# Analytical Algorithm for Oxygen Concentration of Aircraft Fuel Tank in Various Inerting Stages
## 4 Validations and Discussions
### Effect of Model Parameters on the Oxygen Concentration  
In this section, the unlage oxygen concentration is analyzed regarding the duration of the ground-based inerting, the volume inflow rate of NEA \(\dot{v}_{I}\), the oxygen concentration of NEA \(z_{low}\), and the fuel load of the tank. The \(z_{low}\) can be expressed as \(z_{low}=k\cdot z_{Ao}\), with \(k\) the scale coefficient and \(z_{Ao}\) the atmospheric oxygen content. The flight phase conditions are shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Flight Phase** & **Working Condition** \\ \hline Pre-takeoff phase & Started with center wing tank not inerted (20\% O\({}_{2}\)) and empty \\  & The onboard inert gas generation system started 10 mins \\  & before takeoff. \\ Climb phase & Climb from 0 to 11,887.2 m for 20 min \\ Cruise phase & Cruise at 11,887.2 m for 40 min \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Working conditions of the understudied flight phases in [7]. Figure 6: Flight test and calculated results of \(z_{low}\) in climb and cruise phases. According to the proposed analytical method, there is no need to select the initial derivative of the oxygen concentration when solving the dynamic models. Only the initial value of \(z_{Uov}\) and the initial conditions of the unlage space are needed and illustrated in Table 4. Moreover, the understudied ranges of the unlage washing parameters are set according to the flight test in [7] and are also illustrated in Table 4.  
The unlage oxygen concentration \(z_{Uov}\) regarding the ground-based inerting duration is shown in Figure 7, with the boundary condition illustrated in Table 5. In the climb phase, the unlage oxygen concentration decreases with increasing height. It decreases slowly in the cruise phase when the altitude is unchanged at 10,000 m and then during the descent phase, it increases as the altitude decreases. When the ground-based inerting is 0 min, meaning no inerting before takeoff, \(z_{Uov}\) decreases from 20.95% to 9% for 8.493 min. When the ground-based inerting is 10 min, \(z_{Uov}\) = 12.83% when the aircraft takes off, and it costs only 4.268 min to decrease to 9%. In the climb and cruise phases, the pre-takeoff inerting leads to the low oxygen concentration in the fuel tank ullage. Adding 5 min of pre-takeoff inerting duration can shorten the time of decreasing the unlage oxygen concentration to 9% after takeoff by 2.1 min. During the descent phase, the \(z_{Uov}\) increases 6.33%, 6.38%, and 6.42%, with the ground inerting time of 0 min, 5 min, and 10 min. The unlage oxygen concentration \(z_{Uov}\) regarding the volume inflow rate of NEA \(\hat{v}_{l}\) are shown in Figure 8, with the boundary condition shown in Table 6. The \(z_{Uov}\) decreases slower in the pre-takeoff phase than in the climb phase. A high \(\hat{v}_{l}\) can quickly change the \(z_{Uov}\). When \(\hat{v}_{l}\) = 0.1 m\({}^{3}\)/min, 0.15 m\({}^{3}\)/min, and 0.2 m\({}^{3}\)/min, \(z_{Uov}\) drops to 9% for 15.044 min, 9.105 min, and 6.365 min, respectively, since takeoff. During the descent phase, \(z_{Uov}\) increases for 5.01%, 6.02%, and 6.38%, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Initial value of \(z_{Uov}\) & 20.95\% & - \\ Initial value of \(p_{U}\) & 101,325 & Pa \\ Initial value of \(T_{U}\) & 300 & K \\ The volume inflow rate of NEA \(\hat{v}_{l}\) & 0.1, 0.15, 0.2 & m\({}^{3}\)/min \\ Scale coefficient \(k\) & 20\%, 30\%, 40\% & - \\ Fuel load & 40\%, 60\%, 80\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial states and understudied ranges of the controlled parameter of unlage washing. \begin{table}
\begin{tabular}{c c} \hline \hline
**Flight Phase** & **Working Condition** \\ \hline Pre-takeoff phase & Ground-based inerting started 0, 5, or 10 min before takeoff \\ Climb phase & Climb from 0 to 10,000 m for 10 min \\ Cruise phase & Cruise at 10,000 m for 30 min \\ Descent phase & Climb from 10,000 m to 0 for 10 min \\ \hline \hline \end{tabular}
\end{table}