prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/205/1/1680s.pdf · iii development and...

Development and Analysis of Batch andContinuous Crystallization Models

By

Iltaf Hussain

CIIT/SP09-PMT-004/ISB

PhD Thesis

In

Mathematics

COMSATS Institute of Information TechnologyIslamabad - Pakistan

Spring, 2012

ii

COMSATS Institute of Information Technology

Development and Analysis of Batch and ContinuousCrystallization Models

A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad

In partial fulfillmentof the requirement for the degree of

PhD (Mathematics)

By

Iltaf Hussain


Spring, 2012

iii


A Post Graduate Thesis submitted to the Department of Mathematics as partialfulfillment of the requirement for the award of Degree of PhD Mathematics.

Name Registration Number

Iltaf Hussain CIIT/SP09-PMT-004/ISB

Supervisor

Dr. habil. Shamsul Qamar

Associate ProfessorDepartment of MathematicsIslamabad CampusCOMSATS Institute of Information Technology,Islamabad.June, 2012

iv

Final Approval__________________________________

This thesis titled


By

Iltaf Hussain


Has been approved

For the COMSATS Institute of Information Technology, Islamabad

External Examiner 1:___________________________________________Dr. Sohail Nadeem Associate Prof., QAU, Islamabad

External Examiner 2:___________________________________________Dr. Khalid Saifullah Syed, Assistant Prof., BZU, Multan

Supervisor: ________________________________________________Dr. habil. Shamsul QamarAssociate Professor CIIT, Islamabad.

HoD:_____________________________________________________Professor Dr. Moiz-ud-Din KhanHoD (Mathematics/Islamabad)

Dean, Faculty of Sciences: ______________________________Professor Dr. Arshad Saleem Bahatti

v

Declaration

I, Iltaf Hussain, CIIT/SP09-PMT-004/ISB, hereby declare that I have produced the workpresented in this thesis, during the scheduled period of study. I also declare that I havenot taken any material from any source except referred to wherever due that amount ofplagiarism is within acceptable range. If a violation of HEC rules on research hasoccurred in this thesis, I shall be liable to punishable action under the plagiarism rules ofthe HEC.

Date: _________________ Signature of the student:

____________________________Iltaf Hussain


vi

Certificate

It is certified that Iltaf Hussain, registration number CIIT/SP09-PMT-004/ISB has carriedout all the work related to this thesis under my supervision at the Department ofMathematics, COMSATS Institute of Information Technology, Islamabad and the workfulfills the requirement for award of PhD degree.

Date: _________________

Supervisor:

___________________________Dr. habil. Shamsul QamarAssociate Professor

Head of Department:

_____________________________Dr. Moiz-ud-Din Khan

ProfessorDepartment of Mathematics

vii

DEDICATION

I dedicate this thesis tomy parents

ACKNOWLEDGEMENTS

I owe my thanks to the Almighty ALLAH the most Gracious, the most Merciful, the unique

supreme power of the whole universe, Who blessed me the courage, potential and insight

that enabled me to complete this thesis. My all admirations to the last Holy Prophet

Hazrat Muhammad (PBUH) whose life is a candle for us in the darkness of life.

I feel honored to acknowledge my deepest gratitude to my affectionate and devoted super-

visor Dr. habil. Shamsul Qamar whose guidance and keen interest enabled me to complete

this research work. His cooperation and invigorating encouragement will always remain a

source of inspiration for me.

I would like to express my deep thank to our research collaborators Prof. Dr. Andreas

Seidel-Morgenstern and Prof. Dr. Martin Peter Elsner from the Max Planck Institute

for Dynamics of Complex Technical Systems Magdeburg, Germany for their guidance and

research contributions. I would also like to thank Prof. Dr. Moiz ud Din Khan, Head of

Department of Mathematics, CIIT, Islamabad, Pakistan for his cooperation and support.

I pay special thanks to the University of Engineering and Technology, Peshawar

(Mardan Campus), Khyber Pakhtunkhwa, Pakistan for study leave.

Finally, I am very much thankful to Higher Education Commission (HEC), Islam-

abad, Pakistan for financial support under Indigenous PhD 5000 Fellowship

Program Phase-IV under grant number 074-0714-Ps4-213.

Iltaf Hussain


viii

ABSTRACT

Development and Analysis of Batch and Continuous

Crystallization Models

This thesis presents the development and simulation of batch and continuous crystallization

models. Especially, models are derived for simulating batch and continuous enantioselec-

tive preferential crystallization processes in single and coupled crystallizers. Such processes

are highly important in chemical and pharmaceutical industries. The effects of nucleation,

growth, and fines dissolution phenomena on the crystal size distribution (CSD) are inves-

tigated. For the first time continuous preferential crystallization is investigated and the

effects of different seeding and operating strategies on the process are analyzed. To judge

the quality of the process some goal functions are used, such as purity, productivity, yield

and mean crystal size of the preferred enantiomer. The semi-discrete high resolution finite

volume schemes (HR-FVS) and the discontinuous Galerkin (DG) finite element method

are proposed for solving these models. The resulting systems of ordinary differential equa-

tions (ODEs) are solved by using explicit and nonlinearly stable high order Runge- Kutta

method. The schemes satisfy the total variation bounded (TVB) property which guarantees

the positivity of the schemes, for example the non-negativity of CSD in the present case.

The suggested methods have capabilities to capture sharp discontinuities and narrow peaks

of the CSD. In DG-schemes, the accuracy of the method can be improved by introducing

additional nodes in the same solution element and, thus, avoids the expansion of mesh

stencils which is normally observed in high order finite volume schemes. For that reason,

the method can be easily applied up to boundary cells without loosing accuracy. It was

found that the proposed numerical schemes have the capability to solve the given models

more efficiently and accurately. The results support process design and optimization.

ix

Contents

1 Fundamentals of Crystallization Process 1

1.1 Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Types of Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Batch Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Continuous Crystallization . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Enantiomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Preferential Crystallization . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Ternary Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Historical Background and Motivation . . . . . . . . . . . . . . . . 6

1.7 New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 Layout of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 Publications Related to the Research Work . . . . . . . . . . . . . 12

2 Population Balance Models 13

2.1 Population Balance Models . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Growth and Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Breakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Goal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

x

TABLE OF CONTENTS

3 Batch Crystallization 19

3.1 Batch Crystallization Model . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Implementation of Discontinuous Galerkin finite element

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Implementation of Finite Volume Scheme . . . . . . . . . . 27

3.3 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Batch Preferential Crystallization 45

4.1 Batch Preferential Crystallization Model . . . . . . . . . . . . . . . 46

4.2 Coupled Batch Preferential Crystallization Model . . . . . . . . . 48

4.3 Numerical Approximation of the Models . . . . . . . . . . . . . . . 51

4.4 Test Problems for Single Crystallizer . . . . . . . . . . . . . . . . . 54

4.5 Test Problems for Coupled Crystallizers . . . . . . . . . . . . . . . 59

5 Continuous Preferential Crystallization 79

5.1 Continuous Preferential Crystallization Model . . . . . . . . . . . 81

5.2 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.1 Implementation of HR-FVS . . . . . . . . . . . . . . . . . . . 88

5.2.2 Implementation of DG-Scheme . . . . . . . . . . . . . . . . . 90

5.3 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Coupled Continuous Preferential Crystallization 128

6.1 Coupled Continuous Crystallization Model . . . . . . . . . . . . . 131

6.2 Implementation of Finite Volume Scheme . . . . . . . . . . . . . . 138

6.3 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.1 Numerical Test Problem . . . . . . . . . . . . . . . . . . . . . 142

xi

7 Conclusion and Future Recommendations 161

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.2 Future Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 164

8 References 168

xii

List of Figures

1.1 Amino acid enantiomers [61]. . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Ternary phase diagram [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 Batch crystallizer with fines dissolution [61]. . . . . . . . . . . . . . . . . . 21

3.2 Problem 1: Size-independent growth at t = 100 s. . . . . . . . . . . . . . . 32

3.3 Problem: Errors and CPU times of DG-scheme. . . . . . . . . . . . . . . . 32

3.4 Problem 1: Comparison of DG-schemes of different orders . . . . . . . . . . 33

3.5 Problem 2: Size-independent growth at t = 0.5 min. . . . . . . . . . . . . . 34

3.6 Problem 3: Size-dependent growth at t = 1000 s. . . . . . . . . . . . . . . . 37

3.7 Problem 4: Size-dependent growth at t = 180min. . . . . . . . . . . . . . . 39

3.8 Problem 5: CSD without fines dissolution. . . . . . . . . . . . . . . . . . . 43

3.9 Problem 5: Fines dissolution without delay . . . . . . . . . . . . . . . . . . 43

3.10 Test 5: Fines dissolution with delay. . . . . . . . . . . . . . . . . . . . . . . 44

3.11 Problem 5: Solute masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Crystallization in coupled vessels. . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Test 1: Preferred enantiomer CSD with HR-FVS and DG scheme. . . . . . 63

4.3 Test 1: Preferred enantiomer CSD for with and without FD. . . . . . . . . 63

4.4 Test 1: CSD for counter enantiomer with and without FD. . . . . . . . . . 64

4.5 Test 1: Mass fraction for preferred enantiomer with and without FD. . . . 64

4.6 Test 1: Mass fraction for counter enantiomer. . . . . . . . . . . . . . . . . 65

xiii

LIST OF FIGURES

4.7 Test 1: Supersaturation for target enantiomer with and without FD. . . . . 65

4.8 Test 1: Growth rate for target enantiomer with and without FD. . . . . . . 66

4.9 Test 1: Nucleation rate for target enantiomer with and without FD. . . . . 66

4.10 Test 1: Purity and mean crystal size of preferred enantiomer. . . . . . . . . 67

4.11 Test 1: Productivity of preferred enantiomer. . . . . . . . . . . . . . . . . . 67

4.12 Test 2: CSD for target enantiomer with HR-FVS and DG scheme. . . . . . 68

4.13 Test 2: CSD for target enantiomer. . . . . . . . . . . . . . . . . . . . . . . 68

4.14 Test 2: Mass fraction for preferred enantiomer with and without FD. . . . 69

4.15 Test 2: Mass fraction for counter enantiomer. . . . . . . . . . . . . . . . . 69

4.16 Test 2: Supersaturation for preferred enantiomer with and without FD. . . 70

4.17 Test 2: Growth rate for preferred enantiomer. . . . . . . . . . . . . . . . . 70


4.19 Test 2: Purity and mean crystal size of preferred enantiomer. . . . . . . . . 71

4.20 Test 3: CSD for both enantiomers with HR-FVS and DG scheme. . . . . . 72

4.21 Test 3: Mass fraction for both enantiomers with HR-FVS and DG scheme. 72

4.22 Test 3: Purity and mean crystal size for preferred enantiomer. . . . . . . . 73


4.24 Test 4: Preferred CSD with FD for single and coupled crystallizers. . . . . 74

4.25 Test 4: Mass fraction with FD for single and coupled crystallizers. . . . . . 74

4.26 Test 4: Supersaturation with FD for single and coupled crystallizers. . . . 75

4.27 Test 4: Growth rate with FD for single and coupled crystallizers. . . . . . . 75

4.28 Test 4: Productivity for single and coupled crystallizers without FD. . . . . 76

4.29 Test 4: Mean crystal size for single and coupled crystallizers without FD. . 76

4.30 Test 5: Preferred CSD for single and coupled crystallizers. . . . . . . . . . 77

4.31 Test 5: Purity and productivity of p-enantiomer for single and coupled crys-

tallizers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

xiv

4.32 Test 5: Mean crystal size of p-enantiomer for single and coupled crystallizers. 78

5.1 Principle of continuous enantioselective crystallization [66]. . . . . . . . . . 82

5.2 Problem 1 (Case A): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0. 97


5.4 Problem 1 (case A): Error in mass balance and CSDs. . . . . . . . . . . . 99

5.5 Schematic diagram for periodic seeding. . . . . . . . . . . . . . . . . . . . . 100

5.6 Problem 1 (Case B): Periodic seeding: Left τS/τL = 1, Right: τS/τL = 2. . . 103

5.7 Problem 1 (Case B): Periodic seeding: Left τS/τL = 1, Right: τS/τL = 2. . . 104

5.8 Problem 2 (Case A): CSDs for the different mass of seed crystal. . . . . . . 107



5.11 Problem 2 (Case B): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0. . 111


5.13 Problem 2 (Case C): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0. . 114


5.15 Problem 3 (Case A): CSDs for the different mass of seed crystal. . . . . . . 118







6.1 Principle of coupled continuous enantioselective crystallization. . . . . . . . 132

6.2 Case I: LS: Single crystallizer results. RS: Coupled crystallizer results. . . . 151


xv



6.6 Schematic diagram for periodic seeding. . . . . . . . . . . . . . . . . . . . . 154

6.7 Case II: LS: Single crystallizer results. RS: Coupled crystallizer results. . . 155




6.11 Case III: LS: Single crystallizer results. RS: Coupled crystallizer results. . . 159

6.12 Case III: LS: Single crystallizer results. RS: Coupled crystallizer results. . . 160

7.1 Left: without counter enantiomer; Right: with counter enantiomer. . . . . 167

xvi

List of Tables

3.1 Problem 1: L1-errors and CPU times. . . . . . . . . . . . . . . . . . . . . . 31

3.2 Parameters for Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


3.4 Problem 5: Errors in mass balances for size-independent growth rate . . . 42


4.2 Parameters for Problems 1, 2 and 4 . . . . . . . . . . . . . . . . . . . . . . 57


4.4 Parameters for Problems 3 and 5 . . . . . . . . . . . . . . . . . . . . . . . 59


5.1 Physicochemical parameters related to threonine-water system ([17, 18]). . 95

5.2 Problem 1 (Case A): Values of goal functions at τL = 112.41 min. . . . . . 96

5.3 Problem 1 (case B): Periodic seeding: m(p)seeds = 4 g and τL = 112.4 min. . . 102

5.4 Problem 1 (case C): Results with FD (yes) and without FD (no) . . . . . . 105


5.6 Problem 2 (Case B): Results with FD (yes) and without FD (no) . . . . . 110

5.7 Problem 2 (Case C): Periodic seeding: m(p)seeds = 4 g and τL = 112.4 min. . . 116


5.9 Problem 3 (Case B): Results with FD (yes) and without FD (no) . . . . . 121

5.10 Problem 3 (Case C): Periodic seeding: m(p)seeds = 4 g, τL = 112.4 min. . . . . 127

xvii

LIST OF TABLES

6.2 Case I: Goal functions of single crystallizer without FD, τL = 112.41 min. . 149

6.3 Case I: Goal functions of coupled crystallizers without FD, τL = 112.41 min. 149

6.4 Case II: Results of single crystallizer with FD: FD (yes), without FD (no). 149

6.5 Case II: Results of coupled case with FD: FD (yes) and without FD (no). . 149

6.6 Case III : Periodic seeding in single crystallizer: m(p)seeds = 4 g, τL = 112.4 min.150

6.7 Case III: Periodic seeding in coupled case: m(p)seeds = 4 g, τL = 112.4 min. . . 150

xviii

LIST OF ABBREVIATIONS

Abbreviations

CFL courant-friedrichs-lewyCPU central processing unitCSD crystal size distributionDG discontinuous GalerkinFD fines dissolutionFDM finite difference schemeFEM finite element methodFV S finite volume schemeHR − FV S high resolution finite volume schemeLS left sideLPs legendre polynomialsMSMPR mixed suspension mixed product removalODE ordinary differential equationPBE population balance equationPBM population balance modelPC preferential crystallizationPDE partial differential equationPSD particle size distributionRS right sideS solidTV B total variation boundedTV D total variation diminishingPC preferential crystallizationPDE partial differential equationPSD particle size distributionRS right sideTV B total variation boundedTV D total variation diminishing

xix

Subscripts and superscripts

agg aggregationbreak breakagec crystallizercrz crystallizercryst crystallizerdiss dissolutioneff effectiveeq equilibriumexch exchangeGodu Ggodunov fluxin inflow stream to crystallizerk enantiomer k ∈ p, c (p : preferred, c : counter)L liquidLF Lax friedrichs fluxLLF Local Lax friedrichs fluxmin minimummax maximumnuc nucleationout outlet stream from the crystallizerprim primaryR Roe fluxS solidsolv solventsec secondarystd short notation for steady statesat saturated

xx

Symbols units

A1 constant of surface area dependency [m−2]a1 fitted parameter for log-normal distribution function [−]a2 fitted parameter for log-normal distribution function [−]a3 fitted parameter for log-normal distribution function [−]aASL parameter for size dependency of crystal growth [m−1]aeq temperature dependent parameter for solubility [−]aprim lumped constant of primary nucleation in Eq. (5.25) [−]B0 nucleation rate at negligible particle size z = 0 [s−1]

B(k)0 nucleation rate of k-th enantiomer at negligible particle size z = 0 [s−1]

B0,prim nucleation rate for primary heterogeneous mechanism [s−1]B0,prim,α nucleation rate for primary heterogeneous mechanism in tank α [s−1]B0,sec nucleation rate for secondary mechanism [s−1]B0,sec,α nucleation rate for secondary mechanism in tank α [s−1]B0,α nucleation rate at negligible particle size in tank α z = 0 [s−1]b nucleation order [−]

xxi

symbols (continued)

beq temperature dependent parameter for solubility [−]bsec exponent (secondary nucleation) [−]C solubility concentration [g/g of solvent]Ceq solubility concentration at equilibrium temperature [g/g of solvent]Ceq,α solubility concentration at equilibrium temperature for α crystallizer [kg/m3]

C(k)eq solubility concentration of k-th enantiomer at equilibrium temperature [g/g of solvent]

C0 Initial solubility concentration [g/g of solvent]D−

dis dissolution rate [m−1s−1]dASL parameter for size dependency of crystal growth [−]EA,b activation energy for secondary nucleation [J/mol]EA,g activation energy for crystal growth [J/mol]F number density function [m−1]Fi mean value of number density function [m−1]F0 CSD of initial seed crystals [m−1]

F(k)seeds incoming particle number flux due to seeding of k-th enantiomer [m−1s−1]

F(k)out particle number flux due to product removal of k-th enantiomer [m−1s−1]

F(k)pipe particle number flux due to fines dissolution of k-th enantiomer [m−1s−1]

xxii

symbols (continued)

F(k)0 Initial CSD of k-th enantiomer [m−1]

F (k) number density function of k-th enantiomer [m−1]

F(k)α number density of k-th enantiomer in tank α [m−1]

Fseeds number density function for seed crystals [m−1]Fin number density function of incoming particles [m−1]Fout number density function of outgoing particles [m−1]Fnuc number density function of nuclei [m−1]Fh numerical CSD [m−1]f (k) numerical flux for the k-th enantiomer [m−1]

f(k)α numerical flux for the k-th enantiomer in tank α [m−1]G crystal growth rate [m/s]G(k) crystal growth rate of k-th enantiomer [m/s]

G(k)α crystal growth rate of k-th enantiomer in tank α [m/s]

g exponent in crystal growth term [−]h numerical flux function [m−1]hGodu Godunov-flux [m−1]hLF Lax-friedrichs-flux [m−1]hLLF Local Lax-friedrichs-flux [m−1]hR Roe-flux [m−1]Ij jth mesh interval [m]K1 (fitted) constant for density [m3/kg]K2 (fitted) constant for density [m3/(kg]K3 (fitted) constant for density [m3/kg]KT (fitted) parameter for temperature dependence [K]KW (fitted) parameter for mass fraction dependence [−]kg growth rate constant [−]

xxiii

symbols (continued)

kb nucleation rate constant [m−3 s−1]

k(k)b growth rate constant of the k-th enantiomer [−]

kb,prim lumped constant for nucleation [ (kg m−3)−7/3

min K]

kb,sec rate constant for nucleation rate [m−3/min]kb,sec,0 pre-exponent factor for nucleation [m−3/min]kg,eff effective crystal growth constant [m/min]kg,eff,0 pre-exponent factor for growth [m/min]kv volume shape factor [−]m solute mass in the liquid [kg]m(k) solute mass in the liquid of k-th enantiomer [kg]m(p) solute mass in the liquid of preferred enantiomer [kg]m(c) solute mass in the liquid of counter enantiomer [kg]m mass flow rate [g s−1]

m(p)seeds mass flow rate of seeds of preferred enantiomer [kg/min]

m(k)in incoming mass flow rate of k-th enantiomer to the crystallizer [kg/min]

m(k)out outgoing mass flow rate of k-th enantiomer from the crystallizer [kg/min]

msat saturated mass [g]msolv mass of the solvent [g]mseeds mass flow of seeds [kg/min]min incoming mass flow rate to the crystallizer [kg/min]mout outgoing mass flow rate from the crystallizer [kg/min]min,c incoming mass flow rate of fresh solution [kg/min]

xxiv

symbols (continued)

mout,c outgoing stream of exhausted solution [kg/min]min,pipe incoming stream from the fines dissolution unit [kg/min]mL,out,pipe outgoing stream to the fines dissolution unit [kg/min]

m(p)S,out mass of preferred enantiomer [kg]

m(k)L,α solute mass in the liquid for k-th enantiomer in tank α [kg]

mwater mass of water [kg]

m(k)seeds,α mass flow of seeds for k-th enantiomer in tank α [kg/min]

m(k)L,in,α incoming mass flow rate to the crystallizer [kg/min]

m(k)L,out,α outgoing mass flow rate from the crystallizer [kg/min]

m(k)L,in,c,α incoming mass flow rate of fresh solution [kg/min]

m(k)L,out,c,α outgoing stream of exhausted solution [kg/min]

m(k)L,in,pipe,α incoming stream from the fines dissolution unit [kg/min]

m(k)L,out,pipe,α outgoing stream to the fines dissolution unit [kg/min]

m(c)S,out mass of counter enantiomer [kg]

mm minmod limiter function [−]N number of grid points [−]nsec exponent in secondary nucleation [−]Pr(p) productivity per unit volume [kg/(minmin3)]

Pr(p)α productivity per unit volume in tank α [kg/(minm3)]

Pu(p) purity [−]

Pu(p)α purity in tank α [−]

Pl legendre polynomial of order l [−]Q source term [m−1s−1]Qi mean value of source term [m−1s−1]Q+

nuc length based nucleation term [m−1s−1]

xxv

symbols (continued)

Q−diss dissolution rate [m−1s−1]

Q±agg length based aggregation term [m−1s−1]

Q±break length based breakage term [m−1s−1]

Qin inflow rate [m−3s−1]Qout outflow rate [m−3s−1]R gas constant [J/(mol K)]ri+ 1

2(t) upwind ratio of the consecutive flux gradients [−]

S supersaturation degree [−]S(k) supersaturation degree of k-th enantiomer [−]

S(k)α supersaturation degree for k-th enantiomer in tank α [−]

T temperature [K]Teq equilibrium temperature [min]t time [min]tend final time [min]ton on time of periodic seeding [min]tmax maximum time [min]toff off time of periodic seeding [min]tstd the time at which steady state is achieved [min]∆t time step [s]Vpipe volume of the pipe [m3]VL volume of the liquid in the crystallizer [m3]

Vpipe volumetric flow rate to the pipe [m3/min]

V volumetric flow rate [m3/min]

Vin inflow rate [m3/min]

Vout outflow rate [m3/min]

xxvi

symbols (continued)

VS overall volume of the solid phase [m3]Vc volume of the single crystal [m3]Vcrz volume of the crystallizer [m3]

Vexch volumetric exchange flow rate [m3/min]VL,α volume of the liquid in tank α [m3]

VL,α volumetric flow rate in tank α [m3/min]w mass fraction [−]wsat saturated mass fraction [−]w(k) mass fraction of k-th enantiomer [−]

w(k)α mass fraction for k-th enantiomer in tank α [−]

w(p) mass fraction of preferred enantiomer [−]w(c) mass fraction of counter enantiomer [−]weq mass fraction at equilibrium temperature [−]

w(k)eq mass fraction of k-th enantiomer at equilibrium temperature [−]

w(k) mass fraction of k-th enantiomer [−]wtot total mass fraction of preferred and counter enantiomers [−]Y (p) yield [−]

Y(p)α yield in tank α [−]

z particle size [m]∆z mesh size [m]∆zj width of mesh interval Ij [m]z(p) mean crystal size [m]zcrit critical size [m]z0 minimum crystal size [m]zmax maximum crystal size [m]z constant of initial CSD Eq. (3.69) [m]

xxvii

symbols (continued)

µ(p)0 zero moment: total number of crystals [m−1]

µ(p)1 first moment: total length of crystals [m]

ϕl local basis function of order l

ℑ(k)α numerical flux for the k-th enantiomer in tank α [−]

µi,α i-th moment of the CSD in tank α [−]ρL,α density of liquid phase in tank α [kg/m3]ρL,0,α density of fresh liquid at t = 0 in tank α [kg/m3]ψ selection (death) function for fines dissolution [−]φ flux limiting function [m−1]ε small number to avoid division by zero [−]ρc crystal density [g m−3]ρL density of liquid phase [kg/m3]ρL,0 density of fresh liquid at t = 0 [kg/m3]ρS density of solid phase [kg/m3]ρsolu density of solution [kg/m3]ρo density of water [kg/m3]ρf density of fresh solution [kg/m3]τpipe residence time in the dissolution unit [min]τL mean liquid phase residence time [min]τS mean solid residence time [min]τcryst ratio of crystallizer volume to the volumetric flow rate [min]τL,α mean liquid phase residence time in tank α [min]τS,α mean solid residence time in tank α [min]τpipe,α residence time in the pipe for tank α [min]σ constant of initial CSD Eq. (3.69) [m]ρL density of liquid phase [kg/m3]ρL,0 density of fresh liquid at t = 0 [kg/m3]

xxviii

1

Chapter 1

Fundamentals of Crystallization Process

This chapter introduces different crystallization processes which are widely used in chemi-

cal, pharmaceutical and food industries.

1.1 Crystallization

Crystallization is the process of solid crystals formation from a supersaturated homoge-

neous solution. Crystallization is a common process in nature and everyday life. Since

millions of years, it has played a vital role in the formation of stalactites within caves and

precious stones such as diamonds and sapphires. It is the best and cheapest method for the

production of pure solids from impure solutions. In chemical engineering, crystallization

process is one of the oldest unit operation. For several decades, the process is being used

and the production of sodium chloride is one of its examples. It is widely used in other

fields of chemical engineering as well. For example, the process is used in petro-chemical

industry for separation and purification of hydrocarbons, as well as for the manufacturing

of polymers and high valued chemicals and household products.

The crystallization process is driven by supersaturation of the solution [52]. This can be

achieved by cooling or evaporating the solvent. Normally, two processes are active in a

supersaturated solution, i.e., nucleation to form nuclei due to gathering of solute molecules

into clusters and the crystal growth in which the already existing crystals grows [59]. One

very important variable in industrial crystallizers is the crystal size distribution (CSD).

A control of crystal size and shape is important for achieving the desired goals and for

improving the product quality.

