Alpert Ceiling Jet Temperature (far field)

A plume that reaches a ceiling has nowhere to go but sideways. It turns, and spreads outward as a thin, fast layer hugging the underside of the slab — the ceiling jet. It is where sprinklers and heat detectors live, so it is where the arithmetic of detection happens. Ronald Alpert published the correlations for it in Fire Technology in 1972, and they have been in continuous use ever since.

There are two correlations, not one, and the boundary between them is $r/H = 0.18$. Inside that radius you are in the turning region where the plume strikes and spreads, and the applicable form is $\Delta T = 16.9\,\dot{Q}^{2/3}/H^{5/3}$, with no dependence on $r$ at all — the whole impingement zone is at one temperature. Outside it, the far-field form on this page applies. Using the far-field equation inside the turning region overstates the temperature, and the error grows fast as $r$ shrinks, because $\dot{Q}/r$ climbs without bound while the real temperature simply plateaus. This page computes $r/H$ for you and says when you are out of range.

The constant 5.38 is dimensional — kilowatts, metres, kelvin — like almost everything else in this shard.

$H$ is measured from the fuel surface to the ceiling, not from the floor. A fire on a bench two metres up in a five-metre room has $H = 3$ m, and using 5 understates the temperature badly. Where the fire is on top of stored goods, the effective ceiling height can be a small fraction of the deck height, which is exactly why storage height limits appear in sprinkler standards.

The assumptions are strong ones. An unconfined, unobstructed, flat, horizontal ceiling. A fire far enough from walls to entrain freely. And a hot layer that has not yet descended to the detector — once a layer forms and thickens, the jet is running inside hot gas rather than under ambient air, and it runs hotter than the correlation says. Beams and joists channel the jet along the bays and slow its spread across them. A sloped or vaulted ceiling sends it uphill. Ductwork, smoke curtains and open-plan obstructions all break it. In each of those cases the answer stops describing the room, and the honest response is to say so rather than to apply a correction factor pulled from nowhere.

One more distinction, and it is the one that most often causes trouble. Reaching a sprinkler's rated temperature in the gas is a necessary condition for operation and never a sufficient one. The element has thermal mass and lags the gas by a time governed by its response time index. The RTI is defined with a $\sqrt{u}$ in it, so the velocity from the companion page is as much a part of the answer as the temperature here. Gas at the rated temperature moving slowly can take minutes to open a head that a faster jet at the same temperature opens in seconds. Detector spacing and sprinkler design are code matters — NFPA 13, NFPA 72, or whatever the authority having jurisdiction adopts — and these correlations are a way of understanding those requirements, not of replacing them.