Flame speed measurements at the tip of a slot burner: Effects of flame curvature and hydrodynamic stretch

Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 455-461

FLAME SPEED MEASUREMENTS AT THE TIP OF A SLOT BURNER: EFFECTS OF FLAME CURVATURE AND HYDRODYNAMIC STRETCH TAREK ECHEKKI AND M.G. MUNGAL

Mechanical Engineering Department Stanford University, Stanford, CA 94305-3032 The effects of flame curvature and stretch upon the laminar flame speed are investigated experimentally and compared to theoretical predictions. The flame speed at the tip and side of a slot burner is measured using particle tracking velocimetry. Temperatures are measured using Rayleigh scattering. Flame tip curvatures are measured using direct flame photography. For a range of mean exit velocities from 1 to 2.5 m/s and a diffusionally neutral mixture, the flame speed at the tip is found to exceed that at the side by a factor as high as 6.25. The flame speed ratio is found to depend linearly upon the hydrodynamic stretch factor as predicted by the analysis of Matalon and Matkowsky. Equivalently, the flame speed ratio is found to increase nonlinearly with the flame curvature. Comparison with published results from a cylindrical burner shows similar trends.

Introduction The flame tip of the bunsen burner provides a simple geometry of a curved and stretched laminar premixed flame. The flame speed at the tip, SL.,, defined as the cold gas velocity normal to the flame, has been observed to exceed the value of the onedimensional flame speed, S~,,, by a significant factor. One of the first attempts to interpret the flame speed enhancement at the tip of the bunsen burner was by Lewis and von Elbe. 1 An analytical solution for the flame tip based on a constant density approximation was presented by Sivashinsky2 and further advanced with variable density by Buckmaster and Crowley 3 and Buckmaster and Ludford. 4 The analytical results of Buckmaster and Crowley provide implicit solutions of the flame tip curvature as a function of the flame angle at the side. Their results for the geometries of two-dimensional and axisymmetric flame tips show that the flame radius of curvature at the tip for a given flame speed is geometry dependent. The tip of the bunsen burner can be locally modeled as a curved and stretched flame. The effect of curvature on the flame speed was first addressed theoretically by the semi-phenomenoiogical model of Markstein 5 by relating the flame speed to the flame radius of curvature, Rf, using the following relation:

SL, u S ~L,u

=

1 + --

(1)

determined by the thermal and diffusive properties of the combustible mixture. Recently, an integral model by Chung and Law6-8 was used to predict the flame speed dependence on stretch and curvature. At unity Lewis number, their expression is

SL,u - -

=

1 + 8~(V-n),

where 5~- is an integral scale that represents the preheat zone thickness for a one-dimensional, planar flame which depends on the thermal and diffusive properties of the combustible mixture. The curvature parameter (V" n) can be expressed as 1/Rf in a two-dimensional geometry and 2/R1 in an axisymmetric geometry. The value of (V" n) for a convex flame curved toward the fresh mixture is positive. The concept of stretch was first introduced by Karlovitz9 and was later generalized to account for not only flow non-uniformity but also for flame curvature and non-stationarity. 1~ A measure of stretch is the hydrodynamic stretch factor, KH, defined as the fractional area change of a Lagrangian surface element A: ldA Kn =

A dt

KH . . . . 455

(3)

The stretch parameter, Kn, for a stationary flame of radius of curvature Re in a uniform flow and a flame speed SL,u may be derived a s : 11

R/

where ~ is the Markstein length, a constant that is

(2)

S~L,u

SL,u

Rs'

(4)

PREMIXED FLAMES

456 for a two-dimensional flowfield and 2SL, u KH =

-

-

~

as

(5)

for an axi-symmetric flowfield where Rf is positive for a flame convex towards the unburned mixture. 9 Basic relations for small flame surface deformation were derived by Matalon and Matkowsky 1~ who show that S C , tt

....

