# Fire Risk Analysis in Large Multi-Compartment Structures Using a Hybrid Multiscale Approach
## 2 The Hybrid Multiscale Approach
### The Compartment Fire Zone Model  
As illustrated in Figure 2, after ignition, a compartment fire may undergo the phases of growth, fully developed fire, and decay. Flashover is not a fire stage, but the transition from the growth stage to a fully developed fire, involving all the combustible materials within the compartment. Decay occurs when most of the fuel has been consumed and the heat release rate (HRR) begins to decrease. It is generally accepted that this occurs, at time \(t_{de}\), when 70 to 80% of the fuel has been consumed [15; 16; 17]. However, the experiments carried out in [14] have shown that this can take place for values lower than 70% (e.g., 65% for pine wood) or higher than 80% (e.g., 95% for heptane). During the decay phase, the fire evolves from a ventilation-controlled regime to a fuel surface-controlled regime. Design Fires  
In a companion paper [14], the authors present a two-zone model that was developed to predict fire development in a compartment where single- or multi-fuel combustion occurs. This model was validated using data from full-scale experiments involving different combinations of solid and liquid fuels and varying the confinement level of the compartment. In this phase of validation, the pyrolysis or combustion rate, and therefore the HRR, was deduced from the fuel mass loss rate measured over time. Unfortunately, for the present study cases, the HRR time curve in each compartment is unknown. To overcome this issue, the time evolution of the HRR is assumed to follow three phases: a growth phase where it increases as the square of time [18; 19], a steady phase, and a decay phase where it decreases as the square of time:  
\[HRR=\left\{ \begin{array}{ll}\alpha_{gr}t^{2}&\quad\quad\text{$si\ t<t_{max}$ }\\ HRR_{max}&\quad\quad\text{$si\ t_{max}\leq t\leq t_{de}$}\\ \alpha_{de}(t_{ex}-t)^{2}&\quad\quad\text{$si\ t_{de}<t\leq t_{ex}$}\\ 0&\quad\quad\quad\text{$si\ t>t_{ex}$}\end{array} \right. \tag{1}\]  
where \(t\) is the time from established ignition (in s), \(\alpha_{gr}\) and \(\alpha_{de}\) are the respective fire growth and decay rates (in W/s\({}^{2}\)), \(t_{max}\) is the time at which the HRR is maximum (\(HRR_{max}\) in W), \(t_{de}\) is the time when decay starts, and \(t_{ex}\) is the time when fire ends. The calculation of HRR requires knowledge of the initial fuel mass \(m_{0}\), \(HRR_{max}\), \(\alpha_{gr}\), the chemical heat of combustion of fuel, \(\Delta h_{ch}\), and the fraction \(p\) of the initial fuel mass beyond which fire decay begins. The other parameters are deduced from the following relationships:  
\[t_{max}=\left(HRR_{max}/\alpha_{gr}\right)^{0.5} \tag{2}\]  
Figure 2: The successive phases of enclosure fire development and notations used. \(t_{ig}\) is the ignition time, \(t_{fo}\) is the flashover time, \(t_{de}\) is the time of fire decay, \(t_{end}\) is the time when fire ends, \(\tau_{fo}\) is the duration of fire growth phase from ignition to flashover, \(\tau_{fd}\) is the duration of the fully developed phase, and \(\tau_{de}\) is the duration of the decay phase. \[t_{de}=t_{max}+\frac{p\ m_{0}\ \Delta h_{ch}-\frac{1}{3}a_{gr}t_{max}^{3}}{HRR_{max}} \tag{3}\]  
\[\alpha_{de}=\frac{HRR_{max}^{3}}{[3m_{0}(1-p)\Delta h_{ch}]^{2}} \tag{4}\]  
\[t_{ex}=t_{de}+(HRR_{max}/\alpha_{de})^{0.5} \tag{5}\]  
Figure 3a shows the design fires for 50 cm diameter pool fires of heptane and JP5 (kerosene-based fuel used in military aircraft, NATO code: F44) using the following standard set of parameters: for the heptane pool fire, \(\Delta h_{ch}=\)41.2 J/kg [20], \(\alpha_{gr}=\)49 W/s\({}^{2}\), and \(p=0.95\); and for the JP5 pool fire, \(\Delta h_{ch}=\)40.0 J/kg [20], \(\alpha_{gr}=\)30 W/s\({}^{2}\), and \(p=0.95\). A good agreement is obtained with experiments carried out in the DIAMAN device [14]. For larger enclosure fires (\(HRR_{max}>1.5\) MW), experiments are difficult to conduct. To overcome this problem, a scaling procedure is developed.