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PRINCIPLES OF THERMODYNAMICS BY G. A. GOODENOUGH, M.E. PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY OF ILLINOIS SECOND EDITION, REVISED COPYRIGHT, 1911 HY HENRY HOLT AND COMPANY PREFACE THIS book is intended primarily for students of engineering. Its purpose is to provide a course in the principles of thermo- dynamics that may serve as an adequate foundation for the advanced study of heat engines. As indicated by the title, emphasis is placed on the principles rather than on the appli- cations of thermodynamics. In the chapters on. the technical applications the underlying theory of various heat engines is quite fully developed. The discussion, however, is restricted to ideal cases, and questions that involve the design, operation, or performance of heat engines are reserved for a second volume. The arrangement of the subject matter and the method of presentation are the result of some twelve years' experience in teaching thermodynamics. Briefly, the arrangement is as fol- lows : In the first six chapters, the fundamental laws are developed and the general equations of thermodynamics are derived. The laws of gases and gaseous mixtures are dis- cussed in Chapters VII and VIII, and this discussion is fol- lowed immediately by the technical applications in which gaseous media play a part. A discussion of the properties of saturated and superheated vapors is likewise followed by the technical applications that involve vapor media. Some of the features of the book to which attention may be directed are the following : 1. The method of presenting the fundamental laws. In this treatment I have followed very closely the development in. Bryan's thermodynamics. The second law is made identical with the law of degradation of energy, the connection between i vpG-iT-oT'C'iVnliftr ann Inau rr mr-jiln nili'f'.-sr ic nniiTforl rm+: anrl Inr ciioruuv U1 o> iiuu-iauiaucu. oyaiKSLu. CKJ uiio iiiuc^Aax i ffT *& wioiij' derived. By this method of presentation, a definite concep- tion of the meaning and scope of the second law is obtained, and the difficulties that usually surround the definition of entropy are removed. 2. The discussion of saturated and superheated vapors. The experiments in the Munich laboratory and the researches of Professor Marks and Dr. Davis have furnished now and accurate data on the thermal properties of saturated and super- heated steam. In Chapters X and XI a concise but fairly complete account of these important researches is given. Kno- blauch's experiments on specific volumes have been correlated with the experiments on specific heat by means of the Cluusius relation (^A = - AT(~\ and equations for the specific heat, \opj T \oj_ j p entropy, energy, and heat content of superheated steam arc thereby deduced. These results have not hitherto been pub- lished. 3. The discussion of the flow of fluids and of throttling processes. The applications of the throttling process are so important from all points of view that a separate chapter is devoted to them. 4. The treatment of gaseous mixtures, Chapter VIII. An attempt is made to present in concise form the principles and methods required in the accurate analysis of the internal com- bustion engine. 5. The note on the interpretation of differential expressions, Art. 23. This important topic should be discussed fully in calculus, but experience shows that students rarely have a grasp of it. In thermodynamics the exact differential has extensive applications; hence it seems desirable to include a rather complete explanation of exact and inexact differentials and their connection with thermodynamic magnitudes. A f this article should ">aWB the student PREFACE V The text is illustrated by numerous solved problems, and exercises are given at the ends of the chapters and elsewhere. Many of the exercises require only routine numerical solutions, but others involve the development of principles. References are given to the treatment of various topics in standard works and to original articles. It is not expected that undergraduate students will make extensive use of these references, but it is hoped that instructors and advanced students will find them helpful. In writing this book I have consulted many of the standard works on thermodynamics, and have made free use of whatever material suited my purpose. I desire to acknowledge my special indebtedness to the works of Bryan, Preston, Griffiths, Zeuner, Chwolson, Weyrauch, and Lorenz, and to the papers of Dr. H. N. Davis. To Mr. John A. Dent I am indebted for assistance in the construction of the tables and in the revision of the proof sheets. Mr. A. L. Schaller also gave valuable assistance in getting the book through the press. G. A. GOODENOUGH. URBANA, ILL., July, 1911. CONTENTS CHAPTER I ENERGY AB T. PAGE 1. Scope of Thermodynamics 1 2. Energy 1 3. Mechanical Energy 2 4. Heat Energy 3 5. Other Forms of Energy . . . 5 6. Transformations of Energy 5 7. Conservation of Energy 6 8. Degradation of Energy 7 9. Units of Energy 8 10. Units of Heat 9 11. Relations between Energy Units 10 CHAPTER II CHANGE OF STATE. THERMAL CAPACITIES 12. State of a System 15 13. Characteristic Equation 16 14. Equation of a Perfect Gas 17 15. Absolute Temperature 18 16. Other Characteristic Equations ....... 20 17. Characteristic Surfaces 20 18. Thermal Lines 21 19. Heat absorbed during a Change of State 22 20. Thermal Capacity: Specific Heat 24 21. Latent Heat 26 22. Relations between Thermal Capacities 27 23. Interpretation of Differential Expressions 28 CHAPTER III THE FIRST LAW OF THERMODYNAMICS ART. .57 27. External Work . 28. Integration of the Energy Equation ' ' 29. Energy Equation applied to a Cycle Process '>' 30. Adiabatic Processes ' 31. Isodynamic Changes ' " 32. Graphical Representations CHAPTER IV THE SECOND LAW OF THERMODYNAMICS 33. Introductory Statement ' i; | 31 Availability of Energy ' i( j 35. Reversibility '*' 36. General Statement of the Second Law '^ 37. Cavnot's Cycle ; " )( 38. Carnot's Principle '>- 39. Efficiency of the Carnot Cycle ">' 40. Available Energy and Waste <r > ( 41. Entropy ; " lf 42. Second Definition of Entropy <'< 43. The Inequality of Clausius <> 44. Summary l! 45. Boltzmann's Interpretation of the Second Law .... (5: CHAPTER V TEMPERATURE ENTROPY REPRESENTATION 46. Entropy as a Coordinate 47. Isothermals and Adiabatics 48. The Curve of Heating and Cooling 49. Cycle Processes 50. The Rectangular Cycle 51. Internal Frictional Processes 52. Cycles with Irreversible Adiabatics 53. Heat Content CHAPTER VI GENERAL EQUATIONS OF THERMODYNAMICS 54. Fundamental Differentials ' 55. The Thermodynamic Relations 56. General Differential Equations. 57. Additional Thermodynamic Formulas .... CONTENTS ix CHAPTER VII PROPERTIES OF GASES ART. PARE 59. The Permanent Gases 89 60. Experimental Laws 89 61. Comparison of Temperature Scales !)1 62. Numerical Value of B 92 63. Forms of the Characteristic Equation 93 64. General Equations for Gases 94 65. Specific Heat of Gases 90 66. Intrinsic Energy 97 67. Heat Content . 99 G'8. Entropy of Permanent Gases . ' 100 69. Constant Volume and Constant Pressure Changes . . . .101 70. Isothermal Change of State 102 71. Adiabatic Change of State 102 72. Polytropic Change of State 104 73. Specific Heat in Polytropic Changes 106 74. Determination of n . . . 108 CHAPTER VIII GASEOUS COMPOUNDS AND MIXTURES. COMBUSTION 75. Preliminary Statement Ill 76. Atomic and Molecular Weights Ill 77. Relations between Gas Constants 112 78. Mixtures of Gases. Dalton's Law . ' 114 79. Volume Relations 116 80. Combustion :' Fuels ..... , . 117- 81. Air required for Combustion. Products of Combustion . .119 82. Specific Heat of Gaseous Products 123 83. Specific Heat of a Gaseous Mixture 125 84. Adiabatic Changes with Varying Specific Heats .... 126 85. Temperature of Combustion ........ 127 CHAPTER IX TECHNICAL APPLICATIONS. GASEOUS MEDIA 86. Cycle Processes 133 87. The Carnot Cycle 134 88. Conditions of Maximum Efficiency . 135 89. Isoadiabatic Cycles 136 00. r.la.ssifina.f-.inn nf Air "Rnmiifis . . . . . . . .137 ART. 93. Analysis of Cycles 9-4. Heating by Internal Combustion .... 95. The Otto Cycle 96. The Joule, or Brayton, Cycle 97. The Diesel Cycle 98. Comparison of Cycles 99. Closer Analysis of the Otto Cycle .... 100. Air Refrigeration 101. Air Compression 102. Water Jacketing 103. Compound Compression 104. Compressed-air Engines 105. rS-Diagram of Combined Compressor and Engine CHAPTER X SATURATKD VAPORS 106. The Process of Vaporization 107. Functional Relations. Characteristic Surfaces .... 108. Relation between Pressure and Temperature .... 109. Expression for ^ at 110. Energy Equation applied to Vaporization 111. Heat Content of a Saturated Vapor 112. Thermal Properties of Water Vapor 113. Heat of the Liquid 114. Latent Heat of Vaporization 115. Total Heat. Heat Content 116. Specific Volume of Steam 117. Entropy of Liquid and of Vapor 118. Steam Tables IHi 119. Properties of Saturated Ammonia 1H< 120. Other Saturated Vapors IS 121. Liquid and Saturation Curves IS 122. Specific Heat of a Saturated Vapor . . . . , .IS 123. General Equation for Vapor Mixtures IS 124. Variation of x during Adiabatic Changes . .- . . .IS 125. Special Curves on the jTS-plane IS 126. Special Changes of State IS 127. Approximate Equation for the Adiabatic of a Vapor Mixture . 1H CHAPTER XI SUPERHEATED VAPOKS CONTENTS xi AHT. VMF. 130. Equations of van der Waals and Clausius 200 131. Experiments of Knoblauch, Linde, and Klebe .... 201 132. Equations for Superheated Steam 203 133. Specific Heat of Superheated Steam 204 134. Mean Specific Heat 210 135. Heat Content. Total Heat 210 13G. Intrinsic Energy 214 137. Entropy 215 138. Special Changes of State 21(5 139. Approximate Equations for Adiabatic Changes .... 220 140. Tables and Diagrams for Superheated Steam . . . .221 141. Superheated Ammonia and Sulphur Dioxide . . . .225 CHAPTER XII MIXTURES OF GASES AND VAPORS 1 12. Moisture in the Atmosphere 228 143. Constants for Moist Air 230 1 14. Mixture of Wet Steam and Air 232 145. Isothermal Change of State 232 140. Adiabatic Change of State 233 147. Mixture of Air with High-pressure Steam 236 CHAPTER XIII THE FLOW or FLUIDS 148. Preliminary Statement 243 149. Assumptions 214 150. Fundamental Equations 244 151. Special Forms of the Fundamental Equation .... 247 152. Graphical Representation 247 153. Flow through Orifices. Saint Venant's Hypothesis . . .252 154. Formulas for Discharge 255 155. Acoustic Velocity 257 156. The de Laval Nozzle 258 157. Friction in Nozzles 262 158. Design of Nozzles 264 CHAPTER XIV THROTTLING PROCESSES CONTENTS 162. The Expansion Valve . ~7'J 163. Throttling Curves i27!i 164. The Davis Formula for Heat Content 274 165. The Joule-Thomson Effect . . !275 166. Characteristic Equation of Permanent Gases .... 1277 167. Linde's Process for the Liquefaction of Gases .... USD CHAPTER XV TECHNICAL APPLICATIONS, VAVOII MKDIA The Steam Engine 168. The Carnot Cycle for Saturated Vapors . . . . . l2K:{ 169. The Rankine Cycle i>S I 170. The Rankine Cycle with Superheated Steam .... 12 SI 5 171. Incomplete Expansion 288 172. Effect of Changing the Limiting Pressures 12H!) 173. Imperfections of the Actual Cycle !>!)() 174. Efficiency Standards ;2<)1 The Steam Turbine 175. Comparison of the Steam Turbine and Reciprocating Engine . I2!ll 176. Classification of Steam Turbines 'j!>r> 177. Compounding o<)i; 178. Work of a Jet [ ~n } $ 179. Single-stage Velocity Turbine ;}00 180. Multiple-stage Velocity Turbine ;}()o 181. Turbine with both Pressure and Velocity Stages .... Ml 182. Pressure Turbine ; ) () ~ 183. Influence of High Vacuum -my Refrigeration with Vapor Media 184. Compression Refrigerating Machines . . . . ;50 ( s 185. Vapors used iu Refrigeration ' ' ;no 186. Analysis of a Vapor Machine . . . . ' * ' an SYMBOLS NOTE. The following list gives the symbols used in this book. In a magnitude is dependent upon the "weight of the substance, the si letter denotes the magnitude referred to unit weight, the capital letter same magnitude referred to M units of weight. Thus q denotes the ] absorbed by one pound of a substance, Q = Mq, the heat absorbed b; pounds. J, Joule's equivalent. A, reciprocal of Joule's equivalent. M, weight of system under consideration. t, temperature on the F. or the C. scale. T, absolute temperature. p, pressure. v, V, volume. y, specific weight ; also heat capacity. u, U, intrinsic energy of a system, z, 7, heat content at constant pressure. s, S, entropy. W, external work. q, Q, heat absorbed by a system from external sources. h, fl, heat generated within a system by irreversible transformatioj work into heat. c, specific heat. c v , specific heat at constant volume. c p , specific heat at constant pressure. k, ratio c p /c v . B, constant in the gas equation pv = BT. R, universal gas constant. n, exponent in equation for polytropic change, p V n = C. m, molecular weight, oi, 03..., atomic weights. H m , heating value of a fuel mixture. x, quality of a vapor mixture (p. 165). q', heat of the liquid. q", total heat of saturated vapor. iv SYMBOLS v', v", specific volume of liquid and of vapor, respectively, w', u", internal energy of liquid and of vapor, respectively. s', s", entropy of liquid and of vapor, respectively. i', i", beat content of liquid and of vapor, respectively. c', c", specific heat of liquid and of vapor, respectively. <, humidity. w, velocity of flow. w e , acoustic velocity. J 1 , area of cross-section of channel. z, work of overcoming friction in the flow of fluids. p m , critical pressure (flow of fluids). /JL, Joule-Thomson coefficient. 77, efficiency of a heat engine. N, steam consumption per h.p.-hour. PRINCIPLES OF THERMODYNAMICS CHAPTER I ENERGY 1. Scope of Thermodynamics. In the most general sense, thermodynamics is the science that deals with energy. Since all natural phenomena, all physical processes, involve manifes- tations of energy, it follows that thermodynamics is one of the most fundamental and far-reaching of sciences. Thermo- dynamics lies at the foundation of a large region of physics and also of a large region of chemistry ; and it stands in a more or less intimate relation with other sciences. In a more restricted sense, thermodynamics is that branch of physics which deals specially with a form of energy called heat. It deals with transformations of heat energy into other forms of energy, develops the laws that govern such transformations, and investigates the properties of the media by which the transformations are effected. In technical thermodynamics the general principles thus developed are applied to the problems presented by the various heat motors. In this volume the general principles of thermodynamics are developed so far as is essential to give a firm foundation for the technical applications in engineering practice. The scope of the book does not permit a discussion of the methods of inves- tigation that are employed so fruitfully in physics and chem- istry. 2 ENERGY I' :UAI> - l ing to rest, that is, in changing its state as regards velocity; a body in an elevated position can do work in changing ii position; a heated metal rod is capable of doing mechanical work when it contracts in cooling. In each case sonic change, in the state of the body results in the doing of work ; hence, in each case the body in question possesses energy. Energy, like motion, is purely relative. It is impossible to give' a numerical value to the energy of a system without referring it to some standard system, whose energy we ^niay arbitrarily assume to be zero. For example, the energy of the waterman elevated reservoir is considered with re Terence to the energy of an equal quantity at some chosen lower level. The kinetic energy of a body moving with a definite velocity is compared with that of a body at rest on the earth's .surface,, and having, therefore, zero velocity relative to the earth. The energy of a pound of steam is referred to that of a pound of water at the temperature of melting ice. 3. Mechanical Energy is that possessed by a body or system due to the motion or position of the body or system relative to some standard of reference. Mechanical kinetic energy is thai due to the motion of a body and is measured by the product \ mv\ where m denotes the mass of the body and v its velocity relative to the reference system. It should be observed that 2 mv z is a scalar, not a vector, quantity and it must be considered positive in sign. Hence, if a system consists of a number of masses m 15 ra 2 , , m n moving with velocities v v v z , , v, n respectively, the total kinetic energy of the system is the sum -| (ra^ 2 + w 2 v 2 2 4- ... + m n y, 2 ) = | Swy 2 , independently of the directions of the several velocities. The mechanical potential energy of a system is that due to AET. 4] HEAT ENERGY 3 4. Heat Energy. Heat is tlie name given to an active agent postulated to account for changes in temperature. It is ob- served that when two bodies are placed in communication, the temperature of the warmer falls, that of the colder rises, and the change continues until the two bodies attain the same tempera- ture. To account for this phenomenon we say that heat flows from the hotter to the colder body. The fall of temperature of one body is due to the loss of heat, while the rise in tempera- ture of the other is due to the heat received by it. It is to be noted that the change of temperature is the thing observed and that the idea of heat is introduced to account for the change, just as in dynamics the idea of force is introduced to account for the observed motion of bodies. Whatever may be the nature of heat, it is evidently something measurable, something possessing the characteristics of quantity. In the old caloric theory, heat was assumed to be an impon- derable, all-pervading fluid which could pass from one body to another and thus cause changes of temperature. The experi- ments of Rumford (1798), Davy (1812), and Joule (1840) shattered the caloric theory and established the modern me- chanical theory, of which the following is a brief outline. Heat may be generated by the expenditure of mechanical work. Familiar examples are shown in the heating of journals due to friction, the heating of air by compression, the develop- ment of heat by impact, etc. Conversely, work may be ob- tained by the expenditure of heat, as exemplified in the steam engine and other heat motors. Joule's experiments established the fact that a definite relation exists between the heat gener- ated and the work expended ; thus to produce a unit of heat a definite amount of work is required, no matter in what particu- lar way the work is done. Heat and mechanical energy are therefore equivalent in a certain sense. Either may be produced at the expense of the other, and the ratio between the quantity of one produced and the quantity of the other expended is always the same. The conclusion is evident that heat is not a sub- 4 ENERGY t ( ' JIAI> - J Heat energy, like mechanical energy, may bo either of tlm kinetic or the potential form. Denoting the mass of si mole- cule by m and the velocity by v, the kinetic energy of the mole- cule is I- mv z . In a given system the different molecules are moving" with different velocities and in different directions ; nevertheless, the summation 2 -| mv z extended to all the molecules of the system gives the thermal kinetic energy of the system. If we denote by c 2 the mean square of the velocities of the molecules, we have where M denotes the mass of the system. Considerations de- rived from the kinetic theory of gases show that the lempera- ture of the system is a function of 6 >2 ; hence, since the kinetic, energy is directly proportional to 6' 2 , it follows that the tempera- ture of a system is a measure of its thermal kinetie energy. Whenever the temperature of a body rises, wo infer that, the kinetic energy has increased, and that the mean velocity of the molecules is greater than before. Potential thermal energy is due to the relative position of the molecules of the system. The addition of heat to a body usually results in the expansion of the body. The molecules are moving with higher speeds than before the addition of heat, and on the whole they are farther apart. To separate them against their mutual attractions requires the expenditure of work; conversely, in coming back to the original configura- tion the molecules will do work. Hence, the work expended in separating the molecules is stored in the system as potential energy. As long as the body remains in the same state of aggregation, the potential energy it is capable of storing is small. ^ lint if a body changes its state of aggregation, it may, during the pro- cess, store a large amount of potential energy. Consider, for example, the melting- of ice. To nimn ^- QUJL WVjU U.JL V; WJU U1XU DVJJ.1U. JH_/O C1/JLJLIL V^XlCtil tf 14.J.tI J. U \J\J UUclU UJL UliC 11 U L111I. water. The heat is therefore stored as potential energy. In the same manner when water is transformed into steam, work is done in forcing apart the molecules against their cohesive forces, and this work is stored as potential energy. 5. Other Forms of Energy. In addition to heat and mechani- cal energy, there are other forms of energy that require consid- eration. The energy stored in fuel or in explosives may be considered potential chemical energy. Electrical energy is exemplified in the electric current and in the electrostatic charge in a condenser. Other forms of energy are due to wave motions either in ordinary fluid media or in the ether. Sound, for example, is a wave motion usually in air. Light and radiant heat are wave motions in the ether. The vibratory forms of energy are neither kinetic nor potential, but rather periodic alternations between the two. To illustrate this statement, let us consider the motion of a pendulum bob. In its lowest position the bob has zero potential energy and maximum kinetic energy ; as it rises its velocity decreases ; therefore, its kinetic energy also decreases, while its potential energy simultaneously increases and reaches a maximum at the end of the swing when the kinetic energy is zero. This same alternation from kinetic to potential and back occurs in vibrating strings, water waves, and, in fact, in all wave motions. 6. Transformations of Energy. Attention has been called to the generation of heat energy by the expenditure of mechani- cal work. This is only one of a great number of energy changes that are continually occurring. We see everywhere in every- day life one kind of energy disappearing and another form simultaneously appearing. In a power station, for example, the potential energy stored in the coal is liberated and is used up in adding heat energy to the water in the boiler. Part of this heat energy disappears in the engine and its equivalent appears as mechanical work. Finally, this work is expended in driving a generator, and in place of it appears electric energy in the form of the current in the circuit. We say in such cases that \Jil\J J.Uiil-1. ing are a few familiar examples of energy transformations ; many others will occur to the reader. Mechanical to heat : Compression of gases ; friction; im- pact. Heat to mechanical : Steam engine ; expansion and contrac- tion of bodies. Mechanical to electrical : Dynamo ; electric machine. Electrical to mechanical : Electric motor. Heat to electrical : Thermopile. Electrical to heat : Heating of conductors by current. Chemical to electrical : Primary or secondary battery. Electrical to chemical : Electrolysis. Chemical to thermal : Combustion of fuel. 7. Conservation of Energy. Experience points to a general. principle underlying all transformations of energy. The total energy of an isolated system remains constant and cannot be increased or diminished ly any phi/xi.aal prwxm'x whatever. In other words, energy, like matter, can be neither created nor destroyed ; whenever it apparently disappears it has been transformed into energy of another kind. This principle of the conservation of energy was lirst defi- nitely stated by Dr. J. R. Meyer in 1842, and 'it soon received confirmation from the experiments of Joule on the mechanical equivalent of heat. The conservation law cannot be proved by mathematical methods. Like other general principles in physics, it is founded upon experience and experiment. So far, it has never been contradicted by experiment, and it may be regarded as established as an exact law of nature. A perpetual motion of the first class is one that would sup- posedly give out energy continually without any corresponding expenditure of energy. That is, it would create enerU from nothing. ^ A perpetual-motion engine would, therefore, <n've out an unlimited amount of work without fuel or other external supply of energy. Evidently such a machine would violate the conserve law ; and the statement that perpetual motion of ART. SJ DEGRADATION OF ENERGY 7 the first class is impossible is equivalent to the statement of the conservation principle at the beginning of this article. 8. Degradation of Energy. While one form of energy can be transformed into any other form, all transformations are not effected with equal ease. It is only too easy to transform mechanical work into heat ; in fact, it is one duty of the engineer to prevent this transformation as far as possible. Furthermore, of a given amount of work all of it can be trans- formed into heat. The reverse transformation, on the other hand, is not easy of accomplishment. Heat is not transformed into work without effort, and of a given quantity of heat only a part can be thus transformed, the remainder being inevitably thrown away. All other forms of energy can, like mechanical energy, be completely converted into heat. Electrical energy, for example, in the form of a current, can be thus completely transformed. Comparing mechanical and electrical energy, we see that they stand on the same footing as regards transforma- tion. In a perfect apparatus mechanical work can be com- pletely converted into electrical energy, and, conversely, electric energy can be completely converted into mechanical work. We are thus led to a classification of energy on the basis of the possibility of complete conversion. Energy that is capable of complete conversion, like mechanical and electrical energy, we may call high-grade energy; while heat, which is not capable of complete conversion, we may call low-grade energy. There seems to be in nature a universal tendency for energy to degenerate into a form less available for transformation. Heat will flow from a body of higher temperature to one of lower temperature with the result that a smaller fraction of it is available for transformation into work. High-grade energy tends to degenerate into low-grade heat energy. Thus work is degraded into heat through friction, and electrical energy is rendered unavailable when transformed- into heat in the con- ducting system. Even when one form of high-grade energy is substances, the difference being due to tJio Jioat developed dur- ing the reaction. As Griffiths aptly says: "Each time we. alter our investment in energy, we have thus to pay a commis- sion, and the tribute thus exacted can never bo wholly recovered by us and must be regarded, not as destroyed, but us tin-own on the waste-heap of the Universe." The terms degradation of energy, dissipation of energy, and thermodynamic degeneration are applied by different/ writers to this phenomenon that we have just described. We may for- mally state the principle of degradation of energy as follows : Every natural process is accompanied ly a certain rfi't/i'ddiitinn. of energy or tliermodynamic degeneration. The principle of the degradation of energy denies the. possi- bility of perpetual motion of the second class, which may be de- scribed as follows : A mechanism with friction is inclosed in a case through which no energy passes. Let the mechanism be started in motion. Because of friction, work is converted into heat, which remains in the system, since no energy passes through the case. Suppose now that the heat thus produced can be transformed completely into work ; then the work may be used again to overcome friction and the heat thus produeed can be again transformed into work. "We then have a perpetual motion in a mechanism with friction without the addition of energy from an external source. Such a mechanism does not violate the conservation law, since no energy is created. It, however, is just as much of an absurdity as the perpetual mot ion of the first-class because it violates the principle of degradat ion. We shall discuss the degradation principle more at length in a subsequent chapter. 9. Units of Energy. According to the conservation law, the quantity of energy remains unchanged through all trans- formations. Hence, a single unit is sufficient for the measure- ment of energy whatever its form may be. This unit is f urnished by the erg, the absolute unit of work in the C. G. S. system, or by the joule, which is 10* ergs. It would save much confusion ART. 10] UNITS OF HEAT 9 arid annoyance if a single unit, as the joule, were used for all forms of energy. Unfortunately, however, the joule is ordina- rily used in connection with electrical energy only, and other units are used for other forms of energy. The following are the units generally employed. For mechanical energy: 1. The foot-pound (or in the metric system, the kilogram- meter). This is the unit ordinarily employed by engineers. 2. The horsepower-hour, which is equal to 1,980,000 foot- pounds. This unit is most convenient for ex- pressing large quantities of work. It should be noted that although the word " hour " is included in the name, the time element is in reality lacking, and the horsepower-hour is a unit of work, not a unit of power. For heat energy : 1. The British thermal unit (B. t . u.). 2. The calorie. The accurate definition of these thermal units and the means employed in establishing them demand special consideration. 10. Units of Heat. Obviously heat may be measured by observing the effects produced by it upon substances. Two of the most marked effects are : (1) rise of temperature ; (2) change of state of aggregation, as in the melting of ice or vaporization of water. Hence, we have two possible means of establishing a unit of heat : 1. The heat required to raise a given mass of a selected substance, as water, through a chosen range of temperature may be taken as the unit. 2. The quantity of heat required to change the state of aggregation of some substance, as, for example, to melt a given weight of ice, may be taken as the unit. 20 G on the same scale. This thermal unit is called tho ffram- calorie, or the small calorie. If the weight of water in taken an 1 kilogram, the resulting unit is the kilogram-calorie or largo calorie. This is the unit employed by engineers. ^ The British thermal unit is defined as t/w heat rjur.'d to raise the temperature of 1 pound of water from l>3 to C>4' J /*'. The method of establishing thermal units by tho rise <>i tem- perature of water is open to one serious objection, namely : The energy required to raise the temperature of water one degree is quite different at different temperatures. Thu^ the, number of ioules required to raise a given mass of water from 0tol0. or from !>'.)" to 100 C. is considerably larger than the immbc.r of joules required to raise tho same mass from 40 to 41 C. The curve., Fig. 1, shows graphically the energy required per degree riso of tempera- ture from to 100" (I It follows that we may have a number of different thermal units depending upon tho temperature adopted in the definition. By many physicists the 15-calorie is used. This is the heat required to raise the temperature of a gram of water from 14-|- C. to If)! C. In recent years there has been a tendency to unite on the so- called mean calorie, which may be denned as the -^ - ff part, of the heat required to raise a gram of water from (J. to 100" (1. The 17|--calorie, as denned by Griffiths, is practically equal to the mean calorie. Corresponding to the mean caloric, is the mean B.t. u., which is T 7 of the heat required to raise the temperature of one pound of water from 32 to 212 F. This is equal to the B. t. u. at 63-|-. 1.008 L006 1.004 1.002 1.000 0.988 0.996 1 \ -A \ / \ 7* /- \ / , / O u V 4 O u u u X 8 u u 11 \ X FIG. 1, 11. Relations between Energy Units. The relation lwtwo.au. the joule, the absolute unit of energy, and any of the grswita- ART. 11] RELATIONS BETWEEN ENERGY UNITS H hour, is readily derived when the value of the constant g is given. By international agreement g is taken as 980.665 = 32.174^- sec The second value is obtained by means of the conversion factor 1cm. = 0.3937 in. Bearing in mind the definition of the erg, we have 1 kilogram-meter = 98066500 ergs = 9.80665 joules. Now making use of the relation 1 kg. = 2.204622 Ib. and the preceding relation between the units of length, we readily find the relation 1 foot-pound = 1.3558 joules, or 1 joule =0.73756 foot-pound. The numerical relation between the thermal unit and the joule, that is, the number of joules in one gram-calorie, is called the mechanical equivalent and is denoted by J. The determi- nation of this constant has engaged the efforts of physicists since 1843.* In this work two experimental methods have been chiefly employed : (1) The direct method, in which mechanical energy is transformed directly into heat. (2) The indirect method, in which heat is produced by the expenditure of energy in some form other than mechanical. Usually electrical energy is thus transformed. The earliest experiments were those of Joule (1843), using the direct method. Work was expended in stirring water by means of a revolving paddle. From the rise of temperature of the known weight of water, the heat energy developed could be expressed in thermal units; and a comparison of this quan- tity with the measured quantity of work supplied gave imme- diately the desired value of J. Professor Rowland (1878-1879) used the same method, but by driving- the paddle wheel with a petroleum engine he was to the water, and the influence of various corrections was cor- respondingly decreased. Rowland's results are justly tfivcm great weight in deducing the finally accepted value of '/. Another result of the highest value in that [omul by Rey- nolds and Moorby (1897). The work, of a 100 horsepower engine was absorbed by a hydraulic brake. Water entered the brake at or near C. and was run through it at a rale that, caused it to emerge at a temperature of about 1.00" ( . In this way the mechanical equivalent of the heat required to raise (lie. temperature of one pound of water from O u to 100" (!. was determined. Of the experiments by the indirect method those of ( irifliths (1893), Schuster and Gannon (1H!)4), and dallendar and Barnes (1899) deserve mention. In each set of experiments the heat developed by an electric current was measured and compared with the electrical energy expended. From a careful comparison of the results of the most trust- worthy experiments, Griffiths has decided that the, most prob- able value of Jia 4.184. That is, taking the 17 r ^ram-ealorie, 1 gram-calorie = 4.184 joules. By the use of the necessary reduction factors, we, obtain (he following relations : 1 kg. -calorie = 426.65 kilogram-meters. 1 B. t. u. = 777.64 foot-pounds. For ordinary calculations, the values 427 and 77S, respectively, are sufficiently accurate. In writing some of the general equations of thennodynamies it is frequently convenient to use the reciprocal of J. ' This is denoted by the symbol A ; that is, A = X We may re-ard ,1 as the heat equivalent of work; thus 1 ft.-lb. = A B. t. u. When the horsepower-hour is taken as the unit of work, we have As 1980000 ART. 11] RELATIONS BETWEEN ENERGY UNITS 13 Hence, 1 h.p.-hr. = 2546.2 B. t..u., a relation that is frequently useful. EXERCISES 1. If the thermal unit is taken as the heat required to raise the tempera- ture of 1 pound of water from 17 to 18 C., what is the value of / in foot- pounds? '-'.',' '.<."' J ''- 2. In the combustion of a pound of coal 13,200 B. t. u. are liberated. If Y| per cent of this heat is transformed into work in an engine, what is the coal consumption per horsepower-hour? 3. A gas engine is supplied with 11,200 B.t. u. per horsepower-hour. Find the percentage of the heat supplied that is usefully employed. -- - : ? 4. In a steam engine 193 B. t. u. of the heat brought into the cylinder by each pound of steam is transformed into work. Find the steam con- sumption per horsepower-hour. /.'. / 5. The metric horsepower is denned as 75 kilogram-meters of work per second. Find the equivalent in kilogram-calories of a metric horsepower- hour. 6. Find the numerical relations between the following energy unite : () Joule and B.t. u. (1>) Joule and metric h.p.-hr. (c) B. t. u. and kg.-meter (rf) h.p.-minute and B.t.u. 7. A unit of power is the watt, which is defined as 1 joule per second. 1 kilowatt (lew.) is 1000 watts. Find the number of B. t. u. in a kw.-hr. ; the number of foot-pounds in a watt-hour. 8. A Diesel oil engine may under advantageous conditions transform as high as 38 per cent of the heat supplied into work. If the combustion of a pound of oil develops 18,000 B.t.u., what weight of oil is required per h.p.-hr.? REFERENCES THE MECHANICAL THEORY OF HEAT Rumford: Phil. Trans., 1798, 1799. Davy : Complete works 2, 11. Black : Lectures on the Elements of Chemistry 1, 33. Verdet: Lectures before the Chemical Society of Paris, 1862. (See Ront- geu's Thermodynamics, 3, 29.) CONSERVATION AND DEGRADATION ov KNKIUJY Helmholtz : Uber die Erhalfcung der Kraft. Hc.vlin, IS 17. Thomson (Lord Kelvin) : Ediub. Trans. 20, 2(il, iiK!) (1*51); Phil. M;itf. (I) 4 (1852). Griffiths: Thermal Measurement of Energy, Lucluro I. Preston : Theory of Heat, 80, 030. Planck: Treatise on Thermodynamics (Ogg), !<). UNITS OF ENERGY. Tin; MKCIIAWOAI, KQUIVAI.KNT Rowland: Proc. Amer. Acad. 15, 75. 1HSI). Reynolds and Moorby : Phil. Trans. 190 A, Ml . 1 HfW. Schuster and Gannon: Phil. Trans. 186 A, -11 r>. 1K!)5. Barnes: Phil. Trans. 199 A, 140. 1002. Proo. Royal Son. 82 A, :!!><>. 1910. Griffiths : Thermal Measurement of Energy. Chwolson : Lehrbuch der Physik 3, 414. Wiukelmann : Handbuch der Pliysik 2, i5!{7. Marks and Davis : Steam Tables and Diagrams, !)!2. CHAPTER II CHANGE OF STATE. THERMAL CAPACITIES 12. State of a System. A therm odynamic system, may be defined as a body or system of bodies capable of receiving and giving out heat or other forms of energy. In general, we shall assume such a system at rest so that it has no appreciable ki- netic energy due to velocity. As examples of thermodynamic systems, we may mention the media used in heat motors : wa- ter vapor, air, ammonia, etc. We are frequently concerned with changes of state of systems, for it is by such changes that a system can receive or give out energy. We assume ordinarily that the system is a homogeneous substance of uniform density and temperature throughout ; also that it is subjected to a uniform pressure. Such being the case, the state of the substance is determined by the mass, tem- perature, density, and external pressure. If we direct our attention to some fixed quantity of the substance, say a unit mass, we may substitute for the density its reciprocal, the vol- ume of the unit mass ; then the three determining quantities are the temperature, volume, and pressure. These physical quantities which serve to describe the state of a substance are called the coordinates of the substance. In all cases, it is assumed that the pressure is uniform over the surface of the substance in question and is normal to the surface at every point ; in other words, hydrostatic pressure. We may consider this pressure in either of two aspects : it may be viewed as the pressure on the substance exerted by some external agent, or as the pressure exerted ly the substance on whatever bounds it. For the purpose of the engineer, the lat- ter view is the most convenient, and we shall always consider the pressure exerted by instead of on the substance. The pressure is always stated as a specific pressure, that is, pressure per unit 15 pound per square foot. The volume of a unit weight of the substance is the spi'nifit: volume. Ordinarily volumes will bo expressed in cubic, feet, and specific volumes in cubic feet per pound. As it is frequently necessary to distinguish between the specific volume, and the volume of any given weight of the substance, wo shall use v to denote the former and V the latter. Thus, in general, v will denote the volume of one pound of the substance, l r the. volume of M pounds ; hence F= Mv. This convention of small letters for symbols denoting quanti- ties per unit weight and capitals for quantities associated with any other weight M will be followed throughout, the book. Thus q will denote the heat applied to one pound of gas and Q the heat applied to M pounds, u the energy of a unit- weight of substance, 7" the energy of M units, etc. As regards the third coordinate, temperature, wo shall ac- cept for the present the scale of the air thermometer. Later the absolute or thermodynamic scale will be introduced. While the centigrade scale presents great advantages, tins com- mon use of the Fahrenheit scale in engineering practice, compels the adoption of that scale in this book. 13. Characteristic Equation. In general, we may assume the values of any two of the three coordinates p< v, T, and then the value of the third will depend upon values of these two. For example, let the system be one pound of air inclosed in a cylinder with a movable piston. By loading the piston wo may keep the pressure at any desired value ; then by the ad- dition of heat we may raise the temperature to any predeter- mined value. Thus we may fix p and T independently. Wo cannot, however, at the same time give the volume v any value we please ; the volume will be uniquely determined by the assumed values of p and T, or in other words, v is a function of the independent variables p and T. In a similar manner we may take p and v as independent variables, in which ease T will be the function, or we take v and T as independent and p as the function depending on them. or written in the explicit form p -.f ^ JT\ ^2\ The equation giving this relation is called the characteristic equation of the substance. The form of the equation must be determined by experiment. For some substances more than one equation is required ; thus for a mixture of saturated vapor and the liquid from which it is formed, the pressure is a function of the temperature alone, while the volume depends upon the temperature and a fourth variable expressing the relative proportions of vapor and liquid. 14. Equation of a Perfect Gas. Experiments on the so-called permanent gases have given us the laws of Charles and Boyle. Assuming these to be fol- lowed strictly, we may readily derive the charac- teristic equation of a gas as follows. According to the law of Charles, the increase of pressure when the gas is heated at constant volume is proportional to the increase of temperature ; that is, FIG. This equation defines, in fact, the scale of the constant volume gas thermometer. Charles' law is shown graphically in Fig. 2. Point A represents the initial condition (p , ), point JB the final condition (jp, ). Then According to Charles' law, therefore, the points representing the successive values of p and t, with v constant, lie on a straight line through the initial point A, and the slope of this line is the 18 CHANGE OF STATE. TJUUKMALj UArAV-i I ir,r> I I-AI-. u constant k. Evidently k is independent of p and t, but it may depend upoiifl; hence we write A-/00. Substituting this value of k in (1), wo got p -^ =C*-V) /(.<>) In this equation t and i are temperatures measured from Urn Fahrenheit zero ; that is, from the origin (Fig. 2 ). Evidently the difference t-t Q is independent of the position of the as- sumed zero ; hence we may write where ^and T^ denote temperatures measured from some. new zero, assumed at pleasure. Let us choose this new xen> siicli that T when p = 0. This is evidently equivalent to the selection of a new origin 0' (Fig. 2) at the intersee.tion of the line. AB with the i-axis. If we now take the, initial point A at 0' , we have p Q = 0, T Q = 0, and (2) takes the form whence pv=Tvf(v~). (V, ; By hypothesis, the substance follows Boyle's law; that is, the product pv is constant when the temperature T is eonslaut. From (3), therefore, the factor vf(v) is a constant ; and denot- ing this constant by B we have which is the characteristic equation desired. The name perfect gas is applied to a hypothetical ideal gas which strictly obeys Boyle's law, and the internal energy of which is all of the kinetic form, and, therefore, dependent on the temperature only. No actual gas precisely fulfills these conditions; but at ordinary temperatures, air, nitrogen, hydro- gen, and oxygen so nearly meet the requirements that 'they may be considered approximately perfect. 15. Absolute Temperature. The zero of temperature defined in the preceding article is called the absolute zero, ami tempera- tures measured from it are called absolute temperatures. The AWT. 15J AJ3SUJLUTE TEMPERATURE 19 molecules on the containing walls. When this pressure is zero, we infer that molecular motion of translation has entirely ceased, and this is, therefore, the condition at absolute zero. The position of the absolute zero relative to the centigrade zero may be determined approximately by experiments on a nearly perfect gas, such as air. From Eq. (4), Art. 14, we have, assuming that the volume remains constant, whence Pi ~ Pi _ 2 ~" t m \^J Pi -'I Let "us take 2j as the temperature of melting ice, T z that of boiling water at atmospheric pressure. llegnault's experi- ments on the increase of pressure of air when heated at con- stant volume gave the relation .) ) Since for the C. scale 7^-^ O.SOGGp, 100 we have ----- = -m > whence ^ = -- = 272.85. (5) O.oubo That is, using air as the thermometric substance, the abso- lute zero is 272. 85 C. below the temperature of melting ice. Other approximately perfect gases, as nitrogen, hydrogen, etc., give slightly different values for T r The experiments of Joule and Thomson indicate that for an ideal perfect gas, one strictly obeying the law expressed by the equation pv = BT, the value of TI would be between 273.1 and 273.14. The corre- sponding value on the Fahrenheit scale may be taken as 491.6 ; 20 CHANGE OF STATE. THERMAL CAPACITIES [CHAP, u denote ordinary temperatures by t and absolute temperatures by T, we have T- 1 + 273.1, for the C. scale. T=t + 459.0, for the K. .scale. 16. Other Characteristic Equations. The equation jn> = 7/7" gives a close approximation to the changes of state of the. more permanent gases. Other gases, as, for example, carbonic, acid, which are in reality only slightly superheated vapors, show marked deviations from the behavior of the ideally perfect gas, and this equation does not give even a rough approximation to the actual facts. On the basis of the kinetic theory of gases, van der Wauls has deduced a general characteristic equation applicable not only to the gaseous but to the liquid state as well. It has the following form : BT a r v-l> v^ in which J9, a, and I are constants which depend it] ton the nature of the substance. An empirical equation for superheated steam is ' ~.l ' rjln V ~ ' It will be observed that for large values of T and ?>, that, is, when the gas is extremely ratified, the hist term of both equa- tions becomes small and the resulting equation ap- proaches more nearly the equation of the perfect gas. 17. Characteristic Sur- faces. The characteristic. V equation ART. 18] THERMAL LINES 21 by its coordinates p^ v v T, and this state is therefore repre- sented by a point, on the surface. If the state changes, a second point with coordinates p v i> 2 , 5^, will represent the new state. The succession of states between the initial and final states will be represented by a succession of points on the surface. The point representing the state we will call the state-point. 'Hence, for any change of state there will be a corresponding movement of the state-point. The surface representing the equation is shown in Fig. 3. For other characteristic equations the sur- faces are of a less simple form. 18. Thermal Lines. If we impose the restriction that during a change of state the temperature of the substance shall remain constant, the state-point will evidently move on the character- istic surface parallel to the jt?u-plane. Such a change of state is called isothermal, and the curve described by the state-point is an isothermal curve or, briefly, an. isotherm. By taking different constant values for the temperature, we get a complete repre- sentation of the characteristic equation. For the perfect gas, the isotherms consist of a system of equilateral hyperbolas hav- ing the general equation pv const. (1) The restriction may be imposed that the pressure of the sub- stance shall remain constant during the change of state. The state-point will in this case move parallel to the v2 7 -plane, and the projection of the path on the jp-plane will be a straight line parallel to 0V, as AB (Fig. 4). The relation between volume and temperature is found l>y combining the equation p D Substituting this value of p in the characteristic, equation, wo have " ' ' If the substance changes its state at constant volume, the state-point moves parallel to the jp^P-phuie, and the projection of the path on the pv-plnno is a line parallel to the p-axis, as CD (Fig. 4). In the case of a perfect gas, the relation between p and T for a change at constant volume is . . o Lines of constant pressure are called isopiestic linns ; lines of constant volume, isometric lines. Besides the cases just given, others aro of frequent occur- rence, and will be taken up in detail later. Thus we may have changes of state in which the energy of the system remains constant; such changes are called isodynamic. Wo may also have changes in which the system neither receives nor gives out heat ; such are called adiabatic. 19. Heat absorbed during a Change of State. A change of state of a system is generally accompanied by the absorption of heut from external sources. If we denote by q beat thus absorbed pur unit, weight, we may by giving r/ proper signs cover all possible cases ; thus + q indicates heat absorbed, q heat rejected; while if y 0, we have the limiting adiabatle change of state. The heat absorbed may be determined from the changes in two of the three variables j>, t,, t that define the state of the system. As we have seen, any pair may be selected as suits our convenience. For example, let t and be talcen as the independent variables, and let the curve AB f Ki-. frt irnresent FIG. 5. UJtlAJN <JU this curve to be replaced by the broken line PQR, then the segment PQ represents an increment of volume Av with t constant and the segment QR an increment of temperature A with v constant. The rate of absorption of heat along PQ, that is, the heat absorbed per unit increase of volume, is given by the derivative ( - j , the subscript t indicating that t is held \dv/t constant during the process. If the rate of absorption be mul- tiplied by the change of volume v, the product (-) Av is evi- \dvjt dently the heat absorbed during the change of state represented (- \ ) ' dtj-u and the heat absorbed is the product ( --- ) A. The heat ab- 1 \dtjv t sorbed during the change PQR is, therefore, (1) ^ J Jvjt \dtjv and the total heat absorbed along the broken path from A to B is given by the summation )At, + f??)A<|. (2) By taking the elements into which the curve is divided smaller and smaller, the broken path may be made to approach 1 the actual path between A and B. Therefore, passing to the limit, \ve have instead of (1) , (3) and for the heat absorbed during the change of state from A By choosing other pairs of variables as independent, other equations similar to (3) may be obtained. Thus, taking t and or taking p and v as the independent variables, we have dp From (5) and (6) equations corresponding to (!) may bo readily derived. 20. Thermal Capacity. Specific Heat. Of tho partial deriv- atives introduced in the preceding article, two are of special importance, -namely, (^] and ( ( ^) . In general, the heat \vtjv \vtjp required to raise the temperature of a body one, degree under given external conditions is called the thermal capacity. of the. body for these conditions. Hence, if Q denotes the boat, ab- sorbed by a body during a rise of temperature from t t lo / 2 , the quotient - gives the mean thermal capacity of the, body ; ^~ tl O 7 and the quotient - - = - , tho moan thermal capacity 1 of a unit weight. If the thermal capacity varies with tbo tem- perature, then the limiting value of the quotient ---, Miat ^ ~~ *i is, the derivative -J, gives the instantaneous value of (bo (her- Ctu mal capacity. Accordingly, we recognixo in the', derivative -2] the thermal capacity per unit weight of the body under Bt/v the condition that the volume remains constant; and in the derivative f -2 ) the thermal capacity with tho pressure, constant. \ot /p According to the definition of the thermal units (Art. 10), the thermal capacity of 1 gram of water at 17.5 (1. is 1 calorie, and that of one pound of water at 63.5 F. is 1 P>. t. u. The specific heat of a substance at a given temperature t is the ratio of the thermal capacity of the subsisting at this tem- perature to the thermal capacity of an equal muss of water at some chosen standard temperature. If we take, 1T.;V ('. j. JUUIUVJ.V.UJL.U mal capacity per unit weight, then the specific heat c is given by the relation __ 7, (of subtance) 7ir.fi (of water) ' But for water y 17-6 = 1. cal. It follows that the specific heat at the temperature t is numerically equal to the thermal capacity of unit weight at the -same temperature ; thus at 100 C. the thermal capacity of a gram of water is found to be 1.005 cal., 1.005. On account 7l7.fi and the specific heat is 1 -^- = ? 05 1 ' . of this numerical equality, we may consider that the derivative ~jf represents the specific heat, as well as the thermal capacity. It is to bo noted, however, that a specific heat is merely a ratio, an abstract number, and it is determined by a comparison of quantities of heat. The deter- mination of thermal capacity, q on the other hand, involves energy measurements. The specific heat of a sub- stance may be represented geo- metrically, as shown in Fig. (5. Starting from some initial state, let the rise of temperature be taken as abscissa and the heat added to the substance as ordinate. The resulting curve OMvfill represent the equation and the slope of the curve at any point, as P, will give the de- rivative --2, or the specific heat at the temperature correspond- ttv ing to P. With constant specific heat the curve OM is a straight line ; if the specific heat increases with the tempera- ture, the curve is convex to the t-axis. The heat applied to a substance, as will be shown presently, may have other effects than raising the temperature. The tain temperatures the curve temperature; hence, the value of the speeiho heal *i 1 dp, upon the conditions under which the heat is absorbed I! tho substance is in the solid or in the liquid form, the, HJ .or,! he i.'ut s are, practically equal. For substances in the jy.isc.nus bn-m however, the specific heat may have any value from - cc to + oc, depen.lmg upon the external conditions under which the heat is supplied. 21. Latent Heat. If the heat added to a substaneo and the temperature be plotted as in Fig. 0, it may happen that, at, cer- tain tarrmflratures the curve has discontinuities. Knr example, lut boat 1m applied to iei at F. Tbe e.urve is praetieally a st.raighl, lim- until tliu temptTaUuv. .''-' is reaclu-d, but. at, tbis point considerable bea.t is added witliout any cban.^e in ti',inporal.ure. During this addition of ben,t, rcp- resontod l>y Ibe. vi'i'tical sogiuont AB (\ f \\r. 7), the state oi r ji^Ljru^sil ion changes from solid to liquid. As tlu; watcn.- receives beat its temperature rises, as indicated by BC, until tbc temperature 212 F. is reached (assuming atmospheric pressure), \vhere the temperature again remains constant during the, addition of a considerable quantity of heat, and the state of aggregation again changes, this time from the liquid to the gaseous. The beat, that is thus added to (or abstracted from) a substanee during a change of state of aggregation is called latent heat. As pointed out in Art. 4, substantially all of the. latent, heat is stored in the system in the form of potential energy. The specific heat -=i becomes infinite during the changes indicated by AB and CD, since t- constant. The volume of the substance changes, however, and the rate at which heat is FIG. 7. ART. 22J KUJJLjATlUNb iJ WTWliiJUJN TlllliKMAJj (JAl-'AUiTJUUfc) Z/ is a thermal capacity called thu latent heat of expansion and denoted by 1 . If tlie pressure also changes, we have in the derivative ( ) the heat added per unit change of pressure. Tli is thermal capacity is called the latent heat of pressure varia- tion, and is demoted by l p . 22. Relations between Thermal Capacities. Introducing the symbols c v , c^, Z v , and l p in equations (8) and (5) of Art. 19, we have dq = l v dv + c v dT, (1) dy = l lt dp + o v dT. (2) liy means of the characteristic equation of the substance, namely, v =/(?, (3) various relations between the thermal capacities may be de- rived. Some" of the most useful are the following. From (3) we obtain by differentiation, which substituted in (1) gives .^=i,|^+(, + z,||)dr, (5) Comparing (2) and (5), we have dv " In the same way, siibstituting in (2), and comparing the resulting equation with (1), we obtain 7 _ 7 5 , (8) ~ ing thermal capacities when any one is given }' dircd, <'.XJ><M'I- ment, provided the characteristic equation, of the substance is known, so that the derivatives -^ ^ etc., can be determined. For a perfect gas, as an example, ts p is known from experiment and the ratio - has also been (letormimul. From the equation c of the gas pv = BT, we have the partial derivatives 8v _B dp _ It Tf~~p* 52 r "V ; hence from (7) and (9) _ 7 ^ 7 ?' ' - . "\ Vp C = (>v~~ 5 01> ^ ' It^'P '*"' ' and l v =-V(c it -c v }. 23. Interpretation of Diiferential Expressions. In thcnno- dynamics we frecjuently meet with exjtressions of tin; form Mdx 4- Ntly composed of two terms, of which eacli is tin? diffcnMitial of a variable multiplied by a coefficient. The two c.o(>nic.ir.nis may be constants or functions of the two vavitibh-.s iuvolvud. Th proper interpretation of differentials of this form is likely to present difficulties to the student; we shall, lln'ivfori>, dnvott- this article to a discussion of such expressions, their projierties, and their physical interpretations. Let us consider first how such differential cxprossions may arise. Suppose we have given the characteristic equation of a substance in the form jp=/0>, 0; (i) by differentiation according to the well-known methods of cal- cnlus, we obtain the relation (3) where M= ^, and JV=^. aw' d In Art. 19 we derived an equation of similar form, namely, dq^dv+^dt, (4) i dv dt ^ J which may likewise be written in the form dq = M'dv + N'dt. (5) The second members of (3) and (5) are differential expressions of the form Mdx + Ndy, which we have under consideration. Kq. (3) was produced from a known functional relation be- tween p-> V, and i, while Eq. (5) was derived directly from physical considerations by assuming increments AV and A of the independent variables and deducing from them the quantity of heat A<7 that must necessarily be absorbed. No relation between y, v, and t was given or assumed; in fact, it is known that no such relation exists ; that is, q cannot be expressed as a function of the variables v and t. Let us see what is implied by the existence or non-existence of a functional relation between q, v, and t. Referring to Fig. 5, let A and B denote the initial and final states of the system. Since p is a function of v and t \_p=*f(y, t)~\, the pressures at A and B are determined by the values of T and v BT BT at those points ; thus for a perfect gas, p = * and p z = a . v i v z Hence, the change of pressure p 2 p l in passing from A to B is fixed by the points A and B alone and is independent of the path between them. Similarly, if there is a functional rela- tion between q, v, and t, that is, if q = (v, ), we shall have at A, ft = 0( v ii *i)' afc -#' ( 72 = < ^( t; 2' f a)- Therefore, the heat absorbed in passing from A to B will be <? 