Chemical Engineering Science 70 (2012) 135–145

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

sanjeev

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

On the solution of bivariate population balance equations for aggregation:X–discretization of space for expansion and contraction of computationaldomain

Shivendra Singh Chauhan, Abhinandan Chiney, Sanjeev Kumar n

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:

Received 15 January 2011

Received in revised form

5 September 2011

Accepted 5 October 2011Available online 12 October 2011

Keywords:

Population balance

Particulate processes

Mixing

Agglomeration

Mathematical modelling

Discretization methods

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

016/j.ces.2011.10.005

esponding author. Tel.: þ91 80 2293 3110; fa

ail addresses: chesanjeev@gmail.com,

@chemeng.iisc.ernet.in (S. Kumar).

a b s t r a c t

A new structured discretization of 2D space, named X-discretization, is proposed to solve bivariate

population balance equations using the framework of minimal internal consistency of discretization of

Chakraborty and Kumar [2007, A new framework for solution of multidimensional population balance

equations. Chem. Eng. Sci. 62, 4112–4125] for breakup and aggregation of particles. The 2D space of

particle constituents (internal attributes) is discretized into bins by using arbitrarily spaced constant

composition radial lines and constant mass lines of slope �1. The quadrilaterals are triangulated by

using straight lines pointing towards the mean composition line. The monotonicity of the new

discretization makes is quite easy to implement, like a rectangular grid but with significantly reduced

numerical dispersion. We use the new discretization of space to automate the expansion and

contraction of the computational domain for the aggregation process, corresponding to the formation

of larger particles and the disappearance of smaller particles by adding and removing the constant mass

lines at the boundaries. The results show that the predictions of particle size distribution on fixed

X-grid are in better agreement with the analytical solution than those obtained with the earlier

techniques. The simulations carried out with expansion and/or contraction of the computational

domain as population evolves show that the proposed strategy of evolving the computational domain

with the aggregation process brings down the computational effort quite substantially; larger the

extent of evolution, greater is the reduction in computational effort.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Population balance modelling has emerged as a powerful and awidely used tool to capture the dynamics of systems with dispersedentities such as particles, drops, long chain molecules, cells, viruses,bubbles, etc. (Ramkrishna, 2000). It is used by biologists (Mantzaris,2006), chemists (Celnik et al., 2007), bio-chemists (Mantzaris andDaoutidis, 2004), and most extensively by engineers to model andoptimize processes such as liquid–liquid extraction (Attarakih et al.,2006), leaching (Crundwell and Bryson, 1992), polymerization(Immanuel and Doyle, 2003), particle formation in aerosol reactors(Peterson et al., 1978), crystallization (Costa et al., 2007), comminu-tion (Otwinowski, 2006), precipitation (Sheikh et al., 1996), formationof emulsions (Raikar et al., 2009), fluidization (Heinrich et al., 2002),synthesis of nanoparticles (Kumar et al., 2007), etc. In a number ofthese processes, the particles are identified uniquely by specifyingmore than one internal variable. In granulation, a particle is identified

ll rights reserved.

x: þ91 80 2360 8121.

with individual masses of binder and active materials in it. In liquid–liquid extraction, a drop is identified with its volume and concentra-tion of solutes in it. In sintering, a particle is identified with its volumeand surface area (Fox, 2006). Mixing of an active material with aninert in pharmaceutical sector (Matsoukas et al., 2009), evolution offractal aggregates (Kostoglou et al., 2006), anisotropic crystal growth(Briesen, 2006; Ma et al., 2007), multi-component reactions indispersed phase, etc. also belong to the same class of systems.

Although the particles invariably undergo more than one particu-late process simultaneously (from among breakup, aggregation,growth, dissolution, mass transfer, birth, death, etc.), the solution ofPBEs for aggregation of particles with two attributes (bivariate), thefocus of present work, is an important building block for thesimulation of the complete population balance models. A generalbivariate PBE for pure aggregation of particles is given by

@nðx1,x2,tÞ

@t¼

1

2

Z x1

0

Z x2

0nðx1�x01,x2�x02,tÞnðx01,x02,tÞ

Q ðx1�x01,x2�x02; x1,x2Þ dx01 dx02

�nðx1,x2,tÞ

Z 10

Z 10

nðx01,x02,tÞQ ðx1,x2; x01,x02Þ dx01 dx02 ð1Þ

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145136

where x1 and x2 are the masses of two species that constitute aparticle, nðx1,x2,tÞ is the number density, and Q ðx1,x2; x

01,x02Þ is the

frequency of aggregation of particles with attributes (x1,x2) and(x01,x02). The analytical solutions of the above bivariate PBE are farand few (Gelbard and Seinfield, 1978; Fernandez-Dıaz and Gomez-Garcıa, 2007). We need progress on two fronts before bivariate PBEsbegin to be used extensively like 1D PBEs in population balancemodels: availability of bivariate kernels, and efficient techniques foraccurate solution of bivariate PBEs, particularly in the context of on-line control (Immanuel and Doyle, 2003). The quadrature method ofmoments and the discretization of PBEs on pivots have emerged astwo efficient and widely used techniques. The first one provides themoments and the second offers the full particle size distribution. Thepresent work deals with the second approach.

The solution of PBEs by discretization methods is obtainedusing large enough computational domain to contain both theinitial and the final size distribution to avoid mass loss. The pivotsrepresenting large particles thus carry negligible particle popula-tion on them, till the advancing size distribution reaches them.This overhead, although manageable for the solution of 1D PBEs,increases uncontrollably both in terms of memory requirementand computational time for higher dimensional problems. Forexample, solution of a 2D PBEs, with N pivots for each axis of astructured grid has N2 pivots, and aggregation events of the orderof N4. Even for N¼100, it gives rise to more than 50 million typesof aggregation events!

Thus, there is a need to develop techniques that are sufficientlyaccurate with coarse grids and that can be used in new ways tokeep the number of pivots to a minimum. The present workproposes a new discretization of space to solve bivariate PBEs foraggregation (Eq. (1)) with the above objectives in view. The newdiscretization is easy to implement, and leads to more accurateresults. Most importantly, it opens up new methods for keepingthe number of pivots to minimum to reduce the computationaleffort without compromising on the accuracy of the solution.

