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TECHNICAL PHYSICS VOLUME 44, NUMBER 1 JANUARY 1999
The second law of thermodynamics and the limiting capabilities of heat enginesA. M. Tsirlin
Institute of Programming Systems, Russian Academy of Sciences, 152140 Pereslavl’-Zalesski�, Russia~Submitted April 15, 1998!Zh. Tekh. Fiz.69, 140–142~January 1999!
@S1063-7842~99!02801-9#
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INTRODUCTION
The occasion for writing this Note is the appearance oseries of publications by G. V. Skornyakov,1–3 in which heasserts that in thermodynamic systems employing the pnomenon of self-organization it is possible to achieve a netive production of entropy. Hence it immediately follows thit is possible to build a heat engine having an efficiengreater than the Carnot value. But this opens the door totheoretical possibility of bringing the efficiency of a heengine as close as one wants to unity. Thus, glittering pobilities are opened up in the realm of energetics. Skornyaalso proposes a design for a heat engine whose workingdium, with the use of a vortical turbine, is stratified inliquid and vapor, which thereby delivers ‘‘self-organzation.’’ This, he claimes, citing a theorem by Yu. LKlimantovich,4 implies the negative production of entropy
It seems to me that the assertions of G. V. Skornyaare in error, his reference to Yu. L. Klimantovich is unjusfied, and the overturning of the second law of thermodynaics is as improbable as the overturning of the law of consvation of energy. The possibilities of a heat engine with fixpower are limited not only by the Carnot efficiency, but bysubstantially smaller value. Below I will discuss in modetail on these statements and derive an estimate for thficiency of a fixed-power machine.5
SELF-ORGANIZATION AND THE PRODUCTION OFENTROPY IN A SYSTEM
Let us consider a thermodynamic system consistingtwo reservoirs at temperaturesT1 and T2 and a workingmedium ~Fig. 1!. The working medium can have concetrated parameters~the temperature at any instant is the saover the given volume! or distributed parameters~the tem-perature varies across a heat flux!. In the first case, the working medium is in successively in contact with the two resvoirs ~a heat engine!; in the second case, these contactscontinuous and spatially separated~a turbine!. Following thelead of Refs. 1–3, we will consider the second case.
We write the equations of thermodynamic balancethe system of Fig. 1. These are the balance equationsenergy, mass, and entropy. In the given case, mass exchis absent; therefore only energy and entropy balance remThe equations of energy balance and entropy balance osystem between the reservoirs take the form
q12q22p50, ~1!
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s12s21s in5q1
T12
q2
T21s in50. ~2!
Hereq1 andq2 are the heat flux from the hot reservoir anto the cold reservoir, respectively, andp is the power of theheat engine. Analogously,s1 ands2 are the entropy fluxesands in is the production of entropy in the system.
Eliminating the heat fluxq2 and taking as the efficiencythe ratio of the powerp to the heat delivered to the systemq1 , we write
h5p
q15hK2T2
s in
q1. ~3!
HerehK512(T2 /T1) is the Carnot efficiency. For successive contact with the reservoirs with durationst1 andt2 , theproduction of entropy consists of three terms: the productof entropy during heat transfer from each of the reservoi
s15E0
t1q1~T1 ,T1~t!!S 1
T1~t!2
1
T1Ddt, ~4!
s25E0
t2q2~T2~t!,T2!S 1
T22
1
T2~t! Ddt ~5!
and the production of entropyspT due to the creation of aflux of matter and energy in the working medium. For alaw of heat transfer the integrands in Eqs.~4! and ~5! arepositive. In order for the quantityh to exceedhk , the pro-duction of entropy in the working medium must be negatand greater in absolute value thans11s2 . If the workingmedium has concentrated parameters, which may besumed to be only approximately the case for the heat engthenspT50. In all the remaining casesspT.0, and no self-organization will make this quantity negative.
Skornyakov believes thatspT can be made negative ancites a theorem by Klimantovich,4 but does not cite its for-
FIG. 1.
