Solid flame model - Revision history

Jan: Add 4 watermarked SVG illustrations from the Expert visual plan

2026-11-02T16:18:34Z

Add 4 watermarked SVG illustrations from the Expert visual plan

← Older revision		Revision as of 17:18, 2 November 2026
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	== Principle and assumptions ==		== 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.<ref name="NUREG-5-5" /> 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.<ref name="NUREG-5-5" /> In large hydrocarbon fires, the radiation comes mainly from water vapour, carbon dioxide and soot.<ref name="Beyler-270" /> Liquid fuel fires usually have taller and sootier flames than fires of ordinary combustibles, which makes them better radiators.<ref name="NUREG-5-2">Iqbal and Salley (2004), p. 5-2.</ref>		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.<ref name="NUREG-5-5" /> 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.<ref name="NUREG-5-5" /> In large hydrocarbon fires, the radiation comes mainly from water vapour, carbon dioxide and soot.<ref name="Beyler-270" /> Liquid fuel fires usually have taller and sootier flames than fires of ordinary combustibles, which makes them better radiators.<ref name="NUREG-5-2">Iqbal and Salley (2004), p. 5-2.</ref>
			[[File:fse-solid-flame-model-v1.svg\|thumb\|upright=1.5\|alt=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.<ref name="Beyler-269">Beyler (2002a), p. 3-269.</ref> 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.<ref name="Beyler-272">Beyler (2002a), p. 3-272.</ref>		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.<ref name="Beyler-269">Beyler (2002a), p. 3-269.</ref> 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.<ref name="Beyler-272">Beyler (2002a), p. 3-272.</ref>
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	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.<ref name="Beyler-280">Beyler (2002a), p. 3-280.</ref> 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.<ref name="Beyler-280" /><ref name="NUREG-5-10">Iqbal and Salley (2004), p. 5-10.</ref> 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.<ref name="Beyler-277" />		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.<ref name="Beyler-280">Beyler (2002a), p. 3-280.</ref> 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.<ref name="Beyler-280" /><ref name="NUREG-5-10">Iqbal and Salley (2004), p. 5-10.</ref> 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.<ref name="Beyler-277" />
			[[File:fse-solid-flame-model-v4.svg\|thumb\|upright=1.5\|alt=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 ==		== Emissive power ==
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	Mudan proposed a different correlation. It combines a black-body emissive power of 140 kW/m<sup>2</sup> for the luminous flame with 20 kW/m<sup>2</sup> for the smoke, weighted by an extinction coefficient of 0.12 m<sup>−1</sup>.<ref name="Beyler-281">Beyler (2002a), p. 3-281.</ref> Beyler plots both correlations against the pool diameter.<ref name="Beyler-278" /> Evaluating them gives about 57 and 126 kW/m<sup>2</sup> for a pool of 1 m, about 48 and 56 kW/m<sup>2</sup> for 10 m, and about 23 and 20 kW/m<sup>2</sup> 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.<ref name="Beyler-279" /><ref name="Beyler-281" />		Mudan proposed a different correlation. It combines a black-body emissive power of 140 kW/m<sup>2</sup> for the luminous flame with 20 kW/m<sup>2</sup> for the smoke, weighted by an extinction coefficient of 0.12 m<sup>−1</sup>.<ref name="Beyler-281">Beyler (2002a), p. 3-281.</ref> Beyler plots both correlations against the pool diameter.<ref name="Beyler-278" /> Evaluating them gives about 57 and 126 kW/m<sup>2</sup> for a pool of 1 m, about 48 and 56 kW/m<sup>2</sup> for 10 m, and about 23 and 20 kW/m<sup>2</sup> 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.<ref name="Beyler-279" /><ref name="Beyler-281" />
			[[File:fse-solid-flame-model-v3.svg\|thumb\|upright=1.5\|alt=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 ==		== View factor ==
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	For the same fire, the point source model with a radiative fraction of 0.30 gives 1.45 kW/m<sup>2</sup>, less than half the solid flame result.<ref name="NUREG-5-23-24">Iqbal and Salley (2004), pp. 5-23–5-24.</ref> 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/m<sup>2</sup> and the point source model 8.0 kW/m<sup>2</sup>. Only very close to the pool, at 0.25 m, does the point source model give more, 31.4 against 23.2 kW/m<sup>2</sup>. The result of the point source model thus depends strongly on the distance and on the radiative fraction chosen.<ref name="NUREG-5-3">Iqbal and Salley (2004), p. 5-3.</ref><ref name="NUREG-5-9" />		For the same fire, the point source model with a radiative fraction of 0.30 gives 1.45 kW/m<sup>2</sup>, less than half the solid flame result.<ref name="NUREG-5-23-24">Iqbal and Salley (2004), pp. 5-23–5-24.</ref> 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/m<sup>2</sup> and the point source model 8.0 kW/m<sup>2</sup>. Only very close to the pool, at 0.25 m, does the point source model give more, 31.4 against 23.2 kW/m<sup>2</sup>. The result of the point source model thus depends strongly on the distance and on the radiative fraction chosen.<ref name="NUREG-5-3">Iqbal and Salley (2004), p. 5-3.</ref><ref name="NUREG-5-9" />
			[[File:fse-solid-flame-model-v2.svg\|thumb\|upright=1.5\|alt=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 ==		== Use in practice ==

