Detector Response Prediction

When Will the Detector Activate?

Abstract

Heat detectors and sprinklers do not respond to fire; they respond to the hot gas that reaches them. Between the fire and the device lies a physical chain: combustion produces heat, heat drives a ceiling jet, the jet transfers energy to the detector element, and the element — slowed by its own thermal mass — eventually reaches the threshold that triggers activation. Predicting the moment of activation requires modelling every step in that chain. This paper describes SootSafe's implementation of ceiling jet correlations and the RTI thermal-lag equation, and explains how integrating these with the fire room temperature model (see Pseudo-Transient Fire Simulation) produces activation predictions that are more accurate than the original DETACT-T2 tool for confined rooms.

1. Why Activation Time Matters

A detector that activates at 60 seconds into a fire sees roughly one-quarter the heat release rate of one that activates at 120 seconds (for a medium α·t² fire). The difference in outcome — smoke volume produced, suppression effectiveness if a sprinkler is triggered, time before an alarm evacuates occupants — is enormous. A heat detector or smoke-control relay must activate before smoke has migrated beyond the fire compartment, or the evacuation strategy it is meant to trigger becomes moot.

Accurate activation time prediction requires three things:

A model of how hot and fast the ceiling jet gas is at the detector's location.
A model of how the detector element heats up in response to that gas.
A fire room temperature that correctly reflects the accumulated smoke layer — something DETACT-T2, an unconfined-ceiling tool, cannot provide.

2. The Ceiling Jet

When hot combustion gases rise from a fire and impinge on the ceiling, they spread radially outward as a thin, fast-moving layer called the ceiling jet. The gas temperature and velocity in this layer — at any radial distance $r$ from the fire axis and ceiling height $H$ above the fire base — are the primary inputs to the detector thermal model.

2.1 Alpert (1972) Correlations

The widely-used correlations derived by Alpert (1972) express ceiling jet properties in terms of the instantaneous heat release rate $\dot{Q}$ [kW], ceiling height $H$ [m], and radial distance $r$ [m].

Temperature rise above ambient:

$\Delta T_{\text{jet}}(r, H, \dot{Q}) = \begin{cases} \dfrac{16.9\; \dot{Q}^{2/3}}{H^{5/3}} & r/H \leq 0.18 \quad \text{(near-axis)} \\[10pt] \dfrac{5.38\; (\dot{Q}/r)^{2/3}}{H} & r/H > 0.18 \quad \text{(far-field)} \end{cases}$

Gas velocity:

$u_{\text{jet}}(r, H, \dot{Q}) = \begin{cases} 0.96\; \left(\dfrac{\dot{Q}}{H}\right)^{1/3} & r/H \leq 0.15 \quad \text{(near-axis)} \\[10pt] \dfrac{0.195\; \dot{Q}^{1/3}\; H^{1/2}}{r^{5/6}} & r/H > 0.15 \quad \text{(far-field)} \end{cases}$

Reference: Alpert, R.L. (1972). Calculation of response time of ceiling-mounted fire detectors. Fire Technology, 8(3), 181–195.

The near-axis and far-field transition thresholds differ between the temperature and velocity correlations (0.18 and 0.15 respectively), consistent with Alpert's original formulation. Both correlations return zero for $\dot{Q} < 1\,\text{kW}$ to avoid domain errors in the first seconds after ignition.

2.2 Placement Geometry

A detector is characterised by two geometric parameters: its radial distance $r$ from the fire axis (m) and the ceiling height $H$ above the fire base (m). These are user inputs — SootSafe does not infer placement from the network topology. Multiple detectors at different positions can be evaluated in the same simulation run.

3. Detector Thermal Model — The RTI Equation

3.1 Thermal Lag

A detector element (the fusible link of a sprinkler, or the bimetallic strip of a heat detector) has finite thermal mass. It heats up more slowly than the surrounding gas, and this lag determines when it activates. The standard model for this lag was established by Heskestad & Smith (1976):

$\frac{dT_d}{dt} = \frac{\sqrt{u}}{RTI} \cdot (T_g - T_d)$

Symbol	Description
$T_d$	Detector element temperature (°C)
$T_g$	Gas temperature at the detector location (°C)
$u$	Gas velocity at the detector location (m/s)
$RTI$	Response Time Index $\left[(\text{m·s})^{1/2}\right]$

The RTI (Response Time Index) is a device-specific constant that characterises how quickly the element responds to a given gas velocity. A lower RTI means faster response. Typical values range from 10–50 for heat detectors to 50–200 for standard sprinkler heads; fast-response sprinklers fall in the 5–10 range.

Integrating this equation with $\Delta t = 1\,\text{s}$ (forward Euler):

$T_d(t + 1) = T_d(t) + \frac{\sqrt{u(t)}}{RTI} \cdot \bigl(T_g(t) - T_d(t)\bigr) \cdot \Delta t$

Reference: Heskestad, G. & Smith, H.F. (1976). Investigation of a new sprinkler sensitivity approval test: The plunge test. FMRC Technical Report.

