 33-9 
 feet in height ; therefore five atmospheres will balance a 
 column of 169-5 ^ eet - The velocity acquired by falling 
 down this height would be about 104 feet per second. 
 This, therefore, would be approximately the velocity of efflux 
 of the water from the boiler. 
 
 The velocity of efflux of the steam is much greater, 
 although the pressure is the same. It would be impossible 
 in the limits of this chapter to give an account of the theory 
 of the flow of gases and of saturated steam. It must be 
 enough to mention that for the pressures usual in land 
 boilers the velocity of the steam is from 16 to 18 times 
 greater than that of the water. 
 
 Suppose now that some of the steam were discharged 
 from a boiler through a pipe at this high velocity, and that 
 while in the act of discharge it were condensed suddenly by 
 
TJie Steam Engine. 
 
 passing through an intensely cold medium ; the resulting 
 water would travel forward with the same velocity which it 
 had already acquired when in the state of steam ; and if the 
 various particles of water could by any means be gathered 
 together into a continuous stream, they would be more than 
 able to overcome and to force back into the boiler any 
 opposing stream of water of the same size directed against 
 them from the water- room of the boiler. Now the velocity 
 of the condensed steam is so great that it possesses not 
 only energy enough to re-enter the boiler in the face of an 
 opposing stream of water of its own size, but it can also 
 impart energy to a much larger mass of water, so that this 
 larger mass can also enter the boiler. The injector is simply 
 an instrument for allowing steam to rush from a boiler, and 
 to suck up and mix with itself a stream of cold water, by 
 which it is condensed, and to which it imparts so much of 
 its own velocity that the combined mass of cold water and 
 condensed steam enters into and feeds the boiler. 
 
 Fig. 184 shows an elementary form of such an injector. 
 
 A is the section of 
 a boiler, B a pipe 
 leading from the 
 steam space and 
 terminating in a 
 nozzle, C is the 
 cold water pipe 
 leading from the 
 tank, and termi- 
 nating in a hollow 
 cone surrounding 
 the steam nozzle. 
 When the steam 
 
 Fig. 184. 
 
 is turned on, and 
 
 escapes from the lower edge E of the hollow cone, it 
 creates a partial vacuum in the cone and in the pipe C. 
 The water then rushes up the pipe and into the cone sur- 
 
Injectors. 
 
 411 
 
 rounding the nozzle, where it meets with the escaping steam, 
 which it condenses. The particles of condensed steam, 
 impinging on the water surrounding them, communicate 
 their motion to the latter, and the combined mass is de- 
 livered at a high velocity into the feed-pipe F, and through 
 the valve at G into the boiler. Such an injector, if properly 
 proportioned, would work 
 well for a fixed pressure of 
 steam in the boiler, and for a 
 fixed temperature of the feed 
 water. In practice, however, 
 these quantities vary, and in- 
 jectors must be made to suit 
 all such contingencies. For 
 instance, when the pressure 
 of the steam increases, the 
 area of the opening in the 
 steam nozzle must be in- 
 creased, and vice versa. There 
 are very many forms of in- 
 je:tors. Fig. 185 illustrates 
 one which is in common 
 use in this country. The 
 steam and water supply pipes, 
 nozzle, and cone are rendered 
 sufficiently clear by the draw- 
 ing. The steam supply is 
 varied by altering the posi- 
 tion of the conical spindle 
 a, which can be screwed 
 towards or away from the 
 mouth of the nozzle. 
 
 The water chamber CC 
 is so arranged that it com- 
 pletely surrounds the steam nozzle. The supply of the water 
 is varied by contracting or expanding the conical aperture 
 
412 The Steam Engine. 
 
 below the mouth of the steam nozzle. This is accomplished 
 by moving the conical sliding tube E backwards or forwards 
 by means of the hand-wheel D and the rack and pinion. 
 
 If the supply of steam is not properly adjusted to the 
 water, some of the latter will escape at the aperture made 
 in the sliding tube E into the overflow pipe. For instance, 
 if the supply of steam be too small, the current will not 
 have sufficient energy to enter the boiler, and part of it will 
 choke up the sliding tube and escape by the aperture. 
 When this occurs it is only necessary to turn on more 
 steam, or shut off some of the water. 
 
 The efficiency of the injector is measured by the tempe- 
 rature of the current of feed water as it enters the boiler, 
 compared to its temperature before it enters the injector. 
 The less the rise in temperature, the more the energy of the 
 steam is utilised. Theoretically speaking, if we measure the 
 units of heat in the feed water as it enters the boiler over 
 and above the heat before it enters the injector, and sub- 
 tract the amount from the total heat of the steam used, the 
 result ought to give the useful work which the injector does. 
 A great deal of power is wasted in these instruments, 
 as at present constructed, by the formation of eddies. 
 When used for feeding boilers, all the heat represented by 
 the rise of the temperature of the feed water is of course 
 restored to the boiler. As might be expected, the efficiency 
 of an injector increases as the original temperature of the 
 feed water diminishes. These instruments are also used for 
 other purposes besides the feeding of boilers. They have 
 even been employed on a large scale to drain a mine. In 
 this case the work done was represented by about 80 gallons 
 per minute raised through a vertical height of 240 feet. 
 This probably is the greatest amount of work which has 
 ever been accomplished with an injector, and could of course 
 only be undertaken where expenditure of fuel was no 
 consideration. 
 
 Water Gauges. These are used to ascertain the level 
 
Water Gauges and Feed- Heaters. 413 
 
 of the water in the boiler. The simplest sort consist of 
 three cocks screwed into the face of the boiler at different 
 levels, one being usually at the normal level of the water, 
 one above this in the steam space, and a third lower down 
 at a level below which it is dangerous to allow the water to 
 sink. By opening these cocks in succession the position of 
 the water level can be approximately ascertained. 
 
 Another variety in common use consists of a straight 
 glass tube, so fixed that its upper end communicates with 
 the steam, and the lower end with the water space. Cocks 
 are provided for cutting off the connection at either end, 
 and for allowing steam or water to be blown through the 
 glass tube. The latter is fixed at the ends in metal sockets 
 which allow of its being removed and replaced when broken. 
 With this form of gauge the water level is always visible. It 
 is usual to provide a boiler with both forms of gauge, in order 
 that if one gets out of order the other may be available. 
 
 Feed-water Heaters. It is very desirable, whenever it 
 is possible, to feed the boiler with water of the temperature 
 of or about 212. There are three good reasons for this 
 practice. In the first place, the introduction of cold water 
 into the hot boiler tends to produce the strains due to 
 unequal temperature which have been already commented 
 on. In the next place, it has been observed that water 
 which has been previously heated, otherwise than by surface 
 condensers, exercises a far less corrosive effect on the 
 boiler than cold water, the corrosive action taking place 
 in the heater instead, where its injurious effects are not 
 nearly so important. Lastly, there is of course a very con- 
 siderable saving of fuel effected by utilising waste heat to 
 raise the temperature of the fuel. Supposing, for instance, 
 the water were raised from 60 to 212, there would be a 
 saving of 152 units of heat for every pound of water, which 
 is equivalent to about one-seventh of the total heat required 
 to evaporate the water at 212 from the temperature of 60. 
 
 There are three distinct methods in use of heating feed 
 
414 The Steam Engine. 
 
 water. In modern marine engines fitted with surface 
 condensers the steam condensed from the engines is used 
 over and over again in the boilers. The temperature of 
 water coming from surface condensers should be about 
 130. Unfortunately such water is generally more or less 
 charged with fatty acids, generated by the decomposition of 
 the oils used for lubricating the cylinders, and consequently 
 great care has to be exercised to prevent the rapid corrosion 
 of the boilers. Sometimes, also, the lubricant is carried 
 back into the boiler in the shape of a gelatinous, non- 
 conducting substance, which settles on the crowns of 
 furnaces, and prevents the transmission of heat through the 
 plates. The consequence is, the furnace crowns become 
 over-heated and collapse. 
 
 With high-pressure non-condensing engines the exhaust 
 steam is frequently used to raise the temperature of the 
 feed. When the apparatus for utilising the heat of the 
 exhaust steam is properly designed, very excellent results 
 may be obtained with this class of heater, but not unfre- 
 quently the steam vs>. forced through a series of pipes 
 surrounded with cold water, the result being that the back 
 pressure in the cylinder is unduly raised, and much more 
 heat is thus often lost in the engine than is gained by raising 
 the temperature of the feed. 
 
 The third class of feed heater utilises the waste heat 
 from the furnaces before it passes up the chimney, and is 
 admirably adapted to factories where room can be spared. 
 The apparatus usually consists of a series of vertical pipes 
 connected together, through which the feed water is forced. 
 The hot air and gases proceeding from the boiler flues 
 circulate between these pipes, and heat the water contained 
 in them. As, however, the tubes rapidly get covered with 
 non-conducting soot, it is necessary to provide each of 
 them with a scraper driven by machinery, which is con- 
 stantly travelling up and down the tube as long as the 
 apparatus is at work. A feed heater of this description,.. 
 
Chimneys. 415 
 
 applied to the Lancashire boiler illustrated in fig. 159, has 
 sixty tubes, exposing a total heating surface of 600 square 
 feet. It is situated in the base of the chimney. All heaters 
 of this type are, in reality, low-pressure tubulous boilers, 
 and, as such, should invariably be provided with safety 
 valves. 
 
 CHIMNEYS AND OTHER MEANS OF PRODUCING THE 
 DRAUGHT. 
 
 A chimney promotes a flow of air through a furnace, 
 because the hot air contained in the chimney is lighter than 
 the surrounding atmosphere, which consequently endeavours 
 to force its way into the chimney from below in order to 
 restore the balance of pressure. The only way into the 
 chimney is through the fire-bars and furnace, and in passing 
 through these the air maintains the combustion, and at the 
 same time becoming itself heated, makes the action of the 
 chimney continuous. 
 
 In estimating the action of a chimney of a given size in 
 producing a draught, the density, temperature, and volume 
 of the products of combustion must be considered. 
 
 The nitrogen which passes through undergoes no 
 chemical change, and consequently its density and volume 
 are unaltered except by the change of temperature. The 
 oxygen combines partly with the carbon and partly with the 
 hydrogen contained in the fuel, while a great portion goes 
 through unchanged like the nitrogen. The portion which 
 combines with the carbon so as to form carbonic acid 
 undergoes no change of volume except so far as it is affected 
 by temperature, for the volume of the carbonic acid gas is 
 the same as that of the oxygen from which it is formed, but 
 its density is of course increased by the weight of carbon 
 taken up. The portion of the oxygen which combines with 
 the hydrogen forms steam, the volume of which is greater than 
 that of the oxygen, but the proportion of utilisaole hydro- 
 gen in fuel is so small that it is usually left out of account. 
 
416 The Steam Engine, 
 
 Consequently the mixed air and products of combustion which 
 escape from a furnace may be considered as approximately 
 of the same volume as the air which is supplied to the 
 furnace when at the same temperature. One pound of air 
 at 32 has a volume of 12^ cubic feet ; and as we have seen 
 that when the blast is produced by a chimney 24 Ibs. of 
 air are necessary to consume a pound of coal, the volume 
 of furnace gases for every pound of fuel consumed will be, 
 when reduced to 32 = 12 Jx 24 = 300 cubic feet. At 
 any other temperature the volume will equal the volume at 
 32 multiplied by the ratio of the absolute temperature of 
 the new temperature to the absolute temperature of 32 
 Thus, taking 2000 as the temperature of the furnace, the 
 volume in the above case 
 
 , . c t 
 
 = 300 x -- - - =1497 cubic feet ; 
 32 +461 
 
 and, generally, V = V 32 X , where V 32 is the volume at 
 
 7 32 
 
 32, and r and r 32 the absolute temperatures of the gas 
 when at the heat of the furnace and at 32. 
 
 The density of the current depends on the quantity of 
 air supplied per pound of fuel, and on the final temperature 
 of the products of combustion. Thus, if 24 Ibs. of air be 
 supplied per pound of fuel, the volume of this quantity of 
 air at 32 = 24 x 12 J = 300 cubic feet, The weight of the 
 mixture of air and fuel is 24+1 = 25 Ibs. and the volume 
 
 at the temperature r of the furnace gas is = 300 x . 
 
 7 32 
 
 The density, or weight of a cub. ft. = = '083 x ? . 
 
 300 x - 
 
 r 32 
 
 In the general case let w Ibs. be the weight of fuel burned ; 
 V^ the volume of air supplied at 32. 
 
 Then the total volume of the products of combustion= 
 
 ' V 32 T , 
 
 "32 
 
Chimneys. 417 
 
 the weight of the products = ^-^ 2 + w Ibs. ; 
 
 the density or weight per cubic foot equals the total weight 
 in Ibs. divided by the total volume 
 
 , 2 
 
 ! V 32 y T ' 
 
 The quantity varies in value according to the air 
 
 v 32 
 supply. 
 
 The Effect of Height in a Chimney. The difference be- 
 tween the weight of a column of outside air of the height of the 
 chimney above the fire-bars, and standing on a base equal 
 in area to the cross section of the chimney, and that of the 
 column of hot air within the chimney is the measure of the 
 force which produces the draught. Let r l be the outside 
 temperature (absolute measure), and H be the height of the 
 chimney in feet. 
 
 Then, since one cubic foot of air at 32, or T 32 , weighs ^- 
 
 = -08 tb., therefore H ( -oS 1 ^ 32 j= weight of column of out- 
 
 side air of height of chimney standing on an area of one 
 square foot. The corresponding column within the chimney 
 
 weighs Hf *o8+ j -^, and the difference between these 
 
 V V 32/ T 
 
 two weights is the pressure in Ibs. per square foot of chimney 
 section which produces the draught. A column of the hot 
 gas equal in weight to this difference is called the head of 
 the chimney ; and just as in hydraulics the velocity of dis- 
 charge of water from the bottom of a full vertical pipe is 
 proportional to the square root of the height of the pipe, so, 
 in the case of a chimney, the velocity with which air would 
 flow, if unimpeded, into the bottom of the chimney is also 
 proportional to the square root of the height of the head. 
 The height of the head, reckoned in feet of hot gas, is found 
 by dividing the weight of a column of external air as high 
 
 E E 
 
41 8 The Steam Engine. 
 
 as the chimney, as found above, by the weight of one 
 cubic foot of the hot gas (this gives the height of a column 
 of the hot gas weighing as much as the column of the ex- 
 ternal air). If we subtract from this the height of the chim- 
 ney, the difference is the height of the head. 
 
 In actual chimneys the velocity of the discharge of the 
 gas is greatly diminished by the resistance opposed by the 
 fire-grate and layer of fuel to the entrance of the air, and 
 also by the friction of the sides of flues, tubes, &c., and of 
 the internal surface of the chimney itself. 
 
 Peclet gives the following formula for the height of head 
 necessary to produce a given velocity of the gas in the 
 chimney : 
 
 Let / = the length of the chimney + that of the flue 
 
 leading to it. 
 z/ = area of section of chimney divided by its cir- 
 
 cumference. 
 
 /= co-efficient of friction of sides of flues and 
 chimney, which depends for value on the con- 
 dition of the surfaces. 
 G = co-efficient of resistance of grate and layer of 
 
 fuel to entrance of air. 
 u = velocity of gases in chimney. 
 g = acceleration due to gravity ; and 
 h = height of head in feet. 
 
 Then, according to Peclet, 1 h-= ( i + G + \. 
 
 2g\ v ] 
 
 The value of the co -efficients varies according to circum- 
 stances. When the surfaces of the flues are sooty, /= -012. 
 With ordinary grates, burning from 20 to 24 Ibs. of fuel per 
 square foot of grate surface per hour, G=i2, and the 
 formula then becomes 
 
 See Rankine's Manual of the Steam Engine, p. 287. Ninth edition. 
 
Chimneys. 419 
 
 If the head is given, then the velocity of the gas can be 
 calculated from the same formula; and when this is ascer- 
 tained, the weight of fuel which can be consumed in a given 
 time may be calculated on the supposition that each pound of 
 coal requires 24 Ibs. of air = 316 cubic feet at the ordinary 
 temperature (60) of the atmosphere. 
 
 The use of very high chimneys is in many situations a 
 necessity, not in order to create the draught, but in order to 
 discharge the noxious products of combustion at a consider- 
 able distance above animal and vegetable life ; otherwise a 
 forced blast might often be more advantageously employed. 
 It is considered that a chimney is most efficacious in pro- 
 ducing a draught when the temperature inside it is about 
 600, and at this temperature about one-fourth of the avail- 
 able heat of combustion is wasted in creating the draught. 
 
 When a forced blast is produced by means of a fan, 
 blast-pipe, or air injector, the products of combustion may 
 be cooled down as far as is found practicable and con- 
 venient; and as much less air is required to effect combustion, 
 the saving of heat may be very considerable. On the other 
 hand, heat must be expended in order to produce a forced 
 draught. Thus in the case of the blast-pipe the heat ex- 
 pended is represented by the excess of back pressure in 
 the cylinder. 1 In the case of a fan, the heat consumed in 
 driving the fan must be taken into account ; and when an 
 
 1 According to Mr. D. K. Clark's experiments the excess of back 
 pressure over and above the pressure of the atmosphere, caused by the 
 use of the blast -pipe, varies approximately (i) as the square of the 
 speed of piston ; (2) as the pressure of the steam at the commence- 
 ment of the exhaust ; (3) inversely as the square of the area of the 
 nozzle of the blast-pipe. He also found that the back pressure was 
 largely increased by the presence of liquid water in the spray. As 
 to the amount of the back pressure in certain cases the reader can 
 consult the examples of diagrams from locomotive cylinders on page 
 338. Mr. Clark also states the vacuum in the smoke-box to be 
 about 70 per cent, of the blast pressure, while the vacuum in the fire- 
 box is from one-third to one-fourth of the same pressure, and the rate 
 of evaporation varies about as the square root of the vacuum in the 
 smoke-box. 
 
 E E2 
 
420 The Steam Engine. 
 
 air injector is used, the heat expended in producing the 
 draught is represented by the total heat of formation of the 
 steam used in the injector. 
 
 Forced draught. A forced draught is now frequently 
 applied to the furnajces of marine boilers, especially in ships 
 of war. In torpedo boats it is necessary to develope immense 
 power out of a comparatively small boiler, and some sort of 
 artificial draught becomes an absolute necessity. A blast-pipe 
 is of course impossible, as the engines are condensing, and 
 would, moreover, be quite inapplicable on account of the 
 noise and shape of the funnel. The method which has 
 been adopted is to force the air into the furnace by means 
 of a rotary fan, driven either from the main machinery, or 
 else by a separate engine. If the blast were directed solely 
 beneath the fire by means of the device of a closed ash-pit, 
 it would always be leaking outwards through the furnace 
 door, and would, whenever the latter were opened for 
 fresh fuel, cause the smoke and flame to fly out in the face 
 of the stokers. To obviate this, the plan has been adopted 
 of closing the stoke-hold so as to make it air-tight, and 
 then forcing the air into the closed chamber, which can 
 only escape through and over the fuel and boiler tubes to 
 the funnel. When the furnace door is opened, the com- 
 pressed air in the stoke-hold rushes through it to the tubes, 
 thus preventing the escape of flame. In this way most re- 
 markable results have been obtained. The defect of the 
 system is that it is difficult, in a boiler of moderate size, to 
 provide sufficient surface to absorb the heat generated by 
 the large amount of fuel which can be burnt on the grate. 
 The result is that the water evaporated per pound of fuel 
 is necessarily low. 
 
 The application of forced draught is by no means limited 
 to the boilers of torpedo boats. In large men-of-war the 
 system is also applied for the purpose of obtaining a large 
 additional supply of steam when extra speed is required. 
 The general result attained may be stated as follows. With 
 
Forced Draught. 
 
 421 
 
 natural draught about io| horse-power are indicated per 
 square foot of fire-grate at full power ; while with forced 
 draught between 16 and 17 horse-power are obtained. The 
 pressure of air being about two inches. 
 
 In the mercantile marine forced draught is beginning to 
 receive considerable attention, for two reasons. First, its 
 application will enable a considerable saving to be effected 
 in the weight of boilers, provided always that the evapora- 
 tion per pound of coal burnt be not seriously diminished. 
 Second, it enables very inferior and cheap fuel to be 
 utilised. 
 
 In the mercantile marine, hitherto, the closed ash-pit 
 has been used instead of the closed stoke-hold. Fig. 186 
 illustrates the system applied by Mr. Hovvden successfully 
 
 Fig. 186. 
 
 to the boiler of a merchant steamer. The ash-pit is closed 
 in front by a door, and the furnace door is double with a 
 hollow chamber between its outer and inner faces. Air 
 
422 The Steam Engine. 
 
 under pressure is supplied beneath the fuel through the ash- 
 pit, and above the fuel through the holes in the inner furnace 
 door, the air issuing through these holes being under con- 
 siderably higher pressure than that in the ash-pit. The object 
 of this double admission is to secure the complete combus- 
 tion of the fuel with a very moderate supply of air, and thus 
 to increase the temperature of the products of combustion. 
 The air supply is heated by the waste gases from 180 to 
 200 above its ordinary temperature, and whenever the 
 furnace door is opened the current of air is cut off for the 
 moment so as to prevent the sudden cooling down of the 
 furnace. 
 
 At first sight it would appear that the forced draught 
 system is of no special advantage where economy of fuel is 
 a primary consideration ; for, although by its use the grate 
 area may be diminished, nevertheless, the heating surface 
 and the total weight of boiler cannot be reduced. It must 
 however be borne in mind that if the supply of air be pro- 
 perly regulated, the volume of the products of combustion 
 will be greatly diminished, and their temperature increased; 
 consequently, a less heating surface is required to produce 
 the same evaporative results per pound of fuel than when 
 natural draught is employed. Also with forced draught it 
 is possible to make use of tubes of comparatively small 
 diameter, and consequently a considerably increased heat- 
 ing surface can be obtained without increasing the dimen- 
 sions of the boiler. 
 
423 
 
 CHAPTER X. 
 
 CONDENSATION AND CONDENSERS. 
 
 The object and advantages of condensing steam General description of 
 condensers Quantity of water required to condense steam Objects of 
 surface condensation for marine engines Description of a jet condenser 
 for a stationary engine Description of a marine surface condenser 
 Air-pumps Ejector condensers Method of indicating the vacuum. 
 
 THE condenser may, in a certain sense, be described as 
 having the inverse functions of the boiler ; for, whereas the 
 latter is employed to raise the medium with which the 
 engine works to the superior limit of temperature, the 
 purpose of the latter is to reduce the inferior limit of 
 temperature as far as possible. The boiler fulfils its pur- 
 pose by converting the feed water into steam, and the con- 
 denser by re-converting that steam after it has done its work 
 into water. 
 
 The advantages from the thermal point of view of 
 condensing the steam, instead of allowing it to escape into 
 the open air at a little above the atmospheric pressure, are 
 very easily explained by reference to the principles which 
 enable us to calculate the maximum efficiency of heat engines 
 (see p. 88). Suppose, for instance, that we have two pre- 
 cisely similar engines working with steam of 50 Ibs. pressure 
 per square inch absolute, and one provided with a con- 
 denser, while the other discharges the exhaust into the open 
 air. Suppose, also, that the former expands down to a 
 pressure of 3 Ibs. absolute, and the latter down to a pressure 
 of about 3 Ibs. above the atmosphere, say 18 Ibs. absolute. 
 
424 The Steam Engine. 
 
 The relative theoretical efficiencies of the two engines may 
 be expressed as follows. The temperature of steam of 50 Ibs. 
 absolute is 280-5, while that of steam of 18 Ibs. absolute 
 is 222'5, and of 3 Ibs. absolute is 141 -5. Then, accord- 
 ing to the principles of the efficiency of heat engines, the 
 maximum efficiency of the condensing engine is 
 
 280-5 141-5 __ 139 _. 
 461 + 280-5 " 741-5' 
 
 while that of the non-condensing engine is 
 
 280-5 222-5 _ 58 
 461 + 280-5 "741*5' 
 
 Thus the condensing engine is theoretically the more efficient 
 of the two in the ratio of 139 to 58*5, or 2-37 to i. 
 
 From the mechanical point of view, as illustrated by the 
 indicator diagram, the advantages of condensation are most 
 apparent, for it enables the back pressure to be reduced 
 from some three pounds above the atmosphere to, say, ten 
 or eleven pounds below it. Also, as in non-condensing 
 engines it is practically impossible to expand the steam 
 below the atmospheric pressure, it is evident that condensa- 
 tion enables us to make use of much higher grades of 
 expansion than would otherwise be possible. This latter is 
 merely another mode of expressing the advantage explained 
 above by reference to the principles of thermodynamics. 
 
 The condenser is an apparatus into which the steam is 
 discharged when it has done its work, and where it comes 
 in contact either with a jet of cold water, or else with a 
 large area of metallic surface, one side of which is kept 
 cool by contact with cold water. The steam on entering 
 this chamber is instantly condensed, giving up its heat to 
 the water ; and the result would be, if a sufficient quantity 
 of water were used, the formation of a practically perfect 
 vacuum, were it not for the fact that the feed water usually 
 contains a large quantity of air, which passes over with the 
 
Condensation of Steam. 425 
 
 exhaust steam into the condenser, and exerts a back pressure 
 against the piston. In order to get rid of this air, an air- 
 pump, driven by the engine, is fitted to the condenser, and 
 is also made use of to pump away the water into which 
 the steam condenses. Various types of condensers, together 
 with their fittings, are described and illustrated on pages 42 9 
 
 to 437- 
 
 Quantity of water required to effect condensation. Sup- 
 pose the steam is expanded in the cylinder down to a pres- 
 sure of say 4*5 Ibs. per square inch, the temperature corre- 
 sponding to which is 158 ; and suppose, further, that the 
 temperature of the final mixture of condensed steam, and of 
 condensing or injection water is to be 1 10, then for every 
 pound of steam which enters the condenser the injection 
 water will have to absorb the total heat of the steam of 158 
 above the water of IIO D . 
 
 The total heat of steam of 158 is 1,130 thermal units ; 
 subtracting from this the heat of water at 110, which is 
 approximately no thermal units, we have 1,020 units, 
 which have to be absorbed by the injection water. The 
 quantity of the latter required obviously depends upon its 
 initial temperature. Suppose the latter to be 50, each 
 pound of it can absorb 110 50 = 60 thermal units by 
 rising in temperature to no . Therefore the total quantity 
 
 of water required is I0 = 17 Ibs. This, therefore, is the 
 60 
 
 minimum quantity of water required under the given 
 circumstances. 
 
 It is obvious from the foregoing that the quantity of 
 injection water required in any given case depends upon 
 the final pressure of the steam, the initial temperature of the 
 injection water, and the temperature at which it is required 
 that the mixture of injection water and condensed steam 
 should be maintained. 
 
 Let T! = the temperature of the steam when the 
 exhaust opens. 
 
426 The Steam Engine. 
 
 Let L = the latent heat of the steam at this 
 
 temperature. 
 Then T l + L = the total heat in thermal units of i Ib. 
 
 of the steam. 
 
 Let T 2 = the temperature of the injection water. 
 Let T 3 = final mixture. 
 
 Let W = the weight in pounds of the injection 
 
 -water per pound weight of steam. 
 The injection water in rising from T 2 to T 3 absorbs 
 
 (T 3 -T 2 ) W thermal units. 
 
 The pound of steam in falling from the condition of 
 steam at Tj to that of water at T 3 gives out 
 
 T, + L - T 3 thermal units. 
 
 Now the heat lost by the steam must equal that gained 
 by the injection water. Hence we have 
 (T 3 -T 2 )W = T 1 +L-T 3 . 
 
 3~~ 2 
 
 The numerator of this fraction is the expression for the 
 total heat of steam of the temperature T, over and above 
 the heat contained in water at the temperature T 3 , which 
 (see p. 100) 
 
 = 885,200 + 235-46 (T 1 -2i2)-772 (T 3 -32)foot-lbs. 
 Reducing and dividing by 772, so as to obtain thermal units, 
 and substituting the result in the above formula for W, we 
 get 
 
 W = i"4+'3Ti-T 8 _ 
 T 3 T 2 
 
 EXAMPLE. 
 
 Find the amount of injection water required when the exhaust 
 steam has a pressure of lp'5 Ibs. absolute, the injection water a tem- 
 perature of 60, and the required temperature of the mixture 1 10. 
 The temperature T, of steam of the above pressure is 227. Hence 
 
 -3*227-uo, I2 . 4lbs> 
 
 110-60 
 
Surface Condensation. 427 
 
 Surface condensation. In former times, when the pressure 
 of steam rarely exceeded 35 Ibs. per square inch, jet con- 
 densers were universally used for marine engines. The boilers 
 were fed from the hot mixture of condensed steam and in- 
 jection water, which contained nearly as large a per-centage 
 of salt and other solid matters as the sea-water itself. The 
 necessity of making marine engines more economical of fuel 
 led to the abandonment of jet condensers at sea, and the 
 substitution for them of surface condensers, in which the 
 steam is condensed by contact with a cold metallic surface, 
 all mixture of the condensed steam and the injection water 
 being avoided. 
 
