## 2 Model Details and Validation  
In this study, a single-channel HT-PEMFC with a PBI membrane is modeled in three dimensions and isothermal. The schematic of the computational region, including the PBI membrane, the catalyst layers, the gas diffusion layers (GDL) and the gas flow channel, is shown in Figure 1. In this figure, the upper part of the geometry is the cathode and the lower part is the anode. Also, the geometric and physical properties of the fuel cell for modeling and validation are summarized in Table 1. This specification is derived from the experimental study by [PERSON] et al. [27]. To test the results' independency of the mesh, nine different meshes of a structured type have been produced and the results are compared with each other. Table 2 shows the results from different meshes. To check the precision of different meshes, polarization curve is obtained for each mesh; the relative error shown in Table 2 presents the average difference of voltages achieved from numerical simulation of a sample mesh and voltages from an experimental study by [PERSON] et al. [27]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Unit** \\ \hline \(h_{\text{ch}},h_{\text{gdl}},h_{\text{d}},h_{\text{mem}}\) & (1, 0.38, 0.05, 0.1) \(\times 10^{-3}\) & m \\ \(w_{ch},w_{rib}\) & (0.7874, 0.90932) \(\times 10^{-3}\) & m \\ L & 0.02 & m \\ \(V_{\text{Oc}}\) & 0.93 & V \\ \(\varepsilon_{gdl}^{eff},\varepsilon_{cl}^{eff}\) & 0.4 & - \\ \(\varepsilon_{mc}\) & 0.3 & - \\ \(a_{0,t}^{ref}\) & \(1,0\times 10^{9}\) & Am\({}^{-3}\) \\ \(a_{0,c}^{ref}\) & \(3\times 10^{3}\) & Am\({}^{-3}\) \\ \(C_{\text{H}_{2},\text{ref}}\) & 40.88 & mol m\({}^{-3}\) \\ \(C_{\text{O}_{2},\text{ref}}\) & 40.88 & mol m\({}^{-3}\) \\ \(\alpha_{a,c}\) & 1 & - \\ \(\sigma_{m}\) & 9.825 & Sm\({}^{-1}\) \\ \(\sigma_{s}\) & 222 & Sm\({}^{-1}\) \\ \(D_{0,H_{2}},D_{0,\Omega_{2}},D_{0,H_{2}O}\) & (9.15, 2.82, 2.56) \(\times 10^{-5}\) & m\({}^{2}\)s\({}^{-1}\) \\ \(\zeta_{a,c}^{in}\) & 100\% & - \\ \(k_{p}\) & 1.18 \(\times 10^{-11}\) & m\({}^{2}\) \\ F & 96,487 & Cmol\({}^{-1}\) \\ R & 8.314 & J mol\({}^{-1}\)K\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry and main physical parameters. Figure 1: Schematic of 3D single-channel HT-PEMFC. As shown in Table 2, the results are close to each other for the 1,288,763 and 31,624,896 meshes, and finally, to reduce the computational time, the mesh number has been selected 1,288,763. It is important to note that in order to get the best response, it is crucial to consider the number of meshes suitable for the GDL, the catalyst layer, and the membrane along Y. To increase the accuracy of simulation, given that the reaction occurs in the catalyst layers, the mesh of this section is finer, and a multi-grid method is used for meshing (Figure 2). The number of meshes in percent used for gas channels, GDL, catalyst layer, and the membrane are 15, 35, 45, and 5, respectively. For validation, a numerical study has been used from the experimental work of [PERSON] et al. (2017), which is shown in Figure 3. As can be seen, there is an acceptable agreement between empirical work and numerical simulation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Elements** & **Relative Difference with Benchmark Solution (\%)** \\ \hline
104,034 & 1.06549397992631 \\
120,087 & 1.04886832015082 \\
167,684 & 1.0158217386034 \\
253,476 & 0.983903959745763 \\
583,421 & 0.940992880083474 \\
733,403 & 0.932825883514081 \\
1,064,601 & 0.921922105808183 \\
1,288,763 & 0.917297753972291 \\
31,624,896 & 0.88729051065396 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relative difference with benchmark solution for nine different numbers of meshes. Figure 2: Multi-grid mesh for HT-PEMFC (**a**) isometric view, (**b**) right view, (**c**) front view. The flow in the gas channels is laminar, and because the fuel cell works at a temperature above 100 \({}^{\circ}\)C, water will only exist in the steam state.