Virtual origin

The virtual origin of a fire plume is the height of an imaginary point source from which the plume above the flames of a real fire appears to originate. Plume equations are derived for heat released at a single point, while real fires burn over an area, such as a pool of liquid or the top of a cabinet. By measuring heights from the virtual origin instead of from the fuel surface, the point-source equations can describe these real fires.

The virtual origin is usually written as z₀, its height above the base of the fire. It can lie below the fuel surface, which is typical for fires with a modest heat release rate spread over a large area. For fires with a high heat release rate it often lies above the fuel surface. For pool fires, Heskestad's correlation calculates it from the heat release rate and the diameter of the fire. The location matters for predictions close to the fire. High above it, the virtual origin may simply be placed at the fuel surface.

Point sources and real fires

A fire plume is the buoyant flow of hot gases, including any flames, that rises above a burning fuel into the surrounding air. The classic plume theory describes a plume that rises from a point source of heat. Real fires are area sources, such as pool fires, burning cabinets and cable trays, and their plume near the fire does not behave like a point-source plume. Heskestad's solution is to keep the point-source equations but to replace the height above the fire, z, by the height above a virtual source, z − z₀.

The virtual origin is thus the equivalent point source height of a fire with a finite area. Cadorin and co-workers describe it as the position where a virtual point source would produce the same effects as the real fire. In practice, the plume temperature, velocity and mass flow at a given height are calculated as if the fire were a point at this level. The Introduction to Fire Safety Engineering calls it a computational point on the axis of the plume.

The virtual origin is the point from which the plume above the flames appears to originate: below the fuel surface for a wide, low fire (left) and above it for a tall, intense fire (right). Based on Heskestad (2002), pp. 2-6 and 2-11.

Near the fire, this correction is significant, and Heskestad stresses that the location of the virtual origin must then be known for accurate predictions. For a pool fire, it usually lies near the fuel surface. At great heights above the fire, the difference between z and z − z₀ becomes small. In the NFPA 92B plume equations for atria, the virtual origin is set to zero, which Milke justifies by its being small compared with the height. Heskestad does the same in an example with a 20 m high atrium, where deviations of the virtual origin from the fire level are inconsequential. Because the fire is treated as a point source, the shape of the fuel is then of secondary importance.

Calculating the virtual origin

Heskestad derived his correlation from a theoretical model that relies on his flame height correlation. He checked it against virtual origins measured in pool fires with diameters from 0.16 to 2.4 m. Such measurements use the plume temperature law: the temperature rise on the axis, raised to the power −3/5, varies linearly with height above the flames. A straight line through the data crosses the height axis at the virtual origin.

The available data did not show a dependence on the type of fuel within their scatter. Large differences in heat of combustion or convective fraction might still affect the result. In words, the virtual origin rises with the heat release rate to the power 2/5 and falls with the diameter of the fire: $$z_0=0.083{\dot{Q}}^{2/5}-1.02D$$ The mean flame height has the same structure, but with a larger coefficient: $$L=0.235{\dot{Q}}^{2/5}-1.02D$$ For other fire types, such as deep storage fires, Heskestad gives the virtual origin directly from the flame height and the convective heat release rate: $$z_0=L-0.175{\dot{Q}}_{\mathrm{c}}^{2/5}$$

Symbol	Description	Unit
$z_0$	height of the virtual origin above the base of the fire (negative below it)	m
$L$	mean flame height above the base of the fire	m
$\dot{Q}$	total heat release rate	kW
${\dot{Q}}_{\mathrm{c}}$	convective heat release rate	kW
$D$	diameter of the fire, or effective diameter of a non-circular fire	m
$z$	height above the base of the fire	m

The total heat release rate is used for the flame height and the virtual origin. The convective part, which drives the buoyancy, is used for the other plume properties. Radiation typically takes 20 to 40% of the total, so the convective heat release rate is often 0.6 to 0.8 times the total. For a fire that is not circular, D is the diameter of a circle with the same area as the fire. The two flame height and virtual origin equations imply that the virtual origin lies about 0.152 times the total heat release rate to the power 2/5 below the flame tip. With a convective fraction of 0.7, this is almost exactly the 0.175 times the convective rate to the power 2/5 in the storage equation.

