Flame Stability

Stabilization of Premixed Flames on Rotating Bunsen Burners

J. M. CHA and S. H. SOHRAB* Robert R. McCormick School of Engineering and Applied Sciences, Department of Mechanical Engineering,

Northwestern University, Evanston, Illinois 60208

The effect of rotation on stabilization of methane-air premixed Bunsen flames is experimentally investigated. Both the flame blowoff and flashback contours are determined in the fuel mole fraction versus Reynolds number plane (Xr-Re) with the rotational Reynolds number Rer as a parameter. It is found that rotation of the gas increases the flame stabilization area A s = A 8 -A F defined as the difference between the flame blowoff A B and flashback A F areas in the (XF-Re) plane. The flame stabilization efficiency is defined as rls = 1 - Ap /A 8 that approaches unity in either A B ~ ~ or A F --, 0 limit. The experimental results suggest that rotation decreases the flame stabilization efficiency. However, rotation is found to substantially increase the flame stabilization coefficient defined as /3 s = Zs/hst, where ast is the stabilization area of the standard nonrotating burner. The parameters ~s and /3 s may be useful in combustion technology for quantitative evaluation of the stabilization performance of different types of flame holders. In addition, the local hydrodynamics near the center of rotating Bunsen burner is simulated by investigating stabilization of planar laminar premixed flames on rotating porous disks with uniform surface velocity. Physical concepts concerning mechanisms of flame stabilization are discussed in terms of three important parameters namely the translational Reynolds number Re, the rotation Reynolds number Rer, and the fuel mole fraction X F. The results of the experimental findings are shown to be in accordance with prior theoretical investigation.

INTRODUCTION

The understanding of flame stabilization mechanisms is essential for the design of many combustion devices such as ramjets, after- burners, pilot flames among others [1-3]. The problem is how to maintain the flame from being convected out of the combustion cham- ber or away from burner ports because of large velocity of the gas versus that of the flame. Although flame stabilization in ramjets and after-burners involve turbulent fields, studies on stabilization of laminar flames are useful since they help the understanding of the turbulent problem through the change of transport coefficients such as heat and mass diffusivity to turbulent ones [3]. Flame stabilization could be achieved by many mechanisms such as formation of local low-velocity regions by solid ob- jects (bluff body) and hydrodynamics (counter- flows) or by generation of local high tempera- ture regions using pilot flames and ignitors.

The subject of flame stabilization has been extensively discussed by Lewis and von Elbe [1]. Most of the attention has been focused on correlating the flame blowoff data with the

*Corresponding author.

COMBUSTION AND FLAME 106:467-477 (1996) Copyright 1996 by The Combustion Institute Published by Elsevier Science Inc.

rate of flame stretch, Karlovitz number, at the burner rim. It was suggested [1] that a closer correlation of the data with Karlovitz number should be possible when use is made of inverted-flames stabilized on thin wires. This was indeed substantiated by subsequent experimental investigations by Edmondson and Heap [4, 5] performed on inverted-flames stabilized on thin plates within rectangular-slit burners. Another description of premixed flame stabilization is based on the Marble-Adamson problem involving co-flowing streams of cold reactant and hot combustion products [6, 7]. According to this model, flame stabilization requires that the flame development length should be less than the flow recirculation length [3, 6, 7]. For the fundamental understanding of flame stabilization the considerations of the following three questions were found to be essential according to Cheng and Kovitz [7]:

(a) how a flame is stabilized on the bluff body wake under steady state conditions (b) how flame selects its equilib- rium position; and (c) what are the essential roles of chemical kinetics and fluid mechanics.

