Improved Estimates of the Efllciencies of Irreversible Heat Engines (*).

C. TRUESDELL (B~Itimore, Ma~ryland, U.S.A)

Dedicated to DARIO GRAFFI on his 70th birthday

Summary. - Upper and lower bounds ]or the ef]iciency o] a homogeneous, possibly irreversible heat e~gine are derived. The results are illustrated by application to working bodies with linear ]fiction, ]or which also sharper estimates are calculated.

Messrs. FOSDICK and S E ~ I ~ have shown m e an iden t i ty f rom which t hey derive th rough reasoning s impler t h a n mine the es t imates I recent ly gave for the efficiency of i r revers ible hea t engines (~). On the basis of the i r iden t i ty I here obta in sharper es t imates , and a,lso I p u t the resul ts concerning bodies wi th l inear fr ict ion into

more explicit and precise form. They have meanwhi le applied their iden t i ty so as to ob ta in deeper results , which concern cont inua susceptible of general r a the r t h a n

mere ly homogeneous deformat ions (~).

PAR~ I

E S T I M A T E S B A S E D ON T H E T W O L A W S A L O N E

1. - The bas ic ident i ty .

L e t a body be subject a t the t ime t to ne t working (8) W(t) and to hea t ing Q(t).

I n the finite in te rva l [tl, t~], which we denote b y ~ , the body absorbs hea t C +, emits hea t C-, a n d does ne t work U, defined a.s follows:

f f YQ (1) U ~ - - W d t , C + ~ Q d t > O , C - ~ - - d t ~ O ,

(*) Entrata in Redazione il 30 giugno 1975. (i) C. TRUESD:ELL, The e/]ieieney o] a homogeneous heat engine, J. Math. and Phys. Sciences

(Madras), 7 (1973), pp. 349-371; Corrigenda, ibidem, 9 (1975), pp. 193-194. (2) R. FOSDICK - J. SERRIN, Global properties o] continnnm thermodynamic processes, Arch.

Rational Mech. Anal., 59 (1975), pp. 97-109. (a) I f P is the power of the applied forces and K is the kinetic energy of the body,

then W = P - - K.

306 C. TI~UESDELL: Improved estimates of the e]]icieneies, etc.

23+ and 23- being the par t s of 23 on which Q ~ 0 and Q < 0, respectively, and the funct ions W and Q being assumed integrable.

I f Q ~ 0 almost always on 23, the body is said to undergo an adiabatic process. The F i r s t Law of Thermodynamics asserts t h a t a t the t ime t the body has an

in terna l energy E(t) such t ha t

(2) A E = - - U + C + - ~- .

Here and hencefor th we use the nota t ion

(3) A] =/(t~) - - ](ta) .

Let a and b be any posit ive constants ; let 0 be an essentially positive funct ion such t ha t Q[O is integrable on 23; and let H be an absolute ly continuous, differentiable funct ion on 23. Then by use of (2) and (1) we easily ver i fy Fosdick and Serrin 's

iden t i ty :

(4) v = c ~ 1 - - - ( A E - - a A / t ) - - a ~ - Q d t +

73- 73

We shall Mways in te rpre t O(t) as being the temperature of the body at the t ime t. I f O(t)== const, a lmost always on 23, the body is said to undergo an isothermal process. We shall assume always t h a t 0 has a posi t ive essential infimum Omi n and a finite essential sup remum 0m~ ~. I f 23+ has posi t ive measure, 0 has a posi t ive essen- t im supremum 0 + on it ; if 23- has posi t ive measure, 0 has a posi t ive essential in- f imum 0- on it. We shall use (4) only in cases when a and b are chosen f rom the four

numbers 0ram, 0ma x, 0-, and 0+; namely,

0~n ~+ _ (AN-- 0m~n A//) -- 0~n ~ - - ~

( IfC = 1 - - ~ - - ( A E - - O - A I I ) - - O - -~-- Q d t + ~+

(5)

+ f + f , 73-

+

"6"- 23

C. T]~VES])ELL: Improved estimates o/ the e]ficiencies, etc. 307

The identi t ies (5)~ and (5)3 hold always. The fact t h a t wh~t we have called 0 + and 0- are defined only if ~+ and ~ - , respect ively, are not of measure 0, will be t aken account of in any applicat ion we choose to make o~ (5)~.

I f C + > 0, the e]]iciency e of the body in ~ is defined as follows:

U (6) e ~ C+"

Thus, e, whenever i t exists, is eva lua ted by (5). I n the identi t ies (5) appear integrals over ~+ and ~ - . The integrands of these

six integrals are essentially nonnegative. The integrals over ~+ vanish if and only if, respectively, 0 ~ 0 ~ , 0 ~ 0 +, and 0 --~ 0mi n almost always on ~+. The integrals over ~;- vanish if ~nd only if, respect ively, 0----0~l . , 0 : 0-, and 0 = 0 ~ almost always on ~6-.

2. - Interpretation of the identity in terms of the second law.

