Dahoe AE — Explosions — Phenomenology Dynamics And Mitigation

Over-pressure

Over-pressure
blast waves
(side-on)
l properties Reflected Pressure ws a typical al time tA, hat position f overpresBlast wave Stand-off distance o. The presime td, then (creating a Unconfined explosion: Blast-structure interaction [1].
Figure 3: Blast loads on a building ning to amantity Pso is If the exterior building walls are capable of n overpresresisting the blast load, the shock front penetrates merely peak through window and door openings, subjecting the floors, ceilings, walls, contents, and people to sudden pressures and fragments from shattered o are ampliwindows, doors, etc. Building components not ave encouncapable of resisting the blast wave will fracture and cept for spebe further fragmented and moved by the dynamic aves at near pressure that immediately follows the shock front.
are typically Building contents and people will be displaced and adjacent and tumbled in the direction of blast wave propagation.
ish with the In this manner the blast will propagate through the lative to the 2 building.

77

Confined explosion:
Garage beneath building.

Hydrogen versus Gasoline.

Deflagration and Detonation

FLAMES

Co Id reactants
Preflame
Products
.. zone • ,..._....:z:.:.o:..:.:ne=--... ----,-F-r--7"-~ .--=z""on.:.:e_

:

Reactants:

1

c:

.2 ~

a; :J

E~
BOo
c: E

,
,,
To

turbulence

Products

!,,

0.,

()I-

unburnt
mixture

,,
,,
:,,
,,,
,,
,,
,
,,

Temperatu~e

:

Deflagration

disturbance
Figure 3.3 Concentration and
temperature profiles associatedunburnt with a one-dimensional, flame premixed,zone adiabatic flame.
mixture front

on the high temperature side. Beyond the point of inflexion the converse is true; that is, the heat loss rate exceeds the heat gain rate. However, at burnt the reaction rate has increased sufficiently for a this higher temperature of heat to be produced by chemical reaction. The significant amount mixture temperature therefore continues to increase through the flame although at a progressively slower rate. It eventually reaches a constant value when all the fuel has been consumed and reaction has ceased. Parallel behaviour also occurs in the reactant concentration profile, the major pressure waves loss initially being diffusion of fuel into the flame but subsequently being consumption by chemical reaction. The first region is termed the preheat or preflame zone and the second the reaction zone. A third region may also be defined which follows the reaction zone, and is termed the post-flame or product zone.
3 The concentration of all species in the post-flame zone must approach that defined by thermodynamic equilibrium at the prevailing temperature

Detonation

33

Flame Propagation: Laminar and Turbulent

unburnt mixture

fluid
velocity

laminar
flame

turbulent
flame

turbulent
flame

SuL
SuL

SuT

Sf

SuL

smallest length scales
of turbulence are larger than laminar flame thickness

moving
observer

flame
front

largest length scales
of turbulence are smaller than laminar flame thickness

fixed
observer

unburnt
mixture

flame
front

turbulence

burnt
mixture

disturbance
zone

unburnt
mixture

burnt
mixture

pressure waves

4

Gas Explosions versus Dust Explosions

From Ref. [2].

5

Ignition and combustion properties. Initial conditions: 25 ◦C and 101.325 kPa.
Fuel Lower Lower Stoich. Upper Upper Minim. Autoflamm. deton. fuel deton. flamm. ign.
ign.
limit limit conc. limit limit energy temp.
◦ vol. % vol. % vol. % vol. % vol. % mJ C Hydrogen (H2) 4.0 18.3 29.5 59.0 75.0 0.017 585 Methane (CH4) 5.3 6.3 9.5 13.5 17.0 0.274 537 Methanol (CH3OH) 6.0 12.3 36.5 0.174 385 Propane (C3H8) 1.7 3.1 4.0 9.2 10.9 0.240 450 Gasoline (C8H18) 1.0 1.1 1.9 3.3 6.0 0.240 215 •

•

••

•Methanol (CH3 OH)
••Propane (C3 H8 )

• • •Gasoline (C8 H18 )

0

Hydrogen (H2 )
•

• •Methane (CH4 )

•
••

•

10

Flammability range
Detonation range

20
30
40
50
60
70
80
Fuel concentration in air [mol%, vol%]

90

Flammability and detonation limits [3].

100

Domestic hydrogen boiler.
6

Lamin.
burn.
veloc.
cm s−1
225
37
48
47
30

•

••

•

Hydrogen (H2 )
•

• •Methane (CH4 )

•

•Methanol (CH3 OH)

••

••Propane (C3 H8 )

Flammability range
Detonation range

• • •Gasoline (C8 H18 )

41

1.4. Energy Conservation

41

Adiabatic Adiabatic
Flame Temperature
(K)
Flame Temperature (K)

•

1.4. Energy Conservation

2,500

acetylene
ethylene

2,500
2,000

i-octane benzene propane

methane
ethylene

CO

acetylene

H2

methanol
graphite

2,000
1,500

i-octane benzene propane

CO

methane
H2
methanol
graphite

1,500
1,000

1

10

1,000

100

Fuel Mole %

Figure 1.4.2. Adiabatic
flame temperature, Tad , as a function of the mole percent of fuel for 100 several 1 10 fuel–air mixtures at STP. The vertical bars indicate stoichiometric concentrations.

Fuel Mole %

10

•

20
30
40
50
60
70
80
Fuel concentration in air [mol%, vol%]

•
•

•Methanol (CH3 OH)

• •

0.0

Figure 1.4.2.
2,800 Adiabatic flame temperature, Tad , as a function of the mole percent of fuel for several fuel–air mixtures at STP. The vertical bars indicate stoichiometric concentrations.

•Methane (CH4 )

•

••

100

Hydrogen (H2 )
•

•

••

90

AdiabaticAdiabatic
Flame Temperature
(K)
Flame Temperature (K)

0

•

•Propane (C3 H8 )

•
•
Gasoline (C8 H18 )

1.0

2.0

Flammability range
Detonation range

3.0
4.0
5.0
Equivalence ratio, φ [-]

6.0

7.0

2,600
2,800
acetylene

2,400
2,600
CO
acetylene

2,200
2,400

H2
CO
ethylene

2,000
2,200

Hbenzene
2

1,800
2,000

propane
ethylene
benzene
methane

1,600
1,800

propane

0.5

8.0
1,600

1.0

1.5

methane

2.0

2.5

Equivalence Ratio, φ

0.5
1.0
1.5 of fuel equivalence
2.0 ratio, φ, for several 2.5 Figure 1.4.3. Adiabatic flame temperature, Tad Adiabatic flame temperature as, asaa function function of mixture composition [4].
fuel–air mixtures at STP.

Flammability and detonation limits [3].
Initial conditions: 1 bar and 298.15 K.

Initial conditions: 1 bar andEquivalence
298.15 K.
Ratio, φ

Figure 1.4.3. Adiabatic flame temperature, Tad , as a function of fuel equivalence ratio, φ, for several fuel–air mixtures at STP.

7

vol.% hydrogen
0 10 20 30 40
50

vol.% hydrogen
0 40 60 70
80

⊞
•

3000

•
⊞
• ⊞

2500

⊞
•

• ⊞
• ⊞
⊞
• ⊞
•
• ⊞
• ⊞
⊞
• ⊞
•

⊞
•
⊞
•

isochoric

⊞
•

⊞
•

isobaric

⊞
•
⊞
•

⊞
•
⊞
•

⊞
•
⊞
•

⊞
•

2000

⊞
•

1500

⊞
•
⊞
•

⊞
•
⊞
•

⊞
•
⊞
•

H2 -O2
• GASEQ

⊞ CHEMEQUIL

1000
500

Adiabatic flame temperature (K)

4000

3500

3500
3000
2500

⊞
•

2000

⊞
•

⊞
• ⊞
•
⊞
• ⊞
•

⊞
•

⊞
•
⊞
•

500

⊞
•

⊞
•
⊞
•

isochoric
⊞
•
⊞
•

isobaric

⊞
•
⊞
•

H2 -air
• GASEQ

⊞
•

1000

⊞
•

⊞
•
⊞
•

Isobaric Flame Temperature
Isochoric Flame Temperature

⊞
•

• ⊞
⊞
• ⊞
•
⊞
• ⊞
• ⊞
• ⊞
•

⊞ CHEMEQUIL

⊞
•

0
0

0.5 1 1.5 2 2.5
Equivalence ratio, φ

3

3.5

0

10
△
•
△⊞
△
△
⊞
•

8

△△
• ⊞
• △⊞
• ⊞
• ⊞
⊞
• ⊞
•
△
⊞
•
△
⊞
•

△

⊞
•

⊞
•

⊞
•

3.5

Pressure (bar)

