## 2 Overall Analysis Approach and Implemented Methods
### Thermodynamic Analyses
#### 2.5.1 Tank Thermal State Dynamics  
Since the H\({}_{2}\) cryogenic storage condition was considered, the internal tank temperature is always lower than the environmental one. This fact generates a heat flux from the environment to the stored H\({}_{2}\), which leads to a variation in temperature and pressure of the fluid. The laws governing the pressure variation over time in a saturated H\({}_{2}\) homogeneous fluid, in the presence of a heat flux, are reported in [31]. The present paper assumes that the \"Passive thermodynamic venting system\" case is valid (formula n. 8 in [31]). However, for the sake of simplicity, it is assumed that only the gaseous phase is vented, hence neglecting the refrigeration effect due the vented liquid phase. Therefore, the aforementioned formula is simplified in Equation (25). \[\frac{dP}{dt}=2\frac{\Phi}{V}\bigg{[}Q-m_{o}\big{(}h_{g}-h_{l}\big{)}\bigg{(}X _{o}+\frac{\rho_{g}}{\rho_{l}-\rho_{g}}\bigg{)}\bigg{]} \tag{25}\]  
\[m_{o}=m_{fuel}+m_{pent} \tag{26}\]  
\[X_{o}=\frac{m_{vent}}{m_{o}} \tag{27}\]  
\[\Phi=\frac{1}{\rho\Big{(}\frac{\partial u}{\partial P}\Big{)}_{\rho}} \tag{28}\]  
In Equation (25), \(Q\) is the heat flux toward the H\({}_{2}\), \(V\) is the tank volume, \(h_{g}\) and \(h_{l}\) represent the unit mass enthalpy of the gaseous and liquid phase, respectively, whereas \(\rho_{g}\) and \(\rho_{l}\) are the respective densities. Because of the presence of the venting valve, the outgoing mass flow (\(m_{o}\), [kg/s]) is represented by the sum of the mass flow of the liquid phase, headed to the engines (\(m_{fuel}\)), and the mass flow of the gaseous phase, expelled by the venting valve (\(m_{vent}\)), as reported in Equation (26). The quality of the outgoing mass flux (\(X_{o}\)) is the ratio between the gaseous phase mass flow and the total outgoing mass flow (Equation (27)). According to [25; 31], Equation (25) shows a factor 2, in order to take the fluid stratification into account, which is present if proper fluid mixing devices are omitted. The fluid stratification leads to higher pressure variations for a given energy input, if compared to a case in which homogeneous fluid is assumed. The term \(\Phi\) in Equation (25) is the \"energy derivative\", defined as in Equation (28). More details about its estimation are reported in Appendix A. The pressure variation inside the tank is determined using Equation (25) in a Eulerian integration in parallel with the mission analysis simulation, as explained in Section 2.6. The mission simulation determines the environmental conditions and the fuel flow (\(m_{fuel}\)). The venting valve mass flow (\(m_{vent}\)) in Equation (25) is determined for the three conditions summarised in Equation (29):  
* While the tank pressure is lower than the venting pressure \(P<P_{vent}\), \(m_{vent}\) is set to zero. * When \(P\geq P_{vent}\), the venting valve is activated. In this paper, the venting valve mass flow (\(m_{vent}\)) is considered variable in such way that the pressure inside the tank is kept constant. The computation of \(m_{vent}\) is completed imposing \(dP/dt=0\) in Equation (25), obtaining the second case of Equation (29) (details in Appendix B). By substituting the computed \(m_{vent}\) in Equation (25)), the pressure will be kept constant over time. * While \(P\geq P_{vent}\), an increase in fuel consumption, or a decrease in environmental temperature, will cause a pressure decrease; however, since \(dP/dt\) was set to zero, this condition leads to a solution where \(m_{vent}<0\). In this condition, the computed mass flow is clearly non-physical, since the venting valve is a one-way item. For a correctconsideration of this last case, when simultaneously \(P\geq P_{vent}\) and \(m_{vent}\leq 0\), \(m_{vent}\) is set to zero:  
\[m_{vent}= \begin{cases}0&P<P_{vent}\\ \left(-b+\sqrt{b^{2}-4 ac}\right)\ /\left(2a\right)\ if\ P\geq P_{vent}\ \text{and}\ m_{vent}>0\\ 0&P<P_{vent}\ \text{and}\ m_{vent}\leq 0\end{cases}