INDUSTRIAL CRYSTALLIZATION Process Simulation Analysis and Design The Plenum Chemical Engineering Series

Series Editor: Dan Luss, University of Houston, Houston, Texas

COAL COMBUSTION AND GASIFICATION L. Douglas Smoot and Philip J. Smith ENGINEERING FLOW AND HEAT EXCHANGE Octave Levenspiel INDUSTRIAL CRYSTALLIZATION: Process Simulation Analysis and Design Narayan S. Tavare REACTION ENGINEERING OF STEP GROWTH POLYMERIZATION Santosh K. Gupta and Anil Kumar THE STRUCTURE AND REACTION PROCESSES OF COAL K. Lee Smith, L. Douglas Smoot, Thomas H. Fletcher, and Ronald J. Pugmire TRANSPORT MECHANISMS IN MEMBRANE SEPARATION PROCESSES J. G. A. Bitter

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. INDUSTRIAL CRYSTALLIZATION Process Simulation Analysis and Design

Narayan S. Tavare University of Manchester Institute of Science and Technology (UMIST) Manchester, United Kingdom

Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data

Tavare, Narayan S. Industrial crystallization : process simulation analysis and design / Narayan S. Tavare. p. cm. — (The Plenum chemical engineering series) Includes bibliographical references and index. ISBN 978-1-4899-0235-1 1. Crystallization—Industrial applications. I. Title. II. Series. TP156.C7T38 1994 660*.284298--dc20 94-46218 CIP

ISBN 978-1-4899-0235-1 ISBN 978-1-4899-0233-7 (eBook) DOI 10.1007/978-1-4899-0233-7

© 1995 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1995 Softcover reprint of the hardcover 1st edition 1995

10 987654321

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher To My Late Parents PREFACE

There has been a worldwide upsurge in the attention paid and research effort devoted to the field of industrial crystallization, resulting in a publication explo• sion. The initial burst of research activity resulted in rationalizing, modeling, and predicting the crystal size distribution using the concept of population balance of crystals, with their measurable property--crystal size-as the axis. Since then, a considerable volume of information concerning both theoretical and experimen• tal work has appeared throughout the periodical literature. Over the past decades, enormous advances in chemical engineering science, which evolved from the concept of unit operations via the more basic approach of transport phenomena, have paved the way for the analysis and performance evaluation of crystallization configurations. This book presents both the theoretical and experimental material using the following rather unconventional approach. Outlining the more impor• tant aspects of the science and technology of industrial crystallization, together with some closely related topics, I treat the subject matter in as general a manner as possible, so as to emphasize the unit operational nature of the subject and also to keep in close link with the chemical reaction engineering approach. Particular attention has been paid to the more recently developed tech• niques of process simulation and data reduction analysis. Methods of deducing design-oriented crystallization kinetics from experimental responses and their application in process design and performance assessment of industrial crystalliz• ers are considered in great detail. Some of the material and approaches will be valuable in many other interdisciplinary areas involving particulate and solid-liq• uid systems. Crystallization is also becoming increasingly important in many other rapidly expanding areas such as biotechnology, mineral processing, waste treatment, pollution abatement, energy storage, new construction materials, and electronic chemicals.

vii Yin PREFACE

The book takes the balanced functional approach in critically reviewing the research literature. I include several specially designed and solved problems based on real practical situations to illustrate theoretical developments. To bridge the gap between theory and industrial practice, I use an approach analogous to that employed in the field of chemical reaction engineering to present the mate• rial in a different format. The recent switch from large tonnage bulk commodity products manufactured in continuous units to fine chemicals and special-effect high-added-value products manufactured in batch or semibatch units is reflected herein. Emphasis on batch and semibatch operating modes is given in this vol• ume. Several new concepts and techniques employed in process simulation and identification analysis are also presented. Separate chapters on growth rate dis• persion and mixing phenomena are included. Crystallizer design, scaleup, and operation are discussed. The methods and philosophies presented are assessed by the author for best possible applications in practical situations. The volume will, I hope, be a welcome and useful addition to the crystallization literature, of service to practitioners such as process engineers, chemists, technologists and research• ers. It should prove of interest to students of chemical engineering and chemical technology who require a wider appreciation of the subject at an advanced level.

