Hamilton-Jacobi-Bellman theory of dissipative thermal availability

Abstract

We analyze a steady-state problem of maximum work delivered from a finite resource fluid and a bath, as the dissipative, finite-time generalization of the evolutionary Carnot problem in which the temperature driving force between two interacting subsystems varies with the contact time. The thermal capacity of the bath is very large, so its intensive parameters do not change. At the classical, reversible limit, the instantaneous rates do vanish due to the reversibility requirement, whereas in the generalized problem some inherent, rate-related irreversibilities are inevitable, in particular those occurring in boundary layers at interfaces. Methods of the optimal control and variational calculus are suitable to optimize nonlinear dynamics of the process. An analytical formalism, strongly analogous to those in analytical mechanics and optimal control theory, is effective in thermodynamic optimization. A variational theory treats an infinite sequence of infinitesimal Curzon-Ahlborn-Novikov processes as the theoretical model pertinent to develop the theory of a finite-resource fluid interacting with a bath in a finite time, when the active exchange of the energy occurs through the working fluid of participating engines, refrigerators, or heat pumps. The main application is the extension of the classical availability (exergy) beyond the class of reversible processes. The generalized exergy is next discussed in terms of the finite intensity and finite duration of the process. Optimality of a definite irreversible process is an essential feature for a finite duration. A link is shown between the process duration and the optimal intensity measured in terms of a dissipative Hamiltonian. An interesting approach, based on the Hamilton-Jacobi-Bellman equation for the irreversible availability and underlying work functionals (HJB theory), is developed. The HJB formulation is suitable for generation of numerical data of the work potentials, by the standard recurrence equation of Bellman's dynamic programming. Such an equation is, in fact, the sole solving algorithm for functionals with constrained rates and states and with complex boundary conditions. It will certainly be inevitable in the case of the problem generalization to mass transfer and chemical reactions. An essential decrease of the maximal work received from an engine system and an 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 the limits known from the theory of the classical availability 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. The significance of these results for the theory of the structure creation and destruction is underlined.