A Temperature Fourier Series Solution for a Hollow Sphere

In this paper, we derive an analytical solution of a two-dimensional temperature field in a hollow sphere subjected to periodic boundary condition. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel’s theorem is used to solve the problem for a periodic boundary condition. The boundary condition is decomposed by Fourier series. In order to check the validity of the results, the technique was also applied to a solid sphere under harmonic boundary condition for which theoretical results were available in the literature. The agreement between the results of the proposed method and those reported by others for this particular geometry under harmonic boundary condition was realized to be very good, confirming the applicability of the technique utilized in the present work.