Steady symmetrical temperature field in a hollow spherical particle with temperature-dependent thermal conductivity

Abstract

In this work, an exact analytical solution to the axisymmetric heat conduction equation for hollow spherical objects with temperature-dependent thermal conductivity is presented. The nonlinear differential equation is first transformed into a linear one by means of an integral transform method. Then, the separation of variables method is employed to solve the transformed linear equation. Ultimately, we use the inverse transform to obtain the physical temperature field. Furthermore, two examples are worked out, i.e., the one-dimensional heat conduction in the radial direction and the two-dimensional case with axial symmetry. The solution is presented as an infinite series in terms of Legendre functions. The problem with spherical symmetry is also solved by using perturbation methods up to the third-order approximation, and the results are compared with the exact solution.