Rate-ratio asymptotic analysis of the structure and extinction of partially premixed flames

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2007 Fall Technical Meeting

Eastern States Sect
ion of the Combustion Institute

University of Virginia

October 21
-
24, 2007

Rate
-
ratio asymptotic analysis of the structure and extinction of
partially premixed flames

K. Seshadri
1

and X. S. Bai
2

1
Department of

Mechanical and Aerospace Engineering, University of California at San Diego,

La Jolla, California 92093
-
0411, USA

2
Department of Energy Sciences, Lund Institute of Technology, S 221 00 Lund, Sweden

Rate
-
ratio asymptotic analysis is carried out to elucidat
e the structure and mechanisms of
extinction of laminar, partially premixed methane flames. A reduced chemical
-
kinetic mechanism
made up of four global steps is used. The counterflow configuration is employed. This
configuration considers a flame establish
ed between a stream of premixed fuel
-
rich mixture of
methane (CH
4
), oxygen (O
2
), and nitrogen (N
2
) and a stream of fuel
-
lean mixture of CH
4
, O
2
, and
N
2
. The levels of premixing are given by the equivalence ratios
φ
r

of the fuel
-
rich mixture and
φ
l

of the fuel
-
lean mixture. The value of
φ
r

depends on the mass fraction of oxygen,
Y
O2
,
r
, in the fuel
-
rich mixture, and the value of
φ
l

depends on the mass fraction of fuel,
Y
F
,
l

in the fuel
-
lean mixture.
The mass fraction

of the reactants at the boundaries are so chosen that the diffusive flux of
reactants entering the reaction zone is the same for all values of
φ
r

and
φ
l
. The analysis shows that
the value of the scalar dissipation rate at extinction increases with increas
ing
Y
F
,
l

for
Y
O2
,
r

= 0,
while it decreases with increasing
Y
O2
,
r

for
Y
F
,
l

= 0.

{note: your short abstract above should also be submitted electronically so that is may appear in
the bound Program Book}

1.

Introduction

The characteristics of reactive flows

depend on the rate of mixing of reactants, fuel and oxygen,
and on the rates of chemical reactions taking place in the flow
-
fields [1, 2]. In partially premixed
combustion one or both reactant streams of a nonpremixed system is premixed with the other
rea
ctant. Asymptotic flame theories provide valuable insights on mixing and chemical reactions
taking place during combustion [1

3]. Peters [4] analyzed the structure and mechanisms of
extinction of partially premixed flames. An activation
-
energy asymptotic a
nalysis was carried
out and critical conditions of extinction were obtained [4]. An experimental and numerical study
was carried out previously to elucidate various aspects of the structures and mechanisms of
extinction of partially premixed flames [5]. Th
e influence of premixing one reactant stream of a
nonpremixed system with the other reactant on structures and critical conditions of extinction
were examined in detail [5]. The counterflow configuration was employed. The fuel used was
methane [5]. Experim
ental data and numerical calculations performed using a detailed
mechanism show that the value of the strain rate at extinction,
a
q
, increases with addition of fuel
to the oxidizer stream of a nonpremixed system, while the opposite is observed when oxygen
is
added to the fuel stream [5]. Numerical calculations with one
-
step chemistry and results of
activation
-
energy asymptotic analysis show the value of
a
q

to decrease with addition of fuel to
the oxidizer stream and to increase when oxygen is added to the f
uel stream.

The failure of activation
-
energy asymptotic theory [4] and the failure of numerical calculations
using one
-
step mechanism [5] to predict qualitative aspects of the effect of premixing on critical
conditions of extinction is the motivation for t
he present study. Here a rate
-
ratio asymptotic
analysis is performed at conditions similar to those employed in the previous experimental and
numerical study [5]. This permits direct comparison of the results obtained from rate
-
ratio
asymptotic analysis 3
with experimental data and results of numerical calculations with detailed
chemistry.

2.