1 -- aKH,

(6)

S~ L,u

where Sc,= is the laminar flame speed of the stretched flame evaluated at the cold gas state; SL~, u , the flame speed of the one-dimensional, planar flame; and, ~n, is the hydrodynamic stretch normalized by a characteristic stretch parameter and evaluated in the limit of a thin flame sheet (defined later for the geometry of interest); a is a proportionality constant that accounts for heat release and preferential diffusion. In addition, Matalon and Matkowsky provided an expression for a which at unity Lewis number is expressed by:

a

=

amm

=

T/, ln(Tb~, T~,- T,, \ T , , /

-

(7)

where Tu and Tb are the unburned and burned gas temperatures. Equation (6) states that the flame speed decreases linearly with the hydrodynamic 9 13 stretch. Clavin and Wilhams had derived similar expressions for small flame surface deformation. The objective of this study is to correlate the flame speed at the tip of a slot burner with the local parameters of flame curvature and hydrodynamic stretch. The speeds are compared to the analytical results, Eqs. (2) and (6), and the experimental resuits of Wagner and Ferguson 14'1z at the flame tip of a cylindrical bunsen burner.

and 4.91 cm long resulting in a relatively wide 2D region9 The flow exits the burner with a top hat velocity profile over a large portion of the jet center, resulting in a well defined flame angle on the sides of the flame9 A boundary layer is allowed to develop (owing to the extension) at the wall of the burner resulting in an improved capability to stabilize the flame and extend the blowoff limit9 The flow system is shown in Fig. 1. Air is supplied from a compressed air cylinder. Its pressure is regulated upstream of the choked flow orifice. Fuel flow (methane Matheson 99.0% CP) is supplied and regulated with a similar configuration9 A portion of the mixture is passed through the particle seeder and a bypass tube. The seeder can be isolated from the rest of the flow system using two shut off valves for Rayleigh scattering measurements. The final mixture is thoroughly mixed by passing it through a coil before the burner assembly. In the PTV experiments, the beam from a 20 W pulsed copper vapor laser (internal repetition rate 5882 Hz, pulse length 30 ns) is formed into a sheet using an optical train of a spherical and a cylindrical lens. The photographic system is positioned at a 90~ angle from the laser beam and consists of a 35 mm camera (Nikon HP) fitted with bellows (Nikon PB6) and a microlens (Nikon 105 mm f2.8 micro). Kodak Tmax 3200 ASA film was used for Black and White photographs and FUJI 1600 ASA film for color

pressure gage C ~ /eked floworifice regulatingvalve chut-off ~ i ....~ ~ va,ve CompressedAir absoulte

f

[ x.

t / /absoul • tepressure f

|

chokedfloworifice chut-off valve shut-off ( valve

Experimental Set Up and Techniques The propagation of the flame tip is characterized by the following parameters: its laminar flame speed, fame curvature and flame temperature. These parameters are measured using Particle Tracking Velocimetry (PTV) and flame photography. In addition to velocity and radius of curvature measurements, temperature measurements are carried out in some experimental cases using single point Rayleigh scattering. The flame is stablized using a 2-D bunsen burner with a final section consisting of a 2-D nozzle and straight extension. The burner exit is 0.68 cm wide

seed metering valve

seeder~

to burner FIG. 1. Flow system for particle tracking velocimetry and Rayleigh scattering.

FLAME SPEED MEASUREMENTS photographs, m Prints from the experimental cases were digitized and the position of the centroids of the particles were computed from the digitized data and used to infer the velocity. Temperature profiles along the central streamline were determined using Rayleigh scattering. Figure 2 shows the Rayleigh scattering system. The linearly polarized beam of the 5W Ar § laser (h = 488 nm) is focused at the probe volume with a 225 mm lens. The laser beam signal is modulated at the frequency f~ = 920 Hz using a chopper wheel. The scattering signal is collected using the optical train consisting of a polarizer, an imaging lens, an iris, a collimating lens, a narrow band filter and a PMT. The photographic lens (Nikon micro 105 mm f2.8) images the probe volume onto the iris plane. With the lens magnification of 2.1 and the iris size of 0.4 by 0.4 mm the probe volume is approximately 0.2 by 0.2 mm. A correction for the variable scattering cross section of the mixture 17 is made using correlations of the scattering cross section with temperature from computed temperature and concentration profiles in a one-dimensional planar flame.

