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Proceedings of the Combustion Institute 34 (2013) 1861–1868

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of the

CombustionInstitute

Modeling of soot particle size distributionsin premixed stagnation flow flames

R.P. Lindstedt a,⇑, B.B.O. Waldheim a,b

a Department of Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2AZ, UKb Advanced Combustion Systems, Scania CV AB, Sodertalje 151 87, Sweden

Available online 18 June 2012

Abstract

Soot formation in five laminar premixed ethylene flames was studied computationally in order to inves-tigate the ability of a sectional model to reproduce measured soot particle size distributions (PSDs). Theflames were modelled using a burner-stabilised stagnation flow configuration due to the influence of thesampling probe. In the vicinity of the probe, the residence time is relatively long and the temperaturereduced. It was found that under such conditions the coagulation efficiency for small (<10 nm) particlesbecomes important due to the combination of a relatively low nucleation rate with a significant collisionfrequency. In previous studies, the impact of the probe was only accounted for by shifting the positionof the measured PSDs. A model for the collision efficiency was introduced with a particle size dependentinterpolation between a lower limit, based on the Lennard–Jones potential, and an upper limit, based oncollision stabilisation caused by the surrounding gas. The developed model is consistent with a low collisionefficiency for PAH coagulation and explains the measured high number densities of small particles. The sizeof particles affected by a reduced collision efficiency were found to decrease with increasing flame temper-ature. The effect is consistent with temperature dependent internal carbonisation processes changing thepolarisability of particles leading to a strengthening of attractive forces and enhanced collisional stabilisa-tion. The calculated PSDs were found to be in reasonable agreement with measurements.� 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Particle dynamics; Soot; Surface chemistry effects; Premixed laminar flames

1. Introduction

Sectional methods for the calculation of sootformation have been developed in order to obtainthe particle size distribution (PSD) with reasonablecomputational expense. Smooke et al. [1]developed a mass conserving sectional approach[2] and Wen et al. [3] used a moving sectional

1540-7489/$ - see front matter � 2012 The Combustion Instithttp://dx.doi.org/10.1016/j.proci.2012.05.047

⇑ Corresponding author. Fax: +44 207 589 3905.E-mail addresses: p.lindstedt@imperial.ac.uk,

p.lindstedt@ic.ac.uk (R.P. Lindstedt).

technique. Bhatt and Lindstedt [4] developed asectional model that conserves both soot particlenumber densities and soot mass. The model wasapplied to the NIST combined well-stirred (WSR)and plug flow (PFR) reactor configuration [5] andto the PFR system of Kronholm and Howard[6,7]. Subsequent experimental data from the NISTWSR/PFR system [8] was studied by Lindstedtet al. [9] using updated thermodynamic data forpolycyclic aromatic hydrocarbons (PAH). The sen-sitivity to the nucleating species and to a sizedependent coagulation collision efficiency for parti-cles <1 nm was also investigated due to persistent

ute. Published by Elsevier Inc. All rights reserved.

1862 R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868

uncertainties. PAH dimers and clusters have beenobserved experimentally in flames through photoi-onisation mass spectrometry [10]. Moleculardynamics (MD) simulations suggest that the for-mation is reversible [11,12] with a collision effi-ciency <1 and decreasing with a reduction in size[11]. Raj et al. [13] included an empirical modelfor the collision efficiency in a stochastic model ofPAH growth. Collision efficiencies for coagulationwere estimated from premixed laminar flame databy D’Alessio et al. [14]. Values for particles around3 nm were found to be within the range 1 � 10�3 to2 � 10�2 and close to unity for particles >6 nm.Harris and Kennedy [15] estimated the coagulationrate by intermolecular forces from light scatteringmeasurements in a ethylene flame and found anenhancement factor just below 2 for a mean particlediameter ’11 nm. Abid et al. [16–18] measuredsoot PSDs in premixed laminar ethylene flameswith different maximum temperatures. The sam-pling technique was found to cause flow stagnationand to reduce temperatures in the vicinity of theprobe and it was recommended to model the flamesusing a burner-stabilised stagnation (BSS) configu-ration [18]. Stochastic models were applied byCelnik et al. [19] and Singh et al. [20] to study,respectively, the flames of Abid et al. [16] and Zhaoet al. [21] using a burner-stabilised free flame con-figuration. Sampling probe effects were accountedfor by shifting the position of the measured PSDs.The current work investigates the ability of asectional model to reproduce PSDs for the fullrange of laminar ethylene flames measured by Abidet al. [16–18]. The flames are computed using a BSSgeometry. The sensitivity of PSDs to collision effi-ciencies for the coagulation of soot particles<10 nm is explored in detail along with that to prin-cipal modelling assumptions. The oxidation of sootis also extended to include a PAH analogy for theavailability of surface sites.

