the reduced-kinetic description of hydrogen-air premixed...

The reduced-kinetic description of hydrogen-airpremixed-combustion problems relevant for safety

applications5th Meeting of the Spanish Section of the Combustion Institute

A. L. SanchezP. BoivinE. F. TarrazoUC3M, Madrid

D. F. GalisteoC. JimenezCIEMAT, Madrid

A. LinanETSIA, Madrid

F.A. WilliamsUCSD, San Diego

5th Meeting of the Spanish Section of the Combustion Institute

May 23th 2011, Santiago, Spain.

Introduction

Motivation

Context

H2 and Syngas are bound to play a predominant role as energycarriers in the foreseable future.

Safety issues arise concerning hydrogen transport, handling andstorage

2 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications

N

Introduction

Motivation

Context

H2 and Syngas are bound to play a predominant role as energycarriers in the foreseable future.

Safety issues arise concerning hydrogen transport, handling andstorage

Hydrogen combustion characteristics

Li vl δQUENCH Emin δIGNITION

H2 0.3 3 m/s 0.6 mm 0.02 mJ ∼ 50 µm

CH4 1.0 0.45 m/s 1.8 mm 0.21 mJ ∼ 0.8 mm


N

Introduction

Chemistry Reduction

Methodology


N

Introduction

Chemistry Reduction

Methodology

1 Selection of a detailed chemical-kinetic mechanism including acomplete set of chemical species and elementary reactions


N

Introduction

Chemistry Reduction

Methodology


2 Deletion of elementary steps that do not contribute to thechemistry under the conditions of interest


N

Introduction

Chemistry Reduction

Methodology



3 Introduction of steady-state approximations for intermediatespecies with negligible transport rates


N

Introduction

Chemistry Reduction

Methodology



3 Introduction of steady-state approximations for intermediatespecies with negligible transport rates

4 Truncation of the steady-state algebraic expressions to facilitatenumerical computations


N

Introduction

Chemistry Reduction

Methodology

For selection of the test cases for validation one needs toidentify the conditions of interest.

E.g., in gas-turbine combustion the preheated mixture is burnedat elevated pressure.


N

Introduction

Chemistry Reduction

Methodology

Validation for lean premixed systems: laminar deflagrations,homogeneous ignition, nonpremixed ignition

Validation for nonpremixed sytems: laminar strained diffusionflames


N

Introduction

Chemistry Reduction

Methodology

Steady planar adiabatic deflagration (ρv = ρuvl ).

ρuvl

dYi

dx−

d

dx

(ρDT

Li

dYi

dx

)

= Wiωi

ρuvlcp

dT

dx−

d

dx

(

λdT

dx

)

=∑

i

hiωi

Boundary conditions:

x → −∞ : Yi − Yi u = T − Tu = 0

x → −∞ : dYi

dx= dT

dx= 0


N

Introduction

Chemistry Reduction

Methodology

Counterflow diffusion flame (v = −Ay).

ρAydYi

dy+

d

dy

(ρDT

Li

dYi

dy

)

= −Wiωi

ρAcpydT

dy+

d

dy

(

λdT

dy

)

= −∑

i

hiωi

Boundary conditions:

y → −∞ : Yi − Yi−∞ = T − T−∞ = 0

y → −∞ : Yi − Yi∞ = T − T∞ = 0


N

Introduction

Chemistry Reduction

Methodology

Adiabatic ignition history in an homogeneous isobaric reactor:

ρdYi

dt= Wiωi Yi(0) = Yi o

ρcpdT

dt=

∑

i

hiωi T (0) = To


N

Introduction

Detailed H2 chemistry

San-Diego Mechanism: 8 chemical species, 21 reactions, thoroughlytested.

10−1

100

101

0

50

100

150

200

250

300

350

φ

vl[c

m/s]

Law et al.Kwon et al.Dowdy et al.Detailed mech.Detailed mechwithout thermal−diff.


N

Introduction


1. H + O2 ⇋ OH + O2. H2 + O ⇋ OH + H3. H2 + OH ⇋ H2O + H4. H2O + O ⇋ 2OH5. 2H + M ⇋ H2 + M6. H + OH + M ⇋ H2O + M7. 2O + M ⇋ O2 + M8. H + O + M ⇋ OH + M9. O + OH + M ⇋ HO2 + M10. H + O2 + M ⇋ HO2 + M

11. HO2 + H ⇋ 2OH12. HO2 + H ⇋ H2 + O2

13. HO2 + H ⇋ H2O + O14. HO2 + O ⇋ OH + O2

15. HO2 + OH ⇋ H2O + O2

16. 2OH + M ⇋ H2O2 + M17. 2HO2 ⇋ H2O2 + O2

18. H2O2 + H ⇋ HO2 + H2

19. H2O2 + H ⇋ H2O + OH20. H2O2 + OH ⇋ H2O + HO2

21. H2O2 + O ⇋ HO2 + OH

21 elementary reactions from a detailed mechanism(University of California, San Diego)


N

Introduction


1. H + O2 ⇋ OH + O2. H2 + O ⇋ OH + H3. H2 + OH ⇋ H2O + H4.5. 2H + M ⇋ H2 + M6. H + OH + M ⇋ H2O + M7.8.9.10. H + O2 + M ⇋ HO2 + M

11. HO2 + H ⇋ 2OH12. HO2 + H ⇋ H2 + O2

13.14.15. HO2 + OH ⇋ H2O + O2

16. 2OH + M ⇋ H2O2 + M17. 2HO2 ⇋ H2O2 + O2

18. H2O2 + H ⇋ HO2 + H2

19.20.21.

