UNIVERSITY OF CALIFORNIA 
 
 ANDREW 
 
 SMITH 
 
 HALLIDIL: 
 
THE 
 
 THETA-PHI DIAGRAM 
 
 PRACTICALLY APPLIED TO 
 
 STEAM, GAS, OIL, AND AIR ENGINES. 
 
 HENRY A. GOLDING, A.M.I.M.E., 
 
 Chief Draughtsman, Messrs. B. Donkin and Co., an-l Assistant Lecturer, South- 
 western Polytechnic, London. 
 
 PRICE THREE SHILLINGS NET. 
 
 THE TECHNICAL PUBLISHING CO. LIMITED, 
 
 31, WHITWORTH STREET, MANCHESTER. 
 
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 AND 30, SHOE LANE, LONDON; AND RIDGEFIELD, MANCHESTER. 
 And all Booksellers. 
 
HALLIDIE 
 
PREFACE. 
 
 IN the following pages an attempt has been made to present in as 
 simple and practical a manner as possible, the use of the temperature- 
 entropy diagram and the various methods of drawing it for different 
 heat motors. That the subject presented peculiar difficulties, because 
 of its unfitness for presentation in a popular manner, will readily be 
 granted ; but I venture to think that one of the principal reasons for 
 the lack of knowledge upon the subject by draughtsmen, steam 
 students, and others has been the want of an elementary work, not 
 overcrowded with mathematics. Most of the literature upon the 
 subject has presented the mathematical rather than the graphical 
 side of the question, with the result that students have become 
 afraid of tackling what they believe to be an intricate mathematical 
 investigation. 
 
 Of the utility of the temperature-entropy diagram in representing 
 the various thermal changes which take place in all heat motors there 
 cannot be any doubt. To quote only one authority, Mr. Mark H. 
 Robinson, in the discussion on Mr. Willans' last paper, said : " Up to a 
 certain point the practical man might ignore the present paper, and 
 others like it ; but if he aspired to design economical steam engines, he 
 might derive more good from the study of, say, Mr. Macfarlane Gray's 
 t) <j> diagram than from many portfolios of working* drawings." 
 
 Where authorities have been quoted or made use of, the particulars 
 are given in the text, but I will take this opportunity of expressing my 
 indebtedness to Professor Ewing for his work on " The Steam Engine 
 and other Heat Engines," and his Cantor Lectures on the "Mechanical 
 Production of Cold " ; to Professor Boulvin, for his articles in La 
 Revue de Mecanique; and to various papers, principally those by the 
 late Mr. P. W. Willans and Mr. Macfarlane Gray, published in the 
 Proceedings of the Institutions of Civil and Mechanical Engineers. I 
 also wish to thank the Council of the latter Institution for permission 
 to reproduce some of the indicator diagrams and figures given in the 
 reports of the Steam Jacket Research Committee. 
 
 119590 
 
IV. PREFACE. 
 
 I regret that in Chapter IV., page 70, a slight inaccuracy should have 
 occurred. Referring to the late Mr. Willans' method of calculating the 
 thermal efficiency in his central valve engine tests, I find that the 
 lower temperature of 110 deg. Fah. was only assumed for the 
 preliminary calculations of the comparative losses due to incomplete 
 expansion in condensing and non-condensing engines, but for the actual 
 trials the temperature in the exhaust chamber was taken in each trial 
 separately, and used for calculating the thermal efficiency. (See line 
 10 in Appendix, Table I., Willans on " Condensing Steam Engine 
 Trials.") I am also indebted to Captain Sankey for pointing out that 
 the specific heat of gases is not constant at the high temperatures 
 occurring in gas and oil engines, and therefore the calculations which 
 involve the use of C p and C v for a gaseous mixture at high tempera- 
 tures must necessarily be looked upon as approximate only. 
 
 HENRY A. GOLDING. 
 London, 
 
 September, 1898. 
 
CONTENTS. 
 
 PAGE 
 
 INTRODUCTION vii. 
 
 CHAPTER I. ENTROPY. 
 
 ARTICLE 
 
 1. Introduction 1 
 
 2. Entropy Diagrams 2 
 
 3. Entropy 4 
 
 4. Theoretical Entropy Diagram for Steam .... 5 
 
 CHAPTER II. ENTROPY OF WATER AND STEAM. 
 
 5. Method of Constructing the Curves 7 
 
 6. Entropy of Water 10 
 
 7. Entropy of Steam 11 
 
 8. Entropy Diagram for Ice, Water, and Steam 11 
 
 9. Constant Volume Curves 15 
 
 CHAPTER III. CONVERSION OF INDICATOR DIAGRAM TO ENTROPY DIAGRAM. 
 
 10. Calculation of Dryness Fraction 1 
 
 11. Actual Example, Compound Engine 23 
 
 12. Complete Entropy Diagram (Professor Boulvin's Chart) 28 
 
 13. Application of Complete Entropy Diagram 33 
 
 14. Q (ft Diagram for Carnot Cycle 34 
 
 15. Condensation during Adiabatic Expansion 35 
 
 16. Adiabatic Expansion of Wet Steam 37 
 
 CHAPTER IV.-HEAT LOSSES, 
 
 17. Effect of Steam Jacketing 39 
 
 18. Theoretical Entropy Diagram for Superheated Steam 46 
 
 19. Effect of Superheating 49 
 
 20. Effect of Speed 56 
 
 21. Compounding r ' 9 
 
 22. Initial Condensation 63 
 
 23. Measurement of Heat Losses 68 
 
VI. CONTENTS. 
 
 CHAPTER V. APPLICATION TO THE GAS ENGINE. 
 
 PAGE 
 
 24. General Considerations ". 73 
 
 25. Diagram for Theoretical Gas Engine 76 
 
 26. Diagram for Actual Gas-engine Trial 81 
 
 27. Corrected Diagram for Gas-engine Trial 89 
 
 28. Constant Volume Curves 92 
 
 29. Heat Losses in 7 Horse Power Gas Engine 94 
 
 CHAPTER VI. APPLICATION TO OIL AND AIR ENGINES. 
 
 30. Diagram for 20 Horse Power Diesel Motor 97 
 
 31. Stirling's Hot-air Engine.'. 101 
 
 32. Ericsson's Hot-air Engine 104 
 
 33. Entropy Diagram for Refrigerators 105 
 
 APPENDIX. 
 
 Weight of Dry Saturated Steam 109 
 
INTRODUCTION. 
 
 THE following contribution to the temperature-entropy method of 
 graphically solving thermo-dynamic problems marks a further step in 
 advance in the practical application of the system to the every-day 
 questions that arise in the study of the steam engine and other heat 
 motors. Although the method was foreshadowed by William Gibbs 
 in 1873, it is only within the last few years that it has been applied in 
 practice. Unfortunately, information respecting it is scattered about 
 in the Proceedings of the Institution of Civil Engineers, in those of the 
 Institution of Mechanical Engineers, and in various technical journals, 
 both British and foreign. The Author has collected this information 
 together, and has produced a work which treats the matter in a 
 comprehensive manner, bringing it up to date so far as published 
 materials allow. 
 
 Though not prepared to endorse every view expressed, I can fully 
 recommend the book as likely to be eminently helpful to those studying 
 the subject for the first time. 
 
 H. R. SANKEY, R.E. 
 
THE ENTROPY DIAGRAM AND ITS 
 
 APPLICATIONS. 
 
 rE%^ 
 
 ' UNIVERSITY) 
 
 CHAPTER I. ENTR--- 
 1. INTRODUCTION. 
 
 THE representation of various forms of energy by means of 
 a diagram has long been known and used with advantage 
 by engineers and others. It is usually shown as a closed 
 figure, the area of which represents energy (in either work, 
 heat, or other units), and the ordinates, pressure or resistance 
 overcome, and space passed through. The ordinary indicator 
 diagram "is perhaps the most common example of such 
 figures, but it does not show the reception and distribution 
 of heat which takes place in all steam and other heat 
 engines. Considering the great advance made by the science 
 of thermo-dynamics in recent years, it is somewhat sur- 
 prising to find that the representation of the thermal changes 
 which take place in all heat engines, in the form of a " heat 
 diagram," has been so little applied for practical use. The 
 relative advantages of the mathematical and graphical 
 methods of representing the result of any process are so 
 well known, that it will be unnecessary to refer to them 
 here ; beyond stating that where both methods can be 
 employed the graphical very often becomes a useful adjunct 
 to the mathematical, and is usually more easily grasped and 
 understood by draughtsmen, students, and others. 
 
 The introduction of entropy diagrams is mainly due to 
 Mr. J. Macfarlane Gray, who, in a paper read at the Paris 
 meeting of the Institution of Mechanical Engineers in 1889, 
 
 2T 
 
2 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 first showed the method of representing the heat contained 
 by water, steam, and various ideal substances, on what he 
 termed a "theta-phi chart"; the vertical ordinates of which 
 represented temperature, and the horizontal entropy. By 
 denoting all absolute temperatures by the Greek letter 
 (theta), and all quantities of entropy by the letter (phi), 
 the diagram has come to be known in England as the 
 " theta-phi diagram." In this book it will be preferable to 
 adopt the Fahrenheit scale of temperature as that most 
 used by practical men, and therefore absolute temperatures 
 (Fah.) will usually be denoted by the Greek letter r (tau), in 
 accordance with most of the recent works on thermo- 
 dynamics. 
 
 The practical application of the entropy diagram is 
 perhaps more due to the late Mr. P. W. Willans, who, in 
 a paper on " Non-condensing Steam-engine Trials," read 
 before the Institution of Civil Engineers in 1888,* first used 
 the diagram for the representation of steam-engine perform- 
 ance. Since then Capt. H. R. Sankey, R.E., has extended 
 the subject by applying the diagram to the marine engines 
 tested by the Research Committee of the Institution of 
 Mechanical Engineers ;t but there still seems to be a dearth 
 of information on how to draw the diagrams that, it is hoped, 
 the present book will, in a measure, supply. 
 
 2. ENTROPY DIAGRAMS. 
 
 The theta-phi or temperature-entropy diagram is a graphic 
 representation of the thermal changes which take place in a 
 steam-engine cylinder during one cycle. It is plotted on a 
 theta-phi chart, by calculations made from the mean indicator 
 diagram of an experiment. As an aid to the thermo-dynamic 
 study of the steam engine, it is much more useful than the 
 better known indicator diagram, as it shows at a glance the 
 thermal efficiency of the engine. The ordinary indicator 
 
 * See Proc. Inst. of Civil Engineers, vol. xciii., Part III. 
 t See Proc. Inst. of Mechanical Engineers, February, 1894. 
 
ENTROPY. 3 
 
 diagram only shows the amount of work done, independent 
 of the amount of steam used ; but the theta-phi (or as it 
 is better written) diagram shows, by its area, the proportion 
 of heat utilised to the heat received. 
 
 COMPARISON OF 6 DIAGRAM WITH INDICATOR DIAGRAM. 
 
 In the ordinary indicator diagram the area represents the 
 work done on the piston in one stroke, the vertical ordinates 
 being "pressure," and the horizontal abscissae "distance 
 moved through." Similarly, in the diagram, the area 
 
 VOLUME - 
 
 FIG. 1. 
 
 also represents work done, but in heat units; the vertical 
 ordinates being absolute temperature (denoted by the Greek 
 letter 0), and the horizontal dimension is what Clausius 
 termed "entropy," named by Zeuner "heat weight," and 
 represented by the Greek letter <t>. In the indicator diagram 
 the area represents the product of pounds (pressure) x feet 
 (distance), or foot-pounds (work) ; so in the diagram 
 the area represents the product of (absolute temperature) 
 x < (entropy) in thermal units (work). 
 
4 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 3. ENTROPY. 
 
 Entropy is the co-ordinate with temperature of energy ; 
 i.e., it is the length upon a diagram whose height is absolute 
 temperature, and whose area is heat units. 
 
 The meaning of the term will be better understood in the 
 light of Carnot's principle of efficiency applied to a reversible 
 cycle in which the heat is received at various temperatures. 
 Fig. 1 represents such a cycle, where, in the first stage, 
 heat is received at r i and discharged at T O ; in the second 
 stage, a further quantity of heat is received at r 2 and 
 discharged at r , and so on. Let Q x , Q 2 , Q 3 , &c., represent 
 the various quantities of heat received at each stage. The 
 area of the whole figure, which represents in heat units the 
 work done W, will equal 
 
 w = Qi (TI _ TO) + Qt (r - TO ) + 2 ( TS _ TO) + . . I 
 
 T l T 2 T 3 
 
 or, for a general formula, 
 
 W = Q (A r), in heat units ; 
 
 where Q = heat received, 
 
 r = temperature of reception (absolute), 
 A T = difference of temperature between reception 
 
 and rejection of heat. 
 Let R = the heat rejected to the cold body, 
 
 then Q = W + R ; 
 but Q - Q! + Q 2 + Q a + ..... 
 
 and W = Si (T! - r ) + Q (r 2 - r ) + & (r g - r ) + . 
 
 or, by difference, R = QI + 9iT + Q]LL' + 
 
 or, 
 
 ?: 
 
 or (the sum) 
 
ENTROPY. 5 
 
 That is to say, the algebraical sum of the changes of 
 entropy in any complete reversible cycle is nil ; there- 
 fore the entropy diagram for any reversible cycle must be a 
 closed figure ; and the final quantity of entropy will be 
 equal to the initial, no matter where the process be started. 
 
 Entropy, therefore, is the fraction or ratio -*, representing 
 
 the amount of heat taken up or rejected by a body, divided 
 by its absolute temperature at that time. It can be calcu- 
 lated from any arbitrary zero of temperature ; for water ana 
 steam, it will be found most convenient to calculate the 
 entropy above that already possessed by 1 Ib. of water at 
 32 deg. Fah., so as to avoid including the latent heat of 
 water. It is usually plotted with temperature as ordinates 
 to an abscissa of entropy. 
 
 4. THEORETICAL ENTROPY DIAGRAM FOR STEAM. 
 The thermal changes which take place when water is 
 evaporated, expanded (as steam), and condensed, are repre- 
 sented on the 6 diagram in the following way : Take 1 Ib. 
 of water at the condenser temperature, say 7-3 deg. Fah. 
 absolute, and let A represent its position on the diagram, 
 fig. 2, its temperature T S and entropy O a being known. 
 Now pump it into the boiler, where it is heated from r s to 
 TI deg. Fah., and, as its temperature is increased, the process 
 will be shown by an upward curve A F B, to correspond with 
 the vertical temperature scale ; but, as it also receives heat, 
 the curve must progress to the right to indicate its increased 
 entropy. The change in its state by heating it from r 3 to 
 TI deg. is therefore shown by the curve A F B, every incre- 
 ment of temperature being accompanied by a corresponding 
 
 increase in entropy, denoted by , where h is the heat 
 
 contained in 1 Ib. of water at T deg. absolute temperature. 
 Having reached the temperature r of the water in the 
 boiler, it begins to evaporate, its temperature remains 
 constant, but it receives an amount of heat (L^) known as 
 
THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 the latent heat of lib. of steam at r 1 deg. temperature. 
 This is represented in fig. 2 by the horizontal line BC 
 horizontal, because its temperature does not change during 
 the operation ; the amount b c of its increase in entropy 
 
 ENTROPY 
 
 FIG. 2. 
 
 being equal to its latent heat divided by its absolute 
 temperature, or =i. Its total entropy Oc is made up of 
 
 the two quantities O b and b c, representing the heat of 
 formation as water (usually denoted by h), and the latent 
 heat L, respectively ; its total heat H being the sum of the 
 
ENTROPY OF WATER AND STEAM. 7 
 
 two, or h + L. At C the steam is admitted into the cylinder, 
 and allowed to expand. If external heat be added to it 
 during expansion, so as to keep it up to saturation point, it 
 will follow the law of the saturation curve p v l ' os>25 = a con- 
 stant, and its entropy will be denoted by the curve C D, 
 such that all horizontal dimensions from A B to CD 
 are equal to the latent heat of 1 Ib. of dry saturated 
 steam divided by its absolute temperature. If the steam 
 expands adiabatically, the entropy curve will be a straight 
 vertical line C E ; because, as the steam neither receives nor 
 loses heat, its entropy will be unchanged. This also shows 
 clearly the amount of wetness which always accompanies 
 adiabatic expansion, and the ratio of A E to A D represents 
 the dryness fraction of the steam at the end of expansion. 
 To indicate the condensing operation, the curve returns 
 along the horizontal line from D to A, the temperature of 
 the mixture remaining constant at T S deg., and its entropy 
 being reduced from d to O a. 
 
 CHAPTEK II. ENTROPY OF WATER AND STEAM. 
 5. METHOD OF CONSTRUCTING THE CURVES. 
 
 HAVING explained the operations to which the theoretical 
 entropy diagram for steam refers, it is necessary to find a 
 means of drawing the two boundary curves A B and C D of the 
 B chart shown in fig. 2. For ordinary steam engines it is 
 not necessary to refer to temperatures below 100 deg. Fah. 
 (corresponding to a pressure of about lib. absolute), nor 
 higher than 400 deg. Fah., equal to about 260 Ib. absolute 
 pressure. We have therefore 300 deg. Fah. range of tempera- 
 ture to provide for, and a scale of 20 deg. Fah. to 1 in. will 
 be found convenient if the chart be drawn on an ordinary 
 sheet of sectional paper. For the base line, or entropy, 
 starting from water at 100 deg, Fah., the maximum required 
 will be 1'87 ; so that an entropy scale of 01 </> = 1 in. will be 
 ample. These scales will give for area, 1 square inch = 
 
8 
 
 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 20 deg. x 01 = 2'0 British thermal units. It must be 
 distinctly understood that the 6 <j> diagram is always drawn 
 for 1 Ib. of H 2 O, whether it be steam, water, or a mixture of 
 both steam and water. Having plotted the scales, we start 
 at the bottom left-hand corner of the chart with lib. of 
 
 200 
 
 100 
 
 -100 
 
 -200 
 
 UJ 
 
 - 
 
 -1500 
 
 400. 
 
 P -I -Z -3 
 
 FIG. 3. 
 
 water at 100 deg. Fah., or 560 deg. Fah. absolute, and 
 calculate the quantities of heat as above. First construct 
 the curve of entropy of water, or aquene curve as it is some- 
 times called, shown by A B, in fig. 2. As this curve is almost 
 a straight line it is only necessary to calculate the value of 
 
ENTROPY OF WATER AND STEAM. 
 
 entropy for some 10 or 12 points, so commencing with water 
 heated from 100 deg. Fah. to 125 deg. Fah. (see fig. 3), the 
 heat given to 1 Ib. of water to raise its temperature from 
 560 deg. Fah. absolute to 585 deg. will be represented on the 
 
 diagram by the area shaded in fig. 3. The area down to 
 the absolute zero of temperature must be added, so as to 
 include all the heat contained in the water. In this case, 
 the heat received is 25'08 B.T.U. (see tables of Properties of 
 Saturated Steam), and, therefore, the increase of entropy will 
 
 25*08 
 572-5' 
 
 or 0'0438. Similarly, the value of can be calcu- 
 
10 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 lated for any temperature, and should be tabulated as in 
 Table L, both for the purpose of plotting the aquene curve, 
 and for future reference. Table I. gives the values of the 
 entropy of 1 Ib. of water for every 10 deg. Fah. from 32 deg. 
 Fah. to 400 deg. Fah. 
 
 6. ENTROPY OF WATER. 
 
 The aquene curve can also be plotted from calculations 
 made by the aid of the calculus in the following manner : 
 For any increase of temperature from r to r 1 (see fig. 4), 
 
 where A h represents the heat necessary to raise 1 Ib. of water 
 from T- O to TJ_ ; and r is the mean absolute temperature 
 during the operation. Integrating this, we get 
 
 fdh 
 0i =J ; 
 
 T 
 
 and, assuming the specific heat of water as unity, 
 
 or 
 
 rdt 
 
 , / a i . 
 
 01 =/ ; 
 
 and, solving this, we get 
 
 0! = loge TI_ - loge T ; 
 
 or 0! = loge ^ ; 
 
 T o 
 
 that is, d 0, or any small difference in the entropy of 1 Ib. of 
 water at any two temperatures is equal to the difference of 
 the hyperbolic logarithms of the absolute temperatures. If 
 the result be multiplied by the mean specific heat of water 
 between the temperatures r L and T O> the formula becomes 
 
 where s represents the heat necessary to raise 1 Ib. of water 
 1 deg. Fah., between r and T^ as compared with the heat 
 
ENTROPY OF WATER AND STEAM. 11 
 
 required to raise 1 Ib. of water from 39 deg. Fah. to 40 deg. 
 Fab. The values of s are given in Table I., page 12, together 
 with the increase of entropy for every 10 deg. Fah., calculated 
 by the above formulae, and the total entropy above water 
 at 32 deg. Fah. The last column in Table I. gives the 
 difference of entropy per 1 deg. Fah., for the purpose of 
 interpolation. 
 
 7. ENTROPY OF STEAM. 
 
 To draw the entropy curve for steam (C D, in fig. 2), an 
 amount of entropy equal to must be added to the aquene 
 curve, where L is the latent heat of 1 Ib. of steam at r deg. 
 absolute temperature. The values of L and are given in 
 
 Table II, page 13, for every 10 deg. Fah. from 32 deg. to 400 
 deg., together with the difference of entropy of steam per 
 1 deg. Fah. for interpolating, and the total entropy of steam 
 and water (above 32 deg. Fah.) with its difference per 1 deg. 
 It should be noted that fa + s, the entropy of water and steam 
 
 TT 
 
 at any temperature, is not equal to , where H = total heat 
 
 of evaporation from 32 deg. Fah. at absolute temperature r ; 
 because A, the sensible heat of the water, is not all received 
 at temperature r, but at a gradually increasing temperature. 
 
 8. ENTROPY DIAGRAM FOR ICE, WATER, AND STEAM. 
 The entropy diagram for ice, water, and steam is shown in 
 fig. 5, page 14. The curve AB, for ice, is drawn on the 
 assumption that its specific heat is 0'504 at all temperatures. 
 The heat given to lib. of ice per 1 deg. Fah. rise of 
 temperature is, therefore, equal to 0'504 B.T.U., 
 or h = 0-504 d t ; 
 
 .1 = 0= 0-504^; 
 
 T T 
 
 or 2 - ^i = 0-504 x log* ^. 
 
12 
 
 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 TABLE I. 
 
 Entropy of water, from 32 deg. Fah. to 400 deg. Fah., 
 calculated from the formulae, Ti - To = s (loge T I - log e T O .). 
 
 Tempera- 
 ture. 
 Deg. Fab. 
 
 t. 
 
 Absolute 
 tempera- 
 ture. 
 
 r. 
 
 Specific 
 heat. 
 
 S. 
 
 Increase of Total entropy 
 entropy above water 
 per 10 deg. Fah. [ at 32 deg. Fah. 
 
 0r ! ~ 0r * 0^. 
 
 Difference 
 of entropy 
 per 1 deg. Fah. 
 
