lattice boltzmann method for population balance equations with simultaneous growth, nucleation,...

Chemical Engineering Science 69 (2012) 316–328

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Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Lattice Boltzmann method for population balance equations withsimultaneous growth, nucleation, aggregation and breakage

Aniruddha Majumder a, Vinay Kariwala a,n, Santosh Ansumali b, Arvind Rajendran a

a School of Chemical & Biomedical Engineering, Nanyang Technological University, Singapore 637459, Singaporeb Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India

a r t i c l e i n f o

Article history:

Received 4 June 2011

Received in revised form

4 October 2011

Accepted 21 October 2011Available online 4 November 2011

Keywords:

Aggregation

Breakage

Dynamic simulation

Lattice Boltzmann method

Particulate process

Population balance

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.10.051

esponding author. Tel.: þ65 6316 8746; fax:

ail address: [email protected] (V. Kariwala).

a b s t r a c t

Lattice Boltzmann method (LBM) is developed for solution of one-dimensional population balance

equations (PBEs) with simultaneous growth, nucleation, aggregation and breakage. Aggregation and

breakage, which act as source terms in PBEs, are included as force terms in LBM formulation. The force

terms representing aggregation and breakage are evaluated by fixed pivot (FP) method. Multiscale

analysis is used to derive the kinetic equations associated with LBM, whose long-time large-scale

solution provides the solution of the PBE. A coordinate transformation is proposed, which allows the

use of non-uniform grid for LBM to obtain accurate solution of PBE with moderate number of grid

points. The performance of the proposed LBM-FP method is compared with finite volume (FV) and

method of characteristics (MOC) combined with FP (MOC-FP) methods. Using benchmark examples, the

proposed LBM-FP method is shown to be useful for solving PBEs due to its computational efficiency and

ability to handle a wide range of problems.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Population balance modeling has become an important toolto model particulate processes in various disciplines of scienceand engineering, e.g., crystallization (Randolph and Larson, 1962;Ramkrishna, 2000), granulation (Poon et al., 2009), milling (Bilgiliand Scarlett, 2005), emulsion polymerization (Immanuel et al., 2002),aerosols (Ramabhadran et al., 1976) and biological systems(Mantzaris et al., 2001). As the name signifies, population balanceseeks to describe the behavior of the population of the particles bytracking the evolution of the number density function defined in theparticle state space. Population balance equation (PBE) takes intoaccount various mechanisms, such as growth, nucleation, aggregationand breakage, by which particles of a particular state can either formor disappear from the system.

In this work, we assume that the particles are uniformlydistributed in the well-mixed control volume, which implies thatthe particle size distribution (PSD) is independent of the spatialcoordinate. Moreover, only one characteristic property of theparticles is considered. The resulting one-dimensional (1D) PBEtakes the following form:

@

@tnðx,tÞþ

@

@xðGðx,tÞnðx,tÞÞ ¼QnucþQaggþQbreak, ð1Þ

ll rights reserved.

þ65 6794 7553.

where nðx,tÞ is the PSD, Gðx,tÞ represents the growth rate of theparticle at state x and the terms on the right hand side of Eq. (1)denote the rates of nucleation (nuc), aggregation (agg) and break-age (break) given as

Qnuc ¼ B0ðtÞdðx0Þ, ð2Þ

Qagg ¼1

2

Z x

0nðx�x0,tÞnðx0,tÞaðx�x0,x0Þ dx0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Birth term due to aggregation, RBa

�

Z 10

nðx,tÞnðx0,tÞaðx,x0Þ dx0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Death term due to aggregation, RDa

, ð3Þ

Qbreak ¼

Z 1x

bðx,x0ÞGðx0Þnðx0,tÞ dx0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Birth term due to breakage RBb

� GðxÞnðx,tÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}Death term due to breakage RDb

, ð4Þ

where B0 is the nucleation rate, d is the Dirac delta function, x0 isthe size of the nuclei, aðx,x0Þ is the aggregation frequency denotingthe rate at which two particles of size x and x0 are combinedtogether to produce a particle of size xþx0, GðxÞ is the breakagerate and bðx,x0Þ denotes the probability of formation of particle ofsize x due to breakage of particle of size x0. In general, the PBE inEq. (1) has to be supplemented with appropriate equations for thekinetics (e.g., growth and nucleation) as well as mass and energybalances (Lang et al., 1999; Myerson, 2002).

PBEs are hyperbolic partial differential equations. In mostpractical cases, they do not have analytical solution and henceneed to be solved numerically. During the past few decades, anumber of methods have been developed for numerical solution

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A. Majumder et al. / Chemical Engineering Science 69 (2012) 316–328 317

of PBEs. Several reviews of these techniques are available in theliterature (Ramkrishna, 2000; Motz et al., 2002; Nopens et al., 2005;Costa et al., 2007; Kumar et al., 2009; Silva et al., 2010). In thesubsequent discussion, we provide an overview of the relevantmethods for solving PBEs involving simultaneous growth, nuclea-tion, aggregation and breakage.

Some of the available methods are used to track the evolution ofthe moments of the PSD. Various quadrature method of moments(QMOM) are used for this purpose (Marchisio et al., 2003; Marchisioand Fox, 2005; Gimbun et al., 2009; Qamar et al., 2011; Kariwalaet al., in press). However, full PSD is often required for design andcontrol purposes (Nagy, 2009) and reconstruction of the PSD from themoments can be difficult (Aamir et al., 2009). Finite element method(FEM) has been used for solving PBEs with simultaneous growth andaggregation to obtain information about the full PSD (Nicmanis andHounslow, 1998; Mahoney and Ramkrishna, 2002; Alexopoulos et al.,2009). Although accurate, FEM is usually computationally moreexpensive than other methods, e.g., discretized methods. Monte Carlo(MC) techniques are also used for solving PBEs with simultaneousprocesses (van Peborgh Gooch and Hounslow, 1996; Smith andMatsoukas, 1998; Zhao et al., 2007). The major drawbacks of thismethod are that the accuracy largely depends on the number ofsample particles considered (Meimaroglou and Kiparissides, 2007)and the involved computational expense is relatively high for lowdimensional PBEs.

