SIMPLIFIED FORMULATIONS FOR TURBULENT COMBUSTION OF HYDROGEN

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22 Φεβ 2014 (πριν από 3 χρόνια και 10 μήνες)

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1
SIMPLIFIED FORMULATIONS FOR TURBULENT
COMBUSTION OF HYDROGEN
Forman A. Williams
UCSD, La Jolla, CA
General Basic Equations
Kolmogorov
Turbulence Scaling
Regimes of Turbulent Combustion
Approaches to Turbulent Combustion
Chemical-Kinetic Approximations
Transport-Property Approximations
Simplified Conservation Equations
Turbulent Hydrogen Diffusion Flames
2
REACTING NAVIER-STOKES CONSERVATION
EQUATIONS
(
)
.
0
t
=
ρ
ν


+

ρ

()

=
+
ρ
Ρ



=
ν


ν
+

ν

N
1
i
i
i
f
Y
t
()
[]
()

=



+

+

Ρ



+


=






+


+






+


N
i
i
i
i
q
V
v
f
Y
v
pU
t
p
h
v
h
t
1
2
2
.
2
1
2
1
ρ
ν
ρ
ν
ρ
()
[
]
ρ
ρ



ρ
=


+


i
i
i
i
i
V
Y
w
Y
v
t
Y
3
(
)
()
()
[
]
T
v
v
U
v
3
2
p

+

µ















κ

µ
+
=
Ρ
()
R
j
i
ij
i
Ti
j
N
1
j
N
1
i
N
1
i
0
i
i
i
q
V
V
D
W
D
X
T
R
V
Y
h
T
q
+









+
ρ
+

λ

=



=
=
=
()
()
()
N.

1,...,
i

,

1
1
1
=


























+

+










+









=




=
=
=
T
T
Y
D
Y
D
D
X
X
f
f
Y
Y
p
p
p
X
Y
V
V
D
X
X
X
i
Ti
j
Tj
ij
j
i
N
j
N
j
j
i
i
i
i
i
i
j
ij
j
i
N
j
i
ρ
ρ
()()
N

1,...,
i

,
T
R
p
X
e
T
B
W
w
k
,
v
0
j
N
1
j
T
R
/
E
k
k
'
ik
'
'
ik
M 1
k
i
i
'j
0
k
=






ν

ν
=


=

α
=
4
()
i
i
N
1
i
0
W
Y
T
R
p

=
ρ
=

=
=
N
1
i
i
i
Y
h
h
N

1,...,
i

,
dT
c
h
h
T
T
i
,
p
o
i
i
0
=
+
=

(
)
()
N.

1,...,
i

,
W
Y
W
Y
X
N
1
j
j
j
i
i
i
=
=

=
5
6
KOLMOGOROV SCALING
=
'
u
RMS velocity fluctuation
=
l
Integral scale ,
ν
/
'
lu
Rl
=
Turbulence Reynolds Number:
∈=
Rate of dissipation of turbulent kinetic energy
(
)
4
/
1
3

=
ν
k
l
4
/
3
l
l
R
=
Kolmogorov
scale:
dk
k
E
dk
k
k
)
(
1
1

is


and

+
Kinetic energy in eddies of sizes between
(
)
3
/
5
3
/
2
k
k
E

=

for the inertial range
Kolmogorov
scaling:
7
Regimes of turbulent combustion in a diagram of a length scale of the system and a representative
average velocity.
8
l
R
l
D
Regimes of turbulent combustion in a diagram of a turbulence Reynolds number and a Damköhler
number, both based on the integral scale of the turbulence.
,
R
D
D
c
k
K
l
l
=
τ
τ
=
9
Illustration of hydrocarbon-air flamelet
structure as predicted by rate-ratio asymptotics.
10
Regimes of nonpremixed
turbulent comb
ustion in a diagram of a scalar-dissipation parameter related to a
Damköhler
number and the ra
tio of a representative mixture-fraction fluctuation to the mixture-fraction
range that spans a diffusion flamelet.
2
Z
2

