Endo-reversible heat engines coupled to finite thermal reservoirs: A rigorous treatment

TL;DR

This paper rigorously analyzes finite-time endo-reversible heat engines coupled to finite reservoirs, deriving explicit dependencies of maximum power and efficiency on conductance and compression ratio, and clarifying the limits of classical results like Curzon-Ahlborn efficiency.

Contribution

It provides a rigorous treatment of finite-time thermodynamics for heat engines with finite reservoirs, explicitly including adiabatic durations and deriving conditions for maximum power and efficiency.

Findings

Maximum power and efficiency depend explicitly on heat conductance and compression ratio.

Classical Curzon-Ahlborn efficiency is recovered only in specific limiting cases.

The validity regime of endo-reversible models with finite reservoirs is clarified.

Abstract

We consider two specific thermodynamic cycles of engine operating in a finite time coupled to two thermal reservoirs with a finite heat capacity: The Carnot-type cycle and the Lorenz-type cycle. By means of the endo-reversible thermodynamics, we then discuss the power output of engine and its optimization. In doing so, we treat the temporal duration of a single cycle rigorously, i.e., without neglecting the duration of its adiabatic parts. Then we find that the maximally obtainable power output P_m and the engine efficiency \eta_m at the point of P_m explicitly depend on the heat conductance and the compression ratio. From this, it is immediate to observe that the well-known results available in many references, in particular the (compression-ratio-independent) Curzon-Ahlborn-Novikov expressions such as \eta_m --> \eta_{CAN} = 1 - (T_L/T_H)^(1/2) with the temperatures (T_H, T_L) of hot and cold reservoirs only, can be recovered, but, significantly enough, in the limit of a vanishingly small heat conductance and an infinitely large compression ratio only. Our result implies that the endo-reversible model of a thermal machine operating in a finite time and so producing a finite power output with the Curzon-Ahlborn-Novikov results should be limited in its own validity regime.