CFD Modeling of Nucleation, Growth, Aggregation, and Breakage in ContinuousPrecipitation of Barium Sulfate in a Stirred Tank

Jingcai Cheng,† Chao Yang,*,†,‡ Zai-Sha Mao,† and Chengjun Zhao§

Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy ofSciences, Beijing 100190, China; Jiangsu Marine Resources DeVelopment Research Institute,Lianyungang 222005, China; and Shijiazhuang Chemical Fiber Corporation, Hebei 050032, China

In this work, the precipitation of barium sulfate (BaSO4) in a continuous stirred tank reactor (CSTR) is modeled.The flow field is obtained through solving the single-phase Reynolds averaged Navier-Stokes equationswith a standard single-phase k-ε turbulence model. The population balance equation is solved through thestandard method of moments (SMM) and the quadrature method of moments (QMOM) both with and withoutaggregation and breakage terms. In the cases of precipitation simulation without aggregation and breakage,the results predicted from 2-node QMOM, 3-node QMOM, and SMM are very close. Thus, 2-node QMOMcould replace SMM and be well-incorporated into an in-house CFD code to simulate the precipitation inCSTR with acceptable accuracy. The predicted area-averaged crystal size d32 decreases almost linearly withincreasing feed concentration, and the deviation from experimental data becomes significant at high feedconcentration. Numerical simulation using 2-node QMOM with the Brownian motion and shear-inducedaggregation kernels as well as a power-law breakage kernel indicates that the predicted d32 shows goodqualitative agreement with experimental results, and the quantitative agreement is achieved when the appropriatebreakage rate equation is adopted.

1. Introduction

Precipitation or reactive crystallization of sparingly solublesalts is an important industrial operation that is widely used toproduce fine and bulk chemicals, pharmaceuticals, biochemicals,catalysts, pigments, ceramics, etc. Precipitation is a verycomplex process, because it is complicated by several interactingphenomena, and for this reason, it has attracted much attraction.Large-scale production of particulate products is often carriedout in a continuous process whereas batch or semibatch modeis usually for small-scale production. As a result of very lowsolubility of precipitated substances and for economic reasons,high reactant concentrations are commonly applied. Becauseof the high supersaturation levels resulting from high reactantconcentrations, the involved mechanisms of primary nucleation,crystal growth, and aggregation proceed nearly simultaneouslyand eventually breakage takes place. The precipitation of bariumsulfate (BaSO4) has been used as a model reaction and widelystudied in the past few decades. Numerous theoretical andexperimental studies have been reported in the literature, tryingto understand important aspects of the precipitation process.1-8

The previous model-based experimental studies seldom tookinto account the local flow and turbulence in a precipitator.Nevertheless, computational fluid dynamics (CFD) has beenapplied to simulate the precipitation process in simple andcomplex geometries (e.g., stirred tanks) in recent years. This isusually implemented through solving the standard momentumand mass transport equations together with the populationbalance equation (PBE). Successful CFD simulations of pre-cipitation in a stirred tank have been reported in manystudies.9-14 By implementing the standard method of moments(SMM) without considering aggregation and breakage, Jaworski

and Nienow11 successfully simulated the precipitation of BaSO4

in a continuous stirred tank reactor (CSTR). They found thatthe residence time and the crystal shape factor significantlyaffected local supersaturation and volume-averaged crystal size,but in all cases the coefficients of crystal size variation obtainedwere very close to 1.0, which differed considerably from theexperimental results in the range from 0.4 to 0.7.15 The influenceof activity coefficient on the crystallization process of BaSO4

in the coaxial pipe mixer was studied by Oncul et al.,5 and thePBE was modeled using SMM without aggregation and break-age. The precipitation of BaSO4 in a CSTR was also modeledby Wang et al.9 The effects of feeding location, feed concentra-tion, impeller speed, and residence time were investigated, andthe discrepancy between predictions of mean particle size andexperimental results was found to increase with increasing feedconcentrations, which was attributed to the neglect of aggrega-tion. The influence of turbulent mixing on BaSO4 precipitationin a semibatch stirred tank was investigated by Vicum andMazzotti,13 and the PBE was solved using SMM withoutaggregation and breakage. It was found that, at low concentra-tions, the predictions were quantitatively correct, but at highconcentration, the overprediction was evident with the CFD-based mixing-precipitation model.

Most of the above-discussed studies on the modeling of aprecipitation process usually neglected aggregation and break-age, since the PBE was mainly solved using SMM, which hasgreat difficulties in closing nonconstant aggregation and break-age kernels. However, aggregation was clearly evidenced inBaSO4 precipitation.2,16-19 Aggregation and breakage have animportant influence on the quality of particulate products,especially for dense particle systems, and should not beneglected in the modeling of precipitation. In recent years, someworks were reported on modeling BaSO4 precipitation withparticle aggregation incorporated. The PBE is usually solvedusing SMM with a restrictive simplifying aggregation kernelor quadrature method of moments (QMOM) with a morecomplex aggregation term. Marchisio et al.16 studied the

* To whom correspondence should be addressed. Tel.: +86-10-62554558. Fax: +86-10-62561822. E-mail: chaoyang@home.ipe.ac.cn.

† Chinese Academy of Sciences.‡ Jiangsu Marine Resources Development Research Institute.§ Shijiazhuang Chemical Fiber Corporation.

