Two-component Brownian coagulation: Monte Carlo simulation and process characterization

Particuology 9(2011)414 -423Contents lists available at ScienceDirectPARIK LOL06YParticuologyELSEVIERjournal homepage: www.elsevier. com/locate/particTwo-component Brownian coagulation: Monte Carlo simulation and processcharacterizationHaibo Zha0*, Chuguang ZhengState Key Laboratory ofCoal Combustion, Huazhong University of Science and Technology. Wuhan, 430074 Hubei, ChinaARTICLE INFOABSTRACTArticle history:The compositional distribution within aggregates ofa given size is essential to the functionality of com-Received 17 November 2010posite aggregates that are usually enlarged by rapid Brownian coagulation. There is no analytical solutionAccepted 21 April 2011for the process of such two-component systems. Monte Carlo method is an effective numerical approachfor two-component coagulation. In this paper. the differentially weighted Monte Carlo method is usedKeywords:to investigate two-component Brownian coagulation, respectively, in the continuum regime, the free-Multivariate population balancemolecular regime and the transition regime. Itis found that(1)for Brownian coagulation in the continuumAggregationregime and in the free-molecular regime, the mono-variate compositional distribution, ie, the numberStochastic methoddensity distribution function ofone component amount (in the form of volume of the component in aggre-MixingSelf-preservinggates) satisfies self-preserving form the same as particle size distribution in mono-component Browniancoagulation; (2) however, for Brownian coagulation in the transition regime the mono-variate composi-tional distribution cannot reach self-similarity; and (3) the bivariate compositional distribution, i.e.. thecombined number density distribution function of two component amounts in the three regimes satis-fies a semi self-preserving form. Moreovver, other new features; inherent to aggregative mixing are alsohesasemi. sel-demonstrated; e.g. the degree of mixing between components. which is largely controlled by the initialcompositional mass fraction, improves as aggregate size increases.口2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.1. Introductioncomponent coagulation (Lushnikov, 1976):Multi-component Brownian coagulation is ubiquitous innature and in engineering, e.g.. nanocomposite ceramic powders(Al2O3/TiO2). with special properties such as superconductivity,superparamagnetism, or high catalytic activity, produced throughxn(以x- x uy - vy, t)n(%x，y t)duydryBrownian coagulation from a mixture of nanoparticle precursors(1)of functional molecular- scale monomers (Pratsinis, 1998). Insight-n(x,y,t))。"S。into the evolution of compositional distribution is very useful tooptimize the synthesis process of nanoparticles while tailoring thexn(%. uy t)duvgduy.functional particles, typically via the gas phase method at high tem-perature. Brownian coagulation of a two-component non-reactivesystem is obviously the most basic, and the most important case.The underlying nature of Eq, (1) is a two-component coagu-Spatially homogeneous two- component coagulation processes arelation event between particle A of state (x,Uy) and particle B ofdescribed by the following bivariate population balance equationstate (Vx, vW to result in a new particle C of state (Ux +v%,Uy+以(PBE) which is an extension of Smoluchowski's equation for one-and the death of particles A and B. Here Vx and uy are the vol-ume of x-component and y-component, respectively. within anaggregate having volume ofvx +Uy; n(Ux.Vy.t) is the number densityfunction at ti”汇Vy represents the num-ber concentr中国煤化工range of x-component,”Corresponding author, Tel: +86 27 87545526; fax: +8627 87545526.UxtoVx+dVxFMHC N M H Gponent, yy to uy+dvy;E-mail address: klinsmannzhb@163.com (H. Zhao).B(vx, Uy, vx. y,njb .aguialuil laic oefficient between par-1674-2001/$ - see front matter日2011 Chinese Society of Particuology and Institute of Process Engineering. Chinese Academy of Sciences. Published by Elsevier B.V. AlI rights reserved.doi:10.1016/j.partic.201 1.04.003H. Zhao, C Zheng/ Particuology 9(2011)414 -423151_ 8D;.8)Nomenclatureπc'Ctotal coagulation rate of a simulation particle,particle