In recent years, another class of methods, known as discretizedmethods, have become popular for solving PBEs due to theirefficiency and accuracy. These methods include fixed pivot (FP)method (Kumar and Ramkrishna, 1996a, 1997; Lim et al., 2002),moving pivot (MP) method (Kumar and Ramkrishna, 1996b), cellaverage technique (Kumar et al., 2006, 2008b), hierarchical two-tiermethod (Immanuel and Doyle, 2003, 2005), two-level discretizationalgorithm (Pinto et al., 2007, 2008) and finite volume (FV) method(Qamar and Warnecke, 2007; Qamar et al., 2009). The FP method,proposed by Kumar and Ramkrishna (1996a), has been widely usedfor solving PBEs with aggregation and breakage due to its easyimplementation and ability to handle a variety of grids. In thismethod, the computational domain is discretized into small cells.The size of all the particles in each cell is represented by a size thatfalls within that cell and is called the pivotal point. The key idea inthis method is to distribute the newly born particles, which do not fallon the pivotal points, among the neighboring pivotal points such thatsome properties of interest are preserved exactly. When a coarse gridis used, the FP method overpredicts the PSD for aggregation processin the region where the PSD changes rapidly. In order to overcomethis problem, MP method was proposed (Kumar and Ramkrishna,1996b). In this method, the pivotal size in each cell is adjustedaccording to the non-uniformity of the PSD in that cell (Kumar andRamkrishna, 1996b). However, the MP method is more complex thanthe FP method and the resulting ODEs are difficult to solve (Kumaret al., 2006). Since these methods are developed for pure aggregationand breakage processes, they need to be combined with othermethods for handling simultaneous processes, such as growth andaggregation. Such methods include combination of method of char-acteristics (MOC) and FP method (Kumar and Ramkrishna, 1997; Limet al., 2002) and combination of weighted essentially non-oscillatory(WENO) and FP method (Lim et al., 2002). The cell average technique,developed initially for pure aggregation and breakage processes, issimilar to the FP method but with the improved assignment rule forthe newly born particles to the neighboring cells (Kumar et al., 2006,2008a). This method has been extended to solve simultaneousprocesses including growth by reformulating the growth term(Kumar et al., 2008b).

The two-tier technique comprises of two steps. The first stepinvolves the calculation of the rates of nucleation, growth andaggregation. This information is then used in the second step to

update the PSD (Immanuel and Doyle, 2003, 2005). A variant ofthe two-tier technique is the two-level discretization technique(Pinto et al., 2007, 2008). In this technique, a coarse mesh isused for aggregation, while a fine mesh is used for growth andnucleation based on the observation that the effect of aggregationon PSD is less sensitive to mesh size than the effects of nucleationand growth. In FV method, the PBE is reformulated in a massconservation form so that the aggregation and breakage terms canbe written as flux terms (Filbet and Laurencot, 2004; Qamar andWarnecke, 2007; Qamar et al., 2009). The reformulated PBE canbe easily solved using FV method available for conservation laws(LeVeque, 2002). Many methods mentioned earlier are not veryefficient for model based online optimization and control, whererepeated solution of the model equations is required. Efficientsolution of the PBE is also crucial when computational fluid dynamics(CFD) is coupled with PBE to take into account the effect of non-idealmixing (Marchisio and Fox, 2005; Woo et al., 2009).

Recently, we have proposed a mesoscopic particle based method,namely lattice Boltzmann method (LBM), for efficiently solving PBEswith growth and nucleation (Majumder et al., 2010a, in press). Inthis paper, LBM is extended for efficient solution of 1D PBE withsimultaneous growth, nucleation, aggregation and breakage phenom-ena. This extension requires incorporation of aggregation and break-age processes as force terms in the LB formulation. This force term isevaluated by applying the FP method and thus the resulting methodis referred to as LBM-FP method in the subsequent discussion.Detailed multiscale analysis is carried out to derive the kineticequations, whose long-time large-scale solution provides the solutionof the desired PBE.

The traditional LBM is only applicable with uniform grid. Theuse of such grid requires a large number of grid points to capturethe PSD with reasonable resolution for aggregation and breakageprocesses and hence make the numerical scheme computationallyexpensive. In order to overcome this problem in the case of FPmethod, the use of non-uniform grid (e.g., geometric grid) hasbeen proposed by Kumar and Ramkrishna (1996a). However, theimplementation of such non-uniform grid in LBM requires inter-polation of the discrete Boltzmann distribution function (He et al.,1996). In this paper, a coordinate transformation is proposed,which allows the use of non-uniform grid (e.g., logarithmic grid)without performing any interpolation and thus avoids the inac-curacy and computational cost resulting due to the interpolation.

The performance of the proposed LBM-FP technique is assessedfor pure aggregation and breakage processes, as well as simultaneousprocesses, such as processes with simultaneous growth, nucleation,aggregation and breakage phenomena. The accuracy and efficiencyof the proposed scheme is compared with the FP (Kumar andRamkrishna, 1996a), MOC-FP (Kumar and Ramkrishna, 1997) andFV (Qamar et al., 2009) methods. It is found that the LBM-FPtechnique has similar accuracy as FP and FV methods for pureaggregation and pure breakage. However, the proposed technique isfound to be more accurate and more efficient than the FV methodfor the growth dominant simultaneous processes, e.g., simultaneousgrowth and aggregation. In comparison to MOC-FP method, theproposed method is found to be more accurate when nucleation ispresent and when growth rate is constant.

The rest of the article is organized as follows: the principle ofLBM is discussed and detailed derivation of LBM for solvingadvection equation with source term is presented in Section 2. InSection 3, adaptation of the LBM for solving PBEs with simultaneousgrowth, nucleation, aggregation and breakage is discussed byidentifying the analogy between the advection equation and thePBE. Subsequently, some case studies are presented in Section 4,where the performance of the proposed LBM-FP method is com-pared with FP, MOC-FP and FV methods. Finally, some concludingremarks are presented in Section 5.

Right moving

Left moving

Stationary

Fig. 1. 1D lattice model showing fictitious particles with three velocities.

A. Majumder et al. / Chemical Engineering Science 69 (2012) 316–328318

2. Lattice Boltzmann method

Lattice based methods (Frisch et al., 1986; McNamara andZanetti, 1988; Higuera et al., 1989) have received significantattention in the past two decades for simulation of hydrody-namics. The major difference between lattice based and othertraditional methods, e.g., finite difference method, is that most ofthe traditional methods follow a top-down approach, where thegoverning partial differential equations (PDEs) at the macroscopicscale are discretized to obtain ordinary differential equations(ODEs) or algebraic equations. On the other hand, lattice basedmethods adopt a bottom-up approach, where instead of usingdiscretization, pseudo particle kinetics is used in such a way thatthe governing equations are recovered at the appropriate scale.LBM was initially developed as a spin-off of one such methodknown as Lattice Gas Cellular Automata (LGCA) (Frisch et al.,1986; Succi, 2001). The key distinguishing feature of LBM is theintroduction of Boltzmann equation to overcome the problem ofstatistical noise encountered in LGCA. With its roots in kinetictheory, LBM uses a simplified representation of the microscopicstate of the process so that the model is computationally viableand can capture the essential description of the system at themacroscopic level. LBM has now established itself as a popularscheme due to its ability to provide fast, easily implementablecode and versatility (Aidun and Clausen, 2009). It has beenapplied to various problems including hydrodynamics (Chenand Doolen, 1998), turbulence (Chen et al., 2003), multi-phaseflow (Mantle et al., 2001), microflow (Ansumali et al., 2007; Kimet al., 2008; Yudistiawan et al., 2010), non-Newtonian fluids(Singh et al., 2011) and crystallization (Majumder et al., 2010a,in press). Several good reviews of LBM are available in theliterature (Benzi et al., 1992; Chen and Doolen, 1998; Succi,2001; Aidun and Clausen, 2009).

PBE with aggregation and breakage is analogous to the advectionequation with source term. Thus, in the subsequent discussion, wefirst derive LBM for the advection equation with source term and thenapply the developed scheme for solving PBEs with aggregation andbreakage by identifying their similarities later in the paper.

