THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES THE TEMPERATURE-ENTROPY DIAGRAM BY CHARLES W. BERRY Assistant Professor of Mechanical Engineering in the Massachusetts Institute of Technology THIRD EDITION, REVISED AND ENLARGED. FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONPON : CHAPMAN & HALL, LIMITED 1911 Copyright, 1905, 1908, 1911, BY CHARLES W. BERRY THE SCIENTIFIC PRESS tRT OBUMMONO BROOKLYN, TT II PREFACE TO THIRD EDITION THE present revision includes the following additions and changes: Minor insertions have been made in the chapters upon the Flow of Fluids, the Gas Engine Cycles, and the Non-conducting Steam Engine. The chapter on Refrigeration and the Warming Engine has been ex- panded into separate chapters upon each subject. A special chapter has been added upon Entropy Analysis in the Boiler Room. The Tables upon the Efficiency, 'Water and Heat Consumption of the Rankine Cycle have been extended to cover the range of low-pressure turbines as well as high-pressure reciprocating engines. All illustrative problems have been recalculated to agree with the most recent and accurate data upon steam. The second and third editions of this book have so extended its scope that it is now a treatise upon graphical thermodynamics although still abiding by the limitations imposed by its title. CHARLES W. BERRY. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, January, 1911. iii PREFACE TO SECOND EDITION. IN the revised edition of the Temperature-Entropy Diagram a more extended application of the principles of the T^-analysis to advanced problems of thermo- dynamics has been made than was possible in the limited scope of the previous edition. The Chapter on the Flow of Fluids has been entirely rewritten and treats at length various irreversible processes. A graphical method of projecting from the pv- into the T^-plane has been elaborated for perfect gases and its application illustrated in the chapters on Hot-air Engines and Gas-engines. The various factors affecting the cylinder efficiency of both gas- and steam-engines have been thoroughly discussed. One chapter has been devoted to the thermodynamics of mixtures of gases and vapors, and another to the description and use of Mollier's total energy-entropy diagram. CHARLES W. BERRY. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, October, 1908. iv PREFACE TO THE FIRST EDITION. Tms little volume was prepared for the use of stu- dents of thermodynamics, and therefore I have en- deavored to bring together hi logical order certain information concerning the construction, interpreta- tion, and applications to engineering problems of the temperature-entropy diagram, which otherwise would not be readily available for them, as such information is scattered throughout various treatises. The book is not intended for the advanced student, as he is already familiar with its contents, neither is it expected that one entirely ignorant of thermodynamics can use it to advantage, as the reader is assumed to have an ele- mentary knowledge of the fundamental theory and equations. An exhaustive treatment has not been attempted, but it is believed that the graphical presen- tation here given will aid the student to a clearer comprehension of the fundamental principles of thermo- dynamics and make it possible for him to read under- standingly more pretentious works. CHARLES W. BERRY. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, January, 1905. CONTENTS. PACHE INTRODUCTION ix TABLE OF SYMBOLS USED IN TEXT xvii CHAPTER I. GENERAL DISCUSSION. REVERSIBLE PROCESSES AND CYCLES. EFFECT OF IRREVERSIBILITY - , 1 II. THE TEMPERATURE-ENTROPY DIAGRAM FOR PERFECT GASES 13 III. THE TEMPERATURE-ENTROPY DIAGRAM FOR SATURATED STEAM 43 IV, THE TEMPERATURE-ENTROPY DIAGRAM FOR SUPER- HEATED VAPORS 65 V. THE TEMPERATURE-ENTROPY DIAGRAM FOR THE FLOW OF FLUIDS > 78 VI. MOLLIER'S TOTAL ENERGY-ENTROPY DIAGRAM 131 VII. THERMODYNAMICS OF MIXTURES OF GASES, OF GASES AND VAPORS, AND OP VAPORS 139 VIII. THE TEMPERATURE- ENTROPY DIAGRAM APPLIED TO HOT-AIR ENGINES 158 IX. THE TEMPERATURE-ENTROPY DIAGRAM APPLIED TO GAS- ENGINE CYCLES 171 X. THE GAS-ENGINE INDICATOR CARD 206 XI. THE TEMPERATURE-ENTROPY DIAGRAM APPLIED TO THE NON-CONDUCTING STEAM-ENGINE 224 XII. THE MULTIPLE-FLUID OR WASTE-HEAT ENGINE 248 XIII. THE TEMPERATURE-ENTROPY DIAGRAM OF THE ACTUAL STEAM-ENGINE CYCLE 258 vii viii CONTENTS. CHAl PAGE XIV. STEAM-ENGINE CYLINDER EFFICIENCY 276 XV. LIQUEFACTION OF VAPORS AND GASES 294 XVI. APPLICATION OF THE TEMPERATURE-ENTROPY DIAGRAM TO AIR-COMPRESSORS AND AIR-MOTORS 302 XVII. DISCUSSION OF REFRIGERATING PROCESSES 337 XVIII. DISCUSSION OF KELVIN'S WARMING ENGINE 359 XIX. ENTROPY ANALYSIS IN THE BOILER-ROOM 374 TABLE OF PROPERTIES OF SATURATED STEAM FROM 400 F. TO THE CRITICAL TEMPERATURE 388 HYPEBBOLIC OB NAPERIAN LOGARITHMS ... 389 INTRODUCTION. IT seems necessary in a book dealing with the appli- cation of the temperature-entropy diagram to discuss briefly that " ghostly quantity," entropy, although I do not intend to give any new definition of a function already too variously defined, but rather to pick out such of the present ones as are correct. One has but to plot an irreversible adiabatic process hi the temperature-entropy plane to realize once and for all that the entropy does not necessarily remain constant along an adiabatic line. In fact isentropic and adiabatic changes coincide only when the latter process is reversible: and such a change practically never occurs in nature. For example, in one irrevers- ible adiabatic expansion representing the flow through a non-conducting porous plug, the heat added is zero, /7/-) -~- = 0, but nevertheless the entropy of the substance increases. It is even possible to imagine an irreversible process which is at the same time isen- tropic. Suppose a gas to expand through a nozzle iz X INTRODUCTION. losing " heat by radiation and conduction and also undergoing friction losses whereby part of its kinetic energy is dissipated and restored to the gas as heat. The loss of heat by radiation and conduction will re- duce the entropy of the gas, while the gain of heat by friction will increase it. It is possible to consider these two opposing influences as equal, and then the flow will be isen tropic although not adiabatic. The entropy of a substance, just as much as its mtrinsic energy, specific volume, specific pressure, or temperature, has a definite value for each position of the state point upon the characteristic surface, and the increase in the value of the entropy in changing from one point to another is a definite quantity regard- less of the path chosen. The magnitude of this increase is equal to / -~-, taken along any reversible path between these points. This fact has led to the inexact definition of change of entropy, d<}) = -~-, a definition true only for ideal reversible processes and hence utterly wrong when applied to actual irreversible pro- cesses, as in general d>-~-. Since for reversible cycles d< = -~-, it follows that the heat added during any reversible change is equal f 2 to Q = / Tdi}. This is undoubtedly the basis for all the physical analogies attempting to explain entropy as heat-weight, etc., and also for the name "heat diagram" applied to the temperature-entropy diagram. The area under a curve in the 7^-plane is equal to the heat received from, or rejected to, some outside body only when the process is reversible. Similarly if the specific pressure and specific volume of a gas could be ascertained at various points in its passage through a porous plug, these points if plotted would form a pr-curve giving a true history of the movement of the state point, but the area under the curve would not represent work, as no external work has been performed. Preston in his Theory of Heat says: "The entropy of a body being taken arbitrarily as zero in some stand- ard condition A, defined by some standard temperature and pressure (or volume), the entropy in any other /A(~\ -f=f taken along any reversible path by which the body may be brought to B from the standard state A. The path may obviously be an arc AG of an isothermal line passing through the point defining the standard state, together with the arc BC of the adiabatic line passing through B. The entropy in the state B may consequently be measured thus. Let the volume be changed adiabatically " (reversible process) "until the standard temperature T is attained, INTRODUCTION. and then change the volume isothermally until the standard pressure is attained. If the quantity of heat imparted during the latter operation be Q, the entropy Q in the state B is = ip " In this operation the temperature and pressure are supposed uniform throughout the body. ... If, how- ever, any body be subject to operations which produce inequalities of temperature in the mass, there will be a transference of heat from the warmer to the colder parts by conduction and radiation, and although the body may neither receive heat from nor give it out to other bodies (so that the transformation is adiabatic throughout), yet on account of the inequalities of tem- perature, the entropy of the mass will increase, . . . and under these circumstances the transformation will not be isentropic." Swinburne in his Entropy says: "Entropy may be defined thus: Increase of entropy is a quantity which, when multiplied by the lowest available temperature, INT ROD UC TION. xiii gives the incurred waste. In other words, the increase of entropy multiplied by the lowest temperature available gives the energy that either has been already irrevocably degraded into heat during the change in question, or must, at least, be degraded into heat in bringing the working substance back to the standard state. . . . "Thus the entropy of the body in state B is not a function of the heat actually taken in during its change from A to B, as the change must have been partially, and may have been wholly, irreversible; but it can be measured as a function of the heat which would have to be taken in to change from A to B reversibly, or which would have to be given out if the substance were changed from B to A reversibly, which amounts to the same thing. . . . "The entropy of a body therefore depends only on the state, and not on its past history." Planck in his Treatise on Thermodynamics writes (see English translation by Ogg): "A process which can in no way be completely reversed is termed irreversible, all other processes re- versible. That a process may be irreversible, it is not sufficient that it cannot be directly reversed. This is the case with many mechanical processes which are not irreversible. The full requirement is, that it be impossible, even with the assistance of all agents in nature, to restore everywhere the exact initial state when the process has once taken place. . . . The gen- xiv INTRODUCTION. oration of heat by friction, the expansion of a gas without the performance of external work, and the absorption of external heat, the conduction of heat, etc., are irreversible processes. "Since there exists in nature no process entirely free from friction or heat-conduction, all processes which actually take place in nature, if the second law be correct, are in reality irreversible; reversible processes form only an ideal limiting case. They are, however, of considerable importance for theoretical demonstration and for application to states of equilib- rium. "// a homogeneous body be taken through a series of states of equilibrium, that follow continuously from one another, back to its initial state, then the summation of the differential fj*~ extending over all the states of that process gives the value zero. It follows that, if the process be not continued until the initial state, 1, is again reached, but be stopped at a certain state, 2, the value of the summation / ~ depends only on the states 1 and 2, not on the manner of the transformation from state 1 to state 2. ... "The (above) integral . . . has been called by Clausius the entropy of the body in state 2, referred to- state 1 as the zero state. The entropy of a body in a given state, like the internal energy, is completely INTRODUCTION. XV determined up to an additive constant, whose value depends on the zero state. "It is impossible in any way to diminish the entropy of a system of bodies without thereby leaving behind changes in other bodies. If, therefore, a system of bodies has changed its state in a physical or chemical way, without leaving any change in bodies not belong- ing to the system, then the entropy in the final state is greater than, or, in the limit, equal to the entropy in the initial state. The limiting case corresponds to reversible, all others to irreversible, processes. ' ' The restriction . . . that no changes must remain in bodies outside the system is easily dispensed with by including in the system all bodies that may be affected in any way by the process considered. The proposition then becomes: " Every physical or chemical process in nature takes place in such a way as to increase the sum of the entropies of all the bodies taking any part in the process. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged. This is the most general state- ment of the second law of Thermodynamics." SYMBOLS USED IN THE FOLLOWING PAGES, A =y =Heat equivalent of a unit of work. Apw=Heat equivalent of the external work of vaporization. c = General expression for the specific heat during any change, c p =Specific heat at constant pressure. c v = Specific heat at constant volume. J= Change of ... #= Internal (intrinsic) energy in work units =S+I. T) = Thermal efficiency of an engine. F=Area (Flache). G= Weight (Gewichf). g= Acceleration due to gravity. H= Total heat above some arbitrary zero, =q, q+xr, A, A + Cpfc-0. h = Specific heat of dry saturated vapor. / = Internal energy due to separation of molecules. t = Total energy = A (E+pv) = II + Apn. k = for a perfect gas. A = Total heat of dry saturated vapor =q+r. n = Exponent of the poly tropic change pv n =c. p=Specific pressure. <= General expression for entropy. Q=Heat received from or exhausted to some outside body. 5= Heat of the liquid. R=7p, for a perfect gas. r =Total latent heat of vaporization = o + Apu. SYMBOLS USED IN THE FOLLOWING PAGES. /o = Internal latent heat of vaporization. m = Entropy of vaporization. 5 = Internal energy due to vibration of molecules, s = Specific volume of a dry saturated vapor. <7= Specific volume of a liquid. 7 7 = Temperature in degrees absolute, Fahrenheit or Centi- grade. t = Temperature in degrees Fahrenheit or Centigrade. tg = Temperature of superheated vapor. 6 = Entropy of the liquid = J -^r. u= Increase of volume due to vaporization =s a. v = General value of specific volume = a. xu + a, s, etc. V = Velocity in linear units per second. y 2 2~ = Kinetic energy of a jet in work units. W =Work performed by or upon a substance. x = Quality of a unit weight of a mixture of a liquid and its vapor. THE TEMPERATURE-ENTROPY DIAGRAM. CHAPTER I. GENERAL DISCUSSION. REVERSIBLE PROCESSES AND CYCLES. EFFECT OF IRREVERSIBILITY. THE condition of a substance is in general completely denned by any two of its five characteristic properties, specific pressure, specific volume, absolute temperature, intrinsic energy, and entropy. The relations existing between any three of these qualities being expressed by the formula z = f(x, y}. Exceptions to this occur in the case of saturated vapors, where specific pressure and temperature alone do not suffice to define com- pletely the state of the body, and in the case of per- fect gases, where isodynamic and isothermal changes coincide. In the analytical solution of thermodynamic problems those formula? are used which contain the properties most important for the investigation in hand. Similarly in graphical solutions any pair of character- 2 THE TEMPERATURE-ENTROPY DIAGRAM. istics may be used as coordinates as convenient, and then the curves, y = j(x), expressing the relations between the various characteristics drawn in this r?/-plane, assume different forms according to the laws of variation of the five variables p, v, T, E, and <. The pressure-volume diagram, or 7W-diagram, is the one most widely used, as its coordinates are those common to every-day experience, but for some investi- gations of heat-transference, changes of temperature, changes of ' entropy, etc., the temperature-entropy diagram, or the T ^-diagram, lends more ready assist- ance. As a consequence of the fundamental relation 2==/(x, y), any curve p^f^v) in the py-plane has its counterpart ^> = / 2 (7 T ) in the T<-plane; both being but special projections of the same change on the charac- teristic surface. In the following a discussion will be given of the different forms assumed by the curve =-, or dQ = Td$, and therefore or Q rr* I cdT jTi Td!>. THE TEMPERATURE-ENTROPY DIAGRAM. Now I Tdd> represents the area included between J* the curve. AB and its projection upon the -axis. Hence the heat necessary to produce any reversible change is represented by the area under the curve in the T$-plane. The differential form of equation (1) leads at once to the expression At any point n of the curve AB draw the tangent nm and construct the infinitesimal triangle dtd. Then from similar triangles b :dT, i.e., if from any point in a curve representing a re- versible process a tangent be drawn, the length of the subtangent on the <-axis represents the momentary value of the specific heat for that change. In Fig. 2 (A} let AmBnA represent a reversible cycle in which AmB is the forward stroke and the area aAniBb represents the heat received from external sources, and where BnA is the return stroke and the area aAnBb represents the heat rejected. On the completion of the cycle the intrinsic energy has regained its initial value and therefore the difference between CARNOT CYCLE. the heat received and the heat rejected, i.e., the magni- tude of the enclosed area AmBnA, must represent the amount of heat changed into work. Carnot Cycle. In the case of the Carnot engine this choice of coordinates leads to a beautiful simplicity. The cycle Fig. 2 (J5) becomes a rectangle consisting (A) FIG. 2. (B) of (1) the isothermal expansion cd, during which is received the heat Q l = i T l d = T 1 (t-$ l ), repre- J 4>\ sented by the total rectangle facdfa', (2) the adiabatic line de\ (3) the isothermal compression during which is rejected the heat Q 2 = I 2 T 2 d = T 2 ((f> 2 - ( }> l ), repre- */ i sented by i)-^(^-^) = (7 t 1 -7 T 2 )(^-^). The effi- ciency, T) = ^ -, is simply the ratio of the rectangles Vi cdef and ^cd^, and as these have the same base, the 6 THE TEMPERATURE-ENTROPY DIAGRAM. areas are proportional to the altitudes and at once T T the efficiency becomes >j = ~^m " * i Isodiabatic Cycles. In deducing the equation d = -jjf- for reversible cycles, use is made of the Carnot cycle and it is shown that no other cycle can have a greater efficiency. There are, however, a number of other cycles having the same efficiency as the Carnot within the same range of temperature. In Fig. 3, let the reversible cycle abed be formed by FIG. 3. the two isothermals ab and cd and of the two curves fee and da. The curve be is arbitrary, but da is drawn like it, being simply displaced to the left. The heat turned into work during the cycle is abed, or equal to that of the equivalent Carnot cycle abef, but the heat supplied is d'daa' +a f aW, or is greater than that needed to perform the same work with the Carnot cycle by the amount d'daa'. Now the heat rejected from b to d consists of the two parts, cdd'c f , which passes to the ISODIABATIC CYCLES. 7 refrigerator at the lowest temperature T 2 and is of necessity lost, and bb'c'c, which is rejected during a dropping temperature and which is strictly equal to the heat required to carry out the reverse operation da. Instead of being wasted, the heat rejected along be might be stored up and returned to the fluid along da, thus making no further demands upon the source of heat. In this manner the one operation would balance the other and the heat actually required from an external source would be represented by afaW, as in the Carnot cycle. The heat actually rejected would also be represented by d'dcc', equal to a'feb', as in the Carnot cycle. Thus the adiabatic lines of the Carnot cycle would be replaced by two lines be and da, along which the interchanges of heat are compensated. The efficiency of such isodiabatic cycles would thus be equal to that of the Carnot cycle. The operations along the lines be and da may be imagined as follows. Heat only flows spontaneously from one body to another at a lower temperature. Thus the heat given up along be can be stored and utilized to effect the operation da, if the process be subdivided into very small differences of temperature and each portion of the heat rejected at the momentary temperature. Thus the heat rejected along be between the temperatures T +dT and T (Fig. 3) is returned along a portion of da at the same temperature. The difficulty is to find a practical regenerator to 8 THE TEMPERATURE-ENTROPY DIAGRAM. perform this duty. It must have large heat-conducting surfaces, and consist of a number of subdivisions at all temperatures between T and T 2 , and there must be no conduction of heat from any part to the next at lower temperature. To accomplish the operation along be, the working fluid passes successively through these divisions and deposits in each a part of its heat. Dur- ing the process da the fluid passes again through the divisions but in a reverse direction, from the coldest to the hottest. This ideal process can only be roughly realized as the regenerator has a limited conductibility, permits the flow of heat between adjacent sections, and can only store up heat if the fluid is much hotter and refund it if the fluid is much colder than itself. Therefore only the upper portion of the line be can actually be utilized to refund heat gratuitously along the lower portion of da. Hence in practice it should be possible to obtain a higher efficiency with an engine working on the Carnot cycle, because those working on the isodiabatic cycle have the added losses of the regenerator. But theoretical efficiency is only one of several factors entering into the design of an actual engine and may be more than balanced by the increased size, cost, strength, etc., required to attain it. A different type of regenerator is described by Swin- burne. The engine consists of a primary cylinder to perform the external work and a secondary one to act as a regenerator. Each cylinder possesses sides and ISODIABATIC CYCLES. 9 piston absolutely impermeable to heat, but an end which is a perfect heat conductor. The cycle of the working fluid in the primary cylinder consists of two isothermal and two constant-volume processes. Placed 'in contact with a source of heat of constant tempera- ture TI the fluid expands isothermally up to the end of the stroke; then removed from the source of heat, heat is extracted at constant volume until the tem- perature has fallen to T 2 ', next placed in contact with a refrigerator of constant temperature T 2 the working fluid is compressed isothermally until the initial volume is regained. The second cylinder operates only during the constant volume changes. With its working sub- stance compressed into the clearance space and at the temperature TI its conducting end is brought into contact with the corresponding end of the primary cylinder, just as the latter is removed from the source of heat. The two charges in their respective cylinders .are thus maintained separate while the temperature is always equalized. The piston in the secondary cylin- der now moves forward, performing external work at the expense of the internal energy of its own charge and of the heat received from the primary cylinder, c v (TiT 2 }. This process is to occur so slowly that only an infinitesimal drop of temperature exists through- out the combined masses. When not in contact with the main cylinder the auxiliary remains inactive. For its return stroke the secondary is placed in contact with 10 THE TEMPERATURE-ENTROPY DIAGRAM. the primary as the latter is removed from the refrig- erator, and external work of an amount equal to that developed during the forward stroke must be performed upon it, thus gradually restoring its own temperature and that of the primary by returning to it the heat* c*(Ti-T$. Thus while the entropy of the working charge is decreasing along be (Fig. 4) that of the auxiliary charge is increasing an equal amount along be', and while that of the auxiliary charge decreases along d'a (curve c'b a b r \ I \ \ / \ / \ d d' c 1 FIG. 4. moved to the left) the entropy of the working charge undergoes an equal increase at the same temperatures. Thus during the .isodiabatic processes the combined entropies of the two charges possess a constant value. Decrease in Efficiency due to Irreversibility. In all actual engines where the charge enters and leaves the cylinder each cycle, a small difference of pressure and hence of temperature must always exist between the fluid in the supply-pipe and in the cylinder, and between that in the cylinder and in the exhaust-pipe. Thus, let TI and T 2 (Fig. 5) represent the temperatures of IRREVERSIBLE CYCLES. 11 source and refrigerator; then, for a Carnot cycle abed represents the heat utilized, while dcgh represents that rejected. If it be assumed that no loss of heat is experienced by the working fluid during admission, but simply the drop in temperature 7\7Y, then area afb'g'li must equal area dbgh. Similarly, the heat rejected at 7Y from the actual engine (area d'c'g'h) would at the temperature T 2 be equal to the area dee'h, and thus exceeds that rejected in the ideal case by the a Supply Pipe b Ci' Cylinder I,' d Cylinder r' 5T (I Exhaust Pipe C 99' FIG. 5. area cee'g. The efficiency of the actual cycle is there- fore reduced by these irreversible processes. It is important to notice that the increased exhaust subdivides into the two areas cc\g f g and ci'ee'g', which represent the losses incurred during admission and exhaust respectively. Thus if there were no throttling during exhaust the adiabatic expansion would be from b' to C/, and therefore cci'g'g must represent the loss due to throttling at admission alone. The loss is in each case equal to the increase in entropy during the 12 THE TEMPERATURE-ENTROPY DIAGRAM. throttling, multiplied by the lowest available tem- perature. Thus it follows that any irreversibility, which is always accompanied by a growth of entropy, must cause a decrease in the thermal efficiency of a cycle. Of course, the net efficiency is always still fur- ther reduced by actual heat losses due to conduction and radiation. Therefore in working between any two temperatures the highest possible efficiency will not be attained unless all the heat received is taken in at the upper temperature, and all the heat rejected is given out at the lower tem- perature. The only exception to this is in the case of the isodiabatic cycles already considered. CHAPTER II. THE TEMPERATURE-ENTROPY DIAGRAM FOR PERFECT GASES. A PERFECT gas is defined as a substance whose specific volume, pressure, and temperature satisfy the equation pv =RT, and whose specific heat at constant pressure, c p , and at constant volume, c v , are constants. Starting from the equation for the conservation of energy, dQ=A(dE+pdv), (1) for a change of condition at constant volume we obtain dQ = AdE=c v dT; s* whence E=-jT+ constant (2) The internal energy of a perfect gas is thus seen to depend solely upon the absolute temperature, so that an isothermal process is also an isodynamic one. Let us consider next a unit weight of gas in the condition (pvT), and let the temperature increase from T to T\, once at constant volume and a second time at constant pressure. In the first case the condition 13 14 THE TEMPERATURE-ENTROPY DIAGRAM. is changed from (pvT) to (p\vT^) by the addition of the heat, c v (Ti T); in the second case from (pvT) to (pViTi) by the addition of the heat c p (T l -T\ Since the change of temperature is the same in both cases the increase in internal energy is the same and therefore the difference between the heat added in the two cases must represent the difference in the external work performed. Then we have whence c p c v =AR = c v (k 1). . . (3) Substituting, this relation in the expression for in- ternal energy gives The entropy for any condition can now be determined without difficulty. Since dv r r A / v = J ^=c v j ^r + AJ pyr + constant, + AR I [-constant. J v pv ~R -c v ) log e v+constant, c v loge p + ~ loge v + constant, Cv loge pv k + constant i ...... (5a) PERFECT GASES. 15 By use of the characteristic equation pv = RT, either p or v can be eliminated so that the entropy may also be expressed as . . (56) i-fc + constant 3 . . . (5c) THE fundamental heat equations for a perfect gas are . . . . (6) and c p -c v = AR, (7) which for reversible processes give three different expres- sions for entropy : (8) The curves for constant volume and constant pressure, respectively, in the T^-diagram become T, $2-$i = c v loge7jr (9) 16 THE TEMPERATURE-ENTROPY DIACRAM. and &-&-**. a? ...... (10) L i If p = f(v) be the pv-projection of any curve on the characteristic surface pv = RT.', to find the T^-projec- tion of the same curve it is only necessary to find =fr(T) or ^=/ 2 (T) from the above equations and to Vi pi substitute them in the first and second forms, respec- tively, of equations (8). The most general form of p = f(v) with which the engineer is concerned is that of an indicator card, namely, pv n = pjV^ = a constant, . . . . (11) where the exponent is determined from any two points by means of the formula _ log v a -log< The respective projections of equation (11) on the TV- and Tp-planes, as found by combination with the characteristic equation, are T l v l n - l = T^ n ~ l = a constant . . . (12) and fj^j * -Tap, a constant, . . (13) whence it follows that PERFECT GASES. 17 The substitution of equations (14) in the corresponding forms of equations (8) gives 2-l = Cp log e T - (C, -Cj log, i Equations (11), (12), (13), and (15) represent simply different projections of the same curve on the charac- teristic surface upon different planes. From equation (2), p. 4, for perfect gases, I/TF m d = c, or #,-^-clog.r. . . (16) Comparison of equations (15) and (16) shows that the specific heat for the general expansion p l v l n = p 2 v 2 n is '= 2 (f> 1 = = constant, equation (15) gives n = k, ft, and equation (17) becomes c = c v -, 7 =0; i.e., for an K 1 isentropic change the heat capacity of the substance is nil; that is to say, the temperature of the substance can be increased or diminished without the addition or subtraction of heat as heat. Equation (11) becomes pv k = constant. (3) For p = constant, equation (11) becomes vf^vf, hence n equals zero. k c From (14) c = c v ^ T-=C V - = C P , and equation (15) C v T becomes < 2 - ioge^r, as already found in equa- tion (10). (4) For v = constant, the only value of n which will satisfy pA n = p 2 V is n=oo; so that equation (11) becomes v= constant. PERFECT GASES. 19 Equation (17) gives c = r- = c v , whence from T \og e ~, as previously found in -* i equation (15) 2 l equation (9). We are now in a position to transfer any curve from the pv-plane to the 7^-plane as soon as we know the value of c v for the given gas. In Fig. 6 let abed represent a cycle consisting of two curves of constant volume and two of constant pres- sure indicated by a rectangle in the yw-plane. In the T^-plane start with the value of the entropy at a as the zero-point. The curve ab will be of the nature shown, becoming steeper as T increases because the subtangent at any point represents the value of c v and this is a constant for perfect gases. Arriving at 6, the curve of constant pressure will assume some such position as shown, be will not be as steep as ab at the point of intersection because c p >c v , i.e. the subtangent of be 20 THE TEMPERATURE-ENTROPY DIAGRAM. is greater than the subtangent of ab. Two similar curves cd and da complete the cycle. All possible variations of the curves pv n = constant, n k T or 2 -0i = c I , r-log e ^r, are summarized in the fol- n 1 1 1 lowing table and diagrams: pv-coordinates. T^-coordinates. n Form of the Curve. *'i* Form of the Curve. p = constant C P 1 T and increaje or de- 0c p 1 crease together 1 Kn incr'sing = constant fc increase or de- 00 v= constant c v 1 crease together * During an isothermal change of a perfect gas all the heat added performs external work, hence in the diagrams the isodynamic lines are superimposed upon the isothermals. |=fi(T), it will sometimes prove simpler to avoid the analytical solution and to find the desired curve graphically by plotting it point by point at the intersections of the proper constant pressure and constant volume curves. This may be done the more readily since the curves of constant pressure and constant volume and 0=c v log e T+c 2 are of such a character that the intercepts upon iso- thermals made by any two constant pressure or con- stant volume curves are of equal lengths and equal respectively to 4=ARlog e - and 40 = A/Hog* ^ Hence to draw a series of constant pressure or of constant volume curves in the T^-plane the following procedure is necessary and sufficient: (1) Determine accurately the contour of the two curves =Cp logeT and < = c v log e T 7 . (2) Construct a template of both curves. 22 THE TEMPERATURE-ENTROPY DIAGRAM. (3) Determine A$ = AR loge and P* for the necessary pressures and volumes. (4) Along the isothermal passing through piv i} lay off the values of J^ determined in (3). (5) Using the templates draw through these points the corresponding pressure or volume curves. Having these base lines once constructed, any irregular curve, p=f(v), may be at once transferred from the pv- to the !T^-plane by noting the values of pressure and volume at a sufficient number of points to give a smooth curve. Relative Simplicity of the T(j>- and pv -Projections. In graphical work where it may be necessary to plot isothermals and reversible adiabatics as well as con- stant pressure and constant volume curves, it is simpler to use the T- than the pu-plane. This is at once evident from the following tabulation: Curves. pv-plane. T^-plane. Constant pressure Horizontal lines. . . . Vertical lines < < = e p log e T + CI ' The curves are all alike ( = c v log e 7' + c 2 " temperature . . ** entropy I pv = c \ No two curves alike { vv n =c * No two curves alike < The curves are all alike } Horizontal lines } Vertical lines The Logarithmic Curve y =log x. The logarithmic curve forms the basis not only of the graphical method PERFECT GASES. 23 for plotting any polytropic curve in the pv-plane, but also of the method for projecting any curve from the TV- and pu-planes into the T(-plane. Such a curve can be constructed very quickly from a logarithmic table, and in case much of this work must be done a FIG. 8. template can be constructed and the curve thus be drawn whenever needed. In case logarithmic tables are not at hand and even if they are, and some arbitrary value of x is assumed as unity, the following graphical method for construct- ing the logarithmic curve is often of great convenience. 24 THE TEMPERATURE-ENTROPY DIAGRAM. Let Ox Q (Fig. 8) represent the arbitrary unit of x. Through draw any straight line OS. Then starting with as a centre and OXo as the first radius, con- struct the series of arcs and perpendiculars, determining the series of values, From the series of similar triangles these quantities are related as follows: _____.___ m~X n - l ~ '~~X 3 ~X2~Xi~XQ~Xi~X 2 ~X3~' ~ X whence Xi = mr 1 x Xi = mx X n = mX n - 1 = Taking the logarithms and remembering that log =0 and setting log m = D, we obtain logm~ n x =--n-logm= -n-D = log log x 3 =\ogm~ s XQ^ 3 log m= -3-D = log XQ \ogx 2 =logw~ 2 xo=-2 logw--2 Z) = log XQ PERFECT GASES. 25 log x\ = log m~- l x = 1 log m = 1 D = log XQ * = = log XQ v =+1 logw=+l D = log XQ jr =+2-logm=+2 D = log XQ v =+3 logw=+3 D = log- XQ X n logX n =logm n Xo = +n logm = +n D=log XQ This method of construction has thus given a series of points x such that the logarithms of any two succes- sive points of the series differ by a common amount D. Therefore to construct the desired logarithmic curve y=\ogx, starting at XQ, lay off on the vertical lines through Xi; X 2 , X 3 , . . . X n , and Xi, x 2 , x 3 , . . . x n the distances D, 2D, 3D, ...nD and -D, -2D, -3D, ... nD, respectively, and connect them with a smooth curve. In case any multiple of this curve is desired, such as T ya-logx, as, for example, 4(}>=c v log e -^ , the ordi- * o nates of the curve y = log x may each be multiplied by a and the product aD, 2aZ>, . . . , laid off as before. A 26 THE TEMPERATURE-ENTROPY DIAGRAM. simple and more convenient method is that shown in Fig. 9. Lay off from the distance unity (x = l), from x drop a perpendicular equal to a units. Draw OA. Then from similar triangles if OB equal log x, BC will equal a-\ogx. Therefore to determine a\ogx = y for any value of x, take log x from the log-curve with FIG. 9. the dividers and locate B. Then measure BC with the dividers. Graphical Construction of the Polytropic Curves (pv n = c) in the pv-Plane. To construct the polytropic curves choose any convenient volume, as OA (Fig. 10), for unit length, and then through this point draw in any convenient manner the logarithmic curve p = logv. From the origin lay off to the left OA' = OA, and then A'B =n. Draw the straight line OB, and the constant-volume curve through A. Choose any number of points on this line, as 1, 2, 3, . . . , according to the number of curves desired, and connect each point by a straight line to the origin. PERFECT GASES. 27 To determine the pressures on the curves 1, 2, 3, . . . , corresponding to any volume, as v\, take log^i from the log-curve and lay off at the left of 0, as Ovi. Drop a vertical line through vi until it intersects OB at a\. Project horizontally on the log-curve at 61. Through 61 erect a perpendicular intersecting 01, 02, 03, . . . , at hi, h 2 , ha, . . . , respectively. Draw horizontal lines 28 THE TEMPERATURE-ENTROPY DIAGRAM. through these points intersecting v\ at p if p 2 , ps, . . . . These are the desired points. The proof of this is simple. From similar triangles, i'oi : -log i --:-!, Vi'a = n log vi = log v i~ n = W. Again from similar triangles, bhf.Ob = A1:OA, To continue the curves to the left of A, prolong BO into pv-plane, as shown, and lay off log v to the right of 0. Then proceed as before. This same method can be used to plot the TV-pro- jection of the polytrope, Tv n ~ 1 = c, provided the auxiliary line OB is determined by the coordinates [ 1, (n 1)] instead of ( 1, n). This construction may be used backwards to deter- mine n for a given polytropic curve, or if the curve is only approximately polytropic a mean value of n can be determined. Construction of the Isothermals, pv=c, in the pv- Plane. Draw the curves pi=c and Vi =c, which bound that portion of the pu-plane (Fig. 11) in which the isothermals are to be constructed. To construct the isothermal through any point, as A, draw the curve PERFECT GASES. 29 v =v a . Draw the straight lines 01, OH, OIII, . . . , which intersect v =v a in points 1, 2, 3, 4, .... Draw a series of vertical lines through 7, II, HI, . . . , and horizontal lines through 1, 2, 3, . . . , then the points of intersection II, 112, 1 1 13, . . . , are points on the desired isothermal. I IT Til IV V VI VII FIG. 11. The isothermals through the successive points 7, 77, 777, . . . , can be drawn by using the same radial and vertical lines. The only extra construction necessary being the horizontal lines through the points of intersec- tion of 7, 77, 777, . . . , with the radial lines. This construction is very simple if carried out on plotting paper, as then only the radial lines need to be constructed. 30 THE TEMPERATURE-ENTROPY DIAGRAM. The justification for this construction 5s found in the equation of the curve itself. Thus pv=c may be construed graphically to mean that the area of the rectangle included between the axes of p and v and the constant pressure and constant volume lines through any point on the curve is a constant. Thus take the points (A) and (2,11) on the isothermal through A. If these points lie on the same isothermal rectangle OA must equal rectangle 2, II. Now A OHp = A Oil 1 1' and A 022' = A 02v a and A 21 1 A = A 2/7(2, 77), whence rectangle 2' A = rectangle v a (2, 77) and therefore rectangle OA = rectangle 0(2, 77). Slide Rule Construction of Isothermals. If, in place of drawing isothermals through points already known, it becomes necessary to determine them for definite temperatures the work of plotting may be carried on directly from the slide rule. Thus set the slider to mark the product R T, and then the volumes corre- sponding to any desired number of pressures may be read from one setting for each value. The slider must be reset for each new isothermal. Representation of W, Q, and AE in the pv- and T<-Planes. The pv-plane gives at once the external work performed during, and the 7 7 ^-plane the heat interchange involved in, any reversible process. To represent the change of internal energy two methods are available in both planes. (1) Since isothermal and isodynamic lines are coro- PERFECT GASES. 31 cident it is only necessary to determine JE per unit increase of temperature and then, by assuming some arbitrary zero of internal energy, to assign values to the different isodynamics. Thus 4E=- 1 -^- = -, A; 1 A; 1 and AE per unit change of temperature equals r 7 = ~A- Hence the change of intrinsic energy during K 1 A. any process may be found by noting tha initial and final state points with reference to the isodynamic lines. (2) If a substance expands adiabatically, performing work at the expense of its internal energy, and if this process occurs so slowly as to prevent increase of kinetic energy and to permit the maintenance of a uniform temperature, and if there be neither internal nor surface friction, then the expansion will at the same time be isentropic. For such an isentropic expansion the area under the curve in the pv-plane will not only represent the external work performed but will at the same time be a measure of the decrease of internal energy. Thus between any two points a and b upon the same isentrope there must exist the difference of internal energy, where n equals the exponent of the adiabatic curve. 32 THE TEMPERATURE-ENTROPY DIAGRAM. If the point b be removed to infinity there results v F P aVa - Ilia tii^ = --- T 00 n-l' i.e., the decrease in internal energy during frictionless adiabatic expansion from any finite condition a to infinity is represented by . This, however, is not n 1 the total value of the internal energy at condition a, because by definition, Internal energy = vibration energy plus disgregation energy; = kinetic energy plus potential energy; or E=S+I, and at infinity while the vibration or kinetic energy has become zero with the disappearance of both pressure and temperature, the disgregation or potential energy has assumed its maximum value due to infinite increase in volume. Hence # 00 =, X) .+/ 00 =0 + / 00 =/ 00 , and we may write Thus the internal energy of a perfect gas can be deter- mined up to the value of a certain unknown constant, /.