Mathematical Modeling of the Steady-State Processes of Convective Diffusion in Regular Structures Under Mixed Boundary Conditions

Abstract

We study steady-state processes of the mass transfer of admixtures in two-phase regular structures with regard for the periodic character of convective phenomena under mixed boundary conditions. To construct the exact analytic solutions of contact boundary-value problems of this kind, we adapt the method based on the use of different integral transformations in different contacting domains. A relation between these integral transformations is established by using the conditions of imperfect contact. We obtain the analytic solution of the diffusion problem for a two-phase layer of regular structure with regard for convective transfer in one of the phases with preservation of a constant concentration on the surface of this phase and a constant diffusion flow on the boundary of the other phase. The mass flows through the internal interface contact surface are investigated and numerical analysis of the concentration of migrating particles in structural elements of the body is performed. The interpretation of the results of experimental investigations of the processes of mass transfer in polycrystalline bodies and capillary-porous media is based on the solution of the corresponding boundary-value problems of mathematical physics, which take into account the influence of internal structures [9, 15, 16, 18]. In particular, the exact solutions of specific contact boundary-value problems of mass transfer for piecewise homogeneous systems, e.g., spatially regular, are of a certain interest. These media may consist of domains characterized by different diffusion coefficients and the presence or absence of the convective transfer of particles. The mass transfer is realized between these domains. Since the construction of exact solutions of this problems even for the simplest geometric regions is connected with significant difficulties, the researchers, as a rule, apply the approximate analytic [2, 6, 13, 14] or numerical [4, 8, 10, 17] methods. A new method for the construction of the exact solutions of contact boundary-value diffusion problems in bodies with regular structures based on the integral transformations with respect to spatial variables (separate in different contacting domains) was proposed in [12] and then developed in [11]. The analytic solution of the diffusion problem was found for a horizontal layer in the form of a periodic structure formed by two physically different vertical sublayers. On this basis, the transitions to the continuum limits corresponding to the approximations of single inclusion [13] and continual heterodiffusion [7] were investigated. In the present work, this method is generalized to the case where convective transfer is realized in the sublayers of one of the components of periodic structure and constant values of concentration are preserved on the external boundaries and a constant mass flow is specified on one of the boundaries of domains of the other component. For the steady-state mode, we obtain analytic expressions for the concentration of admixtures and mass flows through the internal contact surface.