Numerical investigation of flame propagation in small thermally-participating, adiabatic tubes

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1 Paper # 7MI-181 Topic: Microcombustion and New Combustion Devices 8 th US National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-, 13. Numerical investigation of flame propagation in small thermally-participating, adiabatic tubes G. P. Gauthier G.M.G. Watson J.M. Bergthorson Department of Mechanical Engineering, McGill University, 817 Sherbrooke West, Montréal, QC, Canada, H3A C3 This study investigates the behaviour of flames propagating in small adiabatic tubes where the wall acts as a heat-recirculating medium. First, a simple analytical model is used to map global system behaviour at various inlet flow velocities, tube diameters, and for specific wall conditions and mixture composition. Regimes identified in the analytical model are verified with detailed numerical simulations. Providing a more accurate representation of the system is a two-dimensional model solving the low Mach number Navier-Stokes equations and the wall energy equation, with axisymmetry, implemented with a -step chemical kinetics mechanism. Results from this model are compared to those obtained with a 1-D formulation at similar conditions. Predictions of flame structure and burning and propagation rates are compared, and dissimilarities between model formulations are used to explain discrepancies in the results. As expected, two-dimensional effects are observed to become more significant as tube diameter and inflow velocity are increased, which influences the global behavior of the system. As 1-D models are predominant tools for the study of flames in small flow channels, their accuracy is assessed, and their range of applicability discussed. 1 Introduction There has recently been increased interest in the design of burners that rely on combustion at the small scale. These types of burners offer the compactness of batteries combined with the higher energy densities of hydrocarbon fuels, and an extended operating range when compared to conventional burners [1]. They can take on a number of configurations; however, all rely on combustion in small flow channels which have dimensions that are of the same order as the flame thickness. At these small scales, the surface-to-volume ratio is large enough such that interfacial heat transfer between the combustible gas and the channel walls becomes a dominant phenomenon. At specific conditions, this significant heat transfer allows thermal coupling between the flame and the wall, which can significantly promote burning rate by allowing heat to be recirculated from products to reactants [, 3]. However, these important wall/fluid interactions also affect the thermodiffusive and hydrodynamic characteristics of the flame [4], and make these systems challenging to model. Design and control of micro-scale burners will be facilitated by the development of efficient models which capture the important physics at play in these systems. A predictive model must be capable of obtaining accurate heat and mass transport, hydrodynamics, and should also include detailed chemical kinetics. The most comprehensive models are based on solving the low-machnumber Navier-Stokes equations considering both axial and radial, and occasionally azimuthal, 1

2 directions for the conservation of mass, momentum, energy, and species. There are a few examples of detailed simulations [5 7], but such models are computationally expensive and require parallel processing capabilities over an extended number of hours. Therefore, most studies of this type rely on simplified chemical kinetics models [4, 8]. An alternative formulation is the simplified volumetric model [9], which is orders of magnitude less computationally expensive because it only considers changes in the stream-wise direction through a volumetrically-averaged formulation. It has been shown to satisfactorily predict the qualitative behavior of combustion in small channels [9]. However, quantitative agreement between 1-D model predictions and experiments or detailed simulations has not been observed, nor been compared explicitly. There is a need for characterizing the relative accuracy of the different formulations in order to determine their respective range of applicability and the most important effects that should be captured by the models. This will facilitate the selection of an appropriate compromise between efficiency and accuracy in the modeling effort. This study looks at the problem of flame propagation inside a tube of finite thickness, where conjugate heat transfer, between the fluid and the wall, takes place. For a quiescent mixture, this typically reduces to a quenching problem, where the wall can be considered as isothermal. Fast flames are then observed, as they propagate at a velocity of the same order as the adiabatic flame speed. As the inlet velocity is increased, and the propagation velocity of the flame gradually reduces, thermal coupling between the flame and adjacent wall becomes increasingly significant, up to a point where flame behavior is controlled by wall heat recirculation from products to reactants. This flame regime typically occurs at low-propagation velocities, and is referred to as the slow flame regime [3]. The predicted burning velocities in the slow flame regime can be much greater than those observed in the fast flame regime. In this study, a one-dimensional asymptotic model and a more detailed multi-dimensional model are implemented with conjugate heat transfer, allowing them to capture both the fast and slow regimes. Heat loss of the system is set to zero, in order to concentrate on the effect of tube diameter and inflow velocity on the flames. These two parameters are varied, and the results from differing model formulations are compared at equivalent conditions. The discrepancies between models are assessed, and the two-dimensionality of flames is discussed. This study aims at characterizing the relative accuracy of the models and determining the leading order effects influencing these flames. Numerical Models The system of interest, involving a tube of finite thickness with a circular cross section, is shown in Figure 1. The inflow or unburned velocity is denoted as, and the flame propagation velocity is denoted as S p. As heat is recirculated through the wall, a heat wave travels inside the wall at the same propagation velocity as the flame. In the reference frame of the flame, the burning velocity, S b, is the sum of and S p..1 The asymptotic model As a first step in the investigation of the system, the 1-D asymptotic model is considered. The analytical solution was derived by Ju and Xu (5) [3] for a flame propagating inside a channel.