- In the case of a perfect gas the molecules are sup- posed to be so far apart as to be beyond the sphere of PERFECT GASES. 33 mutual attraction, so that the value of I x is already established at finite distances, and changes in internal energy are directly proportional to changes in tem- perature. (a) To find the difference in internal energy between any two state points a and 6 of a perfect gas we have W _ F a- T _ aa ~k-l +1 k-1 - *.-! k-1 or A (E b - c v T b - c v T a . Interpreted graphically (Fig. 12) this states that the difference between the intrinsic energy of any two state points is represented in the pu-plane by the dif- ference of the areas under the adiabatics drawn through the respective points and extended to infinity; or the 34 THE TEMPERATURE-ENTROPY DIAGRAM. heat equivalent of the difference in kinetic energy is represented in the 7^-plane by the difference of the areas under the constant volume curves drawn through the respective points and extended to infinity. Here again, although both diagrams are infinite, the T(j> is the easier to construct accurately. (6) To avoid the use of the infinite diagram the method shown in Fig. 13 can be used. Suppose it is O M'a' O a' L' P' FIG. 13. desired to find the difference of intrinsic energy between a and 6. Draw through a (pu-plane) the isodynamic E a =c and through 6 the iseritrope b c, interescting at some point N. If the gas be imagined to expand isentropically from 6 to N it will develop the work bNN'b' and suffer a decrease of internal energy EbE N equal to this work. But by construction E N =E a so that the area under bN represents the magnitude of the difference of internal energy between a and 6. An PERFECT GASES. 35 equivalent solution indicated by dotted Fines may also be used. In the T^-plane a slightly different construction must be used. During p. constant volume change no external work is performed and the heat added in- creases the internal energy. Draw through b (T 1 ^- plane, Fig. 13) the constant volume curve Vb = c, and through a the isodynamic E a = c intersecting at P. T f . the gas be imagined to undergo the change Pb it will absorb the heat c t) (7 7 6-7 T p )=area under Pb, and during this change its internal energy will be increased from E p to Eb, that is, from E a to Eb. Hence the difference between the heat equivalents of the internal energy at the points a and 6 is represented by the area under Pb. An equivalent solution indicated by dotted lines may also be used. It follows from the first law of thermodynamics, as applied to reversible processes, Q = A-AE + AW, that (in the pv-plane) since the area under db represents the work performed and the area under bN represents the change of internal energy, the algebraic sum of these, or the area under dbN, must represent in work units the heat received. Similarly, in the TV-plane if the area under ab represents Q a b, and the area under Pb represents A(EbE a ), then the area abPP'a' must represent AW a b. In the case of a perfect gas it is thus possible in both the pv- and the T^-planes to represent by finite areas 36 THE TEMPERATURE-ENTROPY DIAGRAM. all three terms involved in the statement of the first law of thermodynamics. But again the T^-plane proves itself the simpler of the two, in that one of. the two necessary construction curves is a straight line, and the other has only one form for any given gas. Graphical Projection of any Curve from the pv- to T>m the TV- Plan 3. From the laws of a perfect gas v = or vaT if p = constant, it follows that in the TVplane FIG. 14. a constant pressure curve is represented by a straight line passing through the origin. Let 1, Fig. 14, repre- sent the condition T 1} v\, and hence the condition pi. PERFECT GASES. 37 Then the line 01 represents the line of constant pressure pi. Now if the volume of a perfect gas be maintained constant the pressure is proportional to the tempera ture, so that the point 27\, vi, represents the pressure p 2 = 2pi. The line Op 2 therefore represents the con- stant pressure 2pi. Similarly Op 3 represents p = 3pi, etc. Thus to obtain the constant pressure curves, lay off on any convenient constant volume curve a series of equal intervals and draw a set of straight lines from these points to the origin. If the scale of T is already determined the scale of pressure may be determined to correspond, or if these pressures are laid off arbitrarily the temperature scale must be determined to corre- spond. For ordinary use the latter method is more convenient. Let a curve ah be given in the pv-plane which inter- sects the pressure curves p\, 2pi, 3pi, etc., in the points 1, 2, 3, etc., respectively. Each point piVi, p 2 V2, p 3 v 3 , . . . , has an unique location in each plane, namely, at the intersection of the corresponding pressure and volume curves. The position in the TVplane may be found by projecting upward along the constant volume curve from p in one plane to p in the other. For any known weight of gas T is at once deter- mined, but if the quantity is unknown and if the volumes are only known relatively, as in the case of cards taken from hot-air or gas-engines or air-com- pressors, the scale of T cannot be determined, but the 38 THE TEMPERATURE-ENTROPY DIAGRAM. projection still gives relative values of the temperature and may thus throw considerable light upon the nature of the operations. The Graphical Projection of any Curve from the TV-Plane into the T^-Plane. The entropy, temperature and volume of a gaseous mixture are connected by the equation < = d, log e T + AR loge v + constant. Assuming any condition v Q T Q as the reference point "he difference of entropy between this reference point and any other point, vT, is given by T v 4$ = C v log e TfT + AR loge -- 1 o VQ Putting the variable factor R into the left-hand term the expression reduces to In the case of diatomic gases when k= 1.405 this re- duces to 333 5 TV ~^- 4 = 2.47 logio TfT + logio - ft 1 o VQ This equation will apply directly to hot-air engine and air-compressor cards, but for gas-engine work the coefficient may need to be modified for each individual case, as A; is a variable, being about 1.38. PERFECT GASES. 39 It is evident that if we could construct two curves such that the ordinates for any point (T, v) are re- spectively equal to 2.47 logic TJT and logic , then the J-Q ^o 333 5 sum of these two ordinates would equal ^ J<, ZV and by using this value and T a 7^-plot could be con- structed. This coefficient could be eliminated from the 333 5 T-plot by making ' =1 the unit of entropy, and then the abscissae would read directly in units of entropy. Thus suppose we have the curve ab (Fig. 15) in the TVplane and desire to find its ^-projection. Choose any point T v as the reference point, and then on base lines parallel to the T and v axes and located if necessary outside the diagram (the axes themselves may be used if the diagram is not unnecessarily complicated thereby), construct two logarithmic curves (see pp. 23 to 26), such that T v Ay = log Y Q and Ay ' 2 = log Then by means of the auxiliary construction draw also the curve Now, Jjf! = 2.47 log - 333.5 40 THE TEMPERATURE-ENTROPY DIAGRAM. PERFECT OASES. 41 and therefore if from any point n on the curve ab we obtain by projection upon the two logarithmic curves the distances Ay^ and Ay 2 , the sum of these distances laid off to the right of (f> along the isothermal T n will locate the point in the T-plane. For point a, Ay i = Ay a and Ay 2 = Q, and for point b, Ay i = - 4y b l and Ay 2 = + Ay b ,. After the curve is once constructed the scale of 333 5 entropy can be chosen by setting ' = !. It is evident when the logarithmic curves are con- structed graphically that, as D is chosen arbitrarily, the scale of the drawing may be anything. In fact, the scale can be suitably chosen by varying the length D. Furthermore the values Ayi and Ay 2 may be read from the base lines on which the logarithmic curves were constructed or from any convenient parallel line. This would mean increasing or decreasing all the values of J< by the same amount and result simply in a bodily transfer of the T^-projection to the right or left by that amount. This would not affect the value of the diagram, as we are dealing with changes in entropy and not with absolute values. In fact the value of < is infinite so that any point may be taken as the arbitrary zero. 42 THE TEMPERATURE-ENTROPY DIAGRAM. Advantage may thus be taken of these facts to shift the T ^-projection parallel to the $>-axis so as to obtain the most convenient location on the drawing- paper. CHAPTER III. THE TEMPERATURE-ENTROPY DIAGRAM .FOR SATURATED STEAM. DUE to the very slight variations in the volume of water with increasing temperature the heat equiva- lent of the external work is very small, and the differ- ence between the work performed under atmospheric pressure and that which would be performed under a pressure increasing with the temperature according to Regnault's pressure-temperature curve is negli- gible compared with the heat required to increase the temperature. Hence the value of the specific heat c = /(0 as determined by Rowland is taken as equiva- lent to that of the actual specific heat required for the transformation occurring when feed-water is heated in a boiler. Similar statements hold for other fluids. Therefore the heat required to increase the tem- perature of a pound of liquid by the amount dT while the pressure increases by dp is taken as dq = cdT, ....... (1) 43 44 THE TEMPERATURE-ENTROPY DIAGRAM. Whence q, = \dT, ..... (2) and To make the temperature-entropy chart conform to the tables for steam and other saturated vapors, we will plot the increase of entropy and intrinsic energy and the heat added above that of the liquid at freezing-point. Hence for water the zero of the entropy scale will be at 32 F. Furthermore the chart will be constructed for one pound of water. Point a in Fig. 16 represents the position of a pound of water at 32 F. on the !T0-chart. Starting from here and plotting values of 6 and T from the equation 6- 32 as given in the steam- tables, we obtain the "water- line " ab. At any temperature t the value of the specific heat c is shown by the subtangent gf. Further, as we proved in the general case of equatipn (1), the area under the curve aefO represents the heat of the liquid, q; that is, the number of heat-units required to warm up the water from 32 F. to the temperature t. This "water-line" is the same kind of a curve as that repre- sented by equation (15) for perfect gases, except that SATURATED STEAM. 45 the specific heat of a gas is a constant, while that of water is a function of the temperature. If at t the water has reached the temperature corre- sponding to the boiler pressure any further increase of heat will cause the water to vaporize under con- stant pressure and thus there will be an increase of entropy at constant temperature. This will be repre- sented on the chart by the horizontal line ed, and will continue until all the water is vaporized and a 82' -f 9 O f m $. FIG. 16. condition of dry-saturated steam reached. During this change the increase of entropy will be that is, the length of the line ed may be found by taking the latent heat of vaporization r and dividing by the absolute temperature corresponding to t. 46 THE TEMPERATURE-ENTROPY DIAGRAM. The area under the curve ed represents the heat added during vaporization or r. The area under the curve aed therefore represents all the heat necessary to be added to a pound of water at 32 F. to change it in a boiler into dry steam at the temperature t, or area Oaedm = X = q+r. Similarly the location of the "dry -steam" point may be found for any number of temperatures, thus giving the location of the dry-steam line or saturation curve. In the area between the water-line and the dry- steam line ij, or the region of vapor in contact with its liquid, lie all the values given in the steam-tables; to the right of the dry-steam curve lies the region of superheated steam. Having given the chart with the " water-line" and "dry-steam" line located, and knowing the temperature t of the steam, it is simply necessary to draw the hori- zontal line ed and drop the two perpendiculars ef and dm and the tangent eg-, then the diagram gives at once q, r, ;, 0, > T, and c. Let e in Fig. 17 represent the state point of a pound of water in a boiler under the pressure p corresponding to the temperature t. As heat is applied part of the water vaporizes and the state point moves toward d. At any point as M, the area under eM represents the heat already added, while the remaining part, under Md, SATURATED STEAM. 47 represents the heat, which must be added to complete the vaporization. That is, for the complete change from water to dry steam the state point travels the distance ed, and to vaporize one half it would travel one half the distance, etc. Then if M represents the momentary position of the state point, the ratio eM + ed will repre- sent the fractional part of the water already vaporized; O N FIG. 17. that is, the dryness of the mixture, x, is given by eM ed (6) Hence if the line ed is divided into any convenient number of equal parts (say 10) the value x can be read directly from the chart as soon as the position of the state point M is known. In Fig. 17, for example, 48 THE TEMPERATURE-ENTROPY DIAGRAM. Furthermore, the value of the entropy at M is repre- sented by that of the liquid at e and the xih part of the entropy of vaporization, or and the total heat at M equals the heat of the liquid plus the xih part of the heat of vaporization, or and is represented by the area OaeMN. In a similar way the distance -^ for several tem- peratures can be divided into the same number of equal parts, and then if all these corresponding points are connected by smooth curves, each curve will repre- sent a change during which the fractional part of the water vaporized is constant. These are known as the z-curves. In place of having separate scales of pressure and temperature for the ordinates of the !F<-diagram, it is often convenient to take the values of p and t from the steam-tables and to plot Regnault's pressure- temperature curve in the second quadrant, as shown in Fig. 18. Then given any pressure p^ the corresponding tem- perature ij may be found as indicated or vice versa. SATURATED STEAM. 49 Let Pi be the pressure and x t the dryness fraction of a pound of mixture, then on the chart its condi- Fia 18. tion will be indicated by the point 1. Its volume may be found from the steam-tables by use of the formula 50 THE TEMPERATURE-ENTROPY DIAGRAM. To find the location of the state point 3 at any other pressure p 3 , so that v 3 = i\, proceed as follows: Draw the axis of volume opposite to that of tem- perature and lay off along Ov the distance Oa equal to the volume of one pound of water. The variations of a with t may be neglected and o set equal to 0.016 cubic feet. Draw oa. Project 6 V and 6^+-^- on the ^-axis, and from the latter point lay off the distance ms x equal to the volume of one pound of saturated steam as given in the steam-table for t. Prolong the perpendicular from t until it intersects aa at o v . Con- nect this last point with s v This line a l s l shows the increase of the volume and the entropy during vapori- zation. Project x l upon 0 and continue until it intersects the rx^-curve at c. Then be = x^ and ac = x^ + o 1 = v 1 . Through c draw yy parallel to 0$. For any pressure p 3 draw the corresponding t'0-curve <7 3 .? 3 , and where this intersects yy the volume will be V 3 '=v 1 =x 3 u 3 + o. Projected up this intersects the isothermal t 3 in 3, giving the desired dryness fraction x 3 . Points 1 and 3 have the same volume i\. Other points, as 2, etc., may be found and connected with a smooth curve. This will intersect the dry-steam line at some point s 9 = v l . In this manner similar constant vol- ume curves can be constructed to cover the entire diagram. Suppose it is required to find the heat necessary SATURATED STEAM. 51 to cause a change from some point L to an adjacent point L + dL. (See Fig. 19.) , a , xr rcdT xr v = +Y = J ~T~ + ~T' cdT x , r xr OCT -dT + (cxdT - cxdT) (7) To interpret the last term imagine this change to occur along the dry-steam line. Then x = l and dx = Q, whence dQ=\cY+p]dT( = Td). . . (7fl) The comparison of this with eq. (2), p. 4, shows that cTr+Tm ' ls tne "specific heat" of dry-saturated steam, r dr or h^c- F f i + -rFn' (8^ The diagram shows at once that h is negative, i.e. to move along the dry-steam line with increasing tem- perature heat must be rejected. Ether has a positive value for h. This signifies that the saturated-vapor line for ether slants to the right instead of to the left. 52 THE TEMPERATURE-ENTROPY DIAGRAM. Him found that steam condensed upon adiabatic expansion and Cazin that it did not condense upon compression. The reason for this is at once clear from Fig. 19. Expanding from a the reversible adiabatic Water FIG. 19. line for water cuts successively ' ' z-lines " of decreasing value, showing condensation. Compressed adiabatic- ally from a the steam would at once become super- heated. Exactly the reverse occurs with ether, con- densation occurring during adiabatic compression, super- heating during expansion. If water is permitted to expand adiabatically from 6 it is partially vaporized, as shown by the adiabatic line cutting increasing values of x. Similarly if very wet steam is compressed it condenses. SATURATED STEAM. 53 An inspection of the z-curves near the value z = 0.5 shows that they change from convex to concave and that it is thus possible with water for the reversible adiabatic to cut an x-curve twice at different tempera- tures, as at c and d; i.e., it is possible at the end of an isentropic expansion to have the same value of x as at the beginning. Thus t, = 4' 1 =O l +=0,+ Z i , or *,--. 0) Consequently there must exist some adiabatic which is tangent to this z-curve. Above the point of tan- gency the z-curve slants to the right and possesses a positive specific heat, below it the curve slants to the left and has a negative specific heat. The values of the specific heat above and below the point of tan- gency diminish in magnitude as the tangent is ap- proached and at the point of tangency are identical and equal to zero; that is, just there the tempera- ture may be raised without the addition of heat because the change is isentropic. If these points of zero specific heat are determined for all the x-curves and connected by the smooth curve MN, the chart is divided into two portions, such that to the left of MN the specific heats along the z-lines are positive, i.e. isentropic expansion will be accompanied by vaporization, and to the right of MN the specific 54 THE TEMPERATURE-ENTROPY DIAGRAM. heats are negative and isentropic expansion will be accompanied by condensation. The curve MN is known as the zero-curve. So far we have located p, v, T, <, and x, and to make the definition of the point complete it is only necessary to draw a curve of constant intrinsic energy. This could be done by solving E 1 = E 2 ==E 3 = etc., = q l +X 1 p 1 =q 2 +X 2 p 2 = q 3 +x 3 p 3 = etc., for the proper values of x and connecting these by a smooth curve. This would be very laborious, as there does not seem to be any con- venient graphical construction. Fortunately it is pos- sible not to draw the isodynamic curves, but to find an area which represents the value of the intrinsic energy for any state point and at the same time to divide r into two areas proportional to p and Apu respectively. The first fundamental heat equation becomes, when made to conform to the limitations of the first and second laws of thermodynamics, This is applicable to the case of saturated vapors because the state point is uniquely defined by the intersection of the temperature and volume curves. SATURATED STEAM. 55 Equating the coefficient of the last terms, a relationship which holds good for any reversible change. During the process of evaporation T ic a constant and ) = _^ or r_ A7 &, (10) / T s a u dT whence Apu-^- .... (10a) dT 7 pdT Let a (Fig. 20) be the point of which the intrinsic energy is to be obtained, p a is the corresponding pres- sure. From of on Regnault's !Tp-curve draw the tan- gent a'b'. From similar triangles it is evident that jrp the subtangent equals p -T-. Draw b'b parallel to a'a. jm Then the rectangle abed has the area p--j-X-^ = Apu (see eq. 10a). That is, abed represents the external heat of vapori- zation and the rest of r, namely bckf, must equal p. Hence the intrinsic energy of the point a is given by E a = OgdW + kcbf = q +p. 56 THE TEMPERATURE-ENTROPY DIAGRAM. If M represents the state point dMnc = x M Apu, and If for each point a of the dry-steam line a correspond- ing point c, which divides the absolute temperature FIG. 20. into two parts proportional to^Apu and p, is determined the curve ee will be located. The curve ee being located once for all, the intrinsic energy at any point M can be found by drawing the isothermal Md and the adiabatics dk and Mm. At c, the point of intersection of dk with ee, draw the isothermal en. Area OgdcnmO gives the desired value. SATURATED STEAM. 57 The temperature-entropy diagram for steam thus enables one to find directly the following quantities: p, t, v, E, , x, c, h, s, q, r, I, p, 6, ^, and Apu. That is, the diagram is equivalent to a set of steam-tables and in some ways superior to them in that it enables one to obtain a comprehensive idea of the changes taking place between these quantities. Let the curve MN in Fig. 21 represent the T^-pro- FIG. 21. jection of some reversible change. During the change MN there has been added from some external source the amount of heat MNnm. At the same time the intrinsic energy has increased from OgabcmO to OgdefnO, the increase being shown by the area adefnmcba. The area MifnmM is common to both the heat added and the increase of intrinsic energy, and as the remaining part of the intrinsic energy increase, adeiMcba, is greater than the remaining portion of the heat added, iNfi, it follows that the heat added does 58 THE TEMPERATURE-ENTROPY DIAGRAM. not equal the increase of internal energy and hence an amount of work must have been performed upon the substance equal to the difference in area of these two surfaces; i.e., the external work performed upon the substance equals adeiMcbaiMfi. These areas may be found readily with a planimeter. The performance of external work upon the sub- stance might have been foretold at once from the diagram, because the volume of N is less and the pres- sure greater than that for M. Fig. 22 represents another type of reversible change. During the transformation MN, there is added the FIG 22. heat MNnm, and the intrinsic energy changes from OgabcmO at M to OgdefnO at N. The portion OgdeimO is common to both, and the intrinsic energy at N will be greater or less than it was at M according as the area ifnmi is greater or less than the area abcieda. The magnitude and sign of this difference may be de- termined each time by the use of planimeters. SATURATED STEAM. 59 In Fig. 22, E n >E n ] hence this increase of internal energy must have come from the heat added, and subtracting this difference from the total heat added will give the amount remaining for external work. This is positive in the case shown, as was to be expected, as the pressure has fallen and the volume increased. In case there was a decrease in the internal energy that area would need to be added to the heat area in order to obtain the external work performed. Having learned to construct and interpret the T$- diagram for saturated vapors we must now resume once more the main object of our investigation, namely, to find the location in the T^-plane of any curve given in the pv-plaue or vice versa, so that we may eventually be able to investigate the action of a steam-engine from both its indicator and its temperature-entropy diagrams. In the case of perfect gases it was possible to use one of the fundamental heat equations and thus obtain a simple analytical expression which could be easily * plotted; for saturated steam, however, this process is. too cumbersome to be of any service. Fortunately the graphical method offers a solution both simple and elegant. In constructing the T-diagram we have already made use of the first, second, and fourth quadrants to express T, pt, and v variations respectively, and now we have but to take the third or pv-quadrant and the diagram is complete. 60 THE TEMPERATURE-ENTROPY DIAGRAM. The saturation curve, or curve of constant steam- weight ab in the pv-plane, is depicted by the dry-steam curve a'b r in the jT-plane, Fig. 23. The method of obtaining corresponding points aaf and W is shown by the projection through the point a l and b 1 of the pT-curve. Aa and Bb show the increase in volume of a pound of H 2 m vaporizing under the constant pres- sure p a . and pi> respectively. This same increase of vol- ume is represented in the -plane. The problem resolves simply into the location of a sufficient number of state points, through which a smooth curve is finally to be drawn. To locate L, M, and- N project them on to the corresponding volume curves A" a", 7)"d", and B"V of the FIG. 24. with the constant volume curve AC. Draw the per- pendicular El. Then area cBln represents the heat of the liquid q a , and area BAml represents the intrinsic energy of vaporization xp. Since the area OgaAmO represents the total heat H A equal to q a + xp A + xApu A it follows that the difference between these two areas, or the area OgaACnO. must represent the external heat of vaporization. Fig. 24 illustrated the special problem of represent- ing the quantities of the steam tables by assuming the 64 THE TEMPERATURE-ENTROPY DIAGRAM. substance to start from freezing and to reach the state point A by travelling along the curve gaA, and is readily seen to represent a special case of the more general problem illustrated in Fig. 25. Let it be required to find Q AB , AW AB , and 4E ' AB for any general process such as AB. Draw through A the isodynamic curve AC and through B the constant volume curve BC intersecting at C. Then if the vapor be assumed to undergo the process CB, the increase in intrinsic energy E B E A will be represented by area CBbc. And as Q AB is represented by area ABba, the heat equivalent of the external work must be repre- sented by area ABCca. It is true that the isodynamic through g cannot be drawn without knowledge of the properties of water vapor in contact with ice, but this does not affect the validity of this method as applied to the general case just discussed. CHAPTER IV. THE TEMPERATURE ENTROPY DIAGRAM FOR SUPER. HEATED VAPORS. THE analytical treatment of superheated vapors, such as steam, ammonia, sulphurous anhydride, etc., is more complicated than that of perfect gases for two reasons : (1) The characteristic equations are more com- plicated and but imperfectly known; and (2) The specific heat at constant pressure and con- stant volume are no longer constant but follow com- plicated laws, unknown in most cases and but imper- fectly known in others. At the present time two different equations are being used for superheated steam. Thus Stodola in his Steam Turbines makes use of the Battelli-Tumlirz equation p(v + Q.135)=85.1T, while Peabody has based his new Entropy Tables upon the equation of Knob- lauch, Linde., and Klebe, pv = 85.857 1 - p(l + 0.00000976?) A simplified form of the latter, which does riot differ from this by more than 0.8 per cent., has the same 65 66 THE TEMPERATURE-ENTROPY DIAGRAM. form as the Battelli-Tumlirz, but different values for the constants, viz., 2KV + 0.256) = 85.857. Of the manv determinations of c p the results of only two sets of investigators nee:l be noticed. Those obtained by Thomas and Short, because their values are used in Peabody 's Entropy Tables,* and those of Knoblauch and Jakob, as their values vary most closely in accordance with theoretical predictions: THOMAS AND SHORT. (Mean value of c p .) Decree of Superheat Fahr. Pressure Pounds per Square Inch. (Absolute.) 6 15 30 50 100 200 400 20 50 100 150 200 250 300 0.536 0.522 0.503 0.486 0.471 0.456 0.442 0.547 0.532 0.512 0.496 0.480 0.466 0.453 0.558 0.542 0.524 0.508 0.424 0.481 0.468 0.571 0.555 0.537 0.522 0.509 0.496 0.484 0.593 0.575 0.557 0.544 533 0.522 0.511 0.621 0.600 0.581 0.567 0.556 0.546 0.537 0.649 0.621 0.599 0.585 0.574 0.564 0.554 KNOBLAUCH AND JAKOB. (Mean value of c p .) Kg. Si. sq. Cor.t Cen Cor. Fab F. 212 302 392 482 572 662 752 per m... per in.... emp. t temp r. . . C. 100 150 200 LT,O ri 300 350 400 14.2 99 210 0.463 0.462 0.462 0.463 0.464 0.468 0.473 2 28.4 120 248 4 56.9 143 289 6 85.3 158 316 8 113.8 169 336 10 142.2 179 350 12 170.6 :ar 368 14 199.1 194 381 16 227.5 200 392 18 156.0 206 403 20 284.4 211 412 0.478 0.475 0.474 0.475 0.477 0.481 0.515 0.502 0.41)5 I). 11)2 0.492 0.494 0.530 0.514 0.505 . 503 0.504 0.560 0>>32 0.517 0.512 0.512 0.597 0.552 0.530 0.522 0.520 0.635 0.570 0.541 0.520 0.526 0.677 0.5SS 0.550 o.r,:u> 0.531 0.609 0.561 0.513 0.537 0.635 0.572 0.550 0.542 0.664 0.5S5 ) . 557 0.547 * In the 1909 and uses the values of c r later editions of the Entropy Tables Peabody obtained by Knoblauch and Jacob. SUPERHEATED VAPORS. 67 To obtain the heat required to superheat at constant pressure we must either know how c p varies with the temperature at any given pressure and then integrate /*2 the expression Q= I CpdT, or else we may do what is accurate enough for all engineering work take a mean value of c p and assume this value to be main- tained throughout the given temperature range. Making the same assumption with reference to Cp the increase in entropy during superheat is given by dT T 2 In superheated as in saturated steam increase in entropy and internal energy as well as total heat added along a constant pressure curve, are all measured from water at 32 F. as an arbitrary zero. Thus if a pound of water at 32 F. is confined under a pressure of p pounds per square foot, and is then heated to a tem- perature T corresponding to this pressure, next vaporized at constant pressure, and finally superheated at constant pressure to some final temperature T 8 , we may express the changes in its entropy, and internal energy and the total heat added by the formulae, T cdT xr T 8 2 -f- + Y +Cplo ^ e T' = H=q+xr+c p (T s -T) 68 THE TEMPERATURE-ENTROPY DIAGRAM. The first term of each expression refers to the warm- ing of the water, the second to its vaporization, and the rest to the process of superheating. Of course, when the steam is superheated x = l. There is no method of measuring directly the increase in internal energy during vaporization or during super- heating at constant pressure, so that in both cases we are forced to fall back upon the first law of thermo- dynamics Q = A(AE + W), and measure Q and AW, thus obtaining AAE indirectly. Thus p = r Apu, and similarly the increase of internal energy from a condition of dry steam to any degree of superheat equals Q AW = c p (T a -T)-Ap(v-s). The above equations combined with those for deter- mining the specific volume, r = .016 + z(s-.016) for saturated steam and pv = 85.85T 0.256/> for superheated steam, make it possible to solve all heat and work problems involving changes of condition in superheated steam, or between saturated and superheated steam. Construction of the Constant Pressure Curves in the TV-Plane. Starting at the dry-steam line (Fig. 26) at any pressure p corresponding to the temperature T, look up in the c p -table the mean value of c p for this pressure, and, say, 50 degrees superheat. Compute the value of SUPERHEATED VAPORS. 69 Take from the table the mean value of Cp for each successive 50 degrees and compute T + 150 Continue this operation until a sufficient number of points has been determined to locate the curve with FIG. 26. the desired accuracy. This operation must be repeated until a sufficient number of lines has been accurately determined to permit of interpolation for the remainder. If a copy of the entropy tables is at hand it may be used to plot those curves or portions of curves which fall within its limits. In general these curves are about twice as steep as the water-line, as the specific heat of water is about twice that of superheated steam. 70 THE TEMPERATURE-ENTROPY DIAGRAM. Construction of the Constant Volume Curves. It is possible from the fundamental relation AT c v -c v = idT\ idT\ > \dpL\dv /p together with the characteristic equation for super- heated steam and the tables of c p , to derive (1) an ex- pression for the momentary value of c v in terms of the momentary values of p, v, T, and c p ; or (2} an expression for the mean value of c v at any volume for a given increase in tempearture. By such a method a table could be constructed for c v . Until a table of values of c v has been computed the simplest method of constructing the constant volume curves is as follows : Substitute the value of the desired volume in the characteristic equation and then solve the equation to obtain the pressures corresponding to a series of temperatures T s , !F S + 50, !T S 4-100, . . . From the pressures thus obtained look up the corresponding temperatures of saturated steam and the mean values of c p between the dry-steam line and T s , !F S + 50, ... From these data compute, T s . T s J- Plane. There is no graphical method of procedure so that the curve must be plotted point by point by deter- mining the pressure and volume at these points and finding the intersections of the same pressure and volume curves in the T^-plane. The curve obtained by connecting these points will be an approximate t epres- entation of the original. This method must be adopted, for example, in the case of any indicator card where there is superheated steam in the cylinder after cut-off. v n a' I'i FlG. 30. Graphical Method of Representing AE and W. Following the general method outlined for saturated steam on pp. 60-62, we may represent the various heat functions of any state point (as A, Fig. 30) in the super- 70 THE TEMPERATURE-ENTROPY DIAGRAM. heated region. Draw through A the constant pressure curve Aba and the constant volume curve Acd. Draw through b the constant volume curve tv>. Draw through points A, b, a, and g the isodynamic curves E A , Ei, E a , and Eg, cutting the constant volume curve V A in the points A, e, h, and d, respectively. Then we have H represented by area under gab A, E A " " " " dhcA, or by the sum of the areas under ga and under h c A ; W gA represented by area Ogab Acdno,or by the area a' a b Achia'. In Fig. 31 let AB represent any process whatever 6 c FIG. 31. a 9 which in order to be perfectly general is assumed to start with superheated and to end with saturated vapor. Draw through A the constant volume curve SUPERHEATED VAPORS. 77 v and through B the isodynamic E B , intersecting V A at C. Then the heat rejected during the change AB is represented by the area ABba, the decrease in internal energy, E A E B , by area ADCca, and the heat equiva- lent of the work performed upon the substance by area ABbcCDA. To obtain the external work during any change measure the area under the curve which represents the change with a planimeter. Then note by reference to the isodynamic lines the initial and final values o'f the internal energy. Then AW=QAAE. CHAPTER V. THE TEMPERATURE-ENTROPY DIAGRAM FOR THE FLOW OF FLUIDS. IN dealing with fluids in motion when the velocity of any given mass varies from moment to moment it becomes necessary to introduce a term for the kinetic energy of translation into the mathematical expression for the conservation of energy. Suppose an expansible fluid to be flowing continually through a conduit of varying cross-section (Fig. 32). FIG. 32. If no external work is performed by the fluid other than to push itself forward we know that all the energy carried out across section 2 must be equal to that brought in across section 1 plus the heat received by the fluid in its passage from 1 to 2. If further we assume the conduit to be constructed of a perfect non- 78 FLOW OF FLUIDS. 79 conductor of heat then the process is adiabatic and there must exist the following energy balance per pound, The increase in kinetic energy between the two sections is therefore If the process is assumed to be frictionless and there- fore in the thermodynamic sense reversible (see p. xi of Introduction) the expansion occurs at constant entropy. That is, the specific volume, temperature and internal energy are found for any given pressure from the equation for frictionless adiabatic expansion, pv n = constant, where n = k= 1.405 for air and other diatomic gases; n= 1.035 + 0.1002 for wet steam (x= initial quality) ; n= 1.3 to 1.33 for superheated steam. If then the pressure and volume at section 1 are represented by point 1 in Fig. 33, it follows that the specific volume for any lower pressure may be found upon the constant entropy curve through 1. Thus for pressure p 2 the volume is v 2 . This diagram thus enables 80 THE TEMPERATURE-ENTROPY DIAGRAM. us to give a graphical interpretation to the expression for the increase in kinetic energy between 1 and 2. The first term piVi is represented by the rectangle 01, the second term by the rectangle 02. The last two terms together measure the decrease in internal energy between sections 1 and 2 and are represented by the FIG. 33. work performed during this expansion, even if expended in producing self-acceleration. Therefore EiE 2 =J pdv is represented by the area under the curve 12. The algebraic sum of these different areas shows that the increase in kinetic energy during frictionless, adiabatic flow is represented by the area between the expansion line and the pressure axis, or 72 FLOW OF FLUIDS. 81 This discussion, which up to this point has been per- fectly general, must now be carried on independently for each substance. Perfect Gases. If it is possible to project each point of the pv-plane uniquely into the T^-plane it follows that a given area in the one plane can be transferred to the other by simply determining its boundary curves FIG. 34. in the two planes. The area in the T^-plane will then be the heat equivalent of the corresponding area in the pu-plane. Let 12, Fig. 34, be the TV-projection of curve 12 in Fig. 33. The constant-pressure curves through 1 and 2 will intersect at infinity with the curve of zero volume so that the cross-hatched area will represent the heat equivalent of the increase of kinetic energy. A slight transformation of the above formula makes 82 THE TEMPERATURE-ENTROPY DIAGRAM. it possible to replace the infinite diagram by an equiva- lent finite one (Fig. 35). Thus from V 2 it follows that V* V 2 A-A-z- is thus represented by the area under the constant pressure curve between the upper and lower FIG. 35. temperature levels. Thus in expanding from 1 to 2 the heat equivalent of the increase in kinetic energy is shown by the area under 12'. If the expansion is con- tinued to 3 the further increase in kinetic energy is shown under 2'3', while the total amount is shown under 13'. The same result might have been obtained from Fig. 34 by remembering that all the constant- FLOW OF FLUIDS. 83 pressure curves for a perfect gas have exactly the same form, so that the curve 2'oo is an exact reproduction of 2 oo , and therefore the total area under 1 minus that under 2 leaves the area under 12'. An interesting comparison between the flow of a perfect gas and the velocity of a freely falling body can be made from this method of presentation. Imagine the expansion to continue until the final pressure becomes zero, when the volume will be infinite. The kinetic energy will then equal c p Ti. This represents the total energy existing in space at 1 due to the presence of the pound of gas, i.e., suppose one pound of air at zero temperature could have been inserted into this space at 1 under pressure pi and then heated to TI, the total energy thus intreduced would be c p -T\. If then we represent this total heat by H, we notice that the increase in kinetic energy is equal to the decrease in total heat or ~ = 778-4H, whence V 2 = V2g 778 AH if 7i = 0. It is thus seen that the "total heat" head (Erzeu- gungswarme) plays the same role in the acceleration of fluid flow that gravitational head does in the case of a freely falling body. Saturated and Superheated Vapors. Before project- ing into the !T<-plane draw in the liquid (a) and dry vapor (s) lines (Fig. 36). It is then seen that the 84 THE TEMPERATURE-ENTROPY DIAGRAM. expansion line may be wholly in (1) the saturated, or (2) the superheated region, or (3) may pass from the s \ d a e\A A, FlG. 36. superheated to the saturated region. That part of the diagram abed bounded by the liquid line, the adiabatic FLOW QF FLUIDS. 85 expansion, and the two pressure curves, may be at once located in the T(f>-p\ane as indicated in Fig. 37. The little rectangle to the left of the liquid line is retained in the pt'-plane. Area abed in the 7 7 ^>-plane is seen to be equal to HiH 2 , and area klmn, expressed in heat units is Z 2 . Therefore 77 i~ 778 Now H + +E 32 o is the total energy (Erzeugungs- warme) existing in any space due to the presence of the pound of substance under these conditions. The difference in this total space energy differs by the amount ^ Pl ~o from the difference of total heats (#i-# 2 ). 7/o Thus if 7i = V 2 = V2g[778(H 1 - H 2 ) + ( Pl - p 2 }o\ Here again the " total energy " plays the same role as it does with the perfect gases, and thus similar to that played by gravitational heacLin the case of a freely falling body. We can now at once write out the formula required for each of the three special cases. THE TEMPERATURE-ENTROPY DIAGRAM. V 2 (1) A-4- = (2) = qi + r l +c p (T Sl -T 1 )-q 2 -r 2 -c p (T S2 -T 2 )+A(p l -p 2 )a; (3) For the sake of those who may prefer analytical methods to graphical ones the ordinary manner of deriving these formulae will be given next. Starting from the fundamental equation, 72 E 2 , we must substitute in each case the special values for volume and internal energy and then collect terms . V 2 (1) AJ K- = Api (xiUi + a) -Ap 2 (x 2 u 2 + a)+ qi+Xipiq 2 V 2 (2) A4-- -q 2 - X 2 p 2 - x 2 Ap 2 u 2 Sl - T l )-q 2 -p 2 -Ap 2 u 2 -c P (T S2 -T 2 )+A(p l -p 2 )a. FLOW OF FLUIDS. 87 72 (3) A4- ?2 - ^2|02 + Cp(T Sl - TI) - Api (Vi - Si) = qi + pi + ApiUi + c p (T Sl - TI) q 2 - X 2 p 2 If we consider the flow through such conduit or nozzle to be part of a continuous cycle of operations, it becomes necessary to return the water to the boiler, that is, to pump it back again from p 2 to p\, the work required being A(p l p 2 )ff. Hence if this term be subtracted the difference will represent that part of the increase in kinetic energy which is available for external work, such as driving the rotor of a turbine. Thus, V 2 Design of a Turbine Nozzle. Suppose now we wish to design a nozzle to permit the flow of G pounds of steam per second. At any cross-section of area F the necessary and sufficient condition for continuity of flow is that - - For any pressure p x there can be only one definite value for velocity and specific volume, viz., V x and v x , and 88 THE TEMPERATURE-ENTROPY DIAGRAM. hence a unique value of the area F x . The value of v x may be read directly from the constant-volume curves; the value of V x must either be computed from the above formulie or else the area cdAB measured by planimeter and V x computed from that. In Fig. 38 let a'b' be the /^-projection of the fric- tionless adiabatic flow ab. Let MN represent the rela- tive variations of specific volume and velocity during this expansion. Draw the- line yy parallel to Ov and make Oy equal to G. Then at any point of the expan- sion as d, the volume v^ = md r and the velocity V 4 =nd", FLOW OF FLUIDS. 