## 2 Fire Curve
### Fire Development  
Ignition is the phase at which fuel, oxygen, and heat combine to begin combustion. During this phase, the time of ignition can be a few seconds, hours, or even days because of the fact that the HRR is negligible. The fire can self-extinguish or transit to the next phase.  
Figure 1: Four phases of a fire curve.  
Figure 2: (**a**) Fast growth phase, narrow HRR profile characteristic. (**b**) Slow growth phase, broad HRR profile characteristic.  
The calculations used in practice for the growth phase can be conducted with the power law equation:  
\[\dot{Q}=\alpha t^{p}[\text{kW}] \tag{1}\]  
\(\dot{Q}\)--HRR [kW];  
\(\alpha\)--fire coefficient [kW/sP];  
\(t\)--time [s];  
\(p\)--exponent of growth [-].  
Furthermore:  
\[\alpha=\frac{\dot{Q}}{t_{cr}^{p}}\left[\text{kW/s}^{2}\right] \tag{2}\]  
\(t_{cr}\)--the critical time at which a fire reaches a size of 1055 kW [s].  
Two parameters are used for the NFPA file classification (Table 1) [6]:  
* The coefficient \(\alpha\) defines the speed of fire growth;
* The growth phase is also expressed using a critical time, which is the moment when the fire coefficient would be the constant.  
The direct use of data for calculations is acceptable if the enclosure is large enough that the burning behavior is similar to that in the open atmosphere [7].  
The HRR of a burning item in an enclosure causes air vitation effects, generates symmetrical airflow patterns, and, in some cases, enhances the burning rate due to radiation feedback from the hot surfaces and upper gas layer, which cause an increase in the HRR. To overcome these enclosure effects, standard test methods for measuring the HRR are operated in an open configuration, allowing for an unrestricted flow of air to the combustion zone [8].  
The ventilation factor is the controller of the fire growth phase; once the ventilation limitation is reached, fire continues until it reaches its fully developed phase. To simplify experiment data during this phase, an idealized t-squared fire or power law equation can be utilized when there are unknown variables (Equations (1) and (2) and, in Table 2, Equations (3)-(8)).  
The fully developed phase is computed using the HRR and the duration of the phase with Equation (4). A fuel-controlled fire is defined by multiplying the HRRPUA and the  
\begin{table}
\begin{tabular}{c c c} \hline
**Fire Classifications NFPA 72** & \(t_{cr}\)**[s]** & **\(\alpha\) [kW/s2]** \\ \hline Slow & \(t_{cr}\geq 400\) & \(\alpha\leq 0.0066\) \\ Medium & \(150\leq t_{cr}\geq 400\) & \(0.0066<\alpha\leq 0.0469\) \\ Fast & \(t_{cr}<150\) & \(\alpha>0.0469\) \\ \hline \end{tabular}
\end{table}
Table 1: NFPA 72 “National Fire Alarm Code”, 2013 edition.  
\begin{table}
\begin{tabular}{c c c c} \hline  & & **Fire Development** & **Decay Phase** \\ \hline \(t_{growth}=\sqrt{\frac{\dot{Q}}{t_{cr}^{0.05}}}\) & (3) & & \(t_{decay}=\frac{\dot{Q}}{\dot{Q}_{decay}}\) & (5) \\ \(E_{growth}=\frac{1055}{34r}\begin{subarray}{c}{A_{growth}}\\ {E_{growth}}\end{subarray}\) & (6) & \(t_{full\ developed\ fire}=\frac{\left[m-\left(m_{total,growth}+m_{total,decay} \right)\right]*H_{L}}{\dot{Q}_{VL}}\) & (4) & \(E_{decay}=\frac{\dot{Q}^{{}^{\prime}}_{decay}}{\dot{Q}_{decay}}\) & (7) \\ \(m_{total,growth}=\frac{E_{growth}}{H_{L}}\) & (8) & & \(m_{total,decay}=\frac{E_{decay}}{H_{L}}\) & (9) \\ \hline \end{tabular}
\end{table}
Table 2: Simplified calculations for fire development.  
\"heat release rate per unit area\" from the test results, and if the fire is ventilation-controlled, then the limited ventilation methodology can be applied in Equation (4).  
Essentially, the decay phase can be computed following the same calculations as the growth phase using Equations (5), (7) and (9) (Table 2).