## 2 Model Development
### Geometric Model
#### 2.2.2 Governing Equations  
* Mass conservation equation \[\frac{\partial(\varepsilon\rho)}{\partial t}+\
abla\cdot(\varepsilon\rho\overset{ \rightarrow}{u})=S_{m},\] (1) \[S_{m,a}=S_{H_{2}}=-\frac{M_{H_{2}}}{2F}i_{an},\] (2) \[S_{m,c}=S_{H_{2}O}+S_{O_{2}}=\frac{M_{H_{2}O}}{2F}i_{ca}-\frac{M_{O_{2}}}{4F}i_ {ca},\] (3)  
The mass conservation equation is shown in Equation (1). \(\varepsilon\) is the porosity. \(\rho\) is the density of the fluid. \(S_{m}\) is the mass source term, which is zero in reactive gas flow channels and GDL of anode and cathode. \(S_{m,a}\) and \(S_{m,c}\) are the mass source terms in anode and in cathode, as shown in Equations (2) and (3). \(M_{H_{2}}\), \(M_{H_{2}O}\), and \(M_{O_{2}}\) are the molar mass of H\({}_{2}\), H\({}_{2}\)O, and O\({}_{2}\). \(F\) is Faraday constant, 96,485 C/mol. \(i_{an}\) and \(i_{ca}\) are the exchange current densities of anode and cathode. * Momentum conservation equation \[\frac{\partial(\varepsilon\rho\overset{\rightarrow}{u})}{\partial t}+\
abla \cdot(\varepsilon\rho\overset{\rightarrow}{u}\overset{\rightarrow}{u})=- \varepsilon\
abla p+\
abla\cdot(\varepsilon\mu\
abla\overset{\rightarrow}{u })+S_{u},\] (4) \[\varepsilon_{g}u_{g}=\frac{k_{p}}{\mu_{g}}\
abla p_{g},\] (5) \[S_{u}=\frac{\varepsilon^{2}\mu}{k_{p}}\overset{\rightarrow}{u},\] (6)  
The momentum conservation equation is shown in Equation (4), and \(S_{u}\) is the momentum source term. \(p\) is the pressure, and \(\mu\) is the dynamic viscosity of the fluid. Equation (4) could be simplified in different components. \(\varepsilon\) is 1, and \(S_{u}\) is zero in flow channels. Equation (4) is simplified to Equations (5) and (6) in porous medium according to [PERSON]'s law. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Unit** \\ \hline Operation pressure & 200,000 & Pa \\ Operation temperature & 343.15 & K \\ Output voltage at optimization & 0.4 & V \\ Open circuit voltage & 1.1 & V \\ Electrical conductivity of BP & 1.0 \(\times\) 10\({}^{6}\) & S/m \\ Reference concentration & H\({}_{2}\): 56.4 & O\({}_{2}\): 3.39 & mol/m\({}^{3}\) \\ Stoichiometry ratio & Anode: 1.5 & Cathode: 2 & - \\ Porosity of GDL & Anode: 0.4 & Cathode: 0.4 & - \\ Porosity of CL & Anode: 0.3 & Cathode: 0.3 & - \\ Electrical conductivity of GDL & Anode: 5000 & Cathode: 5000 & S/m \\ Relative humidity & Anode: 30\% & Cathode: 50\% & - \\ Reference exchange current density & Anode: 13000 & Cathode: 30 & A/m\({}^{2}\) \\ Concentration exponent & Anode: 0.5 & Cathode: 1 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Key physicochemical parameters in this model. * Energy conservation equation \[\frac{\partial(\varepsilon\rho c_{p}T)}{\partial t}+\
abla\cdot(\varepsilon \rho c_{p}\overrightarrow{u}T)=\
abla\cdot(k^{eff}\
abla T)+S_{Q},\] (7)  
\[S_{Q}=I^{2}R_{ohm}+\beta S_{H_{2}O}h_{reaction}+r_{w}h_{L}+i_{an,ca}\eta, \tag{8}\]  
The energy conservation equation is shown in Equation (7), and \(S_{Q}\) is the energy source term. \(c_{p}\) is the specific heat at constant pressure. \(T\) is the temperature. \(k^{eff}\) is effective thermal conductivity. The calculation of \(S_{Q}\) is shown in Equation (8) considering ohmic heat, chemical reaction heat, phase transition heat, and overpotential. \(I\) is the current, and \(R_{ohm}\) is ohmic resistance. \(\beta\) is the ratio of chemical energy and converted heat. \(S_{H_{2}O}\) is the rate at which the water vapor is generated. \(h_{reaction}\) is the enthalpy of the electrochemical reaction. \(r_{w}\) is the rate of phase transition of water, and \(h_{L}\) is the enthalpy of phase transition of water. \(\eta\) is overpotential.