Crossover Temp.: k1f CO2CH = k10f CMCO2

CH

k1f = k10fp

RoT

{Tc ≃ 1000K at p = 1atmTc ≃ 1500K at p = 100atm


N

Introduction

Skeletal mechanism

Skeletal mechanism12 elementary steps, 8 species

H + O2 ⇋ OH + O (1)

H2 + O ⇋ OH + H (2)

H2 + OH ⇋ H2O + H (3)

H + O2 + M ⇀ HO2 + M (4)

HO2 + H ⇀ 2OH (5)

HO2 + H ⇋ H2 + O2 (6)

HO2 + OH ⇀ H2O + O2 (7)

H + OH + M ⇋ H2O + M (8)

H + H + M ⇋ H2 + M (9)

HO2 + HO2 ⇀ H2O2 + O2(10)

HO2 + H2 ⇀ H2O2 + H(11)

H2O2 + M ⇀ 2OH + M (12)

Justification

Reactions 1-7 describe accurately leanpremixed combustion(ignition and deflagration)at atmospheric pressures

Reactions 8-9 Adding recombinationreactions gives betterpredictions forstoichiometric and richmixtures. Also allows agood description of theequilibrium at hightemperatures.

Reactions 10-12 include the chemistry ofH2O2, important forhigh-pressure flames andlow-temperature ignition.


N

Introduction

Skeletal mechanism


H + O2 ⇋ OH + O (1)

H2 + O ⇋ OH + H (2)

H2 + OH ⇋ H2O + H (3)

H + O2 + M ⇀ HO2 + M (4)

HO2 + H ⇀ 2OH (5)

HO2 + H ⇋ H2 + O2 (6)

HO2 + OH ⇀ H2O + O2 (7)

H + OH + M ⇋ H2O + M (8)

H + H + M ⇋ H2 + M (9)

HO2 + HO2 ⇀ H2O2 + O2(10)

HO2 + H2 ⇀ H2O2 + H(11)

H2O2 + M ⇀ 2OH + M (12)

Validation

100

1010

50

100

150

200

250

300

350

ΦF

lam

e ve

loci

ty (

cm/s

)

detailedskeletal

10atm.

50atm.

1atm.

Laminar flame speed of steady planar flamesT0 = 300K


N

Introduction

Skeletal mechanism


H + O2 ⇋ OH + O (1)

H2 + O ⇋ OH + H (2)

H2 + OH ⇋ H2O + H (3)

H + O2 + M ⇀ HO2 + M (4)

HO2 + H ⇀ 2OH (5)

HO2 + H ⇋ H2 + O2 (6)

HO2 + OH ⇀ H2O + O2 (7)

H + OH + M ⇋ H2O + M (8)

H + H + M ⇋ H2 + M (9)

HO2 + HO2 ⇀ H2O2 + O2(10)

HO2 + H2 ⇀ H2O2 + H(11)

H2O2 + M ⇀ 2OH + M (12)

Validation

0 1 2 3

x 10−4

1500

2000

2500

21-step

12-stepTPEA

K

1/a (s)

Peak temperature as a function of strain ratefor a H2-air counterflow flame


N

Introduction

Skeletal mechanism


H + O2 ⇋ OH + O (1)

H2 + O ⇋ OH + H (2)

H2 + OH ⇋ H2O + H (3)

H + O2 + M ⇀ HO2 + M (4)

HO2 + H ⇀ 2OH (5)

HO2 + H ⇋ H2 + O2 (6)

HO2 + OH ⇀ H2O + O2 (7)

H + OH + M ⇋ H2O + M (8)

H + H + M ⇋ H2 + M (9)

HO2 + HO2 ⇀ H2O2 + O2(10)

HO2 + H2 ⇀ H2O2 + H(11)

H2O2 + M ⇀ 2OH + M (12)

Validation

Induction time of a stoichiometrichomogeneous mixture


N

Introduction

The steady-state approximations

Laminar premixed flame, p = 1atm, Tu = 300K, and φ = 0.8:

0 0.2 0.4 0.6 0.8 1 1.2

−3

−2

−1

0

1

2

3

x 105

x [mm]

OH

−ba

lanc

e

ProductionConsumptionConvection+Diffusion

[mol/m3s]

(a) OH

0 0.2 0.4 0.6 0.8 1 1.2

−1.5

−1

−0.5

0

0.5

1

1.5

x 105

x [mm]

H−

bala

nce


[mol/m3s]

(b) H

ρDYi

Dt−∇ · (ρDi∇Yi)

︸︷︷︸

transport

= ωp,iWi

︸︷︷︸

production

− ωc,iWi

︸︷︷︸

consumption


N

Introduction

The steady-state approximations

Laminar premixed flame, p = 1atm, Tu = 300K, and φ = 0.8:

0 0.2 0.4 0.6 0.8 1 1.2

−3

−2

−1

0

1

2

3

x 105

x [mm]

OH

−ba

lanc

e


[mol/m3s]

(a) OH

0 0.2 0.4 0.6 0.8 1 1.2

−1.5

−1

−0.5

0

0.5

1

1.5

x 105

x [mm]

H−

bala

nce


[mol/m3s]

(b) H

(((((((((((ρDYi

Dt−∇ · (ρDi∇Yi)

︸︷︷︸

transport

= ωp,iWi

︸︷︷︸

production

− ωc,iWi

︸︷︷︸

consumption

→ ωp,i = ωc,i


N

Introduction

Reduced chemistry in H2-air flames

Steady-State Analysis

All intermediates but H are insteady state


N

Introduction




H2 reduced mechanism

3H2 + O2I

⇋ 2H2O + 2H, ωI ≃ w1

2H + MII⇋ H2 + M, ωII ≃ w4f


N

Introduction





3H2 + O2I

⇋ 2H2O + 2H, ωI ≃ w1


Premixed flame

0 2 4 60

50

100

150

200

250

300

350

φ

p=1atm.

p=10atm.

p=50atm. Laminar flame speed of steadyplanar flames. T0 = 300K


N

Introduction





3H2 + O2I

⇋ 2H2O + 2H, ωI ≃ w1


Premixed flame

0 2 4 60

50

100

150

200

250

300

350

φ

p=1atm.

p=10atm.

p=50atm.