Mark: FSE wiki: corrections after independent fact-check (written by Claude Opus 5.5)

2026-11-01T23:31:05Z

FSE wiki: corrections after independent fact-check (written by Claude Opus 5.5)

← Older revision		Revision as of 00:31, 2 November 2026
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	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.<ref name="Beyler-277" /> 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.<ref name="NUREG-5-17">Iqbal and Salley (2004), p. 5-17.</ref>		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.<ref name="Beyler-277" /> 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.<ref name="NUREG-5-17">Iqbal and Salley (2004), p. 5-17.</ref>

	Mudan proposed a different correlation. It combines a black-body emissive power of 140 kW/m<sup>2</sup> for the luminous flame with 20 kW/m<sup>2</sup> for the smoke, weighted by an extinction coefficient of 0.12 m<sup>−1</sup>.<ref name="Beyler-281">Beyler (2002a), p. 3-281.</ref> ~~The two~~ correlations ~~agree for large pools but differ strongly for small ones~~. ~~For a pool of 1 m they give~~ about 57 and 126 kW/m<sup>2</sup>, for 10 m about 48 and 56 kW/m<sup>2</sup>, and ~~for 50 m~~ about 23 and 20 kW/m<sup>2</sup>.~~<ref name="NUREG-5-7" /><ref name="Beyler-281" />~~ Mudan's method also includes the transmissivity of the atmosphere, because water vapour and carbon dioxide absorb part of the radiation over long distances.<ref name="Beyler-279" /><ref name="Beyler-281" />		Mudan proposed a different correlation. It combines a black-body emissive power of 140 kW/m<sup>2</sup> for the luminous flame with 20 kW/m<sup>2</sup> for the smoke, weighted by an extinction coefficient of 0.12 m<sup>−1</sup>.<ref name="Beyler-281">Beyler (2002a), p. 3-281.</ref> Beyler plots both correlations against the pool diameter.<ref name="Beyler-278" /> Evaluating them gives about 57 and 126 kW/m<sup>2</sup> for a pool of 1 m, about 48 and 56 kW/m<sup>2</sup> for 10 m, and about 23 and 20 kW/m<sup>2</sup> 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.<ref name="Beyler-279" /><ref name="Beyler-281" />

	== View factor ==		== View factor ==
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	NUREG-1805 works out a transformer oil spill of 9.0 ft<sup>2</sup>, or 0.84 m<sup>2</sup>, 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.<ref name="NUREG-5-21">Iqbal and Salley (2004), p. 5-21.</ref> For transformer oil the spreadsheet uses a burning rate of 0.039 kg/(m<sup>2</sup>·s), an effective heat of combustion of 46,000 kJ/kg and an empirical constant of 0.7 m<sup>−1</sup>.<ref name="NUREG-5-25">Iqbal and Salley (2004), p. 5-25.</ref> 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/m<sup>2</sup>.<ref name="NUREG-5-26">Iqbal and Salley (2004), p. 5-26.</ref>		NUREG-1805 works out a transformer oil spill of 9.0 ft<sup>2</sup>, or 0.84 m<sup>2</sup>, 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.<ref name="NUREG-5-21">Iqbal and Salley (2004), p. 5-21.</ref> For transformer oil the spreadsheet uses a burning rate of 0.039 kg/(m<sup>2</sup>·s), an effective heat of combustion of 46,000 kJ/kg and an empirical constant of 0.7 m<sup>−1</sup>.<ref name="NUREG-5-25">Iqbal and Salley (2004), p. 5-25.</ref> 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/m<sup>2</sup>.<ref name="NUREG-5-26">Iqbal and Salley (2004), p. 5-26.</ref>

