Fall Technical Meeting
of the Eastern States Section of the Combustion Institute
Hosted by the University of Connecticut, Storrs, CT
Oct 9-12, 2011
Rate-ratio asymptotic analysis of the structure and
extinction of partially premixed flames
K. Seshadri1 and X. S. Bai2
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 elucidate 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
established between a stream of premixed fuel-rich mixture of methane (CH4), oxygen (O2), and nitrogen
(N2) and a stream of fuel-lean mixture of CH4, O2, and N2. The levels of premixing are given by the
equivalence ratios φr of the fuel-rich mixture and φl of the fuel-lean mixture. The value of φr depends on the
mass fraction of oxygen, YO2,r, in the fuel-rich mixture, and the value of φl depends on the mass fraction of
fuel, YF,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 φr and φl. The analysis
shows that the value of the scalar dissipation rate at extinction increases with increasing YF,l for YO2,r = 0,
while it decreases with increasing YO2,r for YF,l = 0.
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
reactant. 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 analysis 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]. The 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]. Experimental data and numerical calculations performed using a detailed
mechanism show that the value of the strain rate at extinction, aq, 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 aq to decrease with addition of fuel to
the oxidizer stream and to increase when oxygen is added to the fuel 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 the 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-rich stream, is made up of a mixture
of methane (CH4) and nitrogen (N2) or a premixed fuel-rich mixture of CH4, oxygen (O2), and
N2. The other stream, called the oxidizer stream or fuel-lean stream is made up of a mixture of
O2 and N2 or a fuel-lean mixture of CH4, O2, and N2 . 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 φr = 4.0 YF,r/YO2,r. Here, Yi 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 in the fuel-lean stream is φl = 4.0
YF,l/YO2,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 φr and φl. The present study is
carried out for values of φr > 3.0 and φl < 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-ratio
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]. This 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/ (ρcp)] |∇ξ|2, where λ is the thermal conductivity, ρ the density, and cp 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 YF,r, YF,l, YO2,r, and YO2,l and the characteristic overall
Damk¨ohler number, Δo. The quantity, Δo, is defined as the ratio between the characteristic
residence time and the characteristic reaction time. For large values of Δo the flow-field
comprises two chemically inert regions separated by a thin reaction zone [1, 3, 6]. In the limit
Δo→∞, the thickness of the reaction zone approaches zero. The stoichiometric mixture fraction,
ξst, is evaluated using the equation
ξst ≡ (YO2,l − 4YF,l)/(YO2,l + 4YF,r − YO2,r − 4YF,l)
(1)
The quantity ξst represents the position of the reaction zone in the limit Δo→∞ with the Lewis
number of the reactants assumed to be equal to unity. The Lewis number is defined as Lei =
λ/(ρcpDi), where Di is the diffusion coefficient of species i. The adiabatic temperature Tst can be
calculated for given values of the mass fraction of reactants at the boundaries. A relation between
Tst 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 YF,r, YF,l, YO2,r, and YO2,l so chosen that the stoichiometric mixture fraction, ξst, and
adiabatic flame temperature, Tst are ξst = 0.1 and Tst = 2000 K, respectively. These conditions
were also employed in previous experimental and numerical study [5]. Since the values of ξst and
Tst are held constant, changes in the values of the critical conditions of extinction can be
attributed to changes in flame structure as a result of premixing. Fixed values of ξst and Tst place
two restrictions on the values of YF,r, YF,l, YO2,r, and YO2,l. It fixes values of two of the four
independent mass fractions at the boundaries. 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 φr−1 = 0, and the studies are carried out for various values of φl. In the second set, φl = 0,
and studies are conducted for various values of φr.
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]
CH4 + 2H + H2O CO + 4H2,
CO + H2O CO2 + H2,
H + H + M H2 + M,
O2 + 3 H2 2 H + 2 H2O.
I
II
III
IV
This four-step mechanism was employed in previous rate-ratio asymptotic analysis of
nonpremixed 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, identify 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 kn = BnTαnexp[−En/(R^T)], where T denotes the
temperature and R^ is the universal gas constant. The quantities Bn, αn, and En 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 Kn. The concentration of the third
body CM is calculated using the relation CM = [pW¯ /(R^T)]∑ni=1 ηiYi/Wi where p denotes the
pressure, ¯W is the average molecular weight and Wi and ηi, are respectively the molecular
weight and the chaperon efficiency of species i. For the elementary reaction 5, the chaperon
efficiencies [M] = 6.5[CH4] + 1.5[CO2] + 0.75[CO] + 0.4[N2] + 6.5[H2O] + 0.4[O2] +
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 wI = w7f − w7b −
w8, wII = w6f − w6b, wIII = w5 + w8, wIV = w1f − w1b. 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 elementary reactions employed in the asymptotic analysis. Units are moles,
cubic centimeters, seconds, kJoules, Kelvin.
Number
1f
Reaction
O2 + H → OH +
Bn
2.000E+14
αn
0.00
En
70.30
1b
2f
2b
3f
3b
4f
4b
5
6f
6b
7f
7b
8
9
O
O + OH → H +
1.568E+13
O2
H2 + O → OH +
5.060E+04
H
H + OH → O +
2.222E+04
H2
H2 + OH → H2O
1.000E+08
+H
H + H2O → OH
4.312E+08
+ H2
OH + OH →
1.500E+09
H2O + O
O + H2O → OH
1.473E+10
+ OH
H + O2 + M →
2.300E+18
HO2 + M
CO + OH → CO2 4.400E+06
+H
H + CO2 → OH
4.956E+08
+ CO
CH4 + H → CH3
2.200E+04
+ H2
CH3 + H2 → CH4 8.391E+02
+H
CH3 + H → CH4 6.257E+23 2.10
k0
8E+14
k∞
CH3 + O →
7.000E+013
CH2O + H
0.00
3.52
2.67
26.3
2.67
18.29
1.60
13.80
1.60
76.46
1.14
0.42
1.14
71.09
-0.80
0.00
1.50
-3.10
1.50
89.76
3.00
36.60
3.00
34.56
-1.80 0.00
0.00 0.00
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 reactions 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 = ξp. The value of ξp 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 structure of the reaction zone.
For convenience, the definition
Xi≡YiWN2/Wi
(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, φr−1 = 0
5. Concluding Remarks
The rate-ratio asymptotic analysis described here elucidates the influence of flame structure on
critical conditions of extinction. The analysis 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 activation 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 activationenergy asymptotic analysis, oxygen is completely consumed and fuel leaks from the reaction
zone for small ξst. Experimental data and numerical calculations using skeletal chemistry show
oxygen leakage, thereby confirming the prediction of rate-ratio 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 Company, 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, New 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, Heidelberg, 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 Flame 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.