89 Draw Od" and prolong if necessary until it intersects yy in k. From the similar triangles Oyk and Od"n it follows that yk :yO = On : nd" yk = y0 .^,=G.^=F. . Conversely to find the pressure which would be obtained at any cross-section, lay off yk = F, draw W > and prolong until it intersects MN in d", from d" drop the perpendicular d"n and project d r back to d. The smallest cross-section, or the throat of the nozzle, will be reached when Ok is tangent to MN, viz., Ok', giving the value F throat = yk'. As Ok cuts the vF-curve between K and N the value of F increases rapidly until at N it becomes infi- nite, which agrees with the initial assumption that F a = 0. It is to be noticed that the throat is reached when the pressure has dropped to about 0.58 p a . From this point on the nozzle flares indefinitely as long as the back pressure is dropped. Constant Heat Curves. The frequency with which V 2 the expression Ad -^- = AH -\- A(pi - p 2 }o must be t7 evaluated in turbine design, and the inconvenience of solving for the final quality by equating the entropies in order to obtain H 2 has made it advisable to plot total-energy curves, H + Apa= constant; In the range 90 THE TEMPERATURE-ENTROPY DIAGRAM. of conditions usually met with the term Apa is negli- gible, so that the total-energy curves practically coincide with the constant heat curves. Care must be taken, however, in cases of extremely high pressures and wet steam to be sure that A pa is really negligible. Thus at the upper pressure quoted in Peabody's Steam Tables for hot water, 336 X 144 X. 016 #=2 = 402.2 B.T.U. and Apo= === = 0.995 B.T.U., or H differs from H+Apa by 0.25 per cent, approxi- mately. Thus we have, in general, the following relation existing between any two points of such a curve H + Apa = qi + Xir i + Ap^a = q 2 + r 2 + c p (T s - T 2 } + Ap 2 a, while for all ordinary conditions reduces to To plot the curve in the saturated region a series of values for x from x = - - must be computed for a sufficient number of different temperatures. In the superheated region we must determine the points of intersection of the desired constant-heat curve with several constant-pressure curves (Fig. 39), by means of the relation FLOW OF FLUIDS 91 As c p is a variable a few trials may be necessary before the correct value of T s is obtained. T s once known, the point can be at once located upon the corresponding constant-pressure curve. The constant-heat curves once located on the dia- 0.2 0..4 0.6 0.8 1.0 1.2 1.4 FIG. 39. 1.6 1.8 2.0 2.2 2.4 gram the solution of nozzle problems becomes very simple. Given any reversible adiabatic expansion, as 72 AB, Fig. 39, the value of ^rp obtained by reading the values of H A and H B directly from the total heat curves. 92 THE TEMPERATURE-ENTROPY DIAGRAM. Peabody's Temperature-entropy Tables. It is at this point that the practical value and great convenience of Peabody's Entropy Tables become manifest. These tables cover that portion of the diagram (Fig. 40) 1.52 1.88 2.18 FIG. 40. between V A . The real significance of the change becomes apparent if we bear in mind Lord Kelvin's statement of the second law of thermodynamics that "it is impossible by means of inanimate material agency to derive mechanical effort from any portion of matter by cool- ing it below the temperature of the coldest of surround- ing objects." Let thz dotted line (Figs. 41 and 42) represent the lowest available temperature, i.e., that of the atmos- 98 THE TEMPERATURE-ENTROPY DIAGRAM. phere in gas-engine work. The minimum available pressure in actual work would likewise be that of the atmosphere. FIG. 42. The work developed during isentropic expansion to atmospheric pressure would be and And since PAVA = PBVB, p a = pb and PA>PB it follows that W Bb < WAa by the amount Therefore k-l\p B / k-l\p FLOW OF FLUIDS. 99 Furthermore, while the work done upon the substance as it enters the cylinder or nozzle is the same (PAVA = PBVB), the work required to exhaust it has increased by the amount The total loss in power resulting from the irreversible operation AB is therefore C p jfc-l/ l-fe A. atm. \ In other words, although the temperature was not changed and although no heat entered or left the body during the change, AB, nevertheless, because of it, heat to the amount c p (TbT a } has been made non-available for actual work. Theoretically the loss in availability is not quite so great. It is possible to conceive of the expansion being carried below the back pressure until the lowest avail- able temperature is reached, i.e., from b to c. Then on the return stroke the charge could be isothcrmally compressed from c to a so that the extra work bca could be gained and the corresponding heat loss avoided (see Fig. 41). The total heat made theoretically non- available would therefore bs represented by the area under ac, i.e., by TJ^^^B^A). 100 THE TEMPERATURE-ENTROPY DIAGRAM. Suppose as a further illustration that the operation AB were introduced into a cycle in which all the other operations were reversible isothermals and adiabatics (Fig. 43). The heat received from some outside source is shown by the area under the reversible isothermal CA The heat rejected is that shown by the area under the reversible isothermal be, while that rejected when the irreversible process AB is eliminated is shown by the area under ac. The heat exhausted during the cycle c' a' FIG. 43. has thus been increased by the amount abb' a', which is equal to the temperature of exhaust multiplied by the increase in entropy during the irreversible process AB. Swinburne generalizes this result in the following words : " The increase of entropy multiplied by the lowest temperature available gives the energy that either has been already irrevocably degraded into heat during the change in question, or must, at least, be degraded into heat in bringing the working substance back to the standard state. ..." (See p. xiii of Introduction.) Transmission of Compressed Air Through Pipes. Air delivered by a compressor is heated above the tern- FLOW OF FLUIDS. 101 perature of the atmosphere. This excess of tempera- ture is soon lost by radiation in the pipe line and from that point on the flow is practically isothermal. There are, however, friction losses which result in a drop of pressure throughout the line. Both of these processes are irreversible and both produce loss of power the first by a direct radiation of heat, the second by a re- duction in availability of the energy remaining in the air. In Fig. 44 let ab represent the cooling at practically constant pressure in the first part of the pipe line, and be represent the irreversible adiabatic expansion caused by friction. If the air operates a motor it cannot be expanded below atmospheric pressure. Then adhf represents the work which the air as delivered by the compressor could have developed during frictionless adiabatic expansion, while cehg represents the work which the air as finally delivered at the motor is capable of producing during frictionless adiabatic expansion. 102 THE TEMPERATURE-ENTROPY DIAGRAM. The total loss of power is thus shown by the area adecgfa. A more complete analysis of this problem will be given in the chapter on air-compressors. Irreversible Adiabatic Expansion of Saturated and Super- heated Vapors. In the case of saturated and super- heated vapors the condition for completely irreversible flow pv + E = constant reduces to H+Apa = constant. The irreversible adiabatic therefore coincides with the constant total-energy curves or approximately with the curve of constant total heat. Such a process occurs whenever steam pressure is lowered through a reducing valve, when the pressure drops as steam passes through the admission and exhaust ports of an engine, or when the pressure drops throughout the length of a pipe line. From its direct application in reducing valves this process is technically known as throttling and the constant total-energy curves are sometimes spoken of as throttling curves (Drosselkurven). Here, as in perfect gases, the adiabatic process is found to be indefinite unless the law of friction loss is specified. When there is no friction and part of the total energy goes into work or kinetic energy it coincides with an isentropic process. When the friction loss is complete so that no acceleration occurs it coincides with a throttling curve. In cases where the friction loss is only partial, as in turbine nozzles, it occupies some intermediate position. It should be noticed that during throttling of satu- rated steam the moisture tends to evaporate and that FLOW OF FLUIDS. 103 if the steam is nearly dry it may even become super- heated during the process. We have found that for perfect gases the throttling curve represents an iso- thermal process; therefore, the further the throttling curves extend into the superheated region the more nearly horizontal they become, approaching the iso- thermal line as a limiting case. Loss of Availability due to Throttling of Steam. Suppose a pound of steam to have undergone an irre- FIG. 45. versible change of condition AB, Fig. 45. In order to restore the steam to its initial condition it must be compressed and heat must be rejected. If it were com- pressed adiabatically from BtoC and then isothermally from C to A there would be rejected an amount of heat equal to the area under AC. Evidently less heat need be rejected if some path falling inside of BCA 104 THE TEMPERATURE-ENTROPY DIAGRAM. were utilized. Evidently, BB'A'A is the path along which the minimum amount of heat would be exhausted" when B'A' represents the lowest available temperature. There would be no advantage in continuing the adia- batic expansion BB' below B', because the steam could not be compressed isothermally at any lower tempera- ture than 2V, i.e., before heat could flow out it would need to be compressed back to B' again adiabatically. The heat which is unavoidably lost in restoring the steam from B to A is therefore equal to the lowest available temperature multiplied by the growth of entropy during the change AB. Problem 1. A throttling- valve reduces steam pres- sure from 150 Ibs. gauge to 80 Ibs. gauge. The steam initially contained 1 per cent, moisture. If the steam is used by an engine running at 2 Ibs. absolute back pressure, find the loss per pound of steam caused by the valve. We have (omitting Apa), ?i64.7-f 0.99r 164 . 7 = g 94 . 7 + r 94>7 or 337.8 +0.99X856.9 =294.4 +890.7 + c P ^-323.9), whence c p (t 8 -323.9) = 1.0 B. From the table of values of c p we obtain 325.7 as the value of t, which satisfies this equation. FLOW OF FLUIDS. 105 Therefore, = 0.5235 + 0.99X1.0381 -1.5513 T T and 2 = 02 + r + ei>loe r 70 r = 0.4696 + 1. 1369 + . 56 X 2.303 Xlogiy /oo = 1.6079. The temperature corresponding to 2 IDS. aosolute is 126.2 F. The loss of available energy caused by the reducing- valve per pound of steam is therefore equal to T(2-i}= (126.2 + 459.5) (1.6079-1.5513) = 33.2 B.T.U. Problem 2. A line of pipe delivers Wi pounds of steam per hour. By means of traps and a separator placed jiist above the throttle w 2 pounds of water are removed per hour. The steam is thus practically dry at both ends of the pipe. The pressure drop from boiler to throttle is pip2 pounds. The steam-engine, using the Wi pounds, runs with p 3 pounds back pressure. Find the total waste produced by transmission through the pipe. Assume that the hot water is returned to the boiler 100 THE TEMPERATURE-ENTROPY DIAGRAM. at the throttle temperature. The radiation loss (see Fig. 46) therefore equals w 2 (H 1 -q 2 ) + w 1 (H 1 -H 2 ') B.T.U. per hour. Besides this direct loss of heat by radiation there is a further indirect loss due to the increase in entropy of the main body of steam. This represents the extra *, J- U T p, / \ / 4- 7 \ "/ \ /. yt r -?* \ v*,-^ / t __ \ (-) Fiu. 40. heat given up to the cooling water of the condenser during exhaust and equals -^i) B.T.U. per hour. The total loss due to pipe line is -32) +wi (H^ H 2 ) +wi - T 3 (fa- ', and E r on the dry saturated curve these differences : H A -H B ,=c p (T B ~T B ,} H A -H c ,=c p (T c -T c ,} H A -H D ,=c p (T D -T D ,\. * Peake, Proc. Roy. Soc. (London), Series A, vol. 76 (1905), pp. 185-205. 118 THE TEMPERATURE-ENTROPY DIAGRAM. The least accurate data in these, as with all other experiments with steam, are at the point A, due to the uncertainty as to its condition. The point A may, however, be eliminated from the observations by com- bining the above equations as follows : H B ,-H c ,=c p (T c -T C '}-c p (T B -TV), H B > H D > = c p (T D TV) c p (T B 7V)> H B , -H E ^c p (T E -T E> ] - Cp (T B - TV), etc. Plotting AR and T as rectangular coordinates, let the line H B represent the total heat of the throttling HE' FIG. 496. curve AE, while the vertical lines BE', CC', DD', etc., represent the heat of superheat at the respective con- ditions B, C, D, E, and then the points B', C', D' ', and E' will represent the total heat of dry steam at the respective temperatures. This avoids the necessity for the determination of the temperature of point A. (Fig. 496.) B' } C f , D', E' thus represents a portion of. the dry FLOW OF FLUIDS. 119 saturated steam curve with the temperature of the points determined absolutely while the total heats are determined only relatively as the value of H A and therefore of H B is unknown. Grindley's data gave seven fragments of the saturated curve, Griessman's data eleven and Peake's six. By skillful combination of these twenty-four fragments Davis obtained the curve showing the most probable relation between the total heat and the temperature of saturated steam from -212 F to ^ = 400 F. This curve is represented within the limits of experi- mental accuracy by the second degree equation, ff = ff 212 + 0.3745(^-212) +0.000550(f-212) 2 . For #212 Davis takes 115C.3 B.T.U., this being the mean of the values obtained by Henning* and Jolyf. Peabody, however, uses 1150 B.T.U., which is the result obtained from Henning's formula. Flow through a Nozzle. In the case of the flow through an actual nozzle the operation is not rever- sible. Heat is lost by radiation; heat is conducted through the metal of the nozzle from the higher to the lower temperatures; and friction occurs in varying amounts hi different parts of the nozzle. The first loss, the rejection of heat as heat, decreases the entropy * Wied. Ann., IV, vol. 21 (1906), pp. 849-78. t In an appendix (p. 322) to paper by Griffiths, Phil. Trans. (London), vol. 186 (1895), pp. 261 et seq. 120 THE TEMPERATURE-ENTROPY DIAGRAM. of the fluid, while the other two losses both increase its entropy. It might happen that these opposing forces just balanced and then the expansion would be isentropic but not reversible. In general, however, the radiation loss may be made small, so that the opera- tion is nearly adiabatic, but with increasing entropy FIG. 50. due to conduction along the nozzle and to friction Thus starting from a, Fig. 50, the actual expansion curve will lie between the isentrope ab and the con- stant-heat curve ab f in some such position as ac. The heat theoretically available is represented by abde. By friction, etc., the portion 6cc 1 6 1 has been returned to the substance at a lower temperature, hence the kinetic energy of the jet at the exit c is equal to area abde area FLOW OF FLUIDS. 121 This loss would make itself noticeable in two ways. Decreased kinetic energy means decreased velocity, and increased entropy means increased volume. That is, if a nozzle were constructed from the dimensions necessary to give frictionless adiabatic flow and drilled at different points so as to measure the pressure, the observed pressure would be found to be greater at any given cross-section than the pressure for frictionless flow. Stodola's experiments show this, and also that the loss is at first slight, being practically negligible down to the throat, but increasing from there onward more and more rapidly as the velocity increases. That is, the curve ac would at first closely approximate ab, but lower down branch off more and more toward the right. As soon as the curve ac has been accurately located it can be projected into the pv-pl&ne and the corre- sponding areas for different cross-sections of the nozzle determined in the manner already indicated for the ideal case of frictionless flow. Adiabatic Expansion with Partial Friction Loss. The better to understand the phenomena of flow in a nozzle let us discuss the flow of steam in a non-con- ducting nozzle in which, however, there is the usual friction loss. Let ab (Fig. 51) represent an isentropic expansion, ai a constant-energy expansion, and ac an adiabatic expansion with some friction loss, all from the same 122 THE TEMPERATURE-ENTROPY DIAGRAM. initial condition a, at pressure pi, clown to the same back pressure p%. At 6 the kinetic energy is greater than that at a by the amount abgk, at i the kinetic FIG. 51. energy is the same as at a, and at c it possesses a value intermediate between that at b and i. Thus and A- n a nc, whence the loss of kinetic energy caused by the friction along ac is shown by the difference, or Vf-Vf A ^ =H c Hb. FLOW OF FLUIDS. 123 That is, the heat equivalent of the loss of kinetic energy is thus represented by the difference in the total heat FIG. 52. for the two final conditions, or by the area under the curve be in the 7^-diagram, Fig. 52. If a constant-heat curve be drawn through c until it intersects the isentrope ab in d we obtain Hd Hb = H c Hb, so that the loss of kinetic energy will also be shown by the areas bdef plus efgh (Figs. 51 and 52). Thus in place of utilizing the total available drop H a Hb, as in the case of isentropic flow, the actual expansion ac has only utilized that portion H a H d which would have been utilized during an isentropic expansion from a to d. As the friction loss increases the expansion line ac 124 THE TEMPERATURE-ENTROPY DIAGRAM. would move more and more to the right, so that the utilized heat H a H d would grow continually smaller, and finally in the limiting case where "the friction loss is complete ac coincides with ai and H a H d reduces to zero. If we analyze closely such an irreversible adiabatic expansion as ac, we notice that it is made up of in- finitely small transformations of heat into kinetic energy and of kinetic energy back into heat, and, taken in that sense, we see that the total area under ac can be inter- preted to mean the heat equivalent of the friction loss. But we have already found that the loss of kinetic energy is shown by the area under be, so the total fric- tion loss exceeds the loss of kinetic energy by the amount represented by area abc. In other words, all of the friction loss is not irretrievable, because although the friction loss has occurred the energy has nevertheless been returned as heat at lower temperature, and is therefore still capable of being partially turned back into work. That is, at any intermediate point on the line ac the total heat is greater than that on the line ab for the corresponding pressure, and therefore more work is theoretically obtainable from it. The efficiency of the excess heat is, however, not as great at this lower temperature as it was originally, but its availability is not wholly destroyed until the expansion is carried down to the lowest possible temperature. Design of a Nozzle for Actual Flow. The design of FLOW OF FLUIDS. 125 a nozzle for actual conditions is fundamentally the same as the case discussed on pp. 87, 88, except that the law of friction loss is now assumed known and pro- vision made for the resulting decrease in velocity and increase in specific volume. Thus if the friction loss at any given section is / per cent, the available kinetic energy at that section is only 1 r^ parts of that available for frictionless flow, or whence the actual velocity is -(Hi-H 2 ) ft. per second. The kinetic energy lost by friction is restored as part of the total heat and results simply in improving the quality of the steam. Thus if the quality were x 2 ' as ffTT _ fj \ the result of isentropic expansion, the heat * J-UU would by further evaporation cause an increase in xy of the amount, , f(H 1 -H 2 ] 100r 2 so that the actual value of the quality would be 120 THE TEMPERATURE-ENTROPY DIAGRAM The actual specific volume would then be v 2 = 0.016 +x 2 (s 2 - 0.016). In case the steam were superheated at the end of the -T(TJ _ TJ \ isentropic expansion the heat would produce 1UU an increase in superheat of the amount, f(H l -H 2 } 100c p ' and the final temperature would be The final volume V2 could then be found from the equation pv=85.85T-0.25Qp, or from the entropy tables. If the weight passing the section per second is G pounds the cross-sectional area is given by The value of G is determined by knowing the power which the nozzle must develop and the total friction loss at the exit section. Thus for n horse-power de- FLOW OF FLUID. 127 livered by the jet and// as the total percentage friction loss at the final section, and rcX550 nX550 Or = Problem. Find the throat and final diameters of a nozzle to develop 10 H.P. in the issuing jet, assuming a friction loss of 3 per cent, and 20 per cent, at these sections respectively. The steam is initially at 150 Ibs. absolute pressure and superheated 100 F., and the back pressure is 4 Ibs. absolute. The actual kinetic energy of the issuing jet per pound is ^-==778X0.80X(1249.6-987.1) = 163,200 ft.-lbs. whence 7 /= \/G4.32X 163,200 = 3240 ft. per second. and irv v- Ken or equals 0.03370X3600 10 - P er secon(l = 12.13 lbs. per H.P. per hour. 128 THE TEMPERATURE-ENTROPY DIAGRAM. The value of ay from tne tables is 0.8617 and the increase in x/ due to friction loss is and the actual final quality is ay= 0.8617 + 0.0522 = 0.9139. The final specific volume is v/= 0.016 + 0.914 (90.4 - .02) -82.7cu.ft. The final cross-sectional area is therefore 0.03370X82.7 F '= -^240- ~ = 0. 1239 sq. ins. The final diameter is therefore 0.3972 inch. The theory of the flow of fluids shows that for isen- tropic expansion the pressure in the throat of the nozzle is about 0.54 of the initial pressure for super- heated steam and about 0.58 of the initial pressure for saturated steam. In case the expansion passed from superheated to saturated steam before the throat was reached the only available method of determining the throat pressure would be to plot the ratio for several values of p 2 and thus determine the pressure corre- y spending to the maximum value of FLOW OF FLUIDS. 129 In this problem the. steam is still superheated at the throat, so that the throat pressure is about 0.54Xl50=811bs. abs. The actual kinetic energy of the jet at the throat per pound is 7.2 ^- = 778 X 0.97(1249.6 - 1 194.5) = 41590 ft.-lbs. whence V,= V64.32X 41590 = 1635 ft. per second. The superheat at the end of isentropic expansion amounts to 23.6 F. This is further increased by the friction loss by the amount 0.03X55.1 1.653 A 1 1 = , C P C P To solve this we neyd to know the momentary value of c P , but not having that we notice in the entropy tables that the total heat at this point increases 7.8 B.T.U. for 14.7 superheating. This gives _ 7.8 Cp ~14.7 whence 130 THE TEMPERATURE-ENTROPY DIAGRAM. The actual superheat at the throat = 26. 6 F. By cross interpolation in the tables the corresponding specific volume is found as v t = ^ (5.678 - 5.550) + 5.609 = 5.636 cu. ft. /.o The cross-sectional area at the throat is ,-, 0.03370X5.636 *'- -1635 Xl44 = 0.01673 sq. ins. The throat diameter is therefore 0.1460 ins. Assuming the flare between throat and exit to be a uniform taper of one in ten the length of the nozzle beyond the throat would be 10(0.3972-0.1460) = 2.51 inch. CHAPTER VI. MOLLIER'S TOTAL ENERGY-ENTROPY DIAGRAM. THE important role played by the total energy, i=H + Apa=E+pv, of steam in the discussion of the phenomena of flow led Mollier * to construct a diagram using i and $ as the coordinates. In that portion of the diagram abed required for turbine nozzles Apo is negligible, so that i=H practically, and the diagram is thus sometimes called the "total heat-entropy dia- gram." Description. The general character of the plot and its relation to the ^-diagram are shown in Fig. 53, where for convenience in projection from the T$- into the i<-plot, or the reverse, the pT- and pz-quadrants are also given. Plot first, q + Apo and 6, and q + r + Apa and 6 + 7F, to obtain the water (w) and dry steam (s) lines. The isothermals representing vaporization at constant pressure are next obtained by laying off the heat of the liquid and the total heat of dry steam, * Neue Diagramme zur technischen Warmelehre, von Prof. Dr. R. Mollier, Dresden. Zeitsch. d. Ver. Deutsch. Ing., Bd. 48 S. 271-274. 131 132 THE TEMPERATURE-ENTROPY DIAGRAM. H w and H a , for the desired pressure, as at a and 6 on the #-axis, finding the corresponding points on the water and steam lines, a and b, and connecting these points by the straight line oh. The same line could MOLLIER'S TOTAL ENERGY-ENTROPY DIAGRAM. 133 of course be determined graphically by projecting upwards the points a and 6 from the T^-plot. The " x-lines " or quality-lines are constructed by sub- dividing the constant-pressure lines between the water and steam lines into any desired number of equal parts, as tenths or hundredths, and connecting . the corresponding divisions on the different lines by smooth curves. To continue the constant-pressure curves beyond the dry-steam line into the superheated region we plot for each pressure the set of values or, starting from the end of any pressure curve at the dry-steam line, lay off the extra values AH = c p (T s -T) and ^ = c p log e ^. The accuracy of this part of the diagram is of course limited by our incomplete knowledge of the laws of variation of c p for superheated steam. Thus the plots in Stodola's Steam Turbines are based upon Regnault's old value, c p = 0.48, while those in Thomas' Steam Turbines are based upon the larger but still constant value, c p = 0.58. To plot the isothermals in the superheated region, start with the temperature corresponding to any pres- sure A on the dry-steam line, then the heat required to 134 THE TEMPERATURE-ENTROPY DIAGRAM. superheat at constant pressure to this same temper- ture from any lower pressure will be c p (Ta T c ) and the point B may then be located on the curve p = c by laying off c p (T A -T p } B.T.U. above the intersection of p = c with the dry-steam line (Fig. 54). The iso- FIG. 54. thermals, provided the 7^-plane is already drawn, may be located graphically before the constant-pressure curves are drawn by projecting the value of < for each value of C P (TBTA) directly from the T-p\a,ne. Reversible Adiabatic Processes. For frictionless adia- batic flow between the two pressures pi and p 2 the kinetic energy of a jet increases by the amount A- Enter the plot at the point p\x\ or piTi, according as the steam is saturated or superheated, and follow down MOLLIER'S 'I'OTAL ENERGY -ENTROPY DIAGRAM. 135 the isentrope thus determined until the constant- pressure line p 2 is reached. The vertical distance between the initial and final points gives at once i\ ii, or the heat equivalent of the kinetic energy of the jet. Assuming the initial velocity to be zero, the final velocity corresponding to any pressure, is given by fJ.A--223.7V32; from which we obtain the following table : M (in B.T.U.) VJl (in ft. per sec.) 0.01 0.1 22.37 0.04 0.2 44.74 0.09 0.3 67.1 0.16 0.4 89.5 0.25 0.5 111.9 0.36 0.6 134.2 0.49 0.7 156.6 0.64 0.8 179.0 0.81 0.9 201.3 1. 1. 223.7 4. 2. 447.4 9. 3. 671. 25. 5. 1119. 64. 8. 1790. 100. 10. 2237. 144. 12. 2684. 225. 15. 3355. 324. 18. 4026. 400. 20. 4474. 441. 21. 4698. 484. 22. 4921. 625. 25. 5593. 136 THE TEMPERATURE-ENTROPY DIAGRAM. An auxiliary plot (Fig. 55) using Ji as ordinates and V as abscissae permits the determination of interme- diate values, and forms a valuable addition not only to the i'0-plot but to the entropy-tables as well. 400 300 100 GOO 6000 Velocity in feet per second FIG. 55. Two plots are advisable, drawn to different scales, one to be used for small values of M, the other for large values. If we project the values of the velocity on to the J{-axis, as indicated in Fig. 55, we obtain two scales, MOLLIER'S TOTAL ENERGY-ENTROPY DIAGRAM. 137 * I 138 THE TEMPERATURE-ENTROPY DIAGRAM. one showing Ji, the other V, both measured from the same origin. If a sufficient number of intermediate values of V are indicated the scale may be plotted on a separate piece of paper, and can then be applied directly to the i^-diagram by placing the scale parallel to the isentropic curves with its zero at the initial point of the expansion. The position of the final pressure line on the scale will indicate the velocity directly in feet per second. Reduction of Pressure by Throttling. Throttling curves represent processes during which the total energy remains constant, and these are therefore represented by horizontal lines in the i-p\ot. Such a plot permits of the ready determination of the quality of steam from the calorimeter observations. Thus if A (Fig. 56) is the point determined by the pressure p c and tempera- ture t c in the calorimeter, proceed from A horizontally to the left until the curve IA intersects the constant- pressure curve corresponding to the boiler pressure PB, say at B. The position of B with reference to the x lines gives the desired quality XB. CHAPTER VII. THE TEMPERATURE-ENTROPY DIAGRAM FOR MIX- TURES (1) OF GASES, (2) OF GASES AND VAPORS, AND (3) OF VAPORS. Mixture of Perfect Gases. If several gases be mixed in the same vessel, the pressure cf the mixture is equal to the sum of the pressures which the gases would exert if they occupied the whole space separately. This result discovered experimentally by Dalton is true of course only so long as the molecules do not sensibly obstruct each other. It may be assumed to hold rigidly in the ideal case of perfect gases. For any given mixture undergoing reversible opera- tions it must also be assumed that heat interchanges occur instantaneously, so that during any change what- ever the pressure and temperature at any given instant are always uniform throughout. The common tem- perature may be measured directly, and if the weights of the different gases confined in the given space are known the specific volumes may be calculated. From these two quantities the respective specific pressures follow at once from the characteristic equations. If the mixture is composed of n constituents possessing the weights mi ... m n , the specific pressures p\ . . . p, 140 THE TEMPERATURE-ENTROPY DIAGRAM. the specific heats c Vl . . . c Vn , and c Pl . . .c Pn , respectively, the values of the specific heats of the mixture are given by and c p = and the specific pressure by p=pi+p 2 + . . . + pn. To apply the 7^-analysis to any process occurring in such a mixture it is sufficient to treat it as a simple gas possessing the specific heats c p and c v and obeying the law pv = RT, where R is determined from some known values of temperature and volume. This is exemplified in gas-engine work, and even in air-compressor work: although air is always treated as a unit and its con- stituents never considered. Mixture of Gases and Vapors. Experiment has again shown where a given gas and liquid are chemically inactive and where the gas is not physically absorbed by the liquid, that when contained in the same vessel the liquid evaporates as if in a vacuum, and the pressure of the vapor is the same whether there is gas in the vessel or not. Common examples of such mixtures of interest to engineers are to be found in the air-pumps of steam-engine condensers and in compressors pumping moist air or those cooled by water injection. MIXTURES OF GASES AND VAPORS. . 141 Let there be w pounds of a gas per pound of satu- rated vapor in a given mixture and let it be desired to trace the relative changes between the two constituents as the mixture undergoes various definite changes. From equation (8), page 15, we have as the change in entropy of w pounds of a perfect gas in going from any condition to any other condition the expression, A$ g = WCp log e ^T-w(Cp Cv) log e Also the change in entropy between any two points in the region of saturated vapor is evidently equal to the difference in the total entropies of the two points, or fi v rT v\ -- m~ i 2 1 1 Hence the total entropy change of the mixture is given by the sum Pit A thermometer will give the common temperatures TI and T 2 and the aid of suitable tables (steam, ammo- nia, etc.) determines the values of the various heat quantities referring to the vapor. Subtracting from the gauge readings the vapor-pressures corresponding to 142 THE TEMPERATURE-ENTROPY DIAGRAM. the observed temperatures from the tables leaves the gas pressures p gi and p g2 . There exist also the further relations between the volumes of the gas and vapor, whence wv gi a x 2 U 2 RT 1 RT 2 W ---- a w --- a U 2 Substituting these values of Xi and x 2 in the expression for 4(j) M gives + WCp log e 7fr-w(c p - C v ] log e * 1 Poi In this equation provided it is 4 possible to measure pressure and temperature there are but two un- knowns 4(j>M an( l w- Hence if w be known J g , and A<}>M, respectively. Lay off P Ml and P Mt equal to the initial and final gage readings, then Aa f and Bb f will represent the pressure MIXTURES OF GASES AND VAPORS, 145 of the gas and AB referred to a'~b f as the axis of volume will represent the pv-curve of the gas, while referred to Ov as the zero pressure line will represent the pu-curve of the mixture. FIG. 57. (2) Heating or Cooling at Constant Volume. This introduces the special conditions, , and RT l RT 2 - = w Poi Po-i By substitution of these values, the general expres- sion for the entropy of a mixture reduces to T 2 a e ^ 146 THE TEMPERATURE-ENTROPY DIAGRAM. The heat received during this process is given by The change of internal energy is of course equal to the heat added. Here again the value of w must be determined by suitable measurements before the expressions can be evaluated. (3) Isentropic Expansion or Compression If the mixture undergoes frictionless adiabatic expansion it is evident that this might occur in one of two ways: either (1) the entropies of both the vapor and the gas remain constant, or (2) if the entropy of the one varies a given amount that of the other varies an equal amount in the opposite direction. Which of these two possibilities is true and what are the character and magnitude of the actual change? The fundamental relation now becomes, since 4s are superheated and the others saturated. The pressures of the superheated vapors may be found from their respective characteristic equations, the pressures of the saturated vapors from the vapor tables. The total heat of such a mixture would no longer be equal to the sum of the total heats of its constituents, after the first one had begun to superheat, but could be found by adding to 2wq the total work performed, Ap M (T.wv ^wa], and the increase of internal energy of each component. The total entropy would be 156 THE TEMPERATURE-ENTROPY DIAGRAM. After all the vapors were superheated the specific pressures of each could be found by inserting the known values of T and v in the various characteristic equa- tions. Then there would exist as a check the relation, PM (observed) = (?1 +JP2+?3 + ) (computed)- This extra condition could, if necessary, be used to determine the weight of one of the constituents. The entropy of the mixture of superheated vapors would be The heat required to raise the mixture at constant pressure from liquid at 32 F. to superheated vapors of temperature T is found by adding the increase of internal energy to the external work, or -. . . + APM(WV 0) = ~S,wE+Ap M (wv 2ci. Let abed (Fig. 62) represent such a cycle. As 4(j) abc = J^> cda it follows that or But C2>ci, and therefore log e TT, 0. There thus exists between the temperatures at the four corners of the cycle .the simple relation, 162 THE TEMPERATURE-ENTROPY DIAGRAM. Operating as a motor the work developed is equal to the difference between the heat received and the heat rejected, or Eliminating T 3 by means of the preceding relation, this reduces to The thermal efficiency is therefore _ ri T{c l (T-T 2 )+c 2 (f 1 -Tn' while its relative efficiency as compared with the Carnot cycle is If the upper and lower temperatures are maintained constant it is evident that by considering point a to be fixed and c to move along the isothermal T\, T can be made to assume all intermediate values. During such a complete change the external work would pass from zero through a maximum back to zero. W is therefore a function of T and the value of T which will make W a maximum can be found by setting -^ = 0. HOT-AIR ENGINES. 163 Thus dAW dT " or The substitution of this value of T in the expression for work gives Under these conditions the heat received becomes and the thermal efficiency The relative value of this cycle as compared with the Carnot is therefore Carnot Special Case. If c 2 = on the cycle consists of two isothermals and two poly tropic curves, as shown 164 THE TEMPERATURE-ENTROPY DIAGRAM. in Fig. 63. The expression for the maximum amount of work (c 2 Ci}(VT~\ \/T 2 ) 2 gives an infinite result, because no limitation has been placed upon the isother- mal expansion. Although the general expressions possess indeterminate forms a special solution can be obtained directly from the T-diagram. 10 T, Ci FIG. 63. The work developed is equal to that developed in a Carnot cycle, W Vb or PC The thermal efficiency of the cyle is therefore HOT-AIR ENGINES. 165 and its relative efficiency as compared with the Carnot cycle is PC PC (a) When c\ = c v this reduces to the Stirling cycle, consisting of two isothermals and two constant-volume curves. In this case it is more convenient to use the ratio of volumes instead of the pressures Vb PC (6) When c\ = Cp this reduces to the Ericsson cycle, consisting of two isothermals and two constant-pressure curves. Of course the above discussion applies to the case when a regenerator is not used. Other special cases will be considered in the Chapter on Gas-engines. The Non-regenerative Stirling and Ericsson Cycles. Inspection of the formula for efficiency or of Fig. 63 shows that the efficiency increases the smaller the relative magnitude of the heat c\(T\ T2) as compared with that received during the isothermal process. That is, the greater the range of pressure in the Ericsson cycle and the smaller the clearance in the Stirling cycle the higher the efficiencies. As the heat received during the isothermal change increases the efficiency of each cycle approaches that of the Carnot as a limit. 166 THE TEMPERATURE-ENTROPY DIAGRAM. Example. The small Ericsson pumping-engine (Stir- ling cycle without regenerator) has a piston 8 ins. in diameter and a stroke of 4 ins. The clearance is 200 per cent, of the piston displacement. Assuming the temperature of the fire to be 2500 F. abs. and that of the water-jacket 600 F., find the theoretical and relative effic ; encies. (2500- 600) X 0.169(2500-600) + = 13.4 per cent. 2500-600 Carnot 2500 17.6 per cent. 76.0 per cent. ^Carnot The Ericsson Pumping-engine. The small Ericsson pumping-engine works but approximately upon the Sterling cycle, as the displacer and working piston are not timed so to operate that the different thermal events are entirely separate. This action, combined with the transference of heat between the air and the cylinder walls, results in a rounding off of the corners of the ideal cycle, giving an indicator card such as is shown in HOT-AIR ENGINES. 167 Fig 64. It is to be further noticed that heating and cooling at constant volume exist only for a moment at the ends of the stroke. This is due to the almost entire absence of any regenerator. The only operation which in any way approximates that of a regenerator is the sudden transference of the air from one end of the cylinder to the other, causing it to- move in a thin sheet through the narrow space between the displacer and the cylinder. It thus comes into contact with metal of varying temperatures, and is thus partly heated FIG 64. or cooled during transmission, the rest of the heating or cooling occurs during the operation of the working piston. This engine in itself is of no great interest to the engineer, but the T^-analysis of its indicator card is of value in that it permits of the direct application of the principles already discussed in the Chapter on Perfect Gases without introducing the various difficulties to be met with in gas-engine cards. The same charge of air is used continuously in the Ericsson engine, the heat being received and rejected through the walls of the cylinder, and the indicator card 168 THE TEMPERATURE-ENTROPY DIAGRAM. thus represents a closed cycle of a definite mass of air. It is true that during the movement of the displacer the air is partly in contact with both the heated and jacketed portions of the cyclinder and is therefore not of uniform temperature, and similarly when the displacer is at rest, although most of the air is in contact with either the source or the refrigerator, part of it is nevertheless in the narrow space outside the displacer and thus at a different temperature from that of the main portion. There is thus at no instant a uniform temperature existing throughout the entire mass, but from the known pressure and volume the average temperature is deter- minable. The range of both pressure and average temperature is so small that the air follows appreciably the laws of a perfect gas, so that c p and c v are constant. The mass of air is unknown, so that only the ratio of the specific volumes and not their absolute values are known for different positions of the piston. The only item definitely known is therefore the pressure as measured from the indicator card. Therefore in the T^-projectioh the pressure will be the only property definitely known, but the ratio of the average tempera- tures is known for any two positions of the piston. Furthermore the indicated work being known the area of the indicator card in the T^-plane represents the heat equivalent of this work, and thus the B.T.U. per unit area are determinable. Evidently if the mean tern- HOT-AIR ENGINES. 