Examining the temperature fields in flat piecewise- uniform structures

Abstract

The paper considers linear and non-linear mathematical models for the thermal conductivity process in designs that are described by a plate and a layered plate with a foreign parallelepiped-shaped through-inclusion on whose one boundary surface heat flux is concentrated. Classic methods do not make it possible to solve the boundary problems of mathematical physics that match these models in a closed form. Given this, in the present work we propose an approach that is based on the fact that the thermal-physical parameters for the piecewise uniform environments are described using generalized functions as a single entity for the whole system. As a result, we obtained one equation of thermal conductivity with generalized derivatives for the entire system with boundary conditions at the boundary surfaces of non-uniform environments. In a classic case, the process of thermal conductivity would be described by a system of equations of thermal conductivity for each of the elements of a non-uniform environment with conditions for an ideal thermal contact at the interface surfaces of non-uniform elements and boundary conditions on boundary surfaces of non-uniform environments. For the case of non-linear models, the condition of temperature equality at the interface surfaces of non-uniform elements of the designs is not applicable. With regard to the aforementioned, this work proposed yet another approach, which is in the introduction of linearizing functions that make it possible to linearize corresponding nonlinear boundary problems for these designs, which, as a result, allows us to solve this kind of boundary problems in mathematical physics. We received calculation formulas for determining the temperature field in the examined thermosensitive systems in the case of linearly variable coefficient of thermal conductivity of design materials. By using the obtained analytical-numerical solutions of linear and nonlinear boundary problems for the given piecewise-uniform structures, we created computational programs that make it possible to obtain the numerical values of temperature distribution and analyze the structures in terms of thermostability. As a result, it becomes possible to improve thermal stability of these designs and thus protect them from overheating, which can cause destruction of separate elements and even entire systems.