The thermodynamics of endoreversible enginesB. H. Lavenda Citation: Am. J. Phys. 75, 169 (2007); doi: 10.1119/1.2397094 View online: http://dx.doi.org/10.1119/1.2397094 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v75/i2 Published by the American Association of Physics Teachers Related ArticlesThe paradox of the floating candle that continues to burn Am. J. Phys. 80, 657 (2012) Origin of the thermodynamic time arrow demonstrated in a realistic statistical system Am. J. Phys. 80, 700 (2012) Thermodynamics of combined-cycle electric power plants Am. J. Phys. 80, 515 (2012) Modeling the exit velocity of a compressed air cannon Am. J. Phys. 80, 24 (2012) The added mass of a spherical projectile Am. J. Phys. 79, 1202 (2011) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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The thermodynamics of endoreversible enginesB. H. Lavendaa�

Università degli Studi, Camerino 62032 (MC), Italy

�Received 4 April 2006; accepted 20 October 2006�

It is shown that the Curzon-Ahlborn engine, a prototype of an endoreversible engine, has the sameefficiency as that of an unequally heated body that produces maximum work when perfectthermodynamic engines equalize its temperature. Maximum power output and finite-time operationsare completely illusory. © 2007 American Association of Physics Teachers.

�DOI: 10.1119/1.2397094�

I. INTRODUCTION

The Carnot efficiency is an idealized abstraction of theefficiency of real engines. So when someone is able to deter-mine the efficiencies of real engines, it is a welcome additionto our knowledge. And if something can be said about thefinite time of the operations in the cycle, we would haveknowledge of the power output, which is beyond the realmof classical thermodynamics.

Notwithstanding the acclaim that finite-time thermody-namics has received, we will show that the condition forreversibility of an endoreversible engine, in which a revers-ible engine is coupled irreversibly to the heat reservoirs, isflawed. The goal of this article is to show that the efficiencyof an endoreversible engine can be determined from the prin-ciple of maximum work.

In Sec. II we discuss the endoreversible engine and showthat there is a flaw in the condition for reversibility. In Sec.III we reduce finite-time thermodynamics to the principle ofmaximum work. In Sec. IV the isochoric Curzon-Ahlbornengine1 will be shown to have the same efficiency as that ofan unequally heated body that produces maximum workwhen perfect thermodynamic engines equalize itstemperature.2 In addition we show that its isothermal coun-terpart has the same efficiency as that of an unequallystressed body, which produces maximum work by perfectthermodynamic engines equalizing its state of stress. A com-parison of the endoreversible engines will be made in Sec. V.We present our conclusions in Sec. VI.

II. THE ENDOREVERSIBLE ENGINE

Carnot was concerned with the way engines behave ide-ally, rather than how they actually behave in practice. Eachstep in the cycle was assumed to be quasi-static so that thesystem passes through a sequence of thermodynamic equilib-rium states.

To determine the efficiencies that are realized by powerplants, Curzon and Ahlborn1 replaced the isothermal expan-sion and compression steps by heat fluxes created by tem-perature differences between the heat reservoirs and theworking fluid. The engine operates reversibly, and all irre-versibilities are relegated to the coupling of the engine to theoutside world. Such an engine has been calledendoreversible.3

The Curzon-Ahlborn engine �see Fig. 1� distinguishes be-tween the temperatures of the reservoirs and the temperaturesof the working fluid at which heat is absorbed and rejected.The temperature of the hotter reservoir is T1 and the enginewould absorb a quantity of heat Q1 at this temperature. How-

ever, the temperature of the fluid is T1,w�T1, which is nec-

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essary if heat is to be transported by thermal conduction. Atthe temperature T1,w the engine absorbs a quantity of heatthat is inferior to Q1. The simplest assumption for the kinet-

ics of the heat transfer is that the heat transfer rate Q1 isproportional to the temperature difference �T1−T1,w� be-tween the reservoir and the working fluid:

Q1 = ��T1 − T1,w� , �1�

where � is the product of the area of over which heat isabsorbed by the engine and the thermal conductance �theinverse of the thermal resistance� per unit area of the surfaceperpendicular to the direction of heat transfer. Because steamengines involve the bulk flow of matter under steady flowconditions, we prefer to use the capacity rate mcV as thecoefficient of proportionality, where cV is the specific heat atconstant volume.4 The units of the capacity rate are the sameas �. If the fluid can be modeled as an ideal gas, the changein energy due to heat transfer is a linear function of thetemperature difference in the heat exchanger.

Curzon and Ahlborn associate the duration of the isother-mal expansion t1 with the time of the flow and multiply bothsides of Eq. �1� by t1 to obtain the total heat transfer:

Q1 = �t1�T1 − T1,w� = m1cV�T1 − T1,w� , �2�

where m1 is the amount of mass transported in the time in-terval t1. In the same way Curzon and Ahlborn assume atemperature difference between the working fluid and thecold reservoir to create an outward heat flux:

− Q2 = �t2�T2 − T2,w� = m2cV�T2 − T2,w� . �3�

The heat rejected is assumed to be proportional to the differ-ence in temperature between the working fluid and the tem-perature of the colder reservoir. We have set �t2 equal to theheat capacity m2cV, where m2 is the product of the mass flowrate m2 and its duration t2. We know only their product, nothow small or large the flow rate is, nor how large or smallthe time interval t2. Only their product is relevant, and m2cVis determined by the ratio of the heat transferred −Q2 to thedifference in temperature �T2−T2,w�. An adiabatic compres-sion brings the working fluid back to its original state.

The crucial step in the Curzon-Ahlborn analysis is theircondition for reversibility:

Q1

T1,w=

Q2

T2,w. �4�

The maximization of the power,

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P =Q1 − Q2

t1 + t2, �5�

with respect to the unknown intermediate temperatures T1,wand T2,w yields the efficiency

� = 1 −�T2

T1, �6�

which is less than the Carnot efficiency,

�C =T1 − T2

T1. �7�

Their condition for maximum power is the first equality in

T2,w − T2

T1 − T1,w=��T2

�T1=

�t1

�t2

T2,w

T1,w=

�t1

�t2�T2

T1. �8�

The second equality uses Curzon and Ahlborn’s condition forreversibility, Eq. �4�, and the last equality expresses the ratioof the intermediate temperatures in terms of the reservoirtemperatures. By equating the second and fourth terms in Eq.�8�, Curzon and Ahlborn obtain the condition

�/� = m2/m1 = �t1/t2�2. �9�

From Eq. �9� there is reason to believe that something isamiss. We would have expected a relation such as

m1 = m1t1 = m2t2 = m2 = m , �10�

which expresses the conservation of mass instead of Eq. �9�.If the latter condition is used to evaluate Eq. �4�, we obtain

2Ti,w = Ti + �T1T2 �i = 1,2� , �11�

so that the sum T1,w+T2,w= 12 �T1+T2�+�T1T2, is the sum of

the arithmetic and geometric average temperatures. If we in-troduce Eq. �11� for the intermediate temperatures in Eqs. �2�and �3�, we obtain

Q1 = 1 mcV�T1��T1 − �T2� , �12a�

Fig. 1. The putative endoreversible engine cycle.

2

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Q2 = 12mcV

�T2��T1 − �T2� , �12b�

given the homogeneity condition �10�. The difference Q1−Q2 yields the maximum work output

W = Q1 − Q2 = 12mcV��T1 − �T2�2

= mcV� 12 �T1 + T2� − �T1T2� . �13�

Equation �13� was found by Curzon and Ahlborn by maxi-mizing the power �5� with respect to the intermediate tem-peratures T1,w and T2,w whose optimal values are given inEq. �11�.