169 perature should be obtained experimentally for any position of the piston, the weight of air in the cylinder could be computed and thus the scales of temperature and entropy become known. Fig. 65 shows the TV- and ^-projections of the pr-card by the methods outlined in Chapter II. Be- cause of the large clearance of the engine it was possible to economize space by placing the T<-projection to the left of the TV-projection. As soon as one becomes familiarized with the methods the three diagrams can be superimposed without detracting any from their value. The indicated power developed per cycle is obtained from the mean effective pressure and the dimensions of the engine. From this, together with the areas of the two diagrams, the foot-pounds and the B.T.U. per square inch may be determined in the pv- and T- projections respectively. The heat received and rejected by the air per stroke may now be measured as the surface scale of the T- diagram is known, although the scale of the coordinates is unknown. The heat theoretically available per cycle can be found in any given case by measuring the fuel and multiplying the weight per cycle by its calorific value. The difference between the heat available and the heat received represents the heat lost up the flue and to the surrounding atmosphere, cooling water, etc. 170 THE TEMPERATURE-ENTROPY DIAGRAM. As a basis of comparison the Stirling cycle between the same temperature and volume limits is also given. CHAPTER IX. THE TEMPERATURE-ENTROPY DIAGRAM APPLIED TO GAS-ENGINE CYCLES. No attempt will be made here to trace the heat losses due to radiation and the cooling water in the jackets, but the cylinder and piston will be considered impermeable to heat in all cases. Thus by a compari- son of the ideal cards for different cycles the gain due to initial compression and the loss from incomplete expansion may be more clearly defined. The Lenoir, Otto, Atkinson, and Diesel Cycles. The Lenoir cycle was introduced in 1860. Its thermo- dynamic principles were retained in the different free- piston engines. These were uneconomical and noisy and have disappeared. The only remaining example is the Bischoff, a simple small vertical engine. The Lenoir cycle consists of the following events: (1) During the first part of the forward stroke a fresh explosive mixture is drawn in by the piston (aA in Fig. 66). (2) A little before half-stroke is reached the supply- valve closes and the explosion occurs. In reality the 171 172 THE TEMPERATURE-EXTROPy DIAGRAM. combustion requires an appreciable time for its com- pletion, and thus the heating takes place while the piston is moving forward, i.e., at increasing volume, but for the ideal case the explosion will be considered instantaneous, and hence the heating will be at con- stant volume; along the line AB. (3) The rest of the stroke represents adiabatic expan- FIG. 66. sion of the heated gas down to initial pressure; along BC. (4) The return stroke, during which the products of combustion are exhausted. Thermodynamically this is equivalent to cooling at constant pressure; along CA. About the time Otto and Langen were experimenting with the free-piston engine, Beau de Rochas described a cycle which would make possible the economical GAS-ENGINE CYCLES. 173 running of a gas-engine. This was embodied by Dr. Otto in his silent engine in 1876 and has thus become asso- ciated, although wrongly, with his name. The "Otto" cycle consists of the following events: (1) The drawing into the cylinder at atmospheric pressure of a new explosive mixture throughout one complete stroke (a A in Fig. 66). The volume of the charge is MA and consists of the burnt products in the clearance space Ma from the last charge plus the fresh charge. (2) The adiabatic compression of this charge on the return stroke of the piston AD. This compression of the gas into the clearance space is done at the expense of the energy in the fly-wheel. (3) The ignition and explosion of the charge while the piston is at rest at the dead-centre, thus increasing the pressure and temperature at constant volume ; along DE. Assuming that the same quantity of mixture is used by both the Lenoir and Otto engine, the heat generated by the explosion will be the same in both cases, i.e., the areas under the curves AB and DE are equal in the T^-diagram. (4) The expansion of the heated gases throughout the entire stroke, assumed adiabatic; along EF. (5) The drop in pressure due to the opening of the exhaust-valve while the piston is at the end of the stroke. This is equivalent to cooling at constant volume; along FA. 174 THE TEMPERATURE-ENTROPY DIAGRAM. (6) The exhaust of the burnt gases during the return stroke Aa. Changes of location are not recorded in the TV-diagram. In the Atkinson engine, now no longer made, the cycle was the same as the Otto up to the point F, and then, instead of releasing the hot gas, the expansion stroke was lengthened by means of an ingenious mechanism permitting the adiabatic expansion down to back pressure, as represented by FG. Then the exhaust stroke was from G to A, which thermo- dynamically is equivalent to cooling at constant pressure. Comparing the Atkinson and Otto cycles it is at once evident that there is a loss of work and of heat equal to AFG in the pv- and T<-planes, respectively, due to incomplete expansion. A comparison of the Atkinson and Lenoir cycles shows that as the heat received in both is the same while that rejected by the Lenoir engine is the greater (compare areas under CA and GA), the efficiency of the Atkinson is the greater. Theoretically, then, the Atkinson engine has the most perfect cycle of the three, but nevertheless it has been entirely superseded by the Otto engine. The reason for this becomes at once apparent from the diagram. Even if the area AFG, rejected by the Otto engine, due to incomplete expansion, were just equal to the area under GC of the Lenoir exhaust stroke, the Otto GAS-ENGINE CYCLES. 175 would still be preferable to the latter because the same amount of power could be developed with a smaller engine. Suppose, now, the clearance space in the above Otto engine were decreased, so that the adiabatic compression would heat the gas to a higher initial tem- perature, as AD'. The explosion would now occur along the constant- volume curve D'E', where the area under D'E' is equal to that under DE, as the same heat is generated in both cases. The adiabatic expan- sion would now be down E'F' and the exhaust would be along F'A. Hence the same Otto engine with increased initial compression due to decreased clearance would .give increased efficiency, as the heat rejected under F'A is less than that rejected under FA. This engine would now be better than the Lenoir, both mechanically and thermodynamically. Furthermore the loss due to incomplete expansion becomes less because the heat thus rejected is reduced from AFG to AF'G'. That is, the higher the initial compression the less the theoretical superiority of the Atkinson over the Otto engine. And in the actual engine the increased complexity, size, friction loss, and danger of the Atkinson more than counterbalanced the theoretical superiority. Hence the Otto engine is practically the most efficient of the three. In the Otto cycle the temperature at the end of compression is not very high relatively, so that during the first part of the combustion the working fluid is 176 THE TEMPERATURE-ENTROPY DIAGRAM. much colder than the source of heat and does not attain this high temperature until the end of the com- bustion is reached. The heat is thus received at con- stant volume and increasing temperature instead of at constant temperature and increasing volume. It was shown in the first chapter that any such deviation from a Carnot cycle means a drop in efficiency. This led Diesel to invent a cycle during which most of the heat should be received at the highest available tem- perature. His method is to compress the air initially up to about five hundred pounds pressure to the square inch, so that its temperature is above the ignition- point of the combustible to be used. The injection of a small quantity of fuel causes the temperature to increase still further at constant volume up to that of the combustion; then as the piston moves forward the temperature of the gas is maintained nearly constant by the injection and combustion of further fuel. This lasts for about one-tenth of the stroke. The indicator- cards taken from such a motor show that the desired regulation is not perfect, the temperature sometimes rising, sometimes falling. This, however, only affects the magnitude of the gain from such a cycle, as all of the heat generated on the forward stroke is trans- mitted to the working fluid at an efficiency corre- sponding to that of the upper part of the Otto cycle. The cards show also that the expansion may or may not be carried down to the back pressure. GAS-ENGLVE CYCLES. 177 Fig. 07 shows the ideal diagrams for the Otto, Atkin- son, and Diesel cycles for the same quantity of heat; that is, the areas under be and b'gc' are the same in the T^-plane. The change from b to b r shows the in- creased compression in the Diesel motor, b' being at or above the temperature of ignition. The heat received g c' c FIG. 67. along b'g is not received under the most efficient con- ditions, but still with an efficiency equal to that of the best part of the Otto cycle, while that received along the " isothermal combustion line " gc' is obtained under conditions of maximum efficiency. The effect, as is clearly shown by the diagram, is to increase the amount of heat changed into work and to dimmish the heat rejected. On the return stroke the conditions of the Carnot cycle are more closely approximated in 178 THE TEMPERATURE-ENTROPY DIAGRAM. the Atkinson and Diesel cycles where the heat is re- jected at constant pressure than in the Otto cycle where it is rejected at constant volume, as lines of constant pressure deviate from isothermals less than do lines of constant volume. The Brayton or Joule Cycle. Up to the present time the Otto cycle has been almost exclusively employed FIG, 68. because of its ease of application in the reciprocating type of motor, but occasional attempts have also been made to heat the fuel under constant pressure. Such a cycle is now assuming importance as offering the most probable solution of the gas-turbine problem. Proposed by Joule this cycle is perhaps better known through its application in the Brayton engine. It consists of the following events (Fig. 68); (1) The charging stroke of the compressor cylinder, ab. (2) Adiabatic compression in the compressor, be. GAS-ENGINE CYCLES, 179 (3) Discharge from the compressor at constant pres- sure, cd. (4) During the transfer from the pump to the working cylinder (or to the nozzle of the turbine) the charge receives heat either from external sources or from combustion under constant pressure, but with increasing volume and temperature. (5) Admission stroke of the working cylinder, de. Processes (3), (4), and (5) occur simultaneously.. (6) Adiabatic expansion in the working cylinder or in the nozzle, ef. (7) Discharge from the working cylinder against a constant back pressure, fa. The net result is thus represented by the cycle beef, consisting of two isentropic and two constant-pressure curves. It is at once evident that the efficiency of the Brayton cycle, like that of the Otto, increases with increased initial compression. It is instructive to compare the three ideal cycles of Carnot, Otto, and Brayton involving as they do heating at constant temperature, constant volume, and constant pressure under different conditions. Let abed, abc'd', and abc"d" (Fig. 69) represent the Otto, Brayton, and Carnot cycles respectively, for the same weight of charge working with the same initial compression. The areas under be, be', and be" will then be equal. 180 THE TEMPERATURi'.-hNTROPY DIAGRAM. The thermal efficiencies of the cycles are found follows : c v (T c -Tb) _ T d -Tg T c -T b = I - Tfr, or 1 - ( Tb' \V a k-l ^Brayton that is, Tj-Tg ZV-n T a * b ^Otto ^Carnof FIG. 69. GAS-ENGINE CYCLES. 181 Such a case might arise when all three engines were using the same combustible, thus necessitating that the initial pressure be limited to prevent pre-ignition. Since the efficiency of all three cycles will be the same, a decision as to the most advantageous to use must be based upon other considerations. The differences between the three cycles are best shown in tabular form : Pressure Range. Volume Range, Temp. Range. Brayton Minimum Intermediate Intermediate Otto Maximum Minimum Maximum Carnot Intermediate Maximum Minimum From the table the Carnot is shown to be the least desirable in that it requires the largest engine, although the fluctuations in its rotative effect are not as great as in the Otto, so that the working parts would not need to be as heavy as in the latter. But besides this the Carnot cycle has to be discarded as unfeasible because although the Diesel motor approximates it on the forward stroke no gas-engine has as yet been in- vented to give isothermal compression. The Brayton cycle has the minimum change of pres- sure and therefore the least fluctuating rotative effect. Its volume range is, however, greater than that of the Otto. But again the maximum temperature reached in the Brayton is less than in the Otto, so that less jacket cooling of the cylinder will be necessary and the inter- change of heat between gas-metal-gas will also be less. 182 THE TEMPERATURE-ENTROPY DIAGRAM. By keeping the gas and air separate, as in the Diesel motor, it is possible to utilize much higher initial com- pression, and then the only limit theoretically is the resistance of the machine to pressure or temperature or both. We will next compare the Brayton and Otto cycles with reference to maximum pressure and maxi- mum temperature. Let bcc' (Fig. 70) represent the maximum safe pressure, FIG. 70. whether it be the sustained pressure of the Brayton cycle or the momentary maximum of the Otto. Fur- ther, let ad be atmospheric pressure, the lowest pressure of exhaust. In the Brayton cycle initial compression is carried up to 6, while in the Otto cycle it must stop at some intermediate point 6', such that the further increase GAS-ENGINE CYCLES. 183 due to heating will carry it just to c'. As the charges are identical the area under be equals the area under b'c'. But the heat rejected in the Otto exceeds that in the Brayton. Whence it follows that given a Brayton and an Otto engine working between the same limits of pressure the former is the more efficient and possesses the smaller temperature range. d FIG. 71. Similarly let cc f (Fig. 71) represent the safe maximum temperature. Then abed and dh'c'd' will represent the Brayton and Otto cycles, respectively, working between atmospheric pressure and temperature and maximum temperature cc'. Again the Brayton cycle shows the greater efficiency, although both possess the same temperature range, and the Otto cycle has a trifle the smaller pressure range. 184 THE TEMPERATURE-ENTROPY DIAGRAM. Deduction from the General Theory of Gas Cycles. On pages 101-163 we discussed the general case of a gas cycle consisting of two pairs of polytropic curves, and FIG. 72. obtained as the condition of maximum work per cycle that This gave for maximum work and the corresponding efficiency the expressions, and Second Special Case. If Ci = the cycle consists of two isentropic and two polytropic curves, as shown in GAS-ENGINE CYCLES. 185 Fig. 72. The maximum work per cycle between given temperature limits reduces therefore to and the corresponding thermal efficiency to _ VT\ - \/T~ 2 VT^Tl -T 2 T-T 2 >5max. W ork= ^ VT^ ~T~ whence as on p. 176, The dimensions of the engine to give the maximum amount of work per cycle between any given tempera- ture limits may be found as follows : The equation of the TV-projection of an adiabatic gives whence v\ k -*^ / Clearance N*" 1 ~~ \Clearance + piston displacement/ m m \rn or P.D.(^r Clearance = 1- 186 THE TEMPERATURE-ENTROPY DIAGRAM. Variation of Efficiency with Load. The efficiency of the internal-combustion engine varies not only with the type of cycle utilized, but also with the form of regulation adopted in any particular engine. A. Engines working upon the Otto cycle are governed by the following methods: I. The omission of an occasional charge, known as hit-or-miss regulation. II. Throttling of the fuel supply, known as quality regulation. III. Variation of the amount of charge, obtained by throttling the mixture or by a variable cut- off, known as quantity regulation. IV. Regulation by varying the time of ignition. V. Various compound methods involving two or more of the above. The effects of these various methods upon the efficiency of the engine under variable load can be clearly illus- trated with the ideal cycle in the T(f> plane. I. Hit-or-miss Regulation. Whenever a charge of fuel is omitted the scavenging action of the air com- bined with the constant action of the jackets serves to cool the cylinder walls and the admission ports to a temperature below the average, so that the next charge will be cooler and of greater density. Therefore the first explosion immediately following a " miss " stroke will be more powerful than the average as it contains a greater quantity of fuel. Thus if abc'd', GAS-ENGINE CYCLES. 187 vig. 73, represent the average cycle, the cycle following the " miss " will be represented by some diagram such as ahc"&" . In case of light load it may happen that .two or more miss strokes occur successively, in which case the walls may be cooled to such an extent that the resulting low temperature of the next charge retards its combustion. This is equivalent to exhausting part FIG. 73. of the charge unburned and results in a smaller heat diagram abed. It will be noticed that with constant port openings and constant speed the throttling action during admission is always the same, so that the pressure at the end of admission is always practically the same. Hence if the minor variations in percentage mixture caused by the varying density and composition of the clearance gases may be overlooked, the effect of a miss stroke is simply to vary the power of the succeeding stroke but not its efficiency. 188 THE TEMPERATURE-ENTROPY DIAGRAM. Variation in load is met automatically by the engine by varying the frequency of the miss strokes. There- fore this type of regulation must give practically the same thermal efficiency for all loads, but it has the mechanical disadvantage of .requiring more massive construction and heavier fly-wheels if good regulation in speed is to be obtained. In practice engines governed by hit-or-miss regulation usually show a somewhat greater fuel economy than is obtained by the other methods. II. Quality Regulation. A full charge of fuel and air is taken each suction stroke, but the percentage mixture is varied by the governor which controls a throttle valve placed in the fuel supply pipe. The initial pres- sure being the same the efficiency should apparently be the same, but experiments made upon explosive mixtures* of fuel and air have shown that the rate of combustion is retarded by dilution, and this dilution may readily be so great as to prevent combustion or at least to delay it until after the charge has left the cylinder. This unburned or wasted heat, combined with the greater relative effect of the cylinder walls upon a smaller heat charge (see next chapter), serves to diminish the efficiency as the load falls off. Thus if abed, Fig. 74, represent the heat available at full load, abc'd' would represent the heat available * See report by Author in Technology Quarterly, Sept., 1900, pp. 248-259. GAS-ENGINE CYCLES. 189 for some smaller load if it were not for the above men- tioned losses. Due to dilution, however, a certain por- tion of this theoretically available heat, say c"c'd'd", either is lost through imperfect combustion or is so much delayed as to be generated after the charge is FIG. 74. exhausted and must therefore be added to- the exhaust heat, as d'efg. III. Quantity Governing. In this system of regula- tion the percentage mixture is adjusted by hand to give the maximum explosive value and is then maintained constant, but the quantity of the mixture admitted to the cylinder is controlled by a valve placed in the pipe leading .from the mixing chamber to the cylinder. The action of the governor being either to throttle the valve throughout the suction stroke or to close it completely at varying percents of the stroke so as to adjust the quantity of charge to the load. 190 THE TEMPERATURE-ENTROPY DIAGRAM. a. Throttling. Regulation by throttling involves a drop of pressure in the throttle valve so that the pressure of admission is less than the pressure of discharge. This produces a negative loop in the indicator card FIG. 75. whose area measures the work required to overcome the friction losses during admission and exhaust. See Fig. 75. b. Automatic Cut-off. Here the valve is kept wide open until cut-off occurs. Hence the charge is admitted FIG. 76. at atmospheric pressure up to cut-off, and from that point expands adiabatically to the end of the stroke. On the compression stroke this adiabatic operation is traversed in the reverse direction until atmospheric pressure is reached. The friction loss of the preceding method is therefore avoided. Figs. 76 and 77. The weight of the burnt gases is practically always the same, as they are discharged under the same pressure, GAS-ENGINE CYCLES. 191 while the weight of the new charge varies with the amount of throttling and with the time of cut-off, so that the proportion between the burnt gases and the new charge is a variable, that is, the burnt gases serve to increase the dilution of the gaseous mixture as the load decreases. Quantity regulation thus possesses not FIG. 77. only the losses due to its throttling action but also to a small extent those due to quality regulation. IV. Regulation by Varying the Time of Ignition. Ignition must occur before the end of the compression stroke so that by the time compression is finished the flame shall have had opportunity to spread throughout the entire charge and produce maximum pressure. For any given load the spark should therefore be adjusted by hand so as to give best results. In quality regulation and to a lesser extent in quantity regulation the rate of flame propagation falls off as the load de- creases, so that if best results are to be attained the spark must be correspondingly advanced. 192 THE TEMPERATURE-ENTROPY DIAGRAM. Advantage is taken of this in many smaller motors, as for example in automobiles and motor boats, to adjust the power of the engine to any desired condition by shifting the spark by hand. Thus practically the same quantity of fuel is admitted each stroke and more and more of it wasted as the load falls off. Assuming for the moment that the explosion is absolutely instantaneous the effect of retarding the FIG. 78. spark is clearly shown in Fig. 78. Let abed represent normal running under full load. If the load falls off somewhat the spark is retarded until the charge has expanded part way down the compression line ba, say to &'; it then burns at constant volume b'c' gener- ating the same amount of heat as at be but with lower temperatures and hence greater entropy increase. The result is therefore an increase in the exhaust heat GAS-E.\GINE CYCLES. 193 by an amount dd'fg. The later the spark the greater the increased exhaust. In other words the effect is similar to that which would be obtained by an in- creasing clearance space. B. Engines working upon the Diesel cycle are governed by varying the time interval during which fuel is admit- ted to the cylinder. Let it be assumed that there is always sufficient oxygen present to insure complete combustion of the FIG. 79. fuel at maximum load. If the fuel is injected so as to supply heat isothermally the shape of the cycle with increasing load is shown by abcdef, abcd'e'f, etc., Fig. 79, where the areas under bed, bed', etc., rep- resent the heat generated by the fuel and de, d'e', etc., represent the adiabatic expansion from the moment the fuel supply is discontinued to the end of the stroke. It is evident that the drop of temperature de, d'e', 194 THE TEMPERATURE-ENTROPY DIAGRAM. etc., experienced by the additional fuel decreases so that the average efficiency falls off as the load increase?. If, for example, injection of fuel could be continued to the end of the stroke, the ideal efficiency would be but slightly greater than one half that of the Carnot efficiency although at ordinary load it is nearly equal to it. Comparison of the Different Gas-cycles. Because of its successful application in commercial use it is necessary to deduce an analytical expression for the efficiency of the Diesel cycle. Certain assumptions must be made at the start in order to give this cycle that simplicity of form which will make the analytical expression useable. . On actual indicator cards taken from Diesel engines the process most vari- able in character is that of the admission of the fuel. This fluctuates more or less in magnitude upon the different cards, but in general may be fairly well repre- sented by a constant pressure line. Such an assumption is justifiable from a theoretical point of view, as the Diesel cycle attempts to utilize the great increase in efficiency which comes from high initial compression. In order that this gain may be a maximum, such com- pression should be carried to the upper pressure limit, so that when the fuel is injected no further increase in pressure is permissible and any decrease in pressure would diminish the temperature range of the heat. This differs from the original Diesel cycle which assumed GAS-ENGINE CYCLES. 195 isothermal combustion, and was based upon the mis- taken idea that isothermal generation of heat was essential to the attainment of maximum efficiency. Assuming, then, that the ideal Diesel cycle may be represented by constant pressure reception of heat and constant volume rejection of heat, these two processes, being connected by adiabatic expansion and com- pression, it follows that the expression of thermal efficiency is equal to C p times the increase in temper- ature during the injection of the fuel, minus C v times the decrease in temperature during release, divided by the heat received. A fundamental distinction between this expression and the common expressions for the Otto, Joule, and Carnot cycles is to be noted in the retention of the specific heats. Thus in the Otto cycle C v appears in all three terms, and similarly in the Joule cycle C p appears in all three terms, so that this common factor cancels in all these expressions making the efficiency absolutely independent of .the character of the working substance, whereas in the expression for the efficiency of the Diesel cycle the ratio of these specific heats remains, so that the efficiency is dependent to a certain extent upon the character of the working fluid used. By means of thermodynamic relations easily established between the temperatures and the volumes of the corners of the cycle it is possible to elim- inate from the expression of the efficiency all the tem- peratures and to retain in their places certain ratios 196 THE TEMPERATURE-ENTROPY DIAGRAM. of volumes. The expression may be given in the follow- ing final form: / Vol. at cut-off \* \Vol. of clearance/ /Vol. of clearance" 1 Mt. = l or / Vol. at cut-off ~\ I Vol. of cl. + pis- 1 AVol.of clearance" A ton dis. / (s&Y-i Fff i ^ CL ' ( CL V" 1 -,/c.o. 1 v\ci.+p.pJ H-cr- 1 ) This expression is of the same form as that of the efficiency of the Otto, Joule, and Diesel cycles, except that a certain coefficient different from unity is intro- duced as a multiplier in the last term. Thus, as we have seen, the expression for the Otto efficiency takes the form : / Q y-i -VCL+RPJ An investigation of this coefficient : Cl. shows us that as the cut-off increases the numerator increases more rapidly than the denominator, since the expression : /c.o.y V~cT/ ' GAS-ENGINE CYCLES. 197 as k is greater than one, increases more rapidly than the simple ratio /C.0.\ rci/- Therefore as the load increases the value of this coeffi- cient increases so that the efficiency of the cycle di- minishes. The maximum values of the efficiency of the Diesel cycle are therefore obtained with the smaller loads, and as the load continues to decrease this value increases and approaches as a limit that of the Otto cycle which theoretically will be attained at zero load. It is evident, therefore, that with the same com- pression the Diesel cycle is not as efficient as either the Otto, Joule, or Carnot cycles, but that its great practical efficiency has been attained by making use of the increase in efficiency in any of these cycles which may be produced by utilizing smaller clearance space. This, as we have seen, is accomplished by keeping the fuel separate from the air during the compression and injecting the fuel whenever the desired pressure has been attained. As soon as this method of separate compression is introduced there is no longer any neces- sity of utilizing such low compressions in the Otto and Joule cycles, so that it would be possible to have in these cycles exactly the same compression as that realized in the Diesel engine, and in such a case this cycle would be inferior to either of the others. It is probable that, as advantage is being taken of this 198 THE TEMPERATURE-ENTROPY DIAGRAM. high compression, the initial compression would be carried to the ultimate upper limit consistent with safety, so that no further increase in pressure would be permitted during the burning of the fuel. In such a case it would no longer be possible to utilize the Otto cycle, and one would be forced back upon the Joule cycle. In case compression is not carried to the maximum upper limit and a still further increase is permissible in the cylinder during combustion, it would then be possible to utilize burning at constant volume and therefore receive the heat in the manner adopted in the Otto cycle. If in connection with this constant volume reception of heat there could at the same time be com- bined a constant pressure rejection of heat, an efficiency could be obtained greater than that of either the Otto or the Joule cycle, A cycle thus constituted would evidently be but the old Atkinson cycle. Proceeding in a manner similar to that adopted in the case of the Diesel cycle, an expression for the efficiency of the Atkinson cycle can be deduced in terms of k and certain volume ratios. This expression may be given the final form: /Vol. after expansion \ , Vol. of X*" 1 \Vol. after admission / / clearance \ ~ /Vol. after expansion^* """' I Vol. after I VVol. after admission/ " \admission / This expression is seen to have the same form as that GAS-ENGINE CYCLES. 199 of the Otto cycle with the exception that the coefficient of the second term is other than unity. An inspection of this coefficient: k /Vol. after expansion \ \Vol. after admission / Vol. after expansion\ k Vol. after admission/ shows that the efficiency is dependent upon the value of k for the given substance and also dependent upon the volume after expansion; this volume necessarily increasing as the quantity of charge increases. As the volume after suction may be considered a fixed quantity, it is evident that as the volume after expansion increases the ratio of these two volumes increases at a slower rate than the same ratio raised to the k power, so that with increasing load the denom- inator of this coefficient increases at a higher rate than the numerator, so that the coefficient as a whole decreases in value as the load increases, a result which is exactly the opposite of the history of the correspond- ing coefficient for the efficiency of the Diesel cycle. An inspection of these two cycles in the temperature entropy plane makes clear this fundamental differ- ence in these two cycles. Start with the same initial compression in both cycles. In the Diesel cycle, on the one hand, the constant pressure curve drawn through the end of compression and the constant volume curve 200 THE TEMPERATURE-ENTROPY DIAGRAM. drawn through the beginning of compression approach indefinitely as the load increases, so that whereas at the beginning or at a very light load the efficiency is equal to that of a Carnot cycle, we find that the suc- cessive quantities of heat taken in as the load increases suffer a smaller drop in temperature than the earlier quantities, and therefore are utilized less efficiently, so that the net result of an increasing load is to bring down the average efficiency at which the heat is utilized. In the Atkinson cycle, on the other hand, since a con- stant volume curve drawn through the end of compres- sion and a constant pressure curve drawn through the beginning of compression diverge as the load increases, the range of temperature experienced by successive quantities of heat increases so that the efficiencies of the final portions are greater than the efficiencies of the earlier portions, and therefore the result of an increasing load is to increase the average efficiency at which the heat is utilized. As the load falls off this coefficient increases in value and approaches unity as a -limit, this value, however, being attained only at zero load, when the efficiency of the Atkinson cycle will be reduced to that of the Otto cycle. This discussion makes it evident that the Carnot cycle is not the most efficient cycle when adapted to gas engine work. The requirement that all the heat should be received at the highest possible temperature, and that all the heat which must be rejected should OAS-EXGIXE CYCLES. 201 be rejected at the lowest possible temperature in order to obtain the maximum amount of work from a given quantity of heat, is necessarily fundamentally true. If the source of heat were at a definite upper temperature, the fulfillment of this fundamental requirement would necessarily require the use of the Carnot cycle, but if the heat is generated by combustion in a confined space the very act of combustion will result in an increase in the temperature of the substance in this confined space, so that any further combustion will occur at a higher temperature. In other words, successive quan- tities of heat will be generated in the confined space at successively higher temperatures. In fact, to prevent the temperature from rising in order to obtain the requirement of the Carnot cycle, would necessitate careful adjustment between the rate of fuel consump- tion and the rate of work production by the piston. In the Diesel cycle the heat is generated at such a rate that the pressure remains practically constant. In the Otto cycle the generation of heat is so nearly instantaneous that the process occurs practically at constant volume. In both of these cases the temperature rises continuously throughout the period of com- bustion, and, other things being equal, the heat ^received along the constant pressure curve is capable of better utilization than that received along the isothermal curve, and similarly the heat received at constant volume is capable of better utilization than that received 202 THE TEMPERATURE-ENTROPY DIAGRAM. at constant pressure. This is very clearly illustrated by the relative slopes of the three curves in the tem- perature entropy plane; as the steeper the curve the smaller the increase in entropy, and therefore the smaller the amount of heat which must eventually be rejected. This analysis shows, therefore, that for the same initial compression the constant volume curve is the one which has the highest theoretical possibili- ties. It might seem at first sight that a curve even steeper than the constant volume could be utilized. A little thought, however, will show that such a process could only be taking place while the piston was on its return stroke. In such a case, therefore, it would be better to wait until this return stroke were com- pletely finished, so that all of the heat might be gener- ated at the higher temperatures consequent upon such an increased compression. As we have seen, a cycle consisting of constant volume production of heat, a constant pressure rejection of heat, combined with adiabatic compression and expansion, gives a higher efficiency than the straight Otto cycle. Such an Atkin- son cycle, however, although it fulfills the requirements for the reception of heat at maximum temperature, does not at the same time fulfill the requirements for the rejection of heat at minimum temperature. It is true that the Atkinson cycle expanding down to initial pressure probably rejects the heat at as low a temperature as is practicable, but there nevertheless GAS-ENGINE CYCLES. 203 exists the theoretical possibility of continuing this expansion down to the initial temperature and then closing the cycle by means of an isothermal com- pression. Possibly such an operation could be realized to a certain extent by the injection of cold water into the cylinder during this compression stroke. It is interesting at least to compare the expression for the thermal efficiency of this cycle with those already considered. The method of obtaining this expression is of the same general character as that adopted for the pre- vious cycles; namely, to draw the cycle in the tempera- ture entropy plane and then to determine the heat quantities as represented by the areas under the proper lines. The only new feature in this particular formula is the heat exhausted during the isothermal compres- sion. This may readily be obtained by multiplying the temperature of exhaust by the increase in entropy, which may be determined during the constant volume generation of heat. Going through the steps involved in such work and eliminating the temperatures from the formula, as in the preceding cases, we arrive eventually at the following expression: Vol. after expansion / Q \* -1 10ge Vol. af tor admissiorf vol. \ /Vol. after expansion^ fc \ Vol. af- / \Vol. after admission/ V tcr ad./ This expression takes a general form similar to those 204 THE TEMPERATURE-ENTROPY DIAGRAM . of the preceding cycles, the only difference being that this coefficient is smaller than any of the preceding for the same compression and tho same amount of fuel. We further see that the denominator increases more rapidly than the numerator as the volume after expansion increases, because this enters the denom- inator as a factor proportional to its kl powe'r, while in the numerator it appears simply as a logarithm. An instructive tabulation of the results obtained from these different cycles may be made by writing them in the general form 1 Eff. =the fractional part of the energy wasted. ao - Diesel , i-Jfe- / Cl X*- 1 otto, i-^-ix( cl+RD ) ; Cl.+P.D-ex fe-l) 7C1. + P.D.1V \C1.+P.D.^/ Atkinson, 1 -B A -k^ + P.^ V AcTTEDJ ad Cl.+P.D. ex Max 1 F 0. CI.+P.D. GAS-ENGINE CYCLES. 205 If the Otto cycle be taken ac the standard of reference, the waste heat of the other cycles may be expressed in terms of the waste heat of the Otto by means of the ratios Diesel '~^~' Otto i ^.v_/. * l ~CT~ Atkinson Cl. +P.D.ad Otto =k ~ -I /Cl.+P.D.ttA* ' VCl.+PJD.J ~ Maximum - Cl 01 +P D ' ' Otto ci.+p.D. ex y- i _ 1 CHAPTER X. THE GAS-ENGINE INDICATOR CARD. Physical Conditions of the Problem. The direct application of the 7^-analysis to an actual gas-engine, although simple in theory, is difficult in practice. To find the various heat losses from the study of an indi- cator card necessitates an exact knowledge of the amount of heat available per revolution. In a steam-engine this may be determined with a fair degree of accuracy, provided the boiler and exhaust pressures remain con- stant and the cut-off varies within narrow limits, as then each pound of steam contains a known quantity of heat and the weight required to develop a given power is measurable. In a gas-engine the operation is more involved. The energy is not drawn from a reservoir but is generated each working stroke in the cylinder. Of course the gas can be metered and the average consumption found per working stroke, but how is the corresponding average indicator card to be determined? It is easy to obtain cards from the same engine showing quick, moderately fast, and slow burn- ing of the fuel. (See Figs. 81 and 82.) How are the 206 THE GAS-ENGINE INDICATOR CARD. 207 corresponding percentage mixtures and the absolute quantity of fuel to be found for each different card? In the hit-or-miss type of governing the " miss " stroke, acfing as a scavenger, cools the cylinder so that the following charge is heated less during admission, is expanded less, and hence a greater weight enters the cylinder than would directly after an explosion. The result is the percentage mixture of gas, air, and burnt products varies throughout the explosions for a com- plete firing cycle. Perhaps the best that can be done is to take each time the complete set of cards for the firing cycle and draw an equivalent average card. If governing is effected by throttling the fuel gas there seems to be no method of determining the relative proportions of gas and air; if the governor controls either by throttling or by cutting off the admission of a mixture of known composition, the relative proportions between it and the burnt products will still be un- known. Possibly in engines of the Korting type, where the gas and air are drawn separately into cylinders of known capacity, it may be possible to attain fairly accurate knowledge of these quantities per stroke. The best that can usually be done is to keep the load as constant as possible and use average cards and average fuel consumption. In any case the gas and air must be measured independently and this in the case of a large gas-engine will require a large gas-meter, or some form of displacement tanks for measuring the 208 THE TEMPERATURE-ENTROPY DIAGRAM. air. If nothing else is at hand, an anemometer can be used. Having determined the average indicator card and the average heat applied per explosion, the next diffi- culty is the determination of the specific heat at constant volume and the ratio . The difficulty is here two- C v fold: first, the character of the mixture is doubtful, due to the varying composition of the burnt gases in the clearance space; secondly, our knowledge of the variations of the specific heats at high temperatures is very incomplete. Thus MM. Mallard and Le Chate- lier found by experiment the values : Carbonic dioxide .... c v = 0.1477 + 0.000176* Water vapor ........ =0.3211 + 0.000219* Nitrogen ... ......... = 0. 170 + 0.0000872* Oxygen ............ =0.1488 + 0.0000763* If the weight of each component present in the charge is known the value of c v for the mixture is given by Due to the chemical changes during combustion the value of c v differs during different parts of the cycle. The value of k is likewise a variable. Clerk * finds that 1.37 and 1.28 are better values for compression * The Engineer, March 1 and 8, 1907, pp. 205 and 236. THE GAS-ENGINE INDICATOR CARD. 209 and expansion respectively than the customary value 1.4. The specific volume of the charge is decreased by combustion due to the rearrangement of the molecules into new combinations. Thus for some samples of Boston illuminating-gas the ratio of final to initial specific volumes was for different percentage mixtures of gas and air as follows: Gas to air 1:6 1:7 1:8 1:9 Final volume Initial volume 0.964 0.969 0.972 0.975 Clerk* states that 1 volume of gas and 5 of air shrink 4 per cent, and 1 volume of gas and 7 of air shrink 3 per cent, and 1 volume of gas and 10 of air shrink 2.2 per cent. To transfer the gas-engine indicator card to the T- plane requires therefore (1) Average percentage mixture of the charge and the corresponding average card. (2) The laws of variation of c v and k. (3) The degree of combustion so that the shrinkage in specific volume may be calculated. If all these conditions are fulfilled the corresponding temperatures and entropies can be calculated and the desired plot obtained. * Min. Proc. Inst. Civ. Eng., vol. CLXII, p. 314. 