Class ~TJz 6^ Book Copight]^" COPVRIGHT DEPOSIT american ^eclianical CDngineering §)erie0 MORTIMER E. COOLEY GENERAL EDITOR PRINCIPLES OF THERMODYNAMICS BY G. A. GOODENOUGH, M.E. M PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY OF ILLINOIS NEW YORK HENRY HOLT AND COMPANY 1911 /^- Copyright, 1911 BY HENRY HOLT AND COMPANY .y II- ^'cuaosfis? 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 irreversibility and loss of availability is pointed out, and by means of the Carnot cycle a measure of availability is obtained. Entropy is then defined in terms of unavailable energy, and iv PREFACE from this fundamental definition the usual definition of the entropy of a non-isolated system as the integral I -~ is easily 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 new 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 Clausius relation [ -^ ] = — AT( -t-^S) and equations for the specific heat, \dp J rp \oI Jp entropy, energy, and heat content of superheated steam are 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 thorough understanding of this article should enable the student to pursue the subsequent mathematical discussions with intel- ligence and ease. 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 Brj^an, 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. Sclialler also gave valuable assistance in getting the book through the press. G. A. GOODENOUGH. Urbana, III., July, 1911. CONTENTS CHAPTER I Energy ART. 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 24. Statement of the First Law 35 25. Effects of Heat 35 26. Intrinsic Energy 36 vii viii CONTENTS ART. PAGE 27. External Work . . . . ' . . . . . .37 28. Integration of the Energy Equation 38 29. Energy Equation applied to a Cycle Process 39 30. Adiabatic Processes 40 31. Isodynamic Changes 42 32. Graphical Representations .42 CHAPTER IV The Second Law of Thermodynamics 33. Introductory Statement 45 34. Availability of Energy » 46 35. Reversibility . .47 36. General Statement of the Second Law .49 37. Carnot's Cycle 50 38. Carnot's Principle 52 39. Efficiency of the Carnot Cycle 54 40. Available Energy and Waste 56 41. Entropy .58 42. Second Definition of Entropy ........ 60 43. The Inequality of Clausius 63 44. Summary 63 45. Boltzmann's Interpretation of the Second Law .... 65 CHAPTER V Temperature Entropy Representation 46. Entropy as a Coordinate 68 47. Isothermals and Adiabatics 69 48. The Curve of Heating and Cooling . . . . . .70 49. Cycle Processes 72 50. The Rectangular Cycle 73 51. Internal Frictional Processes . . . . . . . .74 52. Cycles with Irreversible Adiabatics 75 53. Heat Content 76 CHAPTER VI General Equations of Thermodynamics 54. Fundamental Differentials 79 55. The Thermodynamic Relations 80 56. General Differential Equations 82 57. Additional Thermodynamic Formulas 84 58. Equilibrium 87 CONTENTS ix 4 CHAPTER Vir Properties of Gases ART. PAGE 59. The Permanent Gases 89 60. Experimental Laws .89 61. Comparison of Temperature Scales . 91 62. Numerical Value of B 92 63. Forms of the Characteristic Equation -.93 64. General Equations for Gases 94 65. Specific Heat of Gases 96 66. Intrinsic Energy 97 67. Heat Content 99 68. 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 90. Classification of Air Engines 137 91. Stirling's Engine 138 92. Ericsson's Engine . . . ' . . ' . . . . . 139 X CONTENTS ART. PACK 93. Analysis of Cycles . . 140 94. Heating by Internal Combustion 141 95. The Otto Cycle 142 96. The Joule, or Brayton, Cycle 145 97. The Diesel Cycle . . . . 146 98. Comparison of Cycles 148 99. Closer Analysis of the Otto Cycle .148 100. Air Refrigeration 149 101. Air Compression 152 102. Water Jacketing . 155 103. Compound Compression . . . . . , . .156 104. Compressed-air Engines 158 105. T^AS-Diagram of Combined Compressor and Engine . . .158 CHAPTER X Saturated Vapors 106. The Process of Vaporization .... 107. Functional Relations. Characteristic Surfaces 108. Relation between Pressure and Temperature 109. Expression for dp dt 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 119. Properties of Saturated Ammonia . 120. Other Saturated Vapors . 121. Liquid and Saturation Curves 122. Specific Heat of a Saturated Vapor 123. General Equation for Vapor Mixtures 124. Variation of x during Adiabatic Changes 125. Special Curves on the T/S-plane 126. Special Changes of State 127. Approximate Equation for the Adiabatic of a Vapor Mixture 164 166 167 170 170 173 173 174 175 177 177 179 180 180 181 182 182 184 185 186 188 190 CHAPTER XT Superheated Vapors 128. General Characteristics of Superheated Vapors 129. Critical States 196 197 CONTENTS XI ART. 130. Equations of van der Waals and Clausius < 131. Experiments of Knoblauch, Linde, and Klebe 132. Equations for Superheated Steam . 133. Specific Heat of Superheated Steam 134. Mean Specific Heat 135. Heat Content. Total Heat 136. Intrinsic Energy 137. Entropy .... 138. Special Changes of State 139. Approximate Equations for Adiabatic Changes 140. Tables and Diagrams for Superheated Steam 141. Superheated Ammonia and Sulphur Dioxide PAGE 200 201 203 204 210 210 214 215 216 220 221 223 CHAPTER XII Mixtures of Gases and Vapors 142. Moisture in the Atmosphere . 143. Constants for Moist Air . 144. Mixture of Wet Steam and Air 145. Isothermal Change of State . 146. Adiabatic Change of State 147. Mixture of Air with High-pressure Steam 228 230 232 232 233 236 CHAPTER XIII The Flow of Fluids 148. Preliminary Statement 243 149. Assumptions 244 150. 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 . . . . 2.57 156. The de Laval Nozzle 258 157. Friction in Nozzles 262 158. Desiorn of Nozzles . . 264 CHAPTER XIV Throttling Processes 159. Wiredrawing . .268 160. Loss due to Throttling 269 161. The Throttling Calorimeter 271 xii CONTENTS ART. PAGE 162. The Expansion Valve . . . . . . . . .272 163. Throttling Curves . .273 164. The Davis Formula for Heat Content 274 165. The Joule-Thomson Effect 275 166. Characteristic Equation of Permanent Gases .... 277 167. Linde's Process for the Liquefaction of G-ases ...» 280 CHAPTER XV Technical Applications, Vapor Media The Steam Engine 168. The Carnot Cycle for Saturated Vapors 169. The Rankine Cycle 170. The Rankine Cycle with Superheated Steam 171. Incomplete Expansion 172. Effect of Changing the Limiting Pressures . 173. Imperfections of the Actual Cycle . 174. Efficiency Standards 283 284 286 288 289 290 291 The Steam Turbine 175. Comparison of the Steam Turbine and Reciprocating Engine . 294 176. Classification of Steam Turbines 295 177. Compounding 296 178. Work of a Jet 298 179. Single-stage Velocity Turbine 300 180. Multiple-stage Velocity Turbine 302 181. Turbine with both Pressure and Velocity Stages .... 304 182. Pressure Turbine 305 183. Influence of High Vacuum .307 Refrigeration with Vapor Media 184. Compression Refrigerating Machines 308 185. Vapors used in Refrigeration 310 186. Analysis of a Vapor Machine 311 SYMBOLS Note. The following list gives the symbols used in this book. In case a magnitude is dependent upon the weight of the substance, the small letter denotes the magnitude referred to unit weight, the capital letter the same magnitude referred to M units of weight. Thus q denotes the heat absorbed by one pound of a substance, Q = Mq, the heat absorbed by M pounds. ./, Joule's equivalent. .4, reciprocal of Joule's equivalent. 3/, 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. (p, V, T) = 0, (1) or written in the explicit form p=f(y,T}. (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. 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 (ji?^, ^q), point B the final condition (jt?, ^). Then CB=p-p^,AC t-t^, and—: t -L ^=k. 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. THERMAL CAPACITIES [chap, ii constant h. Evidently k is independent of p and t^ but it may depend upon v, hence we write Substituting this value of k in (1), we get In this equation t and t^ are temperatures measured from the Fahrenheit zero ; that is, from the origin (Fig. 2). Evidently the difference t— t^ 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 zero such that 2^= when p = 0. This is evidentl}^ equivalent to the selection of a new origin 0' (Fig. 2) at the intersection of the line AB with the ^-axis. If we now take the initial point A at 0', we have jo^ = 0, Tq = 0, and (2) takes the form whence pv= Tvf(y). (3) By hypothesis, the substance follows Boyle's law ; that is, the product pv is constant when the temperature T is constant. From (3), therefore, the factor vfQv') is a constant ; and denot- ing this constant by B we have pv = BT, (4) 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, and tempera- tures measured from it are called absolute temperatures. The absolute zero may be physically interpreted as follows : By the kinetic theory, the pressure of a gas is due to the impact of its ART. 15] ABSOLUTE 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 ^ = tT' (1) and P^^'^1^. (2) Pi ^1 Let us take T^ as the temperature of melting ice, T^ that of boiling water at atmospheric pressure. Regnault's experi- ments on the increase of pressure of air when heated at con- stant volume gave the relation ^^^fyp? = 1-3665. (8) jt?i (at 0° C.) ^ ^ Since for the C. scale ^2-1^1 = 100, 0.3665 ». 100 we have ^r^^ = ^^' W 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 slightl}^ different values for Ty 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 T^ would be between 273.1 and 273.14. The corre- sponding value on the Fahrenheit scale may be taken as 491.6 ; that is, the absolute zero is 491. 6°f below the temperature of melting ice, or 459.6° below the ordinary F. zero. If then we 20 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii denote ordinary temperatures by t and absolute temperatures by T^ we have T=t-\- 273.1, for the C. scale. T= t -h 459.6, for the F. scale. 16. Other Characteristic Equations. — The equation pv = BT 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 Waals 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 .-,. in which B^ a, and h are constants which depend upon the nature of the substance. An empirical equation for superheated steam is p(v + c-)^BT-pO- + ap-)^^ (2) It will be observed that for large values of T and v, that is, when the gas is extremely rarified, the last 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 equation . (^ (^, V, T) = 0, having three variables, may be represented geometri- cally by a surface. A state of the substance is defined ART. 18] THERMAL LINES 21 by its coordinates p^^ v^ I\, and this state is therefore repre- sented by a point, on the surface. If the state changes, a second point with coordinates jt?2, v^^ T^^ 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 pv = BT 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 jo?;-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 p?; = 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 t'^-plane, and the projection of the path on the ^v-plane will be a straight line parallel to (9F; as AB (Fig. 4). The relation between volume and temperature is found by combining the equation p = const. with the characteristic equation of the substance c\> (p, V, T) = 0. Thus for a perfect gas, pv = BT, and p = C. Fio. 4. 22 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii Substituting this value of p in the characteristic equation, w& have B T. (2) If the substance changes its state at constant volume, the state-point moves parallel to the ^2^-plane, and the projection of the path on the ^v-plane is a line parallel to the j?-axis, as CD (Fig. 4). In the case of a perfect gas, the relation between p and T for a change at constant pressure is * (3) Lines of constant pressure are called isopiestic lines ; lines of constant volume, isometric lines. Besides the cases just given, others are 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. We 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 heat from external sources. If we denote by q heat thus absorbed per unit weight, we may by giving q proper signs cover all possible cases ; thus + q indicates heat absorbed, ~ q heat rejected ; while if ^ = 0, we have the limiting adiabatic change of state. The heat absorbed may be determined from the changes in two of the three variables p^ v^ 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 v be taken as the independent variables, and let the curve AB (Fig. 5) represent oh the ^v-plane a change of state. Suppose an element PR of Fig. 5. ART. 19] HEAT ABSORBED DURING A CHANGE OF STATE 23 this curve to be replaced by the broken line PQR, then the segment PQ represents an increment of volume ^v 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 ( — ) , the subscript t indicating that t is held \dvjt 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 by PQ. Similarly, the rate of absorption along QR is ( — ) , \dtjv and the heat absorbed is the product [-^j A^. The heat ab- sorbed during the change PQR is, therefore, and the total heat absorbed along the broken path from A to B is given by the summation X dvj \dtj . (2) Ht). By taking the elements into which the curve is divided smaller and smaller, the broken path may be made to approach the actual path between A and B. Therefore, passing to the limit, we have instead of (1) *+(|)/^' (3) and for the heat absorbed during the change of state from A »=c:to,*Ki),]*- . « By choosing other pairs of variables as independent, other equations similar to (3) may be obtained. Thus, taking t and », we have . . ,^ . 24 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii or taking jp and v as the independent variables, we have From (5) and (6) equations corresponding to (4) may be readily derived. 20. Thermal Capacity. Specific Heat. — Of the partial deriv- atives introduced in the preceding article, two are of special importance, namely, ( — -) and (— ) • In general, the heat \dtjv \^tjp 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 heat ab- sorbed by a body during a rise of temperature from t^ to t^^ the quotient — — — gives the mean thermal capacity of the body ; and the quotient — - — — = , the mean thermal capacity of a unit weight. If the thermal capacity varies with the tem- perature, then the limiting value of the quotient , that h ~ h is, the derivative — ^, gives the instantaneous value of the ther- eto mal capacity. Accordingly, we recognize in the derivative -^ ) the thermal capacity per unit weight of the body under the condition that the volume remains constant; and in the derivative f -^ J the thermal capacity with the pressure constant. According to the definition of the thermal units (Art. 10), the thermal capacity of 1 gram of water at 17.5° C. is 1 calorie, and that of one pound of water at 63.5° F. is 1 B. t. u. The specific heat of a substance at a given temperature t is the ratio of the thermal capacity of the substance at this tem- perature to the thermal capacity of an equal mass of water at some chosen standard temperature. If Ave take 17.5° C. (63.5° F.) as the standard temperature, and denote by 7 ther- KatJ ART. 20] THERMAL CAPACITY. SPECIFIC HEAT 25 mal capacity per unit weight, then the specific heat c is given by the relation _ 7^(of subtance) 717.5(0^' water)* But for water 7-^^ 5 = 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., and the specific heat is = 7ioo 1.005 cal. = 1.005. On account Fig. 0. 7i7.5 1 cal. of this numerical equality, we may consider that the derivative -^ represents the specific heat, as well as the thermal capacity. (XTj It is to be 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, ^ on the other hand, involves energy measurements. The specific heat of a sub- stance may be represented geo- metrically, as shown in Fig. 6. Starting from some initial state, O' let the rise of temperature be taken as abscissa and the heat added to the substance as ordinate. The resulting curve OM w'iVl represent the equation and the slope of the curve at any point, as P, will give the de- rivative — ^, or the specific heat at the temperature correspond- (XZ 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 f-axis. The heat applied to a substance, as will be shown presently, may have other effects than raising the temperature. The specific heat, however, is numerically the ratio of the heat supplied, whatever its effects upon the body, to the rise of 26 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii temperature ; hence, the value of the specific heat will depend upon the conditions under which the heat is absorbed. If the substance is in the solid or in the liquid form, the specific heats are practically equal. For substances in the gaseous form, however, the specific heat may have any value from — oc to + oc, depending upon the external conditions under which the heat is supplied. 21. Latent Heat. — If the heat added to a substance and the temperature be plotted as in Fig. 6, it may happen that at cer- tain temperatures the curve has discontinuities. For example, let heat be applied to ice at 0° F. The curve is practically a straight line until the temperature 32° is reached, but at this point considerable heat is added without any change in temperature. During this addition of heat, rep- resented by the vertical Pj(. rj^ segment AB (Fig. 7), the state of aggregation changes from solid to liquid. As the water receives heat its temperature rises, as indicated by BC, until tlie temperature 212° F. is reached (assuming atmospheric pressure), where 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 heat that is thus added to (or abstracted from) a substance 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 -^ 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 added with respect to the volume, that is, the derivative f ^ ART. 22] RELATIONS BETWEEN THERMAL CAPACITIES 27 is a thermal capacity called the latent heat of expansion and denoted by l„. If the pressure also changes, we have in the derivative (|)^ the heat added per unit ehange of pressure. This thermal capacity is called the latent heat of pressure varia- tion, and is denoted by l^. 22. Relations between Thermal Capacities. — Introducing the symbols c^, c^, l^, and l^ in equations (3) and (5) of Art. 19, we have dq = l^dv + CydT^ ' (1) dq = Ipdp + CpdT. (2) By means of the characteristic equation of the substance, namely, v=fCT,p-), (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, dv = ^dT^^-^dp, (4) dT dp ^ ^ ^ which substituted in (1) gives dq=l,^^dp+{c, + l,^dT. (5) Comparing (2) and (4), we have l^^K^, (6) Cp-o^^lvj^' (T) In the same way, substituting dp=%dT+fdv ^ dT dv in (2), and comparing the resulting equation with (1), we obtain K=i/i, (8) 28 CHANGE OF STATE. THERMAL CAPACITIES [chap, n The relations thus obtained enable us to calculate the remain- ing thermal capacities when any one is given by direct experi- ment, provided the characteristic equation of the substance is dv dp known, so that the derivatives — ^, --^, etc., can be determined. o 1 o 1 For a perfect gas, as an example, c^ is known from experiment and the ratio — has also been determined. From the equation of the gas pv = BT^ we have the partial derivatives dv _B dp_B^ hence from (7) and (9) B p c^-€^ = l or l^ = ^(Cp- tO , (10) and h = -^C^p- ^v)' (11) 23. Interpretation of Differential Expressions. — In thermo- dynamics we frequently meet with expressions of the form Mdx 4- JSfd^ composed of two terms, of which each is the differential of a variable multiplied by a coefficient. The two coefficients may be constants or functions of the two variables involved. The proper interpretation of differentials of this form is likely to present difficulties to the student ; we shall, therefore, devote this article to a discussion of such expressions, their properties, and their physical interpretations. Let us consider first how such differential expressions may arise. Suppose we have given the characteristic equation of a substance in the form p=/C<',0; (1) by differentiation according to the well-known methods of cal- culus, we obtain the relation dp = t^dv + %dt, (2) ART. 23] DIFFERENTIAL EXPRESSIONS 29 which may be written in the form, dp = Mdv + Ndt, (3) where M= ^, and ]V=-^' dv dt In Art. 19 we derived an equation of similar form, namely, dq = ^dv+^^dt, (4) which may likewise be written in the form dq = M'dv + JST'dt. (5) The second members of (3) and (5) are differential expressions of the form Mdx + JVd^^ which we have under consideration. Eq. (3) was produced from a known functional relation be- tween j9, V, and ^, 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 Aq that must necessarily be absorbed. No relation between g, 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 |? is a function of v and t [p —f(y^ 0]? the pressures at A and B are determined by the values of T and v at those points ; thus for a perfect gas, p-^ = 1 and p^= ^. Hence, the change of pressure p^ — P\ ^^ passing from Ato 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 ^, that is, ii q = c^) (v, ^), we shall have at A, ^^ = c^(v^, ^j), at B, g2 = <^(^2' ^2)- Therefore, the heat absorbed in passing from Aio B will be 92-9i = 't> (^2' ^2) - ^ (^1^ ^1)' (^) 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 30 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii q = ^('t;, f). As a matter of fact, the lieat absorbed is different for different paths between the same initial and final states ; hence it. is not possible to express q in terms of v and t. The conclusions just given may be stated in general terms as follows. Given an expression of the form du = Mdx-^ Ndy, , (7) where the coefficients M and N are functions of x and ?/, there may or may not exist a functional relation between u and the variables x and y. If ?^ is a function of x and y^ say u = FQc^ ^), then the change in u depends only on the initial and final values of x and y and is independent of the path. This change is found from (7) by integration ; thus r^ du = f "^' ^=^ (Mdx + JVdy^ . (8) In this integration no relation between x and y is required, for since Mdx + JVdy arises from differentiating the function (j) (x^ y), the integral must be (/> (rr, ?/). In this case Mdx -\-Ndy is said to be an exact differential. As an example, consider the equation du — ydx + xdy. Since ydx + xdy is produced by the differentiation of the prod- uct xy^ we have the relation u — xy-\-C^ whence u^— u^z= x^y^ — x^y^ The change of u is represented by the shaded area (Fig. 8), and is evidently not dependent upon the path between the points ioc^rV^) (^1, «/i) and (x^, y^. If, however, no functional rela- tion exists between u and the variables x and ?/, then Mdx + Ndy is said to be an inexact differential. In this case a value of u cannot be found until a relation between x and y is as- ART. 23] DIFFERENTIAL EXPRESSIONS 31 sumed ; in other words, the v^lue 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 Avhich 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 = | dx^^ u=\ \^(^x-{-\)dx — xdx^=l. If we take as the path the parabola y = \x^ -\-l^ we have u = p[(i x^ + l')dx - xMx'] = 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 + Ndy is exact, then u must be a function of x and y^ say /(a;, ^). By differentiation, we have ^ du J du J au= — ax-\ dy. dx dy Hence 31 and W must be, respectively, the partial derivatives du du —- and ^— • By a well-known theorem of calculus, we have dx dy d_fdu\^±fdu\ dy\dxj dx\dyj^ that IS, -_ = -—. (9) 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, f , T of the substance under consideration. From purely physical considerations the energy u of the substance is known to be a function of the state only. 32 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii (See Art. 26.) Hence if u is expressed in terms of two of these coordinates as independent variables, thus, du = Mdv + JYdT, we know at once that du is exact and we can write Furthermore, from the test for an exact differential we must have the relation dT dv' By making use of this test when the differential is known to be exact, many useful relations are deduced. We have also magnitudes that depend upon the coordinates and also upon the method of variation ; that is, upon the path. The heat q absorbed by a system in changing state is one of these. If again we choose v and T as the independent variables, we may write dq = M'dv 4- JST'dT', but since dq is not exact, we cannot write ^^dq = q^- q^. EXERCISES 1. Regnault's experiments on the heating of certain liquids are ex- pressed by the following equations : .^ Ether q = 0.529 t + 0.000296 t'^, - 20° to + 30° C. Chloroform q = 0.232 t + 0.0000507 t\ - 30° to + 60° C. Carbon disulphide q = 0.235 t + 0.0000815 t'^, - 30° to + 40° C. Alcohol q = 0.5476 t + 0.001122 i^+ 0.0000022 t% - 23° to + 66° C. From these equations derive expressions for the specific heat, and for each liquid find the specific heat at 20° C. 2. From the data of Ex. 1, find the mean heat capacity of ether between 0° and 30° C. Also the mean heat capacity of alcohol between 0° and 50° C. 3. If the thermal capacity of a substance at temperature t is given by the relation y = a + ht + ct% show that the mean thermal capacity between 0° and t is given by the relation ART. 23] EXERCISES 33 4. In the investigation of the properties of gases, it is convenient to draw the isothermals {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 pv-p plane the general form of an isothermal of super- heated steam, the characteristic equation being p{v + c)^BT-p{l-^-ap):^^. As an approximate equation for superheated steam, the form p{v + c)^BT, has been suggested by Tnmlirtz. Show the form of the isothermal when this equation is used. 6. Derive relations between c^, c„, Ip, and U, similar to those given by Eq. (10) and (11) of Art. 22, using van der Waal's equation V — v^ as the characteristic equation of the gas. 7. For a perfect gas, as will be shown subsequently, the thermal capacitiy ly is Ap{A — i). Show that c^ — c„ = AB ; also that Ip = — Av. 8. Test the following differentials for exactness : (a) vdp + npdv. (b) v^dp + npv'^-hlv. 9. Find the function u =f(p, T) which produces the differential (c) of Ex. 8. dT r 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' = — ,• dT T 11. For perfect gases, dq = c^dT + Apdo. (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. T REFERENCES Thermal Capacity. Specific Heat Preston : Theory of Heat, 211. Griffiths : Thermal Measurement of Energy, 95. 34 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii Weyrauch : Grundriss der Warme-Theorie 1, 60. Chwolson : Lehrbuch der Physik 3, 172. Exact and Inexact Differentials in Thermodynamics Chwolson : Lehrbuch der Physik 3, 434. Clausius : Mechanical Theory of Heat, Introduction. Preston : Theory of Heat, 597. Weyrauch : Grundriss der Warme-Theorie 1, 28. Townsend and Goodenough : Essentials of Calculus, 245. CHAPTER III TEE 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 IT the work thus obtained, we have, therefore, as symbolic statements of the first law, W^ JQ, ovQ = 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 sj^stem from some external source. Denoting by AIT the work received and hj AQ the heat absorbed, the increment AZ7 of the intrinsic energy of the system is given by the relation AU = JAQ + AW. (1) Ordinarily we take the work done by the system in expanding as positive ; hence the work done on the system during com- pression is negative and (1) takes the form AU = JAQ - AW; (2) 35 36 THE FIRST LAW OF THERMODYNAMICS [chap, iir that is, the increase of energy of the system is equal to the energy received in the form of heat less the energy given to the surrounding systems in the form of work. We may also write (2) in the form JAQ =AU -^ AW, (3) 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) in the performance of external work. Equation (3) is the energy equation in its most general form. Any one of the three terms may be positive or negative. We consider AQ positive when the system absorbs heat, negative when it gives out heat ; as before stated, A TF is positive when work is done % the system, negative when work is done on the system; ACT is positive when the internal energy is increased, negative when the energy is decreased during the change of state. 26. The Intrinsic Energy. — The increase A Uoi the intrinsic energy is, in general, separable into two parts : (1) The in- crease of kinetic energy indicated by a rise of temperature of the system. As we have seen, this is due to an increase in the velocity of the molecules of the system. (2) The increase of potential energy arising from the increase of volume of the system. To separate the molecules against their mutual attrac- tions, or to break up the molecular structure, as is done 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 due to a change of state depends upon the initial and final states only. In Fig. 9, let A represent the initial, and B the final 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 the volume at B determines the potential energy. Hence the final total energy depends upon the coordinates of B and in no way upon the intermediate process represented by the path leading from A ART. 27] EXTERNAL WORK 37 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 U is thus a function of the coordinates only, it follows that d U is always an exact differential. Choosing T and v as the inde- ^ p^^, 9 pendent variables of the system, we may express Z7 as a function of these variables. We have, therefore, U^f(T, v\ whence dU= —-dT -\- dv, dT dv r) TT The term —^dT is the increment of energy due to the in- crease of temperature (il^. The factor-— is the rate at which the energy changes with the temperature when the volume 5 TJ remains constant. Hence ——dT is the change of energy due merely to the rise of temperature, that is, it is the change of kinetic energy. The term ■ dv is the change of energy 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. bv dv 27. The External Work. — In nearly all cases dealt with in applied thermodynamics, the external work A TF 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 AJ^ denote an elementary area on the surface inclosing the system and suppose that during the expansion of the system this area moves in the direction of the normal to it through a distance s. If then p is the 38 THE FIRST LAW OF THERMODYNAMICS [chap, hi normal pressure per unit area, the work done against this pressure is for this one element p^Fs. (1) When all the elements of the surface are taken, the expres- sion for the work is ^W=p^s^F. , (2) But evidently if s be taken sufficiently small, 2s A^ is the increase of volume AF"; hence we may write ATr=jt?AF; (3) from which we have w==^p^v=Qpdv (4) for a change of volume from V^to V^. The external work for a given change of state is represented graphically by the area between the projection of the path of the state-point on the p F'-plane and the F'-axis. Thus in Fig. 10, let the variation of pressure and volume be represented by the curve AB ; this is the projection on the p F-plane of the actual path of the state-point on the characteristic surface. The area A -^ABB^ under AB is clearly given by the integral hence, it represents the work done by the system in passing from the initial to the final state according to the given law. The general energy equa- tion (3), Art. 25, may now be written in the form JAQ=AU^pAV, (5) or using the differential nota- tion, in the form JdQ = dU-{-pdV. (6) For a unit weight of the sub- stance, we have Jdq = du-{- pdv. (6 a) 28. Integration of the Energy Equation. — The heat imparted to a system during a change of state is found by integrating ART. 29] ENERGY EQUATION 39 the equation just derived. Denoting the initial and final states by the subscripts 1 and 2, respectively, we have whence JQ= U^~ U^-\- ^ p dV (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 C/g — U^ The external work W= ^^dV 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 ^^dQ=Q^-Q,. 