Formulation

Steady, axisymmetric, laminar flow of two counterflowing streams toward a stagnation plane is
considered here. One stream, called the fuel stream or fuel
-
r
ich stream, is made up of a mixture
of methane (CH
4
) and nitrogen (N
2
) or a premixed fuel
-
rich mixture of CH
4
, oxygen (O
2
), and
N
2
. The other stream, called the oxidizer stream or fuel
-
lean stream is made up of a mixture of
O
2

and N
2

or a fuel
-
lean mixtur
e of CH
4
, O
2
, and N
2

. The origin is placed on the axis of
symmetry at the stagnation plane. The equivalence ratio of the reactant mixture in the fuel
-
rich
stream is
φ
r

= 4
.
0
Y
F
,
r
/Y
O2
,
r
. Here,
Y
i

is the mass fraction of species
i
. Subscript F refers to the fuel
and subscript r refers to conditions in the fuel stream and the fuel
-
rich stream far from the
stagnation plane. The equivalence ratio of the reactant mixture i
n the fuel
-
lean stream is
φ
l

= 4
.
0
Y
F
,
l
/Y
O2
,
l
. Here subscript l refers to conditions in the oxidizer stream and fuel
-
lean stream far from
the stagnation plane. The flame structure is characterized by
φ
r

and
φ
l
. The present study is
carried out for values o
f
φ
r

>
3
.
0 and
φ
l

<
0
.
45. At these

conditions previous
asymptotic and
numerical studies show that multiple reaction zones present in the reactive flow field of a
partially premixed flame merge at conditions close to extinction [4]. Since the present rate
-
r
atio
asymptotic analysis is focused on flame extinction, the merged structure is analyzed.

In asymptotic analysis of the flame structure, it is convenient to use a conserved scalar quantity
ξ
, called the mixture fraction, as an independent variable [4]. T
his conserved scalar is defined
such that
ξ

= 0 at the oxidizer stream or fuel
-
lean stream far from the stagnation plane and
ξ

= 1
at the fuel stream or fuel
-
rich stream far from the stagnation plane. The scalar dissipation rate,
χ

is given by
χ

= 2[
λ
/
(
ρ
c
p
)]
|

ξ
|
2
, where
λ

is the thermal conductivity,
ρ

the density, and
c
p

the heat capacity per unit mass of the mixture. The quantity
χ

plays a central role in asymptotic
analyses.

The structure of the reactive flow field depends on four independent boundary

values of mass
fractions of fuel and oxygen given by
Y
F
,
r
,
Y
F
,
l
,
Y
O2
,
r
, and
Y
O2
,
l

and the characteristic overall
Damk¨ohler number,
Δ
o
. The quantity,
Δ
o
, is defined as the ratio between the characteristic
residence time and the characteristic reaction tim
e. For large values of
Δ
o

the flow
-
field
comprises two chemically inert regions separated by a thin reaction zone [1, 3, 6]. In the limit
Δ
o
→∞
, the thickness of the reaction zone approaches zero. The stoichiometric mixture fraction,
ξ
st
, is evaluated using

the equation


ξ
st


(
Y
O2
,
l



4
Y
F
,
l
)
/
(
Y
O2
,
l

+ 4
Y
F
,
r



Y
O2
,
r



4
Y
F
,
l
)

(1)

The quantity
ξ
st

represents the position of the reaction zone in the limit
Δ
o
→∞

with the Lewis
number of the reactants assumed to be equal to unity. The Lewis number is defined as
Le
i

=
λ
/
(
ρ
c
p
D
i
), where
D
i

is the diffusion coefficient of species
i
. The adiabatic temperature
T
st

can
be calculated for given values of the mass fraction of reactants at the boundaries. A relation
between
T
st

and mass fraction of reactants at the boundaries
is given later. A systematic study of
the influence of partial premixing of reactants on critical conditions of extinction is carried out
with values of
Y
F
,
r
,
Y
F
,
l
,
Y
O2
,
r
, and
Y
O2
,
l
so chosen that the stoichiometric mixture fraction,
ξ
st
,
and adiabatic fla
me temperature,
T
st

are
ξ
st

= 0
.
1 and
T
st

= 2000 K, respectively. These
conditions were also employed in previous experimental and numerical study [5]. Since the
values of
ξ
st

and
T
st

are held constant, changes in the values of the critical conditions of
e
xtinction can be attributed to changes in flame structure as a result of premixing. Fixed values
of
ξ
st

and
T
st

place two restrictions on the values of
Y
F
,
r
,
Y
F
,
l
,
Y
O2
,
r
, and
Y
O2
,
l
. It fixes values of
two of the four independent mass fractions at the bound
aries. Following previous experimental
and numerical study [5], the remaining two 5 independent mass fractions are chosen as follows.
In one set, the study is carried out with only the reactant stream at the fuel
-
lean boundary
premixed. For this case
φ
r

1
= 0, and the studies are carried out for various values of
φ
l
. In the
second set,
φ
l

=
0, and studies are conducted for various values of
φ
r
.

