## 2 PEM Fuel Cell
The term \(\frac{\bigtriangleup G^{0}}{2F}\) is equal to \(E_{0}=1.229\) V. It varies with the temperature according to the following expression:  
\[-\frac{\Delta G}{2F}=1.229+(T-298)(\frac{\Delta S^{0}}{2F}). \tag{9}\]  
Using the standard thermo-dynamical relations as developed in [63], Equation (8) becomes  
\[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]0  
However, in practice as shown in Figure 2, the potential of the PEMFC is significantly less than the values of the theoretical potential, which is given in the above equation, due to the existence of losses, including polarization and interconnection losses. According to [63], the main voltage losses in a PEMFC are the electric losses, which can be classified into three main polarization losses: the activation polarization losses \(V_{act}\), the ohmic polarization losses \(V_{olm}\), and the concentration polarization losses \(V_{conc}\). Therefore, the voltage of an individual cell can be calculated as [64]  
\[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]1  
\(V_{act}\), \(V_{olm}\), and \(V_{con}\) were developed in [64], and their equations are given, respectively, as  
\[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]2  
\[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]3  
\[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]4  
The defined parameters used in the above equations are listed in \[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]5  
The resulting polarization curve of a typical PEMFC showing the voltage and power characteristics are shown in \[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]6 These curves are obtained by using the PEMFC model developed in [64]. Thus, 10 unit cells are connected and stacked in series to attain the required power. \[2H^{+} + \tfrac{1}{2}O_{2} + 2e^{-} \implies H_{2}O\]7 Ideal and actual V-I characteristics of the PEM fuel cell (PEMFC). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbole** & **Value** \\ \hline Cell operating temperature & T & [K] \\ Cell standard temperature & \(T_{std}\) & 298.15 [K] \\ Cell operating current & I & [A] \\ Universal constant of the gases & R & 83.143 [J\(\cdot\)mol\({}^{-1}\)\(\cdot\)K\({}^{-1}\)] \\ Constant of Faraday & F & 96,485.309 [C\(\cdot\)mol\({}^{-1}\)] \\ Maximum current density & \(J_{max}\) & 0.062 A cm\({}^{-1}\) \\ Current density & J & [A\(\cdot\)cm\({}^{-2}\)] \\ Change in the free Gibbs energy & \(\Delta G\) & [J\(\cdot\)mol\({}^{-1}\)] \\ Change of entropy & \(\Delta S\) & [J\(\cdot\)mol\({}^{-1}\)] \\ Enthalpy of formation & \(\Delta H\) & \(-\)285.84 kJ\(\cdot\)mol\({}^{-1}\) \\ Change in the Gibbs free energy at standard condition & \(\Delta G^{0}\) & [J\(\cdot\)mol\({}^{-1}\)] \\ Change of entropy at standard condition & \(\Delta S^{0}\) & [J\(\cdot\)mol\({}^{-1}\)] \\ Electrochemical thermodynamics potential & E & [V] \\ Standard potential of the fuel cell & \(E_{0}\) & 1.229 [V] \\ Membrane active area & A & [162 cm\({}^{2}\)] \\ Hydrogen and oxygen partial pressures & \(P_{\rm H_{2}},P_{\rm O_{2}}\) & [atm] \\ Oxygen concentration & \(\rm\tilde{C}_{O2}\) & [mol\(\cdot\)cm\({}^{-3}\)] \\ Fuel cell voltage & \(V_{fc}\) & [V] \\ Activation losses & \(V_{act}\) & [V] \\ Ohmic losses & \(V_{ohm}\) & [V] \\ Concentration losses & \(V_{conc}\) & [V] \\ Constant parameters & \(\lambda\) & 0.1 [V] \\ Electric charge & \(Q\) & [coulombs] \\ Equivalent resistance of the electron flow & \(R_{eq}\) & \(\Omega\) \\ Proton resistance & \(R_{p}\) & \(\Omega\) \\ Parametric coefficients & \(k_{1},k_{2},k_{3},k_{4}\) & \(\frac{k_{1}=0.9514\) [V], k\({}_{2}=-0.00312\) [V\(\cdot\)K\({}^{-1}\)] \\ \(k_{3}=-7.4\times 10^{-5}\) [V\(\cdot\)K\({}^{-1}\)], k\({}_{4}=1.87\times 10^{-4}\) [V\(\cdot\)K\({}^{-1}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter explanations of the PEMFC model.