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Availability and irreversibility in thermodynamics

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1951 Br. J. Appl. Phys. 2 183

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O R I GI N A L C O N T R I B U T I O N SAvailability and irreversibility in thermodynamics

By PROFESSOROSEPH H. KEENAN, S.B ,* Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.[Paper received 22 February, 19511

Connexion s made between Gibbssavailable energy of the system and medium and the criterionof stability. The available energy concept is developed for systems which communicateonly withthe uniform medium or atmosphere and for systems which communicate with an additionalreservoir of specified temperature. The treatment is extended to problems of transient andsteady flow. A measure of departure from the ideal, the irreversibility,is defined and examinedfor its significance. Performance coefficients are devised for several classes of processes. Finally,the generality of the method is exhibited by analysisof a variety of thermodynamicphenomena.I N T R O D U C T I O NThe concept of an idealized process with which an actualprocess may be compared is common to all branches ofthermodynamics. I n view of the second law the idealizedprocess is usually so selected as to be one of maximumproduction of work. The obtaining of maximum workinvolves the establishment of some restricting conditionsregarding the possible behaviour of the system. Forexample, the physicist and the chemist have sometimesprescribedan environment of fixed temperaturewithwhichthe otherwise isolated system must be in temperatureequil ibrium initially and finally-the maximum work thenbeing the decrease in the so-called Helmholtz free energy;or they have sometimes prescribed an environment of k edpressure and temperature with which the system must bein both pressure and temperature equil ibrium initiallyand finally-the maximum useful work being the decreasein the so-called Gibbs free energy; the engineer has some-times prescribed adiabatic conditions between somespecified initial state and a specified final volume orpressure-the maximum work being the decrease inenergy or enthalpy (depending upon the nature of theprocess) at constant entropy. In this latter instance it

has often been pointed out that the final state of theidealized process may be quite different from that of theactual process.None of these idealizations has generality-each is anad hoc device of limited utility. The more generalapproach to the statement of the thermodynamicallymost beneficial result and to the evaluation of departuresfrom it has been given limited attention.The foundations of the general approach were laid byJ . W. Gibbs in his second paper on thermodynamics.()Gibbs imposed on the behaviour of the system the con-dition that it should be isolated except for communicationwith a stable environment of uniform pressure and tem-perature. He put no restrictions, other than those im-posed by the nature of thermodynamics, upon theselection of the system or of its initial and final states.Maxwell@ adopted a modified version of Gibbssapproach which omitted from consideration the pressureof the environment. Darrie~s,( ~)tarting from Maxwells* Temporarily at Imperial College of Science and Technology,London.VOL.2, JULY 951 183

method, developed a treatment of engineering problemsin steady flow which was expanded upon by Keenad4)and others. Published quantitative studies of irreversibleprocesses that stem less directly from the work of Gibbsare less general in application. For example, the recentwork of Tolman and Fine() is restricted to cyclic andsteady-flow processes.The purpose of the present paper is to develop themethod of Gibbs more generally than has hitherto beendone. Connexion is made between Gibbss availableenergy of the system and medium and the criterion ofstabil ity. The available energy concept is developed forsystems which communicate only with the uniformmedium or atmosphere and for systems which com-municate with an additional reservoir of specif ied tem-perature. The treatment is extended to problems oftransient and steady flow. A measure of departure fromthe ideal, the irreversibility, is defined and examined forits significance. Performance coefficients are devised forseveral classes of processes. Finally, the generali ty ofthe method is exhibited by analysis of a variety ofthermodynamic phenomena.In this generality may be found the ustification of theavailability concept. I t is through this concept thatprocesses as widely different as the decay of motion in aviscous fluid, the rectification of a binary mixture, andthe dissociation of hydrogen peroxide can be examinedfrom a common basis of comparison and their thermo-dynamic quality compared quantitatively by means ofthe irreversibility or a coefficient o performance as heredetined.S T A B I L I T Y , M A X I M U M W O R K , A N D A V A I L A B I L I T Y

System exposed to an infinite atmosphere only.-Virtually all problems which can be treated adequatelyby the methods of thermodynamics are terrestrial: that is,they relate to the behaviour of systems which are sur-rounded by an essentially infinite atmosphere. A majorexception to this latter generalization is found in thesubject of meteorology wherein the system under con-sideration is the atmosphere itself. For other terrestrialproblems, however, the system considered is small inmass and extent compared with the surrounding atmo-sphere which, for purposes of analysis, may be thought


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J . H ,of as an environment whose temperature and pressureare unaltered by any process experienced by the system.By including within the system as much material ormachinery as is affected by the process (except for theatmosphere) one may consider any process which occursas one in which the system interacts with the atmosphereonly, Gibbda has shown that for any process which canoccur under these circumstanceswhere A@ denotes the increase in the quantity CDwhichisdefined as follows:whereE denotes the energy of the system, V its volume,S its entropy, po the pressure of the atmosphere, andTo ts temperature on the Kelvin scale.Since it is evident that processes or changes can occuruntil the pressure of the system is uniformly po and itstemperature uniformly Toand perhaps even thereafter,then the state from which no spontaneous changes canoccur is the state for which the system has the pressurepoand the temperatureToand for which@ has the minimumof all possible values. This is the state of stable equi-librium if there is only one such state, and of neutralequilibrium and maximum stability if there is morethan one.

A@ 9 0 (1)

@ =B+poV-ToS (2)

Gibbs referred to the difference@ - miwhere @ refers to a state in question and Qmino themost stable state, as the available energy of the bodyand medium for the state in question() (Gibbs uses thetermbody for what is called thesystemhere and the termmediumfor what is called the atmospherehere). By thishe meant the maximum useful work-that is, work inexcess of that done against the atmosphere-whichcould be obtained from the system and atmosphere,without aid from other things, by any possible processes.The point of view of the last paragraph differs fromthat of those which preceded it in that if useful work isdelivered then something other than the system or theatmosphere must be affected by the process through thereception of work. The only other thing considered tobe affected, however, is a work reservoir, such as a coiledspring or a flywheel, which operates adiabatically in thecourse of the process.The proof of Gibbss statement, which he does notgive explicitly, may be given as follows. I f a systemwhich is surrounded by an atmosphere at po and Toexperiences a change from state 1 to state 2 while itreceives net heat @ositive or negative) from the atmo-sphere only, the useful work which will be delivered tothings other than the system and atmosphere cannotexceed that of areversible process between states 1and2.For if this were not true then it would be possible toexecute the hypothetical process from 1 to 2 whichproduces work in excess of that of the reversible processand to complete a cyclic change for the system by means

Keenanof the reverse of the reversible process from 2 to 1.This cycle would havea net production of positive workand, since the atmosphere at Towould be the only heatreservoir engaged in the process, the cycle would con-stitute a perpetual-motion machine of the second kind.The hypothetical process is therefore impossible and noprocess can produce useful work in excess of that of thereversible process.By similar reasoning itmay be shown that all reversibleprocesses between states 1 and 2 having heat transferbetween system and atmosphere only must produceidentical quantities of useful work.In order toevaluate the useful work of an iniinitesimalreversible process one may set up a reversible means ofheat transfer between the atmo-sphere at.Toand the system at T,say, which consists of a reversible

SYS TEM cyclic enghe of small enoughdimensions so that one cyclicoperation will be required toabsorb or deliver an infinitesimalamount of heat (Fig. 1). L etdE,dV, and dS denote respectivelythe energy, volume, and entropyE Rs B L E changes experienced by the system4QOENGtNEig. 1 in going from the first prescribedstate to the second. These quanti-ties would obviously have the same values for this changeof state regardless of the nature of the process or theamount of useful work produced. Let SQ , denote theheat received by the reversible engine at Tofrom theatmosphere. The quantity SQ o (unlike dV, dE, and dS)will have different values for the same change of statein the system, depending upon the nature of the processand the amount of useful work produced. I n whatfollows the symbols d and 6will be used to differentiatebetween quantiti es like dV on the one hand, which arek e d by the end states of the system, and SQo on theother, which arenot so fixed.The magnitude of SQo may be, of course, either greateror less than zero. A magnitude less than zero woulddenote heat flow away from the engine to the atmosphere.Now the work done by the system and the cyclic enginein combination is given by