 There were two principal reasons for this change. The 
 first was that when jet condensers were used the boilers 
 could only be fed with salt water, which during the process 
 of evaporation became constantly more and more saturated 
 with salt, and which would, unless special measures were 
 taken, eventually deposit its solid contents in large masses 
 on the heating surface, and thus destroy the boiler, or 
 render it useless. The only way to avoid this was from 
 time to time to blow off large quantities of the brine in the 
 boilers, and to supply its place with corresponding quantities 
 of feed water. The hot water blown away from the boiler 
 involved, of course, a large loss of heat, the amount of which 
 depended on the state of saturation which the water was 
 allowed to reach before blowing off. If the latter be allowed 
 to reach three times the density of sea- water, the loss of 
 heat in blowing out would be 7 -4 per cent, of the total heat 
 supplied to the boiler ; and if less densities were permitted, 
 the loss was considerably greater. The maximum density 
 permissible was three times that of sea-water. The average 
 loss of heat due to blowing off may fairly be set down as 
 equal to from 12 to 15 per cent, of the total fuel supply. In 
 order to save this loss it was necessary to feed the boilers 
 with fresh water, and to use the condensed steam over and 
 over again as feed water. This of course could only be 
 
428 T/te Steam Engine. 
 
 effected by keeping the steam separate from the condensing 
 water, and hence the introduction of surface condensers. 
 
 The second reason which led to the abandonment of 
 jet condensers was that, in order to effect any considerable 
 economy in the engine as distinguished from the boiler, it 
 was necessary to resort to higher pressures of steam. Now 
 if the temperature of the steam be raised above 280, which 
 is that due to 35 Ibs. per square inch above the atmosphere, 
 the sulphate of lime contained in the sea- water is deposited 
 in hard and insoluble layers all over the boiler surface, and 
 destroys the efficiency of the heating surface. Hence for 
 this reason also the use of fresh water in the boilers, and 
 consequently of surface condensation, became a necessity. 
 
 The amount of water required to condense the steam 
 when surface condensers are used depends upon the 
 efficiency of the cooling surface in abstracting the heat, and 
 this again depends upon the thickness and conductivity of 
 the metallic surfaces, their condition as to cleanliness, and 
 the difference in temperature between the two sides. 
 
 In spite of the fact that the difference between the 
 temperatures is much less than in the case of boilers, the 
 efficiency of the cooling surface of the condenser in abstract- 
 ing heat is far greater than that of the heating surface of the 
 boiler in transmitting it. This is due in part to the fact that 
 the condenser surfaces are much thinner than are the tubes 
 and plates of a boiler, and they are also as a rule much 
 cleaner. For these reasons condensers have usually less 
 than half the surface found necessary for the boilers. Peclet 
 found experimentally that sheet copper backed by water of 
 the temperature of from 68 to 77 was capable of con- 
 densing 2 1 Ibs. of steam per square foot per hour ; while 
 Joule, adopting special means, condensed as much as 100 
 Ibs. per square foot in the same time ; but in practice it is 
 usual to allow one square foot for every 13 Ibs. of steam 
 of the terminal pressure usual in compound marine engines. 
 And even with this large allowance of surface the amount 
 
Surface Condensation. 429 
 
 of cooling water required is about forty per cent, greater than 
 with jet condensers. The usual allowance is about 30 Ibs. 
 of water per pound of steam for vessels which run in the 
 temperate zone, and about 35 Ibs. for the tropics. 
 
 In order to realise the advantage due to maintaining the 
 greatest possible difference of temperature between the two 
 sides of the metallic surfaces, the cooling water must be 
 kept in a constant state of circulation through the condenser 
 by means of a special pump, which removes the water which 
 has been warmed by the condensing steam from contact 
 with the plates. As the air-pump of a surface condenser is 
 only required to pump the condensed steam and air, it may 
 be considerably smaller than the pump of a jet condenser ; 
 but in spite of this advantage surface condensers are much 
 larger, heavier, and more costly than those in which the 
 steam comes into direct contact with the cooling water. 
 
 Examples of condensers. Fig. 187 gives a transverse and 
 longitudinal section of a jet condenser, applied to the quick- 
 running Allen stationary engines. The plunger of the pump 
 
 Fig. 187. 
 
 A is worked direct by the engine piston-rod prolonged back- 
 ward. The exhaust steam enters the condensing chamber 
 B showed in transverse section, by means of the pipe S. 
 It there meets with the jet of water D, which enters by the 
 pipe E, and is condensed. Every time the plunger is with- 
 
430 
 
 The Steam Engine. 
 
 drawn, the pressure in the condensing chamber predominates 
 over that in the pump chamber F, the valves, of which one 
 
 is shown at G, open, and the mixed air, vapour, and water 
 enter F. When the plunger returns, it displaces its own 
 volume of water, the level of the water in F rises, and forces 
 
Condensers. 
 
 431 
 
 out the air and vapour, and a certain amount of water, 
 through the valves HH, the hot water flowing away through 
 the pipe C to a receptacle called the hot well. The valves 
 G and H, which are made of india-rubber discs, are closed 
 
 Fig. 189. 
 
 by spiral springs which exert a pressure equivalent to a 
 quarter of a pound per square inch. With very quick-run- 
 ning engines these springs are found preferable to so 
 arranging the valves that they may^close by their own weight. 
 It will be noticed that the valves G G are seated on a slope. 
 
432 
 
 The Steam Engine. 
 
 This is to allow the air and vapour as they come through 
 to escape easily to the surface of the water in F, and so 
 avoid the possibility of the plunger working in a mixture of 
 air and water, which would injure its efficiency. The surface 
 of the water in this type of condenser is the real air-pump, 
 for it is by its rising and falling that the air is expelled and 
 admitted. The velocity with which the surface rises and 
 
 Fig. 190. 
 
 falls is, on the average, about 30 feet a second, while the 
 velocity of the plunger is 800 feet. 
 
 Figs. 1 88, 189 illustrate longitudinal and transverse sec- 
 tions of a modern marine surface condenser for a compound 
 engine capable of working up to 3430 horse-power. It contains 
 3402 brass tubes of ' inch diameter external and 15 feet 
 long. The total cooling surface is 10,018 square feet, being 
 at the rate of a square foot of surface to -342 horse-power. 
 The tubes are arranged in three horizontal nests, and in them 
 
Siirface Condensers. 
 
 433 
 
 the cooling water circulates, entering the bottom and being 
 discharged after passing through the top nest. The steam 
 enters at the top, passes round the outside of the tubes, and 
 is distributed evenly by means of the perforated plates, of 
 which some are shown in section above the top tubes in 
 fig. 1 88. 
 
 The air-pump which is shown in section in fig. 189 
 is worked by a lever from the cross head of the main engine. 
 It is of the single-acting vertical type, and is similar in 
 principle to the ordinary lift-pump. When the piston or 
 bucket ascends, it draws the condensed steam, air, and 
 vapour through the lower or foot valves, and at the same 
 time lifts whatever has passed through the piston-valves on 
 the down stroke through the head valves shown at the top 
 of the pump, after passing which the water flows away to 
 the hot well. There are two of these air-pumps to the con- 
 denser in question, each having a stroke of 3 feet, a diameter 
 of 34 inches, and a combined discharging capacity of about 
 -^ r of the volume of the low-pressure cylinder. The barrels 
 are of gun-metal, and the valves are indiarubber discs of the 
 type shown in fig. 191, where the full 
 lines represent the disc, above which is 
 a curved metal guard plate which pre- 
 vents the valve rising too high, and by 
 its shape ensures a quick return of the 
 valve to its seat, when the pressure which 
 causes it to open is removed. 
 
 Fig. 190 shows. another section of the 
 air-pump, together with the circulating 
 pump which forces the cold water through 
 the tubes. The two are cast together in 
 one piece. The circulating pump is a double action force- 
 pump of 3 feet stroke and 20 inches diameter. When the 
 engines are making 55 revolutions, it is capable of dis- 
 charging nearly 9 Ibs. of water per square foot of cooling 
 surface per minute. 
 
 F F 
 
 Fig. 191. 
 
434 The Steam Engine. 
 
 The circulating, like the air pump, is worked by a lever 
 from the cross head of the main engine. Very often centri- 
 fugal pumps are used for circulating the water in surface 
 condensers, and not infrequently they are driven by a separate 
 engine. 
 
 Condenser tubes are almost invariably made of brass, 
 which is sometimes tinned on both surfaces. Copper, 
 though a better conductor, is never used, as the fatty acids 
 formed in the condenser from the lubricating materials 
 carried over by the steam from the cylinders attack the 
 metal and form salts of copper, which, becoming dissolved 
 in the condensed steam, are carried back into the boiler 
 where they act most injuriously on the iron plates. 
 
 The packing of the ends of the tubes, so as to make a 
 steam-tight joint, is a troublesome and expensive operation. 
 The method in most general use is 
 shown in fig. 192. In the thick- 
 ness of the tube plates small stuff- 
 ing boxes are formed, the tape pack- 
 ings at the bottom of which are 
 tightened up by means of screwed 
 ferrules. When the tubes are se- 
 vertically, the bottom ferrule is 
 flanged so as to overlap the end of 
 
 the tube to prevent the latter from dropping out, should 
 the packing become loose. 
 
 Air pumps. Nothing is more important in a condenser 
 than the design of the air-pump. If the condenser is of the 
 old-fashioned type the pump has to discharge not only the 
 condensed steam, and the condensing water, but also the 
 large amount of air which is always present in sea water, 
 and which of course expands in volume when raised to the 
 temperature of the condenser. It may be stated that, on 
 an average, when the cooling water has a temperature of 60 
 and the condenser 120, the discharging capacity of the air- 
 pump should be from thirty-six to forty times the volume 
 
Air Pumps. 435 
 
 of the water into which the steam condenses. Hence its 
 theoretical capacity may be calculated when we know this 
 latter quantity, and the number of revolutions made by the 
 engine per minute, and also the nature of the pump, whether 
 double or single acting. The actual dimensions are deter- 
 mined when we know the efficiency of the pump that is to 
 say, the ratio which its actual bears to its theoretical dis- 
 charging capacity. Vertical single-acting air-pumps are by 
 far the most efficient. They are almost always used with 
 vertical engines, but when the latter are of the horizontal 
 type, the use of a double-acting horizontal pump is often 
 unavoidable. The actual efficiency of a single-acting vertical 
 pump is, in the most favourable circumstances, about 60 per 
 cent, of the theoretical, but should not in general be taken 
 as more than 50 per cent. The average efficiency of 
 horizontal double-acting pumps is about 35 per cent., or in 
 other words the actual size of such a pump should be about 
 three times greater than is theoretically necessary. 
 
 In surface condensers the pumps have only to discharge 
 the condensed steam, and any small quantity of air which 
 comes over by leakage ; but as surface condensers are gene- 
 rally arranged to act with a jet in case of necessity, it is 
 usual to make the pumps much larger than is ordinarily 
 necessary, though not so large as if jet injection were the 
 rule. 
 
 Ejector condensers. Fig. 193 illustrates a type of con- 
 denser which has no air-pump nor other moving parts. It 
 is similar in principle to the injector described on page 409. 
 Water having a pressure due to a few feet of head enters by 
 the pipe A and flows through the nozzle B. Exhaust steam 
 from one cylinder enters by the pipe C at a velocity vary- 
 ing with its pressure. If the pressure is 5 Ibs. absolute, the 
 velocity of the steam entering a vacuum of 25 inches of 
 mercury is about 1200 feet per second. The entering steam 
 surrounds the nozzle B, and is condensed on coming in 
 contact with the cold jet issuing from B, and increases ten 
 
 F F 2 
 
436 
 
 The Steam Engine. 
 
 energy of the stream, just as the boiler steam imparts energy 
 to the feed water of an ordinary injector. When the con- 
 denser is used with a two-cylinder engine, the exhaust steam 
 from the second cylinder is ar- 
 ranged to enter by the pipe M, and 
 is condensed by the combined jet 
 issuing from the nozzle F. The 
 steam from the second cylinder 
 further increases the energy of the 
 -!== jet which passes through the nozzle 
 N and into the trumpet- shaped pipe 
 P, where its velocity is gradually 
 reduced. With a steam pressure 
 equal to 147 Ibs. per square inch 
 absolute at the commencement of 
 exhaust, and a double-cylinder en- 
 gine, the energy imparted to the jet 
 is sufficient to raise the discharged 
 water to a height of 6 or 8 feet 
 above the condenser ; or, if there 
 is no head of condensing water to 
 start with, the apparatus can raise 
 the water from a level of 6 to 8 
 feet below the condenser. In the 
 latter cases, however, the apparatus 
 must be started by means of a jet 
 of boiler steam, introduced through 
 the pipe D and down the hollow 
 spindle E. The rate at which the 
 condensing water flows to the nozzle B is regulated by 
 shifting the spindle E up or down by means of a hand- 
 wheel and screw. 
 
 By means of these ejector condensers a perfectly steady 
 vacuum can be maintained of 24 to 25 inches of mercury. 
 The initial temperature of the cooling water should not, 
 in general, exceed 60, and the quantity of water supplied 
 
Ejector Condensers. 437 
 
 should be such that its temperature, on issuing from the con- 
 denser, would not be raised more than 20 to 25. Experi- 
 ments have however been made in which the rise of 
 temperature was 64 and the final temperature 120 ; but 
 the vacuum at this temperature fell about two pounds per 
 square inch. It must be borne in mind that, with con- 
 densers of this type, the whole of the power usually employed 
 in driving the air-pump is saved, the energy of the exhaust 
 steam being alone sufficient for the purpose of discharging 
 the condensing water and the condensed steam. 
 
 Vacuum indications. The amount of vacuum is indicated 
 by a Bourdon pressure gauge, graduated in inches of mercury. 
 It does not show the actual pressure in the condenser, but 
 the difference between that pressure and the external 
 atmosphere. For instance, if the barometer stood at 30 
 inches, and the indication of the vacuum gauge were 26 
 inches, the actual pressure would be 30 26 = 4 inches of 
 mercury. If the pressure in the condenser remained 
 constant, and the outside barometer varied, the indications 
 of the vacuum gauge would vary correspondingly. 
 
 In order to know if the vacuum is good or bad, notice 
 must always be taken of the state of the barometer. The 
 maximum attainable vacuum depends on the temperature at 
 which the condenser is maintained. Thus if the latter be 
 120, the pressure of vapour formed at this temperature is 
 i *68 Ib. per square inch ; then if the barometer stand at 30 
 inches of mercury, or 147 Ibs. per square inch, the maximum 
 vacuum attainable will be 147 1*68 = 12*02 Ibs. per square 
 inch. 
 
438 The Steam Engine. 
 
 CHAPTER XL 
 
 ON SOME OF THE PRINCIPAL CAUSES OF LOSS OF EFFI- 
 CIENCY IN STEAM ENGINES, AND THE METHODS 
 EMPLOYED FOR REDUCING THE LOSS SUPERHEATING 
 STEAM JACKETING COMPOUNDING. 
 
 Early improvement in the steam engine consisted in the separation of the 
 functions of steam generating, steam using, and condensing The 
 cylinder, even in modern engines, still acts as a generator and con- 
 denser Experimental confirmation of the foregoing Cause of con- 
 densation and re-evaporation in engine cylinders Hypothetical 
 example showing the successive stages of condensation and re-evapo- 
 ration Injurious effects of presence of water in cylinders Four 
 principal causes of the presence o* water in cylinders : i, priming ; 
 2, excess of condensation over re-evaporation ; 3. liquefaction duetto 
 work done ; 4, loss of heat by radiation Influence of dimensions of 
 cylinder, pressure of steam, and rate of expansion on the initial con- 
 densation Experiments on steam consumption in conducting and 
 non-conducting cylinders Means employed to diminish the loss due 
 to liquefaction : i, superheating the steam ; 2, steam jacketing The 
 benefits derived from the use of steam jackets Cases in which jackets 
 are useless Precautions to be observed in jacketing Experiments 
 showing the use of jackets in simple and compound engines 3, 
 cushioning, or compressing the exhaust steam ; 4, compounding 
 Variations in temperature of cylinders due to working high-pressure 
 steam expansively Compounding reduces the variation of tempera- 
 ture in each cylinder Tandem compound engines Two-cylinder 
 receiver compounds Three-cylinder ordinary compounds Triple 
 compound, or triple expansive engines Experimental demonstration 
 of the saving in fuel to be effected by compounding The distribution 
 of the steam in the various types of compound engines Actual indi- 
 cator diagrams of the various types of compound engines The 
 manner of reducing the diagrams of compound engines The mechani- 
 cal advantages of compound as compared with simple expansive 
 engines Example of the curve of twisting moments on the crank of a 
 triple compound engine The relative sizes of the cylinders of com- 
 pound engines Means of equalising the power developed in the 
 separate cylinders of compound engines Table of cylinder ratios for 
 various types of compound engines working with different pressures 
 of steam. 
 
 IT was pointed out in the opening chapter that in the 
 elementary form of steam engine the cylinder served the 
 triple purpose of boiler, engine, and condenser. It was 
 
Condensation in Steam Cylinders- 439 
 
 also shown that the principal subsequent improvements 
 consisted mainly in the separation of these functions, the 
 generation and condensation of the steam being now in- 
 variably effected in separate vessels, while the cylinder is 
 reserved for its true purpose the conversion of the energy 
 in the steam into mechanical power. In spite of these 
 improvements, the cylinder, however, still acts in expansive 
 engines, to a great extent both as a steam generator and 
 condenser ; and, moreover, it exercises these functions pre- 
 cisely at the wrong times, for, as will be shown presently, 
 the metal sides and ends abstract heat from the entering 
 steam, causing a partial condensation at the moment when 
 it is most desirable to maintain the temperature ; while, on 
 the other hand, the steam thus condensed is partly re- 
 evaporated during the whole period of expansion, when it 
 can only give out a fraction of its original energy, and partly 
 during the exhaust period, when it not only gives out no 
 useful energy whatever, but also tends to impair the vacuum 
 and to increase the back pressure. It is this peculiarity 
 which accounts, in a great measure, for the wasteful per- 
 formance of many engines which use steam expansively. 
 
 The foregoing statements have been confirmed by 
 numerous experiments on steam engines, in which the feed 
 water supplied to the boiler was carefully measured, and 
 the steam used in the engines was calculated from indicator 
 diagrams for the point before release, in the manner explained 
 in Chapter VIII., p. 343. In engines which work expansively, 
 the amount of steam used, calculated from the diagrams, is 
 nearly always considerably less than that supplied by the 
 boiler, even after every allowance is made for priming and 
 for the condensation due to the mechanical equivalent of 
 the work done. When no special precautions are taken the 
 deficiency is sometimes enormous. This proves that besides 
 the steam which issues at exhaust and which is accounted for 
 by the indicator diagram, there is also present in the cylinder 
 a quantity of water which must continue to evaporate during 
 
440 The Steam Engine. 
 
 the period of exhaust, and which may partly be carried over 
 mechanically into the condenser, and which has no more use- 
 ful effect than if it were passed straight from the boiler to the 
 condenser. 
 
 This peculiarity of engines which use steam expansively 
 is accounted for by the great difference which exists between 
 the initial and final temperatures of the steam in the cylinder. 
 Thus, if the entering steam had a pressure of 150 Ibs. per 
 square inch, and were expanded down to 20 Ibs. absolute, 
 the initial temperature would be 358, and the final 228, 
 showing a difference of 130 ; while, if the engine were 
 condensing, the cylinder would be exposed, during the 
 whole of the exhaust, to the temperature of the condenser, 
 say 158, showing a difference in this case of 200. During 
 the whole period of expansion and exhaust, the surface of 
 the metal forming the sides and ends of the cylinder and 
 piston is being cooled down by the combined effects of 
 radiation to the moist steam which readily absorbs radiant 
 heat and conduction to the condensed steam, which is 
 re- evaporated from the metallic surfaces. The result is, 
 that when the fresh steam re-enters the cylinder, it comes 
 in contact with the relatively cool surfaces, and a large 
 portion of it is condensed ; the latent heat of the steam 
 condensed raising the temperature of the sides up to that of 
 the entering steam. In order to make up for the steam 
 condensed, fresh steam rushes in from the boiler, and thus 
 much more steam enters during the. period of admission than 
 is required to fill the cubic contents of this portion of the 
 cylinder. 
 
 When the steam is cut off and commences to expand, 
 its pressure and temperature fall, but, during the early 
 period of expansion, its temperature is generally still higher 
 than that of the sides of the cylinder, and more condensation 
 takes place. At the same time, however, the reduction in 
 pressure allows a portion of the water formed by condensation 
 during the admission period to re-evaporate ; for this water 
 
Re-evaporation in Steam Cylinders. 44.1 
 
 has, when formed, the temperature of the entering steam, and 
 therefore, when the pressure in the cylinder falls, it possesses 
 too much heat to remain in the state of water, and part of 
 it boils off, the evaporation being aided and increased by 
 the supply of heat forthcoming from the hot sides and end 
 of the admission portion. Thus, during the whole period 
 of expansion, both condensation and re-evaporation go on. 
 During the earlier period of the stroke the condensation 
 predominates, but towards the middle and end of the 
 stroke, as the temperature of the expanding steam approxi- 
 mates more closely to that of the sides of the cylinder, the 
 condensation becomes less and less, and the re evaporation 
 becomes more active, and consequently the latter predo- 
 minates, while, during the exhaust, in consequence of the 
 very low pressure, the circumstances are most favourable 
 for re-evaporation. 
 
 It must be clearly understood that the action of the 
 sides of the cylinder above described goes on, no matter 
 how perfectly the outside of the cylinder may be protected 
 by means of non-conducting coverings. The latter are 
 only useful to prevent radiation of heat from the outer 
 surface of the cylinder, and in no way prevent the condensa- 
 tion due to loss of temperature by the sides during expan- 
 sion and exhaust, though, as will be subsequently shown, 
 the failure to protect the outside of the cylinder may 
 enormously increase the action of the sides. 
 
 The effects due to variation of temperature in the cylinder 
 may perhaps best be understood by examining the case of a 
 cylinder of given dimensions under circumstances which, 
 though they never actually occur in practice, are nevertheless 
 approximated to in very bad cases. 
 
 We will take as an example a cylinder the thickness of 
 the metal of which is only -,-J^ of an inch, and which is 
 supposed to be capable of instantly following the tempera- 
 ture of the steam. The cylinder is further to be lagged 
 with a perfectly non-conducting material ; the ends and 
 
44 2 The Steam Engine. 
 
 piston are supposed to be of the same thickness as the 
 sides. The steam is cut off at one-tenth of the stroke. 
 The following are the remaining data : 
 
 Diameter of cylinder . = 2 feet 
 
 Length of stroke . . = 4 ,, 
 
 Length of admission portion . -4 , , 
 
 Area of ends .... =3*14 square feet 
 
 Area of piston . . . =3*14 ,, 
 
 Area of admission portion of sides = 2-51 ,, 
 
 Initial pressure of steam . = 90 Ibs. per sq. inch absolute 
 
 Initial temperature of steam . = 320 
 
 Exhaust temperature . . = i57'5 
 
 We will next suppose that the temperature of the cylinder 
 when the steam enters is the same as that of the exhaust. 
 The exposed surface is that of the cylinder end, the piston, 
 and the admission portion of the sides, or, in all, 879 square 
 feet, and weighing 3-5 Ibs. If the metal is supposed to be 
 iron, the specific heat will be -n nearly, and the number of 
 thermal units required to heat up this weight of metal by 
 i c " = i62' is- 
 
 3-5 x 162-5 x 'ii = 62-56. 
 
 We must next find out what proportion of the heat con- 
 tained in the entering steam goes towards raising the tempe- 
 rature of the sides, &c., of the cylinder. The volume of the 
 admission portion of the cylinder is 1-256 cubic feet, and 
 the weight of steam contained in this space is -26 Ib. 
 Now the latent heat of i Ib. of steam of 90 Ibs. pressure 
 is about 888 thermal units, and, therefore, 888 x '26 = 231 
 thermal units is the latent heat of the steam which fills the 
 admission space. Now, as we have seen, about 62-5 units 
 must be applied to heating up the metal of the cylinder, 
 that is to say 27 per cent, or over one-fourth of the steam 
 condenses to water of the temperature of 320, and a corre- 
 sponding quantity of fresh steam from the boiler enters the 
 cylinder in order to make good the deficiency. 
 
 Let us next consider the state of things when the steam 
 
Condensation and Re-evaporation. 443 
 
 has expanded to double its bulk. It will then occupy a 
 length of *8 foot of the cylinder. Its final pressure will be 
 about 45 Ibs., its final temperature 275, and the mean tem- 
 perature of the steam during expansion = _Z5j[l3 _ 297^5 
 
 approximately, which is still about 138 above the tempera- 
 ture of the exhaust. Now, to heat up the 2-51 square feet 
 area of side in which the expansion has taken place requires 
 2-51 x -4 X 138 x *ii = 15*25 thermal units, which quantity 
 of heat must be supplied by the condensation of a portion 
 of the expanding steam. But, in the meantime, the tempe- 
 rature imparted to the metal during the period of admission 
 is 22*5 above the mean temperature of the steam during 
 the period of expansion which we have just been considering. 
 The temperature of the 27 per cent, of '26 Ib. of condensed 
 steam is also 22'5 above the average, and is consequently 
 too hot to remain as water at the reduced pressure, and 
 consequently a portion of it re-evaporates. The supply of 
 heat available for re-evaporation is (3*5 x 22^5 x 'ii) + 
 (067 X22 '5x1)=-- 10*17 thermal units ; so that during the 
 first stage of the expansion 15 '25 units are abstracted from 
 the steam, while only 10*17 units are given back to it, and 
 consequently condensation largely preponderates over re- 
 evaporation. 
 
 If the stroke were divided into ten equal portions, and 
 the processes of condensation and evaporation followed in 
 each, in the manner above indicated, it would be found 
 that in the next stage the two processes would be nearly 
 equal, and that throughout the remainder of the stroke the 
 evaporation would preponderate. 
 
 If we suppose that the pressure of the steam before 
 release is about 9 Ibs. per square inch, its temperature will 
 be about 188, or about 30 -5 above the exhaust. The weight 
 of the whole exposed surface of the cylinder and piston will 
 be 1 2 -5 Ibs., and if this fall to the temperature of the con- 
 denser, 1 2 -5 x 30-5 x Ti =42 thermal units will be available 
 
444 The Steam Engine. 
 
 for re- evaporation during the exhaust period, without count- 
 ing the surplus heat in whatever water may be present in 
 the cylinder at the end of the stroke. 
 
 The above description of what goes on in a hypothetical 
 cylinder differs very materially from the actual series of 
 operations as they occur in steam engines. In the first place, 
 the metal of which the cylinder is composed is greatly 
 thicker than in the example given. When once the thick 
 mass of metal has been heated up. it probably exercises a 
 moderating influence on the maximum difference of tempera- 
 ture. Again, metal does not possess the power of instantly 
 taking up the temperature of the steam with which it may 
 be brought in contact. A certain interval of time is necessary, 
 and hence the speed with which the series of operations is 
 repeated, or, in other words, the velocity of the piston may 
 materially affect both the initial condensation and the re- 
 evaporation. Moreover, the dryness or wetness of the sides 
 exercises an immense influence on the condensation. Sup- 
 pose, for instance, that, in the above example, the admission 
 space of the cylinder had been covered with a film of water 
 of one-hundredth of an inch in thickness, of the same 
 temperature as the condenser ; the number of heat units that 
 would have been required to raise it to the temperature of 
 the entering stream would have been 74*25, as against 62-56 
 required to heat up the metal sides to the same degree, and 
 consequently the condensation in the latter case would have 
 been more than double what it would be were the sides dry 
 to start with. 
 
 In addition to the direct condensing effect which water 
 in the cylinder exercises, it also acts as a most excellent 
 medium for the interchange of heat between the steam and 
 the sides. It has been already stated that perfectly dry 
 steam is a very indifferent radiator and conductor of heat, 
 while moist steam is comparatively good in both respects. 
 Now the presence of water in the cylinder, by increasing the 
 moisture of the steam, aids in the rapid transmission of heat 
 
Water in Steam Cylinders. 445 
 
 from the latter to the metal sides ; and, vice versa, the power 
 of the sides in transmitting heat to the exhaust steam is 
 enormously increased by the presence of a film of water, 
 which, as fast as it evaporates, takes up heat from the metal 
 with great rapidity. 
 
 There are four principal causes for the existence of water 
 in the cylinders, viz. 
 
 Priming, or the carrying over of water in the fresh steam 
 from the boiler. 
 
 Excess of initial condensation over re-evaporation, which 
 always occurs when the expansion is carried beyond a certain 
 point. 
 
 The disappearance of heat due to the work done by the 
 steam. 
 
 Loss of heat by radiation from the external surface of 
 the cylinder. 
 
 The latter cause of condensation always exists when the 
 cylinder is uncovered, or but imperfectly protected. The 
 great indirect harm caused by water in the cylinder accounts 
 for the fact already alluded to, that want of proper lagging 
 increases the loss of efficiency in expansive engines, the loss 
 being enormously greater than the amount of heat which 
 escapes by radiation. Of these causes, the first priming 
 is probably not always as harmful as might be supposed. 
 Very often the water comes over from the boiler in the con- 
 dition of minutely fine spray, thoroughly mixed with and 
 having the same temperature as the body of the steam. 
 Under these circumstances priming is not so injurious as 
 when comparatively large bodies of water come over at 
 intervals, which accumulate in the cylinder, or form a film 
 over the surfaces. 
 
