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Solid flame model

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Contents
  1. 1Principle and assumptions
  2. 2Flame geometry
  3. 3Emissive power
  4. 4View factor
  5. 5Worked example
  6. 6Use in practice
  7. 7Limits and uncertainty
  8. 8See also
  9. 9Notes
  10. 10References

The solid flame model is a method for calculating the thermal radiation from a fire to a target outside the flame. The flame is represented by a solid, grey-emitting body with a simple, regular shape, usually a vertical or tilted cylinder.[1][2] The radiation that reaches the target equals the average emissive power of the flame surface multiplied by the view factor between the flame and the target.[3] The model is mainly used for pool fires of flammable liquids, where flame radiation is the main way in which the fire harms people and damages objects at a distance.[4][5]

The model is more detailed than the point source model, which releases all radiation from a single point, and it performs better close to the fire.[6] A version by Shokri and Beyler, also used in the NUREG-1805 spreadsheets, takes the emissive power from a correlation with the pool diameter. Comparisons with large-scale experiments show that the cylinder describes the view factor well, and that most of the uncertainty lies in the emissive power.[7] The same principle, received radiation equals emitted radiation times a view factor, also underlies calculations of fire spread between buildings.[8]

Principle and assumptions

The solid flame model makes three assumptions. The fire can be represented by a solid body of a simple geometrical shape, thermal radiation is emitted from the surface of that body, and non-visible gases do not emit much radiation.[2] The model must account for the volume of the flame, because part of the fire may be hidden from the target by the rest of it.[2] In large hydrocarbon fires, the radiation comes mainly from water vapour, carbon dioxide and soot.[1] Liquid fuel fires usually have taller and sootier flames than fires of ordinary combustibles, which makes them better radiators.[9]

Left: a pool fire drawn as a cylinder of height Hf and diameter D facing a target; right: the same fire as a single point at mid-height.
Two ways to represent a pool fire for radiation: the solid flame model (left) treats the flame as a radiating cylinder, the point source model (right) as one point at mid-height. Based on Beyler (2002a), pp. 3-270 and 3-274–3-276.

A calculation follows three steps. The first defines the geometry of the fire, meaning its burning rate and the time-averaged size of the visible flame. The second defines its radiative properties, usually as an average emissive power. The third calculates the radiation at the location of the target.[10] Beyler stresses that one method must be used for all three steps. The available methods contain empirical elements, and mixing them can lead to unpredictable results.[11]

The incident radiation q̇″ on the target follows from the emissive power E of the flame surface and the view factor F1→2 between the target and the flame:[3][2] q˙″=EF1→2 In principle the emissive power of a surface is its emissivity times the Stefan–Boltzmann constant times the fourth power of its absolute temperature. In a turbulent flame, however, the temperature varies strongly, so the emissive power is taken from measurements instead.[12]

Flame geometry

The flame is idealised as a cylinder with a diameter equal to the pool diameter D and a height equal to the flame height Hf.[3] For a pool that is not circular but has a length-to-width ratio near one, an equivalent diameter D = √(4A/π) is used, with A the pool area.[13] The flame height in metres follows from the heat release rate Q̇ in kilowatts with Heskestad's correlation:[14][1] Hf=0.235Q˙2/5−1.02D The heat release rate of a pool fire can be measured or estimated from the burning rate per unit area, the effective heat of combustion and the pool area. In NUREG-1805 this estimate includes a factor that lowers the burning rate of small pools.[15]

Flame height is not a sharp quantity. Pool fire flames fluctuate, so the tip of the flame differs from the length of the continuous luminous region, and researchers have defined flame height by different criteria.[1] Mudan's method uses a different flame height correlation, developed by Thomas for the mean visible height of turbulent diffusion flames from wood crib fires.[1][16]

Wind tilts the flame. The American Gas Association correlation gives the tilt angle from a dimensionless wind speed. The flame stays vertical up to a value of 1, beyond which the cosine of the tilt angle equals one divided by the square root of that wind speed.[17] For a target at ground level or at the flame height, one cylinder represents the flame. A target above the ground needs two cylinders, one below and one above the height of the target.[17][18] The view factor is symmetric about the half-height of the flame, so a target at one quarter of the flame height sees the same view factor as one at three quarters.[7]

Left: a flame cylinder tilted by wind at angle theta; right: a vertical flame split into two cylinders at the height of a raised target.
Flame geometry in the solid flame model: a tilted cylinder in wind (left) and two cylinders for a target above ground level (right). Based on Beyler (2002a), p. 3-280, and Iqbal and Salley (2004), p. 5-10.