ν
=
χ
11
CATEGORIES OF APPROACHES TO ANALYSIS OF TURBULENT COMBUSTION
1.
Phenomenological
a.
Quasidimensional
b.
Age Theories
c.
Linear Eddy/One-Dimensional Turbulence
2.
Fluids-Based
a.
Direct Numerical Simulation (DNS)
b.
Large-Eddy Simulation (
LES)
c.
Moment Methods (RANS)
i.
Algebraic Closures
ii.
k-
ε
Modeling
iii
Reynolds-Stress Closure
3.
Turbulent Burning Velocity (ST)
a.
Perturbations for Low Intensities and Large Scales
b.
Moment-Method Modeling
of the G Equation
c.
Modeling
Flame-Surface Evolution, such as Coherent-Flamelet
Models (CFM)
d.
Fractals
e.
G-Equati
ons Renormalization
f.
Pseudosolitons
4.
Probability-Density Function (PDF)
a.
Flamelets
b.
Presumed PDF
i.
P(Z) for Diffusion Flames
ii
P(c) for Premixed Flames
iii
P(G) for Premixed Flames
c.
Conditi
onal Moment Closure (CMC)
d.
PDF Transport
i.
Linear Mean-Squar
e Es
tim
at
ion (LMSE),
also call
ed Interaction by Exchange with the Mean (IEM)
ii
Coalescence-Dispersion (CD)
iii.
Mapping Closure (MC)
iv.
Euclidean Minimum Spanning Tree (EMST)
12
FRONT-TRACKING EQUATION
G
S
G
v
t
G
L

=


+


MIXTURE-FRACTION EQUATION
(
)
ρ

ρ


=


+


Z
D
Z
v
t
Z
th
13
TWO-STEP CHEMICAL-KINETIC DESCRIPTION
H
2
O
H
2
O
H
3
2
2
2
+

+
I.
2
H
H
2

II.
O
OH
O
H
2
+

+
1.
H
OH
H
O
2
+

+
2.
H
O
H
H
OH
2
2
+

+
3.
M
H
M
H
2
2
+

+
M
HO
M
O
H
2
2
+

+
+
OH
2
HO
H
2

+
4.
Not Reversible
5.
6.
2
2
2
O
H
HO
H
+

+
7.
14
(
)
5
f
7
f
7
6
f
7
ω
ω
=
ω
+
ω
ω
=
α
(
)
b
7
5
b
1
f
1
I
1
ω
+
ω
α

+
ω

ω
=
ω
5
4
II
ω
+
ω
=
ω
PARTIAL-EQUILIBRIUM APPROXIMATIONS
(
)
2
2
H
3
O
H
H
OH
X
K
X
X
X
=
(
)
2
H
3
2
O
H
2
H
O
2
2
X
K
K
X
X
X
=
O
H
2
/
1
O
2
/
3
H
3
2
/
1
2
2
/
1
1
H
2
2
2
X
X
X
K
K
K
X
=
2
/
1
O
2
/
1
H
2
/
1
2
2
/
1
1
OH
2
2
X
X
K
K
X
=
O
H
O
H
3
1
O
2
2
2
X
X
X
K
K
X
=
15
TRANSPORT-PROPERTY APPROXIMATIONS
()
1
D
Sc
ij
ij
=
ρ
µ
=
1
c
Pr
p
=
λ
µ
=
=
=
p
pi
c
c
=
j
i
D
DTi
= 0,
constant
constant, µ
= constant or C = ρµ
= constant.
and
,
,
,
0

=
κ
16
0
1
2
3
4
5
012345
A
B
C
Schematic illustration of dependences of turbulent burning velocities on turbulence intensity.
17
SIMPLIFIED CONSERVATION EQUATIONS
,
v
t
Dt
D


+


=
(
)
[
]
i
i
i
i
w
Y
S
Dt
DY
=

µ


ρ
i
N
1
i
c
i
T
Y
h
h
h

=

=
()
[]

=






=

µ


ρ
N
1
i
R
i
o
i
T
T
q
w
h
t
p
h
Pr
Dt
Dh
(
)
(
)
[
]
II
'
iII
'
'
iII
I
'
iI
'
'
iI
i
i
v
v
v
v
W
w
ω

+
ω

=
18
(
)
(
)
[
]
i
i
i
i
i
Y
S
Dt
DY
Y
L

µ



ρ
=
(
)
(
)
[
]
T
T
T
T
h
Pr
Dt
Dh
h
L

µ



ρ
=
()
(
)
I
O
2
O
O
O
H
O
H
O
H
2
W
Y
L
2
W
Y
L
2
2
2
2
2
ω
=

=
(
)
I
II
H
H
H
3
W
Y
L
2
2
2
ω

ω
=
(
)
II
I
H
H
H
3
2
W
Y
L
ω

ω
=
19
(
)
II
II
I
I
T
T
Q
Q
h
L
ω
+
ω
=
H
o
H
O
H
o
O
H
I
W
h
2
W
h
2
Q
2
2


=
H
o
H
II
W
h
2
Q
=
()
(
)
(
)
II
O
O
O
H
H
H
O
H
O
H
O
H
2
W
Y
L
2
W
Y
L
W
Y
L
2
2
2
2
2
2
2
2
2
ω
=

=

=
(
)
(
)
,
Q
Q
h
L
II
II
I
T
T
ω
+
=
20
NONPREMIXED TURBULENT BURKE-SCHUMANN
HYDROGEN COMBUSTION
(
)
(
)
[
]
st
F
O
st
st
L
Z
1
L
L
Z
Z
Z

