Development and analysis of mathematical models for the process of thermal conductivity for piecewise uniform elements of electronic systems

Vasily Havrysh,Jaroslav Pelekh,Igor Ovchar,Jaroslav Baranetskyj,Paul Serduik

doi:10.15587/1729-4061.2017.92551

Abstract

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.

Highlights

Of particular importance in the production of electronic devices are composite materials, development of which is one of the leading challenges of modern materials science
Important place is occupied by the structures with foreign inclusions, which are widely used in the designs of sophisticated electronic systems, in particular, in the integrated sensors for monitoring temperature and humidity, light-emitting elements for dynamic light emitting diode lightening, selective optical filters, etc
In paper [4], analytical-numerical solution for a non-stationary problem on thermal conductivity for a hollow ball was received, the thermal-physical parameters of whose material are dependent on temperature

Summary

Introduction

Of particular importance in the production of electronic devices are composite materials, development of which is one of the leading challenges of modern materials science. Important place is occupied by the structures with foreign inclusions, which are widely used in the designs of sophisticated electronic systems, in particular, in the integrated sensors for monitoring temperature and humidity, light-emitting elements for dynamic light emitting diode lightening, selective optical filters, etc. In order to devise a mathematical model, the most adequate to the real process, it is necessary to take into account dependence of thermal-physical parameters of materials on temperature, density of surface flows and intensity of internal heat sources, change in body shape and possible phase and structural transformations [2, 3]

Literature review and problem statement

The aim and tasks of the study

Conclusions