Generic Framework for Crystallization Processes Using the Population

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Generic Framework for Crystallization Processes using population balance model and its Applicability Latif J Shaikh, Atul Harishchandra Bari, Vivek V. Ranade, and Aniruddha B. Pandit Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01421 • Publication Date (Web): 06 Oct 2015 Downloaded from http://pubs.acs.org on October 10, 2015

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Industrial & Engineering Chemistry Research

1

Generic Framework for Crystallization Processes using population balance

2

model and its Applicability

3 4 5 Latif J. Shaikh a,b, Atul H. Bari a, Vivek V. Ranadeb and Aniruddha B. Pandita*

6 7 8 9

a

Chemical Engineering Department, Institute of Chemical Technology, Mumbai 400019, India b

CEPD Division, National Chemical Laboratory, Pune 411008, India

10 11 12 13 14 15 16 17 18 19

*Author to whom correspondence should be addressed

20

Email: [email protected]; Tel: +91-22-3361 2012; Fax: +91-22-33611020

21 22

ACS Paragon Plus Environment


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Abstract

24

A generic modeling framework for batch cooling crystallization processes has been developed to

25

understand the crystallization process from operational and modeling point of view. The generic

26

framework for crystallization process modeling incorporates the characteristic dimensions of

27

crystals, polymorphic transformation as well as the hydrodynamic mixing effects in the

28

crystallizer. This Polyhedral Polymorphic Multizonal Population Balance model (PPMPBM)

29

considers bottom up and top down approach for specific systems with specific targets. PPMPBM

30

framework allows switching between complex and simple models to study different

31

crystallization systems with different scenarios and combination thereof. This framework uses

32

gPROMSTM software (PSE, UK) along with the Microsoft Excel front-end along with Polytope

33

module in Matlab for predicting the crystal size and shape evolution as well as supersaturation

34

profiles inside the crystallizer which can be implemented for various crystallization systems.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53


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54


Introduction

55

Crystallization, an industrially important unit operation is used to produce wide range of

56

materials ranging from bulk chemicals to specialty chemicals with the desired product properties

57

like flow ability, drying time, filterability etc. The important attributes of the crystalline product

58

are specified in terms of its size, distribution of its size and shape, purity etc.

59

The combination of various crystallization mechanisms and population balance (PB)

60

model enables us to simulate temporal crystal size distribution (CSD) evolution in a crystallizer.

61

One of the drawbacks of the use of such a one-dimensional population balance is that changes in

62

the crystal shape cannot be accounted for properly. Taking into account several dimensions and

63

internal shape factors, requires the use of multidimensional population balance equation (PBE).

64

A two-dimensional population balance approach is presented and solved numerically in order to

65

simulate the time variations of two internal sizes of crystals. In many crystallization systems, the

66

crystal properties are specific to a particular face. Therefore, the size and shape of crystal is of

67

significance. For instance, it is desirable to maximize the area of a particular face if it happens to

68

be catalytically highly active surface. Also in cases where dissolution, hydrophobicity and other

69

surface properties create difficulty in downstream processing or in functioning of the product due

70

to a particular face, the area of such a crystal face should be minimized 1.

71

In the field of research on crystal morphology prediction or evolution, the focus has been

72

limited to the shape evolution or prediction of single crystals rather than population of crystals.

73

The population balance models have been used for the single or multiple dimensions for

74

predicting the evolution of CSDs. Together with the crystal morphology, the crystal size

75

distribution (CSD) produced within crystallizer is of crucial importance in determining the ease

76

and efficiency of downstream processing such as solid-liquid separation, the suitability of

77

crystals for further treatments.

78

The imperfect mixing in a crystallizer is a result of hydrodynamic conditions, which leads

79

to different spacial profiles of temperature, supersaturation and CSD inside crystallizer. In order

80

to overcome the deficit of well-mixed models, a multizonal model accounting for spatial

81

variation in the hydrodynamic conditions is needed.

