Numerical treatment of rapidly changing and discontinuous conductivities

doi:10.1016/S0017-9310(01)00089-8

International Journal of Heat and Mass Transfer

Volume 44, Issue 23, December 2001, Pages 4553-4556

https://doi.org/10.1016/S0017-9310(01)00089-8 Get rights and content

Abstract

The Kirchhoff transformation is an effective means for dealing with temperature dependent conductivities. In general numerical applications, however, the use of this approach will produce non-linear discrete equations, which can be costly to solve. This paper introduces a local Kirchhoff approach for approximating the conductivity terms in the discrete equation. This approach results in an efficient solution in terms of temperature alone. Application to a problem with rapidly changing conductivity shows that use of high-order numerical integration in the conductivity approximation leads to very accurate predictions.

Introduction

In the numerical solution of diffusion-controlled heat and mass transfer problems an area that often leads to difficulty is the treatment of conductivities that are strong functions of the dependent variable. In the context of phase change problems, with a sharp front and a discontinuous change in conductivity, e.g., the melting or solidification of a pure material, a number of numerical schemes have been proposed [1], [2]. Voller and Swaminathan [2] develop a scheme based on a local Kirchhoff transformation [3]. The objective of this short note is to extend the local Kirchhoff approach introduced by Voller and Swaminathan [2] to a case where the conductivity is a continuous but strong function of the dependent variable. Such behavior can be associated with phase change problems that exhibit so-called “mushy” regions, e.g., the solid–liquid dendritic region in a binary alloy [4] or unsaturated moisture migration through soil [5].

Section snippets

The local Kirchhoff method

Heat conduction is chosen as an example diffusion process to illustrate the method development. The governing equation for a transient heat conduction process in a given volume is $ρc ∂ T ∂ t =∇· K(T)∇T +S,$ where suitable boundary conditions are applied at the surface of the volume, ρ is the density, c is the specific heat, K(T) is a temperature-dependent conductivity, and S is a source term. Any problems that may occur due to a rapidly changing or discontinues K(T) can be bypassed by using a Kirchhoff

A test problem

The key features of the proposed approach can be sufficiently demonstrated by considering the following steady one-dimensional heat diffusion problem $d d x K(T) d T d x =0, 0⩽x⩽1$ with T(0)=0, T(1)=2 and the conductivity specified by $K= 1 1+T^{6}$ a function which is plotted in Fig. 2. An analytical solution for this problem can be readily obtained in terms of the Kirchhoff transformation $φ=1.041 x$ from which T(x) can be determined by inverting $φ= ∫ 0 T K(ξ) d ξ= tan^{−1} (T) 3 − 312 ln T^{2} − 3 T+1 + tan^{−1} (2T− 3) 6 + 312 ln T^{2} + 3 T+1 + tan^{−1} (2T+ 3) 6 .$

Discussion

In a numerical solution the obvious way of dealing with a variable conductivity is to employ a central difference approximation for K(T). In cases where K(T) changes rapidly or is discontinuous such an approximation may suffer from loss of accuracy. An alternative approach in this situation is to use a Kirchhoff transformation. This will lead to a high accuracy in the treatment of K(T) but its implementation could be awkward and require a costly inversion of the transformation for the treatment

References (5)

S.L. Lee et al.
An enthalpy formulation for phase change problems with a large thermal diffusivity jump across the interface
Int. J. Heat Mass Transfer
(1991)
C.R. Swaminathan et al.
Towards a general numerical scheme for solidification systems
Int. J. Heat Mass Transfer
(1997)

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Technical NoteNumerical treatment of rapidly changing and discontinuous conductivities