Ind. Eng. Chem. Res. 2009, 48, 6992–70036992

10.1021/ie9004282 CCC: $40.75 2009 American Chemical SocietyPublished on Web 07/08/2009

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turbulent precipitation of BaSO4 in a tubular reactor bothexperimentally and computationally and found that the aggrega-tion and growth in an aqueous solution took place simulta-neously with increasing concentrations. The PBE was solvedusing SMM with aggregation and without breakage. Althoughonly an assumed constant laminar aggregation kernel wasconsidered, the agreement with experimental results was muchimproved, suggesting that turbulent aggregation and a noncon-stant kernel were necessary for better prediction. Gavi et al.20

studied numerically the nanoparticle precipitation of BaSO4 ina confined impinging jet reactor (CIJR). The PBE was solvedusing QMOM considering the Brownian motion induced ag-gregation, and the CFD predictions were found to be in goodagreement with experimental data.

Mixing (macro-, meso-, and micromixing) plays an importantrole in precipitation, especially precipitation of nanoparticles,and the role of micromixing in precipitation is generallyrecognized. However, according to Marchisio and Barresi,21 therole of micromixing in fast reactions varies depending on theoperating conditions. In stirred tanks, nucleation takes placemainly at the feeding point, and according to Wijers et al.,22

the meso-mixing time criterion should be used for scale-up.Taguchi et al.23 found that macro-mixing was particularlyimportant for stirred tank arrangements with micromixing beingof lesser significance, which supports an earlier finding byFitchett and Tarbell.24 Van-Leeuwen et al.25 showed that theeffect of impeller speed on crystal size was limited and couldbe negligible for impeller speeds larger than 100 rpm. Wong etal.18 revealed that, in the concentration range applied, impellerspeed had small effect on crystal size and morphology. Also,mixing sensitive reactions has been modeled for a number ofcases via CFD without a micromixing model,26,27 and resultsshowed good agreement with experimental data. In this work,the two feeding nozzles of the CSTR are placed comparativelyfar from each other in order that the chance of direct contact ofboth fresh reactants is reduced; thus, the effects of mixing arereduced. In the case of a double-feed-precipitation process, whenthe feeding nozzles are so close that both fresh reactants contacteach other before any significant dilution with the bulk, mixingbetween fresh reactants becomes more important than a directdilution with the bulk.12 Up to now, there still exists a lack ofunderstanding concerning the relevance of the micromixingmodel in CFD simulations of fast reactions. In considerationof these, micromixing effects have not been included in themodeling at the moment.

To our best knowledge, the full simulation of a precipitationprocess in a CSTR has not been reported yet. This work is anattempt to model the continuous precipitation process includingnucleation, crystal growth, aggregation, and breakage in a stirredtank, so that the design and scale-up of the precipitation reactorcould be better understood. Notwithstanding theoretical analysisand experimental determination are still needed to fully under-stand some fundamental aspects of precipitation, it is beneficialto find some reliable kinetic expressions for aggregation andbreakage and for aggregation efficiency. The investigation of aprecipitation process from a modeling perspective is appealingand promising to derive a fully predictive model.

This work is organized as follows. First, the populationbalance equations and moment transformation methods (i.e.,SMM and QMOM) are presented, and then aggregation andbreakage kernels and aggregation efficiency expression aredescribed. Finally, numerical simulations of a precipitationprocess in the CSTR both with and without aggregation and

breakage are carried out, and comparison of model predictionswith experimental data from the literature is performed.

2. Precipitation Models

2.1. General Population Balance. The Reynolds-averagedform of population balance equation (PBE) expressed in termsof number density function n(L;x,t) is28,29

where G(L) is the growth rate and Ba(L;x,t), Da(L;x,t), Bb(L;x,t),and Db(L;x,t) represent the birth and death rates due toaggregation and breakage, respectively. In the case of the particlevolume and particle length having the relationship υ ∝ L3, moreprecisely υ ) L3, the above length-based birth and death ratescan be rigorously derived from the corresponding volume-basedbirth and death terms.30

The moments of the particle size distribution (PSD) aredefined as follows:

Applying the definition of moments to eq 1 results in

where J(x,t) is the nucleation rate and

where �(L,λ) is the collision kernel, R(L,λ) is the collisionefficiency, a(λ) represents the breakage rate, and b(L|λ) is thefragmentation distribution function.

In the case of size-independent growth without aggregationand breakage, the standard moment method (SMM) can beapplied to compute the moments directly without requiringadditional knowledge of the number density function. Then, thesteady-state set of five equations used for computing five crystalsize distribution moments, from the zeroth to the fourth, isexpressed as28

For dealing with particle secondary processes such asaggregation and breakage, the quadrature method of moments(QMOM) is a good choice.30-33 The QMOM has been tested

∂n(L;x, t)∂t

+ ∇ · [un(L;x, t)] - ∇ · [Γef∇n(L;x, t)] )

- ∂

∂L[G(L)n(L;x, t)] + Ba(L;x, t) - Da(L;x, t) +

Bb(L;x, t) - Db(L;x, t) (1)

mk ) ∫0

+∞n(L;x, t)LkdL (2)

∂mk

∂t+ ∇ · [umk] - ∇ · [Γef∇mk] ) (0)kJ(x, t) +

∫0

+∞kLk-1G(L)n(L;x, t)dL + Bk

a(x, t) - Dka(x, t) + Bk

b(x, t) -

Dkb(x, t) (3)

Bka(x, t) ) 1

2 ∫0

+∞n(λ;x, t)∫0

+∞R(u, λ)�(u, λ)(u3 + λ3)k/3n ×

(u;x, t) du dλ (4)

Dka(x, t) ) ∫0

+∞Lkn(L;x, t)∫0

+∞R(L, λ)�(L, λ)n(λ;x, t) dλ dL

(5)

Bkb(x, t) ) ∫0

+∞Lk∫0

+∞a(λ)b(L|λ)n(λ;x, t) dλ dL (6)

Dkb(x, t) ) ∫0

+∞Lka(L)n(L;x, t) dL (7)

∇ · [umk - Γef∇mk] ) 0kJ(x) + jmk-1G (k ) 0-4)(8)