 The second cause excess of condensation over re- 
 evaporation is a most fruitful source of waste, and should be 
 most carefully guarded against. It results in the continuous 
 accumulation of water in the cylinder, and consequently 
 causes an amount of waste which goes on increasing with each 
 
446 The Steam Engine. 
 
 stroke. It always exists when the expansion is carried 
 beyond a certain limit, and this is one of the reasons why 
 excessive expansion in a non-compound engine is unat- 
 tended by any economy. The exact point to which expan- 
 sion can be safely carried, so as to avoid the accumulation of 
 water, depends largely upon the piston speed and the dimen- 
 sions of the cylinder. It is also greatly affected by the supply 
 of heat to, or its abstraction from, the steam during the progress 
 of expansion. If the cylinder be kept warm artificially, the 
 limit of economical expansion can be considerably extended \ 
 on the other hand, if radiation from the outer surfaces be 
 permitted, the limit will be correspondingly reduced. 
 
 The third cause disappearance of heat due to work 
 done is comparatively insignificant, and as the resulting 
 condensation takes place throughout the mass of the steam, 
 and does not form a film over the surfaces, the mischief due 
 to this cause may be overlooked. 
 
 The fourth cause loss of heat by external radiation 
 can always be effectually guarded against by properly covering 
 the cylinders. 
 
 The initial condensation which takes place is, independ- 
 ently of all other considerations, dependent on the dimen- 
 sions of the cylinder, the ratio of expansion, and the pressure 
 of the steam. Let d be the diameter of the cylinder in feet ; 
 / the length of stroke ; r the ratio of expansion ; V the 
 volume of one pound of steam at its initial pressure ; and E 
 the exposed surface of sides and ends of cylinder and of 
 face of piston per pound weight of steam. Then we have- 
 Exposed surface of end and piston face = - 
 
 2 
 
 admission portion of sides = 
 
 Cubic contents of admission portion = - - 
 
 Weight of steam in 
 
Condensation in Steam Cylinders. 447 
 
 Therefore the exposed surface per pound of steam, or 
 
 ^ Trd' 2 , irdl 
 H, - -}- 
 
 2 r 4/v 
 
 That is to say, the exposed surface per pound weight of 
 admitted steam, on which, other things being equal, the 
 initial condensation depends, varies directly with V, the 
 volume of one pound of the steam, and therefore varies 
 inversely with the pressure. It also varies directly with r, 
 the ratio of expansion, and inversely with / and </, the 
 dimensions of the cylinder. Hence we see that initial 
 condensation is diminished by increasing the pressure of the 
 steam and the size of the cylinder, and by diminishing the 
 expansion. One of the reasons why small steam engines 
 are so uneconomical compared with large ones is the 
 relatively large surface exposed by their cylinders per pound 
 of steam admitted. 
 
 Experiments have been carried out upon steam engines 
 in order to ascertain the loss due to condensation in 
 cylinders, and to test the effects of the various methods of 
 reducing the loss. Some of the results are given on 
 page 453, and prove that in unfavourable cases, with 
 unjacketed cylinders, the steam wasted from this cause may 
 be very nearly equal to the steam accounted for by the 
 indicator diagram. 
 
 Experiments have also been made by Mr. Emery, in 
 America, to test the saving to be effected by using a non- 
 conducting material for the cylinders in place of metal. 
 Two cylinders were used of identical dimensions, one made 
 of glass and the other of iron. They were worked under 
 precisely similar conditions as to steam pressure, valves, 
 ratio of expansion, &c. The steam used by each was 
 condensed and carefully measured. The average of several 
 
448 The Steam Engine. 
 
 series of experiments proved that the iron cylinder con- 
 sumed double as much steam on the average as the glass 
 one. 
 
 Methods employed to diminish the Loss due to Liquefaction 
 in the Cylinders and subsequent Re-evaporation. The great 
 majority of the improvements in the modern steam engine 
 have been directed towards diminishing the losses which 
 come under the above heading. The remedies all consist 
 in methods, either for the addition of heat to the steam before 
 it enters, or while it is in the cylinder ; or else for reducing 
 the range of temperature within the cylinder. These 
 remedies are four in number, two belonging to each category. 
 They may be, however, and often are, used in combination. 
 They are as follows : 
 
 ist, the use of superheated steam. 
 
 2nd, jacketing, or surrounding the cylinder with hot 
 steam from the boiler. 
 
 3rd, cushioning, or compressing the exhaust steam in 
 the cylinder before fresh steam is admitted. 
 
 4th, compounding, or allowing the steam to expand in 
 two or more cylinders successively instead of in one, thus 
 reducing the range of temperature in each. 
 
 Superheated Steam. Steam is said to be superheated 
 when heat is added to it after it reaches the ordinary or satu- 
 rated condition. Superheated steam resembles a perfect gas 
 in its behaviour. Its specific heat at constant volume is -336 
 thermal unit. That is to say, the addition of about one- 
 third of a thermal unit will raise the temperature of one 
 pound of steam by one degree ; so that a comparatively 
 small addition of heat will raise the temperature of the steam 
 very considerably. In practice steam is superheated by 
 passing it, on its way from the boiler to the cylinder, through 
 tubes or vessels surrounded by the hot gases as they pass 
 from the furnace to the chimney. 
 
 If we revert to the example given on page 442, in which 
 62 56 units were abstracted from the entering steam, it will 
 
Superheating. 449 
 
 be seen that we should need to add 62*56-:- '336= 1 86 
 to the steam before entering the cylinder, in order that it 
 might possess heat enough to bring up the temperature of 
 the cylinder sides to that of steam of 90 Ibs. pressure, and 
 still remain in the condition of dry saturated steam. As, 
 however, the natural temperature of steam of this pressure is 
 320, the temperature of the superheated steam would have 
 to be 320 + 186 = 506 in order effectually to prevent initial 
 condensation. Such a high temperature as this, however, 
 is quite unsuited for use in the steam engine, as it burns up 
 and hardens all the lubricants, destroys the packing in the 
 stuffing-boxes, and exercises a chemically corrosive effect on 
 all brass and copper work with which it may come in contact. 
 It is found in practice that whenever steam reaches the 
 temperature of 400, injury to the cylinder, &c., invariably 
 follows. Hence we see that high-pressure steam cannot be 
 sufficiently superheated to thoroughly cure initial condensa- 
 tion. The case is different, however, with steam of com- 
 paratively low pressure, such as 30 to 35 Ibs. absolute per 
 square inch. If steam of 30 Ibs. pressure had been used in 
 the example already given, the difference between the 
 temperature of admission and exhaust would have been 
 259 157*5 = ioi'5. Consequently, only 39 thermal 
 units would have been abstracted from the entering steam, 
 the temperature of which would have needed the addition 
 of 39-7- '336=116, in order to provide for this loss. Thus 
 the temperature of the superheated steam on entering 
 the cylinder would be 259 -f 116 = 375, which falls 
 just within the safe limit. Of course the pressure as well as 
 the temperature of steam is raised by superheating, and in 
 the example, which has been given merely for the sake 
 of illustration, the shape of the diagram and the power 
 indicated up to the point of cut-off would be altered by the 
 process. 
 
 It will be seen from the foregoing that superheating 
 the boiler steam is impracticable for modern high-pressure 
 
 G G 
 
450 The Steam Engine. 
 
 engines, but it is used with great advantage in the old type 
 of low-pressure marine engine. When the boilers are given 
 to priming, a moderate amount of heat may with advantage 
 be added to the steam after it is formed, so as to insure its 
 being thoroughly dry before it enters the cylinder. In this 
 way much of the loss due to accumulation of water in the 
 cylinders may be avoided. 
 
 Steam Jacketing. One of the most common methods of 
 checking the waste due to condensation is to supply heat to 
 the steam while in the cylinder. This is done by surrounding 
 the cylinder proper with a metallic casing, an interstice 
 being left between the two, in which circulates steam of the 
 boiler pressure. An example of a jacketed cylinder has 
 been given on page 206, fig, 49. The effect of the jacket is 
 to supply heat to the inner surface of the cylinder walls, to 
 make up for that abstracted during expansion and exhaust. 
 The result is that the difference between the temperatures 
 of the sides is greatly reduced, and condensation is checked 
 while re-evaporation is promoted, and a preponderance of 
 condensation over re-evaporation is greatly hindered. 
 
 The heat supplied to the cylinder by the jacket causes a 
 certain amount of the jacket steam to liquefy, so that the 
 question may be asked, wherein lies the advantage of this 
 system over that of allowing the entering steam to supply 
 the required heat by its own liquefaction ? It is occasionally 
 urged that in the latter case a portion at least of the con- 
 densed steam does work by subsequent re-evaporation, 
 whereas in the case of the jacket the condensed steam can 
 do no work whatever. The answer to this question is, that 
 the loss when the heat is supplied by the incoming steam is 
 mostly due to the fact that the steam in the cylinder is 
 rendered moist by the first condensation, and consequently 
 becomes a good radiator and conductor of heat, and thus 
 enormously increases the action of the sides. The mere alter- 
 ation of the temperature of the sides would not cause much 
 loss if only the steam could be kept dry even in a comparatively 
 
Steam Jacketing. 451 
 
 cool cylinder, because in this case there would be little inter- 
 change of heat between the steam and sides. Consequently 
 the benefit of a jacket lies in the fact that the condensation 
 which does take place occurs for the most part outside the 
 cylinder, and the inside steam is kept comparatively dry, and 
 thus the interchange of temperature between the sides and the 
 working steam is greatly checked, and the sides themselves 
 are consequently prevented from rapidly taking up the tem- 
 perature of the steam. Thus a jacket operates in two ways in 
 keeping the temperature of the cylinder walls constant. First, 
 by keeping the working steam comparatively dry, it reduces 
 the power of the sides of receiving heat from and giving it 
 out to the former, and thus deprives the sides of the power 
 of taking up the extremes of temperature which would other- 
 wise be possible ; and second, whatever differences of tem- 
 perature would actually occur are further greatly reduced by 
 the flow of heat from the jacket steam to the inner walls of 
 the cylinder. It is only the heat supplied in the latter 
 process which costs the jacket steam anything. The great 
 gain due to the rendering the working steam non-conduct- 
 ing and non-radiating costs nothing whatever, seeing that 
 it is an indirect effect of keeping the sides hot. Thus the 
 steam jacket, though for half its time it is warming the 
 exhaust, has proved in the majority of cases to be an un- 
 doubted source of economy. It would probably be much 
 more perfect in its action were it not for the fact, that it is 
 much more difficult to supply heat to the inner surfaces of 
 the cylinder through the thick sides, than it is for the cool 
 moist steam to abstract heat from these same surfaces with 
 which it is in immediate contact. The sluggish action of 
 the sides in conducting heat is further promoted by the 
 difficulty of keeping up a really effective circulation of steam 
 in the jacket. 
 
 Cases occasionally occur in which steam jackets ha\e 
 been proved to have had little or no effect upon fuel 
 economy. It has been already mentioned th.it when expan- 
 
452 The Steam Engine. 
 
 sion is carried too far, condensation preponderates over 
 re-evaporation. Now one of the effects of steam jacketing 
 would undoubtedly be to increase the profitable range of 
 expansion. If, however, the expansion were carried beyond 
 this point, so that the condensation preponderated, water 
 would of course accumulate in the cylinder, and the steam 
 would become moist in spite of the action of the jacket, 
 which latter would consequently be of little or no use. 
 
 Precautions to be observed in Jacketing Cylinders. Many 
 cases have occurred in practical experience in which jacket- 
 ing has been carried out so as to be positively injurious 
 instead of beneficial. One of the commonest mistakes was 
 to allow the exhaust steam to expand into the jacket, so 
 that the cylinder, instead of being surrounded by a medium 
 hotter than its own mean temperature, was really immersed 
 in steam to which it was in a condition to impart heat. 
 Occasionally the fresh steam from the boiler is allowed to 
 pass through the jacket before it enters the cylinder. This 
 is a bad practice, although it ensures circulation in the 
 jacket, because, as has been shown before, jacket steam to 
 be effective must liquefy, and nothing can be worse than to 
 cause the steam to become moist before it enters the cylinder. 
 Very often it has been found that, in consequence of the 
 absence of good automatic means for drawing away the 
 water caused by condensation in the jackets, the latter have 
 become filled with cold water, thereby aggravating all the 
 evils which they were intended to mitigate. In every case 
 jackets should have their own independent supply of boiler 
 steam, and should be provided with trustworthy automatic 
 means for carrying off the condensed steam. 
 
 Experiments on the effect of Steam Jackets. The follow- 
 ing table is extracted from the record of some experiments 
 made by Mr. Charles Emery on the engines of the United 
 States Coast Survey steamer 'Bache,' in the year 1874. 
 The engines were compound, the high-pressure cylinder 
 having a diameter of 15*98 inches, the low-pressure 25 
 
Experiments on Jacketing and Compounding. 453 
 
 
 
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454 The Steam Engine. 
 
 inches, and the stroke 24 inches. During part of the trials 
 the large engine was used alone as a simple single cylinder 
 engine with and without jacket. During the remainder of 
 the trial the engines were worked compound, the large cylin- 
 der with and without jackets : the high-pressure cylinder had 
 no jacket. 
 
 From an inspection of this table we see that under all 
 circumstances, whether the engine was worked simple or 
 compound, a considerable saving was effected by the use of 
 the jacket, when corresponding rates of expansion are com- 
 pared. For instance, when worked compound, about 3 Ibs. 
 of water were saved per horse-power per hour, being at the 
 rate of about 13 per cent.; while in the case of the simple 
 engine worked at the higher rates of expansion as much as 
 22-5 per cent, was saved. 
 
 The same experimenter carried out tests on other marine 
 engines at about t]je same period, and the result, as given 
 in the published statements, conclusively proved the efficacy 
 of the jacket. 
 
 Cushioning, or compressing the Exhaust Steam in the 
 Cylinder before the Readmission of Fresh Steam. It has 
 been already pointed out that in most engines the exhaust 
 port is closed before the end of the stroke, and the steam 
 then remaining in the cylinder is compressed at the expense 
 of the energy in the steam on the other side of the piston. 
 The advantages of compression in saving the steam required 
 to fill clearance space and steam ports, and in improving the 
 diagram of twisting moment on the crank, have been already 
 made clear. There is, however, another advantage to be de- 
 rived from compression, inasmuch as the steam while under- 
 going forcible diminution of volume rises in temperature, and 
 partly restores the sides and end of the cylinder to their 
 original condition. In condensing engines compression is 
 rarely carried out sufficiently to be of any very great use ; but 
 in high-pressure engines, with link expansion gear, the greater 
 the ratio of expansion the greater must also of necessity be 
 
Compression of Exhaust Steam. 455 
 
 the degree of compression (see fig. 109, and also diagrams, 
 page 338), and it is probable that in these cases compression 
 diminishes very greatly the tendency to initial condensation. 
 Moreover, it is a remedy which, when link gearing is used, 
 automatically proportions itself to the extent of the evil. For 
 the tendency to condensation grows as the rate of expansion 
 increases, and in the same ratio, as has just been pointed out, 
 the compression increases also. This is probably one of the 
 reasons which accounts for the economical performance of 
 locomotive engines. 
 
 Compounding, or allowing the Steam to expand succes- 
 sively in two or more Cylinders. The use of steam of very 
 high pressures, worked at great rates of expansion, causes 
 such severe variations in the temperature of the cylinder, 
 especially in condensing engines, that the remedies above 
 referred to, taken by themselves, fail to cure the evil of 
 initial condensation. Hence it becomes necessary to divide 
 the total expansion between two or more cylinders, so that 
 each cylinder, in an ordinary two-cylindered engine, roughly 
 speaking, ranges through about half the difference of tempe- 
 rature which it would were the expansion carried out com- 
 pletely in a single cylinder. The term compound is applied 
 to all engines in which the steam is expanded partly in a 
 small or high-pressure cylinder, and thence exhausted into 
 a cylinder of larger diameter. In the most modern types of 
 marine engines the pressure used is so great reaching 
 sometimes 160 Ibs. per square inch above the atmosphere 
 that the range of temperature in each of the cylinders of an 
 ordinary compound engine is still too great, and in such 
 cases the steam is expanded successively in three, and even 
 in four cylinders. Such engines are called, according to 
 the number of the cylinders in which successive expansion 
 takes place, triple or quadruple expansive engines. 
 
 There are many different arrangements of compound 
 engines, depending upon the number of cranks used, and 
 upon the angles which these form with one another. 
 
456 
 
 The Steam Engine. 
 
 Tandem Engines. When only one crank is used the 
 two cylinders must obviously be in the same straight line. 
 This arrangement is illustrated in fig. 194, which shows 
 
 the two cylinders connected 
 to each other by a distance 
 piece, or conical bracket. 
 The large and small pistons 
 are connected to the same 
 piston rod, and the slide 
 valves are in the same 
 straight line, so that they 
 can be both worked off the 
 same eccentrics. It is 
 usual to keep the two 
 cylinders some little dis- 
 tance apart, in order that 
 each cylinder cover may have 
 its own stuffing-box, as other- 
 wise direct leakage would in 
 time take place between the cylinders. The pistons, of 
 course, occupy simultaneously the same positions in their re- 
 spective cylinders, and the steam exhausts directly from the 
 one face of the high-pressure on to the opposite face of the 
 low-pressure piston, there being no intermediate reservoir. 
 
 This type of compound engine has the merit of great 
 simplicity. It provides perfectly for the expansion of the 
 steam. It can be repeated as often as is wanted, so as to 
 provide for cases where two or more cranks are required, 
 and old-fashioned simple engines can be readily converted 
 to compounds on this type. As usually made its chief dis- 
 advantage lies in the long steam passages in the high-pressure 
 cylinder, caused by the necessity of bringing the valves of 
 both cylinders into the same straight line. These long 
 passages cause a considerable waste of steam. 
 
 Compound Engines with Cranks at Right Angles, and an 
 Intermediate Receiver. Very often it is found inconvenient 
 
 Fig. 194. 
 
Compound Engines. 457 
 
 to allow the length for horizontal, or the height for vertical 
 engines which is necessary when the tandem arrangement 
 is adopted. Consequently, when the engine has two cranks 
 at right angles to each other, as a rule a high and a low 
 pressure cylinder are placed side by side, each of them 
 driving its own crank. In such cases, however, the pistons 
 cannot occupy simultaneously the same positions in their 
 respective cylinders. For instance, when the high-pressure 
 piston has completed its stroke, the steam, which has just 
 done its work in the small cylinder, is ready to exhaust 
 into the large cylinder. But the large cylinder is not at 
 this moment ready to receive fresh steam, for its piston is 
 only at or about half-stroke. Hence the steam from the 
 high-pressure cylinder must in the first instance be dis- 
 charged into an intermediate receiver, in which it is com 
 pressed by the advancing high-pressure piston, until the large 
 piston has completed its stroke, when the low-pressure valve 
 opens, and admits the steam from the receiver. Figs 195, 
 196 show the general arrangement of a compound receiver 
 marine engine having the cranks at right angles, Fig 195 
 shows a longitudinal section through the cylinders, and fig. 
 196 is an end elevation, showing the arrangement of the 
 engine-frames and of the condenser and air-pumps. The size 
 of the receiver is a matter of some importance, It was 
 formerly the practice to make it a large receptacle, in some 
 cases surrounding the small cylinder. One of the conse- 
 quences of having a large receiver is, that when communi- 
 cation is opened between it and the exhaust ports of the 
 small cylinder there is a considerable drop in the pressure 
 of the issuing steam. This drop of pressure takes place 
 without the performance of any work, and there is a conse- 
 quent loss of efficiency in the steam owing to its expansion 
 not being continuous. This point will be referred to again 
 when we come to deal with the indicator diagrams of com- 
 pound engines. At the present time it is usual to limit the 
 size of the receiver ; in fact t it often consists merely of the 
 
45$ 
 
 The Steam Engine. 
 
 valve-box of the low-pressure cylinder, and the pipe con- 
 necting the latter with the high-pressure exhaust passages, 
 together with that ever-changing portion of the small 
 cylinder which is not cut off by the exhaust face of its valve. 
 
 Fig. 195. 
 
 Whenever the power of the engines is so great that the 
 low-pressure cylinder would become of very large diameter, 
 it is usual to have two low-pressure cylinders, which draw 
 
Compound Engines. 
 
 459 
 
 their steam from a common receiver. The cranks in such 
 cases are usually at angles of 120 with one another, though 
 sometimes, in order to secure a special distribution of the 
 
 Scale */** 
 
 Fig. 196. 
 
 steam, or a more uniform curve of twisting moments on the 
 shaft, the two low-pressure cranks are exactly opposite each 
 other, while the high-pressure crank is at right angles to 
 
460 
 
 The Steam Engine. 
 
 them, and occasionally the high-pressure crank makes a 
 right angle with one low-pressure crank, the remaining one 
 being at 135 with each of the others. Figs. 197, 198 show the 
 general arrangements of a three-cylinder compound engine 
 where two low-pressure cylinders are used, the valves being 
 
 at the sides instead of between the cylinders. In this 
 example the small cylinder is placed between the other two, 
 though occasionally the two low-pressure cylinders are 
 together, and the small cylinder outside. 
 
 Fig. 197 is a sketch of the front elevation, and fig. 198 a 
 section through one of the low-pressure cylinders with its 
 
Triple Expansive Engines. 
 
 461 
 
 piston valve. These engines belong to the well-known 
 Transatlantic mail steamer, the ' Arizona.' 
 
 Triple Expansive Engines. The same arguments which 
 justify expansion in two cylinders successively, when the 
 steam pressure lies between 60 and 90 Ibs. per square inch, 
 render it advisable to expand in three cylinders successively 
 
 Scale 
 
 t ths 
 
 Fig. 198. 
 
 when still higher pressures are used. Such engines are 
 called triple expansive, to distinguish them from three- 
 cylinder ordinary compound engines. They were first intro- 
 duced by Mr. A. C. Kirk. 
 
 The simplest arrangement of triple expansive engines is 
 that illustrated in the frontispiece and figs. 199, 200, in which 
 
462 
 
 The Steam Engine. 
 
 the high, intermediate, and low pressure cylinders are side by 
 side, each working a separate crank. Sometimes the high- 
 
 pressure is placed over the intermediate cylinder, tandem 
 fashion, the two working on to one crank, while the low- 
 pressure cylinder drives a separate crank. Occasionally, as 
 
Triple Expansive Engines. 
 
 463 
 
 in the case of ordinary compound engines, the low-pressure 
 cylinder is divided into two, in order to avoid excessive size. 
 
 In this case the small cylinder is placed over one low-pres- 
 sure cylinder, and the intermediate one over the other low- 
 pressure, the engine resembling a double tandem with four 
 
464 The Steam Engine. 
 
 cylinders and two cranks. The arrangement of the cylinders 
 is, in fact, a matter of convenience, and depends upon the 
 height and length available in the engine-room, and on 
 the approximation to uniformity which it is desired to effect 
 in the curve of twisting moments. Fig. 200 is a vertical 
 longitudinal section through the cylinders, and fig. 199 a 
 plan and half horizontal section through the cylinders, show- 
 ing the positions ot the piston valves, and the arrangement 
 of the steam pipes connecting the cylinders. 
 
 Saving of Fuel effected by Compounding. The saving 
 of fuel effected by the use of steam of the pressure of 
 from 1 20 to 150 Ibs. worked in triple expansion engines 
 may be put down as about fifteen to twenty per cent, over 
 the expenditure in ordinary compound engines using steam 
 of from 70 to 90 Ibs. pressure. One of the first marine 
 triple expansion engines ever made developed 1800 horse- 
 power on the trial trip of four hours, with an expendi- 
 ture of 1-28 Ib. of coal per hour. The ordinary con- 
 sumption at sea may be put down as about 1*5 to r6 Ib. 
 of coal per I.H.P. per hour. The corresponding consump- 
 tion of the older type of engine with the lower pressures 
 given above is generally from r8 to 2*2 Ibs. 
 
 The saving of fuel effected by compounding is well 
 illustrated by the table on page 453. Comparing the simple 
 and compound engines when both working with jackets, it 
 will be noticed that, with a ratio of expansion of between 
 5 and 6, the simple engine consumed 23-15 Ibs. of water 
 and the compound 20-36 Ibs. per I.H.P. per hour, showing 
 a saving in favour of the compound of 12^ per cent. For 
 ratios of expansion of between 7 and 9 the saving in favour 
 of the compound was about 15 per cent, showing, as might 
 be expected, that the greater the difference of temperature 
 in the cylinder, the greater the benefit to be derived from 
 compounding. The difference in the consumption of steam 
 between the simple engine without, and the compound 
 engine with jacket, is still more marked. For instance, 
 
Steam Distribution in Compound Engines. 465 
 
 with a ratio of expansion of 7*6, the simple engine without 
 jacket requires 9*3 Ibs. of water per horse-power more than 
 the compound with jacket, which shows that a saving of 
 nearly 32 per cent, is in this particular instance due to the 
 combination of the remedies of compounding and jacketing. 
 The distribution of the Steam in Compound Engines. 
 Bef9re we examine the actual indicator diagrams of compound 
 engines, or investigate their mechanical as distinguished from 
 their thermal advantages, it is necessary to trace out the 
 theoretical distribution of the steam in some of the types which 
 occur in practice. In doing so, we will assume for the sake 
 of simplicity that the steam is not released till the end of the 
 stroke, that there is no compression of the exhaust steam, 
 and that there is no resistance due to ports and passages. 
 Let v represent the volume of the small cylinder. 
 V volume of the large cylinder. 
 R ratio of the two cylinders. 
 V R volume of the receiver. 
 p ratio of the volumes of the receiver 
 
 and the small cylinder. 
 r rate of expansion in the small 
 
 cylinder. 
 r 1 rate of expansion in the large 
 
 cylinder. 
 
 E total rate of expansion. 
 / absolute initial pressure of the steam 
 
 in the small cylinder. 
 
 Then we have the total rate of expansion E = rR. 
 The pressure of the steam at the end of the stroke in 
 
 the large cylinder = |J, and the volume which it occupies 
 
 is V. 
 
 Hence, as the product of pressures and volumes is equal 
 during hyperbolic expansion, we have 
 
 IT .. P v V v 
 
 H H 
 
466 
 
 The Steam Engine. 
 
 A 
 
 Tandem Engines with a Receiver. This is the first case 
 which occurs in practice, because the passages between the 
 
 face of the high-pressure valve 
 and the back of the low-pres- 
 sure valve, including the valve- 
 box of the low-pressure cylinder, 
 constitute in themselves a re- 
 ceiver the effect of which must 
 not be neglected. 
 
 The steam is admitted into 
 the small cylinder during the 
 portion AB of the stroke, the 
 length of the line AB being 
 determined by the rate of ex- 
 
 O "KT'T) ------- IT"' J 
 
 Fig. 201. 
 
 pansion r. Its pressure at this point being/, and its volume 
 
 - . It then expands to the end of the stroke C, when its 
 volume is v and its pressure CD ~. 
 
 At this point communication is opened with the receiver, 
 the volume of which is V R , and which is filled from the last 
 stroke with steam having the pressure / R . Consequently we 
 have now a total volume # + V R and two masses of steam, 
 
 one having the pressure and the other the pressure / R . 
 
 These pressures at once equalise themselves and attain the 
 value 
 
 Let this pressure be represented by DE. Hence we see 
 that, unless at the moment of the opening of the exhaust the 
 receiver is already filled with steam of the pressure in the 
 
 small cylinder s there will be a sudden drop in the diagram 
 represented by CE. That is to say, expansion will take place 
 
Steam Distribution tn Compound Engines. 467 
 
 without any work being done, and the heat liberated in the 
 process will be expended in superheating the steam in the- 
 receiver. 
 
 The pressure in the receiver is of course the initial pres- 
 sure in the large cylinder. From the point O mark off 
 ON'=DE. In drawing the low-pressure diagram it must be 
 remembered that the volume of the large cylinder is R times 
 that of the small cylinder ; therefore on the line of volumes 
 any given part of the stroke will be represented by a line R 
 times as long as the same part of the stroke in the high- 
 pressure diagram. Thus, for instance, the whole stroke is 
 represented by OP = OD x R. 
 
 The steam now expands in the low-pressure cylinder 
 
 and receiver together till the point of cut-off, which is I of 
 
 the stroke. At this point the volume of the steam is made 
 up of three parts viz. 
 
 The portion of the large cylinder up to the cut-off 
 
 V 
 point = -j 
 
 The volume of the receiver = V R . 
 
 The portion of the small cylinder which yet remains to 
 
 be traversed = v ^ = v(i --, J. 
 
 The pressure is got, as usual, by remembering that the 
 product of the pressure and volume at all points is constant. 
 At the commencement of the stroke of the large cylinder 
 this product was 
 
 Therefore at the point of the stroke now under considera- 
 tion, the pressure FG = fa_ 
 r 
 
 II H 2 
 
468 The Steam Engine. 
 
 This is also the pressure of the point H in the diagram 
 of the small cylinder, where 
 
 As soon as. the steam is cut off in the low-pressure cylinder 
 it expands to the end of the stroke, and as all the steam 
 which is admitted to the high-pressure cylinder must even- 
 tually find its way to the low-pressure cylinder, its final 
 
 pressure PL = f. At the same time the steam in the 
 hi 
 
 receiver is being compressed by the advancing piston of the 
 small cylinder, and finally attains the pressure / R which was 
 assumed to exist in it just before the small cylinder exhausted 
 into it. 
 
 The value of / R can be easily expressed in terms of 
 the known quantities. For, as we have seen, the volume of 
 
 steam in the large cylinder when communication with the 
 
 y 
 receiver was cut off was -, 
 
 and its pressure 
 
 Now at the end of the stroke the volume is V and the 
 pressure is -. Hence the products of these two quantities 
 must be equal, or 
 
 r * V * 
 
 An expression from which p R may be obtained. 
 
