Mathematical models for the determination of temperature fields in thermoactive elements of digital devices with local internal heating and taking thermosensitivity into account

Abstract

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.