Aggregate size distribution evolution for Brownian coagulation—sensitivity to an improved rate constant

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Journal of Colloid and Interface Science 274 (2004) 502–514www.elsevier.com/locate/jcis

Aggregate size distribution evolution for Browniancoagulation—sensitivity to an improved rate constant

M. Zurita-Gotor and D.E. Rosner∗

High Temperature Chemical Reaction Engineering Laboratory, Chemical Engineering Department, Yale University, New Haven, CT 06520-8286, USA

Received 10 April 2003; accepted 25 February 2004

Available online 16 April 2004

Abstract

Brownian motion causes small aggregates toencounter one another and grow in gaseous environments, often under conditions in whicthe coalescence rate (say, spheroidization by “sintering”) cannot compete. The polydisperse nature of the aerosol population formmechanism is typically accounted for by formulating an evolution equation for the joint PDF of the state variables needed for dindividual particles. In the simple case of fractal-like aggregates (prescribed morphology and state, characterized just by theaggregated spherules, or total aggregate volume), we use the quadrature method of moments and Monte Carlo simulations trecent improvements in the laws governingfree molecule regime coagulation frequency (rate “constant”) of these aggregates cause systchanges in the shape of the asymptotic aggregate size distribution, with significant implications for the light-scattering power animpaction behavior of such aggregate populations. 2004 Elsevier Inc. All rights reserved.

Keywords: Fractal-like aggregates; Collision rate; Brownian coagulation; Self-preserving aggregate size distribution; Monte Carlo; Quadrature method omoments; Nanoparticle aerosols

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1. Introduction

A frequent path for aerosol formation involves the nucation (from a supersaturated vapor) of small spherical pcles of approximately uniform size, followed by their growdue to Brownian coagulation, sometimes in the presencnonnegligible sintering [1–7]. As a result of these procesa population of diverse aggregates is produced, each haa complex geometry which can sometimes be describedtistically, using fractal concepts (relating aggregate masits characteristic length):

(1)N = k0(Rg/a)Df .

In this equation,N is the number of “spherules” in the aggrgate, whose radiia are assumed uniform.Rg is the aggregatgyration radius, andDf ,k0 are, respectively, the fractal “dmension” and prefactor.

Simple fractal scaling arguments have been appliedestimate many physical properties of “aggregated” pa

* Corresponding author. Fax: +1-203-432-7387.E-mail address: [email protected] (D.E. Rosner).

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.02.065

-

cles (Brownian diffusivity, thermophoretic diffusivity, vapscavenging power, collision frequency, etc.), for which soexperimental and numerical results are also availablee.g., the review by Friedlander [8]).

Computer simulations mimicking aggregation proceshave led to a better understanding of aggregate geotries and, in turn, assisted the development of fractal ccepts. Relatively simple, often hierarchical algorithmsbe applied to efficiently generate a large number (ensble) of physically relevant fractal-like aggregates (FAswhich then can be used to numerically calculate their phcal properties [9]. Ensemble-averaged properties are nefor the correct interpretation of experimental measuremen(such as elastic light scattering) or, say, for the design/coof particle synthesis reactors. A complete description ofaerosol population (size, shape, composition, etc.) asas of the properties of each individual aggregate thuscomes necessary. Even for the simplest case of particlelution under just Brownian coagulation, this would, in prciple, require following the dynamical evolution of a velarge number of particles in a box over large time intervwhether from the simulation of the Langevin equation [or by Monte Carlo (MC) simulations in which random

http://www.elsevier.com/locate/jcis

M. Zurita-Gotor, D.E. Rosner / Journal of Colloid and Interface Science 274 (2004) 502–514 503

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chosen particles are moved according to some a priorsigned diffusion law [11]. Even with today’s computers susimulations are impractical inmost cases, particularly whea more complicated description of aerosol evolution isquired, such as inclusion of restructuring, or in the preseof nonconstant environments. Furthermore, these simtions lack flexibility, requiring new calculations for differeinitial conditions, changing environments or a different dscription of aggregate evolution.

Fortunately, large simulations of simple coagulatprocesses [12,13] have shown that for low particle densiand after a short transient time, a Smoluchowski desction of aggregation, which neglects spatial correlationsparticle density, accurately predicts the coagulation kineticsSurprisingly, more recent calculations have shown that thmean-field approximation remains valid to describe thegregation kinetics even for dense (higher volume fractisystems, up to approximately the so-called gel point [1Much simpler approaches based on the widely used plation balance equation (PBE) are therefore justified. Thapproaches describe the evolution in time of the joint prability density function (PDF) of the state variables neefor description of individual aggregates. Still complicatesubtle nonlinear partial integrodifferential equations arfor which different methods of solution exist (see, e[15,16]). Accurate modeling of each of the mechanisterms appearing in this PBE is also crucial, and we hfound that the scaling laws proposed/used thus far for ftal aggregates can incur significant systematic error. Perhathe best-studied case, put to the test in this work, isof diffusion-limited cluster–cluster (CC) Brownian aggregtion (with a constant fractal dimension,Df , of ca. 1.8) ina constant property environment. In such cases, sinceaggregate morphology is “specified,” individual aggregsize (typically total volumev = Nv1, wherev1 (= 4πa3/3)

is the volume of each of theN “primary” spherules composing the aggregate) is often the only needed stateable. It is well known that, after a sufficiently long timcoagulation kernels based on simple fractal scaling aments lead to a “self-preserving” aggregate size distribu(ASD) ψ(η) ≡ nv̄/Np (in the dimensionless aggregate vume η ≡ v/v̄(t)), which is nearly log-normal. Yet, it haalso been found that simulation approaches whichpresumea log-normal ASD-shape can fail to accurately predicthigher ASD-moments needed for correctly interpreting etic light scattering measurements [17]. Similarly, we expnonnegligible corrections willresult if a more accurate colision rate expression is used for such aggregates. Herexploit both the quadrature method of moments (hereaQMOM [18]) and Monte Carlo (MC) simulations to accrately compute/compare the resulting ASD and its momwith a coagulation kernel based on (a) earlier simple frascaling estimates and (b) our correlations of accurate Mcomputed collision rates between FAs in the free-molecregime [19]. These new coagulation rate laws were obtaby performing quasi-Monte Carlo simulations to comp

effective collision cross-sections of numerically generaCCAs with prescribed fractal dimension and prefactor [9

As discussed in Section 4, our results (using “consnumber” Monte Carlo simulations as well as an 8-momQMOM implementation, and assumingDf = 1.8, k0 = 1.3)reveal that nonnegligible corrections to the ASD-momeindeed result from introducing the modified coagulation knel, mainly as a consequence of the “penetration” effectcurring when two aggregates of rather disparate size seebe on a “collision path.” This “penetration” reduces the clision rate between such aggregate pairs, which is probthe primary reason for the observed broadening of the A(see Section 4). In terms of dimensionless ASD momewe conclude that asymptotic ASD moments,µk , with k > 1have previously been underestimated, while corresponmoments with 0< k < 1, relevant, for example, in calculations of Brownian diffusion deposition rates across lanar or turbulent boundary layers [20], have previously bslightly overpredicted. We also find (Section 4) that our nFA-coagulation rate kernel leads to systematically differtimes and “paths” to reach self-preservation. These areconveniently tracked using our QMOM simulation methofor different choices of initial conditions.

