Periodic heat transfer through inhomogeneous media part 3: Hollow sphere

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The paper presents the exact analytical solutions for periodic radial heat conduction through an inhomogeneous hollow sphere for a certain class of thermal-conductivity profile. The exact analytical solutions for some of these profiles, (including linear and quadratic) have been compared with those obtained by considering the spherical medium to be made up of a number of homogeneous layers with different thermal conductivities, varying from layer to layer, and using the layered-structure (or matrix-multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. It strengthens the confidence in applying the layered-structure method to the case of periodic heat conduction through an inhomogeneous hollow cylinder. Considering the inhomogeneous conducting medium to be made up of a number of spherical layers with a linear profile of thermal conductivity has been shown to be a more effective alternative method of accounting for any type of inhomogeniety; and it saves computation time, as the rate of convergence is much higher than for the homogeneous-layered structure method. The numerical results have been presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and heat flux of the two surfaces of the hollow sphere.