Variation with strain rate ofthe maximum temperature ina hydrogen-air counterflowdiffusion flame.T0 = 300K , P = 1atm.

Laminar flame speed of steadyplanar flames. T0 = 300K

Diffusion flame

0 1 2 3

x 10−4

1500

2000

2500

1/a (s−1)


N

Introduction

Reduced chemistry and autoignition

2-step reduced mechanism

3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

φ

10−4

00.1 101

detailed chemistry

2-step Induction time (s)

of a homogeneous mixtureT0 = 1200K, p=1atm.


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

φ

10−4

00.1 101

detailed chemistry

2-step Induction time (s)


Steady state approximations

HO2 is not in steady-state during autoignition.

H2 + O2 ⇀ HO2 + H

HO2 + H ⇀ 2OH

HO2 + H ⇀ H2 + O2


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

φ

10−4

00.1 101

detailed chemistry

2-step

3-step including HO2

H2 + O2III

⇋ HO2 + H

Induction time (s)


Steady state approximations

HO2 is not in steady-state during autoignition.

H2 + O2 ⇀ HO2 + H

HO2 + H ⇀ 2OH

HO2 + H ⇀ H2 + O2


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

φ

10−4

00.1 101

detailed chemistry

2-stepPCI 2010

3-step including HO2

H2 + O2III

⇋ HO2 + H

Good agreement is obtained in inductiontime for all φ by including HO2 out ofsteady state and a correction for thebranching time accounting for departuresof O and OH from steady state.

Induction time (s)



N

Introduction

Combustion problems relevant for safetyapplications

Low-Temperature Ignition

0.7 0.8 0.9 1 1.1 1.2 1.310

−6

10−4

10−2

100

102

Tim

e(s)

1000K/T

21-Step

2-Step


N

Introduction

Combustion problems relevant for safetyapplications

Low-Temperature Ignition

0.7 0.8 0.9 1 1.1 1.2 1.310

−6

10−4

10−2

100

102

Tim

e(s)

1000K/T

21-Step

2-Step

Very fuel-lean flames

10−1

100

101

0

50

100

150

200

250

300

350

vl[c

m/s]

φ

CELLULAR FLAMESFLAME BALLS


N

Introduction

Ignition above crossover


3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

ωI = k6bCH2CO2

+ k1f CO2CH

ωII = k4f CMCO2CH


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

ωI = k6bCH2CO2

+ k1f CO2CH

ωII = k4f CMCO2CH

Branched-chain explosion

dCH

dt= 2k6bCH2

CO2+ 2(k1f − k4f CM)CO2

CH; CH(0) = 0


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

ωI = k6bCH2CO2

+ k1f CO2CH

ωII = k4f CMCO2CH


dCH

dt= 2k6bCH2


CH; CH(0) = 0

CH = εCH2

[

e2(k1f −k4f CM)CO2t− 1

]

; ε =k6b

k1f − k4f CM

∼ 10−6


N

Introduction



3H2 + O2I

⇋ 2H2O + 2H

2H + MII

⇋ H2 + M

ωI = k6bCH2CO2

+ k1f CO2CH

ωII = k4f CMCO2CH

0.7 0.8 0.9 1 1.1 1.2 1.310

−6

10−4

10−2

100

102

Tim

e(s)

1000K/T

21-Step

2-Step


dCH

dt= 2k6bCH2


CH; CH(0) = 0

CH = εCH2

[

e2(k1f −k4f CM)CO2t− 1

]

; ε =k6b

k1f − k4f CM

∼ 10−6


N

Introduction

Low-temperature ignition

Initial skeletal mechanism

H + O21

⇋ OH + O

H2 + O2

⇋ OH + H

H2 + OH3

⇋ H2O + H

H + O2 + M4

⇀ HO2 + M

HO2 + H5

⇀ 2OH

HO2 + H6

⇋ H2 + O2

HO2 + OH7

⇀ H2O + O2

H + OH + M8

⇋ H2O + M

H + H + M9

⇋ H2 + M

HO2 + HO210⇀ H2O2 + O2

HO2 + H211⇀ H2O2 + H

H2O2 + M12⇀ 2OH + M


N

Introduction

Low-temperature ignition


H + O21

⇋ OH + O

H2 + O2

⇋ OH + H

H2 + OH3

⇋ H2O + H

H + O2 + M4

⇀ HO2 + M

HO2 + H6

↽ H2 + O2

HO2 + HO210⇀ H2O2 + O2

HO2 + H211⇀ H2O2 + H

H2O2 + M12⇀ 2OH + M

Validation

0.9 1 1.1 1.2

1

1.1 1.15 1.2 1.25 1.3 1.350.1

1

10

100

1000K/T

tim

e(s

)tim

e(s

)

p=1 atm

p=10 atm

10−4

10−2

21 (solid), 8 (dashed), 3 (dot-dashed)


N

Introduction

3-step Reduced Mechanism (Trevino, 1991)


