Stochastic modeling of soot particle sizeand age distributions in laminar premixed flames

Jasdeep Singha, Michael Balthasara, Markus Krafta,*, Wolfgang Wagnerb

a Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UKb Weierstrass Institute of Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Abstract

In the present study, the evolution of size distributions of soot particles in premixed laminar flames isinvestigated. A computationally efficient stochastic approach is used to model the dynamics of the sootparticle ensemble. The stochastic method is enhanced to include a formalism for modeling coagulationat high pressures. A detailed kinetic model is used to describe gas phase reactions and the formation,growth, and oxidation of soot particles. Four fuel rich laminar premixed flames at different pressure areinvestigated. It is found that calculated size distributions vary markedly in shape between the differentflames. The distributions calculated in the two flames at elevated pressure and in the sub-atmospheric flameexhibit log-normal shape in the post-flame zone. In contrast, a bimodal distribution is found for the atmo-spheric flame, which persists throughout the entire flame. The bimodal behavior could be attributed to con-tinuous nucleation. Furthermore, the effect of surface ageing, i.e., the deactivation of sites on the sootparticles� surface available for reaction with gas phase species, is investigated. For this purpose, a definitionof a particle�s age is introduced, and age distributions are calculated. Subsequently, it was investigated ifthe surface reactivity could be correlated with the particle age. Two different correlations were investigated:(a) a step-function attributing a high surface activity to young particles and low activity to old particles and(b) an exponential function giving a smooth transition of surface activity with particle age. Good agree-ment with measured soot volume fractions could be obtained with both approaches. The decay constantin the exponential correlation was found to be a linear function of maximum flame temperature for threeof the four calculated flames. For these three flames, the experimentally established trend that surface deac-tivation proceeds faster at higher temperatures could thus be reproduced.� 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Soot formation; Population balances; Stochastic modeling; Surface ageing

1. Introduction

The formation of carbonaceous particulatematter is often observed in fuel rich premixedflames as well as in diffusion flames [1]. This posesa great challenge in practical devices such as diesel

1540-7489/$ - see front matter � 2004 The Combustion Institdoi:10.1016/j.proci.2004.08.120

* Corresponding author. Fax: +44 1223 334796.E-mail address: markus_kraft@cheng.cam.ac.uk

(M. Kraft).

engines and gas turbines due to the particles� ad-verse health effects. The understanding of theunderlying chemical and physical processes lead-ing to the formation and growth of these particlesis thus of great importance.

To model the evolution of an ensemble of sootparticles, the appropriate population balanceequation [2] has to be solved. Different approacheshave been proposed in the past. Frenklach et al.[3–5] applied the method of moments to study soot

ute. Published by Elsevier Inc. All rights reserved.

1458 J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465

formation in premixed flames using a detailed ki-netic soot model. A major feature of the methodof moments is its computational efficiency, whichenabled its application to more complex problemssuch as soot formation in diesel engines and gasturbines [6–8]. While the method of moments pre-dicts ensemble averaged quantities, the exact shapeof the size distribution is not known explicitly.

Other approaches that have been applied tomodel the dynamics of the size distribution of sootparticles include a discrete sectional method [9], aGalerkin approach [10], and stochastic methods[11,12]. In the latter studies, only atmosphericand low-pressure flames could be investigatedsince the description of coagulation was limitedto the free molecular regime. Stochastic methodshave also been found to be very efficient to studyother aspects of soot formation such as the poly-merization of polycyclic aromatic hydrocarbons[13] and the simulation of trajectories of singleparticles in flames [14].

Besides suitable methods for modeling thedynamics of the soot particle ensemble, detailedknowledge of the processes of particle formation,and growth is essential to accurately model the for-mation of soot. Among the different processes,reactions of gas phase species with sites on the par-ticles� surface have been identified to govern the fi-nal particle mass [15]. It was experimentallyobserved that the reactivity of surface sites de-creases with increasing particle growth or age[1,15–17]. Hence, this process is often called surfaceageing. It was attributed to a decrease of active sur-face sites, i.e., sites that are accessible for reaction.

