Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения

Abstract

Studying heat transfer processes in periodic media containing vacuum interlayers or cavities, heat through which is transferred by radiation, is of significant interest for applications. Direct numerical solution of such problems involves considerable computational efforts and becomes almost impossible for systems containing a large number of heat conducting elements, especially for 2D and 3D structures. Therefore, it is of issue to develop effective approximate solution methods for such problems. This publication continues a series of studies on developing and substantiating special discrete and asymptotic approximations of radiant-and-conduction heat transfer problems in periodic systems of heat conducting elements separated by vacuum. In this study, the stationary radiant-and-conduction heat transfer problem in a system of absolutely black square rods is considered. The sought quantity is the absolute temperature, which is found from the solution of the boundary-value problem for the stationary heat conduction equation with nonlinear nonlocal boundary conditions describing radiant heat transfer between the rods through vacuum interlayers. A special discrete approximation of this problem leading to the system of linear algebraic equations with respect to the fourth power of the temperature is presented. The solution of this system as approximation of the mean temperature over the rod cross-section is described. The discrete approximation error estimate as a function of the square rod side length (the small parameter of the problem) and the thermal conductivity coefficient has been obtained. The obtained error estimate proves applicability of the discrete approximation for materials with a high thermal conductivity coefficient.