2. Problem formulation

2.1. Particle morphology: selection of state variable

A complete and detailed description of an arbitraaerosol population evolution in general requires a lanumber of state variables (to account for the differencephysical properties of particles with diverse morphologcompositions, charges, etc.), as well as consideration omany existing processes leading to creation or destrucof particles at a given state. Consequently, even a simfied Smoluchowski-like description of particle populatievolution results in subtle integro-partial differential equtions which are difficult to solve [15]. Furthermore, maof the relevant phenomena, suchas coagulation, restructuing, nucleation, break-up, etc., are not adequately descby presently available rate laws; this further compromithe accuracy of the solutions. Nevertheless, much caachieved without an excessive computational burden wreasonable selection of both state variables and of the kically relevant processes rate laws. For example, usingtwo state variables, viz., particle surfacearea andvolume,and just approximate rate laws for coagulation and sinterRosner and Pyykönen [7] were able to numerically repduce many essential features of aerosol evolution that wmeasured in a steady laminar counterflow diffusion flaenvironment [21,22].

We consider here a much simpler (yet physically revant) system [23–29] consisting of a population of so-cadiffusion-limited cluster aggregates (DLCAs) evolving inconstant environment. Conditions for the formation of su

504 M. Zurita-Gotor, D.E. Rosner / Journal of Colloid and Interface Science 274 (2004) 502–514

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nanoparticles are often met in combustion-based reaand are typically the result of (1) fast reaction kinetics ofpor precursors, (2) extremely high reaction product supeurations in the gas phase, allowing the nucleation of cluscomposed of just a few molecules, (3) cluster diffusionBrownian motion and collision to form larger clusters, winitially very rapid coalescence rates (spheroidizing pacles between successive collisions), (4) a transition at sthreshold particle size above which the time requiredcoalescence becomes much larger than that required foagulation. This transition to negligible coalescence ocbecause of the much stronger size-dependence of thalescence than that of coagulation [30]. Indeed, usually acrossover, the characteristic times for sintering and coagtion, ts andtc satisfy the conditiondts/da � dtc/da, whichleads to a narrowly distributed primary particle sizea. Af-ter this “onset” of aggregate formation, coalescence can thbe neglected; this effect is further enhanced by the reducin the characteristic time for coagulation that takes placecause of the open, ramified structures that form.

Numerical simulations of initially monodisperse, sphecal particles undergoing coagulation under different rationadiffusion laws, as well as experimental results (see, e.g.review by Friendlander [8, and references cited thereshow that these DLCA aggregates have a fractal dimsion,Df , close to 1.8. Not such good agreement exists invalue of the fractal prefactor, for which values in the ran1.05–3.4 have been reported (higher values are partialltributed to the unavoidable spherule “overlap” and sphepolydispersity found in real aggregates [31,32]). Theparity of values in this parameter is, however, not of rconcern here since most of itseffect can be absorbed inredefined characteristic time for aggregate collision, anddo not expect a large influence on the population charaistics (see [19] and below for a more detailed discussiIn what follows we will consider a population of FAs haing Df = 1.8, k0 = 1.3 undergoing Brownian coagulatioin a constant environment underfree-molecule conditions.In such a system, the physical properties of each of thparticles are well-defined, within a small statistical scatonce its “size” is specified [9]. Different definitions of “sizecould, of course, be used, since fixed morphology also mthat there is (through Eq. (1)) a unique relation betweenferent possible “sizes.” We have chosen to use aggretotal volumev = Nv1 to refer to its “size.”

2.2. Population balance equation: coagulation rate laws

Our starting point is thus the equation describing the elution in time, due to coagulation, of the aggregate sizetribution (ASD) function. Using the continuous form of thSmoluchowski theory of aggregation, such evolution canexpressed through the following balance equation forinstantaneous number densityn(v, t) of aggregates of vol

-

-

umev:

∂n(v, t)

∂t= 1

2

v∫0

β(u, v − u)n(u, t)n(v − u, t) du

(2)− n(v, t)

∞∫0

β(u, v)n(u, t) du.

In this nonlinear IDE, the coagulation kernelβ(u, v) rep-resents the number of collisions per unit time, volume,number densitiesn(u)n(v), between aggregates of volumu andv. Notice that the first term on the RHS represents“birth” rate or number rate of particles of volumev producedfrom coagulation of two smaller aggregates (of volumeu

andv − u). The second term expresses the “death” ratnumber per unit time of particles of volumev that disappeadue to their collision with any other particle in the population.

Extensive discussions exist in the literature aboutproper collision rate law for fractal aggregates. Differregimes can be distinguished according to the ratiostween the three characteristic lengths present in the problemaggregate characteristic size,gas molecule mean-free-paand aggregate “persistence length” [33,34]. We are ccerned in this work with the case in which the first lenis much smaller than either of the last two. Under this “frmolecule” condition, calculation of the coagulation rate between aggregates is analogous tothe calculation of the collision rate of gas molecules, since aggregates behave asmolecules moving in straight trajectories between collisi[35,36]. Using the proper definitions [19], the average coulation rateβ(u, v) can then be expressed as the producan average relative velocity and an effective collision crsection (or effective area swept per unit time; see, e.g., [3That is,

(3)β(u, v) = c̄rσ (u, v).

The average relative velocity between clusters of voluu andv, c̄r (u, v) can be obtained assuming that the agggates are in thermal equilibrium with the backgroundmolecules. In this case the following well-known result imediately follows [37]:

(4)c̄r (u, v) =[(8πkBT )

(1

mu

+ 1

mv

)]1/2

,

wherekB is the Boltzmann constant,T the aggregate themodynamic temperature, assumed to be that of the gas mcules, andmu,mv the mass of aggregates with total sphervolumesu andv, respectively.

Additional theoretical concerns are raised by commoused estimates of the effective collision cross sectσ̄ (u, v). Based on general fractal scaling, it has been argthat

(5)σ̄ (u, v) ∝ (Rg,u + Rg,v)2.


