ACTIVATION ENERGY EFFECT ON FLAME PROPAGATION IN LARGE-SCALE VORTICAL FLOWS

Transcription

1 ACTIVATION ENERGY EFFECT ON FLAME PROPAGATION IN LARGE-SCALE VORTICAL FLOWS L. Kagan (a), P.D. Ronney (b) and G. Sivashinsky (a), Abstract (a) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel (b) Department of Aerospace and Mechanical Engineering, University of Southern California Los Angeles, CA , USA The propagation of a premixed flame through a large-scale vortical flow is studied numerically employing a conventional reaction-diffusion-advection model. It is shown that the response of the flame speed to the flow intensity is strongly influenced by the form of the reaction rate expression that describes the chemical kinetics, in particular the activation energy. For high-activation-energy kinetics typical of gaseous flames this response is characterized by a peculiar non-monotonicity, thereby reflecting the flow induced changes within the flame front structure and, hence, deviation from the classical Huygens propagation. At low activation energies, however, the non-monotonicity vanishes, which also helps to explain its absence in the isothermal autocatalytic reaction waves spreading through strongly stirred liquid solutions where the amplification factor of propagation speed may reach extremely high values compared to gaseous flames. Additionally it is shown that the transition from Huygens to non-huygens propagation occurs at nearly the same Karlovitz number for all activation energies, thereby showing the utility of this parameter for characterizing flame propagating in nonuniform flows when appropriately defined. 1

2 1 Introduction A comprehensive rational understanding of the flame-turbulence interaction remains one of the major challenges of premixed combustion [1]. To gain a better insight into the basic mechanisms involved, a numerical study of the reaction-diffusion-advection model for an equidiffusional premixed flame spreading through a space-periodic array of a large-scale vortices has been recently undertaken [2]. Rather unexpectedly, the turbulent flame speed (V ) was found to be a non-monotonic function of the flow intensity (A). For moderately strong vortices their intensification results in the flame speed enhancement accompanied by shedding of islands of unburned gas. Yet, there is a certain level of stirring at which the flame speed reaches a maximum. A further increase in the flow intensity leads to a drop in the flame speed, followed, for non-adiabatic systems, by the flame extinction. This sequence of behavior is qualitatively consistent with experimental results on turbulent premixed flames as the turbulence intensity is increased, e.g., [9]. This view, however, does not exhaust all the phenomenology contained in the model, as moving to still higher levels of stirring the adiabatic flame was found to speed up again [2]. Although this latter (third) propagation regime is likely out of reach for practical gaseous flames because of quenching considerations, its numerical identification is important for two reasons. First, and this is the main motivation for the current study, the third regime is likely to be of relevance for isothermal autocatalytic reaction waves spreading through strongly stirred liquid solutions which are characterized by a marked persistence of the V (A) monotonicity [4-6]. Secondly, such a study would provide a corroboration of recent analytical work yielding the prediction V A 1/4 at large A [3]. We will show that the basic reason for the marked distinction between the two systems is strong sensitivity of the reaction rate to variations in temperature (i.e., high activation energy) in gaseous flames and its relatively mild response to changes in the concentration of autocatalytic products in liquid flames (analogous to temperature in 2

3 gaseous flames). 2 Model and its Numerical Simulation To demonstrate the influence of chemical kinetics on the flame-flow interaction we consider a conventional constant-density, equidiffusional adiabatic flame-flow system described by a single reaction-diffusion-advection equation, Θ t + u Θ = µ 2 Θ + νω(θ) (1) where in the case of gaseous flames Θ corresponds to the appropriately scaled temperature, and in the case of liquid flames to the concentration of autocatalytic products. u = (2A sin kx cos ky, 2A cos kx sin ky) (2) is the prescribed flow-field, where A = 1 2 < uu > is its intensity, and k is the flow wave-number. The length is scaled with the flame thickness l th to be defined later, velocity with the laminar burning velocity S L, time with l th /S L and temperature with Θ = (T T 0 )/(T b T 0 ), where T is the dimensional temperature and the subscripts b and 0 refer to burned products and fresh reactants, respectively. Equation (1) is considered in the strip 0 < x < π/k, < y < and subjected to insulating and chemically inert boundary conditions, Θ x = 0 at x = 0, π/k (3) The flame is assumed to propagate in the direction of increasing y. Hence the boundary condition at y = ± are, Θ = 0 at y = +, and (4) Θ y = 0 at y = For finite activation energy flames the reaction rate Ω is specified as, 3