Steam Distribution in Compound Engines. 469 
 
 The above equation may be simplified by substituting for 
 V R its equivalent pv 
 V Re; 
 
 E Rr 
 
 And then reducing we get 
 
 An equation which gives the receiver pressure just before 
 the small cylinder exhausts into it, in terms of the initial 
 pressure, the total rate of expansion, the rate of expansion 
 in the low-pressure cylinder, and the ratio which the volume 
 of the receiver bears to that of the small cylinder. 
 
 In the diagram fig. 201 OE' represents the pressure / R , 
 and the curve HE' shows the line of compression as it 
 affects the small piston. 
 
 In most cases the pressure in the receiver is less than 
 the terminal pressure in the small cylinder, and consequently 
 there is the sudden drop shown in the diagram by the line 
 CE. It is obvious, however, from the above equation that 
 the value of the receiver pressure can be varied by varying 
 r', the rate of expansion in the low-pressure cylinder. If it 
 be wished to adjust the cut-off in this cylinder, so that the 
 receiver pressure may just equal the terminal pressure in the 
 small cylinder, we can easily find the required rate of ex- 
 pansion by equating the value just found for the receiver 
 pressure with the terminal pressure of the small cylinder 
 
 viz. ^, and solving for r'. 
 
 Compound Engines with Receivers and Cranks at right 
 angles. In tracing the diagrams of the above type of engines 
 we must remember that, neglecting the effect of the length 
 of the connecting-rod, the large piston is always at mid-stroke 
 when the small piston is at either end of its cylinder ; con- 
 sequently the exhaust steam from the small cylinder enters 
 the receiver when the large piston is at mid-stroke. Whether 
 
47O The Steam Engine. 
 
 it also simultaneously enters the large cylinder depends 
 upon the point of cut-off in the latter ; if this be before 
 half-stroke, the steam cannot enter, but if after half-stroke, 
 it must enter the large cylinder as well as the receiver. 
 
 In the high-pressure cylinder the steam enters during the 
 portion AB of the stroke depending on the point of cut-off, 
 
 and expands to C, the end of the 
 stroke. At C the exhaust takes 
 place, and the pressure falls to 
 E. The next step depends on 
 the rate of expansion in the 
 large cylinder. If this be after 
 half-stroke the steam will expand 
 in receiver and large cylinder 
 ' together till the cut-off takes 
 place. Suppose this to happen 
 when the small piston has tra- 
 velled through DG of its return stroke, and the pressure has 
 then fallen to F. After this point the small piston continuing 
 to advance compresses the steam in the receiver till it reaches 
 the point I corresponding to half-stroke, the pressure having 
 then risen to H. Now when the small piston is at mid-stroke 
 the large one is just about to commence its stroke, and the 
 large cylinder is just commencing to take steam from the 
 receiver. Hence the pressure in the latter will fall till the 
 end of the stroke of the small piston, finally attaining the 
 value OK. 
 
 The diagram of the large cylinder must next be drawn, 
 the line of volumes, as usual, corresponding with the ratio 
 between the two cylinders. The initial pressure is the same 
 as the receiver pressure when the small piston is at half- 
 stroke, viz. HI. Set off, therefore, OL = HI to represent 
 this pressure. From this point the steam expands in receiver 
 and large cylinder till half-stroke, when the small cylinder 
 exhausts into the receiver. At half-stroke, then, the pressure 
 in the cylinder is the same as the pressure in the receiver 
 
Steam Distribution in Compound Engines. 471 
 
 when the small piston has returned to the end of the stroke, 
 viz. OK. Erect therefore SM = OK. At this point the 
 pressure suddenly rises to N, so thatSN = DE, and then the 
 steam again expands in receiver and large cylinder till the 
 point of cut-off T is reached, when the pressure TP is the 
 same as FG. From this point onwards the steam expands 
 in the large cylinder till the end of the stroke R, while the 
 steam in the receiver is compressed as already shown by the 
 line FH. At the end of the stroke the pressure in the 
 large cylinder falls to that of the condenser. 
 
 To express these various pressures in terms of the initial 
 pressure and the volumes of the two cylinders and the 
 receivers, we proceed precisely as in the first example. Let 
 k denote the portion DG of the stroke passed over by the 
 small piston, when steam is cut off in the large cylinder j 
 then GO (i k)vi$ the fraction of the volume of the 
 small cylinder which at that moment is open to the exhaust. 
 The terminal pressures CD and QR are exactly the same as 
 
 in the first example, viz. and respectively. Also the 
 r IL, 
 
 pressure in the large cylinder at the point of cut-off equals 
 the terminal pressure multiplied by the rate of expansion ; 
 
 or TP = 4^j and this, as already shown, is the same as the 
 
 JtLi 
 
 pressure GF. At this point the volume occupied by the 
 steam is made up of two parts, viz. 
 
 The contents of the reservoir = V R 
 
 and the portion GO of the small cylinder = (i k] v 
 
 and consequently the total volume = V R + (i k) v 
 
 As soon as steam is cut off in the low-pressure cylinder, it is 
 compressed in the small cylinder and receiver along the line 
 FH till the high-pressure piston is at half-stroke, when the 
 
 volume occupied by the steam = V R + -, 
 
 2 
 
 and the pressure 
 
472 The Steam Engine. 
 
 m = A = x * - x PI- 
 
 E VR + ? E ? + * 
 
 2 
 
 This is also the value of OL, the initial pressure in the 
 large cylinder. From thenceforward the steam enters the 
 large cylinder and expands in it and the receiver till half- 
 
 stroke, when its volume = V R -\ . 
 
 2 
 
 To obtain its pressure SM at this point we have 
 
 /.SM =- xR i- x 
 
 E -TT . V E . R 
 
 V R + - p + _ 
 
 It now only remains to find the pressure SN = DE, 
 which is that of the receiver when the small cylinder exhausts 
 into it. The resultant pressure and volume is derived from 
 two components, one being that of the steam in the small 
 
 cylinder, whose volume is v and pressure and the other 
 that of the steam in the receiver and half the large cylinder, 
 whose volume is V R + , and whose pressure SM is given 
 above. Compounding these two we get 
 
 2 
 
 Substituting for V R> V and r their values vp, vR, and 
 RE, reducing, we get 
 
Steam Distribution in Compound Engines. 473 
 
 Here, again, there is a considerable drop of pressure in 
 the receiver, and an increase of pressure in the low-pressure 
 diagram at half-stroke. In order that there should be no 
 drop it would be necessary that the terminal pressure in the 
 small cylinder should be equal to the receiver pressure. 
 Then the resultant pressure would be the same as either of 
 its components, and the line CD would equal DE. Equating 
 the values previously given for these two pressures, and 
 solving for R, we can if desired find out what must be the 
 ratio between the two cylinders, with a given rate of expansion 
 in the large cylinder, in order that the receiver pressure may 
 be equal to the terminal pressure in the small cylinder. 
 
 When the cut- off in the low-pressure cylinder takes place 
 before half-stroke the diagrams will differ somewhat from 
 those explained in the preceding example. 
 
 At C, fig. 203, the high-pressure cylinder exhausts into 
 the receiver, the pressure falling to DE. Now when the 
 small piston is at the end of its 
 stroke the large piston is at half- 
 stroke, and therefore the steam 
 is already cut off, and conse- 
 quently the exhaust steam from it 
 the small cylinder only enters the 
 receiver, and not the receiver 
 plus half the large cylinder, as 
 in the previous example. When 
 the small piston makes the Fig. 203 . 
 
 return stroke it compresses the 
 
 exhaust steam before it and in the receiver till mid- stroke, the 
 line EF being the curve of compression. When the small 
 piston is at mid-stroke the large piston is at one end of its 
 cylinder, and consequently draws steam from the receiver till 
 the cut-off is effected and the steam expands in the receiver 
 and the large cylinder along the curve FH. When the small 
 piston occupies the position I the- steam is cut off in 
 the large cylinder, and the steam in the receiver is then 
 
474 Tke Steam Engine. 
 
 compressed by the small piston along the line HK till the 
 end of its stroke. 
 
 The initial pressure OL in the large cylinder is of 
 course equal to the receiver pressure GF. From the point 
 L the steam expands in receiver and large cylinder together 
 till the point of cut-off S is reached, and from this point 
 expansion takes place in the large cylinder alone till the end 
 of the stroke. 
 
 The values of the terminal pressures in the two cylinders 
 
 are the same as in the previous example, viz. *- and . At 
 
 ?* E 
 
 the point of cut-off in the large cylinder, the pressure SM 
 equals the terminal pressure, multiplied by the rate of 
 expansion in the large cylinder ; therefore 
 
 SM = x r' = HI 
 h, 
 
 Immediately the cut-off is effected the volume of the 
 receiver steam is made up of two parts, viz. the volume of 
 the receiver V R and the portion OI of the small cylinder 
 which the piston has yet to travel at the moment of cut-off 
 in the large cylinder. Now OI = (i k)v. Therefore the 
 total volume occupied by the receiver steam is V R + (i - k) v. 
 By the time the small piston reaches the end of its stroke 
 this volume is reduced to V R , and the pressure 
 
 The small cylinder next exhausts into the receiver, and 
 a volume of steam having the above pressure and volume 
 = V R becomes mixed with the contents of the small cylinder 
 
 having a pressure ^ = ^ and volume = v. 
 The resulting pressure 
 
Steam Distribution in Compound Engines. 475 
 
 ^(V. + d-^ + jg /(p+I _, ) + R . 
 V R + v = E p + i 
 
 This body of steam is compressed in the receiver by the 
 advancing high-pressure piston till it reaches mid-stroke, 
 when communication is opened with the large cylinder. 
 Therefore, the volume is reduced at this point by half the 
 
 contents of the small cylinder, and becomes V R + -. 
 While its pressure 
 
 I 
 E 
 
 GF = R - - R 
 
 " 
 
 This is also the initial pressure OL in the large cylinder. 
 
 In order to avoid drop in the receiver when the small 
 cylinder exhausts into it, we should have, as before, to 
 equate the values of DC and DE, and solve for R, which 
 would give us the necessary ratio of the two cylinders for 
 the given rate of expansion. 
 
 The .value of k, the fraction of the stroke traversed by 
 the piston of the small cylinder when steam is cut off in 
 the large cylinder, is not the same in 
 the two examples given. Let CE be -#, 
 the position of the crank of the large 
 cylinder when steam is cut off in it, in 
 the case of the first example, i.e., after A [ 
 half-stroke. Then the corresponding 
 position of the high -pressure crank is 
 found by drawing FC at right angles to 
 EC ; and H and G are the correspond- 
 
 ing positions of the two pistons. Then _ = k is the frac- 
 
 Ar> 
 
 tion of stroke traversed by the small piston when steam is cut 
 off in the large cylinder. 
 
476 The Steam Engine. 
 
 Calling the radius of the circle unity, we have 
 AG = i GC = i cos FCA = i - sin (9 ; 
 ,G i sin 
 
 A Tl 
 
 Also r f = rate of expansion in large cylinder. 
 BH 
 
 " I -f COS 
 
 .%cos0= 2 - r ' 
 
 r 1 
 
 Deducing the corresponding value of sin in terms of 
 ', we have 
 
 k r 2 ^ r 
 
 The case when steam is cut off in the low-pressure 
 cylinder before half-stroke is represented by fig. 205, the 
 same letters being used as in the pre- 
 vious example. 
 
 In this case it can easily be proved 
 
 that 
 
 APr ^ - IB , . 
 
 Space will not permit of a full in- 
 vestigation being given for all the 
 possible arrangements of cylinders, 
 but the principles on which all such calculations proceed, 
 having been fully illustrated in the three examples just given, 
 the student will have no difficulty in applying them to the 
 cases of ordinary compounds with two low-pressure cylinders 
 and cranks at any given angles, or to the case of triple com- 
 pound engines. In the case of ordinary compounds with two 
 low-pressure cylinders and cranks set at angles of 120 with 
 one another it is only necessary to bear in mind that three 
 separate cases may occur. According as the small cylinder 
 exhausts 
 
Steam Distribution in Compound Engines. 477 
 
 1. Into the receiver only ; which it does when the cut- 
 off in the large cylinder takes place before one-quarter stroke 
 
 2. Into the receiver and one of the large cylinders ; 
 which takes place when the cut-off in each low-pressure 
 cylinder takes place between one-quarter and three-quarters 
 stroke. 
 
 3. Into the receiver and both large cylinders ; which 
 takes place when the cut-off in the latter is after three- 
 quarters stroke. 
 
 The latter case never occurs in practice, because the 
 distribution of the steam would be very bad, as the high- 
 pressure cylinder would discharge into one of the large 
 cylinders when its piston was at one-quarter stroke, and into 
 the other at three-quarters stroke, i.e. just before the cut-off 
 took place. Hence, the amount of work done by each of 
 the two large cylinders would be very unequal 
 
 Actual Indicator-diagrams of Compound Engines. ' 
 We give below a few specimens of indicator diagrams of 
 various types of com- 
 pound engines. The 
 ifirst is taken from a 
 tandem, or direct ex- 
 pansion engine. The 
 upper and larger dia- 
 gram is from the small 
 cylinder, while the 
 ower one is from the 
 low-pressure cylinder. Fig. 206. 
 
 The latter diagram, 
 
 however, gives no idea to the eye of the relative work done 
 by this cylinder, for it must be borne in mind that, though 
 the pressures shown are low, the area of piston on which 
 :hey act is, as a rule, from three to four times that of the 
 high-pressure piston. In a subsequent example it will be 
 shown how to combine the diagrams of the several cylinders 
 of a compound engine, so that the work done by each may 
 
47* 
 
 The Steam Engine. 
 
 be directly compared, and the combined work compared 
 with what would be done by the steam, supposing the whole 
 expansion had taken place in the large cylinder only. 
 
 In the tandem type of engine the two cylinders are in 
 direct communication during the whole stroke till com- 
 pression begins in the small cylinder ; consequently the back 
 pressure line of the top diagram is practically identical during 
 the greater part of its length with the steam line of the lower 
 pressure diagram. The gap between them represents loss of 
 pressure due to the receiver formed by the pipes between 
 the two cylinders and to the resistance of the passages. 
 
 Fig. 207 represents a pair of diagrams from a two-cylinder 
 compound engine with receiver, the cylinders having dia- 
 meters of 46 and 87 inches 
 respectively, and the common 
 stroke of 57 inches, and the 
 ratio of the cylinders 3-6 to i. 
 The particulars as to the initial 
 and mean pressures and the 
 horse-power developed by each 
 cylinder are given on the dia- 
 grams. 
 
 It will be noticed that the 
 scales of pressure for each 
 cylinder are quite different, that 
 for the high-pressure being 60 
 Ibs. to the inch, and for the low- 
 pressure 1 6 Ibs. to the inch. In 
 order to make the diagrams 
 comparable, the pressure ordi- 
 nates ought each to be reduced to the same scale, and 
 the volume ordinates should be in the ratio of the volumes- 
 of the two cylinders. This is effected in the following 
 manner Draw the base line AB, fig. 208, corresponding with 
 the zero of pressure. Divide each diagram, fig. 207, by any 
 number of vertical ordinates. Divide the line AB at the 
 
 Steam ores - 76 Mi 
 Vacuum 
 
 Fig. 207. 
 
Diagrams of Compound En t 
 
 point C into two parts, so that AC : AB : : 
 cylinder to vol. of large cylinder : : i : 3-6. Divide AC and 
 AB into the same number of parts as there are divisions 
 in the original diagrams, and draw through these points a 
 series of vertical ordinates, Then measuring from the base 
 line AB mark off on these ordinates, to any convenient 
 scale, the pressures as found from the corresponding ordi- 
 nates on the original diagrams. And through the points 
 thus found draw the new diagrams as shown on fig. 208. 
 
 In order to compare the work done by the 
 steam in the compound engine with the work 
 that would be done if the expansion took 
 place all in the large cylinder, we must know 
 first the rate of expansion in the compound 
 engine. This in the present example is as 
 nearly as possible ten. Hence, allowing an 
 admission line DE of one-tenth the stroke, and 
 drawing in the hyperbolic line of expansion 
 EF, we obtain the approximate diagram for 
 this ratio of expansion 
 in a single cylinder, on 
 the assumption that 
 there is no loss from 
 condensation. It will 
 be seen that the two 
 actual diagrams fit 
 fairly well into the single approximate diagram after allowance 
 has been made for the ' drop,' or fall of pressure at the 
 end of the stroke of the high-pressure cylinder, and for the 
 resistance due to passages between the cylinders. Instead 
 of the hyperbola EF, the curve /# I<o646 =C, which represents 
 the adiabatic curve of expansion of dry saturated steam, is 
 often made use of. Some engineers consider that /e; I-2 =C 
 represents more closely what takes place. 
 
 Fig. 209 is a set of indicator diagrams taken from a three- 
 cylinder compound engine, having two low-pressure cylinders 
 
 Fig. 208. 
 
480 
 
 The Steam Engine. 
 
 fatter prtsiurt. n&U 
 Tteteiver 2' 
 Vacuum in Condense 
 devolutions pri mm 
 
 Top Tur4 Low 
 
 27<i2fit> 
 
 Bottom. 
 
 Aft 
 
 and cranks set at equal angles. The particulars as to boiler 
 and receiver pressure, vacuum, number of revolutions, horse- 
 power, &c., are given on the 
 diagrams. The engines from 
 which these diagrams were 
 taken belong to the Trans- 
 atlantic steamer the ' Arizona ' 
 illustrated in figs. 197, 198. 
 
 Fig. 210 gives the dia- 
 grams taken from a set of 
 modern triple expansive 
 marine engines, together 
 with a combination diagram 
 formed in the manner already 
 explained. The dotted line 
 shows the adiabatic expansion 
 of steam with the same ratio 
 of expansion in a single 
 cylinder, and the gaps between 
 the actual diagrams and the 
 adiabatic line show the losses 
 due to ' drop ' in the two re- 
 ceivers, to the resistance of 
 passages and to the conden- 
 sation which takes place in the 
 cylinders. 
 
 The mechanical advantages of compound engines. In 
 addition to diminishing the loss due to condensation in the 
 cylinders, the compound engine possesses mechanical 
 advantages over the older type of engine, in which the 
 expansion takes place completely in a single cylinder. In 
 the latter case there is a great difference between the initial 
 and mean strains on the piston, whenever the rate of 
 expansion is high ; whereas in the compound engine the 
 difference between the initial and mean strains in each 
 cylinder is much reduced. Similarly the twisting moment 
 
 Indicated Horse tow 
 
 Fig. 209. 
 
 I9?S-<H r nf *ZP 
 2433- IS JJP 
 1327-91 ffftlJ? 
 Tnial 
 
Mechanical Advantages of Compound Engines. 48 r 
 
 on the crank shaft is more nearly uniform in the case of the 
 compound engine. Now the dimensions of the moving parts 
 have to be designed to meet the maximum strains ; hence 
 
 Steam 
 
 Boiler / n^ u. P. reefer 4Z /**, 
 i. P. rece. S /6s 7 M. P. Jacket tflbs? 
 L. P Jacket tS/b* 
 
 feed nater 185' Discharge 100' 
 Sea mter S3 on Deck 30., 
 Cons * tf Coal per day 13 tons 4 cut 
 
 Fig. 210. 
 
 with compound engines a saving may be effected in the 
 weight of these parts. 
 
 As examples we will compare, first, the case of a 
 single cylinder expansive engine with a tandem compound ; 
 and, second, a two-cylinder expansive with a two-crank 
 compound of the receiver type, neglecting clearance and 
 compression and assuming hyperbolic expansion. 
 
 Let the initial pressure of the steam be 100 Ibs. per 
 square inch, absolute ; the ratio of expansion 6, and the 
 area of the piston in the ordinary expansive engine, A. 
 Then, if both engines have the same stroke, the area of the 
 low-pressure cylinder of the compound engine will also be 
 
 i i 
 
482 The Steam Engine. 
 
 A, and if the ratio of the two cylinders be, say, 4, the area 
 
 A 
 
 of the high-pressure cylinder will be . 
 
 4 
 
 Taking first the simple expansive engine, the theoretical 
 mean pressure for hyperbolic expansion is got by the formula. 
 
 Meanpressure=/ +1 g' E = I00 I + I '79i8 6 lbs 
 
 Jli 6 
 
 If the back pressure be 4 lbs. absolute per square inch, 
 then the mean effective pressure on piston = (46-54) A 
 = 42 '5 A lbs., the initial effective pressure on piston = 
 (100 4) A = 96 A lbs. 
 
 Taking now the case of the compound tandem engine, 
 we have, as in the first case, 
 
 The mean effective pressure = 42*5 A lbs. 
 
 The initial pressure on the low-pressure piston equals 
 the terminal pressure on the high, provided there is no loss. 
 
 Now the rate of expansion in the high-pressure cylin- 
 der = = 1-5. 
 4 
 
 Therefore the terminal pressure = - ^ = 66 '6 lbs. 
 Now the effective initial load on the small piston 
 
 = (ioo -66-6)-= 8-32 A. 
 
 4 
 And the effective initial load on the large piston 
 
 = (66-66 - 4) A = 62-66 A. 
 
 Total initial load = 71 A, as against 96 A in the first case, 
 the mean pressure being the same in each instance. 
 
 Take next the comparison between a two-cylinder simple 
 expansive engine and a two- cylinder receiver compound, 
 the area of the low-pressure cylinder of the latter being, as 
 before, denoted by A, while the piston area of each cylinder 
 
 of the simple engine is . The other data are supposed to 
 
 be unchanged. The mean pressure in the simple engine is 
 the same as in the first example, viz. 46-5 lbs. per square inch- 
 
Mechanical Advantages of Compound Engines. 483 
 The mean effective pressure on each piston 
 
 A 
 
 = (46*54) = 21*25 A lb s - 
 The initial effective pressure on each piston 
 
 = (100-4)- = 48 Albs. 
 
 Next take the case of the compound engine, and suppose 
 the ratio of the cylinders to be 3 to i. As the rate of ex- 
 pansion is 6, the cut-off in the small cylinder must be at 
 half-stroke, 
 
 In an engine of this description the receiver pressure 
 would probably be 28 Ibs. Now to find out the point of 
 cut-off in the low-pressure cylinder in order that this may 
 be the receiver pressure at the commencement of the stroke 
 of the large piston, we must remember that the total rate of 
 expansion is 6, and therefore the final pressure in the large 
 
 cylinder is - = i6 Ibs. Therefore the cut-off in this 
 
 cylinder is -- -^ = -59 of the stroke. From these data we 
 
 deduce the following particulars : 
 Mean pressure in small cylinder 
 
 = 100 *? e = 84*65 Ibs. per square inch. 
 
 Mean effective pressure in the small cylinder 
 
 = 84*65 28 = 56*65 Ibs. per square inch. 
 Mean effective load on small piston 
 
 = 56*65 x A = 18-88 A Ibs. 
 
 O 
 
 Initial effective load on small piston 
 
 = (100-28)- = 24 Albs. 
 
 O 
 
 Mean pressure in large cylinder 
 
 - 28 L+1^1^9 = 25 . 2 Ibs. per square inch. 
 
 1*69 
 Mean effective pressure in large cylinder 
 
 = 25*24 = 21*2 Ibs. per square inch. 
 
 I I 2 
 
4 8 4 
 
 The Steam Engine. 
 
 Mean effective load on large piston = 2 1 -2 x Albs. 
 Initial effective load on large piston = (28 4) A = 24 Ibs. 
 
 Hence we see that in the simple engine the initial load 
 is to the mean as 2 - i6 to i. Whereas in the small cylinder 
 of the compound engine the ratio is 1*27 to i, and in the 
 large cylinder 1-13 to i 
 
 It is unnecessary to pursue this investigation further, for 
 it is evident that in the cases of three cylinder ordinary 
 compounds, and triple expansive engines the ratios will be 
 more in favour of the compound system than in the two 
 examples given. 
 
 The foregoing calculations take no account of the strains 
 caused by the inertia of the reciprocating parts, and they 
 
 \ 
 
 Fig 211. 
 
 therefore only apply to the strains when the engine is moving 
 slowly, as, for instance, when stopping and starting. The 
 
Mechanical Advantages of Compound Engines. 485 
 
 
 proper method of investigating the true pressures on the 
 moving parts, when their inertia and the effects of compres- 
 sion are taken into account, is explained 
 in Chapter V. 
 
 The curve of twisting moments on 
 the crank axle is also more uniform in 
 the case of the compound engines, and 
 as the reciprocating parts are compara 
 tively light, the influence exerted by 
 their acceleration and retardation on the 
 curve of moments is less than in the case 
 of simple engines working at the same 
 rate of expansion Fig. 2 1 1 is given as 
 an example of the uniformity of twisting 
 moment which may be obtained with a 
 triple expansive engine having three 
 cranks set at 120, and fig. 212 is the 
 same diagram drawn on a straight 
 instead of a circular base. The indicator 
 diagrams from which the curves were 
 deduced are also given (see fig. 210). 
 
 The relative sizes of the Cylinders 
 in Compound Engines. Within certain 
 limits the engineer possesses consider- 
 able latitude in the choice of the ratio 
 of cylinders in compound engines. 
 The objects to be kept in view are i, 
 to divide the power as equally as pos- 
 sible between the cylinders 2, to avoid 
 an excessive rate of expansion in either 
 cylinder, which produces a high ratio 
 of initial to mean pressure and 3, to 
 avoid excessive drop in the receiver. 
 
 It is always possible to equalise the 
 power developed by the two cylinders) 
 by altering the point of cut-off in the large cylinder, no 
 
486 The Steam Engine. 
 
 matter what the ratio may be between the two. The effect 
 of increasing the rate of expansion in the large cylinder is to 
 increase the receiver pressure and consequently the initial 
 pressure, while at the same time it increases the back pressure 
 in the small cylinder. Consequently the proportion of the 
 power developed in the small cylinder is decreased. 
 
 As all marine engines, no matter what the initial pressure 
 may be, expand down to about the same terminal pressure, 
 the size of the large cylinder is fixed solely by the power to be 
 developed, for it must be large enough to contain the whole 
 volume of the steam at the terminal pressure. Hence the 
 ratio of the two cylinders is determined by the size we 
 choose to give the small cylinder. If we start with the 
 assumption that the low-pressure cylinder is to exert half of 
 the total power developed, then the mean pressure in it for 
 a given power is always the same, and as the terminal 
 pressure is also constant, the point of cut-off depends upon 
 the pressure in the receiver. 
 
 Now the larger the high-pressure piston relatively, the 
 smaller the mean pressure in it per square inch, for a given 
 power. But as the initial pressure is fixed, we must have a 
 more unfavourable ratio of initial to mean pressures with 
 relatively large high-pressure cylinders than with small. 
 Also the larger relatively the high-pressure piston, the 
 smaller must be the terminal pressure in it, and conse- 
 quently the -smaller also the receiver pressure would be, 
 unless it were kept up by an early cut-off in the large 
 cylinder. Thus we see that when the high-pressure cylinder is 
 relatively large, that is, when the ratio of the two cylinders 
 is small, the initial pressure in each must be high, relatively 
 to the mean pressures. While on the other hand the ' drop ' 
 in the receiver will be comparatively small. 
 
 As the boiler pressure increases, the rate of expansion 
 should increase also ; hence, either the cut-off must take 
 place earlier in the small cylinder, or else the ratio of the 
 two cylinders must be increased. If the proportions are 
 
Cylinder Ratios in Compound Engines. 487 
 
 such that the cut-off in the small cylinder must take place 
 before half-stroke, it will be necessary to provide this 
 cylinder with a separate expansion valve. 
 
 The following ! table gives the proportions of cylinders 
 most generally in use so as to avoid the use of expansion 
 valves, and to secure a favourable ratio of initial to mean 
 pressures with a moderate ' drop ' in the receiver. 
 
 
 
 Boiler-pressure absolute 
 
 
 
 85 
 
 95 
 
 105 
 
 "5 
 
 125 
 
 135 
 
 145 
 
 i55 
 
 165! 
 
 Tandem 
 
 Ratio of large to 
 
 4 to 
 
 4 
 
 4'S 
 
 S 
 
 _ 
 
 _ 
 
 .._ 
 
 _ 
 
 2_ 
 
 
 small cylinder 
 
 3 '5 
 
 
 
 
 
 
 
 
 
 3 cylr. receiver . 
 
 Combined ratio of 
 
 3 
 
 V4 
 
 V? 
 
 4 5 
 4 
 
 _ 
 
 _ 
 
 
 
 
 
 
 
 
 low-pressure to 
 
 
 
 
 
 
 
 
 
 
 small cylinder 
 
 
 
 
 
 
 
 
 
 
 Triple expansive 
 
 Ratio of low-pres- 
 
 
 
 
 
 
 
 
 
 S 
 
 S'4 
 
 S'8 
 
 6'2 
 
 6'6 
 
 
 sure to high-pres- 
 
 
 
 
 
 
 
 
 
 
 
 sure cylinder 
 
 
 
 
 
 
 
 
 
 In the case of triple expansive engines the ratio of the 
 large low-pressure to the intermediate cylinder should be 
 half that of the low-pressure to the high. 
 
 1 These figures are 
 Manual of the Marine 
 
 riven on the authority of Mr. A. E. Seaton's 
 Ingijte. 
 
APPENDIX. 
 
 TABLE I. 
 
 THE PROPERTIES OF SATURATED STEAM. 
 
