Foundations of Mathematical Modeling of Reacting Gas Mixtures

Abstract

Here general mass, momentum, and energy balance laws in a multicomponent chemically active gas mixture are formulated to be further used as the basis for a more detailed consideration of the turbulence problems. As it is well known, the most complete and rigorous mathematical description of a multicomponent medium in the case of a regular (laminar) flow can be given in terms of the kinetic theory of multicomponent mixtures of polyatomic ionized gases. The system of generalized integro-differential Boltzmann equations for the distribution functions of particles of each type in the mixture (with the right-hand parts containing the collision and reaction integrals) supplemented by the radiative transfer equation and the Maxwell equations for electromagnetic fields serves as the basic one. In particular, this approach was earlier developed by the authors in the monograph (Marov MY, Kolesnichenko AV (1987) Introduction to planetary aeronomy, Nauka, Moscow (in Russian)), where a generalized Chapman–Enskog method was used to derive the system of differential gas-kinetic equations for a reacting mixture. In addition, from the viewpoint of macroscopic properties, such a multicomponent gas mixture (e.g., the upper atmosphere of a planet) can be considered as a continuous medium and the methods of continuum mechanics for mixtures can be used for its adequate description. Based on the principles of non-equilibrium thermodynamics, these methods allow the system of hydrodynamic equations with all of the necessary closing relations to be obtained. Such a phenomenological approach also allows semiempirical models of turbulent flows in reacting gas media to be developed using extended irreversible thermodynamics.

The formalism of classical non-equilibrium thermodynamics is used to study the mass, momentum, and energy transfer processes. As is well known, a wide class of non-equilibrium transport processes in gases can be described by means of this formalism fully in accord with the experimental data. In particular, the technique for the thermodynamic derivation of generalized Stefan-Maxwell relations for multicomponent diffusion proposed here allows one to obtain a number of algebraic relations for the transport coefficients that relate, for example, the thermal diffusion ratios to the thermal diffusion and multicomponent diffusion coefficients, the true and partial thermal conductivities, and the multicomponent and binary diffusion coefficients. All of such relations that we derived thermodynamically are in complete agreement with the results of the gas-kinetic theory for multicomponent mixtures of monatomic gases obtained in the second approximation of the Chapman–Enskog method. However, in contrast to the latter, the thermodynamic approach is not related to the postulation of a specific microscopic model for the interaction of molecules in the natural medium being investigated, which is indicative of its universality.