 A 3 
 
 32 
 
 492 
 
 
 I 
 
 
 40 
 
 500 
 
 1-0 
 
 0-01613 0-01613 
 
 0-00202 
 
 50 
 
 510 
 
 1-0 
 
 0-01980 0-03593 
 
 00198 
 
 60 
 
 520 
 
 1-0 
 
 0-01942 0-05535 
 
 0-00194 
 
 70 
 
 530 
 
 1*001 
 
 0-01906 0-07441 
 
 0-00191 
 
 80 
 
 540 
 
 rooi 
 
 0-01871 
 
 0-09312 
 
 0-00187 
 
 90 
 
 550 
 
 1-002 
 
 0.01838 
 
 0-11150 
 
 0-00184 
 
 100 
 
 560 
 
 1-002 
 
 0-01806 
 
 0-12956 
 
 0-00181 
 
 110 
 
 570 
 
 1-003 
 
 0-01775 
 
 .0-14731 
 
 0-00177 
 
 120 
 
 580 
 
 1-004 
 
 0-01745 
 
 0-16476 
 
 0-00174 
 
 130 
 
 590 
 
 1-004 
 
 0-01716 
 
 0-18192 
 
 0-00172 
 
 140 
 
 600 
 
 1-005 
 
 0-OK3S9 
 
 0-19881 
 
 0-00169 
 
 150 
 
 610 
 
 1-006 
 
 0-01662 
 
 0-21543 
 
 0-00166 
 
 160 
 
 620 
 
 1-007 
 
 0-01637 
 
 0-23180 
 
 0-00164 
 
 170 
 
 630 
 
 1-008 
 
 0-01613 
 
 0-24793 
 
 0-00161 
 
 180 
 
 640 
 
 1-009 
 
 0-01589 0-26382 
 
 0-00159 
 
 190 
 
 650 
 
 1-010 
 
 0-01566 
 
 0-27948 
 
 0-00157 
 
 200 
 
 660 
 
 1-011 
 
 0-01544 
 
 0-29492 
 
 0-00154 
 
 210 
 
 670 
 
 1-012 
 
 0-01522 
 
 0-31014 
 
 0-00152 
 
 220 
 
 680 
 
 1-013 
 
 0-01501 
 
 0-32515 
 
 0-00150 
 
 230 
 
 690 
 
 1-014 
 
 0-01480 
 
 0-33995 
 
 0-00148 
 
 240 
 
 700 
 
 1*015 
 
 0-01461 
 
 0-35456 
 
 0-00146 
 
 250 
 
 710 
 
 1 017 
 
 0-01443 
 
 0-36899 
 
 0-00144 
 
 260 
 
 720 
 
 1-019 
 
 0-01425 
 
 0-38324 
 
 0-00142 
 
 270 
 
 730 
 
 1-021 
 
 0-01408 
 
 0-3^732 
 
 0-00141 
 
 280 
 
 740 
 
 1-022 
 
 0-01391 
 
 0-41123 
 
 0-00139 
 
 290 
 
 750 
 
 1-024 
 
 0-01374 
 
 0-42497 
 
 0-00137 
 
 300 
 
 760 
 
 1-026 
 
 0*01168 
 
 0-43855 
 
 000136 
 
 310 
 
 770 
 
 1-027 
 
 0-01343 
 
 0-45198 
 
 0-00134 
 
 320 
 
 780 
 
 1-029 
 
 0-01328 
 
 0-46526 
 
 0-00133 
 
 330 
 
 790 
 
 1-031 
 
 0-01313 
 
 0-47839 
 
 0-00131 
 
 340 
 
 800 
 
 1-032 
 
 0-01298 
 
 0-49137 
 
 0-00130 
 
 350 
 
 810 
 
 1-034 
 
 0-012S4 
 
 0-50421 
 
 0-00128 
 
 360 
 
 820 
 
 1-036 
 
 0-01271 
 
 0-51692 
 
 0-00127 
 
 370 
 
 830 
 
 1-038 
 
 0-0125S 
 
 0-52950 ' 0126 
 
 380 
 
 840 
 
 1-040 
 
 0-01245 
 
 0-54195 124 
 
 390 
 
 850 
 
 1-042 
 
 0-01233 
 
 0*55428 
 
 400 
 
 860 
 
 1-044 
 
 0-01221 
 
 0-56649 
 
 0-00122 
 
 
 
 
 
 i 
 
ENTROPY OF AVATER AND STEAM. 
 
 TABLE II. 
 
 Entropy of Dry Saturated Steam, from 32 deg. Fah. 
 to 400 deg. Fah. 
 
 
 
 
 
 Total entropy of 
 
 
 
 
 Difference 
 
 water + steam. 
 
 Tempera- 
 ture. 
 Deg. Fah. 
 
 Absolute 
 tempera- 
 ture. 
 
 Latent 
 heat. 
 
 Entropy of 
 of 1 Ib. entropy 
 steam. per 1 deg. 
 Fah. 
 
 
 Per Ib. 
 
 Difference 
 per 
 
 
 
 
 
 
 
 1 deg. Fah. 
 
 t. 
 
 T. 
 
 L. 
 
 <t>s =-*.' A S . 
 
 <t>i'}, + 0s. 
 
 A 0?{> + S . 
 
 32 
 40 
 50 
 60 
 70 
 80 
 90 
 100 
 110 
 120 
 130 
 140 
 150 
 160 
 170 
 180 
 190 
 200 
 210 
 220 
 230 
 240 
 250 
 260 
 270 
 280 
 290 
 300 
 310 
 320 
 330 
 340 
 350 
 360 
 370 
 380 
 390 
 400 
 
 492 
 500 
 510 
 520 
 530 
 540 
 550 
 560 
 570 
 580 
 590 
 600 
 610 
 620 
 630 
 640 
 650 
 660 
 670 
 680 
 690 
 700 
 710 
 720 
 730 
 740 
 750 
 760 
 770 
 780 
 790 
 800 
 810 
 820 
 830 
 840 
 850 
 860 
 
 1091-7 
 1086-2 
 10793 
 1072-3 
 1065-3 
 1058 '3 
 1051-3 
 1044-86 
 1037-39 
 1030-42 
 1023-40 
 1016-39 
 1009-38 
 1002-37 
 995-33 
 988-30 
 981-24 
 974-18 
 96710 
 960-03 
 952-94 
 945-83 
 938-76 
 931-56 
 924-41 
 917-25 
 910-06 
 902-86 
 895-64 
 888-41 
 881-15 
 873-88 
 866-58 
 859-27 
 851-95 
 844-58 
 837-20 
 829-84 
 
 2-2189 
 2-1724 
 2-1163 
 2-0621 
 2-0100 
 1 -9598 
 1-9115 
 1-8649 
 1-8200 
 1-7766 
 1-7346 
 1-6940 
 1-6547 
 1-6167 
 1-5799 
 1-5442 
 1-5096 
 1-4760 
 1-4434 
 1-4118 
 1-3811 
 1-3512 
 1-3220 
 1-2938 
 1-2663 
 1-2395 
 1-2134 
 1-1880 
 1-1632 
 1-1390 
 1-1154 
 1-0923 
 1-0698 
 1-0479 
 1-0264 
 1-0054 
 0-9849 
 0-9649 
 
 0-00581 
 0-00561 
 0-00542 
 0-00521 
 0-00502 
 0-00483 
 0-00465 
 0-00449 
 0-00434 
 0-00420 
 0-00406 
 0-00393 
 000379 
 0-00368 
 0-00357 
 0-00346 
 0-00336 
 0-00326 
 0-00316 
 0-00307 
 0-00299 
 0-00292 
 0-00282 
 0-00275 
 0-00268 
 0-00261 
 0-00254 
 000248 
 0-00242 
 0-00236 
 0-00231 
 00225 
 0-00219 
 0-00215 
 0-00210 
 0-00205 
 0-00200 
 
 2-2189 
 2-1885 
 2-1522 
 2-1174 
 2-0844 
 2-0529 
 2-0230 
 1-9945 
 1-9673 
 1-9414 
 1-9165 
 1-8928 
 1-8701 
 1-8488 
 1-8278 
 1-8080 
 1 -7891 
 1-7709 
 1-7535 
 1-7369 
 17211 
 1-7058 
 1-6910 
 1-6770 
 1-6636 
 1-6507 
 1-6384 
 1-6265 
 16152 
 1-6043 
 1-5938 
 1-5837 
 1-5740. 
 1-5648 
 1-5559 
 1-5474 
 1-5392 
 1-5314 
 
 0-00380 
 0-00363 
 0-00348 
 0-00330 
 0-00315 
 0-00299 
 00285 
 0-00272 
 0-00259 
 0-00249 
 0-00237 
 0-00227 
 0-00216 
 0-00207 
 0-00198 
 O'OOISO 
 00182 
 0-00174 
 0-00166 
 0-00158 
 0-00153 
 0-00148 
 0-00140 
 0-00134 
 0-00129 
 0-00123 
 0-00119 
 00113 
 0-00109 
 0-00105 
 
 0-00101 
 
 0-00097 
 0-00092 
 0-00089 
 0-00085 
 0-00082 
 0-00078 
 
 Thus becomes infinity when T = ; and the real origin 
 of the entropy scale should be at infinity on the left-hand 
 side of the figure. For convenience of plotting the origin 
 
14 
 
 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 has been taken as for water at 492 deg. Fah. (absolute), 
 the amount of entropy on either side of this point being 
 considered as positive and negative quantities. The break 
 in the curve between B and C is due to the latent heat of 
 water, or the heat required by 1 Ib. of ice at 492 deg. for 
 
 t 
 
 FIG. 5. 
 
 conversion into water at 492 deg. Its amount is equal to 
 
 143 
 
 0-29. 
 
 143 B.T.U., and its entropy is therefore equal to 
 
 The curves for water and steam are drawn as already 
 explained. If the two curves CD and FE be produced, 
 they will meet at a point known as the " critical tempera- 
 ture" for water and steam, above which there can be no 
 
ENTROPY OF WATER AND STEAM. 
 
 15 
 
 liquid.* Mr. Macfarlane Gray* finds this to be at about 
 750 deg. Gen., or about 1,350 deg. Fah. (absolute). The area 
 of the diagram, fig. 5, is such that each square represents 
 100 x 0-5 = 50 B.T.U, 
 
 9. CONSTANT VOLUME CURVES. 
 
 Having constructed the two boundary curves of the 6 
 chart (the most tedious and difficult part of the process), it 
 will be advisable to draw "constant pressure lines" and "con- 
 
 5 1-6 '7 18 '< 
 
 FIG. 6. 
 
 stant volume curves," as shown in fig. 6, in order to facilitate 
 the transfer of indicator diagrams. The former are simply 
 horizontal lines drawn across the chart between the two 
 boundary curves, at a height corresponding to the tempera- 
 ture of dry saturated steam, as given in the steam tables. If 
 a temperature scale of 20 deg. Fah. to 1 in. be adopted, the 
 
 * See Proc. Inst. of Mechanical Engineers, July, 1889, page 419. 
 
16 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 pressure lines may be drawn for every 1 Ib. pressure up to 
 about 20 Ib. absolute, but beyond this they should be re- 
 duced to every 2lb., and afterwards to 51b. intervals, to 
 prevent complicating the chart. These constant pressure 
 lines are shown in fig. 6. 
 
 The constant volume curves present a little more difficulty. 
 They represent the loss of entropy due to a drop in pressure 
 at constant volume, or that which takes place in the ordinary 
 engine cylinder at release, when it opens to the condenser. 
 
 IOO. 
 
 For example, suppose lib. of steam at 100 Ib. pressure to 
 expand, keeping dry, to 10 Ib. absolute, and then to open to 
 the condenser so that the pressure falls to 2 Ib. absolute at 
 constant volume, as shown in fig. 7. The 1 Ib. of dry steam 
 at 10 Ib. pressure occupies 37 '8 cubic feet, and has an 
 entropy of T64 above water at 100 deg. Fah. ; its pressure 
 falls to 2lb. by condensing, without any increase in its 
 volume, thus causing a reduction of entropy in the steam. 
 At 2 Ib. pressure it would occupy 173 cubic feet if it were 
 
CONSTANT VOLUME CURVES. 
 
 17 
 
 all steam; but we know that it only occupies 37'8 cubic 
 
 Q'T'ft 
 
 feet, and therefore there can only be -J - ,or 0'218 Ib. of steam 
 
 17o 
 
 in the cylinder after opening to the condenser, which will 
 possess an entropy represented by the distance E D, in fig. 8, 
 where 
 
 the curve C D forming a part of one of the constant volume 
 curved lines which have to be drawn. To construct these 
 
 curves, divide the horizontal distance or entropy between 
 the water and steam curves, at any particular pressure, into 
 as many equal parts as there are cubic feet of volume to 1 Ib. 
 of dry saturated steam at that pressure. For example, at 
 60 Ib. absolute pressure the specific volume of steam is 
 practically 7 cubic feet ; and therefore the curves for 1, 2, 3, 
 ST 
 
18 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 4, 5, and 6 cubic feet constant volume lines (fig. 6) all cut the 
 60 Ib. pressure line at equal horizontal distances from one 
 another. 
 
 Another method of drawing the constant volume curves is 
 based on the principle that 
 
 dP L 
 
 ^ varies as . or varies as 0. 
 
 dr r 
 
 This is found by equating the work done to the heat 
 expended in a perfect steam engine working on the Carnot 
 
 UJ 
 
 (0 
 
 u) 
 ft: 
 0. 
 
 VOLUME. 
 
 FIG. 9. 
 
 cycle. Take 1 Ib. of water of volume O A, fig. 9, and 
 pressure P^ ; evaporate it to dry steam of volume O B and 
 temperature r x ; expand it adiabatically to C, fig. 9, where 
 its pressure is P 2 , and temperature r 2 ; now condense it at 
 this pressure and temperature until it occupies the volume 
 O D ; and, finally, complete the cycle by compressing it 
 adiabatically until it occupies its initial volume of O A. 
 
 The efficiency of the cycle will be -^ ^ ; and the work done 
 
ENTROPY OF WATER AND 
 
 will be equal to the area shaded in fig. 9, 
 
 in heat units, or J . L ( TI ~ T2 j in work units. Now, if we 
 
 \ T l / 
 
 make P x - P 2 = d P very small, and T I - r 2 = d r also very 
 small, the work done will be equal to the rectangle A B C D, 
 which equals 
 
 (vol. OB - vol. OA). (P x - P 2 ) 
 
 where V and v are the volumes of 1 Ib. of steam and 1 Ib. of 
 water respectively. 
 
 Bat the work done is equal to the heat expended; 
 therefore 
 
 (V -v).d~p = J. 
 
 Transposing J, and dividing all through by d T, we get 
 
 Z^r *--* 
 
 J (ir T 
 
 But v = approximately 0'017 cubic feet, and V is a constant 
 for any pressure, and J a numerical constant; therefore, 
 for any particular pressure, 
 
 -, varies as , or varies as 0. 
 
 dr T 
 
 In other words, the horizontal ordinates of any constant 
 volume curve at various pressures are proportional to the 
 
 value of ~ r for those particular pressures. For example, 
 
 take 1 Ib. dry steam at 150 Ib. absolute pressure, which has a 
 specific volume of 2 '978 cubic feet. The value of 
 
 - 1-91 
 
 d r 
 
 (see Table IA., Appendix of Cotterill on the "Steam 
 Engine"), and or <t> for steam alone (see Table II., page 13) 
 
 81816 
 
20 
 
 THE ENTROPY DIAGRAM AND ITS APPLICATIONS. 
 
 At 100 lb. pressure, 
 
 ^1=1-39, 
 
 dr 
 
 and therefore the ordinate for the 2 '978 constant volume 
 curve will be 
 
 '?-? x 1-052 = 0765. 
 1 Jl 
 
 The values of <f> for 2 -978 cubic feet of steam calculated in 
 this way are given in Table III., and if they are plotted on 
 the chart at their proper pressures, they will be found to 
 form a constant volume curve for 2'978 cubic feet, which 
 will be just inside the 3 cubic feet curve already drawn by 
 the geometrical method previously explained. 
 
 TABLE III. 
 
 Value of Entropy for 2~978 cubic feet of steam, for drawing 
 constant volume line. 
 
 Absolute 
 pressure. 
 Pounds per 
 square inch. 
 P 
 
 Temperature 
 Fahrenheit. 
 Degrees. 
 t 
 
 Ratio of 
 dp 
 dt 
 from steam tables. 
 
 Entropy of 2 -978 
 cubic feet of steam, 
 by proportion. 
 
 150 
 
 358-16 
 
 1-91 
 
 1-05 
 
 100 
 
 327-57 
 
 1-39 
 
 0765 
 
 50 
 
 280-85 
 
 0-79 
 
 0-435 
 
 20 
 
 227-92 
 
 0-376 
 
 0-207 
 
 10 
 
 193-24 
 
 0-213 
 
 0-117 
 
 5 
 
 162-33 
 
 0-119 
 
 0-065 
 
 2 
 
 126-27 
 
 0-055 
 
 0-030 
 
CALCULATION OF DRYXESS FRACTION. 21 
 
 CHAPTER III. CONVERSION OF INDICATOR DIAGRAM TO 
 ENTROPY DIAGRAM. 
 
 10. CALCULATION OF DRYNESS FRACTION. 
 
 To convert the ordinary indicator diagram to the 6 <j> dia- 
 gram, it will be necessary to have the following data 
 furnished by an experiment : 
 
 (a) Exact size of cylinders, and clearance volumes of each 
 end of each cylinder. 
 
 (6) Dryness fraction of the steam in each cylinder at any 
 one period during expansion. 
 
 (c) A mean indicator diagram for each cylinder. 
 
 The diameters of the cylinders should be taken from 
 gauges, and corrected to allow for the expansion of the 
 cylinders when hot. The clearance volumes can be obtained 
 either by calculation or direct measurement by filling the 
 ports, <fec., with water. When the latter method is employed, 
 care must be taken to allow for the escape of the air ; and 
 where possible, it is advisable to calculate the volume as 
 well as measure it, the one method serving as a check upon 
 the other. 
 
 The dryness fraction is calculated from the mean indicator 
 diagram, when the weight of steam used by the engine per 
 s'troke is known, in the following manner : Take any point 
 in the expansion period of the stroke, as A, fig. 10, and scale 
 the pressure Pa shown by the indicator diagrams at that 
 point. Calculate the volume (Va ) occupied by the steam in 
 the cylinder at that portion of the stroke ; i.e., if A be taken 
 at half stroke, 
 
 where A = mean area of front and back of piston in square 
 
 feet; 
 
 L = length of stroke in feet ; 
 
 v = mean clearance volume of front and back in cubic 
 feet. 
 
22 
 
 CALCULATION OF DEYNESS FRACTION. 
 
 Multiply the volume V by the weight of 1 cubic foot of 
 steam at P pounds absolute pressure, and obtain the weight 
 (W) of steam per stroke shown to be present in the cylinder 
 at point A. Subtract from this weight ( W ), the weight of 
 steam ( W& ) left in the cylinder from the previous stroke 
 at the end of compression, as shown at B, fig. 10. This will 
 be equal to Vj x density of steam at ?6 pounds absolute 
 pressure ; and the net weight of steam per stroke accounted 
 for by the indicator at point A will be W - W& pounds. 
 Compare this with the net weight of steam used by the 
 engine per stroke, which can be obtained by measuring 
 
 FJO. 10. 
 
 the feed water and allowing for steam pipe and jacket 
 condensation, or by measuring the water discharged by the 
 air pump if a surface condenser be used, and the proportion 
 
 of 
 
 indicated steam 
 
 is the required dryness fraction at 
 
 steam used 
 
 point A. A table of the weight of dry saturated steam for 
 various pressures will be found in the appendix. 
 
COMPOUND ENGINE AT HAMPTON. 
 
 23 
 
 To obtain an average indicator diagram for each cylinder, 
 all the diagrams taken during the trial should be divided 
 into 20 parts, and the forward and backward pressures 
 marked off on long strips of paper, the means so obtained 
 being tabulated as shown in Table V., page 25, together with 
 the volume of the cylinder plus clearance at that part of 
 the stroke. 
 
 11. ACTUAL EXAMPLE : COMPOUND ENGINE AT HAMPTON. 
 
 As an example, take the trial of a compound pumping 
 engine at Hampton, of the Southwark and Vauxhall Water- 
 works, and tested by Prof. Hudson Beare in January, 1894. 
 Full particulars of the trial will be found in the third report 
 of the Steam Jacket Research Committee of the Institution 
 of Mechanical Engineers, published in October, 1894. The 
 engine is of the vertical, inverted, surface-condensing type, 
 
 TABLE IV. TRIAL OF COMPOUND ENGINE AT HAMPTON. 
 Data from Proc.Inst.Mech.E., October, 1994. 
 
 
 H.P. cylinder. 
 
 L.P. cylinder. 
 
 
 cubic feet 38'41 
 
 105-05 
 
 Volume of cylinder at release 
 
 cubic feet 36 '49 
 
 99-80 
 
 Volume of clearance 
 
 cubic feet O'SG 
 
 1-94 
 
 
 
 
 Total volume of steam at release . . 
 
 ..cubic feet 37 '35 
 
 101-74 
 
 Pressure at release Ibs. 
 
 absolute p, I 26-12 
 
 8'02 
 
 Density of steam at above pressure 
 Weight of steam present at release . . 
 
 J 0-06515 
 Wlbs. 2-4332 
 
 0-02145 
 2-1822 
 
 Pressure at end of compression Ibs. 
 Density of steam at above pressure 
 
 absolute # 2 80 '1 
 ?eJ C'18645 
 
 12-42 
 0-03239 
 
 Weight of steam left in clearance 
 
 zelbs. 0-1609 
 
 0-0628 
 
 Net steam accounted for by indicator . . . 
 
 .(W-w)lbs. 2-2723 
 per cent 92 '1 
 
 2-1194 
 
 85-9 
 
 
 
 
24 
 
 COMPOUND ENGINE AT HAMPTON. 
 
 with cylinders 32 in. and 52f in. diameter ; stroke, 7 ft. ; 
 piston rods, 6 in. diameter. Considering only the trial with 
 all the steam jackets on, made on January 4th, 1894, Table 
 LX., page 578 of the report, gives the particulars from which 
 the figures given in Table IV. are taken. From the mean 
 indicator diagrams published in plate 143 of the committee's 
 report, the pressures at every one-twentieth part of the 
 stroke in both cylinders have been scaled, and tabulated in 
 
 -250 
 
 FIG. 11. 
 
 Tables V. and VI. To calculate the d <j> volume that is, the 
 volume which 1 Ib. of steam of the required dryness would 
 occupy any initial point may be taken where the dryness 
 fraction of the steam is known ; as at release, or 95 per cent 
 of the forward stroke. In this particular trial, the weight 
 of steam used was 6,284 Ib. per hour, or 2'4665 Ib. per stroke 
 of steam and water passing through the cylinders, as found 
 by measuring the air-pump discharge ; and at release, in the 
 high-pressure cylinder, this weight of steam occupies 37 '35 
 
COMPOUND ENGINE AT HAMPTON. 
 