2. Model

2.1. Sectional soot model

The current sectional soot model is based onprevious studies [4,6,9] with soot particles subjectto thermophoretic drift ðvthÞ calculated using thefree-molecular regime limit [22] given in Eq. (1),where t is the kinematic viscosity and T thetemperature.

vth ¼ �0:55t@ lnðT Þ=@z ð1ÞEquation (1) can also be used for part of the tran-sition regime ðKn > 1Þ without significant devia-tions [23]. The Knudsen number ðKn ¼ 2k=dpÞ isbased on the mean free path ðkÞ and the sootdiameter ðdpÞ computed from the particle massand a density ðqsÞ of 1800 kg=m3. Inception ofsoot formation through pyrene ðA4Þ dimerisation

ðdp;A4¼ 0:38 nmÞ and acetylene based surface

growth was sufficient to form larger particles.Abid et al. [16,17] collected and analysed sootparticles using transmission electron microscopy(TEM) and atomic force microscopy (AFM) andfound liquid-like particles, suggesting that car-bonisation [24] is competing with surface growth.The latter promotes a high H/C ratio at the sur-face [16]. Bhatt and Lindstedt [4] used collisionefficiencies for O, OH and O2 from the study byRoth et al. [25] on the oxidation of carbon black.However, for particles that are not fully carbon-ised the reactivity can be expected to be higher.Lindstedt and Louloudi [26] derived the naphtha-lene based analogy given in Eq. (2) to estimate thenumber of radical sites per unit soot surface areaðvSÞ with WðT ;/kÞ derived from truncated steadystate balances.

vS ¼ as �WðT ;/kÞ �vS�H

NAð2Þ

In Eq. (2), N A is Avogadro’s number, vS�H thenumber of hydrogen sites per unit surface area(2.32 � 1019 m�2) and as the fraction of sitesavailable for hydrogen abstraction. A directanalogy for soot particle oxidation via O2 pro-ceeds via Eq. (3) involving the naphthyl radicaland complements the previous oxidation model[4,25].

C10H7 þO2 ! C10H7OþO ð3ÞA soot growth rate based on the addition of acet-ylene ðC2H2Þ to the 4-phenanthryl radical (4-A3�)leading to pyrene [27] was adopted.

4�A3� þ C2H2 ! A4 þH ð4Þ

The soot particle composition and reactivity willevolve [19,20] and the current constant value ofas ð¼ 0:20Þ should be considered a mean. Theassigned value will also depend on other modelaspects and the current choice is lower than thatused by Kazakov et al. [28] ð0:35 6 as 6 0:60Þ inthe context of a method of moments basedapproach.

Narsimhan and Ruckenstein [29] developedmodels for the lower and upper limit of the colli-sion efficiency for particles of the same size. Thelower limit was derived from the Lennard–Jonespotential and the assumption that a moving parti-cle does not experience friction caused by colli-sions with the surrounding gas. The collisionefficiency in the free-molecular regime al

e

� �is then

given by Eq. (5), where U0 is the potential welldepth and kb is the Boltzmann constant. Thedimensionless surface temperature ðhÞ is given byEq. (6) and accounts for the additional energyinvolved in a collision due to the potential. Theequations remain valid for a poly-disperse aerosolin the free-molecular regime as shown in theSupplementary data.

Fig. 1. The parameter D as a function of maximumflame temperature (—) together with (�) optimumvalues for flame sets C1–C5 (right to left) at H = 1.0 cm,except for C2 where the PSD is clearly bimodal atH = 0.8 cm.

