# Analytical Algorithm for Oxygen Concentration of Aircraft Fuel Tank in Various Inerting Stages
## 2. Ullage Washing Model  
Ullage washing technology was used to ensure that the unlage oxygen concentration is below the LOC, thus reducing the hazard and loss of fire and explosion of the fuel tank. The unlage washing uses NEA with low oxygen concentration to replace the oxygen and fuel vapor in the unlage space. The process is shown in Figure 1.  
The dynamic model of unlage washing is established based on the following assumptions. 1. There is only oxygen, nitrogen, and fuel vapor gas in the unlage space, and the distribution is uniform. 2. The NEA can immediately mix with the gas in the unlage space. 3. Dissolved oxygen and nitrogen in the fuel immediately reach equilibrium in response to changes in the partial pressures of oxygen and nitrogen in the unlage space [3]. 4. It is assumed that the fuel flowing out from the tank does not contain dissolved oxygen, nitrogen, and other gases. Figure 1. The principle of aircraft fuel tank usage washing. The mass change of oxygen in the unlage space of the aircraft fuel tank has three sources: the inflow of NEA, the outflow of air, and the overflow of oxygen from the fuel. Therefore, the mass flow rate of oxygen in the unlage space can be expressed as:  
\[\dot{m}_{o}=\dot{m}_{lo}+\dot{m}_{OGo}-\dot{m}_{Oo} \tag{1}\]  
where \(\dot{m}_{o}\) represents the mass flow rate of oxygen in the unlage space, kg/s; \(\dot{m}_{lo}\) the mass inflow rate of oxygen, kg/s; \(\dot{m}_{OGo}\) the mass overflow rate of oxygen from the fuel, kg/s; and the mass outflow rate of oxygen, kg/s. The mass flow rate of oxygen in the unlage space can be obtained by using the ideal gas law as,  
\[\dot{m}_{o}=\left(\frac{p_{U}\cdot v_{U}\cdot z_{Uov}}{R_{o}\cdot T_{U}}\right) ^{\prime} \tag{2}\]  
where \(p_{U}\) is the total pressure of gas mixture in the unlage space, Pa; \(v_{U}\) the volume of the unlage space, m\({}^{3}\); \(z_{Uov}\) the unlage oxygen concentration; \(R_{o}\) the gas constant of oxygen, J/kg\(\cdot\)K; and \(T_{U}\) the temperature of the unlage space, K. The superscript'denotes the time derivative of a variable. The mass inflow and outflow rate of oxygen can be expressed as,  
\[\left\{ \begin{array}{l}\dot{m}_{lo}=\frac{p_{U}\cdot\dot{v}_{T}z_{Uov}}{R_{o }\cdot T_{U}}\\ \dot{m}_{Oo}=\frac{p_{U}\cdot\dot{v}_{O}\cdot z_{Uov}}{R_{o}\cdot T_{U}}\end{array} \right. \tag{3}\]  
where \(\dot{v}_{I}\) represents the volume inflow rate of NEA, m\({}^{3}\)/s; \(\dot{v}_{O}\) the volume outflow rate of gas, m\({}^{3}\)/s; \(\dot{p}_{I}\) the pressure of the inflow, Pa; \(z_{lov}\) the oxygen concentration of NEA; and \(T_{I}\) the temperature of the NEA, K.  
The mass overflow rate of oxygen from the fuel can be expressed by [PERSON]'s law as [3; 19; 20]  
\[\dot{m}_{OGo}=-\left(\frac{\beta_{o}\cdot p_{U}\cdot v_{F}\cdot z_{Uov}}{R_{o }\cdot T_{F}}\right)^{\prime} \tag{4}\]  
where \(v_{F}\) is the volume of fuel, m\({}^{3}\); \(T_{F}\) the temperature of the fuel, K; and \(\beta_{o}\) the Ostwald coefficient of oxygen. Therefore, Formula (1) can be rewritten as  
\[\left(\frac{p_{U}\cdot v_{U}\cdot z_{Uov}}{R_{o}\cdot T_{U}}\right)^{\prime}= \frac{p_{I}\cdot\dot{v}_{I}\cdot z_{lov}}{R_{o}\cdot T_{I}}-\frac{p_{U}\cdot \dot{v}_{O}\cdot z_{Uov}}{R_{o}\cdot T_{U}}-\left(\frac{\beta_{o}\cdot p_{U} \cdot v_{F}\cdot z_{Uov}}{R_{o}\cdot T_{F}}\right)^{\prime} \tag{5}\]  
Similar to oxygen concentration, the mass flow rate of nitrogen can be expressed in as  
\[\left(\frac{p_{U}\cdot v_{U}\cdot z_{Uov}}{R_{n}\cdot T_{U}}\right)^{\prime}= \frac{p_{I}\cdot\dot{v}_{I}\cdot z_{Inv}}{R_{n}\cdot T_{I}}-\frac{p_{U}\cdot \dot{v}_{O}\cdot z_{UInv}}{R_{n}\cdot T_{U}}-\left(\frac{\beta_{n}\cdot p_{U} \cdot v_{F}\cdot z_{UInv}}{R_{n}\cdot T_{F}}\right)^{\prime} \tag{6}\]  
where \(z_{UInv}\) represents the unlage nitrogen concentration, \(z_{Inv}\) the nitrogen concentration of NEA, \(R_{n}\) the gas constant of nitrogen, J/kg\(\cdot\)K; and \(\beta_{n}\) the Ostwald coefficient of nitrogen.