 25 
 
 TABLE V. #0 DIAGRAM FOR COMPOUND ENGINE AT 
 HAMPTON. HIGH-PRESSURE CYLINDER. 
 
 Portion of 
 
 stroke. 
 
 Pressure in pounds absolute. 
 
 Actual volume. 
 Cubic feet. 
 
 e<p 
 
 volume 
 (for 1 Ib. steam). 
 Cubic feet. 
 
 Forward 
 stroke. 
 
 Backward 
 
 stroke. 
 
 
 
 992 
 
 80-1 
 
 0-86 
 
 0-32 
 
 05 
 
 98'3 
 
 32-0 
 
 278 
 
 1-05 
 
 1 ' 
 
 96-6 
 
 22-8 
 
 4-70 
 
 1-78 
 
 15 
 
 93-5 
 
 22-8 
 
 6'62 
 
 250 
 
 2 
 
 89-7 
 
 23-0 
 
 8-54 
 
 3-23 
 
 25 
 
 84'5 
 
 23-4 
 
 ~10-46 
 
 3-96 
 
 3 
 
 74-8 
 
 23-6 
 
 12-38 
 
 4-69, 
 
 35 
 
 63-5 
 
 23-8 
 
 14-30 
 
 5-41 
 
 4 
 
 57-0 
 
 23-8 
 
 16-22 
 
 614 
 
 45 
 
 51-0 
 
 23 '4 
 
 18-14 
 
 6-87 
 
 5 
 
 46-1 
 
 227 
 
 20-06 
 
 7-59 
 
 55 
 
 41-9 
 
 21-9 
 
 21-98 ' 
 
 8-32 
 
 6 
 
 38-6 
 
 21-2 
 
 23-90 
 
 9-05 
 
 65 
 
 35'8 
 
 20-3 
 
 25-82 
 
 9-77 
 
 7 
 
 33-8 
 
 19-8 
 
 27'75 
 
 10-50 
 
 75 
 
 31-6 
 
 19-4 
 
 29-67 
 
 11-23 
 
 8 
 
 29-8 
 
 19-2 
 
 31-59 
 
 11-96 
 
 85 
 
 28-4 
 
 19-1 
 
 33-51 
 
 12-69 
 
 9 
 
 27-2 
 
 19-4 
 
 35-43 
 
 13-41 
 
 95 
 
 26-1 
 
 19-8 
 
 37-35 
 
 14-14 
 
 i-o 
 
 24-0 
 
 22-5 
 
 39-27 
 
 14-87 
 
 cubic feet, the pressure at that point being 2612 Ib. absolute. 
 The specific volume of dry steam at this pressure is 15 '35 
 cubic feet per pound ; but knowing that the steam is not dry 
 at this point, but possesses a dryness fraction of 0'921, and 
 
26 
 
 COMPOUND ENGINE AT HAMPTON. 
 
 therefore the volume of lib. of steam of dryness fraction 
 0*921, and pressure 2612 lb., will be 
 
 15'35 x 0-921 = 1414 cubic feet. 
 
 TABLE VI. 00 DIAGRAM FOR COMPOUND ENGINE AT 
 HAMPTON. LOW-PRESSURE CYLINDER. 
 
 Portion of 
 stroke. 
 
 Pressure in pounds absolute. 
 
 Actual volume. 
 Cubic feet. 
 
 00 
 
 volume 
 'for 1 lb. steam). 
 Cubic feet. 
 
 Forward 
 stroke. 
 
 Backward 
 stroke. 
 
 
 
 23-3 
 
 19-5 
 
 1-94 
 
 0-76 
 
 05 
 
 23-1 
 
 47 
 
 7-19 
 
 2-83 
 
 1 
 
 22*2 
 
 3-5 
 
 12-44 
 
 4-90 
 
 15 
 
 20'S 
 
 3-5 
 
 17-69 
 
 6-96 
 
 2 
 
 19-5 
 
 3'5 
 
 22-95 
 
 9-03 
 
 25 
 
 is-i 
 
 3-5 
 
 28-20 
 
 11-10 
 
 3 
 
 167 
 
 3-5 
 
 33-45 
 
 13-17 
 
 35 
 
 15-6 
 
 2-5 
 
 38-70 
 
 15-23 
 
 4 
 
 14-8 
 
 3-C 
 
 43-96 
 
 1730 
 
 45 
 
 14-0 
 
 3-6 
 
 49-21 
 
 19-37 
 
 5 
 
 13-3 
 
 3-6 
 
 54-46 
 
 21-44 
 
 55 
 
 l2*i 
 
 3-6 
 
 59-72 
 
 23-51 
 
 6 
 
 12-0 
 
 3-6 
 
 64-97 
 
 25-57 
 
 65 
 
 11-2 
 
 3-7 
 
 70-22 
 
 27-64 
 
 "7 
 
 105 
 
 3-7 
 
 75-47 
 
 29-71 
 
 75 
 
 9-8 
 
 3-7 
 
 80-73 
 
 31-78 
 
 8 
 
 9-2 
 
 3-8 
 
 85-98 
 
 33-84 
 
 85 
 
 8-7 
 
 3'S 
 
 91-23 
 
 35-91 
 
 9 
 
 8'3 
 
 3'9 
 
 96 '48 
 
 37-98 
 
 95 
 
 s-o 
 
 4-1 
 
 101-74 
 
 40-05 
 
 1-0 
 
 6-5 
 
 5-5 
 
 106-99 
 
 42-11 
 
COMPOUND ENGINE AT HAMPTON. 27 
 
 This will be the <t> volume for that particular point, 
 and is the specific volume of steam at the pressure shown, 
 reduced according to its dry ness. The volume occupied by 
 the 00791b. of water will be 0'079 x 0'017 cubic feet, and is 
 quite negligible for all practical purposes. Knowing the Q <t> 
 volume at any one point, the < volumes for the other parts 
 of the stroke are calculated by proportion i.e., at 10 per 
 cent of the stroke the actual volume is 4 70 cubic feet, and 
 the volume will therefore be 
 
 470 x 14-14 . 
 
 The pressures and volumes for the low-pressure cylinder 
 are similarly dealt with, and the figures are given in 
 Table VI. The Q <t> diagrams for this trial are given in fig. 11. 
 
 It will be seen that in the example quoted no notice has 
 been taken of the increase or reduction in the volume of 
 steam due to the opening and closing of the passage in the 
 main slide valve. This has not been possible in the present 
 case, because the report does not mention when the main 
 valve cut off ; but if the position of this had been known, 
 a "kick" would appear on the 6 <t> diagram, somewhat as 
 shown dotted at X, in fig. 11, due to the closing of the passage 
 in the main valve at the end of the forward stroke. The 
 ordinary indicator diagram does not show this. 
 
 As a check upon the accuracy with which the $ diagram 
 has been drawn, its area should be measured by a plani- 
 meter, and reduced to heat units according to the scales of 
 temperature and entropy adopted. If the temperature 
 scale ba 20 deg. Fah. to 1 in., and entropy 01 <P = 1 in., then 
 the area of the 6 $ diagram in square inches, multiplied 
 by 20 x O'l = 2'0, will give the work done per stroke 
 expressed in B.T.U. This multiplied by 772, and by the 
 number of strokes made by the engine per minute, and 
 divided by 33,000, should give the same I.H.P. as that calcu- 
 lated from the indicator diagrams for the same cylinder, 
 in the usual way. 
 
28 COMPLETE ENTROPY DIAGRAM. 
 
 12. COMPLETE ENTROPY DIAGRAM. 
 
 Professor Boulvin, of Gand, has investigated and published 
 (see Engineering, January 3rd, 1896 ; and Revue de Mecanique, 
 June, 1897) a complete entropy chart for steam, from which, 
 once the curves are correctly drawn, all the remaining 
 points can be obtained geometrically instead of by calcula- 
 tion. Toe chart is rather complicated, as it includes four 
 separate charts, all drawn from a common centre O (see 
 fig. 12), but it is comprehensive, and saves the continual 
 reference to steam tables. 
 
 Let O in the centre of the diagram be the initial point. 
 Plot a temperature scale vertically upwards on the line W, 
 starting with any convenient minimum temperature, such 
 as 100 deg. Fah. or 32 deg. Fah., and adopting any convenient 
 scale. It will be found advisable to plot the chart on a 
 good-sized sheet of rVin. squared paper. Draw a scale of 
 entropy O X, horizontally to the right to any convenient scale, 
 the entropy given being above that contained in lib. of 
 water at a temperature equal to the zero adopted in the 
 temperature scale. Volumes are plotted vertically down- 
 ward on the axis O Y, to a scale to be afterward determined ; 
 and pressures (absolute) are measured horizontally to the 
 left, as shown by the line O Z, to any convenient scale. 
 
 Starting with the first quadrant (using the word in its 
 trigonometrical sense), the curve O A B, representing the 
 relation between temperature and pressure of dry saturated 
 steam, can be plotted direct from steam tables, using the 
 temperature and pressure scales already decided upon. The 
 entropy curves O C D and E F G, for water and steam 
 respectively, shown in the second quadrant, are drawn as 
 previously described in Chapter II. 
 
 To determine the volume scale O Y, take any point on the 
 temperature pressure curve, say A at 250 deg. Fah., and 
 draw a tangent to the curve at A, making an angle with 
 the vertical axis O W. From C and F in the second 
 quadrant, representing the entropy of lib. of water and 
 
COMPLETE ENTROPY DIAGRAM. 
 
 1 lb. dry steam respectively at 250 deg. Fah., draw the 
 vertical ordinates C H and F J on to the axis O X, and from 
 H draw H L parallel to the tangent at A in the first 
 
 quadrant, to meet F J produced in L. Drop similar perpen- 
 diculars from other temperatures, such as shown at 200 deg., 
 300 deg., &c., and draw the inclined lines M N P Q, &c., 
 parallel to tangents to the pressure temperature curve 
 O A B at their respective temperatures. 
 
30 COMPLETE EXTROPY DIAGRAM. 
 
 These lines in the third quadrant represent the increasing 
 volume of 1 Ib. steam in being generated at constant tem- 
 perature, and are therefore called for brevity " constant 
 temperature lines," in contradistinction to the "constant 
 volume lines," drawn on the 6 chart by Captain Sankey'd 
 process. They should be drawn to start with a volume of 
 G'016 cubic feet, and not from the zero of volume scale, 
 as shown ; but the volume of 1 Ib. water is so small in 
 comparison to the volume of the steam, that it is impossible 
 to show it on the diagram, and is quite negligible. The 
 vertical distances J L, &c., represent to scale the volumes 
 occupied by 1 Ib. of dry saturated steam at the temperatures 
 indicated, and furnish the necessary data for constructing 
 the volume scale Y. 
 
 An alternative method of drawing the constant tem- 
 perature lines, involving a little elementary trigonometry, 
 gives more accurate results than the geometrical method 
 jast described. The angle a made by HL to the vertical 
 axis O Y, is equal to the angle , made by the tangent at A 
 to the vertical axis O W in the first quadrant. Bat the 
 tangents of all such angles as these are proportional to the 
 
 ratios of ^, representing increase in pressure per unit in- 
 
 d t ' 
 
 crease of temperature. This ratio of for steam is given 
 
 d t 
 
 in "Cotterill on the Steam Engine" (see Appendix, Table IA) 
 for every degree of temperature from 93 deg. Fah. to 432 deg. 
 Fah. ; but the ratio given there will only be equal to the 
 tan a when the pressure and temperature scales are equal. 
 If the pressure scale be drawn twice as large as the tem- 
 perature scale (i.e. t lib. pressure = 2 deg. Fah.), as it is in 
 
 fig. 12, the ratio of -f- must be doubled, in order to equal 
 
 Cv t> 
 
 the required tangent. Knowing the values of tan a for 
 various temperatures, the lines HL, M, N, P, Q, <fec., can 
 easily be drawn with the aid of a protractor and table of 
 natural tangents. Table 7 gives the value of these angles 
 
COMPLETE ENTROPY DIAGRAM. 
 
 31 
 
 TABLE VII. CONSTANT TEMPERATURE LINES FOR 
 COMPLETE ENTROPY DIAGRAM FOR STEAM. 
 
 Temperature 
 Fah., 
 t. 
 
 12, 
 
 from "Cotterill." 
 
 Tan a for the 
 particular 
 scales used. 
 
 Corresponding 
 value of a. 
 
 100 
 
 0-02835 
 
 0-0567 
 
 Deg. min. 
 3 15 
 
 110 
 
 0-03675 
 
 0-0735 
 
 4 12 
 
 120 
 
 G'04695 
 
 0-0939 
 
 5 22 
 
 130 
 
 0-05945 
 
 01189 
 
 6 47 
 
 140 
 
 0-07430 
 
 0-1486 
 
 8 27 
 
 150 
 
 0-09185 
 
 0-1837 
 
 10 25 
 
 160 
 
 0-11335 
 
 0-2267 
 
 12 46 
 
 170 
 
 0-1375 
 
 0-275 
 
 15 23 
 
 180 
 
 0-1665 
 
 0-333 
 
 18 25 
 
 190 
 
 0-2005 
 
 0-401 
 
 21 51 
 
 200 
 
 0-2385 
 
 0-477 
 
 25 30 
 
 210 
 
 0-2825 
 
 0-565 
 
 29 28 
 
 220 
 
 0-3325 
 
 0-665 
 
 33 37 
 
 230 
 
 0-391 
 
 0-782 
 
 38 1 
 
 240 
 
 0-455 
 
 0-910 
 
 42 IS 
 
 250 
 
 0-523 
 
 1-047 
 
 46 19 
 
 260 
 
 0-603 
 
 1-206 
 
 50 20 
 
 270 
 
 0-691 
 
 1383 
 
 54 8 
 
 280 
 
 0785 
 
 1-570 
 
 57 30 
 
 290 
 
 0-892 
 
 1-785 
 
 60 44 
 
 300 
 
 1-012 
 
 2-024 
 
 63 42 
 
 310 
 
 1-135 
 
 2-27 
 
 66 14 
 
 320 
 
 1-275 
 
 255 
 
 68 35 
 
 330 
 
 1-42 
 
 2-84 
 
 70 36 
 
 340 
 
 1-58 
 
 3-16 
 
 72 26 
 
 350 
 
 1-75 
 
 3-50 
 
 74 3 
 
32 COMPLETE ENTROPY DIAGRAM. 
 
 for every 10 deg. Fah., from 100 deg. Fah. to 350 deg. Fah. 
 but must only be used when the pressure scale is twice as 
 open as the temperature scale. 
 
 Knowing the volume occupied by 1 Ib. of dry saturated 
 steam at a temperature of 250 deg. Fah. (13'53 cubic feet), 
 represented by J L, the volume scale O Y can be drawn, and 
 the saturation curve for lib. of dry steam plotted in the 
 fourth quadrant by the aid of the pressure scale Z. This 
 completes the construction of the chart as used by Professor 
 Boulvin, which "represents all the transformations of a 
 body by the simultaneous variation of two co-ordinates, 
 which are sufficient to characterise its state, viz., its tem- 
 perature and its entropy." (See La Revue de Mecanique, 
 June, 1897.) 
 
 It is instructive to note that the " constant temperature 
 lines " afford a convenient method of drawing the adiabatic 
 expansion curve for steam on the p. v. diagram, without the 
 aid of any calculations whatever. Take 1 Ib. of steam, dry, at 
 any particular temperature say 350 deg. Fah.; its adiabatic 
 expansion can be shown in the fourth quadrant of fig. 12 by the 
 following graphical method : From R, on the steam entropy 
 curve at 350 deg. Fah., draw R S at right angles to X, and 
 produce it into the third quadrant to T, as shown. Where 
 S T intersects the constant temperature lines will represent 
 the volume occupied by the dry-steam portion of the 
 mixture at various stages of its expansion, and can, there- 
 fore, be transferred directly to the pressure-volume curve 
 in the fourth quadrant. By transferring the temperature 
 into the first quadrant, and where it intersects the curve 
 O A B, drawing ordinates into the fourth quadrant gives 
 various points of interse ction with the volume lines already 
 drawn, thus giving the adiabatic expansion curve on the 
 p. v. diagram for steam initially dry at 350 deg. Fah. The 
 curve is shown dotted in fig. 12, page 20, and marked a, 6, f, d. 
 
APPLICATION OF COMPLETE ENTROPY DIAGRAM. 33 
 
 13. APPLICATION OF COMPLETE ENTROPY DIAGRAM. 
 
 The practical application of the complete entropy diagram 
 will best be explained by taking an actual steam engine 
 test as example. Referring to the trial of a compound 
 engine at Hampton Waterworks, by Professor Beare, already 
 described (see page 23), the mean indicator diagram must 
 first be plotted in the third quadrant to the scales 
 adopted, lengthening or shortening the volume scale as 
 required to correspond with 1 Ib. of mixture present in 
 the cylinder. In the trial under consideration, the weight 
 of steam and water passing through the cylinders was 
 2 '4665 Ib. per stroke, as found by measuring the water dis- 
 charged from the air pump. Add to this the weight of 
 steam left in the high-pressure cylinder after compression, 
 from the previous stroke, which is calculated to be 01609 Ib. 
 per stroke, and the sum, or 2'6274lb. per stroke, is the 
 weight of steam and water present in the high-pressure 
 cylinder during expansion. This figure also represents a 
 kind of volume factor by which the actual volumes must be 
 divided, in order to obtain the volume of 1 Ib. of mixture of 
 similar dryness, for use in plotting on the 4> chart. The 
 actual volume of the high-pressure cylinder at release is 
 37*35 cubic feet, which, divided by 2 '6274, gives 14'215 cubic 
 feet volume at release for an imaginary cylinder containing 
 lib. of mixture of proportionate dryness. Dividing the 
 mean indicator diagram up into any number of equal parts, 
 say 20, of the stroke, the volumes at the other points can be 
 readily transferred to the chart by proportional compasses, 
 or by dividing the actual volume of the cylinder and 
 clearance at any part of the stroke by 2 '6274. The diagram 
 being plotted, take any point e on it, and transfer into the 
 first quadrant to meet the curve O A B, fig. 12, and thence into 
 the second quadrant, as shown by dotted lines. Also transfer 
 the point e into the third quadrant, and where it intersects 
 its proper "constant temperature line" transfer into the 
 second quadrant its intersection with the line previously 
 4T 
 
34 
 
 <t> DIAGRAM FOR CARNOT CYCLE. 
 
 drawn at e 1 , giving a corresponding point on the entropy 
 diagram. The process of transferring the indicator diagram 
 from the fourth to the second quadrants is one of simple 
 projection, and in reality it takes less time to do it than is 
 occupied in reading how to do it. It should be noted that 
 the difficult curves to be drawn are the fixed ones, whereas 
 the new curves required separately for each experiment are 
 simple ones, and easily drawn with a little practice. 
 
 14. e <f> DIAGRAM FOR CARNOT CYCLE. 
 Having shown how to draw the 6 <f> diagram for a steam- 
 engine test, it will be interesting to examine the great 
 facilities afforded by it for explaining some of the various 
 
 -r A B 
 
 I ENTROPY. 
 
 FIG. 13 
 
 phenomena that occur in the steam-engine cycle. In the 
 first place, it should be noted that the <P diagram proves 
 graphically the efficiency of a perfect reversible cycle. 
 Take the well-known Carnot cycle, shown on the pv or 
 
CONDENSATION DURING ADIABATIC EXPANSION. 35 
 
 ordinary indicator diagram in fig. 9, page 18, and on the e <t> 
 diagram in fig. 13. The work done per stroke is represented 
 in fig. 9 by the area of the figure A B C D, but neither the 
 heat received and rejected, nor its efficiency, are shown on 
 the pv diagram. Transfer the same series of changes to 
 the e diagram, shown in fig. 13. The "state point" A 
 represents 1 Ib. of water, of temperature r lt and entropy 
 O E, evaporated into 1 Ib. of dry steam, of the same tempera- 
 ture T I} but with an increase of entropy shown by EF. 
 At B, it expands adiabatically to C, its temperature falls to 
 T 2 , but the amount of heat it contains is neither increased 
 nor diminished, and therefore its entropy is the same at C 
 as it is at B ; or, the expansion is shown on the e diagram 
 by the vertical line B C, representing the fall in temperature 
 from T! to r z . At C it is compressed isothermally, at the 
 same temperature r 2 , but giving up a quantity of heat 
 denoted by its reduced entropy from O F to O E. From D 
 to A, adiabatic compression is represented by the vertical 
 line D A, at constant entropy O E, but with an increasing 
 temperature from r 2 to r x . The work done during the 
 cycle is represented in heat units by the area of the rectangle 
 A B C D, fig. 13, and the heat received by the area of the 
 rectangle A B F E ; thus the efficiency, 
 
 work done 
 
 or 
 
 heat received 
 
 The line O E F corresponds to the absolute zero of 
 temperature ; and should be much lower down than 
 indicated on the diagram. The efficiency of this cycle can be 
 proved mathematically, but it is seen much more clearly on 
 the 6 $ diagram. 
 
 15. CONDENSATION DCTKISTG ADIABATIC EXPANSION. 
 
 The 6 diagram also shows graphically the varying dry- 
 ness of the steam during the expansion period of the stroke. 
 Neglecting the volume of the water present, which (except 
 in the case of very wet steam) is comparatively nil, the 
 
36 
 
 CONDENSATION DURING ADIABATIC EXPANSION. 
 
 proportion ^, shown in fig. 11, page 24, represents the 
 
 OC OC 
 
 dryness fraction of the steam at ?/, and, similarly, the 
 dryness at any time during expansion can be scaled off the 
 <j> diagram. 
 
 It also shows the amount of condensation due to the 
 adiabatic expansion of steam. For instance, 1 Ib. of dry steam 
 at 170 Ib. absolute pressure, expanded adiabatically to 20 Ib. 
 
 UJ 
 K 
 o 
 ~ 
 
 < 
 
 fc 
 ul 
 
 (L 
 
 
 
 uJ 
 
 ENTROPY. 
 
 B 
 
 FIG. 14. 
 absolute, as shown by the line AB in fig. 14, will have a 
 
 dryness fraction of ^^, or 0'8797. But the line C B repre- 
 L- JJ 
 
 sents the latent heat of 0'87971b. steam at 20 Ib. pressure, 
 which should occupy a volume of 
 
 0-8797 x 1972 = 17'35 cubic feet. 
 
 According to the law of adiabatic expansion viz., p.v l ' l3 $ = 
 constant the volume occupied by lib. of steam at 170 Ib. 
 pressure, initially dry, and expanded to 20 Ib., is 17*45 cubic 
 
ADIABATIC EXPANSION OF AVET STEAM. 
 
 37 
 
 feet. This difference may be due to the volume occupied by, 
 and the heat contained in, the 0'12031b. water present, 
 which is not shown by the $ diagram, but which should be 
 allowed for when dealing with wetter steam. 
 
 16. ADIABATIC EXPANSION OF WET STEAM. 
 If a mixture of steam and water be expanded adiabatically, 
 it is shown on the entropy diagram by dividing the higher 
 temperature entropy into two parts in the ratio of the dry- 
 ness of the steam at the commencement of expansion. For 
 example, take question No. 41, given in the Science and Art 
 
 FlG. 15. 
 
