Chemical Engineering Science 57 (2002) 3715–3732www.elsevier.com/locate/ces

On the solution of population balance equations (PBE)with accurate front tracking methods in practical

crystallization processesYoung Il Lima, Jean-Marc Le Lanna, Xuan Mi Meyera, Xavier Jouliaa ; ∗,

Gibaek Leeb, En Sup Yoonc

aLaboratoire de G�enie Chimique (LGC, UMR-CNRS 5503), INPT-ENSIACET, 118 route de Narbonne, 31077 Toulouse, FrancebDepartment of Industrial Chemistry, Chung-Ju National University, Chung-Ju, Chungbuk, 380-702, South Korea

cSchool of Chemical Engineering, Seoul National University, Seoul, 151-742, South Korea

Received 11 May 2001; received in revised form 30 November 2001; accepted 5 March 2002

Abstract

The PBE (population balance equation) containing birth, growth, agglomeration and breakage kinetics is described by a conservationlaw with a moving source term. For the solution of the PBE, we compare two accurate front tracking methods such as a modi8ed methodof characteristics (MOC) and a 8nite di9erence method with the weighted essentially non-oscillatory (WENO) scheme. Both methodsare applied to a potassium sulfate crystallization problem (K2SO4–H2O system) with a discontinuous initial condition. Parameters ofagglomeration and breakage kinetics are estimated on the basis of the experimental data of the K2SO4–H2O system.

Owing to moving axis along a crystal growth rate (i.e. elimination of the growth term), the modi8ed MOC is able to provide a highlyaccurate solution even at discontinuous points without numerical di9usion error. However, in the case of sti9 nucleation which can commonlyappear in practical crystallization processes, it is necessary to adaptively determine time levels to add a mesh of the nuclei size. Forsolving PBEs involving agglomeration and breakage terms, the MOC can take more long computational time than the spatial discretizationmethods like the WENO scheme. It is pointed out that the MOC is not available to solve more than two coupled PBEs in general.

WENO schemes for spatial discretization are 8rstly addressed in this study for the dynamic simulation of batch crystallization processes.The WENO schemes show improvements of accuracy and stability over conventional discretization methods (e.g., backward, central orcommon upwinding schemes). However the WENO schemes on 8xed meshes show, to some extent, the numerical di9usion error neardiscontinuities or steep moving fronts like other 8nite di9erence methods. Hence, they require spatially-adaptive mesh techniques in orderto track more accurately the moving fronts. Even though the WENO schemes are less accurate than the MOC, they are of practical usefor solving complex PBEs owing to a short computational time and little limitation to use.? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Dynamic simulation; Crystallization; PBE (population balance equation); Modi8ed MOC (method of characteristics); WENO (weightedessentially non-oscillatory) scheme; Parameter estimation

1. Introduction

The population balance equation (PBE) has been demon-strated to describe the particle size distribution (PSD) invarious chemical engineering problems such as crystalliza-tion, polymerization, emulsion, L–L=L–G dispersion and mi-crobial culture. Indeed, modeling with the PBE providesa good description for parameter identi8cation, operating

∗ Corresponding author. Tel.: +33-5-6288-5831;fax: +33-5-6288-5600.

E-mail address: xavier.joulia@ensiacet.fr (X. Joulia).

conditions, process control and design. In crystallization pro-cesses, the PBE, which governs the crystal size distribution(CSD), is solved together with mass=energy balances andcrystallization kinetics such as nucleation, crystal growth,breakage and agglomeration. The system, which often leadsto hyperbolic-like IPDEs (integro-partial di9erential equa-tions), is complex due to a lot of feedback relationshipsbetween the equations (Wey, 1985). Fig. 1 shows an infor-mation How diagram in batch suspension crystallizers. Todetermine the CSD, all the equations (e.g., PBE, mass andenergy balances) must be solved simultaneously. An inac-curate solution of a PBE will a9ect particle nucleation andsubsequently particle growth and results in incorrect CSD.

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0009 -2509(02)00236 -1

3716 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

G CSD

BN n(L,t)

n

MT (=kvρ cm3)

∆w m3

∆w w

T(t)T(t)

Total moment

m3=∑(nL3∆ L)

Nucleation

B0=kn(T). fn(∆ w,MT)

Mass balance

dw/dt=-kv c(dm3/dt)

Super saturation

w=w-w*(T)

Growth

G=kg(T). fg(∆ w,L)

Cooling strategy

Temperature control

PBE

nt + (Gn)L = BN . δ (L0)

ρ∆

Fig. 1. Information How diagram of interrelations which determine CSD (crystal size distribution) in the batch crystallization process (Wey, 1985).

Therefore, a numerical procedure to get an accurate solutionof PBEs is necessary.

Litster, Smit, and Hounslow (1995) presented asecond-order spatial discretization for the growth term andan accurate and complex discretization for the agglomer-ation term on the 8ne geometric grid. Muhr, David, andVillermaux (1996) used a 8rst-order upwind scheme (or2-points backward di9erencing) for the spatial discretiza-tion of the growth term. However, an accuracy loss nearsteep fronts or discontinuities has been inevitable becauseof the low-resolution spatial discretization in these methods.

Kumar and Ramkrishna (1997) proposed a uni8eddiscretization technique on a non-uniform grid for the ag-glomeration and breakage terms. To avoid the numericaldissipation error caused by the growth term discretization, amodi8ed method of characteristics in which spatial dis-cretization is no longer required was implemented success-fully to PBEs even with a discontinuous initial condition. Toresolve the troublesome problem of handling a nucleationterm with this method using a moving pivot, a traditionalmethod of characteristics (MOC) is combined with a proce-dure of adding mesh of the nuclei size at given time levels.The ordinary di9erential equations (ODE) integrators (inparticular, a variable-step=variable-order BDF solver) re-cently well developed for sti9 ODEs determine adaptivelyor automatically the time-step for time integration accord-ing to sti9ness of systems (Furzeland, Verwer, & Zegeling,1990). When one of the ODE integrators is used, the in-tegrator stops and restarts at time levels given by user formesh addition of the nuclei size and rearrange of meshes.

Jauner and Jones (2000) employed a second order spa-tial discretization and Kumar’s discretization method for

the agglomeration and breakage terms in order to determinethe kinetic parameters, using a variable-step=variable-orderbackward di9erential formula (BDF) ODE integrator.

To our knowledge, the modi8ed method of characteristicsfor the strictly scalar hyperbolic PBE is very accurate andnumerically stable, if the PBE under consideration is nothighly sti9 with respect to time. In a sti9 PBE which cancommonly appear in practical crystal processes owing tonucleation, determination of time levels to add a mesh ofthe nuclei size becomes an important factor on the solution.However, this has not been detailed in the method postulatedby Kumar and Ramkrishna (1997).

The PBE is a conservation law with a moving source term.Hence high resolution front tracking methods that are of-ten used in the CFD (Computational Fluid Dynamics) 8eldfor spatial discretization can be employed without diNcul-ties. In this paper, we introduce a high resolution spatialdiscretization method based on WENO schemes proposedby Jiang and Shu (1996). The method is applied to a prac-tical crystallization process for the 8rst time. Its numericalresults are compared with those of the modi8ed MOC, us-ing a predictor–corrector BDF ODE integrator, DISCo (Sar-gousse, Le Lann, Joulia, & Jourda, 1999). Advantages anddisadvantages of both the WENO scheme and the MOC arepointed out through numerical tests.

In the Section, the discretized PBE with nucleation,growth, agglomeration and breakage kinetic terms is shortlyreviewed. For its numerical solution, the modi8ed MOCand the spatial discretization methods of high resolu-tion (i.e., WENO schemes with the arti8cial compressionmethod) are introduced. In Section 3, three numerical testsare performed on uniform or nonuniform grid systems. The

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3717

application into a potassium sulfate batch crystallizationproblem (K2SO4–water system) with the discontinuous seeddistribution is presented. Here, the PBE with nucleationand growth is solved by using the MOC and the WENOschemes. Parameters for agglomeration and breakage arethen estimated, minimizing an sum-of-squares of di9erencesbetween simulation results and available experimental data.

