Transient heat conduction in a medium with multiple spherical cavities

This paper considers a transient heat conduction problem for an infinite medium with multiple non-overlapping spherical cavities. Suddenly applied, steady Dirichlet-, Neumann- or Robin-type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so-called addition theorem results in a semi-analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large-time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady-state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented.