- Email: [email protected]
Simulation of soot aggregates formed by benzene pyrolysis
Simulation of Soot Aggregates Formed by Benzene Pyrolysis SHINJI HAYASHI,* YUTAKA HISAEDA, YUUSUKE ASAKUMA, HIDEYUKI AOKI, and TAKATOSHI MIURA Department of Chemical Engineering, Tohoku University, Sendai, 9808579 Japan
HIROYUKI YANO
Nippon Steel Chemical Co., Ltd., Tokyo, 1048263 Japan
YASUHISA SAWA
Nippon Steel Chemical Carbon Co., Ltd., Tokyo, 1030025 Japan Experiments and numerical simulations were carried out to analyze the soot aggregation mechanism during benzene pyrolysis. Soot was formed by the pyrolysis of 1 mol% benzene (in 99 mol% nitrogen) in an alumina tube, which was kept at 1573K with variations of residence time (0.03 to 0.5 sec). A new cluster-cluster aggregation model called the Aggregate Mean free Path (AMP) model was developed. This model simulated the cluster-cluster aggregation between soot particles and aggregates using the particles or aggregates mean free paths. The projected images of simulated soot were compared with the electron micrographs of experimental soot with the same magnification. Simulated shape, peripheral fractal dimension and size distributions of aggregates were in good agreement with experimental data. © 1999 by The Combustion Institute
NOMENCLATURE
Greek Symbols
A A1, A 2, A 3
k l
C deq dp dpa D Dperi f k l m n Na P T u ¯
projected area of aggregate constants of Cunningham’s slip factor constant aggregate volume-equivalent diameter particle diameter average particle diameter diffusion constant peripheral fractal dimension Corrected Stokes friction coefficient Boltzmann constant mean free path of gas mass of aggregate number of particles per aggregate average number of particles per aggregate perimeter of projected aggregate temperature mean thermal velocity of particle or aggregate
*Corresponding author. E-mail: [email protected] COMBUSTION AND FLAME 117:851– 860 (1999) © 1999 by The Combustion Institute Published by Elsevier Science Inc.
m n r
dynamic shape factor particle or aggregate mean free path viscosity of gas kinematic viscosity of gas soot density
INTRODUCTION Carbon black is produced as soot by the incomplete combustion of heavy oils and is mainly utilized in manufacturing tires of automobiles. The aggregate shape of carbon black is essential to the development of high-performance tires because the rolling resistance of tires is influenced by an energy loss generated at the interface between rubber and carbon black. Since the aggregation mechanism of carbon black has not been clarified, basic studies of the aggregation mechanism are now in demand. Lahaye [1, 2] and Tesner [3, 4] have conducted basic studies using tube furnaces. Both researchers focused on the mechanism of formation of primary particles and not on the formation of aggregates. Soot formation mechanism during combustion has been intensively studied by reaction 0010-2180/99/$–see front matter PII S0010-2180(98)00124-2
852 kinetics such as the PAH (Polycyclic Aromatic Hydrocarbon) model, which is the most prevailing procedure [5, 6]. An experimental method called thermophoretic sampling has been developed and enabled the direct sampling of soot from flame [7–10]. Megaridis et al. [8] suggested cluster-cluster aggregation mechanism by the experimental data of thermophoretic sampling. The simulation method of cluster-cluster aggregation has been studied since the 1980’s [11–13] and has been applied to the simulation of aqueous gold colloids [14, 15]. Simulations of the shape of soot aggregates have been carried out by the DLA (Diffusion Limited Aggregation) model [16] and clustercluster aggregation model using Langevin equations [17, 18]. The DLA model calculates the aggregation between an immovable center particle and other particles diffusing from distant points. The shape of the simulated aggregate by the three dimensional calculation was not in good agreement with the actual soot [16]. Samson et al. [17] simulated the Brownian motion of soot particles in an acetylene flame. They adopted the simulation model of clustercluster aggregation based on solutions of the Langevin equations. Although the shapes of the simulated aggregates were in good agreement with the experimental data qualitatively, the fractal dimension of the simulated soot was 1.9 while actual soot had the fractal dimension of 1.5–1.6. Ko ¨ylu ¨ et al. [18] also conducted simulation of cluster-cluster aggregation using Langevin equations. Their procedure included restriction of fractal dimension, which was assumed in advance in order to obtain shape similarity with actual soot. From these points of view, there has been no research study in which simulated soot has the same fractal dimension as the experimental value. This fact keeps the aggregation mechanism of soot still unclear. Samson et al. [17] suggested that there were two reasons why the fractal dimension of simulated soot was different from that of experimental data: 1) Partial oxidation reaction in the flame might affect the aggregate structure in the
S. HAYASHI ET AL. experiment. 2) The initial particle number density in the simulation was assumed to be higher than the actual condition. We considered two additional reasons, which might affect the results: 3) The temperature in the simulation was regarded as a constant at 1500K in their study, while the temperature in the flame was not considered constant. 4) Fractal dimension by the particle counting method [17] was applied for the comparison in their study. The correct particle number per aggregate of the actual soot was difficult to obtain because the particle number per aggregate is about 200 – 600 in the acetylene flame. The error of counting might have affected the results. Specific countermeasures taken in our study against the above limiting factors are as follows; 1) The soot formation experiment is conducted under an inert atmosphere in which no surface oxidation of the soot particles occurs. 2) Particle number density in the simulation is assumed to be the same as in the experiment. 3) The experiment is conducted in an electric furnace in which the temperature is kept constant. 4) Peripheral fractal dimension Dperi is selected as an index for comparison in order to avoid inaccurate particle number counting data. Dperi is obtained only by aggregates’ perimeter and area without particle number data. In this study, soot formation experiments are conducted by pyrolysis of benzene in an inert atmosphere and the shape parameters of the soot are measured. A new simulation model of cluster-cluster aggregation that uses particle and aggregate mean free path is developed. Simulated soot aggregates are compared with experimental soot aggregates in order to obtain information about the soot aggregation mechanism.
