Line plume

A line plume is a buoyant plume of hot gas that rises from a long, narrow source, such as a line of burning material or smoke spilling from a wide opening. Instead of the round column above a compact fire, it forms a flat, wedge-like sheet that draws in air along its two long sides. The Introduction to Fire Safety Engineering describes a line plume as smoke that leaves an adjacent room under a ceiling edge, or under a balcony or gallery, and rises into a larger space.

Because the perimeter of its cross-section is large compared with its area, a line plume mixes in more fresh air than an ordinary round plume. Its mass flow per unit length grows in proportion to the height above the source and to the cube root of the heat release rate per unit length. Near the source, a line plume therefore carries much more smoke than a round plume of the same fire. This matters in smoke control for atria and malls, where the plume mass flow at the smoke layer sets the exhaust rate that the system must provide. The two-dimensional, wedge-like shape lasts only up to a height of about the length of the source.

Forms of line plumes

Any fire plume forms because hot combustion gases are less dense than the surrounding air and therefore rise. The standard plume equations describe an axisymmetric plume above a fire away from walls, which draws in air around its whole perimeter. Cooper adds two plume models for line-like fires, which are described by an effective length. For such fires, the heat release rate and the plume mass flow are expressed per metre of length.

The first form is the free line plume. It rises above a line fire, or above a line-like source of hot smoke, that is far enough from walls to draw in air freely on both sides. Cooper applies it, for example, to smoke that spills over the free edge of a balcony. The second form is the near-wall line plume, which can draw in air on one side only. It occurs above a line fire against a wall, or when smoke from a fire in an adjacent room enters a space through a wide opening and rises along the wall above it.

The book describes the second situation from the point of view of the space that receives the smoke. Smoke flows out of an adjacent room under a ceiling edge or other construction and then rises into a larger space. Its cross-section is more oval than the round section of an ordinary plume. A plume that emerges from under a balcony or gallery has the same oval shape, and the book also calls this a line plume. The SFPE Handbook treats this case separately as the balcony spill plume, with its own entrainment equation.

The flames of a line fire also follow their own rules. For a line fire against a wall, NUREG-1805 gives a flame height that grows with the heat release rate per unit length to the power 2/3. With a heat release rate of 100 kW per metre, this correlation gives a flame height of about 0.37 m. For a fire that spreads over the full height of a wall surface, the same source gives a coefficient twice as large.

Entrainment and smoke production

As hot gases rise in a plume, cold air from the surroundings is mixed into the flow. This entrainment increases the mass flow of the plume with height, so the smoke delivered to a smoke layer consists mostly of entrained air. For an axisymmetric plume, the mass flow grows with the height to the power 5/3. This dependence follows from the work of Heskestad, whose equation for the Heskestad plume is used in NFPA 92B for malls, atria and large areas.

For a free line plume, Cooper gives a simpler relation: the mass flow per metre grows linearly with height. It also grows with the cube root of the convective heat release rate per metre. For a near-wall line plume, the reflection principle applies. The heat release rate per metre is doubled to represent the mirror image behind the wall, and the resulting mass flow is halved. The axisymmetric plume above the flames is given for comparison. $$\dot{m}{\text{'}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}=0.18(z-z_{\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{e}})\dot{Q}{\text{'}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}^{1/3}\text{(free line plume)}$$ $$\dot{m}{\text{'}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}=\frac{0.18}{2}(z-z_{\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{e}}){\left(2\dot{Q}{\text{'}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}\right)}^{1/3}\text{(near-wall line plume)}$$ $$\dot{m}=0.071{\dot{Q}}_{\mathrm{c}}^{1/3}(z-z_{\mathrm{o}}{)}^{5/3}+0.002{\dot{Q}}_{\mathrm{c}}\text{(axisymmetric plume)}$$

Symbol	Description	Unit
$\dot{m}{\text{'}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}$	mass flow in the line plume per metre of length	kg/(m·s)
$\dot{m}$	mass flow in the axisymmetric plume	kg/s
$\dot{Q}{\text{'}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}$	convective heat release rate per metre of source length	kW/m
${\dot{Q}}_{\mathrm{c}}$	convective heat release rate	kW
$z$	height above the floor or reference level	m
$z_{\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{e}}$	height of the base of the line source	m
$z_{\mathrm{o}}$	height of the virtual origin of the axisymmetric plume	m

The difference in the height terms has a large effect. If the height above the source doubles, the line plume carries twice as much mass, but the first term of the axisymmetric equation grows by a factor of about 3.2. Close to the source, the line plume therefore produces far more smoke. Further up, the round plume catches up, and the two become comparable at greater heights. For the balcony spill plume, the handbook estimates that this happens when the height above the balcony is about 12.5 times the width of the plume.

The near-wall equation gives about 63% of the mass flow of a free line plume with the same heat release rate per metre. Doubling the heat release rate raises the flow by the cube root of 2, and halving the result then leaves this fraction. Cooper notes that this estimate assumes small heat losses from the plume to the wall. The same mirror-image idea is used for a round plume against a wall, which behaves like half of a plume with twice the heat release rate.

Worked example

Cooper applies the free line plume equation to smoke that spills from under a balcony. A fire with a total heat release rate of 5 MW burns on 4 m² of floor, 3 m below a balcony with a width of 3 m. The convective part is taken as 65% of the total, or 3.25 MW. The round plume above the fire reaches the underside of the balcony with a mass flow of about 12.0 kg/s. Cooper assumes that the smoke then flows to the edge of the balcony without significant extra entrainment or heat loss and rises from it as a line plume. Its length is estimated at 1.5 times the width of the balcony, which is 4.5 m.