2.1. Principle

The advection equation with a source term can be written as

@r@tþ@ðvrÞ@r¼ F, ð5Þ

where r is the concentration of the species being transported(passive scalar), v is the advection velocity and F is the sourceterm. In order to solve this advection equation, some fictitiousparticles, which resemble groups of molecules, are considered atthe mesoscopic scale. While molecules move randomly in space,these fictitious particles are restricted to move with certainvelocities, which are carefully chosen such that they are consis-tent with the symmetry and isotropy requirement of macroscopicdynamics (Succi, 2001). Finding such a set of velocities involvesa trial and error procedure. One typically starts with a set ofvelocities on a lattice based on computational considerations, andchecks whether it is possible to construct a model, whose long-termlarge-scale solution matches with the macroscopic equation (Succi,2001). In the discrete form, the simplest possible model has threetypes of fictitious particles with velocities ci ¼ f0,c,�cg, i.e., station-ary (0), right moving (þ), left moving (�) (Karlin et al., 2006), asillustrated in Fig. 1.

In order to solve Eq. (5) with LBM, the following kineticequation with force (source) term is used (Dawson et al., 1993):

@f i

@tþci

@f i

@r¼�

1

tðf i�f eq

i ÞþFi, i¼ 0;1,2, ð6Þ

where t40 is the relaxation time, fi is the discrete Boltzmanndistribution function, f eq

i is the equilibrium discrete Boltzmanndistribution function, ci is the discrete velocity and Fi is thediscrete force term. In the simplest form, Fi can be taken asFi¼wiF (Dawson et al., 1993), where wi is the weight associatedwith velocity ci. In Eq. (6), the subscripts 0, 1 and 2 refer tostationary, right moving and left moving particles, respectively.The left hand side of Eq. (6) denotes the free flight, while the righthand side represents the relaxation of the particles to equilibrium(Bhatnagar–Gross–Krook (BGK) approximation (Bhatnagar et al.,1954) for collision). The equilibrium distributions can be found byminimizing appropriate entropy function subject to the con-straints of mass and momentum conservation. The discrete formof the relevant entropy function, also known as H function, isgiven as (Karlin et al., 1999; Ansumali and Karlin, 2002a)

H¼X2

i ¼ 0

f i lnf i

wi�1

� �: ð7Þ

For the 1D advection equation, the weights can be selected asw0 ¼ 4=6, w1 ¼ 1=6, w2 ¼ 1=6 (Karlin et al., 1999; Ansumali andKarlin, 2002b). The constraints for local conservation of mass andmomentum are given as

X2

i ¼ 0

f i ¼ f 0þ f 1þ f 2 ¼ r, ð8Þ

X2

i ¼ 0

cif i ¼ cðf 1�f 2Þ ¼ ru, ð9Þ

where u is the local average velocity. By solving the minimizationproblem, we get the equilibrium values of the discrete Boltzmanndistribution function f eq

i as (Ansumali and Karlin, 2000)

f eq0 ðr,uÞ ¼

2r3

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2

s

q� �, ð10Þ

f eq1 ðr,uÞ ¼

r3ðuc�c2

s Þ=2c2s þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2

s

q� �, ð11Þ

f eq2 ðr,uÞ ¼

r3�ðucþc2

s Þ=2c2s þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2

s

q� �, ð12Þ

where cs ¼ c=ffiffiffi3p

is analogous to the speed of sound in the system.Conserving both mass and momentum during collision willrecover the Navier–Stokes equation in the long-time long-scalelimit, whereas conserving only mass will recover the advectionequation (Karlin et al., 2006; Majumder et al., 2010a). Since weare interested in the advection equation, equilibrium distributionf eq

i ðrðf Þ,uÞ is calculated at the given velocity u¼v (Karlin et al.,2006). It is shown next that breaking this momentum conserva-tion leads to advection–diffusion equation for r in the long-timeand large-scale limit.

2.2. Macroscopic equation from kinetic equation with source term

In this section, multiscale analysis, known as Chapman–Enskoganalysis, is used to verify whether the kinetic equation in Eq. (6) isconsistent with the advection equation at the macroscopic level.


Using Eqs. (8) and (9) the kinetic equation given by Eq. (6) can beequivalently written in terms of moment chain as

0th moment :@r@tþ@ðruÞ

@r¼ F, ð13Þ

1st moment :@ðruÞ

@tþ@P

@r¼�

1

t ðru�rvÞ, ð14Þ

2nd moment :@P

@tþc2 @ðruÞ

@r¼�

1

tðP�Peq

Þþ1

3c2F, ð15Þ

where

P¼X2

i ¼ 0

c2i f i, ð16Þ

Peq¼X2

i ¼ 0

c2i f eq

i : ð17Þ

In order to obtain the macroscopic PDE from the LBM for-mulation, it is necessary to formally separate different timescales. For this purpose, the non-conserved variables, i.e., u andP, are expanded around their equilibrium values in terms of t. Thetime derivative operator is also expanded in terms of t as follows:

u¼ vþtuð1Þ þt2uð2Þ þ � � � , ð18Þ

P¼ PeqþtPð1Þ þt2Pð2Þ þ � � � , ð19Þ

@

@t¼@ð0Þ

@tþt @

ð1Þ

@tþ � � � , ð20Þ

where advection velocity v is the equilibrium value of the localaverage velocity u and the superscripts within the parenthesisdenote the order of the expansion. Upon substituting theseexpansions in Eq. (13), we have

@ð0Þ

@tþt @

ð1Þ

@tþ � � �

� �rþ @

@r½rvþtruð1Þ þ � � �� ¼ F: ð21Þ

Taking Oð1Þ terms

@ð0Þ

@tr¼� @

@rðrvÞþF: ð22Þ

Similarly, by substituting the expansion values in Eq. (14), thefollowing expression is obtained:

@ð0Þ

@tþt @

ð1Þ

@tþ � � �

� �½rvþtruð1Þ þ � � ��þ

@

@r½PeqþtPð1Þ þ � � ��

¼�1

t ½rvþtruð1Þ þ � � � �rv�: ð23Þ

The Oð1Þ terms of Eq. (23) are

�ruð1Þ ¼@ð0Þ

@tðrvÞþ

@

@rPeq: ð24Þ

By substituting the expression for Peq and using Eq. (22),Eq. (24) can be written as

�ruð1Þ ¼ v@ð0Þ

@trþc2

s

@

@rr 1þ

v2

c2s

� �� , ð25Þ

�ruð1Þ ¼ �v@

@rðrvÞþvFþðc2

s þv2Þ@

@rrþ2rv

@v

@r, ð26Þ

�ruð1Þ ¼ rv@v

@xþc2

s

@r@rþvF: ð27Þ

Then, the expression for momentum ru in linear order of tbecomes

ru¼ rvþtruð1Þ þ � � � , ð28Þ

ru¼ rv�trv@v

@r�tc2

s

@r@r�tvF: ð29Þ

By substituting the expression for ru in Eq. (13), we obtain

@

@trþ @

@rðrvÞ ¼ FþD @

2r@r2þt @vF

@rþt @

@rrv

@v

@r

� �: ð30Þ

In Eq. (30), D¼ tc2s is the diffusion coefficient, which can be made

small by choosing sufficiently small relaxation time t. The lasttwo terms on the right hand side of Eq. (30) are the anomalyterms. The last term arises due to the space dependence ofthe advection velocity. When v is a smooth function of spacecoordinate r, this anomaly term is OðtMa2