diameter of raindrop, μm_kgT_ [5+ 4Kn; + 6Kn? + 18Kn}(9)particle massDi= 3πpd; [- 5- Kn, +(8 +π)Kn2.Mtotal particle massnumber density function, m-3total particle number, m-3In Eqs. (2)-(9). kg is the Boltzmann constant; T is the absoluteparticle volumetemperature; u is the gas viscosity. Pp is the particle density; q isnumber weightthe Stokes-Cunningham slip correction factor for considering theslippage of gas molecules around particle i; D is the diffusion coef-Greek lettersficient for particle i; q is the velocity of particle i, 8i is the transitionmultiplicative constantcoagulation kernel, m3/sThe numerical solution of the bivariate PBE (Eq. (1)) is gen-dimensionless particle volumeerally very challenging due to the double integral and nonlineardimensionless particle size distributionbehavior of the equation. There are several analytical solutionscharacteristic coagulation time scalefor the two-component coagulation under considerable simplifi-cations of coagulation kernels (e.g.. constant (Gelbard & Seinfeld,Subscripts1978; Laurenzi, Bartels, & Diamond, 2002; Lushnikov, 1976), suminitial condition(Fernandez-Diaz & Gomez-Garcia, 2007; Laurenzi et al, 2002),i,jindices of simulation particlesand product (Laurenzi et al, 2002)) and initial compositionalminminimum valuedistributions (e.g. an initially monodisperse dstribution of eachmaxmaximum valuecomponent or an initially exponential distribution of each com-p.qa section of compositional distributionponent), However, once Brownian coagulation kernels for realparticlesprocesses are considered, there exists no analytical solution forxycomponent styletwo-component systems and, worse still, the inability of conven-tional numerical methods makes it difcult to simulate the detailedSuperscriptsevolution of compositional distributions. A noteworthy work incocontinwum regime.COthis field was contributed by Matsoukas, Lee. and Kim (2006),imfree-molecular regimeand Lee, Kim, Rajniak, and Matsoukas (2008). They built up the-Irtransition regimeoretical models to obtain the rate and degree of two-componentaggregative mixing as a function of aggregate size and time; andthey also used constant-number Monte Carlo method to solveticles A and B. In this study, the coagulation kernel is consideredthe two-component PBE for Brownian coagulation kernels in thecomposition-independent. With respect to Brownian coagulationcontinuum regime and in the free-molecular regime, respectively.between particles i (volume V, diameter d and mass m;) andj (vol-Their theoretical models and numerical approaches could be usedume U diameter dj and mass mj). its kernel is characterized byto determine the distribution of components within aggregates, tothree widely used formulas, depending on the ratio of the meanquantify the degree of mixing, and to optimize blending of compo-free path (入) of gas molecule to particle radius (d/2), that is, thenents. Nevertheless, some basic features of the mixing process ofKnudsen number, Kn= 2λ/d.two-component Brownian coagulation, especially in the transitionIn the continuum (Kn≤0.1) and near-continuum (0.1 10)(Lee & Chen, 1984): .due to the methodological scheme that does not specify how thesimulation particles should be distributed over the size and compo-sitional spectra. Thus, it is highly necessary to use robust numerical3)methods to investigate two-component Brownian coagulation in向=()()”I+听氐哥Pp,the three regimes and to provide insight into some characteristicsIn the transition regime (1 < Kn≤10) (Fuchs, 1964):inherent to Brownian aggregative mixing of two components.We have proposed a differentially weighted Monte Carlo隋=2m(d,+ d;XD,+ D)4+d8(D.+D)).(4)(DWMC) method (Zhao, Kruis, & Zheng, 2010) for two- component(+4+2(;? +g?Y/2* (d+dX? +引)coagulation processes, which has been proven to be efficient andprecise. Different from conventional MC methods, the DWMCwheremethodtracks differentially weighted simulation particles on-0.435dG=1+金[.493+.4xp(-454)].5)the basis of a new probilistic rule for coagulation betweentwo differential中国煤化工icles. and adopts acomponent-depthe number of sim-4-(8k7)"6)ulation particle:YHC N M H Gh component spacewithin prescribed bounds during simulation. This paper brieflyg:= 3aI[(d+4y -(d+9)921-d,1(7)introduces the DWMC method and then utilizes it to simulate thetwo-component Brownian coagulation.416H. Zhoo, C Zheng/ Prticuology 9(2011)414 4232. Differentially weighted Monte Carlo (DWMC) method forcles having larger weight values than sections where numbertwo- component coagulationdensity is low.2) An adjustable time step is determined from local mean-fieldWhen simulating two or more internal variables of particlescoagulation rate: .