Deflagration index - Revision history

Jan: Remove links to pages that do not exist; retain displayed text (wiki-wide link audit).

2026-11-02T21:32:47Z

Remove links to pages that do not exist; retain displayed text (wiki-wide link audit).

← Older revision		Revision as of 22:32, 2 November 2026
Line 161:		Line 161:
	* [[Explosion]]		* [[Explosion]]
	* [[Inerting]]		* [[Inerting]]
	* [[Hazardous area classification]]		* Hazardous area classification
	* [[Deflagration-to-detonation transition]]		* [[Deflagration-to-detonation transition]]

Jan: Add 5 watermarked SVG illustrations from the Expert visual plan

2026-11-02T16:48:02Z

Add 5 watermarked SVG illustrations from the Expert visual plan

← Older revision		Revision as of 17:48, 2 November 2026
Line 5:		Line 5:
	== Definition and physical meaning ==		== Definition and physical meaning ==
	During a closed-vessel explosion the pressure rises slowly at first, then steeply, and finally levels off at a maximum value. Zalosh characterises the steepest part of this curve with the parameter KG, the maximum rate of pressure rise multiplied by the cube root of the vessel volume.<ref name="Z-409" /> The maximum rate of pressure rise is read at the inflection point of the pressure–time curve, where the slope stops increasing.<ref name="Z-409" /> For dusts the same product is written KSt, and it is obtained with a standardised method for injecting the dust and timing the ignition.<ref name="Z-409" /> The SFPE Handbook describes KSt as the dust reactivity measured in the 20-litre vessel of the American test standard ASTM E1226.<ref name="Z-415" />		During a closed-vessel explosion the pressure rises slowly at first, then steeply, and finally levels off at a maximum value. Zalosh characterises the steepest part of this curve with the parameter KG, the maximum rate of pressure rise multiplied by the cube root of the vessel volume.<ref name="Z-409" /> The maximum rate of pressure rise is read at the inflection point of the pressure–time curve, where the slope stops increasing.<ref name="Z-409" /> For dusts the same product is written KSt, and it is obtained with a standardised method for injecting the dust and timing the ignition.<ref name="Z-409" /> The SFPE Handbook describes KSt as the dust reactivity measured in the 20-litre vessel of the American test standard ASTM E1226.<ref name="Z-415" />
			[[File:fse-deflagration-index-v2.svg\|thumb\|upright=1.5\|alt=S-shaped curve of pressure against time with a tangent at its steepest point and a dashed line at the maximum pressure.\|Pressure history of an explosion in a closed vessel. The deflagration index uses the steepest slope, read at the inflection point, not the final pressure. Based on Zalosh (2002), pp. 3-408 to 3-409.]]

	<math display="block">K_\mathrm{St} = \left( \frac{\mathrm{d}P}{\mathrm{d}t} \right)_\mathrm{max} V^{1/3}</math>		<math display="block">K_\mathrm{St} = \left( \frac{\mathrm{d}P}{\mathrm{d}t} \right)_\mathrm{max} V^{1/3}</math>
Line 14:		Line 15:
	== Measurement in the 20-litre sphere ==		== Measurement in the 20-litre sphere ==
	Dust explosibility is tested today in a near-spherical vessel with a volume of 20 to 30 litres.<ref name="Z-403">Zalosh (2002), p. 3-403.</ref> A weighed dust sample is placed in a small auxiliary chamber and blown into the sphere with air through a perforated dispersion ring.<ref name="Z-403" /><ref name="Z-404" /> After a set delay, an electric spark or a pyrotechnic igniter at the centre of the sphere ignites the dust cloud.<ref name="Z-404" /> A pressure transducer records the pressure history, and the vessel has an exhaust valve and connections for water.<ref name="Z-403" />		Dust explosibility is tested today in a near-spherical vessel with a volume of 20 to 30 litres.<ref name="Z-403">Zalosh (2002), p. 3-403.</ref> A weighed dust sample is placed in a small auxiliary chamber and blown into the sphere with air through a perforated dispersion ring.<ref name="Z-403" /><ref name="Z-404" /> After a set delay, an electric spark or a pyrotechnic igniter at the centre of the sphere ignites the dust cloud.<ref name="Z-404" /> A pressure transducer records the pressure history, and the vessel has an exhaust valve and connections for water.<ref name="Z-403" />
			[[File:fse-deflagration-index-v1.svg\|thumb\|upright=1.5\|alt=Cross-section of a 20-litre steel sphere with a dust chamber below, a dispersion ring inside and an igniter at the centre.\|The 20-litre sphere used to measure the deflagration index: compressed air blows the dust from the dust chamber through a perforated dispersion ring, and an igniter at the centre ignites the cloud after a set delay. Based on Zalosh (2002), pp. 3-403 to 3-404.]]

