**2. Theory and Modeling of Proton Exchange Membrane Fuel Cell**

There are three main components of a fuel cell: anode, cathode, and electrolyte. The fuel oxidation and oxygen reduction take place at anode and cathode, respectively. An electrolyte membrane separates the anode and cathode and allows conduction of protons to complete the electric circuit. The oxidation and reduction reaction are shown by Equations (1) and (2), respectively. The overall reaction is represented by Equation (3) [43,44].

Oxidation: 2H<sup>2</sup> <sup>→</sup> 4H<sup>+</sup> + 4e<sup>−</sup> (1)

$$\text{Reduction: }\mathrm{O}\_{2} + 4\mathrm{H}^{+} + 4\mathrm{e}^{-} \rightarrow 2\mathrm{H}\_{2}\mathrm{O} \tag{2}$$

$$\text{Complete reaction: } 2\text{H}\_2 + \text{O}\_2 \to 2\text{H}\_2\text{O} \tag{3}$$

At open circuit potential the cell voltage can be expressed by Equation (4):

$$V\_{\text{Cell}}^{\text{OCV}} = E\_{\text{O}\_2/\text{H}\_2\text{O}}^r - E\_{\text{H}\_2/\text{H}^+}^r \tag{4}$$

At standard conditions (1.0 atm pressure and 25 ◦C), the fuel cell open circuit voltage (OCV) should be 1.229 V. However, the measured OCV at room temperature is around 1.0 V, due to the losses associated with the fuel cell. The cell voltage (*V*cell) is expressed by Equation (5) when the current (*I*cell) is drawn from the cell.

$$V\_{\text{cell}} = E\_{\text{Nernst}} - V\_{\text{activation}} - V\_{\text{concentration}} - V\_{\text{ohmic}} \tag{5}$$

$$E\_{\text{Nernst}} = 1.229 - 0.85 \times 10^{-4} (T - 298.15) + 4.3085 \times 10^{-5} T \left[ \ln \left( P\_{\text{H}\_2} \right) + 0.5 \ln \left( P\_{\text{O}\_2} \right) \right] \tag{6}$$

The activation overpotential of anode and cathode can be expressed as:

$$V\_{\text{activation}} = -\left[\mathfrak{f}\_1 + \mathfrak{f}\_2 + \mathfrak{f}\_3 \times T \times \ln\left(\mathbf{C}\_{\text{O}\_2}\right) + \mathfrak{f}\_4 \ln(i)\right] \tag{7}$$

where *V*activation is the voltage drop due to the activation of redox processing the anode and cathode. The *ξ*<sup>n</sup> represents the parametric coefficients for each cell model, whose values are defined based on theoretical equations with kinetic, thermodynamic, and electrochemical foundations (Mann et al., 2000). The oxygen concentration at the catalyst layer of the cathode (*C*O<sup>2</sup> , mol/cm<sup>3</sup> ) is given by:

$$\mathcal{C}\_{\rm O\_2} = \frac{P\_{\rm O\_2}}{5.08 \times 10^6 \times e^{\frac{498}{T}}} \tag{8}$$

The mass transport affects the concentrations of hydrogen and oxygen at the anode and cathode, which affects the partial pressures of gases. The change in partial pressure of fuel and reductant rely on the electrical current and on the physical features of the system. The voltage drop due to concentration polarization is represented as:

$$V\_{\text{concentration}} = -b \ln\left(1 - \frac{i}{i\_{\text{max}}}\right) \tag{9}$$

where *b* is a parametric coefficient (*V*) that depends on the cell and its operation state, and *i* represents the actual current density of the cell (A/cm<sup>2</sup> ).

The ohmic drop (*V*Ohmic) in Equation (5) is represented as:

$$V\_{\text{Ohmic}} = i \left( R\_M + R\_c \right) \tag{10}$$

$$R\_M = \rho\_M \frac{l}{A} \tag{11}$$

where *R<sup>M</sup>* is the resistance to the transfer of protons through the membrane (Ω), *R<sup>c</sup>* is the charge transfer resistance, *ρ<sup>M</sup>* is the specific resistivity of the membrane for the electron flow (Ω-m), *A* is the active area of the cell (cm<sup>2</sup> ) and *l* is the thickness of the membrane, which separate electrodes. The following numerical expression for the resistivity of the Nafion membrane is used:

$$\rho\_M = \frac{181.6 \times \left[1 + 0.03 \left(\frac{i\_{\rm FC}}{A}\right) + 0.062 \left(\frac{T}{303}\right)^2 \left(\frac{i\_{\rm FC}}{A}\right)^{2.5}\right]}{\left[\lambda - 0.634 - 3\left(\frac{i\_{\rm FC}}{A}\right) \exp\left(4.18 \left(\frac{T-303}{T}\right)\right)\right]} \tag{12}$$

where 181.6/(<sup>λ</sup> − 0.634) is the specific resistivity (Ω-cm) at OCV at 30 ◦C, the exponential term in the denominator is the temperature factor correction if the cell is operating at different temperature. The parameter *λ* is an adjustable parameter with a maximum value of 24. This parameter is influenced by the preparation procedure of the membrane and is a function of relative humidity and stoichiometry relation of the anode gas.

If '*n*' number of stacks are combined then the cell voltage is defined as:

$$V\_{\text{cell}} = n \times (E\_{\text{Nernst}} - V\_{\text{activation}} - V\_{\text{concentration}} - V\_{\text{ohmic}}) \tag{13}$$

At a given temperature (*T*), the partial pressure of fuel (*P*H<sup>2</sup> ) and oxidant (*P*O<sup>2</sup> ) is given by following equations:

$$P\_{\rm H2} = \frac{0.79}{0.21} P\_{\rm O2} \tag{14}$$

$$P\_{\rm O\_2} = P\_{\rm c} - RH\_{\rm c}P\_{\rm H\_2O}^\* - P\_{\rm N\_2} \exp\left(\frac{0.291 \frac{i}{A}}{T^{0.832}}\right) \tag{15}$$

If H<sup>2</sup> and O<sup>2</sup> are used as reactant then the partial pressure of oxygen and hydrogen is given as:

$$P\_{\rm O\_2} = RH\_c P\_{\rm H\_2O}^\* \left[ \left( \frac{\exp\left(\frac{4.192\left(\frac{i}{A}\right)}{T^{1.334}}\right) \left(RH\_c P\_{\rm H\_2O}^\*\right)}{P\_c} \right)^{-1} - 1 \right] \tag{16}$$

$$P\_{\rm H\_2} = R H\_d P\_{\rm H\_2O}^\* \left[ \left( \frac{\exp\left(\frac{1.635\left(\frac{j}{A}\right)}{T^{1.334}}\right) \left(R H\_d P\_{\rm H\_2O}^\*\right)}{P\_d} \right)^{-1} - 1 \right] \tag{17}$$

where *RH<sup>c</sup>* and *RH<sup>a</sup>* are relative humidity at the cathode and anode, respectively. *P<sup>c</sup>* and *P<sup>a</sup>* are the inlet pressure at cathode and anode, respectively. The *P*N<sup>2</sup> is partial pressure of nitrogen at the cathode. The *P* ∗ H2O is saturated vapor pressure (atm), which is calculated as:

$$\log\_{10}\left(P\_{\rm H\_2O}^{\ast}\right) = 2.95 \times 10^{-2} (T - 273.15) - 9.18 \times 10^{-5} (T - 273.15)^2 + 1.44 \times 10^{-7} (T - 273.15)^3 - 2.18 \tag{18}$$