 Temperature, 
 Fahrenheit, 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P 
 
 Heat required 
 to raise i Ib of 
 Water from 
 32 to t 
 Foot Ibs 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 100 
 
 942 
 
 
 
 
 101 
 
 971 
 
 
 
 
 I O2 
 
 I'OOI 
 
 
 
 
 103 
 
 I-03I 
 
 
 
 
 104 
 
 I-062 
 
 55>6i2 
 
 859,793 
 
 312-8 
 
 i5 
 
 1-094 
 
 
 
 
 106 
 
 127 
 
 
 
 
 107 
 
 160 
 
 
 
 
 1 08 
 
 195 
 
 
 
 
 109 
 
 230 
 
 
 
 
 no 
 
 267 
 
 
 
 
 III 
 
 304 
 
 
 
 
 112 
 
 342 
 
 
 
 
 U3 
 
 381 
 
 62,560 
 
 861,908 
 
 244* 
 
 114 
 
 421 
 
 
 
 
 "5 
 
 462 
 
 
 
 
 116 
 
 '54 
 
 
 
 
 117 
 
 '547 
 
 
 
 
 118 
 
 59i 
 
 
 
 
 119 
 
 637 
 
 
 
 
 1 20 
 
 683 
 
 
 
 
 121 
 
 731 
 
 
 
 
 122 
 
 779 
 
 69,522 
 
 864,024 
 
 I92- 
 
 I2 3 
 
 829 
 
 
 
 
 124 
 
 880 
 
 
 
 
 125 
 
 932 
 
 
 
 
 126 
 
 985 
 
 
 
 
490 
 
 The Steam Engine. 
 
 Temperature, 
 
 Fahrenheit, 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P 
 
 Heat required 
 to raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat o" 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 127 
 
 2-040 
 
 
 
 
 128 
 
 2-096 
 
 
 
 
 I2 9 
 
 2-154 
 
 
 
 
 130 
 
 2-212 
 
 
 
 
 131 
 
 2-273 
 
 76,484 
 
 866,139 
 
 I52-4 
 
 I 3 2 
 
 2-334 
 
 
 
 
 133 
 
 2-397 
 
 
 
 
 !34 
 
 2-461 
 
 
 
 
 135 
 
 2-526 
 
 
 
 
 136 
 
 2-594 
 
 
 
 
 137 
 
 2-663 
 
 
 
 
 138 
 
 2-733 
 
 
 
 
 139 
 
 2-805 
 
 
 
 
 140 
 
 2-878 
 
 83,459 
 
 868,254 
 
 122- 
 
 141 
 
 2-953 
 
 
 
 
 142 
 
 3-030 
 
 
 
 
 *43 
 
 3-108 
 
 
 
 
 144 
 
 3-188 
 
 
 
 
 *45 
 
 3-270 
 
 
 
 
 146 
 
 3-354 
 
 
 
 
 147 
 
 3 '440 
 
 
 
 
 148 
 
 3'5 2 7 
 
 
 
 
 149 
 
 3-616 
 
 90,435 
 
 870,369 
 
 98-45 
 
 150 
 
 3-707 
 
 
 
 
 I 5 l 
 
 3-800 
 
 
 
 
 152 
 
 3-895 
 
 
 
 
 153 
 
 3-992 
 
 
 
 
 154 
 
 4-091 
 
 
 
 
 155 
 
 4-192 
 
 
 
 
 156 
 
 4-295 
 
 
 
 
 157 
 158 
 
 4-401 
 4-508 
 
 97,4" 
 
 872,484 
 
 80-02 
 
 159 
 
 4-618 
 
 
 
 
 1 60 
 
 4-73 
 
 
 
 
 161 
 
 4-844 
 
 
 
 
 162 
 
 4-961 
 
 
 
 
 163 
 
 5-08 
 
 
 
 
 164 
 
 5-20 
 
 
 
 
 165 
 
 532 
 
 
 
 
 1 66 
 
 s-4 1 ; 
 
 
 
 
 167 5-58 
 
 104,387 
 
 874,600 
 
 65^7 
 
 168 571 
 
 
 
 
 169 5-85 
 
 
 
 
Table I. 
 
 491 
 
 Temperature, 
 Fahienheit. 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P- 
 
 Heat required 
 to raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 170 
 
 5-98 
 
 
 
 
 171 
 
 6-12 
 
 
 
 
 172 
 
 6-26 
 
 
 
 
 173 
 
 6-40 
 
 
 
 
 174 
 
 6-55 
 
 
 
 
 J75 
 
 6-70 
 
 
 
 
 176 
 
 6-85 
 
 111,363 
 
 876,715 
 
 53-92 
 
 177 
 
 7-01 
 
 
 
 
 178 
 
 7-17 
 
 
 
 
 179 
 
 7'34 
 
 
 
 
 1 80 
 
 7-5o 
 
 
 
 
 181 
 
 7-67 
 
 
 
 
 182 
 
 7-84 
 
 
 
 
 183 
 
 8-01 
 
 
 
 
 184 
 
 8-19 
 
 
 
 
 185 
 
 8-37 
 
 H8,353 
 
 878,830 
 
 44-70 
 
 1 86 
 
 8-56 
 
 
 
 
 187 
 
 875 
 
 
 
 
 1 88 
 
 8-94 
 
 
 
 
 189 
 
 9-13 
 
 
 
 
 190 
 
 9*33 
 
 
 
 
 191 
 
 9'53 
 
 
 
 
 192 
 
 974 
 
 
 
 
 193 
 
 9 '95 
 
 
 
 . 
 
 194 
 
 io - i6 
 
 I25>357 
 
 880,945 
 
 37-26 
 
 '95 
 
 10-38 
 
 
 
 
 196 
 
 1 0-60 
 
 
 
 
 197 
 
 10-82 
 
 
 
 
 198 
 
 11-05 
 
 
 
 
 199 
 
 11-29 
 
 
 
 
 200 
 
 11-52 
 
 
 
 
 2O I 
 
 11-76 
 
 
 
 
 2O2 
 
 I2-OI 
 
 
 
 
 20 3 
 
 I2-26 
 
 132,360 
 
 883,060 
 
 31-26 
 
 204 
 
 12-51 
 
 
 
 
 205 
 
 12-77 
 
 
 
 
 206 
 
 13-03 
 
 
 
 
 207 
 
 13-30 
 
 
 
 
 208 
 
 I3-57 
 
 
 
 
 209 
 
 13-84 
 
 
 
 
 2IO 
 
 14-12 
 
 
 
 
 211 
 
 14-41 
 
 
 
 
 212 
 
 14-70 
 
 139,363 
 
 885,175 
 
 26-36 
 
492 
 
 The Steam Engine. 
 
 Temperature, 
 Fahrenheit. 
 t. 
 
 Pressure in Ibs. 
 jer square inch 
 at sea level. 
 P- 
 
 Heat requ'.ied 
 to raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 213 
 
 I4-99 
 
 
 
 
 214 
 
 I5-29 
 
 
 
 
 215 
 
 15-60 
 
 
 
 
 216 
 
 15 Qt 
 
 
 
 
 217 
 
 16-22 
 
 
 
 
 
 T6-EJ4 
 
 
 
 
 r.i9 
 
 16-87 
 
 
 
 
 220 
 
 I7-2O 
 
 
 
 
 221 
 
 17-53 
 
 146,380 
 
 887,290 
 
 22-34 j 
 
 222 
 
 17-87 
 
 
 
 
 22 3 
 
 18-22 
 
 
 
 
 224 
 
 18-57 
 
 
 
 
 225 
 
 18-93 
 
 
 
 
 226 
 
 I9-29 
 
 
 
 
 227 
 
 19-66 
 
 
 
 
 228 
 
 20-03 
 
 
 
 
 229 
 
 20-41 
 
 
 
 
 230 
 
 2O -8O 
 
 153,412 
 
 889,405 
 
 I9-03 
 
 231 
 
 21-19 
 
 
 
 
 232 
 
 21-59 
 
 
 
 
 233 
 
 21-99 
 
 
 
 
 234 
 
 22-40 
 
 
 
 
 235 
 
 22-82 
 
 
 
 
 236 , 
 
 23*25 
 
 
 
 
 237 
 
 23-67 
 
 
 
 
 238 
 
 24-II 
 
 
 
 
 239 
 
 24-55 
 
 160,429 
 
 891,520 
 
 I6-28 
 
 240 
 
 25-00 
 
 
 
 
 241 
 
 25-46 
 
 
 
 
 242 
 
 25-92 
 
 
 
 
 243 
 
 26*39 
 
 
 
 
 244 
 
 26-87 
 
 
 
 
 245 
 
 27-35 
 
 
 
 
 246 
 
 
 
 
 
 247 
 
 28-34 
 
 
 
 
 248 
 
 28-85 
 
 167,460 
 
 893,635 
 
 I4-00 
 
 249 
 
 29-36 
 
 
 
 
 250 
 
 29-88 
 
 
 
 
 25 1 
 
 30-41 
 
 
 
 
 252 
 
 30-94 
 
 
 
 
 253 
 
 3I-48 
 
 
 
 
 254 
 
 32-03 
 
 
 
 
 255 
 
 3 2 -59 
 
 
 
 
Table I. 
 
 493 
 
 Temperature, 
 Fahrenheit, 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P- 
 
 Heat required 
 :o raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 2 5 6 
 
 33-I5 
 
 
 
 
 257 
 
 3373 
 
 174,505 
 
 895*75! 
 
 12-09 
 
 2 5 8 
 
 34-3i 
 
 
 
 
 259 
 
 34-90 
 
 
 
 
 260 
 
 35-50 
 
 
 
 
 26l 
 
 36-11 
 
 
 
 
 262 
 
 36-72 
 
 
 
 
 263 
 
 37-35 
 
 
 
 
 264 
 
 37-98 
 
 
 
 
 265 
 
 38-62 
 
 
 
 
 266 
 
 39-27 
 
 181,564 
 
 897,866 
 
 10-48 
 
 267 
 
 39-93 
 
 
 
 
 268 
 
 40-60 
 
 
 
 
 269 
 
 41-27 
 
 
 
 
 270 
 
 41-96 
 
 
 
 
 271 
 
 42-65 
 
 
 
 
 272 
 
 43-35 
 
 
 
 
 273 
 
 44-07 
 
 
 
 
 274 
 
 44-79 
 
 
 
 
 275 
 
 45-53 
 
 188,637 
 
 899,981 
 
 9-124 
 
 2 7 6 
 
 46-27 
 
 
 
 
 277 
 
 47-02 
 
 
 
 
 2 7 8 
 
 47-78 
 
 
 
 
 279 
 
 48-55 
 
 
 
 
 280 
 
 49-33 
 
 
 
 
 28l 
 
 50-13 
 
 
 
 
 282 
 
 50-93 
 
 
 
 
 283 
 
 5 T "74 
 
 
 
 
 284 
 
 52-56 
 
 I95>7H 
 
 902,096 
 
 7'973 
 
 ill 
 
 53-39 
 
 54-24 
 
 
 
 
 287 
 
 55-09 
 
 
 
 
 288 
 
 55-96 
 
 
 
 
 289 
 
 56-83 
 
 
 
 
 290 
 
 5772 
 
 
 
 
 291 
 
 58-62 
 
 
 
 
 292 
 293 
 
 59-53 
 60-45 
 
 202,798 
 
 904,211 
 
 6-992 
 
 294 
 
 61-38 
 
 
 
 
 295 
 
 62-33 
 
 
 
 
 296 
 
 63-29 
 
 
 
 
 297 
 
 64-25 
 
 
 
 
 298 
 
 65-23 
 
 
 
 
494 
 
 TJie Steam Engine. 
 
 Temperatur 
 Fahrenheit 
 t. 
 
 Pressure in Ib 
 per square inc 
 at sea level. 
 P. 
 
 Heat requirec 
 to raise i Ib. o 
 Water from 
 32 to t. 
 Foot Ibs. 
 
 Total heat ol 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 
 
 h. 
 
 
 
 299 
 
 66-22 
 
 
 
 
 300 
 
 67-22 
 
 
 
 
 301 
 
 68-24 
 
 
 
 
 302 
 303 
 
 69-27 
 70-3I 
 
 209,885 
 
 906,327 
 
 6-153 
 
 304 
 
 71-36 
 
 
 
 
 305 
 
 72-42 
 
 
 
 
 306 
 
 73-50 
 
 
 
 
 307 
 
 74-59 
 
 
 
 
 308 
 
 75^9 
 
 
 
 
 309 
 
 76-8I 
 
 
 
 
 310 
 
 77-94 
 
 
 
 
 3" 
 
 312 
 
 79-08 
 80-23 
 
 216,986 
 
 908,442 
 
 S'433 
 
 313 
 
 81-40 
 
 
 
 
 3H 
 
 82-59 
 
 
 
 
 315 
 
 8378 
 
 
 
 
 317 
 
 86-21 
 
 
 
 
 319 
 
 88-70 
 
 
 
 
 320 
 321 
 
 89-97 
 91-25 
 
 224,087 
 
 910,557 
 
 4'8l6 
 
 322 
 
 92-54 
 
 
 
 
 323 
 
 93-85 
 
 
 
 
 324 
 
 95-I7 
 
 
 
 
 325 
 
 96-51 
 
 
 
 
 326 
 
 97-86 
 
 
 
 
 327 
 
 99*23 
 
 
 
 
 328 
 
 100-62 
 
 
 
 
 329 
 
 102-02 
 
 231,216 
 
 912,672 
 
 4-280 
 
 330 
 
 I03-43 
 
 
 
 
 331 
 
 104-86 
 
 
 
 
 332 
 
 I06-3I 
 
 
 
 
 333 
 
 107-77 
 
 
 
 
 334 
 
 109-25 
 
 
 
 
 335 
 
 110-74 
 
 
 
 
 336 
 
 II2-24 
 
 
 
 
 337 
 
 II3-76 
 
 
 
 
 338 
 339 
 
 II5-30 
 116-86 
 
 238,358 
 
 914,787 
 
 3-8I4 
 
 340 
 
 118-43 
 
 
 
 
 341 1 20 -02 
 
 
 
 
Table I. 
 
 495 
 
 Temperature, 
 Fahrenheit, 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P- 
 
 Heat required 
 to raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 342 
 
 I2I-63 
 
 
 
 
 343 
 
 123-26 
 
 
 
 
 344 
 
 124-89 
 
 
 
 
 345 
 
 126-55 
 
 
 
 
 346 
 
 I28-23 
 
 
 
 
 347 
 
 129-93 
 
 245,501 
 
 916,902 
 
 3-410 
 
 348 
 
 131-64 
 
 
 
 
 349 
 
 I33-37 
 
 
 
 
 35 
 
 135'H 
 
 
 
 
 351 
 
 136-87 
 
 
 
 
 35 2 
 
 138-65 
 
 
 
 
 353 
 
 140-45 
 
 
 
 
 354 
 
 142-27 
 
 
 
 
 355 
 
 144-10 
 
 
 
 
 356 
 
 I45-95 
 
 252,658 
 
 919,017 
 
 3-057 
 
 357 
 
 147-82 
 
 
 
 
 358 
 
 149-72 
 
 
 
 
 359 
 
 i5 I-6 3 
 
 
 
 
 360 
 
 I53-56 
 
 
 
 
 361 
 
 i55-5i 
 
 
 
 
 362 
 
 I57-48 
 
 
 
 
 363 
 
 159-46 
 
 
 
 
 364 
 
 161-47 
 
 
 
 
 365 
 
 163-49 
 
 259,829 
 
 921,132 
 
 2-748 
 
 366 
 
 165-53 
 
 
 
 
 367 
 
 167-60 
 
 
 
 
 368 
 
 169-69 
 
 
 
 
 369 
 
 171-79 
 
 
 
 
 370 
 
 173-92 
 
 
 
 
 37i 
 
 176-07 
 
 
 
 
 372 178-23 
 
 
 
 
 373 180-42 
 
 
 
 
 374 182-63 
 
 267,013 
 
 923,247 
 
 2-476 
 
 375 
 
 184-86 
 
 
 
 
 376 
 
 187-11 
 
 
 
 
 377 
 
 189-38 
 
 
 
 
 378 
 
 191-67 
 
 
 
 
 379 
 
 193-98 
 
 
 
 
 380 
 
 196-32 
 
 
 
 
 38i 
 
 198-68 
 
 
 
 
 382 
 
 20 1 -06 
 
 
 
 
 383 
 
 203-46 
 
 274,198 
 
 925,362 
 
 2-236 
 
 384 ; 205-88 
 
 
 
 
The Steam Engine. 
 
 Temperature, 
 Fahrenheit, 
 t. 
 
 Pressure in Ibs. 
 per square inch 
 at sea level. 
 P- 
 
 Heat required 
 to raise i Ib. of 
 Water from 
 32 to t. 
 Foot Ibs. 
 h. 
 
 Total heat of 
 Evaporation. 
 Foot Ibs. 
 H. 
 
 Volume of i Ib. 
 in cubic feet 
 
 385 
 
 208-33 
 
 
 
 
 3 86 
 
 21079 
 
 
 
 
 387 
 
 213-28 
 
 
 
 
 388 
 
 215-79 
 
 
 
 
 389 
 
 218-32 
 
 
 
 
 390 
 
 220-88 
 
 
 
 
 391 
 
 223-46 
 
 
 
 
 392 
 
 226-07 
 
 281,394 
 
 927,478 
 
 2'025 
 
 393 
 
 228-70 
 
 
 
 
 394 
 
 23I-35 
 
 
 
 
 395 
 
 234-02 
 
 
 
 
 396 
 
 236-72 
 
 
 
 
 397 
 
 239-44 
 
 
 
 
 398 
 
 242-19 
 
 
 
 
 399 
 
 244-96 
 
 
 
 
 400 
 
 247-75 
 
 
 
 
 401 
 
 250-57 
 
 288,634 
 
 929,593 
 
 1-838 
 
 In order to reduce the figures contained in columns 2 and 3 of the 
 above table to thermal units, it is only necessary to divide b5\772, the 
 number of foot pounds corresponding to one thermal unit. 
 
 To obtain the latent heat for any temperature we have only to 
 subtract the figures in column 3 from the corresponding figures in 
 column 4. 
 
 To find the pressure for any temperature intermediate to those 
 given in the table, as for example 310 -25. 
 
 Find by the table the difference between the pressures for 310 and 
 311. This is 1-14. Multiplying 1-14 by -25, and adding the result 
 to the pressure corresponding to 310, we get 77-94+ -285 = 78-225 
 Ibs. per square inch as the pressure corresponding to 310 -25. 
 
 To find the temperature corresponding to loo Ibs. per square inch 
 from the tables we find 
 
 Pressure corresponding to 328= 100*62 Ibs. per square inch. 
 327 - 99^3 
 
 Difference = I "39 
 
 Now, if for I -39 Ibs. difference of pressure the difference of tem- 
 perature is i, what will be the difference of temperature for -62 Ib. 
 difference of pressure ? 
 
Table L 497 
 
 1-39 i : -62 : x. 
 
 .-.*=*? -.446. 
 i'39 
 
 Then the temperature corresponding to 100 Ibs. pressure is 328 
 446 = 327-554. 
 
 Similarly to find the value of h. or H. for any temperature inter- 
 mediate between those for which the values are given in the table. 
 For example, find the total heat of evaporation of steam having the 
 temperature 332. 
 
 Total heat of evaporation corresponding to 338 914,787 
 
 329 = 912,672 
 
 Diff. for 9= 2,115 
 
 /. Diff. for i= 235 
 
 Now 332 is 3 degrees above 329, Therefore the total heat of 
 evaporation for 332 = 912,672 + 3 x 235 = 913,377 foot Ibs. 
 
 In a similar manner can be solved such a problem as the following. 
 How much heat is required in order to raise the temperature of feed- 
 water from that of the hot well 122 to that of the boiler 365. The 
 values of h. in this instance are taken from column 3. 
 
 This method of interpolation is only permissible because the values 
 of h. and H. vary so slowly with the temperature. The method would 
 not, for instance, be applicable to column 5. 
 
 K K 
 
498 
 
 The Steam Engine. 
 
 TABLE II. 
 
 HYPERBOLIC LOGARITHMS. 
 
 The hyperbolic logarithm of a number is found by multiplying the 
 common logarithm of the number by 2-30258. 
 
 Example : The common logarithm of 7 is 0-8450980, which 
 multiplied by 2-30258505 gives 1-9459100, the hyperbolic logarithm. 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 I -01 
 
 0099503 
 
 1-26 
 
 2311116 
 
 I-5I 
 
 4I2I095 
 
 7 6 
 
 5653138 
 
 I -O2 
 
 0198026" 
 
 1-27 
 
 2390169 
 
 I-52 
 
 4187103 
 
 77 
 
 5709795 
 
 I-O3 
 
 .0295588 
 
 1-28 
 
 2468601 
 
 i'53 
 
 4252676 
 
 78 
 
 5766133 ; 
 
 I'O4 
 
 0392207 
 
 1-29 
 
 2546422 
 
 i'54 
 
 4317825 
 
 79 
 
 5822156 
 
 1-05 
 
 0487902 
 
 1-30 
 
 2623643 
 
 i'55 
 
 4382549 
 
 80 
 
 5877866 
 
 I -06 
 
 0582690 
 
 31 
 
 2700271 
 
 1-56 
 
 4446858 
 
 81 
 
 5933268 
 
 07 
 
 0676586 
 
 32 
 
 2776317 
 
 i'57 
 
 4510756 
 
 82 
 
 5988365 
 
 08 
 
 0769610 
 
 33 
 
 2851788 
 
 1-58 
 
 4574249 
 
 '83 
 
 6043159 
 
 09 
 
 0861777 
 
 '34 
 
 2926696 
 
 i'59 
 
 4637339 
 
 84 
 
 6097655 
 
 10 
 
 0953102 
 
 '35 
 
 3001046 
 
 i -60 
 
 4700036 
 
 '85 
 
 6151856 
 
 II 
 
 1043600 
 
 '36 
 
 3074847 
 
 1-61 
 
 4762341 
 
 86 
 
 6205764 
 
 12 
 
 1133286 
 
 "37 
 
 3148108 
 
 1-62 
 
 4824261 
 
 87 
 
 6259384 
 
 13 
 
 I222I75 
 
 38 
 
 3220835 
 
 1-63 
 
 4885801 
 
 88 
 
 6312717 
 
 14 
 
 1310284 
 
 '39 
 
 3293037 
 
 1-64 
 
 4946961 
 
 89 
 
 6365768 
 
 15 
 
 1397618 
 
 40 
 
 3364722 
 
 1-65 
 
 5007752 
 
 90 
 
 6418538 
 
 16 
 
 1484200 
 
 41 
 
 3435 8 97 
 
 1-66 
 
 5068176 
 
 91 
 
 6471033 
 
 17 
 
 1570038 
 
 42 
 
 3506568 
 
 1-67 
 
 5128237 
 
 92 
 
 6523251 
 
 18 
 
 l6'55I44 
 
 '43 
 
 3576744 
 
 1-68 
 
 5187938 
 
 i '93 
 
 6575200 
 
 19 
 
 1739534 
 
 '44 
 
 3646431 
 
 1-69 
 
 5247285 
 
 : '94 
 
 6626879 
 
 20 
 
 1823215 
 
 '45 
 
 3715635 
 
 1-70 
 
 5306282 
 
 j -95 
 
 6678294 
 
 21 
 
 1906204 
 
 46 
 
 3784365 
 
 1-71 
 
 5364933 
 
 ! -96 
 
 6729445 
 
 22 
 
 1988508 
 
 '47 
 
 3852623 
 
 1-72 
 
 5423242 
 
 J '97 
 
 6780335 
 
 23 
 
 2070141 
 
 48 
 
 3920420 
 
 i'73 
 
 5481214 
 
 i -98 
 
 6830968 
 
 24 
 
 2151113 
 
 49 
 
 3987762 
 
 1-74 
 
 5538850 
 
 i '99 
 
 6881346 
 
 1-25 
 
 223H35 
 
 50 
 
 4054652 
 
 175 
 
 5596157 
 
 2-OO 
 
 6931472 
 
Table II. 
 
 499 
 
 HYPERBOLIC LOGARITHMS continued. 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 2-01 
 
 6981347 
 
 36 
 
 8586616 
 
 271 
 
 9969486 
 
 3-06 
 
 1-1184148 
 
 2 'O2 
 
 7030974 
 
 37 
 
 8628899 
 
 272 1-0006318 
 
 3'7 
 
 I-I2I6775 
 
 2-03 
 
 7080357 
 
 2-38 
 
 8671004 
 
 273; 1-0043015 
 
 3-08 
 
 1-1249295 
 
 2-04 
 
 7129497 
 
 2-39 
 
 8712933 
 
 274 1-0079579 
 
 3-09 
 
 I-I28I7IO 
 
 2-05 
 
 7178399 
 
 2-40 
 
 8754686 
 
 275 
 
 I-OII6009 
 
 3-10 
 
 I-I3I4O2I 
 
 2-06 
 
 7227059 
 
 2-41 
 
 8796266 
 
 2 7 6 
 
 1-0152306 
 
 3-11 
 
 1-1346227 
 
 2-07 
 
 7275485 
 
 2-42 
 
 8837675 
 
 277 
 
 1-0188473 
 
 3-12 
 
 I-I378330 
 
 2'08 
 
 7323678 
 
 2'43 
 
 8878912 
 
 278 1-0224509 
 
 3-i3 
 
 1-1410330 
 
 2-09 
 
 7371640 
 
 2'44 
 
 8919980 
 
 279 1-0260415 
 
 3'i4 
 
 I-I442227 
 
 2-10 
 
 7419373 
 
 2'45 
 
 8960881 
 
 2-80 
 
 1-0296193 
 
 3-i5 
 
 I-I474024 
 
 2'II 
 
 7466880 
 
 2-46 
 
 9001613 
 
 2-81 
 
 1-0331844 
 
 3'i6 
 
 I-I505720 
 
 2'12 
 
 7514160 
 
 2-47 
 
 9042181 
 
 2-82 
 
 I -0367368 
 
 3-i7 
 
 I-I5373I5 
 
 2-I 3 
 
 7561219 
 
 2-48 
 
 9082585 
 
 2-8 3 
 
 I -0402766 
 
 3'i8 
 
 I-I5688II 
 
 2-I 4 
 
 7608058 
 
 2-49 
 
 9122826 
 
 2-84 
 
 I -0438040 
 
 3-i9 
 
 I-I6OO2O9 
 
 2-I 5 
 
 7654679 
 
 2-50 
 
 9162907 
 
 2-8 5 
 
 1-0473189 
 
 3-20 
 
 1-1631508 
 
 2-16 
 
 7701082 
 
 2-51 
 
 9202827 
 
 2-86 
 
 1-0508215 
 
 3-21 
 
 1-1662708 
 
 2-17 
 
 7747271 
 
 2-52 
 
 9242589 
 
 2-87 
 
 I-0543I20 
 
 3'22 
 
 1-1693813 
 
 2-18 
 
 7793248 
 
 2'53 
 
 9282193 
 
 2-88 
 
 I -0577902 
 
 3-23 
 
 1-1724821 
 
 2-19 
 
 7839015 
 
 2'54 
 
 9321640 
 
 2-89 
 
 1-0612564 
 
 3-24 
 
 I-I755733 
 
 2 '2O 
 
 7884573 
 
 2'55 
 
 9360934 
 
 2-90 
 
 1-0647107 
 
 3-25 
 
 1-1786549 
 
 2-21 
 
 7929925 
 
 2-56 
 
 9400072 
 
 2-91 
 
 I-068I53I 
 
 3-26 
 
 I-I8I727I 
 
 2'22 
 
 7975071 
 
 2'57 
 
 9439058 
 
 2-92 
 
 1-0715836 
 
 3-27 
 
 I-I847899 
 
 2-23 
 
 8O2OOI5 
 
 2-58 
 
 9477893 
 
 '2-93 
 
 I-O75OO24 
 
 3-28 
 
 1-1878434 
 
 2-24 
 
 8064758 
 
 2'59 
 
 9516578 
 
 2'94 
 
 I -0784095 
 
 3-29 
 
 1-1908875 
 
 2-25 
 
 8109303 
 
 2-60 
 
 9555H3 
 
 2-95 
 
 1-0818051 
 
 13-30 
 
 I-I939224 
 
 2-26 
 
 8153647 
 
 2-6 
 
 959350 2 
 
 2-96 
 
 0851892 
 
 3-3i 
 
 1-1969481 
 
 2-27 
 
 8197798 
 
 2-62 
 
 9631743 
 
 ; 2'97 
 
 0885619 
 
 3-32 
 
 1-1999647 
 
 2-28 
 
 8241754 
 
 2-63 
 
 9669838 
 
 2-98 
 
 0919233 
 
 3'33 
 
 I-2029722 
 
 2-29 
 
 8285518 
 
 2-64 
 
 9707789 
 
 2-99 
 
 0952733 
 
 3 '34 
 
 I -2059707 
 
 2 - 3 
 
 8329090 
 
 2-6 
 
 9745596 
 
 3-00 
 
 0986124 
 
 3'35 
 
 I -2089603 
 
 2-3I 
 
 8372467 
 
 2-66 
 
 9783260 
 
 3-01 
 
 IOI9400 
 
 3 '36 
 
 1-2119409 
 
 2-32 
 
 8415671 
 
 2-6 
 
 9820784 
 
 3-02 
 
 1052568 
 
 3'37 
 
 I-2I49I27 
 
 2'33 
 
 8458682 
 
 2-6S 
 
 9858167 
 
 3 '03 
 
 1085626 
 
 3'38 
 
 1-2178757 
 
 2'34 
 
 8501509 
 
 2-65 
 
 9895411 
 
 3'4 
 
 IH8575 
 
 3'39 
 
 I -2208299 
 
 2'35 
 
 8544154 
 
 270 
 
 9932518 
 
 !3*5 
 
 H5HI5 
 
 3-40 
 
 1-2237754 
 
 KK 2 
 
500 
 
 TJie Steam Engine. 
 