H2 -O2 ; isochoric
△ expt. 6 dm3
• GASEQ
⊞ CHEMEQUIL

×

×
×⊞
⊞
×
×△
×××⊞△
×
◦•× ◦•△
×
⊞
×
◦ ◦
◦ ◦•×
△
⊞ × ◦ • △⊞

8

⊞
•

△

2

3

10

⊞
•
△

4

0.5 1 1.5 2 2.5
Equivalence ratio, φ

vol.% hydrogen
0 10 20 30 40
50

vol.% hydrogen
0 40 60 70
80

6

⊞
•

1500

0

Pressure (bar)

Adiabatic flame temperature (K)

4000

◦
•
• ⊞ △⊞
×◦
◦
◦
×
◦ • • ⊞
×
××
△
×
• ⊞
×
×
• ⊞ × ◦ × • ⊞ ◦ ⊞ •△ ◦ ◦•× ◦ × × × × × ×× × × × × H2 -air; isochoric ⊞ × △ • × 3

△

6

4
⊞
•

2

Comparison between the adiabatic flame temperature, Tfad , of hydrogen-oxygen and hydrogen-air mixtures calculated by CHEMEQUIL and GASEQ.
Experimental data: × 120 dm3 explosion vessel [5];
△ 6 dm3 explosion vessel [6]; ◦ 169 cm3 explosion vessel [7].

◦ expt. 169 mm
△ expt. 6 dm3
× expt. 120 dm3
• GASEQ ⊞ CHEMEQUIL

0

0
0

0.5 1 1.5 2 2.5
Equivalence ratio, φ

3

3.5

0

0.5 1 1.5 2 2.5
Equivalence ratio, φ

3

3.5

8

Directives
Regulations
Codes

Stipulations
Standards
Guidelines
Codes of Practice

Directive 2014/34/EU: ATEX 114 [8]
Directive 1999/92/EC: ATEX 153 [9]

LEGISLATIVE HIERARCHY

Optional

Mandatory

Regulations, Codes and Standards

• Directives and regulations are the law. They represent the State’s position on what is acceptable and what is not.
They impose duties, responsibilities and accountabilities on people and organisations. Consequently, regulations are at the top of the hierarchy of things of which you need to take heed. Make sure that you are aware of the regulations pertaining to the work that you are proposing to carry out.

9

• Codes are laws written and enacted by the legislative branch of government. But unlike directives and regulations they require industry consensus or approval by a knowledgeable body to be put into effect.
• Standards are intended to aid designers and engineers with ensuring that the piece of technology needed to fulfil a certain function is appropriate and has been properly designed and manufactured.
• There is an additional, special form of guidance issued by the State: the so-called ‘Approved Code of Practice’ (e.g. COMAH [10]). If you follow this guidance then you will comply with the law. If you don’t do it this way, then the onus is on you to prove that your way assures at least the same level of safety as that achieved through following the guidance.

ATEX 114
Objectives Pertaining to Gas and Dust Explosion Hazards PREVENT FORMATION EXPLOSIVE MIXTURES • Implement a hazardous dust inspection, housekeeping, testing and control program • Use proper dust collection systems and filters • Apply inertization • Minimize the escape of flammable gases and dust from process equipment or ventilation systems • Use surfaces that minimize dust accumulation and facilitate cleaning • Provide access to all hidden areas to permit inspection • Inspect for dust residues in open and hidden areas at regular intervals • If ignition sources are present, use cleaning methods that do not generate dust clouds • Use only vacuum cleaners approved for dust collection • Locate relief valves away from dust deposits

ELIMINATE IGNITION SOURCES

MINIMIZE INJURY AND DAMAGE

• Use appropriate electrical equipment
and wiring methods • Control static electricity, including bonding of equipment to ground • Control smoking, open flames, and sparks • Control mechanical sparks and friction • Use separator devices to remove foreign materials capable of igniting combustibles from process materials • Separate heated surfaces and heating systems from dusts • Separate heated surfaces and heating systems from flammable gas mixtures • Select and use industrial trucks properly • Use cartridge-activated tools properly • Use an ignition preventive maintenance program

• Separation of the hazard (isolate
with distance) • Segregation of the hazard (isolate with a barrier) • Deflagration isolation/venting • Pressure relief venting for equipment • Direct vents away from work areas • Specialized fire suppression systems • Explosion protection systems • Spark/ember detection for suppression activation • Develop an emergency action plan • Maintain emergency exit routes

10

ATEX 114
Equipment Groups and Catergories [8]

11

ATEX 114
Gas Groups and Dust Groups [8]

EXPLOSION SEVERITY CLASSIFICATION
Class KSt -value Severity (bar m s−1 ) St 3 St 2 St 1 St 0

12

KSt > 300
200 < KSt ≤ 300
100 < KSt ≤ 200
0 < KSt ≤ 100 KSt = 0

Extremely explosive
Strongly explosive
Moderately explosive Weakly explosive Not explosive

Approved Code of Practice [10]
Health and Safety Executive. Control of Major Accident Hazards (COMAH). Hazardous Area Classification and Control of Ignition Sources.

ATEX 153
HAZARDOUS AREA CLASSIFICATION
Hazardous places are classified in terms of zones on the basis of the frequency and duration of the occurrence of an explosive atmosphere.
The extent of the measures to be taken in accordance with Annex II, Part A, is determined by this classification.
GAS EXPLOSIONS Zone 0 A place in which an explosive atmosphere consisting of a mixture with air of flammable substances in the form of gas, vapour or mist is present continuously or for long periods or frequently.
Zone 1 A place in which an explosive atmosphere consisting of a mixture with air or flammable substances in the form of gas, vapour or mist is likely to occur in normal operation occasionally.
Zone 2 A place in which an explosive atmosphere consisting of a mixture with air of flammable substances in the form of gas, vapour or mist is not likely to occur in normal operation but, if it does occur, will persist for a short period only.
DUST EXPLOSIONS Zone 20 A place in which an explosive atmosphere in the form of a cloud of combustible dust in air is present continuously, or for long periods or frequently.
Zone 21 A place in which an explosive atmosphere in the form of a cloud of combustible dust in air is likely to occur in normal operation occasionally.
Zone 22 A place in which an explosive atmosphere in the form of a cloud of combustible dust in air is not likely to occur in normal operation but, if it does occur, will persist for a short period only.

13

Zone 0 / 20: Explosive mixture for more than 1000h/yr.
Category 1 equipment. One year equals 8760 hours.
Zone 1 / 21: Explosive mixture for more than 10, but less than 1000 h/yr. Category 1 or 2 equipment.
Zone 2 / 22: Explosive mixture for less than 10h/yr, but still sufficiently likely. Category 1, 2 or 3 equipment. Require controls over ignition sources.
Categorization into the three zones is based upon release calculations according to EN 60079-10-1 [11] and EN 6007910-2 [12].

Codes and Standards Pertaining to Dust Explosion Hazards • NFPA 68. Standard on Explosion Protection by Deflagration Venting, 2013 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2013 [13].
• NFPA 654. Standard for the Prevention of Fire and Dust Explosions from the Manufacturing, Processing, and Handling of Combustible Particulate Solids, 2017 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2017 [14].
• NFPA 61. Standard for the Prevention of Fires and Dust Explosions in Agricultural and Food Processing Facilities, 2017 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2017 [15].
• NFPA 484. Standard for Combustible Metals, 2015 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2015. [16].
• NFPA 664. Standard for the Prevention of Fires and Explosions in Wood Processing and Woodworking Facilities, 2017 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2017 [17].
• NFPA 655. Standard for Prevention of Sulfur Fires and Explosions, 2017 Edition. National Fire Protection Association, Quincy, MA, United States of America, 2017 [18].
• EN 13237. Potentially explosive atmospheres. Terms and definitions for equipment and protective systems intended for use in potentially explosive atmospheres, 2012 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2012 [19] • EN 1437. Explosion suppression systems, 2005 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2005 [20] • EN 14491. Dust explosion venting protective systems, 2012 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2012 [21] • EN 14797. Explosion venting devices, 2006 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2006 [22] 14

