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.

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