2 - <li = $ ( V 2> ^2) ~ < 0>i *i)> ( 6 ) and this will be determined by the points A and B alone. On the other hand, if the heat absorbed by the system depends upon the path between A and B, there can be no relation g = </>(v, ). As a matter of fact, the heat absorbed i different for different paths between the same initial and iimil wtaU'H ; hence it is not possible to express q in terms of v and C. The conclusions just given may be stated in gonural toriiw a.s follows. Given an expression of tho form du = (7) where the coefficients M and TV are funotioiiH of x and //, there mayor may not exist a functional relation between /. and the variables x and y. If u is a function of -x and //, say it . /''('% //), then the change in u depends only on the initial and iinal values of x and y and is independent of tho path. This is found from (7) by integration ; thus In this integration no relation between x and ?/ is required, for since Mdx+'Ndy arises from differentiating the function cf> (a, y), the integral must be (/> (#, ?/). In this ease ^l/f/.c -f N<ltj is said to be an exact differential. As an example, consider the equation du = ydx -f- inly. Since ydx + xdy is produced by the differentiation of the prod- uct xy, we have the relation u = xy + (7, whence u z u^ = # 2 y 2 r.-y.,. The change of u is represented by the -shaded area ( Fitf . K), and is evid(uitly not. dep(Mident, upon the path betwirn t.be points If, however, no functional rela- tion exists between u and tin 1 variables a: and //, then J/r/.r + TV}?// is said to be an inexact differential. In this ease a value sinned ; in other words, the value of u depends upon the path between the initial and final points. For example, let du = ydx 2 xdy and let the initial and final points be respectively (0, 1) and (2, 2). No function of x and y can be found which upon differentiation will produce this differential. If we choose as the path between the end points the straight line y = %x + l, we have (since dy = \ d.ti), u = j" [(J x 4- 1 )dx xdx] = 1. If we take as the path the parabola y = | x 2 4- 1, we have u = f ["(-I x 2 + T)dx x z dx~] ~ 0. The dependence of the value of u upon the path assumed is evident. The test for an exact differential is simple. If the differential du = Mdx -f Ndy is exact, then u must be a function of x and y, say /(a;, v/). .By differentiation, we have j du -, . du ^ au = ax H ay. dx dy y Hence M and ^V" must be, respectively, the partial derivatives O9/ O9/ __ and - By a well-known theorem of calculus, we have ox dy d_(?u dy\dx thati S) y dy dx If relation (9) is satisfied, the differential is exact ; otherwise, it is inexact. As an example, we have from the differential ydx 2 xdy, = 1, = 2 ; therefore, the differential is inexact, as was dy dx shown in the preceding discussion. In thermodynamics we meet with certain functions that de- pend only upon the coordinates p, v, T of the substance under consideration. From purely physical considerations the energy (See Art. 26.) Hence if u is expressed in terms of two of these coordinates as independent variables, thus, we know at once that du is exact and we can write f'A* = >-! =/0 2 < zi) -/O'n Zi ') 'i Furthermore, from the test for an exact differential wo must have the relation T dv By making use of this test when the differential IH known to be exact, many useful relations are deduced. We have also magnitudes that depend upon tho ootirdinatcs and also upon the method of variation ; that in, upon tho path. The heat q absorbed by a system in changing Htato is one of these. If again we choose v and T as the independent variables, we may write but since dq is not exact, we cannot write EXERCISES 1. Regnault's experiments on the heating of cm-tain liquids are ex- pressed by the following equations : Ether q = 0.529 t + 0.000200 * a , - 20 to + :)" ( '. Chloroform q = 0.232 t + 0.0000507 t'\ - '.W> to -I- (it) (!. Carbon disulphide q= 0.235 t + 0.0000815 t", - :}()" to -|- !()" ('. Alcohol q = 0.5476 1 + 0.001122 / a + 0.0000022 /", ~ 2:5" to + lili ' ( 1 . From these equations derive expressions for tho Hpciniu; huat, and for each liquid find the specific heat at 20 C. 2. From the data of Ex.1, find the mean heat oapai-ily of i>iln>r IH-IWIMMI and 30 C. Also the mean heat capacity of alcohol bct/woon ()" and rIV ('. 3. If the thermal capacity of a substance at temp>ratun>. / is given by the relation y = a + U + ct~, AWT. AJJ JUAEKCISES 33 4. In the investigation of the properties of gases, it is convenient to draw the isothermal* (T = const.) on a plane having the pressure p as the axis^ of abscissas and the product pv as the axis of ordinates. Show that the isothermals of a perfect gas are straight lines parallel to the jo-axis. 5. Show on the po-p plane the general form of an. isothermal of super- heated steam, the characteristic equation being As an approximate equation for superheated steam, the form p(v + c)=BT, has been suggested by Tumlirtz. Show the form of the isothermal when this equation is used. 6. Derive relations between c in c m l p , and l v , similar to those given by Eq. (10) and (11) of Art. 22, using van der Waal's equation v b v 2 as the characteristic equation of the gas. 7. For a perfect gas, as will be shown subsequently, the thermal capacity l v is Ap(A .J-). Show that c p - c v = AB ; also that l }> = - Av. B. Test the following differentials for exactness : () vilp + npdv, (J>) v n dp + n])v n - l dv. (0 x + 9. Find the function u f{p, T) which produces the differential (c) of Ex. 8. 10. The differential [c'(l - x)+ c"x] ^- + dx, which appears in the discussion of vapors, is known to be exact, c' and c" may be taken as con- stants, while r is a function of T. Apply the test for exactness and thereby deduce the relation c" c' = -^ ?- 11. For perfect gases, dq - c v dT + Apdn. (See Ex. 7, and Art. 22.) Making use of the characteristic equation pv = BT, show that while dq is not an exact differential, is an exact differential. REFERENCES THERMAL CAPACITY. SPECIFIC HEAT "VVeyraach : Grundriss der Wiinne-Theorie 1, (JO. Chwolson : Lehrbuch der Physik 3, 172. EXACT AND INEXACT DIFFKKENTIALH IN THERMODYNAMICS Chwolson : Lehrbuch der Physik 3, d-34. Clausius: Mechanical Theory of Heat, Introduction. Preston : Theory of Heat, 597. Weyrauch : Gruudriss der Wiirme-Theorie 1, 28. Townsend and Goodenough: Essentials of Calculu.s, 245. CHAPTER III THE FIRST LAW OF THERMODYNAMICS 24. Statement of the First Law. The first law of Thermo- dynamics relates to the conversion of heat into work, and merely applies the principle of conservation of energy to that process. It may be formally stated as follows : When work is expended in producing heat, the quantity of heat generated is proportional to the work done, and conversely, when heat is employed to do work, a quantity of heat precisely equivalent to the work done disappears. If we denote by Q the heat converted into work and by "FT the work thus obtained, we have, therefore, as symbolic statements of the first law, Tf= JQ, or Q = AW. 25. Effects of Heat. When a thermodynamic system, as a given weight of gas or a mixture of saturated vapor and liquid, undergoes a change of state, it in general receives or gives out energy either in the form of heat or in the form of mechanical work. These energy changes must, of course, conform to the conservation law. Suppose in the first place that the system is subjected to a uniform external pressure and that during the change of state the volume is decreased. Mechanical work is thereby done upon the system, or in other words, the system receives energy in the form of work. At the same time heat may be absorbed by the system from some external source. Denoting by ATT the work received and by AQ the heat absorbed, the increment AZ7 of the intrinsic energy of the system is given by the relation AZ7= J&Q + ATT. (1) Ordinarily we take the work done by the system in expanding as positive ; hence the work done on the system during com- _r . 36 THE FIRST LAW OF THERMODYNAMICS [DIAIMII that is, the increase of energy of the system is equal to tho energy received in the form of heat less tho energy tfivi'ii to the surrounding systems in the form of work. Wo may also write (2) in the form and interpret the relation as follows. The heat absorbed by a substance is expended in two ways : (1) in increasing the intrinsic energy of the substance ; (2) iu tho performance of external work. Equation (3) is the energy equation in its most Amoral form. Any one of the three terms may bo positivo or negative. Wo consider A Q positive when the system absorbs boat., negative when it gives out heat ; as before stated, A IK is positivo when work is done by the system, negative whon work is done, on (lio system; A U is positive when the internal energy is increased, negative when the energy is decreased during tho change of state. 26. The Intrinsic Energy. Tho increase A //"of I ho in( rinsio, energy is, in general, separable into two parts: (1) Tho in- crease of kinetic energy indicated by a riso of temperature of the system. As we have seen, this is duo to an increase in tho velocity of the molecules of the system. (l2) Tin; increase of potential energy arising from the inorcaso of volumo of tho, system. To separate the molecules against their mutual at trac- tions, or to break up the molecular structure, as is dono in changing the state of aggregation, requires work, and this work is stored in the system as potential energy. The energy U contained in a body depends' upon the state of the body only, and the change of energy duo to a change of state depends upon the initial and. final' states only. hi Fig. 9, let A represent the initial, and ./>' tho tiual state'. The point B indicates a definite state of the body as regards pres- sure, volume, and temperature. Now the, 'temperature indi- cated by B fixes the kinetic energy and Uo volume at P> determines the potential em>r< r-T,m,,,, +1,., i;,,.,i <..*..i ......... ^ ART. Z/J UA.TJttJtUNAjL WUJKJi. to B. Whether we pass by the path m or the path n, we have the same volume and temperature at B and therefore the same total energy. Since V is thus a function of the coordinates only, it follows that d II is always an exact differential. Choosing T and v as the hide- FlG 9 pendent variables of the system, we may express U as a function of these variables. We have, therefore, ?7 whence dU= --dT ' + dv. dT dv The term ~^,dT is the increment of energy due to the in- r) TT crease of temperature d T. The factor is the rate at which the energy changes with the temperature when the volume n Try remains constant. Hence ^dT is the change of energy due merely to the rise of temperature, that is, it is the change \ 7"7" of kinetic energy. The term - dv is the change of energy dv due merely to the change of volume with the temperature constant ; it is, therefore, the work done against molecular attractions, the work that is stored as potential energy. For a substance in which there are no internal forces between the molecules, the energy is independent of the volume, that is, ~~ = 0, and therefore the term dv is zero. dv dv 27. The External Work. In nearly all cases dealt with in applied thermodynamics, the external work ATT is the work done by the system in expanding against a uniform normal pressure. A general expression for the external work may be deduced as follows. Let AJP denote an elementary area on the surface inclosing the system and suppose that during li JBJ..K.BT normal pressure per unit area, the work donu against this pressure is for this one element p&F-s. ) When all the elements of tlie surface tiro lukim, HIM OXJMVS- sion for the work is But evidently if 8 he taken sui'iiciontly small, A/'' is (In; increase of volume AF"; hence we may writo A If =^ A I 7 ; (-) from which we have for a change of volume from V l to J T 2 . The external work for a given change of staio is n^ircscnUMl graphically by the area between the projuc.tinn of the. initli of the state-point on the j;F-plane and tin; F"-uxis. Thus in Fig. 10, let the variation of pressure and volume lx^ rcjircscnl-i'd by the curve AJB; this is the projection on tlm p f-plaiu-. of tho actual path of the state-point on tho oharautnrislic, surl'ac.o. The area A^BB^ under AB is clearly given by tho inU^nil hence, it represents the work done by tho system in passing from the initial to the final state according to the in von law. n o Tlio gonoral onorgy oqna- A tion (8), Art. U5, may now bo written in tho form or using tlio dilToroiitial nota- tion, in tlii! form B PIG. 10. 98 __ v For a unit weight of tho sub- fil stance, Ave havo Jdq = du + pdt'. (V> a) by the subscripts 1 and 2, respectively, we have whence JQ = Z7 2 - ^ + $p d F (1) It should be noted carefully that since the energy U depends only upon the state of the system and not upon the process of passing from the initial to the final state, the change of energy may be written at once as the difference U 2 U r The external work is evidently dependent upon the path of the state-point between the initial and final states. See Fig. 10. Hence the sum of the change of energy and external work, that is, the heat added to the system, must also depend upon the path. It follows that dQ is not an exact differential, and we cannot write In other words, we cannot properly speak of the heat in a a body in the state 1 or the state 2 ; we can speak only of the heat imparted to the body during the change of state with the reservation, stated or implied, that the quantity thus imparted depends upon the way in which the state is changed. For con- venience we shall denote by $ 12 the heat imparted to the sys- tem in passing from state 1 to state 2 ; and likewise by W lz the corresponding external work done by the system. 29. Energy Equation applied to a Cycle Process. Let a sys- tem starting from an initial state pass through a series of pro- cesses and finally return to the initial state. The path of the state-point on the characteristic surface is a closed curve in space and the projection of the path on the p Fplane is a closed plane curve. See Fig. 11. Let A represent the initial state; then in passing from A to B the external work done by the system is ' p dV (along path m), 40 THE FIRST LAW OF THERMODYNAMICS [THAI-. MI which is represented by area A^AmSS^ while in passing from B back to A along path n the external work is f "0 dV= L p dV (along path ?<.) J Pj J " ' O and this is represented by area B-^BnAA^ Hene.e tlie net external work done by the system is represented by the area inclosed by tho eurve of the cycle. Since tho energy I.' of tho system depends upon the state only, the change of energy for the cycle is /,-/,= <>, y and the energy equation ro- ^ -,.," 01 duces to FIG. 11. That is, for a closed cycle, of processes, the heat imparted to thr. system is the equivalent of the external work, and both are repre- sented graphically by the area of the cycle on the ^r-plane. 30. Adiabatic Processes. When a system in changing its state has no thermal communication with other bodies and therefore neither absorbs nor gives out heat, tho change of state is said to be adiabatic. In general, adiabatie. ehangos arc. possible only when the system is inclosed in a non-oon<imaing envelope. Rapid changes of state are approximately adiabatie, since time is required for conduction or radiation of heat ; thus the alternate expansion and contraction of air during the pas- sage of sound waves is nearly adiabatic; the flow of a gas or vapor through can orifice is practically an adiabatic process. Jor an adiabatic change, the term JQ of tho energy equation reduces to zero, and we have, consequently, During an adiabatic change, therefore, tho extern,! wnrt ,!,, KT. 30] ADIABATIC PROCESSES 41 B The projection on the pT^plane of the path of the state-point .uring an adiabatic change gives the adiabatic curve. See Fig. .2. The area A 1 ABB 1 represents the work TF 12 of the system ,nd from (1) it represents also the decrease of the intrinsic nergy in passing from state 1 epresented by A to state 2 epresented by B. Making ise of this principle, we can rrive at a graphical represen- ation of the intrinsic energy if a system. Suppose the .diabatic expansion to be con- inued indefinitely; the adia- 'atic curve AB will then FIG. 12. pproach the F^axis as an symptote, and the work of the expanding system will be epresented by the area A^A oo between the ordinate AA, the xis OF", and the curve extended indefinitely. The area A^Aca epresents also the change of energy resulting from the expan- ion. Hence if we assume that the final energy is zero, we have i = area A^A oo, \ area A 1 A oo = ( y p d V. It is instructive to compare the adiabatic curve with the isothermal. When the two curves are projected on the pF~-plane, the adiabatic is the steeper. See Fig. 13. This follows from the fact that dur- ing adiabatic expansion the nergy decreases and as a result the temperature falls ; hence or FIG. 13. JblKST JUAW UJf 1U On the other hand, the area under the indefinitely extended isothermal is infinite. 31. Isodynamic Changes. If tho intrinsic, energy of tho system remains unchanged during a change, of slate, the change is called isodynamic or isoenergic. In this case the energy equation reduces to the form For perfect gases, the isodynamic curve is also tho isothermal, but for other substances this is not tho case. 32. Graphical Representa- tions. The throe magnitudes JQiv <r/ 2 ^r :U1( ^ '^1-2 l>11 ^ !1 '- ing into the energy equation can bo represented graphically by areas on the p\ '-plant!. Suppose tho change of state to bo represented by tho curve; m between the initial point, A Y and final point H (Kig. M ). FIG. u. , 'idiabatiu lines be drawn through A and /i siinl ex- tended indefinitely; then from preceding considerations we have F 12 = area = area <x> Hence, JQ 12 = U z - ^ + F 12 = area A l ABB l + area B,B oo - area A,A = area AB oo . That is, the heat imparted is represented on the. p}"-pl<m<> 1, t/,<> area included between the path and two twfc/to/// m.,.,,,!,,?. ',H, latics drawn through the initial and final pointy rmn^hchi Ihrough the initial point A let an Lsodynamie be, drawn, cutting BB,m the point 0, and through let tlio i,ldi,,it,ly extended adiabafao (7=o be drawn. Then the energy r a of tho system m state is equal to U and, therefore, ' ART. 32] 43 It should be noted that the p area representing U z L\ is not influenced by the path m. A second graphical repre- sentation is shown in Fig. 15. Through the initial point A an isodynamic line is drawn, and through the linal point B an adiabatic is drawn, the two lines intersecting at point (7. We have then, denoting the energy in the' state C by ?7 3 , A -B, FIG. 15. L y 2 - ^ = z - z = area W lz = area A^ABB V JQ V1 = W lz + t/2 - /! = area As before, the change of energy is independent of the path w, while botli the external work and the heat imparted depend upon the form of m. EXERCISES 1. Show that the energy equation may be written in the form and that consequently the derivative ( ^ ) must be equal to Jc v . 2. If the energy of a substance is independent of the volume, show that the energy equation reduces to the form Jdq = Jc v dT+pdv. 3. Using the method of graphical representation, show by areas Qi, U'2 - Ui, and Ww () for a change at constant pressure, (b) for a change at 44 l-tUtU JblltoT JLAW UJ. 1 7. Apply the general energy equation to the. procenn of changing ice ; 32 F. to water. What is the effect of greatly incroaidng the pro.s.snrc. o the ice during the process 'I REFERENCES Preston : Theory of Heat, 590. Zeuner: Technical Thermodynamics (Klein) 1, i28. Planck: Treatise on Thermodynamics (Ogg), IkS. THE SECOND LAW OF THERMODYNAMICS 33. Introductory Statement. While the first law of tliermo- ynamics gives a relation that must be satisfied during any iiange of state of a system, and of itself leads to many useful isults, it is not sufficient to set at rest all questions that may rise in connection with energy transformations. It gives no idications of the direction of a physical process ; it imposes no mditions upon the transformations of energy from one form to lother except that there shall be no loss, and thus gives no in- ication of the possibilities of complete transformation of dif- >rent forms; it furnishes no clue to the availability of energy >r transformation under given circumstances. To settle these uestioiis a second principle is required. This principle, called le second law of thermodynamics, has been stated in many ways. i effect, however, it is the principle of degradation of energy, ist as the first law is the principle of the conservation of There are conceivable processes which, while satisfying the jqiurements of the first law, are declared to be impossible be- mse of the restrictions of the second law. As a single ex- mple, it is conceivable that an engine might be devised that mild deliver work without the expenditure of fuel, merely by sing the heat stored in the atmosphere; in fact, such a device as been several times proposed. The first law would not be iolated by such a process, for there would be transformation, ot creation of energy; in other words, such an engine would ot be a perpetual motion of the first class. Experience shows, owever, that a process of this character, while not violating le conservation law, is nevertheless impossible. The statement 34. Availability of Energy. In Art. 8 was noted the (list/mo- tion between various forms of energy with respect to the pos- sibility of complete conversion. Wo shall now consider the point somewhat in detail. Mechanical and electrical energy stand on the same footing as regards possibility of conversion; either can be completely transformed into the other in theory, and nearly so in practice. Either mechanical or electrical energy can bo completely trans- formed into heat. On the other hand, experience shows that heat energy is not capable of complete conversion into mechan- ical work, and to get even a part of heat energy transformed into mechanical energy, certain conditions must bo satisfied. As a first condition, there must bo two bodies of different, tem- perature; it is impossible to derive work from the heal of a body unless there is available a second body of lower temperature. Suppose we have then a source 8 at temperature T { and a re- frigerator R at lower temperature .7! 2 ; how is it possible to derive mechanical work from a quantity of heat energy Q { stored in SI If the bodies $ and R are placed in contact., the heat Q will simply flow from S to R and no work will bo obtained. Hence, as a second condition, the systems *S r and H must be kept apart and a third system M must be nsed to convey energy. This third system is the working fluid or medium. In the steam plant, for example, the boiler furnace is the source /S Y , the con- denser is the refrigerator R at a lower temperature, and the steam is the medium or working fluid M. The medium M is placed in contact with S and receives from it heat Q^ it then by an appropriate change of state (expansion) gives up energy in the form .of work, and delivers to R a quantity of heat $ 2 , smaller than Q v the difference Q l - Q z being tho heat trans- formed into work. The details of this process will be given in following articles, where it will be shown that in no other way can a larger fraction of the heat be transformed into work. The part of the heat Q l that can be thus transformed into work, that is, Q l -Q yt is the available part of Q^ and the purl $ 2 that must be rejected to the refrigerator R, and which is of no further .._. ,-, ' ~ -. . . a. ~ o. cal work. In general, the term availability signifies the fraction of the energy of a given system in a given state that can be transformed into mechanical work. In Art. 8 attention was called to the apparent tendency of energy to degenerate into less available forms. We have now to investigate this point somewhat closely in connection with reversible and irreversible changes of state. 35. Reversibility. The processes described in thermo- dynamics are either reversible or irreversible. A process is said to be reversible when the following conditions are fulfilled : 1. When the direction of the process is reversed, the system taking part in the process can assume in inverse order the states traversed in the direct process. 2. The external actions are the same for the direct and re- versed processes or differ by an infinitesimal amount only. 3. Not only the system undergoing the change but all con- nected systems can be restored to initial conditions. A process which fails to meet these require- ments in any particular is an irreversible pro- cess. The following examples illustrate the above definitions. (1) Suppose a con- fined gas to act on a piston, as in the steam or gas engine. See Fig. 16. If A is the piston area, the pres- sure acting on the face of the piston is pA, and for equilibrium this pressure must be equal to the force F. If now we assume the force pA slightly greater than F, the piston will move slowly to the right and the confined gas will assume a succes- UbLJlJ SJiUJUlNJJ JUa.YV UJ? sion of states indicated by the curve All, It at the slate .# the motion is arrested and I 7 is made infinitesimally greater than pA for all positions of the piston, the .scries of status from B to A will be retraced and the system (tin 1 , confined gas in this case) will be brought back to its original state without leaving changes in outside bodies. The reversed process is accomplished by an infinitely small modification of tins external force F. The process is therefore reversible. (2) Let the force F be removed entirely. Thou the piston will move suddenly and the confined gas will bo thrown into commotion. When the gas finally attains a stato of thermal equilibrium with the volume F" 2 , that state will be represented by some point as B 1 . No path can be drawn between A and Jl' because during the passage from A to Ji' the gas is not in thermal equilibrium, and its state at any instant cannot, there- fore, be determined. Evidently, therefore, the gas cannot be returned to state A by reversing in all particulars the direct change from A to B'. It can be returned to stato A, however, in the following manner : A force F, slightly greater than ;;A, is applied to the piston and the gas is thus compressed slowly, the successive states being indicated by the enrve II' A', say. Then the gas in the state A' is cooled at the, constant volnmo V l until the original state A is attained. The restoration of the^gas to its initial state has, however, left changes in other bodies or systems. Thus the work of compression from tt 1 to A' must be furnished from one external body, and the heat given up by the cooling from A' to A must bo absorbed by another external body. The free expansion of tho gas is, therefore, an irreversible process. It is easy to see that the flow of a fluid through ,-,n online trom a region of high pressure to a region of low pressure is essentially equivalent to the irreversible expansion just de- scribed. Such cases are of frequent occurrence in t'eehnical applications of thermodynamics. The flow of liquid aunnonia through the expansion valve of the refrigerating machine may be cited as an example. AUT. ouj vjrJujiNJtujui/vju >x.t\..i. JIUV.I.JUUN x \jj} and bearing due to the conversion into heat of the work of overcoming friction. A complete reversal of this process would involve turning the shaft in the opposite direction by cooling the bearing. (4) The conduction of heat from one body to another is an irreversible process. There must be a temperature difference to produce the flow of heat, and heat of itself will not flow in the reverse direction ; that is, from the colder to the hotter body. If, however, we take the temperature difference A T in- definitely small and let the transfer take place very slowly, the process can be reversed by changing the sign of A 21 Hence we can conceive of reversible flow as the ideal limiting condi- tion of the actual irreversible flow. Strictly speaking, there are no reversible changes in nature. We must consider reversibility as an ideal limiting condition that may be approached but not actually attained when the processes are conducted very slowly. 36. General Statement of the Second Law. According to the first law, the total quantity of energy in a system of bodies cannot be increased or decreased by any change, reversible or irreversible, that may occur within the system. It is not, how- ever, the total energy, but the available energy of the system that is of importance ; and experience shows that a change within the system usually results in a change in the availability of the energy of the system. It may be considered as almost self-evident that no change of a system which will take place of itself can increase the available energy of the system. On the other hand, experience teaches that all actual changes involve loss of availability. Con- sider, for example, the flow of heat from a body of temperature T v to another at temperature T 2 . For the flow to occur of it- self we must have 5\ > T^ and as a result of the process there is a loss of availability. To produce an increase of availability would require T 2 to be greater than 2j ; in that case, however, the process would not be possible. In the limiting reversible of energy, are based 'entirely on experience: I. No change in a system of bodies that nan take, plane of ifxe/f can increase the available energy of the xi/nl.c.m. II. An irreversible change causes a low of anai/a!>t'!itt/. III. A reversible change doe's not affwt, the. ai>aitat>ility. These statements may be regarded as fundamental natural laws underlying all physical and chemical changes, The seeond and third together constitute the law of degradation of ene.rgy. The first may be taken as a general statement of the .second law of thermodynamics. By considering special processes the general statement of tin; second law here given may be thrown into special forms. Tims if heat could of itself pass from a body of lower to a body of higher temperature, the result of the process would be. an in- crease of available energy, a result that is impossible according to our first statement. We have, therefore, Clausius' form of the second law, viz : It is impossible for a self -acting machine, unaided I if any e.rler- nal agency to convey heat from one body to another at hi<//ier temperature. Again, if we consider the increase of available, unorgy thai, would result from deriving work directly from the heat of tbo atmosphere, we are led to Kelvin's statement, namely : It is impossible by means of inanimate 'material agency to derive mechanical e/ect from any portion of matter bt/ e.ooUng it below th,: temperature of surrounding objects. In order to estimate the available energy ,,f a system in a given state, or the loss of available energy when the system undergoes an irreversible change, it is necessary to know tbo most efficient means of transforming heat into mechanical work under g lv en conditions. This knowledge is furnished by a study of the ideal processes first described by Oarnot in mi. 37 Carnot's Cycle. -Suppose that the conditions stilted in Art. 34 are furnished ; that is, let there be a source of beat fi at temperature ^, a refrigerator R at a lower temperature T v ART. 37] CARNOT'S CYCLE A l 01 FIG. 17. and an intermediate system, the working fluid or medium M. The medium we may assume to be inclosed in a cylinder provided with a piston (Fig. 18). Let the medium initially in a state represented by B (Fig. 17), at the temperature T of the reservoir $, expand adiabatically until its temperature falls to T v the temperature of body R. By this expansion the second state Q is reached, and the work done by the medium is represented by the area S 1 SOO r The expansion is assumed to proceed slowly so that the pres- sures on the two faces of the piston are sensibly equal, and the process is, therefore, re- versible. The cylinder is now placed in contact with R so that heat can flow from Jf.to R, and the medium is compressed. The work represented by the area C l QDD l is done on the medium, and heat Q 2 passes from the medium to the refriger- ator. The process is again assumed to be so slow as to be reversible. From the state D the medium is now compressed adiabatically, the cylinder being removed from R until its tempera- ture again becomes T v that of the source 8. D uring this third process work repre- sented by the area D^DAA^ is done on the fluid. Finally, rjG 18 the cylinder is placed in contact with S and the fluid is allowed to expand at the constant temperature T 52 THE SECOND LAW OF THERMODYNAMICS [CIIAV. iv temperature is kept constant by the flow of heat ^ from StoM. The area ABGD inclosed by tho four curvo.s of the, cycle represents the mechanical work gained; that is, the excess of work done by the medium over that done on tho medium. Denoting this by W, we have from the first, law, The efficiency of the cycle is the ratio of tho work gained to the heat supplied from the source ti. Denoting thu elliciency by 97, we have QI ^i Since all the processes of the Carnot cycle arc revorsiblo, it is evident that they may be traversed in reverse order. Thus starting from B, the fluid is compressed isothermal ly from Ji to A and gives up heat Q 1 to S; from A to .D it expands udiabal- ically, from D to (7 it expands at the constant temperature 7!> and in so doing receives heat Q 2 from Ji ; limilly it is com- pressed adiabatically from Q to the initial state. H. In this ease the work TF represented by area ABCD is done nn tin; lluid ,17, heat Q z is taken from the refrigerator 7, and the sum Q z -\- A \V = Q 1 is delivered to the source 8. This ideal reversed, engine is the basis of our modern refrigerating machines. 38. Carnot's Principle. The efficiency of Carnot's ideal engine evidently depends upon the temperatures 7 r , and r l\ of the source and refrigerator, respectively. Thu question at once, arises whether the efficiency depends also upon thu properties of the substance M used as a working iluid. The answer is contained in Carnot's principle, namely : Of all engines working between the mme sour*: nnJ tlir n<inn> refrigerator, no engine can have an efficiency </r<>at<'r than Unit of a reversible engine. In other words, all reversible engines working Ix-tweei. tins same temperature limits T, and 2!, have the same efficiency; that is. the o-ffimo, ^.3 7 , ",. .-, , . . ._ emcient than, a reversible engine B working between the same temperatures, then A and B can be coupled together in such a way as to produce available energy without a compensating loss of availability. Suppose the two engines A and B (Fig. 19) to take equal quantities of heat Q^ from the source when running direct. Then, since by hypothesis A is the more efficient, and Now let engine B be run reversed. It will take heat Q Z B from R and deliver Q 1 to S. If A and B are coupled together, A will run B reversed and deliver in addition the work W A W B . The source is unaffected since it simultaneously receives heat Q l and gives up heat Q r The re- frigerator, however, loses the heat Q z a Q Z A , which is the equivalent of the work W A W B gained. We have, therefore, an arrangement by which unavail- able energy in the form of heat in the reservoir is transformed into mechanical work. In other words, by a self-acting process the available energy of the system of bodies $, R, A, and B is increased. According to the second law (Art. 36), such a result is impossible ; if such a result were possible, power in any quantity could be obtained from the heat stored in the atmosphere without consumption of fuel. The assumption that engine A is more efficient than the reversible engine B leads to a result that experience has shown to be impossible. We conclude, therefore, that the assumption is not admissible and that engine A cannot be more efficient than engine B. But if engine A is also reversible, B cannot he morfi p.ffimfint than A. and it follows that all reversible FIG. 19. 54 THE SEUUJNU JUAVV vx Ai-Ai^w. "*'" ........ i. engines between the same source and the same refrigerator are equally efficient. 39. Determination of the Efficiency. Since the omoieney of the reversible Carnot engine is independent of the properties of the medium and depends upon the temperatures of .source and refrigerator only, we have i_7"_? = / ( 7\, 7 ! 2 ) , ) I <i/i whence = 1 - iy == JF(2i, 2^) ; (-) that is, the quotient ~f is some function of iho temperatures Vi 2\ and T^. The form of this function in required. So far, we have considered temperatures as given by a mer- cury or air thermometer. The different temperatures of a series of bodies are indicated by sets of numbers which may denote (1) the different lengths of a column of mercury or (2) the different pressures of a mass of confined gas. These sets may or may not precisely agree. Now there are other ways in which such a set of numbers may be chosen. Suppose we take several sources of heat /S^ M v >S' 3 , , *S', ( , whoso tem- peratures are t^ 2 , 8 , , t M as defined by the mercury or gas scale, and let *i>*a>8 >>*,. If we use S l as a source and S z as a refrigerator, a reversible engine will take Q^ from S l and deliver ^ to /S' 2 . vSinuo the bodies S 1 and ^ 2 have definite temperatures T L and 7!,, what- ever the scale adopted, the function .F(T V 2! 2 ) lias some defi- nite value; therefore, from (2) the fraction ^ must have a Vi definite value, and consequently @ a has one and only one value. If ^ 2 is used as a source and S 9 as a refrigerator, a second engine taking Q z from # 2 will give up Q z to N 3 , and so on. Starting with Q v we thus obtain a determinate set of values O n f} Qfn TrrVvi <-,!-> ,^,,.,4. JJICIl J.I 1 . JIT. 39] KELVIN'S ABSOLUTE SCALE 55 rlere we have a set of numbers suitable to define a scale of leruperature. Starting with the heat Q 1 taken from the source 1, to each source there corresponds a number indicating the leat that would be rejected to it if it were used as a refrigerator n connection with S v If we choose these numbers to define a iew scale, then denoting the new temperatures by T T 7 T T -*!' -t-y *& ' 'i J-ni ve have T^kQv T 2 = kQ T 3 = kQ B , ..., T n = kQ n , vhence follows <?i_02_ -&L m m r/j ' " rn ' \ ) Li J-z -'n Returning now to the quotient ~, we have at once lence, using this new scale, the efficiency of the Carnot engine s uid the form of the function is determined. The scale of temperatures arrived at from the investigation )f Caruot's cycle was first proposed by Lord Kelvin in 1848, i/nd is known as the absolute scale because it is independent of he property of any substance. The scale is simply such that my two temperatures on it are proportional to the quantities )f heat absorbed and rejected by a reversible Carnot engine vorking between these temperatures. If in (5) we make Q z = 0, tj = 1 and T z = 0. If we con- lei ve a temperature lower than the zero on the absolute scale, T T hat is, if we assume a negative value for jT 2 , then ~ - > 1, be shown subsequently that tins absolute zero is precisely the same as that derived from the reduction in pressure, of ;i perfect gas, and that the new scale coincides with Hint of a ther- mometer using a perfect gas as a iluid. 40. Available Energy and Waste. Caruot\s ideal eyele gives us a means of measuring the available energy of a system and the waste due to an irreversible change of state. Suppose, that, a quantity of heat A$ is absorbed by the system at a. tempera- ture T, and that we wish to find the part of this heat, that can possibly be transformed into work. As we have seen, no device. can transform a larger portion of A Q into work than the ideal Carnot engine. If T Q is the lowest temperature that, can be T 7 r obtained for a refrigerator, the fraction - '" of A^> can be transformed into work by a Carnot engine, and this is, then-fore, the availability of A$ under the given conditions. The avail- able part of A$ is, therefore, T T / f f r * " / I - T and the waste is A Q -^ . The temperature T Q cannot be lower than that of surrounding objects, i.e. the atmosphere;* for even if a refrigerator could be found with a temperature lower than that of the, atmosphere, it could not be maintained in that state. Ilene.e, the tempera- ture of the atmosphere imposes a -natural limitation on the avail- ability of heat in the performance of work. EXAMPLE. If the absolute temperature of aourc is 1 ()<)()" F. and llisii, of the atmosphere is 520, the available energy in 1000 - 520 n . , = 0.48 of the. total i>ncr'y. Therefore, for every 1000 B. t. u. received from tlu> source i.>t, more Lluui 480B.t.u. can by any means whatever bo transformed into wurk, and at least 520 B. t. u. must be rendered unavailable * Possibly under special conditions a refrigerator whoso t.-mpcratun'. is p,-r- mnently Mow that of the atmosphere may exist; ,,/. the water of the. o,,an or of one of the great lakes. associated with certain important irreversible processes. (1) Conduction of Heat, Suppose a quantity of heat Q to pass by conduction from a source at a temperature T to another at lower temperature T z . At the original temperature the available energy was The same quantity of heat in the second source has the avail- able energy The available energy is, therefore, decreased by the quantity and the unavailable energy is increased by an equal amount. (2) Irreversible Conversion of Work into Heat. A common irreversible process is the conversion of Avork into heat in the interior of a system through the agency of friction. Examples are found in the flow of steam through nozzles and blades, and in the Motional losses due to internal whirls and eddies in fluids. Heat thus produced we shall denote by the symbol H, reserving Q to denote heat brought into the system from outside. If now within the system the small quantity of heat A.H" is generated while the system remains at the temperature T, the part of AJ^Tthat is available is rrr rn / rn A TT J -~ where, as usual, T denotes the lowest available temperature. Of the work Jb.II expended in producing the heat A//, the part may therefore be recovered in the form of work. The re- mainder is rendered unavailable. 05 To obtain the total increase of unavailable energy, when tho quantity of heat .fiTis generated, the temperature of the. system varying in the meantime, we sum the element of tin 1 , type just obtained. Thus if the temperature risen from T t to '1\ during the process, we have for the total waste (3) Free expansion of a {/as. The waste due, to free expan- sion, as described in Art. 85, may be determined by returning the gas to its initial state and observing the changes left in outside bodies. The compression indicated, by B' A' (Fig. 1(5') requires that work W, represented by area B' A' A^ v lie supplied from an outside body $ 2 . Another outside body /V., must receive from the gas heat Q equivalent to the work W. The gas, the the system S v has the same available energy as at first,, being restored to its initial condition; system A' 2 has lost available energy W=JQ; and system ;S' 3 has received energy JQ of which only part is available. On the whole, therefore, there is an increase of unavailable energy. The loss of availability duo to the original irreversible expansion of the ga.s (system A'j) is repaired in this system, but an equal loss is brought about in systems S z and S y It can be shown that the, waste thus in- curred is given by an expression of the form 7|, f'//|- / / 41. Entropy. The expressions for the increase of unavail- able energy derived under various conditions are alike in hav- ing TV the lowest temperature available for a refrigerator, as a factor. It appears, therefore, that the unavailable, energy changes with T ; the lower T () can be taken, the, smaller the waste and the larger the fraction of the heat supplied that can be transformed into work. The other factor in the expression must necessarily, for the sake of consistent units, have the form Q or J "^. To this measure of the change in the unavailable energy of the system ; an increase of entropy involves an increase of unavailable energy, and vice versa. We may formally define entropy as follows : If, from any cause whatever, the unavailable energy of a system is increased and if the increase be divided by T^ the lowest tem- perature available for a cold body, the quotient is the increase of entropy of the system. This definition requires close examination to obviate possible misconception. The " system " spoken of may be either a single substance, as the medium employed in a heat motor, or it may be all the bodies taking part in the process. Now, ac- cording as we take one or the other of these viewpoints we get a particular notion of the significance of the term entropy. To illustrate this point, let us consider a simple example. Suppose we have a fluid medium M and a source of heat S, as described in connection with the Carnot engine. We may direct our attention either to the system M alone or to the sys- tem M+ 8 composed of the medium and source. Let both M and S be at the temperature T and suppose that at this tem- perature heat Q is transferred from S to M. This is the ideal reversible transfer assumed in the description of the Carnot engine. In receiving Q the system M has its available energy f T\ increased by Q 1 1 -9 j and its unavailable energy increased by T Q \ J Q~m = ^oTjfr; hence by the definition just given the entropy of .system M is increased by j," At the same time system S has lost the energy Q and, therefore, the unavailable energy Q ~ ; hence the entropy of S is decreased by ~ It follows that the change of entropy of the system M+ S is zero. As the result of the reversible transfer of heat from 8 to M there is no change in the unavailable energy of the large system S + M and no change in the entropy of this system. Suppose now that sys- tem M. is again at temperature T, but that system S has a higher temperature T', as must be the case in any actual transfer 60 THE SECOND LAW OF THERMODYNAMICS ICUAI'. iv of heat. If now heat Q passes from /S y to M, the unavail- able energy of M is increased by <??}, as before, and tho increase of entropy of system M is ~ The system H has, however, m lost the unavailable energy (?-J, and its entropy has decreased by -- The system tf + M has had its unavailable energy in- creased by tlie amount (?-| - (?|j = ^| - -^)- The irre- versible transfer has therefore resulted in a not loss of available energy of this amount, and this degradation is accompanied by an increase of entropy ^ r -~ The result hero obtained for two systems may be applied to any number of systems. When we apply the notion of increase of entropy to tin; sys- tem composed of all the bodies involved in a process, in other words, an isolated system, we are led to the conception (hat the increase of entropy measures the degradation of energy in- cident to the process. If we combine this notion with that expressed by the second law, we arrive at the following im- portant principles : 1. Any process that can proceed of itself IK cteeoinpanii'd hi/ an increase of the entropy of the system of bodies involved in (he. process. 2. The direction of a process, physianl or eJievu'ral, tJnit own)'* of itself is such as will bring about an increase, of entropy in the system. These principles lie at the foundation of the application of thermodynamics to chemistry. 42. Second Definition of Entropy. While the conception of entronv as thp. ffl.nf-.nr f-.lmt w,n^-,,.,, ART. 42] SECOND DEFINITION OF ENTROPY 61 VA unavailable energy of this single system^involves an increase in the entropy of the system^ but, as we have seen, degradation does not necessarily follow, for the increase of unavailable energy of M may be compensated by an equal loss in some other system taking part in the process. We now inquire by what means the unavailable energy of the single system under consideration can be increased. There are at least three ways that are suggested from the previous discussion of available energy (Art. 40). (1) If energy is added to the system in the form of heat, the total energy of the system is increased, and consequently the unavailable energy is increased. If the heat A Q is thus added when the temperature of the system is T, the resulting increase of unavailable energy is If, as is generally the case, the temperature rises as heat is added, we shall have for the increase -*. / r '*2 ", T' (2) The unavailable energy may be increased by the con- version of work into heat through internal friction. As shown in Art. 40 (2), the increase of unavailable energy from this cause is (3) If the parts of the system are not at the same tempera- ture, there will be an irreversible flow of heat from one part of the system to another, and this will increase the unavailable energy. We may remove this source of unavailable energy by assuming that the system is at all times of uniform temperature throughout, an assumption that is usually justifiable. Neglecting this third effect, we have for the increase of un- available energy from state 1 to state 2, 62 whence by definition, the increase of entropy ia Now while the actual change of the system from state 1 to state 2 may, and usually does, involve Motional effects, wo can r.nn- ceive of a hypothetical change in which thesis internal irroversi- ble effects are entirely absent and in which the, increase, of unavailable energy is due entirely to the addition to the system of heat from some external source. Denoting by Q r tho heat thus added, we have for the increase of entropy involved in this particular process the integral The important question now arises: Does tho increase of en- tropy of the single system under consideration depend only upon the initial and final states or upon the path connecting the states? It is easily shown that the increase of entropy, like the increase of energy, depends upon the initial and final states only. For the change of energy is independent of the path; therefore, the change of the unavailable- part of the en- ergy, as determined by the constant temperature 7 r and the temperatures 2\ and T z at the initial and final states, is also independent of the path; therefore the change of entropy, which is the change of unavailable energy divided by .7 r , is also independent of the path. It follows that the integral T 3 r\ J -~ has the same value whether taken along the path r (Fig. 20) or any other reversible path r' . We may write, there- fore, where S denotes a function of the coordinates of tho system which, may be termed the entropy of the system. We have, then, the following definition : The change of entropy of a system correspond!,/!*/ to a clianye of the system from state. 1 tn st.nto 9 , f o *7, ,1,, /;,,.;/ ,;/,,,.,, 7 C'^'J^' ART. JU.MJliS5UAJLj.LTi: UJT UJUAUSIUS According to this more restricted conception, the entropy of a system, like the energy, pressure, or temperature, is a magni- tude determined by the state of the system, and change of en- tropy has no necessary connection with degradation of energy. It should be noted that entropy as thus denned is like energy purely relative. We are never concerned with the absolute value of the entropy of a system in a given state ; what is desired is the change of entropy associated with a given change of state. For convenience of calculation we assume the zero of entropy to be the entropy of a system in some specified state. Thus, in dealing with vapors we assume the zero of entropy to be the entropy of a unit weight of liquid at C. 43. The Inequality of Clausius. If an actual irreversible change be represented by the path i, Fig. 20 (assuming it to be possible to give such a repre- sentation), a correct value of the change cannot be obtained from .. y, .7 Q the integral ( '-77- taken along the path i. For as we have seen a f T * 'I* "i=J ' 7,7 T. -L -V FIG. 20. where 2 is the increase of en- tropy due to the internal irre- versible changes. For the actual irreversible change we have, therefore, This is the inequality of Clausius. 44. Summary. To present the important principles of this chapter in concise form and in logical order the following sum- mary is added. 1. Experience shows that heat energy is not completely transformable into mechanical work. The ratio of the energy 2. Experience further shows that an irreversible process always decreases the availability of a system. 3. The second law of thermodynamics asserts that tho avail- able energy of an isolated system cannot be increased by any process that takes place of itself. 4. To gain a means'of measuring availability the ideal ( -arnot engine is introduced. By the aid of the second law it is shown that no engine working between the same temperature limits can have an efficiency greater than tho Carnot engine, and as a consequence, that the efficiency of this engine is a function of the temperature limits only. 5. By the introduction of Kelvin's absolute scale of tempera- ture the efficiency of the Carnot engine is found to be given by T T the fraction - 1 2 . T T 6. Having the efficiency fraction 1 2, the available part A of a given quantity of heat Q at temperature T is found to bo $(l o] an( i the unavailable part, Q^ 7. By special examples of irreversible processes it is found that the expression for the loss of available energy in such pro- cesses has the general form ^T)- or 7\ } ( ' '^ . 8. The factor V-i or j"^. which multiplied by 7', gives tho increase of unavailable energy is called the incrcttM of cut ><>]> >/ of the system. 9. Two conceptions of entropy are possible: (a) If atten- tion be directed to all the bodies involved in a process, the increase of entropy of the whole system of bodies measures tho degradation of energy resulting from the process. (/>) If at- tention be directed to a single body, as a medium used in a heat motor, the entropy of this simple system is merely a function of the coordinates of the system. 10. The change of entropy of a simple system is given by / J.N JL/ OJ^XVY tlie initial and final states. The value of this integral is inde- pendent of the path. 11. For an irreversible change of state the change of entropy r 2 ' 2 dO is greater than \ %-. J y\ T 45- Boltzmann's Interpretation of the Second Law. A very clear insight into the real physical meaning of natural irreversible processes and of the second law of thermodynamics is afforded by the researches of Boltzmann and Planck. In this article it is possible to give merely a brief outline of Boltzmanu's contribution ; for a complete exposition the reader is referred to Professor Klein's admirable book, The Physical Significance of Entropy.* According to the molecular theory, the ultimate particles of matter are in a state of incessant motion, the character of the motion depending upon the state of aggregation, solid, liquid, or gaseous. In a gas it is assumed that a particle has a free path and moves along a straight line until it col- lides with another particle or with a restraining surface, as the wall of the containing vessel. To the motion of particles as to the motion of masses we may apply the conception of constraint or control. Thus, in the wave motions that characterize sound, the motion of the particles that constitute the mediums is in some degree controlled or ordered. The molecular motion that constitutes heat is, on the other hand, wholly uncontrolled and disordered. For any given particle of a gas all directions of motion are equally possible and, therefore, equally probable; and the direction of motion and velocity of any particle is independent of the motions of other particles. In a volume of gas particles will be moving in all directions with all possible velocities. However, because of the great number of par- ticles even in a small volume, the values of magnitudes that depend upon the molecular motion, such as pressure and temperature, remain constant notwithstanding the haphazard character of the molecular motion. According to Boltzmann, there is apparently a universal tendency toward the disordered motion that characterizes heat. A motion that is in any degree ordered or controlled tends to become disordered. Thus, as sound waves die out the uniform motion of the particles in the wave changes to disordered motion, and the energy of sound is transformed into heat energy. The relative motion of two bodies in contact is retarded by friction, and the work of overcoming friction is transformed into heat; that is, the constrained motion of the particles in the mass gradually changes to the disordered motion of heat. Since the energy of disordered molecular motion is necessarily less available for direction into any required channel than the energy of constrained or controlled motion, it follows that a change from a less probable state of controlled motion to a more probable state of /3ic.m..3ni.n>3 if.rvJ-i/i-.-. ; ,-.!-> r,-,-.,^ ,./-.iv^ a nnnrli firm nf (TVOflt.PV El.Vni1il.hlft fillftrCW to a condition of less available energy. II<mce, the statement of the. nal ural tendency toward disordered motion is iu reality a broad statement <l the second law of thermodynamics. From the preceding considerations a physical interprelation of entropy is readily deduced. A system of itself passes from a less probable, to a more probable state ; that is, to a state of mure disordered moleeular motion. The entropy of the system during the change must, inornaso. ^Therefore, the entropy of the system may bo associated with the. probability of tins state of the system. From the laws of probability, 1'lanek has shown that the entropy is proportional to the logarithm of the probability of th .slate. The following quotations from Prof. Klein's book indicate in some degree, the significance of this conception of entropy. " Growth of entropy is a passage from a somewhat regulated to a less regulated state." "Entropy is a universal measure of the disonli'r in the mass points of a system." "Entropy is a universal measure of tho spontaneity wit.li whie.h a system acts when it is free to change." "Growth of entropy is a passage from a concentrated (.<> a disl.ribul.cil condition of energy; energy originally concentrated variously in t.lm system is finally scattered uniformly in said system. In this aggregate aspect, it is a passage from variety to uniformity." EXERCISES 1. If a source of heat has an absolute tomporaturo of MOO" F. and tho lowest available temperature is 525 F., what fraction of tho beat drawn from the source is available ? 2. In a boiler 10,000 B. t. n. pass from the hot gases of the. fnrnat'e, tin; temperature of which is 2500 F., through the boiler shell into water at a temperature of 330 F. If the lowest available temperature is 80" F., iind the loss of available energy. 3. Show how the result of Ex. 2 suggests tho superior dlhuency of the gas engine compared with the steam engine. 4. Point out the loss of available energy when heat Hows from steam in a radiator at a temperature of 225 into a room at 70". J)evisu a system of heating that would obviate this loss. 5. A mass of water weighing 60 Ib. at a temperature of 70" F. is churned by a paddle wheel until the temperature rises to 120. Find the increase, of entropy, and the loss of available energy. Take the spec.itu- heat of water as 1. 6. In the demonstration of Garnet's principle, Art. 158, ;iKsnme the two engines A and B to do the same work W. Then show that if. emrine A ART. 45] LITERATURE ON THE SECOND LAW 67 REFERENCES REVERSIBLE AND IRREVERSIBLE PROCESSES Planck : Treatise on Thermodynamics, Ogg's trans., 82. Bryan : Thermodynamics, 34, 40. Klein : Physical Significance of Entropy, 29. Chwolsou : Lehrbuch der Physik 3, 443. Parker : Elementary Thermodynamics, 105. THIS SECOND LAW. ENTROPY Sudi Carnot: Reflections on the Motive Power of Heat. Translated by Tlrarston. Claxisius : Mechanical Theory of Heat. Rankine: Phil. Mag. (4) 4. 1852. Thomson : Phil. Mag. (4) 4. 1852. Franklin : Phys. Rev. 30, 770. 1910. Lorenz : Teehnische Warmelehre, 104. Chwoison : Lehrbuch der Physik 3, 485, 497. Bryan : Thermodynamics, 43, 57. Preston : Theory of Heat, 025. Klein : Physical Significance of Entropy. Magie : The Second Law of Thermodynamics (contains Garnet's " Reflec- tions" and the discussions of Clausius and Thomson). Planck: Treatise on Thermodynamics (Ogg), 86. Parker : Elementary Thermodynamics, 104. CHAPTER V TEMPERATURE ENTROPY REPRESENTATION 46. Entropy as a Coordinate. It was shown in Art. -12 that the entropy of a system measured from an arbitrary /ero is dependent only upon the state of the system ; that, in, tho entropy is a function of the coordinates of the system. It follows that the entropy itself may bo included amon^ tho coordinates used to define a system. We have, therefore, live coordinates, namely, p, v, T, u, and 8, that may bo thus used. From these five, ten pairs may be selected, and the change of state of a system may be represented by ten different curves on ten different planes. Of these possible graphical representa- tions two are of special importance : (1) representation on tho jpF-plane, because the area between the curve and /'-axis repre- sents the external work done by the system; ( L 2) representa- tion on the T$-plane, because with certain restrictions tho area under the curve represents the heat absorbed, by the system from external sources. Graphical representations on the, ^f- plane have been considered in Art. 82. This chapter will be devoted chiefly to representations on the 2W-plano. From the second definition of entropy, we have -%> CO - I J^ from which relation we obtain at once the differential forms and TdS=dQ. Let the curve ATt But from (3) this integral is the heat Q lz absorbed by the system from external sources during the change of state. It follows that the area between T the curve AB and the axis OS represents graphically the heat absorbed along the path AB. One most important restriction must, however, be observed. In defining entropy by means of equation (1) it was expressly stated that the change of state ^ J?i must not involve any internal irreversible effects. If such effects are present, the equation for the change of entropy is where S denotes the increase of entropy due to internal processes, conduction between the parts of the system, trans- C T tdO formation of work into heat through friction, etc., and J r - is the increase of entropy due to the absorption of heat from external bodies. From (4) it follows that in this case whence * a _ a ~ * v dQ<TdS, (5) or the heat absorbed from outside is less than the area between the 2W-curve and the #-axis. This area therefore may be taken as representing the heat absorbed by the system when, and only when, the change of state involves no irreversible effects. Neglect of this restriction has led to many errors. 