EXPERIMENTAL Benzene pyrolysis is carried out in the apparatus shown in Fig. 1. An alumina tube (I. D. f 4 mm, O. D. f 6 mm and length 1500 mm) was used as a reactor tube; 1 mol% of Benzene (purity .99.5%) diluted by nitrogen (purity .99.9999%) is supplied to the tube. The gas
BENZENE PYROLYSIS SOOT AGGREGATES
853 in good agreement with the experimental data. So we intend to develop a simple model where Brownian aggregation in the gas phase can be described. The model called AMP model uses a particle and an aggregate mean free path, which is obtained as follows. The diffusion coefficient (D) of the particle or the aggregate is expressed by Eq. 1 [23]. This is the Stokes-Einstein expression for the diffusion coefficient.
Fig. 1. Experimental apparatus of benzene pyrolysis.
D5 mixture is preheated at 973K by an infrared image furnace (length 400 mm) and then pyrolyzed at 1573K in an electric furnace (length 900 mm). The residence time is defined as the time requires for the gas mixture to traverse the region of uniform temperature (1573 6 10K). Six conditions of residence time are selected (0.03, 0.04, 0.05, 0.1, 0.2, and 0.5 s). A glass filter at the end of the tube collects the reaction products for 20 min. Electron micrographs of the soot are obtained by the transmission electron microscope (JEOL 100CX2). Magnification of 150,000 is applied to the measurement of the particle diameter and magnification of 85,000 is applied to the measurement of the aggregate shape. Diameters of 1000 particles are measured to obtain the average particle diameter of one sample. In this measurement, Spot light method using TGA 10 made by Carl Zeiss is applied [21]. The transmission electron micrographs are scanned and analyzed by the image analyzer (Luzex-F made by Nireco CO., Ltd.). One hundred aggregates are measured for one sample to obtain shape parameters. Maximum Feret’s diameter (L), width (W), area ( A), and perimeter (P) are measured according to ASTM D3849-87. L is defined as the maximum length in a fixed direction [22] and W is defined perpendicular to L. The average particle number per aggregate (N a) is counted using 100 aggregates for one sample. Simulation Method of AMP Model As mentioned in the introduction, simulation results using Langevin equations have not been
kT , f
(1)
where k is Boltzmann constant, T is the temperature, f is the corrected Stokes friction coefficient given by f5
3 pm d eq k, C
(2)
where m is the viscosity of gas, d eq is the equivalent volume diameter of an aggregate, C is the slip correction factor given by Eq. 3. In the case of a single particle, d p is applied instead of d eq. So the following expressions will be focused on an aggregate: C511
2l $A 1 A 2 exp~2A 3d eq/l !%, d eq 1
(3)
where A 1 , A 2 , and A 3 are constants, d eq is obtained by Eq. 4, and l is the mean free path of the gas molecule given by Eq. 5. d eq 5 n 1/3d p
(4)
where d p is the primary particle diameter of the aggregate and n is the particle number per aggregate. l5n
S D
1 p m ⁄2 , 2kT
(5)
where n is the kinematic viscosity of the gas and m is the mass of aggregate. k in Eq. 2 is the dynamic shape factor obtained by Sto ¨ber [24] and expressed by Eq. 6 in the case of the chain-like aggregate whose particle number per aggregate is n. 1 k 5 0.86n ⁄3
In the case of a single particle, k 5 1.
(6)
854
S. HAYASHI ET AL. TABLE 1
Relation Between Particle Number per Aggregate and Aggregate Mean Free Path (l) Particle number per aggregate [part]
l [nm]
1 2 3 4 5 10 20 30
99.7 82.4 67.5 58.6 52.5 37.4 26.7 22.0
The aggregate diffusion coefficient D is obtained by Eqs. 1– 6. The aggregate mean free path l is obtained from
l5
8D , p u#
(7)
where the mean thermal velocity is expressed by Eq. 8 according to the Einstein’s Brownian motion theory [25]: u# 5
Î
8kT . pm
(8)
m is obtained by m 5 nr
p d 3p , 6
(9)
where r is the density of the soot (r 5 1.83) [26]. The aggregate mean free path l is obtained by Eq. 7 incorporating Eqs. 8 and 9. By utilizing Eqs. 1–9, the following information is obtained; The mean free path of nitrogen molecule l is 496 nm at 1573K. The mean free path of a single soot particle whose diameter is 50 nm is 99.7 nm. The Knudsen number (K n 5 2l/d p) is 19.8 and this value means that the particle exists in the condition of the free molecular regime (Kn .10) at 1573K. The relation between particle number per aggregate and l is shown in Table 1. l decreases as the particle number per aggregate increases. In the simulation of diffusion-limited clustercluster aggregation, the clusters are usually assumed to undergo random walks on the lattice according to selected random numbers [27].
Fig. 2. Flow chart of AMP model.
This simulation method has been applied to the cases such as aggregation of gold colloid in aqueous solution in which the mobility of an aggregate is small enough to the particle diameter [14, 15]. This method can not be applied to our study because the mobility of an aggregate is very large and the mean free path of an aggregate is sometimes longer than the particle diameter. A new simulation model is developed for the large mobility of soot particles in high temperature gas in which particles and aggregates undergo random walks according to the length of the mean free path obtained by Eq. 7. The flow chart of simulation is shown in Fig. 2. In this calculation, sticking coefficient is regarded as 1. The Image Analysis Method of Aggregates In order to evaluate complexity of the shape of aggregates, the peripheral fractal dimension (Dperi) proposed by Gerspacher and O’Farrell [28] is applied. Dperi is expressed by P , A Dperi / 2,
(10)
BENZENE PYROLYSIS SOOT AGGREGATES
855
where P is the perimeter of an aggregate and A is the projected area of the aggregate. The logarithmic expression of Eq. 10 is 2 log10P 5 D peri log10A 1 C.
(11)
D peri is obtained as a slope when P and A are plotted in the logarithmic form shown in Eq. 11 [29]. A perfect circle would give a value of D peri 5 1. D peri becomes larger than one if the shape is fractal. The reasons why D peri is selected as an index for evaluation are as follows; 1) The fractal dimension of the particle counting method adopted by Samson et al. [17] needs data of the particle number per aggregate. They counted the particle number per aggregate in the electron micrographs and estimated that undercounting of 20% had occurred near the center of the aggregates. In the case of D peri, data of the particle number per aggregate that contain errors are not necessary. 2) In the case of D peri, the same procedure can be applied to both simulated soot and experimental soot. A projected image of simulated soot and an electron micrograph of experimental soot can be measured by the same image analyzer and the obtained data of perimeter (P) and area ( A) provide D peri. So this method is considered to be better for comparison between simulated soot and experimental soot than the particle counting method. 3) Gerspacher and O’Farrell [28] and Herd et al. [29] showed that commercial carbon blacks have various values of D peri. These facts indicate that D peri is effective as an index to evaluate the fractal property of soot. The three dimensional simulated results of cluster-cluster aggregation are projected to three planes ( xy, yz, and zx) in the magnification of 85,000 which is the same scale as an electron micrograph of experimental soot aggregates. Area ( A) and perimeter (P) are measured by the same procedure as an electron micrograph and the value of D peri is obtained. RESULTS AND DISCUSSION
Fig. 3. Effects of residence time on average particle diameter (d pa) and particle number per aggregate (N a).
