## 2 Approach to Fuel-Cell System Analysis
### Balance of Plant Description  
In Figure 3, the usual layout for the balance of plant for PEMFCs systems is highlighted. In this diagram, the black lines correspond to the cathode input stream (obtained from the ambient air), the red lines relate to the anode input stream (hydrogen), and the orange lines represent the water stream used for the humidification of the fuel cell stack. In particular, two different configurations are introduced: an atmospheric system, where the air stream is supplied at the ambient pressure, and a supercharged system, where an electrical compressor is included upstream the stack. Figure 2: Cathode mass flow and total inlet air mass flow per unit of active PEMFC area as a function of current density and O\({}_{2}\) molar fraction. One of the more important aspects to be taken into account relates to the thermal management of gaseous streams. This is due to the fact that the fuel cell needs to be operated in a temperature range between 60 and 90 \({}^{\circ}\)C in order to reach the best possible efficiency while guaranteeing the presence of liquid water at the proton exchange membrane. In the case of atmospheric systems, electrical heaters can be used to provide the required energy to both H\({}_{2}\) and air up to the working temperature, which, for this study, was assumed to be 80 \({}^{\circ}\)C. When working on the supercharged system, this electrical system is not required in principle due to the temperature increase associated with the compression itself, while a heat exchanger may be required to cool the air depending on the pressure ratio of operation (\(\pi\)). In supercharged fuel cells, \(\pi\) is in the range of 1.2 to 2.2 [38]. This variation suggests that, depending on the compressor efficiency, the air temperature at the heat exchanger inlet could exceed 250 \({}^{\circ}\)C. Additionally, the H\({}_{2}\) line includes an expansion valve to adapt the tank pressure to the stack operating pressure (approximately equal to the pressure on the air side), plus a recirculation circuit to take advantage of the excess hydrogen from the fuel cell. In the air circuit, the outlet gases from the fuel cell cathode, mainly consisting of the excess air and the water produced by the electrolytic reaction, pass through a separator to recover and reuse the water in the humidifier, included to guarantee the correct humidification of the fuel cell membrane (relative humidity between 60 and 90%). Figure 4 shows the balance of plant proposed for the system with on-board O\({}_{2}\) enrichment in the cathode stream. Due to the level of pressure imposed by the O\({}_{2}\)-N\({}_{2}\) separation membrane, it is necessary to incorporate a two-stage boosting system. The first compression stage is achieved with a turbocharger, which operates under a certain pressure ratio \(\pi_{1}\). After this compression, intermediate cooling is carried out using Heat Exchanger 1 to return to a temperature that is as close as possible to the ambient one before entering the second compression stage, so that the efficiency and energy consumption of the later are not negatively impacted. The second compression is carried out through an electric compressor, which is controlled to provide the necessary pressure at the inlet of the polymeric O\({}_{2}\)-N\({}_{2}\) separation membrane. The output of this compression is cooled down again in Heat Exchanger 2, with the objective to bring it down to a temperature of around 80 \({}^{\circ}\)C, close to the optimal operating temperature for both the polymeric O\({}_{2}\)-N\({}_{2}\) separation membrane and the fuel cell stack. To safeguard the N\({}_{2}\) membrane's operation in circumstances where the H\({}_{2}\) stream is unable to effectively cool compressed air to 80 \({}^{\circ}\)C, a third exchanger is employed (utilizing water as the coolant) to ensure that the air enters the separation membrane at the specified temperature.