The important influence of the burner rim hydrodynamics on stabilization of Bunsen flames has been emphasized in recent studies [8, 9]. Also, possible benefits of introduction of

0010-2180/96/$15.00 PII SO010-2180(96)00004-1

468 J .M. CHA AND S. H. SOHRAB

swirl in design of flame stabilizers for indus- trial burners have been recognized [10-13]. Often, swirling flows not only help to increase burning intensity through enhanced mixing and higher residence times, but also help in flame stabilization by the formation of secondary recirculating flows. In a recent theoretical investigation [14] the effect of rotation on stabilization and geometry of Bunsen flames that are situated inside rotating tubes was addressed. In this study, it was found that the rotation of the gas tends to reduce flame stabilization since flame flashback occurs at higher values of the mean flow velocity v~ through the burner. In the present study the effects of rotation on stabilization of Bunsen flames are further investigated. The flame flashback and blowoff contours are determined for methane-air premixed flames under systematic variation of the burner rotation velocity. As an approximate local model valid in the central regions of the rotating Bunsen burner, stabilization of planar flames on rotating porous disks with surface blowing is also discussed. Flame stabilization mechanisms are described in terms of three important parameters, namely the gas velocity at the burner, the rotation velocity of the burner, and the fuel mole fraction in the combustible mixture. The principal findings are summarized in the concluding remarks.

EXPERIMENTAL OBSERVATIONS

The schematic drawing of the experimental apparatus is shown in Fig. 1. The burner con- sists of a long 1.15-m copper tube with R = 1.6 and a = 1.9 cm inner and outer diameters. The burner is rotated by means of a pulley-belt to minimize vibrations from the dc-motor. A piece of honeycomb 0.5 cm thick with a large number of small holes is placed at the base of the tube. Another piece of honeycomb 2 cm thick is placed 71 cm below the burner rim. Two bearings are placed 10 and 30 cm below the rim to insure steady vibration-free rotation of the tube. The combustible gas enters the rotating tube through a base connected to a sealed bearing. A concentric jacket 10 cm in diameter, with multiple screens and 1-mm-diameter glass beads is provided around the burner rim for nitrogen gas flow in order to

AIR NITROGEN

Tt HONEy -COMB

/~ /F taM~ NITROGEN

/ JACKET

El~t'n~ll DETECTOR

BEhJilNG

lOOP BEARING /

Fig. 1. Schematic drawing of the rotating Bunsen burner,

reduce the effects of the diffusion flame that surrounds rich premixed flames. The nitrogen flow velocity is less than a few centimeters per second just sufficient to quench a match flame positioned in front of the jacket. This nitrogen flow is essential for tests on blowoff of rich flames since without it the flame blowoff becomes impossible [1]. Methane and air are metered with conventional rotameters and fully mixed before introduction into the burner base (Fig. 1). The rotation velocity of the burner tube is determined by monitoring electric sig- nals from an emitter to a receiver that are interrupted by a chopper attached to the tube

(Fig. 1). In the tests, a mean flow velocity through

the burner V0 is chosen and a premixed flame is stabilized on the Bunsen burner. The fuel mole fraction X e of the lean (rich) mixture is then gradually decreased (increased) until the flame blowoff from the burner occurs. Simi- larly, for flame flashback tests X e of lean (rich) mixture is gradually increased (decreased) until the flame begins to enter the burner tube. Such tests are then repeated at a larger value of the mean velocity V 0. Thus, the

STABILIZATION OF PREMIXED FLAMES 469

contours of flame flashback and blowoff are determined as a function of the Reynolds number Re = RVo/u, where u = /x ' /p = 0.16 cm2/s is the kinematic viscosity, /x' is the dynamic viscosity, and 0 is the density of the mixture. For rich flames instead of complete flame blowoff a partial blowoff is defined as the condition when at least one eighth of the flame circumference is lifted by 0.5 cm above

the rim. Sufficient time interval between the flame flashback tests is provided to insure cool- ing of the burner rim to its original tempera- ture. Also, lean and rich flashback limits are defined as the point when the flames move into the burner tube completely.

Direct photographs of rich X r = 12.4% methane-air premixed flames with increasing rotation velocity to are shown in Figs. 2a-2d

(a) (b)

() (d) Fig. 2. Direct photographs of methane-air premixed flames. X F = 12.4%, V 0 -- 50 cm/s and w = (a) 0 (b) 10 (c) 16 (d) 20 rps.