47. Isothermals and Adiabatics. If the temperature of the system remains constant during the change of state, the M/V ** ~ D A' FIG. 22. In this case we have merely to divide the heat added to the system (assuming, of course, that the change of state is revers- ible) by the constant tempera- ture T, and the quotient is the change of entropy. If the state point passes from B to A, that is, so as to de- crease the entropy, the area A-^ABB^ represents heat re- jected by the system to outside bodies. For an adiabatie change of state, dQ = ; hence from (1) $, = h\ and tlio adiabatie line on the 5W-plane, if the change of .state involves no irreversible effects, is a straight line parallel to the .'/-axis, as (.11) ( Kig. IW). If the state-point moves from Oio D, indicating a decrease of tem- perature, external work is done by the system, and tlio ehange of state is an adiabatie expansion. If tlio point moves upward from D to (7 the change of state is an adiabatie compression. 48. The Curve of Heating and Cooling. From the equation do G = *- , which defines the specific heat of a substance, we have (1) Substituting this expression for ity in (1), Art. 4(1, we got for a reversible process . T If the specific heat c is constant during tlio change of state, we have for the change of entropy of unit weight of the sub- stance For the weight If, (3 a) If, however, c is variable, it can usually be expressed as a func- tion of the temperature ; that is, we can write whence T, . T (4) The integration can be effected when the function /(2 1 ) is known. EXAMPLK. Let tho specific heat o'f a substance be given by the relation c = a + W = a + &(r-450.G); wo have then r T <1 T rT s a - 8 L = (a - 4-59.0 b) i - ~~ - + I \ dT J r t 1 J ?*[ = ( - 459.0 6) log, ^ +b(T s - J 1 ,). ^i The general form of the curve that represents Eq. (3) is shown in Fig. 28. This curve represents the ordinary pro- cess of heating* a body or sub- stance, as the healing of water iu a boiler or metal in a furnace. It is called by some writers the polytropic curve. The subtan- gent of the curve is constant and numerically equal to the specific heat. Thus from the F E FIG. 23. figure we have ~~ ~d^~ dT It follows that the smaller the value of c, the greater the slope of the curve. The isothermal and adiabatic curves (Fig. 22) ms?y be con- . ^f 4-T->^ V,/->n'f-i-nn. oi-irl nn nil nor miTVP.. T^OT 72 TEMPERATUJRJfi |niAi>. v Cases may arise in which tluj slope of the 2/S'-ourvo is nega- tive, as sliown in Fig. 24. In such cases abstraction of lioat i.s accompanied by a rise in tem- perature or vine ve-rxa. Evidently the speciJic heat ff1 niu.st bo it ,L negative, as is indicated geo- metrically by tho negative sub- tangent. Examples will be shown in the compression of air in the ordinary air compressor, and in the expansion of dry saturated steam with the provision that it remains dry during the expansion. :iy bo series 49. Cycle Processes. Since any reversible process m sliown by a curve in ^-coordinates, it follows that a of such processes forming a closed cycle may be repre- sented by a closed figure on the 2E-plane. In Fig. 25 is shown such a cycle composed of two polytropics AB and DE, an isothermal J3C, and two adiabatics CD and HA. In any such cycle the area included by the cycle repre- sents the net heat added to (or abstracted from) the work- ing fluid during the cycle process. Assuming the cycle to bo traversed in the clockwise sense, we have ab = area Q bc = area 1 BOO V = ABODE. the cycle is traversed in the counterclockwise sense, we have it from the first law, Q is the heat transformed into work; nee for the direct cycle area AS ODE = Q = AW, d for the reversed cycle area AB ODE = -Q**-AW. This reasoning evidently holds for any number of processes, d therefore for a reversible >sed cycle of any form. Thus ? the cycle shown in Fig. 26, > have area F= Q = AW, area F=- = 3ording as the cycle is traversed the clockwise or counter clock- se sense. - FlG _ 2(1> tn later developments it will quently be necessary to show cycle processes on the iZW-plane. )0. The Rectangular Cycle. When the curves representing s four processes of the Carnot cycle are transferred to the 2%'-plane, the cycle becomes the simple rectangle ABQD, Fig. 27. The area A^ABB^ represents the heat Q 1 absorbed by the medium from the source during the iso- thermal expansion AB, and the area B^CDAy the heat Q z rejected to the refrigerator during the isothermal compression CD. The lines BO and DA represent, respectively, the adiabatic expansion and the adia- FIG. 27. batic compression. Tn B 74 TEMPERATURE ENTROPY RKPUKSKNTATION [CHAP, v From the geometry of the figure, we have A IV T-,- T,, whence f] = ^ ~ ~rn " as already deduced in Art. 89. When the cycle is traversed in the counterclockwise souse, the heat Q 2 is received by the medium from the cold body during the isothermal expansion J9(7, and the larger amount of beat Q t is rejected to the hot body during isothermal compression JiA. The difference $ 2 ^ == J. T7 represented K v ^ 10 t! y^ u ari!Jl is the work that must be done on the medium, and must there- fore be furnished from external sources. The reversed heat engine may be used either as a rof rigerating machine or as a warming machine. In the lirst case Uie space, to be cooled acts as the source and delivers then heat Q z = area A 1 DCB 1 to the medium. In the second case the space, to bo warmed receives the heat Q 1 = area B l BAA l from the medium. 51. Internal Frictional Processes. Referring to Art. 4U, the increase of entropy when heat is generated in the interior of a system is seen to be 2 1 ~ ^ j\ ~T J r ~T r ' If $=0, that is, if no heat enters the system from outside sources, the increase of entropy is and is due entirely to the generation of beat in the interior of the system. If it be assumed that this process is steady, so that the system at every instant is approximately in thermal equi- librium, the usual graphical representation may be applied to (2), and the area under the 2^-curve will in this ease repre- sent not the hfifl.t bvnno-ht. infn +!IQ c.irfi 1^,^- 4.1,., 1 4. 77 A FIG. 28. int A (Fig. 28) lias its pressure decreased in passing along the zzle, and as a result the temperature likewise falls. The Dceas is adiabatic, that is, no heat received from external bodies; nee, if there were no internal ction, the drop in temperature iuld be indicated by a motion of 3 state-point along AA r But irk is expended in overcoming 3 friction between the fluid and rale wall. This work is neces- :ily transformed into heat, which retained by the fluid. It follows it there is an increase of entropy, as indicated by the curve AB. om (2) the heat generated is represented by the area A 1 AS r 52. Cycles with Irreversible Adiabatics. In certain cases the >sed cycle of operations of a heat motor may contain an adia- tic irreversible process, the irreversibility arising either from :ernal generation of heat or from the free expansion or wire- awing of the working fluid. Even if it is possible to draw a T&curve representing such a process, the area under that curve does not represent the heat entering the system from an external source. Hence some care is required to inter- pret properly the graphical representations of cycles with such irreversible parts. In the cycle shown in Fig. 29, suppose the process SO to be FlG> 29 ' an irreversible adiabatic, the ler parts of the cycle being reversible. Since AB is revers- .e, the heat absorbed in passing from A to B is represented by 3 area A 1 ABB r Likewise area C 1 ODA 1 represents the heat lected by the system in changing state from to D. The . v 76 TEMPERATURE ENTROPY REPRESENTATION I process DA is adiabatic, hence $ llB Oj and by hypothesis = 0. The value of 2# for the cycle is, therein, V((6 ~i~ VJ " ^' 1 ' 1 r T7 r/r = area ABKD - area /^M- ( r The energy equation applies to any process, reversible or irreversible. Therefore for this cycle, as for those previously considered, we have FIG. 30. It appears, therefore, that, the work derived is less by tho area B- l KOO i than il, would have been il: tho reversible adiabalie .3 BE had been followed. For the reversed, cycle (Fig. r'50) we have as the work required from external sources W=J(Qaa+ Q^ = ~ aron D V DAA 1 + area ./^/iOf^ Comparing this cycle with the cycle A.E<JD having the re.vers- ible adiabatic AZ7, it is seen that the heat absorbed from the cold body is smaller by the heat represented by the area A^EBBy while the work required to drive the machine is greater by an equal amount. In every case tho irreversible process results in a reduction of the useful effect. 53. Heat Content. Since the quantities p, 7', .7 r , H, and s are, function of the state of a system only, it follows that any com- bination of these quantities is likewise a function of the state only. For example, let (V) r. 53] HEAT CONTENT 77 tentials, and are used in certain, applications of thermo- namics to physics and chemistry. The function I has use- L applications in technical thermodynamics. To gain a physical meaning for the function I, let us consider 3 process of heating a substance at constant pressure. If t/p , and p l denote the initial energy, volume, and pressure, jpectively, and 7 2 , V y and p z the final values of the same jrdinates, we have from the energy equation since p z = p 1 = A[U z ~C7 l tat is, the change in I is equal to the heat added to the sys- n during a change of state at constant pressure. For this ison I is called the heat con- it of the system at constant I essure, or, more briefly, the ieat content." In some subsequent investiga- ns, especially those relating to 3 How of fluids, it will be con- :iient to use / and S as the in- pendent variables and to repre- it changes of state by curves on Q . i /xS-plane. The great advantage the /^-representation over the '-representation lies in the fact that in the former quantities heat are represented by linear segments, while in the latter, we have seen, they are represented by areas. A reversible .abatic on the J&plane is a vertical line, as BQ (Fig. 31). t in this diagram segment BO represents a quantity of heat tead of a change of temperature,, FIG. 31. 2. Assuming that the specific heat of water is constant, c 1, plot uu cross-section paper the rS-cuwo reproHonting the heating tif water from 32 to 212. 3. Langen's formula for the .specific heat of CO., ut constant pressure in c = 0.195 4- 0.000066 t. Find tlm increase, of entropy when CO., is healed a^t constant pressure from 500 to 2000 F. ; aim) tlm heat, absorbed. 4. A direct motor operates on a rectangular cycle between temperature, limits ^=840 and T z = 000 and reeeivos from the. source 'J(K) 15. t,. u. per minute. Find the efficiency, and the work don<>, per ininuti 1 . 5. A reversed motor, rectangular cycle., operates between temperature limits of 10 and 130, and receives liOO It. t. u. per minnln from the cold body. Find the heat rejected to the hot body, and the. horsepower required to drive the motor. 6. A direct motor, rectangular cycle, operating between temperatures 2\ = 900 and T 2 = 080, takes 1000 B. t. u. from a boiler. The heat rejected is delivered to a building for heating purposes. This direct, motor driven a reversed motor which operates on a ra'.tangnlar cycle between tempera- tures r 4 = 460 (temperature of outside, atmosphere) and 7'., : <>00. The reversed motor takes heat from tho atmosphere and rejects heat io the. building. Find the total heat delivered to tho building pur 1000 It. t. u. taken from the boiler. 7. In the vaporization of water at atmospheric pressure, the. temperature remains constant at 212 F., and 970.d B. t. vt. arc required for the process. Find the increase of entropy. 8. The expression for the energy U for a given weight of a permanent pV gas is _ | + U m where k and U are constants. Derive an expression for the heat content I of the gas. 9. Combine the energy equation dQ = AdU + AjxlV \\illi tho deiining equation I = A ( U + p 7) and show that d I = d Q + A } 'dp. REFERENCES USE OF TEMPEUATUUE-ENTHOPY COOUPINATKS Berry : The Temperature-Entropy Diagram. Sankey : The Energy Diagram. Boulvin : The Entropy Diagram. Swinburne : Entropy. USE OF HEAT CONTENT AND ENTROPY AS COOUDINATKS Berry : The Temperature-Entropy Diagram, 127. Mollier: Zeit. des Verein. deutscher Ing. 48 271. CHAPTER YI GENERAL EQUATIONS OF THERMODYNAMICS 54. Fundamental Differentials. The introduction of the entropy s and the functions i, F, and $ (Art. 52) permits the derivation of a large number of relations between various thermodynamic magnitudes. While the number of formulas that can be thus derived is almost unlimited, we shall intro- duce in the present chapter only those that will prove useful in the subsequent study of the properties of various heat media. In this article we shall by simple transformations express the differentials of u, i, F, and <J> in terms of the differentials of the variables jp, v, T, and s. We have to start with the fundamental energy equation dq = A(du + pdv), (1) and for a reversible process the relation dq=Tds. (2) Combining (1) and (2), we obtain T du = ds~-pdv, (3) A an equation that gives u as a function of the independent varia- bles s and v. From the defining equation we have di = Adu + Ad (pv) = Adu + Apdv + Avdp. Introducing the expression for Adu given by (8), we get 80 GENERAL EQUATIONS OF THERMODYNAMICS [CHAP, vi Here i is given as a function of s and^> as independent variables. Likewise, from the relation I=Au- Ts, dF '= Adu - Tds - sdT '; whence from (3) - dF = sdT + Apdv. (5) Finally, from the defining relation <E> = Au + Apv Ts, d$> = Adu + Ad(pv) - d(Ts^) = Tds Apdv + Apdv + Avdp - Tds sdT; or d = Avdp - sdT. (6) Now since the functions w, i, JF, and <& depend on the state only, their differentials are exact ; hence the second members of (3), (4), (5), and (6) are all exact differentials. Certain results can be deduced at once from the differential equations (3)-(6). For example, from (6), if a system changes state reversibly under constant pressure and at constant tem- perature, the function $ remains constant. Again from (5), if a change of state occurs at constant temperature, the external work clone is equal to the decrease of the function F. These results are important in the application of thermodynamics to chemistry. 55. The Thermodynamic Relations. The fact that the dif- ferentials in (3), (4), (5), and (6) of the last article are exact gives a means of deriving four important relations. In (3) we have u expressed as a function of the variables s and v; that is, M =/(*> v), whence du = ds+~dv. ds 3v Comparing this symbolic equation with (3), it appears that dv\dsj Bs\dv)' that is, Adv^ J r)8 (If) =-*() (A) The subscripts denote the variables held constant during the differentiations indicated. Relation (A) may be expressed in words as follows : The rate of increase of temperature with respect to the volume along an isentropic is equal to A times the rate of decrease of the pressure with respect to the entropy along a constant vol- ume curve. That is, if the reversible change of state be repre- sented by curves, one on the 2Vplane, another on the jps-plane, the slope of the second curve at a point representing a given state is A times the slope of the first curve at the point that represents the same state. In (4) we have s and p as the independent variables ; and since di is exact, the necessary condition of exactness gives dp dp. \dsp That is, the rate of increase of temperature with respect to the pressure in adiabatic change is A times the rate of increase of volume with respect to the entropy in a constant-pressure change. Since in (5) dF is an exact differential, we have From (6), likewise, we obtain The relations given by (A), (B), (C), and (D) are known as Maxwell's thermodynamic relations. They hold for all (C) and (D) by means of the relation Us = ~ \, aro usolul : (CV) dpT \<Y/'v 56. General Differential Equations. From tho thenno- dynamic relations certain useful general equations arc at oneo deduced. As in Art. 19, we may write according as T and v or 27 and ^ are taken a.s tho indopondont variables. Now replacing (-^-\ and ("--^ ^)} r '' ml. ( ' ; ,i r - \o- yj, yd /. y^ ; spectively, and (~2 ) and (-2 ) by tho exprcission.s i^ivon in VSu/y \dpjy ((7') and (i>') 5 these equations become, rospectivoly, \dv, (I) Eliminating dT between (I) and (II), a third equation having p and v as the variables is obtained. Thus Two other important equations may be derived from (I) and (II). Since from the energy equation du = Jdq pdv, we have from (I) di = c p dT- A - tjp. (V) The general equations (I)-(V) hold for reversible changes I state. The partial derivatives involved may be found from he characteristic equation of the substance under investi- ation. As an application of (IV), we may derive expressions for the lange of energy (a) of a gas that follows the law pv = BT ; b") of a gas that obeys van der Waals' equation ence (a) From the characteristic equation pv J3T, we have *\ =*. dT) v v ' /"ftrji du = Jc v dT+( ^-- \ v = Jc v dT, rid u z u 1 = = ssuming c v to be a constant. (5) From van der Waals' equation, we have B dTj v v-b' , r^rf d P\ 'henoe ^^ 'rom (IV), we have, therefore, du = Jc,dT+dv, v 2 rhence, assuming again that c v is constant, It appears, therefore, that if a gas follows the law jw = IW\ the energy is a function of the temperature only, while ii il. follows van der Waals' law, the energy depends upon the temperature and volume; in other words, the gun possesses 1)uth kinetie and potential energy. . 57. Additional Thermodynamic Formulas. For certain in- vestigations of imperfect gases, especially the superheated vapors, certain formulas involving the specific. heals <> and c v are useful. The most important of these urn (VI;, (VII), and (VIII) following. Since du is an exact differential, wo obtain, upon applying the criterion of exactness to (IV), whence = A (VI) \dvJ T \dT*J v In a similar manner, since di is exact, we have from (V) Equations (VI) and (VII) may be used to show tho depend- ence of the specific heats c v and c v upon tho pressure and vol- ume. For example, if a gas follows the equation pv BT we find - = 0, whence from (VII) <1 \ = 0, and it follows that dpji' c p does not depend upon the pressure, though it may vary with the temperature. Also _JL=: 0, whence it follows that e v does 9T* not vary with the volume. The student may show that the second result follows from van der Waals' equation or from any equation in which p and T appear in the first degree only. If, however, we take the characteristic equation hich applies to superheated steam, we obtain hence (?**} - A <n + V)(l + \ap/r T n+1 itegrating this with 27 constant, we have here 0(27), an arbitrary function of T, is the constant of inte- ation. In this case it is seen that c p is a function of both T Lcl p. An expression for c p c v is obtained as follows : Writing the itropy s as a function of p and v, we have d8 = dp + ~dv. dp dv bis, combined with the familiar equation Adu = Tds Apdv, 3o flo ves the equation Adu = T dp + (T -- Ap~)dv. nee du is an exact differential, we have L( T ?\=*-(T& dv\ dp} dp\ dv at is, dv dp dvdp dp dv dpdv , dTds dTds A f ^ tience ----- = A. (1) dv dp dp dv :om the definition of specific heat, we have G= ^-=T^ dT dT" .d if we express both s and T as functions of p and v, this re- fcion becomes ^-dp+^dv Q ~T dp dV _ (2) *-* dT 7 , dT J ' ^ } If p is constant, c= c v and dp = ; lunico wo huvu from (^'2 i 1 c v ~TlJL. ~dt> Likewise, when v is constant we; have 00 c ~ (4) Combining (3) and (4), we obtain dv dp t Making use of (1), we get finally c n o EXAMPLE. For the character! stio uquatiou;> = 7J7', \vn huvo = ~ dlL = Ji dT p' dT~'v' Therefore, from (8), c p - Cv =A^r ==AB .BT = pv p v That is the difference Cp - Cv is constant (JV( , U jf an(1 ith t , temperature. Taking Zeuuer's equation for superlieatod atoani, vi/,: pv = BT~ Cp", we have j?l _ ^ Jg. .B_ 32 1 j' 32 7 nC>''-HV whence c p - c v = ^5 ^H__ _ y| ^ JIT __ n Cp n + p v ( n _ 1 )c yl ^. 7/7" In this case, therefore, the difference c , - c v varies with 7' and p. By varies substitutions and transformations wo c,mld add Sr md " finitel y to this "at of thermodynamic fc.rnmlas. However the eight formulas (I)-(VIII) arc suflioiont fop the mvestifirationof nnnriwon -^.i... ,-, .apter T must necessarily denote the temperature defined by e Kelvin absolute scale. The coincidence of this scale with e perfect gas scale will be shown in the next chapter. 58- Equilibrium. For irreversible processes the equations of Art. 54 ist be replaced by inequalities. Since for an irreversible process, dq<Tds, (]) [. (3), (4), (5), and (6) of Art. 54 become, respectively, Adu<Tds - Apdv, (2) di < Tds + Avdp, (3) -dF>sdT + Apdu, (4) d$<Avdp-sdT. (5) From the inequalities (4), (5), and (1) the following conclusions are at ce apparent : 1. If the temperature and volume of a system remain constant, then from ), rZZ' T <0. That is, tJF must be negative, and any change in the system, ist result in a decrease of the function F. 2. If the temperature and pressure remain constant, as in fusion, and porization, theii from (5), d$ < 0. Hence any change in the system must such as to decrease the function cfr, 3. If the system be isolated, q = 0, and from (1), tfs>0. Hence in an dated system any change must result in an increase of entropy. The conditions of equilibrium are readily deduced from these conclusions, ider the condition of constant T and v, change is possible so long as F a decrease. When F becomes a minimum, no further change is possible d the system is in stable equilibrium. Likewise, with T and p constant, ible equilibrium is attained when the function $ is a minimum. The fiinctions F and $ are evidently analogous to the potential function in mechanics. A mechanical system is in a state of equilibrium when 3 potential energy is a minimum, and similarly a thermodynamic system in equilibrium when either the function F or the function $ is a minimum. >r this reason F and <$ are called thermodynamic potentials. By the use of thermodynamic potentials, problems relating to fusion, porization, solution, chemical equilibrium, etc., are attacked and solved. EXERCISES 1. From (V) derive an expression for the change of the heat content i len a gas following the law/w = BT changes state. 2. If the gas obeys van der Waal's law, find an expression for the ange of the heat content i. 3. Apply equations (II), (IV), and (V) to the characteristic equation superheated steam, GENERAL EQUATIONS OF THERMODYNAMICS [CHAP, vi 4. Callendar has proposed for superheated steam the equation Apply (VII) to this equation and show that c is a function of p and T. 5. Give geometrical interpretations of the thermodynamic relations (C) and (D). 6. From (I) and (II) derive expressions for dq and also for y for a gas following the law pv - BT. Show that the expressions for ^ are iutegrable, while those for dq are not. 7. Derive (VI) and (VII) by the following method: Divide both mem- bers of (I) and (II) by T, and knowing that ^ = ds is exact, apply the criterion of exactness to the resulting differentials. 8. Deduce the following relation between the specific heats and the functions F and r -v rr& F n\ (a) c.= -r_; (6) c,= 9. Using temperature-entropy coordinates, deduce a system of graphical representation for the three magnitudes Q, U 2 U v and W that appear in the energy equation. Suggestion. Through the point representing one state draw an iso- dynamic, through the other point a constant volume curve. REFERENCES GENERAL EQUATIONS OF THERMODYNAMICS Bryan: Thermodynamics, 107. Preston : Theory of Heat, 637. Chwolson : Lehrbuch der Physik 3, 466, 505. Buckingham : Theory of Thermodynamics, 117. Parker: Elementary Thermodynamics, 239. EQUILIBRIUM. THERMODYNAMIC POTENTIALS Planck: Treatise on Thermodynamics (Ogg), 115. Gibbs : Equilibrium of Heterogeneous Substances. CHAPTER VII PROPERTIES OF GASES 59. The Permanent Gases. The term "permanent gas" survives from an earlier period, when it was applied to a series of gaseous substances which supposedly could not by any means be changed into the liquid or solid state. The recent experimental researches of Pictet and Cailletet, of Wroblewski, Olszowski, and others have shown that, in this sense of the term, there are no permanent gases. At sufficiently low tem- peratures all known gases can be reduced to the liquid state. The following are the temperatures of liquefaction of the more common gases at atmospheric pressure : Atmospheric air - 192.2 C. Nitrogen - 193.1 C. Oxygen - 182.5 C. Hydrogen - 252.5 C. Helium -263.9C. It appears, therefore, that the so-called permanent gases are in reality superheated vapors far removed from temperature of condensation. We shall understand the term " permanent gas " to mean, therefore, a gas that is liquefied with difficulty and that obeys very closely the Boyle-Gay Lussac law. Gases that show considerable deviations from this law because they lie relatively near the condensation limit will be known as super- heated vapors. 60. Experimental Laws. The permanent gases, at the pres- sures usually employed, obey quite exactly the laws of Boyle and Charles, namely : 1. Boyle's Law. At constant temperature, the volume of a given weiaht of aas varies inversely as the pressure. z. v/nanes sure , of a gas is proportional to the change of temperature. By the combination of these laws the characteristic equation pv = BT is deduced. (See Art. 14.) In this equation T denotes absolute temperature on the scale 'of the gas ^ther- mometer, and not necessarily temperature on the Kelvin absolute scale. The classic experiment of Joule showed that permanent gases obey very nearly a third law, namely : 3. Joule's Law. The intrinsic energy of a permanent gas is independent of the volume of the gas and depends upon the temper- ature only. In other words, the intrinsic energy of a gas is all the kinetic form. Joule established this law by the following experiment. Two vessels, a and 6, Fig. 32, connected by a tube were immersed in a bath of water. In one vessel air was compressed to a pres- sure of 22 atmospheres, the other vessel was exhausted. The tem- perature of the water was taken by a very sensitive thermometer. A stopcock G in the connecting tube was then opened, permit- ting the air to rush from a to J, and after equilibrium was es- tablished the temperature of the No change of temperature could be FIG. 32. water was again read, detected. From the conditions of the experiment no work external to the vessels a and 6 was done by the gas ; and since the water remained at the same temperature, no heat passed into the gas from the water. Consequently, the internal energy of the air was the same after the expansion into the vessel 5 as before. Now if the increase of volume had required the expenditure of internal work, i.e. work to force the molecules apart against their mutual attractions, that work must necessarily have come from the internal kinetic energy of the gas, and as a result the temperature would have been lowered. As the temperature remained constant, it is to be inferred that no such internal was required. .a. gas nas, uiereiore, no appreciaoie inter- nal potential energy ; its energy is entirely kinetic and depends upon the temperature only. Joule's law may be expressed symbolically by the relations : The more accurate porous-plug experiments of Joule and Lord Kelvin showed that all gases deviate more or less from Joule's law. In the case of the so-called permanent gases, air, hydrogen, etc., the deviation was slight though measurable ; but with the gases more easily liquefied, the deviations were more marked. The explanation of these deviations is not difficult when the true nature of a gas is considered. Presumably the molecules of a gas act on each other with certain forces, the magnitudes of which depend upon the distances between the molecules. When the gas is highly rarefied, that is, when it is far removed from the liquid state, the molecular forces are van- ish.in.gly small ; but when the gas is brought nearer the liquid state by increasing the pressure and lowering the temperature, the molecules are brought closer together and the molecular forces are no longer negligible. The gas in this state possesses appreciable potential energy and the deviation from Joule's law is considerable. 61 . Comparison of Temperature Scales. Joule's law furnishes a means of comparing the two temperature scales that have been introduced: the scale of the gas thermometer and the Kelvin absolute scale. Since the intrinsic energy u is, in general, a function of T and v, we may write the symbolic equation CD But from the general equation (IV), Art. 56, &\ -p~}dv (2) d-J J paring (1) and (2), we obtain For a gas that obeys Joule's law -- = 0, wlienco from (H) & <3u =. (4) 'A y ^ ; Equation (4) is, however, precisely the equation that expresses Charles' law when T is taken as the absolute temperature on the scale of the constant volume gas thermometer. Thus, if the change of pressure is proportional to the change of tem- perature when the volume remains constant, we have, taking jw as the pressure at C., 1=1- It follows that the value of T is the same whether taken on the Kelvin absolute scale or on the scale of a constant- volume gas thermometer, provided the gas strictly obeys the laws of Boyle and Joule. The fact that any actual gas, as air or nitrogen, does not obey these laws exactly makes the scale of the actual gas thermometer deviate slightly from the scale of the ideal Kelvin thermometer. From the porous- plug experiments of Joule and Kelvin, Rowland has made a comparison between the Kelvin scale and the scale of the air thermometer. 62. Numerical Value of B. The value of the constant B for a given gas can be determined from the values of p, v, and T be- longing to some definite state. The specific weights of various gases at atmospheric pressure and at a temperature of C. are given as follows : Atmospheric air ...... 0.08071 Ib. per cubic foot. Nitrogen ....... O.OT829 Ib. per cubic foot. Oxygen ........ 0.08922 Ib. per cubic foot. Hydrogen ....... 0.00561 Ib. per cubic foot. Carbonic acid ...... 0.12268 Ib. per cubic foot. A pressure of one atmosphere, 760 mm. of mercury, is 10,333 kg. per square meter = 14.6967 Ib. per square inch =2116. 32 Ib. per square foot. Taking as 491.6 the value of T on the F. scale corresponding to C., we have for air 2116 - 32 =5334 T <yT 0.08071x491.6 In metric units the corresponding calculation gives 7? = 10333 __ 9 Q 9 g 273.1 x 1.293 ' ' The values of B for other gases may be found in the same way by inserting the proper values of the specific weight 7. 63. Forms of the Characteristic Equation. In the character- istic equation as usually written, (1) v denotes the volume of unit weight of gas. It is convenient to extend the equation to apply to any weight. Letting M denote the weight of the gas, we have for the volume F~of M Ib. (or kg.), V= Mv, whence instead of (1) we may write : pV=MBT. (2) This equation is useful in the solution of problems in which three of the four quantities, p, v, T, and M, are given and the fourth is required. EXAMPLE. Find the pressure when 0.6 Ib. of air at a temperature of 70 F. occupies a volume of 3.5 cu. ft. From (2) p = MBI = 0.6 x 58.34 x (70 + 469.6) = 484g>7 ^ per square feot V o.o = 33.63 Ib. r>er sauare inch. advantageous in the solution of problems that involve tw states of the gas. If ( p v F r T and (> 3 , F^, T^) are the tw states in question, then ~~m~~ ~~T~ With this equation any consistent, system of units may be usec EXAMPLE. Air at a pressure of 14.7 Ib. per square inch and having temperature of 60 F. is compressed from a volume of 4 cu. ft. to a volun of 1.35 cu. ft. and the final pressure is 55 Ib. per square inch. The fun temperature is to be found. From (3) we have 14.7 x 4 _55x 1.35 60 + 459.6 t 2 + 459.6' whence t 2 = 196.5 F. EXERCISES 1. Find values of B for nitrogen, oxygen, and hydrogen. 2. Establish a relation between the density of a gas and the value of tl constant B for that gas. 3. Find the volume of 13 Ib. of air at a pressure of 85 Ib. per square inc and a temperature of 72 C. 4. If the air in Ex. 3 expands to a volume of 30 cu. ft. and the fin pressure is 20 Ib. per square inch, what is the final temperature ? 5. What weight of hydrogen at atmospheric pressure and a teinperatu of 70 F. will be required to fill a balloon having a capacity of 12,000 cu. ft 6. A gas tank contains 2.1 Ib. of oxygen at a pressure of 120 Ib. p square inch and at a temperature of 60 F. The pressure in the tank shou not exceed 300 Ib. per square inch and the temperature may rise to 100 ! Find the weight of oxygen that may safely be added to the contents of tl tank. 64. General Equations for Gases. The general equatioi deduced in Chapter VI take simple forms when applied 1 perfect gases. From the characteristic equation we obtain by differentiation JTh introducing tnese values or tue derivatives in the general equations (I)-(V) and (VIII), the following equations are obtained : da = c v dT + AS - dv, (Id) < v y dq = c p d T-AB- dp, (II a) P , AB ( T 7 , T , ^ , TTT , dq = G P dv + G V dp , (III a) c p -o v \ v p * J du = J<j v dT, (IV a) di = C] ,dT, (Yd) c p ~c v = AB. (VIII a) The first two equations may be still further reduced by means of the characteristic equation to the forms dq = c v dT 4- Apdv, (I 5) dq=c p dT-Avdp (115) The ratio ^ of tlie two specific heats is usually denoted by c v k. The introduction of this ratio reduces (III a) to the sim- pler form, d q = -A. [kpdv + vdp] . (Ill 5) K JL Equation (IV a) simply expresses symbolically Joule's law that the change of energy of a gas is proportional to the change in temperature. Equation (I 5) follows independently from (IV a) and the energy equation ; thus dq = Adu + Apdv = c v dT+ Apdv, since AJ 1. EXERCISES 1. Deduce (VIII a) from (I 6), (II 5), and the characteristic equation. 2. Derive (V ) from (IV a) and the equation jw = BT. 3. From (I ), (II a), and (III n) derive expressions for -If- 4i From (III&) deduce the equation of the adiabatic curve in _pu-coordi nates. 93 PROPERTIES OF GASES [CHAP, vn 5. From (I a) derive the equation of an adiabatic in TV-coordinates. 6. Using the method of graphical representation explained in Art. 32, show a graphical representation of equation (I fc). 65. Specific Heat of Gases. If a gas obeys the law pv = BT, the specific heat of the gas must be independent of the pressure and also independent of the volume. This principle was shown in Art. 57. The specific heat (c p or c w ) may, however, vary with the temperature, and the results of recent accurate experi- ments over a wide range of temperature show that such a vari- ation exists. As a general rule, the law of variation is expressed by a linear equation ; thus c v = a + bt, Cp = a' + bt. When the range of temperature is large, as in the internal combustion motor, the variation of specific heat with tempera- ture must be taken into account. In the greater number of problems that arise in the technical applications of gaseous media it may be assumed with sufficient accuracy that the specific heat has a mean constant value. For air the value of c p , as determined by Regnault, is 0.2375 from to 200 C. Recent experiments by Swann give the following values : 0.24173 at 20 C. 0.24301 at 100 C. In ordinary calculations we may take c p = 0.24. The value of e p for carbon dioxide (CO 2 ) is usually given as 0.2012. Swann found the values 0.20202 at 20 C., 0.22121 at 100 C. The value of e p for other gases for temperatures between and 200 C. may be taken as follows: Hydrogen .... 3.424-0 Nitrogen .... 0.2438 Values of the ratio k = - have been determined by various c* experimental methods. For air the results obtained range from k = 1.39 'to Tc = 1.42. From the experimental evidence it seems probable that the true value lies between 1.40 and 1.405. In calculations that involve this constant, we shall take the value 1.4 as convenient and sufficiently accurate. For air, there- fore, ^=0.24-1.4 = 0.171. The values of k and of <? for other gases may be taken as follows : k Co Hydrogen ..... 1.4 2.446 Nitrogen ..... 1.4 0.174 Oxygen ..... 1.4 0.155 Carbon dioxide ... 1.3 0.162 Carbon monoxide . . 1.4 0,173 Ammonia ..... 1.32 0.387 If in equation (VIII #), c v is replaced by -A the result is the k relation ,,*=!- A3. Each of the four magnitudes <? p , &, A, and B have been deter- mined experimentally, and this equation serves as a check. 66. Intrinsic Energy. An expression for the intrinsic energy of a gas is obtained by integrating (IV a). Thus O , (1) if c v is assumed to be constant. The constant of integration U Q is evidently the energy of a unit weight of gas at absolute zero. Since, however, we are not concerned with the absolute value of the energy, but the change of energy for a given change of state, the constant M O drops out of consideration when differences are taken, and we need make no assumption as to its value. Hence, if (^ v v T^) and (jt? 2 , v v T^) are the coordinate of the initial and final states, we have tt-M^Jb.CTa-Zi). (2) 98 PROPERTIES OF GASES [CHAP, vn This formula gives the change of energy per unit weight of gas. For a weight M the formula becomes Uz-U^JMc^-TJ. (3) A clear understanding of the physical meaning of formula (2) is of such importance that it is desirable to give a second method of derivation, one based directly upon Joule's law. According to Joule's law the energy of a unit weight of gas is dependent on the temperature only. Hence, if T v Fig. 33, is an isothermal, the energy of the gas in the state A is the same as in the state D; likewise, the energy of the gas at all points on the iso- thermal T a is the FIG. 33. same. It follows that the change of energy in passing from tem- perature jPj to temperature T^ is the same, whether the path is AJB, AC, orDJS. Since the energy is directly proportional to the temperature, the change of energy is directly proportional to the change of temperature. Hence Uz-u^a^-TJ, (4) in which a denotes a proportionality-factor. To determine the factor a, we choose some particular path between the isother- rnals T and T z (Fig. 33). As we have seen, if this constant is established for one path it holds good for every other path. The most convenient path for this purpose is a constant volume line, as A 0. The heat required for a rise in temperature from V'Ziis fca^CZi-Zi)- Since in the constant volume change, the external work is zero, we have from the general energy equation Comparing these equations, we have A loimuia lor tne cnange ot energy in terms of p and Fmay be derived from (3). Multiplying and dividing the second member by JB, k-l ' W In (5) Kj and V l denote the final and initial volumes, respec- tively, of the weight of gas under consideration; consequently it is not necessary to find the weight M in order to calculate the change of energy. It is to be noted, however, that in using (5) pressures must be taken in pounds per square foot. EXAMPLK. Find the change of energy when 8.2 cu. ft. of air having a pressure of 20 Ib. per square inch is compressed to a pressure of 55 Ib. per square inch and a volume of 3.72 cu. ft. Using the value k = 1.40, 55x3.72-20x8.2 -U = = 144 x 67. Heat Content. The change in heat content correspond- ing to change of state of a gas is readily derived from the general equation (Va). Thus, i = c v dT= c p T+ i , (1) and 2 j = c p (T z T-^) . (2) Introducing the factor AB in the second member of (2), For a weight of gas M, (2) and (3) become, respectively, IZ-I^MC^-TJ, (4) and ^-Ji-^^r 68. Entropy. Expressions for the change of entropy are easily derived from the general equations (la), (II a), and (Ilia). Dividing both members of these equations by T, we have dq dT ^-^ (2) ' ^ } dv dp ,-QN ds = c P~+ c -f' < 3 ) Hence for a change of state from (p r v v T{) to (p 2 , v a , T 2 ), s 2 - j = e, log e ^ + J.5 log, ^ (4) g c (5) ^i Pi 8=8 c, log. ^+ c.log e |s. (6) These formulae give the change of entropy per unit weight of gas. For any other weight M, the change of entropy is M (s 2 Sj). Equations (4), (5), and (6) are in reality identi- cal. Each can be derived from either of the other two by means of the relations pv = BT, c p c v = AB. In. the solution of a problem, the equation should be chosen that leads most directly to the desired result. EXERCISES 1. From (4), (o), and (6) deduce expressions for the change of entropy corresponding to the following changes of state : (a) isothermal, (b) tit con- stant volume, (c) at constant pressure. 2. By making s 2 s l = in (4), (5), and (6), deduce relations between Tand v, T and;;, and p and v for an adiabatic change of state. 69. Constant Volume and Constant Pressure Changes. In heating a gas at constant volume the external work is zero. Hence, Q = A( U, - Uj = Mo v (T 2 - zy. (1) (2) \i fin" g,4'-5 is hi'utt-tl at constant prewsure, the external si iruf ul<lrt in, nf rniTjiy is, us in all canon, given by the -, given by the relation C r >) ha\r IHM-U writtmi tlin f entropy in limiM c~> nntl (7) may In* ilrnvril ilirt't'lly from ^jKM'ul fipiutiuiiH for *n- , Art. iJS. f I'ljitnijf.s uf state just Irrnl art- n-pn'sruti'il on fW-jilaiu' by curvi'S of tbe al form sbmvn in Ki.u r . &l* rurvi' ,4/^. wliicli rcp- Is tbe const ant, volume (> ,n\ is MrquT ihsin the ' (6) FIG. 34. that is, area A 1 ABB l < area 70. Isothermal Change of State. If T is made constant in the equation p V=MB T, the resulting equation P ^' r Pi Vi = constant (1) is the equation of the isothermal curve in p F'-coordinates. This curve is an equilateral hyperbola. The external work for a change from state 1 to state 2 is given by the general formula (2) Using (1) to eliminate p, we have (3) For the change of energy, Z7 2 - U^JMc^T^- TJ = 0; (4) hence Tr ^12=^^12 = ^1^^-^, (5) and l 'i Since in isothermal expansion the work done is wholly sup- plied by the heat absorbed from external sources, it follows that if the expansion is continued indefinitely, the work that may be obtained is infinite. This is also shown by (3), thus : JL 71. Adiabatic Change of State. To derive the ^-equation of an adiabatic change of state, we may use the general differen- nun o^ucnuui.1. uujuumumg p uuu v as variaoies. me most con- venient form of this equation is (III a), j_ dq = j T (ydp + /qpcfo) . (1) During an adiabatic process no heat is supplied to or ab- stracted from the system ; hence in (1) dq 0, and therefore vdp + kpdv = 0. (2) Separating the variables, <3/p_ Jcdv __ ^ P v ~ ' whence log e jt? + Tc log e v = log (7, or jpy* = (7. (3) The relation between temperature and volume or between temperature and pressure is readily derived by combining (3) and the general equation pv = BT. Thus from pv* = C, pv = BT, we get by the elimination of p, *-!_ . that is, 2V- 1 = const. (4) Similarly, by the elimination of v, we obtain TDk ,*-! _ - mk . p --gJ. , x--i 7} k that is, - = const. (5) If we choose some initial state, p v v v T r the constants in (4) and (5) are determined, and the equations may be written in the homogeneous forms 37 k-l Since in an adiabatic change the heat Q is zero, the energy equation gives whence using the general expression for the change of energy, By means of the equation the final volume V z may be eliminated from (8). The result- ing equation is EXAMPLE. An air compressor compresses adiabatically 1.2 cu. ft. of free air (i.e. air at atmospheric pressure, 14.7 Ib. per square inch) to a pressure of 70 Ib. per square inch. Find the work of compression; also the final temperature if the initial temperature is 60 F. K . . For the final volume, we have F 2 = 1.2 j~ = 0.3936 cu. ft. The work of compression is piVi-p,V 2 _ 144(14.7 x 1.2 - 70 x 0.3036) _ o ri , Q *. k-l 04 ~ ~ The initial temperature being 60 + 459.6 = 519.6 absolute,, we have for the final temperature 0.4 T 2 = 519.6 (~Y= 811.6 abs., whence z 2 = 352 F. 72. Poiytropic Change of State. The changes of state con- sidered in the preceding sections are special cases of the more general change of state defined by the equation By giving n special values we get the constant volume, constant pressure, and other familiar changes of state. Thus : for n = 0, pv = const., i.e. p = const, for n = co, p^v = const., v = const. for n = 1, pv = const., isothermal, for n = 7c, pv h = const., adiabatic. The curve on the p F~-plane that represents Eq. (1) is called by Zeuner the polytropic curve. By combining (1) with the characteristic equation^? VMBT, as in Art. 71, the following relations are readily derived For the external work done by a gas expanding according to the law p V n p l V{ 1 const., from the volume FJ- to the volume V v we have -tf Pl * The change of energy, as in every change of state, is 77 - IT - P* V<i ~~ Pi T/ i (^ Uz Ul k^T~ w Hence, from the energy equation, we have for the heat absorbed by the gas during expansion JO -U-U + W </Vi 2 u z 1/1+ WM K 1 L n or J0 1n = - - Comparing (3), (4), and (5) we note that the common factor (poV* p,V-,} occurs in the second member of each expression. useful relations : W = ^1 (6) C-ZT 1n ^ These may be combined in the one expression W:U Z - U l :JQ=7c-l:l-n:Jc-n. (9) EXAMPLE. Let a gas expand according to the law j) F 1>2 = const. Taking k 1.4, we have that is, the external work is double the equivalent of the beat absorbed by the gas and also double the decrease of energy. 73. Specific Heat in Polytropic Changes. From Eq. (5), Art. 72, an expression for Q lz in terms of the initial and final temperatures of the gas may be readily derived. Since (5) becomes hence, Q^ = McJ^(T 2 - T^. (1) J- - 7i We have, in general, Qu=Me(T,-T 1 '). (2) where c denotes the specific heat for the change of state under consideration. Comparing (1) and (2), it appears that Hence, for the polytropic change of state, the specific heat is con- ART. 73] SPECIFIC HEAT IN POLYTROPIC CHANGES 107 stant (assuming c v to be constant) and its value depends on the value of n in the equation p V n const. It is instructive to observe from (3) the variation of c as n For ra = 0, c = kc v = and the P FIG. 35. is given different values. change of state is repre- sented by the constant- pressure line a, Fig. 35, 36. For n = 1, c = oo, and the change of state is iso- thermal (line 5). If n = 7c, then c = 0, and the ex- pansion is adiabatic (line cT). For values of n lying between 1 and 7c, the value of c as given by (3) is evidently negative ; that is, for any curve lying between the isothermal b and adiabatic d, rise of temperature accompanies abstraction of heat, and vice versa. This is shown by the curve c. It will be observed that in passing through the region between curves a and 5, n increases from to 1 and c increases from c p to oo ; then as n keeps on increasing from 1 to k, c changes sign at curve b by passing through oo, and increases from oo to 0. As n increases from n = k to n = + co, c increases from c = to <?=<?; and for n oo, the constant volume case, c becomes c v . T O -S FIG. 36. 108 PROPERTIES UF UAbHitt 74. Determination of n. It is frequently desirable in experi- mental investigation to fit a curve determined experimentally _ aSj for example, the compression curve of the indicator diagram of the air compressor by a theoretical curve having the general equation pV n =c. To find the value of the exponent n we assume two points on the curve and measure to any convenient scale p v p v V v and V v Then since wehave =*r g . CO log F! - log F 2 EXAMPLE. In a test of an air compressor the following data were determined from the indicator diagram : At the beginning of compression, p = 14.5 Ib. per square inch. ^ = 2.50 cu. ft. At the end of compression, jo 2 = G8.7 Ib. per square inch. F a = 0.77 cu. ft. Assuming that the compression follows the law p V n const., we have for the value of the exponent ff= log 68.7 -log 1*.5 = 132 log 2.56 - log 0.77 The work of compression is 0.77-15.5x2.56) 1 n The increase of intrinsic energy is " 14 -5 x 2.56) f b k 1 0.4 and the heat absorbed is -5680-7100, . ^12 -- ^g - = - l.od B. t. u. The negative sign indicates that heat is given up by the air during com- pression ; this is always the case with a water-jacketed cylinder. If the initial temperature of the air is 00 F., or 519.6 absolute, the final temperature is EXERCISES 1. A curve whose equation is pV n = C is passed through the points Pi = 40, F! = 6 and jt? 2 = 16, F 2 = 12.5. Find the value of n. 2. Air changes state according to the law pV n C. Find the value of n for which the decrease of energy is one half of the external work; also the value of n for which the heat abstracted is one third of the increase of energy. 3. If 32,000 ft.-lb. are expended in compressing air according to the law^F 1 - 28 = const., find the heat abstracted, and the change of energy. 4. In heating air at constant pressure 35 B. t. u. are absorbed. Find the increase of energy and the external work. 5. A mass of air at a pressure of 60 Ib. per square inch absolute has a volume of 12 cu. ft. The air expands to a volume of 20 cu. ft. Find the external work and change of energy : (a) when the expansion is isothermal ; (&) when the expansion is adiabatic ; (c) when the air expands at constant pressure. 6. If the initial temperature of the air in Ex. 5 is 62 F., what is the weight? Find the heat added and the change of entropy for each of the three cases. 7. Find the specific heat of air when expanding according to the law p v i.25 = const. If during the expansion the temperature falls from 90 F. to 10 F., what is the change of entropy? 8. Find the latent heat of expansion of air at atmospheric pressure and at a temperature of 32 F. 9. The volume of a fire balloon is 120 cu. ft. The air inside has a temperature of 280 F., and the temperature of the surrounding air is 70 F. Find the weight required to prevent the balloon from ascending, including the weight of the balloon itself. 10. A tank having a volume of 35 cu. ft. contains air compressed to 90 Ib. per square inch absolute. The temperature is 70 F. Some of the air is permitted to escape, and the pressure in the tank is then found to be 63 Ib. per square inch and the temperature 67 F. What volume will be occupied by the air removed from the tank at atmospheric pressure and at 70 F. V 11. Air in expanding isothermally at a temperature of 130 F. absorbs 35 B. t. u. Find the heat that must be absorbed by the same weight of air at constant pressure to give the same change of entropy. 12. Air in the initial state has a volume of 8 cu. ft. at a pressure of 75 Ib. per square inch. In the final state the volume is 18 cu. ft. and the pressure is 38 Ib. per square inch. Find: () the change of energy; (b) the change in the heat content ; (c) the change of entropy. 13. Find the work required to compress 30 cu. ft. of free air to a pressure of 65 Ib. per square inch, gauge according to the lawpw 1 - 3 = const. Find the heat n.lisf.ra.fltprl rlnvino- nnmnrfissinri. recourse to general equations. SUGGESTION. Let one pound of gas be heated through the temperature range T 2 - T l (a) at constant volume, (/;) at constant pressure. Find an expression for the excess of heat required for the second case and then make use of the energy equation. 15. Suppose the specific heat of a gas to be given by the linear relation c v = a + bt. Deduce relations between p, v, and T for an adiabatic change. SUGGESTION. Use the general equation dq = c v dT + Ajxlv and the char- acteristic equation pv = BT. REFERENCES CHARACTERISTIC EQUATION OF GASES. DEVIATION FROM TIIK BOYLE-GAY LUSSAC LAW Zeuner: Technical Thermodynamics (Klein) 1, 03. Preston : Theory of Heat, 403. Barus: The Laws of Gases. N.Y. 1809. (Contains the researches of Boyle and Amagat.) Regnault : Relation des Experiences 1. Weyrauch : Grundriss der Wiirme-T heorie 1, 124-, 127. THE POROUS-PLUG EXPERIMENT. THE ABSOLUTE SCALK OF TEMPERATURE Thomson and Joule: Phil. Trans. 143, 357 (1853) ; 144, 321 (1851) ; 152, 579 (1862). Rose-limes: Phil. Mag. (0) 15, 301. 1908. Callendar: Phil. Mag. (0) 5, 48. 1903. Olszewski : Phil. Mag. (6) 3, 535. 1902. Buckingham : Bui. of the Bureau of Standards 3, 237. 1908. Preston : Theory of Heat, 695. Bryan : Thermodynamics, 128. Chwolsou : Lehrbuch der Physik 3, 546. SPECIFIC HEAT OF GASES Regnault: Relation des Experiences 2, 303. Swann : Proc. Royal Soc. 82 A, 147. 1909. Zeuner : Technical Thermodynamics (Klein) 1, 116. Chwolson : Lehrbuch der Physik 3, 22G. Preston : Theory of Heat, 339, 243. Weyrauch : Grundriss der Wsinne-Theorie 1, 146. GENERAL EQUATIONS Zeuner : Technical Thermodynamics 1, 122. Weyrauch : Grundriss der Warme-Theorie 1, 152. Bryan : Thermodynamics, 116. CHAPTER VIII GASEOUS MIXTURES AND COMPOUNDS. COMBUSTION 75. Preliminary Statement. In the preceding chapter we discussed the properties of simple gases with the implied assumption that chemical action was excluded. For many technical applications a knowledge of such properties is suffi- cient for the consideration of all questions that arise. On the other hand, investigations of the greatest importance, those relating to internal combustion motor, have to deal with gaseous substances that enter into chemical combination and (after combustion) with mixtures of inert gases. In the present chapter, therefore, we shall consider some of the pro- perties of gaseous compounds as dependent on chemical com- position, and also the properties of mixtures of gases. 76. Atomic and Molecular Weights. Let U v E^, etc. denote different chemical elements and a r a 2 , etc. their corresponding atomic weights. Then if n^ w 2 , etc. denote the number of atoms of JE r jKj, etc. entering into a molecule of a given combination, the molecular weight of the compound is m = n^ + n z a z + etc. = "^na. (1) For the elements that enter into subsequent discussions the atomic weights (referred to the value 16 for oxygen) are as follows : APPROXIMATE EXACT VALUE INTEGRAL VALUE Hydrogen 1.008 1 Oxygen 16.000 16 Nitrogen . 14.040 14 Carbon 12.000 12 Sulphur 32.060 32 Tlie approximate uj.tegj.tn vo.uuo ,^ **** *j ~ ------ practical purposes, in view of unavoidable errors in experi- mental results. Using these values, we have as the molecular weights of cer- tain important substances the following : Water H 2 m=2x 1 + 1x16 = 18 Carbon monoxide CO 1 x 12 + 1 x 16 = 28 Carbon dioxide C0 2 1 x 12 + 2 x 16 = 44 Ammonia NH 8 1 x 14 + 3 x 1 = 17 Methane CH 4 Ixl2 + 4x 1 = 16 Nitrogen N 2 2x14 = 28 Hydrogen H 2 2x1 = 2 The composition by weight of a compound is readily deter- mined from the value of , a, and m. Thus in a unit weight (pound) of compound there is Ml Ib. of element _EL m lb. of element E v etc. m For example, C0 2 is composed by weight of -| carbon anct If oxygen, NH 3 is composed by weight of \$ nitrogen and -^ hydrogen. 77. Relations between Gas Constants. If in the character- istic equation pv = BT, which holds approximately for any gaseous substance (mixture or compound), we replace v by - we have ^ Here 7 denotes the weight of unit volume of the gas. From this relation it is seen that for a chosen standard pressure and temperature, for example, atmospheric pressure and 0C., the product By is the same for all gases. But since the specific weight 7 of a gas is directly proportional to the molecular weight m, it follows that the product Bm is likewise the same tor all gases. Denoting this product Bm by JK, we have for the characteristic equation of any gas pv = ~T. (2) f m ^ ' From (1) we obtain the relation &; (3) hence the numerical value of R can be found when the values of m and 7 are accurately known for any one gas. From Mor- ley's accurate experiments, we have for oxygen 7 = 0.089222 Ib. per cubic foot at atmospheric pressure and 32 F. ; and for oxygen m = 32. Inserting these numerical values in (3), we obtain 2116.3x32 0.089222x491.6 The constant R is called the universal gas constant. From it ffche characteristic constant B of any gas can be determined at pnce from the molecular weight. Thus for carbonic acid we Ijiave = 1^ = 35.09. j*i 4:4 ^ From the general formula [ CV -C. = AB (4) for the difference between the specific heats of a gas, we have AR 1544 1 1.9855 ,^ / __ i ft ^^_^^. r - _, J _ - . - I > I p v ~~ m ~ 777.64m m ' W This relation gives a ready method of calculating one specific heat from the other when the molecular weight m is known. Thus for C0 2 , Cp - c ,= i^^ = 0.0451, and if e, = 0.2020, we have c v = 0.2020 - 0.0451 = 0.1569. It is convenient to express the specific weight 7 and the specific volume v of a gas in terms of the molecular weight m. These constants are referred to standard conditions, namely, atmospheric pressure and a temperature of 32 F. From (3) we have 7 = JLw, (6) whence inserting tne numerical viuuea, jj ** !F=491 6 7 = 0.002788 w. For the normal specific volume, we have v== l = OTl. 7 p m 358.65 or v = in (7) (8) (9) From the preceding relations, the following values are readily found for the constants of certain gases. Volume per Character- niH'oronco Weight per pound Gas Chemical Symbol Molecular Weight istic Constant of Specific Heats cubic foot nt !!'2 1<\ and Atmo.sphoric at !i V. and Atmos- pheric m B CpC v Pressure Treasure Oxygen .... 2 32 48.249 0.0021 0.089222 11.208 Hydrogen . . . H 2 2.016 765.86 0.9849 0.005621 177.9 Nitrogen . . . N 2 28.08 54.985 0.0707 0.07829 12.773 Carbon dioxide . C0 2 44 35.09 0.0451 0.12268 8.151 Carbon monoxide CO 28 55.142 0.0709 0.078028 12.81 Methane . . . CH 4 16.032 96.314 0.1238 0.04470 22.37 Ethylene . . . CJI 4 28.032 55.079 . 0.0708 0.078036 12.794 Air 53.34 0.0086 0.08071 12.39 78. Mixtures of Gases. Dalton's Law. A mixture of several gases that have no chemical action on each other obeys very closely the following law first stated by Dalton : The pressiire of the gaseous mixture upon the walls of the con- taining vessel is the sum of the pressures that the constituent gases would exert if each occupied the vessel separately. Like Boyle's law, Dalton's law is obeyed strictly by mix- tures of ideal perfect gases only. Mixtures of actual gases show deviations from the law, these being greater with gases most easily liquefied. For the purpose of technical thermodynamics, however, it is permissible to assume the validity of Dalton's law even in the case of a mixture of vapors. Let F denote the volume of a given mixture, M v M z , M& . . the weights of the constituent gases, and J9 2 , J5 3 , the ART. 78] MIXTURES OF GASES. DALTON'S LAW 115 constants for those constituents ; then the partial pressures of the constituents, that is, the pressures they would exert sepa- rately if occupying the volume V-, are : B8 m l i IY\ __ & " m _ " " /I 1 Pi y iPi ' ~Y '"3 Y ' '" ^ - 1 According to Dalton's law the pressure p of the mixture is P=Pi+Pi + PB + -' = ~ r (.M 1 B 1 +M 2 2 +M 3 S 3 + -). (2) Furthermore, if Mis the weight of the mixture, (3) Let us now introduce a magnitude B m defined by the equation MB m = M& + M,B Z + Jf 8 J5 8 +; (4) then (2) takes the form pir=MB m F. (5) The constant B m may be regarded as the characteristic con- stant of the mixture. It is obtained from (4), which may be written in the more convenient form 5m= (6) The partial pressures may readily be expressed in terms of the pressure of the mixture. Thus combining (1) and (5), we obtain , etc. (7) ' ^ EXAMPLE. A fuel gas has the composition by weight given below. The value of the constant B m for this gas is found as indicated by the following arrangement : CONSTITUENTS WEIGHT r,n n ....... 0.04- Since M= 1 and 2MB = 103.24, we have B m = 103.24. The apparent molecular weight of the mixture is 1544 1/( nr m= ios^r 1U)G> and the weight per cubic foot under standard conditions is, therefore, y = 0.002788 x 14.90 = 0.0417 Ib. 79. Volume Relations. Let V v F 2 , F 3 , , denote the vol- ume that would be occupied at pressure p and temperature T by several gaseous constituents; then if B^ B^ J9 3 , -, denote the characteristic constants of these gases, we have pV,= M&T, pV, = M Z B Z T, pV s = M Z B Z T, .... (1) If now the gases be mixed, keeping the same pressure and temperature, the mixture will occupy the volume F-F 1 +F a +F 8 +-, (2) and its weight will be necessarily M=Mi + Mt + M % +.