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However, this estimate shows only the scaling (withconstant prefactor of order unity, not clearly specified) tcorresponds to the asymptotic limit of very large agggates, and therefore neglects finite size corrections. Furmore, serious objections exist for fractal dimensionslow 2, mainly as a result of FA “interpenetration” when twaggregates of rather large size disparity approach each(see, e.g., Meakin [38], Rogak and Flagan [39], or Kazaand Dryer [40]). In a previous work (for details, see [19])have verified that nonnegligible corrections are indeed nessary for collision rates of FAs withDf < 2. In our work, alarge number of aggregates in a size range up toN = 350and with diverse fractal dimensions and prefactors wfirst numerically generated using the Cluster–Cluster aggation algorithm described in detail in Filippov et al. [9Then effective collision cross sections were calculated fensemble-averaged values of individual (also averaged) vaues of the collision rate between pairs of rotating as was translating aggregates. These last values were obtusing a quasi-Monte Carlo algorithm that averaged ovepossible impact parameters (minimum distance of appropolar angle in plane perpendicular to relative motion, agggate relative orientation) as well as over the translationalrotational velocities of the aggregates. Velocities were spled according to a Maxwellian distribution function at tgas temperature. Motivated by asymptotic behaviors, csion rates were finally correlated, to allow for incorporatof the results in future population balance models. Resfor the FAs of interest here are summarized in Fig. 1ter Fig. 15 in [19]), which shows the locii of constant rabetween calculated (and subsequently correlated) collirate constants and previously accepted estimates basthe sum of the aggregate gyration radii. The effect of F“penetration” can be seen as this ratio becomes smallerunity (shaded areas), i.e., when the size disparity betwcolliding aggregates increases. Although differences are nomultidecade (correlated rates are usually within 30% of pdicted ones), it is known that small changes in coagula

Fig. 1. Contours of constant ratio betweennewly calculated correlated fractal aggregate collision rates and previous estimates based on the sumgyration radii of the participating aggregates. Shaded area correspondthe previously overestimated region (after Ref. [19]).Df = 1.8, k0 = 1.3.

-

r

d

,

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e

rates can produce quite different ASDs: For example,exact solution corresponding to a constant coagulationlaw is an exponential size distribution, whereas continuregime estimates of the collision kernel (nearly constanmost of the domain) results in a quite different size distrition, close to lognormal [41].

3. Methods of solution

Different approaches have been used to obtain the numdensity function from Eq. (2), such as the use of “sectial” methods consisting of discretization of the state variaspace [42,43], stochastic approaches using Monte Carlo agorithms [3,5,44–46], and the method of weighted resials [15], discussed later in this section.

However, we are often just interested in the macroscproperties of the population of aggregates, many of whare expressible in terms of particular moments of the aersize distribution [16]. In these cases, both sectional andapproaches are unnecessarily demanding, computatioThis is particularly the case for sectional methods when mtiple state variables and/or a nonconstant environmentto be modeled. In such cases, typical of particle proding crystallizers or reactors, the more numerically efficienmethod of moments [16,47] is advisable. In essence, tmethods assume that the distribution can be convenientlscribed by just a finite set of scalars (moments) evolvintime. The basic set of equations for the evolution in tiof the moments is obtained by multiplying Eq. (2) byvk

followed by integration inv from 0 to ∞. Then, noticingthat the collision kernel is a symmetric function and afa straightforward rearrangement of the domain/variableintegration, one obtains the coupled ODEs

dMk(t)

dt= 1

2

∞∫0

du

∞∫0

dv[(u + v)k − uk − vk

]

(6)× β(u, v)n(u, t)n(v, t).

Equation (6), for 2Nq conveniently chosen values ofk

(see discussion below), represents a much simpler syof ODEs. However, since evaluation of the RHS needs,general coagulation rate expressions, the availability ofments of the distribution not explicitly tracked by the equtions, the former set of equations is generally not closed

Different methods have been developed to deal withdifficulty, such as assuming a lognormal shape of the dibution (e.g., Lee [48]), moment interpolation schemes [4or, more recently, the quadrature method of mome(QMOM [18]). This last approach evaluates the integrin Eq. (6) using nonstandard Gaussian quadratures, wmake it possible to achieve a closed set of equations wout making any a priori assumption about the shape of


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distribution. Equations (6) are then approximated as

(7)

dMk(t)

dt≈ 1

2

Nq∑i=1

Nq∑j=1

[(vi + vj )

k − vki − vk

j

]β(vi, vj )ωiωj .

The 2Nq set of moments determine uniquely at each timeNq weights,ωi , andNq abscissas,vi , through the system o2Nq equations

(8)Mk(t) =Nq∑i=1

vki (t)ωi (t).

Since initial moments are known, inversion of (8) providthe initial weights and moments to start the numerical ingration of (7). After each time increment, momentsMk(t +�t) are obtained and inversion of (8) is again requiredcontinue integration of (7). Direct inversion of Eq. (8) isvery ill-conditioned problem, and even though alternativegorithms have been developed [4,18,50] this step becoone of the major integration difficulties. However, theversion problem can be avoided at all times butt = 0 byusing the Jacobian matrix transformation (JMT) varianthe QMOM [51]. This is done essentially in the followinway: Define

M ≡ {M(1),M(2), . . . ,M(2Nq)},x ≡ {v1,ω1, v2,ω2, . . . , vNq ,ωNq },

(9)J ≡ {Jij }, Jij = ∂Mi

∂xj

,

where the subscript(i) in theM vector elements representhe value of theith moment in the chosen set of{k} momentsthat are tracked by the method. Then

(10)dxdt

= J−1dMdt

.

Notice that once the set of moments{k} is chosen,J isknown in terms ofx and independent of the physical prolem being solved. Analytical calculation ofJ−1 can thus beperformed, and combination of Eqs. (7) and (10) leadsystem of ODEs in which the dependent variables areweights and abscissas instead of the moments. Therefoversion is required only att = 0 to calculate initial weightsand abscissas from initial conditions of the moments.tice that the complementary problem, that of obtainingmoments from the abscissas and weights, can be easilyputed through Eq. (8). The method can be efficiently impmented in a symbolic program such asMathematica. How-ever, if more than 4-point quadratures are attempted, thealytical calculation ofJ−1 becomes a very formidable taswhich we circumvented because of our computationalsources.

To conclude this discussion it may be of interest to remthat moment methods, in conjunction with Gaussian quature techniques, share some features in common withmethod of weighted residuals [52,53]. This latter appro

s

-

-

-

consists of approximating the solutionn(v, t) by itsNq -termtruncated expansion in a complete set of orthogonal polmials. The coefficients in the expansion are then obtaby “minimizing” the error through orthogonalization of th“residual” (i.e., the difference between the actual soluand its expansion) with another set of functions. In mommethods these would be the tracked momentsxk for the 2Nq

values ofk (which need not be integers).The approximated expansion will work better the clo

the solution is to the weight function used to generate theof orthogonal coefficients [53,54]. Since the QMOM takas weight function the solution from a previous integratstep, we can expect this method will provide an optimallection of the basis for expansion. The increased accuis, however, achieved at the expense of a higher comptional cost: in practical terms, both weights and abscissathe Gaussian quadrature approximation need to be updat each integration step.

More computer-intensive MCcalculations have also beeperformed to validate results from QMOM. In essence,simplest MC methods update an initial distribution of pticles (represented by an array whoseith element containthe state—in this case just volume—of particlei) throughcoagulation events chosen according to their relative prbility within the population. The simplest way of achievithis is by randomly selecting two particlesi, j and accepting/rejecting their coagulation with a probability givenβi,j /βmax. The procedure not only allows us to adequaevolve the particle distribution, but we can also estaban easy connection between each coagulation step antime [55]. Since calculation ofβmax at each step is noa straightforward task for the correlated kernel, an upbound was used instead. This way, the adequate probabiliof events is kept and, although the number of rejected culation events is increased, the total amount of calculatis greatly reduced.1 To avoid the depletion of particles ater many coagulation events the “constant number” meof Smith and Matsoukas [55] has been used. Their ideasists of simply adding a randomly sampled particle witthe distribution after each accepted coagulation.