1.2 Types of Crystallization

Crystallization processes are either batch or continuous. The main difference between the

two is consistency of supersaturation and temperature. In batch operation both tempera-

2

ture and supersaturation cannot be kept constant at the same time, but this can be done

in the continuous crystallizers [33].

1.2.1 Batch Crystallization

Batch crystallization is an important unit operation for the production of high-value-added

chemicals at laboratory scale [33]. However, its use in industry is very low. Batch crys-

tallizer is also used if several products are to be crystallized in the same crystallizer. It is

an unsteady process and does not attain steady state [61]. Batch crystallizer is a closed

tank filled with a saturated solution. Afterwards, the solution is made supersaturated by

cooling the crystallizer but still the solution is free of particles. The crystallizer is then

seeded with seed crystals to promptly start the crystallization process. During the process,

new crystals are formed and existing crystals grow, thus, forming a CSD.

1.2.2 Continuous Crystallization

It is the process in which solution is continuously fed to the crystallizer and product is

continuously withdrawn. The continuous processes are usually very lengthy and are used

for the production of large amount of materials. These processes are operated under steady

state condition [70].

1.3 Enantiomers

Enantiomers are organic molecules which are mirror images of each other (chiral com-

pounds). They are stereoisomers just like our left and right hands whose molecules are

non-superimposable mirror images [61]. Figure 1.1 gives the illustration of amino acid

enantiomers as an example. Enantiomers are symmetric organic molecules with identical

chemical and physical properties but with different properties regarding metabolism.

3

Figure 1.1: Amino acid enantiomers [61].

A solution in which both the enantiomers are present in equal amounts is known as a

racemic solution.

1.4 Preferential Crystallization

Separation of enantiomers is an important process in chemical industry because a lot of bio-

organic molecules possess chiral properties. Mostly, one enantiomer has desired properties

while other has undesired. Due to its great importance in chemical and pharmaceutical

industries, there is a need to improve the performance of already established separation

processes [61].

To separate the enantiomeric mixtures, several methods can be applied like chromatog-

raphy, biological and non-biological asymmetric synthesis but one of the most efficient

technique used for separation and purification of enantiomers is preferential crystallization

[3, 11, 12, 13, 17, 18, 31]. In this method, one of the enantiomers is seeded in a super-

saturated racemate solution, resulting in a slight excess of the same isomer to make an

asymmetric environment, so that this isomer is crystallized out from the solution.

4

1.5 Ternary Phase Diagram

This diagram illustrates the principle of preferential crystallization process in which the

saturated solution with initial temperature Tcryst + ∆T is cooled down to Tcryst, see Figure

1.2. Thus, the solution becomes supersaturated. This process takes place in a metastable

zone in which no spontaneous nucleation occurs. In this, two enantiomers and a solvent

are mixed which is represented by point A. At this point the crystals of enantiomer E1 are

seeded. Due to supersaturated solution, the seeds grow and secondary nucleation starts. If

we assume that primary nucleation of enantiomer E2 does not occur, the process would end

at point M, representing the equilibrium point for enantiomer E1 only. But experiments

show that after certain time there is an induction of primary nucleation of E2 and therefore,

after very long time, the common equilibrium point, i.e., point E is achieved for E1 and

E2. At the end, both the enantiomers crystallize out at the same time, but the counter-

enantiomer crystallizes with a higher rate (the desired enantiomer has already been almost

crystallized).

Figure 1.2: Ternary phase diagram [61].

5

1.6 Historical Background and Motivation

Population balance equations (PBEs) are widely used for the simulation of crystallization

processes. These equations are similar to our well known mass and energy balance equa-

tions. They describe a balance law for the number of individuals of a population, such as

crystals, droplets, bacteria etc. These equations are generally important in almost all sci-

entific and engineering disciplines. For instance, biophysicists, pharmacists, food scientists,

chemical engineers, civil (environmental) engineers and many others make use of these bal-

ances for simulating different processes in their respective fields of research. Particularly,

in chemical engineering population balance models (PBMs) are employed for describing a

variety of dispersed processes namely, flocculation, crystallization, granulation, polymer-

ization, and combustion [69]. In their pioneer work, Hulburt and Katz [27] introduced

PBMs in chemical engineering and later on were fully studied by Randolph and Larson

[71]. PBMs are used to simulate different phenomena in the process, such as aggregation,

breakage, growth, nucleation, dissolution, as well as inlet and outlet streams. Mathemat-

ically, the PBEs of these models are nonlinear integro-partial-differential equations of the

number density function. In practical problems, the PBE is normally coupled to one or

more ODEs for the mass and energy balances of the liquid phase. These equations have

no analytical solutions except for few simple cases. Therefore, for most practical problems,

accurate and efficient numerical techniques are applied.

This fact has motivated researchers in this field to develop new and efficient algorithms for

solving population balance equations (PBEs), see for example Ramkrishna [70], Nicmanis

and Hounslow [57], and references therein. Several efficient schemes have been formulated

for the solution of PBEs, such as the quadrature method of moments (QMOM), the method

of characteristics (MOC), the Monte Carlo method (MCM), the finite difference methods

6

(FDMs), and the finite volume methods (FVMs).

Hulburt and Katz [27] were the first who applied the method of moments to PBMs and

pointed out the closure problem associated with this method. Afterwards, different meth-

ods were used to solve the closure problem raised by Hulburt and Katz [27] as considered by

Diemer and Olson [15]. The QMOM was introduced by McGraw [50] for modeling aerosol

evolution. In this method, quadrature approximation was used for solving integrals of PSD

and the required abscissas and weights were obtained by using the product difference (PD)

algorithm of Gordon [22]. Further work in this direction can be found in Fan et al. [20],

Marchisio [48, 49], McGraw [51], Qamar et al. [63], and references therein.

In method of characteristics, a new coordinate system is generated that reduces the PBE

into an ODE with some characteristic curves in the x-t plane [36], where t and x are used

for time and property coordinates. This method gives high accuracy for simple popula-

tion balance equations but for more complex problems the method does not work efficiently.

The Monte Carlo method gives good computational efficiency for complex stochastic PBMs

[70]. However, the computational cost of this method is generally high and accuracy is of

lower order.

In the past few decades, the finite difference methods (FDMs) were frequently used to solve

PBEs [36]. This is a domain discretization method in which the derivatives are replaced

by difference quotients so that the differential equation is represented by algebraic equa-

tions. Various discretization techniques of different accuracy orders have been suggested

by several researchers for PBMs [14, 21, 29, 30, 36, 47, 55].

7

The finite volume schemes (FVS) were initially used for gas dynamics and for the approx-

imation of PBEs, see Ma et al. [44, 45], Gunawan et al. [24], Qamar et al. [62], and

reference therein. The FVS refers to small volumes surrounding the nodal points in the

domain. The method has been used efficiently in different fluid flows and gas dynamics

applications. An advantage of using the FVS is that they can be applied even if the details

of physical characters and behaviors of the flow are not provided. Different researchers have

used these schemes to solve the problems in gas dynamics, astrophysical flows, multi-phase

fluid flows and detonation waves [5, 6, 28, 38, 40, 56, 77].

1.7 New Results

This work is related to the development and simulation of batch and continuous crystal-

lization models. Especially, models are derived for simulating batch and continuous enan-

tioselective preferential crystallization processes in single and coupled crystallizers. Such

processes are highly important in chemical and pharmaceutical industries. The effects of

nucleation, growth, fines dissolution, seeding strategies and residence time characteristics

on the CSD are analyzed.

For the first time, a TVB Runge-Kutta discontinuous Galekrin (DG) finite element method

is applied to solve batch crystallization model [65]. For comparison, the high resolution

finite volume scheme (HR-FVS) is applied to solve the same model [32, 59]. The schemes

satisfy the total variation bounded (TVB) property which assures the positivity of the

schemes, e.g., the non-negativity of CSD in the present case. The suggested methods have

capabilities to capture narrow peaks and sharp discontinuities of the CSD. The perfor-

mance of the DG-scheme can further be improved by inserting additional nodes in the

same solution element and, thus, avoids the expansion of mesh stencils which is normally

observed in high resolution finite volume scheme (HR-FVS). For that reason, the scheme

8

can be easily implemented up to boundary cells without compromising on accuracy and

efficiency. It was found that the proposed numerical schemes have the capability to solve

the given models more efficiently and accurately.

The above schemes were extended to simulate batch preferential crystallization of enan-

tiomers in single and double crystallizers equipped with fines dissolution units. The models

are further elaborated by considering the isothermal and non-isothermal conditions. Firstly,

the crystallization of preferred enantiomer is assumed to take place in a single crystallizer

equipped with a fines dissolution unit. The extracted solution is sieved by filters and

supposed to be free of bigger crystals. Thus, only tiny particles are withdrawn to the dis-

solution unit. To assure a crystal-free liquid exchange, it is assumed that all particles in the

dissolution loop are dissolved before re-entering back to the crystallizer by using a heat ex-

changer. Furthermore, before re-entering into the crystallizer, the liquid in the dissolution

loop is cooled down again. The breakage and agglomeration processes are not considered in

this study. Secondly, the model is extended for a coupled batch preferential crystallization

process with isothermal and non-isothermal conditions. In this setup, the crystallization of

two enantiomers is assumed to take place in two separate crystallizers which are coupled

through exchange pipes. There are two main advantages of considering coupled crystalliz-

ers. The first one is that both enantiomers are crystallized out simultaneously in separate

crystallizers. Secondly, due to the liquid exchange between the crystallizers, the growth

process enhances in both crystallizers and, thus, crystals of large mean size are obtained.

Both HR-FVS and DG-method are used to solve these models. The DG-method is applied

for the first time to simulate such models.

For the first time continuous preferential crystallization is investigated and the effects of

different seeding strategies and residence time characteristics are analyzed on the dynam-

9

ics of a Mixed Suspension Mixed Product Removal (MSMPR) crystallizer equipped with a

fines dissolution unit [66]. The fines dissolution is included as recycle streams around the

MSMPR crystallizer. Moreover, primary heterogeneous and secondary nucleation mecha-

nisms along with size-dependent growth rates are taken into account. The model is then

extended for a coupled continuous preferential crystallization process. In this setup, the

crystallization of two enantiomers is assumed to take place in two separate crystallizers

which are coupled through exchange pipes. To judge the quality of the process some goal

functions are used, such as purity, productivity, yield and mean crystal size of the preferred

enantiomer. Both HR-FVS and DG-method are applied to solve these models [65, 66]. Sev-

eral numerical case studies are carried out. These results could be used to find the optimum

operating conditions for improving the product quality and for reducing the operational

cost of continuous preferential crystallization. Altogether, the process appears to possess

large potential and deserves practical realization.

1.8 Layout of the Thesis

The remaining part of the thesis is organized as follows:

In chapter 2, the population balance model (PBM) is introduced and major phenomena

which include growth, nucleation, dissolution, breakage, aggregation, as well as inflow and

outflow of particles are explained. Furthermore, some goal functions are introduced, such

as purity, productivity, yield and mean crystal size of the preferred enantiomer to judge

the quality of the process. These goal functions give detailed information about the success

and potential of preferential crystallization processes.

Chapter 3 introduces the mathematical model of a batch crystallizer equipped with fines

dissolution unit. This dissolution produces an improvement in the crystal size and product

10

quality. A TVB Runge-Kutta discontinuous Galerkin finite element method is derived to

solve the model. For comparison, the HR-FVS is applied to the same model [59]. The

comparison verifies the robustness of the proposed methods.

Chapter 4 introduces preferential batch crystallization models for single and coupled crys-

tallizers. The same DG-method and HR-FVS are applied to solve these models. Several

case studies are carried out. The comparison verifies the efficiency and accuracy of the

proposed method.

In Chapter 5, a mathematical model is derived for studying the effects of different seeding

strategies and residence time characteristics on the dynamics of a MSMPR crystallizer

equipped with a fines dissolution unit [66]. A semi-discrete HR-FVS is employed for dis-

cretizing the derivatives with respect to the length coordinate. The resulting ordinary

differential equations (ODEs) are solved by a Runge-Kutta method of order four. After-

wards, a DG-method is implemented to solve the same model [65]. Several numerical case

studies are carried out. The results support process design and optimization.

In Chapter 6, a mathematical model is formulated for coupled continuous preferential crys-

tallization which incorporates fines dissolution unit with time-delay. The high resolution

finite volume scheme (HR-FVS) is applied to approximate the model.

Chapter 7 contains the conclusion of the entire research work and future recommendations.

Finally, Chapter 8 contains the references used in the thesis.

11

1.9 Publications Related to the Research Work

Journal papers

Qamar, S., Hussain, I., Morgenstern, A. S., Application of discontinuous Galerkin scheme

to batch crystallization models. Industrial and Engineering Chemistry Research, 50 (2010),

4113-4122.

Qamar, S., Elsner, M.P., Hussain, I., Morgenstern, A. S., Effects of seeding strategies and

different operating conditions on preferential continuous crystallizers. Chemical Engineer-

ing Science, 71 (2012), 5-17.

Mukhtar, S., Qamar, S., Hussain, I., Jan, A., A Quadrature method of moments for solv-

ing volume-based univariate and bivariate population balance models. Positive reviews are

received from the Brazilian Journal of Chemical Engineering, (2012).

Conference proceeding

Qamar, S., Mukhtar, S., Hussain, I., Morgenstern, A. S., An efficient numerical technique

for solving a batch crystallization model with fines dissolution. Proceedings of 4th Interna-

tional Conference on population balance modeling, September 15-17, 2010 Berlin Germany,

pp. 819-833.

12

13

Chapter 2

Population Balance Models

This chapter introduces the length based population balance model which incorporates the

phenomena of crystal nucleation, growth, aggregation, breakage, dissolution, and inflow

and flow streams. The model provides an essential background for understanding the

crystallization process in forth coming chapters. Moreover, the goal functions, such as

purity, productivity, yield, and mean are introduced to analyze the product quality of

preferential crystallization in Chapters 4, 5 and 6.

2.1 Population Balance Models

In this study a well mixed system is considered, i.e., all the particles are homogeneously

distributed in the whole region of study. Thus, the population balance model may be

integrated out over all points in space and the resulting population balance equation (PBE)

describes the dynamics of the size-distribution function F := F (t, z) ≥ 0 of particles of

size z > 0 at time t ≥ 0. The population balance equation (PBE) is expressed as [30, 71]

∂F (t, z)

∂t= − ∂[G(t, z)F (t, z)]

∂z+ Q+

nuc(t, z) −Q−diss(t, z)

+ Q±agg(t, z) + Q±

break(t, z) + Qin(t, z) −Qout(t, z) , (t, z) ∈ R2+, (2.1)

where R+ = (0,∞). The term G(t, z) represents the size-dependent growth rate, Q+nuc(t, z)

is responsible for the nucleation of particles, and Q−diss(t, z) accounts for the dissolution of

particles. The terms Q±agg and Q±

break, represent the birth and death of particles during

the aggregation and breakage processes, respectively. The last two terms Qin(t, z) and

Qout(t, z) on the right hand side of above equation represent the inflow and outflow rates

of particles from the given system, respectively. Mathematically, the inflow and outflow

streams are defined as

Qin(t, z) =Vin

Vcrz

Fin(t, z), Qout(t, z) =Vout

Vcrz

Fout(t, z), (2.2)

where Vin and Vout represent the volumetric inlet and outlet flow rates from the system of

volume Vcrz, while Fin(t, z) and Fout(t, z) are the number densities of incoming and outgo-

14

ing particles.

Generally, a continuous crystallization process has inflow and outflow of particles, while

the batch crystallization process has no inflow or outflow of particles, therefore the last

two terms on the right hand side of Eq. (2.1) does not appear in the batch crystallization

process. This work is concerned with both batch and continuous crystallization processes.

2.2 Growth and Nucleation

During the growth process the existing particles of the dispersed system grow by consum-

ing the solute mass from the solution. Therefore, the total number of particles remains

the same but volume continuously increases. At microscopic level the particle growth rate

is unpredictable but is predictable on macroscopic level. Generally, the growth rate is a

complex phenomenon that is not completely understood when one is dealing with size-

dependent growth rates. Especially, it is difficult to measure the size-dependent growth

rate during the experiments. The adsorption layer theory and mass transfer theory are

usually used for modeling the size-dependent growth rate of particles [78].

The nucleation process introduces new particles in the system by mixing two or more non-

particulate matters. Due to this process, the population of particles increases in the system.

Nucleation can be divided into two categories, such as primary and secondary nucleations,

and the primary nucleation is further subdivided into two categories namely, homogeneous

and heterogeneous nucleations. Homogeneous nucleation occurs in the particle-free liquid

phase. Large amount of small particles can be produced with this type of nucleations. It is

commonly observed in salting out, in precipitation, and in crystallization. The second type

of primary nucleation is heterogeneous nucleation which happens due to several reasons,

for example tubes are not cleaned due to any previous particulate process or any pore

15

in the tube has some dust inside, etc. The secondary nucleation is the heterogeneous

nucleation induced by existing particles, for example due to seeding of batch crystallizer.

The nucleation term is defined as

Q+nuc(t, z) = Fnuc(t, z)B0(t).

Here, Fnuc(t, z) denotes the number density function of nuclei and B0(t) represents the rate

of nucleation. In the case of batch crystallization Fnuc(t, z) = δ(z − z0), where δ is the

Dirac delta distribution and z0 is the minimum crystal size. This means that nucleation

introduces only small nuclei of size z0 in the system. In other words, nucleation terms acts

like a point source at the left boundary of the size domain.

2.3 Dissolution

Generally, particles are not stable below a certain size, usually called critical size. Small

particles with negative growth rate will finally become smaller than the critical size and,

thus, disappear from the population. This phenomenon can be described as

Q−dis(t, z) = D−

dis F (t, z) , (2.3)

where D−dis represents the dissolution rate. If the particles are assumed to disappear im-

mediately when reaching the critical size, the dissolution rate has to be infinite.

2.4 Aggregation

The merging of two or more particles to make a bigger one is called aggregation. In this

process the total volume remains conserved while the number of particles reduces. It takes

place in a variety of processes that includes crystallization, production of dry powders, and

fluidized beds etc. [70].

16

2.5 Breakage

In this process, the bigger particles split into two or more pieces. This results in a rapid

increase of particles. It has a wide range of applications in crystallization, granulation and

environmental sciences. It is obvious to see that the total number of particles increases in

a breakage process while the total volume of the particles remains the same throughout

the process [70].

2.6 Goal Functions

In order to judge the quality of continuous preferential crystallization some goal functions

can be used, such as purity, productivity, yield, and mean crystal size of the preferred

enantiomer [61]. These goal functions give detailed information about the success and

potential of continuous preferential crystallization discussed in Chapters 4, 5 and 6.

Purity: It is a ratio of the mass of preferred enantiomer to the sum of the masses of

preferred and counter enantiomers. It can be described by the following equation

Pu(p) =m

(p)S,out

m(p)S,out + m

(c)S,out

, (2.4)

where m(p)S,out and m

(c)S,out are the masses of preferred and counter enantiomers, respectively.

Productivity: The productivity of a continuous crystallization process can be defined as

the mass flow of solid produced per unit size. It can be described by the following equation

Pr(p) = mass flow of solid produced (preferred) per unit volume of fluid =m

(p)S,out − m

(p)seeds

VL

,

(2.5)

where m(p)S,out, m

(p)seeds and VL denote the mass of preferred enantiomer, mass flow of seeds

and volume of the liquid in the crystallizer, respectively.

17

Yield: The yield can be defined as the ratio of the mass flow of solid target particles (p)

produced over mass flow of this enantiomer in the feed solution introduced

Y (p) =m

(p)S,out − m

(p)seeds

m(p)in.c

, (2.6)

where m(p)S,out, m

(p)seeds and m

(p)in.c denote the mass of preferred enantiomer, mass flow of seeds

and incoming mass of preferred enantiomer, respectively.

Mean crystal size: Besides the purity, the crystal size also plays an important role

in most industrial applications. For that reason, the mean crystal size of the preferred

enantiomer is instructive. It is defined as:

z(p) =µ

(p)1

µ(p)0

, (2.7)

where µ(p)1 is total length of crystals and µ

(p)0 is total number of crystals.

18

19

Chapter 3

Batch Crystallization

Batch crystallization is an important unit operation for the production of high-value-added

chemicals at laboratory scale [33]. Due to its widespread use, finding an efficient working

strategy is mandatory to enhance the product quality and process efficiency. Therefore, it

is essential to understand the process itself and the impact of process variables. The math-

ematical modeling of the process is helpful to attain the required goals and to investigate

the effect of different operating conditions. The observed data can be used to improve the

product quality and to optimize the operational cost.

In this chapter, a new class of scheme is introduced for solving the batch crystallization

models. In 1973, Reed and Hill [73] for the first time introduced the discontinuous Galerkin

(DG) finite element method for solving hyperbolic equations. Since then, various DG meth-

ods were developed for solving hyperbolic, elliptic and parabolic problems [4, 26, 68]. The

technique was further developed by Cockburn and his co-authors by introducing Runge-

Kutta DG-scheme in a series of papers for solving hyperbolic conservation laws [8, 9, 10].

The scheme employs DG-scheme in space-coordinates that converts the given partial dif-

ferential equation (PDE) to a system of ODEs which is computed by using high order

Runge-Kutta method. The scheme satisfies the total variation bounded (TVB) property

that assures the scheme positivity, for instance, the non-negativity of crystal size distribu-

tion (CSD) in the present case. The same DG-scheme is implemented for solving the batch

crystallization models in this dissertation. Besides different classes of schemes, DG-schemes

are more stable and high-order accurate, are capable to handle complex geometries and

irregular meshes with hanging nodes, and can incorporate arbitrary degrees of polynomial

approximations in different elements [8, 68]. The test cases verify the good performance of

the method. For authentication, the numerical results of the suggested method are com-

pared with the flux-limiting HR-FVS of Koren [32, 59]. It was observed that the proposed

method produces better results and is more efficient as compared to other flux-limiting

20

schemes for solving batch crystallization models [59, 64].

3.1 Batch Crystallization Model

Here, a mathematical model is presented for batch crystallizer connected with a fines

dissolution unit. This dissolution process may give an improvement in the crystal size and,

hence, the product quality. The simplified sketch of a batch unit is given in Figure 3.1.

LoopFines Dissolution

dissolution pipe

Settling ZoneAnnular

Figure 3.1: Batch crystallizer with fines dissolution [61].

Assumptions

We make use of the following assumptions while considering this model:

• Ideally mixed crystallizer.

• Constant overall volume.

• Size dependent/independent growth rate.

• No breakage, attrition and agglomeration.

• Isothermal and non-isothermal conditions.

21

• All fines dissolve completely in the dissolution pipe.

• Before liquid comes back to the crystallizer from the pipe, it is cooled again to avoid

negative effect of warm liquid on crystals of the crystallizer.

The rate of change in CSD is described by the PBE [27]

∂F (t, z)

∂t= − ∂[G(t, z)F (t, z)]

∂z+ Q(t, z) , (3.1)

with initial and boundary conditions of the form

F (0, z) = F0(z) , F (t, z0) =B0(t)

G(t, z0). (3.2)

Here, F0(z) represents the CSD of seed crystals added at the beginning of the batch process,

and Q(t, z) is any sink or source term. In addition to the solid phase, a balance law for the

liquid phase is also required in the crystallization process. The expressions for the liquid

balance law, growth rate, and nucleation rate will be presented in the test problems. In

the present case, the term Q(t, z) represents the dissolution of small nuclei below a certain

critical size and is defined in Problem 5.

3.2 Numerical Schemes

In this section, the numerical solution of Eq. (3.1) is presented by using the discontinuous

Galerkin (DG) finite element method and high resolution finite volume scheme (HR-FVS).

3.2.1 Implementation of Discontinuous Galerkin finite elementmethod

For the numerical approximation of PBE in Eq. (3.1), a TVB Runge-Kutta DG-method

[8, 68] is implemented. In this approach, the DG discretization is adopted in the size

22

variable only. The time derivatives are discretized by using a TVB Runge-Kutta method.

Let us define

f(G, F ) = G(t, z)F (t, z). (3.3)

Then, the PBE in Eq. (3.1) can be re-written as

∂F (t, z)

∂t= − ∂[f(G, F )]

∂z+ Q(t, z). (3.4)

To discretize the size-domain [z0, zmax], we proceed as follows. For j = 0, 1, 2, ....N , let zj+ 12

be the cells partitions, Ij =(

zj− 12, zj+ 1

2

)

be the domain of cell j, ∆zj = zj+ 12− zj− 1

2be

the width of cell j, and I =⋃

Ij are the particle size range partitions. We are looking for

an approximation Fh(t, z) to F (t, z) so that for all t ∈ [0, tmax], Fh(t, z) is belonging to the

space of finite dimensions

Vh =

v ∈ L1(I) : v|Ij∈ P k(Ij), j = 0, 1, 2, ....N

, (3.5)

where P k(Ij) stands for the polynomials space in I of at most k degree. Note that in Vh,

discontinuities of the functions are allowed at the cell interface zj+ 12. To find the numerical

solution Fh(t, z), a weak formulation is needed which is usually obtained by multiplying

Eq. (3.4) with a smooth function v(z) and by integrating over the interval Ij. After using

integration by parts, the weak formulation becomes

∫

Ij

∂F (t, z)

∂tv(z)dz = −

(

f(Gj+ 12, Fj+ 1

2)v(zj+ 1

2) − f(Gj− 1

2, Fj− 1

2)v(zj− 1

2))

+

∫

Ij

(

f(G, F )∂v(z)

∂z+ Q(t, z)v(z)

)

dz . (3.6)

Selecting Legendre polynomials (LPs), Pl(z), of order l as local basis functions is one way

to formulate Eq. (4.35). Here, the L2-orthogonality property of LPs can be used, such as

1∫

−1

Pl(s)Pl′(s)ds =

(

2

2l + 1

)

δll′ . (3.7)

23

For each z ∈ Ij, the solution Fh is defined as

Fh(t, z) =k

∑

l=0

F(l)j ϕl(z) , (3.8)

where

ϕl(z) = Pl (2(z − zj)/∆zj) . (3.9)

It can be easily verified that

(

1

2l + 1

)

F(l)j (t) =

1

∆zj

∫

Ij

Fh(t, z)ϕl(z)dz . (3.10)

Due to above Legendre polynomials as local basis functions, the test function ϕl ∈ Vh has

to be used instead of the smooth function v(z) and the approximate solution Fh in place

of the exact solution F . Moreover, the function Fj+ 12

= F (t, zj+ 12) is not known at the cell

interface zj+ 12. Therefore, the flux f(G, F ) has to be approximated by a numerical flux

that depends on two values of Fh(t, z), i.e.,

f(Gj+ 12, Fj+ 1

2) ≈ hj+ 1

2= h(Gj+ 1

2, F−

j+ 12

, F+j− 1

2

) . (3.11)

Here,

F−

j+ 12

= Fh(t, z−

j+ 12

) =k

∑

l=0

F(l)j ϕl(zj+ 1

2) , F+

j− 12

= Fh(t, z+j− 1

2

) =k

∑

l=0

F(l)j ϕl(zj− 1

2) . (3.12)

By utilizing the above expressions, the weak formulation in Eq. (4.37) simplifies to

dF(l)j (t)

dt= − 2l + 1

∆zj

(

hj+ 12ϕl(zj+ 1

2) − hj− 1

2ϕl(zj− 1

2))

+2l + 1

∆zj

∫

Ij

(

f(G, Fh)dϕl(z)

dz+ Q(t, z)ϕl(z)

)

dz . (3.13)

According to Eqs. (3.2) and (4.41), the initial data for this equation is given as

F(l)j (0) =

2l + 1

∆zj

∫

Ij

F0(z)ϕl(z)dz . (3.14)

24

The next step is to select the suitable numerical flux function h. The scheme is monotone

if the flux function h(G, a, b) is consistent, h(G, F, F ) = f(G, F ), and fulfills the Lipschitz

condition [41, 80]. In the literature, the following numerical fluxes are normally used

satisfying the above properties [38, 41, 80].