Figure 3 shows the relation between residence time, average particle diameter (dpa) and average particle number per aggregate (Na). As the residence time increases, dpa and Na increase. In the case of residence time 0.5 sec, soot formation experiments are repeated three times and the obtained values of d pa by 1,000 particles are 48.7, 50.9, and 45.3. The average value of three experiments is 48.3 nm. The uncertainty of d pa is considered to be few nm. Na is counted by the average of 100 aggregates for one sample. We counted two times per one sample and found the differences are less than two particles. This repeatability owes to the fact that N a is less than 40 and easy to be counted. Samson et al. [17] reported that N a is about 200 – 600 in acetylene flame. N a of the soot formed in benzene pyrolysis is very small compared with soot formed in the acetylene flame. Figure 4 shows the relation between residence time, maximum Feret’s diameter (L), width (W), and area ( A). As the residence time increases, L, W, and A increase. The relation between residence time and D peri is shown in Fig. 5. The value of D peri is between 1.4 and 1.6 and is constant regardless of residence time. A discussion about D peri will be held later by comparing the experimental data with simulated ones.
Data Analysis of Soot Formed by Benzene Pyrolysis
Comparison of Simulated Soot and Experimental Soot
In this section, the experimental results of soot formation by benzene pyrolysis are discussed.
Calculation of cluster-cluster aggregation is conducted in order to simulate the aggregates
856
S. HAYASHI ET AL.
Fig. 4. Effects of residence time on measured aggregate size.
obtained by the experiment of residence time 0.5 s. The sample of residence time 0.5 s is selected because the sample has the largest diameter and the data of the particle number per aggregate is easy to obtain and considered to be most reliable. A simulation condition in which the particle diameter of 50 nm and Na is 35.7 particles is selected corresponding to the experimental data of the average particle diameter 48.3 nm and Na is 37.6 particles. Na’s value of 35.7 is obtained assuming that 250 particles become seven aggregates (250/7 5 35.7). The simulation is conducted in a cube whose side length is 4000 nm. In this case, particle number density is about 83 times higher than the experimental condition of benzene 1 mol%. The three-dimensional image Fig. 6. Comparison between initial condition and final results.
Fig. 5. Effects of residence time on peripheral fractal dimension (D peri).
of the initial condition and the final results are shown in Fig. 6. The initial condition shows the random situation of 250 particles and the final results show the situation of seven aggregates. Figure 7 shows the projected image of simulated aggregates to the xy plane. Electron micrographs similar to the simulated soot are selected among the 300 aggregates of the experimental soot. They are shown in Fig. 8. It is found that both projected shapes are very similar and the AMP model can be a reliable
BENZENE PYROLYSIS SOOT AGGREGATES
857
Fig. 9. Relation between projected area and perimeter of simulated soot.
Fig. 7. Projected image of seven simulated aggregates.
method to simulate the shape of soot aggregates. Shapes, sizes, and positions of simulated soot are considered to change depending on the random number generating system in computers. The initial value of the random number generating system is changed to check this effect. Similar aggregates are obtained in the calculation domain.
Fig. 8. Electron micrograph of soot aggregates obtained by benzene pyrolysis (residence time: 0.5 sec).
The AMP model is also applied to the other cases of different residence time and it is found that the shapes of the simulated soot are very similar to those of the experimental soot. Comparison of Peripheral Fractal Dimension and Aggregate Area As mentioned in the introduction, simulation using the actual particle number density is necessary in order to obtain the correct fractal dimension. So a simulation using 5000 particles is conducted in the condition of the actual particle number density assuming that 100% of the carbon content in 1 mol% benzene becomes soot particles. The calculation is conducted on the condition that 5000 particles of diameters 50 nm become 133 aggregates. In this case, Na’s value is 37.6 (5000/ 133 5 37.6) which is equal to the experimental value of 37.6 at the residence time 0.5 sec and the calculation range is a cube whose side length is 4.74 3 104 nm. The total number of projected images to the three planes ( xy, yz and zx) is 133 3 3 5 399 parts and the areas and peripheral lengths of the images are measured by the image analyzer. The plot of Eq. 11 to obtain D peri is shown in Fig. 9. The obtained D peri is 1.47 and it is in good agreement with the experimental data of 1.52. Dperi is also obtained from images which are projected on each plane. Dperi obtained from xy, yz, and zx plane is 1.49, 1.45, and 1.47,
858
S. HAYASHI ET AL.
Fig. 10. Comparison of aggregate area between simulated aggregates and obtained soot aggregates by benzene pyrolysis.