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 Q-^^^ ^^^^ heat imparted to the sys- tem in passing from state 1 to state 2 ; and likewise by W12 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 F^plane 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 c^F" (along path m), 40 THE FIRST LAW OF THERMODYNAMICS [chap, hi ^1 Fig. 11. which is represented by area A^AmBB^, while in passing from B back to A along path n the external work is J^ pdV= — )y P ^F (along path 9^), and this is represented by area B^BnAA^ Hence the net external work done by the system is represented by the area inclosed by the curve of the cycle. Since the energy U of the system depends upon the state only, the change of energy for the cycle is Y and the energy equation re- duces to That is, for a closed cycle of processes^ the heat imparted to the system is the equivalent of the external work^ and both are repre- sented graphically by the area of the cycle on the pF^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, the change of state is said to be adiabatic. In general, adiabatic changes are possible only when the system is inclosed in a non-conducting envelope. Rapid changes of state are approximately adiabatic, 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 an orifice is practically an adiabatic process. For an adiabatic change, the term JQ of the energy equation reduces to zero, and we have, consequently. or w. 12 U,~U,. (1) During an adiabatic change, therefore, the external work done by the system is gained at the expense of the intrinsic energy of the system. ART. 30] ADIABATIC PROCESSES 41 The projection on the joT^-plane of the path of the state-point during an adiabatic change gives the adiabatic curve. See Fig. 12. The area A^ABB^ represents the work W12 of the system and from (1) it represents also the decrease of the intrinsic energy in passing from state 1 represented by A to state 2 represented by B. Making use of this principle, we can arrive at a graphical represen- tation of the intrinsic energy of a system. Suppose the adiabatic expansion to be con- tinued indefinitely ; the adia- batic curve AB will then approach the F^axis as an asymptote, and the work of the expanding system will be represented by the area A^A oo between the ordinate A^A^ the axis OFJ and the curve extended indefinitely. The area A^A 00 represents also the change of energy resulting from the expan- sion. Hence if we assume that the final energy is zero, we have Z7j — = area A^A 00, Ax B, Fig. 12. or JJ^ = area A^A 00 = 1. P dV. A, Fig. 13. 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 energy decreases and as a result the temperature falls ; hence for the same final volume F^, the temperature, and therefore the pressure, is lower for the adiabatic expansion than for the isothermal expansion. If the two curves are produced indefinitely, each will have the axis OF' as an asymptote. The area under the adiabatic, which represents the initial intrinsic energy of the system, is finite. 42 THE FIRST LAW OF THERMODYNAMICS [chap, m On the other hand, the area under the indefinitely extended isothermal is infinite. 31. Isodynamic Changes. — If the intrinsic energy of the system remains unchanged during a change of state, the change is called isodynamic or isoenergic. In this case the energy equation reduces to the form JQ 12 tf;o = 12 dV. For perfect gases, the isodynamic curve is also the isothermal, but for other substances this is not the case. 32. Graphical Representa- tions. — The three magnitudes ^§12. ^2 C^j, and TF^g enter- ing into the energy equation can be represented graphically by areas on the joF^plane. Suppose the change of state to be represented by the curve m between the initial point A and final point B (Fig. 14). Let adiabatic lines be drawn through A and B and ex- tended indefinitely; then from preceding considerations we have TFi2 = area A^ABB^, U^ = area A^A oo , U^ = area B^B oo . Hence, JQ^^ = U^- U^j^ W-^^ = area A^ABB^ + area B^B oo — area A^A oo = area AB oo . That is, the heat imparted is represented on the p V-plane hy the area included between the path and two indefinitely extended adia- batics drawn through the initial and final points, respectively. Through the initial point A let an isodynamic be drawn, cutting BB^ in the point C, and through let the indefinitely extended adiabatic C ao he drawn. Then the energy C/g of the system in state O is equal to ZJ^, and, therefore, ^3 = area B^B oo — area B^O co area oo CB oo . u^-ir,= u. ART. 32] GRAPHICAL REPRESENTATIONS 43 It should be noted that the area representing U^ — U-^ 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 final point B an adiabatic is drawn, the two lines intersecting at point C. We have then, denoting the energy in the state C by ZZg, TJ^ _ jj^= jj^_ jj^^ area B^BCG^, W[^ = area A^ABB^, JQ^^ = W^^ -\-U^-U^ = area A^ABOC^, As before, the change of energy is independent of the path w, while both 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 '"'-[ff)/^-V^i)M dv. and that consequently the derivative f — j must be equal to Jc„. 2. If the energy of a substance is independent of the volume, show that the energy equation reduces to the form Jdq = Jcyd T + pdv. 3. Using the method of graphical representation, show by areas Q12, U2 — Ui, and W12 (a) for a change at constant pressure, (&) for a change at constant volume. 4. Show graphically that y^, the heat capacity of a gas at constant pressure, is greater than y„, the heat capacity at constant volume. 5. Derive an expression for the external work done when a gas (or vapor) is heated at constant pressure. 6. Derive an expression for the external work when a gas expands from a volume V^ to a volume V2 according to the lawpF" = const. 44 THE FIRST LAW OF THERMODYNAMICS [chap, in 7. Apply the general energy equation to the process of changing ice at 32° F. to water. What is the effect of greatly increasing the pressure on the ice during the process ? REFERENCES Preston : Theory of Heat, 596. Zeuner : Technical Thermodynamics (Klein) 1, 28. Planck: Treatise on Thermodynamics (Ogg), 38. CHAPTER IV THE SECOND LAW OF THERMODYNAMICS 33. Introductory Statement. — While the first law of thermo- dynamics gives a relation that must be satisfied during any change of state of a system, and of itself leads to many useful results, it is not sufficient to set at rest all questions that may arise in connection with energy transformations. It gives no indications of the direction of a physical process ; it imposes no conditions upon the transformations of energy from one form to another except that there shall be no loss, and thus gives no in- dication of the possibilities of complete transformation of dif- ferent forms; it furnishes no clue to the availability of energy for transformation under given circumstances. To settle these questions a second principle is required. This principle, called the second law of thermodynamics, has been stated in many ways. In effect, however, it is the principle of degradation of energy, just as the first law is the principle of the conservation of energy. There are conceivable processes which, while satisfying the requirements of the first law, are declared to be impossible be- cause of the restrictions of the second law. As a single ex- ample, it is conceivable that an engine might be devised that would deliver work without the expenditure of fuel, merely by using the heat stored in the atmosphere; in fact, such a device has been several times proposed. The first law would not be violated by such a process, for there would be transformation, not creation of energy; in other words, such an engine would not be a perpetual motion of the first class. Experience shows, however, that a process of this character, while not violating the conservation law, is nevertheless impossible. The statement of its impossibility is, in fact, one form of statement of the second law. 45 46 THE SECOND LAW OF THERMODYNAMICS [chap, iv 34. Availability of Energy. In Art. 8 was noted the distinc- tion between various forms of energy with respect to the pos- sibility of complete conversion. We 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 be 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 be satisfied. As a first condition, there must be two bodies of different tem- perature ; it is impossible to derive work from the heat of a body unless there is available a second body of lower temperature. Suppose we have then a source S at temperature T^ and a re- frigerator H at lower temperature T^; how is it possible to derive mechanical work from a quantity of heat energy Q-^ stored in aS'? If the bodies /S' and M are placed in contact, the heat Q-^ will simply flow from S to H and no work will be obtained. Hence, as a second condition, the systems iS'and R must be kept apart and a third system M must be used to convey energy. This third system is the working fluid or medium. In the steam plant, for example, the boiler furnace is the source a9, the con- denser is the refrigerator i^ 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 i2 a quantity of heat Q^', smaller than $-^, the difference Q^ — Q^ being the 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-^ that can be thus transformed into work, that is, Q^ — §21 is t^® available part of Q-^; and the part §2 ^^^^ must be rejected to the refrigerator i?, and which is of no further use, is the unavailable part of Q-^^ or the waste. The ratio -^^ — ^ ART. 35] REVERSIBILITY 47 is called the availability of Q^ for transformation into mechani- 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 Ave assume the force pA slightly greater than #, the piston will move slowly to the right and the confined gas will assume a succes- F 48 THE SECOND LAYf OF THERMODYNAMICS [chap, iv sion of states indicated by the carve AB. If at the state B the motion is arrested and F is made intinitesimally greater than pA for all positions of the piston, the series of states from B to A will be retraced and the system (the 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 tile external force JF. The process is therefore reversible. (2) Let the force F be removed entirely. Then the piston will move suddenly and the confined gas will be thrown into commotion. When the gas finally attains a state of thermal equilibrium with the volume Fg, that state will be represented by some point as B^. No path can be drawn between A and B because during the passage from A to B' 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 state A, however, in the following manner : A force F^ slightly greater than pA, is applied to the piston and the gas is thus compressed slowly, the successive states being indicated by the curve B'A', say. Then the gas in the state A' is cooled at the constant volume V\ 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 B' to A' must be furnished from one external body, and the heat given up by the cooling from A' to A must be absorbed by another external body. The free expansion of the gas is, therefore, an irreversible process. It is easy to see that the flow of a fluid through an orifice from 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 technical applications of thermodynamics. The flow of liquid ammonia through the expansion valve of the refrigerating machine may be cited as an example. (3) The direct conversion of work into heat is an irreversi- ble process. For example, consider the heating of a journal ART. 36] GENERAL STATEMENT OF THE SECOND LAW 49 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 A2^ in- definitely small and let the transfer take place very slowly, the process can be reversed by changing the sign oiAT. 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 quantit}^ 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 I\, to another at temperature T^. For the flow to occur of it- self we must have T^ > T^^ and as a result of the process there is a loss of availability. To produce an increase of availability would require T^ to be greater than T^ ; in that case, however, the process would not be possible. In the limiting reversible case, 2^2 = ^i» ^he availability remains unchanged. A consid- eration of other physical processes within the range of experi- 50 THE SECOND LAW OF THERMODYNAMICS [chap, iv ence leads to similar results. We may, -therefore, lay down the following general laws, which, like the law of conservation of energy, are based entirel}' on experience: I. No change in a system of bodies that can take place of itself can increase the available energy of the system. II. An irreversible change causes a loss of availability. III. A reversible change does not affect the availability. These statements may be regarded as fundamental natural laws underlying all physical and chemical changes- The second and third together constitute the law of degradation of energy. The first may be taken as a general statement of the second law of thermodynamics. By considering special processes the general statement of the second law here given may be thrown into special forms. Thus 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 oar first statement. We have, therefore, Clausius' form of the second law, viz : It is impossible for a self-acting machine unaided by any exter- nal agency to convey heat from one body to another at higher temperature. Again, if we consider the increase of available energy that would result from deriving work directly from the heat of the atmosphere, we are led to Kelvin's statement, namely : It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of surrounding objects. In order to estimate the available energy of a system in a given state, or the loss of available energy when the system undergoes an irreversible change, it is necessary to know the most efficient means of transforming heat into mechanical work under given conditions. This knowledge is furnished by a study of the ideal processes first described by Carnot in 1824. 37. Carnot's Cycle. — Suppose that the conditions stated in Art. 34 are furnished ; that is, let there be a source of heat S at temperature T^, a refrigerator i^ at a lower temperature T^, ART. 37] CARNOT'S CYCLE 51 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. IS). Let the medium initially in a state represented by B (Fig. 17), at the temperature T^ of the reservoir S^ expand adiabatically until its temperature falls to T^^ the temperature of body R. By this expansion the second state C is reached, and the work done by the medium is represented by the area B^BOC^. 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 M to i2, and the medium is compressed. The work represented by the area C^CDD^ is done on the medium, and heat Q^ passes from the medium to the refriger- ator. The process is again assumed to be so slow as to be reversible. From the state I) the medium is now compressed adiabatically, the cylinder being removed from R until its tempera- ture again becomes T^^ that of the source S. During this third process work rep- resented by the area C^OBI)^ is done on the fluid. Finally, the cylinder is placed in contact with S and the fluid is allowed to expand at the constant temperature T^ to the initial state B. Work represented by the area A^ABB^ is done by the fluid during this process, and the J! \ illllll ill - Fig. 18. 52 THE SECOND LAW OF THERMODYNAMICS [chap, iv temperature is kept constant by the flow of heat Q-^ from S to 31. The area ABCD inclosed by the four curves of the cycle represents the mechanical work gained; that is, the excess of work done by the medium over that done on the medium. Denoting this by W^ we have from the first law, The efficiency of the cycle is the ratio of the work gained to the heat supplied from the source aS'. Denoting the efficiency by 77, we have Q^ - g, _ AW Since all the processes of the Carnot cycle are reversible, it is evident that they may be traversed in reverse order. Thus starting from B^ the fluid is compressed isothermally from B to A and gives up heat Q^io S\ from A to D it expands adiabat- ically, from i> to C it expands at the constant temperature T^^ and in so doing receives heat Q^ from R ; finally it is com- pressed adiabatically from O to the initial state B. In this case the work TF" represented by area ABCD is done on the fluid M^ heat ^2 is taken from the refrigerator R, and the sum Q^-\- AW = Q^ is delivered to the source S. 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 T^ and T^ of the source and refrigerator, respectively. The question at once arises whether the efficiency depends also upon the properties of the substance M used as a working fluid. The answer is contained in Carnot's principle, namely : Of all engines working hetiveen the same source and the same refrigerator^ no engine can have an efficiency greater than that of a reversible engine. In other words, all reversible engines working between the same temperature limits I\ and T.^ have the same efficiency; that is, the efficiency is independent of the working fluid. The proof of Carnot's principle rests on the second law, and consists essentially in showing that if any engine A is more ART. 38] CARNOT'S PRINCIPLE 53 efficient 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 W.>W. Q/ < Q.'- s ftj fft J b - Qt Y R Now let engine B be run reversed. It will take heat Q^^ from R and deliver Q^to S, \i A and B are coupled together, A will run B reversed and deliver in addition the work Wa — Wb- The source is unaffected since it simultaneously receives heat Q^ and gives up heat Q^. The re- frigerator, however, loses the heat $2^ ~ Q^f^^ which is the equivalent of the work W^— W^ 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 aS', i2. 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 be more efficient than A, and it follows that all reversible- Fig. 19. 54 THE SECOND LAW OF THERMODYNAMICS [chap, iv engines between the same source and the same refrigerator are equally efficient. 39. Determination of the Efficiency. — Since the efficiency 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 whence ^=1 - rj = F(T,, T,) ; (2) that is, the quotient j~ is some function of the temperatures T^ and T^. The form of this function is 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'j, S^^ S^, •••, S^, whose tem- peratures are t^, t^, ^3, •••, t,^^ as defined by the mercury or gas scale, and let t^>t^>ts>"->t,. If we use aS'j as a source and S^ as a refrigerator, a reversible engine will take Q^ from S^ and deliver Q^ to S^. Since the bodies S^ and S^ have definite temperatures T^ and T^, what- ever the scale adopted, the function F(^Ty, T^ has some defi- nite value; therefore, from (2) the fraction -^ must have a definite value, and consequently §2 ^^^ one and only one value. If ASg is used as a source and S^ as a refrigerator, a second engine taking §2 fi*oiii ^2 ^^^^ ^^^® ^P ^3 ^^ ^3' ^^^ ^^ ^^* Starting with Q-^, we thus obtain a determinate set of values §2, §3, (?4, etc., which must fulfill the condition ^1 > §2 > ^3 > - > Q.r (3) C^) ART. 39] KELVIN'S ABSOLUTE SCALE 55 Here we have a set of nambers suitable to define a scale of temperature. Starting with the heat Q^ taken from the source aS'j, to each source there corresponds a number indicating the heat that would be rejected to it if it were used as a refrigerator in connection with S^ If we choose these numbers to define a new scale, then denoting the new temperatures by T T T ' ' . T we have whence follows Returning now to the quotient -^, we have at once hence, using this new scale, the efficiency of the Carnot engine is and the form of the function is determined. The scale of temperatures arrived at from the investigation of Carnot's cycle was first proposed by Lord Kelvin in 1848, and is known as the absolute scale because it is independent of the property of any substance. The scale is simply such that any two temperatures on it are proportional to the quantities of heat absorbed and rejected by a reversible Carnot engine working between these temperatures. If in (5) we make Q^ — ^^ 77 = 1 and T^ — 0. If we con- ceive a temperature lower than the zero on the absolute scale, T — T that is, if we assume a negative value for T^^ then ^ ^ — ^> 1, and the engine has an efficiency greater than 1, or transforms more heat into work than it receives from the source. Such an assumption is clearly inadmissible, and it follows that the zero of Kelvin's absolute scale is an absolute zero, and the tempera- ture corresponding to it is the lowest temperature conceivable. We are thus led to the conception of an absolute zero inde- 56 THE SECOND LAW OF THERMODYNAMICS [chap, iv pendent of the properties of any particular substance. It will be shown subsequently that this absolute zero is precisely the same as that derived from the reduction in pressure of a perfect gas, and that the new scale coincides with that of a ther- mometer using a perfect gas as a fluid. 40. Available Energy and Waste. — Carnot's ideal cycle 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$ into work than the ideal Carnot engine. If T^ is the lowest temperature that can be T— T obtained for a refrigerator, the fraction — ^-^ of A^ can be transformed into work by a Carnot engine, and this is, therefore, the availability of A$ under the given conditions. The avail- able part of A§ is, therefore. A<2^^"= A It follows that the smaller the value of ^, the greater the slope of the curve. The isothermal and adiabatic curves (Fig. 22) mpy be con- sidered special cases of the heating and cooling curve. For the isothermal c = oo, and for the adiabatic c = 0. Fig. 23. c. 72 TEMPERATURE ENTROPY REPRESENTATION [chap, v Cases may arise in which the slope of the T)S-Gurve is nega- tive, as shown in Fig. 24. In such cases abstraction of heat is accompanied by a rise in tem- perature or vice versa. Evidently the specific heat -^ must be negative, as is indicated geo- metrically by the 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. 49. Cycle Processes. — Since any reversible process may be shown by a curve in TW-coordinates, it follows that a series of such processes forming a closed cycle may be repre- sented by a closed figure on the TO-plane. In Fig. 25 is shown such a cycle composed of two polytropics AB and 2>^, an isothermal BC^ and two adiabatics 01) and UA. 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 be traversed in the clockwise sense as shown by the arrows, we have Q^ = area A^ABB^, Q,, = SiTe^B,BOO^, Qa, = area C-^BEE^, -. ^ A D E Fig. 25. C, ART. 50] THE RECTANGULAR CYCLE 73 Hence the excess of heat received over the heat rejected is Q=Q.^+Q,c+ Qa, + Q,e + Qea = A,AB B, + B,BCt\ - O^BEA^ = ABODE. If the cycle is traversed in the counterclockwise sense, we have evidently ^ = - area vlS (77)^. But from the first law, Q is the heat transformed into work ; hence for the direct cycle ?.vQii ABODE =Q = AW, and for the reversed cycle area ABODE =- Q = - AW. This reasoning evidently holds for any number of processes, and therefore for a reversible closed cycle of any form. Thus for the cycle shown in Fig. 26, we have area jP= Q = AW or area F = — Q = — AW, according as the cycle is traversed in the clockwise or counter clock- wise sense. In later developments it will frequently be necessary to show cycle processes on the T/S'-plane. 50. The Rectangular Cycle. When the curves representing the four processes of the Carnot cycle are transferred to the 5%^-plane, the cycle becomes the Fig. 2(;. Ty -^SVH -^ B, Fig. 27. simple rectangle ABCD, Fig. 27. The area A^ABB^ represents the heat Q^ absorbed by the medium from the source during the iso- thermal expansion AB, and the area B^ODAy, the heat Q^ rejected to the refrigerator during the isothermal compression OD. The lines BO and DA represent, respectively, the adiabatic expansion and the adia- batic compression. 74 TEMPERATURE ENTROPY REPRESENTATION [chap, v From the geometry of the figure, we have ^2= ^2(^2-^1). whence 7) = — — = ^ ) , ,.^L-^^J^ as already deduced in Art. 39. When the cycle is traversed in the counterclockwise sense, the heat Q^ is received by the medium from the cold body during the isothermal expansion DO^ and the larger amount of heat Q^ is rejected to the hot body during isothermal compression BA, The difference Q^— Q^ = — AW represented by the cycle area 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 refrigerating machine or as a warming machine. In the first case the space to be cooled acts as the source and delivers the heat Q^ = area A^JDCB^ to the medium. In the second case the space to be warmed receives the heat Q-^ = area B^BAA^ from the medium. 51. Internal Frictional Processes. — Referring to Art. 42, the increase of entropy when heat is generated in the interior of a system is seen to be s^- s,= C"^4 + C^. (1) ^^=j.>'h;^- If § = 0, that is, if no heat enters the system from outside sources, the increase of entropy is S, ^i=j*;^' (2) and is due entirely to the generation of heat 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 TS-q\xvyq will in this case repre- sent not the heat brought into the system but the heat H generated in the system through the agenc}^ of friction. An example of such a process is afforded by the flow of air or steam ART. 52] CYCLES WITH IRREVERSIBLE ADIABATICS 75 Fig. 28. through a nozzle. The fluid in the initial state represented by point A (Fig. 28) has its pressure decreased in passing along the nozzle, and as a result the temperature likewise falls. The process is adiabatic, that is, no heat is received from external bodies ; hence, if there were no internal friction, the drop in temperature would be indicated by a motion of the state-point along AA^ But work is expended in overcoming the friction between the fluid and nozzle wall. This work is neces- sarily transformed into heat, which is retained by the fluid. It follows that there is an increase of entropy, as indicated by the curve AB. From (2) the heat generated is represented by the area A^ABB^ 52. Cycles with Irreversible Adiabatics. — In certain cases the closed cycle of operations of a heat motor may contain an adia- batic irreversible process, the irreversibility arising either from internal generation of heat or from the free expansion or wire- drawing of the working fluid. Even if it is possible to draw J, a TS-cvLVVG 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 BO to be an irreversible adiabatic, the other parts of the cycle being reversible. Since AB is revers- ible, the heat absorbed in passing from ^ to jB is represented by the area A^ABB^ Likewise area C^ODA^ represents the heat rejected by the system in changing state from C to B. The Fig. 29. ?6 TEMPERATURE ENTROPY REPRESENTATION [chap, v process IDA is adiabatic, hence Q^a = ; and by hypothesis Q^c — 0' The value of 2 § for tho cycle is, therefore, Qab + Qcd = area A^ABB^ — area C-^CDA-^^ = area ABUI) - area B^ECC^. The energy equation applies to any process, reversible or irreversible. Therefore for this cycle, as for those previously considered, we have W=JQ = J(iQ,,+ Q,,~). It appears, therefore, that the work derived is less by the area B^JECC^ than it would have been if the reversible adiabatic BU had been followed. For the reversed cycle (Fig. 30) w^e have as the work required from external sources W= J(Qaa + QbJ) = - area D^BAA^ + area B^BCB^ = -dvea. B^BCBAA^. Comparing this cycle with the cycle AECB having the revers- ible adiabatic AE. it is seen that the heat absorbed from the cold body is smaller by the heat represented by the area A^EBB^^ while the work required to drive the machine is greater by an equal amount. In every case the irreversible process results in a reduction of the useful effect. 53. Heat Content. — Since the quantities jo, v, T^ w, 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 i = A{u-\-pv)', (1) I^AiU + pV); (la) E=Au-T8; (2) (p = Au-Ts-h Apv. (3) Then ^, jP, and O are magnitudes determined by the state of the system, and if desired, may be used along with p^ v, T as co- ordinates. The functions F and O are called thermodynamic ART. 53] HEAT CONTENT 77 potentials, and are used in certain applications of thermo- dynamics to physics and chemistry. The function I has use- ful applications in technical thermodynamics. To gain a physical meaning for the function I^ let us consider the process of heating a substance at constant pressure. If ZJp F^, and p^ denote the initial energy, volume, and pressure, respectively, and f/g, F^, and p^ the final values of the same coordinates, we have from the energy equation = A[U,^ - L\ + (p^V,^ - pj'i)}, since p^ = p^ = A[{U, + p.V^)~AiL\ + p,r,}] That is, the change in I is equal to the heat added to the sys- tem during a change of state at constant pressure. For this reason / is called the heat con- tent of the system at constant pressure^ or, more briefly, the "heat content." In some subsequent investiga- tions, especially tliose relating to the flow of fluids, it will be con- venient to use I and S as the in- dependent variables and to repre- sent changes of state by curves on the ZS'-plane. The great advantage of the ZS'-representation over the TO'-representation lies in the fact that in the former quantities of heat are represented by linear segments, while in the latter, as we have seen, they are represented by areas. A reversible adiabatic on the ZxS'-plane is a vertical line, sls BO (Fig. 31). But in this diagram segment BC represents a quantity of heat instead of a change of temperature. EXERCISES 1. Find the change of entropy when air is heated at constant pressure from 70" to 200^ F., taking the specific heat as 0.24. 78 TEMPERATURE ENTROPY REPRESENTATION [chap, v 2. Assuming that the specific heat of water is constant, c = 1, plot on cross-section paper the TS -curve representing the heating of water from 32" to 212°. 3. Langen's formula for the specific heat of COg at constant pressure is Cp = 0.195 + 0.000066 t. Find, the increase of entropy when COg is heated at constant pressure from 500° to 2000° F. ; also the heat absorbed. 4. A direct motor operates on a rectangular cycle between tepiperature limits Ti = 840° and T^ = 600° and receives from the source 200 B. t. u. per minute. Find the efficiency, and the work done per minute. 5. A reversed motor, rectangular cycle, operates between temperature limits of 10° and 130°, and receives 600 B. t. u. per minute 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 T^ = 900° and T^ = 680, takes 1000 B. t. u. from a boiler. The heat rejected is delivered to a building for heating purposes. This direct motor drives a reversed motor which operates on a rectangular cycle between tempera- tures T^ = 460° (temperature of outside atmosphere) and T^ = 600. The reversed motor takes heat from the atmosphere and rejects heat to the building. Find the total heat delivered to the building per 1000 B. t. u. taken from the boiler. 