Diffusion and reaction in a two-dimensional multilayer body: Analytical solution and imaginary eigenvalue analysis

Abstract

Heat and mass transfer in multilayer bodies occur commonly in a number of engineering problems, such as thermal management, manufacturing and reacting systems. Much of the past literature on theoretical analysis of multilayer diffusion-reaction problems is based on the assumption of one-dimensional transport. Given the geometrical complexity of practical engineering systems, two- or three-dimensional analysis may be needed for improved accuracy. This work presents theoretical analysis of coupled diffusion and reaction occurring in a general, two-dimensional multilayer body. The transient temperature distribution is written in the form of an infinite series, and the eigenequation for this problem is derived. Adiabatic or isothermal conditions along the side walls are accounted for by appropriate choice of eigenvalues in that direction. Results are shown to be in good agreement with numerical simulations. It is shown that the temperature distribution may converge or diverge, depending on the specific values of various problem parameters, such as reaction coefficients and diffusivities in each layer, as well as the nature of the boundary conditions in both directions. An analysis for the existence of imaginary eigenvalues, which are related to divergence at large times is presented. The theoretical model is used to predict the limits of design parameters to prevent thermal runaway in a two-layer Li-ion cell. This work improves upon the limited past work on multi-dimensional multilayer transport by accounting for the reaction term, and by identifying the possibility of imaginary eigenvalues in such problems. Due to the close relationship between imaginary eigenvalues and thermal runaway, these results may contribute towards improved safety and reliability of practical engineering systems.