(i) The Godunov-flux:

hGodu(G, a, b) =

mina≤s≤b f(G, s) if a ≤ b ,maxa≥s≥b f(G, s) if a > b .

(3.15)

(ii) The Lax-Friedrichs-flux:

hLF (G, a, b) =1

2[f(G, a) + f(G, b) − C(b − a)], (3.16)

C = maxinf F (0)(z)≤s≤sup F (0)(z)

|f ′(G, s)| . (3.17)

(iii) The Local Lax-Friedrichs-flux:

hLLF (G, a, b) =1

2[f(G, a) + f(G, b) − C(b − a)] , (3.18)

C = maxmin (a,b)≤s≤max (a,b)

|f ′(G, s)| . (3.19)

(iv) The Roe-flux with entropy fix:

hR(G, a, b) =

f(G, a), if f ′(s) ≥ 0 for s ∈ [min(a, b), max(a, b)] ,f(G, b), if f ′(s) ≤ 0 for s ∈ [min(a, b), max(a, b)] ,hLLf (G, a, b) otherwise .

(3.20)

The above fluxes were tested and all of them produced the same results for considered test

problems. In this chapter, the local Lax-Friedrichs flux was considered in all test problems.

The 10th order Gauss-Lobatto quadrature formula was used for approximation integrals

on the right-hand-side of Eq. (3.13).

A limiting procedure is required to achieve local maximum principle with respect to the

means. For that reason, it is required to modify the interface values F±

j± 12

in Eq. (4.42) by

some local projection limiter. To this end, Eq. (4.43) can be written as [8, 9]

F−

j+ 12

= F(0)j + Fj , F+

j− 12

= F(0)j − Fj , (3.21)

25

where

Fj =k

∑

l=1

F(l)j ϕl(zj+ 1

2) , Fj = −

k∑

l=1

F(l)j ϕl(zj− 1

2). (3.22)

Next, Fj and Fj can be modified as

F(mod)j = mm

(

Fj, ∆+F(0)j , ∆−F

(0)j

)

, F(mod)j = mm

(

Fj, ∆+F(0)j , ∆−F

(0)j

)

, (3.23)

where ∆± = ±(Fi±1 − Fi) and mm is the minmod function which is given as

mm(a1, a2, a3) =

s · min1≤i≤3

|ai| if sign(a1) = sign(a2) = sign(a3) = s ,

0 otherwise .(3.24)

Then, Eq. (3.21) modifies to

F−(mod)

j+ 12

= F(0)j + F

(mod)j , F

+(mod)

j− 12

= F(0)j − F

(mod)j , (3.25)

and Eq. (4.42) is replaced by

hj+ 12

= h(Gj+ 12, F

−(mod)

j+ 12

, F+(mod)

j− 12

) . (3.26)

This limiter corresponds to adding the minimum amount of numerical diffusion while pre-

serving the stability of the scheme. The DG-method combined with the above stated slope

limiter is proven to be stable [23]. Finally, a Runge-Kutta method is required for solving

the ODE-system. Let us rewrite Eq. (3.13) in a compact form as

dFh

dt= Lh(t, Fh) . (3.27)

Then, the r-order TVB Runge-Kutta method can be used to approximate Eq. (3.27)

(Fh)k =

k−1∑

l=0

[

αkl(Fh)(l) + βkl∆tLh((Fh)

(l), tn + dl∆t)]

, k = 1, 2, · · · , r , (3.28)

where based on Eq. (4.45)

(Fh)(0) = (Fh)

n , (Fh)(r) = (Fh)

n+1 . (3.29)

26

For the second order TVB Runge-Kutta method the coefficient are given as [8]

α10 = β10 = 1 , α20 = α21 = β21 =1

2, β20 = 0; d0 = 0 , d1 = 1 . (3.30)

While, for the third order TVB Runge-Kutta method the coefficient are given as

α10 = β10 = 1 , α20 =3

4, β20 = 0 , α21 = β21 =

1

4, α30 =

1

3,

β30 = α31 = β31 = 0 , α32 = β32 =2

3; d0 = 0 , d1 = 1 , d2 =

1

2. (3.31)

To guarantee stability and numerical convergence of the proposed method, the time step

is chosen according to the following Courant-Friedrichs-Lewy (CFL) condition [8, 41]

∆t ≤(

1

2k + 1

)

min(∆zj)

max(|f ′(G, F )|) , (3.32)

where k = 1, 2 for the second and third order methods, respectively. This time step is

adaptive which reduces for the case of large variations (large slopes) in the solution and

increases otherwise.

Boundary conditions: Consider the boundary at z− 12

= z0. Since, G(t, z) ≥ 0, the left

boundary condition described by Eq. (3.2) can be formulated as [8, 9]

F−

− 12

(t) =B0(t)

G(t, z0), F

(mod)0 = mm

(

F0, ∆+F(0)0 , 2

(

F(0)0 − B0(t)

G(t, z0)

))

,

F(mod)0 = mm

(

F0, ∆+F(0)0

)

. (3.33)

Outflow boundary conditions were used on the right end of the domain,∂F

(l)h

∂z

∣

∣

∣

∣

z=zmax

= 0.

3.2.2 Implementation of Finite Volume Scheme

In this section, the high resolution finite volume scheme of Koren [32, 59] is applied for

discretizing the derivative of the length coordinate in Eq. (3.1).

Before applying the finite volume scheme, we first discretize the computational domain

which is the crystal size z in the present case. Let N be a large integer, and denote by

27

(zi− 12), i = 1, 2, · · · , N + 1 the partitions of cells in the domain [z0, zmax], where z0 is the

minimum and zmax is the maximum crystal length of interest. For each i = 1, 2, · · · , N , ∆z

represents the cell width, the points zi refer to the cell centers, and the points zi± 12

denote

the cell boundaries. The integration of Eq. (3.1) over the cell Ωi =[

zj− 12, zj+ 1

2

]

yields the

following cell centered semi-discrete finite volume schemes for fi± 12

= (GF )i± 12

∫

Ωi

∂F (t, z)

∂tdz = − (fi+ 1

2(t) − fi− 1

2(t)) +

∫

Ωi

Q(t, z)dz. (3.34)

Let Fi and Qi represent the mean values of the number density and source term in each

cell Ωi, i.e.

Fi(t) =1

∆z

∫

Ωi

F (t, z)dz , Qi(t) =1

∆z

∫

Ωi

Q(t, z)dz . (3.35)

Then, Eq. (3.34) can be written as

dFi(t)

dt= −

fi+ 12(t) − fi− 1

2(t)

∆z+ Qi(t), i = 1, 2, · · · , N . (3.36)

Here, the computational domain is divided in N mesh points. The performance of the

finite volume discretization is found by the way in which the cell face fluxes are calculated.

Assuming that the flow is in the positive z−direction, a first order accurate upwind scheme

is found by taking the backward differences [59]:

fi+ 12(t) = (G(t)F (t))i , fi− 1

2(t) = (G(t)F (t))i−1 . (3.37)

To get high order accuracy of the scheme, one has to use a better approximation of the

cell interface fluxes. According to the high resolution finite volume scheme of Koren [32]

the flux at the right boundary zi+ 12

is approximated as

fi+ 12(t) = fi(t) +

1

2φ(ri+ 1

2(t))(fi(t) − fi−1(t)) . (3.38)

Similarly, one can approximate the flux at the left cell boundary. The flux limiting function

φ is defined as [32]

φ(

ri+ 12(t)

)

= max

(

0, min (2ri+ 12(t), min (

1

3+

2ri+ 12(t)

3, 2))

)

. (3.39)

28

Here, ri+ 12(t) is the so called upwind ratio of the consecutive flux gradients:

ri+ 12(t) =

fi+1(t) − fi(t) + ε

fi(t) − fi−1(t) + ε, (3.40)

where ε is a small number to avoid division by zero. It has been observed by Koren [32]

that this method is second order accurate in the coordinates of size. Therefore, the second

order ODE-solver is required for the overall size-time accuracy of the scheme. In this work,

we have used the same TVB Runge-Kutta method as discussed above. In the case of fines

dissolution with time-delay, the residence time in the pipe was taken as an integer multiple

of the time step. This facilitates to keep the old values in memory and avoids the linear

interpolation. The same procedure was also used in the DG-scheme.

The current HR-FVS is not applicable up to the boundary cells because it needs values of

the cell nodes which are not present. To overcome this problem, the first order approx-

imation of the fluxes is used at the interfaces of the first two cells on the left-boundary

and at the interfaces of the last cell on the right-boundary. At the remaining interior cell

interfaces, the high order flux approximation of Eq. (3.38) is used. It should be noted that,

the first order approximation of the fluxes in the boundary cells does not effect the overall

accuracy of the proposed high resolution scheme [32].

3.3 Test Problems

In this section, some case studies are presented. For authentication, the numerical results

of the proposed method are compared with those obtained from the HR-FVS of Koren

[32, 59, 64]. Except the test problem 5, the source term Q(t, z) is set equal to zero. The

linear basis functions are used in all test problems for the DG-scheme which gives second

order accurate results in the length coordinate. The ODE-system was solved by a third-

29

order Runge-Kutta method given by Eqs. (3.28) and (3.31). The adaptive time step was

calculated according to Eq. (3.32) in all test problems.

Problem 1

The initial data are given as

F (0, z) =

0.0 if z ≤ 2.0 µm ,1010 if 2.0 µm < z ≤ 10 µm ,0.0 if 10 µm < z ≤ 18 µm ,

1010cos2(π(z−26)64

) if 18 µm < z ≤ 34 µm ,0.0 if 34 µm < z ≤ 42µm ,

1010

√

1 − (z−50)2

64if 42 µm < z ≤ 58 µm ,

0.0 if 58 µm ≤ z ≤ 66 µm ,

1010e−(z−70)2

2σ2 if 66 µm < z ≤ 74 µm ,0.0 if z > 74 µm .

(3.41)

Here, G = 0.1 µm/s and B0 = 0 s−1. The analytic solution for F (0, z) = F0(z) is given as

F (t, z) = F0(Gt−z). The maximum crystal length is 100µm that is divided into 100 equal

sub-intervals and the simulation time is 100 seconds. The first distribution in the initial

data represents a square step function of width 8µm. This step function is considered

for testing the ability of the schemes in resolving sharp discontinuities. The second one

is a cosine-squared function of width 16µm representing a smooth behavior. The third

one denotes a semi ellipse of width 16 µm which is challenging due to its combined rapid

and slow variations in gradients. The last distribution is a narrow Gaussian distribution

with σ = 0.778∆z for testing the capability of the scheme in resolving sharp peaks. Figure

3.2 shows a comparison of the DG-method with the finite volume schemes. The figure

shows that the DG-method resolves the jump discontinuities and gives a better approxi-

mation of the peak of the Gaussian function as compared to Koren’s scheme. Moreover,

the DG-scheme captures the right discontinuities more efficiently as compared to the left

discontinuities. The first order upwind scheme gives a smeared solution. Table 3.1 presents

30

the absolute errors and CPU times of the schemes. It can be seen that the DG-scheme

gives less error in the solution and its computational cost is comparable to Koren’s scheme.

Figure 3.3 shows absolute errors and CPU times of the DG-scheme at different numbers

of grid points. Finally, Figure 3.4 gives a comparison of different orders of DG-schemes.

There is no drastic improvement in the solution with high order basis functions. This is

due to the more restricted minmod limiter which produces wiggles and more smearing in

the solution of high order basis functions [9]. However, the accuracy of the scheme can be

improved by using WENO limiters [9, 67].

Table 3.1: Problem 1: L1-errors and CPU times.Scheme L1-error CPU (s)DG-scheme 3.57 × 1010 0.046Koren-scheme 4.39 × 1010 0.045First-order FVS 1.32 × 1011 0.039

Problem 2

In this problem the numerical scheme is tested for a stiff nucleation phenomenon. Assume

that a stiff nucleation phenomenon takes place at the minimum crystal size (z0 = 0) as a

function of time [42]

F (t, 0) = 100 + 106 exp(−104(t − 0.215)2) . (3.42)

The crystal size and time ranges are given as 0 ≤ z ≤ 2.0 mm and 0 ≤ t ≤ 0.5 min,

respectively. The initial CSD is taken as

F (0, z) =

100 for 0.4 mm ≤ z ≤ 0.6 mm,0.01 elsewhere .

(3.43)

31

0 20 40 60 80 1000

2

4

6

8

10

12x 10

9

length [µm]

CS

D [1

/µm

]exact DG−scheme FVS first order

Figure 3.2: Problem 1: Size-independent growth at t = 100 s.

0 1000 2000 3000 40000

1

2

3

4

5

6

7

number of grid points

CPU time (s)

absolute error ( × 1010 )

Figure 3.3: Problem: Errors and CPU times of DG-scheme.

32

0 20 40 60 80 1000

2

4

6

8

10

12x 10

9

length [µm]

CS

D [1

/µm

]

DG−schemes of different ordersexact 2nd order 3rd order 4th order

Figure 3.4: Problem 1: Comparison of DG-schemes of different orders .

Here, a constant growth rate, G = 1.0 mm/min, is considered. The analytical solution is

given as [42]

F (t, z) =

100 + 106 exp(−104((Gt − z) − 0.215)2) for 0.0 ≤ z ≤ Gtmm ,100 for 0.4 mm ≤ z − Gt ≤ 0.6 mm,0.01 elsewhere .

(3.44)

In the solution, a square step discontinuity and a narrow peak which is originated from

nucleation move along the propagation path-line, z = z0 +Gt. The numerical test is carried

out on 200 grid points. The numerical results are shown in Figure 3.5. It can be observed

that the DG-method gives a better approximation of the solution than the Koren’s scheme.

Once again, the scheme shows better performance in resolving right discontinuities. The

stiff nucleation at the left boundary, which produces a sharp peak and a second step profile,

makes this problem much harder than the previous problems. The absolute error in the

numerical solution of the DG-scheme is equal to 1.37×104, while in the Koren scheme it is

equal to 1.83 × 104. Moreover, the simulation times of the DG-scheme and Koren scheme

were found to be 0.063 and 0.060 seconds, respectively.

33

0 0.5 1 1.5 210

−2

100

102

104

106

length [mm]

CS

D [1

/mm

]

exactDG−schemeFVS

Figure 3.5: Problem 2: Size-independent growth at t = 0.5 min.

Problem 3

This problem is selected for verifying the accuracy and efficiency of the DG-scheme for a

real batch process. The initial CSD is given as [45, 46]

F (0, z) =

−3.48 × 10−4z2 + 0.136z − 13.3 , if 180.5 µm ≤ z ≤ 210.5 µm ,0 , elsewhere .

(3.45)

A balance law for the liquid phase is needed which takes into account the depletion of

material from the solution due to crystal growth. In this problem, we take as variable

concentration instead of mass

dC(t)

dt= −3ρc

∞∫

0

z2 G(t, z) F (t, z) dz , C(0) = C0 , (3.46)

where C denotes the solute concentration in the solution, ρc is the crystal density and

the volume shape factor is denoted by kv so that the volume of a crystal with length z is

kvz3. The kinetic parameters reported for the crystallization of potassium nitrate (KNO3)

34

crystals were used as given in Table 3.2. The growth rate is given as [2, 53]

G(t, z) = kg [S(t) − 1]g (1 + aASLz)dASL , (3.47)

where kg is the growth rate constant. The exponent g denotes the growth order, while aASL

and dASL are constant given in Table 3.2. Moreover, S(t) denotes the supersaturation of

the dissolved component

S(t) =C(t)

Csat(T ). (3.48)

The saturated concentration quantifying the amount of solute per gram of the solvent is

given as [45]

Csat(T ) = 1.721 × 10−4T 2 + 5.88 × 10−3T + 0.1286 , (3.49)

where the temperature T follows an exponentially decaying trajectory of the form

T (t) = 32 − 4(1 − e−t

18600 ) . (3.50)

The nucleation rate is described as [2, 53]

B0(t) = kb [S(t) − 1]b µ3(t) , (3.51)

where kb and b are the nucleation rate constant and the nucleation order respectively, given

in Table 3.2. The third moment µ3 is defined as

µ3(t) =

∞∫

0

z3F (t, z) dz . (3.52)

The problem was solved by using DG method and Koren’s scheme on 2200 mesh points.

The resulting CSDs after 103 seconds are shown in Figure 3.6. The numerical solutions

of both schemes are in good agreement. However, a discrepancy is visible in the part

of distribution coming from the nucleation phenomenon. The reason is the first order

35

approximation of the Koren’s scheme in the first two left boundary cells. A disadvantage

of the Koren’s scheme is that it cannot be used in these boundary cells. As a result,

use of first order scheme introduces numerical errors in the solution that propagate to

the neighboring cells with the passage of time. However, the DG-scheme is free of such

limitations and, hence, gives better approximation of the solution in the boundary cells.

The absolute error in the mass balances of the DG-scheme is found to be 0.9× 10−4, while

in the Koren’s scheme it is equal to 2.4× 10−3. Moreover, the simulation times of DG and

Koren schemes are 88.6 and 85.4 seconds, respectively.

Table 3.2: Parameters for Problem 3Description Symbol Value UnitMaximum crystal size zmax 1100 µmMesh size ∆z 0.5 µmSimulation time t 1000 sNumber of grid points N 2200 −Growth rate constant kg 1.16 × 102 µm

s

Growth rate exponent g 1.32 −Nucleation rate constant kb 4.64 × 10−7 1

µm3s

Nucleation rate exponent b 1.78 −Volume shape factor kv 1.0 −Initial concentration C(0) 0.493 gDensity of crystals ρc 2.11 × 10−12 g

µm3

Constant (Eq. (3.47)) aASL 0.1 µm−1

Constant (Eq. (3.47)) dASL 1.0 −

Problem 4

The aim of this test problem is to illustrate the applicability of the DG-scheme for the case

of discontinuous crystal growth rate. Here, the simulation of potassium sulfate (K2SO4 −

H2O) is considered. The initial seed distribution is taken as [42, 46]

F (0, z) =

5.472 × 107 , if 5.0 × 10−4m ≤ z ≤ 6.0 × 10−4 ,0 , elsewhere .

(3.53)

36

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

length [µm]

CS

D [1

/µm

]

initial CSDDG−schemeFVS

Figure 3.6: Problem 3: Size-dependent growth at t = 1000 s.

The size range of interest is 5 × 10−5 m ≤ z ≤ 2 × 10−3 m and the final simulation time is

180 minutes. The growth rate, that replaces Eq. (3.47), is described as

G(t, z) =

86.4 exp(

− 40400R(T+273.15)

)

S2(t) (1 + 2(z × 106)2/3) , if z ≤ 7 × 10−4 m,

G(700)(

z×106

700

)

, otherwise ,

(3.54)

where R = 8.314 J K−1 mol−1 is the universal gas constant and S(t) is given by Eq. (3.48).

The nucleation rate, that replaces Eq. (3.51), is given as

B0(t) =

1.56 × 109 exp (−9300/R(T + 273.15)) S(t) µ3(t) , if 0.002 ≤ V (t) ≤ 0.02 ,

1.56 × 109 exp (−9300/R(T + 273.15)) S(t)√

0.02µ3(t) , otherwise .

(3.55)

Here, µ3 is given as

µ3(t) = ρc

∞∫

0

z3kv(z)F (t, z) dz , (3.56)

and ρc = 2660.0 kg/m3. The volume shape factor is defined as

kv(z) =

0.898 exp (168z0.5 − 8234z) , if z ≤ 1 × 10−4 m,4.460 exp (−0.0797z0.5 + 676z) , otherwise .

(3.57)

37

The saturated concentration quantifying solute mass per gram of solvent is given as

Csat(T ) = 6.0 × 10−6T 2 + 2.3 × 10−3T + 6.66 × 10−2 . (3.58)

The temperature profile used to maintain a constant supersaturation (C −Csat = 0.00732)

is given as

T (t) = 70 − 45e−0.008356(180−t) . (3.59)

Finally, the concentration balance in Eq. (3.46) is replaced by the following equation

dC(t)

dt= −3ρc

∞∫

0

z2 kv G(t, z) F (t, z) dz , C(0) = C0 . (3.60)

The numerical results at 400 grid points are shown in Figure 3.7. The reference solution

was obtained from the DG-method on 3000 grid points. It can be seen that the DG-method

gives better approximation of the solution and correct positions of the discontinuities as

compared to the Koren scheme. The absolute error in the mass balances of the DG-scheme

is 7.9× 10−3, while in the Koren scheme it is equal to 8.3× 10−3. Moreover, the simulation

times of DG-scheme and Koren scheme are 2.81 and 2.79 seconds, respectively.

Problem 5

Here, a mathematical model for batch crystallizer loaded with a dissolution loop of fines

is considered [63]. The fines removal and their subsequent dissolution in an external unit

has potential to improve the quality of the product and to facilitate the down stream

processing. For further improving the process, a time delay in the dissolution unit(pipe) is

also included in the model . Here, the source term Q(t, z) of Eq. (3.1) is given as [63]

Q(t, z) = − V

Vcrz

ψ(z)F (t, z) , (3.61)

where V represents the volumetric flow rate and Vcrz is the volume of the crystallizer.

Moreover, ψ(z) represents a dimensionless cutoff (death) function. Thus, ψ(z)F (t, z) de-

notes that part of the size distribution which has been taken out from the crystallizer to

38

0.5 1 1.5 2

x 10−3

104

106

108

1010

1012

length [m]

CS

D[1

/m/k

g]

reference (DG)DG−schemeFVSinitial CSD

Figure 3.7: Problem 4: Size-dependent growth at t = 180min.

the dissolution unit. The following cutoff function is considered in this problem

ψ(z) =1√

2πσ1

e−( z

ασ1)2

, σ1 =1√

2πFmax

, (3.62)

where, α = 1.1547 × 103m and Fmax = 0.6. The balance law for the liquid phase modifies

to

dm(t)

dt= min(t) − mout(t) − 3ρc kv

∞∫

0

z2 G(t, z) F (t, z) dz , m(0) = m0 , (3.63)

where m denotes the mass of the solute in the solvent.The above equation has two liquid

streams due to the fines dissolution unit. The first one, mout, denotes the liquid stream

containing fines which is being taken out from the crystallizer to the pipe. The second one,

min, denotes the incoming pure and particle-free liquid stream to the crystallizer from the

pipe. They are described as

mout(t) = w(t)ρsolu(T )V , (3.64)

min(t) = mout(td) +kvρcV

Vcrz

∞∫

0

z3ψ(z)F (td, z)dz − kvρcz30B0(td) , (3.65)

39

where, td = t − τpipe. Here, w denotes the mass fraction

w(t) =m(t)

m(t) + msolv(T )

, (3.66)

where, msolv(T ) defines the mass of the solvent and ρsolu(T ) is the density of the solution

as a function of temperature T . Moreover, τpipe ≥ 0 denotes the residence time in the pipe.

It is defined as

τpipe =Vpipe

Vpipe

. (3.67)

The size-dependent crystal growth rate and nucleation rate are given by Eqs. (3.47) and

(3.51). Moreover, the supersaturation S(t) of the dissolved component is given as

S(t) =w(t)

weq(t), (3.68)

where, weq(t) is the mass fraction at equilibrium. This definition of supersaturation is dif-

ferent from the one given in Eq. (3.48). It can be observed that the above model reduces

to the case of fines dissolution without time delay when τpipe = 0. Moreover, the model

reduces to the case without dissolution of fines when the second last term on the right-hand

side of Eq. (3.1) and the first two terms on the right-hand side of Eq. (3.63) are zero.

Then Eqs. (3.64) and (3.65) are not needed.

The following initial seed distribution is considered

F (0, z) =mseeds

kv ρc µ3(0)√

2πσexp

(−(z − z)2

2σ2

)

. (3.69)

The smallest and largest crystal sizes are taken as z0 = 0 and zmax = 0.005m, respectively.

The interval [0, zmax] is divided into 100 mesh points and the final simulation time is 800

minutes for size-independent growth rate and 400 minutes for size-dependent growth rate.

For the size-dependent growth rate, aASL = 100m−1 and dASL = 2 in Eq. (3.47). For the

size-independent growth rate, aASL = 0. The constants and other parameters are given

40

in Table 3.3. The crystallizer was kept at a constant temperature of 33 oC. Figures 3.8

and 3.9 show the final CSDs due to size-independent and size-dependent growth rates,

respectively. In these figures, the final CSDs of the models without fines dissolution and

with fines dissolution as well as with and without (τpipe = 0) time-delay are compared.

The reference solution was obtained from the DG-scheme on a refined mesh. The fines

dissolution unit dissolves small crystals below a certain critical size and therefore reduces

the number of small crystals in the crystallizer as shown in Figures 3.9 and 3.10. Due to

dissolution, the solute mass in the solution increases as shown in Figure 3.11. However, the

dissolution has less effect on the growth rate of seeds crystals in the case of fines dissolution

without time-delay as can be observed in Figure 3.9. In this case, small nuclei get no time

to grow and instantaneously dissolve back in the solution as soon as they achieve a stable

size. As a result, the net effect on the solution concentration is negligible. On the other

hand, the fines dissolution with time-delay allows nuclei to grow in the crystallizer for

a certain time and the concentrated solution of the dissolution unit comes back to the

crystallizer with certain time-delay. Therefore, the concentration of the solution increases

in the crystallizer. In other words, the seed crystals and those introduced by nucleation

grow at a faster rate as can be observed in Figure 3.10. The numerical results of the DG-

method and FVS are almost the same. However, the DG-method better resolves the sharp

discontinuities as compared to FVS.

Table 3.4 gives a comparison of absolute and relative errors in the mass balances as well

as CPU time in the case of size-independent growth rate. No significant changes were

observed in the case of the size-dependent growth rate. The observed data show that the

proposed scheme is efficient and preserves the mass balance.

