THERMODYNAMICS C. F. HIRSCHFELD A88T. PROP. OF POWER .: I CORNELL UNIVERSITY Second Edition Revised and Corrected NEW YORK , VAN NO STRAND COM PA N V 23 MURRAY AND 27 WARREN STREET* 1910 THE VAM AMO No. 11. THEORY OF ARCHES. By Prof. W. Allan. No. 12. THEORY OF VOUSSOIR ARCHES. By Prof. Wm. Cain. Third edition, revised and enlarged. THE; VAN NOSTRAND SCIENCE SERIES. No. 13. GASES Mf/r WITH IN COAL, MINKS. By J. J. Atkinson. Third edition, revised and enlarged, to which is added The Action of Coal Dusts by Edward H. Williams, Jr. No. 14. FRICTION OP AIR IN MINES. By J. J. Atkinson. Second American edition. No. 15. SKEW ARCHES. By Prof. E. W. Hyde, C.E. Illustrated. Second edition. No. 16. GRAPHIC METHOD FOR SOLVING Certain Questions in Arithmetic or Algebra. By Pi;of. G. L. Vose. Second edition. No. 17. \VATER AND WATER-SUPPLY. By Prof. W. H. Corfield, of the University Col- lege, London. Second American edition. No. 18. SEWERAGE AND SEWAGE PURIFI- cation. By M. N. Baker, Associate Editor "Engineering News." Second edition, re- vised and enlarged. No. 19. STRENGTH OF BEAMS UNDER Transverse Loads. By Prof. W. Allan, author of "Theory of Arches." Second edi- tion, revised. ^ No. 20. BRIDGE AND TUNNEL, CENTRES. By John B. McMaster, C.E. Second edition. No. 21. SAFETY VALVES. By Richard H. Buel, C.E. Third edition. No. 22. HIGH MASONRY DAMS. By E. Sherman Gould, M. Am. Soc. C. E. Second Edition. No. 23. THE FATIGUE OF METALS UNDER Repeated Strains. With various Tables of Results and Experiments. From the Ger- man of Prof. Ludwig Spangenburg, with a Preface by S. H. Shreve, A.M. No. 24. A PRACTICAL TREATISE ON THE Teeth of Wheels. By Prof. S. W. Robinson. 3d edition, revised, with additions. No. 25. THEORY AND CALCULATION OF Cantilever Bridges. By R. M. Wilcox. No. 2C. PRACTICAL TREATISE ON THE PROP- erties of Continuous Bridges. By Charles Bender, C.E. No. 27. BOILER INCRUSTATION AND CORRO- sion. By F. J. Rowan. Now edition. Re- vised and partly rewritten by F. E. Idell. No. 28. TRANSMISSION OF POWER BY WIRE Ropes. By Albert W. Stahl, U.S.N. Fourth edition, revised. THE VAN NOSTRAND SCI#NC No. 29. STEAM INJECTORS, THEIR THEORY and Use. Translated from the French by M. Leon Pochet. No. 30. MAGNETISM OF IRON VESSELS AND Terrestrial Magnetism. By Prof. Fairman Rogers. No. 31. THE SANITARY CONDITION OF CITY and Country Dwelling-houses. By George E. Waring, Jr. Second edition, revised. No. 32. CABLE-MAKING FOR SUSPENSION Bridges. B. W. Hildenbrand, C.E. No. 33. MECHANICS OF VENTILATION. By George W. Rafter, C.E.. Second edition, re- vised. No. 34. FOUNDATIONS. By Prof. Jules Gaiidarrt, C.E. Translated from the French. Second edition. No. 35. THE ANEROID BAROMETER: ITS Construction and Use. Compiled by G< W. Plympton. Tenth edition, revised and enlarged. No. 36. MATTER AND MOTION. By J. Clerk Maxwell, M.A. Second American edition. No. 37. GEOGRAPHICAL, SURVEYING: ITS Uses, Methods, and Results. By Frank De Yeaux Carpenter, C.E. No. 38. MAXIMUM STRESSES IN FRAMED Bridges. By Prof. William Cain, A.M., C.E. New and revised edition. No. 39. A HANDBOOK OF THE ELECTRO- Magnetic Telegraph. By A. E. Loring. Fourth edition, revised. No. 40. TRANSMISSION OF POWER BY COM- pressed Air. By Robert Zahner, M.E. New edition, in press. No. 41. STRENGTH OF MATERIALS. By Wil- liam Kent, C.E., Asspc. Editor "Engineering News." Second edition. No. 42. THEORY OF STEEL-CONCRETE Arches, and of Vaulted Structures. By Prof. Wm. Cain. Fourth edition, thoroughly re- vised. No. 43. WAVE AND VORTEX MOTION. By Dr. Thomas Craig, of Johns Hopkins University. No. 44. TURBINE WHEELS. By Prof. W. P, Trowbridge, Columbia College. Second edi- tion. Revised. ENGINEERING THERMODYNAMICS C. F. HIRSHFELD PROFESSOR OF POWER ENGINEERING CORNELL UNIVERSITY Second Edition Revised and Corrected NEW YORK D. VAN 1STOSTRAND COMPANY 23 MURRAY AND 27 WARREN STREETS 1910 COPYRIGHT, 1907 BY D. VAN NOSTKAND COMPANY FOREWORD. THERMODYNAMICS is, like most other sciences, built upon and about a very small number of fundamental concepts. The average student, however, in his first attempts to obtain a working knowl- edge of this science is staggered by the enormous superstructure, and fails to observe and fix the few fundamentals upon which that superstructure rests. The following pages represent an at- tempt to develop the principal Thermo- dynamic Properties of Gases and of Vapors in such a manner that the under- lying principles may be clearly recognized. It is thought that a thorough under- standing of the laws developed in the text 360476 IV will, on the one hand, give a working knowledge sufficiently extensive for most engineering purposes, or, on the other, will enable the student to more easily follow the generalized and complicated cases considered in the large standard works on the subject. CONTENTS. CHAPTER I. PAGE HEAT 1 CHAPTER II. GASES 18 CHAPTER III. ENTROPY 44 CHAPTER IV. . CYCLES 50 CHAPTER V. FLOW OF GASES 84 CHAPTER VI. VAPORS 93 CHAPTER VII. EXPANSIONS AND COMPRESSIONS OF VAPORS 118 CHAPTER VIII. VAPOR CYCLES 127 CHAPTER IX. FLOW OF VAPOR 154 v ENGINEERING THERMODYNAMICS. CHAPTER I. HEAT. THE historical experiments of Rumford and Joule proved heat to be a form of energy; and the work of practically all investigators in this field has shown the probability of heat being some form of molecular energy; that is, energy in some way connected with the motions and positions of the constituent particles of the various masses with which it is associated. UNIT OF MEASUREMENT. Heat as a form of energy may be measured in properly chosen units just as any othev form of energy may be measured. In practically all cases of energy measurement the units are based upon one or more of the effects produced upon a given body under given circum- stances by the application of the par- ticular form of energy considered. A rise in temperature is one of the most noticeable phenomena connected with an increase of heat, and, since it happens to be an effect the extent of which is comparatively easily measured, it is generally used as the one upon which to base the measurement of heat energy. The unit used by engineers in English- speaking countries is called the British Thermal Unit and is defined in terms of the amount of heat required to cause a unit temperature change in a unit weight of water. Thus the British Thermal Unit (B.T.U.) is the quantity of heat required to raise the temperature of one pound of water erne degree on the Fahrenheit scale. It has been discovered by experiment that it requires different amounts of heat to produce this effect at different initial temperatures, and the unit is variously given for different points on the tempera- ture scale. The most common is probably based upon a change of temperature from 59 to 60. For exact determinations this temperature rise should be measured by means of a standard thermometer with gas for the expanding substarce, but for all ordinary engineering purposes measure- ments may be made with a mercury thermometer. Further, the amount of heat per degree per pound may, for most engineering purposes, be considered con- stant at all points of the temperature scale. FIRST LAW OF THERMODYNAMICS. Experience