# Analytical Algorithm for Oxygen Concentration of Aircraft Fuel Tank in Various Inerting Stages
## 5 Conclusions  
This paper establishes a dynamic model of unlage washing and proposed an analytical algorithm to solve it without using the initial derivative values. The model has been experimentally validated and used to carry out oxygen concentration analysis in the whole flight mission. The main conclusions are as follows:  
1. Compared with experimental results, the average relative error of the oxygen concentration computed by the proposed model is around 5.78%.
2. Adding 5 min of pre-takeoff inerting duration can decrease the takeoff values of the unlage oxygen concentration and shorten the time of decreasing it to 9% by 2.1 min.
3. Every 10% reduction of NEA oxygen concentration can shorten the time of decreasing unlage oxygen concentration to 9% since takeoff by 1.37 min.
4. In all flight phases, the unlage oxygen concentration changes fast with the increasing volume flow rate of NEA and the fuel load. This is because lifting either of the two factors can directly squeeze the volume of the unlage space.  
The proposed analytical algorithm can solve the unlage washing model without choosing the initial derivative value of oxygen concentration, thus it can eliminate the uncertainty caused by subjective selection. Future studies can be carried out by adding heat transfer inside the unlage washing model.  
Figure 10: The unlage oxygen concentration regarding the fuel load of the tank, with the ground-based inerting duration of 5 min, \(\hat{v}_{I}\) = 0.2 m\({}^{3}\)/min, and \(k\) = 30%.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Ground-based inerting duration & 5 & min \\ The volume inflow rate of NEA \(\hat{v}_{I}\) & 0.2 & m\({}^{3}\)/min \\ Scale coefficient \(k\) & 30\% & - \\ Fuel load & 40\%, 60\%, 80\% & - \\ \hline \hline \end{tabular}
\end{table}
Table 8: The boundary condition of the unlage oxygen concentration analysis regarding the fuel load.  
**Author Contributions:** All the authors conceived the idea and developed the method. [PERSON] contributed to the formulation of methodology and original draft. [PERSON] contributed to the data curation and supervision. [PERSON] contributed to the editing. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\(k\): scale coefficient  
\(\dot{m}\): mass flow rate, kg/s  
\(p\): pressure, Pa  
\(R\): gas constant, J/kg\(\cdot\)K  
\(t\): time, s, min or h  
\(T\): temperature, K  
\(v\): volume, m\({}^{3}\)  
\(\dot{v}\): volume flow rate, m\({}^{3}\)/s  
\(z\): concentration of gas  
\(\beta\): Ostwald coefficient  
**Subscripts**  
\(A\), atm: atmospheric value  
\(f\): fuel vapor  
\(I\): inflow  
\(n\): nitrogen  
\(O\): outflow  
\(o\): oxygen  
\(OG\): overflow  
\(U\): allage  
\(v\): volume  
**Abbreviation**  
\begin{tabular}{l l} FAA & Federal Aviation Administration \\ LOC & Limiting Oxygen Concentration \\ NEA & Nitrogen-Enriched Air \\ \end{tabular}