41

Table 3.3: Parameters for Problem 5Description Symbols Value Unitgrowth rate constant kg 1.37 × 10−5 m

min

growth rate exponent g 0.73 −nucleation rate constant kb 3.42 × 107 1

m3 min

nucleation rate exponent b 2.35 −density of crystals ρc 1250 kg

m3

volume shape factor kv 0.029 −initial solute mass m(0) 0.09915 kgsaturated mass fraction wsat 0.09068 −mass of seeds mseeds 2.5 × 10−3 kgmass of solvent msolv 0.8017 kg

density of solution ρsolu 1000 kgm3

volume of the crystallizer Vcrz 10−3 m3

volumetric flow rate Vpipe 2.0 × 10−5 m3

min

volume of the pipe Vpipe 2.4 × 10−4 mconstant of initial CSD (Eq. (3.69)) σ 3.2 × 10−4 mconstant of initial CSD (Eq. (3.69)) z 1.4 × 10−3 m

Table 3.4: Problem 5: Errors in mass balances for size-independent growth rateScheme Absolute error Relative error CPU time (s)DG-scheme 3.29 × 10−6 3.86 × 10−5 4.12Koren-scheme 4.29 × 10−6 4.21 × 10−5 4.10

42

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5

3x 10

7

length [m]

CS

D [1

/m]

size−independent growth, t=800 minreferenceDG−schemeFVS

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5x 10

7

length [m]

CS

D [1

/m]

size−dependetn growth, t=400 minreferenceDG−schemeFVS

Figure 3.8: Problem 5: CSD without fines dissolution.

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5

3x 10

7

length [m]

CS

D [1

/m]

size−indep. growth, fines, no delayreferenceDG−schemeFVS

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5x 10

7

length [m]

CS

D [1

/m]

size−dep. growth, fines diss., no delayreferenceDG−schemeFVS

Figure 3.9: Problem 5: Fines dissolution without delay

43

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5

3x 10

7

length [m]

CS

D [1

/m]

size−ind. growth, fines, delayreferenceDG−schemeFVS

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5x 10

7

length [m]C

SD

1/m

]

size−dep. growth, fines, delayreferenceDG−schemeFVS

Figure 3.10: Test 5: Fines dissolution with delay.

0 200 400 600 800

0.08

0.085

0.09

0.095

0.1

time [min]

mas

s [k

g]

without fines dissolutionfines diss. without delayfines diss. with delay

0 100 200 300 4000.08

0.085

0.09

0.095

0.1

time [min]

mas

s [k

g]

without fines dissolutionfines diss. without delayfines diss. with delay

Figure 3.11: Problem 5: Solute masses.

44

45

Chapter 4

Batch Preferential Crystallization

This chapter deals with the introduction of single and coupled batch preferential crystal-

lization models along with their numerical solutions by using the DG-scheme. Moreover,

the numerical results of the proposed method are compared with the HR-FVS [59].

4.1 Batch Preferential Crystallization Model

Here, a mathematical model is provided for simulating preferential crystallization of enan-

tiomer in a single batch crystallizer connected with a fines dissolution unit. This dissolution

process may give an improvement in the crystal size and, hence, the product quality. The

simplified sketch of a batch unit is shown in Figure 3.1. This setup preferentially crystal-

lizes only one enantiomer.

We make use of the same assumptions as discussed in section (3.2). In this case, the PBE

for the solid phase is given as [61].

∂F (κ)(t, z)

∂t= −∂[G(κ)(t, z)F (κ)(t, z)]

∂z− 1

τcryst

ψ(z)F (κ)(t, z), κ ∈ p, c , (4.1)

subject to the conditions

F (κ)(0, z) = F(κ)0 (z) , F (κ)(t, z0) =

B(κ)0 (t)

G(κ)(t, z0), (4.2)

where (t, z) ∈ R2+. Here, p and c stand for preferred and counter enantiomers, respectively.

Moreover, F (κ)(t, z) ≥ 0 is the CSD of a corresponding enantiomer with size z ≥ 0 at t ≥ 0,

and G(κ) represents the growth rate. The initial seed distribution is denoted by F(κ)0 , and

B(κ)0 represents the rate of nucleation at smallest size z0. In Eq. (4.1), the time of residence

in the vessel is denoted by τcryst which is defined as a ratio of crystallizer volume Vcrz to

the volumetric flow rate V i.e.

τcryst =Vcrz

V. (4.3)

46

The last term in Eq. (4.1) represents the particles’ death rate below a certain size. The

death function ψ(z) is given as a unit step function in which the critical size zcrit is taken

to be 2 × 10−4m and is defined as

ψ(z) =

0.6 for z ≤ zcrit

0.0 otherwise(4.4)

The moments of CSD are given as

µ(κ)i (t) :=

∞∫

0

zi F (κ)(t, z) dz. (4.5)

The balance laws for the liquid phase are described as

dm(κ)(t)

dt= m

(κ)in (t) − m

(κ)out(t) − 3ρckvG(κ)(t, z)µ

(κ)2 (t), t ∈ R≥0 , (4.6)

where µ(κ)2 denote the second moment, kv > 0 is the volumetric factor and ρc > 0 is the

crystal density. Because of dissolution, there are two mass flow rates given by Eq. (4.6).

The m(κ)out is the outer flux of mass from the vessel whereas m

(κ)in is the re-entering flux from

the pipe. These are given as

m(κ)out(t) = ω(κ)(t)ρsolu(T )V , (4.7)

m(κ)in (t) = m

(κ)out(t − τpipe) +

kvρc

τcryst

∞∫

0

z3 ψ(z) F (κ)(t − τpipe, z)dz. (4.8)

Here, ω(κ) represents the mass fractions and τpipe is the time of residence in the pipe as

expressed below

τpipe =πr2L

Vpipe

=Vpipe

Vpipe

, (4.9)

where the radius of the pipe is denoted by r and the length by L. In this model, the

nucleation is taken at z = z0,

F (κ)(t, z0) =B

(κ)0 (t)

G(κ)(t), κ ∈ p, c . (4.10)

47

The growth rate kinetics for preferential and counter enantiomers are given by [16, 17]

G(κ)(t) = (S(κ)(t) − 1)gkg, (4.11)

where kg ≥ 0 and g ≥ 1. The supersaturation denoted by S(κ) is given by

S(κ)(t) =ω(κ)(t)

ω(κ)eq (t)

. (4.12)

In the above equation, ω(κ) and ω(κ)eq represent mass fraction and mass fraction in equilib-

rium respectively, where

ω(κ)(t) :=m(κ)(t)

m(p)(t) + m(c)(t) + msolv(t), κ ∈ p, c , (4.13)

in which msolv(t) is the solvent mass (water here). Nucleation rates of preferential and

counter enantiomers are given by

B(p)0 (t) = (S(p))b(p)

k(p)b µ

(p)3 (t), (4.14)

B(c)0 (t) = k

(c)b exp

(

− b(c)

(ln(S(c)(t) + 1))2

)

. (4.15)

Without fines dissolution, the model reduces to

∂F (κ)(t, z)

∂t= −∂[G(κ)(t, z)F (κ)(t, z)]

∂z, (4.16)

dm(κ)(t)

dt= −3ρckvG(κ)(t, z)µ

(κ)2 (t) , (4.17)

along with initial and boundary conditions given by Eq. (4.2). Hence, Eqs. (4.7) and (4.8)

are not needed in this case.

4.2 Coupled Batch Preferential Crystallization Model

This section presents a mathematical model for ideally-mixed two-coupled batch prefer-

ential crystallizers connected through exchange pipes and equipped with fnes dissolution

48

units. The residence time and volume are assumed to be the same for both crystallizers.

Similarly, the volume and residence time for the two dissolution pipes are also considered to

be identical. The pipes dissolve the fines completely by providing sufficient heat to them.

Figure 4.1 is showing the coupled preferential crystallization process. In the beginning of

this process, two enantiomers and a solvent are mixed together. Vessel 1 is seeded with

the preferred enantiomer of L-threonine whereas Vessel 2 is seeded with the counter enan-

tiomer of D-threonine. Thus, both enantiomers are simultaneously crystallized in separate

crystallizers [19, 81].

Figure 4.1: Crystallization in coupled vessels.

The PBEs in the coupled case are given below [61]

∂F(κ)α (t, z)

∂t= − ∂[G(κ)

α (t, z)F(κ)α (t, z)]

∂z− 1

τcryst,αψα(z)F (κ)

α (t, z). (4.18)

Here, κ ∈ p, c and α ∈ 1, 2. In Eq. (4.18), τcryst,α can be expressed as

τcryst,α =Vcrz,α

Vα

. (4.19)

The death function ψα(z) is defined as

ψα(z) =

0.6 if z ≤ zcrit ,0 otherwise .

(4.20)

49

The liquid balances are given by

dm(κ)1 (t)

dt= m

(κ)in,2(t) − m

(κ)out,1(t) − 3ρckvG(κ)

1 (t, z)µ(κ)2,1(t), t ∈ R≥0. , (4.21)

dm(κ)2 (t)

dt= m

(κ)in,1(t) − m

(κ)out,2(t) − 3ρckvG(κ)

2 (t, z)µ(κ)2,2(t), t ∈ R≥0. (4.22)

Here, µ(κ)2,1(t) represents second moment of the CSD F

(κ)α (t, z) in Vessel 1. These equations

have four mass fluxes given by Eqs. (4.21) and (4.22) due to fines dissolution. Moreover,

m(κ)in,2(t) denotes the mass flux which is coming from the dissolution unit and going into

vessel 2 whereas m(κ)out,1(t) is the flux which is going out from Vessel 1 and going into the

pipe. Similarly, m(κ)in,1(t) represent the inner mass flux to vessel 1 and m

(κ)out,2(t) represent

the outer flux from Vessel 2. These are given by

m(κ)out,α(t) = ρsolu(T )ω(κ)

α (t)Vα , (4.23)

m(κ)in,α(t) = m

(κ)out,α(t − τpipe,α) +

kvρc

τcryst,α

∞∫

0

z3 sα(z) F (κ)α (t − τpipe,α, z)dz. (4.24)

Here, ω(κ)α denote mass fractions and τpipe,α is expressed as

τpipe,α =πr2

αLα

Vα

. (4.25)

The new particles born during the process are incorporated as

F (κ)α (t, z0) =

B(κ)α (t)

G(κ)α (t)

, κ ∈ p, c , (4.26)

and the growth rate is described as

G(κ)α (t) = (S(κ)

α (t))gkg, (4.27)

where kg ≥ 0 and g ≥ 1. The supersaturation in each crystallizer is given by

S(κ)α =

ω(κ)α

ω(κ)eq,α

− 1, κ ∈ p, c . (4.28)

50

We define the mass fractions for both the enantiomers in each crystallizer as

ω(κ)α (t) =

m(κ)α (t)

m(p)α (t) + m

(c)α (t) + msolv(t)

, κ ∈ p, c , (4.29)

and the nucleation rates by

B(p)α (t) = (S(p)

α (t))b(p)

k(p)b µ

(p)3,α(t), (4.30)

B(c)α (t) = k

(c)b exp

(

− b(c)

(ln(S(c)α (t) + 1))2

)

. (4.31)

4.3 Numerical Approximation of the Models

For the numerical approximation of PBE in Eq. (4.1), a TVB Runge-Kutta DG-method

[8, 68] is implemented. In this approach, the DG discretization is adopted in the size

variable only. The time derivatives are discretized by using a TVB Runge-Kutta method.

For simplicity, it is convenient to consider the simplified PBE for the scheme derivation

∂F (k)(t, z)

∂t= − ∂[G(k)(t, z)F (k)(t, z)]

∂z+ Q(k)(t, z) , (4.32)

where

Q(k)(t, z) = − 1

τcryst

ψ(z)F (κ)(t, z), κ ∈ p, c . (4.33)

Let us define

f(G(k), F (k)) := G(k)(t, z)F (k)(t, z) . (4.34)

To discretize the size-domain [z0, zmax], we proceed as follows. For j = 0, 1, 2, ....N , let zj+ 12

be the cells partitions, Ij =(

zj− 12, zj+ 1

2

)

be the domain of cell j, ∆zj = zj+ 12− zj− 1

2be

the width of cell j, and I =⋃

Ij are the particle size range partitions. We are looking for

an approximation F(k)h (t, z) to F (k)(t, z) so that for all t ∈ [0, tmax], F

(k)h (t, z) is belonging

to the space of finite dimensions

Vh =

v ∈ L1(I) : v|Ij∈ P k(Ij), j = 0, 1, 2, ....N

, (4.35)

51

where P k(Ij) stands for the polynomials space in I of at most k degree. In order to compute

the approximate solution F(k)h (t, z), we require a weak formulation that is usually obtained

by multiplying Eq. (4.32) with a smooth function v(z) and by integrating over the interval

Ij.

∫

Ij

∂F (k)(t, z)

∂tv(z)dz = −

∫

Ij

∂[f(G(k), F (k))]

∂zv(z)dz +

∫

Ij

Q(k)(t, z)v(z)dz . (4.36)

After using integration by parts, the weak formulation has the form

∫

Ij

∂F (k)(t, z)

∂tv(z)dz = −

(

f(G(k)

j+ 12

, F(k)

j+ 12

)vl(zj+ 12) − f(G

(k)

j− 12

, F(k)

j− 12

)vl(zj− 12))

+

∫

Ij

f(Gk, F (k))∂v(z)

∂zdz +

∫

Ij

Q(k)(t, z)v(z)dz . (4.37)

Selecting Legendre polynomials (LPs), Pl(z), of order l as local basis functions is one way

to formulate Eq. (4.35). Here, the L2-orthogonality property of LPs can be used, such as

1∫

−1

Pl(s)Pl′(s)ds =

(

2

2l + 1

)

δll′ . (4.38)

For each z ∈ Ij, the solution F(k)h can be expressed as

F(k)h =

k∑

l=0

F(l,k)j ϕl(z) , (4.39)

where

ϕl(z) = Pl (2(z − zj)/∆zj) . (4.40)

It can be easily verified that

(

1

2l + 1

)

F(l,k)j (t) =

1

∆zj

∫

Ij

F kh (t, )ϕ(z)dz . (4.41)

Then, the test function ϕl ∈ Vh replaces the smooth function v(z) and the approximate

solution F(k)h replaces the exact solution F (k) . Moreover, the function F

(k)

j+ 12

= F (t, z(k)

j+ 12

)

52

is not defined at the cell interface zj+ 12. Therefore, the f(G(k), F (k)) will be replaced with

a numerical flux that further be depends upon two values of F(k)h (t, z), i.e.,

f(G(k)

j+ 12

, F(k)

j+ 12

) ≈ hj+ 12

= h(G(k)

j+ 12

, F(k,−)

j+ 12

, F(k,+)

j− 12

) . (4.42)

Here

F(k,−)

j+ 12

:= F(k)h (t, z−

j+ 12

) =k

∑

l=0

F(l,k)j ϕ(zj+ 1

2) , F

(k,+)

j− 12

) := F(k)h (t, z+

j− 12

) =k

∑

l=0

F(l,k)j ϕl(zj− 1

2) .

(4.43)

Using the Eq. (4.42) and Eq. (4.43), the weak formulation Eq. (4.37) simplifies to

dF(l,k)j (t)

dt= − 2l + 1

∆zj

(

hj+ 12ϕl(zj+ 1

2) − hj− 1

2ϕl(zj− 1

2))

+2l + 1

∆zj

∫

Ij

(

f(G(k), F(k)h )

dϕl(z)

dz

)

dz +2l + 1

∆zj

∫

Ij

Q(k)(t, z)ϕl(z)dz . (4.44)

According to Eqs (5.2) and (4.41), the initial data for this equation is given as

F(l,k)j (0) =

2l + 1

∆zj

∫

Ij

F(k)0 (z)ϕ(z)dz . (4.45)

The next step is to select the suitable numerical flux function h. The scheme is monotone

if the flux function h(G(k), a, b) is consistent, h(G(k), F (k), F (k)) = f(G(k), F (k)), and fulfills

the Lipschitz condition [41, 80]. Here we have used the same numerical fluxes and minmod

limiter which are discussed in Chapter 3. Moreover, for solving the resulting ODE system,

the same Runge-Kutta method is used.

The derivation of the DG-scheme for the coupled batch preferential crystallization model

follows the same procedure as discussed above and is therefore omitted here.

In the following, some case studies are carried out for both single and coupled batch

crystallizers to validate our numerical scheme.

53

4.4 Test Problems for Single Crystallizer

Problem 1

The initial distribution function of preferred enantiomer is

F (p)(0, z) =1√

2πσDa

.1

z. exp

(

−1

2

(

ln(z) − z

σ

)2)

, Da = kvρc

mseedsµ

(p)3 (0) . (4.46)

The counter enantiomer is not seeded initially, i.e.

F (c)(0, z) = 0. (4.47)

Here, it is assumed that σ = 0.3947m, z = −6.8263m, while mseeds is the mass of initial

seeds. Tables 4.1 and 4.2 show the kinetic parameters. For the isothermal case, the

Table 4.1: Parameters for Problem 1Symbols Values Units

zmax 0.005 mz0 1 × 10−8 mt 600 minN 500 −

temperature is kept constant at 33oC whereas the temperature trajectory in the of non-

isothermal case is defined as

T (t)[oC] = −12.407 × 10−8t3 + 45.09 × 10−6t2 − 40.556 × 10−4t + 33, (4.48)

whereas the constants kg(t) and k(k)b (t) are given by

kg(t) = e−EA,g

(T+273.15)R kg,0, k(k)b (t) = e−

EA,b(T+273.15)R k

(k)b,0 . (4.49)

Moreover,

ω(p)eq (t,m(p),m(c)) =

2∑

j=0

T j(t)(aj + bjω(c)(t)), (4.50)

54

ω(c)eq (t,m(p),m(c)) =

2∑

j=0

T j(t)(cj + djω(p)(t)). (4.51)

The values of the constants are listed in Table 4.2. For the isothermal case, the same Eqs.

(4.50) and (4.51) are used.

Figures 4.2 compare the number densities of the preferential enantiomer obtained from the

HR-FVS and the DG-scheme when the fines dissolution is active. Figures 4.3 show the

CSD plots with and without fines dissolution. In the isothermal case, the temperature

is kept constant at 33oC whereas it is given by Eq. (4.48) for non-isothermal condition.

The part of the CSD coming from nucleation is smaller for the fines dissolution case as

compared to the case when fines dissolution is off. This is because the fines are dissolved

in the dissolution unit before they are sent back to the crystallizer, as a result of which

supersaturation of the solution increases which, in turn, diminishes the secondary nucle-

ation.

The non-isothermal case results are totally dependent upon the time depending tempera-

ture. As we reduce the temperature, the number density is increased. This is due to the

increase in nucleation rate. It is observed that the results are better for the non-isothermal

case with fines dissolution as we obtain bigger sized crystals as compared to the other cases.

Figures 4.4 show the CSDs of counter enantiomer which appears because of the nucleation.

The plots for mass fraction of preferred enantiomer are shown in Figures 4.5. Because of

the seeding of preferred enantiomer, its mass fraction decreases sharply in the isothermal

case. With fines dissolution (FD), the curve is slightly above than for without (FD). This

is because of an increase in supersaturation in the fines dissolution case.

55

Figures 4.6 compares the counter enantiomer mass fraction for with and without fines dis-

solution. The plot for supersaturation of the preferred enantiomer is shown in Figures 4.7.

Since the mass fraction of the preferred enantiomer decreases sharply, the supersaturation

of the solution reduces significantly.

The results for the non-isothermal case are influenced by the temperature profile defined

by Eq. (4.48). The change in growth rate and the nucleation rate are due to the change

in mass fraction, see Figures 4.8 and 4.9. Figures 4.10 explain the purity and the mean

crystal size of the crystal during the process, whereas Figure 4.11 shows the productivity

of the preferred enantiomer.

Problem 2

The initial number density for the seeds is

F (p)(0, z) =107

√2πσDa

exp

(

−1

2

(

z − z

σ

)2)

, Da =kv. ρc

mseeds

µ(p)3 (0) , (4.52)

while

F (c)(0, z) = 0. (4.53)

Here we assume σ = z15

, z = 4.10−4m, whereas mseeds is the mass of initial seeds. The

other parameters are given in Tables 4.2 and 4.3 . The comparison of the DG-scheme and

the HR-FVS is given in Figures 4.12. Both the schemes are in accord with each other.

The crystal size distribution of the preferred enantiomer is given in Figures 4.13 which is

compared for both with and without fines dissolution. The first peak in the isothermal

case is due to the nucleation which is very small.

The mass fractions of both the enantiomers (preferential and counter) are displayed in

Figures 4.14 and 4.15.

56

Table 4.2: Parameters for Problems 1, 2 and 4Symbols Values Unitskg,0 4.62 × 108 m

min

k(p)b,0 32.4 × 1024 1

m3min

k(c)b,0 3.84 × 106 1

min

g 1 −EA,g 7.56 × 104 kJ

mol

EA,b 7.87 × 104 kJmol

b(p) 4 −R 8.314 J

mol.K

b(c) 0.6 × 10−1 −ρc 1250 kg

m3

ρsolu 103 kgm3

kv 0.0248 −m(c)(0) 0.100224 kgm(p)(0) 0.100224 kgmseeds 25 × 10−4 kgmsolv 0.7995 kgVcrz 10 × 10−4 m3

τcryst 60 minr 10 × 10−3 mτpipe 10 minL 53 × 10−2 ma1 0.00143838 −a2 −3.41777 × 10−6 −

57

Table 4.3: Parameters for Problem 2Symbols Values Unitszmax 0.002 mz0 1 × 10−8 mt 600 minN 400 −

Figures 4.16 shows the supersaturation of preferred enantiomer with and without fines dis-

solution; the changes in growth rate is shown in Figures 4.17. The results obtained by the

DG-scheme are the same as obtained by the HR-FVS.

The productivity is displayed in Figures 4.18 which increases with time. The purity of the

product in the whole process can be seen in Figures 4.19. It is found that the final product

is 99.7% pure. The mean crystal size is also given in this figure.

Problem 3

The initial number density for the preferred enantiomers is given by

F (p)(0, z) =107

√2πσDa

exp

(

−1

2

(

z − z

σ

)2)

, Da =kv. ρc

mseeds

µ(p)3 (0) , (4.54)

while

F (c)(0, z) = 0. (4.55)

We assume σ = z15

, z = 10−4m. Table 4.4 shows the kinetic parameters for this problem.

In the first plot of Figure 4.20, the CSD of the preferred enantiomer obtained by the DG-

scheme is compared with that of HR-FVS, whereas in the second plot, the CSD of the

counter enantiomer is compared.

The results for mass fraction of the preferential and counter enantiomers are compared

with that of HR-FVS in the case of without fines in Figure 4.21. It is seen that both the

58

Table 4.4: Parameters for Problems 3 and 5Description Symbols Values Unitsmaximum crystal size zmax 0.002 mminimum crystal size z0 10−8 mfinal time tend 700 mintotal mesh points N 400 −growth rate constant kg 15.0 × 108 m

min

exponent of Growth rate g 1 −constant k

(p)b 0.5 × 102 1

m3min

exponent of Nucleation rate b 2 −constant k

(c)b k

(p)b /5 1

min

density of crystals ρc 1250 kgm3

density of liquid ρsolu 1000 kgm3

volume shape factor kv π/6 −solvent mass msolv 0.15004 kginitial preferential mass m(p) 0.01932 kginitial counter mass m(c) 0.01856 kgsaturation mass fraction ωsat 0.05220 kgseeds mass mseeds 50 × 10−5 kgpipe residence time τpipe 10 mincrystallizer residence time τcryst 60.0 minvolume of crystallizer Vcrz 0.001 m3

schemes give the same result. Figure 4.22 shows that the crystal obtained at the end is

99.9% pure, whereas Figure 4.23 displays the productivity with time.

4.5 Test Problems for Coupled Crystallizers

Here, two test cases are presented for the crystallization of enantiomers in coupled crystal-

lizers and the results are compared with those of single batch crystallizers.

Problem 4

The initial CSD of the target enantiomer in both crystallizers is given as

F (p)α (0, z) =

1

z√

2πσDα

exp

(

−1

2

(

ln(z) − z

σ

)2)

, (4.56)

59

with Dα = kv. ρc

mseedsµ

(p)3,α(0). The initial distribution of unwanted enantiomer is taken to be

zero in each crystallizer

F (c)α (0, z) = 0. (4.57)

Here, σ = 0.3947m, z = −6.8263m, while mseeds,α represents the seeds mass in tank

α ∈ 1, 2. Tables 4.2 and 4.5 show the kinetic parameters considered for this problem. It

Table 4.5: Parameters for Problem 4Symbols Values Units

zmax 0.005 mz0 10−8 mt 600 minN 500 −

is to be noted that in crystallizer 1, the preferred enantiomer is denoted by E1 whereas in

crystallizer 2, the preferred enantiomer is denoted by E2. The constants kg(t) and k(k)b (t)

are given by Eq. (4.49). The equilibrium mass fraction of both the enantiomers are defined

as

ω(p)eq,α =

2∑

j=0

T j(aj + bjω(c)α ), (4.58)

ω(c)eq,α =

2∑

j=0

T j(cj + djω(p)α ). (4.59)

The constants of the above equations are presented in Table 4.2.

In Figures 4.24, the number densities of preferred enantiomer are compared for single and

coupled crystallizers. For this test problem, the enantiomer E1 plots in tank 1 and those of

enantiomer E2 in tank 2 are identical. It can be observed that the crystal size distribution

in the coupled crystallizers is better than in the single crystallizer, as large crystals are

60

obtained in the coupled case. For isothermally operated crystallizer, the initially large nu-

cleation rate gives a large peak of preferred enantiomer, while the second curve is small due

to initial crystal growth. However, for the non-isothermal case the peaks are dependent

entirely on the temperature profile.

The mass fractions of the preferred enantiomer are compared for both crystallizers in

Figures 4.25. Because of the seeding of the preferential enantiomer, it crystallizes out

sharply resulting in a decrease in mass fraction. The change in supersaturation and growth

rate is due to the change in mass fraction. These changes can be seen in Figures 4.26 and

4.27 for the isothermal and non-isothermal cases. Comparisons of the productivity and

mean crystal size are made in Figures 4.28 and 4.29 for both single and coupled crystallizers.

Problem 5

The initial distribution functions of the preferred and counter enantiomers are given by the

representations

F (p)α (0, z) =

107

√2πσDα

exp

(

−1

2

(

z − z

σ

)2)

, Dα =kv. ρc

mseeds

µ(p)3,α(0) , (4.60)

F (c)α (0, z) = 0. (4.61)

The values of constants and kinetic parameters considered here are the same as those of

Problem 3.

Here the results of a single crystallizer and for coupled crystallizers are compared. Figure

4.30 makes a comparison of single and coupled crystallizers for the cases of fines and

without FD. It can be observed that the growth in the coupled case is enhanced and, thus,

the distribution sifts to the right. In other words, large size crystals are obtained in the

coupled case. Therefore, the results with the coupled crystallizers are better than with a

61

single crystallizer. Moreover, purity, productivity and mean crystal size are also compared

in Figures 4.31 and 4.32. One can also observe improvements in the purity, productivity

and mean crystal size of the coupled crystallization.