Acknowledgments

I wish to thank Professor Dan Luss, who supported my proposal of writing this volume. It is my pleasure to acknowledge my debt to innumerable friends and colleagues with whom I have worked. Particular thanks are due to Professors Madhav Chivate, John Mullin, and John Garside, who introduced me to the world of industrial crystallization. I also thank Edesio Colonia, Ashok Dixit, Anand Patwardhan, Sylvia Petherick, and Deborah Swift for their assistance in the preparation of the manuscript, as well as the staff of Plenum. Last, but not least, I am indebted to my own family, my wife Vasanti, my first son Aniket, and newly arrived second son Abhijeet, who patiently suffered the inevitable reduc• tion in my attention during the preparation of the manuscript. It is, needless to say, due to Vasanti's understanding, support, and sacrifice that this work is complete. CONTENTS

List of Symbols ...... xv

Chapter 1: Introduction ...... 1 1.1. Crystallization ...... 1 1.2. Crystallization Processes...... 1 1.2.1. Crystallization and Precipitation...... 4 1.3. Crystallizer Systems...... 4 1.4. Further Reading...... 8

Chapter 2: Thermodynamic Aspects ...... 9 2.1. Introduction...... 9 2.2. Solubility Relations...... 9 2.2.1. Theoretical Crystal Yield ...... 12 2.3. Supersaturation...... 14 2.4. Solution Stability...... 15 2.4.1. Experimental Determination of Metastable Zone Width . . . . 18 2.5. Phase Equilibria...... 19 2.6 Two-Component Systems...... 19 2.6.1. Simple Eutectic...... 20 2.6.2. Compound Formation...... 23 2.6.3. Solid Solutions...... 25 2.7. Three-Component Systems...... 29 2.7.1. Eutectic Formation ...... , ...... 30 2.7.2. Aqueous Solutions ...... , 37 2.8. Heats of Solution and Crystallization ...... , 44 2.9. Solubility Product ...... 45

ix CONTENTS

2.10. Problems...... 52 2.11. References...... 56 2.12. Further Reading...... 56

Chapter 3: Crystallization Kinetics ...... 57 3.1. Introduction...... 57 3.2. Crystal Nucleation...... 58 3.2.1. Primary Nucleation...... 58 3.2.2. Induction Period...... 60 3.2.3. Secondary Nucleation...... 64 3.3. Crystal Growth...... 65 3.3.1. Surface Integration Kinetics...... 67 3.3.2. Determination of Surface Integration Kinetics...... 67 3.3.3. Effectiveness Factor ...... 71 3.4. References...... 77

Chapter 4: Crystal Size Distribution ...... 79 4.1. Introduction...... 79 4.2. Crystal Size Distribution Functions...... 79 4.2.1. Representation of Functions ...... 80 4.3. The Population Balance...... 82 4.3.1. Continuity Equation for Crystal Population...... 83 4.3.2. Moment Transformation of Population Balance...... 85 4.3.3. Crystal Size Distribution from Moments ...... 86 4.3.4. The Other Coordinate Systems...... 87 4.4. Summary ...... 89 4.5. Problems...... 89 4.6. References...... 92 4.7. Further Reading...... 92

Chapter 5: Batch Crystallizer ...... 93 5.1. Introduction...... 93 5.2. Process Representation...... 95 5.2.1. Solution-Side Information...... 95 5.2.2. Solid-Side Information...... 98 5.2.3. The Batch Population Density Function...... 102 5.3. Operating Modes ...... 107 5.3.1. Cooling Crystallizers ...... 107 5.3.2. Evaporative Crystallizers ...... 109 5.3.3. Dilution Crystallizers ...... 112 CONTENTS xi

5.3.4. Reactive Crystallizers ...... 118 5.3.5. Crystallization from Previously Supersaturated Solutions .... 124 5.4. Process Analysis ...... 129 5.5. Summary ...... 133 5.6. Problems ...... 134 5.7. References ...... 137