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and compared with other techniques, showing its great potential,and it can solve the PBE with good accuracy by requiring avery small number of equations.30,34 Therefore, this method isvery suitable in computationally demanding CFD codes.32,35

The QMOM is based on the following Gaussian quadratureapproximation:

where abscissas (Li) and weights (wi) are determined throughthe product-difference (PD) algorithm.31 By applying eq 9 toeq 3, the steady-state moment equation is

where �ij ) �(Li,Lj), Rij ) R(Li,Lj), ai ) a(Li) and

2.2. Species Transport Equations. The transport equationsfor all chemical entities, in molar concentration ci (i ) Ba2+,Cl-, Na2+, SO4

2-, BaSO4), are applied for the steady-stateoperation in the following form:

The effective diffusion coefficient, Γef, is computed as a sumof the molecular diffusivity, Γ, and the turbulent diffusivity, Γt

) νt/Sct. The turbulent Schmidt number, Sct, is assumed to be0.7, as in Jaworski and Nienow11 and Oncul et al.36 The sourceterm Si is taken to be equal to the specific crystal growth rate,Sg(eq 13), with a minus sign for Ba2+ and SO4

2-, a plus sign forBaSO4, and zero for other nonreacting ions (Cl- and Na+).36

The specific crystal growth rate Sg is related to the secondmoment of crystal size distribution, m2, and volumetric crystalshape factor, kV:11

2.3. Nucleation and Growth Kinetics. The supersaturationratio Sa and supersaturation ∆c are defined as follows:

where γac is the activity coefficient, cBa2+ and cSO42- are the ion

concentrations of Ba2+ and SO42-, respectively, and Ksp is the

solubility product, which is 1.10 × 10-10 kmol2/m6 at 25 °C.26

The activity coefficient is computed according to Bromley’smethod,37 which provides accurate data up to 6 M ionicstrengths;38 all the ionic strengths considered in this work liewithin this range.

Secondary nucleation is negligible compared with primarynucleation; thus, only primary nucleation is considered here.

The nucleation rate J(x) adopted here is developed by Bałdygaet al.39 and used by other researchers.9,14,16

The local values of growth rate G are calculated from thetwo-step model,1 which includes the surface integration stepand the molecular diffusion step. This model has been used inmany studies for calculating the growth rate of BaSO4

precipitation.1,9,12,14,16

where cAs and cBs are reactant concentrations on crystal surface,kr is akineticconstant that isequal to5.8×10-8 (m/s)(m3/mol)2,1,16

and kd is the mass transfer coefficient with a constant value of10-7 (m/s)/(m3/mol) considered in this work.14,40 For givenvalues of cA and cB, the growth rate G can be found by solvingeq 17 using a Newton-Raphson method.

2.4. Collision Rate. In general, the process of aggregationcomprises two steps. First, particles must be brought into closeproximity by a transport mechanism, giving rise to a collision.Then, an aggregate will be formed if the net interparticle forceis attractive and strong enough to win thermal agitation andhydrodynamic drag in order to make particles adhere or fuse.32,41

There are several mechanisms that can induce relative move-ments among particles and, hence, lead to collisions. Brownianmotion induced collisions (perikinetic coagulation) are thecontrolling mechanism mainly for submicrometer particles(usually smaller than 1 µm in diameter). For particles with adiameter approximately in the range 1-50 µm, collisions arecaused mostly by velocity gradients (orthokinetic coagulation).This shear-induced collision is the prevailing mechanism inparticulate processing systems. Collisions from differentialsedimentation or inertia (fluid acceleration) will become im-portant for aggregates larger than ∼50 µm.42 The collision kernelfunctions for Brownian and shear-induced collision are ex-pressed as43

and

respectively, where L(i) is the characteristic particle size of anaggregate consisting of i primary particles (hereafter referredto as an i-sized aggregate), Rc,i is the collision radius of an i-sizedaggregate, kB is the Boltzmann constant, T is the absolutetemperature, Gsh is the characteristic velocity gradient (shearrate) of the flow field, and ψ is a numerical constant that dependson the type of flow. For a simple shear flow (e.g., laminar flow),ψ ) 4/3 and Gsh refers to the only nonzero component ofvelocity gradient tensor. In an isotropic turbulent flow and forparticles smaller than the Kolmogorov length scale withnegligible inertia, Saffman and Turner43 derived ψ ) 1.29,where the shear rate is related to local energy dissipation rateε and the kinematic viscosity ν through Gsh )(ε/V)1/2. It isassumed that the two collision mechanisms superimpose andthe overall collision rate function is expressed as

mk ) ∫0

+∞n(L;x, t)Lk dL ≈ ∑

i)1

Nd

wiLik (9)

∇ · [umk] - ∇ · [Γef∇mk] ) (0)kJ(x) + k ∑i

Lik-1G(Li)wi +

12 ∑

i

wi ∑j

wjRij�ij(Li3 + Lj

3)k/3 - ∑i

Likwi ∑

j

wj�ij +

∑i

aibji(k)wi - ∑

i

Likaiwi (10)

bji(k) ) ∫0

+∞Lkb(L|Li) dL (11)

∇ · [Fuci - Γef∇(Fci)] ) Si (12)

Sg ) (Si

F) (3m2G)kV

FBaSO4

MBaSO4

(13)

Sa ) γac�cBa2+cSO42-

Ksp(14)

∆c ) (Sa - 1)√Ksp (15)

J(x) )

{2.83 × 1010(∆c)1.775(#m-3s-1), ∆c e 10mol m-3(heterogeneous)

2.53 × 10-3(∆c)15.0(#m-3s-1), ∆c > 10mol m-3(homogeneous)(16)

G ) kr(√cAscBs - √Ksp)2 ) kd(cA - cAs) ) kd(cB - cBs)