© 1999 American Institute of Physics
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130 Tech. Phys. 44 (1), January 1999 A. M. Tsirlin
mulation. In all of this, Klimantovich only proposes a meafor counting up the entropy in a nonuniform~‘‘self-organized’’! flux and asserts that it is less than it wouldunder the same conditions in a uniform flux. Let us dwellthis question in more detail.
The principle of a minimum production of entropy ofPrigogine6,7 asserts that in the stable regime of an open thmodynamic system the production of entropys in it is mini-mal. At first glance, this principle stands in contradictiwith the fact that under certain conditions the stable regof an open system is one in which nonuniformity with rspect to the spatial and/or temporal coordinate is obse~self-organization!.
In fact there is no contradiction if we replaces by s, theproduction of entropy averaged over time and space,take into account that the minimum ofs xmust be soughtunder certain constraints imposed on the system, whicthe self-organization regime are also averaged. For examconsider the production of entropy during heat transthrough a layer of liquid. For some value of the average hflux q the average production of entropys is minimal in theregime in which the liquid becomes nonuniform in evecross section. Thus, the condition for the appearance oforganization reduces the the question, when is
minx
s~x!under conditions in whichf ~x!5 f 0 ~6!
less than
minx
s~x!under conditions in whichf ~x!5 f 0 . ~7!
Herex is the vector of variables on which boths and theseconstraints depend;f is the vector-valued function of constraints~flux rate, heat load, etc.!. This condition is expresseby the following statement8:
s05maxl
minx
@s~x!1l~ f ~x!2 f 0!#,s~x0!5s0, ~8!
wherex0 is the optimal solution of the nonlinear programming problem~7!.
Condition~8! is a necessary and sufficient condition.physical meaning is especially simple for the case in whicfandx are scalars. Then it is possible to construct the depdences( f ). Condition~8! implies that forf 5 f 0 the convexenvelope of this function passes below its graph.
An example: let us consider the process of heat tranwith a prescribed average intensity of the heat flux
q~T!5@k~T2T0!#0.55q0
and entropy production
s~T!5q~T!S 1
T02
1
TDfor T0,T<Tm . After eliminatingT we obtain
s~q!5q3
T~kT01q2!.
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The functions(q) and its convex envelope over the s0,q<qm5q(Tm) are shown in Fig. 2. The minimum of thaverage entropy production is reached whenT takes the val-uesTm andT1 and the heat flux takes the valuesqm andq1 .In this case, the average value ofq should be equal to theprescribed valueq0 . It is not hard to show that according tcondition ~8! the expression
s~T!1l~q~T!2q0!
reaches its maximum with respect tol and its minimum withrespect toT at the pointsT1 andTm .
The production of entropy for some times or at sompoints, allowing for the dynamics of the system, can be netive, but on average it is always greater than zero.
LIMITING POSSIBILITIES OF FIXED-POWER HEAT ENGINES
In the last 30 years a new branch of thermodynamicsbeen actively developed—the thermodynamics of fintimes.9 This approach examines the limiting possibilitiesthermodynamic systems~efficiency indicators, dissipationetc.! with an additional constraint on the duration of a prcess or on the average intensity of the fluxes. The starpoint of the development of finite-time thermodynamican be linked with the appearance of Refs. 10 and 11the limiting power of a heat engine. Let the heat fluxesgiven by
q15a1~T12T1!, q25a2~T22T2!. ~9!
The coefficientsa1 ,a2 reflect the dimensions and material of the heat-exchange surface. The temperaturesT1 andT2 and the timest1 andt2 are subject to choice. The limitingpowerpmax turns out to be equal to10
pmax5a1a2
~Aa11Aa2!2~AT12AT2!2, ~10!
and the corresponding efficiency of the heat engine
hpmax
0 512AT2
AT1
,hk . ~11!
If the power of the enginep0,pmax is assigned, then forsystems with two reservoirs it is possible to pose the probof the limiting value of the efficiency of a heat engine f
FIG. 2.