The Cl~usius-Planck Inequal i ty , which is of ten considered to express the Second Law of Thermodynamics , asserts t h a t each body has an absolutely continuous entropy H(t) such as to sat isfy the inequal i ty

(7) H>_Q- - - 0 '

almost always. I f equal i ty holds in (7) almost always, the body is said to undergo a reversible process; otherwise~ an irreversible process. I f (7) holds, t hen in each of the identi t ies (5) the third integral vanishes i] and only i] the body undergoes a rever- sible process; otherwise it is positive.

3. - Carnot processes and C-processes.

A body is said to undergo a Carnot process if, essentially,

(S) 0 : 0m~ x on ~+ ,

0 = 0m,. on ~ - .

Tha t is, if the body absorbs heat , i t does so only at its greates t t empera tu re ; if it emits heat , i t does so only at its least t empera tu re . The sets ~+ and ~ - m a y huve measure 0. I so the rmal processes and adiubgtic processes are Carnot processes.

A body is said to undergo a C-process if it absorbs heat a t one and only one

308 C. TRVESI)EL~,: Im2)roved estimates o] the e]ficie~wies, etc.

t e m p e r a t u r e 0 +, emits hea t a t one and only one t e m p e r a t u r e 0-. Tha t is, essentially,

(9) 0 : 0 + on "6 + ,

0 : 0 - on "6-,

bo th ~+ and "G- having pos i t ive measure . The t e m p e r a t u r e s 0 + and 0- m a y lie anywhere in the in te rva l [0~n, 0m,x]. I n some

works a Carnot process is defined b y combin ing the r equ i remen t s (8) and (9). Then

0 + = 0max>= 0mjn= 0-. Carnot processes of this k ind are the commones t and the

m o s t i m p o r t a n t . I n general , however , a Carnot process as defined here need not be a C:process,

nor need a C-process be a Carnot process. For a Carnot process the sets ~3+ and ~ - m a y have measure 0, us indeed they do for an ad iaba t ic process; thus such a process

is not u C-process. An i so thermal process is a C~rnot process; an i so the rmal process in which Q is of one sign is not a C-process, for t hen ei ther ~+ or "G- is emp ty . Four examples of C-processes t h a t are not Carnot processes are shown in fig. 1, which refers to the classical var iables : the vo lume V(t) and the t e m p e r a t u r e O(t) of a body, ~ general process being the pa i r of funct ions V, 0.

~max

e+

8 ~

8rain

~ o O > 0 emo x, e-

O<O ~ , 0 8min ' 0 +

(=) V

Q<O

(b)

V

÷ ~, Omox,8 ~O

0rain

Q>O Q<O

25 / Q=O

8 ~

emi~,

8 8max 0 =0

Q = O 0 < o ~ ~ Q<O

Q=O

V (c) (d)

Figure 1. - Examples of C.processes that are not Carnot processes.

"v"

C. TlCUESDELL: Improved estimates o/ the e]/ieiencies, etc. 309

Examples (a) and (b) occur f requen t ly in the s t andard t r ea tments . Example (b) is what is usual ly called a Carnot refr igerator , for which 0 - = 0~a ~ and 0 + = 0~, . Example (c) is possible for a fluid like water , which in a cer ta in range of pressures exhibits t he rma l inversion; failure to perceive tha t in such a process the set ~ - cannot be e m p t y has given rise to a <~ paradox >~ in classical thermodynamics . Example (d) shows a more complicated possibili ty of the same kind, purely hypothet ical .

Finally~ a body is said to undergo a counter-Carnot process if, essentially,

(10) 0 = 0 ~ n o n 25+,

0 = 0ma x o n ~ - .

The Carnot ref r igera tor i l lustrated b y Example (b) in fig. i is a counter-Carnot process. I t is easy to see tha t a Carnot process is also a counter-Carnot process, and conversely, if and only if it is an adiabat ic process or un isothermal process.

4. - T h e m a i n e s t i m a t e s .

We are now ready to app ly to (5) the s ta tements made at the end of § 1 and in § 2. The easiest and perhaps the most useful resul t follows f rom (5)1. I] the Clausius-

Planck inequality holds, then always

(11) U ~ ( 1 - - ~ ) C+--(AE--OmmAH);

equality subsists i I a~d only i] the body undergoes a reversible Carnot process. This is the es t imate I der ived some years ago and published in 1973.

F r o m (11) we see t ha t the four quan t i t i e s 0rain, 0m~x, AE/C +, and AH/C+ deter- mine the efficiency of a reversible Carnot process, and t h a t any irreversible process in which heat is absorbed, including an irreversible Carnot process, if it gives rise to the same values of these fore" quanti t ies has a lesser efficiency t h a n any corresponding reversible Carnot process. This conclusion is less i l luminating t h a n it might seem, for we have no assurance t h a t corresponding processes for a given body exist . For example, as we shall see in §§ 10-11, an irreversible adiabat ic process m a y have l i t t le evident relat ion to a reversible one. Moreover, a typical body is incapable of undergoing any non-tr ivial reversible process whatever . Thus the est imate provided

by (11) m a y be far too weak. On the other hand, even for bodies for which non-tr ivial , 'eversible processes

are possible there are par t icu lar cases in which an es t imate far sharper t h a n t h a t p rovided by (11) m a y be found. To this end we consider (5)2 instead of (5h and thus obtain a second es t imate: I] the Clausius-Planck inequality holds and i/both ~+ and ~ -

have positive measure, then O- A E - - O-AH

(12) e~ l - -0 - - 7 - C+ - - ;

equality subsists i/ and only i] the body undergoes a reversible C-process.