The maximum work output found by Curzon and Ahlborn,Eq. �13�, is an old and well-known result. As early as 1853,William Thomson5 derived the amount of work that could beobtained when perfect thermodynamic engines are intro-duced into an irregularly heated body, thereby equalizing itstemperature. If mi is the mass of the ith element of the ir-regularly heated body with specific heat cV�Ti� at tempera-ture Ti, then a perfect engine is one that will operate suchthat all the heat lost is transformed into work with the bodyreaching a common temperature TM. The condition that noheat be given to any other body at temperature TM is

�i=1

n

mi�Ti

TM cV�T�T

dT = 0. �14�

Thomson used Eq. �14� to determine TM. He then employed

W = �i=1

n

mi�TM

Ti

cV�T� dT �15�

to determine the maximum work done. Observe that Eq. �14�is the condition that thermal equilibrium is achieved underadiabatic conditions, and the work �15� is the negativechange in the internal energy so that Eqs. �14� and �15� areexpressions of the second and first laws when no heat trans-fer takes place.

If we simplify to a body of two equal masses and assumethat the specific heat is independent of the temperature, thefinal common temperature is the geometric mean TM0

=�T1T2, and the work done is given by Eq. �13�. We haveobtained the lowest temperature that the system can achieve,which produces the greatest work output without any re-course to maximizing the power, or any information regard-ing intermediate temperatures.

It has been claimed that the Curzon-Ahlborn engine has anet entropy gain of6

�S = ��C − ��Q1

T2� 0. �16�

This gain would correspond to a portion of the body beingbelow the mean temperature TM0

. So how can there be acriterion of reversibility �4� at the same time that there is anincrease in the entropy? What is the meaning of Eq. �4�?

The flaw lies with Eq. �4�: It is not the condition that theprocess be reversible. According to Thomson,5 a process isperfectly reversible when “… the absolute values of twotemperatures are to one another in the proportion of the heattaken in to the heat rejected in a perfect thermodynamic en-gine working with a source and heat sink at the higher and

lower of the temperatures respectively,” or

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Q1

T1=

Q2

T2. �17�

Equation �17� is the true condition for the cycle to be revers-ible, not the Curzon-Ahlborn criterion of reversibility Eq.�4�.

Instead of Eq. �4� we should assume that an engine ab-sorbs a quantity of heat Q1 at temperature T1 and emits thequantity of heat �TM /T1�Q1 at temperature TM �T1:

Q2 =TM

T1Q1. �18�

Because there is complete symmetry between the hotter andcolder reservoirs, we can equally well write

Q1 =TM

T2Q2, �19�

but with the proviso that TM �T2. If we equate the ratio ofheats absorbed and rejected in Eqs. �18� and �19�, we find thecorrect final temperature, TM0

=�T1T2. We introduce theCurzon-Ahlborn criterion for reversibility �4� and obtain

T1,w

T2,w=

T1

TM=

TM

T2. �20�

Equation �20� fixes the ratio of the intermediate tempera-tures. The difference between their values and those of thetwo reservoirs is determined by substituting Eq. �11� intoEqs. �2� and �3�. We obtain

Q1 = mcV�T1 − T1,w� = 12mcV�T1 − TM� , �21a�

Q2 = mcV�T2,w − T2� = 12mcV�TM − T2� . �21b�

The difference between Q1 and Q2 is the maximum work�13�. The work is always positive definite unless the twotemperatures are equal; the inequality is a consequence of thearithmetic-geometric mean inequality. The arithmetic-geometric mean inequality is one of an infinite number ofinequalities based on the property that means are increasingfunctions of their order, of which the arithmetic-geometric-harmonic mean inequalities are examples.