82

The transformation of metastable to stable form of many systems will only take place in

83

the solution i.e. solution mediated polymorphic transformation. For a monotropic system like L-

84

glutamic acid (LGA), the polymorphic transformation can only take place irreversibly in one



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85

direction 2.

86

and multizonal model for polymorphic system is presented. The present work only considers

87

few cases for validating the usefulness of this framework. The first three cases considered were

88

for 1D, 2D and polyhedral PB model using unseeded linear cooling crystallization of β-form of

89

L-glutamic Acid crystals. Fourth case considered was of polyrmorphic transformation PB model

90

for cooling crystallization followed by constant temperature experiment of L-glutamic Acid. In

91

fifth case, Multizonal model is validated using the literature data on Potassium dihydrogen

92

phosphate (KDP) crystallization.

In this work, a generic model with a combination of population balance model

93 94

Population balance framework

95

The population balance framework is an indispensable tool for dealing with dispersed phase

96

systems. As shown in Figure 1, the generic framework for crystallization process modeling

97

incorporates PPMPB model describing the characteristic dimensions of crystals, polymorphism

98

and its transformation as well as the mixing in the crystallizer can be expressed as follows: r ∂f zk, z ,i ( x, z, k , t ) N ∂ r r  f zk, z ,i ( x, z, k , t ) Gzk, z ,i ( x, z, k , t )  +∑  ∂t i =1 ∂xi (1) k k k k F  Wd f z +1, z +1,i + Wu f z −1, z −1,i − Wu f z , z ,i − Wd f z , z ,i  k k k = Bz , z ( z, k , t ) − Dz , z ( z, k , t ) + Rz , z ( z, k , t ) +   Vz  +Wr f zk, z −1,i + Wl f zk, z +1,i − Wr f zk, z ,i − Wl f zk, z ,i 

99

Where

K

=

Number of polymorphs, 1,2…K

z, z r x

=

Zones in axial and radial directions, 11,12…ZZ

=

Characteristic dimensions, 1,2,…N

F

=

Population density, no./m/m3

G

=

Growth rate, m/s

B

=

Birth Term, no./m3/s

D

=

Death Term, no./m3/s

R

=

Nucleation Rate, no./m3/s

F

=

Flow rate between Zones, m3/s

Vz

=

Volume of zone ‘z’, m3


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Wd,Wu,Wl,Wr

=

Weighing functions for downward, upward, left and right directions of flow from zone ‘z’,-

100

The Equation 1 is described for zzth zone and kth characteristic polymorph of

101

crystallization system. The first term on LHS of equation is the accumulation term of population

102

density f. The second term is the convection of population density f due to growth of the crystals

103

with N dimensions and Gi being the growth rate of face in ith dimensions. The first and second

104

terms on RHS of equation are birth and death of population density f due to breakage,

105

agglomeration, attrition etc. The third term describes birth of population density f due to

106

nucleation. The last term represents the change in the population density f due to exchange of

107

flow of crystals between neighboring zones.

108

The generic growth rate expression considering multiple dimensions and polymorphs is

109

given

110

below

Gi ,k = kgi ,k σ k i ,k g

(2)

111

Where, Gi,k is growth rates of crystal with i characteristic dimensions and k number of

112

polymorphs. kgi,k is growth rate coefficient and σk is supersaturation for kth polymorph. gi,k is

113

growth order for ith dimension of crystal and kth polymorph.

114

Similarly, the nucleation rate is described by the following equation as

(

)

Rk = knk 1 + c * M TK σ knk

(3)

115

Where, Rk is nucleation rates of crystal with kth polymorph. knk is nucleation rate coefficient. c is

116

empirical parameter, MT,k is suspension density and σk is supersaturation for kth polymorph. n,k is

117

nucleation order for kth polymorph.