210 THE TEMPERATURE-ENTROPY DIAGRAM. If the temperature can be obtained independently by direct measurement for any point of the cycle it serves as a check upon the other observations. This would require some thermo-electric device where the circuit is closed at the proper moment by the engine. The difficulties of such a method are considerable, and even when accurately obtained the indicated tempera- tures are not the desired average temperature but the temperature of the charge at the point at which the thermo-electric couple is introduced. Various rough approximations have been given for the temperature of the charge at the end of admission. Boulvin, in the Entropy Diagram, assumes it to be a mean between that of the surrounding air and of the jacket water. He further neglects the chemical change due to combustion and assumes that the values of c p , Cv, and k of the charge hold sufficiently well for the burnt products, and further assumes that c p and c v remain constant at high temperatures. Prof. Burstall, in the Second Report of the Gas-engine Research Committee (Proc. Inst. Mech. Eng., 1901, p. 1083), takes the temperature as generally not differ- ing greatly from the jacket temperature. He finds the computed temperature, however, to be considerably higher than this in all cases, the maximum and mini- mum differences being about 90 and 32 F. He also uses the variable values of c v given on page 175. Prof. Reeve, in The Thermodynamics of Heat-engines, THE GAS-ENGINE INDICATOR CARD. 211 states that he usually assumes the initial temperature at the convenient round number of 600 absolute Fah- renheit. Ideal Indicator Cards. Having settled these prelimi- naries, i.e., having determined the average card, the average heat supplied per cycle, the values of c v and k for the mixture, the shrinkage, and the initial tempera- ture, the next step is to draw the ideal card. Thus if TQ = estimated or measured temperature at admission, po = initial pressure observed from card, ^o = volume of clearance plus piston dis- placement, 1) I = volume of clearance, Qi = B.T.U. received per cycle, Q 2 = B.T.U. rejected per cycle, then, referring to Fig. 73, the following equations give the proper values of the coordinates of the points 1, 2, and 3. A, \ fc- 1 T l = TQ ( ) ,k may usually be assumed as 1 .38 ; (See p. 161.) 212 THE TEMPERATURE-ENTROPY DIAGRAM. The ideal cycle will thus be different for each test as in each case the gaseous mixture is different, so that in each case the special values of c v and k must be determined. This is, as we have seen, a difficult problem, but, fortunately, the variations in c v and k are not great between different tests on the same engine and between the gaseous mixtures arising from different fuels, so that these variations may be overlooked without intro- ducing too great an error in the results. Thus Prof. Lucke, in his Gas-engine Design, establishes a standard reference diagram by considering the work- ing fluid to be air filling the cylinder at the end of the admission stroke under 14.7 Ibs. pressure and at 32 F. The air is supposed 'to pass through the usual Otto cycle, during ignition receiving at constant volume all the heat generated per cycle in the actual engine. Such a device, although recommended by its sim- plicity and admirably adapted to the end in view, viz., the estabishment of diagram factors to assist in the design of engines of any required power and working with any desired fuel, is unsuitable for the purposes of heat analysis. In the opinion of the Committee of the Institution of Civil Engineers (see Min. Proc., 1905, pp. 324 and 326), "it would introduce some uncertainty and difficulty, without adequate compensating advantage, to make the standard engine-cycle depend on a knowledge of THE GAS-ENGINE INDICATOR CARD 213 the physical constants of the exact mixture used. The discussion of the constant for various mixtures of gases already given shows that, apart from the unknown change at high temperatures, these constants do not differ by more than 2 per cent, to 5 per cent, from those of air in such mixtures as are used in gas-engines. The advantages of simplicity and definiteness in the standard are so great that the Committee recommend that the standard engine should be taken to work with a perfect gas of the same density as air. This in no way prevents any one from discussing the distribution of heat losses in any particular trial with constants adjusted -to any particular mixture of gases. But it does render more definite the statement of the relative efficiency (cylinder efficiency), without, in the opinion of the Committee, introducing any error of practical importance. " The standard engine of comparison is therefore a perfect air-gas engine operated between the same maximum and minimum volumes as the actual engine, receiving the same total amount of heat per cycle, but without jacket or radiation loss, and starting from one atmosphere and the selected initial temperature of 139 F. Its efficiency is the same as that of a Carnot engine working through the temperature range T\ T Q or T 2 T 3 (Fig. 80). The efficiency is given by a very simple expression, depending only on the dimensions of the cylinder and independent of the heat supply or the maximum temperature. If the heat supply is 214 THE TEMPERATURE-ENTROPY DIAGRAM. increased by using a richer charge, the work done is increased, but not the efficiency. The pressure-volume or temperature-entropy diagram of the standard cycle can be drawn on the assumptions just stated, which, in the opinion of the Committee, do not essentially differ from those in actual engines." The only effect of using the " standard air cycle " instead of the actual ideal cycle will be in each case to change the values of the ideal and cylinder efficiencies a few per cent. Thus the ideal efficiency varies as shown in the following table: (Clearance + piston displacement) l / clearance \k-l ^Clearance + piston displacement / fc=1.40 k=l.37 2 0.246 0.225 3 0.36 0.332 4 0.43 0.399 5 0.47 0.45 7 0.55 0.51 10 0.61 0.57 20 0.70 0.67 100 0.85 0.82 In ordinary commercial gas-engines of moderate size, the cylinder efficiencies, referred to the air-engine standard, vary from 0.5 to 0.6. A comparision of the efficiencies of actual engines and the air-engine standard is given in Mr. Dugald Clerk's paper, "Recent Develop- ments in Gas-engines," in Min. Proc. Inst. C. E., Vol. cxxiv, p. 96. THE GAS-ENGINE INDICATOR CARD. 215 In discussing the effect of size, speed, clearance, time of firing, etc., upon engines using the same fuel, the air-standard cycle can be used to good advantage, but in studying the heat losses due to different fuels, etc., an error is liable to be introduced unless the proper value of k is used for each mixture. ir o FIG. 80. Cylinder Efficiency. The discrepancy between the ideal and the actual cards is due to three influences : First, the shrinkage due to combustion would cause the expansion line of the ideal card to occur at pressures reduced proportionally to its magnitude. Less work is. thus done during the expansive stroke while the work of compression is undiminished, thus reducing the net work obtainable from the cycle. This new 216 THE TEMPERATURE-ENTROPY DIAGRAM. expansion line would occupy some such position as xx That portion of the ideal card beyond xx is, therefore from the nature of the substance unattainable. Fig. 80. Next, there is the loss due to incomplete combustion. This may be caused by a mixture of such proportions that the fuel cannot all burn, or by one in which the combustion is retarded so much as to be unfinished at exhaust; in either case the full heat value of the charge is not developed in the cylinder. Although such a charge does not burn at constant volume, but is burning throughout part or all of the expansive stroke, and cannot thus be represented on the ideal cycle, yet the magnitude of the lost heat may be represented by drawing yy so that the area between xx and yy equals this loss. The area under ly is thus equal to the heat actually generated in the cycle. Finally, there is the loss due to heat interchange between the gas and the metal which produces the discrepancy between that portion of the ideal card to the left of yy and the actual card. It is almost never possible to locate xx and yy, as the amount of shrinkage in the cylinder has to be estimated from the analysis of the burnt gases in the exhaust- pipe. But with delayed combustion the gas may not be the same in the two places. The best that can be done is to draw xx to 'correspond to the shrinkage for complete combustion. The losses due to incomplete combustion and to heat THE GAS-ENGINE INDICATOR CARD. 217 interchanges with the cylinder walls are thus com- bined, due to our inability to locate yy. But by suitably regulating the percentage mixture it is possible nearly to eliminate the loss due to incomplete combustion, although as the governor controls the percentage mixture either of air and gas, or of mixture and burnt products, slow-burning charges sometimes result. The cylinder efficiency, i.e., the ratio of the actual to the ideal card, is thus limited by two different factors, the physical properties of the fuel shrinkage and rate of flame propagation and the influence of the cylinder walls. It is in tracing out the heat interchanges be- tween the charge and the cylinder waUs that the T(j>- analysis is of greatest value. Actual Indicator Cards. To plot the ^-projection of a gas-engine indicator card the temperature and entropy for a sufficient number of points can be calcu- lated from the equations 10 g? = c v loge PX+CP loge v x constant, where T x and $ x of the desired point are expressed in terms of the observed pressures and volumes as measured on the indicator card. 218 THE TEMPERATURE -ENTROPY DIAGRAM. This computation presupposes a knowledge of T , but if this is unknown it can be assumed, and then the the 7^-plot will give relative values of T and < but not absolute values. As this operation must be re- peated for a number of points large enough to deter- THE GAS-ENGINE INDICATOR CARD. 219 mine the curves with sufficient accuracy, it requires much time and patience. The method of graphical projection from the pv- into the !T<-plane, as explained in Chapter II, was used in drawing Figs. 81, 82, and 83. The indicator 220 THE TEMPERATURE-ENTROPY DIAGRAM. cards were taken from the 36 horse-power gas-engine at the Institute: Boston illuminating gas being used as fuel. Fig. 81 represents an average card with a good Atmospheric FIG. 83. Card taken with weak spring, showing compression line, and negative work during admission and exhaust. mixture and well timed ignition, while Fig. 82 shows the result of using a weak mixture and late ignition. Fig. 83 was taken with a weak spring to show more clearly the character of the exhaust, admission, and compression strokes. The small oscillations are chiefly due to the weak spring. THE GAS-ENGINE INDICATOR CARD. 221 Heat Interchanges between Gas and Cylinder. There is a continuous transmission of heat through the cylinder walls to the jacket water or to the air-cooling surface which may amount to from 25 per cent, to 40 per cent, of the heat of combustion. This prevents the temperature and pressure in the cylinder from ever attaining their theoretical, values. Besides this heat which is thus lost there is a second quantity which passes first to the walls and then back to the gas. We might perhaps expect to find this small compared with the condensation and re-evap- oration in a steam-cylinder due to the less rapid inter- change of heat between gas and metal than between wet vapor and metal, but in the gas-engine we must remember that we are dealing with much greater tem- perature differences. The compression line in the T^-plane slants usually first to the right, showing the transference of heat from the hot metal to the cold charge. Continued compression, however, raises the temperature of the charge until it becomes equal to the mean tempera- ture of the cylinder walls. Beyond this point the com- pression curve slants to the left, as the flow of heat is now from the gas to the metal. As the piston approaches the end of the compressive stroke its speed is slow, so that the rate of increase of tempeature due to com- pression is slow, but the temperature difference between gas and metal being large the loss of heat, arid hence 222 THE TEMPERATURE-ENTROPY DIAGRAM. rate of temperature decrease, due to heat transference, is large. Thus the compression curve approaches more and more closely to the isothermal as the conduction loss approaches the compression gain. It may some- times happen that the loss exceeds the gain and then the temperature actually decreases towards the end of compression. The magnitudes of the heat transfer- ence for any case may be determined by planimetering the areas under the respective curves and expressing in heat units. If the ignition occurs before the end of the com' pression stroke, and if the combustion is not finished until after the piston has passed the dead-center, the combustion line at first approaches the constant-volume curve of the ideal cycle, becomes tangent to it at the moment dead-center is passed, and then proceeds to fall off from it until combustion is finished. Figs. 81 and 82 show the character of the combustion curve for different times of firing. The expansion line varies widely in character, accord- ing to the percentage mixture of the charge, the time of firing, and the speed of the engine. For slow- burning mixtures (slow burning with reference to the time duration of a revolution) the line may slope con- tinuously to the right, showing the constant addition of heat up to the moment of release (Fig. 82). Mix- tures in which the combustion is finished shortly after the beginning of expansion give cards of approximately THE GAB-ENGINE INDICATOR CARD. 223 the character shown in Fig. 81. At first the expansion line frequently shows decreasing entropy due to the rapid interchange of heat between the incandescent gas and the relatively cold metal. The inner layer of metal soon reaches the temperature of the gas and then as the temperature of the gas is decreased by further * expansion the inner surface of the cylinder possesses the highest temperature and heat flows from it in two directions, outwards towards the cold jacket and in- wards to the cold gas. This last change manifests itself by increasing entropy during the latter part of the expansion. Sometimes this second part of the expansion line becomes practically isothermal and thus lends credence to the theory of " after burning " this isothermal process being apparently at the highest temperature at which recombination of the dissociated elements may occur. The character of the exhaust line is of no significance, as it does not represent the history of a fixed quantity of substance. Its sole importance is to close the dia- gram and thus to make the area of the T^-diagram the heat equivalent of the work recorded by the indicator card. CHAPTER XI. THE TEMPERATURE-ENTROPY DIAGRAM APPLIED TO THE NON-CONDUCTING STEAM-ENGINE. THE Carnot cycle for steam gives a very good pv-d\a- gram, and hence there are not the same mechanical objections to its adoption as in hot-air engines. But, due to the physical change in the working fluid, a differ- FIG. S4. ent cycle has proved to be more feasible. In the Carnot engine the steam at condition d, Fig. 84, would be compressed adiabatically to a with the change in 224 NON-CONDUCTING STEAM-ENGINE. 225 condition from x d to x a . The isothermal expansion ab occurring by the application of heat to the cylinder produces the further change in condition to x&. The cycle is finished by the adiabatic expansion be and the isothermal compression cd with the cylinder in con- tact with the refrigerator. The card of the ideal engine differs materially from this. (1) The line ab, Fig. 85, represents the admission a 1,1 FIG. So. of steam of condition xi into the cylinder up to the point of cut-off. This is forced in by the vaporiza- tion of an equivalent amount in the boiler, so that the T<-curve is the same as in the Carnot engine. (2) The adiabatic line be represents the expansion of the steam admitted along ab plus the amount already in the clearance space at the moment of admission a. (3) During exhaust the piston simply forces out into the 226 THE TEMPERATURE-ENTROPY DIAGRAM. condenser all of the steam taken during admission, but the quality of the remaining portion is the same at compression as at the beginning of release. (4) The part confined in the clearance space is then compressed along da in the pr-plane up to the initial pressure, i.e., back along the curve cb in the 7^-plane. Hence, in a non-conducting engine, the amount of steam con- fined in the clearance space is immaterial, as its expan- sion and compression occur along the same adiabatic and do not affect the heat consumption. That part of the steam exhausted during release, however, passes into the condenser and there con- denses and gives up its heat to the cooling water. This is represented by cd. From the condenser the water is forced into the boiler by means of a feed-pump, and is there warmed from d to a and vaporizes from a to 6. The pv-di&- gram gives a history of the work performed per stroke and is confined entirely to the events in the cylinder. The !T-diagram, however, represents the heat cycle, and consists of events occurring ia three different places, da and ab represent the heating of the feed- water and its evaporation at working pressure in the boiler, be represents the adiabatic expansion in the cylinder of the engine, and cd the discharge of heat to the condenser. If it were desired to make this cycle into a Carnot> the condensation would have to stop at d f and the NON-CONDUCTING STEAM-ENGINE. 227 feed-pump arranged to compress the mixture adia- batically to a. Suppose each engine to use one pound of steam of con- dition x^ Fig. 85, per stroke, then the efficiency will be 0) Carnot: ^"^I^'^l area abnm area d'cnm T l T 2 t area abnm T l 6) Non-conducting or Rankine engine: q l +x b r l -q z - An inspection of the diagram shows at once that ^ Carnot > ^ Rankine' It is evident also that for any given boiler pressure, the less the amount of moisture in the steam the smaller the difference between the Carnot and the Rankine cycles. Increased Efficiency by Use of High-pressure Steam. If the same quantity of heat be supplied per pound of steam under constantly increasing pressure the state point, 6, Fig. SO, will assume the successive posi- 228 THE TEMPERATURE-ENTROPY DIAGRAM. tions b f , b", etc., along the constant heat curve W, and at the same time the state point c, representing the condition at the end of the adiabatic expansion to the back pressure, moves towards the left into the successive positions c', c", etc. Now the areas under cd, c'd, c"d, etc., represent the quantity of heat dis- FIG. 83. charged to the condenser under the different condi- tions. Therefore the greater the pressure or the higher the temperature at which a given quantity of heat is supplied to the engine, the smaller the fractional part rejected to the condenser, that is, the larger the por- tien turned into work and the greater the efficiency. The Tp-curvc shows that at high pressures the pressure increases much more rapidly than the tem- perature, and hence the necessary strength, weight, and cost of the engine will increase more rapidly than the gain in efficiency. THE NON-CONDUCTING STEAM-ENGINE. 229 It should be noted, however, that there is also an upper limit to the theoretical gain in efficiency from increasing the initial pressure. Although our knowledge of the properties of saturated steam above 310 pounds pressure is inexact and largely obtained by extra- polation, nevertheless it suffices to define approxi- mately the contour of the water and steam lines up to their junction at the critical temperature (see table on p. 388). The complete diagram for saturated steam is approximately as shown in Fig. 87. There is no doubt that for a while increasing the pressure increases the efficiency, but at the same time the heat of vaporiza- tion is diminishing with increasing rapidity (approach- ing zero as a limit at the critical temperature), so that, as the water and steam lines converge, the discrepancy between the Rankine and the Carnot cycles grows more and more marked. From being nearly identical at lower pressures the Rankine efficiency finally attains to only about one half the Carnot efficiency. For dry steam the point at which the Rankine efficiency reaches its maximum value appears to be at about a pressure of 2000 pounds, the exact point varying somewhat with the back pressure. Table of Rankine Efficiencies. The table of efficiencies on page 231 shows very clearly the relative gains to be expected from increase of boiler pressure or decrease of back pressure. Thus starting, for example, with an absolute boiler pressure of 70 pounds and run- 230 THE TEMPERATURE-ENTROPY DIAGRAM. -| 500 I 450 ft 428 2 400 HSOO 2 5 250 8.200 H 150 100 50 33 CriticaHemperatnre^ Range of Peabody's Steam Tables 0.6 0.8 1.0 1.2 1.4 Units of Entropy. FlO. 87. 1.6 1.8 2.0 2.2 THE NON-CONDUCTING STEAM-ENGINE. 231 RANKINE'S EFFICIENCY WITH DRY SATURATED STEAM FOR DIFFERENT INITIAL AND FINAL PRESSURES. Initial Conditions. Back Pressures, Pounds Absolute. pi h 20.0 14.7 5 4 3 2 l 5.99 170 0.3 1.7 3.4 5.9 9.7 7.51 180 2.7 4.1 5 9 8 2 11 8 9^34 190 4.2 5.6 7.3 9.6 13.1 11.53 200 5.6 6.9 8.6 10.8 14.3 14.13 210 .... 6.9 8.2 9.9 12.1 15.5 14.70 212 7.2 8.5 10.1 12.4 15.6 17.19 220 i.2 8.3 9.5 11.1 12.7 16.6 20.78 230 '6!3 2.6 9.5 10.8 12.3 14.4 17.6 25.0 240 1.7 3.9 10.7 11.9 13.4 15.4 18.6 29.8 250 3.1 5.3 11.8 13.0 14.4 16.5 19.6 35.4 260 4.4 6.5 12.9 14.1 15.6 17.5 20.6 41.8 270 5.6 7.7 14.1 15.2 16.6 18.5 21.5 49.2 280 6.9 8.9 15.1 16.2 17.6 19.4 22.4 57.5 290 8.2 10.0 16.0 17.1 18.5 20.4 23.2 67.0 300 9.1 11.1 17.0 18.1 19.5 21.2 24.0 77.6 310 10.2 12.1 17.9 19.0 20.3 22.1 24.8 89.6 320 11.3 13.2 18.8 19.9 21.1 22.8 25.7 103.0 330 12.3 14.1 19.7 20.7 21.9 23.7 26.3 117.9 340 13.2 15.1 20.5 21.5 22.7 24.4 27.0 134.5 350 14.2 15.9 21.3 22.3 23.5 25.1 27.7 152.9 360 15.1 16.9 22.0 23.0 24.2 25.8 28.4 173.2 370 16.0 17.7 22.8 23.7 25.0 26.5 29.0 195.5 380 16.8 18.5 23.5 24.4 25.5 27.2 29.6 220.0 390 17.6 19.2 24.2 25.1 26.2 27.8 30.2 246.9 400 18.4 20.0 24.8 25.8 26.9 28.4 30.7 276.3 410 19.1 20.8 25.5 26.4 27.5 29.0 31.3 308 . 5 420 19.8 21.4 26.1 27 28.1 29.6 31.8 336.2 428 20.4 22.0 26.6 27.5 28.6 30.0 32.2 422.4 450 21.9 23.3 27.9 28.7 29.7 31.1 33.3 678.5 500 24.8 26.2 30.6 31.1 31.2 33.7 35.4 9.-)() . 1 540 27.3 28.0 31.9 32.7 33.6 34.8 36.7 1212 570 28.0 29.2 32.9 33.6 34.5 35.7 37 . 5 1516 600 28.9 30.0 33.6 34.3 35.2 36.3 38.1 1867 630 29.5 30.6 34.1 34.7 35.5 36.6 38.4 2137 650 29.6 30.7 34.1 34.7 35.5 36.6 38.3 2431 670 29.3 30.3 33.6 34.3 35.0 36.0 37.7 2748 690 21.4 22.7 26.8 27.4 28.4 29.6 31.6 2882 698 20.8 22.0 25.6 26.3 27.1 28.2 30.1 232 THE TEMPERATURE-ENTROPY DIAGRAM. ning non-condensing the Rankine efficiency is found to be 11.4 per cent. If the back pressure is lowered to 1 pound absolute the efficiency becomes 24.4 per cent or is more than doubled, while if the initial pressure is raised to 330.2 pounds absolute the efficiency becomes only 22.0 per cent. The combination of both changes results in an efficiency of 32.2 per cent. Thus in the case cited decreasing the back pressure 13.7 pounds produces a greater gain than that obtained by increas- ing the initial pressure by 266 pounds. This illustrates the great gain in efficiency to be obtained by running an engine condensing. The Low-pressure Turbine. Steam exhausted at atmospheric pressure is still capable of developing 15.6 per cent of its total heat into work if used in a turbine exhausting at 1 pound absolute, which is equal to the work to be obtained from a non-condensing engine taking steam at 128.3 pounds absolute. To utiMze to full advantage the heat in low pressure steam, a reciprocating engine would need to be exceed- ingly large and thus possess large friction losses, where- as a turbine is especially adapted for low pressures as a large part of its friction is due to the windage of the rotor and this diminishes as the density of the working fluid diminishes. The ideal unit for economical power pro- duction would therefore seem to be a reciprocating engine for the high pressures combined with a turbine for the low pressures, thus combining the best conditions for both. NON-COXDUCTIXG STEAM-ENGINE. 233 That is, since the clearance between the blades and guides of a turbine must possess the same absolute value in all stages, the clearance area relatively to the blade area must be larger in the high pressure stages than in the low, so that from this view point alone the efficiency of the high pressure stages might be expected to be less than that of the low pressure stages. But this is just the condition under which steam can be most advantageously used in a reciprocating engine as its small volume permits of small cylinders and high mean effective pressures. Thus a combined unit using a reciprocating engine where the turbine is least efficient and using a turbine where the reciprocating engine is least efficient, may be expected to give a higher total thermal efficiency than either type of motor used alone throughout the entire range of pressure. Effect of Different Substances upon the Rankine Efficiency. The discrepancy between the Rankine and Carnot cycles is caused by the slope of the liquid line. The larger the specific heat of the liquid the greater the growth of entropy during rise of temperature and the more the liquid line inclines from the isentropic. That is, other things being equal, the substance possess- ing the smallest specific heat would give the maximum value of the Rankine efficiency. (See LI and Z/ 2 , Fig. 88.) The effect of the sloping liquid line is further enhanced 234 THE TEMPERATURE-ENTROPY DIAGRAM. or diminished by the relative positions of the liquid and dry vapor lines, V\ and V 2 , Fig. 83. Thus if the latent heat of vaporization were doubled the discrepancy between the Carnot and Rankine cycles would for the same specific heat of the liquids be about halved, etc. That substance possessing the smallest specific heat and the largest latent heat of vaporization will give the maximum value of the Rankine efficiency between any two temperatures. The vapor tables show that these properties are varying simultaneously, so that it becomes \ \ FIG. 88. a special problem for any given temperature range to determine which of the available fluids would give the maximum Rankine efficiency. For the engineer, however, the thermal efficiency is but one of many factors. Thus if the same power is to be developed with the different fluids a different weight of each substance must be used, so that w(Hi H 2 } may be the same. This means a different pressure and volume range, and thus the size and strength of the NON-CONDUCTING STEAM-ENGINE. 235 engine to be used with each fluid would need to be computed. Finally the cheapness and availability of each must be considered. Gain in Efficiency from Decreasing the Back Pressure. If the initial pressure be kept constant (Fig. 89) and the back pressure be diminished by increasing the vacuum, the heat taken up in the boibr by jach pound m, ,m FIG. 89. of steam will be increased from dabnm to, say, d'dbnm', and the heat discharged to the condenser will diminish from dcnmto d'c'nm'; that is, the efficiency increases. Again referring to the pT-curve, it is clear that at low pressure the temperature decreases much more rapidly than the pressure, so that a small decrease in pressure means a considerable increase in efficiency. This is at once evident from an inspection of the efficiency for a Carnot cycle, y = the Rankine cycle, '' =l-^r, but the expression for 236 THE TEMPERATURE-ENTROPY DIAGRAM. does not show the influence of .the upper or lower temperature very clearly. The efficiency may easily be expressed as a function of the upper and lower temperatures by assuming the value of the specific heat of water to be constant and equal to unity. Thus and The value of the last term decreases, hence the efficiency increases, (1) as x b approaches unity, (2) as 7\ increases, and (3) as T 2 decreases. Gain in Efficiency from Using Superheated Steam. To avoid the introduction of excessively high pressures superheated steam is being used more and more. Accord- ing to the Carnot cycle the gain in efficiency is equally great whether superheated or saturated steam of the NON-CONDUCTING STEAM-ENGINE. 237 same temperature is used, but the Rankine cycle shows that the theoretical gain to be expected from super- heated steam is but slight. The portion be of the Rankine cycle, Fig. 90. repre- sents the addition of heat in the superheater, and ec the expansion from superheated to saturated steam FIG. 90. in the cylinder; the rest of the cycle is as previously described. The heat g x 5 2 + r i + C P(^~^I) is received along the line of varying temperatures dabe, while in the Carnot cycle an equal quantity of heat (area e//Zn = area dabenrn) is all received at the upper temperature t 8 . Hence the efficiency of the Rankine is now much less than that of the Carnot cycle working between the same tem- perature limits and the discrepancy increases as the degree of superheating increases. 23S THE TEMPERATURE-ENTROPY DIAGRAM. The analytical formulae for this case are: c p (t g -t 1 ] -x c r 2 x c r 2 7\ r, TV = 1 -^Tf, m l TTf, 7fr\~ (approximately). This shows an increase in efficiency "with increasing T a , but only of small amount. It is evident, then, that the great gain obtained by using superheated steam must be looked for in the overcoming of certain defects inherent in an actual engine. The use of steam expansively entails a cool- ing of the working fluid and hence of the cylinder walls containing it. This effect is increased by release occurring before the expansion has reached the back pressure, and is only partially counteracted in part of the cylinder walls by the heating effect produced during compression. Thus the entering steam under- goes partial condensation before the cylinder walls have been brought up to its temperature; that is, each pound of steam, instead of occupying the volume which it had in the steam-pipe, now occupies a reduced volume proportional to the condensation. And hence, instead of obtaining the total area abed, Fig. 91, only the fractional part akld can be utilized. Thus the area kbcl has been subtracted from the numerator of the expression for efficiency. This condensation may NON-COND VOTING STEA M-ENGINE. 239 range as high as from 20 per cent, to 50 per cent, of the total steam. The addition of superheated steam may result in the superheat bemn being sufficient to supply the heat taken by the cylinder walls and thus preventing the condensation and making available the area kbcl. The econxSmy is further increased as the steam at the FIG. end of expansion has less moisture in it and thus ab- stracts less heat from the cylinder walls during release. That is, the conduction of heat through a vapor occurs but slowly, while water in contact with the metal will abstract large quantities of heat during evaporation. The leakage loss is also less with superheated steam. Loss in Efficiency Due to Incomplete Expansion. If steam be taken throughout the entire stroke the indicator-card is represented by abed (Fig. 92). The 240 THE TEMPERATURE-ENTROPY DIAGRAM. drop in pressure be is equivalent to cooling at constant volume and may be represented on the 7^-diagram by the curve of constant volume be. If the same quantity of steam be taken successively into larger cylinders, so that an increasing degree of expansion is obtained, this will be represented by be, be', be", etc., in both diagrams. The areas B, C, D show the extra work performed per pound in the pi>-plane, and the FIG. 92. extra heat utilized in the 7 7 0-plane respectively, as the expansion progresses from initial to final pressure. As in the gas-engine, so in the steam-engine it seldom pays to carry the expansion completely down to back pressure, because the slight gain from c' to c" is more than counterbalanced by the increased size, cost, and weight of the engine, friction, and radiation losses, etc. For such incomplete expansion the expression for the efficiency of the Rankine cycle is found as follows: NON-CONDUCTING STEAM-ENGINE. 241 = (q c +x c p c ) - (q 2 +x e p 2 ) +x e r 2 = q c + x c r c q 2 Ax c u c (p c p z ), Qi-Qt -, We + q c - q* -Ax c u c (p c - p a ) ' = ^ 1- Loss in Eflriciency from Use of Throttling Governor. The throttling governor acts by wire drawing the steam to a lower pressure. Less steam is thus taken per stroke, as the volume is increased by both the reduced pressure and the increased value of x. A series of cards for dropping pressure is shown in Fig. 93. The FIG. 93. 7V/>-diagram shows the decreased efficiency per pound of steam for the same cases. During wire drawing ths heat remains the same, but the entropy increases, as 242 THE TEMPERATURE-ENTROPY DIAGRAM. the process is irreversible. The heat rejected increases as the initial pressure drops, so that of the total heat brought in a smaller quantity is changed into work and the efficiency of the plant decreased. Pounds of Water per Horse-power-hour for the Ran- kine Cycle. It is customary to quote the economy of an engine in pounds of water per horse-power per hour. In time one acquires a rough knowledge of the varia- tions in water consumption for engines of different sizes under varying conditions, but this is at the best a crude method of comparison between the results of various engines unless one possesses a standard of reference and can show how closely the actual engines have approached the ideal conditions in each case. The Rankine cycle represents the thermal history of the working fluid in a plant containing no heat losses. The maximum amount of work is thus obtained from each pound of steam and, therefore, the steam con- sumption of the Rankine cycle represents, for any given set of conditions, the minimum amount which must be used per hour to develop a horse-power of work. A horse-power is equal to 33,000 foot-pounds of work per minute, so that the heat equivalent of a horse- power is 2545 B.T.U. per hour. The heat utilized per pound in developing useful work is equal to the differ- ence of the total heats at the beginning and end of the isentropic expansion. The pounds of steam required NON-CONDUCTIXG STEAM-ENGIXE. 243 to develop one horse-power-hour of work are therefore 2545 equal to jj TT tiitiz The table on page 244 shows the effect of varying initial anil final pressures upon the water consumption of the ideal engine using dry steam. Pounds of Water per Kilowatt-hour for the Rankine 1000 Cycle. A kilowatt is equal to -=^- H.P., and is there- fore equivalent to 44,240 foot-pounds of work per minute, which is equivalent to the utilization in work of 3412 B.T.U. per hour. The pounds of steam required to develop one kilowatt-hour of electrical energy are 3412 therefore equal to -77 fr- fli fi 2 These values could be readily tabulated if necessary but they may also be obtained by multiplying the corresponding values for pounds of steam per horse- power-hour by the factor 1.341. Use of Entropy Diagram or Tables. The labor involved in computing the ideal water consumption and the Rankine efficiency for any given case can be greatly reduced by using either a temperature-entropy chart containing constant-heat curves or Peabody's entropy tables. The method of using to obtain the water consumption is evident from the expression 2545 Hi- H a ' 214 THE TEMPERATURE-ENTROPY DIAGRAM. POUNDS OF DRY STEAM PER HORSE-POWER-HOUR FOR RANKINE CYCLE. Initial Condition. Back Pressures, Pounds Absolute. Pi h 20.0 14.7 5 4 3 2 i 5.99 170 1340 150.6 72.9 42.0 25.0 7.51 180 94.3 60.8 42.3 29.6 20.2 9.34 190 59.3 44.4 33.6 25.3 18.10 11.53 200 44.8 35.8 28.5 22.4 16.52 14.13 210 36.0 30.0 24.8 20.0 15.28 14.70 212 34.7 29.2 24.2 19.5 15.08 17.19 220 225^2 30.2 25.8 21.9 19.0 14.20 20.78 230 908 ! 8 99.0 26.2 22.9 19.7 16 . 63 13.32 25.0 240 156.2 66.1 23.2 20.6 18.04 15.46 12.55 29.8 250 86.3 49.2 20.8 18.74 16.78 14.40 11.88 35.4 260 59.9 39.4 18.9 17.24 15.48 13.56 11.28 41.8 270 46.4 33.3 17.43 15.99 14.45 12.80 10.76 49.2 280 38.5 28.8 16.19 14.92 13.61 12.14 10.30 57.5 290 32.6 25.5 15.18 14.11 12.91 11.55 9.92 67.0 300 28.4 23.0 14.29 13.33 12.24 11.07 9.57 77.6 310 25.3 21.0 13.52 12.66 11.69 10.61 9.24 89.6 320 22.9 19.3 12.84 12.06 11.20 10.22 8.90 103.0 330 20.9 17.93 12.26 11.57 10.78 9.85 8.67 117.9 340 19.4 16.78 11.75 11.10 10.40 9.52 8.43 134.5 350 18.05 15.83 11.27 10.68 10.02 9.24 8.20 152.9 360 16.96 14.93 10.87 10.32 9.71 8.96 7.97 173.2 370 15.95 14.19 10.47 10.00 9.40 8.72 7.81 195.5 380 15.18 13.54 10.16 9.69 9.15 8.49 7.63 220.0 390 14.42 12.99 9.83 9.40 8.90 8.29 7.46 246.9 400 13.80 12.46 9.57 9.15 8.68 8.09 7.32 276.3 410 13.25 12.00 9.32 8.93 8.48 7.92 7.18 308.5 420 12.74 11.60 9.08 8.71 8.29 7.76 7.05 336.2 428 12.38 11.30 8.91 8.55 8.14 7.64 6.95 422.4 450 11.4 10.55 8.43 8.12 7.76 7.31 6.69 678.5 500 10.1 9.46 7.73 7.52 7.43 6.85 6.31 956.1 540 9.58 9.00 7.52 7.29 7.01 6.66 6.18 1212 570 9.37 8.83 7.45 7.23 6.97 6.65 6.175 1516 600 9.40 8.87 7.54 7.32 7.06 6.74 6.28 1867 630 9.71 9.19 7.82 7.61 7.35 7.02 6.54 2137 650 10.2 9.96 8.23 8.00 7.73 7.38 6 . SS 2431 670 11.2 10.6 9.89 8.76 8.46 8.07 7.50 2748 690 18.1 16.7 13.2 12.7 12.1 11.3 10.3 2882 698 25.4 23.2 18.2 17.4 16.5 15.4 13.9 A'OX-COXDUCTIXG STEAM-ENGINE. 245 To obtain the Rankine efficiency we have which may be regrouped and written in the form, in which all the quantities may be read directly from the chart or the tables. Heat Consumption of the Rankine Cycle in B.T.U. per Horse-power-minute. It is customary to rate the economy of an engine in terms of the heat units required to produce one indicated horse-power per minute. As a standard of comparison the heat consumption of the Rankine cycle using dry saturated steam is given in the table on page 246. The heat equivalent of one horse-power is 42.42 B.T.U. per minute. The heat required by the engine per minute may be determined by dividing the heat utilized by the thermal efficiency, therefore, 4242 B.T.U. per I.H.P. per minute = ^l Or, again, the heat supplied to the engine per horse- power-hour equals the pounds of steam per horse-power- hour multiplied by the hc?at absorbed per pound of water in the boiler^ therefore, Lbs. steam X (Hi g 2 ) B.T.U. per I.H.P. per minute = =. 246 THE TEMPERATURE-ENTROPY DIAGRAM. B.T.U. PER HORSE-POWER-MINUTE FOR RANKINE CYCLE USING DRY STEAM. Initial Conditions Back Pressure, Pounds Absolute. Pi t, 20.0 14.7 5 4 3 2 1 5.9 170 22170 2516 1232 720 439.3 7.5 180 .... 1582 1031 724 515 359.5 9.3 190 1000 755 578 441.3 323.4 11.5 200 758 611 492.5 392.5 296.9 14.13 210 611 514 429.3 351.4 274.3 14.70 212 590 500 420.0 342 . 6 271.4 17. 19 220 3652 514 444.4 380.8 335.0 256.4 20.78 230 14550 1612 448 394 . 8 344.5 294.4 241.4 25.0 240 250S 1079 397 356 . 6 315.9 274.8 228.0 29.8 250 1391 807 359 325.6 294.7 262.6 216.5 35.4 260 969 648 330 300.5 272.9 242.5 206.3 41.8 270 752 549 302 279.4 255.5 229.5 197.4 49.2 280 619 477 281.3 262.2 241.3 218.4 189.5 57.5 290 520 423 264.6 247.7 229.5 208.3 182.9 67.0 300 466 382 249.6 235.1 218.1 200.1 176.8 77.6 310 416 349.5 236.8 223.9 208.8 192.3 171.2 89.6 320 377 322.5 225.6 213.7 200.7 185.7 165.3 103.0 330 345 300.5 215.8 205.5 193.6 179.3 161.4 117.9 340 321 281 . 9 207.3 197.7 187.4 173.7 157.1 134.5 350 299 266.8 199.4 190.5 180.7 169.0 153.2 152.9 360 282 251.9 192.6 184. 5 175.4 164.2 149.1 173.2 370 266 240.1 186.0 179.1 170.0 160.1 146.5 195.5 380 253 229.4 180.7 173.8 166.2 156.1 143.3 220.0 390 241 215.6 175.2 168.9 161.6 152.7 140.4 246.9 400 231 211.9 170.8 164.6 157.9 149.6 138.0 276.3 410 222 204.5 166.5 160.9 154.4 146.3 135.6 308.5 420 214 197.9 162.5 157.0 151.2 143 . 5 133.3 336.2 428 208 193.1 159.7 153.9 14S.6 141.6 131.6 422.4 450 193.7 181.8 152.3 147.8 142.8 136.3 127.4 678.5 500 170.8 162.1 138.9 136.3 136.1 126.0 119.7 956.1 540 158.8 151.7 132.9 129.9 126.5 121.8 115.4 1212 570 151.8 145.5 128.8 126.2 123.0 119.1 113.2 1516 600 146.9 141.1 126.2 123.7 120.6 114.2 111.4 1867 630 143 . 7 138.6 124.5 122.2 119.4 116.0| 110.6 2137 650 143.2 138.0 124 . 5 122.3 119.5 116.1 110.9 2431 670 145.0 139.9 126.4 123.8 112.3 117.8 112.6 2748 690 198.6 1S7.0 159.0 154.6 149.5 143 . 2 134.2 2882 698 203.4 192.7 165 . S 161.6 156.6 150.2 141.1 NON-CONDUCTING STEAM-ENGINE. 247 The heat consumption may therefore be determined from either of the preceding tables, the results are as shown on page 246. B.T.U. per Kilowatt-minute. The heat equivalent of one kilowatt is 56.9 B.T.U. per minute. The heat re- quired by the engine per minute to produce one kilo- watt when working upon the Rankine cycle is found by dividing 56.9 B.T.U. by the thermal efficiency. Or the same value may be obtained by multiplying the corresponding value for B.T.U. per horse-power-hour taken from the table on page 246 by the factor 1.341. CHAPTER XII. THE MULTIPLE-FLUID OR WASTE-HEAT ENGINE. IN the discussion of the Rankine cycle it was shown how the efficiency of the steam-engine could be increased by raising the temperature of the source of heat or by decreasing that of the refrigerator. Due to the course of the pT-curve a practical upper limit is soon reached in the use of saturated steam due to the rapid increase of pressure at upper temperatures, so that recourse has to be taken to superheated steam. Again, in reducing the back pressure a slight drop in pressure means a large drop in the exhaust temperature, but a practical limit is soon reached beyond which it does not pay to carry a vacuum. Theoretically, at least, the efficiency could be increased by using for the higher temperatures some fluid (A") having a smaller vapor pressure than saturated steam, and for lower temperatures some fluid (Y) having a greater vapor pressure than saturated steam at the same temperature. That is to say, the first fluid X could in a saturated condition be heated to the tempera- ture now common for superheated steam, and then be 248 MULTIPLE-FLUID OR WASTE-HEAT EXGL\E. 249 allowed to expand in a cylinder down to some lower temperature at which the pressure of saturated steam would not be excessive. The surface condenser for this X-fluid would be at the -same time the boiler for the stealh. After the steam had expanded through two or three cylinders, it in turn would exhaust into a second surface condenser, which would be the boiler for the next fluid, Y, in the series. Thus the working substance in each case is condensed at the temperature of its own exhaust and fed back to its own boiler at this temperature. The heat of vaporization which the first fluid rejects must warm up the second as it is fed into the boiler and then vaporize part of it, so that heat rejected = heat received, OT ^exhaust - (5 + * r )boiler "^condenser" For the sake of simplicity suppose at first that two such fluids X and Y as described could be found, and further that their liquid and saturated -vapor lines coin- cided in the T<-plane with those for water, but that the Tp-curves are entirely different. Suppose, further, that p t and p 2 (Fig. 94) represent respectively the highest pressure which it is convenient to use, and the lowest pressure which can be obtained in a vacuum. The range of temperatures when satu- rated steam alone is used is limited between ^ and t 2 . If, however, the fluid X is first used the upper tempera- ture can be raised to t x for the same maximum pres- 250 THE TEMPERATURE-ENTROPY DIAGRAM. sure Pi. Suppose in this problem that the substance Y does not solidify at 32 F., its liquid line would extend to the left of the arbitrary zero for the entropy of water, and so the expansion of this fluid down to p 2 would drop the lower temperature from t 2 down to ty. The liquid A" is fed into its boiler at the temperature t l} and is warmed along ab, receiving the heat qb~q a , equal to area ablm; it is then vaporized at the pres- sure Pi and receives the heat r x equal to bcnl. Its ideal cycle is now completed by adiabatic expansion cd down to some pressure p x (in Fig. 94 p x coincides with p 2 ], corresponding to temperature t lt on the X- MULTIPLE-FLUID OR WASTE-HEAT ENGINE. 