Detailed Kinetic Modeling of Carbonaceous Nanoparticle Inception

Energy & Fuels 2006, 20, 547-559

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Detailed Kinetic Modeling of Carbonaceous Nanoparticle Inception and Surface Growth during the Pyrolysis of C6H6 behind Shock Waves J. Z. Wen† and M. J. Thomson* Mechanical & Industrial Engineering, UniVersity of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8

M. F. Lightstone Department of Mechanical Engineering, McMaster UniVersity, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

S. N. Rogak Department of Mechanical Engineering, UniVersity of British Columbia, 2324 Main Mall, VancouVer, BC, Canada V6P 1Z4 ReceiVed March 28, 2005. ReVised Manuscript ReceiVed August 8, 2005

Soot formation in combustion processes is of significant interest due to its influences on both environmental emissions and material synthesis (i.e., the synthesis of fullerences and carbon nanotubes). However, the inception process of the youngest carbonaceous nanoparticles from the gaseous phase is the most poorly understood phenomenon in the study of soot kinetics at the current stage. Recently, researchers have found experimentally the existence of transparent or semi-transparent carbonaceous particles (Krestinin, A. V. Combust. Flame 2000, 121, 513-524) or nanoorganic carbon particles (D’Anna, A.; Rolando, A.; Allouis, C.; Minutolo, P.; D’Alessio, A. Proc. Combust. Inst. 2004, 30, 1449-1456) during soot nucleation, which have not been successfully explained by traditional polycyclic aromatic hydrocarbon (PAH) nucleation mechanisms. Most recently, a more detailed soot kinetic model (Vlasov, P. A.; Warnatz, J. Proc. Combust. Inst. 2002, 29, 2335-2341; Part 2) has been implemented to predict soot formation behind shock waves and to describe the soot nucleation as a combined process of the fast polymerization of supersaturated polyyne vapor and the PAH growth. The lack of a detailed description of fractal particle structures in their aerosol dynamics model, however, restricted the model’s accuracy in predicting the particle coagulation rates and, hence, the particle sizes. In the current study, a new comprehensive kinetic model has been developed to describe soot chemical processes in a heterogeneous phase. The nucleation process is described by the formation of the soot precursors and the transformation from those precursors to solid soot particles. The precursors are represented by six sectional bins, which are formed through the detailed PAH nucleation mechanism and polyyne pathways, respectively. The gaseous reaction mechanism has been validated against measurements of polyynes and the C/C2/C3 carbon radicals. Finally, the aforementioned soot kinetic model has been implemented in an advanced aerosol dynamics model to predict the main parameters of soot particle formation in the pyrolysis of C6H6/Ar mixture. This aerosol dynamics model includes the detailed description of the agglomerate structure of soot particles and calculates the particle coagulation rates according to their sizes and structures. The numerical simulation shows that, during the fuel pyrolysis behind shock waves, both PAH growth and polyynes polymerization play an important role during the soot nucleation process. And the polyynes surface growth model alone is able to predict soot yield as well as averaged particle diameter during the earlier stage of soot formation.

Introduction The formation mechanism of soot particles during the pyrolysis and combustion of hydrocarbons has been an interesting issue for more than one and half centuries.4 Numerous theoretical and experimental studies have been conducted, and * Corresponding author. Phone: 1-416-978-1827. Fax: 1-416-978-7753. E-mail: [email protected]. † Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139. (1) Krestinin, A. V. Combust. Flame 2000, 121, 513-524. (2) D’Anna, A.; Rolando, A.; Allouis, C.; Minutolo, P.; D’Alessio, A. Proc. Combust. Inst. 2004, 30, 1449-1456.

several comprehensive reviews are available.5-7 Currently, there is considerable agreement on the general features of the chemical pathways and physical processes of soot formation, although many important details remain poorly understood. On the basis of their extensive measurements, Homann and Wagner8 sug(3) Vlasov, P. A.; Warnatz, J. Proc. Combust. Inst. 2002, 29, 23352341; Part 2. (4) Porter, G. Proc. Combust. Inst. 1953, 4, 248-252. (5) Kennedy, I. M. Prog. Energy Combust. Sci. 1997, 23, 95-132. (6) Frenklach, M. Chem. Eng. Sci. 2002, 57, 2229-2239. (7) Richter, H.; Howard, J. B. Prog. Energy Combust. Sci. 2000, 26, 565-608.

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548 Energy & Fuels, Vol. 20, No. 2, 2006

gested that, for aliphatic and aromatic fuels, three major contributing groups are responsible for the formation of soot particles: acetylene and polyacetylenes (polyynes) (mass range from 26 to 146 amu), polycyclic aromatic hydrocarbons (PAH) (mass range from 78 to about 300 amu), and larger polycyclic hydrocarbons (mass range from about 150 to 550 amu or more). As a result, all of those precursors should be included in the formation mechanism of soot particles. Richter and Howard7 recently reviewed the chemical pathways of soot formation focusing on the following aspects: the formation of the first aromatic ring, further PAH growth process leading to soot particles, soot growth process attributed to acetylene and PAH, and finally, the synthesis of fullerenes and their relation to PAH and soot formation. While their studies emphasize the roles of large PAHs during the nucleation and mass growth of soot particles, the above overall model excludes the role of polyynes, which have been shown to be important soot intermediates in acetylene-like flames.9 Other researchers have conducted studies on the influence of fuel structure (of aliphatic and aromatic types) on reaction pathways leading to soot8-11 and have suggested that this influence is significant during the early stages of soot formation (where the nucleation process predominates), hence affecting induction time and the initial soot formation rate.11 The aforementioned studies show that, in addition to the lumping of PAH growth and dimerization, which have been both shown to be important in the nucleation process for aromatic fuels, the role of polyynes in soot formation, especially during the nucleation and earlier surface growth stage, requires further clarification in detailed soot modeling. On the basis of the soot formation mechanisms through continuous PAH growth or/and polyynes polymerization1 and by treating soot particles as spherical molecules (thus neglecting the agglomerate structure of soot particles and their effects on coagulation coefficients), we found that some detailed soot models12,13 directly couple soot chemistry into the gas-phase reaction mechanism. The coefficients of particle-particle collisions (coagulation) and of PAH-particle collisions (through the surface condensation process) are empirically assigned within the model. This treatment, however, can cause large discrepancies when the model is implemented for different fuels. Moreover, the nanostructures and the size distribution of soot particles remain undescribed in these soot models. Those properties, however, can be directly measured in experimental studies and are of great interest. In the present study, the following hypotheses are made to address the aforementioned questions on soot chemical and physical processes: (1) Two different kinds of soot precursors are formed during the fuel pyrolysis. One is formed through the continuous growth of larger PAHs, while another is formed through the fast polymerization process of polyynes. (2) Both precursors contribute to the formation of soot particles. A simplified one-step transformation will be used to link between precursors and soot particles. The role of carbonization14 can be included in this transformation. (8) Homann, K. H.; Wagner, H. G. Proc. Combust. Inst. 1967, 11, 371379. (9) Bonne, U.; Homann, K. H.; Wagner, H. G. Proc. Combust. Inst. 1965, 10, 503-512. (10) Lam, F. W.; Longwell, J. P.; Howard, J. B. Proc. Combust. Inst. 1990, 23, 1477-1484. (11) Frenklach, M.; Clary, D. W.; Gardiner, W. C.; Stein, S. E. Proc. Combust. Inst. 1986, 21, 1067-1076. (12) Richter, H.; Granata, S.; Green, W. H.; Howard, J. B. Proc. Combust. Inst. 2004, 30, 1397-1405. (13) Naydenova, I.; Nullmeier, M.; Warnatz, J.; Vlasov, P. A. Combust. Sci. Technol. 2004, 176, 1667-1703.

