# Design and Optimization of Cooling Plate for Battery Module of an Electric Vehicle
## 2 Thermal Analysis of a Single Battery  
Continuity equation  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{1}\]  
X-momentum  
\[\frac{\partial(\rho u)}{\partial t}+\
abla\cdot(\rho u\mathbf{u})=-\frac{ \partial P}{\partial x}+\
abla\cdot(\mu\
abla u)+S_{Mx} \tag{2}\]  
Y-momentum  
\[\frac{\partial(\rho v)}{\partial t}+\
abla\cdot(\rho v\mathbf{u})=-\frac{ \partial P}{\partial y}+\
abla\cdot(\mu\
abla v)+S_{My} \tag{3}\]  
Z-momentum  
\[\frac{\partial(\rho w)}{\partial t}+\
abla\cdot(\rho w\mathbf{u})=-\frac{ \partial P}{\partial z}+\
abla\cdot(\mu\
abla w)+S_{Mw} \tag{4}\]  
Energy equation  
\[mC_{p}\frac{dT_{c}}{dt}=\
abla\cdot(\lambda_{c}\
abla T_{c})+Q_{g} \tag{5}\]  
The main working parameters of the lithium iron phosphate battery [25] are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{1}{c}{**Values**} \\ \hline Nominal voltage & (V) & 3.2 \\ Nominal capacity & (Ah) & 10 \\ Internal resistance & (m\(\Omega\)) & \(\approx\)10 \\ Charging current & (A) & \(\leq 10\) \\ Continuous discharge current & (A) & \(\leq 20\) \\ Maximum discharge current & (A) & 50 \\ Upper cut-off voltage & (V) & \(3.65\pm 0.05\) \\ Lower cut-off voltage & (V) & 2.5 \\ Cycle life & (/) & \(\geq 2000\) \\ Weight & (g) & \(275\pm 5\) \\ Dimension & (mm) & \(131\times 65\times 16\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Working parameters of lithium batteries. The interior part of the battery was simplified as an equivalent solid model and the following assumptions were made for the model:  
(1) The material properties in lithium batteries were uniformly distributed. Because of the multi-layer structure and manufacturing process of lithium batteries, only the thermal conductivity was anisotropic;  
(2) Thermal radiation and convection can be neglected inside the lithium battery;  
(3) The specific heat capacity and thermal conductivity of materials in the lithium batteries were constant and independent of the temperature;  
(4) When the battery was charged and discharged, the current and heat generation were considered uniformly distributed. It was difficult to accurately obtain the heat generation rate of batteries due to the complexity of vehicle operating conditions and environment. The internal resistance of the battery was assumed constant under ambient temperature, and the battery resistance was set to 10 m\(\Omega\). According to the classic model proposed by [PERSON] et al. [26], the heating generation rate of batteries was established below in Equation (6). The polarization heat, chemical reaction heat and the electrode cap heat was not considered in the model. \[q=\frac{I}{V}[(U_{0}-U)-T\frac{\partial U_{0}}{\partial T}] \tag{6}\]  
where, V represents the volume of the battery, in cubic meters, U\({}_{0}\), the open circuit voltage in volts, U, the working voltage of the battery in volts, T, the temperature in Kelvin, \(\frac{\partial U_{0}}{\partial T}\) is measured experimentally, the value is very small at room temperature with a low discharge rate and can be neglected. Therefore, Equation (6) can be simplified and expressed as follows:  
\[\mathrm{q}=\frac{\mathrm{I}}{\mathrm{V}}(\mathrm{U}_{298.15}-\mathrm{U})= \frac{\mathrm{I}^{2}(\mathrm{R}_{\mathrm{P}}+\mathrm{R}_{\mathrm{e}})}{ \mathrm{V}}=\frac{\mathrm{I}^{2}\mathrm{R}}{\mathrm{V}} \tag{7}\]  
U\({}_{298.15}\) represents the open circuit voltage of the battery in the temperature of 298.15 K, in volts; R is total internal resistance which is obtained by the internal resistance, R\({}_{\mathrm{e}}\), and the polarization internal resistance, R\({}_{\mathrm{P}}\), in ohms. In addition to the complex heat generation inside the battery, the heat generation outside the battery will also occur, such as the positive and negative electrodes, the confluent and the welding position of the conductor. These heats can be neglected when studying the heat generation of the battery. Equation (7) was used in this paper to estimate the heat generation of the battery.