Emissive power

Shokri and Beyler fitted measured radiation from pool fires to external targets. They treated the flame as a cylindrical, black, homogeneous radiator with an average emissive power E, in kilowatts per square metre, that depends only on the pool diameter D in metres:[3][7] E=58×10−0.00823D This value is an average over the whole flame and is much lower than the emissive power that can occur locally. It falls with increasing pool diameter, because black smoke outside the flame increasingly hides the luminous flame.[7] The data behind the correlation came from fires of kerosene, fuel oil, gasoline, JP-4, JP-5 and liquefied natural gas, with pool diameters from 1 to 50 m.[19]

Mudan proposed a different correlation. It combines a black-body emissive power of 140 kW/m2 for the luminous flame with 20 kW/m2 for the smoke, weighted by an extinction coefficient of 0.12 m−1.[20] Beyler plots both correlations against the pool diameter.[21] Evaluating them gives about 57 and 126 kW/m2 for a pool of 1 m, about 48 and 56 kW/m2 for 10 m, and about 23 and 20 kW/m2 for 50 m. The two correlations thus agree for large pools but differ strongly for small ones. Mudan's method also includes the transmissivity of the atmosphere, because water vapour and carbon dioxide absorb part of the radiation over long distances.[16][20]

Line chart: emissive power falls with pool diameter, from about 126 (Mudan) and 57 kW/m² (Shokri and Beyler) at 1 m to about 20 kW/m² at 50 m.
Average emissive power of pool fire flames against pool diameter according to Shokri and Beyler and to Mudan. Based on Beyler (2002a), pp. 3-277, 3-278 and 3-281.

View factor

The view factor, also called the configuration factor, is a purely geometric quantity. It gives the fraction of the radiation leaving one surface that strikes another surface directly.[22] It depends on the size of the radiating surface, the distance and the orientation of the receiving surface. Standing close to a radiating surface of only 5 m2 can give a view factor of almost one, whereas the sun, although enormous, gives a small view factor because it is so far away.[8] For a flame, the view factor depends on the position of the target, the flame height and the diameter, and it lies between zero and one.[3]

For a vertical cylinder and a target at ground level, the view factors to a vertical and a horizontal target follow from two dimensionless numbers. These are S = 2L/D, with L the distance from the centre of the flame to the target, and h = 2Hf/D. With A = (h2 + S2 + 1)/(2S), the vertical view factor is:[14][17] F1→2,V=1πStan−1(hS2−1)−hπStan−1S−1S+1+AhπSA2−1tan−1(A+1)(S−1)(A−1)(S+1) A similar expression gives the horizontal view factor F1→2,H. The maximum view factor at a point is the vector sum of the two, Fmax = √(F1→2,H2 + F1→2,V2).[14][18] Close to the flame the view factor approaches one, because the flame fills the whole field of view.[3]

Symbol Description Unit
q˙″ incident radiation on the target kW/m2
E average emissive power of the flame surface kW/m2
F1→2 view factor between target and flame dimensionless
D pool or flame diameter m
Hf flame height m
Q˙ heat release rate of the fire kW
L distance from the centre of the flame to the target m
S, h, A dimensionless geometry parameters dimensionless

Worked example

NUREG-1805 works out a transformer oil spill of 9.0 ft2, or 0.84 m2, in a large compartment. A cabinet stands at ground level 10 ft, or 3.05 m, from the edge of the pool, and there is no wind.[23] For transformer oil the spreadsheet uses a burning rate of 0.039 kg/(m2·s), an effective heat of combustion of 46,000 kJ/kg and an empirical constant of 0.7 m−1.[24] The equivalent diameter is 1.03 m and the heat release rate 772 kW, so the flame height is 2.31 m and the emissive power 56.9 kW/m2.[25]