	The result depends strongly on how turbulent the cloud is at the moment of ignition. Some disturbance is always needed to raise a dust cloud, so every dust explosion is turbulent to some degree.<ref name="Z-409" /> The turbulence is strongest while the dust is being injected and decays as the cloud settles, which makes the ignition delay an indirect measure of turbulence.<ref name="Z-409" /> Tests in a 1 m³ vessel with coal dust and aluminium dust showed that the maximum rate of pressure rise falls steeply as the delay increases, and that the maximum pressure falls as well.<ref name="Z-409" /> A standard method therefore fixes the injection and the ignition delay, so that different vessels and laboratories obtain reproducible results.<ref name="Z-409" /> In the United States this method is ASTM E1226, the standard test method for pressure and rate of pressure rise for combustible dusts, cited in its 1988 edition.<ref name="Z-420">Zalosh (2002), p. 3-420.</ref>		The result depends strongly on how turbulent the cloud is at the moment of ignition. Some disturbance is always needed to raise a dust cloud, so every dust explosion is turbulent to some degree.<ref name="Z-409" /> The turbulence is strongest while the dust is being injected and decays as the cloud settles, which makes the ignition delay an indirect measure of turbulence.<ref name="Z-409" /> Tests in a 1 m³ vessel with coal dust and aluminium dust showed that the maximum rate of pressure rise falls steeply as the delay increases, and that the maximum pressure falls as well.<ref name="Z-409" /> A standard method therefore fixes the injection and the ignition delay, so that different vessels and laboratories obtain reproducible results.<ref name="Z-409" /> In the United States this method is ASTM E1226, the standard test method for pressure and rate of pressure rise for combustible dusts, cited in its 1988 edition.<ref name="Z-420">Zalosh (2002), p. 3-420.</ref>
Line 56:		Line 58:

	Design guidance groups dusts into three bands of KSt values. In the vent equation for low-strength enclosures, dusts with KSt below 200 bar·m/s, between 200 and 300 bar·m/s and above 300 bar·m/s each have their own constant.<ref name="Z-416" /> On that scale, every dust in the table falls in the lowest band except methyl cellulose and aluminium. Methyl cellulose falls in the middle band and aluminium in the highest.<ref name="Z-404" /><ref name="Z-416" />		Design guidance groups dusts into three bands of KSt values. In the vent equation for low-strength enclosures, dusts with KSt below 200 bar·m/s, between 200 and 300 bar·m/s and above 300 bar·m/s each have their own constant.<ref name="Z-416" /> On that scale, every dust in the table falls in the lowest band except methyl cellulose and aluminium. Methyl cellulose falls in the middle band and aluminium in the highest.<ref name="Z-404" /><ref name="Z-416" />
			[[File:fse-deflagration-index-v3.svg\|thumb\|upright=1.5\|alt=Horizontal bar chart of the deflagration index of twelve dusts, from fructose at 28 to aluminium at 515 bar·m/s, with lines at 200 and 300.\|Deflagration index of twelve dusts, with the band limits at 200 and 300 bar·m/s used in vent design. Based on Zalosh (2002), pp. 3-404 and 3-416.]]

	The same test programmes yield other explosibility data that answer different questions. The minimum explosible concentration is the lowest concentration at which a cloud can explode, and the cloud and layer ignition temperatures describe ignition by hot surfaces.<ref name="Z-404" /> The [[minimum ignition energy]] describes ignition by electrical sparks, and for the dusts in the table it ranges from less than 1 mJ to more than 2000 mJ.<ref name="Z-405">Zalosh (2002), p. 3-405.</ref> These properties do not follow the index. Corn starch needs a spark of 300 mJ but has a KSt of 128 bar·m/s, while sugar ignites with 14 mJ and has a KSt of 75 bar·m/s.<ref name="Z-404" /> A dust that is hard to ignite is therefore not necessarily a mild one once it does ignite.		The same test programmes yield other explosibility data that answer different questions. The minimum explosible concentration is the lowest concentration at which a cloud can explode, and the cloud and layer ignition temperatures describe ignition by hot surfaces.<ref name="Z-404" /> The [[minimum ignition energy]] describes ignition by electrical sparks, and for the dusts in the table it ranges from less than 1 mJ to more than 2000 mJ.<ref name="Z-405">Zalosh (2002), p. 3-405.</ref> These properties do not follow the index. Corn starch needs a spark of 300 mJ but has a KSt of 128 bar·m/s, while sugar ignites with 14 mJ and has a KSt of 75 bar·m/s.<ref name="Z-404" /> A dust that is hard to ignite is therefore not necessarily a mild one once it does ignite.
Line 139:		Line 142:

	The result is rounded up because a vent area is a required value; Zalosh reports 1.96 m².<ref name="Z-417" /> The vent is small compared with the size of the precipitator, because coal dust lies in the lowest KSt band and the vessel is relatively strong. The same calculation for corn starch, with a KSt of 128 bar·m/s and a Pmax of 10.2 bar g, gives a vent of 4.6 m².<ref name="Z-404" /> For methyl cellulose, with 209 bar·m/s and 10.1 bar g, it gives 7.4 m².<ref name="Z-404" /> Changing the dust from coal to methyl cellulose thus almost quadruples the vent, a slightly larger step than the ratio of the two KSt values because Pmax rises as well.		The result is rounded up because a vent area is a required value; Zalosh reports 1.96 m².<ref name="Z-417" /> The vent is small compared with the size of the precipitator, because coal dust lies in the lowest KSt band and the vessel is relatively strong. The same calculation for corn starch, with a KSt of 128 bar·m/s and a Pmax of 10.2 bar g, gives a vent of 4.6 m².<ref name="Z-404" /> For methyl cellulose, with 209 bar·m/s and 10.1 bar g, it gives 7.4 m².<ref name="Z-404" /> Changing the dust from coal to methyl cellulose thus almost quadruples the vent, a slightly larger step than the ratio of the two KSt values because Pmax rises as well.
			[[File:fse-deflagration-index-v5.svg\|thumb\|upright=1.5\|alt=Straight rising line of vent area against deflagration index for a 400 cubic metre precipitator, with points for coal, corn starch and methyl cellulose.\|Vent area for the 400 m³ precipitator of the worked example as a function of the deflagration index, with Pmax kept at 9.1 bar g; points show three dusts with their own Pmax. Based on Zalosh (2002), pp. 3-404, 3-416 and 3-417.]]

	The volume correction can be checked with the same coal. In the 20-litre test sphere, a KSt of 59 bar·m/s corresponds to a maximum rate of pressure rise of 59/0.020<sup>1/3</sup>, or about 217 bar/s.<ref name="Z-404" /><ref name="Z-409" /> In a closed compact vessel of 400 m³, the same dust would give only about 8.0 bar/s. The laboratory rate of pressure rise therefore cannot be transferred to a large vessel without the volume correction that KSt contains.		The volume correction can be checked with the same coal. In the 20-litre test sphere, a KSt of 59 bar·m/s corresponds to a maximum rate of pressure rise of 59/0.020<sup>1/3</sup>, or about 217 bar/s.<ref name="Z-404" /><ref name="Z-409" /> In a closed compact vessel of 400 m³, the same dust would give only about 8.0 bar/s. The laboratory rate of pressure rise therefore cannot be transferred to a large vessel without the volume correction that KSt contains.
			[[File:fse-deflagration-index-v4.svg\|thumb\|upright=1.5\|alt=Two falling straight lines on logarithmic axes showing maximum rate of pressure rise against vessel volume for KSt 59 and 209.\|Maximum rate of pressure rise against vessel volume for two dusts, calculated from their deflagration index. Points mark the 20-litre test sphere, a 1 m³ vessel and the 400 m³ precipitator of the worked example. Based on Zalosh (2002), pp. 3-404 and 3-409.]]

	Two cautions from the handbook apply to every such calculation. KSt can vary considerably between samples of one generic material, such as different coals, so the vent area should rest on tests of the actual site material.<ref name="Z-417" /> The calculated area also ignores the inertia of heavy vent panels, which needs an additional allowance.<ref name="Z-417" />		Two cautions from the handbook apply to every such calculation. KSt can vary considerably between samples of one generic material, such as different coals, so the vent area should rest on tests of the actual site material.<ref name="Z-417" /> The calculated area also ignores the inertia of heavy vent panels, which needs an additional allowance.<ref name="Z-417" />

Mark: FSE wiki: align reference entries with the run's canonical reference list (written by Claude Opus 5.5)

2026-11-01T22:03:57Z

FSE wiki: align reference entries with the run's canonical reference list (written by Claude Opus 5.5)

← Older revision		Revision as of 23:03, 1 November 2026
Line 163:		Line 163:

	== References ==		== References ==
	* 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.		* 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.
	* Zalosh, R. (2002). Explosion protection. 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. 3-402–3-421). National Fire Protection Association.		* Zalosh, R. (2002). Explosion protection. 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. 3-402–3-421). National Fire Protection Association; Society of Fire Protection Engineers.

