## 2 Battery Pack Cooling Mathematical Model
### Heat Dissipation Model  
According to the air velocity obtained from the flow resistance model, the faster the airflow velocity, the better the heat dissipation performance, and the difference of airflow velocities between cooling channels can reflect the temperature difference between battery units. In this section, the heat dissipation model was used to calculate the cylindrical battery temperature. In order to simplify the model, the following assumptions were made:  
1. Since the temperature difference in the battery cell was smaller than that in the battery pack, the uneven temperature distribution on battery cell was ignored and we calculated the volume-averaged temperature of battery unit [25].  
2. Due to the complex thermophysical parameters of the materials of the battery components, the weighted average method was used to calculate the physical parameters of the battery [35; 36].  
3. The air and the battery only exchanged heat in the cooling channel [25].  
As shown in Figure 4, when air entered the cooling channel, the heat generated by the battery unit was taken away by the cooling air at the left and right sides, and the temperature of the battery unit was lowered. According to the principle of energy conservation, the heat dissipation equations can be obtained [37]:  
\[\Phi_{i}-h_{i}A\Delta T_{\text{ left},\,i}-h_{i+1}A\Delta T_{\text{ right},\,i}=0 \tag{18}\]  
\[\rho_{\text{ air}}\ C_{p,\text{ air}}\ Q_{cc,i}(T_{\text{air},i}-T_{0})=h_{i}A \Delta T_{\text{ left},\,i}+h_{i}A\Delta T_{\text{ right},\,i-1} \tag{19}\]  
where \(\Phi_{i}\) is the thermal power production by the \(i\)th battery unit, \(h_{i}\) is the convective heat transfer coefficient between the cooling air \(i\) and the surface of \(i\)th battery unit, \(\rho_{\text{ air}}\) and \(C_{p,\text{ air}}\) are the density and the heat capacity of the air, respectively, \(A\) is the contact area between the battery unit and the cooling air on either cooling channel side, \(Q_{cc,i}\) is the airflow rate of the \(i\)th cooling channel, \(T_{0}\) is the initial temperature of battery unit and air, \(T_{\text{air},i}\) is the outlet temperature of the air in the \(i\)th cooling channel, and \(\Delta T_{\text{ left},\,i}\) and \(\Delta T_{\text{ right},\,i}\) are the temperature difference between the battery unit and either side of the air. The latter two can be calculated by [25]:  
\[\Delta T_{\text{ left},\,i}=(T_{\text{ air},\,i}-T_{0})/\ln\biggl{(}\frac{T_{u,i}-T_{0}}{T_{u,i}-T_{\text{ air},\,i}}\biggr{)} \tag{20}\]  
\[\Delta T_{\text{ right},\,i}=(T_{\text{ air},\,i+1}-T_{0})/\ln\biggl{(}\frac{T_{u,i}-T_{0}}{T_{u,i}-T_{\text{ air},\,i+1}}\biggr{)} \tag{21}\]  
Figure 3: Schematic of the flow resistant model.  
where \(T_{u,i}\) is the volume-averaged temperature of the \(i\)th battery unit. The convective heat transfer coefficient is given by Ref. [33]:  
\[h=\frac{\kappa\ \mathrm{air}}{d\ \mathrm{c}}\ \mathrm{YRe}^{y}\mathrm{Pr}^{1/3} \tag{22}\]  
where \(\kappa\ \mathrm{air}\) is the thermal conductivity of the air, \(d\ \mathrm{c}\) is the diameter of battery cell, and \(\mathrm{Pr}\) is the Prandtl Number of air. Y and y are empirical parameters and are given in Table 1. The convective heat transfer coefficient can be obtained by taking the air velocities solved in the flow resistance model into the calculation of Re. Therefore, the convective heat transfer coefficient is a key parameter that links the two models.  
In the heat dissipation model, \(i\) varies from \(1\) to \(N\). There are \(N\) battery units, so there are \(N\) equations for Equation (18) describing the heat dissipation of battery unit. There are \(N+1\) cooling channels, so there are \(N+1\) equations for Equation (19) describing the temperature rise of the air in the cooling channel. There are \(N\) unknown \(T_{\mathrm{u}}\) variables and \(N+1\) unknown \(T_{\mathrm{air}}\) variables in the model, so when the velocities of the air are introduced, the temperature of the battery unit can be obtained.