Alpert Ceiling Jet Velocity (far field)

The companion to the temperature correlation, and the half that is easier to forget. Heat gets into a detector element by convection, and convection needs flow. The velocity of the ceiling jet is therefore not a curiosity — it is half of any response calculation.

The convective coefficient at a small cylindrical element rises roughly with the square root of the gas speed. That is why the response time index is defined as $\tau u^{1/2}$, with units of $\mathrm{m}^{1/2}\mathrm{s}^{1/2}$, and why a quick-response sprinkler and a standard one differ by a factor of several in RTI while sitting in the same gas. Slow gas at the rated temperature can take minutes to open a head that a faster jet at the same temperature opens in seconds.

Note that the velocity far field begins at $r/H = 0.15$, while the temperature far field begins at 0.18. The two thresholds are different and they are not interchangeable, so a point can be inside one correlation's range and outside the other's. Inside 0.15 the applicable velocity form is $u = 0.96(\dot{Q}/H)^{1/3}$, independent of $r$.

The most instructive thing about this correlation is a sign. Ceiling height enters the velocity with a PLUS — $H^{1/2}$ in the numerator — so the jet is faster under a higher ceiling at the same radius. On the temperature page, $H$ makes things cooler. Both are right, and together they say one thing: the plume has further to fall through, so it entrains more air and arrives with more mass and momentum and less temperature. More flow, less heat. That combination is why heat detection gets harder in tall spaces — there is plenty of gas going past the detector and not enough heat in it — and why atria, warehouses and hangars reach for beam detectors, aspirating systems and flame detectors instead of spot heat detectors.

The jet is thin, and how thin matters for mounting. Its maximum velocity sits at roughly one per cent of the ceiling height below the ceiling, and the layer carrying most of the flow is about ten per cent of $H$ deep. A sprinkler deflector or a detector hung too far below the deck is not in the jet at all; it is under it, in slower and cooler air. The deflector distances in sprinkler standards are not arbitrary and they are not a construction tolerance.

Run backwards, this equation estimates a fire size from a measured velocity, and the cube root in the forward direction becomes a cube here: a 10% error in the velocity is a 33% error in the fire. That direction belongs to interpreting an instrumented test, not to design.