Once the virtual origin is known, the plume equations of the Heskestad plume use z − z₀ in place of z. The temperature rise on the plume axis above the flames then falls with (z − z₀) to the power −5/3. The mass flow in the plume, which consists almost entirely of air drawn in by entrainment, grows with (z − z₀) to the power 5/3. The plume radius grows, and the axial velocity falls, with the same distance from the virtual origin. Inside the flames these relations do not apply, and the mass flow is instead taken in proportion to the height divided by the flame height. For normal atmospheric conditions, the temperature rise on the axis above the flames is: $$\mathrm{\Delta }{T}_0=25.0{\dot{Q}}_{\mathrm{c}}^{2/3}(z-z_0{)}^{-5/3}$$ Here ΔT₀ is the temperature rise on the plume axis in kelvin. The constant 25.0 holds for air at 293 K with a density of 1.2 kg/m³.

Sign and size

Negative values of the virtual origin are often found for fires with a low heat release rate and a large diameter. Positive values are common for fires with a high heat release rate. The storage equation shows why: the virtual origin lies a fixed distance, 0.175 times the convective heat release rate to the power 2/5, below the mean flame tip. Short flames over a wide fire therefore place it below the fuel surface, and tall flames above it.

Heskestad illustrates this with a methyl alcohol pool fire of 1.5 m diameter, which releases 884 kW. Its virtual origin lies 0.29 m below the fuel surface. The same pool burning heptane releases about 4,420 kW, and its virtual origin lies 0.85 m above the surface.

Virtual origin divided by fire diameter against Q^2/5/D for four correlations, with the methyl alcohol, heptane and lubricating oil examples. Based on Heskestad (2002), pp. 2-10 to 2-11, and Iqbal and Salley (2004), p. 9-11.

For storage fires, the virtual origin follows from the flame height. In one of Heskestad's examples, a 3 m deep storage burns at 1,500 kW with flames 3.67 m above its base, which places the virtual origin 0.84 m above the base. For a wide range of typical fuel packages, Milke reports virtual origins between 0.5 m and −5 m. Negative values arise when modest heat release rates are spread over a large area, so that the second term of the equation exceeds the first.

The virtual origin of a growing fire need not stay in one place. In tests with rack storage of plastic goods, flames spread upwards and sideways while the fire grew. Yu and Stavrianidis therefore used a virtual origin that moves with the convective heat release rate, from 2.4 m below the base of the fire at the start. Their ceiling jet correlations use the effective ceiling height above this moving point.

Worked example

NUREG-1805 calculates the plume temperature at a steel beam 7.62 m above a pool fire of lubricating oil. The pool covers 3.21 m², and the fire releases 5,000 kW, of which 70% is convective. The ambient temperature is 25 °C and the air density 1.18 kg/m³. The effective diameter is that of a circle with the same area: $$D=\sqrt{\frac{4\times 3.21}{\pi }}=2.02\mathrm{m},z_0=0.083\times 5000^{2/5}-1.02\times 2.02=0.44\mathrm{m}$$ The virtual origin thus lies 0.44 m above the fuel surface, and the beam is 7.18 m above it. With the convective heat release rate of 3,500 kW, the plume temperature at the beam becomes 244 °C.

Without the virtual origin, the same equation gives about 224 °C at the beam. Ignoring the correction would therefore underestimate the temperature rise by about 21 K, or 10%. The effect fades with height: at 15 m above the fire, the difference in temperature rise is about 5%, and at 30 m about 2.5%. This agrees with Heskestad's statement that the virtual origin matters mainly near the fire.

Plume centreline temperature above a 5,000 kW lubricating oil pool fire with the virtual origin (0.44 m) and with the virtual origin set to zero. Based on Iqbal and Salley (2004), pp. 9-10 to 9-11, and Heskestad (2002), p. 2-6.

Use in fire engineering

The virtual origin appears in most methods that use the Heskestad plume. The International Fire Engineering Guidelines (2005) use it in the equation for the centreline temperature of flames and plumes. The same guidelines use it in the mass entrainment equations for smoke filling. NUREG-1805 lists its calculation as the second of four steps in estimating the centreline temperature of a plume.