62

0 1 2 3 4 5

x 10−3

0

2

4

6

8

10

12x 10

8

crystal size [m]

part

icle

den

sity

[1/m

]

Isothermal Case

HR FVSDG scheme

0 1 2 3 4 5

x 10−3

0

1

2

3

4

5

6x 10

9

crystal size [m]pa

rtic

le d

ensi

ty [1

/m]

Non−isothermal case

HR FVSDG scheme

Figure 4.2: Test 1: Preferred enantiomer CSD with HR-FVS and DG scheme.

0 1 2 3 4 5

x 10−3

0

2

4

6

8

10

12x 10

8

crystal size [m]

part

icle

den

sity

[1/m

]

Isothermal Case

without fineswith fines (no delay)with fines (with delay)

0 1 2 3 4 5

x 10−3

0

2

4

6

8

10x 10

9

crystal size [m]

part

icle

den

sity

[1/m

]

Non−isothermal Case


Figure 4.3: Test 1: Preferred enantiomer CSD for with and without FD.

63

0 1 2 3 4 5

x 10−3

0

1

2

3

4

5

6

7x 10

−6

crystal size [m]

part

icle

den

sity

[1/m

]

Isothermal Casewithout fineswith fines (no delay)with fines (with delay)

0 1 2 3 4 5

x 10−3

0

0.5

1

1.5

2

2.5

3

3.5x 10

−3

crystal size [m]

part

icle

den

sity

[1/m

]



Figure 4.4: Test 1: CSD for counter enantiomer with and without FD.

0 100 200 300 400 500 6000.092

0.093

0.094

0.095

0.096

0.097

0.098

0.099

0.1

0.101

0.102

time [min]

mas

s fr

actio

n [−

]

Isothermal Case

without fineswith fines (without delay)with fines (with delay)

0 100 200 300 400 500 6000.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1

0.102

time [min]

mas

s fr

actio

n [−

]



Figure 4.5: Test 1: Mass fraction for preferred enantiomer with and without FD.

64

0 100 200 300 400 500 600

0.1004

0.1006

0.1008

0.101

0.1012

0.1014

0.1016

time [min]

mas

s fr

actio

n [−

]

Isothermal Case


0 100 200 300 400 500 6000.1

0.1005

0.101

0.1015

0.102

0.1025

0.103

0.1035

time [min]

mas

s fr

actio

n [−

]



Figure 4.6: Test 1: Mass fraction for counter enantiomer.

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time [min]

supe

rsat

urat

ion

[−]

Isothermal Case


0 100 200 300 400 500 6000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time [min]

supe

rsat

urat

ion

[−]



Figure 4.7: Test 1: Supersaturation for target enantiomer with and without FD.

65

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time [min]

grow

th r

ate

[m/m

in]

Isothermal Case


0 100 200 300 400 500 6000.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time [m]

grow

th r

ate

[m/m

in]

Non−isothermal Casewithout fineswith fines (no delay)with fines (with delay)

Figure 4.8: Test 1: Growth rate for target enantiomer with and without FD.

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

time [min]

nucl

eatio

n ra

te [m

/min

]

Isothermal Case


0 100 200 300 400 500 6000

2000

4000

6000

8000

10000

12000

14000

time [min]

nucl

eatio

n ra

te [m

/min

]



Figure 4.9: Test 1: Nucleation rate for target enantiomer with and without FD.

66

0 100 200 300 400 500 60099.7

99.75

99.8

99.85

99.9

99.95

100

time [min]

purit

y [%

]

0 100 200 300 400 500 6000.2

0.4

0.6

0.8

1

1.2

time [min]

mea

n cr

ysta

l siz

e [m

m]

isothermal casenon−isothremal case

Figure 4.10: Test 1: Purity and mean crystal size of preferred enantiomer.

0 100 200 300 400 500 6000

2

4

6

8

10

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]

Isothermal Case

0 100 200 300 400 500 6000

5

10

15

20

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]


Figure 4.11: Test 1: Productivity of preferred enantiomer.

67

Figure 4.12: Test 2: CSD for target enantiomer with HR-FVS and DG scheme.

0 0.5 1 1.5 2

x 10−3

0

2

4

6

8

10

12

14

16

18x 10

9

crystal size [m]

part

icle

den

sity

[1/m

]

Isothermal Case

without fineswith fines

Figure 4.13: Test 2: CSD for target enantiomer.

68

0 100 200 300 400 500 6000.092

0.093

0.094

0.095

0.096

0.097

0.098

0.099

0.1

0.101

0.102

time [min]

mas

s fr

actio

n [−

]

Isothermal Case


0 100 200 300 400 500 6000.08

0.085

0.09

0.095

0.1

0.105

0.11

time [min]

mas

s fr

actio

n [−

]



Figure 4.14: Test 2: Mass fraction for preferred enantiomer with and without FD.

0 100 200 300 400 500 600

0.1004

0.1006

0.1008

0.101

0.1012

0.1014

0.1016

time [min]

mas

s fr

actio

n [−

]

Isothermal Case


0 100 200 300 400 500 6000.1

0.1005

0.101

0.1015

0.102

0.1025

0.103

0.1035

0.104

time [min]

mas

s fr

actio

n [−

]



Figure 4.15: Test 2: Mass fraction for counter enantiomer.

69

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time [min]

supe

rsat

urat

ion

[−]

Isothermal Case


0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time [min]

supe

rsat

urat

ion

Sp −1

[−]



Figure 4.16: Test 2: Supersaturation for preferred enantiomer with and without FD.

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time [min]

grow

th r

ate

[m/m

in]

Isothermal Case


Figure 4.17: Test 2: Growth rate for preferred enantiomer.

70


0 100 200 300 400 500 60099.7

99.75

99.8

99.85

99.9

99.95

100

time [min]

purit

y [%

]

0 100 200 300 400 500 6000.2

0.4

0.6

0.8

1

1.2

time [min]

mea

n cr

ysta

l siz

e [m

m]

isothermal casenon−isothremal case

Figure 4.19: Test 2: Purity and mean crystal size of preferred enantiomer.

71

0 0.5 1 1.5 2

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

8

crystal size [m]

part

icle

den

sity

[1/m

]

Without Fines Dissolution

HR FVSDG−scheme

Figure 4.20: Test 3: CSD for both enantiomers with HR-FVS and DG scheme.

0 100 200 300 400 500 600 7000.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

time [min]

mas

s fr

actio

n [−

]


HR FVSDG− scheme

0 100 200 300 400 500 600 7000.098

0.099

0.1

0.101

0.102

0.103

0.104

0.105

0.106

time [min]

mas

s fr

actio

n [−

]

Without Fines (counter)

HR FVSDG−scheme

Figure 4.21: Test 3: Mass fraction for both enantiomers with HR-FVS and DG scheme.

72

0 100 200 300 400 500 600 70099.88

99.9

99.92

99.94

99.96

99.98

100

100.02

100.04

time [min]

purit

y [%

]

0 100 200 300 400 500 600 7000.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

time [min]

mea

n cr

ysta

l siz

e [m

m]

Figure 4.22: Test 3: Purity and mean crystal size for preferred enantiomer.


73

0 1 2 3 4 5

x 10−3

0

1

2

3

4

5

6

7x 10

8

crystal size [m]

part

icle

den

sity

[1/m

]

Isothermal Case

single crystallizercoupled crystallizers

0 1 2 3 4 5

x 10−3

0

2

4

6

8

10

12

14

16

18x 10

9

crystal size [m]pa

rtic

le d

ensi

ty [1

/m]



Figure 4.24: Test 4: Preferred CSD with FD for single and coupled crystallizers.

0 100 200 300 400 500 6000.092

0.093

0.094

0.095

0.096

0.097

0.098

0.099

0.1

0.101

0.102

time [min]

mas

s fr

actio

n [−

]

Isothermal Case


0 100 200 300 400 500 6000.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1

0.102

time [min]

mas

s fr

actio

n [−

]



Figure 4.25: Test 4: Mass fraction with FD for single and coupled crystallizers.

74

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time [min]

supe

rsat

urat

ion

[−]

Isothermal Case


0 100 200 300 400 500 6000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time [min]

supe

rsat

urat

ion

[−]


coupled crystallizerssingle crystallizer

Figure 4.26: Test 4: Supersaturation with FD for single and coupled crystallizers.

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time [min]

grow

th r

ate

[m/m

in]

Isothermal Case


0 100 200 300 400 500 6000.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

time [min]

grow

th r

ate

[m/m

in]



Figure 4.27: Test 4: Growth rate with FD for single and coupled crystallizers.

75

0 100 200 300 400 500 6000

2

4

6

8

10

12

14

16

18

20Isothermal Case

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]


0 100 200 300 400 500 6000

5

10

15

20

25

30

35

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]



Figure 4.28: Test 4: Productivity for single and coupled crystallizers without FD.

0 100 200 300 400 500 6000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2Isothermal Case

time [min]

mea

n cr

ysta

l siz

e [m

m] single crystallizer

coupled crystallizers

0 100 200 300 400 500 6000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2Non−isothermal Case

time [min]

mea

n cr

ysta

l siz

e [m

m]

single crystallizerdoubled crystallizers

Figure 4.29: Test 4: Mean crystal size for single and coupled crystallizers without FD.

76

0 0.5 1 1.5 2

x 10−3

0

2

4

6

8

10

12

14

16

18x 10

7

crystal size [m]

part

icle

den

sity

[1/m

]



0 0.5 1 1.5 2

x 10−3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

7

crystal size [m]

part

icle

den

sity

[1/m

]

With Fines Dissolution

coupled crystallizerssingle crystallizer

Figure 4.30: Test 5: Preferred CSD for single and coupled crystallizers.

77

Figure 4.31: Test 5: Purity and productivity of p-enantiomer for single and coupled crys-tallizers.

mea

n cr

yst

al

size

[mm

]

Figure 4.32: Test 5: Mean crystal size of p-enantiomer for single and coupled crystallizers.

78

79

Chapter 5

Continuous Preferential Crystallization

Preferential crystallization (PC) is an attractive process for gaining pure enantiomers from

racemic mixtures, which has been realized up to now in a discontinuous operation mode,

see for example [3, 11, 12, 13, 17, 18, 31]. For elucidating the principle of a continuous

enantioselective process one might consider a suspension crystallizer revealing MSMPR

characteristics, i.e. a perfectly mixed tank (concerning both phases), which is continuously

fed with a solution possessing a racemic composition of two enantiomers, see Figure 5.1.

Solid particles and liquid phase are continuously withdrawn. By a continuous supply of ho-

mochiral seed crystals of the preferred target enantiomer the preferential crystallization of

only this enantiomer is initialized, i.e. growth of the seed crystals and possibly secondary

nucleation of crystals of the seeded enantiomer, provided the crystallization takes place

within the metastable zone where spontaneous, uncontrolled primary nucleation is kineti-

cally inhibited. During a starting-up period, which strongly depends on the properties of

the system as well as on the process parameters, the concentration of the target enantiomer

is decreasing until a steady state is reached where the composition is determined by the

mean residence time. Due to different kinetic mechanisms and their inherent different time

constants, a different depletion of the supersaturation for each enantiomer can be realized

by an appropriate choice of the process conditions. As long as a critical mean residence

time, where primary nucleation may appear, is not exceeded, the concentration of the un-

desired counter enantiomer remains constant during the whole time. This fact reveals a

benefit of this continuous process in comparison to the batch process. An optimal selection

of the process conditions allows a constant production of the goal enantiomer at a high

purity level. The mathematical modeling of PC as well as the optimization of the essential

operating conditions requires high numerical precision [60].

In this chapter, the semi-discrete flux-limiting finite volume scheme [32] and discontinu-

ous Galerkin finite element method are used to solve a dynamic mathematical model for

80

continuous preferential enantioselective crystallization. The developed model includes the

phenomena of primary heterogeneous nucleation, secondary nucleation and size-dependent

growth. A recycle pipe is attached to the crystallizer for the fines dissolution and it is

assumed that fines are completely dissolved at the other end of the pipe. The effects of

different seeding and operating strategies are investigated in the simulations. The model

and its parameters are based on experimental data obtained from batchwise crystallization

and correspond to the enantiomers of the amino acid threonine and the solvent water, see

[17, 18]. The numerical results demonstrate the high order accuracy, efficiency and potential

of the proposed numerical methods for solving models describing continuous preferential

crystallization of enantiomers and the potential of the process.

5.1 Continuous Preferential Crystallization Model

The model of the continuous crystallization process considers a two-phase dispersed sys-

tem. The first phase is a continuous liquid phase containing the dissolved enantiomers and

the solvent. The conditions in this phase determine the growth and nucleation rates of

crystals during the crystallization process and have significant influence on the crystal size

distributions (CSDs). The second phase is the dispersed solid phase of individual crystals.

The growth of crystals strongly depends on the degree of supersaturation of the liquid

phase.

Description of the CSD for a dispersed system in a crystallizer needs an adequate differential

equation accounting for the effects of nucleation, growth, and other phenomena involving

the change of crystals population, for instance fines dissolution in the current study. The

population balance modeling is a suitable approach for predicting CSD in the solid phase

of a disperse system.

81

(preferred) goal enantiomer: p(unwanted) counter enantiomer: c

dissolution pipe

(p)outS,m&

(p)seedsm&

LV&

LV&

pipeV&

Sτ

pipeτ

Lτ

Residence time in pipe =

Mean residence time of solid =

Mean residence time of liquid =

LV

)()( cp ww =

heat

exc

hang

er

Figure 5.1: Principle of continuous enantioselective crystallization [66].

A one-dimensional mathematical model for an ideally-mixed continuous crystallizer is

given by the following population balance equation (PBE) for the number density function

F (k)(t, z) [2, 27, 54]

∂F (k)

∂t= − ∂(G(k)F (k))

∂z+ F

(k)seeds − F

(k)out − F

(k)pipe , k ∈ p, c , (5.1)

where p stands for the preferred enantiomer and c for the counter enantiomer. In Eq.

(5.1) the term on the left hand side denotes the accumulation of crystals of size z. The

first term on the right hand side represents the convective transport in the direction of the

property coordinate z due to size-dependent crystal growth rate G(k). The term F(k)seeds de-

notes the incoming particle number flux due to seeding, F(k)out is the particle number flux due

to product removal, and F(k)pipe denotes the particle number flux to the fines dissolution unit.

82

The initial conditions for the PBE in Eq. (5.1) are based on the assumption that the

crystallizer initially (at time t = 0) contains only pure liquid phase. In other words

F (k)(t = 0, z) = 0 , k ∈ p, c . (5.2)

Assuming further, that primary nucleation leads to a crystal nuclei of minimum size z0

and that the number density function F (k) vanishes for an arbitrary large crystal size zmax,

holds

F (k)(t, z = z0) =B

(k)0 (t)

G(t, z0), F (k)(t, z = zmax) = 0 , (5.3)

where B(k)0 denotes the nucleation rate. The crystallizer is only seeded with preferred

enantiomer, thus

F(k)seeds(z) =

F(p)seeds(z) for k = p ,

0 for k = c .(5.4)

For the operation of the configuration shown in Figure 5.1, three types of residence times

are relevant:

1. The liquid mean residence time in the perfectly mixed crystallizer, τL = VL/VL, where

VL denotes the volume of the liquid phase in the crystallizer and VL is the volumetric

flow rate of the liquid phase.

2. A characteristic solid mean residence time in a perfectly mixed crystallizer, τS =

VS/(ρS/m(p)seeds), where VS represents the overall volume of the solid phase, ρS is the

density of the solid phase and m(p)seeds is the mass flow rate of the seeds of preferred

enantiomer.

3. The residence time in the pipe (plug flow), τpipe = Vpipe/Vpipe. Here, Vpipe denotes

volume of the pipe and Vpipe is the volumetric flow rate to the pipe.

83

The number density function in the withdrawn crystal stream is assumed to be equal to

the number density function inside the crystallizer. Therefore, the right hand sided second

and third terms of Eq. (5.1) are given as

F(k)seeds(z) =

F(k)seeds(z)

τS

, F(k)out(t, z) =

F (k)(t, z)

τS

, k ∈ p, c . (5.5)

The term corresponding to the dissolution unit in Eq. (5.1) is expressed by

F(k)pipe(t, z) =

ψ(z)F (k)(t, z)

τpipe

, (5.6)

where ψ(z) represents the classification (death) function.

The i-th moment of the CSD is defined as

µ(k)i (t) =

∞∫

0

ziF (k)(t, z) dz , i = 0, 1, 2, · · · , N, k ∈ p, c . (5.7)

According to Figure 5.1, the corresponding mass balances for the liquid phase are given as

dm(k)(t)

dt= m

(k)in (t) − m

(k)out(t) − 3ρS kv

∞∫

0

z2G(k)(t, z)F (k)(t, z) dz , k ∈ p, c (5.8)

with the initial conditions

m(k)(t = 0) = m(k)0 = ρfVL , (5.9)

where ρf = ρ(t = 0) is the density of fresh (initially) supersaturated solution. The incoming

and outgoing mass flow rates are defined as

m(k)in (t) = m

(k)in,c + m

(k)in,pipe(t), m

(k)out(t) = m

(k)out,c(t) + m

(k)out,pipe(t) . (5.10)

The inflow rate m(k)in is the sum of two incoming streams to the crystallizer, the first

one m(k)in,c stands for the incoming flux of fresh solution to the crystallizer and the second

one m(k)in,pipe denotes the incoming flux of particle-free solution from the dissolution pipe.

Similarly, the mass outflow rate m(k)out is the sum of two outgoing streams, i.e. mass flow

84

rate of the utilized solution from the crystallizer and the mass flow rate of the solution

containing the fraction of fines being taken out from the crystallizer to the dissolution

unit. The mass flow rate of the incoming fresh supersaturated solution is defined as

m(k)in,c = w(k)(t = 0)ρf VL , k ∈ p, c , (5.11)

where w(k) represents the mass fraction of the k-th enantiomer which is defined as

w(k)(t) =m(k)(t)

m(p)(t) + m(c)(t) + msolv

, k ∈ p, c . (5.12)

The terms m(k), m(p) and m(c) represent the masses of the k-th, preferred, and counter

enantiomers, respectively. Moreover, msolv is mass of the solvent (water). The composition

dependent solution density is defined as

ρL(t) = 1000(ρ0 + K3wtot(t)) , (5.13)

where ρ0 is the density of water

ρ0 =1

K1 + K2T 2. (5.14)

The parameters K1, K2, K3 are given in the Table 5.1 and T denotes the temperature

which is kept constant at 33 oC. The total mass fraction wtot is given as

wtot(t) = w(p)(t) + w(c)(t) . (5.15)

The outgoing mass fluxes of the solution from the crystallizer are defined as

m(k)out,c(t) = w(k)(t)ρ(t)V , k ∈ p, c , (5.16)

while the outgoing mass flux from the crystallizer to the dissolution unit is given as

m(k)out,pipe(t) = w(k)(t)ρ(t)Vpipe , k ∈ p, c . (5.17)

85

Respecting the fines dissolution and the residence in the tube, the incoming mass flux from

the dissolution unit to the crystallizer is given as

m(k)in,pipe(t) = m

(k)out,pipe(t − τpipe) +

kvρS

τpipe

∞∫

0

z3 h(z) F (k)(t − τpipe, z) dz . (5.18)

Due to the racemic solution, the initial masses of both enantiomers are the same and the

mass fluxes of both enantiomers are equal in the feed stream, so,

m(p)0 = m

(c)0 , m

(p)in,c = m

(c)in,c . (5.19)

A size-dependent growth rate approach is applied ([52, 18])

G(k)(t, z) = kg,eff(T )(S(k)(t) − 1)g(1 + aASLz)dASL . (5.20)

The exponent g denotes the growth order and the constants aASL and dASL represent the

size dependency. The temperature dependence of the growth rate constant kg,eff is given

by an Arrhenius type relation,

kg,eff = kg,eff,0 exp

(

−EA,g

RT

)

. (5.21)

Here, the symbol kg,eff,0 is the pre-exponential factor of the growth rate constant, EA,g

is the activation energy, and R is the universal gas constant. Their values are given in

Table 5.1. The symbol S(k)(t) denotes the supersaturation of the k-th enantiomer which is

defined as

S(k)(t) =w(k)(t)

w(k)eq (t)

, (5.22)

where w(k)eq is the saturated mass fraction of the k-th enantiomer defined as

w(p)eq (t) = aeq + beqw

(c)(t) , w(c)eq (t) = aeq + beqw

(p)(t) . (5.23)

The terms aeq and beq are solubility constants given in the Table 5.1. The nucleation rate

is defined as the sum of primary (heterogeneous) nucleation rate and secondary nucleation

86

rate ([52, 18])

B(k)0 (t) = B

(k)0,prim(t) + B

(k)0,sec(t) , k ∈ p, c . (5.24)

The primary nucleation rate is given by a semi-empirical equation derived from the Mers-

mann model (see [52, 18])

B(k)0,prim(t) = η(k)(t) exp

−aprim ln(

ρS/C(k)eq

)3

(ln S(k)(t))2

, (5.25)

where

η(k)(t) = kb,primTe−KTT exp

(−∑

k w(k)(t)

KW

)

√

√

√

√ln

(

ρS

C(k)eq (t)

)

(

S(k)(t)C(k)eq (t)

)73 . (5.26)

Here, C(k)eq denotes the concentrations of the k-th enantiomer at equilibrium as defined by

C(k)eq (t) = ρ(t)w(k)

eq (t) . (5.27)

The secondary nucleation rate is given by an overall power law expression

B(k)0,sec(t) = kb,sec

(

S(k)(t) − 1)bsec

(

µ(k)3 (t)

)nsec

, (5.28)

where bsec is the secondary nucleation rate exponent and nsec is the third moment exponent.

The secondary nucleation rate constant is given as

kb,sec = kb,sec,0 exp

(

−EA,b

RT

)

. (5.29)

The symbol kb,sec,0 is the pre-exponential factor of the secondary nucleation rate constant

and EA,b is the corresponding activation energy.

Although the steady state results are crucial for an assessment of the process, a better

experimental realization can be achieved by investigating the process dynamically.

5.2 Numerical Techniques

In this section, HR-FVS and DG-schemes are proposed to solve the PBE for continuous

preferential crystallization given by Eq. (5.1).

87

5.2.1 Implementation of HR-FVS

The high resolution finite volume scheme [32] is implemented for discretizing the derivative

of length coordinate in Eq. (5.1). The scheme derivation follows the same procedure as

presented in Chapter 3. However, for completeness, we present the derivation again. It is

convenient to re-write Eq. (5.1) in the following form:

∂F (k)(t, z)

∂t= − ∂[G(k)(t, z)F (k)(t, z)]

∂z+ Q(k)(t, z) , (5.30)

where

Q(k)(t, z) = F(k)seeds(z) − F

(k)out(t, z) − F

(k)pipe(t, z) . (5.31)

In order to apply a numerical scheme, the first step is to discretize the computational

domain which is here the crystal size z. Let N be a large integer, and denote by (zi− 12),

i ∈ ( 1, 2, · · · , N + 1), the partitions of cells in the domain [z0, zmax], where z0 is the

minimum and zmax is the maximum crystal length of interest. For each i = 1, 2, · · · , N , ∆z

represents the cell width, the points zi refer to the cell centers, and the points zi± 12

denotes

the cell boundaries. The integration of Eq. (5.30) over the cell Ωi =[

zj− 12, zj+ 1

2

]

yields the

following cell centered semi-discrete finite volume schemes for f(k)

i± 12

= (G(k)F (k))i± 12

∫

Ωi

∂F (k)(t, z)

∂tdz = − (f

(k)

i+ 12

(t) − f(k)

i− 12

(t)) +

∫

Ωi

Q(k)(t, z)dz. (5.32)

Let F(k)i and Q(k)

i denote the average values of the number density and source term in each

cell Ωi, i.e.

F(k)i (t) =

1

∆z

∫

Ωi

F (k)(t, z)dz , Q(k)i (t) =

1

∆z

∫

Ωi

Q(k)(t, z)dz . (5.33)

Then, Eq. (5.32) can be written as

dF(k)i (t)

dt= −

f(k)

i+ 12

(t) − f(k)

i− 12

(t)

∆z+ Q(k)

i (t), i = 1, 2, · · · , N , k ∈ p, c . (5.34)

88

Here, N denotes the total number of mesh elements in the computational domain. The

accuracy of finite volume discretization is mainly determined by the way in which the cell

face fluxes are computed. Assuming that the flow is in the positive z-direction, a first order

accurate upwind scheme is obtained by taking the backward differences,

f(k)

i+ 12

(t) = (G(k)(t)F (k)(t))i , f(k)

i− 12

(t) = (G(k)(t)F (k)(t))i−1 . (5.35)

To obtain high order accuracy of the scheme, one has to use better approximation of the

cell interface fluxes. According to the high resolution finite volume scheme of Koren the

flux at the right boundary zi+ 12

is approximated as

f(k)

i+ 12

(t) = f(k)i (t) +

1

2φ(r

(k)

i+ 12

(t))(f(k)i (t) − f

(k)i−1(t)) . (5.36)

Similarly, one can approximate the flux at the left cell boundary. The flux limiting function

φ according to [32] is defined as

φ(

r(k)

i+ 12

(t))

= max

0, min (2r(k)

i+ 12

(t), min (1

3+

2r(k)

i+ 12

(t)

3, 2))

. (5.37)

Here, r(k)

i+ 12

(t) is the upwind ratio of the consecutive flux gradients,

r(k)

i+ 12

(t) =f

(k)i+1(t) − f

(k)i (t) + ε

f(k)i (t) − f

(k)i−1(t) + ε

, (5.38)

where ε is a small number to avoid division by zero. This scheme is not applicable up

to the boundary cells because it needs values of the cell nodes which are not present. To

overcome this problem, the first order approximation of the fluxes was used at the inter-

faces of the first two cells on the left-boundary and at the interfaces of the last cell on the

right-boundary. At the remaining interior cell interfaces, the high order flux approximation

of Eq. (5.36) was used. It should be noted that, the first order approximation of the fluxes

in the boundary cells does not effect the overall accuracy of the proposed high resolution

89

scheme.

The resulting system of ordinary differential equations (ODEs) in Eq. (5.34) together with

Eqs. (5.36)-(5.38) can be solved by a standard ODE-solver. In this study a Runge-Kutta

method of order four was used. In the case of fines dissolution with time-delay the residence

time in the pipe was taken as an integer multiple of the time step. This facilitates to keep

the old values in memory and to avoid linear interpolation. The computer program is

written in Matlab 7.9.1 (R2009b).