H + O21→ OH + O

H2 + O2→ OH + H

H2 + OH3→ H2O + H

H + O2 + M4→ HO2 + M

H2 + O25→ HO2 + H

HO2 + HO26→ H2O2 + O2

HO2 + H27→ H2O2 + H

H2O2 + M8→ 2OH + M


N

Introduction



H + O21→ OH + O

H2 + O2→ OH + H

H2 + OH3→ H2O + H

H + O2 + M4→ HO2 + M

H2 + O25→ HO2 + H

HO2 + HO26→ H2O2 + O2

HO2 + H27→ H2O2 + H

H2O2 + M8→ 2OH + M

Steady-state intermediatesH, O, OH


N

Introduction



H + O21→ OH + O

H2 + O2→ OH + H

H2 + OH3→ H2O + H

H + O2 + M4→ HO2 + M

H2 + O25→ HO2 + H

HO2 + HO26→ H2O2 + O2

HO2 + H27→ H2O2 + H

H2O2 + M8→ 2OH + M



2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2


N

Introduction



H + O21→ OH + O

H2 + O2→ OH + H

H2 + OH3→ H2O + H

H + O2 + M4→ HO2 + M

H2 + O25→ HO2 + H

HO2 + HO26→ H2O2 + O2

HO2 + H27→ H2O2 + H

H2O2 + M8→ 2OH + M



2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2


N

Introduction



H + O21→ OH + O

H2 + O2→ OH + H

H2 + OH3→ H2O + H

H + O2 + M4→ HO2 + M

H2 + O25→ HO2 + H

HO2 + HO26→ H2O2 + O2

HO2 + H27→ H2O2 + H

H2O2 + M8→ 2OH + M

Steady-state expression for H

CH =k5CH2

CO2+ k7CH2

CHO2+ 2k8CH2O2

CM

(k4CM − k1)CO2



2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2


N

Introduction



2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2

CH =k5CH2

CO2+ k7CH2

CHO2+ 2k8CH2O2

CM

(k4CM − k1)CO2


N

Introduction


Validation

0.9 1 1.1 1.2

1

1.1 1.15 1.2 1.25 1.3 1.350.1

1

10

100

1000K/T

tim

e(s

)tim

e(s

)

p=1 atm

p=10 atm

10−4

10−2

21 (solid), 8 (dashed), 3 (dot-dashed)


2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2

CH =k5CH2

CO2+ k7CH2

CHO2+ 2k8CH2O2

CM

(k4CM − k1)CO2


N

Introduction


Thermal explosion

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1000

1500

2000

2500

3000

Time (s)

T (K)

αHO2

αH2O2

αi =|CiP

− CiC|

CiP


2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2

CH =k5CH2

CO2+ k7CH2

CHO2+ 2k8CH2O2

CM

(k4CM − k1)CO2


N

Introduction


Thermal explosion

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1000

1500

2000

2500

3000

Time (s)

T (K)

αHO2

αH2O2

αi =|CiP

− CiC|

CiP

HO2 reaches steady state after a shortinitial period


2H2 + O2I∗

→ 2H2O

H2O2 + H2

II∗

⇋ 2H2O

H2 + 2O2III

∗

⇋ 2HO2

ωI∗ = w1 + w6 + w7

ωII∗ = −w6 − w7 + w8

ωIII∗ =

w4 + w5 − 2w6 − w7

2

CH =k5CH2

CO2+ k7CH2

CHO2+ 2k8CH2O2

CM

(k4CM − k1)CO2


N

Introduction

2-step Reduced Mechanism

CHO2= w4 + w5 − 2w6 − w7 = 0


N

Introduction


CHO2= w4 + w5 − 2w6 − w7 = 0


2H2 + O2I

→ 2H2O

2H2OII

→ H2O2 + H2

Global rates

ωI =w5 + w7 + (1 + α)w8

1 − α

ωII =(1 − 1

2α)(w5 + w7) + αw8

1 − α

α =2k1

k4CM4

, w5 = k5CH2CO2

, w6 = k6C2HO2

, w7 = k7CHO2CH2

, w8 = k8CMCH2O2


N

Introduction


CHO2= w4 + w5 − 2w6 − w7 = 0


2H2 + O2I

→ 2H2O

2H2OII

→ H2O2 + H2

Global rates

ωI =w5 + w7 + (1 + α)w8

1 − α

ωII =(1 − 1

2α)(w5 + w7) + αw8

1 − α

α =2k1

k4CM4

, w5 = k5CH2CO2

, w6 = k6C2HO2

, w7 = k7CHO2CH2

, w8 = k8CMCH2O2

Conservation equationsdCH2O2

dt= ωII

ρcp

dT

dt= −2hH2O(ωI − ωII) − hH2O2

ωII

Init. Conditions

CH2O2(0) = T (0) − To = 0


N

Introduction


CHO2= w4 + w5 − 2w6 − w7 = 0 ⇒ CHO2

≃“

(2−α)w5+2w82(1−α)k6

”1/2


2H2 + O2I

→ 2H2O

2H2OII

→ H2O2 + H2

Global rates

ωI =w5 + w7 + (1 + α)w8

1 − α

ωII =(1 − 1

2α)(w5 + w7) + αw8

1 − α

α =2k1

k4CM4

, w5 = k5CH2CO2

, w6 = k6C2HO2

, w7 = k7CHO2CH2

, w8 = k8CMCH2O2

Conservation equationsdCH2O2

dt= ωII

ρcp

dT

dt= −2hH2O(ωI − ωII) − hH2O2

ωII

Init. Conditions

CH2O2(0) = T (0) − To = 0


N

Introduction


Using the approximations w5 = 0 and (w8 − 12w7)α = 0 yields

Reduced global rates

ωI − ωII = =1 + α

1 − αk8CM8

CH2O2

ωII =k7k

1/28

k1/26

CH2CM8

(1 − α)3/2

"

“

1 −α

2

” k5CH2CO2

k8C2M8

+CH2O2

CM8

#1/2


N

Introduction


Using the approximations w5 = 0 and (w8 − 12w7)α = 0 yields

Reduced global rates

ωI − ωII = =1 + α

1 − αk8CM8

CH2O2

ωII =k7k

1/28

k1/26

CH2CM8

(1 − α)3/2

"

