Numerical simulations of soot aggregation in premixed laminar flames

Proceedings

Proceedings of the Combustion Institute 31 (2007) 693–700

www.elsevier.com/locate/proci

of the

CombustionInstitute

Numerical simulations of soot aggregationin premixed laminar flames

Neal Morgan a, Markus Kraft a,*, Michael Balthasar b, David Wong c,Michael Frenklach c,d, Pablo Mitchell c,d

a Department of Chemical Engineering, University of Cambridge, Pembroke street, Cambridge CB2 3RA, UKb Volvo Technology Corporation, Department 6180, CTP, Sven Hultins Gata 9A, SE-412 88 Gothenburg, Sweden

c University of California at Berkeley, CA 94720-1740, USAd Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Abstract

In this paper we make use of a detailed particle model and stochastic numerical methods to simulate theparticle size distributions of soot particles formed in laminar premixed flames. The model is able to capturethe evolution of mass and surface area along with the full structural detail of the particles. The model isvalidated against previous models for consistency and then used to simulate flames with bimodal and uni-modal soot particle distributions. The change in morphology between the particles from these two types offlames provides further evidence of the interplay among nucleation, coagulation, and surface rates. Theresults confirm the previously proposed role of the strength of the particle nucleation source in definingthe instant of transition from coalescent to fractal growth of soot particles.� 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Soot; Laminar premixed flames; Structural evolution; Stochastic modelling

1. Introduction

It has long been known that the size and shapeof nanoparticles directly affect their physical prop-erties. The mechanism for their growth has beeninvestigated under many conditions and for avariety of chemical systems [1–7]. In most casesthere is an initial phase of coalescent growth [8],where coagulation with small particles caused byhigh particle inception and rapid surface growthcause the particles to grow into near spherical‘‘primary’’ particles. The coalescent regime is fol-

1540-7489/$ - see front matter � 2006 The Combustion Institdoi:10.1016/j.proci.2006.08.021

* Corresponding author. Fax: +44 1223 334796.E-mail address: [email protected] (M. Kraft).

lowed by particle aggregation, when the particlestake on the form of fractal aggregates [9–11].

Many numerical techniques have been devel-oped for the simulation of populations of nano-particles [12] and, in particular, soot particles.Here we focus on stochastic particle methods. Inrecent applications of this general approach,particle dynamics were formulated in terms ofone or two state variables like total particle vol-ume and surface area [7,13–15]. In another devel-opment, a collector-particle technique was appliedto investigate the transition between coalescentand aggregate growth [16,17], with the resultsincorporated [18] into the method of moments [6].

In the present study we combine two tech-niques, the efficient stochastic particle collision

ute. Published by Elsevier Inc. All rights reserved.

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694 N. Morgan et al. / Proceedings of the Combustion Institute 31 (2007) 693–700

algorithm [7,13] and the sterically resolved collec-tor-particle simulations [16,17]. In the following,we describe the numerical model, test the accuracyand authenticity of model predictions, and thenperform a series of numerical simulations of thetime evolution of soot particles and their sizedistributions.

2. Model

The population balance is governed by threemain elements: particle nucleation, coagulation,and surface growth. In the present model, eachincepted particle is defined as a sphere with centerof mass in cartesian coordinates (x = 0, y = 0,z = 0), a radius r0, volume v0, and surface areaa0. When an event affects this particle, one ormore of its internal coordinates will change. Asurface event for example will change v, a, and r,whilst a coagulation event will displace the (x, y,z) coordinates of one of the coagulating pair.

The rate of particle nucleation in the model isdetermined by the rate of binary collisions of gas-eous pyrene molecules [19,20]. A detailed gasphase chemistry model that describes the forma-tion of pyrene is taken from Refs. [21,22].

The coagulation is governed by the Smolu-chowski coagulation equation [23] with a suitablekernel, as described in the next section, to calcu-late the rates of collisions of the particles at thetemperatures and pressures investigated.