	[[Category:Explosions & Industrial Fires]]		[[Category:Explosions & Industrial Fires]]

Mark: New FSE wiki article: Deflagration index (written by Claude Opus 5.5)

2026-11-01T21:59:37Z

New FSE wiki article: Deflagration index (written by Claude Opus 5.5)

New page

The '''deflagration index''' ('''KSt''') is a number that describes how violently a cloud of combustible dust explodes in a closed vessel. It equals the maximum rate of pressure rise during the explosion multiplied by the cube root of the vessel volume, and it is expressed in bar·m/s.<ref name="Z-409">Zalosh (2002), p. 3-409.</ref><ref name="Z-415">Zalosh (2002), p. 3-415.</ref> The equivalent parameter for flammable gases and vapours is written KG.<ref name="Z-409" /> In a [[deflagration]] the flame travels through the mixture more slowly than sound, so the pressure rises almost uniformly throughout the enclosure.<ref name="Z-410">Zalosh (2002), p. 3-410.</ref>

The index is measured in a standardised test in a 20-litre sphere.<ref name="Z-415" /> In a table of twenty representative dusts in the SFPE Handbook, the values range from 28 bar·m/s for fructose to 515 bar·m/s for fine aluminium powder.<ref name="Z-404">Zalosh (2002), p. 3-404, Table 3-16.2.</ref> Engineers use KSt to group dusts by explosion violence, to size explosion vents and to judge how hard a suppression system has to work.<ref name="Z-416">Zalosh (2002), p. 3-416.</ref><ref name="Z-418">Zalosh (2002), p. 3-418.</ref> A higher index means a faster pressure rise, a larger vent and a smaller margin for suppression.<ref name="Z-417">Zalosh (2002), p. 3-417.</ref><ref name="Z-418" /> Because the value comes from one sample under fixed laboratory conditions, it does not describe the turbulence, obstructions or partial filling of a real plant.<ref name="Z-416" />

== Definition and physical meaning ==
During a closed-vessel explosion the pressure rises slowly at first, then steeply, and finally levels off at a maximum value. Zalosh characterises the steepest part of this curve with the parameter KG, the maximum rate of pressure rise multiplied by the cube root of the vessel volume.<ref name="Z-409" /> The maximum rate of pressure rise is read at the inflection point of the pressure–time curve, where the slope stops increasing.<ref name="Z-409" /> For dusts the same product is written KSt, and it is obtained with a standardised method for injecting the dust and timing the ignition.<ref name="Z-409" /> The SFPE Handbook describes KSt as the dust reactivity measured in the 20-litre vessel of the American test standard ASTM E1226.<ref name="Z-415" />

<math display="block">K_\mathrm{St} = \left( \frac{\mathrm{d}P}{\mathrm{d}t} \right)_\mathrm{max} V^{1/3}</math>

The cube root of the volume corrects for the size of the vessel. A theoretical model of an explosion in a spherical vessel shows that the rate of pressure rise is inversely proportional to the vessel radius.<ref name="Z-408">Zalosh (2002), p. 3-408.</ref> The cube root of the volume of a sphere is proportional to its radius, so the product of the two no longer depends on vessel size. The same dust therefore gives a high rate of pressure rise in a small test vessel and a low rate in a large silo, while its KSt stays the same. This is why a value measured in a 20-litre sphere can be used in vent calculations for equipment of hundreds of cubic metres.<ref name="Z-415" /><ref name="Z-417" />

KSt is given in bar·m/s because a rate of pressure rise in bar/s is multiplied by a length in metres. Both KG and KSt describe a deflagration, which Zalosh defines as flame propagation at subsonic speed away from the ignition site.<ref name="Z-406">Zalosh (2002), p. 3-406.</ref> NUREG-1805 adds that a deflagration always moves more slowly than sound, which travels at about 330 m/s in air at 20 °C, while a detonation travels at about 2,000 m/s.<ref name="IS-15-3">Iqbal and Salley (2004), p. 15-3.</ref><ref name="IS-F-20">Iqbal and Salley (2004), p. F-20.</ref> Detonations, in which a shock wave drives the reaction, need different methods of protection, and venting and suppression cannot be used against them.<ref name="Z-411">Zalosh (2002), p. 3-411.</ref>

