Unified Approach to Thermodynamic Optimization of Generic Objective Functions in the Linear Response Regime

Yan Wang

doi:10.3390/e18050161

Abstract

While many efforts have been devoted to optimizing the power output for a finite-time thermodynamic process, thermodynamic optimization under realistic situations is not necessarily concerned with power alone; rather, it may be of great relevance to optimize generic objective functions that are combinations of power, entropy production, and/or efficiency. One can optimize the objective function for a given model; generally the obtained results are strongly model dependent. However, if the thermodynamic process in question is operated in the linear response regime, then we show in this work that it is possible to adopt a unified approach to optimizing the objective function, thanks to Onsager’s theory of linear irreversible thermodynamics. A dissipation bound is derived, and based on it, the efficiency associated with the optimization problem, which is universal in the linear response regime and irrespective of model details, can be obtained in a unified way. Our results are in good agreement with previous findings. Moreover, we unveil that the ratio between the stopping time of a finite-time process and the optimized duration time plays a pivotal role in determining the corresponding efficiency in the case of linear response.

Highlights

Classical thermodynamics mainly deals with quasistatic processes in which energy dissipation is negligible, and this leads to the Carnot efficiency ηC = 1 − Tc /Th, which sets an upper bound for extracting work from two heat reservoirs with temperatures Th and Tc, respectively
The power output is often taken as the objective, and the efficiency at maximum power for various kinds of heat engines and heat transfer laws have been investigated following the work of Curzon and Ahlborn [1], in which a bound of efficiency similar to ηC was found to be ηCA = 1 − Tc /Th
We show that within the framework of linear irreversible thermodynamics, there exists a lower bound of dissipation Φ (Equation (14)) for finite-time heat-work conversion processes, which is achieved when there is no direct heat leakage between two heat reservoirs and the tight-coupling condition is fulfilled

Summary

Introduction

Van den Broeck in his seminal work [14] formulated the work-extraction process in terms of generalized fluxes and forces, and by assuming the linear dependence of fluxes on forces, he managed to find the efficiency at maximum power is bounded from above by ηVdB = ηC /2, which is achieved for a finite-time process without heat leakage between two heat reservoirs. This result is universal in the sense that it depends neither on the types of heat engines nor on the heat transfer laws [15,16]. The results obtained in this work are in good agreement with previous findings

Onsager’s Theory for Work-Extraction Processes

Dissipation Bound

Unified Approach to Optimizing Generic Objective Functions by Using Φ

Generic Cases

Conclusions