Mathematical models for the determination of temperature fields in heterogeneous elements of digital devices taking thermo sensitivity into account
Authors:   Lviv Polytechnic National University, Lviv, Ukraine, Havrysh V. I. , Shkrab R. R. ,   Lviv Polytechnic National University, Lviv, Ukraine
Published: October 2023

Linear and nonlinear mathematical models for determining the temperature field and subsequently analyzing temperature regimes in isotropic spatial media with semi-through foreign inclusions subjected to internal and external thermal loads are developed. For this purpose, the heat transfer coefficient for such structures is described as a single unit using asymmetric unit functions, which makes it possible to consider boundary value problems of heat transfer with one linear and nonlinear differential equations of heat transfer with discontinuous and singular coefficients and linear and nonlinear boundary conditions on the boundary surfaces of the media. In the case of a nonlinear boundary value problem, the introduced linearizing function is used to linearize the original nonlinear heat conduction equation and nonlinear boundary conditions, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients is obtained relative to the linearizing function with partially linearized boundary conditions. For the final linearization of the partially linearized differential equation and boundary conditions, the temperature is approximated by one of the spatial coordinates on the boundary surfaces of the inclusion by piecewise linear functions, as a result of which both the differential equation and boundary conditions become fully linearized. To solve the resulting linear boundary value problem, the Hankel integral transformation method is used, which results in an analytical solution that determines the introduced linearizing function. As an example, the linear dependence of the thermal conductivity coefficient of structural materials of a structure on temperature, which is often used in many practical problems, is chosen. As a result, analytical relations in the form of quadratic equations were obtained to determine the temperature distribution in a thermally sensitive layer with a foreign semi-through inclusion under external heating in the form of a heat flux. A numerical analysis of the temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of a foreign inclusion on the temperature distribution is investigated if the VK94-I ceramic is chosen as the material of the medium and the inclusion is silver. To determine the numerical values of temperature in the above structures, as well as to analyze heat transfer processes inside these structures caused by internal and external thermal loads, software tools have been developed that have been used to perform a geometric image of the temperature distribution depending on spatial coordinates. The obtained numerical temperature values indicate that the developed mathematical models for analyzing heat transfer processes in spatially heterogeneous environments with internal and external heating correspond to a real physical process. The software also makes it possible to analyze such environments subjected to internal and external thermal loads in terms of 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.