== Measurement in the 20-litre sphere ==
Dust explosibility is tested today in a near-spherical vessel with a volume of 20 to 30 litres.<ref name="Z-403">Zalosh (2002), p. 3-403.</ref> A weighed dust sample is placed in a small auxiliary chamber and blown into the sphere with air through a perforated dispersion ring.<ref name="Z-403" /><ref name="Z-404" /> After a set delay, an electric spark or a pyrotechnic igniter at the centre of the sphere ignites the dust cloud.<ref name="Z-404" /> A pressure transducer records the pressure history, and the vessel has an exhaust valve and connections for water.<ref name="Z-403" />

The result depends strongly on how turbulent the cloud is at the moment of ignition. Some disturbance is always needed to raise a dust cloud, so every dust explosion is turbulent to some degree.<ref name="Z-409" /> The turbulence is strongest while the dust is being injected and decays as the cloud settles, which makes the ignition delay an indirect measure of turbulence.<ref name="Z-409" /> Tests in a 1 m³ vessel with coal dust and aluminium dust showed that the maximum rate of pressure rise falls steeply as the delay increases, and that the maximum pressure falls as well.<ref name="Z-409" /> A standard method therefore fixes the injection and the ignition delay, so that different vessels and laboratories obtain reproducible results.<ref name="Z-409" /> In the United States this method is ASTM E1226, the standard test method for pressure and rate of pressure rise for combustible dusts, cited in its 1988 edition.<ref name="Z-420">Zalosh (2002), p. 3-420.</ref>

The test is repeated over a range of dust concentrations to find the worst case. For many organic materials the most severe concentration is about 500 g/m³, roughly twice the stoichiometric concentration and five to ten times the minimum explosible concentration.<ref name="Z-407">Zalosh (2002), p. 3-407.</ref> Particle size matters as well. In tests on four dusts, the maximum pressure of three of them did not change as long as the median particle diameter was 100 μm or less.<ref name="Z-407" /> The maximum pressure of polyvinyl chloride, however, fell for all particles above 20 μm, and particles of 150 μm and larger could not be ignited at all.<ref name="Z-407" /> The table of explosibility data shows the same effect on the index: low-density polyethylene with a median size below 10 μm has a KSt of 156 bar·m/s, against 54 bar·m/s at 150 μm.<ref name="Z-404" />

== Typical values ==
The table below lists KSt and the maximum explosion pressure Pmax for a selection of the dusts in the SFPE Handbook. The data were compiled by Eckhoff and were measured with the standardised injection and ignition-delay method.<ref>Eckhoff (1996), as cited in Zalosh (2002), p. 3-404.</ref><ref name="Z-409" /> The dusts are sorted from the lowest to the highest index, and the median particle size shows how fine each tested sample was.

{| class="wikitable"
|+ Deflagration index and maximum explosion pressure of selected dusts<ref name="Z-404" />
! Dust !! Median particle size (μm) !! Pmax (bar g) !! KSt (bar·m/s)
|-
| Fructose || 200 || 7.0 || 28
|-
| Polyvinyl chloride || 25 || 8.2 || 42
|-
| Activated carbon || 18 || 8.8 || 44
|-
| Paper tissue dust || 54 || 8.6 || 52
|-
| Coal, bituminous (high volatility) || 4 || 9.1 || 59
|-
| Sugar || 10 || 8.3 || 75
|-
| Milk powder || 165 || 8.1 || 90
|-
| Corn starch || below 10 || 10.2 || 128
|-
| Rubber || 80 || 8.5 || 138
|-
| Naphthalene || 95 || 8.5 || 178
|-
| Methyl cellulose || 37 || 10.1 || 209
|-
| Aluminium || below 10 || 11.2 || 515
|}

Two features stand out. The maximum pressures of all twenty dusts in the handbook lie between 6.7 and 11.2 bar g, about the same range as for flammable gases.<ref name="Z-407" /> KSt, by contrast, varies by a factor of about 18 between fructose and aluminium.<ref name="Z-404" /> The index, not the final pressure, therefore separates mild dusts from violent ones. Aluminium stands apart, while all other dusts in the table lie between 28 and 209 bar·m/s.<ref name="Z-404" />

For gases and vapours, the handbook reproduces data from NFPA 68. Methane has a KG of 55 bar·m/s and propane 100 bar·m/s, while hydrogen reaches 550 bar·m/s and acetylene 1,415 bar·m/s.<ref name="Z-407" /> Ammonia has the lowest value in that table, 10 bar·m/s.<ref name="Z-407" /> The KSt values of most organic dusts in the table are therefore of the same order as the KG of methane and propane. Hydrogen and acetylene exceed every dust in the table, including aluminium.

