on the solution and applicability of bivariate population balance equations for mixing in particle...

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Chemical Engineering Science 65 (2010) 3914–3927

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Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

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

On the solution and applicability of bivariate population balance equationsfor mixing in particle phase

Shivendra Singh Chauhan, Jayanta Chakraborty, Sanjeev Kumar �

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e i n f o

Article history:

Received 5 December 2009

Received in revised form

4 March 2010

Accepted 10 March 2010Available online 19 March 2010

Keywords:

Population balance

Particulate processes

Mixing

Agglomeration

Mathematical modelling

Discretization methods

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.03.021

esponding author.

ail address: [email protected] (S.

a b s t r a c t

New benchmarks are used to test two classes of discretization methods available in the literature to

solve bivariate population balance equations (2-d PBEs), and the applicability of these mean-field

equations to finite size systems. The new benchmarks, different from the extensions of their 1-d

counterparts, relate to prediction of kinetics of mixing in particle phase under: (i) pure aggregation of

particles, called aggregative mixing, and (ii) simultaneous breakup and coalescence of drops. The

discretization methods for 2-d PBEs, derived from the widely used 1-d solution methods, are first

classified into two classes. We choose one representative method from each class. The results show that

the extensions based on minimum consistency of discretization perform quite well with respect to both

the new and the old benchmarks, in comparison with the geometrical extensions of 1-d methods. We

next revisit aggregative mixing using Monte-Carlo simulations. The simulations show that (i) the time

variation of the extent of mixing in finite size systems has power law scaling with the system size, and

(ii) the mean-field PBEs fail to capture the evolution of mixing for reduced population of particles at

long times. The sum kernel limits the applicability of PBEs to substantially larger particle populations

than that seen for the constant kernel. Interestingly, these populations are orders of magnitude larger

than those at which the PBEs fail to capture the evolution of total particle population correctly.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Modelling of a number of industrially important processes suchas coagulation and growth of aerosols (Gelbard and Seinfeld, 1978),granulation of powders (Iveson, 2002), crystallization (Puel et al.,2003), crystal shape engineering (Briesen, 2006), and synthesis ofnanoparticles (Cushing et al., 2004) require particles to be identifiedwith two or more of their attributes, such as mass and surface area,mass of primary particles and binder volume, two or moredimensions of anisotropic particles, particle volume and uncappedsurface area, etc. A detailed model for any of these processes isexpected to lead to formulation of multidimensional (n-d)population balance equations (PBEs). A simpler bivariate PBE forsimultaneous aggregation and breakup of particles, identified bymasses x and y of their two constituents, is given by

@nðx,y,tÞ

@t¼

1

2

Z x

0

Z y

0nðx�x0,y�y0,tÞnðx0,y0,tÞQ ðx�x0,y�y0; x0,y0Þdx0 dy0

�nðx,y,tÞ

Z 10

Z 10

nðx0,y0,tÞQ ðx,y; x0y0Þ dx0 dy0�Gðx,yÞnðx,y,tÞ

þ

Z 1x

Z 1y

bðx,y; x0,y0ÞGðx0,y0Þnðx0,y0,tÞ dx0 dy0 ð1Þ

ll rights reserved.

Kumar).

Here n(x,y,t) dx dy is the number of particles per unit volume withconstituent masses in ranges x to x+dx and y to y+dy at time t. Theaggregation frequency Q, breakage frequency G and daughterparticle distribution function b are defined in the context ofbivariate particles. The use of population balance models to improveand optimize processes similar to those mentioned above requirescomputationally efficient and accurate methods to solve equationssimilar to Eq. (1).

A number of methods for solving one dimensional version(1-d PBEs) of Eq. (1) have appeared in the literature. Over theyears, some of these have proved to be robust and simple to use,and are being extended to solve bivariate PBEs. Prominent amongthese are: moment-based methods, such as quadrature method ofmoments (McGraw, 1997; Wright et al., 2001), direct quadraturemethod of moments (Marchisio and Fox, 2005; Zucca et al., 2007),and sectional or discretization methods. The moment-basedmethods can be easily combined with CFD simulations and areuseful if the objectives of modeling exercise are served bydetermination of a few lower moments of the size distribution.Monte-Carlo methods can lead to prediction of full size distribu-tion for even quite complex multivariate processes. Thesecomputationally expensive methods are however limited tospatially well mixed systems.

Discretization or sectional methods are in the middle ofthe spectrum spanned by moment and Monte-Carlo methods.

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A variety of discretization methods are available in the literatureto solve 1-d PBEs accurately and efficiently. Some of these areextended to solve 2-d and n-d PBEs for aggregation and breakage.The proposed extensions have been mostly tested for their abilityto predict features of 1-d PBEs extended to n-d PBEs. Comparisonof time evolution of moments and particle population in 1-d binshas been extended to prediction of mixed moments and particlepopulations in 2-d bins, located on the main diagonal alone oreverywhere in the 2-d domain. The features specific to bivariateparticles, often an important reason for developing a bivariatemodel, are yet to be used to compare various solution methods.For example, a model which identifies a particle by its volume andsurface area for aerosol related processes should be finally able topredict synthesis of fully sintered particles; a model for granula-tion should be able to predict formation of agglomerates withbinder content equal to or less than a critical value so that they donot agglomerate any further; a model for liquid–liquid extractionor multiphase reactions should be able to predict completemixing in the dispersed phase by coalescence and dispersion ofdrops when no other process is permitted.

In this work, we propose new benchmarks to test numericalmethods for solving Eq. (1). Instead of subjecting the availablemethods for solving Eq. (1) to the old and the new benchmarks,we first classify them into two classes, on the basis of theprinciples used to develop them. We next consider two repre-sentative techniques, one for each class, for the detailedcomparisons. In order to keep the comparison focused on theprinciples used, two extensions of 1-d fixed pivot technique ofKumar and Ramkrishna (1996a), differing from each other in theprinciples used to derive them, are considered here.

The first of the two new benchmarks is based on thecharacteristics of mixing brought about by pure aggregation ofparticles, called aggregative mixing in the literature. The extent ofaggregative mixing, w2, a measure of the deviation in thecomposition of particles from the expected composition of wellmixed particles (Matsoukas et al., 2006), is defined as

w2ðtÞ ¼

Z 10

dx

Z 10

dy½y�fðxþyÞ�2nðx,y,tÞ ð2Þ

where

f¼R1

0 dxR1

0 dy � ynðx,y,tÞR10 dx

R10 dy � ðxþyÞnðx,y,tÞ

ð3Þ

An interesting and counter-intuitive feature of aggregativemixing, predicted for a number of initial conditions and kernels,is the unchanging value of w2 with time while aggregationproceeds and large size particles are formed (Matsoukas et al.,2006). Prediction of this feature can serve as a new benchmark,along with the old ones.

