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A theoretical analysis of the extinction limits of a methane-air opposed-jet diffusion flame
COMBUSTION
AND FLAME
70: 161-170 (1987)
161
A Theoretical Analysis of the Extinction Limits of a Methane-Air Opposed-Jet Diffusion Flame S. L. O L S O N NASA-Lewis Research Center, Cleveland, OH 44135
and J. S. T ' I E N Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106
A theoretical analysis is described for a methane-air diffusionflame stabilizedin the forward stagnationregion of a porous metalcylinder in a forced convectiveflow. The analysisincludeseffects of radiativeheat loss from the porous metal surface and finite rate kinetics but neglects the effects of gravity. The theoreticallypredicted extinctionlimits compare well with experimentallyobserved extinction limits from the literature. After the predicted limits compared well with the experimental limits, a parametric study of the effect of fuel surface emissivity and Lewis number was conducted with the numerical model. It was found that the computed blowoff limit is independentof radiative heat loss for high fuel blowing velocitiesbut is a strong function of Lewis number. At low fuel blowingvelocities,the extinctionlimit varies with both radiativeheat loss and Lewis number. It is discovered, however, that even if thermal losses from the fuel surface are absent, the flame can extinguishat the fuel surface independentlyof Lewis number due to excessive reaction zone thinning.
INTRODUCTION The counterflow diffusion flame in the forward stagnation region of a cylindrical porous burner is a simple flame to study experimentally because of its stability, and theoretically due to the onedimensional governing equations. This explains its enduring presence in combustion literature. A summary of experimental and theoretical work on the counterflow diffusion flame can be found in Tsuji [1]. The present study focuses on the extinction limits of the counterflow diffusion flame. Two distinct types of extinction have been identified experimentally by Tsuji and Yamaoka [2]. A blowoff limit is believed to be caused by large flame stretch which imposes chemical limitations on the combustion rate within the flame. A thermal This article is in the public domain Published by Elsevier Science PublishingCo., Inc. 52 VanderbiltAvenue, New York, NY 10017
quenching limit at low fuel injection rates is believed to be caused by heat losses. A theoretical analysis by Saitoh [3] showed that a low fuel blowing velocity extinction limit existed as a result of conductive heat losses imposed by maintaining a constant surface temperature. However, in the experiments conducted by Tsuji and Yamaoka [2, 4, 5], no active cooling was used and the surface temperature was allowed to vary. In this case, the low fuel blowing velocity quenching limit has yet to be explained. Recent modeling work by Dixon-Lewis et al. [6] focused on the blowoff or stretch limit of the counterflow diffusion flame using detailed chemical kinetics. They concluded that quenching effects at the fuel surface do not contribute to the observed extinction at blowoff. T ' i e n [7] has focused on the thermal quenching limit of a
0010-2180/87/$00.00
162
S . L . OLSON and J. S. T ' I E N
counterflow diffusion flame and has proposed that the observed extinction at low fuel flow rates can be caused by radiant heat losses from the fuel surface. In this paper the combustion model developed in Ref. [7] is modified to describe a diffusion flame established in the forward stagnation region of an uncooled porous metal cylinder. This is the configuration used by Tsuji and Yamaoka [2, 4, 5]. As in the uncooled experiments, the surface temperature is allowed to vary while the fuel blowing velocity and oxidizer stagnation point velocity gradient are the controlled variables. This paper is organized in four sections. In the first section the modified combustion model is described. In the second section the computed extinction limits are compared with experimentally determined extinction limits. A parametric study of the effects of radiative heat loss from the fuel surface and fuel and oxidizer Lewis number was then conducted. The third section describes the effects of surface emissivity on the predicted extinction limits and the fourth section discusses the Lewis number effects.
axisymmetric geometry to a two-dimensional geometry. Additionally, the fuel feed rate was converted from a solid fuel pyrolysis rate dependent on fuel surface temperature to a fixed feed rate as is the experimental situation with a gaseous fuel. A second order Arrhenius reaction rate for methane and air was used in the model. The reaction rate constants/~ and/~ were taken from Ref. [9]. The reaction rate expressed in nondimensional form is w = D Yf Yo exp( - E/O),
(2)
where
D = (I~P/aR Te)(AIm/Mo)
(3)
is the Damkohler number and a is the stagnation point velocity gradient. There are nine boundary conditions required for this system of equations. These are as follows: forT/ = 0
f--fw
(specified),
(4)
f' =0,
(5)
Yf' = - f w Scf( Yew- 1),
(6)
Y~ = - f w Sco Yow,
(7)
COMBUSTION MODEL A two-dimensional counterflow diffusion flame stabilized in the forward stagnation region of an uncooled porous metal cylinder is described. Fuel is ejected from the cylinder with a specified blowing velocity into an oxidizer flow of specified stagnation velocity gradient. The gas-phase governing differential equations are [7, 8]
f~, +ff,,
= [(f,)2
_
40],
# '÷= - f ,
Pr(O, - 1) + S(0w 4
-
1),
(8)
where
S = ((re Te3/)~e)(#e/(Pea))1/2 is the dimensionless radiative surface heat loss parameter; for ~7 ~ oo
(1/Pr)0" + fO' = -- Qw,
(9)
O=O~= 1,
(l/Scf) Y/' + f Y f ' = w, (l/Sco) Yo' + f Y o =NOW.
-
(1)
The model assumes constant specific heat for all species, and the ideal gas law is assumed for all reactants and products. The product of density and viscosity, p#, is assumed to be constant. Modifications to the model were made to first convert the system of governing equations from an
f' =2
(10)
(u = Ue),
(10
Yf=0, Yo = Yoe = 0.2325
(air).
(12)
Of special note is Eq. (8), which was derived through a heat balance at the surface of the cylinder. Convective heat transfer from the flame
OPPOSED JET DIFFUSION FLAME EXTINCTION through the gas phase to the cylinder surface and radiative heat loss from the surface to the ambient environment are equated with an increase in fuel enthalpy as it is ejected from the cylinder surface. The surface temperature is determined as part of the solution. These equations are solved numerically using a scheme similar to that used in Ref. [7]. Property values used in the computations are contained in Table 1. Table 2 contains nondimensional parameter values used in the model. Definitions of these parameters can be found in the Nomenclature.