R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868 1863

ale ¼ 1� 1þ U0

kbT h

� �exp � U0

kbTh

� �ð5Þ

h ¼ 1þ 2U0

3kbTð6Þ

The upper limit is based on the assumption that aparticle moving away after a collision experiencesfriction and will be trapped in the potential energywell resulting in a collision efficiency of unity. Thelower limit is assumed valid for “sufficiently smallparticles” and the upper limit for “sufficientlylarge particles” (’10 nm) [29]. D’Alessio et al.[14] evaluated a version of the lower limit model[29] without the dimensionless surface tempera-ture and found good to reasonable agreementfor particles around 3 nm with the collisionefficiency under-predicted for particles P6 nm.The collision efficiency ðaeÞ forms part of thecoagulation rate in the free-molecular regime

bfi;j

� �, see Eq. (7), and is here interpolated as a

function of particle size between the lower ale

� �and upper au

e ¼ 1� �

limits according to Eq. (8).The mixing functions ðfiÞ are given in Eq. (9).

bfi;j ¼ Caae

3

4p

� �1=6 6kbT ðvi þ vjÞqsvivj

� �1=2

� v1=3i þ v1=3

j

� �2

ð7Þ

ae ¼ ðfifjÞ1=2aue þ ð1� ðfifjÞ1=2Þal

e ð8Þ

fi ¼1

2ðtanhðA�ðdp;i � DÞÞ þ 1Þ ð9Þ

In Eq. (7), v is the volume of a particle. Multipli-cation of Eq. (7) by the van der Waals enhance-ment factor ðCaÞ and the collision efficiency ðaeÞis necessary in order to account for both phenom-ena [30,31]. The current value ðCa ¼ 3:0Þ is com-parable to the suggestion of 2.2 by Harris andKennedy [15]. In Eq. (9), D is the diameter wherethe upper bound begins to be dominant andA�ð¼ ln½ae;A4

=ð100� ae;A4Þ�=ð2dp;A4

� 2DÞÞ is de-fined so that the contribution of au

e remains insig-nificant ð� 1%Þ for pyrene. The Hamakerconstant, determining the strength of particle–particle potential energy interactions, can beexpected to increase towards the value of graphite(�5 � 10�19) J [14]) during the carbonisation pro-cess. Here the lower limit [29] for non-carbonisedparticles was estimated using a non-retardedHamaker constant of 7.0 � 10�20 J. D’Alessioet al. [14] used a lower limit value of3.0 � 10�20 J and D’Anna and Kent [32] appliedthe value for benzene [33] of 5:0� 10�20 J . Theposition of the potential minimum between atomswas set to 0.388 nm [34]. The upper bound issubject to some uncertainty. Narsimhan andRuckenstein [29] showed that 6 – 7.6 nm particleshave a collision efficiency around unity andD’Alessio et al. [14] estimated ’0.75 for 6 nm soot

particles. The optimal value of D was found tovary in the range 2.5–6 nm for the current flamesand to decrease with increasing maximum flametemperature ðT maxÞ. A possible explanation is thatthe rate of internal carbonisation increases withtemperature and hence causes the polarisabilityof particles to change [14]. This may lead to a fas-ter transition to the upper bound due to strongerattractive forces in combination with friction. Theeffect of internal carbonisation may be qualita-tively taken into account by allowing D to be tem-perature dependent. The functional form of Eq.(10) is used here and is effective within the limits2:0 6 D ðnmÞ 6 10:5 as shown in Fig. 1. The cor-responding collision efficiency function forD = 5.0 nm is shown in Fig. 2.

D ¼ 2:0� 10�9 þ 4:25� 10�9

� tanh27000

T maxðKÞ� 15:57

� �þ 1

� �ð10Þ

2.2. Physical model

A similarity transformed coordinate system [35]was used to compute the current BSS flames withthe strain rate ð23 6 a ðs�1Þ 6 95Þ determined aspart of the solution procedure by specifying a fixeddomain size resolved by 120 volumes correspond-ing to the experimental geometry. The composi-tion of C2H4=O2=Ar was 16.3/23.7/60 mol%ð/ ¼ 2:07Þ. A variation in the inlet velocity (13.0,10.0, 8.0, 6.5 and 5.5 cm/s, for C1 to C5 at stan-dard temperature and pressure) was used tochange the temperature [16]. Temperature mea-surements under stagnation flow conditions areonly available for flame set C3 [18] and were usedas boundary conditions. The temperature was cor-rected ðT cÞ via Eq. (11), where T ad is the adiabaticflame temperature and n ð¼ 0:19Þ is the heat lossfactor, in order to improve agreement withmeasured values [18].