7. In the vaporization of water at atmospheric pressure, the temperature remains constant at 212° F., and 970.4 B. t. u. are 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 ~- — - + Uq, where k and Uq are constants. Derive an expression for the heat content / of the gas. 9. Combine the energy equation clQ = AdU + ApdV v^ith. the defining equation I = A{U -[■ pV) and show that dl = dQ + A vdp. REFERENCES Use of Temperature-Entropy Coordinates Berry : The Temperature-Entropy Diagram. Sankey : The Energy Diagram. Boulvin : The Entropy Diagram. Swinburne : Entropy. Use of Heat Content and Entropy as Coordinates Berry : The Temperature-Entropy Diagram, 127. Mollier: Zeit. des Verein. deutscher Ing. 48, 271. Marks and Davis : Steam Tables and Diagrams, 79. CHAPTER VI GENERAL EQUATIONS OF THERMODYNAMICS 54. Fundamental Differentials. — The introduction of the entropy s and the functions i, jP, and 4> (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 ^ in terms of the differentials of the variables p, 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) an equation that gives tt as a function of the independent varia- bles s and V. From the defining equation i = A(u -h pv) we have di = Adu + Ad (j)v) = Adu -f- Apdv + Avdp. Introducing the expression for Adu given by (3), we get di = Tds + Avdp. (4) 79 80 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi Here i is given as a function of s and p as independent variables. Likewise, from the relation dF = Adu - Tds - sdT; whence from (3) -dF=sdT-\-Apdv. (5) Finally, from the defining relation ^ = Au 4- Apv — Ts, d^ = Adu + Ad(ipv) - dQTs) = Tds — Apdv + Apdv -{- Avdp — Tds — sdT; or d<^ = Avdp - sdT. (6) Now since the functions % ^, F, 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 done 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, whence du = — ds -\ dv. ds dv Comparing this symbolic equation with (3), it appears that du__T du _ _ ds~'A' dv~ ^' ART. 55J THE THERMODYNAMIC RELATIONS 81 We have now from the criterion of exactness (Art. 23), d fdu\ __ d fbu" dvKdsJ ds\dv that is, ii.(r) = |-(_^), Adv f)s 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 2V-plane, another on the ps-plane, — the slope of the second curve at a point representing a given state is — ^ 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 ds (f).=XS). <»> 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 82 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi substances and for all reversible changes of state. For certain transformations the following forms, which are obtained from (C) and (D) by means of the relation ds = -^, are useful : 56. General Differential Equations. — From the thermo- dynamic relations certain useful general equations are at once deduced. As in Art. 19, we may write according as T and v or T and p are taken as the independent variables. Now replacing (— ^ ] and (— ^ ) by c^ and c^, re- spectively, and f--^) and (— ^) by the expressions given in \dvjj, \dpjT (6^') and (i>')? these equations become, respectively, dq=^cJT+AT(^^dv, (I) dq = c^dT-AT{^^^dp. (II) Eliminating dT between (I) and (II), a third equation having p and V as the variables is obtained. Thus AT dq = Two other important equations may be derived from (I) and (II). Since from the energy equation du — Jdq — pdv^ we have iProm (I) du = Jc^dT-\- ^%.-'\ dv ; (IV) ART. 56] GENERAL DIFFERENTIAL EQUATIONS also from the relation di = dq -{- Avdp, we have from (II) 83 di=CpdT-A T dv_ dT dp. (V) The general equations (I)-(y) hold for reversible changes of state. The partial derivatives involved may be found from the characteristic equation of the substance under investi- gation. As an application of (IV), we may derive expressions for the change of energy (a) of ' a gas that follows the Vdw pv = BT; (5) of a gas that obeys van der Waal's equation i' - s) p + ^,](v-b)=^BT. (a) From the characteristic equation pv = BT^ we have dp\ _B^ hence ©.= fBT \ du = Jc^dT -\-{ p\dv = Jc.dT, u^ — u^ = JKc^dT assuming c^ to be a constant. (5) From van der Waal's equation, we have and whence H^l dTj, -P V — B ~v-b' BT a v-b ^ v^' From (IV), we have, therefore, du = Jc^dT-{- — dv^ whence, assuming again that c^ is constant, u^-u^ = Jc^T^ - 7\) + a 84 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi It appears, therefore, that if a gas follows the law pv = BT^ the energy is a function of the temperature only, while if it follows van der Waal's law, the energy depends upon the temperature and volume ; in other words, the gas possesses both kinetic and potential energy. / 57. Additional Thermodynamic Formulas. — For certain in- vestigations of imperfect gases, especially the superheated vapors, certain formulas involving the specific heats Cj, and c^ are useful. The most important of these are (VI), (VII), and (VIII) following. Since du is an exact differential, we obtain, upon applying the criterion of exactness to (IV), dv dT sdT + Apdv, (4) d^0. Hence in an isolated system any change must result in an increase of entropy. The conditions of equilibrium are readily deduced from these conclusions. Under the condition of constant T and v, change is possible so long as F can decrease. When F becomes a minimum, no further change is possible and the system is in stable equilibrium. Likewise, with T and p constant, stable equilibrium is attained when the function $ is a minimum. The functions F and <^ are evidently analogous to the potential function V in mechanics. A mechanical system is in a state of equilibrium when the potential energy is a minimum, and similarly a thermodynamic system is in equilibrium when either the function F or the function $ is a minimum. For this reason F and $ are called thermodynamic potentials. By the use of thermodynamic potentials, problems relating to fusion, vaporization, solution, chemical equilibrium, etc., are attacked and solved. EXERCISES 1. From (V) derive an expression for the change of the heat content i when a gas following the law/>y = BT changes state. 2. If the gas obeys van der Waal's law, find an expression for the change of the heat content i. 3. Apply equations (II), (IV), and (V) to the characteristic equation of superheated steam, p(v + c) = BT-p(l + ap)^. 88 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 4. Calleridar has proposed for superheated steam the equation pi^v-h)=BT-Cpi?^y. 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 -^ for a gas following the law pv = BT. Show that the expressions for -^ are integrable, 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 $ : (a)o.= -rp;(.)c..7f*. 9. Using temperature-entropy coordinates, deduce a system of graphical representation for the three magnitudes Q, U^ — U^, 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. Duhem : Le Potential Thermodynamique. Bryan : Thermodynamics, 91. Preston : Theory of Heat, 668. Chwolson : Lehrbuch der Physik 3, 519. Buckingham : Theory of Thermodynamics, 150. Parker : Elementary Thermodynamics, 325. 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 - 263.9° C. 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 weight of gas varies inversely as the pressure. 90 PROPERTIES OF GASES [chap. VII 2. Charles' Law. With the volume constant, the change of pres- 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 c in the connecting tube was then opened, permit- ting the air to rush from a to 6, 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 5 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 h 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 ART. 61] COMPARISON OF TEMPERATURE SCALES 91 work was required. A gas has, therefore, no appreciable 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 : u=Af), 1^ = 0. dv 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- ishingly 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 du = ^dT-{-^dv. . (1) dT dv ^ -^ But from the general equation (IV), Art. 56, du = Jc^dt -f- ^%) -^ dv (2) 92 PROPERTIES OF GASES [chap, vii in which T denotes temperature on the Kelvin scale. Com- paring (1) and (2), we obtain For a gas that obeys Joule's law — = 0, whence from (8) 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 jOq as the pressure at 0° C, V- t- dp ~dt~ - P _ .P-. T .P ~ T (see Fig. 2); that is. 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 obe}^ 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 0° C. are given as follows : ART. 63] FORMS OF THE CHARACTERISTIC EQUATION 93 Atmospheric air 0.08071 lb. per cubic foot. Nitrogen 0.07829 lb. per cubic foot. Oxygen ........ 0.08922 lb. per cubic foot. Hydrogen 0.00561 lb. per cubic foot. Carbonic acid 0.12268 lb. per cubic foot. A pressure of one atmosphere, 760 mm. of mercury, is 10,333 kg. per square millimeter= 14.6967 lb. per square inch = 2116.32 lb. per square foot. Taking as 491.6 the value of T on the F. scale corresponding to 0° C, we have for air ^ = ^ = JL=__?1M:M_= 53.34. T r^T 0.08071x491.6 In metric units the corresponding calculation gives B = ^^^^^ ^ 29 26 273.1 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, pv = BT, (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 V ol M lb. (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 lb. of air at a temperature of 70^ F. occupies a volume of 3.5 cu. ft. ^ From (2) ^ ^ ^^0-6x53.34x^0 + 459.6) ^ ^^^^^ ,^^ ^^^ ^^^^^^ ,^^^ = 33.63 lb. per square inch. 94 PROPERTIES OF GASES [chap, vii The homogeneous form of the characteristic equation is advantageous in the solution of problems that involve two states of the gas. If (^j, F^, T^ and {p^, F^, T^ are the two states in question, then With this equation any consistent system of units may be used. Example. Air at a pressure of 14.7 lb. per square inch and having a temperature of 60° F. is compressed from a volume of 4 cu. ft. to a volume of 1.35 cu. ft. and the final pressure is 55 lb. per square inch. The final temperature is to be found. From (3) vp^e have 14.7 X 4 ^ 55 X 1.35 60 + 459.6 ^2 + 459-6' whence <<, = 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 the constant B for that gas. 3. Find the volume of 13 lb. of air at a pressure of 85 lb. per square inch and a temperature of 72° C. 4. If the air in Ex. 3 expands to a volume of 30 cu. ft. and the final pressure is 20 lb. per square inch, what is the final temperature? 5. What weight of hydrogen at atmospheric pressure and a temperature 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 lb. of oxygen at a pressure of 120 lb. per square inch and at a temperature of 60° F. The pressure in the tank should not exceed 300 lb. per square inch and the temperature may rise to 100° F. Find the weight of oxygen that may safely be added to the contents of the tank. 64. General Equations for Gases. — The general equations deduced in Chapter VI take simple forms when applied to perfect gases. From the characteristic equation pv=^BT we obtain by differentiation ^ =:? (^\ =^ m dTJ. «' \stJp p ^ -' ART. 64] GENERAL EQUATIONS FOR GASES 95 Introducing these values of the derivatives in the general equations (I)-(V) and (VIII), the following equations are obtained : dq- = c,dT + AB - dv, V (I a) dq = --c^dT-AB - dp, V (11 a} dq = ^B f T . ^ T ' F #), (III a) du = --Je^dT, (IV a~) dl = = CpdT, (V a} (■p-<^v = = AB. (VIII a) The first two equations may be still further reduced by means of the characteristic equation to the forms dq = c^.dT-\- Apdv, (I 5) dq = Cj,dT-Avdp (II J) The ratio ^ of the two specific heats is usually denoted by k. The introduction of this ratio reduces (III a) to the sim- pler form, A dq = [_kpdv 4- vdp'] . (Ill 5) k—1 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 (15) follows independently from (IV a) and the energy equation ; thus dq = Adu + Apdv = c^dT + Apdv, since AJ= 1. EXERCISES 1. Deduce (VIII n) from (I 6), (II 6), and the characteristic equation. 2. Derive (V a) from (IV a) and the equation pv — BT. 3. From (I «), (II a), and (III a) derive expressions for 4. From (III 6) deduce the equation of the adiabatic curve in joy-coordi- nates. 96 PROPERTIES OF GASES [chap, vii 5. From (I a) derive the equation of an adiabatic in rt'-coordinates. 6. Using the method of graphical representation explained in Art. 32, show a graphical representation of equation (I b). 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^ or c^} 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^ = a' -{- ht. 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^, as determined by Regnault, is 0.2375 from 0° 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^ = 0.24. The value of c^ for carbon dioxide (COg) is usually given as 0.2012. Swann found the values 0.20202 at 20° C, 0.22121 at 100° C. The value of Cp for other gases for temperatures between 0° and 200° C. may be taken as follows: Hydrogen .... 3.4240 Nitrogen .... 0.2438 Oxygen .... 0.2175 Carbon monoxide . 0.2426 Ammonia .... 0.5106 ART. 66] INTRINSIC ENERGY 97 Values of the ratio k = -^ have been determined by various experimental methods. For air the results obtained range from ^ = 1.39 to A; = 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 e^ for other gases may be taken as follows ; k Cv 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 a), c^ is replaced by -2, the result is the k relation Each of the four magnitudes c^^ k^ 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 u = J^cJT=Jc,T+u^, (1) if Cy is assumed to be constant. The constant of integration Uq 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 Uq drops out of consideration when differences are taken, and we need make no assumption as to its value. Hence, if (jt?^, v^, 2\) and (jt?2, ^2' ^2) ^^'^ ^^^ coordinate of the initial and final states, we have ^^-u, = JcXT^-T,). (2) 98 PROPERTIES OF GASES CHAP. VII Fig. 33. This formula gives the change of energy per unit weight of gas. For a weight Mthe formula becomes U^-U^^JMcXT^-T{). (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 ;7\, Fig. 33, is an isothermal, the energy of the gas in the state A is the same as in the state i); likewise, the energy of the gas at all points on the iso- thermal T^ is the same. It follows that the change of energy in passing from tem- perature T-^ to temperature T^ is the same, whether the path is-1^, AC, 01 DU. Since the energy is directly proportional to the temperature, the change of energy is directly proportional to the change of temperature. Hence in which a denotes a proportionality-factor. To determine the factor a, we choose some particular path between the isother- mals T^ and T^ (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 AC. The heat required for a rise in temperature from ^1 toy, is q,,=^eXT,-T{). Since in the constant volume change, the external work is zero, we have from the general energy equation Comparing these equations, we have u^-u^ = Jc^(T^-T^^l therefore, the constant a in (4) is Jc^. ART. 67] HEAT CONTENT 99 A formula for the change of energy in terms of p and V may be derived from (3). Multiplying and dividing the second member by B^ - k-1 • <-'^^ In (5) V2 and V\ denote the final and initial volumes, respec- tively, of the weight of gas under consideration; consequently it is not necessary to find the weight iJf 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. Example. Find the change of energy when 8.2 cu. ft. of air having a pressure of 20 lb. per square inch is compressed to a pressure of 55 lb. per square inch and a v&lurae of 3.72 cu. ft. Using the value k = 1.40, _ ^r ^ 144 ^ 55x3.72-20x8.2 ^ ^^^ ^ ^ 0.40 67. Heat Content. — The change in heat content correspond- ing to change of state of a gas is readily derived from the general equation (Y a). Thus, i = I c^dT= c^T+ i, (1) and i,-i, = c^iT,-T,). (2) Introducing the factor AB in the second member of (2), C. (P2^2-Pl^D k = Aj^—jCp^v^-p^v^}. (3) For a weight of gas M, (2) and (3) become, respectively, I,-I, = Mo,iT^-T,), (4) 100 PROPERTIES OF GASES [chap, vii and ^2-^i = ^]^(P2^"2-Pi^i)' (5) 68. Entropy. — Expressions for the change of entropy are easily derived from the general equations (I «), (II a), and (Ilia}. Dividing both members of these equations by T, we have , do dT . T^dv ' ^-.x ds = -^=c,^ + AB-^, (1) ds = '-^=c/-^-AB'-l. (2) ds = Cp \-c^-^. (3) Hence for a change of state from (p^, v^, T^ to (p^, v^, T^, 82-Si=«.l0g.||+^i?l0g.^ (4) C^=o„\og,^-AB\og,^ (5) ^1 Pi = ., log. ;;?+.. log, f. (6) ^1 P\ These formulae give the change of entropy per unit v^-^eight of gas. For any other weight M^ the change of entropy is ilf (§2 — Si). 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^ — c^ = AB. In the solution of a problem, the equation should be chosen that leads most directly to the desired result. EXERCISES 1. From (4), (5), and (6) deduce expressions for the change of entropy corresponding to the following changes of state : (a) isothermal, (h) at con- stant volume, (c) at constant pressure. 2. By making s^— s•^^ = in (4), (5), and (6), deduce relations between Tand v, T and j9, and/> 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(iU^- U,-) = McXT^- T,-). (1) ART. 69] CONSTANT PRESSURE CHANGES 101 The change of entropy is = Jf^,logA (2) Pi When the gas is heated at constant pressure, the external work is W,,=piV,-V,). (3) The increase of energy is, as in all cases, given by the relation rr _ rr ^ pV^j-pVi ^ "^12 ^2 ^1- yt_l -^^Tl* The heat added is, therefore, given by the relation k k-1 or using the relation jt? F^= MBT^ K^2-^i); Cl2 = A<^p-<^v) which might have been written directly. The increase of entropy is Equations (2) and (7) may also be derived directly from the general equations for en- tropy, Art. 68. The changes of state just considered are represented on the ^iS'-plane by curves of the general form shown in Fig. 34. The curve AB^ which rep- resents the constant volume O change, is steeper than the (4) (5) MBi T^ - T{) = Mc,i T^ - Ti), (6) O) 102 PROPERTIES OF GASES [chap, vii curve AC, which represents heating at constant pressure. This follows from the inequality that is, area A^ABB^ < area A^ACC^. 70. Isothermal Change of State. — li T is made constant in the equation ^F= MBT^ the resulting equation pY= p^V^ = 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 TF.o = 12 = f Vf: (2) Jv Using (1) to eliminate j9, we have = MBT\og,^. (3) For the change of energy, U,-U, = JMc, iT,-T,) = 0; (4) hence rr Qn = AW,.^ = Ap,V,log,-^, (5) and •^2 - 'S'l = % = ^*^log, ^ = ABMlog, ^. (6) 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 : 71. Adiabatic Change of State. —To derive the j9v-equation of an adiabatic change of state, we may use the general difPeren- ART. 71] ADIABATIC CHANGE OF STATE 103 tial equation containing p and v as variables. The most con- venient form of this equation is (III a), dq = (ydp + kpdv). (1) K — i During an adiabatic process no heat is supplied to or ab- stracted from the system ; hence in (1) dq = 0, and therefore vdp -f- kpdv = 0. (2) Separating the variables, dp kdv _ r. p V whence loggjo -j- k log^ v = log (7, or pv^ = O. (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^ = (7, pv = RT, we get by the elimination of jt?, v'-^ = — ; BT' that is, Tv^-^=2 const. (4) Similarly, by the elimination of v, we obtain T>k ^k-\ — -P rpk . ^ ~ o ' k-\ that is, =^— = const. (5) If we choose some initial state, p-^^ Vj, 2\, the constants in (4) and (5) are determined, and the equations may be written in the homogeneous forms T fp\ " T, \pj (6) k-i O) 104 PROPERTIES OF GASES [chap, vii Since in an adiabatic change the heat Q is zero, the energy equation gives whence using the general expression for the change of energy, (8) By means of the equation the final volume F^ 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 lb. per square inch) to a pressure of 70 lb. per square inch. Find the work of compression; also the final temperature if the initial temperature is 60° F. For the final volume, we have Fs = 1.2 C—Y' = 0.3936 cu. ft. The work of compression is PiVi-p^V2 ^ 144(14.7 X 1.2 - 70 X 0.3936) ^ _ o.gg ,, ,, k-1 0.4 *" * The initial temperature being 60° + 459.6° = 519.6° absolute, we have for the final temperature T2 = 519.6 /"-^Y' = 811.6 abs., whence t2 = 352° F. 72. Polytropic 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 pV''=: const. (1) ART. 72] POLYTROPIC CHANGE OF STATE 105 By giving n special values we get the constant volume, constant pressure, and other familiar changes of state. Thus : fo^?^ = 0, pv^ = const., i.e. jt? = const, for 71=00, p^v = const., v = const, for n = Ij pv = const., isothermal, for n = k, pv'' = const., adiabatic. The curve on the p F'-plane that represents Eq. (1) is called by Zeuner the poly tropic curve. By combining (1) with the characteristic equation jo V=MBT, as in Art. 71, the following relations are readily derived n-\ n-\ For the external work done by a gas expanding according to the law pV^ =p-^ V^" = const., from the volume F^ to the volume Fg, we have V =i-i^a i_/ J 1 — n The change of energy, as in every change of state, is (3) U,- U.^P^fll^. (4) rC — J. Hence, from the energy equation, we have for the heat absorbed by the gas during expansion K — 1 1 — n Comparing (3), (4), and (5) we note that the common factor (P2 ^2~ Pi ^i) occurs in the second member of each expression. u^-u^ 1- -n w k- -1 JQ k- 1 - n JQ _ _^- - n 106 PROPERTIES OF GASES [chap, vii Hence, dropping out this factor, we may write the following useful relations : W k-1 ^3^ These may be combined in the one expression W: U^- Ui:JQ = k-l:l-n:k-n. (9) Example. Let a gas expand according to the law p V^-^ = const. Taking k = lA, we have W : Uz- Ui'.JQ = OA : - 0.2 :0.2 = 2 : - 1:1; that is, the external work is double the equivalent of the heat 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-^^ in terms of the initial and final temperatures of the gas may be readily derived. Since p^ Fi = MBT^, and p^ V^ = MBT^, (5) becomes ^ _MAB k-n.rp rp^ But AB ^^ k-1 hence, Qn = ^^J^QT^- 2\). (1) We have, in general, Q^^= MciT^- T{) . (2) where c denotes the specific heat for the change of state under consideration. Comparing (1) and (2), it appears that k — n ^o\ c = c^- . (3) 1 — n Hence, for the polytropic change of state, the specific heat is con- ART. 73] SPECIFIC HEAT IN POLYTROPIC CHANGES 107 tfp, and the Fig. 35. stant (assuming c^ to be constant) and its value depends on the value of n in the equation p V^ = const. It is instructive to observe from (3) the variation of ^ as w is given different values. For ti = 0, c = kc change of state is repre- sented by the constant- pressure line aa^ Fig. 35, 36. For n = l^ e = oo, and the change of state is iso- thermal (line 6). li n='k^ then c = 0, and the ex- pansion is adiabatic (line d). For values of n lying between 1 and Ar, the value of c as given by (3) is evidently negative ; that is, for any curve lying between the isothermal h 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 J, n increases from to 1 and c increases from c^, to oo ; then as n keeps on increasing from ^ to 1, c changes sign at curve h by passing through go, and increases from — oo to 0. As n increases from n=^k to n= -{- oo, c increases ^ from c=0 to c— c^; and for n= CO, the constant volume case, c becomes c^. As we proceed further, n changes sign and increases from — oo to 0, the value of c in the meantime increasing from Cv to Cj,. Fig. 36. 108 PROPERTIES OF GASES [chap, vii 74. Determination of n. — It is frequently desirable in experi- mental investigation to fit a curve determined experimentally — as, for example, the compression curve of the indicator diagram of the air compressor — by a theoretical curve having the general equation pV^= c. To find the value of the exponent n we assume two points on the curve and measure to any convenient scale jo-^, p2-> ^v ^^^ ^t Then since we have n = ^ ^ ; ^ — , ?} - (1) log V^- log V^ 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 lb. per square inch. Fj = 2.56 cu. ft. At the end of compression, p2 = 68.7 lb. per square inch. Fa = 0.77 cu. ft. Assuming that the compression follows the law pV* = const., we have for the value of the exponent ^^ log 68.7 -log 14.5 ^-^30 log 2.56 - log 0.77 The work of compression is ^ ^ P^V^-PiVi _ 144(68.7x 0.77- 15.5x2.56) ^ _ ^^^^ ^ j^ ^^ 1-n -0.32 ' * The increase of intrinsic energy is U, - U, ^a-Il^llL^ = 144(68.7x0.77-14.5x2.56) ^ ^gg^ ^^_^^. tC — J. U.4 and the heat absorbed is ^ 5680 - 7100 1 00 T3 X Q12 ;^ = - 1.83 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 60° F., or 519.6° absolute, the final temperature is r, = 519.6(M5y-^^. 763.2, \0.77/ ' whence h = 303.6° F. ART. 74] DETERMINATION OF n 109 EXERCISES 1. A curve whose equation is pV^ = C is passed through the points p^ = 40, Fi = 6 and p^ = 16, V^ = 12.5. Find the value of n. 2. Air changes state according to the law pV^ = 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 p F^-^^ = 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 lb. 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 ; (h) 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 pv^-^ = 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 lb. 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 lb. 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. ? 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 lb. per square inch. In the final state the volume is 18 cu. ft. and the pressure is 38 lb. per square inch. Find: (a) the change of energy; (h) 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 lb. per square inch, gauge according to the lawjoy^-^ = const. Find the heat abstracted during compression. no PROPERTIES OF GASES [chap, vii 14. Deduce the relation Cp — c^ = AB from first principles without recourse to general equations. Suggestion. Let one pound of gas be heated through the temperature range Tg ~ ^\ (^) ^* constant volume, (h) 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 Cy = a -{- bt. Deduce relations between p, v, and T for an adiabatio change. Suggestion. Use the general equation dq = c^dT + Apdv and the char- acteristic equation pv = BT. REFERENCES Characteristic Equation of Gases. Deviation from the Boyle-Gay Lussac Law Zeimer : Technical Thermodynamics (Klein) 1, 93. Preston : Theory of Heat, 403. Barus: The Laws of Gases. N".Y. 1899. (Contains the researches of Boyle and Amagat.) Regnault : Relation des Experiences 1. Weyrauch : Grundriss der Warme-Theorie 1, 124, 127. The Porous-plug Experiment. The Absolute Scale of Temperature Thomson and Joule : Phil. Trans. 143, 357 (1853) ; 144, 321 (1854) ; 152, 579 (1862). Rose-Innes: Phil. Mag. (6) 15, 301. 1908. Callendar: Phil. Mag. (6) 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. Chwolson : 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, 226. Preston : Theory of Heat, 339, 243. Weyrauch : Grundriss der Warme-Theorie 1, 146. General Equations Zeuner : Technical Thermodynamics 1, 122. Weyrauch : Grundriss der Warme-Theorie 1, 152. Bryan: Thermodynamics, 116. CHAPTER YIII 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 E^^ JE^, etc. denote different chemical elements and a^, a^^ etc. their corresponding atomic weights. Then if n-^^ n^, etc. denote the number of atoms of ^j, ^2' ^^^' entering into a molecule of a given combination, the molecular weight of the compound is m Tij^a-^ + n^a^ + ••• 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 32 111 1:12 GASEOUS MIXTURES AND COMPOUNDS [chap, viii The approximate integral values are sufficiently exact for 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.O m=2x 14-1x16 = 18 Carbon monoxide CO 1 X 12 H- 1 X 16 = 28 Carbon dioxide 00, 1x12 + 2x16 = 44 Ammonia NH3 lxl4 + 3x 1 = 17 Methane CH4 lxl2 + 4x 1 = 16 Nitrogen ^2 2 X 14 = 28 Hydrogen H. 2x1 = 2 The composition by weight of a compound is readily deter- mined from the value of n, a, and m. Thus in a unit weight (pound) of compound there is '^h^l lb. of element K, m ^^ lb. of element U,, etc. m For example, COg is composed by weight of ^^ carbon and II oxygen, NHg is composed by weight of ^|- nitrogen and ^j 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 '^ ^7 = 1- (1) 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 0°C., 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 w, it follows that the product Bm is likewise the same ART. 77] RELATIONS BETWEEN GAS CONSTANTS 113 for all gases. Denoting this product Bm by i?, we have for the characteristic equation of any gas pv^^T, (2) m From (1) we obtain the relation B = Bm = P^^; (3) hence the numerical value of B 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 lb. per cubic foot at atmospheric pressure and 32° F. ; and for oxygen m = 32. Inserting these numerical values in (3), we obtain P^ 2116.3x32 ^-.r.. 0.089222 X 491.6 The constant B is called the universal gas constant. From it the characteristic constant B of any gas can be determined at once from the molecular weight. Thus for carbonic acid we have ^ = 1^=35.09. 44 From the general formula c^-c, = AB (4) for the difference between the specific heats of a gas, we have ^ ^AB^ 1544 1 ^ 1.9855 .^. ^ ^ m 777.64 m m This relation gives a ready method of calculating one specific heat from the other when the molecular weight m is known. Thus for CO2, c^-c,= ^-^^^^ = 0.0451, and if t?^ = 0.2020, we 44 have c, = 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 = -^ w, (6) BT 114 GASEOUS MIXTURES AND COMPOUNDS [chap, viii whence inserting the numerical values, jt? = 2116.3, 72 = 1544, ^=491.6, 7 = 0.002788 m. (7) For the normal specific volume, we have or V = V = 7 p m 3 58.65 m (8) (9) From the preceding relations, the following values are readily found for the constants of certain gases. Gas Chemical Symbol Molecular Weight m Character- istic Constant Difference of Specific Heats Cp-C^ Weight per cubic feet at 32° F. and Atmospheric Pressure Volume per pound at 32° F. and Atmos- pheric Pressure Oxygen .... Hydrogen . . . Nitrogen . . . Carbon dioxide . Carbon monoxide Methane . . . Ethylene . . . Air O2 CO2 CO CH, C,H, 32 2.016 28.08 44 28 16.032 28.032 48.249 765.86 54.985 35.09 55.142 96.314 55.079 53.34 0.0621 0.9849 0.0707 0.0451 0.0709 0.1238 0.0708 0.0686 0.089222 0.005621 0.07829 0.12268 0.078028 0.04470 0.078036 0.08071 11.208 177.9 12.773 8.151 12.81 22.37 12.794 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 pressure of the gaseous mixture upon the walls of the con- taining vessel is the sum of the pressures that the constituent gases wovld 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 T^ denote the volume of a given mixture, ilffj, J^, iHfg, . . . the weights of the constituent gases, and B^^ B^, B^, . . . 