Prediction of the quenching effect of various surface geometries

728

KINETICS OF COMBUSTION REACTIONS

of reaction. Hydrogen and carbon monoxide flames which show only slight nonthermal behavior will certainly have to be considered in such detail, since they do not easily fit into the simplified picture presented above. Acknowledgments Acknowledgment is made to the Chief Scientist, British Ministry of Supply, for permission to publish this paper. Crown copyright Reserved Reproduced with the permission of the Controller of Her Brittanic Majesty's Stationery Office.

REFERENCES 1. PARKER, W. G., AND WOLFHARD, H. G.: Fourth Symposium (International) on Combustion, p.

420. Baltimore, The Williams & Wilkins Co., 1953. 2. ADAMS,G. K., PARKER, W. G., AND WOLFHARD, H. G.: Faraday Soc., t3, 97 (1935). 3. WOLFHARD,H. G., AND PARKER, W. g.: Proc. Phys. Soc., A65, 2 (1952). 4. GAYDON,A. G., AND WOLFHARD, IX. G.: Third Symposium on Combustion Flame and Explosion Phenomena, p. 504. Baltimore, The

Williams & Wilkins Co., 1949. 5. GAYDON, A. G., AND WOLFHARD,H. G.: Proc. Roy. Soc., A205, 118 (1951).

82

PREDICTION OF THE QUENCHING EFFECT OF VARIOUS SURFACE GEOMETRIES By A. L. BERLAD AND A. E. POTTER, JR.

Introduction Recent flame quenching research I has indicated that there should be a set of simple relations among the various channel geometries which are capable of just quenching a given flame at a given pressure. The diffusional quenching mechanism proposed by Simon et al. is used there to predict that the quenching effect of a cylindrical tube of a given diameter dc is equivalent to that displayed by infinitely long plane parallel plates of separation dp when dp = ~/i2/32 dc An adequate check of this relation was somewhat hindered at that time by the facts that (1) in practice, quenching data associated with infinitely long plane parallel plates can be at best only approximated to by rectangular channels of large length to width ratio, and (2) rectangular slot quenching data ~ are usually obtained with a downward propagating flame, whereas cylindrical tube quenching data ~ are usually obtained with an upward propagating flame. The objectives of this investigation were essentially twofold: (1) To derive, on the basis of the average active particle chain length criterion of Simon et aI. ~, or on an equivalent thermal basis, a set of equations

which predict the relations among the dimension of a number of simple geometries which are capable of just quenching a given flame at a given pressure, and (2) To test several of these relations by determining experimentally the wall quenching of downward propagating propane-air flames as a function of fuel-air ratio and pressure for rectangular slots, cylinders, and cylindrical annuli.

Nomenclature The following symbols are used in this paper: fraction of molecules present in gas phase which must react for flame to continue to propagate a inside annulus diameter BI an arbitrary constant B2 an arbitrary constant be ellipse major axis b, rectangular slot length Cr total number of active particles c numerical concentration of active particles c average numerical concentration of active particles Co number of active particles created per unit time per unit volume Di diffusion coefficient for active particles of one kind A

729

QUENCHING EFFECT OF SURFACE GEOMETRIES

d d~ dc d, d~ dr dt G ki

quenching distance outside annulus diameter cylinder diameter minor axis of ellipse plane parallel plate separation rectangular slot width side length of equilateral triangle quenching geometry factor specific rate constant for reaction of active particles of one kind with fuel molecules NI average number of fuel molecules per unit volume of reaction zone n power describing the temperature dependence of the diffusion coefficient, D ~ T" p pressure r plane polar coordinate To initial burner wall temperature and temperature of unburned gas TR reaction temperature u average number of effective collisions of active particle with gas phase molecules before the particle collides with and is destroyed at a wall r time between effective colisions ~o equivalence ratio SUBSCRIPTS

a c e G i p r t

The diffusional quenching equation of Simon et al. ~ may be written

where do is the quenching distance for ~nyg!ven geometry and G is a factor associated with that geometry. It is assumed that d2/G is constant for all quenching gometries and that a flame will be quenched when the average chain length for a given active particle species falls below some critical value a. This critical value of uo is then the same for all geometries and may be written 3 d~

