C O M B U S TI ON A N D F L A M E 81:7385 (1990)
73
A Model for Soot Formation in a Laminar Diffusion Flame I A N M. K E N N E D Y and W O L F G A N G K O L L M A N N Department of Mechanical Engineering, University of California, Davis, CA 95616
and J.Y. C H E N Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551
A simple model has been developed for the prediction of soot volume fractions in a laminar diffusion flame. Measurements and computations of a counterftow flame have been used to evaluate the correlation between soot surface growth rates and the mixture fraction or fuel atom mass fraction. An average particle number density was used to permit the determination of the aerosol surface area. Equations for the momentum, mixture fraction, and soot volume fraction were solved numerically for an axisymmetric laminar diffusion flame. Good agreement was obtained with the measurements for two different experimental conditions.
INTRODUCTION The formation of soot in a round, laminar diffusion flame has received a great deal of attention. Laser light scattering measurements of the soot aerosol [1, 2] have yielded a detailed picture of the formation process and its dependence on parameters such as the flame temperature [3]. Experiments in other configurations, e.g., the WolfhardParker burner, have confirmed the overall picture [4]. Soot particles are formed on the fuel side of the flame close to the reaction zone. The earliest particles are of the order of 1 nm in diameter and they form in large numbers. Rapid coagulation serves to reduce the particle number density while intense surface growth increases the soot volume fraction. High soot volume fractions welldownstream in the flames can lead to significant loss of energy by radiation. As a result, the temperatures near the tip of a laminar diffusion flame may be many hundreds of degrees below the Copyright (~) 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010
adiabatic flame temperature. The eventual emission of soot from the flame occurs when the flame temperature is reduced sufficiently by radiation so that the burnout reactions are inhibited [5]. Vandsburger et al. [6] and Axelbaum et al. [7] have made measurements of the soot aerosol in counterflow diffusion flames. By following the flow along the stagnation streamline it was possible to determine the rates of the aerosol processes. The experiments have yielded the soot surface growth rates. These data are potentially useful for the modelling of soot formation. Gore and Faeth [8] attempted to model soot production in a turbulent ethylene diffusion flame by developing an empirical correlation between soot volume fraction and the mixture fraction. The latter quantity is the fuel atom mass fraction and is used widely in the modeling of turbulent diffusion flames [9]. The correlation between soot volume fraction and mixture fraction was obtained from measurements of these quantities in a laminar, axisymmetric diffusion flame. However, this
00102180/90/$3.50
74 approach ignores the effect of the soot particle history and residence time in the flame. It also ignores the possible effect of mixing rates on the soot chemistry. Magnussen and Hjertager [10] and Magnussen et al. [11] incorporated a model for soot formation into an eddy dissipation model for turbulent combustion. They used the kinetic scheme of Tesner et al. [12] for soot formation in acetylene flames by calculating the mass of soot and the mean particle number density. The empirical constants were adjusted to give good agreement between experiment and computation. Recently Moss et al. [ 13] proposed a model that incorporates the essential physics, i.e., a nucleation rate, a surface growth rate and a burnout or oxidation rate. Their rate constants were not known and were determined by fitting their model to measurements in a laminar diffusion flame. The model that they proposed may also be somewhat difficult to implement in a turbulent flow calculation because of the usual closure problems for chemical source terms in conventional models of reacting flows. Methods that are based on transport equations for probability density functions should be capable of dealing with this problem. It is the intent of this research to utilize the present understanding of the soot formation process to develop a simple model that will predict the soot field in laminar and, eventually, turbulent diffusion flames. In the absence of adequately reduced chemical schemes for soot formation, the approach has been to use the available empirical information that light scattering measurements have provided. A further aim of the modeling has been to develop a satisfactorily accurate scheme that does not add substantially to the already significant computational burden that is faced in predicting practical turbulent flow fields. SOOT MODEL The mixture fraction is the primary quantity that is calculated. The temperature, density, and the gas composition are determined as functions of mixture fraction. The mixture fraction is also used to determine the soot volume fraction indirectly. The soot volume fraction is not taken to be a func
I . M . KENNEDY ET AL. tion of the mixture fraction, as Gore and Faeth [8] did, but, rather, the rates of nucleation, surface growth, and oxidation are functions of the mixture fraction. The conservation equation for soot volume fraction, ~b, is then
o¢ 0¢ pu ~~ + o(v + vr)~ur
(
1 0 rpDs~r r cgr
+pWn +pWg

PWo (1)