• EN 13673-1. Determination of the maximum explosion pressure and the maximum rate of pressure rise of gases and vapours - Part 1: Determination of the maximum explosion pressure, 2003 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2003. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection [23] • EN 13673-2. Determination of the maximum explosion pressure and the maximum rate of pressure rise of gases and vapours - Part 2: Determination of the maximum rate of explosion pressure rise, 2005 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2005. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection [24] • EN 14034-1. Determination of explosion characteristics of dust clouds - Part 1: Determination of the maximum explosion pressure pmax of dust clouds, 2004 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2004. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection [25] • EN 14034-1. Determination of explosion characteristics of dust clouds - Part 1: Determination of the maximum explosion pressure pmax of dust clouds, 2011 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2011. European Standard, Prepared by Committee EXL/23 Explosion and fire precautions in industrial and chemical plant. Amendment July 2011. Amends and replaces EN 14034-1:2004. [26] • EN 14034-2. Determination of explosion characteristics of dust clouds - Part 2: Determination of the maximum explosion rate of explosion pressure rise (dp/dt)max of dust clouds. 2006 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2006. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection [27] • EN 14034-3. Determination of explosion characteristics of dust clouds - Part 3: Determination of the lower explosion limit LEL of dust clouds, 2006 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2006. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection [28] • EN 14034-4. Determination of explosion characteristics of dust clouds - Part 4: Determination of the limiting 15

oxygen concentration LOC of dust clouds, 2004 Edition. European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2004. European Standard, Prepared by Technical Committee CEN/TC 305 Potentially explosive atmospheres - Explosion prevention and protection. [29] • EN 60079-10-2. Explosive atmospheres. Classification of areas. Explosive dust atmospheres, 2015 Edition.
European Committee for Standardization (CEN), Brussels, Belgium, Europe, 2015 [12] • ISO 6184-1. Explosion Protection Systems - Part 1: Determination of explosion indices of combustible dusts in air. The International Organization for Standardization (ISO), 1985. [30]

16

Regulations, Codes and Standards
ATEX Methodology Hazardous Area Classification:
Determine whether hazardous places are present and, if so, where they are located.
      Likelihood of Likelihood Likelihood of ×  = explosive of ignition source atmosphere explosion

(1)

Combustible gas and dust clouds in the explosive region are categorized into 3 zones, based upon the release (as per EN 60079-10-2 [12]):
Zone 0 or 20: A continuous release inside a containment enclosure gives rise to Zone 0 (gas) or 20 (dust). A place in which an explosive mixture is continuously present (or for long periods, or, frequently for short periods).
E.g. a mill or pneumatic conveying system.
Zone 1 or 21: A primary instantaneous release gives rise to Zone 1 or 21. A place in which an explosive atmosphere is likely to occur occasionally under normal operating conditions. E.g., bagging points and inspection ports that are frequently opened, ventilation systems for flammable gases.
Zone 2 or 22: A secondary instantaneous release gives rise to Zone 2 or 22. A place where an explosive mixture is unlikely to occur under normal operation but, if it does occur, will persist for a short period only. E.g., leaks from incorrectly fitted lids, spillages.

17

Transition of the energy mix. From [31].
18

Applications and accident scenarios

Phenomena, hazards, and risks
Standards,
Development and Safety assessment Hydrogen release, Thermal and pressure guidelines, and validation of and missile effects from mixing and and risk analysis legal requirements fires and explosions mitigation techniques distribution

V4

V5

H2

H2

H2

H2

H2

H2

H2

H2

H2

H2

H2

H2

Tunnels, parking
and garages

H2

H2

H2

H2

H2

Utilisation, portable
and stationary
applications

H2

H2

H2

H2

H2

V1

V2

Production

H2

H2

Transport and
distribution,
refueling stations

H2

Hydrogen storage
(LH2,CGH2)

H1
H2
H3
H4
H5

V3

The HySafe Activity Matrix. From [32].

19

The Risk Analysis Fishbone Diagram.
20

Coverage: 200000 km/yr

Diesel: 35 kg/100 km → 1 refueling/year

LNG: 25 kg/100 km → 4 refuelings/year

Hydrogen calamity (transportation)

Hydrogen: 10 kg/100 km → 35 refuelings/year

Impact energy transition on refueling operations. From [33].

21

losie en brand bij een Noors waterstoftankstation:
dracht en leerstof swaterstaat

swaterstaat

GASEOUS HYDROGEN STORAGE – EXPLOSION HAZARDS

UARI 2020
EMBER 2019

Kjorbo, Norway
Monday 10 June 2019 at 17:30 h
Hydrogen Explosion (3.0 – 3.5 kg H2 ) Glass damage at 60 – 70 m distance LESSEN KJØRBO-INCIDENT OfficeVANatHET60 m distance Houses at > 225 m distance

Figuur 15: Impressie schade intern.

De bovenstaande foto toont de eindtoestand. In het volgende hoofdstuk gaan we in op de toedracht en op de processen direct na de lekkage tot aan de eindtoestand.

3.2

22 door de brandweer
Repressie

Door de commandant, de heer Sverre Junker van de lokale brandweer, is een uitgebreide terugkoppeling gegeven tijdens de persconferentie op 28 juni 2019 in Oslo [zie referentielijst]. Wij beschikken over de videoopname, maar de spreker heeft zijn verhaal in het Noors verteld. Uit de sheets van hem zijn diverse zaken

DELFT UNIVERSITY OF TECHNOLOGY – GREEN VILLAGE Explosion involving 3.5 kg hydrogen Blast calculation according to Ref. [34].
70

>7
6
5

35

4.5

70

52.5

35

17.5

-17.5

-35

-52.5

-70

17.5

Distance [m]

-17.5

Overpressure [kPa]

Distance [m]

52.5

4
3.5
3
2.5
2

-35

1.5
-52.5

<1

Deflagration

-70
70

> 250
200
150

35

100

Distance [m]

-17.5

70

52.5

35

17.5

-17.5

-35

-52.5

-70

17.5

Overpressure [kPa]

Distance [m]

52.5

70
50
40
30

-35

20
15

-52.5

< 10

Detonation

-70

23

PRESSURE EFFECTS ON PEOPLE, STRUCTURES AND EQUIPMENT

Entry
1

Overpressure
(kPa)
190

2

69–76

3

70

4
5

60
55

6
7
8
9
10
11

50–55
50
40–45
35–40
20–40
35

12
13
14
15

30–35
30
20–28
24

16
17

15–20
20

18

18

Damage/injury
Occupants of unstrengthened building will be killed by direct action of blast, by being struck by building debris, or by impact against hard surfaces [35].
Occupants of unstrengthened building are expected to suffer severe injuries or death from direct blast, building collapse, or translation [35].
Probable total destruction of buildings; heavy machine tools moved and badly damaged [36].
Loaded train boxcars completely demolished [36].
Occupants of unstrengthened building will be killed or incur serious injuries by blast to eardrums and lungs, by being blown down or struck by fragments and building debris [35].
Unreinforced brick panels, 25–35 cm thick, fail by shearing or flexure [36].
Loaded tank cars/train wagons overturned; brick walls, 20–30 cm thick, collapse [36].
Collapse of pipe bridge [36].
Displacement of pipe bridge, failure of piping [36].
Large trees fall down [36].
Breakage of wooden telephone poles; most buildings destroyed, except for concrete reinforced shear wall buildings; platting of cars and trucks pressed inwards [36].
Near-complete destruction of houses [36].
Cladding of light industrial buildings ruptured. Panelling torn-off [36].
Frameless, self-framing steel panel building demolished; rupture of oil storage tanks [36].
Personnel injuries of a serious nature or possible death are likely from fragments, debris, firebrands or other objects. There is a 10% chance of eardrum rupture [35].
Unreinforced concrete or cinderblock walls shattered [36].
Heavy machines in industrial buildings suffer little damage; steel frame building distorted and pulled away from foundations [36].
Lower limit of serious structural damage; 50% destruction of brickwork of houses [36].

24

Table 1: Continued.

19

16

20
21

15
7–15

22
23

10
5.9–8.3

24
25
26
27
28
29

8
7
3.5–7
5
3
2

30
31
32
33
34

1
0.7
0.3
0.2
0.15

Personnel may suffer temporary hearing loss or injury from secondary blast effects such as building debris and the tertiary effect of displacement. No fatalities or serious injuries from direct blast are expected [35].
Partial collapse of walls and roofs of houses [36].
Corrugated asbestos shattered. Corrugated steel or aluminum panels fastenings fail, followed by buckling; wood panel fastenings fail, panels blown in [36].
Steel frame of clad buildings slightly distorted [36].
Personnel are provided a high degree of protection from death or serious injury with injuries that do occur being principally caused by glass breakage and building debris [1].
Partial demolition of houses, made uninhabitable [36].
Collapse of roof of a tank [36].
Windows usually shattered; occasional damage to window frames [36].
Minor damage to house structures [36].
Limited minor structural damage [36].
Probability of 0.95 of no serious damage beyond this value; some damage to house ceilings;
50% of window glass broken [36].
Typical threshold for glass breakage [36].
Breakage of small windows under strain [36].
Loud noise similar to sonic boom; occasional glass failure [36].
No structural damage; occasional breaking of large window panes already under strain [36].
Annoying noise [36].

