## 2 Electrochemical Model of PEM Fuel Cell
### Output Voltage of PEM Fuel Cell  
The output voltage at the terminals of a single PEM fuel cell, \(V_{cell}\), has resulted from thermodynamic potential, \(E_{Nernst}\), and the potential losses within the conversion process [16; 41],  
\[V_{cell}=E_{Nernst}-(All\ losses\ in\ a\ fuel\ cell). \tag{4}\]  
The thermodynamic potential \(E_{Nernst}\) can be determined from Equation (5) for the fuel cell. It depends on hydrogen/oxygen at real functioning conditions, the partial pressure of hydrogen (\(P_{H_{2}}^{*}\)) and oxygen (\(P_{O_{2}}^{*}\)), and temperature (\(T\) in Kelvin). \[E_{Nernst}=1.229-(8.5\times 10^{-3})(T-298.15)+(4.308\times 10^{-5})T\ln(P_{H_{2 }}^{*}\times\sqrt{P_{O_{2}}^{*}}) \tag{5}\]  
\[P_{H_{2}}^{*}=\frac{RH_{a}\cdot P_{H_{2}O}}{2}\left[\left(1/\frac{RH_{a}\cdot P _{H_{2}O}}{P_{a}}e^{(1.635(\frac{1}{A})/T^{1.334})}\right)-1\right] \tag{6}\]  
\[P_{O_{2}}^{*}=RH_{c}\cdot P_{H_{2}O}\left[\left(1/\frac{RH_{c}\cdot P_{H_{2}O }}{P_{c}}e^{(4.192(\frac{1}{A})/T^{1.334})}\right)-1\right] \tag{7}\]  
where, \(P_{H_{2}O}\) is the saturation pressure of water vapor in atm, given in Equation (8). \(RH_{a}\) and \(RH_{c}\) are relative humidity of vapor at anode and cathode respectively [41]. \[\log_{10}(\mathrm{P_{H_{2}O}})=2.95\times 10^{-2}(\mathrm{T-273.15})-9.18\times 10^ {-5}(\mathrm{T-\ 273.15})^{2}+1.44\times 10^{-7}(\mathrm{T-273.15})^{3}-2.18 \tag{8}\]  
The losses associated with PEM fuel cells are classified into three categories; activation, ohmic, and concentration losses. These losses caused the variation of the output voltage of the fuel cell with the load current [18]. The relation between cell terminal voltage and the load current is known as the polarization curve. Figure 1 depicts the voltage losses of PEM fuel cells at different current levels. The PEM fuel cell has three categories of losses; activation loss, ohmic voltage loss, and concentration loss. Activation cell loss is the first category and the most dominant loss, especially at the beginning of the reaction. This is because of the slowness of the reaction resulting from electronic barriers that needed to be overcome [17]. This loss is defined as the portion of voltage, lost to enable ions to flow from one electrode to another [26; 27; 28]. It can be determined by Equation (9). \[V_{activation}=-\Big{[}\xi_{1}+\xi_{2}T+\xi_{3}T\Big{(}\ln\Big{(}C_{O_{2}}^{*} \Big{)}\Big{)}+\xi_{4}T\big{(}\ln(I)\big{)}\Big{]} \tag{9}\]  
where the factors \(\xi_{1}-\xi_{4}\) are coefficients for activation loss voltage varied due to load current, \(I\), at actual temperature, \(T\)(K), and oxygen concentration, \(C_{O_{2}}^{*}\). The value of \(C_{O_{2}}^{*}\) depends on the partial pressure of oxygen \(P_{O_{2}}^{*}\) and the actual temperature. \[C_{O_{2}=}^{*}[P_{O_{2}}^{*}/(5.08\times 10^{6})]e^{(498.15/T)} \tag{10}\]  
The second category of losses is the ohmic voltage loss. It has resulted from resistance to electron transfer through cell electrodes and the resistance to proton flow in the solid polymer membrane. \[V_{ohmic}=V_{ohmic}^{electronic}+V_{ohmic}^{proton}=IR_{c}+IR_{m} \tag{11}\]  
where the value of electronic resistance, \(R_{c}\), is unknown. It is considered approximately constant over the cell operating conditions. The value of proton membrane resistance, \(R_{m}\), depends on the membrane characteristics; specific resistivity, \(\rho_{m}\) (\(ohm.cm\)), area \(A\) (cm\({}^{2}\)), and thickness \(l\) (\(cm\)). For Nafion membrane, the thickness considered maybe [115:5 mil (127 \(\mu\)m)] [26; 27; 41].