## 3 Discussions and Results
### Application to a Fuel Tank Inerting Process  
The phenomenon under study is related to the interaction between substances, i.e., the oxygen and the nitrogen in an aircraft fuel tank to generate an inert gas usage. The model in Problem \(P\) has three different parameters \((n,m,a)\) that need to be determined, making use of real flight testing activities. Such flight test data are extracted from [14] and summarized in Figure 6. Note that for the particular oxygen sampling location, we refer the reader to the discussion in the sourced reference [14]. The diffusion coefficients between the interacting substances can be considered as constant during the flight evolution:  
\[\rho=\sigma=0.196\ \text{cm}^{2}/\text{s}, \tag{28}\]  
which corresponds to a flight profile at a total environmental (static and dynamic) temperature of \(-7\). Note that the diffusion coefficients can be determined at any temperature [23]:  
\[\frac{\rho_{T_{1}}}{\rho_{T_{2}}}=\left(\frac{T_{1}}{T_{2}}\right)^{3/2} \tag{29}\]  
where \(\rho_{T_{1}}\) and \(\rho_{T_{2}}\) are the diffusion coefficients at temperatures \(T_{1}\) and \(T_{2}\) expressed in \(K\). The calibration exercise to assess the parameters \((n,m,a)\) requires the measurement of the nitrogen (\(N\)) or the oxygen concentration (\(\Theta\)). Aiming to calibrate on \((n,m)\) and for the sake of simplicity, the air is assumed to be an homogeneous mixture formed of nitrogen and oxygen. The flight test data measure the oxygen concentration in the centre tank of an Airbus A320 [14]. Figure 6 considers the case of an empty tank and one nitrogen gas generator acting (single membrane). Figure 5: TW profiles for \(a+\lambda=4\) with \(n=0.2\), \(m=0.9\). Note that the solution to (20) is represented by \(f_{1m}\) in the graph. An empty tank is defined as that with low level of fuel and high level of fuel vapours. It is usual that the inerting system is constituted by several inert gas generators or air separation modules (ASM). This is important to increase the inerting capacity and reduce risks. An empty tank has a higher level of oxygen that shall be replaced by nitrogen to provide an inert condition. In addition, a test case with a single acting ASM is considered, which means that only one ASM is available to filter the nitrogen and remove the oxygen. This is conservative and leads to the most demanding configuration for the single inerting ASM which, in turn, will lead to provide higher lead times to get the inert condition. The flight test data compiled in Figure 6 are interpreted over homogeneous mean oxygen values to define the parameters \(n\) and \(m\). Hence, the flat problem \(P\) is:  
\[ \begin{array}{c}N_{t}=\Theta^{n},\\ \Theta_{t}=-N^{m},\\ N_{0}>0,\Theta_{0}>0\,\ \ 0<n,m<1.\end{array}  \tag{30}\]  
For each time step, the time derivatives \(N_{t}\) and \(\Theta_{t}\) are determined. For this purpose, it is considered that the substance concentrations are expressed between the interval \((0,1)\) and the time is expressed in minutes. Such time derivatives are obtained for different discrete times and represented versus the concentration of each substance. Afterwards, the optimal curve fitting is determined as  
\[N_{t}\sim\Theta^{0.586}. \tag{31}\]  
Therefore, \(n=0.586\). The same process for \(\Theta_{t}\) provides:  
\[\Theta_{t}\sim-N^{0.025}. \tag{32}\]  
Then, \(m=0.025\). The parameters obtained, based on flight test data, are indeed between \((0,1)\), as initially appointed. Then, the forced convective parameter \(a\) is obtained. For this purpose, the basic continuity equation is used:  
\[a=\frac{G}{D\,A_{T}}. \tag{33}\]  
where \(G\) is the inerted gas flow, rich in nitrogen, and \(D\) is the air density within the fuel tank, mainly constituted of nitrogen in stabilized flight. Note that the nitrogen flow is kept stable during the flight and increases during the descend phase. For the calibration  
Figure 6: Inerting flight test tank oxygen concentration measurements for an empty tank in different positions (samples).