Simultaneous breakup and coalescence of drops leads tosteady state size distribution for a number of combinations of b,G, and Q functions (Vigil and Ziff, 1989). When bivariate dropswith initially different compositions undergo the same processes,the system first reaches a pseudo steady state with respect to thesize distribution, and finally a true steady state at which all thedrops attain same composition, equal to the mean composition ofdrops. We propose prediction of dynamics of mixing in dropletphase as the second new benchmark.

The time invariant value of w2 for aggregative mixing, aspredicted by the mean-field PBEs, cannot hold at long times forreal systems. The value of w2 must eventually decrease to zero asthe particles aggregate to finally form one single particle. Wetherefore revisit aggregative mixing. We carry out Monte-Carlosimulations to explore the applicability of mean field PBEs topredict aggregative mixing at long times. We will see towards the

end that our results bring out new features of aggregative mixingin systems of finite sizes.

The rest of the manuscript is organized as follows. We firstcritically review the extensions of the 1-d techniques to solve 2-dPBEs, with special focus on those chosen as representativetechniques. This is followed by the results on aggregative mixing,mixing brought about by breakup and coalescence of drops, andthe limitations of the bivariate mean field PBEs.

2. Previous work

The rapid increase in the use of population balance models andthe progress made in solving PBEs have moved hand-in-hand inthe last two decades. The latter, confined mostly to themonovariate PBEs, is reviewed in a number of excellent sources(Ramkrishna, 2000; Vanni, 2000; Costa et al., 2007). We provide,in this section, a brief account of the 1-d discretization methods,and follow it up with a review of extensions of these methods tosolve bivariate PBEs.

In discretization methods for solving 1-d PBEs, the solutiondomain is divided into contiguous size ranges to develop anautonomous set of coupled discrete PBEs, one for each size range.Two approximations have been used in the literature to obtain anautonomous set of equations from the original integro-partialdifferential equation. The first one assumes that all the particles ina discrete size range are represented through a representative sizein it, called pivot. The second one assumes particles to beuniformly distributed in a size range. These approximations aregrouped as M-I and M-II in the literature (Kumar and Ramkrishna,1996a). As the aggregation process represents linear addition ofparticle volumes/masses to form new particles, linear discretiza-tion of the solution domain is natural (Hidy, 1965). The large sizerange spanned by aggregating particles and resolution of smallsize particles mandate the use of a large number of discretizedequations, e.g., of the order of hundreds of thousands, equal to thedegree of polymerization in polymer processing.

In order to reduce the number of equations with acceptableresolution of small size particles, uniform discretization onlogarithmic scale (popularly known as geometric discretizationof space) was tried with both M-I (Batterham et al., 1981) andM-II approximations (Bleck, 1970; Gelbard et al., 1980). Theseefforts reduced the number of equations dramatically but intro-duced new issues with respect to the accuracy of the solution dueto the inherent difficulty in representing both, birth of one particleand birth of particle volume v in a size range. A birth event in a sizerange is easily represented as addition of one particle to itspopulation or addition of volume v to it, but not both.

Sastry and Gaschignard (1981) appear to be the first to havepreserved both the number and the volume of a new particle borninto a bin by aggregation by incorporating non-uniform numberdensity of particles within a bin, with M-II approximation. Theresulting set of equations became quite complex to implementand computationally expensive to solve. Hounslow et al. (1988)achieved the same in a much simpler manner by using acombination of M-I and M-II approximations and a kernelindependent correction factor for a fixed geometric grid vi +1/vi¼2. The same approach was later extended to uniformgeometric grids of type viþ1=vi ¼ 21=q where q is an integer.

Kumar and Ramkrishna (1996a) proposed the concept ofinternal consistency of discretization. An equation for theevolution of an integral property of particles can be obtained intwo ways: (i) by multiplying the discretized PBEs with theappropriate property and then summing them up, and (ii) bydiscretizing the continuum equation for the same integralproperty. According to the concept of internal consistency, the

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two approaches should lead to identical results. The internalconsistency of discretization with respect to the desired proper-ties is achieved by representing new particles in discretizedequations in a manner that preserves these properties. Kumar andRamkrishna (1996a) used M-I approximation, i.e., the particlepopulation is represented at pivots xi as

nðv,tÞ ¼XMi ¼ 1

NiðtÞdðv�xiÞ

The number and volume v of a new particle, formed byaggregation of particles represented at pivots xj and xk, arepreserved if fractions a and b assigned to pivots xi and xi + 1,adjoining the new particle, as shown in Fig. 1(a), satisfy thefollowing relationships:

aþb¼ 1; axiþbxiþ1 ¼ v¼ xjþxk

Their final set of discretized equations for pure aggregation is

dNi

dt¼Xi

j ¼ 1

Xj

k ¼ 1

Zij,kQj,kNjNk 1�

1

2dj,k

� ��Ni

XMj ¼ 1

NjQi,j ð4Þ

The elements of Z matrix, generated at the time of discretizationof space, contain fractions a and b. This technique can use anarbitrary discretization of 1-d space, and preserve an evennumber of particle properties. More importantly, it can be easilyextended to solve n-d PBEs by treating discretization in eachdimension identically. We term here such an extension asgeometric extension of 1-d technique. Fig. 1(b) shows treatmentof a new particle born in 2-d space for particle populationrepresented on a rectangular grid, so that

nðx,y,tÞ ¼XMx

i ¼ 1

XMy

j ¼ 1

Ni,jðtÞdðx�xiÞdðy�yjÞ

A new particle can be first split along one direction using theabove method, followed by splitting of the fragments in thesecond direction (Kumar and Ramkrishna, 1995). Thus,

rþs¼ 1; rxiþsxiþ1 ¼ x

a b

xi+1xixi−1 vi+1vi v = xj + xk

b d

ca

x

y

xi1 xi xi+1

yi

yi+1

Fig. 1. Assignment of a particle in 1-d and 2-d space (with rectangular elements).

aþb¼ r; ayiþbyiþ1 ¼ y

cþd¼ s cyiþdyiþ1 ¼ y

The four fractions a, b, c, and d assigned to pivots (xi,yi), (xi,yi +1),(xi +1,yi), and (xi + 1,yi + 1) can also be obtained by preserving fourproperties: number, x mass, y mass, and product of masses x � y

(Vale and McKenna, 2005).

