## 3 Modeling
### Protontic Ceramic Fuel Cell
#### 3.1.3 Concentration Polarization  
The concentration polarization due to the gas concentration differences in the anode and cathode can be given as follows [8,40]:  
\[\eta_{\text{conc,a}}=\frac{\text{RT}}{2\text{F}}\ln\left(\frac{\text{P}_{\text {H}_{2}}}{\text{P}_{\text{H}_{2},\text{TPB}}}\right) \tag{14}\]  
\[\eta_{\text{conc,c}}=\frac{\text{RT}}{2\text{F}}\ln\left[\left(\frac{\text{P} _{\text{O}_{2}}}{\text{P}_{\text{O}_{2},\text{TPB}}}\right)^{0.5}\left(\frac{ \text{P}_{\text{H}_{2},\text{O},\text{TPB}}}{\text{P}_{\text{H}_{2}}\text{O }}\right)\right] \tag{15}\]  
The partial pressure of hydrogen, oxygen, and water in the electrode-electrolyte interface can be evaluated from the following equations [8,40]:  
\[\text{P}_{\text{H}_{2},\text{TPB}}=\text{P}_{\text{H}_{2}}-\left(\text{P}- \text{P}_{\text{H}_{2}}\right)\exp\left(\frac{\text{i}_{\text{c}}\text{RT} \delta_{\text{a}}}{2\text{P}\text{D}_{\text{a}}\text{eff}}\right) \tag{16}\]  
\[\text{P}_{\text{O}_{2},\text{TPB}}=\text{P}_{\text{O}_{2}}-\frac{\text{i}_{ \text{c}}\text{RT}\delta_{\text{c}}}{2\text{P}\text{D}_{\text{c}}\text{eff}} \tag{17}\]  
\[\text{P}_{\text{H}_{2}\text{O},\text{TPB}}=\text{P}_{\text{H}_{2}\text{O}, \text{a}}+\frac{\text{i}_{\text{c}}\text{RT}\delta_{\text{c}}}{4\text{P} \text{D}_{\text{c}}\text{eff}} \tag{18}\]  
The diffusion coefficient of the gas mixture at the anode can be calculated from the effective diffusion of H\({}_{2}\), while the diffusion coefficient of the gas mixture at the cathode is calculated from the effective diffusion of O\({}_{2}\) and the effective diffusion of H\({}_{2}\)O, as written as follows [8]:  
\[\frac{1}{\text{D}_{\text{a}}^{\text{eff}}}=\frac{1}{\text{D}_{\text{H}_{2}- \text{mix}}^{\text{eff}}}+\frac{1}{\text{D}_{\text{H}_{2},\text{k}}^{\text{ eff}}} \tag{19}\]  
\[\frac{1}{\text{D}_{\text{O}_{2}}^{\text{eff}}}=\frac{1}{\text{D}_{\text{O}_{2},\text{k}}^{\text{eff}}}+\frac{1}{\text{D}_{\text{O}_{2}-\text{N}_{2}}^{\text{ eff}}}+\left(\frac{1}{\text{D}_{\text{O}_{2}-\text{H}_{2}\text{O}}^{\text{ eff}}}-\frac{1}{\text{D}_{\text{O}_{2}-\text{H}_{2}\text{O}}^{\text{ eff}}}\right)\left(1-\text{y}_{\text{N}_{2}}\right) \tag{20}\]  
\[\frac{1}{\text{D}_{\text{H}_{2}\text{O}}^{\text{eff}}}=\frac{1}{\text{D}_{ \text{H}_{2}\text{O},\text{k}}^{\text{eff}}}+\frac{1}{\text{D}_{\text{H}_{2} \text{O}-\text{N}_{2}}^{\text{eff}}}+\left(\frac{1}{\text{D}_{\text{O}_{2}- \text{H}_{2}\text{O}}^{\text{eff}}}-\frac{1}{\text{D}_{\text{H}_{2}\text{O}- \text{N}_{2}}^{\text{eff}}}\right)\left(1-\text{y}_{\text{N}_{2}}\right) \tag{21}\]  
\[\text{D}_{\text{c}}^{\text{eff}}=\left(\frac{\text{P}_{\text{H}_{2}\text{O}}}{ \text{P}}\right)\text{D}_{\text{O}_{2}}^{\text{eff}}+\left(\frac{\text{P}_{ \text{O}_{2}}}{\text{P}}\right)\text{D}_{\text{H}_{2}\text{O}}^{\text{eff}} \tag{22}\]  
where the effective diffusion of a single component can be calculated using the Knudsen diffusion formula as follows:  
\[\text{D}_{\text{i,k}}^{\text{eff}}=\frac{2}{3}\frac{\epsilon}{\tau}\sqrt{\frac {\text{8 RT}}{\pi\text{M}_{\text{i}}}}\text{r}_{\text{P}} \tag{23}\]  
and the effective diffusion of the binary components is calculated using the Chapman-Enskog formula as follows:  
\[\text{D}_{\text{ij}}^{\text{eff}}=0.0018583\frac{\epsilon}{\tau}\sqrt{\text{T} ^{3}\left(\frac{1}{\text{M}_{\text{i}}}+\frac{1}{\text{M}_{\text{j}}}\right)} \frac{1}{\text{P}\text{o}_{\text{ij}}^{2}\Omega_{\text{D},\text{ij}}} \tag{24}\]\(\sigma\) and \(\Omega\) are Lennard-Jones parameters and collision integral parameters, respectively.  
The value can be calculated using the following formulas.  
\[\sigma_{\rm ij}=\frac{\sigma_{\rm i}+\sigma_{\rm j}}{2} \tag{25}\]  
\[\Omega_{\rm D,ij}=\frac{1.06036}{\Gamma^{0.15610}}+\frac{0.19300}{\exp(0.47635 \Gamma)}+\frac{1.03587}{\exp(1.52996\Gamma)}+\frac{1.76474}{\exp(3.89411\Gamma)} \tag{26}\]  
\[\Gamma=\frac{\rm kT}{\epsilon_{\rm ij}} \tag{27}\]  
\[\epsilon_{\rm ij}=\sqrt{\epsilon_{\rm i}\epsilon_{\rm j}} \tag{28}\]