 Department's Examination in Steam, 1898, where lib. of 
 stuff of 0'6 dryness is expanded adiabatically from 311 deg. 
 Fah. to 230 deg. Fah., and it is required to find the weight 
 of water present at the end of expansion, the entropy of 1 Ib. 
 of water being given as 0'339 and 0*451, and that of 1 Ib. of 
 dry steam as 1716 and 1'612 at the lower and higher 
 temperatures respectively. Draw the approximate water 
 and dry steam entropy curves A B and C D from the data 
 given, as shown in fig. 15, and divide B C into two parts 
 
38 ADIABATIC EXPANSION OF WET STEAM. 
 
 representing to scale the dryness of the steam (0'6) at 
 311 deg. Fah. ; that is, make 
 
 BE n 
 
 BC = 6 ' 
 and drop the perpendicular EF meeting AD in F at the 
 
 A F 
 
 lower temperature. Then =- will represent the weight of 
 
 A -L' 
 
 Tjl T~v - 
 
 steam, or ~ *he weight of water present at the end of 
 A. \J 
 
 expansion. 
 
 Using italic letters a, 6, c, &c., to represent the projection 
 of their corresponding state points on the zero tempera- 
 ture line, the result may be expressed numerically thus : 
 
 bc = Oc - Ob, 
 
 = 1-612 - 0-451 = 1161 ; 
 If = (be) x 0-6 = 1-161 x 0'6 = 0-6966 ; 
 Of=0b + bf, 
 
 = 0-451 + 0-6966 = 11476 ; 
 ad = Od-Oa 1 
 
 = 1716 - 0-339 = 1'377 ; 
 fd = 0d- O/, 
 = O d - O e, 
 
 = 1716 - 11476 = 0'5684 ; 
 fd = 0^684 
 ad 1-377 
 
 Answer = 0'41281b. water at end of expansion. 
 It is clearly seen from fig. 15, that if the mixture be 
 very wet to start with (say, consisting of 60 per cent of 
 water and only 40 per cent by weight of steam), adiabatic 
 expansion will produce a dryness, some of the water 
 present becoming re-evaporated at the expense of the 
 sensible heat in the hot water. This would be shown by 
 
 "Cl T) "C 1 
 
 the ratio -v- t ^ being greater than -^ . 
 
EFFECT OF STEAM JACKETING. 39 
 
 CHAPTER IV. HEAT LOSSES. 
 
 17. EFFECT OF STEAM JACKETING. 
 
 IN all steam engines a comparatively large proportion of 
 the steam which enters the cylinder is condensed imme- 
 diately upon its admission, and its work lost to the engine 
 for that period of the stroke. A part of the heat repre- 
 sented by this condensation is returned to the steam towards 
 the end of the expansion period, when its capacity for doing 
 work is considerably diminished ; or it may only be returned 
 during the exhaust stroke, when (except in the case of more 
 than cne cylinder) it is not only of no use, but impedes the 
 free discharge of the exhaust by increasing the volume of 
 the steam. Various methods have been adopted to reduce 
 this initial condensation to a minimum, the principal of 
 which are steam jacketing, superheating, compounding, 
 and high speed. The various effects of these on the internal 
 working steam of the engine, as shown by the 6 <P diagram t 
 form a very interesting and instructive study. 
 
 The effect of steam jacketing is very clearly shown by 
 drawing the 6 diagrams for the same engine, working with 
 and without steam in the jackets. Figs. 16 and 17 are the 
 diagrams for two trials of a triple-expansion engine at the 
 Wapping Pumping Station of the London Hydraulic Power 
 Company, made by Mr. Bryan Donkin in 1892, full particulars 
 of which will be found in the third Report of the Steam 
 Jacket Research Committee of the Institution of Mechanical 
 Engineers, already referred to (see Proceedings of the Insti- 
 tution of Mechanical Engineer?, October, 1894, page 536). 
 The cylinders of the engine are 15 in., 22 in., and 36 in. 
 diameter, with a stroke of 2 ft., with pistons coupled direct 
 to water plungers 5 in. diameter. The steam pressure was 
 about 135 Ib. absolute, the speed about 56 revolutions per 
 minute, and the total I.H.P. about 175. Comparing the two 
 trials c and d, the former with boiler steam in all the three 
 barrel jackets (the covers are not jacketed), and the latter 
 
40 
 
 EFFECT OF STEAM JACKETING. 
 
EFFECT OF STEAM JACKETING. 
 
 41 
 
42 
 
 EFFECT OF STEAM JACKETING. 
 
 without steam in the jackets, the consumption of steam per 
 I.H.P. per hour was 1514lb. and 17*17 lb. respectively, or 
 11*8 per cent less in the trial with steam in the jackets. 
 
 I 
 
 8 
 
 Kl 
 
 Fio. 18. 
 
EFFECT OF STEAM JACKETING. 
 
 43 
 
 The mean indicator diagrams for the two trials are 
 shown in figs. 18 and 19, the former with steam in all 
 jackets and the latter without. The saturation curves for 
 
 FIG. -19. 
 
44 EFFECT OF STEAM JACKETING. 
 
 the known weights of the mixture of steam and water in 
 each cylinder per stroke have been plotted, in order to 
 show the comparative ivork losses in the two cases. These 
 lines do not form one continuous curve, because the 
 weight of compression or " dead steam " left in the cylinder 
 at the end of each stroke is different for each cylinder ; 
 but it is more correct to plot the weight of the mixture 
 than to plot the "live steam" admitted per stroke only, 
 as the comparison of the actual indicator diagram with 
 the former gives the amount of steam present at all points 
 of the stroke. The saturation curves for lib. of dry 
 saturated steam can be plotted if the volumes be taken 
 as the volumes already tabulated to construct the 6 <j> 
 diagram, the table of densities given in the appendix being 
 used for giving the pressures. The curves of dryness 
 fraction shown underneath the mean indicator diagrams 
 have been taken directly from the 6 $ diagrams, figs. 16 
 and 17, and show very clearly the gradual condensation 
 and re-evaporation during the progress of the steam from 
 the high-pressure cylinder to the condenser. At certain 
 periods of the stroke both condensation and re-evaporation 
 may be taking place at the same time ; in which case, the 
 curves only show the excess of the one action over the other. 
 The 6 diagrams for the trial "c," with steam in the jackets, 
 are shown in fig. 16, and those for trial " d" without steam 
 in jackets, in tig. 17. In comparing the two diagrams, the 
 most noticeable difference is the enormously-increased area 
 of those in tig. 17, especially in the case of the intermediate 
 and low-pressure cylinders. The reason for this is, the 
 water formed during admission to the high-pressure cylinder 
 is gradually re-evaporated by the live steam in the jackets 
 during its passage through the three cylinders, until, when 
 it leaves the low-pressure cylinder for the condenser, it 
 consists of 96 per cent dry steam and 4 per cent of water in 
 the trial with all the jackets on, as compared with 65 per 
 cent dry steam and 35 per cent water in the non- jacketed 
 trial. 
 
EFFECT OF STEAM JACKETING. 45 
 
 The comparative areas of the diagrams in figs. 16 and 17 
 are given in Table VIIL, which gives the areas of the 9 <p 
 diagrams, expressed in B.T.U. per stroke, as compared with 
 the actual I.H.P. for each cylinder, also expressed in B.T.U. 
 per stroke. The mechanical equivalent of heat ( J) has been 
 taken as 772 foot-pounds. The volume factor is found by 
 dividing the specific volume of 1 Ib. of dry saturated steam, 
 at the pressure shown where the dryness fraction has been 
 calculated, by the actual volume occupied by the steam at 
 that moment. For example, at release in the high-pressure 
 cylinder of trial C, the pressure shown by the indicator 
 diagrams was 59'3 Ib. absolute, and the specific volume of 
 dry steam at this pressure is 
 
 m cubic feet - 
 
 Multiply this by 855, the known dryness of the steam at 
 that point, and the result is 6 '08 cubic feet per pound 
 occupied by the steam. But the volume of the high-pressure 
 cylinder at release is 2*455 cubic feet ; therefore the volume 
 factor for the high-pressure cylinder in trial C will be 
 
 608 
 
 2-455 
 
 = 2-476. 
 
 The volume factor may be looked upon as the reciprocal of 
 the weight of dry steam passing through the cylinder, and 
 the area given in column three of Table VIII. will be the 
 work done in B.T.U. per stroke by 1 Ib. weight of steam, 
 which, divided by the volume factor in column 4, gives the 
 work done in B.T.U. per stroke for the known weight of 
 dry steam present in each cylinder. The difference between 
 the figures given in columns 5 and 6 is accounted for by 
 drawing the diagrams in figs. 16 and 17 from a mean 
 indicator diagram which was constructed from nine sets of 
 actual indicator diagrams, taking one set per hour; whereas 
 the figures given in column 6 are calculated from the actual 
 I.H.P. developed in each cylinder, taken from all the 
 indicator diagrams (some forty sets). 
 
46 
 
 ENTROPY DIAGRAM FOR SUPERHEATED STEAM. 
 
 TABLE VIII. AREA OF d<t> DIAGRAMS FOR TRIPLE-EXPANSION 
 ENGINE, WITH AND WITHOUT STEAM IN JACKETS. 
 
 Trial 
 and 
 conditions. 
 
 Cylinder. 
 
 Area of 
 9 diagram 
 in B.T.U , 
 measured 
 by 
 planimeter. 
 
 Volume 
 factor 
 for each 
 cylinder. 
 
 Heat utilised per stroke. 
 
 Calculated 
 from d <f> 
 diagrams. 
 B.T.U. 
 
 Calculated 
 from 
 ordinary 
 indicator 
 diagrams. 
 
 Trial C. 
 Steam in all < 
 
 jackets. 
 
 V 
 
 H.P.C 
 I.P.C 
 
 51-0 
 68-1 
 58-3 
 1674 
 
 2-47 
 2-865 
 
 2'82 
 
 20-5 
 23-8 
 20-7 
 65-0 
 
 20 -S 
 23-05 
 20-45 
 64-3 
 
 L.P.C 
 
 Total 
 
 Trial D. 
 No steam in -{ 
 
 jackets. 
 I 
 
 H.P.C 
 I P C 
 
 43-5 
 58-45 
 34-52 
 13647 
 
 1-835 
 1-985 
 2-06 
 
 23-8 
 29-4 
 16-7 
 69-9 
 
 23-6 
 29-4 
 16-3 
 69-3 
 
 LP.C 
 Total 
 
 18. THEORETICAL ENTROPY DIAGRAM FOR SUPERHEATED 
 
 STEAM. 
 
 When saturated steam is removed from the water from 
 which it is generated, and heated beyond the temperature 
 that corresponds to its pressure, it becomes superheated, the 
 additional amount of heat received being 
 Qi =Vp X ( TI -r); 
 
 where Q ! = heat received as superheat per pound of steam ; 
 C p = specific heat of superheated steam at constant 
 
 pressure (usually taken as 0'48) ; 
 r : = temperature after superheating ; 
 r temperature of saturated steam of the same 
 
 pressure. 
 
 Its entropy is therefore greater than that of corresponding 
 saturated steam, and is equal to 
 
 <f>s + 0-48 (loge r l - loge r\ 
 
 <p s being the entropy of 1 Ib. of dry saturated steam of the 
 
ENTROPY DIAGRAM FOR SUPERHEATED STEAM. 
 
 47 
 
 same pressure (see Table II., page 13). This increased heat 
 is represented on the entropy diagram, fig. 20, by the exten- 
 sion curve C D, the area under which C D d c (extended to 
 the absolute zero of temperature) is equal to Q 15 or 0*48 
 (T! - r\ where C is taken at temperature r, and D at r lm 
 If expansion be assumed adiabatic that is, with a non- 
 conducting cylinder and piston the diagram is completed 
 by the vertical line D E from r 1 to r 2> which may intersect 
 
 FIG. 20. 
 
 the dry saturated steam curve CFG at K, in which case the 
 steam will pass from the superheated to the saturated con- 
 dition at K, and at the end of expansion be wet steam, with 
 
 a dryness of 
 
 AE 
 AF' 
 
 In order to have dry steam at F, the 
 
 steam must be superheated up to the point H, the latter 
 being found by drawing the vertical FH to intersect the 
 superheated steam curve CDM. If the superheating be 
 carried beyond this point, as, for instance, to M, the steam 
 remains superheated throughout the whole of the expansion 
 
48 
 
 ENTROPY DIAGRAM FOR SUPERHEATED STEAM. 
 
 period (assumed adiabatic), and at release it still possesses an 
 amount of superheat represented on the temperature scale 
 by the amount N O, the curve F O being drawn by the same 
 equation as C D M. Table IX. gives the entropy for 1 Ib. 
 of superheated steam, starting with dry saturated steam 
 
 TABLE IX. ENTROPY OF SUPERHEATED STEAM FROM 
 DRY SATURATED STEAM AT 320 DEG. FAH. 
 
 Temperature. 
 
 Amount of 
 superheat. 
 Deg. Fah. 
 
 Increase of 
 entropy. 
 d<f>. 
 
 Total entropy 
 above water at 
 32 deg. Fah. 
 <P. 
 
 Deg. Fah. 
 t. 
 
 Absolute. 
 
 T. 
 
 320 
 
 780 
 
 
 
 
 
 1-6043 
 
 330 
 
 790 
 
 10 
 
 0-0061 
 
 1-6104 
 
 340 
 
 800 
 
 20 
 
 0-0121 
 
 1-6164 
 
 350 
 
 810 
 
 30 
 
 0-0181 
 
 1-6224 
 
 360 
 
 820 
 
 40 
 
 0-0240 
 
 1-6283 
 
 370 
 
 830 
 
 50 
 
 0-0299 
 
 1-6342 
 
 380 
 
 840 
 
 60 
 
 0-0356 
 
 1-6399 
 
 390 
 
 850 
 
 70 
 
 0-0413 
 
 1-6456 
 
 400 
 
 860 
 
 80 
 
 0-0469 
 
 1-6512 
 
 410 
 
 870 
 
 90 
 
 0-0524 
 
 1-6567 
 
 420 
 
 880 
 
 100 
 
 0-0579 
 
 1-6622 
 
 430 
 
 890 
 
 110 
 
 0-0634 
 
 1-6677 
 
 440 
 
 900 
 
 120 
 
 0-0687 
 
 1-6730 
 
 450 
 
 910 
 
 130 
 
 0-0740 
 
 1-6783 
 
 460 
 
 920 
 
 140 
 
 0-0792 
 
 1-6835 
 
 470 
 
 930 
 
 150 
 
 0-0844 
 
 1-6887 
 
 480 
 
 940 
 
 160 
 
 0-0896 
 
 1-6939 
 
 490 
 
 950 
 
 170 
 
 0-0947 
 
 1-6990 
 
 500 
 
 960 
 
 180 
 
 0-0997 
 
 1-7040 
 
 510 
 
 970 
 
 190 
 
 0-1046 
 
 1-7089 
 
 520 
 
 980 
 
 200 
 
 0-1096 
 
 1-7139 
 
EFFECT OF SUPERHEATING. 49 
 
 at 320 deg. Fah. (corresponding to about 90 Ib. absolute 
 pressure), for every 10 deg. up to 520 deg. Fah., or 200 deg. 
 of superheat. 
 
 19. EFFECT OF SUPERHEATING. 
 
 The effect of superheating the steam, as shown by 
 the diagram, has been very ably treated by Prof. Ripper 
 in his recent paper read before the Institution of Civil 
 Engineers (see Proceedings, vol. cxxviii., January, 1897), and 
 partially reproduced in The Practical Engineer (see vol. xvi. , 
 page 112). 
 
 The main object of using superheated steam is to minimise 
 initial condensation by reducing the exchange of heat 
 between the working steam and the metal walls. When 
 using ordinary saturated steam, which usually contains a 
 small quantity of suspended moisture, condensation during 
 admission must produce a deposit of water upon all the 
 clearance surfaces, which is re-evaporated as the temperature 
 of the steam falls (principally during exhaust), at the expense 
 of the heat stored up in the cylinder walls. The result ig r 
 the inner skin of the wall in the clearance surfaces is cooled 
 considerably with every stroke, and condenses a compara- 
 tively large proportion of the incoming steam at the next 
 stroke. With superheated steam, the heating up of the 
 walls during admission is done at the expense of the super- 
 heat (providing that is sufficiently high) ; and at cut-off 
 there should still be a little superheat left to evaporate the 
 condensation formed by work being done, so that the steam 
 leaves the cylinder at release just dry, but not superheated. 
 The walls will then be absolutely dry during the exhaust 
 stroke, and consequently require very little heating up at 
 the commencement of the next stroke. 
 
 The comparative effect of jacketing and superheating is 
 shown by the four 6 diagrams superposed in fig. 21, which 
 represent four different trials made by Mr. Bryan Donkin 
 on a small vertical engine (see Proceedings of the Institution 
 of Mechanical Engineers, January, 1895, page 132). The 
 
 5T 
 
50 
 
 EFFECT OF SUPERHEATING. 
 
EFFECT OF SUPERHEATING. 
 
 51 
 
 conditions ana results of the four trials are shown by the 
 following Table X. 
 
 TABLE X. 
 
 Trial. 
 
 Steam in jackets. 
 
 Steam in cylinder. 
 
 Steam 
 consumpti'n 
 
 Kemarks. 
 
 A 
 B 
 C 
 D 
 
 No steam in jackets 
 No steam in jackets 
 Saturated steam in jackets 
 Superheated steam in jackets 
 
 Saturated steam 
 Superheated steam 
 Saturated steam 
 Superheated steam 
 
 Lb. 
 
 45-6 
 
 28-4 
 27-25 
 20-9 
 
 All on the 
 same engine, 
 at same steam 
 pressure, same 
 cut-off, sime 
 speed, and 
 ' same I.H.P. 
 
 The mean indicator diagrams for the four trials are shown 
 in figs. 22 to 25, inclusive, and form a very instructive object 
 lesson to steam users, as showing the different amounts of 
 work which 1 Ib. of steam can be made to do if the conditions 
 under which it is admitted to the cylinder are properly 
 arranged. The weight of steam used per I.H.P. per hour, as 
 given in table, is not an exact comparison when using 
 superheated steam, because 1 Ib. of the latter contains more 
 heat than 1 Ib. of saturated steam of the same pressure. A 
 better standard of comparison is that recommended by the 
 Thermal Effisiencies Committee of the Institution of Civil 
 Engineers, viz., the number of British thermal units of heat 
 used per I.H.P. per minute, taking the total heat contained 
 in the steam on the boiler side of the stop valve, leas the 
 sensible heat contained in the steam (as water) in the exhaust 
 pipe. This quantity of heat, for the four trials A, B, C, D, 
 is 813, 500, 493, and 368 B.T.U.'s per I.H.P. respectively, and 
 they represent the heat used per minute in producing one 
 I.H.P., independently of the pressure or condition of the 
 steam. 
 
 The results of the four trials are shown graphically by 
 their 9 <t> diagrams in fig. 21. The smallness of the diagram 
 in trial A is most noticeable, and shows the enormous loss by 
 condensation during admission which is always produced in 
 
52 
 
 EFFECT OF SUPEEHEATING. 
 
 non- jacketed cylinders, especially small ones. The partial 
 re-evaporation during expansion, at the expense of the heat 
 
 FIG. 22. 
 
 in the walls, tends to counterbalance the initial loss ; but 
 even at release the weight of steam present in the cylinder 
 
EFFECT OF SUPERHEATING. 
 
 53 
 
 is only about one-half what it should have been. The amount 
 of the loss in this particular case was very much increased 
 
 0-12 , (j-lAtukft 
 
 Fio. 23. 
 
 by the cut-off taking place very early (at one-sixteenth of 
 the stroke), and the cylinder being so very small (Gin. 
 
54 
 
 EFFECT OF SUPERHEATING. 
 
 diameter, Sin. stroke); so that the clearance , surf ace, and 
 surface of cylinder exposed to steam during admission, per 
 
 FIG. 24. 
 
 pound of steam present, would be very high indeed with a 
 small engine working under these conditions. The conden- 
 
EFFECT OF SUPEKHEATING. 55 
 
 sation in a larger engine would be relatively much smaller, 
 as shown by figs. 11 and 17. 
 
 ^ " 
 
 7~r,at JJ 
 
 In trial B, the diagrams for which are given in fig. 23, 
 with the steam superheated only 31 deg. Fab., and still 
 
56 EFFECT OF SPEED. 
 
 without steam in the jackets, the reduction effected in the 
 initial condensation is most marked. Instead of there being 
 only 24 per cent of the steam which was admitted present in 
 the cylinder at cut-off, as in trial A, there is now 41 per 
 cent, and the drier walls have produced a larger amount of 
 re-evaporation. 
 
 In trial C (see fig. 24), with saturated steam in the 
 cylinder and jackets, there is an initial dryness of 0'53 at 
 cut-off, and the expansion is continued with a gradually- 
 increasing dryness of steam, until at release all the water 
 has been re-evaporated, and it is discharged to the condenser 
 as all dry steam. 
 
 In trial D, fig. 25, using steam superheated 59 deg. Fah. 
 in both the cylinder and the jackets, all the water has 
 disappeared by the time the piston has made 0'8 of its stroke, 
 after which the curve crosses the dry-steam boundary, and 
 is slightly superheated. The " toe " of the diagram in trial 
 D has been calculated by assuming that superheated steam 
 follows the law of expansion of a perfect gas, viz., that the 
 product of the pressure and volume varies directly as the 
 absolute temperature, or 
 
 2^ = a constant. 
 
 T 
 
 20. EFFECT OF SPEED. . 
 
 The effect of increased speed of rotation on the 6 $ diagram 
 is not very marked. Fig. 28 shows the diagram for two 
 trials on the same engine, at the same steam pressure, cut- 
 off, and general conditions, but in the first trial (shown by 
 the full line in fig. 28) the speed was 216 revolutions per 
 minute, and in the second (shown dotted) at 115 revolutions, 
 or about one-half. With the increased speed there is a little 
 less initial condensation, but the re-evaporation during 
 expansion is greater in the half -speed trial, probably because 
 the time taken by the engine to complete any given portion 
 of its cycle is longer in the latter case, and allows the 
 jackets to make their influence more effective. These 
 
EFFECT OF SPEED. 
 
 57 
 
 diagrams are taken from Mr. Bryan Donkin's experiment, 
 Nos. 121 and 122 (see Proceedings of the Institution of 
 
 FIG. 2t 
 
 Mechanical Engineers, January, 1895). The mean indicator 
 diagrams and quality curves for the two trials are shown in 
 
58 EFFECT OF SPEED. 
 
 figs. 26 and 27 ; the former at full speed, and the latter at 
 half speed. 
 
 )0 40 SO SO 
 
 fer- cenfcye of Jtrote 
 FiO. 27. 
 
 The trials of the late Mr. P. W. Willans on his central- 
 valve engine, recorded in the Proceedings of the Institution 
 
EFFECT OF SPEED. 
 