5.2.2 Implementation of DG-Scheme

The derivation of the Runge-Kutta discontinuous Galerkin (DG) scheme for approximat-

ing Eq. (5.1) follows the same procedure as presented in Chapter 3. Moreover, the same

Runge-Kutta method is used to solve the resulting ODE system. Therefore, we omit the

derivation of the scheme in this chapter.

5.3 Test Problems

In this section the simulation results of a MSMPR preferential crystallizer are presented

for different operating strategies. Firstly, the crystallizer is operated without fines disso-

lution unit and is either seeded continuously or periodically with seeds of the preferred

enantiomer. In this case the last terms on the right hand side of Eqs. (5.31) and (5.10) are

neglected. Secondly, the crystallizer is equipped with a fines dissolution unit and is contin-

uously seeded with seeds of the preferred enantiomer. In this process, it is assumed that

all fines are dissolved in the pipe before re-entering the solution back to the crystallizer. In

order to judge the quality of the process some goal functions can be used such as product

purity, productivity, yield and mean crystal size of the preferred enantiomer. These goal

90

functions give detailed information about the success and potential of continuous prefer-

ential crystallization.

The mean residence time of the liquid phase remained fixed at 112.41 min, while the res-

idence time of the solid phase was considered as either τS = τL or τS = 2τL. In an actual

operation, it may be difficult to decouple the residence times of the liquid and solid phases.

Thus, it is most likely to set τS = τL . However, larger residence times of the solid phase

compared to the liquid phase could eventually be realized by installing a filter for the solid

particles at the outlet, thus, allowing a longer residence τS of the solid phase.

In the numerical calculations, the steady state was identified when the relative deviation

error in the supersaturation was below 5 × 10−7.

Both the HR-FVS and the DG-scheme were applied to solve the model. However, due to

the smoothness of the solution and because of the arrival of steady state both schemes were

found to produce almost the same results. Therefore, to avoid unnecessary repetitions, only

the results of one scheme are presented at a time.

Problem 1

Case A: Continuous seeding without fines dissolution

In this case, the crystallizer is continuously seeded with the preferred enantiomer. The

corresponding seeds size distribution is given as

F(p)seeds =

a1

zAa

exp

(

−0.5

(

1

a3

ln

(

z

a2

))2)

, (5.39)

91

where the values of constants a1, a2 and a3 are given in the Table 5.1. Moreover, the

normalization factor is given as

Aa =kvρS

m(p)seeds · τS

µ(p)3 . (5.40)

The initial masses of the preferred and counter enantiomers in the crystallizer are taken

as m(p)0 = m

(c)0 = 0.0478 kg. Both HR-FVS and DG-scheme produced the same results.

Therefore, the results of HR-FVS are presented only.

In this test problem, the minimum and maximum crystal sizes of interest are taken as

z0 = 1.0 × 10−10 m and zmax = 1.0 × 10−2 m, respectively. The corresponding computa-

tional domain is subdivided into 200 grid points. The remaining parameters are given in

Table 5.1. The physicochemical parameters correspond to the enantiomers of the amino

acid threonine and the solvent water, see [17, 18]. The temperature of the crystallizer was

assumed to be constant at 33 oC.

The simulation results corresponding to steady state conditions are displayed in Table 5.2

for different mass flow of seeds. It can be observed that an increase in the ratio of solid to

the liquid residence times, τS/τL, diminishes the purity of the preferred enantiomer due to

the production of an increasing amount of counter enantiomer as an impurity, while the

productivity, yield, and mean crystal size of the preferred enantiomer are improved. As τS

increases, the crystals of the preferred enantiomer have more time to grow and, hence, the

overall productivity, yield and the mean crystal size are enhanced. The productivity and

yield along with the investment of seeds are increased by increasing the mass flow of seeds.

It is clear that with large amount of seeding more crystals are produced, providing the

potential for high productivity. However, the mean crystal size reduces on increasing the

mass of seeds. The reason is obvious, by increasing the mass flow (amount) of seeds the

92

existing supersaturation is consumed by a large number of crystals and, therefore, the mean

crystal size becomes smaller. However, by investing less seeds, the existing supersaturation

is utilized by only a fewer crystals, leading to larger mean size of the crystals. Finally, the

steady-state is reached after a longer time when the ratio τS/τL is larger. However, the

time needed for steady state reduces by increasing the mass flow of seeds.

Figures 5.2 and 5.3 also justify and support the above discussion. They show a compari-

son of goal functions for different mass flows of seeds and different residence times under

isothermal condition. It can bee seen that supersaturation decreases with increasing mass

flow of seeds. As explained above, an increase in the mass flow (amount) of seeds enhances

the consumption rate of the solute mass (in the solution). Moreover, the time needed to

achieve steady state becomes shorter by increasing the mass flow of seeds which is also

clear from the Table 5.2. The left and right hand side figures show the same behavior but

an increase in the residence time of the solid phase reduces the steady state value of the

supersaturation. As the residence time increases, the crystals remain in the crystallizer for

a longer time and the surface area of the crystals increases, resulting in a high depletion

of the supersaturation. It can be further observed that purity increases by increasing the

mass flow of seeds. The mean crystal size, productivity, and yield plots also agree with

the data in the tables. Initially, overshoots can be observed in the results when the ratio

of residence time is 0.3. It means that, the productivity increases for a certain time and

then bounces back. The reason can be the high initial supersaturation which is depleted

with time before reaching to the steady state condition. As the yield depends on the

productivity, it shows a similar behavior. At the startup, the values of productivity and

yield remain negative for a certain time and then become positive. During that period,

the crystallizer is seeded with the seeds of preferred enantiomer which stay for a certain

time before being taken out as a product. In other words, the productivity does not start

93

instantaneously due to the residence time of the solid phase. Moreover, in Figure 5.2 purity

drops temporarily before reaching the steady state due to the stronger relative (negative)

impact of primary nucleation of counter enantiomer. This effect reduces gradually when a

steady state condition is approaching.

Finally, Figure 5.4 presents the percentage relative errors in the mass balances at steady

state condition and the crystal size distributions for different mass flows of seeds. It can

be observed that the error in the mass balances is moderately increasing until m(p)seeds ≈

0.04 g/min and then bounces back. Moreover, bigger crystals are produced for lower mass

flow rate of seeds which also justifies the results of Table 5.2.

In summary, the purity diminishes by increasing the ratio between residence times of the

solid to the liquid phases, while the productivity, yield, and mean crystal size are enhanced.

An increase in this ratio reduces the mass flow of seeds and increases the time needed to

achieve steady state. Moreover, the supersaturation and mean crystal size are reduced

with increasing the mass flow of seeds, while productivity, yield and purity are improved.

The errors in mass balances are below 0.15% for a wide range of mass flow of seeds.

94

Table 5.1: Physicochemical parameters related tothreonine-water system ([17, 18]).

Parameters Symbols Value UnitVolume shape factor kv 0.122 [−]Density of solid phase ρS 1250 [kg/m3]Crystal growth rate exponent kg,eff,0 2.98 × 109 [m/min]Nucleation rate exponent kb,sec,0 2.38 × 1026 [m−3/min]Volume of the liquid phase VL 4.496 × 10−4 [m3]

Volumetric flow rate VL 0.4 × 10−5 [m3/min]Volume of the pipe Vpipe 4.0 × 10−5 [m3]

Volumetric flow rate Vpipe 0.4 × 10−5 [m3/min]Mass of the solvent msolv 0.3843 [kg]Constant for density K1 1.00023 [cm3g−1]Constant for density K2 4.68 × 10−6 [cm3g−1K−2]Constant for density K3 0.3652 [cm3g−1]Density of water ρ0 0.9947 [g cm−3]Density of fresh solution ρf 1.067 × 103 [kg m−3]Parameter for temperature dependence KT 1874.4 [K]Parameter for mass fraction dependence KW 0.290 [−]Universal gas constant R 8.314 J mol/KGrowth rate exponent g 1.1919 [−]Parameter for crystal growth aASL 2.0209 × 104 [m−1]Parameter for crystal growth dASL −4.066 × 10−1 [−]Activation energy for crystal growth EA,g 75.54 × 103 [J mol−1]Activation energy for nucleation EA,b 63.83 × 103 [J mol−1]Secondary nucleation exponent bsec 4.80 [−]Exponent for third moment nsec 3.0258 [−]

Nucleation constant kb,prim 3.847 × 10−2 [ (kg m−3)−7/3

min K]

Constant for exponential law aprim 4.304 × 10−3 [−]Constant of surface area dependency A1 -6.9584 [m−2]Solubility constant (Eq.(5.23)) aeq 9.83 × 10−2 [−]Solubility constant (Eq.(5.23)) beq −7.45 × 10−2 [−]Seeds distribution constant (Eq.(5.39)) a1 0.014 [−]Seeds distribution constant (Eq.(5.39)) a2 0.0009 [m]Seeds distribution constant (Eq.(5.39)) a3 0.288 [−]

95

Table 5.2: Problem 1 (Case A): Values of goal functions at τL = 112.41 min.τSτL

tstd [min] m(p)seed [ g

min] Pu(p) [%] Pr(p) [ kg

min m3 ] Y (p) [%] z(p) [mm]

1.0 646 0.0089 99.74 0.037 3.91 1.312.0 1411 0.0044 98.95 0.043 4.54 1.541.0 676 0.0356 99.92 0.060 6.29 1.132.0 1315 0.0178 99.54 0.062 6.60 1.271.0 721 0.0623 99.94 0.067 7.08 1.082.0 1318 0.0311 99.68 0.069 7.29 1.18

.

96

0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]

m(p)seeds = 0.0089 g/min



0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time[min]

prod

uctiv

ity [k

g/m

in m

3 ]




0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]




Figure 5.2: Problem 1 (Case A): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0.

97

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 30000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

time [min]

mea

n cr

ysta

l siz

e [m

m]




0 500 1000 1500 2000 2500 30000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

time [min]

mea

n cr

ysta

l siz

e [m

m]





98

0 0.05 0.1 0.15 0.20.1

0.11

0.12

0.13

0.14

0.15

m(p)seeds

rela

tive

erro

r in

mas

s ba

lanc

e [%

]

0 1 2 3 4 5

x 10−3

0

1

2

3

4

5

6

7

8

9

10x 10

6

crystal size [m]

CS

D [1

/m]

seeds distribution





Figure 5.4: Problem 1 (case A): Error in mass balance and CSDs.

Case B: Periodic seeding without fines dissolution

In this test case, continuous crystallization along with periodic seeding of preferred enan-

tiomer is analyzed. The model equations are exactly the same as for the continuous seeding

case in Case A. In Table 5.3, ton represents the time at which seeding is switched on and

toff denotes the period of time at which seeding is switched off. The seed flow rate can

be varied in contrast to the continuous seeding. In this table, a comparison of simulation

results for continuous and periodic seeding strategies are presented. A schematic diagram

of the periodic seeding is shown in Figure 5.5. In that figure, the shaded areas represent the

seeding times (ton) and the non-shaded areas depict the periods when seeding is switched

off (toff ).

In Table 5.3, the simulation results are presented for different ratios of the residence times.

The increase in τS reduces the purity of the preferred enantiomer in the periodic case. As

residence time increases, more nuclei of the counter enantiomer are produced as an impurity

of the product and, thus, the purity is lower. It can also be observed that the purity has a

99

seeding off

seeding on

time [min]

mas

s flo

w o

f see

ds [g

/min

]

Figure 5.5: Schematic diagram for periodic seeding.

decreasing behavior for increasing toff . On the other hand, as τS increases the crystals of

the preferred enantiomer gets more time to grow and, thus, the overall productivity and

yield enhances. Additionally, the productivity and yield are decreased by increasing toff .

It is obvious that when seeding is switched off for a longer time, the mean crystal size will

increase by increasing the residence time τS due to longer stay of seeds in the crystallizer

and because of utilizing the existing supersaturation by fewer crystals. The mean crystal

size also increases by increasing toff for all considered residence times due to the utiliza-

tion of existing supersaturation by fewer crystals. Table 5.3 also shows that the feed flow

rate of seeds decreases by increasing the residence time. A comparison of the continuous

seeding case in Table 5.2 with the current periodic case in Table 5.3 shows that continuous

seeding gives better purity, productivity and yield, while the mean crystal size is smaller

and investment (mass flow of seeds) remains the same in all considered residence times.

On the other hand, the periodic seeding gives larger crystals at low investment (feed flow

rate) of seeds per residence time. Finally, the time needed to achieve steady-state becomes

longer on increasing τS, while keeping the liquid residence time τL fixed. This behavior is

exactly similar to constant, time independent seeding case.

100

Figures 5.6 and 5.7 show the goal functions at different periods of seeding and for different

ratios of residence times. It can be observed that supersaturation decreases with reducing

time period of seeding, because a decrease in toff of seeds speeds up the consumption of

solute mass. Both figures show the same trends. However, a comparison of the plots in

Figures 5.6 and 5.7 shows that, an increase in the residence time of solid phase gives a

decrease in the steady state supersaturation. The reason is clear, as the solid residence

time increases, the stay of crystals in the crystallizer for a longer time and, thus, the surface

area of crystals increases by consuming more supersaturation. Further, the steady-state

time delays as the residence time of the solid phase increases. Oscillations are also visible

which indicate the conditions of periodic seeding. These oscillations are less prominent in

the right hand side plots of Figures 5.6 and 5.7 due to an increase in the residence time

of the solid phase. The plots show that the purity decreases by increasing toff . Initially,

the decrease in purity is due to increasing the amount of counter enantiomer but after

a certain time the amount of preferred enantiomer starts increasing which improves the

purity. It can also be observed that an increase in toff produces large size crystals which

is also clear from Table 5.3. The behavior of productivity is also similar to that in Table

5.3. It is evident that larger toff gives lower productivity. Yield is directly related to the

productivity, thus, it shows a similar behavior as the productivity. Moreover, in Figure 5.6

the purity drops temporarily before reaching the steady state due to the stronger relative

impact (negative) of primary nucleation of the counter enantiomer. This effect reduces

gradually when the steady state conditions are approached.

In summary, the periodic seeding reduces the productivity and yield, while it improves the

mean crystal size. The purity is slightly reduced with periodic seeding (approx. 0.1%).

Moreover, the investment of seed crystals is reduced.

101

Tab

le5.3:

Prob

lem1

(caseB

):Perio

dic

seedin

g:m

(p)

seeds=

4g

and

τL

=112.4

min

.τS

τL

seedin

gty

pe

tstd

[min

]ton

[min

]toff

[min

]m

(p)

seed[

gm

in]

Pu

(p)[%

]P

r(p

)[kg

min

m3 ]

Y(p

)[%]

z(p

)[mm

]

1.0P

67010

50.0244

99.870.051

5.501.18

1.0P

67010

100.0189

99.840.045

4.671.23

1.0P

67010

200.0126

99.790.040

4.261.27

1.0C

676-

-0.0356

99.920.060

6.291.13

2.0P

132010

50.0125

99.410.044

4.671.34

2.0P

132110

100.0089

99.300.033

3.471.40

2.0P

132110

200.0051

99.100.021

2.251.46

2.0C

1315-

-0.0178

99.540.062

6.601.27

102

0 500 1000 1500 20001.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 2000 2500 3000 3500 40001.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 200099

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

time [min]

purit

y [%

]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 2000 2500 3000 3500 400099

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

time [min]

purit

y [%

]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 20000

0.01

0.02

0.03

0.04

0.05

0.06

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 2000 2500 3000 3500 40000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

Figure 5.6: Problem 1 (Case B): Periodic seeding: Left τS/τL = 1, Right: τS/τL = 2.

103

0 500 1000 1500 20000

1

2

3

4

5

6

time [min]

yiel

d [%

]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time [min]

yiel

d [%

]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 20000.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

time [min]

mea

n cr

ysta

l siz

e [m

m]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

0 500 1000 1500 2000 2500 3000 3500 4000

1

1.1

1.2

1.3

1.4

1.5

1.6

time [min]

mea

n cr

ysta

l siz

e [m

m]

ton

=10, toff

=5

ton

=10, toff

=10

ton

=10, toff

=20

Figure 5.7: Problem 1 (Case B): Periodic seeding: Left τS/τL = 1, Right: τS/τL = 2.

104

Case C: Continuous seeding with fines dissolution

In this problem, a continuous seeding with fines dissolution is investigated. Fines particles

below the critical size zcrit = 6.0 × 10−4 m are taken out from the crystallizer along with

the solution into the dissolution unit (the pipe). After a certain time delay the particle-free

solution re-enters to the crystallizer. The fines dissolution unit is usually equipped with a

heat exchanger for dissolving fines followed by a heat sink which brings back the solution

temperature to that of the crystallizer. A complete model Eq. (5.30) is employed in this

problem. The selection (death) function is taken as

ψ(z) =

0.6, if z ≤ zcrit ,0, otherwise .

(5.41)

In this case, the simulation results with continuous seeding and fines dissolution are ana-

lyzed. Table 5.4 shows a comparison of fines dissolution (yes) and without fines dissolution

(no) by considering different ratios of residence times and fixed mass flow of seeds. It can

be seen that the purity is higher in the case of fines dissolution compared to that with-

out fines dissolution, as the fines dissolution diminishes the number of counter enantiomer

crystals in the crystallizer. From the Table 5.4, it can be realized that the productivity and

yield are slightly lower in the case of fines dissolution for both ratios of residence times.

The fines dissolution enhances the supersaturation of the solution which in turn increases

the growth rate and thus, improves the mean crystal size.

Table 5.4: Problem 1 (case C): Results with FD (yes) and without FD (no)τSτL

FD tstd [min] m(p)seed [ g


min m3 ] Y (p) [%] z(p) [mm]

1.0 no 676 0.0356 99.92 0.060 6.29 1.131.0 yes 648 0.0356 99.99 0.060 6.25 1.152.0 no 1315 0.0178 99.54 0.062 6.60 1.272.0 yes 1303 0.0178 99.97 0.062 6.56 1.29

105

Problem 2

The idea behind choosing this problem is its practical considerations, see [16-19] and ref-

erences therein. Here, a different seeds distribution of preferred enantiomer is considered,

F(p)seeds(0, z) =

1√

(2π)σIa

.1

zexp(−0.5)

(

ln (z) − z

σ

)2

, (5.42)

where

Ia =kv.ρc

mseeds

µ(p)3 (0) (5.43)

We do not seed the crystallizer with counter enantiomer, thus

F(c)seeds(0, z) = 0. (5.44)

Here we assumed σ = 0.3947m, z = −6.8263m, while mseeds is the mass of seeds. The

maximum crystal size that was expected is zmax = 0.005m which is subdivided into 400

grid points. The final simulation time was taken as 3000 minutes. Both, HR-FVS are

applied to solve the problem and they produced the same results. Here, we present the

results of the DG-scheme only.


Figure 5.8 shows a comparison of CSDs for different mass flows of seeds. The goal functions

are presented in Figures 5.9 and 5.10 and show the same behavior that was seen in Case

A of Problem 1. After analysis of the figures we conclude that the purity diminishes by

increasing the ratio between residence times of the solid to the liquid phases, while the

productivity, yield, and mean crystal size are improved. An increase in this ratio reduces

the mass flow of seeds and increases the time needed to achieve steady state. Moreover,

the supersaturation and mean crystal size are reduced with increasing mass flow of seeds,

while productivity, yield and purity are improved.

Moreover, bigger crystals are produced for greater masses of the seed crystals. The figures

also justify the results of Table 5.5

106




min m3 ] Y (p) [%] z(p) [mm]

1.0 872 0.0089 99.03 0.0005 0.051 3.762.0 1331 0.0044 93.30 0.0008 0.084 3.491.0 832 0.0356 99.74 0.0014 0.0145 5.152.0 1420 0.0178 98.05 0.0015 0.154 5.291.0 841 0.0623 99.84 0.0021 0.225 5.422.0 1503 0.0311 98.82 0.0022 0.234 5.39

0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5x 10

4

Crystal Size [m]

CS

D [1

/m]

Preferred Enantiomer Number Density

mseeds(p) = 1g

mseeds(p) = 4g

mseeds(p) = 7g

0 0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Crystal Size [m]

CS

D [1

/m]

Counter Enantiomer Number Density

mseeds(p) = 1g

mseeds(p) = 4g

mseeds(p) = 7g

Figure 5.8: Problem 2 (Case A): CSDs for the different mass of seed crystal.

107

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30002.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.5

2

2.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


108

0 500 1000 1500 2000 2500 300099

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 300093

94

95

96

97

98

99

100

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−16

−14

−12

−10

−8

−6

−4

−2

0

2

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−8

−7

−6

−5

−4

−3

−2

−1

0

1

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


109

Case B: Continuous seeding with fines dissolution

In this case, a continuous seeding with fines dissolution is investigated. Fines particles






problem. In this case, the simulation results with continuous seeding and fines dissolution

are analyzed. Table 5.6 shows a comparison of fines dissolution (yes) and without fines

dissolution (no) by considering different ratios of residence times and fixed mass flow of

seeds. It can be seen that the purity is greater in the case of fines dissolution compared

to that without fines dissolution, as the fines dissolution reduces the number of counter

enantiomer crystals in the crystallizer. From Table 5.6, it can be seen that the productivity

and yield are slightly enhanced in the case of fines dissolution for both ratios of residence

times. Supersaturation enhances in the fines dissolution that is the reason why we see

that as increase in the growth rate improves the mean crystal size. Figures 5.11 and 5.12

present the plots of goal function with respect to time which also justify the results of the

Table 5.6.

Table 5.6: Problem 2 (Case B): Results with FD (yes) and without FD (no)τSτL



min] Y (p) [%] z(p) [mm]

1.0 no 832 0.0356 99.74 0.0014 0.145 5.151.0 yes 962 0.0356 99.77 0.0014 0.147 5.172.0 no 1315 0.0178 98.05 0.0015 0.154 5.292.0 yes 1700 0.0178 98.28 0.0015 0.157 5.32

110

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30002.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.5

2

2.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

Figure 5.11: Problem 2 (Case B): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0.

111

0 500 1000 1500 2000 2500 300099.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

100.1

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 300093

94

95

96

97

98

99

100

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time [min]

Pro

duct

ivity

[kg/

min

m3 ] m

seeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−16

−14

−12

−10

−8

−6

−4

−2

0

2

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−8

−7

−6

−5

−4

−3

−2

−1

0

1

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


112

Case C: Periodic seeding without fines dissolution

In this case, continuous crystallization along with periodic seeding of preferred enantiomer

is examined. The model equations are exactly the same as for the continuous seeding case

in test problem 1. In Table 5.7, ton represents the time at which seeding is switched on and



results for continuous and periodic seeding strategies are presented.

The same result as seen in case C of test problem 1 can be observed here. In Table 5.7,

the simulation results are presented for different ratios of the residence times. The increase

in τS reduces the purity of the preferred enantiomer in the periodic case. The longer the

residence time, the larger the formation of the counter enantiomer as an impurity of the

product and, thus, the lower the purity. It can also be observed that the purity has a

decreasing behavior for increasing toff . On the other hand, as τS increases the crystals of

the preferred enantiomer get more time to grow and, thus, the overall productivity and

yield enhances.

Figures 5.13 and 5.14 show the goal functions at different periods of seeding and for dif-

ferent ratios of residence times. It can be observed that supersaturation decreases with

reducing time period of seeding, because a decrease in toff of seeds speeds up the con-

sumption of solute mass. Both figures show the same trends. However, a comparison of

the plots in Figures 5.13 and 5.14 shows that, an increase in the residence time of the solid

phase results in a decrease in the steady state supersaturation.

Keeping all the discussions, we can conclude that the periodic seeding reduces the produc-

113

tivity and yield, while it improves the mean crystal size. The purity is slightly reduced

with periodic seeding (approx. 0.1%). Moreover, the investment of seed crystals is reduced.

0 500 1000 1500 20001.09

1.091

1.092

1.093

1.094

1.095

1.096

1.097

Time [min]

Sup

ersa

tura

tion

[−]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 40001.09

1.091

1.092

1.093

1.094

1.095

1.096

1.097

Time [min]

Sup

ersa

tura

tion

[−]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 20001.5

2

2.5

3

3.5

4

4.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

Figure 5.13: Problem 2 (Case C): Goal functions: Left τS/τL = 1.0, Right: τS/τL = 2.0.

114

0 500 1000 1500 200097.5

98

98.5

99

99.5

100

Time [min]

Pur

ity [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 400082

84

86

88

90

92

94

96

98

100

Time [min]

Pur

ity [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000−0.055

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 4000−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

Time [min]

Pro

duct

ivity

[kg/

min

m3 ] t

on = 10 , t

off = 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

Time [min]

Yie

ld [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 4000−3

−2.5

−2

−1.5

−1

−0.5

Time [min]

Yie

ld [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20


115

Tab

le5.7:

Prob

lem2

(Case

C):

Perio

dic

seedin

g:m

(p)

seeds=

4g

and

τL

=112.4

min

.τS

τL

seedin

gty

pe

tstd

[min

]ton

[min

]toff

[min

]m

(p)

seed[

gm

in]

Pu

(p)[%

]P

r(p

)[kg

min

m3 ]

Y(p

)[%]

z(p

)[mm

]

1.0P

67010

50.0244

99.430.0198

2.0884.45

1.0P

67010

100.0189

99.070.021

2.243.71

1.0P

67010

200.0126

98.080.018

1.922.59

1.0C

832-

-0.0356

99.740.0014

0.01455.15

2.0P

132010

50.0125

95.750.011

1.0774.33

2.0P

132010

100.0089

92.660.010

1.0693.33

2.0P

132010

200.0051

84.200.007

0.7752.282

2.0C

1420-

-0.0178

98.050.0015

0.1545.29

116

Problem 3

The parameter of this problem has been used in the experiments, see [16-19] and references

therein. Another seed distribution of preferred enantiomer is considered:

F(p)seeds(0, z) =

107

√

(2π)σIa

.1

zexp

(

ln (z) − z

σ√

2

)2

, (5.45)

where

Ia =kv.ρc

mseeds

µ(p)3 (0) (5.46)

Crystals of the counter are not initially present, i.e.

F(c)seeds(0, z) = 0. (5.47)

Here we assumed σ = z15

, z = 4.10−4m, while mseeds is the mass of the initial seeds.

The maximum time simulation is 3000 minutes for crystal size zmax = 0.002m which is

subdivided into 400 grids points.


Figure 5.15 shows a comparison of CSDs for different mass flow of seeds. Analysis of the

figures 5.16 and 5.17 the same behavior of the graphs is seen as in case A of test problems 1

and 2. After analysis of the figures we conclude that the purity diminishes by increasing the

ratio between residence times of the solid to the liquid phases, while the productivity, yield,

and mean crystal size are improved. An increase in this ratio reduces the mass flow of seeds

and increases the time needed to achieve steady state. Moreover, the supersaturation and

mean crystal size are reduced with increasing the mass flow of seeds, while productivity,

yield and purity are improved.

Moreover, bigger crystals are produced for greater mass of the seed crystals both for the

preferential and counter which also justifies the results of Table 5.8.