The distance from the centre of the pool to the cabinet is 3.05 + 1.03/2 = 3.56 m. This gives S = 6.91 and h = 4.47, from which the view factor equations give F1→2,V = 0.051, F1→2,H = 0.016 and Fmax = 0.054.[14][25] The incident radiation is therefore 56.9 × 0.054 = 3.05 kW/m2, the same value as the NUREG-1805 spreadsheet, rounded up to 3.1 kW/m2.[26] This is far below the critical heat flux of typically 15 to 25 kW/m2 measured for representative cable samples. It is also below the screening values of 10 and 5 kW/m2 that NUREG-1805 gives for qualified and unqualified cables.[9] The simple screening correlation of Shokri and Beyler gives 15.4 × (3.56/1.03)−1.59 = 2.2 kW/m2 for the same position. With the safety factor of 2 that Beyler recommends for this correlation, that becomes 4.3 kW/m2.[13]

For the same fire, the point source model with a radiative fraction of 0.30 gives 1.45 kW/m2, less than half the solid flame result.[27] Repeating both calculations at other distances shows how the two models differ. At 1 m from the pool edge the solid flame model gives 10.7 kW/m2 and the point source model 8.0 kW/m2. Only very close to the pool, at 0.25 m, does the point source model give more, 31.4 against 23.2 kW/m2. The result of the point source model thus depends strongly on the distance and on the radiative fraction chosen.[28][14]

Line chart: radiation from a 1 m oil pool fire falls with distance; beyond about 0.5 m the solid flame curve lies above the point source curve.
Incident radiation on a vertical target at ground level against distance from the edge of a 0.84 m² transformer oil pool fire, calculated with the solid flame model and the point source model. Based on Iqbal and Salley (2004), pp. 5-3–5-10 and 5-21–5-27.

Use in practice

The solid flame model is used to estimate safe separation distances between a fire and objects that radiation could damage or ignite.[11] In large open spaces no uniform hot gas layer forms, so radiation from the flame becomes the main mode of heat transfer to a target beside the fire. NUREG-1805 gives the example of a floor-level fire next to an electrical cabinet or a vertical cable tray in a large compartment.[5] The methods in NUREG-1805 are taken from the third edition of the SFPE Handbook and are provided as spreadsheets for fire hazard analyses at nuclear power plants in the United States.[9]

Beyler describes a graded approach. Simple screening methods come first, such as a correlation by Shokri and Beyler for the radiation at ground level, q̇″ = 15.4 (L/D)−1.59, and the point source model. If a simple method shows a separation far larger than needed, a more rigorous analysis may be unnecessary.[11][13] For design, a safety factor of 2 is recommended for the detailed Shokri and Beyler method above 5 kW/m2, and for the point source model below 5 kW/m2. Where a realistic result is required, no safety factor is applied.[6][21]

For fire spread between buildings, the view factor determines how much of the radiation from a burning building reaches a neighbouring one.[8] In New Zealand, C/VM2 (second edition, 2025) treats the unprotected areas in an external wall as radiating surfaces. Their emitted flux is 83, 103 or 144 kW/m2 in an unsprinklered firecell, depending on the fire load energy density, with an emissivity of 1.0.[29] The received radiation may not exceed 30 kW/m2 at the boundary and 16 kW/m2 at 1 m beyond it.[30] The calculation takes into account the distance to the boundary and the size and shape of the unprotected areas.[31]

Limits and uncertainty

All these methods are semi-empirical and therefore uncertain.[10] With the best-fitting emissive power for each experiment, the cylinder model reproduced the measured heat fluxes well. Beyler concludes that the main uncertainty lies in the emissive power rather than in the view factor.[7] NUREG-1805 considers the emissive power correlation suitable for most fuels, because apart from liquefied natural gas the test flames were quite luminous. The underlying data covered pool diameters of 1 to 50 m.[19] Correlations for flame geometry in wind are partly based on liquefied natural gas fires only, and Beyler advises caution in using them.[11]