The rest of the paper is organized as follows. Section 2discusses the relevant previous work, followed by the section onnew discretization of space. The proposed discretization is thentested for its accuracy on fixed grid. The next section presents asimple strategy to expand and contract the computational domainwith time. The same strategy is used in the final section tosimultaneously expand and contract the computational domainat its two boundaries to keep the number of pivots to a minimumto control the computer memory and processing requirements.

2. Previous work

Starting with Hidy’s (1965) uniform discretization of 1D spaceinto bins and representation of particles in bins by representativesizes called pivots, the discretization techniques to solve PBEs foraggregation have evolved extensively. A number of reviews coverthese developments in detail (Hounslow et al., 1988; Kumar andRamkrishna, 1996a; Vanni, 2000; Attarakih et al., 2004; Kumaret al., 2006; Chakraborty and Kumar, 2007; Chauhan et al., 2010).We refer an interested reader to them. In the present section, webriefly review the earlier work with focus on discretization ofspace to solve PBEs.

The uniform discretization of space to solve 1D PBEs is intuitiveand closer to the physical processes in some cases. As it requires alarge number of representative sizes, called pivots, to cover theentire size range with acceptable resolution, geometric discretiza-tion of space was developed (Batterham et al., 1981). The correctpreservation of population (numbers) and mass of newly formedparticles in a computationally efficient strategy was first achievedby Hounslow et al. (1988). Kumar and Ramkrishna (1996a)

proposed the concept of internal consistency of discretizationand developed a flexible method for solving PBEs for breakupand aggregation of particles. The new method also offers problemspecific discretization of 1D space, such as in the case of poly-merization–depolymerization that requires discrete population ofmonomer, dimer, trimers, etc. and the continuous populations oflarger mers. This technique has come to be widely known as thefixed pivot technique (FPT). A new particle, formed through breakupor aggregation processes, is represented through discrete popula-tions on adjoining pivots. The fractions assigned to pivots preservetwo pre-chosen particle properties, often taken to be number andmass. Kumar and Ramkrishna (1996b) and Kumar et al. (2006)have developed more accurate but slightly difficult to implementtechniques.

The early investigators (Kumar and Ramkrishna, 1995; Valeand McKenna, 2005; Alexopoulos and Kiparissides, 2007)extended the 1D FPT to solve 2D PBEs by treating the two axesidentically. A newly born bivariate particle is represented throughfour neighbouring pivots in rectangular grid. The solution of a n-dPBE thus requires a particle to be represented through 2n

neighbouring pivots, corresponding to preservation of 2n proper-ties of particles. These geometric extensions, named GEFPT, areimplemented easily, but produce somewhat inaccurate predic-tions. For examples, mixing in dispersed phase for simultaneousbreakup and coalescence of drops is predicted to proceed extre-mely slowly at long times (Alexopoulos and Kiparissides, 2007),which is later shown (Chauhan et al., 2010) to be due to the highnumerical dispersion associated with GEFPT.

Chakraborty and Kumar (2007) argued that an n-d PBEdescribes the evolution of particles with n internal attributes.An internally consistent discretization requires preservation of n

particle attributes and its population (number). A new particlethus needs to be represented through only (nþ1) instead of 2n

neighbouring pivots in n-d space. The space filling naturalelements for n-d space also require only (nþ1) vertices, e.g.,triangles with three vertices in 2D space and tetragons with fourvertices in 3D space. The set of discretized equations, nearlyidentical in appearance to those obtained for the 1D FPT, are

dNi

dt¼�Ni

Xj

Q ijNjþXjZk

j,k

1�1

2djk

� �Zi

jkQ jkNjNk ð2Þ

Here, Ni is the particle population in ith bin and Qjk is thefrequency of aggregation for particles in jth and kth bins. Theterm Zi

jk represents the fractional particles assigned to particlepopulation on ith pivot for aggregation of particles at jth and kthpivots. The non-zero values of Zi

jk are computed for a given gridand stored along with the values of indices i, j, and k in Z-matrixfor re-use. Considering that aggregation of particles at pivots j andk produces a particle surrounded by pivots r, s, and t, then Zr

jk, Zsjk,

and Ztjk, equal to the values of a, b, c are obtained by solving the

following simultaneous equations:

aþbþc¼ 1

ax1rþbx1sþcx1t ¼ x1jþx1k

ax2rþbx2sþcx2t ¼ x2jþx2k

The framework of minimal internal consistency has beencombined with various ways to discretize 2D computationaldomain. Chakraborty and Kumar (2007) bisected the rectanglesused with GEFPT along the diagonals to triangulate the 2Ddomain. Although a better solution was obtained by merelychanging the connectivity among the pivots, this grid lacked theoption of its selective refinement in a region of 2D space. Theauthors also used a commercial mesh generation software toproduce space filling, randomly oriented triangles, selectively

pivotsmean composition

x1

x2

Fig. 1. The radial grid.Fig. 2. The X-grid.

bisectors

pivots

x1

x2

t and r lines

Fig. 3. A triangulated X-grid.

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145 137

refined in a region of computational space. Complex algorithmshad to be used to locate a new particle in randomly oriented andspace filling collection of triangles and tetragons.

Nandanwar and Kumar (2008a) proposed new radial discreti-zation of 2D space to overcome the shortcomings associated withrandomly oriented triangles. The 2D space was discretized usingcircular arcs separated linearly on a logarithmic scale, and radiallines (constant composition lines) from the origin, either equallyspaced or selectively refined. The resulting quadrilaterals arebisected to obtain triangles to use with the framework of minimalinternal consistency. The authors showed that the numericaldispersion is significantly less when the bisecting lines pointtowards the radial line representing the mean composition ofparticles, rather than away from it. A schematic of radial dis-cretization of space is presented in Fig. 1. The efficacy of n-d FPTon radial grid to capture breakup and aggregation of bivariateparticles is established in the literature (Nandanwar and Kumar,2008a,b). Chauhan et al. (2010) used the same approach to studydynamics of mixing brought about by simultaneous aggregationand breakup of drops and showed that complete mixing occurs infinite time. The authors also studied the dynamics of aggregativemixing and found that n-d FPT on radial grid captures mixing inparticle phase in good agreement with the analytical results. Theradial grid must be refined if the predictions are to remain in goodagreement with the expected results for large extents of evolution.