has shown that by means of proper apparatus any form of energy can be converted into other forms, and that during this process absolutely nothing is lost. Heat energy and mechar.ical energy are thus interconvertible and, since nothing is lost in the conversion, one unit of the one must always give a certain number of units of the other. The ratio of conversion has been deter- mined by Joule and other experimenters, and the results of their work may be expressed as a law; thus, Heat and Mechanical Energy may be converted the one into the other, and the ratio of con- version is approximately one heat-unit to 778 mechanical units. That is: 1 B.T.U. = 778 foot-pounds. (1) The value 778 is called Joule's equiva- lent and is represented by J. This law is generally called the " First Law of Thermodynamics." It is a very simple matter to completely convert mechanical energy into heat energy, as is exemplified in the heat generated by friction of moving bodies and in other similar phenomena, but the change in the opposite direction is by far more difficult of accomplishment. As will be seen later, the apparatus necessary to convert heat into mechanical forms of energy is complicated and at best does not even theoretically convert all of the heat energy supplied it. That which is converted, gives of course 778 foot-pounds per unit, but much of it is rejected by the apparatus still in the form of heat, and, what is worse, under such conditions as to be waste heat. SECOND LAW OF THERMODYNAMICS. Study of all the known heat phe:;ome:"a of the universe has shown that heat passes readily from bodies of high temperature to bodies of low temperature, but that external energy of some kind is necessary to make it pass in the reverse direction. So important is this fact that it is called the " Second Law of Thermodynamics." HEAT-ENGINES. The apparatus previously mentioned for the conversion of heat into mechanical forms of energy takes various shapes, dependent upon the circumstances of its use, but they may all be called Heat- engines. They are of particular im- portance in the economics of this world, because, while man finds immerse stores of heat tied up in latent form in the various fuels and easily liberated by the chemical processes of combustion, the mechanical motion necessary in his various industries does not occur just where and in just the manner needed. He is therefore driven by necessity to con- vert that which he has into that which he requires and he does it in some form of heat-engine. The theoretical performances of these heat-engines are studied by means of the - laws of Thermodynamics, a science built upon the two laws just given. It has for its object the determination of all the phenomena connected with the transfor- mations of heat into mechanical energy and vice versa. The conversion taking place in all heat-engines depends upon the behavior of various materials when sub- jected to different kinds of heat treat- ment. These materials are called work- ing substances and are usually contained in some sort of confining vessels with proper mechanism to transmit the motions which they * make available. They are generally gaseous substances for practical reasons, though so far as the theoretical side of the operations is concerned they might be solid, liquid, or gaseous. DISTRIBUTION OF HEAT ENERGY ASSO- CIATED WITH ANY SUBSTANCE. It is a matter of common experience that when substances are heated various phenomena take place : temperatures rise, volumes change, solids melt, liquids vaporize, and all sorts of other familiar changes occur. When these are all studied side by side it is found that the results of changing the amount of heat associated with any substance may be grouped under three heads, irrespective of what its 8 characteristics are and what changes it may experience, provided it is not set in motion as a whole: 1. The temperature may change, de- scribed as a change of Sensible Heat. 2. Change may take place within the substance, necessitating the doing of Internal Work of a positive or negative character against internal forces. This would mean the ab- sorption or giving out of heat energy equivalent to work represented by the change. 3. The volume may change with a cor- responding displacement of the surrounding or inclosing medium, necessitating the doing of External Work of a positive or negative character and the transfer of an equivalent amount of heat. These may be easily pictured mentally if a body of any kind be assumed to be made up of molecules in rapid motion and under certain circumstances exerting some sort of interattraction. Then a change 9 of temperature may be imagined as a change in the speed of vibration of the molecules faster motion meaning higher temperature. An internal change may be imagined as a change of distance be- tween the molecules, necessitating a supply of energy *to overcome their interattraction when they are moved apart, and a supply of energy being liberated when they are allowed to approach. The change of volume so far as external or enveloping media are concerned will mean simply that these media are dis- placed under whatever pressure they exert upon the surfaces of the body chang- ing size. Since this means motion under a certain definite pressure, work is done. When the motion is such as to oppose the pressure, energy is absorbed, and energy is liberated when motion is in the same direction as the pressures exerted by the surrounding media. That is, when the volume increases, energy is absorbed to displace the enveloping medium. When the volume changes from large to small, 10 energy is given out to the same extent as it was absorbed at some previous time when the volume attained the larger value. Since these changes are due only to the change of the amount of heat associated with the substance, the sum of all the energy changes must represent the amount of heat charge; thus, algebraically, Q = S + I + E, (2) where Q = amount of heat absorbed or given out, and, according to the molecular theory, $=the amount of heat necessary to make the molecules move faster when a rise in tempera- ture takes place, or the amount given out when their motion becomes less rapid during a drop of temperature; 7= amount of energy absorbed or given out by internal changes of molecular distances; E = amount of energy absorbed or given out by the external 11 media during an increase or decrease of the volume of the substance under considera- tion. Since the amounts of energy represented by S and 7 are really stored within a substance when heated, the total amounts of such energy present in the body are called its intrinsic energy. Then S+I in a change may be called the change of intrinsic energy. It should be clearly understood that under certain circumstances all of these energy changes may not occur, while under other conditions some may be positive while others are negative. Whatever the conditions, unless the condition of motion of the body as a whole is changed, ex- pression (2) is always true if the proper signs and numerical values be used. This idea of the triple use of heat will become more evident and take on more meaning as the subject is further de- veloped. 