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 : M^B^T _ M^B,T _M^B^T Pi— Y ' i^2 — pr" ' i^3 — jr ' '" v^J According to Dalton's law the pressure p of the mixture is i' = /'i-rft + ft+ - =^(.M^B^ + 3L,B^+M^B^+ ■■■). (2) Furthermore, if Jf is the weight of the mixture, M=M-^-[-M^-hM^-{- ••• =^Mi. (3) Let us now introduce a magnitude B^ defined by the equation MB^ = M,B, + M^B^ + M,B, + • • ; (4) then (2) takes the form pV=MB„T. (5) The constant B^ 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 The partial pressures may readily be expressed in terms of the pressure of the mixture. Thus combining (1) and (5), we obtain Pi_Mi^i ^_^2^2 etc m Example. A fuel gas has the composition by weight given below. The value of the constant Bm for this gas is found as indicated by the following arrangement : CONBTITUENTR WEIGHT B MB CO2 0.04 35.09 1.4036 CO 0.27 55.142 14.8883 H2 0.067 765.86 51.3126 N2 0.585 54.985 32.1652 CH4 0.033 96.314 3.1784 C2H4 0.005 55.079 0.2754 1.000 103.24 116 GASEOUS MIXTURES AND COMPOUNDS [chap, viii Since ikf = 1 and 2 MB = 103.24, we have Brr, = 103.24. The apparent molecular weight of the mixture is 1544 -,. (.n. m = = 14.96, 103.24 and the weight per cubic foot under standard conditions is, therefore, y = 0.002788 X 14.96 = 0.0417 lb. 79. Volume Relations. — Let V^, 1^, Fg, •••, denote the vol- ume that would be occupied at pressure p and temperature T by several gaseous constituents ; then if B^, B^, B^, - • •, denote the characteristic constants of these gases, we have pV^ = M^B^T, pV^ = M,B,T, pV^ = M^B^T, ••. (1) If now the gases be mixed, keeping the same pressure and temperature,' the mixture will occupy the volume V= V,+ V^+ V,+ ..., (2) and its weight will be necessarily M=M^ + M^-\-M^+ '- (3) Taking B^ as the characteristic constant of the mixture, we have pV=MB„T. (4) Comparing (1) and (4), we obtain the relations V MBj V-MB^'" ^^^ 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 7 = ^ = f. (6) For the several constituents of a mixture, we have, therefore, Jfi = 7iF;, M^ = 7,F2, M, = 73 F3, -, (7) and for the mixture 7» = f=-^(7iFi + 72r, + 73n+-)- (8) ART. 80] COMBUSTION: FUELS 117 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^+ ... = ^Sm^Ti. Therefore, -^ = vr^ ' ^ = ^ ^ t^ , • • • (10) If further we denote by m^ the product ^^7^, we have from (8) ^m = yS^iF;. (11) The constant w^ we ma}^ regard as the apparent molecular weight of the mixture, and from it we may determine the con- stants B^^ Cp— From (6) and (7) we have }^=Y^. (8) vJ r; and for the adiabatic DA the relation ART. 88] CONDITIONS OF MAXIMUM EFFICIENCY 135 Introducing in (3) the results given by (4), (5), and (8), we obtain K AW=MB(T,-T,)log,-^ whence AW MB(iT,-T,)log, V = Qo MBT.log,^ T. (9) (10) T, dY- To 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. 88. Conditions of Maximum Efficiency. — On the T/S'-plane the Carnot cycle is the simple rectangle ABOI) (Fig. 38), hav- ing the isothermals AB and CD at the temperatures T^ and T^ 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 EBCD. For the ^ rectangular cycle the efficiency is heat transformed into work _ heat supplied For the trapezoidal cycle, likewise, the efficiency is area BEBC A, E, Fig. 38. area ABCD area A^ABB^ But DEBC area A^DEBB^ ABOD-AEB < ABCD A^BEBB^ A^ABB^ - AED A^AB^B that is, the efficiency of the trapezoidal cycle is less than that of the rectangular cycle. In the same way it can be shown 136 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix "^ that any cycle lying wholly within the rectangular cycle ABCD 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- A/r-, 7iB perature different from either the source or refrigerator. 89. Isoadiabatic Cycles. — Let a cycle be formed with the iso- thermals AB and OB as in the Carnot cycle, but with the adiabatics replaced by similar curves BO and AB (Fig. 39) ; that is, curve BO is simply curve BA shifted horizontally a distance AB. Then AB = BO, as in the Carnot cycle. If the cycle is traversed in the clockwise sense, the heat entering the medium is Qda -^ Qab = area B^BAA^ + area A^ABB^, while the heat rejected by the medium is • Qbc + Qcd = area B^B 00^ + area O^ OBB^. The heat transformed into work is the same as in the Carnot cycle, for the area of the figure ABOB is equal to that of the Carnot rectangle. Now if the heat Q^^ represented by area Fig. 39. ART. 90] CLASSIFICATION OF AIR ENGINES 137 D^DAA^ is taken from the source of heat, the efficiency of the cycle is _ heat transformed _ area ABOD heat taken from source area D^DABB^ ' and this is manifestly smaller than the efficiency of the Carnot cycle. Let it be observed, however, that Qhc = Qdai that is, area B^B CC^ = area D^BAA^. If the heat rejected by the medium during the process BC 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^^. In this case we should have the following series of steps : 1. Medium absorbs heat Q^ from source. 2. Medium rejects heat Qj,^, which is stored. 3. Medium rejects heat Q^^ to refrigerator. 4. Medium absorbs the heat Q^^ (= Qbc} stored during step 2. Since in this case the source furnishes only the heat Qab-, the efficiency is area ABCB V = area A^ABB^' 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 may make a second division based on the manner in 138 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 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 iur 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 Fig. 40. A displacer piston Q works in a cyl- inder Q, Between Q 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 ilf, com- posed of a pipe coil through which cold water is made to circulate. At the lower end is the fire 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 engine there are two displacer cylinders, one for each end of the working cylinder, which is double acting. Fig. 40. ART. 92] ERICSSON'S AIR ENGINE 139 The action of tlie engine is as follows : Assume the working 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 R and the perforations in 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 BQ. Lastly, during the return stroke the air is compressed isothermally, as represented by (7i), 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- doned because of the failure of the regenerators. This engine had an efficiency of 0.3 and consumed 1.7 lb. 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. 140 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix stroke and ran at 9 r.p.m. They developed 300 h.p. with a fuel consumption of 1.87 lb. 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 isothermally, 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 F^diagram is shown in Fig. 41. The pump cycle is DCFE, the motor cycle pj^ ^j ' EABF. The operations are as follows: (1) Compression in pump from C to i>, 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 B^ during which air receives heat from furnace. (5) Dis- charge of air, represented by BF, (6) Suction of air into pump, represented by FQ. Deducting the work of the pump from that of the motor, the effective work is given by the diagram ABCD 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 ^>S'-plane we have Q^ = Ap^V^ log, 5 = ABT^M log, -p '^ a 'a ART. 941 HKATING BY INTERNAL COMBUSTION 141 a, = Ap, F, log, ^' = - AMBT^ log, ^■ = AMB '" r^iog.-p-r.iog,^ But since T'„ = V^ and T' = V,,, AW= AMB (T,~T^) log, ^. Tlie heat Q^^ is taken from a regenerator, and therefore the heat Q^ alone is supplied from the source ; hence the efficiency- is For the Ericsson cycle DA and BO are constant-pressure lines and the analysis is essentially the same except that c^ is replaced by Cp. 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 cylinder, the outer surface of the metal walls can be kept at * For an excellent discussion of this topic see Clerk's The Gas, Petrol, and Oil Engine, Revised Edition, Chapter I. 142 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 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 the cylinder. Represented by EB, 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- inder expand adiabatically as shown by BC. 5. The burned gases are expelled in part. Represented hy BE. 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 cycle is referred to as the air standard. Fig. 42. ART. 95] THE OTTO CYCLE 143 On the TO'-plane, the ideal cycle has the form shown in Fig. 43, AB 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 e^ is constant) For the adiabatic expansion represented by BC, ^'^~ k-1 Ar X" -B Fig. 43. For the cooling at constant volume, represented by CD^ we have Q^ = Mc,{T^-T:)=-Mc,iiT,-T^-), Finally the medium is compressed adiabatically from B to A^ and for this change of state w. k-1 The heat changed into work is The work of the cycle is ir= w^+ Tn„+ W^,+ W^= (F.n-F.T-e)-(y„F„-F.F.) k—\ It is easily shown that these results are identical. The efficiency is W Jc,\(iT,-T:)-iT,-T,y, (1) (2) t) = or (8) 144 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix This expression for tj may be simplified as follows : From Fig. 43 we have hence, Therefore, ^ T. or — ^ = ^ T. T.-T, Ta T,-T, or v=i- T C4) It appears, therefore, that the Otto cycle has the same efficiency as a Carnot cycle having the extreme temperatures T^, and T^ or the extreme temperatures T^ and T^ of the adiabatics, but a smaller efficiency than a Carnot cycle having 75, and T^ 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 or v = l til pj (5) It appears from the last expression that the higher the com- pression pressure p^, the greater the ideal efficiency. If the ratio of volumes -^ be denoted by r, we have for the ideal efficiency the expression 1 (6) Example. If the air is compressed from 14.7 lb. to 45 lb., the ideal efficiency is 1-(1M)" = 0.274. ART. 96] THE JOULE OR BRAYTON CYCLE 145 If the compression is increased to 80 lb., the ideal efficiency is (Wf-- 384. ?i = o„(li-rj whence Since T;=r„ we have The temperature and pressure represented by the point B are readily calculated for this ideal case. Let q^ denote the heat absorbed per pound of air during the process AB ; then (7) (8) The value of q^ 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 Brayton engine the mixture of air and gas was compressed into a reservoir to a pressure of per- haps 60 lb. 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 A 7> compressor to motor is heated at constant pres- sure and the volume is thereby increased as in- dicated by AB. 4. Adiabatic expansion, BO^ after cut off. 5. Expulsion of burned gases, OF. The area FBAF repre- sents the negative work of the compressor, the area FBCE the work obtained from the motor ; hence, area ABCD repre- sents the net available work. On the T^-plane, the ideal Joule cycle has the same form as the Otto cycle (Fig. 43). The curves AB and CD, however, represent, respectively, heating and cooling at constant pressure. We have, therefore. 2 J-c -l-d _ -[ _c _ 2^ Also, ?i c,T,. + 1. r. (1) (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 lb. per square inch. The temperature at the end of compression is consequently higher than the ignition tempera- ture of the fuel. At the end of the compression stroke the fuel is injected into the air and at once burns. By proper regulation of the fuel supply, the air may be made to ART. 97] THE DIESEL CYCLE 147 Fig. 45. expand at practically constant pressure, or if desired, with falling pressure and nearly constant temperature. As in the Bray ton engine, govern- ing is effected by cutting off the fuel injection earlier or later. The ideal cycle of the Diesel engine is shown in Fig. 45. It resembles the Otto cycle except that the process AB in this case represents a constant pressure rather than a constant volume combustion. It was the original B aim of Diesel so to regulate the injection of fuel that a short period of combustion AM would be followed by isother- mal expansion MJV, the fuel being cut off at the point iV. On the IW-plane the ideal Diesel cycle is shown in Fig. 46, in which AB is ?i constant-pressure curve and CJ) a constant- volume curve. We have then M^- iV A G D o Fig. 46. J7 = (T,-Tj k\n-z (1) (2) (3) (4) If the cycle includes an isothermal process, as il/TV, we have Qam = Mc^{T^-T,\ (5) Q^, = AMBT^\oo-^, and 7) = m I ^%mn i Xc Vawi+ Vmn T.-T, hiT„-T:)^{h-\)T„\o^, (6) (7) 148 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 98. Comparison of Cycles. — The three principal cycles are shown superimposed in Fig. 47. The minimum temperature at I) and maximum temperature at B are the same for all three. With this assumption it is seen that the Brayton cycle A' BCD has the largest area, the Otto cycle ABOD^ the smallest. Hence, between the same temperature limits and with the same maximum pressure pi,, the Brayton 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 F^, while the Brayton cycle requires a greater volume, as indicated by the point C. 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. Fig. 47. 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 analysis of any one of the cycles discussed in the preceding articles. The following example, the data for which are fur- ART. 100] AIR REFRIGERATION 149 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 69,550 ft.-lb., the work of expansion 283,600 ft. -lb. These results refer to 1 lb. 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 283,600 - 69,550 = 214,050 ft.-lb. Therefore, the efficiency is ' J X 725.4 The "air standard" efficiency depends upon the ratio of initial and final volumes, which ratio was found to be — = 0.1887. Hence, for this efficiency ' 1 we have 77 = 1 - 0.18870-4 = 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 tiie 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 jt? F^diagram, and Fig. 50 the TO'-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 Fig. 48. 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 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^- Referring to Fig. 49, the actual compression diagram is ABFE., while the diagram FODE taken clockwise is the diagram of the expan- sion cylinder. The net work done on the air is, therefore, given by the diagram ABCB. 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 BA (Fig. 49) is perhaps 40 to 60, and the upper pressure, say 200 lb. 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 we shall assume ideal condi- tions. In the actual machine these conditions are to some extent modified. The compression and expansion are not truly adiabatic, and there is a drop in pressure between the cylinders due to frictional resistances in the coils. Let Q denote the heat absorbed from the cold body per Fig. 49, 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^ to T,, (Fig. 50), we have Q=^Me,iT,-T,-). (1) Let jOj denote the suction pres- sure of the compressor cycle (atmospheric pressure, in the case of the open cycle) and p^ the pressure at the end of com- prefesion ; then, assuming adiabatic compression, we have and if the pressure at cut-off in the expansion cylinder is also ^2 (as in the ideal case), we have also By Fig. 50. (2) T. \pj whence Ic T, The work required per minute is area ABCD W= JQ X JQ n-y. (3) (4) (5) area C^DAB^ and the heat rejected to the cooling water, represented by the area B^BCC^ (Fig. 50), is Q + W ^ rp J- n (6) The compressor cylinder draws in per minute iJf pounds of air having the pressure f^ and temperature T^. Denoting by N the number of working strokes per minute and by F^ the volume displaced by the compressor piston, we have for the ideal case 152 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix or Likewise, the volume V^ of the expansion cylinder is given by the relation Example. An air-refrigerating machine is to abstract 600 B.t. u. per rainnte from a cold chamber. The pressure in the cold room is 14.7 lb. per square inch, and the air is compressed adiabatically to 65 lb. per square inch absolute. The temperature in the cold room is 36^ F. and the air leaves the cooling coils at 80° F. The machine makes 120 working strokes per minute. Required 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^ at the end of expansion. From the relation d \pi- ^^^^^^ r, = 539.6 (i|^y~':= 352.9. ' From (1) we obtain for the weight of air that must be circulated per minute M = . ^ = - ^00 ^17.52 lb. Cp( Ta - To) 0.24(495.6 _ 352.9) The work required per minute is W = JQ^^i-:^-^ = 778 X 600 x 539.6 - 352.9 ^ 246,950 ft. lb. , Ta 352.9 and the horsepower under these ideal conditions is therefore 246950 ^ y g 33000 For the volume of the compressor cylinder, we have J. 172.9 X 53.34 x 495.6 . ^ .. Vc = = l.o CU. It., 120 X 14.7 X 144 and for the volume of the expansion cylinder V,= F,^==1.8 X §^z= 1.29 CU. ft. Ta 495.6 101. Air Compression.^ — ^Air at a pressure greater than that of the atmosphere is used extensively in engineering operations, ART. 101] AIR COMPRESSION 153 especially in mining, tunneling, and metallurgical processes. The compression of air may be effected by rotary fans and blowers or by piston compressors. In the piston compressor, air at atmospheric pressure is drawn into a cylinder through in- let valves and is then compressed upon the return stroke of the piston. When the desired pressure is attained, the outlet valves are opened and the air is discharged into a receiver. The ideal indicator diagram of an air compressor has, therefore, the form shown in Fig. 51. The line DA represents the drawing in of the air ; the curve AB rep- resents the compression from tlie lower pressure p^ to the receiver pressure p^; and BO represents the expulsion of the air at the higher pressure. It should be noted that the curve AB represents a change of state, while lines DA and BO represent merely change of locality ; thus BO represents the passage of the air (in the same state} from the compressor cylinder to the receiver. Let F^ denote the volume denoted by point -A, and F^ the volume after compression ; then the work of compression (area A.ABB,) is nn-P^V. ^ n — 1 Fig. 51. TFl; assuming that the compression curve follows the law jt?F" = const. The w^ork of expulsion (represented by area B^BOO) is evidently Wb, = -p^V^, and the work done by the air during the intake (area ODAA^^ is Hence, the total work represented by the area of the diagram -^ (f 1^1 -ft ^2)- (1) 154 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix From the relation p-^ ¥{" = p^, ^2"' ^^^ have Fo -'-■©■ (2) whence combining (1) and (2) we get w-l W n i^>^{^-(S)"]' , "-'' a formula that does not contain the final volume V^. For the temperature at the end of compression we have the usual formula n-\ T, Px (4) The action of the air com- pressor may be studied advanta- geously by means of the TS- 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 ^2 is drawn, intersecting at F an isothermal through A. It can be shown that the area A^ABFF^ represents the work W given by (1). Denoting by T^ the final temperature corre- sponding to point B^ we have Fig. 52. area A^ABB-^ = Me n 5|(T. area B^BFF^ = Mc^, (2\ - T^), &TesiA^ABFF^=M(cp n k T{). (T.-T,-) = M- n n—1 c^ — c 71 1 n Jn-1 ^ — ^^ (^MBT^ - MBT^) 1 B iPiV.-p^v^y (5) ART. 102] WATER-JACKETING . 155 Comparing (5) with (1), it is seen that the area under the curves AB and BD represents the heat equivalent of the work W. 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 Z/^ - C^i = Mo, (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 AU (Fig. 53) represents adiabatic o' ;; ^ compression. If the compres- sion could be made isothermal, the curve would be AF, less steep than AF^ and the work of the engine would be reduced per stroke by the area AEF, The water jacket gives the curve AB 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 jt? F" = const, lies somewhere between 1 and 1.40. Under usual working conditions, n is about 1.3. Yon: any value of n the relation between the heat abstracted, work done, and change of energy is given by the proportion JQ:(U^- U^) : W= (k - n} : (1 - n) : (k - 1). This applies only to the compression AB not to the expulsion of the air represented by B O. The influence of the water jacket is shown more clearly by the TW-diagram, Fig. 52. The vertical line AF indicates adia- batic compression from p^ to jpg^ the horizontal line AF, isother- 156 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix mal compression, and the intermediate curve AB, compression according to the law pV^= const., with n between 1 and 1.4. The area A^ABB^ represents the heat abstracted from the air during compression, and the area ABB 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 B. In the case of the isothermal compression represented by AF, the area A^AFF^ 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. Fig. 54. 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^ to an intermediate pressure jt?', it is then passed through a cooler where the temperature (and con- sequently the volume) is reduced, and finally it is compressed from p' to the desired pressure p^^ In Fig. 54, DA represents the entrance of air into the cylinder, and AG, which lies between the adiabatic AB and the isothermal AF, the compression in the first cylinder. From Gr to ^the air is cooled at constant pressure in the intercooler. The curve HL shows the compression in the second cylinder, and the line LO the expulsion into the receiver. In a single cylinder the diagram would be ABCB; hence compounding saves the work indicated by the area BGHL. The saving is shown even more clearly if we use the TS- plane (Fig. 55). During the first compression AG the heat represented by the area A^AGG-^ is absorbed by the water jacket. Then the heat G^GHH^ is abstracted by the inter- cooler. During the second compression the heat H^HLL^ is ART. 103] COMPOUND COMPRESSION 157 abstracted by the water jacket, and finally the heat L^LFF^ is radiated from the receiver and main. As shown in the preceding article, the area A^AGrHLFF^ gives the work of the compressor. Evidently area BGrHL represents the work saved by compounding. If we take (3) of Art. 101, we find for the work done in the first cylinder T B ^i ! 1 1 1 O ^\ Ly -Hi Gi ^x ' Fig. 55. n-l w,= n XPi'^y :)■} and for the work done in the second cylinder ^2= T 2 n—1 p'V where V is the volume indicated by point H (Fig. 54). But since point J^is on the isothermal AF^ we have and, therefore, TK = n Vi n-l n The total work is, consequently. Tf,+ TFi PxV, 2 — (1) p'J Ui. The work required is numerically a minimum when the expression (?) ■ -&;) has a maximum value. Note that p^ and p^ are fixed, while p^ 158 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix is variable. Using the ordinary method of the calculus, we find that this expression is a maximum when P' = ^PiPt (2) 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 -1 n—1 n-1 "2' and likewise, Hence, ^i = T^ ; that is, for a minimum work of compression the points Cr 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 p as steam. In fact, compressed air ^^ B has in some instances been used in ordinary steam engines. The indicator diagram for the motor should approach the form shown in Fig. 5Q. With clearance and •^' ^' ^ compression, A'U' will replace o' '^ AJE. The work per stroke is Fig 56 readily calculated in either case. The expansion curve BO may be taken as an adiabatic. 105. TS-diagram of Combined Compressor and Engine. — The ^>S^-diagram shows clearly the losses in a compressed-air system and the effects of various expedients employed to reduce such A' B \ \ \ \ \ \^ \ \ \C ^, E' D ART. 105] COMBINED COMPRESSOR AND ENGINE 159 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, as- 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 ^>iS'-plane. However, the passage of the air along the main is usually accompanied by a cooling, and this is represented by ^O' (Fig. 58), the final point O 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 Fig. 58. The air discharged from the motor in the state I) is now heated at the constant pressure of the atmos- phere until it regains its original temperature T^- This heating is represented by DA. The complete process is a cycle of four distinct operations, two of which are adiabatic and two at constant pressure ; that is, the cycle is a reversed Joule cycle. The question now arises: 160 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix what does the area ABCD of the cycle represent — something useful or something wasteful ? To answer this question let us recur to the original energy equation JQ= U^- U^+ 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 — — Wm- Work done by air = area of diagram n = -{- W^. Total work = W^ - TT^. Heat absorbed by air = area under DA. Heat rejected by air = area under BO. Total heat put into system = — area ABCD. Change of energy = Z7„ — 1/^=0. Hence, Jx 'dre^ABOD = W^ tf: that is, the area ABOD 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 (5) shows the effect of reheating the air before it enters the motor. In the main the air cools, as indicated by BC^ 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 heating EA. This reheating saves work Fig. 59. ART. 105] COMBINED COMPRESSOR AND ENGINE 161 represented by the area CDEF. It would be possible to carry I) to the right of B^ in which case the waste would become zero or even negative. The area CDEF does not, however, represent clear gain, as account must be taken of the heat expended in the process OD. In Fig. 59 ((?) 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 lb. 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^-^ = const. 3. If 200 cu. ft. of air at 14.7 lb. is compressed to a pressure of 90 lb. 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 lb. per square inch absolute. 5. Draw a curve showing the relation between the efficiency of the Otto cycle and the compression pressure. Take values of p from 40 to 200 lb, per square inch. 6. An air-refrigerating machine takes air from the cold chamber at a pressure of 40 lb. per square inch and a temperature of 20° F., and com- presses it adiabatically to a pressure of 200 lb. per square inch. The air is then cooled at this pressure to 80° F. and expanded adiabatically to 40 lb. 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, (h) If the machine makes 80 working strokes per minute, find the necessary cylinder volumes. 7. Air is to be compressed from 14.7 lb. per square inch to 300 lb. per square inch absolute. If a compound compressor is used, find the interme- diate pressure that should be chosen. 162 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 8. In Ex. 7, the compression in each cylinder follows the law pV^-^ = const. Find the saving in work effected by compounding, expressed in per cent 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 T^-plane draw accurately an ideal Diesel cycle from the fol- lowing data: Adiabatic compression of air from 14.7 to 500 lb. per square inch absolute ; heating at constant pressure to a temperature of 2200° F. ; adiabatic 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 constant-pressure heating at 1600° F. and continuing with an isothermal expansion (as shown by MN, Fig. 46). Calculate the efficiency of this modified cycle. 12. The ideal Lenoir cycle has three operations, as follows : heating of air at constant volume, adiabatic expansion to initial pressure (atmospheric), and cooling at constant pressure. Show the cycle on pV- and T^S-planes, and derive 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 TS-planes and derive an expression for the efficiency. REFERENCES Hot-air Engines Innes : Applied Thermodynamics for Engineers, 129. Zeuner : Technical Thermodynamics (Klein) 1, 340. Rankine : The Steam Engine (1897), 370. Ewing : The Steam Engine, 402. Gas-engine Cycles Clerk : Gas, Petrol, and Oil Engines, 67. Carpenter and Diederichs : Internal Combustion Engines, 65. Levin : The Modern Gas Engine, 43. Berry : The Temperature Entropy Diagram, 167. Innes: Applied Thermodynamics, 154. Peabody : Thermodynamics of the Steam Engine, 5th ed., 304. Lorenz : Technische Warmelehre, 421. Weyrauch : Grundriss der Warme-Theorie, 277. ART. 105] REFERENCES 163 Air Refrigeration Innes: Applied Thermodynamics, 396. Ewing : The Mechanical Production of Cold, 38. Peabody : Thermodynamics of the Steam Engine, 5th ed., 397. Zeuner: Technical Thermodynamics, 384. Air Compression Peabody : Thermodynamics, 5th ed., 358. Innes : 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 may be 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 ART. 106] THE PROCESS OF VAPORIZATION 165 So long as any liquid is present the vapor has a constant maxi- mum density and a constant temperature. The vapor in this case is said to be saturated and the constant temperature corre- 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?T^-plane. See Fig. 60. Consider a unit weight of liquid subjected to a pressure jt? represented by the ordinate of the line A' A" ; and let the volume of the liquid (de- noted by v'} 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" represents the volume v" of the saturated vapor at the completion segment A' A" represents Any point between A' and of a mixture of liquid and point depends on the ratio of the weight of the vapor to the weight of the mixture. Denoting this ratio by x^ we have A'M appears of the vaporization increase of therefore, volume v'^ the X = AA"' whence it A'\ as M^ represents the state vapor, and the position of the that at A', a; = 0, while at J.", x = l. 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 jB'' is smaller than that indicated by A", The curve v", giving the specific volumes of the satu- 166 SATURATED VAPORS [chap. X 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 E^ 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 p^ 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 : t = fip-),oTp = F(t), (1) v^^ip,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 f(p, V, = 0. (3) The characteris^tic surface of a saturated vapor is shown in Fig. 61. It is a cylindrical surface jS whose generating elements cut the p^-plane in the curve p = F(t). These ele- ments are limited by the two space curves v' and v" ^ which when pro- jected on the jov-plane give the curves v\ v" of Fig. 60. The space curve v" is the intersection of the surface S and the surface for the Fig. 61. superheated vapor. ART. 108] PRESSURE AND TEMPERATURE 167 108. Relation between Pressure and Temperature. — The rela- tion p = F{t^ between the pressure p and temperature ^ 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 vi^ater vapor have been made by Ramsey and Young, by Battelli, and very recently by Hol- 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. Biofs Formula. — As used by Regnault, Biot's equation has the form log ^ = a — 5a" + ^/3% (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 21.2° F., p For Steam from 212° to 428° F., p IN Pounds per Sqcare Inch. in Pounds per Square Inch. a = 3.125906 a = 3.743976 log h = 0.611740 log h = 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 yS= 0.0042454 n = f-S2 n = t-2U 168 SATURATED VAPORS [chap, x 2. Rankine^s Formula. — Rankine proposed an equation of the form 7^ p logp=A + f+±, (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 1^ = ^ + 460: ^ = 6.1007; 5 = -2719.8; (7=400125. 3. The DuprS-Hertz formula has the form \ogp = a-h\og T- ^. (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). a h c 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. BertrancCs Formulas. — Bertrand has suggested two equa- tions, namely : ^^ P=^-^ ^ (4) and p^k (-^)"- (5) The latter may be written in the more convenient form T log p = \ogk-n log rp _^ ' (^) ART. 108] PRESSURE AND TEMPERATURE 169 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= ^ + 459.6 n = 50. Fkom From 90° - 237° From 238° - 420° F. 5 = 140.1 5 = 141.43 5 = 140.8 log^= 6.23167 log^= 6.30217 log k = 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. PeessTtre in Mm. of Mercury Bertrand's Formula Experiments of Holborn and Henning 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 170 SATURATED VAPORS [chap, x 5. Marks' Equation. — Professor Marks has deduced an equation that gives with remarkable accuracy the relation between jp and 2^ throughout the range 32° F. to 706.1° F., the latter temperature being the critical temperature, as established by the recent experiments of Holborn and Banmann. The form of the equation is log p=. a- ~-cT+eT\ (7) The constants have the following values: a = 10.515354, b = 4873.71, c = 0.00405096, e = 0.000001392964. 