respectively. This fact indicates that random aggregation happens isotropically in the simulation. The effect of particle number on D peri is examined by comparing the results of 2000 particles with the results of 5000 particles. In the case of 2000 particles, particles are assumed to become 53 aggregates. In this case, Na’s value is 37.7 (2000/53 5 37.7) which is almost equal to the experimental value of 37.6 and the calculation range is a cube whose side length is 3.49 3 104 nm. Dperi obtained from each plane (xy, yz, and zx) is 1.44, 1.57, and 1.42. The average value is 1.47 and it is the same as the obtained value using 5000 particles, while the accuracy of Dperi obtained from each plane becomes worse than the obtained value using 5000 particles (1.47 6 0.02). Considering the accuracy of Dperi, the particle number of 5000 is necessary for the calculation of Dperi. The area distribution of aggregates obtained by the calculation of 5000 particles is shown in Fig. 10, compared with the experimental data. As the distributions are considered to be lognormal type, the comparison data in logarithmic form is shown in Table 2. It is found that the TABLE 2 The Comparison of the Area Distribution Data in Logarithmic Form
Average Standard deviation
Simulated
Measured
10.03 0.63
9.77 0.78
area distribution of simulated soot is also in good agreement with that of the experimental soot. All of the above-mentioned results show that the AMP model describing the Brownian motion provides simulated soot which has similar shape, similar fractal dimension and similar size distribution to the experimental soot. This fact means that the soot aggregation in an inert atmosphere can be described as Brownian aggregation mechanism. In the calculation of the Langevin model, Brownian particle motions are described in detail compared with the AMP model [19]. Four reasons are considered in the introduction as to why the fractal dimension of the Langevin model is not in good agreement with the experimental data. Particle number may be another reason why the AMP model gives similar fractal dimension to the experimental data, while the Langevin model gives worse results. In the calculation of the Langevin model, stochastic force is assumed [19]. More than 5000 particles are considered to be necessary for the Langevin model to obtain mean properties of aggregates, because the AMP model without stochastic term needs 5000 particles. If the particle number is small in the Langevin model, the number of final aggregates becomes small and stochastic term may disturb obtaining mean properties of aggregates. However, the calculations using only 500 [20] or 4000 [30] particles have been reported for the Langevin model because the calculations of stochastic force need excessive calculation time with present computers. A study of the Langevin model using many particles is expected. As mentioned above, the reasons become clear why the former studies using the Langevin model in which particle motions are strictly calculated have not presented good agreement with experimental data. The Relation Between Average Particle Number per Aggregate (Na) and Peripheral Fractal Dimension (Dperi) N a increases as the residence time increases in the experimental data in Fig. 3. The relation between N a and D peri is examined in the simu-
BENZENE PYROLYSIS SOOT AGGREGATES
859 ment with experimental data. It has been also shown that peripheral fractal dimension is constant regardless of particle number per aggregate at 1573K if the residence time is between 0.03 and 0.5 s.
REFERENCES 1.
2. 3. Fig. 11. Relation between N a and D peri.
4. 5.
lation and compared with the experimental data as shown in Fig. 11. It is found that the simulation data of Dperi is independent of Na except in the case of Na 5 1. The experimental data of Dperi is also constant regardless of Na. In this study, experiments are conducted by changing the residence time from 0.03 to 0.5 s while keeping the temperature at 1573K. It can be concluded that Dperi is constant if the temperature is kept constant and the atmosphere is kept inert. In order to obtain different value of Dperi, it will be necessary to change the temperature or the atmospheric condition. If the temperature changes, l is changed and this effect may change Dperi. If oxygen or carbon dioxide is applied as an atmosphere, the sticking coefficient of particles may become less than one owing to the surface oxidation reaction of soot and Dperi may be also changed.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
CONCLUSIONS In order to inspect the aggregation mechanism of soot, soot formation experiments by benzene pyrolysis were conducted and the image analysis data of the soot were compared with those of simulated soot. A new simulation method of the AMP model based on particle and aggregate mean free path has been developed. Simulated shape, peripheral fractal dimension, and size distributions of aggregates are in good agree-
20.
21.
22.
23.
24.
Prado, G., and Lahaye, J., in Particulate Carbon (D. C. Siegla and G. W. Smith, Eds.) Plenum Pub., New York, 1981, p. 143. Lahaye, J., Prado, G. and Donnet, J. B., Carbon 12:27 (1974). Tesner, P. A. and Shurupov, S. V., Combust. Sci. Tech. 92:61 (1993). Tesner, P. A. and Shurupov, S. V., Combust. Sci. Tech. 105:147 (1995). Frenklach, M. and Wang, H., in Soot Formation in Combustion (H. Bockhorn, Ed.) Springer-Verlag, New York, 1994, p. 165. Wang, H. and Frenklach, M., Combust. Flame 110:173 (1997). Dobbins, R. A., and Megaridis, C. M., Langmuir 3:254 (1987). Megaridis, C. M., and Dobbins, R. A., Combust. Sci. Technol. 71:95 (1990). ¨. O ¨ ., and Faeth, G. M., J. Heat Transfer Ko ¨ylu ¨, U 116:971 (1994). Fang, T. C., Megaridis, C. M., Sowa, W. A., and Samuelsen, G. S., Combust. Flame 112:312 (1998). Meakin, P., J. Colloid. Int. Sci. 102:505 (1984). Meakin, P., Phys. Rev. B, 29:2930 (1984). Meakin, P., Phys. Rev. A, 29:997 (1984). Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung., J. Phys. Rev. Lett. 54:1416 (1985). Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung., J. Phys. Rev. Lett. 57:2037 (1986). Richter, R., Sander, L. M., and Cheng, Z., J. Colloid. Int. Sci. 100:203 (1984). Samson, R. J., Mulholland, G. W. and Gentry, J. W., Langmuir 3:272 (1987). ¨. O ¨ ., Faeth, G. M., Farias, T. L., and CarKo ¨ylu ¨, U valho, M. G., Combust. Flame 100:621 (1995). Mountain, R. D., Mulholland, G. W., and Baum, H., J. Colloid. Int. Sci. 114:67 (1986). Mountain, R. D. and Mulholland, G. W., in Kinetics of Aggregation and Gelation (F. Family and D. P. Landau, Eds.) Elsevier, Amsterdam, 1984, p. 83. Donnet, J. B., Bansal, R. C. and Wang, M. J., in Carbon Black, Science and Technology (Second Edition) Marcel Dekker, New York, 1993, p. 108. Perry, R. H. and Chilton, C. H., in Chemical Engineer’s Handbook (Fifth Edition) McGraw Hill, London, 1973, 8-4. Friedlander, S. K., Smoke Dust and Haze: Fundamentals of Aerosol Behavior, Wiley, New York, 1977, p. 24. Sto ¨ber, W., in Fine Particles (B. Y. H. Liu, Ed.), Academic Press, New York, 1976, p. 352.
860 Einstein, A., Ann D Physik 17:549 (1905). Donnet, J. B., and Voet, A., Carbon Black, Marcel Dekker, New York (1976), p. 103. 27. Vicsek, T. Fractal Growth Phenomena, World Scientific Publishing, Singapore, 1989, p. 209. 28. Gerspacher, M., and O’Farell, C. P.: Elastomerics 123:35 (1991). 25. 26.
S. HAYASHI ET AL. 29. 30.
Herd, C. R., McDonald, G. C. and Hess, W. M., Rubber Chem. Tech. 65:107 (1992). Mountain, R. D. and Mulholland, G. W., Langmuir 4:1321–1326 (1988).