~o cori

G1 -

(3)

G~

I t is thus possible to predict the relative quenching behavior of any two quenching geometries on the basis of the geometry factors obtained from the average chain length calculations. It should be noted here that a thermal criterion exists for quenching which is fully equivalent to the average active particle chain length criterion of Simon et al?. Equation (2) indicates that va is proportional to ~o. Also, the concentration term of the differential equation of diffusion is entirely analogous to the temperature excess term of the differential equation of conduction. Thus, it follows that the fully equivalent thermal criterion is one which requires that a given flame be quenched when its average temperature excess falls below some critical value. However, the specific cases treated below will be computed on the basis of the active particle chain length criterion. Q U E N C H I N G BY CYLINDRICAL TUBES AND PLANE PARALLEL PLATES OF I N F I N I T E E X T E N T

Vc

d~o ~o - = - 32D~r~ cor~

vp

12Diri

(4)

d~

~p -

cori

(5)

These two geometries will each just quench a given flame when vp = vr At this condition, then Ec --- ~p and dp = ~ de .

Ns ~., (p,/DO

GOir~

d~

and

T h e o r y a n d Derivations

vo

d~

On the basis of the average chain length calculations of Semenoff3, a relation between the quenching distances associated with cylindrical tubes and plane parallel plates of infinite extent is given by Simon et al. 1. For these two geometries, equation (2) yields

annulus cylinder ellipse geometry type active particle species plane parallel plates of infinite extent rectangular slot equilateral triangle

\To/

For any two geometries, equation (2) gives

(2)

Q U E N C H I N G BY RECTANGULAR SLOTS

In order to calculate the flame quenching effects of rectangular channels, one may proceed by first solving the diffusion equation subject to the appropriate boundary conditions. Thus, con~ sider a rectangle with center at the origin of the x-y plane. The rectangle is of length br and of width dr. Let this rectangle correspond to a typical cross section of a rectangular channel of infinite extent in which active particles are being

730

KINETICS OF COMBUSTION REACTIONS

generated uniformly throughout the volume and destroyed on collision with the walls. The differential equation of diffusion describing this case is 02Cr --

~x ~

02Or +

D~

(6)

subject to the boundary conditions

and

c,=0{ ~=x y •

\

-- 0.047 \ b ~ ] . J d "

~

(9)

12D~ri QUENCHING BY CYLINDRICAL ANNULI

CO

0y2

1 -- 0.300 v~ =

(7b)

Proceeding in a similar manner, consider an annulus of infinite extent that is defined by two concentric cylinders of diameters a and d~, where a < d~. As before, let this annulus represent a region in which active particles are uniformly generated throughout the volume and destroyed on collision with the walls. The differential equation of diffusion describing this case is given in plane polar coordinates as r dr

r

+

-D~

= 0

(10)

The boundary conditions are C=0

(11a)

r = a/2 and c----0

r = do~2

(llb)

I t can be shown ~ t h a t equation (10) h~s the solution

1 r2co c = B, + B2 log, r - - - 4

ADJUST SCREWS

(12)

Application of the boundary conditions gives B, = ~

FIG. 1. Slot burner with annular insert. Considering a unit height, the total number of active particles in the rectangular channel Cr.r is given by br/2

Cr., = 4 f

D~

dr/2

f

crdxdy = crb,dr

(8)

The formal mathematical problem associated with equations (6) and (7) corresponds to other physical problems which have already been treated and which are presented by Jakob 4and by Purday ~. The results given by these authors may be used to calculate relative values of c~ for a range of length-to-width ratios of the rectangular slot. The factor G~ m a y be evaluated as a function of (d~/b~) through the use of these values. The results may then be fitted to a quadratic equation and used with equation (2) to give