COMPARISON OF COMPUTED EXTINCTION LIMITS W I T H EXPERIMENTALLY DETERMINED LIMITS
163 TABLE 2
Nondimensional Parameter Values Used in the Numerical Computation Parameter
Value
E No Q e
54.14 4.0 122.0 Variable
0~
1.00
Lef Leo Pr Yo~
Variable Variable 0.70 0.2325
sional blowing velocity as
The numerical analysis was performed for a number of fuel feed rates and stagnation velocity gradients for the case where the surface is a grey body radiator. Table 3 contains the identified extinction limits and corresponding limit burning solution flame conditions. In order to compare these limiting solutions to the experimentally determined limits, the relationships between the nondimensional fuel blowing velocity, -fw, defined in this work and those defined by Tsuji must be determined. Tsuji and Yamaoka [2] define their nondimen-
- f*ww= ( v * / V ) ( R e / 2 )
(13)
°'5,
where o* is the methane ejection velocity at ambient temperature. V is the upstream air velocity far from the cylinder surface, and Re is the Reynolds number based on cylinder radius and mean flow properties. In the present work we use (14)
- f w = ml(oo~tz~a} 1/2,
where r r / = ,OwVw.
TABLE 1 Dimensional Quantities Used in the Numerical Computation Property
C~ P t~ Mo.f.m R T~ pe urn t~e
Value 5.06 X I013 5.80 32.3 1.00 212.8 32, 16, 30 1.5 1.987/82.05 300 1.177 × 10 -3 1.168 1.846 × 10 -5 or 10 -4
Unit cm3/gmol s cal/gmol K kcal/gmol atm kcal/gmol g/gmol cm (cal/gmol K)/(cm 3 atm)/(gmol K) K g/cm 3 cm2/s Pa s g/cm s
Reference 9 14 9 1 15 14 1 14 1 16 16 16
164
S. L. OLSON and J. S. T ' I E N TABLE 3
2.5
Limit BurningSolutionFlame ConditionsDetermined fromthe Numerical Computationfor ~ = 0.25, Lef = Leo = 0.833
2.0
a 10 20 50 100 200 301 333 331 313 304 297
-jr, o. 157 o. 181 0.233 0.309 0.473 0.750 0.97 1.0 1.5 2.0 2.5
One, 4.95 5.20 5.43 5.65 5.76 5.89 6.04 6.14 6.59 6.58 6.55
O.~l, 3.75 4.01 4.34 4.55 4.61 4.39 4.01 3.97 3.11 2.33 1.32
fw 0.096 o. 104 o. 126 o. 161 0.242 0.389 0.512 0.525 0.820 1.190 2.020
Comparing these relations using a = 2 V / I ~ , it can be shown that
-J~w=
- fw(21Omean)( M m l M f ) .
(15)
Figure 1 shows a comparison between the computed and experimentally determined extinction limits. The model developed here clearly predicts the nature of both extinction by blowoff and extinction by quenching. For high stagnation point velocity gradient a blowoff limit is experimentally observed and theoretically predicted. For very low fuel blowing velocity a quenching boundary is again experimentally observed and theoretically predicted. In the transition region from quenching to blowoff there is some discrepancy between experiment and prediction. It should be noted that for comparative purposes a Lewis number of 0.833 and a fuel surface emissivity of 0.25 were used. The Lewis number is reasonable for a methane-air system and the emissivity represents an approximation for a dull bronze surface [10]. In later sections of this paper the effects of both Lewis number and surface emissivity on the computed extinction limits will be discussed in greater detail. It is also of interest to note that although the experimental data show a slight increase in the limiting - f * at very low stretch rates, the computed values of the limiting - f * continue to
1.5
Experiment Tsuji and Yamaoka (1) .... Present computation ( Lef = L e o = 0.833, ( - - 0 . 2 5 )
~: U.
-fw *
o
1,0 FLAME 0.5
o I----~ lO
i
i
i
i
i
i|1
100 a, sec -1
|
t
|
500
Fig. 1. A comparison of the computed extinction boundary with experimentally measured extinction limits.
decrease well beyond the range of stretch rates used in the experiment. We suspect that this slight increase in the limiting - f * is due to buoyant flows which become important in very low convective flows. As our model neglects the effects of gravity, it would not predict this behavior. Dixon-Lewis et al. [6] have recently done some computational work for comparison with Tsuji and Yamaokas' data also. They focused on one flame which was well documented in Ref. [5]. Using complex chemical reaction mechanisms and detailed transport flux information they compared species concentrations, velocity profiles, and temperature distributions. Because of the simple onestep second-order chemical reaction scheme used in this work, a detailed comparison is not in order. However, the flame temperature computed by this model is within the same range of temperatures as reported by Dixon-Lewis et al. for a flame at a = l O O s - l a n d - f * = 1.5.
E F F E C T S OF S U R F A C E R A D I A T I O N ON THE EXTINCTION LIMITS After the model demonstrated good agreement with experiment, a parametric study was conducted to determine the effects of surface radiative heat loss on the extinction limits predicted by the model. For this study the Lewis numbers of the
OPPOSED JET DIFFUSION FLAME EXTINCTION
165
limit. Near this limit the flame is positioned right next to the fuel surface. To illustrate this, Fig. 3 shows a series of temperature profiles for three different values of - f , at a given velocity gradient. For - f w = 0.20 the temperature peak is located in the gas phase away from the wall. As - f w decreases, the flame moves close to the porous cylinder surface and the temperature peak reaches the cylinder surface at - f w = 0.11. As - f w is decreased further the flame zone is compressed against the wall and the flame finally extinguishes at - f w < 0.079. The cause of the extinguishment is that as the reaction zone is pressed closer to the fuel surface it does not have enough volume to generate the heat required to sustain the flame. This occurs because the porous metal surface acts as a wall to the oxidizer and does not permit the oxidizer to diffuse into the fuel. The wall oxidizer concentration then builds to a substantial level just before extinguishment. Therefore, this lower extinction branch is a surface extinction branch and the extinction is caused by excessive reaction zone thinning. To our knowledge, no one has observed this type of extinguishment experimentally for diffu-
fuel and oxidizer were set equal to unity. The effect of Lewis number is discussed in the last section of this paper. Most of the work focused on three cases: a blackbody surface, an intermediate greybody surface, and a perfectly nonradiating surface. Emissivity values for these cases are 1, 0.25, and 0, respectively. A few other intermediate graybody surfaces were studies also so that trends could be determined more accurately. Figure 2 shows three extinction limit curves found for emissivities of 1.0, 0.25, and 0.0. Each curve will be discussed in detail. The extinction curve where surface radiation is neglected is labeled in Fig. 2 as A B C . This curve is clearly two separate limit branches A B and CB connected at B by a branch point. The upper branch A B corresponds to a blowoff limit. Near the blowoff the flame temperature drops until extinction occurs. The temperature drop is caused by limitations of the time available for chemical reaction to occur. The flame extinguishes because it cannot react fuel and oxidizer fast enough. The lower branch CB of the zero surface emissivity curve is a different type of extinction
3.0 Leo = Lef = 1 2.5 2.0
NO FLAME
3" 1.5 FLAME
j/~ B"
1.0 0.5
2
5
~0
20
50
~00
200
400
a (sec -1) Fig. 2. Effect of radiative heat loss from the porous metal surface on the extinction boundaries.