(17)

�Br(L(i), L(j)) )2kBT

3µ(Rc,i + Rc,j)

2

Rc,iRc,j(18)

�Fl(L(i), L(j)) ) ψGsh(Rc,i + Rc,j)3 (19)

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The collision radius of an aggregate is proportional to thenumber of primary particles in the aggregate, i, according tothe following relation

where R0 is the radius of the primary particles forming theaggregate, df is the mass fractal dimension, and kg is a constantusually taken to be unity.44,45 According to Lattuada et al.,45

the scaling relation of eq 21 is considered to hold for theaggregates consisting of more than three primary particles,however, it is assumed that it can be extended to all aggregatesizes in order to simplify the calculations. Hence, the solidvolume υi for a fractal aggregate is related to its characteristicsize Rc,i by Jiang and Logan46

where υ0 is the volume of the primary particles.2.5. Collision Efficiency. Solid particles are subject to

hydrodynamic interaction when approaching each other in asheared fluid, as the viscous fluid layer between them producesresistance forces. Furthermore, a newly formed aggregate willbe split into its original two parts if it does not have sufficienttime for restructuring after collision, thus resulting in nullaggregation. All these are reflected in the collision efficiency,R(L,λ), which is the ratio of the actual aggregation rate to thetheoretical collision rate given by eq 20. Collision efficiency isa function of hydrodynamic interactions, interparticle forces,and the structure of aggregates (porosity and permeability).Several models have been developed in the literature44,47,48 tocalculate the collision efficiency of porous aggregates (perme-able floc models). Since the first step in the formation of clustersfrom monomers is a doublet formation, the collision efficiencyfor doublets can be estimated by investigating collisions betweenimpermeable solid particles (impermeable flocs model).35,42,49,50

In a stirred precipitation system, aggregates are compactbecause they are subject to strong flow shear. Thus, the collisionefficiency calculated from the impermeable flocs model isadopted in this study. The collision efficiency is then expressedas35

where Fl is the flow number computed by

The prefactor kce ) 0.43 was obtained by fitting the particlenumbers with time in the doublet formation stage.35 TheHamaker constant AH ranges approximately between 10-20 and10-19 J for solid-liquid systems,51 and the value of 5.0 × 10-20

J is used in all simulations.2.6. Breakage Kernel. The breakage of aggregate can occur

by several mechanisms; however, only the hydrodynamic stressinduced fragmentation, which is most commonly encounteredin precipitation systems, is considered. Several expressions forthe breakage rate kernel have been developed by differentauthors, and they are summarized by Marchisio et al.32 Asemitheoretical expression that has found applications to a widevariety of fragmentation phenomena is the power-law breakagekernel:52,53

where c1 is a dimensionless empirical constant. By fitting themodel to experimental data, Peng and Williams54 found thatthe exponent γ can be assumed to be between 1 and 3 and γ )2 is usually used.

The fragment distribution function used in this work is theuniform distribution. Some other fragment distribution functions,such as symmetric fragmentation and erosion, have been used:30

3. Computational Details

A continuous flow stirred tank reactor (CSTR) with basicdimensions identical to those assumed by Jaworski and Nie-now11 is modeled in this study. Figure 1 shows the schematicdiagram of the tank and the impeller. The cylindrical tank hasa diameter of T ) 0.27 m with a flat bottom and is filled witha solution up to H ) 0.27 m. It is equipped with four standardbaffles of width B ) 0.1T and a standard six-bladed Rushtonturbine impeller of diameter D ) T/3, located at C ) T/2 fromthe bottom. The impeller speed is set to N ) 120 rpm. Twotubes are simulated, and through each a solution containingeither BaCl2 or Na2SO4 is fed. Five concentrations are applied,i.e., 0.05, 0.1, 0.2, 0.3, and 0.4 mol/L. The modeling is alwayscarried out with stoichiometric quantities of two reactants beingintroduced to the precipitator. Thus, the feed rates of the twosolutions are always equal and set to 18 mL/s. With the networking volume of 15.4 L, the mean residence time, τ, is then∼430 s. The feed tubes are located at the free surface, on theopposite sides of the shaft, midway between two neighboringbaffles and roughly at the radial location of 5T/12. The reactionmixture exit is located in the tank bottom with the same radiallocation as that of the feed tubes.

All the simulations undertaken in this work were programmedwith FORTRAN language. The three-dimensional flow filed inthe stirred tank is first obtained through solving the Reynoldsaveraged Navier-Stokes equations (RANS). Since the solidparticles are smaller than 20-30 µm, the solid concentration islow, the solid particles follow closely the liquid, and theinfluence of the dilute solid phase on the flow field can beneglected, the single-phase RANS and a standard single-phase k-ε turbulence model can be applied.11,14,32 The formof the single-phase RANS in the cylindrical coordinates systemand the numerical procedure have been detailed elsewhere.55

Figure 2 shows three predicted velocity components at differentradial positions using four different grid numbers. These radialpositions are 0.75H from the bottom and midway between twoneighboring baffles. It is shown that the simulation results with36 × 72 × 75 are very close to those with 36 × 72 × 90, andthis can also be found in other positions of the tank. Thus, thegrid of 36 × 72 × 90 (radial, azimuthal, and axial) is adoptedhere. The action of impeller is modeled using a modified“inner-outer” iterative procedure.55 Its main advantage is thatthe calculated flow parameters on the surface of the “inner”and “outer” regions are not averaged in the present procedure,thus preserving the pseudoperiodical turbulent properties. Wallfunctions are applied to all solid surfaces. After reachingconvergence, with the sum of normalized residuals well below10-4, the velocity and turbulence fields are saved and keptunchanged for the subsequent simulations of reactive precipitation.