Þ, where Ma¼ v=cs isthe Mach number. The contribution from this term can beneglected for small Ma, e.g., Ma� 0:01, and vanishes when v isindependent of r. Note that Ma cannot be chosen to be very small,as smaller Ma (i.e., larger cs) increases the diffusion coefficient D.The anomaly term involving the force ðtð@vF=@rÞÞ can be nullifiedby considering the following correction term in the kineticequation:

@f i

@tþci

@f i

@r¼�

1

tðf i�f eq

i ÞþwiFþwiciC, ð31Þ

where C is the unknown parameter included in the correctionterm. Now, following the same steps for the Chapman–Enskoganalysis as used earlier, the following equation is recovered at themacroscopic level:

@

@trþ @

@rðrvÞ ¼ FþD @

2r@r2þt @vF

@r�tc2

s

@C@rþt @

@rrv

@v

@r

� �: ð32Þ

In Eq. (32), the third term on the right hand side is the anomalyterm involving F and the fourth term is the correction term. Thus,the condition to nullify this anomaly term is

@

@r½tvF�tc2

sC� ¼ 0, ð33Þ

) tvF ¼ tc2s CþC, ð34Þ

) C¼v

c2s

F, ð35Þ

where the integration constant C has been taken as zero. Withthis choice, the kinetic equation with the correction termbecomes

@f i

@tþci

@f i

@r¼�

1

t ðf i�f eqi ÞþwiF 1þ

civ

c2s

� �: ð36Þ

Effectively, to nullify the anomaly term due to F in Eq. (30),instead of choosing the force term in the kinetic equation as wiF

as suggested by Dawson et al. (1993), one needs to use

Fi ¼wiF 1þciv

c2s

� �: ð37Þ

2.3. Discretization of the kinetic equation

In order to solve the kinetic equation numerically, we integrateit over Dt using the trapezoidal rule (which is OðDt2Þ accurate) toobtain

f iðrþciDt,tþDtÞ � f iðr,tÞ�Dt

2t ½ðf iðr,tÞ�f eqi ðr,tÞÞ

þðf iðrþciDt,tþDtÞ�f eqi ðrþciDt,tþDtÞÞ�

þDt

2½FiðrþciDt,tþDtÞþFiðr,tÞ�: ð38Þ

It can be noted that the scheme given by Eq. (38) is not explicitas f iðrþciDt,tþDtÞ appears on both sides. In order to obtain an


explicit numerical scheme, we define a transformation as

f i ¼ f iþDt

2tðf i�f eq

i Þ�Dt

2Fi: ð39Þ

With this transformation, Eq. (38) can be rewritten in termsof f i as

f iðrþciDt,tþDtÞ � f iðr,tÞ�Dt

tðf iðr,tÞ�f eq

i ðr,tÞÞþDtFiðr,tÞ: ð40Þ

Based on Eqs. (8) and (39) the density r in terms of f i can befound as

X2

i ¼ 0

f i ¼ r�Dt

2

Xi

Fi, ð41Þ

) r¼X2

i ¼ 0

f iþDt

2F: ð42Þ

Moreover, from Eqs. (10)–(12), f eqi is a functional of fi and depends

on the particle population only via its lower order moments r andru. Substituting for r in Eqs. (10)–(12) with the expression inEq. (42), it is found that at the same local average velocity u, theequilibrium distributions of f eq

i and feq

i are related as

feq

i ¼ f eqi �

Dt

2Fi: ð43Þ

Now, from Eqs. (39) and (43)

f i�f eqi ¼

2t2tþDt

ðf i�feq

i Þ: ð44Þ

Substituting for ðf i�f eqi Þ from Eq. (44) in Eq. (40), we obtain

f iðrþciDt,tþDtÞ � 1�2Dt

2tþDt

� �f iðr,tÞþ

2Dt

2tþDtf

eq

i ðr,tÞþDtFi,

ð45Þ

) f iðrþciDt,tþDtÞ � ð1�abÞf iðr,tÞþabfeq

i ðr,tÞþDtFi, ð46Þ

where a¼ 2 and b¼Dt=ð2tþDtÞ. Thus, we arrive at an explicitexpression in terms of f i. Although f i are not exactly particlepopulations, they serve the purpose of solving the advection equationsince the quantity r can be recovered from f i using Eq. (42).

Numerical implementation of the LBM scheme developed foradvection equation requires solving Eq. (46). In practice, this equationis split into two consecutive steps: (1) collision; and (2) streaming.The collision step involves evaluation of the right hand side ofEq. (46), while streaming step involves shifting the values of thedistribution functions to the next lattice node based on their discretevelocities.

3. Adaptation of LBM for solving PBEs

The PBE given by Eq. (1) is analogous to the advection equationgiven by Eq. (5), where the PSD n is analogous to the fluid density r,the growth rate G is analogous to the local fluid velocity v, the spacecoordinate r is analogous to the characteristic size x and the terms onthe right hand side of the PBE (Qnuc , Qagg and Qbreak) are analogousto the source term F. Thus, the LBM scheme discussed in the previoussection for advection equation with source term F can be applied tosolve PBEs with simultaneous growth, nucleation, aggregation andbreakage. While solving such a PBE, the nucleation term is imple-mented as a boundary condition as the size of the nuclei is assumedto be represented by the smallest grid considered.

The integrals related to aggregation and breakage can be evaluatedusing numerical integration techniques easily, if a uniform grid isused. However, a uniform grid requires very large number of gridpoints in order to cover the entire size range of the particles withreasonable accuracy and is thus computationally expensive (Kumar

and Ramkrishna, 1996a). On the other hand, a non-uniform grid suchas geometric or logarithmic grid can provide better resolution of thesize distribution by covering the entire size range of the particles withmoderate number of grid points (Batterham et al., 1981; Kumar andRamkrishna, 1996a). However, in such a grid, the newly born particlesdue to aggregation and breakage may have sizes which do not fall onthe grid points. This problem can be overcome by redistributingthe newly born particles among the neighboring grid points so thatsome properties of interest are preserved. We use the fixed pivot (FP)method proposed by Kumar and Ramkrishna (1996a) for thispurpose, where the zeroth and the first order moments of the PSD,i.e., total number and total size of the particles are considered to beconserved when particle size is the independent variable.

The LBM discussed earlier for solving the advection equationrequires a uniform grid for implementation, as the grid spacingshould be such that during the streaming step all the particlesfrom one node reach the next node. Non-uniform grid can beimplemented with an interpolation based LBM, where the popula-tions that do not reach the next node in the streaming step areinterpolated (He et al., 1996; Chikatamarla and Babu, 2005). In orderto avoid the inaccuracy and computational cost due to interpolation,we propose a coordinate transformation, which usually leads to alogarithmic grid in the original coordinate system (Majumder et al.,2010b). Similar to geometric grid, the logarithmic grid allowscapturing the PSD with good resolution with moderate number ofgrid points.

