3.4.2. Hydrogen Fuel Cell

Proton exchange membrane (PEM) fuel cells offer a higher energy density than batteries, around 500 Wh/kg [28,29], in a unit that is still clean and hydrocarbon-free, mechanically simple, operates near ambient temperature, and produces no harmful emissions. In terms of fuel cell power density, there are several works estimating its improvement for a value of 800 W/kg [30]. The problems with hydrogen storage and the boil-off are also less significant in aviation compared to cars, and even lesser even for eVTOL aerial vehicles, because of the shorter duration missions, typically a few hours compared to weeks. Thus, the significant progress made in the past decade toward lighter gaseous hydrogen storage can be exploited to full advantage. A PEM pack consists of identical cells, each with a voltage *Ecell* (V) given by [30,31]:

$$E\_{cell} = E\_0 + \frac{R \cdot T}{2F} \cdot \ln(P\_{H\_2} \cdot P\_{O\_2}^{-0.5}),\tag{17}$$

where *E*0, *R*, *T*, *F*, *PH*<sup>2</sup> , and *PO*<sup>2</sup> are, respectively, the thermodynamic reversible voltage based on the higher heating value (HHV) of hydrogen (1.23 V), the universal gas constant (8.314 J/molK), the operating temperature, the Faraday constant (96,485 C/mol), the partial pressure of hydrogen (Pa), and the partial pressure of oxygen (Pa). The nominal voltage of the PEM hydrogen fuel cell stack *Vstack* (V) is given by

$$V\_{stack} = \aleph\_{cell} \cdot E\_{cell}.\tag{18}$$

The fuel cell area is defined as:

$$A\_{cell} = \frac{P\_{FC}}{(p\_{cell} \cdot N\_{cell})} \,\prime \tag{19}$$

where *PFC* and *pcell* are, respectively, the required electrical power and the power density of a single cell. For the battery case, the fuel cell output power required for the flight mission *PFC* and the corresponding hydrogen consumption *HC*(*kg*/*h*) can be estimated by

$$\begin{cases} P\_{\text{FC}} = N\_m \cdot \frac{P\_m}{\eta\_{\text{fc}} \cdot \eta\_{\text{FC}}},\\ HC = \frac{P\_{\text{FC}}}{LHV \cdot \eta\_{\text{FC}}}, \end{cases} \tag{20}$$

where *ηFC* and *LHV* are, respectively, the fuel cell stack efficiency and the low heating value of hydrogen (33.3 Wh/g). At the current technology level, the efficiency of the PEMFC is approximately 40∼50%, and if there is no information about the polarization curve of a single cell, this value can be used for sizing. Thus, the hydrogen fuel cell mass *mFC* is given by [30]:

$$m\_{\rm FC} = \frac{Ncell \cdot k\_A \cdot \rho\_{\rm cell} \cdot A\_{\rm cell}}{1 - \eta\_{\rm ov}} \cdot (1 + f\_{\rm BOP}) \,, \tag{21}$$

where *kA*, *ρcell*, *fBOP*, and *ηow* are, respectively, the ratio of the cross-sectional area to the electrode area of a single cell (fixed at a value of five), the area density of a single cell (fixed at 1.57 kg/m), the ratio of the BOP weight to the HFC weight (with a value that varies depending on the HFC configuration; in this paper, a value of 0.2 is considered), and the overhead fraction to account for gaskets, seals, connectors, and endplates (fixed at 0.3). The flight time in the case of a fuel cell is given by the following expression:

$$\text{At}\_{flight} = \frac{60 \cdot LHV}{P\_{FC}}.\tag{22}$$

For the hydrogen tank, a type 4 tank was selected among the gaseous hydrogen tanks. Liquid hydrogen is 800 times less in volume and has a higher energy density than gaseous hydrogen, but it must be kept at a low temperature, which limits its use in HFC UAVs [31,32]. A regression model that estimates the hydrogen tank mass *Mtank* (kg), based on the amount of hydrogen *mH*<sup>2</sup> (g) required for the flight mission, is established as shown in Figure 7d. The data are based on the tank type, e.g., such as types 3 or 4 [33], and this model can be expressed as:

$$M\_{tank} = -0.000047 m\_{H\_2} ^2 + 0.0367 m\_{H\_2} - 0.126. \tag{23}$$
