# A Study on a Modified Model for the Hold Time of a Clean Extinguishing Agent
## 2 Theoretical Background
### Validation Method  
The modified outflow volumetric flow \(\dot{V}_{mo}\) can be validated through two methods. First, it can be verified by comparing the theoretical hold time predicted through \(\dot{V}_{mo}\) with the hold time observed in the test enclosure. This can be verified directly by measuring the concentration of the agent-air mixture observed at a specific height. The second method is to compare the predicted value and the measured value of outflow volumetric flow. However, it is difficult to measure outflow volumetric flow directly in a test enclosure. Therefore, in this study, the outflow volumetric flow was calculated using the agent concentration data observed at a specific height and compared with the existing model's \(\dot{V}_{vo}\) and the modified model's \(\dot{V}_{mo}\). The prediction of outflow volumetric flow according to the agent concentration can be confirmed through a theoretical model for the hold time of the CEA. Figure 2: Schematic diagram of \(\dot{V}_{mo}/\dot{V}_{o}\) change with change of leakage exponent \(n\). A lighter shade indicates that the density of the agent-air mixture increases. The theoretical model for the prediction of hold time is expressed in one dimension through zone model analysis, as shown in Figure 3.  
Figure 3a shows a schematic diagram of the initial agent concentration distribution inside the enclosure. All theoretical models for the hold time of the CEA start from the initial condition, as shown in Figure 3a. The air outside the enclosure and the agent-air mixture inside the enclosure are formed with a boundary of \(H_{max}\). The z-direction driving force is formed inside the enclosure by the outflow volumetric flow from the enclosure, and the incoming air descends downward. This is _the descending interface theory_. Figure 3b-d shows the concentration distribution inside the enclosure at a specific time \(t=t_{h}\) for each model. The difference in the concentration distribution of each model is the difference in interpretation of diffusion. _The sharp model_, which does not consider species diffusion, has zero thickness of the interface between the agent-air mixture and air. Therefore, the concentration distribution inside the enclosure is only \(c_{mix}\) and \(c_{air}\), and the hold time is interpreted optimistically. _The wide model_ interprets species diffusion as a function of height \(c=f(H)\). Here, it is assumed that species diffusion can form from the ceiling to the floor of the enclosure. Therefore, the interface thickness can be widely distributed, and the hold time is interpreted conservatively. _The thick model_ is a combination of _the sharp model_ and _the wide model_. This model also interprets species diffusion as a function of height \(c=f(H)\). However, it can be interpreted that the interface thickness does not grow beyond a certain thickness, so it is assumed that the interface thickness descends after a certain time while maintaining the same value, and the hold time has a value between that of _the sharp model_ and that of _the wide model_. However, the interpretation of advection for these three models is the same. According to the diffusion theory of fluids, if there are two fluids, as shown in Figure 3a, they diffuse up and down around the center of concentration, \((c_{mix}+c_{air})/2\)[25, 26]. If there is also advection, the interface formed by diffusion around the axis of \((c_{mix}+c_{air})/2\) moves in the direction in which the advection is formed [27, 28]. This is also confirmed in Figure 3b-d. For _the wide model_, in Figure 3c, the descent by advection of the inflow air is expressed as a value of \((c_{mix}+c_{air})/2\). This is the same height as the interface of _the sharp model_ in Figure 3b, which does not consider species diffusion. This can also be verified in Equation (10). In Equation (10), if \(c_{th}=0.5c_{mix}\), then \(H_{e,uMi}=H_{e,sdi}\) (in this equation, it is interpreted as \(c_{air}=0\)). That is, \(c_{th}=0.5c_{mix}\) only expresses the descent by advection without diffusion.