Entrainment

Entrainment is the drawing of surrounding air into a rising stream of hot gases from a fire. The added air mixes with combustion products, increases the transported mass and cools the rising mixture. In a fire plume, air enters across an irregular boundary that continually changes shape. The mass reaching a ceiling can therefore greatly exceed the mass released directly by the burning fuel.

Entrainment connects the fire to the smoke layer above it. Its prediction helps determine the layer's mass, temperature and rate of growth. Different correlations describe different source shapes and heights above the fire. Their results depend on the surrounding conditions and on whether the calculation lies within or above the flames. A free plume correlation also has a boundary of application: it describes the rise through surrounding air, rather than continued mixing inside an existing smoke layer. This distinction matters when calculating the flow entering that layer.

How surrounding air enters the plume

Hot combustion products rise because their density differs from that of the cooler surroundings. Their motion draws ambient air into the flow as it develops upwards. The interface between plume and surroundings is sharply defined at an instant, but its folded shape changes continuously. A time averaged plume outline therefore represents a fluctuating flow rather than a fixed physical surface.

Surrounding air enters the rising plume and increases its mass flow. Based on Heskestad (2002), chapter 2-1, pp. 2-1–2-2 and 2-11.

Turbulence creates mixing and fluctuations within the plume. The flame region contains both the incoming air needed for combustion and combustion products leaving the reacting region. Above the flames, additional air continues to mix with the hot gases. The plume can therefore become cooler while its total mass flow increases with height.

Mass flow and temperature describe different aspects of this process. Mass flow measures the quantity crossing a horizontal plane each second, whereas temperature describes the thermal state of that flow. A larger flow does not by itself indicate a larger fire. It can indicate that the plume has travelled farther and mixed with more surrounding air.

Heskestad explains that the fuel's direct mass contribution is generally insignificant compared with the entrained flow at an elevated cross section. This explains why smoke management calculations cannot use the fuel burning rate as their exhaust mass flow. The system receives a mixture dominated by air that entered after combustion began. Fuel consumption remains relevant to energy and species production, but it is a different input.

The distinction also prevents an apparent contradiction in smoke calculations. Adding air can dilute the concentration of combustion products while increasing the volume that needs accommodation or extraction. Dilution therefore does not necessarily produce a smaller smoke layer. Its effect must be assessed together with the layer temperature, density and outgoing flow.

Calculating the mass flow above flames

The Heskestad plume correlation relates entrained flow above the mean flame height to convective heat release and distance from a virtual source. Convective heat release is the energy carried upwards by the gas flow. The virtual source is a calculated origin that represents the effect of a fire with finite area. It need not coincide with the fuel surface.

Van Rossum gives the following additive form for the plume mass flow above the mean flame height. The first term increases with height, while the second provides a correction proportional to convective heat release. The numerical coefficients apply when heat release is entered in kilowatts and lengths in metres.

ṁ_p = 0.071 Q̇_c^1/3 (z − z₀)^5/3 + 0.00192 Q̇_c

For a pool fire, the virtual origin can be estimated from total heat release and fire diameter. Total heat release is used here even though the mass flow equation uses its convective part. Mixing these two inputs would alter both the predicted source position and the resulting flow.

z₀ = 0.083 Q̇^2/5 − 1.02 D

Symbol	Description	Unit
ṁp	Plume mass flow at the calculation height	kg/s
Q̇c	Convective heat release rate	kW
Q̇	Total heat release rate	kW
z	Height above the fire source	m
z0	Virtual origin above the same reference level	m
D	Diameter of the fire source	m

The two heights must share a reference level. A negative virtual origin increases the effective rise distance above the source, whereas a positive origin reduces it. The virtual origin is a mathematical representation of the developed plume. It is not an additional fire located below or above the actual fuel.

These equations describe mean properties of an approximately axisymmetric plume in suitable ambient conditions. They do not describe each instantaneous eddy or each local velocity across the plume. Nor does the calculated mass flow establish the average smoke layer temperature. That requires a separate energy balance, including energy losses and flow leaving the layer.

A calculated example

Heskestad's worked examples describe a methyl alcohol pool fire with a diameter of 1.5 m and heat release of about 884 kW. The source heat release per area is 500 kW/m². The chapter adopts a convective fraction of 0.8 for this fuel, which produces relatively little luminous radiation. These values describe the selected example, rather than every liquid pool fire.

Using the circular area without intermediate rounding gives a total heat release of 883.6 kW. Its convective part is 706.9 kW. Substitution in the virtual origin equation gives approximately −0.28 m. The same chapter's flame height calculation places the mean flame tip at about 1.8 m, so evaluation at 5 m lies above the flames.

At a height of 5 m, the effective rise distance is about 5.28 m. Substituting the unrounded inputs into the mass flow equation gives 11.48 kg/s, or about 11.5 kg/s. Evaluating the same idealised fire at 10 m gives 32.09 kg/s, or about 32.1 kg/s. These are calculated extensions of the source example using the published correlation.

Calculated plume flow increases with height for an unchanged fire. Based on Heskestad (2002), chapter 2-1, pp. 2-4, 2-8 and 2-10, and Van Rossum (2027), §6.7.

The increase illustrates the effect of entrainment during continued rise. The fire's heat release remains unchanged between the two calculation planes. Nevertheless, the flow at the higher plane is much larger because additional ambient air has entered the plume. The comparison assumes that both planes remain below any smoke interface and within the free plume's application conditions.

The results are mass flows, not volumes per second. Conversion to a volume flow requires the gas density at the location considered. The temperature of gases in an exhaust duct or smoke layer can differ from that of the surrounding room air. A consistent smoke control calculation therefore keeps mass and energy balances separate until the relevant gas state is established.

