# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 2 Theoretical Background  
### Theoretical Models for Retention Time  
For an inert agent, the retention time can be determined by various parameters such as temperature, pressure, density, agent concentration, leakage area size and location, and agent discharge conditions. It is challenging to predict the retention time considering all these variables. Therefore, the theoretical models for the retention time of an inert agent minimize the following variables to simplify the prediction [1; 2; 3; 4; 13; 14; 15; 16; 17]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Agent** & **N\({}_{2}\) (\%)** & **Ar (\%)** & **CO\({}_{2}\) (\%)** & **Molecular Mass (g/mol)** \\ \hline IG-01 & 0 & 100 & 0 & 39.948 \\ IG-55 & 50 & 50 & 0 & 33.980 \\ IG-541 & 52 & 40 & 8 & 34.066 \\ IG-100 & 100 & 0 & 0 & 28.013 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Type and material characteristics of inert agents. Agent discharge steps are not considered. * The protected room has a constant cross-sectional area according to its height. * Leakages through the protected room boundaries occur only at the upper and lower extremes of the room elevation. * The internal and external environments of the protected room have a standard temperature and pressure. * All species diffusivity is assumed to be the same or ignored. * The thermal effects are ignored. * The initial state of the agent in the protected room is a homogeneous mixture. Simplification through these assumptions may reduce the accuracy of the retention time, but because it improves the accessibility of the analysis, the designer or manager of the fire extinguishing equipment in the enclosure has the advantage of easily predicting the retention time. The models for the retention time based on these assumptions can be classified into four models, as shown in Figure 1.  
* _The continuous mixing model:_ NFPA 2001, Annex C, Figure 1a
* _The sharp descending interface model:_ NFPA 2001, Annex C, Figure 1b
* _The wide descending interface model:_ ISO 14520-1, Annex E, Figure 1c
* _The thick descending interface model:_ Considered by [PERSON] [34], Figure 1d  
Figure 1 shows the agent concentration for the enclosure height. Therefore, it is expressed in four models depending on the degree of diffusion of O\({}_{2}\) flowing from the outside. When 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, the air-agent mixture gas flows out through the leakage area at the bottom at a volumetric flow rate of \(\dot{V}_{o}(t)\), and the outside air flows in through the leakage area at the top at a volumetric flow rate of \(\dot{V}_{i}(t)\). (If \(\rho_{mix}\) is smaller than \(\rho_{air}\), such as IG-100, air flows in from the lower part and the air-agent mixture gas flows out from the upper part.) At this time, an interface is formed between the incoming air and the air-agent mixture gas inside the enclosure. Figure 1: Schematic diagrams of the theoretical models for the retention time: (**a**) The continuous mixing model; (**b**) the sharp descending interface model; (**c**) the wide descending interface model; (**d**) the thick descending interface model. Therefore, the theoretical models are classified according to the degree of mixing between air and air-agent mixture gas, that is, the size of the interface [1; 2; 3; 4]. _The continuous mixing model_ can be applied when an inert material such as the IG-100 has a small difference in density inside and outside the enclosure; furthermore, the incoming O\({}_{2}\) diffuses throughout the enclosure. However, because of the insignificant difference in the internal/external density, very few O\({}_{2}\) are introduced inside the enclosure. Therefore, the retention time for this model tends to be very long. This model is not applicable to gaseous agents, except for IG-100.