166
S . L . OLSON and J. S. T ' I E N
8.0
7.o
Leo=Lef=l
Limit - f w = O . 0 7 9 Flame just t o u c h e s wall - f w = 0 . 1 1 0
....
~
~mFlame
I
,
t
6.0
a=20.1/s ~=0.0
a w a y from
wall - f w = 0 . 2 0 0
,
t I l
0
a o
1.o 0
014
1 J2
018
' 1.6
2.0
2.4
~7 Fig. 3. Temperature profiles of flame as it approaches the surface extinction limit for zero surface radiation.
sion flames presumably because of the high surface temperatures required and the difficulty of eliminating radiative loss at these temperatures. The phenomena bear a certain resemblance to the symmetric premixed flame extinction of Sato [11], where it was found that when the Lewis number of the deficient species is greater than unity, the flame is quenched at the stagnation surface. For the present diffusion flame in the absence of thermal losses from the fuel surface, the surface extinction mode always exists independently of fuel or oxidizer Lewis number. To illustrate further the difference between blowoff and surface extinction, the peak flame and wall temperature profiles are plotted in Fig. 4 as a function of fuel blowing velocity for a stretch rate a = 300 s-1 from the surface extinction to the blowoff limit near the branch point B on Fig. 2. The flame temperature is much higher than the wall temperature at blowoff. This indicates that the flame is far from the wall. As the fuel blowing
velocity is decreased from blowoff toward surface extinction the flame and wall temperatures both increase to a maximum near the surface extinction limit. The flame temperature then quickly drops to the wall temperature as the flame is pressed against the wall. With further reduction in the fuel blowing velocity the flame extinguishes. The case of a blackbody surface is shown in Fig. 2 in curve A B " C " . The blowoff branch A B " and a quenching limit B " C" flow together smoothly and appear to be a single curve. This is in contrast to the zero emissivity curve where the two branches were quite distinct. At both limits the flame (the peak temperature position) is located away from the wall but as extinction is approached the flame temperature drops until extinction occurs. The reason behind the temperature drop near the limit is different, however, for the two limits. Near the quenching limit the surface radiation becomes important and heat transfer from the flame via gas-phase conduction to the surface to offset the radiative losses becomes excessive. Near the blowoff the temperature drop is caused again by limitations on the time available for chemical reaction to occur. The limiting temperature is higher at the blowoff limit than at the surface extinction limit. To understand better the different appearance of the two extinction curves described above, the effect of intermediate graybody surface emissivity was explored. Curve A B ' C ' in Fig. 2 is for an emissivity of 0.25. It is observed that the branch point B ' falls on the zero emissivity blowoff curve but intercepts the blowoff curve above the zero emissivity branch point B. Following the same trend the blackbody branch point B" falls even further up the zero emissivity blowoff curve. Other values of intermediate graybody emissivity were used and all showed the same trend. As the emissivity is increased from 0 to 1, the branch point migrates along the upper branch of the zero emissivity curve. From this it can be concluded that for low fuel blowing velocity there is a strong effect of surface emissivity on the extinction limits. The migration of the branch point also is very nonlinear. That is, at low emissivities there is a large shift in the branch point location up the curve, but at higher emissivities there is not much
OPPOSED JET DIFFUSION FLAME EXTINCTION
167
7.0
a = 3 0 0 , sec -1
6.8 6.6
~
6.4
0 6.2
Blowoff
Surface Extinction O wal~l \
6.0
\ \\\
5.8
\
\ N
5.6
5.41 0.2
o13
oi,
o15
o16
-fw
Fig. 4. Maximumflame and wall temperaturesas a function of fuel blowing velocityat a = 300s-t for e = O.
change. Thus it can be concluded that even low emissivity fuel surfaces can act as significant contributors to the extinction phenomenon until the fuel blowing velocity becomes high enough that radiative heat losses do not influence the blowoff in any way. Similarly, the transition of the surface extinction branch BC to the quenching extinction B" C " was explored. Figure 5 shows the limiting fuel blowing velocity as a function of surface emissivity for a constant stagnation velocity gradient. A similar nonlinear transition is found. For high surface emissivities there is only a slight change in the limiting fuel blowing velocity with decreasing emissivity, but as the surface emissivity decreases the limiting - f w drops significantly. From this it can again be said that even for low emissivity surfaces the effect of surface radiation should not be ignored. E F F E C T OF L E W I S N U M B E R O N
EXTINCTION LIMITS The effect of varying fuel and oxidizer Lewis number was explored for the case of no radiation
from the surface and blackbody radiation from the fuel surface. The Lewis numbers were changed independently for fuel and oxidizer by varying the Schmidt number while holding the Prandtl number constant. For this work we define: Le = Sc/Pr = a/:D. The Lewis number was varied from 0.833 to 1.25 during this part of the parametric study. It should be noted at this point that other studies have been conducted [12] varying the Prandtl number and holding the Schmidt number constant which find similar results over the same range of values. For the thermal quenching limit for a blackbody radiating surface, the results are plotted in Fig. 6. Decreasing the Lewis number of the oxidizer from 1.25 to 0.833 greatly decreases the limiting fuel blowing velocity. This indicates that the limiting flames exist closer to the cylinder surface as the oxidizer Lewis number decreases. Varying the fuel Lewis number over the same range has a smaller but complementary effect on the limits. This result can be explained in light of the fact that decreasing the Lewis number increases the flame temperature. The fuel Lewis number has a lesser
168
S . L . OLSON and J. S. T ' I E N 0.30
a=20.
sec - 1
i
t
0.25
0.20
"~ 0.15
T 0.10 0.05 0.0
i
t
i
i
i
I
0.5
0
I
1.0
Fig. 5. Effect of surface emissivity (~) on the limiting fuel blowing velocity (-fw) at a stretch rate of a = 20 s ~. influence on the limiting blowing velocity than that of the oxidizer because the flame is on the fuel side of the stagnation plane and the stoichiometric methane/oxygen mass ratio is 0.25. In the case of no radiative loss from the fuel surface we find no appreciable change in the
0.30
a = 20 ~=1
sec -1
/ / /
// / /
0.29
/;."Z /
0.28
-
;//
._E •~
//
0.27
/5 / ///
T 0.26
SUMMARY
"/I 0.25
'7 m --
Lef = 0.833 L e f = 1.0 Lef = 1 . 2 5
,
I 1.0
0.24 I 0.8
extinction limit with either fuel or oxidizer Lewis number. This is reasonable when we consider that the flame is pressed against the fuel surface and extinguishes by reaction zone thinning as a result of a short supply of fuel. Since the reaction zone is at the cylinder surface where fuel is supplied, the reaction rate becomes insensitive to the fuel diffusivity. Although the premixed flame extinction and the diffusion flame extinction discussed here are very different, it is interesting to note some similarities. The diffusion flame configuration at the surface extinction limit is similar to that of the opposedflow premixed flames when the Lewis number of the deficient species is greater than unity [13]. In that case, two symmetric premixed flames are compressed toward each other and incomplete combustion is observed to be the cause of extinction. This is similar to the reaction zone thinning mechanism of the present adiabatic surface extinction of the diffusion flame. On the other hand, for the diffusion flame the surface extinction mode has been found to be independent of the value of either fuel or oxidizer Lewis number, so the comparison is not exact. For the blowoff limits at high fuel blowing velocity both the case with radiation and the case neglecting radiation from the surface gave almost identical results. Figure 7 presents the results for the blackbody surface case. Decreasing the fuel and oxidizer Lewis number increases the limiting stretch rate. The trends in Fig. 7 show that again the oxidizer Lewis number has a larger influence on the limit than that of the fuel for the same reasons as described for Fig. 6.