On the basis of the already known flow field, the concentra-tion distribution of chemical species and lower-order moments

�(L(i), L(j)) ) �Br(L(i), L(j)) + �Fl(L(i), L(j)) (20)

Rc,i ) R0(i/kg)1/df, i > 3 (21)

υi ) υ0(Rc,i/R0)df (22)

Rij ) kceFl-0.18, 10 < Fl < 105 (23)

Fl ) 12πµRc,iRc,j(Rc,i + Rc,j)2/8AHR0 (24)

a(L) ) c1νxεyLγ (25)

b(L|λ) ) {6L2/λ3 if 0 < L < λ0 otherwise

(26)

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are obtained through simultaneous solution of five equationsfor the chemical entity, eq 12, and the set of moment equations.For SMM, five moment equations, eq 8, are used, and forQMOM, the set of moment equations, eq 10, are applied. Inthe second stage of these simulations, a zero flux at the walland the free surface are enforced for the variables such as ci

and mk. The area-averaged crystal size d32, which is of greatinterest in precipitation processes, is computed by

where Vi is the volume of cell i and N is the total number ofgrid cells. It should be noted that both SMM and QMOM cannottrack the PSD directly, which is also an important criterion foraccessing the properties and qualities of crystallization products.However, starting from the first six moments, the PSD can bereconstructed, and first results are promising56 though it is notan easy task.

Concerning the shape factor of BaSO4 crystals, kV, it is veryhard to determine its value since a number of morphologieshave been observed and many different values have beenreported or used.2,7,11,16 According to Jones,57 kV ) π/6 forspherical particles and kV > π/6 for other shapes. As mentionedin Section 2.1, the length-based birth and death terms areoriginally derived from the volume-based PBE with the as-sumption of particle volume and particle length having therelationship υ ∝ L3.30 Actually, when defining υ ) L3 (i.e., kV)1.0), the length-based PBE is rigorously deduced from thevolume-based PBE.30,35 On the basis of these, kV) 1.0 wasadopted in this paper.

The first five moments of PSD, from m0 to m4, are computedwith SMM. This method is presented and implemented heremainly for the purpose of comparing the results of QMOMwithout aggregation and breakage with those of SMM.

The QMOM is implemented with either two nodes (Nd ) 2)or three nodes (Nd ) 3) when taking no account of aggregationand breakage processes (only the first two terms on the right of

eq 16 are used). By applying QMOM with Nd nodes, the first2Nd moments will be tracked. For example, if Nd ) 3, the firstsix moments, from m0 to m5, are calculated.

In the full simulations of BaSO4 precipitation with QMOM,some parameters in the expressions for collision rate, collisionefficiency, and breakage rate should be determined first. The

Figure 1. Sketch of the stirred tank precipitator and impeller.

d32 ) ∑i)1

N

m3Vi/ ∑i)1

N

m2Vi (27)

Figure 2. Velocity components at different radial positions using fourdifferent grid numbers (radial positions are 0.75H from the bottom andmidway between two neighboring baffles): (a) radial component, (b)azimuthal component, and (c) axial component.

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fractal dimension of the particles df ) 2.8 is used in this work.It should be noted here that the QMOM in the form presentedin this work has assumed Euclidean (nonfractal) aggregates;hence, it will present an inconsistency when introducing a fractaldimension. However, in a stirred precipitation system, particlesare subject to strong shear stresses and/or hydrodynamicbreakage causes particles restructuring and ripening, which makeparticles become compact (df is very close to 3). Therefore, theapplication of the QMOM (assuming df ) 3) and the calcula-tions of the collision rate and collision efficiency (assuming df

) 2.8) seem acceptable. This approach was also validated else-where.32,35,58

As already mentioned above, the exponent for the particlesize in the power-law breakage kernel in eq 25 can assumevalues between 1 and 3, and 2 is usually taken.35,54 Therefore,by forcing a parabolic dependence on the particle size (γ ) 2),other exponents, x and y, can be found through a dimensionalanalysis. The resulting exponent for the turbulent dissipationrate is y ) 1 and that for the kinematic viscosity is x ) -2.

As mentioned, c1 is a dimensionless constant and its valuecan be determined empirically. Hence, a best value for c1 amongthree adopted values will be recommended when best agreementbetween CFD simulations and experimental data is obtained.On the basis of the experimental disruption kernel of CaCO3

during precipitation obtained by Wojcik and Jones,59 theapproximate order of magnitude of c1 or an approximate rangeof c1 was estimated prior to simulations. This was found to bevery helpful to find a roughly reliable breakage rate whenstarting full simulations. The power-law breakage law used inthis paper is semiempirical. Several other theoretical-basedbreakage laws have been developed and summarized byMarchisio et al.32 Since some parameters (e.g., interparticle forceand distance between primary particles, etc.) for these theoreticalexpressions are hard to obtain or estimate, their applicationsand practicability have been limited compared with the semi-theoretical power-law breakage kernel, although an empiricalconstant needs to be determined.