 59 
 
 of Civil Engineers (vol. xciii., 1888, and vol. cxiv., 1893), 
 prove that the percentage of steam not accounted for by 
 the indicator diagram at cut-off' varies inversely as the 
 square root of the number of revolutions per minute. In 
 fact, the amount of initial condensation can be approxi- 
 mately calculated by the following formula : 
 W \oger 
 
 I D x J N 
 
 where W = the weight of steam per hour not accounted for 
 
 by the indicator at cut-off ; 
 I = the weight of steam per hour accounted for by 
 
 the indicator at cut-off; 
 r = number of expansions ; 
 D = diameter of cylinder in feet ; 
 N = revolutions of engine per minute ; 
 c = a numerical coefficient, depending on the design 
 
 of cylinders and conditions of working. 
 For ordinary slide-valve engines, with an average amount 
 of clearance surface, c can be taken at about 3 or 4 for 
 jacketed cylinders and 6 for non-jacketed cylinders. 
 
 The value of the coefficient c for the two trials just quoted 
 (see figs. 26 and 27) is shown by the accompanying Table XI. 
 to be 2'07 at full speed, and 211 at half speed. 
 
 TABLE XL CONDENSATION AT Two SPEEDS : OTHER CON- 
 DITIONS REMAINING CONSTANT. 
 
 
 
 
 Steam condensed 
 
 ^ 
 
 Revs, 
 per 
 minute. 
 
 Indicated steam 
 at cut-off I. 
 Pounds per hour. 
 
 Actual steam used, 
 from experiment. 
 A. 
 Pounds per hour. 
 
 during admission. 
 
 |g 
 
 'II" 
 fl o 
 
 W. 
 
 Pounds 
 
 Ratio 
 W 
 
 
 
 
 per hour. 
 
 I 
 
 6 
 
 216-3 
 
 130-1 
 
 179-4 
 
 49-3 
 
 0-379 
 
 2 '07 
 
 114'9 
 
 68-0 
 
 104-1 
 
 36-1 
 
 0-531 
 
 211 
 
 21. COMPOUNDING. 
 
 The use of high-pressure steam, requiring a greater number 
 of expansions for economical working, has necessitated the 
 
COMPOUNDING. 
 
COMPOUNDING. 61 
 
 introduction of multi-cylinder engines, in order that the 
 range of temperature in each cylinder shall not be too large. 
 The question therefore arises, What is the most economical 
 number of expansions i.e., what is the point beyond which 
 the extra work gained per pound of steam by expansive 
 working is more than neutralised by the loss due to initial 
 condensation 1 The late Mr. Willans' trials seemed to point 
 to the law 
 
 r - P + 10 . 
 ~25~ 
 
 where r = most economical number of expansions ; 
 
 p initial steam pressure (absolute) in pounds per 
 square inch, 
 
 for high-speed non-condensing engines ; but the author thinks 
 that with moderate-sized engines, well jacketed, the expan- 
 sions may be greater than those given by the above law, 
 and would substitute the following rule, 
 
 r = P_+_20 
 20 
 
 ??*+& 
 
 as . being more in accordance with experimental results 
 obtained with different engines. The enormous loss of heat 
 which accompanies very early cut-off is clearly seen by 
 comparing figs. 31 and 32. The diagrams given in fig. 31 
 are for a compound non-jacketed engine of about 400 I.H.P., 
 tested by Mr. Michael Longridge on October 21st, 1896, the 
 full particulars of which are recorded in the Report of the 
 Engine, Boiler, and Employers' Liability Association for 
 1896 ; partly re-published in The Practical Engineer for 
 October 15th, 1897. The diagrams indicate a fairly economical 
 performance, as they almost fill up the available area between 
 the steam and water boundary curves. Comparing them 
 with the <t> diagram given in fig. 32 for a compound non- 
 jacketed engine of the same type, and indicating about the 
 same power, but which was made much too large for the 
 
62 
 
 COMPOUNDING. 
 
 work required, it will be seen how the automatic expansion 
 controlled by the governor destroyed the efficiency of the 
 engine in the latter case (fig. 32). The mean indicator 
 diagrams and quality curves for the two trials are shown 
 in figs. 29 and 30. The cut-off in the high-pressure 
 cylinder for Mr. Longridge's trial was 0'31 of the stroke ; 
 
 <*^ 
 
 FIG. 29. 
 
 and for L the diagrams given in fig. 32 the mean cut- 
 off in the high-pressure cylinder was 0'054 of the stroke. 
 The result is, that the steam consumption is increased from 
 13'91b. per I.H.P. per hour in the former, to 27'21b. in the 
 
INITIAL CONDENSATION. 
 
 63 
 
 latter case. The importance of this waste is realised when 
 one considers that a difference of 13 Ib. of steam per I. H.P. 
 per hour for an engine of this size, running night and day, 
 represents a loss of some 15 a week in the coal bill alone. 
 
 FIG. 30. 
 
 22. INITIAL CONDENSATION. 
 
 The weight of steam missing at cut-off may be accounted 
 for in two ways. In the first place, it may be assumed to 
 have entered the cylinder as hot water i.e., due to priming 
 and condensation in pipes ; in which case it would part 
 with its heat during expansion by evaporating a very 
 small portion of the water formed by adiabatic expansion. 
 Assuming a non-conducting cylinder, the expansion curve 
 would then theoretically follow the dotted line B r, shown 
 
64 
 
 INITIAL CONDENSATION. 
 
INITIAL CONDENSATION. 
 
 65 
 
 6T 
 
66 INITIAL CONDENSATION. 
 
 on the high-pressure and intermediate cylinder diagrams in 
 figs. 16, 17, <fec. This line was called by the late Mr. Willans 
 the "priming water heat-recovery curve" (see discussion by 
 Capt. Sankey of Prof. Beare's paper on "Marine-engine 
 Trials" Proceedings, Institution of Mechanical Engineers, 
 February, 1894). On the other hand, the quantity of steam 
 missing at cut-off may be assumed to have entered the 
 cylinder as dry steam, and been condensed during admission ; 
 in all probability the larger portion of it would be condensed 
 before the stroke had commenced. Assuming the admission 
 valves to be tight, and the cylinder non-conducting, the 
 latent heat of this condensed steam will be stored up in the 
 cylinder walls for use at a later period. If all the heat be 
 returned to the steam during expansion, the diagram 
 from the point of cut-off should follow the dotted line B R 
 shown in figs. 16 and 17. This line is called the "condensation 
 water heat recovery line." The actual expansion curve of 
 the engine should lie somewhere between these two extreme 
 cases, assuming no external heat to be furnished from a 
 j acket or similar source ; in which case the position of the 
 actual curve relatively to B r and B R will show the nature 
 and extent of heat exchange from the metal to the steam 
 after cut-off. Should the cylinder be of small dimensions, 
 or the cut-off very early, this exchange of heat becomes a 
 considerable factor in determining where all the heat of the 
 steam has gone to. 
 
 METHOD OF DRAWING B r AND B R LINES. 
 
 Let x represent the fractional dryness of the steam at 
 cut-off, scaled from the <t> diagram ; then (1 - x) will be the 
 weight of water in pounds present in the cylinder, as 
 the diagram is drawn for 1 Ib. of the mixture. But the 
 heat which (l-.e) pounds of water will evolve in cooling 
 from a temperature T I to a temperature r is 
 
 (1 - X) . S . (TJ_ - r), 
 
 where s is the mean specific heat of water between r and r L 
 
INITIAL CONDENSAT 
 
 Let r 1 equal the temperature of the steam at cut-off, and r the 
 minimum temperature of steam in the cylinder. Draw- 
 horizontal lines across the 8 <p chart at these temperatures 
 (see AC and D E, fisj. 16), and drop a vertical ordinate from 
 A, meeting D E in F, as shown. Then the triangle D AF, 
 extended to the absolute zero of temperature, represents 
 the heat given to 1 Ib. of water in raising it from r to T L ; 
 and vice versd, the heat evolved by 1 Ib. of water in cooling 
 from T L to r. Divide D F at G, in the proportion of the 
 
 dryness fraction at cut-off i.e. t make ^-=, equal to -r-^ ; then 
 
 the triangle G AF, extended to the zero of temperature, will 
 represent the heat given up by (l-#) pounds of water in 
 cooling from r x to r. This is not mathematically correct, 
 because the curve D A is not quite a straight line; but 
 the error introduced in assuming D A straight is quite 
 negligible, and does not materially affect the length of the 
 line G F. Now turn to the point of cut-off, B, and drop a 
 vertical ordinate from B meeting D E in H, and make the 
 triangle B H K equal to the triangle A G F. The line B H 
 represents adiabatic expansion, assuming a non-conducting 
 cylinder ; and the line B K (or B r as it is usually called) 
 will represent the theoretical expansion curve, assuming the 
 (1 - x) pounds of water present at B to give up its sensible 
 
 TT T7" 
 
 heat to the steam as the temperature falls ; also will 
 
 represent, to scale, the fractional weight of water re- 
 evaporated by the heat received from the hot water. 
 
 To draw the B R line of condensation water-heat recovery, 
 multiply the weight of water present at cut-off by the latent 
 heat of steam at the same temperature, and divide the 
 product by the mean absolute temperature of the cycle. 
 This will give the amount of increased entropy which would 
 be added to that possessed by the steam present at cut-off, 
 should all the heat of condensation be returned by the walls 
 during expansion. 
 
68 MEASUREMENT OF HEAT LOSSES. 
 
 Using the same notation as before, the increase of entropy 
 will be 
 
 where L x is the latent heat of 1 Ib. steam at^r x ; 
 
 Tm is the mean absolute temperature between cut-off 
 
 and exhaust ; 
 X - is the increase in entropy obtained from the 
 
 re-evaporation. 
 
 Referring again to fig. 16, make H R equal to (0 X - 0), to 
 the scale of entropy used, and join B R, which will represent 
 the theoretical expansion curve, assuming the walls to give 
 back to the steam during expansion all the heat they received 
 from the liquefaction of (1 - x] pounds of steam up to cut-off. 
 The lines B r and B R are shown on the various diagrams, 
 so that the position of the actual expansion line relative 
 to the two theoretical lines can be noted. 
 
 23. MEASUREMENT OF HEAT LOSSES BY[THE AID OF THE 
 DIAGRAM. 
 
 THE most important application of the diagram is, that 
 all the various losses and exchanges of heat can be easily 
 represented graphically, and to a fixed scale of heat units. 
 Referring to fig. 28, and considering only the trial at full 
 speed (shown by the full line), the area of the diagram 
 ABODE represents in heat units the net work done on the 
 piston per stroke. Draw the maximum and minimum steam 
 temperature lines, F Gr and K H respectively, and drop the 
 vertical ordinate G M, meeting K H in M ; then, if k and m 
 be the ordinates K and M respectively projected on the 
 absolute zero of temperature, the area fcKFGMm will 
 represent to scale the total heat supplied to the steam used 
 per stroke, assuming the feed water to be returned to the 
 boiler at the temperature denoted by the line KH. The 
 weight of steam used per stroke is assumed to be lib., for 
 
 "simplicity of calculation ; but in reality it is 
 
 volume factor 
 
MEASUREMENT OF HEAT LOSSES. 69 
 
 pounds per stroke, as explained previously. The ratio of 
 the two areas 
 
 ABODE 
 
 will therefore represent the absolute thermal efficiency of 
 the engine under these conditions. If the engine consists of 
 more than one cylinder, the line F G should be drawn at a 
 temperature corresponding to the pressure in the steam pipe 
 near the high-pressure cylinder, and the line K H at the 
 temperature of the hot-well discharge from the condenser. 
 The work done will also be the sum of the areas of the 
 diagrams for each cylinder. 
 
 The work done by a perfect heat engine working between 
 the same limits of temperature can be represented by the 
 rectangle L F G M i.e., the Carnot reversible cycle with 
 adiabatic expansion and compression, and no clearance. The 
 efficiency of the perfect heat engine will therefore be 
 represented on the <P diagram by the ratio of the two areas 
 LFGM 
 
 , 
 
 or . 
 
 The Rankine cycle, which some engineering professors now 
 prefer to use as the standard steam engine of comparison 
 (see report of the Thermal Efficiency Committee of the 
 Institution of Civil Engineers, 1898), includes the heating 
 up of the feed water from the temperature of the exhaust, 
 and is represented in fig. 28 by the area K F G M ; the upper 
 limit of temperature, F G, being that corresponding to the 
 pressure of the steam on the boiler side of the stop valve, 
 and the lower limit of temperature, KM, being that of the 
 steam in the exhaust pipe, near the engine. If superheated 
 steam be used, the extra work theoretically obtained should 
 be included, as shown in fig. 20 (page 47). The efficiency of 
 the standard engine of comparison (Rankine cycle) is there- 
 fore the ratio of 
 
 KFGM 
 
70 MEASUREMENT OF HEAT LOSSES. 
 
 Consequently, the ratio of the actual thermal efficiency to 
 that of the standard heat engine (Rankine cycle) will be 
 represented by 
 
 ABODE 
 KFGM ' 
 
 The ratio of the actual thermal efficiency to that of a 
 Carnot engine working between the same limits of tempera- 
 ture is ABODE 
 ~LF~(TM~ ' 
 
 Mr. Willans adopted a fixed minimum pressure for 
 condensing engines, corresponding to a temperature of 110 
 deg. Fah., in calculating the thermal efficiency of his central- 
 valve engine in his trials, published by the Institution of 
 Civil Engineers (see vol. cxiv., p. 9, Proc.Inst.C.E ), and called 
 the thermal efficiency of the engine the ratio of the areas 
 ABCDE 
 KTFGM ' 
 
 with F G drawn to correspond to the boiler pressure, and 
 K G drawn at 1'267 lb., corresponding to 110 deg. Fah. This 
 represents the thermal efficiency of the engine compared 
 with a perfect steam engine, without clearance, receiving 
 steam at boiler pressure and expanding it adiabatically 
 completely down to the condenser pressure of 1'267 lb., and 
 discharged at that pressure. Mr. Willans' argument was 
 that it was unfair to credit a steam engine with the heat 
 contained by the steam at a temperature (110 deg. Fah.) 
 below which it was useless for power purposes. He there- 
 fore compared his engine with a perfect steam engine, and 
 obtained a thermal efficiency of some 50 to 60 per cent ; 
 whereas the absolute thermal efficiency of steam engines, or 
 the ratio of work done to heat supplied, is only about 12 to 
 15 per cent. 
 
 For the particular trial under consideration, the diagram 
 of which is given in fig. 28, the areas measured by a 
 planimeter gave the following results : Area ABCDE, 
 equal to the net work done per stroke, 
 
 = 94-85 B.T.U. for lib. steam. 
 
MEASUREMENT OF HEAT LOSSES. 71 
 
 Area k K F G M ra = heat received per stroke = 1050-2 B.T.U. 
 Thermal efficiency, or ratio of work done to heat supplied, 
 
 1050-2 
 
 Area L F G M = work done by a perfect heat engine 
 working between same limits of temperature on the Carnot 
 cycle = 171 6 B.T.U. Thermal efficiency of perfect heat 
 engine (Carnot cycle) 
 
 Z^Lli = m5 = 0190. 
 902-2 
 
 Ratio of actual thermal efficiency to thermal efficiency of 
 perfect heat engine (Carnot cycle) 
 
 0,53. 
 
 Work done by perfect steam engine (Willans' method), 
 expanding done to 110 deg. Fah., - 249 6 B.T.U. 
 
 Ratio of actual thermal efficiency to the thermal efficiency 
 of a perfect steam engine (Willans' standard) 
 
 _ 
 
 The difference is thus seen between the various methods 
 of expressing the thermal efficiency of a steam engine, 
 according to the way in which the term " efficiency " is used 
 and understood. 
 
 The areas representing the heat losses in fig. 18 are : 
 
 1. Loss due to clearance, equal to area k K F A E N n* 
 (extended to absolute zero of temperature), = 30"0 B.T.U. 
 
 2. Loss by heat given to walls during admission = area 
 b B A G M m ; less the heat given by the walls to the steam 
 during expansion, represented by b B C c = 3877 - 3047 = 
 83-0 B.T.U. 
 
 3. Heat carried away by exhaust steam = area d D C c ; 
 plus compression n N E D d 
 
 = 801-8 + 40-6 = 842 4 B.T.U. 
 
 * The italic letters k, m, n, &c., are the extremities of the ordinates K, M, N, 
 &c., on the line of absolute zero of temperature (- 460 deg. Fah.) 
 
72 MEASUREMENT OF HEAT LOSSES. 
 
 The sum of these quantities (1 + 2 + 3), plus the heat 
 represented by the useful work done (area ABODE = 
 94'85 B.T.U.), should be equal to the total heat received by 
 the engine, represented by the area /jKFGMw (1,050'2 
 B.T.U.) 
 
 Accounting for all the heat in a balance sheet, according 
 to Hirn's method of treatment, the quantities expressed in 
 B.T.U.'s per stroke for 1 Ib. of steam are : 
 
 1 HEAT BALANCE SHEET. 
 
 Heat received. To Heat accounted for. 
 
 By steam received. ! In useful work 94 -85 B.T.U. = 9 '03 per cent. 
 
 Area A- K F G M m. 1. Loss by clearance 30 '() B.T.U. = 2 '86 per cent. 
 
 = 1050-2 B.T.U. I 2. Loss by walls 83'0 B.T.U. = 7 -90 per cent. 
 
 = 100 per cent. i 3. Loss by exhaust steam 842 '4 B.T.U. = 80'21 per cent. 
 
 Total = 1050-25 B.T.U. = 100'Opercent. 
 
APPLICATION TO THE GAS ENGINE. 73 
 
 CHAPTER V. APPLICATION TO THE GAS ENGINE. 
 
 24. GENERAL CONSIDERATIONS APPLICABLE TO ALL 
 
 PERMANENT GASES. 
 
 IF 1 Ib. of any gas be heated through a small rise of tem- 
 perature denoted by d t, the amount of heat received can be 
 calculated from its increased energy by the usual formula : 
 
 city = Cv x dt + A x P x dv (1) 
 
 where 
 
 d Q = the amount of heat received in B.T.U. ; 
 C v = the specific heat of the gas at constant volume ; 
 
 A = -= , or the reciprocal of Joule's mechanical 
 
 equivalent of heat ; 
 
 P = thepressure of the gas in Ibs. per square foot ; 
 d v = the increase of volume in cubic feet. 
 
 The first part of the equation (Cv x dt) represents the 
 additional internal energy of the gas ; and the second part 
 (A x P x dv) that corresponding to the external resistance 
 overcome. 
 
 Bat the combination of Bayle's and Charles's laws proves 
 that the product of pressure and volume of any permanent 
 gas varies directly as the absolute temperature ; or 
 
 P x V = R x r (2) . 
 
 where V = volume of the gas in cubic feet ; 
 
 R = some numerical constant (Kp - Kv) ; 
 T = absolute temperature. 
 
 Substituting the value of P so found in formula (1), we 
 get 
 
 dQ = (Cv x dt) + (A x R x r x ^) . . (3) 
 and, dividing all through by T, 
 
 A9 = C* x ^ + A. xR.x-t? . (4) 
 
 T 
 
74 APPLICATION TO THE GAS ENGINE. 
 
 But -* is the required small increase of entropy corres- 
 ponding to the increase of temperature d t. 
 
 The external work done by 1 Ib. of gas in heating 
 
 = P . d v = d t (C p - C v) J (in work units) ; 
 
 therefore Cp - C v = A . P . ^, 
 
 d t 
 
 but also Cp - C v = A . R . ; 
 
 and, substituting this value f or A . R in equation (4), we get 
 
 d L^QTd<t> = Gv d - t - + (Cp - Cv)^ . (5) 
 
 T T V 
 
 This is the general formula from which the change of 
 entropy of a gas can be calculated under the various con- 
 ditions of change. For example, during explosion in a gas 
 engine the volume is constant, or very nearly so, and 
 therefore d v = 0. Under these circumstances, 
 
 d<f> = Vv.d-*; 
 which, integrated, gives 
 
 = C.loge^ ...... (6) 
 
 ~0 
 
 On the other hand, if the pressure remains constant, as is 
 sometimes the case just after explosion in a gas engine, the 
 volume and the temperature increase in accordance with 
 Charles's law, viz., 
 
 ^-i^Eg*..'-o.^. 
 
 In this case, equation (5) becomes 
 
 P .-; 
 which, reduced, 
 
 = dt = Cp.loge ^ . . . (7) 
 
APPLICATION TO THE GAS ENGINE. 75 
 
 Where both the pressure and the volume of the gas 
 change, as during expansion in a gas engine, the change of 
 entropy must be calculated from the formula representing 
 the law of the particular expansion or compression, viz., 
 
 P . V* = P! V/' (8) 
 
 But, from equation (2), 
 
 P . V = R . r, 
 and, substituting the value of 
 
 P = ^ in (8), 
 we get 
 
 7, 1 7. 
 
 (9) 
 
 or, taking the value of ~ from (3), and substituting in (5), 
 we get 
 
 j n k - y d t 
 
 ' Ic - 1 ' 
 which, integrated, gives 
 
 do) 
 
 where 0! = initial entropy ; 
 02 = final entropy ; 
 r 1 = initial temperature ; 
 r 2 = final temperature. 
 
 This is the general formula for finding the change in 
 entropy in all changes represented by the law 
 
 P . V* = P! Vi*. 
 
 Given any expansion or compression curve of an indicator 
 diagram for a gas engine, k is found by measuring any two 
 co-ordinates, thus : 
 
 P . V* = P 1 V/ ; 
 . , = log P - log P! 
 logVi - logV 
 
 In the particular case where k = y that is to say, for 
 
76 DIAGRAM FOE THEORETICAL GAS ENGINE. 
 
 adiabatic expansion or compression the change of entropy 
 becomes nil, as k - y = 0, and therefore 
 
 01 - 2 = C v x *I_? x log, - 2 = 0. 
 & l T i 
 
 The entropy diagram, therefore, for any adiabatic change. 
 is a vertical straight line. 
 
 25. ENTROPY -DIAGRAM FOR THEORETICAL GAS ENGINE, 
 WITH ADIABATIC EXPANSION AND COMPRESSION. 
 
 The indicator diagram for a perfect gas engine, working 
 on the Otto or Beau de Rochas cycle, is shown in fig. 33, 
 with pressure as ordinates and volume as abscissa. In this 
 diagram AB represents the compression of the charge, 
 assumed to be adiabatic that is, following the law p . v y = a 
 
 constant, where y is the ratio ^^ of the mean specific heats 
 
 w V 
 
 of the gaseous mixture. BC shows the explosion, with 
 instantaneous increase of pressure at constant volume, and 
 C D the expansion period, also assumed to be adiabatic. 
 DA represents the exhaust, and AE the discharge and 
 suction strokes. The entropy diagram for this cycle is 
 shown in fig. 34, and is drawn for 1 Ib. of the gaseous 
 mixture, the vertical ordinates representing absolute 
 temperatures (r) and the base, entropy or 0. Starting with 
 the mixture at A, it is compressed from volume V a to V 6 
 adiabatically, without receiving or rejecting any heat. The 
 process is therefore represented in fig. 34 by the vertical 
 line AB, the entropy at B being the same as at A. The 
 pressure at B can be found by the general formulae 
 
 or, taking 7 = 1*4 (its approximate value), 
 
 log P b = log P a + 1'4 log Va - 1'4 log V6. 
 The temperature at B is found from the general formulse 
 
DIAGRAM FOR THEORETICAL GAS ENGINE. 
 
 where r b = absolute temperature at B ; 
 
 P b = pressure at B in Ibs. per square foot ; 
 
 V b = specific volume of the mixture at B ; 
 
 Kp = specific heat of the mixture at constant pressure; 
 
 K v = specific heat of the mixture at constant volume. 
 