12 SPECIFIC HEAT. Equation (2) shows that if a substance is caused to undergo a certain change by the addition of heat, the amount of heat required to cause that change in the same body will always be the same, just as, in any other kind of a change which takes place in the universe, it always takes the same amount of energy to do the same amount of work. It is con- venient to have names for the amounts of heat required to cause a unit amount of change of various kinds, and these are called thermal capacities. Thus there is v a thermal capacity for unit rise in tem- perature, and this is commonly called the Specific Heat. Similarly, there is a thermal capacity for unit change of volume and it is commonly called the Latent Heat of Expansion. There are several other thermal capacities, but they are of minor importance and have no commonly ac- cepted names. As just indicated, the Specific Heat of a 13 body is the amount required to cause unit temperature change in unit weight, and, if the specific heat is constant at all temperatures, SrT=C, (3) where T% and T\ represent final and initial temperatures; W= weight of substance in pounds; and C= Specific Heat. This expression may be written Q = C7TF(r 2 -r 1 ); (4) and if the specific heat is variable, then /" T Q=W *CdT. (5) J*v From equation (2) it can be seen that it would be possible for Q to have different values for the same change in tempera- ture if that temperature change could be accompanied by different amounts of internal and external work. Therefore 14 in equation (4) the value of C may vary with the conditions of heating; that is, the value of the specific heat may vary according to the conditions of heating, and every substance may have several specific heats. This is actually the case, as will be shown in the development of the equations for the behavior of gases. Of all the numerous possible values for the specific heat of any single substance, it would seem possible to find one which represented the heat necessary to cause a change in S only. That is, one which represents the quantity of heat necessary to cause a change of temperature without any other heat-absorbing phenomena. Such a one would be called the True Specific Heat and would represent only the heat necessary to increase the velocity of vibration of the molecules during a rise in temperature of one degree. Such a specific heat will be considered in dealing with gases. 15 EXTERNAL WORK DURING VOLUME CHANGES. It is often necessary or desirable for purposes of analysis to divide the heat associated with changes in the condition of a bocly; that is to say, how much of it represents change of sensible heat, how much represents internal work, and how much external work. For this purpose an expression for the value of the external work is very necessary, and such an ex- pression may be easily developed. Assume any substance enclosed, as in Fig. 1, in the end of a cylinder behind FIG. 1. a moving piston of area a square feet and under a pressure of P pounds per 16 square foot. If the body be allowed to expand while the pressure is maintained constant, it will drive the piston out through a distance which may be called L feet. The expanding substance will the i be doing external work, that is, work upon the confining piston, and the value of the external work in B.T.U. will be aPL but aL is the volume through which the piston moves, that is, the change of volume of the expanding substance, and if V\ and 7 2 be initial and final volumes in cubic feet, P(F 2 -y!) E= -778 ' Similarly, if the pressure be variable, each infinitesimal volume change being multiplied by the pressure existing while it took place and then all the small values of E being added together. 17 It is readily seen that equations (6) and (7) will apply equally well to cases where the volume change takes place in all three possible directions and they are therefore expressions for external work done during any kind of volume changes. CHAPTER II. GASES. EXPERIMENTAL investigations into the behavior of gases under different condi- tions show that they all approximately obey very simple laws. None of them exactly obey these laws, but the farther the gases are removed from their points of liquefaction the nearer do the laws express their characteristics. Such being the case, it has become customary to consider an ideal " Perfect Gas" and to develop the laws which it would obey, and then so far as necessary to study the variations of the real gases from those laws. For all practical engineering pur- poses all of the ordinary so-called " per- manent gases " may be assumed as perfect gases. 18 19 LAWS OF THE PERFECT GAS. A Perfect Gas may be defined as a substance existing in such a physical state that its constituent particles (mole- cules) exert no interattraction. Such a body would perfectly obey the laws of Boyle and Charles. Boyle's Law states that when the tem- perature of a gas is constant, its volume mill vary inversely as the pressure upon it; that is, %-% or ViPi = P 2 V 2 = Si constant. (9) Charles' Law states that with the pressure or the volume constant, the volume or the pressure of a gas will vary 1-$ of its volume at 32 F. for every Fahrenheit degree change of temperature, increasing in volume or pressure when the temperature increases. If a gas be imagined with unit volume at 32 F., it would lose T part of that volume for every degree drop in tempera- ture at constant pressure, and when it had dropped 493 degrees it would have zero volume if the law held at such low tem- peratures. Such a point on the temperature scale, which is (493-32 = ) 461 degrees below the Fahrenheit zero, is called the Absolute Zero, and temperatures measured from it are called Absolute Temperatures. If the absolute temperature be represented by T, Charles' Law may be very simply stated in algebraic form: When pressure is constant, =; do) \2 *2 When volume is constant, "rr == T>~~* (1-U JT2 2 This idea of absolute temperature is very useful in dealing with heat-engines. The absolute temperature of a body in Fahrenheit degrees may be obtained from 21 the ordinary temperature above Fahren- heit zero by adding 461 : !F = *.