(e.g.. in this paper, two chemical compositions). conventionalpNdeterministic methods such as the method of moments (McGraw△t=z高(VC)& Wright, 2003) and sectional method (Kim & Seinfeld, 1990) areformulated by complicated mathematical equations, but could notwhere empirical parameter p is set around 2/N;t- 0.05; Nst isdeal with the innate fluctuations for multi-component coagulationthe total number of simulation particles in the system; C (with(Laurenzi et al. 2002). And more unfortunately, the conventionalunit of m -3 s-1)is the total coagulation rate of simulation parti-deterministic methods may not be valid for long time periods, whencle i. C is calculated from the probabilistic coagulation rule foronly several particles acquire enough mass to become larger thancoagulation event between two differentially weighted simu-that of the rest of the population (Alfonso, Raga, & Baumgardner,2008) for complete coagulation to occur (Laurenzi et al, 2002).lation particles (Zhao et al, 2009; Zhao & Zheng, 2009). In thisFortunately, stochastic methods (Kruis, Maisels, & Fissan, 2000;rule, for a coagulation event between simulation particles i andLee et al, 2008; Maisels, Kruis, & Fissan, 2002; Matsoukas et al,j, it is imagined that each real particle from i undergoes a realcoagulation event with a probability of min(W,Wj)/Wi, and each2006; Sun, Axelbaum, & Huertas, 2004), which directly simulatereal particle fromj does so with a probability of min(W,Wj)/Wj,the dynamic evolution of a finite sample of the particle populationwhere W; and Wj are the private weights of i andj, respectively.using Monte Carlo(MC) technique, are capable of simulating multi-q is thus calculated ascomponent population balances in a simple and straightforwardmanner.We have proposed the DWMC method for particle coagulation[2ByWy max(W, w2]6y， (12) .in monovariate systems (Zhao, Kruis, & Zheng, 2009; Zhao & Zheng,W:+ Wj2009), and then extended it to two-component coagulation pro-j=1j#icesses (Zhao et al, 2010). The robust MC is briefly presented aswhere By is the coagulation kernel between particle i and par-follows:ticle, j, m3/s; B; is a normalized kernel that relates not only tothe states (like volumes) but also to the weights of the two(1) Simulation particles are dferentially weighted according tosimulation particles.initial compositional distributions. First, two individual com-It is noteworthy that DWMC evolves in either event drivenpositional spectra are respectively divided into intervals bymode or time-driven mode according to the value of empiricallaws which can be freely adapted to the problems to be solved,parameter p. If p=2/Nst. the resultant time-step, 2/(VS CC)is just the waiting time between two successive coagulationpositional distribution. With respect to a section (p,q) in theevents, and DWMC evolves in the event driven mode, wherespace. it represents a state (Uxp,UYy,q) of particles having xonly one coagulation event occurs within this time-step. Ifcomponent volumes between V%x,p and城p and y-componentp> 2/Nst, there are pNst/2 coagulation events within the time-volumes between y.q and yg, and the number concentrationstep and DWMC evolves in the time-driven mode. Generallyof these particles is n(ux,p, Vy,q, 0)(味p - vxp)(5$,p - Vy,p).Thesespeaking, the event-driven version is more accurate becausereal particles are considered to have similar dynamic behaviorevents are fully uncoupled among different time steps, whileand are represented by a certain number of weighted simu-the time- driven mode is faster because many events are simu-lation particles. The mean weight of simulation particles forlated within one time step.section (p.q) is thus calculated as:) Within the time step the interacting particle pair(s) is (are)selected with probability ;/2 i2 i+ ;B[% Either the cumula-tx.p,. y.q.0X喊p一吃X咕q一5g)V，(10)tive probility method or the acceptance-rejection method isWp(xp, y.q)=N<(x.p,Vy,)adopted to determine coagulated pair(s) in either event-drivenmode or time-driven mode, as described in reference (Zhaowhere N.