i
I
1.2
Le o Fig. 6. Effect of fuel and oxidizer Lewis number on the limiting fuel blowing velocity.
A theoretical analysis is described for a methaneair diffusion flame stabilized in the forward stagnation region of a porous metal cylinder in a forced convective flow. Predicted extinction limits compared well with experimentally observed extinction limits for a system Lewis number of 0.833 and a surface emissivity of 0.25. The effect of surface radiative heat loss on the predicted extinction limits was examined. For the low fuel blowing velocity limit, radiative heat loss
OPPOSED JET DIFFUSION FLAME EXTINCTION
320
There was no effect of either Lewis number on the new surface extinction limit.
= 2.0
-fw
J S T would like to acknowledge support from N S F Grant MEA-8115339. The authors wish to thank John deRis f o r a discussion on Tsuji and Yamoaka's papers and to John Haggard f or commenting on the manuscript.
300
280
\
260
A
169
NOMENCLATURE
To
a
240 v
E
B
,. ,, \
:, ',,\
220
Cpm
,.,),\
200
"\',, \ \,
180
D
\ ~LeI ---Lef
X \\
= 0 . 8 3 3 `\ , \ = 1.0 \'x
160 _ - - - Lef = 1.25 I
0.8
t
I
1.0
"\ t
I
1.2
Le o Fig. 7. Effect of fuel and oxidizer Lewis number on the limiting flame stretch rate.
was seen to have a strong effect on the limit even with low emissivity fuel surfaces. At the blowoff limit, surface radiative heat loss did not play an important role for high fuel blowing velocities. However, at lower fuel blowing velocities even low emissivities caused a substantial change in the limiting stretch rate. A new extinction mechanism due to excessive reaction zone thinning is described for the case of the zero emissivity fuel surface. To the authors' knowledge no one has observed this type of surface extinction for a diffusion flame because of the high surface temperatures required and the difficulty of eliminating radiative heat loss at these temperatures. The effect of variable fuel and oxidizer Lewis number on the extinction limits was also examined. The oxidizer Lewis number had a strong influence on both the limiting fuel blowing velocity and the limiting stretch rate. The fuel Lewis number plays a smaller complementary role.
E
f Le M m
V
No P Pr
Q
stagnation velocity gradient, spreexponential factor in reaction rate heat capacity evaluated at the mean temperature, cal/gmol K Damkohler number diffusion coefficient, cm2/s activation energy in Arrhenius expression of reaction rate modified stream function Lewis number, a / ~ molecular weight mass flow rate of fuel, g/cm 2 s velocity of air far from cylinder, cm/s ratio of oxidizer/fuel in stoichiometric reaction pressure Prandtl number nondimensional heat of combustion, q/ CpmTe
R /~ Re S Sc T u V v w Y e
X # ~,
heat of combustion of fuel, kcal/gmol ideal gas constant radius of porous cylinder, cm Reynolds number radiant heat loss parameter Schmidt number temperature, K velocity in y direction, cm/s velocity of air far from cylinder, cm/s velocity in x direction, cm/s reaction rate of fuel mass fraction emissivity of the porous metal surface nondimensional y coordinate as defined in Ref. [8] thermal conductivity, W/m K viscosity, g/cm s kinematic viscosity, cm2/s
170 p 0 o a
S . L . OLSON and J. S. T'IEN density, g / c m 3 dimensionless temperature, Stefan-Boltzmann constant thermal diffusivity, cmZ/s
5.
T/Te 6.
Subscripts e f m o w
environment fuel mean oxidizer wall, porous metal surface
Superscripts
' *
dimensional quantity differentiated with respect to rt as defined by Ref. [2]
REFERENCES 1. Tsuji, H., Prog. Energy Comb. Sci. 8:93-119 (1982). 2. Tsuji, H., and Yamaoka, I., 11th Syrup. (lnt'l) on Comb,, 1967, pp. 979-984. 3. Saitoh, T., Combust. Flame 36:233-244 (1979). 4. Tsuji, H., and Yamaoka, I., 12th Symp. (Int'l) on Comb., 1969, pp. 997-1005.
7. 8. 9. 10. 11. 12. 13. 14.
15. 16.
Tsuji, H., and Yamaoka, I., 13th Symp. (Int'l) on Comb., 1971, pp. 723-731. Dixon-Lewis, G., David, T., Gaskell, P. H., Fukutani, S., Jinno, H., Miller, J. A., Kee, R. J., Smooke, M. D., Peters, N., Effelsberg, E., Warnatz, J., and Behrendt, T., 20th Symp. ([nt'l) on Comb., 1984, pp. 18931904. T'ien, J. S., Combust. Flame 65:31-34 (1986). T'ien, J. S., Singhal, S. N., Harrold, D. P., and Prahl, J. M., Combust. Flame 33:55-68 (1978). Lee, S. T., and T'ien, J. S., Combust. Flame 48:273286 (1982). Handbook on Chem. Phys., 60th Ed., 1979-80. Sato, J., 19th Syrup. (lnt'l) on Comb., 1982, pp. 15411548. Beyerle, R., Unpublished Report, Case Western Reserve University, 1985. Ishizuka, S., and Law, C. K., 19th Syrup. (Int'l) on Comb., 1982, pp. 327-335. Smith, J. M., and Van Ness, H. C., Intro. to Chem. Engr. Thermo., 3rd Ed., McGraw-Hill, New York, 1959. Doss, M. P., Phys. Constants o f the Principal Hydrocarbons, 4th Ed., The Texas Co., 1943. Welty, J. R., Wicks, C. E., and Wilson, R. E., Fund. o f Momentum, Heat, and Mass Transfer, 2nd Ed., Wiley, New York, 1976.