Chapter 6: Characterization of Crystallization Kinetics from Batch Experiments ...... 141 6.1. Introduction ...... 141 6.2. Method ofIsolation ...... 141 6.2.1. Crystal Growth ...... 141 6.2.2. Nucleation...... 159 6.3. Simultaneous Estimation ...... 160 6.3.1. Solution-Side Information ...... 161 6.3.2. Solid-Side Information ...... 175 6.4. Consistency Checks ...... 188 6.5. Problems...... 192 6.6 References...... 195

Chapter 7: Semibatch Crystallizer ...... 199 7.1. Introduction...... 199 7.2. Reactive Precipitation Systems ...... 201 7.3. Semibatch Crystallization Studies ...... 213 7.4. Ostwald Ripening ...... 220 7.5. Characterization of Rate Processes ...... 227 7.6. Agglomerating Reactive Precipitation Systems...... 229 7.6.1. Silica Precipitation...... 230 7.6.2. Population Balance in Crystal Volume Coordinate System .. 232 7.6.3. Moment Transformation ...... 234 7.6.4. Crystallization and Agglomeration Kinetics ...... 237 7.7. Problems ...... 242 7.8. References...... 243

Chapter 8: Continuous CrystaUizers ...... 247 8.1. Introduction ...... 247 8.2. Continuous MSMPR Crystallizer ...... 248 8.2.1. Steady-State Population Balance...... 248 8.2.2. Significance of the Model Parameters ...... 250 8.2.3. Moments of Product Crystal Size Distribution ...... 252 xli CONTENTS

8.2.4. The Steady-State Mass Balance ...... 255 8.2.5. Washout Curves ...... 256 8.3. Process Modifications...... 258 8.3.1. Size-Dependent Growth Rate ...... 259 8.3.2. Growth Rate Dispersion...... 260 8.3.3. Size-Dependent Residence Time Distribution ...... 262 8.4. Dynamics...... 266 8.4.1. Transients of an MSMPR Crystallizer ...... 266 8.4.2. CSD Dynamics and Control ...... 269 8.5. Agglomeration ...... , 273 8.5.1. Population Balance in Crystal Volume Coordinates ...... , 273 8.5.2. Moment Transformation...... 275 8.5.3. Analytical Solution...... 276 8.5.4. Population Density Plots...... 277 8.6. Plug Flow Crystallizer...... 279 8.6.1. Population Balance and Moments Transformation .... , .... 279 8.7. Process Identification...... 280 8.7.1. Crystallization Kinetics...... 280 8.7.2. Crystallization and Agglomeration Kinetics ...... , 288 8.8. Problems...... 292 8.9. References ...... 297

Chapter 9: Growth Rate Dispersion ...... 303 9.1. Introduction...... 303 9.2. Experimental Evidence ...... 303 9.3. The Dispersion Model ...... , 307 9.4. Parameter Characterization...... 309 9.4.1. Time Domain Methods ...... 309 9.4.2. Laplace Transform Domain Methods ...... "...... 317 9.4.3. Frequency Domain Methods...... 324 9.4.4. Growth Rate Activity Distribution...... 329 9.5. Continuous Crystallizers ...... 337 9.5.1. Size-Dependent Growth Rates ...... 337 9.5.2. Growth Rate Dispersion...... 337 9.6. Concluding Remarks...... 342 9.7. Problems ...... 346 9.8. References...... 349

Chapter 10: Mixing ...... 353 10.1. Introduction ...... 353 CONTENTS xiii

10.2. Macromixing ...... 355 10.2.1. Residence Time Distribution ...... 355 10.2.2. Laplace Transform Domain Formulation...... 359 10.2.3. Flow Models...... 359 10.2.3. Multistage Configurations ...... " 363 10.2.5. Macromixing Models ...... 370 10.3. Micromixing ...... 372 10.3.1. Limits of Micro mixing ...... 372 10.3.2. Models for Limits of Micromixing ...... 374 10.3.3. Degree of Segregation ...... 386 10.3.4. Mixing Space ...... 388 10.3.5. Micromixing Models ...... 391 10.4. Elurian Mixing ...... 397 10.4.1. Crystal Suspensions ...... 397 10.4.2. Mass Transfer ...... 401 10.4.3. Heat Transfer ...... 402 10.5. Problems ...... 403 10.6 References ...... 406