The notion of active sites on the soot particlesurface was introduced into kinetic soot modelingby Frenklach and Wang [18]. In conjunction witha decrease in concentration of H-radicals [5,18], itwas used as an explanation for the experimentalobservation of surface ageing. On a mechanisticbasis, Frenklach and co-workers [19,20] attributedsurface ageing to the formation of defects on theparticles� surface generated during surface growth.Surface ageing was also attributed [21–23] to thereversibility of the acetylene addition step in ahydrogen abstraction–carbon addition (HACA)[5,18] surface growth scheme.

While initially a constant fraction of activesites was used with the kinetic soot model [5,18],it was expressed as a function of flame tempera-ture [24], and subsequently as a function of flametemperature and mean particle size [25]. A globalexpression for the mean fraction of active surfacesites for a series of laminar premixed flames wasestablished [25].

The aim of the present study is twofold. First,the stochastic approach presented in [11] is en-hanced to include coagulation at high pressures.Previous investigations of size distributions[11,12] are extended to a series of laminar pre-mixed flames varying in pressure.

Second, a definition of particle age is proposed,and particle age distributions are calculated. Thisis possible because a stochastic particle method isused, which can carry different properties for eachstochastic particle. Hence, individual fractions ofactive surface sites can be assigned to each parti-cle. Using this approach it is investigated if thenumber of active sites could be correlated withparticle age. The fact that each particle can carryits individual fraction of active surface sites meansthat not only laminar premixed flames but alsoother classes of problems can be described wherethe history of each particle might differ. The pres-ent paper is a first attempt to investigate thisapproach.

2. Method

The evolution of the soot particle size distribu-tion functions in laminar premixed flames wassimulated in two steps. In the first step, the flamestructure and moments of the soot size distribu-tion [4] were simulated using detailed kinetic mod-els for gas phase chemistry [25,26] and sootformation [5,25]. In the second step, the soot sizedistribution was calculated with a stochastic ap-proach using results from the first step as aninput.

2.1. Flame simulations

The flame structure was calculated with thePREMIX computer code [27]. A chemical mecha-nism describing the oxidation of the fuel and theformation of polycyclic aromatic hydrocarbons(PAHs) was applied [25,26]. The PREMIX codeused in the present study had been modified [28]to include transport equations for the soot mo-ments [29]. The processes involved in the forma-tion of soot particles were described on the basisof the detailed kinetic soot model developed byFrenklach and co-workers [5,25] and the methodof moments [4]. Rates of soot formation includednucleation, coagulation, and condensation ofPAHs onto the surface of soot particles, surfacegrowth, i.e., the addition of acetylene to radicalsites on the surface, and oxidation by hydroxylradical and molecular oxygen.

Using the calculated velocity profile of the pre-mixed flame, the spatial coordinate, i.e., heightabove burner, was transformed into a residencetime. Temperature and species concentrations asfunction of residence time were saved and usedas input to the simulations calculating the dynam-ics of the particle size distribution function.

2.2. Simulation of the soot particle size distribution

The dynamics of an ensemble of soot particlesis described by the Smoluchowski equation [30]

J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465 1459

with additional source terms for nucleation andsurface reactions:

o

otNðt; iÞ ¼RðtÞd�in þ

1

2

Xi�1

j¼1

bi�j;jNðt; i� jÞNðt; jÞ

�X1j¼1

bi;jNðt; iÞNðt; jÞ

þX4

l¼1

kli�dðlÞNðt; i� dðlÞÞ � kliNðt; iÞh i

;

ð1Þ

where N(t, i) is the number density of particles ofsize i at time t, R(t) is the rate of nucleation, d�inis the size of the incepted particles, bi, j is the coag-ulation kernel, and kli is the rate of the lth surfacereaction for particles of size i. The smallest parti-cles were assumed to consist of 32 carbon atomscorresponding to a dimer of pyrene molecules [31].

Four different surface reaction processes wereconsidered: (1) particle growth by the additionof C2H2 to the surface (d(l) = 2); (2) particlegrowth by the addition of pyrene to the surface(d(l) = 16); (3) oxidation of particles by O2

(d(l) = �2); and (4) oxidation of particles byOH, (d(l) = �1).

