Analytical Solution of Temperature Field in Hollow Cylinder under Time Dependent Boundary Condition Using Fourier series

The objective of this study is to derive an analytical solution of one dimensional heat conduction equation applied in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface while the inner surface is insulated. The material is assumed to be homogenous 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 periodic boundary condition is decomposed by Fourier series. The obtained temperature distribution contains two characteristics, the dimensionless amplitude and the dimensionless phase difference. These results were plotted with respect to Biot and Fourier numbers. The agreement between our results and the former work that was related to one dimensional solution of infinite, solid cylinder, under simple harmonic condition was realized to be very good.