“

1 −α

2

” k5CH2CO2

k8C2M8

+CH2O2

CM8

#1/2

k8 ∝ e−

E8RoT ,

k7k1/28

k1/26

∝ e−

E7+ 12

E8−12

E6RoT

with β =E8

RoTo

≃E7 + 1

2E8 − 1

2E6

RoTo

≃ 30 for To = 800K


N

Introduction

Dimensionless ProblemDimensionless variables

ϕ =h

(1 − α)1/2(1 + α)βqi2/3

„

k7

(k6k8)1/2

«

−2/3 „

CH2

CM8

«

−2/3 CH2O2

CM8

τ =(1 + α)1/3

(1 − α)4/3(βq)1/3k8CM8

„

k7

(k6k8)1/2

«2/3 „

CH2

CM8

«2/3

t

θ = βT − To

To

, q =−2hH2OCM8

ρcpTo


N

Introduction

Dimensionless ProblemDimensionless variables

ϕ =h

(1 − α)1/2(1 + α)βqi2/3

„

k7

(k6k8)1/2

«

−2/3 „

CH2

CM8

«

−2/3 CH2O2

CM8

τ =(1 + α)1/3

(1 − α)4/3(βq)1/3k8CM8

„

k7

(k6k8)1/2

«2/3 „

CH2

CM8

«2/3

t

θ = βT − To

To

, q =−2hH2OCM8

ρcpTo

Conservation equations

dϕ

dτ= (a + ϕ)1/2eθ

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ

Init. Conditions

ϕ(0) = θ = 0


N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0


N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0

a =“

1 −α

2

”1/3(1−α)1/3(1+α)2/3(βq)2/3 k5k

1/36

(k7k8)2/3

„

CH2

CM8

«1/3 „

CO2

CM8

«

∼ 10−5

Initiation counts for τ ∼ a1/2 when ϕ ∼ θ ∼ a but it is negligible at later times


N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0

dθ

dϕ= Λ + ϕ1/2

θ = (2/3)ϕ3/2 + Λϕ

a =“

1 −α

2

”1/3(1−α)1/3(1+α)2/3(βq)2/3 k5k

1/36

(k7k8)2/3

„

CH2

CM8

«1/3 „

CO2

CM8

«

∼ 10−5



N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0

dθ

dϕ= Λ + ϕ1/2

θ = (2/3)ϕ3/2 + Λϕ

a =“

1 −α

2

”1/3(1−α)1/3(1+α)2/3(βq)2/3 k5k

1/36

(k7k8)2/3

„

CH2

CM8

«1/3 „

CO2

CM8

«

∼ 10−5


τi =∫∞

0dϕ

ϕ1/2 exp( 23 ϕ3/2+Λϕ)


N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0

dθ

dϕ= Λ + ϕ1/2

θ = (2/3)ϕ3/2 + Λϕ

a =“

1 −α

2

”1/3(1−α)1/3(1+α)2/3(βq)2/3 k5k

1/36

(k7k8)2/3

„

CH2

CM8

«1/3 „

CO2

CM8

«

∼ 10−5


τi =∫∞

0dϕ

ϕ1/2 exp( 23 ϕ3/2+Λϕ)

Λ =

[k7/(k6k8)

1/2

(1 − α)1/2(1 + α)

]2/3

(βq)1/3

(CH2

CM8

)2/3hH2O2

2hH2O

≃ 0.1


N

Introduction

Ignition time

dϕ

dτ= (a + ϕ)1/2eθ; ϕ(0) = 0

dθ

dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0

dθ

dϕ= Λ + ϕ1/2

θ = (2/3)ϕ3/2 + Λϕ

a =“

1 −α

2

”1/3(1−α)1/3(1+α)2/3(βq)2/3 k5k

1/36

(k7k8)2/3

„

CH2

CM8

«1/3 „

CO2

CM8

«

∼ 10−5


τi =∫∞

0dϕ

ϕ1/2 exp( 23 ϕ3/2+Λϕ)

= (2/3)2/3Γ(1/3) ≃ 2.0444

Λ =

[k7/(k6k8)

1/2

(1 − α)1/2(1 + α)

]2/3

(βq)1/3

(CH2

CM8

)2/3hH2O2

2hH2O

≃ 0.1


N

Introduction

Ignition timeExplicit analytic prediction

ti = 2.0444(1 − α)4/3

(1 + α)1/3(βq)−1/3(k8CM8

)−1

„

k7

(k6k8)1/2

«

−2/3 „

CH2

CM8

«

−2/3


N

Introduction

Ignition timeExplicit analytic prediction

ti = 2.0444(1 − α)4/3

(1 + α)1/3(βq)−1/3(k8CM8

)−1

„

k7

(k6k8)1/2

«

−2/3 „

CH2

CM8

«

−2/3

0.8 1 1.2 1.4 1.6

1000K/T

φ=0.5

φ=1

φ=2

10−6

10−6

10−6

10−4

10−4

10−4

10−2

10−2

10−2

1

1

1

102

102

102

104

104

104

21-step (solid curves)ti for p= 1 atm (squares)ti for p= 10 atm (triangles)ti for p= 50 atm (circles)


N

Introduction

Very lean flames and flammability limit

10−1

100

101

0

50

100

150

200

250

300

350

vl[c

m/s]

φ



N

Introduction

Skeletal mechanism for very lean flames


H + O21

⇋ OH + O

H2 + O2

⇋ OH + H

H2 + OH3

⇋ H2O + H

H + O2 + M4

⇀ HO2 + M

HO2 + H5

⇀ 2OH

HO2 + H6

⇋ H2 + O2

HO2 + OH7

⇀ H2O + O2

H + OH + M8

⇋ H2O + M

H + H + M9

⇋ H2 + M

HO2 + HO210⇀ H2O2 + O2

HO2 + H211⇀ H2O2 + H

H2O2 + M12⇀ 2OH + M


N

Introduction

Skeletal mechanism for very lean flames


H + O21

⇋ OH + O

H2 + O2

⇋ OH + H

H2 + OH3

⇋ H2O + H

H + O2 + M4

⇀ HO2 + M

HO2 + H5

⇀ 2OH

HO2 + H6

⇋ H2 + O2

HO2 + OH7

⇀ H2O + O2

Simplification

Reactions 1-7 describe accurately leandeflagrations atatmospheric andmoderately elevatedpressures