Fig. 2. The collision efficiency for coagulation as afunction of particle diameter and for different temper-atures: (—) 600 K, ð Þ 1050 K, (- -) 1500 K, (- -)1950 K. A value of 5.0 nm of the parameter D was used.

1864 R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868

T c ¼ T 1� nT

T ad

� �2 !

ð11Þ

2.3. Gas phase chemistry

The gas phase chemistry of Lindstedt et al. [36],leading to the formation and oxidation of naphtha-lene, was combined with the mechanism of Wangand Frenklach [27] for higher PAHs up to pyreneand updated using the indene and cyclopentadienechemistry outlined by Lindstedt et al. [6] and Rob-inson and Lindstedt [37]. The acetylene additionsteps [27] were augmented by parallel paths leadingto pyrene via addition of C3 species such as thepropargyl and allyl radicals. The C3-growth mech-anism was developed using reaction class basedestimates with naphthalene and acenaphthaleneas starting points. Important intermediates includephenalene, cyclopenta[a]naphthalene and cyclo-penta[def]phenanthrene. The mechanism makesonly a minor contribution to soot formation inthe current flames and is listed for completenessin the Supplementary data along with species struc-tures and thermodynamic properties. The completegas phase mechanism includes 357 species and 1785reactions. It is, however, recognised that furtherimprovements in the PAH inception chemistryremain desirable.

3. Results and discussion

Measured [18] and calculated temperature pro-files are shown for three C3 flames in Fig. 3. Theagreement is arguably reasonable and also appliesto flames not shown, although the calculated tem-perature is somewhat higher than measurements inparts of post flame zone. A possible reason is thatthe positional uncertainty of the thermocoupleincreases near the stagnation plate [18]. The PSDsof the other studied flame sets were measured usinga tubular probe [16,17] and use of the recom-mended BSS geometry is an approximation.

However, the T max values are approximately thesame as for the plate probe, though temperaturesin the post flame zone were somewhat higher forthe tubular probe [18]. The calculated soot PSDsobtained using 43 sections are shown in Figs. 4–6for flame sets C3–C5 with soot assumed to bedeposited on the probe due to thermophoreticforces. Reasonable to good agreement is obtainedand some of the differences may be due to theapproximate nature of the BSS approximationclose to the stagnation plane. However, the currentcollision efficiency model does offer an explanationfor the high concentration of smaller particles inthe vicinity of the probe, where the nucleation rateis relatively low due to reduced temperatures andthe small particles already formed are scavengedby collisions with larger particles during the rela-tively long residence time caused by flow stagna-tion. Larger particles are subject to a collisionefficiency below unity as the flame temperaturedecreases with the position of the local minimumin the soot PSD moving to larger diameters. Thecurrent work suggests that the cause is consistentwith an increased rate of internal carbonisationcaused by increased flame temperatures leadingto a change in the polarisability and an earliertransition to the upper bound limit due tostronger attractive forces enhancing collisionalstabilisation.

The sensitivity of the model to the fraction ofsites available for hydrogen abstraction ðasÞ wasinvestigated through ±50% changes as shown inFig. 4. A significant sensitivity is observed withthe variation encompassing the measured PSDs.The sensitivity of the PSDs to variations in theparameter D, see Eq. (9), was investigated through±1 nm changes with results shown in Fig. 5. Thevariation results in a change of position of thelocal minima in the PSDs. Reasonable agreementis obtained for all cases. By contrast, the assump-tion of unity collision efficiency ðae ¼ 1Þ results inexcessive depletion of small particles as alsoshown in Fig. 5. The sensitivity of the PSDs tointer-molecular forces is shown in Fig. 6. The barsshow calculations with the value of the Hamakerconstant ð5:0� 10�20 JÞ set to that of benzene[33] as used by D’Anna and Kent [32]. The con-centration of small particles increases due to theweaker attractive forces, which can be explainedfor this particular flame set (C5) by the lowertemperatures.

