## 4 Results and Discussion
### Results of the Simplified Fire Model
#### 4.2.1 Oscillatory Flow through Ceiling Vents  
Figure 5a shows, for Case G-4 (\(T_{\text{in}}=1500\) K), the FDS results for the mass flow rates leaving and entering the compartment through the west vent (\(\dot{m}_{w,\text{out}}\) and \(\dot{m}_{w,\text{in}}\), respectively), the net mass flow rate through this vent (\(\dot{m}_{w,\text{out}}-\dot{m}_{w,\text{in}}\)) and the mass flow rate entering the compartment through the east vent (\(\dot{m}_{e,\text{in}}\)). During the whole simulation, the outward mass flow rate through the east vent is negligible compared to the inward one. On the other hand, hot gases flow outward through the north side of the west vent (closer to the fire focus; see Figure 1b) and cool air at ambient conditions (\(T_{a}=300\) K) flows inward through the south side. The bidirectional flow predicted through the west vent is widely described in the literature (see, for example, [36; 37; 38]). In the problem considered, whether the flow across the vents is induced by buoyancy or by a fire-generated pressure difference is determined by the Richardson number, which can be expressed as [38; 39]  
\[\text{Ri}=\text{Gr}/\text{Re}^{2}=gD\Delta\rho/\Delta p,\]  
where Gr and Re are the Grashof and Reynolds numbers, \(D\) is the vent size, \(\Delta\rho\) is the variation of density across the vent and \(\Delta p\) is the pressure increment due to thermal expansion, calculated as  
\[\Delta p=\frac{1}{2\rho_{e}}\left(\frac{\text{HRR}}{c_{p}T_{e}AC_{d}}\right)^ {2},\]  
where \(\rho_{e}\) and \(T_{e}\) are the density and temperature in the vicinity of the vent, \(C_{d}\) is a discharge coefficient and \(A\) is the area of the vent. In the scenario of Figure 5a, using the mean values of density and temperature obtained from the simulation at \(t=45\) s over the west vent (of area \(A=11.38\times 7.15\) m\({}^{2}\)) yields \(\Delta p_{p}\approx 6\) mPa, \(\Delta\rho_{e}\approx 30\) g m\({}^{-3}\) and \(\text{Ri}\approx 500\), a value indicating that the flow is dominated \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \multicolumn{4}{c}{**West Vent**} & \multicolumn{4}{c}{**Forced Extraction Grilles**} \\ \cline{2-7}
**Case** & \(\mathcal{I}_{h}\) & \((\Delta)\) & \(\mathcal{I}_{c}\) & \((\Delta)\) & \(\mathcal{I}_{h}\) & \((\Delta)\) & \(\mathcal{I}_{c}\) & \((\Delta)\) \\ \hline \multicolumn{7}{c}{\(t=90\) s} \\ \hline G-1 & 38.90 & 40.16 & 17.76 & 19.19 \\ G-2 & 38.87 (\(-0.1\)) & 40.01 (\(-0.4\)) & 17.77 (\(+0.0\)) & 19.22 (\(+0.2\)) \\ G-3 & 39.20 (\(+0.8\)) & 40.07 (\(-0.2\)) & 17.46 (\(-1.7\)) & 18.70 (\(-2.6\)) \\ G-4 & 39.37 (\(+1.2\)) & 40.12 (\(-0.1\)) & 17.31 (\(-2.6\)) & 18.54 (\(-3.4\)) \\ \hline \multicolumn{7}{c}{\(t=720\) s} \\ \hline G-1 & 61.20 & 58.36 & 28.70 & 30.18 \\ G-4 & 60.67 (\(-0.9\)) & 58.28 (\(-0.1\)) & 28.06 (\(-2.2\)) & 29.86 (\(-1.0\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fractions (in %) of heat (\(\mathcal{I}_{h}\)) and mass of CO\({}_{2}\) (\(\mathcal{I}_{c}\)) extracted by the natural and forced ventilation systems (Equation (6)). In parentheses, variation (in %) with respect to Case G-1. Values obtained with FDS and the simplified fire model after the first 90 s (cases G-1 to G-4) and 720 s (cases G-1 and G-4) of the fire. by buoyancy. As the mean temperature around the vent increases, Ri reaches values above \(10^{3}\) for \(t>135\) s.  
It can be observed from Figure 5a that the inward and outward mass flow rates across the west vent oscillate. Note that the outward mass flow rate oscillation wave has a slight and approximately constant delay of about 10 s with respect to the inward flow wave. Also note that the resulting oscillating net mass flow rate across the west vent determines the oscillating behavior of the inward flow through the east vent, with the same frequency (of about 0.03 Hz, calculated over a simulation of 750 s) and approximately the same wave shape and time-dependent amplitude. This behavior was observed for all the simulations run.