Diffusion flame extinction at small stretch rates: The mechanism of radiative loss

C O M B U S T I O N A N D F L A M E 6 5 : 3 1 - 3 4 (1986)

31

Diffusion Flame Extinction at Small Stretch Rates: The Mechanism of Radiative Loss J A M E S S. T ' I E N Department o f Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio 44106

A theoretical analysis of diffusion flame extinction in the stagnation point region of a condensed fuel has been made including radiative heat loss from the fuel surface. In addition to the blowoff limit normally found when flame stretch rate is large, an extinction limit has been identified when the flame stretch rate becomes sufficiently small. This limit occurs as a result of flame temperature reduction when the rate of radiative loss becomes substantial compared with the rate of combustion heat release. A flammability map using oxygen mass fraction and stretch rate as coordinates shows that the extinction boundary consists of a blowoff and a radiative extinction branch. The merging point of the two branches defines a fundamental low oxygen flammability limit.

INTRODUCTION Studies on the mechanism of diffusion flame extinction have mostly neglected the radiant heat losses from the flame zone. This is probably justified when the flame size is small and the combustion is intense. As will be demonstrated later, when combustion rate is slowed down due to a reduced oxidizer supply rate, radiation can no longer be ignored in studying the flame extinction behavior. Although the relevance of radiative losses to the extinction of gaseous premixed [ 1] and solidpropellant flames [2-4] was studied extensively some time ago, the first suggestion of the importance of radiation to diffusion flame extinction did not appear until the work by Bonne [5]. Using simulated experimentation and theoretical estimation, he found that when convection became negligible, radiative loss could reduce the flame temperature substantially, and hence contributed to extinction. In a study of the stagnation-point diffusion flame of solid fuel [6], this author and his colleagues had included surface radiative heat loss and gain in their Copyright © 1986 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue. New York, NY 10017

theoretical model, but no numerical computation was made other than the case of zero heat loss. Sibulkin et al. [7, 8] included both gasphase and surface radiative losses in their combustion analysis of a vertical flat fuel surface in a natural convective flow. They found that while the effect of gas-phase radiation is negligible, the surface radiative loss is very important to extinction. Using asymptotic analysis, Sohrab et al. [9] studied radiant loss from the gaseous zone of a diffusion flame in a stagnation-point boundary layer geometry. They also concluded that the radiative contribution to extinction was through the reduction of flame temperature. Another study by Sohrab and Williams [10] dealt with a diffusion flame of condensed fuel with radiative loss from the fuel surface. Its emphasis was more on improving the accuracy of the overall kinetic constants determined by the blowoff experiment rather than on the issue of radiative extinction. In the present work, a condensed-fuel diffusion flame in the stagnation-point region of a forced flow is also employed with radiative loss from the condensed-fuel surface. The emphasis

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32

JAMES S. T ' I E N

is to clarify the extinction behavior when the flame stretch rate becomes small. THEORETICAL MODEL The combustion model is basically the same as that presented in Ref. [6] except for minor changes of the surface boundary condition. The analysis assumes a one-step forward overall gasphase chemical reaction of second order occurring in the axisymmetric stagnation-point region adjacent to the fuel surface. The mathematical formulation follows conventional analysis and the readers are referred to Ref. [6] for details. Here, only the essential definitions and the new element will be outlined. Damkohler number, D, is defined by

D=B/a,

(1)

where B is the modified preexponential factor for the gas-phase reaction, which has the unit of 1/time, and a is the stagnation-point velocity gradient or the flame stretch rate, which also has the unit of 1/time. The gas-phase radiation and absorption are assumed to be negligible with radiative exchanges occurring only on the fuel surface. The energy balance there is given nondimensionally by

Ow'=-fw Pr (Ow--~p+l)+SOw 4,

(2)

where the first term on the right-hand side of Eq. (2) represents heat transferred to the condensed phase for pyrolyzing the fuel and the second term represents radiative heat loss from the hot surface to the surrounding. The background radiation to the surface is neglected here and the heat conduction in the solid is assumed to be one-dimensional without lateral heat loss. The nondimensional burning rate ( - f w ) is related to the dimensional mass burning rate by m = (pe#ea) 1/2( -fw)-