3 Figure 1: System of interest: tube of circular cross-section of diameter D with finite wall thickness t w. The inflow velocity is denoted as, which can be represented as a corresponding parabolic profile. Propagation speed is denoted as S p, and burning velocity, as S b The model consists of normalized, 1-D governing equations for the gas and solid phase and are solved in the limit of the flame sheet assumption, β, with constant properties and a simplified chemistry scheme which consists only of the deficient reactant specie.the conservation equations are: Gas Energy: Gas Species: Gas Energy: ρc p ( S p ) dt dx = λ d T dx 4h D (T T w) + Q c ω, (1) ρ( S p ) dy dx = α d Y ω, () dx dt w ρ w c w S p dx = λ d T w w dx 4h D (T T w), (3) Equations (1) to (3) are written for an isolated system with no heat loss to the ambient environment, and constant properties are assumed. The term 4h D (T T w) is a simplified model for interfacial heat transfer between the gas and the wall, where h is an average volumetric heat transfer coefficient. Unity Le is assumed as heat diffusivity, α, is used to model species diffusion. This formulation leads to an eigenvalue problem which can be solved analytically. More details of this analytical formulation can be found in [3].. The detailed model A detailed model for this problem is implemented in OPENFOAM based on the Navier-Stokes equations for low-mach-number reacting flows, and on the heat conduction equation for the solid wall. The multi-dimensional governing equations for mass, momentum, energy, and species are shown below: Gas Continuity: ρ t + (ρ u) =, (4) 3

4 Gas Momentum: Gas Energy: Wall Energy: Gas Species: ρh t ρ u t + (ρ u u) τ = g ρ p, (5) + (ρ u h) (ρα h) = p t u p + S h, (6) ρy i t (ρ w c w T w ) t (λ w T w ) =, (7) + (ρ u Y i ) (ρα Y i ) = ω i, (8) where τ is the viscous stress tensor and ω i is the net production rate of species i. The energy equation is written in terms of sensible enthalpy; therefore, chemical reactions are included in the heat release term, S h, as it includes the enthalpies of formation. Unity Le is assumed as heat diffusivity, α, is used in the species equation. This formulation is solved using a finite-volume discretization scheme in OPENFOAM. Fluid equations are appropriately formulated in order to use a PISO (Pressure-Implicit Splitting of Operators) algorithm [1] that solves the transient solution. Detailed chemistry is handled using an operator-split method [11] that calculates a reaction rate based upon an Arrhenius kinetics integration over the time-step in each cell. The reaction mechanism is chosen to be a reduced -steps mechanism by Westbrook and Dryer [1] that includes 5 species. To obtain accurate two-dimensional results, precautions are taken to ensure sufficient grid refinement. The grid size in the flame region is 1 µm in the stream-wise direction. In the radial direction, non uniform node spacing is employed; cell size is minimized close to the wall to to a minimum of less than 5 µm. Heat release profiles are found to be satisfactorily resolved at the refinement level used. Roughly, cells are typical for the calculations reported here. Sufficient computational power is provided by cluster resources that allow for use of parallel processing capabilities. Each simulation made use of 3 processors for around 5 hours in order to provide accurate velocity measurements of a steady propagation. Velocity values are obtained by tracking flame position along a series of time-steps, until steady propagation is observed. A 3 cm long grid is used for most of the simulations. This detailed model accounts for most hydrodynamic, chemical and transport phenomena which produce the two-dimensional structure of these flames. The detailed model has few assumptions and is, therefore, expected to give the most accurate representation of these flames for the assumed boundary conditions. For small channels, thermal quenching and radical recombination could also affect the -D structure of a flame. These shall be neglected from the present formulation, but for hot channel walls, homogeneous chemical reactions are assumed to overcome the effect of radical removal at the wall so that the flame becomes resistant to quenching [13]. 4

