## 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
\tag{19}\]  
Referring to Equation (19):* \(y^{\prime}\) is a dimensionless parameter given by \[y^{\prime}=\frac{r+(h_{t}-h_{base})+z^{\prime}}{L_{h}+(h_{t}-h_{base})+z^{\prime}}\] (20)
* \(r\) is the \(x\)-component of the horizontal distance (in m) between the equivalent vertical axis of the fire and the structural member axis, given by the Equation (20) (see Figure 2) \[r= \begin{cases}\left(\begin{matrix}Backside_{x}+\frac{D}{2}\end{matrix}\right)- A_{x}&if\ A_{x}<Backside_{x}+\frac{D}{2}\\ 0&if\ Backside_{x}+\frac{D}{2}\leq A_{x}\leq Frontside_{x}-\frac{D}{2}\\ A_{x}-\left(Frontside_{x}-\frac{D}{2}\right)&if\ Frontside_{x}-\frac{D}{2}<A_{x} \end{cases} \] (21)
* \(h_{t}-h_{base}\) is the distance, in m, between the fire source basis and the ceiling
* \(z^{\prime}\) is the vertical position of the virtual heat source, in m, given by \[ \begin{split} z^{\prime}=\begin{cases}2.4D\left(Q_{D}^{*2/5}-Q_ {D}^{*2/3}\right)&if\ Q_{D}^{*}<1.0\\ 2.4D\left(1.0-Q_{D}^{*2/5}\right)&if\ Q_{D}^{*}\geq 1.0\end{cases}\\ \text{with}\ \ Q_{D}^{*}=Q_{loc}/(1.11\times 10^{6}.D^{2.5})\end{split} \] (22)
* \(L_{h}\) is the horizontal flame length, in m, given by \[ \begin{split} L_{h}=2.9(ht-h_{base})(Q_{H}^{*})^{0.33}-(h_{t}-h_{ base})\\ \text{with}\ \ Q_{H}^{*}=Q_{loc}/(1.11\times 10^{6}.(h_{t}-h_{ base})^{2.5})\end{split} \] (23)  
The determination of r implies the equivalent vertical axis of the fire. [PERSON]'s model is a localised fire model whose thermal action is computed while considering a circular based fire. Any distance from such fire source is then computed from the axis of the circle. In the present methodology, the fire source is assumed to be represented by a rectangular prismatic solid flame, implying a rectangular (or square) based fire. The [PERSON] model is generalised as presented in Equations (19)-(23), i.e., by stretching the axis of the fire source onto a whole rectangle. Referring to Figure 7, with b and F being, respectively, the width and the length of the burning area, three cases can be encountered: F > b, F = b, F < b. Actually, when applying [PERSON]'s model, the results are only a function of the horizontal distance x (i.e., parallel to the length of the compartment) and the schemes should be represented in 1 dimension, implying that this situation applies whatever the value of y (coordinate parallel to b). As described on Figure 7, in case where F > b, the parameter r is considered equal to zero from b/2 from the fire back and until b/2 before the front. \(F_{rs}\)--radiative heat flux emitted by the structural member  
The radiative heat flux emitted by the steel member is computed by Equation (24), where \(T_{s}\) is the steel temperature at height z\({}_{s}\), in \({}^{\circ}\)C.  
\[F_{r,s}=-\sigma\epsilon_{m}(T_{s}+273.15)^{4} \tag{24}\]  
\(F_{c}\)--resulting convective heat flux received by the steel member from its surrounding environment  
The resulting convective heat flux received by the steel member from its surrounding environment is computed by Equation (25) where \(\alpha_{c}\), considered equal to 35 Wm\({}^{-2}\)K\({}^{-1}\) when natural fire models are used (EN1991-1-2 [13]), is the coefficient of heat transfer by convection and \(T_{g}\) is the gas temperature in the vicinity of the steel member at height z\({}_{s}\), in \({}^{\circ}\)C. In zones 1, 2, 5, and 6: \(T_{g}\) is taken as the ambient background temperature (assumed to be at 20 \({}^{\circ}\)C), while in zones 3 and 4: \(T_{g}\) is taken as \(T_{A2}\) (the local temperature of the flame at height zs, as detailed in previous sub-section, computed through Heskestad model). \[F_{c}=\alpha_{c}(T_{g}-T_{s}) \tag{25}\]  
\(F_{tot}\)--total resulting heat flux received by the structural member at a certain height  
The total resulting heat flux \(F_{tot}\) received by the structural member at height zs is computed by summing all the previously detailed contribution (see Equation (26)). Equations (24)-(26) are in accordance with EN1991-1-2 [13]. \[F_{tot}=F_{r,f,tot}+F_{r,b}+F_{r,s}+F_{c} \tag{26}\]