## 2 Overall Analysis Approach and Implemented Methods
### Parametric Definition of Tanks Geometry
#### 2.3.1 Full-Section Layout
The choice of limiting the maximum number of tanks to 7 was set for practical reasons, both  
Figure 8: Full-section layout: steps for definition of the space available for the tanks within the fuselage cross-section: (**a**) definition of the main tank cross-section, (**b**) introduction of mid-size side tanks, (**c**) introduction of an additional couple of tanks on the left-side, (**d**) introduction of an additional couple of tanks on the right-side. related to the technological limitations in manufacturing and using vessels with small radii, and to the computation time of the coupled mission-thermodynamic simulation, which is almost proportional to the number of tanks. By implementing this algorithm in a MATLAB code, some geometric performance analyses were carried out varying the \(t_{0}\) parameter, in order to assess the capability of covering the available area while limiting the external perimeter, since it affects the heat exchange between the LH\({}_{2}\) at cryogenic temperature and the external atmosphere. For this purpose, two geometric performance indicators were defined: \(e_{a}\), the fraction of area covered by all the circles (\(A_{TOT}\)) defined with respect to the ellipse area as shown in Equation (11), and \(e_{p}\), the normalised area-perimeter ratio defined as in Equation (12), where the denominator is \(A/p^{2}\) for the circle, i.e., the maximum achievable value for a plane figure. \[e_{a}=\frac{A_{TOT}}{\pi f} \tag{11}\]  
\[e_{p}=\frac{A_{TOT}/p_{TOT}{}^{2}}{1/(4\pi)} \tag{12}\]  
Figure 9 shows the results of \(e_{a}\) and \(e_{p}\) calculated for \(f\) = 0.75 and various \(t_{0}\) values, both for the ideal case of no constraints to the minimum external radius \(R\) (a) and for the case in which the minimum value was set to 10% of the fuselage half-width (b). Figure 10 provides some examples of cross-sections to better visualise the configurations. Figure 10: Examples of full-section layouts without minimum radius constraints for three different \(t_{0}\) values: (**a**) \(0^{\circ}\), (**b**) \(70^{\circ}\), (**c**) \(90^{\circ}\). Figure 9: Geometric performance of full-section layout for \(f\) = 0.75 and various \(t_{0}\) values: (**a**) no constraints on minimum radius values, (**b**) minimum radius set to 10% of fuselage half-width. Whereas in Figure 9a each configuration with \(t_{0}\
eq 0\) is composed of 7 tanks (see Figure 11), the minimum radius constraint reduces this number to 4 for a significant part of \(t_{0}\) range. Understandably, further increasing the radius threshold extends the reduced tanks region and can bring to configurations with only 1 or 2 tanks. Considering the trends shown in Figure 9, the main remarks concern:  
* The low sensitivity of \(e_{a}\) to \(t_{0}\), for which the best solution is given by the case at \(t_{0}=90^{\circ}\) (\(e_{a}=85.7\%\)) but lower angles show a small reduction in \(e_{a}\) hence still providing a good area coverage (see Figure 12a for an example);
* The previous observation is valid both with and without the minimum radius constraint;
* \(e_{p}\) is quite insensitive to \(t_{0}\) if the minimum radius constraint is not applied;
* The suppression of small tanks due to the minimum radius constraint causes better values of \(e_{p}\).