25

Confined deflagrations: explosion severity parameters

26

Confined deflagrations: explosion severity parameters

10
8
pressure (bar)

Pmax

6


4

dP
dt



max

2
0
0

50

100

150
time (milliseconds)

200

250

Experimental explosion pressure curve of a 500 g m−3 cornstarch-air mixture [37].

27

300

explosion, the accompanying rate of pressure rise also decreases progressively. Although the role of buoyancy is negligible in the 169 ml vessel, there is still the effect of a progressively decreasing flame surface area in the final stage of the explosion. Initially, the flame ball grows with a progressively increasing flame surface area, until it reaches the wall of the vessel. From that moment onwards, the flame

explosion has completed. The cause of this phenomenon is described by Garforth and Rallis (1976) and Lewis and von Elbe (1961), Chapter 15, and will be discussed in Section 3.
To enable a comparison with results presented by other researchers, the maximum explosion pressure, Pmax, and the maximum rate of pressure rise, (dP/dt)max, were determined as illustrated by the upper part of Fig. 3. Since the experimental

Confined deflagrations: explosion severity parameters 470

A.E. Dahoe, L.P.H. de Goey / Journal of Loss Prevention in the Process Industries 16 (2003) 457–478

pressure (bar)

8

φ=1.0

1 bar

6
4
φ=0.5

2
0

pressure (bar)

8

φ=1.25

1 bar

6
4
φ=3.0
2
0
0

4

8

12

16

20

time (ms)

Fig. 3. An illustration of the determination of the maximum explosion pressure, Pmax, and the maximum rate of pressure rise, (dP/dt), from the measured

Hydrogen-air

Fig. 6. Measured explosion pressure curves of fuel lean to stoichiometric methane–air mixtures, the maximum explosion pressure, the maximum explosion curve (top), theoretical values of the maximum explosion pressure (middle), and the behavior of the maximum explosion pressure and the maximum rate of pressure rise, and KG values (× our results, 䊏 Cashdollar & Hertzberg, 1985).

Methane-air

and Hertzberg for off-stoichiometric mixtures, but differences may be observed with near-stoichiometric mixtures. With stoichiometric mixtures, discrepancies of 6% and 13% are observed in Pmax and (dP/dt)max. Bartknecht (1981) measured the explosion behavior of stoichiometric methane–air mixtures in a 20-l explosion sphere and found a Pmax of 8.4 bar and a KG value5 of 55 bar

rate of pressure rise as a function of the equivalence ratio (bottom).

m s⫺1. Our measurements show that for stoichiometric methane–air mixtures, Pmax is 8.7 bar and KG is 80 bar m s⫺1.
The occurrence of an inflection point in all our press28 ure–time curves is attributed to the effect of buoyancy.
Due to the density difference between the hot combustion products and the cold unburnt mixture, a flame ball

Confined deflagrations: explosion severity parameters Scaling an explosion: a matter of invariance across size?



1
V3
KSt = dP
dt max

laboratory
test vessel

Pmax



dP
dt

industrial
apparatus

cube-root-law


Pmax ?

max

n0 Reactants −→ np Products :

np Tf
Pmax =
P0
n0 T0

29

and



dP
dt

KSt =





max

dP
dt

?



max

V 1/3

Deflagration parameters (hydrogen-air, laminar)

Comparison between Pmax - and KG -values: (◦) 169-ml cylindrical vessel [7], (×) 120-litre explosion g. 3. An illustration of the determination of the maximum explosion pressure, Pmax, and the maximum rate of pressure rise, (dP/dt), from the measu sphere [5].
plosion curve (top), theoretical values of the maximum explosion pressure (middle), and the behavior of the maximum explosion pressure and the maxim e of pressure rise as a function of the equivalence ratio (bottom).
30

Confined deflagrations: explosion severity parameters Effect of compression on laminar burning velocity

Sf

Su

Se

Sf
rf

rf

unburnt
mixture

rr

SuL

moving
observer

unburnt
mixture

laminar
flame

burnt mixture

unburnt mixture

velocity
profile

burnt
mixture

burnt
mixture

fixed
observer

DUST EXPLO
A Study of Flame

=
D
a

10

well stirred reactor
Re > 1, Da < 1, Ka > 1

1

Sf = Se + Su

31
uL

S f = flow velocity + S u

distributed reaction
zones Re > 1,

Laminar burning velocity: effect of pressure and temperature Methane-air [38]

burning velocity (cm/s)

5
50
45
40
35
30
25
20
15
10
5
0

160

vol.% methane
9
11

7
curve 1

13

A.E. Dahoe / Journal of Loss Prevention in the Process Industries 18 (2005) 152–166 Hydrogen-air [7]

15

curve 4

⊠ ⊠

◦ ⊠∗◦
⊠
N ∗♦
◦N⊠××
♦◦
⋆N ∗
∗ ×∗
♦ ⊠∗
◦N×∗∗♦⋆ ⊞▽ ▽▽ ⊞× ▽ ×N ▽◦ ∗ curve 2 × ⊞ ⊚⊗ ▽ • ⊕ ⊕ ⊞ ⊕ ⊟ ⋆▽ ⊠∗  ♦ ⊚ ⊕  ⊕ ⊗ • N⋆ ⊕∗♦ ⊕ ⊟ ⊞ ⊞ ⊟ ▽ △ ▽  ◦ ⊠ • △ ⊕×⊗ ⊞ ⊚ ⊕ ⊕ ×⊚ ∗  curve 5 ∗ • • ∗ ♦ ▽ ⊞ ▽∗ ⊗ ⊟ • × N⊠ ⊕ × N  ◦ ⊟ • ⋆× ⊞ ⊗ •  ⋆△ curve 3 ⊠ • • ⊕ ∗ ⊚ ⊞ ∗ ⊟ N × ⊞△ ⊠ •• ♦⊕ ▽◦ • ⊗ N • ∗ ⊠ •⊗∗⊞ ⊕ ♦ ⊞ N ⋆ ∗•  ⊗ • ⊕ ⊚ ⋆▽ •⊕△ ♦ ⊠ • ⊞ ∗ ∗ • ⊚ •∗ ⊞ N♦ ∗ ∗ N⊕ ▽∗ ⊠ • ⊚ △⊞ ⊗ •⊚ ⊕ ⊗ ⊞

⊚
⊞

• ⊕

⊚
⊗

⊚⊗ ⊕

⊚

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 equivalence ratio, φ

burning velocity (cm/s)

100



×
×
×


×
×

model 1
×
×
×

N N
⊕ ⊞
⊕
⊞
×
⊕
♦ ♦ 
×
N
♦♦
♦
♦♦⊕
♦
♦
N
♦
H ♦♦ ♦  ⊞ ♦ ♦ ♦♦ H  ▽ ♦⊠ H △ 

H

◦

◦•

×
×


⊞
⊠ ▽

◦

⊞

curve 1

▽


◦


▽
⊠

10

▽ ▽

•◦

▽

◦△•


⊠



⊠

model 2


⊠



•

△

1
0.1

1

10

100

pressure (bar)

•

burning velocity (cm/s)

250

⊞

model 1

200

⊞

•

150
•

100
50
△

0
100

△

200

•

×
×
▽
△
♦⊞×


◦

300

◦

♦
△▽

×

◦×

⊞×△ ♦
▽

⊞

△

◦ ◦♦△

⊞

▽△ 

400
500
temperature (K)

model 2

600

700

800

Fig. 5. Effect of equivalence ratio (upper part), pressure (middle-left part), and temperature (lower-left) on the laminar burning velocity of hydrogen–air mixtures (!: Takahashi, Mizomoto, & Ikai, 1983; ,: Dowdy et al., 1990; %: Koroll, Kumar, & Bowles, 1993; $: Vagelopoulos, Egolfopoulos, & Law, 1995; †: Iijima & Takeno, 1986; ?: Wu & Law, 1984; 6: Egolfopoulos & Law, 1990; B: Lamoureux, Djebaili-Chaumeix, & Paillard, 2002, stretched; ::
Lamoureux et al., 2002, unstretched; ;: Kwon & Faeth, 2001; 0: Law, 1993; @: Aung et al., 1997; 4: Tse et al., 2000; solid line: Marinov et al., 1996).

