Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method

In order to better manage computational requirements in the study of thermal conduction with short-scale heterogeneous materials, one is motivated to arrange the thermal energy equation into an accurate and efficient form with averaged properties. This should then allow an averaged temperature solution to be determined with a moderate computational effort. That is the topic of this paper as it describes the development using multiple-scale analysis of an averaged thermal energy equation based on Fourier heat conduction for a heterogeneous material with isotropic properties. The averaged energy equation to be reported is appropriate for a stationary or moving solid and three-dimensional heat flow. Restrictions are that the solid must display its heterogeneous properties over short spatial and time scales that allow averages of its properties to be determined. One distinction of the approach taken is that all short-scale effects, both moving and stationary, are combined into a single function during the analytical development. The result is a self-contained form of the averaged energy equation. By eliminating the need for coupling the averaged energy equation with external local problem solutions, numerical solutions are simplified and made more efficient. Also, as a result of the approach taken, nine effective averaged thermal conductivity terms are identified for three-dimensional conduction (and four effective terms for two-dimensional conduction). These conductivity terms are defined with two types of averaging for the component material conductivities over the short-scales and in terms of the relative proportions of the short-scales. Numerical results are included and discussed.