0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

φ

vl[c

m/s

]


N

Introduction

One-step reduced mechanism

1. H+O2 ⇋ OH+O2. H2+O ⇋ OH+H3. H2+OH ⇋ H2O+H4f. H+O2+M → HO2+M5f. HO2+H → OH+OH6f. HO2+H → H2+O2

7f. HO2+OH → H2O+O2

CH2= −ω2 − ω3 + ω6f

CO2= −ω1 − ω4f + ω6f + ω7f

CH2O = ω3 + ω7f

CO = ω1 − ω2

COH = ω1 + ω2 − ω3 + 2ω5f − ω7f

CH = −ω1 + ω2 + ω3 − ω4f − ω5f − ω6f

CHO2= ω4f − ω5f − ω6f − ω7f


N

Introduction




CH2= −ω2 − ω3 + ω6f

CO2= −ω1 − ω4f + ω6f + ω7f

CH2O = ω3 + ω7f

CO = ω1 − ω2

COH = ω1 + ω2 − ω3 + 2ω5f − ω7f

CH = −ω1 + ω2 + ω3 − ω4f − ω5f − ω6f

CHO2= ω4f − ω5f − ω6f − ω7f

CH2+

{

CO + 12 COH + 3

2 CH − 12 CHO2

}

= −2ω4f

CO2+

{

CO + 12 COH + 1

2 CH + 12 CHO2

}

= −ω4f

CH2O −

{

CO + CH − CHO2

}

= 2ω4f


N

Introduction




CH2= −ω2 − ω3 + ω6f

CO2= −ω1 − ω4f + ω6f + ω7f

CH2O = ω3 + ω7f

CO = ω1 − ω2 = 0

COH = ω1 + ω2 − ω3 + 2ω5f − ω7f = 0

CH = −ω1 + ω2 + ω3 − ω4f − ω5f − ω6f =

CHO2= ω4f − ω5f − ω6f − ω7f = 0

CH2+

((((((((((((((({

CO + 12 COH + 3

2 CH − 12 CHO2

}

= −2ω4f

CO2+

((((((((((((((({

CO + 12 COH + 1

2 CH + 12 CHO2

}

= −ω4f

CH2O −��{

CO + CH − CHO2

}

= 2ω4f

One-step reaction among the main chemical species

2H2 + O2 → 2H2O (ω4f = k4f CMCO2CH)


N

Introduction


CO = ω1 − ω2 = 0

COH = ω1 + ω2 − ω3 + 2ω5f − ω7f = 0

CH = −ω1 + ω2 + ω3 − ω4f − ω5f − ω6f = 0

CHO2= ω4f − ω5f − ω6f − ω7f = 0


N

Introduction


CO = ω1 − ω2 = 0 = k1f CO2CH − k1bCOHCO − k2f CH2

CO + k2bCOHCH

COH = ω1 + ω2 − ω3 + 2ω5f − ω7f = 0

CH = −ω1 + ω2 + ω3 − ω4f − ω5f − ω6f = 0

CHO2= ω4f − ω5f − ω6f − ω7f = 0


N

Introduction


CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)


N

Introduction


CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

COH =k2f CH2

Hk1b

(k1f

αk4f CM

− 1

)


N

Introduction


CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

COH =k2f CH2

Hk1b

(k1f

αk4f CM

− 1

)

CO =αk3f CH2

Gk1b

(k1f

αk4f CM

− 1

)


N

Introduction


CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

COH =k2f CH2

Hk1b

(k1f

αk4f CM

− 1

)

CO =αk3f CH2

Gk1b

(k1f

αk4f CM

− 1

)

CHO2=

k3f

(f + G)k7f

CH2


N

Introduction


CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

COH =k2f CH2

Hk1b

(k1f

αk4f CM

− 1

)

CO =αk3f CH2

Gk1b

(k1f

αk4f CM

− 1

)

CHO2=

k3f

(f + G)k7f

CH2

f =k5f + k6f

k7f

k3f

k4f CM

CH2

CO2

H =1

2+

1

2

»

1 + 4γ2b f1

α

„

k1f

αk4f CM

− 1

«–1/2

G =1 + γ3b

2+

f

2

n

[1 + 2(3 + γ3b)/f + (1 + γ3b)2/f 2]1/2 − 1o

α =k6f f /(k5f + k6f ) + G

f + Gγ3b =

k3bCH2O

k4f CMCO2

γ2b =k7f

k5f + k6f

k2bk2f

k1bk3f


N

Introduction


CH =1

�HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

COH =k2f CH2

�Hk1b

(k1f

αk4f CM

− 1

)

CO =αk3f CH2

Gk1b

(k1f

αk4f CM

− 1

)

CHO2=

k3f

(f + G)k7f

CH2

f =k5f + k6f

k7f

k3f

k4f CM

CH2

CO2

H =1

2+

1

2

»

1 + 4γ2b f1

α

„

k1f

αk4f CM

− 1

«–1/2

≃ 1

G =1 + γ3b

2+

f

2

n

[1 + 2(3 + γ3b)/f + (1 + γ3b)2/f 2]1/2 − 1o

α =k6f f /(k5f + k6f ) + G

f + Gγ3b =

k3bCH2O

k4f CMCO2

γ2b =k7f

k5f + k6f

k2bk2f

k1bk3f≪ 1


N

Introduction


−20 −10 0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x [mm]

Xi

200

300

400

500

600

700

800

900

1000

1100

1200

1300

T [K]

6 8 100

1

2

x 10−5

x [mm]

Xi

H2O

H2

103×H

10−3×H2

T

Tcω4f = k4f CMCO2

CH

CH =1

HG

k2f k3f C2H2

k1bk4f CMCO2

(k1f

αk4f CM

− 1

)

H + O21f→ OH + O

H + O2 + M4f→ HO2 + M

The concentration of the radicals H, O and OH vanish at acrossover temperature Tc defined by k1f = αk4f CM.