Received 6 June 1986; revised 30 April 1987
AND FLAME
70: 161-170 (1987)
161
A Theoretical Analysis of the Extinction Limits of a Methane-Air Opposed-Jet Diffusion Flame S. L. O L S O N NASA-Lewis Research Center, Cleveland, OH 44135
and J. S. T ' I E N Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106
A theoretical analysis is described for a methane-air diffusionflame stabilizedin the forward stagnationregion of a porous metalcylinder in a forced convectiveflow. The analysisincludeseffects of radiativeheat loss from the porous metal surface and finite rate kinetics but neglects the effects of gravity. The theoreticallypredicted extinctionlimits compare well with experimentallyobserved extinction limits from the literature. After the predicted limits compared well with the experimental limits, a parametric study of the effect of fuel surface emissivity and Lewis number was conducted with the numerical model. It was found that the computed blowoff limit is independentof radiative heat loss for high fuel blowing velocitiesbut is a strong function of Lewis number. At low fuel blowingvelocities,the extinctionlimit varies with both radiativeheat loss and Lewis number. It is discovered, however, that even if thermal losses from the fuel surface are absent, the flame can extinguishat the fuel surface independentlyof Lewis number due to excessive reaction zone thinning.
INTRODUCTION The counterflow diffusion flame in the forward stagnation region of a cylindrical porous burner is a simple flame to study experimentally because of its stability, and theoretically due to the onedimensional governing equations. This explains its enduring presence in combustion literature. A summary of experimental and theoretical work on the counterflow diffusion flame can be found in Tsuji [1]. The present study focuses on the extinction limits of the counterflow diffusion flame. Two distinct types of extinction have been identified experimentally by Tsuji and Yamaoka [2]. A blowoff limit is believed to be caused by large flame stretch which imposes chemical limitations on the combustion rate within the flame. A thermal This article is in the public domain Published by Elsevier Science PublishingCo., Inc. 52 VanderbiltAvenue, New York, NY 10017
quenching limit at low fuel injection rates is believed to be caused by heat losses. A theoretical analysis by Saitoh [3] showed that a low fuel blowing velocity extinction limit existed as a result of conductive heat losses imposed by maintaining a constant surface temperature. However, in the experiments conducted by Tsuji and Yamaoka [2, 4, 5], no active cooling was used and the surface temperature was allowed to vary. In this case, the low fuel blowing velocity quenching limit has yet to be explained. Recent modeling work by Dixon-Lewis et al. [6] focused on the blowoff or stretch limit of the counterflow diffusion flame using detailed chemical kinetics. They concluded that quenching effects at the fuel surface do not contribute to the observed extinction at blowoff. T ' i e n [7] has focused on the thermal quenching limit of a
0010-2180/87/$00.00
162
S . L . OLSON and J. S. T ' I E N
counterflow diffusion flame and has proposed that the observed extinction at low fuel flow rates can be caused by radiant heat losses from the fuel surface. In this paper the combustion model developed in Ref. [7] is modified to describe a diffusion flame established in the forward stagnation region of an uncooled porous metal cylinder. This is the configuration used by Tsuji and Yamaoka [2, 4, 5]. As in the uncooled experiments, the surface temperature is allowed to vary while the fuel blowing velocity and oxidizer stagnation point velocity gradient are the controlled variables. This paper is organized in four sections. In the first section the modified combustion model is described. In the second section the computed extinction limits are compared with experimentally determined extinction limits. A parametric study of the effects of radiative heat loss from the fuel surface and fuel and oxidizer Lewis number was then conducted. The third section describes the effects of surface emissivity on the predicted extinction limits and the fourth section discusses the Lewis number effects.
axisymmetric geometry to a two-dimensional geometry. Additionally, the fuel feed rate was converted from a solid fuel pyrolysis rate dependent on fuel surface temperature to a fixed feed rate as is the experimental situation with a gaseous fuel. A second order Arrhenius reaction rate for methane and air was used in the model. The reaction rate constants/~ and/~ were taken from Ref. [9]. The reaction rate expressed in nondimensional form is w = D Yf Yo exp( - E/O),
(2)
where
D = (I~P/aR Te)(AIm/Mo)
(3)
is the Damkohler number and a is the stagnation point velocity gradient. There are nine boundary conditions required for this system of equations. These are as follows: forT/ = 0
f--fw
(specified),
(4)
f' =0,
(5)
Yf' = - f w Scf( Yew- 1),
(6)
Y~ = - f w Sco Yow,
(7)
COMBUSTION MODEL A two-dimensional counterflow diffusion flame stabilized in the forward stagnation region of an uncooled porous metal cylinder is described. Fuel is ejected from the cylinder with a specified blowing velocity into an oxidizer flow of specified stagnation velocity gradient. The gas-phase governing differential equations are [7, 8]
f~, +ff,,
= [(f,)2
_
40],
# '÷= - f ,
Pr(O, - 1) + S(0w 4
-
1),
(8)
where
S = ((re Te3/)~e)(#e/(Pea))1/2 is the dimensionless radiative surface heat loss parameter; for ~7 ~ oo
(1/Pr)0" + fO' = -- Qw,
(9)
O=O~= 1,
(l/Scf) Y/' + f Y f ' = w, (l/Sco) Yo' + f Y o =NOW.
-
(1)
The model assumes constant specific heat for all species, and the ideal gas law is assumed for all reactants and products. The product of density and viscosity, p#, is assumed to be constant. Modifications to the model were made to first convert the system of governing equations from an
f' =2
(10)
(u = Ue),
(10
Yf=0, Yo = Yoe = 0.2325
(air).
(12)
Of special note is Eq. (8), which was derived through a heat balance at the surface of the cylinder. Convective heat transfer from the flame
OPPOSED JET DIFFUSION FLAME EXTINCTION through the gas phase to the cylinder surface and radiative heat loss from the surface to the ambient environment are equated with an increase in fuel enthalpy as it is ejected from the cylinder surface. The surface temperature is determined as part of the solution. These equations are solved numerically using a scheme similar to that used in Ref. [7]. Property values used in the computations are contained in Table 1. Table 2 contains nondimensional parameter values used in the model. Definitions of these parameters can be found in the Nomenclature.