118

The solute mass balance for kth polymorph is given by the following equation as

+ 119

,,.,

=0

(4)

Where, C is the solute concentration and MT,k is the suspension density for kth polymorph

120

Thus, PPMPB model is the generic crystallizer model describing almost all physical and

121

geometrical aspect of the crystallization process, with N characteristic dimensions including

122

crystal morphology, polymorphic transformation and fluid mixing effect. The present work thus

123

can be elaborated as different cases of PPMPB model. One can select the relevant models based



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124

on the requirements such as study of 2D crystal growth, study of spatial variations in the

125

crystallizer with polymorphism, study of simultaneous crystal shape evolution of polymorphic

126

system etc.

127

Several numerical schemes have been developed for the solution of population balance

128

models such as method of moments3, the method of characteristics, the method of weighted

129

residuals4, the Monte Carlo method5, finite difference scheme6, 7, and the high resolution finite

130

volume schemes8, 9.

131

In the present work, the numerical solution is based on transformation of model equations

132

using the form of population density and form of grid (linear or logarithmic) and solving these

133

equations using finite difference method10. Literature already reports the framework for 1D/2D

134

PBM11, Compartmental modeling using 2D PBM12, Polymorphic transformation using 1D

135

PBM13, Crystal shape evolution using multidimensional PBM14,

136

independent and the link between these subparts of generic population balance model is missing.

137

To understand the effect of required parameter and operating conditions on the particle size and

138

shape, it is essential to study all these parameters simultaneously for easy comparison. The MS-

139

Excel is chosen for generating such platform in order to study generic PPMPBM framework as it

140

is being user friendly platform and can be connected to available computational softwares easily

141

(gPROMS, MatLab, Ansys).

15

. All these frameworks are

142 143

MSEXCEL Front-End for Generic PPMPBM Framework

144 145

Batch crystallizer model in gPROMS is linked to Excel where the input parameters can

146

be varied and the results of simulation are updated in the form of plot (e.g. supersaturation

147

profile, CSD etc.) which helps in studying the influence of different parameters on desired

148

variables at a time and reduce the number of simulations to check the influence of parameters

149

(Figure 2). Snapshot of same is given in figure below. This linking is found to be very useful for

150

monitoring the important variables like the crystal length and width distribution, their growth

151

rates, concentration in polyhedral population balance model. The normal distances from origin to

152

the crystal face are updated in Excel worksheet and these distances are imported in Polytope

153

module in MatLab for the reconstruction of crystal shape and its evolution. During the crystal

154

shape evolution, if face disappears from the crystal, the inputs for faceted kinetics in polyhedral


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155

PB model are changed to account for solute mass balance. For multizonal model, the number of

156

zones is entered in the worksheet to get the updated concentration and supersaturation profiles

157

for the respective zones.

158

The following section will showcase different scenarios for crystallization system

159

considering crystal size, crystal shape evolution, polymorphic transformation and mixing effects

160

on supersaturation profiles within crystallizer. Five Case Studies on PPMPBM framework are

161

described in following sections.

162 163

Case Studies

164

(1) 1D PBM of LGA cooling crystallization

165

In order to study one dimensional L-glutamic acid crystals, PPMPB model is reduced to one

166

dimension without agglomeration and breakage and no polymorphic transformation and uniform

167

mixing (Equations 5-8). For this, stable needle shaped β-form crystals of L-Glutamic acid are

168

considered. In the PPMPB, these crystals can be approximated for needle shaped morphology by

169

equivalent shape factor. Using the operating conditions and kinetics parameters (Table 1)

170

L-Glutamic Acid cooling crystallization; the reduced model is simulated. ∂f ∂ ( f G ) + =R ∂t ∂x

(5)

G = kgσ g

(6)

R = kn (1+ c * MT ) σ n

(7)

dC dM T + =0 dt dt

(8)

16

for

171

Experiment was performed in a crystallizer which was a jacketed glass vessel of 500 ml capacity.