1

a~co --

co(as -- d~) og, (a/2) -i-o~?a~)

(13a)

co(a s -- d]) 16Di loge (a/da)

(13b)

and B~ = Thus

co Fa' - 4r~ + ( d ~ - a0 loge (a/2r)] c =~L loge (a/da) J

(14)

When a unit height is considered, the total number of active particles in the annulus is given by I " da]2

Cr.~ = / Jal2

c(21rr)dr

05)

or

oF,~

i sE,

I a' + d,~

i

a/Z)

a" 1

(16)

QUENCHING EFFECTOF SURFACE C.E~)METRIES dr (OBS.), in.

. 7 -dp(REF.), in. .6 -

jo

dr / b r

0.150

.5-

in.

dr,

.5

~

731

jl.020

~

. 4 - 0.255

.4

.255

_ " 2 ~ ~ , o

3

_ 3

3

~

.

~

~

340

o.-.2

E.3 o Q.

"

/

.680 .340 .051

o~1.020

o3

u~ .2 w

.51o

~ . 7

o_

34o

.494

.I

.510

I

I0

I

12

II

14

16

I

c

I

18

I

20

I0

AIR-FUEL RATIO, A/F

I.'65

~

r

_,

t

i

It

t

I

I

II

1

. 102

t

12 14 16 18 AIR-FUEL RATIO, A/F

I

L I II I I 1.30 I.II .976 .867 .780 EQUIVALENCE RATIO, (a)

k

20 I

1.56 1.30 I.II .975 .867 EQUIVALENCE RATIO, r

.780

(b) FIG. 2. l,imiting pressure curves for various series of r e c t a n g u l a r slots. (a) Full length (5 in.) slots of various widths. (b) Slots of various lengths and widths. .8

o, in.

~

.7 .6

.80 .70

0.498

.64

tY

.~--

~

~

a, in. 0.750

~

.48

E .4--

0

.56

.5 ~ o

-

E

.626

.40

. .30

~NS:k%.~.~

.250

tE

U) r LU r Q.

if) (/3 w r n

.040

.2--

.498

o.

i.u" .24

.250 .16

.100 .040

,12 .000 "110 I

I

I

I I

I

I

t

I

II

I

I

12 14 16 18 20 AIR-FUEL RATIO, A/F

1.56 1.30 1.11 .975 .897 .780 EQUIVALENCE RATIO, r

I

.0810

I

II

I

l

12 14 16 18 20 AIR-FUEL RATIO, A/F I

i

i

II

i

I

1.56 1.30 I. II .975 .867 .780 EQUIVALENCE RATIO, ~P

(a) (b) FIG. 3. L i m i t i n g pressure curves for two series of annuli. (a) Outside diameter, 0.750 in. (t)) Outside diameter, 1.000 in.

732

KINETICS O F C O M B U S T I O N

On the basis of the relations given for the reaction velocity3 it follows that Cr,=

-

-

vi

c0~r (d~

a=)v,

42=

length is given by ,,,

(17)

1 -f-

-4~

log~ (a/d~).]

(18)

To compare the predicted quenching effect of an .56

(20)

+ 1 Diri As expected, this equation reduces to equation (4) for the case where d, = b,. Similarly, for a channel with a cross section defined by an equilateral triangle of side length dr, the results of Purday s may be employed to show that

d c ~ irl.

vt =

q~~,,~0.252

.44

d ~.