3.

Reduced Mechanism

A reduced chemical
-
kinetic mechanism made up of four global steps is employed to describe the

combustion of methane. The four
-
step mechanism is [7]


CH
4

+ 2H + H
2
O CO + 4H
2
,

I


CO + H
2
O CO
2

+ H
2
,

II


H + H + M H
2

+ M,

III


O
2
+ 3 H
2

2 H + 2 H
2
O.

IV

This four
-
step mechanism was employed in previous rate
-
ratio asymptotic analysis of
nonpre
mixed methane flames [7]. Table 1 shows the elementary reactions which are presumed
to be the major contributors to the rates of the global steps of the reduced mechanism. The
symbols
f
and
b
appearing in the first column of Table 1, respectively, identif
y the forward and
backward steps of a reversible elementary reaction
n
. The rate data for the elementary steps are
the same as those employed in the previous rate
-
ratio asymptotic analysis of nonpremixed
methane flames [7]. This allows direct comparison of

the results obtained for partially premixed
flames with those for the nonpremixed flame. The reaction rate coefficients
kn
of the elementary
reactions are calculated using the expression
k
n

=
B
n
T
α
n
exp[

E
n
/
(
R^T
)], where
T
denotes the
temperature and
R^
is
the universal gas constant. The quantities
B
n
,

α
n
, and
E
n

are the frequency
factor, the temperature exponent, and the activation energy of the elementary reaction
n
. The
equilibrium constant for a reversible reaction is represented by
K
n
. The concentration

of the third
body
C
M

is calculated using the relation
C
M

= [
p
W
¯
/
(
R^T
)]

n
i
=
1

η
i
Y
i
/W
i

where
p
denotes the
pressure, ¯
W
is the average molecular weight and
W
i

and
η
i
, are respectively the molecular
weight and the chaperon efficiency of species
i
. For the el
ementary reaction 5, the chaperon
efficiencies [M] = 6.5[CH
4
] + 1.5[CO
2
] + 0.75[CO] + 0.4[N
2
] + 6.5[H
2
O] + 0.4[O
2
] +
1.0[Other]. The rate constant for reaction 8 is calculated using a formula given in Ref. [8]. The
reaction rates of the global steps
wk
in
the four
-
step mechanism (
k
= I

IV) are
w
I

=
w
7f



w
7b



w
8
,
w
II

=
w
6f



w
6b
,
w
III

=
w
5

+
w
8
,
w
IV

=
w
1f



w
1b
. The procedure used for evaluating the rates of
the global steps in the four
-
step mechanism is described in Ref. [7].

Table 1: Rate data for elemen
tary reactions employed in the asymptotic analysis. Units are moles,
cubic centimeters, seconds, kJoules, Kelvin.


Number

Reaction

B
n

α
n

E
n




1f

O
2

+ H


OH + O

2.000E+14

0.00

70.30

1b

O + OH


H + O
2

1.568E+13

0.00

3.52

2f

H
2

+ O


OH + H

5.060E+04

2.
67

26.3

2b

H + OH


O + H
2

2.222E+04

2.67

18.29

3f

H
2

+ OH


H
2
O + H

1.000E+08

1.60

13.80

3b

H + H
2
O


OH + H
2

4.312E+08

1.60

76.46

4f

OH + OH


H
2
O + O

1.500E+09

1.14

0.42

4b

O + H
2
O


OH + OH

1.473E+10

1.14

71.09

5

H + O
2
+ M


HO
2

+ M

2.300E+18

-
0
.80

0.00

6f

CO + OH


CO
2

+ H

4.400E+06

1.50

-
3.10

6b

H + CO
2



OH + CO

4.956E+08

1.50

89.76

7f

CH
4

+ H


CH
3

+ H
2

2.200E+04

3.00

36.60

7b

CH
3

+ H
2



CH
4

+ H

8.391E+02

3.00

34.56

8

CH
3

+ H


CH
4

k
0


k


6.257E+23

2.108E+14

-
1.80

0.00

0.00

0.00

9

CH
3

+ O


CH
2
O + H

7.000E+013

0.00

0.00


4.