~ (3) Taking B m as the characteristic constant of the mixture, we have pV=MB m T. (4) Comparing (1) and (4), we obtain the relations V 1 _M 1 B, 3_J^ _ V~ MBj V~MB m ' 1 " ^ } It will be seen that the volume ratios given by (5) are equal to the pressure ratios given by (7) of Art. 78. If 7 denotes the weight of a unit volume (1 cu. ft.) of gas, then 1 M xnx 7 = - = y (6) For the several constituents of a mixture, we have, therefore, M l = 7l V v M, = 72 F 2 , M z = 73 F 3 , . .., (7) and for the mixture Similarly, we have for the specific volume of the mixture Since 7 = 0.002788 m = km (see Art. 77), we have from (7) and M M + M,+ -. m j; - m i I ., W 9 , Therefore, -^ = =-*-_*= , _1 = 2_A (10) J(K? STO^ jf Sm^JY If further we denote by w ro the quotient ^, we have from (8) /c i m m = -y2.m i V i . (11) The constant w m we maj 7 " regard as the apparent molecular weight of the mixture, and from it we may determine the con- stants B m , c p <?, 7, and v of the mixture. Equations (10) and (11) are useful in the investigation of a mixture when the composition by volume is given. The follow- ing example shows the method of procedure. EXAMPLE. A producer gas has the composition by volume given below. .Required the composition by weight and the constants of the mixture. EU . . . .... 0.08 2 0.16 2m f F"{ 0.006 CO .... .... 0.22 28 6.16 0.2308 CH 4 .... .... 0.024 16 0.384 0.0144 CO 2 .... .... 0.066 44 2.904 0.1088 N 2 .... .... 0.61 28 17.08 0.64 1.000 26.688 1.000 According to (10) the last column gives the composition by weight. The constant m m is 26.688; hence we have = 57.85. v = 0.002788 x 26.688 = 0.07441. ' 26.688 1.9855 pv 2688 80. Combustion : Fuels. The elements that chiefly combine with oxygen to produce reactions characterized by the evolution rvf Tioof QVQ T-nrrJy^rrcm onrl n.a.rlnrm . rinmnrmnrls that are made up largely ot these elements are lueis; ror example, metnane CH 4 , benzol C 6 H 6 , alcohol C 2 H 6 0. The product of complete combustion of hydrogen is H 2 0, water ; that of complete com- bustion of carbon is C0 2 , carbon dioxide. Sulphur is a possible constituent of fuels, and the product of combustion is SO 2 , sul- phur dioxide. Chemical reactions are, in general, characterized by the evo- lution or absorption of heat. The union of a combustible with oxygen is accompanied by the evolution of a considerable quantity of heat, and the heat evolved by the combustion of a unit weight of the combustible is called the heating value of the combustible. The heating value of hydrogen alone or car- bon alone must be determined by experiment, but the heating value of a compound of C and H may be calculated, at least approximately. Hydrogen and compounds containing hydrogen have two heating values, called respectively the higher and the lower. This arises from the fact that the product H 2 O may be either water or steam. If the temperature after combustion is above 212, the product exists as vapor, and the heat necessary to keep it in the vapor form is not set free ; hence, we have the lower heating value. If, however, the vapor condenses, the heat of vaporization is recovered, and we have the higher heating value. The heating values of various fuels are given in the follow- ing table. B. T. U. 1 KK POUND 15. T. u. VKH Ounic High " Low Low Hydrogen .... H, 02100 52230 2!M Carbon c 14(500 1 '1 (if)O Carbon monoxide . ^2 CO 4380 4380 312 Methane .... CH 4 23842 21385 951) Ethyleue .... C 2 II 4 2142.9 20025 15(i3 Acetylene .... C 2 H 2 21429 20073 14!)!) The heating value of a fuel mixture is determined from the heating values of the separate constituents. Denoting bv M,. ART. 81] AIR REQUIRED FOR COMBUSTION 119 My , the weights of the constituents, by H v jET 2 , H^ , the corresponding heating values per pound, and by H m the heat- ing value of the mixture, we have whence H m = . (1) By a similar procedure the heating value per cubic foot may be obtained when the composition by volume is given. EXAMPLE. Required the lower heating value of the producer gas de- scribed in the example of Art. 79. For the heating value per pound we have Jf IT MIT H a ........ 0.006 52230 313.38 CO ....... 0.2308 4380 1010.9 CH 4 ....... 0.0144 21385 307.94 2 MH = 1632.2 Since M = 1, we have // = 5 MH = 1632.2 B. t. u. per Ib. The heating value per cubic foot (at 32 F. and atmospheric pressure) is evidently the product II m j = 1632.2 x 0.07441 = 121.5 B. t. u. Or from the composition, we have Ho .... r . . . . 0.08 IT 294 r/7 23.52 CO ... " 342 75.24 CH 4 . . . . . . . 0.024 956 22.94 121.7 B. t. u. per cu. ft. The difference in the two results is due to approximations in the calculation, and is of no importance. 81. Air required for Combustion and Products of Combustion. The oxygen required for the complete combustion of a given fuel is determined from the equation of the reaction. For example, the combustion of methane, CH 4 , is given by the equation CH 4 + 2 2 = C0 2 + 2 H 2 ; 120 GASEOUS MIXTURES AND COMPOUNDS [CHAP, vin and two molecules of H 2 O. Since by Avogaclro's law the volumes are proportional to the numbers of molecules enter- ing into the equation, we may also read the preceding chemical equation as follows : two volumes of oxygen combine with one volume of CH 4 , producing one volume of C0 2 and two volumes of H 2 0. Taking the molecular weights of the four gases into con- sideration, we may write the equation 16 + 2x32 = 44 + 2x18. From this it appears that one pound of CH 4 requires for com- plete combustion f f = 4 Ib. of oxygen and the products are if = 2.75 Ib. of CO 2 and ff = 2.25 Ib. of H 2 O. Since oxygen is 23 per cent of air by weight, the weight of air required for the complete combustion of one pound of CJT 4 4. is = IT. 4 Ib. The volume of air required for the burning 2 of one cubic foot of CBL is ^ = 9.52 cu. ft. 4 0.21 We may generalize the process illustrated by the preceding example as follows : Let the gaseous fuel have the composition C ni H Ba O n ,, and let !, 2 , 3 denote the atomic weights of C, H, and 0, respectively. Then the molecular weight of the fuel in question is m = a 1 ?i 1 + a z n z + a 3 n 3 . The equation of the reaction may be written where $, ?/, and z indicate the number of molecules of the respective substances. Comparing the two members of the equation, we find whfinp.fi ART. 81] AIR REQUIRED FOR COMBUSTION 121 for alcohol C 2 H 6 O, x = 2 + | = 3, y = 2, and g = 3, showing that for the combustion of one cubic foot of alcohol vapor, 3 cu. ft. of oxygen are required, and the resulting products are 2 cu. ft. of CO 2 and 3 cu. ft. of H 2 0. To get the relations between the weights of the substances under consideration we must introduce the molecular weights in the reaction equation. Thus we obtain m + 2 a B x = y(ai + 2 3 ) + z (2 a 2 + a 3 ), from which follow the ratios : , weight of oxvgen 2 ax , , -, N x - - -1+111 = = -3(2 n 1 +l~n z -n B ) ; weight of fuel m m i weight of OCX, yCa-, + 2 a^) n, , . . y = .^ a = &^LZ: az _ _i ( a + 2 a ) ; weight of fuel m m g ; _ weight of H 2 O _ g(2 a 2 + 8 ) _ M 2 XQ g + a ). weight of fuel m 2m 2 3 If we make use of the integral values of the atomic weights, namely, ^ = 12, a 2 = 1, a s = 16, we have for the complete com- bustion of one pound of the combustible : w /* x' = oxygen required = (2 n^ + -|-w 2 n s) lb. ; y' = C0 2 produced = 44 ^1 Ib. ; wi ' = HoO produced = 9^ Ib. 2 ^ m Taking alcohol, C 2 H 6 0, for example, we have ! = 2, Wjj = 6, TC S = 1, w = 2 x 12 + 6 x 1 + 16 = 46, whence x' = -||(2 x 2 + i- x 6 - 1) = 2.08T; , 44 x 2 1 Q1 o . =-- =1.913; = 1.174. 46 The weight of air required per pound of alcohol is and the weight of nitrogen appearing among the products of combustion is, therefore, 9.075 - 2.087 = 6.988 Ib. If a gaseous fuel is a mixture of several combustible con- stituents, the values of x\ y ! , and z' may be found for the indi- vidual constituents separately. Then if M-^ M v M z , -, are the weights of the constituents respectively, we have , y , __ _ . at' __ */__,_ fy' - ' y ~ ' __ M ' ~ M ' M EXAMPLE. For the producer gas heretofore investigated, we have the following values : jir Q)' y' a' Ufa' Mi/' Ms,' H, 0.006 8 9 0.04-8 0.05-1- CO 0.2308 0.571 1.571 0,1 -'5 18 ().3(!2G CH 4 0.0144 4 2.75 2.25 0.0570 O.OSOG 0.0324 C0 2 0.1088 1 0.1088 N 2 0.64 1.00 0.287-1 0.511 0.08(31 One pound of the gas requires 0.2374 Ib. of oxygen for complete combustion. The weight of air required is, therefore, 0.2374 -f- 0.23 = 1.032 Ib., and this air brings with it 1.032 - 0.2374 = 0.7940 Ib. of nitrogen. We have then the following balance : CONSTITUENTS PRODUCTS Fuel gas 1.00 Ib. C0 2 0.511 Ib. A ir 1.0^2 H,Q 0.08(54 2.032 Ib. No 0.64 + 0.7040 = l.ljWn. (W2 Ib. Taking the composition by volume, the following results are found : V te y z Fin Vff r~ H Q 0.08 0.5 1 0.04 0.08 CO 0.22 0.5 1 0.11 0.22 CH 4 0.024 o 1 o 0.048 0.024 0.048 C0 2 0.066 1 O.OOli N 2 0.61 1.00 0.198 0.31 0.128 Since 0.198 cu. ft. of oxygen is required per cubic foot of gas, the volume of air required is 0.198 -*- 0.21 = 0.943 cu. ft., and the volume of nitrogen corre- sponding is 0.943 - 0.198 = 0.745 cu. ft., which is added to the O.(il cu. ft. in the fuel gas. The volume of gas and air before combustion is 1 + 0.943 = 1.943 cu. ft., and the volume of the products is 0.31 + 0.128 + 0.01 + 0.745 82. Specific Heat of Gaseous Products. In deducing the special equations for gases we assumed that the specific heat of any gas remains constant at all pressures and temperatures. In many technical applications this assumption is sufficiently near the truth and is justified by the simplicity of the analysis based upon it ; but when a very wide range of temperature is encountered, as in the case of the internal combustion motor, the assumption of constant specific heat is no longer permissible. The gaseous products that come under consideration may be separated into two classes. (1) The simple or diatomic gases, as nitrogen, oxygen, air, etc. ; (2) the compounds, like carbon dioxide (CO 2 ) and steam (H 2 0), which may be regarded as superheated vapors rather than as gases. For the products in the first group, the law pv = B T holds quite exactly, and, there- fore (see Art. 57), the specific heat must be independent of the pressure, but may vary with the temperature. The substances in the second group, which are comparatively near the liquid state, do not follow the gas law closely, and for these the specific heat may vary with the pressure as well as with the temperature. The character of the variation of the specific heat of steam is shown in Fig. 71, Art. 133. At the lower tempera- tures the specific heat increases with the pressure, but as the tem- perature rises the influence of the pressure becomes negligible and the specific heat rises with the temperature. It is probable that the specific heat of CO 2 varies in somewhat the same manner. Experiments on the specific heats of various gases show that in general the specific heat rises with the temperature, and that the law governing the variation is expressed sufficiently well by the simple linear equation G = a -f- bt. The formulas, as usually stated, give molecular specific heats, the molecular specific heat being numerically equal to the thermal capacity of a weight of the substance expressed by the molecular weight. Thus, since the molecular weight of carbon monoxide (CO) is 28, the molecular specific heat of CO is numerically equal to the thermal capacity of 28 pounds of CO. We mav denote molecular specific heat bv the product me. It gases are quite different, the molecular specific heats are sub- stantially identical. The results of Langen's experiments are given by the follow- ing formulas, in which * denotes temperature in degrees C. For all simple gases me,** 4. 8 + 0.0012*. (1) For carbon dioxide we, = 6. 7 + 0.0052*. (2) For water vapor we, = 5.9 + 0.0043*. (3) Dividing by the appropriate value of the molecular weight m, the heat capacity of a gas per unit weight is readily found. Thus for oxygen m = 32, and from (1) we have c,= 0.15 + 0.0000375*; for C0 2 , m = 44, and from (2) we obtain c w 0.1523 + 0.0001182*. Formulas (1), (2), and (3) give molecular specific heats at constant volume. From the relation m(c p <?)= 1. 1)855 (see Art. 77), we have approximately mc p = mc v + 2. Therefore, from the preceding equations we obtain corresponding equa- tions for Op, namely : mc p = 6.8 + 0.0012 *; (4) ; (5) CO For temperatures F. the preceding formulas become respec- tively: 1. For simple gases <? = - (4. 77 + 0.000667*) = -(4.48 + 0.00066720 m < = 1(6.75 + 0.0006670 m ' = 1(6.46 + 0.000667 T\ m J 2. 1< or carbon dioxide c,= 0.15 + 0.000066* = 0.12 + 0.000066 I 7 . c p = 0.195 + 0.000066 = 0.165 + 0.000066.^1' ^ ^ 3. For superheated water vapor c v = 0.324 + 0.000133S = 0.263 + 0.000133 T c p = 0.435 + 0.000133S = 0.374 + 0.000133^] 83. Specific Heat of a Gaseous Mixture. Let M^ M z , respectively, denote the weights of the constituents of a mix- ture and e Vi , <? V2 , , the corresponding specific heats. It is assumed that for a given temperature rise each constituent requires the same quantity of heat when mixed with other constituents as it would if separated from them. Hence, the heat Q required for a temperature change T z T^ is But we have also where M=M 1 -{-M z + , and c v denotes the specific heat of the mixture. Combining these expressions, we obtain __ or c v TM Likewise, c p = -., EXAMPLE. Find the specific heat c v of a mixture of 1 Ib. of the pro- ducer gas described in the example of Art. 79 and 1.25 Ib. of air, which, is about 20 per cent in excess of the air required for complete combustion. Find also the specific heat c v of the products of combustion. Of the 1.25 Ib. of air furnished 0.2875 Ib. is oxygen and 0.9625 Ib. is nit'.i'ii. .Mintti;: ntt- nut. ,:> n i-> MM- IM.I m. jjj tJu ^UH, y u , |, t.lKKJ.'. Hi. \Vi- ]iu\" UK-II v - ^ .v... ....... 'HJ ' "'""'(l.-iH + O.OOCHJOT 7') n 'J'U m ;"' '7^ ( l.'JH -|- O.OOOliliT 7') rn ...... , n.in^ u,j OHM (0.12 j- o.ooiKKm 7') N, ....... l.Uii;-;, ','' tjjll ""''( 1,1-S -|. I).()(){)()li7 7') ys ? 7') 7' o (i . 1;i y- Fur the jirniluri- nf ,.tu!.n-f;,.n, lu,v' t * Art. -SI -V.-,, tj.r.ll (O.lL* | O.ODOOlili 7') ]:j;j 7') t! - n: i '" 1 ( I.1S -1- O.OIHHiiiT 7') n.:U7 l.'i i- II.ODOOS.I.M 7' i> i.,|| . .,<(.. ..;';..:' 7V 84. Adiabutic Chan^rs with VitryinK Specific Heats. AVhon tlu i sjiri-ifir In-iU ni ,i ; ^ s , 5 -, ul^n ,i- .1 fniu'tion of Uunporaiurc, tints ,-. - .1 / 'A ,- s ./' * /?* lh- siuipli' ivlaiJMiis .lr-ri\ t ,l m V:t. 71 i lt>n^i!r apply. We have as lii'fntv, ltu\vr\rj-, For an adiabatic change dq = ; hence from (T), we have c v dT= Apdv, or v From (4) we obtain upon integration . (5) M z From the characteristic equation pv=BT, we have 2 =-^2, ^i therefore (5) becomes or alog. + J(2',-y 1 ) = (^5 + a)log.-. (6) jrl "2 Finally, if in (5) we substitute for -1 its equivalent ^- 2 , we , . v z obtain . a . - . Pl T whence For the external work of adiabatic expansion, we have TF=.Z7 _ TJ . (8) Equations (5), (6), and (7) are readily applied when the initial and final temperatures are given. When, however, the final temperature is required, the equation in T is tran- scendental and its solution requires a process of successive approximations. The illustrative example of the following article shows the method of procedure. 85. Temperature of Combustion. A close analysis of' the pro-- cess of burning a fuel gas under given conditions involves com- plicated equations, especially when the specific heat is taken as variable. The temperature and pressure at the end of the pro- cess are the results usually uesireu, least approximately, by a simple method. Let ^ denote the temperature of the gaseous mixture at the beginning of combustion and T z the desired final temperature ; H the lower heating value of the fuel per pound, and M the combined weight of one pound of fuel and of the air furnished for combustion (M is evidently also the weight of the products of combustion). It is assumed that the combustion is complete, and that the heat His all expended in raising the temperature of the products from ^ to T v As a matter of fact, the com- position of the mixture during the combustion process is con- tinually changing, but as the specific heat changes but little, it is considered permissible to base the calculation on the final products alone. We have then H=M( T \a + bT)dT, (1) Tl where a + bT is the expression for the variable specific heat of the products. From (1) we obtain upon integration from which T z may be calculated. EXAMPLE. The mixture of producer gas and air in the example of Art. 83 is compressed adiabatically from an initial pressure of 14.7 Ib. pel- square inch to a pressure of 150 Ib. per square inch absolute. The initial temperature is 530 absolute. The mixture is then burned at constant volume and the products of combustion expand adiabatically to the initial volume. Required the temperature and pressure after compression, after combustion, and after expansion. Also the work of compression, and the work of expansion. The characteristic constants of the fuel mixture and of the mixture of the products, respectively, are first required. Tor the fuel mixture we have M i) j//; H 2 ....... 0.006 765.86 4.5!);-)! (5 CO ...... 0.2308 55.142 12.72077 CH 4 ...... 0.0144 90.31-1 1.3861)2 C0 2 ...... 0.1088 85.00 3.81770 N 2 ....... 1.6025 54.985 88.11340 2 ....... 0.2875 48.249 13.8715!) 2.25 124,512 B = 124.512 -s. 2.25 = 55.34 ; AB = 0.07116. .r or me mixture 01 products, we ODtain JJ 51.50; AB 0.06621. For the fuel mixture, the expression for the specific heat is c v = 0.1618 + 0.00002643 T. We have, therefore, from (7), Art. 84 0.23296 log e ^= 0.07116 log^ - 0.00002643 (T z - Ti). i | JLur. I To solve this equation for T 2 let us assume as a first approximation T 2 - I\ = 500. Then l T* = 0.16529 -O.ni3215 =0 h 2\ 0.2329(3 and -^=1.921. Therefore, T 2 = 1.921 x 530 = 1018.1, and r 2 - 7 7 1 = 488.1. As a second approximation, we assume Tz T l = 490. We obtain T, 0.16529-0.012951 = Ot6539 ^^ 0.23296 ^ = 1.9231, T z = 1.9231 x 530 = 1019.2, TI T 2 - T v = 489.2. As the assumed value of T 2 - T\ is so nearly attained, we may take the value Tz = 1020 as sufficiently exact. The ratio of initial and final volumes is now readily found from the relation Thus, V l p 2 Ti 150 530 For the external work required to compress one pound of the mixture, we have W= J . 1 (0.1618 + 0.00002643 T)dT - 69460 ft.-lb. If T s denotes the temperature after combustion, we have from (2), taking c v = 0.1544 + 0.00003753 T for the products of combustion, 8 .- 1020") = whence T z = 3949. To find the pressure j> 3 , we must take account of the change of composi- tion during combustion. For the initial state, p 2 V = 55.34 T 2 , at the end of combustion p s V = 51.50 T 3 . Hence, we have 130 GASEOUS MIXTURES AND COMPOUNDS [CHAP.VIII For the adiabatic expansion, the ratio of volumes is the same as for the adiabatic compression. That is, r =0.1887. From (5) Art. 84, we have which may be written in the form Inserting the known values AB = 0.06021, a = 0.1544, 6 = 0.00003753, T s = 3949, ^ = 0.1887, we get log 7*4 = 3.7028 - 0.000105(5 T v This equation may be .solved graphically, as shown in Fig. 37. As the value of !T, evidently lies between 2500 and 3000 we plot the curves 3.45 3.44 8.42 3.41 25X) 3.40 / 2600 2700 FIG. 37. 2800 ?/ = ^g T, and y = 3.7028 - 0.0001056 T^ between these limits. The intersec- tion gives the desired value, T = 2049. The external work of expansion is / row W = J\ (0.1544 + 0.00003753 T)dT Jwa ^ ' =287,940 ft.-lb. EXERCISES The following are the compositions by volume of two gases, one a rich natural gas, the other a blast furnace gas : NATURAL GAS (Indiana) H 2 0.02 CO 0.007 CH, 0.931 BLAST FURNAOM GAS II Z 0.05 CO 0.27 CH 0.015 Work the following examples for each of these gases : 1. Find the composition, by weight. 2. Find the heating value : (a) per cubic foot under standard conditions; (6) per pound. 3. Calculate the constants B m , y, v, and c p c v . 4. Find the volume of air required for the combustion of one cubic foot. 5. Find the weight of air required for the combustion of one pound. 6. Find the products of combustion, by weight. 7. Find the specific heat c,, of a mixture of the gas with air, the weight of air being 15 per cent in excess of that required for complete combustion. 8. Find c v for the products of combustion, assuming that 15 per cent excess of air is used. 9. Find the constants B m , y, and v of the mixture of Ex. 7; also of the products of combustion. 10. The mixtiire of Ex. 7 is compressed adiabatically from a pressure of 14.7 Ib. per square inch to a pressure of 120 Ib. per square inch in the case of the natural gas and to a pi-essure of 175 Ib. per square inch in the case of the blast furnace gas. The initial temperature in each case is 80 F. Find the temperature at the end of compression in each case. 11. Find the work of adiabatic compression. 12. Find the ratio of initial to final volume. 13. If at the end of adiabatic compression the mixture is ignited and burns at constant volume, find the temperature at the end of the process, assuming that no heat is lost by radiation. 14. After combustion the products expand adiabatically to the initial volume. Calculate the final temperatures. 15. Find the work of adiabatic expansion. 16. Assume that the adiabatic compression follows the law p 7" = const. Find the values of n. Find also the values of n for the adiabatic expansion. REFERENCES GAS MIXTURES Preston : Theory of Heat, 350. Bryan : Thermodynamics, 121. Zeuner: Technical Thermodynamics (Klein) 1, 107. Wevrannh : Grnndriss cler Wanne-Theorie 1, 137, 140. FUELS. COMBUSTION. HEATING VALUES Levin : Modern Gas Engine and Gas Producer, 80. Carpenter and Diederichs : Internal Combustion Engines, 129. Zeuner : Technical Thermodynamics 1, 405, 410. Weyrauch : Grundriss der Wiirme-Theorie, 216, 255. Jones : The Gas Engine, 293. Poole : The Calorific Power of Fuels. In the field of thermochemistry reference may be made to the extei sive researches of Favre and Silbermann, Berthelot, and J. Thomson. Fc tables of heating values see Landolt and Bornstein : Physik.-chemiscb TabeUen. VARIABLE SPECIFIC HEAT OF GASES Mallard and Le Chatelier : Annales des Mines 4. 1883. Vieille: Comptes rendus 96, 1358. 1883. Langen : Zeit. des Verein. deutsch. Ing. 47, 022. 1903. Haber : Thermodynamics of Technical Gas Reactions, 208. Clerk: Gas, Petrol, and Oil Engines, 341, 301. Zeuner: Technical Thermodynamics 1, 146. Carpenter and Diederichs : Internal Combustion Engines, 220. THERMODYNAMICS OF COMUUSTION Zeuner: Technical Thermodynamics 1, 423, 428. Lorenz : Technische Wiirmelehre, 392. Stodola: Zeit. des Verein. deutsch. Ing. 42, 1045, 1086. 1898. CHAPTER IX TECHNICAL APPLICATIONS. GASEOUS MEDIA 86. Cycle Processes. In any heat motor, heat is conveyed from the source of supply to the motor by some medium, which thus simply acts as a vehicle or carrier. In practically all cases the medium is in the liquid or gaseous state, though a motor with a solid medium is easily conceivable. The perform- ance of work is brought about by a change in the specific volume of the medium due to the heat received from the source. By a proper arrangement of working cylinder and movable pis- ton this change of volume is utilized in overcoming external resistances. (In the steam turbine another principle is em- ployed.) The medium must pass through a series of changes of state and return eventually to its initial state, the series of changes thus forming a closed cycle. To use a crude illustra- tion, the medium taking its load of heat from the source at high temperature, delivering that heat to the working cylinder and to the cold body (condenser) and returning to the source for another supply may be compared with an elevator taking freight from an upper story to a lower level and returning empty for another load. Where the medium is expensive it is used over and over, and thus passes through a true closed cycle. Examples are seen in the ammonia refrigerating machine and in the engines and boilers of ocean steamers, in which fresh water must be used. In such cases we may speak of the motor as a closed motor. If the medium, on the other hand, is inexpensive or available in large quantities, as air or water, open motors are quite generally used. In these the working fluid is discharged into the atmos- phere and a fresh supply is taken from the source of supply. Even in this case the medium mav pass through a closed cycle, but all the changes of state are not completed in the organs of the motor. In this chapter we shall take up the analysis of several cycles that are of importance in the technical applications of gaseous media. In general, we shall assume ideal conditions, which cannot be attained in actual heat motors. However, the con- clusions deduced from the analysis of such ideal cycles are usually valid for the modified actual cycles ; furthermore, the ideal cycle furnishes a standard by which to measure the effi- ciency of the actual cycle. 87. The Carnot Cycle. Although the Carnot cycle is of no practical importance, it possesses the greatest interest from a theoretical point of view. Hence an analysis of it is included. Referring to Fig. 18, the heat absorbed from the source dur- ing the isothermal expansion AB is given by the equation a log e , (1) 'a and the heat rejected to the refrigerator is 77" , = Av V loo- -LA (v\ Vcd -"-jfc ' c lu be rr " \^} ' c The heat transformed into work is, therefore, A W= Q a(i + Q cd = A( PU V a log fi -p - p. V c log. IT). (3) \ ' a I <;/ Since in the state A the temperature is T v we have p a r a = MBT v (4) and likewise p c V c = MB T 2 . (."> ) Furthermore, for the adiabatic BO we have the relation and for the adiabatic DA the relation ~~ T' K K ART. 88] CONDITIONS OF MAXIMUM EFFICIENCY 135 Introducing in (3) the results given by (4), (5), and (8), we obtain whence AW Q t ab (9) (10) f n This expression for the efficiency is identical with that deduced from the Kelvin absolute scale of temperature. We have in Eq. (10) a proof, therefore, that the Kelvin absolute scale coin- cides with the perfect gas scale. E D r, 88. Conditions of Maximum Efficiency. On the SW-plane the Carnot cycle is the simple rectangle ABCD (Fig. 38), hav- ing the isothermals AB and CD at the temperatures T-^ and T 2 of the source and refrigerator, respec- tively. This geometrical rep- resentation affords an intuitive insight into the property of maxi- mum efficiency. Between the same isothermals let us assume some other form of cycle, as the trapezoidal cycle EB CD, For the rectangular cycle the efficiency is heat transformed into work_ area ABCD heat supplied area A 1 ABB 1 For the trapezoidal cycle, likewise, the efficiency is area DEBC A, #1 FIG. 38. But DEBC area A l DEBB l ABCD-AED ~~ A^ABB^ - AED ABCD that any cycle lying wholly within the rectangular cyle AB CD has a smaller efficiency than the rectangular cycle. With a given source and refrigerator, the conditions of maxi- mum efficiency, which may be approached but never actually attained, are the following : 1. The medium must receive heat from the source at the temperature of the source. No heat must be received at lower temperature. 2. The medium must reject heat to the refrigerator at the temperature of the refrigerator. 3. Provided the medium, source, and refrigerator are the only bodies involved in the transfer of heat, it follows from 1 and 2 that the intermediate processes must be adiabatic, as any departure from the adiabatic would mean passage of heat to or from some body at a tem- perature different from either the source or refrigerator. 89 . Isoadiabatic Cycles . Let a cycle be formed with the iso- thermals AB and CD as in the Carnot cycle, but with the adiabatics replaced by similar curves BC and AD (Fig. 39) ; that is, curve BC is simply -^r ^ g s curve DA shifted horizontally FIG. 39. a distance AB. Then AB = DC, as in the Carnot cycle. If the cycle is traversed in the clockwise sense, the heat entering the medium is Qda +Qab = area D 1 DAA 1 + area while the heat rejected by the medium is Qbc + Qcd = area B^B CC l + area 1 CDD r The heat transformed into work is the same as in the Carnot cycle, for the area of the figure ABCD is equal to that of the r>. j. j. i ^T . ,. ., _ " D 1 DAA 1 is taken from the source of heat, the efficiency of the cycle is _ heat transformed __ area ABQD heat taken from source ~~ area D l DABB l ' and this is manifestly smaller than the efficiency of the Carnot cycle. Let it be observed, however, that V&c Qdal that is, area B l BOO l = area D^AA^ If the heat rejected by the medium during the process BO could be stored instead of thrown away, then this heat might be used again during the process DA, thus saving the source the heat Q da . In this case we should have the following series of steps : 1. Medium absorbs heat Q^ from source. 2. Medium rejects heat Q be , which is stored. 3. Medium rejects heat Q cd to refrigerator. 4. Medium absorbs the heat Q da (= Q b J) stored during step 2. Since in this case the source furnishes only the heat Q&, the efficiency is area ABCD 77 area which is the same as that of the Carnot cycle. A cycle in which the adiabatics of the Carnot cycle are replaced by similar curves, along which the interchanges of heat are balanced, is called an isoadiabatic cycle. Any such cycle has the same ideal efficiency as the Carnot cycle. 90. Classification of Air Engines. Heat motors that employ air or some other practically perfect gas as a working fluid may be divided into two chief classes : (1) Motors in which the fur- nace is exterior to the working cylinder, so that the medium is heated by conduction through metal walls. (2) Motors in which the medium is heated directly in the working cylinder by the combustion of some gaseous or liquid fuel. These are called internal combustion motors. We mav make a, second division based on the manner in which the working fluid is used. In the closed-cycle type of motor, the same mass of air is used over and over again, fresh air being supplied merely to replace leakage losses. In the open-cycle type a fresh charge of air is drawn in at each stroke, and after passing through its cycle is discharged again into the atmosphere. Air motors of the first class, namely, those with the furnace exterior to the working cylinder, are usually designated as hot- air engines. Motors of this class are no longer constructed except in small sizes for pumping and domestic purposes ; they are, however, of historical interest, and besides they furnish in- structive illustrations of the application of the regenerative principle. We shall, therefore, describe briefly the two leading types of hot-air engines and give an analysis of the cycles. 91. Stirling's Engine. The engine designed by Robert Stirling in 1816, and bearing his name, is of the external fur- nace closed-cycle type. The general features of the engine are shown in IP Fig. 4-0. A displacer piston Q works in a cyl- inder Q. Between and an outer cylinder D is placed a regenerator RR, made of thin metal plates or wire gauze. At the upper end of the cylinder is a refrigerator M, com- posed of a pipe coil through which cold water is made to circulate. At the lower FIG. 40. end is the lire F. The piston Q is filled with some non-conducting material. The working cylinder A has free communication with the displacer cylinder. In the actual piston P to be at the beginning of its upward stroke and the displacer piston at the bottom of its cylinder. The air is, therefore, all in the upper part of the cylinder in contact with the refrigerator, and its state may be represented by the point D (Fig. 39). Now let the displacer piston be moved suddenly to the upper end of its cylinder. The air is forced through JK and the perforations in O into the lower end of the cylinder. The air remains at constant volume, since the piston P has not yet moved, and has received heat in passing through R. Hence the change of state is a heating at constant volume represented by DA in the diagram. The air now receives heat from the furnace and expands at constant temperature during the up- ward working stroke of piston P, This process is represented by AB. When the piston P reaches the upper end of its stroke, the displacer piston Q is suddenly moved to the bottom of the cylinder, thus forcing the air back through R into the refrigerator M. This again is a constant volume change and is represented by BO. Lastly, during the return stroke the air is compressed isothermally, as represented by (7Z>, and heat is re- jected to the refrigerator. The ideal cycle is seen to be an isoadiabatic cycle with the adiabatics of the Carnot cycle replaced by constant-volume curves. The cycle given by the actual engine deviates consid- erably from the ideal cycle on account of the large clearance necessary between the two cylinders. A double acting Stirling engine of 50 i. h. p. was used for some years at the Dundee foundry, but was eventually aban- d.oned because of the failure of the regenerators. This engine had an efficiency of 0.3 and consumed 1.7 Ib. of coal per i. h. p. 92. Ericsson's Air Engine. The Swedish engineer Ericsson made several attempts to design hot-air engines of considerable power. His large engines proved failures, however, because of their enormous bulk and the rapid deterioration of the regener- ators. The engines for" the 2200-ton vessel Ericsson had four single-acting working cylinders 14 ft. in diameter and 6 ft. stroke and ran at 9 r.p.m. They developed 300 li.p. with a fuel consumption of 1.87 Ib. of coal per h.p.-hour. The working of the Ericsson engine was substantially as fol- lows : A pump compressed air at atmospheric temperature into a receiver, whence it passed through the regenerator into a working cylinder. The pump was water-jacketed so as to act as a refrigerator. During the passage through the regenerator the air was heated at constant pressure. After the air was cut off in the working cylinder, it expanded isotherimilly, the nec- essary heat being furnished by a furnace external to the working cylinder. On the return stroke the air was dis- charged through the regener- ator at constant pressure. The p /^diagram is shown in Fig. 41. The pump cycle is DCJ?E, the motor cycle JEAJBF. The operations are as follows: (1) Compression in pump from to D, heat abstracted by pump water-jacket. (2) Discharge from pump to regenerator, represented by DE. (3) Suction of air into working cylin- der represented by EA. (4) Isothermal expansion from A to J9, during which air receives heat from furnace. (5) Dis- charge of air, represented by BF. (G) Suction of air into pump, represented by FO. Deducting the work of the pump from that of the motor, the effective work is given by the diagram AB CD composed of the two isothermals and two constant-pressure lines. 93. Analysis of Cycles. The ideal cycles of the Stirling and Ericsson engines are isoadiabatic cycles. In the Stirling cycle the constant-volume lines DA and BO (Fig. 39) replace the adiabatics of the Carnot cycle. Using the iW-plane we have Q* = Ap a V a log e ^ = ABTJf log. !J ' o I' a n 7I/T, / m m \ ART. 94] HEATING BY INTERNAL COMBUSTION ' 141 Q cd = Ap c V c log, -p = - A MBT Z log, |f But since F a = V d and F c = F 6 , The heat ^ is taken from a regenerator, and therefore the heat Qa alone is supplied from the source ; hence the efficiency s " ft* " *i ' For the Ericsson cycle Z>J. and .#(7 are constant-pressure lines and the analysis is essentially the same except that c v is replaced by c p . 94. Heating by Internal Combustion.* While the hot-air engine with exterior furnace should apparently be an efficient heat motor, experience has proved the contrary. The difficulty lies in the slow rate of absorption of heat by any gas. Even with high furnace temperatures and comparatively large heat- ing surfaces it has been found impossible to get a high tempera- ture in the working medium. Furthermore, if the air could be effectively heated, the metal surfaces separating the furnace from the hot medium would be destroyed; hence, while high tempera- ture of air is necessary for high efficiency, low temperature is necessary to secure the durability of the metal. These contradictory conditions are completely obviated by the method of heating by internal combustion. The rapid chemical action supported by the medium itself makes possible the rapid heating of large quantities of air to a very high temperature. The medium and the furnace being within the t.>io nn-t-o-p onvFunA rvf t.Tip. mftta.l walls can be keT)t at low temperature by a water jacket, and consequently the inner surface can be exposed to the high temperature desired without danger of destruction. .Furthermore, the low conductivity of gases becomes here an advantage as it prevents a rapid flow of heat from the medium to the cylinder walls. The low gas temperature of the hot-air engine results in a small effective pressure and makes the engine very bulky for the power obtained. The high temperature possible in the internal combustion motor, on the other hand, permits high effective pressures, and therefore gives a relatively small bulk per horsepower. 95. The Otto Cycle. The cycle of the well-known Otto gas engine has five operations as follows : 1. The explosive mixture is drawn into tho cylinder. Represented by HI), Fig. 42. 2. The mixture is com- pressed, as represented by DA. 3. The charge is ignited, causing a rise of temperature and pressure, as shown by AB. 4. The gases in the cyl- FIG. 42. inder expand adiabatically as shown by BQ. 5. The burned gases are expelled in part. Represented by DE. In the case of the four-cycle Otto engine, each of the opera- tions 1, 2, 4, and 5 occupies one stroke of the piston, while operation 3 occurs at the beginning of a stroke. The cycle is completed in four strokes, whence the term four-cycle. It is customary in the analysis of gas-engine cycles to assume in the first instance that the medium is pure air throughout the cycle and that the air receives during the process AB an amount of heat equal to that developed by the combustion of the fuel in the actual cycle. This assumed ideal CVfilfi is rpfprrorl f V-T, , _._ , _ ART. 95] THE OTTO CYCLE 143 On the Titf-plane, the ideal cycle has the form shown in Fig. 43, AS and CD being constant volume curves. The medium in the state repre- sented by point A is heated at constant volume, as shown by the curve AB. For this pro- cess we have (assuming that c v is constant) For the adiabatic expansion represented by BO, W - ~ For the cooling at constant volume, represented by (7.Z), we have Q cd = Jfc,( T d - T c ) = - Finally the medium is compressed adiabatically from D to J., and for this change of state W - rr 7 - K 1 The heat changed into work is (1) The work of the cycle is W bc + W cd + W^ It is easily shown that these results are identical. The efficiency is / TJBJUJlJNUJAJj Arr JUJLOA A This expression for rj may be simplified as follows : From Fig. 43 we have S b - S a =S. -S d = Mc v log. ? = Mc v log. ', , hence, Therefore, rn T T Jt c ^v, L c _ -*rf -- or _ , J. a -*& x a or It appears, therefore, that the Otto cycle has the same efficiency as a Carnot cycle having the extreme temperatures T a and T d or the extreme temperatures T b and T of the adiabatics, but a smaller efficiency than a Carnot cycle having T b and T d as extreme temperature limits. The expression for the ideal efficiency may be written in another convenient form. Since the curve DA represents an adiabatic process, we have whence 1-1, or (5) It appears from the last expression that the higher the com- pression pressure^, the greater the ideal efficiency. If the ratio of volumes -~ be denoted by r* we have for the T' * a ideal efficiency the expression 11 1 (6) EXAMPLE. If the air is compressed from 14.7 Ib. to 45 ll>., the ideal The temperature and pressure represented by the point B are readily calculated for this ideal case. Let q l denote the heat absorbed per pound of air during the process AB; then whence ^A + l. (7) -* a Cv-*-a Since F.= F 6 , The value of q l for a given fuel depends upon the heating value of the fuel and the weight of air required for the com- bustion of a unit weight of the fuel. 96. The Joule or Brayton Cycle. In the Otto type of motor, the fuel gas is mixed with air previous to compression, and when the mixture is ignited the combustion is so rapid as to produce an explosion; the heat is supplied, therefore, at practically constant volume. Another type of motor was first suggested by Joule and was developed in working form by Brayton (1872). In the Bray ton engine the mixture of air and gas was compressed into a reservoir to a pressure of per- haps 60 Ib. per square inch and from the reservoir flowed into the working cylinder, where it was ignited by a flame. A wire gauze diaphragm was used to prevent the flame from striking back into the reservoir. The mixture was thus burned quietly in the working cylinder during about one half the stroke of the piston, and by proper regulation of the admission valve the rate of combustion was so regulated as to give practically con- stant pressure during the period of admission. The ideal cycle of operations is as follows: 1. Charge drawn, into compressor cylinder, ED (Fig. 44). 2. Adiabatic compression, DA. 146 TECHNICAL APPLICATIONS. GASEOUS MEDIA [CHAP, ix 3. Expulsion at constant pressure from compressor, AF; simultaneous admission to motor cylinder, FB, The charge during the passage from compressor to motor is heated at constant pres- sure and the volume is thereby increased as in- dicated by AB, 4. Adiabatic expansion, BC, after cut off. 5. Expulsion of burned T , A , gases, OE. FIG. 44. b ' The area JEDAF repre- sents the negative work of the compressor, the area FBQJH the work obtained from the motor ; hence, area ABCD repre- sents the net available work. On the T/S'-plane, the ideal Joule cycle has the same form as the Otto cycle (Fig. 43). The curves AB and (72), however, represent, respectively, heating and cooling at constant pressure. We have, therefore, = Q ab + Q cd = .,__?'' \ T: Also, ~ = . 0) (2) (3) (5) 97. The Diesel Cycle. The principle of gradual and quiet combustion as opposed to explosion was seized upon by Diesel in the design of the Diesel motor. In this motor air without fuel is compressed in the working cylinder to a pressure ap- proximating 500 Ib. per square inch. The temperature at the end of compression is consequently higher than the ignition tempera- FIG. 45. expand at practically constant pressure, or if desired, with falling pressure and nearly constant temperature. As in the Brayton engine, govern- ing is effected by cutting off the fuel injection earlier or later. The ideal cycle of the Diesel o engine is shown in Fig. 45. It resembles the Otto cycle except that the process AS in this case represents a constant pressure rather than a constant volume combustion. It was the original aim of Diesel so to regulate the injection of fuel that a short period of combustion AM would be followed by isother- mal expansion Jf-ZV, the fuel being cut off at the point N. On the 2!$-plane the ideal Diesel cycle is shown in Fig. 46, in which AB is a constant-pressure curve and CD a constant-volume curve. We have then (1) (2) ' (3) FIG. 46. c p (T b -T a ) If the cycle includes an isothermal process, as MN, we have Q am = Mc p ( T m - T a }, (5) V, f~\ A ]\/rj2 fT* Inrr . -_ *Vmn == jti.JLrj.JL> J- m. J- U ^P tr 1 <,*, o* Y^ ?n+ <?, and 77 = -^ :=1 (6) a) FIG. 47. 98. Comparison of Cycles. The three principal cycles are shown superimposed in Fig. 47. The minimum temperature at J) and maximum temperature at B are the same for all three. With this assumption it is seen that the Bray ton cycle A'BC'D has the largest area, the Otto cycle ABGD, the smallest. Hence, between the same temperature limits and with the same maximum pressure jp 6 , the Bray ton cycle is the most efficient, the Otto cycle the least efficient. Com- $ paring the maximum volumes, it is seen that the Otto and Diesel cycles have the same maximum volumes V& while the Bray ton cycle requires a greater volume, as indicated by the point O 1 '. The Diesel cycle, therefore, combines the advantages of the high efficiency of the Brayton cycle due to the high compression pressure and the smaller cylinder volume of the Otto cycle. 99. Closer Analysis of the Otto Cycle. In the preceding analysis of gas-engine cycles two assumptions have been made : (1) That the medium employed has throughout the cycle the properties of air. (2) That the specific heat of the medium is constant. While the approximate analyses based on these assumptions are of value in giving the essential characteristics of the various cycles, and an idea of their relative efficiencies, they give misleading notions regarding the absolute magnitudes of those efficiencies. To obtain the true value of the maximum possible efficiency of a gas-engine cycle, it is necessary to take account of the properties of the fuel mixture entering the cylin- der and of the mixture of the products of combustion after the fuel is burned. Making use of the principles and methods laid down in Chapter VIII, we may thus make an accurate nislied by the example of Art. 85, shows such an analysis for the Otto cycle. EXAMPLE. Determine the ideal efficiency of an Otto cycle in which the compression, combustion, and expansion follow the course described in the example of Art. 85. Compare this efficiency with the "air standard" efficiency under the same conditions. In the example quoted, the work of adiabatic compression was found to be 69,460 ft.-lb., the work of expansion 287,940 ft.-lb. These results refer to 1 Ib. of the fuel mixture. The heating value of the fuel per pound was found to be 1632.2 B. t. u. ; hence the heating value per pound of fuel mix- ture is 1632.2 - 2.25 = 725.4 B. t. u. The net work derived from the cycle per pound of mixture is 287,940 - 69,460 = 218,480 ft.-lb. Therefore, the efficiency is Q.387. J x 72o.4 The "air standard" efficiency depends upon the ratio of initial and final Vz volumes, which ratio was found to be =- = 0.1887. Hence, for this efficiency 1 i we have -rj = 1 - 0.1887-* = 0.487. The discrepancy between the two efficiencies is in a large measure due to the assumption of constant specific heat in. the analysis of Art. 95. 100. Air Refrigeration. The term refrigeration is applied to the process of keeping a body permanently at a temperature lower than that of surrounding bodies. Since heat naturally flows from the surroundings to the body at lower temperature, this heat must be continually removed if the body is to remain permanently at its lower temperature. Hence a refrigerating machine has the office of removing heat from a body of low temperature and depositing it in some other convenient body of higher temperature. The operation of a refrigerating machine is thus precisely the reverse of the operation of the direct-heat motor ; and if the cycle of a heat motor be traversed in the reverse direc- tion, it will give a possible cycle for a refrigerating machine. When air is used as a medium for refrigeration, the reversed Joule cycle is employed. Fig. 48 shows diagrammatically the arrangement of the refrigerating machine, Fig. 49 the ideal j? ^diagram, and Fig. 50 the ^-diagram. Air in the state A in the cold room is drawn into the compressor c and is com- 150 TECHNICAL APPLICATIONS. GASEOUS MEDIA [CHAP, ix pressed adiabatically as indicated by AB. It then passes into the cooling coils, about which cold water circulates, and is cooled at constant pressure, as indicated by BO. In the state the > ff- f U ' < 'Jill IB 3 f It C Cooling 1 Cold J w 3 s~ N / > C Coils Room 4 1" - / 1 J f III! Illl Illllllll 1 J , Ij, ..^ U. J_ _Li j FIG. 48. air passes into the expansion cylinder e and is permitted to ex- pand adiabatically down to the pres- sure in the cold room, i.e. atmos- pheric pressure. The final state is represented by point D. Finally the air absorbs heat from the cold room, and its temperature rises to the original value T a . Referring to Fig. 49, the actual compression diagram is ABFE, while the diagram JFCDJE taken clockwise is the diagram of the expan- sion cylinder. The net work done on the air is, therefore, given by the diagram ABOD. The Allen dense-air machine has a closed cycle and the air is always under a pressure much higher than that of the atmos- phere. Thus the pressure DA (Fig. 49) is perhaps 40 to t>0, and the upper pressure, say 200 Ib. per square inch. The air, after expanding to the lower pressure, is led through coils immersed in brine and absorbs heat from the brine. In the following analysis of the air-refrigerating machine E we shall assume ideal condi- tions. In the actual machine " JT IO . 49. these conditions are to some ART. 100] AIR REFRIGERATION 151 minute, and M the weight of air circulated per minute. Then since in passing through the cold body the temperature of the air is raised from T d to T a (Fig. 50), we have Q = Mc p (T a -T^. (1) Let p-L denote the suction pres- sure of the compressor cycle (atmospheric pressure, in the case of the open cycle) and p z the pressure at the end of com- pression ; then, assuming adiabatic compression, we have FIG. 50. ~T ~~ ( ~rT I ' (Q a \ J f'l / and if the pressure at cut-off in the expansion cylinder is also p z (as in the ideal case), we have also (3) (4) (5) $\) whence _ = . The work required per minute is rn m rn J -b~ -La " ~ " v " area O l DAB l * T a ' and the heat rejected to the cooling water, represented by the area B l BGO l (Fig. 50), is W T/, The compressor cylinder draws in per minute M pounds of air having the pressure p l and temperature T a , Denoting by N the number of working strokes per minute and by V c the volume displaced bv the comnressor mston. we have for the ideal case 152 TECHNICAL APPLICATIONS. GASEOUS MJSJUJLA ICHAP.IX or Likewise, the volume V e of the expansion cylinder is given by the relation EXAMPLE. An air-refrigerating machine is to abstract GOO B. t. u. per minute from a cold chamber. The pressure in the cold room is 14.7 Ib. per square inch, and the air is compressed acliabatically to 05 Ib. per square inch absolute. The temperature in the cold room is 30 F. and the air leaves the cooling coils at 80 F. The machine makes 120 working strokes per minute. Kequired the ideal horsepower required to drive the machine, and the volumes of the compression and expansion cylinders. The first step is the determination of the temperature T d at the end of expansion. From the relation 0.4 we have T d = 539.0 (iM ) " = 352.9. From (1) we obtain for the weight of air that must be circulated per minute M = Q _ ^ __ = 17r,'Ml, c P (T a -T d ) 0.24(405.0 - 352.9) The work required per minute is W = JQ T "~ Td = 778 x GOO x 5:j9 - 6 ~ 352 - <} = 240,950 ft. Ib. , T d 352.9 ' ' and the horsepower under these ideal conditions is therefore 246950 _ 7 P 33000 '' For the volume of the compressor cylitider, we have v 17.52 x 53.34 x 495.0 . U0 .. Fc= 120x14.7x144 = Lb2c " ft -> and for the volume of the expansion cylinder ny in mining, tunneling, ana metallurgical processes, impression of air may be effected by rotary fans and s or by piston compressors. In the piston compressor, itmospheric pressure is drawn into a cj'linder through, in- ves and is then compressed upon the return stroke of the When the desired pressure is attained, the outlet valves sued and the air is discharged into a receiver. The ideal or diagram of an air p ;ssor has, therefore, the hown in Fig. 51. The c 4. represents the drawing le air ; the curve AB rep- i the compression from wer pressure p 1 to the sr pressure jt? 2 ; and BO D jnts the expulsion of the _ the higher pressure. It FlG> 51 _ be noted that the curve ipresents a change of state, while lines DA and BO nit merely change of locality ; thus BQ represents the D of the air (in the same state} from the compressor 31* to the receiver. V^ denote the volume denoted by point A, and V 2 the } after compression ; then the work of compression (area B is '"* n-l ng that the compression curve follows the law pV n = const. )rk of expulsion (represented by area B^BOO) is evidently 3 work done by the air during the intake (area ODAA^) is the total work represented by the area of the diagram n (1) 1 V V I I 2 ~ x W ' whence combining (1) and (2) we get (2) (3) a formula that does not contain the final volume Y v For the temperature at the end of compression we have the usual formula (4) The action of the air com- pressor may be studied advanta- geously by means of the T8- diagram. Let the point A (Fig. 52) represent the state of the air at the beginning of com- pression, and suppose that AB represents the compression pro- cess. Through B a line repre- senting the constant pressure p z is drawn, intersecting at F an isothermal through A. It can be shown that the area A l ABFF l represents the work W given by (1). Denoting by T 2 the final temperature corre- sponding to point B, we have area A l ABS l = Mo v ^- (T 2 - TJ, FJG> 52> area Mc area A l ABFF l = n c ~ n 1 1 n B 102. Water- jacketing. Unless some provision is made for withdrawing heat during the compression, the temperature will rise according to the adiabatic law. Ordinarily the energy stored in the air due to its increase of temperature, that is, the energy U 2 - U,= Mc^T.-T^ is never utilized because during the transmission of the air through the mains heat is lost by radiation and the temperature falls to the initial value. Hence a rise in the temperature during compression indicates a useless expenditure of work. The water jacket prevents in some degree this rise in temperature and decreases the work required for compression. The curve AE (Fig. 53) represents adiabatic . -TP , compression. If the compres- sion could be made isothermal, the curve would be AF, less steep than AE, and the work of the engine would be reduced per stroke by the area AEF. The water jacket gives the curve AS lying between AE and AF, and the shaded area represents the saving in work. Because of the water jacket the value of the exponent n in the equation p V n = const, lies somewhere between 1 and 1.40. Under usual working conditions, n is about 1.8. For any value of n the relation between the heat abstracted, work done, and change of energy is given by the proportion JQ:(U 2 - ZTj) : TF= (k - n) : (1 - n) : (k - 1). This applies only to the compression AB not to the expulsion of the air represented by B 0. The influence of the water jacket is shown more clearly by the ^-diagram, Fig. 52. The vertical line AE indicates adia- batic compression from p l to jp 2 , the horizontal line AF, isother- mal compression, and the intermediate curve .&., compression according to the law p V n const., with n between 1 and 1.4. The area A-^ABB^ represents the heat abstracted from the air during compression, and the area AEB represents the work saved by the use of the jacket. A more efficient jacket would give a compression curve with its extremity lying nearer the point F. In the case of the isothermal compression represented by AF, the area A l AFF l represents the heat absorbed from the air and also the work done on the air. These must necessarily be equivalent, since there is no change in the internal energy. 103. Compound Compression. The excess of work required by the increase of temperature during compression may be obvi- ated in some measure by dividing the compression into two or more stages. Air is compressed from the initial pressure p 1 to an intermediate pressure p', it is then passed through a cooler where the temperature (and con- sequently the volume) is FlGt 54 ' reduced, and finally it is compressed from p' to the desired pressure p T In Fig. 54, DA represents the entrance of air into the cylinder, and A 6r, which lies between the adiabatic AE and the isothermal AF, the compression in the first cylinder. From Gr to Jt the air is cooled at constant pressure in the intercooler. The curve HL shows the compression in the second cylinder, and the line LQ the expulsion into the receiver. In a single cylinder the diagram would be ABQD ; hence compounding saves the work indicated by the area B CrHL. The saving is shown even more clearly if we use the TS- plane (Fig. 55). During the first compression AGr the heat represented by the area A 1 AGrGr 1 is absorbed by the water jacket. Then the heat G-^GrHH^ is abstracted by the inter- cooler. During the second compression the heat HMLL, is ART. 103] COMPOUND COMPRESSION 157 abstracted by the water jacket, and finally the heat is radiated from the receiver and main. As shown in the preceding article, the area A l AG-HLFF l gives the work of the compressor. Evidently area BGrSL represents the work saved by compounding. If we take (3) of Art. 101, we find for the work done in the first cylinder FIG. 55. and for the work done in the second cylinder n-l n p -i S where V is the volume indicated by point H (Fig. 54). But since point .ffis on the isothermal AF> we have and, therefore, n-l P The total work is, consequently, n-l l\ n (1) The work required is numerically a minimum when the is variable. Using the ordinary method of the calculus, we find that this expression is a maximum when P 1 = Equation (2) is useful in proportioning the cylinders of a com pound compressor. Referring to Fig. 55, we have With the condition expressed by (2) we have n-l n1 and likewise, n-l n~l \PiPzJ \Pi, Hence, T l = T a ; that is, for a minimum work of compression the points G- and L should lie on the same temperature level. The same statement applies to three-stage compression. 104. Compressed-air Engines. Compressed air may be used as a working fluid in a motor in substantially the same way as steam. In fact, compressed air has in some instances been used in ordinary steam engines. The indicator diagram for the motor should approach the form shown in Fig. 50. With clearance and compression, A. 1 12' will replace FlG 56 AE. The work per stroke is readily calculated in either case. The expansion curve BO may be taken as an adiabatic. 105. TS-diagram of Combined Compressor and Engine. The 2W-diagram shows clearly the losses in a compressed-air system losses. In the following discussion we shall take up first an ideal case and afterwards several modifications that may be made. In Fig. 57, m represents the compressor diagram, n the motor diagram, both without clearance. Air in the state repre- sented by point A is taken into the com- pressor at atmos- pheric pressure and temperature. The compression, a s- sumed here to be adiabatic, is repre- sented on the TS- plane by the vertical line AB (Fig. 58). The expulsion of the air into the receiver and thence into the main is merely a change of locality and does not itself involve any change of state ; hence, it is not represented on the ^fW-plane. However, the passage of the air along the main is usually accompanied by a cooling, and this is represented by BQ (Fig. 58), the final point representing the state of the air at the beginning of expansion in the motor. The adiabatic expansion to atmos- pheric pressure in the motor is represented by CD. This is accompanied by a drop in tem- perature which is given by the equation k-l A- T FIG. 58. The air discharged from the motor in the state D is now heated at the constant pressure of the atmos- phere until it regains its original temperature T a . This heating is represented by DA. The complete process is a cycle of four distinct operations, L ^ ,-,,, ,, . (-l-.r.-f- what does the area AJUJJU ot tne cycle represent sometnmg useful or something wasteful ? To answer this question let us recur to the original energy equation JQ = Z7 2 - Z/i + W, and apply it to the air which passes through the cycle process just described. We have Work done on air = area of diagram m = W m . Work done by air = area of diagram n + W n . Total work = W n - W m . Heat absorbed by air = area under DA. Heat rejected by air = area under SO. Total heat put into system = area ABQD. Change of energy = U a U a 0. Hence, j x ^ AB Q]) = ^ _ ^ that is, the area ABQD represents the difference between the work done by the compressor and the work delivered by the motor. Consequently it represents a waste, which is to be avoided as far as possible. Various modifications of the simple cycle of Fig. 58 are shown in Fig. 59. The effect of using a water jacket is shown at (a). The shaded area represents the saving. Figure 59 (7>) shows the effect of reheating the air before it enters the motor. In the main the air cools, as indicated by BO, but in passing through the reheater it is heated again at constant pressure, and the state point retraces its path, say to D. Then follows adiabatic expansion DE, and constant-pressure heatins- EA. This rpVmntincr RH.VPR work B (d) FIG. 59. vy LUC area \JJJJUM. xo wuuiu ue pussiiuie TO carry D to the right of B, in which case the waste would "become zero or even negative. The area CDJ3]? does not, however, represent clear gain, as account must be taken of the heat expended in the process CD. In Fig. 59 (c) is shown the effect of compound compression, and in Fig. 59 (c?) the effect of compound compression with a compound motor. In each case the shaded area represents the saving. It would not be difficult to represent also the loss of pressure in the main due to friction. EXERCISES 1. Find the efficiency of a Stirling hot-air engine "working under ideal conditions between the temperatures 1340 F. and 140 F. Find the weight of air that must be circulated per minute per horsepower. 