117




min m3 ] Y (p) [%] z(p) [mm]

1.0 874 0.0089 99.03 0.0005 0.052 3.762.0 1332 0.0044 93.29 0.0008 0.085 3.491.0 832 0.0356 99.74 0.0014 0.145 5.152.0 1650 0.0178 98.03 0.0015 0.158 5.291.0 842 0.0623 97.42 0.0021 0.225 5.222.0 15768 0.0311 98.81 0.0023 0.237 5.39

0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5x 10

4

Crystal Size [m]

CS

D [1

/m]

Preferred Enantiomer Number Density

mseeds(p) = 1g

mseeds(p) = 4g

mseeds(p) = 7g

0 0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Crystal Size [m]

CS

D [1

/m]

mseeds(p) = 1g

mseeds(p) = 4g

mseeds(p) = 7g

Figure 5.15: Problem 3 (Case A): CSDs for the different mass of seed crystal.

118

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30002.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

] mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.5

2

2.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


119

0 500 1000 1500 2000 2500 300099

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

Time [min]

Pur

ity [%

] mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 300093

94

95

96

97

98

99

100

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time [min]

Pro

duct

ivity

[kg/

min

m3 ] m

seeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [min]

Pro

duct

ivity

[kg/

min

m3 ] m

seeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−16

−14

−12

−10

−8

−6

−4

−2

0

2

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−8

−7

−6

−5

−4

−3

−2

−1

0

1

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


120

Case B: Continuous seeding with fines dissolution

In this case, a continuous seeding with fines dissolution is investigated. Fines particles






problem. In this case, the simulation results with continuous seeding and fines dissolution

are analyzed. Table 5.9 shows a comparison of fines dissolution (yes) and without fines

dissolution (no) by considering different ratios of residence times and fixed mass flow of

seeds. The same behavior is seen as in case C of test problems 1 and 2, concluding that the

purity is greater in the case of fines dissolution compared to that without fines dissolution,

as the fines dissolution reduces the number of counter enantiomer crystals in the crystallizer.

From Table 5.9, it can be seen that the productivity and yield are slightly lower in the case

of fines dissolution for both ratios of residence times. Supersaturation enhances in the fines

dissolution that is the reason why we see increase in the growth rate improves the mean

crystal size. Figures 5.18 and 5.19 present the plots of goal function with respect to time

which also justify the results of the Table 5.9.

Table 5.9: Problem 3 (Case B): Results with FD (yes) and without FD (no)τSτL



min] Y (p) [%] z(p) [mm]

1.0 no 832 0.0356 99.74 0.0014 0.145 5.151.0 yes 961 0.0356 99.77 0.0014 0.147 5.182.0 no 1650 0.0178 98.03 0.0015 0.158 5.292.0 yes 1700 0.0178 98.28 0.0014 0.157 5.32

121

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.075

1.08

1.085

1.09

1.095

1.1

Time [min]

Sup

ersa

tura

tion

[−]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30002.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 30001.5

2

2.5

3

3.5

4

4.5

5

5.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


122

0 500 1000 1500 2000 2500 300099.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100

100.1

Time [min]

Pur

ity [%

] mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 300093

94

95

96

97

98

99

100

Time [min]

Pur

ity [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time [min]

Pro

duct

ivity

[kg/

min

m3 ] m

seeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−16

−14

−12

−10

−8

−6

−4

−2

0

2

Time [min]

Yie

ld [%

]]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g

0 500 1000 1500 2000 2500 3000−8

−7

−6

−5

−4

−3

−2

−1

0

1

Time [min]

Yie

ld [%

]

mseeds(p) =1g

mseeds(p) =4g

mseeds(p) =7g


123

Case C: Periodic seeding without fines dissolution

In this case, continuous crystallization along with periodic seeding of preferred enantiomer

is examined. The model equations are exactly the same as for the continuous seeding case

in Problem 1. In Table 5.10, ton represents the time at which seeding is switched on and



results for continuous and periodic seeding strategies are presented.

The figures 5.20 and 5.21 show results for the comparison of different residence times

taken. Keeping all the discussions, we can conclude that the periodic seeding reduces the

productivity and yield, while it improves the mean crystal size. The purity is slightly

reduced with periodic seeding (approx. 0.1%). Moreover, the investment of seed crystals

is reduced.

124

0 500 1000 1500 20001.09

1.091

1.092

1.093

1.094

1.095

1.096

1.097

Time [min]

Sup

ersa

tura

tion

[−]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 40001.09

1.091

1.092

1.093

1.094

1.095

1.096

1.097

Time [min]

Sup

ersa

tura

tion

[−]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 20001.5

2

2.5

3

3.5

4

4.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 40000.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [min]

Mea

n C

ryst

al S

ize

[mm

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20


125

0 500 1000 1500 200097.5

98

98.5

99

99.5

100

Time [min]

Pur

ity [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 400082

84

86

88

90

92

94

96

98

100

Time [min]

Pur

ity [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000−0.055

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 4000−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

Time [min]

Pro

duct

ivity

[kg/

min

m3 ]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

Time [min]

Yie

ld [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20

0 500 1000 1500 2000 2500 3000 3500 4000−3

−2.5

−2

−1.5

−1

−0.5

Time [min]

Yie

ld [%

]

ton

= 10 , toff

= 5

ton

= 10 , toff

= 10

ton

= 10 , toff

= 20


126

Tab

le5.

10:

Pro

ble

m3

(Cas

eC

):Per

iodic

seed

ing:

m(p

)se

eds=

4g,τ L

=11

2.4m

in.

τ S τ Lse

edin

gty

pe

t std

[min

]t o

n[m

in]

t off

[min

]m

(p)

seed

[g

min

]P

u(p

)[%

]P

r(p)[

kg

min

m3]

Y(p

)[%

]z(

p)[m

m]

1.0

P67

010

50.

0244

99.4

30.

0195

2.06

4.45

61.

0P

670

1010

0.01

8999

.07

0.02

12.

243.

721.

0P

670

1020

0.01

2698

.07

0.18

1.92

2.59

1.0

C83

2-

-0.

0356

99.7

40.

0014

0.14

55.

152.

0P

1320

105

0.01

2595

.75

0.01

01.

077

4.33

2.0

P13

2010

100.

0089

92.6

50.

010

1.07

03.

332.

0P

1320

1020

0.00

5184

.20

0.00

70.

775

2.28

2.0

P13

20−

−0.

0178

98.0

30.

0015

0.15

85.

29

127

128

Chapter 6

Coupled Continuous Preferential Crystallization

Separation of chiral molecules is of great interest in chemical and pharmaceutical indus-

tries as many (bio)organic molecules are chiral. In most of the cases only one enantiomer

has desired properties for therapeutic activities or metabolism, while the other one may

be inactive or may produce undesired effects. Several special techniques can be applied

to separate enantiomeric mixtures, such as chromatography, classical non-biological reso-

lutions via the formation of diastereomers, biological methods, non-biological asymmetric

synthesis, and membrane technologies. An attractive alternative and energy efficient way

to separate chiral substances is the enantioselective preferential crystallization. This tech-

nique is normally applied to conglomerates, a physical mixture of enantiomerically pure

crystals. A lot of work has been done on preferential crystallization in batch mode, see for

example [3, 11, 12, 13, 17, 18, 31].

Recently, Qamar et al. [66] have adopted the concept of continuous mixed-suspension,

mixed product removal (MSMPR) crystallizers to obtain pure enantiomers from racemic

mixtures in a single crystallizer. The drawback of decoupled mode (single crystallizer) is

that only a small amount of pure substance can be recovered from the racemic solution.

To overcome this limitation, in this work, two MSMPR crystallizers are connected through

the liquid phase and operated under continuous exchange of crystal free solution. These

perfectly mixed tanks are continuously fed with the racemic solutions of two enantiomers

and solid particles and liquid phase are continuously withdrawn, see Figure 6.1. On seeding

both enantiomers, one in each vessel, the preferential crystallization of only those enan-

tiomers could be initiated in the corresponding vessels, provided that crystallization takes

place within the metastable zone where spontaneous, uncontrolled primary nucleation is

kinetically diminished. Due to the exchange, the liquid phase shows a higher overall con-

centration of the preferred enantiomer in the vessel in which it is seeded. As a result,

the supersaturation level is higher during the whole process in comparison to the case

129

without an exchange. Additionally, the concentration of the respective counter-enantiomer

in the liquid phase decreases in each tank. For the theoretical case of an infinitely high

exchange flow rate, racemic composition is reached in the liquid phase of both vessels

which corresponds to the maximum supersaturation for both preferred enantiomers. Since

the concentration of the respective counter-enantiomer is decreased, the probability for its

crystallization through primary nucleation is lowered. This leads to higher product purity

and enhances the productivity. During a starting-up period, depending on the properties

of system and process parameters, the concentration of the target enantiomer is decreasing

until a steady state is arrived where the composition is determined by the mean residence

time. Due to different kinetic mechanisms and their inherent different time constants, a

different depletion of the supersaturation for each enantiomer can be realized by an appro-

priate choice of the process conditions. As long as a critical mean residence time, where

primary nucleation may appear, is not exceeded, the concentration of the undesired counter

enantiomer maintains constant during the whole time. This fact reveals a benefit of this

continuous process in comparison to the batch one. An optimal selection of the process

conditions allows a constant production of the goal enantiomer at a high purity level.

The population balance based models are frequently used for simulating crystallization

processes. The theory of population balances began in 1960s when [27] introduced it in the

field of chemical engineering. A comprehensive overview on population balance modeling,

nucleation and growth kinetics terms and methods of solution can be found in the books

by [52, 58, 70]. During the last decades many efficient methods were developed for solv-

ing population balance models (PBMs) such as the method of characteristics introduced

in various fields by [74] and adopted for PBMs by [42, 62, 70], the method of weighted

residual or orthogonal collocation by [72], the Monte Carlo simulation by [75, 76], the fixed

and moving pivot techniques by [34, 35], and the high resolution finite volume schemes by

130

[24, 59, 66].

This work is an extension of our recent work on continuous preferential crystallization in a

single vessel, see [66]. A dynamic mathematical model is derived for simulating two-coupled

continuously operated ideally mixed MSMPR crystallizers applied for continuous prefer-

ential enantioselective crystallization. Both crystallizer are connected through exchange

pipes and are equipped with fines dissolution loops for dissolving small crystals below cer-

tain critical size. It is assumed that fines are completely dissolved at the other end of the

dissolution pipes. The developed model includes the phenomena of primary heterogeneous

nucleations, secondary nucleations, and size-dependent growth rates in each crystallizer.

The effects of different seeding and operating strategies are investigated in simulations.

The model and its parameters are based on experimental data obtained from batchwise

crystallization, see [13, 17, 18]. The semi-discrete flux-limiting finite volume scheme of

[32] is implemented to solve the model equations. To judge the quality of process some

goal functions are used, such as purity, productivity, yield and mean crystal size of the

preferred enantiomer. These goal functions provide useful information about the success

and potential of the process. To realize the advantages of coupled process, its results are

compared with those of single crystallizer obtained by [66]. It was found that coupled

process has considerably improved the values of goal functions and crystallizes both enan-

tiomers simultaneously in separate vessels. The numerical results also demonstrate the

high order accuracy, efficiency and potential of the proposed numerical method for solving

such models.

6.1 Coupled Continuous Crystallization Model

A mathematical model is presented for simulating ideally-mixed two-coupled continuous

preferential crystallizers connected through exchange pipes and equipped with fines disso-

131

L,AV&

L,AV&

L,BV&

L,BV&

Fin

es d

iss.

Fin

es d

iss.

Tank A Tank B

Figure 6.1: Principle of coupled continuous enantioselective crystallization.

lution units. The model contains four population balance equations (PBEs) for the solid

phase, two PBEs for two enantiomers in each tank, coupled with four ordinary differential

equations for the balance of solute masses of both enantiomers in both tanks, a set of al-

gebraic equations for liquid and solid mass flow rates, as well as the growth and nucleation

kinetics. The balance law for the solid phase is represented by PBEs of the form [2, 27, 54]

∂F(k)α

∂t= − ∂(G

(k)α F

(k)α )

∂z+ F

(k)seeds,α − F

(k)out,α − F

(k)pipe,α , k ∈ p, c , α ∈ A,B , (6.1)

where p stands for preferred enantiomer, c denotes counter enantiomer, and A and B rep-

resent crystallizers. The CSD of the k-th enantiomer in tank α is represented by F(k)α . In

Eq. (6.1) the term on the left hand side denotes the accumulation of crystals of size z.

The first term on the right hand side represents the convective transport in the direction

of the property coordinate z due to size-dependent crystal growth rate Gα(k). The term

F(k)seeds,α denotes the incoming particle number flux due to seeding, F

(k)out,α is the particle

number flux due to product removal, and F(k)pipe,α denotes the particle number flux to the

fines dissolution unit in each crystallizer.

132

The initial conditions for the PBEs in Eq. (6.1) are based on the assumption that initially

(at time t = 0) the solutions of both vessels are particle free. In other words

F (k)α (t = 0, z) = 0 , k ∈ p, c , α ∈ A,B . (6.2)

Assuming further, that primary nucleation leads to a crystal nuclei of minimum size zmin

and that the number density function F (k) vanishes for a arbitrary large crystal size zmax,

holds

F (k)α (t, z = zmin) =

B(k)0,α(t)

Gα(t, zmin), F (k)

α (t, z = zmax) = 0 , (6.3)

where B(k)0,α denotes the nucleation rate in the corresponding vessel. Each crystallizer is

only seeded with preferred enantiomer, thus

F(k)seeds,α(z) =

F(p)seeds,α(z) for k = p ,

0 for k = c .(6.4)

For the operation of configuration shown in Figure 6.1, the following assumption are taken

into account.

1. Both tanks are of MSMPR type and are equipped with fines removal systems.

2. The volumetric flow rates of crystallizers and dissolution units are kept constant.

3. Both vessels have the constant temperature and volume.

4. The particle-free liquid exchange flow rates to both crystallizers are the same and

constant.

5. Both primary and secondary nucleations take place at size zmin.

6. The feed flows are crystal-free.

7. All fines are dissolved in the dissolution units.

133

8. The removal of the fines are size-dependent and can be described by the product of

a classification function ψ(z) and the crystal size distributions in each crystallizer.

9. The liquid mean residence time is given by τL,α = VL,α/VL,α, where VL,α denotes the

volume of the liquid phase in the tank α and VL,α is the volumetric flow rate of the

liquid phase to that tank.

10. The characteristic solid mean residence time is given as, τS,α = VS,α/(ρS/m(p)seeds,α),

where VS,α represents the overall volume of the solid phase, ρS is the density of the

solid phase and m(p)seeds,α is the mass flow rate of the seeds of preferred enantiomer to

tank α.

11. The residence time in the pipe (plug flow) is defined as, τpipe,α = Vpipe,α/Vpipe,α. Here,

Vpipe,α denotes volume of the pipe and Vpipe,α is the volumetric flow rate to the pipe

for tank α.

For each vessel, the number density function of withdrawn crystals as a product is assumed

to be equal to the number density function inside that vessel. Therefore, the right hand

sided second and third terms of Eq. (6.1) are given as

F(k)seeds,α(z) =

F(k)seeds,α(z)

τS,α

, F(k)out,α(t, z) =

F(k)α (t, z)

τS,α

, k ∈ p, c , α ∈ A,B . (6.5)

The term corresponding to the dissolution unit in Eq. (6.1) is expressed as

F(k)pipe,α(t, z) =

ψ(z)F(k)α (t, z)

τpipe,α

, (6.6)

where ψ(z) represents the classification (death) function. The i-th moment of the CSD is

defined as

µ(k)i,α(t) =

∞∫

0

ziF (k)α (t, z) dz , i = 0, 1, 2, · · · , N, k ∈ p, c , α ∈ A,B. (6.7)

134

According to Figure 6.1, the corresponding mass balances for the liquid phase are given as

dm(k)L,α(t)

dt= m

(k)L,in,α(t) − m

(k)L,out,α(t) − 3ρS,α kv

∞∫

0

z2G(k)α (t, z)F (k)

α (t, z) dz , (6.8)

with initial data

m(k)L,α(t = 0) = m

(k)L,0,α = w(k)

α (0)ρL,0,αVL,α , k ∈ p, c , α ∈ A,B , (6.9)

where ρL,0,α = ρL,α(t = 0) is the density of fresh (initially) supersaturated solution.

Each crystallizer has six liquid streams. The incoming liquid streams include the continues

feeding of fresh solution, the solution coming from exchange pipe, and the solution coming

from the dissolution pipe. The outer streams include the outgoing liquid stream of the

utilized solution, and the solutions going to exchange and dissolution pipes. The inner

fluxes to both tanks are defined as

m(k)L,in,A(t) = m

(k)L,in,c,A(t) + m

(k)L,out,pipe,A(t) + m

(k)L,in,exch,B(t) , (6.10)

m(k)L,in,B(t) = m

(k)L,in,c,B(t) + m

(k)L,out,pipe,B(t) + m

(k)L,in,exch,A(t) , (6.11)

where the incoming flux of fresh solution to each crystallizer is given by

mL,in,c,α(t) = w(k)α (0)ρL,0,αVL,α , (6.12)

and the mass fluxes from the dissolution pipes and exchange pipes are given as

m(k)L,out,pipe,α(t) = w(k)

α (t)ρL,α(t)Vpipe,α , (6.13)

m(k)L,in,exch,B(t) = w

(k)A (t)ρL,A(t)Vexch , m

(k)L,in,exch,A(t) = w

(k)B (t)ρL,B(t)Vexch . (6.14)

Here, w(k)α denotes the mass fraction of k-th enantiomer in tank α and is defined as

w(k)α (t) =

m(k)α (t)

m(p)α (t) + m

(c)α (t) + mwater

, (6.15)

where m(k)α , m

(p)α , m

(c)α denote the mass of k-th, preferred and counter enantiomers, respec-

tively. Moreover, mwater denotes the mass of water, VL,α is the flow-rate to tank α, Vpipe,α

135

is flow-rate to the dissolution pipe of tank α, Vexch is flow-rate to the exchange pipes of

both tanks, and ρL,α represents the solution density at any time t which is defined as

ρL,α(t) = 1000(ρwater + K3wtotα (t) . (6.16)

Here

wtotα (t) = w(p)

α (t) + w(c)α (t) , (6.17)

and ρwater is defined by

ρwater =1

K1 + K2T 2. (6.18)

The symbols K1,K2 and K3 are the density parameter given in Table 6.1. The outer mass

flux from tank α is defined as

m(k)L,out,α(t) = m

(k)L,out,c,α(t) + m

(k)L,in,pipe,α(t) + m

(k)L,out,exch,α(t) . (6.19)

Here, the outer flux to the dissolution pipe is formulated as

m(k)L,in,pipe,α(t) = m

(k)L,out,pipe,α(t − τpipe,α) +

kvρcVα

Vpipe,α

∞∫

0

z3ψ(z)F (k)α (t − τpipe,α, z) dz , (6.20)

and the outer fluxes of the utilized solution from the tank α and to the exchange pipe are

given as

m(k)L,out,c,α(t) = w(k)

α (t)ρL,α(t)Vα , mL,out,α,exch(t) = w(k)α (t)ρL,α(t)Vexch . (6.21)

Because of racemic solution, the mass of both enantiomers in the feed stream of each vessel

are the same. The size-dependent growth rates are defined as

G(k)α (t, z) = kg(S

(k)α (t) − 1)g(1 + aASLz)dASL , k ∈ p, c , α ∈ A,B . (6.22)

The exponent in above equation denotes the growth order and the constants aASL and

dASL represent the size dependency. The growth rate constant, kg, is given by

kg = kg,0 exp

(

−EA,g

RT

)

. (6.23)

136

Here kg,0 denote the pre-exponent growth rate constant, EA,g represent the activation

energy, and R is the universal gas constant. There values are given in Table 6.1 which

are assumed to be the same for both vessels. Here, S(k)α (t) is the super saturation of k-th

enantiomer

S(k)α (t) =

w(k)α (t)

w(k)sat,α(t)

, (6.24)

where w(k)sat,α is the saturated mass fraction which under isothermal condition is defined as

w(p)sat,α = asat + bsatw

(c)α , w

(c)sat,α(c) = asat + bsatw

(p)α . (6.25)

Here, asat and bsat are the constant of the solubility given in Table 6.1. The nucleation rate

is defined as the sum of primary (heterogeneous) and secondary nucleation rates ([18, 52])

B(k)0,α(t) = B

(k)0,prim,α(t) + B

(k)0,sec,α(t) , k ∈ p, c , α ∈ A,B . (6.26)

The primary nucleation rate is given by a semi-empirical equation derived from Mersmann

model ([18, 52])

B(k)0,prim,α(t) = η(k)

α (t) exp

−aprim ln(

ρS/C(k)eq,α

)3

(ln S(k)α (t))2

, (6.27)

where

η(k)α (t) = kb,primTe−

KTT exp

(

−∑

k w(k)α (t)

KW

)

√

√

√

√ln

(

ρS

C(k)eq,α(t)

)

(

S(k)α (t)C(k)

eq,α(t))

73 . (6.28)

Here, C(k)eq,α denotes the concentrations of the k-th enantiomer at equilibrium as defined

below

C(k)eq,α(t) = ρL,α(t)w

(k)sat,α(t) . (6.29)

The secondary nucleation rate is given by an overall power law expression

B(k)0,sec,α(t) = kb,sec

(

S(k)α (t) − 1

)bsec(

µ(k)3,α(t)

)nsec

, (6.30)

137

where bsec is the secondary nucleation rate exponent and nsec is the third moment exponent.

The secondary nucleation rate constant is given as

kb,sec = kb,sec,0 exp

(

−EA,b

RT

)

. (6.31)

The symbol kb,sec,0 is the pre-exponential factor of the secondary nucleation rate constant

and EA,b is the corresponding activation energy.

Although the steady state results are crucial for an assessment of the process, a better

experimental realization can be achieved by investigating the process dynamically.

6.2 Implementation of Finite Volume Scheme

In this section, the high resolution finite volume scheme of [32] is implemented for discretiz-

ing the derivative of length coordinate in the population balance model. For the derivation

and explanation of the scheme, it is convenient to re-write Eq. (6.1) in the following form

∂F(k)α (t, z)

∂t= − ∂[G

(k)α (t, z)F

(k)α (t, z)]

∂z+ Q(k)

α (t, z) , (6.32)

where

Q(k)α (t, z) = F

(k)seeds,α(z) − F

(k)out,α(t, z) − F

(k)pipe,α(t, z) . (6.33)

Before implementing the proposed numerical scheme, the first step is to discretize the

computational domain which is the crystal size z in the present study. Let N be a large

integer, and denote by (zi− 12), i = 1, 2, · · · , N + 1, the partitions of cells in the domain

[zmin, zmax], where zmin is the minimum and zmax is the maximum crystal length of interest.

For each i = 1, 2, · · · , N , ∆z represents the cell width, the points zi refer to the cell centers,

and the points zi± 12

denotes the cell boundaries. The integration of Eq. (6.32) over the cell

Ωi =[

zj− 12, zj+ 1

2

]

yields the following cell centered semi-discrete finite volume schemes for

138

f(k)

i± 12,α

= (G(k)α F

(k)α )i± 1

2

∫

Ωi

∂F(k)α (t, z)

∂tdz = −

(

f(k)

i+ 12,α

(t) − f(k)

i− 12,α

(t))

+

∫

Ωi

Q(k)α (t, z)dz. (6.34)

Let F(k)i,α and Q

(k)i,α denote the average values of the number density and source term in each

cell Ωi, i.e.

F(k)i,α (t) =

1

∆z

∫

Ωi

F (k)α (t, z)dz , Q

(k)i,α(t) =

1

∆z

∫

Ωi

Q(k)α (t, z)dz . (6.35)

Then, Eq. (6.34) for i = 1, 2, · · · , N can be written as

dF(k)i.α (t)

dt= −

f(k)

i+ 12,α

(t) − f(k)

i− 12,α

(t)

∆z+ Q

(k)i,α(t) , k ∈ p, c, α ∈ A,B . (6.36)

Here, N denotes the total number of mesh elements in the computational domain. The

accuracy of finite volume discretization is mainly determined by the way in which the cell

face fluxes are computed. Assuming that the flow is in positive z-direction, a first order

accurate upwind scheme is obtained by taking the backward differences

f(k)

i+ 12,α

(t) = (G(k)α (t)F (k)

α (t))i , f(k)

i− 12,α

(t) = (G(k)α (t)F (k)

α (t))i−1 . (6.37)

To get high order accuracy of the scheme, one has to use better approximation of the cell

interface fluxes. According to the high resolution finite volume scheme of [32] the flux at

the right boundary zi+ 12

is approximated as

f(k)

i+ 12,α

(t) = f(k)i,α (t) +

1

2φ(r

(k)

i+ 12,α

(t))(f(k)i,α (t) − f

(k)i−1,α(t)) . (6.38)

Similarly, one can approximate the flux at left cell boundary. The flux limiting function φ

according to [32] is defined as

φ(

r(k)

i+ 12,α

(t))

= max

0, min (2r(k)

i+ 12,α

(t), min (1

3+

2r(k)

i+ 12,α

(t)

3, 2))

. (6.39)

139

Here, r(k)

i+ 12,α

(t) is the so-called upwind ratio of the consecutive flux gradients

r(k)

i+ 12,α

(t) =f

(k)i+1,α(t) − f

(k)i,α (t) + ε

f(k)i,α (t) − f

(k)i−1,α(t) + ε

, (6.40)

where ε is a small number to avoid division by zero. This scheme is not applicable up

to the boundary cells because it needs values of the cell nodes which are not present. To

overcome this problem, the first order approximation of the fluxes was used at the inter-

faces of the first two cells on the left-boundary and at the interfaces of the last cell on the

right-boundary. At the remaining interior cell interfaces, the high order flux approximation

of Eq. (6.38) was used. It should be noted that, the first order approximation of the fluxes

in the boundary cells does not effect the overall accuracy of the proposed high resolution

scheme.

The resulting system of ordinary differential equations (ODEs) in Eq. (6.36) together with

Eqs. (6.37)-(6.40) can be solved by a standard ODE-solver. In this study a Runge-Kutta

method of order four was used. In the case of fines dissolution with time-delay the residence

time in the pipe was taken as the integer multiple of the time step. This facilitates to keep

the old values in memory and to avoid the linear interpolation. The computer program is

written in the Matlab 7.9.1 (R2009b).

6.3 Test Problems

In this section the simulation results of two-coupled MSMPR preferential crystallizers are

presented for different operating strategies. Firstly, both crystallizers are operated without

and with fines dissolution units and are seeded continuously. Afterwards, both tanks are

seeded periodically with seeds of the preferred enantiomers. In the case of no fines disso-

lution, the last terms on the right hand side of Eq. (6.1) is neglected.

In order to analyze the quality of process some goal functions can be used, such as product

140

purity, productivity, yield and mean crystal size of the preferred enantiomers. These goal

functions give detailed information about the success and potential of continuous prefer-

ential crystallization.