2. Discretized PBEs

The crystal size distribution (CSD) is usually expressedas the crystal number (N , no.) or number density (n, no=mor no=m3) with respect to the crystal size (L; m) or volume(v;m3). It is noted that when the crystal size (L) instead ofthe crystal volume (v) is used in the PBE, the density func-tion (n) is de8ned as crystal numbers per unit size (e.g., n:no.=m) instead of the volume based density (e.g., n: no.=m3).A simple relation between the crystal number (N ) and thecrystal number density (n) is given as follows, using the8nite volume approach:Ni =

∫ Li+1

LindL ≈ ni(Li+1 − Li)

Ni =∫ vi+1

vindv ≈ ni(vi+1 − vi):

(1)

For simplicity or practical purposes, the crystal number isapproximated like the right hand sides of Eq. (1), as usedby Kumar and Ramkrishna (1997). The two values are thusreversible easily.

Both bases (i.e., N and n) can give a good description ofthe CSD. However, the CSD based on the number (N ) isoften preferred, for conservation of the mass and the num-ber of crystals, in the cases involving agglomeration andbreakage kinetics.

A PBE as a governing equation of the CSD is usually ex-pressed in terms of birth of nuclei (BN ), their growth (G),and birth (BA, BB) or death (DA, DB) caused by agglomer-ation and breakage, respectively, as follows:

nt + (nG)L = BN · �(L− L0) + BA − DA + BB − DB; (2)

where the subscripts t and L denote the time and space par-tial derivatives. The kronecker delta, �(L− L0), means thatnucleation takes place only at a minimum nuclei size L0 (orvolume v0). It is noted that when the crystal size (L) in-stead of the crystal volume (v) is used in the PBE, the den-sity function (n) is de8ned as crystal numbers per unit size(e.g., n= no:=m) instead of the volume based density (e.g.,n=no:=m3). Eq. (2) is a scalar hyperbolic PBE with sourceterms and it involves only a convection term (@(nG)=@L)without any other spatial derivatives.

For conservation of the total number and mass, a num-ber (N )-based PBE suggested by Kumar and Ramkrishna(1996) is substituted for Eq. (2) from a 8nite volume ap-proach in space (i.e., L). The discretized number-based

PBE is given as

dNi

dt=(

dNi

dt

)nucleation

+(

dNi

dt

)growth

+(

dNi

dt

)agglomeration

+(

dNi

dt

)breakage

: (3)

These four kinetic terms are often explicitly expressed:

(dNi

dt

)nucleation

=

{BN ; i = 0

0; i �= 0(4)

(dNi

dt

)growth

= −d(Gi · ni) (5)

(dNi

dt

)agglomeration

=

i¿j¿k∑j; k

Li−16L6Li+1

�(

1 − 12�(Lj − Lk)

)· �(Lj; Lk) · Nj · Nk

−Ni

imax∑k=1

�(Li; Lk) · Nk (6)

(dNi

dt

)breakage

=imax∑j=i

bij · �(Lj) · Nj − �(Li) · Ni (7)

with

�=

Li+1 − LLi+1 − Li

; Li6L6Li+1

L− Li−1

Li − Li−1; Li−16L6Li

(8)

bij =∫ Li+1

Li

Li+1 − LLi+1 − Li

· fb(L; Lj) dL

+∫ Li

Li−1

L− Li−1

Li − Li−1· fb(L; Lj) dL (9)

where Lj and Lk in Eq. (6) denote a capturing crystal sizeand a captured crystal size in binary collisions, respectively.L= Lj + Lk in Eqs. (6) and (8). � is called the agglomera-tion eNciency or the agglomeration kernel. In the case of auniform 8xed grid, Eq. (6) reduces to

(dNi

dt

)agglomeration

=12

i∑j=0

�(Lj; Li−j+1) · Nj · Ni−j+1

−Ni

imax∑k=0

�(Li; Lk) · Nk: (10)

3718 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

Applying a uniform-binary breakage function (fb = 2=Lj)and a power-law breakage kernel (Hill & Ng, 1995) depend-ing on the size (i.e., �(Li) = �0Lai ) to Eqs. (7) and (9), thediscretized breakage term is analytically derived accordingto the size level.(

dNi

dt

)breakage

=

�0(Li+1 − Li) ·imax∑

j=i+1

LajLj

· Nj; i = 0

�0(Li − Li−1) · LaiLi· Ni + �0(Li+1 − Li−1)

·imax∑

j=i+1

LajLj

· Nj − �0Lai · Ni; i = 1 · · · (imax − 1)

�0(Li − Li−1) · LaiLi· Ni − �0Lai · Ni; i = imax

(11)

where �0 and a are parameters to be determined. Conserva-tion of the total number (m0) and mass (m3) was veri8ed forEqs. (6) and (11) by Kumar and Ramkrishna (1997). Here,the zeroth and third moments (m0 and m3) of the crystaldensity (n) are de8ned as

m0 =∫ ∞

0L0 · n dL ∼=

imax∑i=0

ni · PLi =imax∑i=0

Ni (12)

m3 =∫ ∞

0L3 · n dL ∼=

imax∑i=0

L3i · ni · PLi

=imax∑i=0

L3i · Ni (13)

The 8nite integral of Eq. (2) only for the growth term leadsto∫ Li+1(t)

Li(t)

@n@t

dL= −∫ Li+1(t)

Li(t)

@(nG)@L

dL (14)

The discretized growth term is derived by the Leibnitz rule.(dNi

dt

)growth

= +[n(Li+1; t) · Li+1 − n(Li; t) · Li]

− [n(Li+1; t) · Gi+1 − n(Li; t) · Gi] (15)

The above equation is handled di9erently in uniform ornon-uniform grid systems. On the uniform 8xed grid (Li =0and 8xed PL), it becomes a 8rst-order derivative:(

dNi

dt

)growth

= −d(GiNi)dL

(16)

On the non-uniform 8xed grid (Li = 0 and changeablePLi) such as geometric meshes, Eq. (5) is obtained and its

right hand side can be approximated in the upwindingsense:

(dNi

dt

)growth

= −d(Gi · ni) = −(f i+1=2 − f i−1=2) (17)

where f i±1=2 is called the numerical Hux. For the twoabove equations, a great number of 8nite di9erence dis-cretization methods such as central, backward or upwindingschemes have been presented in the literature (e.g., Wu,Fan, & Erickson, 1990; Litster et al., 1995; Muhr et al.,1996). However, they can produce much numerical erroror=and numerical instability near a steep front (Shu, 1997).In the next section, stable methods with accuracy will beintroduced.

In the case of moving grid systems, Eq. (15) is usedbecause meshes move with time (i.e., Li �=0). The PBE ofEq. (2) has only one scalar conservation law equation withsource terms and there is an explicit and unique How speed,@(nG)=@n = G, that is, the crystal growth rate. In this case,since the mesh velocities (Li) are equal to the growth rates(Gi), Eq. (15) reduces to the following equation called themethod of characteristics (MOC).

dNi

dt= 0

dLidt

= Gi

(18)

2.1. Modi;ed method of characteristics

It is well known that for the scalar linear conservation law(e.g., PBE considered here) there usually exists a uniquecharacteristic curve along which information propagates. Ifthe solution moves along the pathline of propagation, theconvection term (@Gn=@L in the PBE) disappears like in Eq.(18).

The total derivative of n depending on L (or v) and t leadsto the following equation:

nt =dndt

− G ·(

dndL

)(19)

When the growth rate (G) is independent of the size orvolume (L or v), substituting Eq. (19) for Eq. (2), the PBEis split into two coupled ODEs based on the crystal density(n):

dn=dt = BN · �(L− L0) + BA − DA + BB − DB (20)

dL=dt = G (21)

In the case of a nonlinear convection term, @(Gn)=@L, whichmeans that the growth rate (G) depends on the size (L),

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3719

Fig. 2. Renumbering of meshes (Li) and numbers (Ni) at t = tj+1 in themodi8ed MOC (M : the number of meshes).