Regarding the primary particle radius R0, the accurateestimation of its value is very difficult. An assumed value wasthus adopted in this work. In precipitation, nuclei are usuallyof the size of several molecules. Nuclei will grow intocrystallites rapidly. Since there are a huge number of nucleiand crystallites, collision and aggregation among these tinyparticles would inevitably take place. Therefore, it seemsreasonable to assume that the primary particle is of the size ofseveral nuclei. Heineken et al.3 assumed the radius of bariumsulfate nuclei being 0.5 × 10-9 m in their model based on theanalysis of continuous precipitation process of BaSO4. Actually,we have conducted test simulations using different values ofR0, i.e., 2.0 × 10-9, 1.0 × 10-8, and 5.0 × 10-8 m, in the caseof aggregation and breakage, and both 2-node and 3-nodeQMOM were performed. Since the order of magnitude of c1

estimated from the experimental disruption kernel of CaCO3

during precipitation59 was approximately 100, for convenience,c1 ) 1.0 was applied for the test simulations. The feedconcentration was 0.1 mol/L. As shown in Figures 6 and 7, thebiggest error committed by three adopted R0 values is ∼4%(for m0). Therefore, R0) 2.0 × 10-9 m was adopted insimulations incorporating aggregation and breakage. It has beenshown that the QMOM with 2 nodes can work with acceptableaccuracy for aggregation and breakage problems.32,60 Also, itcan be found in Figure 6 that the predictions from 2-node and3-node QMOM are very close (the biggest error is <0.4% form0). Moreover, the simulations with 3-node QMOM took around

one and a half times more time than those with 2-node QMOMin the case of aggregation and breakage. Therefore, 2-nodeQMOM was adopted for simulations with aggregation andbreakage terms.

4. Results and Discussion

4.1. Features of Simulated Turbulent Tank Flow. Anexample map of the mean velocity vectors in the vertical cross-section between the baffles of the stirred tank is presented inFigure 3a, which shows a typical mean flow pattern driven bya Rushton turbine impeller in a baffled stirred tank. Figure 3bshows the distribution of ε in a vertical plane between thebaffles. On the whole, the distribution shows good symmetrywith the shaft given the radial location of the exit tube. Figure3c presents the distribution of ε in three horizontal cross sectionsat different distances from the tank bottom.

On the average, the energy dissipation rate in the impellerregion is several orders of magnitude higher than that in thebulk region. The presence of flow structures in the turbulentflow shows that the turbulent energy dissipation rate, andtherefore the shear rate approximated by Gsh )(ε/ν)1/2 inturbulent flows for particles smaller than Kolmogorov scale, ishighly nonuniform throughout the tank. Some studies tried todescribe the collision rate as a function of average shear rateestimated by (εj/ν)1/2 in stirred tanks.61-63 Wang et al.35 usedspatial average shear rate to calculate aggregation and breakagerates in the CFD modeling of aggregation and breakageprocesses in a Taylor-Couette laminar flow. Since the collisionrate is a highly nonlinear function of ε1/2 in eq 19 and thebreakage rate is a nonlinear function of ε in this work, the spreadin collision and breakage rates will be very large. Hollander etal.64 found that using volume-averaged energy dissipation rateto model the agglomeration process in a stirred tank will yieldunrealistic prediction of reactor performance. Although thereis less spread in the distribution of ε over a Taylor-Couettereactor, using the volume-averaged values of turbulent propertieswould introduce significant differences in the predictions ofhomogeneous and spatially heterogeneous turbulent properties.32

Thus, the modeling of aggregation and breakage processes musttake into account the local turbulent information in order toaccurately predict the PSD and better understand the scale-upbehavior of precipitation processes.

4.2. Comparison of QMOM with SMM. Figure 4 showsthe moments predicted by two kinds of moment methods atdifferent feed concentrations, i.e., SMM and QMOM. Thepredictions of d32 at different feed concentrations using SMM,2-node QMOM, and 3-node QMOM, and the comparisonbetween the predicted d32 and experimental data taken from theliterature,65 are presented in Figure 5. The experimental datawere obtained for BaSO4 continuous precipitation at differentfeed concentrations in a Rushton-driven stirred tank. The resultsshowed that the particle size decreased with increasing feedconcentrations, which has also been confirmed in many subse-quent studies, both experimentally and numerically.2,5,12,14 Theexperimental stirring speed (150 rpm) was a little larger thanthat in this study; however, as discussed in the introduction,impeller speed has little effect on mean particle size whenexceeding 100 rpm. Moreover, this work is more focused onthe qualitative comparison with experimental data at highconcentrations that have seldom been investigated in moststudies (the reagent concentrations below 0.05 mol/L). Theseexperimental data have also been used in other work forcomparison with simulation results.9,14 The QMOM is imple-mented with both two and three nodes and without taking into

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account aggregation and breakage. It is found in Figure 4 thatthe differences in the predicted lower-order moments, m0-m3,from SMM and QMOM are very small. Also, QMOM with twoand three nodes get very close results. Considering the enormouscomputational load, especially when aggregation and breakageprocesses are included with very fine grid, 2-node QMOM isrecommended in the case without aggregation and breakagewhereas 3-node QMOM offers better accuracy.

It is shown in Figure 5 that the predicted mean crystal sized32 declines almost linearly with increasing feed concentrations.However, experimental results show that d32 indeed decreaseswith increasing feed concentrations, but more modestly at higherfeed concentration. The highest concentration of the available

experimental data is 0.2 mol/L, but it may be inferred from thedecreasing trend in experimental data that d32 would stabilizeor decline more slowly when the concentration is larger than0.2 mol/L. In reactive crystallization, competition exists betweennucleation and growth. Higher feed concentration produceshigher supersaturation and, therefore, higher nucleation rate,making nucleation more favored over growth. In fact, smallparticles are produced if nucleation is favored over growth.Therefore, both simulations and experimental data illustrate adecreasing trend in d32 with increasing feed concentrations inFigure 5. However, aggregation and breakage also have a stronginfluence on crystal size. Very often aggregation leads to largeparticles. As aggregates become larger, eventually particle

Figure 3. Features of the simulated turbulent flow of the CSTR: (a) mean velocity vectors in a vertical cross section between the baffles, (b) distribution ofturbulent energy dissipation rate in a vertical cross section between the baffles, and (c) distribution of turbulent energy dissipation rate in the horizontal crosssections of the CSTR; the distances of three horizontal cross sections from the tank bottom are H/15, H/2, and 14H/15 from down to up, respectively.