The surface processes include chemical growthand oxidation as well as physisorption of pyrene[24,20]. The chemical growth is modelled as theaddition of acetylene to a radical site on the parti-cle’s surface through the hydrogen abstraction car-bon addition (HACA) mechanism [20]. Thecondensation of pyrene onto the surface of a sootparticle is modelled as a coagulation event betweena pyrene molecule and the particle of interest[24,19,25]. Oxidation of a particle is considered asthe reaction of O2 with surface radicals or the reac-tion of OH with the surface of the particle [24,20].The surface growth kinetics are taken from [22].

3. Numerical treatment

The model defined above is solved using a sto-chastic particle algorithm [7,13]. The efficiency ofthe algorithm has been vastly improved by useof a time-splitting technique for the fast surfacegrowth processes, and by introducing a majorantkernel [13,26] for the coagulation term. Anotherimportant feature of the new code is the abilityto handle essentially an infinite number of internalcoordinates, thereby tracking the positions andradii of every primary particle of the aggregatestructure. This, in turn, allows us to determinethe particle volume, surface area, collision diame-

ter, and fractal dimension distributions through-out the flame.

There are two input files required for the sim-ulation. The first supplies species concentrationsand flame temperature as functions of reactiontime, while the second file specifies details of thesimulation, such as the final time, and the timefor splitting the processes.

The time between events, s, is generated as anexponential distribution with parameter equal tothe sum of inception rate and the total coagulationrate. The calculation of the total coagulation raterequires a summation over all possible collisionpairs. This summation normally takes of the orderN2 operations, where N is the number of stochasticparticles, because of the mathematical form of thecoagulation kernel, K(x, y), where x and y are rep-resentative particle properties. However, if thecoagulation kernel can be represented as the prod-uct of two independent terms, we can reduce thiscomplexity to the order of N. Such a majorant ker-nel, Kðx; yÞ, must satisfy the inequality

Kðx; yÞP Kðx; yÞ 8x; y: ð1Þ

This results, however, in an over-prediction ofthe coagulation rate. To correct for this, we intro-duce ‘‘fictitious jumps’’, when time advances, butno coagulation event is performed. A fictitiousjump is performed with probability K=K. For fur-ther details on the form of the majorant kernel seeRef. [13].

A coagulation event, is performed by summingthe volumes and surface areas of the two aggre-gates picked. This is accomplished with the codeof Mitchell [27], which performs a ballistic colli-sion in R3 and, based on the point of contact ofthe newly formed pair, translates the coordinatesof the primary particles of one of the aggregatesrelative to the other particle.

Particle nucleation, adds a spherical particle ofsize 32 carbon atoms.

Surface growth, is performed over an intervaldt which is typically a value of 1/100th to 1/1000th of the final simulation time. The rates ofthe surface processes are calculated for each parti-cle using the gas-surface rates [22] and the surfaceareas of the particles. From these rates a meannumber of events for each particle for interval dtis calculated. This number is then used as aparameter for a poisson random process to deter-mine the actual number of events that should haveoccurred (for more details on this see [15]). Thecorresponding number of carbon atoms is addedto the aggregate and by considering the densityof soot and the surface area of the aggregate, eachprimary particle is also increased in radius. Thesurface area of the particle, its radius of gyration,and fractal dimension are then recalculated withMonte-Carlo integration routines over the wholeaggregate [27]. If through oxidation the number

1012 10-8

N. Morgan et al. / Proceedings of the Combustion Institute 31 (2007) 693–700 695

of carbon atoms in an aggregate falls to below 32then this particle is considered ‘‘oxidized’’ and isremoved from the system [6,7].

108

109

1010

1011

0 0.005 0.01 0.015 0.02 0.025 0.03

Spherical modelSpherical modelFull Aggregation modelFull Aggregation model

10-12

10-11

10-10

10-9

Num

ber

dens

ity /

cm-3

time / s

Soo

t vol

ume

frac

tion

Fig. 2. Comparison of number density and volumefraction between the aggregation model and the coales-cent-sphere model.

030.

040.

050.

