NON-LINEAR DYNAMICS OF THE GENERALIZED CARNOT PROBLEM: MAXIMUM WORK RECEIVED IN A FINITE TIME FROM A SYSTEM OF TWO CONTINUA WITH DIFFERENT TEMPERATURES

Abstract

A finite time extension of the classical Carnot problem of maximum work extracted from a system of two continua with different temperatures is a good example of the problem where non-linear thermodynamic models are linked with ideas and methods of the optimal control. In this work we restrict ourselves to a somewhat special but important case when the amount or flow of continuum 2 is very large so that its intensive parameters (T2, μ2i, etc.) do not change (ambient or environmental fluid, T2 = Te). In this context we consider applications of the optimization theory based on a classical (energy- like) Hamiltonian for various active continuous and cascade processes associated with the theory of a body in a bath, when the indirect exchange of the energy occurs through the working fluid of the participating engine, refrigerator or heat pump. These applications refer in particular to extension of the classical thermodynamic problem of minimal work (exergy) supplied to the system of a finite area of heat (mass) exchange or with a finite contacting time. Non-linear thermodynamic models are obtained for the purpose of work optimization. The optimal work functionals (continuous and discrete) are optimized by calculus of variations, dynamic programming and maximum principle methods. An extended exergy function can next be discussed in terms of the finite process intensity and finite duration. A discrete canonical formalism strongly analogous to those in analytical mechanics and the optimal control theory of continuous systems is an effective tool for thermodynamic optimization of cascade systems. The optimality of a definite irreversible process for a finite-time transition of a controlled fluid is pointed out as well as the connection between the process duration, optimal dissipation and the optimal process intensity measured in terms of a hamiltonian. A decrease of the maximum work received from an engine system and an increase of work added to a heat pump system is revealed in the high-rate regimes and for short durations of thermodynamic processes. The results show that the criteria known from the classical availability theory should be replaced by limits obtained for finite time processes which are closer to reality. Hysteretic properties which arise as the difference between the work supplied and the work delivered are effective.