III. REDUCTION OF FINITE-TIMETHERMODYNAMICS TO THE PRINCIPLEOF MAXIMUM WORK

Where do maximum power considerations enter? The

multiplication of the rates of heat transfer Q1 and Q2 by thefinite times t1 and t2, respectively, seems innocuous, but thesubsequent maximization of their difference requires an in-determinate amount of time. The heat capacities mcV in Eqs.�2� and �3� are given in terms of the energies transferred dueto their respective differences in temperature. All concept oftime has disappeared. Even if we were willing to hold ontothe interpretation of �t1 and �t2 as products of thermal con-ductances and time intervals, their units are not power perdegree Kelvin. So by starting from the rate of heat transferand converting it into the heat transferred, Curzon and Ahl-born give the illusion that the times t1 and t2 are independent

parameters. It therefore seems reasonable to maximize the

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power output in a finite time, and this procedure has resultedin a new branch of thermodynamics known as finite-timethermodynamics.7

Curzon and Ahlborn’s principle of maximum work can bereexpressed as follows. Maximize the work,

W = a�T1 − T1,w� − b�T2,w − T2� = ax − by , �22�

given the condition that

ax

T1 − x=

by

T2 + y, �23�

which is Eq. �4� with a=�t1 and b=�t2. The condition for anextremum in y is

y0 =a

a + b��T1T2 − T2� = T2,w − T2, �24�

which is Eq. �21b� for a=b; the geometric mean �T1T2is the final mean temperature TM. That Eq. �24� is a maxi-mum follows from the fact that ��2W /�y2�y=y0

=−2b�a+b� /a�T1T2�0.What Curzon and Ahlborn rediscovered was that the final

mean temperature at which adiabatic equilibration occursyields the maximum work. This principle can be traced as farback as Ref. 5 and8

“Let �TM0� be the temperature to which the whole

body can be brought by means of perfect engines,so that all the heat lost is converted into work.”

The time it takes to reach the geometric mean temperature isindeterminate so no maximum power output can be attrib-uted in a finite time interval.

In a little known paper,9 Cashwell and Everett derived thefirst and second laws from the properties of means. A meanof order r is defined as10

XMr= ��

i=1

n

pixir1/r

, �25�

for an n-tuple of numbers x1 ,x2 , . . . ,xn with correspondingprobabilities p1 , p2 , . . . , pn. Following in the footsteps ofThomson5 without realizing it, Cashwell and Everett draw ananalogy to a thermodynamic system comprising a set of nisolated cells or heat reservoirs with initial temperatures Tithat are not all equal; the xi are the specific heats of thedifferent cells, cV�Ti�, and the probabilities pi are the corre-sponding mass fractions, mi. This set of cells is what Thom-son considered to be an “unequally heated space.”5

Conservation of energy implies that when these isolatedcells are placed in thermal contact, a single uniform meantemperature TM will be reached in an indeterminate amountof time, such that there is no overall change in the internalenergy

�E = �i=1

n

mi�Ti

TM

cV�T� dT = 0. �26�

It follows that the change in entropy is

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�S = �i=1

n

mi�Ti

TM cV�T�T

dT � 0. �27�

The inequality is a consequence of the fact that the means aremonotonically increasing functions of their order. The role ofthe integrating denominator is to lower the order of themean; the inequality follows from the property that meansare increasing functions of their order:11

TM�� TM�

�28�

�����, unless all the Ti are equal, where M� is the mean oforder �.

To determine the maximum work output, all parts of thebody must come to a common temperature, which is thelowest possible temperature. There can be no heat exchangewith another body at the same or lower temperature. In thisway we are assured that all the heat is converted into work.5

This conversion of lost heat into work requires adiabaticequilibration at a lower temperature than the temperature thatwould have been reached if equilibration had come aboutthrough energy conservation �26�. For adiabatic equilibra-tion, the decrease in the internal energy determines the maxi-mum work. This interpretation is akin to Carnot’s assump-tion of heat conservation and the universal maximumefficiency of his engine.12 Here it is conservation of entropyand not heat that is being used, so there is a distinction be-tween adiabatic and isentropic processes.