 
 FIG. 
 
 [NOTE. K p and K v are in ft.-lb. unit?, = J x C p and 
 J x Cv] 
 
 where J = Joule's mechanical equivalent of heat = 772 
 ft.-lbs. 
 
78 DIAGRAM FOR THEORETICAL GAS ENGINE. 
 
 If the temperature of the mixture at A be known, and 
 the ratio of the compression, the temperature at B can be 
 calculated direct from the general formula for adiabatic 
 compression of a gas 
 
 rb = ra(r) y ~ l 
 
 where r I and r a are the absolute temperatures at B and A 
 respectively, 
 
 r = ratio of compression = =^~ ; 
 
 7 = ratio of specific heats. 
 
 From B to C the pressure is increased instantaneously 
 before the piston has moved, and therefore the increase of 
 temperature will be directly proportional to the increase of 
 pressure, or 
 
 rb X Pr. 
 -P^ ; 
 
 PC and fb being the absolute pressures in Ibs. per square 
 foot at C and B respectively. When the pressure at C is 
 not known, its theoretical temperature can be calculated 
 from the calorific value of the gas, assuming perfect com- 
 bustion ; thus 
 
 where H = total heat of combustion of 1 Ib. of the particular 
 
 gas used ; 
 R = ratio (by weight) of the gas, air, and diluent to 
 
 the gas. 
 
 In actual practice, this theoretical rise of temperature is, 
 for various reasons, never obtained. An average value of H 
 for London lighting gas is about 19,000 B.T.U/s per pound. 
 The ratio R varies very considerably with different types of 
 engines and conditions of working, but is about 20 to 30 
 for lighting gas. The increase of entropy during explosion 
 will be represented in fig. 34 by the logarithmic curve B C, 
 whose equation is re 
 
 d * -fCv.11', 
 
 rb 
 
DIAGRAM FOR THEORETICAL GAS ENGINE. 79 
 
 which, integrated, gives 
 
 <j>c - <j>b = Cv x loge r -^ ; 
 
 rb 
 
 that is, the increase of entropy from B to C is equal to the 
 
 specific heat of the mixture at constant volume (about 018 
 to 0*19), multiplied by the hyperbolic logarithm of the ratio 
 of the two temperatures. Knowing T c and (0 c - 6), the 
 
80 DIAGRAM FOE THEORETICAL GAS ENGINE. 
 
 curve B C can be plotted as in fig. 34, all intermediate points 
 such as B x being calculated from the formula by sub- 
 stitution, thus 
 
 0& x - 06 =Cv x log e T A. 
 
 rb 
 
 A short geometrical method of constructing such curves as 
 B C and D A will be explained later. 
 
 The adiabatic expansion is denoted on the entropy diagram 
 by the vertical line C D, fig. 34, the gas neither receiving nor 
 evolving heat, and the temperature falling from r c to rd ; 
 the latter being obtained from the formula, 
 7 Pd x Vd 
 = K P -Kv> 
 and P d being obtained from the usual formula, 
 
 Pd x Vrf? = PC x Vc^; 
 or, as Vd = Va, and Vc = V6, 
 
 Pd x Va 7 =Pc x V6^; 
 or log P d = log P c + y . log V b y . log V a. 
 
 From D to A, or exhaust at constant volume, the entropy 
 diagram again assumes a logarithmic curve D A, fig. 34, the 
 temperature at A falling to its initial point, and its change 
 of entropy being equal to 
 
 <t>d-<pa = Cvx log e ~. 
 T d 
 This change of entropy must always be negative, because 
 
 T . is less than unity that is to say, the increase of entropy 
 T d 
 
 from D to A is a minus quantity or a loss, the gas giving up 
 its heat to the exhaust. The entropy for any intermediate 
 point D x on the curve D A can be calculated by substitution : 
 
 0rf x _ <}>d = Cv x log* ^-. 
 
 T a^ 
 
 The exhaust and suction strokes A E do not have any 
 effect upon the entropy diagram, as the temperature during 
 those strokes is assumed constant. 
 
 The diagram is completed by drawing O X at the absolute 
 zero of temperature, when the work done per cycle will be 
 equal to the area enclosed by A B C D, fig. 34, in heat units ; 
 
DIAGRAM FOR AN ACTUAL GAS ENGINE TRIAL. 81 
 
 the heat received per cycle will be equal to the area B C X ? 
 and the thermal efficiency or 
 
 work done _ ^ ratio rf ^ twQ areag ABC D 
 heat received O B C X 
 
 The heat given to the exhaust gases will be equal to the 
 area O A D X. 
 
 It is evident from the entropy diagram that the two 
 quantities (<t> c - 06) and (<t> d - <t> a} being equal, and the 
 two curves B C and A D following the same law, the ratio 
 of the two temperatures is a constant quantity, and depends 
 entirely upon the amount of compression that is, 
 
 rb _ re 
 TO, rd* 
 
 and the higher this ratio, the higher will be the thermal 
 efficiency. 
 
 26. ENTROPY DIAGRAM FOR ACTUAL GAS-ENGINE TRIAL. 
 
 For example, take the trial of a 7 horse power Crossley- 
 Otto engine, made by Professor Capper, at King's College, 
 London, on December 7th, 1892, full particulars of which are 
 given by Mr. Bryan Donkin, in his book on " Gas, Air, and 
 Oil Engines," from which the following data and mean 
 indicator diagram (fig. 35), are taken. 
 The principal particulars of the trial are : 
 Cylinder, 8^ in. diameter by 18 in. stroke. 
 
 Revolutions per minute 162'5 
 
 Explosions 71*2 
 
 Net I.H.P 13-32 
 
 Cylinder volume '591 cubic foot. 
 
 Clearance volume 0'2467 
 
 Total volume 0'8377 
 
 Gas used (by meter) 27975 cb. ft. per hour. 
 
 Gas used per explosion, at atmo- 
 spheric pressure and temperature 0'06544 cubic foot. 
 Gas used at pressure and tempera- 
 ture in cylinder at A 0'0822 
 
 Air used (from cylinder volume)... 07556 cubic foot 
 7T per explosion. 
 
82 DIAGRAM FOR AN ACTUAL GAS ENGINE TRIAL. 
 
 The mean indicator diagram, fig. 35, furnishes the 
 following pressures and volumes : 
 
 p a = 13-8 Ib. per square inch absolute. 
 Pb = 67-8 
 PC = 240 
 P* - 240 
 
 re = 48-71 
 
 v,i = G'8377 cubic foot. 
 
 vt> = 0*2467 
 
 v c = 0-2467 
 
 va = 0-2617 
 
 v e = 0-8377 
 
 The point E is taken on the ideal expansion curve, or the 
 actual expansion line continued to the end of the stroke 
 without exhausting. From the above pressures and volumes, 
 the index k in the equation p a x v a k = pb x vb k is calculated 
 
 to be 
 
 For expansion, k = T3707 ; 
 compression, k = 1*3022. 
 
 During compression, the mixture consists of air, gas, and 
 exhaust products, in known proportions, and of known 
 chemical analysis ; therefore K p and K v for the mixture 
 can be calculated in the same way as for any compound gas. 
 The proportions of air, gas, and diluent present during 
 compression are calculated thus : 
 
 1. Gas, '06544 cubic foot per explosion at atmospheric 
 pressure and temperature, with a specific volume of 34'87 
 cubic feet per pound, 
 
 = '^ 6 y = 0-001877 Ib. per explosion. 
 
 34 '87 
 
 2. Exhaust products left in cylinder at end of previous 
 discharge stroke = 0'2467 cubic foot at 605 deg. Fah. absolute 
 temperature, and 14"8lb. per square inch pressure. 
 
 From Table XIIL, the exhaust gas was of 0'08201b. per 
 cubic foot average density. 
 
DIAGRAM FOE AN ACTUAL GAS ENGINE TEIAL. 
 
 .*. G'2467 cubic foot weighed 0'2467 x 0'0820, 
 
 = 0-02023 Ib. at 492 deg. Fah. and 14'71b. pressure, 
 
 = 02023 x ^ x M| = 0-01656 Ib. 
 bOo 14 7 
 
 at 605 deg. Fah. and 14'8 Ib. pressure. 
 
 83 
 
 FIG. 35. 
 
 3. Air admitted per explosion occupied the total volume 
 of the cylinder, less the volume of the exhaust products and 
 gas. This was 
 
 0-8377 - (0-2467 + 0'0822) = 0'5088 cubic feet, 
 and weighed 
 
 = 0-03131 Ib. per explosion. 
 
84 
 
 DIAGRAM FOE AN ACTUAL GAS ENGINE TRIAL. 
 
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DIAGRAM FOR AX ACTUAL GAS ENGINE TRIAL. 
 
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86 DIAGRAM FOR AN ACTUAL GAS ENGINE TRIAL. 
 
 The mixture during compression therefore consisted of 
 0-001877 lb. gas ; 
 0*01656 lb. exhaust products ; 
 0-03131 lb. air. 
 
 Total = 0-049747 lb. per explosion. 
 
 The average specific heats of the mixture will be the 
 specific heat .of each constituent part multiplied by its 
 relative weight. For the gas, the mean specific heats C v and 
 C p must be calculated from the chemical analysis, as shown 
 in Table XII, from which C v = 0'5279, and Cp = 0'6961 ; or 
 multiplying by 772, Kv = 407'5, and Kp = 537'4. Calcu- 
 lating similarly for the exhaust products, as shown in 
 Table XIIL, 
 
 C v = 0-1716, and Cp = 0'2385, 
 or K v = 132-5, and Kp = 1841. 
 
 For air, the figures are Kv = 130'20, and Kp = 183'55. 
 The mean specific heat of the mixture can therefore be 
 found as shown in Table XIV., page 88 to be, 
 
 Kp = 199-09, and Kv = 141 '43 foot-pounds, 
 
 n = Kp - Kv - 57 '66 foot-pounds, 
 and C p = 0*25788, 
 
 Cv = 01832, 
 
 7 = JL = 1-4077. 
 C v 
 
 Adopting these values, the temperatures will be 
 
 Tc = 840 * 240 = 2973 deg. Fah. absolute ; 
 
 OYO 
 
 3154 deg. Fah. abaolute ; 
 
 - 58 * 
 
DIAGRAM FOR AN ACTUAL GAS ENGINE TRIAL. 87 
 
 It will be noticed that the temperature at A found by 
 calculation (580 deg. Fah. absolute) is 25 deg. less than that 
 (605 deg. Fah. absolute) previously assumed for calculating 
 the relative weights of gas, air, and exhaust products in the 
 mixture ; but this difference does not affect the result, as all 
 the three constituents will be increased in weight by the 
 same amount, and their relative weights for the corrected 
 temperature will be the same. 
 
 To draw the entropy diagram for the trial (see fig. 36), 
 start with the mixture at B as the zero of entropy, 
 when the entropy at C will be found from the formula, as 
 explained previously 
 
 (f>c - 06 = C V X loge 
 
 - 01832 x log 
 
 = 0-23158. 
 
 4>d - <t>c = C p X loge 
 T c 
 
 = 0-25788 x loge = 0-01524 
 
 <t>e ~ <t>d = C V X 
 
 = 000790. 
 
 * - <t>e = C V loge ^ 
 
 - 01832 xlog.^ 
 - 0-23112 
 
 i = C V X 1 y X loge - 
 K 1 TI 
 
 = - 0-02369. 
 
88 DIAGRAM FOR AN ACTUAL GAS ENGINE TRIAL. 
 
 Balancing up the quantities of entropy, thus 
 
 C - 06 0-23158 
 
 - C 0-01524 
 
 e - 0c* 0-00790 
 
 0a - 0-23112 
 
 06 - 0a ....- 0-02369 
 
 0-25481 
 
 The result shows a numerical error of about 0'04 per cent, 
 which is practically negligible, probably due to slight 
 inaccuracies of calculation. 
 
 TABLE XIV. SPECIFIC HEATS OF MIXTURE, FOR TRIAL 
 OF 7 HORSE POWER CROSSLEY ENGINE. 
 
 Constituent. 
 
 Weight 
 used 
 per cvcle, 
 Ibs. 
 
 Kv 
 for lib., 
 ft. -Ibs. 
 
 Kv 
 
 for weight 
 used, 
 ft. -Ibs. 
 
 Kp 
 for 1 lb., 
 ft. -Ibs. 
 
 Kp 
 
 for weight 
 used, 
 ft. -Ibs. 
 
 Gas 
 
 0-001877 
 
 407-5 
 
 0*7649 
 
 537-4 
 
 1-0087 
 
 Exhaust products . . 
 Air 
 
 0-01656 
 0-03131 
 
 132-5 
 130-2 
 
 2-1942 
 4-0765 
 
 184-1 
 183-55 
 
 3-1487 
 5-7469 
 
 
 
 
 
 
 
 Totals 
 
 0-049747 
 
 .. 
 
 7-0356 
 
 
 9-9043 
 
 K v for mixture = 
 
 K p for mixture = 
 
 7-0356 
 0-049747 
 0-9043 
 0-049747 
 
 = 141-13 ft. -Ibs. perlb. 
 = 199-09 ft. -Ibs. per lb. 
 
 = Kj-Kw- 57-66 ft. -Ibs. per lb. 
 
 The e diagram for the ideal cycle is shown by A B C D E, 
 in fig. 36, and if the area enclosed be measured by a 
 planimeter, it will be found to equal 171 '875 B.T.U. This 
 represents the work which would be done by 1 lb. of gaseous 
 mixture, and has to be corrected by multiplying by 049747 
 
CORRECTED DIAGRAM. 
 
 89 
 
 to get the work done by the known weight of mixture in 
 the cylinder per explosion. Thus, 
 
 171'875 x 049747 = 8 55 B.T.U. 
 work done per explosion ; or, expressed in work units, 
 
 8 '55 x 772 = 6600 foot-pounds per explosion, 
 which corresponds within 01 per cent with the value given 
 
 "1 
 
 1000 ;. _ _.._ 
 
 Fio. 36. 
 
 by Mr. Donkin (6,594 foot-pounds), as found by measuring 
 the ideal cycle ABCDE of the pressure volume diagram 
 shown in fig. 35. 
 
 27. CORRECTED DIAGRAM FOR GAS ENGINE TRIAL. 
 This ideal cycle has now to be corrected for the actual 
 diagram, and, starting with the explosion period, the curves 
 
90 
 
 CORRECTED DIAGRAM. 
 
 BC and CD, in fig. 36, are correct for the actual diagram, 
 but during the expansion period it will be seen that the 
 actual pressure curve shown by the full line D G H J K L, 
 fig. 35, differs very materially from the ideal expansion 
 curve D E, shown dotted. It will therefore be necessary to 
 take various additional points between D and F upon the 
 actual pressure curve, such as G, H, J, K, and L, and 
 calculate the temperatures and additional entropy at each of 
 
 TABLE XV. ACTUAL EXPANSION CURVE : 7 HORSE 
 POWER GAS ENGINE. 
 
 Position. 
 
 Pressure. 
 Pounds 
 per 
 square 
 inch. 
 
 Volume. 
 Cubic 
 feet. 
 
 Tempera- 
 ture. 
 Fah. 
 absolute. 
 
 Index of 
 expansion. 
 
 Increase of 
 entropy. 
 i+. 
 
 Total 
 entropy, 
 0. 
 
 
 
 deg. 
 
 
 D 
 
 240 
 
 0-2617 
 
 3154 
 
 
 ! 0-24682 
 
 
 
 
 
 1-3965 
 
 + 0-000521 
 
 G 
 
 170 
 
 0-335 
 
 2858 
 
 0-24734 
 
 
 
 
 
 1-4668 ! - 0-00175 
 
 H 
 
 134 
 
 0-394 
 
 2050 
 
 
 
 0-24559 
 
 
 
 
 
 1-4798 
 
 - 0-00185 
 
 
 J 
 
 109 
 
 0-453 
 
 2478 
 
 
 
 0-24374 
 
 
 
 
 
 1-2995 
 
 + 0-00463 
 
 
 K 
 
 80-5 
 
 0-572 
 
 2312 
 
 
 
 0-24837 
 
 
 
 
 
 1-3526 
 
 + 0-00190 
 
 
 L 
 
 62-5 
 
 0-6897 
 
 2164 
 
 
 
 0-25027 
 
 these points. For example, at G the pressure is 170 Ib. per 
 square inch, and the volume 0'335 cubic feet ; the tempera- 
 ture at this point will therefore be 
 
 ^fg&SiS -*******+ 
 
 and the entropy at G above that at D will be 
 
 <t>g - (pd C V X ~ y X loge . 
 k - I r d 
 
 The value of &, the index of the expansion between D and 
 
 G, will be 
 
 , _ log 240 - log 170 _ 1<3%5 
 * log 335 - log 0-2617 6 ' 
 
CORRECTED DIAGRAM. 91 
 
 when (p g - <t>d becomes equal to 
 
 = G'000521. 
 
 The values of &, together with the temperatures, and 
 entropy at the various points G, H, J, K, &c., are given in 
 the accompanying Table XV., from which the actual 
 expansion curve on the 6 <i> diagram, D G H J K, fig. 36, can 
 be plotted and drawn. 
 
 For the exhaust period, the temperature and entropy can 
 be calculated by the formula? already given, as follows : At 
 M, just after release, where the actual pressure shown by the 
 indicator diagram is 53 '8 Ib. per square inch, and the 
 pressure F on the ideal expansion curve at the same volume 
 is 56'79 Ib., the two temperatures will be 
 
 The loss of entropy due to a drop in temperature from 
 2135 deg. to 2023 deg. at constant volume is equal to 
 
 91 ^Pi 
 
 d = 0/ - 0m = 01832 x log e |j| = 0-00991. 
 
 JUZo 
 
 But this amount of entropy must be taken from that which 
 the mixture would possess at F on the dotted curve. The 
 entropy at F above that at D will be 
 
 = 0-00713 
 
 add C + 0<z = Q'24682 
 
 then 0/= 0-25395 
 
 less (0/ - m ) = -00991 
 
 then 0m = 0-24404 
 
 Similar calculations at other points in the exhaust curve of 
 
 the indicator diagram, such as N and O, are made, as shown 
 
92 
 
 CONSTANT VOLUME CURVES. 
 
 in Table XVI., when the actualTtemperature and entropy at 
 O, coinciding with the theoretical curve E A, finishes the 
 process. 
 
 TABLE XVI. EXHAUST PERIOD : 7 HORSE POWER 
 GAS ENGINE TRIAL. 
 
 Position. 
 
 Pressure. 
 Ibs. aba. 
 
 Volume, 
 cub. ft. 
 
 Actual 
 Tempera- 
 ture. 
 Fab. ab=. 
 
 Theoretical 
 Tempera- 
 ture. 
 Fah. abs. 
 
 Entropy. 
 
 M 
 
 53-8 
 
 0-749 
 
 de ff . 
 2023 
 
 dej?. 
 2135 
 
 0-25395 - 0-00991 = 0'24404 
 
 N 
 
 38-0 
 
 0-808 
 
 1541 
 
 ?076 
 
 0-255 - 0-055 = 0-200 
 
 
 
 24-0 
 
 0-8377 
 
 1009 
 
 2048 
 
 0-2557 - 0-134 = 01217 
 
 28. CONSTANT VOLUME CURVES. 
 
 The two principal curves in the entropy diagram, fig. 36, 
 viz., B C and E A, are seen to follow the general law of a 
 logarithmic curve 
 
 2 - 1 = (J . loge T ^, 
 T l 
 
 and may be very easily drawn by the following geometrical 
 method. Plot a temperature curve A B C D (fig. 37) to 
 a base of its hyperbolic logarithms, on sectional paper, and 
 between any two temperatures, say at B = 840 deg. Fah. 
 absolute and C = 2,973 deg. Fah. absolute, draw the ordinates 
 B M and C N on to any arbitrary base line X Y. From M 
 draw the inclined line MOP, making an angle a with the 
 base X Y, such that tan a = C, the constant in the above 
 formulae. Produce M O to intersect the higher temperature 
 ordinate C N at P ; then P N will represent to the same 
 scale as the base the change of entropy < 2 ~ 0i> where 
 r 2 = 2,973 deg. and T X = 840 deg. 
 
 If C be taken = C v for the trial just considered, viz., 01832, 
 then the tan a = 01832, or a = 10 deg. 23 min., and P N will 
 be found to scale 0'231. By drawing ordinates from the 
 intersection of the curve with various temperatures between 
 840 deg. and 2,973 deg., such as 1,000 deg. Fah., 1,200 deg. 
 
CONSTANT VOLUME CURVES. 
 
 93 
 
 Fah., tkc., the change of entropy from 840 deg. Fah. to these 
 temperatures can be scaled direct from the line M F, being 
 equal to the intersected portions of the ordinates at 1,000 
 deg. and 1,200 deg., or R S and T U respectively. By 
 adopting this method of graphically finding the intermediate 
 values of entropy for points between B and C, much time 
 and calculation will be saved, and if the curve be drawn 
 accurately and to an open scale, the result will be sufficiently 
 
 FIG. 37. 
 
 correct for all practical purpose?. It should be noted that 
 the most laborious part of the process, viz., drawing the 
 logarithmic curve accurately, is only done once for all, and 
 all other temperatures and values of C p and C v can be used 
 on the same chart by drawing various inclined lines M P. 
 The line V W, in fig. 37, is drawn for the period E to A of the 
 7 horse-power cycle just considered, the entropy at E being 
 G'23112 above that at A. 
 
-94 HEAT LOSSES. 
 
 29. HEAT LOSSES IN 7 H.P. OTTO GAS ENGINE. 
 
 In the entropy diagram for the trial of this engine, shown 
 in fig. 36, the heat usefully employed as work is represented 
 by the area enclosed by ABGDJLO; but it does not 
 represent the total heat evolved by the explosion of the gas. 
 Knowing the weight of gas used per explosion (0'C01877 lb.), 
 and its calorific value (19,200 B.T.U. per lb.), the total 
 available heat will be 
 
 0-001877 x 19200 = 36'04 B.T.U. 
 
 This must be represented on the entropy diagram, as in fig. 
 38, by producing the explosion line B C to P, so that the 
 area 6 B P p shall be equal to 
 
 724-5 B.T.U. per lb. of mixture. 
 
 The theoretical temperature at P, due to complete com- 
 bustion, can be calculated thus : 
 Let x = rise in temperature from B, 
 
 and Cv = specific heat at constant volume ; 
 then x x Cv = 724'5, 
 or x = 3955 deg. Fah., 
 
 and temperature at 
 
 P = 3955 + 840 = 4795 deg. Fah. absolute. 
 
