## 2 Materials and Methods
The governing equations to describe the transports of electrons, protons, and oxygen ions, and their mutual transfers processes can be resolved by the following equations. \[\
abla\cdot i_{\mathrm{e}^{-}}=\
abla\cdot\left(-\sigma_{\mathrm{e}^{-}}^{ \mathrm{eff}}\
abla\Phi_{\mathrm{e}^{-}}\right)=\left\{ \begin{array}{ll}i_{ \mathrm{e}^{-}-\mathrm{O}^{2-},\mathrm{LSCF}}^{\mathrm{V,c}}&\text{Cathode}\\ -i_{\mathrm{e}^{-}-\mathrm{H}^{+},\mathrm{TPB}}^{\mathrm{V,a}}&\text{Anode} \end{array} \right. \tag{4}\]  
\[\
abla\cdot i_{\mathrm{H}^{+}}=\
abla\cdot\left(-\sigma_{\mathrm{H}^{+}}^{ \mathrm{eff}}\
abla\[PERSON]_{\mathrm{H}^{+}}\right)=\left\{ \begin{array}{ll}-i_ {\mathrm{e}^{-}-\mathrm{H}^{+},\mathrm{L-B}}^{\mathrm{V,c}}&\text{Cathode}\\ 0&\text{Electrolyte}\\ i_{\mathrm{e}^{-}-\mathrm{H}^{+},\mathrm{TPB}}^{\mathrm{V,a}}&\text{Anode} \end{array} \right. \tag{5}\]  
\[\
abla\cdot i_{\mathrm{O}^{2-}}=\
abla\cdot\left(-\sigma_{\mathrm{O}^{2-}}^{ \mathrm{eff}}\cdot\
abla\Phi_{\mathrm{O}^{2-}}\right)=-i_{\mathrm{e}^{-}- \mathrm{O}^{2-},\mathrm{LSCF}}^{\mathrm{V,c}}+i_{\mathrm{O}^{2-}-\mathrm{H}^ {+},\mathrm{L-B}}^{\mathrm{V,c}}\text{Cathode} \tag{6}\]  
where \(\sigma_{\mathrm{e}^{-}}^{\mathrm{eff}}\), \(\sigma_{\mathrm{H}^{+}}^{\mathrm{eff}}\), and \(\sigma_{\mathrm{O}^{2-}}^{\mathrm{eff}}\) are the effective conductivities. (\(\Phi_{\mathrm{e}^{-}}\), \(\hat{\Phi}_{\mathrm{H}^{+}}\), \(\hat{\Phi}_{\mathrm{O}^{2-}}\)) are the local electronic, protonic, and oxygen ionic electric potentials, respectively. (\(i_{\mathrm{e}^{-}}\), \(i_{\mathrm{H}^{+}}\), and \(i_{\mathrm{O}^{2-}}\)) are their corresponding electric current densities, respectively. \(i_{\mathrm{e}^{-}-\mathrm{H}^{+},\mathrm{TPB}}^{\mathrm{V,a}}\) is the conversion rate between \(\mathrm{e}^{-}\) and \(\mathrm{H}^{+}\) per unit volume within composite anode. \[i_{\mathrm{e}^{-}-\mathrm{H}^{+},\mathrm{TPB}}^{\mathrm{V,a}}=j_{\mathrm{TPB }}^{\mathrm{a}}\Lambda_{\mathrm{TPB,eff}}^{\mathrm{V}},\ \left[\mathrm{A}\ \mathrm{m}^{-3}\right] \tag{7}\] \begin{table}
\begin{tabular}{l l l l} \hline  & **Cathode** & **Electrolyte** & **Anode** \\ \hline Composition & LSCF-BZCY & BZCY & Ni-BZCY \\ Thickness (\(\mathrm{\SIUnitSymbolMicro m}\)) & 25 [34] & 55 [34] & 500 [34] \\ Mass ratio & 70/30 [34] & & 65/35 [34] \\ Particle diameter (\(\mathrm{\SIUnitSymbolMicro m}\)) & 1/1 [34] & 1 [34] & 1/1 [34] \\ \(\hat{\Phi}_{\mathrm{g}}\) & 30\% [34] & & 40\% [34] \\  & 2.54 \(\times\) 104 750 \({}^{\circ}\)C & & 3.32 \(\times\) 105 750 \({}^{\circ}\)C \\ \(\sigma_{\mathrm{e}^{-}}^{\mathrm{eff}}(\mathrm{S}\ \mathrm{m}^{-1})\) & 2.81 \(\times\) 104 650 \({}^{\circ}\)C & & 3.48 \(\times\) 105 650 \({}^{\circ}\)C \\  & 2.92 \(\times\) 104 550 \({}^{\circ}\)C & & 3.64 \(\times\) 105 550 \({}^{\circ}\)C \\  & 0.0127 50 \({}^{\circ}\)C & 2.457 750 \({}^{\circ}\)C & 0.064 750 \({}^{\circ}\)C \\ \(\sigma_{\mathrm{H}^{+}}^{\mathrm{eff}}(\mathrm{S}\ \mathrm{m}^{-1})\) & 0.007 650 \({}^{\circ}\)C & 1.538 650 \({}^{\circ}\)C & 0.040 650 \({}^{\circ}\)C \\  & 0.004 550 \({}^{\circ}\)C & 0.708 550 \({}^{\circ}\)C & 0.022 550 \({}^{\circ}\)C \\  & 0.452 750 \({}^{\circ}\)C & & \\ \(\sigma_{\mathrm{O}^{2-}}^{\mathrm{eff}}(\mathrm{S}\ \mathrm{m}^{-1})\) & 0.121 650 \({}^{\circ}\)C & & \\ \(r_{\mathrm{g}}\) (\(\mathrm{\SIUnitSymbolMicro m}\)) & 0.023 550 \({}^{\circ}\)C & & \\ \(r_{\mathrm{g}}\) (\(\mathrm{\SIUnitSymbolMicro m}\)) & 0.476 & & 0.556 \\ \(\Lambda_{\mathrm{TPB,eff}}^{\mathrm{V}}(\mathrm{m}^{-2})\) & 0.80 \(\times\) 1012 & & 1.69 \(\times\) 1012 \\ \(S_{\mathrm{LSCF}}^{\mathrm{V}}(\mathrm{m}^{-1})\) & 1.74 \(\times\) 106 & & \\ \(\tau\) & 3 & & 3 \\ \hline \end{tabular}
\end{table} 
Table 1: Microstructure parameters and the effective properties of each PCFC component. where \(\lambda^{\rm V}_{\rm TPB.eff}\) is the Ni-BZCY-pore TPB lengths per unit volume.