In brief, we have no knowledge of how long it will take toreach the final, uniform, temperature to which the systemequilibrates. The mean temperature is always greater than thefinal temperature of the coldest reservoir, or coldest initialpart of the system before it thermally equilibrates. It is thisdifference that leads to an illusory increase in the entropy�16�. Equation �18� is a statement of reversibility accordingto the second law. The optimality of the Carnot efficiency isa consequence of the fact that the temperature of adiabaticequilibration, the geometric mean temperature, is greaterthan the temperature of the coldest reservoir.

IV. MAXIMUM WORK OUTPUT AND HEATUPTAKE

We now establish that the final mean temperature and vol-ume, as determined from adiabatic equilibration, yield themaximum work output and maximum heat uptake with re-spect to all other admissible sets of temperatures and vol-umes.

Let us first consider the case of thermal interactions. Con-sider n parts of an irregularly heated body all at differenttemperatures Ti with Ti,max=T1 and Ti,min=T2. FollowingCashwell and Everett,9 we divide the reservoirs into twogroups: � for those with Ti�TM, and u for those reservoirshaving temperatures Ti�TM. Denote by ��T� a continuous,monotonically decreasing function of T. On the strength ofthe adiabatic constraint and the monotonicity of ��T�, the

following inequalities result

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��TM����

Ti

TM

CV�T� dT

� ���

Ti

TM

��t�CV�T� dT

= �u�

TM

Ti

��t�CV�T� dT � ��TM��u�

TM

Ti

CV�T� dT .

�29�

From these inequalities we conclude that

�i=1

n �TM

Ti

CV�T� dT 0, �30�

with the equality holding if T1=T2. If the heat capacitiesCV�T� are independent of temperature, as they are for anideal gas, then Eq. �30� implies that the arithmetic mean isgreater than the final common mean temperature. If we placetwo cells in thermal contact and allow them to come to auniform final temperature with no work being done, then themean temperature would be the arithmetic mean of the twoinitial temperatures, and Eq. �30� would vanish.

Let Ti denote any admissible set of final temperatures ofthe reservoirs. Because

�i=1

n �Ti

TiCV�T� dT = �

i=1

n �Ti

TM

CV�T� dT + �i=1

n �TM

TiCV�T� dT ,

�31�

and Eq. �30� guarantees that the last term will be positive,regardless of the nature of the set of temperatures, it followsthat

�i=1

n �TM

Ti

CV�T� dT � �i=1

n �Ti

Ti

CV�T� dT . �32�

Consequently, the work output will be maximum; the meantemperature at which there is maximum work output is givenby the revised Curzon-Ahlborn condition Eq. �18�.

The principle of maximum heat uptake in an isothermalprocess can be established similarly by considering a nonuni-formly stressed body. Instead of the cells being at differentinitial temperatures and not in thermal contact, consider cellsat different volumes V1 ,V2 , . . . ,Vn and not in mechanicalcontact. Suppose that Vi,max=V2 and Vi,min=V1. When placedin mechanical contact work will be produced by perfect ther-modynamic engines equalizing the state of stress of the body.We are interested in determining the maximum heat uptakeand the largest final volume.

Let �V� be a continuous, monotonically increasing func-tion of the volume. We again divide the cells into two groupsdepending on whether their volumes Vi are less than orgreater than the mean value VM and obtain the reverse in-

equalities

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�VM����

Vi

VM

LV�V� dV

���

Vi

VM

�V�LV�V� dV

= �u�

VM

Vi

�V�LV�V� dV �VM��u�

VM

Vi

LV�V� dV .

�33�

It follows that

�i=1

n �Vi

VM

LV�V� dV 0, �34�

where the equality holds if V1=V2. If we consider any other

set of final volumes Vi, then

�i=1

n �Vi

ViLV�V� dV = �

i=1

n �Vi

VM

LV�V� dV − �i=1

n �Vi

VM

LV�V� dV .