2. An air compressor with 18 in. by 24 in. cylinder makes 140 working strokes per minute and compresses the air to a pressure of 52 Ib. per square inch, gauge. Assuming that there is no clearance, find the net horsepower required to drive the compressor. Take the equation of the compression curve as p V 1 - 3 = const. 3. If 200 cu. ft. of air at 14.7 Ib. is compressed to a pressure of 90 Ib. per square inch, gauge, find the saving in the work of compression and expulsion by the use of a water jacket that reduces the exponent n from 1.4 to 1.27. 4. Find the efficiency of the ideal Otto cycle (air standard) when the compression is carried to 120 Ib. per square inch absolute. 5. Draw a cxirve showing the relation between the efficiency of the Otto cycle and the compression pressure. Take values of p from 40 to 200 Ib. per square inch. 6. An air-refrigerating machine takes air from the cold chamber at a pressure of 40 Ib. per square inch and a temperature of 20 F., and com- presses it adiabatically to a pressure of 200 Ib. per square inch. The air is then cooled at this pressure to 80 F. and expanded adiabatically to 40 Ib. per square inch, whence it passes into the coils in the cold chamber. The machine is required to abstract 45,000 B. t. u. per hour from the cold room, (a) Find the net horsepower required to drive the machine. () If the machine makes 80 working strokes per minute, find the necessary cylinder volumes. 7. Air is to be compressed from 14.7 Ib. per square inch to 300 Ib. per square inch absolute. If a compound compressor is used, find the interme- diate pressure that should be chosen. 8. In Ex. 7, the compression in each cylinder follows the law p F 1 - 3 = ;onst. Find the saving in work effected by compounding, expressed in per uent of the work required of a single cylinder. 9. Using the results of Ex. 10-15 of Chapter VIII, find the efficiencies of the Otto cycle with the natural gas and the blast furnace gas, respectively, under the conditions stated. Compare these efficiencies with corresponding air standard efficiencies. 10. On the TS-plane draw accurately an ideal Diesel cycle from the fol- lowing data: Adiabatic compression of air from 14.7 to 500 Ib. per square inch absolute ; heating at constant pressure to a temperature of 2200 F. ; idiabatic expansion to initial volume ; cooling at constant volume to initial state. Calculate the ideal efficiency of the cycle. 11. Modify the Diesel cycle of the preceding example by stopping the jonstant-pressure heating at 1600 F. and continuing with an isothermal jxpansion (as shown by MN, Fig. 40). Calculate the efficiency of this modified cycle. 12. The ideal Lenoir cycle has three operations, as follows : heating of air it constant volume, adiabatic expansion to initial pressure (atmospheric), and jooling at constant pressure. Show the cycle on pV- and TS-planes, and lerive an expression for its efficiency. 13. Let the expansion in the Otto cycle be continued to atmospheric pressure. Show the resulting cycle on pV- and T'S-planes and derive an jxpression for the efficiency. REFERENCES HOT-AIR ENGINES Snnis : Applied Thermodynamics for Engineers, 129. 5euner : Technical Thermodynamics (Klein) 1, 340. Stankine: The Steam Engine (1897), 370. Swing : The Steam Engine, 402. GAS-ENGINE CYCLES "lerk : Gas, Petrol, and Oil Engines, 67. Carpenter and Diederichs : Internal Combustion Engines, 65. ^evin : The Modern Gas Engine, 43. terry: The Temperature Entropy Diagram, 107. Sum's : Applied Thermodynamics, 154. 'eabody: Thermodynamics of the Steam Engine, 5th ed., 304. jorenz : Technische Warmelehre, 421. Veyrauch : Grundriss der Warmp-TViPmMa 077 AIR REFRIGERATION Ennis : Applied Thermodynamics, 396. Ewing : The Mechanical Production of Cold, 38. Peabody : Thermodynamics of the Steam Engine, 5th ed., 397. Zeunev: Technical Thermodynamics, 384. AIR COMPRESSION Peabody : Thermodynamics, 5th ed., 358. Ennis : Applied Thermodynamics, 96. CHAPTER X SATURATED VAPORS 106. The Process of Vaporization. The term vaporization may refer either (1) to the slow and quiet formation of vapor at the free surface of a liquid or (2) to the formation of vapor by ebullition. In the latter case, heat being applied to the liquid, the temperature rises until at a definite point vapor bubbles begin to form on the walls of the containing vessel and within the liquid itself. These rise to the liquid surface, and breaking, discharge the vapor contained in them. The liquid, meanwhile, is in a state of violent agitation. If this process takes place in an inclosed space as a cylinder fitted with a movable piston so arranged that the pressure maybe kept constant while the inclosed volume may change, the following phenomena are observed: 1. With a given constant pressure, the temperature remains constant during the process ; and the greater the assumed pres- sure, the higher the temperature of vaporization. The tempera- ture here referred to is that of the vapor above the liquid. As a matter of fact, the temperature of the liquid itself is slightly greater than that of the vapor, but the difference is small and negligible. 2. At a given pressure a unit weight of vapor assumes a definite volume, that is, the vapor has a definite density; and if the pressure is changed, the density of the vapor changes correspondingly. The density (or the specific volume) of a vapor is, therefore, a function of the pressure. 3. If the process of vaporization is continued at constant pressure until all the liquid has been changed to vapor, then if heat be still added to the vapor alone, the temperature will rise and the specific volume will increase ; that is, the density will decrease. 164 uaoo 10 ociiv^. i/u uo oenuiaicu. emu. line uuJJ.UttiJU lit;m]Jt;ra/UUI.e sponding to the pressure at which the process is carried on is the saturation temperature. If no liquid is present, and through absorption of heat the temperature of the vapor rises above the saturation temperature, the vapor is said to be superheated. The difference between the temperature of the vapor and the saturation temperature is called the degree of superheat. The process just described may be represented graphically on the jt?F"-plane. See Fig. 60. Consider a unit weight of liquid subjected to a pressure p represented by the ordinate of the line A' A 1 ' ; and let the volume of the liquid (de- p noted by ') be represented by A'. As vaporization proceeds at this constant pressure, the volume of the mixture of liquid and vapor increases, and the point representing the state of the mixture moves along the line A' A" . The point A' r represents the volume v" of the saturated vapor at the completion segment A' A" represents FIG. 60. of vaporization ; therefore, the the increase of volume v" v'. Any point between A' and A'\ as M, represents the state of a mixture of liquid and vapor, and the position of the point depends on the ratio of the weight of the vapor to the weight of the mixture. Denoting this ratio by x, we have x f n , whence it appears that at A', 3 = 0, while at A", 3 = 1. This ratio x is often called the quality of mixture. If the mixture is subjected to higher pressure during vapor- ization, the state-point will move along some other line, as B'B". The specific volume indicated by B" is smaller than that indicated by A". The curve v", giving the specific volumes of the satu- rated vapor for different pressures, is called the saturation curve ; while the curve v\ giving the corresponding liquid volume, is the liquid curve. These curves v', v" are in a sense boundary curves. Between them lies the region of liquid and vapor mixtures, and to the right of v" is the region of superheated vapor. Any point in this latter region, as -Z7, represents a state of the superheated vapor. 107. Functional Relations. Characteristic Surfaces. For a mixture of liquid and saturated vapor, the functional relations connecting the coordinates jp, v, and t are essentially different from the relation for a permanent gas. As explained in the preceding article, the temperature of the mixture depends upon the pressure only, and we cannot, as in the case of a gas, give p and t any values we choose. The volume of a unit weight of the mixture depends (1) upon the specific volume of the vapor for the given pressure and (2) upon the quality x. Hence we have for a mixture the following functional relations : * = /GO, orp = ^(0, (1) v= ( f}(p,x'). (2) With superheated steam, as with gases, p and t may be varied independently, and consequently the functional relation between p, v, and t has the general form , , = 0. (3) The characteristic surface of a saturated vapor is shown in Fig. 61. It is a cylindrical surface iS whose generating elements cut the pi-plane in the curve p = F(t')> These ele- t ments are limited by the two space curves v f and v", which when pro- jected on the jw-plane give the curves v', v" of Fig. 60. The space curve v" is the intersection of the iuo. Relation oetween Pressure and Temperature. The rela- tion p = ]?() between the pressure p and temperature t of a saturated vapor must be determined by experiment. To Reg- nault are due the experimental data for a large number of vapors. Further experiments on water vapor have been made by Ramsey and Young, by Battelli, and very recently by Hoi- born and Henning. These last-mentioned experiments were made with the greatest accuracy and with all the refinements of modern apparatus; they may, therefore, be regarded as furnishing the most reliable data at present available on the pressure and temperature of saturated water vapor. Experi- ments on other saturated vapors of technical importance, carbon dioxide, sulphur dioxide, ammonia, etc., have been made by Amagat, Pictet, Cailletet, Dieterici, and others. It is likely, however, that further experiments must be made before the data for these vapors are as reliable as those for water vapor. If the experimentally determined values of p and t be plotted, they will give the curve whose equation is p = f(t) (Fig. 61), To express this relation many formulas have been proposed, some purely empirical, some having a more or less rational basis. A few of these formulas are the following : 1. Siot's Formula. As used by Regnault, Biot's equation has the form log p = a ba n + c/3 n , (1) where n = t C. This formula is purely empirical. Having five constants, the curve may be made to pass through five experimentally determined points.; hence, the formula may be made to fit the experimental values very closely throughout a considerable range. The follow- ing are the values of the constants as given by Prof. Peabody : FOR STEAM FROM 32 TO 212 F., p FOR STEAM FROM 212" TO 428 F., p IN POUNDS PER SQUARE INCH. IN POUNDS PER S<JUAUE Iscu. a =3.125906 a = 3.743976 log b == 0.611740 log 5 = 0.412002 log c = 8.13204 - 10 log c = 7.74168 - 10 log a = 9.998181 - 10 log a = 9.998562 - 10 log /3 = 0.0038134 log/3= 0.0042454 n = t - 32 n = t-212 2. Rankings Formula. Rankine proposed an equation of the form 7? ,7 log^=JL + |+^ 2 , (2) in which T denotes the absolute temperature. This formula has been much used in calculating steam tables, especially in England. Having but three constants, it is not as accurate as the Biot formula. The following are the values for the constants, when p is taken in pounds per square inch, and .4 = 6.1007; B = -2719.8; (7=400125. 3. The Dupr-Hertz formula has the form a-blogT~ (3) This equation has been derived rationally by Gibbs, Bertrand, and others, and gives, with a proper choice of constants, results that agree well with experiment. Using the results of Reg- nault's experiments, Bertrand found the following values of the constant for various vapors (metric units). ale Water 17.44324 3.8682 2795.0 Ether 13.42311 1.9787 1729.97 Alcohol 21.44687 4.2248 2734.8 Chloroform 19.29793 3.9158 2179.1 Sulphur dioxide .... 16.99036 3.2198 1604.8 Ammonia 13.37156 1.8726 1449.8 Carbon dioxide .... 6.41443 - 0.4186 819.77 Sulphur 19.1074 3.4048 4684.5 4. Bertrand 's Formulas. Bertrand has suggested two equa- tions, namely : , * Wo. - 7 - ^ eo and p^k^T-iy. (5) The latter may be written in the more convenient form log p = log k n log (6) Bertrand's second formula (6) has the advantage over the others suggested of lending itself to quick and easy computa- tion. Furthermore, although it has but three constants, it gives results that agree remarkably well with the experiments of Holborn and Henning on water vapor. The constants are as follows (English units) : T=t + 459.6 n = 50. FROM 32 - 90 F. FROM 00 - 23T F. FROM 238 420 F. 6 = 140.1 6 = 141.43 6 = 140.8 log&= 6.23167 log& = 6.30217 log 7c = 6.27756 The agreement between observed and calculated values is shown in the following table. The maximum difference is one tenth of one per cent. TEMPERATURE, C. PRESSURE IN MM. OF MERCURY Bertrand's Formula tWflj // tsf^e Experiments of Holborn and Kenning 'LjLS^e-t^s.fft**. 4.577 4.579 10 9.208 9.205 20 17.511 17.51 30 31.682 31.71 40 55.121 55.13 50 92.325 92.30 60 149.21 149.19 70 233.55 233.53 80 354.97 355.1 90 525.64 525.8 100 760 760 110 1075.2 1074.5 120 1489.7 1488.9 130 2025.2 2025.6 140 2708.3 2709.5 150 3566.7 3568.7 160 4631.1 4633 170 5935.2 5937 180 7515 7514 190 9409.1 9404 200 11658 11647 5. Marks' Equation. Professor Marks nas deduced an equation that gives with remarkable accuracy the relation between |? and ^throughout the range 32 F. to 706.1 F., the latter temperature being the critical temperature, as established by the recent experiments of Holbom and Baumami. The form, of the equation is log p = a -t-cT+eT*. (7) The constants have the following values: a = 10.515354, 1 4873.71, c = 0.00405096, e = 0.000001392964. 109. Expression for ^- In the Clapeyron-Clausius formula dt /Jin for the specific volume of a saturated vapor, the derivative -_- dt is required. An expression for this derivative is obtained by differentiating any one of the equations (1) to (7) of Art. 108. Thus from (6), dp _ ( 1 1 \ _ nip . , -i N dt ~ np \T^b ~ Tj ~ T(T -b)' ^ ) whence log & = log nb + logp - log T - log (S 7 - 5). dt Values of -^ are readily calculated since the terms log T, dt log (_T 5), and log p appear in the calculation of p from (6). 110. Energy Equation applied to the Vaporization Process. It is customary in estimating the energy, entropy, heat content, etc., of a saturated vapor to assume liquid at 32 F. (0C.) as a datum from which to start. Thus the energy of a pound of steam is assumed to be the energy above that of a pound of "water at 32 F. Suppose that a pound of liquid at 32 is heated until its temperature reaches the boiling point corresponding to the pressure to which the liquid is subjected. The heat required is given by the equation where c' denotes the specific heat of the liauid. This process LRT. 110] VAPORIZATION PROCESS 171 s represented on the ^fitf-plane by a curve AA' (Fig. 62). Che ordinate OA represents the initial absolute temperature 52 + 459.6 = 491.6, the ordinate A^A! the temperature of va- )orization given by the relation = /(j?), and the area OAA'A l ,he heat q' absorbed by the liquid. This heat q' is called the ieat of the liquid.* When the temperature of vaporization is reached, the liquid )egins to change to vapor, the temperature remaining constant luring the process. A definite quantity of heat, dependent ipon the pressure, is required to change the liquid completely nto vapor. This is called the ieat of vaporization and is de- loted by the symbol r. In Fig. 52, the passage of the state- )oint from A' to A" represents ;he vaporization, and the heat is represented by the area ^A'A'Ay For a higher pres- iure the curve AB' represents .he heating of the liquid and ihe line B' B" the vaporization. During the heating of the iquid the change in volume is small and may be neg- A'f- M A 1 B 1 MT FIG. 62. ected ; hence, the external work done is negligible also, and ubstantially all of the heat q f goes to increase the energy of he liquid. During the vaporization, however, the volume ihanges from v' (volume of 1 Ib. of liquid) to v" (volume of . Ib. of saturated vapor). Since the pressure remains constant, he external work that must be done to provide for the increase >f volume is I f = p (v" - v')- ( 2 ) According to the energy equation, the heat r added during vaporization is used in increasing the energy of the system and is the heat required to increase the energy of the unit weight of substance when it changes from liquid to vapor. This heat is denoted by p and is called the internal latent heat. Since during the vaporization the temperature is constant, there is no change of kinetic energy ; it follows that p is expended in in- creasing the potential energy of the system. The heat equiva- lent of the external work, namely, Ap (y" v'), is called the external latent heat, and for convenience may be denoted by ^. We have then . . ^.N r = p + -^. (4) The total heat of the saturated vapor is evidently the sum of the heat of the liquid and the heat of vaporization. Thus, q" = q' + r, or q" = q' + p + "f (5) Comparing (5) with the general energy equation, it is evident that the sum q' + p gives the increase of energy of the saturated vapor over the energy of the liquid at 32 F. Denoting this by w", we have , , , ,. J Au" = q' +p. (6) If the vaporization is not completed, the result is a mixture / A'M\ of saturated vapor and liquid of qviality x f x = ), as indi- \ A A j cated by the point M (Fig. 60 and 62). In this case the heat required to vaporize the part x is xr heat units and the total heat of the mixture, which may be denoted by q x , is given by q x = q' + xr = q f + xp + x-^r. (7) The energy of the mixture (per unit weight) above the energy of water at 32 F. is, therefore, given by the relation Au x = q' + zp, (8) and the external work done is L x = Jx^. (9) If heat is added at constant pressure, after the vaporization is completed, the vapor will be superheated. The state-point will thft P.nrvP 4"7fJ Cff\rr d.8"\ nnrl +.1-.0 liocif. /> ft f. n ~\ epresented by the area A 2 A" ' EE^ will be added. Here c p de- lotes the mean specific heat of the superheated vapor, t e the inal temperature, and t" the saturation temperature correspond- ng to the pressure p. The total heat corresponding to the )oini E and represented by the area OAAA'EE-^ is, therefore, q e =q'+r + c p (t e -t"). (10) f v e denotes the final volume, and u e the energy above liquid ,t 32 F., then the external work for the entire process is L=p(y e -v<-), (11) ,nd, therefore, Au>. = q e -Ap (y e -v'). (12) 111. Heat Content of a Saturated Vapor. By definition we iave for the heat content of a unit weight of saturated vapor i" = A(u" +pv"~) = q' + p + Apv". (1) iince the total heat is <? = q' + p + Ap(v"-^ (2) ; appears that i" is larger than q" by the value of the term \.pv'. As v', the specific volume of water, is small compared ,dth v", the term Apv' may be neglected except for very high ressures, and q" and i" may be considered equal. In most of the older steam tables values of q" were given ; i the more recent tables, the values of i" instead of q" are .sually tabulated. 112. Thermal Properties of Water Vapor. From the relation q" = q' + r, -> appears that if any two of the three magnitudes q", q', r are de- srmined by experiment, the third may be found by a combina- ion of those two. Various experiments have been made to etermine each of these magnitudes for the range of temperature rdinarily employed, and as a result several empirical formulas ave been deduced. Naturally the greatest amount of attention as been given to water vapor, and we may consider the proper- ies of this medium as quite accurately known at the present ime. Ammonia, sulphur dioxide, and other vapors have not UtitJUL BUUUJ.CU Wltii UHC aclillC cm AM. UJJL&J.J. are as yet only imperfectly known. In the sections immediately following we shall give briefly the results of the latest and most accurate experiments on water vapor. 113. Heat of the Liquid. Denoting c r the specific heat of water, the heat of the liquid above 32 F. is given by the re- IatioQ j- -/*. CD If the specific heat c' were constant at all temperatures, this equation would reduce to the simple form q' = c'(t 32). As a matter of fact, however, c' is not constant, and its variation with the temperature must be known before (1) can be used to calculate q'. Between C. and 100 C. (32-212 F.) the experiments of Dr. Barnes may be regarded as the most trust- worthy. Taking c' 1 at a temperature of 17.5 C., the fol- lowing values are given by Griffiths as representing the results obtained by Barnes. TEMPERATURE SPECIFIC HEAT TKMl'KKATUUE Si'KOinc HKAT C. F. C. F. 32 1.0083 55 131 0.9981 5 41 1.0054 60 140 0.9987 10 50 1.0027 65 149 0.9993 15 59 1.0007 70 158 1.0000 20 68 0.9992 75 1C7 1.0007 25 77 0.9978 80 176 1.0015 30 86 0.9975 85 185 1.0023 35 95 0.9974 90 194 1.0031 40 104 0.9973 95 203 1.0040 45 113 0.9974 100 212 1.0051 50 122 0.9977 These values are shown graphically in Fig. 63. From them values of q' may be obtained by means of relation (1). In the actual calculation of the tabular values of q', the fol- lowing method may be used advantageously. Since the specific heat c' does not differ greatly from 1, let c'= 1 + &, ART. 114] LATENT HEAT OF VAPORIZATION 175 1.008 1.000 1.004 1.002 1.000 0.908 0.096 \ \ \ / \ f "7 \ , / \ / \ / U u ^ iO L 4 X 8 w \ S ' N. *~ FIG. 63. where k is a small correction term. Then for q' we have q' = f c'dt = t - 32 + f kdt. 1 JM J32 If now values of Jc are plotted as ordinates with correspond- ing temperatures as abscissas, the values of the integral (kdt may easily be determined by graphical integration. For temperatures above 212 F. the only available experi- ments giving the heat of the liquid are those of Regnault and Dieterici. The results of these ex- periments are somewhat discordant and unsatis- factory. Fortunately, we have for the range 212 to 400 F. reliable formulas for the total heat q" and the latent heat r, and we may therefore determine q' from the relation q' = q" r. 114. Latent Heat of Vaporization. The latent heat of water vapor for the range to 180 C. (32-356 F.) has been accu- rately determined by direct experiment. The results of the experiments of Dieterici at C., Griffiths at 30 and 40 C., Smith over the range 14-40 C., and Henning over the range 30-180 C. show a remarkable agreement, all of the values lying on, or very near, a smooth curve. The observed values are given in the third column of the following table. As the thermal units employed by the different investigators were not precisely the same, all values have been reduced to a common unit, the joule. It is readily found that a second-degree equation satis- factorily represents the relation between r and t. Taking r in joules, the following equation gives the values in the fourth 176 SATURATED VAPORS [CHAP, x LATENT HEAT OF WATER, IN JOULES IiATENl llKAT TEMPKKA- DlFFKKHNOK PUB CUNT Observed Calculated 2493.8 2495.8 -0.08 30.00 2429.3 2430.8 - 0.06 Griffiths 40.15 2403.6 2407.5 0.16 13.95 2467.6 2406.3 + 0.05 21.17 2451.2 2450.5 + 0.03 . 28.06 2435.0 2435.2 U.U1 39.80 2405.8 2408.3 -0.10 30.12 2424,8 2430.6 - 0.24 Henning, First Series .... 40.14 64.85 77.34 2385.3 2343.0 2313.7 2386.2 2347.7 2316.0 -0.04 - 0.20 -0.10 89.29 2285.6 2284,6 + 0.05 100.59 2254.2 2254,0 + 0.01 102.34 2248.7 2249.2 - 0.02 Henning, 120.78 2200.2 2197.2 + 0.14 Second Series . . . 140.97 2134.2 2137.6 -0.10 100.56 2077.0 2077.2 -0.01 180.72 2018.6 2012.3 + 0.31 The differences between the observed values and those calcu- lated from this formula are shown in the last column. The mean calorie is equivalent to 4.184 joules ; hence, divid- ing the constants of Eq. (1) by 4.184, the resulting equation gives r in calories. This equation is readily changed to give r in B. t.u. with t in degrees F. We thus obtain finally r = 970.4 - 0.655 (* - 212) - 0.00045 (t - 212) 2 . (2) This formula may be accepted as giving quite accurately the latent heat from 32 F. to perhaps 400 F.* 115. Total Heat. Heat Content. For the temperature range 32 to 212 F. the total- heat q" is obtained from the relation q" = q' + r. As has been shown, values of q' and of r can be accurately determined for this range. For temperatures be- tween 212 and 400, we are indebted to Dr. H. N. Davis for the derivation of a formula for the heat content of saturated vapor of water. The earlier experiments of Regnault led to the formula q n ^ 1091 . 7 + 0>305 ^ __ 33^ which has been extensively used in the calculation of tabu- lar values. By making use of the throttling experiments of Grindley, Griessmann, and Peake, Dr. Davis* has shown that Regnault's linear equation is incorrect, and that a second-degree equation of the form q" = a + b (t - 212) + e (t - 212) 2 may be adopted. Dr. Davis obtains for the heat content i" the formula i" = 1150.4 + 0.3745(15- 212) - 0. 00055 (t - 212) 2 . (3) From this formula the total heat q" is readily determined from the relation q" = i" Apv' . It is found, however, that slight changes in the constants are desirable in view of Henning's sub- sequent experiments on latent heat. The modified formula i" = 1150.4 + 0.35 (t - 212) - 0.000333 (t - 212) 2 (4) may be accepted as giving with reasonable accuracy values of i" for the range 212 to 400 F. 116. Specific Volume of Steam. The specific volumes" of a saturated vapor at various pressures may be determined experimentally. For water vapor accurate measurements of v" for temperatures between 100 and 180 C. have been made by Knoblauch, Linde, and Klebe. It is possible, however, to calculate the volume v" from the general equations of thermo- dynamics ; and the agreement between the calculated values and those determined by experiment serves as a valuable check critical temperature, 689 F. At the higher temperatures it doubtless gives more accurate values than the second-degree formula. See Proceedings of the Amer. Acad. of Arts and Sciences 45, 284. * Trans. Am. Soc. of Mecli. Engs. 30, 1419, 1908. See Art. 104 for a dis- cussion of thfi inRf-.bnrl mrmlnvfirl in t.hfi rlflviva.tion of formula (3}. on the accuracy with which the factors entering into the theo- retical formula have been determined. The general equation (Art. 56) do c v dT-\- AT( -- ) dv (1) \dtjv applies to any reversible process. Let us apply it to the pro- cess of changing a liquid to saturated vapor at a given constant temperature. For a saturated vapor, the partial derivative is simply the derivative -|-, and this is a constant for any dtjv " " dt given temperature (Art. 10T). Hence, for the process in ques- tion, we have (since dT 0) (2) But in this case q is the heat of vaporization r ; hence we have .. , r 1 Jr 1 .ox 1)" v' = -- --.. {&) dt dt This is the Clapeyron-Clausius formula for the increase of vol- ume during vaporization. Having for any temperature the derivative -- (Art. 109) and the latent heat r, the change of volume v" v' is readily calculated. The following table shows a comparison between the values of v" determined experimentally by Knoblauch, Linde, and Klebe, and those calculated by Henning from the Clapeyron equation, using the values of r determined from his own experiments. The third line gives values of v" calculated from the characteristic equation of superheated steam. (See Art. 132.) SPECIFIC VOI.UMBB, Cu. MKTBKB I'KII Ko. 100 120 140 160 180 C. Experimental .... Hennin r 1.674 1.073 1.073 0.8922 0.8912 0.8915 0.5001 0.5078 0.5084 0.3073 0.3071 0.3071 0.1043 0.1947 0.1945 o From the equation for superheated steam . . ART. 117] ENTROPY OF LIQUID AND OF VAPOR 179 The relation between the pressure and specific volume v" of saturated steam may be represented approximately by an equa- tion of the form // _. 0- ,^ Zeuner, from the values of v" given in the older steam tables, deduced the value n = 1.0646. Taking the more accurate values of v" given in the later steam tables, we find 91 = 1.0631, (7=484.2. 117. Entropy of Liquid and of Vapor. During the process of heating the liquid from its initial temperature to the tem- perature of vaporization the entropy of the. liquid increases. Thus, referring to Fig. 62, if the initial temperature be 32 F., denoted by point -4, and if the temperature be raised to that denoted by A', the increase of entropy of the liquid is repre- sented by OA}, the heat of the liquid by area OAA'A V Since dq' = c'dT, we have as a general expression for the entropy s 1 of the liquid corresponding to a temperature T, - c T ^L- C T m J 491.6 T ~ J 491-6 T If the specific heat c' is given as a function of T, the inte- gration is readily effected. In the case of water, where the specific heat varies somewhat irregularly, as shown by the table of Art. 115, the following expedient may be used. Put c' = 1 + k ; then k is a small correction term that is negative between 63 and 150 F. and positive elsewhere. From (1) we have, therefore, The first term is readily calculated and the small correction term may be found by graphical integration. This method was used in calculating the values of s' in table I. The increase of entropy during vaporization, represented by v A' A!' (Fig. 62), is evidently the quotient -= Hence the en- tropy of the saturated vapor in the state A" is For a mixture of quality x, as represented by tlie point M, the entropy is = *' + f. (4) 118. Steam Tables. The various properties of saturated steam considered in the preceding articles are tabulated for the range of pressure and temperature used in ordinary tech- nical applications. Many such tabulations have appeared. The older tables based largely upon Regnault's data are now known to be inaccurate to a degree that renders them value- less. The recent tables of Marks and Davis * and of Peabody, f however, embody the latest and most accurate researches on saturated steam. Table I at the end of the book has been calculated from the formulas derived in Arts. 108-116. The values differ but little from those obtained by Marks and Davis. The first col- umn gives the pressures in inches of mercury up to atmospheric pressure, and in pounds per square inch above atmospheric pressure ; the second column contains the corresponding temperatures. Columns 3 and 4 give the heat content of the liquid and saturated vapor, respectively. The values in col- umn 3 may be taken also as the heat of the liquid q' ; similarly, column 4 may be considered as giving the total heat q" of the saturated vapor. As we have seen, the difference between i" and q" is negligible except at high pressures. 119. Properties of Saturated Ammonia. Several tables of the properties of saturated vapor of ammonia have been pub- lished. Among these may be mentioned those of Wood, Pea- body, Zeuner, and Dieterici. The values given by the different tables are very discordant, as they are for the most part obtained by theoretical deductions based on meager experimental data. For temperatures above 32 F. the values obtained by Dieterici as the result of direct experiment are most worthy of confidence. Dieterici determined experimentally the specific volume v" of the saturated vapor for the temperature range to 40 C. * Marks and Davis, Steam Tables and Diagrams, Longmans, 1908. (32 to 104 F.) and also for the same range the specific heat c f of the liquid ammonia. The formula deduced by Dieterici for specific heat is, for the Fahrenheit scale, c' = 1.118 + 0.001156 (t - 32). (1) From this formula, the heat of the liquid q r and the entropy of the liquid s' are readily calculated by means of the relations / 1 / 7 r /* / 0> X q' = I c at, s' = ( c'-- J 32 Jfi>l.G T The relation between pressure and temperature is given by the experiments of Regnault. The results of these experiments are expressed quite accurately by Bertrand's formula log p = 5.87395 - 50 log m T QA . (2) Above 32, having Dieterici's experimental values of v" and from (2) the derivative -. we dT the Clapeyron-Clausius formula from (2) the derivative JL we may find the latent heat r from Ct ~L r = A(v"-v'~)T. (See Art. 116.) (3) Gv JL For temperatures below 32 we have neither v" nor r given experimentally; hence for this region values of various prop- erties can only be determined by extrapolation, and the ac- curacy of the results thus obtained is by no means assured. In calculating the values of table III the following method was used. The values of r for temperatures above 32 were calcu- lated by means of (3). It was found that these values may be represented quite accurately by the equation log r = 1.7920 + 0.4 log (266 - ), ( 4 ) in which 266 is the critical temperature of ammonia. (See p. 176, footnote.) Formula (4) was assumed to hold for the range 32 to 30 ; and from the values of r thus obtained values of v" were calculated by means of the Clapeyron relation (3). 120. Other Saturated Vapors. Several saturated vapors in addition to the vapors of water and ammonia have important technical aDplications. Sulphur dioxide and carbon dioxide in particular are used as media for refrigerating machines. The properties of the former fluid have been investigated by Cailletet and Mathias, those of the latter by Amagat and M oilier. The results of these investigations are embodied in tables.* The properties of several vapors of minor importance have also been tabulated, the data being furnished for the most part by Regnault. These include ether, chloroform, carbon bisul- phide, carbon tetrachloride, aceton, and vapor of alcohol, f 121. Liquid and Saturation Curves. If for various tem- peratures the corresponding values of s', the entropy of the liquid, be laid off as abscisses, the result is a curve s', Fig. 62. This is called the liquid curve. If, likewise, values of be laid off as abscissa;, a second curve s' f is obtained. This is called the saturation curve. As already stated (Art. 106), any point between the curves s' and s" represents a mixture of liquid and vapor, the ratio x de- pending upon the position of the point. It is possible, there- fore, to draw between the curves s' and s" a series of constant-a; lines. Each of the horizontal segments A' A", B'B", etc., is divided into a convenient number (say 10) of equal parts and corresponding points are joined by curves. The successive curves, therefore, are the loci of points for which x = 0.1, #=0.2, etc. The form of the saturation curve has an important relation to the behavior of a saturated vapor. For nearly all vapors, the curve has the general form shown in Fig. 62 ; that is, the entropy s" decreases with rising temperature. In the case of ether vapor, however, the entropy increases with rising tem- perature and the curve has, therefore, the same general direc- tion as the liquid curve s'. 122. Specific Heat of a Saturated Vapor. Kef erring to the saturation curve of Fig. 62, suppose the state-point to move * For tables of the properties of saturated vapor of S0 2 and C0 2 in English units, see Zeuner's Technical Thermodynamics, Klein's translation, Part II. t See Peabody's Steam and Entropy Tables, or Zeuner's Technical Thermo- " ART. 122] SPECIFIC HEAT OF A SATURATED VAPOR 183 from A" to B" '. This represents a rise of temperature of the saturated vapor during which the vapor remains in the satu- rated condition. The process must evidently be accompanied by the withdrawal of heat represented by the area A^Al'IP'S^ ; and the reverse process, fall in temperature from B" to A", is accompanied by the addition of heat represented by the same area. It appears, therefore, that along the saturation curve the ratio -^ is negative (except in the case of ether) ; that is, ZA the specific heat of a saturated vapor is, in general, negative. An expression for the specific heat c" of the saturated vapor may be obtained as follows. The entropy of the saturated vapor is given by the equation hence the change of entropy corresponding to a change of temperature is obtained by differentiating (1), thus (2) But <fo' = ^fr < 3 ) and similarly for the saturation curve, *" = ^. (4) Substituting these values ds' and ds" in (2), the result is ' m "> j_ 1 dT\T ] ' But since c' = -T|J, (5) may be written ^d(q'+r*) r c ~ dT T a IS Known. JLIIUS lur wauor vtipui ciuuvo *JJ-A , wo JU.O/VD 2 " = a + 6(< - 212) - c(t - 212) 2 ; whence where 5 = 0.35 and c = 0.000333. At 212, we have, for example, r 970-4 T 212 + 459.6 123. General Equation for Vapor Mixtures. Let heat be added to a unit weight of mixture of liquid and saturated vapor, of which the part x is vapor and the part 1 x is liquid. In general, the temperature T and quality x will change ; hence the heat added is the sum of two quantities : (1) the heat required to increase the temperature with x remaining constant; (2) the heat required to increase x with the temperature constant. The first is evidently c'(l x~)dT -\-c"xdT; and the second is rdx ; hence we have dq = c'(l - x)dT+ c"xdT + rdx (1) as the general differential equation for the heat added to a mixture. From (1) the general expression for the change of entropy of a mixture is given by 7 dq c'fl x^ -4- c"x im , rj /-o\ rts=x-! = _A t 2 T -f_ (&. C2) M/J. ( _ \^ J The fact that ds is an exact differential leads at once to the rekti n arV(i-*) + tf dx\_ T whence c = c _., dT T the relation that was obtained in Art. 122. F A" 124. Variation of x during Adiabatic Changes. Let the point A" (Fig. 64) represent the state of saturated vapor as regards pressure and temperature. Adiabatic expansion will then be represented by a vertical line A" E, the final point H being at lower temperature. Adiabatic compression will be shown by a vertical line A" Gr. With a saturation curve of the form shown, it appears that during adiabatic expansion some of the vapor .condenses, while adiabatic compression results in super- heating. If the state-point is originally at M so that x is some- what less than 1 (say O.T or 0.8), then adiabatic expansion is ac- companied by a decrease in #, adiabatic compression by an in- crease of x. If the saturation curve slopes in the other direction, as in the case of ether, the conditions just stated will, of course, be reversed. Adiabatic expansion of the liquid is represented by the line A'F ; evidently some of the liquid is vaporized during the process. If the mixture is originally mostly liquid, as indicated by a point .ZVnear the curve *', then adiabatic expansion results in an increase of #, adiabatic compression in a decrease of x. For a given pressure there is some value of x for which an indefinitely small adiabatic change produces no change in x ; in other words, at this point the constant-a? curve has a vertical tangent. For this point we have evidently dq = and dx 0, and the general equation (1), Art. 123, becomes FIG. 64. whence or x = c'-c"' (1) (2) (3) The locus of the points determined by (3) is a curve n (Fig. 64), dq = rax ; (4) that is, all the heat entering the mixture is expended in vapor- izing the liquid. The zero curve is of little practical importance. The change of the quality x during the adiabatic expansion of a' mixture is readily calculated by means of the entropy equation. In the initial state, the entropy of the mixture is and in the final state it is z But for an adiabatic change s 2 = s 1 ; therefore, we have the relation s/ + -^ = s 2 ' + '-j^- 2 , (5) in which # 2 is the only unknown quantity. 125. Special Curves on the TS-plane. The region between the liquid and saturation curves may be covered with series of curves in such a way that the position of the point represent- ing a mixture indicates at once the various properties of the mixture. In the first place, horizontal lines intercepted between the curves s' and s" are lines of constant temperature, also lines of constant pressure ; while vertical lines are lines of constant entropy. Lines of constant quality, z v # 2 , # 3 , . . . may be drawn as explained in Art. 121. Curves of constant volume may be drawn as follows : The volume of a unit weight of mixture whose quality is x is given by the equation v = x(v" v'} + v', (1) whence x- V ~ v V V Suppose that the curve for some definite volume (say v 5 cu. ft.) is to be located. For different pressures p^ p v p y . . . the saturation volumes v/', v/, v a ", . . . are known from the tables, substituting successively these values of v" in (2), values of #, as x v x v a; 8 , . . . corresponding to the pressures Pv> Pv> P& ' w ^ be f un d. The value of v' may be taken as constant for all pressures. The value of x l locates a definite point on the p l line, that of x 2 a point on the p z line, etc. The locus of these points is evidently a curve, any point of which represents a mixture having the given volume v ; hence it is a constant- volume curve. In a similar manner curves of constant energy u may be located. Since u = q'+xp, (3) u q' we have x = . (4) P For given pressures p v p 2 , . . . f f ry *! fv* _ *% 0,4-p JU-\ ' 3 *Vn ^ uUV_/ Pi Pz Values of q' and p for different pressure are given in the table, and therefore for a given w, values of x v # 2 , . . . are readily calculated. These locate points on the corresponding ^>-lii .and the locus of the points is the desired constant-w curve. T By the same process may be drawn curves of constant total heat, q = q' -f- xr const. or curves of constant heat content i = i' -\-xr- const. In Fig. 65, the various curves are shown drawn through the ~ FlG G5 same point P. From the general course of the curves the behavior of the mixture during a given change of state may be traced. Thus : (1) If a mixture expands adiabatically, v increases but p, T, u, and i decrease. The quality x decreases as long as the state-point lies to the right of the zero curve. ' (2) If a mixture expands isody- namicallv (u= const.), v, s, and x increase, p, T, and i decrease. for water vapor, taking values of s' and s" from the steam table. Then draw the curves v = 2, v = 10, v - 40 cu. ft. Also draw the curves u = 600 B. t. u., M = 800 B. t. u. 126. Special Changes of State. Certain of the curves de- scribed in preceding articles represent important changes of state of the mixture of saturated vapor and liquid. The prin- cipal relations governing some of these changes will be de- veloped in this article. It is assumed that the system remains a mixture during the change, that is, that the path of the state- point is limited by the curves s' and s". (a) Isothermal, or Constant Pressure, Change of State. Let x 1 denote the initial quality, x z the final quality. Then the initial volume is and the final volume is / n t~\ i t 1)n = Xn(V V ) + V . The change in volume is therefore v v = (x x'}(v"v'') 00 and the external work is The change of energy is and the heat absorbed is q^rtxt-xj. (4) These equations refer to a unit weight of mixture. EXAMPLE. At a pressure of 140 lb., absolute, the volume of one pound of a mixture of steam and water is increased by 0.8 cu. ft. The change of quality is 2L = = 0.2514. The external work is 140 x 144 x 0.8 = 16,128 ft.-lb. The increase of energy is Jp(x z - xj = 778 x 786.1 x 0.2514 = 153850 ft.-lb. ; and the heat absorbed is r (x z - xj = 869 x 0.2514 = 218.5 B. t. u. (6) (Jliange oj /state at Constant Volume. Since the volumes > l and v% are equal, we have *i(V'-"0=z 2 (< -<;'), (5) ivhere v^" and v z " are the saturation volumes corresponding to ihe pressures p- and p^ respectively. From (5) the quality x z .n the final state may be determined. The external work TTis Hero ; hence we have for the heat absorbed - ( ? / - x lPl ) . (6) ' EXAMPLE. A pound of a mixture of steain and water at 120 Ib. pressure, quality 0.8, is cooled at constant volume to a pressure of 4 in. of mercury. Required the final quality and the heat taken from the mixture. From (5) ^ = *,-* > ) = 0.8(3.724-0.017) = ^ v 2 " - v' 176.6 rherefore q = 311.9 + 0.8 x 795.8 - (93.4 + 0.0167 x 959.5) = 839.2 B. t. u. (c) Adiabatio Change of State. For a reversible adiabatic change the entropy of the mixture remains constant ; hence we have 'i' + % L = '*' + *> CO -L\ J-2, from which equation the final quality z 2 can be found. Having z; 2 , the final volume v 2 per unit weight is v = z- 2 <> 2 "-^)+^. (8) Since the heat added is zero, the external work is equal to the decrease in the intrinsic energy of the mixture. That is, (9) EXAMPLE. Three cubic feet of a mixture of steam and water, quality 0.89, and having a pressure of 80 Ib. per square inch, absolute, expands adiabatically to a pressure of 5 in. Hg. The final quality, final volume, and the external work are required. From the steam tables we find the following values : fl P . T For^SOlb. 281.8 819.6 0.4533 1.1667 5.464 Forp = 5in. Hg. 101.7 953.7 0.1SSO 1.7170 143.2 The weight of the mixture is M - 3 = = 0.6167 Ib. m ~ Xi ( v _ ') + ' 0.89(5.464 - 0.017) + 0.017 From (7), the quality x 2 in the second state is given by the relation 0.4533 + 0.89 x 1.1667 = 0.1880 + 1.7170 x a , whence x % = 0.759. The volume in the second state, neglecting the insignificant volume of the liquid, is V 2 = 0.6167 x 0.759 x 143.2 = 67.02 cu. ft. Finally, the external work is W = 778 x 0.6167 [(281.8 + 0.89 x 819.6) - (101.7 + 0.759. x 953.7)] = 89,080 ft.-lb. (dT) Isodynamic Change of State. If the energy of the mix- ture remains constant, we have Wj = Up or ft' + x lPl = qj + x z p 2 . (10) From (10) the final value of x is determined, and the final volume is then found from (8). For the isodynamic change, the heat added to the mixture is evidently equal to the external work. There is no simple way of finding the work. As an approximation, an exponential curve p 1 v 1 n =pv n (11) may be passed through the points p^ v^ and j? 2 , v 2 , and the value of n can be found. This curve will approximate to the true isodynamic on the j?v-plane, and the external work will then be approximately ytri v i~P2 v <2 (12^) n-l ' ^ J In practice the isodynamic of vapor mixtures is of little importance. 127. Approximate Equation for the Adiabatic of a Vapor Mix- ture. In certain investigations, especially those relating to the flow of steam, it is convenient to represent the relation between p and v during an adiabatic change by an equation of the form RT. 127] APPROXIMATE EQUATION OF ADIABATIC 191 ?he value of the exponent n is not constant, but varies with the litial pressure, the initial quality, and also with the final ressure ; and at best the equation is an approximation, tankine assumed for n the value !- for all initial conditions. a ieuner, neglecting the influence of initial pressure, gave the ormula n = 1.035 + 0.1 x. (2) Ir. E. H. Stone,* using the tables of Marks and Davis, has .erived the relation n = 1.059 - 0.000315 p + (0.0706 + 0.000376^>. (3) The following table gives values of n calculated from (3). nitial iuuli- ty INITIAL PRESSURE IN POUNDS PUR SQUAKE INCH, ABSOLUTE 20 40 60 80 100 120 140 160 180 200 220 240 1.00 1.131 1.132 1.133 1.134 1.136 1.137 1.138 1.139 1.141 1.142 1.143 1.145 0.95 1.127 1.128 1.128 1.130 1.131 1.131 1.132 1.133 1.134 1.135 1.136 1.137 0.90 1.123 1.123 1.124 1.124 1.125 1.125 1.126 1.126 1.127 1.127 1.128 1.129 0.85 1.119 1.119 1.119 1.119 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.121 0.80 1.115 1.115 1.114 1.114 1.114 1.114 1.113 1.113 1.113 1.113 1.112 1.112 0.75 1.111 1.110 1.110 1.109 1.109 1.108 1.107 1.106 1.106 1.105 1.104 1.104 0.70 1.108 1.106 1.105 1.104 1.103 1.102 1.101 1.100 1.099 LOSS 1.097 1.096 0.65 1.104 1.102 1.101 1.099 1.098 1.096 1.095 1.093 1.092 1.091 1.089 1.088 0.60 1.100 1.098 1.096 1.094 1.093 1.091 1.089 1.087 1.085 1.083 1.081 1.080 0.55 1.096 1.093 1.092 1.089 1.087 1.085 1.083 1.080 1.078 1.076 1.074 1.072 0.50 1.092 1.089 1.087 1.084 1.082 1.079 1.077 1.074 1.071 1.069 1.066 1.064 Having the initial values p v V v and x# and the final pressure > 2 , the final volume V 2 is found approximately from (1), the ppropriate value of n being taken from the table. The exter- lal work is found approximately by the usual formula for the hange represented by (1), namely, w= (4) n EXAMPLE. Taking the data of the example of Art. 126 (c), we have _ on T7 a . n so lionpp 1.193. The final tiressure is 5 in. Hff. and W = 144 x = 88 OT4 b " lb ' Comparing these results with the results obtained by the exact method, it appears that the volume F 2 is about 0.36 per cent smaller and the work W about 0.13 per cent smaller. Hence the approximation is sufficiently close for all practical purposes. EXERCISES 1. From Bertrand's equation calculate the pressure of steam corre- sponding to the following temperatures : 60, 250, 400 F. 2. Find the values of the derivative ( -P for the same temperatures. at 3. Using the results of Ex. 1 and 2, find the specific volumes for the given temperatures. 4. Find (a) the latent heat, (&) the total heat of saturated steam, at a temperature of 324 F. 5. Calculate the latent heat of steam, (a) by the quadratic formula (2), Art. 114; (b) by the exponential formula (see footnote, p. 170) for the tem- peratures 220 F. and 380 F. Compare the results. In the following examples take required values from the steam table, p. 315. 6. Find the entropy, energy, heat content, and volume of 4.5 Ib. of a mixture of steam and water at a pressure of 120 11). per square inch, quality 0.87. 7. Find the quality and volume of the mixture after adiabatic expan- sion to a pressure of 16 Ib. per square inch. 8. Find the external work of the expansion. 9. Using the data of the preceding examples, calculate the volume and work by means of the approximate exponential equation p V n = C. 10. A mixture, initial quality 0.97, expands adiabatically in a 12 in. by 12 in. cylinder from a pressure of 100 Ib. per square inch, gauge, to a pressure of 10 Ib. per square inch, gauge. Find the point of cut-off. 11. The volume of 6.3 Ib. of mixture at a pressure of 140 Ib. per square inch is 17.2 cu. ft. Find the quality of the mixture ; also the entropy and energy of the mixture. 12. The mixture in Ex. 11 is cooled at constant volume to a pressure of 20 Ib. per square inch. Find the final value of x and the heat abstracted. 13. At a pressure of 180 Ib. per square inch the volume of 2 Ib. of a mixture of steam and water is increased by 0.9 cu. ft. Find the increase of quality, increase of energy, heat added, and external work. 14. A mixture of steam and water, quality 0.85, at a pressure of 18 Ib. per square inch, is compressed adiabatically. Find the pressure at which tne water is completely vaporized, .eina aiso tne woric 01 compression, per pound of mixture. 15. Steam at a pressure of 80 Ib. per square inch expands, remaining sat- urated until the pressure drops to 50 Ib. per square inch. Find approxi- mately the heat that must be added to keep the steam in the saturated condition. 16. Water at a temperature of 352 F. and under the corresponding pressure expands adiabatically until the pressure drops to 30 Ib. per square inch. Find the per cent of water vaporized during the process. Find the work of expansion per pound of water. 17. Two vessels, one containing M t Ib. of mixture at a pressure p 1 and quality x\, the other M 2 Ib. at a pressure p and quality x 2 , are placed in communication. No heat enters or leaves while the contents of the vessels are mixing. Derive equations by means of which the final pressure ps and final quality x s may be calculated. 18. Let 1 Ib. of mixture at a pressure of 20 Ib. per square inch, quality 0.96, enter a condenser which contains 20 Ib. of mixture at a pressure of 3 in. Hg., quality 0.05. Assuming that no heat leaves the condenser during the process, find the pressure and quality after mixing. REFERENCES PRESSURE AND TEMPERATURE OF SATURATED VAPORS Kegnault: Mem. de 1'Inst. de France 21, 465. 1847. Rel. des exper. 2. Henning : Wied. Ann. (4) 22, 609. 1907. Holborn and Henning : Wied. Ann. (4) 25, 833. 1908. Holborn and Baumann: Wied. Ann. (4) 31, 945. 1910. Risteen : The Locomotive 26, 85, 183, 246 ; 27, 54 ; 28, 88. These articles contain a very complete account of the experiments of Regnault, Holborn and Henning, and Thiesen. Chwolson : Lehrbuch der Physik 3, 730. Gives comprehensive discussion of the many formulas proposed for the relation between the pressure and temperature of various vapors. Preston : Theory of Pleat, 330. Marks and Davis : Steam Tables and Diagrams; 93. Peabody: Steam and Entropy Tables, 8th ed., 8. Marks : Jour. Am. Soc. Mech. Engrs. 33, 563. 1911. PROPERTIES OF SATURATED STEAM (a) Specific Heat of Water. Heat of Liquid Regnault : Mem. de 1'Inst. de France 21, 729. 1847. Dieterici : Wied. Ann. (4) 16, 593. 1905. Barnes : Phil. Trans. 199 A, 149. 1902. Rowland : Proc. Amer. Acaa. oi Arcs ana sciences **, < u ; j-o, oo. j.oov- 1881. Day: Phil. Mag. 46, 1. 1898. Griffiths : Thermal Measurement of Energy. Marks and Davis : Steam Tables and Diagrams, 88. (6) Latent Heat Regnault: Mem. de 1'Inst. de France 21, 635. 1847. Griffiths : Phil. Trans. 186 A, 261. 1895. Henning: Wied. Ann. (4) 21, 849, 1906; (4) 29, 441, 1909. Dieterici: Wied. Ann. (4) 16, 593. 1905. Smith : Phys. Rev. 25 145. 1907. (c) Total Heat Davis: Proc. Am. Soc. of Mech. Engrs. 30, 1419. 1908. Proc. Amer. Acad. 45, 265. Marks and Davis : Steam Tables and Diagrams, 98. (!) Specific Volume Fairbairn and Tate : Phil. Trans. (I860), 185. Knoblauch, Linde, and Klebe : Mitteil. liber Forschungsarbeit. 21, 33. 1905 Peabody : Proc. Am. Soc. Mech. Engrs. 31, 595. 1909. Peabody : Steam and Entropy Tables, 8th ed., 12. Marks and Davis : Steam Tables and Diagrams, 102. Davis : Proc. Am. Soc. Mech. Engrs. 30, 1429. PROPERTIES OF REFRIGRATING FLUIDS (a) Ammonia Dieterici : Zeitschrift fur Kalteindustrie. 1904. Jacobus: Trans. Am. Soc. Mech. Engrs. 12, 307. Wood : Trans. Am. Soc. Mech. Engrs. 10, 627. Peabody : Steam and Entropy Tables, 8th ed., 27. Zeuner: Technical Thermodynamics (Klein) 2, 252. Lorenz : Technische W'armelehre, 333. (I) Sulphur Dioxide Cailletet and Mathias : Comptes rendus 104, 1563. 1887. Lange: Zeitschrift fur Kalteindustrie 1899, 82. Mathias : Comptes rendus 119, 404. 1894. Miller : Trans. Am. Soc. Mech. Engrs. 25, 176. Wood: Trans. Am. Soc. Mech. Engrs. 12. 137. Zeuner : Technical Thermodynamics 2, 256. (c) Carbon Dioxide iagat: Comptes rendus 114, 1093. 1892. llier : Zeit. fur Kalteindustrie 1895, 66, 85. Liner: Technical Thermodynamics 2, 262. GENERAL EQUATIONS FOR VAPORS. CHANGES OF STATE .iner: Technical Thermodynamics 2, 53. >,yrauch : Grundriss der Warme-Theorie 2, 33. *Hlon : Theory of Heat, 650. L*ry : Temperature Entropy Diagram, 43. CHAPTER XI SUPERHEATED VAPORS 128. General Characteristics of Superheated Vapors. The nature of a superheated vapor has been indicated in Art. 106, describing the process of vaporization. So long as a vapor is in immediate contact with the liquid from which it is formed it remains saturated, and its temperature is fixed by the pressure according to the relation t = /"(#>). When vaporization is com- pleted, or when the saturated vapor is removed from contact with the liquid, further addition of heat at constant pressure results in a rise in temperature. If t s denotes the saturation temperature given by t t =/Q?) and t the temperature after su- perheating, the difference t t s is the degree of superheat. Thus for steam at a pressure of 120 Ib. per square inch, t s = 341.3^; hence if at this pressure the steam has a temperature of 460, the degree of superheat is 460 - 341.3 = 118.7. As soon, therefore, as a vapor passes into the superheated state, the character of the relation between the coordinates p, v, and t changes. The temperature is freed from the rigid con- nection with the pressure that obtains in the saturated state, and p and t may be varied independent!}' . The volume v of the superheated vapor depends upon both p and t thus taken as independent variables ; that is, as in the case of a perfect gas. The form of the characteristic equation (1) for a superheated vapor is, however, less simple than that of the gas equation pv = BT. The state described by the term " superheated vapor " lies between two limiting states ; the saturated vapor on the one hand, and the perfect gas, obeying the laws of Boyle and Joule, on the other. The characteristic equation therefore should 196 be of such form as to reduce to the equation of the perfect gas, as the upper limit is approached and to give the proper values of p, v, and t of saturated vapor when the lower limit is reached. In the case of compound substances like water or ammonia, however, one disturbing element is introduced at very high temperatures. The vapor may to some extent dissociate ; thus steam may in part split up into its components hydrogen and oxygen, ammonia into nitrogen and hydrogen. Nernst has found for example that at a pressure of one atmos- phere 3.4 per cent of water vapor is dissociated at a temperature of 2500 C. Manifestly the existence of dissociation must in- fluence the relation between the variables p, i>, and t. However, at the temperatures and pressures with which we are concerned in the technical applications of thermodynamics, the amount of dissociation is entirely negligible, and the characteristic equation may be assumed to hold for all temperatures within the range of ordinary practice. 129. Critical States. The region between the limit curves v', v" (Fig. 60) or s', s" (Fig. 62) is the region of mixtures of saturated vapor and liquid. The fact that these two curves approach each other as the tem- perature is increased suggests that a temperature may be reached above which it is im- possible for a mixture of liquid and vapor to exist. Let it be assumed that the two limit curves merge into each other at the point S (Fig. 66), and 0' thus constitute a single curve, of which the liquid and saturation curves, as we have previously called them, are merely two branches. The significance of this assumption may be gathered from the following considerations. Let superheated vapor in the initial state represented by point A (Fig. 66 and 67) be compressed isothermally. Under usual conditions, the pressure will rise until it reaches the pres- FIG. 6(3. sure of saturated vapor corresponding to the given constant temperature *, and the state of the vapor will then be represented by point B on the saturation curve. Further compression at constant temperature results in condensation of the saturated vapor, as indicated by the line B 0. If the liquid be compressed isothennally, the volume will be decreased slightly as the pres- sure rises, and the process will , / \B' A > be represented by curve CD. ' \ The isothermal has therefore three distinct parts : along AB the. fluid is superheated vapor, along BO a mixture, and along QD a liquid. If the initial tem- perature be taken at a higher -I 1 s value ', the result will be similar FIQ 67 l except that the segment B' O' will be shorter. If the limit curves meet at point IT, it is evident that the temperature may be chosen so high that this horizontal segment of the isothermal disappears ; in other words, the isothermal lies entirely outside of the single limit curve. In Fig. 66 the segment BO represents the difference v" v' between the volume v" of saturated vapor and the volume v 1 of the liquid; and in Fig. 67, the area B 1 B00 1 represents the la- tent heat r of vaporization. For the isothermal t a that passes through J?, the segment BO reduces to zero; hence, for this temperature and all higher temperatures, we have v" v' = 0, or v" = i>', and r = 0. The second result also follows from the first when we consider the Clapeyron equation v - v ' = Jr ^L Tdp. dT The experiments of Andrews show that the condition just dioxide as determined oy Andrews are snown in Jfig. t>o. Jb or t= 13.1 and 21. 5 C. the horizontal segments corresponding to condensation are clearly marked. For *= 31.1 the horizontal segment disappears and there is merely a point of inflexion in the curve. At 48.1 the point of inflexion dis- appeared, and the iso- thermal has the general form of the isothermal for a perfect gas. The temperature t c was called by Andrews the critical tempera- ture. It has a definite value for any liquid. The pressure p c and volume v c indicated by the point S are called respectively the critical pressure and critical volume. Values of t c and p c for various substances are given in the following table: 50 FIG. 68. SUBSTANCE t c , DEOUEES C. PC, ATMOSPHERES Water . . . ... 365.0* 200.5 Ammonia ... .... 