Generalized Carnot problem of maximum work in finite time via Hamilton–Jacobi–Bellman theory

Abstract

In this work an irreversible extension of the Carnot problem of maximum mechanical work delivered from a system of two fluids with different temperatures is analysed. We consider the active heat exchange between the fluid of a limited thermal capacity and an ambient or environmental fluid whose thermal capacity is so large that its intensive parameters do not change. In the classical problem the rates vanish due to the reversibility requirement, in the extended problem irreducible irreversibilities occurring in boundary layers are admitted.Our thermodynamic modelling is linked with ideas and methods of the optimal control. A variational theory treats an infinite sequence of infinitesimal Curzon–Ahlborn–Novikov processes (CAN processes) as the pertinent theoretical model leading to a finite time exergy of the system with a finite exchange area or a finite contact time. This dissipative exergy is discussed in terms of the process intensity and finite duration. An analytical formalism, analogous to that in analytical mechanics is effective. A novel approach is worked out based on the Hamilton–Jacobi–Bellman equation for the dissipative exergy and related work functionals (HJB theory). The HJB formulation is helpful to deal with work potentials by numerical methods which use Bellman’s recurrence equation as practically sole method of extremum seeking for functionals with constrained rates and states and for complex boundary conditions.It is pointed out that only an irreversible process can be optimal for a finite duration. A link is shown between the process duration, dissipation and optimal intensity measured in terms of a Hamiltonian. Hysteretic properties are effective which cause the difference between the work supplied and delivered, for inverted end states of the process. A decrease of 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. The results prove that bounds of the classical availability should be replaced by stronger bounds obtained for finite time processes.