+ 461. (12) Apparently if real gases obey the law of Charles, they would have no volume at the temperature of absolute zero, but this absurdity need not be considered because they would all liquefy before reaching such a low temperature. That is, they would cease to even approxi- mately obey the law which leads to this absurdity. Boyle's and Charles' Laws may be com- bined to give a third and more general Law of Perfect Gases. If a volume of gas, Vi, at pressure PI and temperature TI have its temperature maintained constant while its pressure is changed to P', its new volume, V, will be given by equation (8): F'=^. (13) If then its pressure be kept constant while its temperature is varied to T2, 22 its new volume, V 2 > will be given by equa- tion (10) : and substituting for V from (13), Since P r is the same as P 2 , an expres- sion is derived which gives the value of the volume change for variations of both temperature and pressure: 2 -- p T , -TV 1 which may be written P V P V PV -jr^ =-|r-^ or -^-=a constant. (17) Where the value of V above is for one pound of gas, the constant is called R, and this gives PV ~r=R. (18) 23 Any of the equations under (17) or (18) is called the Law of Perfect Gases. SPECIFIC HEAT OF PERFECT GASES. A material constituted as is this ideal gas offers absolutely no resistance to sepa- ration of 'its constituent particles, since they exert no attraction upon one an- other, and therefore when it is heated no energy is absorbed for the purpose of doing internal work. Hence in this case one pound of mate- rial when heated would require an amount of heat (19) If such material is heated at constant volume, the pressure rises directly as the absolute temperature, as shown by the law of perfect gases, but no external work is done because the enveloping medium is not displaced , and under such condi- tions Q=S. (20) 24 In such a case the " Real or True " Specific Heat can be found by dividing Q by the change of temperature, provided that specific heat is constant with variable pressure and variable temperature. If it varies, the quotient will only be the average real specific heat. The specific heat obtained under the conditions of constant volume as above is designated by C v when expressed in Thermal Units, and by Ky when ex- pressed in foot-pounds. Then (21) In the case of the ideal gas there is no theoretical reason why C* should not be constant, that is, why it should not re- quire exactly the same amount of heat to raise the temperature of one pound of a given gas one degree with volume con- stant no matter what the temperature, pressure, and volume. If this quantity, C v , obtained by di- viding S by the change of temperature of 25 the body while receiving heat S, is a constant, it follows that the total sensible heat or intrinsic energy possessed by a gas must depend upon its temperature only. Thus if a gas have its tempera- ture increased n degrees, its intrinsic energy will have been increased by nC v heat-units, and if it have its temperature increased n f degrees, its sensible heat or intrinsic energy is increased by n'C v heat-units. That the intrinsic, energy of real gases may be considered to depend upon temperature only, and to be inde- pendent of volume and pressure, is shown by an experiment devised and performed by Joule and Thomson. Their experiment was performed by means of two vessels connected by a passage containing a valve. One vessel was filled with gas at high pressure and the other w r as devoid of material content, that is, contained a vacuum. These two vessels were immersed in a larger vessel containing water, and when everything had reached the same temperature, the 26 valve between the two was opened so that the gas expanded in volume until it filled both vessels at an equal but lower pressure. Thermometers stowed that no heat was given to or taken from the surrounding water, that is, the gas neither lost nor gained energy, and that after equilibrium had been established the temperature of the gas within the two vessels was sensibly what it had been when occupying only one vessel at a higher pressure. Here, then, was a change of pressure and volume with no loss or gain of heat energy by the gas within the vessel and the tem- perature remaining constant. It should be noted that no energy could have been lost by the doing of external work because under the circumstances the gas could do none. It therefore requires exactly the same amount of sensible heat or intrinsic energy to maintain a certain temperature no matter what the pressure and volume, and therefore the intrinsic energy of a gas may be said to depend upon the tempera- 27 ture only and to be independent of pressure and volume. In describing this experiment the tem- perature before and after expansion was said to be " sensibly the same/ 7 and ad- visedly. Very accurate measurements showed slight variations of temperature with the different gases tested, because in actual gases the molecules do exert interattractions. The experiments when considered with others of similar nature indicated, however, that these tempera- ture changes would disappear when the attractions disappeared. If one pound of gas be heated in such a way that the pressure remains constant, the volume increases directly as the absolute temperature, T. Increase of volume under external pressure means the doing of external work, and the heat necessary to cause a change of tempera- ture under these conditions will be Q = S+E. (22) For a given temperature range the 28 specific heat may be found by dividing Q by the temperature range, providing that specific heat be constant. If it is not constant, the quotient as before would be its average value through the particular temperature range chosen. The specific heat under constant pressure conditions is designated by C p when measured in thermal units, and by K p when measured in mechanical units. Then 77SC P =K P . (23) Since the value of Q given by equation (22) is larger than the value of Q given by equation (20) by just the amount of the external work done, it is immediately seen that the specific heat at constant pressure is greater than the specific heat at constant volume by the amount of ex- ternal work done by volume increase in the first case. This is easily seen to be so by imagining a change of one degree -in temperature under the two different conditions. Then if the material being heated weighs exactly one pound, the 29 quantity of heat, Q, is the specific heat in each case. C P C V = E expressed in thermal units (24) and V Vi). (25) The fact that C p may be taken as prac- tically constant, that is, independent of the pressure, for the ordinary gases has been proved by experiment, and it can readily be seen that it must be constant for the perfect gas by the following line of reasoning. C p is larger than the constant C v by just the amount of the external work done during a change of temperature, and since the value of C v is the same for any o::e particular gas no matter what the pressure, the value of C p must be independent of the pressure if it can be shown that the external work done in raising the tem- perature from a value T\ to a value T% is the same at any two different pressures. 