172

The temperature of the solution is controlled by circulating the bath fluid through jacket of the

173

crystallizer using heating/cooling circulator (Julabo FP50). Universal PMDC RQG-126D motor

174

is used for stirring with a pitched blade impeller. Experimental run was carried out under linear

175

cooling mode from 600C to 280C. Five Samples were withdrawn at different temperatures. All

176

samples were filtered and filtrate was analyzed for solute concentration using UV-vis

177

spectrophotometry. Crystals collected on filter paper were weighed and were analyzed for crystal

178

size distribution using offline image analysis. The experiment was carried out in duplicate. This



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179

experiment has been referred hereafter as run III with operating conditions are mentioned in

180

Table 1.

181 182 183

Figure 3 shows the solute concentration for the linear cooling run without seeds (run III).

184

A constant solute concentration is observed for long period as the driving force is not sufficient

185

enough for homogeneous nucleation to occur. As can be seen in Figure 4, the

186

supersaturation peak is achieved in third quarter of a batch time corresponding to the

187

primary nucleation.

188

The CSD profile for linear cooling is also quite wide spread due to the absence of seeds

189

in run III (Figure 5, experimental). The spontaneous nucleation in the later period leads to a

190

number of fine crystals which compete with each other for growth and creates a wide distribution

191

of crystal sizes.

192

In order to study the effect of breakage & agglomeration on CSD, PPMPB model is

193

reduced to one dimension and no polymorphic transformation and uniform mixing. Since

194

breakage & agglomeration are mass conserving (consequentially volume conserving) processes,

195

it is worthwhile to represent their equation in volume co-ordinates. The birth & death term due to

196

agglomeration & breakage in volume co-ordinates is given as: v

Bagg (v) =

1 kagg (v ', v − v ') * f (v ', t ) * f (v − v ', t )dv ' 2 ∫0

(9)

∞

Dagg (v) = f (v, t ) ∫ kagg (v ', v) * f (v ', t )dv '

(10)

0

∞

BBreak (v) = ∫ b(v, v ') * S Break (v ') * f (v ', t )dv '

(11)

DBreak (v) = SBreak (v)* f (v, t )

(12)

v

197

where kagg is agglomeration kernel, Sbreak is specific breakage rate & b(v, v') is breakage

198

distribution function. Breakage distribution function b(v, v') is defined as the probability of

199

formation of particle of size (v) after breakage of particle of size (v'). Uniform binary breakage

200

distribution function is widely used & is given as: b ( v, v ' ) = 2 / v '

(13)


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201


The breakage rate was given as:

S Break (v) = kbreak v 202

(14)

Also for the sake of simplicity, agglomeration kernel was considered to be size independent.

203

PBE involving agglomeration & breakage is partial-integro differential equation. These

204

equations can be reduced to set of ordinary differential equations (ODEs) by discretization. Here

205

fixed pivot technique of discretization developed by Kumar & Ramkrishna17 was applied to

206

reduce PBE in to ODE & then these equations were solved in gPROMS platform. Simulations

207

were carried out for the LGA cooling crystallization with the process parameters same as that

208

used in run III. The agglomeration & breakage parameters were estimated to fit the CSD data

209

well. These parameters were estimated using gPROMS parameter estimation tool by optimizing

210

sum of squared residuals (SSR) of CSD. The estimated parameters are given in table 2.

211

Figure 5 shows the CSD profile for LGA cooling crystallization. We can see the

212

improvement in simulated CSD in coarse size range after considering breakage & agglomeration.

213

But for the finer size range, simulated CSD with no agglomeration & breakage matches well

214

with that of experimental. Also, number of particles in finer range are less for crystallization

215

considering breakage & agglomeration than that for without agglomeration & breakage,

216

suggesting that agglomeration is dominant than breakage.

