Nonlinear thermokinetics of maximum work in finite time

Abstract

We show that an extended exergy of the kinetic rather than static origin can be formulated for finite-time transitions, which simplifies to the classical thermal exergy in the limiting case of infinite duration. The extended exergy can be derived either by gauging a functional of the energy dissipation by the classical exergy or as a finite-time extension of the classical thermodynamic work extracted from a system of a body and its environment. With this exergy we consider optimization applications for various active continuous and cascade processes encountered in the theory of the energy exchange through working fluid of an engine, a refrigerator or a heat pump. They refer e.g. to systems with finite exchange area or with a finite contact time, and they clearly show that a complementary economic theory is possible.This theory is a good example of formulation where nonlinear thermodynamic models are linked with ideas and methods of optimal control. Variational approaches strongly analogous to those in analytical mechanics and the optimal control theory of continuous and discrete systems are effective tools in thermodynamics optimization. Nonlinear equations of dynamics, which follow as combinations of the energy balance and transfer equations, are constraints in the problem of work optimization. Functionals, which describe the maximized work, can effectively be optimized by various optimization methods; in this work variational calculus is used along with some aspect of the canonical transformation theory. In particular one can discuss the role of the finite process intensity and finite duration. The optimality of a definite irreversible process for a finite-time transition of a controlled fluid is pointed out as well as a connection between the process duration, optimal dissipation and the optimal process intensity measured in terms of a Hamiltonian. Discrete processes can be analyzed by methods extending those known for continuous ones.The results show that limits of the classical availability theory should be replaced by better limits which are obtained for the finite-time processes and which are closer to reality. A hysteretic property is discovered for the generalized exergy that describes a decrease of the maximum work received from an engine system and an increase of work added to a heat pump system, the features of which are particularly important in high-rate regimes (for short durations of thermodynamic processes).