An exact solution to steady heat conduction in a two-dimensional slab on a one-dimensional fin: application to frosted heat exchangers

The thermal conditions for frost forming on a metallic fin are considered, and a numerical solution to the two-dimensional conduction problem is used to identify the parametric space for which the problem reduces to a two-dimensional slab on a one-dimensional fin. An analytical solution gives rise to an eigenvalue problem that requires an unusual scalar product definition. A one-term approximation to the new analytical solution provides fin efficiency calculations useful for a range of conditions, including most frosted-coated metallic fins. The series solution and the one-term approximation are of general applicability to low-conductivity coatings on high-conductivity fins.