0

- d E + s Q owheredE, he increase in energy of the system, is also theincrease in energy of the system and the cyclic enginecombined. Of this work the amount podV must beexpended in displacing the atmosphere. Therefore theuseful work of the reversible process which is the maxi-mum useful work (SW,),, of all possible processes isgiven byBy the definition of the temperature scale, however,

(8Wu )m a x =- - odV 4- S Q owhere SQ denotes the heat received by the system from

184 BRITISH JOURNAL OF APPLIED PHYSI CS


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Availability and irreversibility in thermodynamicsthe reversible engine. Moreover, the change of entropydS of the system in the course of this reversible processis given by dS=SQITI t follows upon substitution into the previous expres-sion for (SW,),, that(6 WJ,, =- dE- odV+TodS

=- @ (3)SW,


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J . H. Keenansatisfactory answer in terms of change in @ of the reser-voir and the system. For convenience, however, i t wouldbe better to put the answer in terms of the quantity ofheat withdrawn from the reservoir. This may be doneas follows:-For a change of state in the system while it receivesnet heat from the atmosphere and from a reservoir R,one may, from the point of view of the preceding para-graph, write from (10)where the last term refers to the reservoir R and thenext-to-the-last to the system as usual. This may berewritten in the form

Wu< A@-A@R (16)

which is dQR(TR - O)/TR. The sum is as indicate1in (20).I t should be noted that the heat from the reservoir itreated differently here as compared with the heat fronthe atmosphere. For example, the former appearexplicitly in (19) and (20), whereas the latter does notThe implication is that QR, ike the change of state of thlsystem, is prescribed. The heat from the atmosphereon the other hand, is not prescribed and varies in facwith the magnitude of W,. For this reason the symbodeR s used for the prescribed infinitesimal quantity 0.heat from R to correspond with the prescribed change:in properties of the system dE, dV, and dS.

(W,)- =- @- ER - P o ~ ~RToASR (17)The term reservoir generally implies a system whichpasses only through stable states and which if it expands

sphere. Letting QR denote the heat f low from thereservoir to the system, one may therefore writeQR =- ER- ~AVR=- RASR (18)Substituting from (18) into (17), one getsTR -To

F L O W PROCESSES A N D M A X I M U M S H A F T W O R KFlow across a control surface.--(=onsider a closec

exchanges heat Only with the atmosphere. When themass element dm crosses the surface U from outside tcinside, the useful work of the process executed by asystem consisting of all the fluid finally inside U is given bj

or contracts does so slowly in the presence of the atmo- control surface U (Fig. 3) in a field of fluid flow whid

SW, Q - d@(19) or swuQ - .b'+ (a); +4 , h ) (21)u


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Availability and irreversibil ity in thermodynamicsIf the fluid has appreciable velocity and it flows in agravitational field of strength g, then one has

where U denotes the energy per unit mass of fluid at restat the datum level, c the velocity of the fluid, z its heighlabove the datum plane, g the acceleration of gravity,andgothe acceleration given to unit mass by unit force.Substituting (25) into (24) one gets

orwhere b G h- o$and h=u+pv (27)the latter being the enthalpy.Itmay readily be seen that if a reservoir R is consideredto be outside U and to supply heat dQR to the fluidinside U whiledmflows in, then

