Development and analysis of mathematical models for the process of thermal conductivity for piecewise uniform elements of electronic systems - Havrysh, Vasily; Ovchar, Igor; Baranetskyj, Jaroslav; Pelekh, Jaroslav; Serduik, Paul (Eastern-European Journal of Enterprise Technologies, 2017)
We examined linear and non-linear mathematical models for the thermal conductivity process in the elements of electronic systems, which are described by a layer and a piecewise uniform layer with a through foreign cylindrical inclusion, with a concentrated heat flow at one of their boundary surfaces. Classical methods cannot resolve boundary problems of mathematical physics, which correspond to these models, in a closed form. In this connection, thermophysical parameters for piecewise uniform media are described by using generalized functions as a single entity for the entire system. As a result of this approach, we obtain one equation of thermal conductivity with generalized derivatives for the entire system with boundary conditions at the boundary surfaces of inhomogeneous media. In the classic case, the process of thermal conductivity would be described by a system of equations on thermal conductivity for each of the elements of heterogeneous medium under conditions of perfect thermal contact at the conjugating surfaces of dissimilar elements and boundary conditions at the boundary surfaces of non-uniform media. For a case of nonlinear models, which are more accurate than the linear ones, one of the conditions of a perfect thermal contact, namely equality of temperatures at the conjugating surfaces of dissimilar elements of the structure, cannot be used in the process of linearization of nonlinear boundary problems that correspond to these models. In this regard, in the present study we propose approaches that make it possible to solve such type of boundary problems in mathematical physics.