Luminous heights of turbulent diffusion flames

Fire Safety Journal, 5 (1983) 103 - 108

103

Luminous Heights of Turbulent Diffusion Flames GUNNAR HESKESTAD

Factory Mutual Research Corporation, 1151 Boston-Providence Turnpike, Norwood, MA 02062 (U.S.A.) (Received in revised form July 30, 1982)

SUMMARY

L/D = fn (N/s)

A general analytical relation for predicting mean luminous heights o f buoyancy-controlled, turbulent diffusion flames is established. The relationship is based on a plot o f experimental flame heights in correlation coordinates proposed previously, including extensive data recently published.

where L is the mean flame height, D is the diameter of the flame source (or the effective diameter if the source is not circular, such that 7rD2/4 = the area of the source), ~ is the convective fraction of the heat-release rate, and N represents the non-dimensional group:

1. INTRODUCTION

Previously, the author proposed nondimensional variables for correlating flame heights of turbulent diffusion flames [ 1 ]. The variables were tested, with fair success, on a number of experimental observations. More recently, a considerable body of additional observations has become available, contributed by Zukoski et al. [2, 3]. Some of these observations incorporate flame heights determined from flame intermittency measurements near the flame tip, providing rather more objective flame-height determinations than those of the past which were based nearly always on visual estimates of average flame height. The additional body of data now makes it possible to propose, with a good degree of confidence, a general mathematical relation for predicting the mean luminous height of buoyancy-controlled, turbulent diffusion flames. It is the purpose of this paper to propose and justify such a relation.

(1)

N - cp T Oth 2r 3 gp2oHcD 5 .

(2)

Here, cp is the specific heat of air, To and P0 are the ambient temperature and density, respectively, g is the acceleration of gravity, rh the mass burning rate, r the stoichiometric mass ratio of air to volatiles, and Hc is the heat of combustion per unit mass. On theoretical grounds it is indicated that eqn. (1) becomes inaccurate for fire sources with substantial in-depth combustion. In-depth combustion would be "substantial" if a significant fraction of the volatiles are oxidized within the combustible array by air entering the array, as might be the case for very openly constructed wood cribs [1]. In the previous test [ 1] of the validity of eqn. (1), it was assumed that the effects of the convective fraction, ~, could be ignored, leading to the expectation that

L / D = fn(N)

(3)

to sufficient accuracy. The group N in eqn. (2) can be written in the following alternative form:

[ c. T0 ]Q2 2. CORRELATING VARIABLES

N = [gp2(Hjr)a

Ds

(4)

where Q is the total heat-release rate: Under conditions where flames are controlled by b u o y a n c y (gas efflux velocities at the source sufficiently low that the source Froude number is small), the following functional relationship has been derived [ 1] : 0379/7112/83/$3.00

Q = rhHc.

(5)

The brackets in eqn. (4) contain variables of the environment; they also contain the fuel parameter, H J r , which does not vary appre© Elsevier Sequoia/Printed in The Netherlands

104

/

./

/< .-/~

\Eq

(8)

1.2 1.0 •J

0.8 g

~'

0.6 0.4 /

0.2

-0.2 -5

J

/+

0

/

4

4 IOglo

-', N

Fig. 1. Correlation of flame height data from measurements by Vienneau [ 4 ] (o, methane; o, methane plus nitrogen; v, ethylene; v, ethylene plus nitrogen; •, propane; ~, propane plus nitrogen; ~, butane; ~, butane plus nitrogen; <>,hydrogen); D'Souza and McGuire [5] (. . . . . . , natural gas); Blinov and Khudiakov [7 ] (O, gasoline); I-L~gglundand Persson [8] (+, JP-4 fuel); and Block [6] (. . . . , wood cribs, underventilated).

ciably among c o m m o n fuels. Consequently, for normal ambient conditions, important variations in N and the non-dimensional flame height can be expected to be associated primarily with variations in the heat-release rate, Q, and the e x t e n t of the flame source, D. Zukoski's recent flame-height data [ 2, 3 ], which will be considered in Section 3, were plotted against the following non-dimensional parameter instead of N: QD = Q / ( P o C p T o ~ --~ D2} •

(6)

A simple relationship is readily established between N and Q* : N = [cpTo/(Hc/r)] 3Q~2.

