## 2 Conditions and Methods for Fire Simulation
### Fire Detector Numerical Models Based on Fire Models  
Depending on the type of fire model, FDS and CFAST have differences in the numerical methods applied to predict the activation times of smoke and heat detectors. FDS employs the Heskestad and Cleary models to predict the activation times of smoke detectors. The \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Combustion Properties**} & \multicolumn{2}{c}{**Fire Detector Properties**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Effective fuel formula & C\({}_{7}\)H\({}_{16}\) & Cleary model (\(a_{e}\), \(\beta_{e}\), \(a_{c}\), \(\beta_{c}\)) & 0.86, \(-\)1.09, 1.56, \(-\)1.04 \\ Heat of combustion & 44,600 kJ/kg & Activation obscuration & 30.21\%/m \\ CO yield & 0.010 kg/kg & RTI & 37.51 \(\surd(m\cdot s)\) \\ Soot yield & 0.037 kg/kg & Activation temperature & 65.134 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of Simulation Conditions for Combustion and Fire Detector Properties. Figure 2: Fire conditions considered for the predictive performance comparison of FDS and CFAST. Heskestad model [24] is a first-order accuracy model that calculates the characteristic time (\(L/v\)) when smoke generated by a fire travels a distance \(L\) through the detector at velocity \(v\) to reach the detector. \[\frac{dY_{c}}{dt}=\frac{Y_{e}(t)-Y_{c}(t)}{L/v} \tag{2}\]  
Here, a delay exists in the characteristic time (\(L/v\)) between the smoke concentration at the detector (\(Y_{e}\)) and the smoke layer (\(Y_{c}\)). The detector is triggered when \(Y_{c}\) exceeds a threshold. In addition, the Cleary model [25] adds the mixing time inside the detector to the Heskestad model, which considers delays that may be caused by the geometry of the detector or internal flow. \[\delta t_{e}=\alpha_{e}v^{\beta_{e}},\delta t_{c}=\alpha_{c}v^{\beta_{e}},\frac {dY_{c}}{dt}=\frac{Y_{e}(t-\delta t_{e})-Y_{c}(t)}{\delta t_{c}} \tag{3}\]  
Here, \(t_{e}\) represents the time at which smoke enters the smoke detector, and \(t_{c}\) represents the mixing time in the smoke detector chamber. Additionally, empirical constants derived through experimentation, such as \(\alpha\) and \(\beta\), are applied considering the inflow and mixing of smoke particles according to the shape of the smoke detector. \(O_{u}\) (obscuration per meter, %/m)--a common input in both numerical models--is an experimentally measured value. When it exceeds the set value, the detector is activated, as given by Equation (4) [2,9]:  
\[O_{u}=\left(1-e^{-K_{m}pY_{c}l}\right)\times 100(\%), \tag{4}\]  
where \(K_{m}\) is the specific extinction coefficient (m\({}^{2}\)/kg), \(\rho Y_{c}\) represents the density of smoke particulate matter (mg/m\({}^{3}\)), and \(l\) represents the distance. As \(K_{m}\) is sensitive to variations in the type of fuel and ventilation conditions, the application of empirically measured values is crucial. However, in this study, an average value of 8700 m\({}^{2}\)/kg was applied, considering various fuels and conditions [2,8]. The activation time of a heat detector in FDS is determined using Equation (5), which considers factors such as the air velocity (\(v\)) at the detector location, response time index (RTI), gas temperature (\(T_{g}\)), and predicted temperature at the detector (link temperature, \(T_{L}\)). These elements all contribute to the prediction of the activation time of the heat detector [26,27]. Users can specify the RTI and activation temperature according to the installation location of the heat detector and an experimentally measured database [2,8,28]. \[\frac{dT_{L}}{dt}=\frac{\sqrt{|v|}}{RTI}\left(T_{g}-T_{L}\right) \tag{5}\]  
The algorithm applied by CFAST to predict the activation time of a smoke detector is given by Equation (6), and it is relatively simpler than FDS. In contrast to FDS, it does not account for factors such as the delay in smoke reaching the detector location, temperature, or airflow within the detector chamber as smoke moves from the fire source to the detector [13].