30 If a pound of gas b.e assumed to nave a volume V\ at pressure PI and tem- perature T 7 !, and to have a volume V\ at pressure P\ but the same temperature TI, and if in both cases the temperature be raised to T 2 while the pressure remains constant, Boyle's Law gives PI Vi = Pi'VV at temperature T l , (27) and Pi7 2 = Pi / 7 2 / at temperature T 2 . (28) If now (27) be subtracted from (28), Pi(V 2 -V 1 ) = P l '(V 2 '-Vi'); (29) shov.ing the external work to be the same in each case. It is therefore independent of - the pressure during the change, provided that pressure is constant and depends only on the value of the initial and final temperatures. It therefore follows that the value of C p is independent of the 31 pressure and is a constant for a perfect gas. RELATIONS EXISTING BETWEEN SPECIFIC HEATS. Returning to equation (25) for a change of temperature cf one degree, K P -K V = PdV, JVi and for any change of temperature rv 2 / PdV, (30) JVi which for constant pressure changes is, by equation (6), K p (T 2 -T l )-K t (T 2 ^T l ) =Pi(V 2 -Vi). (31) From equation (18), P l V l = R l T l and P 2 V 2 =RT 2 , 32 and when Pi=P 2 , that is, when the pressure during a -change is constant, so that (31) may be written K P -K V = R; (32) showing that R, the constant in the perfect-gas law, is really the value of the external work done by one pound of gas when undergoing a change of tem- perature of one degree at constant pres- sure. In many cases the ratio of the two -rr specific heats ~ occurs, and this is repre- A,, sented by ?. Writing (32) K v = K p R, a useful expression in terms of K v , R, and 7- may be obtained by dividing 33 by K v , substituting j- for ^- p , and trans- posing. This gives R TYPE VOLUME CHANGES. The facts so far developed may now be made use of .in deriving the laws governing expansions and compressions of gases. By expansion is meant an in- crease of volume, and by compression is meant a decrease of volume. There are an infinite number of possible expansions and compressions dependent upon different laws of heat change, but two are of par- ticular importance in the study of heat- engines. These are called (a) Isothermal volume changes, and (6) Adiabatic volume changes. Isothermal changes are those taking place while the temperature remains con- stant. Adiabatic changes are those taking place without reception or rejection of heat, as heat, by the body changing volume. ISOTHERMAL VOLUME CHANGES. An isothermal change is easily analyzed by means of Boyle's Law, equation (9). If the temperature remains constant, constant. Plotting this equation with horizontal distances equal to volumes, and vertical distances equal to pressures, a curve is obtained as shown in Fig. 2. The law of expansion may be regarded as a logarithmic equation of the form P7 n =a constant, in which n is equal to unity. 35 // external work is done during an isother- mal increase of volume, an amount of heat energy equal to the work done must be Volume FIG. 2. consumed. Since the expanding gas by definition is maintained at constant tem- perature, the work cannot be done at the expense of its own intrinsic energy. The 36 only alternative is that heat equal in amount to external work done be supplied from some outside source. Equation (18) shows that if the temperature of a gas remains con- stant, the pressure must decrease in pro- portion to the increase of volume. The external work can then be expressed by equation (6): and since PI V\ = P7= a constant, equation (34) may be written r v *P,V, fVidV 778E-/ ZldV-.p l V l % i/F, JV l V integrating which gives 77SE=P 1 V 1 (log. F 2 -loge Fj) (log. I?). (35) 37 Calling ~ the ratio of expansion, r, 778E=P 1 Vi log* r=RT l loge r, (36) and the heat supplied during the ex- pansion will be ! F T log e r RT^ lo ge r ~778~~ ~~~ If a decrease of volume, that is, a com- pression, be considered, an amount of work equal to that shown in equation (36) would have to be expended upon the gas, and an equal amount of heat would be generated which in some way would have to be carried off by external means to keep the temperature constant. ADIABATIC VOLUME CHANGES. In the case of an Adiabatic Expansion any work done by an increase of volume would have to be done at the expense of heat already contained within the expanding 38 gas, because during such an expansion no heat can be received from external sources. That is, the work would be done at the expense of its intrinsic energy and therefore its temperature would drop. Experience has shown that an adiabatic expansion can be represented by the same form of loga- rithmic equation as that used for the isothermal; in this case, however, n would have a different value. p l Vt* = P 2 V 2 n = PV n = a constant (38) may then be taken as the equation of adiabatic expansion. The external work would be, as before, = f V Jv, and substituting from (38), 39 integrating which gives Pi FiVa i 778E= lVl ( \_ n l (39) which by means of equation (38), Pl can be written '. (40 ) n-l n-l Since the amount of external work done must be equal to the change of intrinsic energy, and since in a perfect gas this is equal to the change of sensible heat, S, it follows that 2 ,. 1N t , : - > (41) 71 1 and substituting for K v by means of (33), R(T l -T 2 ) _R(T 1 -T 2 ) 71 n-l which shows that for adiabatic expan- sion of a perfect gas 40 and the equation for such an expansion is PiV l r =P 2 V 2 r = PV r = const. (43) The work being done at the expense of sensible heat, there must be a drop of temperature, and this can be found by combining equation (43) with the law of perfect gases, equation (17). These two give w_ ~ which multiplied together and simplified give Il-(Y*\- 1 ', (44) 1*1" w and representing ratio of expansion by r, ^ = r- 1 . (45) ^2 It is often convenient to have the ratio of the temperatures expressed in terms of the pressures, and this can easily be done by substituting for (T^) in equa- 41 tion (44) its value derived from equation (43), giving Adiabatic Compressions are the exact reverse .of adiabatic expansions. The work expended by external bodies to compress the gas will all be converted into sensible heat in the gas, since by definition none can be given out by the material under compression. Such being the case, there will be a rise of tempera- ture equal to the drop during expansion between corresponding volume limits. The equations previously developed for external work done by the gas during expansion also give the value of work done upon the gas during compression. AREA AS MEASURE OF EXTERNAL WORK. It has been shown that both Isothermal and Adiabatic changes may be represented by means of the general equation p l V l n = P 2 V 2 n = PV n = a constant, 42 and that the external work in each case can be represented by the equation 77SE = f V *PdV. Such volume changes may be plotted to PV coordinates and give curves of ap- proximately similar appearance, though with sl'ghtly different curvatures and slopes. For the following demonstration the curve of Fig. 3 may be assumed to represent any such volume change. Since the area abed under this curve is A= f V2 PdV, Jv l and since this is the same as the expression A=77SE. (47) That is, the area under a curve 0:1 a PV diagram represents external work done by a body expanding according to the law represented by that curve, or repre- sents work done upon a body compressed according to the same law. 43 In the case of isothermal expansion this area must then equal the heat sup- plied from external sources during ex- FIG. 3. pansion, and in the case of adiabatic expansion it must represent the loss of intrinsic energy. CHAPTER III. ENTROPY. IN dealing with heat-engines the ex- pression , or the change of quantity of heat divided by the absolute tempera- ture at which the change takes place, is of very frequent occurrence and has received a name. This expression is said to represent the change of Entropy of the body. If Entropy be denoted by <, d^, (48) and dQ=Td, (49) which may be integrated as indicated in the following equation: Td. (50) 01 44 45 It can easily be seen from equation (49) above that the entropy of a body must