	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 F<sub>1→2,V</sub> = 0.051, F<sub>1→2,H</sub> = 0.016 and F<sub>max</sub> = 0.054.<ref name="NUREG-5-9" /><ref name="NUREG-5-26" /> The incident radiation is therefore 56.9 × 0.054 = 3.05 kW/m<sup>2</sup>, the same value as the NUREG-1805 spreadsheet, rounded up to 3.1 kW/m<sup>2</sup>.<ref name="NUREG-5-27">Iqbal and Salley (2004), p. 5-27.</ref> This is far below the critical heat flux of 15 to 25 kW/m<sup>2</sup> measured for representative cable samples.<ref name="NUREG-5-2" /> The simple screening correlation of Shokri and Beyler gives 15.4 × (3.56/1.03)<sup>−1.59</sup> = 2.2 kW/m<sup>2</sup> for the same position. With the safety factor of 2 that Beyler recommends for this correlation, that becomes 4.3 kW/m<sup>2</sup>.<ref name="Beyler-273" />		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 F<sub>1→2,V</sub> = 0.051, F<sub>1→2,H</sub> = 0.016 and F<sub>max</sub> = 0.054.<ref name="NUREG-5-9" /><ref name="NUREG-5-26" /> The incident radiation is therefore 56.9 × 0.054 = 3.05 kW/m<sup>2</sup>, the same value as the NUREG-1805 spreadsheet, rounded up to 3.1 kW/m<sup>2</sup>.<ref name="NUREG-5-27">Iqbal and Salley (2004), p. 5-27.</ref> This is far below the critical heat flux of typically 15 to 25 kW/m<sup>2</sup> measured for representative cable samples. It is also below the screening values of 10 and 5 kW/m<sup>2</sup> that NUREG-1805 gives for qualified and unqualified cables.<ref name="NUREG-5-2" /> The simple screening correlation of Shokri and Beyler gives 15.4 × (3.56/1.03)<sup>−1.59</sup> = 2.2 kW/m<sup>2</sup> for the same position. With the safety factor of 2 that Beyler recommends for this correlation, that becomes 4.3 kW/m<sup>2</sup>.<ref name="Beyler-273" />

	For the same fire, the point source model with a radiative fraction of 0.30 gives 1.45 kW/m<sup>2</sup>, less than half the solid flame result.<ref name="NUREG-5-23-24">Iqbal and Salley (2004), pp. 5-23–5-24.</ref> 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/m<sup>2</sup> and the point source model 8.0 kW/m<sup>2</sup>. Only very close to the pool, at 0.25 m, does the point source model give more, 31.4 against 23.2 kW/m<sup>2</sup>. The result of the point source model thus depends strongly on the distance and on the radiative fraction chosen.<ref name="NUREG-5-3">Iqbal and Salley (2004), p. 5-3.</ref><ref name="NUREG-5-9" />		For the same fire, the point source model with a radiative fraction of 0.30 gives 1.45 kW/m<sup>2</sup>, less than half the solid flame result.<ref name="NUREG-5-23-24">Iqbal and Salley (2004), pp. 5-23–5-24.</ref> 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/m<sup>2</sup> and the point source model 8.0 kW/m<sup>2</sup>. Only very close to the pool, at 0.25 m, does the point source model give more, 31.4 against 23.2 kW/m<sup>2</sup>. The result of the point source model thus depends strongly on the distance and on the radiative fraction chosen.<ref name="NUREG-5-3">Iqbal and Salley (2004), p. 5-3.</ref><ref name="NUREG-5-9" />
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	== Limits and uncertainty ==		== Limits and uncertainty ==
	All these methods are semi-empirical and therefore uncertain.<ref name="Beyler-269" /> 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.<ref name="Beyler-277" /> ~~The~~ emissive power correlation ~~rests on luminous hydrocarbon flames and~~ liquefied natural gas~~, so its use for other fuels or for pools outside~~ the ~~range~~ of 1 to 50 m ~~is less certain~~.<ref name="NUREG-5-17" /> Correlations for flame geometry in wind are partly based on liquefied natural gas fires only, and Beyler advises caution in using them.<ref name="Beyler-272" />		All these methods are semi-empirical and therefore uncertain.<ref name="Beyler-269" /> 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.<ref name="Beyler-277" /> 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.<ref name="NUREG-5-17" /> Correlations for flame geometry in wind are partly based on liquefied natural gas fires only, and Beyler advises caution in using them.<ref name="Beyler-272" />

