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Ceiling Jet and Sprinkler Activation — Alpert
Temperature and velocity in the ceiling jet, and the activation time of a sprinkler or detector against a t² fire, using Alpert's correlations.
What is being calculated
Geometry
Zero is valid and describes a point directly above the fire.
The fire
This can be taken from the pool fire calculator or the t² fire growth calculator.
Two transition points, not one
Validity range
The conditions that determine whether the correlation applies. Crossing them changes the state of the result.
- The temperature changes branch at r/H = 0.18 and the velocity at r/H = 0.15. These are not the same number: between the two ratios the temperature is still on the near branch while the velocity is already on the far one, which is why every value in the result states the branch it was calculated from.
- The two branches do not meet exactly at the transition point. For temperature the difference is 0.14% and for velocity 1.29% — the far branch is below the near one in both cases. The tool does not smooth the step: smoothing would produce numbers Alpert never published, and it would hide a real difference.
Model limits
What the model does not describe, even within its validity range. These always hold.
- The correlations hold for a flat, unobstructed ceiling, and for the period before a developed smoke layer has formed. These are qualitative limits rather than a numeric range, which is why they are stated here and not enforced as a domain.
The equation
r/H ≤ 0.18: ΔT = 16.9 · Q̇^(2/3) / H^(5/3) · r/H > 0.18: ΔT = 5.38 · (Q̇/r)^(2/3) / H · T_g = T∞ + ΔT
20 + 82.11
- ΔT [°C]
- Temperature rise above ambient
- T_g [°C]
- Gas temperature beneath the ceiling
- Q̇ [kW]
- Heat release rate
- H [m]
- Ceiling height above the fire
- r [m]
- Radial distance from the plume axis
- T∞ [°C]
- Ambient temperature
Alpert — SFPE Handbook 5th ed., Ceiling Jet Flows
Result
ResultThe calculation completed within the model's range of validity.
- Gas temperature beneath the ceiling
- 102.1 °C
- ΔT above ambient — far branch
- 82.1 °C
- Jet velocity u — far branch
- 2.199 m/s
- r/H ratio
- 0.6
- r/H ratio3 / 50.6
- ΔT — far branch (r/H > 0.18): 5.38·(Q̇/r)^(2/3)/H5.38 · (2,000/3)^(2/3) / 582.11 °C
- u — far branch (r/H > 0.15): 0.195·Q̇^(1/3)·H^(1/2)/r^(5/6)0.195 · 2,000^(1/3) · 5^(1/2) / 3^(5/6)2.199 m/s
- Gas temperature beneath the ceiling20 + 82.11102.11 °C
Source of the equations
Alpert, SFPE Handbook of Fire Protection Engineering, 5th ed. — the "Ceiling Jet Flows" chapter.
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