5 .3 Investigated Conditions Dirichlet boundary conditions are specified at the tube inlet. A stoichiometric methane-air mixture enters at a specified velocity. For the -D model, a parabolic profile corresponding to a specified mean velocity is imposed at the inlet to eliminate entrance length effects. The inlet temperature and the initial wall temperature are set to 9 K. This elevated temperature was chosen as microburners typically involve high preheating temperatures. 9 K is very close to the threshold temperature predicted by the -step chemistry model above which the mixture auto-ignites. Gas properties are constant in the asymptotic model, and based on air properties at 9 K. Flame parameters are chosen to mimic the stoichiometric methane-air flame predicted by the -step mechanism. The chemical heat release per unit mass of fuel is J/kg, and the activation energy of chemical reaction is E a = J/mol/K. The reaction frequency factor of the chemical reaction is 47 kg/m /s. With this chosen set of inputs, both models predict comparable adiabatic flame speeds and identical adiabatic flame temperatures. However, the asymptotic model predicts a thinner flame. As diameter is varied, wall thickness is always equal to half of the radius, as shown on Figure 1. Wall properties are the same as quartz, except for density and heat capacity that are reduced by a factor of ten (k w = 1.1 W/m/K, ρ w = 65 kg/m 3, C w = 75 J/kg/K). This effectively reduces thermal inertia of the wall, which allows flames to propagate at higher velocities, facilitating the assessment of propagation speed. Adiabatic conditions are set on all external surfaces of the wall, and heat is allowed to transfer between gas and fluid on the wall inner surface. 3 Results and Discussion As a first step in comparing models, the quenching behavior at isothermal wall conditions is shown in Figure. This condition is obtained when thermal inertia of the wall is brought towards infinity. Burning velocity is seen to decrease with diameter up to a quenching diameter, d q. The analytical solution also captures an unstable branch at lower velocities, which is not captured in the -D model results. Both quenching curves behave similarly, but the predicted d q s differ significantly. This was partly expected as the two models predict different flame thicknesses, and interfacial heat transfer is treated differently in each formulation. In this study, d q is chosen as a characteristic length scale of the problem. It is used as a scaling parameter for diameter values to be tested, which allows the 1-D and -D model results to be compared at equivalent conditions, despite the disagreement observed in the respective quenching curves. It should be mentioned, however, that d q is not unique in the -D case, as seen on the right hand side of Figure for the case of U =.5. Hence, quenching phenomenon cannot be fully captured by a single scaling factor, as two-dimensional effects related to inflow velocity can come into play. This effect of inflow velocity on quenching is not captured by the 1-D model. For the purpose of getting conditions in both models to be equivalent in terms of the regimes and physics at play, using d q obtained for a quiescent mixture is deemed adequate. Two-dimensional effects captured by the -D model are seen to influence not only the quenching diameter prediction, but also the predicted burning rate at larger diameters. For the quiescent mixture, the 1-D model fails at predicting burning rate that overshoots laminar flame speed in the 5

