Derivation and numerical validation of a homogenized isothermal Li-ion battery model

Lithium-ion batteries are multiscale systems with processes occurring at different scales. We start from a mathematical model derived on the microscale level of a single battery cell [Latz et al., NMA’10, 2011, pp. 329–337] where we can resolve the porous structure of the electrodes. Direct numerical simulations on this scale lead to a huge number of degrees of freedom and, consequently, very high computational costs. From an application perspective, it is often sufficient to predict the macroscopic properties of the electrodes. Therefore, we derive homogenized macroscopic equations with effective transport coefficients for the concentration of lithium ions in the electrolyte, potential in the electrolyte, and potential in the solid particles. These upscaled equations are coupled via the Butler–Volmer interface conditions to a microscale equation for the concentration of lithium ions in the electrode particles. We follow the idea developed by Ciucci and Lai [Transport Porous Med 88(2):249–270, 2011] and extend their analysis by computing the asymptotic order of the interface exchange current densities, which is an important factor in the homogenization study of the problem. We also perform a numerical homogenization and run extensive numerical simulations in order to validate the derived upscaled model. The numerical experiments show very good agreement between the homogenized model and the microscale one. Hence we are able to predict the macroscopic properties of the porous electrodes and capture the small-scale effects on the large scales without fully resolving all the microscale features.