## 2 Overall Analysis Approach and Implemented Methods
### Parametric Definition of Tanks Geometry
#### 2.3.1 Full-Section Layout  
The design concept behind a full-section layout consists of embarking large quantities of LH\({}_{2}\) by replacing the fuselage volume occupied by passengers and containers with fuel tanks. Such a solution has the potential advantage of allowing long-range flights, with a penalisation in terms of maximum transportable payload. The full-section layout is defined for an elliptic fuselage internal section, whose width and height are given by the semi-major axis (\(w\)) and the semi-minor axis (\(h\)), respectively. These parameters also determine the aspect ratio \(f\), according to Equation (3). \[f=\frac{h}{w} \tag{3}\]  
Normalising the semi-major axis (\(w=1\)), the following analytical treatment can refer to non-dimensional variables. The space available for the main tank (or \"Tank 0\") is defined as the biggest circle (C0) inscribed in the ellipse, fulfilling the condition of tangency in the point \(T_{0}\equiv(cos\ t_{0},f\cdot sin\ t_{0})\), where \(t_{0}\) is the arbitrary chosen polar coordinate indicated in Figure 8a. The picture provides the graphical representation of C0, showing its symmetry to the horizontal axis. The centre of C0 has coordinates \(C_{0}\equiv(x_{0},\ y_{0})\) given by Equation (4), whereas the radius \(r_{0}\) is given by Equation (5). \[C_{0}:\ \left\{ \begin{array}{c}x_{0}=cos\ t_{0}\cdot\left(1-f^{2}\right)\\ y_{0}=0\end{array} \right. \tag{4}\]  
\[r_{0}=f\sqrt{f^{2}\cdot cos^{2}t_{0}+sin^{2}t_{0}} \tag{5}\]  
The space available for side tanks, \"Tank 1\" and \"Tank 2\", is then defined setting the conditions that describe the circles C1 and C2, respectively, at left and right of C0 as Figure 8b shows. As Equations (6)-(9) show, once C0 is known, C1 and C2 centres and radii can be defined univocally. \[C_{1}:\ \left\{ \begin{array}{c}x_{1}=min\left(\frac{x_{0}-r_{0}-1}{2},f^{2}-1 \right)\\ y_{1}=0\end{array} \right. \tag{6}\]  
\[r_{1}=min\left(\frac{x_{0}-r_{0}+1}{2},f^{2}\right) \tag{7}\]  
\[C_{2}:\ \left\{ \begin{array}{c}x_{2}=max\left(\frac{x_{0}+r_{0}+1}{2},1-f^{2} \right)\\ y_{2}=0\end{array} \right. \tag{8}\]  
\[r_{2}=min\left(\frac{1-x_{0}-r_{0}}{2},f^{2}\right) \tag{9}\]  
The following two steps consist of defining the space available for two additional couples of tanks (\"Tanks 3\" and \"Tanks 4\") through the construction of the circles C3 (Figure 8c) and C4 (Figure 8d), which are tangent to the ellipse and to C0 as well as to C1, if in the case of C3, or C2, if in the case of C4. Considering the C3 circle belonging to the upper semi-plane, its centre \(C_{3}\equiv\left(x_{3},~{}y_{3}\right)\) and radius \(r_{3}\) can be found by solving the system given by Equation (10), in which the first line introduces the condition of tangency of C3 to both C0 and C1 as a constraint for \(r_{3}\), the second line constrains the centre \(C_{3}\) to belong to the line perpendicular to the ellipse in an unknown point defined by the polar coordinate \(t_{3}\), and the third line sets the tangency of C3 to the ellipse through an additional constraint for \(r_{3}\). Equation (10) was solved numerically using the MATLAB _pasolve_ function and an analogous system was solved to determine C4 centre and radius. \[\left\{ \begin{array}{l}r_{3}=\sqrt{\left(x_{3}-x_{0}\right)^{2}+\left(y_{3}- y_{0}\right)^{2}}-r_{0}=\sqrt{\left(x_{3}-x_{1}\right)^{2}+\left(y_{3}-y_{1} \right)^{2}}-r_{1}\\ \frac{sin\,t_{3}\cdot\left(x_{3}-cos\,t_{3}\right)-f\cdot cos\,t_{3}\cdot\left( y_{3}-f\cdot sin\,t_{3}\right)=0}{\sqrt{\left(x_{3}-x_{0}\right)^{2}+\left(y_{3}-y_{ 0}\right)^{2}}-r_{0}=\sqrt{\left(x_{3}-cos\,t_{3}\right)^{2}+\left(x_{3}-f \cdot sin\,t_{3}\right)^{2}}}\end{array} \right. \tag{10}\]  
The procedure here described constitutes an algorithm which allows to define the layout of a large number of tanks by varying just one parameter, \(t_{0}\), in the range \(\left[0,~{}~{}\pi\right]\).