(xp,y,) is the number of simulation particles locatedet al, 2010).at sectionV is the volume of the simulated system. In theDWMC method, Ns is prescribed to be more than a fixed mini-simulation particles with new states and weights according tomum number Ns,min but less than a maximum number Ns,max.the probabilistic coagulation rule. As for the ij coagulationevent, two new simulation particles replace the *old" particlesi and j, as formulated by:( W; = max(W;, W)- min(W;W), mi = mlwe max(w,wy), 竹= Vklws=max(w,.Wy),if Wi≠W,收，= vx,kIwp -max(wy,Wy)+ y,i = Uy.kIwp= -max(wy,Wy),W"j = min(W,Wy), m=mi+m, y=V+切，. .=x;i+Uxj， *,:=y.1+ Uys,(13)w°=气，m'=mi+mj, 听=片+切，呢;=Ux.+Uxj， =Vy, +ys, .if Wi=W,中国煤化工mj=mi+m，I=U+步Dxj， W=y,+y,MHCNM HGSections where number density of real particles is high can thuswhere the asterisk indicates a new value of weight or state afterbe designated to have a certain number of simulation parti-the coagulation event; m; and Vi are the total mass and volumeH. Zhao, c Zheng/ Particuology9 (2011)414 423417of simulation particle i; Vxi and ys; are the mass of component3. Two-component Brownian coagulationx and component y in simulation particle i. It is obvious thatEq. (13) satisfles the law of conservation of mass. and DWMCis3.1. Self-preserving distributioncapable of keeping simulation particle number constant. Par-ticle diameter d is obtained from particle volume, providedIt is well known that the size distribution of mono-componentthat the aggregates are spherical due to faster sintering in highparticles satifes the self- preserving form after a time-lag intemperature.cases of Brownian coagulation in the continuum regime or in(5) The total coagulation rate of each particle is updated accordingthe free-molecular regime (Friedlander, 2000). In this study, two-to the involved coagulation event(s)within Ot using the smart-component Brownian coagulation processes in the continuumbookkeeping technique (Zhao et al., 2009).regime, in the free-molecular regime, and in the transition regime(6) When certain conditions are reached, for example, whenare respectively simulated to study whether the compositional dis-the number concentration of real particles is halved, thetribution in these cases is still self-preserving or not. Althoughcomposition-dependent shift action is performed whichno analytical solution of compositional distribution exists. therestricts the number of simulation particles in predefined sizediscrete-sectional models (Landgrebe & Pratsinis, 1990; Vemuryintervals of each component space within these prescribed& Pratsinis, 1995) provide classical benchmark solutions of self-bounds (between Ns min and Ns,max). The action is realized aspreserving particle size distributions for Brownian coagulation infollowing: first, the distribution of each component is sectional-the continuum and free-molecular regimes.ized by some prescribed laws (e.g. logarithmical discretization)In the self-preserving formulation, the dimensionless particleand the number of simulation particles in the chosen inter-volume is defined as η = Nv/M = v/D, and the dimensional distri-vals of each component space is counted (e.g.. the numbersbution as ψ = Mn(v, t)/N2 (Friedlander & Wang, 1966). where N isof simulation particles in size interval p of x component spacethe total number concentration of particles. M is the total mass (orand size interval q of y-component space are Nsx.p and Nsy.q.volume) concentration, vis the average volume, n(v,t) is the particlerespectively). Then, with respect to a section (p.q) of thesize distribution function at time t such that n( v,t)dv represents thetwo-component space, if min(Nsx.p,.Nsyq)< Ns,min, a simulationnumber concentration of particles in the volume range ofvtov+dv.particle. in the section(p.g)is equally split into new simulationsAs for two-component coagulation, the dimensionless volume andparticles with an integer number {Ns, min/min(Nsx.p,Nsy.q)].Thesenumber density distribution of x-component are defined asnew particles have the same internal variables as thnewpaent particle i and have a weight of w/[N.min/min(Nsxp.Nsy.q)J.