SteadyJ70w.-Application to steady flow results in twoimportant modifications of the more general equationswhich were developed above. The first is thatdrI, =0

wherel-i, denotes the total value of any property, such asV,S, or @, of the mass of fluid found within surface Uat any instant. The second isthat the mass flow acrossacan be subdivided into two equal mass flows, one in andone out. The most general of the flow equations (28)then becomes

where Py denotes the shaft power flowing .out of a,Ci , a summation over all the streams flowing into U,CO, a summation over all the streams flowing outof U, and dQR/dt the rate of heat flow from reservoir Rto material inside a. This last equation may be com-pared with the so-called energy equation of steady flowin the form

where SQ,/dt denotes the rate of heat flow from theatmosphere to material inside a.VOL.2, JULY1951 187

I R R E V E R S I B I L I T YA quantitative definition.-From (15) a quantitativedefinition of irreversibil ity can be devised. Letting Idenote the irreversibility of a process which the system-atmosphere combination executes, one may write

I=Wu)mUx - wu (31)=-M-W,, (32)= - A @ - W,, (33)

Substituting into (33) the definition (2) of @ and for W,the integrated form of (S), and noting from (7) thatone getswhich, in accordance with (3, ay be reduced to

AV =- V,I=TOAS+aE, f P&Va

I =T04(Sa+S) (34)A similar result may be had by using (9) without (7).Thus, the irreversibility becomes equal to the increase inentropy of everything involved in the process multipliedby the temperature of the atmosphere. I t is evident fromboth (31) and (34) that

I > O (35)If heat is received from a reservoir R then the irreversi-bility is given, in accordance with the definition (31), by

Considering the system and atmosphere combined as asystem, one may writeW"= -&--E,+ QR

Upon substituting this into (36) and proceeding as in theprevious paragraph, one getsI=ToA(Sa+5 +SR)

Irreversibility in OW across a control surface.-According to (31) one gets for the flow of dmacross airreversibility of the magnitudeSI =(SW,),, - W,

Since (SW,),, differs from (SW,),, by the amount(px- o)vxdm and 8W,,differs from 6W, by the sameamount, it follows that(37)I=6W,),, - w

This is a consequence, of course, of eliminating byhypothesis shear stress at the boundary of the mass dmI t is not implied, however, that the process of flowacrossU is reversible. For example, a may coincide witha thin porous. wall through which the fluid flows irre-versibly. This would involve shear between the fluidand the wall , but not between the fluid dmand thesurrounding fluid which pushes itacross a.


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J . H. KeenanI t has been shown(@ rom the fi rst law of thermo- per unit mass of a single stream passing in steady flowacross a control surface would beynamics that

I =TO( S~I +Asp.+AS,)for a control surface which is of invariable volume.Substituting into (37) the expression for (ST,) givenby (28) and that for SW, in (38), one gets

Since dV, is zero by hypothesis, the last equation becomesSI =To(dS, - xdm - (4)TR To

or, since the increase in entropy d S of al l the fluid finallyfound within U isdS=dS, - Xd m61=To(dS+dSR +SS,) (41)

If the control surface is permitted to change in volumeby amount dVu during this process and 6W, is made toinclude the useful part of the work done at the movingboundary, or Cp, - ,,)dV,, then the right-hand memberof (38) is altered by adding a term -podVC, the right-hand member of (39) by cancelling out the term- odVC, nd(40)and (41) remain unchanged. Examplesof this case would be the intake and discharge processesin areciprocating machine.The equations developed above for flow processes areapplicable to processes involving flow in or out. Theconvention adopted requires that for flow in the quantitydm should be a positive number, and for flow out anegative number. In both instances the subscript xrefers to states just outside the control surface. For flowin, the state outside is essentially independent of thestate inside. For flow cut, on the other hand, the stateoutside may be a direct consequence of a state im-mediately on the other side of the surface U. Forexample, if no heat ilows across the surface, then theenthalpy h, will be identical with the enthalpy h im-mediately on the other side of U.Irreversibility in steady flow.-For steady flow (40)reduces toSI=To(-Xsxdm---%)~ Q R