The proposed coordinate transformation is defined as z¼ zðxÞ.As the number of particles in a specified size interval in both thecoordinate systems are the same, the PSDs in terms of z and x arerelated as (Roussas, 2003)

hðz,tÞ ¼ nðx,tÞdx

dz

��, ð47Þ

where hðz,tÞ is the number density in the transformed coordinatesystem. We consider the case of size-dependent growth rateof the particles, which is usually modeled as (Abegg et al., 1968;Lim et al., 2002; Gunawan et al., 2004)

Gðx,tÞ ¼FðtÞOðxÞ, ð48Þ

where FðtÞ and OðxÞ characterize the size and time dependence ofthe growth rate. We assume that OðxÞ is continuous and positiveover the domain ½xmin,xmax� to make sure that the mapping fromone coordinate system to another is unique and to avoid singu-larity (Majumder et al., 2010b). Substituting for nðx,tÞ and Gðx,tÞon the left hand side of Eq. (1), we obtain

@

@th

dx

dz

� ��1 !

þFðtÞ@

@xOðxÞh

dx

dz

� ��1 !

¼QnucþQaggþQbreak:

ð49Þ

Without loss of generality, we may define

dx

dz¼OðxÞ, ð50Þ

) z¼cðxÞ, ð51Þ

where OðxÞ is a function of x that defines the coordinatetransformation and

c¼Z

dx

OðxÞþk, ð52Þ

where k is the constant of integration. Thus, in the transformedcoordinate z, Eq. (1) becomes

dx

dz

� ��1 @h

@tþFðtÞ

@

@z

Oðc�1ðzÞÞ

Oðc�1ðzÞÞ

h

! !¼QnucþQaggþQbreak: ð53Þ


Now, substituting for nðx,tÞ and x on the right hand side ofEq. (53), we get

Qnuc ¼ B0dðc�1ðz0ÞÞ: ð54Þ

Similarly, other terms on the right hand side of Eq. (53) become

RBa ¼1

2

dx

dz

� ��1 Z z

0aðc�1

ðzÞ�c�1ðz0Þ,c�1

ðz0ÞÞhðz�z0,tÞhðz0,tÞ dz0,

ð55Þ

RDa ¼dx

dz

� ��1 Z 10

aðc�1ðzÞ,c�1

ðz0ÞÞhðz,tÞhðz0,tÞ dz0, ð56Þ

RBb ¼dx

dz

� ��1 Z 1z

bðc�1ðzÞ,c�1

ðz0ÞÞGðc�1ðz0ÞÞhðz0,tÞ dz0, ð57Þ

RDb ¼dx

dz

� ��1

Gðc�1ðzÞÞhðz,tÞ: ð58Þ

Finally, by combining Eqs. (53)–(58), we obtain

@h

@tþFðtÞ

@

@z

Oðc�1ðzÞÞ

Oðc�1ðzÞÞ

h

!¼ B0dðc

�1ðz0ÞÞOðc

�1ðzÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Qnuc

þ1

2

Z z

0aðc�1

ðzÞ�c�1ðz0Þ,c�1

ðz0ÞÞhðz�z0,tÞhðz0,tÞ dz0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}RBa

�

Z 10

aðc�1ðzÞ,c�1

ðz0ÞÞhðz,tÞhðz0,tÞ dz0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}RDa

þ

Z 1z

bðc�1ðzÞ,c�1

ðz0ÞÞGðc�1ðz0ÞÞhðz0,tÞ dz0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

RBb

�Gðc�1ðzÞÞhðz,tÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}RDb

:

ð59Þ

For processes with size-dependent growth, we can chooseOðxÞ ¼OðxÞ. In this case, we usually have a logarithmic grid inthe original coordinate system, since OðxÞ is often modeled asOðxÞ ¼ 1þgx, where g40 is a parameter that determines theextent of size dependence (Gunawan et al., 2004). With thistransformation, Eq. (59) becomes a PBE with size-independentgrowth rate, i.e.,

@h

@tþFðtÞ

@h

@z¼QnucþRBaþRDaþRBbþRDb: ð60Þ

The developed LBM scheme now can be used efficiently forsolving Eq. (60).

On the other hand, if the growth rate is size independent,e.g., OðxÞ ¼ 1, we can define OðxÞ ¼ 1þgx. In such case, Eq. (59)simplifies as

@h

@tþ@

@zðGhÞ ¼QnucþRBaþRDaþRBbþRDb, ð61Þ

where

Gðz,tÞ ¼FðtÞ

expðgzÞ: ð62Þ

Note that the growth rate in Eq. (62) is size dependent. Thus withthe appropriate coordinate transformation, the proposed LBM-FPmethod can be applied for efficient simulation of simultaneousprocesses with growth, nucleation, aggregation and breakage.Nucleation term is interpreted as the following boundary condi-tion for the PBE in Eq. (60) (Ramkrishna, 2000):

hð0,tÞFðtÞ ¼ B0ðtÞdðc�1ðz0ÞÞOðc

�1ðzÞÞ: ð63Þ

This is a Dirichlet type boundary condition and is similar to a fluidflow problem with specified inlet flow rate. This type of boundary

condition can be implemented by considering a buffer pointbefore the first grid point in the computational domain andcontinually refilling this point with the equilibrium populationscorresponding to the number density at size z0 as (Succi, 2001;Majumder et al., 2010a)

f i,buff ¼ feq

i ðhðz0,tÞÞ, ð64Þ

where f i,buff is the population density in the buffer pointconsidered.

4. Numerical examples

In this section, numerical examples are considered for purebreakage, pure aggregation, simultaneous growth–aggregationand simultaneous growth–nucleation–aggregation–breakage pro-cesses. The problems are solved using the proposed LBM-FPmethod and the results are compared with FP (Kumar andRamkrishna, 1996a), MOC-FP (Kumar and Ramkrishna, 1997)and FV methods (Qamar et al., 2009) in terms of error normsand computation time. The L1- and L2-norms of the error aredefined as follows:

L1-norm¼XN

i ¼ 1

9nei�ni9Dxi, ð65Þ

L2-norm¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i ¼ 1

ðnei�niÞ

2Dxi

vuut , ð66Þ

where xi are the discrete points of the chosen grid, Dxi is width ofthe ith cell and ni

e is the exact solution.For simulation of pure aggregation and breakage processes,

the LBM parameter c is chosen such that the time step Dt¼

Dx=c� 0:01 s. Selecting the relaxation parameter b closer to 1 pro-vides smaller diffusion coefficient but it may introduce oscillations,especially for nonsmooth distributions. Thus b is chosen such that thediffusion coefficient is sufficiently small and oscillations are minimal.A typical value of b used here is 0.99. For FV, MOC-FP and pure FPmethods, semidiscrete formulation is used and the Matlab ODE solverode45 is applied to solve the resulting ODEs. The default values ofabsolute tolerance of 10�6 and relative tolerance of 10�3 are usedfor the ODE solver unless otherwise stated. All computations areperformed using a Windows Vista PC with an Intel CoreTM 2 DuoProcessor 6600 (2.40 GHz, 4 GB RAM) using Matlabs 2007a.