加=M一家NxUx。y. = Mxnx(Ux,t),(14)N及One daughter particle replaces the position of its parent par-ticle i, and other daughter particles are added to the arraywhere Nx is total number concentration of particles containing x-of simulation particles; if max(Nxp,Nsy,)> Ns.max, a simula-component (in the case. there may be some particles not containingtion particle j in the section (p.q) can be randomly removedx-component due to the initial discrete distribution, thus Nx maywith a probability of [max(NsxpNsy.q)- Ns.max ]/max(Nsx.p.Nsy.q).be unequal to N); Mx is total mass concentration of x-componentA random process is used to decide whether simulation(it may not be equal to the total mass concentration of particles);particle j is removed or not. If a random number r from或x is the average x-component volume; and nx(Ux,t) is the numbera uniform distribution in the interval [0.1] is less thandensity distribution function of x-component at time t such that[max(Nx.p.Ny,q)- Ns,max ]/max(Nsxp,Nsy,.).j is removed and itsnx(Ux,t )dvx represents the number concentration of particles in theopen position is taken by the last particle in the simulationx-component volume range of Ux to Vx + dVx. Similarly:particle array. If not, the number weight of j is corrected bymy=Nyp_y: =Myny(y.t).(15)a multiple factor max(Nsxp,.Nya)INs.max. .In fact, the action shifts some simulation particles fromdensely populated regions of the two dimensional componentThe dimensionless distribution is obviously subjected to thespace to less populated regions by splitting some simula-integral constraints:tion particles in less populated regions into more simulationparticles and randomly removing some simulation parti-ψdn= 1,η4dn=1,cles in densely populated regions from the simulation. Thecomposition-dependent shift action overcomes the drawback4xdnx=1,nx4xdnx= 1,(16)of a stochastic approach as far as possible, and at the same timethe computational cost can be limited.4ydny=1,Iny4ydny=1.(7) Steps (2)-(6) are repeated until 2 ; Ot reaches the prescribedJoend time of process simulation.The characteristic coagulation time scales for the three Brown-The DWMC method is especially efective for two componentian coagulation cases are defined as follows:coagulation because dferentially weighted simulation par-In the free-molecular regime:ticles can be specified to distribute as homogeneously aspossible over high-dimensional joint space of internal vari-ables, which will greatly reduce statistical noise inherent to墙=[(3kTo)y)"()"°No17)the MC method and determine full Ccompositional distribu-tions in multi-component coagulation processes. We utilize theDWMC method to describe two-component Brownian coagula-In the continuum regime:tion, which are commonly encountered in industrial processes中国煤化工(18)such as nanoparticle synthesis, coal combustion, incineration,2kgTNGgranulation, and crystallization.In the transiJMHCNMH G(19)Cog = Br(dxo, dyp)No*418H. Zhao, C. Zheng/ Particuology 9 (2011)414-423Table 1Simulation conditions for three two-component Brownian coagulation cases (per unit volume of computational domain).Coagulation caseNxo (m-3)No (m-3)dxo (um)dso (um)T(K)μ (Pa.s)Pp (kg/m3)3x 10210.5001.81x 10-7x 10210.0010.00218004200Intrasitinresimese4x 10216x 10210.020.011 8005.65x 10-5a 10 4(m)~7| 4(7)-7,()-n,10"f10文1010- Vemury et al, 1995+ MC results (=1000_MC results (=10000.10- 10 10 1010' 10*7b,4(7)~n,|7),1o. Vemury et al, 1995MC results 1000.nMC results (-10000.. )10 10η"n°102 r(m)n,~4(n)n~]MC results (t=1000rMC results (=0000中国煤化工190YHC NMHG10 10xo:Nyo-3:o: 1:1; (b)in the free molecular regimne: No:Nyo-7:3. Vo:Y -1:8; and (c)in the tanstion regime: No:No- 4:6. B:y08:1:1H. Zhao, C. Zheng/ Particuology 9 (2011)414-423419a2t=1000...=10000r。-1-10.001-0-6.500飞0--4.750(og()-log,(,)-3.000骂.-2里-1.2500.50002-I log,(n)-log.()2.2504.0003t一--2 -log,(4)log,()log,(双)t=1000r.=10000..."coagCoag-8.2500---1多log/()-log()2一3-| lg,(.)1og.,)-4-" 4.000-4 -3-2-10123-54-3 -2. -1 0 1 2log,(Y)log。()2f=1000."cog=10000τ-8.500og,()-log,.