Chapter 11: Crystallizer Design and Operation ...... 415 11.1. Introduction ...... 415 11.2. Crystallizer Selection...... 417 11.3. Design Illustrations ...... 418 11.4. Crystallizer Scaleup ...... 454 11.4.1. Heresies of Scaleup ...... 455 11.4.2. Approaches to Scaleup ...... 455 11.4.3. Crystallizer Design Interactions ...... 457 11.4.4. Research Scenario ...... 458 11.4.5. Planning the Work ...... 459 11.5. Conclusions ...... 459 11.6. Problems ...... 460 11. 7 . References and Further Reading...... 463

Chapter 12: Crystallization Techniques and Phenomena ...... 465 12.1. Introduction ...... 465 12.2. Adductive or Extractive Crystallization ...... 465 12.3. Dissociation Extractive Crystallization ...... 468 12.4. Hydrotropy ...... 472 12.5. Freeze Crystallization ...... 478 12.6. Emulsion Crystallization ...... 480 I1v CONTENTS

12.7. Solid Phase Reactions ...... 482 12.8. Encrustation...... 485 12.9. Crystal Habit Modification...... 486 12.10. Phase Transformations ...... 488 12.11. Summary ...... 490 12.12. References...... 491

Appendix ...... 501 Units ...... 501

Name Index...... 505

Subject Index ...... 515 LIST OF SYMBOLS

a Diameter of ion, m a Width parameter in gamma distribution a Constant a Temperature coefficient in linear solubility relation, kg/kg K a Coefficient in growth rate correlation a' Empirical constant (in Eq. (8.53), 11m) ai Coefficients in polynomial aA Activity of A A Crystal surface area, m2/kg; flow area, cross-sectional area of crystal- lizer, m2 A Preexponential factor (Eq. (3.1» A 1 Modified surface area, m3/s kg Ah Heat transfer area, m2 AT Total crystal surface area, m2/kg solvent AR Amplitude ratio [Eqs. (9.39),(9.41),(9.42)] b Nucleation order b Size parameter in gamma distribution, equal to mode b True estimate of ~ b' Empirical constant bi Coefficients in polynomial B Nucleation rate, no.lkg solvent s, no.1L s B Birth rate function, no.lm3 kg solvent s, no.lm3 L s B Brine feed flowrate, tlh B System constant B(L) Crystal birth function at size L, no.lm3 kg solvent s, no.lm3 L s Bo Nucleation rate at L =0, no.lkg solvent s, no.lL s Bo Initial impulse of seeds, no.lkg m s

xv xvi LIST OF SYMBOLS

Birth rate function due to aggregation, no.lm3 L s Nucleation rate, kg solute/kg solvent s Nucleation rate, no.1L s, no./kg s Birth rate function, no.lm3 L s, no./m3 kg solvent s Birth rate function due to aggregation, no./m3 L s, no./m3 kg solvent s Step length bound Bed pressure, mm water c Concentration, kg/kg, kg solute (or hydrate)/kg (free) solvent, kg/L, mol/kg, mollL, wt % c* Equilibrium concentration, kg/kg, kg/kg free solvent, kg hydrate/kg solution Empirical constant, pre-exponential factor, kg solute/kg solvent Pre-exponential constant, no./kg solvent m Interfacial concentration, kg/kg Liquid phase concentration, mollL Threshold metastable concentration limit, kg/kg Specific heat, kJ/kg K, kcal/kg K Solid phase concentration, mollL Concentration driving force, kg/kg, kg solute (or hydrate)/kg (free) solvent, kg hydrate/kg solution ~co Exit solution supersaturation, kg/kg free solvent Il.cp Inlet solution or working supersaturation, kg/kg free solvent /lcmax Maximum supersaturation allowed by the system, kg/kg free solvent C Number of components C Dimensionless exit concentration to an impulse input C Fourier cosine transform of population density with respect to size