470 J. M. CHA AND S. H. SOHRAB

similar to those of butane-air premixed flames reported earlier [14]. The experimental results are shown in Figs. 3a-3c for the conditions to = 0, 10, and 20 rps corresponding to the rotational Reynolds numbers Rer = toR2/v = 0, 161, and 322, respectively. For high rotation velocity of to = 20, the value of Xe(> 18%) for blowoff of rich flames at Re = 1000 is found to exceed the rich flammability limit of about 15% (Fig. 3c). This effect is due to mixing of the reactants with the outside atmo- spheric air such that the local X~ actually experienced by the flame are lower than 15%. The influence of the rotation on the stability contours for flame flashback and blowoff are shown in Figs. 4 and 5, respectively. The maximum Re for flashback increases from about 800 to 1600 when rotation Reynolds number increases from 0 to 322 (Fig. 4). According to Fig. 5 larger rotation velocities increase the maximum Re for flame blowoff that occurs near the stoichiometric point XF, = 9.5% from 3000 to about 4000. Also, at lower velocities Re = 250, flame blowoff contours become wider changing from Xe = (6-14)% to Xe = (4-17)% when Re~ is changed from 0 to 322 (Fig. 5).

In view of the above discussions, it is clear that rotation of the Bunsen burner results in enlargement of the area between the flashback and the blowoff contours thus permitting flame stabilization over a wider range of X~-Re parameters. The area within the blowoff and the flashback contours shown in Figs. 3a-3c are denoted by Ae and A s, respectively. The total flame stabilization area is then defined as the difference between the blowoff and the flashback areas A s =-An-A e on the (Xe-Re) plane. Two phenomenologieal parameters will be introduced that may help to characterize the stabilization efficiency of different types of flame holders. First, is the f l ame stabil ization efficiency of a burner which is defined as

rls = 1 - Ap /A B =As /A ~ (1)

that approaches unity in the limits A n ~ oo or A F ~ O. Therefore, the stabilization efficiency of flame holders could be increased by decreasing (increasing) the flashback (blowof0 area. For example, if a flame holder can prevent

XF

O.2~

0.20

O. 18

0.16

O.14

O.12

O.I0

0.08

I)05

004

OO2

0.00

O..22

O.20

O.IJ

O.1~

0,14

O.12

XF O.,o O.08

O.06

O00

++ + +~ + + 4-+ +

2' " oo~ % *** i t * o e

i . , - , , L . * . , - . - . . I~ 1~ ~ ZJ00 30W 3 JW 4000 ~ 50~

Re (a)

+*+ ++4#1~.+~++++ + + +

+ %" . . - . . . "o ,t

i . i . c , i . i , i i I - ' - ' -

Re (b)

3~0

0.22

O.20

+

0.16 ++4* +++++.+++++

GI4

X O.12 %re ~mooeeee 4- .4. 17 KN oe +

: +

coo: ++++* GOB oe lneeee :ee .~.4., .t..4.4.4-

I lG l +

Re (c)

Fig. 3. Flame blowoff and flashback stabilization contours from experiments on methane-air premixed flames with to : (a) 0 (b) 10 (c) 20 rps.

flame flashback under all equivalence ratios then A F = 0 and hence *?s = 1.

The flame stabilization efficiency defined in Eq. 1 is useful for quantitative evaluation of flame stabilization characteristics of a particu-


0.16

0.15

0.14

0.13

0.12

0.11

0.10

XF o.o9

0.08

0.117

0.116

0.05

0.114

0.113

! I ! I I | I

o

A*

OOA 0 **AOoAa ~ O ~ s . * .S** 0 ~ o O o

A

~ o

oo cl oo 2 . . ,A .~$ $06~0 e * si *

G

* R~=0 I A 1~ ffi 161 * P,e, = 322

I I I I I I I I I I I

200 400 600 800 1000 1200 1400 1600 1800 2000

Re Fig. 4. Comparisons between the flashback contours for methane-air premixed flames with to =(1) 0 (2) 10 (3) 20 rps.

lar flame holder under various operating conditions and/or comparative evaluation of different types of flame holders. For example, Eq. 1 is now employed for the analysis of the influence of rotation on stabilization of Bunsen flames being investigated herein. Using the

trapezoidal method the areas (AB, A F) in Figs. 3a, 3b, and 3c, corresponding to Re r = 0, 161, and 322 are approximately determined as (84, 14), (130,29), and (159,36), respectively. Therefore, the total flame stabilization areas for Re r = (0, 161,322) are A s = (70, 101,123)

