Characteristics of an open diffusion flame

COMBUSTION A N D F L A M E 25, 137-139 (1975)

137

Characteristics of an Open Diffusion Flame K. ANNAMALAI and P. DURBETAKI School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332

with the boundary conditions:

1. Introduction

In recent analytical and experimental investigations on the ignition of fabrics [1-4] the effects of convective and radiative heat flux sources have been considered. Particularly, the convective heat flux to the fabric depends upon the height of the premixed or diffusion flames. Diffusion flames are simple to treat. However, this simple problem is complicated by the presence of both free and forced convection. The literature survey revealed results only for flame heights [5], which unfortunately do not show any dependency on ambient and stoichiometric conditions as observed experimentally [6], and no solutions have come to our attention for the flame profiles. Here then are presented the complete results for flame heights and flame profiles.

7=1

at ~ = 0

and 0 < ~ < 1 ;

3' = - a

at ~ = 0

and 1 < ~ < o o ;

7=-c~

at ~ 0

and ~-+0.

Equation (1) is transformed into Laplace coordinate after dividing the domain into two regions, I: 0 < ~ < 1 and II: 1 < ~ < ~. Using matching conditions, the solutions are obtained after inverse Laplace transform. Then 7i = 1 - (12-a-) Iexp - (1 + ~2)/4r/1 I lo(~/2r~)

"

2. Equations and Solutions

It is assumed that (i) pv = constant, (ii) the flame occurs at the stoichiometric surface, (iii) a very rich mixture or pure fuel flows through the burner, (iv) pD = constant, and (v) the flow through the burner is axisymmetric. Instead of obtaining extension of results for outer burner diameter tending to infinity in the BurkeSchumann flame [7, 8], the adopted procedure follows the mathematical technique for heat transfer through composite cylinders [9]. Conservation equation for normalized coupling function is written as

(2)

=

j=O

_1(1 + ~2)/4~7]/ I J! - - ,

0~<~<1 •

(3)

1 <~j
(4)

/

71i =71 -1 -o~,

7 = (1~2~) - (--~2-1) [exp - (1 + ~J2)/4r~] t to (~/2 r~)

- ( - 1 - 4 ~ ) ~o (~/2r/) + I, (~/2 r/)] 1, (~j- 1)2 ~ O,

(s)

and

(1)

7 = 1 - (a + 1) [exp - 1/4r/I,

~ -+ O.

(6)

Copyright © 1975 by The Combustion Institute Published by AmericanElsevierPublishing Company, Inc.

138

BRIEF COMMUNICATION

By assuming the flame to occur at the stoichiometric surface [10], the flame profiles are obtained. Figure 1 shows the flame profiles of diffusion flames with the stoichiometric quantity ~ as a parameter. Comparing with experimental flame profiles for a = .0575, it is found that the theory gives considerable overhang near the burner rim. This discrepancy arises because radial flow of oxygen due to free convection has been neglected, which is not justifiable near the burner rim. By setting 3' = 0 in Eq. (6), the flame height is obtained as: r~/= 1/[4ln (1 + a ) ] .

(7)

Figure 2 shows the plot of ~7t. as a function of a. In terms of the real physicalvariables Eq. (7) can be represented as

x t. QF, o/[47r(p-D/Po)In (1 + a)] =

(8)

The mean mass diffusion coefficient pD in a multicomponent system is estimated by using the weighted average method [11, 12] at mean tern perature 1220°K. With PDo-m = .792 × 10 .3 g/cm sec and a = .0575, the result for flame height 1.C

2.5

2.0

.

-

-

.

1.5

-

-

-

-

-

----

0

.5

1.0

1.5

Fig. 2. Variation of the flame height with the overall stoichiometric parameter.

is plotted on Fig. 3. There is an excellent agreement regarding variation of flame height with respect t o QF o and flame height thus predicted is at the most'25% higher than the measured values. The agreement is due to compensating effects of radial convection and axial acceleration of the species involved. Comparing with Edelman's numerical results [6], which include the effects of momentum transfer and parabolic velocity distribution in the burner, there does not appear to be appreciable deviation. However, their numerical results for flame height show some small dependency on burner diameter. Also, the flame width and flame height are shown to decrease with increasing oxygen concentration, which is in agreement with the above numerical results,

3

I Explriment I

I

/

/

O Burner Dll. .1360:~mm

.E

• " ..... ..