increase when heat is added to it and decrease when heat is taken from it, and therefore remain constant when no heat is received or rejected. ENTROPY DIAGRAMS. If coordinates be chosen with vertical distances representing absolute tempera- ture and horizontal distances representing entropy, it is possible to plot the various entropy changes which bodies undergo when subjected to heat treatment. Thus in the case of an isothermal ex- pansion the temperature remains constant, and hence the line representing such a change upon the entropy-temperature diagram must be horizontal as in Fig. 4. Heat having been added to the body at constant temperature, the entropy has increased. During an adiabatic expansion no heat is received or rejected, and therefore the 46 entropy must remain constant, that is ? the line representing such change must be vertical as in Fig. 4. This shows a drop Is sothermal Cha age 50 a diagram represents the heat change; area under a line traced from left to right represents heat supplied to the body, and under a line traced from right to left, heat rejected by the body. 48 In a general case of heat change of any kind, equation (48), may be written CdT where C represents the proper specific heat for the conditions governing the heat change. If C be assumed constant, integration gives = Clo ge p, (53) -* i so that the total change of entropy for a given temperature change can be cal- culated whenever the proper value of the specific heat is known. The change of entropy of a perfect gas 49 when being heated or expanded under ' constant pressure is then < 2 - equal to Qi E. When reversed the same amount of heat, Q 2 , would be absorbed from the cold body, the same * amount of work, E } would be received from outside sources, and the sum of the two, Q, would be given to the hot body. In this form the apparatus may be considered a heat-pump, since it car- ries heat from a lower to a higher tem- perature. The full extent of the meaning of the term "reversible" should be understood. In the sense in which the word is used here, no process is reversible unless at its completion it can be immediately carried through in the opposite direction with no change of apparatus. Thus in the Carnot method of applying heat along an isother- mal or at constant temperature, the process is reversible because the heat could again be rejected with all changes 60 exactly opposite to those during reception by a simple reversal of the direction of motion of the piston. As an example of a non-reversible process, imagine the heating of a gas by bringing it into con- tact with a body at a higher temperature. No method of doing this can be imagined by a reversal of which the heat can be again extracted from the gas and given back to the source of supply while that gas is in contact with the high-temperature body; that is, no method which will result in leaving the body at the initial high temperature and the gas at the initial low temperature, and which will not in- volve the use of additional apparatus. An example of a non-reversible process in actual practice is the heating of the working fluid in an internal-combustion engine by the chemical process of com- bustion. An engine, then, which is reversible in this thermodynamic way, when run as a pump absorbs the same amount of work as it previously made available, takes 61 from the cold body the same amount as it previously rejected to it, and gives to the hot body the same amount as it pre- viously absorbed from it. It is a simple matter to prove that such a reversible engine has the maximum possible theo- retical efficiency. For this purpose imagine a reversible engine and another engine of any kind which is assumed to be capable of de- veloping exactly the same amount of useful or external work, but is more efficient, that is, absorbs less heat from the hot body for the purpose of de- veloping that work. Since both en- gines are equally powerful, either one can drive the other, and it may therefore be assumed that the more efficient may be made to drive the reversible engine as a heat-pump, removing an amount of heat, Q 2 , from the cold body and giving an amount of heat, Qi, to the hot body. But since the driving engine which is developing the external work which the pump is absorbing is more efficient than 62 the pump, it is removing from the hot body an amount of heat, Q\, less than Qi> and rejecting to the cold body an amount of heat, Q 2 ', less than Q 2 . The result of assuming the driving' engine more efficient than the driven is to develop a system which, without the reception of any energy from outside sources, is capable of conveying more heat from a low to a high temperature than it carries from the high to the low temperature in keeping itself in operation. This action is contrary to common ex- perience as expressed in the Second Law of Thermodynamics, and is regarded as an absurdity. Therefore the driving en- gine cannot have an efficiency greater than that of the reversible engine; the best that it can do is to have an equal efficiency. Since the driving engine was not specified as to type, the proof is absolutely general, depends only on common experience as stated in the Second Law, and shows the reversible engine to have as high an efficiency as is attainable. 63 It is only one step farther to show that all engines, reversible in the same way as the Carnot engine is reversible, must have the same efficiency when operating between the same temperature limits no matter what their cycle of operations. If this were not so, the more efficient could be. made to drive the less efficient, giving the absurdity described above. /rr rp The Carnot efficiency ^= - is then ^i the maximum possible theoretical effi- ciency which may be obtained with heat- engines. For this reason it is often used as a standard with which to compare the performances of actual engines. CARNOT CYCLE ON T(j> DIAGRAM. The shape of the Carnot cycle when drawn on the T(f> diagram must be as shown in Fig. 6, because the lines ab and cd are lines of constant temperature, and the lines be and ad are adiabatics, that is, lines of constant entropy. 64 Equation (51), which applies to this case, gives Q-T^fa-fa) and Q2 = W2-0i), so that the efficiency becomes Qi -Q 2 _^i ( 2 -fa) = l ~ 2 as before. (59) -* i Since the area under ab represents heat received from the hot body and the area under cd represents heat rejected to the cold body, the area abed, which is the difference between them, must equal the heat converted into external work. It can readily be seen from the diagram of Fig. 6 and the following demonstration that the criterion of maximum efficiency is the reception at the highest tempera- ture of all the heat received from the hot body and the rejection at lowest tempera- ture of all heat rejected to the cold body. If the heat is received at temperatures varying between TJ and 7\, the line 65 showing entropy change will be given by equation (53) and will have a shape ap- proximately that shown by dotted line d'a' of Fig. 6. This would result in the I-'IG loss of area d'aa' ', which is a much larger proportion of area abed than it is of area abef, and therefore the efficiency, which" win now be Eff.