Finite time generalization of thermal exergy

Abstract

An extended exergy allowing some inevitable dissipative properties (e.g. those occurring in boundary layers) can be formulated for finite rate transitions, which lead a system not in equilibrium with the environment (or “dead state”) to equilibrium with it in a finite time. This exergy is a function of the end states and the process duration or a process intensity index. This exergy simplifies to classical thermal exergy in the limiting case of infinite duration, when the property of reversibility is recovered due to the vanishing rates. The functional of the extended exergy can be derived either as a finite time extension of the classical thermodynamic work extracted from a system and its environment or by evaluating a functional of the dissipated classical exergy. With finite time exergy, one can effectively optimize various continuous and multistage processes encountered in the theory of energy conversion and energy exchange in systems with a finite exchange area or with a finite contact time. However, the most important application of the dissipative exergy is the enhanced bounds predicted for the work delivery (consumption) in finite rate processes. These bounds are stronger than classical thermostatic bounds. In our formulation, nonlinear thermodynamic models are linked with ideas and methods of variational calculus. The variational formalism, with certain energy like and momentum like quantities strongly analogous to those known from analytical mechanics and optimal control theory, is an effective tool in the optimization of work. In this theoretical framework, one can easily discuss the role of finite process intensity and finite duration. The optimality of a specific irreversible process for the finite time transition of a fluid from one thermodynamic state to another is pointed out as well as a connection between the process duration, optimal dissipation and the optimal process intensity measured in terms of a Hamiltonian function. Our analysis proves that finite time exergy is different for processes approaching equilibrium with the environment, in which work is released (engine mode) and for processes departing equilibrium in which work is supplied (heat pump mode). However, the coincidence between the dissipative exergies and classical exergy is shown for the quasistatic case. The hysteretic properties of the finite time exergy cause a decrease in the maximum work received from a system in engine mode and an increase of work added to a system in heat pump mode, features which are particularly important in high rate regimes or for short duration thermodynamic processes. These results prove that the limits implied by the classical exergy theory should be replaced by the more realistic, stronger limits which are obtained for finite time processes.