=

4

Equations (16) and (17) then give

~ -- 32D,r~

REACTIONS

d ~,

(21)

80Dirl

.40

GENERAL RELATIONS

.32-

The relations among the dimensions of the six quenching geometries treated may be expressed by the following set of equations

E .24 -

.499

1-"2 = 32 -- 1--'2

a,(/) oo I.U

.J

32

~- . 1 6 -

~

.12-

I0

(22)

.750

32

~

> . . . ~ ~

1.000

I 12

I 14

I 20

II 16

~ 18

I

I

II

I

I

1.30

I.II

.975

.8fi7

.780

EQUIVALENCE RATIO,

annulus and that of a cylinder, set ra = r, to get 1 +

( a ) 2 + 1 - (a/d~)" ~ log, (alga,)

(19)

As expected, dc = d= for the case where a = 0. QUENCHING BY

~d,/b,)~

Apparatus

Fio. 4. Limiting pressure curves for series of cylinders.

r

1+

d~ 8O

AIR-FUEL RATIO, A/F

i 1.56

d, d-~ =

-- 0.0470 \ ~ ]

d~ [ 1+ ( a ) 21-=(a/d~ q~ log. (a/da)J

n,-

.08

~,

CHANNELS OF ELLIPTICAL AND

EQUILATERALLY TRIANGULAR CROSS SECTION

Calculations for geometries other than those discussed previously may be carried out in a similar fashion. Thus, when a channel of elliptical cross section, where d. and b~ are the minor and major axes, respectively is considered, it follows from Purday s that the average chain

The apparatus consisted of a fuel-air meteringand-mixing system, a rectangular slot burner enclosed in a pressure-controlled tank, and various inserts which were placed in the opening of the slot burner so as to alter the geometry of the burner channel. Sonic orifices were employed to meter the propane and the air separately. The slot burner is described elsewhereL Two types of slot burner insert were used--one type was designed to produce rectangular channels, the other to produce annular or cylindrical channels. The first type of insert consisted of two watercooled brass blocks placed in the opening of the slot burner in such a way as to produce a rectangular channel. Blocks of various dimensions were made in order to provide various rectangular channels. The kind of insert used to form channels of annular cross section is shown in Figure 1. The cylindrical opening in the hollow brass block

733

QUENCHING EFFECT OF SURFACE GEOMETRIES

forms the outer diameter of the annulus. The removable center body forms the inner diameter of the annulus. Both the hollow block and center-

cylindrical channels of different diameters, brass sleeves were slipped into the cylindrical opening. The lips of such sleeves were water cooled9 .6

.8

~

.6 c

~.4 kt~ ~J O3

o o

GEOMETRY CYLINDER ANNULUS

GEOMETRY o J~ [3

.4

.=E

CYLINDER ANNULUS RECTANGULAR SLOT REFERENCE PLANE PLATES

"

~

PARALLEL PLATES

g

r162

.2

o~ bJ o3

I I I .2 .4 .6 PRESSURE, p, otto

I"%1

.8

I I .2 .4 PRESSURE, p~ otto

hO

(a)

I~1 .6

.8

(b)

.8

96 ~ c' ~ &.4 '~

GEOMETRY o CYLINDER ~NNULUS RECTANGULARSLOT

r ~ ~

z _o

REFERENCE PLANE PARALLEL PLATES

r~

9I

L .2

.4

.6

I .8

PRESSURE, p, atrn

(c) FIG. 5. Calculated plane parallel plate quenching distance as a function of observed ]imiting"pressure for various geometries. (a) Air fuel ratio, 19.0. (b) Air-fuel ratio, 15.6. (c) Air-fuel ratio, 12.0. body were water cooled as shown. The position of the centerbody with respect to the cylindrical opening could be adjusted by means of the screws. Very small centerbodies were made by joining 0.04- or 0.10-in stainless-steel tubing to a short brass tube of large enough diameter to be conveniently held by the positioning screws. These small centerbodies were aircooled, so that it was necessary to extend the centerbody 10 to 12 in above the burner lip to avoid disturbance of the flame by the air blast from the centerbody tip. The insert used to form annular burner channels was also used to form cylindrical channels simply by removal of the centerbody. To obtain