Received 20 March 1998; revised 5 August 1998; accepted 20 August 1998
HIROYUKI YANO
Nippon Steel Chemical Co., Ltd., Tokyo, 1048263 Japan
YASUHISA SAWA
Nippon Steel Chemical Carbon Co., Ltd., Tokyo, 1030025 Japan Experiments and numerical simulations were carried out to analyze the soot aggregation mechanism during benzene pyrolysis. Soot was formed by the pyrolysis of 1 mol% benzene (in 99 mol% nitrogen) in an alumina tube, which was kept at 1573K with variations of residence time (0.03 to 0.5 sec). A new cluster-cluster aggregation model called the Aggregate Mean free Path (AMP) model was developed. This model simulated the cluster-cluster aggregation between soot particles and aggregates using the particles or aggregates mean free paths. The projected images of simulated soot were compared with the electron micrographs of experimental soot with the same magnification. Simulated shape, peripheral fractal dimension and size distributions of aggregates were in good agreement with experimental data. © 1999 by The Combustion Institute
NOMENCLATURE
Greek Symbols
A A1, A 2, A 3
k l
C deq dp dpa D Dperi f k l m n Na P T u ¯
projected area of aggregate constants of Cunningham’s slip factor constant aggregate volume-equivalent diameter particle diameter average particle diameter diffusion constant peripheral fractal dimension Corrected Stokes friction coefficient Boltzmann constant mean free path of gas mass of aggregate number of particles per aggregate average number of particles per aggregate perimeter of projected aggregate temperature mean thermal velocity of particle or aggregate
*Corresponding author. E-mail: [email protected] COMBUSTION AND FLAME 117:851– 860 (1999) © 1999 by The Combustion Institute Published by Elsevier Science Inc.
m n r
dynamic shape factor particle or aggregate mean free path viscosity of gas kinematic viscosity of gas soot density
INTRODUCTION Carbon black is produced as soot by the incomplete combustion of heavy oils and is mainly utilized in manufacturing tires of automobiles. The aggregate shape of carbon black is essential to the development of high-performance tires because the rolling resistance of tires is influenced by an energy loss generated at the interface between rubber and carbon black. Since the aggregation mechanism of carbon black has not been clarified, basic studies of the aggregation mechanism are now in demand. Lahaye [1, 2] and Tesner [3, 4] have conducted basic studies using tube furnaces. Both researchers focused on the mechanism of formation of primary particles and not on the formation of aggregates. Soot formation mechanism during combustion has been intensively studied by reaction 0010-2180/99/$–see front matter PII S0010-2180(98)00124-2
852 kinetics such as the PAH (Polycyclic Aromatic Hydrocarbon) model, which is the most prevailing procedure [5, 6]. An experimental method called thermophoretic sampling has been developed and enabled the direct sampling of soot from flame [7–10]. Megaridis et al. [8] suggested cluster-cluster aggregation mechanism by the experimental data of thermophoretic sampling. The simulation method of cluster-cluster aggregation has been studied since the 1980’s [11–13] and has been applied to the simulation of aqueous gold colloids [14, 15]. Simulations of the shape of soot aggregates have been carried out by the DLA (Diffusion Limited Aggregation) model [16] and clustercluster aggregation model using Langevin equations [17, 18]. The DLA model calculates the aggregation between an immovable center particle and other particles diffusing from distant points. The shape of the simulated aggregate by the three dimensional calculation was not in good agreement with the actual soot [16]. Samson et al. [17] simulated the Brownian motion of soot particles in an acetylene flame. They adopted the simulation model of clustercluster aggregation based on solutions of the Langevin equations. Although the shapes of the simulated aggregates were in good agreement with the experimental data qualitatively, the fractal dimension of the simulated soot was 1.9 while actual soot had the fractal dimension of 1.5–1.6. Ko ¨ylu ¨ et al. [18] also conducted simulation of cluster-cluster aggregation using Langevin equations. Their procedure included restriction of fractal dimension, which was assumed in advance in order to obtain shape similarity with actual soot. From these points of view, there has been no research study in which simulated soot has the same fractal dimension as the experimental value. This fact keeps the aggregation mechanism of soot still unclear. Samson et al. [17] suggested that there were two reasons why the fractal dimension of simulated soot was different from that of experimental data: 1) Partial oxidation reaction in the flame might affect the aggregate structure in the
S. HAYASHI ET AL. experiment. 2) The initial particle number density in the simulation was assumed to be higher than the actual condition. We considered two additional reasons, which might affect the results: 3) The temperature in the simulation was regarded as a constant at 1500K in their study, while the temperature in the flame was not considered constant. 4) Fractal dimension by the particle counting method [17] was applied for the comparison in their study. The correct particle number per aggregate of the actual soot was difficult to obtain because the particle number per aggregate is about 200 – 600 in the acetylene flame. The error of counting might have affected the results. Specific countermeasures taken in our study against the above limiting factors are as follows; 1) The soot formation experiment is conducted under an inert atmosphere in which no surface oxidation of the soot particles occurs. 2) Particle number density in the simulation is assumed to be the same as in the experiment. 3) The experiment is conducted in an electric furnace in which the temperature is kept constant. 4) Peripheral fractal dimension Dperi is selected as an index for comparison in order to avoid inaccurate particle number counting data. Dperi is obtained only by aggregates’ perimeter and area without particle number data. In this study, soot formation experiments are conducted by pyrolysis of benzene in an inert atmosphere and the shape parameters of the soot are measured. A new simulation model of cluster-cluster aggregation that uses particle and aggregate mean free path is developed. Simulated soot aggregates are compared with experimental soot aggregates in order to obtain information about the soot aggregation mechanism.
EXPERIMENTAL Benzene pyrolysis is carried out in the apparatus shown in Fig. 1. An alumina tube (I. D. f 4 mm, O. D. f 6 mm and length 1500 mm) was used as a reactor tube; 1 mol% of Benzene (purity .99.5%) diluted by nitrogen (purity .99.9999%) is supplied to the tube. The gas
BENZENE PYROLYSIS SOOT AGGREGATES
853 in good agreement with the experimental data. So we intend to develop a simple model where Brownian aggregation in the gas phase can be described. The model called AMP model uses a particle and an aggregate mean free path, which is obtained as follows. The diffusion coefficient (D) of the particle or the aggregate is expressed by Eq. 1 [23]. This is the Stokes-Einstein expression for the diffusion coefficient.
