## 2 The Model
### Governing Equations  
Following the two-zone modeling concept, the gas in each layer k has attributes of mass, density, temperature, volume, and pressure denoted, respectively, \(m_{k},\ \rho_{k},\ T_{k},\ \ V_{k},\)  
Figure 1: Schematic of the two-zone model in the plane \(y=y_{fire}\).  
Figure 2: Schematic used for the ten-surface model [12] and notations.  
and \(p_{k}\), where \(k=u\) for the upper layer and \(k=l\) for the lower layer. Some relationships exist between these variables. For example:  
\[V_{u}+V_{l}=V_{room}\ (total\ volume) \tag{1}\]  
\[\rho_{k}=m_{k}/V_{k}\ (density) \tag{2}\]  
\[E_{k}=c_{p}m_{k}T_{k}\ (internal\ energy) \tag{3}\]  
\[p_{k}V_{k}=m_{k}RT_{k}\ (ideal\ gas\ law) \tag{4}\]  
The mass conservation equations for each species s of the gas mixture, which is composed of \(N_{s}\) species, can be written in the upper and lower zones as:  
\[\frac{dm_{su}}{dt}=\dot{m}_{s}^{ent}+\dot{m}_{s}^{pc}+\dot{m}_{s,u}^{ov}\ \ \ \ \ \ s=1,N_{s} \tag{5}\]  
\[\frac{dm_{sl}}{dt}=-\dot{m}_{s}^{ent}+\dot{m}_{s,l}^{ov}\ \ \ \ \ \ s=1,N_{s} \tag{6}\]  
where \(\dot{m}_{s}^{ent}\) is the mass flow rate of species s in the air entrained by the fire plume, \(\dot{m}_{s}^{pc}\) is the net rate of production/destruction of species s due to pyrolysis and combustion, and \(\dot{m}_{s,k}^{ov}\) is the net mass flow rate of species s entering or leaving the layer k through vents due to natural (e.g., doors or windows) or mechanical ventilation.  
The energy equations in the two layers are expressed as:  
\[\frac{dE_{u}}{dt}=Q_{c}+c_{p}\dot{m}^{b}T_{\infty}+c_{p}\dot{m}^{ent}T_{l}+ \dot{h}_{u}^{ov}-\sum_{w=1}^{5}\dot{q}_{w}^{conv}A_{w}+\dot{h}_{u}^{rad}-p_{u} \frac{dV_{u}}{dt} \tag{7}\]  
\[\frac{dE_{l}}{dt}=-c_{p}\dot{m}^{ent}T_{l}+\dot{h}_{l}^{ov}-\sum_{w=6}^{10} \dot{q}_{w}^{conv}A_{w}+\dot{h}_{l}^{rad}-p_{l}\frac{dV_{l}}{dt} \tag{8}\]  
where \(Q_{c}\) is the convective fraction of the heat release rate (HRR) \(Q\), \(\dot{m}^{b}\) is the actual pyrolysis rate, \(A_{w}\) is the area of the wall segment w, \(\dot{h}_{k}^{ov}\) is the enthalpy source terms due to ventilation flows entering or leaving the layer k, \(\dot{h}_{k}^{rad}\) is the net enthalpy due to radiation into the layer k, \(\dot{q}_{w}^{conv}\) is the convective heat flux at the wall segment w, \(T_{\infty}\) is the ambient temperature.  
As previously mentioned, a momentum equation is added to calculate the displacement rate of the smoke layer interface \(u_{sli}\). It is obtained by applying the variable-mass [PERSON]'s second law to the control volume extending from the interface to the ceiling (i.e., the upper layer). The [PERSON]'s law states that the sum of all forces that act upon the control volume is equal to the net rate of mechanical momentum relative to the control volume, which leads to:  
\[\frac{dm_{u}u_{sli}}{dt}=\overline{u}^{p}(z_{i})\dot{m}^{p}(z_{i})-(p_{l}-p_{u })A_{room}-gm_{u} \tag{9}\]  
where \(z_{i}\) is the interface height, \(m_{u}=\sum_{s=1}^{N_{s}}m_{s,u}\), \(A_{room}=x_{room}\ y_{room}\), \(\overline{u}^{p}(z_{i})\) is the average plume velocity at the interface height, and \(\dot{m}^{p}=\dot{m}^{b}+\dot{m}^{ent}\) is the plume mass flow rate. The terms on the left-hand side of Equation (9) correspond, respectively, to the forces exerted on the interface by the plume, the pressure gradient, and gravity, respectively.  
The volumes of the two layers are calculated from the following relations:  
\[\dot{V}_{l}=-\dot{V}_{u}=u_{sli}A_{room} \tag{10}\]  
The layer interface height is thus given by: \(z_{i}=V_{l}/A_{room}\).  
Due to the CFL limitation of the time step (typically about 50 \(\upmu\)s), the ordinary differential Equations (5)-(10) are solved by using the simple explicit Euler method. An iterative procedure is also used, strengthening the coupling between equations, and ensuring that the mass and energy conservation are satisfied at each time step.