## 2 Materials and Methods
### Model Approach  
The simplest model to describe the thermal behaviour of the battery cell is a first-order Cauer model. This kind of model corresponds with an RC element in system theory. Often the scope of that model is enough to feedback on the system response to a specific current signal of an electric model. Here, the model is adjusted for the system in Figure 5. The approach requires assumptions. For instance, heat generation \(\dot{Q}\) and the heat capacity's \(C_{p}\) location occur in the cell's volumetric centre. Also, the heat transfer in both cross-plane directions, meaning towards the large-area surfaces, is equally distributed. This simplifies the overall model setup, like dividing the internal thermal resistance \(R_{in}\) into two similar parts. Polystyrene plates cover the outer boundaries of the battery, as shown in Figure 4. In the model, these also simplify as single resistances, \(R_{poly}\), excluding the heat capacity of the plates assuming the impact is negligible. The value \(R_{poly}\) can be calculated since all material constants and dimensions are known. Considering Figure 4, the temperature measurement points lie between the cell surface and the individual polystyrene plates. In Figure 5, these points are marked as \(T_{sur,up}\) and \(T_{sur,down}\), respectively. The conditions inside the climate chamber are assumed to be homogeneous in the whole chamber. Therefore, the ambient temperature \(T_{amb}\) on all sides of the setup is set to be equal.  
The measured spectra need to be fitted to extract the cell's heat capacity and thermal conductivity in a cross-plane direction. This approach implements a least-squares fitting algorithm utilising an RC element as the model equation. Therefore, two unknown values, the time constant \(\tau\) and the resistance value \(R\), are fitted. When neglecting the impact of additional heat capacities, like the polystyrene plates and the effect of the temperature sensors, as discussed later, the cell's heat capacity is calculated by \(C=\tau/R\). Dividing the resulting value by the cell mass gives the specific heat capacity \(c_{p}=C/m_{cell}\).  
The fitted value \(R\), representing the overall thermal resistance, is adapted to obtain the cross-plane thermal conductivity \(\lambda_{\perp}\). As described above, the system assumes an equal heat distribution at the top and the bottom. This postulation is required to simplify the  
Figure 4: Depiction of test setup.  
Figure 5: Representation of utilised model configuration.  
model's calculation to only one dimension by two resistive elements (\(R_{in}\) and \(R_{poly}\)). At the measurement point \(T_{sur}\), both \(R_{in}\) and \(R_{poly}\) are connected in parallel:  
\[R_{in}=\frac{R}{\left(1-\frac{R}{R_{poly}}\right)} \tag{4}\]  
Therefore, as shown in Equation (4), it is necessary to calculate \(R_{in}\) from the fitted \(R\) by subtracting the known thermal resistance of the polystyrene plate. Afterwards, the resulting \(R_{in}\) is divided by two, assuming an equal heat dissipation in both cross-plane directions of the cell. Finally, converting the final thermal resistance by the dimensions results in the cell's cross-plane heat conductivity \(\lambda_{\perp}\) using \(l\) as the cell thickness and \(A\) as the cell area in the following Equation (5):  
\[\lambda_{\perp}=\frac{l}{\frac{R_{in}}{2}*A} \tag{5}\]