3. X-discretization of 2D space

We propose in this work a new discretization space as shownin Fig. 2. The 2D space is discretized using lines of slope �1, calledt-lines, and radial lines of different slopes passing through theorigin, called r-lines. If particles are identified with masses ofindividual species in them, the t-lines represent constant masslines and r-lines represent constant composition lines. A pivot isassigned a region in 2D space around it, obtained by intersectionsof lines mid-way between t-lines and r-lines (shown by dottedlines) for discretization of initial condition. This grid is namedX-grid as every intersection point resembles the letter X.

The quadrilaterals formed by the intersection of t and r linesare divided into triangles by using lines that point to the meancomposition line (thick line) to use the framework of minimalinternal consistency. A triangulated X-discretization of space isshown in Fig. 3.

The X-grid retains all the advantages of the radial grid. Whenparticles on the same radial line aggregate, a new particle isformed on the same radial line and is assigned to pivots locatedon it, with no numerical dispersion in the perpendicular direction,a shortcoming that is the source of high numerical dispersionwith GEFPTs. Similarly, when a homogeneous particle breaks, thedaughter particles lie on the same radial line, thereby convertingthe 2D breakup problem into a series of 1D problems. The X-gridcan be selectively refined by introducing more radial lines in theregion of interest.

Similar to the procedure followed by Nandanwar and Kumar(2008a) for the radial grid, the non-pivot particles formed onX-grid are represented through the three vertices of the trianglethey belong to. The fractions are obtained by solving the equa-tions that preserve number, and the two constituent masses ofparticle, as originally outlined by Chakraborty and Kumar (2007).The preservation of these three quantities ensures preservationof concentrations of the constituent species as well when theparticles formed lie on t-lines.

Although the radial discretization of space also leads to astructured grid, the presence of the circular arcs leads to non-linear equations that need to be solved to determine the locationof pivots. The procedure followed for the X-grid is quite simpledue to the linear nature of the governing expressions. The most

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145138

important advantage of X-grid over radial grid however lies indetermination of elements of Z matrix (Eq. (2)). The rows of Zmatrix contain information on aggregating pair of particles andtheir contribution to the evolution of particle population. The firstelement of a row contains index i of the pivot that gets acontribution when particles at pivots j (second element) and k

(third element) aggregate and contribute fraction Zijk (fourth

element) to pivot i. We next explain how Z matrix is generated.The aggregation of particles located on different radial lines of

a X-grid produces a new particle in the region bound by the tworadial lines, beyond both the t lines on which the aggregatingparticles are located. This feature does not hold if the t-lines arereplaced by the arcs used in the radial discretization of space. Inview of the monotonicity of this behaviour, an efficient method togenerate Z matrix starts with sequential indexing of M pivotsalong t-lines, starting at the smallest particle on one of the axes.We next consider j¼ 1, . . . ,M and allow k to vary from 1 to j, anddetermine for each k index i of the pivot to which the aggregationof particles at pivots j and k contributes. As k increases for fixed j,index i increases monotonically. If aggregation of particle at pivotj with a particle at pivot kmax produces a new particle that liesoutside the computational domain, the monotonicity of the X-gridguarantees that the new particles formed by aggregation ofparticles at pivots j and k, for all the values of k from kmax to j,will also lie outside the computational domain. The value of kmax

increases monotonically with increase in j. The monotonicity ofvarious relationships for X-grid thus allows Z matrix to begenerated very efficiently and progressively as index j increasesfrom 1 to M. It also allows the matrix to be easily augmented orpruned for the addition or removal of t-lines at the boundaries ofthe domain. The lack of similar monotonic relationships for theradial grid due to the non-constancy of particle mass on an arcrequires extensive testing of all possible combinations for aggre-gation events. Furthermore, addition or removal of arcs at theboundaries of the computational domain is best handled byrecomputing the Z matrix from scratch, which can be quiteintensive computationally. These considerations explain whythe radial discretization of space has not been used for expan-sion/contraction of computational domain, demonstrated in detailin a later section.

4. Results

The following initial condition was used in the investigationsreported in this section:

nðx1,x2, . . . : :,xm0Þ ¼N0

Ymi ¼ 1

1

xi0exp�xi

xi0

� �ð3Þ

with N0 ¼ 1000, m¼2, and x10 ¼ x20 ¼ 1. The above expression isintegrated over each bin (designated by the shaded regionmarked Aeff in Fig. 2) to obtain Nið0Þ. The total particle populationN(t) at time t is obtained from

NðtÞ ¼XMi ¼ 1

NiðtÞ ð4Þ

The results presented in this section are obtained by solvingthe discretized PBEs obtained using (i) GEFPT (Kumar andRamkrishna, 1995; Vale and McKenna, 2005; Alexopoulos andKiparissides, 2007) with preservation of four properties (number,two masses, and the product of two masses) on rectangularelements, (ii) n-d FPT with radial grid of Nandanwar and Kumar(2008a), and (iii) n-d FPT with the newly proposed X-grid. The lasttwo are based on the preservation of number and two masses ontriangular elements. Henceforth, the solution obtained using the

rectangular, and the radial and X-grids will be understood to beobtained using GEFPT, n-d FPT respectively.

It is difficult to visually compare 3D plots of bivariate particlepopulations. One can compare particle populations along certaint-lines (Vale and McKenna, 2005). The data on the left out t-lines,which are large in number, is not compared. We have thereforeused 2D flat representation method (Chakraborty and Kumar,2007) in this work. The particle population on pivots for one ofthe solutions, typically the analytical solution, is sorted indecreasing order, and the pivots are indexed. The pivot with thesmallest index has the highest population, and so on. The pivotindex is mapped to the location of pivot in 2D space. The particlepopulations obtained by using the two methods are next plottedagainst this index (denoted by k). A single 2D plot thus provides acomparison of the two solutions. The quantitative differencesbetween the two bivariate solutions are quantified using Di,j, ameasure of error used earlier in the literature (Nandanwar andKumar, 2008a), and defined below:

Di,j ¼

Pmp ¼ 1 9Nana�Nnum9xi

1,pxj2,pPm

p ¼ 1 Nanaxi1,pxj

2,p

ð5Þ

D0;0 is a measure of the accuracy of the technique and representserror in distribution of particle population over the entire range;D1;0 and D0;1 represent error in distribution of x1 and x2 massesrespectively; D1;1 amplifies error in the advancing front region. Allthe simulations reported here were carried out using a 1.66 GHzIntel(R)-Core(TM)2 Duo CPU Processor T5450 with 1.89 GB Ran-dom Access Memory (RAM) and 2048 Kb cache size. The CPU timerefers to the time dedicated by the CPU to carry out calculations,and does not include the time taken to read/write files.