4 Ω = 1 2 β2 (1 Θ) [ exp ( ) ( β(θ 1) exp β )] σ + (1 σ)θ σ (5) Here β = E(T b T 0 )/RTb 2 - Zeldovich number and σ = T 0 /T b. Other notations are conventional. The term exp ( β/σ) is introduced to avoid the so-called cold boundary difficulty, i.e. to suppress reaction ahead of the advancing flame. For autocatalytic reaction Ω is specified as Ω = 2Θ 2 (1 Θ), (6) which pertains to the so-called KPP-type kinetics [7]. Such kinetics are relevant to some autocatalytic fronts [8] 1. The factor 2 is introduced for convenience of onedimensional calculations (see below). µ and ν are the normalizing factors to keep the burning velocity of the planar flame as well as its diffusive width at unity, and hence defined by the following eigen-value problem, Θ η + µθ ηη + νω(θ) = 0, (7) Θ(+ ) = 0, Θ( ) = 1, (8) subjected to the additional requirements to maintain S L and l th at unity, l 2 th = (η η) 2 Θ η dη = 1, η = ηθ η dη (9) The flame width is defined as a length of the interval covering the bulk of the temperature change. The adopted formal definition (9) is borrowed from the probability theory and is analogous to the dispersion, provided Θ is interpreted as the distribution function. To evaluate µ and ν it is helpful to make the following transformations, µ = 1/λΛ, ν = λ/λ, η = ξ/λ, (10) 1 Actually, in Ref. [8] the reaction rate is specified as Ω Θ(κ + Θ)(1 Θ) with κ = In the current study κ is set at zero, which is fully validated by a comparative numerical study conducted for A=0, 18, 90. 4

5 bringing the problem (7)-(9) to a more tractable form, ΛΘ ξ + Θ ξξ + Ω(Θ) = 0, (11) Θ( ) = 0, Θ( ) = 1, (12) (ξ ξ) 2 Θ ξ dξ = λ 2, ξ = ξθ ξ dξ (13) Here one first calculates Θ(ξ) and Λ, and thereupon λ. Some representative values of λ, Λ µ and ν for σ = 0.2 and 0.4 β 16 are given in Table 1. Table 1 β λ Λ ν µ KPP An important point is that for a certain small Zeldovich number (β=1.6) the Ω vs. Θ plot defined by the Arrhenius kinetics (5) becomes rather close to the KPP rate (6) (Fig.1). In this sense the KPP kinetics may well be perceived as a special case of the Arrhenius kinetics normally associated with gaseous flames. By comparison, high activation energy Arrhenius kinetics yields a much thinner reaction zone that is heavily biased toward the high temperature side of the front (Fig. 1). The problem (1)-(4) was solved numerically for a wide range of Zeldovich numbers (β) and flow intensities (A) using the parameters ν and µ from Table 1. The computational method employed is described in Ref. [2]. The results obtained show (Fig.2) that non-monotonicity of the V (A) dependency reported previously [2] is indeed a result of the high activation energy (β), and vanishes as β decreases, thereby also explaining its 5

6 absence in liquid flames. For the range of flow intensities covered by Fig. 2 the double change of monotonicity is observable only for β = 1.6 and β = 4. For higher β the second upswing occurs beyond the figure s frame. The non-monotonicity of the V (A) - curve occurring at large Zeldovich numbers may be interpreted in terms of the classical theory of counter-flow flames, as discussed in Ref. [2]. The flow-field (2) near the stagnation points x = nπ/k, n = 0, ±1, ±2,... and y = mπ/k, m = 0, ±1, ±2,... coincides with the counter-flow with the strain 2Ak. The steady counter-flow flame is realized only provided the strain falls below a certain threshold value. Above this value flame-holding becomes unfeasible and one ends up with the flame spreading through the vortical flow-field at a reduced speed. The threshold value increases significantly with β. Moreover, for small β (KPP) kinetics the counter-flow flame exists at all strains. This explains the absence of the V (A) non-monotonicity for the KPP kinetics. For sufficiently small A, V is nearly independent of β. This is likely because at low A, the characteristic strain rate, given in dimensional terms as the vortex intensity (AS L ) divided by the vortex length scale d = (π/k)l th is significantly smaller than the characteristic chemical rate S L /l th (or equivalently that the Karlovitz number Ka defined as Ak/π is sufficiently small), then the Huygens propagation mode of combustion applies in which the front can be treated as an interface propagating normal to itself with constant velocity S L relative to the flow-field [1] [5] [9]. In this case the internal front structure is not affected by the front and thus V is the same for KPP and/or Arrhenius kinetics at any β. For the current computations Fig. 2 shows that Huygens propagation occurs for A < 15, and since k = for these calculations, the corresponding non-dimensional criterion is Ka < 0.6, which is similar to that found for gaseous flames in turbulent flows [1] [9]. At Ka higher than that corresponding to Huygens propagation, for a given A, V (A) is higher for lower β. This is probably a 6

