Sprinkler Activation TimeCeiling-jet temperature rise and RTI-based activation

What this calculates

This calculator estimates when a sprinkler will activate above a t²-growth fire, based on Alpert's ceiling-jet correlations and the sprinkler's Response Time Index (RTI). The fire grows as Q(t) = α t², generating a ceiling jet at the height H above the fire. The sprinkler, located at radial distance r from the plume axis, heats up at a rate that depends on the ceiling-jet temperature and velocity and on the bulb's RTI; it activates when its element reaches the rated activation temperature.

How to use it

Enter the ceiling height, the sprinkler spacing on each side (the radius is computed automatically), the ambient temperature, and the sprinkler activation temperature. Choose a fire growth rate (Slow / Medium / Fast / Ultrafast — α values per NFPA) and an RTI category (Quick Response fast/slow or Standard).

The activation time is solved iteratively. The Q at activation is reported alongside.

Variables

t_act
Sprinkler activation time [s]
Q(t)
Heat release rate = α t² [kW]
α
Fire growth coefficient (NFPA) [kW/s²]
H
Ceiling height above the fire [m]
r
Radial distance from fire plume axis to sprinkler [m]
T_a
Ambient temperature [K]
T_act
Sprinkler activation temperature [°C]
RTI
Sprinkler Response Time Index [m½·s½]

Equations

$$ r = \sqrt{ (S_{1}/2)^{2} + (S_{2}/2)^{2} } $$Sprinkler radius from spacing S₁ × S₂
$$ \Delta T_{cj} = \dfrac{ 5.38 \, ( Q / r )^{2/3} }{ H } \quad (r/H > 0.18) $$Maximum ceiling-jet temperature rise (Alpert)
$$ \dfrac{ d T_{b} }{ d t } = \dfrac{ \sqrt{ u_{cj} } }{ RTI } \, ( T_{cj} - T_{b} ) $$Sprinkler bulb response (Heskestad / Schifiliti)
$$ T_{b}(t_{act}) = T_{act} $$Activation criterion

Discussion

The original spreadsheet used a manual goal-seek on the activation time. The web version solves the same equations numerically (root-find on the bulb temperature crossing the activation set-point), so no iteration is required.

Inputs are entered in imperial units to match the spreadsheet, but the equations are evaluated in SI internally.

Worked example

Ceiling height 17′, sprinkler spacing 6′ × 10′ (radius ≈ 5.83′), ambient 65 °F, activation 165 °F, Fast growth rate, Standard RTI.

Expected result: activation at about 140 s, fire size at activation ≈ 920 kW.

References

  1. Babrauskas, V. — "Burning Rates," SFPE Handbook of Fire Protection Engineering.
  2. NFPA 13 — Standard for the Installation of Sprinkler Systems, for RTI categories.
  3. Evans, D. — "Ceiling Jet Flows," SFPE Handbook of Fire Protection Engineering.
  4. Alpert, R. — "Ceiling Jet Flows," SFPE Handbook of Fire Protection Engineering.
  5. Schifiliti, R., Meacham, B., and Custer, R. — "Design of Detection Systems," SFPE Handbook of Fire Protection Engineering.