## 2 Overall Analysis Approach and Implemented Methods
### Thermodynamic Analyses
#### 2.5.2 Tank Heat Flow  
The heat flow across the tank walls (\(Q\) in Equation (25))) is computed considering three main thermal exchange stages: the external heat exchange, the heat conduction across the tank insulation, and the heat exchange inside the fluid. For the sake of simplicity, the following assumptions were introduced:  
1. Each tank has the same structure shown in Figure 7 and it is assumed that the foam insulation is in direct contact with the external air, which flows at the aircraft flight speed. 2. The heat exchange across the tank bulkheads occurs in the same way as for the external diameter. This assumption is less conservative, because the temperature inside the fuselage, in cruise, could be higher than the external one (non-conservative). Moreover, the heat exchange phenomena at the boundary of the insulation inside the fuselage is natural convection, that is different from the forced convection occurring at the boundary with the external environment (conservative). The heat exchange across the bulkheads, however, is strongly dependent upon the passenger compartment wall design, which is out of the scope of this work, so the assumption is considered valid. 3. The computation of thermal exchange surfaces was simplified, assuming that the boundaries of every heat exchange phenomenon have the shape of a cylinder of the same overall length of the tank with plane bulkheads. The external heat exchange is characterised by forced convection and thermal radiation between the external skin of the insulation and the environment; these phenomena are considered by means of the equations reported in [25]. For the convection, we have:  
\[Nu=\frac{h_{conv}\cdot L}{k_{air}}=0.03625\cdot Pr^{0.43}\cdot Re^{0.8} \tag{33}\]  
\[Pr=\frac{\mu_{air}\cdot Cp}{K_{air}} \tag{34}\]  
\[Re=\frac{\mu_{air}\cdot p\cdot L}{K_{air}} \tag{35}\]  
where \(Nu\), \(Pr\), and \(Re\) are the Nusselt, Prandtl, and Reynolds number typical of this phenomenon, \(h\) is the heat transfer coefficient [W/(m\({}^{2}\) K)], \(L\) is the tank length, \(k_{air}\) is the air thermal conductivity, \(\mu_{air}\) is the dynamic viscosity of the air, \(Cp\) is the air specific heat at constant pressure, and \(v\) is the airspeed. The radiation to the environment is characterised by the following formula [25; 41]:  
\[h_{rad}=\sigma\ \varepsilon\left(T_{skin}^{2}+T_{atm}^{2}\right)\left(T_{skin} +T_{atm}\right) \tag{36}\]where \(h_{rad}\) is the heat transfer coefficient of the phenomenon, \(T_{skin}\) is the aircraft skin temperature, \(T_{atm}\) is the environment temperature, \(\sigma\) is the Stefan-Boltzmann constant, and \(\varepsilon\) is the emittance of the radiating surface, which is set to 0.95, corresponding to a white painted surface [25]. As for the assumptions (a), (b), and (c), the total external resistance is computed as in Equation (37), where \(S_{ext}\) is the total external surface of the tank at the external foam interface. \[TR_{ext}=\frac{1}{S_{ext}\ (h_{conv}+h_{rad}\ )} \tag{37}\]  
The thermal resistance of a cylindrical foam layer \(TR_{ins}\) (tube-like problem) is given by the Equation (38), as reported in [25; 42]:  
\[TR_{ins}=\frac{1}{2\pi l}\frac{ln(\frac{r_{0}}{r})}{K_{ins}} \tag{38}\]  
where \(ri\) and \(ro\) are the foam internal and external radius, respectively, \(l\) is the cylinder length, and \(K_{i}\) is the thermal conductivity of the foam. Equation (38) assumes that the foam thermal conductivity is constant along the radius, but this condition is not verified due to the thermal gradient inside the foam, and the foam's thermal conductivity dependence upon temperature. Because of this fact, it is necessary to split the foam thickness into a number of concentric shells which were set to 40, according to [25]. The heat exchange inside the fluid is characterised by the natural convection in the liquid and gaseous phases.