0.22

0.20

0.18

0.16

0.14

0.12

XF 0.1o

O.O8

O.O6

0.04

0.02

I I I I I I

o o

oO 8 oooo o o

I I I

L " P~ '= 161 o R~ -3z21

*~. o o Ooo Do a o an o a a o o a ~ ,~ 0

o o o A O

o o o, o is A m o o

,

0 .OOi J J I I i I * I * I ~ I I

0 500 1000 lf~0 2000 2500 3000 3500 4000 4500 5000

Re Fig. 5. Comparisons between the blowoff contours for methane-a i r premixed flames with w =(1) 0 (2) 10 (3) 20 rps.

472 J .M. CHA AND S. H. SOHRAB

units and the corresponding stabilization efficiency calculated from Eq. 1 are ~Ts = (83, 78, 77)%. Although rotation of burner increases the flame stabilization area A s, because the corresponding increase in the total blowoff area A B is even more the absolute flame stabilization efficiency r/s actually decreases with rotation. This tendency is reason- able since while flow recirculations that often accompany rotating flows tend to help flame stabilization by hindering flame blowoff they also tend to promote flame destabilization by enhancing flame flashback in accordance with the previous theory [14].

The second parameter that is useful in eval- uating the stabilization characteristics of flame holders and is a measure of the actual increase of flame stabilization area A s is the f lame stabilization coefficient defined as

t3, = A JA , , , (2)

where hst is the stabilization area under standard operating condition. In the present para- metric study of the effect of rotation on stabilization of Bunsen flames A~t refers to the flame stabilization area of Bunsen burner without rotation Ast = 70. With A s = (70, 101, 123) the calculated flame stabilization coefficients using Eq. 2 are /3 s = 1.44 and 1.76 for Rer = 161, and 322, respectively. According to this result the flame stabilization capacity of the Bunsen burner is improved by 44% and 76% in the presence of rotation velocities to = 10 and 20 rps as compared to the standard nonrotating Bunsen burner. Thus, in the present study increasing /3 s means that the burner rotation tends to increase the total flame stabilization area As, and decreasing 7/s means that the increase of A s is actually weaker than that of A B. Clearly, from the design point of view one would like to have both r/3 and [33 to be as large as possible. It is noted that while r/3 is bounded 0 < ~/s < 1,/3 s assumes a value larger (smaller) than unity when A s is larger (smaller) than the standard stabilization area Ast.

FLAME STABILIZATION CRITERIA

The principles of flame stabilization in laminar streams have been described by Lewis and von

Elbe [1]. The data are correlated according to the criteria that flame blowoff should occur at a critical value of the Karlovitz number K b based on the burner velocity gradient gb that is close to unity

K b = rgb / [ pcpv?l ---- 1, (3)

where r , cp, and v[ are thermal conductivity, specific heat at constant pressure, and laminar flame propagation velocity. The validity of the above relation was subsequently verified by experiments performed on inverted-flames stabilization on thin plates situated within rectangular-slit burners [4, 5].

Another phenomenological description of flame stabilization is based on the Marble-Adamson problem [6, 7] according to which, stable flames occur when the flame development length is less than the flow recirculation length. For flame stabilizers with very thin plates or wires the length of flow recirculation L is close to the characteristic stabilizer dimension such that the velocity gradient at the stabilizer gb = Vo/L is the inverse of the residence time of the gas flowing at the mean velocity of V 0 across the length of recirculation zone L as suggested by Zukoski and Marble [15]

go = Vo/L = 1/Tres" (4)

On the other hand, the ignition-delay time is approximately the time taken for reactive gases moving at the velocity vy to travel across the flame thermal thickness I r = ot /u f ,

% = ( lpcp)/v 7 = t lv I = /v7 (s)

when a = r / (pcp) is the thermal diffusivity. In view of the Eqs. 4 and 5, the criteria of equality of Tig and %~s is equivalent to the Karlovitz number criteria in Eq. 3 expressed as

