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Journal of Non-Equilibrium Thermodynamics

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Ed. by Michaelides, Efstathios E. / Rubi, J. Miguel

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Volume 42, Issue 3 (Jun 2017)


Universal Optimization Efficiency for Nonlinear Irreversible Heat Engines

Yanchao Zhang / Juncheng Guo
  • College of Physics and Information Engineering, Fuzhou University, Fuzhou 350116, People’s Republic of China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Guoxing Lin / Jincan Chen
Published Online: 2017-03-08 | DOI: https://doi.org/10.1515/jnet-2016-0065


We introduce a multi-parameter combined objective function of heat engines under the strong coupling and symmetry condition and derive the universal expression of the optimization efficiency. The results obtained show that the optimization efficiency derived from the multi-parameter combined objective function include a variety of optimization efficiencies, such as the efficiency at the maximum power, efficiency at the maximum efficiency-power state, efficiency at the maximum ecological or unified trade-off function, and Carnot efficiency. It is further explained that these results are also suitable for the endoreversible cycle model of the Carnot heat engines operating between two heat reservoirs.

Keywords: irreversible thermodynamics; nonlinear; master equation; heat engine; universal optimization efficiency


  • [1]

    A. Bejan and E. Mamut, Thermodynamic Optimization of Complex Energy Systems, Kluwer Academic, Dordrecht, 1999.Google Scholar

  • [2]

    R. S. Berry, V. A. Kazakov, S. Sieniutycz, Z. Szwast and A. M. Tsirlin, Thermodynamics Optimization of Finite-Time Processes, Wiley, Chichester, 2000.Google Scholar

  • [3]

    L. Chen and F. Sun, Advances in Finite-Time Thermodynamics: Analysis and Optimization, Nova Science, New York, 2004.Google Scholar

  • [4]

    F. Angulo-Brown, M. Santillán and E. Calleja-Quevedo, Thermodynamic optimality in some biochemical reactions, Nuovo Cimento Soc. Ital. Fis. D 17 (1995), 87–90.CrossrefGoogle Scholar

  • [5]

    K. H. Hoffmann, J. Burzler, A. Fischer, M. Schaller and S. Schubert, Optimal process paths for endoreversible systems, J. Non-Equilib. Thermodyn. 28 (2003), 233–268.Google Scholar

  • [6]

    N. Nakpathomkun, H. Q. Xu and H. Linke, Thermoelectric efficiency at maximum power in low-dimensional systems, Phys. Rev. B 82 (2010), 235428.CrossrefGoogle Scholar

  • [7]

    S. A. Amelkin, B. Andresen, J. M. Burzler, K. H. Hoffmann and A. M. Tsirlin, Thermo-mechanical systems with several heat reservoirs: Maximum power processes, J. Non-Equilib. Thermodyn. 30 (2005), 67–80.Google Scholar

  • [8]

    M. H. Rubin, Optimal configuration of a class of irreversible heat engines. I, Phys. Rev. A 19 (1979), 1272–1276.CrossrefGoogle Scholar

  • [9]

    B. Andresen, P. Salamon and R. S. Berry, Thermodynamics in finite time, Phys. Today 37 (1984), 62–70.CrossrefGoogle Scholar

  • [10]

    T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett. 98 (2007), 108301.CrossrefGoogle Scholar

  • [11]

    F. Curzon and B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43 (1975), 22–24.CrossrefGoogle Scholar

  • [12]

    F. Angulo-Brown, An ecological optimization criterion for finite-time heat engines, J. Appl. Phys. 69 (1991), 7465–7469.CrossrefGoogle Scholar

  • [13]

    B. Sahin, A. Kodal and H. Yavuz, Efficiency of a Joule-Brayton engine at maximum power density, J. Phys. D: Appl. Phys. 28 (1995), 1309–1313.CrossrefGoogle Scholar

  • [14]

    A. Bejan, Entropy generation minimization: The new thermodynamics of finite-size devices and finite-time processes, J. Appl. Phys. 79 (1996), 1191–1218.CrossrefGoogle Scholar

  • [15]

    A. Calvo Hernández, A. Medina, J. M. M. Roco, J. A. White and S. Velasco, Unified optimization criterion for energy converters, Phys. Rev. E 63 (2001), 037102.Google Scholar

  • [16]

    A. Durmayaz, O. S. Sogut, B. Sahin and H. Yavuz, Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics, Prog. Energy Combust. Sci. 30 (2004), 175–217.CrossrefGoogle Scholar

  • [17]

    Y. Izumida and K. Okuda, Work output and efficiency at maximum power of linear irreversible heat engines operating with a finite-sized heat source, Phys. Rev. Lett. 112 (2014), 180603.CrossrefGoogle Scholar