4. Results and discussion

4.1. Dimensionless aggregate size distribution

An important property of the PBE, Eq. (2), is that whthe collision kernel is a homogeneous function2 the change

1 More computationally demanding models could, however, use mooptimized algorithms, such as, for example, the “majorant kernel” approacrecently described by Eibeck and Wagner [56]. In essence, their mereduces the number of rejected coagulation events at the expense ofcomplicated particle selection.

2 A function β(u,v) is said to be “homogeneous of degreeλ” if it satis-fies the scaling relationβ(γ u,γ v) = γ λβ(u, v).


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of variables

η ≡ v/v̄ = v/(φ/Np),

(11)ψ(η) ≡ n(v, t)(φ/N2

p(t))

transforms the original equation (2) into a nonlinearordi-nary integrodifferential equation [57]. In Eqs. (11),η andψ

are the new independent and dependent variables,φ is thevolume fraction o accupied by aerosol particles (= M1), andNp is the time-dependent total number of aggregates pervolume(= M0).

Dimensional moments and their dimensionless coterparts are easily interrelated. Using the definitionsEqs. (11) it is straightforward to show that thekth moment,µk , of the dimensionless size distributionψ can be obtainedas

(12)µk = Mk

M0

(M0

M1

)k

.

Free-molecule coagulation rate kernels based on appmate fractal scaling are homogeneous functions. CombiEqs. (3)–(5) and the definition of the degree of homogenits value is immediately found to be

(13)λ = 2

Df

− 1

2.

The possibility of the similarity transformation does nhowever ensure the existence of such solutions for the trformed equation. In other words: homogeneity is a necsary but not sufficient condition for the existence of a timinvariant “self-preserving” dimensionless size distributψ(η) [58]. Many such distributions have nevertheless bfound, both by direct solution of the transformed PBEthe new variables as well as by solving the original PBElarge times until a self-preserving, time-independent disbution is found in the transformed variables. This is indethe case for a system of particles coagulating under alaw based on the theoretical considerations just discusas amply demonstrated previously [3,5,7,41,59]. In the cwork, the attainment of an unique self-preserving sizetribution was shown after short interval, irrespective ofdifferent initial conditions considered. Different morphogies, as well as simple laws for restructuring were studwith approximate self-preservation in the bivariate casedemonstrated/reported by Tandon and Rosner [3] andner and Yu [5].

The MC-computed/correlatedcollision kernel used in twork is certainlynot homogeneous and, in principle, the eistence of self-preserving solutions is precluded. Howefor large times—i.e., large aggregates—small size cortions are lost and asymptotically the kernel becomesmogeneous. This is so because penetration and rotateffects—only corrections to previous scaling—were shofor large aggregates to depend just on aggregate sizethus not affecting the “homogeneity” of the kernel. Indeit was shown that effective collision diameters could be scessfully correlated as

-

,

-

l

,

(14)deff,ij = [Reff,if (α) + Reff,j

]ηrot,

whereα is the aggregate size ratio,α ≡ vi/vj .It was also shown that the effective collision radii for clisions of equally sized aggregates,Reff, and the rotationacorrections,ηrot, asymptotically behave as

Reff ∝ Rg, N � 1,

(15)ηrot(α,Nj ) ∝ fct(α), Nj � 1,

which means that once finite size corrections are lostcorrelated kernel becomes a homogeneous function withsame degree of homogeneity given by Eq. (13). This hawe will see below, important consequences on the long-“averaged” coagulation rate.A more immediate implicationis that we can hope to asymptotically reach a self-preservindistribution of sizes, but only at much larger coagulattimes. This was indeed found to be the case, as stronglygested by Fig. 2, which shows how two particular momeof the dimensionless distribution approach a constant vfor large coagulation timesτ ≡ t/tc, where the characteristicoagulation timetc is defined as

(16)tc ≡(

8πkBT

ρp

)1/2(3v1

4π

)2/3

Np(0).

In this figure, results for the new, MC-based collision knel are shown as the continuous line (see below for m

(a)

(b)

Fig. 2. Time evolution of dimensionless moments of order (a)k = 1/2 and(b) k = 3/2. The continuous curves show results for the new kernel [whereas dashed lines represent previous estimates. Notice the muchtime required for self-preservation as well as the nonmonotonic evolutiofor the new kernel.Df = 1.8, k0 = 1.3.


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lyhed isrep-).r the

t thements)

details about the calculations), whereas those corresping to theoretical estimates based on the sum of the gyraradii are shown as the dashed line. As can be noticed, rdifferent timesτ are required to reach self-preservationeach case, for the reason discussed above. Note, alsnonmonotonic approach to the asymptotic values in theof MC-based collision rates, possibly due to finite sizefects on the collision rate. Although self-preservationobtained—see below—at mean sizes much larger thanrange of sizes for which collision rates were calculatedcorrelated (up to aggregates with ca. 350 spherules),relations were based on asymptotic behaviors and thupected to remain reasonably accurate for much larger aggates. Indeed, nonstatistically significant random checkcollision rates for aggregates formed by as many as 4spherules were performed by the authors, showing agment to within a few percent with the correlated values (viations were consistent with population spread).

4.2. Practical issues in QMOM implementation

Implementation of the QMOM first requires addresstwo main questions [16]:

• Number of quadrature points needed for adequate aracy?

• Particular set of moments to be tracked?

Unfortunately, no general answer can be given to thquestions, and each problem will require careful examtion of both aspects. A few general, qualitative remarksply, however, to all cases. First, notice that, numerical incuracies aside, the source of error in the method residthe quadrature formula, Eq. (7). Since weights and abscare such that the quadrature formula is exact for a seleset of moments, it is immediately clear that the appromation will work for smooth coagulation kernels but wfail to provide accurate results otherwise. It can also besumed that the higher the moment that needs to be trathe higher the error. This will be the result of putting tmuch emphasis on the tail of the usually slowly decayASD, which therefore needs to be “computed” more accurately.3 These qualitative arguments can be more formunderstood in the light of the dual character of the QMOas a weighted residual method: an adequate selectionof a set of functions for expansion of the solution and oset of functions (i.e., tracked moments) to orthogonalizeresiduals [53,54], is required.

With regard to the second question, a particularly sable set of moments would be that comprising magnitudephysical interest. However, we expect that any convenie

3 This motivates the use of fractional rather than integral momentleast if volume and not radius is taken as state variable. Moreover, mphysically significant quantities can be expressed in terms of fractionaments rather than integral ones [16].

-

r

e

--

-

-

s

,

h

spaced4 set of moments spanning the range of orders whcalculation is sought would be acceptable (see, e.g., [16fact, Eq. (8) allows us to immediately compute any momonce the weights and abscissas are determined. Such apimations should indeed work quite well, as long as (trackmoment evolution is adequately described, as just discuand only moments smaller (or slightly larger) than the mimum one tracked are calculated. Notice that in this cwe would be performing Gaussian quadratures of a smfunction.