where wn, Wg, and Wo are the rates of soot volume formed by nucleation and surface growth and removed by oxidation, respectively. The axial velocity component is u and the radial velocity component is v. The equation contains a soot diffusivity, Ds, which is actually negligible but in the calculations is taken to be 1% of the gas diffusivity. This was found to be desirable in order to reduce the numerical oscillations which result from the absence of diffusivity. The results were not substantially affected by variations in this term. A thermophoretic radial velocity is introduced in the soot volume fraction equation; it is calculated as
vozr vr =  0 . 5 5 m l ~rr'
(2)
where v is the kinematic viscosity of the gas. It is simply added to the convective radial velocity. The axial thermophoretic velocity is negligible. Equation 2 is the appropriate form of the thermophoretic velocity equation for a free molecular aerosol such as soot [14]. The soot volume fraction equation is integrated along with the other flow field equations that are the standard, axisymmetric, boundary layer forms of the momentum equation and the mixture fraction equation that yield the axial and radial velocity components and the mixture fraction. Full details of the numerical procedure are available in Chen et al. [15]. The thermochemistry of the flame has been determined from the results of a detailed laminar counterflow diffusion flame code [16] in which the fuel was ethylene. The code was
SOOT FORMATION MODEL
75
run with a velocity gradient of 200 s  l , which is far from extinction conditions for ethylene. The results for the temperature and major species concentrations may be presumed to apply to the flow conditions in the laminar diffusion flame which is modeled. Most of the soot in a flame is produced by surface growth on a particle. The surface growth rate depends on the local temperature, the gas composition, and the available aerosol surface area. The latter quantity is determined by the number density and the appropriate moment of the particle size distribution. An accurate evaluation of the surface area would require at least a knowledge of the particle number density. This would require the solution of an additional equation for the particle number density, which is undesirable in a lengthy code. The examination of available data reveals that it may not be necessary to consider the number density equation in light of the other uncertainties associated with the modeling. Most measurements of the soot aerosol show surprisingly similar number densities in different flames. This is a result of the rapid coagulation that takes place. In addition, the surface area is a weak function of number density so that the rate Wg may be written in terms of an average number density as
W g  7rl /362/3 Navgl /3 dp2/3k( ~).
(3)
This equation was derived by considering the aerosol to be monodisperse. The surface area of the particles is N(Trdp2), where dp is the particle diameter. The volume fraction is N(lr/6dp3). Elimination of the diameter in terms of N and ~b gives the aerosol surface area which appears in Eq. 3. In this equation k(~) is the specific surface growth rate which is the rate of increase in soot volume normalized by the total particle surface area (msoot s  l ) ; it is a function of the mixture fraction, ~. These observations suggest that it may be possible to neglect the variations in number densities and to adopt a suitable average number density, Navg. In the present calculations the number density has been taken to be 1.0 × 1016 m 3, which is an average number density measured by Axel
baum et al. [7] in their ethylene diffusion flame. The specific surface growth rate has been taken from their measurements. A correlation between the specific surface growth rate and the mixture fraction is required. In order to do this it is necessary to find the mixture fraction profile through the flame. A detailed counterflow diffusion flame code [16] has been run with ethylene kinetics to provide the necessary mixture fraction profile. Two calculations of the laminar flame structure have been performed with this code. The first was for a nonluminous flame without soot, which had a maximum flame temperature of 2040 K. The second calculation was for a flame with soot for which an adjustable radiative loss term was added to the energy equation in the code as if the soot volume fraction were uniform throughout the flame. The radiation loss term was chosen to give a maximum flame temperature of 1716 K. The peak specific surface growth rate that was measured by Axelbaum et al [7] was about 3 × 10 6 msoot s  I • Vandsburger et al. [17] found similar values in their study of a counterflow diffusion flame of ethylene. Harris and Weiner [18] reported values for the specific growth rate that were about an order of magnitude less; their measurements were made in a premixed ethylene flame. They concluded that acetylene was the dominant growth species, with a mole fraction of about 0.02. The maximum acetylene mole fraction was calculated with the detailed counterflow flame code to be about 0.046 on the fuel side of the ethylene diffusion flame; this tends to suggest that acetylene may not be the only important growth species in a diffusion flame. A number of other hydrocarbon species are present in significant concentrations in the soot growth region of a diffusion flame. It has been possible to reproduce the soot growth rates of Axelbaum et al. [7] within a factor of 2 with the simple approach that is embodied in Eq. 3. The comparison between the model and the measurements is shown in Fig. 1 for the soot surface growth rate as a function of the time of flight of a particle from the flame front. The correlation between specific surface growth rates and the mixture fraction is shown in Fig. 2. The correlations between mixture fraction and temperature for the
76
I. M. K E N N E D Y ET AL. 1.40
2250
1.20
2000 1750
1.00 v' o UJ
0.80
I,,U ,"t:::1 I', ,,¢ ,nI,,U a.