251 curve, and condensation along da. The condensed fluid is at the proper temperature to be returned to its boiler. The heat rejected along da, viz., x d r d , must warm up the water fed into the X-condenser at temperature t 3 up to ^ and then vaporize part of it at the upper temperature. Assuming no heat lost, area m'gaen' = area madn. The steam describes the ideal cycle gaef, rejecting in turn the heat under fg, equal to x t r f at some pressure p 3 corresponding to temperature t 3 . In the steam-condenser the fluid Y is first warmed up from the temperature t Y} corresponding to p 2 , to t 3 and then enough vaporized to make area m"kghn" = area m'gfn'. From the diagram the following conclusions can be drawn : Heat received from fuel equals mabcn. Heat utilized by X-engine equals abed. -rffi c v Efficiency of X-engme, ^ Heat rejected by X-engine = area wadn = area m'gaen' = heat absorbed by steam-engine. Heat utilized by steam-engine equals gaef. 252 THE TEMPERATURE-ENTROPY DIAGRAM. Efficiency of steam-engine fj st = -~ - -,. Efficiency of X- and steam-engines combined equals abed +gaef mabcn Heat received by F-engine = heat rejected by steam- engine = m"kghn" = m'gfn'. Heat utilized by F-engine = kghi. Heat rejected by Y-engine = w"A;in". Efficiency of F-engine = ////. Efficiency of all three engines together, abed +gaef +kghi ' The heat rejected has been reduced from madn to m ff kthfednm rf . The great gain in efficiency shown in this assumed case is deceptive. The exhaust temperature has been taken far below freezing. This could not be done unless some cooling mixture could be employed in order to condense the exhaust fluid Y. This would be expensive and probably represent as great an expenditure of work as the increased gain MULTIPLE-FLUID OR WASTE-HEAT ENGINE. 253 recorded by the F-piston, possibly more. In prac- tice the lowest temperature t Y will be governed by that of the cheapest available condensing substance, i.e., the temperature of the cooling water at the power- station . The upper temperature t x will be governed by the materials used in construction. Since x ex r ex = q B q c +x B r B , it follows that x B -p\ot of the actual card. The Indicator-card. Considered as a whole, the indi- cator-card furnishes the following information. The admission line and the exhaust line simply represent the pressure of part of the steam, but do not give any information regarding the specific volume. On the other hand, both the expansion and the compression lines give the history of definite quantities of steam. Thus both the expansion and compression of the clearance steam are recorded, while only the expan- sion of the cylinder feed appears, the compression of the latter occurring in the boiler. The entire card, if plotted directly, can at the best be considered only as the heat equivalent of the work done, but not as the !T<-history of any closed cycle. Thus, for example^ while the projection of the exhaust line gives some such curve as ef'g', which on the !T<-chart indicates con- densation, it is probable that the value of x of the con- fined steam is actually increasing. The difficulties involved in the proper interpretation of the irreversible portions of the indicator-card have led different investigators to make certain assumptions as to the influence of the clearance eteam and as to the possibility of replacing the actual curves by equivalent reversible processes. 268 THE TEMPERATURE-ENTROPY DIAGRAM. The Clearance Steam Considered as an Elastic Cushion. During expansion the clearance steam follows the same laws and variations as the cylinder feed, and this hi general is not the reverse of its history during com- pression. Thus the cycle of the clearance steam, if it could be drawn, would enclose an area representing either positive or negative work. This cycle would then be of especial interest in determining losses ex- perienced by the clearance steam, but as these losses must eventually be charged against the entering steam, the total effect upon the efficiency would be the same if the clearance steam were considered as an isolated elastic cushion expanded and compressed along the same adiabatic. If, then, an adiabatic is drawn through the point of compression on the indicator-card, the horizontal distance from any point on this adiabatic to the corresponding point on the indicator-card shows the volume which the cylinder feed would occupy under the above assumptions. Taking this adiabatic as the line of zero volume, a diagram can thus be con- structed which shows only the variations of the cylinder feed. It is then only necessary to draw on the satura- tion-curve for the weight of steam fed to the cylinder per stroke and the card can be at once projected into the T$-plane. This is, in its essential points, the method adopted by Prof. Reeve in his book on the Thermodynamics of Heat Engines, although he changes volumes so that the reconstructed card represents ACTUAL STEAM-ENGINE CYCLE. 269 the pr-history of one pound of cylinder feed instead of the actual weight in the cylinder. To project his .reconstructed card into the T^-plane, it is only nec- essary to draw the saturation-curve for one pound of steam instead of that for the pounds fed per stroke. Inasmuch as the T ^-projection will be the same in either case it seems somewhat simpler to adopt the first method, viz., to construct the saturation-curve for the number of pounds in the cylinder rather than to redraw the diagram to correspond to one pound of cylinder feed. This method undoubtedly makes possible a deter- mination of the general magnitude and character of the various heat interchanges, but is open to the following objections. The compression line of the card refers to the clear- ance steam alone, so that the deviations from the adia- batic thus obtained refer to itself and not to the cylin- der feed. Furthermore, the reconstructed curve may actually pass to the left of the water line, assuming imaginary values on the T<-plane, and thus give a wrong conception of the condition of this steam, which, instead of being wet, is usually dry or superheated, and lies to the right of the expansion line. Again, the ex- pansion line no longer represents the actual pv- or T<-history of the steam, but an imaginary history which the cylinder feed might have if the clearance steam expanded adiabatically. The entire card thus becomes 270 THE TEMPERATURE-ENTROPY DIAGRAM. to a certain extent imaginary, and in so far is unde- sirable. The Indicator-card Considered as a Reversible Cycle. The area of the card gives the heat changed into work, and this same result may be attained by assuming the clearance steam and cylinder feed to remain in the cylinder, being heated and cooled by external means and thus caused to expand and contract along a rever- sible cycle coincident in shape with the actual card. The original card may thus be projected into the T(f>- plane as soon as the saturation-curve for the total weight of steam has been drawn on it. The expansion line represents the actual history of the substance, but the compression line is entirely imaginary. This method was adopted by Prof. Boulvin in his book, The Entropy Diagram, from which the following two illustrations are copied, with but slight alterations. ABCDEF, in Fig. 97, represents the ^-projection of a certain indicator-card. The line DE represents the actual expansion history of the total steam; the other lines give more or less imaginary values. The small diagram at the right is used to interpret the compres- sion line AB. If W and w represent the pounds of cylinder feed and clearance steam respectively per revolution, the large diagram represents the cycle of W +w pounds of the mixture, while the small diagram represents the entropy of w pounds. Assuming dry steam at compression, the heat rejected by w pounds ACTUAL STEAM-ENGINE CYCLE. 271 during compression is shown by area abb^. The entropy a a = A A is that due to the vaporization of w pounds of water, so that AG must represent that for the W pounds of cylinder feed. Through A and G it is then possible to draw the water line AL and the dry-steam line GX for W pounds. The cycle for the ;<*, FIG. 97. Rankine engine using W pounds per revolution is rep- resented by ALXH, so that the efficiency of the actual engine as compared with that of the ideal engine is ABCDEF ALXH ' In Fig. 98, let L' and L be the liquid lines of W and W +w pounds respectively, and let L" be drawn through 272 THE TEMPERATURE-ENTROPY DIAGRAM. the point of compression a parallel to L' ; and let S and S f be the dry-steam lines corresponding to the water lines L and L" ' . As is evident from Fig. 97, the hori- zontal distance between the liquid line A C and the compression line AB is equal to the entropy of vapori- zation of the clearance steam (for example, A A=atfi } BoB = b b}. In Fig. 98 the horizontal distance between L' and L represents the entropy for W+w W or w FIG. 98. pounds of water. Therefore the horizontal distance from L' to the compression-curve db represents the total entropy of the clearance steam. a'U is thus the curve of zero entropy for the clearance steam, and any curves, such as nb and aL", parallel to a'U represent isentropic changes. The heat rejected along ab is thus shown by abnn^ without the help of any auxiliary diagram. a&U'S'r represents the heat received per cycle in the ACTUAL STEAM-EXGIXE CYCLE. 273 W pounds of cylinder feed, abcdef represents the heat utilized, so that the difference, M, represents the total heat losses. These consist of the exhaust -heat a t aee 1} the heat lost during compression abnn^, and the heat lost during admission. The heat refunded during ex- pansion is represented by dee^. Hence the heat lost during admission equals M a i aee 1 abnn^ +dee^d = cL If the condition of the steam is desired at any point of the expansion de, it is found by reference to the lines L and S and not with reference to L" and S'. Hence the initial condensation changed the state point from S to d f , so that this loss is represented by the area under Sd' and not by that under S'd'. Subtracting this from the total loss during admission there is left cL"gcdd' +&S 1 nS' -njibgaa^ as the heat losses due to wire-drawing and friction. This method is simple and easy of application, as it requires the construction of only two extra curves, L' and S'. Furthermore it gives a very complete account of the clearance steam. The only objection, appar- ently, is that, with the exception of the expansion line de, it gives an entirely false idea of the cycle of the cyl- inder feed, as this in reality is entirely condensed and then heated along aL" in the boiler. 274 THE TEMPERATURE-ENTROPY DIAGRAM. If the cylinder is jacketed, the heat given out by the jacket steam may be indicated at the right of XH (Fig. 97), as shown under XJ. Further, if the radiation loss is known, it may be represented under JR, and then the remaining area under XR will represent the heat given by the jackets to the steam in the cylinder. Separate Cycles for the Cylinder Feed and the Clear- ance Steam. Whatever the assumptions made with reference to the card, its total area must not be changed. Thus, even if an attempt is made to draw separate cycles for both cylinder feed and clearance steam, the compression line of the resultant cylinder-feed card will never exactly coincide with the water line of the T- diagram, so that this line must always remain imagi- nary hi its readings. It is, however, possible to have the reconstructed expansion line represent the true pv- history of the steam. Thus, in place of drawing an isentropic line through the point of compression, draw the polytropic curve pv n = C, where n has the value found for the expansion-curve. Assuming that the clearance steam is expanded and compressed along this line, the card for the cylinder feed can be constructed by assuming this curve to represent zero volume. The 7^-projection of the expansion-curve will thus repre- sent the true average history of the cylinder feed, and the compression line will follow more closely the water line than it does when an adiabatic curve is taken as the new base line. ACTUAL STEAM-ENGINE CYCLE. 275 The cycle for the clearance steam can be found be- tween cut-off and release, and between compression and admission, but the rest of it would be entirely imagi- nary. Possibly it might be continued from admission up to the attainment of initial pressure by assuming adiabatic compression. As may be seen from Fig. 96, the clearance steam contains less moisture at admission and compression than at cut-off and release respec- tively, so that whatever its exact path between admis- sion and cut-off and between release and compression, it must at least show decreasing and increasing values of X respectively, so that its history is the reverse of that shown by the indicator-card itself. It is doubtful if the increased labor involved in the making of such a plot would be recompensed by any added information which could not be obtained by a proper interpretation of the simple method used by Prof. Boulvin. CHAPTER XIV. STEAM-ENGINE CYLINDER EFFICIENCY. THE thermal efficiency of an engine, that is, the frac- tional part of the heat energy received which it changes into useful work, is limited first by the ideal efficiency which is fixed as soon as the initial quality and pres- sure and the back pressure are known, secondly, by the cylinder efficiency which is a measure of the heat losses in the cylinder, and finally, by the mechanical efficiency which is a measure of the mechanical fric- tional losses of the whole engine and is equal to the brake horse-power divided by the indicated horse- power. The thermal efficiency is thus the product of these three separate efficiencies, or 'Jtotal = 'jRankine ' 'Jcylinder ' ^mechanical- In order to make the total thermal efficiency a maximum the engineer must so design the engine that each of the component efficiencies possesses its maximum value.- The factors influencing the Ran- kine efficiency have already been discussed in Chapters XI and XII, and this is, in any case, determined by 276 STEAM-ENGINE CYLINDER EFFICIENCY. 277 the running conditions in the plant. The mechanical efficiency is controlled by workmanship and the choice of proper metals for bearing surfaces and of suitable lubricants. It is thus a mechanical and not a thermal problem, and so falls beyond the scope of this treatise. The cylinder efficiency being the result of purely thermal processes can, as shown in the preceding chapter, be submitted to the temperature-entropy analysis. In order to minimize the heat losses in a steam- engine it becomes necessary to understand the various factors which produce them. The losses are of three kinds : (1) Loss of availability due to throttling during admission and exhaust; (2) External radiation and conduction loss (3) Condensation and re-evaporation inside the cylinder. The throttling losses depend upon the relative areas of piston and steam ports, as well as upon the speed of the piston and the length and crookedness of the ports. In the case of compound engines the length and cross-sectional areas of the receivers and piping between the cylinders must also be considered. For a given engine the higher the speed the greater the throttling loss. The external loss depends upon the difference be- tween the mean temperature of the cylinder walls 278 THE TEMPERATURE-ENTROPY DIAGRAM. and the outside air, upon the area of the radiating surface and the conductivity of the materials used. A high-pressure cylinder will radiate more heat per unit area than a low-pressure cylinder; a steam- jacketed cylinder will lose more heat externally than a non-jacketed cylinder, because of its higher tem- perature and larger surface. This loss is small in comparison with the condensation loss and can be minimized by lagging the cylinder with some suitable insulating material. The condensation and re-evaporation loss is caused by the tendency of heat to equalize the temperature of bodies in thermal contact. Thus the cylinder and piston are exposed to steam fluctuating in tempera- ture between that corresponding to the pressures at admission and exhaust. Investigations have shown that the fluctuations in temperature of the metal extend only a slight distance, two or three hundredths of an inch, into the mass of the cylinder. It is thus almost a surface phenomenon and the mass of metal participating in it is proportional (practically) to the surfaces exposed to the steam. The quantity of heat transferred is proportional to the difference of tem- perature between the steam and metal, the area, and the time. Of course if the time interval is long enough the metal will be raised from the temperature of ex- haust to that of admission, and then the heat required will equal the weight of metal involved times its specific STEAM-ENGINE CYLINDER EFFICIENCY. 279 heat times the change in temperature, but at higher speeds the temperature of the steam fluctuates so rapidly that there is not sufficient time for the metal to arrive at either the temperature of admission or exhaust, and it thus fluctuates between some inter- mediate temperatures. Another way of looking at this is to assume that the metal directly in contact with the steam assumes its temperature immediately and then by conduction successive layers are brought to the same temperature. This requires time, so that the higher the speed the thinner the layer of metal which is involved in the operation. From either point of view the only way to eliminate this heat transfer- ence would be to run the engine at infinite speed. Comparative tests made upon various engines by setting the governors so that they could run at different speeds with the same conditions of cut-off, and of initial and back pressures always show that the condensation loss is nearly proportional to the time required for a revolution. Increased speed means, however, increased port area to prevent increased throttling loss and thus increased area of radiating surface with its consequent loss. The number of revolutions is also limited by various mechanical factors, so that the speed in any given case may be considered as fixed. It therefore remains to investigate the effect upon initial con- densation of the other elements of design at the dis- posal of the designer, viz., 280 THE TEMPERATURE-ENTROPY DIAGRAM. (1) Size of unit; (2) Point of cut-off; - (3) Compounding; (4) Steam jacketing; (5) Use of superheated steam. (1) The mass of steam contained in a cylinder, other things being the same, increases as the cube of the dimensions, while the area exposed to the steam Increases only as the square of the dimensions. Thus the radiating surface per unit weight of steam dimin- ishes as the first power with increasing size. That is, doubling the dimensions, or increasing the volume eight times, reduces the initial condensation per pound of steam in the larger cylinder to one-half its amount in the smaller cylinder; trebling the diameter and stroke reduces the condensation to one-third its origi- nal amount. (2) The position of cut-off in the high-pressure cylin- der controls the weight of steam admitted to the cylinder and therefore the power developed per stroke. The longer the period of admission the greater the horse-power. Does the thermal efficiency or economy increase at the same time, or does it fluctuate, and if so, at what point does it attain its maximum value? Roughly speaking, the cylinder head and the piston and that portion of the cylinder exposed to the steam up to the point of cut-off have their temperatures raised to that of the entering steam, while the tempera- STEAM-ENGINE CYLINDER EFFICIENCY. 281 ture of the rest of the cylinder is raised varying amounts as the temperature of the steam drops during expan- sion. As cut-off changes the heat required to warm the metal also changes slightly, because the portion of the cylinder heated to admission temperature changes with the cut-off, but a large part of the surface, viz., the cylinder-head, the piston, and the inside of the steam-ports, remains the same, so that although the total variation in cooling surface increases and de- creases with the cut-off it is at a much slower rate. The heat required to warm the cylinder per stroke is thus but slightly variable, being nearly the same for wide variations in cut-off, but the weight of steam increases with the cut-off so that the loss of heat per pound diminishes as the cut-off increases. The maxi- mum condensation loss is thus found at early cut-off and the minimum loss when steam is taken through out the entire stroke. Apparently from this point of view the economy of an engine increases with the power developed. If no heat losses occurred in the cylinder, expansion from early cut-off would carry the pressure at the end of the stroke below back pressure, then as cut-off increased the final pressure would also increase, and at some definite point of cut-off would just equal the back pressure. From this point on the final pressure would exceed the back pressure, and, finally, when cut off occurred at the end of the stroke there would 282 THE TEMPERATURE-ENTROPY DIAGRAM. be no drop of pressure whatever. Due to the con- densation during the first part of the expansion the pressure in the actual cylinder drops more rapidly than during adiabatic expansion, so that the cut-off which will give a pressure at the end of expansion just equal to the back pressure is somewhat later in the actual than in the theoretical case. If cut-off occurs later than this the pressure at release drops, without the performance of work, down to the back pressure; that is, the availability of its internal energy is wasted. FIG. 99. Thus the more cut-off exceeds that value at which the expansion would bring the pressure down to exhaust pressure the greater the wasted internal energy.' Con- sidered from this aspect alone the engine would first run with a negative loop in the card, so that its economy would increase up to the point at which the loop dis- STEAM-ENGINE CYLINDER EFFICIENCY. 283 appeared, and from this point onwards the economy would decrease slowly as the power increased. There are thus two factors exerting opposing in- fluences: initial condensation which diminishes, and loss of internal energy which increases with the load. The mutual variations of these factors cannot in the present state of our knowledge be predicted from point to point, but it is at least evident that there is some cut-off at which the sum of these two losses will prove a minimum (Fig. 99). To determine this point of most economical cut-off for any given engine economy tests must be run at varying loads for the same boiler and back pressures. Then the efficiency and cut-off must be plotted as coordinates and the minimum point of the curve thus established will be the desired cut-off. (3) The heat absorbed by the cylinder metal each revolution is proportional to the change of tempera- ture which the metal undergoes; this in turn depends upon the difference in temperature of the entering and leaving steam. Therefore the greater the range of pressure, or better, of temperature, in a cylinder the greater the initial condensation. Halving the tempera- ture range halves the condensation loss. Thus if an engine is to work between fixed temperature limits the initial condensation loss may be reduced by sub- dividing this temperature drop among several cylinders, that is, by compounding the engine. The steam condensed in the first cylinder is re-evaporated during 284 THE TEMPERATURE-ENTROPY DIAGRAM. exhaust and it can thus be utilized in the second cylinder to produce power or to heat the walls. The same steam could be used over and over in successive cylinders to heat the walls, so that the condensation losses are not additive. Therefore the greater the number of cylinders the smaller the condensation loss. But the addition of each new cylinder entails an extra throttling loss in its admission- and exhaust-ports, an extra external radiation loss from its walls and from the piping and receiver which connect it to the pre- ceding cylinder. These radiation and transference losses are practically proportional to the number of cylinders. / \ L = FIG. 100. In compounding there are thus two opposing factors to be considered: the initial condensation loss which diminishes, and the transference and radiation losses which increase with the number of cylinders. As STEAM-ENGINE CYLINDER EFFICIENCY. 285 before, the theoretical development of the subject proves inadequate and the proper degree of compound- ing for different pressure ranges must be determined experimentally. Fig. 100 sh:nvs the character of these losses for simple, compound, and triple expansion, between the same initial and final conditions. (4) Watt stated that the function of a steam-jacket was to heat the engine cylinder to the temperature of the entering steam. If a jacket entirely fulfilled this requirement initial condensation of steam inside a cylinder would be entirely eliminated. This would mean that the cooling of the cylinder walls by low- temperature steam during expansion and exhaust must be made up before admission by heat taken from the jacket. During expansion the steam would receive heat from the hotter walls and its pressure would thus be maintained at a higher value than that correspond- ing to adiabatic expansion. Some work would thus be performed by the heat received from the jacket. But this heat, thus received at low temperatures, could not be as efficient as the high-temperature heat contained i:i the entering steam. During exhaust the walls, being at the temperature of live steam, give heat directly to the steam which heat is carried away without per- forming any work. In fact by increasing the volume of the exhaust steam it may actually increase the throttling loss during exhaust. The steam-jacket by 286 THE TEMPERATURE-ENTROPY DIAGRAM. increasing the external temperatures and dimensions of the cylinder also increases the external radiation and conduction losses. The steam-jacket is thus not only an inefficient method for supplying heat to the inside of the cylinder for the development of work, but it also increases the external losses. The increased economy obtained by the use of a jacket indicates that the decrease in initial condensation must more than offset the losses inherent in the jacket. The reason for this is readily found. Although steam is ordinarily nearly dry at the throttle the moisture at cut-off may range any where from 25 per cent, to 50 per cent. The steam at release usually contains somewhat less moisture than at cut-off, but not to any great amount. The re- evaporation of this moisture during exhaust extracts a large quantity of heat from the metal which must be be made up by the entering steam. If now the jacket by preventing initial condensation permits of dry steam at cut-off, a lesser weight of steam would be re- quired and at release this would not contain more moisture per pound than that naturally incident to adiabatic expansion. There would thus be a smaller quantity of water to be evaporated during exhaust, and consequently correspondingly less heat would be required from the cylinder walls. Further, all the heat .lost to the cylinder walls through initial condensation and re-evaporation is entirely wasted, and only the STEAM-ENGINE CYLINDER EFFICIENCY. 287 heat of the liquid at exhaust temperature can be returned to the boiler, but some of the heat given to the cylinder by the jacket does work during expansion and~ the heat of the liquid can be restored to the boiler at boiler temperature. Experience has shown that in most cases CB FIG. 101. a net saving is obtained from the use of a jacket. Natu- rally, the engine having the largest condensation loss would be most benefited by the addition of a jacket. Thus a jacket would produce a greater saving on a small engine than on a large one, on a slow-speed engine than on a high-speed (speed refers to number of revolutions and not the distance traversed by the 288 THE TEMPERATURE-ENTROPY DIAGRAM. piston in feet per minute), at early cut-off than at late cut-off. There is of course no expression for ideal efficiency involving jacket steam, as heat conduction losses do not exist in an ideal engine, but in any actual engine the thermal efficiency of the indicated horse-power is given by __ 2545XI.H.P. I?I.H.P.->? R . ^cylinder - gteam per hour- (#1 -g 2 ) + jacket steam per hourX?"i 2545 where w c and w } - are cylinder feed and jacket steam per I.H.P. per hour respectively. If the card is projected into the T^-plane every- thing is established upon the basis of 1 Ib. of cylinder feed, so that the expression for the efficiency would be Area of card in T^-plane It is thus possible to indicate the heat received from the jacket steam by laying off a rectangle ab, Fig. 101, equal to -r\. If the external radiation loss is known W c this can be further subdivided into ac and cb, repre- senting heat given to the cylinder feed and that lost in radiation respectively. STEAM-ENGINE CYLINDER EFFICIENCY. 289 The cylinder efficiency can be obtained from the above by dividing by the Rankine efficiency. When jackets are used on intermediate and low- pressure cylinders the jacket steam is at a higher temperature than the cylinder feed during admission and can thus more readily perform its function. The use of jackets on and reheating coils in inter- mediate receivers has the primary object of drying the exhaust from the preceding cylinder before it passes on to the next. If the steam is not very wet, or if the moisture is removed by a separator, they may serve to superheat the steam. Certain tests seem to indicate that unless they superheat the steam they fail in their purpose.* (4) As mentioned on pp. 236-239 the use of super- heated steam has but slight effect upon the Rankine efficiency. Thus with steam pressure of 200 Ibs. absolute and exhaust at 1 Ib. absolute the Rankine efficiency for dry saturated steam is 30.0 per cent., while with steam superheated 300 F. the efficiency is only 31.3 per cent. The large saving actually effected by superheated steam must therefore be due to its effect upon the cylinder efficiency. The explana- tion is found in the observed fact that heat is conducted less rapidly between a gas and a metal than between a liquid and a metal. Thus Ripper (Steam, Engine Theory and Practice, p. 149) found in a certain small * Trans. A. S. M. E., vol. xxv, pp. 443-501. 290 THE TEMPERATURE-ENTROPY DIAGRAM engine "that for each 1 per cent, of wetness at cut-off, 7.5 F. of superheat must be present in the steam on admission to the engine to render the steam dry at FIG. 102. cut-off." The loss of a small amount of superheat prevents a larger condensation due to the decreased rate of flow. And furthermore the steam containing less moisture at release, less heat is removed from the STEAM-ENGINE CYLINDER EFFICIENCY. 291 cylinder metal, and consequently less heat is required from the next admission steam. Let 1, Fig. 102, represent the expansion line when dry-saturated steam is supplied to the engine and 2 when sufficient superheat is present to give dry steam at cut-off. The cross-hatched portion represents the extra heat utilized per pound because of the addition of the small amount of superheat. This represents a case where the heat received per pound, H 1 q 2 , is increased by about one-fifth or one-sixth, while the heat utilized per pound is nearly doubled. Comparison between Reciprocating Engines and Tur- bines. In the steam turbine there is nothing analogous to the initial condensation and re-evaporation losses of the reciprocating engine. After the turbine is once in operation a constant temperature gradient is established from the admission to the exhaust. That is, the steam is continually diminishing in temperature as its pressure drops in the successive nozzles and stages, and the adjacent metal soon acquires the same tempera- ture as the steam. Thus the heat conduction losses are of two kinds: transference of heat through the metal from the steam to the atmosphere and trans- ference of heat through the metal by conduction from steam of high temperature to steam of lower tempera- ture. Both of these losses are probably relatively small as compared with the condensation and re-evaporation losses of the reciprocating type. 292 THE TEMPERATURE-ENTROPY DIAGRAM. Expansion in the reciprocating engine must not be carried beyon I the point at which the difference of pressure upon the two faces of the piston becomes less than the frictional resistance of the engine, as such an operation would mean the wasting of power already realized. In other words, part of the heat theoretically available for work must be sacrificed to avoid a greater friction loss. In the turbine, however, FIG. 103. no such limitation is placed upon the expansion. In fact the greater the expansion in a turbine the smaller the friction loss, as the. major part of the friction is clue to windage, and this decreases as the density of the steam surrounding the rotor diminishes. It would thus seem t'hat the turbine has a distinct advantage over the reciprocating type, as complete advantage may be taken of the theoretical possibilities by expand- ing completely down to back pressure with a net saving of the heat area abc, Fig. 103. STEAM-ENGINE CYLINDER EFFICIENCY. 293 In the reciprocating engine the throttling losses during admission and exhaust are comparatively small, and exist primarily because of the impossibility of making the ports large enough consistent with good design to prevent drop in pressure. In the turbine type, on the other hand, wire drawing occurs because of the impossibility of making the clearance spaces about the blades small enough to prevent it. Thus in each stage a considerable portion of the steam passes by the blades without performing work upon them. Each type of motor thus possesses its own method of dissipating the availability of the heat in the entering steam, and, judging from economy tests, the total losses are about the same in both cases. CHAPTER XV. LIQUEFACTION OF VAPORS AND GASES. SUPERHEATED steam of the condition shown at a in Fig. 104 might be changed to saturated steam by one FIG. 104. of two methods. First, it could be kept in a hot bath at constant temperature and the pressure increased from p a to 7)5, so that its state point would move from a to 6. Secondly, it might be kept under constant pres- sure p a (as a weighted piston) and its temperature allowed to drop by radiation from t a to t c) so that the state point travels from a to c. Liquefaction mil LIQUEFACTION OF VAPORS AXD GASES. 295 begin by either process as soon as the state point reaches the dry-steam line. It is at once evident that the second method is the only one always applicable, for the isothermal change might take place above the critical temperature and then no increase of pressure, however great, could result in liquefaction. The critical temperature of steam is beyond the upper limit of temperature used hi engineering, so that this is not as clear here as in the case of some vapor which superheats at ordinary temperatures, as, for example, carbon dioxide. Fig. 105* shows the T$-, pv-, Tp-, and v^-diagrams for C0 2 . The chief differences between this diagram and that for steam lie in the fact that the critical tempera- ture is included, thus showing the intersection of the liquid- and saturated-vapor curves, and further, that the volume of the liquid is now appreciable with refer- ence to that of the vapor and its variation with increas- ing pressure and temperature no longer negligible. If a (Fig. 105) represent the state point of the super- heated C0 2 vapor at the pressure and temperature under consideration, isothermal compression will fail to produce condensation, although cooling at constant pres- sure will produce liquefication as soon as c is reached. * This diagram is only approximately correct, being based upon somewhat discrapant data given by Amagat, Regnault, and Zcuner. 296 THE TEMPERATURE-ENTROPY DIAGRAM. The ordinary gases, hydrogen, oxygen, nitrogen, etc., have their critical points as much below ordinary FIG. 105. engineering temperatures as that of water is above them. LIQUEFACTION OF VAPORS AXD GASES. 297 Superheated steam at ordinary temperatures could be liquefied isothermally because its temperature is less than that of the critical point; carbonic dioxide at ordinary atmospheric temperatures could also ordinarily be liquefied isothermally because the usual atmospheric temperature is less than 31. 9 C. (89.4 F.); hydrogen, oxygen, etc., cannot be liquefied isothermally simply because the atmospheric temperature is far above their critical temperatures. The essential factor in liquefaction, then, is to reduce the temperature and then simply to compress the gas isothermally until lique- faction commences. Suppose it is desired to liquefy some carbon dioxide some summer day when the temperature of the atmos- phere is above its critical temperature. Let the com- pression be carried on slowly, so that the heat generated may be dissipated by radiation and the process be isothermal. Liquefaction will not occur. It will be necessary to cool the gas down to the critical tempera- ture by some means, physical or chemical. Possibly a coil containing cold water will suffice in this case. Proceeding to other substances possessing lower and lower critical temperatures, cooling mixtures giving lower and lower temperature would be required. That is, by this method it would never be possible to liquefy any substance, however great the pressure applied, unless there already existed some source of cold as low as its critical temperature. When the "permanent " 298 THE TEMPERATURE-EXTROPY DIAGRAM. gases are reached the sources of artificial cold fail, and unless the gas may be made to cool itself investigation must cease. Any new gas thus liquefied of course in turn becomes a new source of cold to aid in further investigation. Let Fig. 106 represent the T^-diagram of some gas having a low critical temperature. Consider the FIG. 106. throttling-curve abc. By definition this represents an adiabatic change, during which no work is performed; i.e., the heat contained in the substance is a constant. For this curve the first law of thermodynamics gives which becomes LIQUEFACTION OF VAPORS AND GASES. 299 that is, the curve represents an irreversible isodynamic process in the case of a perfect gas and differs but slightly from it for ordinary gases. Now, the internal energy of any substance is defined as the summation of the sensible heat and the dis- gregation work, or the kinetic energy of the molecules due to their own vibrations plus the potential energy due to their mutual positions. Representing these quantities by S and / respectively, it follows that Si +A -S 2 -L = p 2 v 2 -piVi. The work necessary to separate the molecules against their mutual attraction must increase with the distance between them although the rate of increase is inversely as the square of the distance between them. As the volume increases the value of / increases and hence the value of 8 must decrease an amount equal to that is, the temperature of the substance decreases.* With increasing volume the rate of temperature drop decreases so that the curve abc approaches T= const. as an asymptote. Near the saturation curve, in the region of the "superheated" vapors, this drop is con- siderable, but far to the right of this curve and above the critical temperatures these " throttling" curves become almost parallel to the ^-axis. The "perfect *For hydrogen at ordinary temperatures ;y 2 -p,f, is negative and greater than 7 a I lt so that the temperature increases. 300 THE TEMPERATURE-ENTROPY DIAGRAM. gas" is simply the limiting condition in which the potential energy has attained its maximum value, or rather where any limited change in volume does not affect the total value of / appreciably. In such a case and that is, the disgregation change being negligible, the isodynamic curves become coincident with the iso- thermals. In Fig. 106 let a t , a 2 , a 3 , etc., represent a series of such throttling-curves. Let a pound of air be taken from its initial condition P (representing atmospheric pres- sure and temperatifre) and be compressed isothermally to 1. This may be effected by jacketing the cylinder walls of the compressor with cold water. If the air is then permitted to expand along the throttling -curve a 1} the temperature will drop, say, from 1 to 2. The air at reduced pressure is fed back to some interme- diate stage of the compressor. In some forms of lique- fiers the expansion is carried at once down to atmos- pheric pressure, thus securing a somewhat greater drop in temperature, and the air is then returned to the first stage of the compressor. If this cooled, expanded air be made to flow back outside the pipe containing more air from the compressor, this in turn will be LIQUEFACTION OF VAPORS AND GASES. 