Edge-Flames in Homogeneous Mixtures

Edge-Flames in Homogeneous Mixtures T. G. VEDARAJAN and J. BUCKMASTER*

Department of Aeronautical and Astronautical Engineering, University of Illinois, Urbana, IL 61801, USA We examine a premixed flame located in a plane counterflow of fresh cold mixture and hot inert. The temperature of the inert is significantly less than the adiabatic flame temperature, so that the flame response to strain-rate variations is multivalued for a finite interval of strain rates. In this interval there are two stable one-dimensional solutions, one characterized by vigorous combustion, the other by weak combustion. Numerical solutions are constructed that describe an unsteady two-dimensional evolution between the two solutions in the direction normal to the straining-flow plane. Following an initial transient, this evolution consists of a wave of permanent form, traveling at constant speed in the out-of-plane direction. This wave is an edge-flame whose speed is positive or negative, depending on the value of the strain rate. The role of edge-flames in the failure of upward rising methane/air flames in sublimit mixtures is briefly discussed. © 1998 by The Combustion Institute

PREAMBLE: EDGE-FLAMES IN NON-PREMIXED COMBUSTION A precise definition of an edge-flame is elusive but, roughly speaking, it is the propagating structure in the neighborhood of a flame-sheet edge. Edge-flames are commonplace in non-premixed combustion. Consider, for example, a flame spreading over a fuel bed (Fig. 1). Nominally, the combustion field consists of a diffusion flame separating the fuel gases from the air but, because of quenching by the cool bed surface, there is a dead-space between the bed and the flame, so that the latter has an edge. Mixing occurs in the neighborhood of the edge, and the local structure is neither a diffusion flame nor a premixed flame, but is hybrid in nature. The propagating nature of edge-flames is clearly revealed (as for deflagrations without edges) when we examine flames that are well removed from surfaces. Consider, for example, the plane counterflow diffusion flame (Fig. 2) cut at the plane z 5 0, so that there is, instantaneously, a flame-sheet in the region x 5 x 0 , z . 0, with an edge at z 5 0 [1]. This edge will move as a reactive-diffusive wave with a well-defined speed that is controlled by the Damko ¨hler number (i.e., the rate of strain). If the Damko ¨hler number is large the edge will move in the negative z-direction, but if the *Corresponding author. COMBUSTION AND FLAME 114:267–273 (1998) © 1998 by The Combustion Institute Published by Elsevier Science Inc.

Damko ¨hler number is small the edge will move in the positive z-direction. The attachment of premixed flames to burner rims has long been discussed in the context of a propagating flame that positions itself near the rim so that its velocity, reduced in magnitude by heat losses to the rim, is balanced by the local gas flow. A detailed discussion can be found in Chapter 5 of Lewis and von Elbe [2]. What is now clear (but was not when Lewis and von Elbe wrote their book) is that diffusion flame attachment has similar ingredients. A pure diffusion flame does not propagate but its edge does, and the edge structure positions itself so that its velocity is balanced by the local flow. It is useful to think of the edge-flames that we have been discussing so far in the context of the S-shaped response that is characteristic of diffusion flames. Thus the one-dimensional counterflow flame has variations of maximum temperature with Damko ¨hler number that define an S-shaped curve with stable lower and upper branches (see Fig. 3, interpreting uequil as the maximum temperature and ignoring f for the moment). Then the edge of a cut flame, as in Fig. 2, can be thought of as an evolution in space from the equilibrium point 1 to the equilibrium point 2. For the unbounded flame, this evolution takes place over the infinite z-interval (2`, 1`). If the Damko ¨hler number is close to D i , the hot gases characteristic of equilibrium state 2 act as an ignition source for the cold mixture ahead of the edge, and the edge advances. If D 0010-2180/98/$19.00 PII S0010-2180(97)00283-6

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T. G. VEDARAJAN AND J. BUCKMASTER

Fig. 1. Flame spread over a fuel bed.

is close to D e , the cold mixture characteristic of the equilibrium state 1 quenches the post-edge flame, and the edge retreats. Ignition waves and failure waves of this type can be modeled using the one-dimensional equation [3]: du d2u V 2 2 5 2~u 2 uW! dz dz 1