The sources describe the error of the simpler point source model differently. Beyler reports that it gives systematically non-conservative results above about 5 kW/m2 and underpredicts the radiation close to the fire.[6] NUREG-1805 states that it overestimates the radiation at locations close to the fire, because flame size, shape, tilt and the orientation of the target dominate there.[19] Both agree that a theoretical analysis by Modak found the point source model within 5% of the correct value when L/D exceeds 2.5.[6][19]

The radiative fraction adds further uncertainty. NUREG-1805 quotes values from about 0.15 for low-sooting fuels such as alcohol to 0.60 for high-sooting hydrocarbons.[28] For pool fires, Beyler gives 0.21 − 0.0034D, about 0.21 for the example pool, whereas the NUREG-1805 spreadsheet uses 0.30.[32][27]

A common misconception is that the flame radiates like a black body at the flame temperature. The correlated emissive power is an average over the whole flame, much lower than local peak values, and smoke reduces it further for large pools.[7] Another misconception is that the radiation is greatest at ground level. The radiation from a flame is greatest near the mid-height of the radiating source, and a target facing the centre of the flame receives the most.[13]

See also

Notes

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Beyler (2002a), p. 3-270.
  2. ↑ 2.0 2.1 2.2 2.3 Iqbal and Salley (2004), p. 5-5.
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 Beyler (2002a), p. 3-276.
  4. ↑ Beyler (2002a), p. 3-268.
  5. ↑ 5.0 5.1 Iqbal and Salley (2004), p. 5-1.
  6. ↑ 6.0 6.1 6.2 6.3 Beyler (2002a), p. 3-275.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 7.5 Beyler (2002a), p. 3-277.
  8. ↑ 8.0 8.1 8.2 Van Rossum (2027), §5.4.
  9. ↑ 9.0 9.1 9.2 Iqbal and Salley (2004), p. 5-2.
  10. ↑ 10.0 10.1 Beyler (2002a), p. 3-269.
  11. ↑ 11.0 11.1 11.2 11.3 Beyler (2002a), p. 3-272.
  12. ↑ Iqbal and Salley (2004), pp. 5-6–5-7.
  13. ↑ 13.0 13.1 13.2 13.3 Beyler (2002a), p. 3-273.
  14. ↑ 14.0 14.1 14.2 14.3 14.4 Iqbal and Salley (2004), p. 5-9.
  15. ↑ Iqbal and Salley (2004), p. 5-4.
  16. ↑ 16.0 16.1 Beyler (2002a), p. 3-279.
  17. ↑ 17.0 17.1 17.2 Beyler (2002a), p. 3-280.
  18. ↑ 18.0 18.1 Iqbal and Salley (2004), p. 5-10.
  19. ↑ 19.0 19.1 19.2 19.3 Iqbal and Salley (2004), p. 5-17.
  20. ↑ 20.0 20.1 Beyler (2002a), p. 3-281.
  21. ↑ 21.0 21.1 Beyler (2002a), p. 3-278.
  22. ↑ Iqbal and Salley (2004), p. 5-7.
  23. ↑ Iqbal and Salley (2004), p. 5-21.
  24. ↑ Iqbal and Salley (2004), p. 5-25.
  25. ↑ 25.0 25.1 Iqbal and Salley (2004), p. 5-26.
  26. ↑ Iqbal and Salley (2004), p. 5-27.
  27. ↑ 27.0 27.1 Iqbal and Salley (2004), pp. 5-23–5-24.
  28. ↑ 28.0 28.1 Iqbal and Salley (2004), p. 5-3.
  29. ↑ Ministry of Business, Innovation and Employment (2025b), p. 44.
  30. ↑ Ministry of Business, Innovation and Employment (2025b), p. 43.
  31. ↑ Ministry of Business, Innovation and Employment (2025b), p. 45.
  32. ↑ Beyler (2002a), p. 3-274.

References

  • Beyler, C. L. (2002a). Fire hazard calculations for large, open hydrocarbon fires. 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. 3-268–3-314). 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.
  • Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
  • Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
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