310 C. TRUESDELL: Improved estimates o / the eHicieneies, etc.

Final ly , by using (5)3 we m a y read off a lower bound for the mot ive power:

and equality subsists i] and only i~ the body undergoes a counter-Carnot process; more-

over, in a eounter-Carnot process

(14) - - U ~ ( ~ - - 1 ) C + -[-(ALE--Omax A"),

and equality subsists i f and only i f the process is reversible.

5. - Illustrations of the main estimates.

Some simple cases i l lustrate the difference between (11) and (12). For bodies t ha t absorb no heat or undergo isothermal processes, general ly (12) fails to apply bu t (11) yields U g -- (AE--OminLJH). In a reversible C-process the sign of equal i ty holds in (12), bu t in general not also in (11). Comparison of the two results shows t h a t in a reversible C-process

O- Omt, > (0- - - Om~) A H (15) 0 + Omax= C + '

and equa l i ty subsists if and only if the C-process is also a Carnot process. More generally, no t res t r ic t ing a t t en t ion to any special k ind of process, let us

assume tha t bo th ~+ and ~ - have posi t ive measure, so tha t we m~y apply (11) as well as (12) to obta in upper bounds for the efficiency. The difference d of these bounds is given by

O- Ore,. (O---Omin) A g (16) (5 =

O+ Omax C +

Hence we conclude tha t in order/or (12) to provide a sharper upper bound than (11), it is necessary and su]ficient that

(17) (0- -- 0~,~,) AlH 0- 0mi. c÷ < O:a:"

I f equal i ty holds ins tead of inequali ty, the two bounds are equal; if the inequMity is reversed, (11) is a sharper bound than (12).

We r emark in passing t h a t in classical the rmodynamics A H = 0 for all ~(reversible cycles~. Then (17) shows t h a t if e i ther 0->0mi . or 0 + < 0 ~ , the bound (12) is sharper t han (11). We pause to notice two str iking examples, bo th of which refer to the classical theory of ~ reversible cycles 5 in which AE = 0 and AlH = 0.

C. TI~UESDELL: In~prvved estimates o] the e]]iciencies, etc. 311

T he first e x a m p l e is p r o v i d e d b y S k e t c h (b) in fig. t , wh ich i l lustr~tes a C~rnot

refxigerator . Of cottrse (11) y ie lds t h e classical e s t ima te

{}rain (18) ~ < i - g £ : ~ .

H o w e v e r , t h e b o d y unde rgoes g revers ib le C-process in which 0 + = 0mi ~ a n d

0 - = 0=,=, so (12) y ie lds

{}max ( 1 9 ) e = 1 - - {}mi-~ '

of course g n e g a t i v e q u a n t i t y .

A n even m o r e s t r ik ing e x a m p l e is p r o v i d e d b y Ske t ch (c) in fig. 1. Aga in (18)

holds , because t h e b o d y is sot u n d e r g o i n g ~ C~rnot process . Of course (18) is correc t ,

b u t i t is v e r y c rude . On t he c o n t r a r y , t h e b o d y does u n d e r g o ~ revers ib le C-process

in wh ich 0 + = {}-, so t h e e s t i m a t e (12) gives t h e exgc t eff iciency:

(20) e = O.

The <~ p a r a d o x ~> conce rned w i t h use of wa te r us a Ca.rnot eng ine a b o v e a n d be low

its invers ion t e m p e r a t u r e is ea, si ly reso lved b y cons ide ra t ions of this k i n d (4).

(~) The << paradox ~> was raised by A. 8OMM~RF~LD, Exercise 1.6 of his The~'modynamik und Statistils, ed. F. BoPP and J. M~IxN~R, Wiesbaden, 1952. The << solution >> given there asserts tha t there are no adiabats connecting the isotherms above and below the hlversion temperature for a given pressure. Such a claim could be just only if the inversion temper- ature were independent of pressure; experiment shows that it is not, nor need we assume it is in order to explain the <, paradox ~>, which merely reflects misunderstanding of classical thermodynamics. Only reversible cycles according to classical thermodynamics need be con- sidered. Sketch (A) in fig. 2 represents the particular case in which H has the same value

8 8 8

8max,~

8mlnfl:

Q > O Q<O

Q / , I ~ 0

" ,~ <°~ i'Q >o*- ~min"

Q> 0 Q<O

/ " Omo~,e~ O-

O=O Omln

Q>O Q<O

0=0

V V V (A) (B) (v)

Figure 2. - The <~ pa radox , regarding Carnot cycles near a thermal inversion, in the ease when the expansion at the higher temperature does not change the entropy of the body. The dotted line descending from the isotherms is the locus of points at which a thermal

inversion occurs.