Fig. 1. Experimental apparatus of benzene pyrolysis.
D5 mixture is preheated at 973K by an infrared image furnace (length 400 mm) and then pyrolyzed at 1573K in an electric furnace (length 900 mm). The residence time is defined as the time requires for the gas mixture to traverse the region of uniform temperature (1573 6 10K). Six conditions of residence time are selected (0.03, 0.04, 0.05, 0.1, 0.2, and 0.5 s). A glass filter at the end of the tube collects the reaction products for 20 min. Electron micrographs of the soot are obtained by the transmission electron microscope (JEOL 100CX2). Magnification of 150,000 is applied to the measurement of the particle diameter and magnification of 85,000 is applied to the measurement of the aggregate shape. Diameters of 1000 particles are measured to obtain the average particle diameter of one sample. In this measurement, Spot light method using TGA 10 made by Carl Zeiss is applied [21]. The transmission electron micrographs are scanned and analyzed by the image analyzer (Luzex-F made by Nireco CO., Ltd.). One hundred aggregates are measured for one sample to obtain shape parameters. Maximum Feret’s diameter (L), width (W), area ( A), and perimeter (P) are measured according to ASTM D3849-87. L is defined as the maximum length in a fixed direction [22] and W is defined perpendicular to L. The average particle number per aggregate (N a) is counted using 100 aggregates for one sample. Simulation Method of AMP Model As mentioned in the introduction, simulation results using Langevin equations have not been
kT , f
(1)
where k is Boltzmann constant, T is the temperature, f is the corrected Stokes friction coefficient given by f5
3 pm d eq k, C
(2)
where m is the viscosity of gas, d eq is the equivalent volume diameter of an aggregate, C is the slip correction factor given by Eq. 3. In the case of a single particle, d p is applied instead of d eq. So the following expressions will be focused on an aggregate: C511
2l $A 1 A 2 exp~2A 3d eq/l !%, d eq 1
(3)
where A 1 , A 2 , and A 3 are constants, d eq is obtained by Eq. 4, and l is the mean free path of the gas molecule given by Eq. 5. d eq 5 n 1/3d p
(4)
where d p is the primary particle diameter of the aggregate and n is the particle number per aggregate. l5n
S D
1 p m ⁄2 , 2kT
(5)
where n is the kinematic viscosity of the gas and m is the mass of aggregate. k in Eq. 2 is the dynamic shape factor obtained by Sto ¨ber [24] and expressed by Eq. 6 in the case of the chain-like aggregate whose particle number per aggregate is n. 1 k 5 0.86n ⁄3
In the case of a single particle, k 5 1.
(6)
854
S. HAYASHI ET AL. TABLE 1
Relation Between Particle Number per Aggregate and Aggregate Mean Free Path (l) Particle number per aggregate [part]
l [nm]
1 2 3 4 5 10 20 30
99.7 82.4 67.5 58.6 52.5 37.4 26.7 22.0
The aggregate diffusion coefficient D is obtained by Eqs. 1– 6. The aggregate mean free path l is obtained from
l5
8D , p u#
(7)
where the mean thermal velocity is expressed by Eq. 8 according to the Einstein’s Brownian motion theory [25]: u# 5
Î
8kT . pm
(8)
m is obtained by m 5 nr
p d 3p , 6
(9)
where r is the density of the soot (r 5 1.83) [26]. The aggregate mean free path l is obtained by Eq. 7 incorporating Eqs. 8 and 9. By utilizing Eqs. 1–9, the following information is obtained; The mean free path of nitrogen molecule l is 496 nm at 1573K. The mean free path of a single soot particle whose diameter is 50 nm is 99.7 nm. The Knudsen number (K n 5 2l/d p) is 19.8 and this value means that the particle exists in the condition of the free molecular regime (Kn .10) at 1573K. The relation between particle number per aggregate and l is shown in Table 1. l decreases as the particle number per aggregate increases. In the simulation of diffusion-limited clustercluster aggregation, the clusters are usually assumed to undergo random walks on the lattice according to selected random numbers [27].
Fig. 2. Flow chart of AMP model.
This simulation method has been applied to the cases such as aggregation of gold colloid in aqueous solution in which the mobility of an aggregate is small enough to the particle diameter [14, 15]. This method can not be applied to our study because the mobility of an aggregate is very large and the mean free path of an aggregate is sometimes longer than the particle diameter. A new simulation model is developed for the large mobility of soot particles in high temperature gas in which particles and aggregates undergo random walks according to the length of the mean free path obtained by Eq. 7. The flow chart of simulation is shown in Fig. 2. In this calculation, sticking coefficient is regarded as 1. The Image Analysis Method of Aggregates In order to evaluate complexity of the shape of aggregates, the peripheral fractal dimension (Dperi) proposed by Gerspacher and O’Farrell [28] is applied. Dperi is expressed by P , A Dperi / 2,
(10)
BENZENE PYROLYSIS SOOT AGGREGATES
855
where P is the perimeter of an aggregate and A is the projected area of the aggregate. The logarithmic expression of Eq. 10 is 2 log10P 5 D peri log10A 1 C.
(11)
D peri is obtained as a slope when P and A are plotted in the logarithmic form shown in Eq. 11 [29]. A perfect circle would give a value of D peri 5 1. D peri becomes larger than one if the shape is fractal. The reasons why D peri is selected as an index for evaluation are as follows; 1) The fractal dimension of the particle counting method adopted by Samson et al. [17] needs data of the particle number per aggregate. They counted the particle number per aggregate in the electron micrographs and estimated that undercounting of 20% had occurred near the center of the aggregates. In the case of D peri, data of the particle number per aggregate that contain errors are not necessary. 2) In the case of D peri, the same procedure can be applied to both simulated soot and experimental soot. A projected image of simulated soot and an electron micrograph of experimental soot can be measured by the same image analyzer and the obtained data of perimeter (P) and area ( A) provide D peri. So this method is considered to be better for comparison between simulated soot and experimental soot than the particle counting method. 3) Gerspacher and O’Farrell [28] and Herd et al. [29] showed that commercial carbon blacks have various values of D peri. These facts indicate that D peri is effective as an index to evaluate the fractal property of soot. The three dimensional simulated results of cluster-cluster aggregation are projected to three planes ( xy, yz, and zx) in the magnification of 85,000 which is the same scale as an electron micrograph of experimental soot aggregates. Area ( A) and perimeter (P) are measured by the same procedure as an electron micrograph and the value of D peri is obtained. RESULTS AND DISCUSSION
Fig. 3. Effects of residence time on average particle diameter (d pa) and particle number per aggregate (N a).