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(3) The fast polymerization process contributes to the earlier surface growth of soot particles as well. Because the concept of this process1 includes the fast breeding procedure of radical sites when polyynes attached to soot surface, the availability of surface reactive sites (for both hydrogen activated and polyyne attached) is expected to be larger than that specified in the standard HACA mechanism.15 In this article, we describe a new kinetic soot model that has been developed on the basis of published mechanisms and includes both the PAH continuous growth and polyyne polymerization mechanisms. It is proposed that the structure and size of soot precursors are different from those presented in previous studies.12,13 After being validated against a series of species measurements, this mechanism was implemented in a fixed sectional aerosol dynamics model16 to predict soot formation during the pyrolysis of benzene (C6H6) in a shock tube. The soot model used includes the agglomerate description of soot particles, with coagulation coefficients calculated for nonspherical agglomerates in all three possible Knudsen regimes. The time evolution of soot particle nucleation and surface growth will be investigated on the basis of the model’s prediction. Experimental Measurement The soot formation process during benzene pyrolysis was extensively studied by Roth et al.17,18 The experiments were conducted behind reflected shock waves in a conventional stainless steel diaphragm-type shock tube with an inner diameter of 80 mm. The driver section had a length of 2.50 m, and the driven section was 6.30 m in length. Evacuation was accomplished with an Edwards dry star vacuum pump, which enabled pumping of particle loaded gas mixture down to a pressure of 4 × 10-2 mbar. Test gas mixtures were prepared manometrically in a stainless steel mixing vessel. The experimental apparatus is described in the literature.18 Four mixtures of 0.25, 0.5, 1, and 2% C6H6 diluted in Ar (99.998%) were measured. The experiments were carried out in a temperature range of 1750-2600 K, at nearly constant pressure (0.1-0.13 MPa) behind a reflected shock wave. Soot formation properties were measured using the CW-laser extinction technique, laser induced incandescence (LII), and a transmission electron microscopy (TEM) grid. In the CW-laser extinction measurement, a conventional 20-mW HeNe laser (λ ) 632.8 nm) was used. The refractive index and density of soot particles were taken as m ) 1.90-1.0i and 1.86 g/cm2, respectively. Measurement uncertainties caused by the selection of different refractive indices have been reported in ref 17. This is important since the measured soot volume fraction can actually vary by a factor of 2 if different indices are used. A typical CW-laser signal measured from a mixture of 1% benzene in argon at a temperature of 2095 K and a pressure of 0.123 MPa showed that soot extinction began after around 80 µs and increased continuously until a final constant value was reached at about 1000 µs. These observations are in agreement with other experimental findings19,20 on the soot aging behavior, which in turn corresponds to sharply declined soot particle reactivity. The LII setup used consisted of a pulsed Spectron Nd:YAG laser, for which the light path is perpendicular to the HeNe laser beam and the receiver optics. This laser had a half-width pulse time of about 9 ns at the wavelength of 1064 nm. It was triggered by a pressure transducer (14) Vander Wal, R. L. Combust. Flame 1998, 112, 607-616. (15) Appel, J.; Bockhorn, H.; Frenklach, M. Combust. Flame 2000, 121, 122-136. (16) Park, S. H.; Rogak, S. N. J. Aerosol Sci. 2004, 35, 1385-1404. (17) Starke, R.; Roth, P. Combust. Flame 2002, 127, 2278-2285. (18) Starke, R.; Kock, B.; Roth, P. Shock WaVes 2003, 12, 351-360. (19) Harris, S. J.; Weiner, A. M. Combust. Sci. Technol. 1983, 32, 267275. (20) Frenklach, M.; Wang, H. Proc. Combust. Inst. 1990, 23, 15591566.

Detailed Kinetic Modeling of Nanoparticle Formation with a time delay generator, allowing LII measurements in time steps of 1 µs during the available time interval after the arrival of the reflected shock wave. The LII signals were used to calculate soot particle diameters. A detailed error calculation with individual uncertainties of 1.5% for the gas temperature and gas pressure and assumed uncertainties of 10% for all other values including signal fitting resulted in an estimated overall uncertainty for measured particle diameter between -36 and +56%.18 Gas-Phase Kinetics. Recently, a detailed kinetic model3,13 has been developed to predict soot formation behind shock waves. The model was able to describe soot nucleation as a combined process of fast polymerization of supersaturated polyyne vapor1,21 and PAH growth.7,22 Four gas-phase mechanisms are included in this kinetic model: a complete set of the PAH formation pathways for laminar premixed acetylene and ethylene flames23 with all the appropriate modifications presented in a later work,15 a reaction mechanism for acetylene pyrolysis,24,25 a formation mechanism for polyyne molecules,21,26-28 and a formation mechanism for small, pure carbon clusters.29-31 The overall proposed mechanism consists of approximately 1700 elementary reactions among 141 gaseous phase species. This large set of reaction mechanisms has not, however, been validated against species profiles of shock tube measurements, which probably results from the unavailability of the measured data. For the current study, a new comprehensive gas-phase kinetic model is developed to describe the formation of soot particles in a heterogeneous phase during fuel pyrolysis behind a shock wave. The pathways for acetylene pyrolysis and the formation of polyynes are adopted from the same literature and reduced as much as possible. The formation mechanism for small pure carbon clusters is based on a very recent research.32 To verify the combination model of PAH formation mechanism23 and polyyne pathway,33 the profiles of polyynes and carbon radicals are validated against species measurements of various shock tube experiments from the literature.24,25 The mechanism is first used to predict the molar concentration of carbon radicals (i.e., C, C2, and C3), which have been measured during the high-temperature pyrolysis of acetylene.25 Those carbon radicals play significant roles in the acetylene pyrolysis mechanism24,25 which links the polyyne formation pathways and the PAH kinetic model. To use the CHEMKIN Code34 to model the fuel pyrolysis under constant temperature and pressure, the thermodynamic data for all the participation species had to be found and converted to an appropriate, usable format. It is interesting to note that, in a previous work,25 the proposed thermodynamic data for large carbon radicals (especially C6 and C6H) were not in agreement with two commonly used thermodynamic data sets.34,35 Figures 1 and 2 compare the thermodynamic enthalpy and entropy for several (21) Krestinin, A. V. Chem. Phys. Rep. 1998, 17, 1441-1461. (22) Frenklach, M.; Clary, D. W.; Gardiner, W. C.; Stein, S. E. Proc. Combust. Inst. 1984, 20, 887-901. (23) Wang, H.; Frenklach, M. Combust. Flame 1997, 110, 173-221. (24) Kiefer, J. H.; Sidhu, S. S.; Kern, R. D.; Xie, K.; Chen, H.; Harding, L. B. Combust. Sci. Technol. 1992, 82, 101-130. (25) Kruse, T.; Roth, P. J. Phys. Chem. A 1997, 101, 2138-2146. (26) Krestinin, A. V. Khim. Fiz. 1994, 13, 121-131. (27) Krestinin, A. V.; Kislov, M. B.; Raevskii, A. V.; Kolesova, O. I.; Stesik, L. N. Kinet. Katal. 2000, 41, 90-98. (28) Zhil’tsova, I. V.; Zaslonko, I. S.; Karasevich, Y. K.; Wagner, H. G. Kinet. Katal. 2000, 41, 76-89. (29) Agafonov, G. L.; Nullmeier, M.; Vlasov, P. A.; Warnatz, J.; Zaslonko, I. S. Combust. Sci. Technol. 2002, 174, 185-213. (30) Sojka, J.; Warnatz, J.; Vlasov, P. A.; Zaslonko, I. S. Combust. Sci. Technol. 2000, 158, 439-460. (31) Wagner, H. G.; Vlasov, P. A.; Dorge, K. J.; Eremin, A. V.; Zaslonko, I. S.; Tanke, D. Kinet. Katal. 2001, 42, 583-593. (32) Wen, J. Z.; Thomson, M. J.; Lightstone, M. F. Combust. Theory Modell., in press. (33) Krestinin, A. V. Chem. Phys. Rep. 1994, 13, 191-210. (34) Kee, R. J.; Rupley, F. M.; Miller, J. A.; Coltrin, M. E.; Grcar, J. F.; Meeks, E.; Moffat, H. K.; Lutz, A. E.; Dixon-Lewis, G.; Smooke, M. D.; Warnatz, J.; Evans, G. H.; Larson, R. S.; Mitchell, R. E.; Petzold, L. R.; Reynolds, W. C.; Caracotsios, M.; Stewart, W. E.; Glarborg, P.; Wang, C.; Adigun, O.; Houf, W. G.; Chou, C. P.; Miller, S. F. CHEMKIN Collection; Reaction Design, Inc.: San Diego, CA, 2002.

