## 2 The Model
### Source Terms
#### 2.3.2. Vent Flow  
Natural Flow through Vertical Vents (e.g., Doors or Windows)  
Natural flow at a vent is governed by the pressure difference \(\Delta p\) between the two sides of the vent. In the present model, the vent flow is calculated by integrating [PERSON]'s equation along the vertical direction with the correct number of neutral planes (i.e., the elevation at which \(\Delta p=0\) or at which flow reversal occurs). Figure 3 depicts an example of vent flow through an opening with a neutral plane below the interface. The approach to calculating the vent flow consists of partitioning the opening in one or more vertical slabs where each slab is bounded by the interface, the neutral plane, and the vent boundaries [15]. In the example of Figure 3, there are three slabs, one between  
Figure 3: Example of vertical-vent configuration. the lower vent boundary and the neutral plane, one between the neutral plane and the interface, and one between the interface and the higher vent boundary. The mass flow for each slab is determined from [15]:  
\[\dot{m}_{slab}=\frac{\sqrt{8\rho^{*}}}{3}C_{vv}A_{slab}\frac{x^{2}+xy+y^{2}}{x+y} \tag{13}\]  
where \(x=\sqrt{|\Delta p_{t}|}\), \(y=\sqrt{|\Delta p_{b}|}\), \(C_{vv}\) is the constriction (or flow) coefficient (here, \(C_{vv}=0.68\)), and \(A_{slab}\) is the cross-sectional area of the slab. \(\Delta p_{t}\) and \(\Delta p_{b}\) are the pressure differences at the top and bottom elevations of the slab, respectively. The pressure difference at a height z is given by:  
\[\Delta p(z)=\left\{ \begin{array}{rl}p_{l}+0.5\rho_{l}gz_{i}-\rho_{l}gz-p_{ ext}(z)\;if\;z\leq z_{i}\\ p_{l}-0.5\rho_{l}gz_{i}-\rho_{u}g(z-z_{i})-p_{ext}(z)\;if\;z>z_{i}\end{array} \right. \tag{14}\]  
where \(p_{ext}(z)=p_{0}-\rho_{\infty}gz\). The pressure \(p_{0}\) represents the base (reference) pressure at the floor (\(z=0\)), outside the compartment. The density \(\rho^{*}\) is calculated as follows, depending on the slab location \(z_{slab}=(z_{t}+z_{b})/2\) and the sign of \(\Delta p_{slab}=(\Delta p_{t}+\Delta p_{b})/2\):  
\[\rho^{*}=\left\{ \begin{array}{rl}\rho_{u}\;\;if\;z_{slab}>z_{i}\;and\;\Delta p _{slab}>0\\ \rho_{l}\;\;if\;z_{slab}\leq z_{i}\;and\;\Delta p_{slab}>0\\ \rho_{\infty}\;\;if\;\Delta p_{slab}\leq 0\end{array} \right. \tag{15}\]  
Natural Flow through Horizontal Vents (e.g., Ceiling Hatches or Holes)  
For a ceiling vent, the standard vent-flow model is used:  
\[\dot{m}_{vv}=C_{vv}A_{vv}\bigg{\{} \begin{array}{rl}\rho_{u}\sqrt{2\Delta p/ \rho_{u}}\;\;if\;\Delta p>0\\ \rho_{\infty}\sqrt{2|\Delta p|/\rho_{\infty}}\;\;if\;\Delta p\leq 0\end{array}  \tag{16}\]  
where \(\Delta p=p_{u}-p_{ext}(z_{room})\), \(C_{vv}\) is the flow coefficient, and \(A_{vv}\) is the vent area. A similar expression is used for a floor vent. Note that the standard model always yields a unidirectional flow through the vent. Further development is required for horizontal vents in which the upward and downward mass flow rates depend on both pressure and density differences. [PERSON]'s correlation [16] could be used to model possible bidirectional flow through a horizontal vent. Mechanical ventilation is considered. Through the vent, smoke can be extracted out of the compartment to ambient, or ambient air can be supplied into the compartment. The mass flow rate due to forced convection through a vent located in the layer k, neglecting the effects of natural convection, can be deduced from the general equation for subsonic flow of an ideal gas [15]:  
\[\dot{m}_{fv}=C_{fv}A_{fv}\left\{ \begin{array}{rl}\rho_{k}\Big{(}\frac{p_{fv} }{p_{k}}\Big{)}^{1/\gamma}\Big{\{}2 RT_{k}\frac{\gamma}{\gamma-1}\bigg{[}1- \Big{(}\frac{p_{fv}}{p_{k}}\Big{)}^{(\gamma-1)/\gamma}\bigg{]}\bigg{\}}^{1/2} \;\;if\;\Delta p>0\\ \rho_{fv}\Big{(}\frac{p_{k}}{p_{fv}}\Big{)}^{1/\gamma}\Big{\{}2 RT_{fv}\frac{ \gamma}{\gamma-1}\bigg{[}1-\Big{(}\frac{p_{k}}{p_{fv}}\Big{)}^{(\gamma-1)/ \gamma}\bigg{]}\bigg{\}}^{1/2}\;\;if\;\Delta p\leq 0\end{array} \right.