Kumar and Ramkrishna (1997) convert the PBE into a PBEbased on the crystal number (N ):

dNdt

=(

dNdt

)nucleation

+(

dNdt

)agglomeration

+(

dNdt

)breakage

(22)

dLdt

= G (23)

For the agglomeration and breakage terms, Eqs. (6) and(7) are used. In Eqs. (20) and (22), there is no convectionterm which would cause much numerical error and instabil-ity. Hence the solution obtained by the MOC is very accu-rate and stable. To overcome the nucleation problem, a newmesh of the nuclei size is added at given time levels. Thesystem size can be kept constant by deleting the last meshat the same time levels. As a result, all variables (Ni and Li)are reinitialized at these time levels and the time integratorrestarts.

Apparently, the total number and mass are conserved inthe modi8ed MOC. However, it is still to be shown whethermesh addition or deletion a9ects, in particular, mass conser-vation. Suppose that a number-based PBE with nucleationand growth is solved by the modi8ed MOC on M meshes.Fig. 2 depicts crystal sizes (Li) and crystal numbers (Ni)in the last part of mesh points. They are renumbered at anew time level (tj+1) due to an adding mesh of the nucleisize (Lj+1

0 ). The crystal numbers and sizes are not changed,i.e., Nj+1

M−i = NjM−(i+1) and Lj+1

M−i = LjM−(i+1), in this 8gure.

To keep constant the system size, the last mesh (Lj+1M+1) is

deleted at the new time level (tj+1). To conserve the totalcrystal number, the crystal number (Ni+1

M+1) correspondingto the last mesh deleted at tj+1 is added to the previous point(Nj+1

M ), i.e., (Nj+1M )after deletion =Nj

M−1 +NjM . But it does not

imply the conservation of the total mass. In fact, one can-not simultaneously conserve the crystal number and mass inthe deleting procedure but rather choose only one property.

0

50

100

150

200

250

300

350

400

1.0E-05 6.0E-05 1.1E-04 1.6E-04 2.1E-04 2.6E-04

Crystal volume (v)

Cry

stal

num

ber

dens

ity (

n)

Analytic solution

One-mesh addition

Two-mesh addition

Fig. 3. The e9ect of the frequency of mesh addition on the crystal numberdensity (n).

In the mesh adding procedure, the same problem occurs.When the crystal number is conserved, di9erence of the thirdmoment (m3) is calculated, using Eq. (13) in the adding anddeleting procedures at tj+1:

Pm3 = +N0 · (L31 − L3

0) − NM+1 · (L3M+1 − L3

M ) (24)

where L0 is an added mesh of the nuclei size and LM+1 is thelast mesh size to be deleted. Mesh addition leads to a massincrease, but mesh deletion results in a mass loss. In the caseof small step sizes (PLi = Li+1 − Li) or=and small numbersof crystals (N0 and NM+1), the e9ects of mesh addition anddeletion on the total mass are negligible. Normally NM+1

can be small if the last crystal size is set to be suNcientlylarge. If so, the second term of Eq. (16) is suNciently small.However, in practical crystallization processes the value ofN0 is usually large. Hence the 8rst step size (PL0) must besmall for mass conservation. That is, small time step sizes(Ptj) for a new mesh addition (L0) are desirable.

Another reason to set a small time step size is foundin the case of sti9 nucleation. For example, consider adensity-based PBE with nucleation and growth terms. Sup-pose that no initial seed lies within 1:0× 10−56 v6 2:6×10−4, and the growth rate is constant (G = 1). Given adensity-based nucleation with time, n = 1 × 1010 · t2, theproblem is solved by the MOC, adding one mesh or twomeshes for 06 t6 2:0 × 10−4. As shown in Fig. 3, themass is much overestimated for one mesh addition withrespect to the analytic solution. Indeed, solution quality ofthe MOC varies according to the frequency of mesh addi-tion. That is, addition of a signi8cant number of meshesis required. When the nucleation rate is sti9 with time andthe growth rate is nonlinear (e.g., size-dependent growthrate), it is not trivial to determine the time steps (Ptj) orthe frequency for adding a minimum mesh.

It is worth noting that the method of characteristics hassome restrictions to use. If there is another PDE (not ODE)coupled with a PBE and mesh velocities (Li) are not equalto the growth rates (Gi), the MOC cannot be used (see

3720 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

Eq. (15)). For example, in the case of several crystal specieswith di9erent growth rates, there is no unique characteristiccurve.

Moreover, when an adaptive time-step solver (such as apredictor–corrector BDF integrator) is used for sti9 ODEsystems, two basic problems arise. One is how to choose thetime levels to add a new mesh and to reinitialize variables.The short time step is compulsory for sti9 PBEs originatedeventually from nucleation, as mentioned above.

The second problem is the calculation time. Since onePDE becomes two coupled ODEs in the MOC and thetwo ODEs discretized on M spatial grid points provide a(2M × 2M) Jacobian matrix. The MOC will likely takelonger calculation time than 8nite di9erence methods whichhave only a (M ×M) Jacobian matrix. To reduce the com-putational time, a relatively small number (M) of meshes isdesirable. However, this conHicts with the short time steprequirements.

In this study, the number of crystals is only conserved inthe mesh addition and deletion levels, when the modi8edMOC is used. An adaptive mesh method, which can reducethe error in the mesh addition step, is presented in a subse-quent paper of Lee, Yoon, Lim, Le Lann, and Joulia (2001).

2.2. High resolution spatial discretization method forgrowth terms

Alternative methods for sti9 PBEs are, without any mod-i8cation of the original PBE, the high resolution spatialdiscretization methods such as ENO (essentially nonoscil-latory) or WENO (weighted ENO) schemes. These TVD(total variation diminishing) methods guarantee a stablesolution even near discontinuities and give more accurateresults than traditional 8nite di9erence methods (backward,central or common upwinding schemes).

Letfx be the convection term, corresponding to (@Gn=@L)or (@Gn=@v), which can be discretized into kth-order accu-rate conservative form

(fx)i =(f i+1=2 − f i−1=2)

Px(25)

where f i+1=2 and f i−1=2 are numerical upHux and down-Hux, respectively, and the uniform mesh size, Px, is usedover the discretization procedure. For the kth accuracy-orderreconstruction of the numerical Hux (f i+1=2), three di9erentapproaches, that is, 8xed stencil (FS), adaptive stencil (AS)and weighted stencil (WS), are proposed by Lim, Le Lann,and Joulia (2001). In the FS approach (i.e., traditional meth-ods), the stencil (Si) taken to approximate spatial deriva-tives is 8xed with both its number and its position over thediscretization procedure. The AS approach (ENO schemes)uses just one selected stencil out of some candidate sten-cils when doing the reconstruction, retaining the same ac-curacy order. Whereas the WS (WENO schemes) approachuses all the candidate stencils, each being assigned a weightwhich depends on the local smoothness of the numerical

solution. Finally, the WS method is a technique combiningthe FS method with the AS method and it is usually morerobust than the others (Lim et al., 2001). In Appendix A, thethird-order WENO scheme with the arti8cial compressionmethod (Jiang & Shu, 1996) is presented in detail.

The discretized PBE based on the crystal number (N ), Eq.(3), is solved, employing the WENO schemes to Eq. (16) or(17). For these schemes, considerable numerical di9usionerror still exists near discontinuities or steep fronts in thecase of insuNcient number of meshes. Therefore, the crystalnumber and the mass (m0 and m3) could not be conserved,like in the modi8ed MOC.