1 D~1 2 u !2 e e ~121/u! ; f~u !

(1)

where u represents temperature. The terms on the left account for convection and diffusion in the z-direction; the first term on the right is a simple linear model of transverse heat fluxes (in the x-direction in Fig. 2) from the flame to the

Fig. 3. S-shaped response.

remote boundaries; and the second term on the right is a model reaction term. For this model, equilibrium is defined by setting f 5 0, and then the graph of uequil versus D is S-shaped (Fig. 3), and for D fixed at the value D* in the interval (D i , D e ), f( u , D*) has three simple zeros (Fig. 4). Integrating Eq. 1 from z 5 2` (where u 5 u1) to z 5 1` (where u 5 u2) yields the formula [4] V

E

1`

2`

~d u /dz! 2 dz 5

E

u2

f~ u ! d u

(2)

u1

so that it is clear from Figs. 3 and 4 how V changes from negative values to positive values as D is increased.

Fig. 2. Counterflow edge-flame.

Fig. 4. f( u , D*) versus u.

EDGE-FLAMES IN HOMOGENEOUS MIXTURES satisfies the boundary-value problem

EDGE-FLAMES IN HOMOGENEOUS MIXTURES We now turn to the central question of our paper, one that does not appear to have been addressed before: Can edge-flames occur in homogeneous mixtures? To examine this question we consider the plane counterflow of fresh mixture and an inert.1 We suppose that the adiabatic flame temperature for the mixture is T a , and the supply temperature of the inert is T i . The Lewis number is assumed to be 1. Then the response of the flame to variations in strain in this configuration depends on the relative magnitudes of T i and T a . Roughly speaking if T i $ T a the response is monotonic, but if T i , T a a multivalued response occurs for a range of strain rates [5, 6], which raises the possibility that edge-flames can be described that effect a transition between two of these multiple values. We start with a framework in which multiple solutions can be constructed. THE ONE-DIMENSIONAL COUNTERFLOW FLAME Consider the equations 2C pra x

dT d 2T 5 l 2 1 QBYe 2E/RT, dx dx

dY l d Y 5 2 BYe 2E/RT, dx C p dx 2

(3)

corresponding to a constant density model with one-step Arrhenius kinetics. Boundary conditions are: x 3 2` T 3 T i, Y 3 0; x 3 1` T 3 T f, Y 3 Y f.

(4)

The Schvab-Zeldovich variable QY w;T1 Cp 1

d 2w dw 50 2 1 j dj dj

(6)

j 3 2` w 3 T i; j 3 1` w 3 T f 1

QY f ; Ta Cp

where j 5 x =ra C p / l is a nondimensional distance. Its solution is

w 5 Ta 1

1 ~T 2 T a!erfc~ j / Î2!. 2 i

(7)

Then Eq. 3a is an equation for T alone, 2j

S D

dT d 2T E 5 21D dj dj RT a

where D;

S D

Be 2E/RTa RT a ra E

2

~ w 2 T!e E/RTa2E/RT (8)

2

(9)

is a Damko ¨hler number controlled by a. With DT 5 T a 2T f , we have made the choices Ti Tf E 5 16, 5 0.9, 5 0.2, RDT DT DT

(10)

so that

2

2ra x

269

(5)

This raises the semantic question of whether it is correct to talk of homogeneous mixtures, but we wished to draw a sharp distinction between the structures that we describe and edge-flames that can arise in stratified mixtures that contain a stoichiometric level surface.

Ta E 40 5 1.2, u 5 5 ; DT RT a 3

(11)

and have solved Eq. 8 numerically using Newton iteration and an arc-length continuation method [7]. The nondimensional stand-off distance of the flame from the stagnation plane is j f (D) where

ÎD j f 5 x f

Î

1

BC p e 22 u ; ˜x f l u

(12)

and ˜x f is also nondimensional, but is constructed without the use of a as a scaling factor (x f is the dimensional distance, the location of the temperature maximum). Variations of ˜x f with 1/D are shown in Fig. 5. Figure 6 shows the two possible stable temperature profiles when 1/D 5 0.329. Note that for our parameter choices there is very little heat generated by chemical reaction for solution 1.