Design guidance groups dusts into three bands of KSt values. In the vent equation for low-strength enclosures, dusts with KSt below 200 bar·m/s, between 200 and 300 bar·m/s and above 300 bar·m/s each have their own constant.<ref name="Z-416" /> On that scale, every dust in the table falls in the lowest band except methyl cellulose and aluminium. Methyl cellulose falls in the middle band and aluminium in the highest.<ref name="Z-404" /><ref name="Z-416" />

The same test programmes yield other explosibility data that answer different questions. The minimum explosible concentration is the lowest concentration at which a cloud can explode, and the cloud and layer ignition temperatures describe ignition by hot surfaces.<ref name="Z-404" /> The [[minimum ignition energy]] describes ignition by electrical sparks, and for the dusts in the table it ranges from less than 1 mJ to more than 2000 mJ.<ref name="Z-405">Zalosh (2002), p. 3-405.</ref> These properties do not follow the index. Corn starch needs a spark of 300 mJ but has a KSt of 128 bar·m/s, while sugar ignites with 14 mJ and has a KSt of 75 bar·m/s.<ref name="Z-404" /> A dust that is hard to ignite is therefore not necessarily a mild one once it does ignite.

== Theory: burning velocity and turbulence ==
Zalosh explains the meaning of KG with a simple model of an explosion in a closed sphere.<ref name="Z-408" /> The model assumes that the pressure is uniform in the vessel at any moment and that the pressure rise at any time is proportional to the burned fraction of the total mass.<ref name="Z-407" /><ref name="Z-408" /> It also makes the most common assumption for the unburned gas ahead of the flame, namely that it is compressed isentropically.<ref name="Z-408" /> For a spherical flame of radius rb in a vessel of radius a, the rate of pressure rise is then:<ref name="Z-408" />

<math display="block">\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{3 \chi S_\mathrm{u}}{a} \left( \frac{P}{P_0} \right)^{1/\gamma} \left( P_\mathrm{m} - P_0 \right) \left( \frac{r_\mathrm{b}}{a} \right)^2</math>

In words, the pressure rises faster when the burning velocity is high, when the flame is large and when the vessel is small. In theory the rate is greatest at the end of the explosion, when the flame reaches the wall and the pressure reaches its maximum value.<ref name="Z-409" /> Setting the flame radius equal to the vessel radius and the pressure equal to the maximum, and then multiplying by the cube root of the sphere volume, gives the theoretical index:<ref name="Z-409" />

<math display="block">K_\mathrm{G} = 4.84 \, \chi S_\mathrm{u} \left( \frac{P_\mathrm{m}}{P_0} \right)^{1/\gamma} \left( P_\mathrm{m} - P_0 \right)</math>

The constant 4.84 in this equation equals <math>3 (4 \pi / 3)^{1/3}</math>, the factor that links the cube root of a sphere's volume to its radius.<ref name="Z-409" /> The vessel radius has disappeared from the result, so in this model KG depends only on the mixture and on its turbulence. A fast-burning mixture with a high final pressure therefore has a high index in any vessel. The symbols in the three equations above are listed in the following table.

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>K_\mathrm{G}</math>, <math>K_\mathrm{St}</math> || deflagration index of a gas or of a dust || bar·m/s
|-
| <math>(\mathrm{d}P/\mathrm{d}t)_\mathrm{max}</math> || maximum rate of pressure rise in a closed vessel || bar/s
|-
| <math>V</math> || vessel volume || m³
|-
| <math>P</math> || pressure at time t || bar
|-
| <math>P_0</math> || initial pressure || bar (absolute)
|-
| <math>P_\mathrm{m}</math> || final pressure of the closed-vessel explosion || bar (absolute)
|-
| <math>S_\mathrm{u}</math> || laminar burning velocity of the unburned mixture || m/s
|-
| <math>\chi</math> || ratio of turbulent to laminar burning velocity || none
|-
| <math>\gamma</math> || ratio of specific heats of the unburned gas, 1.4 for most hydrocarbon–air mixtures || none
|-
| <math>a</math> || vessel radius || m
|-
| <math>r_\mathrm{b}</math> || flame radius || m
|}

The turbulence factor χ cannot be predicted in advance.<ref name="Z-409" /> The measured KG values of gases rise with the laminar burning velocity but are not proportional to it, because χ differs from one gas to another.<ref name="Z-409" /> Zalosh shows how to work backwards from a measurement. For n-butane, with KG = 92 bar·m/s, a burning velocity of 0.45 m/s, a pressure ratio of 9.0 and a pressure rise of 8 bar, the equation gives χ = 1.10.<ref name="Z-410" /> The flame in that test was therefore not much faster than a laminar flame.<ref name="Z-410" />