130.0 115.0 Ether 197.0 35.77 155.4 78.9 30.92 77.0 277.7 78.1 -146.0 35.0 Oxygen -118.0 50.0 Hydrogen -220.0 20.0 Air -140.0 30.0 * According to the recent experiments of Holborn and Baumann, the critical temperature of water is 706.1 F (374.5 C) and the critical pressure is 3200 11). per square inch. See article by Prof. Marks, Jour. A. S. M. E., Vol. 33, p. 563. Although at sufficiently high pressure the fluid may be in the liquid state, the closest observation fails to show where the gaseous state ceases and the liquid state begins. As stated by Andrews, the gaseous and liquid states are to be regarded as widely separated forms of the same state of aggregation. It has been proposed to make the critical temperature the basis of a distinction between gases and vapors. Thus, air, nitrogen, oxygen, nitric oxide, etc., whose critical temperatures are far below ordinary temperature, are designated as gases, while steam, chloroform, ether, etc., whose critical temperatures are above ordinary temperature are designated as vapors. The determination of the critical values c , p c , and v c by ther- modynamic principles is a problem of great theoretical interest, but lies beyond the scope of this book. 130. Equations of van der Waals and Clausius. Many attempts have been made to deduce rationally a single charac- teristic equation, which with appropriate change of constants will represent the properties of various fluids in all states from the gaseous condition above the critical temperature to the liquid condition. Such a general equation is that of van der Waals, namely, v - a v which was deduced from certain considerations derived from the kinetic theory of gases. As van der Waals' equation does not accurately represent the results of Andrew's experiments on carbon dioxide, Clausius suggested a modification of the last term of the equation and ultimately arrived at an equation of the form where /( 2") is a function of the absolute temperature that takes the value 1 at the critical temperature. The equations of van der Waals and Clausius are constructed with special reference to the behavior of fluids in the vicinity >f the critical state ; hence they apply more particularly to such fluids as carbon dioxide, the critical temperature of which .s within the range of temperature encountered in the practical implications of heat media. The critical temperatures of most mportant fluids, as water, ammonia, and sulphur dioxide are, lowever, far above the ordinary range, and for these media ihe general equations do not give as good results as certain purely empirical equations deduced from experiments covering i relatively small region. For some fluids, notably ammonia, :here is unfortunately a lack of experimental data; for the .nost important fluid, water, we have, however, reliable data tarnished by the recent experiments at Munich. 131. Experiments of Knoblauch, Linde, and Klebe. The sxperiments made at the Munich laboratory were so con- iucted that three important relations could be obtained simultaneously. These were : 1. Relation between pres- sure and temperature of saturated steam. 2. Relation between spe- sific volume and temperature of saturated steam. 3. Relation between pres- sure and temperature of superheated steam with the volume remaining constant. The experiment covered the range 100 to 180 C. The apparatus employed is shown diagrammatically in Fig. 69. An iron vessel a contains a smaller glass vessel 5 to which is attached a glass tube c. A similar glass tube d leads B a tube/ leading to a mercury manometer, oteam is mi/ruuuoeu. into vessel a from a boiler, and suitable provision is made for returning the condensed steam to the boiler. A given weight of water is put into the glass vessel b and is evaporated gradually by the heat absorbed from the steam surrounding it. As long as vessel b contains a saturated mix- ture, the pressure within b must be the same as that within a, since the temperature is the same throughout. Hence the mercury levels m, m in tubes o and d will be at the same height. When the water in b is all vaporized and the pressure and temperature of the steam in a is further increased, the steam in b becomes superheated. While the temperature is still the same in vessels a and 5, the pressures in the two vessels are not equal. This may be shown by the ^-diagram (Fig. 70). Let point A on the saturation curve s" denote the state of the steam in vessel b just at the end of vaporization ; it also repre- sents the state of the saturated steam in the outer vessel a. As the temperature rises from ^ to t z the state of the steam in a changes as represented by the curve A -, that is, the steam in a is saturated at the pressure p v The apparatus is so manipulated, however, that the mercury level m in tube o is held constant, thus keeping a constant volume of steam in vessel b. The point representing the state of the steam in b moves along the constant volume curve AS in the superheated region, and the final pressure p 3 given by the point JS is smaller than the pressure p 2 of the saturated steam in a. As a result the mercury level in the tube d will be depressed to the level n. A comparison of the mercury level in the manometer with the level m gives the relation between the pressure and temperature of superheated steam at the given constant volume v\ and a comparison witli the level n gives the relation between the pressure and temperature of saturated steam. L FiG. 70. .RT. 132] EQUATIONS FOR SUPERHEATED STEAM 203 132. Equations for Superheated Steam. To represent the esults of the Munich experiments, Linde deduced the empiri- al equation - JZ>. (1) n metric units with p in kilogram per square meter, the con- tants have the following values : J5 = 47.10 tf= 0.031 =3. a = 0.0000002 D= 0.0052 English units and pressures in pounds per square inch, the iquation becomes : pv = 0.5962 T-p(l + 0.0014^?) A 5030 ^ 000 _ o.0833\ (2) Fhe form of Eq. (1) is such as to make it inconvenient for he purpose of computation ; and the constant D in the last ,erm leads to complication in the working out of a general heory. A modified form of the equation, namely, * + ' = -(l+*)fi ( 3 ) s free from these objections and with constants properly chosen epresents the results of the Munich experiments as accurately is Linde's equation. The constants are as follows : METRIC UNITS ENGLISH UNITS B = 47.113 B 85.87, p in pounds per square foot = 0.5963, p in pounds per square inch ,ogm = 11.19839 log TO = 13.67938 n = 5 n = 5 c = 0.0055 . c- 0.088 a = 0.00000085 a = 0.0006, p in pounds per square inch. Fhe final equation with constants inserted is therefore T f-i . A nnnfl m \ 47795 x 10 _ SA-*. An equation o tne simple lorm v + c=^- (5) P has been proposed by Tumlirz on the strength of Battelli's experiments. Lincle has shown that this equation may be made to represent with fair accuracy the results of the Munich ex- periments. For English units and with p in pounds per square inch, the equation becomes v + 0.256 = 0.5962. (G) For moderate pressure this formula is quite accurate, but at high pressures and superheat the volumes given by it are con- siderably smaller than those indicated by the experiments. Two other characteristic equations deserve mention. For many years Zeuner's empirical equation pv = BT- Cp n (7) has been extensively used. The results of the Munich experi- ments have shown that the form of this equation is defective, and that it cannot accurately represent the behavior of super- heated steam over a wide range. Callendar, from certain theo- retical considerations, has deduced the equation, which in form resembles Eq. (3), but lacks the factor p in the last term. While this equation is somewhat simpler than Eq. (3), it is less accurate. 133. Specific Heat of Superheated Steam. The experimental evidence on the specific heat of superheated steam may be clas- sified as follows : 1. The early experiments of Regnault at a pressure of one atmosphere and at temperatures relatively close to saturation. 2. The experiments of Mallard and Le Chatelier, Langen, and others at very high temperatures. 3. The experiments of Holborn and Henning at atmospheric pressure and at temperatures varying from. 110 to 1400 0. 4. Recent experiments with steam at various pressures and with temperatures close to the saturation limit. Of these, the experiments of Knoblauch and Jakob are considered the most reliable. Regnault concluded from his experiments that at a pressure f one atmosphere the specific heat of superheated steam has he constant value 0.48 for all temperatures. This value has een largely used for all temperatures and for all pressures as rail. Experiments by Mallard and Le Chatelier and by Langen at igh temperatures agree in making the specific heat a linear auction of the temperature. Thus, according to Langen, c p = 0.439 + 0. 000239 t, (1) rhere t is the temperature on the C. scale. The earlier experiments of Holborn and Henuing at much Dwer temperatures than those of Langen lead to the formula c p = 0.446 + 0.0000856 t. (2) ?his is again a linear relation, but the coefficient of t is smaller han that in Langen's formula. Equations (1) and (2) show hat the specific heat varies with the temperature at least, and hat the convenient assumption of the constant value 0.48 is Lot permissible. Finally, the experiments of Knoblauch and Mollier show con- lusively that c p depends also upon the pressure. In these experiments, steam was run through a first superheater in diich all traces of moisture were removed. It was then run hrough a second superheater consisting of coils immersed in m oil bath. The heat was applied by means of an electric lurrent and could be measured quite accurately, and a com- >arison of the heat supplied with the rise of the temperature of lie steam gave a means of calculating the mean specific heat over .he temperature range involved. Experiments were conducted ii, -m-Assm-As of 9, 4. fi. a.nrl 8 ICQ-. TtBT sauare centimeter. The 206 SUPERHEATED VAPORS [CHAP, xi results are shown by the points in Fig. 71. From these results the following conclusions may be drawn : (1) The specific heat varies with the pressure, being higher the higher the pressure at the same temperature. (2) With the pressure constant, the specific heat falls gradually from the saturation limit, reaches a minimum value, and then rises again. Starting with the characteristic equation (3), Art. 132, it is possible to deduce a general equation for the specific heat c p that will give results substantially in accord with the experi- mental results of Knoblauch and Mollier. For this purpose we make use of the general relation From the characteristic equation, BT ^ 00 in x ' -* s 'J. we obtain by successive differentiation dv B mn , . C 1 + op). (6) Substituting in (3), the result is dc p \ Amn(n "" Talcing T as constant and integrating (7) with p as the in- dependent variable, the result is Amn(n + 1) / a \ , , , . J C P = - jrs+i -p(^ + nP )+ const, of integration. Now since T was taken as constant, the constant of integration may be some function of T; hence we may write (8) 60 160 240 480 560 3SO 400 Temperature le groups of points represent the results of experiments at 2, 4, 6, and 8 kg. per sq. cm. respectively, beginning with the lowest group. FIG. 71. increased. From JLangen s experiments, it is seen tnat at very high, temperatures c p is given by an equation of the form hence we are justified in assuming that where and /3 are constants to be determined from experi- mental evidence. Equation (8) thus becomes 0) This is the general equation for the specific heat of superheated steam at constant pressure. It may be seen at once that this equation gives results agree- ing in a general way with those of Knoblauch and Mollier. At a given temperature T the specific heat increases with the pres- sure ; furthermore for a given pressure, c p has a minimum value as appears by equating to zero the derivative a rn ' Wn+Z The following values of the constants have been found to make Eq. (9) fit fairly well the experimental results of Knob- lauch and Mollier : a = 0.367 /3 = 0.00018 for the C. scale. /3 = 0.0001 for the F. scale Replacing the product Amn(n + 1) by a single constant (7, we have as the final formula for the specific heat c p = 0.367 + 0.0001 T+p(l + 0.0003 j?) ~, (10) where log (7=14.42408 (pressure in pounds per square inch). Figure 71 shows the curves representing this formula for the pressures of the Knoblauch and Mollier experiments. The agreement between the points and curves is satisfactory, con- sidering the difficulty of the experiments. In Fig. 72 the <?p-curves for various pressures in pounds per square inch are UJb' SUPERHEATED STEAM 209 134. Mean Specific Heat. Formula (10), Art. 133, gives he specific heat at a given pressure and temperature. For ome purposes it is desirable to have the mean specific heat be- ween two temperatures, the pressure remaining constant. ?his is readily calculated by the mean value theorem ; thus L enoting by (c p ~) m the mean specific heat, we have \. c p)m~~7jn rfi" ^ J J 2 -M Jsing the general expression for c p , we have, therefore, / N 1 f^f , 0/77, Amn(n-}-V) ( ., , a ' (2) The calculation, while straightforward is rather long, and if ^-curves are available, it is usually preferable to determine he mean c p by Simpson's rule or by the planimeter. Curves of mean specific heat are shown in Fig. 73. For any degree of superheat the mean specific heat between the satura- ion state and the given state is given by the ordinate corre- ponding to the given degree of superheat and the given iressure. For example, at a pressure of 150 Ib. per square ach the mean specific heat for 240 superheat is 0.529. 135. Heat Content. Total Heat. Having a formula for the pecific heat at constant pressure, equations for the heat con- ent and the intrinsic energy of a unit weight of superheated team at a given pressure and temperature are readily derived. for this purpose the general equation dq = c p dT- AT dp (see Art. 54) (1) ^ 300 400 Superheat, Deg. F. 500 GOO i=A(u+pv^ we have di = A [du + or di = dq + Avdp. (2) Hence, making use of (1), From the characteristic equation we have dv _ B n 4- ^ TO ^ ___- + rc( a P)7jwi whence T -7^ v = (w + 1) (1 + ap') 7 ~~ + c. Introducing in (3) this expression for T - v and the general expression for c p , the result is Since z depends upon the state of the subtance only, the second member of (4) must be an exact differential. The integral is readily found to be i Q . (5) The constant of integration i Q is determined by applying Eq. (5) to the saturation state. For a given pressure and cor- responding saturation temperature the second member of (5) exclusive of can be calculated. The first member is the value of i for the assumed pressure as given in the steam table. Hence i Q is found by subtraction. By this method the mean value i =886.7 is obtained. Introducing known constants, Eq. (5) becomes i = ^(0.367 + 0.00005 T) - p (1 + 0.0003^)^ -0.0163^ + 886.7. ' (6) Here log (7= 13.72511 when p is taken in pounds per square inch. The total heat of a unit weight of superheated vapor is the heat required to raise the tem- perature of the liquid to the boiling point at the given con- stant pressure, evaporate it, and then superheat it, still at con- stant pressure, to the tempera- ture under consideration. On the ^ZW-plane, the process is shown by the line ABCD (Fig. 74). The area OABCO l rep- resents the total heat of the saturated vapor, which has been denoted by q" , The area A FIG. 74. represents the heat added to superheat the vapor. This heat is evidently given by the integral taken between the saturation temperature T 8 at point and the final temperature T at point D. This integral is, in fact, the product (c p ) OT (2 7 T a ~), where (e p ) w is the mean specific heat for the temperature range T T t . The total heat of a unit weight of superheated steam is given therefore by the expression q= <? +(c,^T - Tj. (7) The term (c p ^) m (T T s ) is easily found from the mean specific heat curves (Fig. 73), and gr"(=i") is given in the steam table. Hence with the aid of the curves, an approxi- mate value for the heat content may be calculated. EXAMPLE. Find the heat content of one pound of steam at a pressure of 150 Ib. per square inch superheated 200. From the steam table t"(= <?") for this pressure is 1194.6 B.tu.; and from Fig. 73 the mean specific heat from saturation to 200 superheat is 0.534. Hence i = 1194.6 + 200 x 0.534 = 1301.4 B. t. u. Thp roc-lllf. rnirar. K,T -frvTTYiTila. f(\\ il 1 2f)1 .7 "R. t. 11. 214 SUPERHEATED VAPORS [CHAP, xi 136. Intrinsic Energy. For the intrinsic energy we have from the defining equation i = A(u + pv), Au = i Apv. (1) Using the expressions for i and v heretofore derived, we obtain the equation * . (2) This expression gives the intrinsic energy in B. t. u. of a unit weight of superheated steam. Introducing the proper constants, we have, when p is taken in pounds per square inch, AM = 2 T (0.2566 + 0.00005 T^~ -(1 + 0.00024 p) + 886. 7, (3) where log (7=13.64593. The intrinsic energy may also be found quite exactly by the following method. For the given pressure p the energy of one pound of saturated steam is Au" = q' + p, and the increase of energy due to the superheat is where (c^) m denotes the mean specific heat at constant volume. The difference (c^) m (c v ) m varies somewhat with the pressure and superheat, but 0.13 may be taken as a mean value. Hence the energy of one pound of superheated steam is given by the equation A U =q'+p + [(*,) - 0.13](^- r a ). (4) Values of q' and p are given in the steam table and the proper value of (c p ) m may be found from the curves of Fig. 73. EXAMPLE. Find the intrinsic energy of one pound of steam at a pres- 137. Entropy. From the general equation ntroducing in this equation the expressions previously derived or c p and ( ^) (see Art. 133), the result is \dJTJp ds = + dT+ Amnp( - n + ["his is necessarily an exact differential since s is a function of he state only. The integral is found to be +ir (3) nserting the known constants and passing to common loga- ithms, (3) becomes s = 0.8451 log T+ 0.0001 T- 0.2542 logp 0.0003^) - 6 - 0.3964. (4) 11 using (4), p is taken in pounds per square inch, and og (7=13.64593. The constant 0.3964 is determined by >assing to the saturation limit, as was done in finding the ralue of . Equation (4) gives the entropy of one pound of superheated iteam at any given pressure and temperature. The entropy may also be found as follows. Let the point D 'Fig. 74) represent the state of the fluid and assume CD ;o be a constant pressure line cutting the saturation curve it 0. Then OO 1 gives the entropy s" of saturated steam it the same pressure as the superheated steam, and en! ropy is > the i-iiii ,-." ! thi' MIMW in ih' '*;!*' /' fit , i Tf .4 * ?'' t . * ?' ! /if f _, /,. 1 ., ulili*, h\vr\T, .i.' if 'l"' rr. ) 1. Pillil llli rtitj'i'p.^, 'I-U;>. as.-l )i'--' >' -! Hlcain ul ;i jur-otuv { >1 t! ISi, J-<-i v|n-,r.- :..! .'(- '!'' ;t'. ; I ; t";^| - ' ;if-s " tif -I in- F. 2. S.tiiirat'-'l f.l":nit ;il 11 j.jr-.-.m.- .< Hull; ]t ---la^.- -,;...!; V*-".. -'nf<- '- f Nll|i('r!l';t(f'il (u U fria|"-i :s!n' i-f .'>' I a- -"" '*.*.' j--..v.j: J rS:'l (,i) h-:vt iwlili-.l; (A) rh.ur,:" { >-u-^:\ f. l .!..; .; - i' s.-'i^j . J-- J-- ' 5 -^ S el' fili'aJn. 3. A'.HUIMI- i|.U;i ;uiS l 'Uij-4'.." !l. n. -n"i-. .M.v,s.-I 1--. v^,:.,; M ai.| t'/} nf Art. i:i;. 4. Av,uitH' il.ii.i tuil iskuhit*- !?:.)> 1;. ?"!: tM .-."! * *. ^ !t ' i '*'* Coiuj'iUo ti-'.nU 1 '. 138. Special Change of State. Uv ju.-.ti- ( > !,.u.i. !;> i*' I'tpUtttUll (o) % Art. l:'L', UUll llir i'rjtrj.l! r,,'i.s!l..:r. jL.if L.t^*" "iicrn (li'iltti'i'il for tin* li-.it < inti-ul, !;,"* , .UP! : j IMJ.*, , iu"'-t- pnilili'ins (luit arise in fujun'fijMU \%j?h lit'- *-h.iH .** "f '-".t- * stiiti'rhcali'tl sti-.un may } Ml\nl *,Mfis ..jir, .&:;-'' .-. { * !l Ht'l'iiUnl, l|i>W(M'r, [ the f.l.'I ii,U fhr ',j.,- ( :!:- L. ,if '! ?. S H given by a snnu\\-hat fiiupli-.tt-'l f.isjr.t'.;, it > .un;"! i'* - *'^" ju'rti'tl lliat tin- rclatiuns Is.'r.- .i-rn ! will h.r, ? ; H- -:?in-*^ < Iiin nf tlmsr fur pcrfrrl ;j;t^-s. In fhr }.;!.!!:.' '! :' U'<'4' >H' *'* spi'i'ial chanin-s ttf ;-taf\ v> c sh.t'.l ',;IM* ini-;.'., .tu in!an- "i <^*' jiroct'ssi'.s involved, Iraviii:,^ the ilf-Mil'' I" ) I;'/.-'l in ^'> ^*'' slutU'iit. 1. Constant Pressure. Let superheated stearn change state i constant pressure from an initial temperature ^ to a final mperature t z . For the heat added we have - Amp (n + 1) (l + ^V-L _ 1 \ (1) \ it j \jt 2 J-i J he external work is given by the relation W=p(v z -vJ=B(T,-TJ-mp (I + ap) [~-L _ 1 1 (2 ) L-f-z - L l J he change of energy may be found from the energy equation u z u-j, = Jq W, 1 independently by calculating from the general formula the lergies in the initial and final states. The change of entropy may be obtained, likewise, from the 3neral equation for entropy or from the relation /Q-N (3) The preceding equations apply to a unit weight of the lid. 2. Constant Volume. If T and T 2 denote, as before, the itial and final temperatures, respectively, we have from the laracteristic equation l + (4) om which p z may be found. Having T v p v and T z , p v the itial and final values of the energy and entropy may be de- rmined from the general formulas. Since the external work zero, the heat added is equal to the increase of energy. 3. Isothermal Expansion. Let the initial and final pressures characteristic equation. For the change of entropy per unit weight we have from the general equation for entropy (6) The heat added during the expansion per unit weight is therefore For the external work, taking dv from the characteristic equa- tion, we have fl + S!L(f l *-pf). (7) Pz * The change of energy may be found by combining (6) and (7) or from the general equation of energy. It is found to be % - % = |{0>i -JPa) + f ( - 1) Oi 2 - ?)] (8) It should be noted that in the case of superheated steam con- stant temperature does not, as with perfect gases, indicate con- stant intrinsic energy. 4. Adiabatic Change of State. For an adiabatic change the entropy remains constant ; hence, for the relation between the final pressure p z and temperature T z , we have from the general equation for entropy where is a constant determined from the initial state. The pressure p 2 is generally given ; therefore, we have the tran- scendental equation z =C', (9) Having the initial and final values of p and T, the initial and nal values u^ and w 2 of the intrinsic energy may be calculated, 'he external work per unit weight is then W=u l u z . (10) In problems connected with the flow of steam the change of .eat content resulting from an adiabatic expansion is required. ?his difference is found by calculating from the general equation or the heat content the initial and final values i t and z 2 . If the adiabatic expansion is carried far enough, the expansion Ine, as >JE (Fig. 74), will cross the saturation curve s", and the tate-point will enter the region between the curves s' and s". lliis means that at the end of the expansion the fluid is a mix- ure of liquid and vapor. The investigation of this case presents to difficulties. The entropy and energy at the initial point D ,re calculated from the general equation. Knowing the pressure or the final state JS, the quality x is readily determined from he equation xr n vhere s denotes the entropy in the initial state. Having x, the snergy in the final state is calculated from the equation u 2 = J^' + z/> 2 ). (12) Flien the external work per unit weight is given by the equation (13) EXAMPLE. Steam at a pressure of 150 Ib. per square inch absolute and superheated 100 F. expands adiabatically to a pressure of 5 in. of mercury. Required the final condition of the fluid and the external work per pound; ilso the pressure at which the steam becomes saturated. From the general equation the entropy in the initial state is found to be L.6346. From the steam table we obtain for the final pressure s' = 0.1880, -= 1.7170; hence T 1.6346 = 0.1880 + 1.7170 x, )r x = 0.8425. [n the initial state the energy in B. t. u. is 4wi = 918.1(0.2566 + 0.00005 x 918.1) - JfjrrgC 1 + 0-00024 x 150) + 880.7 = 1153.9 B. t. u. In tne janal state tne energy is Au z - q z ' + x 2 p 2 = 101.7 + 0.8425 x 953.7 = 905.2. Hence, the external work per pound of steam is W= MI - Ma = 778(1153.9 - 905.2) = 193,4-90 ffc.-lb. The initial entropy 1.6346 is the entropy of saturated steam at a pressure of 66.6 Ib. per square inch. Hence the steam becomes saturated at this pressure. 139. Approximate Equations for Adiabatic Change of State. Exact calculations that involve adiabatic changes of superheated steam are tedious on account of the transcendental form of the .equation for entropy ; and it is therefore desirable to introduce simplifying approximations, provided the results obtained by them are sufficiently accurate. An investigation of a number of cases covering the range of values ordinarily used in the technical applications of superheated steam shows that a set of equations similar in form to the equations for a perfect gas may be obtained, and that the error involved in using these approximate equations does not in general exceed one or two per cent. The relation between pressure and volume during an adiabatic change may be represented approximately by the equation p (v + o) n = const. (1) The value of c is taken the same as in formula (4), Art. 131, namely, c = 0.088. The value of n probably varies slightly with the initial pres- sure and with the degree of superheat ; however, it appears that the value n 1.31 gives quite accurate results for the range of pressure and superheat found in practice. If now we take the approximate characteristic equation p(y + c) = BT, (Art. 132) (2) we get by combining (1) and (2), or Given the initial state of the fluid, the volume in the final ;ate may be found from (1), the final temperature from (4), ad the external work from (5). EXAMPLE. A pound of superheated steam at a pressure of 200 Ib. per [uare inch and superheated 200 expands adiabatically to a pressure of ) Ib. per square inch. Kequired the final condition and the external work. The initial volume is found to be 2.973 cu. ft., and the initial entropy 6657. Using the formula for s (Art. 137), the final temperature is found r trial to be 752.5 absolute ; and taking this value of T, the exact value ; the final volume is found to be 8.6S1 cu. ft. From (3), Art. 136, the energy in the initial state is found to be 1200.57 . t. u., that in the final state 1098.82 B. t. u. ; hence the external work is '8 (1200.57 - 1098.82) = 79,262 ft.-lb. Taking the approximate formulas, we have i _i_ v 2 + c = ( Vl + c) ( iV= (2.973 + 0.088) f?2V- a = 8.819: \2 } 'i' \50 J hence v 2 = 8.819 - 0.088 = 8.731 cu. ft. H4 It will be seen that for practical purposes the results obtained from the iproximate equations are satisfactory as regards accuracy. 140. Tables and Diagrams for Superheated Steam. The lead- g properties of superheated steam volume, entropy, and tal heat for various pressures and degrees of superheat ive been calculated and tabulated by Marks and Davis and r Peabody. The values in the Marks and Davis tables are srived from specific heat curves that differ somewhat from the irves of Fig. 72, and they therefore differ from the values itained from the equations of Arts. 135-137. However, .roughout the range of ordinary practice, the difference does t exceed one half of one per cent. The Marks and Davis tables are accompanied by graphical .arts that may be used to great advantage in the approximate 22 SUPERHEATED VAPORS jlution of numerical problems. The principal chart has the eat content i as ordinate and the entropy s as abscissa. The turated steam at various pressures. The region above this irve is the region of superheat, and the lines running approxi- ately parallel to the saturation curve are lines of constant igree of superheat. Below the saturation curve is the region wet steam, and the lines running parallel to the saturation irve are lines of constant quality. The lines that cross the ituration curve obliquely are lines of constant pressure. The first conception of the heat content-entropy chart is .ie to Dr. E. Mollier of Dresden, hence we shall refer to it as ie Mollier chart. In addition to the chart published by "arks and Davis, one is contained in Stodola's Steam Turbines id one in Thomas' Steam Turbines. In the light of the icently acquired knowledge of the properties of saturated and iperheated steam, the Marks and Davis chart must be regarded j the most accurate. The Mollier chart may be used for the approximate solution : many problems that involve the properties of saturated and iperheated steam, and it is specially valuable in problems on le flow of steam. The following examples illustrate some of le uses of the chart : Ex. 1. Steam at a pressure of 150 Ib. per square inch superheated 200 F. rpands adiabatically to a pressure of 3 Ib. per square inch. The point representing the initial condition lies at the intersection of the instant-pressure line marked 150 and the line of 200 superheat. Locating ds point on the chart, it is found at the intersection of the lines i 1300 id s = 1.087. The heat content and entropy in the initial state are thus itermined. The line 5 = 1.687 intersects the constant-pressure curve p = 3 i the line i = 1002 ; hence the heat content after adiabatic expansion is )02 B. t. u. The quality in the final state is found to be 0.88. Ex. 2. When steam is wire-drawn by flowing through a valve from a igiou of higher pressure j t to a region of lower pressure p, the heat content mains constant. Steam at a pressure of 200 Ib. per square inch and lality 0.95 flows into the atmosphere ; required the final condition of the earn. Dra-wing a line of constant-heat content from the initial point to the irve p = 14.7, it is found that the final point lies above the saturation curve id that the steam is superheated about 12 at exit. The entropy increases om s = 1.498 to s = 1.766. 141. Sunftrheated Ammonia and Sulphur Dioxide. Experi- 224 SUPERHEATED VAPORS [CHAP, xi other than that of water is very scant, and our knowledge of such properties is accordingly imperfect. For superheated ammonia Ledoux has proposed the characteristic equation pv = BT Qp m , (1) and this form has been accepted by Peabody, who derives the following values of the constants (English units) : =99, (7=710, m = |. For sulphur dioxide Peabody uses the same equation with the constants : 5 = 26.4, (7=184, TO = 0.22. According to Regnault the specific heat of superheated ammo- nia has the constant value 0.52. It is very likely that this specific heat is no more constant than that of superheated steam and that it varies with pressure and temperature. How- ever, experimental evidence on this point is lacking. Lorenz finds that for superheated sulphur dioxide c v = 0.329. The problem that most frequently arises in connection with the use of these fluids as refrigerating media is the determi- nation of the state of the superheated vapor after adiabatic compression. It may be assumed that the relation between pressures and temperatures for an adiabatic change follows approximately the law for perfect gases, namely: . (2) _ r Zeuner found that for superheated steam the exponent in (2) is equal to the exponent m in the characteristic equation (1). Hence, using the values of m assumed by Peabody, we have: For ammonia n = - = = 1.333. l_ m 1-Q.25 For sulphur dioxide n = = 1.282. BT. vapor, juet A. ^rig. 10; represent tne initial state, id B the final state after adiabatic compression. EA and 'B are constant-pressure curves. Denoting by TJ the satura- on temperature correspond- ig to the pressure p r the icrease of entropy from E T ) A is Cploge^, and the >tal entropy in the jstate A is s/'-f-Cplog.^l. ikewise, the entropy in the ;ate B is T. II FIG. 76. ince AB is an adiabatic, the entropies at A and B are equal, id therefore i this equation s/', s 2 ", 2V, and 3Y' ^ re tabular values corre- jonding to the given pressures p 1 and [ence, !T 2 is the only unknown quantity. and 2j is given. EXERCISES 1. Calculate by Eq. (2), (4), and (6), respectively, of Art. 132 the vol- ne of one pound of superheated steam at a pressure of 180 Ib. per square .ch and a temperature of 430 F. Compare the results. 2. If the products pv are plotted as ordinates with the pressures p as >scissas, show the general form of the isothermals T = C when Eq. (3), rt. 132 is used ; when Eq. (6) is used. 3. For ammonia, Peabody gives the following equations for the latent ?at of vaporization : r = 540 0.8 (t 32) . If at the critical temperature = 0, find t c for ammonia by means of this formula and compare with the ilue of t c given in Art. 129. Explain the discrepancy. 4. Following the method of Art. 133, deduce an equation for c p , using le approximate equation (5), Art. 132; also using Calendar's equation (8). 5. By means of Eq. (3), Art. 132, calculate the specific volume of satu- ,ted steam at the following pressures : 5 in. Hg., 20, 50, 150 Ib. per square men. USB pare the results with the values of v" given in the table. 6. Calculate the mean specific heat of superheated steam at a pressure of 140 Ib. per square inch between saturation and 250 superheat. Compare the result with the curves of Fig. 73. 7. Using the mean specific heat curves, Fig. 73, find the heat content and energy of one pound of superheated steam at a pressure of 85 Ib. per square inch and a temperature of 430 F. 8. A pound of saturated steam at a pressure of 120 Ib. per square inch is superheated at constant pressure to a temperature of 386 F. Find the heat added, the external work, and the increase of energy. 9. The steam after superheating expands adiabatically tmtil it again be- comes saturated. Find the pressure at the end of expansion and the external work. 10. The following empirical equation has been proposed for the value of c p very close to the saturation limit : ^=.----, (Jc la) in which t c is the critical temperature, 689 F., and t a is the saturation tem- perature corresponding to an assumed pressure. Using the curves of Fig. 72, calculate the value C for several assumed pressures, and thus test the validity of the formula for these curves. 11. The following equation has also been proposed for the value of c p at saturation : (c p ) Ba t = a + bt s . Test this equation, and if it holds good within reasonable limits determine the constants o and &. 12. In the initial state 6.4 cu. ft. of superheated steam has a temperature of 420 F. and is at a pressure of 160 Ib. per square inch. By the approxi- mate equations of Art. 139 find the temperature and volume after adiabatic expansion to a pressure of 80 Ib. per square inch ; also the work of expansion. 13. Assume for the initial state of superheated steam p^ = 80 Ib. per square inch, v : = 20 cu. ft., ^ = 350 F. Plot the successive pressures and volumes for an isothermal expansion to a pressure of 30 Ib. per square inch. Compare the expansion curve with the isothermal of air under the same conditions. 14. With the data of Ex. 13 find the external work, heat added, and change of energy (a) for the superheated steam ; (fc) for air. REFERENCES THE CRITICAL STATE. EQUATIONS OF VAN DER WAALS AND CLAUSIUS The literature on these subjects is very extensive. For comprehensive discussions, reference may be made to the following works : Preston: Theory of Heat, Chap. V, Sections 6 and 7. euner: Technical Thermodynamics (Klein) 2, 202-229. hwolson : Lehrbuch de Physik 3, 791-841. CHARACTERISTIC EQUATIONS allendar : Proc. of the Royal Soc. 67, 266. 1900. inde : Mitteilungen iiber Forschungsarbeiten 21, 20, 35. 1905. euner : Technical Thermodynamics 2, 223. /"eyrauch : Grundriss der Warme-Theorie 2, 70, 87. SPECIFIC HEAT OF SUPERHEATED STEAM [allard and Le Chatelier : Annales des Mines 4, 528. 1883. angen : Zeit. d. Ver. deutsch. Ing., 622. 1903. olborn and Henning : Wied. Annalen 18, 739. 1905. 23, 809. 1907. egnault: Mem. Inst. de France 26, 167. 1862. noblauch and Jakob : Mitteilungungen iiber Forschungsarbeiten 35, 109. noblauch and Mollier : Zeit. des Ver. deutsch. Ing. 55, 665. 1911. horaas : Proc. Am. Soc. Mech. Engrs. 29, 633. 1907. A most complete discussion of the work of various investigators is given f Dr. II. N. Davis, Proc. Am. Acad. of Arts aud Sciences 45, 267. 1910. GENERAL THEORY OF SUPERHEATED VAPORS allendar : Proc. of the Royal Soc. 67, 266. r eyrauch : Grundriss der "Warme-Theorie 2, 117. 3uner : Technical Thermodynamics 2, 243. CHAPTER XII MIXTURES OF GASES AND VAPORS 142. Moisture in the Atmosphere. Because of evaporation of Welter from the earth's surface, atmospheric air always con- tains a certain amount of water vapor mixed with it. The weight of the vapor relative to the weight of the air is slight even when the vapor is saturated. Nevertheless, the moisture in air influences in a considerable degree the performance of air compressors, air refrigerating machines, and internal com- bustion motors ; and in an accurate investigation of these ma- chines the medium must be considered not dry air but rather a mixture of air and vapor. The study of air and vapor mixtures is also important in meteorology and especially in problems relating to heating and ventilation. Finally, it has been pro- posed to use a mixture of air with high-pressure steam as the working medium for heat engines, and the analysis of the action of an engine working under this condition demands a special investigation of air and steam mixtures. Experiment has shown that Dalton's law holds good within permissible limits for a mixture of gas and vapor. The gas has the pressure p' that it would have if the vapor were not present, and the vapor has the pressure p" that it would have if the gas were not present. The pressure of the mixture is P=p'+p". (1) If the vapor is saturated, the temperature t of the mixture must be the saturation temperature corresponding to the pressure p". If the temperature is higher than this, the vapor must be superheated. The water vapor in the atmosphere is usually superheated. Let point A, Fig. 77, represent the state of the vapor, and let A.B be a Constant DresSlirH P.nrvo nnt.-f-.inrr f.lm unfn vrH-irm p.nrvft RT. 142] MOISTURE IN THE ATMOSPHERE 229 b B. Further, let m denote the weight per cubic foot of the apor in the state A, and m-^ the weight per cubic foot of satu- xted vapor at the same temperature, that is, in the state 0. 'he ratio is called the humidity of the air under the given onditions. If the mixture of air and vapor is cooled at constant ressure, the vapor will follow the ath AB and at B it will become iturated. Upon further cooling 3me of the vapor will condense. ?he temperature T Q at which con- ensation begins is called the dew oint corresponding to the state A. The humidity may be expressed pproximately in terms of pressures. <et p a " denote the pressure of the apor in the state A and p c n the ressure of saturated vapor at the nine temperature, hence in the state represented by 0. At the )w pressures under consideration we may assume that the vapor allows the gas law p V MET. Hence, taking V- 1, we have FIG. 77. 'herefore, denoting the humidity by <, we have m ">" p (2) 'hat is, the humidity is the ratio of the pressure corresponding D the dew point to the saturation pressure corresponding to le temperature of the mixture. For investigations that involve hygrometric conditions, the ata ordinarily required may be found in table II, page 319. At 70 the saturation pressure is, irom taoie JLI, U.MO incnes 01 ug, while at 52 the saturation pressure is 0.13905 inches of Hg. The humidity is therefore 03905 = v 0.738 If the air were saturated at 70, it would contain 8.017 grains of vapor per cuhic foot. Hence with 52.9 per cent humidity the weight of vapor per cubic foot is 8.017 x 0.529 = 4,211 grains. EXAMPLE 2. Atmospheric air has a temperature of 90 F. and a humidity of 80 per cent. It is required that air be furnished to a building at 70 F. and with 40 per cent humidity. From table II, the pressure of saturated vapor at 70 is 0.738 inches of Hg; hence from (2) the pressure corresponding to the dew point is 0.40 x 0.738 = 0.2952 inches of Hg, and the dew point is 44.5. In the initial state one cubic foot of air contains 0.80 x 14-.85 = 11.88 grains of vapor. The air is cooled to 44.5 by proper refrigerating apparatus and in this state contains 3.39 x 459 ' 6 + M ' 5 = 3.11 grains, the difference 11.88 - 3.11 = 8.77 459.6 + 90 grains being condensed. The air freed from the condensed vapor is now heated to the required temperature, 70. 143. Constants for Moist Air. The constants B, c v , <?, etc., given in Chapter VII apply only to dry air. For ail- containing water vapor the constants must bo changed some- what, the magnitude of the change depending, of course, upon the relative weight of vapor present. An expression for the constant B of the mixture may be obtained by the following method. Let the volume V contain M 1 Ib. of air at the pressure p' and M z Ib. of water vapor at the pressure p". Then assuming that the gas law may be applied to the vapor, we have , (1) (2) Let |Ta = 3, an d ^2 = e ; then from (3) (4) hence *>"=.Pff^ ^f' dding the members of (1) and (2), we obtain 'he constant m of the mixture is, however, given by the ^nation pV=(M^M^B m T. (7) [ence, comparing (6) and (7), we have Taking the molecular weight of water vapor as 18, we have =85.72, . j jD 9 85. 1 2 1 PI ld '"^-so*- 1 - 81 - EXAMPLE. Find the value of B for air at 90 F. completely saturated Lth water vapor. The pressure of the mixture is 14.7 Ib. per square inch. From the table the pressure p" of the vapor is 0.691 Ib. per square inch ; erefore the pressure p 1 of the air is 14.7 0.691 = 14.009 Ib. per square ch. From (5), 1 + ez = = -^- = 1.0493, ez = 0.0493, and z = '/) J.'x.UUty 1 D4.Q3 0.0306. Therefore, B m = 53.34 x ~^ = 54.31. l.OoOo The specific heat of the mixture is found by applying the w deduced in Art. 83. If cj and c p " denote respectively .e specific heats of the air and steam, then the specific heat of .e mixture is given by the equation EXAMPLE. Taking c p for air as 0.24, and for steam at 90 as 0.43, the seine heat of the mixture given in the preceding example is 0.24 + 0.0306 x 0.43 _ 1 4- 0.0306 144. Mixture of Wet Steam and Air. In a given volume V let there be M Ib. of air and M z Ib. of saturated vapor mixture of quality x. The absolute temperature of the entire mixture is T, and the total pressure p. The pressure p is the sum of the partial pressures p' and p" of the air and steam, respec- tively. This follows from Dalton's law, which whithin reason- able limits holds good for the case under consideration. We have then p' + p" = p, (1) p'V^MJBT, C2) F= Jf 2 [>(*/' - ') + <], (3) where, as usual, v' r and v' denote, respectively, the specific volumes of steam and water at the saturation temperature T. The energy of the mixture is the sum of the energies of the two constituents ; hence, we have AU= M lCv T+ M z (j + xp) + Z7 . (4) Likewise, the entropy of the mixture is S = M l [> log, T+ (c 9 - O log e F] + M z + + ff . (5) By means of these equations various changes of state may be investigated. 145. Isothermal Change of State. Since ^remains constant, we have from (4) A( U z - Z7i) = Mf(zt - xj, (1) and from (5) %-S^ M,AB log e ^ + M 2 Lfa - xj. (2) Hence, the heat added is given by the equation Q = T(S Z - SJ = MtABTlog. -p + Jf 2 r(^ 2 - ^. (3) The external work is (4) 1 neglecting the small water volume v', VZ = hile in the initial state (6) ence, combining (5) and (6), _ JF From (7) it appears that isothermal expansion is accompanied r an increase of the quality #, that is, by evaporation, while ^thermal compression involves condensation. 146. Adiabatic Change of State. In the case of an adiabatic ange the final total pressure j9 2 is usually given. Assuming at the steam in the mixture does not become superheated, e final temperature T z of the mixture must be the saturation cnperature corresponding to the partial pressure p 2 " of the jam. The determination of the final state of the mixture solves the determination of two unknown quantities ; namely, 3 partial pressure p 2 " and the quality # 2 of the saturated por. Hence two relations are required. One is given by 3 condition that the entropy of the mixture shall remain astant during the change, the other by the condition that 3 final volume V z may be considered as occupied by each istituent of the mixture independently of the other. En the application of the first condition it is convenient to 3 an expression for the entropy of the mixture of a form ferent from that given by (5), Art. 144. In terms of the nperature and pressure, the entropy of a unit weight of air ^iven by the expression s = c p log, T-AB log e p + s ; ice for the mixture we have S=M, (c Joer, T-AB loer. '") + A IVJLJL^VJ. \JJ As the constant $ disappears when the difference of entropy between two states is taken, it may be ignored in the calculation. Let 8} denote the entropy in the initial state. Then since the entropy remains constant, we have ^ = M l (c v log fl T z - AB log a K) + JfV + (2) \ j.% / In Eq. (2), S r M v M. 2 , and the coefficients c v and AB are known, as is the final total pressure p z . The partial pressures p 2 r and p z ", the quality a; 2 , and temperature T z are unknown. However, T z depends upon |? 2 ", and p z ' is found from the relation p z + p z " = p 2 when p z ' f is determined. Denoting the final volume by V v we have whence W Inserting this expression for x z in (2), we have finally SMIo T-ABlo Mfs + - (4) \ M z Pz V 2 / In this equation p z is the only unknown. The solution is most easily effected by assuming several values of p z " and calculating for these the values of the second member. These calculated values are then plotted as ordinates with the corre- sponding values of p z " as abscissas and the intersection of the curve thus obtained with the line ^ = const, gives the desired value oip z "' The external work of expansion or compression is equal to the change of energy. Hence, using the general expression for the energy of the mixture, we have (5) EXAMPLE. In a compressor cylinder suppose water to be injected at the beginning of compression in such a manner that the weight of water and water vapor is just equal to the weight of the air. Let the pressure of the mixture be normal atmospheric pressure 20.92 in. of mercury, and let the temperature be 79.1 F. The mixture is compressed to a pressure of 120 Ib. ing to 79.1 is 1 in. Pig, hence the partial pressure of the air is 28.92 in. Hg. The initial quality x is found from the relation whence 53 ' 34 x 538 ' 7 = 0.0214. M 2 pi'vi" 28.92 x 0.4912 x 144 x 656.7 The factor 0.4912 x 144 is used to reduce pressure in inches of mercury to pounds per square foot. For l.he entropy of the mixture we obtain from (1) (neglecting the con- stant So) Si = 0.24 log e 538.7 - 0.0686 log e (28.92 x 0.4912) +0.0916 + 0.0214 x 1.9482 = 1.4587. Since the ratio of the final to the initial pressure of the mixture is =1 = 8.2, we assume that the pressure pz" of the vapor after compression will be approximately 8 times the initial pressure pi". Hence we assume p 2 " 7, 8, and 9 in. of mercury, respectively, and calculate the corresponding values of the second member of (2). Some of the details of the calculation are given. FROM STEAM TABLE (in. I !")" ' n> I T z **' j- o V*' Pz & 2 sq. in. 7 3.43 116.57 146, .0 606.5 0.2007 1011.1 104.4 ] 8 3.92 116.08 152. 3 611.9 0.2186 1007.9 92.18 \ Data 9 4.41 115.59 157. 1 616.7 0.2265 1005.0 82.57 J I p a ABlog 1.5378 1.5400 1.5418 0.3264 0.3262 0.3259 0.0308 0.0349 0.0390 1.4519 1 1.4673 I Results 1.4814 J The pressure p 2 " that gives the value S = 1.4587 lies between 7 and 8 in. Hg and by the graphical method or by interpolation we find p 2 " = 7.44 in. Hg, or 2>a" = 3.65 Ib. per square inch. Therefore p 2 ' = 120 - 3.65 = 116.35 Ib. per square inch. From the steam table the following values are found for the pressure p a " = 7.44 in. Hg : t = 149.3, T 2 = 608.9. qj = 117.3, r 2 = 1000.4, p 2 = 942.8, v 2 " = 99. The final quality is 53.34 x 608.9 p 2 'v 2 " 116.35 x 144 x 99 The external work per pound of air is W= ,7[0.17(149.3 - 79.1) + 117.3 - 47.2 + 0.0214 x 989.8 - 0.01958 x 942.8] = 61566 ft. Ib. 236 MIXTURES OF GASES AND VAPORS [CHAP, xn The volume of the mixture at the end of compression is V = * = 6*8421608.0 = 1<08ao cu< f t p a ' 110.85 x 144 and the work of expulsion is therefore 1.9380 x 120 x 144 = 33408 ft. Ib. Hence, the work of compression and expulsion is 950(5'! ft. Ib. The effect of injecting water into a compressor cylinder may be shown by a comparison of the result just obtained with the work of compressing and expelling 1 Ib. of dry air under the same conditions. The initial volume of 1 Ib of air is f- = la - C74 cu - fti - The final volume after adiabatic compression to 120 Ib. per squaro inch is JL 13.574 (^pY' 4 = 3.0290 cu. ft. \ .l^U / The work of compression is ,7 x 13.574 - 120 x 3.0290) = 59044 ft. Ib., the work of expulsion is 3.0290 x 120 x 144 = 52350 ft. Ib., and the sum is 111394 ft. Ib. The effect of water injection is therefore to reduce the volume and temperature at the end of compression and the work of com- pression and expulsion. The reduction of work in this case is about 17 per cent. 147. Mixture of Air with High-pressure Steam. In the pre- ceding articles, we have dealt with mixtures of steam and air in which the pressure of the vapor content was small. The suggestion has been made that a mixture of air at relatively high temperature and pressure mixed with steam either super- heated, saturated, or with a slight amount of moisture be used as a medium for heat engines. An analysis of the action of such a medium in a motor demands in the first place a discussion of the process of mixing, afterwards a discussion of the change of state of the mixture. Let M^ Ib. of air compressed to a pressure p l and having a temperature T^ be mixed with M z Ib. of wet steam having a pressure w~ and miali-hv 1- Tl-io fomT\ovof.iivo T 1 nf g the air into a receiver which contains steam, or vice versa. nee under these conditions the pressure of the mixture can- it be raised above the pressure of the constituents, the volume the mixture cannot be taken as the original volume V l of e air. We assume, on the other hand, that the conditions e such that the volume of the resulting mixture is the sum the volumes of the constituents ; that is, F=F 1+ F 2 . (1) s a second condition, the internal energy of the mixture is Tial to the sum of the energies of the constitutents ; hence e have the equation of condition U=U l+ U r (2) Let T denote the temperature after mixing p' the partial pres- .re of the air and p" the partial pressure of the steam. Then, ovided the steam does not become superheated, the tempera- .re T must be the saturation temperature corresponding to the essurep". The following relations are readily obtained. (3) 1 since the quality x 2 is nearly 1, (4) (5) here x denotes the quality after mixing, and v" is the specific )lume of steam corresponding to the pressure//'. (6) z (q ! + xp). (8) com (2) we have = M,e n T, + MM + a; 2 P 2 ). (9) (10) Pi Having V calculated from (10), we obtain from (5) and this expression for x substituted in (0) gives finally (12) In (12) the second member is known from the initial condi- tions. In the first member q', p, and v" are dependent on T \ hence T is the one unknown. As usual, tho solution is ob- tained by taking various values of T and plotting the resulting values of the first member of (12). EXAMPLE. Let 1 Ib. of wefc steam, quality 0.85, at a pressure of 200 Ib. per square inch, be mixed with 2 Ib. of air at a pressure of 220 Ib. per square inch and a temperature of 400. Required the condition of the mixture. From the data given, the following values are readily found : Vi = 2.895 cu. ft. ; V z = 1.948 cu. ft. ; V = 2.81)5 + 1.948 = 4.843 cu. ft. U=Ui+U>2 = 1273.8 B. t. u. Equation (12) becomes 0.34 T + q' + 4.843 ^ = 1273.8. v" We now assume for p" the values 50, 75, and 100 Ib. per square inch; from the tables we find the corresponding values of q', p, v", and T, and calculate the values of the first member. The results are : For /'= 50, OSlB.tu. p"= 75, 1 222.3 B. t.u. p" = 100, 1451 B. t. u. Plotting these results, we find p" = 81 Ib. per square inch very nearly. The temperature of the mixture is therefore 313 F. and the quality of the 4 843 steam is x = ~ = 0.897. (5.4 is the specific volume v" corresponding to a pressure of 81 Ib.) The partial pressure p' of the air is found from (5) to be 130 Ib. per square inch. Hence the pressure of the mixture is 130 + 81 = 211 Ib. per square inch. JLU is seen mia,\j, tis L.UW IBSUIL UL mixing, trie temperature is considerably vered, the pressure takes a value between pi and jo 2 , and the quality of 3 steam is increased. If the steam is initially superheated, the preceding equations list be modified by inserting for V 2 arid V z the appropriate :pressions for the volume and energy, respectively, of super- ated steam. To reduce as far as possible the complication the formulas we shall take the approximate equation (5), L-t. 132, for the volume. We have then F 2 = M z v z = M 2 (^ - A (13) ae constant B is written with a prime merely to distinguish from the constant for air. The intrinsic energy of the steam given by Eq. (2), Art. 136. This equation can be simplified ith a small sacrifice of accuracy by dropping the term con- ining a. The modified equation then takes the form Au = !F(e + fT) -& + 886.7, (14) which e = 0.2566, /= 0.00005, and log 0= 13.64593. From (6) and (14) the energies of the constituents before ixing can be calculated, and the sum of these gives the Lergy If of the mixture. We have then as one equation of >nditioii + fT) - + 886.7] =AU. (15) nee p" and T are here independent, there are two unknowns id a second condition is required. From (3) and (13) the itial volumes V^ and V 2 are found and the sum gives the >lume V of the mixture. Then .P'- P" = ~V Jf 2 :om (15) and (16) the unknowns p" and STcan be found. 240 MIXTURES OF GASES AND VAPORS [CHAP, xn EXAMPLE. Let 5 Ib. of air at 60 F. be compressed adiabatically from atmospheric pressure to a pressure of 200 Ib. per square inch and mixed with 1 Ib. of steam at 200 Ib. per square inch superheated 100. The con- dition of the mixture is required. The temperature of the air after compression ^=519.6(^)^=1095. The saturation temperature of steam at 200 Ib. per square inch is 381.8 F ; hence T z = 381.8 + 100 + 459.0 = 941.4. The energy of the air is 5 x 0.17 x 1095 = 930.75 B. t. u. and that of the steam is, from (14), 941.4 (0.2566 + 0.00005 x 941.4)- C-^- + 880.7 = 1100.0 B. t. u. v ' 941. 4 5 Hence A ( U l + Z7 2 ) = A U = 030.75 + 11GO.O = 2001.35. B. t. u. We have then from (15). 0.85 T + 7X0.2566 + 0.00005 T)-Q~ = 1204.65. To derive an expression for the partial pressure p" the total volume V must be found. Before mixing, the volume of the air is v MZ. = 5x63.34x1095 = 1 p l 144x200 ' and the volume of the steam is 0-5062 x 941.4 - _ pz 200 Hence V= 10.14 + 2.55 = 12.09 cu. ft. After mixing the superheated steam at the partial pressure p" and tem- perature T occupies this volume ; hence, we have (since M% = 1) _ B'T 0.5062 T V+c 12.69 + 0.256 Introducing this expression for p" in the term ~~-, that term becomes C 1 , where log C' - 12.30919. The equation in T then becomes 1.1066 T + 0.00005 T z - - = 1204.65. T 1.1060 T 0.00005 T* - Sum JT 4 J"' 1 " 1000 1106.6 50. 2.04 1154.56 1050 1161.93 55.13 1.68 1215.38 1100 1217.26 60.5 1.39 1276.37 By interpolation it is readily found that T= 1041. The pressure of the :am is 0.5962 x 1041 Afr nA ,, . , p - TTTai - = 47.94 Ib. per square inch, lile the pressure of the air is . 53.34x1041x5 ,_, no .. P' 777 77 = 151.93 Ib. per square inch. terefore p =p' + p" = 199.9 Ib. per square inch. The total pressure p should evidently be 200 Ib. per square inch ; hence } result may be regarded as a check on the calculation. Having now the initial condition of the mixture, the condi- >n after adiabatic expansion to any assumed lower pressure d the work of expansion may be found by the methods of ?t. 146. The discussions of Arts. 146 and 14T furnish the necessary uations for the analysis of the action of a motor that uses a .xture of air and steam as its working fluid. EXERCISES 1. Find the humidity and the weight of vapor per cubic foot when the nperature is 85 and the dew point is 70. 2. The humidity is 0.60 when the atmospheric temperature is 74 F. id the dew point. 3. Find the value of B for air at 80 with 70 per cent humidity. Fiud the specified heat c p of the mixture. 4. A mixture of air and wet steam has a volume of 3 cu. ft. and the nperature is 240 F. The weight of the air present is 1 Ib., that of the am and water 0.4 Ib. Find the partial pressures of the air and vapor, the al pressure of the mixture, and the quality of the steam. 5. Let the mixture in Ex. 4 expand isothermally to a volume of 5 cu. ft. id the external work, the heat added, the change of entropy, and the inge of energy. 6. Let the mixture expand adiabatically to a volume of 5 cu. ft. Find 1 condition of the mixture after expansion, and the external work. 7. Let 1 Ib. of steam, quality 0.87, at a pressure of 150 Ib. per square inch, be mixed with 4 Ib. of air at a pressure of 100 Ib. per square inch and a temperature of 340 F. Find the condition of the mixture. 8. Let the mixture in Ex. 7 expand adiabatically to a pressure of 40 Ib. per square inch. Determine the final state of tho mixture and calculate the work of expansion. 9. Let 1 Ib. of steam at a pressure of 150 Ib. per squaro inch and super- heated 140 be mixed with 6 Ib. of air at a pressure of 100 11). per square inch and a temperature of 340 F. Find the condition o the mixture. 10. Let the mixture in Ex. 9 expand adiabatically until the pressure drops to 14.7 Ib. per square inch. Required the final state of the mixture and the work of expansion. REFERENCES Berry: The Temperature Entropy Diagram, 130. Zeuner : Technical Thermodynamics, 3'20. Lorenz : Technische Warmelehre, 306. ' ' CHAPTER XIII THE FLOW OF FLUIDS f> " ' 148. Preliminary Statement. Under the title " flow of fluids " are included all motions of fluids that progress continu- ously in one direction, as distinguished from the oscillating motions that characterize waves of various kinds. Important examples of the flow of elastic fluids are the following : (1) The flow in long pipes or mains, as in the transmission of illuminat- ing gas or of compressed air. (2) The flow through moving channels, as in the centrifugal fan. (3) The flow through orifices and tubes or nozzles. The recent development of the steam turbine has made especially important a study of the last case, namely, the flow of steam through orifices and nozzles, and it is with this problem that we shall be chiefly concerned in the present chapter. Of the early investigators in the field under discussion, mention may be made of Daniel Bernoulli (1738), Navier (1829), and of de Saint Venant and Wantzel (1839). The latter de- duced the rational formulas that to-day lie at the foundation of the theory of flow ; they further stated correctly conditions for maximum discharge, and advanced certain hypotheses regard- ing the pressure in the flowing jet which were at the time dis- puted but which have since been proved valid. Extensive and important experiments on the flow of air were made by Weisbach (1855), Zeuiier (1871), Fleigner (1874 and 1877), and Him (1844). These served to verify theory and afforded data for the determination of friction coefficients. In 1897 Zeuner made another series of experiments on the flow of air through well-rounded orifices. Experiments on the flow of steam were made by Napier (1866), Zeuner (1870), Rosenhain (1900), Rateau (1900), Gutermuth and Blaess (1902, 1904). 243 Most of the experimental worK nere notcu .^, - ^ flow of fluids through simple orifices or through short con- vergent tubes. The more complicated relations between veloc- ity, pressure, and sectional area that obtain for How through relatively long diverging nozzles have been investigated experi- mentally by Stodola, while the theory has been developed by H. Lorenz and Prandtl. The flow of steam through turbine nozzles has also been discussed by Zeuner. 149 Assumptions. -In order to simplify tlio analysis of fluid flow and render possible the derivation of fundamental equations, certain assumptions and hypotheses must necessarily be made. 1. It is assumed that the fluid particles move in non-inter- secting curves stream lines which in the case of a prismatic channel may be considered paral- ^^rp^rrrrr:p^&r3?- lul to tlxo tlxi ' S f th clianne1 ' -"i^ir^^^^^tt We mav imagine surfaces '-^^^ ' r '" '-^^^C^- stretched across the channel, as FlG> 78 ' JF, .F', I", etc., Fig. 78, to which the stream lines are normal. These are the cross sections of the channel. They are not necessarily plane surfaces, but they may usually be so assumed with sufficient accuracy. 2. The fluid, being elastic, is assumed to fill the channel completely. From this assumption follows the equation of con- tinuity, namely : in which I denotes the area of cross section, w the mean veloc- ity of flow across the section, M the weight of fluid passing in a unit of time, and v the specific volume. 