# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 2 Theoretical Background
### Existing Governing Equation  
The theoretical equation for the retention time is derived based on mass conservation through _the sharp descending interface model_ and is extended to _the wide descending interface model_ and _the thick descending interface model_[1; 2; 3; 4; 8; 9; 10; 11]. This is because the analysis becomes significantly complicated when species diffusion is added to the retention time. Figure 2 shows a schematic diagram of the O\({}_{2}\) concentration distribution profile for _the sharp descending interface model_.  
From Figure 2, the theoretical equation for the retention time can be calculated. The theoretical equation for the retention time is a function of height, where a one-dimensional analysis is performed in the enclosure; proof of this analysis has already been presented in several studies [1; 2; 3; 4; 9; 17]. In the case of Figure 2, the density \(\rho_{mix}\) of the air-agent mixture gas inside the enclosure is greater than the density \(\rho_{air}\) of the air surrounding the enclosure. If \(\rho_{air}\) is larger than \(\rho_{mix}\), the direction of the inflow and outflow volumetric flow is reversed to the above Figure 2.  
Fresh air introduced into the enclosure by the difference between the density \(\rho_{mix}\) of the air-agent mixture gas inside the enclosure and the density \(\rho_{air}\) of the external air moves to the bottom of the enclosure along the z-direction by the total flux \(J_{tot}\). At this time, the total flux \(J_{tot}\) can be expressed by Equation (1) as follows [12; 17; 18]:  
\[J_{tot}=J_{a}+J_{d} \tag{1}\]  
When analyzing the retention time upon applying _the sharp descending interface model_, which does not reflect species diffusion, Equation (1) is interpreted as \(J_{tot}=J_{a}\) by reflecting \(J_{d}\) as zero. Therefore, when the retention time is calculated by changing the z-direction advection flux \(J_{a}\) to a function of the specified height \(H_{e}\), it is expressed as shown in Equations (2) and (3) [3]:  
\[J_{tot}=J_{a}=v_{a}\rho_{mix}=\varphi_{mix}\frac{\partial\dot{H_{e}}}{\partial t }=-\varphi_{mix}\frac{\dot{V}_{o}}{A_{f}} \tag{2}\]  
\[\frac{\partial\dot{H_{e}}}{\partial t}=-\frac{\dot{V}_{o}}{A_{f}} \tag{3}\]  
In the aforementioned equations, if \(\dot{V}_{o}\) is converted to a function of \(\dot{H_{e}}\) and both sides are integrated, the theoretical retention time can be derived as shown in Equation (5):  
\[\dot{V_{o}}=A_{o}C_{d}C_{u}\left(\frac{2\Delta P_{o}}{\rho_{mix}}\right)^{n}= A_{o}C_{d}C_{u}\left(\left(\frac{2}{\rho_{mix}}\right)\left[\frac{(\rho_{mix}- \rho_{air})gH_{e}}{1+\left(\frac{\rho_{air}}{\rho_{mix}}\right)\left(\frac{A_ {e}}{A_{i}}\right)^{\frac{1}{n}}}\right]\right)^{n} \tag{4}\]  
\[t_{f}=t_{0}-\frac{A_{f}}{A_{o}C_{d}C_{u}(1-n)}\left[\frac{1+\left(\frac{\rho _{air}}{\rho_{mix}}\right)\left(\frac{A_{e}}{A_{i}}\right)^{\frac{1}{n}}}{2 \left(1-\frac{\rho_{air}}{\rho_{mix}}\right)\delta}\right]^{n}(H_{e}^{1-n}-H_ {max}1^{-n}) \tag{5}\]  
All values except \(H_{e}\) in Equation (5) are determined by the enclosure integrity test (EIT) and inert agent design [1; 2; 8; 9]. Therefore, when species diffusion is not included, the retention time is determined by the interface height of the air flowing from the outside and the agent-air gas mixture  
Figure 2: Schematic diagram of the O\({}_{2}\) concentration distribution of _the sharp descending interface model_.  
inside. In other words, when expanding the theoretical equation with _the wide descending interface model_, the height of the interface between air and air-agent mixture gas is adjusted.