A Linear Approach Suitable for a Class of Steady-State Heat Transfer Problems with Temperature-Dependent Thermal Conductivity

R M S Gama,R Pazetto

doi:10.1155/2021/7004581

Abstract

This work presents an useful tool for constructing the solution of steady-state heat transfer problems, with temperature-dependent thermal conductivity, by means of the solution of Poisson equations. Specifically, it will be presented a procedure for constructing the solution of a nonlinear second-order partial differential equation, subjected to Robin boundary conditions, by means of a sequence whose elements are obtained from the solution of very simple linear partial differential equations, also subjected to Robin boundary conditions. In addition, an a priori upper bound estimate for the solution is presented too. Some examples, involving temperature-dependent thermal conductivity, are presented, illustrating the use of numerical approximations.

Highlights

In order to avoid the barriers arising from nonlinear mathematical descriptions, heat transfer problems with temperature-dependent thermal conductivity and/or nonlinear boundary conditions are, usually, represented by means of somewhat inaccurate approximations
A Dirichlet boundary condition represents a prescribed temperature while the Neumann boundary condition represents a prescribed heat flux
Both are mathematically interesting but lie far from the physical reality. e use of a Robin boundary condition provides a more realistic description

Summary

Introduction

In order to avoid the barriers arising from nonlinear mathematical descriptions, heat transfer problems with temperature-dependent thermal conductivity and/or nonlinear boundary conditions are, usually, represented by means of somewhat inaccurate approximations. Among these approximations, we have the constant thermal conductivity as well as the well-known Dirichlet and Neumann boundary conditions [1]. It is clear the strong relationship between the thermal conductivity and the temperature in this case. 413 386 352 ermal conductivities for some materials at 1 atm

The Kirchhoff Transformation and Its Inverse

An Illustrative Example