Figure 3 shows the relation between residence time, average particle diameter (dpa) and average particle number per aggregate (Na). As the residence time increases, dpa and Na increase. In the case of residence time 0.5 sec, soot formation experiments are repeated three times and the obtained values of d pa by 1,000 particles are 48.7, 50.9, and 45.3. The average value of three experiments is 48.3 nm. The uncertainty of d pa is considered to be few nm. Na is counted by the average of 100 aggregates for one sample. We counted two times per one sample and found the differences are less than two particles. This repeatability owes to the fact that N a is less than 40 and easy to be counted. Samson et al. [17] reported that N a is about 200 – 600 in acetylene flame. N a of the soot formed in benzene pyrolysis is very small compared with soot formed in the acetylene flame. Figure 4 shows the relation between residence time, maximum Feret’s diameter (L), width (W), and area ( A). As the residence time increases, L, W, and A increase. The relation between residence time and D peri is shown in Fig. 5. The value of D peri is between 1.4 and 1.6 and is constant regardless of residence time. A discussion about D peri will be held later by comparing the experimental data with simulated ones.
Data Analysis of Soot Formed by Benzene Pyrolysis
Comparison of Simulated Soot and Experimental Soot
In this section, the experimental results of soot formation by benzene pyrolysis are discussed.
Calculation of cluster-cluster aggregation is conducted in order to simulate the aggregates
856
S. HAYASHI ET AL.
Fig. 4. Effects of residence time on measured aggregate size.
obtained by the experiment of residence time 0.5 s. The sample of residence time 0.5 s is selected because the sample has the largest diameter and the data of the particle number per aggregate is easy to obtain and considered to be most reliable. A simulation condition in which the particle diameter of 50 nm and Na is 35.7 particles is selected corresponding to the experimental data of the average particle diameter 48.3 nm and Na is 37.6 particles. Na’s value of 35.7 is obtained assuming that 250 particles become seven aggregates (250/7 5 35.7). The simulation is conducted in a cube whose side length is 4000 nm. In this case, particle number density is about 83 times higher than the experimental condition of benzene 1 mol%. The three-dimensional image Fig. 6. Comparison between initial condition and final results.
Fig. 5. Effects of residence time on peripheral fractal dimension (D peri).
of the initial condition and the final results are shown in Fig. 6. The initial condition shows the random situation of 250 particles and the final results show the situation of seven aggregates. Figure 7 shows the projected image of simulated aggregates to the xy plane. Electron micrographs similar to the simulated soot are selected among the 300 aggregates of the experimental soot. They are shown in Fig. 8. It is found that both projected shapes are very similar and the AMP model can be a reliable
BENZENE PYROLYSIS SOOT AGGREGATES
857
Fig. 9. Relation between projected area and perimeter of simulated soot.
Fig. 7. Projected image of seven simulated aggregates.
method to simulate the shape of soot aggregates. Shapes, sizes, and positions of simulated soot are considered to change depending on the random number generating system in computers. The initial value of the random number generating system is changed to check this effect. Similar aggregates are obtained in the calculation domain.
Fig. 8. Electron micrograph of soot aggregates obtained by benzene pyrolysis (residence time: 0.5 sec).
The AMP model is also applied to the other cases of different residence time and it is found that the shapes of the simulated soot are very similar to those of the experimental soot. Comparison of Peripheral Fractal Dimension and Aggregate Area As mentioned in the introduction, simulation using the actual particle number density is necessary in order to obtain the correct fractal dimension. So a simulation using 5000 particles is conducted in the condition of the actual particle number density assuming that 100% of the carbon content in 1 mol% benzene becomes soot particles. The calculation is conducted on the condition that 5000 particles of diameters 50 nm become 133 aggregates. In this case, Na’s value is 37.6 (5000/ 133 5 37.6) which is equal to the experimental value of 37.6 at the residence time 0.5 sec and the calculation range is a cube whose side length is 4.74 3 104 nm. The total number of projected images to the three planes ( xy, yz and zx) is 133 3 3 5 399 parts and the areas and peripheral lengths of the images are measured by the image analyzer. The plot of Eq. 11 to obtain D peri is shown in Fig. 9. The obtained D peri is 1.47 and it is in good agreement with the experimental data of 1.52. Dperi is also obtained from images which are projected on each plane. Dperi obtained from xy, yz, and zx plane is 1.49, 1.45, and 1.47,
858
S. HAYASHI ET AL.
Fig. 10. Comparison of aggregate area between simulated aggregates and obtained soot aggregates by benzene pyrolysis.