3.2 The C-Factor — Conduction Loss for Sprinklers

For sprinkler heads, the fusible link or glass bulb is in thermal contact with the fitting and pipe, which act as a heat sink. At low temperatures this conduction loss is significant and causes the sprinkler to respond more slowly than the basic RTI equation predicts. The enhanced model adds a conduction term:

$\frac{dT_d}{dt} = \frac{\sqrt{u}}{RTI} \cdot (T_g - T_d) \;-\; \frac{C}{RTI} \cdot (T_d - T_{\text{amb}})$

where $C$ $\left[(\text{m/s})^{1/2}\right]$ is the C-Factor, also called the conduction factor. The second term represents heat conducted away from the element toward ambient through the mounting; it is proportional to how far the element has risen above ambient and inversely proportional to the velocity-normalised response time.

Setting $C = 0$ recovers the classic DETACT-T2 behaviour. For sprinklers, published C-Factor values typically range from 0 to 1.5 $(\text{m/s})^{1/2}$ ; heat detectors generally use $C = 0$ .

3.3 Activation Criteria

Two activation criteria are supported:

Fixed-temperature activation (most heat detectors and sprinklers):

$T_d \geq T_{\text{act}}$

where $T_{\text{act}}$ is the rated activation temperature (e.g. 68 °C for a standard sprinkler).

Rate-of-rise activation (combination detectors):

$\frac{dT_d}{dt} \geq \dot{T}_{\text{limit}}$

where $\dot{T}_{\text{limit}}$ [°C/s] is the rate-of-rise threshold. This criterion can trigger before the element reaches $T_{\text{act}}$ , allowing early activation in fast-growing fires. Both criteria can be active simultaneously on the same device; activation occurs on whichever is satisfied first.

4. The Hybrid Gas Temperature — SootSafe's Key Differentiator

4.1 The Limitation of Unconfined Ceiling Models

DETACT-T2 was designed for an unconfined ceiling: a large open space where the hot gas layer does not accumulate. In such a space, the ceiling jet temperature at the detector is well-described by the Alpert correlations alone.

In a real building, however, the fire room has walls. Hot gases accumulate in the upper layer of the room as the fire grows. In a small or poorly-ventilated compartment, this accumulated smoke layer can become substantially hotter than the ceiling jet at the detector's radial distance — particularly for detectors located far from the fire axis or for fast-growing fires. DETACT-T2 would predict a lower gas temperature at the detector than actually exists, and therefore a later (non-conservative) activation time.

4.2 The Hybrid Model

SootSafe resolves this by comparing the Alpert ceiling jet temperature against the room node temperature from the pseudo-transient model and using the higher of the two as the effective gas temperature at the detector:

$T_g^{\text{eff}}(t) = \max\!\left(T_{\text{amb}} + \Delta T_{\text{jet}}(t),\; T_{\text{room}}(t)\right)$

where $T_{\text{room}}(t)$ is the fire room temperature computed by the adiabatic well-stirred model at each timestep (see Pseudo-Transient Fire Simulation, §4.1).

This hybrid approach is only possible because the detector model runs inside the same pseudo-transient loop that computes $T_{\text{room}}(t)$ . The Alpert model and the room thermal model share the same $\dot{Q}(t)$ at each second, so the comparison is always at consistent fire conditions.

The practical effect is that in small or confined rooms, the detector is exposed to the hotter of the jet (relevant for near-axis placement) and the smoke layer (relevant for far-axis placement or late in the fire). The result is a more conservative and more realistic activation time than DETACT-T2 alone would produce.

5. Integration with the Transient Loop

Detector probes are updated once per simulated second inside the pseudo-transient fire loop, after the pressure solve and before the snapshot is stored:

Retrieve $\dot{Q}(t)$ from the current fire timestep.
Compute $\Delta T_{\text{jet}}$ and $u_{\text{jet}}$ from the Alpert correlations using the detector's $r$ and $H$ .
Compute $T_g^{\text{eff}} = \max(T_{\text{amb}} + \Delta T_{\text{jet}},\; T_{\text{room}}(t))$ .
Call the detector's update(T_g^{\text{eff}}, u_{\text{jet}}, T_{\text{amb}}, 1.0).
If the activation criterion is now met, record a DetectorActivationEvent with the current time, detector index, element temperature, and HRR.

Once triggered, a detector stops updating — it cannot re-arm within a simulation run. The detector element temperature at every sampled snapshot is included in the output, allowing the engineer to observe the full heating trajectory, not just the activation instant.

Multiple detectors at different positions can be evaluated in a single simulation run at no additional solver cost, since the Alpert evaluation and RTI integration are negligible compared to the pressure solve.