Asymptotic Analysis

The Damköhler numbers constructed from the ratio of the characteristic residence time to the
characteristic chemical time obtained from the rates of elementary reacti
ons of the four
-
step
mechanism are presumed to be large. At conditions close to extinction, the reactive flow field is
presumed to be made up of a thin reaction zone where chemical reactions take place. This
reaction zone is presumed to be located at
ξ

=
ξ
p
. The value of
ξ
p

depends on
χ
. The
chemically inert regions outside this thin reaction zone is called the outer zone. The structure of
the outer zone is analyzed first. The analysis gives boundary conditions for differential equations
that describe the s
tructure of the reaction zone.

For convenience, the definition


X
i

Y
i
W
N2
/W
i


(2)

is introduced.

CONTINUED TEXT……..


Figure 1: Schematic illustration of the outer structure of a partially premixed methane flame
established between counterflowing streams
of methane mixed with nitrogen and fuel
-
lean
mixture of oxygen, nitrogen and methane,
φ
r

1

= 0

5.

Concluding Remarks

The rate
-
ratio asymptotic analysis described here elucidates the influence of flame structure on
critical conditions of extinction. The ana
lysis shows that premixing the reactant streams of a
nonpremixed system alters the outer structure. This has a significant influence on critical
conditions of extinction. The analysis also illustrates a fundamental difference between rate
-
ratio
and activat
ion energy asymptotic analysis. In the former, the characteristic Damköhler numbers
for reactions that consume fuel are larger than those for reactions that consume oxygen.
Therefore fuel is completely consumed and oxygen leaks from the reaction zone. In a
ctivation
-
energy asymptotic analysis, oxygen is completely consumed and fuel leaks from the reaction
zone for small
ξ
st
. Experimental data and numerical calculations using skeletal chemistry show
oxygen leakage, thereby confirming the prediction of rate
-
ra
tio asymptotic analysis. Thus the
present analysis overcomes an important limitation of activation
-
energy asymptotic analysis.

Acknowledgments

The research at the University of California at San Diego was supported by the U. S. Army
Research Office, grant
#W911NF
-
04
-
1
-
0139. Program manager Dr. Kevin McNesby. The
research at Lund Institute of Technology was supported by the Swedish research council (VR)
and CeCOST.

References

[1]

F. A. Williams,
Combustion Theory
, 2nd Edition, Addison
-
Wesley Publishing Compa
ny, Redwood City, CA,
1985.

[2]

N. Peters,
Turbulent Combustion
, Cambridge University Press, Cambridge, England, 2000.

[3]

A. Liñán, F. A.Williams,
Fundamental Aspects of Combustion
, Vol. 34 of
Oxford Engineering Science Series
,
Oxford University Press, N
ew York, 1993.

[4]

N. Peters,
Proceedings of the Combustion Institute
20 (1984) 353

360.

[5]

R. Seiser, L. Truett, K. Seshadri,
Proceedings of the Combustion Institute
29 (2002) 1551

1557.

[6]

K. Seshadri,
Proceedings of the Combustion Institute
26 (1996)
831

846.

[7]

X. S. Bai, K. Seshadri,
Combustion Theory and Modelling
3 (1999) 51

75.

[8]

N. Peters, in: N. Peters, B. Rogg (Eds.), Reduced Kinetic Mechanisms for Applications in Combustion
Systems, Vol. m15 of
Lecture Notes in Physics
, Springer
-
Verlag, Hei
delberg, 1993, Ch. 1, pp. 1

13.

[9]

M. D. Smooke (Ed.),
Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane
-
Air Flames
,
Vol. 384 of
Lecture Notes in Physics
, Springer Verlag, Heidelberg, 1991.

[10]

K. Seshadri, N. Ilincic,
Combustion and F
lame
101 (1995) 69

80.

[11]

J. S. Kim, F. A. Williams,
SIAM Journal on Applied Mathematics
53 (1993) 1551

1566.

[12]

A. Liñán,
Acta Astronautica
1 (1974) 1007

1039.