# A Study on a Modified Model for the Hold Time of a Clean Extinguishing Agent
## 2 Theoretical Background
### Consideration of the Theoretical Models  
In the previous section, the existing theoretical model was derived under the assumption that the inflow and outflow volumetric flow rates of the enclosure are the same. However, the interpretation of the neutral plane height \(\dot{H}_{np}\) in terms of hydrodynamics is derived in terms of mass flow [12; 23; 24]. Therefore, the theoretical model derived from the point of view of the volume flow needs to be reinterpreted from the point of view of the mass flow. The inflow and outflow mass flow through the leakage area can be expressed as:  
\[\dot{m}_{o}=\rho_{mix}\dot{V}_{o} \tag{13}\]  
\[\dot{m}_{i}=\rho_{air}\dot{V}_{i} \tag{14}\]  
If the outflow and inflow mass flows are the same (\(\dot{m}_{o}=\dot{m}_{i}\)), the hydrostatic pressure difference at the top of the enclosure can be expressed as Equation (15).  
\[\Delta P_{UL}=\bigg{(}\frac{\rho_{air}}{\rho_{mix}}\bigg{)}^{\frac{1-n}{n}} \bigg{(}\frac{A_{o}}{A_{i}}\bigg{)}^{\frac{1}{n}}\Delta P_{LL} \tag{15}\]  
where \(A_{i}\) and \(A_{o}\) are the summed orifice areas of all upper and lower leakages, respectively.  
The sum of the pressure difference between the top and bottom of the enclosure is the same as Equation (16), and by substituting Equation (15) into Equation (16), \(\Delta P_{LL}\) can be expressed as Equation (17).  
\[\Delta P_{LL}+\Delta P_{UL}=(\rho_{mix}-\rho_{air})gH_{i} \tag{16}\]  
\[\Delta P_{LL}=\left[\frac{(\rho_{mix}-\rho_{air})gH_{i}}{1+\bigg{(}\frac{\rho _{air}}{\rho_{mix}}\bigg{)}^{\frac{1-n}{n}}\bigg{(}\frac{A_{o}}{A_{i}}\bigg{)} ^{\frac{1}{n}}}\right] \tag{17}\]  
Therefore, the modified outflow volumetric flow \(\dot{V}_{mo}\) analyzed from the point of view of the mass flow can be derived as Equation (18).  
\[\dot{V}_{mo}=A_{o}C_{d}C_{u}\Bigg{\{}\bigg{(}\frac{2}{\rho_{mix}}\bigg{)} \left[\frac{(\rho_{mix}-\rho_{air})gH_{i}}{1+\bigg{(}\frac{\rho_{air}}{\rho_{ mix}}\bigg{)}^{\frac{1-n}{n}}\bigg{(}\frac{A_{o}}{A_{i}}\bigg{)}^{\frac{1}{n}}} \right]\Bigg{\}}^{n} \tag{18}\]  
When comparing Equations (6) and (18), the difference between \(\dot{V}_{o}\) and \(\dot{V}_{mo}\) is the presence or absence of a power of \((1-n)/n\) in \(\rho_{air}/\rho_{mix}\) of the denominator of the equation, and \(\dot{V}_{mo}/\dot{V}_{o}\) can be expressed as Equation (19).  
\[\frac{\dot{V}_{mo}}{\dot{V}_{o}}=\bigg{(}\frac{\rho_{air}}{\rho_{mix}}\bigg{)} ^{\frac{2n-1}{n}} \tag{19}\]In Equation (19), if the leakage exponent \(n\) is 0.5, it can be seen that \(\dot{V}_{mo}\) and \(\dot{V}_{o}\) are the same value. However, if \(n\) changes, \(\dot{V}_{mo}/\dot{V}_{o}\) is calculated as shown in Figure 2.  
In Figure 2, it can be seen that the value of \(\dot{V}_{mo}/\dot{V}_{o}\) decreases as the leakage exponent \(n\) increases. In other words, when \(n\) increases to more than 0.5, \(\dot{V}_{mo}\) has a value that becomes smaller than \(\dot{V}_{o}\). Therefore, because the hold time is inversely proportional to outflow volumetric flow, it can be predicted that the modified hold time \(t_{mh}\) will increase compared to the existing hold time \(t_{h}\). The theoretical equations for \(t_{mh}\) are identical to Equations (4) and (5) from the existing model. Only the values of the simplifying constants \(k_{3}\) and \(k_{4}\) are different, and these values can be defined as follows:  
\[k_{m3}=\frac{2g|\rho_{mix}-\rho_{air}|}{\rho_{mix}\left[1+\left(\frac{\rho_{ air}}{\rho_{mix}}\right)^{\frac{1-n}{2n}}\left(\frac{F}{1-F}\right)^{1/n}\right]} \tag{20}\]  
\[k_{m4}=\frac{2|P_{bh}|}{\rho_{mix}\left[1+\left(\frac{\rho_{air}}{\rho_{mix}} \right)^{\frac{1-n}{2n}}\left(\frac{F}{1-F}\right)^{1/n}\right]} \tag{21}\]  
Therefore, the difference between the existing and modified hold time will eventually show a difference according to the calculation of the simplifying constants \(k_{m3}\) and \(k_{m4}\). Also, it is expected that the difference in the predicted hold time will become larger as the leakage exponent \(n\) increases and the density of agent-air mixture increases. This was validated through experimentation.