Theoretical analysis of a two-dimensional multilayer diffusion problem with general convective boundary conditions normal to the layered direction

Abstract

Theoretical analysis of transient thermal conduction in a two-dimensional multilayer structure has been limited to problems with isothermal or adiabatic boundary conditions along the walls normal to the layered direction. Due to mathematical difficulties pointed out in past work, an analytical solution has not been possible so far for the general case where each layer has a distinct convective boundary condition. This work presents an analytical technique to solve this general problem using Laplace transforms followed by derivation of a sufficient number of linear algebraic equations based on given boundary conditions to determine the coefficients of an eigenfunction-based series solution. This technique makes it possible to solve the problem when each layer may have a different convective heat transfer coefficient along the walls normal to the layered direction. Results from this general analysis are shown to correctly reduce to past work for the special cases of very small or very large Biot number. Good agreement with specific results presented in a past paper is also demonstrated. The technique is used to investigate the impact of key dimensionless parameters on the temperature field. This work significantly generalizes past work that was limited only to adiabatic or isothermal boundary conditions. Results presented here may help improve the theoretical understanding of multilayer diffusion problems, and make theoretical models much more representative of realistic conditions.