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Figure 1. Comparison of thermodynamic enthalpy data for large carbon radicals. “_KR” data are taken from ref 25, “_Burcat” data are taken from the database in ref 35, and “_Chem” are taken from the database in ref 34.

Figure 2. Comparison of thermodynamic entropy data for large carbon radicals. “_KR” data are taken from ref 25, “_Burcat” data are taken from the database in ref 35, and “_Chem” data are taken from the database in ref 34.

carbon radicals. In a previous study,25 it has been found that the discrepancies of thermodynamic data could result in larger errors in the prediction of species concentration. To match the originally developed acetylene pyrolysis mechanism,25 which was adopted in the present mechanism, the thermodynamic data in this study were chosen to be as close as possible to the proposed data in the previous study (shown in bold italic within the aforementioned figures). A special case exists for C6 where the available polynomial data set35 cannot provide the desired enthalpy data25 at two specific temperatures: 300 and 3500 K. As such, the thermodynamic data used in the previous study25 may need more validations. This disagreement of enthalpy data of C6 will inevitably result in the discrepancy in predicting species concentrations in various validation cases in this study, as shown later. On the basis of the chosen thermodynamic data shown in Figures 1 and 2, we used the new mechanism developed in this study, as well as the previously published one,25 to calculate the species concentration for two C2H2/Ar mixtures that were measured in the literature.25 Note that the species predictions based on the previously published mechanism could be different from the originally published predictions,25 since a different thermodynamic dataset was used in the study. Predictions of molar concentration of carbon radicals for two mixtures are shown in Figures 3 and 4. These figures indicate that the new mechanism can provide the same species prediction for three carbon radicals (C, C2, and C3) just as the previous mechanism does, even though a large set of reactions has been added. In comparison with a previous numerical model, the prediction of C is significantly improved. This results from a more accurate kinetics description which was included in the PAH mechanism and developed for the formation of smaller hydrocarbons. The difference between measured and predicted data results from the inherent uncertainty within the thermodynamic data set, as previously discussed, and is beyond the scope of this study. (35) Burcat, A. Technion Aerospace Engineering Report, Third Millennium Ideal Gas and Condensed Phase Thermochemical Database for Combustion, 2001.


Figure 3. Predictions of carbon radicals in a mixture of 10 ppm C2H2 in Ar at 3450 K and 2.20 bar,25 where symbols are for measured data: 9, C; 4, C2; ], C3; light lines are for the numerical results taken from ref 25: solid, C; dashed, C2; dashed dot, C3; and heavy lines are for this study: solid, C; dashed, C2; dashed dot, C3.

Figure 4. Predictions of carbon radical C2 in a mixture of 20 ppm C2H2 in Ar at 3380 K and 2.20 bar,25 where symbols are for measured data; solid line is for the numerical result taken from ref 25 and dashed line is for this study.

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Figure 6. Species profiles of polyynes predicted for a mixture of 3% C4H2 and 7.5% H2 in Ne at 2067 K and 0.45 atm,24 where symbols are for measured data: 0, C2H2; 4, C4H2; ], C6H2; O, C8H2; lines for the numerical predictions in this study: solid, C2H2; dashed, C4H2; dashed dot, C6H2; dashed dot dot, C8H2.

C4H2 decomposes into C2H2 very quickly. At larger residence times, the concentration of individual polyynes can differ by a factor up to 10. Both Figures 5 and 6 demonstrate that the new mechanism is capable of predicting the formation of polyynes during the pyrolysis of acetylene and 1,3-butadiene. Formation Mechanism for Soot Precursors. The existence of transparent or semi-transparent soot precursors27 or nanoorganic carbon particles (NOC)2 has been experimentally observed during the early stages of soot formation. However, detailed descriptions of the structure and formation of these precursors are currently unavailable. Kronholm studied the molecular weight growth pathways of fuel-rich combustion and suggested that the distinction between what constitutes the largest PAH molecule and the smallest soot particle is arbitrary.36 In that study, Kronholm used the concept that PAH and soot can be treated analogously in a general formulation of molecular weight growth to develop a model of PAH-growth and soot nucleation. He did this by treating large PAHs similarly to soot aerosols, lumping them into averaged property bins with a molecular weight between 100 and 1600 amu. The same expression was used for the rate coefficients of reactions between PAH molecules, PAH radicals, PAH molecules and radicals, and between PAH and soot particles. These coefficients may be written as: k ) γ‚kfreq

(1)

where γ is collision efficiency that has been empirically optimized in the previous study.36 The quantity kfreq is the collision frequency between molecules:

(

kfreq ) NAv‚d2‚

Figure 5. Species profiles of polyynes predicted for a mixture of 3.2% C2H2/Ne at 2348 K and 0.45 atm,24 where symbols are for measured data: 0, C2H2; 4, C4H2; ], C6H2; light lines for the numerical results taken from ref 24: solid, C2H2; dashed, C4H2; dashed dot, C6H2; heavy lines for this study: solid, C2H2; dashed, C4H2; dashed dot, C6H2.

Next, the reaction mechanism was used to predict the molar concentration of polyynes (C2H2 to C8H2) which have been measured during the pyrolysis of C2H2 and C4H2 in another study.24 The same thermodynamic data set was used. Figures 5 and 6 show the predictions for two mixtures. During the pyrolysis of C2H2 in neon, the molar concentration of polyynes with large molecular weight is smaller than that of polyynes with small molecular weight. After a short induction time (0.0003 s in the figure), the concentration of those species remains unchanged with increasing residence time. For the C4H2/Ne mixture with H2 added, a large amount of

)

8π‚kBoltz‚T µ

1/2

(2)

where µ is the reduced mass, NAv is Avogadro’s number, T is temperature, d is the collision diameter, and kBoltz is Boltzmann’s constant. To distinguish large PAHs and soot particles, a specific molar weight value of 1600 amu was taken as the boundary. In this study, the aforementioned approach has been adopted. Soot precursors are represented by large hydrocarbon molecules that have molar weights ranging from about 200 amu to 1600 amu. In addition to soot precursors (BINs and BINJs) with structures similar to large PAH and PAH radicals, pyrolytic soot precursors (PINs) are included within the model to account for fast polymerization processes. The size of the largest soot precursors is taken to be the same as those used in previous studies.12,36 The previously mentioned BINs and PINs are represented by a given average (36) Kronholm, D. F. Molecular Weight Growth Pathways in Fuel-Rich Combustion. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 2000.

Detailed Kinetic Modeling of Nanoparticle Formation


Table 1. Description of Soot Precursors No.

mass (amu)

formula

σ (nm)

H/C

BIN1 BIN2 BIN3 PIN1 PIN2 PIN3

201-400 401-800 801-1600 195-390 391-778 779-1558

C24H12 C48H24 C96H48 C24H4 C48H8 C96H16

0.85 1.07 1.34 ∼0.85 ∼1.07 ∼1.34

0.5 0.5 0.5 0.167 0.167 0.167

number of carbon and hydrogen atoms and have the characteristics shown in Table 1. The formation and growth pathways of BINs are similar to the mechanism proposed by Richter et al.12 where eqs 1 and 2 were used to determine the reaction rates. These BINs are formed via reactions between smaller BINs, PAHs, PAH radicals (of which the number of aromatic rings is greater than two in this study), and HACA mechanism. A detailed description of this model can be found in the literature.12 Table 2 shows pathways contributing to the formation of sectional precursors through reactive collisions of large PAHs, where rate calculations refer to a previous study.36 The formation and growth pathways of PINs are similar to those published in another work13 based on polyyne polymerization kinetics.21,26 Table 2 also shows the formation mechanism of the first sectional PIN. Larger PINs (PIN2 and PIN3) are formed as a result of the coagulation of smaller PINs and their surface growth. Interactions between PINs and PAHs are not included due to the lack of experimental data. All soot precursors contribute to the creation of the earliest forming solid soot particles, as will be discussed below. Soot Nucleation. In contrast to the previous nucleation model,37 which uses the dimerization rate of specific PAH molecules to calculate the nucleation rate of soot particle, this model considered the possible transition processes between the gaseous phase and the particulate phase by using simplified one-step reactions that link soot precursors to the smallest soot particle. The effect of carbonization, which has been shown as an important process during the nucleation,14 is included in this simplified transformation. The smallest solid particles, represented by the first section in the aerosol dynamics model, are assumed to have a size within BIN3 (around 1.34 nm). The transformation of soot precursors into soot particles may then be expressed as: k

b1 BIN1(C24H12) 98 41 S1(C96H48)