respectively. This fact indicates that random aggregation happens isotropically in the simulation. The effect of particle number on D peri is examined by comparing the results of 2000 particles with the results of 5000 particles. In the case of 2000 particles, particles are assumed to become 53 aggregates. In this case, Na’s value is 37.7 (2000/53 5 37.7) which is almost equal to the experimental value of 37.6 and the calculation range is a cube whose side length is 3.49 3 104 nm. Dperi obtained from each plane (xy, yz, and zx) is 1.44, 1.57, and 1.42. The average value is 1.47 and it is the same as the obtained value using 5000 particles, while the accuracy of Dperi obtained from each plane becomes worse than the obtained value using 5000 particles (1.47 6 0.02). Considering the accuracy of Dperi, the particle number of 5000 is necessary for the calculation of Dperi. The area distribution of aggregates obtained by the calculation of 5000 particles is shown in Fig. 10, compared with the experimental data. As the distributions are considered to be lognormal type, the comparison data in logarithmic form is shown in Table 2. It is found that the TABLE 2 The Comparison of the Area Distribution Data in Logarithmic Form
Average Standard deviation
Simulated
Measured
10.03 0.63
9.77 0.78
area distribution of simulated soot is also in good agreement with that of the experimental soot. All of the above-mentioned results show that the AMP model describing the Brownian motion provides simulated soot which has similar shape, similar fractal dimension and similar size distribution to the experimental soot. This fact means that the soot aggregation in an inert atmosphere can be described as Brownian aggregation mechanism. In the calculation of the Langevin model, Brownian particle motions are described in detail compared with the AMP model [19]. Four reasons are considered in the introduction as to why the fractal dimension of the Langevin model is not in good agreement with the experimental data. Particle number may be another reason why the AMP model gives similar fractal dimension to the experimental data, while the Langevin model gives worse results. In the calculation of the Langevin model, stochastic force is assumed [19]. More than 5000 particles are considered to be necessary for the Langevin model to obtain mean properties of aggregates, because the AMP model without stochastic term needs 5000 particles. If the particle number is small in the Langevin model, the number of final aggregates becomes small and stochastic term may disturb obtaining mean properties of aggregates. However, the calculations using only 500 [20] or 4000 [30] particles have been reported for the Langevin model because the calculations of stochastic force need excessive calculation time with present computers. A study of the Langevin model using many particles is expected. As mentioned above, the reasons become clear why the former studies using the Langevin model in which particle motions are strictly calculated have not presented good agreement with experimental data. The Relation Between Average Particle Number per Aggregate (Na) and Peripheral Fractal Dimension (Dperi) N a increases as the residence time increases in the experimental data in Fig. 3. The relation between N a and D peri is examined in the simu-
BENZENE PYROLYSIS SOOT AGGREGATES
859 ment with experimental data. It has been also shown that peripheral fractal dimension is constant regardless of particle number per aggregate at 1573K if the residence time is between 0.03 and 0.5 s.
REFERENCES 1.
2. 3. Fig. 11. Relation between N a and D peri.
4. 5.
lation and compared with the experimental data as shown in Fig. 11. It is found that the simulation data of Dperi is independent of Na except in the case of Na 5 1. The experimental data of Dperi is also constant regardless of Na. In this study, experiments are conducted by changing the residence time from 0.03 to 0.5 s while keeping the temperature at 1573K. It can be concluded that Dperi is constant if the temperature is kept constant and the atmosphere is kept inert. In order to obtain different value of Dperi, it will be necessary to change the temperature or the atmospheric condition. If the temperature changes, l is changed and this effect may change Dperi. If oxygen or carbon dioxide is applied as an atmosphere, the sticking coefficient of particles may become less than one owing to the surface oxidation reaction of soot and Dperi may be also changed.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
CONCLUSIONS In order to inspect the aggregation mechanism of soot, soot formation experiments by benzene pyrolysis were conducted and the image analysis data of the soot were compared with those of simulated soot. A new simulation method of the AMP model based on particle and aggregate mean free path has been developed. Simulated shape, peripheral fractal dimension, and size distributions of aggregates are in good agree-
20.
21.
22.
23.
24.
Prado, G., and Lahaye, J., in Particulate Carbon (D. C. Siegla and G. W. Smith, Eds.) Plenum Pub., New York, 1981, p. 143. Lahaye, J., Prado, G. and Donnet, J. B., Carbon 12:27 (1974). Tesner, P. A. and Shurupov, S. V., Combust. Sci. Tech. 92:61 (1993). Tesner, P. A. and Shurupov, S. V., Combust. Sci. Tech. 105:147 (1995). Frenklach, M. and Wang, H., in Soot Formation in Combustion (H. Bockhorn, Ed.) Springer-Verlag, New York, 1994, p. 165. Wang, H. and Frenklach, M., Combust. Flame 110:173 (1997). Dobbins, R. A., and Megaridis, C. M., Langmuir 3:254 (1987). Megaridis, C. M., and Dobbins, R. A., Combust. Sci. Technol. 71:95 (1990). ¨. O ¨ ., and Faeth, G. M., J. Heat Transfer Ko ¨ylu ¨, U 116:971 (1994). Fang, T. C., Megaridis, C. M., Sowa, W. A., and Samuelsen, G. S., Combust. Flame 112:312 (1998). Meakin, P., J. Colloid. Int. Sci. 102:505 (1984). Meakin, P., Phys. Rev. B, 29:2930 (1984). Meakin, P., Phys. Rev. A, 29:997 (1984). Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung., J. Phys. Rev. Lett. 54:1416 (1985). Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung., J. Phys. Rev. Lett. 57:2037 (1986). Richter, R., Sander, L. M., and Cheng, Z., J. Colloid. Int. Sci. 100:203 (1984). Samson, R. J., Mulholland, G. W. and Gentry, J. W., Langmuir 3:272 (1987). ¨. O ¨ ., Faeth, G. M., Farias, T. L., and CarKo ¨ylu ¨, U valho, M. G., Combust. Flame 100:621 (1995). Mountain, R. D., Mulholland, G. W., and Baum, H., J. Colloid. Int. Sci. 114:67 (1986). Mountain, R. D. and Mulholland, G. W., in Kinetics of Aggregation and Gelation (F. Family and D. P. Landau, Eds.) Elsevier, Amsterdam, 1984, p. 83. Donnet, J. B., Bansal, R. C. and Wang, M. J., in Carbon Black, Science and Technology (Second Edition) Marcel Dekker, New York, 1993, p. 108. Perry, R. H. and Chilton, C. H., in Chemical Engineer’s Handbook (Fifth Edition) McGraw Hill, London, 1973, 8-4. Friedlander, S. K., Smoke Dust and Haze: Fundamentals of Aerosol Behavior, Wiley, New York, 1977, p. 24. Sto ¨ber, W., in Fine Particles (B. Y. H. Liu, Ed.), Academic Press, New York, 1976, p. 352.
860 Einstein, A., Ann D Physik 17:549 (1905). Donnet, J. B., and Voet, A., Carbon Black, Marcel Dekker, New York (1976), p. 103. 27. Vicsek, T. Fractal Growth Phenomena, World Scientific Publishing, Singapore, 1989, p. 209. 28. Gerspacher, M., and O’Farell, C. P.: Elastomerics 123:35 (1991). 25. 26.
S. HAYASHI ET AL. 29. 30.
Herd, C. R., McDonald, G. C. and Hess, W. M., Rubber Chem. Tech. 65:107 (1992). Mountain, R. D. and Mulholland, G. W., Langmuir 4:1321–1326 (1988).
Received 20 March 1998; revised 5 August 1998; accepted 20 August 1998