Upload
dinhnga
View
225
Download
5
Embed Size (px)
Citation preview
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
2 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
3 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
1 Selection of a detailed chemical-kinetic mechanism including acomplete set of chemical species and elementary reactions
3 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
1 Selection of a detailed chemical-kinetic mechanism including acomplete set of chemical species and elementary reactions
2 Deletion of elementary steps that do not contribute to thechemistry under the conditions of interest
3 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
1 Selection of a detailed chemical-kinetic mechanism including acomplete set of chemical species and elementary reactions
2 Deletion of elementary steps that do not contribute to thechemistry under the conditions of interest
3 Introduction of steady-state approximations for intermediatespecies with negligible transport rates
3 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
1 Selection of a detailed chemical-kinetic mechanism including acomplete set of chemical species and elementary reactions
2 Deletion of elementary steps that do not contribute to thechemistry under the conditions of interest
3 Introduction of steady-state approximations for intermediatespecies with negligible transport rates
4 Truncation of the steady-state algebraic expressions to facilitatenumerical computations
3 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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.
4 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Chemistry Reduction
Methodology
Validation for lean premixed systems: laminar deflagrations,homogeneous ignition, nonpremixed ignition
Validation for nonpremixed sytems: laminar strained diffusionflames
5 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
6 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
7 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
8 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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.
9 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Detailed H2 chemistry
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)
10 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Detailed H2 chemistry
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
10 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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.
11 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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)
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
11 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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)
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
11 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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)
Validation
Induction time of a stoichiometrichomogeneous mixture
11 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
ProductionConsumptionConvection+Diffusion
[mol/m3s]
(b) H
ρDYi
Dt−∇ · (ρDi∇Yi)
︸ ︷︷ ︸
transport
= ωp,iWi
︸ ︷︷ ︸
production
− ωc,iWi
︸ ︷︷ ︸
consumption
12 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
ProductionConsumptionConvection+Diffusion
[mol/m3s]
(b) H
(((((((((((ρDYi
Dt−∇ · (ρDi∇Yi)
︸ ︷︷ ︸
transport
= ωp,iWi
︸ ︷︷ ︸
production
− ωc,iWi
︸ ︷︷ ︸
consumption
→ ωp,i = ωc,i
12 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Reduced chemistry in H2-air flames
Steady-State Analysis
All intermediates but H are insteady state
13 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Reduced chemistry in H2-air flames
Steady-State Analysis
All intermediates but H are insteady state
H2 reduced mechanism
3H2 + O2I
⇋ 2H2O + 2H, ωI ≃ w1
2H + MII⇋ H2 + M, ωII ≃ w4f
13 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Reduced chemistry in H2-air flames
Steady-State Analysis
All intermediates but H are insteady state
H2 reduced mechanism
3H2 + O2I
⇋ 2H2O + 2H, ωI ≃ w1
2H + MII⇋ H2 + M, ωII ≃ w4f
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
13 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Reduced chemistry in H2-air flames
Steady-State Analysis
All intermediates but H are insteady state
H2 reduced mechanism
3H2 + O2I
⇋ 2H2O + 2H, ωI ≃ w1
2H + MII⇋ H2 + M, ωII ≃ w4f
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)
13 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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.
14 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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.
Steady state approximations
HO2 is not in steady-state during autoignition.
H2 + O2 ⇀ HO2 + H
HO2 + H ⇀ 2OH
HO2 + H ⇀ H2 + O2
14 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
3-step including HO2
H2 + O2III
⇋ HO2 + H
Induction time (s)
of a homogeneous mixtureT0 = 1200K, p=1atm.
Steady state approximations
HO2 is not in steady-state during autoignition.
H2 + O2 ⇀ HO2 + H
HO2 + H ⇀ 2OH
HO2 + H ⇀ H2 + O2
14 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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-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)
of a homogeneous mixtureT0 = 1200K, p=1atm.
