Steady Conduction With Space-Dependent Conductivity

Abstract

The steady diffusion equation has long been used to describe heat conduction and the flow of an incompressible fluid through a porous medium. Unfortunately, even for regions of simple shape and simple boundary conditions, practical interest is not limited to the case of constant properties for which exact solutions are available. For example, if the temperature inside a manufactured part is to be accurately determined, the variation of thermal conductivity with position should be taken into account. Accurate determination of the hydraulic head at the site of a water well in a water resources problem requires accounting for the spatial variation of the permeability of the porous medium. In many cases, it is the spatial variation of thermal conductivity or permeability that causes the difficulty. However, lumped problem formulations with constant properties in which temperature variation across the thickness of a plate or velocity across the thickness of a confined porous medium is ignored can have the same sort of difficulty when the thickness varies with position. The accuracy of numerical solutions obtained by finite difference and finite element methods can be improved by employing increasingly finer grids or finite elements, of course. Still, exact solutions against which the numericalmore » results can be compared are advantageous to evaluate the accuracy of both the numerical results and the programming that implements the algorithms that lead to them. In the following, exact solutions for steady heat conduction are developed for this purpose for a class of continuous thermal conductivity variations in space.« less