Constitutive Relation for Nonlinear Response and Universality of Efficiency at Maximum Power for Tight-Coupling Heat Engines

TL;DR

This paper derives a universal nonlinear constitutive relation for tight-coupling heat engines and identifies conditions under which the efficiency at maximum power is universally quadratic, resolving a paradox in thermodynamics.

Contribution

It introduces a generic nonlinear response relation for tight-coupling heat engines and clarifies the conditions for universality of maximum power efficiency.

Findings

Universal quadratic efficiency at maximum power under symmetry or characteristic energy conditions

Derivation of a nonlinear constitutive relation from symmetry and stall conditions

Resolution of the paradox regarding universality in different heat engine models

Abstract

We present a unified perspective on nonequilibrium heat engines by generalizing nonlinear irreversible thermodynamics. For tight-coupling heat engines, a generic constitutive relation of nonlinear response accurate up to the quadratic order is derived from the symmetry argument and the stall condition. 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. As a result, we solve the following paradox: On the one hand, the universal quadratic term in the efficiency at maximum power for tight-coupling heat engines proved as a consequence of symmetry [M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 102, 130602 (2009); S. Q. Sheng and Z. C. Tu, Phys. Rev. E 89, 012129 (2014)]; On the other hand, two typical heat engines including the Curzon-Ahlborn endoreversible heat engine [F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975)] and the Feynman ratchet [Z. C. Tu, J. Phys. A 41, 312003 (2008)] recover the universal efficiency at maximum power regardless of any symmetry.