Estimated coagulation kernels in both the continuumfree-molecule regime are quite smooth functions andvious studies (e.g., [7,51,60,61]) have been successfulviding accurate low order moments with just three quadture points, particularly for the almost-constant “continuukernel. The same behavior should be expected for ourgenerated/correlated kernel. However, in this work weinterested in high order moments (for example, correctterpretation of light scattered by a polydisperse populaof aggregates (Rayleigh–Debye–Gans theory) requiresments as high as 2+ (2/Df ) ≈ 3.1 for a population ofDLCAs [17]) so that further validation is required for thQMOM. To this end we have also performed extensivesimulations (20 runs of 75,000 particle populations). Tis enough to accurately provide the ASD and its momeas can be seen from the comparison (Fig. 3 and Tabwith previously available results for infinitely fast coalecence [59,62].

Fig. 3. Self-preserving PSD corresponding to coagulation with infinitefast coalescence, i.e. coagulation of spherical particles. Shown dasthe sectional result from Vemury and Pratsinis [59]. Continuous curveresents MC results (average over 20 runs of 75,000 particle populationsExcellent agreement exists between the two solutions (as well as fomoments, Table 1).Df = 3.

4 Spacing of the selected moments should also, in principle, affecaccuracy of the results. We have, however, tested a few sets of mowith uniform as well as nonuniform spacing (e.g., geometrically locatedand no significant differences in performance were detected.


Ve-

ith

spec

ac-n in

entso-lues-

y-et o

ger.re-intthe-

ith

spec-

ill

lf-itialonsd, in

osents

se-tingre-ivid-al-

hissult-withes-tiontlyess-. 6.

gerbu-by

Table 1Comparison of asymptotic dimensionless moments,µk , obtained byQMOM (Set 3 in Fig. 4), and MC simulations with values reported bymury et al. [62] forDf = 3

k Vemury et al. [62] Monte Carlo 3p-QMOM

−1/2 1.5641 1.5652 1.5606−1/3 1.2937 1.2940 1.2939−1/6 1.1155 1.1155 1.11560 1.0001 1.0000 1.00001/6 0.9296 0.9295 0.92981/3 0.8929 0.8931 0.89391/2 0.8836 0.8844 0.88552/3 0.8984 0.8996 0.90095/6 0.9360 0.9380 0.93881 0.9998 1.0000 1.00002 2.0873 2.1192 2.1158

Fig. 4. Percent difference in moments computed by the QMOM wNq = 3 and by an average over 20 MC-simulation runs withNp = 75,000.Sets 1, 2, and 3 denote the particular set of tracked moments, which, retively, are{ k−1

5 }, { k−15 ·2}, { k−1

5 ·3}, { k−13 }, for k = 1,2, . . . ,6.Df = 1.8,

k0 = 1.3.

As expected, even a three-point QMOM can providecurate results for the low order moments. This can be seeFig. 4, which shows the percent difference between momcalculated by the QMOM with different sets of tracked mments and the (assumed correct) corresponding MC vaMoments in the rangek = [0,1] are very accurately calculated using the set of momentsk = {0,1/5,2/5,3/5,4/5,1}(i.e., using each ofk = 0, k = 1/5, etc.) or even by thesetk = {0,2/5,4/5,6/5,8/5,2}, which remains acceptablaccurate up to momentsk = 2. If, however, higher order moments are sought errors become large, ca. 5% for the smomentsk = {0,3/5,6/5,9/5,12/5,3}. Notice also (dot-ted line) how the method completely fails if moments larthan the maximum one tracked are calculated by Eq. (8)

The QMOM can nevertheless provide fairly accuratesults for moments of order ca. 3 with the use of four-poquadratures, as shown in Fig. 5. However, notice that ifrange of moments is extended tok = 3.5 errors become con

-

.

f

Fig. 5. Percent difference in moments computed by the QMOM wNq = 4 and by an average over 20 MC-simulation runs withNp = 75,000.Sets 1, 2 and 3 denote the particular set of tracked moments, which, retively, are{ k−1

4 }, { k−17 ·3}, { k−1

2 }, for k = 1,2, . . . ,8.Df = 1.8, k0 = 1.3.

siderably larger (ca. 4%). Once again, “extrapolation” wgrossly fail.

Initial conditions cannot be expected to modify the sepreserving solution and hence the selection of the indistribution becomes irrelevant unless small time solutiare sought (see also the discussion below). Unless statewhat follows all QMOM results presented (as well as thpreviously shown in Fig. 2) correspond to the set of momek = {0,3/7,6/7,9/7,12/7,15/7,18/7,3} and to either aninitial lognormal ASD given by

(17)n(v,0) = 1√2πvgv

exp

[− ln2(v/vg)

2ω2

],

wherevg = v1√

3/2 andw = √ln(4/3), as in [61], or to an

exponential distribution of sizes with a mean valuev̄ = v1.

4.3. Self-preserving distribution

The principal goal of this paper is to analyze the conquences of an improved collision rate law on the resulASDs. Although implications in more complex systemsquiring more state variables/processes need to be indually determined, many interesting conclusions can beready drawn for the described population of DLCAs. To tend we have compared the long-time behavior of the reing dimensionless-ASD and its moments as obtained(a) the correlated collision kernel [19] and (b) with antimated collision kernel based on the sum of the gyraradii, Eq. (5). MC simulations show that after a sufficienlong time, both kernels lead to time-invariant dimensionlASDs, which did not dependon the initial conditions chosen (see below). Both asymptotic ASDs are shown in FigA first observation is that the new kernel results in lardeviations from the frequently assumed lognormal distrition. This conclusion can be easily extracted from Fig. 6a


r thernel

nhichduelesareae

6an ae-te oisused-tion

redon

s,25%

ertyorof aex-less

op-hus,e twoele-,

llentnts)or-

inglcu-r or%.

sions-n-

im-ag-y ofy

SD

r-n

(a)

(b)

Fig. 6. MC-results for the self-preserving aggregate size distributions fotwo kernels considered. Notice the broader distribution for the new ke[19] and its effect on the moments (Fig. 7).Df = 1.8, k0 = 1.3.

just realizing that a plot ofηψ(η) versus log(η) would yielda Gaussian ifψ(η) were lognormal. Notice also that, givethe logarithmic scale used for the aggregate size axis (wbecomes the natural way of representing the distributionto its nearly lognormal character), the number of particbelonging to a size range can be directly related to theunder the curveηψ vs log(η) and not to that under the curvψ–log(η), which is nevertheless also shown (Fig. 6b).

