Contents
A Heskestad plume is an engineering representation of the buoyant flow above a localised fire, developed from plume theory and experiments. Its correlations estimate mean flame height, source position, gas temperatures, velocities and entrained mass flow. They describe an approximately axisymmetric turbulent flow rising through surrounding air. The representation is widely used to connect a prescribed fire with the conditions developing above it.[1]
The fire plume includes the flames and the hot gases above them. Heskestad's equations distinguish these regions because combustion and mixing behave differently within and beyond the visible flame. They also use a calculated source position to account for the finite dimensions of the fire.[2] A calculation therefore requires compatible information about fire power, burning geometry and the height being considered. Its free plume assumptions cease to apply when the rising flow enters an existing smoke layer.[1]
The flow represented by the model
A localised fire releases hot gases into cooler surroundings. The resulting density difference produces a rising flow, while surrounding air mixes into its sides. This entrainment increases the mass carried upwards and dilutes the combustion products. Above the flames, continued mixing also reduces the temperature difference between the plume and the ambient air.[3]
The model treats this motion through mean quantities. Real plume boundaries fold and move, and local temperatures and velocities fluctuate. A calculated centreline temperature is consequently a time averaged value at a particular position. It is not a uniform temperature throughout the plume cross section or a prediction of each passing turbulent structure.[4]
Axisymmetry means that the simplified plume develops around a central vertical axis. This is an appropriate starting representation for certain compact sources in otherwise unobstructed surroundings. It does not directly represent every fire spreading along a wall or every stream spilling beneath a balcony. Those arrangements alter the boundaries through which surrounding air can enter.[1][5]
The atmosphere surrounding the modelled plume is also part of its definition. Heskestad generally assumes air uncontaminated by fire products and uniform in temperature. Temperature stratification is treated separately because it can weaken the buoyancy of a rising plume. An existing smoke layer is therefore more than an arbitrary upper boundary drawn on an otherwise unchanged flow.[1][6]
The mass transported at an elevated plane is dominated by entrained air rather than by fuel vapour alone. This makes plume flow much larger than the direct mass supplied by the source. The model's purpose in smoke calculations is to estimate that mixed flow. A fuel mass loss measurement and a plume mass flow prediction describe different quantities.[7]
Fire power and source coordinates
Total heat release and convective heat release enter different parts of the calculation. Total heat release describes energy actually liberated by combustion. The convective part is carried by the rising gases, while radiation transfers another part away from the flame region. Confusing these inputs changes both the predicted flame dimensions and the transported flow.[8]
A convective fraction of 0.7 is a common assumption in the introductory treatment. Heskestad also uses 0.8 for a methyl alcohol example because that flame produces relatively little luminous radiation. These fractions are selected approximations, rather than universal constants for every fuel or burning condition. The selected value therefore belongs with the other stated inputs to the calculation.[9][5]
The flame height establishes the approximate boundary between the reacting region and the nonreacting plume. For suitable buoyancy dominated fires under normal atmospheric conditions, the representative correlation uses total heat release and fire diameter. A noncircular source can be assigned the diameter of a circle with equal area, within the correlation's source assumptions.[10]
The virtual origin is a calculated point from which the developed plume appears to originate. For a pool fire, Heskestad gives the following representative relationship. Its value can lie below or above the actual fuel surface. All subsequent heights must use the same reference level, so the relevant plume distance is z minus z0.[11]
For appropriate deep storage fires, the chapter gives a further relationship using mean flame height and convective heat release. In the documented storage application, both flame height and virtual origin are referenced to the base. This avoids treating a three dimensional fuel array as though its burning behaviour were confined to a horizontal pool surface.[11]
Mass flow within and above the flames
Above the mean flame height, the mass flow increases strongly with distance from the virtual origin. Van Rossum writes the relationship in an additive form, with a height dependent term and a correction proportional to convective power. The coefficient values below require power in kilowatts, heights in metres and the resulting mass flow in kilograms per second.[5]