 HYPERBOLIC LOGARITHMS- continued. 
 
 No 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No 
 
 Logarithm 
 
 3'4i 
 
 I-2267I22 
 
 376 
 
 1-3244189 
 
 4-11 
 
 1-4134230 
 
 4-46 
 
 1-4951487 
 
 3-42 
 
 2296405 
 
 377 
 
 I -3270749 
 
 4-12 
 
 I-4I5853I 
 
 4-47 
 
 4973883 
 
 3'43 
 
 2325605 
 
 378 
 
 I -3297240 
 
 4-13 
 
 1-4182774 
 
 4-48 
 
 4996230 
 
 3'44 
 
 23547H 
 
 379 
 
 I -3323660 
 
 4-14 
 
 I -4206957 
 
 4-49 
 
 5018527 
 
 3'45 
 
 2383742 
 
 3'8o 
 
 I-33500IO 
 
 4-15 
 
 1-4231083 
 
 4-50 
 
 5040773 
 
 3-46 
 
 2412685 
 
 3*i 
 
 1-3376291 
 
 4-16 
 
 1-4255150 
 
 4-5I 
 
 5062971 
 
 3'47 
 
 2441545 
 
 3-82 
 
 I -3402504 
 
 4-17 
 
 I -4279161 
 
 4-5 2 
 
 5085119 
 
 3-48 
 
 2470322 
 
 3-83 
 
 I -3428648 
 
 4-18 
 
 1-4303112 
 
 4'53 
 
 5107219 
 
 3-49 
 
 2499017 
 
 3-84 
 
 I -3454723 
 
 4-19 
 
 I -4327007 
 
 4-54 
 
 5129269 
 
 3-50 
 
 2527629 
 
 3-85 
 
 1-3480731 
 
 4-20 
 
 I -4350845 
 
 4'55 
 
 5151272 
 
 3'S 1 
 
 2556160 
 
 3-86 
 
 I -3506671 
 
 4-21 
 
 I -4374626 
 
 4-56 
 
 5173226 
 
 3-52 
 
 2584609 
 
 3-87 
 
 I 3532544 
 
 4-22 
 
 1-4398351 
 
 4-57 
 
 5I95I32 
 
 3'53 
 
 2612978 
 
 3-88 
 
 I-355835I 
 
 4-23 
 
 I -4422020 
 
 4-58 
 
 5216990 
 
 3'54 
 
 2641266 
 
 3% 
 
 I -3584091 
 
 4-24 
 
 I -4445632 
 
 4-59 
 
 5238800 
 
 3'55 
 
 2669475 
 
 3-90 
 
 I -3609765 
 
 4-25 
 
 1-4469189 
 
 4-60 
 
 5260563 
 
 3-56 
 
 2697605 
 
 3'9i 
 
 I-3635373 
 
 4-26 
 
 1-4492691 
 
 4'6i 
 
 5282278 
 
 3-57 
 
 2725655 
 
 3-92 
 
 1-3660916 
 
 4-27 
 
 1-4516138 
 
 4-62 
 
 5303947 
 
 3-58 
 
 2753627 
 
 3 '93 
 
 I -3686395 
 
 4-28 
 
 1 -4539530 
 
 4-63 
 
 5325568 
 
 3'59 
 
 2781521 
 
 3-94 
 
 I -3711807 
 
 4-29 
 
 I -4562867 
 
 4-64 
 
 5347143 
 
 3-60 
 
 2809338 
 
 3*95 
 
 I'3737I56 
 
 .4-30 
 
 1-4586149 
 
 4-65 
 
 5368672 
 
 3'6i 
 
 2837077 
 
 3-96 
 
 I -3762440 
 
 4-3i 
 
 I -4609379 
 
 4-66 
 
 5390154 
 
 3'62 
 
 2864740 
 
 3 '97 
 
 1-3787661 
 
 4-32 
 
 I-4632553 
 
 4-67 
 
 54"590 
 
 3-63 
 
 2892326 
 
 3-98 
 
 1-3812818 
 
 4'33 
 
 I-4655675 
 
 4-68 
 
 5432981 
 
 3-64 
 
 2919836 
 
 3 '99 
 
 1-3837912 
 
 4'34 
 
 I -4678743 
 
 4-69 
 
 5454325 
 
 3-65 
 
 2947271 
 
 4'OO 
 
 I -3862943 
 
 4-35 
 
 I-470I758 
 
 ,4-7o 
 
 5475625 
 
 3-66 
 
 2974631 
 
 4-01 
 
 1-3887912 
 
 4-36 
 
 I -4724720 
 
 4-71 
 
 5496879 
 
 3-67 
 
 3001916 
 
 4'O2 
 
 1-3912818 
 
 4-37 
 
 I -474763 
 
 4-72 
 
 5518087 
 
 3-68 
 
 3029127 
 
 4 '03 
 
 I-3937763 
 
 438 
 
 I ^770487 
 
 473 
 
 5539252 
 
 3-69 
 
 3056264 
 
 4-04 
 
 I -3962446 
 
 4-39 
 
 I ^793292 
 
 :474 
 
 5560371 
 
 370 
 
 3083328 
 
 4-05 
 
 1-3987168 
 
 4-40 
 
 I -4816045 
 
 |475 
 
 5581446 
 
 371 
 
 3110318 
 
 4-06 
 
 1-4011829 
 
 4*41 
 
 1-4838746 
 
 1-76 
 
 5602476 
 
 372 
 
 3137236 
 
 4-07 
 
 I -4036429 
 
 4-42 
 
 1-4861396 
 
 477 
 
 5623462 
 
 373 
 
 3164082 
 
 4-08 
 
 I -4060969 
 
 4'43 
 
 I-4883995 
 
 478 
 
 5644405 
 
 374 
 
 3190856 
 
 4-09 
 
 I -4085449 
 
 4-44 
 
 I -4906543 
 
 4-79 
 
 5665304 
 
 375 
 
 3217559 
 
 4'io 
 
 1-4109869 
 
 4-45 
 
 I -4929040 
 
 4-80 
 
 5686159 
 
Table II. 
 
 501 
 
 HYPERBOLIC LOGARITHMS -continued. 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 4'8l 
 
 1-5706971 
 
 5-16 
 
 I -6409365 
 
 5-51 
 
 I -7065646 
 
 5-86 
 
 17681496 
 
 4-82 
 
 I-5727739 
 
 
 1-6428726 5-52 
 
 1-7083778 
 
 5-87 1-7698546 
 
 4-83 
 
 I -5748464 
 
 5-i8 
 
 1-6448050 5-53! 1-7101878 
 
 5-88 I-77I5567 
 
 4-84 
 
 1-5769147 
 
 5 -I 9 
 
 1-6467336 5-54 1-7119944 
 
 5-8 9 
 
 I-7732559 
 
 4-85 
 
 1-5789787 
 
 5-20 
 
 I -6486586 
 
 5-55 
 
 17137979- 
 
 5-90 
 
 I-7749523 
 
 4-86 
 
 1-5810384 
 
 5-21 
 
 1-6505798 
 
 5-56 
 
 I7I5598I 
 
 5*91 
 
 1-7768458 
 
 4-87 
 
 I -5830939 
 
 5-22 
 
 1-6524974 5-57 
 
 I7I73950 
 
 5-92 
 
 17783364 
 
 4-88 
 
 1-5851452 
 
 5-23 
 
 I-6544II2 
 
 5-58 
 
 1-7191887 
 
 5-93 1-7800242 
 
 4-89 
 
 1-5871923 
 
 5-24 
 
 1-6563214 
 
 5'59 
 
 1-7209792 5-94 1-7817091 
 
 4-90 
 
 I-5892352 
 
 5-25 
 
 1-6582280 5-60 
 
 1-7227655 
 
 ;5'95 
 
 1-7833912 
 
 4-91 
 
 I-59I2739 
 
 5-26 
 
 1-6601310 5*61 
 
 1-7245507 
 
 5-96 
 
 I -7850704 
 
 4-92 
 
 1-5933085 
 
 5-27 
 
 i -6620303 
 
 i5-62 1-7263316 
 
 
 I -7867469 
 
 4-93 
 
 I-5953389 
 
 5-28 
 
 i -6639260 
 
 5-63 1-7281094 
 
 5-98 
 
 I 7884205 
 
 4'94 
 
 I-5973653 
 
 5-29 
 
 6658182 5-64 17298840 5-99 
 
 1-7900914 
 
 4-95 
 
 I-5993875 
 
 5-30 
 
 6677068 
 
 5-65 
 
 I73I6555 6-00 
 
 I-79I7595 
 
 4-96 
 
 1-6014057 
 
 5 . 3I 
 
 6695918 
 
 5-66 
 
 17334238 6-01 
 
 I -7934247 
 
 4'97 
 
 1-6034198 
 
 5-32 
 
 67H733 
 
 5-67 
 
 1-7351891 6-02 
 
 1-7950872 
 
 4-98 
 
 I '6054298 
 
 5-33 
 
 6733512 
 
 5-68 
 
 1-7369512 6-03 
 
 I -7967470 
 
 4-99 
 
 I -6074358 
 
 5-34 
 
 6752256 
 
 5-69 
 
 1-7387102 
 
 6-04 
 
 I -7984040 
 
 5-00 
 
 I -6094379 
 
 5-35 
 
 6770965 
 
 570 
 
 1-7404661 
 
 6-05 
 
 I -8000582 
 
 5-01 
 
 1-6114359 
 
 5-36 
 
 "6789639 
 
 57i 
 
 1-7422189 
 
 6-06 
 
 1-8017098 
 
 5-02 
 
 1-6134300 
 
 5-37 
 
 6808278 
 
 572 1-7439687 
 
 6-07 
 
 1-8033586 
 
 5-03 
 
 1-6154200 
 
 5-38 
 
 6826882 
 
 573 I-7457I55 
 
 6-08 
 
 I -8050047 
 
 5-04 
 
 1-6174060 
 
 539 
 
 6845453 
 
 574 
 
 1-7474591 6-09 
 
 1-8066481 
 
 
 1-6193882 
 
 5-40 
 
 i -6863989 
 
 575 
 
 17491998 
 
 6-10 
 
 I -8082887 
 
 5-06 
 
 1-6213664 
 
 5'4i 
 
 i -6882491 
 
 576 
 
 1 7509374 
 
 6-ii 
 
 I -8099267 
 
 5-07 
 
 I '6233408 
 
 5-42 
 
 1-6900958 
 
 577 
 
 i -7526720 
 
 6-12 
 
 1-8115621 
 
 5-08 
 
 1-6253112 
 
 5-43 
 
 1-6919391 
 
 578 
 
 1-7544036 6-13 
 
 1-8131947 
 
 5-09 
 
 1-6272778 
 
 5'44 
 
 i 1-6937790 
 
 579 
 
 1-7561323 
 
 6-14 
 
 I -8148247 
 
 5-10 
 
 I '6292405 
 
 5-45 
 
 1-6956155 
 
 5-80 
 
 17578579 
 
 6;i 5 
 
 1-8164520 
 
 5'H 
 
 1-6311994 
 
 5-46 
 
 i -6974487 
 
 5-81 
 
 1 7595805 
 
 6-16 
 
 1-8180767 
 
 5-12 
 
 1-6331544 
 
 5-47 
 
 i -6992786 
 
 5-82 
 
 1-7613002 
 
 6-17 
 
 1-8196988 
 
 5-i3 
 
 1-6351057 
 
 
 1-7011051 
 
 5-83 1-7630170 
 
 6-18 
 
 1-8213182 
 
 
 I '6370530 
 
 5'49 
 
 1-7029282 
 
 5-84, 1-7647308 
 
 6-19 
 
 1-8229351 
 
 5-15 
 
 I -6389967 
 
 5-50 1-7047481 
 
 5-85! 1-7664416 
 
 6-20 
 
 I -8245493 
 
502 
 
 The Steam Engine. 
 
 HYPERBOLIC LOGARITHMS contimied. 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 6'2I 
 
 1-8261608 6-56 
 
 I -8809906 
 
 6-91 
 
 1 -9329696 
 
 7-26 
 
 1-9823798 
 
 6'22 
 
 1-8277699 :6-57 
 
 1-8825138 
 
 6-92 
 
 I-9344I57 
 
 7-27 1-9837562 
 
 6-23 
 
 1-8293763 6-58 
 
 I -8840347 
 
 6"93 
 
 I-9358598 
 
 7-28 1-9851308 
 
 6-24 
 
 1-8309801 6-59 
 
 1-8855533 
 
 6-94 
 
 I-93730I7 
 
 7-29 1-9865035 
 
 6-25 
 
 1-8325814 
 
 6'6o 
 
 1-8870697 6-95 
 
 1-9387416 
 
 7'30| 1-9878743 
 
 6-26 
 
 1-8341801 6-61 
 
 1-8885837 ;6- 9 6 
 
 1-9401794 
 
 7-31 
 
 I -9892432 
 
 6-27 
 
 I-8357763 6-62 
 
 1-8900954 6-97 
 
 1-9416152 
 
 7-32' 1-9906103 
 
 6-28; 1-8373699 6-63 
 
 1-8916048 6-98 
 
 I -9430489 
 
 7-33 I-99I9754 
 
 6-29' 1-8389610 6-64 
 
 1-8931119 
 
 6-99 
 
 I -9444805 
 
 7-34 I-9933387 
 
 6-30 
 
 i -8405496 
 
 6-65 
 
 1-8946168 
 
 7-00 
 
 I -9459100 
 
 7-35 I-9947002 
 
 6-31 1-8421356 
 
 6-66 
 
 1-8961194 
 
 7-01 
 
 I-9473376 
 
 7-36 1-9960599 
 
 6-32 1-8437191 6-67 
 
 1-8976198 7-02 
 
 I -9487632 
 
 737 I-9974I77 
 
 6-33 1-8453002 6-68 
 
 1-8991179 7-03 
 
 1-9501866 
 
 7-38 I-9987736 
 
 6-34 
 
 1-8468787 6-69 
 
 1-9006138 
 
 7-04 
 
 1-9516080 
 
 J7-39J 2-0001278 
 
 6-35 
 
 1-8484547 6-70 
 
 1-9021075 
 
 7-05 
 
 I-9530275 
 
 7-40 2-0014800 
 
 6-36 
 
 1-8500283 6-71 
 
 i -9035989 
 
 7-06 
 
 I '9544449 
 
 7-41 2-0028305 
 
 6-37 
 
 1-8515994 6-72 
 
 i -9050881 
 
 7-07 
 
 1-9558604 
 
 7-42 2-0041790 
 
 6-38 
 
 1-8531680 673 
 
 1-9065751 
 
 7-08 
 
 I-9572739 
 
 7-43 2-0055258 
 
 6-39 1-8547342 6-74 
 
 i -9080600 
 
 ,7-09 
 
 I-9586853 
 
 7'44 
 
 2-0068708 
 
 6-40 
 
 i -8562979 
 
 675 
 
 i -9095425 
 
 7-10 
 
 I -9600947 
 
 7-45 
 
 2-0082140 
 
 6-41 
 
 i -8578592 
 
 1 
 
 6-76 1-9110228 
 
 7-11 
 
 1-9615022 
 
 7-46 
 
 2-0095553 
 
 6-42 
 
 1-8594181 
 
 6*77 
 
 1-9125011 
 
 7-12 
 
 1-9629077 7-47 
 
 2-0108949 
 
 6-43: 1-8609745 
 
 6-78 
 
 1-9139771 
 
 
 1-9643112 
 
 7-48 2-0122327 
 
 6-44 1-8625285 
 
 6-79 
 
 1-9154509 
 
 7-14 
 
 1-9657127 
 
 7-49 
 
 2-0135687 
 
 6-45! 1-8640801 
 
 6-80 
 
 I 
 
 1-9169226 
 
 
 1-9671123 
 
 7-50 
 
 2-0149030 
 
 6-46 
 
 i -8656293 
 
 6-81 
 
 1-9183921 
 
 7-16 
 
 I -9685099 
 
 7-51 2-0162354 
 
 6-47 
 
 1-8671761 
 
 6-82 
 
 1-9198594 
 
 17*17 
 
 I -9699056 
 
 7-52 2-0175661 
 
 6-48 
 
 i -8687205 
 
 6-83 
 
 1-9213247 
 
 J7-i8 
 
 1-9712993 
 
 7-53! 2-0188950 
 
 6-49 
 
 i -8702625 
 
 6-84 
 
 1-9227877 
 
 |7-i9 
 
 I -9726911 
 
 7'54 2-0202221 
 
 6-50 
 
 1-8718021 
 
 6-85 
 
 i -9242486 
 
 7-20 
 
 I -9740810 
 
 7-55: 2-0215475 
 
 6-51 
 
 I-8733394 
 
 ,6-86 
 
 9257074 
 
 7-21 
 
 I -9754689 
 
 7- 5 6 
 
 2-0228711 
 
 6-52 
 
 1-8748743 
 
 6-87 
 
 9271641 
 
 7-22 
 
 I -9768549 
 
 7'57 2-0241929 
 
 
 i -8764069 
 
 6-88 
 
 9286186 
 
 7-23 
 
 I -9782390 
 
 7-58: 2-0255I3I 
 
 6-54 1-8779371 
 6-55! 1-8794650 
 
 6-89 
 6-90 
 
 9300710 
 1 93 I 52i4 
 
 7-24 
 
 ,7-25 
 
 I-97962I2 
 I -9810014 
 
 7'59 2-0268315 
 7'6O 2-0281482 
 
Table IL 
 
 503 
 
 HYPERBOLIC LOGARITHMS continued. 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 No. 
 
 Logarithm 
 
 7-61 
 
 2-0294631 
 
 7- 9 6 
 
 2-0744290 
 
 8-31 
 
 2,174596 
 
 8-66 
 
 2-I587I47 
 
 7-62 
 
 2-0307763 7-97 
 
 2-0756845 8-32 2-II86622 
 
 8-67: 2-1598687 , 
 
 7-63 
 
 2-0320878 
 
 7-98 
 
 2-0769384 8-33 
 
 2-1198634 
 
 8-68 2-1610215 
 
 7-64 
 
 2-0333976 
 
 7-99 
 
 2-0781907 ;8-34 
 
 2-1210632 
 
 8-69 2-1621729 
 
 7-65 
 
 2-0347056 
 
 8-00 
 
 2-0794414 
 
 8-35 
 
 2-I2226I5 
 
 8-70' 2-1633230 
 
 7-66 
 
 2-0360119 
 
 8-01 
 
 2 -0806907 
 
 8-36 
 
 2,234584 
 
 8-71! 2-1644718 
 
 7-67 
 
 2-0373166 8-02 
 
 2-0819384 
 
 8-37 
 
 2-1246539 
 
 8-72 
 
 2-1656192 
 
 7-68 
 
 2-0386195 ^8-03 
 
 2-0831845 
 
 8-38 
 
 2-1258479 
 
 873 
 
 2-1667653 
 
 7-69 
 
 2-0399207 8-04 
 
 2 -0844290 
 
 8-39 
 
 2-1270405 
 
 8-74 
 
 2-1679101 
 
 770 
 
 2-0412203 
 
 8-05 
 
 2*0856720 
 
 8-40 
 
 2-1282317 
 
 875 
 
 2-1690536 
 
 7-71 
 
 2-0425181 
 
 8-06 
 
 2-0869135 
 
 8-41 
 
 2-1294214 
 
 8-76 
 
 2-1701959 
 
 772 
 
 2-0438143 
 
 8-07 
 
 2-0881534 
 
 8-42 
 
 2 ' 1 306098 
 
 8-77 
 
 2-1713367 
 
 773 
 
 2-0451088 
 
 8-08 
 
 2-0893918 
 
 8-43 
 
 2-1317967 8-78 
 
 2-1724763 
 
 774 
 
 2-0464016 
 
 8-09 
 
 2-0906287 
 
 8-44 
 
 2-1329822 
 
 8-79 
 
 2-1736146 
 
 775 
 
 2-0476928 
 
 8,0 
 
 2-0918640 
 
 8-45 
 
 2-1341664 
 
 8-80 
 
 2-1747517 
 
 7-76 
 
 2-0489823 
 
 8-ii 
 
 2-0930984 
 
 8-46 
 
 2-I35349I 
 
 8-81 
 
 2-1758874 : 
 
 777 
 
 2-0502701 
 
 8-12 
 
 2-0943306 8-47 
 
 2-1365304 
 
 8-82 
 
 2-1770218 
 
 7-78 
 
 2-0515563 
 
 8-13 
 
 2-0955613 18-48 
 
 2T377I04 
 
 8-83 
 
 2-1781550 
 
 779 
 
 2-0528408 
 
 8-14 
 
 2-0967905 
 
 8-49 
 
 2-1388889 
 
 8-84 
 
 2-1792868 
 
 7-80 
 
 2-0541237 
 
 8-15 
 
 2-0980182 
 
 8-50 
 
 2T4OO66I 
 
 8-85 
 
 2-1804174 
 
 7-81 
 
 2-0554049 
 
 8-16 
 
 2-0992444 
 
 8-51 
 
 2-I4I24IO 
 
 8-86 
 
 2-1815467 
 
 7-82 
 
 2-0566845 
 
 8-17 
 
 2-1004691 
 
 8-52 
 
 2-1424163 
 
 8-87 
 
 2-1826747 
 
 7^3 
 
 1 2-0579624 
 
 8-iS 
 
 2-1016923 
 
 8-53 
 
 2-1435893 
 
 8-88 
 
 2-1838015 
 
 7-84 
 
 2-0592388 
 
 8-19 
 
 2-IO29I4O 
 
 8'54 
 
 2-1447609 
 
 8-89 
 
 2-1849270 
 
 7-85 
 
 2-0605135 
 
 8-2C 
 
 2-104134! 
 
 8-55 
 
 2-I4593I2 
 
 8-90 
 
 2-1860512 
 
 7-86 
 
 2-0617866 
 
 8-21 
 
 2T053529 
 
 8-56 
 
 2-I47IOOI 
 
 8-91 
 
 2-1871742 
 
 7-87 
 
 2-0630580 
 
 8-22 
 
 2-IO657O2 8'57 
 
 2-1482676 
 
 8-92 
 
 2-1882959 
 
 7-88 2-0643278 
 
 8-23 
 
 2-I07786I 8-58 
 
 2-1494339 
 
 8-93 
 
 2-1894163 
 
 7-89 2-0655961 
 
 8-24 
 
 2-1089998 :8-59 
 
 2-1505987 
 
 8-94 
 
 2-1905355 
 
 7 - 9 c 
 
 2-0668627 
 
 8-25 
 
 2-1102128 '8-60 
 
 2-1517622 
 
 8-95 
 
 2-1916535 
 
 7-91 
 
 2-0681277 
 
 8-26 
 
 2-1114243 
 
 '8-61 
 
 2-1529243 
 
 8-96 
 
 2-1927702 
 
 7-92 2-0693911 
 
 8-27 
 
 2-1126343 
 
 8-62 
 
 2-1540851 
 
 8-97 
 
 2-1938856 
 
 7-93 2-0706530 
 
 8-28 
 
 2-1138428 
 
 8-63 
 
 2-1552445 
 
 8-98 
 
 2-1949998 
 
 7-94 2-0719132 
 
 8-29 
 
 2-1150499 
 
 8-64 
 
 2 ' I 564026 
 
 8-99: 2-1961128 
 
 7-95 2-0731719 
 
 8-30 
 
 2-1162555 
 
 8-65 
 
 2-1575593 
 
 9-00 2-1972245 
 
EXAMPLES. 
 
 1. Give reasons for the supposition that heat is not a substance. 
 
 2. Define the meaning of the terms ' work ' and ' energy.' 
 
 3. Give an account of Davy's reasons for believing that heat is a 
 form of energy. 
 
 4. Define a horse-power. What is the distinction between 33,000 
 foot-pounds and one horse-power ? 
 
 .5. A coal-mine 250 feet deep must, in order to keep the workings 
 dry, have 108,000 gallons of water pumped out of it every hour. 
 What horse-power must the engines exert merely to raise this water 
 without taking any account of the friction of the machinery, &c. 
 
 6. State the distinction between potential ' and ' kinetic ' energy, 
 and give an example of each. 
 
 7. Describe any experiment with which you are acquainted which 
 proves that work may be done by the expenditure of heat. 
 
 8. State what is meant by ' temperature.' Describe how temperature 
 is commonly measured. How many scales of temperature are there 
 in common use in Europe? A thermometer registers 364 on the 
 Fahrenheit scale : what would be the corresponding numbers on the 
 Centigrade and Reaumur scales ? 
 
 9. What is meant by the term ' specific heat ' ? 
 
 10. Describe an experiment, which proves that the same quantity 
 of heat imparted to equal weights of different substances affects the 
 temperatures of the substances unequally. 
 
 n. A pound of cast-iron (specific heat ='130) is made red-hot 
 and plunged into two gallons of water of the temperature 60. In 
 quenching the iron the temperature of the water rises 9 : what was the 
 original temperature of the hot iron ? 
 
 12. What is meant by the mechanical equivalent of heat ? What 
 is the equivalent in foot-pounds of the British thermal unit ? Work at 
 the rate of a horse-power is expended for an hour in creating friction, 
 the heat generated by which is all communicated to 10 cubic feet of water 
 
506 The Steam Engine. 
 
 contained in a non-conducting tank. The original temperature of the 
 water was 60 : what will be its temperature at the end of the hour ? 
 
 13. State Boyle's law connecting the pressure and volume of gas. 
 Show how the law may be represented graphically. Prove that the 
 curve which represents the varying pressures of a portion of gas when 
 the volume is changed and the temperature is kept constant is a rect- 
 angular hyperbola. 
 
 14. A cylinder containing air is fitted with a gas-tight piston by 
 means of which the contained air is compressed to one- fourth of its 
 original volume. Will the final pressure be four times the original 
 pressure immediately after the compression takes place ? Give your 
 reasons for your conclusion. 
 
 15. State what is meant by an isothermal line of a gas. 
 
 1 6. What is the general effect of raising the temperature of a por- 
 tion of gas, first, when the volume is kept unchanged, and second, 
 when the pressure is kept constant ? State Charles' law, and give the 
 formula which expresses it. A cylinder of 18 inches diameter and of 
 indefinite length contains a cubic foot of air enclosed by a gas-light 
 piston. The cylinder is plunged into water which is kept at the tem- 
 perature of 175 : to what height above the bottom of the cylinder will 
 the piston be moved after the inclosed air has attained the temperature 
 of the water ? 
 
 17. State the distinction between Charles' and Dalton's laws. 
 
 1 8. Describe the air-thermometer, and state what is meant by the 
 term 'absolute temperature.' Show how to deduce the number of 
 degrees which the absolute zero is below the zero of the Fahrenheit 
 scale on the assumption that Charles' law is true. 
 
 19. Show how to deduce from Boyle's and Charles' laws a formula 
 connecting the volume, the pressure, and the absolute temperature of 
 a portion of gas. 
 
 A pound of air of the temperature 32 and atmospheric pressure is 
 heated up to 100 : what is the product of its pressure and volume at 
 the latter temperature ? 
 
 20. Establish the ratio between the specific heat of air heated, first, 
 at constant volume, and, second,. at constant pressure. 
 
 21. What is meant by the term ' latent ' as distinguished from 
 ' sensible ' heat ? When wa'ter is turned into steam, state the various 
 ways in which heat is expended. 
 
 22. Is Boyle's law applicable to the case of expanding steam ? 
 State your reasons for your aswer. Make a sketch of the isothermal 
 line of steam of, say, 212, and explain what the different portions of 
 the line represent. 
 
Examples. 507 
 
 23. A portion of gas is enclosed in a cylinder under pressure, and 
 is expanded so as to do work. How can you secure that the curve of 
 expansion shall be a rectangular hyperbola ? 
 
 24. Describe what is meant by a ' cycle of operations. ' 
 
 25. Give a numerical expression for the quantity of heat required 
 to raise the temperature of air from temperature /, to t., : first, the volume 
 of the air being kept constant ; and, second, the pressure being kept 
 constant. 
 
 26. A portion of gas is expanded isothermally from volume #, to -z/ 2 , 
 the initial and final pressures being / t and /.,, and the temperature t. 
 State how much heat is expended in doing external work ; how much 
 in doing internal work ; and how much heat must be supplied to the 
 gas in order that the condition of isothermal expansion may be fulfilled. 
 
 27. A portion of gas is expanded from volume z/ to v-, the equation 
 of the curve of expansion being pv n = constant. Deduce an expression 
 for the total quantity of heat expended during the operation. 
 
 28. When gas expands adiabatically, prove that the equation of the 
 curve of expansion is p V Y = constant. 
 
 29. What will be the final temperature T 2 of gas expanded adia- 
 batically, the original temperature being T, and the ratio of expansion r ? 
 What will be the total loss of heat by the gas during the expansion ? 
 
 30. As a numerical application of the above, find the final tempera- 
 ture of air expanded adiabatically to double its volume, the initial tem- 
 perature being 539 Fahrenheit. Find also the final temperature when 
 the ratios of expansion are 3, 4, and 5. (A table of logarithms will 
 be required for the solution.) The student should notice the great fall 
 in temperature of gas expanding adiabatically as compared with steam, 
 and draw his own conclusions as to the suitability of air as a medium 
 for the working of heat-engines. 
 