(7)

3. PREVIOUS DATA CORRELATION Figure 1 has been taken from ref. 1 and shows literature data available at that time on flame heights, plotted in the coordinates of eqn. (3}. The long-dashed line is a reference curve which has been added and obeys the expression: L/D

= --

1.02 + 1 5 . 6 N 'is

(8)

This is the flame height relation which, eventually, an a t t e m p t will be made to justify. Note that the relation is asymptotic to

L / D c c N 1is at large values of N, as predicted

theoretically [ 1]. In Fig. 1, the data of Vienneau [4] pertain to 51 and 102 mm diameter gas jets of various fuels as indicated in the caption. Vienneau's data for two smaller gas jets, 25 m m and 6 mm in diameter, have n o t been included, but would scatter about the curve from eqn. (8) to logl0 N = 5 {N = 10s), approximately, The data of D'Souza and McGuire [5] (shortdashed line) are for flames above circular " s a n d b o x " burners operating on natural gas and varying in diameter from 102 to 279 mm. Block's extensive data [6] for wood cribs (longer-dashed line) pertain to underventilated cribs (see ref. 1). The four data points attributed to the measurements of Blinov and Khudiakov [7] represent pool fires of gasoline varying in diameter from 0.3 to 23 m. The data attributed to Hiigglund and Persson [8] are for square pools o f JP-4 fuel varyingin size from 1 m X 1 m to 10 m X 10 m ; t h e s e data are based on flame heights averaged over time from motion pictures, ignoring detached flame fragments at the flame tip. In evaluating N for these data, heats of combustion, H c, and stoichiometric ratios, r, were selected corresponding to complete combustion. These values may differ from the actual values, depending on the degree of completeness of the combustion:

105

A brief digression on previous work in correlating flame heights is useful. Beginning with the pioneering work of Thomas and coworkers [9 - 11], several investigators have correlated their flame-height data with an equation of the form: L I D o: (rh2/gDS) n.

(9)

The coefficient n has been variously quoted in the range from 0.20 to 0.30 [1], depending on the fire source. It may be verified that, for a given fuel system and environment, the parameter rh21gD s is proportional to the parameter N (eqn. (2)), Zukoski's parameter Q~2 (eqn. (6)), as well as Steward's flameheight parameter known as the "combustion n u m b e r " [12]. A power relation between L / D and any of these other parameters implies a relation of the form of eqn. (9) for a given fuel system. Steward's work [12] implies a power n = 1/5. Zukoski proposed a two-power representation [3], implying a lower power (n = 0.20) for Q~ greater than a given value (Q~ = 1) and a greater power (n = 0.38) for Q~ smaller than the given value. The flame-height relation, eqn. (8) as depicted in Fig. 1, implies a continuously varying power, n, depending on the magnitude of N. Provided H / r is approximately constant for fuels under consideration and the ambient temperature does not vary appreciably, the three flame-height parameters which incorporate fuel characteristics and ambient conditions (N, Q~ and Steward's combustion number) can be considered equally capable. This can be verified from eqn. (7) for N vs. Q~ and by reference to the mathematical expression for the combustion number [ 12]. Experimental evidence is lacking for choosing among the three parameters when H / r and T O cannot be considered constant. The author believes N will be the choice after the parameters have been put to an appropriate experimental test.

4. NEW D A T A

Zukoski et al. have contributed t w o sets of data based on measurements on natural-gas flames (essentially methane) above circular, porous-bed burners. The first, more extensive set [2] incorporates flame heights averaged