301 cooled by conduction to some lower temperature and so will escape from the throttling- valve at pressure p t at some lower temperature (3). This will now expand along the throttling-curve a 2 to a still lower tempera- ture (4). This process is continued until the tempera- ture of the issuing jet has fallen below the critical temperature, when liquefaction will ensue. This will evidently occur first in the nozzle, as the temperature outside will need to be still further de- creased before the air will remain liquid at the reduced pressure. The vaporization of the liquid first formed tends to decrease still further the temperature of the air-tubes, etc. It is probable that at first all of this liquid vaporizes as soon as ejected, but the vaporiza- tion of part soon cools down the rest and its surround- ings, so that a small portion remains liquid at the lower pressure. The back pressure may thus in time be reduced to that at P and then the temperature will be found at which air vaporizes at atmospheric pres- sure. The expansion of the air thus provides in itself the cooling process needed to reduce the temperature below the critical point so that sufficient increase of pressure may cause liquefaction. CHAPTER XVI. APPLICATION OF THE TEMPERATURE-ENTROPY DIA- GRAM TO AIR-COMPRESSORS AND AIR-MOTORS. The Reversed, Power-absorbing, Thermodynamic Pro- cesses. In the direct power producing thermody- namic processes already considered, whether the simple theoretically perfect Carnot cycle, or the various prac- tical cycles such as the Rankine cycle for saturated and superheated vapors, or the Ericsson and Stirling cycles for air, or the Otto, Joule or Diesel cycles for internal- combustion engines, the fundamental principle of operation is the same; namely, during the expan- sion of the working fluid heat is absorbed at highest possible temperatures, and during the compressive stroke heat is rejected at the lowest possible tempera- tures, the difference between the amount of heat absorbed and the heat rejected being transformed into mechanical energy. In all of these processes the amount of work produced by means of the cycle is always much less than the energy supplied to the working substance, because a necessary condition for the transference of heat into mechanical work is a temperature difference. The greater this temperature difference the greater the 302 AIR-COMPRESSORS AND AIR-MOTORS. 303 fractional part of the heat which can be transferred into work. This temperature difference is therefore limited by the range in temperature between that at which the heat is supplied and that of the coldest of the surrounding bodies, which ordinarily is that of the atmosphere or of the cooling water. In all these cycles the maximum efficiency seldom ranges above 30 per cent, more ordinarily 25 per cent, but under exceptional conditions the Carnot efficiency may reach 50 per cent. In the actual engine working approximately on any of these cycles the fractional part of the heat received which is changed into work is naturally considerably less than the fractional part theoretically available, due to heat losses of varying kinds and magnitude. In other words, we may sum up the properties of these cycles as follows: Heat is taken in at high temperature; a portion of this heat is changed into work, but most of it is rejected as heat at lower temperatures. In the reversed power absorbing thermodynamic cycles the working fluid is made to traverse its ther- modynamic history in a reverse direction; namely, it takes in heat at low temperatures while expanding, is then compressed and gives out heat at high tempera- tures, the power required to perform this compression being equal to the power which in the direct cycle is produced by the expansion of the working fluid. In the usual definition for efficiency the efficiency 304 THE T EM PER AT V RE-ENTROPY DIAGRAM. is stated as output divided by input. For the direct cycles this magnitude is always less than unity, but for the indirect cycles its value, theoretically at least, is always greater than unity, although in the actual working mechanism heat and friction losses may serve to reduce it to practically unity, and sometimes even less. As we analyze this reversed cycle we find that the output may be of two different kinds, i.e., the cycle may be utilized to remove heat at low temperatures from some confined space (as for example the ordinary refrigerating machinery), or may be utilized to deliver heat at high temperatures to some confined space, as in the ordinary heating and ventilating systems. In both of these cases the input represents the work required to drive the compressor, the output, however, corresponds in the first case to the exhaust heat of the direct cycle, and in the second case to the ht at received in the direct cycle. Thus, for example, in the use of steam, the efficiency of the Rankine cycle under ordinary conditions is approximately 20 per cent. Therefore the exhaust heat represents 80 per cent of the total heat available for supply to the engine. In other words, the exhaust heat is four times the magnitude of the utilized heat. Therefore in the reversed cycle, if we consider the cooling effect as useful output, it is evident that this cooling effect is four times the mag- nitude of the force required to drive the compressor, AIR-COMPRESSORS AND AIR-MOTORS. 305 while if we consider the heating effect of the exhaust steam at high temperature, it would have five times the magnitude of the power required to drive the com- pressor. Again the suction of the compressor might be used for refrigeration and its discharge for heating, and in such a case the effective thermal output or useful effort would be nine times that required to drive the compressor. A third use of the compressor is the production of gas under high tension for future use in power produc- tion, physical experimentation, and industrial use. The Compressor. In all of these systems the com- pressor plays an essential part, and it is therefore well to make a study of the compressor cycle first, before entering into the details of the different cycles embody- ing this -reversed process. The compressor is really an engine driven backward, with all of its parts, valve gears, etc., operating in a reverse order. The usual cycle of operation is something as follows: Consider the piston to be at the beginning of the stroke; a certain small quantity of air is compressed in the clearance space under the pressure prevailing in the delivery pipe. As the piston moves forward this clearance air expands behind it until the pressure has decreased slightly below that in the suction pipe. The small difference of pressure then prevailing between the suction pipe and the inside of the cylinder suffices to lift the suction valve off its seat against the spring 306 THE TEMPERATURE-ENTROPY DIAGRAM. which usually holds it. As the piston continues to move forward new air will rush in behind it to the end of the stroke. It is quite possible that the inertia of the air in the suction pipe will be such that even after the piston slows down at the end of the stroke the air will continue to rush into the cylinder, and may even bring the pressure in the cylinder up to that prevailing in the suction pipe. The admission valve now closes under the action of the spring. As the piston starts FIG. 107. to retrace its motion it crowds the air ahead of it into the further end of the cylinder, so that the pressure continually rises until it reaches such a magnitude as to slightly exceed that in the delivery pipe; when the small difference of pressure thus created suffices to life the discharge valve from its seat, and then during the rest of the stroke the air will be delivered into the discharge pipe. This cycle repeats itself indefinitely. Referring to Fig. 107, the cycle is as follows : From a to d the gas (or vapor) in the clearance space expands until the pressure at d falls sufficiently below that in the supply pipe to permit the admission valve to open AIR-COMPRESSORS AND AIR-MOTORS. 307 and admit a new supply from d to c. On the return stroke, the entire quantity is compressed along cb, until the pressure becomes sufficient to lift the release valve, when discharge occurs from & to a against the upper pressure. The indicator-card thus shows the entire cycle of the clearance gas, but only one portion, the compression, of that of the charge. The expansion and compression of a gas in a cylinder is more nearly adiabatic than that of a saturated vapor, and hence, as the temperature of the gas at the end of admission is nearly that existing at the end of expansion, the expansion and compression of the gas in the clearance space may be considered to neutralize each other thermodynamically; although mechanically the greater the clearance the greater the size of the cylinder neces- sary to compress a given amount of gas. Therefore all discussion of the effects of clearance will be omitted for the present. During the expansion of the clearance air and the compression of the total charge, the heat interchanges between air and cylinder walls are much smaller than the corresponding changes between the steam and cylinder walls in steam engines, i.e., the operations are nearly adiabatic. If atmospheric air were simply a mixture of oxygen and nitrogen the equation of such a compression or expansion would be pv l ' 405 = C, but due to the fact that a certain amount of moisture is always present in the air the value of the exponent 308 THE TEMPERATURE-ENTROPY DIAGRAM. is somewhat different. Thus diatomic gases, as #2, #2, etc., possess a ratio of specific heats = 1.4, approximately, but as the complexity of the molecular composition increases the value of this ratio decreases. Thus for superheated H^O the value is approximately 1.31 to 1.33, so that the mixture of atmospheric air and super- heated water vapor gives a value of this ratio which is something less than 1.4. It is of course evident that this theoretical value is not actually realized in the air compressor cycle, because certain heat inter- changes must inevitably take place, so that in place of adiabatic compression the compression line shows slightly decreasing pressures as the volume decreases, i.e., decreasing below the values which you would expect from adiabatic compression. Nevertheless, the devia- tion from the true adiabatic compression is so small that most engineers consider any marked deviation to be indicative of leakage rather than of thermal interchanges. In case the compressor forms part of a heating engine the increase in the temperature during compression is an essential feature, and every effort should be made to pro- vent loss of temperature through external radiation and conduction, but if the temperature at which the heat is delivered is of no importance and the total value of AIR-COMPRESSORS AND AIR-MOTORS. 309 the operation coasists either in removing heat for refrigerative purposes or in storing up compressed fluids for future power purposes, the work required for adiabatic compression represents a real loss, because the same result could theoretically be accomplished by isothermal compression to the upper pressure limit. Thus, in the case of air-compressors supplying air for air-motors such as rock-drills, etc., the hot air delivered by the compressor must, during its transmission through the pipe-line, be reduced in temperature to that of the surrounding atmosphere and therefore the air occupies, at the pipe-line pressure, that volume which it would attain provided it underwent isothermal compression from the initial condition, and therefore the difference between the work of adiabatic compression and the work of isothermal compression represents useless expenditure of power. In order to obtain in a motor power equal to that required for compression, it would be necessary to pre-heat the air up to the final temperature of com- pression before it entered the motor. Actually without pre-heating the expansion of the air in the motor cools the temperature of the exhaust to such a point that the water entrained in the atmosphere frequently freezes in the exhaust ports. In Fig. 108, let ab and ac represent isothermal and frictionless adiabatic compression respectively from some lower pressure p v to a higher pressure p z . If the 310 THE TEMPERATURE-ENTROPY DIAGRAM. temperature of the cooling water is the same as that of the atmosphere, the minimum possible expenditure of work in compressing from a to 6 equals that under the isothermal ab, or . If, however, the FIG. 108. compression is adiabatic it will be necessary for the delivered air to contract at constant pressure, losing by conduction and radiation the heat c p (t c t b ). That is, the work T H ( K 1 L \pa 1 is performed upon the gas during compression, as shown under ac, and the work pb(v c ~ v b) during contraction in the storage tubes or coil, as shown by cb in the ^-diagram. The wasted work is thus shown by the area abc, and has the value AIR-COMPRESSORS AND AIR-MOTORS. 311 Jfc-l = Ac p [T c - To] -AT a [a - . i.e., when p t : p x = p x ' p^ r Px = ^pip2- Fig. 110 shows similar diagrams for a three-stage compressor with intercoolers. The saving thus intro- duced is shown by the irregular-shaped figure defghc. This again varies in magnitude with the values of FIG. 110. p* and p a , and in a manner similar to the above may be shown to have its maximum value when Pi ' or >Pv=Py P2J and p y = V 314 THE TEMPERATURE-ENTROPY DIAGRAM. Naturally the greater part of the saving is obtained by the use of one extra cylinder, the gain from successive extra cylinders becoming correspondingly smaller. Taylor Hydraulic Air-compressor. To make the compression line coincide with the isothermal through- out, would require an indefinite number of stages, . but as the introduction of each new stage increases the total friction loss of the compressor the number of stages actually feasible is limited to three, or possibly four. There is, however, one type of compressor essentially simple in construction which realizes almost perfectly this ideal condition. This is what is known as the Taylor hydraulic air-compressor, consisting mainly of a well, sunk at some point where a drop of water is available, as at any waterfall, and containing a submerged bell-shaped chamber which is connected through the top by a pipe-line leading to the upper level of the water, this pipe-line extending downward in the bell part way from the top. Water from the upper level flows through this pipe into the bell, keeping the lower end of the pipe submerged. To introduce the compressive feature there is a series of small tubes with the lower ends inserted in the water entering the top of this tube. The rush of the water serves to draw bubbles of air down through these tubes, which bubbles are caught in the descending water and carried downward into the bell. Here the stream is deflected by horizontal aprons, thus giving AIR-COMPRESSORS AND AIR-MOTORS. 315 the air a chance to separate from the water by gravity ; the air collecting in the top of the bell, the water passing out underneath the lower edge of the bell up around the entire mechanism to the lower level of the fall. By this means the air in the top of the bell is subjected to a pressure equal to the weight of water correspond- ing to the head, measured from the top of the water in the submerged bell up to the lower level of the water at the fall. It is evident that the energy input of this compressor is equal to the weight of water passing through the tube, multiplied by the height of the water- fall. The useful output, on the other hand, is repre- sented by the work necessary to compress the actual quantity of air delivered isothermally from atmospheric pressure to the pressure prevailing in the submerged bell. Evidently to obtain a maximum efficiency the in-rush of air should be so regulated that the maximum quantity of air possible should be carried down by the water. The actual loss in this instrument is occasioned by the buoyancy effect due to gravity. The air particles as they pass down through the tube are subjected to ever greater pressures, decrease in volume, and thereby assume greater density, so that the buoyancy effect diminishes as they descend in the tube. The buoyancy effect, however, tends to maJce-the bubbles move upward relatively to the falling water, so that part of the work already accomplished in moving them downward is wasted by this retrograde action. 316 THE TEMPERATURE-ENTROPY DIAGRAM. ' The Influence of the Clearance Space. Returning once more to the piston compressor it is next necessary to consider the effect of the clearance space. As we have seen, the air under high pressure, stored in the clearance space at the beginning of the stroke, expands as the piston moves outward, and fills a certain portion of the piston displacement, thereby decreasing the space actually available for new air during the succion stroke. In other words, the actual effective displace- ment of a compressor is less than its apparent displace- ment, as determined from its dimensions and length of stroke. The ratio of the actual displacement to the apparent displacement is known as the displacement efficiency of the compressor. Sometimes the actual air delivered per cycle divided by the apparent dis- placement is also called displacement efficiency, but as this ratio also includes the effect of leakage by the piston and valves its use should be avoided. It is better to have separate ratios: one defining the influence of the clearance, the other the effect of leakage upon the delivered air. Therefore, in calculating the size of the compressor to perform a stated amount of work so as to deliver a definite volume of free air, the piston displacement calculated directly from the volume of the air to be compressed and 'delivered must be divided by this displacement efficiency in order to obtain the actual piston displacement. The power required to drive the actual compressor (neglecting friction losses) is AIR-COMPRESSORS AND AIR-MOTORS. 317 the same as the power that would be required to drive a compressor having no clearance space, because, as the heat interchange is small between air and cylinder walls, the expansion and compression lines are practically the same, so that the extra work required to compress this clearance air is returned by expansion of the air during the suction stroke. If x = the increase in volume of the clearance air dur- ing expansion from delivery (/?2) to suction (pi) pressure, and n = the exponent of the expansion curve, while Cl. and P.D. represent the volume of the clearance and piston displacement respectively, there exists the simple relation J^V-ilci. L\W whence From this it follows that the actual displacement equals P.D.-X The displacement efficiency thus becomes equal to " 318 THE TEMPERATURE-ENTROPY DIAGRAM. Determination of the Clearance of an Air-compressor from the Indicator Card. (a) It may be assumed without much error that the values of the exponent \ Atmospher Discharge pressure \ Atmospheric Lino FIG. 111. n for the expansion and compression curves are the same. Let these curves be intersected by any two straight lines parallel to the atmospheric line in the points 3 and 4 and 1 and 2, respectively (in Fig. Ill the atmospheric line is utilized), so that ps = p\ and p4 = p2- Determine v\, v 2 , v 3 , and v 4 in per cent. of the piston displacement, and let c represent the volume of the unknown clearance in per cent, of the piston displacement. Then and AIR-COMPRESSORS AXD AIR-MOTORS. 319 whence or c As neither p nor n enter into the value of c, neither the scale of the indicator spring nor the actual law of expansion and compression need to be known. As the percentage errors are much larger in the deter- mination of v 3 and v 4 than in v\ and v 2 , the accuracy of the method may be increased by assuming a prob- able value for the clearance and adding this to v\ t v 2 , v 3 , and v 4 .- The resulting value of c thus obtained will be not the clearance itself, but the difference be- tween the actual and assumed value, and may be either positive or negative according to the error in the assumption. (6) When, as is usually the case, the scale of the card is known greater accuracy may be attained by using only the compression line. Intersect the curve at pi and p 2 = %pi, and also at p 3 and p 4 = $p 3 . Deter- mine the corresponding volumes v\, v 2 , v 3 , and v^ Then and 320 THE TEMPERATURE-ENTROPY DIAGRAM. whence Actual Air-compressor Indicator Cards. Fig. Ill shows indicator cards from the two cylinders of a small single-stage air-compressor in the laboratories of the Institute. In the lower card the discharge- valve was functioning properly, in the upper card it opened and closed spasmodically. The admission- ports of the lower card offered more resistance than those in the upper card, as shown by the dropping suction line. The indicator card does not represent a closed cycle, and hence if projected into the 2*<-plane only por- tions of it have any significance, namely, the com- pression and expansion lines and the enclosed area. It is, however, instructive to study the compression line, as this gives some clue to the effectiveness of the jacketing. In any case, the compression line should not deviate much from the adiabatic. The presence of a large deviation is indicative of a leak instead of a heat interchange. Various Efficiencies. As we have seen, part of the work of compression is, as far as useful output is con- cerned, wasted energy, so that it has become customary to consider the power required for isothermal compres- sion of the air to represent the useful output of the AIR-COMPRESSORS A\D AIR-MOTORS. 321 compressor and to define the ratio of this isothermal work to the actual indicated work as the air efficiency, or sometimes as compressor efficiency. In the case of the ideal engine the air efficiency would equal the ratio of the isothermal work to the adiabatic work for the proper number of stages. Usually such compressors are driven either by direct connected motors or steam engines, or are belted to such motors. There is usually no attempt made to separate the friction losses in such combined units of the motor and of the compressor, but the total friction loss is obtained by determining the difference in the indicated power of the steam cylinders and the air cylinders, and the mechanical efficiency of such a unit is defined as the ratio of the indicated air power to the indicated steam power. Of course in the case of an electric motor operating such a compressor the efficiency of the unit would be represented by the ratio of the indicated air power to the electrical input at the terminals of the motor. The values of these various efficiencies actually attained in practice vary between very wide limits due to different forms of compressors and the varying degrees of workmanship. Thus the displacement effici- ency ranges between 78 per cent and 94 per cent, i.e., the effect of the clearance is to decrease piston displace- ment by 22 to 6 per cent of its actual value. The air efficiency or the compression efficiency is 322 THE TEMPERATURE-ENTROPY DIAGRAM. influenced by the action of the admission and delivery ports and also by the heat interchanges. The heat losses are greater in compound than simple compressors because the air comes in contact with two cylinders instead of one. In certain of the better types of com- pressors the valves are not operated by difference of pressure, but by mechanical means. In these latter compressors the throttling loss during admission and exhaust is much diminished. Numerically the efficiency of compression varies from about 50 to 90 per cent. On page 371, Peabody, there is a table giving the values found in tests on a variety of compressors. The mechanical efficiency of the unit is ordinarily in the neighborhood of 85 per cent, but occasionally reaches higher values as, for example, page 372, Pea- body, mechanical efficiency for blowing engines at Creusot is quoted at 92 per cent. From the values given for the air and mechanical efficiencies it is evident that the total efficiency of the compressor unit, which may be defined as the isothermal work of com- pression divided by the indicated power of the steam cylinders, ranges from 40 to 75 per cent. It is interesting to compare these values with those obtained with hydraulic compressors. Thus reports of tests made upon the Taylor Hydraulic Compressor show that the ratio of the isothermal work on the air to the water work ranges from 60 to 70 per cent. This corresponds to the total efficiency of a steam- AIR-COMPRESSORS AND AIR-MOTORS 323 driven air- compressor unit and shows that the results obtained for Taylor compressors always exceed the minimum values of the steam-driven compressor, and usually the values very nearly equal to the maximum one obtained with steam-driven compressors. It is of course possible to use water power to operate a water turbine and to connect this turbine to a piston air-compressor. Such water turbines range in efficiency from 20 to 80 per cent. If for a basis of comparison we take this maximum efficiency of the water turbine as 80 per cent and multiply it by a high value for the efficiency of compression (say 80 per cent) this would give as the total efficiency of a water turbine air-com- pressor unit a value of 64 per cent, which is readily exceeded by the Taylor compressor. Applications of Compressed Air. Compressed air is used in almost all branches of mechanical and chemical engineering, the scope of this utilization being indicated briefly by the following headings: (1) Mechanical Draft, Cooling Towers, Ash and Coal Conveyors, (2) Heating and Ventilating, (3) Air Drying, Lumber Drying Kilns, etc., (4) Steel Production and Manipulation, as Forges, Hammers, Drills, Riveters, etc., (5) Air Motors and Rock Drills, (6) Exhausters to Promote Increased Speed of Evaporation, etc., (7) Pneumatic Conveying Systems, Planing Mill Exhausters, (8) Sand Blast for Cleaning Iron, Stone, etc., (9) Paint Blast for Painting Bridges, Freight Cars, etc., (10) Pumps of both Displacement and 324 THE TEMPERATURE-ENTROPY DIAGRAM. Expansion type, of which latter the Pohle* air lift is a good example, and the combination of both methods, such as the Starrett air pump. Types of Compressors. These various applications call for air pressures varying from a fraction of an ounce to the square inch up to'possibly ten pounds to the square inch. In the case of the air-motors and for the production of liquid air, and some few other devices, much higher pressures extending up to the hundred or thousand, or several thousands, of pounds to the square inch are required. To produce air of such varying pressures a variety of mechanisms has been devised. For low pressures extending" up to one ounce or less per inch, a disk or propeller fan running at slow speed \vill handle large volumes of air at small cost. For pressures up to one pound per square inch centrifugal fan blowers consisting of steel wheels running at high speeds in conical casings will handle large volumes of air at comparatively small cost. In both these types of fans there is always the possibility for air to leak back from the delivery side to the suction side through the clearance space, so that for higher pressures some more positive action or mechan- ism must be utilized. Thus for pressures ranging to ten pounds we have a rotary type of blower or exhauster engine which consists essentially of two geared wheels so intermeshing that on the upward stroke or motion air is carried forward between the teeth, but on the down- ward stroke the teeth intermeshing prevent the escape AIR-COMPRESSORS AND AIR-MOTORS. 325 of the air, so that it is delivered into the discharge pipe. These blowers handle capacities ranging anywhere from five cubic feet to over 15,000 cubic feet per minute The piston compressor is designed to deliver air at any pressure up to three or four thousand pounds per square inch. The Taylor hydraulic compressor occupies a unique position in that it attains most closely to isother- mal compression. It is capable of supplying large vol- umes of air at ordinary commercial pressures. Use of Compressed Air for Pumping. As a simple, economical, and reliable power for pumping purposes, compressed air for many applications cannot be sur- passed. The ordinary types of air pumps fall under two general headings, (1) those using the pressure of the air alone to displace water from a suitable chamber, (2) those using the expansive force of the air by interming- ling it with the column of ascending water, as in the Pohle air lift. There is a third type which is really a combination of these two, a good illustration being the Starrett pump. In the displacement type of pumps the compressed air is led into the top of the pump chamber and the water is forced out through a pipe from the bottom. When the water is completely displaced the supply of air is shut off and the air in the chamber is either exhausted to the atmosphere or in some types is fed back during the suction stroke of the compressor. The weight of water in the delivery pipe closes the check valve and 326 THE TEMPERATURE-ENTROPY DIAGRAM. prevents water flowing back into the chamber. The decrease in pressure in the chamber, however, permits water to enter through an admission pipe either under the influence of gravity (in case the chamber is sub- merged) or under the influence of atmospheric pressure in case the air pressure is reduced by drawing it into a compressor. Ordinarily such pumps are made with two cylinders, so that one is discharging while the other is filling, thus giving a fairly constant delivery of the water. Such a method of pumping utilizes simply the direct pressure of the air and fails to utilize any of its expansive force or internal energy. The economy obtained by such a method must be comparable with that obtained in direct-acting steam pumps where the steam is used non- expansively. The Pohle" Air Lift consists of two tubes, one a large tube with a bell-shaped enlargement at the bottom, sub- merged to a considerable extent in the well from which water is to be pumped. The other is a small pipe for supplying the compressed air which it delivers under- neath the bell-shaped opening of the larger tube. The result is that the compressed air rising upward under the influence of gravity tends to carry with it slugs of water. We have thus in this tube a rising column of alternating layers of water and air, or a column of water interspersed with numerous air bubbles. This results in a dimin- ished de'nsity of the combined mass, so that the weight AIR-COMPRESSORS AND AIR-MOTORS. 327 of water outside the pipe will balance a considerably higher head of this mixture inside the tube. The greater the height to which the water must be raised the less the density of the mixture in the tube. It is to be noticed that this Pohle air lift is simply .the reverse of the Taylor air-compressor. Thus, as the bubbles rise in the pipe, they are subjected to ever decreasing pressure, namely, the weight of the water above them, and therefore ex- pand. The tendency to cool, due to expansion, is offset by the rapid interchange of heat from the water to the air, due to the intimate mixture. The expansion of the air is therefore practically isothermal, and the work developed is thus the maximum which could possibly be attained without the use of an external source of heat. The sources of loss in this pump are (1), in the absorption of air by water so that the whole of the air supplied cannot be utilized for pumping purposes, and (2), in the fact that water from the bottom of one slug tends to leak downward to the top of the one following, so that there is a continual backward flow of water down the pipe, i.e., water once lifted falls back and must be pumped up again, thereby wasting part of the power of the air. Experiments on certain Pohle air lifts have shown an efficiency of about 50 per cent. The Starrett Air Pump utilizes a combination of the displacement pump and the Pohle air lift. Essentially it consists of two displacement chambers connected to a common discharge pipe and supplied through an auto- 328 THE TEMPERATURE-ENTROPY DIAGRAM. matic valve from a common air pipe. One cylinder is filling under the influence of gravity, while the other is being emptied by means of the compressed air. By means of a rather complicated valve action a portion of the air used for displacement escapes into the column of ascending water, the rest of it being discharged directly to the air. The valve also bleeds a certain quantity of compressed air directly into the discharge pipe. It is evident that all the air discharged to the atmosphere escapes with its internal energy unutilized, and there- fore fails to give as good economy as the air which passes into the delivery pipe. Theoretically, at least, such a pump cannot have as high a thermal efficiency as the simple Pohle air lift. It has, however, certain other advantages which in many installations would make it more advantageous than the Pohle air lift. Thus the Pohle air lift requires that the pipe be submerged to such a distance beneath the level of the water that the weight of the water outside the tube shall overbalance the longer column of water and air inside the tube. In other words, such a method could not be utilized unless the well was sunk far below the level of the water. The Starrett pump, on the other hand, may be placed just below the surface of the water, as the water flows into it under the influence of gravity, and therefore does not need such a deep excavation. It is quite possible that the decreased cost of installation might offset any ther- mal advantages inherent in the Pohle air lift. Tests AIR-COMPRESSORS A\'D AIR-MOTORS. 329 made upon the Starrett pump installed in the labora- tories of the Institute have shown efficiencies of about fifty per cent, which for some strange reason seem to be higher than the efficiencies which are quoted for the Pohle air lift. There is one interesting feature common to both the Pohle and the Starrett pump, and that is that during the practically isothermal expansion of the air in the delivery pipe heat is flowing from the water to the air, therefore increasing the work developed by the air; or, in other words, part of the heat energy of the water is being utilized to help pump it. Compressed Air Used as a Source of Power. Return- ing to the first general case where the air is used for power, it is necessary first of all to discuss the influence of the pipe line which conducts the air from the com- pressor to the machine to be operated by the compressed air, such as a rock-drill, a penumatic riveter, a com- pressed-air motor, etc. The temperature of the pipe may be considered as equal to that of the surrounding atmosphere and hence constant. The heat generated by friction is thus at once dissipated by conduction, and there thus results an isothermal drop of pressure and increase of volume. Thus, if abcde, Fig. 112, represent the passage of the air through the compressor, ef shows the lo?s experienced by the air in flowing from the compreasor to the engine. Suppose the air at /to expand adiabatically in the motor 330 THE TEMPERATURE-ENTROPY DIAGRAM. down to back pressure, the amount of work performed will be equal to the area under fg in the pv-diagram. The exhaust air is now warmed to the initial tempera- ture along the constant-pressure curve ga, thus per- FIG. 112. forming upon the atmosphere the work under ga in the pv-plane, and receives from the atmosphere the heat under ga in the T-plane. The maximum amount of work could be obtained from such an engine if the expansion were along the isothermal fa. This can be partially attained by jacketing with water at atmos- pheric temperature, so that the actual expansion-curve lies somewhere between these two limiting cases, as at }g / . A further gain could be made by compounding the engine and heating the air up to atmospheric tem- perature in the intermediate receiver, as indicated by fhkla. Effect of Pre-heating the Air. The amount of vrork theoretically obtainable from a pound of air in a non- conducting motor is given by the formula : AIR-COMPRESSORS AND AIR-MOTORS. 331 0.405 ,1.405 pjO-2883 and is thus shown to be directly proportional to its ini- tial temperature. In the operation of motors by com- pressed air it is thus possible to bring about a consid- erable saving in the amount of air required as well as an improvement in the thermal efficiency in any given case by heating the air. This heating, or pre-heating, as it is generally termed, may be accomplished either by : 1. The direct application of heat in some form of com- bustion heater, or 2. The admixture of steam. This latter method may operatively be effected by one of two methods, viz., by blowing steam into the air main or by passing the air through a boiler, in which latter operation it becomes heated to the temperature of the steam. 332 THE TEMPERATURE-ENTROPY DIAGRAM. Air-motor Economy. In air-motor work it is custom- ary tor quote the consumption in terms of cubic feet of free air per indicated horse-power per hour. This result may be derived from the preceding equation as follows : 33000X00 Lbs. of air per I.H.F. hour = 1072 Cubic feet free air per I.H.P. hour = lbs. air X vol. of 1 Ib. air at atmospheric temperature 1 079 270.27\ The maximum air consumption evidently occurs when the air in the pipe-line has been cooled to atmos- pheric temperature, and is equal to ' cu.ft. per I.H.P. hour. AIR-COMPRESSORS AND AIR-MOTORS. 333 The cubic feet for any amount of pre-heating may then T be obtained by multiplying this result by = 1 1 Air-motor Efficiencies. Assuming the air to be sup- plied by a single-stage compressor, and that before entering the motor it is cooled to atmospheric temper- FIG. 113. ature,then the work of adiabatic compression would be (Fig. 113): so that the efficiency of the compressor-motor unit would be: If the compressor is driven by a steam engine, the effi- ciency of the motor is equal to the combined efficiencies of boiler, Rankine cycle, cylinder, mechanical efficiency of engine, compressor unit, efficiency of motor compres- sor, or 334 THE TEMPERATURE-ENTROPY DIAGRAM. thermal efficiency of motor = ^boiler ' r /Rankine ' ^cylinder ' ^mechanical ' ^compressor ' r /'motor The value of this thermal efficiency lies well inside ten per cent. Influence of Pressure and Temperature upon the Indi- cated Work. The expression for the work developed in an air motor: and the corresponding expression for the work required to compress air : may be transformed the one into the other by making use of the expression for an adiabatic expansion in terms of temperature and pressure : IZL* i-fc *~ = These must of course be of the same magnitude, as they represent the same cycle of operations simply in a re- versed direction. In both formula? the subscript 1 applies to the conditions at the upper pressure level, and subscript 2 the conditions at the lower pressure level. These different forms for the same quantity are AIR-COMPRESSORS AND AIR-MOTORS. 335 used because ordinarily one knows the initial condition of the air supplied to a motor and would not be liable to know the exhaust temperature, and similarly, one knows the condition of the air supplied to the compres- sor and would not be liable to know its exhaust temper- ature. A comparison of these two formulae makes evident the fact that the amount of work produced or consumed is absolutely independent of the actual values of the upper arid lower pressures, but is dependent solely upon the ratio of these values. Thus the amount of work required to compress a pound of air from 1 Ib. to 15 Ibs. pressure is equal to that which would be required to compress it, say from 15 Ibs. to 225 Ibs. Similarly the work required to compress a pound of air from .01 of a pound per sq. in. to one pound per sq. in. is as great as the work required to compress it from, say 10 Ibs. per sq. in. to 1000 Ibs. per sq. in. On the other hand the amount of work obtained or absorbed per pound of air is directly proportional to the initial temperature of the charge supplied to the motor or to the compressor. If therefore it is essential and necessary to water-cooi the cylinders of an ordinary air-compressor which is compressing from atmospheric pressure to two or three hundred pounds, it is just as necessary to cool the cylin- ders of the dry air pump, whose function it is to take the 'urn* pressure air from a vacuum chamber and discharge it into the atmosphere. This is clearly demonstrated in 336 THE TEMPERATURE-ENTROPY DIAGRAM. the temperature-entropy plane from the fact that the constant pressure cur.ves all possess the same form and are simply displaced from one another parallel to the entropy axis, the displacements being proportional to the logarithms of the pressure. Thus the pressure curves for 1000, 100, 10, 1, Vio, Vioo Ibs., etc., would form a FIG. 114. series of curves separated by equal distarces (Fig. 114). Therefore the work of compression from any one curve to the succeeding higher curve would be the same as .hat from any other curve to its succeeding curve; assuming the compression in all cases to start at the .^ame temperature, the increase in temperature would be the same in all cases, and finally the heat absorbed by the cooling water must also be the same in all cases. CHAPTER XVII. DISCUSSION OF REFRIGERATING PROCESSES. Refrigerative Units. Various branches of engineering have their own units, which in a sense are historical records of the functions formerly exercised by other devices or systems, and which have been usurped by these particular branches of engineering. Thus in power development the name of the horse is preserved in horse- power, Pferdstaerke, cheval vapeur, etc., which is the unit for all steam- and gas-engine measurements ; even in boiler rating the name still sticks, although its numer- ical value has long ceased to possess any relation with its origin. In refrigeration engineering it is not the horse but natural ice which is being replaced. As heat conduction is a continuous operation, the refrigerating agent must operate twenty-four hours a day, so that when we fill our refrigerators with ice the supply must be sufficient to last the time interval from one filling to the next. In popular or commercial parlance the refrigerating needs of any plant would be represented by the rate of melt- ing of the ice, as so many units of weight of ice per day. It is but natural, then, to find the capacity of mechanical refrigerating plants expressed as the equivalent of so 337 338 THE TEMPERATURE-ENTROPY DIAGRAM, much ice melted daily. The unit of refrigeration adopted by the A.S.M.E. is the heat absorbing capacity due to the melting of one ton of ice in twenty-four hours. As the latent heat of fusion of ice is 144 B.T.U. per Ib. this unit corresponds to the absorption of 288,000 B.T.U. per 24 hours, or 12,000 B.T.U. per hour, or 200 B.T.U. per minute. A report of the committee to the society may be found in the records of the A.S.M.E. in Vol. 28, year 1906, page 1249. The output of a refrigerating plant is expressed in various ways, according to the feature it is desired to accentuate. Thus to the purchaser or user, the chief item of interest is the ice equivalent, so the output is quoted in tons of ice melted in twenty-four hours. For the power engineer the important question is the horse-power re- quired to produce a certain number of units of refrig- eration, so that the output is usually quoted in B.T.U. refrigeration per steam horse-power per minute. Finally, from the standpoint of cost, reference must eventually be made to the coal pile, so that the rcfrigerative effect is also quoted in terms of pounds of ice melted per pound of coal burned. Refrigerative Systems. Refrigerating- plants group naturally under one of three headings : 1. Those using air with either the natural or dense air system. 2. Those using some volatile Substance, as ammonia, with a compressor. . DISCUSSION OF REFRIGERATING PROCESSES. 