## 2 Methods and Modeling
### Conservation of Species  
Reduction Equation (1) and oxidation Equation (2) describe chemical reactions that occur at the TPBs within the porous cathode and anode:  
\[\frac{1}{2}\text{O}_{2}+2\text{e}^{-}=\text{O}^{2-} \tag{1}\]  
\[\text{H}_{2}+\text{O}^{2-}=\text{H}_{2}\text{O}+2\text{e}^{-} \tag{2}\]  
Since oxygen (O\({}_{2}\)) in air and hydrogen (H\({}_{2}\)) in fuel are consumed when SOFC operates as the electric source, their mass fractions (\(y_{\text{i}}\)) within the pores are different from the mass fractions in the gas channels, so the conservation equation of gas species is written in the following form:  
\[\frac{\partial(\rho y_{\text{i}})}{\partial t}+\overset{\rightarrow}{\
abla} \cdot(\rho y_{\text{i}}\overset{\rightarrow}{v})+\overset{\rightarrow}{\
abla} \cdot\overset{\rightarrow}{j_{\text{i}}}=S_{\text{i}} \tag{3}\]  
Gas density is denoted with \(\rho\), \(y_{\text{i}}\) is mass fraction of the \(i\)-th gas species (index \(i\) or \(j\) denotes H\({}_{2}\), H\({}_{2}\)O, O\({}_{2}\) or N\({}_{2}\)), \(v\) is the average velocity of mass flux, \(j_{\text{i}}\) is the \(i\)-th component of the diffusive mass flux, \(S_{\text{i}}\) is source or sink of the \(i\)-th gas species. Diffusion of gas species is modeled by considering the effective diffusion coefficient \(D_{\text{i,j,eff}}\). The \(j_{\text{i}}\) is obtained by the following equation:  
\[\overset{\rightarrow}{j_{\text{i}}}=-\sum_{\text{j=1}}^{\text{N}-1}\rho\cdot D _{\text{i,j,eff}}\overset{\rightarrow}{\
abla}y_{\text{j}}=-\sum_{\text{j=1 }}^{\text{N}-1}\rho\cdot\mu^{\frac{3}{2}}\cdot D_{\text{i,j}}\cdot\overset{ \rightarrow}{\
abla}y_{\text{j}} \tag{4}\]  
\(N\) denotes the number of gas species in gas mixture (\(N\) = 2 since only two gas species are present). The diffusion coefficient \(D_{\text{i,j}}\) of a binary mixture of gas species \(i\) and \(j\) can be estimated as follows [14]:  
\[D_{\text{i,j}}=\frac{10^{-7}\cdot T^{\frac{7}{4}}}{p\cdot M_{\text{i,j}}^{\frac {1}{2}}\cdot\left(V_{\text{i}}^{\frac{1}{2}}+V_{\text{j}}^{\frac{1}{2}}\right) ^{2}} \tag{5}\]  
\[M_{\text{i,j}}=\left(\frac{1}{M_{\text{i}}}+\frac{1}{M_{\text{j}}}\right)^{-1} \tag{6}\]  
\(T\) is the temperature in Kelvin (K), \(p\) is the pressure in bar (1 bar = 10\({}^{5}\) Pa), \(M_{\text{i,j}}\) is the molar mass (g mol\({}^{-1}\)) of gas species in the binary mixture, and \(V_{\text{i,j}}\) is the diffusion volume in cm\({}^{3}\). Detailed values of \(M_{\text{i}}\) and \(V_{\text{i}}\) can be found in one of our previous papers [15]. The \(S_{\text{i}}\) is calculated at the TPBs, where all three phases (i.e., solid, electrolyte, and gas) meet, otherwise it is zero. The fuel (H\({}_{2}\)) is consumed and water (H\({}_{2}\)O) is produced at the TPBs in the anode as the SOFC operates, so the sink/source term (\(S_{\text{H}_{2}}\)/\(S_{\text{H}_{2}\text{O}}\)) depends on anode current (\(i_{\text{a}}\)). Faraday constant is denoted with \(F\) = 96,485 As mol\({}^{-1}\).  
\[S_{\text{H}_{2}}=-\frac{M_{\text{H}_{2}}\cdot i_{\text{a}}}{2\cdot F} \tag{7}\]  
\[S_{\text{H}_{2}\text{O}}=+\frac{M_{\text{H}_{2}}\text{O}\cdot i_{\text{a}}}{2 \cdot F} \tag{8}\]  
The oxygen (O\({}_{2}\)) is consumed at the TPBs in the cathode as the SOFC operates, so the sink term (\(S_{\text{O}_{2}}\)) depends on cathode current (\(i_{\text{c}}\)):  
\[S_{\text{O}_{2}}=-\frac{M_{\text{O}_{2}}\cdot i_{\text{c}}}{2\cdot F} \tag{9}\]  
It should be noted that nitrogen (N\({}_{2}\)) is neither consumed nor produced, but its mass fraction depends on the mass fraction of oxygen, which changes within the pores as the SOFC operates atdifferent output current. The mass fraction of water (\(y_{\rm H_{2}O}\)) is calculated as \(1-y_{\rm H_{2}}\), whereas the mass fraction of nitrogen (\(y_{\rm N_{2}}\)) is calculated as \(1-y_{\rm O_{2}}\).