= T ig /T re s = 1 (6)

that is inverse of the Damk~ihler number [3]. It is noted, however, that the flame stabilization criteria that are based on flame stretch should also address the strong Lewis number depen- dence of flame stretch effects [16, 17]. It is also


emphasized here that correlation of flame quenching with the classical expression of the rate of flame stretch K = d(ln A) /d t involving flame surface area A may require reexamina- tion. This is because flame quenching could be more accurately described in terms of the more general concept of voluminal rate of flame stretch involving flame volume V and defined as E = d(ln V) /dt since E includes the influence of hydrodynamics on the flame thermal thickness [18, 19].

In the previous theoretical study [14] the effects of rotation on stabilization of Bunsen flames was discussed in terms of a positive parameter A > 0 related to the rotation velocity of the burner w as

A = Tw2R4/4oe 2 (7)

and another parameter related to the uniform velocity of the combustible gas through the burner v~ and the normal flame propagation velocity v I expressed as

I J, = (V~ -- v f )u fR2/oe 2. (8 )

The quantities y and R are the thermal expan- sion coefficient and the tube radius. Since steady flame configurations in the absence of dissipative effects are governed by [14]

d~') v/[A(2~'2 - 1) + 2/,1, (9)

where ff = r /R, z is the axial coordinate, F(r, t) = - z is the flame front, and ff = F + (v~- vs)t. Therefore, stationary flame fronts are only possible over the entire range of the dimensionless tube radius 0 < ~" < 1 when

-A + 2~ ~ O. (10)

Hence, stable flames occur when the rotation velocity is smaller than a critical value [14]

A~ O. (131}

Hence the following inequality

2 vf - v~vf + A/c < O, (14)

with c = 2R2/~ 2 needs to be examined. Thus, the flame propagation velocity should be

474 J. M. CHA AND S. H. SOHRAB

bounded from below and above according to the expression

(vy)I=Iu~-(v.2-4A/c)~/2]/2 (4A/c ) 1/2 = (y /2 )x /2 toR (16)

for a given rotation velocity to, or an upper bound on the rotation velocity to c < (2 /y )a /2 (v=/R) for a given gas velocity v=.

It is now clear that for a particular rotation velocity oJ and v= there exists a minimum (maximum) value of flame propagation velocity below (above) which the criteria in Eq. 14 is violated and stationary flames become impossible. Therefore, if the fuel concentration X F of lean (rich) mixture is decreased (increased) thereby decreasing the flame propagation velocity vf below the lower critical value (vy) t given in Eq. 15 the criteria in Eq. 14 will be violated and flame blowoff occurs. On the other hand, if the fuel concentration X F of lean (rich) mixture is increased (decreased) thereby increasing the flame propagation velocity vf above the upper critical value (v[) u given in Eq. 15 again the criteria in Eq. 14 will be violated and flame flashback occurs. Thus, the loss of flame stabilization is associated with blowoff or flashback depending on whether the critical conditions just described occur when decreasing or increasing the flame propagation velocity. These predictions are in qualitative agreement with the experimental observations shown in Figs. 3b-3c. It is important to remark therefore that the previous model [14] does predict flame blowoff when the gas velocity v= is indefinitely increased (/z ~ ~) as is experimentally observed.

FLAME STABILIZATION ON ROTATING POROUS DISKS

In the previous theoretical model [14] flame was stabilized within the rotating tube thereby avoiding the complex hydrodynamic interac- tions with the surrounding atmosphere. On the

other hand, the Marble-Adamson model of flame stabilization requires the presence of a shear layer between recirculating hot combustion products and cold unburned mixture [6, 7]. In the following, a simple hydrodynamic model will be discussed that is based on the similarity solutions for cylindrically symmetric flow through rotating porous disks with uniform blowing velocity V 0. Although the gas velocity at the exit of the Bunsen burner used in the experiments (Fig. 1) is parabolic, in the central regions r

STABIL IZATION OF PREMIXED FLAMES 475

The following dimensionless parameters

u = u ' /G , . = v ' /Vo ,

r ---- r 'e l /Vo , z = z'o)/Vo,

p = p ' /oVo 2

W~W '/v0,

(22)

have been defined. Because only the central region of the tube r

476 J .M . CHA AND S. H. SOHRAB

z /< I r and hence is in thermal communication with the burner surface. The flame stabilization is therefore expected to be dependent on parameters that directly or indirectly influence either zf or I r to be discussed next.