()-7.000-5.500-4.000-2.5001----1.000--3log(,)log。(,)2.000-243-2 -10log,(.)logr。(办)Fig 2. The dimensionless bivariate compositional dstributions at dferent time points. (a)in the continumreg i; and (c)in the transtionregime.中国煤化工With respect to any simulation, each particle has only oneFig. 1(a)-(c) :YHC N M H Gule size dstributionchemical component at the initial stage, either x-component orand dimensionless mono-variate compositional distributions (they-component. The detailed simulation conditions for three two-number density distribution function of one component amount, incomponent Brownian coagulation cases are listed in Table 1.the form of volume of the component in aggregates) for Brownian420H. Zhao, c Zheng/ Pricuology 9(2011)414- 423a 10'10°NJIN-3:7;vx/"=l:1;M./M=3:710'10*=0co*≥10*2* 1010=10=10*0.0 0.10.20.30.40.50.60.7 0.8 0.9100.00.1 0.2 0.3 0.4_0.6 0.70.8 0.9 1.”v/vb 10°r10°N.IN-=7:3;vd0_=1:8;MM.=7:24t 10'o°1o°o*]=10r之* 10*o*}10*]D0.10.20.30.40.50.60.70.80.9100.00.10.20.30.40.50.60.70.80.91.0” vivC 10t=10o"10°1/100* 1o*+o*ff10|f=10=10'N IN -2:3;v/v =8:1;M.JM =16:310中00.10.20.30.40.50.60.70.80.91000.10.20.3yJ/v中国煤化工Hg3 Copstioal number desity dsrbutors for wocomeno Brownian ogulato ()IntTYHCNMHGo- 1:1.Mo:Myo-3:7;(b)in; No:Nyo-2:3, Vo:Hyo-8:1, Ma:Mo- 16:3.H. Zhao, C Zheng/ Particuology 9(2011)414 4234210°厂10°[NIN=3:7;d0=:1;M。M。=3:7100o102o*j=10ro*+E 10110”1=10'10*。0.00.10.20.30.40.50.60.70.80.9100.00.1 0.20.3 0.40.50.60.70.8 0.9v/vNJN,=7:3;vdw=l:8;M/M. 7:241=107t。 。o"=10'τo12 I'cosg=10°0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0vx/vvy/vt=10'r10"110~f 10*10=10*τCogg|=10r。NJIN,=2:3;v/v=8:1;M/M,=16:30.00.10.20.30.40.50.60.70.8 0.910000.1 0.2 0.3中国煤化工THCNMHGFig. 4. Compositional mass density distributions for two-component Brownian coagulation. (a) In the continuum regime: Nxo:Nyo =3:7, Vso:Uyo = 1:1, Mxo:Myo = 3:7; (b)in thefree-molecular regime: No:Nyo = 7:3. vo0:yo = 1:8, Mx:Myo = 7:24: and (C) in the transition regime: Nxo:Nyo = 2:3. Vxo:Uyo = 8:1, Mxo:Myo= 16:3.422H. Zhao, C. Zheng/ Particuology 9 (2011)414-423Table 2Peak values of compositional number (and mass) density distribution.In continuum regimeIn free molecular regimeIn transition regimet-102rco8 t=10'τc03 t=10*tcom t-102rcog t-10Prtou t=104ro8_ t=102rcog t- 10°tcosg_ t- 10*tconmax(N/N)0.09570.26120.45280.27810.55060.75100.1194 .0.32680.5369max(N,IN)0.09410.26060.45340.27800.11930.3269max(N./N):max(N,IN)1.01701.00230.99871.00011.00001.00030.999(v/)(max(N/N)0.3042030420.29580.22920.8458(vy)0)(max(N,/N)0.70420.77080.1542(u/])a)(N)}({(/)(max(Ny,N)| 0.43200.43200.29735.4863 .5.4863max(Mx/[M)0.04100.1055 .0.08730.18510.13970.35450.48580.09480.29450.48260.02560.06490.0901max(M,/M);max(M,/M)0.43250.43260.42420.29630.29320.29185.46935.45865.3939(n/v)( max(M,[/M))(u/10)(max(M,/M)04330029730 297302973.; 48635 4863{v/v)(max(M,/M)}({(y1v)(max(M,/M)} 0.4320coagulation in the three regimes, respectively. It is found that not tional distributions (like Fig. 1) but also the semi self-preservingonly the size distribution (ψ vs n) but also the mono-variate com-bivariate compositional distributions (like Fig. 2) are independentpositional distributions (4x vs ηx and ψy vs ny) are self-preserving of initial compositional distributions.forms for Brownian coagulation cases in the continuum regimeand in the free-molecular regime. Also, both the size distribution3.2. Aggregative mixingand the mono-variate compositional distributions correspond tothe same self-similar formulation for the tWo cases. However, thesize distribution and the mono-variate compositional distributionsWith respect to the three two-component Brownian coagula-in the transition regime cannot reach the self-preserving form,tion cases described in Section 3.1, Fig. 3 shows the mono-variateas expected in the mono-component Brownian coagulation in thecompositional distribution in terms of the compositional num-transition regime.ber density ditribution (Nx/N or Ny/N) vs. the component massWith respect to the combined number density dstribution of rai(0o/10 or w1at sveral reseatite time points, The corex-component and y-component (the bivariate compositional dis- sponding compositional mass density dstribution(Ms/M or My/M)against