109. Expression for — — In the Clapej^ron-Clausius formula for the specific volume of a saturated vapor, the derivative -^ CtJ/ 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), ^ = np(-^ ly ""^P ; (1) whence log ^ = log nh + log JO - log 7^ _ log (2^ - h). at d'l Values of -^ are readily calculated since the terms log T, 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. (0°C.) 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 q' = gc^dt, (1) where c' denotes the specific heat of the liquid. This process ART. 110] VAPORIZATION PROCESS 171 M is represented on the 2W-plane by a curve AA' (Fig. 62). The ordinate OA represents the initial absolute temperature 32 + 459.6 = 491.6, the ordinate A^A' the temperature of va- porization given by the relation t=f(^p}^ and the area OAA'A^ the heat q' absorbed by the liquid. This heat q' is called the heat of the liquid.* When the temperature of vaporization is reached, the liquid begins to change to vapor, the temperature remaining constant during the process. A definite quantity of heat, dependent upon the pressure, is required to change the liquid completely into vapor. This is called the heat of vaporization and is de- noted by the symbol r. In Fig. 62, the passage of the state- point from A' to A'^ represents the vaporization, and the heat r is represented by the area A^A'A"A^. For a higher pres- sure the curve AB' represents the heating of the liquid and the line B' B" the vaporization. During the heating of the liquid the change in volume is very small and may be neg- lected ; hence, the external work done is negligible also, and substantially all of the heat q' goes to increase the energy of the liquid. During the vaporization, how^ever, the volume changes from v' (volume of 1 lb. of liquid) to v" (volume of 1 lb. of saturated vapor). Since the pressure remains constant, the external work that must be done to provide for the increase of volume is L=p(iv"-v'). (2) According to the energy equation, the heat r added during vaporization is used in increasing the energy of the system and in doing external work. Hence, the difference r- AL = r-Ap (v" - v'') (3) * In this chapter symbols with primes, c', q', v', s', etc., are used for the liquid ; symbols with double primes, c'', q", v" , s", etc., for the saturated vapor. A,B, Fig. 62. B2A2E1 172 SATURATED VAPORS [chap, x 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 i/r. We have then , , .^ r^p-h'f. (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' -hr, or q" = qf -^ p -[- yjr. (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 u", we have ... , , ,^. -^ Au" = q' +/0- (o) If the vaporization is not completed, the result is a mixture of saturated vapor and liquid of quality x ix= J, as indi- cated by the point Jf (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^^ is given by q^=q' -{- xr = q' + xp + xyjr, (7) The energy of the mixture (per unit weight) above the energy of water at 32° F. is, therefore, given by the relation Au^ = q' -{- xp, (8) and the external work done is L, = Jxylr. (9) If heat is added at constant pressure, after the vaporization is completed, the vapor will be superheated. The state-point will move along the curve A"^ (Fig. 48), and the heat Cp(t^—t") ART. 112] THERMAL PROPERTIES OF WATER VAPOR 173 represented by the area A^A" EE^ will be added. Here c^ de- notes the mean specific heat of the superheated vapor, t^ the final temperature, and t" the saturation temperature correspond- ing to the pressure p. The total heat corresponding to the point ^and represented by the area OAA' A" EE^ is, therefore, qe=-q' + r-{-e^(t,-t"). (10) If Vg denotes the final volume, and u^ the energy above liquid at 32° F., then the external work for the entire process is X = ^(^;,-^;'), (11) and, therefore, Au,=^q,-Ap(:v,-v'). (12) 111. Heat Content of a Saturated Vapor. — By definition we have for the heat content of a unit weight of saturated vapor i^' = A(u" +pv"^ = q' ^ p + Apv^\ (1) Since the total heat is q" = q' + p + Ap(y"-v'), (2) it appears that i" is larger than q'' by the value of the term Apv' . As v\ the specific volume of water, is small compared with v'\ the term Apv' may be neglected except for very high pressures, and q" and i" may be considered equal. In most of the older steam tables values of q" were given ; in the more recent tables, the values of i" instead of q" are usually tabulated. 112. Thermal Properties of Water Vapor. — From the relation q" =q' ^-r, it appears that if any two of the three magnitudes q'\ q\ r are de- termined by experiment, the third may be found by a combina- tion of those two. Various experiments have been made to determine each of these magnitudes for the range of temperature ordinarily employed, and as a result several empirical formulas have been deduced. Naturally the greatest amount of attention has been given to water vapor, and we may consider the proper- ties of this medium as quite accurately known at the present time. Ammonia, sulphur dioxide, and other vapors have not 174 SATURATED VAPORS [chap. X been studied with the same completeness, and their properties 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 d the specific heat of water, the heat of the liquid above 32° F. is given by the re- l-«°" 5' = £.'di. ' (1) If the specific heat c?' were constant at all temperatures, this equation would reduce to the simple form g' = e'(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 0° 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 Temperature Specific Heat Specific Heat 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 75 167 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 + k, ART. 114] LATENT HEAT OF VAPORIZATION 175 -^here ^ is a small correction term. Then for c^ we have If now values of k 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 5-" and the latent heat r, and we may therefore determine 5' from the relation 1.008 1.006 1.004 .1.002 1.000 0.998 1 r : \ . r / 5 y , v /" 5 : /_ , ^ / ' 0° \§0° 40° 60°/ 80° lio \_ ^^""^ ^^ _--■' Fig. 63. ?' = r. 114. Latent Heat of Vaporization. — The latent heat of water vapor for the range 0° to 180° C. (32°-356° F.) has been accu- rately determined by direct experiment. The results of the experiments of Dieterici at 0° C, Griffiths at 80° 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 column of the table : r= 2265.6 - 2.7405(f - 100) - 0.003389(^ - 100)2. (1) 176 SATURATED VAPORS [chap, x LATENT HEAT OF WATER, IN JOULES Tempera- ture, C. Latent Heat Difference Pee (Jent Observed Calculated Dieterici 2493.8 2495.8 -0.08 Griffiths 30.00 40.15 2429.3 2403.6 2430.8 2407.5 -0.06 -0.16 Smith 13.95 21.17 28.06 39.80 2467.6 2451.2 2435.0 2405.8 2466.3 2450.5 2435.2 2408.3 + 0.05 + 0.03 - 0.01 -0.10 Henning, First Series .... 30.12 49.14 64.85 77.34 89.29 100.59 2424.8 2385.3 2343.0 2313.7 2285.6 2254.2 2430.6 2386.2 2347.7 2316.0 2284.6 2254.0 -0.24 -0.04 -0.20 -0.10 + 0.05 + 0.01 Henning, Second Series . . . 102.34 120.78 140.97 160.56 180.72 2248.7 2200.2 2134.2 2077.0 2018.6 2249.2 2197.2 2137.6 2077.2 2012.3 -0.02 + 0.14 -0.16 -0.01 + 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 (t - 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.* * Henning has proposed an exponential formula for r. As modified by Dr. Davis, this formula becomes in English units r = 139(689 - ty-^^^, or log r := 2.14302 + 0.315 log (689 - 0- The exponential formula has the advantage of making the value of r = at the ART. 116] SPECIFIC VOLUME OF STEAM 177 115. Total Heat. Heat Content. — For the temperature range 32° to 212° F. the total heat q" is obtained from the relation q" z= 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 f = 1091.7 + 0.305 (t - 32), 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-{-h(t- 212) + c(t- 212)2 may be adopted. Dr. Davis obtains for the heat content i" the formula f = 1150.4 + 0.3745(^-212) - 0.000550-212)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 - 212) - 0.000333 (t - 212)2 (4) may be accepted as giving with reasonable accuracy values of {" for the range 212° to 400° F. 116. Specific Volume of Steam. — The specific volume v" 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 Mech. Engs. 30, 1419, 1908. See Art. 164 for a dis- cussion of the method employed in the derivation of formula (3). N 178 SATURATED VAPORS [chap. X on the accuracy with which the factors entering into the theo- retical formula have been determined. The general equation (Art. 56) / \dtjv (1) 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 otjv at given temperature (Art. 107). Hence, for the process in ques- tion, we have (since dT= 0^ dp _ AT^ dt Jv dt v'), (2) (3) But in this case q is the heat of vaporization r ; hence we have " — ' _ ^ 1 _ Jr 1 ^ ^ ~ ATdp~ Tdp' 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 Volumes, Cu. Meters per Kg. 100° 120° 140° 160° 180° C. Experimental .... Henning From the equation for superheated steam . . 1.674 1.673 1.673 0.8922 0.8912 0.8915 0.5091 0.5078 0.5084 0.3073 0.3071 0.3071 0.1913 0.1947 0.1945 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 /^ ^ q^ ^^^ Zeuner, from the values of v'^ given in the older steam tables, deduced the value m = 1.0646. Taking the more accurate values of v" given in the later steam tables, we find ^=1.0681, (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 J., and if the temperature be raised to that denoted by J.', the increase of entropy of the liquid is repre- sented by Ovlp the heat of the liquid by area OAA'A^. Since dq' = c'dT, we have as a general expression for the entropy s' of the liquid corresponding to a temperature T, J 491.6 T J 491.6 T ' If the specific heat c' is given as a function of jP, 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 ^ 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 i" = «' + f • (3) 180 SATURATED VAPORS [chap, x For a mixture of quality x, as represented by the 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 0° to 40° C. * Marks and Davis, Steam Tables and Diagrams^ Longmans, 1908. t Peabody, Steam and Entropy Tables^ J. Wiley and Sons, 1908. ART. 119] PROPERTIES OF SATURATED AMMONIA 181 (32° to 104° F.) and also for the same range the specific heat / 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' and the entropy of the liquid s' are readily calculated by means of the relations q' = Ce'dt, s' = C<^'~ 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. 87S95 - 50 log ^Z^^U' ^^^ Above 32°, having Dieterici's experimental values of v'^ and d[ the Clapeyron-Clausius formula from (2) the derivative — ^, we may find the latent heat r from r = A (y'^ - y') T^ . (See Art. 116.) (3) 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 - OA log (266 - 0, (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 applications. Sulphur dioxide and carbon dioxide in 182 SATURATED VAPORS [chap, x 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 Mollier. 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 abscissse, the result is a curve s'. Fig. 62. This is called the liquid curve. If, likewise, values of T be laid off as abscissae, a second curve s" 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, x= 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. — Referring to the saturation curve of Fig. 62, suppose the state-point to move * For tables of the properties of saturated vapor of SO2 and COo in English units, see Zeuner's Technical Thermodynamics^ Klein's translation, Part IL t See Peabody's Steam and Entropy Tables^ or Zeuner's Technical Thermo- dynamics^ Part II. 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^A'^ B"B^ ; 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, 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 b" = »'+^; (1) hence the change of entropy corresponding to a change of temperature is obtained by differentiating (1), thus <^- (2) But _ rf«' = ^. (3) and similarly for the saturation curve, , „ e"dT ds = — — -. T Substituting these values ds' and ds" in (2), the result is (4) •"-'■+ '■/j.' IW' or ''''*jt- r T- But since c' = dq ~ dT' (5) may be written d(q' + r) r " - dT T or dT r whei re q" -- -/ -h r is tlie total heat of the saturated vapor, (5) (6) 184 SATURATED VAPORS [chap, x The derivative -^y- is readily found when an expression for q^' is known. Thus for water vapor above 212°, we have qf =a-\-b{t- 212) - e(t - 212)2 . whence g=5-2<^-212), where b = 0.S5 and c = 0.000333. At 212°, we have, for example, c" = 0.35 - - = 0.35 ^^IM__= _ 1.095. 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 — a; 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'0. — x)dT + c"xdT ; and the second is rdx ; hence we have dq=cX^-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 ^. = I = lO^-^^:L^dT + ldx. (2) The fact that ds is an exact differential leads at once to the relation ^ r- ... . „ ^ n / \ = U^\> (8) dx V(l — a;) 4- e"x T whence dT\T T dT\T/ or c" = c'-^ — --. (4) the relation that was obtained in Art. 122. ART.124] VARIATION OF X DURING ADIABATIC CHANGES 185 'If 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 E being at lower temperature. Adiabatic compression will be shown by a vertical line A" G-. 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 0.7 or 0.8), then adiabatic expansion is ac- companied by a decrease in x, 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 iVnear the curve «', then adiabatic expansion results in an increase of x, 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^O and dx = 0, and the general equation (1), Art. 123, becomes Fig. 64. lc'(l-x)-\-c'x'\dT=(), whence or X = 0, (1) (2) c' c' - c"' (3) Tlie locus of the points determined by (3) is a curve n (Fig. 64), 186 SATURATED VAPORS [chap, x called the zero curve. Along this curve we have from the general equation dq = rdx ; (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 x.r and in the final state it is 1^ But for an adiabatic change s^ = s^; therefore, we have the x^r relation ^^^ + :^ =. .^^ + ^, (5) in which x^ 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, Xy, x^^ 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) V — v^ whence a; =-77 7- C^^ v" — v' ^ ^ Suppose that the curve for some definite volume (say v = 5 cu. ft.) is to be located. For different pressures p^, p^^ p^, . . . the saturation volumes v^^', v^", Vg", . . . are known from the ART. 125] SPECIAL CURVES ON THE T/S-PLANE 187 tables. Substituting successively these values bi v'^ in (2), values of x^ as x^, x^^ x^^ . . . corresponding to the pressures Pv Pv Ps^ ' ' ' ^^^^ ^® found. The value of v' may be taken as constant for all pressures. The value of x^ locates a definite point on the p^ line, that of X2 a point on the p^ 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) we have x = For given pressures pp p^^ . (*) X. = _u-q{ Xn etc. Pi P2 Values of q' and p for different pressure are given in the table, and therefore for a given ^t, values of x^ x^^ , , . are readily calculated. These locate points on the corresponding j?-lines, and the locus of the points is the desired constant-'i* curve. By the same process may be drawn curves of constant total heat, q= q' -\- xr = const. or curves of constant heat content i — i' -{■ xr = const. In Fig. Q^^ the various curves are shown drawn through the ^ p^^ g^ 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 jo, T^ ^t, 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- namically (u— const.), v, s, and x increase, j9, T^ and i decrease. 188 SATURATED VAPORS [chap, x (3) If heat is added to a mixture at constant volume, p^ T^ s, rr, w, and i all increase. Exercise. On cross-section paper draw liquid and saturation curves for water vapor, taking values of s' and s'l from the steam table. Then draw the curves v = 2, v = 10, y = 40 cu. ft. Also draw the curves w = 600 B. t. u., u = 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'L (a) Isothermal^ or Constant Pressure^ Change of State. — Let x-^ denote the initial quality, x^ the final quality. Then the initial volume is v^ = x^(v" — v'y + v' and the final volume is The change in volume is therefore ^2 - ^1 = (^2 - ^i)(^"- v'), (1) and the external work is W=p(y^ - v^) = p(v" - v^)(x^ - x{). (2) The change of energy is u^-u^ = Jp(x^-x^}, (3) and the heat absorbed is q = r(x^-x^). (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 ^'' ~ ^^ = — ==0.2514. The external work is ^ -^ v"-v> 3.199-0.017 140 X 144 X 0.8 = 16,128 ft.-lb. The increase of energy is Jp(x2 - x^) = 778 x 786.1 x 0.2514 = 153850 ft.-lb. ; and the heat absorbed is r (x^ — x{) = 869 x 0.2514 = 218.5 B. t. u. ART. 126] SPECIAL CHANGES OF STATE 189 (5) Change of State at Constant Volume, — Since the volumes v^ and ^2 are equal, we have ^ii< - ^') = ^M' - ^')^ (5) where v^' and v.^' are the saturation volumes corresponding to the pressures p^ and p^, respectively. From (5) the quality x^ in the final state may be determined. The external work TF'is zero ; hence we have for the heat absorbed q = A(u^ - u{) = (q^' + x^p^) - (^/ - x^p^) . (6) Example. A pound of a mixture of steam and water at 120 lb. 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) ^^ ^ x,(v," - V') ^ 0.8(3.724 - 0.017) ^ ^ ^^^^^ v^" - v' 176.6 Therefore q = 311.9 + 0.8 X 795.8 - (93.4 -f 0.0167 x 959.5) = 839.2 B. t. u. (c) Adiahatic Change of State. For a reversible adiabatic change the entropy of the mixture remains constant ; hence we have from which equation the final quality x^^ can be found. Having x^^ the final volume v^ per unit weight is v = x^{v^'-v')-{-v'. (8) Since the heat added is zero, the external work is equal to the decrease in the intrinsic energy of the mixture. That is, W= u^-u^ = JlCq^' + x^p^) - {q^ + x^p^)^ . (9) Example. Three cubic feet of a mixture of steam and water, quality 0.89, and having a pressure of 80 lb. 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 : Q P s' T V" For /> = 80 lb. 281.8 819.6 0.4533 1.1667 5.464 For p = o in. Hg. 101.7 953.7 0.1880 1.7170 143.2 190 SATURATED VAPORS [chap, x The weight of the mixture is M = = ^ = 0.6167 lb. x^{v" - v') + v' 0.89(5.464 - 0.017) + 0.017 From (7), the quality x^ in the second state is given by the relation 0.4533 + 0.89 x 1.1667 ^ 0.1880 + 1.7170^:2, whence a;2 = 0.759. The volume in the second state, neglecting the insignificant volume of the liquid, is Fg = 0.6167 X 0.759 x 143.2 = 67.02 cu. ft. Finally, the external work is PF = 778 X 0.6167 [(281.8 + 0.89 x 819.6) - (101.7 + 0.759 x 953.7)] = 89,086 ft.-lb. (c?) Isodynamic Change of State. If the energy of the mix- ture remains constant, we have or q^ + x^p^ = q^ + x^p^. (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^v^ = pv"^ (11) may be passed through the points jt?^, v^, and jOg^ ^'2' ^^^ ^^^ value of n can be found. This curve will approximate to the true isodynamic on the p?;-plane, and the external work will then be approximately W=P-Jh.::ilA. (12) n — 1 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 pV-=a (1) ART. 127] APPROXIMATE EQUATION OF ADIABATIC 191 The value of the exponent n is not constant, but varies with the initial pressure, the initial quality, and also with the final pressure ; and at best the equation is an approximation. Rankine assumed for n the value J^ for all initial conditions. Zeuner, neglecting the influence of initial pressure, gave the formula n = 1.035 + 0.1 X. (2) Mr. E. H. Stone,* using the tables of Marks .and Davis, has derived the relation n = 1.059 - 0.000315 p + (0.0706 + 0.000376^>. (3) The following table gives values of n calculated from (3). Initial Initial Pressure in Labor per Square Inch, Absolute Quali- ty 20 40 60 80 100 120 140 160 180 200 220 240 100 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 1127 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 1.098 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-^^ F^, and a;^, and the final pressure jt?2i the final volume F^ is found approximately from (1), the appropriate value of n being taken from the table. The exter- nal work is found approximately by the usual formula for the change represented by (1), namely, ^i^i-i^2^2 W n — \ (4) Example. Taking the data of the example of Art. 126 (c), we have p^ = 80, Fj = o, x^ = 0.89, whence n = 1.123. The final pressure is 5 in. Hg. = 2.456 lb. per square inch. Hence from (1) 80 \iAZi 66.78 cu. ft.. .2.456/ * Graduating thesis, University of IlHnois, 1910. 192 SATURATED VAPORS [chap, x and Tr = 144 x§^iil^^Mi2i^i56^ 88,974 ft.-lb. 0.12o Comparing these results with the results obtained by the exact method, it appears that the volume Fg 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 -^ for the same temperatures. dt 3. Using the results of Ex. 1 and 2, find the specific volumes for the given temperatures. 4. Find (a) the latent heat, (b) 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. 176) 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 lb. of a mixture of steam and water at a pressure of 120 lb. per square inch, quality 0.87. 7. Find the quality and volume of the mixture after adiabatic expan- sion to a pressure of 16 lb. 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 jo F" = C. 10. A mixture, initial quality 0.97, expands adiabatically in a 12 in. by 12 in. cylinder from a pressure of 100 lb. per square inch, gauge, to a pressure of 10 lb. per square inch, gauge. Find the point of cut-off. 11. The volume of 6.3 lb. of mixture at a pressure of 140 lb. 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 lb. per square inch. Find the final value of x and the heat abstracted. 13. At a pressure of 180 lb. per square inch the volume of 2 lb. 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 lb. per square inch, is compressed adiabatically. Find the pressure at which ART. 127] EXERCISES 193 the water is completely vaporized. Find also the work of compression per pound of mixture. 15. Steam at a pressure of 80 lb. per square inch expands, remaining sat- urated until the pressure drops to 50 lb. 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 lb. 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^ lb. of mixture at a pressure p^ and quality X]_, the other M, lb. at a pressure p^ and quality x^^ 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 xg may be calculated. 18. Let 1 lb. of mixture at a pressure of 20 lb. per square inch, quality 0.96, enter a condenser which contains 20 lb. 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 Regnault: Mem. de I'lnst. 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. Kisteen : 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 Heat, 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 I'Inst. de France 21, 729. 1847. Dieterici : Wied. Ann. (4) 16, 593. 1905. P>arnes : Phil. Trans. 199 A, 149. 1902. o 194 SATURATED VAPORS [chap, x Rowland : Proc. Amer. Acad, of Arts and Sciences 15, 75 ; 16, 38. 1880- 1881. Day: Phil. Mag. 46, 1. 1898. Griffiths: Thermal Measurement of Energy. Marks and Davis : Steam Tables and Diagrams, 88. (b) Latent Heat Regnault : Mem. de ITnst. 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. (d) Specific Volume Fairbairn and Tate : Phil. Trans. (1860), 185. Knoblauch, Linde, and Klebe : Mitteil. iiber 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 fiir Kalteindustrie. 1904. Jacobus: Trans- Am. Soc. Mech. Engrs. 12, 307. Wood: Trans. Am. Soc. Mech. Engrs. 10, 677. Peabody : Steam and Entropy Tables, 8th ed., 27. Zeuner: Technical Thermodynamics (Klein) 2, 252. Lorenz : Technische Warmelehre, 333. (b) Sulphur Dioxide Cailletet and Mathias : Comptes rendus 104, 1563. 1887. Lange : Zeitschrift fiir 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. ART. 127] REFERENCES ' 195 (c) Carbon Dioxide Amagat : Comptes rendus 114, 1093. 1892. Mollier : Zeit. fur Kalteiiidustrie 1895, 66, 85. Zeauer: Technical Thermodynamics 2, 262. General Equations for Vapors. Changes of State Zeuner : Technical Thermodynamics 2, 53. Weyrauch : Grundriss der Warme-Theorie 2, 33. Preston : Theory of Heat, 650. Berry : 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 ^ = /*(p). 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^ denotes the saturation temperature given by t^ —f(^p^ and t the temperature after su- perheating, the difference ^ — ^^ is the degree of superheat. Thus for steam at a pressure of 120 lb. per square inch, tg= 311. 3° J5^; 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 jt?, 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 independently . The volume v of the superheated vapor depends upon both p and t thus taken as independent variables ; that is, 2; = (/>(^, 0, (1) 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 ART. 129] CRITICAL STATES 197 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 jt?, 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^ v, 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 IT (Fig. 66^, and o" 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. (JG. 198 SUPERHEATED VAPORS [chap. XI 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 BC If the liquid be compressed iso thermally, the volume will be decreased slightly as the pres- sure rises, and the process will be represented by curve CD. The isothermal has therefore three distinct parts: along AB the fluid is superheated vapor, along BQ ^ mixture, and along OD a liquid. If the initial tem- perature be taken at a higher value t\ the result will be similar except that the segment B'C will be shorter. If the limit curves meet at point H^ 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. QQ the segment BO represents the difference v^' — v' between the volume v^' of saturated vapor and the volume v' of the liquid; and in Fig. 67, the area B^BCO^ represents the la- tent heat r of vaporization. For the isothermal t^ that passes through IT^ the segment BO reduces to zero; hence, for this temperature and all higher temperatures, we have Fig. G^ V — V 0, or v' and r = 0. V' = The second result also follows from the first when we consider the Clapeyron equation JrJ_ Tdp. dT The experiments of Andrews show that the condition just described may be actually attained. The isothermals for carbon ART. 129] CRITICAL STATES 199 dioxide as determined by Andrews are shown in Fig. 68. For 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 % was called by Andrews the critical tempera- ture. It has a definite value for any liquid. The pressure p^ and volume Vc indicated by the point H are called respectively the critical pressure and critical volume. Values of t^ and p^ fo^ various substances are given in the following table : p 110 Atm. \ 100 \ \ 48.f 90 uV \ vg5.5° ^ \ 80 v^ — ^o N \ 81.1"N ^ \ \ 70 \ ^ N^ \ N ^ \ \ 60 i 21.5° ~1V \ 50 \ 13.1° "^ — ?> Fig. 68. Substance tc, Degrees C. Pc Atmospheres Water 365.0* 130.0 197.0 155.4 30.92 277.7 -146.0 -118.0 -220.0 -140.0 200.5 115.0 35.77 78.9 77.0 78.1 35.0 50.0 20.0 30.0 Ammonia Ether Sulphur dioxide Carbon dioxide Carbon disulphide Nitrogen Oxygen Hydrogen Air * According to the recent experiments of Holborn and Baumann, the critical temperature of water is 706.1° F (874.5° C) and the critical pressure is 3200 lb. per square inch. See article by Prof. Marks, Jour. A. S. M. E., Vol. 83, p. 563. 200 SUPERHEATED VAPORS [chap, xi It appears from the definition of the critical temperature that it is possible for a mixture of liquid and vapor to exist only for temperatures below %. At higher temperatures the mass re- mains homogeneous throughout the entire range of pressure. 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 regal'ded 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., w^hose critical temperatures are above ordinary temperature are designated as vapors. The determination of the critical values f^, p^-^ and v^ 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, which was deduced from certain considerations derived from the kinetic theory of gases. As van der Waal's 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 ^ V - a {v^cy^ ^ ^ where /( T) is a function of the absolute temperature that takes the value 1 at the critical temperature. ART, 131] THE MUNICH EXPERIMENTS 201 The equations of van der Waals and Clausius are constructed with special reference to the behavior of fluids in the vicinity of the critical state ; hence they apply more particularly to such fluids as carbon dioxide, the critical temperature of which is within the range of temperature encountered in the practical applications of heat media. The critical temperatures of most important fluids, as water, ammonia, and sulphur dioxide are, however, far above the ordinary range, and for these media the general equations do not give as good results as certain purely empirical equations deduced from experiments covering a relatively small region. For some fluids, notably ammonia, there is unfortunately a lack of experimental data; for the most important fluid, water, we have, however, reliable data furnished by the recent experiments at Munich. 131. Experiments of Knoblauch, Linde, and Klebe. — The experiments made at the Munich laboratory were so con- ducted that three important relations could be obtained simultaneously. These were : 1. Relation between pres- sure and temperature of saturated steam. 2. Relation between spe- cific 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 h to which is attached a glass tube e. A similar glass tube d leads from the outer vessel a^ and the two are connected at h with 7V- To Manometer 202 SUPERHEATED VAPORS [chap, xi a tube /leading to a mercury manometer. Steam is introduced 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 h and is evaporated gradually by the heat absorbed from the steam surrounding it. As long as vessel h contains a saturated mix- ture, the pressure within h must be the same as that within a, since the temperature is the same throughout. Hehce the mercury levels m, m in tubes c and d will be at the same height. When the water in h is all vaporized and the pressure and temperature of the steam in a is further increased, the steam in h 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 hj the 2^s-diagram (Fig. 70). Let point A on the saturation curve s^^ denote the state of the steam in vessel h 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 t^ to t^ the state of the steam in a changes as represented by the curve AC \ that is, the steam in a is saturated at the pressure f^. The apparatus is so manipulated, however, that the mercury level m in tube c is held constant, thus keeping a constant volume of steam in vessel h. The point representing the state of the steam in h moves along the constant volume curve AB in the superheated region, and the final pressure ^3 given by the point B is smaller than the pressure 'p^ 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 with the level n gives the relation between the pressure and temperature of saturated steam. ART. 132] EQUATIONS FOR SUPERHEATED STEAM 203 132. Equations for Superheated Steam. — To represent the results of the Munich experiments, Linde deduced the empiri- cal equation pv = BT- p(\ +.«/?) 