3. Numerical results

Three examples are presented in this section. To morereasonably measure the solution accuracy of PBEs, the localand global errors are considered, since the former is relatedto the total mass conservation and the latter to the total num-ber conservation. The local error is the L1 error (L1-error)for accuracy measurement on each mesh points, using thetrapezoidal rule:

L1-error =imax−1∑i=0

12

(|(ni)reference − (ni)numerical|

+ |(ni+1)reference − (ni+1)numerical|)×(Li+1 − Li) (26)

The global error is the zeroth moment error (m0-error) whichmeans the di9erence in total crystal numbers between thereference solution (if possible, analytic solution) and thenumerical solution.

m0-error = |(m0)reference − (m0)numerical| (27)

where (m0)reference =∫ LfL0

(n)reference dL, and (m0)numerical =∑imax−1i=0 ni · (Li+1 − Li). Lf is a 8nal crystal size. Both of

the two errors are expressed in the crystal number. Thus, thetotal error (total-error) can be de8ned as

total-error = (L1-error) + (m0-error) (28)

In the next section, the MOCs (MOC-5p, MOC-20p orMOC-50p) that di9er according to the number of meshesadded or deleted (5p, 20p or 50p, respectively) aretested. We also test the WENO schemes of third-orderor 8fth-order accuracy without the arti8cial compres-sion method (WENO-3 or WENO-5) and with the ACM(WENO-3A or WENO-5A).

All of the numerical tests are performed on the PC (oneAMD Athelon 750 MHz processor). The computational timeis measured for a given single-task on the processor.

3.1. Simple PBE with nonsti? nucleation

In the case of a PBE only containing a growth term,the MOC gives a quasi-analytic solution. When a 8nite

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3721

0.5 1 20

0.5

1

1.5

v

n

(a) FS-upwind-1

0.5 1 20

0.5

1

1.5

v

n

(b) FS-central-2

0.5 1 20

0.5

1

1.5

v

n

(c) WS-Roe-5

0.5 1 20

0.5

1

1.5

v

n

(d) WS-Roe-5A

Fig. 4. CSDs on the uniform 8xed grid, obtained by the four spatial discretization methods at t = 1:0 (solid line: analytic solution).

di9erence approximation (e.g., central, upwinding andWENO schemes) is used for the growth term, its solutionoften su9ers from the numerical di9usion (or dissipation)error near discontinuities or steep fronts. Suppose G = 1within 10−36 v6 2 and n0(v0;∀ t) = 1 at v0 = 10−3, asa boundary condition. The particle density values (ni) areconverted to the particle number values through a simple re-lation, Ni = ni(vi+1 − vi) of Eq. (1). After the number-basedPBE is integrated with time, the results are reversed intodensity values with the same relation function. The PBEbased on the particle number is solved on uniform ornonuniform grids, using the BDF ODE integrator.

3.1.1. Finite di?erence approximations on uniform ;xedgrids

The PBE are solved numerically on uniform 8xed grids(100 meshes, Pv= 0:01999) with ATOL = RTOL = 10−6.Two traditional 8nite di9erence methods, 8rst-order back-ward method (FS-upwind-1) and second-order centralmethod (FS-central-2), are compared with the WENOschemes (WENO-Roe-5 or WENO-5 and WENO-Roe-5Aor WENO-5A). From Eq. (16),

N i = −(

dNi

dvi

)= − (f i+1=2 − f i−1=2)

Pv(29)

For the two WENO schemes, a parameter for the non-zerodenominator, �, is set to 10−6. In the ACM, the compressionfactor, � = 40.

Table 1Comparison of accuracy and computational time according to the numer-ical methods in the case of constant nucleation (t = 1:0 s)

L1-error CPU time

FS-upwind-1 0.11255 1:1 sUniform 8xed grid FS-central-2 0.10026 1:2 s(101 meshes) WS-Roe-5 0.02745 3:4 s

WS-Roe-5A 0.01163 73 s

MOC-5p (20 meshes) 0.0 1:5 s

Fig. 4 depicts the crystal size distributions (CSDs) ob-tained by the four discretization methods at t = 1:0. TheFS-upwind-1 scheme produces inaccurate results. The cen-tral scheme (FS-central-2) shows spurious oscillations neara shock. The solutions of the two WENO schemes are ac-curate and stable. But they take longer computational time,as shown in Table 1 that reports performance of the spatialdiscretization methods on accuracy and temporal eNciency.

3.1.2. Modi;ed MOC on nonuniform 20 meshesApplying the modi8ed MOC, the numbers (Ni) and the

sizes (vi) are initialized as follows at t = 0:

N0 = 0; N1 = n0 · (v2 − v1); Ni = 0

for i = 1; : : : ; 20 (30)

3722 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

0

0.5

1

1.5

0 0.5 1 21.5Volume (v)

Den

sity

(n)

Analytic solution

WENO-5 on 20 uniform meshes

MOC-5p on 20 nonuniform meshes

Fig. 5. CSDs obtained on 20 meshes by the modi8ed MOC and the WENO-5.

v0 = v1 = 10−5; vi = v0 · 1:5(i−1) for i = 1; : : : ; 20(31)

The discretized PBE is expressed for t ¿ 0:

N0 = n0 · (v1 − v0);dNi

dt= 0 for i = 1; : : : ; 20

dv0

dt= 0;

dvidt

= 1 for i = 1; : : : ; 20 (32)

At 8ve given time levels (e.g., Pt=1=5), the smallest meshsize (v0) is added and the largest mesh size is deleted tokeep the system size constant. At t = j · Pt for j = 1; : : : 5,the system is renumbered and is reinitialized.

N0 = 0 and v0 = 10−3 (33)

Ni = Ni−1 and vi = vi−1 for i = 1; : : : ; 20 (34)

In Fig. 5, it is found that the MOC provides a solution freeof the numerical dissipation error, when the initial grid iswell given. The WENO-5 scheme used to approximate thegrowth term gives a stable solution. But near a discontinuouspoint (v = 1) considerable error is shown because of bothnumerical dissipation error and insuNcient mesh points. TheMOC-5p is thought of as the best method among the 8vemethods tested, as reported in Table 1 for the simple PBEwith non-sti9 nucleation.

3.2. Simple PBE with sti? nucleation

Suppose that sti9 nucleation takes place only at a mini-mum crystal size (L0 = 10−5) as a function of time:

n(t; L0) = 100 + 106 exp(−104(t − 0:215)2) (35)

Within the size range, 10−56L6 2:0, the nuclei grow andthe crystals aggregate as well as break for 0:06 t6 0:5.The square initial condition as seeds is also given: for0:46L6 0:6; n(0; L) = 100, elsewhere n(0; L) = 0:01.The kinetic parameters are given: G = 1 (linear growthrate), � = 1:5 × 10−5 (constant agglomeration kernel) and� = 1:0 × L2 (breakage kernel). The discretized PBE basedon the crystal number (Ni) or the crystal density (ni) issolved, using the adaptive time-step BDF ODE solver. Inthe density-based PBE, the absolute and relative integra-tion tolerances (ATOL and RTOL) are set to 1 × 10−2

and 1 × 10−5, respectively. In the number-based PBE,ATOL = 10−4 and RTOL = 10−4.

3.2.1. Nucleation and growthThe PBE with the nucleation and growth terms is consid-

ered on the basis of the crystal density. Its analytic solutionwith Eq. (35) is known as

n(t; L) = 100 + 106 exp(−104((Gt − L) − 0:215)2)

for 0:06L6Gt (36)

n(t; L) = 100 for 0:46 (L− Gt)6 0:6 (37)

elsewhere

n(t; L) = 0:01 (38)

In the solution, a discontinuous shock (square seed) and anarrow wave (originated from nucleation) move along thepropagation pathline, L = L0 + Gt. The numerical tests arecarried out on the 8xed-200 grids for both the MOC and theWENO schemes.

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3723

Table 2Error and calculation time according to the numerical methods in the case of sti9 nucleation without agglomeration and breakage (t = 0:5 s)

Error Calculation time

L1-error m0-errora Total-error

MOC-5p 31.56 7189.1 7221 8:7 sMOC MOC-10p 15.78 12456 12472 8:8 s(201 grids) MOC-20p 26.28 5893.2 5919 12:5 s

MOC-50p 22.01 9.42 31 16:7 s

WENO-3 22381 0.69 22382 40:7 sWENO WENO-5 16738 0.51 16739 18:4 s(200 grids) WENO-3A 17789 0.36 17789 571:2 s

WENO-5A 13308 0.47 13308 597:4 sa(m0)analytic = 17794:55.