217 218

(2) 2D PBM of LGA cooling crystallization

219

In order to study two characteristic dimensions of L-glutamic Acid crystals (stable β-

220

form which is needle shaped), PPMPB model is reduced to two dimensions without

221

agglomeration and breakage and no polymorphic transformation and uniform mixing (Equations

222

15-18). Using the kinetics parameters values (Table 3) for L-Glutamic Acid crystallization

223

for 2D crystals of β-form which are already estimated in the published work by Author16; the

224

reduced model is simulated using initial and boundary conditions represented by the

225

operating conditions reported in Table 1. 2 ∂f ∂ +∑ [ f Gi ] = R ∂t i =1 ∂xi

(15)

Gi = kgi σ gi

(16)



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Page 10 of 34

R = kn (1+ c * MT ) σ n

(17)

dC dM T + =0 dt dt

(18)

226

Figures 6-7 show the comparison of concentration and supersaturation profiles for the run

227

III along with the profiles obtained using 1D and 2D PB models. The simulations show better

228

agreement with the measured concentration and supersaturation profiles when two characteristic

229

dimensions are used (2D PB model) for describing the crystals of L-glutamic acid. The growth

230

kinetics for length and width dimensions thus play an important role for the crystal size

231

increment in the said characteristic dimensions. The linear cooling run (RUN III) without

232

seeding (Figures 6-7) shows a lowering of the concentration profile as compared to concentration

233

profile obtained using 1D PB model. The higher reduction in concentration for 2D PB model (as

234

compared to1D PB model) is attributed to the surface area for needle-shaped LGA crystals as

235

they consume more solute than spherical crystals considered in 1D PB model. A similar

236

observation has been made for the relative supersaturation profile following experimental

237

data (RUN III) more closely (Figure 7) than that by 1D PB model.

238

As the CSD profile observed during the unseeded run III was unimodal with 1D PB

239

model, the CSD considering the length and width dimensions for 2D PB model is also unimodal

240

in nature (Figure 8). The unseeded run shows a wider distribution of CSD in both width and

241

length directions.

242 243

(3) Polyhedral PBM of L-glutamic acid

244

In order to study the shape evolution of L-glutamic Acid crystals, PPMPB model is

245

reduced to three dimensions without agglomeration and breakage and no polymorphic

246

transformation and uniform mixing (Equations 19-22). 3 ∂f ∂ +∑ [ f Gi ] = R ∂t i =1 ∂xi

(19)

Gi = kgi σ gi

(20)

R = kn (1 + c * MT ) σ n

(21)

dC dM T + =0 dt dt

(22)


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247

The methodology explained in author’s previous research paper18 can be used to simulate

248

the shape evolution of β-LGA crystal. L-glutamic acid with β-form is represented schematically

249

as in Figure 9. β-LGA has 2 main faces, {100} and {110} family. With the help of a

250

description of crystal morphology mapping19 and growth rate data, the crystal shape evolution

251

has been yielded. The crystal shape evolution of β-LGA crystal is shown in Figure 10. For

252

additional details, the reader is referred to above mentioned research article by Authors18.

253

(4) Polymorphic transformation of LGA from α-form to β-form

254

In order to study polymorphic transformation of L-glutamic Acid crystals from α-form to

255

β-form, PPMPB model is reduced to one dimensional polymorphic transformation without

256

agglomeration and breakage and uniform mixing (Equations 23-26). ∂f k ∂ f k G k + = Rk ∂t ∂x

(23) (24)

Gi ,k = kgi ,k σ k i ,k g

(

)

(25)

Rk = knk 1 + c * M TK σ knk + =0 ,

= 3

!"

(26)

,

# $ %& ' ()*, %+* *

257

(27)

258

To study the polymorphic transformation, L-glutamic acid was dissolved in distilled water and

259

was cooled from 70 °C to 44 °C by rapid cooling (1°C/min) with initial concentration

260

corresponding to 24.3 g/Kg, which was then held constant for 2h. This experiment is referred as

261

Expt P1. As per the phase diagram for L-glutamic acid polymorphs, metastable α-form will

262

nucleated first and will grow. At 44°C, the grown α-form crystals will start to dissolve as the

263

solution will be supersaturated with respect to β-form. Then, stable β-form will be nucleated and

264

grown over the time. The reduced polymorphic transformation model is simulated using these

265

operating conditions along with kinetic parameters for L-glutamic acid polymorphs mentioned in

266

the literature.13 The concentration profile matches well with simulation as shown in Figure 11.

