# Analytical Algorithm for Oxygen Concentration of Aircraft Fuel Tank in Various Inerting Stages
## 1 Introduction  
Fire and explosion of the fuel tank are the main causes of aircraft loss [1, 2]. Expanded metal mesh filling, polymer foam materials filling, and fuel tank inerting are often used to reduce these losses [3, 4]. Fuel tank inerting is the long-term maintenance of an inert atmosphere in the vapor space of the fuel tank during normal operation. The goal is to keep the oxygen concentration in the unlage lower than the Limiting Oxygen Concentration (LOC), which are 12% and 9% for commercial and military aircraft, respectively [5, 6].  
To predict the behavior of inert gases, unlage washing models were established according to experiment results. Flight tests on the A320 were conducted in 2003 by Federal Aviation Administration (FAA) to study the dynamics of center tank inerting [7]. Since the flight tests are very expensive, more studies were conducted through ground-based inerting experiments. Based on the experiments data, [PERSON] [8] analyzed the influence of the quantity and purity of nitrogen-enriched air (NEA) on the unlage oxygen concentration. The conventional model of unlage washing was established based on the volume balance of inerting gases. [PERSON] [9] investigated the inerting effectiveness regarding flow rate of NEA, inert gas concentration, and fuel load of the tank. [PERSON] [10, 11] tested the influence of the composition of N\({}_{2}\), CO\({}_{2}\), and O\({}_{2}\) on the inerting time of the fuel tank. [PERSON] [12] tested the variation trend of oxygen concentration with time and found that concentration gradients within the fuel significantly increase the total amount of outgassing during depressurization. [PERSON] and [PERSON] [13, 14] modeled the catalytic inerting system according to ground-based experiments and found that the fuel type can change the gas inflow rate. In practice, the ground-based inerting mainly happens before takeoff. However, studies on other flight phases are very few, for it is difficult to control the pressure and temperature in the unlage.  
Mathematical methods could possibly involve dynamic factors into the study of fuel tank inerting [8]. Based on the conventional fuel scrubbing model, [PERSON] [15] proposed an iterative model of fuel tank inerting in the battle environment by calculating the heat and oxygen transfer brought by projectiles. [PERSON] [16] modified the conventional model by considering the ratio of the dissolved inert gases to analyze the relationship between bubble diameter and scrubbing efficiency. [PERSON] [17] developed a numerical method that can assess the performance of the dissolved oxygen evolution in various conditions. Polynomial expressions of oxygen evolution time constant were fitted regarding temperature, pressure, and fuel load. [PERSON] [18] developed that model by using the traveling wave approach to consider the differences in the gas concentration of the fuel tank. The model was verified by the flight test data in [7]. However, when solving the differential equations of the inerting model, the initial derivatives of gas concentration are usually needed and selected subjectively by researchers. The accuracy of oxygen concentration results strongly depends on how much experience the researcher has.  
To solve the above problems, this paper develops an analytical algorithm that needs no initial derivative of unlike gas concentration to solve the differential equations of the unlage washing model regarding various flight phases. By using this new algorithm, the uncertainty caused by inexperience can be controlled. In Section 2, the dynamic model of fuel tank inerting in form of an implicit ordinary differential equation is established based on the mass conservation law. In Section 3, the analytical algorithm of solving the unlage model without selecting the initial derivative is proposed. Finally, the method is experimentally verified and the influence of factors is analyzed in Section 4.