4.1. Constant kernel

The analytical solution for the constant kernel for the initialcondition earlier is given by (Gelbard and Seinfield, 1978)

nðx1,x2, . . . : :,xm,tÞ ¼ 4N0

ðtþ2Þ2

Ymi

exp�xi

xi0

� �xi0

X1k ¼ 0

ttþ2

Qmj ¼ 1

xj

xj0

� �k

ðk!Þm

ð6Þ

where non-dimensional time t is defined as t¼ b� N0 � t.Fig. 4(a) shows a flat comparison of the results obtained withGEFPT using a rectangular grid with 625 (25�25) pivots att¼0.01, corresponding to NðtÞ=Nð0Þ ¼ 0:17. The similar compar-isons for the radial and the X-grid are presented in Fig. 4(b) and(c) for identical grid sizes. The relative errors represented throughthe D variables are presented in Table 1. Fig. 5 and Table 2 showthe same comparisons for t¼0.1 (NðtÞ=Nð0Þ ¼ 0:019). The resultspresented in the figures show that the radial and the X-gridproduce quite accurate solution as compared to the GEFPT. Therelative errors shown in Tables 1 and 2 also confirm it. The tablesalso show that the X-grid performs better than the radial grid forthe same number of grid points; the error represented by D1;1 issignificantly smaller for the X-grid in comparison with thatobtained with the radial grid.

Eq. (3) dictates that the particle population evolves along thecentral diagonal of the grid, as it represents the mean concentra-tion of particles. The X-grid was selectively refined aroundthe main diagonal, while keeping the number of pivots thesame (625) as for the unrefined grid. The results are comparedin Fig. 6 and Table 3 for refined and unrefined grids for t¼0.1(NðtÞ=Nð0Þ ¼ 0:019). The results clearly establish that the refine-ment of X-grid is a powerful tool to increase the accuracy of thesolution.

Table 1D values for aggregation at time t¼0.01 (NðtÞ=Nð0Þ ¼ 0:17) for the constant kernel

obtained by GEFPT, the unrefined radial grid, and the unrefined X-grid for the

same number of pivots.

D GEFPT Unrefined

radial grid

Unrefined

X-grid

D0;0 0.1682 0.0619 0.0634

D0;1 0.3143 0.1018 0.0976

D1;0 0.3140 0.1018 0.0956

D1;1 0.4299 0.1931 0.1720

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 4. A comparison of the analytical and the numerical solution for particle

population at pivots at t¼0.01 (NðtÞ=Nð0Þ ¼ 0:17) for the constant kernel with

GEFPT, and n-d FPT with unrefined radial and X-grids. Both the grids used 625

pivots and a geometric ratio of 1.63. (a) GEFPT, (b) unrefined radial grid and

(c) unrefined X-grid.

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145 139

4.2. Sum kernel

The analytical solution of Eq. (1) for the sum kernel, Q ðx1,x2;x01,x02Þ ¼ b½ðx1þx2Þþðx

01þx02Þ�, for the initial condition given by Eq. (3)

is given by (Fernandez-Dıaz and Gomez-Garcıa, 2007)

nðx1,x2, . . . : :,xm,tÞ ¼N0ð1�tÞ exp �x

x0t

� ��

Ymi ¼ 1

1

xi0exp�

xi

xi0

� � !

�X1k ¼ 0

1

ðkþ1Þ!

tx

x0

� �k

�Ymi ¼ 1

ðxi=xi0Þk

Gðkþ1Þð7Þ

where t¼ 1�expð�bftÞ, x¼Pm

i xi and x0 ¼Pm

i xi0, and f is thetotal particle mass in the system,

Fig. 7(a) shows a flat comparison of the results obtainedby using the GEFPT (25�25) with the analytical solution attime t¼ 1� 10�3 (NðtÞ=Nð0Þ ¼ 0:14). Fig. 7(b) and (c) shows thesame comparisons for the unrefined radial and X grids. Therelative errors represented through the D variables are shown inTable 4. Fig. 8 and Table 5 show the same comparisons at t¼ 2�10�3 (NðtÞ=Nð0Þ ¼ 0:023).

The X-grid was selectively refined around the main diagonalfor this case as well while keeping the total number of gridpoints unchanged at 625. The results obtained for t¼ 2� 10�3

(NðtÞ=Nð0Þ ¼ 0:023) are presented in Fig. 9 and Table 6. As notedearlier with the constant kernel case, the accuracy of the solution issignificantly improved by introducing more radial lines around themain diagonal while making the radial grid coarse near the axes.

5. Expanding grid

The results presented thus far, including others in the litera-ture for 2D PBEs, are obtained with a fixed grid. Since thecomputational domain that can contain the final size distributionis not known a priory, a large enough fixed domain is taken toensure that mass is not lost through the formation of particleslying outside the solution domain. The particles initially populateonly the small particle size region of the grid. As aggregationcontinues and size distribution expands, pivots in the largeparticle size region also get populated. These pivots do notcontribute to evolution process in the beginning though, butsince the equations for them are also solved at every step, theycontribute to the computational cost. If the grid size is over-estimated to prevent mass loss, avoidable computational burdenis carried all through the simulation. The presence of pivots,dormant all through or part of the evolution process, makes thefixed grid techniques computationally quite demanding for higherdimensional PBEs.