A stochastic approach is used to solve Eq. (1).Only a brief outline of the method will be givenfollows, for a more detailed description, see,[11,32–34].

The ensemble of soot particles is approximatedby a stochastic particle system. Rates of nucle-ation and surface reactions were calculated usingthe kinetic soot model [5,25], and the temperatureand species profiles as function of residence timewere obtained with the PREMIX simulation asdescribed above. With these rates, the size of theindividual time steps or the waiting time is deter-mined according to an exponentially distributedrandom variable [32]. Subsequently, one of thepossible events is chosen probabilistically relativeto their respective rates. Depending on which ofthe events has been selected, the appropriate num-ber of particles needed to perform this event ischosen according to the particles� individual rates.Once a step is performed, the state of the particlesystem is updated, and a new time step isdetermined.

Table 1Boundary values of the premixed laminar flames simulated

Flame designation p (bar) C/O-ratio

WBF.12.3 0.12 1.3JW1.69 1 0.69JW10.673 10 0.673JW10.68 10 0.68

Flame designations are taken from [25]. Flames were studied

To enhance the performance of the algorithmas compared to a direct Monte-Carlo simulation[32], the concept of majorant kernels and fictitiousjumps [33] is used. A similar approach was used tocalculate the size distribution in a low pressurepremixed flame in [11]. However, the method usedwas valid for coagulation in the free molecular re-gime only. In the present study, an efficient majo-rant kernel [35] valid over the entire regime ofKnudsen numbers was used. It is based on theharmonic mean of the kernels for free molecularand continuum regime [36]. Also, a new methodto accommodate surface reactions was introducedto account for the different timescales between thenumerous processes. It was thus feasible to calcu-late size distributions in highly sooting premixedflames at elevated pressure.

3. Results and discussion

Four different laminar premixed flames differ-ing mainly in pressure and C/O ratio were simu-lated, see Table 1. Measured volume fraction andtemperature for the WBF.12.3 flame were avail-able from the literature [37,38]. These flames werechosen since they were subjected to previous stud-ies byAppel et al. [25] andKazakov et al. [24,36]. Amore detailed description of the JW-flames can befound in [24,36]. Two flames, i.e., flames JW10.673and JW1.69, were investigated in more detail asthey differ markedly in their particle size distribu-tion and soot growth characteristics.

3.1. Flame structure

Soot volume fractions and number densitiescalculated with the method of moments for flamesJW10.673 and JW1.69 are depicted in Fig. 1. Alsoshown are experimental measurements of sootvolume fraction [38].

From Fig. 1, it can bee seen that the measuredsoot volume fractions are well reproduced by thekinetic soot model. Final soot volume fractionsin the JW10.673 flame exceed those in theJW1.69 markedly. Number density drops by twoorders of magnitude in the high pressure flame,while its decrease is much less pronounced in theatmospheric flame.

Fuel Cold gas velocity (cm/s)

C2H2 20.1C2H4 5.9C2H4 3.0C2H4 6.0

experimentally in [37,38].

Fig. 1. Comparison of measured [38] (symbols) andcalculated (lines) soot volume fractions. Also shown aresimulated number densities (lines) and temperatureprofiles (dashed lines) as function of residence time forflames JW10.673 and JW1.69.

Fig. 2. Calculated evolutions of the soot particle sizedistributions with residence time for flames JW10.673and JW1.69.

1460 J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465

3.2. Particle size distributions

The temporal evolutions of the size distribu-tion of soot particles for all flames were simulatedusing the stochastic algorithm. Calculated size dis-tributions as function of residence time for flamesJW10.673 and JW1.69 are depicted in Fig. 2 interms of n(d) = dN/dlog(d). In the high pressureflame, JW10.673, there are two distinct peaks onlyearly in the flame. Subsequent to its peak value,nucleation ceases and the distribution is governedby surface growth and coagulation leading to log-normal shape. For this flame, the transition fromcoalescent to aggregate growth of soot particleswas investigated in [39]. In this study, the particlesize distribution was calculated by direct integra-tion of Eq. (1) in the initial part of the flame.The distribution shown in Fig. 2 agrees well withthe one calculated in [39]. Small differences appearto be caused by a slightly different choice of thefraction of active surface sites a in the respectivestudies.