aþbþcþd¼ 1

axiþbxiþcxiþ1þdxiþ1 ¼ x

ayiþbyiþ1þcyiþdyiþ1 ¼ y

axiyiþbxiyiþ1þcxiþ1yiþdxiþ1yiþ1 ¼ xy

The two methods lead to identical values of a, b, c, and d. The finaldiscretized equations obtained are

dNi,r

dt¼Xi

j ¼ 1

Xj

k ¼ 1

Xr

s ¼ 1

Xs

t ¼ 1

Zi,rj,s;k,tQj,s;k,tNj,sNk,t 1�

1

2dj,kds,t

� �

�Ni,r

XMx

j ¼ 1

XMy

s ¼ 1

Nj,sQi,r;j,s ð5Þ

A similar extension to 3-d PBEs has been carried out by writingequations for preservation of eight properties, number, x, y, z, xy,yz, xz, and xyz for eight unknown fractions a, b, y, h and so on(Chakraborty and Kumar, 2007). The 1-d moving pivot techniqueof Kumar and Ramkrishna (1996b) in which the location of a pivotin a bin is moved to simultaneously preserve number and mass ofparticles born into it, cannot be geometrically extended to thesolution of 2-d PBEs with preservation of four properties asa moving pivot in 2-d space has only three degrees of freedom:x and y locations of a pivot and the population it carries.

Immanuel and Doyle (2003) followed M-II approximation ofBleck (1970) and Gelbard et al. (1980). The latter represented atypical aggregation term of a 1-d PBE for discretization asZ viþ 1

vi

vnZ v

0nðv�v0Þnðv0ÞQ ðv�v0,v0Þdv0 ¼

Xj

Xk

njnk

ZAi

jk

vnQ ðv00

,v000

Þdv00

dv000

where nj and nk are average number densities in jth and kth bins.Area Ai

jk represents those aggregation events among particlesbelonging to jth and kth size ranges that lead to formation ofparticles in ith size range. The presence of size dependentfrequency Q in the integral on the rhs ensures that when onlyparts of jth and kth ranges contribute new particles to ith range,aggregation frequency corresponds to only these sub-ranges. Theparticles formed in the ith size range in this manner typicallybelong to only a part of the ith size range but are representedassuming that they are formed uniformly in this size range. Thus,only one, vn, property of new particles is preserved. The value of nis taken to be unity by Bleck (1970) and zero by both Gelbard et al.(1980) and Immanuel and Doyle (2003). In order to reduce thecomputational load involved with one time evaluation of doubleintegrals, the latter further approximated

Xj

Xk

njnk

ZAi

jk

Q ðv00

,v000

Þdv00

dv000

�X

j

Xk

njnkQj,kAijk

where Qj,k is aggregation frequency corresponding to representa-tive sizes xj and xk for the respective bins. Thus, the aggregation ofparticles belonging to only part or full jth and kth size rangesoccurs at one frequency, Qjk, both for the size independent andsize dependent kernels. The extent of error introduced by thisapproximation is however not known. The geometric extension of

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x

y

Main diagonal

Fig. 2. Radial grid with triangles oriented along the main diagonal.


M-II approximation based 1-d technique to n-d PBEs is straight-forward as the multiple axes can be handled identically andindependently (Immanuel and Doyle, 2005; Gelbard and Seinfeld,1980), similar to the procedure followed for the 1-d fixed pivottechnique. There is one major difference, though. The extensionsof M-II approximation-based methods to n-d PBEs can preserveeither the population of new particles or the masses of individualconstituents of particles, but not the both. Simultaneous pre-servation of population and masses of individual constituents ofparticles is possible only in the limit of fine grid.

Kumar et al. (2006) developed a two step cell averagetechnique in which first the center of mass of newly bornparticles in a bin is determined, similar to that in the moving pivottechnique (Kumar and Ramkrishna, 1996b). The incomingparticles are next represented through the pivots adjoining thecenter of mass. This technique produces quite accurate solutions.The geometrical extension of this technique to bivariate PBEs(Kumar et al., 2008) however leads to an inconsistency. Asdiscussed earlier, a moving pivot (center of mass) in 2-d space hasonly three degrees of freedom, hence, in the first step onlythree properties of new particles are preserved. In the secondstep, four properties of the new particles are preserved, similar tothe extension of Vale and McKenna (2005). The results obtainedare quite accurate. The implementation of the scheme howeverrequires care as a new bivariate particle is represented through atotal of 9 pivots, four at a time; a trivariate particle is expected tobe represented by a total of 27 pivots, eight at a time. The effortsto preserve more than two properties of 1-d particles are alsoreported in the literature (Alopaeus et al., 2006; Kostoglou, 2007).

Chakraborty and Kumar (2007) proposed a new framework fordiscretization of multidimensional PBEs based on the concept ofminimal internal consistency of discretization. As a n-d PBE is astatement of how n internal attributes of newly formed particlesevolve with time, the representation of a new particle, accordingto this framework, needs to preserve its population and its n

attributes. Thus, only n+1 attributes need to be preserved incomparison with 2n attributes preserved in the geometricextensions discussed above. The space filling discretization ofn-d space also requires elements with a minimum of n+1 vertices.These natural elements are triangles for 2-d space, tetrahedronsfor 3-d, and so on. The population of n-variate particles isrepresented through pivots located at positions xi (a vector in n-dspace). Thus,

nðv,tÞ ¼XMi ¼ 1

NiðtÞdðv�xiÞ

A bivariate particle with internal attributes represented by vectorv (components x and y), and surrounded by pivots xr , xs, and xt in2-d space is represented through these pivots in a manner thatpreserves number and its two masses x and y. Thus, fractions a, b,and c assigned to the neighbouring pivots are obtained by solving

aþbþc¼ 1

axrþbxsþcxt ¼ x

ayrþbyrþcyt ¼ y

simultaneously. The fractions a, b, and c are guaranteed to be non-negative for an arbitrarily oriented triangle in space. It isimportant to note that the non-negativity of fractions does nothold for arbitrarily oriented rectangles/quadrilaterals for preser-vation of four properties in geometric extensions of 1-d PBEs(Nandanwar and Kumar, 2008a). The final set of discretized PBEs

obtained using this framework is

dNi

dt¼XMj ¼ 1

XMk ¼ 1

Zij,kQj,kNjNk 1�

1

2dj,k

� ��Ni

XMj ¼ 1

NjQi,j ð6Þ

The elements of matrix Zij,k contain indices i, j, k, and non-zero

values of fraction of particle assigned to pivot i when particles atpivots j and k aggregate. The matrix Z is generated once, at thetime of generation of a triangulated 2-d grid. This class oftechniques, requiring preservation of n+1 particle properties forn-d PBEs, is named as n-d fixed pivot techniques (n-FPT).Correspondingly, the class of techniques which treat each newaxes of n-d space identically and require preservation of 2n

particle properties, is named as geometrically extended fixedpivot techniques (GEFPT). A mere change in the connectivityamong the pivots, which allows a GEFPT to be replaced by a n-FPT,was shown to improve the numerical predictions substantially(Chakraborty and Kumar, 2007). The same study also showed thatthe orientation of triangles and tetrahedrons influences theaccuracy of the solution significantly.

