Rate-Ratio Asymptotic Analysis of the Structure and Extinction

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

1Department of Mechanical and Aerospace Engineering, University of California at San Diego,
La Jolla, California 92093-0411, USA
2Department 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, I
CO + H2O CO2 + H2, II
H + H + M H2 + M, III
O2 + 3 H2 2 H + 2 H2O. 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.

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 activation-energy 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.

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