by eye, using burners with diameters of 0.50, 0.30, 0.20, and 0.10 m, each operated at several gas rates. The second set [3] incorporates flame heights determined as the 50 percent intermittency elevation of the luminous flame according to video records, using three of the four burners employed in the first set and considerably fewer gas rates. Although significantly extensive laminar regions may have been present in the flames of the two smaller burners (0.10 and 0.20 m diam.), the flame-height data from these burners, as plotted by Zukoski et al. [2, 3], hardly bear evidence of this when compared against flame heights of the larger burners in overlapping and abutting regions of Q~. The first set of data from Zukoski et al. [2], based on visual averaging of flame heights, is represented in Fig. 2 together with the reference curve based on eqn. (8). The circles correspond to a curve faired through the original data; the surrounding envelope would enclose all but one of the approximately 80 data points. Clearly, the reference curve is an excellent representation of the data. To interpret the second set of flame-height measurements by Zukoski et al. [3], consider the schematic plot of intermittency, I, vs. distance above the burner, z, in Fig. 3. As used by the authors [3], the intermittency, /, is defined as "the fraction of time during which at least part of the flame lies above a horizontal plane located at elevation, z, above the burner". The intermittency decreases from unity deep in the flame to smaller values in the intermittent-flame region, eventually to reach zero at higher elevations. The mean flame height, L, is taken as the distance above the burner where the intermittency has declined to 1/2. A length scale for the intermittent flame region, LI, has been defined by Zukoski et al. [3], corresponding to the difference in elevation between the z-intercepts at I = 0 and I = 1 of the maximum slope to the/-curve. Flame heights from the intermittency measurements are compared with the reference curve according to eqn. (8) in Fig. 4 (open symbols). The correspondence is remarkably good. Further, it is seen that L~, the intermittency scale (solid symbols), increases in ratio to the flame height as the flame height decreases.

106

i.6

/

-J

z<

1.4

......

"Eq.(8)

~'

1.2 ~

1.0 /

9

o.s i 0.6 0.4 4

O.2

i

0 -0.2 -5

_ I

_ ~

-4

. . . . . .

"3

J

[.__

-2 lOglo

"1

.

i

1

_±

0

I

2

.........

N

Fig. 2. Flame height data of Zukoski et al. [ 2 ] according to visual averaging, compared with reference curve. Circles correspond to curve faired through original data; envelope would enclose all but one of the approximately 80 data points.

,0

~

1 0.5

~-

L

0 z (Arbitrary

Units]

Fig. 3. Definition of mean flame height (L) and intermittency scale (LT) from flame intermittency measurements.

H e n c e , it has been s h o w n t h a t the m e a n flame height is well r e p r e s e n t e d b y eqn. (8) f o r N in t h e range 1 0 - s . 10 s ' a p p r o x i m a t e l y . A set o f d a t a w h i c h e x t e n d s d o w n to even l o w e r values o f N has just c o m e t o the a u t h o r ' s a t t e n t i o n * . W o o d et al. [13] b u r n e d fixed a m o u n t s o f a c e t o n e and m e t h a n e in a sand-filled a l u m i n u m pan and studied burning rates, h e a t - f l u x d i s t r i b u t i o n s on the fuel surface, and flame heights {visually averaged, p r e s u m a b l y ) . T h e pan was 1.52 m in d i a m e t e r

*Pointed out by a referee.

a n d 20 m m high. In each e x p e r i m e n t , the sand was s a t u r a t e d with liquid fuel, ignited and allowed t o b u r n dry. F o l l o w i n g an initial s t e a d y b u r n i n g interval, the burning rate decreased m o n o t o n i c a l l y . During the d e c a y in burning rate, the c o h e r e n t flames were said to change t o d i s t r i b u t e d flamelets [ 1 3 ] . Figure 5 shows the flame height d a t a c o n v e r t e d to the c o o r d i n a t e s L I D vs. N, t o g e t h e r with a p l o t o f eqn. (8). It appears t h a t eqn. (8) is quite reliable even at v e r y low values o f N, nearly t o values w h e r e L / D b e c o m e s zero according t o eqn. (8), w h i c h o c c u r s at log10 N = --5.92. T h e significance o f L / D = O, if any, is n o t clear in the c o n t e x t o f eqn. (8), a l t h o u g h negative values o f L / D w o u l d be meaningless. W o o d et al. [13] r e p o r t e d t h a t c o h e r e n t flaming changed t o d i s t r i b u t e d flamelets w h e n L I D b e c a m e smaller t h a n a b o u t 0.5 (log10 L / D = - - 0 . 3 0 ) , which o c c u r s near logl0 N = - 5 ( N = 1 0 - s ) . It might be significant t h a t the n o n - d i m e n s i o n a l i n t e r m i t t i n g scale, L~/L, is close t o u n i t y also at log10 N = --5 {Fig. 4).