1.E+06

1.E+05

1.E+04

1.E+03

1.E+02

1.E+01

1.E+00

1.E-01

1.E-020.0 0.5 1.0 1.5 2.0

Crystal size (L)

Num

ber

dens

ity (

n)

Analytic solutionWENO-3WENO-5AMOC-50p

Fig. 6. CSDs for the sti9 nucleation case without agglomeration andbreakage.

Table 2 shows results qualitatively close to the previ-ous example. However, it is important to choose carefullytime-steps to add meshes. If not, error increases abnormallyas in the MOC-10p case. When suNcient grid addition (50points) is taken in the MOC, the sharp pro8le from sti9 nu-cleation is caught accurately. The MOCs are more eNcienton the computational time and the accuracy than the WENOschemes. Even though the system size of the MOC is fourtimes larger than that of the WENO scheme, the MOC doesnot take more computational time due to the simple nucle-ation and growth equations. In this problem, it is not diNcultto determine time-steps to add meshes, since growth rate isconstant and birth rate equation is not complex.

In the results of the WENO schemes, the global error(m0-error) is relatively small with respect to the local error(L1-error), which means that the total number (m0) is wellconserved but the total mass (m3) related to the L1-errorcannot be conserved. Note that the WENO-5=5A convergesmore rapidly than the WENO-3=3A respectively. In Fig. 6,the numerical results of the WENO-3=5A and MOC-50p arecompared with the analytic solution. The WENO schemeswith the ACM usually tracks better right-shocks thanleft-shocks.

3.2.2. Nucleation, growth, agglomeration and breakageIn this example, the agglomeration and breakage kinetics

are added to the previous PBE. The discretized PBE basedon the crystal number, Eq. (3) is employed. The nucleationrate is calculated analytically from Eq. (35) at L=L0, usingthe Leibnitz rule:

BN = (dn0=dt) = −2 × 1010(t − 0:215)

×exp (−104(t − 0:215)2) (39)(dN0

dt

)nucleation

=∫ L1

L0

dn0

dtdL+ (n1L1 − n0L0) (40)

Since in the 8xed grid there is no mesh movement, the aboveequation reduces to(

dN0

dt

)nucleation

= BN · (L1 − L0) for 8xed grids: (41)

For the moving grid system of the MOC, Eq. (40) isde8ned as(

dN0

dt

)nucleation

= BN (L1 − L0) + n0L1

for the MOC: (42)

Here, L0 is an inserted mesh size 8xed with time (L0 = 0)with the empty crystal number (N0 = 0). L1 starts to growat L0. Thus, N1 = n0 · (L1 − L0).

At 8rst, the numerical solution of Eq. (3) incorporatedwith Eqs. (10), (11), (16) and (41) is obtained on 101points of the uniform grid, using the WENO-5 scheme forthe growth term. Fig. 7 depicts e9ects of the growth, ag-glomeration, and breakage terms on the CSD. The solidline is the analytic solution for pure growth problem with-out agglomeration and breakage. Considerable numericalerror is found in steep regions (or discontinuities) in Fig.7(a). The agglomeration term causes peaks of the CSD tobe smeared and increases the population of large crystalsizes. In contrast, the breakage term increases the popula-tion of small crystal sizes. Therefore, the CSD of the PBEwith the four kinetics is dispersed more broadly as shown inFig. 7(d).

3724 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

0 0.5 1 1.5 2

10-2

100

102

104

106

crystal size, L

den

sity

,n

(a) Pure growth

0 0.5 1 1.5 2

10-2

100

102

104

106

crystal size, L

den

sity

,n

(b) Growth+Agglomeration

0 0.5 1 1.5 2

10-2

100

102

104

106

crystal size, L

den

sity

,n

(c) Growth+Breakage

0 0.5 1 1.5 2

10-2

100

102

104

106

crystal size, L

den

sity

,n

(d) All kinetics

Fig. 7. CSD changes obtained by the WENO-Roe-5 scheme on 101 meshes according to the growth, agglomeration, and breakage terms (solid line:analytic solution for pure growth).

Next, the MOC-20p (inserting=deleting of 20 meshes) isemployed under the same condition, to solve Eq. (22) andthe following mesh equations:

dL0

dt= 0;

dLidt

= 1 for i = 1; : : : ; 101 (43)

Terms in Eq. (22) are taken from Eqs. (6), (11) and (42).It is noted that for the agglomeration term Eq. (6) is usedrather than Eq. (10), since the MOC is usually performed onnon-uniform meshes. For mesh size variables (Li), ATOL=RTOL = 10−5. The Jacobian matrix is calculated numeri-cally, too.

In Fig. 8, CSD changes are depicted according to the ki-netics used. Since numerical di9usion error is little, highresolution is examined on the corner of shocks or waves.Comparing Fig. 8(d) with Fig. 7(a), the e9ects of agglom-eration and breakage on the CSD are similar, to some ex-tents, to those of the numerical di9usion error in the WENOscheme.

Table 3 reports comparisons of the error and the computa-tional time for the WENO schemes (WENO-Roe-5=5A) andthe MOCs (MOC-20p on 102 meshes and MOC-15p=20pon 42 meshes). The MOC-20p on 102 meshes is consid-ered as a reference solution, as the analytic solution for thisexample does not exist. This method takes a long compu-tational time because of the large system size and the com-plex kinetic terms such as agglomeration and breakage. In

particular, the long calculation time is mainly caused by theagglomeration term coupled highly with the mesh move-ment. To reduce the computational time, the MOC-15p=20pon 42 meshes are performed. The L1-error is measured bythe cubic spline interpolation on the basis of the solution ofthe MOC-20p on 102 meshes.

The reduction of the number of meshes decreases the com-putational time. The MOC-20p on 42 meshes gives eNcientresults on accuracy and temporal performance. However,the MOC-15p on 42 meshes shows poor results because ofinsuNcient mesh addition. As a result, the MOC does notalways guarantee an accurate solution nor eNcient compu-tational performance. For the WENO schemes, the L1-errordominates the m0-error in the total-error. Their computa-tional times are relatively short owing to the small systemsize and a uniform mesh structure.

In Fig. 9, the population density distributions consideringthe four kinetics are shown within a semilog coordinate. Forthe same kinetic parameters, the results of the WENO-5Ascheme are more broadly dispersed than the reference solu-tion of the MOC-20p on 102 meshes. In other words, whenagglomeration and breakage parameters are estimated withthe WENO schemes, these values would be underestimated.The numerical di9usion error absorbs e9ects of agglomera-tion and breakage on the CSD.

Again, the MOC-15p on 42 meshes produces much er-ror due to insuNcient mesh addition. As mentioned above,

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3725

0 0.5 1 1.5 2

10-2

100

102

104

106

crystal size, L

crystal size, L crystal size, L

crystal size, L

dens

ity, n

dens

ity, n

dens

ity, n

dens

ity, n

(a) Pure growth

0 0.5 1 1.5 2

10-2

100

102

104

106

(b) Growth+Agglomeration

0 0.5 1 1.5 2

10-2

100

102

104

106

(b) Growth+Breakage

0 0.5 1 1.5 2

10-2

100

102

104

106

(d) All kinetics

Fig. 8. CSD changes obtained by the MOC-20p on 102 meshes according to the growth, agglomeration, and breakage terms (solid line: analytic solutionfor pure growth).

Table 3Error and calculation time according to the numerical methods in the case of sti9 nucleation with agglomeration and breakage (t = 0:5 s)

Error Calculation time

L1-error m0-error Total-error

MOC-15p on 42 meshes 8978 6863 15841 413 sMOC-20p on 42 meshes 24 14 38 508 sMOC-20p on 102 meshesa — — 4 h 53 min

WENO WENO-5 11480 542 12022 44 s(101 grids) WENO-5A 10711 541 11252 191 s

aMOC-20p on 102 meshes is the reference solution (m0 = 11539).

suNcient mesh addition is desirable in the sti9 nucleationcase.

