Bounds of efficiency at maximum power for normal-, sub- and super-dissipative Carnot-like heat engines ∗ Yang Wang and Z. C. Tu Department of Physics, Beijing Normal University, Beijing 100875, China (Dated: January 5, 2012) The Carnot-like heat engines are classified into threetypes(normal-, sub- and super-dissipative) according to relations between the minimum irreversible entropy production in the “isothermal” processes and the time for completing those processes. The efficiencies at maximum power of 2 normal-, sub- and super-dissipative Carnot-like heat engines are proved to be bounded between 1 0 ηC/2andηC/(2−ηC),ηC/2andηC,0andηC/(2−ηC),respectively. Theseboundsarealsoshared 2 by linear, sub- and super-linear irreversible Carnot-like engines [Tu and Wang, arXiv:1110.6493] although thedissipative engines and theirreversible ones are inequivalent to each other. n a PACSnumbers: 05.70.Ln J 4 I. INTRODUCTION tropy production in the “isothermal” (The quote marks ] on the word “isothermal” merely indicate the working h c The issue of efficiency at the maximum power (EMP) substancetobeincontactwithareservoir)processwith- e hasbeendrawnmuchattentionsincethepioneerachieve- outintroducingtheeffectivetemperatureofworkingsub- m ments make by Yvon[1], Novilov[2], Chambadal[3], Cur- stance [27], which inspired us that the upper and lower - zon and Ahlborn[4]. The emerging theoretical advances bounds of EMP might be straightly derived from the re- at inthisfield[5–28]improveourunderstandingoftheissue lationbetweenthe irreversibleentropyproductioninthe t of EMP for heat engines and irreversible thermodynam- “isothermal” process and the time for completing that s . ics. Recently, Esposito et al. investigated the Carnot- process. Here, we classify the Carnot-like engines into t a like engines working in the low-dissipation region [27] threetypes(normal-,sub-andsuper-dissipative)accord- m and obtained the lower bound η− ≡ ηC/2 and the up- ing to the relations between the minimum irreversible - per bound η ≡ η /(2−η ) of EMP for this kind of entropy production in the finite-time “isothermal” pro- + C C d engines, where η is the Carnot efficiency. In addition, cesses and the time for completing those processes. The n GaveauandhiscCoworkersproposedaconceptofsustain- EMPsofnormal-,sub- andsuper-dissipativeCarnot-like o c able efficiency and proved that it has the upper bound heatenginesareprovedtobeboundedbetweenηC/2and [ 1/2,basedonwhichthey alsoobtainedthe upper bound ηC/(2−ηC), ηC/2 and ηC, 0 and ηC/(2−ηC), respec- η = η /(2−η ) for the efficiency of Carnot-like en- tively. 1 + C C gines at maximum power [28]. Seifert argued that the v 8 upperbound1/2forthesustainableefficiencyholdsonly 4 in the linear nonequilibrium region [29]. Esposito et al. II. THEORETICAL MODEL 8 [30] provided a nice example of single level quantum dot 0 where the upper bound can vary from 1/2 to 1 with in- The engines perform the following Carnot-like cycle . 1 creasingthe thermodynamic force, whichto some extent which is similar to the conventionproposed by Schmiedl 0 supports Seifert’s argument. Therefore the upper bound and Seifert [12]. 