# A Study on the Effect of O\({}_{2}\) Diffusion on the Retention Time of Inert Agents
## 1 Introduction  
The most important aspect to consider when using gaseous agents in a fire extinguishing system (FES) is the ability of the system to maintain the concentration of the agents in the enclosure for an extended period of time [1; 2]. Therefore, when designing an FES-gaseous system, the structure of the space to be protected, the sealing ability of the agent in the space to be protected, and the design concentration of the agent used, among others, must be considered. The most useful way to evaluate the capabilities of this FES-gaseous system is to measure its retention time or hold time. The retention time is defined as the time until the concentration of the FES-gaseous agent drops below the specified threshold at the equivalent specified height for a protected room. At this time, the specified height is usually set as the point of maximum combustibles or 75% to 80% of the maximum height of the enclosure, and the specified threshold is set to approximately 80% of the designed concentration [1; 2; 3; 4]. In particular, the extinguishing ability of the inert agent is the ability of the O\({}_{2}\) concentration in the protected space to maintain a nonflammable concentration that cannot be burned. Therefore, as the concentration of the inert agents refers to the concentration of O\({}_{2}\), the retention time of the inert agents can eventually be determined by the threshold of the concentration of O\({}_{2}\)[5]. The design standards for inert agent systems published by the National Fire Protection Association 2001 (NFPA 2001) [6] and the International Standards Organization 14520-1 (ISO 14520-1) [7] provide simplified physical models that can predict retention time. These models are well-established theories for orifice flow, which model the decrease in the agent concentration assuming the worst case in the space to be protected where the FES-gaseous system is designed [1; 2; 3; 4; 8; 9; 10; 11]. However, retention time predictions using these simplified models are often inaccurate; namely, they present an overly optimistic or conservative approximation of the retention time [3]. Previous studies have shown that the NFPA and ISO models provide almost the same input data to analyze output data; however, the retention time prediction using the NFPA model differs by up to twice that of the ISO model [3; 4; 12]. This is because when interpreting the models, it is assumed that there is no species diffusion (NFPA2001, _the sharp descending interface model_), or that species diffusion is rapid (ISO 14520-1, _the wide descending interface model_). Thus, species diffusion is a significant factor for the retention time. To solve this problem, [PERSON] [3; 4] proposed a new model (_the thick descending interface model_) by measuring the interface thickness by species diffusion. The assumption for species diffusivity in this model is reformulated as a combination of _the sharp descending interface model_ and _the wide descending interface model_. However, in this model, the theoretical equation for the evaluation of retention time reflects the halocarbon-compound agent and inert agent according to the same criteria. The purpose of this study is to improve the accuracy of retention time predictions when using an inert agent. The O\({}_{2}\) diffusion, which affects the retention time in an inert agent, is measured, and a new method is proposed to reflect the measured O\({}_{2}\) diffusion in the theoretical equation for the retention time. The proposed method improves the accuracy of the retention time predictions by including the diffusion flux caused by the difference in concentration of O\({}_{2}\), in addition to the advection flux generated by the difference in density between the air-agent mixture gas inside the enclosure and the air outside. To reflect the diffusion flux, we define the interface slope \(\alpha\) and interface thickness \(\omega\).