Numerical Treatment of Diffusion Coefficients at Interfaces

Abstract

The main purpose of this work is to evaluate the values of diffusion coefficients at controlvolume faces, using the finite-volume method, in transport problems that include dwusion processes. A heat conduction problem in a composite slab with linearly temperaturedependent conductivity, which has a closed-form solution, is selected for demonstrmion. It is found that the harmonic mean operation performs better than the arithmetic mean operation in evalunfr'ng the interface diffusion coeffieints. Moreover, if we intend to obtain the informotion at some specific interfaces with abrupt changes of diffusion coefficients, the recommended grid layout is arranged such that each specific interface contains a grid node. This arrangement can give more accurate predictions at these unusual inter faces.

A New Method for Numerical Treatment of Diffusion Coefficients at Control-Volume Surfaces

“…This sometimes may make us feel 150 uncomfortable, since we have become used to the harmonic mean method, which usually requires that no nodes should be placed on the abrupt interface. However, more recent work [6] has proved that if the nodes are set to the position of abrupt-change interfaces with a finer modification in calculating the node values of the diffusion coefficient, then a better result can be obtained with the harmonic 155 mean method. Hence it is suggested that the abrupt-change interfaces should not be used as control-volume surfaces to avoid unnecessary troubles.…”

“…repeatedly proved, is not a very good approach due to its inability to handle a discontinuity of diffusion coefficient, and is now very rarely used [2, 4,6].…”

“…He obtained a better result with the harmonic mean method and thus the method was recommended [5]. Since then the method has been so popular, perhaps due to Patankar's famous book [3], that it is almost accepted as a standard interpolation method and prevails in the literature [2, 6,7]. Later, Chang 45 and Payne reexamined the method [6].…”

“…Since then the method has been so popular, perhaps due to Patankar's famous book [3], that it is almost accepted as a standard interpolation method and prevails in the literature [2, 6,7]. Later, Chang 45 and Payne reexamined the method [6]. Their work again approved Patankar's conclusion, with a suggestion that the abrupt change interface of the diffusion coefficient should be superposed with a grid node.…”

“…Among them the harmonic mean is perhaps the most widely used [2, 6,7]. The Kirchhoff approximation [4,8] is perhaps the newest as far as the present authors know.…”