Heskestad Flame Height

Gunnar Heskestad's correlation is two terms and it is worth reading them separately. The first, $0.235\,\dot{Q}^{2/5}$, is the height a fire of that size would reach over a base of vanishing width. The second, $-1.02D$, is the penalty for the base being wide. A broad fire entrains air along its whole perimeter, the combustion finishes lower down, and the flame is shorter. Concentrate the same energy into a narrower base and the flame climbs.

That sign is the physically interesting part, and it explains a great deal. A burning waste bin throws flame far higher than the same energy spread over a floor. A trench fire and a pool fire of equal area and equal heat release rate do not look alike. The geometry of a fuel array matters as much as how much fuel is in it — which is why storage arrangement is a fire protection question and not just a logistics one.

The constants are dimensional. Heskestad fitted 0.235 and 1.02 with the heat release rate in kilowatts and lengths in metres, and they are correct in no other units. This site converts your entry before applying them and converts the answer back, so an imperial input gives a correct answer — but 0.235 carried into a spreadsheet working in BTU and feet gives a number that is simply wrong, and looks fine.

What the correlation returns is the MEAN height of the luminous flame, and a real flame does not have one height. It pulses. The tip oscillates over roughly half the mean to about one and a half times it, at a frequency near $1.5/\sqrt{D}$ hertz — about 1.5 Hz for a one-metre fire, which is slow enough to watch. The mean is defined as the height where flame is present half the time. For fire spread, and for the question of whether flame reaches a ceiling, a cable tray or a structural member, the intermittent tip matters more than the mean, and the mean will understate the reach.

Three conditions the correlation assumes. The fire is axisymmetric and free-burning, so a non-circular base takes an equivalent diameter $D = \sqrt{4A/\pi}$, which is fair for a square and increasingly poor for a long trench. The fire is in the open: against a wall, entrainment is halved on one side and the flame runs taller for the same heat release rate; in a corner, taller still. And the air is still — any cross-draught tilts the flame and changes both its height and where it reaches.

Run backwards, the equation estimates a fire size from a measured flame height, which is a standard investigative move from photographs or witness accounts. The 5/2 power makes it brutal: a 20% error in the height becomes a 50% error in the fire, and a flame height read off a video frame is easily 30% out because the tip is moving. Treat the answer as an order of magnitude.

If the correlation returns a flame height at or below the fuel surface, it is not saying the fire has gone out. It is saying the fire is too small for a base that wide, and that what is actually there is a patchy sheet of low flame rather than a coherent plume — a condition Heskestad's correlation does not describe.