Nandanwar and Kumar (2008a) used these findings to proposea new discretization of space for n-FPT. They divided 2-d spaceusing radial lines and arcs as shown in Fig. 2 and located pivots atthe intersection points. The resulting quadrilaterals wereconverted into triangles, required for n-FTP, by using thediagonal lines pointing towards the direction of evolution ofthe solution. Unlike the rectangular grid, which requires the gridto be refined in the entire 2-d space, it is possible to increase thedensity of radial lines only in the region in which the solutionevolves. The radial discretization also reduces breakup of bivariateparticles into a series of 1-d problems, and allows particles ofsame composition to aggregate and form new particles of thesame composition (Nandanwar and Kumar, 2008b). A comparisonof the numerical and the standard results for aggregation andbreakup of particles shows that radial discretization of space isquite effective in improving the accuracy of the numericalsolution. The redistribution of radial lines (more in the region ofevolution) without increasing the total number of pivots and thecomputational effort was also shown to improve the accuracy ofthe numerical solutions.

Filbet and Laurenc-ot (2004) have recently represented neteffect of birth and death of particles due to aggregation in theform of a flux term, which converts the 1-d PBE for pureaggregation into a hyperbolic equation. This equation is thensolved by using finite volume techniques. The complex imple-mentation it entails does produce quite good predictions at theend, though. The same approach has been extended to solve

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bivariate PBE as well (Qamar and Warnecke, 2007) with a furtherincrease in complexity in its implementation.

We choose 2-d discretization technique of Kumar andRamkrishna (1995) and Vale and McKenna (2005) to representthe class of GEFPT’s and the technique of Chakraborty and Kumar(2007) with radial refined discretization of space (Nandanwar andKumar, 2008a) to represent the class n-FPT’s. These tworepresentative techniques are referred to as GEFPT and n-FPT,respectively, in the rest of this paper. The results obtained withthese techniques for the new and the old benchmark tests arepresented next.

3. Results

3.1. Aggregative mixing

Pure aggregation of particles of different compositions bringsabout mixing, as in granulation of powders, coagulation ofparticles in mixed colloids, and precipitation. The evolution ofbivariate number density and the extent of mixing brought aboutby pure aggregation process is given by (Eq. (1) without the termsrepresenting particle breakup)

@nðx,y,tÞ

@t¼

1

2

Z x

0

Z y

0nðx�x0,y�y0,tÞnðx0,y0,tÞQ ðx�x0,y�y0; x0,y0; tÞdx0 dy0

�nðx,y,tÞ

Z 10

Z 10

nðx0,y0,tÞQ ðx,y; x0y0Þdx0 dy0 ð7Þ

Gelbard et al. (1980) have solved Eq. (7) analytically for theconstant kernel. Their solution for the following initial condition

nðx1,x2, . . . ,xm,0Þ ¼N0

Ymi ¼ 1

1

xi0exp�xi

xi0

� �ð8Þ

is given by

nðx1,x2, . . . ,xm,tÞ ¼ 4N0

ðtþ2Þ2

Ymi

1

xi0exp�xi

xi0

� � X1k ¼ 0

1

ðk!Þmt

tþ2

Ymj ¼ 1

xj

xj0

24

35

k

ð9Þ

where t, the non-dimensional time, is defined as t¼ QN0t.Fernandez-Diaz and Gomez-Garcia (2007) have recently

developed analytical solution for the sum kernel ½Q ðx,y; x0,y0Þ ¼bðxþyþx0 þy0Þ� for the same initial condition.

nðx1,x2, . . . ,xm,tÞ ¼N0ð1�tÞexp �x

x0t

� � Ymi ¼ 1

1

xi0exp �

xi

xi0

� � !

�X1k ¼ 0

1

ðkþ1Þ!

tx

x0

� �k Ymi ¼ 1

ðxi=xi0Þk

Gðkþ1Þð10Þ

where x¼Pm

i xi, x0 ¼Pm

i xi0, t¼ 1�expð�bftÞ, b is prefactor forthe sum kernel, and f is the total mass in the system.

Eq. (7) has been analyzed in detail to unravel a rich set ofscaling and self-similar solutions associated with it (Lushnikov,1976; Krapivsky and Ben-Naim, 1996; Vigil and Ziff, 1998).Matsoukas et al. (2006) have summarized these findings andfurther expanded the applicability of scaling solutions to include awider set of initial conditions and aggregation kernels. We referan interested reader to their work (Matsoukas et al., 2006;Lee et al., 2008) for many more interesting features observed withthese systems. In below, we highlight the findings which we seekto use in the new benchmarks.

Matsoukas et al. (2006) show that for composition indepen-dent kernels such as the constant and the sum kernel, the state ofmixing of a partially mixed collection of particles remains

unchanged with time (constant value of w2) while the particlesaggregate to form bigger particles. A partially mixed collection ofparticles is the one in which mean composition of all the particlesof any fixed size (x+y) is the same as the mean composition of theentire population of particles. It is easily shown that the initialcondition expressed by Eq. (8) for bivariate number densityindeed satisfies this requirement. Matsoukas et al. (2006) alsoshowed that for the sum type kernels, i.e., Q ðv1,v2Þ ¼ kðv1Þþkðv2Þ,where v1 ¼ x1þy1 and v2 ¼ x2þy2, the state of mixing ascharacterized by w2 remains unchanged with time for an arbitraryinitial distribution of particles as well. The constant and the sumkernels belong to this class of kernels.

We compare in this section the results obtained for thesolution of Eq. (7) using GEFPT and n-FPT for the constant andthe sum kernels, for the initial condition shown in Eq. (8), with theanalytical results. The variables compared are: the state of mixingas characterized by w2 and the bivariate size distribution asrepresented through particle population on pivots. The numericalsolution was obtained for GEFPT on geometrically uniform grid inx and y directions with rectangular elements, and n-FPT ongeometrically uniform grid in r direction and non-uniform grid inangular direction with triangular elements, as shown in the insets.