3. It is assumed that the motion is steady. The variables p, v, T giving the state of the fluid and also the mean velocity w remain constant at any cross section J? ; in other words, these variables are independent of the time and depend only upon the position of the cross section. 150. Fundamental Equations. The general theory of flow of elastic fluids is based upon two fundamental equations^which MTP. dp.rived hv aDDlvinsr the mlneii)le of conservation of energy to an elementary mass of fluid moving in the tube or channel. Let w l denote the velocity with which the fluid crosses a section F of a horizontal tube, Fig. 79, and w the velocity at some second section F. A unit weight of the fluid at section an 2 F l has the kinetic energy of motion ^- due to the velocity w^ ; hence if u is the intrinsic energy of the fluid at this section, the nn 2 total energy is w x + ~. Likewise, the energy of a unit weight w 2 of fluid at section F is u -f- -. In general, the total energy at ^9 section F is different from that at section F l and the change of energy between the sections must arise : (1) from energy entering or leaving the fluid in . . the form of heat during the - passage from F to F \ (2) from \ work done on or by the fluid. The heat entering the fluid per unit of weight between the two sections we will denote by q. Evidently work must be done against the frictional resistance between the fluid and tube ; let this work per unit weight of fluid be denoted by z. The heat equivalent Az necessarily enters the flowing fluid along with the heat q from the outside. Aside from the friction work, the only source of external work is at the sections F and F. As a unit weight of fluid passes section F v a unit weight also passes section F. Denoting by p l and v 1 the pressure and specific volume, respectively, at F-^ the work done on a unit weight of fluid in forcing it across section J^ is the product p^ ; simi- larly, the product pv gives the work done ~by a unit weight of fluid at section F on the fluid preceding it. For each unit weight flowing the net work received at the section F l and F is, therefore, Equating the change of energy between F l and F to the energy received from external sources, we obtain 7T- " U 2 n 2\ ^ )= z a J or G) This is the first fundamental equation. It will be observed that the friction work z drops out of the equation; the effect of friction is to alter the distribution 7/1** between internal energy u and kinetic energy - at section jP, .t-j f/ leaving the sum total unchanged. Differentiation of (1) gives ^ + du + d(pv-) = Jdq, (2) u/ a form of the fundamental equation that is useful in subsequent analysis. Equation (1), as is apparent, takes account only of initial conditions at section jf\ and final conditions at section _F, and gives no information of anything that occurs between these sections. A second fundamental equation taking account of internal phenomena between the two sections is derived as fol- lows. Consider a lamina of the fluid moving along the channel. This element receives from external sources the heat dq and also the heat Adz, the equivalent of the work done against frictional resistances. Independently of its motion, the lamina of fluid may increase in volume and thereby do external work against the surrounding fluid, and its internal energy may increase. According to the first law we have, therefore, J(dq + Adz) = du + p dv . (3) The first member represents the energy entering the lamina during the passage from J^ to F, du is the increase of energy, and pdv the external work done. Combining (3) with (2), we get wdw 7 , , A si\ - + vdp + dz = 0, (-i) 9 whence by integration we obtain ART. 151] FORMS OF THE FUNDAMENTAL EQUATION 247 The fundamental equations (1) and (5), or the equivalent dif- ferential equations (2) and (4), are perfectly general and hold equally well for gases, vapors, and liquids. 151. Special Forms of the Fundamental Equation. In nearly all cases of flow the heat entering or leaving the fluid is so small as to be negligible, and we may, therefore, assume that q = 0. The sum u+pv will be recognized as the work equiva- lent of the heat content i ; that is, u +pv = Ji. (See Art. 52.) Equation (1) of Art. 150 may, therefore, be written in the form For a perfect gas pv, (2) K 1 whence, .TON (3) If the fluid is a mixture of liquid and saturated vapor, the heat content i is practically equal to the total heat. (See Art. 86.) Hence we may put i = q' + xr, (4) and (1) becomes ^-JW + W-W + xrX. (5) %g For a superheated vapor, the general form (1) is used, the values of ^ and i being calculated from formula (6), Art. 135. Equations (3) and (5) being derived from the first funda- mental equation hold equally well for frictionless flow and for flow with friction. 152. Graphical Representation. A consideration of the fundamental equations developed in Art. 150 leads to several convenient and instructive graphical representations, in which jp-axis is given by In the case of frictionless flow, however, the second FlGr 80> fundamental equation [(5), Art. 150] becomes (i) Hence for frictionless flow, the increase of kinetic energy is given by the area between the jp-axis and the. curve representing the expansion. 2. If the flowing fluid is a saturated vapor of given quality, the representation just given applies but the equation of the expansion line AS must be expressed in the form pv n const. It is, therefore, more convenient to use the tem- perature T and entropy S as coordinates. If the flow is frictionless and adiabatic, the expansion curve AB is the vertical isentropic, Fig. 81. The area OHCAA^ rep- resents the total heat of the mixture in the initial state A, FIG. 81. A* and the area OHDBA l the total heat in the final state B ; hence the difference of these areas, namely, the area ABDC, represents the difference q^ + x^ - (</ + ar), and from (5), Art. 151, this area, therefore, represents the increase of energy If the initial point is at A' in the superheated region, we have ^ = area OJETOAA'A^ i = area OSDB'A^ \ - i = area A'B'D OAA'. 3. The work z expended in overcoming friction may be shown on either the pv- or the ^ZW-plane. When friction is taken into account, the heat Az, the equivalent of the friction work z, reenters the fluid, and consequently. the heat content i and the volume v are both greater at the lower pressure p than they would be were there no friction. Hence the expan- sion curve AB' , Fig. 82 and 83, for flow with friction must lie to the right of the curve AS for flow without friction. This statement applies to both figures. Let ij denote the heat content in the initial state JL, i the heat content in the state B, and i' the heat content in the final state B' when friction enters into consideration. Then i' > *, whence ^ i' < *! i' It follows from (1) Art. 151, that the change of kinetic energy 01) ^2 _ an 2 - 1- for flow with friction is less than the change 2<7 * in the case of frictionless flow. Friction, therefore, causes a loss of kinetic energy given by the relation 2 w' 2 Tr ., . N /-ON -~ = J (* -0- ( 2 ) On the ^W-plane, Fig. 83, this loss is represented by the area A.BB'B'; for i' = area OHDB' -/, i = area OHDBA V ;'-&'= area A.BB'BJ. fluid ; hence as explained in Art. 50, the increase of entropy is jf '*:, and the area A^B'BJ under the curve AB repre- sents (in heat units) the friction work z. On the jpf-plane let a constant i line be drawn from point B', Fig. 82, cutting the frictionless expansion line in the point Cr. Then since the heat content i' is the same at Gr as at B f , the difference i 1 i' in pass- ing from A to B' along the actual curve is the same as in passing from A to Gr along the ideal frictionless expan- sion curve AB. .Hut the change of i between the mo. 82. states represented by points A and Gr, which in work units represents the increase of kinetic energy between A and Cr, is given by the area AGf-.EO. Hence we have : For frictionless flow, For the actual flow, = area ARDO. . n -f-fsy 1- = area A GrJEC. Hence the loss of kinetic energy due to friction is given by the area BDEG-. From the fundamental equation (5), Art. 150, we have C Pl j w 2 Wi 2 /Q-N z=\ vdp L, (3) J P 2(/ in which the integral refers to the actual expansion curve. Referring to Fig. 82, V \dp is given by the area AB'DC ART. 152] GRAPHICAL REPRESENTATION 251 T ,Z \ FIG. 83. while the change of energy for the actual flow is, as just shown, given by the area AGrECi hence the difference, the area AB'DEG-A, represents the work of friction z. The friction work z (area A-iAB'B-l, Fig. 83) is greater than the loss of kinetic energy (area A 1 BS'B 1 1 '). The reason for this lies in the fact that part of the heat Az entering the moving fluid is capable of being transformed back into mechanical energy. As shown or in Chapter IY, the loss of available energy, represented by area A^B'B^, is the increase of entropy multiplied by the lower temperature. The triangular area ABB' represents, therefore, the part of the friction work that is recovered. 4. The most convenient graphical representation for practi- cal purposes is obtained by taking the heat content i and entropy s as coordinates. On this is- plane a series of constant pres- sure lines are drawn, Fig. 84; then a vertical segment AB represents a Motionless adia- batic change from pressure p l to a lower pressure jp, while a curve AB' between the same pressure limits represents an expansion with increasing en- tropy, that is, one with fric- tion. The segment AB, there- of 2 w, z ..-i , without o' FIG. 84. fore, represents the increase of jet energy friction, the segment AG-, the smaller increase with which the pressure is p^ through an orifice or short tube, Jbig. 85, into a region in which exists a pressure p 2 lower than p v If we take the section F l in the reservoir, the velocity w l will be small and may be assumed to be zero. The second section F will be taken at the end of the tube, and the pressure at this section will be denoted by p. Assuming the flow to be frictionless and adiabatic, we have, since w^ = 0, FIG. 85. The law of the expansion is given by the equation m V n = pyn C2^\ where for air n = k, while for saturated or superheated vapor it has a value depending on the conditions existing. In any case, n can be determined, at least approxi- mately. Making use of (2) to evaluate the definite integral of (1), we get n _j w n - p\ I ' p l j J If F denotes the area of the orifice or tube, and M the weight of fluid discharged per second, the law of continuity is expressed by the equation Mo = Fw, (4) whence eliminating w between (3) and (4), we obtain 71-1 From (2), we have w r / n \ n n -\ 1 - (^ _l ^ \p^J J (5) \P which substituted in (5) gives ART. 153] SAINT VENANT'S HYPOTHESIS 253 If now various values be assigned to the lower pressure p and the values of w and M be found from (3) and (6), respec- tively, the relations be- tween p, w, and M will be as shown in Fig. 86. The *^11P!WS\ ^^I^P^ initial pressure p 1 is rep resented by the ordinate n OQ-, the lower pressure p by the ordinate OH, and the curve AS represents the change of state of the moving fluid starting from the initial state A. The shaded area G-ABff rep- * FlG - 86< /PI resents the integral J vdp and, therefore, the kinetic energy w of the jet at the section J?. The abscissa HE represents *ff the velocity w found from the equation w = V2 g x area &ABH (in ft. lb.), while the abscissa HD represents to some chosen scale the weight of fluid discharged per second, as found from (4) or directly from (6). Inspection of (6) shows that the discharge M reduces to zero when p=p-^ and also when p = 0. It fol- lows that the curve CrDO must have the general form shown in the figure and that the discharge JjTinust have a maximum value for some value of p between p = and p p z . Let this value of p be denoted by p m . Evidently from (6), M is a maximum when 2 nl is a maximum. Placing the first derivative of this expression r> This ratio is called the critical ratio, and^ is called the critical value of the lower pressure p. For air, taking n = Tc 1.4, this ratio is 0.5283 or approximately 0.53 ; for saturated or slightly wet steam, taking n= 1.135, the ratio is 0.5744. The question now arises as to the relation between the pres- sure p in the jet at section F and the pressure p z of the region into which the jet discharges. If it be assumed that p and p 2 are always equal, then p = when p z = 0, and from (6) M = 0. This can only mean that no fluid can be discharged into a perfect vacuum, a result manifestly absurd. It follows that under certain conditions, p must be different from p z . Saint Venant o.i 0.3 0.5 FIG. 87. 0.9 and Wantzel, to whom equations (3) and (6) are due, asserted that the discharge into a vacuum must be a maximum and advanced the hypothesis that for all values of p% lower than the critical pressure p m the discharge is the same. We have, there- fore, two distinct cases : (1) If p% is greater than p m , the pres- sure p in the jet takes the value p 2 , and w and M are found from. (3) and (6), respectively. (2) If p 2 is equal to or less than p m the pressure p assumes the constant value p m given by (7), and the velocity and discharge remain the same for all values of p z between p 2 =p m and p z = 0. The hypothesis of Saint Venant has been fully confirmed by the experiments of Fleigner, Zeuner, and G-utermuth. Figure 87 shows the results of Gutermuth's experiments on the flow of steam through a short tube with rounded entrance, usinp- dif- f erent initial pressures p r In each case the discharge becomes constant when the lower pressure reaches a definite value p m . 154. Formulas for Discharge. Since for all values of p z less than p m the discharge remains constant and the pressure at the plane of the orifice or tube takes the value p m , we may obtain P the maximum velocity and discharge by substituting for in _n_ Pi (3) and (6) of Art. 153 the critical value ( - ^f 1 . The U- 4.' \+ V resulting equations are : and *_ ,-. (2) \n+\J * n + 1 V-L ^ ' These equations give w and Jf for p z <p m i if jp 2 >p m the ratio must be substituted for in the original equations. Pi Pi By easy transformations (1) and (2) may be given simpler forms. The following are some of the well-known formulas that have been thus derived. 1. Fliegners Equation for Air. From the general equation which applies to the air in the reservoir from which the flow proceeds, we have Substituting this expression for ^J in (2), and taking n*=lc, v i the result is Inserting the numerical values of k and B for air, we get in English units J!f= 0.53 ^^i=. (4) v y x'^vv VX ' JJX.U11JS [CHAP. Xlli This is the equation given by Fliegner as representing the re- suits of his experiments on the flow of air from a reservoir into the atmosphere. It holds good when the pressure in the reser- voir is greater than twice the pressure of the atmosphere When the pressure in the reservoir is less than twice the at- * f 110Wing em P irical equation is given by J. 2. Grashofs Equation for Steam. In formula (2), Pl and v refer to the fluid in the reservoir. If this fluidSs seated steam, then^ and v, are connected by an approximate relation in which for English units, m = 1.0631 and (7= 144 x 484 2 From (6) we readily obtain b4 ' 2 ' m+i Pl = Q~bn and substituting this in (2), the resulting equation is If now we take for steam the value n = 1.135, (7) reduces to the simple form ^ w auuces to 1 m ^=0.01911^0.97. In this formula, F is taken in square feet and in pound, npr square foot. When the area is taken in square^ iX a td the pressure in pounds per square inch, (8) becomes ^"=0.0165^0-97. This formula is applicable for values of p 9 below the criH^T backpressure^. ^nwoai 3. Rateau^K T^nvmiiJ^ r>_j. i TO-I modified the Grashof the results of his experiments : ABT. 155] ACOUSTIC VELOCITY 257 4. Napier's equations. The following simple, though some- what inaccurate, equations based upon the experiments of Mr. R. D. Napier, are due to Rankine. When the pressure in the reservoir exceeds of the back pressure when it is less than of the back pressure ~ 42 EXAMPLE. Find the discharge in pounds per minute of saturated steam at 100 Ib. pressure (absolute) through, an orifice having an area of 0.4 sq. in. The back pressure is less than the critical pressure, 57 Ib. per square inch. 1. By Grashof s formula M = 60 x 0.0165 x 0.4 x lOO -^ = 34.493 Ib. 2. By Kateau's formula 6 X X 10 (16-367 - 0.96 x 2) = 34.673 Ib. 3. By Napier's formula M = ' 4 * 10 x 60 = 34.286 Ib. 4. The discharge may be found from the two fundamental formulas to = VSflr/ (ij - f a ) = 223.7 V^ - i g , and M=^. v The critical pressure p m is 57.44 Ib. per square inch. From the steam table (or more conveniently, and with sufficient accuracy, from the is-chart) we find : ii (for 100 Ib.) = 1186.5 B. t. u. i m (for 57.44 Ib.) = 1142.7 B. t. u. x m = 0.964. t'm = arm Om" - v') + / = 7.07 cu. ft. Then w = 223.7 VI 186.5 - 1142.7 = 1480 ft. per second, 0.4 1480 144 7.07 and M = 60 x M x ^2 - 34.89 Ib. K tne ratio -? 01 tne specmc neats. JLJUBJU. c v in the medium is given by the relation w = -\Jgkpv. (See any textbook in Physics). (1) If we denote by p m the critical back pressure, we have k (2) p m __f 2 V-i which combined with the adiabatic equation a- ft VW* C8) gives w / 2 \*=I ^1=(__) . (4) Vm \k + 1J ^ J Combining (2) and (4), we have W," 2 The velocity through the orifice is and by the use of (5) this becomes w = -Vgkp m v m . (6) Comparing (6) with (1), it appears that the maximum velocity of flow from a short convergent tube is the same as the velocity of sound in the fluid in the state it has at the critical section. This result is due to Holtzmann (1861). 156. The de Laval Nozzle. The character of the flow through a simple orifice depends largely upon the pressure jt? 2 in the region into which the jet passes. There are two cases to be discussed : 1. When^? 2 is equal to or greater than the critical pressure p m given by the ratio p m _f 2 Y& Pi \k -I- 1 2, When p 2 is less than p m . In the first case the pressure a f as wo have seen, takes' and therefore, t he jet the for Fig. 88, jet, the , ,, constant cross section. Furthermore 8 . e> -. dAio u. remains practically con- stant at successive cross sections. Ihis velocity is gi veu by m Art. 151. W ' In the second case the pres- sure at section a takes the critical v is greater than the pressure of the sun As a result of the pressure difference - jet will expand laterally, as shown Fig furthermore, along the axis of the jet the the 89. i successive sections are passed. itial velocity at section a is FIG. 90. at is, the acoustic velocity. The lateral spreading of the jet may prevented by adding to the orifice )roperly proportional tube, as shown Fig. 90. The orifice and tube to- ^er constitute a de Laval nozzle. Pressurr/ 6XPanSi0n f the pressuie from^ at section to ^ 2 at section 3. The area tne end section 5 depends upon the final pressure Pv At aon a the jet has the acoustic velocity w 3 as if the added The tube must diverge its velocity increases and at the end section b takes the value w 2 given by the relation 2 W. T/ N x-i N - = J(l m -^ z ). (1) The general character of the flow through the de Laval nozzle may be seen from the following analysis. Assuming frictionless adiabatic flow, the fundamental equa- tions (6) and (7), Art. 150, become, respectively, du+pdv = Q, (2) wdw j ^ON . - = vdp. (3) & We have also the equation of continuity Fw = Mv, (4) from which by differentiation we obtain dw^ dF__dv _ + IT- . W For perfect gases, pv ni -f _, % ~A;-1' while for superheated or saturated vapors, nl Therefore, (2) becomes or Jcpdv + vdp = 0, (6) , dv dp whence ___.. v Icp Combining this relation with (5), we obtain dw dF dp __ ^r + T + ^- u - w Now from (3), W hence (7) becomes p w/ F By introducing the equation for the acoustic velocity w? = kffpv, (8) may be readily reduced to the form ld p== kw 2 I dF pdx~~ w 2 w 2 F Ax (8) (9) (10) The variable x may be used to denote the distance of a nozzle section from some fixed origin, Fig. 90. For vapors, k may be replaced by n. The nozzle has two distinct parts: the rounded orifice ex- tending from to A, Fig. 91, and the diverging tube extend- ing from A to B. As the cross sections decrease in area from to A, the deriva- tive is negative for this dx part ; for the diverging part (JF from A to J5, is positive; dx for the throat A it has the value zero. The pressure drops continuously from to B as shown by the curve _: , dp . FIG. 91. of pressure ; hence - is negative throughout. Referring to (10) we have the following results : For orifice OA, is ; -- is ; dx dx For tube AB, ~ is + ; ^ is - ; dx dx kw z is ; w < iv a . is -f \ w>w a . rl V For throat A, ~ = ; kw 2 -= co ; w ==' Hence the velocity steadily increases until at the throat it attains the acoustic velocity; then in the diverging tube it further increases. Inspection of (10) shows that divergence is necessary if the velocity w is to exceed the acoustic velocity w s . 157. Friction in nozzles. In the case of flow through a simple orifice or through a short convergent tube with rounded en- trance, the friction between the jet and orifice, or tube, is small and scarcely demands attention. With the divergent de Laval nozzle, on the contrary, the friction may be considerable and must be taken into account. As explained in Art. 152, the *iJ) effect of friction is to produce a decrease in the jet energy ^9 at the end section. Referring back to Fig. 83, suppose A to denote the initial state of the fluid entering the nozzle, B' the final state at exit, and B the final state that would have been attained with frictionless flow ; then the area A-jBB'B^' repre- sents the increase in the final heat content i z due to friction and it likewise represents the decrease in jet energy at exit. Let ij, z 2 , and i z ' denote, respectively, the heat content of the fluid in the states represented by the points A, B, and B'. Without friction, we have 2 y*2 .... while with friction The loss of kinetic energy due to friction is, therefore, It is customary to take as a friction coefficient the ratio of the loss of energy to the kinetic energy without friction. Denoting this ratio by y we have, therefore, whence ' -- (2) The experiments that have been made on the flow of steam through nozzles indicate that the value of y may lie between 0.08 and 0.20. EXAMPLE. Steam in the dry saturated state flows from a boiler in -which the pressure is 120 Ib. per square inch absolute into a turbine cell in which the pressure is 35 Ib. absolute. A de Laval nozzle is used, and the value of y is 0.12. Find the velocity of the jet, and the loss of kinetic energy ; also the final quality of the steam. For the given initial state, i 1190.1 B. t. u. At the end of adiabatic ex- pansion to the lower pressure, xz is found to be 0.925, and i a is found to be 1095.8 B. t. u. The exit velocity on the assumption of frictionless flow is, therefore, w = 223.7 V1190.1 - 1095.8 = 2172 ft. per second, while the actual velocity is w' = 223.7 V(l - 0.12) (1190.1 - 1095.8) = 2038 ft. per second. The loss of kinetic energy is, 0.12 x 778 x 94.3 = 8804 ft.-lb., or in B. t. u., 0.12 x 94.3 = ll.SB.t.u. This heat is represented by the rectangle A\BB'Bi!, Fig. 83. Hence, for the quality x>z in the actual final condition J3', we have x j - X2 = ?/0'i - *'Q = HA = 0>012 r z 938.4 and, therefore, xj = 0.925 + 0.012 = 0.937. The effects of friction are : (1) to decrease the velocity of flow at a given section ; (2) to increase the specific volume v of the fluid passing the section. The latter effect is seen in the case of steam in the increased quality or increased degree of superheat due to the heat generated through friction reenter- ing the moving fluid. From the equation of continuity F=M~, (3) w it appears that the effect of friction is to increase the numera- tor v and decrease the denominator w of the fraction of the second member ; hence for a given discharge M> the cross sec- tion F must be larger the greater the friction, that is, for the same lower pressure p 2 . The effect of friction may be viewed from another aspect. In Fig. 92, let the curve OMAE represent the pressures along the axis of a de Laval nozzle on the assumption of no friction. This curve is readily found for a given value of p 1 by finding for various lower pressures^? -%_ J?> the proper cross section F by means of the two equations, T Mo FIG. 92. Ag w Let A be a point on the pres- sure curve obtained in this manner. If- now friction is taken into account, the sec- tion I" associated with the lower pressure p has a larger area than the section F calcu- lated on the assumption of no friction ; therefore, the point A is shifted by friction to a new position A' underneath the new section F' . Similarly the end section F e must be increased in area to JFJ, and the point E on the frictionless pressure curve is shifted to a new position U 1 . The effect of friction, therefore, is to raise the pressure curve as a whole, that is, to increase the pressure at any point in the axis of the nozzle. 158. Design of Nozzles. The data required in the design of a nozzle are the initial and final pressures, the weight of steam that must be delivered per hour or per minute, and the coef- ficient y. Two cross section areas must be calculated, that at the throat, and that at the end of the nozzle. The following example illustrates the method. EXAMPLE. Kequired the dimension of a nozzle to deliver 450 Ib. of steam per hour, initially dry and saturated, with an initial pressure of 175 TU nVici/^1 4-n n-nA -fi v n 1 -rvwcic< n l*k /-x-p 1ft 1V\ Q V\D<-*1 Tt + l T" !*- II f} 1Q The critical pressure in the throat is 175 x 0.57 = 100 Ib. approx. Then r frictionless adiabatic flow t, = 1196.4 B. t. Ti., i m (at throat) = q m ' + x m r m = 298.1 + 0.962 x 888.4 = 1152.9, a = ^2' + Wz = 181.1 + 0.863 x 909.7 = 1017.5, ^ - i m - 43.5 ; i, - 12 = 178.9. nee the throat is near the entrance, the effect of friction between entrance id throat is practically negligible ; hence the velocity at the throat is w m = 223.7 V41T5 = 1475 ft. per second, iking account of the loss of energy (37 = 0.13), the velocity of exit is w 2 = 223.7 V0.87 x 178.9 = 2791 ft. per second. ae quality of steam at the throat was found to be 0.962, and that at exit, ithout friction, 0.863. Because of friction, the quality at exit is increased ' the amount 178.9 x 0.13 * 969.7 = 0.024, thus giving a final quality 363 + 0.024 = 0.887. Neglecting the volume v' of a unit weight of water, ice x is large, the specific volumes at throat and exit are respectively 4.42 x 0.962 = 4.252 cu. ft. :d 26.23 x 0.887 = 23.26 cu. ft. om the equation of continuity Fw = Mv, we have, since M = 45 = 0.125 Ib. per second, 60 x 60 i ' F m = - 125 >< 4.252 _ aoo036 sq> ftf = 0.0519 sq. in. the area of the cross section at the throat. The area at exit is F z = 0-125 x 23.26 _ 0>001042 sq . f t- _ alg sq> in> w i J J. the cross section of the nozzle is made circular, the diameters at throat d exit are respectively d m = 0.251 in., d a = 0.437 in. ; d taking the taper of the nozzle as 1 to 10, the length, of the conical part 10(0.437 - 0.257) = 1.8 in. EXERCISES 1. .Find the weight of air discharged per minute through an orifice inch in diameter from a reservoir in which the pressure is maintained at orifice having an area of 0.4 sq. in. into a region in which the pressure is 55 Ib. per square inch. Find (a) the velocity; (b) the weight discharged per minute. Compare the results obtained by using Grashof's, Napier's, and Rateau's formulas, respectively. 3. If in Ex. 2 the back pressure is 80 Ib. per square inch, what in the weight discharged? Assume the steam to be initially dry and saturated. 4. If for superheated steam the exponent n in the adiabatic equation Pm pv n = const, is taken as 1.30, find the critical ratio 5. A de Laval nozzle is required to deliver 080 Ib. of steam per hour. The steam is initially dry and saturated at a pressure of 110 Ib. per square inch and the final pressure is 8 in. of mercury. Find the necessary areas of the throat section and end section of the nozzle, assuming frictionless ilow. 6. In Ex. 5 find the areas of the two sections when the loss of kinetic energy is 0.15 of the available energy. 7. Find the area of an orifice that will discharge 1000 Ib. of dry steam per hour, the initial pressure being 150 Ib. per square inch and the back pressure 105 Ib. per square inch. 8. In an injector, steam flows through a diverging nozzle into a combin- ing chamber in which a partial vacuum is maintained, due to the condensa- tion of the steam in a jet of water. If the initial pressure is 80 Ib. per square inch and the pressure in the combining chamber is 8 Ib. per square inch, find the velocity of the steam jet. Assume y = 0.08. 9. Steam at 160 Ib. pressure superheated 100 flows through a nozzle into a turbine cell in which the pressure is 70 Ib per .square inch. Find the area of the throat of the nozzle for a discharge of 36 Ib. per minute. 10. Let steam at 1GO Ib. pressure, superheated 100, expand adiabatically without friction. Take values of the back pressure p% as abscissas, and plot curves showing (a) the available drop in heat content i\ i ; (?) the veloc- ity of the jet ; (c) the area of cross section required for a discharge of one pound per second. SUGGESTION. Find i z for the following pressures: 140, 120, 100, 80, 60, 40, 20, 10, 5 Ib. per square inch. Then find w from the formula V} 223.7 Vt'i - i z , and the cross section from the equation of continuity. 11. Steam at 160 Ib. pressure superheated 100 is discharged into a region in which the pressure is p through an orifice having an area of 0.25 sq. in. Take the values of p 2 given in Ex. 10 and plot a curve showing the weight discharged for different values of p. 12. Show that if the loss of kinetic energy is y per cent of the available energy, the decrease in the velocity of the jet is approximately \y per cent of the ideal velocity. Zeuner : Technical Thermodynamics 1, 225 ; 2, 153. Lorenz : Teclmische Warmelehre, 99, 122. Weyrauch : Grundriss der Warme-Theorie 2, 303. Peabody: Thermodynamics, 5th ed., 423. Stodola : Steam Turbines, 4, 45. Rateau : Flow of Steam through Nozzles. ORIGINAL PAPERS GIVING EXPERIMENTAL RESULTS OR DISCUSSIONS Weisbach : Civilingeineur 12, 1, 77. 1866. Eliegner : Civilmgenieur 20, 13 (1874) ; 23, 443 (1877). De Saint Venant and Wantzel : Journal de I'E'cole polytechnique 16. 183 Comptes rendus 8, 294 (1839) ; 17, 140 (1843) ; 21, 366 (1845). Gutermuth : Zeit. des Verein. deutsch. Ing. 48, 75. 1904. Emden : Wied. Annallen 69, 433. 1899. Lorenz : Zeit. des Verein. deutsch. Ing. 47, 1600. 1903. Prandtl and Proell : Zeit. des Verein. deutsch. Ing. 48, 348. 1904. Biichner : Zeit. des Verein. deutsch. Ing. 49, 1024. 1904. Rateau: Ann ales des Mines, 1. 1902. Rosenhain : Proc. Inst. C. E. 140. 1899. Wilson : London. Engineering 13. 1872. CHAPTER XIV THROTTLING PROCESSES 159. Wiredrawing. The flow of a fluid from a region of higher pressure into a region of much lower pressure through a valve or constricted passage gives rise to the phenomenon known as wiredrawing or throttling. ' Examples are seen in the passage of steam through pressure-reducing valves, in the throttling calorimeter, in the passage of ammonia through the expansion valve in a refrigerating machine, and in the flow through ports and valves in the ordinary ' steam engine. Wire-drawing is FIG. 93. . , & . ., . & evidently an irreversible process, and as such, is always accompanied by a loss of available energy. The fluid in the region of higher pressure is moving with a velocity w^ Fig. 93. As it passes through the orifice into the region of lower pressure jt? 2 , the velocity increases to w 2 ac- cording to the general equation for flow, viz : !lfi!..JXh-S>- (1) The increased velocity is not maintained, however, because the energy of the jet is dissipated as the fluid passing through the orifice enters and mixes with the fluid in the second region. 9/J 2 ___ nn 2 Eddies are produced, and the increase of energy ^ - is re- turned to the fluid in the form of heat generated through in- ternal friction. Utimately, the velocity w z is sensibly equal to the original velocity w 1 ; therefore from (1), we obtain L = L, (2) ART. 160] LOSS DUE TO THROTTLING 269 as the general equation for a wiredrawing process. The initial and final points lie, therefore, on a curve of constant heat content. 160. Loss due to Throttling. Let steam in the initial state denoted by point A, Fig. 94, be throttled to a lower pressure, the final state being denoted by point B on the constant-z curve AB. Also let TQ denote the lowest available temperature. The increase of entropy during the change AB is represented by A-iB-p and this increase multi- plied by the lowest available temperature 2* gives the loss of available energy. Evidently this loss is represented by the area EXAMPLE. In a steam engine the pressure is reduced by a throttling "valve from 160 Ib. per square inch to 90 Ib. per square inch absolute. The initial quality is x 0.99 and the absolute back pressure is 4 in. of mercury Required the loss of available energy per pound of steam. From the steam table the initial heat content is 1187.2 B. t. u. At a pres- sure of 90 Ib. the heat content of saturated steam is 1184.5 B. t. u., therefor* in the second state the steam is superheated. As the degree of superheat ii evidently small, it may be determined -with sufficient accuracy from th( curves of mean specific heat. At a pressure of 90 Ib. the mean specific heai near saturation is 0.55 ; hence the superheat is 1187.2 - 1184.5 0.55 = 5, nearly The entropy in the second state is the sum of the entropy at saturation 1.6107 for a pressure of 90 Ib., and the entropy due to superheat, "which is approximately. 0.55 log e LJ? = 0.55 log, f = 0.0035. Hence, .93 = 1.6107 + 0.0035 = 1.6142. The entropy in the initial state replaced by a corresponding reversible change with the condi- tion that the heat content i remains constant. The general equation di = Tds + Avdp, then becomes, = Tds + Avdp, and approximately we have, therefore, A S = -^, (1) in which As is the increase of entropy corresponding to the change of pressure Ap. Since Ajp is intrinsically negative, it follows that As must be positive. Equation (1) may be written in the more convenient form For perfect gases (2) reduces to the simple form For steam having the quality x, we have i} = x(v" v'~) -f t/, and Apv Apx(v" v'~) -f Apv' ; or neglecting the small specific volume v r of the water, Apv = x-fy. Eq. (2) therefore takes the form Aff- (4) Mean values for p, T, and -^ should be taken. EXAMPLE. If in passing into the engine cylinder the pressure of steam is reduced by wii'edrawing from 125 Ib. to 120 Ib. per square inch, what is the loss of available energy ? The initial value of x is 0.98 and the pressure at exhaust is 16 Ib. per square inch. Taking the two pressures 125 and 120, the following mean values are fouud from the table : p = 122.5, T = 802.4, if, = 82.5. Also, A p = - 5- Hence, A 5 = x = 0.00398. For T we take the temperature corresponding to the 16 Ib., namely, 675.9. Therefore the loss of available energy is 6.75.9 x 0.00398 = 2.7 B. t. u. approx. 161. The Throttling Calorimeter. A valuable application of the throttling process is seen in the calorimeter devised by Professor Peabody for determining the quality of steam. In the operation of the calorimeter steam from the main is led into a small vessel in which the pressure is maintained at a value slightly above atmospheric pressure. The steam is thus wiredrawn in passing through the valve in the pipe that con- nects the main and the vessel. For successful operation the amount of moisture in the steam must be small so that, as the result of throttling, the steam in the vessel is superheated. In Fig. 94, let point A represent the state of the steam in the main and point B the observed state of the steam, in the calorimeter ; then i A = i s . (1) But i A = z y + xr v (2) where ij and r-^ refer to the pressure p 1 in the main ; and ijB = h" + c P (t' Z '~t z ), (3) where t 2 f is the observed temperature of the steam in the calorimeter, t z is the saturation temperature corresponding to the pressure p z in the calorimeter, i z " is the saturation heat content corresponding to the pressure p z , and c p is the mean specific heat of superheated steam for the temperature range t 2 ' 2 . Combining the preceding equations, we obtain EXAMPLE. The initial pressure of the steam is 140 Ib. per square inch, the observed pressure in the calorimeter 17 Ib. per square inch, and the temperature in the calorimeter 258 F. Required the initial quality. The temperature of saturated steam at 17 Ib. pressure is 219.4 F. ; hence the steam in the calorimeter is superheated 258 219.4 = 38.6. From the curves of mean specific heat the value 0.477 is found for the pressure 17 Ib. and the degree of surperheat in question ; and from the steam table we have t," = 1153, ijf = 324.2, n = 869. Hence, 1153 + 0.477 x 38.6 - 324.2 n Q7 - x = __ _ 0.975. The Mollier chart, Fig. 75, may be used conveniently in the solution of problems that involve the throttling of steam. Since the heat content remains constant during a throttling process, the points representing the initial and final states lie on the same horizontal line. In the preceding example the final point is located from the observed superheat 38.6 and the observed pressure 17 Ib. in the calorimeter. A horizontal line drawn through this point intersects the constant pressure line p = 140 Ib., and from this point of intersection the quality x = 0.975 is read directly. 162. The Expansion Valve. In the compression refrigerat- ing machine the working fluid after compression is condensed and the liquid under the higher pres- sure p is permitted to flow through the so-called expansion valve into coils in which exists a much lower pressure p v Let point A, Fig. 95, on the liquid curve represent the initial state of the liquid. The point that represents the final state must lie at the intersection of a constant -i curve through A and -S line of constant pressure p z . Evidently we have and i b = i 2 ' + # 2 r 2 , where x% denotes the quality of the mixture in the final state. Therefore r _ t , 1 2 ^ t 2'2' l-i la or (2) The increase of entropy (represented by s (3) and the loss of refrigerating effect due to the expansion valve, which is represented by the area A t G-BB V is -h'-h'-W -,')- (4) The following equalities between the areas of Fig. 95 are evident : area ^ = area area F6rA = area 163. Throttling Curves. If steam initially dry and saturated be wiredrawn by passing it through a small orifice into a region of lower pressure, then, as has been shown, it will be super- heated in its final state. If the lower pressure p z is varied, the tempera- 400, 350 soo S50 200 50 100 150 Pressure, Ib. per sq. in. FIG. 96. ture t z will also vary, and the successive values of p% and t 2 will be rep- resented by a series of points lying on a curve. By taking various initial pressures a series of such curves may be obtained. Sets of throttling curves for water vapor have been obtained by Grind- ley, Griessmann, Peake, and Dodge. The curves deduced from Peake's experiments are shown in Fig. 96. Abscissas represent pressures, ordinates, temperatures. The curve from which the throttling curves start is the curve t=f(p~) that represents the relation between the pressure and temperature of saturated steam. It was the original purpose of Grindley, Griessmann, and Peake to make use of the throttling curves in finding the specific heat of superheated steam. The theory upon which this determination rests is simple. From Eq. (4), Art. 161, we readily obtain . / . ,/ *i ~r ^i^i ? 2 /-IN The temperature difference t z ' 2 for any lower pressure p z is the vertical segment between the throttling curve and the satu- ration curve and is given directly by the experiment. Hence if the initial quality x is known, and if i^' and z* 2 " are accurately given by the steam tables, the mean value of c p is readily cal- culated. The results obtained were, however, discordant and of no value. The form of Eq. (1) is such that a slight error in any of the terms of the numerator of the fraction produces a large error in the calculated value of c p . The impossibility of deriving consistent values of o p by the method just described led to the belief that Regnault's formula for the total heat of saturated steam, hitherto regarded as authoritative, must be incorrect. The experiments of Kno- blauch and Jakob on the specific heat having appeared, Dr. H. N. Davis of Harvard University discerned the possibility of reversing the method and deriving by it a new formula for total heat. 164. The Davis Formula for Heat Content. The method employed by Dr. Davis in deriving from the throttling curves a formula for the heat content of steam may be described as follows : Let AD, Fig. 97, be one of the series of throttling curves, and AD' the saturation curve. The heat content is constant along the throttling curve, that is p FIG. 97. Let p 2 be the lower pressure cor- responding to the points B, J5', and let A* denote the temperature difference indicated by the segment B'B. If the steam were made to pass from the satura- tion state B' to the superheated state B at the constant pres- sure > 2 , the heat absorbed during the process would be c p Ai, c p denoting the mean specific heat between B' and B. It follows that *J3 IB> ~ C P A, that is, IA ia, = c- A. ART. 165] THE JOULE-THOMSON EFFECT 275 In a similar manner the differences i A i cr) i A i Dl , etc. are ob- tained. The result is a relation between the heat content of saturated steam at the original pressure p l (state A) and the values of the saturation heat content for various lower pressures. The temperatures corresponding .to these pressures are now laid off on an arbitrarily chosen line MN, Fig. 98, and from the points A, J5', (7', etc., the segments etc. are laid off. A curve through the points A, B n ', <7 ; , o [ D", etc. is an isolated segment of * 10 ' y8 ' the curve giving the relation between the heat content i and the temperature t. Necessarily only relative values are thus obtained. From the individual throttling curves Dr. Davis thus obtained twenty-four overlapping segments of the it-curve, and by properly coordinating these segments he obtained finally a smooth curve covering the range 212 to 400 F. The curve was found to be represented by the second degree equation i=a + 0.3745(^-212) -0.00055 (-212) 2 ; and from the experiments of Henning and Joly on the latent heat of steam at 212 F., the value of the constant a was found to be 1150.4. 165. The Joule-Thomson Effect. The classical porous plug experiments of Joule and Lord Kelvin were undertaken for the purpose of estimating the deviation of certain actual gases from the ideal perfect gas. The gases tested were forced through a porous plug and the temperatures on the two sides of the plug were accurately determined. In the case of hydrogen the tem- perature after passing through the plug was slightly higher than on the high pressure side ; air, nitrogen, oxygen, and car- bon dioxide showed a drop of temperature. 276 THROTTLING PROCESSES [CHAP, xiv For an ideal perfect gas, MJ = Jc^jf. U p and pv = BT' } hence, (Jc v + ) ^ = (Jb, + JB) !F a or ^1=^2- It follows that a perfect gas would show no change of tempera- ture in passing through the plug, and that the change of temper- ature observed in the actual gas is, in a way, a measure of the degree of imperfection of the gas. The results of the experi- ments have been used to reduce the temperature scale of the air thermometer to the Kelvin absolute scale. The ratio of the observed drop in temperature to the drop in pressure, that is, the ratio , is called the Joule-Thomson coefficient and is denoted by /*. According to the experiments of Joule and Kelvin //, varies inversely as the square of the absolute temperature. That is, It may be assumed that this relation holds good for air, nitro- gen, and other so-called permanent gases within the region of ordinary observation and experiment. At very low tempera- tures it seems probable that p varies with the pressure as well as with the temperature. An expression for p in the case of superheated steam can readily be derived from the formula for the heat content, namely: Since i is constant in a throttling process, we may define the Joule-Thomson coefficient more accurately as the derivative fdT\ { ] . From calculus, we have ART. 166] EQUATION OF THE PERMANENT GASES and from the definition of the heat content i, 277 Hence dT *' dp ji e f or The following table contains values of Eq. (2). (2) calculated from PUESSTTKE Lit. PER Sij. IN. 250 F. 300 850 400 450 500 550 000 15 0.668 0.492 0.369 0.282 0.220 0.176 0.143 0.119 100 0.327 0.261 0.208 0.169 0.140 0.118 300 0.191 0.162 0.138 0.118 It will be observed that the value of /j, varies with the pres- sure ; however, as the temperature rises, the influence of pressure decreases ; hence for gases far removed from the satu- ration limit, such as were used in the porous plug experiments, it seems probable that p is a function of the temperature only, as found by Joule and Kelvin. Dr. Davis has deduced from, the throttling experiments of Grindley, Griessmann, Peake, and Dodge values of p for super- heated steam.* These were found by direct measurement of the mean slopes of the throttling curves. The values thus obtained agree very closely with those calculated from (2) and shown in the preceding table. 166. Characteristic Equation of the Permanent Gases. From the cooling effect shown in the Joule-Thomson's experiments for all gases except hydrogen, it appears that those gases do not follow precisely the law expressed by the equation pv = BT. By making use of the relation /*. = ~ it is possible to derive a characteristic equation that represents more nearly the behavior of bince the heat content i is constant during a throttling process, the gen- eral equation di = c dT takes the form c <*T = A(T^-V\ 0) * dp V dT I Differentiating both members of (1) with respect to T, we obtain dT \ r ~dp 1 ~ y \dT dT 2 ~dT> = AT. (2) H HF$i But we have _ . dp I 2 and from the general thermodynamic relations, \ p T Substituting these expressions in (2), we obtain 3 whence This is a partial differential equation, the general solution of which is the equation c p = T*<t>(T*-3ap). (4) Here ^> denotes an arbitrary function which must be determined from physical considerations. Since at high temperatures c p for permanent gases is given by the linear relation c p = a + bT, we have from (4), whence (j>(T 3 ) = -J-+ j,, and Since the term 3 otp is small in comparison with the term T 3 , we have approximately ART. 166] EQUATION OF THE PERMANENT GASES 279 Introducing these expressions in (5) and substituting the resulting expres- sion for <f>(T 8 3 op) in (4), we obtain finally (6) It appears from (6) that the specific heat of the permanent gases varies with the pressure and temperature. At very high temperatures the term containing/) is small and the specific heat is given simply by a + bT; at low temperatures, however, this term becomes appreciable and the specific heat increases with the pressure. The specific heat curves have, therefore, somewhat the form shown in Fig. 71. From (6), we have by differentiation _ AT 9 '_ ~ _ /2a ,\ ~**A~r '' Integrating, we obtain (7) Vy Introducing in (1) the expression for -^ given by (7), we obtain after dJ- reduction To determine the function /O), we assume that the perfect gas equation pv = BT holds when T is very large. Hence f(p) = , and (8) becomes Since the last term in the bracket is very small, it may be neglected, and (9) may be written The equation given by Joule and Thomson, namely , (ID Joule-Thomson effect has been employed by Linde in a very ingenious machine for the liquefaction of gases. A diagram- matic sketch of the machine is shown in Fig. 99. Air (or any other gas that is to be liquefied) is com- pressed to a pressure of about 65 atmos- pheres and is dis- charged into a pipe leading to the cham- ber c. A current of cold water in the vessel b cools the air during its passage from the compressor to the receiving cham- ber. From c the air passes through a valve v into a vessel d, in which a pressure of about 22 atmospheres is maintained. As a result of the throttling the temperature of the air is lowered. Thus, if p 1 is the pressure in the chamber c and j9 2 the pressure in the vessel cZ, the drop in temperature is (1) The air now passes from vessel d at temperature t 8 into the space enclosing the chamber c and thence back to the compressor. In passing back, the air absorbs heat from the air in c, and the temperature rises from t B to the final temperature 4 , which is nearly the same as the initial temperature t v Due to this cooling, the air in c arrives at the valve v with a temperature t z , which becomes lower and lower as the process continues. As the temperature 2 sinks the temperature 3 also sinks, but as shown by (1), t% sinks more rapidly than t z . Ultimately, the value of t 3 drops below the critical temperature of the gas, FIG. 99. PRINCIPLES OF THERMODYNAMICS BY G. A. GOODENOUGH, M.E. PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY OF ILLINOIS SECOND EDITION, REVISED Griessmann : Zeit. des Verein. deutsch. Ing. 47, 1852, 1880. 1903. Grindley : Phil. Trans. 194 A, 1. 1900. Peake : Proc. Royal Society 76 A, 185. 1905. THE JOULE-THOMSON EFFECT Thomson and Joule : Phil. Trans. 143, 357 (1853) ; 144, 321 (1854) ; 152, 579 (1862). Natanson : Wied. Annallen 31, 502. 1887. Preston : Theory of Heat, 699. Bryan : Thermodynamics, 128. Lorenz : Technische Wiirmelehre, 273. Davis : Proc. Amer. Acad. 45, 243. CHARACTERISTIC EQUATION OF GASES Zeuner: Technical Thermodynamics 2, 313. Lorenz: Technische Warmelehre, 296. Plank : Physikalische Zeit. 11, 633. Bryan : Thermodynamics, 138. CHAPTER XY TECHNICAL APPLICATIONS, VAPOR MEDIA THE STEAM ENGINE 168. The Carnot Cycle for Saturated Vapors. Since the constant pressure line of a saturated vapor is also an iso- thermal, three of the processes of the Carnot cycle are ap- proximately attainable in a vapor motor, namely: isothermal expansion, adiabatic expansion, and isothermal compression. The adiabatic compression might also be accomplished by a proper arrangement of the organs of the motor, though in practice this is never attempted. Hence, the Carnot cycle is D \ -s o FIG. 100. FIG. 101. discussed in connection with vapor motors merely for the pur- pose of furnishing an ideal standard by which to compare the cycles actually used. The Carnot cycle on the T/S-plane and p F-planes, respec- tively, is shown in Fig. 100 and 101. The isothermal expan- sion AB occurs in the boiler, the adiabatic expansion BO in the engine cylinder, the isothermal compression CD in the con- denser. To effect the adiabatic compression DA, the mixture of liquid and vapor in the state D would have to be compressed aaiabaticaily in a separate cylinder ana delivered to tne Doiier in the state represented by point A. The heat received from the boiler per unit weight of fluid is 2i = r i( x t> x <d ( area A-^B-^i) 00 that rejected to the condenser is fe = r t (x a - x d } ; (area S 1 QDA^) (2) and the heat transformed into work, represented by the cycle area, is AW=q,- & = 5^^ - xj. (3) *! The efficiency is and the weight of fluid used per horsepower-hour is 2546 2546 g, ?i-?. nc^-^yj-r,- v - 1 If point J. lies on the liquid line s' and point on the satu- ration curve s", then x a =0, # 6 = 1, and (3) and (5) become, respectively, 2546 IV _ EXAMPLE. Let the upper and lower pressures "be respectively 125 Ib. per square inch absolute, and 4 in. of mercury. Then from the steam table TI = 804, T z = 585.1, n = 875.8 B.tu. From (4), the efficiency is 804 - 585.1 804 = 0.272. The heat transformed into work is 875.8x0.272 = 238.2 B.t.u., and the 2546 238.2 steam consumption is ' = 10.7 Ib. per h. p.-hour. r 9:-*R.9 - 1 r 169. The Rankine Cycle. In the actual engine the iso- thermal compression is continued until the vapor is entirely condensed, thus locating the point D on the liquid curve s', Fig. 102, The liquid is then forced into the boiler by a pump and is there heated to the boiline 1 temnerature -. This heat- JLU UO clSBUJUULCU. UHit U liUt! ! IJLttB LLU CM.BiUilLl.UO, liilt) V V - diagram necessarily takes the form shown in Fig. 103. D O D l B\ FIG. 102. FIG. 103. The heat supplied from the boiler per pound of steam is in this case (1) (2) (3) and the heat rejected to the condenser is Hence, the heat transformed into work is and the efficiency of the cycle is 77 = gi (4) It is evident that this efficiency is less than that of the Carnot cycle. The steam, consumption per h. p. -hour is ,r 2546 2546 (5) (6) EXAMPLE. Using the data of the example of the last article, determine the efficiency and steam consumption of an engine running in a Rankine cycle with dry steam. The quality at point C is determined from the relation end of adiabatic expansion is _ 0.4957 - 0.1739 + 1.0893 _ *' ~ L7497 ~ The available heat is 315.2 - 93.4 + 875.8 - 0.806 x 1023.7 = 272.4 B. t. u. ; while the heat supplied in the boiler is 315.2 - 93.4 + 875.8 = 1097.6 B.t. u. Hence the efficiency is = i = 0.248, 7 1097.6 which may be compared with the efficiency 0.272 of the Carnot cycle under similar conditions. The steam consumption is 2o46 272.4 170. Rankine's Cycle with Superheated Steam. If super- heated steam is used, the Rankine cycle has the form shown in Fig. 104. The heat q l supplied from the source is increased by the heat represented by the area B^BEEy which comes from the superheater; and the heat avail- able for transformation into work is increased by the amount repre- sented by the area FBEO. Evi- dently the efficiency of the ideal cycle is increased by the use of superheated steam, but the in- ~ s crease is small. The advantage of FIG. 104. superheated steam lies in another direction. If T e denote the temperature of the superheated steam (i.e. at point E\ the heat required for the superheating process BE is i o p dT where c p is the specific heat of superheated steam. for c p given by Eq. (9), Art. 133. Then the heat represented by the area D^DA'BEE^ is given by the expression 2i = ft'-fc / + ''i+Vz r . (1) m However, as has been shown, the sum 5'/+r 1 + ( c p dT is practically equal to the heat content of the steam in the state E. Hence we may write n ,* i /-n\ zl ( J2 \^J and calculate i e from the general formula (5) Art. 135. If the point O at the end of adiabatic expansion lies in the saturated region, as is usually the case, we have, as in the first case, g<2 = r z x c . The heat transformed into work is, therefore, and the efficiency is '? = l-T^ r - (4) The value of x c is determined from the relation where s is the entropy for the state J, and is calculated from the general equation (3), Art. 137. EXAMPLE. Find the efficiency of the Ranldne cycle, using the data of the previous examples, but assuming the steam to be superheated to 1000 absolute. From (6), Art. 135, the heat content of the superheated steam is i = 1000(0.367 + 0.00005 x 1000) - 125(1 + 0.0003 x 125) -- -0.0163 x 125 + 886.7 = 1294.8 B. t. u. ; and from (4), Art. 137, the entropy is s = 0.8451 log 1000 + 0.0001 x 1000 - 0.2542 log 125 - 125(1 + 0.0003 x 125) r~ - 396i = L7002 - Hence -= 1.7497 Heat supplied = i - q z > = 1294.8 - 93.4 = 1201.4 B. t. u. Heat rejected = r z x c = 1023.7 x 0.872 = 892.7 B. t. u. Available heat = 1201.4 - 892.7 = 308.7 B. t. u. Efficiency = ^=0.257. Steam consumption = = 8.25 Ib. per h. p.-hour. 308.7 171. Incomplete Expansion. Because of the very large specific volume of saturated steam at low pressures, it is usu- ally impracticable to continue the adiabatic expansion down to the lower pressure p z . The exhaust valve opens and re- leases the steam at a pressure somewhat higher than p 2 . The passage of the steam from the cylinder is an irreversible pro- cess in the nature of a free expansion and is indicated on the pF-diagram by the drop in pressure EF (Fig. 106). The O D, F, B i FIG. 105. F FIG. 106. actual process may be replaced by an assumed reversible pro- cess, cooling at constant volume. On the 5W-diagrain the cooling is represented by a constant volume line EF (Fig. 105) drawn as described in Art. 125. Evidently this " cutting off the toe " of the diagram results in a decrease in the ideal efficiency, but it is justified by the smaller cylinder volume required (JDF instead of DC*) and by other considerations. Denoting by p 3 the pressure at E, the end of adiabatic expansion, we have: *' 1' Heat rejected by medium Heat transformed into work Si - ft = ?i' + Vi ~ (&' + ^s) - s/fo ~ Pa) The qualities x e and x f are found from the equations and (2) (3) (4) If the steam is admitted throughout the stroke without cut- off, the adiabatic expansion is lacking, and the diagram takes the form ABGrJ) (Figs. 105 and 106). The equations for this case are readily derived from the preceding equations by 172. Effect of changing the Limiting Pressures. If the upper pressure p 1 be raised to p-^ while the lower pressure p z is kept the same, the effect is to increase both q v the heat absorbed, and q q z , the available heat, by an amount represented by the area AAIB'B (Fig. 107). Evidently the ideal efficiency is thus in- creased. If p z be lowered to p%, keeping p l the same, q z is decreased and q 1 q% increased without any change in q r For the ideal Rankine cycle the increase of avail- able heat would be that represented FIG. 107. by the area D'DCC'. For the modified cycle with incomplete expansion, however, the in- crease is represented by the relatively small area We may draw the conclusion that in the actual steam engine the limitation imposed by the cylinder volume prevents us from realizing much improvement in efficiency by lowering the back pressure p v Herein lies one important difference be- tween the steam, engine and steam turbine. With the turbine, as will be shown, a lowering of the condenser pressure results in a marked increase of efficiency. 173. Imperfections of the Actual Cycle. In the discussion of the ideal Rankine cycle the following conditions are assumed: 1. That the wall of the cylinder and piston are non-conduct- ing, so that the expansion after cut-off is truly adiabatic. 2. Instantaneous action of valves and ample port area so that free expansion or wiredrawing of the steam may not occur. 3. No clearance. In the actual engine none of these conditions is fulfilled. The metal of the cylinder and piston conducts heat and there is, consequently, a more or less active interchange of heat, between metal and working fluid, thus making adiabatic expansion im- possible. The cylinder must have clearance, and the effect of the cushion steam has to be considered. The valves do not act instantly and a certain amount of wiredrawing is inevitable. It follows that the cycle of the actual engine deviates in many ways from the ideal Rankine cycle, and that the actual efficiency must be considerably less than the ideal efficiency. We must regard the Rankine cycle as an ideal standard unattainable in practice but approximated to more and more closely as -the im- perfections here noted are gradually eliminated or reduced in magnitude. The effects of some of these imperfections may be shown quite clearly by diagrams on the T$-plane. In Fig. 108 is shown the cycle of a non-condensing steam engine. The feed water enters the boiler in the state represented by point 6r and is changed into dry saturated steam at boiler pressure, represented by point B. When this dry steam is transferred to the engine cylinder, which has been cooled to the temperature of the exhaust steam, it is partly condensed, OY-I^ 4-Tia efa-f-a rv-F fVio mivHi r>a in fVio oirlivirl ov o4: mik-riff ia T-e>T->va_ sentea 07 point (J. me neat trnis aosorDeci by tne cylinder walls is represented by the area 1 BOO 1 . CD represents the adiabatic expansion, DE the assumed constant-volume cooling of the steam, and JEF the condensation of the steam at the tem- perature corresponding to the back pressure, which is slightly above atmospheric pressure. To close the cycle, the water at the temperature represented by F (somewhat above 212) must be cooled to the original tempera- ture of the feed water ; this process is represented by FG-. The heat supplied is repre- sented by the area G-^G-ABB-^ the heat transformed into work by the area FAODE. It will be observed that two segments of the cycle, namely, G-F and CB, are traversed twice, and the effect is a serious loss of effi- ciency. The loss due to starting the cycle at point G- instead of at point F may be obviated to a large extent by the use of a feed water heater. The heat rejected in the exhaust is used to heat the feed water to a temperature represented by point If, which is only a little lower than the temperature of the ex- haust. The area Cr-^G-H^ represents the saving in the heat that must be supplied. The loss due to cylinder condensation, which is shown by the segment BO, cannot be wholly obviated ; it may be reduced, however, by superheating and jacketing. Losses due to wiredrawing and clearance are not shown on the diagram. The drop of pressure in the steam main and in the ports may be taken into account roughly by drawing a line A'O' somewhat below the line AB, which represents full boiler pressure. 