\\
0.60
i
0.40
0 n0
0.20
1500
v
k
I,
X w
/\.X I
1250 1000
:z
t,U I
750 500 250
0.00
0.0
2.0 4.0 6.0 8.0 1 0 . 0 1 2 . 0 1 4 . 0 1 6 . 0 TIME (ms)
'
0.0
I
0.2
'
I
0.4
,
I
~
0.6
I
0.8
'
1.0
MIXTURE FRACTION
Fig. 1. Measured soot volume fraction growth rates as functions of particle time of flight from flame front (, Axelbaum et al. 17]) and rates determined by Eq. (3) (. . . . ).
Fig. 3. Temperature as a function of mixture fraction (, with radiation; . . . . , without radiation).
nonluminous and for the luminous flame are presented in Fig. 3; these results were obtained with the counterflow diffusion flame code. Nucleation produces the initial aerosol surface area; in this model initial particles are taken to be 1 nm in diameter and produced at the rate of
1018 m 3 s  l . Oxidation has been allowed to occur through the action of molecular oxygen and through the action of OH radicals. The former rate is described by a NaglerStricklandConstable expression [19] and the latter rate is determined by the kinetic theory collision rate with an efficiency of 5 % [20]. Both the concentrations of oxygen and OH radicals are evaluated as functions of the mixture fraction through the results of the counterflow flame code [16]; in conjunction with the calculated temperatures, the oxidation rates can be evaluated as functions of the mixture fraction. The oxidation rate expression, Wo, takes a form similar to the growth rate expression in Eq. 3. Computations were also performed in which an equation for the particle number density was included. This was done to test the effect of the assumption of an average number density that was described above. The equation incorporated terms for the convection o f particles, their generation by a nucleation mechanism, and their reduction in number by coagulation. The particle inception term was modeled in the same manner as Kennedy [21], i.e., as a Gaussian distribution in mixture fraction space with a peak rate of 10 t8 m 3 s  t . The rate of coagulation was calculated from the expression for the collision rate of free molecular
4.00
3.00
tO
•
X ttl
2.00
n,. 1.00 0 t~ 0.00
i
0.0

0.1
.
0
0.2
0.3
0.4
0.5
MIXTURE FRACTION Fig. 2. Specific surface growth rates of soot as a function of mixture fraction.
SOOT F O R M A T I O N M O D E L
77 1.50
aerosols with an enhancement factor for van der Waals forces of 2.2 [22]; a monodisperse aerosol was assumed to exist for the purposes of this calculation. The equation for the particle number density, N (particles/m3), is
 I"
' "%
1.oo
ON
ON puff~ + p(v + or) Or
1 0 (rpDs~r ) + pwi  pWc, r Or
(4)
0.50
~\\'~i.