Finally, it is also immediately clear from both Figs.and 6b that the improved collision rate also results ibroader ASD, presumably as aconsequence of the abovmentioned penetration factor, which decreases the racollisions involving pairs of small with large particles. Thbroadening can be measured in terms of the frequentlynumber-based geometric standard deviation of the dimensionless number density distribution considered as a funcof the dimensionless volumeη [3]. Asymptotically, a valueof 5.97 is obtained for the improved kernel, to be compawith a value of 4.52 that would be obtained with collisirates based on previous estimates.

f

Fig. 7. Asymptotic dimensionless moments for both aggregation kernelusing both QMOM and MC methods. Notice a systematic error of ca.for k near 3.Df = 1.8, k0 = 1.3.

Evaluation of any population-averaged physical propwill involve integration over particle volume of the ASD. Fthe common case of a power-law dependence in sizephysical property, the population-averaged value can bepressed in terms of the corresponding order dimensionmoment of the ASD times the value of that physical prerty evaluated at the mean size of the population [16]. Tconsequences on the averaged physical properties of thdifferent ASDs are best discussed by examining their rvant dimensionless moments,µk . These are shown in Fig. 7for both collision kernels. As already discussed, an exceagreement between MC results (lines) and QMOM (poiis obtained. Not surprisingly, the broadening of the nmalized distribution implies that momentsµk , with k < 1,were previously overestimated, whereas those withk > 1were systematically underestimated. Corrections involvthe first group of moments—needed, for example, to calate Brownian diffusion deposition rates across laminaturbulent boundary layers [20]—remain small, within 5Nonnegligible corrections (ca. 25% fork = 3) are, how-ever, needed in the second group. The improved collirate kernel thus noticeably affects the interpretation of elatic light scattering experiments [17] or inertial-impactioinduced aggregate deposition predictions [63].

The attainment of a self-preserving distribution alsoplies a well-defined asymptotic behavior of the mean coulation rate (as defined, for example, by the rate of decathe particle number densityNp , see below), which is mainlcharacterized by the homogeneity factor,λ. Indeed, it caneasily be shown [64] that after attainment of the SP-Athe number density evolution in time (Eq. (6) withk = 0)can be integrated to yield

(18)Np =[Nλ−1

p,0 + 1− λ

2β̂φλ(t − t0)

]1/(λ−1)

,

where the initial conditionNp,0 is the number density of paticles at a timet0 for which the self-preserving distributio


ime.r

um-

r-e,

d as

lineach

theum-eatde-

rsumofcallycomasetaldra-w)d by

gn the

trib-n-as-rs,ses

reachere-iate-es

ron-dente

vene ofof-pop-hised,not-so-pori-togly

urlt af-mo-ilini-foxi-the

ove

Fig. 8. Decay rate of the particle number density as a function of tNotice that for both kernels the expected asymptotic power law behavio(exponent 1/(λ− 1) ≈ −2.571, Eq. (18)) is recovered.Df = 1.8, k0 = 1.3.

has already been attained, andβ̂ is a constant for each ASDdefined as

(19)β̂ ≡∞∫

0

dηi

∞∫0

dηjβ(ηi, ηj )ψiψj .

The equations above imply that for large times the total nber density of particles in the population scales asNp ∝t1/(λ−1). A log–log plot of the number of particles vesus time will therefore ultimately become a straight linwith a slope given by 1/(λ − 1) � 2.571, for Df = 1.8,for both coagulation kernels. These slopes, calculated logNp/d logτ from the 4p-QMOM time evolution of aninitial exponential distribution of particles with v̄ = v1 areplotted as a function of the dimensionless timeτ in Fig. 8,for both coagulation kernels. Also shown by the dashedis the expected slope towards which both solutions approat long times. As an interesting corollary, notice thatdecay rate per particle number density squared of the nber of particles (or its inverse, the mean cluster volum̄vgrowth rate) resulting from the two kernels will differlarge times by just a constant factor; this quantity alsofines a population-averaged collision rate constantβ̄ throughthe expression

−2

N2p

dNp

dt= 1

N2p

∞∫0

dvi

∞∫0

dvjβ(vi, vj )ninj

(20)= −2v̄λβ̂ ≡ β̄.

The ratio between the population-averaged collision rate fothe correlated collision kernel and that based on theof the gyration radii is shown in Fig. 9 as a functionaggregate mean size. As expected, the ratio asymptotiapproaches a constant whose value, obtained from theputations, is about 0.91. Hence, asymptotic estimates bon the gyration radius will globally overestimate the tocollision (aggregate mean size grow) rate, although notmatically. Collision rates in the more important (see beloearly stages of coagulation are, however, underestimateRg-based values, in this case incurring larger errors.

-d

Fig. 9. Ratio between the mean coagulation rates (Eq. (19)) correspondinto the computed/correlated collision kernel and to estimates based osum of the gyration radius of the aggregates.Df = 1.8, k0 = 1.3.

4.4. Times and sizes required to achieve self-preservation

Results just discussed pertain to self-preserving disutions attained under conditions of free-molecular Browian coagulation. While they represent the interestingymptotic limit corresponding to large Knudsen numbedirect application of the results to practical environmentmay be limited. This is a consequence of the long timand therefore large aggregate sizes that are required toself-preservation. Moreover, even if self-preservation wachieved under free-moleculeconditions, further coagulation would soon lead to the corresponding intermedand ultimately continuum regimedistributions. Purely freemolecule conditions will, however, prevail at early stagof coagulation in many atmospheric-pressure flame enviments [22,34], with results that therefore become depenon initial conditions for our model, i.e., those at the timat which coagulation starts dominating over sintering. Ein constant, uniform environments, the stochastic naturcoagulation by Brownian motion originating the growththe initial molecular clusters, will lead to an “initial” distribution of aggregates rather than to a monodisperseulation of spherical particles. Accurate prediction of tdistribution is well beyond the goal of this paper. Indesuch a task would, in principle, require a model whichonly considers coagulation and sintering, but also the notwell-known mechanisms of nucleation as well as the vaprecursor chemical reactions and associated kinetics. Addtionally, results will not likely be as easily generalizeddifferent environments/products, but rather will be strondependent on particular combinations of both.

This difficulty is, however, somewhat overcome by oclaim that initial information is “partially” forgotten welbefore self-preservation o accurs. By this we mean thater a short time, and at still small aggregate sizes, thements will follow a similar, “quasi-universal” evolution untthe attainment of self-preservation, particularly sincetial distributions will usually not differ much from that omonodisperse spherules. While undoubtedly only apprmate, this statement is supported by Fig. 10 which showsevolution of two representative moments (below and ab


gate

as

aryrsepo-ndlso

reheperra

th

ton-notex-cal

gate

as

esusteas-

or ales.atevia-

wellnce

ionion,, we

tely

(a)

(b)

Fig. 10. Evolution of dimensionless moments as a function of aggrepopulation mean size. In (a), moments of orderk = 0.3 are shown, whilein (b) k = 3. Results correspond to three different initial distributions,described in Section 4.4.Df = 1.8, k0 = 1.3.

k = 1) as a function of aggregate mean number of primparticles, for three different initial conditions: monodispe(evolution calculated from MC simulations), and the exnential and lognormal distributions used for the QMOM adescribed above. Notice that, although the QMOM is aamenable to a discrete formulation [16], we have used heit in the continuous form; this can partially account for tdifferences found for a small mean number of particlesaggregate. We have also considered the evolution from otheinitial distributions, particularly considering cases withmuch broader initial distribution of sizes: monodisperse wiN = 1, uniform distribution in the size rangeN = 1–20,and the asymptotic self-preserving distribution “scaled”a mean size〈N〉 = 20. Evolution of the same two represetative moments is shown in Fig. 11. Although there isan universal curve, still in all cases a common trend ishibited, with dimensionless moments going through a lomaximum (minimum) for moments of orderk > 1 (k < 1),which is very similar for all cases plotted.