(R-1)

k

b2 BIN2(C48H24) 98 21S1(C96H48)

kb3

BIN3(C96H48) 98 S1(C96H48) kp1

PIN1(C24H4) + 8H 98 41 S1(C96H48)

(R-2) (R-3) (R-4)

k

p2 PIN2(C48H8) + 16H 98 21S1(C96H48)

kp3

PIN3(C96H16) + 32H 98 S1(C96H48)

(R-5) (R-6)

where S1 is the first soot section, and reaction constants kb,1-3 and kp,1-3 are adjusted in the model according to a previous study.13 The above treatment for the nucleation mechanism is an approximate treatment. However, since the formation of soot precursors is accurately modeled and the transition from gaseous precursors into solid particles follows an unknown process, this approximation is reasonable and could actually be an improvement on the single PAH dimerization nucleation model. After careful sensitivity analysis, the conversion rates from soot precursors, BINs, to solid particles were set to 1 × 106 mol/cc/s. The same value is also used in the recently mentioned study.13 To account for the (37) Wen, J. Z.; Thomson, M. J.; Park, S. H.; Rogak, S. N.; Lightstone, M. F. Proc. Combust. Inst. 2005, 30, 1477-1484.

effect of fast polymerization during the nucleation process, the conversion rates of soot precursors, PINs, into solid particles were set to 1 × 108 mol/cc/s. In the soot model, the numbers of carbon and hydrogen atoms are conserved during the transition from precursors into solid particles. Heterogeneous Surface Growth Model. As mentioned previously in this article, the heterogeneous surface reactions described in the polyyne polymerization model1 could be very important during the early stage of particle formation. Moreover, a previous study36 based on detailed molecular weight growth pathways found that the soot growth rate increases and oscillates within a plug flow reactor and sharply declines within one-dimensional flames during soot growth after initial particle inception. This behavior is in agreement with PAH formation characteristics which follow the same basic process, validating the role of PAH during the surface growth of soot particles. To account for these processes, the surface growth submodel within this study is composed of three parts: PAH surface condensation, addition of pure carbon radicals (C4 to C12), and fast polyyne polymerization (including C2H2). To prevent the excessive depletion of large PAH molecules, only PAH radicals and three BIN radicals are included in the PAH condensation model. The condensation rates are calculated from the product of an effective factor (γ) and the collision frequencies between soot particles and PAH/BIN radicals. Instead of using eq 2, the collision frequencies between PAHs and soot particles are calculated in the same way as the frequencies between soot particles are calculated. The characteristics of soot agglomerates are also included in the model. Mass addition rates by pure carbon radicals are the same as those used in the literature.13 A simplified polyyne polymerization model, which is similar to the one developed in previous studies,13,28,33 is used in this model. In contrast to the aforementioned studies, which used the number density of soot particles to directly calculate the appropriate reaction rates, this study expresses surface reactions using the definition for the availability of surface reactive sites, as discussed in the literature.1 However, only one type of surface reactive sites, Csoot•, is used here. Its number density is determined by assuming that [C-H] maintains a state of equilibrium, as shown in the standard HACA surface growth model.15,38 The steric parameter, R, defined in the literature15 as RHACA, is also used to determine the surface reaction rate. Detailed heterogeneous surface reaction kinetics are shown in Table 3, where the surface reaction rate (g/cm3/s) is calculated by a product of the surface area of primary particles (cm2/cc), number density of reactive sites (sites/cm2), steric parameter R, molar weight of carbon (g/mol), and deposition rate of carbon atoms on the surface of soot particles (mol/site/s). To accommodate the effects of the fast polymerization process, the availability of reactive sites, χCsoot-H‚R, is increased to 1 × 1017‚RHACAsites/cm2. This value is greater than that used in the HACA mechanism, 2.3 × 1015‚RHACAsites/cm2.15 Because the value of RHACA was determined by interpolating it as the function of local temperatures15,38 and has been often adjusted according to the real flame situations,39 it is reasonable to adjust the availability of reactive sites in this study to predict a correct soot induction time behind the shock wave. The collision coefficient of PAH surface condensation, γ, is investigated and will be discussed later. Numerical Simulation. The species profiles and main parameters of soot formation during the pyrolysis of C6H617,18 are investigated using the new gaseous phase reaction mechanism and kinetics soot nucleation and heterogeneous surface growth model. The nucleation, coagulation, and surface growth processes in the particulate phase are modeled within an aerosol dynamics soot code. This study uses a modified three-point fixed sectional method developed by Park and Rogak16 to calculate surface growth and PAH condensation rates of soot agglomerates. The sectional model uses the classical sectional description of the aerosol dynamics equations based on the fixed pivot approach.16,40 The size range of soot agglomerates (38) Kazakov, A.; Frenklach, M. Combust. Flame 1998, 114, 484-501. (39) Kim, C. H.; El-Leathy, A. M.; Xu, F.; Faeth, G. M. Combust. Flame 2004, 136, 191-207.


Wen et al. Table 2. Formation Pathways of Soot Precursorsa k ) AT**b exp(-E/RT)

reactions considered*

A (mol cm s K)