COMPARISON OF COMPUTED EXTINCTION LIMITS W I T H EXPERIMENTALLY DETERMINED LIMITS
163 TABLE 2
Nondimensional Parameter Values Used in the Numerical Computation Parameter
Value
E No Q e
54.14 4.0 122.0 Variable
0~
1.00
Lef Leo Pr Yo~
Variable Variable 0.70 0.2325
sional blowing velocity as
The numerical analysis was performed for a number of fuel feed rates and stagnation velocity gradients for the case where the surface is a grey body radiator. Table 3 contains the identified extinction limits and corresponding limit burning solution flame conditions. In order to compare these limiting solutions to the experimentally determined limits, the relationships between the nondimensional fuel blowing velocity, -fw, defined in this work and those defined by Tsuji must be determined. Tsuji and Yamaoka [2] define their nondimen-
- f*ww= ( v * / V ) ( R e / 2 )
(13)
°'5,
where o* is the methane ejection velocity at ambient temperature. V is the upstream air velocity far from the cylinder surface, and Re is the Reynolds number based on cylinder radius and mean flow properties. In the present work we use (14)
- f w = ml(oo~tz~a} 1/2,
where r r / = ,OwVw.
TABLE 1 Dimensional Quantities Used in the Numerical Computation Property
C~ P t~ Mo.f.m R T~ pe urn t~e
Value 5.06 X I013 5.80 32.3 1.00 212.8 32, 16, 30 1.5 1.987/82.05 300 1.177 × 10 -3 1.168 1.846 × 10 -5 or 10 -4
Unit cm3/gmol s cal/gmol K kcal/gmol atm kcal/gmol g/gmol cm (cal/gmol K)/(cm 3 atm)/(gmol K) K g/cm 3 cm2/s Pa s g/cm s
Reference 9 14 9 1 15 14 1 14 1 16 16 16
164
S. L. OLSON and J. S. T ' I E N TABLE 3
2.5
Limit BurningSolutionFlame ConditionsDetermined fromthe Numerical Computationfor ~ = 0.25, Lef = Leo = 0.833
2.0
a 10 20 50 100 200 301 333 331 313 304 297
-jr, o. 157 o. 181 0.233 0.309 0.473 0.750 0.97 1.0 1.5 2.0 2.5
One, 4.95 5.20 5.43 5.65 5.76 5.89 6.04 6.14 6.59 6.58 6.55
O.~l, 3.75 4.01 4.34 4.55 4.61 4.39 4.01 3.97 3.11 2.33 1.32
fw 0.096 o. 104 o. 126 o. 161 0.242 0.389 0.512 0.525 0.820 1.190 2.020
Comparing these relations using a = 2 V / I ~ , it can be shown that
-J~w=
- fw(21Omean)( M m l M f ) .
(15)
Figure 1 shows a comparison between the computed and experimentally determined extinction limits. The model developed here clearly predicts the nature of both extinction by blowoff and extinction by quenching. For high stagnation point velocity gradient a blowoff limit is experimentally observed and theoretically predicted. For very low fuel blowing velocity a quenching boundary is again experimentally observed and theoretically predicted. In the transition region from quenching to blowoff there is some discrepancy between experiment and prediction. It should be noted that for comparative purposes a Lewis number of 0.833 and a fuel surface emissivity of 0.25 were used. The Lewis number is reasonable for a methane-air system and the emissivity represents an approximation for a dull bronze surface [10]. In later sections of this paper the effects of both Lewis number and surface emissivity on the computed extinction limits will be discussed in greater detail. It is also of interest to note that although the experimental data show a slight increase in the limiting - f * at very low stretch rates, the computed values of the limiting - f * continue to
1.5
Experiment Tsuji and Yamaoka (1) .... Present computation ( Lef = L e o = 0.833, ( - - 0 . 2 5 )
~: U.
-fw *
o
1,0 FLAME 0.5
o I----~ lO
i
i
i
i
i
i|1
100 a, sec -1
|
t
|
500
Fig. 1. A comparison of the computed extinction boundary with experimentally measured extinction limits.
decrease well beyond the range of stretch rates used in the experiment. We suspect that this slight increase in the limiting - f * is due to buoyant flows which become important in very low convective flows. As our model neglects the effects of gravity, it would not predict this behavior. Dixon-Lewis et al. [6] have recently done some computational work for comparison with Tsuji and Yamaokas' data also. They focused on one flame which was well documented in Ref. [5]. Using complex chemical reaction mechanisms and detailed transport flux information they compared species concentrations, velocity profiles, and temperature distributions. Because of the simple onestep second-order chemical reaction scheme used in this work, a detailed comparison is not in order. However, the flame temperature computed by this model is within the same range of temperatures as reported by Dixon-Lewis et al. for a flame at a = l O O s - l a n d - f * = 1.5.
E F F E C T S OF S U R F A C E R A D I A T I O N ON THE EXTINCTION LIMITS After the model demonstrated good agreement with experiment, a parametric study was conducted to determine the effects of surface radiative heat loss on the extinction limits predicted by the model. For this study the Lewis numbers of the
OPPOSED JET DIFFUSION FLAME EXTINCTION
165
limit. Near this limit the flame is positioned right next to the fuel surface. To illustrate this, Fig. 3 shows a series of temperature profiles for three different values of - f , at a given velocity gradient. For - f w = 0.20 the temperature peak is located in the gas phase away from the wall. As - f w decreases, the flame moves close to the porous cylinder surface and the temperature peak reaches the cylinder surface at - f w = 0.11. As - f w is decreased further the flame zone is compressed against the wall and the flame finally extinguishes at - f w < 0.079. The cause of the extinguishment is that as the reaction zone is pressed closer to the fuel surface it does not have enough volume to generate the heat required to sustain the flame. This occurs because the porous metal surface acts as a wall to the oxidizer and does not permit the oxidizer to diffuse into the fuel. The wall oxidizer concentration then builds to a substantial level just before extinguishment. Therefore, this lower extinction branch is a surface extinction branch and the extinction is caused by excessive reaction zone thinning. To our knowledge, no one has observed this type of extinguishment experimentally for diffu-
fuel and oxidizer were set equal to unity. The effect of Lewis number is discussed in the last section of this paper. Most of the work focused on three cases: a blackbody surface, an intermediate greybody surface, and a perfectly nonradiating surface. Emissivity values for these cases are 1, 0.25, and 0, respectively. A few other intermediate graybody surfaces were studies also so that trends could be determined more accurately. Figure 2 shows three extinction limit curves found for emissivities of 1.0, 0.25, and 0.0. Each curve will be discussed in detail. The extinction curve where surface radiation is neglected is labeled in Fig. 2 as A B C . This curve is clearly two separate limit branches A B and CB connected at B by a branch point. The upper branch A B corresponds to a blowoff limit. Near the blowoff the flame temperature drops until extinction occurs. The temperature drop is caused by limitations of the time available for chemical reaction to occur. The flame extinguishes because it cannot react fuel and oxidizer fast enough. The lower branch CB of the zero surface emissivity curve is a different type of extinction
3.0 Leo = Lef = 1 2.5 2.0
NO FLAME
3" 1.5 FLAME
j/~ B"
1.0 0.5
2
5
~0
20
50
~00
200
400
a (sec -1) Fig. 2. Effect of radiative heat loss from the porous metal surface on the extinction boundaries.
