Thomas plume

A Thomas plume is a simplified representation of the mass flow rising from a broad fire. Its commonly used correlation relates the flow to the fire perimeter and the height above the source. It was originally developed for the flaming region of large fires, where the supply of surrounding air can be related to the source boundary.

The method is used to estimate the mixed flow entering a smoke layer. It is particularly associated with fires whose flames are relatively short compared with the source diameter. Van Rossum presents this relationship as a rough basis for choosing between Thomas and Heskestad plume formulations. The absence of an explicit heat release term makes the formula convenient, but does not make it independent of burning conditions. Its coefficient incorporates assumptions about the atmosphere and flame gases. Applications above the flames also require care because the original physical basis concerned the flaming region.

Physical basis and source shape

A fire plume carries combustion products mixed with surrounding air. As the hot gases rise, entrainment adds ambient air through the sides of the flow. At an elevated cross section, this added air generally accounts for most of the transported mass. The plume flow is therefore different from the rate at which fuel is consumed.

The Thomas formulation emphasises the perimeter of the burning area. For a broad source with relatively short flames, that boundary provides a useful geometric measure of access for surrounding air. The simplified expression predicts more entrainment for a larger perimeter at unchanged rise height. It also predicts more flow as the gases travel farther above the source.

The Thomas formulation relates mass flow to source perimeter and rise height. Based on Heskestad (2002), chapter 2-1, p. 2-12, and Van Rossum (2027), §6.7.

The perimeter is the boundary of the burning source, rather than the perimeter of the room. It is also different from the burning area. Two sources with equal areas can have different perimeters, so retaining only their area can discard information used by this correlation. A circular equivalent diameter appropriate to another equation does not automatically preserve the perimeter required here.

Source geometry must nevertheless remain compatible with the physical plume represented. Smoke emerging from beneath a balcony can form an elongated spill flow, while a wall restricts access for surrounding air. Van Rossum describes these as distinct arrangements with different mixing behaviour. The perimeter term alone does not demonstrate that every obstructed or elongated source can be treated as an unrestricted Thomas plume.

The mass flow equation

The commonly used expression multiplies source perimeter by height raised to the power of three halves. In the stated units, its coefficient is 0.188. The result is the mass flow at the selected elevation, commonly the point where the plume enters the underside of a smoke layer.

ṁ_p = 0.188 W_f z^3/2

Heskestad also gives a form retaining gravity and the densities of ambient air and flame gases. The simplified coefficient incorporates normal atmospheric conditions and an assumed flame temperature. This fuller expression explains why the numerical coefficient cannot be transferred unchanged between arbitrary gas states and unit systems.

ṁ_p = 0.096 (gρ_∞ρ_fl)^1/2 W_f z^3/2

Symbol	Description	Unit
ṁp	Plume mass flow at height z	kg/s
Wf	Perimeter of the burning source	m
z	Rise height above the source reference level	m
g	Acceleration of gravity	m/s²
ρ∞	Ambient air density	kg/m³
ρfl	Gas density in the flames	kg/m³

The output is a mass flow rather than a volume flow. Conversion to volume per second requires the density of the gases at the relevant location. In a compartment calculation, the smoke layer temperature is influenced by incoming energy, mixing, extraction and heat losses. It cannot be determined from the Thomas mass flow equation alone.

The height input refers to the rise before the plume enters the layer. It therefore changes when the smoke interface descends towards the source. Using the ceiling height throughout a filling calculation would represent a different rise distance. A dynamic model must keep its plume input consistent with the evolving layer position.

Selecting the representation

Van Rossum compares the Thomas formulation with the Heskestad plume method through the ratio of mean flame length to source diameter. In the described approach, a flame shorter than the diameter favours Thomas. A flame at least as long as the diameter favours Heskestad. This is presented as a rough division, rather than a sharply defined physical transition.

Mean flame length can be estimated from total heat release and source diameter. The introductory source uses the following representative relationship, with power in kilowatts and lengths in metres. This provides information for selecting the plume model even though the final Thomas mass flow expression contains no explicit power term.

L = 0.235 Q̇^2/5 − 1.02 D

Symbol	Description	Unit
L	Mean flame length	m
Q̇	Total heat release rate	kW
D	Source diameter used in the flame correlation	m

Heskestad's review notes that the Thomas formula was also tested against flows above the flames. The cited work reported satisfactory comparisons over an extended height range, but the chapter stresses the limited theoretical justification there. A reported comparison range is therefore different from proof that the formula is valid for every source within similar dimensions.

For calculations in a building, the selected method must also match the assumed fire development. A stationary assessment selects a fire and geometry at a particular time. A dynamic calculation may involve changing area and power, while retaining one chosen plume formulation. Van Rossum identifies this limitation in models that cannot switch their plume treatment during the scenario.