 The several losses can now be estimated direct from the 
 entropy diagram, fig. 38, by measuring the areas they repre- 
 sent, as follows : 
 
 The net work done =ABCDJLO = 8'20 B.T.U. per 
 explosion, or 22 '8 per cent of the total available heat. The 
 heat given to the walls during compression, represented by 
 the area a A B6, is equal to 0'77 B T.U. per explosion ; that 
 given to the exhaust gases a A O M L I = 13*63 B.T.U. ; and 
 the remainder, or I L J D C P^ = 13 '44 B.T.U., is transmitted 
 through the cylinder walls. The total heat, therefore, given 
 to the walls is 13'44 + 077 = 14 '21 B.T.U., and this will be 
 equal to the heat given to the jacket water, plus the radiation 
 of the cylinder and piston. The former of these was measured 
 during the trial, and found to be 14 '02 B.T.U. per explosion, 
 
HEAT LOSSES. 
 
 95 
 
 thus leaving 0'19 B.T.U. for radiation. It is probable that 
 some of the heat represented by a A LI will have passed 
 through the cylinder wall, and be included in that measured 
 by the jacket water, so that the actual radiation will be 
 
 ENTROPY PXH CB of/narvtte 
 
 FIG. 38. 
 
 considerably in excess of 019 B.T.U. The details of the heat 
 balance sheet, as measured from the entropy diagram, fig. 38, 
 are given in Table XVII., and compare very favourably with 
 the values given by Mr. Donkin in his report (Table V., 
 Appendix A, of "Gas, Oil, and Air Engines"). 
 
96 
 
 HEAT LOSSES. 
 
 With an ideal engine, assuming a non-conducting cylinder, 
 complete combustion, and exhaust at constant volume, with 
 adiabatic expansion and compression, the work done per 
 explosion would have been that represented by the area 
 
 TABLE XVIL HEAT BALANCE SHEET FOR 7 H.P. GAS- 
 ENGINE TRIAL. 
 
 Area. 
 
 (See fig. 27.) 
 
 Description. 
 
 B.T.U. per 
 explosion. 
 
 Per cent. 
 
 b B C D d 
 dDGKLl 
 
 Heat received during explosion . . . 
 Heat received during expansion . . 
 
 21-98 
 0-62 
 
 
 bBCVLl 
 bEPp 
 
 Total received shown on diagram . . 
 
 Total received, calculated from 
 weight of gas used 
 
 22-60 
 36-04 
 
 62-7 
 lOO'O 
 
 
 
 
 
 LDCPjp 
 aAOMLZ 
 
 Difference = heat to jacket water.. 
 Heat to exhaust gases 
 
 13-44 
 13-63 
 
 37-3 
 37-8 
 
 aAEb 
 
 Heat to walls during compression. 
 
 0-77 
 
 21 
 
 
 Total heat lost 
 
 27 '84 
 
 77'2 
 
 ABCD JLO 
 
 
 8 -20 
 
 22'8 
 
 
 
 
 
 6BP 
 
 Total received 
 
 36-04 
 
 100-0 
 
 
 
 
 
 R B P Q, instead of A B C D L. The maximum work theoreti- 
 cally possible under these conditions is therefore equal to 
 
 Tb ~ ra of the total heat evolved, amounting in this case to 
 
 Tb 
 
 840_- 580 = o-3095 of 36 . 04 . 
 
 840 
 
 or 1115 B.T.U. per explosion. The net work actually 
 obtained in the cylinder, being 8'2 B.T.U., was only 73 5 per 
 cent of this. The Carnot cycle of maximum theoretical 
 efficiency, when applied to a gas engine, is therefore not only 
 misleading, but incorrect, because the entropy diagram for a 
 gas engine can never become a rectangle, as in the case of 
 that for the steam engine. 
 
DIAGRAM FOR DIESEL OIL MOTOR. 97 
 
 CHAPTER VI. APPLICATION TO OIL AND AIR ENGINES. 
 
 30. ENTROPY DIAGRAM FOR 20 H.P. DIESEL OIL MOTOR. 
 IN general, the method of drawing the entropy diagram 
 for an oil-engine test is similar to that employed for the 
 gas engine ; the temperature being calculated by the usual 
 formula, P . V = R . T, and the entropy from the various 
 equations used in the gas-engine tria], already described 
 (see chapter v., page 75, &c.). As an illustration of a 
 different cycle to the Otto, take that of the Diesel oil 
 motor, described in The Practical Engineer, for May 6th, 
 1898. In this engine the compression stroke is followed by 
 the admission of finely-sprayed oil injected by an air pump, 
 and lasting from 5 to 10 per cent of the explosion stroke, 
 according to the amount of power required. The ignition is 
 effected by heating the air by adiabatic compression to a 
 sufficiently high temperature as to cause the oil to explode. 
 In other respects the cycle is similar to that of an ordinary 
 gas engine. Taking the first trial made by Professor Schroter 
 on a 20 horse power Diesel motor, the following particulars 
 are taken from the Zeitschrift des Vereines Deutscker 
 Ingenieure for July 24tb, 1897 : 
 
 Diameter of motor cylinder inches 9'856 
 
 Stroke of motor cylinder inches 15 725 
 
 Capacity of motor cylinder cubic feet 695 
 
 Volume of clearance (assumed 6 per cent) 
 
 cubic feet 0*045 
 
 Revolutions per minute 171*8 
 
 Explosions per minute 85'9 
 
 Oil used per explosion Ibs. 002122 
 
 Air used per explosion Ibs. 0'039529 
 
 Total mixture per explosion Ibs. 0'041651 
 
 Professor Schroter estimates the mean specific heat of the 
 gases at constant pressure at 0'264 ; and assuming the value 
 of y to be T408, the value of C v will be 01875 ; or, expressed 
 
 8T 
 
98 
 
 DIAGKAM FOR DIESEL OIL MOTOR. 
 
 in foot-pound units, K p = 203'Sl, and K v = 14475, and R = 
 K p - K v = 59 06 foot-pounds per pound of gas. 
 
 From this, knowing the weight of mixture present, and 
 its pressure and specific volume, its temperature can be 
 calculated in the usual way 
 
 r= P X V - 
 
 P = pressure in pounds per square foot (absolute) ; 
 
 V = specific volume of the mixture ; 
 
 w = weight of mixture per explosion ; 
 
 R = K p - K v for 1 Ib. = 59 '06 foot-pounds. 
 
 VOLUME 
 
 0"]S C** ft 
 
 Fio. 39. 
 
 Thus, at point 1, at the beginning of the power stroke 
 (see fig. 39), the temperature is 
 
 T = 515 x 144 x 0-045 = u j F h 
 
 0-039529 x 59-06 " ' 
 
 and at point 2, where the maximum pressure occurs, 
 
 - = ^ = ^-. 
 
DIAGRAM FOR DIESEL OIL MOTOR. 
 
 99 
 
 assuming the^whole of the oil to have been injected during 
 this portion of the stroke. 
 
 From 1 to 2 the increase of entropy can best be calculated 
 by the following formula : 
 
 2 - 
 
 + 00 X loge 
 
 ENTROPY 
 
 FIG. 40. 
 
 that is to say, calculate the difference of entropy due to the 
 increase of volume, as at constant pressure, and add to it the 
 difference of entropy due to the increase of pressure, as at 
 
100 
 
 DIAGRAM FOE DIESEL OIL MOTOR. 
 
 constant volume. After point 2 this second quantity 
 
 p 
 becomes negative, because the ratio of ^ is less than unity. 
 
 "2 
 
 TABLE X VII L ENTROPY DIAGRAM : 20 H.P. DIESEL 
 MOTOR. 
 
 a 
 
 Pressure 
 
 
 Absolute 
 
 Difference of entropy. 
 
 Total 
 
 a 
 
 Lbs. per 
 square 
 
 - Volume. 
 Cubic feet, 
 
 tempera- 
 ture. 
 
 
 entropy 
 
 from 
 
 
 
 o 
 
 Oi 
 
 inch. 
 
 
 Deg. Fah. 
 
 Positive. 
 
 Negative. 
 
 position 1. 
 
 1 
 
 515 
 
 0-015 
 
 1430 
 
 0-09098 
 
 
 ' 
 
 2 
 
 558 
 
 0-060 
 
 1960 
 
 
 
 0-09098 
 
 
 
 
 
 0-06863 
 
 
 
 3 
 
 539 
 
 0-0797 
 
 2516 
 
 
 
 0-15961 
 
 
 
 
 
 0-06413 
 
 
 
 4 
 
 456 
 
 0-1145 
 
 3056 
 
 
 
 0-22374 
 
 
 
 
 
 0-03380 
 
 
 
 5 
 
 376 
 
 0-1492 
 
 3285 
 
 
 
 0-25754 
 
 
 
 
 
 0-02062 
 
 
 
 6 
 
 245 
 
 0-2187 
 
 3137 
 
 
 
 0-27816 
 
 
 
 
 
 0-02876 
 
 
 
 7 
 
 165 
 
 0-323 
 
 3120 
 
 
 .... 
 
 0-30692 
 
 
 
 
 
 0-00974 
 
 
 
 8 
 
 105 
 
 0-462 
 
 2S40 
 
 .... 
 
 .... 
 
 0-31666 
 
 
 
 
 
 0-OT128 
 
 
 
 9 
 
 77 
 
 0-601 
 
 2709 
 
 .... 
 
 
 0-32794 
 
 
 
 
 
 
 0-00478 
 
 
 10 
 
 56 
 
 0-740 
 
 2426 
 
 
 
 0-32316 
 
 
 
 
 
 
 0-24699 
 
 
 11 
 
 15 
 
 0-740 
 
 684 
 
 .... 
 
 .... 
 
 0-07617 
 
 
 
 
 
 
 0-00099 
 
 
 12 
 
 20 
 
 0-601 
 
 741 
 
 
 .... 
 
 0-07518 
 
 
 
 
 
 0-01273 
 
 
 
 13 
 
 31 
 
 0-462 
 
 883 
 
 
 .... 
 
 0-08791 
 
 
 
 
 
 0-00250 
 
 
 
 14 
 
 52 
 
 0-323 
 
 1036 
 
 
 .... 
 
 0-09041 
 
 
 
 
 
 
 0-02686 
 
 
 15 
 
 78 
 
 0-2187 
 
 1052 
 
 
 .... 
 
 0-06355 
 
 
 
 
 
 
 0-01250 
 
 
 16 
 
 125 
 
 0-1492 
 
 1150 
 
 .... 
 
 .... 
 
 0-05105 
 
 
 
 
 
 
 0-00160 
 
 
 17 
 
 180 
 
 0-1145 
 
 1271 
 
 .... 
 
 .... 
 
 0-04945 
 
 
 
 
 
 0-00340 
 
 
 
 18 
 
 305 
 
 0-0797 
 
 1500 
 
 .... 
 
 .... 
 
 0-05285 
 
 
 
 
 
 
 0-05285 
 
 
 1 
 
 515 
 
 0-045 
 
 1430 
 
 .... 
 
 .... 
 
 
 
 Totals 
 
 
 
 0-34657 
 
 0-34657 
 
 
 This formula has been used for calculating the entropy 
 throughout the whole of the cycle, because the expansion 
 and compression curves on the p . v diagram do not approach 
 
STIRLING'S HOT-AIR ENGINE. 101 
 
 sufficiently to a simple curve of the nature Pj. V x n = P 2 V 2 n . 
 The mean value of n for the expansion period, from 15 per 
 cent of the stroke to the end (points 5 to 10, fig. 39), is 11894 ; 
 but its value varies very much during the expansion. The 
 mean value of n for the compression period (from points 11 
 to 1) is 1*2629, but that also varies a good deal. The mean 
 indicator diagram from the motor cylinder, shown in fig. 39, 
 is reproduced from the translation of Professor Schroter's 
 trial, as published in the Engineer of October 15th, 1897. 
 All the figures for calculating the entropy are given in 
 Table XVIII., and it is satisfactory to note that in this case 
 the positive entropy exactly counterbalances the negative, 
 a good proof of the accuracy of the calculations. The mean 
 effective pressure of the diagram in fig. 39 is 114 '5 Ib. per 
 square inch, which equals 12,200 foot-pounds of work done 
 per explosion. The area of the entropy diagram in fig. 40 is 
 equivalent to 11,900 foot-pounds of work per explosion, or 
 2 per cent less than that shown by the p . v diagram. This 
 difference is probably due to the various assumptions made 
 in calculating the specific heats. These calculations do not 
 include the power absorbed by the air pump, which must be 
 taken into account when calculating the net L.H.P., to 
 obtain the consumption of oil. 
 
 31. APPLICATION TO STIRLING'S HOT-AIR ENGINE. 
 
 Air engines may be divided generally into two classes 
 (a) those in which the air is heated at constant volume, 
 and (6) those in which it is heated at constant pressure. 
 Of the former class, Stirling's engine is perhaps the 
 most common example ; and the cycle of operations for 
 such an engine is shown on the pressure volume diagram 
 in fig. 41, and on the temperature entropy diagram 
 in fig. 42. In this engine the first operation is to 
 admit a quantity of heated air at a temperature r x from 
 the regenerator to the motor cylinder, and expand it 
 isothermally at the higher temperature r L from A to B. 
 The loss of heat due to the work done during expansion is 
 
102 
 
 STIRLING'S HOT-AIR ENGINE. 
 
 repaired by an external furnace, so as to keep its temperature 
 constant during the stroke. At B the air is^passed through 
 the regenerator, where it deposits some of its heat ; its tem- 
 perature falling from r^ to r 2 , and its pressure from B to C 
 at constant volume. AtfC communication withlthe regene- 
 rator is closed, and the cooled air is compressed! isothermally 
 at the lower temperature r 2 , indicated by the curve C D ; 
 and, finally, it is passed through the regenerator again, to 
 take up its deposited heat at constant volume ; its tempera- 
 ture rising from r 2 to r lf and'its pressure from D to A. The 
 
 FIG. 41. 
 
 thermal changes which take place will be better understood 
 by reference to fig. 42, which shows the entropy diagram 
 for the same cycle lettered similarly to fig. 41. Here the 
 straight lines A B and C D represent the isothermal expan- 
 sion and compression at r r and r 2 respectively ; and B C and 
 D A the constant volume curves, whose equation is : 
 
 = C v x log " 
 
 where = change of entropy between any two 
 
 temperatures T I and r 2j 
 and C v = specific heat of air at constant volume 
 
 = 01686 B.T.U. per Ib. 
 
 The heat supplied to the air during expansion is repre- 
 sented in fig. 42 by the area A B b a ; and that deposited by 
 
STIRLING S HOT-AIR ENGINE. 
 
 103 
 
 the air in the regenerator by c C B 6. That rejected during 
 compression is shown by the area d D C c ; and that taken 
 up in the regenerator by the air at the end of the cycle by 
 the area dDAa. The net work done during one cycle is 
 represented in heat units by the enclosed area A B C D, and 
 the net heat supplied by the furnace (excluding the regenera- 
 tor, which only acts as a reservoir of heat) being A B 6 a, the 
 efficiency of the cycle will be the ratio 
 
 ABCD 
 
 Atiba ' 
 
 But the two curves D A and C B are similar that is, they 
 are both logarithmic curves of the same equation between 
 
 > I fro/try 
 
 the same temperature limits, and therefore the area D A F 
 will be equal to the area C B E. This must be so, because 
 the amount of heat received from the regenerator is equal to 
 the amount deposited in it, radiation being neglected. The 
 area ABCD, representing the work done, will therefore be 
 equal to the area of the rectangle F A B E ; and the efficiency 
 becomes equal to 
 
 FABE r- 
 
 Thus it can be geometrically proved without any mathe- 
 matics that the Stirling hot-air engine has theoretically a 
 maximum possible efficiency, being equal to that of the 
 Carnot reversible cycle. 
 
104 
 
 ERICSSON'S HOT-AIR ENGINE. 
 
 32. APPLICATION TO ERICSSON'S HOT-AIR ENGINE. 
 
 The other class of air engines, viz., those that receive heat 
 at constant pressure, can be similarly treated. For example, 
 take the cycle of Ericsson's engine, shown with pressure and 
 volume as ordinates in fig. 43, and temperature entropy 
 ordinates in fig. 44. In this case, the first stage is admission 
 from A to B of hot air at constant pressure, with its 
 consequent reception of heat, represented in fig. 44 by 
 AB6. Then expansion isothermally at the higher tem- 
 perature T I from B to C, followed by the exhaust at constant 
 pressure from c to D, the heat rejected being equal to 
 
 FIG. 43. 
 
 c C D d ; and, finally, isothermal compression from D to A 
 completes the cycle. The curves AB and CD, fig. 44, 
 follow the equation 
 
 = Cp x loge , 
 
 T 2 
 
 where Gp = specific heat of 1 Ib. air at constant pressure = 
 0-2375. 
 
 The net work done per cycle and per pound of air is the 
 area enclosed by A B C D, and the net heat supplied, exclusive 
 of that received from and given to the regenerator, is 
 
 6 B C c ; therefore the efficiency of the cycle is 
 
 ABCD 
 
 But 
 
 6BC.c 
 the area a A B J must be equal to the area d D C c ; and the 
 
DIAGKAM FOB, REFRIGERATORS. 
 
 105 
 
 work done, or A B C D, is therefore equal to E B C F ; or the 
 efficiency is 
 
 EBCF = r, - r 3 
 
 6BCc T i 
 
 as in the case of the Stirling engine. 
 
 JSflt/vyby 
 
 FIG. 44. 
 
 ^33. ENTROPY DIAGRAM FOR REFRIGERATORS. 
 The cycle of operations in refrigerators is exactly the 
 reverse of that in the Garnot hot-air engine. Instead of 
 taking in heat at a high temperature r it and transforming 
 part of it into work, and rejecting the remainder at a lower 
 temperature r 2 , as in the heat engine, the working substance 
 in the refrigerator receives its heat at the lower temperature 
 r 2 , and discharges it at a higher temperature r 1 , the extra 
 energy required being obtained from external work done on 
 the gas. The theoretically perfect cycle that is reversible is 
 shown in fig. 45 with pressure volume ordinates, and in fig. 
 46 with temperature entropy ordinates. The first stage of 
 the cycle, A to B, consists of the adiabatic expansion of a 
 certain quantity of air, the temperature falling from T I to r 2 . 
 From B to C the expansion is continued isothermally at 
 constant temperature r 2 , the air receiving heat from the body 
 which it is desired to cool, the amount of heat abstracted 
 
106 
 
 DIAGRAM FOR REFRIGERATORS. 
 
 being equal to the area E B C F, fig. 46. Compression com- 
 mences at C, and is at first carried on adiabatically at 
 constant entropy (or isentropically) from C to D, the 
 
 Volume 
 
 FIG. 45. 
 
 temperature rising from r 2 to r lf and is finally completed by 
 isothermal compression from D to A, at constant temperature 
 T lt a quantity of heat being rejected to the water jacket 
 equal to F D A E. The heat expended in the process is the 
 equivalent of the work done on the gas, and is equal to the 
 area A B C D in both diagrams. The heat absorbed from 
 the substance to be cooled is equal to the rectangle E B C F, 
 fig. 46, and the efficiency, therefore (in its thermo-dynamic 
 sense), is equal to the ratio 
 
 EBCF = _jrj_ 
 
 AtiCD r x - r 2 ' 
 
 It is thus seen clearly how the efficiency is increased bv 
 reducing the difference of temperature between ^ and r 2 y 
 and as the ratio 
 
 may sometimes be greater than unity, it is better known as 
 the "coefficient of performance" (see Howard Lectures, by 
 Professor Ewing, on " The Mechanical Production of Cold," 
 Society of Arts, 1897). 
 
DIAGRAM FOR REFRIGERATORS. 
 
 107 
 
 The series of operations in air refrigerators with an open 
 cycle is somewhat different, and is shown in figs. 47 and 48. 
 In this case the air is taken from the cold room and com- 
 
 FIG. 46. 
 
 pressed adiabatically [from A to B. It is then cooled at 
 constant pressure, the temperature falling from B to C, 
 
 fig. 48, and contracting in volume from B to C, fig. 47, after 
 which it is passed into the expansion cylinder, where it 
 expands adiabatically from C to D, and is discharged to the 
 
108 
 
 DIAGRAM FOE, REFRIGERATORS. 
 
 cold room again. The work done on the air in the compres- 
 sion cylinder is equal to the area E B A F, fig. 47, or G C B H, 
 fig. 48, and that done by the air in the expansion cylinder is 
 equal to ECDF, fig. 47, or GDAH, fig. 48; so that the 
 net external work required is the difference of these two 
 quantities, represented by the area enclosed by A B C D in 
 
 FIG. 48. 
 
 both diagrams. The efficiency of the process will be repre- 
 sented by the ratio of the two areas, 
 
 ECDF fig 47- 
 EB~AF }t1 ' 
 
 but, as A B and C D are similar adiabatic curves, this will be 
 equal to the ratio 
 
 EC FD 
 
 E B Or F A 
 
APPENDIX. 109 
 
 APPENDIX. 
 
 TABLE OF THE WEIGHT OF DRY SATURATED STEAM. 
 
 HAVING been engaged for some time past in carrying out 
 and calculating the results of steam engine and boiler trials,, 
 the author has keenly felt the want of a more complete table 
 of the properties of saturated steam than those hitherto 
 published ; one which would give the temperature, density, 
 and total and latent heats for very slight intervals of 
 pressure, in order to save the inconvenience of so much 
 interpolation. In furtherance of this object, he compiled 
 some time ago a new Table of the Density of Dry Saturated 
 Steam, taking the well-known table of Prof. Dwelshauvers 
 Dery, of Liege, as a basis, and extending it so as to give 
 directly the density for every T Vlb. difference of pressure, 
 with a list of additional weights which may be added to give 
 the density for every i^lb. difference in pressure; the 
 latter, of course, only being used when extreme accuracy is 
 required. It is published now for the first time, with the 
 hope that it may be of some service to others engaged in 
 similar work. 
 
110 
 
 APPENDIX. 
 
 
 P 
 
 CN ^ SS 
 
 
 S 
 
 S 
 
 3 
 
 
 
 
 1 
 
 S 
 
 
 S 
 
 g 
 
 S 
 
 
 t 
 
 cp 
 
 * 
 
 
 r^ 
 
 S 
 
 S 
 
 
 #c5 
 
 
 t>- - <o 
 
 
 
 
 u- 
 
 ^J 
 
 - 
 
 si 
 
 9 
 
 r- rH i-H 
 
 
 
 
 1-1 
 
 1 
 
 si 
 
 | 
 
 rS S 3 
 
 
 - 
 
 S 
 
 <M 
 
 
 rs, 
 
 
 
 i-H i-l O 
 
 
 o 
 
 o 
 
 O 
 
 
 4 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 | 
 
 9 
 
 OO CO OO 
 
 
 00 
 
 * 
 
 fc- 
 
 
 P 
 
 S 
 
 <0 iO 
 
 
 o 
 
 iT, 
 
 
 
 
 <1 
 
 
 
 
 
 
 
 
 
 9 
 
 CO CO CO 
 
 
 M 
 
 (N 
 
 <M 
 
 
 
 05 
 
 S S S 5 
 
 I 
 
 a 
 
 i S 
 
 S S S 
 
 ill 
 
 
 O 
 
 99999 
 
 9 
 
 9 
 
 9 9 
 
 999 
 
 999 
 
 
 
 00 
 
 b 
 
 (M ^ Oi i-H Oi 
 
 10 r~ o co 10 
 
 
 
 00 
 
 9 
 
 | 
 
 1 1 
 
 O5 t- fM 
 111 
 
 o c-. i 
 
 ill 
 
 a/ 
 
 
 
 
 
 
 
 
 & 
 
 .0 
 
 ^ 
 
 1 1 I S 
 
 00 
 
 -Z 
 
 CO O 
 
 sis 
 
 <M ITS f~ 
 
 S i* 
 
 p p p p p p p 
 
 O S 00 
 
 9 S o S 
 
 9 
 
 1 1 
 
 ^ e 2 
 
 S 5 ? 
 
