Optimal Control Framework for Multistage Endoreversible Engines with Heat and Mass Transfer

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Abstract
We develop a general optimal control framework for a difficult class of problems of work maximization in endoreversible multistage processes which yield mechanical work with finite rates and are characterized by multiple (vectorial) efficiencies. Bellman's method of dynamic programming is used either to construct his recurrence equation or to arrive at a discrete maximum principle of Pontryagin's type, in which a Hamiltonian is maximized with respect to controls. Both these algorithms are powerful computational tools which serve to maximize the power output and evaluate optimal controls. Equations of dynamics which follow from energy and matter balances and transfer equations are difference constraints for optimizing work. Irreversibilities caused by the energy and mass transport are essential. Variation of efficiencies is analyzed in terms of heat and mass fluxes as natural control variables. Enhanced bounds for the work released from an engine system or added to a heat-pump system are evaluated. Lagrangians and Hamiltonians of work functionals and discrete canonical equations are effective; they reach their continuous counterparts in the limit of an infinite number of stages. For a finite-time passage of a resource fluid between two given thermodynamic states, an optimal process is shown to be irreversible. Its optimal intensity is characterized well by the Hamiltonian H. Characteristic functions which describe extremal work are found numerically in terms of final states, process duration and number of stages. An extension of classical exergy to nonisothermal separation systems with a finite number of stages and finite holdup time of the resource fluid is one of the main results. This extended exergy 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. This work is a significant step towards a realistic theory of nonisothermal chemical engines.