The zone model CFAST uses the same correlation in dimensionless form, also based on the total heat release rate. For a fire against a wall or in a corner, CFAST mirrors the fire in the wall faces. It doubles or quadruples the convective heat release rate, enlarges the fire diameter and divides the resulting entrainment by two or four. The virtual origin then changes as well, because it depends on the effective diameter.

The concept also appears in ceiling jet and structural calculations. Alpert recommends correcting the ceiling height for the location of the virtual point source when ceiling jet heat transfer correlations are applied. For steel members near a localised fire, the OZone tool uses Hasemi's model, in which a virtual heat source is placed relative to the fire surface. The heat flux to a beam then depends on distances measured from this virtual source.

The same idea can be applied to sources that are not points. In a balcony spill example, Cooper places an imaginary line source 1.7 m below the balcony edge, so that a line plume equation can describe the smoke that spills over the edge. As with a point source, the imaginary source is chosen so that the equation reproduces the known mass flow at a reference height.

Limits and uncertainty

The virtual origin is hard to measure. It is found by extrapolating temperatures measured along the plume axis, and small errors in these temperatures have a large effect on the result. Such errors can come from sensors placed off the axis, from radiation affecting the readings or from too little averaging of the signal. Heskestad's correlation rests on pool fires of 0.16 to 2.4 m diameter.

Different correlations exist, and they do not always agree. Heskestad's chapter compares his correlation with those of Hasemi and Tokunaga and of Cetegen and co-workers, finds them surprisingly similar and recommends his own for its simplicity, theoretical basis and central position. The correlation of Hasemi and Tokunaga, as given by Heskestad, can also be applied to the lubricating oil fire of the worked example. The calculated virtual origin then lies about 0.28 m below the fuel surface, instead of 0.44 m above it.

Cetegen and co-workers based their correlation on entrainment measurements above gas burners of 0.10 to 0.50 m, with and without a floor flush with the burner. Their constant is −0.80 without such a floor and −0.50 with one, so a surrounding floor raises the virtual origin by 0.3 times the diameter. Sign conventions also differ. Heskestad counts the virtual origin as positive above the fire surface, while in the Hasemi model used by OZone the virtual source is drawn below it.

Two sign conventions: in Heskestad's correlations z0 is positive above the fuel surface; in the Hasemi model used by OZone the virtual source distance is drawn below it. Based on Heskestad (2002), p. 2-6, and Cadorin et al. (2003), pp. 443 to 444.

The plume equations that use the virtual origin have their own limits. They are not valid at or below the mean flame height. For a fire with very low flames, a 2.44 m silicone fluid pool with a virtual origin of −1.5 m, Heskestad predicted a centreline temperature rise near the surface of 580 K. The rise of roughly 440 K derived from the measurements was fairly close to this prediction. It is not yet clear, however, whether such predictions are generally valid for fires with very low flames.

According to the International Fire Engineering Guidelines, the centreline temperature equation is accurate up to a temperature rise of 500 K and overestimates it between 500 and 900 K. NUREG-1805 adds that it does not apply to jet fires and underestimates temperatures when the surrounding air is already hot.

Care is also needed with the constants in some guidance documents. In the International Fire Engineering Guidelines, the flame height and virtual origin equations reuse the symbols C₁ and C₂ of the entrainment equation above them. The values listed for these symbols, 0.071 and 0.027, differ from the coefficients 0.235 and 0.083 in Heskestad's chapter.

See also

• Thomas plume
• Dimensionless heat release rate
• Alpert correlations
• Line plume

References

• Alpert, R. L. (2002). Ceiling jet flows. 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. 2-18–2-31). National Fire Protection Association; Society of Fire Protection Engineers.
• Cadorin, J.-F., Pintea, D., Dotreppe, J.-C., & Franssen, J.-M. (2003). A tool to design steel elements submitted to compartment fires–OZone V2. Part 2: Methodology and application. Fire Safety Journal, 38, 429–451. https://doi.org/10.1016/S0379-7112(03)00015-8
• 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.
• Heskestad, G. (2002). Fire plumes, flame height, and air entrainment. 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. 2-1–2-17). 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.
• 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.
• National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• 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.