Analytical exact solutions of heat conduction problems for anisotropic multi-layered media

Analytical exact solutions of a fundamental heat conduction problem in anisotropic multi-layered media are presented in this study. The steady-state temperature and heat flux fields in multi-layered media with anisotropic properties in each layer subjected to prescribed temperature on the surfaces are analyzed in detail. Investigations on anisotropic heat conduction problems are tedious due to the presence of many material constants and the complex form of the governing partial differential equation. It is desirable to reduce the dependence on material constants in advance of the analysis of a given boundary value problem. One of the objectives of this study is to develop an effective analytical method to construct full-field solutions in anisotropic multi-layered media. A linear coordinate transformation is introduced to simplify the problem. The linear coordinate transformation reduces the anisotropic multi-layered heat conduction problem to an equivalent isotropic ones without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, explicit closed-form solutions of the specific problems are presented in series forms. The numerical results of the temperature and heat flux distributions for anisotropic multi-layered media are provided in full-field configurations.