the component mass ratio is plotted in Fig. 4. The first directtribution), its dimensionless form is defined asobservation from Fig. 3 is that the number density distribution4w = MxMyn(w, y.5),(20) of x-component closely mirrors the number density distributionNxNyNxyof y-component. The simulation results obviously meet the intu-where Noy (with unit of m-3)is total mumber concentration of par- itive understanding of agregative mixing of the two-componentticles smuitaneousy cntaining xcomponent and y component. sysem. The ele miror should be asried to that the dfeMxy (with unit ofm3 x m3 x m-3)is total mass concentration of par-entially weighted MC is capable of capturing the details of not onlyticles simultaneously containing x-component and y-component,the homogeneously mixed particles, which occupy a larger numberand ixy = Mxy/Nxy = M.1/Mo,o. It is found that(and mass) share in particle population (for example, the particleswhose vx/Uy is 3:7), but also the inhomogeneously mixed parti-cles, which are much less-populated (for example, the particlesNxyM,M,"口导*1.at the two edges of compositional distributions). We also observe(21)that the geometric standard deviation of both x-component andy-component number (and mass) density distributions decreases asObviously, the dimensionless bivariate compositional distri- time evolves, that is, as aggregate size increases. The results showbution does not satisfy the normalizing condition and is thus the degree of mixing between components improves as aggregatenot self-preserving from the classical Friedlander's point of viewsize increases, much in agreement with the numerical simulation(Friedlander, 2000). However, from the dimensionless bivariate and theoretical predictions of Matsoukas et al. (2006) and Lee et al.distribution at different time points, as shown in Fig. 2, we can008)find that the contour plot of the logarithm of the normalizedThe peak of component number (and mass) density distribu-two-dimensional compositional distribution function, log1o(4xy), tion clls for particular atention, as shown by the maximum valuesis nearly symmetric about the line logro(ny)= log1o(nx) on the of Nx/N (or Ny/N) and M,/!M (or My/M) as well as the correspond-plane log1o(ny)- log1o(nx). The white line in the three sub-figures ing vx/v (or y/U) listed in Table 2. It is found that the componentrepresents the line log1o(ny)= log1o(nx). Rather, as time evolves,number (and mass) density distributions are peaked at the corre-log1o( ψxy) stretches along with the line log1o( ny)= log1o(nx). .sponding mass fraction of x-component中x = Mxo/(Mxo + Myo), andresulting in a slimmer and slimmer contour plot. We thus argueφy= Myol(Mxo + Myo) when reaching the self-preserving size and/orthat the bivariate compositional distribution satisfies a semi self-compositionaldistributions, implying that the two-componentpreserving form for Brownian coagulation cases in the continuum mixing is largely controlled by the initial degree of segre-regime, in the free-molecular regime, and also in the transition gation in the feed. These results also agree with those ofregime.Matsoukas et 21 (2006) and IP et a1 (2008). It is interestingWe also simulated the coagulation processes of two-component that max(中国煤化工: moments is approxi-systems having different initial component distributions. for exam-mately equMy/M)approaches φx:中yple, different Nxo:Nyo and Vx0:Uyo. Similar results, which are not(3:7= 0.428YHCN M H Gthe continum regime,shown here because of limited space, are obtained for the thr7:24=0.2917 for the case in the free-molecular regime, andBrownian coagulation cases. It is worth emphasizing that not only16:3= 5.3333 for the case in the transition regime). And, as aggre-the self-preserving size distribution and mono-variate composi- gate size increases the maximum values ofNx/N(or Ny/N) and Mx/MH. Zhao, C Zheng/ Porticuology 9(2011)414 423423(or My/M)increase accordingly.The above observations also hold inFernandez- Diaz,j.M. &Comez-Garcia, G.J.