�35�

Because the inequality Eq. �34� ensures that the last term ispositive, whatever the initial states, we conclude that

�i=1

n �Vi

VM

LV�V� dV �i=1

n �Vi

ViLV�V� dV . �36�

If the final state of the initially unequally stressed bodyhas a final mean volume of order ��0,

VM�= �1

n�i=1

n

Vi�1/�

, �37�

then the maximum heat absorbed by a perfect gas, whoselatent heat LV= p in the isothermal expansion, when the cellsare placed in mechanical contact, will be

Qmax = �i=1

n �Vi

VM�

p dV

= mRT�i=1

n �Vi

VM� dV

V= nmRT ln �VM�

VM0

� 0, �38�

where

VM0= �

i=1

n

Vi1/n

�39�

is the geometric mean volume, and T is the uniform tempera-ture of the body.

The inequality �38� follows from the fact that means aremonotonically increasing functions of their order, and ��0.

Table II. Comparison of the efficiencies of real heat eendoreversible, C�Carnot, and Obs�Observed.

Power source T2 �K� TM0�K� T1 �

West Thurrock �Ref. 14� 298.15 499.9 838.CANDU �Ref. 15� 298.15 413.38 573.Larderello �Ref. 16� 353.15 429.82 523.

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The geometric mean volume �39� would be the final meanvalue when n cells, with finite volumes V1 ,V2 , . . . ,Vn, arebrought into mechanical contact, and left alone indefinitely,without any absorption of heat.

V. COMPARISON OF ENDOREVERSIBLE ENGINES

We now want to compare the isochoric engine of Curzon-Ahlborn and the isothermal engine of Sec. IV. The efficiencyof the isothermal engine is

� = 1 − �VM0

VM�

�

. �40�

The maximum work output

W = �Q = nmRT�1 −VM0

�

VM�

� ln �VM�

VM0

�41�

is the product of Eqs. �40� and �38�.In order to compare the two engines, we must use the

adiabatic relation Vi�Ti�const because there is an equivalent

cycle to that of Carnot obtained by replacing isothermals byisochores.13 Then for n=2 the mean volume �37� becomesthe harmonic mean temperature

VM�

� =1

2�V1

� + V2�� =

1

2� 1

T1+

1

T2 = TM−1

=TM1

TM0

2 , �42�

and the geometric mean of the volume, Eq. �39�, becomesthe inverse of the geometric mean of the temperature,

VM0

� = �V1�V2

� =1

�T1T2

=1

TM0

. �43�

Because we will be concerned only with their ratio, we havedispensed with the arbitrary constant in Eqs. �42� and �43�.

In terms of the temperature, the efficiency �40� is

� = 1 −TM0

TM1

, �44�

and the amount of heat released to the cold reservoir is

Table I. The efficiencies of the Carnot, Curzon-Ahlborn, and isothermalengines, where TM0

=�T1T2, the geometric mean, and TM1= 1

2 �T1+T2�, thearithmetic mean.

Engine �

Carnot 1−T2 /T1

Curzon-Ahlborn 1−TM0/T1

Isothermal 1−TM0/TM1

s, with CA�Curzon-Ahlborn engine, IE�isothermal

TM1�°C� � �CA� � �IE� � �C� � �Obs�

568.15 40.36% 12.01% 64.43% 36%162.5 27.9% 5.1% 48% 30%165 17.84% 1.9% 32.5% 16%

ngine

K�

151515

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Q =TM0

TM1

Qmax. �45�

Consequently, the maximum heat uptake and work are

Qmax = 2mcVTM1ln �TM1

TM0

, �46a�

Wmax = �Qmax = 2mcV�TM1− TM0� ln �TM1

TM0

, �46b�

respectively, where we have introduced cV=R /�. We nowcompare the efficiencies of the Carnot and two endorevers-ible engines listed in Table I.