It is difficult to visually compare solutions of 2-d PBEs(requires 3-d plots). We therefore use the flat comparison method(applicable for n-d PBEs as well) available in the literature(Chakraborty and Kumar, 2007). In this method, first theanalytical solution is integrated over n-d bins to obtain analyticalprediction for particle population on pivots; the bin boundariesare located by drawing bisecting lines and arcs. The particlepopulations on pivots obtained using one of the methods(analytical solution, unless stated otherwise) is next sorted indecreasing order of population and the pivots are re-indexed. Thepredictions obtained using the two methods are next plottedagainst the new pivot index. A single plot thus enables the overallcomparison of two solutions. In order to quantify the differencebetween two solutions, we use measures Dij (Nandanwar andKumar, 2008a), defined as

Dij ¼

PMp ¼ 1 jNp,ana�Np,numjxi

pyjpPM

p ¼ 1 Np,anaxipyj

p

ð11Þ

D00, D10, and D01 quantify overall error in predicting distributionof particle population, x mass, and y mass, respectively. It isimportant to note that even if moments M00 and M10 and M01 arepredicted exactly, which is often the case with internallyconsistent discretization methods (including GEFPT and n-FPT),the variables D00, D10, and D01 can be quite different from zero.

3.1.1. Constant kernel

Fig. 3(a) shows a comparison of numerical results, obtainedusing GEFPT with a uniform geometric grid consisting of 25�25grid points (pivots), with the analytical solution at t¼ 0:01, andNðtÞ=Nð0Þ ¼ 0:166. The same comparison for the numerical resultobtained using n-FPT with a radial refined grid (shown in theinset) and the same number of grid points is presented in Fig. 3(b).Although, both the techniques make identical and exactpredictions for mixed moments M00, M10, and M01, theprediction of distribution of particle population is in significanterror for GEFPT for bins carrying both small and large particlepopulations. In comparison, n-FPT makes quite good predictions.Fig. 4 shows a similar comparison at a longer time, t¼ 0:1 andNðtÞ=Nð0Þ ¼ 0:019. A comparison of Figs. 3 and 4 shows that anincrease in the extent of evolution increases the error in thesolution obtained with GEFPT substantially more than that forn-FPT. The reason for this behaviour is that sharply decreasingnumber density in directions perpendicular to the direction of

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k

AnalyticalNumerical

1e-05

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0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 3. Flat comparison of numerical solution obtained using GEFPT and n-FPT for

the constant kernel at t¼0.01 unit corresponding to N(t)/N(0)¼0.166 using

25�25 pivots. (a) GEFPT, (b) n-FPT.

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Fig. 4. Same as that for Fig. 3 at t¼0.1 unit corresponding to N(t)/N(0)¼0.019.

(a) GEFPT, (b) n-FPT.

Table 1D values for aggregation at times t¼ 0:01 and 0.1 for the constant kernel.

D t¼0.01 t¼0.1

Rectangular Radial Rectangular Radial

D0,0 0.168 0.061 0.591 0.097

D0,1 0.314 0.100 0.812 0.146

D1,0 0.314 0.100 0.812 0.146

D1,1 0.430 0.189 0.895 0.233


main evolution is captured very poorly by coarse bins (Kumar andRamkrishna, 1996a; Chakraborty and Kumar, 2007).

These conclusions are also supported by the numerical valuesof the D variables, presented in Table 1. The error in prediction ofdistribution of population, indicated by D00, is 0.168 for GEFPTand 0.061 for n-FPT at t¼ 0:01. These values at t¼ 0:1 rise to0.591 and 0.097, respectively. A comparison of D variables forGEFPT and n-FPT shows that n-FPT leads to about three timessmaller error at short time and about six times smaller error atlong time.

We next compare predictions for the variation of w2 with timefor the two techniques with the analytical results. As discussedearlier, the value of w2 for the present case is predicted to remainconstant with time, on the basis of moment-based analysis(Matsoukas et al., 2006). The full solution for number density(Eq. (9)) was also introduced in Eq. (2) to further validate it. Theinitial set of predictions of w2 obtained using GEFPT with 25�25grid showed that w2 did not remain constant even for quite shortextents of evolution. The grid was therefore refined to 101�101,and the same size grid was used for both the techniques. Fig. 5shows that the value of w2 predicted by using GEFPT remainsconstant with time for about NðtÞ=Nð0Þ ¼ 0:16 at t¼ 0:01, andthen begins to increase rapidly. The results obtained by usingn-FPT for nearly the same computational effort show that w2

remains constant up to much larger extent of evolution,NðtÞ=Nð0Þ ¼ 0:002 at t¼ 1. The wide difference in the behaviourof the two techniques is attributed to preservation of fourproperties (22) in GEFPT vs. preservation of minimum threeproperties (2+1) in n-FPT. This is taken up in detail at a laterpoint.

3.1.2. Sum kernel

Figs. 6 and 7 show comparisons of size distributions, similar tothose presented in the previous section, for the sum kernel with25�25 grid points at t¼ 1� 10�3 and 2�10�3), corresponding toNðtÞ=Nð0Þ ¼ 0:135 and 0.018, respectively. Although both thetechniques predict time variation of moments M00, M10, and M01

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AnalyticalRadial

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Fig. 5. Comparison for w2 predicted using GEFPT and n-FPT (101�101 grid points)

with the analytical result ðw2 ¼ constantÞ for the constant kernel.

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10

100

1 10 100 1000

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k

AnalyticalNumerical

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0.001

0.01

0.1

1

10

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Nk

k

AnalyticalNumerical

Fig. 6. Flat comparison of numerical solution obtained using GEFPT and n-FPT for

the sum kernel at t¼ 1� 10�3 unit corresponding to NðtÞ=Nð0Þ ¼ 0:135 using

25�25 pivots. (a) GEFPT, (b) n-FPT.

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k

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AnalyticalNumerical

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1

10

Nk

AnalyticalNumerical

Fig. 7. Same as that for Fig. 6 at t¼ 2� 10�3 unit corresponding to

NðtÞ=Nð0Þ ¼ 0:018. (a) GEFPT, (b) n-FPT.

Table 2

D values for aggregation at time t¼ 1� 10�3 and 2� 10�3 for the sum kernel.

D t¼ 1� 10�3 t¼ 2� 10�3


D0,0 0.147 0.066 0.193 0.063

D0,1 0.645 0.221 0.867 0.313

D1,0 0.641 0.221 0.864 0.313

D1,1 0.977 0.483 1.168 0.510


perfectly, the figures show that n-FPT predicts the size distributionof particles quite well-compared with the predictions of GEFPT.Table 2 further supports this conclusion—the predicted value

of D variables for n-FPT are 2–2.5 times smaller than thosefor GEFPT.

