## 2 Methodology
### Hydrogen Vessel Design  
We propose that a fuel cell system can be used as an alternative for an auxiliary power unit (APU) in the commercial aircraft Boeing 777, in which the APU provides power of 440 kW when the aircraft is on the ground and the main engines are not engaged for an hour [34]. A proton-exchange membrane fuel cell (PEMFC) with 60% efficiency will operate for one hour. The hydrogen required to produce 440 kW for one hour of continuous operation can be determined to design the storage vessel using Equation (1) [35; 36].  
\[Hydrogen\;amount\;required\;(\text{Kg})=\frac{P\times t}{n\times LHV} \tag{1}\]  
where P is the power in kW, \(t\) is the time in hours, \(n\) is the efficiency of the fuel cell system, and LHV is the lower heating value of hydrogen (33.33 kwh/kg or 120,000 kJ/kg).  
\[Hydrogen\;required\;(\text{Kg})=\;\frac{440\;\times\;1}{0.60\;\times\;33.33}=22\; \text{kg}\]  
Compressing 1 kg of hydrogen to 35 MPa will give a volume of 45 litres and 26 litres for a pressure up to 70 MPa [37]. Therefore,  
\[22\;\text{Kg}\;of\;hydrogen=22\;\times\;45=990\;\text{L}\;=\;0.99\;\text{m}^{3}.\]  
The volume of the vessel is equal to the volume of the required hydrogen, and for a cylindrical vessel with a diameter of 0.6 m, the volume of the vessel can be determined by Equation (2) [36]:  
\[Volume\;of\;cylindrical\;vesel\;(V)=\;\pi\;r^{2}\;h \tag{2}\]  
where h is the height of the vessel (in m), V is the volume (in m\({}^{3}\)), and r is the radius (in m); thus, h will be equal to 3.5 m. This cylindrical hydrogen vessel is assumed to be designed with hemispherical ends. The volume of a hemisphere is half the volume of the sphere and can be calculated as given in Equation (3) [36]:  
\[Volume\;of\;the\;Hemisphere\;=\;\frac{1}{2}*\frac{4}{3}*\pi*(r)^{3} \tag{3}\]  
\[Volume\;of\;the\;hemisphere\;for\;one\;side=\;\frac{1}{2}\times(0.11)=0.057\; \text{m}^{3}\]  
\[Volume\;of\;the\;middle\;part=overall\;volume\;-\;(2\;\times\;one\;side\;volume\;of\;hemisphere)\]  
\[Volume\;of\;the\;middle\;part=0.99\;-\;(2\;\times\;0.057)=0.876\;\text{m}^{3}\]  
\[Height\;of\;the\;middle\;part=\;\frac{0.876}{\left(\pi\right)\left(0.3\right)^{ 2}}=3.09\;\text{m}\]  
The overall height of the vessel is 3.5 m, and the height of the middle part is 3.09 m, with a 0.3 m radius. ANSYS was used to design 2D axisymmetric vessels with a capacity of22 kg of hydrogen and with wall thicknesses of 25, 50, 100, and 150 mm. The 2D and 3D designs of the hydrogen vessel are presented in Figure 1.  
Titanium, 304 stainless steel, and aluminium were used in the ANSYS design and analyses at internal pressures of 25, 50, 60, and 70 MPa, thus confirming the structural integrity of these materials; the properties of these materials are given in Table 1. Mesh generation in ANSYS is vital for breaking down the complex geometry into manageable elements to accurately analyse structural behaviour. The 2D body is meshed with quadrilateral elements for improved accuracy. Element quality was maintained close to 1.0, indicating good mesh quality. The total element after meshing is 224 and 817 nodes. The element size was set to 30 mm for optimal analysis. The mesh generation can be seen in Figure 2a. The boundary conditions are crucial in pre-processing. Frictionless support was applied at the vessel's edge to prevent deformation. Uniform pressure loads of 35, 50, 60, and 70 MPa were applied inside the vessel. These loads simulate real-world scenarios and are crucial for evaluating structural performance. The boundary conditions applied can be seen in the Figure 2b.