where wi is the particle inception rate and Wc is the particle coagulation rate (particles/m3/s). 0.00
RESULTS The calculations have been carried out for the axisymmetric, laminar et.hylene diffusion flame of Santoro et al. [23]. Two conditions have been studied viz. their case 1 flame with a flow rate of 2.3 ml s  t and their case 2 flame with a flow rate of 3.85 ml s ~ . The nozzle diameter in both cases was 11 ram. Most attention will be given to the latter flame in which greater amounts of soot are formed. The calculations were carried out for two different mixture fractiontemperature correlations that either incorporated or ignored radiative energy loss. The axial component of velocity with both thermochemistries is shown in Fig. 4 for the flame 2 case at an axial station of 20 ram; also shown are the measurements of Santoro et al. [23]. Figure 5 shows the axial velocities at 70 mm. Good agreement is evident in both cases, although the nonradiative thermochemistry is apparently superior at locations near the nozzle where soot volume fractions are low whereas the radiative thermochemistry is superior higher in the flame. Figure 6 shows the predicted and measured temperature profiles at 20 mm. A similar comparison at 70 mm is presented in Fig. 7. Once again it is clear from these comparisons that the nonradiative thermochemistry is superior in predicting the flow and temperature field close to the nozzle whereas the inclusion of radiative loss improves the predictions further up in the flame. The soot contours for the case 2 flame are
. 0.0
: 0.2
:
: 0.4
o _
,
:
0.6
:
:
0.8
1.0
1.2
1.4
Y/D Fig. 4. Axial velocity profile for flame 2 at 20 mm from the nozzle (, with radiation; . . . . , without radiation; 0, Ref. 23). shown in Fig. 8. These results were obtained with a radiative loss included in the temperaturemixture fraction correlation so that the maximum temperature was 1716 K. On the basis of this calculation no soot is predicted to be emitted from 2.50
%%
2.00
1.5o E
~ o
1.00
J
0.50
0.00
"~ 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
YID
Fig. 5. Axial velocity profile for flame 2 at 70 mm from the nozzle (, with radiation; . . . . , without radiation; 0, Ref. 23),
78
I . M . KENNEDY ET AL. 2250 2000
./"\.
1750
~
1500
m
1250
O
/
\ o
./ 0
\
/'I0~ \" /"Of
\
\,o
lOOO 750
~\
500 250
,
: : : : : : : . :
:
0.0
0.2
0.4
0.6
0.8
1.0
: : : : : , 1.2
1.4
1.6
Y/D Fig. 6. Temperature profile for flame 2 at 20 m m from the nozzle (   , with radiation; . . . . , without radiation; 0, Ref. 23).
the flame; all the contours are closed curves on this plot. A comparison with the corresponding contour plots from Santoro et al. [23] indicates that a reasonable agreement has been achieved although the predicted peak soot volume fraction is about a factor of 2 larger than the experiments.
The general distribution of soot in the flame is in agreement with the experiments. A better evaluation of the soot model can be obtained by examining the radial profiles of soot volume fraction at 20 and 40 mm from the nozzle (Figs. 9 and 10). Good agreement with the experiments has been achieved, particularly at 20 mm from the nozzle. At both locations the use of a radiative loss in the calculation of the thermochemistry results in somewhat poorer predictions of the soot profile as a result of diminished oxidation rates at lower temperatures. Good agreement with the measurements has also been obtained for the low flow rate flame, i.e., flame 1 [21]. This is apparent in Fig. 11, which is a radial profile of predicted and measured soot volume fractions at 20 mm from the nozzle and in Fig. 12 at 30 mm from the nozzle. The model is evidently capable of yielding reasonable results for different conditions. Inclusion of the equation for the particle number density did not change the predicted structure of the sooting region greatly. Radial profiles of the soot volume fractions at 20 and 40 mm from the nozzle for flame 2 of Santoro et al. [23] are shown in Figs. 13 and 14, respectively, along with the measurements.
2000 ~ _ _
DISCUSSION 1750
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uJ I=:
1500
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1250
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O
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0.0
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, : ~ : 0.0
1.0
'%
: : ; ,, , , 1.2
1.4
1.6
Y/D
Fig. 7. Temperatureprofile for flame 2 at 70 mm from the nozzle (, with radiation; . . . . without radiation; 0, Ref. 23).
The measurements of Santoro et al [23] indicated that the sooting behavior of a laminar diffusion flame was determined, in part, by the residence time of soot in the flow. It is clear that this should be an important parameter. The model that has been developed here is capable of handling this effect quite well as a result of the successful prediction of the flow field. The ability to predict the appropriate variation in soot volume fractions with the fuel flow rate (i.e., the reasonable agreement obtained with both case 1 and case 2 flames) supports this assertion. Another important parameter is not handed as well. This is the effect of temperature. As a flame forms more and more soot, the loss of energy by radiation reduces the flame temperature. Temperatures may become many hundreds of degrees below the nonluminous values; the same flame may
79
SOOT FORMATION MODEL 10.0
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X (mm) Fig. 8. Soot volume fraction contours for flame 2 with no radiation.
