## 2 Conditions and Methods for Fire Simulation
### Fire Detector Numerical Models Based on Fire Models
\[O_{u}=100\frac{10^{D_{u}-1}}{10^{D_{u}}},\ D_{u}=\frac{K_{m}m_{s}^{\prime \prime\prime}}{ln10} \tag{6}\]  
Here, \(D_{u}\) represents the smoke concentration (optical smoke density, m\({}^{-1}\)), derived from the relationship between \(K_{m}\) and \(m_{s}^{\prime\prime\prime}\) (mass of soot per unit volume, kg/m\({}^{3}\)). Additionally, \(O_{u}\) relatively simplistically predicts \(D_{u}\) compared with FDS, and if \(D_{u}\) exceeds a predefined threshold, the detector is activated [13]. The method of predicting the smoke concentration and temperature in CFAST depending on the utilization of the \"compartments\" function is illustrated in Figure 3. From the perspective of smoke concentration, utilizing the \"compartments\" feature allows the implementation of smoke movement in defining the calculation domain. As shown in Figure 3a, when the \"compartments\" functionality is utilized, a boundary separating high-temperature upper layers and lower-temperature, adequately oxygenated lower layers leads to similar physical quantities in zones of uniform volume centered around that bound ary. Additionally, as depicted in Figure 3b, the use of the \"compartments\" feature allows mass transfer to be set up between adjacent compartments. CFAST applies the principles of mass and energy conservation. If the domain is divided into \"compartments\" accounting for zones with and without the fire source, the smoke concentrations in both the upper and lower regions can be calculated using the Bernoulli equation for predicting airflow velocities in the ventilation section. Temperature and airflow were not considered when determining the smoke concentration. Hence, if the fire and area of interest are distant, this function can be used to predict suitable smoke concentrations. Heat detectors are predicted using the same physical quantities and algorithms as FDS, but the approach for obtaining the velocity differs; Equations (7) and (8) are employed [13]. \[v_{cj}(r)=\sqrt{gH}\dot{Q_{H}^{*}}^{1/3}3.61\ (r/H\leq 0.17) \tag{7}\]  
\[v_{cj}(r)=\sqrt{gH}\dot{Q_{H}^{*}}^{1/3}1.06(r/H)^{-0.69}\ (0.17<r/H\leq 4.0) \tag{8}\]  
Here, \(v_{cj}\) represents the ceiling jet flow rate (m/s), which depends on the distance (\(r\)) from the fire to the heat detector within the fire compartment and the height (\(H\)) of the calculation area, and \(g\) represents the gravitational acceleration (m/s\({}^{2}\)), which is outlined schematically in Figure 3a for clarity. CFAST employs an empirical correlation to derive \(v_{cj}\), considering the relationship between the radial distance (\(r\)) from the fire center and the specified height (\(H\)) of the calculation area. It is categorized according to the ratio \(r/H\) from the fire center axis: when \(r/H\) is \(\leq\)0.17 and when it exceeds 0.17 but remains \(\leq\)4.0. In other words, if the distance between the fire and the area of interest is relatively close, the velocity is calculated using Equation (7), and if they are distant, Equation (8) is applied. Moreover, as shown in Figure 3b, even when the fire is extremely close, if the heat detector is not within the fire compartment, \(v_{cj}\) is fixed at 0.1 m/s [13]. Consequently, to accurately predict smoke transport, it is crucial to appropriately segment the fire and the area of interest. Temperature prediction focuses on configuring the calculated area to consider \(v_{cj}\). Figure 2:0 Comparison of the numerical models for smoke and heat detection used in CFSAT. In Equations (7) and (8), \(\dot{Q}_{H}^{*}\)[27] is a dimensionless number related to the HRR of the fire, which is given by Equation (9). The terms in this equation represent the density of the surrounding air (\(\rho_{\infty}\), kg/m\({}^{3}\)), specific heat (\(c_{p}\), kJ/kg\(\cdot\)K), absolute temperature (\(T_{\infty}\), K), gravitational acceleration (\(g\), m/s\({}^{2}\)), maximum HRR of the fire (\(\dot{Q}\), kW), and height (\(H\), m) of the calculation area.