TR (42)

where subscripts 1 and 2 refer to entrance and exitrespectively.An alternative, though entirely equivalent, develop-ment of (42) consists of getting the rate of production ofirreversibility

by subtracting the right-hand member of (30) from thatof (29) and then multiplying through by dt.COEFFICIENTS OF PERFORMANCE

Various coefficients of performance in the nature ofefficiencies can be devised in view of the first and secondlaws of thermodynamics. I t would seem to be desirableto define these coefficients in such a way that they wouldnot exceed unity for any processes to which they mayproperly be applied.Starting from the relationwu


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Availability and irreversibility in thermodynamicsprovided that (Wu),, (and, therefore, Wu) s lessthan zero.The coefficient C1 is appropriate for processes whichare intended to produce useful work: for example, theexpansion process in a heat engine or the combustionprocess in a Diesel engine. I t compares the useful workproduced with the sum of the expenditures of @ and ofthe work-producing capacity of heat from a reservoir;thus,

The coefficient C2 is appropriate for processes whichconsume useful work: for example, compression com-bined with heat rejection in a heat pump. I t comparesthe sum of the gain of availability for the system andreservoir with the useful work consumed; thus,

Again c, Q 1when A@


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J . H. Keenanas distinguished from the more general quantity perature. Ths would be true for a change of phasebetween two saturation states. If, on the other hand,no such series of intermediate equilibrium states existsE +POW - gn e roperty o is, for any fixed atmospheric conditions,a function of two independent properties such aswand .Expanding (55), one gets

at that Pressure and temperature, then (58) is not valid,because the integration of (58) implies the integrationof (56) which can be valid only for a path throughequil ibrium states. Thus, (58) will not hold for a change6WuQ - u- odv f To& from a supersaturated vapour state to a liquid state atthe same pressure and temperature. For such a changeFor idni tesimal changes between equil ibrium states the (57) may be integrated over any path which consistsfist and second laws of thermodynamics give entirely of equil ibrium states, such as the constant-

- - U = -Tds$pdv (56) temperature path from the supersaturated-vapour stateto saturated vapour to saturated liquid to the liquidso thatUpon integration of (57) over a path passing throughequilibrium states, one gets for the maximum useful workthe difference between an areaA on the pressure-volumediagram and an area B on the temperature-entropydiagram (Fig. 4).

(57) state(wgfZ,Fig. 7) .W,Q Cp - o)dv- T- o)&

Fig. 7The pure substance has been treated by Gibbs atsome length.() His discussion omits capil larity, gravity,electricity, magnetism and chemical change, but includesmotion and non-homogeneity of state. H e representsstates by points in a co-ordinate space in which thevertical axis is the energy axis and the two horizontalaxes are those of entropy and volume. A ll equil ibriumhomogeneous states in this space lie on a primitivein this space and all non-homogeneous equili-brium states on -derived surfaces, The availabil ity,

as defined here, is the vertical distance the state pointlies above a plane which is tangent to the primitivesurface at the point representing the most stable state.Other representations of availabili ty can be devised, butfew observations could be made from them that are notexpressed or implied by Gibbs.Combustion.-Equations (15), (31), (32), and (33) maybe applied to any thermodynamic system that may beconsidered to be enclosed within a stable and uniformatmosphere. They may, therefore, be applied to processesinvolving, mixing, solution, or chemical reaction. Inorder to determine the change in @ or A and the valueof (Wu)ma t is necessary only to know the change in

P

V SFig. 4I n Particular, for C o O b at constant-volume (h


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Availability and irreversibility in thermodynamicsreactants and products are, say, diatomic. The problemmay be further simplified by the assumption that reactantsand products are perfect gases having identical molalspecific heats which are independent of temperature.For an adiabatic reaction at constant volume theenergy of the system is unaltered. Letting subscriptsP and R refer to products and reactants respectively,one may writeIn terms of the energy of reaction

u p - =0 (59)

where CO s the specific heat at constant volume of thesystem either as products or as reactants, its value beingby hypothesis the same constant for both. From (61) itis clear that the temperature rise in combustion (Tp- TR)is constant and, therefore, independent of the tempera-tureTRat which the reaction begins.The maximum useful work for this process-or thedecrease in availability-is given by

and W, s in this case zero; therefore,I = (Wu )m

and the irreversibility is seen to decrease with increase inthe temperature at which the reaction begins. This is amajor element in the justification for compression of acharge in an internal-combustion engine before ignition.If the combustion occurs adiabatically at constantpressure p, instead of at constant volume, the result issimilar. Once more the temperature rise is independentof the temperature of the reactants. The maximumuseful work becomes