Three distinguishable mechanisms of flame stabilization on the rotating porous-disk Bun- sen burner are schematically shown in Figs. 7a-7c in terms of three important hydrodynamic and thermo-chemical parameters V 0, to, XF, respectively. When both X F (hence vf) and to are fixed, increasing the gas velocity

t J /

(a)

(b)

(c) Fig. 7. (a) Stabilization of premixed flame at constant to and v[ with I/o2 > Vol showing how a stable flame zfl < lrl becomes unstable zfz > lr2 by increasing the gas velocity V 0. (b) Stabilization of premixed flame at constant V 0 and vf with to2 < to1 showing how a stable flame zfl < l r becomes unstable zf2 > l r by decreasing the rotation velocity to. (c) Stabilization of lean (rich) premixed flame at constant V 0 and to with vf2 < vfl and X w < XF~ (v/-2 < vfl and X,~ 2 > XF1) showing how a stable flame zf l < 1 r becomes unstable zf2 > l r by decreasing (increasing) X F.

Voz > Vol results in lr2 < It1 and a new flame location zf2 > zll as shown in Fig. 7a. How- ever, at this new location the flame is no longer in thermal communication with the burner since zf2 > It2 and hence could blowoff from this surface upon any perturbation of either //o2 or vf. Similarly, when I'1o and X v (hence vf) are fixed and the burner rotation velocity is decreased to2 < to1 the flame axial position increases zf2 > Ztl while l r = lr2 = lrl remains constant (since V 0 is fixed) until eventually the flame becomes unstable zf2 > I r and blowoff occurs as shown in Fig. 7b. Finally, when V 0 and co are fixed, decreasing (increasing) X F of a lean (rich) flame decreases vf and results in a new flame location zi2 > Zfl and eventually when z f2 > l r the flame becomes unstable and flame blowoff occurs as shown in Fig. 7c. In the same spirit, it can be shown that flame flashback (flame collapse onto the surface of porous disk) occurs with decreasing V 0 (to and X F fixed), increasing to (V 0 and X F fixed), or increasing (decreasing) X v in lean (rich) mixtures (V o < Ufmax and to fixed).

CONCLUDING REMARKS

The effect of rotation on stabilization of methane-air premixed Bunsen flames was experimentally investigated. Both the flame blowoff and flashback contours were determined in fuel mole-fraction versus Reynolds number plane (XF-Re) with the rotational Reynolds number Re, = (0, 161, 322) as a parameter. It was found that rotation of the gas increased the flame stabilization area A s = A s -A F defined as the difference between the flame blowoff A B and flashback A F areas in (XF-Re) plane. The study also led to the introduction of f lame stabilization efficiency defined as r/$ = 1 -AF /A B that approaches unity in either A B ~ oo or A F ~ 0 limit. Using the experimental data the flame stabilization efficiency ~ = (83, 78, 77)% was calculated for Re, = 0, 161, 322, respectively. An additional parameter called the flame stabilization coefficient defined as /3 s = AJAst was introduced, where Zst is the stabilization area of the standard (nonrotating) burner. The values of/3 s = (1.44, 1.76) were found at Re, = (161, 322), respectively. The parameters rts and /3 s may


have applications in combustion technology since they could be used for quantitative and comparative evaluation of the stabilization efficiency of different types of flame holders.

