Analysis of a diffusion-reaction heat transfer problem in a finite thickness layer adjoined by a semi-infinite medium

Abstract

Multilayer diffusion-reaction problems are of much interest for both heat and mass transfer. While past work in this direction has mainly addressed multilayer bodies of finite size, practical problems such as immersion cooling of Li-ion cells and thermal runaway in semiconductor devices necessitate considering one of the layers to be semi-infinite. This work presents theoretical analysis of a diffusion-reaction problem in a finite layer surrounded by an infinite medium on both sides, where heat generation proportional to the local temperature occurs in the finite layer. Transient temperature distributions in the two bodies are determined using Laplace transformation technique. Through analysis of the poles of the solution in the Laplace domain, it is proved that this problem is unconditionally unstable, in that temperature in the finite thickness layer is predicted to always diverge at large times. However, the time taken to reach the unstable regime is shown to depend strongly on the key non-dimensional parameters of the problem. Temperature in the finite layer is shown to exhibit non-monotonic behavior at small times, particularly for large values of the heat generation coefficient, which is explained on the basis of the balance between temperature-dependent heat generation, heat dissipation into the semi-infinite medium and eventual slowdown of heat dissipation due to temperature rise. The impact of thermal properties on thermal behavior of the system is examined. A practical problem related to thermal safety design of a Li-ion cell is solved. Results from this work expand the state-of-the-art in theoretical analysis of diffusion-reaction problems, and also offer practical tools for thermal design of engineering problems including Li-ion cells and semiconductor devices.