Two-dimensional analytical solution for temperature distribution in FG hollow spheres: General thermal boundary conditions

The paper aimed at obtaining an analytical solution for the steady-state heat transfer in a hollow sphere made of functionally graded material. Two-dimensional distribution of temperature is considered to be in both radial and peripheral directions and the conductivity coefficients in both directions are a function of radius. The general boundary conditions for the interior and exterior surface of the sphere are considered so that the resulting solution can be extended to treat a wide range of functional cases. The obtained solutions are in the form of Bessel and Legendre functions. Unknown solution coefficients are achieved by applying boundary conditions and orthogonal Legendre functions' relations. The paper additionally provides results for two practical test cases to assess the robustness of the achieved solution. The influence of material constants and conductivity ratio are investigated in order to shed a light on material selection. The results confirm that the achieved general exact solution is able to adequately calculate the distribution of temperature. The validated findings of the current paper could be considered as a clue for tailoring of functionally graded spheres, like spherical vessels, based on the actual thermal boundary conditions in the manufacture process.