(W,>mu=@ - 0)FP - R) -k TO(SP- R )This time, however, the useful work is not zero but

W,=@ - 0)FP- R)and the irreversibility is, as before,

I =To(Sp- ,)which again is a quantity which decreases with increasein the temperature at which reaction begins.Flow into achamber.-The adiabatic flow of a perfectgas fromaregion of constant conditions into an evacuatedchamber may be used to illustrate a problem in non-steady 00w. Referring to conditions in the outsideregion by subscript x one may write for (38)and for (28) SW,=O= -dE,+h,dm (65)(SW,),, =-do, +b,dm

because both ( Up- UR)and (Vp- R)are zero. But Upon integrating each of these between the initial con-dition of nofluid inside the chamber to the final condition,for which subscriptfwill be used to denote the conditionin the chamber, one gets( SP- d= Sp - PO)- SR- RO)+ S ~p ) ~TP=Cv(1n ToTP=C In- SRP)OT R- n> sRp) O -muf+ mh,=0 (67)

(63) and (8W,,),u =-my5f+ Qj+mb, (68)where (SRP)o s the entropy of reaction at TO.stituting (63) into (62), one gets Sub-

Since, from (61),(uRP)O- I - -PTR TRc,

the temperature ratio Tp/TRdecreases with increase intemperature (for URp ess than zero-that is, for atemperature rise in combustion). It follows from (64)that the maximum useful work decreases with increasein the temperature of the reactants.The irreversibility is defined by

_ -

wheremdenotes the mass which flows into the chamber.I t is necessary to introduceai ,he initial value of 0 nthe chamber, into (68) because, unlike Vi , t is not zero.Specifically,@ j =uj i pov - OS= o6=PomvfFrom (67) one gets

U/ = h,or Tf=kT,where k=Ccfrom (68) and (69)


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J . H.w>*. IC \ien-of c@).educes to

(i*;b,72=ZdSj- Sx)=To(cp n k - lnpf/pJ (72)

F.5xmi: auldhsve been anticipated by noting that ther.-m--....,...-.sksfiw r k per unit of mass entering is also thek v ~ x 3 i I i r yf ~s proclss and is therefore the productG~T:-d b e ncrme n entropy of the fluid.1 -...&.&ybe noted from (72) that the maximum shaftn,-k- - ..-id the irreversibility per unit mass are infinite for&e 5s elementary mass which enters the chamberc~=01md that just before flow ceases ( pf=px)these~x=ti& s e educed to c,T, In k.T?< s.cam poii.er-pIant cycle.-A steady-flow steampn-er-pIam cycle may be used to il lustrate severalq p f application of the availability, maximum work,and irrelersibi lity concepts. The various processesinvoh-ed may be classified according to whether they are(e) adiabatic-that is, employing no heat reservoirsenema1 to the water passing through the cycle, (b) ex-chan@ng heat with the atmosphere only, and (c) exchang-ing heat with a hot reservoir.Examples of (a)are expansion in a turbine, compressionin a feed pump, and heating feed water with steam bledfrom the turbine. An example of (b) s the condensationprocess. Examples of (c) are the steam-generatorprocesses: namely, heating of feed water, evaporation,superheating, and reheating.The change in availability and the maximum usefulwork (W,) of a unit mass of fluid may be evaluated foreach process in the cycle. These are, however, of lessengineering interest than the maximum shaft work(W,),, for each process. For the cycle as a whole thesummations of (W,),, and (W,),, are identicalbecause their difference, the useful work done on adjacentfluid, sums up to zero. The irreversibility summed upfor the cycle will be equal to the difference between thesummation of the maximum shaft work (or maximumuseful work) and the net work of the cycle.I n dealing with the (c) type of process-namely, thatinvolving heat exchange with a hot reservoir-severalchoices are open to the analyst. He may, for example,choose to employ as a reservoir a stream of products ofcombustion such as might be found in an actual powerplant. In that event (29),with the last term equal to zero,must be applied to the stream of water and to the streamof products of combustion as well in order to find themaximum shaft work. This method brings into con-sideration the characteristics and behaviour of the streamof products of combustion. A logical extension of itwould be to include in the application of (29) the entireprocess of the air-fuel stream from its entrance to thepower plant in the form of reactantsto its exit in the formof cooled products of combustion. Any analysis whichincludes the behaviour of the stream of hot gases is, ofcourse, concerned with a non-cyclic process.It might be desired, however, to study acyclic process