5. CONCLUSIONS T h e new, extensive d a t a c o n t r i b u t e d by Zukoski e t al. [2, 3] have s u p p o r t e d the validity o f c o r r e l a t i o n variables p r o p o s e d previously for l u m i n o u s heights o f b u o y a n c e c o n t r o l l e d , t u r b u l e n t d i f f u s i o n flames f r o m fire sources w i t h no substantial in-depth com-

107 1.8

I

I

I

I

[

I

0 • D=O. lOm '~ • D=O.19m [] • D=O, 50m

1.6

I

/

/

/ /

/

1.4 -J

12

/

/

L/D

~k, Eq. (8)

-J

I0

2 r~ o

//o

(18

•

•

y

0.6 Q4 0.2'

~

oF -0.2 I - 4 / -5

V'YO

S" I -4

•

• L[/L

l -.%

l -2

l

I 0

-I

I I

l 2

IOgio N

Fig. 4. Flame heights and intermittency scales according to flame-intermittency data of Zukoski et al. [ 3 ].

t

I

I

I

t

i

0 Acetone g7 Melhonol

L LI rh g Q Q~ r

ol- J \ 0 , / A::~ /

gF

© /

,/

g Hc I

Eq.(8)

V

(:Y

// / /

To z

f I

-2 -6

t -5

I

I -4

I

t -3

loglON

P0

A c c e l e r a t i o n o f gravity Heat of combustion per unit mass Flame intermittency (fraction of time f l a m i n g exists a b o v e o b s e r v a t i o n level) M e a n f l a m e height I n t e r m i t t e n c y scale Mass b u r n i n g r a t e {ep T o / [ g p ~ ( H c / r ) 3 ] }Q2/DS thHc, t o t a l h e a t release rate D e f i n e d in eqn. (6) S t o i c h i o m e t r i c m a s s ratio, air t o volatiles Ambient temperature Height above flame source C o n v e c t i v e f r a c t i o n o f heat-release r a t e Ambient density

Fig. 5. Flame heights measured by Wood et al. [ 13] over sandfilled pan initially saturated with liquid fuel. b u s t i o n . With t h e aid o f t h e n e w data, it has b e c o m e possible to establish and verify a general m a t h e m a t i c a l r e l a t i o n f o r p r e d i c t i n g m e a n f l a m e heights, eqn. (8). T h e e q u a t i o n has b e e n verified f o r a range in t h e c o r r e l a t i n g variable N f r o m l 0 s d o w n t o 1 0 - s, w i t h s o m e e x p e r i m e n t a l s u p p o r t even f o r N < 10 - s w h e r e flames b r e a k u p into d i s t r i b u t e d flamelets. NOMENCLATURE

Cp D

Specific h e a t o f air D i a m e t e r (or e f f e c t i v e d i a m e t e r ) o f flame source

REFERENCES 1 G. Heskestad, Eighteenth Symp. (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 951. 2 E. E. Zukoski, T. Kubota and B. Cetegen, Fire Safety, 3 (1980/81) 107. 3 E. E. Zukoski, T. Kubota and B. Cetegen, Entrainment in the near field of fire plumes, California Institute of Technology, Daniel and Florence Guggenheim Jet Propulsion Center, August 1981. 4 H. Vienneau, Mixing controlled flame heights from circular jets, BSc Thesis, Dept. Chem. Eng., Univ. New Brunswick, Fredericton, N. B., 1964. 5 M. V. D'Souza and J. H. McGuire, Fire Technol., 13 (1977) 85. 6 J. A. Block, A theoretical and experimental study

108 of nonpropagating free-burning fires, Ph.D. Thesis, Harvard University, 1970. 7 V. I. Blinov and G. N. Khudiakov, Dokl. Acad. Nauk SSSR, 113 (1957) 1094. 8 B. H//gglund and L. E. Persson, The heat radiation from petroleum fires, F6rsvarets Forskningsanstalt, Stockholm, FDA Rep. C 20126-D6 (A3), 1976. 9 P. H. Thomas, Combust. Flame, 4 (1960) 381.

10 P. H. Thomas, C. T. Webster and M. M. Raferty, Combust. Flame, 5 (1961) 359. 11 P. H. Thomas, Ninth Syrup. (Int.) on Combustion, The Combustion Institute, Pittsburgh, 1963, p. 844. 12 F. R. Steward, Combust. ScL Technol., 2 (1970) 203. 13 B. D. Wood, P. L. Blackshear, Jr. and E. R. G. Eckert, Combust. Sci. Technol., 4 (1971) 113.