Choosing a plume representation

The Thomas plume correlation offers a different representation for broad fires with relatively short flames. In its simplified form, mass flow depends on fire perimeter and rise height. Van Rossum describes a rough selection based on the relationship between mean flame length and source diameter. The choice concerns the geometry and development of the plume, rather than fire power alone.

A source with a large perimeter presents a different entraining boundary from a compact circular source. Smoke spilling from beneath an obstruction can also form an elongated plume. Van Rossum describes this shape as having more perimeter relative to its cross section, allowing greater mixing with surrounding air. Applying a circular free plume correlation to that flow can miss the relevant geometry.

A nearby wall or corner changes the paths available for incoming air. However, a universal rule that simply halves or quarters free plume flow is not adequate for every arrangement. The introductory book distinguishes a round burner merely touching a wall from a semicircular source with its straight edge against the wall. The observed interaction depends on how the source and boundary are arranged.

The surrounding temperature distribution matters as well. A weak plume rising into progressively warmer ambient air can lose the temperature difference that provides its buoyancy. It may then cease to rise before reaching the ceiling. A calculation based on uniform cool surroundings cannot represent that behaviour merely by increasing the ceiling height.

These distinctions explain why correlations with similar numerical results are not automatically interchangeable. Their inputs represent particular physical arrangements, and their empirical support comes from particular experiments. Model selection therefore accompanies the arithmetic. The resulting uncertainty includes uncertainty in the chosen representation, as well as uncertainty in fire size and geometry.

Connection to smoke extraction and layer growth

A smoke reservoir receives mass and energy from the plume. If incoming mass exceeds the mass leaving, the amount stored in the reservoir increases. Temperature and density then influence how that mass occupies the available volume. Layer descent therefore depends on both entrainment and the other terms in the compartment balance.

Layer growth depends on incoming flow, outgoing flow and the gas state. Based on Van Rossum (2027), §§9.1–9.3.

Smoke and heat exhaust ventilation calculations use plume flow to estimate the supply that extraction must accommodate at the layer interface. Van Rossum describes experiments in which extraction from a collecting hood was adjusted to maintain a stationary layer. Under that balance, the extracted mass flow equalled the flow entering from the plume. The measurement principle concerns a steady layer under the experimental conditions.

A real compartment can also lose heat through its boundaries and exchange gases through openings. These exchanges change layer temperature and density, even when the prescribed fire remains constant. A dynamic zone calculation updates the accumulated quantities over time. A stationary calculation instead investigates whether a balance can exist for a selected fire and set of conditions.

Extraction capacity alone does not guarantee that only smoky gases are removed. An individual outlet with excessive extraction can draw clean air through a thin smoke layer, a process called plugholing. Part of its nominal capacity then removes air from below the layer. This is a separate limitation from an error in the plume entrainment calculation.

Measurement and interpretation

A plume cross section contains a distribution of temperatures and upward velocities. The centreline generally differs from positions farther from the axis. Heskestad distinguishes a temperature radius from a velocity radius because their distributions need not have identical widths. A single measurement at the centre therefore cannot establish the complete mass flow without additional information about the surrounding profile.

Experimental determination must also distinguish mean quantities from fluctuations. The chapter describes substantial fluctuations of both centreline temperature and velocity in turbulent plumes. Measurements averaged over time represent the statistical behaviour used by the correlations. An instantaneous high velocity or a passing flame tip represents a different quantity and cannot simply replace that mean input.

The virtual origin introduces a further measurement challenge. It can be inferred by transforming centreline temperature measurements above the flames and extrapolating their trend. Small temperature errors can produce large changes in the inferred intercept. Heskestad identifies sensor positioning away from the axis, radiation errors and inadequate averaging as possible causes of such inaccuracies.

Consequently, agreement on a measured temperature does not necessarily establish agreement on the inferred source position or mass flow. The chapter compares correlations obtained from different measurements and theoretical treatments. Their broadly similar behaviour supports practical use, but their differences remain relevant near the source. At greater height, a small difference in virtual origin becomes less influential relative to the total rise distance.

The definition of the reported flow also needs attention. A plume correlation usually represents the total mixture crossing the selected plane, dominated by entrained ambient air. A soot production rate instead concerns a particular product released by the fuel. Keeping these quantities distinct prevents a measurement of smoke particles from being mistaken for the entire gas flow that fills a reservoir.

Limits of the prediction

The flame region and the nonreacting plume require different correlations. Heskestad reports approximately linear growth of entrained mass with height inside sufficiently large turbulent flames. Smaller burners can follow a different relationship, and momentum driven jet flames introduce further differences. Continuing the above flame equation down towards the fuel does not resolve these changes.

Laboratory conditions also affect measured flow. Heskestad reports experiments conducted in a deliberately quiet atmosphere, where small disturbances increased measured plume mass flows by 20–50%. This observation identifies sensitivity to ambient movement. It does not establish a universal correction factor for every building or every plume correlation.

A smoke interface marks another limit. The free plume relations assume surroundings that have not already become contaminated by fire products. Once the plume enters a hot layer, its surrounding temperature and composition change. Treating the ceiling as the calculation height after the layer has descended would therefore misrepresent the free rise distance.

Finally, a spatially uniform layer is itself a modelling approximation. Local temperatures, species concentrations and interface motion can differ from the zone average. Obstructions, adjoining rooms and complex geometry can introduce additional mixing stages. The entrainment estimate must remain consistent with the compartment representation used to interpret its consequences.

See also

• Diffusion flame
• Smoke filling
• Balcony spill plume

References

• Heskestad, G. (2002). Fire plumes, flame height, and air entrainment. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-1–2-17). National Fire Protection Association.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.