4.1. Pure aggregation and pure breakage

Two case studies are considered with pure breakage and pureaggregation processes. It is assumed that particle growth andnucleation are negligible. A logarithmic grid is used to capture thePSD in a wide range of size domain.

4.1.1. Pure breakage

First, we consider a pure breakage problem. The initial dis-tribution is taken as the following exponential distribution:

nðx,0Þ ¼ expð�xÞ: ð67Þ

The breakage rate and breakage probability are GðxÞ ¼ x andbðx,x0Þ ¼ 2=x0, respectively, which corresponds to a binary break-age problem. The analytical solution of this problem is given byZiff and McGrady (1985) as

nðx,tÞ ¼ expð�xð1þtÞÞð1þtÞ2: ð68Þ

10−2 100 10210−10

10−5

100

Particle size, x (µm)

PSD

, n (

#/µ

m)

Initial

LBM

FP

FV

Analytical

Fig. 3. Final distribution for pure aggregation problem at tf¼20 s.

Table 1Simulation results for pure aggregation and pure breakage processes.

Cases Parameter LBM-FP FP FV

Breakage L1-norm 3.86�10�2 3.82�10�2 7.68�10�2

L2-norm 9.93�10�2 8.86�10�2 0.17

CPU time (s) 0.04 0.06 0.43

Aggregation L1-norm 1.44�10�3 3.18�10�4 2.57�10�4

L2-norm 3.07�10�4 5.58�10�5 6.25�10�5

CPU time (s) 1 0.06 0.37


4.1.2. Pure aggregation

The next problem is a pure aggregation problem with aconstant aggregation kernel aðx,x0Þ ¼ 1, where the initial PSD istaken as

nðx,0Þ ¼N0

xaexpð�x=xaÞ: ð69Þ

Analytical solutions for this kernel are given by Scott (1968) as

nðx,tÞ ¼4N0

xaðtþ2Þ2exp

�2x0

tþ2

� �, ð70Þ

where t¼N0at and x0 ¼ x=xa. The parameters N0 ¼ 1, xa¼1 mm arethe total number of particles per unit volume and mean size ofthe particles, respectively.

4.1.3. Results and discussion

All the problems are solved using proposed LBM-FP method andthe results are compared with the pure FP (Kumar and Ramkrishna,1996a) and FV (Qamar et al., 2009) methods. The simulationsare performed using a logarithmic grid obtained by defining thetransformation as OðxÞ ¼ x, so that z¼ logðxÞ. Final time ðtf Þ for thesetwo problems are taken as 10 and 20 s, respectively. Final distribu-tions for the problems obtained with N¼100 grid points are shownin Figs. 2 and 3. In Fig. 2, it can be seen that as the particles break,more smaller sized particles are produced. As a result, most of theparticles are concentrated in the smaller size region. However, asexpected, an opposite scenario is seen in Fig. 3, where the particlesize increases significantly due to aggregation and the number of thesmaller sized particles decreases. From these plots, it can be notedthat the solutions obtained by the proposed method are in very goodagreement with the analytical solutions. It is able to capture the PSDin the whole range of size domain.

The L1 and L2 error norms for all the methods are summarizedin Table 1. The performance of pure FP and FV methods is found tobe similar in terms of accuracy for the aggregation example, whileFP method is more efficient. However, the proposed method,although based on FP method for evaluation of the aggregationand breakage terms, is found to be less accurate as compared topure FP method. This happens as LBM scheme was primarilydeveloped for simulation of pure growth processes (i.e., advectiontype). For aggregation and breakage processes (without growth),the streaming step in LBM introduces undesired numerical diffu-sion as can be seen in the larger size region in Fig. 3. Whencompared with FV method, the proposed LBM-FP method is more

10−4 10−2 100

10−5

100


PSD

, n (

#/µ

m) Initial

LBM−FP

FP

FV

Analytical

Fig. 2. Final distribution for pure breakage problem at tf¼10 s.

accurate in the case of the breakage example, although it is notvery obvious from Fig. 2. The reason is that LBM-FP methodcaptures the CSD more accurately in size range where CSD ishigher. This issue is further discussed in Section 4.2.

In order to explore how these methods perform with varyinggrid size, the trade-off plots obtained using different grid sizes areshown in Figs. 4 and 5 for breakage and aggregation problems,respectively. These plots provide the useful information about thecomputational cost required to obtain a solution with tolerableerror. In these plots, curves that are located near the bottom-leftcorner correspond to methods which can provide more accuratesolution with less computation time. It is clear from these plotsthat FP method performs better than LBM-FP method for bothaggregation and breakage. Thus, for pure breakage and pureaggregation no significant advantage is provided by the proposedLBM-FP method. It is shown in the next section that for growthdominant simultaneous processes e.g., processes with simultaneousgrowth and aggregation, the proposed method performs better thanFV technique. The proposed method is also found to be advantageousfor constant growth and processes with nucleation as compared toMOC-FP method (Kumar and Ramkrishna, 1997).

4.2. Simultaneous growth and aggregation

In this section, we consider some processes with simultaneousgrowth and aggregation. The various growth rates and aggrega-tion kernels used for the case study are shown in Table 2. Theinitial distribution is given by Eq. (69) with N0 ¼ 5, xa ¼ 0:01 mmfor Cases 1 and 2, and N0 ¼ 1, xa ¼ 1 mm for Case 3. The analyticalsolutions are available for the first two cases (Ramabhadran et al.,1976). Analytical solution is not available for Case 3 and a solutionobtained with refined grid is treated as a reference solution for

10−1 100 10110−3

10−2

10−1

Computation time (s)

L1−

norm

LBM−FPFPFV

10−1 100

10−2

10−1


L2−

norm

LBM−FPFPFV

Fig. 4. Trade-off plots for pure breakage. Solid curves represent the fitted curves assuming power law relation. (a) L1-norm. (b) L2-norm.

100 10210−5

10−4

10−3


L1−

norm

LBM−FPFVFP

10−1 100 101

10−5

10−4


L2−

norm

LBM−FPFVFP

Fig. 5. Trade-off plots for pure aggregation. Solid curves represent the fitted curves assuming power law relation. (a) L1-norm. (b) L2-norm.

Table 2Parameters for simultaneous growth and aggregation processes where g¼ 0:6 and

growth exponent g¼0.7.

Case G(x) a L

1 1 100 0.20

2 x xþx0 20

3 ð1þgxÞg 0:1xx0 10

10−2 100 102

10−10

10−5

100


PSD

, n (

#/µ

m)

Initial

Analytical

LBM−FP

FV

MOC−FP

Fig. 6. Final distribution for simultaneous growth and aggregation at tf¼1 s,

N¼200 (Case 1).


comparing the accuracy of different methods. Two different pro-cesses, i.e., growth and aggregation, are involved in these studies.These rate processes have fundamental characteristic time scales.Ramabhadran et al. (1976) defined a dimensionless variable L as theratio of the characteristic time scales for growth and aggregation.For constant growth rate G and constant aggregation kernel a, L canbe written as

L¼G

aN0xa: ð71Þ

This dimensionless number L gives us an idea of the governing rateprocess. For example, if L41, the growth process is faster thanaggregation and dominates the dynamics of the system. The valuesof L for the different cases are shown in Table 2. For the lineargrowth rate function in Case 2, the coordinate transformation isdefined as dx=dz¼ x. This transformation results in a logarithmicgrid that is suited for the aggregation process. In Case 3, thetransformation is defined as dx=dz¼ ð1þgxÞg . With these transfor-mations, the growth rate in the transformed coordinate becomessize-independent. On the other hand, for the constant growth rateG¼ 1 mm s�1 in Case 1, similar coordinate transformation is per-formed as Case 2 to obtain the logarithmic grid for which the growthrate GðzÞ ¼ expð�gzÞ is size-dependent, as shown in Eq. (62).