	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/m<sup>2</sup> and underpredicts the radiation close to the fire.<ref name="Beyler-275" /> 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.<ref name="NUREG-5-17" /> 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.<ref name="Beyler-275" /><ref name="NUREG-5-17" />		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/m<sup>2</sup> and underpredicts the radiation close to the fire.<ref name="Beyler-275" /> 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.<ref name="NUREG-5-17" /> 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.<ref name="Beyler-275" /><ref name="NUREG-5-17" />

Mark: FSE wiki: replace unsupported 'widely used' with source-based wording (written by Claude Opus 5.5)

2026-11-01T22:40:19Z

FSE wiki: replace unsupported 'widely used' with source-based wording (written by Claude Opus 5.5)

← Older revision		Revision as of 23:40, 1 November 2026
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	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.<ref name="Beyler-270">Beyler (2002a), p. 3-270.</ref><ref name="NUREG-5-5">Iqbal and Salley (2004), p. 5-5.</ref> 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.<ref name="Beyler-276">Beyler (2002a), p. 3-276.</ref> The model is mainly used for [[pool fire]]s of flammable liquids, where [[flame radiation]] is the main way in which the fire harms people and damages objects at a distance.<ref name="Beyler-268">Beyler (2002a), p. 3-268.</ref><ref name="NUREG-5-1">Iqbal and Salley (2004), p. 5-1.</ref>		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.<ref name="Beyler-270">Beyler (2002a), p. 3-270.</ref><ref name="NUREG-5-5">Iqbal and Salley (2004), p. 5-5.</ref> 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.<ref name="Beyler-276">Beyler (2002a), p. 3-276.</ref> The model is mainly used for [[pool fire]]s of flammable liquids, where [[flame radiation]] is the main way in which the fire harms people and damages objects at a distance.<ref name="Beyler-268">Beyler (2002a), p. 3-268.</ref><ref name="NUREG-5-1">Iqbal and Salley (2004), p. 5-1.</ref>

	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.<ref name="Beyler-275">Beyler (2002a), p. 3-275.</ref> A ~~widely used~~ version by Shokri and Beyler 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.<ref name="Beyler-277">Beyler (2002a), p. 3-277.</ref> The same principle, received radiation equals emitted radiation times a view factor, also underlies calculations of [[fire spread between buildings]].<ref name="VR-5.4">Van Rossum (2027), §5.4.</ref>		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.<ref name="Beyler-275">Beyler (2002a), p. 3-275.</ref> 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.<ref name="Beyler-277">Beyler (2002a), p. 3-277.</ref> The same principle, received radiation equals emitted radiation times a view factor, also underlies calculations of [[fire spread between buildings]].<ref name="VR-5.4">Van Rossum (2027), §5.4.</ref>

	== Principle and assumptions ==		== Principle and assumptions ==

Mark: New FSE wiki article: Solid flame model (written by Claude Opus 5.5)

2026-11-01T22:40:08Z

New FSE wiki article: Solid flame model (written by Claude Opus 5.5)

