Mathematical Models of the Temperature Field in Elements of Electronic Devices with Foreign Inclusion

Abstract

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic media with semi-transparent foreign inclusions that are also subject to external heat load, have been developed. To solve the nonlinear boundary-value problem, a linearizing function was introduced, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients relative to the linearizing function and partially linearized boundary conditions was obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the approximation of the temperature according to one of the spatial coordinates on the boundary surfaces of the inclusion was performed by piecewise constant functions. To solve the obtained linear boundary value problem, the Hankel integral transformation method was used, as a result of which an analytical solution was obtained, which determines the introduced linearizing function. For the numerical analysis of temperature behavior and heat exchange processes caused by external heat load, software tools have been developed, which are used to create a geometric image of temperature distribution depending on spatial coordinates. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The developed models make it possible to analyze environments with foreign elements under external thermal loads regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements, but also the entire structure.