## 2 Modeling of Fuel Cell Systems
### Fuel Cell Single Cell Modeling
#### 2.1.2 Bipolar Plate Temperature Model
The parameters of each layer of the cathode and oxygen pole are shown in Table 1. Because the heat source of a fuel cell is mainly concentrated at the cathode, in order to simplify the model, only the cathode side of the reactor is modeled in this paper. The calculation results show that when the air flow rate is 1 L/min, the Reynolds number of the fluid in the flow channel is 6778, much higher than 2300. Therefore, the fluid has turbulent flow, and the influence of friction should be considered, when calculating the Nusselt coefficient. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Name** & **Value** \\ \hline \(\rho_{\rm{L1}}\) & The end plate density & 1300 kg/m\({}^{3}\) \\ \(\rm{A_{L1}}\) & The end plate area & 0.0064 m\({}^{2}\) \\ \(\rm{h_{L1}}\) & The end plate thickness & 0.01 m \\ \(c_{\rm{p,L1}}\) & Specific heat capacity & 1200 J/(kg\(\cdot\)K) \\ \(\rm{U_{air}}\) & Heat transfer coefficient & 16.44 W/m\({}^{2}\cdot\)K \\ \(\rm{U_{L2}}\) & Heat transfer coefficient & 481.18 W/m\({}^{2}\cdot\)K \\ \(c_{p,avg}\) & Specific heat capacity of mixing (air, liq) & 1395 J/(kg\(\cdot\)K) \\ \(\rm{\rho_{L2}}\) & Density of the bipolar plate & 8000 kg/m\({}^{3}\) \\ \(\rm{A_{L2}}\) & Area of the bipolar plate & 0.0049 m\({}^{2}\) \\ \(\rm{h_{L2}}\) & Thickness of the bipolar plate & 0.005 m \\ \(\rm{c_{p,L2}}\) & Specific heat capacity of & 500 J/(kg\(\cdot\)K) \\ \(\rm{\rho_{L3}}\) & The MEA layer density & 1300 kg/m\({}^{3}\) \\ \(\rm{A_{L3}}\) & The MEA layer area & 0.0036 m\({}^{2}\) \\ \(\rm{h_{L3}}\) & The MEA layer thickness & 0.0015 m \\ \(\rm{c_{p,L3}}\) & The bipolar plate & 864 J/(kg\(\cdot\)K) \\ \(\rm{c_{p,cl}}\) & Specific heat capacity (water) & 4200 J/(kg\(\cdot\)K) \\ \(\rm{L_{P,gas}}\) & Length of gas passage & 1768.8 mm \\ \(\rm{P_{Cs,gas}}\) & Gas passage section perimeter & 5.69 mm \\ \(\rm{A_{gas}}\) & Gas passage section area & 2.00 mm\({}^{2}\) \\ \(\rm{\rho_{gas}}\) & Gas density (air) & 1.24 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter table. The calculation formula of the friction coefficient is:  
\[\mathrm{f}=\big{(}0.79\mathrm{ln}\mathrm{Re}_{\mathrm{gas}}-1.64\big{)}^{-2} \tag{11}\]  
The calculation formula of Nusselt number \(\mathrm{Nu}_{\mathrm{gas}}\) and air convection coefficient \(\mathrm{hc}_{\mathrm{gas}}\) is as follows:  
\[\mathrm{Nu}_{\mathrm{gas}}=\frac{(\mathrm{f}/8)\big{(}\mathrm{Re}_{\mathrm{gas} }-1000\big{)}\mathrm{Pr}_{\mathrm{gas}}}{1+12.7(\mathrm{f}/8)^{1/2}\big{(} \mathrm{Pr}_{\mathrm{gas}}{}^{2/3}-1\big{)}} \tag{12}\]  
\[\mathrm{hc}_{\mathrm{gas}}=\frac{\mathrm{Nu}_{\mathrm{gas}}\mathrm{kg}_{ \mathrm{gas}}}{\mathrm{D}_{\mathrm{h},\mathrm{gas}}} \tag{13}\]  
In Formulas (12) and (13), \(\mathrm{Pr}_{\mathrm{gas}}\) is the Prandtl constant of air at 373 K; and \(\mathrm{k}_{\mathrm{gas}}\) is the conductivity coefficient. The enthalpy changes of the gas and liquid flowing through the plate layer is \(\mathrm{\Delta H}_{\mathrm{gas}}\) and \(\mathrm{\Delta H}_{\mathrm{liq}}\), which can be expressed by the following formula:  
\[\mathrm{\Delta H}_{\mathrm{c}}=\mathrm{H}_{\mathrm{c,in}}+\mathrm{H}_{\mathrm{ c,L3}}-\mathrm{H}_{\mathrm{c,out}}-\mathrm{H}_{\mathrm{c,L2}} \tag{14}\]  
In Formula (14), \(\mathrm{H}_{\mathrm{c,in}}\) is the enthalpy of substance c, entering the layer, \(\mathrm{H}_{\mathrm{c,L3}}\) is the enthalpy of substance c flowing into the bipolar plate, from the MEA layer, and \(\mathrm{H}_{\mathrm{c,out}}\) is the enthalpy of substance c flowing out of the layer. \(\mathrm{H}_{\mathrm{c,L2}}\) is the enthalpy of substance c flowing into the MEA layer; \(\mathrm{n}_{\mathrm{c}}\) is the molar flow rate of substance c; and \(\mathrm{h}_{\mathrm{c}}\) is the specific enthalpy of component c at the current temperature. \(\mathrm{T}_{\mathrm{c}}\) indicates the temperature in the current state.