New page

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.<ref name="Beyler-270">Beyler (2002a), p. 3-270.</ref><ref name="NUREG-5-5">Iqbal and Salley (2004), p. 5-5.</ref> 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.<ref name="Beyler-276">Beyler (2002a), p. 3-276.</ref> The model is mainly used for [[pool fire]]s of flammable liquids, where [[flame radiation]] is the main way in which the fire harms people and damages objects at a distance.<ref name="Beyler-268">Beyler (2002a), p. 3-268.</ref><ref name="NUREG-5-1">Iqbal and Salley (2004), p. 5-1.</ref>

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.<ref name="Beyler-275">Beyler (2002a), p. 3-275.</ref> A widely used version by Shokri and Beyler 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.<ref name="Beyler-277">Beyler (2002a), p. 3-277.</ref> The same principle, received radiation equals emitted radiation times a view factor, also underlies calculations of [[fire spread between buildings]].<ref name="VR-5.4">Van Rossum (2027), §5.4.</ref>

== 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.<ref name="NUREG-5-5" /> 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.<ref name="NUREG-5-5" /> In large hydrocarbon fires, the radiation comes mainly from water vapour, carbon dioxide and soot.<ref name="Beyler-270" /> Liquid fuel fires usually have taller and sootier flames than fires of ordinary combustibles, which makes them better radiators.<ref name="NUREG-5-2">Iqbal and Salley (2004), p. 5-2.</ref>

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.<ref name="Beyler-269">Beyler (2002a), p. 3-269.</ref> 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.<ref name="Beyler-272">Beyler (2002a), p. 3-272.</ref>

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:<ref name="Beyler-276" /><ref name="NUREG-5-5" />
<math display="block">\dot{q}'' = E\,F_{1\to2}</math>
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.<ref name="NUREG-5-6-7">Iqbal and Salley (2004), pp. 5-6–5-7.</ref>

== 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.<ref name="Beyler-276" /> 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.<ref name="Beyler-273">Beyler (2002a), p. 3-273.</ref> The flame height in metres follows from the heat release rate Q̇ in kilowatts with Heskestad's correlation:<ref name="NUREG-5-9">Iqbal and Salley (2004), p. 5-9.</ref><ref name="Beyler-270" />
<math display="block">H_f = 0.235\,\dot{Q}^{2/5} - 1.02\,D</math>
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.<ref name="NUREG-5-4">Iqbal and Salley (2004), p. 5-4.</ref>

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.<ref name="Beyler-270" /> 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.<ref name="Beyler-270" /><ref name="Beyler-279">Beyler (2002a), p. 3-279.</ref>

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.<ref name="Beyler-280">Beyler (2002a), p. 3-280.</ref> 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.<ref name="Beyler-280" /><ref name="NUREG-5-10">Iqbal and Salley (2004), p. 5-10.</ref> 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.<ref name="Beyler-277" />

== 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:<ref name="Beyler-276" /><ref name="Beyler-277" />
<math display="block">E = 58 \times 10^{-0.00823\,D}</math>
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.<ref name="Beyler-277" /> 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.<ref name="NUREG-5-17">Iqbal and Salley (2004), p. 5-17.</ref>

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.<ref name="Beyler-281">Beyler (2002a), p. 3-281.</ref> The two correlations agree for large pools but differ strongly for small ones. For a pool of 1 m they give about 57 and 126 kW/m2, for 10 m about 48 and 56 kW/m2, and for 50 m about 23 and 20 kW/m2.<ref name="NUREG-5-7" /><ref name="Beyler-281" /> Mudan's method also includes the transmissivity of the atmosphere, because water vapour and carbon dioxide absorb part of the radiation over long distances.<ref name="Beyler-279" /><ref name="Beyler-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.<ref name="NUREG-5-7">Iqbal and Salley (2004), p. 5-7.</ref> 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.<ref name="VR-5.4" /> 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.<ref name="Beyler-276" />