2 ηC/(2−ηC)mightnotexistforEMPofCarnot-likeheat “Isothermal” expansion. The working substance is in 1 engines arbitrarily far from equilibrium [31]. In recent contact with a hot reservoir at temperature T and the : 1 v work, we classified irreversible Carnot-like heat engines constraint on the system is loosened according to some i into three types (linear, sublinear and superlinear), and external controlled parameter λ (τ) during the time in- X 1 derived the corresponding bounds to be between ηC/2 terval 0 < τ < t1 where τ is the time variable. It is ar and ηC/(2−ηC), ηC/2 and ηC, 0 and ηC/(2−ηC), re- in the sense of loosening the constraint that this step is spectively[32]. Inparticular,itisfoundthattheEMPof called expansion process. A certain amount of heat Q 1 sublinear irreversible heat engines can reach the Carnot isabsorbedfromthe hotreservoir. Thenthe variationof efficiency [32]. entropy can be expressed as Itisnecessarytointroducetheconceptofeffectivetem- perature of working substance for most of models men- ∆S =Q /T +∆Sir, (1) 1 1 1 1 tionedabove. However,thedefinitionofeffectivetemper- ature is debatable in some cases. Esposito et al. started where ∆Sir ≥0 is the irreversible entropy production. 1 from the time-dependent behavior of the irreversible en- Adiabatic expansion. The adiabatic expansion is ide- alizedastheworkingsubstancesuddenly decouplesfrom the hot reservoir and then comes into contact with the coldreservoirinstantly attime τ =t . During this tran- 1 ∗ [email protected] sition, the constraint on the system is loosened further. 2 Thereisnoheatexchangeandentropyproductioninthis process, i.e., Q =0 and ∆S =0. 2 2 “Isothermal” compression. The working substance is in contact with a cold reservoir at temperature T and 3 the constraint on the system is enhanced according to the external controlled parameter λ (τ) during the time 3 interval t < τ < t +t . It is in the sense of enhancing 1 1 3 the constraint that this step is called compression pro- cess. A certainamountofheatQ is releasedto the cold 3 reservoir. Thevariationofentropyinthisprocesscanbe expressed as ∆S =−Q /T +∆Sir, (2) 3 3 3 3 where ∆S3ir ≥0 is the irreversible entropy production. FIG. 1. Schematic diagram of three types of dissipative en- Adiabatic compression. Similar to the adiabatic ex- gines. pansion,the workingsubstancesuddenly decouples from the cold reservoir and then comes into contact with the hot reservoir instantaneously at time τ = t +t . The 1 3 constraint on the system is enhanced further. There is in our previous work [32]. The first one is called normal noheatexchangeandentropyproductioninthisprocess, dissipative type which is representedby the straightline i.e., Q =0 and ∆S =0. in Fig. 1. This type of Carnot-like engines is also called 4 4 Having undergone this Carnot-like cycle, the system the low-dissipation engines by Esposito et al. [27]. The comes back to its initial state again. Thus there are no second one is called sub-dissipative type which is repre- net energy change and variation of entropy in the whole sented by the convex curve in Fig.1. The third one is cycle. Thenwehave∆S =−∆S andandthe network called super-dissipative type which is represented by the 3 1 output W =Q −Q . The power can be expressed as concave curve in Fig.1. The behavior of three kinds of 1 3 characteristics can be mathematically expressed as Q −Q (T −T )∆S −T ∆Sir−T ∆Sir P = 1 3 = 1 3 1 1 1 3 3 . ′ t t +t xL =L, normal-dissipative tot 1 3  (3) xL′ >L, sub-dissipative (5) According to the equation above, maximizing the power ′ xL <L, super-dissipative means minimizing the irreversible entropy production  with respect to the protocols λ (τ) and λ (τ) for given 1 1 where L, x, and L′ represent L (or L ), x (or x ) and time intervals t and t first, then Eq. (3) can be trans- 1 3 1 3 1 3 dL /dx (or dL /dx ), respectively. formed into 1 1 3 3 (T −T )∆S −T L −T L 1 3 1 1 1 3 3 P = , (4) t +t IV. OPTIMIZATION 1 3 where L and L represent min{∆Sir} and min{∆Sir} 1 3 1 1 The heat absorbed or released by the engines can be for given time intervals t and t , respectively. In fact, 1 3 expressed as they reflecttowhatextenttheenginesdepartfromequi- librium for given time intervals t and t . 