Plume Centreline Temperature Rise

Above the flame tip, the fire stops being a chemical reaction and becomes a fluid mechanics problem: a column of hot gas rising through cooler air, entraining as it goes, cooling and slowing and widening. That column is the plume, and it is what fills a room with smoke, what triggers detectors, and what heats structure long before any flame reaches it.

The correlation is a similarity solution. Far enough above the source, a buoyant plume forgets the details of what made it and depends only on the buoyancy flux — which is why the temperature rise on the axis goes as $\dot{Q}_c^{2/3}(z-z_0)^{-5/3}$ and nothing else appears. The 25.0 is not dimensionless. It is Heskestad's coefficient evaluated for air at about 20 °C, with the heat release rate in kilowatts and heights in metres, and it has the ambient density, the specific heat and gravity baked into it. It is not valid at altitude, in an already-hot room, or in any other unit system.

$\dot{Q}_c$ is the convective heat release rate, not the total. Radiation leaves the fire sideways and does not drive the column, so the plume only feels what is left. For most fuels the convective fraction is 0.6 to 0.8, and 0.7 is the usual default. Entering the total instead overstates the temperature by about 27%, and it is the single commonest error on this page.

The virtual origin $z_0$ is a bookkeeping device rather than a place. The tidy similarity solution assumes a point source; a real fire is a finite region of flame. $z_0$ is the location of the point source that would produce the same plume far above, and it is usually BELOW the fuel surface — a negative number — for a small, intense fire, and above it for a wide, cool one. Heskestad's own expression is $z_0 = 0.083\,\dot{Q}^{2/5} - 1.02D$, which echoes the flame height correlation closely: the same $1.02D$, and a leading coefficient about a third of the 0.235.

Three limits are worth stating plainly. First, this is a FAR-FIELD solution and it describes the column above the flame tip. Inside the flame it returns fiction — the equation will happily report two thousand kelvin, while the real gas in continuous flame sits near 900 °C and goes no higher, because it is still burning rather than simply rising. Find the flame tip first. Second, the answer is a CENTRELINE value, and the plume is hottest on the axis, falling off roughly as a Gaussian across it; a target a metre off-axis at the same height sees much less. Third, this is a plume in a quiet room. A supply diffuser, an open door, a fan — any cross-flow tilts the plume, and the correlation stops applying at the moment it does.

Used backwards, the plume equation is a reconstruction tool: a thermocouple record, or the melting point of something at a known height, gives a fire size. The 3/2 power on temperature is tolerable, but the height it comes paired with carries a 5/2 power, and heights at a fire scene are rarely known that well.