312 C. TRUESDELL: Improved estimates of the ef/ieiencies, etc.

6. - Converses to the first m a i n es t imates .

Suppose t h a t (ii) holds . T h e n f r o m (5)1 we conc lude t h a t i n M1 Carno t processes

(21) ~3

I n p~r t i cu lu r , in an isothermal process at temperature 0

C + ~ C- (22) A H > - - y - - ;

in an adiabatic process

(23) A H _--_ 0.

on each of the adiabatic processes of the cycle. The cycle itself does not correspond either to a Carnot process or to a U-process. However, according to the classical thermodynamics of reversible processes.

Net Work of (A) ~- Net Work of (C) = Net Work of (B) ;

both (B) and (C) represent U-processes in which 0 + = 0% and (20) applies to both. Hence

Net Work of ( A ) = 0. In sketch (A) the value of H on the two adiabatic parts of the cycle is the same. A cycle

of similar form but using adiabats with different values of H can be obtained from sekteh (A) by adjoining a Carnot cycle on the left-hand side or a Caruot refrigerator on the right-hand side. In the former case the total work done is tha t of the Carnot cycle; in the latter case, the total work consumed is that of the Carnot refrigerator.

These same results may be obtained easily ~-om the equations. If H = H I on the left- hand adiabat and H = H 2 on the right-hand one, from (7) we see that

c+ -- C- = (o~=-- O~=)(H2 -- H,)

and hence from (2) tha t

= ( 0 m , = - - 0=~o)(H2 - - H I ) .

If Hf<--_H l, the classical estimate (18) holds trivially, while if Hf>H1, then (7) and (1) 2

show that C+> Om,,(H~-- HI)

and so again (18) follows. The ~ paradox ~ was explained correctly by J. S. THOI~SE~- - T. J. HA~TI~A, Strange Carrot

cycles, Amer. Journal of Physics, 30 (1962), pp. 26-33, 388-389. The cycles themselves had been noticed and analysed correctly earlier by J. E. T~Evo~, Carnot cycles o] u~]amiliar types, Sibley J. Engr., 42 (1928), pp. 274-278. In these papers the cycles look ((strange ~) indeed because the authors begin by describing them in the plane of pressure and volume, which is altogether unsuited to fundamental studies.

I ment ion the mat ter here part ly because the solution does not seem to be well known and par t ly because it is produced ins tan t ly and effortlessly by the present, far more general

estimates of efficiency.

C. TI%UESDELL: Improved estin~ates o] the e]ficienvies, etc. 313

Likewise, if (12) holds whenever "6+ and ~ - h~ve posi t ive measur% then f rom (5)2 we conclude t ha t (21) holds in all C-processes. These s ta tements are among those f rom which physicists inier, and b y which physicists s ta te the Second Law, va- r iously according to tas te . We have shown tha t t hey follow from the Firs t Law and the est imates (11) and (12). To this ex ten t , the Second Law in Planck ' s form (7) is a consequence of es t imates of efficiency.

I f t he funct ion H is such as to yield (11) and (12), so is any funct ion H* such tha t ~lways AH* ~ AH. The same m a y be said of (21), (22), and (23). Thus these relations themselves do not mean much as t h e y stand. I t is only through ~ class of const i tu t ive relat ions t ha t t he y can be pu t to sharp use. In Pa r t I I of this note we shall i l lustrate the concrete results t h a t follow when a const i tu t ive class is prescribed.

7. - C las s i ca l e f f i c i ency .

The classical efficiency of a Carnot cycle is 1 - 0min/0nl~x. We m a y character ize as follows the circumstances tha t give rise to this efficiency.

THEOREM 1

A) I!

(24)

then

(25)

AE - - 0mi n AH ~ 0,

B) (F0SDICK and SER~I~). For a body undergoing a reversible Carnot process, a necessary and su]ficient condition that

(26) U : ( 1 - - ~ ) C +

is

(27) LIE-- Omm AH ~- O .

C) I] (26) holds, then AE--Or~nAH ~ O. For both (26) and (24) to hold, it is necessary that the body undergo a reversible Carnot process.

314 C. TI~UESDELL: Improved estimates o / t h e e]fieieneies, etc.

Conclusions (A) and (B) lollow f rom (11). To p rove (C), we note f rom (5h t h a t a necessary and sufficient condit ion for (26) to hold is

(2s) Om,~ - Q dt + ~ -- (-- Q) dt ÷

dt} : -- (zJE -- Omin AH) •

This ~ssert ion contradic ts (24) unless bo th sides of the equat ion, and hence all three

integTats on the l e f t -hand side of it, are zero.

THEOt~EM 2. -- _Let it be supposed that both v~+ and ~ - have positive measure.

A) Then the inequality

(29)

is su]ficient that

A E - - O - A H ~ O,

_

(30) e_<l - - - - 0 + ,

B) For a body undergoing a reversible C-process, a necessary and su]fieient con~

dition that

(31)

is

(32)

O _

e ~ l - - - - 0 ÷

A E -- O- A H = 0 .