As mentioned earlier, similar issues arise in the context of 1DPBEs as well. Although the additional computational effortrequired is not substantial, some attempts are reported in theliterature to address these issues. A conservative choice of solu-tion domain to reduce the number of equations leads to errors,named as finite domain error in the literature. Attarakih et al.(2003) have used the moving pivot technique (Kumar andRamkrishna, 1996b) for 1D particle breakup in batch mode, anddeveloped an optimal moving grid with fixed number of gridpoints to minimize finite domain error. Azizi and Al Taweel(2010) have solved continuous PBE for breakup and coalescenceof drops for number density at grid points using cubic splineinterpolation. The negative and unstable solutions obtained using

Table 2D values for aggregation at time t¼0.1 (NðtÞ=Nð0Þ ¼ 0:019) for the constant kernel

for GEFPT, the unrefined radial grid, and the unrefined X-grid for the same number

of pivots.

D GEFPT Unrefined

radial grid

Unrefined

X-grid

D0;0 0.5910 0.1387 0.1345

D0;1 0.8117 0.1920 0.1815

D1;0 0.8124 0.1920 0.1804

D1;1 0.8954 0.2716 0.2398

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 6. A comparison of the analytical and the numerical solution for particle

population at pivots at t¼0.1 (NðtÞ=Nð0Þ ¼ 0:019), for the constant kernel with the

X-grid selectively refined along the main diagonal of the grid for the same number

of pivots as those used for Fig. 5(c).

Table 3D values for aggregation at time t¼0.1 (NðtÞ=Nð0Þ ¼ 0:019) for the

constant kernel for the unrefined X-grid and the selectively refined

X-grid for the same number of pivots.

D Unrefined X-grid Refined X-grid

D0;0 0.1345 0.0971

D0;1 0.1815 0.1399

D1;0 0.1804 0.1390

D1;1 0.2398 0.2056

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 5. A comparison of the analytical and the numerical solution for particle

population at pivots at t¼0.1 (NðtÞ=Nð0Þ ¼ 0:019) for the constant kernel with

GEFPT, and n-d FPT with unrefined radial and X-grids. Both the grids used 625

pivots and a geometric ratio of 1.63. (a) GEFPT, (b) unrefined radial grid and

(c) unrefined X-grid.

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145140

this technique were stabilized using additional measures, such astracking rates of all the individual particulate processes at eachgrid point, and then forcing the rates of these individual processesto approach zero asymptotically from a critical size, where aninstability is expected, to the largest particle size.

A simple expanding grid based on X-discretization of space isproposed here to alleviate the computational inefficiency causedby the dormant pivots. The simulations are started on a smallgrid, large enough to accommodate the initial population. As thepopulation evolves with time and attempts to move into newerdomains of large size particles, the grid is expanded by adding onenew t-line (constant mass line) at a time as shown in Fig. 10. Thistechnique is named t-expansion technique.

The implementation of the proposed strategy is as follows. Thepopulation on pivots located on the last t-line is checked at everytime-step (monitored by the adaptive Runge–Kutta routine, Presset al., 1998). If population on any one of the pivots exceeds a pre-specified value, it indicates marching of the advancing sizedistribution to the end of the computational domain. If thedomain is not expanded, further aggregation of particles resultsin mass loss through formation of particles outside the solutiondomain. A new t-line of pivots is therefore added to expand thecomputational domain. The Z matrix needs to be updated. Thereare two ways to proceed at this stage: (i) recompute the Z matrixfor the current grid, or (ii) augment the existing Z matrix for the

1e-05

1e-04

0.001

0.01

0.1

1

10

100

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 7. A comparison of the analytical and the numerical solution for particle

population at pivots at t ¼ 1� 10�3 (NðtÞ=Nð0Þ ¼ 0:14) for the sum kernel with

GEFPT, the unrefined radial grid, and the unrefined X-grid. (a) GEFPT, (b) unrefined

radial grid and (c) unrefined X-grid.

Table 4

D values for aggregation at time t¼ 1� 10�3 (NðtÞ=Nð0Þ ¼ 0:14) for the sum kernel

obtained by GEFPT, the unrefined radial grid, and the unrefined X-grid.

D GEFPT Unrefined

radial grid

Unrefined

X-grid

D0;0 0.1473 0.0728 0.0814

D0;1 0.6454 0.3006 0.2799

D1;0 0.6410 0.3006 0.2782

D1;1 0.9768 0.6397 0.4772

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145 141

addition of new pivots. While the Z matrix for the radial grid ispreferably updated with the first option due to its limitations(discussed in an earlier section), the X-gird, due to its simple andmonotonic structure, permits the second option as well. We haveused the second option here. The computations are carriedforward in this manner till the target time at which the sizedistribution desired is reached.

We now compare the size distributions predicted with thefixed and the expanding X-grid. The starting grid for latter had 27uniformly spaced r-lines and 15 t-lines with a geometric ratio of1.63. The new t-lines were added when the population at any oneof the pivots on the last t-line became more than 10�15 times ofthe total initial population. These results were compared with theresults obtained using fixed grid of the size that was reached atthe final time for the t-expanding grid.

5.1. Constant kernel

The focus of this section is to show how well an expanding gridcan capture results obtained with a fixed grid. Fig. 11 shows a flatcomparison of the two results for the constant kernel at twoextents of evolution. The figure shows that the results areidentical both at short and large extents of evolutions. Thecomputational effort required is presented in Table 7. The tableshows that the saving achieved with an expanding grid continuesto increase as the simulations are carried out for larger extents ofevolution. It is further interesting to note that even for thesmallest extent of evolution, with addition of only seven new t-lines, the cost incurred in checking the population and augment-ing Z matrix is more than compensated by the savings effected byreducing the number of equations that are solved. As the gridused is made finer, perhaps to obtain grid independent results orotherwise, the absolute values of CPU times and the savingachieved with the expanding increase proportionally.

5.2. Sum kernel

Fig. 12 shows a comparison similar to that shown in Fig. 11 forthe size dependent sum kernel. The t-expansion of grid leads topredictions identical to those obtained with the fixed grid. Table 8shows a comparison of the computational effort required for thesum kernel. A large increase in CPU time when the constantkernel is replaced by the sum kernel for comparable extents ofevolution is reported in the literature earlier as well (Kumar andRamkrishna, 1996b; Kumar et al., 2008). The reason for thisbehaviour appears to be the qualitatively different nature of thesolution in the tail region for the sum kernel with a power lawdecrease in number density over a particle size range (Kumar andRamkrishna, 1996b), which makes the system of equations stiff. Itis clear from the table that the saving effected by t-expansion ofgrid for this case is even greater.

