## 2 Approach to Fuel-Cell System Analysis
### Fuel Cell Stack  
The first of the components to be analyzed is the fuel cell stack. First, the hydrogen consumption in g/(hr/cm\({}^{2}\)) can be calculated as a function of the current density, \(i\), in the stack according to the following expression:  
\[\hat{n}_{H_{2},est}=\frac{iMW_{H_{2}}}{2F}\times 3600\times 1000 \tag{1}\]  
where \(i\) is the current density in A/cm\({}^{2}\); \(F\) is the Faraday constant, with a value of 96,485.3 C/mol; and \(MW_{H_{2}}\) is the molecular weight of H\({}_{2}\). It must be noted that while the previous expression evaluates H\({}_{2}\) based on pure reaction stoichiometry, it is typical that the PEMFC anode is fed with an excess of H\({}_{2}\) between 10 and 50% with respect the stoichiometric reference in order to avoid local starvation phenomena in the catalytic layer. However, as it described later on, in production PEMFC systems, the excess H\({}_{2}\) is recirculated, so the stoichiometric flow is a reasonable estimation of real H\({}_{2}\) consumption. In order to convert H\({}_{2}\) consumption into the efficiency of the PEMFC stack, it is necessary to evaluate the relationship between voltage (or power) and current density as a function of the O\({}_{2}\) enrichment level. For this purpose, the experimental information available in the study by [PERSON] [26] was considered. In their study, the influence of the O\({}_{2}\) molar fraction was explored for a 35-cell stack with an active surface area of 232 cm\({}^{2}\). During their study, the sensitivity of the O\({}_{2}\) molar fraction was evaluated up to pure oxygen. However, due to the characteristics of the N\({}_{2}\)-O\({}_{2}\) separation membrane, which is describes afterwards, only the range up to 40% was considered in the present analysis. Utilization of the comprehensive dataset from the Fournier study [26] enabled superior resolution in calculating the influence of O\({}_{2}\) enrichment levels. This dataset provides complete polarization curve data for O\({}_{2}\) molar fractions at 21%, 40%, 60%, 80%, and 100% levels. With the aid of these data, second-order polynomial functions were defined, each corresponding to a distinct current density level. These functions streamline the correlation process between the resulting voltage and the specific O\({}_{2}\) molar fraction deployed in this study--from 21% to 40%, evaluated in steps of 1%. The results from this methodology are depicted in Figure 1 for five of these levels, both in terms of the voltage delivered by the stack as well as its efficiency, calculated considering the H\({}_{2}\) flow and its higher heating value (assuming that water can exit the stack in the liquid phase). The efficiency was calculated assuming that the H\({}_{2}\) consumption is only a function of the current density, as expressed in Equation (1). As it can be seen, a higher O\({}_{2}\) molar fraction helps to increase both voltage (i.e., power) and efficiency, especially at higher current density values, as previously discussed in the Introduction. Finally, to evaluate the balance of the plant, it was necessary to also compute the normalized mass flow consumed in the PEMFC cathode in g/(hr/cm\({}^{2}\)). This parameter can be calculated according to the following expression:  
\[\hat{n}_{c}=\frac{\lambda_{O_{2}}}{X_{O_{2}}}\frac{\hat{n}_{H_{2},est}}{2}= \frac{\lambda_{O_{2}}}{X_{O_{2}}}\frac{iMW_{cutth}}{4F}\times 3600\times 1000, \tag{2}\]  
where \(\hat{n}_{c}\) is the molar flow reaching the cathode, while \(\lambda_{O_{2}}\) is the excess ratio with respect to the stoichiometric value, expressed in terms of O\({}_{2}\). The variable \(X_{O_{2}}\) represents the O\({}_{2}\) molar fraction in the stream arriving at the cathode, and \(MWcath\) stands for the average molecular weight of this stream. During the balance of plant evaluation performed in the current study, it was assumed that the same excess O\({}_{2}\) ratio coefficient would be used regardless the degree of enrichment used.