# An Analysis of the Altitude Impact on Roots Compressor Operation for a Fuel Cell System
## 2 Materials and Methods
### Compressor Selection and Modeling  
As previously introduced, the fuel cell can be supercharged by either radial or Roots compressors. In general, a radial compressor map exhibits pressure ratio curves that reach higher values with decreasing corrected flow at a constant corrected speed. However, this increasing trend disappears when approaching the surge conditions, which are associated with low-flow-velocity environments and high-pressure-ratio conditions, inducing the appearance of backflows. These backflows induce an instability into the flow that first induces a drop in efficiency and eventually the appearance of severe vibrations and mechanical damage, limiting their boosting capability for a given mass flow value. Instead, since Roots compressors are volumetric machines with no direct contact between low-pressure and high-pressure flow areas, surge is much less prone to appear, and the maximum pressure ratio is mainly limited by the maximum temperature reached after compression. Therefore, a wider range of pressure ratios is possible for a given mass flow value, which is controlled by the rotational speed. Also, the efficiency (although it has a lower peak) is less sensitive to the flow rate compared to radial compressors. In the context of altitude operation, being less impacted by surge limitations can be a significant advantage due to the capability to compensate for the low ambient pressure by a higher-pressure-ratio operation, making it possible to maintain more similar conditions inside the fuel cell stack with a limited impact on efficiency and power consumption.  
Figure 3 summarizes the characteristic maps of the roots compressor used for the current study in terms of isentropic efficiency (A) and compressor corrected speed (B) as a  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (Units)** \\ \hline Number of cells in stack & 1000 \\ Cell area & 280 cm\({}^{2}\) \\ Membrane thickness & 125 \(\upmu\)m \\ Gas diffusion layer (GDL) thickness & 160 \(\upmu\)m \\ Catalytic layer (CL) thickness & 10 \(\upmu\)m \\ Symmetry factor & 0.5 \\ Density of dry membrane & 2000 kg/m\({}^{3}\) \\ Equivalent weight of dry membrane & 1.1 kg/mol \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fuel cell parameters.  
Figure 2: Validation of the fuel cell stack model.  
function of corrected mass flow and pressure ratio, which are provided by the compressor manufacturer. As can be seen, in this kind of compressor technology, the speed mainly controls the air mass flow provided, since the constant speed lines are practically vertical. Instead, at a given compressor speed, there is a wide variation in the pressure ratio, with a maximum value around 2.8 achieved up to a corrected mass flow of 0.35 kg/s (corresponding to \(\sim\)15,000 rpm), from which the maximum achievable pressure ratio starts going down. Finally, it can also be noted that the peak isentropic efficiency reaches 70%, achieved in a wide area ranging from 0.15 to 0.35 kg/s and from 1.4 to 2.1 for the pressure ratio. From that point, the main efficiency deterioration is produced when the corrected mass flow reduces below 0.05 kg/s.