Approximations for the Stationary Problem of Radiative-conductive Heat Exchange in a System of Rods of Circular Cross Section

Abstract

In applications, it is of great importance to study the heat transfer process in periodic media containing vacuum interlayers or cavities through which the heat transfer is realized by radiation. A numerical solution of such problems requires considerable computational efforts and becomes, in fact, impossible for a large number of heat transferring elements, especially in the case of two-dimensional and three-dimensional structures. Therefore, it is important to construct effective approximation methods. This article continues a series of papers devoted to the construction and substantiation of special semi-discrete and asymptotic approximations of radiation-conductive heat transfer problems in periodic systems of heat-conducting elements separated by a vacuum. In this paper, we consider a stationary problem of radiation-conductive heat transfer in a periodic system consisting of absolutely black heatconducting rods of circular cross section with radius ε packed in a square box. We constract two new approximate methods based on special discrete and asymptotic approximations of the original problem. The results of computational experiments are given, which make it possible to draw conclusions about the efficiency of methods and the nature of the dependence of their relative errors on the coefficient of thermal conductivity λ and the radius of the rods ε. The seeking value is the absolute temperature. This function is the solution of the boundary-value problem for the stationary heat equation with nonlinear nonlocal intregral conditions describing the heat exchange by a radiation between the rods. Assuming that the temperature on each of the rods is approximately equal to the mean value over the cross section, we comstruct a discrete approximation of the original problem, which is a system of linear algebraic equations with respect to the fourth power of a temperature. Since the matrix of the system is symmetric and positive definite, the method of conjugate gradients can be used to solve it numerically. By its structure, the discrete problem is such that it can be considered as a difference approximation of the homogenized boundary-value problem for the Poisson equation with non-standard boundary conditions. This problem, which we consider as an asymptotic approximation of the original problem, is linear with respect to the fourth power of temperature. A series of computational experiments is carried out, which confirms the operability of the proposed methods for materials with a large coefficient of thermal conductivity. The relative errors of the methods tend to zero as the parameter tends to zero. For a fixed value of radius ε, the relative errors of the methods decrease with increasing values of the thermal conductivity, reaching values in the tenths of a percent. As expected, both methods are practically unsuitable for poorly conductive materials and require substantial modification.