burning velocity of hydrogen–air mixtures (see the middle 32 and lower part of Fig. 5). Because of adiabatic compression of the unburnt mixture, the pressure and the temperature in Eq. (20) do not behave independently, but change simultaneously. This simultaneous change in pressure and

Eq. (23) was incorporated into Eq. (19) and the laminar burning velocity was determined by fitting the latter by means of the Levenberg–Marquardt method (Marquardt, 1963; Press et al., 1992) to the initial part of the pressure– time curves shown in Fig. 2 with SouL as the only degree of

Governing equations of a compressible reacting flow Governing equations:

Assumptions:
∂ρ
+ ∇ · (ρv) = 0
∂t

(2)

N
X
∂ (ρv)
+ ∇ · (ρvv) = −∇p + ∇ · τ +
ρYi g
∂t i=1 ∂ (ρYi ) + ∇ · (ρvYi ) = −∇ · [ρYi V i ] + ẇi ∂t ∂ (ρh) ∂p + ∇ · (ρvh) = + v · ∇p + τ ..∇v − ∇ · [λ∇T ] + ∇ · q ∂t ∂t " N #  N X N  N X X X Xj α i +∇ · ρYi hi V i + RT (V i − V j ) + ρYi g · V i M D i ij i=1 i=1 j=1 i=1   T Z N γ−1X  ◦ p = Yi hf i + CPi (T ) dT  ρ γ i=1

(3)
(4)

(5)

(6)

and q = ϵσT

(7)

4

(8)

33

• Non-uniform pressure field
• Viscous dissipation in energy equation included • Neglect effect of body forces in momentum and energy equation

T◦

i
h


†
where τ = µ ∇v + (∇v) + κ − 32 µ [∇ · v] I

• Steady state

• Soret and Dufour effect in species and energy equation included

4
O + H2 ⇋ OH + H
1.80 · 10
1.0
8826.
5
H + O2 + M ⇋ HO2 + Ma 2.10 · 1018 -1.0 0.
19 6 H + O2 + O2 ⇋ HO2 + O2 6.70 · 10 -1.42 0.
7 H + O2 + N2 ⇋ HO2 + N2 6.70 · 1019 -1.42 0.
8 OH + HO2 ⇋ H2 O + O2 5.00 · 1013 0.0 1000.
9 H + HO2 ⇋ 2OH 2.50 · 1014 0.0 1900.
13 10 O + HO2 ⇋ O2 + OH 4.80 · 10 0.0 1000.
11 2OH ⇋ O + H2 O 6.00 · 108 1.3 0.
12 H2 + M ⇋ H + H + Mb 2.23 · 1012 0.5 92600.
13 O2 + M ⇋ O + O + M 1.85 · 1011 0.5 95560.
c 23 14 H + OH + M ⇋ H2 O + M 7.50 · 10 -2.6 0.
15 H + HO2 ⇋ H2 + O2 2.50 · 1013 0.0 700.
16 HO2 + HO2 ⇋ H2 O2 + O2 2.00 · 1012 0.0 0.
17 H2 O2 + M ⇋ OH + OH + M 1.30 · 1017 0.0 45500.
12 18 H2 O2 + H ⇋ HO2 + H2 1.60 · 10 0.0 3800.
19 H2 O2 + OH ⇋ H2 O + HO2 1.00 · 1013 0.0 1800.
a Third body efficiencies: k5 (H2 O) = 21 k5 (Ar), k5 (H2 ) = 3.3 k5 (Ar).
b Third body efficiencies: k12 (H2 O) = 6 k12 (Ar), k12 (H) = 2 k12 (Ar), k12 (H2 ) = 3 k12 (Ar). c Third body efficiency: k14 (H2 O) = 20 k14 (Ar).

Governing equations of a compressible reacting flow: laminar flame solution



N

N

H

H

H

H

•

H

H

H

H

















∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

◦

◦

◦

YH2 O /10
◦ YN /100
2

∗

∗

YOH /10
∗ YH O /10−3
2 2

N N N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • N • • • • • • N • • • • • • N • • • • • • N • • • • • N • • • • • N • • • • • N • • • • • N • • • • • N • • • • N • • N N N N •••••••••••••••••• N N N N N N ••••••••••••••••••••• N N N N N ••••••••••••• ◦N ◦N ◦N ◦◦◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦N ◦◦N ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦N NN •••••  • N •  N •• N N•• N••  •  N •  N •  N  •  N •   N •     N •  ∗∗N   ∗•  N H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H • H  H H ∗ H  N H ∗• 

Flame structure

Velocity [m s−1 ]
∂T /∂x [K m−1 ]







YH2 /10−1
YO2 /100
∗ YHO /10−3
2

H



 












                                                                                                                                            

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

Solution of the governing equations for the laminar flame structure. From [40].

2

2

YHO2 /10

2400
2100
1800
1500
34
1200
900



16.0
2400
N N N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N Flame structure N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N 14.0 2100 N N N •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • N N N N N N N N N N ••••••••••••••••••••••••••••••••••••••••••••••••••••••• N • • • • N • • • • • N • • • • N • • • • N • • • N • • • • N • • • • N N N •••••••• N N N N N ••••••••••••••• 12.0 ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦N◦N◦N◦N◦•N◦•N◦•N◦•N◦•N◦•N◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦•◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ 1800 N••• N • N N•• 10.0 1500 • N • N • N • 8.0 1200 N • N • N • 6.0 900 ∗∗N ∗• N H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H • H H H ∗N H H ∗•• H N 4.0 600 ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗N ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ H ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ H ∗ H ∗ ∗ ∗ •H ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ H ∗ ∗H H∗∗H ∗∗∗∗∗∗∗∗∗∗ ∗H •∗ N ∗N HH H N H ••∗ H∗ H 2.0 N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N•N∗∗∗∗∗∗∗∗HH∗∗HH∗H∗H∗∗∗∗∗∗∗∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ 300 H H H ∗ H H H H H H H H H ∗H H HH H H H H H H H H HH H H H ∗H H H H H H H H H H H H ∗H H H H H H H H H H H H H H ∗H H H H ∗H H H H H H H H ∗∗∗∗∗H H H H H H H H H H H H H H H ∗H H H H H H H H H H H H H H H H H H H H H H H H H ∗HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH H H H H H H H H H H H H H H H H 0.0 H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗∗H∗H∗HH∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗H∗ H∗ H∗ 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Distance [10−3 m] 2400 6.0·106 Flame thickness • Temperature N 2100 N ∂T /∂x, 2nd -order finite NNN difference • • • • • • • • • • • 5.0·106 • • • • • • • N N • • • • • rd • 3 1800 filtered, -degree polynomial N • • • • N • • • • QR decomposition using Householder N 4.0·106 • 1500 N • N• transformations, 5 pts (2 left,2 right) • N 1200 • N 3.0·106 ad • T −T 0 N δL = (∂Tf/∂x)max = 402.9 µm • 900 N 6 • N 2.0·10 • N N •• 600 • N N 1.0·106 • • • • • • • • • •••N••••••• N N 300 N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N 0 0.0·106 N N N N N N N N N N 2.5 3.0 3.5 4.0 4.5 5.0 Distance [10−3 m]

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3

Scaled species mass fra
Y ,YH ,YO ,YO2 , YOH
YHO2 , YH2 O2 ,YN

•

H2

elocity [m s−1 ]

N

•

Temperature [K

16.0
14.0
12.0
10.0
8.0
6.0

N

•



Temperature [K]

P0 =1.0 bar, T0 =298.15 K, φ=1.0, H2 :O2 :N2 = 2.0:1.0:3.76 Detailed kinetic scheme forTemperature hydrogen-air combusVelocity YH /10−1 −2 −2 Y /10 Y /10 YO /100 tion. FromH [39].
O −1 0 −3



Scaled species mass fractions [-]
YH2 ,YH ,YO ,YO2 , YOH ,YH2 O , YHO2 , YH2 O2 ,YN2

P0 =1.0 bar, T0 =298.15 K, φ=1.0, H2 :O2 :N2 = 2.0:1.0:3.76 N N N N Temperature H H H • • • • Velocity H H H H Y /10−2 YO /10−2 H ∗ ∗ ∗ ∗ YOH /10−1 ∗ ∗ ∗ ∗ YH O /100 ∗ ∗ ∗ 2 ◦ ◦ ◦ ◦ YN /100 ∗ ∗ ∗ ∗ YH O /10−3 2 2 2