3.3. Potassium sulfate batch crystallization process(K2SO4–H2O system)

In the previous sections, the basis studies for the dis-cretized PBEs were performed. Here a practical crystalliza-tion problem is solved. First of all, a PBE for the potassiumsulfate crystallization is described. Parameters of the ag-glomeration and breakage kinetics are estimated, minimiz-ing the summation of di9erences between experimental andnumerical results. To diminish the numerical error, the highresolution spatial discretization methods and the MOCs areemployed for the solution of the PBE.

3.3.1. Problem descriptionWith programmed cooling in a MSMPR (Mixed-Suspen-

sion and Mixed-Product-Removal) vessel for the potassiumsulfate crystallization, the PBE is described for the crystaldensity distribution (n, number of crystals=kg-solution=m):

@n(L; t)@t

+@(G(L; t) · n(L; t))

@L= BN�(L− L0) (44)

Assuming that an e9ective size (L0) of crystal birth is equalto 50 �m, the kinetic data according to Jones, Buds, andMullin (1986) are obtained as follows:

dN0

dt= B0(L0; t) = 1:56 × 109 e−9300=RT ·MT · !

for 0:0026MT 6 0:02 (45)

3726 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

0.0 0.5 1.0 1.5 2.0

Crystal size (L)

Num

ber d

ensi

ty (n

)

Analytic solution for pure growthMOC-20p on 102 meshesMOC-20p on 42 meshesMOC-15p on 42 meshesWENO-5A on 101 meshes

Fig. 9. Population density distribution for the sti9 nucleation case withgrowth, agglomeration and breakage.

G(L6 700 �m)

=86:4 e−40400=RT · (1 + 2L2=3) · !2 (46)

G(L¿ 700 �m) = G(700) ·(

L700

)0:5

(47)

Here the magma density (MT , kg=kg-solution) calculated bythe third moment (m3) of the CSD, and the relative super-saturation (!) are de8ned as

MT = "c

∫kv · L3 · n(L; t) dL

≈ "cM∑i=0

kv(Li) · L3i · ni · (Li+1 − Li) (48)

! =w − w∗w∗ ; (49)

where the crystal density "c = 2660 kg=m3 and M is thenumber of meshes. The volumetric shape factor (kv) is pro-posed according to the crystal size (L) by Budz, Jones, andMullin (1987b).

for 0¡L¡ 100 �m;

kv = 0:898 exp(0:168L0:5 − 8:234 × 10−3L) (50)

for 100 �m¡L¡ 2000 �m;

kv = 4:460 exp(−0:0797L0:5 + 6:76 × 10−4L) (51)

Unfortunately, the above equations are discontinuous at L=100 �m. Since discontinuous equations with respect to thesize (L) provoke a serious problem in Eqs. (21) or (23) ofthe MOC, the kv function is reformulated with a sigmoidfunction which does not a9ect the original functions butconverts them into a continuous function.

kv = (1 − &) · [equ: (50)] + & · [equ: (51)]; (52)

where &= exp(0:2(L− 100))=(1 + exp(0:2(L− 100))). Theconcentration, w (kg−K2SO4=kg-solution) is the prevailingsolution concentration. The equilibrium solubility (w∗) isexpressed as the function of temperature (T;◦ C) .

w∗(T ) = 6:66 × 10−2 + 2:3 × 10−3T + 6:0 × 10−6T 2

(53)

The temperature pro8le (Jones & Mullin, 1974) for constantsupersaturation cooling (Pw=0:00732) is programmed withtime for 180 min.

T (t) = 70 − 45 e−0:008356(180−t): (54)

To obtain larger crystals, the seed (ws = 0:00236 kg=kg) isput initially. The seed is assumed as a discontinuous initialcondition of the rectangular distribution in the crystal num-ber (Ns, no.=kg).

Ns =ws

"c∑

i (kv)i · L3i

for 500 �m6Li ¡ 600 �m: (55)

The crystal number density (ns, no.=kg=m) of the seed isobtained from the above equation:

ns(Li) =Ns

(Li+1 − Li)for 500 �m6Li ¡ 600 �m (56)

The boundary condition (n0 or N0) at L0 = 50 �m is alsode8ned as

n0 =dN0

dt· dtdL0

=B0(L0; t)G(L0; t)

N0 = n0(L1 − L0) (57)

For the mass conservation of solute in the liquid and solidphases, a governing equation must be added in the PBE.

dw=dt = −"c(

d(kv · m3)dt

)(58)

d(kv · m3)=dt = 3∫ Lmax

Lmin

kv · L2(t) · n(L; t) · G(L; t) dL (59)

where m3 denotes the third-moment of the CSD. Lmin =50 �m and Lmax = 1900 �m. For the initial condition of Eq.(58), w0 =w∗(T0)+7:32×10−3 (kg=kg). Finally, the batchcrystallization model aims to solve one PBE as well as onemass balance equation which are associated with the birthand growth kinetics subject to the cooling strategy like Eq.(54), as shown in Fig. 1.

3.3.2. Solution of the PBE with nucleation and growthOn the uniform 8xed grid, the number based discretized

PBE is solved on 101 meshes (orM=100), using the WENOschemes (WENO-5 and WENO-5A):

dNi

dt=(

dN0

dt

)nucleation

−(

dNi

dt

)growth

for i = 0; : : : ; M

dwdt

= "cd(kvm3)

dt(60)

where Eq. (45) for the nucleation term and Eq. (25) forthe growth term are used. The number of unknowns or

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3727

equations is (M + 2). As the initial condition, Eqs. (55)and (57) are used. The constant seed number (Ns, no.=kg)within 500 �m6Li ¡ 600 �m is set to Ns = 1067:11. Theinitial nucleation number is assumed to N0 = 61436:76 atL0 = 50 �m. The numerical solution during the integrationtime of 180 min is obtained for the absolute and relativetolerances of 1 × 10−3 and 1 × 10−6, respectively.

For the modi8ed MOC, the number based PBE can beexpressed:

dN0

dt=(

dN0

dt

)nucleation

;(

dNi

dt

)= 0 for i = 1; : : : ; M

dL0

dt= 0;

dLidt

= G(Li) for i = 1; : : : ; M

dwdt

= −"c d(kvm3)dt

(61)

Here, the number of unknowns is (2M + 3). The nu-merical solution of the MOC-30p on 102 mesh points(M = 101) is considered as the reference solution.In the MOC-30p, the mesh points are uniformly ini-tialized within 50 �m6Li6 1250 �m. Thus, N0 = 0,N1 = 39850:87 and Ns = 671:12. To reduce the computa-tional time, the MOC-15p on 42 meshes (M = 41) is alsotested. The meshes are also uniformly set at t = 0 within50 �m6Li6 1250 �m. In this case, N1 = 99627:18 andNs = 1357:38, as the initial condition. As the crystal num-ber (Ni) is a sum of particles within the corresponding sizeinterval, a larger interval produces more particles, as on the42 meshes.

New meshes of the nuclei size are inserted regularly. Thatis, the mesh is inserted at each constant time step Pt =180 min=30 or 180 min=15. The last mesh (LM+1) is deletedat these time levels, adding the corresponding crystal number(NM+1) into its neighboring mesh (LM ).

