# A Study on a Modified Model for the Hold Time of a Clean Extinguishing Agent
## 2 Theoretical Background  
### Deviation of the Theoretical Equations for the Hold Time  
An existing theoretical model for the hold time of a CEA starts from _the sharp model_ proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (1986). _The sharp model_ was extended to _the wide model_ proposed by [PERSON] and [PERSON] and modified to _the thick model_ proposed by [PERSON] and [PERSON] (1986); [PERSON] and [PERSON] (1986); [PERSON] (1986). The theoretical model is derived from the mass conservation equation for the agent inside the enclosure through several assumptions. The assumptions for the hold time of a CEA are as stated below [PERSON] and [PERSON] (1986); [PERSON] (1986); [PERSON]  
Equation (2) shows that the descending velocity of the leading edge of the interface \(\dot{H}_{i}\), which is the boundary between the agent-air mixture and air, is ultimately determined by the outflow volumetric flow \(\dot{V}_{o}\) discharged from the enclosure. Therefore, the time it takes for \(\dot{H}_{i}\) to descend to the required protected height \(H_{p}\) for which the hold time is determined is called the hold time, and \(H_{p}\) is usually set as the point of maximum combustibles or 75% of the maximum height of the enclosure [11; 18]. The theoretical equation for the outflow volumetric flow \(\dot{V}_{o}\) is expressed in the form of a power law function as in Equation (3). \[\dot{V}_{o}=k_{1}\Delta P^{n} \tag{3}\]  
As shown in Equation (3), \(\dot{V}_{o}\) can be determined from the pressure difference inside and outside the enclosure. The pressure difference inside and outside the enclosure can be seen in Equations (4) and (5). \[\Delta P_{LL}=(\rho_{mix}-\rho_{air})g\dot{H}_{np} \tag{4}\]  
\[\Delta P_{UL}=(\rho_{mix}-\rho_{air})g(\dot{H}_{i}-\dot{H}_{np}) \tag{5}\]  
where \(g\) is the acceleration due to gravity. The pressure difference inside and outside the enclosure can be predicted through the analysis of the height of the neutral plane \(\dot{H}_{np}\), and \(\dot{H}_{np}\) can be predicted through the conservation of mass within the enclosure (\(\rho_{mix}\dot{V}_{o}=\rho_{air}\dot{V}_{i}\)). However, the existing theoretical model was derived by assuming that the inflow and outflow volume flows are the same (\(\dot{V}_{o}=\dot{V}_{l}\)). The outflow volumetric flow \(\dot{V}_{o}\) and hold time \(t_{h}\) are calculated through the same volume flow as follows:  
\[\dot{V}_{o}=A_{o}C_{d}C_{u}\left(\frac{2\Delta P_{LL}}{\rho_{mix}}\right)^{n}= A_{o}C_{d}C_{u}\left\{\left(\frac{2}{\rho_{mix}}\right)\left[\frac{(\rho_{mix}- \rho_{air})g\dot{H}_{i}}{1+\left(\frac{\rho_{air}}{\rho_{mix}}\right)\left( \frac{A_{o}}{A_{i}}\right)^{\frac{1}{n}}}\right]\right\}^{n} \tag{6}\]  
\[t_{h}=\frac{V}{H_{max}}\left[\frac{(k_{3}H_{max}+k_{4})^{1-n}-(k_{3}H_{e}+k_{4 })^{1-n}}{(1-n)k_{2}Fk_{3}}\right]\ \ (\rho_{mix}>\rho_{air}) \tag{7}\]  
\[t_{h}=\frac{V}{H_{max}}\left[\frac{(k_{3}H_{max}+k_{4})^{1-n}-(k_{3}(H_{max}- H_{e})+k_{4})^{1-n}}{(1-n)k_{2}(1-F)k_{3}}\right]\ \ (\rho_{mix}<\rho_{air}) \tag{8}\]  
where  
\[k_{2}=k_{1}\left(\frac{\rho_{air}}{2}\right)^{n},\ k_{3}=\frac{2g|\rho_{mix}- \rho_{air}|}{\rho_{mix}+\rho_{air}\left(\frac{F}{1-F}\right)^{1/n}},\ k_{4}= \frac{2|P_{bh}|}{\rho_{mix}+\rho_{air}\left(\frac{F}{1-F}\right)^{1/n}}\]  
Through the EIT, the leakage exponent \(n\), leakage constant \(k_{1}\), bias pressure \(P_{bh}\), and lower leakage fraction \(F\) are measured or estimated, and the hold time \(t_{h}\) can be predicted by calculating \(k_{2}\), \(k_{3}\), and \(k_{4}\) using these values [8; 9]. The hold time of _the wide and thick model_, which considers diffusion as well as advection, is predicted by modifying the equation above. _The wide and thick model_ evaluates the effect on diffusion by changing the specified height \(H_{e}\). That is, by setting the value of \(H_{e}\) to be larger than in _the sharp model_, a shorter hold time is predicted. The specified height \(H_{e}\) for each model is as follows [7; 8; 9; 11; 12].