267

Due to inherent issues associated with sample withdrawal procedure, the resulting errors are



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Page 12 of 34

268

acceptable. . Figure 12 shows that the CSD obtained using simulation are in good agreement

269

with the experimental CSD. Since it is solution-mediated polymorphic transformation

270

experiment, the metastable form i.e. α-LGA is transformed into stable form i.e. β-LGA. At the

271

end of batch time, it is confirmed by XRD analysis that final form obtained is β-LGA.

272 273

(5) Multizonal PB Model

274

In order to study the effect of mixing on supersturation profiles and crystal size distribution

275

inside the crystallizer, PPMPB model is reduced to two dimensions with multizonal model

276

without agglomeration and breakage and no polymorphic transformation (Equations 28-31). ∂f z , z ∂t

2

+∑ i =1

F = R+ Vz

∂  f z , z Gi  ∂xi  (28)

 Wd f z +1, z +1 + Wu f z −1, z −1 − Wu f z , z − Wd f z , z     +W f   r z , z −1 + Wl f z , z +1 − Wr f z , z − Wl f z , z 

Gi = kgi σ gi

(29)

R = kn (1+ c * MT ) σ n

(30)

dC dM T + =0 dt dt

(31)

277

The weighing functions (Wd, Wu,Wl, Wr) in axial and radial directions are functions of constants

278

(α1, α2, β1, β2) mentioned in Table 4 and are given as;

279

- = 1.0 − )0 1 − 0 1 +/max)1 , 1 +

280 281

(32)

-6 = 1.0 + )0 1 + 0 1 +/max )1 , 1 +

(33)

-8 = 1.0 − )9 1 − 9 1 +/max )1 , 1 +

(34)

282

-: = 1.0 + )9 1 + 9 1 +/max )1 , 1 +

283

The constants α1, α2, β1 & β2 are constants which can be determined from CFD and

284

experiments, and L1 and L2 are the characteristic length scales for the crystals.

285

The reduced model is simulated with operating conditions and kinetic parameters (Table 4) for

286

well studied Potassium dihydrogen phosphate (KDP) system.

287

12

can be found out in literature.

(35)

12

The description of crystallizer

288

The simulation considers the variation in radial and axial direction. The crystallizer is

289

divided into six zones in axial direction and radially into two zones i.e. central and peripheral


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290

zones. Figure 13 shows different zones considering M zone in axial direction & two zones in

291

radial direction. As can be seen from Figure 14, the concentration profiles in the central zone are

292

more or less similar to the axial multizonal model but for peripheral zone, the profiles show

293

variation due to an effect of existing velocity profile in the crystallizer. The high supersaturation

294

level exists in the top zones as compared to the bottom zones in the peripheral region. The

295

supersaturation level is different in various zone in the central region due to the different flow

296

exchange in the lower and upper part of the crystallizer.

297

Similarly, for final crystal size distribution in peripheral zone (Figure 15), as we move

298

down from top to bottom zone, the crystal size distribution varies such that the larger size

299

crystals are more in the bottom zone as compared to the top zone. For central region, the CSD

300

variation is non-linear as a result of the exchange of flow between the zones being different for

301

lower part and upper part of the crystallizer with regard to the velocity profile in the crystallizer.

302

This present framework not only helped to gain better insight into the crystallization

303

process but also it is easier and faster to switch between models depending upon the process

304

requirement. Though this framework is not tested on phenomena like aggregation and breakage

305

for multidimensional crystals, it can be improved upon considering the basic framework is based

306

on the generic PB model. This framework will form the guidelines which will facilitate the

307

selection of models or combimnations thereof to predict the crystal size and shape distributions

308

under variety of conditions and configurations.