  • [18]

    A. E. Allahverdyan, K. V. Hovhannisyan, A. V. Melkikh and S. G. Gevorkian, Carnot cycle at finite power: Attainability of maximal efficiency, Phys. Rev. Lett. 111 (2013), 050601.CrossrefGoogle Scholar

  • [19]

    L. Chen, C. Wu and F. Sun, Finite time thermodynamic optimization or entropy generation minimization of energy systems, J. Non-Equilib. Thermodyn. 24 (1999), 327–359.Google Scholar

  • [20]

    J. Yvon, The Saclay reactor: two years experience on heat transfer by means of a compressed gas, in: Proceedings of the International Conference on Peaceful Uses of Atomic Energy, Geneva, 1955.Google Scholar

  • [21]

    I. I. Novikov, The efficiency of atomic power stations (a review), J. Nuclear Energy II 7 (1958), 125–128.Google Scholar

  • [22]

    A. De Vos, Efficiency of some heat engines at maximum-power conditions, Am. J. Phys. 53 (1985), 570–573.CrossrefGoogle Scholar

  • [23]

    L. Chen and Z. Yan, The effect of heat-transfer law on performance of a two-heat-source endoreversible cycle, J. Chem. Phys. 90 (1989), 3740–3743.Google Scholar

  • [24]

    J. Chen, The maximum power output and maximum efficiency of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 27 (1994), 1144–1149.CrossrefGoogle Scholar

  • [25]

    S. Velasco, J. M. M. Poco, A. Medina and A. C. Hernández, Feynman’s ratchet optimization: Maximum power and maximum efficiency regimes, J. Phys. D: Appl. Phys. 34 (2001), 1000–1006.CrossrefGoogle Scholar

  • [26]

    C. Van Den Broeck, Thermodynamic efficiency at maximum power, Phys. Rev. Lett. 95 (2005), 190602.CrossrefGoogle Scholar

  • [27]

    M. Esposito, K. Lindenberg and C. Van Den Broeck, Universality of efficiency at maximum power, Phys. Rev. Lett. 102 (2009), 130602.CrossrefGoogle Scholar

  • [28]

    J. Guo, J. Wang, Y. Wang and J. Chen, Universal efficiency bounds of weak-dissipative thermodynamic cycles at the maximum power output, Phys. Rev. E 87 (2013), 012133.Google Scholar

  • [29]

    L. Chen, Z. Ding and F. Sun, Optimum performance analysis of Feynman’s engine as cold and hot ratchets, J. Non-Equilib. Thermodyn. 36 (2011), 155–177.Google Scholar

  • [30]

    J. Wang, J. He and Z. Wu, Efficiency at maximum power output of quantum heat engines under finite-time operation, Phys. Rev. E 85 (2012), 031145.Google Scholar

  • [31]

    J. Guo, J. Wang, Y. Wang and J. Chen, Efficiencies of two-level weak dissipation quantum Carnot engines at the maximum power output, J. Appl. Phys. 113 (2013), 143510.CrossrefGoogle Scholar

  • [32]

    R. Uzdin and R. Kosloff, Universal features in the efficiency at maximal work of hot quantum Otto engines, Europhys. Lett. 108 (2014), 40001.CrossrefGoogle Scholar

  • [33]

    U. Seifert, Efficiency of autonomous soft nanomachines at maximum power, Phys. Rev. Lett. 106 (2011), 020601.CrossrefGoogle Scholar

  • [34]

    N. Golubeva and A. Imparato, Efficiency at maximum power of interacting molecular machines, Phys. Rev. Lett. 109 (2012), 190602.CrossrefGoogle Scholar

  • [35]

    C. Van Den Broeck, N. Kumar and K. Lindenberg, Efficiency of isothermal molecular machines at maximum power, Phys. Rev. Lett. 108 (2012), 210602.CrossrefGoogle Scholar

  • [36]

    Y. Izumida and K. Okuda, Efficiency at maximum power of minimally nonlinear irreversible heat engines, Europhys. Lett. 97 (2012), 10004.CrossrefGoogle Scholar

  • [37]

    Y. Wang and Z. C. Tu, Bounds of efficiency at maximum power for linear, superlinear and sublinear irreversible Carnot-like heat engines, Europhys. Lett. 98 (2012), 40001.CrossrefGoogle Scholar

  • [38]

    Y. Wang and Z. C. Tu, Efficiency at maximum power output of linear irreversible Carnot-like heat engines, Phys. Rev. E 85 (2012), 011127.Google Scholar

  • [39]