06

y de

nsity

a

4. Validation

The numerical implementation of the modelwas compared to a previously validated one-di-mensional (1D) stochastic code in order to verifythat the surface growth routines had been imple-mented correctly. Both simulations were run using218 stochastic particles, with particle inception,coagulation, and surface growth included. Thenew routines produced results in very close agree-ment to the previous code, the error in the firstmoment arising from the numerical splitting rou-tine being only 1%.

The code was also run in pure coagulation modeto verify that the fractal dimensions calculated arein agreement with previous values in the literature.Figure 1 shows the results of this simulation. The x-axis shows the ratio of collision diameter, Dc, to theprimary particle diameter, Dp, while the y-axisshows the number of primary particles, np. In thispaper, Dc is defined to be twice the radius of gyra-tion of the particle. The circles on the diagram rep-resent individual particles in the simulation. Theprimary particle diameter is calculated from [28]

Dp ¼ 6v=a; ð2Þ

1

10

100

1000

1 10 100

num

ber

of p

rimar

y pa

rtic

les,

np

Collision diameter / primary diameter, Dc / D

p

np = ( D

c / D

p)1.93

Fig. 1. A plot of Dc/Dp vs np for a pure coagulationsimulation.

Table 1Summary of flame conditions [30]

Flame code Mol% gasesC2H4/O2/Ar

Cold gasvel. (cm/s)

Tmax (K)

A1 24.2/37.9/37.9 7.0 1790A3 24.2/37.9/37.9 10.0 1920

0.00

0.01

0.02

0.

volume equivalent diameter / nm

prob

abili

t

706050403020100

706050403020100

0.00

0.05

0.10

0.15

0.20

volume equivalent diameter / nm

prob

abili

ty d

ensi

ty

b

Fig. 3. PSDFs of Dv from flames A1 and A3: simula-tions (solid lines) and experiments (circles). (a) PSDF ofDv for flame A1. (b) PSDF of Dv for flame A3.


while the number of primaries can be determinedby dividing the volume by the volume of aprimary

np ¼6v

pD3p

¼ a3

36pv2; ð3Þ

where v and a are the volume and surface area ofthe aggregate particle, respectively. The calculatedfractal dimension for this simulation was 1.93, inaccordance with the result from the simulationsin Ref. [29], which reported a value of1.91 ± 0.03 using a similar ballistic cluster–clusteraggregation model.

5. Results

Two sooting laminar premixed flames weresimulated. The flame properties were taken from

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

5

10

15

20

0 282420161284

Dia

met

er /

cm

time / ms

Dc

np

Dp

num

ber

of p

rimar

y pa

rtic

les

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

5

10

15

20

0 84 12 16

Dia

met

er /

cm

time / ms

num

ber

of p

rimar

y pa

rtic

les

Dp

Dc

np

a

b

Fig. 4. Temporal evolution of particle properties forflames A1 and A3. (a) Temporal evolutions of averageDc, Dp, and npin flame A1. (b) Temporal evolutions ofaverage Dc, Dp, and np in flame A3.

Zhao et al. [30] and are summarized in Table 1.These flames were chosen for their short residencetimes and because they included both unimodaland bimodal particle size distributions.

A simulation was performed to compare thefull three-dimensional (3D) particle aggregationmodel with the simpler, particle coalescence-onlymodel. A comparison of the number density andsoot volume fraction of the particle distributionis shown in Fig. 2.

As expected, at the early stages of the flame,when the particles are in the coalescent growthregime, the two simulations agree closely. After4 ms the aggregation phase of growth beginsand the number of particles falls more rapidly inthe particle aggregation simulation than in thecoalescence-only simulation. This is entirely dueto the fact that the collision diameters of the par-ticles are greater in the aggregation simulationsand hence the rate of coagulation is greater. Itcan be seen also that the total volume of particlesin the system is increased when using the aggrega-

Fig. 5. Renderings of particles most likely to be found inflames A1 and A3. (a) 3D rendering of a particle fromflame A1. Dc = 49.5 nm, Df = 2.160, (b) 3D rendering ofa particle from flame A3. Dc = 50.1 nm, Df = 2.025,Ds = 0.731.


tion simulations. This is due to the fractal aggre-gate particles having greater surface area thanthe equivalent spherical particles, hence the sur-face growth processes occur at a much greaterrate. These features are similar to previous model-ling studies [5,18].

