Effective thermal conductivity of two-dimensional anisotropic two-phase media

The effective thermal conductivity of anisotropic two-phase media is studied. Using Chang’s unit cell, a new analytical solution is developed for anisotropic materials based on the self-consistent field concept. The structure comprises randomly distributed aligned elliptical inclusions embedded in a continuous medium. Inclusions have arbitrary aspect ratio and arbitrary orientation relative to the coordinate system of interest. The temperature distribution is solved and averaged in the unit cell to obtain all the components. The model shows correct limiting properties in all its independent variables. In particular, it yields Maxwell’s theory in the limit where the inclusion aspect ratio approaches unity. Compact expressions for the components of the effective thermal conductivity are presented. The present model is compared with available expressions for anisotropic systems based on an equivalent inclusion model. To assess the accuracy of these, the closure problem associated with the volume averaging method with periodic boundary condition is numerically solved. The present model agrees well with the result of the periodic unit cell compared with equivalent inclusion based methods, particularly for low aspect ratios and moderate particle volume fractions. This is consistent with Ochoa-Tapia’s analysis for isotropic systems that Chang’s unit cell can accurately approximate spatially periodic models in a wider range of porosities compared to Maxwell’s theory. The present solution can serve as a general 2D model for anisotropic structures with dilute to moderate inclusion concentrations.