Formation of PAH Precursors12 904. A2R5-+A2R5- f BIN1+H2 1.55E+13 905. A2R5-+A2C2HA* f BIN1+H2 1.55E+13 906. A2R5-+A2C2HB* f BIN1+H2 1.55E+13 907. A2R5-+A2C2H2 f BIN1+2H2 1.55E+13 908. A2R5-+A3-1 f 1.0833333333BIN1+1.5H2 1.57E+13 909. A2R5-+A3-4 f 1.0833333333BIN1+1.5H2 1.57E+13 910. A2R5-+A4- f 1.1666666667BIN1+H2 1.60E+13 911. A2C2HA*+A2C2HA* f BIN1+H2 1.55E+13 912. A2C2HA*+A2C2HB* f BIN1+H2 1.55E+13 913. A2C2HA*+A2C2H2 f BIN1+2H2 1.55E+13 914. A2C2HA*+A3-1 f 1.0833333333BIN1+1.5H2 1.57E+13 915. A2C2HA*+A3-4 f 1.0833333333BIN1+1.5H2 1.57E+13 916. A2C2HA*+A4- f 1.1666666667BIN1+H2 1.60E+13 917. A2C2HB*+A2C2HB* f BIN1+H2 1.55E+13 918. A2C2HB*+A2C2H2 f BIN1+2H2 1.55E+13 919. A2C2HB*+A3-1 f 1.0833333333BIN1+1.5H2 1.57E+13 920. A2C2HB*+A3-4 f 1.0833333333BIN1+1.5H2 1.57E+13 921. A2C2HB*+A4- f 1.1666666667BIN1+H2 1.60E+13 922. A2C2H2+A2C2H2 f BIN1+3H2 1.55E+13 923. A2C2H2+A3-1 f 1.0833333333BIN1+2.5H2 1.57E+13 924. A2C2H2+A3-4 f 1.0833333333BIN1+2.5H2 1.57E+13 925. A2C2H2+A4- f 1.1666666667BIN1+2H2 1.60E+13 926. A3-1+A3-1 f 1.1666666667BIN1+2H2 1.59E+13 927. A3-1+A3-4 f 1.1666666667BIN1+2H2 1.59E+13 928. A3-1+A4- f 1.2500000000BIN1+1.5H2 1.61E+13 929. A3-4+A3-4 f 1.1666666667BIN1+2H2 1.59E+13 930. A3-4+A4- f 1.2500000000BIN1+1.5H2 1.61E+13 931. A4-+A4- f 0.6666666667BIN2+H2 1.62E+13 932. A2R5+A2R5- f BIN1+H2+H 1.06E+12 933. A2R5+A2C2HA* f BIN1+H2+H 1.06E+12 934. A2R5+A2C2HB* f BIN1+H2+H 1.06E+12 935. A2R5+A2C2H2 f BIN1+2H2+H 1.06E+12 936. A2R5+A3-1 f 1.0833333333BIN1+1.5H2+H 1.08E+12 937. A2R5+A3-4 f 1.0833333333BIN1+1.5H2+H 1.08E+12 938. A2R5+A4- f 1.1666666667BIN1+H2+H 1.10E+12 939. A2C2HA+A2R5- f BIN1+H2+H 1.06E+12 940. A2C2HA+A2C2HA* f BIN1+H2+H 1.06E+12 941. A2C2HA+A2C2HB* f BIN1+H2+H 1.06E+12 942. A2C2HA+A2C2H2 f BIN1+2H2+H 1.06E+12 943. A2C2HA+A3-1 f 1.0833333333BIN1+1.5H2+H 1.08E+12 944. A2C2HA+A3-4 f 1.0833333333BIN1+1.5H2+H 1.08E+12 945. A2C2HA+A4- f 1.1666666667BIN1+H2+H 1.10E+12 946. A2C2HB+A2R5- f BIN1+H2+H 1.06E+12 947. A2C2HB+A2C2HA* f BIN1+H2+H 1.06E+12 948. A2C2HB+A2C2HB* f BIN1+H2+H 1.06E+12 949. A2C2HB+A2C2H2 f BIN1+2H2+H 1.06E+12 950. A2C2HB+A3-1 f 1.0833333333BIN1+1.5H2+H 1.08E+12 951. A2C2HB+A3-4 f 1.0833333333BIN1+1.5H2+H 1.08E+12 952. A2C2HB+A4- f 1.1666666667BIN1+H2+H 1.10E+12 953. A3+A2R5- f 1.0833333333BIN1+1.5H2+H 1.08E+12 954. A3+A2C2HA* f 1.0833333333BIN1+1.5H2+H 1.08E+12 955. A3+A2C2HB* f 1.0833333333BIN1+1.5H2+H 1.08E+12 956. A3+A2C2H2 f 1.0833333333BIN1+2.5H2+H 1.08E+12 957. A3+A3-1 f 1.1666666667BIN1+2H2+H 1.09E+12 958. A3+A3-4 f 1.1666666667BIN1+2H2+H 1.09E+12 959. A3+A4- f 1.2500000000BIN1+1.5H2+H 1.10E+12 960. A3C2H+A2R5- f 1.1666666667BIN1+H2+H 1.06E+12 961. A3C2H+A2C2HA* f 1.1666666667BIN1+H2+H 1.06E+12 962. A3C2H+A2C2HB* f 1.1666666667BIN1+H2+H 1.06E+12 963. A3C2H+A2C2H2 f 1.1666666667BIN1+2H2+H 1.06E+12 964. A3C2H+A3-1 f 1.25BIN1+1.5H2+H 1.08E+12 965. A3C2H+A3-4 f 1.25BIN1+1.5H2+H 1.08E+12 966. A3C2H+A4- f 0.6666666667BIN2+H2+H 1.10E+12 967. A4+A2R5- f 1.1666666667BIN1+H2+H 1.10E+12 968. A4+A2C2HA* f 1.1666666667BIN1+H2+H 1.10E+12 969. A4+A2C2HB* f 1.1666666667BIN1+H2+H 1.10E+12 970. A4+A2C2H2 f 1.1666666667BIN1+2H2+H 1.10E+12 971. A4+A3-1 f 1.25BIN1+1.5H2+H 1.10E+12 972. A4+A3-4 f 1.25BIN1+1.5H2+H 1.10E+12 973. A4+A4- f 0.6666666667BIN2+H2+H 1.11E+12

b

E (cal/mol)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 111.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0 4308.0



Table 2. (Continued) k ) AT**b exp(-E/RT) reactions considered*

A (mol cm s K)

Formation of Polymeretic Precursors13 1055. C8H2+C8H2 f 0.66666667PIN1+0.66666667H2 4.00E+13 1056. C10H2+C10H2 f 0.8333333PIN1+0.3333333H2 4.00E+13 1057. C12H2+C12H2 f PIN1 4.00E+13 1058. C12H2+C10H2 f 0.9166667PIN1+0.1666667H2 4.00E+13

b

E (cal/mol)

0.0 0.0 0.0 0.0

38958.0 20076.0 4063.0 4063.0

a The notations used in this table are the same as those published in previous studies,12,13 except for PIN1, which is the first sectional precursor formed through the fast polymerization process.

Table 3. Simplified Polyyne Surface Growth Mechanism, k ) ATn exp(-E/RT) No.

reaction Csoot•

Csoot + H f H S + H2 Csoot• + H f Csoot - H Csoot• + C2H2 f Csoot - Hn + H Csoot• + C2H2 f Csoot - C2 + H2 Csoot• + C2H f Csoot - C2 + H Csoot• + C4H2 f Csoot - C4 + H2 Csoot• + C4H f Csoot - C4 + H Csoot• + C6H2 f Csoot - C6 + H2 Csoot• + C6H f Csoot - C6 + H Csoot• + C8H2 f Csoot - C8 + H2 Csoot• + C8H f Csoot - C8 + H Csoot• + C10H2 f Csoot - C10 + H2 Csoot• + C10H f Csoot - C10 + H Csoot• + C12H2 f Csoot - C12 + H2 Csoot• + C12H f Csoot - C12 + H Csoot• + C4H4 f Csoot - C4 + 2H2

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16

dt

)

|

dN(i),(p,i) dt

+

|

dN(i),(p,i)

in

dt

+

|

dN(i),(p,i)

co

dt

+

|

dN(i),(p,i) dt

sg

sc

(3)

where the first term on the right-hand side is the inception rate, which normally only includes the contribution from the first section where inception occurs:

|

dN1 dt

) in

|

dNp,1 ; dt in

|

dNi *1 dt

) in

|

dNp,i*1 dt

in

)0

(4)

The coagulation rate, the second term on the right-hand side of eq 3, is calculated based on the collision kernel, βj,k, which corresponds to the entire Knudsen number regime for nonspherical agglomerates:16 dNi dt

|

co

kejei

)

∑

mi-1emj+mkemi+1

( ) 1-

δj,k 2

ηβj,kNjNk - Ni

M

∑β k)1

n

E (kcal/mol)

4.2 × 2.0 × 1013 8.0 × 107 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 4.0 × 1013 2.0 × 1012

0.0 0.0 1.56 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

13.0 0.0 3.8 31.8 0.0 12.0 0.0 8.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 27.7

1013

is divided into a fixed number of sections, and the discrete aerosol dynamics equations are solved for each section. The particle population in each section is represented by a representative mass, mi, and the number concentration of clusters, Ni. To model the structure of fractal agglomerates, another property for each section is introduced: the number concentration of total primary particles, Np,i. Primary particles in each section (agglomerate) are assumed to have the same size. The above properties are calculated by assuming that the density of soot particle is constant and all agglomerates have the same mass fractal dimensions (Df).16 A constant fractal dimension has been used extensively in modeling the simultaneously occurring particle nucleation, coagulation, and surface growth processes. In sectional models, the state-of-the-art coagulation rate of fractal aggregates is calculated using very precise collision diameters and mobility diameters, as presented in the previous section. Therefore, the collision rate can be accurately calculated even for small clusters that may have higher fractal dimensions. For more details, refer to the work of Park and Rogak.41 The discrete expressions of governing equations for the number concentrations of clusters (Ni) and primary particles (Np,i) are: dN(i),(p,i)

A (cm3 mol-1 s-1)

i,kNk

(5)

where δ is the delta function, and M is the number of the sections. The parameter η weights the new formed mass into two adjacent sections:

{

mi+1 - (mj + mk) mi e mj + mk e mi+1 mi+1 - mi η) mi-1 - (mj + mk) mi-1 e mj + mk e mi mi-1 - mi

(6)

The change in the number of primary particles by coagulation is expressed as:

|

dNp,i dt

kejei

) co

∑

( ) 1-

mi-1emj+mkemi+1

δj,k 2

M

ηηpβj,kNjNk - Np,i

∑β

i,kNk

(7)

k)1

where ηp )

mi (n + np,k) mj + mk p,j

(8)

The factor mi/(mj + mk) assigns primary particles to two adjacent sections in such a way that the primary particle size is conserved. The representative masses, mi, are defined by m1 )

2 × mnuc 1 + fs

mi ) fs × mi-1

i ) 2, M + 1

(9) (10)

where mnuc is the mass of a nuclei, and fs is the spacing factor (2 for the baseline model in this study). The third and fourth terms on the right-hand side of eq 3 are the surface growth and condensation rates, respectively. These terms can cause instability and numerical diffusion when computationally (40) Kumar, S.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1311-1332. (41) Park, S. H.; Rogak, S. N. Aerosol Sci. Technol. 2003, 37, 947960.