Figure 3 shows the particle size distributionfunctions (PSDFs) of the volume equivalentsphere diameter, Dv, for flames A1 and A3 at 28and 18 ms, respectively, (solid lines) plottedagainst the corresponding experimental data ofZhao et al. [30] (circles). We can see in Fig. 3a thatflame A1 exhibits a bimodal PSDF whilst Fig. 3bshows that flame A3 exhibits a unimodal distribu-tion. This behavior is captured by the model how-ever it predicts a broader PSDF. This is due inpart to the original model being optimized for aspherical particle description, rather than onewhere the full morphology of the particles is cal-

0.66

0.68

0.7

0.72

0.74

Flame A1Flame A3

Sha

pe d

escr

ipto

r

2

2.2

2.4

2.6

2.8

3

2824201612840

Flame A1Flame A3

Frac

tal d

imen

sion

time / ms

2824201612840

time / ms

a

b

Fig. 6. Temporal evolution of mean fractal dimensionand shape descriptor flames A1 and A3. (a) Temporalevolution of average Ds in flames A1 and A3. (b)Temporal evolution of average Df in flames A1 and A3

culated. This leads to an overprediction of thecoagulation and surface growth rates.

Figure 4 shows how various mean particleproperties evolve throughout the flames. It isobserved that the average collision diameter andthe average primary particle diameter are smallerin flame A1 than in flame A3, furthermore the cal-culated number of primary particles is higher inflame A1 than A3. It can be seen in Fig. 5 thatthe aggregate from flame A1 (Fig. 5a) appears tohave larger primary particles than the aggregatefrom flame A3 (Fig. 5b). However, the primaryparticles from the aggregate of flame A1 are con-siderably ‘‘bumpier’’ and hence have greater sur-face area than those of flame A3. Since theprimary particle diameter was calculated usingEq. (2) it is clear that the calculated primary diam-eter will be smaller. This indicates that one of thedisadvantages of using Eq. (2) to calculate aprimary particle diameter with particles of a mor-phology observed in Fig. 5a.

Figures 7 and 8 show renderings of the largestparticles obtained in the simulations of flames A1and A3. The TEM-style rendering is of the sameparticle shown in the 3D rendering, but from adifferent angle. Note that the primary particle size

Fig. 7. Renderings of largest particle from flame A1.Dc = 169 nm, Df = 1.826, Ds = 0.799. (a) 3D renderingsof largest particle from flame A1. (b) TEM-stylerenderings of largest particle from flame A1.


for both particles is unchanged from the render-ings seen in Fig. 5.

Figure 6 shows the temporal evolution of theaverage fractal dimension, Df, and the averageshape descriptor, Ds, defined by

Ds ¼logða=a0Þlogðv=v0Þ

: ð4Þ

It can be seen that the average shape descriptorstarts at a value of 0:66 _6, quickly rises, and thenslowly asymptotes to a value of about 0.735. Thisvalue is approximately the same for both flames.The average fractal dimension for both flamesstarts from a value of 3, which then falls off rapid-ly and eventually asymptotes to a final value. Inthe case of flame A1, the final value isDf = 2.348, while for flame A3 the final value isDf = 2.160. This is consistent with the observa-tions of Fig. 5 which show that the aggregate fromflame A1 is made up of a smaller number of moredensely packed and bumpy primary particles thanthe aggregate from flame A3.

Fig. 8. Renderings of largest particle from flame A3.Dc = 183 nm, Df = 1.838, Ds = 0.784. (a) 3D renderingsof largest particle from flame A3. (b) TEM-stylerenderings of largest particle from flame A3.

