Mathematical Models of Local Heating of Elements of Electronic Devices

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 sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems 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, analytical relations for determining the temperature distribution in this envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence 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 and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.