Steady-state heat conduction in multilayer bodies: An analytical solution and simplification of the eigenvalue problem

This paper studies steady-state heat conduction in multilayer bodies when the system is subject to excitations in the direction perpendicular to the junction planes (transverse) and in the directions parallel to the junction planes (longitudinal) as well. In such situations, the eigenvalue problem can be quite complicated if attempting an analytical solution of the thermal distribution. In this study, we propose a method to transform longitudinal excitations into a combination of only transverse effects. In the case of only transverse excitations, the eigenconditions of the layers can be readily determined; thus, the associated eigenproblems can be simplified, and the derivation of an analytical solution becomes less complicated. As a further advantage, the presented method can also be applied to certain forms of internal heat generation, thereby providing a useful tool for modeling electrical machines, e.g., solenoid actuators. The underlying model is linear, and an analytical solution is derived using separation of variables (SOV). We consider steady-state, multidimensional heat conduction for slabs with boundary conditions of the third kind on exposed surfaces. A numerical example is also included, and the results are compared to a FEM (finite element method) solution.