All the case studies are solved using the proposed LBM-FPtechnique and the results are compared with those obtainedusing the FV and MOC-FP methods. Final distributions obtainedwith N¼200 grid points along with the analytical solutions areshown in Figs. 6–8. A common feature of the final distributionsshown in these figures is that the distributions have a sharp frontmoving from the smaller sized region. This front is attributed tothe growth process which results in the translation of the PSDalong the computational domain with time. In addition, thepresence of the aggregation process causes the shape of the PSDto stretch by increasing the number of larger particles at the cost

10−4 10−2 10010−10

10−5

100


PSD

, n (

#/µ

m)

Initial

Analytical

LBM−FP

FV

MOC−FP

10−4 10−3 10−20

10

20

30

40

50


PSD

, n (

#/µ

m)

Initial

Analytical

LBM−FP

FV

MOC−FP


N¼200 (Case 2). (a) Final distribution. (b) Zoomed view.

10−2 100 10210−15

10−10

10−5

100


PSD

, n (

#/µ

m)

Initial

LBM−FP

FV

MOC−FP


N¼200 (Case 3).

Table 3Simulation results for simultaneous growth and aggregation.

Cases Parameter LBM-FP FV MOC-FP

1 L1-norm 6.94�10�4 9.42�10�4 6.09�10�4

L2-norm 6.60�10�4 8.82�10�4 6.45�10�4

CPU time (s) 4.05 2.04 31.82

2 L1-norm 3.64�10�2 0.30 1.62�10�3

L2-norm 0.18 0.67 1.96�10�3

CPU time (s) 0.20 1.57 0.15

3 L1-norm 0.13 0.42 6.74�10�3

L2-norm 0.10 0.66 4.07�10�3

CPU time (s) 0.22 0.29 3.89


of smaller particles. It can be noted that the final distributionsobtained using all the three methods match well. However,diffusive solutions are obtained for LBM-FP and FV methods inthe small size region where sharp moving fronts are encountered.The diffusion is more pronounced for LBM-FP technique. The errornorms and the computation times for all the case studies arepresented in Table 3.

We see that for Case 1, all the methods have similar accuracywhile MOC-FP method is most expensive. This is due to the fact thatin the MOC-FP method, the newly born particles due to aggregation,which do not fall on a grid, have to be redistributed among theneighboring grids so that some properties of interest are preserved(the first two moments of PSD in this case). Since, the grids are alsomoving at the characteristic velocity, this redistribution rule has tobe updated frequently (Kumar and Ramkrishna, 1997). On the otherhand, MOC-FP method is most accurate and efficient for Case 2 wherea linear growth rate is considered. Here the redistribution rule for theFP method need not be updated since at any time t the grid pointsare given as xiðtÞ ¼ xið0Þ expðG0tÞ (Kumar and Ramkrishna, 1997).The LBM-FP method is found to be more accurate than FV methodeven though this is not apparent in Fig. 7(a) where log–log plot of thefinal distribution is shown. However, from Fig. 7(b) where zoomedplot is shown on a semi-log axis, it is clear that in the size rangex¼ 10�4

210�2 mm the PSD corresponding to LBM-FP method ismore close to the analytical solution than FV method. The largerdeviations of the PSD due to FV method from the analytical solutionin that size range dominate the error norms and thus error norms forFV method are greater than LBM-FP method. In Case 3, nonlineargrowth and product kernel are considered in order to test theapplicability of the proposed technique to more complex processes.Since analytical solution is not available for this example, we take thesolutions obtained with N¼1000 as the reference solutions tocalculate the error norm. It can be seen from Table 3 that LBM-FPis again more accurate than the FV method, while MOC-FP method ismost accurate with largest computation time among the threemethods. The accuracy of the MOC-FP method is because of itsnegligible numerical diffusion and large computation time is causedby the update of the redistribution rule for aggregation.

In order to ensure that the presented results are not coincidentaland to have a better idea of the performance of these methods withdifferent number of grid points, the trade-off plots between theerror norms and the computation times are shown in Figs. 9–11.From Fig. 9 it can be seen that for the aggregation dominatedproblem (Lo1), the trade-off curves for LBM-FP and FV schemesalmost overlap each other which implies that for a given error norm,the computation time for both methods will be similar. Thus, therelatively higher computation time in LBM-FP method for theaggregation dominated process is compensated by the more accu-rate solution. No significant improvement is found for MOC-FP

100 10210−4

10−3


L1−

norm

LBM−FPFVMOC−FP

101 102 10310−4

10−3


L2−

norm

LBM−FPFVMOC−FP

Fig. 9. Trade-off plots for simultaneous growth and aggregation (Case 1 in Table 2). Solid curves represent the fitted curves assuming power law relation. (a) L1-norm.

(b) L2-norm.

10−1 100 101

10−3

10−2

10−1


L1−

norm

LBM−FPFVMOC−FP

10−1 100 101

10−3

10−2

10−1

100


L2−

norm

LBM−FPFVMOC−FP


(b) L2-norm.

10−1 100 101 102 10310−4

10−3

10−2

10−1

105


L1−

norm

LBM−FPFVMOC−FP

10−1 100 101 102 10310−4

10−3

10−2

10−1

100


L2−

norm

LBM−FPFVMOC−FP


(b) L2-norm.


method with the refinement of grids for this case study. On theother hand, for the growth dominated problems (L41) in Cases2 and 3, it can be noted in Figs. 10 and 11 that for a given errornorm, MOC-FP method is the most efficient followed by LBM-FP,while FV method is least efficient. As discussed earlier, the advan-tage of MOC-FP is more evident for linear growth rate. It is shown inthe next section that when nucleation is present, LBM-FP performsbetter than the MOC-FP method as implementation of nucleation inMOC-FP method is not straight forward and requires special effort(Kumar and Ramkrishna, 1997; Lim et al., 2002).

4.3. Simultaneous growth, nucleation, aggregation and breakage

In this section, it is shown that the developed method canbe applied to practically relevant case of simultaneous growth,

nucleation, aggregation and breakage. We consider an exampletaken from Lim et al. (2002), where the initial distribution is

nðx,0Þ ¼100 for 0:4rxr0:6,

0:01 elsewhere:

(ð72Þ

The expression for stiff nucleation as a function of time is given as

nðt,0Þ ¼ 100þ106 expð�104ðt�0:215Þ2Þ: ð73Þ

The particles are assumed to be growing with the growth rate ofG¼ 1:0 mm s�1 and the computational domain is taken as 0rxr2:0 mm. The parameters for aggregation and breakage are taken asaðx,x0Þ ¼ 1:5� 10�5, bðx,x0Þ ¼ 2=x0 and GðxÞ ¼ x2.

