t-Squared Fire Growth

Almost every performance-based fire analysis begins by drawing a curve of heat release rate against time, and almost every one of those curves is a parabola: $\dot{Q} = \alpha t^{2}$. The choice is not arbitrary. Fire spreading over a flat fuel surface tends to grow radially, the burning area goes as the square of a spreading radius, and the heat release rate follows the area. The square falls out of the geometry.

The four named growth rates — slow, medium, fast and ultra-fast — are not four kinds of fire. They are four values of one coefficient, and they come from a single convention: the time taken to reach 1055 kW, which is 1000 BTU per second. Six hundred seconds gives $\alpha = 0.00293$, three hundred gives 0.01172, one hundred and fifty gives 0.0469, seventy-five gives 0.1876. That is the whole of the definition. The names are a filing system, and the association of particular occupancies with particular rates is a matter of judgement and of code guidance, not of physics.

The coefficient is unit-bound and does not convert. It is kilowatts per second squared, and this site types it as dimensionless with a printed label because there is no unit type for a power over a squared time and there should not be one — nothing else on the site would ever use it. The consequence is exactly the one that afflicts the Taylor constant in machining: a bare number copied out of a source is only meaningful in the units that source was written in, and no calculator can guess which.

What the curve leaves out is more interesting than what it contains. It has no incubation period. A cigarette in an armchair, a pyrolysing cable, a smouldering bale — these can run for many minutes, sometimes hours, producing smoke and carbon monoxide and no significant heat, and the t² clock has not started. Many fatal fires do their killing in that phase. The curve starts at what the literature calls effective ignition, and the gap between real ignition and effective ignition is not modelled here at all.

It also has no ceiling, and every real fire has two. The first is the fuel: the curve will happily climb past what the material present can deliver, and the fuel-area page is where that gets checked. The second is the air: a compartment fire cannot burn faster than its openings admit oxygen, and the ventilation-limited page gives that cap. The smaller of the two governs, and an unchecked t² curve carried into a room calculation is one of the commonest ways to produce a confidently wrong answer.

Finally, a real fire's growth is nothing like smooth. It jumps when it finds new fuel, plateaus when it does not, and can decay and revive. The t² curve is an envelope drawn through that for design purposes, and it is intended to be a reasonably conservative one. It is a design tool, not a prediction, and the distinction matters most when someone starts quoting seconds off it as though they were measured.