The transformation from Celsius or any other scale of tem-perature to the Kelvin scale poses no problem if the tempera-tures are additive, because the scales differ by an additiveconstant. However, the distinction between empirical and ab-solute temperatures becomes crucial when a nonlinear tem-perature is introduced, like the geometric mean temperature.In this article, we have used the condition of reversibilityinvolving the ratio of heat absorption to rejection. This useautomatically necessitates using Kelvin’s definition of tem-perature, which does not depend on the substance to measurethe temperature. If the final temperature is taken as the geo-metric mean of the Kelvin temperatures of the hottest andcoldest parts of the body, the efficiencies of the endorevers-ible engines will be given as in Table II. The isochoric en-doreversible engine approximates the real efficiencies quitewell, while the isothermal endoreversible engine is much lessefficient.

The Curzon-Ahlborn endoreversible engine has the sameefficiency as that calculated from the principle of maximumwork. Hence, if we start with a system with cells at empiricaltemperatures t1 , t2 , . . . , tn and allow them to equilibrate ther-mally to a common final temperature, the latter is given bythe principle of maximum work is �t1 , t2 , . . . , tn�1/n, the geo-metric mean of the initially measured temperatures. The ab-solute temperature T must be determined on the thermody-namic scale, which corresponds to the reading t, measured byany ordinary thermometer, for example, a centigradethermometer.17 Because the final temperature of equilibra-tion, which corresponds to the geometric mean of the initialtemperatures of the unequally heated parts of the body, isaccessible to measurement, it must then be converted into the“work scale �which� depends on the amount of work ob-tained from a given supply of heat to a heat engine.”18 Be-cause these geometric mean temperatures when convertedinto Kelvin are lower than those calculated from the productof the absolute temperatures, the efficiencies of the endor-eversible engines are necessarily increased, as shown in

Table III. Comparison of the efficiencies of real hemeasured in degrees Celsius, is converted into degre

Power source T2 �°C� TM0�°C� T1

West Thurrock �Ref. 14� 25 118.85 5CANDU �Ref. 15� 25 86.6 3Larderello �Ref. 16� 80 141.4 2

Table III. The isothermal endoreversible engine now be-

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comes a viable competitor of the Curzon-Ahlborn engine inbeing able to determine the efficiencies of real heat engines.

VI. CONCLUSION

The fate of the Curzon-Ahlborn engine, where the cou-pling of the engine to the environment occurs through differ-ences in the temperature between the heat source and heatsink and the working fluid, is either that it will absorb andreject heat at the temperatures of the working substance,thereby reducing the motive power of heat according to Car-not, or it will correspond to the maximum work a system cando when different parts of a body are irregularly heated andsubsequently isolated so that the thermal interaction amongits parts produces a state of thermal equilibrium character-ized by a uniform temperature. This uniform temperaturewill necessarily be greater than the temperature of the coldestpart of the body or the heat sink. The thermal interactionsinvolving dissipative processes, such as thermal conduction,require a temporal description in which the body passesthrough nonequilibrium states. Such a temporal descriptionis beyond the scope of thermodynamics, and the time it takesto establish thermal equilibrium, where all parts of the bodyreach a common temperature, is indeterminate. All that canbe done is to compare a more and a less constrained state ofequilibrium, and whether the final state has occurred throughenergy or adiabatic equilibration.

The finite time thermodynamics that the Curzon-Ahlbornendoreversible engine supposedly introduces is illusory. Theirreversibilities that supposedly arise from the coupling ofthe engine to the external world are fictitious. These state-ments are supported by the fact that Curzon and Ahlbornobtain the lowest final mean temperature achievable bywhich the system has done maximum work. Maximum workoutput and heat uptake occur when parts of the irregularlyheated body, initially at different temperatures or volumes,are allowed to interact, thermally or mechanically, and whosefinal mean values are determined by adiabatic equilibration.Because no statement can be made about the time the bodywill reach thermal or mechanical equilibrium, power cannotbe defined. And because the final mean values of the tem-perature and volume are necessarily greater than the coldestand smallest parts of the body, the efficiencies are necessarilyless than the efficiency determined by Carnot. The Carnotefficiency singles out the extreme parts of the irregularlyheated or stressed body, and nothing can be hotter and colderor larger and smaller than these parts. The illusory increasein entropy in Eq. �16� is due to the fact that the heat given tothe coldest part of the body is divided by its temperature,which is lower than the mean value obtained from adiabatic

gines when the final geometric mean temperature,lvin.