6. Output

For each detector that activates, a DetectorActivationEvent records:

Field	Description
`t_s`	Activation time (s)
`detector_index`	Index in the input detector array
`trigger_temp_c`	Element temperature at the moment of activation (°C)
`hrr_at_trigger_kw`	Fire heat release rate at activation (kW)

For detectors that do not activate within the 600-second simulation window, no event is recorded — the absence of an event is itself informative: the fire room temperature never reached the threshold within the design fire duration.

The detector element temperature trajectory is available at each snapshot second via detector_temps_c, enabling visualisation of the heating curve and comparison across multiple detector specifications.

7. Assumptions and Limitations

Assumption	Consequence	Direction
Steady-state ceiling jet	Alpert correlations assume a fully-developed, quasi-steady jet; they are less accurate in the first seconds of a fire and for very low HRR	Mildly conservative in the very early seconds — Alpert slightly over-predicts a still-developing jet, causing the model to heat the detector element faster than reality; negligible once the fire is established
Point source fire	The correlations assume a point source at floor level; plume height and flame geometry are not modelled	Mildly non-conservative — the real plume's virtual origin is above floor level, making the effective ceiling clearance shorter and the ceiling jet hotter than modelled with the full floor-to-ceiling $H$
Unobstructed ceiling	Beams, joists, or partitions that disrupt the ceiling jet are not accounted for	Context-dependent — beams can trap hot gas near the detector (conservative) or deflect the jet away from it (non-conservative); no consistent direction, requires case-by-case assessment
Single ceiling height	Each detector uses a single $H$ value; sloped or multi-level ceilings require per-detector adjustment	Input-dependent — conservative if $H$ matches the actual ceiling height at the detector; non-conservative if the actual ceiling at the detector location is lower than the specified $H$ (e.g. using ridge height for a sloped ceiling when the detector sits at a lower point)
Well-stirred room layer	The room temperature $T_{\text{room}}$ is uniform (no stratification); in tall rooms the lower half may be cooler than the model suggests	Mildly non-conservative for detectors where $T_{\text{room}}$ is the dominant term (far-field placement, late in the fire) — the well-stirred average under-estimates the actual hot ceiling layer temperature
Fixed ambient temperature	$T_{\text{amb}}$ is constant at 20 °C throughout the simulation	Neutral at the 20 °C design condition; cold-climate installations (higher $\Delta T$ needed to reach activation) are mildly conservative, hot-climate installations mildly non-conservative
No post-activation suppression feedback	Activation is predicted but any suppression effect (sprinkler cooling, alarm-triggered ventilation changes) is not modelled; HRR continues on its prescribed curve after activation	Neutral for single-detector activation time; mildly non-conservative in multi-detector scenarios where early suppression would have reduced HRR before a second detector activates
Forward Euler RTI integration	First-order accurate; adequate at 1-second resolution for the RTI time scales of interest	Mildly non-conservative in principle — forward Euler under-estimates a monotonically rising heating rate; negligible at 1-second resolution given RTI time scales of tens to hundreds of seconds

The primary conservative mechanism in this model is the hybrid temperature formula (§4.2): using $\max(T_{\text{amb}} + \Delta T_{\text{jet}},\; T_{\text{room}})$ as the effective gas temperature ensures the detector is never under-exposed. This offsets the mildly non-conservative assumptions above, particularly for far-field detectors late in the fire where the well-stirred room temperature under-estimates the hot ceiling layer.

8. Standards and References

Source	Scope
Alpert, R.L. (1972). Fire Technology, 8(3), 181–195	Ceiling jet temperature and velocity correlations
Heskestad, G. & Smith, H.F. (1976). FMRC Technical Report	RTI equation and C-Factor for sprinkler thermal lag
Evans, D.D. & Stroup, D.W. (1986). DETACT-T2 (NIST)	Original fixed-temperature detector activation program
EN 12259 / UL 199	Sprinkler RTI and C-Factor classification
ISO 10294-3	Air physical constants used in the companion thermal model

9. Summary

SootSafe's detector response model combines three components:

Alpert (1972) ceiling jet correlations — two-branch empirical formulas that give the gas temperature and velocity at any point beneath an unconfined ceiling, parameterised by HRR, ceiling height, and radial distance.
The enhanced RTI equation (Heskestad & Smith, 1976) — a first-order ODE that models the thermal lag of a sensing element, with an optional C-Factor term that corrects for conduction loss in sprinkler heads. Setting $C = 0$ reproduces the original DETACT-T2 behaviour.
The hybrid temperature model — SootSafe's key extension: the effective gas temperature at the detector is the maximum of the Alpert ceiling jet temperature and the fire room node temperature from the pseudo-transient model. This ensures that in confined rooms, where an accumulated smoke layer becomes hotter than the ceiling jet, the model does not under-predict the thermal exposure — and therefore does not over-predict activation time.

The result is activation time predictions that are strictly more conservative than DETACT-T2 for confined spaces, and identical to DETACT-T2 in the limit of a large unconfined room.