ω =1

HG

(k1f

αk4f CM

− 1

)k2f k3f

k1b

C 2H2

if k1f > αk4f CM

ω = 0 if k1f < αk4f CM


N

Introduction

Kinetically-controlled lean flammability limit

0.2 0.3 0.4 0.5 0.6 0.7800

1000

1200

1400

1600

1800

2000

φ

T[K

]

φl = 0.251, p = 1atm, Tu = 300K

T∞

(Tc)l

k1f = αk4f CM → Tc

α =k6f f /(k5f + k6f ) + G

f + G

f =k5f + k6f

k7f

k3f

k4f CM

CH2

CO2

The crossover temperature at the lean flammability limit (Tc)l isdefined by k1f = k4f CM because α = 1 for CH2

≪ 1.

Flames can not exist for values of the equivalence ratio φ < φl , suchthat T∞ < (Tc)l .


N

Introduction

Numerical computation of planar flames

H2-air at p = 1 atm and Tu = 300 K

0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

φ

vl[c

m/s]

Solid curve: 21-step mech.Dashed curve: 7-step mech.

ω =1

HG

(k1f

αk4f CM

− 1

)k2f k3f

k1b

C 2H2


N

Introduction

Numerical computation of planar flames

H2-air at p = 1 atm and Tu = 300 K

0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

φ

vl[c

m/s]

Solid curve: 21-step mech.Dashed curve: 7-step mech.Thick dot-dashed: 1-stepThin dot-dashed: 1-step (H = 1)

ω =1

HG

(k1f

αk4f CM

− 1

)k2f k3f

k1b

C 2H2


N

Introduction

Very lean flames and flammability limit

10−1

100

101

0

50

100

150

200

250

300

350v

l[c

m/s]

φ



N

Introduction

Lean hydrogen-air flame ballsRonney’s experiments

on space shuttle (1997)

10−1

100

101

0

50

100

150

200

250

300

350

vl[c

m/s]

φ


r [m]

ωH

2[K

g/m

3s]

0 0.01 0.02 0.030

0.5

1

0

0.05

0.1

T/Tmax

ωH2

H2O

O2

H2

DIFFUSION+SORET

CONDUCTION+

RADIATION

fr


N

Introduction

Detailed numerical description of steady flameballs

1r2

ddr

[λr2 dTdr

] = QR −∑

i hoi mi

1r2

ddr

[ρDi r2(dYi

dr+ αiYi

TdTdr

)] = mi

dTdr

= dYi

dr= 0 at r = 0

T (∞) − T∞ = Yi(∞) − Yi∞ = 0

mi : San Diego 21-step mechanism with 8 reacting species (O2, H2,H2O, O, H, OH, HO2, H2O2)

Molecular diffusion: Fick’s Law with Smooke’s model:(λ/cp)/(λ/cp)0 = (T/T0)

0.7, Lei = constant

Thermal diffusion with αH = −0.23 and αH2= −0.29

QR : Statistical Narrow Band model (SNB)


N

Introduction

Detailed numerical description of steady flameballs

Detailed chemistry + SNB radiation model

φ

r f[m]

0 0.1 0.2 0.30.00

0.01

φ

Tmax[K]

0 0.1 0.2 0.3900

1000

1100

1200

1300


N

Introduction

One-step chemistry description

For H2-air mixtures near the lean flammability limit(Fernandez-Galisteo et al, C&F 156, 985-996, 2009) all chemicalintermediates have very small concentrations and are in steadystate, while the main species react according to

2H2 + O2 → 2H2O

with a rate given by{

IFk1f > αk4f CM : ω = 1GH

(k1f

αk4f CM− 1

)k2f k3fk1b

(ρYH2/WH2

)2

IFk1f ≤ αk4f CM : ω = 0

The crossover temperature, Tc , is defined from k1f = αk4f CM in

terms of the rates of the elementary reactions H + O21⇋ OH + O

and H + O2+M4f→ HO2 + M with a factor 1/6 ≤ α ≤ 1 that

depends on the local hydrogen content. Nondimensional activationenergy β ∼ 10 for k1f /(αk4f CM).


N

Introduction

One-step chemistry description

φ

r f[m]

0 0.1 0.2 0.30.00

0.01

φTmax[K]

0 0.1 0.2 0.3900

1000

1100

1200

1300


N

Introduction

Radiation models

φ

r f[m]

0 0.05 0.1 0.15 0.20.00

0.01SNB

Opt. thin

Gray

h=10 %

Dry air, h=0

h=20 %

Characteristic absorption length α−1a ∼ 10cm (αa ≡ absorption

coefficient

Optically thin approximation is accurate enough for description offlame-balls near extinction

QR = 4σαa(T4 − T 4

∞)


N

Introduction

Steady flame balls

The identity ∇YH2+ αH2

YH2∇T/T = T−αH2∇(TαH2YH2

)enables thermal diffusion to be incorporated in a single Fickian-likediffusion term as a function of Y = (T/T∞)αH2YH2

withincreased diffusivity D = (T/T∞)−αH2DH2

, so that

1r2

ddr

(λr2 dT

dr

)= 4κH2Oσp(W /WH2O)YH2O(T 4 − T 4

∞) − 2WH2qω

1r2

ddr

(ρDO2

r2

WO2

dYO2

dr

)