1 3 Q =T ∆S −T L , (6) 1 1 1 1 1 and III. CLASSIFICATION FOR THREE TYPES OF DISSIPATIVE HEAT ENGINES Q =T ∆S +T L . (7) 3 3 1 3 3 Intuitionally, the irreversible entropy production de- Given the two equations above, we can obtain the effi- creases as the increase of time for completing the ciency “isothermal”processes,thusL andL canbe expressed 1 3 asmonotoneincreasingfunctionwithrespectto1/t and Q η ∆S −L −(1−η )L 1 3 C 1 1 C 3 η =1− = , (8) 1/t3, respectively. If we introduce a transformation of Q1 ∆S1−L1 variables x = 1/t (i = 1,3), the minimum irreversible i i entropy production in each “isothermal” process can be and the power expressed as L =L (x ) (i =1,3). We can also defined i i i threekindsoftypicalcharacteristicsaccordingtothebe- [(T1−T3)∆S1−(T1L1+T3L3)]x1x3 P = . (9) havior of Li = Li(xi), which is similar to the analysis (x1+x3) 3 Maximizing the power with respect to x and x , we C. Super-dissipative engines 1 3 can obtain ′ The super-dissipative engines satisfy x L < L and [(T1−T3)∆S1−(T1L1+T3L3)]x3 =T1x1(x1+x3)(L1′10,) x3L′3 < L3 which imply x1L′1 +(1−ηC)1x31L′3 <1L1 + (1−η )L . Giventhat L +(1−η )L <L +L , then C 3 1 C 3 1 3 and we finally derive the upper bound of EMP to be η = + [(T −T )∆S −(T L +T L )]x =T x (x +x )L′. ηC/(2−ηC)forthesuper-dissipativeengines. The above 1 3 1 1 1 3 3 1 3 3 1 3 3 inequalitiesgivenoconfinementonthelowerbound,thus (11) we may take η− =0 as a conservative estimate. Dividing Eqs. (10) and (11), we derive T x2L′ =T x2L′. (12) 1 1 1 3 3 3 D. Examples for three types of engines Adding Eqs. (10) and (11) with the consideration of Eq. (8), we find that the EMP satisfies For examples, we consider the relation of power-law profile, L =Γ xn,(i=1,3) where Γ >0 and n>0 are η i i i i ∗ C given parameters. It is easy to see that a heat engine is η = , (13) 1+ (1−ηC)(L1+L3) ofnormal-,sub- or super-dissipativetype if n=1, n>1 x1L′1+(1−ηC)x3L′3 or0<n<1,respectively. Substitutingthisrelationinto which is the key equation in our paper. Eqs.(12)and(13),wecanexplicitlyderivetheexpression of EMP as η ∗ C η = , (14) V. BOUNDS OF EMP FOR THREE TYPES OF 1+ 1 − ηC DISSIPATIVE HEAT ENGINES n n[1+(T3Γ3/T1Γ1)1/(n+1)] from which we can find In this section, we will discuss the bounds of EMP for η η three types of Carnot-like dissipative engines in terms of C ∗ C <η < . (15) the characteristics of the relation between the minimum 1+1/n 1+1/n−ηC/n entropy production in each “isothermal” processes and the time for completing those processes. For normal-dissipative engines, n = 1, Eq. (15) implies ∗ that η /2 < η < η /(2−η ). For sub-dissipative en- C C C gines,n>1,Eq.(15)impliesthatη /2<nη /(n+1)< C C ∗ η < nη /(n+1−η ) < η where the upper bound A. Normal-dissipative engines C C C η can be reached for sufficiently large n while the C lower bound can be reached for n → 1 . For super- The minimum entropy production in the “isothermal” + dissipative engines, 0 < n < 1, Eq. (15) implies that processes and the time for completing those processes of ∗ 0<nη /(n+1)<η <nη /(n+1−η )<η /(2−η ) normaldissipativeenginessatisfyx L′ =L andx L′ = C C C C C ′ 1′ 1 1 3 3 wherethelowerbound0canbereachedforsmallenough L which imply x L +(1−η )x L =L +(1−η )L . 3 1 1 C 3 3 1 C 3 n while the upper bound can be reached for n→1−. Considering 0< η <1, we derive (1−η )(L +L ) < C C 1 3 L +(1−η )L < L +L . Considering Eq. (13), we 1 C 3 1 3 can derive the EMP of normal-dissipative engines to be VI. CONCLUSION boundedbetweenη− ≡ηC/2andη+ ≡ηC/(2−ηC)which arethesameastheboundsobtainedbyEspositoandhis coworkers[27]. A major difference is that here we derive The heat engines are classified into three dissipative theboundsdirectlyfromEq.