# GoZone: A Numerical Model for Travelling Fires Based on Cellular Automata Concept
## 4 Radiation toward the Cells
### Radiation from the Flames  
The semi-empirical model known as the point source model (PSM) [51] was used here to compute the radiation from each burning cell to the fuel cells. PSM assumes the fire to radiate isotropically from a point \"P\" at the centre of the flame located at half of the flame length (Figure 4). The angle \(\theta\) between the normal of the receiving surface and the line of sight (target to point source) allows for considering the orientation of the receiving target with respect to the point source. The incident radiant heat flux on the target surface is evaluated according to Equation (3), where \(RHR(t)\) is the fire power as a function of time and \(\chi_{r}\) is the radiative fraction coefficient set as 0.3 [52]. \[q^{\prime}(t)=\frac{\chi_{r}\ RHR(t)}{4\pi R(t)^{2}}\ \cos\theta(t)\ \left[\frac{W}{m^{2}}\right] \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test ID**} & **Distance from** & **Device** & \multirow{2}{*}{**\(t_{ign,test}\) [s]**} & \multirow{2}{*}{**\(t_{ign,model}\) [s]**} & \multirow{2}{*}{**Error [s]**} \\  & **the Origin [m]** & & & & \\ \hline LA1v1 & 0.5 & V & 741 & 800 & 59 \\ LA1h1 & 0.5 & H & 1024 & 800 & \(-\)224 \\ LA1v2 & 0.5 & V & 784 & 800 & 16 \\ LA1h2 & 0.5 & H & 1039 & 800 & \(-\)239 \\ LA1v3 & 0.95 & V & 1210 & 1254 & 44 \\ LA2v1 & 0.5 & V & 886 & 910 & 24 \\ LA2h1 & 0.5 & H & 1045 & 910 & \(-\)135 \\ LA3v1 & 0.5 & V & 1312 & 1195 & \(-\)117 \\ LA3h1 & 0.5 & H & 1373 & 1195 & \(-\)178 \\ LA4v1 & 0.5 & V & 533 & 608 & 75 \\ LA4h1 & 0.5 & H & 680 & 608 & \(-\)72 \\ LA4v1 & 0.95 & V & 701 & 710 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ignition strategy tested on test series LA. Opinions on PSM are controversial in the literature. [PERSON] stated that PSM overestimates the intensity of the thermal radiation for a target close to the fire [53]. [PERSON], in contrast, wrote that PSM underestimates thermal radiation for close targets [52]. [PERSON], after having tested several semi-empirical models for the incident heat flux evaluation toward a wide range of target orientations and fire powers, concluded that PSM was the most accurate model among the ones analysed in his work [54]. PSM was tested here with respect to real experimental tests on localised pool fires and to CFD simulations. Figure 5 shows a comparison between the PSM and 18 different pool fire tests conducted within the framework of the RFCS project LOCAFI [13]. The flux measurements during the tests were taken at a fixed distance of approximately 3 m from the centre of the fire and each test had a different steady state fire power created by different diameters. Gas burners of different constant powers (200, 500, 750, 1000 kW) have been modelled in FDS [15]. The incident radiative heat flux has been computed at 1 m, 2 m, and 3 m from the fire centre by varying the target orientation. Figure 6 shows the results for vertical orientation facing the fire source, represented with black dots, whereas the white filled dots are for the horizontal orientation. Figure 4: Point source model. Figure 5: PSM compared with 18 pool fires; data from the LOCAFI project. length \(L_{CH}\) of 2 m was found appropriate here through calibration of the model presented hereafter with circular shaped fires with a radius varying from 1 to 5 m and rate of heat release densities of 250 and 500 kW/m\({}^{2}\). An equivalent fire area was first evaluated for each burning cell \(i\) with central coordinates (\(x_{i},y_{i}\)) using Equation (5) where the surface integration was conducted on the whole burning domain. \[\Omega_{eq,i}(x,y)=\int\limits_{x,y}\exp(-d(x,y)/L_{CH})\ dxdy\]  
\[d(x,y)=\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}}\]  
An associated equivalent diameter was computed from the equivalent fire area (see Equation (6)). \[D_{eq,i}=2\ (\ \Omega_{eq,i}/\pi)^{0.5}\]  
The flame length for the cell is then computed using Equation (7).