(a)

(b)

Fig. 11. Evolution of dimensionless moments as a function of aggrepopulation mean size. In (a), moments of orderk = 0.3 are shown, whilein (b) k = 3. Results correspond to three different initial distributions,described in Section 4.4.Df = 1.8, k0 = 1.3.

This similar behavior of the moment evolution curvallows us to estimate details of the distribution from jthe value of the mean aggregate size, which can beily obtained from elastic light scattering measurementsfew TEM images of thermophoretically deposited particIt is worth noticing that for these more relevant aggregsizes (rather than those found at self-preservation), detions from previous estimates become much higher.

5. Conclusions

We have used the quadrature method of moments asas Monte Carlo simulations to solve the population balaequation for a system of fractal-like aggregates (Df = 1.8,k0 = 1.3) undergoing free-molecular Brownian coagulatin a constant environment. For this regime of aggregatand using aggregate volume as the only state variablehave demonstrated the ability of the QMOM to accura


et o

preg-tingteder

htasenaliblewecol-D-

esti-the

las-atecor-,ossvi-ionys-

elf-for

tita-tionri-uretionticleousr

thees-orear

and,

ng-eac-withaleful

Fer-urseur

tte,

r-

at-

n

ize

compute high-order moments (up to aboutk = 3 in volume)using four-point quadrature and an adequately chosen smoments.

We have also studied the effects that a more accuratediction of the FA–FA collision frequency [19] has on the agregate size distribution of a population of such coagulaparticles. While we make no claim that the new correlacollision rates comprise the “last word,” and many othsimplifications inevitably still included in our model migcause systematic errors, we believe these calculations, bon MC-computed FA-collision rates, have a more ratiojustification than previous results based only on plausscaling laws for the collision rates. With this premise,have, in particular, shown that introducing the computedlision frequency systematically shifts the asymptotic ASshape, concluding that asymptotic moments,µk, with k > 1(see Fig. 7) have previously been systematically undermated. This is expected to have a noticeable effect onpredicted behavior of such populations in the case of etic light scattering, and inertial-impaction-induced aggregdeposition. While perhaps a less dramatic corollary, theresponding moments with 0< k < 1, relevant, for examplein calculations of Brownian diffusion deposition rates acrlaminar or turbulent boundary layers [20,65], have preously been slightly overpredicted. Our new FA-coagulatrate kernel and QMOM simulation method also lead to stematically different times, and even “paths,” to reach “spreservation.” The much longer times and sizes requiredstrict self-preservation compromises their practical quantive importance. We have, however, shown that informaon initial conditions is soon “lost” so that details of the distbution can be estimated at each time from just the measment of the mean particle size. The nonmonotonic evolutowards self-preservation indicates that, for the usual parsizes found in our flames [7,21,22], deviations from previestimates are expected to be (sometimes significantly) largethan those shown in Fig. 7.

We conclude by remarking that this work now setsstage for the incorporation of this collision frequency exprsion [19] into the modeling of aggregate populations in mcomplex flow environments, including counterflow lamindiffusion flames (see, e.g., Rosner and Pyykönen [7]),ultimately,turbulent flame synthesis reactors.

Acknowledgments

These aggregate coagulation calculations, part of a lorange fundamental study to improve the accuracy of sol rtion engineering predictions, have been made possiblethe financial support of NSF under Grant 998 0747 at YUniversity. It is also a pleasure to acknowledge the helpsuggestions/correspondence of Drs. R.L. McGraw, J.nández de la Mora, and C.M. Sorensen during the coof this investigation. A preliminary verbal account of o

f

-

d

-

findings was presented at AAAR ’02 (October 7, CharloNorth Carolina) as Paper 1E4.

Appendix A. Nomenclature

a mean “primary” particle (spherule) radiusc̄r average relative velocity between aggregatesDf fractal exponent (“dimension”) describing mo

phology of aggregatepopulationdeff effective collision diameterf penetration factorJ Jacobian matrix (Eq. (9))k0 fractal “prefactor” defined byN = k0(Rg/a)Df

kB Boltzmann constantM moment vector (Eq. (9))Mk kth moment of the pdfn(v; t); i.e.,

∫ ∞0 vkn(v; t) dv

m aggregate massn aggregate size distribution functionN number of “spherules” in an aggregateNp total number of particles per unit volumeReff effective collision radius of a single aggregateRg gyration radius of aggregate containingN spherulesT absolute temperature (K)tc characteristic time for coagulationts characteristic time for sinteringu,v or i, j aggregate sizev1 volume of each “primary” spherule,(4π/3)a3

v̄ mean particle volume,̄v = φ/Np

x abscissas and weights vector (Eq. (9))

Greek

α size ratio (< 1) between two aggregates participing in a collision

β(u, v) or βij coagulation rate constantβ̄ population-averaged collision rate (Eq. (20))β̂ ASD constant factor, Eq. (19)η dimensionless particle volumeηrot ratio between ballistic and total effective collisio

diametersλ degree of homogeneity of a functionµk kth moment of the dimensionless aggregate s

distributionψ(η); i.e.,∫ ∞

0 ηkψ(η) dη

σ collision cross sectionτ dimensionless time,t/tcφ particle volume fractionψ dimensionless aggregate size distributionωi weight in the quadrature formula

Subscripts

c pertaining to coagulationeff effective valueij pertaining to collision partnersi andj

i, j or u,v particle sizesi, j or u, v


g

.

36

99.

rosol

29

ce

9..t. 89

r-00.003)

10.

7)

Sci-

.5.

st.

33

,

s,

.as

.of

ce

76

) 63.

)

.:,

al

.

