## 2 Materials and Methods
### Dynamics and Thermal Behavior of PEM Fuel Cell  
The mechanism underlying the generation of cell voltage is explicated through the movement of hydrogen from the anode to the cathode, concomitant with the transfer of electrons. However, the intricate computation of fuel cell voltage is complicated by the inclusion of voltage losses within the system. Therefore, it becomes crucial to consider these voltage losses and the diverse factors influencing cell behavior by formulating empirical equations, serving as pivotal tools for accurately predicting cell performance. The voltage losses within fuel cells encompass a spectrum of phenomena, including activation, ohmic, and concentration polarizations. In practice, empirical equations account for specific parameters such as temperature, pressure, and reactant flow rates, all of which significantly impact the magnitude of these voltage losses. Optimizing the values of these parameters and understanding their intricate interplay become crucial for enhancing overall fuel cell performance and efficiency. The relationship between cell voltage and current density is graphically illustrated in Figure 7, depicting the characteristic discharge behavior of a fuel cell. This graphical representation elucidates the voltage response across different operational conditions, facilitating a comprehensive grasp of the performance limitations and potential avenues for optimization. By leveraging a combination of empirical equations and experimental data, a comprehensive understanding of the complex dynamics governing fuel cell voltage is attainable, thereby driving advancements in the design and operational efficiency of fuel cell systems. Details of the fuel cell stack are depicted in Appendix A.3. Figure 6: Fuel cell porosity. Figure 7: PEM fuel cell polarization curve. In Equation (2), the cell open-circuit voltage is expressed as a Nernst voltage equation without voltage losses in a fuel cell, considering pressure and temperature. \[\mathrm{V_{F0}}=\mathrm{V_{F0}^{0}}-0.85\times 10^{-3}(\mathrm{T_{FC}}-298.15)+ \frac{RT_{FC}}{2F}\mathrm{ln}\left(\frac{P_{H2}P_{O2}{}^{0.5}}{P_{H2O}}\right) \tag{2}\]  
where \(\mathrm{V_{F0}}\) is the open circuit voltage, \(\mathrm{V_{F0}^{0}}\) is the initial open circuit voltage, \(\mathrm{T_{FC}}\) is the fuel cell temperature, \(R\) is the universal gas constant, and \(F\) is the Faraday constant. \(P_{H2}\), \(P_{O2}\), and \(P_{H2O}\) denote hydrogen pressure, oxygen pressure, and water vapor pressure, respectively. \(P_{H2}\) hydrogen fuel pressure is equal to 0.5 bar gauge pressure. The \(P_{O2}\) value changes from 0 to 0.5 bar gauge pressure. The voltage losses can be found using Equation (3). Here, activation \(V_{\mathrm{act}}\) and concentration voltage \(V_{\mathrm{con}}\) losses are calculated empirically. \[V_{\mathrm{loss}}=\{V_{\mathrm{act}}\}+\{V_{\mathrm{ohmic}}\}+\{V_{\mathrm{ con}}\}=\left\{ \begin{array}{l}\frac{RT_{FC}}{n\pi F}\mathrm{ln}\left( \frac{I}{I_{0}}\right)\end{array} \right\}+\left\{1\frac{t_{m}}{\sigma} \right\}+\left\{1.1\times 10^{-4}-12\times 10^{-6}(\mathrm{T_{FC}}-273)e^{0.008I}\right\} \tag{3}\]  
where \(V_{\mathrm{ohmic}}\) is the ohmic losses, \(I\) is the current, \(I_{0}\) is the reference current, \(t_{m}\) is the membrane thickness, and \(\sigma\) is the conductivity. The transfer coefficient \(\alpha\) equals to 0.5. If all cells are connected in a series, the stack voltage equals the cell voltage multiplied by the cell number \(N_{fc}\). Equation (4) expresses consumed hydrogen \(m_{H2}\) in the fuel cell reaction. \[m_{H2}=\frac{IM_{H2}N_{fc}A_{FC}}{2F} \tag{4}\]  
Here, \(M_{H2}\) is the molecular weight of hydrogen and \(A_{FC}\) is the active area of the fuel cell. Figure 8 shows the hydrogen consumption rate theoretically and experimentally with respect to power. Experimental consumption results are higher than theoretical calculation results due to unused or leaked hydrogen in the system.