Discrete Hamiltonian Analysis of Endoreversible Thermal Cascades

Endoreversible multistage processes which yield mechanical work are optimized by a relatively little-known discrete maximum principle of Pontryagin’s type. A discrete optimization approach extends the classical method, well known for continuous systems in which a Hamiltonian is maximized with respect to controls. Equations of dynamics which follow from energy balance and transfer equations are difference constraints for optimizing work. Irreversibilites caused by the energy transport are essential. Variation of efficiency is analyzed in terms of the heat flux. Enhanced bounds for the work released from an engine system or added to a heat-pump system are evaluated. Lagrangians of work functionals, canonical equations, and structure of the Hamiltonian function are all discrete characteristics which reach their continuous conterparts in the limit of an infinite number of stages. For a finite-time passage of a resource fluid between two given temperatures, optimality of an irreversible process manifests itself as a connection between the process duration and an optimal intensity expressed in terms of the Hamiltonian. Extremal performance functions that describe extremal work are found in terms of final states, process duration, and number of stages. A discrete extension of classical thermal exergy to systems with a finite number of stages and a finite holdup time of a resource fluid is one of the main results. This extended exergy, that has an irreversible component, simplifies to the classical thermal exergy in the limit of infinite duration and an infinite number of stages. The extended exergy exhibits a hysteretic property as a decrease of maximum work received from a multistage engine system and an increase of minimum work added to a heat-pump system, two properties which are particularly important in high-rate regimes.