165

k pertaining to moment exponentk, wherek neednotbe an integer, nor positive

max maximum value (in the distribution)s pertaining to sintering

Other

〈〉, ¯( ) mean value

Abbreviations and acronyms

ASD aggregate size distribution,n(v; t)

CC cluster–clusterDLCA diffusion-limited cluster–cluster aggregatesFA fractal-like aggregatefct( ) function of indicated argumentHTCRE high-temperature chemical reaction engineerinIDE integrodifferential equationMC Monte CarloMOM method of momentsPDF probability density functionQ quadrature-basedODE ordinary differential equationPDE partial differential equationPBE population balance equationSP self-preservingSRE sol reaction engineeringTEM transmission electron microscopy

References

[1] W. Koch, S.K. Friedlander, J. Colloid Interface Sci. 140 (1990) 419[2] Y. Xiong, S.E. Pratsinis, J. Aerosol Sci. 24 (1993) 283.[3] P. Tandon, D.E. Rosner, J. Colloid Interface Sci. 213 (1999) 273.[4] D.L. Wright, R. McGraw, D.E. Rosner, J. Colloid Interface Sci. 2

(2001) 242.[5] D.E. Rosner, S.Y. Yu, AIChE J. 47 (2001) 545.[6] M. Kostoglou, A.G. Konstandopoulos, J. Aerosol Sci. 32 (2001) 13[7] D.E. Rosner, J.J. Pyykönen, AIChE J. 48 (2002) 476.[8] S.K. Friedlander, Smoke, Dust, and Haze: Fundamentals of Ae

Dynamics, Oxford Univ. Press, New York, 2000.[9] A.V. Filippov, M. Zurita, D.E. Rosner, J. Colloid Interface Sci. 2

(2000) 261.[10] R.D. Mountain, G.W. Mulholland, H. Baum, J. Colloid Interfa

Sci. 114 (1986) 67.[11] T. Vicsek, P. Meakin, F. Family, Phys. Rev. A 32 (1985) 1122.[12] R.M. Ziff, E.D. Mcgrady, P. Meakin, J. Chem. Phys. 82 (1985) 526[13] G.W. Mulholland, R.D. Mountain, J. Chem. Phys. 84 (1986) 4109[14] D. Fry, T. Sintes, A. Chakrabarti, C.M. Sorensen, Phys. Rev. Let

(2002) art 148301.[15] D. Ramkrishna, Population Balances: Theory and Applications to Pa

ticulate Systems in Engineering, Academic Press, San Diego, 20[16] D.E. Rosner, R. McGraw, P. Tandon, Ind. Eng. Chem. Res. 42 (2

2699.[17] C.M. Sorensen, Aerosol Sci. Technol. 35 (2001) 648.[18] R. McGraw, Aerosol Sci. Technol. 27 (1997) 255.

[19] M. Zurita-Gotor, D.E. Rosner, J. Colloid Interface Sci. 255 (2002)[20] D.E. Rosner, M. Tassopoulos, AIChE J. 35 (1989) 1497.[21] Y. Xing, D.E. Rosner, U.O. Köylü, P. Tandon, AIChE J. 43 (199

2641.[22] Y.C. Xing, U.O. Köylü, D.E. Rosner, Appl. Opt. 38 (1999) 2686.[23] R. Jullien, R. Botet, Aggregation and Fractal Aggregates, World

entific, Singapore, 1987.[24] P. Meakin, Rev. Geophys. 29 (1991) 317.[25] S.R. Forrest, T.A. Witten, J. Phys. A Math. Gen. 12 (1979) L109.[26] R.J. Samson, G.W. Mulholland,J.W. Gentry, Langmuir 3 (1987) 272[27] C.M. Megaridis, R.A. Dobbins, Combust. Sci. Technol. 71 (1990) 9[28] J. Cai, N. Lu, C.M. Sorensen, Langmuir 9 (1993) 2861.[29] U.O. Köylü, G.M. Faeth, T.L. Farias, M.G. Carvalho, Combu

Flame 100 (1995) 621.[30] Y. Xing, D.E. Rosner, J. Nanoparticle Res. 1 (1999) 277.[31] C. Oh, C.M. Sorensen, J. Colloid Interface Sci. 193 (1997) 17.[32] A.M. Brasil, T.L. Farias, M.G. Carvalho, Aerosol Sci. Technol.

(2000) 440.[33] J.H. Seinfeld, Atmospheric Chemistry and Physics of Air Pollution

Wiley, New York, 1986.[34] C. Oh, C.M. Sorensen, J. Aerosol Sci. 28 (1997) 937.[35] G.M. Hidy, J.R. Brock, The Dynamics of Aerocolloidal System

Pergamon, Oxford, 1970.[36] N.A. Fuchs, The Mechanics of Aerosols, Pergamon, Oxford, 1964[37] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of G

Flows, Oxford Univ. Press, New York, 1994.[38] P. Meakin, Phys. Scripta 46 (1992) 295.[39] S.N. Rogak, R.C. Flagan, J. Colloid Interface Sci. 151 (1992) 203[40] A. Kazakov, F.L. Dryer, in: 2nd Joint Meeting of the U.S. Sections

the Combustion Institute, 2001, Paper 123.[41] F.S. Lai, S.K. Friedlander, J. Pich, G.M. Hidy, J. Colloid Interfa

Sci. 39 (1972) 395.[42] F. Gelbard, Y. Tambour, J.H. Seinfeld, J. Colloid Interface Sci.

(1980) 541.[43] J.D. Landgrebe, S.E. Pratsinis, J. Colloid Interface Sci. 139 (1990[44] L.A. Spielman, O. Levenspiel, Chem. Eng. Sci. 20 (1965) 247.[45] T. Matsoukas, S.K. Friedlander, J. Colloid Interface Sci. 146 (1991

495.[46] J.R.P. Gooch, M.J. Hounslow, AIChE J. 42 (1996) 1864.[47] H.M. Hulburt, S. Katz, Chem. Eng. Sci. 19 (1964) 555.[48] K.W. Lee, J. Colloid Interface Sci. 92 (1983) 315.[49] M. Frenklach, S.J. Harris, J. Colloid Interface Sci. 118 (1987) 252[50] W.H. Press, S.A. Teukolsky, W.T.Vetterling, Numerical Recipes in C

The Art of Scientific Computing,Cambridge Univ. Press, New York1992.

[51] R. McGraw, D.L. Wright, J. Aerosol Sci. 34 (2003) 189.[52] D. Ramkrishna, Chem. Eng. Sci. 26 (1971) 1134.[53] B.A. Finlayson, The Method of Weighted Residuals and Variation

Principles, Academic Press, New York, 1972.[54] D. Ramkrishna, Chem. Eng. Sci. 28 (1973) 1362.[55] M. Smith, T. Matsoukas, Chem. Eng. Sci. 53 (1998) 1777.[56] A. Eibeck, W. Wagner, SIAM J. Sci. Comput. 22 (2000) 802.[57] S.K. Friedlander, C.S. Wang,J. Colloid Interface Sci. 22 (1966) 126[58] E. Ruckenstein, J.C. Chi, J. Colloid Interface Sci. 50 (1975) 508.[59] S. Vemury, S.E. Pratsinis, J. Aerosol Sci. 26 (1995) 175.[60] D.L. Marchisio, J.T. Pikturna, R.O. Fox, D.R. Vigil, A.A. Barresi,

AIChE J. 49 (2003) 1266.[61] J.C. Barrett, N.A. Webb, J. Aerosol Sci. 29 (1998) 31.[62] S. Vemury, K.A. Kusters, S.E. Pratsinis, J. Colloid Interface Sci.

(1995) 53.[63] D.E. Rosner, P. Tandon, Chem. Eng. Sci. 50 (1995) 3409.[64] M.K. Wu, S.K. Friedlander, J. Aerosol Sci. 24 (1993) 273.[65] D.E. Rosner, AIChE J. 35 (1989) 164.

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