## 2 Calculation
### Geometry and Mesh Generation  
As shown in Figure 2a, the PEMFC geometry includes the proton exchange membrane, diffusion layers, catalyst layers, flow channels, and collector plates. The difference is that [PERSON] et al. [31] the geometry does not include collector plates, in order to shorten the simulation time. The entire mesh generation is shown in Figure 2b. The anode and cathode flow channel mesh generation is respectively listed in Figure 2c,d for better observation. Non-conformal mesh is used at the interface to reduce the workload, and to improve mesh quality. The number of elements is almost up to 1,600,000. The number of anode and cathode flow channels elements is 159,653 and 138,246, respectively. The body mesh generation is non-uniform as [PERSON] [32] presented, because of the dimensional difference. Collector plates and channels have the same elements Tet/Hybrid and type TGrid, and other parts use the same elements Hex and type Map. It is assumed that the gas mixture is ideal and incompressible, and that the reactions are homogeneous, resulting from low flow rate and low pressure. The model is isothermal and single-phase. The parameters of geometrical configurations, transport properties, and electrochemical properties in the ARH model are listed in Table 1. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Active area & 0.0034 & m\({}^{2}\) \\ Thickness of the membrane & 0.00005 & m \\ Thickness of the diffusion layer & 0.0002 & m \\ Thickness of the catalyst layer & 0.0001 & m \\ Length of the flow channel & 0.05 & m \\ Width of the flow channel & 0.0012 & m \\ Depth of the hydrogen channel & 0.0006 & m \\ Depth of the oxygen channel & 0.0008 & m \\ Thickness of the collector plate & 0.002 & m \\ Number of serpentine turns & 5 & \\ Length of the single channel turn & 0.05 & m \\ Thermal conductivity of the membrane & 0.4 & W (m K)\({}^{-1}\) \\ Thermal conductivity of the gas diffusion layer & 1.2 & W (m K)\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters in the anode relative humidity (ARH) model. Figure 1: The fuel cell numerical analysis schematic. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Thermal conductivity of the catalyst layer & 1.5 & W (m K)\({}^{-1}\) \\ Thermal conductivity of the current collector & 20 & W (m K)\({}^{-1}\) \\ Electrical conductivity of the gas diffusion layer & 2500 & (ohm m)\({}^{-1}\) \\ Electrical conductivity of the catalyst layer & 2500 & (ohm m)\({}^{-1}\) \\ Electrical conductivity of the current collector & 20,000 & (ohm m)\({}^{-1}\) \\ Porosity of the gas diffusion layer & 0.5 & \\ Porosity of the catalyst layer & 0.28 & \\ Membrane equivalent weight & 1100 & kg kmol\({}^{-1}\) \\ Hydrogen reference exchange current density & 4000 & A m\({}^{-2}\) \\ Anode reference concentration & 1 & kmol m\({}^{-3}\) \\ Anode transfer coefficient & 0.5 & \\ Oxygen reference exchange current density & 5.75 & A m\({}^{-2}\) \\ Cathode reference concentration & 1 & kmol m\({}^{-3}\) \\ Cathode transfer coefficient & 0.5 & \\ Open circuit voltage & 0.95 & V \\ Leakage current & 0 & A \\ Electrochemical equivalent of hydrogen & \(1.05\times 10^{-8}\) & kg (A s)\({}^{-1}\) \\ Electrochemical equivalent of oxygen & \(8.29\times 10^{-8}\) & kg (A s)\({}^{-1}\) \\ Reference diffusivity of hydrogen & \(9.15\times 10^{-5}\) & m\({}^{2}\) s\({}^{-1}\) \\ Reference diffusivity of oxygen & \(2.2\times 10^{-5}\) & m\({}^{2}\) s\({}^{-1}\) \\ Reference diffusivity of water & \(2.56\times 10^{-5}\) & m\({}^{2}\) s\({}^{-1}\) \\ Anode catalyst layer surface/volume ratio & \(2\times 10^{6}\) & m\({}^{-1}\) \\ Cathode catalyst layer surface/volume ratio & \(1\times 10^{7}\) & m\({}^{-1}\) \\ Operating temperature & 70 & \({}^{\circ}\)C \\ Anode relative humidity & 40, 55, 70, 85, 100 & \% \\ Cathode relative humidity & 100 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 2: The proton exchange membrane fuel cell (PEMFC) geometry and mesh.