## 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  
#### 2.3.1 Flame Temperature  
A vertical discretisation is applied to the virtual solid flame to evaluate its temperatures. The floor-to-ceiling distance (along \(z\) coordinate) is divided into several layers of identical thickness. The symbol d\({}_{x}\) represents the thickness of the layers, while the index k is used to denote the layer number. Layer (1) extends from \(z=0\) to \(z=\) d\({}_{x}\), layer (k) extends from \(z=(\)k\(-\)1).d\({}_{x}\) to \(z=\) k.d\({}_{x}\) and so on (see Figure 3). The flame temperature is computed applying the equations from EN1991-1-2 [13], given below in Equations (10)-(12). These equations are based on the model developed by [PERSON] [16; 17] and [PERSON] [21] and are applied for each layer to evaluate the flame temperature, which is considered as uniform within a given layer.  
\[T(z)=900\;\text{for}\;z<h_{base} \tag{10}\]  
\[T(z)=20+0.25\;Q_{c}\lx@overaccentset{{ 2/3}}{(z\text{-}h_{base}\text{-}z_{0})^{-5/3}}\leq 900\; \lx@overaccentset{{ 0}}{\circ}C\;\text{for}\;h_{base}\leq z \tag{11}\]  
\[\text{where}\;Q_{c}=0.8\;Q_{loc}\;\text{by default}\]  
\[z_{0}=-1.02\;D+0.00524\;Q_{loc}\lx@overaccentset{{ 2/5}}{2} \tag{12}\]  
The temperature in the solid flame only depends on the height \(z\). At a given height, it is assumed to be constant in the whole horizontal section of the solid flame. In the developed methodology, the thickness d\({}_{x}\) of the layers has to be defined as input, and is rounded down (if needed) to generate layers that all have the same thickness. Knowing the flame length (i.e., height, along \(z\) coordinate), the flame is therefore also discretised, following d\({}_{x}\) (see Figure 3).  
When evaluating the fire geometry and its position in the compartment (see previous section), a time-step dt was introduced, for the only purpose of calculating the related out  
Figure 3: Vertical discretisation of the virtual solid flame.  
puts. To evaluate the flame temperature in the compartment, a new time-step \(\Delta\)t is needed, for the purpose of calculating the outputs related to this step and the next one (evaluating the steel temperature of a structural member placed in the compartment). Indeed, the evolution of the flame temperature will be used, along with heat fluxes as described in subsequent sections, to evaluate the evolution of the steel temperature of an (unprotected) structural member following the equation provided in EN1993-1-2 SS4.2.5.1 [22], and such equation requires a dedicated time-step of five seconds maximum.  
The SFPE handbook [18] states that \"fires with very low flame height-to-diameter ratios may generally be expected to produce lower maximum mean temperatures than other fires\" but also states that \"it is not yet clear whether the type of prediction attempted here for a particular low L/D fire is generally valid\". As the present methodology is intended to evaluate the heating of structural members, the upper bound of the gas temperatures is set to 900 \({}^{\circ}\)C, following EN1991-1-2 [13].