The characteristics of the distribution in flameJW1.69 are markedly different. The distribution isbimodal throughout the entire flame. A distinctpeak is located at small diameters while the largesize region of the distribution is characterized bya log-normal shape. Soot particle size distribu-tions exhibiting a similar shape have recently beenmeasured by Zhao et al. [12] in an atmosphericlaminar premixed flame.

The differences in shape observed for the distri-butions in the two flames are caused by discrepan-cies in nucleation rates. Nucleation peaks ataround 20 ms in the JW10.673 flame and ceasesa few of milliseconds later. In contrast, nucleationis continuous throughout the entire JW1.69 flame.

3.3. Particle age distributions

Experimental studies have indicated that thefraction of active surface sites decreases withparticle growth or age [1,15–17]. Frenklach andco-workers used the concept of active sites tocapture the effect of surface ageing in the kineticsoot model. They found that the fraction ofactive sites can be correlated to maximum flametemperature and mean particle size [24,25]

J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465 1461

a ¼ tan hðaðT Þ=l1 þ bðT ÞÞ; ð2Þ

where a(T) and b(T) are parameters depending onthe maximum flame temperature, l1 is the firstreduced moment of the particle size distributiondefined as l1 ¼ M1=M0 ¼ ðR1

i¼1iN iÞ=ðR1i¼1NiÞ, rep-

resenting the average mass per particle. In a studyof a series of laminar premixed flames, it wasshown that parameters a(T) and b(T) could becorrelated to the maximum flame temperature[25]. Thus, a global expression for a in premixedlaminar flames could be derived giving good re-sults for all calculated premixed flames in termsof soot volume fraction. However, a mean a isused for all particles in a given flame.

In the stochastic simulations, the size and ageof each particle in the ensemble is known explic-itly. One aim of the present study was thus tomake use of this feature to investigate if the

Fig. 3. Calculated soot particle age distributions atdifferent residence times for flames JW1.69 andJW10.673.

number of active sites on the particles� surfaceaccessible for surface reactions could be corre-lated with particle age. With the stochastic algo-rithm, it is possible to assign an individual a foreach particle.

To investigate correlations between a and theage of individual particles, the age distribution ofsoot particles was obtained. The age of a particleis defined as follows. If a particle i is incepted, itsage is initialized as zero. After every event, ortime step, the age of the particle is incrementedwith the actual waiting time. If the particle is se-lected for coagulation with particle j, the age ofthe new particle of size i + j is set to the massweighted average of the ages of particles i andj. The reactivity of large particles can thus be al-tered by collisions with young particles. Calcu-lated age distributions are shown for flamesJW10.673 and JW1.69 in Fig. 3. The age distri-butions reflect the basic features of the size distri-butions shown in Fig. 2. The age distribution forflame JW1.69 is bimodal throughout the entireflame due to continuous nucleation while it is bi-modal in the JW10.673 flame for a short periodin the main nucleation zone only. The main dif-ference between size and age distributions, how-ever, is the width in the large size or in case ofthe age distribution in the large age regions.While the size distribution develops a log-normalshape, the width of the age distribution decreaseswith increasing time. Thus, the range of particleages in flame JW10.673 at large residence timesis rather small.

3.4. Age dependence of active sites

To find a suitable correlation function betweenthe age and the fraction of active surface sites of aparticle, the expression in Eq. (2) used with themethod of moments [25] was investigated. Figure4 shows the evolution of the mean a with residencetime for both flames. It can be seen that a is unityinitially and decreases markedly once soot parti-cles start to grow. It levels off to a final value thatis different for the investigated flames, i.e., 0.4 forflame JW1.69 and 0.2 for flame JW10.673.