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0.0
= 0.1
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=
,
0.2
I 0.3
,
: 0.4
, 0.5
Y/D
Fig. 10. Radial profile of soot volume fraction in flame 2 at 4 0 nun (   , with radiation; . . . . without radiation; 0, Ref. 23).
80
I . M . KENNEDY ET AL. 1.0E05
1.0E05
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Fig. 11. Soot volume fraction profile in flame 1 at 20 mm (no radiation; 0, Ref. 23).
show a substantial variation in temperature between locations near the nozzle, where there is little soot, and near the flame tip, where there may be significant amounts of soot [1, 5]. The reduction in temperature can affect a number of different chemical processes that are relevant to soot formation. The particle formation rate and
Fig. 13. Soot volume fraction profiles in flame 2 at 20 mm from the nozzle obtained by solving for the particle number density; 0, Ref. 23.
the oxidation rate will be reduced. In fact, Kent and Wagner [5] showed that the oxidation reaction can be frozen as temperatures drop below about 13130 K at the flame tip due to radiation losses. This results in the emission of soot from a flame.
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IO < n'IJ.
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0.2
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Y/D
Fig. 12. Soot volume fraction profile in flame 1 at 30 mm (no radiation; 0, Ref. 23).
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0.0
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0.1
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0.3
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Fig. 14. Soot volume fraction profiles in flame 2 at 40 mm from the nozzle obtained by solving for the particle number density; 0, Ref. 23.
SOOT FORMATION MODEL
81
1.0E05
O Z o tO IZ IJ.
O
1.0E06
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, 0.0
t 0.1
,
t 0.2
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, 0.5
Y/D Fig. 15. Soot volume fraction profile for flame 2 with corrected growth rate at 40 mm from the nozzle; 0. Ref. 23.
perature was about 2000 K. As a result, the specific surface growth rates that have been used are likely to be upper bounds of those pertaining in flames in which there is significant radiation. An attempt has been made to introduce the temperature/radiation effect in a relatively crude, ad hoc manner without the additional difficulty of solving the energy equation. An examination of the available experimental data [ 1, 4, 6] indicates that flame temperatures are around 2000 K when the volume fraction is about 10 6 and it decreases typically to about 1600 K when the volume fraction is about 10 5. A linear correlation between temperature and soot volume fraction has been implemented. As a result, for the purposes of estimating the surface growth rates, the temperature is estimated to decrease to 1600 K when the volume fraction is 10 5 , i.e., the corrected growth rate, Wg tort, is estimated from the following equation:
Wgc o r r Although the surface growth reactions are characterized by comparatively low activation energies [12], they too will be diminished as the temperatures are reduced by radiation. A typical activation energy for these reactions is 20 kcal/mol. A reduction in temperature from 2000 to 1600 K would then lead to a reduction in the surface growth rates by a factor of about 3 or 4. Reduced growth rates in a flame as a result of decreased temperatures limit the amount of soot that can form. However, in the model as it has been presented up to this point, there is no limit on the formation of soot because temperatures are not coupled to the soot volume fraction in the calculation. In an actual flame this coupling between the soot volume fraction, the temperature and the rates of growth are significant. Neglect of this effect in the present model contributes to the overprediction of the soot volume fraction. The model would be most accurate for very lightly sooting flames where there was not a substantial radiation loss. The specific surface growth rates that have been used in the calculations were derived from measurements on the counterflow flame of Axelbaum et al. [7] in which the maximum soot volume fraction was of the order of 10 6 and the flame tern
=
Wg exp{5  10,000/[2,000 400(~b
106)/(10
5

106)]}
(5)
An activation temperature for the growth reactions of 10,000 K is assumed; this value is in accord with the measurements of Tesner et al. [12]. With the corrected growth rates, Fig. 15 shows the predicted soot volume fraction profile at 40 mm from the nozzle. The peak soot volume fraction is now in better agreement with the measured values. The distribution of soot is basically unchanged. By using the correction to the surface growth rate, improved agreement with the measurements of Santoro et al. [23] is obtained. The predictions do depend on the choice of the surface growth activation energy or temperature. A value of the activation temperature of 15,000 K reduces the peak soot volume fraction by two orders of magnitude. These results show the importance of the temperature in determining the rate of soot formation. A more reliable method of accounting for this effect is desirable but it could only be achieved by solving for the energy equation. This approach introduces uncertainties of its own (such as the calculation of the radiation term) and adds significantly to the computational burden. A further empirical input that is required in the
82
I . M . KENNEDY ET AL.
