Hamilton-Jacobi-Bellman analysis of irreversible thermal exergy

Abstract

A dissipative extension of the classical Carnot problem of maximum work extracted from a system of two bodies with different temperatures is analysed. In the classical problem the instantaneous rates do vanish due to a reversibility requirement imposed on the process, in the extended problem some inevitable, rate-related irreversibilities are allowed, in particular those occurring in boundary layers. In our analysis, nonlinear thermodynamic modeling is inherently linked with ideas and methods of the optimal control. In this paper, we consider a somewhat special but important case in which the thermal capacity of the second body is very large and its intensive parameters do not change. This case may also be referred to the active energy exchange between the fluid of a limited thermal capacity and ambient or environmental fluid. Our variational theory treats an infinite sequence of infinitesimal Curzon-Ahlborn-Novikov processes (CAN processes), as the pertinent theoretical model to develop a finite time theory of a body in a bath, when the indirect exchange of the energy occurs through the working fluid of participating engines, refrigerators or heat pumps. These applications refer, in particular, to the extension of the classical thermodynamic problem of maximum work (exergy) delivered from the system of a finite exchange area or of a finite contact time. The dissipative exergy is discussed in terms of the finite process intensity and finite duration. An analytical formalism, strongly analogous to those in analytical mechanics and optimal control theory, is an effective tool in the thermodynamic optimization. In this paper, a novel approach is worked out which is based on the Hamilton-Jacobi-Bellman equation for the dissipative exergy and related work functionals (HJB theory). The HJB formulation is important for finding the work potentials by numerical methods which use the related Bellman's recurrence equation. The latter is practically the sole method of extremum seeking for functionals with constrained rates and states, and for complex boundary conditions. It will certainly be inevitable in the case of the problem generalization to mass transfer and chemical reactions. The optimality of a definite irreversible process is pointed out for a finite duration. The connection is shown between the process duration, optimal dissipation and the optimal intensity measured in terms of a Hamiltonian. An essential decrease of the maximal work received from an engine system and increase of minimal work added to a heat pump system is shown in the high-rate regimes and for short durations of thermodynamic processes. The results prove that criteria known from the classical availability theory should be replaced by stronger limits obtained for finite time processes, which are closer to reality. Hysteretic properties are effective, which cause the difference between the work supplied and delivered, for the inverted end states of the process.