As already mentioned above, the shapes of thesize distributions in the two flames are noticeablydifferent. In flame JW1.69, small, newly inceptedparticles are present throughout the entire flame.These particles should have a high surface reactiv-ity as compared to the larger, older particles. It isthus intriguing to attribute the difference in final ato the continuous presence of young particles inflame JW1.69. As a first attempt, the dependenceof a on the particle age Ap was thus expressed interms of a step-function

aðApÞ ¼1 for Ap 6 h;

�ð3Þ

0:2 for Ap > h;

Fig. 4. Upper panel: the fraction of active sites on thesoot particle surface, a, as function of residence times forflames JW10.673 and JW1.69. Shown are valuesobtained with Eq. (2) [25] used in the method ofmoments (dashed lines). Also shown is the surfaceweighted a obtained from the age distribution (lines)depicted in Fig. 3 using the step function defined in Eq.(3) with h = 12 ms. Lower panel: comparison of calcu-lated volume fractions (lines) using the step-functionapproach with experimental values (symbols) [38].

Table 2Parameter C used in correlation a = 0.2 + 0.8exp(�CAp) and maximum flame temperature for allsimulated flames

WBF.12.3 JW1.69 JW10.673 JW10.68

C 150 10 10 100Tmax (K) 1980 1711 1895 1880

1462 J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465

where h is the critical age defining the boundarybetween the two step values.

In the step-function, a high reactivity is as-signed to young particles and a low reactivity tolarge particles. Surface weighted values of a ob-tained with the step-function approach usingh = 12 ms are compared to the a used in the sim-ulations with the method of moments in Fig. 4.The evolution of a obtained with the step-functionclosely resembles the original approach in bothflames. The sudden decrease in a is well captured.

The simulations of the particle size distribu-tions were repeated using the step-function ap-proach to express the age dependence of a. Acomparison of measured and calculated soot vol-ume fractions using the step-function approach in

combination with the stochastic algorithm isshown in the lower panel of Fig. 4. It can be seenthat the calculated volume fractions are within afactor of 2 as compared to the measured valuesdespite the simplicity of the correlation betweenage and a.

It was also suggested that the fraction of activesurface sites decreases exponentially with its age[15,40]. This idea was adopted and instead of astep function an exponential function was usedto correlate age and fraction of active sites

a ¼ aþ b exp�CAp ; ð4Þwhere a and b are constants, C is the inverse of atypical decay time, and Ap is the age of a particle.

Using Eq. (4) simulations were performed forall four flames stated in Table 1. The constant Cwas adjusted individually for each flame such thatthe calculated soot volume fraction matched themeasured values. Values of C obtained in thismanner are shown in Table 2. A comparison be-tween calculated and measured volume fractionsalong with surface weighted values of a is depictedin Fig. 5. From Table 2, it can be seen that param-eters C vary markedly between the four calculatedflames. Large values of C were obtained for theWBF.12.3 and JW10.68 flames corresponding toa strong decrease in a with increasing particleage. In contrast, a much smaller value of C wasfound to give a good match to measured volumefractions for flames JW1.69 and JW10.673. Thisimplies that for these two flames the decrease ofa with particle age is much slower.

It was suggested that the process of surfacedeactivation depends on flame temperature [15].Also, in their study Appel and Frenklach [25] cor-related the parameters used in Eq. (2) successfullywith maximum flame temperature. Thus, the val-ues of parameter C used in Eq. (4) were plottedas function of maximum temperature of therespective flames, as shown in Fig. 6. It can beseen that C can be correlated to the maximumflame temperature for three out of the four calcu-lated flames, i.e., flames JW1.69, JW10.68, andWBF.12.3. For these flames, parameter C in-creases linearly with increasing maximum flametemperature. A large value of C corresponds toa fast depletion of active surface sites with increas-ing particle age. With the exception of flameJW10.673, the results thus reproduce the trendthat deactivation of surface reactivity proceedsfaster at higher temperatures.

Fig. 5. Upper panel: the fraction of active sites on thesoot particle surface, a, as function of residence times.Shown are values obtained from the age distributiondepicted in Fig. 3 using an exponential correlationbetween a and the particle age as shown in Eq. (4).Parameter C in Eq. (4) was chosen individually for eachflame, see Table 2. Lower panel: comparison of calcu-lated (lines) and measured (symbols) soot volumefractions as function of residence time.

Fig. 6. Parameter C used in the exponential correlationbetween the fraction of active surface sites and particleage [a = 0.2 + 0.8 exp(�CAp] as function of maximumflame temperature. The line is a linear fit through thethree data points for flames JW1.69, JW10.68, andWBF.12.3.