309 310 311

CONCLUSIONS •

The PPMPB model is found to be useful when the generic model for crystallizer is to be

312

formulated. The combination of two or more models can help to understand some

313

crystallization systems. It offers the user along with the MS excel linking, the choice to

314

use the model which fits the requirement.

315

•

316 317 318

Depending upon the operating conditions and properties desired, one can switch between different models with just few parameters to be entered in Excel spreadsheet.

•

Excel interface is easy to use, efficient and flexible tool for understanding the crystallization process and crystal morphology evolution.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

•

319

Page 14 of 34

Multizonal model helped to understand that the classification of crystals under real

320

operating conditions is different than the assumption of uniform crystal size distribution

321

inside the crystallizer. •

322 323

Polymorphic transformation PB model helped to choose the operating regime to get the desired polymorph.

•

324 325

Crystal shape evolution is described in transient manner from spherical-like nuclei to particular shape crystals depdending upon the growth rates of different faces on crystals.

326 327

Acknowledgements

328

One of the authors (LJS) is grateful for the CSIR fellowship. The work was partially supported

329

by CSIR project OLP3026.

330 331

Nomenclature: σ

Suspension Density

B

Birth rate, no./m3/s

b(v, v')

breakage distribution function

c

empirical parameter of secondary nucleation

D

Death rate, no./m3/s

f

Population density, no./m/m3

F

Flow rate between Zones, m3/s

G

Growth rate, m/s

g

Growth order

K

Number of polymorphs, 1,2…K

kagg

agglomeration kernel

kg

Growth rate coefficient

kn

nucleation rate coefficient

kv

Volumetric shape factor

MT

Suspension Density

R

Nucleation Rate, no./m3/s


Page 15 of 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Sbreak

specific breakage rate

T

Time

v, v'

Volume of particle, m3

Vz

Volume of zone ‘z’, m3

Wd,Wu,Wl,Wr

Weighing functions for downward, upward, left and right directions of flow from zone ‘z’,

r x

Characteristic dimensions, 1,2,…N

z, z

Zones in axial and radial directions, 11,12…ZZ

Suffix i

Characteristic dimension

k

Type of polymorph

332 333 334

References

335

1.

336

growth and dissolution. AIChE J. 2007, 53, 1510.

337

2.

338

glutamic acid in the absence of additives. J. Cryst. Growth 2004, 273, 258.

339

3.

340

method of moments for population-balance equations. AIChE J. 2003, 49, 1266.

341

4.

342

Comput. Chem. Eng.1977, 1, 23.

343

5.

344

for the solution of population balances in dispersed systems. Powder Technol.2007, 173, 38.

345

6.

346

processes through multidimensional population balance equations. Part 1: a resolution

347

alogorithm based on the method of classes. Chem. Eng. Sci.2003, 58, 3715.

348

7.

349

population balance. AIChE J. 2008, 54, 2321.

350

8.

351

population balance equations. AIChE J. 2004, 50, 2738.

Snyder, R. C.; Studener, S.; Doherty, M. F., Manipulation of crystal shape by cycles of

Cashell, C.; Corcoran, D.; Hodnett, B. K., Control of polymorphism and crystal size of L-

Marchisio, D. L.; Pikturna, J. T.; Fox, R. O.; Vigil, R. D.; Barresi, A. A., Quadrature

Singh, P. N.; Ramkrishna, D., Solution of population balance equations by MWR.

Zhao, H.; Maisels, A.; Matsoukas, T.; Zheng, C., Analysis of four Monte Carlo methods

Puel, F.; Fevotte, G.; Klein, J. P., Simulation and analysis of industrial crystallization

Ma, C. Y.; Wang, X. Z., Crystal growth rate dispersion modeling using morphological

Gunawan, R.; Fusman, I.; Braatz, R. D., High resolution algorithms for multidimensional



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

352

9.

353

dimensional batch crystallization models with size independent growth rates. Comput. Chem.

354

Eng. 2009, 33, 1221.

355

10.

356

Experiments and modelling. Can. J. Chem. Eng. 2013, 91, 47.