0[^\ -B (1) In metric units with p in kilogram per square meter^ the con- stants have the following values : ^ = 4T.10 6^=0.031 n=^. a = 0.0000002 i)= 0.0052 With English units and pressures in pounds per square inch^ the equation becomes : pv = 0.5962 2^-^9(1 + 0.0014^) /'l^M^^^^i _ O.SSsY (2) The form of Eq. (1) is such as to make it inconvenient for the purpose of computation ; and the constant I) in the last term leads to complication in the working out of a general theory. A modified form of the equation, namely, v-^c=-—--(l-^ctp)— (3) is free from these objections and with constants properly chosen represents the results of the Munich experiments as accurately as 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 log m = 11.19839 log m = 13.67938 n = 5 n = 5 c = 0.0055 c = 0.088 a = 0.00000085 a = 0.0006, p in pounds per square inch. The final equation with constants inserted is therefore V + 0.088 = 0.5963 ^ - (1 + 0.0006 j^) ^^^^^f ^^^ • (4) This equation is the one that will be used in the subsequent developments. 204 SUPERHEATED VAPORS [chap, xi An equation of the simple form v-^c=-^^ (5) P has been proposed by Tumlirz on the strength of Battelli's experiments. Linde 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 i; + 0.256 = 0.5962—. (6) P 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"" (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, , BT ^/273Y .Q. ^-^ = — -^o(-^j (8) 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. ART. 133] SPECIFIC HEAT OF SUPERHEATED STEAM 205 3. The experiments of Holborn and Henning at atmospheric pressure and at temperatures varying from 110° to 1100° C. 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 of one atmosphere the specific heat of superheated steam has the constant value 0.48 for all temperatures. This value has been largely used for all temperatures and for all pressures as well. Experiments by Mallard and Le Chatelier and by Langen at high temperatures agree in making the specific heat a linear function of the temperature. Thus, according to Langen, c^ = 0.439 + 0.000239 t, (1) where t is the temperature on the C. scale. The earlier experiments of Holborn and Henning at much lower temperatures than those of Langen lead to the formula c^ = 0.446 -H 0.0000856 t. (2) This is again a linear relation, but the coefficient of t is smaller than that in Langen's formula. Equations (1) and (2) show that the specific heat varies with the temperature at least, and that the convenient assumption of the constant value 0.48 is not permissible. Finally, the experiments of Knoblauch and Jakob show con- clusively that Cp depends also upon the pressure. In these experiments, steam was run through a first superheater in which all traces of moisture were removed. It was then run through a second superheater consisting of coils immersed in an oil bath. The heat was applied by means of an electric current and could be measured quite accurately, and a com- parison of the heat supplied with the rise of the teaiperature of the steam gave a means of calculating the mean specific heat over the temperature range involved. Experiments were conducted at pressures of 2, 4, 6, and 8 kg. per square 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 Avith 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^, that will give results substantially in accord with the experi- mental results of Knoblauch and Jakob. For this purpose we make use of the general relation From the characteristic equation, ^H-^ = ^ (l + «i?)^, (4) we obtain by successive differentiation dv B BT p - + mn ^^i(l + «p), (5) d'^v mn(n 4- 1) .., , . ..,. Substituting in (3), the result is <¥)• a) Taking T as constant and integrating (7) with p as the in- dependent variable, the result is Cp = -^^ — '-p[l + ^i^J-f const, of mtegration. Now since T was taken as constant, the constant of integration may be some function of T\ hence we may write .,=Ky)+ ^"y^> i>(i+|i>) (8) Inspection of (8) shows that as T is increased the last term grows smaller ; in fact, c^ approaches (pCT) as 7^ is indefinitely ART. 133] SPECIFIC HEAT OF SUPERHEATED STEAM 207 100 150 300 350 300 Temperature C. Fig. 71. 350 400 208 SUPERHEATED VAPORS [chap, xi increased. From Langen's experiments, it is seen that at very high temperatures c^ is given by an equation of the form hence we are justified in assuming that where a and /3 are constants to be determined from experi- mental evidence. Equation (8) thus becomes ., = «+^2'+^^^^|^^(l+|4 (9) 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 Jakob. At a given temperature T the specific heat increases with the pres- sure ; furthermore for a given pressure, c^ has a minimum value as appears by equating to zero the derivative The following values of the constants have been found to make Eq. (9) fit fairly well the experimental results of Knob- lauch and Jakob : a = 0.367 /? = 0.00018 for the C. scale. /S= 0.0001 for the F. scale Replacing the product Amn(^n -f- 1) by a single constant (7, we have as the final formula for the specific heat e^ = 0.367 + 0.0001 T-^pQl + 0.0003 p} ^, (10) where log 0= 14.42408 (pressure in pounds per square inch). Figure 71 shows the curves representing this formula for the pressures of the Knoblauch and Jakob experiments. The agreement between the points and curves is satisfactory, con- sidering the difficulty of the experiments. In Fig. 72 the Cp-cnryes for various pressures in pounds per square inch are ART. 133] SPECIFIC HEAT OF ' SUPERHEATED STEAM 209 100 .OoU ~~ "~ P- i \ ' \ \ \ ^ \ \ .610 \ \ \ \ .600 \ \ \ \ — — .590 ^ ^ \ \ .580 \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ .500 \ \ \ \ ^ \ \ \, \ \ B .550 \ \ \ \ V ^, \ \ \ '' K^r \ \ \ \ s \ % '^^ \ \ \ s ^ \ ^ k^ p. 530 \ \ \ \ \ \ . ^ s \ \ \ \, \, \ N, \ 1 .520 .510 , \ \ \ \ A \ \ \ X \ ^ \ \ \ \ \ \ ■\ -^ -— — -^ ^ \ \ s \ \ ^^c \ \ \ X ^ ^ . ___ — ■ "^ ^ -^- "^ -^ \ \ \ ^ \ ^ \ -^ _ — — — ' " ^ =q >■ ^ :^ \ \ \ -^ \ \ V. ^ ^ -^ _ — - -— ^ ^ — ^ ;:: ^-■ <^ "^ \ ^ s \ V ^n X ^ ^ _ — — ■ ■ ^-- -^ -^ "C^ -^ ^ — ■^ .490 \ \ \ \ 'to X ^~ — — — — — ^ -- -^ ^-^ "C^ i> -^ \ \ \ — ^^ \ X- iv^ ^ " ■ — — — -^ _- ^ '^^ ^ ^ ■^ ^ ^ ^80 \ \ s ^ ■^~- — m. — ' — ;1^ '- '^ ■^ -^ .^ -^ V- \ "x X \ " ^ -^ ^ ■;;;;:^ "^^ ^ "^^^ ^ ::^ ^ b^ ^ — .470 \ s \ \ ^-^ "^^ — — — nil -- ^ ^ ^ ^ ^ .^ ^ ^ ^ " ^>. ^ ^ — ^ -^ ^ ^ ^ ^ X \^ ~~~- 4l — " ' __^ — ^ ^ ■^ ^ -^ ^ .460 "" ■ — - m ^ ^ :> ■^ .450 \ r _^ ^ ^ :^ ^^ ^ " — — -5- -^ ^ ^ ^ ^ ^_ S ^ -^ .440 , — ' ^ > -^ ~" .430 — — .420 .410 ann 200 300 400 Superheat, Hq^. E. Fig. 72. 500 600 210 SUPERHEATED VAPORS [chap, xi shown. The abscissas are, however, not temperatures but degrees of superheat. 134. Mean Specific Heat. — Formula (10), Art. 133, gives the specific heat at a given pressure and temperature. For some purposes it is desirable to have the mean specific heat be- tween two temperatures, the pressure remaining constant. This is readily calculated by the mean value theorem ; thus denoting by (^Cp)„^ the mean specific heat, we have Using the general expression for c,,, we laave, therefore, ^ A 1 f^'^r , om , Amn(n-^V) (-, , a Wjm Amp(n + l)(l + lp){jrn - ^„)- rp n f— ' rr" ^ ' ^^^ (2) The calculation, while straightforward is rather long, and if Cp-curves are available, it is usually preferable to determine the mean e^, 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- tion state and the given state is given by the ordinate corre- sponding to the given degree of superheat and the given pressure. For example, at a pressure of 150 lb. per square inch the mean specific heat for 240° superheat is 0.529. 135. Heat Content. Total Heat. — Having a formula for the specific heat at constant pressure, equations for the heat con- tent and the intrinsic energy of a unit weight of superheated steam at a given pressure and temperature are readily derived. For this purpose the general equation dq = c^dT- At(~) dp (see Art. 54) (1) ART. .650 1341 MEAN i 5P E( :!i PI c H E. A.1 1 2 11 \ .640 \ .630 \ \ \ ,620 \ s \ \ \ .610 \ \^ \ \ s \ .600 \ \ \^ \ \ ^ \ Knrt \ \ [ \ \ \ \ \ \ \ \ s \ \ \^ \ \, \ s \ V \ \ \ <% . \ \ \ ' \ ^ <* .560 \^ \ \ \ ^J ^,s> \ s \ ^ \ \ ^ \ 1 ^ te ^f^ .550 \ \, \, \ \^ 1 h b \ s \ \ > \ ■--^ ^ c Heat \ \ ^ N \ 1 \ \ -^ ~~^ \, \ N \ N ^^ ^ «^__ ^^ — ■ —J ■— , ^ \ s \ \ k 1 v^ .^ -^ ;C .oou \ \ \ \ \ ^ ^-81 "^ ^ ^ -^ w'.520 k^ \j \ \ \ -^ "" — — — — ^ s \ s \ N 4 --^ --, — ~~] a .510 \ ^ \ \ \ ^ \ ■^ 7^ ^ ^ -— — ___ N \ ^ -^ ^ ~~ ~ \ \ \ ^ \ no ^ ^ ^ — — — ^^ s ^ k ^ -^ ^ ~~iL --^ ■ — — — — — — — ^ V. ^ ^ ^ ■^ 0(1 --.^ " . 490 — — — ; ._ \ ^ ^ -^ ^ ^ 4U_ _ -n — — — — — — — — — -^ 480 \ ^ ^ ^^ -^ -_ sa ~ — — — — . — \ ^ .^ ^~- ^ 25 — — — — — — — ~~ I^ — — — — — Z- ^ .^^ ■ 20 — — . — — ^ — 15 — — — — '" ^- -- — .460 ~^ -^ 10 — -— — —^ "^ ^^ ^ " — — __. _— ■^ '^ _^ -^ -^ '^ ^ .450 - , .A — ■ — ■ _, ^ -^ — — . ^ — 3 — — ^ 1- ■ ' , . ^ .440 — — — — zz J^ ^ . -^ -^ — — — ■ AOn — — — — ■ .430 .410 .400 ) 1( K) 2( X) g up erl I 3( lea X) t,I . 7 )eg 3. .F 4( )0 5 X) 60 212 SUPERHEATED VAPORS [chap, xi is most convenient. Since by definition i= A(lu -{-pv'), we have di = A \_du + d (^v)], or di = dq-\- Avdp. (2) Hence, making use of (1), di = c^dT-A[T^-vyp. (3) From the characteristic equation we have whence rg,- „ = („+ 1)(1 + »^)^^. Introducing in (3) this expression for T -—— v and the general expression for Cp^ the result is di = (a + fiT)dT-^ Amnin + r)p{\ + \v\-^^ - ^^y^^ (1 + «i>) * - ^^^i>. (4) Since i 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 = «y+|r^-^f« + i)p(i+^;>)^-^«P + v (5) The constant of integration i^ 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 i^ can be calculated. The first member is the value of i for the assumed pressure as given in the steam table. Hence i^ is found by subtraction. By this method the mean value 2q= 886.7 is obtained. Introducing known constants, Eq. (5) becomes i = ^(0.367 + 0.00005 2^) - ^ (1 + 0.0003 jt>)-^ -0.0163jt?-|- 886.7. (6) ART. 135] HEAT CONTENT. TOTAL HEAT 21S Here log C = 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 TO'-plane, the process is shown by the line ABCD (Fig. 74). The area OABCC^ rep- resents the total heat of the saturated vapor, which has been denoted by q" . The area C^CDB. represents the heat added to superheat the vapor. This heat is evidently given by the integral C Fig. 74. ^c^dT^^ a-\-^T Jin -P (-1^)] dT taken between the saturation temperature T^ at point C and the final temperature T at point D. This integral is, in fact, the product ((?p)^(^— T^), where (e^)^ is the mean specific heat for the temperature range T— T^. The total heat of a unit weight of superheated steam is given therefore by the expression ^ ^ ^„ ^ (O-C^ - T,). (7) The term (^Cp)^(^T — T^) is easily found from the mean specific heat curves (Fig. 73), and q"Q=i"^ is given in the steam table. Hence Avith 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 lb. per square inch superheated 200°. From the steam table {"(= q") for this pressure is 1194.6 B. t. u. ; and from Fig. 73 the mean specific heat from saturation to 200° superheat is 0.534. Hence { = 1194.6 + 200 x 0.534 = 1301.4 B. t. u. The result given by formula (6) is 1301.7 B. t. u. 214 SUPERHEATED VAPORS [chap, xr 136. Intrinsic Energy. — For the intrinsic energy we have from the defining equation ^ = A(u + pv}, Au = i — Apv, (1) Using the expressions for i and v heretofore derived, we obtain the equation Au= Tia-hi/BT-AB)-^^'^ Jfr n+(n—l)-p 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, ^M = 2^(0.2566 + 0.00005 2^)- ^(1 + 0.00024 jt?) + 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 (,^). (7) The change of energy may be found by combining (6) and (7) or from the general equation of energy. It is found to be (^l-i'2)+|(»-l)(l'l'-^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. Adiahatic Change of State. For an adiabatic change the entropy remains constant ; hence, for the relation between the final pressure p^ and temperature T^^^ we have from the general equation for entropy « log, T^ + ^T^-AB log,p2 - Anp^ (l + Ip^-^, = O, where (7 is a constant determined from the initial state. The pressure p^. i^ generally given; therefore, we have the tran- scendental equation «log, r, + pT^-p.^{l + Ip^f^, = 0+AB\og,p^ = C", (9) from which the value of T^ may be found by successive trials. ART. 138] SPECIAL CHANGES OF STATE 219 Having the initial and final values of p and T^ the initial and final values u-^ and u^ of the intrinsic energy may be calculated. The external work per unit weight is then W=u^-u^^. (10) In problems connected with the flow of steam the change of heat content resulting from an adiabatic expansion is required. This difference is found by calculating from the general equation for the heat content the initial and final values ^J and i^. If the adiabatic expansion is carried far enough, the expansion line, as DE (Fig. 74), will cross the saturation curve s^'^ and the state-point will enter the region between the curves s' and s". This means that at the end of the expansion the fluid is a mix- ture of liquid and vapor. The investigation of this case presents no difficulties. The entropy and energy at the initial point I) are calculated from the general equation. Knowing the pressure for the final state E^ the quality x is readily determined from the equation «i = V + ^> (11) where s^ denotes the entropy in the initial state. Having x, the energy in the final state is calculated from the equation Then the external work per unit weight is given by the equation W= u^-u^== Wj - J(q^' + x^p^}. (18) Example. Steam at a pressure of 150 lb. 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 ; also the pressure at which the steam becomes saturated. From the general equation the entropy in the initial state is found to be 1.6346. From the steam table we obtain for the final pressure s' — 0.1880, -= 1.7170; hence ^ 1.6346 = 0.1880 + 1.7J70X, or x = 0.8425. In the initial state the energy in B. t. u. is Aux = 918.1(0.2566 + 0.00005 x 918.1) _ J-|^(l + 0.00024 x 150) + 886.7 = 1153.9 B. t. u. 220 SUPERHEATED VAPORS [chap, xi In the final state the energy is Au2 = qoj + X2P-2 = 101.7 + 0.8425 x 953.7 = 905.2. Hence, the external work per pound of steam is W=ui-U2= 778(1153.9 - 905.2) = 193,490 ft.-lb. The initial entropy 1.6346 is the entropy of saturated steam at a pressure of 66.6 lb. 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 -{- c')"' = 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 w^th the degree of superheat ; however, it appears that the value w = 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), T = ^P\ • (3) n-l or ART. 140] TABLES FOR SUPERHEATED STEAM 221 For the external work, we have w= rpdv= r ^"^ ^^= yi(''i + o-;>2(''2 + o . (5) Given the initial state of the fluid, the volume in the final state may be found from (1), the final temperature from (4), and the external work from (5). Example. A pound of superheated steam at a pressure of 200 lb. per square inch and superheated 200° expands adiabatically to a pressure of 50 lb. per square inch. Required the final condition and the external work. The initial volume is found to be 2,973 cu. ft., and the initial entropy 1.6657. Using the formula for s (Art. 137), the final temperature is found by trial to be 752.5° absolute ; and taking this value of- 7^2, the exact value of the final volume is found to be 8.681 cu. ft. From (3), Art. 136, the energy in the initial state is found to be 1200.57 B. t. u., that in the final state 1098.82, B. t. u. ; hence the external work is 778 (1200.57 - 1098.82) = 79,262 ft.-lb. Taking the approximate formulas, we have 1 j_ v^ + c= (I'l + c) (^y'^= (2.973 + 0.088) (^V'' = 8.819; whence v^ = 8.819 - 0.088 = 8.731 cu. ft. To =Ti(^] •" = 1041.4 (—] '" = 749.8°. j^ = Pi(vi + c)-p2(v2+c) ^ M4 ,^00 X 3.061 - 50 x 8.819) = 79,550 ft.-lb. n-1 0.31^ ^ It will be seen that for practical purposes the results obtained from the approximate equations are satisfactory as regards accuracy. 140. Tables and Diagrams for Superheated Steam. — The lead- ing properties of superheated steam — volume, entropy, and total heat — for various pressures and degrees of superheat have been calculated and tabulated by Marks and Davis and by Peabody. The values in the Marks and Davis tables are derived from specific heat curves that differ somewhat from the curves of Fig. 72, and they therefore differ from the values obtained from the equations of Arts. 135-137. However, throughout the range of ordinary practice, the difference does not exceed one half of one per cent. The Marks and Davis tables are accompanied by graphical charts that may be used to great advantage in the approximate 222 SUPERHEATED VAPORS CHAP. XI solution of numerical problems. The principal chart has the heat content i as ordinate and the entropy s as abscissa. The \ \ J v \ \ h \f \ ) h j^ M ■A ^'^ \ o 1 0/ \ \ Y \l «/ I K ]\ '' \ l\ \ c? V v' 0/ ^ \ / v^ . \ K^ j\ \ h \ ^ \ \ \7 \i v\ W y ^ N \ J\ , l/j k") \\( \ X V \ M Y ^v A \ \\ \/ y \ \\ s^ \/\ (\ ^\ s 1 %{ ^ r\ \\ \V) \ k ^\ |\\ ^y ^y\ \ ^^ ' N V) \\ Y A^ w V ^ X \\ u A ^ J\\ \\ )v \V \ \\ \\ \\ v\\ ^ A \\ V \^ \X\ \\ \\ \\ g-5 Fig. 75. general character of the chart is shown in Fig. 75. The curve marked " saturation curve " represents the condition of dry ART. 140] MOLLIER'S CHART 223 saturated steam at various pressures. The region above this curve is the region of superheat, and the lines running approxi- mately parallel to the saturation curve are lines of constant degree of superheat. Below the saturation curve is the region of wet steam, and the lines running parallel to the saturation curve are lines of constant quality. The lines that cross the saturation curve obliquely are lines of constant pressure. The first conception of the heat content-entropy chart is due to Dr. R. MoUier of Dresden, hence we shall refer to it as the MoUier chart. In addition to the chart published by Marks and Davis, one is contained in Stodola's Steam Turbines and one in Thomas' Steam Turbines. In the light of the recently acquired knowledge of the properties of saturated and superheated steam, the Marks and Davis chart must be regarded as the most accurate. The Mollier chart may be used for the approximate solution of many problems that involve the properties of saturated and superheated steam, and it is specially valuable in problems on the flow of steam. The following examples illustrate some of the uses of the chart : Ex. 1. Steam at a pressure of 150 lb. per square inch superheated 200° F. expands adiabatically to a pressure of 3 lb. per square inch. The point representing the initial condition lies at the intersection of the constant-pressure line marked 150 and the line of 200° superheat. Locating this point on the chart, it is found at the intersection of the hues i = 1300 and s = 1.687. The heat content and entropy in the initial state are thus determined. The line s = 1.687 intersects the constant-pressure curve p = ^ on the line { = 1002 ; hence the heat content after adiabatic expansion is 1002 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 region of higher pressure jo^ to a region of lower pressure p2, the heat content remains constant. Steam at a pressure of 200 lb. per square inch and quality 0.95 flows into the atmosphere ; required the final condition of the steam. Drawing a line of constant-heat content from the initial point to the curve p = 14.7, it is found that the final point lies above the saturation curve and that the steam is superheated about 12° at exit. The entropy increases from s = 1.498 to s = 1.766. 141. Superheated Ammonia and Sulphur Dioxide. — Experi- mental evidence regarding the properties of superheated vapors 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- Cp^, (1) and this form has been accepted by Peabody, who derives the following values of the constants (English units) : ^=99, C=710, m = l. For sulphur dioxide Peabody uses the same equation with the constants : j5=26.4, (7=184, m = 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 Cp= 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: ii)=(r- Zeuner found that for superheated steam the exponent n 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. 1 — m 1 — 0.25 For sulphur dioxide n = — — = 1.282. J. — u. zz Regnault, however, gives for ammonia w = 1.32. A second method of finding the temperature T^ at the end of adiabatic compression makes use of the properties of the n — \ ART. 141] SUPERHEATED AMMONIA 225 saturated vapor. Let A (Fig. 76) represent the initial state, and B the final state after adiabatic compression. UA and FB are constant-pressure curves. Denoting by TJ the satura- tion: temperature correspond- ing to the pressure p^, the increase of entropy from U T to A is tfplog^-J^, and the total entropy in the state A is s/' 4- Cp log. TJ Likewise, the entropy in the state B is TJ'' ?2'^+^pl0ge B. T2 p„t: — \F' \^ T, P.X \ ^ A V" Fig. 76. Since AB is an adiabatic, the entropies at A and B are equal, and therefore j/' + c^ log, 4r7 = s^" + <^v lo^e -^, • (8) In this equation s^\ s^\ TJ ^ and TJ' are tabular values corre- sponding to the given pressures p^ and p^^ and 7^^ is given. Hence, T^ is the only unknown quantity. EXERCISES 1. Calculate by Eq. (2), (4), and (6), respectively, of Art. 132 the vol- ume of one pound of superheated steam at a pressure of 180 lb. per square inch and a temperature of 430° F. Compare the results. 2. If the products pv are plotted as ordinates with the pressures p as abscissas, show the general form of the isothermals T = C when Eq. (3), Art. 132 is used ; when Eq. (6) is used. 3. For ammonia, Peabody gives the following equations for the latent heat of vaporization : r = 540 — 0.8 (t — 32). If at the critical temperature r = 0, find ?c for ammonia by means of this formula and compare with the value of tc given in Art. 129. Explain the discrepancy. 4. Following the method of Art. 133, deduce an equation for Cp, using the approximate equation (.5), Art. 132 ; also using Callendar's equation (8). 5. By means of Eq. (3), Art. 132, calculate the specific volume of satu- rated steam at the following pressures : 5 in. Hg., 20, 50, 150 lb. per square Q 226 SUPERHEATED VAPORS [chap, xi inch. Use the saturation temperatures given in the steam table, and com- 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 lb. 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 lb. per square inch and a temperature of 430° F. 8. A pound of saturated steam at a pressure of 120 lb. 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 until 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 Cp very close to the saturation limit : (c^)sat=0.4H--— ^, in which tc is the critical temperature, 689° F., and tg 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 Cp at saturation : (c^)gat = a -\- htg. Test this equation, and if it holds good within reasonable limits determine the constants a and h. 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 lb. per square inch. By the approxi- mate equations of Art. 139 find the temperature and volume after adiabatic expansion to a pressure of 80 lb. per square inch ; also the work of expansion. 13. Assume for the initial state of superheated steam jOj = 80 lb. per square inch, v^ = 20 cu. ft., t^ = 350° F. Plot the successive pressures and volumes for an isothermal expansion to a pressure of 30 lb. 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 ; (b) 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. ART. 141] REFERENCES 227 Zeuner: Technical Thermodynamics (Klein) 2, 202-229. Chwolson : Lehrbuch de Physik 3, 791-841. Characteristic Equations Callendar : Proc. of the Royal Soc. 67, 266. 1900. Linde : Mitteilungen iiber Forschungsarbeiten 21, 20, 35. 1905. Zeuner : Technical Thermodynamics 2, 223. Weyrauch : Grundriss der Warme-Theorie 2, 70, 87. Specific Heat of Superheated Steam Mallard and Le Chatelier : Annales des Mines 4, 528. 1883. Langen : Zeit. d. Ver. deutsch. Ing., 622, 1903. Holborn and Henning : AVied. Annalen 18, 739. 1905. 23, 809. 1907. Regnault : Mem. Inst, de France 26, 167. 1862. Knoblauch und Jakob : Mitteilungungen iiber Forschungsarbeiten 35, 109. 1906. Thomas : Proc. Am. Soc. Mech. Engrs. 29, 633. 1907. A most complete discussion of the work of various investigators is given by Dr. H. N. Davis, Proc. Am. Acad, of Arts and Sciences 45, 267. 1910. General Theory of Superheated Vapors Callendar : Proc. of the Royal Soc. 67, 266. Weyrauch : Grundriss der Warme-Theorie 2, 117. Zeuner : Technical 'I^hermodynamics 2, 243. CHAPTER XII MIXTURES OF GASES AND VAPORS 142. Moisture in the Atmosphere. — Because of evaporation of water 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 jo' 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 Jt?'^ If the temperature is higher than this, the vapor must be superheated. The water vapor in the atmosphere is usually superheated. Let point J., Fig. 77, represent the state of the vapor, and let AB be a constant pressure curve cutting the saturation curve 228 ART. 142] MOISTURE IN THE ATMOSPHERE 229 at B, Further, let m denote the weight per cubic foot of tlie vapor in the state A^ and m^, the weight per cubic foot of satu- rated vapor at the same temperature, that is, in the state 0. The ratio — is called the humidity of the air under the given conditions. If the mixture of air and vapor is cooled at constant pressure, the vapor will follow the path AB and at B it will become saturated. Upon further cooling some of the vapor will condense. The temperature T^ at which con- densation begins is called the dew point corresponding to the state A. The humidity may be expressed approximately in terms of pressures. Let pj' denote the pressure of the vapor in the state A and pj' the pressure of saturated vapor at the same temperature, hence in the state represented by C. At the low pressures under consideration we may assume that the vapor follows the gas Iaw pV= MBT. Hence, taking V= 1, we have pj' = p^" = mBT, - ^'o ^^ ^o and ^ pJ'=m^BT. / Therefore, denoting the humidity by (^, we have Fig. 77. -'o^To m p^" m (2) That is, the humidity is the ratio of the pressure corresponding to the dew point to the saturation pressure corresponding to the temperature of the mixture. For investigations that involve hygrometric conditions, the data ordinarily required may be found in table II, page 319. This table gives the more important properties of water vapor at low temperatures. Example 1. Air is at 70"" and the dew point is found to be 52"^ F. Required the humidity and the weight of vapor in a cubic foot of the mixture. 230 MIXTURES OF GASES AND VAPORS [chap, xii / At 70° the saturation pressure is, from table II, 0.738 inches of Hg, while at 52° the saturation pressure is 0.3905 inches of Hg. The humidity- is therefore 0,390_5^,52, 0.738 If the air were saturated at 70°, it would contain 8.017 grains of vapor per cubic foot. Hence with 52.9 per cent humidity the weight of vapor per cubic foot is 8.017 X 0.529 = 4,241 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 ^59.6 + 44.5 ^ 3 -j^^ 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^, 3.34 Example. Find the value of B for air at 90° F. completely saturated with water vapor. The pressure of the mixture is 14.7 lb. per square inch. From the table the pressure p" of the vapor is 0.691 lb. per square inch; therefore the pressure j^' of the air is 14.7 — 0.691 = 14.009 lb. per square inch. From (5), 1 Jr ez = ^ = ^^^ = 1.0493, ez = 0.0493, and z = ^i^^ ^ ^ p' 14.009 ' 1.61 = 0.0306. Therefore, 5^ = 53.34 x i^^^ = 54.31. 1.0306 The specific heat of the mixture is found by applying the law deduced in Art. 83. If cj and ej' denote respectively the specific heats of the air and steam, then the specific heat of the mixture is given by the equation i-\-z Example. Taking Cp for air as 0.24, and for steam at 90° as 0.43, the specific hsat of the mixture given in the preceding example is 0.24 + 0.0306 X 0.43 1 + 0.0306 0.2456. 232 MIXTURES OF GASES AND VAPORS [chap, xii 144. Mixture of Wet Steam and Air. — In a given volume V let there be M^ lb. of air and M^ ^^' 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=M^BT, C2) V=M^lx(y" -v'^ + v'-], (3) where, as usual, v" 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^c^T^-M^{q + xp) + U^. (4) Likewise, the entropy of the mixture is *S' = Jfi[^,log,y+(<;,-01og.F] + Jf2(«' + f) + '^o- (5) By means of these equations various changes of state may be investigated. 145. Isothermal Change of State. — Since 2^ remains constant, we have from (4) AiU^-U{) = 3£,p<:x^-x{), (1) and from (5) 8^~S,= M,AB log. ^ + Mj-ix, - x{). (2) Hence, the heat added is given by the equation Q = TiS, - S\) ^ M.ABTlog, ^ + M,r(x, - x,-). The external work is Tf = J^ and p^' ^ the quality x*^, and temperature T^ are unknown. However, T^ depends upon p^' ^ and p^ is found from the relation p^ -f p^' = p^ when p^'^ is determined. Denoting the final volume by F^, we have whence ^9= ^^ , „ ' (^) Inserting this expression for x^^ in (2), we have finally 8, = M, (., log, T, - AB log, ft') + i(fi V + ^ -^) • (4) In this equation p^ is the only unknown. The solution is most easily effected by assuming several values of p<^' 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<^' as abscissas and the intersection of the curve thus obtained with the line S-^ = const, gives the desired value oi p^' . 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 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 29.92 in. of mercury, and let the temperature be 79.1° F. The mixture is compressed to a pressure of 120 lb. per square inch absolute. Required the final state of the mixture and the work of compression per pound of air. ART. 146] ADIABATIC CHANGE OF STATE 235 From tlie steam table the partial pressure of the water vapor correspond- ing to 79.1° is 1 in. Hg, hence the partial pressure of the air is 28.92 in. Hg. The initial quality x^ is found from the relation Vi = Ml^Il = M2xivi", Pi' , MxBTi 53.34x538.7 ^noi/i whence xi = - — ^ = = 0.0214. Mo^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 the entropy of the mixture we obtain from (1) (neglecting the con- stant So) Si = 0.24 loge 538.7 - 0.0686 log^ (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 -^=^ = 8.2, we assume that the pressure p-r' of the vapor after compression will be approximately 8 times the initial pressure pi". Hence we assume p2" = 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 Pa" (in.ng)jt)2"-i^ p,' t, T, s,' r, v," 7 3.43 116.57 146.9 606.5 0.2097 1011.1 104.4 ] 8 3.92 116.08 152.3 611.9 0.2186 1007.9 92.18 I Data 9 4.41 115.59 157.1 616.7 0.2265 1005.0 82.57 J p," c.logT, ABlogp,' • -^, S P2^-2 7 1.5378 0.3264 0.0308 1.4519] 8 1.5100 0.3262 0.0349 1.4673 [ Results 9 1.5418 0.3259 0.0390 1.4814] The pressure p2" that gives the value aS^ = 1.4587 lies between 7 and 8 in. Hg and by the graphical method or by interpolation we find P2" = 7.44 in. Hg, or P2" = 3.65 lb. per square inch. Therefore P2' = 120 - 3.65 = 116.35 lb. per square inch. From the steam table the following values are found for the pressure />," = 7.44 in. Hg : fg = 149.3, ^2 = 608.9, (72' = 117.3, r2 = 100-9.4, po = 942.8, V2" = 99. The final quality is ^ ^ BT2 ^ 53.34 X 608.9 ^^^ P2'v2" 116.35 X 144 X 99 The external work per pound of air is W= /[0.17(149.3 - 79.1) + 117.3 - 47.2 + 0.0214 x 989.8 - 0.01958 x 942.8] = 61566 ft. lb. = 0.01958. 236 MIXTURES OF GASES AND VAPORS [chap, xii The volume of the mixture at the end of compression is y^BT,^ 53.34x608.9 ^ ^^33^ ^^ ^ p^' 116.85 X 144 ' and the work of expulsion is therefore 1.9386 X 120 X 144 =: 33498 ft. lb. Hence, the work of compression and expulsion is 95064 ft. lb. 