Weighted reciprocal of temperature, weighted thermal flux, and their applications in finite-time thermodynamics

Abstract

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.

Constitutive relation for nonlinear response and universality of efficiency at maximum power for tight-coupling heat engines

“…The contribution of interactions between the heat engine and the two reservoirs is explicitly included in this picture since the engine operates in a finite period or at a finite rate rather than in a quasistatic state. The reasonability of this picture and the significance of the weighted thermal flux were fully discussed in our previous work [32], which will not be repeated here. The generalized thermal force conjugated to J t may be expressed as…”

“…One proposal is the rate of heat absorbed from the hot reservoir [27]; another choice is the mean rate of heat absorbed from the hot reservoir and that released into the cold reservoir [33]. The present authors resolved this controversy by introducing the weighted ther-mal flux in recent work [32]. However, the generic constitutive relation remains unknown for nonequilibrium heat engines in the regime of nonlinear response.…”

“…Generic model.-Above all, we briefly revisit a generic model for tight-coupling heat engines proposed in our previous work [32], which lays a solid theoretical foundation for the solution to the paradox. A heat engine may be simplified as the following schematic setup.…”

“…First, λ = 0 + O(η C ) is called symmetry condition, which represents that the heat engine interacts symmetrically with both heat reservoirs. Second, αβξ = 1+O(η C ) is called energy-matching condition, which indicates that the elementary thermal energy (ξ) flowing through the heat engine matches the characteristic energy (1/β) of the heat engine since 1/β may be interpreted as the effective temperature [32] of the heat engine and the Boltzmann constant has been set to unit. More precisely, the ratio of the characteristic energy of the heat engine to the elementary thermal energy flowing through the heat engine equals to α, the second master coefficient of constitutive relation.…”

“…According to equations (F2)-(F9) in Ref. [32], this engine may be mapped into the generic model. The main results are as follows:…”