## 2 The Model
### Source Terms
#### 2.3.5. Convection
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometry** & **Correlation** & **Comments** \\ \hline Side walls & \(Nu_{L}^{1/2}=0.825+\frac{0.387 Ra_{L}^{1/4}}{\big{[}1+(0.492/Pr)^{/1/\alpha} \big{]}^{3/2}}\) & Lower wall segments : \(L=z_{room}-z_{i}\) \\  & & \(\overline{T}_{f}=T_{w}-0.25(T_{w}-T)\) \\ \hline Ceiling for \(\Delta T_{w}>0\) & \(Nu_{L}=0.54 Ra_{L}^{1/4}\) for \(10^{4}\leq Ra_{L}\leq 10^{7}\) & \(L=A_{w}/P\)  where \(P\) is the perimeter of \\ and floor for \(\Delta T_{w}\leq 0\) & \(Nu_{L}=0.15 Ra_{L}^{1/3}\) for \(10^{7}\leq Ra_{L}\leq 10^{11}\) & the wall segment and \\ \hline Ceiling for \(\Delta T_{w}\leq 0\) & \(Nu_{L}=0.27 Ra_{L}^{1/4}\) & \(\overline{T}_{f}=(T+T_{w})/2\) \\ and floor for \(\Delta T_{w}>0\) & \(Nu_{L}=0.27 Ra_{L}^{1/4}\) & \(\overline{T}_{f}=(T+T_{w})/2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Empirical correlations used for convection heat transfer [18; 19; 20]. \[\vec{h}=\rho_{u}c_{p}(gH)^{1/2}Q_{H}^{*\,1/3} \tag{31}\]  
\[f(r^{*})=\Big{(}1-1.1r^{*0.8}+0.808r^{*1.6}\Big{)}/\Big{(}1-1.1r^{*0.8}+2.2r^{*1.6} +0.69r^{*2.4}\Big{)} \tag{32}\]  
with  
\[z_{source}^{\prime}=\left\{ \begin{array}{c}z_{i}-\big{(}z_{i}-z_{\text{eq}} \big{)}\alpha^{3/5}\dot{M}^{*2/5}[(1+\sigma)/\sigma]^{1/5};\ \ z_{i}>z_{fire}\\ z_{fire};\ \ z_{i}\leq z_{fire}\end{array} \right. \tag{33}\]  
\[Q^{\prime}=\left\{ \begin{array}{c}Q_{c}\sigma\dot{M}^{*}/(1+\sigma);\ \ z_{i}>z_{fire}\\ Q_{c};\ \ z_{i}\leq z_{fire}\end{array} \right. \tag{34}\]  
\[\sigma=\Big{(}1-\alpha+9.115Q_{eq}^{*\,2/3}\Big{)}/(\alpha-1) \tag{35}\]  
\[\alpha=T_{u}/T_{l} \tag{36}\]  
\[Q_{eq}^{*}=\Big{[}0.21Q_{c}/\Big{(}c_{p}T_{l}\vec{m}^{p}\Big{)}\Big{]}^{3/2} \tag{37}\]  
\[z_{eq}=z_{i}-\Big{[}Q_{c}/\Big{(}Q_{eq}^{*}\rho_{l}c_{p}T_{l}g^{1/2}\Big{)} \Big{]}^{2/5} \tag{38}\]  
\[\dot{M}^{*}=\left\{ \begin{array}{c}\big{(}1.04599\,\sigma+0.360391\,\sigma^{ 2}\big{)}/\big{(}1+1.37748\,\sigma+0.360391\,\sigma^{2}\big{)}\ \ if\ \sigma>0\\ 0\ \ \ if-1<\sigma\leq 0\end{array} \right. \tag{39}\]  
In these relations, \(z_{fire}\) is the elevation of the base of fire (Figure 1), and the kinematic viscosity \(v_{u}\) and Prandtl number are evaluated at temperature \(T_{u}\). The average convective flux is obtained by integrating the above relationship over the ceiling area:  
\[\dot{q}_{w=1}^{cj}=\frac{1}{x_{room}y_{room}}\int\limits_{0}^{x_{room}}\int \limits_{0}^{y_{room}}\dot{q}_{ceil}(x,y)dxdy \tag{40}\]  
As mentioned previously, the ceiling jet can be blocked by the relatively cool wall surfaces, which can increase the rate of heat transfer to the side wall surfaces. The model of [PERSON] [16] is then used to calculate the ceiling-jet-induced convective heat flux for the upper and lower wall segments, \(\dot{q}_{w=2t09}^{cj}\). The following strategy is adopted for estimating the average rate of convective heat transfer to the wall segments of the compartment:  
* For all wall segments, calculate \(\dot{q}_{w}^{st}\) from Equation (22). * For all wall segments, except the floor (\(1\leq w\leq 9\)): \[\begin{array}{c}\bigcirc\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad  
yields of soot, CO and HCN, \(y_{s},\ y_{CO}\) and \(y_{HCN}\), which are related to their stoichiometric coefficients as follows:  
\[ \begin{array}{ccc}v_{s}&=&y_{s}M_{f}/M_{C}\\ v_{CO}&=&y_{CO}M_{f}/M_{CO}\\ v_{CO_{2}}&=&n_{C}-(v_{CO}+v_{HCN}+v_{s})\\ v_{H_{2}O}&=&0,5\ \ [n_{H}-(v_{HCI}+v_{HCN})]\\ v_{O_{2}}&=&v_{CO_{2}}+0,5\ \ (v_{H_{2}O}+v_{CO}-n_{O})\\ v_{HCI}&=&n_{CI}\\ v_{HCN}&=&\min\Big{(}n_{N}\ ;\ y_{HCN}M_{f}/M_{HCN}\Big{)}\end{array}  \tag{42}\]  
where \(M_{f}\), \(M_{CO}\), \(M_{C}\) and \(M_{HCN}\) are the molar masses of fuel, CO, soot (soot are assumed to be pure carbon), and HCN. The heat released by the fire has a convective and a radiative component, respectively \(Q_{c}=(1-\chi_{r})\ Q\) and \(Q_{r}=\chi_{r}\ Q\), where \(\chi_{r}\) is the fraction of the heat release rate emitted by radiation [21].