This test problem is solved using LBM-FP, FV and MOC-FPmethods for final time tf¼0.5 s with the number of grid points


being N¼200. It is necessary to adaptively determine the timelevels to add a grid to accommodate the newly born particleswhile dealing with nucleation in MOC-FP method. For implemen-tation of nucleation, we take the approach of Lim et al. (2002)which is to add a new grid at a given time level and delete the lastgrid at the same time level to keep the number of grid pointsconstant. For MOC-FP method, the absolute tolerance of 10�8 andrelative tolerance of 10�4 are used in the ODE solver, since thedefault tolerance settings are not able to resolve the stiff nuclea-tion. The final distributions obtained are shown in Fig. 12. Thegrowth process causes the PSD to translate along the size axiswhile the aggregation and breakage processes produce larger andsmaller particles, respectively. The combined effect of all theseprocesses along with nucleation is seen in the final distribution.

0 0.5 1 1.5

100

105


PSD

, n (

#/µ

m)

Initial

LBM−FP

FV

MOC−FP

Fig. 12. Final distribution for simultaneous growth, nucleation, aggregation and

breakage after tf¼0.5 s with N¼200.

Table 4Simulation results for 1D process with simultaneous growth, nucleation, aggrega-

tion and breakage, N¼200.

Parameter LBM-FP FV MOC-FP

L1-norm 1.13�104 1.98�104 1.44�104

L2-norm 4.31�104 7.74�104 7.37�104

CPU time (s) 0.68 3.99 72.28

100 102

104

105


L1−

norm

LBM−FPFVMOC−FP

Fig. 13. Trade-off plots for simultaneous growth, nucleation, aggregation and breakage.

(b) L2-norm.

The plateau in the final distribution seen near x¼ 1 mm is due to theinitial distribution. The spread of this square pulse is attributed to theaggregation and breakage processes. The part of the curve beforethis plateau is mostly owing to the newly born particles due tonucleation. However, in the final distribution the nucleation part fromthe FV method is slightly shifted to the left. This is likely due to thereformulation of the PBE used in FV method where the original PBE iswritten in mass conservation form.

The analytical solution for this problem is not available. In orderto compare the accuracy of the proposed technique with the FVmethod, we take the final distributions obtained by all the methodswith a finer grid, i.e., N¼1000, as the reference solutions. The errornorms and computational time for these methods are presented inTable 4 which suggest that the proposed method is the mostaccurate among the three methods, while it is significantly efficient.The trade-off plots showing the error norms and the computationtimes, which have been obtained by solving the problem withdifferent number of grid points, are shown in Fig. 13. In this figure,the curves for the proposed LBM-FP method are situated at thebottom-left corner indicating that for a given accuracy, the compu-tation time required for the proposed method is much lower (aboutan order of magnitude) than the other methods. The MOC-FPmethod is computationally expensive since at some time levels anew grid is added to accommodate nucleation. Consequently, thesolver has to initialize the variables and evaluate the redistributionrule used in the FP method for the newly born particles dueto aggregation and breakage. Thus, it can be concluded that theproposed scheme is more suitable for simulation of simultaneousprocesses involving growth, nucleation, aggregation and breakagedue to its higher accuracy and significant advantage of computationtime as compared to the FV and MOC-FP methods.

5. Conclusions

Lattice Boltzmann method (LBM) has been introduced for solu-tion of 1D population balance equations (PBEs) with simultaneousgrowth, nucleation, aggregation and breakage. Aggregation andbreakage, which act as source terms in PBE, have been incorporatedas force terms in the LBM formulation. These force terms can beevaluated by the techniques available in the literature for solvingPBEs with aggregation and breakage. We have used fixed pivot (FP)method by Kumar and Ramkrishna (1996a) for its easy implemen-tation and flexibility. The performance of the proposed method iscompared with pure FP, finite volume (FV) and method of char-acteristic (MOC) combined with FP (MOC-FP) methods in terms oferror norms and computation time. It has been found that FPmethod performs best for pure aggregation and breakage. Therelatively less accurate solution for the proposed method is due to

100 102 104

104.4

104.6

104.8


L2−

norm

LBM−FPFVMOC−FP

Solid curves represent the fitted curves assuming power law relation. (a) L1-norm.


the fact that LBM was primarily developed for growth processes(advection type) and for pure aggregation and breakage processes,the growth term is zero. For such cases, the streaming step in LBMintroduces numerical diffusion.

For simultaneous processes with growth and aggregation, theproposed LBM-FP method is found to have similar performance asthe FV method for aggregation dominated processes, and betterperformance when the growth process is dominant in comparisonto the aggregation process. The MOC-FP method is found to bemost accurate for these simultaneous growth and aggregationprocesses. However, when nucleation is present, the LBM-FPmethod performs better than the MOC-FP method, since imple-mentation of nucleation in this technique is difficult. The use ofthe MOC-FP method has some limitations as solution qualityvaries according to the frequency of mesh addition and computa-tion time can be large for stiff nucleation. In addition, MOC cannotbe used to solve some more complex processes, e.g., where morethan one species with different growth rates are present in thesystem and no unique characteristic curve exists (Lim et al.,2002). Overall, the proposed LBM-FP method is found to besuitable for solution of PBEs for its wide range of applicabilityand efficiency. Theoretical analysis of the observation that LBM-FPhas better efficiency and accuracy for growth dominant processcompared to the aggregation dominant process, will be pursued infuture. In addition, LBM will be extended to multi-dimensional PBEswith growth, nucleation, aggregation and breakage, where themost challenging part to be addressed is the multi-componentaggregation.

Nomenclature

aðx,x0Þ
rate of aggregation of particles with size x and x0, s�1
bðx,x0Þ
probability of formation of particles with size x throughbreakage of particles with size x0, s�1
B0
rate of nucleation, # s�1
c
nonzero velocity component of the particles along thecoordinate axes, mm s�1
cs
speed of sound, mm s�1
D
diffusivity, mm2 s�1
fi
discrete Boltzmann distribution function, # mm�1
f eqi
equilibrium discrete Boltzmann distribution, # mm�1
f i
re-defined discrete distribution function, # mm�1
G
growth rate of particles, mm s�1
h
distribution function in transformed coordinate system,
# mm�1

Ma
Mach number, dimensionless n particle size distribution, # mm�1
N
number of grid points, dimensionless P pressure, kg m�1 s�2
Q
various rate processes, e.g., aggregation, #mm�1 s�1
r
space coordinate, mm t time, s u local average velocity, mm s�1
v
advection velocity, analogous to G in PBE, mm s�1
x
characteristic size of particles, mm z size in the transformed coordinate system, mm
Greek letters

b
relaxation parameter, dimensionless
L
ratio of the characteristic time scales for growth andaggregation, dimensionless
F
time-dependent part of growth rate, mm s�1
r
concentration of the transported species, g mm�3
t
relaxation time, s O size-dependent part of growth rate, mm s�1
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