Summary
The first time you meet the fire duration formula, the 5.5 is what snags you.
T = \frac{q \cdot A_f}{5.5 \cdot A\sqrt{H}}
The ventilation factor is A\sqrt{H}, so why is there a 5.5 sitting in front of it? It looks like part of the ventilation factor, which is exactly the confusion.
It is not. The 5.5 is an empirical coefficient that turns the ventilation factor into a burning rate. Once that clicks, the formula collapses into a single division.
The formula is one division
It looks busy. The skeleton is only this.
\text{duration} = \frac{\text{how much there is to burn}}{\text{how much burns per minute}}
If you have 100 kg of wood and it burns at 5 kg per minute, it lasts 20 minutes. The fire duration formula is no more than that.
All that remains is substituting real quantities into the numerator and the denominator.
Numerator — total fuel present
Fire load is fuel mass per unit floor area. Multiply by floor area and you have the total fuel in that compartment.
\text{total fuel} = q \cdot A_f
One multiplication, done.
Denominator — how much can burn in a minute?
This is where the 5.5 enters.
In a ventilation-controlled fire, the air coming through the opening limits the burning rate. However much fuel is present, only as much can burn as the incoming oxygen allows.
The quantity that represents that incoming air is the ventilation factor A\sqrt{H}, built from opening area and opening height. The square root sits on the height because a taller opening produces a larger pressure difference between top and bottom, so more air flows in.
But the ventilation factor is not itself in kg/min. It needs an experimentally derived constant to become a mass burning rate.
\dot m \approx 5.5\, A\sqrt{H} \quad [\mathrm{kg/min}]
So 5.5 times the ventilation factor is the fuel mass that can burn in one minute.
Which gives the formula
Put the two pieces in and you are finished.
T = \frac{\text{total fuel}}{\text{burn rate per minute}} = \frac{q A_f}{5.5\, A\sqrt{H}}
That is the answer to why the 5.5 is there. It is the coefficient that converts a ventilation factor into a burning rate, which is why it lives in the denominator.
Check it with units
This check will outlast memorizing the number.
| Symbol | Meaning | Unit |
|---|---|---|
| q | Fire load | kg/m² |
| Af | Floor area | m² |
| A | Opening area | m² |
| H | Opening height | m |
| 5.5 | Burning rate coefficient | kg/(min·m^2.5) |
The numerator is kg/m² times m², which is kg. The denominator is m² times m^0.5, giving m^2.5, and multiplying by the units of 5.5 gives kg/min. Divide and only minutes remain.
If a time unit falls out, the formula was assembled correctly.
5.5 and 0.09 are the same constant
Textbooks disagree on this coefficient, which is easy to misread as a contradiction. Some print 5.5, others print 0.09.
5.5\ \mathrm{kg/min} \div 60 = 0.0917\ \mathrm{kg/s}
It is one constant expressed per minute or per second. There is no second number to memorize.
Whenever an unfamiliar coefficient appears, matching units is the fastest way to identify it. Usually it is either a unit conversion or an experimentally fitted proportionality.
Memorize the slots, not the number
Memorizing a bare number breaks the moment the formula is rearranged. Three lines hold up better.
- Ventilation factor = opening area × square root of opening height
- Burning rate = 5.5 × ventilation factor
- Duration = total fuel ÷ burning rate
Conditions on using it
This is not a derived theoretical result. It is an empirical relation from experiments, so conditions come with it.
- It assumes a ventilation-controlled fire. In a fuel-controlled fire the opening does not set the burning rate
- The original experiments centered on wood. A different fuel can shift the coefficient
- Multiple openings, or glass failing mid-fire, change the ventilation factor itself
- It assumes all the fuel burns
The value it produces is an order-of-magnitude figure for design and comparison, not a prediction of a real fire.
What I took away
- The 5.5 is not part of the ventilation factor; it converts it into a burning rate
- The skeleton of the formula is total fuel divided by burn rate per minute
- The square root on height comes from the pressure difference a tall opening creates
- 5.5 kg/min and 0.09 kg/s are the same constant
- When a coefficient looks strange, check units first
- It is an empirical relation with ventilation-controlled assumptions baked in
On the strength of this evidence
The proportionality between mass burning rate and ventilation factor in compartment fires is the classic empirical relation associated with Kawagoe’s compartment fire experiments, and it is widely cited in that form. This article summarizes it at textbook level; I did not check the original paper directly. Confirm the exact coefficient and its range of validity in the relevant sections below.
- SFPE Handbook of Fire Protection Engineering : ventilation-controlled burning in compartments
- Drysdale, An Introduction to Fire Dynamics : ventilation factor and mass burning rate
- Karlsson and Quintiere, Enclosure Fire Dynamics : assumptions and limits of compartment fire models
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