270

T. G. VEDARAJAN AND J. BUCKMASTER

Fig. 5. Flame stand-off distance versus rate of strain.

TRANSVERSE WAVES (EDGE-FLAMES) Figure 7 shows the location of two possible flames in the x-y plane when D has a value in the interval for which multiple solutions exist. We now examine an evolution in the z-direction (Fig. 7) in which the solution 2 describes the equilibrium at z 5 1`, and solution 1 describes the equilibrium at z 5 2`. Throughout this evolution w does not change and so is defined by the formula (7), and the equation for T is ­T ­T ­T rCp 1 rCpU 2 rCpax ­t ­z ­x

S

5l

D

­2T ­2T 1 1 BCp~w 2 T!e2E/RT , ­x2 ­z2

Î

ra C p , t 5 a t, V 5 U l

Î

rCp , al

the equation becomes ­T ­T ­T ­ 2T ­ 2T 1V 2j 5 21 2 ­t ­h ­j ­j ­h

(13)

where U is the gas speed in the z-direction. Defining

h5z

Fig. 7. Edge-flame evolution.

(14)

Fig. 6. Stable temperature profiles, 1/D 5 0.329.

1D

S D E RT a

2

~ w 2 T!e E/RTa2E/RT.

(15)

We set V 5 0, impose initial conditions which describe a smooth (but arbitrary) interpolation between solutions 1 and 2, and integrate the equation using the method of lines. This can be carried out for any value of (1/D) in the interval (0.294, 0.517). A differential-algebraic system solver DDASSL is used for this purpose [8]. Figure 8 shows the temperature profiles for t 5 0 and t 5 27 when 1/D 5 0.494, together with temperature contours. They reveal a failure wave, an edge which retreats. Decreasing 1/D to the value 0.329 generates an ignition wave, an edge which advances (Fig. 9). The reaction rate contours shown in Fig. 9 show how, for the value of activation energy we have chosen, significant reaction is confined to a thin curved flame sheet. Our solutions reveal that, in all cases, and following an initial transient, a steadily propa-

EDGE-FLAMES IN HOMOGENEOUS MIXTURES

Fig. 8. Extinction wave, 1/D 5 0.494. Top: temperature profiles at t 5 0, 27. Bottom: temperature contours.

gating wave emerges of constant form. Temperature variations with z at different times, with x fixed at the value 1.175 and 1/D 5 0.494, are shown in Fig. 10. (This value of x is the location of the temperature maximum for the type 2 solution, so that x 5 1.175, z 5 ` is the point of maximum temperature for the entire combustion field). By calculating the shift of the profile in h with t, a speed (nondimensional) of wave propagation can be calculated where (cf. Eq. 14c)

Î

dh rCp 2 5 V * ~D! 5 U * dt al

(16)

and U * is the dimensional edge-speed. Equation 16 can be written as

271

Fig. 10. Time evolution of the temperature profile at x 5 1.175 when 1/D 5 0.494. t 5 0, 1, 3, 9, 15, 21, 27.

V*

ÎD

5 U*r

Î

C p u/2 ˜ ue ; U * lB

˜ is a dimensionless speed related to where U * ˜ with D U * independently of a. Variations of U * are shown in Fig. 11. Note that ˜ 5 U *

rU*

Î2T a

M ad` DT

5 1.697

rU* M ad`

(18)

where M ad` is the adiabatic mass flux for the deflagration in the asymptotic limit u 3 `. Since the exact value of M ad calculated numerically is M ad 5 ~1.556!

1 2u/2 e u

Î

lB , Cp

we see that

Fig. 9. Ignition wave, 1/D 5 0.329. Top: temperature contours at t 5 0, 6. Bottom: reaction rate contours (values 1, 3, 5, 7, 9).

(17)

˜ versus 1/D. Fig. 11. Edge-speed U *

(19)

272

T. G. VEDARAJAN AND J. BUCKMASTER

Fig. 12. Failure waves. Fig. 13. Ronney’s experimental configuration.

