# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 2 Theoretical Background
### Theoretical Models for Retention Time
Therefore, because this is a model to which the existing theoretical equation is not applied, experiments and numerical analyses on IG-100 were not performed in this study. _The sharp descending interface model_ assumes no transport by species diffusion. As shown in Figure 1b, the interface slope \(\alpha\) and interface thickness \(\omega\) are zero because the air flowing from the top of the enclosure does not mix with the internal agent-air gas mixture. That is, it is assumed that the retention time is determined only by fluid flow transport. Therefore, this model results in an overly optimistic prediction of retention time. _The wide descending interface model_ includes transport by species diffusion and fluid flow. It is assumed that the incoming air is mixed with the internal agent-air gas mixture at a known proportion [3; 7]; half of the incoming air is mixed with the agent-air gas mixture, and the other half remains at the top. Therefore, the concentration of the agent decreases linearly from the leading edge of the interface, \(H_{i}\), as shown in Figure 1c. This model results in a shorter retention time than that obtained by _the sharp descending interface model_ because it includes transport by species diffusion. However, it significantly reflects the species diffusion, namely, the interface thickness, which increases over time and finally appears over the entire height of the enclosure. The interface slope \(\alpha\) and thickness \(\omega\) are determined by the concentration of the inert agent and the height of the enclosure [3]. Therefore, the retention time was evaluated conservatively. The two aforementioned models present extreme results of the theoretical retention time; therefore, _the thick descending interface model_ compensates for these shortcomings [3; 4]. It reflects the species diffusion (the interface thickness) measured by experiments; it is composed of the recombination of _the sharp descending interface model_ and _the wide descending interface model_. As shown in Figure 1d, _the thick descending interface model_ assumes that the maximum interface thickness is formed after a certain period of time, after which the interface is maintained at a constant thickness and moves under the enclosure over time. In this model, the degree of species diffusion is expressed as a dimensionless interface thickness, which is expressed as a function of the enclosure height. However, species diffusion is caused by differences in temperature, concentration, pressure, etc. at the boundary of the species facing each other, and it has been shown that the influence on the height of the enclosure will not be significant [12]. The retention times of the four models mentioned here are compared in Figure 1; the specified height \(H_{e}\), which determines the retention time of the b, c, and d models, indicates the value at the same time \(t=t_{r}\). As shown in Figure 1, _the sharp descending interface model_ indicates the slowest retention time and _the wide descending interface model_ indicates the fastest. _The thick descending interface model_, which complements the two models, indicates the retention time between the values of the other models.