Dust explosions are harder to model, because the particles need a finite time to burn, which gives the flame a finite thickness.<ref name="Z-409" /><ref name="Z-414">Zalosh (2002), p. 3-414.</ref> Some dust models use the burning velocity and flame area of the gas theory, while others use the measured KSt directly as the effective burning rate.<ref name="Z-414" /> Because some turbulence is always present when a dust cloud is raised, every measured KSt includes an effect of turbulence, which the standard test keeps fixed.<ref name="Z-409" />

== Use in explosion protection ==
The main use of KSt is in the design of protection against a [[dust explosion]]. NUREG-1805 lists the hazards of a deflagration: catastrophic failure of equipment, ejection of flame and unburned material, secondary explosions and injury.<ref name="IS-15-3" /> Zalosh groups the protective measures into three approaches. The first prevents a flammable mixture, for example by ventilation or inerting, and the second removes ignition sources, for example by grounding equipment against static discharges.<ref name="Z-402">Zalosh (2002), p. 3-402.</ref> The third limits the pressure rise by venting or suppression, and KSt enters directly into the sizing of vents and the assessment of suppression systems.<ref name="Z-402" /><ref name="Z-416" /><ref name="Z-418" />

For [[explosion venting]], the handbook defines a dimensionless vent parameter in which KSt appears in the denominator, together with the two-thirds power of the enclosure volume.<ref name="Z-415" /> In the United States, NFPA 68 (1998 edition) used the same dust equation for high-strength enclosures as the German guideline VDI 3673. The following equation for compact enclosures had been proposed for the 2002 edition when the handbook was written:<ref name="Z-416" />

<math display="block">A_\mathrm{v} = 8.535 \times 10^{-5} \left( 1 + 1.75 P_\mathrm{stat} \right) K_\mathrm{St} V^{0.75} \sqrt{\frac{1 - \Pi}{\Pi}}, \qquad \Pi = \frac{P_\mathrm{red}}{P_\mathrm{max}}</math>

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>A_\mathrm{v}</math> || vent area || m²
|-
| <math>V</math> || enclosure volume || m³
|-
| <math>K_\mathrm{St}</math> || deflagration index of the dust in the 20-litre test || bar·m/s
|-
| <math>P_\mathrm{stat}</math> || pressure at which the vent opens || bar g
|-
| <math>P_\mathrm{red}</math> || reduced pressure, the highest pressure the vented enclosure may reach || bar g
|-
| <math>P_\mathrm{max}</math> || maximum pressure of the unvented explosion || bar g
|-
| <math>\Pi</math> || pressure ratio || none
|}

The required vent area in this equation is directly proportional to KSt, so doubling the index doubles the vent.<ref name="Z-417" /> The equation also contains Pmax through the ratio Π, so a dust with a higher maximum pressure needs a somewhat larger vent for the same index. The equation applies to compact enclosures; for elongated ones, with a length-to-diameter ratio above 2, an additional vent area was being proposed.<ref name="Z-416" />

Weak enclosures such as rooms and buildings, with a damage threshold below about 0.1 bar, are treated with the Swift–Epstein equation.<ref name="Z-410" /><ref name="Z-415" /> In this equation the vent area is proportional to the internal surface area of the enclosure and to a fuel constant C.<ref name="Z-416" /> For dusts, C is 0.10 psig1/2 for KSt below 200 bar·m/s, 0.12 psig1/2 between 200 and 300 bar·m/s and 0.20 psig1/2 above 300 bar·m/s.<ref name="Z-416" /> Consider a room of 4 m × 4 m × 3 m, with 80 m² of internal surface and a damage threshold of 1.0 psig, about 6.9 kPa.<ref name="Z-416" /><ref name="IS-15-8">Iqbal and Salley (2004), p. 15-8.</ref> The three constants then give vent areas of 8.0 m², 9.6 m² and 16.0 m², so the most reactive band needs twice the vent of the least reactive one.

Suppression systems detect an incipient explosion and inject an extinguishing agent before the pressure reaches the damage threshold.<ref name="Z-417" /> The pressure that still develops while the agent travels to the flame front rises with KSt and with the turbulence of the application.<ref name="Z-418" /> Aluminium dust is the highest-KSt material that has been tested with suppression, and early tests showed the limits of first-generation systems against it.<ref name="Z-418" /> Later tests reported in the handbook reduced the suppressed pressure of aluminium dust to 10 to 20 % of Pmax, compared with usually less than 5 % for organic fuels.<ref name="Z-418" />