C) I] (31) holds, then A E - - O - A H ~ O . For both (31) and (29) to hold, it is

necessary that the body undergo a reversible C-process. The proof is paral lel to t h a t of Theo rem 1. F r o m these theorems follow all

the classical asser t ions abou t Carnot cycles. Indeed , the a s sumpt ions A E ~ 0 and A H ~ 0 are sufficient t h a t (24) shall hold always and t h a t (29)shal l hold whenever '

0- exists. Thus the two foregoing theorems imply t h a t the e s t ima te (25) holds always, while J~he es t imate (30) holds whenever ~+ and ~ - have pos i t ive measure .

I n a s imilar wa, y we can use (5)3 to ob ta in bounds for the work t h a t m u s t be done on a body in order to m a k e it absorb a given a m o u n t of hea t in a given range

of tempera~tures.

TttEO~E~ 3.

A) if

(33) A E - - Om~ X A H ~ 0 ,

O. TI~UESDELL: Improved estimates o] the e]]icieneies, etc. 315

then

(34) -

B) For a body undergoing a reversible counter-Carnot process, a necessary and suHicient condition that

(35)

is

(36)

) - - U - = ~O~l= 1 C +

AE -- 0ma x AH ~ 0 .

C) I ] (35) holds, then

- _ [ t Q (37) (AE--OmaxAH)~Oma~ f ( ---~) dt, 2~

equality being achieved i/ and only i] the body undergoes a counter-Carnot process.

Of course (37) reduces to a t r iv ia l i ty if

A/iT - - 0 ~ A H > 0 .

PART I I

B O D I E S W I T H L I N E A R F R I C T I O N AS AN E X A M P L E

OF T H E E F F E C T OF C O N S T I T U T I V E R E L A T I O N S

8. - Bodies w i th l inear fr ict ion.

Hencefo r th we consider only the s implest appl icat ions of the results p roved above. As is c u s t o m a r y in classical t he rmodynamics , we suppose now t h a t the con-

di t ion of a body is described b y its t e m p e r a t u r e O(t) and a cer ta in k-dimensional vec tor /¢(t). A process is a pa i r of cont inuous piecewise C 1 funct ions 0, T. A cycle is a process such t h a t A0 ~ 0 a n d A T = 0. We suppose also t h a t the in te rna l en-

e rgy and the e n t r o p y of the b o d y in a process are the values of Cl-functions of the presen t va lue of t h a t process:

(38) E ----- E(0 , /¢ ) , H = / J ( 0 , F)

in some domain of (k ~ 1)-dimensional space. Thus for the body with which the

316 C. TRUESDELL: Improved estimates o] the e/]icieneies, etc.

const i tu t ive functions E and / I are associated the fur~ctions E and H are deter-

mined by assignment of a process. These assumptions imply tha t A E-~ 0 and AH ~ 0 in ~ny cycle, and we m a y

apply Theorem 1 in §7. I f C + > 0 , the efficiency of a cycle cannot exceed 1 - - 0 ~ J 0 m ~ ; reversible Carnot cycles, and t h e y alone among cycles, do achieve tha t l imit. However , as we shall see now, such cycles general ly fail to exist for assigned 0m~., 0 ~ , and in m a n y cycles no heat a t all is absorbed.

We proceed to lay down a const i tu t ive relat ion for the working:

(39) w = - ~ ( 0 , r ) . / ¢ - ~ . n ( 0 , r )

almost always. The vector-valued funct ion ~ is the equilibrium pressure; the tensor- valued funct ion 11, the values of which we ma y without, loss of general i ty assume

to be symmetr ic , is the frictional pressure; we assume tha t ~ and H are continuous over the domain of E a n d / 7 . The const i tu t ive class defined by (38) and (39) is the

class of bodies with linear ]fiction. As is shown in Chapter 1 of my Rational Thermodynamics, by imposing Coleman

and ~ot l ' s r equ i rement that, the Second L~w be an identical res t r ic t ion upon the funct ions E, H , ~ , and / / w e m a y obtain necessary and sufficient conditions tha t

these be admissible as const i tu t ive f lmctions:

(40) / t - - ~0 ' D = - - -~--, X.II(O, :F)X<O for eve ry X .

Here ~ is the free-energy funct ion: ~ ~ E - 0/7.

I f we set

(~1) t _-- - ~ - n ~ ' ,

t hen l ~ 0 because of (40h, and almost always

(42) Q = O R - I ,

= - 0 ( - :~-O-F bO b l e' 1~ -- .

In the preceding sections we did not define a. process in general, bu t we did define isothermal processes, adiabat ic processes, reversible and irreversible processes, Carnot processes, eounter-Carnot processes and C-processes. These now appear, as their names suggest t hey should, as special kinds of processes. For example, a Carnot cyle is a cycle t h a t is also a Carnot process. F r o m (42)1 we see t h a t a process in which H = const, general ly fails to be adiabatic , and t h a t a process is reversible if and

only if 1 = 0 almost always.