Figures 9 and 10 show the temporal evolutionof the joint probability distribution functions ofthe shape descriptor against the collision diameterfor flames A1 and A3. With both flames, theshape of this distribution is set up in the earlystages of particle formation and merely expandsalong the vertical axis of collision diameters. Withflame A1, it can be seen that at the end of theflame (Fig. 9b) there exists a secondary peak inthe bottom left corner where the smallest sphericalparticles exist, which is not present at the end offlame A3 (Fig. 10b).

The particle seen in Fig. 5a has a collisiondiameter of 49.5 nm, a fractal dimension of2.160 and a shape descriptor of 0.763. This placesit in the middle of the contour of maximum valueon Fig. 9b. The particle seen in Fig. 5b has a col-lision diameter of 50.1 nm, a fractal dimension of2.025 and a shape descriptor of 0.731, placingnear the contour of maximum value on Fig. 10b.The larger particles (Figs. 7 and 8) by contrastlie to the top right of Figs. 9 and 10, respectively.

shape descriptor

Dc

/ nm

0.68 0.70 0.72 0.74 0.76 0.78 0.80

050

100

150

shape descriptor

Dc

/ nm

0.68 0.70 0.72 0.74 0.76 0.78 0.80

050

100

150

b

a

Fig. 9. Joint distributions of shape descriptor andcollision diameter for flame A1. The numbers on thecontours represent the log of the distribution function.(a) Contour plot at 7.9 ms. (b) Contour plot at 28 ms.

shape descriptor

Dc

/ nm

0.68 0.70 0.72 0.74 0.76 0.78

050

100

150

shape descriptor

Dc

/ nm

0.68 0.70 0.72 0.74 0.76 0.78

050

100

150

a

b

Fig. 10. Joint distributions of shape descriptor andcollision diameter for flame A3. The numbers on thecontours represent the log of the distribution function.(a) Contour plot at 5.9 ms. (b) Contour plot at 18 ms.


6. Conclusions

It is usually assumed that inception and evolu-tion of particles in a reactive system, such as sootformation in fossil-fuel flames, can be separatedinto essentially non-intersecting regimes of parti-cle nucleation, particle coagulation and surfacegrowth, and particle aggregation. Recent studies,using numerical simulations of single-particle tra-jectories [16,17] and of ‘‘lumped’’ properties withthe method of moments [18], suggested a criticalimportance of interplay among all these individu-al processes—nucleation, coagulation, and surfacegrowth. Moreover, these results suggested that thetransition of spherical, coalescent growth intofractal-like objects is linked directly to thenucleation, as it supplies the abundance of verysmall primary particles.

The present modelling results, now per-formed with a numerical model that incorpo-rates all physical processes at a detailed levelalong with the resolution of individual particlesand their distributions, fully supports the earliersuggested mechanistic features. The specific sim-

ulations were performed for two flames: withcharacteristically different bimodal and unimodalsoot particle distributions. The present model isable to capture the evolution of soot particleproperties observed experimentally in theseflames [30]. These results confirm thus that theorigin of the differences in the PSDFs is thestrength of the nucleation, suggested theoretical-ly two decades ago [4,31,32].

Acknowledgments

This work was funded by the EPSRC (GrantNo. GR/R85662/01) for the financial support ofNeal Morgan under the title Mathematical andNumerical Analysis of Coagulation–DiffusionProcesses in Chemical Engineering. The work atBerkeley was supported by the Director, Officeof Energy Research, Office of Basic Energy Sci-ences, Chemical Sciences, Geosciences and Biosci-ences Division of the US Department of Energy,under Contract No. DE-AC03-76F00098.

References

[1] D. Lindackers, M.D.G. Strecker, P. Roth, C.Janzen S.E. Pratsinis, Combust. Sci. Tech. 123(1997) 287–315.

[2] P.T. Spicer, O. Chaoul, S. Tsantilis, S.E. Pratsinis,Aerosol Sci. 33 (2002) 17–34.

[3] Z. Sun, R. Axelbaum, B. Chao, Proc. Combust.Inst. 29 (2002) 1063–1069.

[4] M. Frenklach, S.J. Harris, J. Colloid Interface Sci.118 (1987) 252–261.

[5] A. Kazakov, M. Frenklach, Combust. Flame 114(1998) 484–510.