2

° .... o.. :

"'¢

t / >

--'-- Theory

"• ~

~"~4 1O ~f" 1r

.4

I

.2

• f/+'B

/ %/

'

o,'b

++.<++of 0 •5

1.0

Fig. 1. Flame profiles of an open diffusion flame.

40

~uE,.

80

FLOW

RA'r~,

120

QF,o (~m3/m,nl

Fig. 3. Variation of the flame height with the fuel flow rate.

BRIEF COMMUNICATION Nomenclature a radius of burner D mass diffusivity coefficient zeroth and first order modified Bessel I 0 , 11 functions mass flow rate of fuel volume flow rate of fuel r radial coordinate velocity WF, W0 molecular weights of fuel and oxygen x axial coordinate mass fractions of fuel and oxygen YF' YO overall stoichiometric parameter,

m.F QF

139

2.

3.

4.

5.

/3

( Y o ,~VF WF)/(YF o VO WO) (1 - C~b) richness parameter', YO o PF WF/YF oVO WO 6. coupling function, (YF/VFWF)

3'

- (Yo/~oWo) nondimensional coupling function,

Otb

{JVFWF/(YF,o) (1 - C~b) nondimensional axial coordinate,

7.

7rx p D / & f

P

stoichiometric coefficients of fuel and oxygen nondimensional radial coordinate, r/a density

Subscripts f I, II m o

flame regions one and two mixture burner condition

VF , V0

Supetscript mean conditions

8. 9. 10.

I1.

12.

ical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, NSF (RANN Program), Grant No. GI-31882 #1, 1974. Wulff, W., and Durbetaki, P., Study of Hazards from Burning Apparel and the Relation of Hazards to Test Methods, Proceedings of the Flammability Characteristics of Materials, Polymer Conference Series, University of Utah, salt Lake City, Utah, June 11-15, 1973. Wuiff, W., and Durbetaki, P., Fabric Ignition and Burn Injury Hazard, Proceedings, 1973 International Seminar on Heat Transfer from Flames, Trogir, Yugoslavia, Scripta Publishing Co., 1974. Heskestad, G., Ease of Ignition of Fabrics Exposed to Flaming Heat Sources, Final Report, l~actory Mutual Research, Boston, Massachusetts, 1973. Lewis B., and Von Elbe, G., Combustion, Flames and Explosions of Gases, Academic Press, New York, 1971. Edeiman, R. B., Fortune, O. F., Weilberstein, G., Cochran, T. H., and Haggard, J. B., An Analytical and Experimental Investigation of Gravity Effects upon Laminar Gas Jet Diffusion Flames, Fourteenth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, 1973, pp. 399-412. Burke, S. P., and Schumann, T. E. W., Diffusion Flames, Ind. and Eng. Chem. 20, 998-1004 (1928). Williams,F. A., Combustion Theory, Addison Wesley, Reading, Massachusetts, 1965. Carslaw, H. S., and Jaeger, J. C.,Combustion of Heat in Solids, Oxford University Press, London, 1959. Spalding, D. B., Theory of Mixingand Chemical Reactions in Opposed Jet Diffusion Flames, AIAA 31,763 (1961). Annamalai, K., Alkidas, A., and Durbetaki, P., On Droplet Burning: Concentration Dependent Transport Properties, Proceedings of the Eighth Southeastern Seminar on Thermal Sciences, School of Engineering, Vanderbilt University, Nashville, Tennessee, 1972, p. 101. Annamalai, K., Rao, V. K., and Sreenath, A. V., Distribution of Products Around a Burning Droplet: An Analytical Approach, Combust. Flame 16, 287 (1971 ).

References 1. Durbetaki, P., Wuiff, W., Acree, R. L., Champion, E. R., Lee, C., Naveda, O. A. A., Wedel, G. L., and Williams, P. T., Third Annual Report, School of Mechan-

Received 6 September 1974; revised 12 December 1974