= − 1r2

ddr

(ρDH2Or2

2WH2O

dYH2O

dr

)

= 1r2

ddr

(ρDr2

2WH2

dYdr

)

= ω

with boundary conditions{

r = 0 : dT/dr = dYi/dr = 0r = ∞ : T − T∞ = Y − YH2∞

= YO2− YO2∞

= YH2O = 0


N

Introduction

Steady flame balls

The identity ∇YH2+ αH2

YH2∇T/T = T−αH2∇(TαH2YH2

)enables thermal diffusion to be incorporated in a single Fickian-likediffusion term as a function of Y = (T/T∞)αH2YH2

withincreased diffusivity D = (T/T∞)−αH2DH2

, so that

1r2

ddr

(λr2 dT

dr

)= 4κH2Oσp(W /WH2O)YH2O(T 4 − T 4

∞) − 2WH2qω

��XXXXXXXXXX1r2

ddr

(ρDO2

r2

WO2

dYO2

dr

)

=((((((((((((hhhhhhhhhhhh− 1

r2ddr

(ρDH2Or2

2WH2O

dYH2O

dr

)

= 1r2

ddr

(ρDr2

2WH2

dYdr

)

= ω

with boundary conditions{

r = 0 : dT/dr = dYi/dr = 0r = ∞ : T − T∞ = Y − YH2∞

= 0

Assuming for simplicity DH2O ∝ DO2∝ D leads to

YH2O = 2WH2O

WO2

DO2

DH2O(YO2∞

− YO2) =

WH2O

WH2

DDH2O

(YH2∞− Y )


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Introduction

The thin reaction-layer description

φ = 0.15

r [m]

wH2[Kg/m

3s]

0 0.01 0.02 0.030

0.5

1

0

0.05

0.1

T/Tmax

wH2

H2O

O2

H2

For β ≫ 1 the reaction occursin a thin layer whereYH2

/YH2∞∼ (T − Tc)/Tc ∼ β−1


N

Introduction

The reaction-sheet approximation withTmax = Tc

φ

r f[m]

0 0.1 0.2 0.30

0.01


N

Introduction

Characteristic scales near turning point1r2

ddr

(λr2 dT

dr

)= ��ZZQR − 2WH2

qω1r2

ddr

(ρDr2 dY

dr

)= 2WH2

ω

}

1r2

ddr

(λr2 dT

dr+ ρDqr2 dY

dr

)= 0

Integrating with boundary conditionsdTdr

= dYdr

= 0 at r = 0 and T − T∞ = Y − YH2∞= 0 as r → ∞

leads to (λ ∝ T ν , ρD ∝ T γ)

LH2cp∞T∞

1 + ν − γ

(T

T∞

)1+ν−γ

+ qY =LH2

cp∞T∞

1 + ν − γ+ qYH2∞

at the flame (Y = 0):(

Tf

T∞

)1+ν−γ= 1 +

(1+ν−γ)qYH2∞

LH2cp∞T∞

.

Integrating for r > rf with ω = 0:

(∂Y∂r

)

f += 1+ν−γ

1+ν(Tf /T∞)1+ν−1

(Tf /T∞)γ [(Tf /T∞)1+ν−γ−1]

YH2∞

rf

Across the flame: ρD4WH2

(∂Y∂r

)2

f +=

∫ ∞0 ωdY


N

Introduction

Characteristic scales(

Tf

T∞

)1+ν−γ= 1 +

(1+ν−γ)qYH2∞

LH2cp∞T∞

.

(∂Y∂r

)

f += 1+ν−γ

1+ν(Tf /T∞)1+ν−1

(Tf /T∞)γ [(Tf /T∞)1+ν−γ−1]

YH2∞

rf

ρD4WH2

(∂Y∂r

)2

f +=

∫ ∞0 ωdY ω = 1

GH

(k1f

αk4f CM− 1

)k2f k3fk1b

(ρYH2

WH2

)2

As Tf → Tc ,k1f

αk4f CM→ 1,

(∂Y∂r

)

f +→ 0, rf → ∞

φlo φ

rf

Q =0R

rc

φl

Extinction occurs for Tf − Tc ∼ β−1Tc (φl − φol ∼ β−1)

rc =(

β3DcGcHc

2(k2f k3f /k1b)c (ρcYH2∞/WH2

)

)1/2

ε = O[QR ]O[(∇(λ∇T )]

= 4κH2Ocσp Wair

WH2O

YH2OrT 3

c r2c

λc≪ 1


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Introduction

Extinction limit analysis for β−1 ∼ ε ln(ε−1)

RADIATION−FREE

8

T f

e −1

*

O(1)

YH2

T88

T0.5

1

rf/ r

c

ΦΦl

∆=0.5

∆=2

∆=5

The effect of far-field radiation introduces an apparent ambienttemperature T ∗

∞ < T∞ such that

(T∞ − T ∗∞)/T∞ ∼ ε ln(ε−1) ∼ β−1

Rf = rf /rc , Φ = β(φ − φol ), ∆ = β(T∞ − T ∗

∞)/T∞ ∼ O(1)


N

Introduction

Extinction limit results

φl = φol + β−1Φl

0 0.2 0.40

0.1

0.2

0.3φl

YO2

YN2

/( )∞

numerical results

analytic results

0 0.1 0.20

0.1

0.2φl

YH2O ∞

numerical results

analytic results


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Introduction

(Some) Conclusions

Reduced-kinetic mechanisms appropriate for low-temperatureignition and ultra-lean premixed combustion have been derived andused to develop explicit analytic expressions for quantities ofpractical interest in connection with safety applications (i.e., ignitiontimes and flammability limits).

The reduced-kinetic descriptions can be used to shortencomputational times in numerical calculations and can also aidfurther analytical work on deflagration and flame-ball stability.


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the reduced-kinetic description of hydrogen-air premixed...

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