## 3 Theoretical Analysis
### Heat Transfer Process  
All heat generated in this process is transferred to and taken out by a coolant. The heat transferred from cell to cell and from cell to ambient is neglected. A schematic structure of the single cell cooling model is shown in Figure 4a, including the heat flow transferred from conduction in the cell, conduction in the silicon layer, conduction in the aluminum pipe wall, and convection at the solid-liquid interface. The heat transfer process from the cell-silicon interface to the coolant can easily be simplified to a one-dimensional heat transfer problem and represented as thermal resistance, as in Figure 4b. The heat transfers from the coil to the pipe by conduction. According to [PERSON]'s law of heat conduction, the heat flow is:  
\[\Phi=kA\frac{\Delta T}{t} \tag{4}\]  
and the conduction heat resistance, \(R\), is defined as:  
\[R=\frac{t}{kA} \tag{5}\]  
where \(k\), \(A\), and \(t\) are the heat conductivity, heat conduction area, and thickness of the medium, respectively. Thus, for \(R_{1}\), \(R_{2}\), and \(R_{3}\), we find:  
\[R_{1}=\frac{t_{prc}}{k_{prc}A};\,R_{2}=\frac{t_{pd}}{k_{pul}A};\,R_{3}=\frac{t_ {uull}}{k_{uull}A} \tag{6}\]  
From pipe to fluid, the convection heat flow, using the equation from [PERSON]'s law of cooling, is:  
\[\Phi=hA(T_{w}-T_{f}) \tag{7}\]  
Thus, the convection heat resistance at the pipe-fluid interface, \(R_{4}\), is:  
\[R_{4}=\frac{1}{hA_{f}} \tag{8}\]  
For a smooth rectangular pipe, the convective heat transfer coefficient at the inner wall of the pipe can be obtained from the Nusselt number Nu, its hydraulic diameter \(d_{H}\), and the heat conductivity of fluid \(k_{f}\), using Equation (8):  
\[h=Nu\frac{k_{f}}{d_{\rm H}} \tag{9}\]  
Figure 4: A single cell heat transfer model. (**a**) Schematic of the heat transfer pass of a single cell (**b**) presented as heat resistance. The laminar flow in pipes is calculated according to the Sieder and [PERSON] correlation [39], where Nu is  
\[Nu=1.86\bigg{(}RePr_{f}\frac{d_{\rm H}}{L_{\rm b}}\bigg{)}^{\frac{1}{3}}\bigg{(} \frac{\mu_{f}}{\mu_{w}}\bigg{)}^{0.14} \tag{10}\]  
\(L_{\rm b}\) is the length of the branch pipe, and \(\mu\) is the dynamic viscosity (subscript f and w represent the fluid temperature and wall temperature, respectively). The Reynolds number Re and Prandtl number (for the temperature of the fluid) Pr\({}_{f}\) are:  
\[Re=\frac{Vd_{\rm H}}{\
u},\ Pr_{f}=\frac{\mu C_{\rm p}}{k}. \tag{11}\]  
For pipes with inner support, assuming the temperature on the supporting walls is uniform and neglecting the heat transfer at the opposite surface, R\({}_{4}\) can also be calculated from Equations (7)-(10). Additionally, inside the cell, ignoring the heat transfer in the axis direction, we find a two-dimensional conducting problem with the heat source. Since the temperature distribution within each cell is not investigated in this article, heat flow can be written as heat resistance:  
\[\Phi=\frac{T_{c}-T_{\rm s}}{R_{c}} \tag{12}\]  
where \(R_{c}\) is the equivalent heat resistance within the battery cell and is determined by the cell's materials, components, and dimensions. \(T_{c}\) is the maximum temperature in the cell, \(T_{s}\) is the temperature of the battery's steel shell, and \(\Phi\) is the heat flow at the contact surface. The bending characteristic is neglected in the thermal calculation. The serpentine pipe is considered to be straight.