Within the flame region, the same introductory source gives a linear relationship between height and mass flow. This branch uses the distance above the source divided by mean flame height. It does not use the virtual rise distance employed in the expression above the flames.[5]
| Symbol | Description | Unit |
|---|---|---|
| ṁp | Plume mass flow at the selected height | kg/s |
| Q̇ | Actual total heat release rate | kW |
| Q̇c | Convective heat release rate | kW |
| D | Source diameter or equivalent area diameter | m |
| L | Mean flame height above the defined source level | m |
| z | Calculation height above that level | m |
| z0 | Virtual origin above that level | m |
The lower branch requires particular care with small sources. Heskestad reports linear growth within flames for turbulent sources with diameters of 0.3 m and greater. Smaller burners did not consistently follow that behaviour, and jet flame measurements showed other relationships. The simplified branch therefore does not establish a general law for all flames regardless of source size or release momentum.[12]
The numerical presentation also varies between the supplied sources. The equations above follow Van Rossum's coefficient of 0.0056 for the lower branch. The SFPE chapter prints 0.0056 at mean flame height and 0.0059 in its following lower region equation. Combining those alternatives without identifying the chosen formulation would conceal a difference in the source material.[5][12]
The power dependence is different in the two regions. Within the stated lower branch, mass flow is directly proportional to convective heat release for fixed relative height. Above the flames, the main term combines a cube root of convective power with a stronger dependence on rise distance. Changes in smoke interface height can consequently alter inflow even when the fire power is unchanged.[5]
A calculation for a pallet fire
Heskestad's storage example describes a wood pallet stack with a footprint of 1.07 m by 1.07 m and a height of 1.2 m. The total heat release is 2,600 kW under normal atmospheric conditions. The equivalent area diameter is approximately 1.21 m. The representative flame correlation then gives about 4.23 m above the base of the stack.[13]
For this illustrative extension, a convective fraction of 0.7 gives 1,820 kW. Applying the storage virtual origin relationship gives about 0.703 m above the same base. The positive result means that the effective plume origin lies above the base. It does not imply that the actual fuel has moved or that combustion begins only at that elevation.[9][11]
Consider a calculation plane 5 m above the stack base. It lies above the predicted mean flame height, so the upper mass flow branch applies. The effective rise distance is approximately 4.30 m. Substitution with unrounded intermediate values gives 13.34 kg/s, reported as about 13.3 kg/s.[13][11][5]
The calculation is an extension of the handbook inputs, rather than a reported experimental measurement at that plane. It assumes an unobstructed plume and sufficiently cool, uncontaminated surrounding air up to the selected height. A smoke layer whose underside lies below 5 m would invalidate that free rise distance. Its interface would instead determine where the plume enters the layer.[1]
Repeating the calculation at 10 m gives approximately 39.1 kg/s for the same idealised source. This increase is caused by additional entrainment during the longer rise. It is not an increase in the fuel burning rate or total heat release. The larger mass flow would also need its temperature and density before conversion to a volume flow.[5][14]
Coupling the plume to a compartment
Smoke filling calculations use the plume as a source of mass and energy entering an upper layer. A zone model then tracks the amount stored and the amount leaving through openings or extraction. It also accounts for heat transferred to surrounding surfaces within its formulation. The plume correlation supplies part of that balance rather than the complete compartment prediction.[14]
The calculation height is generally the underside of the smoke layer, measured relative to the fire. As that interface descends, the free rise distance changes. In a dynamic model, plume inflow therefore interacts with the evolving layer conditions. Assuming that the plume always rises freely to the ceiling would miss this feedback between layer position and incoming mass.[5][14]
A mean layer temperature must also be distinguished from the plume centreline temperature. The plume has nonuniform temperature and velocity profiles, while a basic zone represents uniform properties within each layer. Mixing its centreline temperature directly into a whole layer calculation would confuse a local value with a mass and energy balance. The zone approximation needs its own consistent treatment of incoming energy.[4][14]
Van Rossum contrasts the model with the Thomas plume representation for relatively broad fires with shorter flames. The described selection uses a rough comparison between mean flame length and source diameter. It reflects different plume development rather than an absolute threshold in megawatts. A growing fire may change that relationship as both its burning area and power evolve.[5][14]