Cj Cosine Fourier transform of population density function at size L j CN Cumulative undersize number, no. CV Coefficient of variation, % CW Cumulative undersize weight percent d Order of the diffusion process d Diameter, equivalent diameter of the annulus for fluid flow, (d= 2e), m D Diameter, impeller diameter, m D Dispersion coefficient, m2/s D Diagonal scaling matrix D Death rate function, no./m3 L s D(L) Crystal death function at size L, no./m3 kg solvent s, no./m3 L s D Overall linear dissolution rate, mls Da Death rate function due to aggregation, no./m3 L s Dac Damkohler number De Dispersion number Dv Death rate function, no./m3 L s, no./m3 kg solvent s LIST OF SYMBOLS xvii

Dva Death rate function due to aggregation, no./m3 L s, no.!m3 kg solvent s DCR Downcomer resistance, mm water DC Effective growth rate diffusivity, m2/s, Ilm2/S e Annular gap (e =R - R;), m e Boiling point elevation, K, °C e Exponent of solid voidage to solid fraction (Eq. (3.13» ej Jth coordination direction E Activation energy of the rate process, kJ/mol M Activation energy, J/mol Ec Activation energy of growth process, kJ/mol E(8) Dimensionless residence time distribution function; dimensionless exit age distribution f Friction factor f Exponent of crystal size (Eq. (3.13» f Dimensionless crystal size distQ.bution, n/nv0 f Dimensionless growth rate, GI G j(L) Initial population density function at size, L, no.!m kg solvent j(L) Seed CSD as a function of size L, no.!m kg solvent j(L) Normalized one-dimensional distribution function in crystal size, L j(T) Modified dimensionless nuclei population density function F Number of degrees of freedom F Transfer function in Laplace transform and frequency domain F Ratio of surface to volume shape factor F(~) Objective function used for optimization F(L) Normalized one-dimensional distribution function in crystal size, L F(9) Dimensionless cumulative residence time distribution younger than 9; dimensionless exit concentration to a step input g Growth rate order g Acceleration due to gravity, mls2 g* Growth rate activity, J.lmls g* Average growth rate activity, J1mIs gv Free energy change of the transformation Ivolume g(T) Dimensionless nuclei population density function G Overall linear growth rate, mls G(L) Size-dependent linear crystal growth rate, J.lmls 3 Gv Overall crystal volume growth rate, m Is GD Overall linear dissolution rate, mls IlG Overall excess free energy, J IlGs Surface excess free energy, J IlGv Volume excess free energy, J h Hydrostatic head, m h Heat transfer coefficient, W/m2K mil LIST OF SYMBOLS

H Height of crystal bed, m H Enthalpy, kJ/kg H(9) Response of the system as a function of dimensionless age (or time) (Ml) Enthalpy change, kJ/mol (-Ml) Heat of crystallization, kJ/mol Relative nucleation order (i = big) Index variable 1 Impurity concentration, kg/kg 1 Ionic strength, mollL I Identity matrix 1(9) Dimensionless internal age distribution 11 (X) Modified Bessel function of the first kind of order one j Exponent of magma concentration in nucleation rate correlations j Index variable j Imaginary coefficient J Nucleation rate (Eq. (3.1 », no./kg s J Degree of segregation J Jacobian matrix JT Transpose of J k Coefficient of impurity, kglkg k Index variable k Supersaturation generation rate, kg/kg s k Stage number k Reaction rate constant, Umol s k Boltzmann constant, JIK k2 Second order reaction rate constant, kglkmol s ka Surface shape factor kb Nucleation rate constant, no./[kg s (kglkg)b+j], no./[s kg (mollkg)b] kbm Nucleation rate constant, no./[s k~ (mollkg)b] kbt Nucleation rate constant, no./[s KD] kB Nucleation rate constant, kg/[kg s (kglkg)b+j] kc Constant kd Diffusional mass transfer coefficient, m/[s (kg/kg)], kg/[m2s (kglkg)] kd Dissolution rate constant, m/s (mollkg)d, kg/[m2s (kglkg)d] kd Rate coefficient for the decomposition process, s-1 kD Constant ke Constant kg Overall linear growth rate constant, m/[ s (kglkg)g] kgm Overall linear growth rate constant m/[s (mol/kg)8] kgt Overall linear growth rate constant, m2/s K kG Overall growth rate constant, kg/[m2s (kg/kg)8] km Enzymatic reaction rate constant, s-1 LIST OF SYMBOLS xix