Keenanas such in which heat exchange is with conventionalreservoirs. I n that event, the temperature of the hotreservoir is arbitrary. For a given cycle the irreversi-bility will be greater the higher the temperature selectedfor the hot reservoir, and the lowest magnitude of theirreversibility will correspond to the lowest possibletemperature of the reservoir. For example, in a steamcycle the lowest possible temperature for a single hotreservoir which supplies heat to the cycle without the aidof intermediate cyclic machinery would be the highesttemperature attained by the steamin the courseof heating.If a reservoir temperature is selected higher than thisminimum, the increase in the value of the irreversibilitywill be equal to the maximum useful work which couldbe obtained by virtue of heat transfer from the highertemperature and to the lower one-the magnitude of theheat being that taken from either reservoir by the cycle.Thus the increase in irreversibility is attributable to heatflow across the finite difference between the temperaturesof the two reservoirs.

C O N C L U S I O N SQuantitative concepts of maximum useful work,availabil ity, irreversibility, and quality of performance ofa thermodynamic task may be defined from considera-tion of the first and second laws of thermodynamics forall processes between equil ibrium states of a systemoperating within an infinite stable atmosphere. Theseconcepts may be extended to cover flow across a controlsurface and, asamore special case, to steady flow througha control surface. They may be applied to as wide arange of processes and as great a variety of systems asthe science of thermodynamics itself.

REFERENCES(1) The Collected Worksof J . Willard Gibbs,p.39 (L ondon:Longmans, Green and Co. L td., 1931).(2) MAXWELL,. C . Theory of Heat, 10th edition, p. 195(London: L ongmans, Green and Co. Ltd., 1891).(3) DARRIEUS. ev. Gen. de IElect., 27, p. 963 (1930); seealso Engineering,130, p. 283 (1930).(4) KEENAN,. H. Mech. Engng, 54, p. 195 (1932);Thermodynamics(New York: John Wiley and SonsInc., 1941).( 5 ) TOLMANndFINE.Rev. Mod. Phys., 20, p.51 (1948).(6) The Collected Works of J . Willard Gibbs, 1, p. 40(London: L ongmans, Green andCo. Ltd., 1931).(7) The Collected Works of J . Willard Gibbs, 1, p. 53(London: Longmans, Green and Co. Ltd., 1931).(8) KEENAN,. H. Thermodynamics, p. 34 (New Y ork:

J ohn Wiley and Sons Inc., 1941).(9) THOMSON,IR WILLIAM.Trans. Roy. Soc. Edinb., 16,(1849); see also Math. and Phys. Papers, p. 152(L ondon: Cambridge University Press, 1882).KEENAX,. H. Mech. Engng,54, p. 195(1932).(10) DARRIEUS.ngineering,130,p. 283 (1930).(1 1) The Collected Worksof J . Willard Gibbs,p.33 (London:Longmans, Green and Co. L td., 1931).192 BRI TI SH JOURNAL OF APPLIED PHYSICS

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