model is the rate of soot volume increase due to particle nucleation, Wn. Values of this term were obtained from estimates of the particle inception rate (from 1018 to 1022 m 3 s 1) and particle diameters of about 1 nm. The predictions were insensitive to this term; variations in wn of four orders o f magnitude resulted in less than 0.1% change in the predicted soot volume fractions. This behavior results from the assumption of a constant particle number density. The effect of this assumption can be seen by considering the behavior of the soot aerosol along a streamline in the absence of molecular diffusion. Following the aerosol along a streamline without thermophoresis, diffusion, or oxidation and considering the particle inception to occur in a narrow region that creates an initial volume fraction prior to surface growth, Eq. 3 shows that (6)
dd) = C , 2 / 3 . tit
The term C incorporates the number density, N, and specific surface growth rate. In general, C would be a function of time but for our present purposes we may consider the specific surface growth rate to be fixed. With N as a constant, the solution to Eq. (6) is ¢ = [ 1 C ( t  to) +
001/313,
(7)
where ~0 is the initial volume fraction at time to. When either the constant C or the elapsed time are sufficiently large the ultimate volume fraction is not sensitive to the initial volume fraction. Variations in the rate wn cannot change the number density in this model but can change only the effective initial particle diameter. With rapid surface growth, memory of the initial particle size is soon lost and, as a result, the predictions of soot volume fraction are not at all sensitive to assumptions regarding the rate wn. The behavior of Eq. 7 in the limit of zero initial soot volume fraction is worth examination. At first inspection it appears that this equation predicts soot to form from surface growth when $0 is zero. However, when ~b0 is zero the number density N must also be zero. Consequently, the factor C in Eq. 7 becomes zero in this limit and the correct behavior of the model is preserved.
A more complete description of the aerosol dynamics has been obtained by solving the equation for particle number density with finite rates of particle inception and coagulation. In this case a more physically reasonable response to a variation in the particle inception rate is obtained. That is, a change in the particle inception rate from 10 is to 1022 m 3 s 1 caused the peak soot volume fraction at 40 mm from the nozzle to change from 1.0 × 10 5 to 2.9 x 10 4. Such a large variation in the inception rate may not be expected in a flame. For example, in a flame with substantial radiative heat loss the drop in temperature from near the nozzle to well downstream is of the order of 400K. With a typical activation energy of about 40 kcal/mol this would give rise to an order of magnitude decrease in the particle inception rate. The response to the rate of particle nucleation that has been predicted in this flame is in agreement with the earlier work of Kennedy [24], where soot formation in a laminar stagnation point diffusion flame was modeled. The flow field, the numerical method, and the modeling of surface growth reactions and coagulation were quite different from the present formulation. In Kennedy's model a reduction in the particle inception rate by an order of magnitude gave rise to a drop in the final soot volume fraction by a factor of 2; this occurred at the lower soot loadings which he considered. It is illuminating to consider the response to the particle inception rate at different soot loadings. As the amount of soot formed in the flame increases the effect of particle inception rates is diminished, as can be seen Fig. 5 of Ref. 24. A simple extension to the mathematical model of Eq. 7 can serve to demonstrate an important point in this regard. If the assumption of a constant average number density is relaxed, Eq. 3 can give the rate of change of soot volume fraction as dO _ ~.l/362/3 kN(t)l/3 dp(t)2/3. dt
(8)
The very minor contribution of particle inception to the soot loading of the flame has been neglected. There are two unknowns in this equation, 4~and N. A second equation for N is given by the Smoluchowski equation of particle coagulation, which
SOOT F O R M A T I O N M O D E L
83 TABLE 1
The Effect of Initial Number Density on Final Soot Volume Fraction
Specific Surface G r o w t h Rate
k=4
Initial N u m b e r Density ( m  3 ) 10 TM 1019 1018 1019 1018 10t9
X 10 `4 (kg m 2 s  l )
k = 1 X 10 3 (kg m 2 s 1) k=4
x 10 `3 ( k g m  2 s
i)
will suffice for our present purposes although it neglects the development of a size distribution. The second equation is then
dN _/3N(t)2, dt
(9)
where/3 is the collision rate term. These two equations may be solved straightforwardly to yield an expression for the soot volume fraction at any instant. There is, o f course, no limit on the ultimate soot volume fraction that is predicted; such a limit has been observed in premixed flames and is believed to result from a loss of particle reactivity. If we assume that the freshly formed particle volume at inception is fixed and is given by Vp, then the solution to Eqs. 8 and 9 may be expressed in terms of the initial particle number density No as ~1/3
: '/"IV•/3NO/3 b
7rl/362/3 k ~N02/3
x [(NoBt + 1)2/3  1].
