An Existence Theorem for a Model of Temperature Within a Lithium-Ion Battery

TL;DR

This paper proves the local existence of continuous temperature solutions in a complex coupled PDE model for Lithium-Ion batteries, addressing nonlinearities from electrochemical interactions.

Contribution

It establishes a mathematical existence theorem for a nonlinear PDE model of battery temperature, a problem previously lacking rigorous solutions.

Findings

Proves local existence of temperature solutions.

Handles nonlinear coupling with electrochemical potentials.

Addresses challenges from exponential nonlinearities.

Abstract

In this article we investigate a model for the temperature within a Lithium-Ion battery. The model takes the form of a parabolic PDE for the temperature coupled with two elliptic PDE's for the electric potential within the solid and electrolyte phases. The primary difficulty comes from the coupling term, which is given by the Butler-Volmer equation. It features an exponential nonlinearity of both the electric potentials and the reciprocal of the temperature. Another difficulty arising in the temperature equation are the gradients of the electric potentials squared showing up on the right-hand side. Due to the nonlinearity, meaningful estimates for the temperature are currently not known. In spite of this, our investigation reveals the local existence of continuous temperature for the Lithium-Ion Battery.