357

11.

358

dimensional model-based system for batch cooling crystallization processes. Chem. Eng. Sci.

359

2011, 35, 828.

360

12.

361

crystallization. Int. J. Mod. Phy. B2002, 16, 383.

362

13.

363

model describing crystallization of polymorphs.Ind. Eng. Chem. Res. 2010, 49, 4940.

364

14.

365

crystal growth in face directions. AIChE J.2008, 54, 209.

366

15.

367

Morphology Distributions with Population Balances. Cryst. Growth Des.2013, 13, 1397.

368

16.

369

University, Mumbai, 2012.

370

17.

371

discretization—I. A fixed pivot technique. Chem. Eng. Sci. 1996, 51, 1311.

372

18.

373

Balance Model. Ind. Eng. Chem. Res. 2014, 53, 18966.

Page 16 of 34

Qamar, S.; Seidel-Morgenstern, A., An efficient numerical technique for solving multi-

Shaikh, L.; Pandit, A.; Ranade, V., Crystallisation of ferrous sulphate heptahydrate:

Fazli, N. A. A. S.; Singh, R.; Sin, G.; Gernaey, K. V.; Gani, R., A generic multi-

Ma, D. L.; Braatz, R. D.; Tafti, D. K., Compartmental modeling of multidimensional

Qamar, S.; Noor, S.; Seidel-Morgenstern, A., An efficient numerical method for solving a

Ma, C. Y.; Wang, X. Z.; Roberts, K. J., Morphological population balance for modeling

Singh, M. R.; Doraiswami, R., A Comprehensive Approach to Predicting Crystal

Shaikh, L. J. Studies in Crystallization Process and Crystal Morphology. Mumbai

Kumar, S.; Ramkrishna, D., On the solution of population balance equations by

Shaikh, L.; Ranade, V.; Pandit, A., Crystal Shape Evolution using Polyhedral Population

374 375

List of figures:

376

Figure 1: The framework for generic PPMPB model

377

Figure 2: Snapshot of MSExcel viewing the results from model implemented in gPROMS

378

Figure 3: Experimental and simulated concentration profiles for run III

379

Figure 4: Experimental and simulated supersaturation profiles for run III

380

Figure 5: Experimental and simulated CSD profiles for run III

381

Figure 6: Concentration profile for run III without seeding

382

Figure 7: Supersaturation profile for run III without seeding


Page 17 of 34

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383

Figure 8: CSD profiles for linear cooling run-III

384

Figure 9: (a) β-LGA crystal morphology with Miller Indices (b) real crystal

385

Figure 10: The predicted temporal evolution of β-L-glutamic acid crystals

386

Figure 11: Concentration profile for polymorphic transformation experiment

387

Figure 12: CSD profile for polymorphic transformation experiment

388

Figure 13 : Multizonal model with M zones in axial direction and two zones in radial direction

389

Figure 14: Relative supersaturation profiles when 6x2 zones are used (A: Central, B: Peripheral)

390

Figure 15: Final crystal size distribution when 6x2 zones are used.

391 392

List of Tables:

393

Table 1: Operating Conditions and Kinetic Parameters for L-glutamic Acid Crystallization for

394

Run-III17

395

Table 2. Kinetic parameters for agglomeration & breakage for L-glutamic Acid Crystallization

396

Table 3: Kinetic Parameters for crystallization of LGA (2D PBM)

397

Table 4: Parameters used in the simulation of reduced model

398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413



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Page 18 of 34

414 415 Table 1: Operating Conditions and Kinetic Parameters for L-glutamic Acid Crystallization for Run-III17 Initial Temperature oC

Initial

60

kG, m/s

Solute 30

Concentration, g/L Temperature Profile oC

T=58.77-.007*time

Initial

0

Suspension

density,(MT)0 g/L Boundary

condition

population density

Eq. conc., g/L

for

()*+ = 0; (

Generic Framework for Crystallization Processes Using the Population

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