Secondary explosions add to the hazard. When a primary explosion bursts a piece of equipment, the blast can lift dust that has settled on walls and floors, and the emerging flame ignites the new cloud.<ref name="Z-409" /> Zalosh estimates the effect of a coal dust layer of 1000 g/m², only about 2 mm thick, in an unvented enclosure with a 4 m ceiling.<ref name="Z-410" /> If that layer is dispersed into a cloud of 500 g/m³ in the lower part of the room, the pressure could rise by 4.55 bar g.<ref name="Z-410" />

== Worked example ==
This example follows a case in the SFPE Handbook and extends it with other dusts from the same table. A 400 m³ electrostatic precipitator in a power plant that burns bituminous coal is to be fitted with lightweight explosion vents.<ref name="Z-416" /> The precipitator is damaged at 1.2 bar, its length-to-diameter ratio is 1.5 and the vents open at 0.4 bar.<ref name="Z-416" /> For the coal, KSt is 59 bar·m/s and Pmax is 9.1 bar g.<ref name="Z-404" /> The pressure ratio is therefore Π = 1.2/9.1 = 0.132, and the vent equation gives:<ref name="Z-417" />

<math display="block">A_\mathrm{v} = 8.535 \times 10^{-5} \times (1 + 1.75 \times 0.40) \times 59 \times 400^{0.75} \times \sqrt{\frac{1 - 0.132}{0.132}} \approx 1.97\ \mathrm{m^2}</math>

The result is rounded up because a vent area is a required value; Zalosh reports 1.96 m².<ref name="Z-417" /> The vent is small compared with the size of the precipitator, because coal dust lies in the lowest KSt band and the vessel is relatively strong. The same calculation for corn starch, with a KSt of 128 bar·m/s and a Pmax of 10.2 bar g, gives a vent of 4.6 m².<ref name="Z-404" /> For methyl cellulose, with 209 bar·m/s and 10.1 bar g, it gives 7.4 m².<ref name="Z-404" /> Changing the dust from coal to methyl cellulose thus almost quadruples the vent, a slightly larger step than the ratio of the two KSt values because Pmax rises as well.

The volume correction can be checked with the same coal. In the 20-litre test sphere, a KSt of 59 bar·m/s corresponds to a maximum rate of pressure rise of 59/0.0201/3, or about 217 bar/s.<ref name="Z-404" /><ref name="Z-409" /> In a closed compact vessel of 400 m³, the same dust would give only about 8.0 bar/s. The laboratory rate of pressure rise therefore cannot be transferred to a large vessel without the volume correction that KSt contains.

Two cautions from the handbook apply to every such calculation. KSt can vary considerably between samples of one generic material, such as different coals, so the vent area should rest on tests of the actual site material.<ref name="Z-417" /> The calculated area also ignores the inertia of heavy vent panels, which needs an additional allowance.<ref name="Z-417" />

== Limitations and uncertainty ==
KSt is a property of a test as well as of a material. It depends on the turbulence at ignition, on the igniter and on the particle size and concentration of the sample.<ref name="Z-404" /><ref name="Z-407" /><ref name="Z-409" /> The standard test fixes these conditions so that results can be compared, but the dust in a plant may have a different size distribution from the tested sample. A fine laboratory sample can therefore give a value that differs from the behaviour of the dust as it occurs in the process.

The vent guidelines based on KSt assume that the turbulence and the obstructions in the protected equipment resemble those in the tests behind the guidelines.<ref name="Z-416" /> With an extremely high level of turbulence, as in a hammer mill, the recommended constants can underestimate the pressure in a vented explosion.<ref name="Z-416" /> With lower turbulence than in the tests, the design is conservative to an ill-defined degree. Most guidelines also do not account for how much of the enclosure is filled with the combustible mixture.<ref name="Z-416" />

The data in this article come from a compilation published in 1996 and from the 1998 edition of NFPA 68, as reproduced in the 2002 SFPE Handbook.<ref name="Z-404" /><ref name="Z-407" /> The equations show how KSt enters a design calculation, including one equation that was only proposed for NFPA 68 at the time. They are not necessarily the equations of later editions of the American standard or of guidelines in other countries.<ref name="Z-416" />

Finally, KSt describes only deflagrations. A detonation produces a sudden, spatially non-uniform pressure rise that rules out both venting and suppression.<ref name="Z-411" /> The deflagration index is therefore no guide to the loads from a detonation, which are assessed with other methods.<ref name="Z-411" />

== See also ==
* [[Explosion]]
* [[Inerting]]
* [[Hazardous area classification]]
* [[Deflagration-to-detonation transition]]

== Notes ==
<references />

== References ==
* 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.
* Zalosh, R. (2002). Explosion protection. 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. 3-402–3-421). National Fire Protection Association.

[[Category:Explosions & Industrial Fires]]