Basic Considerations In this experiment, two regions of the slot burner flame are of interest: the flame tip and the flame side. At the tip, the incoming velocity, t~, is equal to the flame speed, SL,',. The choice of a uniform burner exit velocity eliminates the stretch effects due to flow nonuniformity on the sides of the flame. The resulting flame on the side behaves as a onedimensional planar, adiabatic flame with the normal velocity component equal to S~,.',. In the flame tip, flow divergence, flame curvature and preferential SPAT~Fi~ERS

/

N

457

diffusion effects may be present. The composition of the combustible mixture is chosen to be stoichiometric methane/air to eliminate preferential diffusion effects, m-m The effect of stretch and flame curvature on the flame speed are quantified using the following measurable quantities: (1) The radius of curvature, Rf, measured at the middle of the luminous zone and (2) the hydrodynamic stretch, Kn, defined earlier. Rf and K H Can be non-dimensionalized using the characteristic length scale, S~, and the one-dimensional, planar flame speed, to yield the following curvature and stretch factors

~', o

--

S,, SL,', o

and

~n =

(8)

The length scale, S~ is defined by: so

~

puCp,uSo L,u

(9)

where )t is the heat conductivity of the gas mixture, cp, is the heat capacity, the subscript u denotes the unburned gas state and the superscript ~ refers to the one dimensional, planar flame geometry. Two other length scales are used in this study. They are: ~ defined by: )t b

ST, p',cp,/,S~,','

(10)

where the subscript b denotes the burned gas state. S} which is a measure of the preheat zone thickness and is defined as: S} ~-

T b - T', -

-

(11)

BEAMSTOP

POJ~aZER flOgGINGLENS

where the subscript ig denotes the ignition surface which defines the onset of chemical reactions. The preheat zone thickness can be approximated using a two-zone model to give:

NARROWBANDFILTER

s} = 7~ - T%~,, T ~ - r', SIGNAL ILOCK-IN AMP. ), ~

COMPUTERINTERFACE

FIG. 2. Optical system for Rayleigh scattering.

(m)

where T~ is the ignition temperature. It is important at this stage to note that one of the difficulties of correlating the experimental data with analytical and other experimental data is the fact that some of the properties are normalized by different scales that may vary from one study to the other. This is particularly a problem when analytical solutions based on constant thermodynamic and

458

PREMIXED FLAMES

transport properties are used. These normalization issues are addressed below.

~= I

Experimental Results Figure 3 shows a typical luminous flame. The luminous zone at the tip is used to define the radius of curvature R.f. The particles tracks illustrate the geometry of the flowfield. Along the centerline, particle tracks show a net streamtube area expansion. Along, the side the streamlines remain essentially parallel, thereby confirming the general onedimensional behavior at the side. Eight cases were studied at ranges of mean exit velocities between 1 and 2.5 m/s using a stoichiometric methane/air mixture. For each case the flame speed at the tip and radius of curvature of the luminous zone were measured. Figure 4 shows typical velocity profiles (components normal to the flame) at the side and tip of the flame at a mean exit velocity ti = 1.5 m/s. The velocity along the central streamline is everywhere greater than the one-dimensional flame speed (S~c,u = 0.4 m/s). In comparison to the one-dimensional, planar flame a different mechanism of flame stabilization is present2~ at the tip and is based on two basic processes in-

FIG. 3. Direct photograph of flame and particle tracks.

'

+

'

'

,

,

1 5 0 c m / s , C H 4 / A i r 0 = 1.0) I

'

'

'

'

I

'

'

'

'

I

.

~

,

,

I

'

'

I

,

,

tip

1

~;~ u 0

I 0

,

,

I

.

,,~

. . . .

5

10

15

xn (mm)

FIc. 4. Velocity profiles at the tip and the side of the flame (0 = 150 cm/s and 4' = 1.0). volving area expansion (increase in the reaction Volume) and reactant diffusion towards the flame sides. Figure 5 shows the velocity and temperature profiles along the central streamline at the same exit velocity. The maximum temperature recorded on the central streamline is 2150~ K which is within the error bounds of the adiabatic flame temperature (Tad ~ 2230~ K). Since the deviation of the flame temperature from the adiabatic value is insignificant, the results confirm that the stoichiometric mixture of methane/air is diffusionally neutral. Therefore, preferential diffusion effects on the propagation of the flame tip are eliminated. The velocity in the central streamline changes in accordance with the temperature suggesting that thermal expansion effects due to density changes are more dominant than streamtube area expansion effects which tend to decrease the velocity,zl Along the central streamline, the flame speed, which is the normal component of the gas velocity upstream of the flame, is equal to the burner exit centerline velocity. Upon reaching the luminous zone, the velocity increases rapidly to a maximum value. The point of maximum velocity is also the point of lowest pressure in the flowfield. 21 The gas velocity, eventually, decreases sharply to approach the cold gas velocity. This process is accompanied by still continuing area expansion as the streamlines curve toward the centerline, and a modest drop in the gas temperature. The fast deceleration of the gas downstream of the tip is a result of the flow's attempt to adjust to the higher back pressure of the surroundings. 21 Figure 6 shows the laminar flame speed ratio as a function of the hydrodynamic stretch for all cases studied. The characteristic lenght, ~o, used for normalization is calculated to be 0.0557 mm. The flame speed increases linearly with the magnitude of the