The proposed t-expanding X-grid thus produces solutionsidentical to those obtained with the fixed X-grid at substantiallyreduced computational effort.

6. Contracting grid

Just as the pivots corresponding to large size particles do notget populated till the advancing solution reaches them, the pivotspopulated initially may also lose most of their population in finitetime and stop contributing to the evolution of size distribution.

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 8. A comparison of the analytical and the numerical solution for particle

population at pivots at t¼ 2� 10�3 (NðtÞ=Nð0Þ ¼ 0:023) for the sum kernel with

GEFPT, the unrefined radial grid, and the unrefined X-grid. (a) GEFPT, (b) unrefined

radial grid and (c) unrefined X-grid.

Table 5

D values for aggregation at time t¼ 2� 10�3 (NðtÞ=Nð0Þ ¼ 0:023) for the sum

kernel for GEFPT, the unrefined radial grid, and the unrefined X-grid.

D GEFPT Unrefined

radial grid

Unrefined

X-grid

D0;0 0.1934 0.0864 0.0836

D0;1 0.8667 0.5997 0.5274

D1;0 0.8640 0.5997 0.5268

D1;1 1.1681 0.8553 0.7715

1e-05

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000

Nk

k

AnalyticalNumerical

Fig. 9. A comparison of the analytical and the numerical solution for particle

population at pivots at t ¼ 2� 10�3 (NðtÞ ¼Nð0Þ ¼ 0:023) for the sum kernel with

the X-grid selectively refined along the main diagonal of the grid.

Table 6

D values for aggregation at time t ¼ 2� 10�3 (NðtÞ=Nð0Þ ¼ 0:023)

for the sum kernel for the unrefined and the selectively refined

X-grids for the same number of pivots.

D Unrefined X-grid Refined X-grid

D0;0 0.0836 0.0703

D0;1 0.5274 0.3002

D1;0 0.5268 0.2998

D1;1 0.7715 0.4630

Site for violation

Existing t−line

Newly added t−line

Newly added pivot

x2

x1

Fig. 10. The t-expansion of X-grid.

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145142

The computational efficiency can be increased by removing thesepivots and contracting the computational domain. We test thisidea by removing an entire t-line in small particle size range fromthe simulation if the particle population and particle mass carriedby all the pivots on it became smaller than the correspondingthreshold values. These values were taken to be 10�6 and 10�4

times of the initial population and the total mass respectively.The threshold values for expansion and contraction were deter-mined by using a trial and error approach for zero mass loss forexpansion and less than 0.01% mass loss for contraction. The same

1e-10

1e-08

1e-06

1e-04

0.01

1

100

1 10 100 1000

Nk

k

Fixed GridExpanding Grid

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

100

1 10 100 1000

Nk

k

Fixed GridExpanding Grid

Fig. 11. A comparison of the particle population on pivots for the fixed X-grid and

the t-expansion of X-grid for different extents of evolution for the constant kernel.

(a) t¼0.01 (NðtÞ=Nð0Þ ¼ 0:17) and (b) t¼10.0 (NðtÞ=Nð0Þ ¼ 1:9� 10�4).

Table 7A comparison of the computation times for the fixed X-grid and the t-expansion of

X-grid for different extents of evolution for the constant kernel.

Evolution

time

(NðtÞ=Nð0Þ) Fixed grid

CPU time (s)

Expanding

grid CPU

time (s)

Final

number of

t-lines

0.01 0.17 7 5 22

0.1 1:9� 10�2 16 10 26

1.0 1:9� 10�3 32 15 30

10.0 1:9� 10�4 48 21 33

1e-15

1e-10

1e-05

1

100000

1 10 100 1000

Nk

k

Fixed GridExpanding Grid

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

100

10000

1 10 100 1000

Nk

k

Fixed GridExpanding grid

Fig. 12. A comparison of particle population on pivots for the fixed X-grid and t-

expansion of X-grid for different extents of evolution for the sum kernel. (a)

t¼0.001 (NðtÞ=Nð0Þ ¼ 0:14) and (b) t¼0.002 (NðtÞ=Nð0Þ ¼ 0:035).

Table 8A comparison of the computation times for the fixed X-grid and the t-expansion of

X-grid for different extents of evolution for the sum kernel.

Evolution

time

(NðtÞ=Nð0Þ) Fixed grid

CPU time (s)

Expanding

grid CPU

time (s)

Final

number

of t-lines

1� 10�4 0.83 6 5 21

5� 10�4 0.38 301 60 27

1� 10�3 0.14 4052 453 31

2� 10�3 0.023 61 082 4962 38

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145 143

threshold values were then used for other kernels. Given themonotonicity of X-grid, and the exponential initial condition(Eq. (3)), a t-line can be removed from the simulations by settingparticle population of pivots on it to zero. The simulations werecarried out for two kernels: the constant and the Brownian kernel.The initial population was discretized on computational domainlarge enough to test the effect of contraction of computationaldomain alone.

Figs. 13 and 14 show a comparison of the particle sizedistribution along the main diagonal (14th out of a total of 27radial lines) obtained with and without the contraction of thecomputational domain for the constant and the Brownian kernelrespectively. Additional comparisons between the two approaches

for the two kernels are presented in Table 9. The figures show thatover a 500-fold reduction in particle population, 162 pivots for theconstant kernel and 243 pivots for the Brownian kernel (out of 756initial pivots) become dormant to the extent that their removaldoes not affect evolution at larger particle sizes. The number ofpivots removed for the Brownian kernel is far more those for theconstant kernel because the small particles aggregate rapidly withlarge particles for the Brownian kernel. Table 9 shows thatremoval of t-lines leads to slight mass loss from the system,compared to the expansion of domain which conserves massnearly perfectly. The mass loss from the system depends on the‘minimum’ mass that a t-line can have and the number of t-linesremoved. The mass loss for the cases demonstrated is less than0.01% and is not expected to influence the evolution of the rest ofthe population, also shown by Figs. 13 and 14.

