Conservation laws as given constraints for processes at mechanical equilibrium

Abstract

In this chapter we initiate the treatment of the problems dealing with a minimum of dissipated energy or entropy. The related criteria of extremum behavior are sometimes called the power criteria; they involve the time change of a thermodynamic potential as a integrand of the functional extremized subject the balance or conservation laws and/or certain conditions imposed on intensive parameters, e. g., the isothermicity condition. Here we investigate a subclass of the processes in mechanical equilibrium, i. e., the situation where any possible convective motion is restricted to a constant, uniform velocity field or the fluid is at rest. These processes may involve transport phenomena only, or a combination of such phenomena with accompanying chemical or electrochemical reactions. The latter case is quite general and complex, and its successful analysis requires appropriate self-consistent description of chemical or electrochemical reaction kinetics with nonlinear resistances or conductances. Such description has been worked out only recently, and it is the key for the successful variational formulation. Therefore, prior to any variational analysis, the basic aspects of the chemical and electrochemical kinetics with their resistances should be consulted (Chap. 4). Only then can one pass to the next crucial issue —the formulation of appropriate lagrangian. The relevant lagrangian L_σ differs significantly from that of the Hamilton’s principle. In an adiabatic representation L_σ involves the four-dimensional functional representing the generated entropy which has to be extremized at the constraints expressing conservation or balance laws for the energy, species and momentum. With this approach the conservation laws prescribe differential constraints, which are recovered during the variational procedure. While the procedure seems to exclude by assumption any search towards varying the structure of the balance or conservation laws, it has the virtue of great generality, that is probably no less than that of Hamilton’s principle. This generality follows from the second law of thermodynamics which requires monotonic growth of the entropy of an isolated system under the constraints determined by the first integrals of the motion of the system. The most important virtue of this approach lies in that it naturally leads to the kinetic equations and the equations of change in their thermodynamically suitable form (related to a given entropy source) and that it allows for search towards the structure of very complex kinetic schemes.