Wen et al.

solving the aerosol dynamics equations. In addition, the inclusion of the surface growth submodel increases the nonlinearity of the system equations since the mass growth rate is a function of the surface area of the particles. In this study, the adjusted-point fixed sectional method is used to reduce both the instability and numerical diffusion. Detailed information on this method can be found in ref 16. Only the final expressions are given here. dNi dt

|

) sg/sc

Ai-1Ii-1Ni-1 BiIiNi Ci+1Ii+1Ni+1 + + Vi-1 Vi Vi+1

(11)

where the appropriate parameters A, B, and C are defined to conserve three moments of particle size distribution.16 The surface growth and condensation processes described above do not increase the number of primary particles in each section. The above aerosol dynamics model covers all three possible Knudsen number regimes. The density for solid soot particles is assumed to be constant (1.86 g/cc), and all agglomerates are assumed to have the same mass fractal dimensions (Df ) 3.0 in this study corresponding to compact structures observed during the shock tube measurement18). The gas phase chemistry is completely coupled to the aerosol dynamics model. Thus, the consumption rates of soot precursors during particle nucleation and the feedback of surface reaction rates for gaseous species are all included in the species balance equations.

Simulation Results Prediction of Species Profiles. Accurate prediction of species profiles is a very important part of a good soot model. However, time-resolved species measurements are unavailable for the particular shock tube study being considered because of experimental difficulties. Though the existing PAH growth mechanism15 has not been fully validated against the shock tube measurements, its prediction can still provide a guideline for species profiles. The comparison of species predictions has been made using two different gas-phase mechanisms, a previous PAH growth mechanism,15 and the new developed mechanism. The major aliphatic and aromatic species were calculated behind a shock wave with a 1% C6H6/Ar mixture under constant temperature (2095 K) and pressure (1.23 bar). The simulation shows that the pyrolysis of C6H6 (A1) occurs extremely quickly, whereby its molar fraction declines to a value of almost 1 × 10-6 at a residence time of 1 ms. Large amounts of polyynes (C4H2 and C6H2) are formed, and the concentrations of naphthalene (A2) and phenanthrene (A3) quickly increase to their maximum values, then decline to smaller amounts at a residence time of 1 ms. The concentration of pyrene (A4) continuously increases and plateaus at an even greater residence time. Upon comparison of predictions using two gas-phase mechanisms, one can conclude that the inclusion of soot precursors significantly affects the chemical pool of gaseous phase species, especially polyynes. The influence of mechanisms on the concentration of aromatic species is small. The proposed gaseous-phase mechanism has been validated for major carbon radicals and polyyne species against several measurements in the pyrolysis of C2H2 and C4H2, as was demonstrated previously in this study. However, this mechanism requires further validation for a heavily sooting environment. One such environment is the pyrolysis of C6H6 behind a shock wave. Unfortunately, the concentrations of species interested have not been measured in Roth et al.’s studies.17,18 The numerical predictions of polyyne concentrations measured in another study44 show that, for a (42) Zhao, B.; Yang, Z.; Johnston, M. V.; Wang, H.; Wexler, A. S.; Balthasar, M.; Kraft, M. Combust. Flame 2003, 133, 173-188. (43) Vander Wal, R. L.; Tomasek, A. J. Combust. Flame 2004, 136, 129-140.

Figure 7. Induction time predicted by another detailed soot model.12 The dash dot line is for the measured soot volume fraction.17 BIN5 and BIN6 are soot sections defined in the soot model.

constant temperature and pressure case (2190 K and 0.48 atm, respectively, for the 2.1% C6H6 in Ne), the full mechanism described in this study results in a discrepancy of around 40%. Prediction of Soot Parameters. The characteristics of soot measurements behind shock waves, including induction time, observable growth rate, time history of soot yield, total particle number density, and averaged particle diameter, are predicted for a 1% C6H6/Ar mixture using the sectional aerosol dynamics soot model. This model also provides a detailed description of the particle size evolution, which is of great interest in understanding the roles of specific soot formation processes. Before analyzing the predicted results from the aforementioned model, another recently published PAH growth soot model12 is used to simulate the same C6H6/Ar mixture. Soot Formation Predicted by the Other Model. Most recently, a complicated kinetics soot model12 has been implemented to predict PAH and soot formation in a premixed benzene/oxygen/argon flame. The model focuses on describing molecular weight growth pathways.36 It includes the formation of PAH up to C30H10 as well as the growth of carbon nanostructures with particle sizes up to 7 nm. The formation and consumption of PAH and soot are described by reactions involving acetylene, PAH, PAH radicals, and BINs (i.e., soot sections) and their radicals. The implementation of this model in CHEMKIN requires a modification of the gas-phase interpreter to handle large molecules with carbon and hydrogen atoms which correspond to soot particles of increasing size. For this study, the aforementioned kinetic model has been implemented into a commercial version of the CHEMKIN code34 to predict fuel prolysis behind a shock wave. Restricted by the requirements of CHEMKIN input format, the kinetic model12 has been shortened to include only the six smallest BINs. Among these BINs the first four hold large PAHs while the last two (BIN5 and BIN6) contain the earliest forming soot particles. Because a set of reactions describing the further growth of larger soot particles has been cut off, it is clear that the soot yield predicted by this modified model will be much smaller than that of the full model. However, the soot induction time, which is particularly determined by the appearance of the smallest soot particles (corresponding to the BIN5), should be close to the value predicted by the full model. Figure 7 shows measured soot volume fraction and the predictions of molar concentration for BIN5 and BIN6. (44) Kern, R. D.; Singh, H. J.; Esslinger, M. A.; Winkeler, P. W. Proc. Combust. Inst. 1982, 19, 1351-1358.


Figure 8. Predictions for soot volume fraction of a 1% C6H6/Ar mixture.17 The simulation cases refer to Table 4.


Figure 9. Predictions for averaged particle diameter of a 1% C6H6/Ar mixture.17 The simulation cases refer to Table 4.

Table 4. Soot Model Parameters in Numerical Simulations of 1% Mixture

No. 1 2 3

inception rate (mol/cc/s) PINs ) 1E8 BINs ) 1E6

availability of surface reactive sites in polyyne surface growth model (sites/cm2)

PAH condensation coefficient

χCsoot-H‚R ) 1E17‚R

γ ) 0.0 γ ) 0.5 γ ) 0.1

Figure 7 demonstrates that the kinetic soot model developed by Richter et al.12 overpredicts the induction time for the pyrolysis of C6H6 in Ar behind a shock wave. Predictions are about a factor of 5 times greater than the measured data. Since detailed descriptions for the formation of PAH and soot have been included in the model and all possible molecular growth pathways have been carefully analyzed and evaluated, the underprediction values may result from the absence of polyynes within the formation and growth mechanism of soot particles. Comparison with Measurements. The kinetic model developed in this study has been implemented to investigate the pyrolysis of a C6H6/Ar mixture within a shock tube. The model used includes detailed PAH and soot precursor growth submodels and a polyynes growth submodel. A single measurement for a mixture of 1% C6H6 in Ar at a temperature of 2095 K and a pressure of 1.23 bar has been studied carefully and used to optimize the model parameters in light of the incomplete datasets. In ref 18, Starke and Kock discussed the influence of refractive index, m, on the measured soot volume concentration. They found that using alternative soot refractive indices (i.e., m1 ) 3.25 - i2.07, chosen for the final result in the previous article, and m2 ) 1.9 - i1) resulted in measured soot volume concentrations that differed by a factor of 2. Figure 8 shows the predicted time history of soot formation in a 1% C6H6/Ar mixture. This figure also shows an error bar range of (25% for soot measurements, which is a typical experimental estimation in this situation. The three numerical simulation cases plotted in the figure were performed using the model parameters stated in Table 4. Figure 8 shows that, for all residence times, the predictions of soot volume fraction are located well within the error bar range. This explicitly indicates that the induction time and observable growth rate (defined by the slope of soot yield) are reasonably predicted in all three simulation cases. The figure also shows that using the largest value of collision coefficients between PAH radicals and soot particles (simulation 2) brings about a greater soot growth rate at a very early residence time (before 0.0004 s) and a smaller final yield of soot (2.2 × 10-6 at 0.0015 s). On the contrary, the smaller rate of PAH surface condensation (simulations 1 and 3) results in a