Procedure

The limiting pressure for flame propagation through a burner channel of any given geometry was measured in the same manner as described by Berlad 7. First, a flame was established on the burner. Then, after the pressure in the tank was stabilized, flow to the burner was suddenly interrupted. The flame would either die on the burner lip, or flash back through the burner channel. Generally, the difference between the two pressures which defined the transition from the flashback region to the quenching region could be determined to about 0.02 in of mercury. Flash back

734

KINETICS OF COMBUSTION REACTIONS

occurred at the upper pressure, quenching at the lower. The limiting pressure was taken to be the average of these two pressures. The question of whether or not air cooling was sufficient for the small (0.04 and 0.10 in) centerbodies was resolved by decreasing stepwise the flow through the centerbody, measuring the limiting pressure after each step. The limiting pressure remained unchanged until the air flow was decreased to an exceedingly small fraction of the normally used flow. I t was concluded that the method of cooling was satisfactory.

Experimental Results Experimental propane-air quenching data were obtained for three quenching geometries: recTABLE 1. PERCENTAGE DEVIATION OF CALCULATED FROM ]:~EFERENCE PLANE PARALLEL PLATE QUENCHING DISTANCES FOR VARIOUS GEOMETRIES

predicted by theory, in the following way. The full length (br, 5 in) rectangular slot quenching distances need only small end corrections for conversion to quenching distances for plane parallel plates of infinite extent. The appropriate portion of equations (22) was used to calculate dp values (reference plane parallel plate separations) from the rectangular slot width dr values for which the length-to-width ratio was greater than 10. The dr values used and the reference dp values calculated from them are indicated in Figure 2a. For a given air-fuel ratio, a logarithmic plot of these dp values against limiting quenching pressure was then made, giving a straight line. For a given airfuel ratio, this line (Fig. 5) defines the reference dp values with which the quenching behavior of all other geometries are compared. The calculated z

CYLINDER

A/F mass ratio

Equivalence Rectangularl Annuluratio, ~, I devSl~ n deviation

Cylinder deviation

LAMINAR FLAMES

',

per cent

19.0 18.0 15.6 13.0 12.0

0. 822 .867 1.00 1.2O 1.30

3.63 3.96 6.16 8.50 11.12

1.77 2.39 3.99 5.99 11.33

7.48 7.91 7.07 6.57 5.07

~ 1 range . . . . . . . . > 1 range . . . . . . . .

4.58 9.81

2.72 8.66

7.48 5.82

Total range . . . . . . .

6.67

5.09

6.86

tangular slots, cylinders, and cylindrical annuli. Curves of limiting quenching pressure as a function of air-fuel ratio are presented in Figures 2, 3 and 4 for rectangular slots, cylindrical annuli, and cylinders, respectively. In these Figures, an equivalence ratio scale ~, as well as a mass air-fuel ratio scale is given. Stoichiometric air-fuel ratio is indicated in each Figure by a short vertical line placed just above the air-fuel scale at 15.6. The range of ~ values examined, 0.82 to 1.3, was deemed sufficiently large to serve the aims and purposes of this research and was actually narrower than the range that could have been obtained with the quenching apparatus for the 0.11.0-atm pressure range investigated. Discussion of Results The observed quenching behavior of the geometries investigated was compared with that

I

I

ANNULUS

,

, II

I

II

I I I I

I I I I

I I I I I

t I

' I I I I I

I

FIG. 6. Relative atom concentration (or reaction temperature excess) as function of position in cylinder and annulus for typical case of laminar flame propagation (qualitative representation). plane parallel plate quenching distance for any geometry was then defined to be that wall separation which would, according to equations (22), be as effective a quenching system as the geometry under consideration. These calculated values are indicated in Figure 5 as data points. A comparison of reference and calculated plane parallel plate quenching distances as a function of observed limiting pressures for cylinders, annuli, and rectangular slots is presented in Figure 5 for air-fuel ratios of 19.0, 15.6, and 12.0. In Figure 5, the straight lines are exclusively based on the reference dp values of Figure 2a. The agreement of theory and experiment may be seen from Table 1 which presents the percentage deviation of calculated from reference plane parallel plate quenching distances for various geometries. For rectangular slots and for cylindrical annuli, the agreement between theory and experiment improves systematically as the flame varies from