C. TRUESDELL: Improved estimates oj the e]]ieieneies, etc. 317

The ]fictional loss L is the work done on the body b y the f r ic t ional pressures :

(43) Z - ~ f t d t ~ - - - - f ~ . I I ~ d t ~ O.

We m a y regard as trivial a n y process such t h a t /e remains cons tan t in ~ ; we shall refer to o ther processes as nontrivial. So as to set aside t roub lesome except ions we

shall a s sume hencefor th t h a t the values o f / / a r e negat ive tensors . A body of this k ind will be said to have definite linear ]riction. For it, L ~ 0 in every non-trivial process, and Z ~-- 0 in a trivial one. F o r such a body a process is revers ible if and only if i t is t r iv ia l . Many of the resul ts ob ta ined below will hold also if the values o f / / a r e m e re l y non-posi t ive , as (40)a requires.

9. - Sharpened e s t i m a t e s o f ef f ic iency.

For a n y process the re la t ion (42)1 shows t h a t

/? Q

Compar ison wi th (43) yields

(4~) ___L< L

B y app ly ing (45)1 to (5)~ and (45)~ to (34) we conclude t h a t for a n y cycle

(46) \0~l~ 0 ~ "

B o t h the uppe r bound and the lower one are achieved for an i so thermal cycle because, as we see f rom (42)1 and (2), for such a cycle U : C + - - C - : - - L . Con-

verse]y, if some cycle does achieve the uppe r bound in (46), b y looking back a t (5)1 we see t h a t t h a t cycle is a Carnot cycle such t h a t l(~m~-- 0) ~ 0 a lmos t always. Hence

1 ~ 0 a t a lmos t all t imes when O(t)~= Omx. Now since 0 is ~ cont inuous funct ion, its range is [O~n , 0ma~]. I f 0min<0max, there is a t i m e at which 0 assumes the va lue 0~in; hence the re is a non-nul l sub in te rva l of ~ on which 0¢0m~ ~ and the process is not i so thermal . On t h a t sub- in terval , then , 1 ~ 0 a lmos t a lways ; for a body wi th definite l inear fr ict ion, the process m u s t be t r iv ia l on t h a t subinterval .

Then (42) reduces to Q ~ K(O, ~')~, the func t ion K being the specific hea t a t con- s t an t /¢. I f K > 0, we have found a set of t imes of pos i t ive measure on which the process is ne i ther ad iaba t ic nor i so thermal . We are a r r ived a t a contradic t ion with

2 1 - A n n a l i dt Matemat ica

318 C. TRVESDELL: Improved estimates of the e/fieiencies, etc.

the requi rement established at the outset : t h a t the cycle be a Carnot cycle. In just the same way we can use (5)s to see t h a t t he lower bound is achieved only by counter- Carnot cycles such t h a t l(O--0mi.)-~ 0 almost always; again for a body with posi- t ive K we can derive a contradic t ion unless the cycle is isothermal. Thus we have shown tha t when a body with de/inite linear friction and positive speci/ic heat under- goes a cycle, equality is achieved in (46) /or isothermal cycles, and /or them only. In part icular , if 0nS~< 0~a~ neither bound provided by (46) is achieved, not even by Carnot cycles or counter-Carnot cycles.

B y using (45)~ in (5)~ we can obta in also a much sharper lower bound for the

ne t work done by a Carnot cycle:

Thus for Carnot cycles

(48) O~in L ~ ( 1 - 0 ~ ) 0 ~ --

C + - - L .

C + - U g L .

By reasoning similar to tha t we followed in connect ion with (46) we m a y show tha t equal i ty is achieved on the lef t -hand side if and only if it is achieved on the right- hand side; t ha t i s , /o r isothermal cycles but /or no other Carnot cycles. I t is interest ing to notice t h a t if C +--- 0, then (48) reduces to

0 (49) \ ~ i n z g - - U ~ L . ( ~ m a x - - - -

The upper bound is obvious f rom first principles, bu t the lower one does not seem to be. The theorems s ta ted in § 7 are valid, of course, for bodies with l inear fr ict ion;

we have sharpened, as far as cycles are concerned, the es t imate affirmed b y P a r t A of Theorem 1. For bodies with definite l inear fr ict ion and posi t ive specific hea t K the only reversible Carnot or counter-Carnot processes are constant processes. Hence for such bodies P a r t B of Theorem 1 is t r ivial and Par t s A and C of t h a t theorem show tha t for non-constant cycles, s tr ict inequal i ty holds in (25).

lO. - The reversible process corresponding to a given class of irreversible processes.