Figure 10. Predictions for total particle number density (#/cc) of a 1% C6H6/Ar mixture.17 The simulation cases refer to Table 4.

flat soot yield at greater residence times, which is in agreement with shock tube observations. In general, the difference among the three simulations is quite small. All of them are thus capable of predicting the soot volume fraction. Those models are also compared with measured soot diameter and particle number density. Figure 9 shows the prediction of averaged particle diameters for the same measurement of a 1% C6H6/Ar mixture. The measurements are plotted with an error bar range of -36 and +56%, as suggested by ref 18. As shown in the figure, the new soot kinetics and aerosol dynamics model are capable of predicting the mean particle diameter well within experimental error. Specifically, after a residence time of 0.0003 s, the predictions are within or nearly within the given error bar range. In agreement with the prediction of soot yield, using the larger collision coefficients for the PAH surface condensation model (simulation 2) results in larger soot particles and an obvious positive slope at the greatest residence time (0.0012 s). This is not in agreement with the experimental finding which shows a leveling-off behavior. These results demonstrate that, for the fast-reacting environment behind a shock wave, the fast polyyne polymerization model alone is able to predict the soot yield and particle size very well. Note that at very small residence times (less than 0.0003 s, in Figure 9) particle size is overpredicted. This particle size discrepancy is directly correlated with the underprediction of particle number density or the experimental uncertainties. Figure 10 shows the predictions of total particle number density, which all have similar maximums at around 0.00015 s. This particular residence time could be viewed as a boundary between the rapid nucleation process and the dominating coagulation/surface growth process.32 The total particle number


Wen et al.

Figure 11. Evolution of particle size distribution of a 1% C6H6/Ar mixture.17

density flattens out at greater residence times, which is in agreement with previous experimental findings. Figure 10 also shows that, if a larger collision coefficient of PAH surface condensation is used (simulation 2 in the figure), smaller peak particle number densities are obtained. This is likely a result of smaller nucleation rates due to the depletion of PAH-formed soot precursors. The role of PAHs and polyynes are discussed later in this article. The model also predicts particle size distribution at individual residence times. When a great heterogeneity of soot aggregates exists in the flame, global properties such as total number density and averaged particle diameter cannot provide an accurate description of soot yield. Studying size distribution, however, can help clarify the individual roles of different chemical and physical processes. Figure 11 shows predictions for soot particle evolution in simulation case 1. As it is in agreement with the previous numerical findings for the plug flow reactor,37 the particle size curve evolves from a power law shape (at 0.01 ms) to a well-developed bimodal shape (from 0.4 to 1 ms). The same evolution of particle size distribution has been found experimentally in laminar premixed flames.42 In summary, combined with the newly developed gas-phase mechanism, the aerosol dynamics soot model is capable of predicting all the essential characteristics governing the formation of soot during the pyrolysis of a 1% C6H6/Ar mixture behind a shock wave. The model is also used to predict soot formation in several other mixtures within a shock tube. Sensitivity Analysis. A sensitivity analysis has been carried out to investigate the effect of the availability of surface reactive sites, χCsoot-H‚R, on these measured soot parameters. These parameters include the soot induction period, soot volume fraction at 1 ms, and particle diameters at 1 ms, which have been measured during the pyrolysis of a mixture of 1% C6H6 in Ar at a temperature of 2095 K and a pressure of 1.23 bar. The comparisons are shown in Figure 12. Figure 12a shows that, when increasing the availability of surface reactive sites, the model predicts a shorter induction period and the effect is significant (with a variation of 30%). Figure 12b shows that increasing the availability of surface reactive sites does not bring about a large difference on the prediction of soot volume fraction. The model predicts fairly consistent soot volume fractions, which are located within the experimental error bar range. Figure 12c shows that, when increasing the fraction of surface reactive sites, the model predicts larger particle sizes that are still located in the experimental error bar range. In summary, the above analysis

Figure 12. Effects of the availability of surface reactive sites, χCsoot-H‚R, on soot predictions of the induction period (a), volume fraction (b), and particle size (c). Table 5. Parameters Used in Simulation Cases mixture No.

temperature (K)

pressure (atm)

0.25% 0.5% 1% 2%

1980 2030 2095 1980

1.21 1.18 1.21 1.18

indicates that the effect of the availability of surface reactive sites is significant in determining the soot induction period. Soot Predictions for Other Shock Tube Fuel Mixtures. The soot volume fraction, averaged particle diameter, total number density, and evolution of size distribution are predicted for several fuels with different C6H6/Ar mixtures: 0.25, 0.5, 1, and 2%, respectively. The numerical simulations were carried out using the temperature and pressure measured in previous studies.17,18 All the different cases are shown in Table 5. Among the predicted soot properties, only the measured data for time dependence of the particle diameter is available for three other mixtures, 0.25, 0.5, and 2%, in the literature.17,18


Figure 13. Predictions for soot volume fraction of various C6H6/Ar mixtures.17,18


Figure 15. Comparison of averaged particle diameter for a 0.25% C6H6/Ar mixture,17 where CHI ) χCsoot-H‚R/RHACA. Lines are for numerical predictions in this study and correspond to the different fraction of surface reactive sites.

Figure 14. Comparison of averaged particle diameter for a 0.5% C6H6/ Ar mixture.17

Figure 13 shows predictions of soot volume fraction. The soot model parameters are the same as those shown for simulation case 1 in Table 4. When increasing the fuel composition within the mixture, it is found that the mixture with a larger fuel molar fraction produces more soot within the entire soot formation process. The induction time decreases to a minimum for the 1% mixture, and then the induction time increases. In the case of the 2% mixture, its induction time is longer but its growth rate is the greatest. Generally there are three major causes for this difference. First, as will be shown in the prediction of averaged particle diameter for the 0.25% mixture, the fraction of surface reactive sites could be a function of fuel composition. Using a constant fraction may result in an underprediction of soot formation. Second, the underprediction can result from the density effects on both the shock tube measurements and numerical simulation. Density effects are generally more pronounced for mixtures with greater fuel compositions. This is because a high fuel/Ar ratio cannot be viewed as a well-diluted mixture, which is a required condition for providing a constant temperature and pressure environment during shock tube measurements. The predictions of soot-averaged particle diameter for various mixtures exhibit behaviors similar to that of soot yield. When increasing the fuel composition of the mixture, it is found that particle size increases. Once again, an exception exists in the case of the 2% mixture, where soot particle size is greatly underpredicted. Figures 14 and 15 show a comparison between predicted particle diameters and measured particle sizes for a 0.5% mixture and a 0.25% mixture, respectively. Figure 14 indicates that the soot model correctly predicts particle sizes for the 0.5% mixture. Figure 15 demonstrates that the model tends to overpredict particle size for the 0.25% mixture. This

Figure 16. Predictions for soot number density of various C6H6/Ar mixtures.17,18

latter discrepancy is most likely due to the large fraction of reactive sites used in the polyyne surface growth model. Another simulation using a lower availability, χCsoot-H‚R ) 5E16‚RHACA, shows more reasonable size predictions, as demonstrated by a solid line in Figure 15. This dependency of particle size suggests that the availability of surface reactive sites, χCsoot-H‚R, or the parameter, R, could be a function of fuel composition. Figure 16 shows predictions for soot number density. When increasing the fuel composition within the mixture, it is found that the total number density increases at earlier residence time. The more fuel contained within the mixture, the faster and earlier the number density increases. It is interesting to note that at larger residence times, the total number densities of soot particle climb to the same value for three (0.25, 0.5, and 1%) of the mixtures. Yet again, an exception exists in the case of the 2% mixture, where the number density increases more slowly than the 1% mixture. Discussion The model predictions presented in the previous section provide valuable insight regarding the soot formation process, especially in its earlier stage of development. Relative Rates of Inception and Surface Growth. There are generally two different approaches for studying nucleation and surface growth during the early stages of soot formation. One is by making sophisticated experimental measurements on the nanostructure of soot particles. This method was used to discover that a complex phase transition exists during the nucleation process. This transition is not easily modeled with chemical kinetics alone.43 Another approach is to use detailed