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131Tech. Phys. 44 (1), January 1999 A. M. Tsirlin
p5p0 . This problem was solved in Ref. 5. It turns out ththe efficiency of a heat engine does not exceed the valu
hp0
0 52dk
dk112A~12k!~12kd2!, ~12!
wherek5p0 /pmax andd5(AT12AT2)/(AT11AT2).It is clear from Eq.~3! that the problem of the limiting
efficiency is equivalent to the problem of the minimum prduction of entropy. In the derivation of estimate~12! it wasassumed thatspT50 and that for the processes of heat echange the conditions of minimum dissipation are fulfilled12
which for an arbitrary heat-transfer lawq(Tp ,T) and for anypossible variation of the temperature of the reservoirTp withthe contact timet have the form
q2~Tp ,T!5Cdq
dTT2, ;t. ~13!
The constantC depends on the prescribed averagedtensity of the heat flux.
CONCLUSION
In the above-mentioned works of G. V. Skornyakov tassumption of the possibility of negative entropy productin a nonuniform thermodynamic system is in error. Nonuformity of the system can lower the spatiotemporal averof the entropy production if the constraints imposed onsystem are also averaged. But this average entropy protion will always be positive.
The proof of a theorem in mathematics, if it is logicalderived from the fundamental axioms, does not allow it tosubjected to doubt without subjecting the axioms themseto doubt. For physical laws it is the same. They should onot contradict one another and be confirmed in any correformulated experiment.
The works of G. V. Skornyakov do not contain anproof of a contradiction by the second law of thermodyna
t
-
-
n-eec-
es
yly
-
ics of any other fundamental law of physics and arebased on any experimental material. All of the accumulaexperience of mankind to date confirms the validity of thlaw, and the probability that it is in error is as small as tprobability that the particles of a gas in a closed vessel wsimultaneously collect in one of its halves.
1G. V. Skornyakov, Pis’ma Zh. Tekh. Fiz.15, No. 22, 12~1989! @Tech.Phys. Lett.15, 833 ~1989!#.
2G. V. Skornyakov, Zh. Tekh. Fiz.65, 35 ~1995! @Tech. Phys.40, 18~1995!#.
3G. V. Skornyakov, Zh. Tekh. Fiz.66, 3 ~1996! @Tech. Phys.41, 1 ~1996!#.4Yu. L. Klimantovich, Usp. Fiz. Nauk158, No. 1, 59~1989! @Sov. Phys.Usp.32, 416 ~1989!#.
5L. I. Rozonoer and A. M. Tsirlin, A. i T., No. 1, 70~1983!; ibid., No. 2, 88~1983!; ibid., No. 3, 50~1983!.
6V. N. Orlov and L. I. Rozonoe´r, in Tenth All-Union Conference on Problems of Control@in Russian#, Nauka, Moscow, 1986.
7G. Nicolis and I. Prigogine,Self-Organization in Non-Equilibrium Systems~Wiley, New York, 1977!.
8A. M. Tsirlin, Dokl. Akad. Nauk SSSR323, 271 ~1992! @sic#.9A. M. Tsirlin, Methods of Averaged Optimization and Their Applicatio~Nauka-Fizmatlit, Moscow, 1979!.
10I. I. Novikov, Atomnaya Energiya, No. 3, 409~1957!.11F. L. Curzon and B. Ahlborn, Am. J. Phys.43, 22 ~1975!.12V. Mironova, A. Tsirlin, V. Kazakov, and R. J. Berry, J. Appl. Phys.76,
629 ~1994!.
From the Editorial BoardThe second law of thermodynamics is one of the fundamental axiomphysics. So far, not one experiment has been found to be in contradicwith this axiom. Therefore, some very extraordinarily serious reasons wobe in order to justify a consideration of the question of the non-observaof the second law. We are not able to discover any such reasons inpublished literature. In light of this, the editorial board of the JournalTechnical Physics deems it inexpedient to continue the discussion ovalidity of the second law of thermodynamics on the pages of this jour
Translated by Paul F. Schippnick