## 2 The Model
### Source Terms
#### 2.3.5. Convection  
The transfer of heat between the gas and walls is handled differently at the ceiling, upper and lower wall segments, and floor. It depends on the position of the fire in the compartment, the orientation of the wall segment and the presence of the ceiling jet. Standard Convection (No Ceiling Jet Effect)  
The convective heat flux at the inner surface of the wall segment w is given by:  
\[\dot{q}_{w}^{st}=h(T-T_{w})=h\Delta T_{w} \tag{22}\]  
where \(T\) is the gas temperature and \(T_{w}\) is the temperature of the inner or outer surface of the wall segment. The convection coefficient is defined as \(h=kNu_{L}/L\), where \(k\) is the thermal conductivity of the gas and \(L\) is the characteristic length of the geometry. The Nusselt number \(Nu_{L}\) is based on the Rayleigh number \(Ra_{L}=g\beta|\Delta T_{w}|L^{3}/v\alpha\), where \(\beta=1/\overline{T}_{\beta}\) is the volumetric expansion coefficient evaluated at the temperature \(\overline{T}_{\beta}=(T+T_{w})/2\), \(v\) isthe kinematic viscosity, and \(\alpha\) is the thermal diffusivity. The typical correlations applicable to the problem at hand are available in the literature [18; 19; 20]. Table 1 gives the correlations used in the model in the absence of ceiling jet. The Prandtl number is defined as \(Pr=\
u/\alpha\). Except \(\beta\), gas properties are assumed to be those of air at the film temperature \(\overline{T}_{f}\):  
\[\
u=0.04128\times 10^{-7}\overline{T}_{f}{}^{5/2}/\left(\overline{T}_{f}+110.4\right) \tag{23}\]  
\[k=2.72\times 10^{-4}\overline{T}_{f}{}^{4/5} \tag{24}\]  
\[\alpha=10^{-9}\overline{T}_{f}{}^{7/4} \tag{25}\]  
Convective Heating Due to Ceiling Jet Effect  
In case of a fire, the flame and plume spread vertically upward and can impinge the ceiling, forming a ceiling jet that extends radially and, when it is blocked by the walls, forms a downward-spinning wall jet flow that is eventually turned back inward and upward by its own buoyancy [16]. The heat transfer at the ceiling is then driven by the temperature and velocity of the ceiling jet. The Cooper correlation [16] is used to evaluate the local convective heat flux between the plume and the inner ceiling surface:  
\[\dot{q}_{ceil}=h(T_{ad}-T_{ceil}) \tag{26}\]  
where \(T_{ceil}\) is the temperature of the inner ceiling surface and \(T_{ad}\) is a characteristic temperature that would be measured adjacent to an adiabatic inner ceiling surface. In Equation (26), \(h\) and \(T_{ad}\) satisfy:  
\[\frac{h}{\overline{h}}=\left\{ \begin{array}{cc}8.82 Re_{H}^{-1/2}Pr^{-2/3} \big{[}1-\big{(}5-0.284 Re_{H}^{0.2}\big{)}r^{*}\big{]};&0\leq r^{*}<0.2\\ 0.283 Re_{H}^{-0.3}Pr^{-2/3}r^{*-1.2}(r^{*}-0.0771)/(r^{*}+0.279);&0.2\leq r^{* }\end{array} \right. \tag{27}\]  
\[(T_{ad}-T_{u})/\left(T_{u}Q_{H}^{*}{}^{2/3}\right)=\left\{ \begin{array}{cc} 10.22-14.9r^{*};&0\leq r^{*}<0.2\\ 8.39f(r^{*});&0.2\leq r^{*}\end{array} \right. \tag{28}\]  
where \(r^{*}=r/H\) is a geometric parameter defined as the ratio between the radial distance from fire plume axis, \(r=\left[\left(x-x_{fire}\right)+\left(y-y_{fire}\right)\right]^{1/2}\), and the vertical distance between the ceiling and the (presumed) point source fire, \(H=z_{room}-z_{source}^{*}\). The correlation involves the following terms:  
\[Re_{H}=(gH)^{1/2}HQ_{H}^{*\ 1/3}/\
u_{u} \tag{29}\]  
\[Q_{H}^{*}=Q^{\prime}/\left[\rho_{u}c_{p}T_{u}(gH)^{1/2}H^{2}\right] \tag{30}\]