Finite-time exergy with a finite heat reservoir and generalized radiative heat transfer law

Abstract

The problem of the maximum work that can be extracted from a system consisting of one finite heat reservoir and one subsystem with the generalized radiative heat transfer law [$q \propto \Delta (T^n)$] is investigated in this paper. Finite-time exergy is derived for a fixed duration and a given initial state of the subsystem by applying optimal control theory. The optimal subsystem temperature configuration for the finite-time exergy consists of three segments, including the initial and final instantaneous adiabatic branches and the intermediate heat transfer branch. Analyses for special examples show that the optimal configuration of the heat transfer branch with Newton's heat transfer law [$q \propto \Delta (T)$] is that the temperatures of the reservoir and the subsystem change exponentially with time and the temperature ratio between them is a constant; The optimal configuration of the heat transfer branch with the linear phenomenological heat transfer law [$q \propto \Delta (T^{ - 1})$] is such that the temperatures of the reservoir and the subsystem change linearly and non-linearly with time, respectively, and the difference in reciprocal temperature between them is a constant. The optimal configuration of the heat transfer branch with the radiative heat transfer law [$q \propto \Delta (T^4)$] is significantly different from those with the former two different heat transfer laws. Numerical examples are given, effects of changes in the reservoir's heat capacity on the optimized results are analyzed, and the results for the cases with some special heat transfer laws are also compared with each other. The results show that heat transfer laws have significant effects on the finite-time exergy and the corresponding optimal thermodynamic process. The finite-time exergy tends to the classical thermodynamic exergy and the average power tends to zero when the process duration tends to infinitely large. Some modifications are also made to the results from recent literatures.