Temperature [K]

The reaction rate coefficients are in the form kf = AT β exp (−Ea /RT ).
Units are in moles, cubic centimetres, seconds, Kelvins and calories.
Reaction A β Ea 1 H2 + O2 ⇋ 2OH 1.70 · 1013 0.0 47780.
2 OH + H2 ⇋ H2 O + H 1.17 · 109 1.3 3626.
3 H + O2 ⇋ OH + O 5.13 · 1016 -0.816 16507.
4 O + H2 ⇋ OH + H 1.80 · 1010 1.0 8826.
5 H + O2 + M ⇋ HO2 + Ma 2.10 · 1018 -1.0 0.
6 H + O2 + O2 ⇋ HO2 + O2 6.70 · 1019 -1.42 0.
7 H + O2 + N2 ⇋ HO2 + N2 6.70 · 1019 -1.42 0.
8 OH + HO2 ⇋ H2 O + O2 5.00 · 1013 0.0 1000.
14 9 H + HO2 ⇋ 2OH 2.50 · 10 0.0 1900.
10 O + HO2 ⇋ O2 + OH 4.80 · 1013 0.0 1000.
11 2OH ⇋ O + H2 O 6.00 · 108 1.3 0.
b 12 H2 + M ⇋ H + H + M 2.23 · 1012 0.5 92600.
13 O2 + M ⇋ O + O + M 1.85 · 1011 0.5 95560.
c 14 H + OH + M ⇋ H2 O + M 7.50 · 1023 -2.6 0.
15 H + HO2 ⇋ H2 + O2 2.50 · 1013 0.0 700.
16 HO2 + HO2 ⇋ H2 O2 + O2 2.00 · 1012 0.0 0.
17 H2 O2 + M ⇋ OH + OH + M 1.30 · 1017 0.0 45500.
18 H2 O2 + H ⇋ HO2 + H2 1.60 · 1012 0.0 3800.
19 H2 O2 + OH ⇋ H2 O + HO2 1.00 · 1013 0.0 1800.
a Third body efficiencies: k5 (H2 O) = 21 k5 (Ar), k5 (H2 ) = 3.3 k5 (Ar).
b Third body efficiencies: k12 (H2 O) = 6 k12 (Ar), k12 (H) = 2 k12 (Ar), k12 (H2 ) = 3 k12 (Ar). c Third body efficiency: k14 (H2 O) = 20 k14 (Ar).

Confined deflagrations: a simple model
Objective:
Find a dynamic equation for the pressure evolution

unburnt
mixture

Approximate expression relating the unburnt mixture mass fraction to the fractional pressure rise [38, 41, 42]:
(9)

dP
Pmax − P0 dmu
=−
dt
mu0
dt

(10)

rf

mu
Pmax − P
=
mu0
Pmax − P0

Differentiation with respect to time:

Se

Sf

burnt
mixture

35

A Stu

A relationship can be established between the rate of pressure rise and the burning velocity. By substitution of equation (11) into equation (10):
10

dt

mu0

f

=

1

well stirred reactor
dP
P
− P0 2
Re > 1, Da <=1,4π Ka >max 1 r ρu S u D a

(11)

DUS

,
vrms
/Su L

The combustion wave propagates with a velocity equalling the sum of the expansion velocity, Se , the conversion velocity, Sn , and the burning velocity Su . Since the unburnt mixture immediately ahead of the flame front moves with velocity Se +Sn , the velocity at which the unburnt mixture enters the combustion wave is minus the burning velocity.
Therefore, the mass consumption rate of the unburnt mixture can be expressed as dmu = −4πrf2 ρu Su dt

Su

distributed reaction
zones Re > 1,
Da > 1, Ka > 1

(12)

Confined deflagrations: a simple model
The next step is to express the density of the unburnt mixture, ρu , and the location of the flame front, rf , in terms of known variables. For adiabatic compression of the unburnt mixture, P ρ−γ = constant and hence   γ1 ρu0 P0 (13) = ρu P Furthermore, Vb = Vvessel − Vu , which can be rewritten as

4 3
mu RTu
πrf = Vvessel −
3
P

(14)

Since ρ−1 = R̂T /P where R̂ denotes the specific gas constant in J kg−1 K−1 , the volume of the unburnt mixture can be expressed as   Pmax − P −1 ρu0 Pmax − P mu R̂Tu = Vvessel ρu0 ρu = Vvessel (15) P Pmax − P0 ρu Pmax − P0   γ1 P0 Pmax − P = Vvessel P Pmax − P0 and equation (14) yields the following expression for the location of the flame front:
"

rf = Rvessel 1 −



P0
P

36

 γ1

Pmax − P
Pmax − P0

# 13

(16)

Confined deflagrations: a simple model
By inserting equations (16) and (13) into equation (12) and by noting that mu0 = ρu 0 Vvessel , the following ordinary differential equation is obtained for the rate of pressure rise:
"

3 (Pmax − P0 )
dP
=
1−
dt
Rvessel



P0
P

 γ1

Pmax − P
Pmax − P0

# 23 

P
P0

 γ1

Su

(17)

From equation (17) it can be seen that (dP/dt) increases monotonically with P and hence the maximum rate of pressure rise is attained when P = Pmax . By substituting P = Pmax into equation (17) and multiplying both sides by the cube root of the vessel volume, the following expression is found for the KG/St -value:


KG/St = dP
dt



V

1/3

1/3

= (36π)



max
(Pmax − P0 ) PP
0

 γ1

Su

which is a normalization of the maximum rate of pressure rise with respect to the vessel volume.

37

(18)

Burning velocity: determination by solving inverse problem

Determination of the burning velocity by solving the inverse problem for the pressure-time curve of a 500 g m−3 cornstarch-air mixture measured in the strengthened 20-litre explosion sphere.
From [42].
    Su = P α T β Su0 P0 T0

38

Vented deflagrations: pressure relief venting

39

A model for vented deflagrations
Bradley D. and Mitcheson A. The venting of gaseous explosions in spherical vessels. I - Theory. Combustion and Flame, 32:221-236, 1978 [43].
Bradley D. and Mitcheson A. The venting of gaseous explosions in spherical vessels. II - Theory and experiment.
Combustion and Flame, 32:237-255, 1978 [44].
Model developed for centrally ignited fuel-air mixtures in a spherical vessel with vent flow expressions for the rate of change in the mass of unburnt mixture:
 # "    1+γ 1/2   γP γ + 1 1−γ 2   −4πrf ρu Su − Cd Av ρ ρ 2  dmu    = " # 1+γ 1/2  γ−1 1−γ    2   γ  dt  2γP ρ Pa γ  Pa 2  −4πr ρ S − C A 1 −  u u d v f  P    γ − 1 P  

if

if


 γ
γ−1
Pa ≤
2
γ+1
P


Pa >
2
γ+1
P

 γ

γ−1

(sonic)
(19)
(sub-sonic)

where ρ is the density of the vented mixture, Pa the ambient pressure, Cd a discharge coefficient and Av the vent area. Substitution into (10), and repeating steps (11) to (17) results in  # " #2 "  1+γ 1/2 3  1 1        3 (Pmax − P0 ) P γ S + C A ρ γP γ + 1 1−γ  0 γ Pmax − P  1− P u d v  ρ 2 R P P − P P  vessel max 0 0  dP   = " #2 " # 1+γ 1/2  γ−1  3  dt    γ1 1  2   γ 1−γ    3 (Pmax − P0 ) 2γP ρ Pa γ Pmax − P P0 Pa P γ S +C A  1 − 1 −  u d v  γ − 1 R P P − P P P P  vessel max 0 0     

40

(20)

A model for vented deflagrations
Assume that the density of the vented mixture, ρ, is equal to that of the burnt mixture, ρb . With this assumption, the density of the vented mixture, ρ, in equation (20) can be computed from Tu Tu0 ρ ≡ ρb = ρu = ρu0 Tf Tf

28



P
P0

 γ−1

γ

1
P γ
(21)
P0BRADLEY and A. MITCHESON
D.

)0
A.0 004

"~

p,.zo~ . e' ,,,,,' / / ~ o

o~

The figure shows the solution of equations
(20) and (21) for two area ratios (vent area divided by total surface area of enclosure) and four different vent opening pressures.

- 2 0

!

