**1. Introduction**

Lithium ion battery technology is more and more widespread due to its high energy density and good cyclability. However, this technology can suffer from safety issues. Most Li-ion battery electric vehicles catch fire due to thermal runaway of the battery. Therefore, safety is a key issue in the development of this technology.

Thermal runaway of the battery results from a chain of reactions, some of which are exothermic. Chemical reactions occur one after another, raising the temperature and triggering new reactions. The different exothermic reactions are as follows: first, the decomposition of the solid electrolyte interphase (SEI); then, the reaction of the anode active material and the electrolyte and the reaction between the anode active material and the binder. Next, the cathode active material decomposes and can release oxygen as the battery temperature goes higher. The released oxygen can oxidise the electrolyte or react with the anode active material bringing the battery temperature to a higher value. Combustion of the flammable gases, such as the gas-state solvents and the gases released from the chemical reactions, would happen at a high temperature if an ignition source occurred (short-circuit), resulting in fire and explosion [1].

Most of the studies concerning safety are conducted on new/fresh cells. However, different ageing mechanisms inside the cell can induce physical and chemical modifications of the internal components that could influence the initiation of possible combustion. Ageing can be an important factor that must be taken into account for safety. Few studies investigate the influence of ageing on safety and most of them are conducted from an experimental point of view. Cells are cycled and tested in order to exhibit different ageing mechanisms that impact the safety. For example, Friesen et al. [2] studied the impact of low-temperature cycling on safety, and Borner et al. [3] or Larson et al. [4] showed the correlation between ageing and thermal stability of batteries. All existing studies use the runaway temperature and the heat rate to determine the thermal stability. This article proposes to better understand the correlation between ageing and safety. Several

**Citation:** Grandjacques, M.; Kuntz, P.; Azaïs, P.; Genies, S.; Raccurt, O. Thermal Runaway Modelling of Li-Ion Cells at Various States of Ageing with a Semi-Empirical Model Based on a Kinetic Equation. *Batteries* **2021**, *7*, 68. https://doi.org/10.3390/ batteries7040068

Academic Editor: Kai Peter Birke and Duygu Kaus

Received: 31 July 2021 Accepted: 9 October 2021 Published: 18 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

accelerating rate calorimeter (ARC) tests on cells ageing in different conditions have been realised. The results of the experiment have been crossed with a model.

Thermal runaway models are based on the kinetic equation parametrised by the temperature (*T*) and *α* (0 ≤ *α* ≤ 1), the fractional conversion degree of the reactants. The kinetic model has different forms. Avrami–Erofeev introduced the semi-empirical model parametrised by three constants (*m*, *n*, *p*). Frequently, the more general Sestak and Berggren [5] model is used.

It is believed that this kinetic equation can describe any reaction. The choice of the parameters (*m*, *n*, *p*) is often made without justifications and the parameters (*m* = 1, *n* = 1, *p* = 0) are classically chosen.

We assumed it necessary to develop a method to select the parameters; the parameters are calculated from the ARC test data. The test protocol will be described in Section 2 and the model in Section 3. Section 4 will expose the results of the crossing of the test results with the model. The data-crossing proves that ageing mechanisms have a major impact on the parameters that is confirmed by the post-mortem analysis.