6 -D case, which arises due to flame curvature and increased flame area allowing higher burning rates. It also fails in capturing the relative reduction in burning rate when inlet velocity is increased, which reflects an increase in heat loss due to hydrodynamic effects. These trends in -D predictions are consistent with previous findings for flame propagation in tubes at isothermal conditions [4]. S b D (U=) 1 D d q =1.16 mm d q =.6 mm S b D (U=) 1 D D (U=.5) d (mm) Figure : Quenching curves for both model formulations. On the left-hand side, normalized burning velocity is represented as a function of diameter, and respective quenching diameters, d q, are shown. On the right hand side, diameter values are normalized by the respective quenching diameters, d q, obtained for each model. The significance of d q as a characteristic length of the problem can be seen from inspection of Figure 3, which is obtained from the asymptotic model as heat recirculation is turned on. Predicted burning rates are seen to vary with diameter and inflow velocity. For inlet velocities smaller than S L (U < 1), as the diameter is decreased, the transition from fast to slow flames is seen to occur at d/d q 1. d q is thus seen to be physically relevant not only for the quenching of flames, but also to describe the transition from fast to slow flames. Both these phenomenon depend on the rate of heat exchange between the flame and the wall. Slow flames propagate at very small velocities, such that U = is very close to S b at these conditions. As inflow velocity is increased, for a specific tube diameter, the slow flame propagating upstream progressively slows down, up to a stationary point, where the flame transitions from upstream to downstream propagation. The predicted stationary points, where S b =, are shown in Figure 3. It is relevant to test the system behavior at different diameter values, above and below d q, and to see how 1-D and -D predictions compare to one another. Figure 4 shows the dependence of burning velocity and propagation velocity on inflow velocity predicted by both models. At d/d q = 1.64, a transition from the fast to the slow flame regime is observed. As found in the isothermal case, the 1-D model fails at predicting the reduction of burning rate with increasing inflow velocity in the fast flame regime. As the inflow velocity is further increased, the slow flame regime is reached, and both models show much closer agreement in terms of burning rate. This is due to the fact that propagation velocity becomes very small, making burning rate very close to inflow velocity. However, when looking more closely at propagation velocity curves, it can be seen that the slopes 6

7 .5 Downstream Propagation U=.5 U=. U= =S b U=1.5 S b 1 Upstream Propagation U=.75 U=1.5 U=1. U=.5.5 U=.5 Isothermal wall U=.1 U= Figure 3: Dependence of normalized burning velocity on normalized diameter, for different normalized inflow velocities, where U =. Transition from upstream to downstream propagation in denoted by the line S b =. differ significantly at larger inflow velocities. At d/d q =.41, flames can only exist in the slow flame regime. Propagation velocity predictions show best agreement at low inflow velocity, where both models capture a peak in propagation velocity. However, as inflow velocity is increased, predicted propagation velocities are seen to diverge from one another. At even smaller diameters, where d/d q =.8, propagation velocity predictions are not in close agreement at small velocities, but they both capture an inflection in the velocity dependence. As the velocity is increased, the slopes of the propagation velocity curves show better agreement than for the larger diameters. The point at which a flame becomes stationary (S b = ) is an important parameter for burner operation and control, as stationary flames are advantageous for steady state operation. Figure 4 shows the predicted stationary points for both models. It can be seen that the 1-D model significantly underestimates the velocity at which this steady state condition occurs, especially for large diameters. Not only do the results disagree, but the trends in the results do not match. Two-dimensional effects thus play a significant role in flame predictions, especially for large diameters. From equation (7), it can be seen that the stationary flame prediction, which is a steady state condition, only depends on wall thermal conductivity, λ w. The small values of ρ w and C w chosen for this study have no effect on the stationary flame condition. Stationary results are thus applicable to any wall material having a thermal conductivity of 1.1 W/m/K, such as quartz. 7

8 =.8 =.41 = S b S p Figure 4: Dependence on normalized burning and propagation velocities on normalized inflow velocity, for three different tube diameters. 1-D results are shown as thick lines, while -D results are shown as circles. In order to better explain discrepancies in the results, -D contour plots of temperature and volumetric heat release are shown in Figure 6, for various diameters and inlet velocities. As the inlet velocity is increased, flame shape becomes increasingly elongated, and shows a tulip shape, which is consistent with previous numerical observations [5]. The elongation can become significantly more important than the flame thickness, which disagrees with the flat flame assumption made in the 1-D model. This elongation allows the flame to increase its burning rate by allowing greater burning area. This effect of curvature cannot be captured by the 1-D model, and may explain the discrepancies observed in propagation speeds at high velocities. Diameter is also seen to play a role on flame structure. As the diameter is increased, curvature increases, particularly at high inlet velocities. This is consistent with the results in Figure 4, where propagation speed is badly predicted by the 1-D model at high velocities, for the larger diameters. This shows significant limitations of the 1-D model as it fails to account for flame curvature effects. At small diameters and velocities, flame thickness is seen to increase when compared to the more or less constant thickness observed at other conditions. This cannot be captured by the 1-D model that assumes heat release zone to be infinitely thin, implying that interfacial heat transfer cannot occur within the reaction zone. This assumption is in disagreement with -D observations where flame thickness is significant when compared to other length scales of the problem. This inaccurate estimation of reaction zone thickness may explain the discrepancies observed in Figure 4 for d/d q =.8, for small inlet velocities. At these conditions, the -D flame is particularly thick as opposed to the assumed thin 1-D flame. 8