 31. A cubic foot of air of the pressure 100 pounds per square inch 
 and temperature 539 Fahrenheit is expanded adiabatically till its vol- 
 ume is doubled : construct a table showing the pressures and corre- 
 sponding temperatures for the volumes 1-1, 1-2, 13, 1-4, &c. up to 2. 
 
 32. What are the essential conditions of working to realise a theo- 
 retically perfect heat-engine ? Prove that if the essential conditions are 
 realised the efficiency of the engine is represented by the fraction 
 
 -J-H^ where T, and r 2 are the absolute temperatures of the sources of 
 
 heat and cold respectively. 
 
 33. A pound of coal will generate during combustion 14,000 units 
 of heat. Supposing that a theoretically perfect heat-engine consumed 
 one pound of coal per minute and that its limits of working temperature 
 
5cB The Steam Engine. 
 
 were T, = 2,440 and T 2 = 60, what would be the horse-power developed 
 by the engine ? 
 
 34. What is the meaning of the terms ' specific volume ' and ' relative 
 volume ' of steam ? How may the relative volume be calculated when 
 the specific volume is given ? State Zeuner's law connecting the pies- 
 sure and volume of dry steam. 
 
 35. State an approximate formula for the total heat of steam formed 
 from water of the temperature /, the temperature of the steam when 
 formed being T. Of the above a certain quantity is expended in doing 
 external work : give Zeuner's approximate formula for the heat which 
 thus disappears. 
 
 36. If/ be the pressure of steam, and/ b be the back pressure in 
 pounds per square inch ; also, if H be the total heat of formation of a 
 pound of steam, and v its specific volume : deduce expressions for the 
 heat expended, the external work done, and the heat rejected per cubic 
 foot of the contents of the cylinder in a non-expansive steam-engine. 
 
 37. A condensing non-expansive engine uses steam of the pressure 
 of 60 pounds absolute ; the back pressure is 2 pounds per square inch. 
 How many pounds of water must be evaporated for this engine per 
 effective indicated horse-power per hour ? 
 
 N.B. The specific volume of steam of 60 pounds pressure is 7*037 
 cubic feet. Ans. 337. 
 
 38. Supposing the feed-water in the above engine is taken from 
 the condenser and has the temperature of 100, what is the ratio of heat 
 expended to work done ? 
 
 N. B. The total heat of formation of steam of 60 pounds pressure is 
 1171-3 thermal units. 
 
 39. In an engine using steam expansively the pressure during ad- 
 mission is P, pounds per square inch, the volume when steam is cut off 
 is V,, the rate of expansion is r, and the back 'pressure, which is sup- 
 posed to be uniform, is P f) . Find an expression for the effective work 
 done, and also for the mean pressure, on the assumption that the ex- 
 pansion takes place in accordance with Boyle's law, and that there is 
 no clearance. 
 
 40. In an engine using steam expansively the pressure duiing ad- 
 mission is P, = 95 pounds per square inch absolute. The back pressure 
 P b = 3 pounds per square inch. The volume of one pound of steam of 
 pressure/, is v\ cubic feet. The rate of expansion r=2. 
 
 Find out the effective work per pound of steam in thermal units ; 
 and the total weight of steam supplied to the cylinder per effective 
 indicated horse-power per hour. 
 
Ex a inple.s. 509 
 
 The expansion is supposed to be hyperbolic, and clearance neglected 
 and the steam dry at the end of the stroke. 
 
 Solution, Let p be the final and p m the mean pressure in 
 pounds per square inch. Also, let v 2 = the specific volume of steam 
 of pressure /_. Then the total work done 
 = 144/2^(1 +log e r) 
 
 The effective work done equals the foregoing minus the work done 
 in overcoming the back pressure 
 
 ^ 
 
 Now, multiplying v.,, i.e. the specific volume of dry steam of 42-5 
 pounds per square inch by 144/0, and reducing to thermal units, 
 we obtain the number 777. Hence the effective work done 
 
 = 77 '7 ( : '696 | = 126-3 thermal units per pound of steam. 
 V 4 2 '5/ 
 
 Now one horse-power per hour = 33 = 2565 thermal 
 units per hour ^256 = ounds 
 
 I2O'3 
 
 equals the weight of steam which must be supplied to the cylinder. 
 41. Find out what quantity of heat must be added to the steam during 
 expansion in the above example, in order that the condition may be 
 realised that it should be dry, and saturated at the end of the stroke. 
 
 Solution. The symbol P is used throughout to denote the 
 pressure per square foot corresponding with the pressure p per 
 square inch. If steam of pressure p 2 be dry at the end of the stroke, 
 it must have had imparted to it not only the heat of formation H 2 
 of dry steam of this pressure, but also the equivalent of heat cor- 
 responding to the work done over and above whatever work done 
 is included in H.,. Now the work done included in H 2 equals 
 the pressure 144 P._, multiplied by the corresponding volume v., (see 
 page 99). But the total work done equals the mean pressure P m 
 multiplied by the final volume v v Therefore the difference, or 
 P m 2 - P 2 2 = (P m - P 2 ) 2 , must be added' to H 2 . Hence the tota 
 heat in the steam at the end of the expansion, provided it be then 
 dry and saturated, 
 
 = H 2 +(P m -P,>, 
 
 Now the steam when admitted into the cylinder was at the pressure 
 P 1} and its total heat of formation is H r 
 
 must be added to the steam during expansion. 
 
510 77/6' Steam Engine. 
 
 Substituting for P m its value, the above expression becomes 
 P^log^H,-!!, 
 
 The value of H 2 H, can be obtained from Table I. ; or, if no table 
 be at hand, its approximate value is -305 thermal units for every 
 degree of difference of temperature between steam of the pressures 
 #, and/. 2 . 
 
 Applying these results to the case in hand, we have 
 
 P,zUog e r+H,-H 1 = 777x '696- -305(324 -271) 
 37 '9 thermal units per pound of steam used. 
 
 ut, as we proved in the previous example, 20-3 pounds of steam 
 are used per effective horse-power per hour ; 
 
 .'. 20-3 x 37'9 = 769-37 thermal units per horse-power per hour 
 
 must be supplied to the cylinder from a steam-jacket in order to 
 keep the steam dry at the end of the stroke. This would be sup- 
 plied by the liquefaction in the jacket of about "J pound of steam 
 per hour. Hence the theoretical consumption of water in this 
 engine working under the above conditions would be 21 pounds 
 per hour, as against 33-7 in the case of Example 37. 
 
 42. What would be the theoretical quantity of water required in 
 the above example if the steam-engine were a perfect engine working 
 between the limits of temperature corresponding to the initial and the 
 back pressures of the steam ? 
 
 43. State in what respects the action of a steam-engine differs from 
 that of a perfect heat-engine. 
 
 44. State Navier's formula for the expansion of steam, giving the 
 numerical constants. 
 
 45. Investigate an expression for the work done during expansion, 
 using Navier's formula (instead of, as heretofore, assuming hyperbolic 
 expansion), and taking account of the effect of clearance. 
 
 46. State De Pambour's theory of the steam-engine, and reduce it 
 to mathematical form. 
 
 47. Analyse the nature of the resistances to the motion of stationary 
 and locomotive engines. 
 
 (For numerical examples of the application of De Pambour's theory 
 see pages 136 and 139). 
 
 48. Steam of 75 Ibs. pressure above the atmosphere is used in a 
 cylinder. It expands down to 4 Ibs. absolute at the point of release, 
 What is the ratio of expansion, supposing the clearance to occupy a 
 space equivalent to 5 per cent, of the stroke, and the release to take 
 place at a point 7 per cent, of the stroke, before the end ? 
 
Examples. 5 1 1 
 
 49. The stroke of a piston is 3 feet 6 inches, the ratio of expansion 
 is 3*5 : at what pressure must the steam be admitted in order that at 
 the release, which is supposed to take place at the end of the stroke, 
 the steam may have expanded down to 5 Ibs. absolute, the clearance 
 being equal to 6 per cent, of the stroke ? 
 
 50. A crane is employed to lift a maximum weight of one ton. 
 The chain is wound round a barrel 2 feet in diameter, to which is 
 made fast a spur wheel 4 feet in diameter, driven by a 9 inch pinion. 
 The pinion is keyed to the crank axle of a two-cylindered engine, the 
 diameter of each cylinder being 6 inches, and the stroke I foot. What 
 mean pressure of steam will be required in order to lift the weight, 
 without taking any account of the other resistances to the motion of 
 the piston ? 
 
 51. A locomotive weighing 32 tons is drawing a train of 150 tons 
 up an incline of I in 180, at a speed of fifteen miles an hour. The 
 diameter of each cylinder is 18 inches, the stroke 24 inches, and the 
 diameter of the driving-wheel 6 feet. What is the mean pressure of 
 the steam required in order to overcome the resistance of the engine 
 and train, without taking account of the other resistances to the motion 
 of the piston ? 
 
 52. How is mass measured, and what units of mass are adopted in 
 practice ? 
 
 For examples on the application of the laws of motion see pages 
 148 to 153. 
 
 For examples on fly-wheels see page 155. 
 
 53. Calling the weight of a body w, v the velocity with which it 
 
 moves in a circle of radius r, prove that the centrifugal force F = 
 
 gr ' 
 
 Also, deduce an expression for the centrifugal force when you are given 
 the number of revolutions per minute (N), instead of the circular 
 velocity. 
 
 54. Explain what is meant by the twisting moment on a crank 
 shaft, and show how the variation in the twisting moments during a 
 revolution may be represented graphically by a curve on a straight base. 
 
 55. Show by the principle of work that there is no loss of power 
 in converting rectilinear into circular motion by means of a crank and 
 connecting-rod. 
 
 The mean pressure in the cylinder is P Ibs. per square inch : what 
 is the mean tangential pressure on the pin of a crank of radius = r ? 
 
 56. Supposing the motion of the crank -pin in its circle to be prac- 
 tically uniform, what influence has the fact that the connecting-rod is 
 
5 1 2 T/ie Steam Engine. 
 
 finite in length, on the velocity of the piston in each of the four quarters 
 of a revolution ? 
 
 57. Show how to obtain the twisting moment on the crank graphic- 
 ally when the pressure on the piston is known, and the ratio of length 
 of connecting-rod to length of crank is given, for any position of the 
 crank-arm. 
 
 58. Explain the general effect of the inertia of the reciprocating 
 parts in modifying the twisting moments on the crank. Explain the 
 nature of a graphic diagram for exhibiting the pressures absorbed and 
 restored by the reciprocating parts at different parts of the stroke, 
 stating how you would calculate the initial and final pressures, taking 
 no account of the length of the connecting-rod. Also explain how 
 this graphic diagram would be altered, first, in the case of vertical 
 engines by the effect of gravity on the reciprocating masses ; and 
 second, in the case of horizontal engines, when the ratio of the length 
 of the connecting-rod to that of the crank is given. 
 
 59. Explain in detail the various steps to be taken in order to con- 
 struct an exact curve of twisting moments, when you are provided with 
 a pair of indicator diagrams, and the necessary data concerning weights 
 and dimensions. 
 
 60. Explain the methods by which in practice uniformity of twisting 
 moment is approximated to. 
 
 61. What are the essential data necessary in order to determine 
 the weight of the fly wheel of an engine ? 
 
 62. Explain the objects of cushioning the exhaust steam. 
 
 63. What area of passage would you give to a steam port for a 
 cylinder of 24 inches diameter, 36 inches stroke, the engine making 
 45 revolutions per minute ? 
 
 64. What are the advantages of making the working barrel of a 
 cylinder of a separate detachable piece, called a liner ? 
 
 65. Under what circumstances would you fit a steam-jacket to a 
 cylinder, and what precautions would you adopt in designing the jacket? 
 
 66. Make a sketch of the general arrangements of the cylinder of a 
 locomotive engine, showing a section through the cylinder and valve 
 box. 
 
 67. Give a description, with sketch, of the piston-packings of a 
 marine engine. 
 
 68. A locomotive piston of 18 inches diameter is provided with 
 three half-inch packing-rings, so adjusted as to exert a pressure on the 
 piston sides of 3 Ibs. per square inch. The stroke is 24 inches, and 
 the diameter of the driving-wheels 6 ft. 6 in. What horse-power is 
 exerted in overcoming the friction of the pistons when the engine is 
 
Examples. 5 1 3 
 
 running at the speed of 48 miles an hour, the co-efficient of friction 
 between the rings and sides of the cylinder being taken as '085 ? 
 
 69. Prove that so long as an engine runs in one direction pressure 
 is only exerted upon one of the slide bars. 
 
 70. In designing motion blocks and slide bars, what should be the 
 maximum pressure per square inch of bar surface allowed for ? 
 
 71. Given the steam pressure, the diameter of cylinder, and the 
 ratio of length of crank to connecting-rod, what is the maximum pres- 
 sure on the slide bar ? 
 
 72. What are the principal disadvantages of making the connecting- 
 rod short relatively to the crank arm ? 
 
 73. The diameter of a cylinder is 30 inches, the steam pressure 40 
 Ibs. per square inch. What is the maximum strain in the connecting- 
 rod when the latter is 4 times the length of the crank ? 
 
 74. Make sketches of the big ends of connecting-rods (i) when the 
 brass steps are held in place by a strap, and (2) when the end is solid. 
 What effect on the length of the rod is produced driving in the cotter 
 in each of these cases ? 
 
 75. You are required to drive a slide valve having a travel of four 
 inches from a main shaft of the diameter of seven inches : make a sketch 
 of the method you would adopt, giving dimensions. 
 
 76. Investigate the moment of resistance of a hollow shaft, the ex- 
 terior diameter of which is R and the interior r, and prove that with a 
 given weight of metal you can turn out a stronger shaft by making it 
 hollow rather than solid. 
 
 77. The indicated horse-power of an engine is 500, the number of 
 revolutions 50 per minute. What should be the diameter of the crank 
 shaft, the metal being steel having a shearing strength of 80,000 Ibs. 
 per square inch, and the factor of safety being 6 ? 
 
 78. Explain the action of Watt's governor and prove that the speed 
 of revolution of the engine is inversely proportional to the square root 
 of the height of the cone of revolution. What is the object of crossing 
 the arms of a governor ? 
 
 79. Explain the nature of the objections to extreme sensitiveness in 
 a governor. 
 
 80. Make a sketch showing how the rate of expansion of an engine 
 may be controlled by the governor. 
 
 81. A fly-wheel has a mean radius of 10 feet, and weighs 10 tons, 
 the whole of which is supposed to be concentrated at the mean radius. 
 The section of the rim is 160 square inches. What is the maximum 
 safe speed the wheel can be run at, on the assumption that the tensile 
 
 L L. 
 
514 The Steam Engine. 
 
 strength of cast-iron is 15,000 Ibs. per square inch, and that the factor 
 of safety is 5 ? Also at what speed would the wheel burst asunder ? 
 
 82. Sketch a D slide valve, and explain the action of outside and 
 inside lap on the valve. 
 
 83. What is the meaning of the term ' lead ' ? Given a slide valve 
 with a travel of 2\ inches, outside lap of \ inch and lead of ^ inch, 
 what will be the throw and the angle of advance of the eccentric ? 
 
 84. Make a sketch of an arrangement for reversing an engine fitted 
 with a slide valve. 
 
 85. Make a sketch of Meyer's valve gear, and state under what 
 circumstances it is desirable to fit a separate expansion valve to an 
 ordinary slide valve. State clearly all the functions of the main and 
 the expansion valves. 
 
 86. What are the disadvantages of slide valves ? and make a sketch 
 showing how these disadvantages are obviated in the Corliss engine. 
 What is the object of the separate exhaust valves in the latter engine ? 
 
 87. Make a sketch of the piston valve of a marine engine, and state 
 what its advantages are. 
 
 88. Explain how the slide valves of marine engines are usually 
 relieved of a portion of the pressure on their backs. 
 
 89. The back of the slide valve of a locomotive exposes 1 80 square 
 inches of area. The pressure in the valve box is 140 pounds per square 
 inch, which is partly balanced by the pressures acting on the under sur- 
 faces of the valve so that the average net force pressing the valve down 
 is 130 pounds. The coefficient of friction has been experimentally 
 proved to be '22. The travel of the valve is 4 inches. All the other 
 data of the engine are the same as in Example 68. What is the 
 horse-power absorbed when the engine is running at 48 miles an hour 
 in overcoming the friction of the valves ? 
 
 90. Make a sketch of Joy's valve gear, and explain some of the 
 advantages which it possesses over the ordinary eccentric gear. 
 
 91. Explain the principle of Zeuner's valve diagrams, and show, 
 choosing any dimensions of valves &c. you like, how the diagram may 
 be made to indicate the positions of the piston at which the steam 
 admission, the cut-off, the release, the compression, take place. 
 
 92. Prove that with an eccentric valve motion the point at which 
 compression commences must vary with the rate of expansion. 
 
 93. The travel of a slide valve is 8 inches, the outside lap 2 inches, 
 the inside lap \ inch, the angle of advance 40. Construct a Zeuner's 
 diagram showing the positions of the crank when the admission takes 
 place, the steam is cut off and released, the exhaust closed, and also 
 the amount of the lead. 
 
Examples. 5 1 5 
 
 94. The travel of a slide valve is 4 inches, the angle of advance is 
 40, the ratio of expansion is I '25, the steam is released when the 
 piston has still 3 per cent of the stroke to travel. Find the outside 
 and inside lap, the lead, and the position of the crank when the steam 
 is admitted and the exhaust closed. The ratio of length of connecting- 
 rod to crank is to be neglected. 
 
 95. In an engine with a 3-feet stroke the length of the connecting- 
 rod is 6 feet, the steam is cut off when the crank is at angles of 60 
 from the line of dead centres : what is the ratio of expansion in the 
 forward and in the back stroke ? 
 
 Numerous other problems in simple valve setting and designing 
 will be found on pages 281 to 290. 
 
 96. In a Stephenson's link motion with open arms, you are required 
 to fix the positions of the notches in the reversing lever quadrant by a 
 geometrical construction, so that steam may be cut off when the crank 
 makes angles of 45, 60, 90, I2cf and 135, with the dead centres, 
 choosing any dimensions you think proper for the various parts of 
 the valves and gear. 
 
 97. What is the general effect on the lead and the point of com- 
 pression of increasing the rate of expansion in Stephenson's link motion, 
 (i) when the arms are open, (2) when the arms are crossed? Illus- 
 trate your answer by means of Zeuner's diagrams, the angles of advance 
 of the virtual eccentrics corresponding with the various rates of expan- 
 sion being found by geometrical method. Choose any convenient 
 dimensions for the gear. 
 
 98. Show that when Zeuner's diagram is applied to the elucidation 
 of Meyer's valve gear, a resultant -circle can always be found the 
 chords of which represent the distances apart of the centres of the two 
 valves. 
 
 99. Is Meyer's valve gear suitable for use with engines which have 
 to be reversed frequently ? Illustrate your reply by means of a Zeuner's 
 diagram, and state the best position for the eccentric of the expansion 
 valve so as to secure the most uniform steam distribution for running in 
 both directions. 
 
 100. Describe Richards' indicator with the help of illustrative 
 sketches. 
 
 101. What points connected with the working of steam engines are 
 revealed by indicator diagrams ? 
 
 102. Being given the diagram of an expansive engine, state how 
 you would estimate the mean pressure. \Vhat data in addition to the 
 diagram would you require before you could calculate the power 
 exerted by the engine ? 
 
 L L 2 
 
516 The Steam Engine. 
 
 103. Make a sketch of the theoretical diagram of a condensing 
 engine, and show what modifications in the outline of the latter are to 
 be expected in practice. 
 
 104. What effect has clearance on the shape of the diagram ? 
 
 105. What are the leading characteristics of the diagrams of loco- 
 motive engines working at high rates of expansion ? 
 
 106. Do the pressures recorded by indicator diagrams give the 
 actual forces urging the piston ? Give your reasons for your reply. 
 
 107. Why is it that when every care is taken in the valve-setting, 
 the diagrams from the two ends of a cylinder often differ considerably 
 in area and shape ? Under what circumstances may this peculiarity be 
 turned to account ? 
 
 108. Explain exactly how you would draw the combined diagram 
 of the two cylinders of a compound engine when you are provided with 
 a diagram from each cylinder. 
 
 109. What are the principal causes which affect the back -pressure 
 line of the diagram ? 
 
 no. What is the meaning of the terms ' gross ' and ' net ' indicated 
 power ? 
 
 in. Are high rates of expansion economical in non-condensing 
 engines ? Give your reasons for your reply. 
 
 112. State how to ascertain if the valves and piston of an engine are 
 steam-tight. 
 
 113. Explain how to measure the expenditure of steam accounted 
 for by the diagram. 
 
 114. How many units of heat are obtained by the combustion of 
 one pound of carbon with sufficient air to form, (i) carbonic oxide, 
 (2) carbonic anhydride ? 
 
 115. What is the minimum weight of air necessary to effect the 
 complete combustion of one pound of carbon, and what should be the 
 temperature of the products of combustion ? 
 
 116. When a chemical combination of carbon and hydrogen is 
 burnt in oxygen how would you estimate the heat of combustion ? 
 
 117. What are the principal constituents of fuel ? 
 
 118. A firegrate is 4 feet 6 inches long, and 3 feet wide ; twenty 
 pounds of coal are burnt on it per square foot of area per hour, with 
 a supply of 24 Ibs. of air per Ib. of fuel. What is the temperature, 
 and what the volume in an hour of the products of combustion, (i) as 
 formed in the furnace ; and (2) in the chimney, supposing the latter 
 to be maintained at the temperature most suitable for draught- 
 creation ? 
 
 119. State the principal causes of the waste of fuel in boilers. 
 
Examples. 5 1 7 
 
 1 20. Describe a modern Lancashire boiler, and give illustrative 
 sketches. 
 
 121. What are the principal ends gained by the use of Galloway 
 tubes ? 
 
 122. What precautions would you observe in placing the gusset 
 stays in the flat ends of Lancashire boilers ? 
 
 1 23. What are the principal peculiarities of locomotive boilers ? 
 
 124. The firegrate area of a locomotive is 20-5 square feet. 
 It is intended to burn on the average 50 Ibs. of fuel per square 
 foot of grate-surface per hour. How much heating-surface would you 
 provide ? 
 
 125. A modern marine high-pressure boiler has to supply steam to 
 an engine indicating 560 horse-power, and which consumes 18 pounds 
 of water per I.H.P. per hour. How much grate-area would you think 
 it necessary to provide, and how much heating-surface, ordinary draught 
 being used ? 
 
 126. Give sketches showing the general arrangements and the 
 approximate dimensions of the boiler which you would provide for the 
 above purpose, the pressure being 90 Ibs. absolute. 
 
 127. If the shell-plates were made of steel, what thickness would 
 you employ, having reference to the Board of Trade rules ? 
 
 128. What area of opening of safety-valves would you allow? 
 
 129. Why are the ends of tubes furthest from the furnace or com- 
 bustion chamber of comparatively little use in absorbing heat when a 
 boiler is new? and why are they likely to be more useful after the 
 boiler has been worked for a time ? 
 
 130. An engine is required to give out very varying quantities of 
 power during the course of every hour. Would you provide for it a 
 boiler of comparatively large or of comparatively small cubic contents ? 
 State your reasons. 
 
 131. Investigate the strength of a hollow cylinder with flat ends 
 pressed from within, and prove that the strain in the plane of the axis 
 is double that in a plane at right angles to it. Does the shape of the 
 ends affect the strain transmitted by them to the boiler body ? 
 
 132. How are internal furnaces and flues constructed so as to allow 
 for expansion and contraction, and to provide against collapse ? 
 
 133. On what does the strength of a hollow cylinder to resist 
 collapse principally depend ? Why are hollow cylinders pressed from 
 without more liable to destruction than the same cylinders pressed with 
 equal force from within ? 
 
 134. State what you consider to be the principal advantages of 
 Fox's corrugated flues. 
 
518 TJie Steam Engine. 
 
 135. Explain the action of, and illustrate by sketches the Bourdon 
 pressure gauge. 
 
 136. Describe any of the structural arrangements with which you 
 are acquainted for attaching the ends to the body of a boiler, and for 
 strengthening them ; also for attaching the furnace tube to the ends. 
 State fully what precautions must be adopted in strengthening the flat 
 ends. 
 
 137. A cylindrical land boiler is 7 feet 6 inches in diameter, and has 
 to sustain a pressure of steam of 90 Ibs. by the gauge. What thickness 
 of mild steel shell-plates would you adopt for the shell ? 
 
 138. The effect of punching rivet-holes is to compress a thin layer 
 of the metal all round the hole, and to greatly increase its tensile 
 strength, and, at the same time, to diminish its stretching power. 
 When the holes are drilled the metal remains in its normal condition. 
 What conclusions would you draw from these facts as to the relative 
 strength of punched and riveted joints, and state your reasons ? 
 
 139. Make sketches, including sections, of single and double riveted 
 lap-joints. 
 
 140. State the methods in which a single riveted lap-joint may give 
 way when subjected to tensile strain. 
 
 Ans. i. The plate may tear asunder where its area is reduced, 
 between the rivet-holes. 2. The rivets may shear asunder. 3. The 
 metal between the rivet-holes and the edge of the plate may be crushed. 
 4. The plate may break across in front of the rivet-holes and at right 
 angles to the edge. (N.B. The two latter causes of fracture may be 
 provided against by giving the plates a sufficient depth of lap. As a 
 rule, the portion of the plates which overlap should be not less than 
 three times the diameter of the rivet-hole.) 
 
 141. Show how to proportion a single-riveted lap-joint so that the 
 resistance of the plates to tearing may just equal the resistance of the 
 rivets to shearing. 
 
 Ans. Let d - diameter of rivet in inches, t= thickness of plates, 
 
 ^ = area of a rivet, / = pitch of rivets, i.e. distance apart from centre 
 
 4 
 
 to centre, S= shearing strength of rivets per square inch, T = tensile 
 strength of plates per square inch. Then area of section of plate 
 between any two holes, multiplied by tensile strength of plate, must 
 equal area of one rivet multiplied by the shearing strength of the 
 material. 
 
Examples. 
 
 As a general rule, the tensile and shearing strengths are equal for iron 
 plates and iron rivets. 
 
 '.' f -(f-f),^ .-.t-'-g+d-flfl+j. 
 
 Which formula gives the pitch in terms of the thickness of plates and 
 the diameter of the rivets. 
 
 142. In a single-riveted joint the plates are inch thick, the rivets are 
 i inch diameter. What should be the pitch on the suppositions ( I ) that 
 the shearing and tensile strengths are equal, and (2) that the safe tensile 
 strength is 25 per cent, greater than the safe shearing strength ? 
 
 143. What must be the diameter of the rivet in a single-riveted 
 lap-joint so that the resistance of the rivet to crushing and shearing 
 may be equal ? 
 
 Ans. The resistance of the rivet to crushing equals its diameter 
 multiplied by the thickness of the plate, multiplied by the resistance of 
 the metal to crushing per square inch. The resistance of iron rivets in 
 iron plates to crushing is double the resistance to shearing. The resis- 
 tance of the rivet to shearing equals its area multiplied by the shearing 
 strength of the metal per square inch. 
 
 4 7T 
 
 The diameter thus obtained is, however, far larger than is admissible 
 in practice. 
 
 N. B. A practical rule for the diameter of the rivet in terms of the 
 thickness of the plate is 
 
 d=\'2 Vt. 
 
 If the diameters of rivets progress by i6ths of an inch, then for 
 single-riveted joints we may take the number of i6ths next above the 
 diameter, as given by the formula, and for double-riveted joints the 
 number of i6ths next below. 
 
 144. In a double-riveted lap-joint, find the pitch of the rivets so 
 that the shearing and tensile strength of the joints may be equal, for 
 iron plates and rivets. 
 
 Ans. Referring to Ex. 141, when the joint is double-riveted, we 
 have the area of two rivets to shear instead of one. 
 
 N.B. By making use of this equation d=i'2*/T, and substituting 
 this value of / in the above equation, we can obtain an expression for 
 the pitch in terms of the thickness of the plate alone. When the pitch 
 
520 The Steam Engine, 
 
 and diameter of the rivets are known the strength of the plate between 
 the rivet-holes can be ascertained, and the strength of the joint com- 
 pared with that of the whole plate can be calculated. 
 
 145. Make sketches, including sections of single-and double-riveted 
 butt-joints, in both single and double shear. 
 
 146. Is there any difference between the strength of a single-riveted 
 lap-joint and the corresponding butt-joint in single shear ? 
 
 147. Show how to proportion a double-riveted butt-joint in double 
 shear, so that the tensile strength of the plate between the rivet-holes 
 may be equal to the shearing strength of the rivets (iron plates and 
 rivets). 
 
 Ans. In this case (referring to Ex. i) we have four rivet areas 
 to shear instead of one. 
 
 t \(p-d}t = Trd~ and p=-^- + tt. 
 
 N.B. In practice it is not found safe to calculate on the area of 
 four rivets, and by the Board of Trade rules only 3! are allowed. 
 Hence, taking this into consideration, and making use of the formula 
 d=i'2 >//, we get 
 
 For plates of less than ^ inch thickness, the results got by the above 
 formula are greater than those adopted in practice, because the diameter 
 of the rivets is generally less than what would be indicated by theory. 
 
 148. Enumerate the fittings required for a Lancashire boiler. 
 
 149. Describe the principle on which the injector works, and make 
 a sectional sketch of an injector, showing how the steam and water 
 supply can be adjusted. 
 
 What are the objections to creating a draught in the ordinary way by 
 means of a funnel or chimney ? 
 