The experimental observations were also compared with the results of the previous theoretical study. It was shown that the prior theoretical model could in fact predict a particular type of flame-blowoff that is associated with the reduction of flame propagation velocity cf induced by reduction of X e or by other mechanisms. The concepts of flame stabilization were discussed for planar laminar flames that are stabilized on rotating porous disk with uniform surface velocity of combustible mixture. This model approximately corresponds to the hydrodynamics in the central regions of the rotating Bunsen burner used in the experiments. The mechanism of flame stabilization was discussed in terms of three important parameters namely the translational Reynolds number Re, the rotation Reynolds number Rer, and the fuel mole fraction Xv. It was found that rotation could enlarge the flame stabilization area A s in (Xe-Re) plane because flames could be anchored to the burner at larger values of Re (for a given Xe) or smaller (larger) values of XF in lean (rich) mixtures (for a given Re). The experimental observations of the flame behavior in the vicinity of the center of the Bunsen burner were found to be consis- tent with the hydrodynamics based on analyti- cal similarity solutions of the inviscid flow through rotating porous disk with surface blowing.

This research has been supported by the Na- tional Science Foundation grant No. CTS- 8820077. The authors also thank Professors A. A. Kovitz and S. Lichter for stimulating discussions.

REFERENCES

1. Lewis, B., and von Elbe, G., Combustion, Flames and Explosion of Gases, Academic Press, New York, 1987, pp. 233-274, 457-466.

2. Williams, F. A., in Applied Mechanics Surveys (H. N. Abramson, H. Liebowitz, J. M. Crowley, and S. Juhasz,

Eds.), Spartan Books, Washington, D. C., 1966, pp. 86-91.

3. Williams, F. A., Combustion Theory, Addison-Wesley, New York, 1985, pp. 503-516.

4. Edmondson, H., and Heap, M. P., Combust. Flame 14:191-194 (1970).

5. Edmondson, H., and Heap, M. P., Combust. Flame 15:179-187 (1970).

6. Marble, F. E., and Adamson, T. C., Jet Propul. 24:85 (1954).

7. Cheng, S. I., and Kovitz, A. A., Seventh Symposium (International) on Combustion, Butterworths Scien- tific Publications, London, 1959, pp. 681-691.

8. Kawamura, T., Asato, K., and Mazaki, T., Combust. Sci. Technol. 22:211-216 (1980).

9. Sohrab, S. H., and Law, C. K., Combust. Flame 62:243-254 (1985).

I0. Grover, J. H., Kessler, M. G., and Scurlock, A. C., Jet Propul. 30:386-391 (1957).

11. Chigier, N. A., and Chervinsky, A., Eleventh Sympo- sium (International) on Combustion, The Combustion Institute, Pittsburgh, 1967, pp. 489-499.

12. Lilley, D. G., A/A4 J. 15:1063-1078 (1977). 13. Oven, M. J., Gouldin, F. C., and McLean, W. J.,

Seventeenth Symposium (International) on Combus- tion, The Combustion Institute, Pittsburgh, 1979, pp. 363-374.

14. Sheu, W. J., Sohrab, S. H,, and Sivashinsky, G. I., Combust. Flame 79:190-198 (1990).

15. Zukoski, E. E., and Marble, F. E., Gas Dynamic Symposium on Aerotherrno-chemistry, Northwestern University, Evanston, IL, 1955, p. 205.

16. Sivashinsky, G. I., Acta Astronaut. 3:889-918 (1976). 17. Buckmaster, J. D., Seventeenth Symposium (Interna-

tional) on Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 835-842.

18. Buckmaster, J. D., Acta Astronaut. 6:741-769 (1979). 19. Pearlman, H. G., and Sohrab, S. H., Combust. Sci.

Technol. 107:155-164 (1995). 20. Clarke, J. F., and Mclntosh, A. C., Proc. R. Soc.

Lond. A 372:367-392 (1980). 21. Buckmaster, J. D., and Takeno, T., Combust. Sci.

Technol. 25:153-158 (1981). 22. Sheu, W. J., and Sohrab, S. H., Combust. Sci. Technol.

66:39-57 (1989). 23. von Karman, T., ZAMM 1:233 (1921). 24. Sivashinsky, G. I., and Sohrab, S. H., Combust. Sci.

Technol. 53:67-74 (1987). 25. Pearlman, H. G., and Sohrab, S. H., Extinction of

diffusion flames and viscous hydrodynamics around rotating porous spheres with surface blowing. To appear.

Received 15 September 1995; revised 23 December 1995

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Flame Stability