(2007). Exact solution of Smoluchowski'sother cases where the initial size and number of two componentscontinuous multi-component equation with an additive kernel. Europhysics Ler-ters, 78(5), 56002.are different, and are also true for two-component Brownian coag-Friedlander, S. K (2000). Smoke, dust and haze: Fundamentals of aerosol dynamicsulation in the continuum regime, in the free-molecular regime, also. (2nded.). NewYork: Oxtord ryrtrss......s.di.couoin the transition regime.tion for coagulation by Brownian motion. Jourmal ofColloid and interface Science.4. ConclusionsFuchs, N. A (1964) The mechanics of aerosols. New York: Pergamon Press.Gelbard, F. M. & Seinfeld,J. H.(1978). Ccoagulation and growth of a multicomponentThe two-component Brownian coagulation is simulated by theKim, Y. P.. & Seinfeld. J. H. ( 1990). Simulation of multicomponent aerosol conden-differentially weighted MC method. It is found that: (1) the self-preserving formulations ofboth size distribution and mono-variateKruis, F. E Maisels, A. & Fissan. H. (2000) Direct simulation Monte Carlo methodcompositional distributions are obtained for Brownian coagulationfor particle coagulation and aggregation, AIChE Journal. 46(9)。1735-1742.cases in the continuum regime and in the free-molecular regime,Landgrebe,J D.. & Pratsinis. S. E (1990). A discrete-sectional model for particulatewhere the values of the self-preserving mono-variate composi-production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime. journal of Colloid and Interface Science, 139(1). 63-86.tional distributions are the same as those of the self-preserving sizeLaurenzi, I. J Bartels.J. D. & Diamond,S. L (2002). A general algorithm for exactdistributions; however Brownian coagulation in transition regimecannot reach self-preserving size distribution and mono-variatecompositional distributions. (2) The bivariate compositional dis-ee. K. W.. & Chen, H. (1984). Coagulation rate of polydisperse particles. AerosolScience & Technology. 3(3). 327-334.tribution of two components satisfies a semi self-preservingLee, K. Kim, T.. Rajniak, P. & Matsoukas, T. (2008). Compositional distributions inform for any Brownian coagulation case, that is, the normalizedmulticomponent aggregation. Chemical Engineering Science, 63(5).1293-1303.two-dimensional compositional distribution is symmetrically dis-Lushnikov, A. A. (1976) Evolution of coagulating systems: I. Coagulating mixtures.tributed independent of initial compositional distributions; (3) theMaisels.A. Kruis,F.E & Fssan, H.(2002) Mixing selectivity in bicomponent, bipolardegree of mixing between components improves (i.e.. the widthaggregation. Journal of Aerosol Science, 33(1),35- -49.of the compositional distributions is narrower and narrower) asMatsoukas, T. Lee, K., & Kim, T. (2006). Mixing of components in two-componentaggregate size increases; (4) the compositional number (and mass)McGraw.R.& Wright,D. L(2003). Chemically resolved aerosoldynamics for intemaldensity distribution is largely controlled by the initial composi-mixtures by the quadrature method of moments. Joumal of Aerosol Science, 34,tional mass fraction.andCombustionScience 24(31 197-219AcknowledgementsSun, z, Axelbaum, R & Huertas. J. (2004). Monte Carlo simulation of multicompo-nent aerosols undergoing simultaneous coagulation and condensation. AerosolScience and Technology. 38( 10).963 -971.H. Zhao was supported by funds from "The National Natural Sci-Vemury, s., & Pratsinis, s. E. (1995) Self-preserving size distrbutions of aggomer-ence Foundation of China" (50876037 and 50721005), "Programates. Journal of Aerosol Science, 26, 175-185.for New Century Excellent Talents in University" (NCET-10-0395),Zhao. H.KruisE&EZhenenc.cCoOReuegstststaoasisfeindsand "National Key Basic Research and Development Program"Monte Carlo simulation of coagulation. Aerosol Science and Technology. 43(8).(2010CB227004).781-793Zhao, H. Kruis, F.E & Zheng, C. A (2010). A iferentially weighted MonteReferencesCarlo method for two-component coagulation. Joumnal of Computational Physics,229(19). 6931-6945.Alfonso. L. Raga, G. B.. & Baumgardner, D. (2008). Monte Carlo simulations of twa-Zhao, H. & Zheng C (009) Anew event-driven constant volume method for solu;component drop growth by stochastic coalescence. Atmospheric Chemisory andpo2525 402212 pridre sie itsiudionon Joumal oCompuatoraoalPhysics Discussions, 8(2). 7289-7313.中国煤化工MHCNMH G