## 2 Development of a New Analytical Model Based on the Virtual Solid Flame Concept
### Evaluation of the Flame Temperature and of the Heat Fluxes in the Compartment
#### 2.3.2 Heat Fluxes  
Two situations can be encountered: the flame does not impact the ceiling of the compartment, i.e., \(L_{f}<h_{t}\) (see Figure 4), or the flame does impact the ceiling of the compartment, i.e., \(L_{f}\geq h_{t}\) (see Figure 5). Depending on the situation and depending on the location of the target (i.e., where the heat transfer computation for the structural element needs to take place), different zones are defined (see Figures 4 and 5). The following heat fluxes components will be used (all expressed in W/m\({}^{2}\)):  
* \(F_{rf}\) refers to the radiative heat flux emitted by the fire surface and received by the structural member;
* \(F_{r,b}\) refers to the radiative heat flux emitted by the surrounding background and received by the structural member;
* \(F_{rs}\) refers to the radiative heat flux emitted by the structural member;
* \(F_{c}\) refers to the convective heat flux received by the structural member from its surrounding environment;
* \(F_{tot}\) refers to the total net heat flux at the surface of the structural member at a certain height. Tables 1 and 2 summarize the heat flux components which are considered in the different zones, whilst the formulas to calculate these heat flux components are detailed below. Figure 4: Zones for fire model if the flame does not impact the ceiling. Figure 5: Zones for fire model if the flame impacts the ceiling. \(F_{r,f}\)-radiative heat flux emitted by the fire surface and received by the structural member. The fire is schematised as a rectangular virtual solid flame, and the following equations are based on EN 1991-1-2 Anne G [13] and the outcomes of the RFCS project \"LOCAFi\" [8]. Referring to the schematic plan view (x,y) depicted on Figure 6, on each face l--with l = 1 to 4--of the rectangular envelope around the steel member at height zs, the radiative heat flux from an infinitesimal area of the fire surface by an infinitesimal area located at the height of interest on the steel member is computed by Equation (13):  
\[dF_{r,f,dA_{1}}=\frac{\cos\theta_{1}\cos\theta_{2}}{\pi S_{1-2}^{2}}\sigma \varepsilon_{f}\varepsilon_{m}\big{(}T_{A_{2}}+273.15\big{)}^{4}dA_{2} \tag{13}\]  
where  
\(d\Phi_{dA_{1}-dA_{2}}=\frac{\cos\theta_{1}\cos\theta_{2}}{\pi S_{1-2}^{2}}dA_{2}\) is the view factor between two infinitesimal areas \(dA_{1}\) and \(dA_{2}\) (see EN1991-1-2 equation G.1 [13]);  
\(dA_{1}\) refers to an infinitesimal area on the face l--located at the height of interest on the structural member--which receives the radiative heat flux;  
\(dA_{2}\) refers to an infinitesimal area on the surface of the solid flame which emits the radiative heat flux; \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Scenario 1–The Flame does not Impact the Ceiling of the Compartment**} \\ \hline
**Heat Flux Component** & **Zone 1** & **Zone 2** & **Zone 3** & **Zone 4** & **Zone 5** & **Zone 6** \\ \hline \(F_{r,f}\) & ✔ & ✔ & & & ✔ & ✔ \\ \(F_{r,b,ambient}\) & ✔ & ✔ & & ✔ & ✔ & ✔ \\ \(F_{r,b,Hessatal}\) & & & ✔ & ✔ & ✔ & ✔ \\ \(F_{r,s}\) & ✔ & ✔ & ✔ & ✔ & ✔ & ✔ \\ \(F_{c}\) & ✔ & ✔ & ✔ & ✔ & ✔ & ✔ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Heat flux components in Scenario 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Scenario 2–The Flame Impacts the Ceiling of the Compartment**} \\ \hline
**Heat Flux Component** & **Zone 1** & **Zone 2** & **Zone 3** & **Zone 4** & **Zone 5** & **Zone 6** \\ \hline \(F_{r,f}\) & ✔ & & & & ✔ & \\ \(F_{r,b,ambient}\) & ✔ & & & ✔ & & ✔ \\ \(F_{r,b,Hessatal}\) & & & ✔ & ✔ & & ✔ \\ \(F_{r,b,Hessatal}\) & & & ✔ & ✔ & ✔ & ✔ \\ \(F_{r,b,Hessemi}\) & & & ✔ & ✔ & ✔ & ✔ \\ \(F_{r,s}\) & ✔ & ✔ & ✔ & ✔ & ✔ & ✔ \\ \(F_{r,s}\) & ✔ & ✔ & ✔ & ✔ & ✔ & ✔ \\ \(F_{c}\) & ✔ & ✔ & ✔ & ✔ & ✔ & ✔ \\ \hline \hline \end{tabular}  
* Both [PERSON]’s and Hasemi’s components are computed but only the maximum value is kept, following the [prEN1991-1-2].  
\end{table} 
Table 2: Heat flux components in Scenario 2.