Purity: It is typically of key importance in enantioselective crystallization

Pu(p)α =

m(p)S,out,α

m(p)S,out,α + m

(c)S,out,α

, where m(k)S,out,α = kvρS

∫ ∞

0

z3F (k)α (t, z)dz . (6.41)

It is a ratio of the mass flow rate of preferred enantiomer to the sum of the mass flow rates

of preferred and counter enantiomers.

Productivity: The productivity of a continuous crystallization process can be defined as

the mass flow of solid produced per unit size. It can be described by the following equation

Pr(p)α = mass flow of solid produced (preferred) per unit volume =

m(p)S,out,α − m

(p)seeds,α

VL,α

.

(6.42)

Yield: The yield can be defined as the ratio of the mass flow of solid target particles (p)

produced over mass flow of this enantiomer in the feed solution introduced

Y (p)α =

m(p)S,out,α − m

(p)seeds

m(p)L,in,c,α

, α ∈ A,B . (6.43)

Mean crystal size: Besides purity, the crystal size also plays an important role in most

industrial applications. For that reason, the mean crystal size of the preferred enantiomer

is instructive. It is defined as:

z(p)α =

µ(p)1,α

µ(p)0,α

, (6.44)

where µ(p)1,α is total length of crystals and µ

(p)0,α is total number of crystals in the correspond-

ing crystallizers (c.f. Eq. (6.7)).

141

In the following three test problems, the minimum and maximum crystal sizes of interest

are taken as zmin = 1.0×10−10 m and zmax = 1.0×10−2 m, respectively. The corresponding

computational domain is subdivided into 200 grid points. The remaining parameters are

given in Table 6.1 and are assumed to be the same for both crystallizers. Moreover, the

volumes of both tanks and liquid volumetric flow rates are taken the same. The physico-

chemical parameters correspond to the enantiomers of the amino acid threonine and the

solvent water, see [17, 18]. The temperatures of both crystallizers were assumed to be

constant at 33 oC.

The mean residence time of the liquid phase was kept fixed at τL,α = 112.41 min in both

tanks, while the residence time of the solid phase was either τS,α = τL,α or τSτL,α= 2τL,α. In

real process, it may be difficult to decouple the residence times of liquid and solid phases.

Thus, it is most likely to set τS,α = τL,α. However, a larger residence times of the solid

phase compared to the liquid phase could eventually be realized by installing a filter for

the solid particles at the outlet to allow a longer residence τS,α of the solid phase. In the

numerical simulations, the steady state was identified when the relative deviation error in

the supersaturation was below 5 × 10−7.

6.3.1 Numerical Test Problem

This case study deals with the continuous enantioselective preferential crystallization pro-

cess in two-coupled vessels under isothermal condition. Both vessels are seeded with the

seeds of corresponding preferred enantiomers. The seeds distributions are taken to be the

same in both crystallizers

F(p)seeds,α =

a3

zAa,α

exp

(

−0.5

(

1

a4

ln

(

z

a5

))2)

, (6.45)

142

where a4, b4 and c4 are constants and there values are given in the Table 6.1. The normal-

ization factor is defined as

Aa,α =kvρS

m(p)seeds,α

µ(p)3,α . (6.46)

The initial masses of the preferred and counter enantiomers in each crystallizer are taken

as m(p)0,α = m

(c)0,α = 0.0478 kg for α ∈ A,B.

Case I: Continuous seeding without fines dissolution

In this case both vessels are seeded continually and the fines dissolution units are not

connected to them. The simulation results at steady state conditions are shown in Tables

6.2 and 6.3 for both single and coupled continuous crystallization models at different mass

flows of seeds and ratios of residence times τs,α/τL,α. The results of single crystallizer are

taken from the recent article by [66].

Moreover, Figures 6.2-6.5 show the comparison of results for single and coupled crystallizers

at different mass flow-rates of seeds and different ratios of residence times. In the coupled

case, only the results of tank A are presented which are exactly the same for tank B due

to the same operating conditions. It is evident from tables and figures that an increase

in the ratio of solid to the liquid residence times, τs,α/τL,α, reduces the purity of preferred

enantiomer due to the production of increasing amount of counter enantiomer as an im-

purity, while the productivity, yield, and mean crystal size of the preferred enantiomer are

enhanced. As τs,α increases, the crystals of the preferred enantiomer have more time to

grow and, hence, the overall productivity, yield and the mean crystal size are improved.

The productivity, yield and investment of seeds increases by increasing the mass flow rate

of seeds. It is understood that large amount of seeding produces more crystals and, thus,

enhances productivity. However, the mean crystal size reduces on increasing the mass of

seeds because the existing supersaturation is consumed by large number of crystals. On the

143

other hand, by low investment of seeds, the available supersaturation is consumed by only a

fewer crystals to obtain larger mean size of the crystals. Finally, the arrival of steady-state

delays when the ratio of residence times is larger and arrives earlier on increasing the mass

flow of seeds.

A comparison of the results for single and coupled crystallizers shows that all goal functions

have improved values in the coupled case. Thus, coupled process not only improves the

product quality but also crystallizes both enantiomers simultaneously in separate crystal-

lizers. This is a big advantage of the couple process over the decoupled (single) one.

Case II: Continuous seeding with fines dissolution

In this case both vessels are connected with fines dissolution units and are seeded continu-

ously. Fines particles below the critical size zcrit = 6.0×10−4 m are removed from the vessel

along with the solution into the recycle unit (the pipe). The selection (death) function is

given as

ψ(z) =

0.6 if z ≤ zcrit ,0 otherwise .

(6.47)

It is assumed that all particles in the dissolution unit are dissolved before re-entering the

solution to the vessel. Tables 6.4 and 6.5 show the results of single and coupled crystalliz-

ers with and without fines dissolution. Moreover, Figures 6.7-6.10 show the comparison of

results for single and coupled crystallizers at different mass flow rates of seeds and different

ratios of residence times. It can be seen that fines dissolution has improved the mean crys-

tal size of the crystals. However, effects on other goal functions are negligible. The results

of the figures reflect the same behavior observed in the tables. Moreover, the advantages of

coupled process are clearly visible these figures and tables. The results show that all goal

function have improved values in the coupled case. Thus, coupled process has enhanced

the process potential and product quality.

144

Case III: Periodic seeding without fines dissolution

In this case continuous crystallization process along with the periodic seeding is investi-

gated for both single and coupled crystallizers. The model equations are exactly the same

to Case I for continuous seeding. The numerical results at steady-state level are given in

Tables 6.6 and 6.7 for single and coupled crystallizers at different ratios of residence times.

In these tables ton represents the time at which seeding is switched on and toff denotes the

period of time at which seeding is switched off. The seed flow rate varies compared to the

continuous seeding.

A schematic diagram of the periodic seeding is shown in Figure 6.6. In that figure, the

shaded regions represent the seeding times (ton) and the non-shaded regions depict the pe-

riods when seeding is switched off (toff ). In both tables it can be observed that an increase

in τS reduces the purity of the preferred enantiomer. As residence time increases, more

nuclei of the counter enantiomer are produced as an impurity of the product and, thus, the

purity is lower. It can also be observed that the purity has decreasing behavior for increas-

ing toff . On the other hand, as τS increases the crystals of the preferred enantiomer get

more time to grow and, thus, the overall productivity and yield increases. Additionally, the

productivity and yield are decreased by increasing toff . It is obvious that when seeding is

switched off for a longer time, the mean crystal size will increase by increasing the residence

time τS due to longer stay of seeds in the crystallizer and because of utilizing the existing

supersaturation by fewer crystals. The mean crystal size also increases by increasing toff

for all considered residence times due to the utilization of existing supersaturation by fewer

crystals. It is also evident from the tables that the seeds flow rate decreases by increasing

the residence time.

145

A comparison of continuous seeding case with the current periodic case shows that con-

tinuous seeding gives better purity, productivity and yield, while the mean crystal size is

smaller and investment (mass flow of seeds) remains the same in all considered residence

times. On the other hand, the periodic seeding gives larger crystals at low investment (feed

flow rate) of seeds per residence time. The arrival of steady-state time delays on increasing

τS, while keeping the liquid residence time τL fixed. This behavior is exactly similar to the

continuous seeding case. Moreover, a comparison of the results in Tables 6.6 and 6.7 shows

that all goal functions are improved in the coupled case.

Figures 6.11-6.12 show the goal functions at different periods of seeding. It can be observed

that supersaturation decreases with reducing time period of seeding, because a decrease in

toff of seeds speedup the consumption of solute mass. All figures show the same trends.

Oscillations are also visible which indicate the conditions of periodic seeding. The purity

plots show that it decreases by increasing toff . Initially, a decrease in purity is due to

increasing amount of counter enantiomer but after a certain period of time the amount of

preferred enantiomer starts increasing which improves the purity. It can also be observed

that an increase in toff produces large size crystals which is also clear from the Tables.

The behavior of productivity is also similar to the corresponding Tables. It is evident that

larger toff gives lower yield and productivity. Moreover, a temporary drop in the purity,

before reaching the steady state, is due to the stronger relative impact (negative) of pri-

mary nucleation of counter enantiomer. This effect reduces gradually when steady state

condition is approaching.

In summary, the periodic seeding diminishes the productivity and yield, while improves

the mean crystal size. The purity is slightly reduced with periodic seeding (approx. 0.1%).

146

Moreover, the investment of seed crystals is reduced. In overall, the periodic seeding

has improved the process and productivity, if one carefully compare the investment and

productivity. In other words, more product is achieved at low investment. Once again,

the coupled process has improved the values of goal functions and, thus, justify the use of

coupled process.

147

Table 6.1: Physicochemical parameters of threonine-water system ([17]).

Parameters Symbols Value UnitVolume shape factor kv 0.122 [−]Density of solid phase ρS 1250 [kg/m3]Crystal growth rate exponent kg,0 2.98 × 109 [m/min]Nucleation rate exponent kb,sec,0 2.38 × 1026 [m−3/min]Volume of the liquid phase VL,α 4.496 × 10−4 [m3]

Volumetric flow rate VL,α 0.4 × 10−5 [m3/min]Volume of the pipe Vpipe,α 4.0 × 10−5 [m3]

Volumetric exchange flow rate Vexch 0.4 × 10−5 [m3/min]Mass of water mwater 0.3843 [kg]Constant for density K1 1.00023 [cm3g−1]Constant for density K2 4.68 × 10−6 [cm3g−1K−2]Constant for density K3 0.3652 [cm3g−1]Density of water ρwater 0.9947 [g cm−3]Density of fresh solution ρL,0,α 1.067 × 103 [kg m−3]Parameter for temperature dependence KT 1874.4 [K]Parameter for mass fraction dependence KW 0.290 [−]Universal gas constant R 8.314 J mol/KGrowth rate exponent g 1.1919 [−]Parameter for crystal growth aASL 2.0209 × 104 [m−1]Parameter for crystal growth dASL −4.066 × 10−1 [−]Liquid mean residence time τL,α 112.4116 [min]Activation energy for crystal growth EA,g 75540 J

mol

Activation energy for nucleation EA,b 63830 Jmol

Secondary nucleation exponent bsec 4.80 [−]Exponent for third moment nsec 3.0258 [−]

Nucleation constant kb,prim 3.847 × 10−2 [ (kg m−3)−7/3

min K]

Constant for exponential law aprim 4.304 × 10−3 [−]Seeds distribution constant a1 0.014 [−]Seeds distribution constant a2 0.0009 [m]Seeds distribution constant a3 0.288 [−]

148

Table 6.2: Case I: Goal functions of single crystallizer without FD, τL = 112.41 min.τSτL



min m3 ] Y (p) [%] z(p) [mm]

1.0 646 0.0089 99.74 0.037 3.91 1.312.0 1411 0.0044 98.95 0.043 4.54 1.541.0 676 0.0356 99.92 0.060 6.29 1.132.0 1315 0.0178 99.54 0.062 6.60 1.271.0 721 0.0623 99.94 0.067 7.08 1.082.0 1318 0.0311 99.68 0.069 7.29 1.18

Table 6.3: Case I: Goal functions of coupled crystallizers without FD, τL = 112.41 min.τS,A

τL,Atstd [min] m

(p)seed,A [ g

min] Pu

(p)A [%] Pr

(p)A [ kg

min m3 ] Y(p)A [%] z

(p)A [mm]

1.0 642 0.0089 99.99 0.045 4.79 1.352.0 1410 0.0044 99.99 0.054 5.71 1.621.0 676 0.0356 99.99 0.079 8.32 1.182.0 1313 0.0178 99.99 0.084 8.90 1.331.0 720 0.0623 99.99 0.091 9.62 1.112.0 1316 0.0311 99.99 0.095 10.02 1.24

Table 6.4: Case II: Results of single crystallizer with FD: FD (yes), without FD (no).τSτL



min m3 ] Y (p) [%] z(p) [mm]

1.0 no 676 0.0356 99.92 0.060 6.29 1.131.0 yes 648 0.0356 99.99 0.060 6.25 1.152.0 no 1315 0.0178 99.54 0.062 6.60 1.272.0 yes 1303 0.0178 99.97 0.062 6.56 1.29

Table 6.5: Case II: Results of coupled case with FD: FD (yes) and without FD (no).τS,A

τL,AFD tstd [min] m

(p)seed,A [ g

min] Pu

(p)A [%] Pr

(p)A [ kg

min m3 ] Y(p)A [%] z

(p)A [mm]

1.0 no 676 0.0356 99.99 0.079 8.32 1.181.0 yes 650 0.0356 99.99 0.078 8.27 1.202.0 no 1314 0.0178 99.99 0.084 8.90 1.332.0 yes 1301 0.0178 99.99 0.084 8.85 1.36

149

Tab

le6.6:

Case

III:Perio

dic

seedin

gin

single

crystallizer:

m(p

)seed

s=

4g,τL

=112.4

min

.τS

τL

seedin

gty

pe

tstd

[min

]ton

[min

]toff

[min

]m

(p)

seed[

gm

in]

Pu

(p)[%

]P

r(p

)[kg

min

m3 ]

Y(p

)[%]

z(p

)[mm

]

1.0P

67010

50.0244

99.870.025

2.591.18

1.0P

67010

100.0189

99.840.0071

0.721.23

1.0P

67020

100.0252

99.870.025

2.611.19

1.0C

676-

-0.0356

99.920.060

6.291.13

2.0P

132010

50.0125

99.410.04

4.21.34

2.0P

132110

100.0089

99.320.028

2.981.40

2.0P

132120

100.0102

99.40.0372

3.931.34

2.0C

1315-

-0.0178

99.540.062

6.601.27

Tab

le6.7:

Case

III:Perio

dic

seedin

gin

coupled

case:m

(p)

seeds=

4g,τL

=112.4

min

.τS

,A

τL

,Aseed

ing

type

tstd

[min

]ton

[min

]toff

[min

]m

(p)

seed,A

[g

min

]P

u(p

)A

[%]

Pr(p

)A

[kg

min

m3 ]

Y(p

)A

[%]

z(p

)A

[mm

]

1.0P

67210

50.0244

99.990.041

4.291.24

1.0P

67210

100.0189

99.990.021

2.1671.27

1.0P

67220

100.0252

99.990.04

4.261.24

1.0C

678-

-0.0356

99.990.079

8.321.18

2.0P

131810

50.0125

99.990.062

6.551.42

2.0P

131810

100.0089

99.990.049

5.191.49

2.0P

131920

100.0102

99.990.062

6.531.43

2.0C

1312-

-0.0178

99.990.084

8.901.33

150

0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 3000−15

−10

−5

0

5

10

time [min]

yiel

d [%

]




Figure 6.2: Case I: LS: Single crystallizer results. RS: Coupled crystallizer results.

151

0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time[min]

prod

uctiv

ity [k

g/m

in m

3 ]




0 500 1000 1500 2000 2500 3000−0.15

−0.1

−0.05

0

0.05

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 30000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

time [min]

mea

n cr

ysta

l siz

e [m

m]




0 500 1000 1500 2000 2500 30000.9

1

1.1

1.2

1.3

1.4

time [min]

mea

n cr

ysta

l siz

e [m

m]





152

0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 300098.8

99

99.2

99.4

99.6

99.8

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 3000−10

−5

0

5

10

time [min]

yiel

d [%

]





153

0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time [min]

prod

uctiv

ity [k

g/m

in m

3 ]




0 500 1000 1500 2000 2500 3000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 30000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

time [min]

mea

n cr

ysta

l siz

e [m

m]




0 500 1000 1500 2000 2500 30000.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

time [min]

mea

n cr

ysta

l siz

e [m

m]





seeding off

seeding on

time [min]

mas

s flo

w o

f see

ds [g

/min

]

Figure 6.6: Schematic diagram for periodic seeding.

154

0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 300099.98

99.985

99.99

99.995

100

time [min]

purit

y [%

] m(p)seeds = 0.0089 g/min



0 500 1000 1500 2000 2500 300099.98

99.985

99.99

99.995

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 3000−15

−10

−5

0

5

10

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 3000−15

−10

−5

0

5

10

time [min]

yiel

d [%

]




Figure 6.7: Case II: LS: Single crystallizer results. RS: Coupled crystallizer results.

155

0 1000 2000 3000−0.15

−0.1

−0.05

0

0.05

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 3000−0.15

−0.1

−0.05

0

0.05

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 30000.9

1

1.1

1.2

1.3

1.4

time [min]

mea

n cr

ysta

l siz

e [m

m]




0 500 1000 1500 2000 2500 30000.9

1

1.1

1.2

1.3

1.4

time [min]

mea

n cr

ysta

l siz

e [m

m]





156

0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 30001.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]




0 500 1000 1500 2000 2500 300099.93

99.94

99.95

99.96

99.97

99.98

99.99

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 300099.93

99.94

99.95

99.96

99.97

99.98

99.99

100

time [min]

purit

y [%

]




0 500 1000 1500 2000 2500 3000−10

−5

0

5

10

time [min]

yiel

d [%

]




0 500 1000 1500 2000 2500 3000−10

−5

0

5

10

time [min]

yiel

d [%

]





157

0 500 1000 1500 2000 2500 3000−0.1

−0.05

0

0.05

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 3000−0.1

−0.05

0

0.05

0.1

time [min]

prod

uctiv

ity [%

]




0 500 1000 1500 2000 2500 30000.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

time t [min]

mea

n cr

ysta

l siz

e [m

m]




0 500 1000 1500 2000 2500 30000.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

time [min]

mea

n cr

ysta

l siz

e [m

m]





158

0 500 1000 1500 2000 2500 30001.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 30001.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

time [min]

supe

rsat

urat

ion

[−]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 300099.8

99.85

99.9

99.95

100

time [min]

purit

y [%

]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 300099.8

99.85

99.9

99.95

100

time [min]

purit

y [%

]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 3000−10

−5

0

5

time [min]

yiel

d [%

]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 3000−10

−5

0

5

time [min]

yiel

d [%

]

ton

=10, toff

=5

ton

=10, toff

=10

Figure 6.11: Case III: LS: Single crystallizer results. RS: Coupled crystallizer results.

159

0 500 1000 1500 2000 2500 3000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

time [min]

prod

uctiv

ity [%

]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 3000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

time [min]

prod

uctiv

ity [%

]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 30000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

time [min]

mea

n cr

ysta

l siz

e [m

m]

ton

=10, toff

=5

ton

=10, toff

=10

0 500 1000 1500 2000 2500 30000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

time [min]

mea

n cr

ysta

l siz

e [m

m]

ton

=10, toff

=5

ton

=10, toff

=10

Figure 6.12: Case III: LS: Single crystallizer results. RS: Coupled crystallizer results.

160

161

Chapter 7

Conclusion and Future Recommendations

7.1 Conclusion

In this thesis, batch and continuous crystallization models were developed and numerically

investigated. Especially, models were derived for simulating batch and continuous enan-

tioselective preferential crystallization processes in single and coupled crystallizers. The

effects of nucleation, growth, fines dissolution, seeding strategies and residence time char-

acteristics on the CSD were analyzed.

A TVB Discontinuous Galerkin (DG) finite element method was proposed for solving batch

crystallization models. The method is explicit in time and, hence, can be coupled with

a high-order TVB Runge-Kutta time discretization method. The local projection limiter

avoids spurious oscillations in the vicinity of a discontinuity that does not affect the accu-

racy of the scheme in smooth regions. Thus, the limiter preserves the non-negativity of the

CSD and assures the TVB property of the scheme. In contrast to the finite difference and

finite volume methods, the high order accuracy of the scheme can be achieved by using

more information within a cell rather than widening the mesh stencil. For that reason,

the method can be used to boundary cells without loosing accuracy and the application of

the boundary conditions are easier. On the other hand, the high resolution finite volume

schemes (HR-FVS) are not applicable up to the boundary cells and, hence, the order of

the scheme has to be lowered in the boundary cells. This produces significant errors in the

part of distribution coming from nucleation as illustrated in the numerical test problems

of Chapter 3. Therefore, the DG-scheme is more flexible and accurate as compared to the

HR-FVS. The obtained results support the process design and optimization.

Mathematical models were derived for batch preferential single and coupled crystallizers

connected with pipes (fines dissolution units). The models were further elaborated by

162

considering the isothermal and non-isothermal conditions. Firstly, the crystallization of

preferred enantiomer was assumed to take place in a single crystallizer equipped with a

fines dissolution unit. In this setup, the extracted solution was sieved by filters and was

supposed to be free of bigger crystals. Therefore, only small particles were withdrawn

to the fines dissolution loop. In order to assure a crystal-free liquid exchange, the with-

drawn liquid in the fines dissolution loop was heated for dissolving small particles. Before

re-entering into the crystallizer, this liquid was cooled again. The breakage and agglom-

eration processes were not considered in this study. Secondly, the model was extended

for a coupled batch preferential crystallization process with isothermal and non-isothermal

conditions. In this setup, the crystallization of two enantiomers was assumed to take place

in two separate crystallizers which were coupled through exchange pipes. There were two

main advantages of considering two coupled crystallizers. The first one was that both

enantiomers were crystallized out at the same time in separate crystallizers. Secondly,

because of liquid exchange between the crystallizers, the growth process was enhanced in

both crystallizers and, thus, crystals of large mean sizes were obtained. Both HR-FVS

and DG-method were used to solve these models. The DG-method was applied for the

first time to model such processes. The results of both schemes were found comparable.

However, the DG-scheme produces better approximations in the part of the distribution

coming from nucleation. Moreover, the results are in good agreement with the experimental

results of our collaborating colleagues from the Max Planck Institute Magdeburg, Germany.

For the first time continuous preferential crystallization was investigated and the effects of

different seeding strategies and residence time characteristics were analyzed on the dynam-

ics of a Mixed Suspension Mixed Product Removal (MSMPR) crystallizer equipped with

a fines dissolution unit. The fines dissolution was included as recycle streams around the

MSMPR crystallizer. Moreover, primary heterogeneous and secondary nucleation mech-

163

anisms along with size-dependent growth rates were taken into account. The model was

then extended for a coupled continuous preferential crystallization process. In this setup,

the crystallization of two enantiomers was assumed to take place in two separate crystal-

lizers which were coupled through exchange pipes. To judge the quality of the process

some goal functions were used, such as purity, productivity, yield and mean crystal size of

the preferred enantiomer. The HR-FVS and DG-method were applied to solve the models.

The results of the DG-method for the single continuous preferential crystallization model

were compared with the results of HR-FVS. The supersaturation and mean crystal size

were reduced on increasing the mass flow of seeds, while productivity, yield and purity

were improved. The errors in mass balances were found below 0.15% for a wide range

of mass flow of seeds. The periodic seeding slightly diminished the purity, productivity

and yield but improved the mean crystal size and the investment (mass flow) of seeds was

also smaller. The fines dissolution improved the mean crystal size and purity but slightly

reduced the productivity and yield. These results could be used to find the optimum oper-

ating conditions for improving the product quality and for reducing the operational cost of

continuous preferential crystallization. For the case study carried out in a single crystallizer

(resolution of an amino acid enantiomers) a production of 100 kg/(day ·m3) appears to be

feasible.

7.2 Future Recommendations

In this dissertation, a DG-method was used to solve the models of different crystallization

processes. However, the accuracy of the DG-scheme can be further improved by using basis

functions of higher order and by using better slope limiters, for example WENO limiters

[67]. In future, we also intend to implement adaptive mesh refinement techniques in the

current DG-scheme. This technique is particularly effective when the PBEs are coupled

with computational fluid dynamic (CFD) codes for non-ideally mixed systems. Moreover,

164

many crystallization processes require more than one property variable to characterize the

crystals. Besides that, the multivariate population balance model (PBM) has various ap-

plications in many scientific, medical, and industrial research disciplines. Thus, it will be

interesting to extend the current DG-scheme for solving bivariate and multivariate PBMs.

The current models can be further extended by including the attrition, aggregation and

breakage phenomena during the crystallization process. In such cases, one has to solve the

resultant integro-differential equations by using HR-FVS or DG-scheme. These schemes

are expensive for such simultaneous processes, especially on regular grids. However, the

use of the adaptive mesh refinement technique and the combination of the current schemes

with other schemes can considerably reduce the overall computational cost for the same

desired accuracy.

In the current study of preferential crystallization, a model system of amino acid-H2O

was considered for our computational study. In this model no significant interdependence

between the preferred and counter enantiomers on crystallization kinetics has been ob-

served. The same was also experimentally observed by the process engineering group of

Max Planck Institute Magdeburg, Germany. However, in their experiments for the man-

delic acid-H2O model system, they observed a strong influence of the counter-enantiomer

on the growth rate of the preferred-enantiomer, see [43] for further details. Moreover, it

can be shown that in the latter case the counter-enantiomer may even change growth rates

of particular crystal faces which results in different crystal shapes, see Figure 7.1. Hence,

a detailed study on the influence of the counter-enantiomer on the growth rate and mor-

phology of the seeded enantiomer is required. To model such processes and to study the

different evolution of crystal faces a multidimensional population balance model has to be

derived along with correct growth and nucleation rate kinetics whose results matches with

165

the experimental results. This in turn also leads inevitably to a higher requirement of the

efficient numerical discretization techniques. Currently, work is in progress in this direction.

Furthermore, this work can be extended to the study and design of control strategies in

crystallization processes. In this direction, it will be more interesting to concentrate on the

less restrictive models of crystallization processes and to derive control strategies for both

batch and continuous crystallizers.

Also, this work can be extended to the study when temperature varies in the vessels, then

the solutions may well be inhomogeneous and will require an energy equation determining

the temperature and crystallization rates for the p and c crystals.

Finally, the current schemes can be extended to other particulate processes, such as pre-

cipitation, polymerization, food processes, pollutant formation in flames, particle size dis-

tribution of crushed material and rain drops, and growth of microbial and cell populations.

166

Figure 7.1: Left: without counter enantiomer; Right: with counter enantiomer.

167

168

Chapter 8

References

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