 999 
 
 999 
 
 * O T^" 
 
 Ok eS 3 
 
 p o p p b 
 
 8 
 
 CO OO t- 
 
 S 1 
 
 999 
 
 99999 
 
 S S 2 
 
 999 
 
 o o o o 
 
 co S S i f2 S 
 
 ^ Oi I-H -f -O OS 
 
 <M ^1 CO CO CO CO 
 
 p o o o 
 
 9 9 
 
 9 P 
 
 9999 
 
 I i I 1 I 
 
 99999 
 
 P- r- to --6 To >o LO 
 
 (M CN <M (M <M (M f-1 
 
 S O O p p p O 
 
 00 i- 10 >*< 
 
 ^ S S S 
 
 9 P 9 9 
 
 S CO" S 
 
 9999 
 
 -H ,-H C5 
 
 999 
 
 . >O 
 
 aad 
 
 -ut 'bs 
 ' 
 
 10 O 1- CO 05 
 
APPENDIX. 
 
 Ill 
 
 S S 
 
 < Oi CO 
 
 9 3 
 
 S 
 
 S c 
 
 S co 
 
 p p 
 
 p p 
 
 O CO CO r-i CO 
 
 S r 
 
 P P P 
 
 P P 
 
 I P I 
 
 1- H OO i-i 
 
 'p 'p 'p S 
 
 S 
 
 i I I 
 
 O CO i- rH -* l~-- 
 
 i 1 I I i 
 
 p P 
 
 m <s> eo 
 
 ^ o co co J-H ^ 
 i co o co o co 
 ? P P P P 
 
 CO 00 
 
 S 
 
 P 
 
 o co <=> 55 
 
 ?O O l~ 1~ 
 
 p p p p 
 
 K g g 
 
 p p p 
 
 p p p 
 
 i 1 e i g 
 
 p p p p p p 
 
 P P 9 
 
 S S S 
 
 S ii s 
 p p p p 
 
 O Oi CO CO ^ -*i 
 
 I 1 1 R S S 
 
 50 ?D O O 1- I- 
 
 P P P P P P 
 
 g i i 
 
 US O 
 
 S 
 
 p p p 
 
 3 3 
 
 o . ' ' 
 
 co io So 
 
 ? p p 
 
 
 
 
 
 S S 
 
 
 
 O <M "* 
 
 5 % 
 p P 
 
 1^ CO O 
 
112 
 
 APPENDIX. 
 
 o i 
 
 g 
 
 I 
 
 O CC 
 
 ^ 
 
 S 
 
 g 8 
 S S 
 
 
 
 :- 
 <j 
 CQ "S 
 
 ^ 
 
 . . CQ O CO O G^l 
 
 2SSSS^ 
 
 i i I ? i 
 
 
 CO C-. 
 
 
 
 S ff 
 
 2 2 
 
 I g 2 S 
 
 S fe 
 
 <M <M i-i O 
 
 i- TT< o-i 
 
 S 
 
 So 5 3 
 
 o o 
 
 CO O 
 
 O CO 
 r^ CO 
 
 s g 
 
 M (M <N 
 
 
 
 sqt? 'ui -bs 
 asd -sqa 
 
 t-- CC <T. O rH (M 
 
APPENDIX. 
 
 113 
 
 CO I-H iO 1~ O 1 - i-H CO 
 
 S S fe 2 5! S 8 
 
 * O Oi 1-1 
 i-H CO O CO 
 CM <M CM CM 
 
 CO --O 
 
 CO >O CO 00 rH 
 
 lO ' 1^- Oi rH <* 
 
 70 O CO O 
 
 TTt Tt^ TJ1 TJ1 O 
 
 O IO CO CO 1~ 
 04 TTI 50 01 ^< 
 
 <M (M CM 
 
 O 1- O 
 CO O l~ 
 
 S S 
 
 CM Tt CO CO 
 
 00 <M -H I- O 
 
 g S S5 
 
 O 1- O3 r- 
 
 S S? S 
 
 t- 35 CM O CO i-l 
 CM ^* i Oi r-* ^t 1 
 
 CO CO CO CO -^ 
 
 e s ^2 
 
 3 co- ^ 53 5! 
 
 CM CM OJ CM CM (M 
 
 II I I I I 
 
 CM CM CM CM (M 
 
 s g i - a 
 
 I-H -^ 
 
 i- S 
 
 00 Oi O (M 00 -^ O 
 
 OOOiOiOOfcOiOiO 
 
 9T 
 
114 
 
 APPENDIX. 
 
 I i 
 
 1- I 
 
 ? I 
 
 j 2 
 
 BI 
 
 sq ui-bs 
 aad -sqq 
 
 ao oo oo 
 
 *M i< O 
 
 <N <M 
 
 1- CC CO 
 
 S VI 5 
 
 1^ CO CO OO 
 
 CO OO 
 
 
 ^ ^ 
 
 CO O CC O TJ * 
 1- t~ 1- CO GO CO 
 
 s s 
 
 C-. ^H 
 
 to 1- 
 
 s s 
 
 i- O C: CO I- 'O 
 CO <" O C3O O "T-l 
 
 ?! {: {^ S S 
 
 01 01 01 C. 
 
 2 S S S S 
 
 00000 
 
 p <p p p p 
 
 C^ (M <M ^t 
 
 i -M CO 
 
APPENDIX. 
 
 115 
 
 5 2 5 2 
 
 O CO i-l 
 
 !O CM O5 
 
 O J- GO 
 
 g S 
 
 9 
 
 <N (M (M CM 
 
 ".D *f rH :TJ t 
 
 S 5 3 S8 3 
 
 
 
 I 2 i 
 
 3 1 
 
 IF ^ ^ 
 
 i 
 
 01 O -X' 
 
 s i 
 & 
 
 01 CM CM 
 
 2 
 
 3 3 
 
 e* CM 
 
 2 ?! ^ 3 
 
 GO LO CM 
 
 s g 
 
 2 2 S 
 
 s s 
 
 
 
 C^l C^l (M 
 
 ?2 22 
 
 c^ S ^ 
 
 CO 01 -. 
 
 H i- CO 
 
 I s 
 
 M CM CM 
 
 3 
 
 ^ 
 
 CM * 
 
 
 GO r. r- 
 
 
 
 ^ O O T-l O ^H 
 
 g s 
 
 CO CO CO 
 
 '.O SO 5O O O 
 
 S 8 
 
 0<I CM <M CM Cq 
 
 o o o o o 
 
 <? 9 <P 9> <? 
 
 CO I-H CO 
 
 sS5S^Ss 
 
 C^ g . 
 
 S 8 S 
 
 - S ,_, ,_, 
 
 CM CM (N 
 
 s g 
 
 T- '> CO 
 
 'a a 
 
 8 g 
 
116 
 
 APPENDIX. 
 
 z> I 
 
 9 I 
 
 1 
 
 
 C-ICq 
 
 C^JC^ 
 
 WEIGHT OF DRY SATURATED STEAM cont 
 
 - _. 
 
 Weight in pounds per cubic foot for each ,' Ib. presbure. 
 
 CO 
 
 o 
 
 s s 
 
 CM 04 
 
 s ? S i i 
 
 04 04 01 04 ?i 04 
 
 04 04 
 
 CO 
 
 s 
 
 (N 
 
 i 
 
 cS 
 
 i 
 
 
 
 
 P P 
 
 in o o o L- o 
 
 04 CO CO 
 
 
 W CM 
 
 CO 
 04 
 
 1 
 
 i 
 
 04 
 
 1 
 
 o 
 
 1 1 
 
 < * & ~* ft 
 
 ro C: i i 01 ^t 1 u~ 
 
 P s g P P 3 
 
 * CO 
 
 1 
 
 o 
 
 CO 
 
 1 
 
 04 
 
 P 
 
 o 
 
 s S 
 
 <N t <N t <N - 
 
 o i- Oi o <ri co 
 
 1 s 
 
 cS 
 
 CO 
 
 1 
 
 P 
 
 o 
 
 i 1 
 
 04 04 
 
 1 i S 1 I 1 
 
 -H 
 
 s 
 
 i 
 
 2 
 
 i 
 
 04 04 04 04 04 O4 
 
 04 04 
 
 04 
 
 9 
 
 r- 
 
 
 
 1 1 
 
 ^ ^J ^ ^ fi ^ 
 
 O4 Ol O4 O4 O4 O4 
 
 O CM 
 
 CO 
 
 04 04 
 
 CO 
 ^3 
 
 1 
 
 > 
 
 i 
 
 01 
 
 o 
 
 i I 
 
 CO CO CO CO CC CO 
 
 CO CO 
 
 P ? 
 
 P 
 
 I 
 
 ? 
 
 u? 
 
 1 
 
 b 
 
 S 
 
 04 04 
 
 C; i i O i i O I-H 
 1- Ci O 01 CO >O 
 
 i i i p p 1 
 
 i 3 
 
 1 
 
 1 
 
 s 
 
 1 
 
 ^1 
 
 Ji 
 
 .- o 
 
 >-- ,- 
 
 * 
 
 CO 
 
 CO 
 
 < 
 
 1 i 
 
 i i | 1 1 
 
 i 1 
 
 I 
 
 i 
 
 I 
 
 i 
 
 "5 
 
 1 i 
 
 fcO O O O C? O 
 
 ^ ^ 
 
 1 
 
 i 
 
 s 
 
 s 
 
 S 
 
 1 
 
 sq'B - ut 'bs 
 aad -sqq 
 
 cc cr. 
 
 Oi C". 
 
 -J 04 CO TT 
 O O O 
 
 s s 
 
 CO 
 
 o 
 
 9 
 o 
 
 o 
 
 3 
 
APPENDIX. 
 
 117 
 
 CO CO 
 
 1? 0> 
 
 8 g g g 
 
 ^ I'- 
 
 00 OO 
 
 to oc 
 
 P r* 1 CO 
 
 l i 
 
 53 
 
 
 CO O t- 
 Ci O I-H 
 
 <M O rH * 00 i-i 1< f~ O <M ^t< O 00 C 
 
 OS O CN CO *J O 1- GO O r I <M CO Tt< CC 
 
 GO i- CO O 1^ O rH CO SO GO O C 1 ! ^J< SC 
 
 S S S S S S S5 ^ feJ ^ S |i a S S3 ? 
 
 g 
 
 SSSS Sg^S^SSi2 
 
 iOl>-O-H COiO.t'-OfN'VOCXJOrM 
 
 O^O'O^OCOcoi'. J^-t 1--OOGOCOCOOO 
 
 C^ (N 71 <N C^ ?1 01 <N CQ (>1 ^ GVI (^ Cl y 
 
 ? 1 s i 1 
 
 3 
 
 -M >O 1- 
 
 rH >0 OS CO !>. 
 
 t- CO Oi i <M 
 
 10 l^ o.. fN TH 
 
 iO O iO <tO ^O 
 
 <>l CN <N (>) 01 
 
 Sc5 c5cNSl<:<ie^c<i7<ic<i<>)<fi 
 
 o ~i 
 
 i p 
 
 (M CM O4 
 
 S 
 
 CO <M <M 
 
 ill 
 
 C^l "* 
 
 S 5 
 
 (N <M 
 
 ^feS;^i5SS??S 
 
118 
 
 APPENDIX. 
 
 A Weight for each T J n lb. 
 pressure. 
 
 ^ 
 
 9 - 
 
 i- 2 
 
 2 
 
 o 
 
 - 
 
 s 
 
 2 
 
 p , -* 
 
 P 
 
 CN 
 
 Weight in pounds per cubic foot for each T ' a Ib. pressure. 
 
 b 
 
 g g | | | 
 
 *>J C 5 -^ 
 
 2 2 
 
 s 
 
 I 
 
 yf 
 S3 
 
 Ill 
 
 CO CO CO 
 
 g CO CO 
 
 CO 
 
 CO 
 
 CO 
 
 
 
 
 
 
 
 oo 
 b 
 
 | 3 | S g | 
 
 O .-< ?] 
 
 o 2 S 
 
 i 
 
 CO 
 
 i 
 
 8i M jg 
 
 
 co 
 
 CO 
 
 p 
 
 b 
 
 o O5 Oi O O O 
 
 C>1 <T1 <N CO CO CO 
 
 00 03 O 
 1- 00 
 1- OJ CM 
 
 O i 
 CO CO CO 
 
 
 
 1 
 
 CO 
 
 3 
 
 Cl j_l T'l 
 
 b 
 
 oo o >-t (h co ^M 
 
 Oi Oi Ci O O O 
 
 n o, co co co 
 
 111 
 
 1 
 
 CO 
 
 (M 
 
 00 
 
 5? 
 
 CM CO Tt< 
 
 b 
 
 ^ <^> OO rH CO kO 
 (M (M <M CO CO CO 
 
 11 = 
 
 CO CO CO 
 
 00 
 
 00 
 CO 
 
 o 
 Ic 
 
 CO 
 
 I 
 
 S 2 S 
 
 O 9? s 
 
 s s ^ 
 
 
 
 
 
 
 
 b 
 
 1 1 i 11 1 
 
 i 1 1 
 
 CO 
 
 p 
 
 CO 
 
 s 
 
 to 
 p 
 
 r-H fM Ol 
 
 es co co 
 
 T 5 
 b 
 
 CO O t^ Oi O CM 
 
 1 1 1 1 1 i 
 
 o 1 5 
 
 p 
 
 i 
 
 00 
 
 Cf> O i-l 
 03 ^ CO 
 
 P co 
 
 7 s 
 
 CO 
 
 V 
 
 p f- p 
 
 M 
 
 b 
 
 i 1 1 1 1 I 
 
 C-l CO < 
 O 00 O 
 
 i 
 
 
 
 g 
 
 00 O O 
 
 CO CO CO 
 
 CO 
 
 CO 
 
 CO 
 
 CO c. CO 
 
 b 
 
 1III11 
 
 i-H (M CO 
 S 00 
 
 1 
 
 1 
 
 CO 
 
 o 
 
 g 
 
 CO 
 
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APPENDIX. 
 
 119 
 
 
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120 
 
 APPENDIX. 
 
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APPENDIX. 
 
 121 
 
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122 
 
 APPENDIX. 
 
 A Weight for each T J n Ib. 
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APPENDIX. 
 
 123 
 
 t- (M 
 
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 2 
 
 g^'iissii 
 
 O O O O i I- t- 1- 1- 
 
 CO W CO S ^ 2? 
 O <M ^ ^) OO O 
 
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INDEX. 
 
 PAGE 
 
 Adiabatic Expansion Curve 32 
 
 Adiabatic Expansion Condensation 35 
 
 Adiabatic Expansion of Wet Steam 37 
 
 Air Calculations, Gas Engine 83 
 
 Air Engines, Diagrams for 101, 104 
 
 Aquene Curve 8 
 
 Areas of Diagrams, Comparison of 46 
 
 Balance Sheet, Heat, Steam Engine 72 
 
 Balance Sheet, Heat, Gas Engine 96 
 
 Boulvin'a (Professor) Complete Entropy Diagram .' . . 28 
 
 Carnot Cycle . 4, 34 
 
 Chart, Theta-Phi 2 
 
 Clearance Volumes 21 
 
 Clearance Surfaces 54 
 
 Coefficient ot Performance 106 
 
 Compari-on of Areas 46 
 
 Complete Entropy Diagram 28 
 
 Compound Engine, Diagram for '23 
 
 Compounding, Effect of 59 
 
 Condensation Coefficient 59 
 
 Condensation during Adiabatic Expansion 36 
 
 Condensation during Expansion 44 
 
 Condensation. Initial 49, 56, 63 
 
 Constant Volume Curves 15, 92 
 
 Constant Temperature Lines 30 
 
 Conversion of Indicator Diagram 21 
 
 Cotterill, Professor, on Steam Engine 19, SO 
 
 Critical Temperature 14 
 
 Cut-off, Most Economical 61 
 
 Density of Steam '. '22, 109 
 
 Diesel Oil Motor, Professor Schroter'a Test 97 
 
 Diesel Oil Motor, Temperature Calculations 98 
 
 Diesel Oil Motor, Entropy Diagram 99 
 
 Donkin, Mr. B., on "Experiments on Small Vertical Engine " 49, 57 
 
 Donkin, Mr. B., on "Gas, Air, and Oil Engines" 81 
 
 Dryness Fraction, Calculation 21 
 
 Dryness Fraction, Comparisons of 56 
 
 Efficiency, Thermal, Steam Engine 51, 69 
 
 Efficiency, Thermal, Stirling's Air Engine 103 
 
 Efficiency, Thermal, Refrigerators 106, 108 
 
 Engine, Compound, Diagrams for 23 
 
 Engine, Single-cylinder, Diagrams for 46 
 
 Engine, Triple-expansion, Diagrams for 39 
 
126 INDEX. 
 
 PAGE 
 
 Entropy 3 
 
 Entropy of Water 7, 10 
 
 Entropy of Steam 11 
 
 Entropy of Superheated Steam 46 
 
 Entropy of Gases 74 
 
 Entropy Diagram 5 
 
 Entropy Diagram for Ice, Water, and Steam 11 
 
 Entropy Diagram for Gas Engine 87 
 
 Entropy Diagram for O il Engine 97 
 
 Entropy Diagram for Air Engines 101, 104 
 
 Ericsson's Air Engine 104 
 
 Ewing, Professor, on "Mechanical Production of Cold " lOo 
 
 Exchanges of Heat 68 
 
 Exhaust Period, Gas Engine 91 
 
 Exhaust Products, Gis Engine 85 
 
 Exhaust Waste, Gas Engine 94 
 
 Expansion, Adiabatic 32, 35 
 
 Expansion, Most Economical 61 
 
 Expansion Period, Gas Engine 90 
 
 Gas, Coal, Analysis ' t 84 
 
 Gas, Coal, Specific Heat 84 
 
 Gases, Entropy of 73 
 
 Gases, Specific Heat of 73 
 
 Gas Engine, Theoretical 76 
 
 Gas Engine, Temperature Calculations 78 
 
 Gas Engine, Actual, 7 Horse Power hi 
 
 Gas Engine, Ideal Diagram 87 
 
 Gas Eugine, Corrected Diagram 89 
 
 Gray, Mr. M. F., on " Theta-PM Chare " 1 
 
 Heat Balance Sheet, Steam Engine 72 
 
 Heat Balance Sheet, Gas Engine 96 
 
 Heat, Latent 6 
 
 Heat Losses, Steam Engine 39 
 
 Heat Losses, Measurement of 68 
 
 Heat Losses, Gas Engine 94 
 
 Heat Recovery Lines 66 
 
 Heat Weight 3 
 
 High-pressure Cylinder, 6 Diagram 25 
 
 Indicator Diagrams Compared with & . . , , 3 
 
 Indicator Diagrams Converted to 6 21 
 
 Initial Condensation 4 ( J, f>5, 63 
 
 Introduction of Diagrams 1 
 
 Jacketing, Steam, Effect of 
 
 Latent Heat 6 
 
 Logarithmic Curves 79 
 
 Longridgtj Mr. M., 011 "Trials of a Compound Engine " 61 
 
 Losses of Heat 39, (58, 94 
 
 Low-pressure Cylindtr, <f> Diagram 26 
 
 Main Valve, Passage in 27 
 
 Mixture in Gas-engine Cylinder 82 
 
 Oil Engines, 6 Diagram for 97 
 
 Priming Water 63- 
 
INDEX. 127 
 
 PAGE 
 
 Quality Curves 21 
 
 Radiation, Gas-engine !4 
 
 Re-evaporation during Expansion 44 
 
 Refrigerators, Closed Cycle 105 
 
 Refrigerators, Efficiency 106, 108 
 
 Refrigerators, Open Cycle 107 
 
 Regenerator, Effect of 103, 104 
 
 Ripper, Prof. , on " Superheated Steam-engine Trials " 49 
 
 Sankey, Captain, on " Marine-engine Trials " 2 
 
 Scales for Kntropy Diagram 7 
 
 Schroter, Prof., on " Diesel Oil Motor " 97 
 
 Specific Heat, Water 10 
 
 Specific Heat, Superheated Steam 46 
 
 Specific Heat, Coal Gas 84 
 
 Specific Heat, Gas-engine Mixture 86 
 
 Specific Volume, Steam 17 
 
 Speed, Effect of 50 
 
 Standard Engine of Comparison 09 
 
 State Points 35 
 
 Steam, Entropy Diagram for r,, 11 
 
 Steam Jacketing, Effect of 3V) 
 
 Stearn, Superheated, Diagram 46 
 
 Steam, Superheated, Effect of 49 
 
 Hteam, Wet, Expansion of 37 
 
 Stirling's Air Engine 101 
 
 Superheated Steam, 6 $ Diagram for 40 
 
 Superheated Steam, Entropy of 48 
 
 Superheated Steam, Effect of 49 
 
 Surfaces, Clearance 54 
 
 Temperature of Combustion 94 
 
 Temperature, Constant, Lines SO 
 
 Temperature, Critical 14 
 
 Temperature in Gas Engine 80 
 
 Temperature in Oil Engine 98 
 
 Theoretical Gas Engine 76 
 
 Theoretical Temperature, Gas Engine 94 
 
 Thermal Efficiency, Steam Enurine 51, 69 . 
 
 Thermal Efficiency of Carnot Cycle 69 
 
 Thermal Efficiency of Rankine Cycle 69 
 
 Therm*! Efficiency of Stirling's Air Engine 103 
 
 Thermal Efficiency of Refrigerators 100, 108 
 
 Triple-expansion Engine' <f> Diagram 39 
 
 Volume Factor 27, 33, 45 
 
 Volume, Specific, Steam 17 
 
 Wall Action, Steam Engine 52 
 
 Wall Action, Gas Engine 94 
 
 Water, Entropy of 7, 10 
 
 Water, Specific Heat of ' 10 
 
 Weight of Steam . 22, 10i> 
 
 Wet Steam, E xpansioii of 37 
 
 Willans, Mr. P. W., on " Non-condensing Steam-engine Trfals " 2 
 
 Willans, Mr. P. W., on " Condensing Steam-engine Trials " 58, 70 
 
 Work Losses . . .44 
 
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