## 3 Theoretical Analysis
### Thermal-Hydraulic Performance  
In a parallel flow system with n branches, the pressure loss or head loss in each branch is equal according to the Bernoulli equation [40]:  
\[H_{1}=H_{2}=\ldots=H_{i}\ldots=H_{n} \tag{17}\]For each cooling module, n = 5, and for branch \(i\), there is [41]:  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]  
The head loss consists of two parts, the local loss and the friction loss. For the \(i\)th branch, the local loss contains the loss at two three-way connectors and at 24 bending positions on the serpentine pipe (in contact with the 24 battery cells, see Figure 3c) along the flow pass. A representation of the local loss factor \(\xi\) of the three-way connectors is shown in Figure 5. The loss for the \(i\)th branch is represented in Figure 6.  
Then, the sum of the local head loss between in and out for branch \(i\) is:  
\[H_{\text{local},i}=H_{1-2,1}+H_{1-2,2}+\ldots+H_{1-2,j-1}+H_{1-3,j}+24H_{bend, i}+H_{3-1,j}+H_{2-1,1}+H_{2-1,2}+\ldots+H_{2-1,j-1} \tag{19}\]  
where the local three-way loss is: (a-b represent 1-2, 1-3, 3-1, 2-1)  
\[H_{\text{a-b},i}=\frac{\xi_{\text{a-b},i}}{2g}V_{\text{main},i}^{2} \tag{20}\]  
Figure 5: Schematic of the three-way connector. (**a**) Separate flow; (**b**) convergence flow. Figure 6: Schematic of the head loss for the \(i\)th branch. where the local loss factor \(\xi_{\rm a-b}\) is calculated from the experimental equations in [42; 43]:  
\[\xi_{\rm 1-2}=0.35\bigg{(}1-\frac{V_{2}}{V_{1}}\bigg{)}^{2},\xi_{\rm 1-3}=0.5 \bigg{(}\frac{V_{3}}{V_{1}}\bigg{)}^{2}+1,\xi_{\rm 2-1}=0.5\bigg{(}1-\frac{V_{2}}{V_{1}} \bigg{)},\xi_{\rm 3-1}=\bigg{(}\frac{V_{3}}{V_{1}}\bigg{)}^{2}-\bigg{(}\frac{V_{2}}{V_ {1}}\bigg{)}^{2}+0.5\bigg{(}1-\frac{V_{2}}{V_{1}}\bigg{)} \tag{21}\]  
The local loss at each bending position on the serpentine pipe is:  
\[H_{bend,i}=\frac{\xi_{bend}}{2g}V_{i}^{2} \tag{22}\]  
where the local loss factor \(\xi_{bend}\) is calculated from the experimental equations in [42; 43]. L/2 is the bending radius (half of the cell's center distance L in Figure 3c). \[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]0  
The sectional average speed V can be represented by the volumetric flow rate Q divided by the cross-sectional area \(Sa\) and \(Sb\), so:  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]1  
where  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]2  
From Equations (20)-(25), the local head loss through the \(i\)th branch is:  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]3  
The friction loss \(H_{\rm fri,i}\) comprises the loss along each segment in the main pipe and branch pipe for the cooling plate in Group Set A:  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]4  
For the cooling plate in Group Set B, there are two segments of additional main silicone tube inlets and outlets with a length of \(L_{\rm C}\), so:  
\[H_{i}=\sum\xi\frac{V^{2}}{2g}\ \ \ +\ \ \sum\lambda\frac{l}{d}\frac{V^{2}}{2g} \tag{18}\]5  
Thus, a mathematical relationship between H\({}_{\text{i}}\) and Q\({}_{\text{i}}\)\(\sim\)Q\({}_{\text{n}}\) has been established, and the flow of different channels (the bias flow rate \(\beta\)) can be obtained by iterative calculations according to Equation (17). According to the heat transfer formula introduced in Section 3.2, the maximum temperature and temperature difference corresponding to different flows can also be calculated. In this section, based on the cooling structure proposed in Section 2, heat transfer calculations and thermal hydraulic calculations are performed using theoretical Formulas (1)-(31). The design parameters of the cooling plate and the reasonable values of the boundary conditions are thus obtained, thereby providing theoretical guidance for the simulation and reducing the number of combinations required for the simulation.