4.2.4. Simulation

It is often considered that the combination of all the parameters (*m*, *n*, *l*, *Ea*, *A*) describes the kinetics of the reaction.

To illustrate this, a simulation using the parameters estimated previously (*m*, *n*, *l*, *Ea*, *A*) was realised for each ageing process (Figure 10b). To do so, the initialisation of the temperature and *α* is required. Two algorithms are proposed.

The simulations in Figure 10 were performed with the true *α*<sup>0</sup> and *T*0. In Figure 10b, the simulation performed with Algorithm 2 is compared to the true data. The parameters estimated previously (*m*, *n*, *l*, *Ea*, *A*) are representative of the kinetics. The kinetics seem to be well reproduced except for the cells aged at 0 ◦C. One possible explanation is that only one energy of activation is considered. In fact, several reactions occur at the same time. The kinetics can change over the time due to a new group of reactions, which will impose their kinetics. This is why the kinetics must be studied on a range of temperatures to improve the results, which verifies an isokinetic relationship.

By applying the second Algorithm 3, it is possible to evaluate the time of reaction. In this case no extra reaction occurs. The parameters (*m*, *n*, *l*, *Ea*, *A*) represent the whole reaction. This is the most optimistic case. The most reactive cells are the cells aged at 25 ◦C and cells aged in open circuit voltage (OCV). They are considered more thermally stable because the onset temperature is high and yet the time of reaction is shorter. Cells aged at 0 ◦C and −20 ◦C are less stable (onset temperature small) but their time of reaction is longer. Therefore, mitigation solutions can be set up more easily. The most thermally stable cells are the fresh cells and the cells aged at constant voltage.

#### **Algorithm 2** Calculation of T.

**Require:** *n*, *m*, *Ea*, *A*0, *Tmax*, *T*0,*α*0, *dt T*<sup>0</sup> = *T*<sup>0</sup> *α*<sup>0</sup> = *α*<sup>0</sup> **while** T ≤ *Tmax* **do** *<sup>α</sup>i*+<sup>1</sup> = *<sup>A</sup>*<sup>0</sup> exp(−*Ea*/*RTi*)*α<sup>m</sup> <sup>i</sup>* (<sup>1</sup> − *<sup>α</sup>i*)*<sup>n</sup>* ∗ *dt* + *<sup>α</sup><sup>i</sup> Ti*+<sup>1</sup> = *ETot Ctot* (*αi*+<sup>1</sup> − *<sup>α</sup>i*) + *Ti* **end while**