(3)

The radiative surface heat loss parameter is given by s

Xe

'

(4)

where tr is the Stefan-Boltzmann constant and es is the emissivity of the fuel surface. For a given Prandtl number, the nondimensional number S can be interpreted as the ratio of the rates of surface radiative heat loss to gas-phase heat flux to the surface. Instead of a constant surface temperature used in Ref. [6], in the present work an Arrhenius relationship between burning rate and surface temperature is assumed. In nondimensional form, it is given by

-fw=A exp(-Ew/Ow),

(5)

where A = b/(Pe#ea)1/2 is the nondimensional preexponential factor for surface pyrolysis. We see from Eqs. (1), (4), and (5) that varying the flame stretch rate, a, not only affects the Damkohler number, D, but also changes the values of the radiative heat loss parameter S and the preexponential factor A for the surface pyrolysis. RESULTS A N D DISCUSSION Modified by Eqs. (2) and (5), the mathematical model in Ref. [6] was solved numerically using an IBM PC. Since the main objective is to investigate the effect of flame stretch rate, a, on the extinction limit, the computation has been performed with a fixed set of fuel properties. The fuel chosen in PMMA with property values given in Tables 1 and 2. Some of these values are slightly different from those used in Ref. [6], reflecting a more realistic estimate of flame temperature. The gas-phase kinetic parameters (E, B) deduced are based on no radiative heat loss. Therefore, the realistic kinetics for PMMA will be faster than the ones used here. TABLE 1

Property Values B b Te pe ~ ~e

5.047 x 107 1/s 2.32 x 107 kg/m2s 300K 1.176 kg/m 3 1.85 x 10 - S k g / m s 2.93 x 10 -5 kJ/m sec K

DIFFUSION FLAME EXTINCTION AT SMALL STRETCH TABLE 2 Values of the Nondimensional Parameters

cJcp

1.17 2.79 69. 1.92 0.7 0.7 45.3 50.3

L q No Pr Sc E E~

Figure 1 presents the dimensionless fuel burning rate ( - f w ) , the peak flame temperature (0max), and the distance of the temperature peak from the fuel surface (r/f) as a function of the stretch rate (a). The oxygen mass fraction Yoe is fixed at 0.2324 (air). Two cases are shown in the figure: Es = 0, corresponding to no radiative loss, and ~s = 1, corresponding to a blackbody radiative surface. Figure 1 shows that for ~ = 0, the dimensionless burning rate, the flame standoff distance, and the peak temperature are all essentially constant at small stretch rate and decrease only in the neighborhood of the blowoff limit (large a). With surface radiation loss (~

0.6 8

1.0

.8 -fw

i"

.6

~S=I

Blowoff

.4 .2 ~ Extinction I 0 .5 1 10 a (1/see) I 10 7

I 10 e

I

I

.3

.2

I

I 100 I 10 s

D I

I

| 300 I 5X10

5

I

.1 .05 .03 .02 S (for ~s=l)

Fig. I. Nondimensional flame standoff distance O/f), maximum flame temperature (0max), and fuel burning rate ( - f , )

as a function of flame stretch rate (a) in air.

