## 2 Development of a New Analytical Model Based on the Virtual Solid Flame Concept
### Evaluation of the Fire Geometry and Its Position in the Compartment
#### 2.2.3 Flame Length  
The flame length is evaluated based on the localised fire theory, and more precisely on the model developed by [PERSON] [16; 17]. This model allows evaluating the mean flame height of a turbulent diffusion flame in buoyancy regime and was validated against experimental data in the frame of the research conducted by [PERSON] et al. [14]. The flame length actually refers to the mean luminous flame height \(L_{f}\), which is the distance between the fire source and the point where the intermittency has decreased by half. The Heskestad correlation [17] is given in Equation (6), considering the heat of combustion of the fuel as 3 kJ/g (air) (value approximately constant for most of the fuels [18]) and normal atmospheric conditions (T\(\infty\) = 20 \({}^{\circ}\)C and p = 101325 Pa), with \(L_{f}\) being the flame length (m), \(D\) the diameter of the flame source (m) and \(Q_{loc}\) the rate of heat release of the localised fire (kW). An analogous correlation (Equation (7)) is given in EN 1991-1-2 [13] where \(Q_{loc}\) is now expressed in (W), and this correlation was considered in the present analytical model.  
\[L_{f}=-1.02\;D+0.23\;{Q_{loc}}^{2/5} \tag{6}\]  
\[L_{f}=-1.02\;D+0.0148\;{Q_{loc}}^{2/5} \tag{7}\]  
It has to be noted that the correlation linking the flame length with the diameter of the flame source and the rate of heat release is valid if the surrounding air is uncontaminated by fire products and is uniform in temperature. Such conditions are not met in travelling fire scenario, nevertheless the predictions given by the Heskestad correlation are in good agreement with the experimental results (as shown in this paper), even though this empirical equations was not precisely conceived for that purpose.  
To apply Heskestad correlation, a diameter \(D\) of the fire basis needs to be defined. It is here taken as the minimum of b (width of the compartment and therefore width of the combustion area) and the length of the combustion area F (see Figure 2).  
The flame height \(H_{f}\) is then computed based on Equation (7), simply adding the height of the fire load basis \(h_{base}\) to allow the consideration of a fire load which does not lay on the  
Figure 2: Length of the combustion area.  
floor (see Equation (8)). The power released by the fictitious localised fire \(Q_{loc}\) inscribed in the circular combustion area is computed following Equation (9). The equation for \(L_{f}\) (and \(H_{f}\)) may lead to negative values, as well as values greater than the floor to ceiling distance ht. In such cases, \(L_{f}\) is bounded to 0 and h\({}_{t}\), respectively.  
\[H_{f}=L_{f}+h_{base}=-1.02\;D+0.0148\;Q_{loc}\lx@overaccentset{{ 2}}{5}+h_{base} \tag{8}\]  
\[Q_{loc}=R\times\pi\frac{D^{2}}{4} \tag{9}\]  
These correlations detailed above were derived for the following range of application: the diameter of the fire is limited by \(D\leq 10\) m and the rate of heat release of the fire by \(Q\leq 50\) MW, but these limitations were disregarded, in the lack of more precise equations. It is nevertheless worth to point out that the virtual solid flame model (project LOCAFI [8]) was compared against experimental data of kerosene pool fires as large as 50 m and characterised by RHR in the order of 3000 MW ([PERSON] [19]). [PERSON] [20] has reviewed effects on flame height of placing fire sources next to a wall or in a corner but these effects were reported to be small, and no such consideration is made in the present analytical procedure. Also, windblown flames (i.e., bent or deflected flames) were not considered.