[6] M. Frenklach, Chem. Eng. Sci. 57 (2002) 2229–2239.

[7] M. Balthasar, M. Kraft, Combust. Flame 133 (2003)289–298.

[8] B.S. Haynes, H.G. Wagner, Prog. Energy Combust.Sci. 7 (1981) 167–229.

[9] U.O. Koylu, G.M. Faeth, T.L. Farias, M.G.Carvalho, Combust. Flame 100 (1995) 621–633.

[10] D.B. Kittelson, J. Aerosol Sci. 29 (1997) 575–588.[11] C.M. Megaridis, R.A. Dobbins, Combut. Sci. Tech.

63 (1989) 152–167.[12] M. Kraft, Kona, Powder and Particle 23 (2005) 18–35.[13] M. Goodson, M. Kraft, J. Comput. Phys. 183

(2002) 210–232.[14] J. Singh, R. Patterson, M. Balthasar, M. Kraft, W.

Wagner, Combust. Flame 145 (3) (2006) 638–642.[15] R. Patterson, J. Singh, M. Balthasar, M. Kraft, J.

Norris, SIAM J. Sci.Comput. 28 (1) (2006) 303–320.[16] P. Mitchell, M. Frenklach, Proc. Combust. Inst. 27

(1998) 1507–1514.[17] P. Mitchell, M. Frenklach, Phys. Rev. E 67 (2003)

061407.[18] M. Balthasar, M. Frenklach, Combust. Flame 140

(2005) 130–145.[19] M. Frenklach, H. Wang, in: H. Bockhorn (Ed.),

Soot Formation in Combustion. Mechanisms andModels, Springer Verlag, 1994, pp. 165–192.


[20] M. Frenklach, Phys. Chem. Chem. Phys. 2 (2002)2028–2037.

[21] H. Wang, M. Frenklach, Combust. Flame 110 (1-2)(1997) 173–221.

[22] J. Appel, H. Bockhorn, M. Frenklach, Combust.Flame 121 (2000) 122–136.

[23] M. von Smoluchowski, J. Phys. Z 17 (1916) 557571,585599.

[24] M. Frenklach, H. Wang, Proc. Combust. Inst. 23(1990) 1559–1566.

[25] F. Mauss, B. Trilken, H. Breitbach, N. Peters, SootFormation in Combustion. Mechanisms and Models,Springer Verlag, Berlin, 1994, p. 325.

[26] A. Eibeck, W. Wagner, SIAM J. Sci. Comput. 22(3) (2000) 802–821.

[27] P. Mitchell, Ph.D. Thesis, University of California,Berkeley, 2001.

[28] S. Tsantilis, S.E. Pratsinis, Langmuir 20 (2004)5933–5939.

[29] R. Jullien, J. Phys. A 17 (1984) L771–L776.[30] B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang,

Proc. Combust. Inst. 30 (2005) 1441–1448.[31] R.A. Dobbins, G.W. Mulholland, Combust. Sci.

Technol. 40 (1984) 175–191.[32] J.D. Landgrebe, S.E. Pratsinis, Ind. Eng. Chem.

Res. 28 (1989) 1474–1481.

Comment

B.A. Dobbins, Brown University, USA. The progressin this model is remarkable. Two effects that remain tobe discussed are the development of classification andcrystallinity that is apparent in dark field TE.

Reply. Currently the physical knowledge/under-standing of the soot particle structure at an elementa-ry/molecular level is sufficient to allow it to bemodeled within the context of a population balanceapproach. However, in the future we plan to extendour model to include such details as soon as they be-

come available, and the numerical method which wehave developed in this paper is sufficiently powerfulto permit this. In particular, the incorporation of fur-ther particle properties such as the chemical composi-tion of soot particles is possible. Also, to ourknowledge, while there are indications of some struc-tural re-arrangements, there is no experimental evi-dence for the development of a crystalline latticewithin the soot particle—the structures reported forthe past three decades or so are consistently elec-tron-amorphous.

Numerical simulations of soot aggregation in premixed laminar flames

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