## 2 Battery Pack Cooling Mathematical Model
### Flow Resistance Model  
In previous studies, the CFD simulation of heat dissipation was computationally intensive [27; 28; 29]. Therefore, the flow resistance model was used to quickly estimate the flow field in the battery pack. The flow resistance model is shown in Figure 3. Since the cooling air diverged first and then converged, the divergence points and the convergence points are used as nodes to calculate each section. The blocks in the Figure 3 represent the static pressure lost when the air flowed. The static pressure loss consists of friction loss and local loss. The friction loss was caused by the friction of the air as it flowed. The local loss was caused by the air divergence and convergence. Since the air velocity was much less than acoustic sound, it is assumed that the air is an incompressible Newtonian fluid [30]. In the dotted box in Figure 3, the static pressure loss equation similar to Kirchhoff's voltage law can be derived as [31]:  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]  
Figure 1: Three-dimension of the parallel air-cooled BTMS. where \(\Delta P_{\text{loss, DP }}\), \(\Delta P_{\text{loss, CC }}\), \(\Delta P_{\text{loss, CP }}\), respectively correspond to the static pressure loss of the divergence plenum, the cooling channel, and the convergence plenum. The total static pressure loss of each channel is equal to the sum of the friction pressure loss and the local pressure loss [31]. \[\Delta P_{\text{loss }}=\Delta P_{\text{local }}+\Delta P_{\text{ friction}} \tag{2}\]  
Local loss is caused by the large number of eddies that air produces during divergence and convergence. The local loss of each channel was calculated as follows [31]:  
\[\Delta P_{\text{local,DP,}i}=\frac{\xi_{\text{DP},i}}{2}\rho U_{\text{DP},i-1}^ {2} \tag{3}\]  
\[\Delta P_{\text{local, CP }i}=\frac{\xi_{\text{CP},i}}{2}\rho U_{\text{CP},i+1}^ {2} \tag{4}\]  
\[\Delta P_{\text{local,CC,}i}=\frac{\xi_{\text{DP}\rightarrow\text{CC},i}}{2} \rho U_{\text{DP},i}^{2}+\frac{\xi_{\text{CC}\rightarrow\text{CP},i}}{2} \rho U_{\text{CP},i}^{2} \tag{5}\]  
Figure 2: Diagram of the parallel air-cooled BTMS. where e \(\xi\) is the local loss coefficient, and \(U\) is the average velocity in each channel. The friction loss was calculated as follows [31]  
\[\Delta P_{\text{friction, DP,}i}=\lambda_{\text{DP},i}\frac{l_{\text{DP},i}}{2D_{\text{DP},i}} \rho U_{\text{DP},i}^{2} \tag{6}\]  
\[\Delta P_{\text{friction, CP},i}=\lambda_{\text{CP},i}\frac{l_{\text{CP},i}}{2D _{\text{CP},i}}\rho U_{\text{CP},i}^{2} \tag{7}\]  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]0  
where \(\lambda\) is the dimensionless friction factor, \(l\) is the length of the segment and \(D\) is the equivalent diameter of the segment. The local loss coefficient is related to the geometric dimensioning of the battery pack and the direction of air. Based on the research of [PERSON] et al. [32], the local loss coefficient of each divergence and convergence point can be expressed as:  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]1  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]2  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]3  
\[\Delta P_{\text{loss, DP }j+1}+\Delta P_{\text{loss, CC }j+1}-\Delta P_{\text{loss, CP }j}-\Delta P_{\text{loss, CC }j}=0 \tag{1}\]4  
where \(\psi\) is the ratio of the sectional area of the convergence plenum and the divergence plenum to the cooling channel and \(q\) is the ratio of the airflow rate of the sub-channel to the main channel. The friction factor is related to the flow state of the air and can be expressed as [33]:  
\[\lambda=\left\{