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145144

7. Simultaneous expansion and contraction of X-grid

Addition and removal of pivots in the form of a whole t-line toresize the computational domain suggests that the number ofpivots can be kept to a minimum when the domain is permittedto simultaneously expand or contract, as demanded by theevolving solution. Since this process can keep the number of

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

100

0 5 10 15 20 25 30

Pop

ulat

ion

t-line

Contracting gridFixed Grid

Fig. 13. A comparison of particle populations on pivots located on the main

diagonal line (14th radial line) for the fixed X-grid and the contracting X-grid at

t¼1.0 (NðtÞ=Nð0Þ ¼ 1:9� 10�3) for the constant kernel.

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

100

0 5 10 15 20 25 30

Pop

ulat

ion

t-line number

Contracting GridFixed Grid

Fig. 14. A comparison of particle populations on pivots located on the main

diagonal line (14th radial line) for the fixed X-grid and the contracting X-grid at

t¼1.0 (NðtÞ=Nð0Þ ¼ 1:8� 10�3) for the Brownian kernel.

Table 9A comparison of parameters related to simulations carried out using th

kernel. The initial grid for each simulation consisted of 28 t-lines with a

were carried out for t¼1.0 for both the kernels.

Parameter Constant kernel

Fixed Co

Final number of pivots 756 59

t-Lines removed – 6

Eta matrix size 797,121 48

N(t) 1.9 1.9

M10ð1:0Þ=M10ð0Þ 1.0 0.9

M01ð1:0Þ=M01ð0Þ 1.0 0.9

equations to minimum possible, the small size of Z matrix shouldallow simulations to be carried out for longer evolution time withfinite processing power and random access memory. (The laterlimits the maximum number of equations that can be solved.)

We present here results for the case reported in the previoussection for the Brownian kernel, but with simultaneous expansionand contraction of the computational domain. The initial gridcontained 15 t-lines at a geometric ratio of 1.7 and 27 uniformlyspaced radial lines. The solution obtained was identical to thatobtained with the fixed grid, except that the latter had negligibleparticle population at pivots which were not present in simulta-neously expanding and contracting grid. While the fixed gridsimulation starts and ends with 729 pivots, the simulations withcomputational domain resized with the aggregation processstarted with 405 pivots and ended with 486 pivots. During thisevolution, 324 new pivots (12 new t-lines) were added and 243pivots (9 t-lines) were removed.

8. Conclusion

A new discretization of space, named X-discretization, isdeveloped in this work to solve bivariate population balanceequations for aggregation of particles by discretization methods.The 2D space is discretized by using lines of constant composition(r-lines) and constant particle mass (t-lines). The new structureddiscretization of space combined with the framework of mini-mum internal consistency of discretization is easy to implementand computationally efficient. It also produces more accuratesolution than those obtained with the other reported discretiza-tions in the literature, as demonstrated through a number ofsimulation studies.

The new X-discretization facilitates development of a newclass of techniques in which the computational domain is alsoevolved by adding new pivots and removing dormant pivots. Asimple and easy to implement strategy is developed to resize thecomputational domain—addition or removal of a whole t-line ofpivots at the boundaries of the computational domain. A new t-line is added when the population at the boundary begins to rise,and a whole t-line is removed from the other boundary when theparticle population and mass contained on it become negligiblysmall. The simulation of bivariate aggregation for the Browniankernel shows that even for modest evolution, a simulation can bestarted with 15 t-lines and continued with addition of 12 andremoval of 9 t-lines. A fixed grid simulation requires a minimumof 27 t-lines. The addition of more than one t-line at a time forrapid aggregation of particles, such as for gelling kernels, is aninteresting possibility.

Thus, the resizing of the computational domain as the sizedistribution evolves can possibly keep the computation require-ments from blowing up. The strategies developed here are quite

e fixed and contracting X-grids for the constant and the Brownian

geometric ratio of 1.7 and uniformly spaced 27 r-lines. Simulations

Brownian Kernel

ntracting Fixed Contracting

4 756 513

– 9

1,950 797,121 353,889

1.8 1.8

999766 1.0 0.9999164

999768 1.0 0.9999164

S.S. Chauhan et al. / Chemical Engineering Science 70 (2012) 135–145 145

simple and effective. Another, more complex technique thatallows pivot-wise expansion of the computational domain (addi-tion of one pivot at a time) is the focus of part II of this work.

As pointed out by one of the anonymous reviewers, represen-tation of a continuous number density through discrete popula-tions and representation of a particle through pivots adjoining itis in use in another field as well: simulation of problems inplasma physics (Birdsall and Langdon, 2004). There are howeveressential differences between the two fields. Calculation of forceon a particle by using Maxwell’s equations, in place of pairwiseinteractions among particles, permit the order of computation forplasma physics related problems to be reduced from N2 to N,where N is the total number of particles. Whether such areduction is possible with regard to the solution of PBEs foraggregation is to be investigated.

References

Alexopoulos, A.H., Kiparissides, C., 2007. Solution of the bivariate dynamicpopulation balance equation in batch particulate systems: combined aggrega-tion and breakage. Chem. Eng. Sci. 62, 5048–5053.

Attarakih, M., Bart, H., Faqir, N., 2003. Optimal moving and fixed grids for thesolution of discretized population balances in batch and continuous systems:droplet breakage. Chem. Eng. Sci. 58, 1251–1269.

Attarakih, M., Bart, H., Faqir, N., 2004. Numerical solution of the spatiallydistributed population balance equation describing the hydrodynamics ofinteracting liquid–liquid dispersions. Chem. Eng. Sci. 59, 2567–2592.

Attarakih, M.M., Bart, H.J., Faqir, N.M., 2006. Numerical solution of the bivariatepopulation balance equation for the interacting hydrodynamics and masstransfer in liquid–liquid extraction columns. Chem. Eng. Sci. 61, 113–123.

Azizi, F., Al Taweel, A., 2010. Algorithm for the accurate numerical solution of PBEfor drop breakup and coalescence under high shear rates. Chem. Eng. Sci. 65,6112–6127.

Batterham, R.J., Hall, J.S., Barton, G., 1981. Pelletizing kinetics and simulation of fullscale balling circuits. In: Proceedings of the 3rd International Symposium onAgglomeration, pp. A136.

Birdsall, C., Langdon, A., 2004. Plasma Physics via Computer Simulation. Instituteof Physics Publishing, Bristol, UK.