33

= 1), a radiative extinction limit is found to exist at small stretch rate (a = 1.2 1/s in Fig. 1). The dimensionless burning rate ( - f w ) peaks at a = 90 l/s, decreases continuously as a decreases, and reaches a minimum at the radiative extinction limit. Similarly, the flame temperature peaks at an intermediate stretch rate (a 15 l/s). The flame temperature at the radiation extinction limit is lower than that at the blowoff limit (therefore, there is no unique extinction flame temperature for a given fuel). Nondimensionally, the flame moves closer to the fuel surface near the radiative extinction limit as shown by r/f in Fig. 1. The dimensionless heat transfer rate 0" increases monotonically when the stretch rate is decreased (not shown in the figure). The reason for the existence of the radiative extinction limit can be found from the definition of the nondimensional radiative heat loss parameter S given in Eq. (4). As the stretch rate, a, decreases, S increases as 1/a 1/2. From Eq. (2), we see that nondimensionally this increases the heat transferred from the gas to the surface of the condensed phase, which lowers the flame temperature and leads to flame extinguishment. As shown in Fig. 1, radiative extinction occurs at large but finite Damkohler number. Also shown in Fig. 1 is the change of heat loss parameter S with a for the case c, = 1. Physically, as stretch rate decreases, the dimensional flame standoff distance increases (Ref. [6]), and the fuel burning rate decreases [m ~ -,fa, Eq. (3)]. As a result, the combustion heat generated per unit time per unit burning area of the condensed fuel surface decreases and a low-power flame is produced. The rate of radiative loss, however, is decreased much slower than that of the heat generation (for sufficiently large value of E,0. Hence by comparison, the radiative heat loss becomes significant as the stretch rate is decreased to small value. Figure 2 gives a plot of flammability boundary as a function of oxygen mass fraction and flame stretch rate. Without radiative loss, a flame blowoff boundary (e~ = 0) exists at sufficiently large value of a. But there is no

34

JAMES S. T'IEN

extinction branch probably occurs in a range of stretch rate too small to be easily realized in normal gravity due to buoyancy-induced flow. Its practical importance to combustion in a reduced gravitational environment is likely to be significant, however.

.25 .24

o~ .2a >-

~-air~

g .22 -= o .21

Blowoff branch / I /

~s=1mmab/

.2o

~ .19 ~ .18 ~ ~

Radiati.ve ~ p . . . / r extinction

branch

A

/

--/ - /

I would like to thank Kurt R. Sacksteder, who has been instrumental in the selection o f this problem. This work has been supported by N S F Grant MEA-8115339, Dr. Win Aung, Program Director.

BIowoff boundary

(~s=O)

.17 .16 .15

I

.5

1

II

I

lO lOO Stretch Rate a ( t / s e c )

300

Fig. 2. Flammability map.

REFERENCES 1.

extinction limit when a is small. There appears to be no fundamental low oxygen limit either: flame can be established at low oxygen fraction when the value of a is sufficiently small. With surface radiative loss, the picture is quite different. Figure 2 shows that the flammable region for es = 1 is bounded by two branches of extinction boundaries: a blowoff branch (AC) and a radiative extinction branch (AB). For a fixed value of Yoe, if we start with a flame, blowoff can be reached when a is increased and radiant extinction can be reached when a is decreased. The two branches merge at point A, which defines a fundamental low oxygen limit. For Yoe below the value specified by point A, a steady stagnation-point diffusion flame with properties given by Tables 1 and 2 cannot be established regardless of the values of the stretch rate a. The author is not aware of any experimental finding of the radiative extinction boundary as shown in Fig. 2. For most common fuels, this

2.

3.

4.

5. 6. 7.

8. 9. IO.

Spalding, D. B., Proc. Roy. Soc. (London) A240:83 (1957). Friedman, R., Levy, J. B., and Rumbel, K. S., Atlantic Research Corporation, AFOSR TN-59-173, AD NO 211-213, 1959. Johnson, W. E., and Nachbar, W., Eighth Symposium (International) on Combustion, Williams and Wilkins, Baltimore, 1961, pp. 618-689. DeLuca, L., Chapter 12 in Fundamentals o f SolidPropellant Combustion (K. K. Kuo and M. Summerfield, Eds.) Vol. 90, Progress in Astronautics and Aeronautics, American Institute of Aeronautics and Astronautics, 1984. Boone, U., Combust. Flame 16:147-159 (1971). T'ien, J. S., Singhal, S. N., Harrold, D. P., Prahl, J. M., Combust. Flame 33:55-68 (1978). Sibulkin, M., Kulkarni, A. K., and Annamalai, K., Eighteenth Sym. (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, pp. 611617. Silbulkin, M., Kulkarni, A. K., and Annamalai, K., Combust. Flame 44:187-199 (1982). Sohrab, S. H., Linan, A., and Williams, F. A., Comb. Sci. and Tech. 27:143 (1982). Sohrab, S. H., and Williams, F. A., J. Polymer Sci., Polymer Chem. Ed. 19:2955-2976 (1981).

Received 9 October 1985; revised 30 January 1986