14 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
15 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
15 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition above crossover
H2 reduced mechanism
3H2 + O2I
⇋ 2H2O + 2H
2H + MII
⇋ H2 + M
16 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition above crossover
H2 reduced mechanism
3H2 + O2I
⇋ 2H2O + 2H
2H + MII
⇋ H2 + M
ωI = k6bCH2CO2
+ k1f CO2CH
ωII = k4f CMCO2CH
16 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition above crossover
H2 reduced mechanism
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
16 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition above crossover
H2 reduced mechanism
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
CH = εCH2
[
e2(k1f −k4f CM)CO2t− 1
]
; ε =k6b
k1f − k4f CM
∼ 10−6
16 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition above crossover
H2 reduced mechanism
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
Branched-chain explosion
dCH
dt= 2k6bCH2
CO2+ 2(k1f − k4f CM)CO2
CH; CH(0) = 0
CH = εCH2
[
e2(k1f −k4f CM)CO2t− 1
]
; ε =k6b
k1f − k4f CM
∼ 10−6
16 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
17 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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 + 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)
17 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
Initial skeletal mechanism
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
18 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
Initial skeletal mechanism
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
18 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
Initial skeletal mechanism
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
3-step reduced mechanism
2H2 + O2I∗
→ 2H2O
H2O2 + H2
II∗
⇋ 2H2O
H2 + 2O2III
∗
⇋ 2HO2
18 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
Initial skeletal mechanism
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
3-step reduced mechanism
2H2 + O2I∗
→ 2H2O
H2O2 + H2
II∗
⇋ 2H2O
H2 + 2O2III
∗
⇋ 2HO2
ωI∗ = w1 + w6 + w7
ωII∗ = −w6 − w7 + w8
ωIII∗ =
w4 + w5 − 2w6 − w7
2
18 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
Initial skeletal mechanism
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
Steady-state intermediatesH, O, OH
3-step reduced mechanism
2H2 + O2I∗
→ 2H2O
H2O2 + H2
II∗
⇋ 2H2O
H2 + 2O2III
∗
⇋ 2HO2
ωI∗ = w1 + w6 + w7
ωII∗ = −w6 − w7 + w8
ωIII∗ =
w4 + w5 − 2w6 − w7
2
18 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
3-step reduced mechanism
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
19 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
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)
3-step reduced mechanism
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
19 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
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
3-step reduced mechanism
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
19 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
3-step Reduced Mechanism (Trevino, 1991)
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
3-step reduced mechanism
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
19 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
CHO2= w4 + w5 − 2w6 − w7 = 0
20 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
CHO2= w4 + w5 − 2w6 − w7 = 0
2-step reduced mechanism
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
20 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
CHO2= w4 + w5 − 2w6 − w7 = 0
2-step reduced mechanism
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
20 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
CHO2= w4 + w5 − 2w6 − w7 = 0 ⇒ CHO2
≃“
(2−α)w5+2w82(1−α)k6
”1/2
2-step reduced mechanism
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
20 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
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
21 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
2-step Reduced Mechanism
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
21 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
22 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
22 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Ignition time
dϕ
dτ= (a + ϕ)1/2eθ; ϕ(0) = 0
dθ
dτ= ϕeθ + Λ(a + ϕ)1/2eθ; θ(0) = 0
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
Initiation counts for τ ∼ a1/2 when ϕ ∼ θ ∼ a but it is negligible at later times
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
Initiation counts for τ ∼ a1/2 when ϕ ∼ θ ∼ a but it is negligible at later times
τi =∫∞
0dϕ
ϕ1/2 exp( 23 ϕ3/2+Λϕ)
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
Initiation counts for τ ∼ a1/2 when ϕ ∼ θ ∼ a but it is negligible at later times
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
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
Initiation counts for τ ∼ a1/2 when ϕ ∼ θ ∼ a but it is negligible at later times
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
23 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
24 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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)
24 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
CELLULAR FLAMESFLAME BALLS
25 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Skeletal mechanism for very lean flames
Skeletal mechanism12 elementary steps, 8 species
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
26 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
Skeletal mechanism for very lean flames
Skeletal mechanism12 elementary steps, 8 species
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
]
26 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
27 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
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
27 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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 = 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)
27 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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 = 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)
27 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
CH =1
HG
k2f k3f C2H2
k1bk4f CMCO2
(k1f
αk4f CM
− 1
)
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
CH =1
HG
k2f k3f C2H2
k1bk4f CMCO2
(k1f
αk4f CM
− 1
)
COH =k2f CH2
Hk1b
(k1f
αk4f CM
− 1
)
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
)
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
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
28 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
One-step reduced mechanism
−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
29 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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 .
30 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
31 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
31 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
CELLULAR FLAMESFLAME BALLS
32 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
CELLULAR FLAMESFLAME BALLS
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
33 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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)
34 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
35 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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).
36 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
37 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
∞)
38 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
39 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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 )
40 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
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
41 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
Introduction
The reaction-sheet approximation withTmax = Tc
φ
r f[m]
0 0.1 0.2 0.30
0.01
42 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
43 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
44 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
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)
45 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
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
46 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N
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.
47 / 47The reduced-kinetic description of hydrogen-air premixed-combustion problems relevant for safety applications
N