9 S b 8 6 D 4 1 D Figure 5: Normalized burning rate as a function of normalized diameter, at the stationary flame condition (S b = ). -D data is interpolated between upstream and downstream propagation velocity results. 4 Conclusion In this study, flame propagation in small thermally-participating, adiabatic tubes is studied with models of differing formulations. A simple analytical model is implemented as a first step, and a detailed two-dimensional model is used as a benchmark for comparison. Results from both models are compared at equivalent conditions. The analytical model is shown to predict most of the relevant trends of the system. However, in the fast flame regime, the 1-D model fails at capturing the effect of inflow velocity on burning rate and quenching. In the slow flame regime, significant discrepancies in predicted propagation speeds are observed at high inlet velocities, especially for large tube diameters. These discrepancies are associated to curvature effects observed in the - D flame contours, which cannot be captured by the 1-D model. This has important implications on the prediction of stationary flame conditions, which should be predicted accurately for steady operation of the system. The 1-D model predicts stationary flames at inlet velocities which are significantly lower than those predicted by the -D model, and the disagreement increases with tube diameter. At diameters much smaller than the quenching diameter, the flames are observed to be more flat, which allows better agreement with the 1-D model. However, as the -D flame thickness is observed to become more significant a small diameters and small inflow velocities, such that the thin flame assumption prevents the 1-D model from modeling the system with quantitative accuracy even in conditions where the flame is flat. The use of the 1-D model is advantageous in a design approach where a large number of conditions need to be simulated efficiently in order to come up with optimal design and operating parameters. In order to come up with optimal values for diameter and inflow velocity, one would have to proceed with great care with the 1-D model, as it fails to predict all the trends observed in -D data, especially at large values of diameter and inflow velocity. A -D model would be more 9

10 / S l =.94 D =.1 mm.5.5 / S l = / S l = D =.5 mm D = 1. mm.5.5 x x f (mm) x x f (mm) x x f (mm) Figure 6: Contour plots of temperature and volumetric heat release profiles. Iso-contours are shown for T Tw T a T w and Q Q max at values of.,.4,.6,.8, where T a = 77 K, and Q max is calculated for each simulation. appropriate for such an optimization approach, but significantly increased computational times are required. Acknowledgments We are grateful for the assistance of Gideon Balloch in the development of numerical tools. Computing resources for this study were provided by the Consortium Laval, Université du Québec, McGill and Eastern Quebec (CLUMEQ). Finally, funding from the Natural Sciences and Engineering Research Council of Canada is acknowledged. References [1] I Schoegl and JL Ellzey. Combust. Flame, 151 (7) [] DR Hardesty and FJ Weinberg. Combust. Sci. Technol., 8 (1974) [3] Y Ju and B Xu. Proc. Combust. Inst., 3 (5) [4] NI Kim and K Maruta. Combust. Flame, 146 (6) [5] I Schoegl and J Ellzey. Combust. Sci. Tech., 18 (1) [6] GG Pizza, CE Frouzakis, J Mantzaras, AG Tomboulides, and K Boulouchos. J. Fluid Mech., 658 (1) [7] G Gauthier, G Watson, and J Bergthorson. Combust. Sci. Technol., Accepted (1) journal. [8] DG Norton and DG Vlachos. Combust. Flame, 138 (4) [9] K Maruta, T Kataoka, NI Kima, S Minaev, and R Fursenko. Proc. Combust. Inst., 3 (5) [1] SV Pantakar. Numerical heat transfer and fluid flow, First Edition. McGraw-Hill, New York, 198. [11] NN Yanenko. The Method of Fractional Steps. Springer, New York,

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