= a'bcdd' a'befd' 66 will be less than the previous efficiency, which was abed Mt. = r -, abef and a similar proof holds for rejection of heat at any temperature other than 7\ as along the line b'c'. STIRLING CYCLE. There is an apparent exception to the above criterion of maximum efficiency in the case of certain cycles which have been proposed and of which the so-called u Stirling " cycle is an example. In this cycle, which is graphically compared with a Carnot cycle for same temperature range in Fig. 7, the working substance is expanded isothermally with reception of heat from the body at tem- perature Tij until reaching point b, when some of its heat is transferred to a device known as a regenerator. The apparatus is so arranged that the pressure drops with the temperature at just the rate necessary to keep the volume constant. 67 \ \ \ c' FIG. r. 68 The regenerator is a device which can receive, store, and restore heat. For purposes of illustration it may be assumed to be a long pipe, with temperature graded from T\ to T 2 . through which the material can flow, giving up heat at just such a rate as to keep its volume constant while its pressure decreases, arriving at the temperature T 2 when its pressure has dropped to that shown at point c'. Then the material is compressed in contact with a cold body, giving an iso- thermal compression at temperature T 2 untjl the point d r is reached. The mate- rial is then passed through the regenerator in the reverse direction, taking up the heat which it previously rejected, until arriying at the pressure shown at a with the ! corresponding temperature TI . Tjhe efficiency of such a cycle would be Heat Supplied from hot body RT l \og c r-RT 2 \o&r _ T l -T 2 RT l log,r T l ' 69 which is seen to be the same as the Carnot cycle. This does not contradict the criterion of maximum efficiency previously arrived at, because, although the body receives heat at varying temperatures along the line d'a and rejects it similarly along the line be', all ihe heat received into the system from the hot body is received at the highest temperature, and all that rejected from the system to the cold body is rejected at the lower temperature. The T(j> diagram for such a cycle is shown in Fig. 8, where it is the figure abc'd' superimposed upon the Carnot cycle abed. The lines d'a and c'b are obtained by means of equation (55) and are readily seen to be parallel curves, so that the areas d'ad and c'bc are equal and therefore the areas abc'd' and abed are equal. The efficiency in each case is the area of the cycle divided by the area abef, which is the heat received along the line ab, and therefore the two efficiencies are equal. The heat received by the hot body along d'a and rejected along the 70 line be' never really leaves the system or apparatus and may be considered as simply short-circuited during part of the cycle. FIG. 8. OTTO CYCLE. Another cycle of interest is the " Otto " cycle, which is the theoretical cycle of most of the gas-engines built at the present time. Drawn on the PV diagram, it has the shape shown in Fig. 9. Heat is added 71 to the gas at conditions shown at point a, causing a rise of pressure to P* while the volume is maintained constant. Then adiabatic expansion takes place with a fall of temperature and pressure until the volume equals that shown at c, where heat is rejected at const-ant volume until the pressure has fallen to P*, where adiabatic compression is used to bring the material back to initial pressure and volume. In actual practice the working substance is changed at even- cycle, a charge erf mixed air and combustible being drawn into the cylinder along the line diagram. The letters of Figs. 9 and 10 designate corresponding points. The line ab in the T diagram repre- sents the addition of heat *4 constant 74 volume, and the change of entropy is given by the heat added is Adiabatic expansion with < constant is given by the line be. The rejection of heat to the cold body while volume re- mains constant is given by cd with entropy change : T (f>=C v \og el ~, and the heat rejected is Writing efficiency as M -^4p Eff.=- / equal to Q% plus the work of compression from d to c. The cycle as drawn to T

diagram for this same process is very useful and is given in Fig. 15. As in the case of the tables and the diagram of Fig. 14, an arbitrary is chosen; in this case the entropy, (, is figured above 32 F. Using equation (53) and assuming the specific heat of liquid constant, 02-& = 2 i = v = change of (j> during vapori- zation=^, (77) ' t and the horizontal line be is readily ob- tained. If the process of vaporization is not complete, that is, if the final quality is less than unity, heat added during vapori- zation will be xr and then the entropy change will be ^ = ^- (78) L v The horizontal would then be shorter jT and would really be - of the total 100 length of be, as can readily be seen by comparing (76) and (77). The length of the line for any case divided by the length be must then be a measure of the quality of the vapor. 112 For the line cd representing superheat- ing, the change of entropy can be found by means of equation (53), which re- written to suit these conditions is 2 co- ordinates it has the appearance shown in Fig. 17, having exactly the same shape as that obtained when using gas as working substance. The area under the lineafr representing heat supplied, and the 133 area of the cycle representing heat con- verted into work, the efficiency It will immediately be noticed that this efficiency is the same as that obtained when using a perfect gas, and, although the proof of such a statement is beyond the limits of this book, the Carnot effi- ciency is the same for all working sub- stances no matter what their properties. Since Work = Heat supplied X Efficiency, the work resulting from the operation of the Carnot cycle with vapor is (93) The Carnot cycle has not been used in any practical vapor engine, but several such engines operate with cycles very 134 similar to that of Carnot. The so-called "Clausius Cycle/' which is that of the modern steam-turbine, is an illustration of a practical cycle obtained by slight modification of Carnot's. CLAUSIUS CYCLE. Drawn on the PV diagram, this cycle has the shape shown in Fig. 18. Assume the volume, V aj to represent that of one pound of liquid at the temperature of vaporization, corresponding to the pres- sure P a . The addition of heat will cause isothermal vaporization or expan- sion until some point, fr, is reached, at which time the vapor will have a quality, x b , equal to or less than unity. Adia- batic expansion will then bring the mate- rial to V c with a pressure, P c , at which point condensation may be made use of to bring the material back to the liquid state at the same pressure, P c or P d , and with volume, V&, practically equal to V a , since the expansion of liquids through 135 ordinary temperature ranges is negligible when compared to the vapor volume changes. The line da may then be a b FIG. 18. assumed a constant volume or vertical line, and represents the raising of pressure and temperature of the liquid from con- ditions at d to