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 lb. of dry air under the same conditions. The initial volume of 1 lb of air is —^—^ zrvi- = 13.574 cu. ft. 14.7 X 144 The final volume after adiabatic compression to 120 lb. per square inch is 13.574 ( IM V"^ = 3.0296 cu. ft. 1 120 The work of compression is 5^(14.7 X 13.574 - 120 x 3.0296) = 59044 ft. lb., the work of expulsion is 3.0296 x 120 x 144 = 52350 ft. lb., and the sum is 111394 ft. lb. 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 ifc/j lb. of air compressed to a pressure ^^ and having a temperature T^ be mixed with M^ lb. of wet steam having a pressure p^ ^^^ quality x^- The temperature T2 of the steam is, of course, the saturation temperature corresponding to the pressure p^. Let F^ denote the volume of the air and F^ the volume of the steam. The mixing is accomplished by discharg- ART. 147] MIXTURE OF AIR AND STEAM 237 ing the air into a receiver which contains steam, or vice versa. Since under these conditions the pressure of the mixture can- not be raised above the pressure of the constituents, the volume of the mixture cannot be taken as the original volume Fj of the air. We assume, on the other hand, that the conditions are such that the volume of the resulting mixture is the sum of the volumes of the constituents ; that is, V= V, + F,. (1) As a second condition, the internal energy of the mixture is equal- to the sum of the energies of the constitutents ; hence we have the equation of condition U= U, + U^. (2) Let T denote the temperature after mixing p' the partial pres- sure of the air and p" the partial pressure of the steam. Then, provided the steam does not become superheated, the tempera- ture T must be the saturation temperature corresponding to the pressure p^' , The following relations are readily obtained. V, = ^^^. (3) Pi or since the quality X2 is nearly 1, V, = M,x,v,'^, (4) V= Mi^.^ M.xv'\ (5) P where x denotes the quality after mixing, and v" is the specific volume of steam corresponding to the pressure^". U,=^M,e,T,. (6) U=M^e,T+M^iq' + xp). (8) From (2) we have M^e,T + M^iq' + xp) = M,c, ?! + Bl,(iq^ + x^^). (9) 238 MIXTURES OF GASES AND VAPORS [chap, xii From (1), Having V calculated from (10), we obtain from (5) and this expression for x substituted in (9) gives finally M,c,T + M^{q' + ^) = M^c,T, + M^iq^ + x^^y (12) In (12) the second member is known from the initial condi- tions. In the first member q\ /?, and v" are dependent on T\ hence T is the one unknown. As usual, the solution is ob- tained by taking various values of T and plotting the resulting values of the first member of (12). Example. Let 1 lb. of wet steam, quality 0.85, at a pressure of 200 lb. per square inch, be mixed with 2 lb. of air at a pressure of 220 lb. per square inch and a temperature of 400°. Required the condition of the mixture. From the data given, the following values are readily found : Fi = 2.895 cu. ft. ; Vi = 1.948 ca. f t. ; V = 2.895 + 1.948 = 4.843 cu. ft. U=Ui-\- C72 = 1273.8 B.t.u. Equation (12) becomes 0.34 T^q' + 4.843 J° = 1273.8. v" We now assume for p" the values 50, 75, and 100 lb. per square inch; from the tables we find the corresponding values of q', p, v", and 2\ and calculate the values of the first member. The results are : For p" = 50, 981 B. t. u. p" = 75, 1222.3 B.t.u. p" = 100, 1451 B. t. u. Plotting these results, we find p" = 81 lb. per square inch very nearly. The temperature of the mixture is therefore 313° F. and the quality of the 4 843 w steam is a: == — — - = 0.897. (5.4 is the specific volume v" corresponding to 5.4 a pressure of 81 lb.) The partial pressure p' of the air is found from (5) to be 130 lb. per square inch. Hence the pressure of the mixture is 130 + 81 = 211 lb. per square inch. ART. 147] MIXTURE OF AIR AND STEAM 239 It is seen that, as the result of mixing, the temperature is considerably lowered, the pressure takes a value between pi and p2y and the quality of the steam is increased. If the steam is initially superheated, the preceding equations must be modified by inserting for F^ and U^ the appropriate expressions for the volume and energy, respectively, of super- heated steam. To reduce as far as possible the complication of the formulas we shall take the approximate equation (5), Art. 132, for the volume. We have then V^ = M,v^ = M^ (^ - .). (13) The constant B is written with a prime merely to distinguish it from the constant for air. The intrinsic energy of the steam is given by Eq. (2), Art. 136. This equation can be simplified with a small sacrifice of accuracy by dropping the term con- taining a. The modified equation then takes the form Au=^T(^e+fT)-^^ + ^m.l, (14) in which g = 0.2566, /= 0.00005, and log (7= 13.64593. From (6) and (14) the energies of the constituents before mixing can be calculated, and the sum of these gives the energy U of the mixture. We have then as one equation of condition M^c,T+M^l Tie + fT)-^+ 886.7] = A U. (15) Since p" and T are here independent, there are two unknowns and a second condition is required. From (3) and (13) the initial volumes V^ and V^ are found and the sum gives the volume V of the mixture. Then B'T or y' = _^i_. (16) From (15) and (16) the unknowns p'' and I^can be found. 240 MIXTURES OF GASES AND VAPORS [chap, xii Example. Let 5 lb. of air at 60^^ F. be compressed adiabatically from atmospheric pressure to a pressure of 200 lb. per square inch and mixed with 1 lb. of steam at 200 lb. per square inch superheated 100°. The con- dition of the mixture is required. The temperature of the air after compression r, = 519.6 (||)'^=1095». The saturation temperature of steam at 200 lb. per square inch is 381.8° F ; hence T^ = 381.8 + 100 + 459.6 = 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-^^+ 886.7 -: 1160.6 B.t.u. 941.4^ Hence A(U^+U2) = AU = 930.75 + 1160.6 = 2091.35. B. t. u. We have then from (15). 0.85 T + r(0.2566 + 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 y_ M^BT, ^ 5x53.34x1095 ^ ^^^^ ^^ ^ p^ 144 X 200 ' and the volume of the steam is F. = ^^ - c = ^:^^^^xMi _ 0.256 = 2.55 cu. ft. P2 200 Hence V= 10.14 + 2.55 = 12.69 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 0.5962 T F+c 12.69 + 0.256 Introducing this expression for p" in the term —^, that term 'becomes C — , where log C = 12.30919. The equation in T then becomes 1.1066 T + 0.00005 7^2 - -^ == 1204.65. As the value of T evidently lies between 1000° and 1100°, we assume the three values 1000, 1050, 1100 and calculate the first member of this equa- tion. The results are : AKT. 147] MIXTURE OF AIR AND STEAM 241 T 1.1066 T 0.00005 T^ Sum 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 steam is -" ^'^^^^ ^ ^^^^ = 47.94 lb. per square inch, ^ ~ 12.946 while the pressure of the air is Therefore P^ 53.34 X 1041 X 5 144 X 12.69 P=p'+p" 151.93 lb. per square inch. 199.9 lb. per square inch. The total pressure p should evidently be 200 lb. per square inch ; hence the result may be regarded as a check on the calculation. Having now the initial condition of the mixture, the condi- tion after adiabatic expansion to any assumed lower pressure and the work of expansion may be found by the methods of Art. 146. The discussions of Arts. 146 and 147 furnish the necessary equations for the analysis of the action of a motor that uses a^ mixture of air and steam as its working fluid. EXERCISES 1. Find the humidity and the weight of vapor per cubic foot when the temperature is 85° and the dew point is 70°. 2. The humidity is 0.60 when the atmospheric temperature is 74° F. Find the dew point. 3. Find the value of B for air at 80° with 70 per cent humidity. Find also the specified heat Cp of the mixture. 4. A mixture of air and wet steam has a volume of 3 cu. ft. and the temperature is 240° F. The weight of the air present is 1 lb., that of the steam and water 0.4 lb. Find the partial pressures of the air and vapor, the total 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. Find the external work, the heat added, the change of entropy, and the change of energy. 6. Let the mixture expand adiabatically to a volume of 5 cu. ft. Find the condition of the mixture after expansion, and the external work. 242 MIXTURES OF GASES AND VAPORS [chap, xii 7. Let 1 lb. of steam, quality 0.87, at a pressure of 150 lb. per square inch, be mixed with 4 lb. of air at a pressure of 160 lb. 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 lb. per square inch. Determine the final state of the mixture and calculate the work of expansion. 9. Let 1 lb. of steam at a pressure of 150 lb. per square inch and super- heated 140° be mixed with 6 lb. of air at a pressure of 160 lb. per square inch and a temperature of 340° F. Find the condition of the mixture. 10. Let the mixture in Ex. 9 expand adiabatically until the pressure drops to 14.7 lb. per square inch. Required the final state of the mixture and the work of expansion. REFERENCES Berry : The Temperature Entropy Diagram, 136. Zeuner : Technical Thermodynamics, 320. Lorenz : Technische Warmelehre, 366. CHAPTER XIII THE FLOW OF FLUIDS 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), Zeuner (1871), Fleigner (1874 and 1877), and Hirn (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). 243 244 THE FLOW OF FLUIDS [chap, xiii Most of the experimental work here noted relates to the 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 flow 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 the 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- lel to the axis of tlie channel. We may imagine surfaces ^ „c - stretched across the channel, as Fig. 78. ' F, F\ F\ 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 : Fw = Mv, (1) in which F 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 jo, V, T giving the state of the fluid and also the mean velocity w remain constant at any cross section F ; 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 are derived by applying the principle of conservation of ART. 150] FUNDAMENTAL EQUATIONS 245 energy to an elementary mass of fluid moving in the tube or channel. Let Wj denote the velocity v^ith 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 F^ 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 total energy is u^ + ^. Likewise, the energy of a unit weight of fluid at section F \^u-\- —- . In general, the total energy at section F is different from that at section F^ 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 ^^ to i^; (2) from | ^^"i work done on or by the fluid. ""P 1 1 The heat enterinsf the fluid per 1 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^^ a unit weight also passes section F. Denoting by jo^ and v-^ the pressure and specific volume, respectively, at F^^ the work done on a unit weight of fluid in forcing it across section F-^ is the product jp^^ ; simi- larly, the product fv gives the work done hy 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^ and F is, therefore, f\^\ ~ P'^- Equating the change of energy between i^^ and F to the energy received from external sources, we obtain pv, 246 THE FLOW OF FLUIDS [chap, xiii or 1-= J"^ + (^^+p^r;^)_(w+j9y). (1) 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 2 between internal energy u and kinetic energy -— at section F^ leaving the sum total unchanged. Differentiation of (1) gives \- du-{- d(pv^ = Jdq^ (2) 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 F^ 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 -h Adz') = du+ pdv. (3) The first member represents the energy entering the lamina during the passage from F^ to #, du is the increase of energy, and pdv the external work done. Combining (3) with (2), we get "^^vdp + dz^O, (4) whence by integration we obtain — - — L = - vdp-z, (5) 2g JPx ART. 151] FORMS OF THE FUNDAMENTAL EQUATION 247 The fandamental 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 ^ = 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 vP" — w •2g For a perfect gas J(.H-iy (1) Tc Jl = u-\-pv = j—^P^^ (2) Avhence, - ^^ = .4tO^:^x-^^0. (B) If the fluid is a mixture of liquid and saturated vapor, the heat content ^ is practically equal to the total heat. (See Art. 86.) Hence we may put i= q^ -\- a:r, (4) and (1) becomes w ^9 For a superheated vapor, the general form (1) is used, the values of i^ 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 the change of kinetic energy and the effect of friction on this change are clearly shown. 248 THE FLOW OF FLUIDS [chap. XIII 1. Using p and v as coordinates, let AB^ Fig. 80, be the curve representing the relation between pressure and volume during the passage of the fluid from section F^ to sec- tion F. The area ABDC between this curve and the jt?-axis is given by In the case of frictionless flow, however, the second fundamental equation [(5), Art. 150] becomes Fig. 80. w"- W 1_ — 9 — r^ vdp. (1) Hence for frictionless flow, the increase of kinetic energy is given by the area between the jt?-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 AB must be expressed in the form pv"^ = 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 OHOAA^ rep- resents the total heat of the mixture in the initial state A^ and the area OHBBA^ the total heat in the final state B ; hence the difference of these areas, namely, the area ABBC^ represents the difference q^ -f- x^r^ — (^q' + xr')^ and from (5), Fig. 81. ART. 152] GRAPHICAL REPRESENTATION 249 Art. 151, this area, therefore, represents the increase of energy 'Up' — tv ^ ^9 " If the initial point is at A' in the superheated region, we have zi = area OHCAA'A^, i = area OHDB'A^, ^\ - ^ = area A'B'BOAA'. 3. The work z expended in overcoming friction may be shown on either the pv- or the ^/S'-plane. When friction is taken into account, the heat Az, the equivalent of the friction work 25, 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 AB for flow without friction. This statement applies to both figures. Let ^^ denote the heat content in the initial state J., i the heat content in the state B^ and i' the heat content in the final state B^ when friction enters into consideration. Then whence i' > ^, ^l — i' < ^l — It follows from (1) Art. 151, that the change of kinetic energy -1- for flow with friction is less than the change 2^ " 2g in the case of frictionless flow. Friction, therefore, causes a loss of kinetic energy given by the relation = J(i'-i). (2) 2d On the ^xS'-plane, Fig. 83, this loss is represented by the area A^BB'B^; for i' = area OHBB'B^, i = area OHDBB^, ^BB'B^ i' -{='dre^A,BB'B' 250 THE FLOW OF FLUIDS [chap, xiii When, therefore, on account of friction the entropy of the fluid is increased by BB' , the area representing the increase of kinetic energy is the original area ABDO iov frictionless flow minus the rectangular area A^BB'B-^ , The increase of en- tropy in this case is due entirely to the heat Az entering the fluid ; hence as explained in Art. 50, the increase of entropy is ^ r — , and the area AyAB' B^ under the curve AB' repre- 2 J- sents (in heat units) the friction work z. On the j92;-plane let a constant i line be drawn from point B\ Fig. 82, cutting the frictionless expansion line in the point G-. Then since the heat content i' is the same at G- as at B'^ the difference i-^ — i' in pass- ing from A to B' along the actual curve is the same as in passing from Aio G- along the ideal frictionless expan- sion curve AB. But the change of i between the FIG. 82. states represented by points A and (r, which in work units represents the increase of kinetic energy between A and G-^ is given by the area AGEO. Hence we have : For frictionless flow, ^ = area ABBO, For the actual flow, ^'^ " ^' l = area A QEC. Hence the loss of kinetic energy due to friction is given by the area BBBG. From the fundamental equation (5), Art. 150, we have \ vdp — fl-wl ^3^ ^9 in which the integral refers to the actual expansion curve. Referring to Fig. 82, j vdp is given by the area AB'BO AKT. 152] GRAPHICAL REPRESENTATION 251 while the change of energy for the actual flow is, as just shown, given by the area ACrUO; hence the difference, the area AB'DUGA, represents the work of friction z. The friction work z (area A^AB'B^', Fig. 83) is greater than the loss of kinetic energy (area A^BB'B^^. 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 in Chapter IV, the loss of available energy, represented by area A^BB'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 frictionless adia- batic change from pressure p^ to a lower pressure p^ 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- FiG. 84. fore, represents the increase of jet energy w^ — w friction, the segment AGr^ the smaller increase 2^ i- without W-, 2g with friction, and the segment GrB^ the decrease in final kinetic energy due to friction. 252 THE FLOW OF FLUIDS [chap. XIII 153. Flow through Orifices and Short Tubes. Saint Venant's Hypothesis. — Let the elastic fluid flow from a reservoir in which the pressure is p-^ through an orifice or short tube, Fig. 85, into a region in which exists a pressure p^ lower than p^ If we take the section F-^ in the reservoir, the velocity w^ 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 hj p. Assuming the flow to be frictionless and adiabatic, we have, since w^ = 0, j Wp. (1) 2^ The law of the expansion is given by the equation p^v^'' =pv' (2) Fig. 85. 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-l 2^ Pi' ■J (3) 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 Mv = Fw, (4) whence eliminating w between (3) and (4), we obtain F n-l From (2), we have which substituted in (5) gives (5) M=F^\2g n— 1 v^ f-if) (6) 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 ^, w^ and M will be as shown in Fig. 86. The initial pressure jp-^ is rep- resented by the ordinate ^^' ^ OQ^ the lower pressure jt? by the ordinate OK^ and the curve AB represents the change of state of the moving fluid starting from the initial state A. The shaded area GrABH rep- resents the integral \ vdp and, therefore, the kinetic energy of the jet — at the section F, The abscissa HE represents the velocity w found from the equation w=^2gx area GABR (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 =Pi and also when jt? = 0. It fol- lows that the curve G-DO must have the general form shown in the figure and that the discharge i)[f must have a maximum value for some value of p between ^ = and p = p^- Let this value of p be denoted by p^. Evidently from (6), Mis sl maximum when 2 n+1 is a maximum. Placing the first derivative of this expression equal to zero and solving, we find for the ratio — that makes Msi maximum _ji_ 9 \n-l i) ■ C^) Pm Pi n H- 254 THE FLOW OF FLUIDS [chap. XIII This ratio is called the critical ratio, andj?^ is called the critical value of the lower pressure p. For air, taking n= k = 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^ of the region into which the jet discharges. If it be assumed that p and p^ are always equal, then ^ = when p^ = 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^. Saint Venant 0.3 , 0.5 Fig. 87. 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^ the discharge is the same. We have, there- fore, two distinct cases : (1) If p^ is greater than p^, the pres- sure p in the jet takes the value |?2, and w and Mare found from (3) and (6), respectively. (2) If p^ is equal to or less than p^ the pressure p assumes the constant value p^ given by (7), and the velocity and discharge remain the same for all values of ^2 between p^ =Pm and p^— 0. The hypothesis of Saint Yenant has been fully confirmed by the experiments of Fleigner, Zeuner, and Gutermuth. Figure 87 shows the results of Gutermuth's experiments on the flow of steam through a short tube with rounded entrance, using dif- ART. 154] FORMULAS FOR DISCHARGE 255 ferent initial pressures p-^. In each case the discharge becomes constant when the lower pressure reaches a definite value p^. 154. Formulas for Discharge. — Since for all values of jOgless than p^ the discharge remains constant and the pressure at the plane of the orifice or tube takes the value p^^ we may obtain P the maximum velocity and discharge by substituting for — in ^1 Pi (3) and (6) of Art. 153 the critical value ( ^ ] "^ . The resulting equations are : 1 9 \^{^ and M=F[-^] ^\2g-^^. (2) \n+lj ^ ^ n-\-l v^ ^ ^ These equations give w and M for p2ZF~^. (4) 256 THE FLOW OF FLUIDS [chap, xin This is the equation given by Fliegner as representing the re- sults 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- mosphere pressure the following empirical equation is given by Fliegner : M= 1.06i^V-^^^%— ^ (5) 2. Grashof's Equation for Steam. In formula (2), p^ and v^ refer to the fluid in the reservoir. If this fluid is saturated steam, then p^ and v-^ are connected by an approximate relation PiV{-= (7, (6) in which for English units, m = 1.0631 and 0= 144 x 484.2. From (6) we readily obtain V 2m C2m and substituting this in (2), the resulting equation is m+l ^"^^^ V(«+i)c™ If now we take for steam the value n = 1.135, (7) reduces to the simple form Jf= 0.01911 ^^"-^^ (8) In this formula, F is taken in square feet and p in pounds per square foot. When the area is taken in square inches and the pressure in pounds per square inch, (8) becomes iHf =0.0165 J^/-97. (9) This formula is applicable for values of p^ below the critical back pressure p^^. 3. Bateau's Formula. Rateau has modified the Grashof formula and gives the following as more nearly agreeing with the results of his experiments : ART. 1551 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 M^^; ■ (11) when it is less than |^ of the back pressure ^^^^3(^-^^ (12) Example. Find the discharge in pounds per minute of saturated steam at 100 lb. pressure (absolute) through an orifice having an area of 0.4 sq. in. The back pressure is less than the critical pressure, 57 lb. per square inch. 1. By Grashof s formula M = 60 X 0.0165 X 0.4 x lOO^-^? = 34.493 lb. 2. By Bateau's formula M= 60 X 0.4 X 100 n 6.367 - 0.96 x 2) = 34.673 lb. 1000 ^ ^ 3. By Napier's formula M = ^-^ ^ ^^^ X 60 = 34.286 lb. 70 4. The discharge may be found from the two fundamental formulas w = V2^77?7^r^ = 223.7 V/^ - «2, and AI = ^. V The critical pressure p,,, is 57.44 lb. per square inch. From the steam table (or more conveniently, and with sufficient accuracy, from the is-chart) we find : ii (for 100 lb.) = 1186.5 B. t. u. {^ (for 57.44 lb.) = 1142.7 B. t. u. x„, = 0.964. r,n = x,„ (vj' - vj) + vj = 7.07 cu. ft. Then w = 223.7 V1186.5 - 1142.7 = 1480 ft. per second, and M = 60 X — X i^ = 34.89 lb. 144 7.07 155. Acoustic Velocity. — Let p and v denote, as usual, the pressure and specific volume of a medium in a given state, and 258 THE FLOW OF FLUIDS [chap, xiir c. k the ratio -^ of the specific heats. Then the velocity of sound in the medium is given by the relation w = Vgkpv. (See any textbook in Physics). (I) If we denote by p^ the critical back pressure, we have Pi which combined with the adiabatic equation Prn_f^ Pi ~ \^r gives 'fAi^r- ® (3) Combining (2) and (4), we have p^v^ _ k-\-\ PmVm ~ 2 The velocity through the orifice is ^-1=(-^Y\ (4) (5) ^^V^^rv^i^i^i' k_ k + l and by the use of (5) this becomes w = ^gkp^v^. (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 p<^ in the region into which the jet passes. There are two cases to be discussed : 1. When jt?2 is equal to or greater than the critical pressure p^ given by the ratio Pm Pi 2. When p^ is less than p^. U + iy ART. 166] THE DE LAVAL NOZZLE 259 In the first case the pressure at the cross section a, Fig. 88, as we have seen, takes the value p^ of the surrounding region, and, therefore, the jet experiences no change of pressure as it passes into the region beyond the nozzle. There is no tendency, consequently, for the jet to spread laterally, and for some distance beyond the orifice it will have practi- cally constant cross section. Furthermore, since there is no drop in pressure along the axis of the jet, the velocity remains practically con- stant at successive cross sections. This velocity is given by (1)^ Art. 151. In the second case the pres- sure at section a takes the critical value jt?^, which is greater than the pressure of the surroundings. As a result of the pressure difference p^ — p^^ the jet will expand laterally, as shown in Fig. 89. Furthermore, along the axis of the jet the pres- sure drops continuously from its initial value p^ until at same distance from the orifice it attains the pressure p^. Hence, due to this pressure drop, the velocity of the jet in the direction of the axis will increase as successive sections are passed. The initial velocity at section a is Fig. 88. Fig. 90. that is, the acoustic velocity. The lateral spreading of the jet may be prevented by adding to the orifice a properly proportional tube, as shown in Fig. 90. The orifice and tube to- gether constitute a de Laval nozzle. The tube must diverge so as to permit the expansion of the fluid required by the drop of pressure from p^ at section a to p^ at section h. The area of the end section h depends upon the final pressure p^. At section a the jet has the acoustic velocity w^ as if the added tube were not present. As the jet proceeds along the tube 260 THE FLOW OF FLUIDS [chap, xiri its velocity increases and at the end section h takes the value «^2 given by the relation 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 = 0, (2) wdw 7 ^o\ (4) (5) 9 — — vajj. We have also the equation of continuity Fw = if?;, from which by differentiation we obtain For perfect gases, dw dF w F u _ dv V pv 7 1 1 k-V while for superheated or saturated vapors, pv u = — — r • n— 1 Therefore, (2) becomes or kpdv -f- vdp = 0, (6) ■ dv dp whence — = — -i- . V kp Combining this relation with (5), we obtain ^ + ^ + ^ = 0. (7) w F kp ^ Now from (3), dw qv 7 — =^-^dp\ ART. 156] THE DE LAVAL NOZZLE 261 hence (7) becomes \kp wV ^ F By introducing the equation for the acoustic velocity (8) may be readily reduced to the form 1 dp hiiP' p dx 1 dF vP" — w^ F dx (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, h 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- dF tive — is negative for this — ^ dx part ; for the diverging part from A to B^ — — is positive ; dx for the throat A it has the value zero. The pressure drops continuously from to B as shown by the curve J. , dp . Fig. 91. 01 pressure ; hence -^ is negative throughout. Referring to (10) we have the following results : For orifice OA^ — - is — dx For tube AB, ^ is + dx dp dx IS k'uP' IS — dp . For throat A^ dl[ dx 0; dx hvP' IS w^—w^ W — Z^/ hnP' 'Up' — W^ wWs' 262 THE FLOW OF FLUIDS [cHAt. xm 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^. 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 effect of friction is to produce a decrease in the jet energy — - 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^BB'B^' repre- sents the increase in the final heat content ^ due to friction and it likewise represents the decrease in jet energy at exit. Let 2\, 2*2, and ^y denote, respectively, the heat content of the fluid in the states represented by the points J., B^ and B'. Without friction, we have ^ = J(.h-id> while with friction r,=^(^-^v). _^. 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, %9 ti to ART. 157] FRICTION IN NOZZLES 263 Avhence ^=J-(l-2/)0-j-V,). (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 lb. per square inch absolute into a turbine cell in which the pressure is 35 lb. 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, x% is found to be 0.925, and ^2 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 = 11.3 B.t. u. This heat is represented by the rectangle A\BB'Bi, Fig. 83. Hence, for the quality x%' in the actual final condition B', we have X,' -x,= y(kj^J^ = HA :. 0.012; r^ 938.4 and, therefore, x^' = 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 264 THE FLOW OF FLUIDS CHAP. XIII 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^. The effect of friction may be viewed from another aspect. In Fig. 92, let the curve CMAE 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^ by finding for various lower pressures p F. J.'e the proper cross section F by means of the two equations, — — = JU — ^), and F= 1g "^ w Let ^ be a point on the pres- sure curve obtained in this manner. If now friction is taken into account, the sec- tion F^ 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 FK Similarly the end section F^ must be increased in area to jP/, and the point E on the frictionless pressure curve is shifted to a new position W . 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. Required the dimension of a nozzle to deliver 450 lb. of steam per hour, initially dry and saturated, with an initial pressure of 175 lb. absolute and final pressure of 15 lb. absolute. Let y — 0.13. Fig. 92. ART. 158] DESIGN OF NOZZLES 265 The critical pressure in the throat is 175 x 0.57 = 100 lb. approx. Then for frictionless adiabatic flow /, = 1196.4 B. t. u., C (at throat) = qj + a:^7-^ = 298.1 + 0.962 x 888.4 = 1152.9, i2 = q.2' + X2r2 = 181.1 + 0.863 x 969.7 = 1017.5, i^ — i^ = 43.5 ; i^ — io = 178.9. Since the throat is near the entrance, the effect of friction between entrance and throat is practically negligible ; hence the velocity at the throat is w^ = 223.7 V43^ = 1475 ft. per second. Taking account of the loss of energy (y = 0.13), the velocity of exit is iV2 = 223.7 Vo.87 x 178.9 = 2791 ft. per second. The quality of steam at the throat was found to be 0.962, and that at exit, without friction, 0.863. Because of friction, the quality at exit is increased by the amount 178.9 x 0.13 -^ 969.7 = 0.024, thus giving a final quality 0.863 + 0.024 = 0.887. Neglecting the volume y' of a unit weight of water, since x is large, the specific volumes at throat and exit are respectively 4.42 X 0.962 = 4.252 cu. ft. and 26.23 x 0,887 = 23.26 cu. ft. From the equation of continuity Ftv = Mv, we have, since M = _i^^ ^ 0.125 lb. per second, 60 X 60 ^ j.^^ 0.125 X 4.252 ^QQQQ3g^^_^^^ 1475 = 0.0519 sq. in. as the area of the cross section at the throat. The area at exit is p = 0.125 X 23.26 ^ q 001042 sq. ft. = 0.15 sq. in. 2791 ^ ^ If the cross section of the nozzle is made circular, the diameters at throat and exit are respectively dra = 0.251 in., d^ = 0.437 in. ; and taking the taper of the nozzle as 1 to 10, the length of the conical part is 10(0.437 - 0.257) = 1.8 in. EXERCISES 1. Find the weight of air discharged per minute through an orifice J inch in diameter from a reservoir in which the pressure is maintained at 80 lb. per square inch absolute. The air is discharged into the atmosphere. 266 THE FLOW OF FLUIDS [chap, xiii 2. Steam at a pressure of 120 lb. per square inch flows through an orifice having an area of 0.4 sq. in. into a region in which the pressure is 55 lb. per square inch. Find (a) the velocity; (6) 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 lb. per square inch, what is 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 pyw = const, is taken as 1.30, find the critical ratio ^ Pi 5. A de Laval nozzle is required to deliver 630 lb. of steam per hour. The steam is initially dry and saturated at a pressure of 110 lb. 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 flow. 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 lb. of dry steam per hour, the initial pressure being 150 lb. per square inch and the back pressure 105 lb. 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 lb. per square inch and the pressure in the combining chamber is 8 lb. per square inch, find the velocity of the steam jet. Assume y = 0.08. 