Whethe r or not a process O, ~ be reversible, depends upon the const i tu t ive

relat ions of the body t ha t undergoes it. The example provided by bodies with l inear fr ict ion allows us to cont ras t rever-

sible and irreversible processes in a specific way. I f / / ~ 0, the bodies defined by (39) and (38) reduce to those considered in classical thermodynamics . For such bodies, as (4:2)1 shows, all processes are reversible. I f we consider a given process 0, r ,

C. TI~UESD:ELL: Improved estimates o/ the e//ieieneies, etc. 319

we can calculate the various quant i t ies to which it gives rise in the body whose free- energy funct ion is ~ , first when H ~ 0 and second w h e n / / n e e d not vanish iden- t ically. Thus we m a y compare the proper t ies of a given process for a given body with the corresponding reversible process, namely, the same process when undergone by the body wi th the same free-energy funct ion ~ bu t devoid of fr ict ional pres- sure: H ~ 0. Quanti t ies associated with the la t te r possibili ty we shall denote by the subscript <~rev ~>.

Accordingly, we wri te

v ,~ ~f~(o , r)- ~ dr,

(50) C~+v ~ f 0117dt, e tc . ,

+ ~rev

and so obtain f rom (1), (39), (42)~, and (43) the relat ions

(51) C + -- C- = Cre v --

Exis tence of the decomposit ion (51) is the main advantage gained by considering bodies with l inear fr ict ion, for more general bodies do no t en joy it.

The subscript <,rev ~ is in tended to suggest not only the formal definition of t he rmodynamic reversibil i ty, namely, On ~ Q, bu t also the rule of sign corresponding to reversal of the process 0, T. Namely , if we write

(52) rev ] (t) ==_/(tl + t~ - t), tl <= t < t~,

and use the same abbrevia t ion for the values of functionals of / evaluated for reel, t h en

rev U~.,~ = -- ~ , (53)

revC~v = c;~,

bu t

(5~) rev L = L .

Urev, C+v, and C~v are independent of the ra te at which a process traverses its

pa th , bu t L is not . I f I~I is made large enough th roughout ~3, t hen L becomes grea ter t han any assigned quant i ty , and in par t icu lar L > C+v and L > [~,. . Then

the mot ive power of the process is negative. On the o ther hand, if I~I is made small enough th roughout ~3, then it is easy to show t h a t L becomes arbi t rar i ly small.

Using (42)1, we shall now relate C + to + Crew. The definitions of 23+ and its rever-

320 C. TRUES:DELL: Improved estimates ol the effieieneies, ete.

sible coun te rpa r t .6~+~ are

(55)

so if we set

(56)

then

(57)

Therefore

(58)

I f

(59)

"6 + = {t: 0 / ~ - / > 0 } ,

.6~o~- {t. 0B > o},

. 6 f ~ { t : O < O B < : Z } ,

.6rev- .6f"

¢ + = f (Ot:I-t)dt, ~+o~-~

=C~v-fOI:ldt-- f ~at,

f td , t hen (58)1 becomes

(60) C + + C~e ~ - L - ~ A ;

also, because l ~ 0, by use of (56) and (59) we see t h a t

(61) O ~ _ A ~ L .

Therefore

(62)

For bodies of this + +

.6f .6re~, A Z - - C + > 0 . Then e exists, and because of (51)1 and (60)

(63) e : - C+v-- L - ~ A "

F r o m (60) and (61) we see thu t C~ v ~ 0, so ere v exists, and of course

Ul"e'~ (64) e~o~= c+o ~ .

+ C + >_-- Uro ~ - L .

k ind it is possible thu t .6+ be e mp t y ; then C + = O We shall consider only processes sufficiently slow tha t

C. TI~UESDELL: Improved estimates of the e/]ieieneies, ete. 321

Hence

L - - ( L - - A ) ere~ (65) e = e ~ +

C~-- L ÷ A

I f 0 ~ ere v ~ 1, t hen (65) and (61) show t h a t

(66) erev-- ~ ~< e ~ erev.

I f Cr+~ -- L 3> 0, t hen (60) allows us to obtain ~ lower bound for e in t e rms of nothing bu t L a n d quanti t ies associated with the corresponding reversible process:

L (67) e~ - c ~ _ L _ _ < e .

The est imates (46) and (66) refer to different classes of processes: the former to cycles only, bu t with no restr ic t ion on e ~ aaad C+, the la t te r to all processes such t ha t C + > 0 and 0 g e ~ g 1.

11 . - S l o w p r o c e s s e s , r e t a r d a t i o n .

In a process such t ha t ITt is kept sufficiently small, the fr ict ional loss L is like-

wise arbi t rar i ly small. Nevertheless, in a fami ly of processes such t h a t sup l iel-+0 the reversible work Ure v and heat absorbed C+~ ma y approach limits, which we m a y denote by (~ev)o and (C+~)0. I f so, b y (51)1, (60), and (61) we see t h a t

(68) U --> ( Urev)0, C + --> (C+v)0.

In the limit, then, the results are jus t t he same as if we had supposed t h a t H ~-0 in (39). Something of this sort mus t lie behind the claim, as a rule urged zealously by physicists, t h a t (( the rmodynamics is valid only for quasistat ie processes. ~>

In m a n y eases, one of these being t h a t of bodies with l inear friction, Coleman and ~ol l ' s concept of re ta rda t ion serves to give concreteness to this otherwise vague as well as unsuppor ted claim. I f ] is given on the in terval [0, to] , then its retardation ]r is defined for a given posi t ive r as follows:

(69) ] , ( t ) ~ / ( r t ) , 0 ~ t ~ to/r .