We now compare predictions for the variation of w2 with timewith the expected behaviour. The value of w2 for this case is alsopredicted to remain constant with time. The recently developedfull solution for number density (Eq. (10)) was also introduced inEq. (2) to validate this result. Fig. 8 shows that the predictions ofGEFPT for w2 (even with a refined grid of size 101�101) remainconstant with time (at 0.5) for very short extent of evolution,NðtÞ=Nð0Þ ¼ 0:82 at t¼ 4� 10�4, and then begin to increaserapidly to attain w2 ¼ 100 at t¼ 2� 10�3. The n-FPT on theother hand predicts w2 to remain constant all the way up toNðtÞ=Nð0Þ ¼ 0:018 at t¼ 2� 10�4. Interestingly, for the similar

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Fig. 8. Comparison for w2 predicted using GEFPT and n-FPT with the analytical

result ðw2 ¼ constantÞ for the sum kernel.

0.01

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10

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Nk

k

’MC_simulation’’numerical’

0.1

1

10

1 10 100 1000

Nk

k


Fig. 9. Flat comparison of numerical solution obtained using GEFPT and n-FPT

with standard result obtained using Monte-Carlo simulations for simultaneous

breakup and coalescence of drops at t¼0.01 unit corresponding to N(t)/N(0)¼0.23

using 25�25 pivots, and the initial condition considered for Case A. (a) GEFPT,

(b) n-FPT.


extent of evolution for the constant kernel ðNðtÞ=Nð0Þ � 0:019Þ, thevalue of w2 predicted using GEFPT increases from 0.5 to about 10only. The effectiveness of GEFPT in capturing aggregative mixingthus worsens for a size dependent kernel.

3.2. Simultaneous coalescence and breakup of bivariate drops

Unlike the previous case where deviation from the well mixedstate remains unchanged with time and no steady state sizedistribution is possible, the presence of breakup along withcoalescence of drops (in agitated liquid–liquid dispersions) leadsto formation of well mixed drops and a steady state distribution ofdrop sizes. In order to quantify mixing in the context of agitateddispersions so as to test GEFPT and n-FPT for their ability tocapture dynamics of mixing, we define a new variable D here,quite similar to w2 defined earlier (Eq. (2)), but in terms of thevariables more relevant to this sub-field

DðtÞ ¼

PMp ¼ 1 jðCp�CavgÞjNp,numðtÞPM

p ¼ 1 Np,numðtÞð12Þ

where Cp is xp=ðxpþypÞ for particle population on p th pivot, andCavg ¼M10=ðM10þM01Þ.

We simulate two cases of different types of initial populationsof segregated drops undergoing mixing for bðv0,vÞ ¼ 2=v, GðvÞ ¼ v2,and Q ðv,v0Þ ¼ 1:0, where v¼x+y. The simulation procedurefollowed is described in detail in Nandanwar and Kumar (2008b).

3.2.1. Case A

We chose for this case two identical sets of polydisperse dropsin terms of their sizes, but different in composition. One setconsists of drops with only one species (x¼0). The other setconsists of both the species in ratio x/y¼10. Alexopoulos andKiparissides (2007) used GEFPT to simulate simultaneous breakupand coalescence of drops for this initial condition with a slightlydifferent breakup frequency kernel ðGpvÞ. It is not clear ifsimulations were carried out in (x,y) or (v,c) domain. At steadystate, the well mixed drops must have ratio x/y equal to 5. Theirsimulations did not capture the expected steady state. Theyconjectured that it perhaps takes extremely long time for thedrops to get well mixed.

Simulations were carried out in this work by using N0¼400 inEq. (8) for both sets of drops, one of which is located on the y-axisand other on a radial line of slope of 0.1 in (x�y) plane. The steady

state distribution of drops is expected to fall on a radial line ofslope 1.2. Numerical solutions were obtained for both GEFPT andn-FPT using 25�25 pivots. The standard results were obtained byusing Monte-Carlo simulation algorithm (Gillespie, 1976; Shahet al., 1977). A large number of simulations (up to 10 000) wereaveraged to predict D, accurate to the last significant digit.

Fig. 9 shows a flat comparison of numerical solutions obtainedwith GEFPT and n-FPT with the standard results at t¼0.01corresponding to N(t)/N(0)¼0.23. The same comparison at t¼2,well into the steady state, Nð1Þ=Nð0Þ ¼ 0:16, is shown in Fig. 10. Acomparison of the D variables for both the times is presented inTable 3. The time variation of variable D as predicted by GEFPT,n-FPT, and Monte-Carlo simulations is presented in Table 4. Theresults show that in comparison with GEFPT, n-FPT capturespopulation on pivots quite well. GEFPT predicts variation of D

with time for short extents of evolution well (decrease from 0.45to 0.26 at t¼0.01), but approaches a wrong steady state value of0.12. Interestingly, this wrong steady state is reached at a time ofabout 0.06 time units, while the Monte-Carlo simulations andn-FPT continue to predict further decrease in D, in agreement witheach other till nearly well mixed state (D¼0.0161) is reached att¼0.1 time units. Subsequently, Monte-Carlo simulations predicta rapid approach to well mixed state ðDo10�10

Þ at t¼2 and n-FPTtakes a little longer. Interestingly, for this case also, both GEFPT

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Nk

k


Fig. 10. Same as that for Fig. 9, at t¼2 unit corresponding to N(t)/N(0)¼0.16

(at steady state). (a) GEFPT, (b) n-FPT.

Table 3D values obtained for GEFPT and n-FPT for simultaneous breakup and aggregation

at times t¼0.01 and 2 units, for the initial condition considered in Case A.

D t¼0.01 t¼2


D0,0 1.119 0.072 1.333 0.020

D0,1 0.975 0.080 1.126 0.019

D1,0 0.967 0.070 1.656 0.021

D1,1 0.968 0.087 1.295 0.020

Table 4Variation of D, representing deviation from well mixed state (as per Eq. (12)), with

time as predicted by GEFPT, n-FPT, and the Monte-Carlo simulations, for the initial

condition considered in Case A.

Time D (GEFPT) D (n-FPT) D (Monte-Carlo)

0 0.450 0.455 0.455

0.001 0.417 0.416 0.416

0.01 0.267 0.263 0.262

0.02 0.194 0.123 0.126

0.03 0.159 0.112 0.114

0.04 0.148 0.087 0.083

0.06 0.128 0.048 0.046

0.08 0.124 0.028 0.026

0.1 0.122 0.017 0.016

1 0.122 9.53�10�5

2 0.122 4.26�10�5

5 0.122 1.56�10�5

40 0.122 3.55�10�16


and n-FPT predict the time variation of M00, M01, and M10

perfectly.It is possible that the poor quality of results obtained with

GEFPT is due to the absence of any pivots in rectangular grid on aradial line of slope 1.2, on which the final solution is expected tolie. Pivots located on such a line do exist in solution obtainedwith n-FPT. In such a situation, the steady state solution obtainedwith GEFPT can be represented only through pivots locatedaround this line, leading to a large non-zero value of D at steadystate, and substantially large deviations in distribution ofpopulation on pivots. In order to provide the best possiblescenario for GEFPT, a different initial condition is considered inthe next section.

