MQH Hot Gas Layer Temperature

McCaffrey, Quintiere and Harkleroad's 1981 correlation is the workhorse of enclosure fire analysis by hand. It answers the question a room fire poses: how hot does the upper layer get? And it does it by dimensional analysis over roughly a hundred experiments, rather than by solving anything.

The structure is a heat balance in disguise. The numerator is what the fire puts in, squared. The denominator has two loss paths: $A_0\sqrt{H_0}$ is the ventilation factor, governing how much hot gas the opening carries out, and $h_kA_T$ is the conductance of the boundaries, governing how much the walls, ceiling and floor absorb. The cube root is what dimensional analysis produced, and the 6.85 is what the data fitted. It is dimensional — kilowatts, square metres, metres, and $h_k$ in kW/(m²·K).

The square root on the opening height is buoyancy-driven flow through a vertical opening. Hot gas leaves through the top, cool air enters through the bottom, and between them is a neutral plane where the pressures match. The driving head is the density difference times the height available, so the velocity goes as $\sqrt{H}$ — the same square root that appears in every orifice and weir equation. It is why a tall narrow doorway ventilates a fire better than a short wide one of equal area, and why a transom above a door matters more than its size suggests.

$h_k$ is not a film coefficient, and it depends on time. It is an effective conductance lumping everything the boundaries absorb. Before the heat wave has reached the far face of a thermally thick wall, $h_k = \sqrt{k\rho c/t}$ — it FALLS as the fire goes on, so the layer gets hotter with time even at constant fire size. Once the boundary has warmed through, $h_k = k/\delta$. The changeover is at the thermal penetration time $\delta^{2}\rho c/4k$, and which side of it you are on decides which expression to use. A gypsum-lined room and a concrete one behave nothing alike, and one $h_k$ for a whole enclosure is already an average over surfaces that differ. And $A_T$ is ALL interior surfaces less the opening — walls, ceiling and floor — not the floor area and not the walls alone.

Where the correlation stops. One opening. Natural ventilation. A quasi-steady fire — long enough at this size for the layer to settle. And a temperature rise below roughly 600 K, which is the range the data covered. Above that it drifts badly, and past flashover it is meaningless outright, because there is no longer a hot upper layer over a cool lower one for it to describe. Mechanical ventilation breaks it. Two openings at different heights break it; summing ventilation factors is a defensible approximation for openings at similar heights and a poor one otherwise. A fire growing faster than the boundaries can respond breaks the quasi-steady assumption.

Two things to check any answer against. Whether the fuel present can deliver that heat release rate at all, and whether the opening can supply the air — roughly $3000A_0\sqrt{H_0}$ kilowatts. A fire calculated above the ventilation limit does not burn at that rate inside the room; the excess leaves unburned and finds its air outside, which is a different and usually worse problem.

Finally, the answer is a RISE above ambient, and it is an AVERAGE over the upper layer. The gas in the plume is far hotter, the ceiling surface somewhat cooler than the gas it touches, and the lower layer is a different world.