339 3. Those using some volatile substance, as ammonia, with some absorbent, as water, with its necessary ad- juncts, absorber, generator, analyser, rectifier, inter- changer. Air Refrigeration. Air refrigeration at the present time is used chiefly under such circumstances as make the use of ammonia refrigeration impossible or inexpe- dient, the advantage of this method being that leakage of the refrigerative fluid causes no inconvenience. The disadvantage lies in the small heat capacity of the air, which is only 0.2375 B.T.U. per pound per degree dif- ference of temperature, so that large volumes of air have to be handled by the compressor and the rest of the system in order to affect any appreciable refriger- ation. The apparatus used consists of several members, each possessing the peculiarities and performing the functions described in the following. A compressor which may if desirable be two-staged, the function of which is to increase the pressure of the air a consider- able amount. The involved increase of temperature must next be eliminated by carrying the delivery pipe of the compressor through a suitable water bath, so that the air is restored to atmospheric temperature. The only way in which the temperature of the air can lie diminished other than by direct radiation to some colder body, is to utilize the internal energy of the air in the performance of work. This is most effectively accomplished by expanding the air in a motor or expan- 340 THE TEMPERATURE-ENTROPY DIAGRAM. dor which ordinarily is either direct connected to the compressor piston rod or indirectly through a common shaft and thus serves to help drive the compressor. The difference between the power absorbed by the compres- sor and that developed by the motor must be supplied from some external source. Because of the practically adiabatic expansion in the motor cylinder the final tem- perature of the air is expressed by the formula /PatmA \ 2 I i-k k i.e., the air during expansion undergoes the same frac- tional diminution in temperature as its increase during compression. The air leaving the compressor is there- fore much colder than the atmosphere and may be used as a refrigerative agent. The rest of the system consists of a pipe conducting the cold air from the exhaust of the motor to the refrigerating rooms or brine tank, etc. As the air circulates through these various chambers heat flows into it from the surrounding hotter atmo- sphere and it therefore serves to cool these respective chambers. Eventually the air will regain atmospheric temperature, so that the theoretical refrigerative effect is equal to 0.2375 X(7 T atm .-7 7 2 ). Actually, however, the full value of this refrigerative effect cannot be real- ized as the latter portion of the heat is received at tem- peratures greater than that of the desired refrigeration, so that this would be transferred through the piping DISCUSSION OF REFRIGERATING PROCESSES. 341 after the air left the rcfrigerative . chamber and is on its return to the compressor. Ordinarily, therefore, this final section of the pipe is omitted and the air is wasted, a new supply being drawn in at the compressor. In air refrigerating plants the work performed during expan- sion is no longer tho main object, as in the case just discussed, but is simply a means of obtaining the desired end, viz., a sufficient drop in the temperature of the air. The essential parts of such a system arc shown in Fig. 115. The compressor A takes its supply of air from the atmosphere and discharges the compressed air into the cooling coil B. where temperature and volume are both decreased at constant pressure. The cold air now 342 THE TEMPERATURE-ENTROPY DIAGRAM. passes into C, and in expanding helps to operate the compressor. The expanded air is delivered at low tem- perature and atmospheric pressure to the refrigerator room, and as it passes through D its temperature in- creases and reaches that of D by the time it leaves at the right. The cycle of the air is completed by warming to the initial atmospheric temperature outside of the o ,'/ FIG. 116. refrigerator. The expansion in C being used to attain low temperature instead of work, care is taken not to heat the air during expansion, so that the expansion may be as nearly adiabatic as possible. In Fig. 116 let anb represent the passage of the air through a two-stage compressor A and the cooling tank B. During the expansion be, in the working cylinder C, the temperature of the air drops below that maintained in the refrigerator T r , the air being delivered at pressure DISCUSSION OF REFRIGERATING PROCESSES. 343 p (l . As the temperature of the air increases along the constant-pressure curve ca, it extracts from the refrig- erator the heat under the curve cd, in the T^-plane The heat under da, necessary to complete the cycle, is obtained from the atmosphere. That is, the refrigera- ting effect cdef is attained by the expenditure of the work abcda. If the compressor has but one stage, amb, the efficiency will be correspondingly less. When we realize that a pound of air at ordinary at- mospheric conditions occupies about 13 cu.ft. and for a range of temperature of, say 100 degrees, has a refriger- ative capacity of approximately 24 B.T.U., we see that- to accomplish any appreciable amount of refrigeration a large number of pounds of ajr must be circulated and therefore the volume of air to be handled is practically prohibitive. It is at this point that advantage is taken of the fact previously mentioned that the work of com- pression is independent of the actual pressures, and only dependent upon the ratio of pressures in what is known as the Allen Dense Air Machine. In this mechanism the air is never expanded down to atmospheric pressure, but reaches, as it exhausts from the motor, a pressure of about five atmospheres, or roughly 00 Ibs. gage, and when discharged from the compressor is ordinarily about 15| atmospheres, or roughly 210 Ibs. gage. Assuming that the air entering the compressor is practically of atmospheric temperature, we see that the volume; of one pound has been reduced from approx- 344 THE TEMPERATURE-ENTROPY DIAGRAM. imately 13 cu.ft. to approximately 2.6 cu.ft. In ot hoi- words, the volume of every part of the system is reduced to about one-fifth of that which would be needed if atmos- pheric air had been used. The temperatures ordinarily attain, with the Allen Dense Air Machine, from 70 to 90 F. below zero. This system is mostly used on board ship, and the air is used to a certain extent in a regenerative manner, i.e., the coldest air just leaving the motor passes through the, ice-making box, and then somewhat warmer next enters the meat chamber, and upon leaving this is still cold enough to be used in chilling the drink- ing water butt. Finally, as it passes on its way to the compressor, it is used to chill the water-cooled air about to enter the expander. The amount of work theoretically is the same in the dense air as in the natural air system, but because of the reduced volume the compressor and motor friction losses may be smaller. The cost is less, and this, combined with the decreased floor space, makes it feasible to util- ize such a system. Such plants are still used in pb'vs where it is more essential to guard against danger aris- ing from the leakage of fluids, such as ammonia, than to install the most economical plant, as, for example, on war vessels. Ammonia Refrigerating Plant. A complete cycle of the working fluid in the second class of refrigerating plants, where a saturated vapor is used, differs mate- rially from the above, as the substance is condensed DISCUSSION OF REFRIGERATING PROCESSES. 345 and vaporized during the process. The more com- monly used fluids are ammonia, carbon dioxide, and sulphur dioxide; ammonia being used most generally. Fig. 117 shows diagrammatically the essential features of an ammonia refrigerating plant. It consists of (1) a FIG. 117. compressor which takes the low-pressure vapor from the refrigerator coils and delivers it at some higher pressure, (2) a condenser consisting of a series of coils in which the hot gas is cooled until it liquefies, (3) a storage-tank containing a supply of ammonia which remains liquefied under the high pressure at atmos- pheric temperature, (4) an expansion-valve from which the liquid emerges under reduced pressure, and (5) the 346 THE TEMPERATURE-ENTROPY DIAGRAM. refrigerator coils in which the liquid, under reduced pressure, is vaporized by withdrawing the necessary heat from its surroundings. The high pressure pre- vails from the delivery-valve of the compressor to the expansion- valve, and low pressure from the lower side of the reducing-valve to the admission-valves of the compressor. The refrigerator coils may be used directly, thus bringing the temperature of the surroundings down near the boiling temperature of the liquid, or, if such a low temperature is not desired, the coils may pass through a bath, as of brine, and reduce this to the desired temperature. The cold brine is then circulated through the refrigerator. This latter method gives a more nearly constant temperature. The least move- ment of the expansion-valve causes variations in the back pressure, and hence in the boiling -temperature of the ammonia, which would affect the surrounding air if used directly, but which would be absorbed by the large heat capacity of the brine. The direct sys- tem is, however, simpler and less expensive to install and to maintain. As long as any liquid remains unvaporized in the refrigerator coils these will remain at the temperature of vaporization, but afterwards the pipes assume the higher temperature of the bath or surrounding atmos- phere, and then begin to superheat the vapor at con- stant pressure. This superheating is still further in- DISCUSSION OF REFRIGERATING PROCESSES. 347 creased in the pipe leading back to the compressor. After leaving the compressor the highly superheated vapor passes to the condenser, losing part of its super- heat at constant pressure on the way. This process is finished in the condenser, and then the vapor begins to liquefy at the temperature corresponding to the high pressure. The liquid finally emerges cooled to the temperature of the cooling water and collects in the storage-tank above the expansion-valve ready for a new cycle. The corresponding pv- and T^-cycles are shown in FIG. 118. Fig. 118, ab represents the passage of the liquid through the expansion-valve along a constant-heat curve, dur- ing which a portion of the liquid r~ is vaporized and the refrigerative power of its liquefaction destroyed; 348 THE TEMPERATURE-ENTROPY DIAGRAM. be represents the vaporization of the remainder of the liquid; cd, the superheating of the vapor during the last part of the refrigerator coils and the return pipe to the compressor; de' represents the compression, and e'h, the loss of superheat by conduction, radiation, etc., as the hot gas flows along the pipe to the condenser. If the compressor were two-stage, with intermediate cooling down to atmospheric temperature, the path followed would be defgh. The liquefaction is repre- sented by hi, and the further cooling of the liquid down to atmospheric temperature by w. To decrease the work required to compress the gas, attempts are made in various types of compressors to cool it during compression by the use of water-jackets or by the direct injection into the cylinder of either liquid ammonia or oil. In such cases the compression is no longer adiabatic, but of the form pv n =p l r l n , where n will depend in any given case upon the amount of heat extracted by the jackets or absorbed by the injected fluid. The temperature of the entering vapor is usually considerably lower than that of the cooling water, so that heat is radiated only during the latter part of the stroke, when the temperature of the vapor greatly ex- ceeds that of the water. The compression line of indicator-cards from such compressors should there- fore approximate closely the adiabatic curve drawn through the commencement of the compression stroke DISCUSSION OF REFRIGERATING PROCESSES. 349 and should begin to fall below it more and more only as the discharge pressure is approached. If oil of the same temperature as the entering vapor is injected into the cylinder, it can affect the temper- ature only by absorbing heat as the gas is compressed. The specific heat of the oil is greater than that of the cylinder walls, and possibly conduction occurs some- what more rapidly from vapor to oil and then oil to metal than it would directly from vapor to metal, especially if the oil is in a finely divided state. This can only result in changing slightly the exponent n of the compression curve pv n = p l v 1 n . The effect of injecting liquid ammonia is difficult to describe in general terms, as the results will differ according to the quantity injected and the various temperatures of the liquid, vapor, and cylinder walls. If the cylinder walls are assumed to be non-conduct- ing and the injected liquid is previously cooled to the temperature of the refrigerator, the superheated vapor will then lose its superheat and in so doing suffer a drop in pressure, as the cylinder volume is momen- tarily constant. At the same time, however, the in- jected liquid will be partially vaporized, increased in volume, and thus effect an increase in pressure. The resultant effect in this case would undoubtedly be a net reduction of pressure and thus decrease the work of compression. As, however, the cylinder walls are good conductors when in contact with a liquid, enough 350 THE TEMPERATURE-ENTROPY DIAGRAM. of the ammonia might thus be vaporized to produce a net increase in pressure. If the liquid is injected at the same temperature as the entering vapors, it is at a higher temperature than that of saturated vapor at the prevailing pressure, and hence will partially vapor- ize until a condition of equilibrium is established. What the final conditions of pressure and temperature will be will evidently depend upon the relative weights of liquid and vapor, the pressure, and the temperatures of vapor, liquid, and cylinder. In either of the above assumptions, if the resultant pressure is less and part of the liquid still remains unevaporated, the work of compression would be still further decreased, as satu- rated vapors transmit heat to the cylinder walls more rapidly than superheated vapors. If the liquid is injected into the suction-pipe of the compressor, it will expand at the prevailing back pressure and reduce the temperature down to that corresponding to that pressure. Whether or not there results a net diminution in volume must depend upon the amount injected and the quantity of heat received from external sources. Although the work may or may not be decreased, according to circumstances, the temperature will at least be decreased, and thus the amount of necessary cooling water diminished. It is theoretically possible to effect a further saving in liquid refrigerating plants by changing from the throttling-curve ab (Fig. 118) to a frictionless adiabatic DISCUSSION OF REFRIGERATING PROCESSES. 3ol expansion ab' ' that is, by replacing the expansion- valve with an auxiliary cylinder and thus utilize the expansive force of the ammonia to help run the com- pressor. The refrigerative power of the ammonia would be increased at the same time, since the amount vaporized during expansion would be decreased from kb to kb'. It is possible that the mechanical com- plications thus introduced would more than counter- balance the thermodynamic savings. It has also been suggested that the loss in refriger- ative power occasioned by the expansion ab could be diminished by reducing the temperature of the liquid at the point a. Thus the gas in passing from the refrigerator to the compressor absorbs from the atmos- phere the heat represented by the area under Id. If such a loss is unavoidable, it could be neutralized by jacketing the return pipe with the liquid ammonia about to be fed to the refrigerator, thus reducing the temperature of the latter from a to a t and so decreas- ing the amount vaporized by expansion from kb to W. Conditions for Maximum Efficiency. The ammonia, after leaving the expansion coil, must be as cold as the lowest temperature desired in the refrigerating plant, and after condensation in the cooling coils cannot have a temperature less than that of the cooling water. Evi- dently the minimum temperature range is covered by these two factors, so that the minimum amount of com- 352 THE TEMPERATURE-ENTROPY DIAGRAM. prossion work will be obtained when the ammonia is cooled just to the minimum temperature and is com- pressed to a pressure corresponding to the temperature of the cooling water. Such conditions necessitate the ammonia entering the compressor as soon as it is com- pletely vaporized. If we call TI the upper temperature and T 2 the lower temperature, the refrigcrative effect per pound of ammonia will be represented by H 2 q\ and the work of compression will be represented by H\ H. Thus ENTROPY ANALYSIS IN THE BOILER ROOM. 377 while the entropy of the hot gases in the furnace has decreased by the amount ^>=-m- t the entropy of the m r\ steam has increased by the amount ^f- $ = ^- , and J- 8 1 8 there has therefore resulted a net increase in the entropy of the system of the amount ( - 1 J . If now T c represent the temperature of the condenser, i.e., the coldest temperature practically attainable, the area efdc represents the heat theoretically unavailable for work, while the heat Q is still contained in the gases, while efd'c represents the heat no longer available after the heat Q has entered the steam. The increase in non- availability or the loss in motivity, is thus represented by the a,rea,ff f d'd, and is equal to the increase in entropy multiplied by the lowest of available temperatures, or X5H- This increase in non-availability is thus seen to be equal to the heat unavoidably lost under the original coneli- m tions, T c -(h, multiplied by a factor 7/r^-l, which in- L 8 creases in value as T a decreases, and only disappears when T 8 = T F , i.e., when the heat is used at the tem- perature of the fire. Loss of Availability Due to the Liquid Line. Due to the fact that two out of the four operations constituting 378 THE TEMPERATURE-ENTROPY DIAGRAM. the Rankinc cycle occur in the boiler a further loss must be charged against it. The deviation of the Rankine cycle from the Carnot cycle has already been discussed upon pp. 233-234, and was found to be due to the slope of the liquid line &s determined by the magnitude of the specific heat of the liquid used. The full motivity of the heat as determined by the boiler pressure is not possible of attainment, as part of the heat absorbed by the water is received at temper- atures less than that corresponding to this pressure. This can be remedied only by heating the feed water to boiler temperatures before it enters the boiler. The deviation between the Carnot and Rankinc cycles for any given case can be readily obtained by cornpu- T T ting the motivity ^^ and comparing it with the * corresponding Rankine efficiency taken from the table; on p. 231. The losses due to the boiler may therefore be sum- marized as (1) Heat lost through the setting and up the stack. (2) Loss of availability due to reduced temperature. (3) Loss of availability due to use of Rankine cycle in place of the Carnot cycle. The Regenerative Principle. In the discussion of iso- diabatic cycles on pp. 6-10 it was shown that the effici- ency of such cycles was equal to that of the Carnot cycle, and it was pointed out that in place of an adiabatic ENTROPY ANALYSIS IN THE BOILER ROOM. 379 expansion there should be an extraction of heat by conduction during decrease in temperature exactly equal to the heat required during the rise-in-temperature op- eration. (See be and da, Fig. 3.) The application of this principle is further illustrated on pp. 158-160 in the discussion of the Stirling and Ericsson cycles for hot air engines. The application of this principle to the cycle of water in a power plant necessitates a modification of the Fiu. 123. Rankine cycle whereby the adiabatic expansion line is replaced by a curve whose contour is a duplicate of the water line. That means that as the- steam expands an amount of heat must be transferred from it to the feed water just equal in amount to that required to heat the water at eacli temperature. Thus in Fijr. 123 the isen- tropic curve be' is replaced by the curve be posse>srng the same slope as the water line ad. If it were possible to extract heat from the steam in 380 THE TEMPERATURE-ENTROPY DIAGRAM. this fashion during its expansion it would then be pos- sible to realize theoretically at least the maximum avail- ability of the heat in the boiler. A complete utilization of this principle is not feasible with steam engines as constructed, but in the case of compound engines it is possible to approximate it by taking sufficient steam from each receiver to warm the FIG. 124. feed water up to the temperature of the receiver steam. Thus for example in a triple expansion engine operating with cold feed water, Fig. 124, the heat A can be taken from the first receiver and used to warm the feed water from b to a, and the heat B can be taken from the second receiver and serve to warm the feed water from G (the temperature of exhaust) to b, and finally the heat C can be taken from the exhaust steam and used to warm the feed water from d (the temperature in the supply pipe) to c. ENTROPY ANALYSIS IN THE BOILER ROOM. 381 Although this regenerative principle as thus applied may seem startling, the first time it is considered a moment's thought will show that one is already familiar with part of it under different names. Thus the quantity of heat C is saved in all economical plants with a primary heater, while part of B at least is saved by utilizing the exhaust of the auxilliary engines which ordinarily discharge at atmospheric pressure. And finally the equivalent of the rest of B and part of A is obtained by saving some of the heat of the exhaust gases with an economizer. The use of the economizer is not truly a part of this regenerative principle, but is the utilization of heat otherwise wasted, and this opens up the interesting question as to whether it is better to use small heating surface in the main boiler combined with a good econ- omizer and no regenerative action from the interme- diate cylinders, or to make the heating surface of the boiler so extensive that nothing remains for the econo- mizer to do, combined with regenerative action at the various intermediate receivers.* The Primary Feed-water Heater. Ideally it would be perfectly feasible to return the condensate or an equal amount of new feed water to the boiler at the temperature of the exhaust steam; practically the tem- perature is -always considerably less than this because * Engineering, 1895, pp.63, 97, 191. Feed-Water Heaters, by Professor A. C. Elliott. 382 THE TEMPERATURE-ENTROPY DIAGRAM. of heat losses due to radiation from the feed pipe between the heater and the boiler, or between the primary heater and any secondary device which may be employed. The use of a primary heater is based upon the impos- sibility or rather the practical unfeasibility of reducing the temperature of the exhaust steam to that of the cooling water. That is, the small increase in power is offset by greater increased loss in the condenser plant. FIG. 125. In the case of engines using vapors other than steam where the pressure at exhaust is still above atmospheric, the gain in power from the lowering of the back pres- sure line justifies the maintenance of a lower temper- ature in the exhaust condenser, so that the question of a primary feed-fluid heater is eliminated both from the thermal as well as the practical point of view. (See temperatures quoted on pp. 255-257 for the S02 and ^Eihylamine engines. ENTROPY ANALYSIS IN THE BOILER ROOM. 383 The saving to be credited to the primary heater is the decrease in the amount of heat absorbed by the water in the boiler occasioned by its use. Thus if the heater raises the temperature from i f to h (Fig. 125) the decrease in the amount of heat absorbed in the boiler per pound of steam is while the total heat absorbed per pound without the heater is and therefore the efficiency of the heater is equal to The denominator of this expression, assuming the steam to be dry and the cold feed water to be at an average temperature of 70 F., fluctuates in value from 1130 B.T.U. per Ib. at 40 Ibs. gage to 1165 B.T.U. per Ib. at 330 Ibs. gage, or undergoes a variation of less than four per cent for a temperature range of 140 F. Between 100 Ibs. gage and 150 Ibs. gage the variation is from 1147 to 1156 B.T.U. per Ib., or less than one per cent. For rough calculations the average value of the de- nominator may be assumed as 1160 B.T.U. per Ib. The value of the numerator shows the effectiveness of the feed water heater, and it is evident that the efficiency is directly proportional to the increase in the 384 THE TEMPERATURE-ENTROPY DIAGRAM. temperature of the feed water and equals approximately * J;; so that on an average 11.0 F. increase in the J. J.OU temperature of the feed water, coriesponds to one per cent saving in coal, and an increase of 100 F. corre- sponds to a saving of about nine per cent in coal. This thermal saving is but one of many economies instituted by a feed-water heater. The heating of the feed water serves to precipitate many of the impurities in the boiler feed in the heaters, whence they may be easily removed, instead of in the less easily accessible boiler. This results in less impairment of the conduc- tivity of the heating surface and effects a saving in fuel varying, according to the manufacturers, anywhere from fifteen per cent to fifty per cent with hard feed waters, while the average saving with soft waters is about ten per cent. A further saving is obtained from the decreased chill- ing of the boiler from the introduction of cold water, thus diminishing the internal stresses due to temper- ature changes, reducing the repair bills upon the boiler, and prolonging its life. The Secondary Feed-water Heater. The primary heater operating with exhaust steam at one, two or three pounds pressure can raise the temperature of the feed water only to 102, 126, or 142 F. With con- densing engines there is a steam consumption by the auxiliaries of about ten per cent that of the main ENTROPY ANALYSIS IX THE BOILER ROOM. 385 engine which is usually exhausted to the atmosphere at 212 F., or slightly in excess of this temperature. It is customary to utilize part of the waste heat in this exhaust to raise the temperature of the feed coming from the primary heater to as near 212 F. as possible. Thus in the case of three pounds back- pressure it is possible to heat the feed water from 142 F. to 212 F. or through 70 F. On the assumption of ten per cent consumption by the auxiliaries, 11 pounds of feed water must be heated for each pound used by the auxiliaries, i.e., 770 B.T.U. would need to be supplied by the secon- dary heater. Each pound of steam at 212 F. possesses a heat of condensation equal to 970 B.T.U.,. so that, assum- ing the exhaust from the auxiliaries to be fifteen per cent condensed, this leaves 970 X. 85 -823 B.T.U. avail- able for heating, which is more than ample. In case the main engine exhausts at lower pressure a greater heat demand is made upon the secondary heater, but this adjusts itself automatically to a considerable extent by the increased steam consumption of the auxiliaries. The total saving due to primary and secondary heaters expressed in per cent of the heat required by the steam without such heaters is given by which, upon the assumption of 70 F. as the average temperature of the supply water, reduces to 386 THE TEMPERATURE-ENTROPY DIAGRAM. 142 .122, 1160 or 12.2 per cent at ordinary boiler pressures. Saving Effected by an Economizer. Advantage may be taken of the unavoidable loss of heat up the stack to offset, at least partially, the loss caused by cold feed water. Although the hot gases cannot be cooled below the temperature of the boiler and must in fact be con- siderably hotter than the steam to insure rapid flow of heat, they are still hot enough to impart considerable heat to the colder feed water. This is accomplished by passing the feed water on its way to the boiler through suitable piping installed in the path of the hot gases. Such an economizer will not deliver the water to the boiler at boiler temperature, but will succeed in raising it far above the temperatures obtained in primary and secondary heaters. For example, tests on Green econ- omizers show an increase in the feed temperature of about 120 F., whether the water enters the economizer at 40 F. or 200 F. making but little difference. The gases leaving the economizer possessed temperatures varying from 254 F. to 293 F. Any such increase in temperature above that of the secondary heater represents a direct saving of heat which would otherwise be wasted. It might seem pos- sible at first sight to bring the feed water up to the boiler temperature, but this is impracticable as a cer- tain temperature of the exhaust gases is required to EXTROPY ANALYSIS IN THE BOILER ROOM. 387 produce the necessary draft in the chimney. If cooled beyond this point it becomes necessary to install a forced draft and then its steam cost must be charged against the saving produced by the economizer. In many installations economizers are installed in place of secondary and sometimes even in place of primary heaters, and in such case they obtain credit for savings due to other devices, but apparently they produce about the same increase in temperature re- gardless of the initial temperature of the water. TABLE OF PROPERTIES. PROPERTIES OF SATURATED STEAM FROM 400 F. TO THE CRITICAL TEMPERATURE.* Increase in Heat of Temper- ature, Degrees Absolute Pressure in Lbs. Specific Volume Due to the Liquid Latent Heat in B.T.U. Total Heat. Entropy Water Entropy xr f Vapor- Fahren- heit. per Sq P In. Vapori- zation in Cu. Ft. per Lb. per Lb. above 32 F. ization. t P a a 1 r fl+r 4-o *-* 420 308.7 1.495 390.1 818.9 1209.0 .5912 .9254 430 343.6 1.347 405.6 805.4 1211 .6031 .9055 440 381.5 1.215 416.1 796.2 1212.3 .6118 .8853 450 422.4 1.099 426.7 786.5 1213.2 .6266 .8648 460 466.2 .9953 437.3 777.4 1214.7 .6381 .8455 470 513.5 .9024 447.9 765.2 1213.1 .6495 .8233 480 564.4 .8194 458.5 753.5 1212.0 .6610 .8021 490 619.5 .7436 469.2 741.3 1210.5 .6723 .7808 500 678.5 .6752 479.9 728.2 1208.1 .6835 .7590 510 741.7 .6134 490.6 714.4 1205.0 .6946 .7369 520 808.5 .5565 501.4 699.8 1201.2 .7058 .7145 530 889.1 .5050 512.1 684.4 1196.5 .7166 .6917 540 956.1 .4585 523.0 668.1 1191.1 .7275 .6685 550 1036. .4174 533.8 650.8 1184.6 .7383 .6448 560 1122. .3798 544.7 632.4 1177.1 .7490 .6204 570 1212. .3457 555.6 612.9 116S. 5 .7597 .5955 580 1308. .3131 566.5 592.2 1158.7 .7702 .5698 590 1409. .2828 577.5 570.0 1147.5 .7807 .5432 600 1516. .2547 588.5 546.3 1134.8 .7912 .5157 610 1628. .2283 599.6 520.8 1120.4 .8015 .4871 620 1745. .2037 610.6 493.2 1103.8 .8118 .4569 630 1867. .1809 621.7 463.2 1084.9 .8221 .4252 640 2000. .1610 632.9 430.3 1063.2 .8322 .3914 650 2137. .1416 641.0 396.7 1037.7 .8422 .3549 660 2281. .1217 655.2 352.2 1007.4 .8527 .3146 670 2431 . . 1031 666.5 304.1 970.6 .8634 .2693 680 2586. .0800 677.8 245.2 923.0 .8723 .2152 690 2748. .0464 689.1 164.3 853.4 .8825 .1429 698 2882. * See Engineering (London), Jan. 4, 1907. HYPERBOLIC OR NAPERIAN LOGARITHMS. THE log of a number is the exponent of the power to which it Is necessary to raise a fixed number, called the base, to produce the given number. In the com- mon logs the base is 10 and in the Naperian logs the base, e, is 2.71828 . . . The mathematical relations between these two systems and any third are given by the equations, lQlog ]0 where a is any number and b is the third base. Whence log e a = log e 10-logio a = 2.3026 logic a and logio a = logio e-loge a = 0.4343 loge a and ea or = In general, the log of a number in any system equals either the reciprocal of the Naperian log of the base of that system times the Naperian log of the number, or equals the reciprocal of the common log of the base of that system times the common log of the number. In any system the base of which is greater than 1, the logs of all numbers greater than 1 are positive and the logs of all numbers less than 1 are negative. 389 390 THE TEMPERATURE-ENTROPY DIAGRAM. NAPERIAN LOGARITHMS. e=2.7182818 log e= 0.4342945= M. 1 2 3 4 5 6 7 8 9 0.08618 1.0 0.0000 0.00995 0.01980 0.02956 0.03922 0.04879 0.05827 0.06766 0.07696 '.2 1.3 0.09531 0.1823 0.2624 0.1044 0.1906 0.2700 0.1133 0.1988 0.2776 0.1222 0.2070 0.2852 0.1310 0.2151 0.2927 0.1398 0.2231 0.3001 0.1484 0.2311 0.3075 0.1570 0.2390 0.3148 0.1655 0.2469 0.3221 0.1739 0.2546 0.3293 1.4 1.5 1.6 0.3365 0.4055 0.4700 0.3436 0.4121 0.4762 0.3507 0.4187 0.4824 0.3577 0.4243 0.4886 0.3646 0.4318 0.4947 0.3716 0.4382 0.5008 0.3784 0.4447 0.5068 0.3853 0.4511 0.5128 0.3920 0.4574 0.5188 0.3988 0.4637 0.5247 17 1.8 1.9 0.5306 0.5878 0.6418 0.5365 1 -,((;; L6471 0.5423 0.5988 0.6523 0.5481 0.6043 0.6575 0.5539 0.6098 0.6627 0.5596 0.6152 0.6678 0.5653 0.6206 0.6729 0.5710 0.6259 0.6780 0.5766 0.6313 0.6831 0.5822 0.6366 0.6881 2.0 0.6931 0.6981 0.7031 0.7080 0.7129 0.7178 0.7227 0.7275 0.7324 0.7372 2.1 2.2 2.3 0.7419 0.7884 0.8329 0.7467 0.7930 0.8372 0.7514 0.7975 0.8416 0.7561 0.8020 0.8459 0.7608 0.8065 0.8502 0.7655 0.8109 0.8544 0.7701 0.8154 0.8587 0.7747 0.8198 0.8629 0.7793 0.8242 0.8671 0.7839 0.8286 0.8713 2.4 1 5 2.6 0.8755 0.9163 0.9555 0.8796 0.9203 0.9594 0.8838 0.9243 0.9632 0.8879 0.9282 0.9670 0.8920 0.9322 0.9708 0.8961 0.9361 0.9746 0.9002 0.9400 0.9783 0.9042 0.9439 0.9821 0.9083 0.9478 0.9858 0.9123 0.9517 0.9895 2'S 2 ) 0.9933 1.0296 1.0647 0.9969 1.0332 1.0682 1.0006 1.0367 1.0716 1.0043 1.0403 1.0750 1.0080 1.0438 1.0784 1.0116 1.0473 1.0818 1.0152 1.0508 1.0852 1.0188 1.0543 1.0886 .0225 .0578 1.0919 1.0260 1.0613 1.0953 3.0 1.0986 1.1019 1.1053 1.1086 1.1119 1.1151 1.1184 1.1217 1.1249 1.1282 LI 3^3 1.1314 1.1632 1.1939 1.1346 1.1663 1.1969 1.1378 1.1694 1.2000 .1410 .1725 .2030 1.1442 1.1756 1.2060 1.1474 1.1787 1.2090 1.1506 1.1817 1.2119 1.1537 1.1848 1.2149 .1569 1.1878 .2179 1.1600 1.1909 1.2208 !V5 .56 1.2238 1.2528 1.2809 1.2267 1.2556 1.2837 1.2296 1.2585 1.2865 .2326 .2613 1.2892 1.2355 1.2641 1.2920 1.2384 1.2669 1.2947 1.2413 1.2698 1.2975 1.2442 1.2726 1.3002 .2470 .L'754 1.3029 1.2499 1.2782 1.3056 3.7 3 8 3.'> 1.3083 1.3350 1.3610 1.3110 1.3376 1.3635 1.3137 1.3403 1.3661 1.3164 1.3429 1.3686 1.3191 1.3455 1.3712 1.3218 1.3481 1.3737 1.3244 1.3507 1.3762 1.3271 1.3533 1.3788 1.3297 .3558 1.3813 1.3324 1 .3584 1.3838 1.0 1.3863 1.3888 1.3913 1.3938 1.3962 1.3987 1.4012 1.4036 1.4061 1.4085 1.1 4.2 4.3 1.4110 1.4351 1.4586 1.4134 1.4375 1.4609 1.4159 1.4398 1.4633 1.4183 1.4422 1.4656 1.4207 1.4446 1.4679 .4231 .4469 .4702 1.4255 1.4493 1.4725 1.4279 1.4516 1.4748 1.4303 1.4540 1.4770 1.4327 1.4563 1.4793 4.4 1 5 4.6 1.4816 1.5041 1.5261 1.4839 1.5063 1.5282 1.4861 1.5085 1.5304 1.4884 1.5107 1.5326 1.4907 1.5129 1.5347 .4929 .5151 .5369 .4951 .5173 .5390 1.4974 1.5195 1.5412 1.4996 1.5217 1.5433 1.5019 1.5239 1.5454 4.7 4.8 4.< 1.5476 1.5686 1.5892 1.5497 1.5707 1.5913 1.5518 1.5728 1.5933 1.5539 1.5748 1.5953 1.5560 1.5769. 1.5974 .5581 .5790 1.5994 .5602 .5810 1.6014 1.5623 1.5831 1.6034 1.5644 1.5851 1.6054 1.5665 1.5872 1.6074 5.( 1.6094 1.6114 L6134 1.6154 1.6174 1.6194 1.6214 1.6233 1.6253 1.6273 5. 5.2 5.. 1.6292 1.6487 1.6677 1.6312 1.6506 1.6696 1.6332 1.6525 1.6715 1.6351 1.6544 1.6734 1.6371 1.6563 1.6752 1.6390 1.6582 1.6771 1.6409 1.6601 1.6790 1.6429 1.6620 1.6808 1.6448 1.0639 1.6827 1.6467 1 .66.58 1.6845 5, 5.5 56 1.6864 1.7047 1.7228 1.6882 l.Tonr, 1.7246 1.6901 1.7084 1.7263 1.6919 1.7102 1.7281 1.6938 1.7120 1.7299 1.6956 1.7138 1.7317 1.6974 1.7156 1.7334 1.6993 1.7174 1.7354 1.7011 1.7192 1.7370 1.7029 1.7210 1.7387 HYPERBOLIC OR NAPER1AN LOGARITHMS. 391 XAPERIAN LOGARITHMS Continued. 5.7 5.8 5.9 1 2 3 4 5 6 7 8 9 1.7405 1.7579 1.7750 1.7422 1.7596 1.7766 1.7440 1.7613 1.7783 1.7457 1.7630 1.7800 1.7475 1.7647 1.7817 1.7492 1.7664 1.7834 1.7509 1.7681 1.7851 1.7527 1.7699 1.7867 1.7544 1.7716 1.7884 1.7561 1.7733 1.7901 6.0 1.7918 1.7934 1.7951 1.7967 1.7984 1.8001 1.8017 1.8034 1.8050 1.8066 6.1 6.2 6.3 1.8083 1.8245 1.8405 1.8099 1.8262 1.8421 1.8116 1.8278 1.8437 1.8132 1.8294 1.8453 1.8148 1.8310 1.8469 1.8165 1.8326 1.8485 1.8181 1.8342 1.8500 1.8197 1.8358 1.8513 1.8213 1.8374 1.8532 1.8229 1.8390 1.8547 6.4 6.5 6.6 1.8563 1.8718 1.8871 1.8579 1.8733 1.8886 1.8594 1.8749 1.8901 1.8610 1.8764 1.8916 1.8625 1.8779 1.8931 1.8641 1.8795 1.8946 1.8656 1.8810 1.8961 1.8672 1.8825 1.8976 1.8687 1.8840 1.8991 1.8703 1.8856 1.9006 6.7 6.8 6.9 1.9021 1.9169 1.9315 1.9036 1.9184 1.9330 1.9051 1.9199 1.9344 1.9066 1.9213 1.9359 1.9081 1.9228 1.9373 1.9095 1.9242 1.9387 1.9110 1.9257 1.9402 1.9125 1.9272 1.9416 1.9140 1.9286 1.9430 1.9155 1.9301 1.9445 7.0 1.9459 1.9473 1.9488 1.9502 1.9516 1.9530 1.9544 1.9559 1.9573 1.9587 7.1 7.2 7.3 1.9601 1.9741 1.9879 1.9615 1.9755 1.9892 1.9629 1.9769 1.9906 1.9643 1.9782 1.9920 1.9657 1.9796 1.9933 1.9671 1.9810 1.9947 1.9685 1.9824 1.9961 1.9699 1.9838 1.9974 1.9713 1.9851 1.9988 1.9727 1.9865 2.0001 7.4 7.5 7.6 2.0015 2.0149 2.0281 2.0028 2.0162 2.0295 2.0042 2.0176 2.0308 2.0055 2.0189 2.0321 2.0069 2.0202 2.0334 2.0082 2.0215 2.0347 2.0096 2.0229 2.0360 2.0109 2.0242 2.0373 2.0122 2.0255 2.0386 2.0136 2.0268 2.0399 7.7 7.8 7.9 2.0412 _Mi:>41 2.0668 2.0425 2.0554 2.0681 2.0438 2.0567 2.0694 2.0451 2.0580 2.0707 2.0464 2.0592 2.0719 2.0477 2.0605 2.0732 2.0490 2.0618 2.0744 2.0503 20631 2.0757 2.0516 2.0643 2.0769 2.0528 2.0656 2.0782 8.0 2.0794 2.0807 2.0819 2.0832 2.0844 2.0857 2.0869 2.0881 2.0894 2.0906 8.1 82 8.3 2.0919 2.1041 2.1163 2.0931 2.1054 2.1175 2.0943 2.10M 2.1187 2.0956 2.1078 2.1199 2.0968 2.1090 2.1211 2.0980 2.1102 2.1223 2.0992 2.1114 2.1235 2.1005 2.1126 2.1247 2.1017 2.1138 2.1258 2.1029 2.1150 2.1270 8.4 8.5 8.6 2.1282 2.1401 2.1518 2.1294 2.1412 2.1529 2.1306 2.1424 2.1541 2.1318 2 n: j ,r, LM ,-,.->:.' 2.1330 2.1448 2.1564 2.1342 2.1459 2.1576 2.1353 2.1471 2.1587 2.1365 2.1483 2.1599 2.1377 2.1494 2.1610 2.1389 2.1506 2.1622 8.7 KM 8.9 2.1633 2.1748 2.1861 2.1645 2.1759 2.1872 2.1656 2.1770 2.1883 2.1668 2.1782 2.1894 2.1679 2 17M 2.1905 2.1691 2.1804 2.1917 2.1702 2.1815 2.1928 2.1713 2 1827 2.1939 2 1725 2.1838 2.1950 2.1736 2.1849 2.1961 9.0 2.1972 2.1983 2.1994 2.2006 2.2017 2.2028 2.2039 2.2050 2.2061 2.2072 9.1 92 9.3 2.2083 2.2192 2.2300 2.2094 2.2203 2.2311 2.2105 2.2214 2.2322 2.2116 2.2225 2.2332 2.2127 2.2235 2.2343 2.2138 2.2246 2.2354 2.2148 2.2257 2.2364 2.2159 2.2268 2.2375 2.2170 2.2279 2.2386 2.2181 2.2289 2.2396 9.4 9.5 9.6 2.2407 2.2ol3 2.2618 2.2418 2.2523 2.2628 2.2428 2.2534 2.2638 2.2439 2.2544 2.2649 2.2450 2.2555 2.2659 2.2460 2.2565 2.2670 2.2471 2.2576 2.2680 2.2481 2.25M 2.2690 2.2492 2.2597 2.2701 2.2502 2.2607 2.2711 9.7 9.8 9.9 2.2721 2.2824 2.2925 2.2732 2.2834 2.2935 2.2742 2.2844 2.2946 2.2752 2.2854 2.2956 2.2762 2.2865 2.2966 2.2773 2.2875 2.2976 2.2783 LV2.S.S.-, 2.2986 2.2793 2.2895 2.2996 2.2803 2.2905 2.3006 2.2814 j NIB 2.3016 10.0 2.3026 392 THE TEMPERATURE-ENTROPY DIAGRAM. LOGARITHMS. Nat. Nos. I 2 3 4 5 6 7 8 9 Proportional Parts. 1 23 456 789 10 11 12 0000 0414 0792 0043 0453 0828 0086 0492 0864 0128 0.531 0899 0170 0569 0934 0212 0607 0969 0253 064.5 L004 0294 0682 103s 0334 071!) 1072 0374 07.5.5 1106 4 8 12 4 8 11 3 7 10 17 21 25 15 19 23 14 17 21 29 33 37 26 30 34 24 28 31 13 1139,1173 1206 1239 1271 1303 1 :;:,.-, 1367 1:599 3 6 10 13 16 19 23 26 29 14 1461 1492 1523,1553 1584 1614 1644 1673 1703 1732 369 12 15 18 21 24 27 15 1761 1790 18181847 187.5 1903 1931 1959 1987 2014 368 11 14 17 20 22 25 16 2041 2068 2095:2122 2148 217.5 2201 2227 22.53 22", 9 3 5 8 11 13 16 18 21 24 17 230412330 23552380 240,5 2430 245512480 2504 2,529 257 10 12 15 17 20 22 18 2553 2,577 2601 2625 264s 2:172 2695 271s 2742 276,5 257 9 12 14 16 19 21 19 2788 2810 2833 2856 2878 2900 292:i 294,5 2967 2989 247 9 11 13 16 18 20 20 3010 3032 3054 3075 509f 3118 3139 3160 3181 3201 246 8 11 13 15 17 19 21 3222 32 n 326213284 5304 3324 334,5 :-',365 ;::s,5 3404 246 8 10 12 14 16 18 22 1424 3411 34643483 5,502 3522 3.541 :!,560 ',.579 3.59s 246 8 10 12 14 15 17 23 3617 3636 3655 ! 3674 I'll): 3711 3729 3717 3700 3784 246 7 9 11 13 15 17 24 3802 3820 3838 3856 3874 3892 3909 3927 394,5 3962 245 7 9 11 12 14 16 25 3979 3997 4014 4031 4048 406,5 4082 4099 4116 4133 235 7 9 10 12 14 15 26 11,50 4166 4183 4200 12 ic 4232 1219 1205 1281 4298 235 7 8 10 11 13 15 27 1314 i:i:;o f.'.Kl 4362 l:;7s 4393 4409 1425 4410 4 t,5(l 2 3 5 689 11 13 14 28 1172 1187 4502 4518 !.i:;:; 1.54s 4564 4579 1591 4609 235 689 11 12 14 29 4624 1639 4654 4669 1683 4698 4713 4728 4742 4757 1 3 4 679 10 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 3 4 6 7 9 10 11 13 31 4914 4928 4942 4955 4969 19S.:; 4997 ,5011 502.1 .5038 3 4 678 10 11 12 32 5051 S005 5079 5092 5105 .5119 5132 51 1.5 51.59 5172 3 578 9 11 12 33 518,5 519s 5211 5224 5237 52(1:; 527(1 5289 ,5302 3 568 9 10 12 34 5315 5328 5340 5353 5366 .5378 5391 5403 541(1 5428 3 568 9 10 11 35 5441 5453 5465 5478 5490 5.502 5514 5,527 5.5:59 5551 2 5 6 7 9 10 11 36 6603 5575 5S87 :,599 5611 5623 563,5 5647 56.58 5670 2 567 8 10 11 37 5682 5994 5705 5717 5729 57.10 5752 5763 577.5 .578(1 2 3 567 8 9 10 38 5798 5809 5821 5832 5843 58.55 5866 5877 5S88 5S99 2 3 567 8 9 10 39 5911 5922 5933 5944 59,5.5 5966 5977 59XS 5999 6010 2 3 457 8 9 10 40 6021 5031 6042 6053 6064 607,5 608.5 H096 6107 6117 2 3 5 6 8 9 10 41 612x 1138 1149 6160 6170 5180 1191 1201 0212 (1222 2 3 5 6 789 42 43 44 0232 0335 (ll.V, 121:; 1345 6444 125:; ;:;-,5 14.54 6263 r,.;r,5 6464 ,271 137.5 6474 6284 i:;s.-, 1484 1291 639.5 6493 6304 1405 6314 1415 6,513 !i:;25 6425 6522 2 3 2 3 2 3 5 6 5 6 5 6 789 789 7 8 9 45 46 0632 6628 1542 66:57 6,5.51 6646 6561 iiti.56 6571 i(i(,5 (1.580 167.5 6.590 K1S! 6,599 6693 6609 6702 6618 6712 2 3 2 3 5 6 5 6 789 778 47 6721 1730 6739 6719 67.5S 6767 177H 678.5 1791 f,S03 2 3 5 5 678 48 49 (IS 12 (1902 1821 1911 1830 1920 6839 6928 1S|S 1937 is:, 7 194-1 1800 1955 687.5 19(11 1SS! 1972 (189:5 6981 2 3 2 3 4 5 4 5 6 7 '8 678 50 51 6990 7076 6998 70S1 7007 7093 7016 7101 7024 7110 7033 7118 7042 7126 70.50 713.5 70.59 711:; 7067 7152 2 3 2 3 3 4 5 345 678 678 52 7160 7168 7177 7185 719:; 7202 721017218 722d 7235 2 2 345 677 53 72415 72.51 7259 7267 -275 7281 729217300 7:lds 7316 2 2 345 667 54 7324 7332 7340 7348 7356 7364 73727380 738S 7396 2 2 345 667 COMMON LOGARITHMS. 393 LOGARITHMS Continued. "6 "jr. Proportional Parts. "ej % 1 2 8 9 1 23 456 789 "55" 7404 ^7412 7419 7427 7435 7443 7451 7459 7466 7474 122 345 567 56 7482 7490 7497 7505 7513 7520 7:,2S 7536 7543 7551 1 22 345 567 57 7.-1.59 7566 7574 7582 7589 7597 7604 7612 7619 7627 122 345 567 58 7634 7642 7649 7657 7664 7672 7679 7686 76M 7701 1 1 2 344 567 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 12 344 567 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 344566 61 7853 7860 7v, x 7875 7882 7889 7896 7903 7910 7917 1 1 2 344 566 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7 9917 9961 9786 9vr, 9969 9795 9841 9880 9930 9974 9800 984{ 9891 !)4 9978 9805