The convective heat release rate per metre of this equivalent line source is 3,250 kW divided by 4.5 m, or 722 kW/m. At the balcony edge, the line plume must carry the same mass as the round plume, 12.0 kg/s spread over 4.5 m, or 2.7 kg/(m·s). Solving the free line plume equation for the height difference places an imaginary line source 1.7 m below the balcony edge: $$z_{\mathrm{b}\mathrm{a}\mathrm{l}}-z_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}=\frac{2.7}{0.18\times 722^{1/3}}=1.7\mathrm{m}$$ The design level for the smoke layer lies 3 m above the balcony, so 4.7 m above this imaginary source. The line plume there carries 0.18 × 4.7 × 722^1/3 = 7.6 kg/(m·s), or 34.2 kg/s over the full length. For comparison, the axisymmetric plume equation gives about 27.3 kg/s at the same height of 6 m above the fire. This assumes that the virtual origin lies at the base of the fire. The spill over the balcony edge therefore increases the smoke flow at the design level by about a quarter. A smoke exhaust system that holds the smoke layer at this level must remove at least this mass flow.

Cooper's energy balance gives the average temperature of the plume gases as they enter the smoke layer. It equals the convective heat release rate divided by the plume mass flow and the specific heat of air. With a specific heat of 1.0 kJ/(kg·K), the line plume reaches the design level about 95 K above ambient and the round plume about 119 K. The extra entrainment of the line plume thus gives more smoke at a lower temperature, as the Australian handbook also notes.

Use in smoke control design

In atrium smoke control, the smoke layer stops descending when the exhaust rate equals the rate at which the plume supplies smoke to the layer. The required exhaust capacity therefore depends directly on the type of plume. A line plume that forms under a balcony or gallery can increase the volume of smoke for a given fire. The same fire placed in the centre of the atrium gives a hotter and more concentrated plume with less volume.

The handbook to the Australian Fire Safety Verification Method illustrates this with an example of a large enclosure with overhanging projections and escape routes above floor level. Because the worst fire position could not be found by reasoning alone, design fires had to be evaluated at several positions. One of these positions lies under horizontal projections where line plumes may form. There the larger smoke volume may affect more escape routes in less time, although temperatures and concentrations of toxic species are lower.

In vent design, the flow out of the ceiling vents must match the plume mass flow at the chosen height of the smoke layer. Cooper's design procedure lists the free and near-wall line plume equations among the plume models that can supply this mass flow. For a near-wall line plume, the plume also loses heat to the wall it touches.

Zone models reproduce the same effect for smoke that flows between rooms. In the zone model CFAST, hot gas that leaves the upper layer of one room and enters the lower layer of the next rises into the upper layer as a spill plume. Its entrainment follows a correlation of Poreh and co-workers, in which the mass flow grows with the width of the opening to the power 2/3. The book points out that a fire in a room next to an atrium, such as a shop in a covered mall, is often the governing scenario. The smoke then passes through more mixing steps and forms a larger volume. A calculation with only one room is then not realistic if the adjacent room is not separated by fire- and smoke-resisting construction.

Limits and uncertainty

The line plume equations hold only near the source. Cooper states that a line plume cannot keep its two-dimensional, wedge-like shape much beyond a height above the source equal to its length. At greater heights, the mass flow is better estimated with an axisymmetric plume equation and an effective fire diameter. The near-wall equation is limited in the same way and also requires that heat losses to the wall stay small.

Line plume sub-models in fire models have not always been tested against experiments. In the 2007 validation study NUREG-1824, the line plume model of the zone model MAGIC was not evaluated, while its axisymmetric plume model was. The axisymmetric model predicted plume temperatures within the experimental uncertainty of 14% in one test series.

The constants in these correlations come from experiments, and published values differ. For the spill plume in CFAST, the suggested values of the entrainment constant range from 0.44 to 0.66. The constant used in CFAST, 0.44, lies at the low end of this range. The balcony spill plume equation in the SFPE Handbook rests on small-scale experiments with a single opening and a walkway. Its use is questionable when the smoke reaches the atrium through a long corridor and loses heat on the way.

The length of an equivalent line source is often uncertain as well. In Cooper's balcony example, the spill length was expected to lie between one and two times the width of the balcony, and 1.5 times was chosen. Because the mass flow scales with the length to the power 2/3 for a given total heat release rate, such choices change the result noticeably. The Australian example therefore evaluates several fire positions instead of one assumed worst case.

See also

• Thomas plume
• Virtual origin
• Flame height
• Turbulence

References

• Australian Building Codes Board. (2020b). Handbook: Fire Safety Verification Method (Print version 1.1).
• Cooper, L. Y. (2002b). Smoke and heat venting. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-219–3-242). National Fire Protection Association; Society of Fire Protection Engineers.
• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• Joglar, F., Gautier, B., Gay, L., & Texeraud, J. (2007). Verification and validation of selected fire models for nuclear power plant applications: Vol. 6. MAGIC (NUREG-1824; EPRI 1011999). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Milke, J. A. (2002b). Smoke management in covered malls and atria. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-292–4-310). National Fire Protection Association; Society of Fire Protection Engineers.
• Peacock, R. D., McGrattan, K. B., Forney, G. P., & Reneke, P. A. (2026). CFAST – Consolidated Fire and Smoke Transport: Vol. 1. Technical reference guide (Version 7, NIST Technical Note 1889v1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.TN.1889v1
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.