Ventilation-Limited Burning Rate (Kawagoe)

Kunio Kawagoe published this in 1958, and it contains the strangest fact in compartment fire science: in a fully developed compartment fire, how much fuel is present does not set how fast it burns. The opening does.

Double the fuel load in a room and the fire burns for twice as long at the same rate. Halve the doorway and the same fuel takes twice as long. Nothing about the fuel's own burning characteristics enters the equation at all — only the area and height of the opening. The reason is that the fire has long since consumed all the oxygen in the room and is burning everything the doorway brings in; the volatiles are available in excess, and the air is the constraint.

This is why fire resistance ratings are expressed in TIME. A fuel load in a compartment does not translate into an intensity, it translates into a DURATION — how long the fire has to run before the fuel is gone. The whole apparatus of standard fire tests and equivalent fire severity rests on this observation.

The 0.5 is dimensional — kilograms per second, square metres, metres — and it is specifically for CELLULOSIC fuel, wood and paper, which is what Kawagoe burned. Underneath it is an air flow of about $0.5A_0\sqrt{H_0}$ times 5.7 kg/s, the 5.7 being roughly the stoichiometric air-to-fuel mass ratio for wood. Fuels with different stoichiometry give a different coefficient. Taking the effective heat of combustion of wood near 16.4 MJ/kg gives the familiar rule of thumb $\dot{Q} \approx 3000A_0\sqrt{H_0}$ kilowatts, which is the cap that every other calculation in this shard has to be checked against.

The square root on the height is the same buoyancy argument as the MQH page: hot gas out the top, cool air in the bottom, a neutral plane between, and a driving head proportional to the height available. A tall narrow opening feeds a fire better than a short wide one of the same area.

What happens when a fire is calculated to burn harder than this is important and it is not that the equation is wrong. The excess fuel leaves the compartment unburned and finds its air outside the opening. That is why flames come out of windows, why fire spreads up a façade, and why an under-ventilated compartment fills with a flammable mixture that will ignite the moment someone provides air. A door opened onto that room is a backdraught.

The correlation applies to the FULLY DEVELOPED phase only. A growing fire is fuel-controlled and this rate is an upper bound it has not yet reached; a decaying one has run out of volatiles and falls below it. And ventilation is not a fixed property of a room during a fire. Glass fails, a door burns through, a lightweight partition collapses, and $A_0$ jumps. The burning rate jumps with it, and a compartment can go from quietly smouldering to fully involved in a very short time on nothing more than a window breaking.