    J. Wang and J. He, Efficiency at maximum power output of an irreversible Carnot-like cycle with internally dissipative friction, Phys. Rev. E 86 (2012), 051112.Google Scholar

  • [40]

    M. Moreau, B. Gaveau and L. S. Schulman, Efficiency of a thermodynamic motor at maximum power, Phys. Rev. E 85 (2012), 021129.Google Scholar

  • [41]

    H. Yan and H. Guo, Efficiency and its bounds for thermal engines at maximum power using Newton’s law of cooling, Phys. Rev. E 85 (2012), 011146.Google Scholar

  • [42]

    C. Van Den Broeck, Efficiency at maximum power in the low-dissipation limit, Europhys. Lett. 101 (2013), 10006.CrossrefGoogle Scholar

  • [43]

    Z. C. Tu, Efficiency at maximum power of Feynman’s ratchet as a heat engine, J. Phys. A: Math. Theor. 41 (2008), 312003.CrossrefGoogle Scholar

  • [44]

    T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochastic heat engines, Europhys. Lett. 81 (2008), 20003.CrossrefGoogle Scholar

  • [45]

    M. Esposito, K. Lindenberg and C. Van Den Broeck, Thermoelectric efficiency at maximum power in a quantum dot, Europhys. Lett. 85 (2009), 60010.CrossrefGoogle Scholar

  • [46]

    M. Esposito, R. Kawai, K. Lindenberg and C. Van Den Broeck, Quantum-dot Carnot engine at maximum power, Phys. Rev. E 81 (2010), 041106.Google Scholar

  • [47]

    M. Esposito, R. Kawai, K. Lindenberg and C. Van Den Broeck, Efficiency at maximum power of low-dissipation Carnot engines, Phys. Rev. Lett. 105 (2010), 150603.CrossrefGoogle Scholar

  • [48]

    C. Van Den Broeck and K. Lindenberg, Efficiency at maximum power for classical particle transport, Phys. Rev. E 86 (2012), 041144.Google Scholar

  • [49]

    O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, et al., Single-ion heat engine at maximum power, Phys. Rev. Lett. 109 (2012), 203006.CrossrefGoogle Scholar

  • [50]

    R. Wang, J. Wang, J. He and Y. Ma, Efficiency at maximum power of a heat engine working with a two-level atomic system, Phys. Rev. E 87 (2013), 042119.Google Scholar

  • [51]

    F. Wu, J. He, Y. Ma and J. Wang, Efficiency at maximum power of a quantum Otto cycle within finite-time or irreversible thermodynamics, Phys. Rev. E 90 (2014), 062134.Google Scholar

  • [52]

    Z. Yan, η and P of a Carnot engine at maximum ηλP, J. Xiamen Univ. 25 (1986), 279–286.Google Scholar

  • [53]

    Z. Yan and J. Chen, A generalized Rutgers formula derived from the theory of endoreversible cycles, Phys. Lett. A 217 (1996), 137–140.CrossrefGoogle Scholar

  • [54]

    T. Yilmaz, A new performance criterion for heat engines: Efficient power, J. Energy Inst. 79 (2006), 38–41.CrossrefGoogle Scholar

  • [55]

    T. Yilmaz and Y. Durmuşoǧlu, Efficient power analysis for an irreversible Carnot heat engine, Int. J. Energ. Res. 32 (2008), 623–628.CrossrefGoogle Scholar

  • [56]

    G. Maheshwari, S. Chaudhary and S. K. Somani, Performance analysis of a generalized radiative heat engine based on new maximum efficient power approach, Inter. J. Low-Carbon Technol. 4 (2009), 9–15.Google Scholar

  • [57]

    G. Maheshwari, S. Chaudhary and S. K. Somani, Performance analysis of endoreversible combined Carnot cycles based on new maximum efficient power (MEP) approach, Inter. J. Low-Carbon Technol. 5 (2010), 1–6.Google Scholar

  • [58]

    C. Y. Cheng and C. K. Chen, The ecological optimization of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 30 (1997), 1602–1609.CrossrefGoogle Scholar

  • [59]

    L. Chen, J. Zhou, F. Sun and C. Wu, Ecological optimization for generalized irreversible Carnot engines, Appl. Energ. 77 (2004), 327–338.CrossrefGoogle Scholar

  • [60]

    Z. Yan and G. Lin, Ecological optimization criterion for an irreversible three-heat-source refrigerator, Appl. Energ. 66 (2000), 213–224.CrossrefGoogle Scholar

  • [61]

    R. Long and W. Liu, Ecological optimization for general heat engines, Physica A 434 (2015), 232–239.CrossrefGoogle Scholar

  • [62]

