# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 4 Results
### Numerical Analysis  
The interface slope \(\alpha\) for each agent was measured through the experiments and was confirmed to be unaffected by the z-direction advection flux \(J_{a}\). Then, the edge of the interface \(H_{i}\) was calculated by measuring the interface slope \(\alpha\) for each agent. In addition, by measuring the interface descending velocity \(\partial\dot{H}_{i}/\partial t\), it was also possible to confirm the difference between the calculated and measured retention times. However, it was challenging to obtain all the information regarding the retention time of the inert agent through experiments. Information regarding the outflow volumetric flow \(\dot{V}_{o}\) and turbulence mixing zone of each agent was insufficient. Therefore, numerical analysis was performed to obtain additional information that was lacking in the experiment. First, Figure 12 presents the retention time at a specified height \(H_{e}\) of 1.5 m and the outflow volumetric flow \(\dot{V}_{o}\) for each agent. The comparison between the experiment and numerical analysis for the retention time was optimized through the mesh-sensitive test in the previous chapter. Figure 12: Comparison of experimental and numerical results for the retention time of each agent: (**a**) O\({}_{2}\) concentration over time; (**b**) outflow volumetric flow \(\dot{V}_{o}\) over time. Figure 12a shows the numerical and experimental results of the O\({}_{2}\) concentration over time. In Figure 12a, the numerical results of the O\({}_{2}\) concentration over time also presented the same results as those of the experiment. For the 8 holes, the retention time of IG-01 was the fastest and that of IG-541 was the slowest. For 12 holes, the results of the O\({}_{2}\) concentration over time also yielded similar results. Figure 12b shows the results for the outflow volumetric flow \(\dot{V}_{o}\). As the outflow volumetric flow \(\dot{V}_{o}\) is proportional to the density of the air-agent mixture gas, IG-01 showed the largest value. In addition, the increase in leakage holes resulted in an increase in the outflow volumetric flow \(\dot{V}_{o}\). It can be seen that this is the same result as the theory in the previous chapter. In order to define the newly proposed theoretical equation for the retention time, it is necessary to determine the diffusion velocity \(v_{d}\). The interface descending velocity \(\partial\dot{H}_{j}\partial t\) was measured through experiments; thus, the diffusion velocity \(v_{d}\) can be calculated using Equation (6) if the advection velocity \(v_{a}\) is known. Therefore, the advection velocity \(v_{a}\) can be calculated using the outflow volumetric flow \(\dot{V}_{o}\) measured through numerical analysis. The outflow volumetric flow \(\dot{V}_{o}\) and advection velocity \(v_{a}\) are listed in Table 6.  
Table 6 indicates that the z-direction advection velocity \(v_{a}\) is proportional to \(\rho_{mix}\) because it is a function of the outflow volumetric flow \(\dot{V}_{o}\), and the z-direction diffusion velocity \(v_{d}\) is proportional to the interface slope \(\alpha\) (i.e., diffusion flux \(J_{d}\)). Therefore, it was confirmed that the diffusion velocity \(v_{d}\) of each agent was not affected by the outflow volumetric flow \(\dot{V}_{o}\). In addition, species diffusion is determined by the difference in species concentration and the binary diffusion coefficient \(D_{ij}\), as can be seen from Fick's law [18]. As all of the theoretical models for the retention time, including this study, do not consider the effect of temperature, the binary diffusion coefficient \(D_{ij}\) of each agent can also be treated as a constant value [24; 25]. That is, the interface slope \(\alpha\) and the binary diffusion coefficient \(D_{ij}\) can be kept constant regardless of the outflow volumetric flow \(\dot{V}_{o}\) (i.e., advection flux \(J_{a}\)) and the geometry of the enclosure.