(10)
This equation may be shown to have the correct limiting behavior as No goes to zero, i.e., q~ becomes zero. It should be noted that Dasch [25] developed a similar formulation but he also included the effect of particle reactivity. Equation 10 can be used to examine the development of the soot loading in a flame under different conditions of surface growth rates and particle inception rates. The specific surface growth rate in Eq. 10 has been assumed to be constant with time. The measurements of Harris and Weiner [26] in
Final Soot V o l u m e Fraction at 20 ms 1.4 3.1 1.4 2.2 7.0 8.5
X x x X X x
10 10 10 10 10 10
7 7 6 6 5
a premixed ethyleneoxygenargon flame over a range of equivalence ratios indicated a decay in the specific surface growth rate with time. For the sake of argument, however, we can choose the upper and lower limits of the specific surface growth rates that they measured at 5 ms from the reaction zone. The upper limit is about 10  4 g/cm 2 s  t and the lower limit is about 4 × 10 5 g cm 2 s  I . The soot volume fraction at 20 ms has been calculated using Eq. 10 with initial particle number densities of 10 TM m 3 and 1019 m 3. The collision parameter, /3, was taken to be 5 × 10 15 m 3 s  l , which is typical of flame conditions. The resuits of this calculation are presented in Table 1. Two factors are apparent after examination of the data. Firstly, the sensitivity to the initial conditions diminishes with increasing growth rate and soot loading. Secondly, there is a strong dependence of the soot volume fraction on the specific surface growth rate so that a factor of 2.5 in growth rate yields a variation in the predicted soot volume fraction of about 10; a similar variation in the soot volume fraction was measured by Harris and Weiner [26] for their two limiting conditions. The third row of data in Table 1 presents the results of a computation using a specific surface growth rate that is typical of a diffusion flame [7, 17], i.e., 4 × 10 3 g cm 2 s  l . Because the growth rate is a constant and the temperature gradients and their concomitant effect on rates are not accounted for, the predicted soot volume fractions are somewhat larger than those typically found in ethylene diffusion flames (of the order of 105). Nevertheless, one may observe that the sensitivity of the soot volume fraction to the initial particle
84 number density is diminished compared to the resuits that were presented in the first two rows and that were obtained with lower growth rates. As the surface growth rate and the ultimate soot loading increases, the sensitivity to initial conditions appears to diminish. The aerosol surface area is formed mostly by surface growth itself, and if it is intense enough the volume fraction is not particularly sensitive to the particle formation step. These conditions pertain to diffusion flames of sooty fuels. Further support for this observation may be found in a recent article by Megaridis and Dobbins [27], who formulated an analytical model for a soot aerosol. In Fig. 5 of their article they compare predicted volume fractions for different initial particle number densities. Again it is apparent from the modeling that the sensitivity to the initial conditions decreases with increasing soot loading. The choice of the particle inception rate in the calculation is arbitrary in the sense that there is little direct information on the rate; the inception rate is chosen within acceptable bounds to give reasonable agreement with the measurements of soot volume fraction and number density [13]. The results that were presented in Figs. 13 and 14 indicate that the qualitative nature of the predictions is unchanged by the inclusion of the number density equation (compare Figs. 9 and 13 and Figs. 10 and 14). Some difference in the volume fractions is apparent, but no attempt has been made in these calculations to adjust the particle inception rate to optimize agreement with the measurements. The exercise of including the additional particle number density equation has suggested that acceptable predictions of the soot volume fractions may be obtained by adopting the average number density approach and dispensing with the additional computing penalty of the number density equation. CONCLUSIONS A simple model of soot formation in a diffusion flame has been developed. The results show reasonable agreement with the available measurements for two flow conditions. The effect o f temperature on the kinetics of surface growth rates was found to be an important factor in determin
I . M . K E N N E D Y ET AL. ing the soot volume fractions. A crude correction to the surface growth rates for the temperature drop due to radiation losses improved the prediction of the soot concentrations. However, the need to account more precisely for the effect of radiation on temperature in sooting flames is indicated. The proposed modeling for the soot aerosol dynamics has the advantage that it is simple enough to be easily implemented in a conventional turbulent flame calculation and yet it incorporates most of the essential physics. The contribution o f I. M . K . to this w o r k was s u p p o r t e d b y an N S F A w a r d C B T  8 8 5 7 4 7 7 a n d a Unocal F o u n d a t i o n A w a r d . A t Sandia Laboratories the w o r k is s u p p o r t e d b y the U.S. D e p a r t m e n t o f Energy, Office o f Basic E n e r g y Sciences, Division o f C h e m i c a l Sciences.
