MODELING OF THE HEAT TRANSFER PROCESS IN A MULTI-LAYER SPHERICAL CONSTRUCTION TAKING INTO ACCOUNT OF INTERNAL HEAT SOURCES

Abstract

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a multilayer hollow spherical structure in the presence of internal (distributed) heat sources. It is assumed that there is an ideal thermal contact between the layers, and that the laws of changes in the temperatures of the media that wash the surface (inner and outer) layers of the structure are arbitrary functions of time and are evenly distributed over the surfaces. That is, convective heat exchange with the environment is assumed and the boundary conditions of the third are satisfied. Therefore, the isotherms inside this structure are concentric circles. The coefficients of the thermal conductivity equation are considered to be lump-constant functions with respect to the spatial coordinate.

This problem is solved by applying the reduction method, when the original problem is divided into two or more simple ones. Analytical studies were conducted using the method of reduction, the concept of quasi-derivatives, modern theory of systems of linear differential equations, the Fourier method and a modified method of eigenfunctions with the active use of computer mathematical environments. The numerical implementation of the method was performed using the Maple 13 computer algebra system.

To illustrate the proposed method, a model example of finding the distribution of the temperature field in a four-layer spherical structure with available internal heat sources under the influence of the temperature of the external fire is solved. The results of the calculations are presented as a graph of temperature change depending on time and spatial coordinate. It should be noted that 30 first roots of the characteristic equation were used to achieve the result with the given accuracy.

The results obtained are directly applicable in a number of applications. This task describes the processes of heat exchange both heating and cooling.