Generalized exergy for finite-time heat transfer processes

Abstract

The problem of the maximal work that can be extracted from a system consisting of one infinite heat reservoir and one subsystem with a generalized heat transfer law [q ∝ (Δ(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> )) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ], which includes the generalized convective heat transfer law [q ∝ (ΔT) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ] and the generalized radiative heat transfer law [q ∝ Δ(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> )], is investigated in this paper. Finite-time exergy is derived for the fixed process duration by applying optimal control theory. Effects of heat transfer laws on the finite-time exergy and the corresponding optimal thermodynamic process are analyzed. The optimal thermodynamic process for the finite-time exergy with the heat transfer laws that the power exponents m and n satisfy the inequality n(m +1)/(mn − 1) &#60; 0 is that the temperature of the subsystem switches between two optimal values during the heat transfer process, while that with other heat transfer laws is that the temperature of the subsystem is a constant, and the temperature difference between the reservoir and the subsystem is also a constant during the heat transfer process. Numerical examples for the cases with some special heat transfer laws are given, and the results are also compared with each other. The finite-time exergy tends to the classical thermodynamic exergy and the average power tends to zero when the duration tends to infinite large. Some modifications to the results of recent literatures are also performed. The finite-time exergy is a more realistic, stronger limit compared to the classical thermodynamic exergy.