Mass gain and loss in the addition and deletion proce-dures of the MOCs can be obtained from Eq. (24). From theexperimental results (Jones and Mullin, 1974), the crystaldensity (nf) of the last mesh size (Lf = 1900 �m) is esti-mated. Total mass gain and loss (wgain=loss, kg=kg) is roughlycalculated for 180 min:

wgain=loss = 30 × ("c · kv · Pm3)

= +3:23 × 10−4 − 9:11 × 10−6

for the MOP-30p on 102 meshes (62)

wgain=loss = 15 × ("c · kv · Pm3)

= +1:45 × 10−3 − 5:78 × 10−5

for the MOP-15p on 42 meshes (63)

Recall that ws = 2:36 × 10−3 as the seed mass andwf = 116:61 × 10−3 as a 8nal yield from the experimentaldata. The e9ect of mesh deletion on mass loss is relativelysmall, as mentioned above. However, the modi8ed MOCs

1.E+03

1.E+05

1.E+07

1.E+09

1.E+11

0 0.0005 0.001 0.0015 0.002

Crystal size (m)

Num

ber

dens

ity (

no/m

/kg)

Experiment dataMOC-30p on 102 gridsWENO-5 on 101 gridsWENO-5A on 101 gridsMOC-15p on 42 grids

Fig. 10. Population density distribution of the K2SO4–H2O system withoutagglomeration and breakage terms.

introduce intrinsically an error in mass of about 0.5–1.5%during the mesh addition procedure, when mass gain iscompared with the 8nal yield (wf). On the other hand, inthe 8fth-order WENO schemes, the truncation error for thegrowth term approximation is proportional to (PL)5, wherePL is the uniform mesh interval. The error in mass origi-nated from the truncation error can be greater than that ofthe modi8ed MOC, as illustrated in the previous section.

In Table 4, the errors and the calculation times are re-ported for the four methods tested. The MOCs give accuratesolutions. But the calculation times increase a lot because ofthe highly nonlinear PBE model. The numerical results ofthe WENO-5=5A and the MOC-15p=30p are compared withthe experimental data in Fig. 10. Its real solution will be thata square wave from seeds and a steep front from nucleationmove along the crystal size with time.

It is sure that the solution of the PBE involving only nu-cleation and growth kinetics does not agree with the experi-mental data. Moreover, the growth rate and birthrate are toomuch elevated with regards to the experimental data. In fact,these kinetic data were obtained by correlation to experimen-tal data without taking agglomeration and breakage into ac-count. Budz, Jones, and Mullin (1987a) suggested reductionof about 10% for the growth rate, when the agglomerationkinetic is considered. It is also inadequate to use directly thebirthrate of Eq. (45) obtained within 0:0026MT 6 0:02,because magma density variation is ten times larger in thisproblem, i.e., 0:0026MT 6 0:2. In this study, the growthrate and the birthrate are corrected by a reduction of 20% inthe kinetic constants. That is, Eqs. (45) and (46) are revisedas follows:

G = 73:44 e−40400=RT · (1 + 2L2=3) · !2 (64)

{B0 = 1:326×109e−9300=RT ·MT ·!; 0:0026MT¡0:02;

B0 = 1:326×109e−9300=RT ·√

0:02MT ·!; 0:026MT :(65)

3728 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

Table 4Error and calculation time according to the numerical methods in the case of the K2SO4–H2O system without agglomeration and breakage terms(t = 180 min)

Logarithmic error Calculation time

L1-error m0-errora Total-error

MOC MOC-15p on 42 grids 6.2756 0.0538 6.329 24 sMOC-30p on 102 grids — — — 184 s

WENO WENO-5 6.5681 0.0952 6.663 10:6 s(101 grids) WENO-5A 6.5339 0.0853 6.619 251 s

alog10(m0)reference = log10∑

i12 (ni+1 + ni)(Li+1 − Li) = 7:1788.

The new kinetic equations diminish the growth rate of crys-tals. In Eq. (65), the crystal density (n0) or number (N0) ofnuclei at L0 have the same values as those in Eq. (45) untilMT = 0:02, since they are de8ned by Eq. (57).

In the next section, the agglomeration and breakage termsare taken into account in the discretized PBEs. Their param-eters are estimated, minimizing di9erences between experi-mental data and simulation results.

3.3.3. Parameter estimationThe agglomeration and breakage kinetics are added into

the PBE, incorporating with Eqs. (6) and (11). After Budz,et al. (1987a), the agglomeration degree depends on thecrystal size (L), the magma density (MT ), the relative su-persaturation (!) and so on. For simplicity, we propose forthis example an agglomeration sum kernel (�) containingonly one tuning parameter (�0). For the breakage terms, avolume proportional kernel (�) is used as follows:

�(MT ; !; Lj; Lk) = �0("sMT!)0:36(Lj + Lk)3 (66)

�(Lj) = �0L3j (67)

The solution density, "s (kg=m3), is obtained in Rohani andBourne (1990), as a function of the potassium sulfate con-centration (w, kg=kg) and the temperature (T;◦ C):

"s = 1015:6 + 644:6w − 0:33T (68)

This agglomeration kernel is originally presented for thecollisions of particles entrained within the turbulent eddiesdue to a di9usion mechanism (Hartel & Randolph, 1986)and is con8rmed in Jauner and Jones (2000).

For parameter identi8cation, an SQP (sequential quadraticprogramming) optimization algorithm (DONLP2, Spellucci,1998) is used. The parameters are optimized on sum of areasbetween the experimental and numerical data in the com-mercial logarithmic crystal density (log10 n). The ODE in-tegrator is called for sensitivities of the objective function(F) to the parameters (�0 and �0). The optimization strat-egy is similar to the single shooting technique for solvingboundary value problems (BVPs). One shoots trajectories ofthe same ODEs (e.g., discretized PBEs) with di9erent ini-tial values (e.g., optimization parameters) until one hits thecorrect given boundary values (e.g., experimental data) atthe 8nal integration time (Ascher & Petzold, 1998, p. 177).

For the eNcient shootings of optimization parameters, theSQP algorithm is used.

The objective function is de8ned by the trapezoidal rule.

min�0 ;�0

F =∑i

12(|log(ni+1)exp − log(ni+1)num|

+ |log(ni)exp − log(ni)num|)(Li+1 − Li) (69)

In Table 5, the simulation results and the calculation timeare reported at arbitrary parameter values which are howevertaken near the optimum points. From the preliminary tests,it is found that the MOC-30p on 102 meshes consumes anexcessive computational time like in the previous example.

The optimization is performed on the WENO-5 withoutthe ACM and the MOC-15p on 42 meshes, which take a rel-atively short computational time in Table 5. The optimiza-tion starts at the parameter values used in the preliminarytests. In Table 6, the optimization results are shown for thetwo numerical methods. Given the same level of the objec-tive function values, it may be said that the estimation withthe MOC-15p are better, since the MOC often produces lessnumerical error than the WENO-5 scheme. Therefore, thevalues of the agglomeration and breakage parameters areunderestimated in the WENO-5 scheme because of numer-ical di9usion error, as shown in Table 6. It is worth notingthat the parameter values optimized are concerned only withthe kernel Eqs. (66) and (67).

Fig. 11 shows a comparison of the CSDs between the ex-perimental data and the numerical results with the optimizedparameters. The simulation results are improved when ag-glomeration and breakage terms are considered, and boththe growth rate and the birthrate are corrected. However,in some regions a discrepancy between simulation resultsand experimental data could still be two orders of magni-tude. The discrepancy is mainly due to the model, espe-cially, growth rate, nucleation rate as well as kernels of ag-glomeration and breakage. These must be modeled carefullywith respect to some related variables such as temperature,amount of impurities, crystal size, magma density, etc.

The model considered for parameter estimation in theK2SO4–H2O system, which is corrected somewhat by therule of thumb, is not veri8ed rigorously. Since one of theobjectives of this study is not in modeling but is to reduce

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3729

Table 5Preliminary tests for parameter estimation of agglomeration and breakage kinetics in the case of the K2SO4–H2O system

Simulation results Calculation time

�0 �0 Obj. function

MOC MOC-15p on 42 grids 0.55 7 × 106 0.00125 9 min 9 sMOC-30p on 102 grids 0.6 9 × 106 0.00445 5 h 42 min

WENO WENO-5 0.4 3 × 106 0.00117 2 min 45 s(101 grids) WENO-5A 0.5 5 × 106 0.00125 29 min 27 s

Table 6Parameter estimation of agglomeration and breakage kinetics in the case of the K2SO4–H2O system

Optimization results

�0 �0 Obj. function Calculation time

MOC-15p on 42 grids 0.57 8:4 × 106 0.00123 155 minWENO-5 on 101 grids 0.55 6:5 × 106 0.00109 47 min

1.E+03

1.E+05

1.E+07

1.E+09

1.E+11

0 0.0005 0.001 0.0015 0.002Crystal size (m)

Num

ber

dens

ity (

no/m

/kg)

Experiment data

MOC-15p on 42 grids

WENO-5 on 101 grids

Fig. 11. Comparison of the CSDs between the experimental data and theoptimization results.

the numerical error using an existing modeling equation,the discussion on modeling is not progressed further in thisstudy. Through the parameter estimation, the WENO schemeshows a relative advantage (i.e., short CPU time) and po-tential weakness (i.e., underestimation of the parameters).

