## 2 Methodology  
### Mathematical Equations  
The electrochemical reactions in the cell unit are shown in Equations (1) and (2) [29; 35]:  
\[\text{Anode side of cell}:\;2H_{2}\to 4H^{+}+4e^{-} \tag{1}\]  
\[\text{Cathode side of cell}:\;O_{2}+4H^{+}+4e^{-}\to 2H_{2}O \tag{2}\]  
The charge-transfer kinetics are calculated in Equations (3) and (4) [32]:  
\[i_{a,ct}=i_{0,a}\Bigg{(}\frac{c_{h2}}{c_{h2,ref}}exp\left(0.5\frac{F}{RT}\eta \right)-\frac{c_{h_{2}O}}{c_{h_{2}O,ref}}exp\left(\frac{-1.5F}{RT}\eta\right) \Bigg{)} \tag{3}\]  
\[i_{i,ct}=i_{0,c}\Bigg{(}exp\left(3.5\frac{F}{RT}\eta\right)-x\alpha_{2}\frac{ c_{t}}{c_{O_{2},ref}}exp\left(\frac{-0.5F}{RT}\eta\right)\Bigg{)} \tag{4}\]  
where \(\text{c}_{i}\) is the molar concentration, \(\text{c}_{i,ref}\) is the reference concentration, \(\text{F}\) is the [PERSON]'s constant, \(\text{T}\) is absolute temperature, \(\text{R}\) is the universal gas constant, \(\eta\) is the over-potential, and \(\text{x}_{\text{O}2}\) is the molar fraction of oxygen.  
The conduction equation is shown in Equation (5) [36; 37]:  
\[\
abla\cdot\left(-k^{eff}\
abla T\right)=\left[\frac{A}{V}\right]_{eff}S_{T} \tag{5}\]  
where \(k^{eff}\) is the effective thermal conductivity, \(\text{A}\) is the active area, \(\text{V}\) is the unit volume, and \(\text{S}_{\text{T}}\) is the source term (1 atm, 300 K).  
The equations for calculating the concentration band diffusion are shown in Equations (6) and (7), respectively.  
\[-\
abla\cdot\left(c_{\phi}^{eff}\Delta C_{\phi}\right)=\left[\frac{A}{V} \right]_{eff}S_{\phi} \tag{6}\]\[D_{\phi}^{eff}=\left(\frac{1}{D_{m,\phi}^{eff}}+\frac{1}{D_{k,\phi}^{eff}}\right)^{ -1} \tag{7}\]  
where \(\emptyset\) = O\({}_{2}\), H\({}_{2}\), and VW, representing oxygen, hydrogen, and water vapor, respectively.  
\(D_{\phi}^{eff}\) is the effective diffusivity, \(D_{k,\phi}^{eff}\) is Knudsen diffusion, and \(D_{m,\phi}^{eff}\) is molecular diffusion.  
The [PERSON]'s law equations are shown in Equations (8) and (9):  
\[-\
abla\cdot(\sigma_{e}^{eff}\
abla_{\varphi_{e}})=\left[\frac{A}{\dot{V}} \right]_{eff}S_{e} \tag{8}\]  
\[-\
abla\cdot(\sigma_{i}^{eff}\
abla_{\varphi i})=\left[\frac{A}{\dot{V}} \right]_{eff}S_{i} \tag{9}\]  
where \(\sigma_{i}^{eff}\) and \(\sigma_{e}^{eff}\) are the ionic conductivity and effective electron conductivity, respectively.  
The species conservation could be calculated using Equation (10):  
\[\
abla\cdot\left(\rho\omega_{i}\overrightarrow{u}\right)=\
abla\cdot\left( \rho\frac{\omega_{i}}{x_{i}}D_{i}^{eff}x_{i}\right)+\dot{R}_{i} \tag{10}\]  
where \(\dot{R}_{i}\) is the mass production rate of species \(i\), \(\omega_{i}\) is the mass fraction of species \(i\), and \(x_{i}\) is the molar fraction of species \(i\).  
The mass conversion equation is shown in Equation (11):  
\[\
abla\cdot\left(\rho\overrightarrow{u}\right)=\dot{W} \tag{11}\]  
where \(\dot{W}\) is the mass source.  
The momentum conservation is calculated in Equation (12):  
\[\
abla\cdot\left(\frac{\rho\overrightarrow{u}\overrightarrow{u}}{\varepsilon }\right)=-\varepsilon\
abla p+\
abla\cdot\left[\mu(\
abla\overrightarrow{u }+\
abla\overrightarrow{u}^{T})-\frac{2}{3}\mu\
abla\overrightarrow{u} \right]+\varepsilon\overrightarrow{F}_{Da} \tag{12}\]  
where \(\overrightarrow{F}_{Da}\) is the Darcy's friction force, \(\varepsilon\) is the porosity, and \(\mu\) is the dynamic viscosity of the species.  
The Navier-Stokes equation is shown in Equations (13) and (14) [35; 38; 39]:  
\[\
abla\cdot(\rho v)=S_{mass} \tag{13}\]  
\[\rho v\cdot\
abla v=-\
abla p+\
abla\cdot\left[\mu(\
abla v+v\
abla)-\frac{2 }{3}\mu\
abla v\right]-\frac{\varepsilon\mu v}{k} \tag{14}\]  
\[S_{mass}=\frac{\left(M_{H_{2}O}-M_{H_{2}}\right)i}{2F} \tag{15}\]  
\[S_{mass}=\frac{M_{O_{2}}i}{4F} \tag{16}\]  
where \(S_{mass}\) is the mass source term (calculated using Equations (15) and (16)), \(v\) is the velocity vector, \(\varepsilon\) is the porosity, \(p\) is the pressure, \(k\) is the specific permeability, and \(\mu\) and \(\rho\) are the gas dynamic viscosity and total gas density.