The concepts of weighted reciprocal of temperature and weighted thermal flux are proposed for a heat engine operating between two heat baths and outputting mechanical work. With the aid of these two concepts, the generalized thermodynamic fluxes and forces can be expressed in a consistent way within the framework of irreversible thermodynamics. Then the efficiency at maximum power output for a heat engine, one of key topics in finite-time thermodynamics, is investigated on the basis of a generic model under the tight-coupling condition. The corresponding results have the same forms as those of low-dissipation heat engines [M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010)]. The mappings from two kinds of typical heat engines, such as the low-dissipation heat engine and the Feynman ratchet, into the present generic model are constructed. The universal efficiency at maximum power output up to the quadratic order is found to be valid for a heat engine coupled symmetrically and tightly with two baths. The concepts of weighted reciprocal of temperature and weighted thermal flux are also transplanted to the optimization of refrigerators.
We present a unified perspective on nonequilibrium heat engines by generalizing nonlinear irreversible thermodynamics. For tight-coupling heat engines, a generic constitutive relation for nonlinear response accurate up to the quadratic order is derived from the stalling condition and the symmetry argument. By applying this generic nonlinear constitutive relation to finite-time thermodynamics, we obtain the necessary and sufficient condition for the universality of efficiency at maximum power, which states that a tight-coupling heat engine takes the universal efficiency at maximum power up to the quadratic order if and only if either the engine symmetrically interacts with two heat reservoirs or the elementary thermal energy flowing through the engine matches the characteristic energy of the engine. Hence we solve the following paradox: On the one hand, the quadratic term in the universal efficiency at maximum power for tight-coupling heat engines turned out to be a consequence of symmetry [M. Esposito, K. Lindenberg, and C. Van Introduction.-Energy-transduction devices such as heat engines [1][2][3][4][5][6][7][8][9][10][11][12][13][14], nano-motors [15][16][17][18], and biological machines [19][20][21][22][23] are crucial to our human activities. It is important to investigate their energetics in our times of resource shortages. Since they usually operate out of equilibrium, we need to develop some concepts of nonequilibrium thermodynamics to understand their operational mechanism. Finite-time thermodynamics is a branch of nonequilibrium thermodynamics. One of its most profound findings in recent years is the universality of efficiency at maximum power. Up to the quadratic order of η C (the Carnot efficiency), the efficiencies at maximum power for the Curzon-Ahlborn endoreversible heat engine [1], the stochastic heat engine [24], the Feynman ratchet [25], and the quantum dot engine [26], were found to coincide with a universal form
Abstract. A unified χ-criterion for heat devices (including heat engines and refrigerators) which is defined as the product of the energy conversion efficiency and the heat absorbed per unit time by the working substance [de Tomás et al 2012 Phys. Rev. E 85 010104(R)] is optimized for tight-coupling heat engines and refrigerators operating between two heat baths at temperatures Tc and T h (> Tc). By taking a new convention on the thermodynamic flux related to the heat transfer between two baths, we find that for a refrigerator tightly and symmetrically coupled with two heat baths, the coefficient of performance (i.e., the energy conversion efficiency of refrigerators) at maximum χ asymptotically approaches to √ ε C when the relative temperature difference between two heat baths ε −1 C ≡ (T h − Tc)/Tc is sufficiently small. Correspondingly, the efficiency at maximum χ (equivalent to maximum power) for a heat engine tightly and symmetrically coupled with two heat baths is proved to be η C /2 + η 2 C /8 up to the second order term of η C ≡ (T h −Tc)/T h , which reverts to the universal efficiency at maximum power for tight-coupling heat engines operating between two heat baths at small temperature difference in the presence of left-right symmetry [Esposito et al 2009 Phys. Rev. Lett. 102 130602].
The χ-criterion is defined as the product of the energy conversion efficiency and the heat absorbed per unit time by the working substance [de Tomás et al., Phys. Rev. E, 85 (2012) The χ-criterion for Feynman ratchet as a refrigerator operating between two heat baths is optimized. Asymptotic solutions of the coefficient of performance at maximum χ-criterion for Feynman ratchet are investigated at both large and small temperature difference. An interpolation formula, which fits the numerical solution very well, is proposed. Besides, the sufficient condition for the universality of the coefficient of performance at maximum χ is investigated.
Typical heat engines exhibit a kind of homotypy: the heat exchanges between a cyclic heat engine and its two heat reservoirs abide by the same function type; the forward and backward flows of an autonomous heat engine also conform to the same function type. This homotypy mathematically reflects in the existence of hidden symmetries for heat engines. The heat exchanges between the cyclic heat engine and its two reservoirs are dual under the joint transformation of parity inversion and timereversal operation. Similarly, the forward and backward flows in the autonomous heat engine are also dual under the parity inversion. With the consideration of these hidden symmetries, we derive a generic nonlinear constitutive relation up to the quadratic order for tight-coupling cyclic heat engines and that for tight-coupling autonomous heat engines, respectively. c h CA with T h and T c being the temperatures of the hot reservoir and the cold one, respectively. This result has attracted much attention from physicists and engineers . The previous researches reveal that the Curzon-Ahlborn efficiency (η CA ) is also recovered, or at least approximately recovered, in a lot of heat engines such as the stochastic heat engine [30], the Feynman ratchet [31], the single-level quantum dot engine [32], and the symmetric low-dissipation heat engine [33].The connection between finite-time thermodynamics and linear irreversible thermodynamics was proposed by Van den Broeck [34]. The constitutive relation, which is defined as the relation between the generalized thermodynamic fluxes and forces, is one of the central formulas in irreversible thermodynamics. The constitutive relation is assumed to be linear and restricted by the Onsager reciprocal relation [38] in linear irreversible thermodynamics. Using the linear constitutive relation, Van den Broeck found that the efficiency at maximum power for tight-coupling heat engines is half of the Carnot efficiency. However, we observed that many heat engines, such as the Curzon-Ahlborn endoreversible heat engine [3], the Feynman ratchet as a heat engine [31] and the single-level quantum dot heat engine [32], can not be precisely described by the constitutive relation for linear response. There exist higher order terms in the relations between generalized thermodynamic fluxes and forces for these heat engines. This conflict may lead to inappropriate results when we try to investigate the energetics of heat engines in a higher precision [35][36][37]. In our recent work [37], in order to solve the OPEN ACCESS In this paper, we focus on tight-coupling heat engines, in which the heat-leakages vanish so that the thermal flux is proportional to the mechanical flux:t m
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