9. Steam at 160 lb. pressure superheated 100° flows through a nozzle into a turbine cell in which the pressure is 70 lb per square inch. Find the area of the throat of the nozzle for a discharge of 36 lb. per minute. 10. Let steam at 160 lb. pressure, superheated 100°, expand adiabatically without friction. Take values of the back pressure p2 as abscissas, and plot curves showing (a) the available drop in heat content ii — h ; (h) the veloc- ity of the jet ; (c) the area of cross section required for a discharge of one pound per second. Suggestion. Find i^ for the following pressures: 140, 120, 100, 80, 60, 40, 20, 10, 5 lb. per square inch. Then find w from the formula w — 223.7 Vi'i — ^2, and the cross section from the equation of continuity. 11. Steam at 160 lb. 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^ given in Ex. 10 and plot a curve showing the weight discharged for different values of p2- 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. ART. 158] REFERENCES 267 REFERENCES General Theory of Flow of Fluids Zeuner : Technical Thermodynamics 1, 225 ; 2, 153. Lorenz : Technische, Warmelehre, 99, 122. Weyrauch : Grundriss der Warme-Thebrie 2, 303. Peabody: Thermodynamics, 5th ed., 423. Stodola • Steam Turbines, 4, 45. Rateau : Flow of Steam through Il^ozzles. Original Papers giving Experimental Results or Discussions Weisbach : Civilingeineur 12, 1, 77. 1866. Fliegner: Civilingenieur 20, 13 (1874); 23, 443 (1877). De Saint Yenant and Wantzel : Journal de I'Ecole polytechnique 16. 1839. 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: Annales des Mines, 1. 1902. Rosenhain : Proc. Inst. C. E. 140. 1899. Wilson : London Engineering 13. 1872. ^_=-^=^ ^^^ ---^-F-- ^^^^^^J rp^ — ''^^ ="^^W^ CHAPTER XIY 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 'p^^ the velocity increases to w^ ac- cording to the general equation for flow, viz : ^-i^^jii^-i^y (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 w^ith the fluid in the second region. If) ^ in) " 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^ is sensibly equal to the original velocity w^ ; therefore from (1), we obtain H = H-' (^) 268 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^B^, and this increase multi- plied by the lowest available temperature Tq gives the loss of available energy. Evidently this loss is represented by the area A^BCB^. Example. In a steam engine the pressure is reduced by a throttling valve from 160 lb. per square inch to 90 lb. per square inch absolute. The initial quality is a; = 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 lb. the heat content of saturated steam is 1184.5 B. t. u., therefore in the second state the steam is superheated! As the degree of superheat is evidently small, it may be determined with sufficient accuracy from the curves of mean specific heat. At a pressure of 90 lb. the mean specific heat 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 lb., and the entropy due to superheat, which is approximately. 0.55 log« TiAl = 0.55 loge ^ = 0.0035. Hence, s^ = 1.6107 + 0.0035 = 1.6142. The entropy in the initial state is 1.5553, and the lowest available temperature Tq corresponding to the back pressure 4 in. Hg. is 585.1°. The loss of available energy is therefore 585.1 (1.6142 - 1.5553) = 34.46 B.t.u. If the drop in pressure is small, the following approximate method gives a simpler solution. 270 THROTTLING PROCESSES [chap, xiv While the process is irreversible, we may assume that it is replaced by a corresponding reversible change with the condi- tion that the heat content i remains constant. The general equation di = Tds 4- Avdp, then becomes, = Tds + Avdp, and approximately we have, therefore, i^> = -^i^P, (1) in which As is the increase of entropy corresponding to the change of pressure Ap. Since A^ is intrinsically negative, it follows that As must be positive. Equation (1) may be written in the more convenient form ^^ = -^% (2) For perfect gases (^) reduces to the simple form As=-^^^. (3) For steam having the quality x^ we have V — x(^v" — v'') + v'^ and Apv = Apx(y" — v') + Apv' ; or neglecting the small specific volume v' of the water, Apv = x-yjr. Eq. (2) therefore takes the form Mean values for jo, T^ and -yjr should be taken. Example. If in passing into the engine cylinder the pressure of steam is reduced by wiredrawing from 125 lb. to 120 lb. 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 lb. per square inch. Taking the two pressures 125 and 120, the following mean values are found from the table : ^ = 122.5, T = 802.4, i/a = 82.5. Also, A/> = - 5. ART. 161] THE THROTTLING CALORIMETER 271 Hence, As = ^'^^ ^ ^^'^ x ^— = 0.00398. 802.4 122.5 For Tq we take the temperature corresponding to the 16 lb., namely, 675.9°. Therefore the loss of available energy is 675.9 X 0.00398 = 2.7 B.t.n. 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 ^^ = ^V (1) But ^ ^A = h' + ^^r (2) where i/ and r-^ refer to the pressure p^ in the main ; and ij, = i^" + e^(_t^' - t^}, (3) where t^ is the observed temperature of the steam in the calorimeter, t^ is the saturation temperature corresponding to the pressure p^ ^^ ^^^ calorimeter, ^y is the saturation heat content corresponding to the pressure jt^g, and c^ is the mean specific heat of superheated steam for the temperature range t^ — ^2- Combining the preceding equations, we obtain ^2 "^ ^p\*2 ^2/ ^1 X= - (4) Example. The initial pressure of the steam is 140 lb. per square inch, the observed pressure in the calorimeter 17 lb. per square inch, and the temperature in the calorimeter 258° F. Required the initial quality. The temperature of saturated steam at 17 lb. 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 lb. 272 THROTTLING PROCESSES [chap. XIV and the degree of surperheat in question ; and from the steam table we have iV = 1153, h' = 324.2, ri = 869. Hence, 1153 + 0.477 X 38.( 324.2 = 0.975. The MoUier 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 lb. in the calorimeter. A horizontal line dravrn through this point intersects the constant pressure line p = 140 lb., 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 pi is permitted to flovr through the so-called expansion valve into coils in which exists a much lower pressure P2' 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 ^j B^ '^ line of constant pressure jOg- Evidently Fig. 95. we have and "1 ' 2' 2' where x^ denotes the quality of the mixture in the final state Therefore .- r _ - r , ^ ^ -2' 2' or Xn — The increase of entropy (represented by A^B^ is As 1 . -^2' 2 „ ! (1) (2) (3) ART. 163] THROTTLING CURVES 273 and the loss of refrigerating effect due to the expansion valve, which is represented by the area A^ GrBB^^ is 2'2(V-<) ,' - T^isl - V). (4) The following equalities between the areas of Fig. 95 are evident : ^^^^ OEAA^ = area OEFBB^, area FCrA = area A^ GrBB^. 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^ is varied, the tempera- ture ^2 will ^1^^ vary. 350 300 goO 200 50 100 150 Pressure, lb. per sq. in. Fig. 96. 200 and the successive values of jt?2 a^^ h ^m ^^ ^^P" 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 =/(j?) 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 (1) Cr,= ^1 ' •^l^l ^2 t' - L 274 THROTTLING PROCESSES [chap, xiv The temperature difference t^ — f^ for any lower pressure p2 ^^ 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 ^y and i^" are accurately given by the steam tables, the mean value of c^ 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^. The impossibility of deriving consistent values of e^ 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. 9T, be one of the series of throttling curves, and AB' the saturation curve. The heat content is constant along the throttling curve, that is z^ = ^^ = ^^ = etc. Fig. 97. Let p^ ^^ ^^^ lowcr pressure cor- responding to the points B, B\ 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^A^, Cp denoting the mean specific heat between B' and B. It follows that that is, ^A — iBt = Cp ^i' ART. 165] THE JOULE-THOMSON EFFECT 275 In a similar manner the differences i^ — i^,^ i^ — ij)„ etc. are ob- tained. The result is a relation between the heat content of saturated steam at the original pressure p-^ (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 MW, Fig. 98, and from the points J., B', C\ etc., the segments etc. are laid off. A curve through the points A, B" , 0" , B'\ etc. is an isolated segment of 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 z^-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 (t - 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. From the general law of throttling, the heat content remains constant during the process ; that is. 276 THROTTLING PROCESSES [chap, xiv For an ideal perfect gas, u-^^ = Jc^T-{- Wq, and pv = BT; hence, {Jc, ^ B)T^ = {Jc, + B) T^ or ' T^=T^' 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 ^l. According to the experiments of Joule and Kelvin ^ varies inversely as the square of the absolute temperature. That is, i^-f,- a) 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 /x varies with the pressure as well as with the temperature. An expression for /a in the case of superheated steam can readily be derived from the formula for the heat content, namely : i = «r+ I /32 2 _ ^ mCy 1) ^^ ^ pyAcp + V Since ^ is constant in a throttling process, we may define the Joule-Thomson coefficient more accurately as the derivative From calculus, we have 'dT\ _ _ _^ dp Ji ~ di/ dT ART. 166] EQUATION OF THE PERMANENT GASES and from the definition of the heat content z, Hence or 277 _(^T\ __1 di \dpji Cpdp' A ["%l'\l + «i»+^] (2) The following table contains values of /^ calculated from Eq. (2). Pkessure Lh. per Sq. In. 250° F. 300° 350° 400° 450° 500° 550° 600° 15 100 300 0.668 0.492 0.869 0.327 0.282 0.261 0.220 0.208 0.191 0.176 0.169 0.162 0.143 0.140 0.138 0.119 0.118 0.118 It will be observed that the value of /^ 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 /i. 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 /i 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 coohng- 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 /x = -^ it is possible to derive a characteristic equation that represents more nearly the behavior of such gases. * Proceedings of the Amer, Acad, of Arts and Sciences, Vol. 45, p. 261, 278 THROTTLING PROCESSES [chap, xiv Since the heat content i is constant during a throttling process, the gen- eral equation di = CpdT - A It p- ~ v\ dp takes the form c,^ = a(t^-v]. (1) "^ dp \ dT I Differentiating both members of (1) with respect to T, we obtain dT\ dp J \d7' dT^ dTl = AT^^, C2) But we have dT _ _ a ~dp-^~Y^' and from the general thermodynamic relations, AT^ = -[^M . dT^ \dp)T Substituting these expressions in (2), we obtain d [ a\ , (dc. whence 'dcp\ , a fdc„\ 2 ac^ '/> This is a partial differential equation, the general solution of which is the equation Cp=T^(T^-dap). (4) Here <^ denotes an arbitrary function which must be determined from physical considerations. Since at high temperatures Cp for permanent gases is given by the linear relations Cp — a -{- bT, we have from (4), a+bT= T^cf>(T^), w^hence <^(r3) = |^^ + y,, and (T^-3ap)^ ^ -+ ^ ^. (5) (T^-dap)^ {T^-^apY Since the term 3 ap is small in comparison with the term T% we have approximately {T^-^ap) ~' = T-^{\+^^. ART. 166] EQUATION OF THE PERMANENT GASES 279 Introducing these expressions in (5) and substituting the resulting expres- sion for {T^ — Sap) in (4), we obtain finally '^.=«(i + ^) + "K^+^,) = aOr + ffi(^^ + *). (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 JO 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 dT'^ AT^\T I Integrating, we obtain dv a ('2 a , 1 6 \ , ^ . >, .-v •J Introducing in (1) the expression for — ^ given by (7), we obtain after reduction V T '-=f(p) ^[l^ + h + ^(^ + b)\ (8) To determine the function /(i?), we assume that the perfect gas equation TO pv = BT holds when T is very large. Hence /(j») = — , and (8) becomes P ^ ATUST 2 ^ T^\ T /J ^^ Since the last term in the bracket is very small, it may be neglected, and (9) may be written p,= BT-^(^^ + ^-h]. (10) ^ AT\ST 2 I ^ ^ The equation given by Joule and Thomson, namely, p.^BT-^^. (11) is equivalent to eq. (10) if the constant b is taken as zero. Equation (10) may be taken as the general characteristic equation of such gases as air, oxygen, and nitrogen. It is useful in certain investigations relating to the liquefaction of gases. 280 THROTTLING PROCESSES [chap. XIV 167. Linde's Process for the Liquefaction of Gases. — The 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 e, A current of cold water in the vessel h 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^ is the pressure in the chamber e and p^ ^^^^ pressure in the vessel d, the drop in temperature is Fig. 99. f2-h = Y'^(pi-P2)' (1) The air now passes from vessel d at temperature t^ into the space enclosing the chamber c and thence back to the compressor. In passing back, the air absorbs heat from the air in (7) and by other considerations. Denoting by p^ the pressure at FJ, the end of adiabatic expansion, we have: ART. 172] CHANGING THE LIMITING PRESSURES 289 — /v ' q^ + ^6^1- Heat absorbed by medium ^1 = q\ Heat rejected by medium % = qef + qfd Heat transformed into work ^1-^2= qi-^Vi-Cqs-^^ePs) -^fCr^-P2)' The qualities x^ and :cy are found from the equations (?2' ^/Pi) + ^iW (1) (2) (S) ?l'+^=?3'+^> and (4) (5) If the steam is admitted throughout the stroke without cut- off, the adiabatic expansion is lacking, and the diagram takes the form ABai> (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^ be raised to p-^ while the lower pressure p^ is kept the same, the effect is to increase both q^^ the heat absorbed, and q^ — q^^ the available heat, by an amount represented by the area AAIBB (Fig. 107). Evidently the ideal efficiency is thus in- creased. If ^2 be lowered to p^^ keeping p^ the same, q^ is decreased and q^ — q^ increased without any change in q^ For the ideal Rankine cycle the increase of avail- able heat would be that represented by the area B' DCC . For the modified cycle with incomplete expansion, however, the in- crease is represented by the relatively small area B' DFF', Fig. 107. 290 TECHNICAL APPLICATIONS [chap, xv 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^. 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/S'-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 G- 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, and the state of the mixture in the cylinder at cut-off' is repre- ART. 174] EFFICIENCY STANDARDS 291 sented by point G. The heat thus absorbed by the cylinder walls is represented by the area B^BCC^. CD represents the adiabatic expansion, i>£^ the assumed constant-volume cooling of the steam, and EF 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 FACDE. It will be observed that two segments of the cycle, namely, GF and CB^ are traversed twice, and the effect is a serious loss of effi- O G,H^ A Fig. 108. 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 jff", which is only a little lower than the temperature of the ex- haust. The area G^GHH^ 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^ C 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 292 TECHNICAL APPLICATIONS [chap, xv 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, ^^= heat transformed into work by an engine working in an ideal Rankine cycle (Art. 169), q^ = heat transformed into work by actual engine under the same conditions, Wa = work equivalent of heat q^^ the indicated work, Wb = the work obtained at the brake. We have then 77^ = — = thermal efficiency of ideal Rankine engine, Q 7]^ = — = thermal efficiency of actual engine, V Q T/i == — = — = efficiency ratio (based on indicated work), AW ^ — ~ = brake efficiency ratio (based on work at brake), W T}^ = —^ = mechanical efficiency. The ratios rji and t;^ 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 %. We have Vb = Vi X Vm- Of the heat q supplied, only the heat g^ could be trans- formed into work by the ideal engine using the Rankine cycle ; hence the heat q^^ rather than the total heat q should be charged Q to the engine. The ratio 77^ = — is a measure of the extent to ART. 174] EFFICIENCY STANDARDS 293 which the engine transforms into work the heat ^^ that may possibly be thus transformed ; it may be called the cylinder efficiency. The ratio rj^ measures the mechanical perfection of the engine. Hence, the product rji x rj^ measures the perform- ance of the engine both from the thermodynamic and the mechanical standpoints. The efficiencies rji and rji, may be given other equivalent defi- nitions that are frequently useful. Let iV^ = steam consumption of ideal Rankine engine per h. p.-hour. iV^ = steam consumption per h. p.-hour of actual engine. N't, = steam consumption per b. h. p.-hour of actual engine. Then .. = |, .. = |. Example. An actual engine operating under the conditions defined 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 Vi = ^= 0.663, and r], = ^ = 0.52. EXERCISES In Ex. 1 to 5 find the heat transformed into work, efficiency, and steam consumption per h. p.-hoar. 1. Carnot cycle, p^ = 110 lb., jOg = 15 lb. absolute, Xf, = 0.85. 2. Rankine cycle, same data as in Ex. 1 . 3. Rankine cycle, p^ = 110 lb., P2 = ^ in. of mercury, steam superheated to 450" F. 4. Rankine cycle p^ = 110 lb., J02 = 15 lb., Xi, = 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 oif . 6. Let P2 be fixed at 5 in. of mercury. Take x^, = 1 and draw a curve showing the relation between r) and jOi. 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 J02 is lowered to 5 in. of mercury. 8. Make the same calculation for the cycle with incomplete expansion,, Ex. 4, and compare the results. 294 TECHNICAL APPLICATIONS [chap, xv . 9. The efficiency rji of an engine is 0.65 and the mechanical efficiency is 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 lb. per h. p.- hour, and the efficiency ratio rji is 0.70. Find the heat transformed into work by the actual engine per pound 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 energy 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 p^. It Hkewise 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^ into 2 a region in which it is p^. Hence if this kinetic energy — 2g 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 tlie nozzle is then directed against the buckets of the turbine wheel. Fig. 109, and the impulse of the let produces rotation. It will be noted Fig. 109. that with this type of turbine only a part of the buckets are filled with steam at any instant, even if several nozzles are used. In turbines of the second class, the steam flows through guide 296 TECHNICAL APPLICATIONS [chap, xv vanes in a stationary ring s, Fig. 110, and then through blades in the circumference of the moving wheel m. The guides and wheels " run full," that is, the stationary and moving blades are filled with steam throughout the entire circumference. The pressure of the steam is reduced during the 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. 110. 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. Thus 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 efliciency of the turbine. With velocity turbines three methods of compounding are employed. 1. J^ressure Compounding. The total drop of pressure ^^ —jt?2 may be divided among several wheels, thus reducing the jet velocity at each wheel. If, for example, the change of heat content is ii — i^ and the expansion takes place in a single nozzle, the ideal velocity of the jet is w = V^gJi^i-^^ — i^) ; if, however, i^ — i^ is divided equally among n wheels, the jet velocity is reduced tow— \-^— (i^ — i^. The general arrange- n ment of a turbine with several pressure stages is shown in Fig. 111. Steam passes successively through orifices m-^^ m^^ etc. in partitions h^, h^, etc., which divide the interior of the ART. 177] COMPOUNDING 297 Fig. 111. turbine into wheel chambers. The pressure drops from p^ to P2 in the first cell and the jet acts on the first wheel ; then in passing through the orifice Wg the pres- sure drops from p^ to p^; 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^, 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^ 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. ^r=^ ^ 9i Fig. 112. Evi- dently the two methods of compounding may be combined in a 298 TECHNICAL APPLICATIONS [chap. XV variety of ways. The Curtis turbine, which is a well-known representative type, has usually four or five pressure stages with two velocity stages to each. That is, there are four or five sets of nozzles delivering steam to a corresponding number of wheels running in separate chambers, 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 a large diverging nozzle. The steam velocity with this type of turbine is, however, relatively low even in the last stages. Fig. 113. 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 e 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 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. Referring to Fig. 1 20, let w-^ denote the absolute velocity of the steam entering the 306 TECHNICAL APPLICATIONS CHAP. XV stationary blade Sj, and W2 the absolute exit velocity. If there were no change of pressure, W2 would be smaller than w-^ be- cause of friction ; but the drop in pressure Ap causes a decrease in heat content Ai, and as a result, there is an increase of velocity given by the relation ^ w„ w. w '^0 = -VJ(1 - ^)A? ai w^ Fig. 120. Thus the exit velocity w^ is greater than the entrance velocity w-^^. Combining w^ with c, the velocity of the moving blade, we obtain a^, 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^ is greater than a^, just as Wg' ^^ greater than w^ Combining a^ with , and the heat abstracted by the area CjCjDjDj. Then heat is further removed at the constant tem- perature ^2 (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 U 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 jt?2' 3-nd one including the brine coils and the suction pipe at the lower pressure p^. These are separated by a valve called the expansion valve. The liquid in the state represented by point ^ 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^ABC^ The position of the point A is determined as follows : The passage of the liquid through the expansion valve is a case of throttling or wiredrawing of the character discussed in Art. 162. Fig. 123. 310 TECHNICAL APPLICATIONS [chap, xv Hence, the heat content at A must be equal to the heat content at U, that is, Graphically, the area OUGrAA^ is equal to the area OHEE^ ; or taking away the common area OHGrFE^^ the rectangle E^FAA^ is equal to the triangle QEF. (See Art. 162). Since the throttling process represented by EA is assumed to be adiabatic, the work that must be done on the medium is the difference between §i, the heat absorbed, and ^21 the heat rejected to the condenser. We have then (32 = area C^ODEE^, Q^ = area A^ABC^, W = area C^CDEE^ - area A^ABQ^ = area BODEE^A^AB = area BODEaB. If the expansion valve be replaced by an expansion cylinder, permitting a reversible adiabatic expansion from p^ to jt?^, as in- dicated by the line EF^ we have (32 = area C^CBEE^, q\ = area E^FBC^, W= area BQDEFB. The effect of using the expansion valve rather than the expansion cylinder is thus to decrease the heat removed by the area E^FAAl^ 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^. 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 temperature is fixed by the temperature of the cooling water NH3 S02 CO, 41.5 14.75 385 124 47.61 826 4.4 12 1 ART. 186] CALCULATION OF A VAPOR MACHINE 311 available. (2) The volume of the medium required for a given amount of refrigeration. This determines the bulk of the machine. If the upper temperature be taken as 68° F. (^2 = 528) and the lower temperature as 14° F., the pressures and the volume ratios for the three vapors mentioned are about as follows: Suction pressure, lb. per sq. in. Discharge pressure, lb. per sq. in. Volume, taking that of CO2 as 1 It appears that carbon dioxide requires for proper working very high pressures, so high, in fact, as to be practically prohib- itive except in machines of small size. With sulphur dioxide the pressures are low, but the necessary volume of medium is high, being nearly three times that required by ammonia and twelve times that required by carbon dioxide. With ammonia, the pressures are reasonable and the volume of medium is not excessive; hence from these considerations, ammonia is seen to be most advantageous. From the point of view of economy, ammonia and sulphur dioxide are about equal. Carbon dioxide shows a somewhat smaller efficiency than the others under similar conditions be- cause, on account of the small latent heat of carbon dioxide, the losses due to superheating and the passage through the expan- sion valve are a larger per cent of the total effect. 186. Calculation of a Vapor Machine. — -The following analysis applies to the ideal cycle shown in Fig. 123. Denoting by T^ the temperature at the end of compression indicated by the point (7, the heat that must be removed per minute from the superheated vapor to bring it to the saturation state (the heat represented by the area (7j ODB^ is in which c^ denotes the specific heat of superheated vapor, and iff", the weight of the medium required per minute. The heat rejected by the vapor during condensation (area D^DEE^) is Mr^. Hence the heat rejected by the medium per minute is Q^ = Mlr^ + c,(Z-T^^-\. (1) 312 TECHNICAL APPLICATIONS [chap, xv Denoting by x^ 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^ABC^) Q, = Mr,a-^i)' (2) But area OHGiAA^ = area OHEE^, that is, 9.1 + Vi = 92 5 (3) whence combining (3) and (2), Q, = MCr, - q^' + j/) = M(q," - q,'). (4) The work required per minute is, therefore, W= Ji Q fa < o g o Tempera- ture, F. OlOOkOOlOOlOOlOOlOOIOOlOOlOOlOOlOOiOOiOO CO O O 1 1 1 1 1 ^ Vt>LUME OF One Pound v" GO^COCOOOCO^:OOOCOOi'-HT-HOO^r-^coO'-^Tt^l-lOrHTf^OOTt^,-^0 t^coc^^'NtHoqiococoTt^^coo^oO'*ococooi>iocoi-HOii>coLOTj^ a5^>^l6co^doio6t>^o:dlOT^Tl^^o6c6coc^c^'(^^(N'-^T-^T--^r-^^ {H a, o 1 Vaporization r T Ot>>O(MO00»OC0T-(0iC0'stH(MO00<:O'*(MO00CD-^(Mi-i05l>iO ^C0C0C0C0(NCq(M(MT-iT-H^^^OOOOO0505C3i05C5000000 cocorHa5oocoTtHcqT-io5i:^coioi>ooo5rH(McoTjoooooiO -*CO(NOaiOOt>COiOCO(NrHOOrH(NTt^i001>00050rHCqcOiO ^^^^OOOOOOOOOOOOOOOOOO^^rHrHT^ 1 1 1 i 1 1 1 1 1 1 1 1 1 ■3 1 - X! % (NI>COOOCOOOC005COOOCOOOCOOO(Mt^C^OOiO »O'*TtH'*C0C0(M(M'-HT-HOO0i0i0000I>l><:OC0iOiOTtHTtHC0(r0CQ of Vapori- zation r coC5LO'--it>cooO"^'*5:itoo<:OrHi>c^i>c^i>c^t>cooOl-^(^^coT}^locD^>oooiO'-H(^^(^^(^OTt^Ti^loto^:ol>l>l>oo 1 t>Oqt^^CO^iOO"*a5COOO(NC005iOOCO(Ml>COC5iO'-H>COC55 OCOiOiO'^^COCOtNrH^ 1 1 rH (M (M CO CO 'st^ ^ »0 CD CO l> l> 1 1 1 1 1 1 1 1 1 1 1 ' ' Pressure Lb. per Sq. In. i-Hrt^lMCOOOlMGOOqO^COOOOOa ioiooocO'-Hcoi>iooort^'^aio^cococDiOi-Hiocoiooqoq(N(>q col0^^dc6ddco^-^c^'^>^(^idLd(^qdo6^>^^>^^>^odd(^idr-^dco '-H^rHOCOt>0000050'-H(M'*»OC00005i-H rHi-HT-HT-Hr-li-Hi-HT-HC^ « 0100100»00»00»00»00»OOIOOIOOU30100»00»00 COC4C4r-l«H i tH tH C4 C4 CO CO ^ ^ lO lO CO CO t> t« 00 00 O) O) O Mill' "^ 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 r*S-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. 323 324 INDEX Dalton'slaw, 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 oi u, i, F and *, 79 . Dissociation, 197. Dupre-Hertz formula, 168. Efficiency, conditions of maximum, 135. cyHnder, 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. Stirhng's, 138. Engines, gas, 142. hot-air, 138. steam, 283. Entropy, as a co5rdinate, 68. first definition of, 59. of gases, 100. of hquid, 179. of superheated steam, 215. of vapor, 179. principle, application of, 60. second definition of, 62. Equation of continuity, 244. 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. Heat absorbed during change of state, 22. conduction, waste in, 57. content, Davis formula for, 177, 274. content, defined, 77. INDEX 325 Heat content of gases, 99. of saturated vapor, 173, 177. of superheated steam, 210. Heat, effects of, 35. latent, 26. mechanical equivalent of, 11. mechanical theory of, 3. of liquid, 171, 174. of vaporization, 171, 175. specific, 24. total, 172, 177, 213. units of, 9. Heating of air by internal combustion, 141. Heating value of fuels, 118. Henning's formula for latent heat, 176. Holborn and Henning's experiments, 205. Hot-air engines, 138. Humidity, 229. Inequality of Clausius, 63. Internal combustion, heating by, 141. Intrinsic energy, 36. of gases, 97. of superheated steam, 214. of vapors, 172. Irreversible adiabatics, 75. processes, 47. processes, waste in, 57. Isoadiabatic cycles, 136. Isodynamic change of vapor, 190. processes, 42. Isometric lines, 22. Isopiestic lines, 22. Isothermal, definition of, 21. expansion of gases, 102. of superheated steam, 217. of vapor mixture, 188. on TS-plsine, 70. of steam and air mixture, 232. ' Jet, work of, 298. Joule's cycle, 145. experiments, 11. law, 90. Joule-Thomson coefficient, 276. effect, 275. Kelvin's absolute scale, 55. statement of the second law, 50. Knoblauch's experiments, 201. Knoblauch and Jakob's experiments, 205. Langen's equations for specific heat, 124, 205. Latent heat, 26. external, 172. Henning's formula for, 176. Latent heat, internal, 172. of expansion, 27. of pressure variation, 27. of vaporization, 171, 175. Laws of gases, 89. 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, 80 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. 326 INDEX Rankine's cycle, 284. effect of changing pressure, 289. incomplete expansion, 288. with superheated steam, 286. formula, 168. Rateau's formula, flow of steam, 286. Rectangular cycle, 73. Refrigerating machine, analysis of, 311. Refrigeration, air, 149. vapors used in, 310. with vapor media, 308. Reversed heat engine, 74. Reversible processes, 47. Reynolds and Moorby's experiments, 12. Rowland's experiments, 11. Rotary engine, 294. Saint Venant's hypothesis, 254. Saturated 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. Saturation curve, 166, 182. temperature, 165. Second law of thermodynamics, 50. Boltzmann's interpretation of, 65. Specific heat, 24. curves, 209, 211. in poly tropic changes, 106. Langen's formulas for, 124. mean, 210. Specific heat, molecular, 123. of gaseous mixture, 125. of gaseous products, 123. of gases, 96. of saturated vapor, 182. of superheated steam, 204, 273. Specific volume of vapors, 177. Steam and air, mixture of, 232, 236. •critical temperature of, 199. specific volume of, 177. tables, 180. thermal properties of, 173. total heat of, 172, 177. Steam turbine, 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. Thermodynamic 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. INDEX 327 Vacuum, influence of, on steam turbine, 307. Van der Waals' equation, 20, 200. Vapor, energy of, 172. entropy of, 179. heat content of, 173, 177. latent heat of, 171, 175. Vapor mixture, adiabatic expansion of, 189. constant volume change, 189. curves on T*S-plane, 186. general equation of, 184. isodynamic of, 190. isothermal expansion of, 188. Vapor refrigerating machine, 311. superheated, 196. total heat of, 172, 177. Vaporization, 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. SEP 21 19H One copy del. to Cat. Div. W 21 \9n SEP 21 19n One copy del. to Cat. Div. fJpp 21 \9fi