This presents a practical limitation in dynamic calculations. The introductory text notes that some models require a plume choice before the calculation or provide a fixed internal choice. A changing fire does not guarantee that the software changes its plume representation correspondingly. The selected correlation and its range therefore remain part of the model assumptions throughout the scenario.[14]
Temperature and velocity profiles
The mass flow relationship describes the whole cross section, while other Heskestad correlations describe conditions along the plume axis. The centreline temperature rise decreases above the flames as entrainment dilutes the hot gases. The centreline upward velocity also decreases with increasing height in this region. These changes occur while the plume spreads and its total mass flow increases.[3][4]
The temperature radius and velocity radius provide different measures of plume width. Each identifies where the corresponding quantity falls to a specified fraction of its centreline value. Heskestad discusses measured differences between these radii and the difficulties associated with their determination. A single drawn plume edge therefore cannot represent every temperature, velocity and species contour simultaneously.[4]
Profiles are often represented by Gaussian shapes, although the chapter does not claim a theoretical requirement for that form. Such an assumed distribution allows local values to be related to the axis and characteristic width. Its use introduces another modelling choice beyond the source power and diameter. A structural element crossing the plume can consequently experience conditions that vary along its length.[4]
Within the reacting region, the temperature does not continue the same above flame power law towards the source. The lower portion of a relatively tall flame can have nearly constant centreline temperature, followed by an intermittent upper region. The velocity maximum lies slightly below the mean flame height. Separate treatment of these regions prevents an extrapolated plume equation from implying unrealistic conditions close to the fuel.[3][4]
Experimental basis and uncertainty
The correlations combine theoretical relationships with experimentally determined coefficients. Heskestad compares several methods for estimating virtual origin from temperature and entrainment measurements. Their predictions show broadly similar behaviour but do not coincide exactly. The chapter recommends a practical expression while recognising limitations in the measurements available to distinguish alternatives.[11]
Determining virtual origin experimentally can be sensitive to small errors. A transformed centreline temperature profile is extrapolated to obtain an intercept. Off-axis sensor placement, radiation effects on temperature measurements and insufficient averaging can alter that intercept. This sensitivity is particularly relevant near the source, where the origin correction is large relative to the rise distance.[11]
Ambient disturbances introduce another uncertainty. Heskestad reports laboratory observations in which small disturbances increased plume mass flow by 20–50% compared with deliberately quiet conditions. The reported range describes those experiments. It is evidence of sensitivity to the environment, rather than an automatic multiplier for every design calculation.[12]
Weak plumes in a temperature stratified atmosphere can stop rising before reaching a ceiling. As the surrounding air becomes warmer with height, the temperature difference driving buoyancy can vanish or reverse. The model's separately discussed stratification treatment addresses that behaviour. Uniform ambient equations alone cannot establish the maximum rise in such a space.[6]
Finally, very low flame height to diameter ratios and substantial geometric interference require special consideration. Heskestad identifies uncertainty near the flames of very broad, short flame sources. Van Rossum describes different mixing behaviour at walls, corners and spill edges. These physical limitations remain even when a numerical implementation evaluates every equation without error.[12][5]
See also
Notes
- ↑ 1.0 1.1 1.2 1.3 1.4 Heskestad (2002), chapter 2-1, p. 2-1.
- ↑ Heskestad (2002), chapter 2-1, pp. 2-6–2-12.
- ↑ 3.0 3.1 3.2 Heskestad (2002), chapter 2-1, pp. 2-1–2-2.
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 Heskestad (2002), chapter 2-1, pp. 2-6–2-8.
- ↑ 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 Van Rossum (2027), §6.7.
- ↑ 6.0 6.1 Heskestad (2002), chapter 2-1, pp. 2-8–2-9.
- ↑ Heskestad (2002), chapter 2-1, p. 2-11.
- ↑ Heskestad (2002), chapter 2-1, p. 2-2.
- ↑ 9.0 9.1 Heskestad (2002), chapter 2-1, p. 2-8.
- ↑ Heskestad (2002), chapter 2-1, p. 2-3.
- ↑ 11.0 11.1 11.2 11.3 11.4 11.5 Heskestad (2002), chapter 2-1, pp. 2-9–2-10.
- ↑ 12.0 12.1 12.2 12.3 Heskestad (2002), chapter 2-1, p. 2-12.
- ↑ 13.0 13.1 Heskestad (2002), chapter 2-1, p. 2-4.
- ↑ 14.0 14.1 14.2 14.3 14.4 14.5 Van Rossum (2027), §9.3.
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.