Stability analysis of a multilayer diffusion-reaction heat transfer problem with a very large number of layers

Abstract

The stability of thermal conduction problems is of much practical interest in the context of Li-ion batteries, power electronic devices, chemical reactors and other engineering systems. While past work in predicting the thermal stability of a multilayer diffusion-reaction problem has been carried out using pole analysis or by determining imaginary eigenvalues of the problem, such techniques are impractical when the number of layers in the problem is very large. This work presents an exact analysis of the stability of a multilayer diffusion-reaction problem, in which the number of layers may be very large. Using homogenization and Heaviside functions based representation of thermophysical properties and parameters, it is shown that the stability of such a system may be easily predicted by computing the determinant of a matrix containing contributions from each layer/interface. Results from the present work are shown to correctly reduce to results from past work for pure-diffusion and single-layer special cases, as well as agree well with numerical simulations. Threshold for thermal stability of a representative Li-ion battery pack containing 1000 Li-ion cells is determined. It is shown that stability of the system is governed by a balance between temperature-dependent heat generation, diffusion and heat removal from the boundary. A key result of practical interest is that even small improvement in thermal diffusivity of Li-ion cells may greatly help reduce the risk of thermal instability. Additionally, practical guidelines for the maximum permissible operating time of a battery pack to avoid thermal runaway are discussed. This work advances the theoretical understanding of multilayer diffusion-reaction problems, and helps understand the thermal stability of Li-ion battery packs containing a very large number of cells, which is very difficult to analyze using traditional techniques.