VARIATIONAL DESCRIPTION OF HEAT AND MASS EXCHANGE FOR A HETEROGENEOUS SYSTEM WITH THE ACCOUNT OF ELECTRO-SORPTION PROCESSES IN THE CONTINUUM APPROXIMATION

The variational description of heat and mass exchange processes for a heterogeneous system with the account of electro-sorption processes in the continuum approximation has been carried out with the attraction of the basics of analytical thermodynamics. Analytical thermodynamics is a new trend in the development of classical phenomenological thermodynamics. Having the investigation subject and method being common with the classical thermodynamics, analytical thermodynamics differs from it at least by two peculiarities.

In the first place, it is based on the variational principle, which consequences are basic macrosystem laws — the first and the second laws of thermodynamics, and therefore has a broader scientific basis as compared with the classical thermodynamics. In the second place, the analytical thermodynamics differs from the classical thermodynamics by the analysis method, i. e. the vector analysis. The use of vector analysis reduces a body of mathematics of the classical thermodynamics to the generally accepted one for other macrophysical theories.

The outstanding works by Onsager have laid down the beginning of the deductive theory of irreversible processes and the establishment of variational principles of non­equilibrium thermodynamics [96]. The variational formalism provides a possibility of construction of the whole phenomenological theory of thermodynamics based on the variational principle. The heat and mass exchange problems in the system in question are characterised by an unsteady nature, a considerable non-linearity, the interconnection of heat and mass transfer processes, multi-dimensionality, non-homogeneity and electrochemical activity of heat-insulating system.

To main advantages of the variational description of irreversible processes, one normally relate the following [97-104]:

— a high degree of generality of the physical content of variational principles, as they express fundamental physical laws, and differential equations of irreversible processes in macrosystems and boundary conditions of their implementation can be obtained from variational principles,

— the use of direct methods, by means of which accurate, approximate analytical and numerical solutions in the problems formulated in the variational form can be obtained,

— an opportunity to obtain rough approximation to the accurate solution of the problem, which is especially important in engineering calculations, at that most various information on the process can be used for the selection of a trial solution, including empirical data or accurate solutions of simpler problems of this class;

— the use of extreme values of variational principles functionals for obtaining of integral estimates of the approximate solution accuracy;

— a property of functionals to express important characteristics of irreversible processes such as energy dissipation or entropy production, heat taking part in the process.

The variational principle developed by my teacher V. V. Chikovani [44,45] has been called basic variational principle of the classical phenomenological thermodynamics. Such name of the variational principle is connected with the fact that the basic laws (principles) of classical thermodynamics follow from it — the first and the second laws of thermodynamics.

The main differences of the V. V. Chikovani’s variational principles from the well-known principles of thermodynamics of irreversible processes [100, 105, 106, 108, 109] as well as from the conditions of steadiness of functionals having a formal mathematical nature [101, 110-120] is that they, though being an expression of the basic variational principles of the classical phenomenological thermodynamics, have a simple and clear physical meaning expressed in terms of fundamental macrosystem properties. These principles have an integral form both by spatial variables and by time, which provides the use of not only the Kantorovich’s method but also the most effective direct method of variational calculus, i. e. the Ritz method.

As it is well-known, at the description of heat and mass exchange processes in continua, an assumption on the local equilibrium is used, in accordance with which any differential volume of the continuum is an internal equilibrium thermodynamic system [45].

Each differential continuum volume can be considered as a multi-phase system characterised not only by thermodynamic parameters of state being equilibrium over the whole differential volume but also by parameters determining irreversible processes of interphase interaction inside the differential volume being non-equilibrium within the limits of the differential volume (but being equilibrium within the limits of each phase being a part of the differential volume.

A peculiarity of the proposed superinsulation model as a continuum is the concept of any differential volume in the form of a two-phase system (V. V. Chikovani,

N. V. Dolgorukov,1991). The mass exchange occurs between the gaseous and solid phases due to sorption processes. The temperature within the limits of the differential volume is considered identical for both phases. We will describe the sorption system "adsorbent — adsorbate (solid phase)” by parameters characterising it on the whole, i. e. without account of the real structure of the adsorption phase. Such approach allows to use the gas release characteristics of the superinsulation materials being determined experimentally and obtain a mathematical model being suitable for the description of heat and mass exchange processes, both at the availability of gas adsorption in the surface layer and in the material micropores (V. V. Chikovani, N. V. Dolgorukov, 1991).

In order to obtain the variational formulation of the mathematical model of molecular heat and mass exchange processes in the superinsulation, one can use the basic principle of the classical phenomenological thermodynamics [44,45]:

bQ = 8j Pdr =8j ^Pn(xj, х2,Хз,…,xN)dxn = 0, (1)

1-2 1-2 n=l

‘where P is a vector in the N-dimensional space of macrophysical parameters of state Xn(n=1,…, N) of a thermodynamic system characterising its interaction with the environment; dr is the radius-vector differential in the same space.

The variational description of the heat and mass exchange processes for a heterogeneous system with the account of electro-sorption processes will be given in detail in the second part of the review.