On the well-posedness of a mathematical model for Lithium-ion batteries

TL;DR

This paper examines the mathematical well-posedness of a Li-ion battery model, correcting a common boundary condition error and providing a rigorous analysis to ensure accurate simulations.

Contribution

It identifies and corrects a boundary condition error in a Li-ion battery model and analyzes its well-posedness from both physical and mathematical perspectives.

Findings

Corrected a widely used boundary condition in the model

Proved the well-posedness of the corrected mathematical problem

Clarified the physical and mathematical consistency of the model

Abstract

In this article we discuss the well-posedness of a mathematical model that is used in the literature for the simulation of Lithium-ion (Li-ion) batteries. First, a mathematical model based on a macro-homogeneous approach is presented, following previous works. Then it is showed, from a physical and a mathematical point of view, that a boundary condition widely used in the literature is not correct. Although these errors could be just sign typos (that can be explained as carelessness over d/d$x$ versus d/d$n$, with $n$ the outward unit vector) and authors using this model probably use the correct boundary condition when they solve it in order to do simulations, readers should be aware of the right choice. Therefore, the deduction of the correct boundary condition and a mathematical study of the well-posedness of the corresponding problem is carried out here.