# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 4 Results
### Experiment Analysis
Therefore, the analysis of the interface slope \(\alpha\) of this part was excluded. Figure 9 shows the interface slope \(\alpha\) and the maximum interface thickness \(\omega\) for each inert agent. That is, this figure shows how, when the amount of O2 introduced from the outside is changed, the interface slope \(\alpha\) and the interface thickness \(\omega\) change when the O2 concentration threshold value is reached at the specified height \(H_{e}\) and ceiling of the enclosure \(H_{max}\). Therefore, the theoretical equation of the newly proposed model can be completed in Figure 9.  
Figure 8: Change in the interface slope \(\alpha\) over time for each agent (average value of 5 data). Figure 9a shows a comparison of the measured O\({}_{2}\) concentration for each height of the enclosure for each agent when the specified height \(H_{e}\)(1.5 m) has an O\({}_{2}\) concentration of 0.15. The O\({}_{2}\) concentration value is the average value of the data measured 5 times. In this figure, the difference in the interface slope \(\alpha\) according to each agent is shown, and it can be seen that the interface slope \(\alpha\) (i.e., diffusion flux \(J_{d}\)) is not significantly affected by the outflow volumetric flow \(\dot{V}_{o}\). The interface slope \(\alpha\) of each agent was measured as \(\alpha_{541}=0.062\), \(\alpha_{01}=0.071\), and \(\alpha_{55}=0.083\). Figure 9b shows the maximum interface thickness \(\omega_{max}\) and the leading edge of the interface \(H_{i}\) for each agent. From this figure, it can be seen that if the interface slope \(\alpha\) is known, the interface thickness \(\omega\) and the leading edge of the interface \(H_{i}\) can be obtained. The relationship between the interface slope \(\alpha\) and the maximum interface thickness \(\omega_{max}\) can be inferred by Figure 9b as follows:  
\[\alpha=\frac{\partial wO_{2}}{\partial z}=\frac{\left(wO_{2,air}-wO_{2,mix} \right)}{\omega_{max}} \tag{10}\]  
\[\omega_{max}=\frac{\left(wO_{2,air}-wO_{2,mix}\right)}{\alpha} \tag{11}\]  
The leading edge of the interface \(H_{i}\) can also be calculated using the above equation. First, if the thickness from the specified height \(H_{e}\) to \(H_{i}\) is \(\omega_{r}\), \(H_{i}\) can be expressed as follows:  
\[\alpha=\frac{\partial wO_{2}}{\partial z}=\frac{\left(wO_{2,th}-wO_{2,mix} \right)}{\omega_{r}}=\frac{\left(wO_{2,th}-wO_{2,mix}\right)}{H_{e}-H_{i}} \tag{12}\]  
\[H_{i}=H_{e}-\frac{\left(wO_{2,th}-wO_{2,mix}\right)}{\alpha} \tag{13}\]  
As shown in Equation (13), if the specified height \(H_{e}\) and the O\({}_{2}\) threshold concentration \(wO_{2,th}\) are determined to measure the retention time, \(H_{i}\) would have a fixed value. Therefore, if the interface slope \(\alpha\) and the z-direction velocity \(v_{d}\) can be inferred, the retention time including the species diffusion can be obtained using Equation (14). \[\int_{H_{max}}^{H_{i}}\partial\dot{H_{i}}=-\int_{t_{0}}^{t_{r}}(v_{a}+v_{d}) \partial t=-\int_{t_{0}}^{t_{f}}\!\!\left(\frac{\dot{V}_{o}}{A_{f}}+v_{d} \right)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!To obtain the retention time using Equation (14), it is necessary to know the information about the remaining factors. Here, \(H_{max}\) and \(A_{f}\) are design factors, and the outflow volumetric flow \(\dot{V}_{o}\) can be obtained through calculation or EIT. Therefore, if the interface slope \(\alpha\) and diffusion velocity \(v_{d}\) are known, a theoretical calculation of the retention time is possible. The interface descending velocity \(\partial\dot{H}_{i}/\partial t\) can be calculated using the measured O\({}_{2}\) concentration data according to the enclosure height. This can be seen in Figure 10.  
Figure 10 shows the time to reach an O\({}_{2}\) concentration of 0.15 per 0.1 m from 1.8 to 1 m of enclosure height for the IG-55, IG-01, and IG-541. In other words, the slope of this figure shows the interface descending velocity \(\partial\dot{H}_{i}/\partial t\).