## 2 Fire Curve
### Ventilation-Limited Fire Curve  
Flow occurs as a result of fire because of buoyancy-driven convection. Buoyancy causes a pressure difference that results in a velocity. Buoyant pressure and momentum push the flow [9]. Consider an enclosure with an opening, as shown in Figure 3. The pressure difference, \(\Delta p\), is related to the buoyancy:  
\[\Delta p=\big{(}\rho_{a}-\rho_{g}\big{)}g\;H\;[\mathrm{Pa}] \tag{10}\]  
Velocity due to pressure caused by buoyancy:  
\[V=\sqrt{\frac{2(\rho_{a}-\rho_{g})g\;H}{\rho_{g}}}\;[\mathrm{m/s}] \tag{11}\]  
\(\rho_{a}\)--density of air (cold) \(\left[\mathrm{kg/m^{3}}\right]\);  
\(\rho_{g}\)--density of smoke (hot) \(\left[\mathrm{kg/m^{3}}\right]\);  
\(g\)--gravitational acceleration: \(9.81\left[\mathrm{m/s^{2}}\right]\); \(H\)--height of opening \([\mathrm{m}]\).  
The exact mass flow rate \(\dot{m}\) depends on the temperature difference and size of the opening [10]. The maximum air flow rate was first introduced by [PERSON], the Japanese fire research pioneer [11]:  
\[\dot{m}_{a,max}=0.52\;A_{o}\sqrt{H}\;[\mathrm{kg/s}] \tag{12}\]  
\(A_{o}\)--area of opening \(\left[\mathrm{m^{2}}\right]\).  
The degree of ventilation, in principle, is the ratio of oxygen required over the oxygen available for complete stoichiometric combustion. This so-called equivalence ratio, \(\varphi\), is defined as:  
\[\varphi=\frac{\dot{m}_{fuel}/\dot{m}_{o_{2}}}{\left(\dot{m}_{fuel}/\dot{m}_{o_{2 }}\right)_{stechometric}} \tag{13}\]  
Excess fuel produces a rich mixture (\(\varphi>1\)) and excess air a lean mixture (\(\varphi<1\)). Since the overall combustion process is studied in this work, \(\varphi\) is defined as the equivalence ratio for the overall combustion process. This is usually referred to as the global equivalence ratio [12].  
Figure 3: Buoyant flow across the vent of an enclosure fire.  
The stoichiometric limit for the maximum HRR in natural ventilation conditions can be calculated as [13]:  
\[\dot{Q}_{VL,nat}=1500\frac{kW}{m^{5/2}}A_{o}\sqrt{H}[kW] \tag{14}\]  
\(\dot{Q}_{VL,nat}\)--HRR stoichiometric limit of natural ventilation \([kW]\).  
The total ventilation of the enclosure is computed as the sum of individual vents:  
\[A_{o}\sqrt{H}=\sum_{i=1}^{n}A_{oi}\sqrt{H_{i}} \tag{15}\]  
\(n\)--number of openings.  
Multiple doors and windows, ceilings, or floor vents complicate the process. For more complex vent configurations, a simple formula computation should be prohibited. Those can only be appropriately computed with a more involved analysis, such as FDS (CFD) simulations, which can generate data on those complex processes.  
The prediction of the entire range of the flow conditions can be expressed by Equation (16), and this has shown a degree of accuracy with the data within \(\mp 20\%\)[14].  
\[\frac{\dot{m}}{A_{o}\sqrt{H}}=0.52\frac{kg/s}{m^{5/2}}e^{-Cy}\left[1+0.6(1+(Cy) ^{2})\left(1-e^{-\frac{3H_{w}}{H}}\right)^{2}\right] \tag{16}\]  
where:  
\[y=\left(\frac{W_{o}}{H}\right)^{2/3}\left(\frac{T_{a}}{T-T_{a}}\right)^{1/2} \tag{17}\]  
\(W_{o}\)--area of (window) opening \([\mathbf{m}^{2}]\);  
\(H_{w}\)--height from floor to opening \([\mathbf{m}]\);  
\(C\)--fire location coefficient (0.55 center; 0.85 near a wall; 1 corner).  
The peak of the HRR is in correlation with the ventilation or fuel (Figure 4).  
In a real fire, ventilation can undergo random changes due to doors being left open by an occupant fleeing the fire, fire department personnel operations, or a window failure as a result of fire.  
For a ventilation-controlled fire, the activation of the smoke management system will have a larger impact on the burning behavior since the air mass flow rate into the compartment changes; thus, the equation cannot be used directly under such conditions. The consequences of these differences are beyond the scope of this work and thus are not discussed.  
Figure 4: Idealized fire curve: (**a**) fuel-controlled; (**b**) ventilation-controlled.