TM1�°C� � CA � IE � C � Ob

295 53.23% 31% 64.43% 36%162.5 37.23% 17.42% 48% 30%165 20.76% 5.4% 32.5% 16%

at enes Ke

�°C�

650050

equilibration of all the parts of the body.

174B. H. Lavenda

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a�Electronic mail: bernard.lavenda@unicam.it1F. L. Curzon and B. Ahlborn, “Efficiency of a Carnot engine at maximumpower output,” Am. J. Phys. 43, 22–24 �1975�.

2W. Thomson, “On thermodynamic motivity,” Philos. Mag. VII, 348–352�1879�.

3M. Rubin, “Optimal configuration of a class of irreversible heat engines,”Phys. Rev. A 19, 1272–1276 �1979�; 19, 1277–1289 �1979�.

4See, for instance, E. G. Cravalho and J. L. Smith, Jr., Engineering Ther-modynamics �Pitman, Marshfield, MA, 1981�, p. 429. The specific heat atconstant pressure is commonly used instead of the specific heat at con-stant volume because, in general, the heat transfer rate is proportional tothe enthalpy difference. In the Curzon-Ahlborn model, there is no energytransfer due to the displacement of the boundary, that is, pV work trans-fer. The mass flow arising from this motion is accounted for by the pVpart of the enthalpy.

5W. Thomson, “On the restoration of mechanical energy from an un-equally heated space,” Philos. Mag. 5, 102–105 �1853�.

6A. De Vos, Endoreversible Thermodynamics of Solar Energy Conversion�Oxford U. P., Oxford, 1992�, p. 36.

7Finite-Time Thermodynamics and Thermoeconomics, edited by S. Sien-iutycz and P. Salamon �Taylor & Francis, New York, 1990�.

8P. G. Tait, Sketch of Thermodynamics �Edmonston & Douglas, Edin-

175 Am. J. Phys., Vol. 75, No. 2, February 2007

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burgh, 1868�, p. 101.9E. D. Cashwell and C. J. Everett, “Means of order t, and the laws ofthermodynamics,” Am. Math. Monthly 74, 271–274 �1967�.

10See, for instance, B. L. Burrows and R. F. Talbot, “Which mean do youmean?” Int. J. Math. Educ. Sci. Technol. 17, 275–284 �1986�.

11 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed. �Cam-bridge U. P., Cambridge, 1952�, p. 26.

12P. Lervig, “On the structure of Carnot’s theory of heat,” Arch. Hist. ExactSci. 9, 222–239 �1972�.

13B. H. Lavenda, “Thermodynamics of an ideal generalized gas: I. Ther-modynamic laws,” Naturwiss. 92, 516–522 �2005�.

14D. B. Spalding and E. H. Cole, Engineering Thermodynamics, 2nd ed.�Edward Arnold, London, 1966�, p. 209.

15G. M. Griffiths, “CANDU–A Canadian Success Story,” Phys. Can. 30,2–6 �1974�.

16A. Chierici, Planning of a Geothermal Power Plant: Technical and Eco-nomic Principles �U.N. Conference on New Sources of Energy, NewYork, 1964�, Vol. 3, pp. 299–311.

17J. Clerk Maxwell, Theory of Heat �Longmans, Green, London, 1904�, p.213.

18J. H. Poynting and J. J. Thomson, A Text-book of Physics: Heat �Griffin,London, 1908�, p. 9.

175B. H. Lavenda

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