REFERENCES 1. Santoro, R. J., and Semerjian, H. G., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 997. 2. Flower, W. L., and Bowman, C. T., Combust. Sci. Technol. 37:93 (1984). 3. Glassman, I., and Yaccarino, P., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, p. 1175. 4. Kent, J. H., Wagner, H. Gg., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1007. 5. Kent, J. H., Wagner, H. Gg., Combust. Sci. Technol. 41:245 (1984). 6. Vandsburger, U., Kennedy, I. M., and Glassman, I., Combust. Sci. Technol. 39:263 (1984). 7. Axelbaum, R. L., Flower, W. L., and Law, C. K., in Combust. Sci. Technol. (in press). 8. Gore, J. P., and Faeth, G. M., Twentyfirst Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1521. 9. Bilger, R. W., Prog. Ener. Combust. Sci. l:87 (1976). 10. Magnussen, B. F., and Hjertager, B. H., Sixteenth Symposium (lnternationaO on Combustion, The Combustion Institute, Pittsburgh, 1976, p. 719. 11. Magnussen, B. F., Hjertager, B. H., Olsen, J. G., and Bhaduri, D., Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1978, p. 1383. 12. Tesner, P. A., Snegiriova, T. D., and Knorre, V. G., Combust. Flame 17:253 (1971). 13. Moss, J. B., Stewart, C. D., and Syed, K. J., Pre
SOOT FORMATION MODEL sented at the Twentysecond Symposium (International) on Combustion, University of Washington, Seattle, 1988. 14. Talbot, L., Cheng, R. K., Schefer, R. W., and Willis, D. R., J. Fluid Mech. 101:737 (1980). 15. Chert, J.Y., Kollmann, W., and Dibble, R. W., Presented at the Eighteenth Annual Pittsburgh Conference, Pittsburgh, 1987. 16. Miller, J. A., Kee, R. J., Smooke, M. D., and Grcar, J. F., Paper WSS/CI 8410, presented at the Western States Section Meeting of the Combustion Institute, University of Colorado, Boulder, CO, April 23, 1984. 17. Vandsburger, U., Kennedy, I. M., and Glassman, I., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 18. 19.
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Neoh, K. G., Howard, J. B., and Sarofim, A. F., in Particulate Carbon Formation During Combustion (D. Siegla and G. Smith, Eds.), 1981, p. 26. Kennedy, I. M., Combust. Flame 68:1 (1986). Harris, S. J., and Kennedy, I. M., Combust. Sci. Technol. 59:443 (1988). Santoro, R. J., Yeh, T. T., Horvath, J. J., and Semerjian, H. G., Combust. Sci. Technol. 53:89 (1987). Kennedy, I. M., Twentieth Symposium (lnternationaO on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1095. Dasch, C. J., Combust. Flame 61:219 (1985). Harris, S. J., and Weiner, A. M., Combust. Sci. Technol. 32:267 (1983). Megaridis, C. M., and Dobbins, R. A., Combust. Sci. Technol. 63:153 (1989).
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Harris, S. J., and Weiner, A. M., Combust. Sci. Technol. 31:155 (1983). Nagle, J., and StricldandConstable, R. F., Proceedings o f the Fifth Conference on Carbon, Pergamon, London, 1962, p. 154.
Received 13 April 1989