We study the efficiency at maximum power, $\eta^*$, of engines performing
finite-time Carnot cycles between a hot and a cold reservoir at temperatures
$T_h$ and $T_c$, respectively. For engines reaching Carnot efficiency
$\eta_C=1-T_c/T_h$ in the reversible limit (long cycle time, zero dissipation),
we find in the limit of low dissipation that $\eta^*$ is bounded from above by
$\eta_C/(2-\eta_C)$ and from below by $\eta_C/2$. These bounds are reached when
the ratio of the dissipation during the cold and hot isothermal phases tend
respectively to zero or infinity. For symmetric dissipation (ratio one) the
Curzon-Ahlborn efficiency $\eta_{CA}=1-\sqrt{T_c/T_h}$ is recovered.Comment: 4 pages, 1 figure, 1 tabl

A. Bejan

Christian Van den Broeck

H. B. Callen

I. I. Novikov

Katja Lindenberg

Massimiliano Esposito

P. Chambadal

R. S. Berry

Ryoichi Kawai

English

arXiv

Crossref

Physical Review Letters

Efficiency at Maximum Power of Low-Dissipation Carnot Engines

We study the efficiency at maximum power, η*, of engines performing finite-time Carnot cycles between a hot and a cold reservoir at temperatures Th and Tc, respectively. For engines reaching Carnot efficiency ηC=1-Tc/Th in the reversible limit (long cycle time, zero dissipation), we find in the limit of low dissipation that η* is bounded from above by ηC/(2-ηC) and from below by ηC/2. These bounds are reached when the ratio of the dissipation during the cold and hot isothermal phases tend, respectively, to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency ηCA=1-√Tc/Th is recovered.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Esposito, Massimiliano

