On the Lump Model in a Composite Sphere

This study investigates transient heat conduction in a two-layer composite sphere, where one layer is assumed to maintain a spatially uniform but time-dependent temperature, while the other exhibits both spatial and temporal variations. The resulting formulation leads to a nonstandard Sturm-Liouville problem with eigenvalue-dependent boundary conditions. Exact analytical solutions are obtained and validated against a full model that resolves the complete spatiotemporal behavior in both layers. The comparison provides a refined framework for assessing the applicability of lumped thermal models. In particular, the results demonstrate that both the aspect ratio and the thermal conductivity ratio significantly influence model accuracy, indicating that the classical Biot number criterion alone is insufficient for composite geometries.