Semi-discrete and Asymptotic Approximations of the Non-stationary Problem of Radiation-conductive Heat Transfer in a System of Heat-conductive Shields with Alternating Thermophysical Properties

Of significant interest for applications is a study of radiation-conductive heat transfer in periodic media containing vacuum layers or cavities, heat through which is transferred by radiation. Direct numerical solution of such problems involves significant computational costs and becomes practically unrealistic with a large number of heat-conducting elements, especially for 2D and 3D structures. Therefore, it is relevant to construct effective approximate methods for solving these problems, which, in particular, can be based on special averaging of the initial problems. This article continues a series of works devoted to the construction and analysis of special discrete, semi-discrete and asymptotic approximations of problems involving complex (radiation-conductive) heat transfer in small-scale structures, consisting of a large number of heat-conducting elements separated by vacuum layers or cavities. We consider a nonstationary problem that describes the heat propagation process in a system consisting of  parallel layers of heat-conducting materials (shields) with alternating thermophysical properties, separated by vacuum layers. The function to be found is the absolute temperature . The heat transfer inside each of the shields is described by the heat conduction equation. At the interfaces between the shields, the conditions of radiation heat transfer between the adjacent shields are specified. At the shield system boundaries, the conditions desribung radiation heat transfer from the outer shields to the external environment are specified. Three semi-discrete and two asymptotic approximations of the problem under consideration are proposed, which make it possible to find its approximate solutions. Each of the semi-discrete approximations represents the Cauchy problem for a system of nonlinear differential equations, where the unknowns are the approximations  to the mean values of temperature  on the -th shield. Each of the asymptotic approximations represents a nonlinear initial-boundary value problem for a partial differential equation with non-standard boundary conditions, the solution  of which is considered as an approximation to the solution  of the original problem. The results of computational experiments are presented, which demonstrate fairly high accuracy of the proposed methods.