174. Efficiency Standards. The ratio of the heat transformed into useful work to the total heat supplied is usually termed the thermal efficiency of the engine. The thermal efficiency, how- ever, does not give a useful criterion of the good or bad qualities of an. engine for the reason that it does not take account of the conditions under which the engine works. It has become cus- tomary, therefore, in estimating engine performance to make use of certain other ratios. Let q = heat supplied to the engine per pound of steam, q R = heat 'transformed into work by an engine working in an ideal Rankine cycle (Art. 169), q a = heat transformed into work by actual engine under the same conditions, W a = work equivalent of heat q a , the indicated work, W b = the work obtained at the brake. We have then r] R = = thermal efficiency of ideal Rankine engine, ?7 a = = thermal efficiency of actual engine, 3 77 Q 77.- = = = efficiency ratio (based on indicated work), VR ?.R _ = brake efficiency ratio (based on work at 9.R , , , brake), r) m - mechanical efficiency. WA The ratios ??, and % are sometimes called the potential efficiencies of the engine, the first the indicated potential efficiency, the second the brake potential efficiency. When the term efficiency is used without qualification it usually means the efficiency ratio or potential efficiency rather than the thermal efficiency. It is clear that the useful criterion of the performance of an engine is the ratio ?? 6 . We have % = i?i X t] m . Of the heat q supplied, only the heat q R could be trans- formed into work by the ideal engine using the Rankine cycle ; hence the heat q R rather than the total heat q should be charged to the engine. The ratio T?,- = is a measure of the extent to which the engine transforms into work the heat q R that may possibly be thus transformed ; it may be called the cylinder efficiency. The ratio -r) m measures the mechanical perfection of the engine. Hence, the product ^ x rj m measures the perform- ance of the engine both from the thermodynarnic and the mechanical standpoints. The efficiencies ??< and % may be given, other equivalent defi- nitions that are frequently useful. Let N R = steam consumption of ideal Rankine engine per h. p. -hour. N a steam consumption per h.p.-hour of actual engine. N b = steam consumption per b. h. p.-hour of actual engine. &R N R Ihen ^, % = ^- EXAMPLE. An actual engine operating under the conditions denned in the example of Art. 169 shows a steam consumption of 14. 1 lb. per i. h.p.- hour and 18 lb. per b. h. p.-hour. Since for the ideal engine the steam consumption is 9.35 lb. per h. p.-hour, we have 17,- = ||f= 0.663, and ^ = ^ = 0.52. EXERCISES In Ex. 1 to 5 find the heat transformed into work, efficiency, and steam consumption per h. p.-hour. 1. Carnot cycle, p^ = 110 lb., p 2 = 15 lb. absolute, x b = 0.85. 2. Rankine cycle, same data as in Ex. 1. 3. Rankine cycle, p l = 110 lb., p 2 5 in. of mercury, steam superheated to 450 F. 4. Rankine cycle p^ = 110 lb., p z = 15 lb., x b = 0.85 and adiabatic ex- pansion carried to 27 lb. per square inch. 5. Data the same as Ex. 4 except that steam is not cut off. 6. Let jt? 2 be fixed at 5 in. of mercury. Take x b = 1 and draw a curve showing the relation between 17 and p\. Rankine's cycle. 7. Taking the data of Ex. 2, find the increase of available heat and effi- ciency when a condenser is attached and p^ is lowered to o in. of mercury. 8. Make the same calculation for the cycle with incomplete expansion, y. JLne emciency -rji or an engine is u.uo anu nue meciminc:<u muuiemj.y JN 0.85. If the heat transformed into work by the ideal Rankine engine is 190 B. t. u. per pound, what is the steam consumption of the actual engine per b. h. p.-hour? 10. The steam consumption of a Rankine engine is 9.2 11). per h. p.- hour, and the efficiency ratio 77,- is 0.70. Find the heat transformed into work by the actual engine per poxmd of steam. THE STEAM TURBINE 175. Comparison of the Steam Turbine and Reciprocating En- gine. The essential distinction between the two types of' vapor motors turbines and reciprocating- engines lies in the method of utilizing the available energy of the working- fluid. In the reciprocating engine this energy is at once util- ized in doing work on a moving piston ; in the turbine there is an intermediate transformation, the available energy being first transformed into the energy of a moving jet or stream, which is then utilized in producing motion in the rotating element of the motor. While the turbine suffers from the disadvantage of an added en-ergy transformation with its accompanying loss of efficiency, it has a compensating advantage mechanically. With any motor the work must finally appear in the rotation of a shaft. Hence, intermediate mechanism must be employed to transform the reciprocating motion of the piston to the rotation required. Evidently this is not the case with the turbine, which is thus from the point of view of kinematics a much more simple ma- chine than the reciprocating engine. Many attempts have been made to construct a motor (the so-called rotary engine) in which both the intermediate mechanism of the reciprocating en- gine and the intermediate energy transformation of the turbine should be obviated. These attempts have uniformly resulted in failure. With ideal conditions it is easily shown that the two methods of working produce the same available work and, therefore, give the same efficiency with the same initial and final con- ditions. Thus the Rankine ideal cycle, Fig. 102, gives the maximum available work per pound of steam of a reciprocating ART. 176] CLASSIFICATION OF STEAM TURBINES 295 engine with the pressures p- and jP 2 . It likewise gives (Art. 152) the kinetic energy per pound of steam of a jet flowing without friction from a region in which the pressure is p 1 into o a region in which it is p 2 . Hence if this kinetic energy - g is wholly transformed into work, the work of the turbine per unit weight of fluid is precisely equal to that of the reciprocat- ing engine. Under ideal conditions, therefore, neither type of motor has an advantage over the other in point of efficiency. Under actual conditions, however, there may be a consider- able difference between the efficiencies of the two types. Each type has imperfections and losses peculiar to itself. The re- ciprocating engine has large losses from cylinder condensation ; the turbine, from friction between the moving fluid and the passages through which it flows. It is a question which set of losses may be most reduced by careful design. Aside from the question of economy, the turbine has certain advantages over the reciprocating engine in the matters of weight, cost, and durability (associated with certain disadvan- tages) and these have been sufficient to cause the use of tur- bines rather than reciprocating engines in many new power plants and also in some of the recently built steamships. 176. Classification of Steam Turbines. Steam turbines may be divided broadly into two classes in some degree analogous to the impulse water wheel and the water tur- bine, respectively. In the first class, of which the de Laval turbine may be taken as typical, steam expands in a nozzle until the pressure reaches the pressure of the region in which the turbine wheel rotates. The jet issuing from the nozzle is then directed against the buckets of the turbine wheel, Fig. 109, and the impulse of the iet produces rotation. It will be noted that with this type of turbine only a part of the r*n rC\r}4-c* r ri -fill arl TIT-T -f-T~ o4-an v*-* n 4- o-rtr i -ri o4-o r^-f- atran 1 T The pressure of the steam is reduced during the I passage through the blades both in the guide and turbine wheels. In the turbine of the first type all the available internal energy of the steam is trans- formed into kinetic energy of motion before the steam enters the turbine wheel, while in the turbine of the second type part of the internal energy is transformed into work during the passage of the fluid through the wheel. The terms impulse and reaction have been used FIG. no. to designate turbines of the first and second class, respectively. Since, however, impulse and reaction are both present in each type, these terms are somewhat mis- leading, and the more suitable terms velocity and pressure have been proposed. Tims a de Laval turbine is a velocity turbine ; a Parsons turbine is a pressure turbine. 177. Compounding. The high velocity of a steam jet result- ing from a considerable drop of pressure renders necessary some method of compounding in order that the peripheral speed of the turbine wheels may be kept within reasonable limits without reducing the efficiency of the turbine. With velocity turbines three methods of compounding are employed. 1. Pressure Compounding. The total drop of pressure jt^ p z may be divided among several wheels, thus reducing the jet velocity at each wheel. If, for example, the change of heat content is % 2 and the expansion takes place in a single nozzle, the ideal velocity of the jet is w = V2 g J^ ^' 2 ) ; if, however, ^ i z is divided equally among n wheels, the jet velocity is reduced to w = \' ^~- (^ i' 2 ) . The general arrange- ment of a turbine with several pressure stages is shown in Fig. 111. Steam passes successively through orifices m v w 2 , etc. in partitions 5 r 5 2 , etc., which divide the interior of the FIG. 111. passing through the orifice m z the pres- sure drops from p 2 to p s ; as a result the velocity is again increased and the jet passes through the second wheel. The pressure and velocity changes are shown roughly in the diagram at the bottom of the figure. The method of compounding here described is called pressure compound- ing. Each drop in pressure constitutes a pressure stage. 2. Velocity Compounding. The steam may be expanded in a single stage to the back pressure p 2 , thus giving a rela- tively high velocity ; and the jet may then be made to pass through a suc- cession of moving wheels alternating with fixed guides. This system is shown diagrammatically in Fig. 112. The jet passes into the first moving wheel, where it loses part of its absolute velocity, as indicated by the velocity curve w. It then passes through the fixed guide g 1 with practically con- stant velocity and has its direction changed so as to be effective on enter- ing the second moving wheel. Here the velocity is< again reduced and the decrease of kinetic energy appears as work done on the wheel. This process may be again repeated, if desired, by adding a second guide and a third wheel. However, the work obtainable from a wheel is small after the second moving wheel is passed, and a third wheel is not usually employed. 3. Combination of Pressure and Velocity Compounding. Evi- dently the two methods of compounding may be combined in a FIG. 112. live sets oi nozzles delivering steam to a corresponding numoer of wheels running in separate cliambers, and each wheel has two sets of blades separated by guide vanes. Pressure turbines are always of the multiple pressure-stage type, and the number of stages is large. The arrangement is that shown in Fig. 113. The steam flows through alternate guides and moving blades, its pressure falling gradually as indicated by the curve pp. The absolute velocity of flow increases through the fixed blades and decreases in the moving blades as indicated by the velocity curve ww. This curve, it will be observed, rises as the pressure falls much as if the turbine were i a large diverging nozzle. The . steam velocity with this type of turbine is, however, relatively low even in the last stages. 178. Work of a Jet. While the problems relating to the impulse and reaction of fluid jets belong to hydraulics, it is desirable to introduce here a brief discussion of the general case of the impulse of a jet on a moving vane. Let the curved blade have the velocity c in the direction in- dicated, Fig. 114, and let w- denote the velocity of a jet directed against the blade. The velocity w^ is resolved into two compo- nents, one equal to c, the velocity of the blade, the other, there- fore, the velocity a-^ of the jet relative to the blade. The angle of the blade and the velocity c should be so adjusted that the direction of a is tanent to the ede of the blade at entrance. The jet leaves the blade with a relative velocity a z equal in magnitude to a^ neglecting friction, but of less magnitude if friction is taken into account. This velocity a 2 combined with the velocity c of the blade gives the absolute exit velocity w 2 . It is convenient to draw all the velocities from one point as shown in the velocity diagram. The absolute entrance and exit velocities w l and w 2 may be resolved into components w-[ and w z ' in the direction of the motion of the vane and w-J 1 and w 9 " at I M right angles to this direction, that is, parallel to the axis of the wheel that carries the vane. These latter may be termed the axial components, the former the pe- ripheral components. The driving impulse of the jet depends upon the change in the peripheral component only. To deduce an expression for the impulse we proceed as follows : Let Am denote the mass of fluid flowing past a given cross .section in the time At ; then the stream of fluid in contact with the blade may be considered as made up of a number of mass elements Am, and in the time element At one mass ele- ment enters the vane with a peripheral velocity w-^ and another leaves it with a peripheral velocity w z '. The effect is the same as if a single element Am by contact with the blade had its velocity decreased from 10^ to w z ' in the time At. From the fundamental principle of mechanics, the force required to pro- duce the acceleration FIG. 114. At Am is W-, At (1) ; an equal ana opposite xorce is, tnereiore, wie impulse or Am on the vane. If M denotes the weight of steam flowing per second, then Aw = At, and we have for the force exerted by the jet on 9 the vane in the direction of the velocity 0, = C '->'") (2*) Evidently this equation holds equally well when the weight M flowing from the nozzle is divided among several moving vanes. The product pc of the peripheral force and peripheral veloc- ity of vane gives the work per second ; therefore, work per second = - (w^ w 2 '), (3) y and work per pound of fluid = - (w/ w 2 ')- 00 y When, as is usually the case, the direction of w 2 r is opposite to that of w^, the sign of w 2 ' must be considered negative and the algebraic difference tv^ w 2 in (2), (3), and (4) becomes the arithmetic sum w-/ + w 2 '. 179. Single-stage Velocity Turbine. In analyzing the action of the single-stage velocity turbine, it is convenient to start with an ideal frictionless tur- bine and then take up the case of the actual turbine. Let the jet emerge from the nozzle with the velocity w v Fig. 115, at an angle a with the plane of the wheel. Combining w 1 with <?, the periph- eral velocity of the blade, the velocity ^ of the jet relative to the blade is obtained. The angle /3 between the direction of &J and the plane of the wheel determines the angle of the blade at entrance. If the blade is symmetrical, the exit relative velocity a 2 makes the same angle /3 with the plane of the wheel, and since the frictionless case is assumed, a z = a r Combining 2 and <?, the result is the absolute exit velocity w z . an 2 The energy of the jet with the velocity w^ is -i- per pound rt/j 2 oi medium flowing; and the jet at exit has the energy -^-. ^9 The work absorbed by the wheel per unit weight of steam in this ideal frictionless case is, therefore, W = W ? ~ W *\ (1) 2 ff and the ideal efficiency is (2) w 2 i From the triangle OAE, Fig. 115, we have W 2 2 =w 1 2 + (2c) 2 -2w 1 (2c) cos a; (3) whence w-f m 2 2 = ^(w^ c cos a c 2 ). (4) Combining (2) and (4), we get, --\ ' (5) l iv \J Equation (5) shows that the efficiency is greater the smaller the angle oc ; and that with a given constant angle , the effi- ciency depends upon the ratio . It is readily found that vj W-^ rt takes its maximum value ?? max = cos 2 when the ratio takes w l the value |- cos a. As an example, let K = 20, whence cos a =0.9397 and cos 2 = 0.883. If w = 3600 ft. per second, then to get the nmximnm efficiency 0.883, the ratio must be - cos a = 0.47, whence c = 0.47 x 3600 = 1692 ft. per second, a wi 2 value too high for safety. If c be given the permissible value 1200 ft. per second, we have = J, and 77 = 4 x - (0.9397 - 0.3333) = 0.809. zi>j o 3 . In the actual turbine, friction in the nozzle and blades reduces the efficiency considerably below the value given by (5). The velocity diagram with friction is shown in Fig. 116. The ideal >* actual jeu veiOCliy W-^ uumumeu. wiuu veiuui^y u^ivoo imc iciauivc velocity a v as before. The exit relative velocity a z is smaller than G&! because of friction in the blades, and as a result the absolute exit ve- locity w% is smaller than in the ideal case. The work per pound of 4- ^ f 1 steam may be found from the velocity diagram either by calculation or by direct measure- ment. Having the components wj and wj, the work per pound of steam is given by the expression \s (6) This work may be compared with the work obtained from the ideal Motionless turbine given by (1) or with the energy of Alt 2 the jet per pound of steam, namely, ~ . * 9 180. Multiple-stage Velocity Turbine. In the Rateau turbine and in others of similar construction, the principle of pressure compounding is employed. The turbine consists essentially of several de Laval turbines in series, running in separate cham- bers. See Fig. 111. The action of this type of turbine is con- veniently studied in connection with a Mollier diagram, Fig. 117. Let the initial state of the steam entering the turbine at the pressure p 1 be that indicated by the point A. If p 2 is the pressure in the first chamber, a Motionless adiabatic expansion from p^ to p z is represented by AB, and the decrease in the heat content ^ 2 is represented by the length of the segment AB. Under ideal conditions, this drop in the heat content would all be transformed into kinetic energy of the jet of steam flowing into the chamber, and this in turn would be given up to the wheel. Actu- ally, however, friction losses are encountered and the jet has an exit velocity w z , thereby carrying away the kinetic an A energy -. The velocity diagram for the single wheel under consideration is similar to that shown in Fig. 116. The work lost in overcoming friction in the nozzles and blades and the nn exit energy ^- are transformed into heat, and this heat, except ^9 a small fraction that is radiated, is expended in further super- heating (or raising the quality of) the steam. Hence, instead of the final state B, we have a final state on the same con- stant-pressure curve. Referring to Fig. 117, AC' represents FIG. 117. the part of the heat drop that is utilized by the wheel, while O'B represents the part that is rendered unavailable by internal losses of various kinds. The steam in the state flows into the second chamber where the pressure is p y Frictionless adiabatic expansion would give the second state D, but the actual state is represented by the point E. Again CE' represents the effective drop of heat con- tent in this stage, while E'D represents the part of the drop going back into the steam. The same process is repeated in succeeding stages until finally the steam drops to condenser pressure in the last stage. The final state is represented by the point K, and the curve AEK represents the change of state of the steam during its passage through the turbine. The final state under ideal fric- tionless conditions is represented by point M, The segment AM represents the ideal heat drop, which, as has been shown, is AN represents the heat drop utilized. The ratio AM de- pends upon the magnitude of the internal losses, such as friction in nozzles and blades, leakage from stage to stage, windage, exit velocity, etc. Roughly, this ratio may lie between 0.50 and 0.80. 181. Turbine with both Pressure and Velocity Stages. In certain turbines, notably the Curtis turbine, velocity compound- ing is employed. There are relatively few (three to seven) pressure stages, but in each cham- ber there are two or three rows of moving blades at- tached to the F IG . us. wheel rim and these are sepa- rated by alternate rows of guide blades, as shown in Fig. 112. The velocity diagram for a single pressure stage with two velocity stages is shown in Fig. 118. The velocities in relation to the successive sets of blades are shown in Fig. 119. The jet emerges from the nozzle with an absolute velocity w v which is smaller than the ideal W Q because of friction in the nozzle. Combining w 1 with the peripheral velocity c of the first moving blade m p the result is the velocity a-^ of jet relative to blade m v The angle a between a^ and the plane of rotation is the proper entrance angle of the blade m v The exit relative velocity 2 , which is smaller than a v due to friction in the blade, is combined with the velocity c, giving the absolute exit velocity w 2 which makes the angle /3 with the plane of rotation. The jet enters the stationary guide blade s with the velocity w z and emerges with a smaller FIG. 119. blade m v Combination of w 3 -with c gives the relative velocity a z and the entrance angle y for the blade m z . The exit velocity 4 is determined from a s and the friction in the blade, and by combining a 4 and c, the absolute exit velocity w 4 is obtained. In the diagram, Fig. 119, the blades have been taken as symmetrical. Sometimes, however, the exit angles of the last sets of blades are made smaller than the entrance angles. The diagram can easily be modified to suit this condition. The work per pound of steam for this wheel is readily deter- mined from the velocity diagram. From the first set of blades m 1 the work -(w-^ w z '~) and from the second set of blades m 2 i/ the work - (wJ w /) is obtained. Hence the total work per 9 pound of steam is W= c - (w / - < + wj - O . (1) 9 Care must be taken that w z ' and w/ be given their proper alge- braic signs. The state of the fluid as it passes through the turbine may be shown by the Mollier diagram precisely similar to that shown in Fig. 117. Starting with an initial state indicated by point .A, the available drop from the initial pressure p to the pressure p z in the first chamber is represented by AB. The heat utilized in useful work Tfas given by (1) is represented by AC'. Hence projecting C' horizontally to on the line of constant pressure p v we get the state of the steam as it enters the second stage nozzles. 182. Pressure Turbine. In the pressure type of turbine there is always a large number of stages, the guide blades and moving blades alternating in close succession. The fact that the pressure falls continuously, both through the guide blades and the moving blades, makes the velocity diagram essentially different from that of the velocity turbine, lief erring to Fig. 120, let w l denote the absolute velocity of the steam entering the stationary Diaae S 1? ana w z tne aosoiute exit veiocicy. JLI unere were no, change of pressure, w z would be smaller than w l be- cause of friction ; but the drop in pressure Ajt? causes a decrease in heat content Ai, and as a result, there is an increase of velocity given by the relation FIG. 120. Thus the exit velocity w 2 is greater than the entrance velocity w^. Combining w z with c, the velocity of the moving blade, we obtain 2 , the velocity of entrance relative to the moving blade. Now the pressure drops through the moving blades also ; hence as a result the velocity of exit a z is greater than a v just as e# 2 , is greater than w. Combining a z with (?, the result is zv^, the absolute velocity of entrance into the next row of fixed blades. The work done in any single stage, consisting of one set of stationary blades and one set of moving blades, is obtained from the velocity diagram for that stage in the usual way. Thus, if we have the diagram shown in Fig. 120 for a particular stage, the work per pound of steam for that stage is given by the product G -(w' z -w'J. u If the fixed and moving blades have the same entrance angles and exit angles, it may be as- sumed that the velocity diagram has the symmetrical form shown in Fig. 121 ; that is, w^ = j and w 2 a z . In this case, the work may be obtained by a simple graphical construction. Using point B as a center and with a radius BA let a circular arc ADC be described and from ^let a perpendicular be dropped cutting this arc in _D. Denoting the length JED by h, we have G FIG. 121. It follows that the work per pound of steain is given by the h 2 expression provided h is measured to the same scale as the y velocity vectors w v w z . 183. Influence of High Vacuum. In Art. 172 it was pointed out that the reciprocating engine is unable to take advantage of a very low back pressure for the reason that the cylinder volume cannot be made sufficiently large to permit the expan- sion of the steam to the condenser pressure. No such restriction applies to the steam turbine. The blades in the final stages may be made long enough to pass the required volume of steain at the lowest pressures obtainable. The advantage of the tur- bine in this respect is shown graphically in Fig. 107. Since the cylinder volume of the reciprocating engine is limited to the volume indicated by the point _Z?, the effect of lowering the back pressure from p 2 to p z ' is the addition of the area D'DFF' to the area of the original cycle. The turbine, however, can accomodate volumes indicated by points and 0' ; hence if the pressure is lowered from p 2 to p 2 ', the area of the ideal cycle is increased by the area D'DCG 1 . It is evident, therefore, that high vacuum is much more effective in the case of the steam turbine than in the case of the reciprocating engine. The superior efficiency of the steain turbine at low pressures and the ability of the turbine to make effective use of high vacuum has led to the introduction of the low-pressure turbine in combination with the reciprocating engine. The engine takes steam at boiler pressure and exhausts into the turbine at about atmospheric pressure. In general, the combination is more efficient than either the engine alone or the turbine alone using the entire range of pressure. EXERCISES 1. In a single-stage velocity turbine the jet emerges from the nozzle -with a velocity of 3150 ft. per second and the direction of the jet makes an angle that will give maximum efficiency, (b) Find the efficiency if the circum- ferential velocity is 1100 ft. per second. 2. Find the work per pound of steam in case (b) of Ex. 1. 3. Using the data of Ex. 1 and 2 assume that the exit relative velocity is reduced 10 per cent by friction in the blades. Draw a velocity diagram and by measurement or calculation find the work done per pound of steam, Compare this result with that found for the ideal frictionless case. 4. A reciprocating engine receives steam at a pressure of 160 Ib. per square inch, superheated 120. The steam expands adiabatically to a pres- sure of 16 in. of mercury and is then discharged into a low pressure turbine where it expands adiabatically to a pressure of 2 in. of mercury. Find the percentage by which the efficiency is increased by the addition of the tur- bine. Assume ideal conditions. 5. A turbine of the Curtis type has three pressure stages. The initial pressure is 140 Ib. with the steam superheated 120 F., and the condenser pressure is 3 in. of mercury, The loss of energy due to friction, etc., is 30 per cent of the total available energy, (a) Find the condition of the steam entering the condenser. (&) Find the consumption per h. p.-hour. (c) Determine the intermediate pressures in the cells on the assumption that the work developed in each stage shall be approximately the same. REFIGERATION WITH VAPOE MEDIA 184. Compression Refrigerating Machines. The essential organs of a compression machine using vapor as a medium are shown in Fig. 122. The action of the machine may be studied to advan- tage in connection with the ^-dia- gram, Fig. 123. The medium is drawn into the compressor cylinder through the suction pipe from the coils in the brine tank. It may be assumed that the medium entering is in the saturated state at the temperature T^ which may be taken equal to the i f -P 4-1 "K >* Tl "4-4-" 4-11 "j. 7~> Expansion Vulvo FIG. 122. M Jbig. 126. Ihe vapor is compressed adiabatically to a final pressure p z , which is determined by the upper temperature T z that may be obtained with the cooling water available. The adiabatic compression is represented by B Q, The superheated vapor in the state (7 is discharged into the coils of the cooler or condenser, where heat is abstracted from it. The coils are surrounded by cold water which flows continuously. First the gas is cooled to the state of saturation ; this process is rep- T E{ ^ YD .i resented by the curve CD, and the heat abstracted by the area r /j\A_T l CiCDDv Then heat is further removed at the constant tern- II\ perature T z (and pressure p%) and the vapor condenses. At the end of the process, the medium is liquid and its state - is represented by the point E Fl{} 123> on the liquid curve. It should be noted that there are two parts of the fluid circuit : one including the discharge pipe and coils at the higher pres- sure p z , and one including the brine coils and the suction pipe at the lower pressure p r These are separated by a valve called the expansion valve. The liquid in the state represented by point H is allowed to trickle through the valve into the region of lower pressure. The result of this irreversible free expan- sion is to bring the medium to a new state represented by point A. In this state the medium, which is chiefly liquid with a small percentage of vapor, passes into the coils in the brine tank or in the room to be cooled. The temperature of the brine being higher than that of the medium, heat is absorbed by the medium, and the liquid vaporizes at constant pressure. This process is represented by the line AB and the heat absorbed from the surrounding brine by the area A l ABQ r The position of the point A is determined as follows : The passage of the liquid through the expansion valve is a case of throttline- or wiredrawing of the character discussed in Art. 162. Hence, the heat content at A must be equal to the heat content at E, that is, ^ 2 = ^i + ^ r i Graphically, the area OHG-AA l is equal to the area OHEE^, or taking away the common area OHG-FE^ the rectangle E l FAA l is equal to the triangle G-EI. (See Art. 162). Since the throttling process represented by EA is assumed to be adiabatic, the work that must be done 011 the medium is the difference between Q^ the heat absorbed, and Q z , the heat rejected to the condenser. We have then Q 1 = area W = area O 1 ODEE 1 - area A 1 AB0 1 If the expansion valve be replaced by an expansion cylinder, permitting a reversible adiabatic expansion from p z to p v as in- dicated by the line JEF, we have = area W= area BCDEFB. The effect of using the expansion valve rather than the expansion cylinder is thus to decrease the heat removed by the area E-^FAA-^ and to increase the work done by an equivalent amount. 185. Vapors used in Refrigeration. The three vapors that are used to any extent as refrigerating media are ammonia, sulphur dioxide, and carbon dioxide. Of these, ammonia is used almost exclusively in America and largely in Europe. The other two are used to a small extent chiefly in Europe. The choice of vapor to be used depends chiefly upon two things : (1) The suction and discharge pressures that must be employed to give proper lower and upper temperatures T^ and T z . The lower temperature must be such as to keep the proper temperature in the brine or the space to be kept cool, while the upper temnerature is fixed bv the temoerature of the ooolino- ailable. (2) The volume of the medium required for a given louiit of refrigeration. This determines the bulk of the iichine. If the upper temperature be taken as 68 F. (T z = 528) and 13 lower temperature as 14 F., the pressures and the volume lion for tlio three vapors mentioned are about as follows: Nils SO. COj ol.ion proHHuni, 11). pur sq. in. 41.5 14.75 385 s<:]i:ir^(! pressure, II). per sq. in. 124 47.61 826 >lumi!, taking Uuiti of (.X) 3 an 1 4.4 12 1 It appears that carbon dioxide requires for proper -working !iy high pressures, so high, in fact, as to be practically prohib- ve except in maehines of small size. With sulphur dioxide e pressures are low, but the necessary volume of medium is gh, being nearly three times that required by ammonia and ,'elvo times that required by carbon dioxide. With ammonia, c pressures are reasonable and the volume of medium is not :<:essive; hence from these considerations, ammonia is seen to ! most advantageous. From the point of view of economy, ammonia and sulphur oxide are about equal. Carbon dioxide shows a somewhat Killer el'l'ieiem:/ than the others under similar conditions be- ,use, on account of the small latent heat of carbon dioxide, the sses due to superheating and the passage through the expan- jn valve are a larger per cent of the total effect. 186. Calculation of a Vapor Machine. The following analysis >plies to the ideal, cycle shown in Fig. 123. Denoting by T ie temperature at the end of compression indicated by the >int (7, the heat that must be removed per minute from the iperheated vapor to bring it to the saturation state (the heat presented by the area O^DD^ is v . i Avlm-.li * p denotes the specific heat of superheated vapor, and E the weight of the medium required per minute^ i he heat ejected by the vapor during condensation (area 1)^^) is fr a . Hence the heat rejected by the medium per minute is ^ -*rr , .. rrn _ TV1. (1) Denoting by x l the quality of the mixture of liquid and vapor in the state represented by point A, we have for the heat ab- sorbed by the medium from the brine or cold room (repre- sented by the area A-^ABO-^) Q^Mr^l-xJ. (2) But area OHGf-AA^ area OHEU V that is, 2i' + r i x i = &' ; ( 3 ) whence combining (3) and (2), Q, = M^ - qj + ft ') = M( qi - 2a '). 00 The work required per minute is, therefore, TT= JX<? a -0i) = ^[2 a "-<?i" + 'p(^- 2*)], (5) and the net horsepower required to drive the machine is, Combining (6) and (4), we have 778^[^-^+ gp (r e2 83000( ?1 "- 2a ') ' . ^ } To the horsepower thus calculated should be added perhaps 10 to 20 per cent to allow for imperfections of the cycle, and to the gross horsepower must be added 10 to 20 per cent to allow for the friction of the mechanism. Assuming the vapor entering the compressor to be dry and saturated, as indicated by point 33, Fig. 123, the volume of vapor entering the compressor per stroke is Mv " V _1_ /^ v c ft- (0) where v^' is the specific volume of vapor at the pressure p l and N the number of working strokes per minute. If the medium enters the compressor as a mixture of quality x m as in- dicated by point M, then approximately The net cylinder volume as determined by (8^ or (9) must JL The weight of cooling water required per minute is readily found from (1) when the initial and final temperatures of the water are fixed. Denoting this weight by Cr and the initial and final temperatures by t" and t\ respectively, we have <?(*" - O = M fa + c p (T - 5i)]. (10) To determine the value of Q z from (1) the temperature T c at the end of compression must be obtained. For adiabatic .com- pression T c may be found by the following method. Eeferring to Fig. 123, the decrease of entropy in passing from to D is the same as passing from B to D. If e p , the specific heat along curve CD, is assumed to be constant, we have T UUt Sjj Sd S-[ -\- -fif i "2 ' rn -LI \ J- 2 . hence c p log e - = s^ -f ~ Since <? p , 2j, jP 2 , s/, s 2 , r 15 and r 2 are known quantities, ^ is easily calculated. EXAMPLE. Required the dimensions and the horsepower of an ammonia refrigerating machine that is to abstract 15,000 B. t. u. per minute from a cold chamber which is to be kept at a temperature of 30 3?. The tempera- ture of the ammonia in the condenser may be taken as 85 F. and that of the ammonia in the brine coils 20 F. Assume one double-acting com- pressor making 75 r. p. m. From the table of the properties of saturated ammonia, we have the fol- lowing values corresponding to ti = 20 and fe = 85 : pi = 47.46 Ib. per square inch, ri = 500 B. t. u., q\ = 13 B. t. u., ji" = 547 B. t. u., si' = - 0.027, -g- = 1.168, v{' = 6.01 cu. ft., -M. p z = 166.8 Ib. per square inch, r 2 = 496 B.t. u., q z ' = 01 B. fc.u., 0," = 557 B.t.u., s 2 ' = 0.118, ^.= 0.910, t? 2 " = 1.78. end of compression, we have, from (11), 0.51 log e p-^4 = - 0.027 + 1.168 - (0.118 + 0.910)= 0.113, whence log T c = log 544.6 + 0.4343 x 9^ = 2.83231, T = 679.7, and 4 = 679.7 - 459.6 = 220.1 F. The weight of ammonia that must be circulated per minxtte is, from (4), isooo o47 - 61 The net horsepower is, from (6), 778 x 80.86 r 33000 _ 547 0<51(22 o.l _ 85)] = 57.4. V ^ Adding 15 per cent for cycle imperfections, the compressor will requii about 66 horsepower. The steam engine required to drive the compressc should develop, say, 80 horsepower. The volume of the compressor cylinder is, from (S), 30.86 x 6.01 = mcu . ft . 2 x 75 Adding 15 per cent for clearance, etc., the required volume is 1.43 cu. f This is given by a stroke of 20 in. and a cylinder diameter of 12 in. TABLE I PROPERTIES OF SATURATED STEAM PRESSURE t3 INCHES OP HG. TEMP. FAHU. t HEAT CONTENT LATENT HEAT ENTROPY VOLUME OF ONE POUND V of Liquid i' of Vapor i" Total r Internal P of Liquid s' of Vapor- ization r T of Vapor s" 0.5 58.81 26.9 1087.1 1060.2 1002.9 .0532 2.0431 2.0963 1259.3 1.0 79.12 47.2 1096.7 1049.5 989.8 .0916 1.9482 2.0398 656.7 1.6 91.90 59.9 1102.5 1042.6 982.2 .1150 1.8905 2.0055 443.0 2.0 101.27 69.2 1106.6 1037.4 975.9 .1317 1.8497 1.9814 338.3 2.5 108.81 76.7 1109.9 1033.2 970.8 .1451 1.8178 1.9629 274.3 3.0 115.15 83.1 1112.7 1029.6 966.5 .1561 1.7915 1.9476 231.2 3.5 120.63 88.5 1115.0 1026.5 962.8 .1656 1.7692 1.9348 200.1 4.0 125.48 93.4 1117.1 1023.7 959.5 .1739 1.7497 1.9236 176.6 4.5 129.85 97.7 1118.9 1021.2 956.5 .1813 1.7325 1.9138 158.1 5.0 133.81 101.7 1120.6 1018.9 953.7 .1880 1.7170 1.9050 143.2 6 140.83 108.7 1123.4 1014.7 94S.8 .1997 1.6901 1.8898 120.7 7 146.90 114.8 1125.9 1011.1 944.7 .2097 1.6672 1.8769 104.4 8 152.28 120.2 1128.0 1007.9 940.9 .2186 1.6473 1.8659 92.2 9 157.12 125.0 1130.0 1005.0 937.5 .2265 1.6296 1.85G1 82.6 10 161.52 129.4 1131.7 1002.3 934.3 .2336 1.6138 1.8474 74.8 11 165.57 133.4 1133.3 999.9 931.5 .2401 1.5994 1.8395 68.38 12 169.31 137.2 1134.7 997.6 928.8 .2460 1.5862 1.8322 63.03 13 172.80 140.6 1136.0 995.4 926.3 .2516 1.5739 1.8254 58.48 14 176.07 143.9 1137.3 993.4 924.0 .2568 1.5627 1.8195 54.55 15 179.16 147.0 1138.5 991.5 921.8 .2616 1.5522 1.8138 51.13 16 182.08 150.0 1139.6 989.6 919.6 .2662 1.5423 1.8085 48.11 17 184.84 152.7 1140.6 987.9 917.6 .2705 1.5330 1.8035 45.46 18 187.47 155.4 1141.6 986.2 915.7 .2746 1.5242 1.7988 43.09 19 189.99 157.9 1142.5 984.6 913.8 .2785 1.5158 1.7943 40.96 20 192.38 160.3 1143.4 983.1 912.1 .2822 1.5079 1.7901 39.04 21 194.69 162.6 1144.2 981.6 910.4 .2857 1.5003 1.7860 37.29 22 196.91 164.8 1145.0 980.2 908.8 .2891 1.4931 1.7822 35.68 23 199.04 167.0 1145.8 978.8 907.2 .2923 1.4862 1.77S5 34.22 24 201.10 169.0 1146.5 977.5 905.7 .2955 1.4796 1.7751 32.88 25 203.09 171.0 1147.2 976.2 904.2 .2985 1.4732 1.7717 31.65 26 205.01 173.0 1147.9 974.9 902.7 .3014 1.4670 1.7674 30.52 27 206.87 174.9 1148.6 973.7 901.4 .3042 1.4611 1.7653 29.46 28 208.68 176.7 1149.2 972.5 900.0 .3069 1.4554 1.7623 28.47 29 210.43 178.4 1149.8 971.4 898.8 .3095 1.4499 1.7594 27.55 PHEHIUIKK TUMP. JL1KAT V . Ln. J'Klt S. In. FA.UU. )(' Liquid of VHIHT Toliil Internal ,1 i,,,,,.i.l Vupon- /ution <>! Viijior P t '' 7." r ' K 1 7' K" 14.7 212.0 180.0 1150.4 970.4 85)7.7 .3121 1.44-15) 1.7570 15 213.0 181.1 1.150,8 5)09.7 85)0.8 .3137 1.4417 1.755-1 16 21(5.3 184,4 1152.0 5)07.0 85)4,5 .318(5 1.4315 1.7501 17 219.4 187.5 1153.0 5X55.5 892.1 .323 1 1.4220 1. 745 1 4 O ,),)>) A 1 1 54 1 5)03.0 8X9 9 3275 1 1 125) 1 7404 lo 19 225/2 1SKU iibr>!o 887.7 .3317 1.4043 1.7300 20 227.9 15)0.1 1 155.9 5)59.8 885.7 .3357 1.3901 1.73 IX 21 230.5 198.7 1 150.8 5)58.1 883.8 .335)5 1 .3X83 1.727X 22 233.0 201.2 1 157.7 5)5(5.5 882.0 .3431 1.3X09 1.72-10 23 235.4 203.0 1 158.5 5)54,8 880.0 .3407 1 .3738 1.7205 24 237,8 20(i'.() 1 159.2 5)53.2 878.0 .3501 1.30(55) 1.7170 25 240.0 208.3 1100.0 5)51.7 87(5.2 .3533 1.31504 1.7137 26 242 2 210.5 1 1(50.7 5)50.2 874.5 .35(55 1.3540 1.7105 27 244,3 212.7 11(51.5 5)48.8 Uto .355)5 1.347!) 1.707-1 28 24(5.4 214,8 11(52.1 /fl.{7.3 \7l.2 , .3(525 1.341!) 1.704-1 29 30 248.4 250.3 21(5,8 218,8 1JW2.N HS3.4 kj-ibfcO ' SOi).S 8(58/2 .3(553 .3(581 1.330'J 1.3307 1.7015 1.0988 31 252/2 220.7^ 11(54,0 5)-13.3 800.H .3708 1 .3253 1.05)01 32 254.0 222.5 \ V 104*0 5)42. 1 805.-1 .373.1 1.3'JOl l.( 55)35 33 255,8 224,3 N&.1 5)40,8 8(54.0 .37(50 1.3151 1.155) 11 34 257.0 22(5.1 V 105.7 5)39.0 8(52.7 .3784 1.3102 1.15X8(5 35 259.3 227.8 \1(5(5.2 5)38.- 1 ,8(51.3 .3808 1.3054 1 .0X03 36 2(51.0 229.5 1.1(5(5,8 5)37.3 8(50.1 .3832 1.300X 1. (58-10 37 2(52.0 231.2 11(57.3 93(5.1 858.8 .3855 1 .251(53 1. (5,8 1 8 38 2(54/2 232.8 11(57.8 5)35.0 857.0 .387S 1.25)18 1.075 Hi 39 2(55.7 234.4 1108.3 5)33.5) 85(5.4 .35)00 1.287(5 1 .(577(5 40 2(57.3 23(5.0 1 1 (58.8 5)32.8 855. 1 .35)21 1 .2834 1.0755 41 268.8 237.5 11(59.3 931.8 854.0 .39-12 1 .275)3 1.0735 42 270.2 239.0 1109.7 5)30.7 8513,8 .35)02 1 .2753 1. 07 1 5 43 271.7 240.5 1170.1 92!).7 851.7 .35)82 1.2714 1.005X5 44 273.1 241.9 1170.0 5)28.7 S50.0 .-1002 1.2(57(5 1 .(5078 45 274.5 243.3 1171.0 927.7 849.5 .4021 1 .2(538 1. (5(559 46 275.8 244.0 11.71.4 5)2(5.8 8-18.5 ..10-10 1.2(502 1. (5(542 ' 47 277.2 24(5.0 1.171.8 5)25,8 847.4 .4059 1.25(5(5 1. (5(525 48 278.5 247.3 1172.2 5)24,9 84(5.4 .-1077 1 .253 1 1.0008 49 279.8 248.7 1172.0 923.!) 8-15.3 .-1095 1.2-19(5 1.055)1 50 281.1 250.0 1173.0 923.0 8-14,4 .4112 1/2-103 1.0575 51 282.3 251.3 1173.4 5)22.1 843.4 .4130 1.2425) 1.0555) 52 283.5 252.0 1.173,8 5)21.2 842.4 .4147 1.235)7 1.05-14 53 284.8 253.8 1174,2 5)20.4 8-11.5 .41(5-1 1.23(55 1.0525) 54 280.0 255.0 1174.5 919.5 840.5 .-1180 1 .2333 1.0511? 55 287.1 250.2 1174,5) 918.7 835U5 .4151(5 1/230: 5 1. (5-15)5) 56 288.3 257.4 1175/2 917,8 838.7 .4212 1 .2272 1. (548-1 57 289.4 258.0 1175.0 917.0 837.8 .4228 1.22-13 1.0-171 58 290.6 259.7 1175.5) 910.2 83(5.5) .4243 1.2213 1. (54 5(5 59 291.7 2(50.8 1170/2 5)15.4 83(5.0 .-1258 1.218-1 1.0442 JIi.vi- f 'n,\r.sr LATKN r HKAT ENTKOPV Tl.MI'. i _ VOLUME OF ONE I''AUH. if Liuuiit u( Vn|n.r Tot ul Intonml of Liquid Vapori- zation of Vapor POUND t T' i" r P ' r T s" v" 21)2.8 2(52.0 117(i.(i 014.0 S85.2 .4273 1.215(5 1.6429 7.168 1208.0 2(58. 1 1 170.0 013.S 834,3 .4288 1.2128 1.0416 7.057 201.0 12(5 4.2 1177.2 1)13.0 833.4 .4302 1.2101 1.0403 6.949 12015.0 120,"). 3 1 177."' 012 2 S3'' '"5 481(5 1 2074 1 (i'-WO K VAK 207.0 1177.8 S31.S .4330 j. *\j t T: 1.2047 JL .IJOt/V/ 1.6378 U.OTrU 6.744 2118.0 2(57.4 117S.1 010.7 S80.0 .4344 1.2021 1.6365 6.646 120!>.l 17S.-I 5)10.0 830.2 .4358 1.105)5 1.6353 6.551 800. 1 17K.7 000.2 820.8 .4372 1.1069 1.6341 6.459 801.1 27<Y.f> 170.0 U08.5 S2s!r> .4385 1.1044 1.6329 6.370 802.0 271.5 170.8 007.S S27.7 .4308 1.1020 1.6318 6.283 308.0 1272.4 170.5 007.1 S27.0 .441 1 1.1895 1.6306 6.198 308J> 273.4 170.S 000..4 82(5.2 .4424 1.1871 1.6295 6.115 301.!) 1274.4 ISO.] !)05?7 S25.5 .4437 1.1847 1.6284 6.035 305.8 127.">. 4 1S0.4 005.01 S24.7 .4450 1.1823 1.6273 5.957 8015.S 27(5.3 ISO. (5 i)o4 !n 823.9 .44(52 1.1800 1.0262 5.882 307.7 277.8 ISO.!) 008.15 823:2- .4474 1.1777 1.6251 5.809 308.15 27S.2 181. 1 002.0 S22'.4 . .448(5 1.1755 1.6241 5.737 305). 5 27! I.I LSI 4 <)()'> 'i S'M S .445)8 1.1732 1.6230 5.666 310.4 12SIU) IS 1.0 001.15 821.0* .4510' 1.1710 1.6220 5.597 311.2 12SO.O IS 1.0 001 S20.4 .4522 1.1688 1.6210 5.530 812.1 281.8 1 1S2.1 000.3 SI 0.0 .4533 1.1667 1.6200 5.464 313.S 1 1S2.7 SO!) 81 8.2 .455(5 1.1625 1.6181 5.338 315.5 2sr>'3 1 183.1 S07!s 81(5.1) .4578 1.1584 1.0162 5.219 317.2 12S7 .4(500 1.1543 1.6143 5.104 31S.S 288.7 IS'LO SOfvl 814.3 .4022 1.1503 1.0125 4.995 320.4 ><)() 3 1 1 S4 r > S04.2 818.0 .4042 1.1465 1.6107 4.890 321.0 323.4 25) L5) 1S4!<) S03.0 811.7 810.4 .4(503 .4(583 1.1427 1.1390 1.6090 1.6073 4.789 4.692 824.0 320.5 LJOOJi ish!7 1815.1 siwhV S00.3" SOS.O .4703 .4723 1.1353 1.1317 1.6056 1.6040 4.599 4.511 327.0 320.8 330.7 882.1 833.5 2! )!>.(! 801.0 802.5 18(5.5 1S7.7 188.1 888.4. SSf>!2 SS4.2 80(5.8 S05.0 803!4 S02.3 .4742 .4701 .4770 .4707 .4815 1.1282 1.1248 1.1214 1.1181 1.1149 1.6024 1.6009 1.5993 1.5978 1.5964 4.425 4.343 4.264 4.188 4.114 334.8 83(5.2 887.5 338.S 840.1 80S.O 310/7 ISS.4 1SS.S 1 8' ) 5 1 S! ) S SS3.1 SS2. 1 881.1 SSO.l 801.1 800.0 7!)7!l) 70(5.8 .4833 .4850 .4807 .4884 .4901 1.1117 1.1085 1.1054 1.1024 1.0994 1.5950 1.5935 1.5921 1,5908 1.5895 4.043 3.975 3.909 3.845 3.784 341.3 342.(5 343.S 345.0 il 1 00.1 100.5 inn .1 S7S.2 S77.2 875!3 705.S 704.8 703.S 71)2.7 701.S .4917 .4033 .4040 .40(55 .4980 1.0965 1.0936 1.0907 1.0879 1.0851 1.5882 1,5869 1.5856 1.5844 1.5831 3.724 3.666 3.610 3,555 3,502 Ll). I'BU SQ. IN. TEMP. FAUK. HKAT GONTKNT LATKNT HKAT KNTHOl'Y >f Liquid of Vupor Total Internal of Liiiuid Vfipori- zutioii of Vupor P t i' i" r P ' T 130 347.4 318.2 1191.7 873.5 700.S .4995 1.0824 1.581!) 132 348.0 319.4 1192.0 872.15 78!).!) .5010 1.0797 1.5807 134 349.7 320.0 1102.3 871.7 788.9 .5025 .0770 1 .5795 136 350.8 321.8 1192.0 870.8 788.0 .5039 .0744 1 .5783 138 352.0 323.0 1192.9 800.0 787.0 .5054 .071!) 1 .5773 140 353.1 324.2 1193.2 8(59.0 78(5.1 .50(58 .0(593 ! 1.57(51 142 354.2 325.3 1193.5 808.2 785.2 .5082 .0(5(58 1.5750 144 355.3 32(5.5 1103.X 8(57.3 784,3 .5090 .0(544 1.5740 146 350.4 327.0 1104.0 8(5(5.5 783.4 .5110 .0(519 1.572!) 148 357.4 328.7 1194.3 805.0 782.4 .5123 1.0595 1.5718 150 358.5 329.8 1104.0 8(54,8 781. (i .5137 1.0571 1.5708 160 303.0 335.0 1195.8 8(50.8 777.4 .5202 1.045(5 1.5(558 170 308.5 340.2 1107.1 85(5.9 773.2 .5203 1.034!) 1.5(512 180 373.1 345.0 1108.2 853.2 7(59.3 .5321 1.024(5 1.55(57 190 377.0 349.0 1199.3 849.0 7(55.5 .5377 1.014!) 1.552(5 200 381.8 354,1 1200.3 84(5.2 7(52.0 .5430 1.0057 1.5487 210 385.9 358.4 1201.3 842.!) 758.5 .5481 .9908 1 .544!) 220 389.5) 3(52.5 1202.2 839.0 755. 1 .5530 .9884 1.5414 230 393.7 3(5(5.5 1203.0 83(5.5 751.8 .5577 .9803 1.5380 240 397.4 370.4 1203.9 833.5 748.7 .5(522 .972(5 1.5348 250 401.0 374,1 1204,7 830.0 745.7 .5(500 .9(551 1.5317 260 404.5 377.8 1205.5 827.7 742.7 .5708 .9579 1 .5287 270 407.8 381.3 120(5.2 824.!) 730.S .574!) .9510 1.5259 280 411.1 384,7 120(5.9 822.2 737.0 .5788 .9443 1.5231 290 414.3 388.1 1207.0 819.5 734.2 .582(5 .9378 1.5204 300 417.4 391.3 1208.3 817.0 731.5 .58(53 .9315 1.5178 TABLE II tt OF SATURATED STEAM BELOW 212 F. mini: '(IMIMK OF )NK 1'OIINl) ((Ju. FT.) VBIOHT OF ONIJ CUBIC FOOT TOTAL HEAT 1" LATENT HEAT r 'BMP. t Pounds 7 Grains 0.1 S02 32XX 0.000304 2.129 1073.7 1073.7 32 0. 1955 3047 0.000328 2.297 1074.7 1072.7 34 ().2\'\l) 282(5 0.000354 2.477 1075.8 1071.7 36 0.22!) 1 2(523 0.000381 2.669 1076.8 1070.8 38 0.2-17X 2-137 0.000410 2.872 1077.8 1069.8 40 0.2(577 22(55 0.000442 3.091 1078.8 1068.8 42 0.2X91 2107 0.000475 3.322 1079.8 1067.7 44 0.311!) Mil 0.000510 3.570 1080.8 1066.7 46 0.33(5(5 1 827 0.000547 3.831 1081.8 1065.7 48 0.3(527 1703 0.000587 4.110 1082.8 1064.7 50 0.3905 15S7 0.000(530 4.411 1083.8 1063.7 52 0.-1202 14X3 0.000(574 4.720 1084.7 1062.7 54 1 3X5 0.000722 5.054 1085.7 1061.6 56 o''lX r (i 1 204 0.000773 5.410 1086.7 1060.6 58 (K5217 1210 0.000827 5.785 1087.6 1059.6 60 0.55i)X OJ5-I4 1132 10(50 993 0.000883 0.000943 0.001007 6.043 6.604 7.048 1088.6 1089.6 1090.5 1058.5 1057.5 1056.4 62 64 66 ()'.(5X!) 0.73X 931 X73 0.001074 0.001145 7.52 8.02 1091.5 1092.4 1055.4 1054.3 68 70 0.7X9 O.X.M O.!)03 0.9()-l 1.02!) X20 770 723 (5X0 (53!) 0.001220 0.001300 0.001383 0.001471 0.0015(54 8.54 9.10 9.68 10.30 10.95 1093.3 1094.3 1095.2 1096.1 1097.1 1053.3 1052.2 1051.2 1050.1 1049.0 72 74 76 78 80 1.0! )X 1.171 I. IMS 1.32!) (501.4 5(55.7 532.2 500.X 471.4 0.001603 0.001708 0.001879 0.001997 0.002121 11.64 12.37 13.15 13.98 14.85 1098.0 1098.9 1099.8 1100.7 1101.6 1048.0 1046.9 1045.8 1044.7 1043.6 82 84 86 88 90 1.50(5 1.1502 1 .704 LSI 2 1.925 443.9 41X.2 394.2 371.X 350.9 0.002253 0.002391 0.002537 0.00265)0 0.002850 15.77 16.74 17.76 18.79 19.95 1102.5 1103.4 1104.3 1105.2 1106.1 1042.5 1041.4 1040.3 1039.2 1038.1 . 92 94 96 98 100 TEMP. FAHH. t PHESSUUE VOLUME OF ONE POUND (Cu. FT.) v" WEIGHT OF UNB UUBIC FOOT TOTAL HEAT q" LATENT HEAT r Lb. per Sq. In. P Inches of Eg. Pounds 7 Grains 102 1.004 2.044 331.4 0.003017 21.12 1107.0 1037.0 104 1.066 2.171 313.2 0.003193 22.35 1107.9 1035.9 106 1.131 2.303 296.2 0.003376 23.63 1108.7 1034.8 108 1.199 2.441 280.4 0.003566 24.96 1109.6 1033.7 110 1.271 2.588 265.6 0.003765 26.36 1110.5 1032.5 120 1.689 3.439 203.4 0.004916 34.42 1114.8 1026.9 130 2.219 4.518 157.5 0.00635 44.45 1119.0 1021.1 140 2.885 5.874 123.1 0.00812 56.86 1123.1 1015.2 150 3.714 7.56 97.2 0.01029 72.0 1127.1 1009.3 160 - 4.737 9.64 77.4 0.01293 90.5 1131.1 1003.2 170 5.988 12.19 62.09 0.01611 112.7 1135.0 997.1 180 7.506 15.28 50.23 0.01991 139.4 1138.8 990.9 190 9.335 90.01 40.94 0.02443 171.0 1142.5 984.6 200 11.523 23.46 33.60 0.02976 208.3 1146.2 978.2 210 14.122 28.75 27.77 0.03601 252.1 1149.7 971.7 212 14.697 29.92 26.75 0.03738 261.7 1150.4 970.4 FQ W . PH H s a o fc O a I J H ^ * >o eO<N(M THrH I 1 I r-H T-l (N CJ CO CO -^ * U3 1O O ? l> t- 00 00 <T ThiCOCOCOCOC^C<l<MC<li IT-HI it li iO OOO>C^O3O3OG ooooooooooooooooooooooooc I I I I I I I I I I I I I - CD CD lO O * >* CO CO <N i 1 II 1 I 1 1 II rH (N (M CO CO * -^ O C3 T-H * (M CO CO iM 00 <M O I-H CD 00 O <N INDEX [The numbers refer to pages] Absolute scale, Kelvin's, 55. temperature, 18. zero, 18. Acoustic velocity, 257. Adiabatic change, defined, 40. expansion of gas, 103. of vapor mixture, 185, 189. of superheated steam, 218. irreversible, 75. of air and steam mixture, 233. of superheated steam, approximation to, 220. of vapor mixture, approximation to, 190. on TS-plane, 70. with variable specific heat, 126. Air and steam, mixture of, 232, 236. compression, 152. engine cycles, analysis of, 140. engines, classification of, 137. moist, constants for, 230. moisture in, 228. refrigeration, 149. required for combustion, 119. Allen dense-air refrigerating machine, 150. Ammonia, saturated, 180. superheated, 223. Andrews' experiments, 198. Atomic weights, 111. Availability of energy, 46. Available energy of a system, 56. Bertrand's formulas, 168. Biot's formula, 167. Boltzmann's interpretation of the second law, 65. Boyle's law, 89. Brayton cycle, 145. Callendar's equation for superheated steam, 204. Calorimeter, throttling, 271. Caloric theory, 3. Carbon dioxide, saturated, 182. Carnot cycle, 50, 134. for saturated vapors, 283. on TS-plane, 73. engine, efficiency of, 54. Carnot's principle, 52. Characteristic equation, 16. of gases, 93, 277. surface, 20. Charles' law, 90. Chemical energy, 5. Clapeyron-Clausius formula, 178. Clausius' equation, 200. inequality of, 63. statement of the second law, 50. Combustion, 117. air required for, 119. products of, 119. temperature of, 127. Compound compression of air, 156. Compounding of steam turbines, 296. Compressed air, 152. engines, 158. Compression, compound, 156. refrigerating machine, 308. Conduction of heat, waste in, 57. Conservation of energy, 6. Constant energy curve of mixture, 187. Constant volume curve, 186. Continuity, equation of, 244. Coordinates defining state of system, 15. Critical states, 197. temperature, volume, and pressure, 199. Cycle, Carnot, 50, 134. Diesel, 146. Joule, 145. Lenoir, 162. Otto, 142. processes, 72, 133. Rankine, 284. rectangular, 73. Cycles, isoadiabatic, 136. of actual steam engine, 290. of air engines, analysis of, 140. of gas engines, comparison of, 148. with irreversible adiabatics, 75. Cylinder efficiency, 293. Curtis type of steam turbine, 304. Curve, constant volume, of steam, 186. of heating and cooling, 70. polytropic, 71. saturation, 166, 182. Curves, specific heat, superheated steam, 209, 211. 324 INDEX Dalton's law, 114, 228. Davis formula for heat content, 177, 274. Degradation of energy, 7. Degree of superheat, 165, 196. De Laval nozzle, 258. Derivative ^- 170. Design of nozzles, 264. Diesel cycle, 146. Differential equations of thermodynam- ics, 82, 84. expressions, interpretation of, 28. inexact, 30. Differentials of u, i, F and $, 79 . Dissociation, 197. Dupre-Hertz formula, 168. Efficiency, conditions of maximum, 135. cylinder, 293. of Carnot engine, 54. potential, 292. ratio, 292. thermal, 291. standards, 291. Electrical energy, 5. Energy, availability of, 46. chemical, 5. conservation of, 6. degradation of, 7. dissipation of, 8. electrical, 5. Energy equation, 36. applied to cycle process, 39. applied to vaporization, 170. integration of, 38. Energy, heat, 3. high grade, and low grade, 7. mechanical, 2. of gases, 97. of saturated vapor, 172. of superheated steam, 214. relativity of, 2. transformations of, 5. units of, 8. units, relations between, 10. Engine, compressed air, 158. Ericsson's, 139. Stirling's, 138. Engines, gas, 142. hot-air, 138. steam, 283. Entropy, as a coordinate, 68. first definition of, 59. of gases, 100. of liquid, 179. of superheated steam, 215. Equation of Clausius, 200. of perfect gas, 17. of van der Waals, 20, 200. of vapor mixture, 184. Equations for gases, 94. for discharge of air and steam, 255. for superheated steam, 203. general, of thermodynamics, 79. Equilibrium of thermodynamics systems, 87. Ericsson's air engine, 139. Exact differentials, 30. Expansion of gases, adiabatic, 103. at constant pressure, 101. isothermal, 102. Expansion valve, 272, 309. Exponent n, determination of, 108. External work of a system, 37. First law of thermodynamics, 35. Fliegner's equations for flow of air, 255. Flow of air, equations for, 255. Flow of fluids, assumptions, 244. experiments on, 243, 254. formulas for discharge, 255 fundamental equations, 244. graphical representation, 247. through orifices, 252. Flow of steam, Grashof's equation, 256. Rateau's equation, 256. Napier's equation, 257. Free expansion of gases, 58. Friction in nozzles, 262. Frictional processes, 74. Fuels, 118. Gas, characteristic equation of, 93, 277. constant B, value of, 92. constant, universal, 113. constants, relations between, 112. free expansion of, 58. permanent, 89. Gas-engine cycles, comparison of, 148. Gases, entropy of, 100. general equations for, 94. heat content of, 99. intrinsic energy of, 97. laws of, 89. mixtures of, 114. specific heat of, 96, 124. Graphical representation of energy equa- tion, 43. of flow of fluids, 247. Grashof's equation, flow of steam, 256. iat content 01 gases, yy. of saturated vapor, 173, 177. of superheated steam, 210. iat, effects of, 35. Intent, 20. mechanical equivalent of, 11. mechanical theory of, 3. af liquid, 171, 174. sf vaporization, 171, 175. specific, 24. total, 172, 177, 213. units of, 9. jilting of air by internal combustion, 141. sating value of fuels, 118. inning's formula for latent heat, 176. >lborn and Hcnning's experiments, 205. >t-air engines, 138. imidity, 229. equality of Clausius, 63. (xsrnul combustion, heating by, 141. trinsic energy, 30. of gases, 97. of superheated steam, 214. of vapors, 172. evorsiblo adiabatics, 75. processes, 47. processes, waste in, 57. Kidiabatic cycles, 130. nlynamic change of vapor, 190. processes, 42. >rnetric lines, 22. >piestio lines, 22. >thermal, definition of, 21. sxpansion of gases, 102. of superheated steam, 217. of vapor mixture, 188. 311 7Y>-plane, 70. sf steam and air mixture, 232. ' i, work of, 298. ulc's cycle, 145. experiments, 11. law, 90. tile-Thomson coefficient, 276. effect, 275. ilvin's absolute scale, 55. statement of the second law, 50. loblauch's experiments, 201. loblauch and Jakob's experiments, 205. ngcn's equations for specific heat, 124, 205. tent heat, 26. external, 172. Hemline's formula for, 176. Latent heat, internal, 172. of expansion, 27. of pressure variation, 27. of vaporization, 171, 175. Lenoir cycle, 162. Linde's process for liquefaction, 280. Liquefaction of gases, 280. Liquid curve, 166. Mallard and Le Chatelier's experiments, 205. Marks' formula, 170. Maxwell's thermodynamic relations, SO Mean specific heat, 210. Mechanical energy, units of, 9. Mechanical equivalent of heat, 11. theory of heat, 3. Mixture of gases and vapors, 228. of gases, specific heat of, 125. of steam and air, 232, 236. Moist air, constants for, 230. Moisture in atmosphere, 228. Molecular specific heat, 123. weights, 111. Mollier's chart, 223. use in flow of fluids, 251. use in steam turbines, 302. Munich experiments, 201. Napier's equations, flow of steam, 257. Nozzle, De Laval, 258. Nozzles, design of, 264. friction in, 262. Otto cycle, 142, 148. Peake's throttling curves, 273. Perfect gas, definition of, 18. equation of, 17. Permanent gas, explanation of term, 89. Perpetual motion of first class, 6. of second class, 8. Polytropic change of state, 104. changes, specific heat in, 106. curve, 71. Potential efficiency, 292. thermodynamic, 77, 87. Pressure and temperature, relation be- tween, 167. Pressure compounding, 296. critical, 199. turbines, action of, 298, 305. Products of combustion, 119. Quality of mixture, 165. variation of, 185. 's cycle, 284. effect of changing pressure, 289. incomplete expansion, 288. with superheated steam, 286. a, 168. i formula, flow of steam, 286. alar cycle, 73. iting machine, analysis of, 311. ition, air, 149. i used in, 310. apor media, 308. I heat engine, 74. le processes, 47. 3 and Moorby's experiments, 12. .'a experiments, 11. ingine, 294. :nant's hypothesis, 254. d vapor, 165. energy of, 172. entropy of, 179. heat content of, 173, 177. latent heat of, 171, 175. specific heat of, 182. surface representing, 166. total heat of, 172, 177. >n curve, 166, 182. nature, 165. iw of thermodynamics, 50. tiann's interpretation of, 65. ticat, 24. curves, 209, 211. in polytropic changes, 106. Langen's formulas for, 124. mean, 210. heat, molecular, 123. of gaseous mixture, 125. of gaseous products, 123. of gases, 96. of saturated vapor, 182. of superheated steam, 204, 273. volume of vapors, 177. id air, mixture of, 232, 236. I temperature of, 199. 3 volume of, 177. 180. il properties of, 173. teat of, 172, 177. irbine, 294. classification of, 295. compared with reciprocating engine 294, compounding, 296. Curtis type, 304. impulse and reaction, 296. influence of high vacuum, 307. low pressure, 307. Steam turbine multiple stage, 302. pressure type, 298, 305. single stage, 300. velocity and pressure, 296, Stirling's engine, 138. Sulphur dioxide, saturated, 182. superheated, 223. Superheat, degree of, 165, 196. Superheated ammonia, 223. Superheated steam, 165, 196. changes of state, 216. energy of, 214. entropy of, 215. equations for, 203. heat content of, 210. specific heat of, 204, 273. tables and diagrams, 221. total heat of, 213. Superheated sulphur dioxide, 223. vapor, characteristics of, 196. Surface, characteristic, 20. representing saturated vapor, 166 System, defined, 15. state of, 15. Temperature, absolute, 18. and pressure, relation between, 167. critical, 199. Kelvin scale of, 55. Temperature of combustion, 127. saturation, 165. scales, comparison of, 91. Temperature entropy representation, 68. Thermal capacities, relation between, 27. capacity defined, 24. efficiency, 291. energy, 4. lines, 21. properties of steam, 173. Thcrmodynamic degeneration, 8. potentials, 77, 87. relations, 80. Thermodynamics, first law of, 35. general equations of, 84. scope of, 1. second law of, 50. Throttling calorimeter, 271. curves, 273. loss due to, 269. processes, 268. Total heat of saturated vapor, 172, 177. of superheated steam, 213. Transformations of energy, 5. Tumlirtz equation for superheated steam, 204. Turbine, steam, see Steam turbine. Units of energy, 8. of heat, 9. Universal gas constant, 113. uum, influence of, on steam turbine, 307. i der Waals' equation, 20, 200. ior, energy of, 172. itropy of, 179. [tit content of, 173, 177. itent heat of, 171, 175. ior mixture, acliabatic expansion of, 189. instant volume change, 189. jrvos on TiS-planc, 186. cm-nil equation of, 184. lodynamic of, 190. lothurniiil expansion of, 188. >or refrigerating machine, 311. jporhotitcd, 196. :>tal heat of, 172, 177. )orization, heat of, 171, 175. Vaporization, process of, 164. Vapors used in refrigeration, 310. Velocity compounding, 297. Volume, critical, 199. specific, of vapor, 177. Waste in irreversible processes, 57. Water, critical temperature of, 199. jacketing, 155. vapor, thermal properties of, 173. Wiredrawing, 268. Work, conversion of, into heat, 57. external, of expansion, 37. of a jet, 298. Zero curve, 186. Zeuner's equation for superheated steam, 204.