In this way we m a y define the re ta rda t ion 0,, 1¢~ of a given process 0, T defined on [0, to] and calculate quant i t ies associated with it.

Using the prefix (( re t )) to denote quant i t ies so obtained, we easily show f rom (50) and (43)

re t Ur~ v =: U~e~,

(70) re t C+~ -= C+~,

re t L -~ rL .

322 C. TRL~SDELL: Improved estimates o] the e/]iciencies, etc.

Also~ f rom (61) and (70)s we see t h a t

(71) 0 ~ r e t A ~ r L .

Hence r e t A - > 0 as r-->0. I t follows f rom (51)~ and (60) t h a t

+ (72) re t U--> U ~ , r e t C + -~ C~v, r e t e - ~ e ~ as r-->O,

subs tan t ia t ing (68). In my first paper on this subject I used (47) in conjunct ion with (46)~ so as to

conclude t ha t for a Carnot cycle

0rain (73) r e t e --> 1 -- 0m~ as r -~ 0 .

The reasoning there is not correct , nor is the resul t as there s ta ted, for re ta rda t ion does not general ly car ry a Carnot cycle for a given body into another Carnot cycle for t ha t body, so we m a y not use (47) for the re ta rded cycle. Indeed , f rom (42)1 and (38)~ we see t ha t an adiabat ic process for a body with l inear fr ict ion is a pair of funct ions 8, /" such as to sat isfy the differential equat ion

all). 2+ K0 : 0 (74) n £ + 0 ~

The adiabat ic par t s of a given Carnot cycle are par t icu lar solutions of (74). I f we call one of these solutions 0o, To, t hen ret0o, re t / 'o is not general ly a solution of (74) and hence is not an adiabat ic process. Not only tha t , the corresponding reversible process as defined in § 10 is not an adiabat ic process for the corresponding body devoid of fr ict ional pressure, for i t does not generally sat isfy the differential equa-

t ion for the adiabat ic processes of t ha t body:

(75) 0 ~ . .

Thus the corresponding reversible process general ly is not a Carnot cycle. Accordingly,

~min (76) erev< 1 - 0ma----~ "

Application of (72)~ shows t ha t (73) does not general ly hold. Tha t is, as the process de]thing an irreversible Carnot cycle ]or a given body is traversed more and more slowly~ it ceases to be a Carnot cycle ]or that body~ and its e]]icieney generally approaches a value less than that o] a reversible Carnot cycle with the same extreme temperatures.

C. TRUESDEL]~: Improved estimates o] the e]fieiencies, etc. 323

To obtain (73) we mus t consider for the body with friction not ~ Carnot cycle

bu t a cycle whose corresponding reversible process is a Carrier cycle for the body

wi thout friction, so tha t

0rain ( 7 7 ) ere v = 1 - - On~"-~ "

Application of (72)~ then yields (73). We m a y phrase this result as follows. ]Set a

body with linear ]fiction describe a cycle which ]or the corresponding body without ]rio- tion is a Carnot cycle. In the limit o] retardation the e]ficiency o] the body with friction approaches the value appropriate to a reversible Carnot cycle with the same extremes o] temperature.

The cont ras t between the two s ta tements in italics suggest tha t claims ~bout

quasi-static processes should be phrased more precisely if t hey are designed to be

correct. Perhaps the physicists mean by a Carnot process something different f rom

w l ~ t h~s been defined in § 3. If , indeed, by a Carnot process the physicists mean

one in which H ~ const, except on its isothermal parts , then the second s ta tement

corresponds to their views, expressed precisely for bodies with linear friction (5).

I am indebted to Dr. S. BItARATItA for criticism of the first draught of this paper,

for providing the present form of the argtunent leading to (66) a.nd (67), and for

pointing out the usefulness of considering (5)2 and (5)3 ~s well as (5h:

The research presented here was suppor ted by a grant f rom the U.S . l~ational

Science Founda t ion to The Johns Hopkins Universi ty.

(~) Of course it is easy to obtain a neat limit theorem if, abandoning the idea of slowing the process, we employ the concept of a ]amily o] bodies with diminishing friction. We need only replace 1I by ell, fix distinct 0m~ x and 0rain; and consider particular corresponding Carnot cycles for small values of e.

By (74) we see that the adiabatic parts of one of those cycles must satisfy the differential equation

gO o ÷ + =

Thus the processes depend upon g, and we compare the efficiencies of a family of irreversible Carrier cycles 0n, T~. We may expect that as e ~ 0 the adiabatic parts of those cycles tend to solutions of (75). If they do, then

0rain e ~ 1 - - ~ as ~-+0 .

For this purpose it is sufficient also that --ST~.II~'~dt be bounded, for then L ~ 0, so the desired result follows from (48). 3%

This comparison does not seem to represent what the physicists have in mind, since it says nothing directly about the speed with which the processes are traversed.