DI-fusion

Kawai, Ryoichi

Lindenberg, Katja

Van den Broeck, Christian

Open Repository and Bibliography - Luxembourg

Efficiency at Maximum Power of Low-Dissipation Carnot EnginesMassimiliano EspositoCenter for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles,CP 231, Campus Plaine, B-1050 Brussels, BelgiumRyoichi KawaiDepartment of Physics, University of Alabama at Birmingham,1300 University Boulevard, Birmingham, Alabama 35294-1170, USAKatja LindenbergDepartment of Chemistry and Biochemistry and BioCircuits Institute, University of California,San Diego, La Jolla, California 92093-0340, USAChristian Van den BroeckHasselt University, B-3590 Diepenbeek, Belgium(Received 14 August 2010; revised manuscript received 3 September 2010; published 7 October 2010)We study the efficiency at maximum power, , of engines performing finite-time Carnot cyclesbetween a hot and a cold reservoir at temperatures Th and Tc, respectively. For engines reaching Carnotefficiency C ¼ 1 Tc=Th in the reversible limit (long cycle time, zero dissipation), we find in the limitof low dissipation that  is bounded from above by C=ð2 CÞ and from below by C=2. These boundsare reached when the ratio of the dissipation during the cold and hot isothermal phases tend, respectively,to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency CA ¼ 1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃTc=Thpis recovered.DOI: 10.1103/PhysRevLett.105.150603 PACS numbers: 05.70.Ln, 05.20.yThermal machines performing Carnot cycles transforma certain amount of heat Qh from a hot reservoir at tem-perature Th into an amount of work W, with the remain-ing energy being evacuated as heat Qc ¼ Qh W to acold reservoir at temperature Tc. We adopted here the usualconvention that heat and work absorbed by the system arepositive. By assuming that there is no perpetuum mobile ofthe second kind, more precisely that heat does not sponta-neously flow from a cold to a hot reservoir, Carnot was ableto show that the efficiency of the heat-work transformation ¼  WQh¼ 1þ QcQh(1)is universally bounded by a maximum value, the so-calledCarnot efficiencyC ¼ 1 TcTh : (2)This insight lies at the heart of thermodynamics, since it ledClausius to the introduction of the entropy, the state functionwhich is central for the formulation of the Second Law. Theentropy change of a system is given by S ¼ Rqs dQ=T,where the integral is over the infinitesimal amounts ofabsorbed heat dQ for a quasistatic transformation of thesystem (i.e., a succession of equilibrium states) [1]. The totalentropy production during a Carnot cycle is given byStot ¼ QhTh QcTc¼ ðC  ÞQhTc (3)since the auxiliary work-performing system returns to itsinitial state (hence no change in its entropyS ¼ 0) and theheat reservoirs are assumed to undergo a quasistatic heatexchange while preserving their temperature. The fact thatthe total entropy cannot decrease,Stot  0, is equivalent tothe statement that efficiency is bounded by Carnot effi-ciency,   C. The latter is reached for a reversible pro-cess,Stot ¼ 0, which can only be achieved for a quasistatictransformation of the system implying infinitely slow Carnotcycles.While the concept of Carnot efficiency is of paramountimportance in the derivation of thermodynamics, its prac-tical implications are more limited: to reach the reversiblelimit, one needs in principle to work with infinitely slowcycles; hence, the power of such a thermal machine is zero.This leaves open the question of efficiency at finite power.Although this issue was first addressed by Chambadal [2]and Novikov [3], it is often associated with the later workof Curzon and Ahlborn (CA) [4]. Using an approximateanalysis of a finite-time Carnot cycle, they observed thatthe power goes through a maximum, and that the corre-sponding efficiency at maximum power is given by theappealing expressionCA ¼ 1ﬃﬃﬃﬃﬃTcThs: (4)Unfortunately, the CA efficiency turns out to be neither anexact nor a universal result, and it is neither an upper norPRL 105, 150603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending8 OCTOBER 20100031-9007=10=105(15)=150603(4) 150603-1  2010 The American Physical Societya lower bound [5]. Yet it describes the efficiency of actualthermal plants very well [1,4,6], and is reasonably close tothe efficiency at maximum power for several model sys-tems [7–18]. How does this agreement come about?As a first explanation, we note that the underlying time-reversibility of the laws of physics under some conditionsimplies universal properties for the efficiency at maximumpower. More precisely, let us consider the expansion of theefficiency at maximum power in terms of the Carnotefficiency C. For CA efficiency, one has CA ¼1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 Cp ¼ C=2þ 2C=8þ    . It was provenfrom the symmetry of the Onsager coefficients that thecoefficient 1=2 is actually an upper bound for the linearresponse at maximum power, and that the bound is reachedfor strong coupling between the heat-performing and thework-performing fluxes [19]. Using the equivalent ofOnsager symmetry at the level of nonlinear response,one can show that the coefficient of 2C is also universal,i.e., equal to 1=8, for strongly coupled fluxes possessing inaddition a left-right symmetry [20].In this Letter, we further clarify the special status of CAefficiency: it turns