 150. Make a sketch illustrating the ordinary weighted lever safety- 
 valve. The diameter of opening of a safety-valve is 4 inches ; the 
 distance from the fulcrum to the centre of the valve is 5 inches ; the 
 lever is 21 inches long, weighs 3^ Ibs., and its centre of gravity is 8 
 inches from the fulcrum ; the valve weighs 5 Ibs. What weight must 
 be hung one inch from the end of the lever so that steam may blow 
 off at 50 Ibs. absolute per square inch ? 
 
 151. Explain the reasons which led to the abandonment of jet con- 
 densers for marine engines, and show how surface condensation has 
 rendered possible the use of high-pressure steam in marine boilers. 
 
 152. Describe fully, and illustrate with sketches, a modern surface 
 condenser for a marine engine, together with its air and circulating 
 pumps. 
 
Examples. 521 
 
 1 53. A marine engine indicating 4,000 H. P. uses 1 7 pounds of steam 
 per I.H.P. per hour, and expands down to 5 pounds absolute : what 
 surface would you provide in the condenser ; how much cooling water 
 of initial temperature of 60 would be required per hour ; and what 
 should be the capacity of the air-pumps (single-acting), the engine run- 
 ning at 56 revolutions per minute ? 
 
 154. Why must the capacity of the air-pump of a surface condenser 
 be so largely in excess of the volume of the water into which the 
 steam condenses per stroke ? 
 
 155. How is the condition of the vacuum in a condenser recorded ? 
 and what is the relation between the vacuum as recorded and the 
 absolute pressure in pounds per square inch ? 
 
 156. What precautions must be adopted in making the tubes of 
 surface condensers, so as to avoid the possibility of carrying over 
 corrosive metallic salts into the boiler ? 
 
 157. Make a sketch of any of the packings for condenser tubes in 
 common use. 
 
 158. Describe a jet condenser for a high-speed stationary engine, 
 and state how the use of a separate air-pump can be avoided. 
 
 159. An engine indicating 60 H.P. uses 20 pounds of steam per 
 I.H.P. per hour, and expands down to 6 pounds absolute : how much 
 injection water will be required per hour, the original temperature of 
 which is 60, the temperature of the hot well being 106 ? 
 
 160. Make a sketch, and give a description, of an ejector condenser 
 for a two-cylinder engine. 
 
 161. The metal of a cylinder is capable of rapidly receiving and 
 transmitting heat : explain why this property is the cause of serious loss 
 of efficiency in the expansive steam engine. How does it come to pass 
 that the use of the steam-jacket greatly diminishes this loss of efficiency 
 in spite of the fact that, to keep the cylinder always hot, steam is con- 
 stantly being condensed in the jacket, the greater part of the heat 
 thus liberated being uselessly transmitted to the exhaust. 
 
 162. What is the object of superheating steam? Is it possible to 
 obtain any practical good effect in modern high-pressure engines by 
 superheating the steam ? Give your reasons for your reply. 
 
 163. Examine the effect of the dimensions of the cylinder, the 
 initial pressure of the steam, and the rate of expansion on the initial 
 condensation which takes place in the cylinder. 
 
 164. What are the four principal causes for the presence of water in 
 the cylinders of steam engines ? 
 
 165. What is the object of compounding or expanding the steam 
 successively in two or more cylinders instead of in one ? Why is it 
 
522 The Steam Engine. 
 
 that in modern marine engineering triple and quadruple expansive 
 engines are now so largely used ? 
 
 1 66. Illustrate by outline diagrams the principal methods of 
 arranging the cylinders in ordinary and triple expansive engines. 
 
 167. A simple and a compound engine work at the same rate of 
 expansion, and develope the same power. What is the size of the low- 
 pressure cylinder of the compound compared with the cylinder of the 
 simple engine ? Give your reasons for your reply. 
 
 1 68. State the mechanical advantages of compound over simple 
 expansive engines, and investigate the ratio of maximum to mean pres- 
 sures, in (i) a pair of simple expansive engines, and (2) a compound 
 engine of the same power, and working with the same initial pressure 
 and the same ratio of expansion, on the supposition that the steam 
 expands hyperbolically and that the effects of early release, compression, 
 and clearance are neglected. The initial pressure is 115 Ibs. absolute. 
 The ratio of expansion = 12, the area of each of the high-pressure cylin- 
 ders = A. The ratio of area of low to area of high -pressure cylinder in 
 the compound engine -=4-5. The received pressure 24 Ibs. per square 
 inch. 
 
 169. Explain how in practice the powers developed in the two 
 cylinders of a compound engine may be made approximately equal. 
 
 170. Make a sketch diagram illustrating the distribution of the 
 steam in both cylinders of an ordinary two-cylinder receiver com- 
 pound engine, choosing any symbols you like to represent the govern- 
 ing data. The steam in the large cylinder is supposed to be cut off 
 before half stroke, and the expansion to take place hyperbolically. The 
 effects of clearance, early release, and compression are to be neglected. 
 
 171. A triple expansive engine works at a consumption of 1*3 Ib. 
 of coal per I.H.P. per hour. The boilers evaporate 8 pounds of water 
 per hour per pound of coal from the temperature of the feed 105, and 
 at the temperature of the initial pressure of the steam, viz. 165 Ibs. 
 above the atmosphere. (Corresponding temperature 366.) What is 
 the consumption of steam per horse-power per hour ? What would it be 
 if the engines were theoretically perfect and working between the above 
 limits of temperature ? What is the efficiency of the engine compared 
 to a perfect engine ? What is its absolute efficiency ? What is the 
 efficiency of the boiler compared to that of a perfect boiler which cools 
 the products of combustion down to the temperature of the feed, 
 the heat of combustion of one pound of coal being put down as 14,000 
 thermal units ? Finally, what proportion of the total heat-supply is 
 wasted by the boiler and what by the engine ? 
 
INDEX. 
 
 ABS 
 
 A BSOLUTE temperature, 50 
 r* Adhesion of locomotives, 141 
 Adiabatic expansion of gas, 68, 84 
 
 of steam, 70 
 
 Advance angle of eccentric, 255 
 Air pumps, 433, 434 
 
 supply required for combustion of 
 fuel, 353 
 
 thermometer, 49 
 Appendix, 489 
 
 Area of fire grate, 384 
 
 Areas bounded by curves, calculation 
 
 of, 82, 83 
 Axle boxes, 238 
 
 T3 ACK pressure, 134, 141, 419 
 
 lines in diagrams, 328 
 Balanced slide valves, 270 
 Bearings, 236 
 Blast pipes, 375, 419 
 Boiler construction, materials used in, 
 399 
 
 description of vertical, 5 
 
 ends, staying of, 365 
 
 fittings, 401 
 
 internal flues, strengthening of, 363, 
 35 
 
 shells, strength of cylindrical, 389, 
 39 1 
 
 Boilers, Cornish, 361 
 
 cubic capacities of various types 
 of, 387 . 
 
 cylindrical, with external firing, 359 
 
 description of various types of, 358 
 
 draught production, 415 
 
 effects of unequal expansion on 
 strength of, 398 
 
 essential parts of, 358 
 
 evaporative powers of fuel in different 
 types of, 383 
 
 feed apparatus for, 408 
 
 Lancashire, 363 et seqq. 
 
 locomotive, 371 et seqq. 
 
 marine, 377 et seqq. 
 
 proportions of, 383 
 
 CON 
 
 Boilers, strength of, 389 
 
 tubulous, 369 
 
 Bourdon pressure gauge, 407 
 
 Boyle's and Charles' laws, combination 
 
 of, 51 
 Boyle's law, 43, 74 
 
 graphic representation of, 44 
 Brine discharge, 427 
 British thermal unit, 38 
 
 (CAPACITY for heat, 38 
 ^ Carbonic acid, 348 
 
 oxide, 348 
 Centrifugal force, 159 
 Charles' law, 46, 74 
 
 and Boyle's laws, combination of, 51 
 Chimneys,action of, in creating draught, 
 
 f - 415 , 
 
 Circular motion, 157 
 
 Clearance, 120, 131, 203 
 Closed ash pits, 421 
 
 stoke holds, 420 
 
 Combination of Boyle's and Chaiks 
 
 laws, 51 
 Combustion of fuel, 347^ 
 
 air supply required for, 353 
 Combustion, weight and temperature 
 
 of products of, 353 
 Compound engines, distribution of 
 
 steam in, 465 
 
 indicator diagrams of, 477 
 rrechanical advantages of, 480 
 
 relative sizes of cylinders of, 485 
 
 various types of, 456 
 Compounding as a means of diminishing 
 
 condensation in cylinders, 455 
 
 economic advantages of, 453, 464 
 
 experiments on value of, 452 
 Compression of exhaust steam, 330, 454 
 Corliss' valve gear, 264 
 
 Cornish boiler, 361 
 Corrugated furnaces, 365 
 Condensation and re-evaporation in 
 cylinders, 439 et seqq. 
 
52 4 
 
 Index. 
 
 :es acting on, 220 
 
 CON 
 
 Condensation and re-evaporation in 
 cylinder, shown by indicator dia- 
 gram, 324 
 
 in cylinders, compounding as a 
 means of diminishing, 455 
 
 quantity of water required t.o effect, 
 
 425 
 
 surface, 427 
 Condensers, 423 
 
 ejector, 435 _ 
 
 for land engines, 429 
 
 Connecting rod, influence of, in modify- 
 ing curve of twisting moments, 172, 
 183 
 
 no loss of power due to use of, 222 
 Connecting rods, 219 
 
 examples of, 224 
 
 resolution of forces 
 
 Cranks, 219, 226 
 
 Crank pins, safe working pressure on, 
 229 
 
 Crank shafts, 232 
 - twisting moments on, 162 et seqq. 
 
 influence of connecting rod in 
 
 modifying twisting moments on, 172, 
 183 
 
 - influence of inertia of recipro- 
 cating parts in modifying twisting 
 moments on, 176, 189 
 
 strains in, 232 
 Cross heads, 213 
 
 Curve of twisting moments, how to form 
 
 complete, 194 
 
 Cushioning, exhaust steam, 330, 454 
 Cycle of operations, 78 
 Cylinder, causes of presence of water 
 
 in, 445 
 
 pressed from within, strength o f , 389 
 
 without, strength of, 395 
 
 Cylinders, compounding as a means of 
 
 diminishing condensation in, 455 
 
 condensation and re-evaporation in, 
 439 et seqg. 
 
 details of, 201 
 
 experiments on steam consumption, 
 in conducting and non-conducting, 
 
 of compound engines, relative sizes 
 of, 485 
 
 precautions t > be observed in jacket- 
 ing, 452 
 
 Cylindrical boiler with external firing, 
 
 359 
 shells, strength of, 389, 391 
 
 "TJALTON'S law, 47 
 
 ^ Davy on modern theory of heat, 22 
 
 Dead plate, 402 
 
 De Pambour's theory of steam engine, 
 
 130 
 
 Details of engines, 201 et seqg. 
 Diagrams, indicator, 317 
 Zeuner's slide valve, 273 
 
 FLA 
 
 Double-ported slide valves, 269 
 
 Draught, artificial, 375, 420 
 
 production, 415 
 
 Dynamics, elementary principles of, 145 
 
 gCCENTRIC, angle of advance of, 
 
 rods, 231 
 Eccentrics, 229 
 
 Effective pressure on piston, 194, 340 
 Efficiency of engine, definition of, 89 
 
 heating surface, 385 
 
 perfect heat engines, 88 et seqq. 
 
 in steam engine, losses of, 113 et 
 seqq. 
 
 Ejector condensers, 435 
 Energy, definition of, 23 
 
 examples of, 25, 154 f 
 
 heat a form of, 22, 28 
 
 - various forms of, 26 
 
 Engine (see Compound engine, Loco- 
 motive engine, Marine engine, Steam 
 engine, and Triple expansive engine). 
 
 Engines, efficiency of perfect, 88 et seqq. 
 
 - ideally perfect heat, 85 et seqq. 
 Equivalent pressures, how to represent 
 
 expenditure of heat by, 101 
 Evaporation, external and internal work 
 of, 58. 62, 100 
 
 of water, heat absorbed in, 57, 60, 99 
 Evaporative power of fuel, 352 
 
 in various types of boilers, 383 
 Examples and examination questions, 
 
 136, 151, 158, 281, 313, 324, 505 
 Exhaust steam, cushioning of, 330, 454 
 Expansion line in indicator diagrams. 
 
 326 
 
 means of varying rate of, 257 
 
 - of gas, 65, 74 
 
 heat expended in, 81 
 
 of steam, 67, 105, 128 
 
 work done during, 106, 128 
 
 Expenditure of heat in steam engines, 
 
 101 
 
 represented by equivalent pres- 
 sure, 101 
 
 when condition of steam at end 
 of stroke is known, 107 
 
 Experiments on heat, Joule's, 41 
 
 on steam consumption in conducting 
 and non-conducting cylinders, 447 
 
 on value of jacketing and compound- 
 ing, 452 
 
 External work done in vaporising 
 water, 62, 100 
 
 "P ACTORS of safety, 235, 391 
 
 * Feed apparatus for boiler, 408 
 
 heaters, 413 
 
 Firebars, 403 
 
 Fittings of boilers, 401 
 
 Flat surfaces, staying of, 374, 396 
 
Index. 
 
 525 
 
 FLU 
 
 Flues, strengthening of internal, 363, 
 
 Fly-wheel, graphic representation of 
 action of, 196 
 
 uses of, 14, 196 
 
 construction of, 247 
 
 how to proportion weight of, 198 
 - strength of, 246 
 
 theory of action of, 196 
 Force, centrifugal, 159 
 
 definition of, 146 
 
 Forced draught, 375, 419, 420 
 Formulae of thermodynamics, summary 
 
 of, 1 20 
 
 Fox's corrugated furnaces, 365 
 Friction of steam engines, 134, 141 
 Fuel, chemical constituents of, 346 
 
 combustion of, 347 
 
 evaporative power of, 352 
 
 heat of combustion of, 349, 352 
 
 saving effected by compounding, 
 453. 464 
 
 waste of, 355 
 Furnace doors, 401 
 
 flues, strength of, 395 
 Furnaces, Fox's corrugated, 365 
 
 C* ALLOW AY tubes, 364 
 ^^ Gas, adiabatic expansion of, 63, 
 84 
 
 connection between pressure and 
 volume of, 43 
 
 connection between temperature and 
 volume and pressure of, 46 
 
 expansion of, 65, 74 
 
 heat expended in expanding, 81 
 
 isothermals of, 65, 74 
 
 nature of. 42 
 
 specific heat of, at constant pressure, 
 5 2 . 75 
 
 specific heat of, at constant volume, 
 5 2 > 75 
 
 Gases, effect of application of heat to, 
 42 
 
 specific heat of, 52, 75 
 Geometrical representation of motion 
 
 of slide valves, 273 
 Geometry of link motions, 294 
 Governor, Watts', 239 
 Governors, 238 
 
 acting on link motion, 244 
 
 high speed, 242 
 
 marine, 245 
 
 theory of, 240 
 
 with crossed arms, 241 
 Gradient, resistance due to, 142 
 Graphic diagram of twisting moment 
 
 on crank shaft, 166 et seqq. 
 
 representation of action of fly-wheels, 
 196 
 
 - representation of heat expended in 
 evaporating water, 63 
 
 representation of work, 79 
 
 IND 
 
 Grate area, 384 
 
 ratio of, to heating surface, 386 
 Gusset stays, 365, 397 
 
 l_I EAT, a form of energy, 22 
 
 absorbed in evaporating water, 
 57, 60, 99 
 
 - and work, relation between, 40 
 
 application of, to water and ice, 55, 
 97 
 
 capacity for, 38 
 
 conversion of, into work, 29 
 
 effect of application of, to gases, 42 
 
 ngines, ideally perfect, 85 et seqq. 
 
 xpended in expanding gas, 81 
 
 xpenditure of, in steam engines, 101 
 
 - xpenditure of, represented by equi- 
 alent pressures, 101 
 
 xpenditure of, when condition of 
 earn at end of stroke is known, 107 
 
 --- general ideas of nature of, 21 
 
 Joule's experiments on, 41 
 
 material theory of, 21 
 
 measurement of, 31 
 
 mechanical equivalent of, 40 
 
 of combustion of fuel, 349, 352 
 
 of steam, total, 60, 99 
 
 - old notions regarding, 21 
 
 quantity of, 37 
 
 -- specific, 38, 52, 75, 98 
 
 specific, of gas at constant pressure, 
 
 52 ' 7? C r 
 
 specific, of gas at constant volume, 
 
 52..7S 
 Heating surface, efficiency of, 385 
 
 ratio of, to grate area, 386 
 High speed governors, 242 
 Horse-power defined, 23, 135 
 
 how to measure from indicator dia- 
 grams, 323 
 
 Howden's system of forced draught, 
 
 421 
 Hyperbolic logarithms, table of, 498 et 
 
 'seqq. 
 
 curve, how to draw, 333 
 
 1 CE, application of heat to, 55 
 
 1 Ideally perfect heat engines, 85 et 
 
 seqq. 
 
 Indicated power, gross and net, 339 
 Indicator diagrams, condensation and 
 
 re-evaporation shown by, 334 
 
 examples of, 332 
 
 expansion curve in, 326 
 
 expenditure of heat accounted for 
 
 |jy 343 
 
 from faulty engines, examples of, 
 336 
 
 from locomotive, examples of, 338 
 
 general character of, 321 
 
 how defects are revealed by, 325 
 
526 
 
 Index. 
 
 IND 
 
 Indicator diagrams, how they reveal 
 leaking pistons, and slide valves, 336 
 
 how to deduce effective pressure 
 on piston from, 194, 340 
 
 how to measure horse-power from, 
 
 323 
 of compound engines, 477 
 
 point of release, 328 
 
 the exhaust line of, 336 
 Richards', 318 
 
 Indicators and indicator diagrams, 317 
 
 Injectors, 409 
 
 Internal work done in vaporising water, 
 
 62, 100 
 
 Isothermal curve, definition of, 45 
 Isothermals of steam, 67 
 
 of gas, 65, 74 
 
 JACKETING cylinders, precautions 
 to be observed in, 452 
 Jackets, experiments on value of, 452 
 
 steam, 108, 205, 450 
 Joule's experiments on heat, 41 
 Journals, 236 
 
 Joy's valve gear, 271 
 
 * T ANCASHIRE boiler, 363 et seqq. 
 
 Lap, definition of, 254 
 Law. Boyle's, 43, 74 
 
 Charles', 46, 74 
 
 connecting pressure and temperature 
 of steam, 60, 97 
 
 connecting pressure volume and 
 density of steam, 61, 97 
 
 Dalton's, 47 
 
 Laws and formulae of thermodynamics, 
 summary of, 120 
 
 combination of Boyle's and Charles', 
 
 5'; 74 . 
 
 of motion, 148 
 
 of thermodynamics, 40, 92, 100 
 Lead, definition of, 254 
 
 Leaking pistons, and slide valves, how 
 
 to detect. 336 
 Link motion, geometrical method of 
 
 fixing centres of valve circles, 294 
 governors acting on, 244. 
 
 Zeuner's diagrams applied to, 
 290 
 
 motions, 258 
 
 analytical method of fixing centres 
 
 of valve circles; 292 
 
 suspension of, 300 
 Links, various forms of, 261 
 Liquefying ice, heat absorbed in, 55 
 ' Load' of steam engines defined, 135 
 Locomotive axle boxes, 238 
 
 boilers, 371 et seqq. 
 
 crank shaft, 228 
 - cylinder, 201 
 
 engine, examples of diagrams from. 
 338 
 
 PIS 
 
 Locomotive engines, 139 
 - - pistons, 208 
 
 adhesion of, 141 
 
 back pressure in, 141, 419 
 
 resistances to motion of, 141 
 Logarithms, table of hyperbolic, 498 et 
 
 seqq. 
 Losses of efficiency of steam engines, 
 
 113 et seqq. 
 Lubricators, 207 
 
 IV/T ARINE boilers, 377 et seqq. 
 lVA engine connecting rods, 225 
 governors, 245 
 
 pistons, 211 
 
 valves, 268 
 
 various types of compound,. 456 
 
 et seqq. 
 
 Mass, definition of, 145 
 
 )iler 
 399 
 
 Materials used in boiler construction, 
 
 Measurement of heat, 31 
 Mechanical advantages of compound 
 engines, 480 
 
 equivalent of heat, 40 
 Mechanics of the steam engine, 145 et 
 
 seqq. 
 Mechanism of steam engines, 201 ei 
 
 seqq. 
 
 Metals, shearing strengths of, 235 
 Meyer's valve gear, 263 
 
 application of Zeuner's dia- 
 grams to. 3<">2 et seqq. 
 
 problems in, 313 
 -- reversing with, 311 
 
 Motion blocks, 214 
 
 definition of 146 
 
 laws of, 148 
 
 --of bodies in circles, 157 
 
 under action of constant force 
 
 TVJAVIER'S law connecting pressure 
 ^ ' and volume of steam, 127 
 
 QPERATIONS, cycle of, 78 
 
 PACKINGS for pistons, 209 
 for stuffing boxes, 202 
 
 for surface condenser tubes, 435 
 Pedestals, 236 
 
 Perfect heat engines, 85 et seqq. 
 
 application of theory of, to 
 steam, no 
 
 efficiency of, 88 et seqq. 
 Piston packings, 209 . 
 
 valves, 268 
 Pistons, 208 
 
 effective pressure on, deduced from 
 indicator diagrams, 194 340 
 
Index. 
 
 527 
 
 PIS 
 
 Pistons, how to detect leaky, 336 
 Pressure and temperature of steam, law 
 connecting, 60, 97 
 
 on crank pins, 229 
 
 on pistons, effective, 194, 340 
 
 on slide bars, 214 
 Pressure gauges, 407 
 
 Problems, 136, 151, 158, 281, 313, 324, 
 
 505 
 Products of combustion, weight and 
 
 temperature of, 353 
 
 Properties of steam, table of, 489 etseqq. 
 Proportions of boilers, 383 
 
 UANTITY of heat, 57 
 
 UAMS BOTTOM'S lubricators, 207 
 ** piston rings, 210 
 
 reversing gear, 261 
 
 Rankine's formula for expenditure of 
 heat in steam engines, no 
 
 law connecting pressure and volume 
 of steam, 97 
 
 Receiver compound engines, 456 
 Reciprocating parts, example of action 
 
 of, on curve of twisting moments, 189 
 -- influence of inertia of, in modifying 
 
 twisting moments, 176, 189 
 -- influence of steam distribution on 
 
 action of, 192 
 --- influence of weights of, in vertical 
 
 engines, 181 
 
 Relative volume of steam, 61, 127 
 Release, point of, in indicator diagrams, 
 
 328 
 
 Resistance due to gradient, 142 
 Resistances to motion of steam engines, 
 
 !33, M 1 
 Reversing, means of, 257 
 
 with Meyer's valve gear, 311 
 
 gear, Ramsbottom's, 261 
 Richards' indic3fcor, 318 
 
 Riveted joints, strength of, 392, 519 
 
 C AFETV, factors of, 235, 391 
 
 *^ Safety valves, 404 
 
 Shaft bearings, 236 
 
 Shearing strength of metals, 235 
 
 Slide bars, 214 et seqq. 
 
 Slide valve, distribution of steam by, 
 
 12, 249, 273 
 with lap and lead, and driven by 
 
 single eccentric, 254 et seqq. 
 
 setting problems in, 281 
 Slide valves, 249 et seqq. 
 
 and ports, how to design, 289 
 
 balancing of, 270 
 
 double ported, 269 
 
 geometrical representation of 
 action of, 273 
 
 leaking, how to discover, 336 
 
 STE 
 
 Slide valves, without lap or lead and 
 driven by a single eccentric, 249 et 
 
 Zeuner's diagrams for, 273 et seqq. 
 Specific heat, 38, 52, 75, 98 
 
 of gas at constant pressure, 52, 
 
 75 
 
 of gas at constant volume, 52, 75 
 
 of steam, 98 
 
 of superheated steam, 98 
 
 of water, 39, 98 
 
 Specific volume of steam, 61 
 Staying boiler ends, 365 
 
 of flat surfaces, 374, 396 
 Stays, gusset, 365, 397 
 
 Steam, adiabatic expansion of, 70 
 
 consumption, in conducting and non- 
 conducting cylinders, 447 
 
 cushioning exhaust, 330, 454 
 
 distribution, influence of, on action 
 of reciprocating parts, 192 
 
 distribution of, by slide valve, 1 2, 249 
 273 
 
 distribution of, in compound engines 
 465 et seqq. 
 
 expansion of, 67, 105, 128 
 
 isothermals of, 67 
 
 law connecting pressure and tem- 
 perature of, 60, 97 
 
 law connecting pressure and volume 
 of, 61, 97 
 
 -- specific and relative volumes of, 6r, 
 127 
 
 table of properties of, 489 et seqq. 
 
 total heat of, 60, 99 
 
 work done during expansion of, 106, 
 128 
 
 Steam engine (see also Compound, 
 Locomotive, Marine, and Triple 
 expansive engines). 
 
 application of theory of perfect 
 engine to, no 
 
 De Pambour s theory of, 130 
 
 description of simple form of, 9 
 elementary conception of, 2 
 
 essential elements of, 3 
 
 mechanics of, 145 
 
 subdivisions of study of, 16 
 
 expenditure of heat in, 101 
 friction of, 134, 191 
 
 influence of weights of recipro- 
 cating parts in, 176 et seqq. 
 
 ' Load ' of, defined 135 
 
 locomotive, 139. 
 
 losses of efficiency of, 113 et seqq. 
 
 mechanism of, 201 et seqq. 
 
 purposes for which used, 15 
 
 resistances to motion of, 133, 141 
 Steam jackets, 108, 205, 450 
 
 experiments on, 452 
 
 Steam passages, proportions of, 203 
 Steam, superheated, used to diminish 
 
 condensation in cylinders, 448 
 Stephensun's link motion, 258 
 
5-28 
 
 Index. 
 
 STR. 
 
 Strength of boilers, 389 
 
 effects of unequal expansion on, 
 
 39 8 
 
 of cylindrical boiler shells, 389, 391 
 
 of furnace flues, 395 
 
 of hollow cylinder, pressed from 
 without, 395 
 
 of riveted joints, 392, 518 
 Stuffing boxes, 202 
 
 Summary of laws and formulae of ther- 
 modynamics, 120* 
 
 Superheated steam, 98 
 
 : .used to diminish condensation in 
 
 cylinders, 448 
 
 Surface condensation, 427 
 
 condenser, example of, 432 
 Suspension of link motions, 300 
 
 H^ABLE of experiments on jacketing 
 and compounding, 453 
 
 hyperbolic logarithms, 498 et seqq. 
 
 properties of steam, 489 et seqq. 
 
 strength of riveted joints, 393 
 Tandem, compound engines, 456 
 Temperature, 31 
 
 absolute, 50 
 
 and pressure of steam, law connect- 
 ing, 60, 97 
 
 of products of combustion, 353 
 - scales of, 34 
 
 Theory of action of fly wheel, 196 
 
 of governors, 240 
 
 of perfect heat engine applied to 
 steam, no 
 
 of steam engine, De Pambour's, ' 
 130 
 
 Thermal unit, British, 38 
 Thermodynamics, first law of, 40, 100 
 
 second law of, 92, 120 
 
 summary of laws and formulae, 120 
 Thermometer, the air, 49 
 
 . Thermometers, 32 
 Total heat of steam, 60, 99 
 Triple expansive engines, 461 
 
 curve* of twisting moment on 
 crank shaft of, 48^ 
 Tubulous boilers, 369 
 Twisting moment, on crank shaft, 162 
 et seqq. 
 
 how to approximate to uniformity 
 of, 195 
 
 j- how to form complete curve of, 104 
 influence of connecting rods in 
 
 1. 
 
 ZEU 
 
 Twisting moment, on crank shaft of 
 triple expan>ive engine, 484 
 influence of inertia of recipro- 
 cating parts in modifying, 176, 189 
 
 T T NIT of heat (British), 38 
 
 *-' Units of mass, weight, .velocity, 
 
 and force, 147 
 Vacuum indications, 437 
 Valve boxes, 205 
 Valve gear, Corliss', 264 
 Joy's, 271 
 Meyer's, 263 
 Valves, safety, 404 
 
 simple slide. 249 et seqq (see also 
 Slide valves). 
 
 varieties of, 268 
 Velocity, definition 0^145 
 
 Vertical engines, influence of weights of 
 
 reciprocating parts in, 181 
 Volumes 6f steam, specific and relative, 
 
 61, 127 
 
 \\TATER, application of heat to, 55, 
 
 97 
 
 -- heat absorbed in evaporating, 57, 60, 
 9.9 
 
 in cylinders, causes of presence of, 
 445 
 
 Water gauges, 412 
 Waste of fuel, 355. 
 Watt's governor, 239 
 t Weight, definition -of, .145 
 Work and heat, relation between, 40 
 Work done during expansion of steam, 
 
 166, 128' 
 -- in evaporation, 58, 62, 100 
 
 when gas is heated at constant 
 
 pressure, 53, 84 ' 
 
 external, done in vaporising water, 
 62, loo 
 
 - graphic representation of, 79 
 
 internal, done in vaporising water, 
 62, 100 
 
 rate of doing, 33 
 
 /EUNER'S valve diagrams, 273 et 
 
 ** seqq. 
 
 applied to link motion, 290 
 ' applied to Meyer's valve gear, 
 302 et*seqq. 
 
 law connecting pressure and volume 
 of steam, 97 
 
 
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