Figure 17. Comparison of individual soot formation rates calculated for simulation cases 1 and 2.

numerical modeling, which often treats soot nucleation as the continuous growth of large PAH molecules.12 Soot predictions made by this model rely on the accuracy of PAH molecular growth pathways and the soot surface growth model. The aggregate structures of soot particles are excluded from the model. In this study, the aerosol dynamics soot model provides a powerful tool that can couple comprehensive physical process (phase transition, thermal restructure, agglomeration, etc.) into detailed chemistry. As discussed in the previous section, though PAH surface condensation had not been accounted for during the surface growth of soot particles in simulation case 1, PAHs contributed to the formation of BINs, which determined the nucleation rate. The inclusion of the PAH surface condensation model would affect soot volume fraction and the total particle number density. This would, in turn, significantly affect the soot nucleation process by depleting soot precursors and reducing the nucleation rate. Surface growth via the addition of more PAH molecules on particle surfaces would not be as significantly affected. Figure 17 shows a comparison between individual soot formation rates for simulation cases 1 and 2, where PAH surface condensation rates are either zero or very large, respectively. Figure 17 shows the influence of PAH surface condensation on the nucleation and surface growth rates. The figure demonstrates that the influence of PAH surface condensation is more significant at earlier residence times (from 0 to 0.0004 s in both figures). It is also found that depletion of the PAH pool can significantly alter the nucleation rate and, hence, particle number density. This indicates why the underprediction of averaged particle diameter occurs at very small residence times (less than 3 ms in Figure 9). The underpredictions are caused by corresponding underpredictions of concentrations of PAH soot precursors (BINs). It may be the case that the gas-phase mechanism cannot correctly predict the formation of soot precursors. Recall that this mechanism is based on an existing PAH growth model that has been validated against the formation of PAH in various experimental combustion environments. However, if these PAHs directly contribute to the formation of soot, being consumed to form soot precursors, then their concentrations must decrease. Thus, if the feedback from soot precursors is accounted for in the kinetic model, PAH species will be underpredicted. The above analysis shows that, to obtain a more accurate prediction of particle diameter at very early residence times, the PAH growth model has to be improved so that accurate species profiles for all PAHs, polyynes, and soot precursors may be calculated when the polyyne formation and soot formation mechanisms are fully coupled. Unfortunately, the validation of such a model is impossible for the shock tube study being

Wen et al.

Figure 18. Predicted coagulation rates of different mixtures.

investigated since no species concentration data are available. Further validation of the nucleation model also requires the measurement of the total particle number density or a detailed size distribution for earlier residence times. Figure 17 presents a comparison between the nucleation rate and surface growth rate predicted for simulation case 1 as well. The nucleation rate increases very quickly and reaches its peak at around 0.1 ms. It then declines dramatically and reaches a stable value of 4 × 1015 #/cc/s. This level rate is maintained because of the low concentration of soot precursors. The surface growth rate also increases during the rapid nucleation process. This particular process gives rise to a great gain of soot particle mass during the early stages of soot formation. The surface growth rate reaches its peak value at a residence time when soot particles have the maximum number of reactive sites. Finally, the rate decreases to a value so small that it cannot change soot yield and particle diameter. These results are in good agreement with the experimental finding in shock tube, where soot yield and particle size decline to a constant value at greater residence times. In summary, nucleation and the surface growth processes are highly coupled during the early stages of soot formation. This implies that the same species (PAHs and polyynes) contribute to the formation of both soot precursors and solid particles. Coagulation in Shock Tubes. Under normal circumstances, it is understood that the coagulation process does not adversely affect shock tube measurements because the fuel mixtures used are highly diluted and soot volume fraction is relatively low. However, this study found that the coagulation process contributes to the rapid surface growth of soot particle by increasing particle size and providing more surface reactive sites. Figure 17 shows the prediction of the coagulation rate in simulation case 1. The nucleation and surface growth rates are also plotted for comparison. The figure shows that the particle-particle coagulation rate has the same order of magnitude as the nucleation rate, and its peak is close to the peak of surface growth rate, where the maximum particle number density is reached. Finally, the coagulation and nucleation rates converge to the same value at larger residence times, where the change in particle number densities is nearly balanced. Figure 18 shows the coagulation rates calculated by different C6H6/Ar mixtures. The figure indicates that smaller mixture fuel fractions, such as the 0.25% C6H6 in Ar, result in a very small coagulation rate. Therefore, soot particles are influenced primarily via the nucleation and surface growth processes. An increasing fuel molar fraction in the original mixture gives rise to increasing coagulation rates with peak values moving toward earlier residence times. Generally, coagulation rates slowly


Figure 19. Density changes of different mixtures.

decline at greater residence times. In contrast, soot particle number densities remain almost unchanged. Density Effects in Shock Tubes. In the well-defined combustion environment of a shock tube, it is not difficult to keep the mixture density relatively constant. This is accomplished by using large amounts of diluting gas (Ar) and a spontaneous energy balance. However, if a large amount of fuel is included within the mixture, reaction heat is released very quickly. As a result, a significant change in mixture density may be observed. This change in density contributes to a decrease in species concentration, which ultimately leads to an underprediction of soot formation. In the 2% C6H6/Ar mixture, density decreases quite dramatically, as shown in Figure 19. In contrast to the 0.25% mixture, the 2% mixture suffers from varying density which in turn affects molar species concentrations. These concentrations, which are actually not very welldefined (in the numerical simulation or the experimental measurement) behind a shock wave, have a large influence on reaction rates and can severely alter model predictions. For these reasons, the 2% mixture is excluded from any further investigations made in this study. Conclusions This article focuses on the development of soot kinetics in a heterogeneous phase where the particle nucleation, surface growth/condensation, and coagulation take place. Reaction mechanisms for the gaseous-phase chemistry obtained from the literature were used. Coupling with an existing gas-phase PAH


growth mechanism, a chemical kinetic soot model has been developed to describe the formation and growth of carbonaceous precursors and soot particles. The formation pathways of soot precursors proposed in this article are different from those published previously. Two different types of precursors, formed through either the continuous growth from PAHs or the fast polymerization process, are described as sectional bins. Both the PAH growth pathway and the polyyne fast polymerization process contribute to the early surface growth of soot particles. The developed kinetic soot model has been implemented in an aerosol dynamics code to simulate the major characteristics of soot particle formation during the pyrolysis of C6H6 behind a shock wave. This particular model has also been validated against measured data such as induction time, observable soot growth rate, soot yield, and averaged particle diameters. Further analysis on the relative importance of individual processes inferred the following conclusions: (1) With respect to the fuel pyrolysis experiments behind a shock wave, both PAH growth and polyynes polymerization play an important role during the soot nucleation process. This nucleation process can be described by the formation of two different types of precursors through the aforementioned processes and the transformation from these precursors to the solid soot particle. (2) The polyyne surface growth model is able to describe the short induction time and fast mass growth rate of soot formation. Fuel composition plays an important role in the calculation of polyyne surface growth rate. As a result, the fraction of surface reactive sites may be a function of fuel composition. (3) The effect of the PAH pool on the nucleation and surface condensation rates of soot particle is significant at earlier residence times. Inclusion of the PAH surface condensation model affected the soot nucleation process more significantly by depleting precursors and reducing the nucleation rate than it did in surface growth by adding more PAH molecules on the surface of particles. Acknowledgment. This work has been supported by Auto21 Canada program and the Natural Sciences and Engineering Research Council of Canada (NSERC). We also wish to thank Dr. H. Richter, MIT, and Dr. A. V. Krestinin, Russian Academy of Sciences, for discussions on soot models. EF050081Q

Detailed Kinetic Modeling of Carbonaceous Nanoparticle Inception

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