## 2 Experimental
### Numerical Simulation  
In this work, numerical simulations are carried out according to an ageing and temperature dependent electrochemical-thermal modelling of the battery cells. The electrochemical simulations  
Figure 5: **(Left)** Graphite potential versus discharge capacity; **(Right)** Graphite incremental capacity during lithiation processes. Figure 6: Battery cell schematic and regions for disposition of temperature sensors. are performed with a pseudo two dimensional (P2D) modelling [39; 40]. Wherein the electrodes are assumed to be porous materials composed of spherical particles immersed in the electrolyte. As part of physico-chemical models, electrochemical models are well documented in the literature [41; 42; 43]. These models are capable of accurately predicting the performance of lithium-ion batteries, which allows the battery manufacturers to optimise and improve the cell design and materials properties chosen for their products. With its wide set of physical parameters, porous electrochemical models are useful tools for design purposes of battery cells since they allow predicting and understanding the physical processes occurring inside the battery during its lifetime. For the electrochemical modelling the internal structure of the battery is considered to be composed of multiple layers (electrodes, separator, current collectors) which is reduced (using a superposition approach) to a single unit layer made of negative current collector, negative electrode, separator, positive electrode and positive current collector. The coupling between the electrochemical (EC) and the thermal (T) parts of the developed temperature and ageing dependent electrochemical-thermal (ECT) modelling of the studied battery cells, is illustrated in Figure 7.  
The heat equation which describes the heat transfer phenomenon occurring between the cell and its environment, for the coupling between both electrochemical and thermal modelling parts, is given in Equation (5):  
\[\rho c_{\mathrm{P}}\frac{\partial T}{\partial t}=\
abla(k\
abla T)+Q \tag{5}\]  
where \(\rho\) represents the density, \(c_{\mathrm{P}}\) the heat capacity, \(T\) the temperature, \(k\) the thermal conductivity, and \(Q\) represents the heat generation. The heat generation \(Q\) inside the cell is derived from Equation (6):  
\[ \begin{split} Q&=\underbrace{j^{Li}T(\frac{\partial U }{\partial T})}_{\text{reaction heat}}+\underbrace{j^{Li}(\phi_{s}-\phi_{e}-U)}_{ \text{active heat}}\\ &+\underbrace{[\sigma_{s}^{eff}\
abla\phi_{s}\
abla\phi_{s}+ \sigma_{e}^{eff}\
abla\phi_{e}\
abla\phi_{e}+\kappa_{D}^{eff}\
abla ln(c_{e}) \
abla\phi_{e}]}_{\text{ohmic heat}}\end{split}  \tag{6}\]  
where \(j^{Li}\) is the transfer current due to the intercalation or deintercalation of lithium ions, \(\phi_{s}\) is the potential of the solid phase, \(\phi_{e}\) is the potential of the electrolyte phase, \(U\) is the open circuit voltage, \(T\) is the temperature, \(\sigma_{s}^{eff}\) is the effective value of the electrical conductivity of the solid phase, \(\sigma_{e}^{eff}\) is the effective value of the electrical conductivity of the electrolyte phase and \(c_{e}\) represents the concentration of the electrolyte [44]. The first term in Equation (6) defines the reaction heat or the irreversible heat generation, the second term defines the reversible heat generation or active heat generation, while the third one (in brackets) defines the ohmic heat generation [45; 46]. By definition, the reaction and the  
Figure 7: Representation of the coupling between the Electrochemical and the Thermal modelling parts. reversible heat generations are caused by the negative and the positive electrodes, while the ohmic heat generation comes from the electrolyte and both negative and positive electrodes.