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:<ref name="NUREG-5-9" /><ref name="Beyler-280" />
<math display="block">F_{1\to2,V} = \frac{1}{\pi S}\tan^{-1}\!\left(\frac{h}{\sqrt{S^2-1}}\right) - \frac{h}{\pi S}\tan^{-1}\!\sqrt{\frac{S-1}{S+1}} + \frac{A\,h}{\pi S\sqrt{A^2-1}}\tan^{-1}\!\sqrt{\frac{(A+1)(S-1)}{(A-1)(S+1)}}</math>
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).<ref name="NUREG-5-9" /><ref name="NUREG-5-10" /> Close to the flame the view factor approaches one, because the flame fills the whole field of view.<ref name="Beyler-276" />

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>\dot{q}''</math> || incident radiation on the target || kW/m2
|-
| <math>E</math> || average emissive power of the flame surface || kW/m2
|-
| <math>F_{1\to2}</math> || view factor between target and flame || dimensionless
|-
| <math>D</math> || pool or flame diameter || m
|-
| <math>H_f</math> || flame height || m
|-
| <math>\dot{Q}</math> || heat release rate of the fire || kW
|-
| <math>L</math> || distance from the centre of the flame to the target || m
|-
| <math>S</math>, <math>h</math>, <math>A</math> || 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.<ref name="NUREG-5-21">Iqbal and Salley (2004), p. 5-21.</ref> 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.<ref name="NUREG-5-25">Iqbal and Salley (2004), p. 5-25.</ref> 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.<ref name="NUREG-5-26">Iqbal and Salley (2004), p. 5-26.</ref>

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.<ref name="NUREG-5-9" /><ref name="NUREG-5-26" /> 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.<ref name="NUREG-5-27">Iqbal and Salley (2004), p. 5-27.</ref> This is far below the critical heat flux of 15 to 25 kW/m2 measured for representative cable samples.<ref name="NUREG-5-2" /> 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.<ref name="Beyler-273" />

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.<ref name="NUREG-5-23-24">Iqbal and Salley (2004), pp. 5-23–5-24.</ref> 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.<ref name="NUREG-5-3">Iqbal and Salley (2004), p. 5-3.</ref><ref name="NUREG-5-9" />

== 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.<ref name="Beyler-272" /> 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.<ref name="NUREG-5-1" /> 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.<ref name="NUREG-5-2" />

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.<ref name="Beyler-272" /><ref name="Beyler-273" /> 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.<ref name="Beyler-275" /><ref name="Beyler-278">Beyler (2002a), p. 3-278.</ref>

For fire spread between buildings, the view factor determines how much of the radiation from a burning building reaches a neighbouring one.<ref name="VR-5.4" /> 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.<ref name="CVM2-44">Ministry of Business, Innovation and Employment (2025b), p. 44.</ref> The received radiation may not exceed 30 kW/m2 at the boundary and 16 kW/m2 at 1 m beyond it.<ref name="CVM2-43">Ministry of Business, Innovation and Employment (2025b), p. 43.</ref> The calculation takes into account the distance to the boundary and the size and shape of the unprotected areas.<ref name="CVM2-45">Ministry of Business, Innovation and Employment (2025b), p. 45.</ref>

== Limits and uncertainty ==
All these methods are semi-empirical and therefore uncertain.<ref name="Beyler-269" /> 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.<ref name="Beyler-277" /> The emissive power correlation rests on luminous hydrocarbon flames and liquefied natural gas, so its use for other fuels or for pools outside the range of 1 to 50 m is less certain.<ref name="NUREG-5-17" /> Correlations for flame geometry in wind are partly based on liquefied natural gas fires only, and Beyler advises caution in using them.<ref name="Beyler-272" />

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.<ref name="Beyler-275" /> 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.<ref name="NUREG-5-17" /> 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.<ref name="Beyler-275" /><ref name="NUREG-5-17" />

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.<ref name="NUREG-5-3" /> For pool fires, Beyler gives 0.21 − 0.0034D, about 0.21 for the example pool, whereas the NUREG-1805 spreadsheet uses 0.30.<ref name="Beyler-274">Beyler (2002a), p. 3-274.</ref><ref name="NUREG-5-23-24" />

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.<ref name="Beyler-277" /> 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.<ref name="Beyler-273" />

== See also ==
* [[Flame height]]
* [[Radiative fraction]]
* [[Burning rate]]
* [[Heat release rate]]

== Notes ==
<references />

== 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.

[[Category:Fire Dynamics]]