out to be an exact property for Carnotmachines operating under conditions of low, symmetricdissipation. The argument is very simple, as can be ex-pected from its claim of generality. Our starting pointis a Carnot engine which operates under reversible con-ditions when the durations of the cycles become very large,i.e., when the system always remains infinitesimally closeto equilibrium all along the cycle. While in contact with thehot reservoir, the work-performing auxiliary system ab-sorbs an amount of heat Qh, resulting in a system entropychange S ¼ Qh=Th. During the heat exchange with thecold reservoir, the entropy of the system returns to itsoriginal value, decreasing by an amount S ¼ Qc=Tc.From the equality Qh=Th ¼ Qc=Tc we recover Carnotefficiency  ¼ 1þQc=Qh ¼ 1 Tc=Th. We next con-sider finite-time cycles which move the engine awayfrom the reversible regime.Let c (h) be the time durations during which thesystem is in contact with the cold (hot) reservoir along acycle. In the weak dissipation regime, the system relaxa-tion is assumed to be fast compared to h and c. Theentropy production per cycle along the cold (hot) part ofthe cycle is expected to behave as c=c (h=h) since thereversible regime is approached in the limits h ! 1 andc ! 1 (for a further comment on this assumption, see[21]). As a result, the amount of heat per cycle entering thesystem from the cold (hot) reservoir will beQc ¼ TcS ccþ . . .;Qh ¼ ThShhþ . . .:(5)Note that we did not specify the details of the procedureby which we deviate from the reversible scenario. Thisinformation is contained in the coefficients c and h.They express how dissipation increases as one moves awayfrom the reversible limit. We also do not need to assumethat the temperature difference between Tc and Th is small;hence the expansion is not limited to the linear responseregime.We now consider the power generated during this Carnotcycle. Using (5), we getP ¼ Wh þ c ¼Qh þQch þ c¼ ðTh  TcÞS Thh=h  Tcc=ch þ c : (6)The maximum power is found by setting the derivatives ofP with respect to h and c equal to zero. We find a uniquephysically acceptable solution ath ¼ 2 ThhðTh  TcÞS1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃTccThhs c¼ 2 TccðTh  TcÞS1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃThhTccs : (7)Using (5) with (7) in the efficiency (1) leads to the mainresult of this paper, namely, the following expression forthe efficiency at maximum power: ¼C1þﬃﬃﬃﬃﬃﬃﬃﬃTccThhq 1þﬃﬃﬃﬃﬃﬃﬃﬃTccThhq 2 þ TcTh1 ch : (8)This result was previously obtained by Schmiedl andSeifert using a Fokker-Plank formulation of stochasticthermodynamics [10]. We present it here in a broadercontext by arguing that the expansion (5) is generic inthe weak dissipation limit if a reversible long time limitexists. For symmetric dissipation,h ¼ c, we recover theCurzon-Ahlborn efficiency: ¼ CA ¼ 1ﬃﬃﬃﬃﬃTcThs¼ 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 Cp : (9)We also note that (8) can be expanded in C as ¼ C2þ 2C4þ 4 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃc=hpþOð3CÞ (10)The coefficient of the second order term lies between 0 and1=4 and for symmetric dissipation we recover the 1=8, asdiscussed in [20]. Symmetric dissipation for time-dependent cycles is thus similar to the left-right symmetryon the fluxes [see Eq. (20) of Ref. [20]] which is required torecover the universal value of the quadratic coefficient forsteady-state problems.We now turn to the main focus of the result (8). In thelimits c=h ! 0 and c=h ! 1, the efficiency atPRL 105, 150603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending8 OCTOBER 2010150603-2maximum power converges to the upper bound þ ¼C=ð2 CÞ and to the lower bound  ¼ C=2,respectively:C2     þ  C2 C : (11)In Fig. 1 we plot the efficiency (8) as a function of Ccomparing the CA result with the upper and lower bounds(11). We note that these bounds were previously derived byassuming a specific form of heat transfers in [25]. Theupper bound þ, which is reached in the completelyasymmetric limit c=h ! 0, is particularly interesting.It coincides with a reported universal upper bound that wasderived in [23] [cf. Eq. (16)] using a very different ap-proach. It also agrees with the upper bound obtained byoptimizing with respect to the temperature of the hotreservoir [24]. Finally, it also arises in a model for theFeynman ratchet [26] [cf. Eq. (25)].In order to identify the regime of operation of a particu-lar engine other than via the ratio of coefficientsc=h, weevaluate the ratio of the contact times at maximum power:ch¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃTccThhs: (12)We conclude that symmetric dissipation corresponds to thecase when this ratio is equal to the square root of the ratioof the temperatures,ch¼ﬃﬃﬃﬃﬃTcThs; (13)whereas maximum and minimum efficiency are reachedfor the highly asymmetric casesch! 