Efficiency of a thermodynamic motor at maximum power

TL;DR

This paper investigates the efficiency of thermodynamic motors at maximum power, clarifying controversies around the Curzon-Ahlborn efficiency and analyzing how different maximization approaches affect efficiency outcomes.

Contribution

It clarifies the conditions under which the Curzon-Ahlborn efficiency applies and introduces a comprehensive analysis of the sustainable efficiency for motors with multiple sources.

Findings

Maximum power with respect to system temperatures yields CA efficiency.

Maximum power with respect to transition durations shows CA as a lower bound.

Sustainable efficiency has an upper bound of 1, often not exceeding 1/2.

Abstract

Several recent theories address the efficiency of a macroscopic thermodynamic motor at maximum power and question the so-called "Curzon-Ahlborn (CA) efficiency." Considering the entropy exchanges and productions in an n-sources motor, we study the maximization of its power and show that the controversies are partly due to some imprecision in the maximization variables. When power is maximized with respect to the system temperatures, these temperatures are proportional to the square root of the corresponding source temperatures, which leads to the CA formula for a bi-thermal motor. On the other hand, when power is maximized with respect to the transitions durations, the Carnot efficiency of a bi-thermal motor admits the CA efficiency as a lower bound, which is attained if the duration of the adiabatic transitions can be neglected. Additionally, we compute the energetic efficiency, or "sustainable efficiency," which can be defined for n sources, and we show that it has no other universal upper bound than 1, but that in certain situations, favorable for power production, it does not exceed 1/2.