# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 2 Theoretical Background
### Consideration of the Theoretical Models  
The theoretical model with and without transport by species diffusion is compared in Figure 3. This figure shows the profile of the agent concentration for the height of the enclosure with and without O\({}_{2}\) species diffusion. As previously indicated, _the sharp descending interface model_ in Figure 3a does not reflect diffusion. Therefore, the total flux \(J_{tot}\) can be expressed as \(J_{tot}=J_{a}\). Based on the leading edge of the interface \(H_{i}\) between air and air-agent mixture gas, the upper part is filled with air (\(\rho_{air}\), agent concentration \(c=0\)) and the lower part is interpreted to be filled with the initial air-agent mixture gas (\(\rho_{mix}\), agent concentration \(c=c_{0}\)).  
However, Figure 3b reflects the species diffusion, and the total flux \(J_{tot}\) can be expressed as \(J_{tot}=J_{a}+J_{d}\). This is because when reflecting species diffusion, the diffusion flux \(J_{d}\) is no longer zero. At this time, diffusion occurs between the air and the air-agent mixture gas due to the difference in concentration between each species [18]. Therefore, the interface has a thickness, and the concentration and density of the agent at the interface decrease as a function of the height. This is a combination of the advection-diffusion equation for the descending interface model with species diffusion. The equation expresses the physical phenomena of how particles, energy, or other physical quantities are transferred inside a physical system due to the two processes of diffusion and advection [19].  
In the existing theoretical equation, the retention time is calculated as the interface descending velocity \(\partial\dot{H_{i}}/\partial t\), as shown in Equation (3). The interface descending velocity \(\partial\dot{H_{i}}/\partial t\) reflects the advection velocity \(v_{a}\), but not the diffusion velocity \(v_{d}\). Therefore, if species diffusion is reflected in the interface descending velocity \(\partial\dot{H_{i}}/\partial t\), it can be expressed as follows:  
\[\frac{\partial\dot{H_{i}}}{\partial t}=-(v_{a}+v_{d})=-\left(\frac{\dot{V}_{o} }{A_{f}}+v_{d}\right) \tag{6}\]  
The purpose of this study is to calculate the interface descending velocity reflecting the diffusion velocity \(v_{d}\) of the inert agent through experiments and numerical analyses. Therefore, this study proposes a method to quantify the leading edge of the interface \(H_{i}\) by measuring the interface slope \(\alpha\) through an experimental study. In addition, a method for calculating the retention time including species diffusion by measuring diffusion velocity \(v_{d}\) through experiments and numerical analysis is also proposed. In addition, the retention time calculated using the newly proposed method is compared  
Figure 3: Schematic diagram of the z-direction diffusion flux with and without species diffusion: (**a**) Without species diffusion; (**b**) with species diffusion.  
and verified with the measured retention time, and the difference with the existing theoretical models is compared.