(13)withoutcalculatingthe types according to characteristics of the relations be- explicit expression of EMP. tween the minimum irreversible entropy production in the “isothermal” processes and the time for completing those processes. The bounds of EMP for three types of B. Sub-dissipative engines dissipative Carnot-like engines can be summarized as ∗ x3TL′3he>sLu3b-wdhisiscihpaimtivpelyexn1gLin′1e+s (s1a−tisηfCy)xx31LL′3′1>>L1L+1 (a1n−d ηηCC//22<<ηη∗ <<ηηCC,/(2−ηC), snuobrm−adl−issdipisastiipvaetive ∗ η )L . GiventhatL +(1−η )L >(1−η )(L +L ), 0<η <η /(2−η ), super−dissipative C 3 1 C 3 C 1 3 C C then we finally derive the lower bound of EMP to be  η− =ηC/2 for the sub-dissipativeengines fromEq. (13). which display certain universality for each types of dis- The above inequalities give no confinement on the up- sipative Carnot-like engines. It is interesting that the per bound, thus we may take η = η as a reasonable bounds of EMPs for normal-, sub- and super-dissipative + C estimate. Carnot-like engines correspond respectively to those for 4 linear, sub- and super-linear irreversible Carnot-like en- bounds. gines discussed in our previous work [32]. However, a simpleconsiderationimpliesthatthesetwokindsofclas- sifications aredifferent fromeachother [24]. It is still an VII. ACKNOWLEDGEMENT open question to understand why they share the same The authors are grateful to the financial support from Nature Science Foundation of China (Grant NO.11075015). [1] J. Yvon, Proceedings of the International Conference [18] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den on Peaceful Uses of Atomic Energy, (United Nations, Broeck, Phys. Rev.E 81, 041106 (2010). Geneva,1955), p387. [19] Y. Izumida and K. Okuda, Phys. Rev. E 80, 021121 [2] I.Novikov,Atommaya Energiya 11,409(1957) (2009). [3] I.Chambadal, Les Centrales Nuclearies (Armand Colin, [20] X. Wang, J. Phys. A:Math. Theor. 43, 425003 (2010). Paris, 1957) [21] N. S´anchez-Salas, L. L´opez-Palacios, S. Velasco, and A. [4] F.L.CurzonandB.Ahlborn,Am.J.Phys.43,22(1975). C. Hern´andez,Phys. Rev.E 82, 051101 (2010). [5] P. Salamon, A. Nitzan, B. Andresen, and R. S. Berry, [22] S. Abe, Phys.Rev.E 83, 041117 (2011). Phys.Rev.A 21, 2115 (1980). [23] L. Chen, Z. Ding, and F. Sun, J. Non-Equil. Thermody. [6] L. Chen and Z. Yan,J. Chem. Phys. 90, 3740 (1989). 36, 155 (2011). [7] J. Chen, J. Phys. D:Appl. Phys.27, 1144 (1994). [24] Y. Wang and Z.C. Tu,arXiv:1108.2873 (2011). [8] A.Bejan, J. Appl.Phys.79, 1191 (1996). [25] J. H.Wang and J. Z.He, arXiv:1111.5077 (2011). [9] L. Chen, C. Wu, and F. Sun, J. Non-Equil. Thermody. [26] Y. Izumidaand K.Okuda,EPL 97, 10004 (2012). 24, 327 (1999). [27] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den [10] C. Van den Broeck, Phys.Rev.Lett. 95, 190602 (2005). Broeck, Phys. Rev.Lett. 105, 150603 (2010). [11] B.J.deCisnerosandA.C.Hern´andez,Phys.Rev.Lett. [28] B. Gaveau,M. Moreau and L. S. Schulman, Phys. Rev. 98, 130602 (2007). Lett. 105, 060601 (2010). [12] T. Schmiedland U.Seifert, EPL 81, 20003 (2008). [29] U. Seifert, Phys.Rev.Lett. 106, 020601 (2011). [13] T. Schmiedland U.Seifert, EPL 83, 30005 (2008). [30] M. Esposito, N.Kumar,K.Lindenberg,and C.Van den [14] Z. C. Tu, J. Phys. A:Math. Theor. 41, 312003 (2008). Broeck, arXiv:1201.0669 (2012). [15] B.LinandJ.Chen,J.Phys.A:Math.Theor.42,075006 [31] U. Serfert, privite communications, obergurgl, Austria (2009). (2011) [16] M. Esposito, K. Lindenberg, and C. Van den Broeck, [32] Y. Wang and Z.C. Tu,arXiv:1110.6493 (2011). EPL 85, 60010 (2009). [17] M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys.Rev.Lett. 102, 130602 (2009).