166
S . L . OLSON and J. S. T ' I E N
8.0
7.o
Leo=Lef=l
Limit - f w = O . 0 7 9 Flame just t o u c h e s wall - f w = 0 . 1 1 0
....
~
~mFlame
I
,
t
6.0
a=20.1/s ~=0.0
a w a y from
wall - f w = 0 . 2 0 0
,
t I l
0
a o
1.o 0
014
1 J2
018
' 1.6
2.0
2.4
~7 Fig. 3. Temperature profiles of flame as it approaches the surface extinction limit for zero surface radiation.
sion flames presumably because of the high surface temperatures required and the difficulty of eliminating radiative loss at these temperatures. The phenomena bear a certain resemblance to the symmetric premixed flame extinction of Sato [11], where it was found that when the Lewis number of the deficient species is greater than unity, the flame is quenched at the stagnation surface. For the present diffusion flame in the absence of thermal losses from the fuel surface, the surface extinction mode always exists independently of fuel or oxidizer Lewis number. To illustrate further the difference between blowoff and surface extinction, the peak flame and wall temperature profiles are plotted in Fig. 4 as a function of fuel blowing velocity for a stretch rate a = 300 s-1 from the surface extinction to the blowoff limit near the branch point B on Fig. 2. The flame temperature is much higher than the wall temperature at blowoff. This indicates that the flame is far from the wall. As the fuel blowing
velocity is decreased from blowoff toward surface extinction the flame and wall temperatures both increase to a maximum near the surface extinction limit. The flame temperature then quickly drops to the wall temperature as the flame is pressed against the wall. With further reduction in the fuel blowing velocity the flame extinguishes. The case of a blackbody surface is shown in Fig. 2 in curve A B " C " . The blowoff branch A B " and a quenching limit B " C" flow together smoothly and appear to be a single curve. This is in contrast to the zero emissivity curve where the two branches were quite distinct. At both limits the flame (the peak temperature position) is located away from the wall but as extinction is approached the flame temperature drops until extinction occurs. The reason behind the temperature drop near the limit is different, however, for the two limits. Near the quenching limit the surface radiation becomes important and heat transfer from the flame via gas-phase conduction to the surface to offset the radiative losses becomes excessive. Near the blowoff the temperature drop is caused again by limitations on the time available for chemical reaction to occur. The limiting temperature is higher at the blowoff limit than at the surface extinction limit. To understand better the different appearance of the two extinction curves described above, the effect of intermediate graybody surface emissivity was explored. Curve A B ' C ' in Fig. 2 is for an emissivity of 0.25. It is observed that the branch point B ' falls on the zero emissivity blowoff curve but intercepts the blowoff curve above the zero emissivity branch point B. Following the same trend the blackbody branch point B" falls even further up the zero emissivity blowoff curve. Other values of intermediate graybody emissivity were used and all showed the same trend. As the emissivity is increased from 0 to 1, the branch point migrates along the upper branch of the zero emissivity curve. From this it can be concluded that for low fuel blowing velocity there is a strong effect of surface emissivity on the extinction limits. The migration of the branch point also is very nonlinear. That is, at low emissivities there is a large shift in the branch point location up the curve, but at higher emissivities there is not much
OPPOSED JET DIFFUSION FLAME EXTINCTION
167
7.0
a = 3 0 0 , sec -1
6.8 6.6
~
6.4
0 6.2
Blowoff
Surface Extinction O wal~l \
6.0
\ \\\
5.8
\
\ N
5.6
5.41 0.2
o13
oi,
o15
o16
-fw
Fig. 4. Maximumflame and wall temperaturesas a function of fuel blowing velocityat a = 300s-t for e = O.
change. Thus it can be concluded that even low emissivity fuel surfaces can act as significant contributors to the extinction phenomenon until the fuel blowing velocity becomes high enough that radiative heat losses do not influence the blowoff in any way. Similarly, the transition of the surface extinction branch BC to the quenching extinction B" C " was explored. Figure 5 shows the limiting fuel blowing velocity as a function of surface emissivity for a constant stagnation velocity gradient. A similar nonlinear transition is found. For high surface emissivities there is only a slight change in the limiting fuel blowing velocity with decreasing emissivity, but as the surface emissivity decreases the limiting - f w drops significantly. From this it can again be said that even for low emissivity surfaces the effect of surface radiation should not be ignored. E F F E C T OF L E W I S N U M B E R O N
EXTINCTION LIMITS The effect of varying fuel and oxidizer Lewis number was explored for the case of no radiation
from the surface and blackbody radiation from the fuel surface. The Lewis numbers were changed independently for fuel and oxidizer by varying the Schmidt number while holding the Prandtl number constant. For this work we define: Le = Sc/Pr = a/:D. The Lewis number was varied from 0.833 to 1.25 during this part of the parametric study. It should be noted at this point that other studies have been conducted [12] varying the Prandtl number and holding the Schmidt number constant which find similar results over the same range of values. For the thermal quenching limit for a blackbody radiating surface, the results are plotted in Fig. 6. Decreasing the Lewis number of the oxidizer from 1.25 to 0.833 greatly decreases the limiting fuel blowing velocity. This indicates that the limiting flames exist closer to the cylinder surface as the oxidizer Lewis number decreases. Varying the fuel Lewis number over the same range has a smaller but complementary effect on the limits. This result can be explained in light of the fact that decreasing the Lewis number increases the flame temperature. The fuel Lewis number has a lesser
168
S . L . OLSON and J. S. T ' I E N 0.30
a=20.
sec - 1
i
t
0.25
0.20
"~ 0.15
T 0.10 0.05 0.0
i
t
i
i
i
I
0.5
0
I
1.0
Fig. 5. Effect of surface emissivity (~) on the limiting fuel blowing velocity (-fw) at a stretch rate of a = 20 s ~. influence on the limiting blowing velocity than that of the oxidizer because the flame is on the fuel side of the stagnation plane and the stoichiometric methane/oxygen mass ratio is 0.25. In the case of no radiative loss from the fuel surface we find no appreciable change in the
0.30
a = 20 ~=1
sec -1
/ / /
// / /
0.29
/;."Z /
0.28
-
;//
._E •~
//
0.27
/5 / ///
T 0.26
SUMMARY
"/I 0.25
'7 m --
Lef = 0.833 L e f = 1.0 Lef = 1 . 2 5
,
I 1.0
0.24 I 0.8
extinction limit with either fuel or oxidizer Lewis number. This is reasonable when we consider that the flame is pressed against the fuel surface and extinguishes by reaction zone thinning as a result of a short supply of fuel. Since the reaction zone is at the cylinder surface where fuel is supplied, the reaction rate becomes insensitive to the fuel diffusivity. Although the premixed flame extinction and the diffusion flame extinction discussed here are very different, it is interesting to note some similarities. The diffusion flame configuration at the surface extinction limit is similar to that of the opposedflow premixed flames when the Lewis number of the deficient species is greater than unity [13]. In that case, two symmetric premixed flames are compressed toward each other and incomplete combustion is observed to be the cause of extinction. This is similar to the reaction zone thinning mechanism of the present adiabatic surface extinction of the diffusion flame. On the other hand, for the diffusion flame the surface extinction mode has been found to be independent of the value of either fuel or oxidizer Lewis number, so the comparison is not exact. For the blowoff limits at high fuel blowing velocity both the case with radiation and the case neglecting radiation from the surface gave almost identical results. Figure 7 presents the results for the blackbody surface case. Decreasing the fuel and oxidizer Lewis number increases the limiting stretch rate. The trends in Fig. 7 show that again the oxidizer Lewis number has a larger influence on the limit than that of the fuel for the same reasons as described for Fig. 6.