J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465 1463

It is, however, unclear why the value of C inflame JW10.673 differs so markedly to flameJW10.68 although maximum flame temperaturesare similar. Aggregation, which is not includedin the current model, could affect the age depen-dence of a. Total surface areas might be altereddifferently depending on the degree of aggregationin the different flames. For future work, it is thuscrucial to include aggregation and to investigate alarger set of premixed flames, i.e., different typesof fuels, pressure, and C/O-ratios.

4. Conclusions

Particle size distributions were calculated inlaminar premixed flames at different pressures.

A stochastic approach to model the populationbalance of soot particles was used. The methodfeatures a probabilistic description of all processesaffecting the size distribution. Using a new majo-rant kernel and a technique to treat surface reac-tions, the size distribution of soot particles inhigh-pressure flames could be calculated.

With this approach soot formation was simu-lated in four laminar premixed flames at differentpressures. Reactions in the gas phase were de-scribed by a detailed chemical mechanism andsoot formation by a detailed kinetic soot model.

It was found that the distributions in the fourflames differ in shape. In three of the calculatedflames, the distributions exhibited log-normalshape in the post-flame zone. The distribution inthe atmospheric flame was found to be bimodalwith a distinct peak at sizes smaller than 3 nm anda lognormal part in the large size region. The peakat small diameters is due to continuous nucleation.

The capabilities of the stochastic approachwere used to calculate particle age distributions.With the calculated age distributions it was inves-tigated if particle age could be correlated with thefraction of active surface sites. Correlations of twodifferent functional forms were employed. First, astep function was used to allocate a high surfaceactivity to young particles and a low activity toold particles. In the second approach, surfaceactivity was expressed as an exponential functionof particle age. Measured soot volume fractionscould be well reproduced with both methods.The decay constant, which corresponds to the in-verse of a typical deactivation time, used in theexponential approach was found to be a linear

1464 J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465

function of maximum flame temperature, forthree of the four calculated flames. For the flamewith the highest maximum temperature, the frac-tion of active sites was found to decrease fastestwith particle age. For the flame with the lowestmaximum flame temperature the decay of surfacereactivity was calculated to be markedly slower.This compares well to experimental findings.

The results indicate that the fraction of activesurface sites could indeed be expressed as a func-tion of particle age. However, one of the calcu-lated flames did not fit the trend found for theother flames. It is thus necessary to extend thepresent investigation to a larger set of flames.Also, aggregation of soot particles affecting theparticles� surface should be accounted for.

Acknowledgments

The support of Churchill College, Cambridge,UK, is gratefully acknowledged by J.S. andM.B. W.W. was partly supported by the EPSRC(Grant No. GR/R85662/01).

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Comment

Daniel E. Rosner, Yale University, USA. On the faceof it, MC methods do not appear to be as well-suited(as, say, moment methods) to multi-state aerosol prob-

lems in environments in which there is not a ‘‘time-like’’ variable (e.g., in the presence of state-specificstream-wise diffusion in physical space [1]). Can your

J. Singh et al. / Proceedings of the Combustion Institute 30 (2005) 1457–1465 1465

present MC-approach be extended into this importantdomain?

Reference

[1] D.E. Rosner, et al., Ind. Eng. Chem. Res (ACS) 42(2003) 2699–2711.

Reply. The method of moments as well as mostsectional models should be used whenever possible.However there are situations where both methodsreach their limitations. This is normally the case whenthe number of internal coordinates is larger than fouror the approximations made in the method of mo-ments or sectional methods are unknown. We are cur-

rently working on including spatial coordinates andthe effects of diffusion into population balance prob-lems. In [1] we developed the �single particle method�to model a rotating disc contactor used for liquid-li-quid extraction. Although the physical problem isquite different from flame synthesis of nano-particlesthe governing equations have a very similar structure.We are now applying the same method to modelingthe synthesis of nano-particles in counter flow diffu-sion flames.

Reference

[1] A. Vikhansky, A. Kraft, Chem. Eng. Sci. 59 (2004)2597–2606.