~

From: Bradley D. and Mitcheson A. The
venting of gaseous explosions in spherical vessels. I - Theory. Combustion and Flame, 32:221-236, 1978 [43].

OQIli

i~, * 1 ~Iki* | I 0 ' ~

m

I0

I
Vli~

ires)

I:ig. 3. C o m p u t e r p r e s s u r e * t i m e solution,~ for t~t) at'ca ratio~ at four d i f f e r e n t vent pressures.

is and cause the pressure to increase, thereby oducing a second peak. With much smaller values

41

curves which are similar to those recorded experimentally in Refs. 18 and 24.

Explosion severity
Pressure relief venting (VDI methodology)

From [45, 46].

42

Explosion severity
Pressure relief venting (NFPA methodology)

The NFPA 68 standard [13] proposes the following expression for the required venting area to protect an enclosure against a dust deflagration:
r   Pmax 4/3 Av = 10−4 1 + 1.54 Pstat × KSt × V 3/4 × −1 Pred

43

Explosion severity
Pressure relief venting (Other methodologies)

Tamanini F. Scaling parameters for vented gas and dust explosions. Journal of Loss Prevention in the Process Industries, 14:455–461, 2001. [47] r γ + 1 Av Pmax − P0 P − P0 and Γ = C c Define Π = d u vention in the Process Industries 14 (2001) 455–461 Pmax − P0 2 V 2/3 KG

venrate
vent
prormal
ench
nges
nifions,
alue
ized Eq.
tion

Fig. 3. Normalized reduced explosion pressure variation as a function of the vent parameter, ⌫, for a constant value of 44the fraction, iK.

Explosion severity
Peculiarities of vented deflagrations Flame distortion by vent ducts

Flow pattern inside a vent duct [48–50].

45

Explosion severity
Peculiarities of vented deflagrations Flame distortion by vent ducts

Flow patterns in vessel before and after combustion in the duct, effect of flame front distortion; a) narrow ducts, b) wide ducts. After Ponizy & Leyer (1999) [48, 49].

46

Explosion severity
Peculiarities of vented deflagrations Flame distortion by vent ducts

Experimental setup used by Ponizy & Leyer (1999) [48, 49]. IGN, ignition (electrically heated wire); VP, vacuum pump; MI, mixture inlet; V, valves; PM, photomultipliers; PC, P1, P2, . . . Pn, pressure gauges; IIGN, I0, I1, . .
. Im, ionization gauges. The tube diameter is 53 mm.

47

Explosion severity
Peculiarities of vented deflagrations Flame distortion by vent ducts

Deflagration to detonation transition (detonation onset at 1.7 m from duct entrance) [48, 49].

48

Proceedings of the 5th International Seminar on Fire and Explosion Hazards, Edinburgh, UK, 23-27 April 2007

RESULTS AND
DISCUSSION
Explosion
severity
Figure 2 shows an example of the simulation results as a sequence of 3D representations of the flame Peculiarities of vented deflagrations

surface. Data was extracted from the B0X simulation in two forms: pressure traces and rate of heat release, dH+/dt, traces in time. The pressure traces were generated for a number of virtual pressure transducers evenly spaced within the enclosure. The pressure within the enclosure was averaged over all the transducers, eliminating high frequency local pressure oscillations.

• Enclosure geometry/aspect ratio
• Ignition location

• Varying pressure and temperature
• Varying turbulence

• Changes in flame shape

• Flame stretch and flame curvature
• Darrieu-Landau instability

• Thermo-diffusive instability
• Raleigh-Taylor instability

• Richtmeyer-Meshkov instability

• Flame distortion at outflow boundaries
• Vent cover inertia

From:
[51].
Figure
2 3D Representations of the flame propagation for a trial with center ignition, AR = 8, Av = 5.76 m2, ST = 14.8 m/s shown at 4 times (0.03, 0.07, 0.10, and 0.14 s) in the development of the explosion.
2

AR = 4, Av = 1.44m , ST = 8m/s
3000

49

8
AR = 4.0, dH+/dt
AR = 4.0, Pressure

Laminar burning velocity/flame thickness: effect of pressure/temperature Objective:
Find an expression for the pressure and temperature dependence of the laminar burning velocity and the laminar flame thickness Governing equations:

Assumptions:
∂ρ
+ ∇ · (ρv) = 0
∂t

(22)

N
X
∂ (ρv)
+ ∇ · (ρvv) = −∇p + ∇ · τ +
ρYi g
∂t i=1 ∂ (ρYi ) + ∇ · (ρvYi ) = −∇ · [ρYi V i ] + ẇi ∂t ∂ (ρh) ∂p + ∇ · (ρvh) = + v · ∇p + τ ..∇v − ∇ · [λ∇T ] + ∇ · q ∂t ∂t # " N  N N X N  X X X Xj α i ρYi g · V i +∇ · (V i − V j ) + ρYi hi V i + RT Mi Dij i=1 i=1 i=1 j=1   ZT N X p γ−1 = Yi h◦f i + CPi (T ) dT  ρ γ i=1

(23)
(24)

(25)

and q = ϵσT

(26)

(28)

50

• Neglect viscous dissipation in energy equation

• Neglect Dufour effect in energy equation
• Keep Soret effect in species and energy equation

(27)

4

• Uniform pressure field

• Neglect effect of body forces in momentum
and energy equation

T◦

h
i


†
where τ = µ ∇v + (∇v) + κ − 32 µ [∇ · v] I

• Steady state

Step 1:

Make use of the fact that h =

P

Yi hi to restate equations (24) and (25) as:

∇ · [ρYi (v + V i )] = ẇi
" N
#
X
∇·
ρYi hi (v + V i ) − λ∇T = 0

(29)
(30)

i=1

Step 2:

Use
hi = h◦f i +

ZT

CPi dT

(31)

T◦

to rewrite equation (30) as


T
Z
N
N
X
X
∇· ρYi (v + V i ) h◦f i + ρYi (v + V i ) CPi dT − λ∇T  = 0 i=1

i=9

T◦

and apply equation (29) to the first term of the left hand side to obtain   ZT N N ZT N X X X ∇ · ρv V i Yi CPi dT − λ∇T  = − h◦f i ẇi Yi CPi dT + ρ i=6 T ◦

i=1

(32)

i=1

T◦

51

(33)

Step 3:

Assume a background fluid so that
Since

ρYi V i = −ρID∇Yi

(34)

X

(35)

CP =

Yi CP

it is possible to rewrite equation (33) as

ZT ZT N X (∇Yi ) Yi CPi dT − ρID ∇ · ρv CP dT + ρID i=1

T◦

T◦

and because of the equality,
∇

ZT

T◦

CP dT = ∇
=

N
X
i=1

N
X
i=1

Yi

ZT

CPi dT =

(∇Yi )

(∇Yi )

i=1

T◦

ZT

N
X

CPi dT +

N
X
i=1

T◦

ZT



N
X
λ
h◦f i ẇi
CP ∇T  = −
ρCP ID
i=1

CPi dT +

i=1

T◦

Yi CPi ∇T =

N
X

N
X
i=6

Yi ∇

(∇Yi )

ZT

T◦

ZT

CPi dT

(37)

CPi dT + CP ∇T

(38)

T◦

equation (36) may be rewritten as


ZT
ZT
N
X h◦f i ẇi ∇ · ρv CP dT + ρID∇ CP dT − ρID [Le − 1] CP ∇T  = − T◦

T◦

52

(36)

i=1

(39)

Step 4:
velocity
profile

laminar
flame

temperature
profile

SuL
,,

ṁ
ρu

Sf

ẇF
ẇOx

v

T = Tu
dT /dx = 0

Assume unity Lewis number in equation (39):

T = Tf
dT /dx = 0



∇ · ρv

reaction
zone
preheat
zone

δL

ZT

CP dT + ρID∇

T◦

ZT

T◦



CP dT  = −

N
X

h◦f i ẇi

(40)

i=1

If Le = 1 then ρID may be replaced by λ/CP ! From figure with simplified flame structure: ρu v = ρu S uL = ṁ” = constant.
N X ” Finally ṁ CP ∇T + ∇ · [λ∇T ] = − h◦f i ẇi (41) i=1

53

Step 5:

Step 6:

Express the combustion reaction in terms of a mass balance: Integrate the one-dimensional form of equation (45) from x → −∞ to x → ∞ for the simplified flame 1 kg Fuel + ν kg Oxidizer −→ (ν + 1) kg Products (42) structure,   Then dT dT d 1 1 λ = −∆c H ẇF (46) ṁ”CP + − ẇF = − ẇOx = ẇPr (43) dx dx dx ν ν+1 and hence, subjected to the boundary conditions, N X

h◦f i ẇi = h◦f F ẇF + h◦f Ox ẇOx + h◦f Pr ẇPr i=1   ◦ ◦ ◦ = hf F + ν hf Ox − (ν + 1) hf Pr ẇF = ∆c H ẇF

where ∆c H denotes the fuel’s heat of combustion.
Equation (39) may then be restated as:
ṁ”CP ∇T + ∇ · [λ∇T ] = −∆c H ẇi

x → −∞ :

T = Tu

x→∞:

T = Tf (47)

and the subsidiary conditions,
dT
=0
(44)
dx
Inside the flame zone, x → −∞ :

(45)

x ∈ δL :

Tf − Tu
dT
=
dx
δL

(49)

Integration of equation (46) gives:
∂T
T
ṁ”CP T |Tfu + λ

dT /dx=0

∂x dT /dx=0

54

dT
= 0 (48)
dx

x→∞:

= −∆c H

Z∞

−∞

ẇF dx (50)

Step 8:

Step 7:

Apply a change of variables to the right hand side of equation Find a second algebraic equation by integrating equation (46) from x → −∞ to x → δL /2, subjected to the boundary con(50) on the basis of the assumed temperature profile (49):
ditions, dT Tf − Tu δL = ⇐⇒ dx = dT (51) δL Tu + Tf dx δL Tf − Tu x → −∞ : T = Tu x= : T = (54) 2 2 Hence, and the subsidiary conditions, T f Z dT δL dT Tf − Tu δL ∆c H ” x → −∞ :
=0 x→ :
= (55) ṁ CP (Tf − Tu ) = − ẇF dT = −δL ∆c H ẇF (52) dx 2 dx δL Tf − Tu Tu

Integration of equation (41) results in

or, equivalently,
”

ṁ CP (Tf − Tu ) + δL ∆c H ẇF = 0
=⇒ A single algebraic equation with two unknowns:
• the mass flux ṁ” (≡ ρu SuL ), • and the laminar flame thickness δL .

(53)

”

ṁ CP T

∂T
(Tu +Tf )/2
+
λ
Tu
∂x

(Tf −Tu )/δL
dT /dx=0

= −∆c H

δZL /2

ẇF dx

−∞

(56)

and hence,
Tf − Tu
1
C ṁ” (Tf − Tu ) − λ
=0
2 P δL since ẇF is practically zero in the preheat zone.

55

(57)

Step 9:

Step 10:

Solve equations (53) and (57) for SuL and δL :

Assume a generalized overall reaction,

1
∆c H
ẇF 2
λ
SuL = −2
ρu CP CP (Tf − Tu ) ρu  1 λ CP (Tf − Tu ) ρu 2 δL = −2 ρu CP ∆c H ẇF 

(58)

i=1

(59)

Express the heat of combustion of the fuel as
∆c H = (ν + 1)CP (Tf − Tu )

N
N
X
X
,
νi Mi −→
νi,, Mi

with a an overall reaction order n =

1
ẇF 2
λ
(ν + 1)
SuL = −2
ρu CP
ρu

1
λ
1 ρu 2 δL = −2 ρu CP ν + 1 ẇF

PN

,
i=1 νi

Mass consumption rate of each individual species:
  N  ν , d (ρYi /Mi ) Ea Y ρYj j ,, , m = (νi − νi ) BT exp − (64) dt RT j=1 Mj

(60)

and substitute this into equations (58) and (59):


(63)

i=1

(61)
(62)

=⇒
Since



Ea
ẇF ∝ ρ BT exp −
RT
n

m

ρu ∝ Tu−1 P

Y
N

,

(Yj /Mj )νj

(65)

j=1

(66)

and most of the combustion occurs in the reaction zone,   Ea −n n m ẇF ∝ Tf P Tf exp − (67) RTf

56

Step 11:
Substitute equations (66) and (67) into equations (61) and (62) to obtain:
s   n−2  − n   m    2 2 λ(Tu ) Tu P 1 SuL Tf Tf 2 Ea 1 ∝ − exp − Su ◦L λ(Tu0 ) Tu0 P0 Tf◦ Tf◦ 2R Tf Tf◦ s   − n   n  − m   2 2 λ(Tu ) P Tf 2 Tf Ea 1 δL 1 exp + − ∝ δL◦ λ(Tu0 ) P0 Tf◦ Tf◦ 2R Tf Tf◦

(68)
(69)

where Su ◦L , δL◦ and Tf◦ are the laminar burning velocity, the laminar flame thickness and the flame temperature of the unburnt mixture at a reference state P0 and Tu0 .
When the variation in Tu is caused by adiabatic compression, equations (68) and (69) can be approximated by:
c+ γ−1 γ −1+α P P0  c−α δL P = δL◦ P0

SuL
=
Su ◦L



57

(70)
(71)

Laminar burning velocity: effect of flame morphology H2/O2/N2 3.0 ms P= 1 atm f = 2.25

5.5 ms

8.0 ms

10.5 ms

13.0 ms

H2/O2/N2 3.0 ms
P= 1 atm
f = 0.70

4.0 ms

5.0 ms

6.0 ms

7.0 ms

H2/O2/He 14.0 ms
P= 60 atm
f = 0.70

24.0 ms

34.0 ms

44.0 ms

54.0 ms

Flame morphology of propagating hydrogen-air flames.
From: Tse S.D., Zhu D.L., and Law C.K. Morphology and burning rates of expanding spherical flames in H2/O2/inert mixtures up to 60 atmospheres. In Proceedings of the Twenty-Eighth Symposium (International) on Combustion, pages 1793-1800, Pittsburgh, 2000. The Combustion Institute.

58

Laminar burning velocity: effect of flame stretch and flame curvature

unburnt mixture

Su L < Su ◦L

convective heat
transfer

convective heat
transfer

flam

e

e
tiv
uc
d
r
con nsfe
tra

at
he

burnt mixture

Su L > Su ◦L

burnt mixture

unburnt mixture

convective heat
transfer

global
curvature

compression
streamline 2
stretch

flam

local
curvature

e

burnt mixture

unburnt mixture

streamline 1

streamline 3

Effect of flame stretch and flame curvature on the laminar burning velocity.

59

Laminar burning velocity: effect of flame stretch and flame curvature Stretch rate of a surface element in a strained fluid:
1 dA (72) A dt Universal expression which relates the stretch rate of a flame surface element to the velocity field, v, of a nonuniform flow-field:
(73) ṡ = −nn:∇v + ∇ · v + Su ◦L ∇ · n {z } | {z } | ṡs ṡc ◦ When a planar laminar flame with a thickness δL is distorted into a bulge of size Λ, the local laminar burning velocity at each point can be related to the local stretch rate as  
Su ◦L − SuL L 1 dA 2 = + O ϵ (74) Su ◦L Su ◦L A dt ṡ =

where ϵ = δL◦ /Λ. When Λ ≫ δL◦ , ϵ is a small number so that:
 
dA 1 + O ϵ2 Su ◦L − SuL = L A dt
= Lṡ + O ϵ2

60

(75)
(76)

Laminar burning velocity: effect of flame stretch and flame curvature Express Lṡ as a linear combination of quantities that account for the separate effects of the strain rate, flame curvature, pressure, etc., each having its own Markstein length:
Lṡ ≡ Ls ṡs + Lc ṡc + . . .
=⇒ Su ◦L − SuL = (Ls ṡs + Lc ṡc + . . .) + O ϵ2

(77)



(78)

Combine equations (73), (76) and (78):
SuL = Ls [nn:∇v − ∇ · v] + [1 − Lc ∇ · n] Su ◦L   Lc = Ls [nn:∇v − ∇ · v] + 1 + Su ◦L κ1 + κ2  Lc = Ls [nn:∇v − ∇ · v] + 1 + Su ◦L R

The quantities denoted by Ls and Lc are called Markstein lengths.

61

(79)
(80)
(81)

Unstretched laminar burning velocity

4
3.5
3
2.5
2
1.5
1
0.5
0



20

0.5

30



×× × × 
×
•
×
⊗
××
⊕◦
⊕ ⊕
• ⊕ × ×



×
⊕ ×
◦   
⊕  × 
 ◦   ◦ ⊗ ◦   ×⊕  •◦ ◦
⊗  •
⊕ 
 ×    ⊕    ◦ ⊗ •◦