    N. Sánchez Salas and A. Calvo Hernández, Unified working regime of nonlinear systems rectifying thermal fluctuations, Europhys. Lett. 61 (2003), 287–293.CrossrefGoogle Scholar

  • [63]

    G. Özel, E. AçIkkalp, A. F. Savas and H. YamIk, Comparative analysis of thermoeconomic evaluation criteria for an actual heat engine, J. Non-Equilib. Thermodyn. 41 (2016), 225–235.Google Scholar

  • [64]

    B. Jiménez De Cisneros and A. Calvo Hernández, Collective working regimes for coupled heat engines, Phys. Rev. Lett. 98 (2007), 130602.CrossrefGoogle Scholar

  • [65]

    B. Jiménez De Cisneros and A. Calvo Hernández, Coupled heat devices in linear irreversible thermodynamics, Phys. Rev. E 77 (2008), 041127.Google Scholar

  • [66]

    Y. Zhang, C. Huang, G. Lin and J. Chen, Universality of efficiency at unified trade-off optimization, Phys. Rev. E 93 (2016), 032152.Google Scholar

  • [67]

    L. A. Arias-Hernandez, F. Angulo-Brown and R. T. Paez-Hernandez, First-order irreversible thermodynamic approach to a simple energy converter, Phys. Rev. E 77 (2008), 011123.Google Scholar

  • [68]

    R. Long and W. Liu, Unified trade-off optimization for general heat devices with nonisothermal processes, Phys. Rev. E 91 (2015), 042127.Google Scholar

  • [69]

    N. Sánchez-Salas, L. López-Palacios, S. Velasco and A. Calvo Hernández, Optimization criteria, bounds, and efficiencies of heat engines, Phys. Rev. E 82 (2010), 051101.Google Scholar

  • [70]

    C. De Tomas, J. M. M. White, A. Calvo Hernández, Y. Wang and Z. C. Tu, Low-dissipation heat devices: Unified trade-off optimization and bounds, Phys. Rev. E 87 (2013), 012105.Google Scholar

  • [71]

    R. Long, Z. Liu and W. Liu, Performance optimization of minimally nonlinear irreversible heat engines and refrigerators under a trade-off figure of merit, Phys. Rev. E 89 (2014), 062119.Google Scholar

  • [72]

    E. Bonet, M. M. Deshmukh and D. C. Ralph, Solving rate equations for electron tunneling via discrete quantum states, Phys. Rev. B 65 (2002), 045317.CrossrefGoogle Scholar

  • [73]

    G. Schaller, Open Quantum Systems Far from Equilibrium, Springer, New York, 2014.Google Scholar

  • [74]

    L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev. 37 (1931), 405–426.CrossrefGoogle Scholar

  • [75]

    S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984.Google Scholar

  • [76]

    T. E. Humphrey, R. Newbury, R. P. Taylor and H. Linke, Reversible quantum Brownian heat engines for electrons, Phys. Rev. Lett. 89 (2002), 116801.CrossrefGoogle Scholar

  • [77]

    Y. Wang, S. Su, B. Lin and J. Chen, Parametric design criteria of an irreversible vacuum thermionic generator, J. Appl. Phys. 114 (2013), 053502.CrossrefGoogle Scholar

  • [78]

    Y. Zhang, Y. Wang, C. Huang, G. Lin and J. Chen, Thermoelectric performance and optimization of three-terminal quantum dot nano-devices, Energy 95 (2016), 593–601.CrossrefGoogle Scholar

  • [79]

    J. Chan and Z. Yan, Unified description of endoreversible cycles, Phys. Rev. A 39 (1989), 4140–4147.CrossrefGoogle Scholar

  • [80]

    A. De Vos, Is a solar cell an endoreversible engine?, Sol. Cells 31 (1991), 181–196.CrossrefGoogle Scholar

  • [81]

    A. Bejan, Theory of heat transfer-irreversible power plants-II. The optimal allocation of heat exchange equipment, Int. J. Heat Mass Transfer 38 (1995), 433–444.CrossrefGoogle Scholar

  • [82]

    J. Chen, Z. Yan, G. Lin and B. Andresen, On the Curzon–Ahlborn efficiency and its connection with the efficiencies of real heat engines, Energy Convers. Mgmt 42 (2001), 173–181.CrossrefGoogle Scholar

About the article

Received: 2016-08-29

Accepted: 2017-02-15

Revised: 2016-12-21

Published Online: 2017-03-08

Published in Print: 2017-06-27

This work has been supported by the National Natural Science Foundation (No. 11405032), People’s Republic of China.

Citation Information: Journal of Non-Equilibrium Thermodynamics, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2016-0065.

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