those at a. The heat above 32 F. contained in the 136 liquid at V a will be g a , corresponding to the pressure P a . That in the material at b will be q a + x b r b , and r b will of course correspond with the pressure P b or P a . At the end of the expansion the heat con- tained will be x c r c + q c , and after condensa- tion the heat remaining in the material will be qa equal to q c . During the process of raising the pressure from d to a, an amount of heat equal to the difference between q a and q d must be added, and when it is observed that this heat is added at temperatures varying from the lowest to the highest, the reason why this cycle is less efficient than the Carnot is immediately evident. The energy developed by the engine, as in previous cases, is b -x c r c , (94) and the efficiency of the engine is c . iyO) q a - 3d) + x b r b - x c r c - - - ; (q a -q 145 so that the quality may be very approxi- mately found for any point on the line cd of Fig. 20. As was developed in connection with Fig. 15, the point c of Fig. 21 divides the line fg in such a way that jjj=x c ; (103) and since the quality at any point of the line cd can be found by the method out- lined in the last paragraph, the length of the line dg'' for any position is easily determined, because f -x* giving fc'=x-f'g' (104) for any position which may be selected. Therefore by choosing enough positions for the point c', a sufficient number of lengths c'g' may be calculated to rriake possible the plotting of the line cd, after 146 which the cycle continues along the same horizontal line as is used in the Clausius Cycle. The Rankine Cycle is often modified in practice by adding more heat along the line ab than is required to form dry and saturated vapor. The result is the superheating of the material after com- plete evaporation and then the ac!iabatic expansion starts with this superheated vapor. The T$ diagram for such a case is shown in Fig. 22, in which, v/ith the exception of the curve bb', all the lines are similar to those of Fig. 21. The curve W, which represents the super- heating of the vapor at const ant pressure, is obtained by means of equation (78). The adiabatic expansion is a vertical line from the point b' down to the point which gives the desired temperature and corre- sponding pressure. The point, n, in which b'c crosses bg gives the temperature at which the expanding material will just reach the dry and saturated condition. If expansion continues beyond this point 147 the quality will constantly decrease until the partial condensation at constant volume begins at c. It is readily seen I from the dotted lines on the figure what an enormously high degree of superheat would have to be used to just give dry 148 and saturated steam at the end of ex- pansion. VAPOR REFRIGERATING-MACHINES. Many refrigerating-machines are oper- ated with vapors as the working substance. The reversed Carnot Cycle could be used for such purposes, but it introduces certain complications of apparatus and has been considered undesirable in prac- tice. The most common cycle is one having the shape of the Clausius but operated of course in the opposite direc- tion. Starting with a volume of liquid, V d in Fig. 18, behind a piston, evaporation is allowed to take place as the piston moves out until the vapor or mixture of vapor and liquid finally occupies a volume V c . During this process the latent heat of vaporization is absorbed from the cold body, the one which is to be cooled in the ordinary practical case. The vapor, dry or wet, is then compressed adiabat- 149 ically until some pressure, P b , is reached, when condensation at constant pressure is effected. This is done by removing the latent heat of vaporization corresponding to the pressure Pb or P a , that is, dis- charging it into the hot body. From a to d the pressure is dropped at practically constant volume, which means that the heat of the liquid) q a diagram given in Fig. 19 is a very simple matter. Starting at d, heat is absorbed during the vaporization dc. The quantity of heat absorbed during this process is repre- sented by the area dcef. Then adiabatic compression raises the temperature from T c to Tfc, after which condensation gives to the hot body the amount of heat represented by the area bage. The liquid must then be lowered in temperature from T a to T d and this is done by allow- ing it to give to the cold body an amount of heat adfg. 150 The result of the operation of the cycle is then the removal from the cold body of an amount of heat dcef and the relurn to it of an amount adfg, so that the actual refrigeration or net heat removal is the amount of heat Q 2 = dcef-adfg. (105) This is accomplished by an expenditure of work equal in thermal units to the area dcbad. Expressing these areas in terms of heat quantities gives Area dcef represents x c r c ] Area adfg represents q a q diagram, as already shown. Drawing a vertical line on such a diagram from any initial conditions will give the final conditions of an adiabatic expansion down to any set of terminal or final values which may be selected. 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Translated from the French of M. Argand by Prof. A. S. Hardy. No. 53. INDUCTION COILS: HOW MADE AND How Used. Eleventh American edition. No. 54. KINEMATICS OF MACHINERY. By Prof. Alex. B. W. Kennedy. With an intro- duction by Prof. R. H. Thurston. No. 55. SEWER GASES: THEIR NATURE AND Origin. By A. de Varona. Second edition, revised and enlarged. No. 56. THE ACTUAL LATERAL PRESSURE of Earthwork. By Benj. Baker, M. Inst., C.E. No. 57. INCANDESCENT ELECTRIC LIGHTING. A Practical Description of the Edison Sys- tem. By L. H. Latimer. To which is added the Design and Operation of Incandescent Stations, by C. J. Field; and the Maximum Efficiency of Incandescent Lamps, by John W. Howell. No. 58. VENTILATION OF COAL MINES. By W. Fairley, M.E., and Geo. J. Andre'. THE VAN NOSTRAND SCIENCE No. 59. RAILROAD ECONOMICS; OR, NOTES With Comments. By S. W. Robinson, C.E. No. 60. STRENGTH OF WROUGHT-IRON Bridge Members. By S. W. Robinson, C.E. No. 61. 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A Field-book for Engineers, Containing Rules and Tables for Laying out Transition Curves. By Wal- ter G. Fox, C.E. Second edition. No. 111. GAS-LIGHTING AND GAS-FITTING. Specifications and Rules for Gas-piping. Notes on the Advantages of Gas for Cook- ing and Heating, and Useful Hints to Gas Consumers. Third edition. By Wm. Paul Gerhard, C.E. No. 112. A PRIMER ON THE CALCULUS. By E. Sherman Gould, M. Am. Soc. C. E. Third edition, revised and enlarged. No. 113. PHYSICAL PROBLEMS and Their So- lution. By A. Bourgougnon, formerly As- sistant at Bellevue Hospital. Second ed. No. 114. USE OF THE SLIDE RULE. By F. A. Halsey, of the "American Machinist." Fourth edition, revised and enlarged. THE) VAN NOSTRAND SCIENCE SERIES. No. 115. TRAVERSE TABLE. Showing the Dif- ference of Latitude and Departure for Dis- tances Between 1 and 100 and for Angles to Quarter Degrees Between 1 Degree and 90 Degrees. (Reprinted from Scribner's Pocket Table Book.) No. 116. WORM AND SPIRAL, GEARING. 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