3.2.2. Case B

The two sets of otherwise identical polydisperse drops arechosen for this case. One set contained only x and the other only y

in them. The initial population of drops thus lie on x and y axes.Since the two groups of drops are identical otherwise, the wellmixed drops formed at steady state lie on the main diagonal(slope unity). Pivots are located on this diagonal in equal numberfor both the techniques. All the other parameters and detailsremain the same as those described in the previous section.Figs. 11 and 12 and Tables 5 and 6 present comparisons similar tothose provided in the previous section. The results show thatthere is no improvement in predictions obtained using GEFPT.Expectedly, n-FPT leads to very good predictions for this case aswell. Nandanwar and Kumar (2008b) earlier reported results for asimilar problem but with partially pre-mixed drops at initial time.Their predictions of D variables for GEFPT and n-FPT are similar tothose obtained in the present work. They however did not predictand compare time evolution of mixing in dispersed phase.

3.3. Discussion

The shortcomings of GEFPT—variable D does not approachzero for finite size grid, and variable w2 deviates from the expectedconstant value quite early on—are generic. This is due to thelimitations of the principle used to develop extensions belongingto this class. A new particle formed on the diagonal of a rectangle,either by breakup or aggregation, is represented though twopivots on the diagonal and the two others located off-diagonallyto preserve four properties. In n-FPT, which preserve only threeproperties, such a particle is represented through pivots locatedonly on the diagonal; the third off diagonal pivot does not receiveany fraction (Nandanwar and Kumar, 2008b). This is the majorcause of numerical dispersion with the techniques belonging tothe class represented by GEFPT.

The 1-d techniques based on M-II approximation considerparticles to be uniformly distributed in a bin. Although thesetechniques can preserve only one property of particles (Bleck,1970; Gelbard et al., 1980; Immanuel and Doyle, 2003) andtherefore lead to quite inaccurate solutions for coarse grids, it mayappear that these should not suffer from the numerical dispersionthat appears in GEFPT on account of splitting of particles topreserve four properties. To illustrate that the situation is nobetter (in fact worse than GEFPT), Fig. 13 shows a typical grid usedin geometric extensions of 1-d techniques. Rectangle 0 representsthe domain on which the initial population of bivariate particles ispresent. When these particles aggregate among themselves, the

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Fig. 12. Same as that for Fig. 11, and the initial condition considered for Case B. (a)

GEFPT, (b) n-FPT.

0.001

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k

1 10 100 1000k


0.001

0.01

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10

100

Nk


Fig. 11. Same as that for Fig. 9, and the initial condition considered for Case B.

(a) GEFPT, (b) n-FPT.

Table 5Same as that for Table 3, and the initial condition considered for Case B.

D t¼0.01 t¼2


D0,0 0.273 0.097 1.159 0.053

D0,1 0.220 0.099 1.145 0.054

D1,0 0.220 0.101 1.143 0.054

D1,1 0.139 0.131 1.071 0.077

Table 6Same as that for Table 4, for the initial condition considered for Case B.

Time D for rectangular D for radial D for Monte-Carlo

0 0.500 0.500 0.500

0.001 0.476 0.458 0.457

0.01 0.286 0.286 0.285

0.02 0.207 0.193 0.190

0.03 0.161 0.134 0.129

0.04 0.143 0.098 0.097

0.06 0.125 0.053 0.053

0.08 0.119 0.032 0.032

0.1 0.117 0.019 0.017

1 0.116 8.77E�05

2 0.116 3.86E�05

5 0.116 9.42E�06

40 0.116 1.55E�16


new particles are expected to belong to rectangle 1, and ifthese are further permitted to aggregate, the second generationof particles is expected to lie in rectangle 2. The M-IIapproximation (uniform number density in a bin) dispersesparticles formed in rectangle 1 to rectangle 10, much biggerthan the expected one. Aggregation of particles in 10 amongthemselves makes them appear in rectangle 2

0

, which likerectangle 1, covers some bins only partially. The application ofM-II approximation to represent these particles further dispersesthem to rectangle 2

00

, which is substantially bigger than the actualrectangle 2 to which the second generation of aggregatedparticles should have belonged. The numerical dispersion isquite large for a coarse grid. The results presented by Bleck(1970) and others (Kostoglou and Karabelas, 1994) corroborate itfor 1-d.

Thus, if one starts with particles located initially in a bin on themain diagonal such as the one shown in the figure, particles withlarge variation in composition, located far off the main diagonal,soon appear in the numerical solution. A similar analysis forGEFPT and n-FPT for the aggregation of particles of uniformcomposition reveals the superiority of the latter to both the GEFPTand 2-d extensions of M-II approach. In the limit of bin sizeapproaching zero, the predictions of all the above discussedtechniques can however be expected to match with the analyticalsolution, in all respects.

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0.01

0.1

1

1e-04 0.001 0.01

χ2

Time

20001000400

Fig. 14. Comparison for w2 for different initial population of particles for the

constant and the sum kernel. (a) Constant kernel, (b) sum kernel.

7 1812

712

18

26

35

47

63

84

x

y

0

1

109

109

1’

2’

2’’

2

3526 47 63 84

Fig. 13. Schematic showing numerical dispersion for extension of M-II approx-

imation based numerical techniques to solve 2-d PBEs.


3.4. Monte-Carlo simulations of aggregative mixing

Pure aggregation of particles for the cases considered inSection 3.1 leads to time invariant value of w2. We have arguedearlier that for a finite size system, as the particle populationapproaches unity, w2 must become zero. The prediction of timeinvariant value of w2 is thus seen as a consequence of the bivariatePBE being mean-field approximation of the stochastic evolution ina finite size system. The approximation must break down at somestage during the evolution due to the particle populationbecoming small. Matsoukas et al. (2006) carried out constantnumber Monte-Carlo simulation in which the number of particlesin the simulation box is kept constant. These simulations thusrepresent aggregative mixing in systems of infinite size. Expect-edly, the results of constant number Monte-Carlo simulationsvalidated the prediction of mean-field PBEs that w2 is timeinvariant.

