## 3 Discussions and Results
### Application to a Fuel Tank Inerting Process
Note that the increase in oxygen concentration at \(t=78\) min is due to air ingestion in the fuel tanks during the descent phase (source [14]). exercise, it is considered the flight cruising phase (at \(t=40\) min), in which the nitrogen flow value is given by Figure 7:  
\[4.55\,SCFM=0.044\,\text{kg/min}. \tag{34}\]  
The gas density (\(D\)) is obtained based on the predominant nitrogen and remaining oxygen. It is considered an average value of 95% of nitrogen and 5% of oxygen during the cruise phase (stabilized flight):  
\[D=0.95D_{nitrogen}+0.05D_{oxygen}=0.391\,\text{kg/m}^{3}. \tag{35}\]  
The A320 centre fuel tank is almost of rectangular shape with a mean height of 1.5 m and a mean wide of 6 m:  
\[A_{T}=9\,\text{m}^{2}. \tag{36}\]  
Then, \(a\) can be determined as per (33) \(a=0.0125\) m/min. After calibration, the problem \(P\) is summarized as  
\[ \begin{array}{c}N_{t}=\rho\Delta N+a\cdot\
abla N+\Theta^{n},\\ \Theta_{t}=\sigma\Delta\Theta+a\cdot\
abla\Theta-N^{m},\\ n=0.586\,\text{ }m=0.025\,\text{ }a=0.0125\text{ m/min},\\ \rho=\sigma=0.196\text{ cm}^{2}/\text{s},\\ N_{0}(x),\Theta_{0}(x)>0\in L^{1}_{loc}(\text{R}^{3})\cap L^{\infty}(\text{R}^{ 3}).\end{array}  \tag{37}\]  
Once the model parameters have been obtained, TW solutions are provided for the initial concentrations given by the level of nitrogen and oxygen in the standard atmosphere:  
\[N_{0}=0.8;\,\,\,\Theta_{0}=0.2. \tag{38}\]  
The TW solutions are formed of a front and a tip. The front carries the information transition in the media from one state to the other. This kind of behaviour exhibited can be applied to the propagation of the nitrogen gas within the fuel tank. The nitrogen TW constitutes a propagating front that shifts the tank toal  
pose, it is considered that \(t=78\) min (according to Figure 6 and previously to starting the descent phase) is sufficiently large time for the stationary solutions:  
\[N=r=0.98;\quad\Theta=0.02. \tag{39}\]  
The problem \(P_{T}\), analysed in the travelling wave domain, adopts the form:  
\[ \begin{array}{c}N_{t}=\rho\Delta N+a\cdot\
abla N-\Theta^{n}(N-0.98),\\ \Theta_{t}=\sigma\Delta\Theta+a\cdot\
abla\Theta-N^{m}\Theta,\\ 0<m,n<1,\end{array}  \tag{40}\]  
where the parameters \(n,m,\rho,\sigma\) and \(a\) are given in (37). In this case, a corrected speed \(\lambda^{*}=\lambda-a\) is introduced to account for the advection term. Then:  
\[\xi=x-\lambda^{*}t. \tag{41}\]  
To determine the TW propagation speed, we first consider that the TW front and tip move with corrected speed \(\lambda^{*}=\lambda-a\), so that when the variable \(\xi=0\), the wave has propagated along the whole domain:  
\[0=x-(\lambda-a)t\to x=(\lambda-a)t. \tag{42}\]  
Once the TW has finished the propagation along the tank, the inerted level corresponding to \(N=0.98\) is reached. At \(t=80\) min, the TW has reached all tank areas to provide the lowest value, measured in flight, for the oxygen concentration. In addition, \(x\) is a typical tank dimension obtained as the geometric mean of the tank shape:  
\[\lambda=\frac{x}{t}+a=\frac{3.88}{80}+0.0125=0.061\,\text{m}/\text{min}. \tag{43}\]  
After assessing the TW speed, the model (40) is solved with the function bvp4c in Matlab, as shown in Section 3.2. The results, in the TW variable \(\xi\), are compiled in Figure 8.  
The time required to reach a certain level of nitrogen \(N\) that ensures an inerted condition is then obtained. As an example, a concentration of 9% of oxygen in a fuel tank is sufficient to get the inert status [24]. A value of \(\Theta=0.09\) corresponds to the following value of \(\xi\) (Figure 8):  
\[\xi=0.004\,\text{m}. \tag{44}\]  
Figure 8: Travelling wave solutions. The blue solution represents the evolution of the nitrogen within the tank evolving from 0.8 (80% in volume) up to filling the whole tank 1 (100%). The red solution represents the oxygen density that evolves from 0.2 (20% in volume) down to zero. Note that the horizontal axis represents the variable \(\xi\), while \(\varphi_{1}=f_{1}=N\) and \(\varphi_{2}=f_{2}=\Theta\).