459

FLAME SPEED MEASUREMENTS

~u= 1.5 m/s, CH 4 / Air 0 = 1 ) 3.0 ~ 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

velocity (tip) temperature

''''1'''

2500

\

2000

\

~e

2.5

~soo o~ A

g == 2.0

E

1000

j

500

1.5

I,,,,I,,ll[l~, 0

5

,,,,I,,~ 10

15

0

20

25

x n (ram)

FIG. 5. Velocity and temperature profiles at the central streamline (fi = 150 m/s and ~b = 1.0). The locations of the maxima in temperature and velocity correspond approximately to the outer edges of the 0.2 mm thick luminous zone. negative stretch. The experimental slope is found to be ae~ = 6.46. The Matalon and Matkowsky proportionality constant, a,,~, can he approximated to 2.32 using Eq. (7) and values of Tb = 2230~ K and

T, = 300~ K. The difference between the values of a~ (experimental) and atom (Eq. (7)) is due to the length scales used to normalize the radius of curvature in each case. In the calculations used for the evaluation of ae=, the length scale used for norrealization is 8~,. In the Matalon and Matkowsky analysis, the length scale used to normalize the stretch is more appropriately approximated by the preheat zone thickness, 8]-. A resulting approximate value of 8~- is 0.155 mm with a value of T~g = 1600~ K for the ignition temperature. Therefore, ae, should scale as 0.155/0.056 = 2.77 times atom which compares with an observed ratio of 2.81. Therefore, the experimental correlation of flame speed vs hydrodynamic stretch correlates very well with the linear profile predicted by Matalon and Matkowsky. This is a surprising result as Eq. (6) was derived for relatively weak stretch. We note that a correlation with stretch cannot be extended to the general case of mixtures with Le ~ 1 because preferential diffusion greatly alters the structure of the flame, and thereby its propagation properties. Figure 6 also shows the experimental data of Wagner and Ferguson14'15 for a cylindrical bunsen burner at <~ = 1.0. The data was extracted from Reference 15 which gives the radius of curvature as a function of the flame angle. Equation (5) was used for the hydrodynamic stretch. The radii of curvature in the Wagner and Ferguson study14'15 were normalized by the same length scale 8~ to compare to the present slot burner results. These experimental results correlate on a different linear curve. In fact, the slope of the cylindrical burner stretch curve is approximately one half of the slot burner.

Figure 7 shows the flame speed ratio SLs,/ S~L, as a function of the curvature parameter 8~ The radius of curvature Rf decreases with an increased (OH 4 / A i r at r = 1.0)

( C H 4 / A i r at ~ = 1.0) '

'

'

'

I

. . . .

I

'

'

'

'

I

'

'

'

'

I

'

'

'

7~

'

....

I ....

I ....

I ....

I ....

I']'''

d 6

~ 9 ~ "

| Slot Burner ~. C y l i n d r i c a l B u m e r

/

/

1 ~~

0,2

0.4 Hydrodynamk:

0.6

0.B

1.0

0

,

,

0.025

~1

0.050

,

,

I

0.075

,

~

,

,

I

0.100

. . . .

I

0.125

,

,

,

,

0.150

Stretch, ~H}

FIG. 6. Variation o f the l a m i n a r flame w i t h h y d r o d y n a m i c stretch.

speed ratio

FIG. 7. Variation of the laminar flame speed ratio with curvature parameter.

PREMIXED FLAMES

460

flame speed at the tip. The curvature parameter, 5~ can be correlated with the flame speed ratio using the relations by Mataion and Matkowsky (see Eq. (6)) with Ktt defined by Eq. (8). By manipulation of the terms, the correlation of SL,./S~ with (5~ can be expressed by:

havior at the slot tip, as they would correspond to straight lines in Fig. 7. In this regard, we note that Eq. (2) can be viewed as a limiting form of Eq. (13) for weak flame curvature.

Summary and Conclusions 1

SL, u - -

sL.