7 consequence of the wider reaction zone for lower β as discussed below. Also note that, as one might expect judging from Fig. (1), the V (A) dependency evaluated for the KPP kinetics is nearly identical to that based on Arrhenius kinetics with β = 1.6 for all values of A examined. Figure 3 shows the effect of Ka on the burning rate V for one fixed value of the flow intensity A. Note that again for high Ka, V is higher for KPP than Arrhenius kinetics at β = 16, but for lower Ka, V is virtually independent of Ka. The threshold Ka is close to that inferred from Fig. 2. At sufficiently low Ka, where Huygens propagation applies, the results are independent of the chemistry model. At sufficiently high Ka, KPP provides higher propagation rates, which is consistent with Figure 2. Figure 3 also shows the effect of activation energy on the threshold for transition from Huygens to non-huygens propagation. This transition can be seen to occur at slightly lower Ka for KPP than Arrhenius kinetics. The results obtained are qualitatively in line with the experimental data on liquid flames [4]. In order to achieve a better quantitative agreement one presumably has to employ a more sophisticated model involving a set of coupled reaction-diffusion equations with different diffusivities (see footnote in Ref [8] p.3843) rather than trying to capture all of the physics with a single equation. This issue will be addressed in future studies. Figure 4 plots F = A 1/4 V (A) against A. The behavior seen in Fig. 4 is consistent with analytical findings [3] that predict saturation of F (A) at large A. Figures 5 and 6 show typical distributions of the reaction rate (Ω) and the associated temperature/concentration fields (Θ) for fronts with high-β Arrhenius kinetics and KPP kinetics spreading through an array of high-intensity eddies. With KPP kinetics the front has a relatively wide reaction zone that is spread across more of the diffusive width of the front (see Fig. 1) compared to Arrhenius kinetics. The broader reaction zone maintains its structure more robustly against the flow induced deformations and 7

8 strains. Moreover, the front speed is less affected for low activation energy since in this case the integrated reaction rate is less sensitive to temperature, and thus less sensitive to temperature fluctuations caused by the stirring. Consequently, depending on the stirring intensity, the low activation energy flame is capable of building up a virtually unlimited amplification of its propagation speed. 3 Acknowledgments The authors gratefully acknowledge the support of the US-Israel Binational Science Foundation under Grant No , the Israel Science Foundation under Grants Nos , 67-01, and , The Gordon Foundation of Tel-Aviv University, the European Community Program TMR-ERBF MRX CT180201, and the NASA-Glenn Research Center under Grant NAG The numerical simulation were performed at the Israel Inter-University Computer Center. References [1] Ronney, P.D. in Modelling in Combustion Science (J. Buckmaster and T. Takeno, Eds). Lecture Notes in Physics, Springer-Verlag, Berlin, 449:3-22 (1995) [2] Kagan, L., and Sivashinsky, G., Combust. Flame, 120: , (2000) [3] Audoly, B., Berestycki, H. and Pomeau, Y. C.R. Acad. Sci. Paris, Ser IIB, 328: (2000) [4] Shy S.S., Ronney P.D., Buckley S.G., and Yakhot V. Proc. Comb. Inst.,24:543 (1992). [5] Ronney, P.D., Haslam, B.D., and Rhys, N.O. Phys. Rev. Lett., 74:3804 (1995) [6] Shy, S.S., Jang, R.H., and Ronney, P.D. Combust. Sci. Techn., 113:329 (1996) 8

9 [7] Zeldovich, Y.B., Barenblatt, G.I., Librovich, V.B., and Makhviladze, G.M. Mathematical Theory of Combustion and Explosion, Plenum, New York (1985) [8] Hanna, A., Saul, A. and Showalter, K. J. Am. Chem. Soc., 104:3838 (1982) [9] Bradley, D., Proc. Combust. Inst., 24:247 (1992) 9

10 Figure captions Figure 1 Reaction rate (νω) versus Θ. Bold line corresponds to the Arrhenius kinetics at β = 1.6, σ = 0.2; thin line corresponds to the Arrhenius kinetics at β = 16, σ = 0.2; dashed line corresponds to the KPP kinetics. Figure 2 Flame speed (V ) versus flow-intensity (A) evaluated for σ = 0.2, k = and 0.4 β 16. Dashed line corresponds to the KPP kinetics. Figure 3 Flame speed (V ) versus Karlovitz number (Ka = Ak/π) evaluated for the KPP (dashed line) and the Arrhenius kinetics (bold line) at β = 16, σ = 0.2 and A = 50. Figure 4 Scaled flame speed (F = A 1/4 V ) versus flow-intensity (A) evaluated for σ = 0.2, k = and 0.4 β 16. Dashed line corresponds to the KPP kinetics. Figure 5 Reaction rate (Ω) and temperature (Θ) distributions for the Arrhenius kinetics at β = 16, σ = 0.2, A = 100, k = The vertical arrow indicates direction of propagation. Darker shading corresponds to higher levels of Ω and Θ. Figure 6 Reaction rate (Ω) and concentration (Θ) distributions for the KPP kinetics at A = 100, k = Darker shading corresponds to higher levels of Ω and Θ. The arrow indicates direction of propagation. Note an increase in the number of trailing islands compared to the case of high-β Arrhenius kinetics (Fig. 5) 10

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