˜ 5 1.556 U *

rU* M ad

(20)

and r U * /M ad 5 0.213 at the maximum Damko ¨hler number, r U * /M ad 5 20.511 at the minimum Damko ¨hler number.

aspect ratio slots deliver the two flows, but the slots are misaligned by a few degrees so that the rate of strain varies across the slot width. In this way an edge-flame is generated, positioned at that point where the rate of strain is compatible with (essentially) a zero edge speed.

THE EXPERIMENTAL RECORD Edge-flames (failure waves) have been observed in methane/air flames rising in a standard inflammability tube containing sublimit mixtures [9]. For such mixtures, the rise speed of the flame and the hot bubble of gases behind it is buoyancy controlled—following extinction of the flame, the hot gases continue to rise with unchanged speed until they have cooled and buoyancy forces have weakened. Consequently, in a reference frame tied to the rising bubble, the flame is close to a stagnation plane and experiences a straining flow generated by the displacement by the bubble of the cold fluid ahead of the flame. The post-flame gases are cooled by heat losses to the tube walls so that in due course, after the supercombustion initiated by the spark has subsided, the flame is quenched at the centerline by reason of the mechanism identified in Fig. 5 [10]. Once the quenching is initiated in this way, it continues via an axisymmetric failure wave that sweeps down the flame from the tip to the skirt (Fig. 12). This failure wave propagates both because of its innate property, and because of convection by the flow parallel to the flame. The only other experimental record of edgeflames in homogeneous mixtures that we are aware of is recent experimental work by Ronney [11] in which a standard counterflow apparatus is used with an ingenious twist (Fig. 13). High

CONCLUDING REMARKS In this paper we have examined edge-flames in homogeneous mixtures, and provided a plausible explanation of the failure waves that have long been part of the record of inflammability limits for upward propagation. It is characteristic of these edge-flames, as it is for their diffusion flame counterparts, that they are more vulnerable than the underlying one-dimensional flame. Thus, for the calculations that we have described here, a one-dimensional flame undergoes the 2-1 quenching transition when 1/D 5 0.517 (Fig. 5), the static quenching point. But the corresponding edge-flame quenches via a failure wave for values of 1/D down to 0.482. As for ignition, there the situation is also without surprises. Ignition in the absence of vigorous burning requires a value of 1/D 5 0.294, but once a strong flame has been established it will ignite the cooler gases at its edge for values of 1/D up to 0.482. This work was supported by the Air Force Office of Scientific Research and by the NASA–Lewis Research Center. REFERENCES 1.

Dold, J. W., Hartley, L. J., and Green, D., in Dynamical Issues in Combustion Theory, IMA Volumes in

EDGE-FLAMES IN HOMOGENEOUS MIXTURES Mathematics and its Applications (P. C. Fife, A. Lin ˜´an, and F. A. Williams, Eds.) Springer, New York, 1991, p. 83. 2. Lewis, B., and von Elbe, G., Combustion, Flames and Explosion of Gases, 3rd ed., Academic Press, Orlando, FL, 1987. 3. Buckmaster, J., J. Eng. Math. 31:269 –284 (1997). 4. Grindrod, P., Patterns and Waves, Clarendon Press, Oxford, UK, 1991, p. 36. 5. Buckmaster, J., and Mikolaitis, D., Combust. Flame 47:191–204 (1982). 6. Darabiha, N., Candel, S. M., Giovangigli, V., and Smooke, M. D., Combust. Sci. Technol. 60:267–285 (1988). 7. Giovangigli, V., and Smooke, M. D., Appl. Numer. Math. 5:305–331 (1989).

8.

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Brenan, K. E., Campbell, S. L., and Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier Science Publishing, New York, 1982. 9. Jarosinski, J., Strehlow, R. A., and Azarbarzin, A., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, pp. 1549 – 1557. 10. Buckmaster, J., and Mikolaitis, D., Combust. Flame 45:109 –119 (1982). 11. Ronney, P., Personal communication.

Received 7 April 1997; accepted 25 August 1997