An illustrative calculation

Heskestad reports comparisons involving heat release rates between 1,000 and 8,000 kW and burning intensities between 250 and 1,000 kW/m². Consider an illustrative circular source with 4,000 kW and 250 kW/m², selected within those reported ranges. Its burning area is 16 m². This is a constructed calculation example, rather than an individual experiment reported by the source.

The circular area gives a diameter of approximately 4.51 m and a perimeter of 14.18 m. The representative flame equation gives a mean length of approximately 1.88 m. The ratio of flame length to diameter is therefore about 0.42. The example lies on the broad source side of Van Rossum's rough model selection approach.

At a selected rise height of 5 m, the Thomas expression gives 0.188 multiplied by 14.18 and by 5 raised to the three halves power. Calculation without intermediate rounding gives 29.80 kg/s, reported as about 29.8 kg/s. The selected plane lies above the predicted flames, so the earlier qualification about extension beyond the flaming region applies.

For the same idealised source at 10 m, the result becomes approximately 84.3 kg/s. Doubling height increases the result by about 2.83 because height enters to the three halves power. This comparison illustrates the mathematical sensitivity to rise distance. It does not establish that either plane remains below the smoke layer in a particular building.

Calculated flow grows with rise height for a fixed source perimeter. Based on Van Rossum (2027), §6.7, and Heskestad (2002), chapter 2-1, p. 2-12.

Geometry produces a separate change. A square burning area of 16 m² has a perimeter of 16 m, rather than the circle's 14.18 m. Inserting that perimeter at 5 m gives about 33.6 kg/s. This arithmetic demonstrates the role of perimeter, without constituting experimental validation for arbitrary source shapes.

Equal burning areas can have different perimeters in the Thomas calculation. Based on Van Rossum (2027), §6.7.

Use in smoke management

A smoke reservoir accumulates the mass and energy delivered by the plume. If incoming mass exceeds outgoing mass, the stored quantity grows. The layer temperature and density determine how that mass occupies the available volume. Consequently, a plume calculation supplies one term in a wider compartment balance.

Smoke and heat exhaust ventilation can remove gases from the upper layer. In a stationary balance, extraction must accommodate the plume supply while respecting the assumed layer conditions. Van Rossum describes collecting hood experiments where a constant layer was maintained by adjusting extraction. Their operating principle illustrates the relationship between plume supply and exhaust mass flow.

A dynamic calculation also accounts for the changing interface and heat transfer to the enclosure. These effects can alter the required volume flow even when a mass flow estimate remains unchanged. Openings may supply fresh air or discharge hot gases, depending on their location and pressure conditions. The Thomas correlation does not replace those opening and energy calculations.

Individual exhaust outlets can also draw clean air through a shallow smoke layer, known as plugholing. In that condition, part of the nominal extraction removes air from below the reservoir. A correct estimate of plume inflow therefore does not guarantee effective removal of the smoke layer. Outlet behaviour constitutes an additional part of the smoke management assessment.

Software implementation can introduce a further geometry assumption. The published OZone V2 description includes Thomas among four selectable entrainment models and assumes a circular fire when deriving diameter from area. A model using that assumption represents the perimeter of the equivalent circle. It does not independently retain the longer boundary of every possible noncircular burning footprint.

The OZone paper also distinguishes the entrained mass from the energy transported with it. Its zone formulation accounts for the temperature of air transferred from the lower layer. This reinforces the distinction between selecting a plume mass flow model and completing the compartment energy calculation. The software context does not supply a separate validation dataset for every Thomas plume application.

Limits of the available evidence

The simplified model does not predict the detailed temperature or velocity distribution across the plume. Its primary output is a mean mass flow for the selected geometry and height. A numerical result therefore provides no direct map of local conditions experienced by a detector, structural member or occupant. Those quantities require additional physical relationships and compatible assumptions.

Heskestad compares Thomas and Heskestad mass flow predictions over a range of fires and heights. In those comparisons, Thomas results ranged from 0.64 to 1.38 times the alternative predictions. This spread demonstrates that one formulation does not consistently reproduce the other. It is not a statistical confidence interval or a universal design safety factor.

The same review reports that small ambient disturbances increased measured plume flows by 20–50% in experiments conducted under otherwise quiet conditions. This sensitivity highlights the influence of surrounding movement. It does not justify automatically multiplying every Thomas result by a chosen percentage. The experimental setting and the actual building environment remain relevant to interpreting the calculation.

Walls, corners, spill edges and pre-existing hot layers can all change the mixing process. A source surrounded by unrestricted air differs from a flow whose sides are blocked or whose surroundings are already hot. The introductory text treats those geometries separately. Applying the short formula therefore requires retaining its physical setting alongside the selected perimeter and height.

References

• Cadorin, J.-F., & Franssen, J.-M. (2003). A tool to design steel elements submitted to compartment fires: OZone V2. Part 1: Pre- and post-flashover compartment fire model. Fire Safety Journal, 38, 395–427.
• 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.