4. Conclusion

In this study, the high resolution spatial discretizationmethods (WENO schemes) and the modi8ed method ofcharacteristics (MOC) were applied to the dynamic simula-tion of the batch crystallization including nucleation, growth,agglomeration and breakage kinetics. Steep moving frontsor discontinuities appearing in the solution of the crystalliza-tion models are well captured by the two methods withoutany spurious oscillation.

Though the MOC gives more accurate results than theWENO scheme, there are some limitations to use it such ascareful determination of time-steps, obliged scalar modelingin the PBE and an excessive calculation time when complexand practical systems are dealt with.

Using the spatial discretization methods to surmount theselimitations, attentions must be paid to discretization of thegrowth term (convection term) which causes much numer-ical error and instability in the presence of steep fronts ordiscontinuities. In this case, high resolution spatial dis-cretization methods like the WENO schemes are stronglyrecommended to obtain suNciently accurate results.

Owing to the accurate numerical procedures presented inthis study, parameters of agglomeration and breakage ki-netics are estimated correctly. They allow us to generatenumerically the experimental crystal size distribution withgood accuracy.

Acknowledgements

We would like to thank the ministry of France of foreigna9airs and the embassy of France in Korea, which supportin part this work in the framework of the Korea–France sci-enti8c cooperation. This work was also partially supportedby the Korea Science and Engineering Foundation (KOSEF995-1100-001-2).

Appendix A

The third order WENO scheme (named WENO-3 in thisstudy) uses two candidate stencils to de8ne the numericalupHux (f i+1=2). For the upwinding scheme, the Roe speed(ai+1=2) is introduced, which is relatively simple to achievethe Hux splitting.

ai+1=2 =fi+1 − fi

ui+1 − ui(A.1)

3730 Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732

In the case of positive Roe speeds (in the crystallizationprocess, the speeds are always positive, because the growthrate, G, is positive), the numerical Hux is

f i+1=2(fi−1; fi; fi+1) =r−1∑m=0

wmqm(fi−m+1; fi−m)

=w0q0 + w1q1 (A.2)

where

w0 =�0

�0 + �1and w1 =

�1

�0 + �1

qm are obtained like the FS approach in the case of theaccuracy order k = 2:

q0 =r−1∑j=0

c0jfi+j = c00fi + c01fi+1 = (fi + fi+1)=2

(A.3)

q1 =r−1∑j=0

c1jfi+j−1 = c10fi−1 + c11fi

= (−fi−1 + 3fi)=2 (A.4)

The two possible stencils from Eqs. (A.3) and (A.4) aretaken into account in the numerical Hux with weightingfactors (w0 and w1). The question now is how to de8nethe weighting parameters (�0 and �1), while ENO prop-erty is achieved. The weighting parameters are calculatedby smoothness indicators (indicators of solution variation),ISi. The weights generate the least oscillatory reconstructionso that the maximum order of accuracy (third order accu-racy) in smooth regions is guaranteed. But in the presenceof a discontinuity they are automatically switched to the bestone-side stencil:

�0 =2=3

(� + (ISi+1=2)2)p(A.5)

�1 =1=3

(� + (ISi−1=2)2)p(A.6)

where ISi+1=2=(fi+1−fi)=Px and ISi−1=2=(fi−fi−1)=Px.Here � is a positive real number that is introduced to avoid thedenominator to become zero (in our case, we take �=10−10).It is suggested in Jiang and Shu (1996) that the power p=2is adequate to obtain ENO approximations. If a Roe speedis negative (a), the numerical Hux is de8ned in the reverseorder as f i+1=2(fi+2; fi+1; fi). For details and the 8fth orderWENO scheme named by the WENO-5, refer to Shu (1997).

The TVD (total variation diminishing) schemes such asENO or WENO schemes avoid spurious oscillations. How-ever, they cannot entirely avoid the poor resolution near dis-continuity. To sharpen discontinuities, we introduce Yang’sarti8cial compression method (Yang, 1990) which can beemployed in the Gear-type ODE integrator using a adaptive

time-step:

fYi+1=2 = f i+1=2 + ci+1=2 (A.7)

where f i+1=2 is the numerical Hux obtained by the methodsintroduced above, and

ci+1=2 = minmod[�i

2minmod

×[fRi+1=2 − f i+1=2; f

Ri+1=2 − f i−1=2]

fi+1 − f i+1=2; fRi+1=2 − fi−1

](A.8)

Here fRi+1=2 is obtained by the same method for f i+1=2 pre-

tending the Roe speed a¡ 0. The usual minmod function,minmod[b1; : : : ; bn], is de8ned as

minmod[b1; : : : ; bn]

=

{s · min

16j6n|bj| if s = sign(b1) = · · · = sign(bn);

0 otherwise;(A.9)

and �i is given by

�i = �( |fi+1 − 2fi + fi−1|� + |fi+1 − fi| + |fi − fi−1|

)2

(A.10)

where � is a positive parameter. � is also a positive realnumber to avoid the denominator becoming zero (in ourcase, we take �=10−6). �i of Eq. (A.10) is then very smallexcept near a discontinuity in either fi or its derivative,hence over-compression in smooth regions is avoided. Yang(1990) suggested �=33, though this parameter can be tunedto optimize the results for individual problem. The case ofa¡ 0 can be treated symmetrically. The modi8cation ofthe numerical Hux does not a9ect the accuracy in smoothregions.

Notation

a Roe speed, m=s or constant for the power-lawbreakage kernel

ATOL absolute tolerance for time integrationB0 birth rate, no.=min or no.=min=kgBA birth term for agglomerationbij breakage kinetic factor for uniform binary

breakageBN birth rate function, no.=min=m or

no.=min=m=kgDA death term for agglomerationfa function for agglomerationfg function for growth

Y. I. Lim et al. / Chemical Engineering Science 57 (2002) 3715–3732 3731

fn function for nucleationf i+1=2 numerical Hux

fRi+1=2 numerical Hux pretending the negative How

speed

fYi+1=2 numerical Hux with the arti8cial compression

methodG growth rate, m=minimax the number of meshesISi smoothness indicator on a ith meshkg kinetic constant of growthkn kinetic constant of nucleationkv volume shape factorL crystal size or capturing crystal size, m or �mL0 minimum crystal size, mLf last crystal size, mLt time partial derivative of crystal sizem0 Oth-order moment of crystal size distribution

or total number of crystalsm3 third order moment of crystal size distributionMT magma density, kg=kgn crystal density, no.=m or no.=m=kgN crystal number, no. or no.=kgn0 crystal density of the nuclei size, no.=m or

no.=m=kgN0 crystal number of the nuclei size, no. or

no.=kgNs seed crystal number, no. or no.=kgnt time partial derivative of crystal densityNt time partial derivative of crystal numberqm numerical Hux in FS schemesR gas constant, 8:3143 J=mol=KRTOL relative tolerance for time integrationt time, s or minT temperature, ◦C or Kv crystal volume, m3

w∗ equilibrium solubility, kg=kgw solute concentration, kg=kgw0 initial concentration of w, kg=kgwm weighting factors for WENO schemesws seed concentration, kg=kg

Greek letters

� constant of the arti8cial compression method�i weighting values� agglomeration kernel�0 constant of the agglomeration kernel� breakage kernel�0 constant of the breakage kernel� little value for nonzero denominator

� agglomeration factor on nonuniform meshes"c crystal density, lg=m3

"s solution density, kg=m3

! relative supersaturation

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