## 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
Figure 6: Plan view (x,y) of the compartment: faces of the steel element (rectangular envelope). \(T_{A2}\) is the local temperature of the flame at height z\({}_{\rm s}\) [\({}^{\circ}\)C] (as detailed in previous sub-section);
* \(\sigma\) = 5.67 \(\times\) 10\({}^{-8}\) is the Stefan-Boltzmann constant [Wm\({}^{-2}\)K\({}^{-4}\)];
* \(\varepsilon_{m}\) (generally = 0.7) is the surface emissivity of the steel member [-];
* \(\varepsilon_{f}\) (generally assumed to be = 1) is the surface emissivity of the flame [-]. The view factor \(d\Phi_{A1-A2}\) measures the fraction of the total radiative heat leaving a given radiating surface that arrives at a given receiving surface. Its value depends on the size of the radiating surface \(A_{2}\), on the distance from the radiating surface to the receiving surface \(S_{1-2}\) and on their relative orientation (through angles \(\theta_{1}\) and \(\theta_{2}\) between the line that joins both surfaces and their respective normal). The view factor to a member face from which the fire is not visible is taken equal to zero. In the present methodology, the view factor is calculated assuming that each face l is shifted to be localised on the section axis. The total radiative heat flux received by \(dA_{1}\) from all the fire surfaces is obtained by Equation (14). As the present model assumes that the radiative heat flux \(F_{r,f,A1}\) is homogeneous on a face l (at the height of interest), the total radiative heat flux received by face l (at the height of interest) is thus computed by Equation (15), where \(\Delta A_{2}\) = \(\Delta\)x. \(\Delta\)y or \(\Delta\)y. \(\Delta\)z depending on the location on the fire surface. The total resulting radiative heat flux \(F_{r,f,tot}\) received from the fire by the steel member at height z\({}_{\rm s}\) is computed as the average of the \(F_{r,f,l}\) weighted by the dimensions \(C_{x}\) and \(C_{y}\) of the edges of the steel section envelope, as detailed in Equation (16). \[F_{r,f,dA_{1}}=\int_{fire\ surface}\frac{\cos\theta_{1}\cos\theta_{2}}{\pi S_{ 1-2}^{2}}\sigma\varepsilon_{f}\varepsilon_{m}\big{(}T_{A_{2}}+273.15\big{)}^{ 4}dA_{2} \tag{14}\]  
\[F_{r,f,l}=\sum_{fire\ surface}\frac{\cos\theta_{1}\cos\theta_{2}}{\pi S_{1-2}^ {2}}\sigma\varepsilon_{f}\varepsilon_{m}\big{(}T_{A_{2}}+273.15\big{)}^{4} \Delta A_{2} \tag{15}\]  
\[F_{r,f,tot}=\frac{C_{x}\Big{(}F_{r,f,2}+F_{r,f,4}\Big{)}+C_{y}\Big{(}F_{r,f,1} +F_{r,f,3}\Big{)}}{2\big{(}C_{x}+C_{y}\Big{)}} \tag{16}\]  
\(F_{r,b}\)--radiative heat flux emitted by the surrounding background and received by the structural member  
The radiative heat flux received from the surrounding background by the structural member at height z\({}_{\rm s}\) is computed in a different manner depending on the zone where it lies. If the point of interest is situated in ambient air, Equation (17) applies. It corresponds to zones 1, 2, 5, 6 when the flame does not impact the ceiling (see Figure 4), and to zone 1, 5 when the flame impacts the ceiling (see Figure 5). If the point of interest is within the solid flame (i.e., in zone 3 or zone 4), Equation (18) applies, while if it is in the horizontal layer underneath the ceiling when the flame impacts the ceiling (i.e., in zone 2, 4, 6 in Figure 5), Equation (19), as described in EN1991-1-2 [13], applies. These equations, commonly referred as the Hasemi model, are based on experimental tests [23; 24; 25; 26]. Finally, when the flame impacts the ceiling, if the point of interest is in the top layer near the ceiling (zone 4 in Figure 5): the maximum between \(F_{r,b,Heskatad}\) and \(F_{r,b,Husemi}\) is considered, following prEN 1991-1-2:2021 E [27]. \[F_{r,b,ambient}=\sigma\varepsilon_{f}\varepsilon_{m}\big{(}20+273.15\big{)}^{4} \tag{17}\]  
\[F_{r,b,Heskatad}=\sigma\varepsilon_{f}\varepsilon_{m}\big{(}T_{A_{2}}+273.15 \big{)}^{4} \tag{18}\]  
\[F_{r,b,Husemi}=\left\{ \begin{array}{cl}100,000&if\ y\prime\leq 0.30\\ 136,300-121,000\ y\prime&if\ 0.30<y\prime<1.0\\ 15,000\ y^{-3.7}&if\ y\prime\geq 1.0\end{array} \right.