In this section, we present results of simulations carried outon finite size systems of constant volume, in which the numberof particles decreases with time due to aggregation but thetotal mass of each species remains constant with time.The simulations were carried out using the Interval of Quiescencetechnique (Gillespie, 1976; Shah et al., 1977). The initialparticle distribution was given by Eq. (8). Simulationswere carried out for both the constant and the sum kernel. Alarge enough number of simulations were carried out in each caseto ensure that the results obtained are converged for a givensystem size.

Figs. 14(a) and (b) show variation of w2 with time for the twokernels for three initial populations of particles: 2000, 1000, and400. Both the figures show that the value of w2 initially remainsconstant and then begins to decrease to zero. The deviation fromthe constant value of w2 begins at a larger time for a larger sizesystem. This confirms the expectation that mean fieldapproximation (MFA) must breakdown as population ofparticles in the system decreases with time.

To further explore if the inability of MFA to capture thebehaviour of w2 is also accompanied by its inability to predict

variation of particle population with time, we plotted (Fig. 15)time variation of w2 and total population of particles. The verticalbars show the extent of run to run variation around the mean inthe quantity of interest (represented through the converged valueof the standard deviation). The figures show two interestingfeatures. First, the MFA predicts time variation of total populationquite well for particle populations decreasing from an initial valueof 1000 to populations as small as about 2 to 3. In comparison, thew2 value predicted by MFA begins to deviate from the expectedvalue when the number of aggregated particles reduces to 20 forthe constant kernel and about 120 for the sum kernel. The secondfeature shown by the simulations is that the two predictionsof w2 begin to deviate from each other when run-to-run variationbecomes larger than about 40% of the mean value. The reasonsfor the relationship between the increased variation amongsimulation results and the failure of the mean fieldapproximations to predict correct mean field results are underinvestigation.

To test if the breakdown of MFA shows a scaling with systemsize, t, the time of evolution for a system of size Vn is scaled withrespect to a reference system of size Vref using the followingrelation:

ts ¼ tVref

Vn

� �m

ð13Þ

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MC-SimulationAnalytical




0.001

0.01

0.1

1

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1e-0

4

0.00

1

χ2

Time

Fig. 15. Time variation of number of particles and w2 with error bars for the two kernels for an initial population of 1000 particles. (a) Constant kernel, (b) sum kernel.


where ts is the scaled time. The results obtained with this scalingare presented in Fig. 16. The figure shows that the scaling isalmost perfect for both the kernels. A value of exponent m¼0.88for the constant kernel and m¼0.15 for the sum kernel results inw2 vs. t curves for different system sizes to collapse on top of eachother. The exponent m¼0.88 for the constant kernel suggeststhat the reduced population of particles at which MFA starts tomake erroneous predictions of w2 scales with No, the initialpopulation of particles (same as system size) as N0

0.12. This israther weak dependence and suggests that irrespective of theinitial population of particles, the final population must decreaseto nearly the same value before MFA breaks down. Alternatively,an increase in system size increases the time of evolution overwhich MFA remains valid as No

0.88, almost a linear increase withrespect to system size.

In comparison, the exponent m¼0.15 for the sum kernelsuggests that the reduced population of particles at which MFAstarts to make erroneous predictions of w2 scales as No

0.85. This isquite a strong dependence, and suggests that a decrease in thenumber of particles to nearly a constant fraction of the initialpopulation is enough for the breakdown of MFA. This is alsoreflected by the quite small increase in time scale of evolutionover which MFA holds for increase in system size.

Fig. 17 shows the variation of w2 against the survivingpopulation of particles for the two kernels for initial populationof 1000 particles. The figure shows that for the same number ofaggregation events, resulting in same decrease in population ofparticles, deviation in w2 from the expected value is different forthe two kernels, and quite large for the sum kernel, in agreementwith the discussion presented above. We are currently exploringthe reasons for the different behaviour of the two kernels withrespect to aggregative mixing.

4. Conclusions

Extensions of 1-d discretization techniques to solve n-d PBEsare classified into two classes on the basis of the principle used indeveloping them. Two representative techniques, GEFPT andn-FPT, one for each class, are subjected in this work to newbenchmark tests designed around features specific to bivariatePBEs—kinetics of mixing under (i) pure aggregation and(ii) simultaneous break up and coalescence of drops. Comparisonsof numerical and standard results for a number of cases show thatfor same grid size, n-FPT, based on the framework of minimuminternal consistency of discretization and radial discretization ofspace, provides good predictions, while GEFPT is seen to leadto incorrect distribution of solute at steady state. This is largelyon account of the substantially reduced numerical dispersionassociated with preservation of three properties on triangularelements (natural elements for 2-d space) vs. preservation of fourproperties on rectangular elements, and the ability of radialdiscretization of space to densely populate 2-d region in whichpopulation evolves, without increasing the number of pivots. It isalso argued that the aggregation of particles of uniform composi-tion soon leads to formation of bigger particles of quite differentcompositions for geometrical extensions of M-II type 1-ddiscretization techniques.

Constant volume Monte-Carlo simulations for aggregativemixing show that mean field approximation, used in bivariatePBEs, breaks down for prediction of mixing characteristics fororders of magnitude larger population than that required for thebreakdown of MFA in predicting total population of particles.While the applicability of MFA for constant kernel holds tillparticle population reduces to nearly the same value irrespectiveof the initial population, the same for the sum kernel holds for

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1e-04

0.001

0.01

0.1

1

10

1 10 100

Constant kernelSum Kernel

χ2

Number of particles

Fig. 17. Variation of w2 with number of particles for the two kernel for an initial

population of 1000 particles.

0.01

0.1

1

1e-04 0.001 0.01 0.1 1

ptcl-2000ptcl-1000

ptcl-400

χ2

Time

0.01

0.1

1

1e-04 0.001 0.01

ptcl-2000ptcl-1000

ptcl-400

χ2

Time

Fig. 16. Variation of w2 with scaled time for different initial particle populations.

(a) Constant kernel, (b) sum kernel.


reduction in particle population equal to nearly a constantfraction of the initial population of particles. The evolution timeover which the MFA holds thus increases nearly linearly with

an increase in system size for the constant kernel and onlymarginally for the sum kernel. The latter suggests that theapplicability of bivariate PBEs for even somewhat large sizesystems and size dependent kernels should not be taken forgranted. Additional work is required to fully appreciate thedifferent nature of aggregation brought about by the two kernels.

Acknowledgements

Funding from the Department of Science & Technology, NewDelhi is gratefully acknowledged. The authors would like to thankDr. S. Venugopal for his many useful suggestions all through thework.

References

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on the solution and applicability of bivariate population balance equations for mixing in particle...

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