0 and ch! 1: (14)In conclusion, we have presented a simple and generalargument for estimating efficiency of a thermal engine atmaximum power. The main bonuses of this analysis are thederivation of the Curzon-Ahlborn efficiency in the case ofsymmetric dissipation, and the prediction of an upper andlower bound reached in the limits of extremely asymmetricdissipation. While actual plants usually operate understeady-state conditions rather than as a Carnot cycle, and0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0CCAC 2C 2 CFIG. 1 (color online). Efficiency at maximum power as afunction of C. The upper and lower bounds of the efficiencygiven by Eq. (11) are denoted by a black solid line and a (blue)dotted line, respectively. The Curzon-Ahlborn efficiency is the(red) dashed line. The dots represent the observed efficiencies ofthe various thermal power plants reported in Table I. Observedefficiencies above and below the bounds could result from powerplants not operating at maximum power.TABLE I. Theoretical bounds and observed efficiency obs of thermal plants.Plant ThðKÞ TcðKÞ C  þ obsDoel 4 (Nuclear, Belgium) [6] 566 283 0.5 0.25 0.33 0.35Almaraz II (Nuclear, Spain) [6] 600 290 0.52 0.26 0.35 0.34Sizewell B (Nuclear, UK) [6] 581 288 0.5 0.25 0.34 0.36Cofrentes (Nuclear, Spain) [6] 562 289 0.49 0.24 0.32 0.34Heysham (Nuclear, UK) [6] 727 288 0.60 0.30 0.43 0.40West Thurrock (Coal, UK) [1] 838 298 0.64 0.32 0.48 0.36CANDU (Nuclear, Canada) [1] 573 298 0.48 0.24 0.32 0.30Larderello (Geothermal, Italy)[1] 523 353 0.32 0.16 0.19 0.16Calder Hall (Nuclear, UK) [6] 583 298 0.49 0.24 0.32 0.19(Steam/Mercury,USA) [6] 783 298 0.62 0.31 0.45 0.34(Steam, UK) [6] 698 298 0.57 0.29 0.40 0.28(Gas Turbine, Switzerland) [6] 963 298 0.69 0.35 0.53 0.32(Gas Turbine, France) [6] 953 298 0.69 0.34 0.52 0.34PRL 105, 150603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending8 OCTOBER 2010150603-3while the assumptions of low dissipation and maximumpower may not hold, one feels compelled to compare theupper and lower bounds with observed efficiencies, as isdone in Table I and in Fig. 1.M. E. is supported by the Belgian Federal Government(IAP project ‘‘NOSY’’) and by the European UnionSeventh Framework Programme (FP7/2007-2013) undergrant agreement 256251. This research is supported inpart by the NSF under Grant No. PHY-0855471.[1] H. B. Callen, Thermodynamics and an Introduction toThermostatistics (Wiley, New York, 1985), 2nd ed..[2] P. Chambadal, Les Centrales Nuclaires (Armand Colin,Paris, 1957).[3] I. I. Novikov, At. Energy (N.Y.) 3, 1269 (1957); J. Nucl.Energy 7, 125 (1958).[4] F. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975).[5] K. H. Hoffmann, J. Burzler, and S. Schubert, J. Non-Equilib. Thermodyn. 22, 311 (1997); R. S. Berry, V. A.Kazakov, S. Sieniutycz, Z. Szwast, and A.M. Tsvilin,Thermodynamic Optimization of Finite-Time Processes(Wiley, Chichester, 2000); P. Salamon, J. D. Nulton, G.Siragusa, T. R. Andersen, and A. Limon, Energy 26, 307(2001).[6] A. Bejan, Advanced Engineering Thermodynamics(Wiley, New York, 1997), p. 377.[7] A. De Vos, Am. J. Phys. 53, 570 (1985).[8] A. Bejan, J. Appl. Phys. 79, 1191 (1996).[9] B. JimenezdeCisneros and A. C. Hernandez, Phys. Rev.Lett. 98, 130602 (2007).[10] T. Schmiedl and U. Seifert, Europhys. Lett. 81, 20003(2008).[11] H. Then and A. Engel, Phys. Rev. E 77, 041105 (2008).[12] Y. Izumida and K. Okuda, Europhys. Lett. 83, 60003(2008); Phys. Rev. E 80, 021121 (2009); Prog. Theor.Phys. Suppl. 178, 163 (2009); arXiv:1006.2589.[13] Z. C. Tu, J. Phys. A 41, 312003 (2008).[14] A. E. Allahverdyan, R. S. Johal, and G. Mahler, Phys. Rev.E 77, 041118 (2008).[15] M. Esposito, K. Lindenberg, and C. Van den Broeck,Europhys. Lett. 85, 60010 (2009).[16] B. Rutten, M. Esposito, and B. Cleuren, Phys. Rev. B 80,235122 (2009).[17] M. Esposito, R. Kawai, K. Lindenberg, and C. Van denBroeck, Phys. Rev. E 81, 041106 (2010).[18] Y. Zhou and D. Segal, Phys. Rev. E 82, 011120(2010).[19] C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005);Adv. Chem. Phys. 135, 189 (2007).[20] M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys.Rev. Lett. 102, 130602 (2009).[21] In general, reconnecting the system to one of the reser-voirs will induce dissipation because the system is not inequilibrium with this reservoir. This leads to an additionalcontribution to the entropy production in (5), which canhave a complicated time dependence and does not vanishin the very slow cycle limit. This unavoidably results in adecrease of the efficiency at maximum power. By consid-ering cycles with a reversible limit, we assume this dis-sipation to be negligible or absent. This assumption isreasonable in large systems but might require a largenumber of external control parameters for small systems(see, for example, [22]). One can then use a calculationsimilar to the adiabatic theory in quantum mechanics toshow that the entropy production related to the change ofthe system during the times  while in contact with thereservoir behaves as 1=.[22] K. Sato, K. Sekimoto, T. Hondou, and F. Takagi, Phys.Rev. E 66, 016119 (2002).[23] B. Gaveau, M. Moreau, and L. S. Schulman, Phys. Rev.Lett. 105, 060601 (2010).[24] A. Bejan and H.M. Paynter, Solved Problems inThermodynamics (MIT, Cambridge, MA, 1976), problemVII-D.[25] L. Chen and Z. Yan, J. Chem. Phys. 90, 3740 (1989)[26] S. Velasco, J.M.M. Roco, A. Medina, and A. CalvoHerna`ndez, J. Phys. D 34, 1000 (2001).PRL 105, 150603 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending8 OCTOBER 2010150603-4

http://orbilu.uni.lu/bitstream/10993/11949/1/10EspoKawaiPRL.pdf

Efficiency at maximum power of low dissipation Carnot engines

Efficiency at maximum power of low dissipation Carnot engines

Abstract

Similar works

Full text

Available Versions

Crossref

DI-fusion

Open Repository and Bibliography - Luxembourg