Thermal Plume Analysis
Theory: a rising flow that entrains air
Overview
I want to simulate smoke spreading under a ceiling in a fire. What should I understand first?
The rising flow above a heat source — the thermal plume. As it rises it entrains surrounding air, so mass flow keeps growing and temperature falls quickly. Hitting the ceiling, it spreads sideways as a ceiling jet, which triggers detectors and sprinklers. Experimental correlations exist for these flows and serve as yardsticks for CFD. Let's compute plume and ceiling-jet values for a 100 kW fire, plus grid guidance.
Centreline temperature and velocity (Heskestad)
$\dot Q_c$ is the convective part of the heat release (about 70% of the total), $z_0$ the virtual origin. Temperature falls steeply as the −5/3 power of height and velocity slowly as the −1/3 power. These apply above the flame.
Ceiling jet (Alpert)
$H$ is the fire-to-ceiling height, $r$ the horizontal distance from the plume axis. It gives the hot layer just under the ceiling, used to estimate detector and sprinkler activation times.
Characteristic fire diameter
A guide for grid resolution in fire CFD: the larger $D^*$ divided by cell size, the better the plume is resolved.
The entrainment hypothesis
In 1956 Britain's Morton, Taylor and Turner proposed a simple hypothesis: a plume entrains surrounding fluid at a rate proportional to its rising velocity. From this 'entrainment hypothesis' follow the −5/3 power for temperature and −1/3 power for velocity in a point-source plume. Fire engineers such as Heskestad and Zukoski later confirmed it experimentally and extended it to finite fire sources. A hypothesis over sixty years old still underpins smoke-control design.
Worked examples
Example 1: 100 kW fire (0.5 m diameter)
Ambient 20 °C. Flame height is 0.97 m and the virtual origin is near the floor (0.01 m).
| Height | Centreline rise | Centreline velocity | Plume mass flow |
|---|---|---|---|
| 1.5 m | 219 K | 3.72 m/s | 0.70 kg/s |
| 2 m | 135 K | 3.38 m/s | 1.05 kg/s |
| 3 m | 69 K | 2.95 m/s | 1.94 kg/s |
| 5 m | 29 K | 2.49 m/s | 4.39 kg/s |
| 8 m | 13 K | 2.13 m/s | 9.47 kg/s |
From 2 m to 8 m, the temperature rise falls to a tenth but mass flow grows ninefold. Smoke volume depends more on how far the plume rises than on fire size.
Example 2: ceiling jet temperature
| Condition | Above axis | 1 m away | 3 m away | 5 m away |
|---|---|---|---|---|
| 3 m ceiling, 100 kW | 58 K | 39 K | 19 K | 13 K |
| 3 m ceiling, 500 kW | 171 K | 113 K | 54 K | 39 K |
| 6 m ceiling, 100 kW | 18 K | 18 K | 9 K | 7 K |
| 6 m ceiling, 500 kW | 54 K | 54 K | 27 K | 19 K |
With a 6 m ceiling, even a 100 kW fire raises only 18 K overhead. Detection could be slow.
Exactly. With a high ceiling, the plume entrains much more air before arriving and is diluted. Temperature falls as the −5/3 power of height, so doubling ceiling height cuts the rise to about a third. Many heat detectors need tens of kelvins above ambient, so tall atriums and warehouses use smoke or flame detectors or closer spacing. And with thermal stratification (warm air aloft), the plume may stall before reaching the ceiling.
Example 3: CFD grid guidance
| Heat release | D* | D*/δx = 4 | D*/δx = 10 | D*/δx = 16 |
|---|---|---|---|---|
| 100 kW | 0.38 m | 9.6 cm | 3.8 cm | 2.4 cm |
| 500 kW | 0.73 m | 18 cm | 7.3 cm | 4.6 cm |
| 2 MW | 1.27 m | 32 cm | 13 cm | 7.9 cm |
For small sources, a person (100 W) gives a 1.4 K rise and 0.34 m/s at 2 m above, and a 1 kW heater 6.7 K and 0.74 m/s. In room ventilation, such weak plumes still shape the air movement.
Analysis tips
- First estimate plume and ceiling-jet values from correlations, then compare with CFD.
- Set cells by D*/δx and check results don't change across at least two grids.
- Transient turbulence approaches such as LES suit resolving entrainment.
- Include heat loss to ceiling and walls (conduction and radiation); it lowers ceiling-jet temperature.
- Openings and ventilation tilt the plume; match boundary conditions to reality.
“Smoke stopped mid-way in an atrium”
In smoke tests in a 30 m glazed atrium, on a summer afternoon the smoke did not reach the ceiling; it spread sideways and stalled partway up. Solar gain had warmed the upper atrium, creating stratification, so the smoke's temperature matched its surroundings mid-way and it lost buoyancy. The ceiling detectors never activated. Detectors were added at intermediate heights and smoke extraction revised. Plume calculations miss such behaviour unless the ambient temperature profile is included.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Grid too coarse | Entrainment wrong | Use D*/δx |
| All heat as convective | Temperature too high | About 70% convective |
| Plume formula inside flame | Wrong temperatures | Check flame height |
| Ignoring ambient stratification | Smoke reach misjudged | Include temperature profile |
| Ignoring wall heat loss | Ceiling jet too hot | Include conduction and radiation |
I'd like to learn related topics.
Try the sprinkler coverage tool and tunnel fire ventilation tool. Related pages include fire simulation, thermal stratification, mixed convection and natural convection–conduction coupling.
Related Topics
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