i
I
1.2
Le o Fig. 6. Effect of fuel and oxidizer Lewis number on the limiting fuel blowing velocity.
A theoretical analysis is described for a methaneair diffusion flame stabilized in the forward stagnation region of a porous metal cylinder in a forced convective flow. Predicted extinction limits compared well with experimentally observed extinction limits for a system Lewis number of 0.833 and a surface emissivity of 0.25. The effect of surface radiative heat loss on the predicted extinction limits was examined. For the low fuel blowing velocity limit, radiative heat loss
OPPOSED JET DIFFUSION FLAME EXTINCTION
320
There was no effect of either Lewis number on the new surface extinction limit.
= 2.0
-fw
J S T would like to acknowledge support from N S F Grant MEA-8115339. The authors wish to thank John deRis f o r a discussion on Tsuji and Yamoaka's papers and to John Haggard f or commenting on the manuscript.
300
280
\
260
A
169
NOMENCLATURE
To
a
240 v
E
B
,. ,, \
:, ',,\
220
Cpm
,.,),\
200
"\',, \ \,
180
D
\ ~LeI ---Lef
X \\
= 0 . 8 3 3 `\ , \ = 1.0 \'x
160 _ - - - Lef = 1.25 I
0.8
t
I
1.0
"\ t
I
1.2
Le o Fig. 7. Effect of fuel and oxidizer Lewis number on the limiting flame stretch rate.
was seen to have a strong effect on the limit even with low emissivity fuel surfaces. At the blowoff limit, surface radiative heat loss did not play an important role for high fuel blowing velocities. However, at lower fuel blowing velocities even low emissivities caused a substantial change in the limiting stretch rate. A new extinction mechanism due to excessive reaction zone thinning is described for the case of the zero emissivity fuel surface. To the authors' knowledge no one has observed this type of surface extinction for a diffusion flame because of the high surface temperatures required and the difficulty of eliminating radiative heat loss at these temperatures. The effect of variable fuel and oxidizer Lewis number on the extinction limits was also examined. The oxidizer Lewis number had a strong influence on both the limiting fuel blowing velocity and the limiting stretch rate. The fuel Lewis number plays a smaller complementary role.
E
f Le M m
V
No P Pr
Q
stagnation velocity gradient, spreexponential factor in reaction rate heat capacity evaluated at the mean temperature, cal/gmol K Damkohler number diffusion coefficient, cm2/s activation energy in Arrhenius expression of reaction rate modified stream function Lewis number, a / ~ molecular weight mass flow rate of fuel, g/cm 2 s velocity of air far from cylinder, cm/s ratio of oxidizer/fuel in stoichiometric reaction pressure Prandtl number nondimensional heat of combustion, q/ CpmTe
R /~ Re S Sc T u V v w Y e
X # ~,
heat of combustion of fuel, kcal/gmol ideal gas constant radius of porous cylinder, cm Reynolds number radiant heat loss parameter Schmidt number temperature, K velocity in y direction, cm/s velocity of air far from cylinder, cm/s velocity in x direction, cm/s reaction rate of fuel mass fraction emissivity of the porous metal surface nondimensional y coordinate as defined in Ref. [8] thermal conductivity, W/m K viscosity, g/cm s kinematic viscosity, cm2/s
170 p 0 o a
S . L . OLSON and J. S. T'IEN density, g / c m 3 dimensionless temperature, Stefan-Boltzmann constant thermal diffusivity, cmZ/s
5.
T/Te 6.
Subscripts e f m o w
environment fuel mean oxidizer wall, porous metal surface
Superscripts
' *
dimensional quantity differentiated with respect to rt as defined by Ref. [2]
REFERENCES 1. Tsuji, H., Prog. Energy Comb. Sci. 8:93-119 (1982). 2. Tsuji, H., and Yamaoka, I., 11th Syrup. (lnt'l) on Comb,, 1967, pp. 979-984. 3. Saitoh, T., Combust. Flame 36:233-244 (1979). 4. Tsuji, H., and Yamaoka, I., 12th Symp. (Int'l) on Comb., 1969, pp. 997-1005.
7. 8. 9. 10. 11. 12. 13. 14.
15. 16.
Tsuji, H., and Yamaoka, I., 13th Symp. (Int'l) on Comb., 1971, pp. 723-731. Dixon-Lewis, G., David, T., Gaskell, P. H., Fukutani, S., Jinno, H., Miller, J. A., Kee, R. J., Smooke, M. D., Peters, N., Effelsberg, E., Warnatz, J., and Behrendt, T., 20th Symp. ([nt'l) on Comb., 1984, pp. 18931904. T'ien, J. S., Combust. Flame 65:31-34 (1986). T'ien, J. S., Singhal, S. N., Harrold, D. P., and Prahl, J. M., Combust. Flame 33:55-68 (1978). Lee, S. T., and T'ien, J. S., Combust. Flame 48:273286 (1982). Handbook on Chem. Phys., 60th Ed., 1979-80. Sato, J., 19th Syrup. (lnt'l) on Comb., 1982, pp. 15411548. Beyerle, R., Unpublished Report, Case Western Reserve University, 1985. Ishizuka, S., and Law, C. K., 19th Syrup. (Int'l) on Comb., 1982, pp. 327-335. Smith, J. M., and Van Ness, H. C., Intro. to Chem. Engr. Thermo., 3rd Ed., McGraw-Hill, New York, 1959. Doss, M. P., Phys. Constants o f the Principal Hydrocarbons, 4th Ed., The Texas Co., 1943. Welty, J. R., Wicks, C. E., and Wilson, R. E., Fund. o f Momentum, Heat, and Mass Transfer, 2nd Ed., Wiley, New York, 1976.
Received 6 June 1986; revised 30 April 1987