=

-

-

5~'

(13)

1 -- ae~-~f

where the best fit parameter aex is 6.46 (Fig. 6). This correlation predicts the behavior of the flame at the limiting condition of (SL.,/S~ ---) 1) leading to a planar flame geometry (Rf---) oo). The expression given by Eq. (13) imposes a lower bound on the radius of curvature. This is seen by expressing 5~ in terms of SL,u/S~ 5,,

1

Rf

ae~

1

L.,

(14)

At higher flame speeds 8~ approaches I/am. This limiting value is approximated to 0.155 which corresponds to RI = 0.36 mm. The existence of the asymptotic value of the curvature parameter, 8~,/ RI, can be interpreted physically by the notion that the flame thickness (flame and preheat zone) cannot exceed the value of the radius of curvature. To illustrate this fact, consider the situation where the radius of curvature is smaller than the flame thickness. At this condition heat from the curved flame is focused toward the center of curvature resulting in a rise in temperature. Sharp concentration gradients of active radicals result in the upstream diffusion of the flame. The resulting effect is the ignition of the mixture upstream of the initial ignition plane, thereby, the motion of the flame upstream. The radius of curvature, therefore, increases as the flame tip flattens. Figure 7 also shows the experimental data of Wagner and Ferguson. la'ls There is also a rising trend in the data and a better agreement with the slot burner data. In fact, Eq. (13) derived from the original stretch equation (Eq. (6)) for the two-dimensional geometry, seems to correlate better with both the two-dimensional and cylindrical geometries than Eq. (6). The analytical results by Buckmaster and Crowleya do show that the radius of curvature for a given flame speed at the tip is geometry dependent and that Rf is higher for the round tip compared to the slot tip. Finally, given that Eq. (6), or equivalently Eq. (13), appears to correlate the slot data, this also implies that the analytical expressions by Markstein5 and Chung and Laws-s relating the flame speed to the flame radius of curvature cannot accurately predict the flame be-

The flame speed at the tip and side of a slot burner is measured for a range of mean exit velocities for a diffusionally neutral mixture. The flame speed at the tip is found to exceed that at the side by a factor as high as 6.25. The flame speed ratio is found to depend linearly upon the hydrodynamic stretch factor (Eq. (6)) as predicted by Matalon and Matkowsky. Equivalently, the flame speed ratio is found to increase nonlinearly with the flame curvature (Eq. (13)). Comparison with published results from a cylindrical burner show similar trends.

Acknowledgment We would like to thank Drs. Uri Vandsburger and Greg Lewis for their helpful assistance and suggestions in the Rayleigh scattering and the particle tracking velocimetry experiments, and Terrance Wagner for useful discussions on flame tip propagation.

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Symposium (International) on Combustion, p. 613, The Combustion Institute, 1953. 10. STREHLOW, R. A. AND SAVAGE, L. D.: Comb. Flame 31, 209 (1978).

FLAME S P E E D MEASUREMENTS 11. MATALON, M.: Comb. Sci. Tech. 31, 169 (1983). 12. MATALON, M., AND MATKOWSKY, B. J.: J. Fluid Mech. 124, 239 (1982). 13. CLAVIN, P. AND WILLIAMS, F. A.: J. Fluid Mech. 116, 251 (1982). 14. WAGNER, T. C. AND FERGVSON, C. R.: Comb. Flame 59, 267 (1985). 15. WAGNER, T. C. : An Investigation of the Structure of a Bunsen Flame Using Laser Velocimetry, M.S. Thesis, Purdue University (1983). 16. ECHEKKI, T. ANt) MUNGAL, M. G.: Particle Tracking in a Laminar Premixed Flame, to appear in Physics of Fluids A, Sept. (1990). 17. NAMER, I., AND SCHEFER, a. W.: Experiments in Fluids 3, 1 (1985).

461

18. MIZOMOTO, M., ASAKA, Y., Iral, S., AND LAW, C. K. : Twentieth Symposium (International) on Combustion, 1933, The Combustion Institute, (1985). 19. LAW, C. K., CHO, P., MIZOMOTO, M. AND YOSHIDA, H.: Twenty-First Symposium (International) on Combustion, 835, The Combustion Institute, (1988). 20. POINSOT, T., ECHEKKI, T. AND MUNGAL, M. G.: A study of the laminar flame tip and implications for premixed turbulent combustion, submitted to Comb. Sci. and Tech. (1990). 21. ECnEKrd, T. AND MUNGAL, M. G.: HTGL Memo. Report MR 4-89, Mechanical Engineering Department, Stanford University (1989).