QUENCHING EFFECT OF SURFACE GEOMETRIES

rich to lean. For the case of quenching by cylinders, systematic improvement occurs when the flame is varied from lean to rich. In practically all cases, however, the deviation of the calculated from the reference plane parallel plate quenching distance appears to depend upon the air-fuel ratio. This deviation may be interpreted to mean that the term

d' A(T~')"/\~IP,)

of equation (1) varies slightly with geometry and that the amount of this variation is mildly airfuel-ratio dependent. Examination of the data of Figure 5 indicates t h a t the individual deviations for any geometry practically always have the same algebraic sign. Allowing for a small amount of experimental scatter, it is apparent t h a t the calculated plane parallel plate quenching distances for annuli are somewhat larger than anticipated and that those for rectangular slots and cylinders are somewhat smaller than anticipated. This may again be interpreted as meaning that the term d2/G varies slightly with geometry. Good agreement between calculated and reference dp values is obtained even for values of > 1 ( A / F < 15.6). Because equation (1) does not appear to be entirely suitable for values of > 17, this agreement may be attributed to the fact that all terms other than d and G essentially cancel out in the course of the calculations. The fact that a relatively small cold surface may exhibit a large quenching effect, under proper circumstances, is illustrated by the fact that when a 0.04-in outside diameter cold tube is inserted along the axis of a 1-in inside diameter tube, the unit is found to quench as effectively as a 0.75-in inside diameter cylindrical tube. This large effect may be explained by the fact that even a small surface can serve as a large sink for

735

active particles (Fig. 6). Thus, a small centerbody changes the active-particle concentration field from one with a relative maximum at the center for the cylinder to zero throughout the centerbody for an annulus (Fig. 6). The average concentration of active particles in the channel is thereby sufficiently lowered to cause a large increase in the quenching effect of the unit. As indicated earlier, the effect of geometry on quenching predicted on the basis of the average chain length calculations may also be predicted through the use of a thermal quenching equation in which the propagation or nonpropagation of a flame is determined by some critical value of the difference between the average reaction temperature and the cold gas temperature. Of course, whether or not such an equation would be generally as satisfactory as equation (1) has yet to be answered. Thus, Figure 6 may also represent the change in the reaction temperature field from one with a maximum at the center for the cylinder to cold gas temperature throughout the centerbody for the annulus. REFERENCES 1. SIMON, D. M., BELLES, F. E., AND SPAKOWSKI, A. F.: Fourth Symposi~Lm (International) on Com,~:stion, p. 126 Baltimore, The Williams & W:lkins Co., 1953. 2. FP*IEDMAN,R., AND JOHNSTON, W. C.: J. Appl. Phys., 21, 791 (1950). 3. SEMENOFF, N.: Chemical Kinetics and Chain Reactions. Oxford, Clarendon Press, 1935. 4. JAKOB, M.: Heat Transfer, vol. 1, pp. 177-180. New York, John Wiley & Sons, Inc., 1949. 5. PUP*DAY, H. F. P.: A n Introduction to the Mechanics of Viscous Flow, pp. 16-18, 28. Dover

Pub., Inc., 1949. 6. CARSLAW,H. S., AND JAEGER, J. C. : Conduction of Heat in Solids, p. 168. Oxford, University

Press, 1950. 7. BERLAD, A. L. : Flame Quenching by a VariableWidth Rectangular-Channel Burner as a Function of Pressure for Propane-Oxygen-Nitrogen Mixtures. J. Phys. Chem., 58, 1023 (1954).