The full coagulation model of Narsimhan andRuckenstein [29] was also applied to model flameset C5 assuming that mean values of modelparameters can be formed for particles of differentsize classes as described in the Supplementarydata. The resulting PSDs were found to be almostunchanged under the current conditions, though itwas observed that coagulation rates for particlesof different size classes larger than 26 nm wereincreased. Reasons include the treatment of the

Fig. 3. Calculated (–) and measured ð�Þ [18] temperature profiles for three C3 flames. Temperature measurementuncertainties and in one case the positional uncertainty are shown with bars.

Fig. 4. Measured and calculated soot PSDs for flame set C3. Measured soot PSDs are shown with open circles ð�Þ [16]and dots ð�Þ [18]. Calculated soot PSDs are shown with solid lines (—) and soot PSDs calculated with as in Eq. (2)changed by ±50% are shown with dashed thick bars and solid thin bars, respectively.

R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868 1865

transition regime and the assumption of amono-disperse aerosol [29]. The concentration ofparticles subject to a collision efficiency belowunity is sensitive to temperature as shown inFig. 7 for flame set C1. The bars show the effectof varying the heat loss coefficient in Eq. (11) asn ¼ 0:19 0:05 resulting in T max changes of�48/+35 K. The variation is similar to measure-ment uncertainties [16,18]. The agreementobtained for the C2 data set is similar to the C1flames and results are therefore not shown.

4. Conclusions

Soot formation in five premixed laminarethylene stagnation flow flames [16,18] have beeninvestigated computationally. Flow stagnation in

the vicinity of the sampling probe is causing theresidence time to become relatively long, withthe collisions of soot particles still significant,while the reduced temperature causes the nucle-ation rate to become comparatively low. Hence,the measured PSDs are sensitive to the coagula-tion collision efficiency for smaller ð< 10 nmÞ sootparticles and the flames are particularly useful forexploring this aspect. A model for the collisionefficiency of coagulation was introduced andbased on a soot particle size dependent interpola-tion between a lower limit derived from theLennard–Jones potential and an upper limitinvolving collisional stabilisation by the surround-ing gas. The decreasing collision efficiency withdecreasing soot particle size does explain the highnumber densities of small particles in the vicinityof the probe. By contrast, a constant (unity)

Fig. 5. Measured and calculated soot PSDs for flame set C4. Measured soot PSDs are shown with open circles ð�Þ [16].Calculated soot PSDs are shown with solid lines (—), soot PSDs calculated with the parameter D in Eq. (9) changed by±1 nm are shown with solid thin bars and dashed thick bars, respectively, and the dashed lines (- -) are PSDs calculatedusing a unity collision efficiency ðae ¼ 1Þ.

Fig. 6. Measured and calculated soot PSDs for flame set C5. Measured soot PSDs are shown with open circles ð�Þ [16].Calculated soot PSDs are shown with solid lines (—) and with bars showing corresponding PSDs calculated with thevalue of the Hamaker constant set to 5:0� 10�20 J [32].

1866 R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868

collision efficiency does cause excessive depletion.The local minima in the measured PSDs move tosmaller particle sizes when the flame temperatureincreases and a possible explanation is that therate of internal carbonisation increases, resultingin a changed polarisability that strengthens the

attractive forces between particles and cause, incombination with friction, an earlier transitionto the upper limit collision efficiency. The effectwas taken into account and calculated soot PSDswere found to agree reasonably well withmeasurements.

Fig. 7. Measured and calculated soot PSDs for flame set C1. Measured soot PSDs are shown with open circles ð�Þ [16]and open squares ð�Þ [17]. Calculated soot PSDs are shown with solid lines (—). The impact of changing the heat losscoefficient in Eq. (11) by n ¼ 0:19 0:05 leading to changes in the peak temperature of +35/�48 K are shown with solidthin bars and dashed thick bars, respectively.

R.P. Lindstedt, B.B.O. Waldheim / Proceedings of the Combustion Institute 34 (2013) 1861–1868 1867

Acknowledgments

The authors are grateful for the financial sup-port of Scania CV AB and the Swedish EnergyAgency and to Madelaine Nordqvist, OlaStenl _a _as, Jonas Holmborn and Rolf Egnell fortheir support. The authors would also like tothank Roger Robinson for advice on calculationsof thermodynamic data.

Appendix A. Supplementary data

Supplementary data associated with this articlecan be found, in the online version, at http://dx.doi.org/10.1016/j.proci.2012.05.047.

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