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breakage becomes important and a steady state is approachedwhen aggregation and breakage reach dynamic equilibrium. Fordilute particle systems (low feed concentrations), the effect ofaggregation and breakage is small and can be neglected, butthis is not the case for dense particle systems. Since aggregationand breakage are not taken into account in the current simulationin this section, the differences between the predicted d32 andexperimental results become more significant at higher feedconcentrations, as shown in Figure 5.

4.3. QMOM Incorporating Aggregation and Breakage.Figure 8 shows the variation of lower-order moments (m0-m3)with feed concentrations predicted from 2-node QMOM incor-porating aggregation and breakage. Simulations were conductedby solving the complete form of eq 10 and using different valuesof the power-law breakage kernel constant c1, i.e., 0.1, 0.3, and

0.8. The predictions of moments using 2-node QMOM withoutaggregation and breakage are also presented in Figure 8. Themodel predictions of lower-order moments, m0, m1, and m2,increase with increasing feed concentrations, but more modestlycompared with those without aggregation and breakage, espe-cially at higher feed concentrations. The total particle numberdensity m0 is reduced compared with the predictions withoutaggregation and breakage due to crystal aggregation andrelatively low breakage rate (c1 ) 0.1). When increasing thevalue of c1, therefore increasing the breakage rate, the predictedm0, m1, and m2 all increase. There is little difference in the modelpredictions of m3 with and without aggregation and breakageterms due to conservation of mass.

Figure 9 shows the numerical predictions of mean particlesize d32 using 2-node QMOM both with and without secondaryterms. Predicted results with different values of c1 and experi-mental data65 are also presented in Figure 9. Unlike theunrealistic predictions in Figure 5, predictions from 2-nodeQMOM incorporating aggregation and breakage show a moremodest decrease in d32 with increasing feed concentrations. Fordifferent values of c1, the decreasing trends in d32 are allqualitatively in good agreement with that inferred from experi-mental data. The best agreement between simulations andexperimental data (both qualitatively and quantitatively) is foundwhen c1 ) 0.1. It is also shown in Figure 9 that the mean particlesize d32 declines with increasing values of c1, thus increasingbreakage rate. Secondary processes, such as aggregation andbreakage, are important especially at high reactant concentra-tions, so they should not be neglected in experimental and CFDmodeling for dense particle systems.

The mechanism of aggregation and breakage is not thoroughlyunderstood yet, and the CFD simulation on aggregation and

Figure 4. Predicted lower-order moments from m0 to m3 using SMM, 2-node QMOM, and 3-node QMOM without aggregation and breakage at differentfeed concentrations: (a) m0, (b) m1, (c) m2, and (d) m3.

Figure 5. Comparison between experimental data and predicted area-averaged particle size d32 at different feed concentrations using SMM, 2-nodeQMOM, and 3-node QMOM without aggregation and breakage.

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breakage processes without any adjustable or estimated param-eters is still very difficult so far.32,35 Aggregation and breakagemodels used in the literature mostly include several adjustableparameters (usually 1-4) or estimated model parameters (mainlyfor theoretically based models), and none of the parameters isuniversally applicable to different particulate systems. Thepower-law breakage rate adopted in this work is a semiempiricalexpression, and the collision efficiency expression is moretheoretically based, in which several model parameters areestimated based on the recommended values in the liter-ature together with some theoretical analysis. Actually, a “morereliable” set of parameters could be obtained through fittingthese parameters to experimental data; however, this is notfeasible due to high computational burden and will also losesome predictive nature of CFD. The main point of this work isfocused on BaSO4 precipitation in CSTR with relatively highconcentrations. In this case, the complicated secondary processesplay an important role in determining the product quality, andthis was seldom considered in CFD simulations in the past. Forall the three adopted values of c1, good qualitative agreementwith experimental data was obtained especially at higher feedconcentrations, and quantitative agreement can also be achievedwhen adjusting c1 to 0.1. It should be noted that c1 ) 0.1 wasreliable just in this case and of no universal applicability. It ishoped that the quantitative agreement could be reached withoutadjusting c1 or even without any estimated parameter; however,as stated above, this is still not feasible at the moment. Of course,qualitative agreement is the prerequisite to quantitative ones.The successful full simulation and the good qualitative agree-

ment give us a hint that the approach can be used. Moreover,with more reliable experimental data and large numbers ofcomputations together with the improved aggregation andbreakage models, a reliable set of parameters (the less the better)that are of more universal applicability could be obtainedthrough this method.

5. Conclusions

The turbulent precipitation of BaSO4 with different feedconcentrations in a CSTR has been successfully simulated usingSMM and QMOM without aggregation and breakage processes.Meanwhile, the full simulation of BaSO4 precipitation usingQMOM (all the involved mechanisms, nucleation, growth,aggregation, and breakage, being incorporated) was also con-ducted, and the predictions of area-averaged particle size d32

are compared with data from the literature.In the case without considering aggregation and breakage,

2-node QMOM, 3-node QMOM, and SMM are applied tomodeling the continuous precipitation process. The predictedlower-order moments and d32 show that the difference betweenpredictions by three modeling approaches is very small.Therefore, QMOM can replace SMM and be well incorporatedinto CFD codes to simulate reactive crystallization in the CSTR.Furthermore, the simulation with 2-node QMOM can achieveacceptable accuracy, making a good trade-off between thecomputational cost and overall accuracy for a stirred tank witha huge number of grids, especially when aggregation andbreakage are included. The model prediction of d32 withoutaggregation and breakage decreases almost linearly with in-creasing feed concentrations, and the deviations from experi-mental data are more significant at high concentration.

In the case with aggregation and breakage, the simulationresults with 2-node and 3-node are very close. Therefore, 2-node

Figure 6. Predicted lower-order moments (m0-m3) for different primaryparticle radii R0 using both 2-node and 3-node QMOM in the case ofaggregation and breakage (feed concentration is 0.1 mol/L and c1 ) 1.0).