+
=
Pr
S
L
2
O
O
=
Pr
S
L
2
H
F
=
for Z < ZL; L = LF
for Z > ZL
(
)
Z
D
Dt
LDZ
th

ρ


=
ρ
(
)
Pr
µ
ρ
=
th
D
O
L
L
=
Everywhere:
21
(
)
()
L
L
H
II
I
T
H
F
Z
Z
H
Z
Y
Q
Q
h
W
L
2
2
2
+
+
=

(
)
(
)
L
L
H
H
Z
1
Z
Z
Y
Y
2
2


=

(
)(
)
(
)
[
]
L
L
H
II
I
T
H
F
Z
1
Z
1
H
Z
Y
Q
Q
h
W
L
2
2
2


+
+
=

(
)
H
D
Dt
NDZ
Dt
DH
th

ρ


=
ρ
+
ρ
()
L
O
Z
L
1
N

=
for Z < ZL;
(
)
(
)
L
F
Z
1
1
L
N


=
for Z > ZL
(
)
L
O
O
Z
Z
1
Y
Y
2
2

=

For Z < Z
L:
For Z > ZL:
Everywhere:
22
FINITE-RATE CHEMISTRY NONPREMIXED
TURBULENT HYDROGEN COMBUSTION (1)
H
2
O
H
2
O
H
3
2
2
2
+

+
temperature.
2
H
H
2

O
2H
O
2H
2
2
2

in a thin reaction zone at the highest
I.
in thicker recombination
+
and/or
II.
zones on each side.
Partial equilibrium of
O
OH
O
H
2
+

+
occurs at the reaction zone of I.
3
2
/
1
2
2
/
1
1
H
K
K
K
K
=
O
H
1/2
O
H
H
2
2
/X
X
K
X
=
2
/
1
2
2
/
1
1
OH
K
K
K
=
1/2
O
1/2
H
OH
OH
2
2
X
X
K
X
=
O
H
O
H
O
O
2
2
2
/X
X
X
K
X
=
KO
= K1K3
23
THE PARTIAL-EQUILIBRIUM QUANTITIES KH, KOH
and KO
AS FUNCTIONS OF TEMPERATURE
24
FINITE-RATE CHEMISTRY NONPREMIXED
TURBULENT HYDROGEN COMBUSTION (2)
(
)
()
2
2
2
2
2
2
O
O
O
O
H
O
H
O
H
W
Y
L
2
W
Y
L

=
()
(
)
(
)
2
2
2
2
2
2
H
H
H
O
O
O
H
H
H
W
Y
L
2
W
Y
L
4
W
Y
L

=
()
(
)
(
)
[]
(
)
2
2
2
2
2
2
2
2
2
O
O
O
I
O
O
O
H
H
H
II
T
T
W
Y
L
Q
W
Y
L
3
W
Y
L
Q
h
L


=
()
(
)
5
4
O
O
O
H
H
H
2
2
2
2
2
2
W
Y
L
3
W
Y
L
ω
+
ω
=

,
0
Y
2
O
=
0
Y
2
H
=
At the stoichiometric
surface of
H
2
O
H
2
O
H
3
2
2
2
+

+
(Z
st=0.036, not 0.024)
25
CONCLUSIONS
Turbulent hydrogen combustion obeys the reacting Navier-Stokes equations with
fluid-mechanical turbulence.
There are four broad categories of approaches to the description
of turbulent
combustion, with optimal approaches differing in different regimes of regime
diagrams.
Of the flamelet
and distributed-reaction limiting regimes, most applications lie
closer to the flamelet
regime.
A formulation specific to hydrogen can be based on two-step reduced chemistry
and account for finite-rate recombination in nonpremixed
turbulent combustion.
It is important to account for Schmidt and Prandtl
numbers different from unity in
treating turbulent hydrogen combustion.