Briesen, H., 2006. Simulation of crystal size and shape by means of a reduced two-dimensional population balance model. Chem. Eng. Sci. 61, 104–112.

Celnik, M., Patterson, R., Kraft, M., Wagner, W., 2007. Coupling a stochastic sootpopulation balance to gas-phase chemistry using operator splitting. Combust.Flame 148, 158–176.

Chakraborty, J., Kumar, S., 2007. A new framework for solution of multidimen-sional population balance equations. Chem. Eng. Sci. 62, 4112–4125.

Chauhan, S.S., Chakraborty, J., Kumar, S., 2010. On the solution and applicability ofbivariate population balance equations for mixing in particle phase. Chem.Eng. Sci. 65, 3914–3927.

Costa, C.B.B., Maciel, M.R.W., Filho, R.M., 2007. Considerations on the crystal-lization modeling: population balance solution. Comput. Chem. Eng. 31,206–218.

Crundwell, F.K., Bryson, A.W., 1992. The modelling of particulate leachingreactors—the population balance approach. Hydrometallurgy 29, 275–295.

Fernandez-Dıaz, J., Gomez-Garcıa, G., 2007. Exact solution of Smoluchowski’scontinuous multi-component equation with an additive kernel. Europhys. Lett.78, 56002.

Fox, R., 2006. Bivariate direct quadrature method of moments for coagulation andsintering of particle populations. J. Aerosol Sci. 37, 1562–1580.

Gelbard, F.M., Seinfield, J.H., 1978. Coagulation and growth of a multi-componentaerosol. J. Colloid Interface Sci. 63, 472–479.

Heinrich, S., Peglow, M., Ihlow, M., Henneberg, M., Morl, L., 2002. Analysis of thestart-up process in continuous fluidized bed spray granulation by populationbalance modelling. Chem. Eng. Sci. 57, 4369–4390.

Hidy, G.M., 1965. On the theory of the coagulation of non-interacting particles inBrownian motion. J. Colloid Interface Sci. 20, 123–144.

Hounslow, M.J., Ryall, R.L., Marshall, V.R., 1988. A discretized population balancefor nucleation, growth, and aggregation. AIChE J. 34, 1821–1832.

Immanuel, C.D., Doyle, F.J., 2003. Computationally efficient solution of populationbalance models incorporating nucleation, growth and coagulation: applicationto emulsion polymerization. Chem. Eng. Sci. 58, 3681–3698.

Kostoglou, M., Konstandopoulos, A.G., Friedlander, S.K., 2006. Bivariate populationdynamics simulation of fractal aerosol aggregate coagulation and restructur-ing. J. Aerosol Sci. 37, 1102–1115.

Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., Morl, L., 2006. Improved accuracyand convergence of discretized population balance for aggregation: the cellaverage technique. Chem. Eng. Sci. 61, 3327–3342.

Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., 2008. The cell average techniquefor solving multi-dimensional aggregation population balance equations.Comput. Chem. Eng. 32, 1810–1830.

Kumar, S., Gandhi, K.S., Kumar, R., 2007. Modeling of formation of gold nanopar-ticles by citrate method. Ind. Eng. Chem. 46, 3128–3136.

Kumar, S., Ramkrishna, D., 1995. A general discretization technique for solvingpopulation balance equations involving bivariate distributions. In: AIChEAnnual Meeting, pp. 139c.

Kumar, S., Ramkrishna, D., 1996a. On the solution of population balance equationsby discretization—I. A fixed pivot technique. Chem. Eng. Sci. 51, 1311–1332.

Kumar, S., Ramkrishna, D., 1996b. On the solution of population balance equationsby discretization—III. A moving pivot technique. Chem. Eng. Sci. 51,1333–1342.

Ma, C.Y., Wang, X.Z., Roberts, K.J., 2007. Multi-dimensional population balancemodeling of the growth of rod-like L-glutamic acid crystals using growth ratesestimated from in-process imaging. Adv. Powder Technol. 18, 707–723.

Mantzaris, N., 2006. Stochastic and deterministic simulations of heterogeneouscell population dynamics. J. Theor. Biol. 241, 690–706.

Mantzaris, N.V., Daoutidis, P., 2004. Cell population balance modeling and controlin continuous bioreactors. J. Process Control 14, 775–784.

Matsoukas, T., Kim, T., Lee, K., 2009. Bicomponent aggregation with composition-dependent rates and the approach to well-mixed state. Chem. Eng. Sci. 64,787–799.

Nandanwar, M., Kumar, S., 2008a. A new discretization of space for the solution ofmulti-dimensional population balance equations. Chem. Eng. Sci. 63,2198–2210.

Nandanwar, M., Kumar, S., 2008b. A new discretization of space for the solution ofmulti-dimensional population balance equations: simultaneous breakup andaggregation of particles. Chem. Eng. Sci. 63, 3988–3997.

Otwinowski, H., 2006. Energy and population balances in comminution processmodelling based on the informational entropy. Powder Technol. 167, 33–44.

Peterson, T., Gelbard, F., Seinfeld, J., 1978. Dynamics of source-reinforced, coagu-lating, and condensing aerosols. J. Colloid Interface Sci. 63, 426–445.

Press, W., Teukolsky, S., Verllerling, W., Flannery, B., 1998. Numerical Recipes inFORTRAN The Art of Scientific Computing. Cambridge University Press, NewDelhi.

Raikar, N.B., Bhatia, S.R., Malone, M.F., Henson, M.A., 2009. Experimental studiesand population balance equation models for breakage prediction of emulsiondrop size distributions. Chem. Eng. Sci. 64, 2433–2447.

Ramkrishna, D., 2000. Population Balance: Theory and Applications to ParticulateSystems in Engineering. Academic Press, San Diego.

Sheikh, A.Y., Jones, A.G., Graham, P., 1996. Population balance modeling of particleformation during the chemical synthesis of zeolite crystals: assessment ofhydrothermal precipitation kinetics. Zeolites 16, 164–172.

Vale, H.M., McKenna, T.F., 2005. Solution of the population balance equation fortwo-component aggregation by an extended fixed pivot technique. Ind. Eng.Chem. Res. 44, 7885–7891.

Vanni, M., 2000. Approximate population balance equations for aggregation-breakage processes. J. Colloid Interface Sci. 221, 143–160.