Figure 7. Predicted lower-order moments (m4 and m5) for different primaryparticle radii R0 using 3-node QMOM in the case of aggregation andbreakage (feed concentration is 0.1 mol/L and c1 ) 1.0).

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QMOM is a good choice for the simulations incorporatingcomputationally time-consuming secondary terms. The 2-nodeQMOM considering aggregation and breakage was performedwith different values of the power-law breakage kernel constantc1. The trend of increase in m0, m1, and m2 with increasing feedconcentration becomes more modest compared with that withoutaggregation and breakage. The predictions of m0, m1, and m2

increase with increasing value of c1 (higher breakage rates). Thepredicted values of d32 are qualitatively and quantitatively,

depending on the value of c1, close to experimental resultscompared with the model predictions without aggregation andbreakage. When increasing c1, thus the breakage rate, the meanparticle size d32 decreases. Since aggregation and breakage arevery important at high concentration, they should be includedin the full simulation of a precipitation process. With betterunderstanding of the involved aspects of a precipitation process,such as nucleation, growth, aggregation, and breakage kernels,the precipitation model in combination with CFD techniquesmay become a powerful tool for performance prediction anddesign and scale-up of practical precipitators.

Nomenclature

Roman Letters

a(L) ) breakage kernel, s-1

AH ) Hamaker constant, Jb(L|λ) ) fragmentation distribution function, m-1

bji(k) ) kth moment of the fragmentation distribution function

for L ) Li, mk

B ) width of baffle, mBa(L;x,t) ) birth rate due to aggregation, # m-4 s-1

Bb(L;x,t) ) birth rate due to breakage, # m-4 s-1

Bjka(x, t) ) kth moment transform of birth term due to aggrega-tion, # mk m-3 s-1

Bjkb(x, t) ) kth moment transform of birth term due to breakage,# mk m-3 s-1

c1 ) dimensionless constant in power-law breakage kernelci ) molar concentration of chemical entity i, mol m-3

Figure 8. Lower-order moments (m0-m3) at different feed concentrations obtained using 2-node QMOM in the cases both with and without aggregation andbreakage. Combinations of the aggregation kernel and three breakage kernels (three values of c1) were applied: (a) m0, (b) m1, (c) m2, and (d) m3.

Figure 9. Comparison between experimental data and the predicted meanparticle size d32 at different feed concentrations using 2-node QMOM inthe cases both with and without aggregation and breakage. Combinationsof the aggregation kernel and three breakage kernels (three values of c1)were applied.

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cAs ) reactant concentration on crystal surface, mol m-3

∆c ) supersaturation, mol m-3

C ) clearance of impeller to tanker bottom, md32 ) mean particle size, mdf ) mass fractal dimensionD ) diameter of Rushton turbine impeller, mDa(L;x,t) ) death rate due to aggregation, # m-4 s-1

Db(L;x,t) ) death rate due to breakage, # m-4 s-1

Dj ka(x, t) ) kth moment transform of death term due toaggregation, # mk m-3 s-1

Dj kb(x, t) ) kth moment transform of death term due to breakage,# mk m-3 s-1

Fl ) flow numberG ) molecular growth rate, m s-1

Gsh) fluid shear rate, s-1

H ) height of the tank, mJ(x, t) ) nucleation rate, # m-3 s-1

kB ) Boltzmann constant, J K-1

kce ) prefactor of collision efficiency expressionkd ) mass transfer coefficient, mol m-2 s-1

kV ) volumetric shape factorKsp) solubility product, mol2 m-6

L ) particle size, mLi ) abscissa (or node) of quadrature approximation, mL(i) ) particle size of an i-sized aggregate, mmk(x, t) ) kth moment of crystal size distribution, # mk m-3

MBaSO4) molar mass of BaSO4, kg mol-1

n(L;x,t) ) number density function, # m-4

N ) stirred speed, s-1

Rc,i ) collision radius of an i-sized aggregate, mR0 ) radius of primary particle, mSa ) relative supersaturation, dimensionlessSg ) specific crystal growth, mol m-3 s-1

Sct ) turbulent Schmidt numbert ) time, sT ) absolute temperature, Ku ) Reynolds-averaged velocity vector, mwi ) weight of quadrature approximationx ) space vector, m

Greek Symbols

R(L,λ) ) collision efficiency�(L, λ) ) overall collision rate, m3 s-1

�Br(L, λ) ) Brownian motion-induced collision rate, m3 s-1

�Fl(L, λ) ) fluid shear-induced collision rate, m3 s-1

γ ) exponent for particle size in power-law breakage kernelΓ ) molecular diffusivity, m2 s-1

Γef ) effective diffusivity, m2 s-1

Γt ) turbulent diffusivity, m2 s-1

ε ) turbulent energy dissipation rate, m2 s-3

εj ) average energy dissipation rate, m2 s-3

λ ) particle size, mµ ) dynamic viscosity of the fluid, Pa sν ) kinematic viscosity, m2 s-1

νt ) turbulent kinematic viscosity, m2 s-1

F ) density, kg m-3

τ ) mean residence time, sυi ) volume of an i-sized aggregate, m3

υ0 ) volume of primary particle, m3

ψ ) numerical constant in shear-induced collision kernel� ) property vector specifying the state of particlei ) ith component of property vector�j ) jth component of flux in �-space

Acknowledgment

The authors acknowledge the financial support from theNational Natural Science Foundation of China (Nos. 20676134,20576133), 973 Program (2007CB613507), and the NationalProject of Scientific and Technical Supporting Program(2008BAF33B03).

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ReceiVed for reView November 23, 2008Accepted June 21, 2009

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