kn Nucleation rate constant, no./[kg s (kg/kg)n] kM Constant kNH Ammonia addition rate, kg ammonia/(kg water s) k 3 Coefficient of diluent concentration, kg/kg P kr Surface~ntegration rate constant, kgl[m2(kglkg)r] ks Constant temperature coefficient in solubility relation, kglkg K kv Volume shape factor kw Rate constant for solid phase formation, s-I, (mol/L)l-j-n s-I kR Relative nucleation rate constant, no./sl-i kg mi kr Constant cooling rate, Kls K Constant (Eq.(3.2)) K Skewness K Equilibrium constant, mol/kg Ke Solubility product (kmol/m3)2 Ke Constant to account for contraction losses Ks Solubility product Kse Stability constant KA,Kp Equilibrium constants KN Relative nucleation rate constant KR Relative nucleation coefficient, no./[kg s (mls)i(kglkg~] L Crystal size, J.lIIl, m L Characteristic crystal size or equivalent sieve size of crystals, m L Characteristic dimension, m L Liquor flowrate, tlb L Length of Couette flow device, cm Mean mass particle size, m Arithmetic mean size in normal distribution, m L' Geometric mean size in log normal distribution, m L* Gibbs-Thomson critical size, m LI Modified total length, m2/s kg L2,1 Length weighted average size, m, IJ.Ill Le Critical size of a nucleus, m Lm Mass mode size, m Ln Nucleus size, m LM Product mass median size, m IlL Difference between successive sieve or channel size, m, IJ.Ill Le Lewis number LT Laplace transform of Constant Exponent of stirrer speed in a rate correlation jth moment of population distribution with respect to size about origin, no. milkg solvent xx LIST OF SYMBOLS m(L) Mass density distribution function of crystal size M Concentration of diluent, kg diluent/kg (diluent + solvent) M Solid concentration, kg/kg M(L) Solid concentration between Land L + dL, kg/kg M Molecular weight, kg/kmol MA Molecular weight of A, kg/kmol MT Suspension density, kg crystal/kg solvent, gIL, kg/m3, kg/kg, kmollkg MT Dimensionless solid deposition rate

M j ith moment of the distribution M"'P nth weighted moment at p in Laplace domain with respect to time n Population density, no'/m kg solvent, no'/m L, no'/~m mL ii (p, L) Laplace transform of response population density with respect to time n(t, L) Population density function at size L and time t, no'/m kg solvent ii (t, s) Laplace transform of response population density with respect to size ii (t, iro) Fourier transform of response population density with respect to size ii (t, L; g*)Population density of subpopulation with growth rate activity, g* at size L ii (iro, L) Fourier transform of response population density with respect to time I ii(iro, L)I Modulus representation of (iro, L), i. e., magnitude ratio Lii (iro, L)Argument representation of (iro, L), i. e., phase shift ii (t, s) Laplace transform of response population density with respect to size ii (t, iro) Fourier transform of response population density with respect to size nO Nuclei population density at zero size, no'/m kg solvent nv Crystal volume population density, no'/m3 L, no'/m3 kg nvo Nuclei population density, no'/m3 kg N Number of crystals, no,/kg solvent N Rotational speed, Hz; stirrer speed, Hz, rev/s N Number of grids AN Number of crystals retained over AL AN Difference in cumulative number over AL N] Modified total crystal population, no. m/kg s N p Power number for impeller NT Total number of crystals, no., no./kg Nu Nusselt number 00 Order of magnitude p Laplace transform variable with respect to dimensionless time, with respect to time, lis p Length parameter, m p Direction of search variable P Number of phases P Power input, W P Dimensionless parameter (Eq. (8.39» LIST OF SYMBOLS xxi

P Product flowrate, tlh AP Pressure drop, N/m2, Pa Pc Mass production rate of crystals, kg/s, tlh Pr Prandtl number P(g *) Growth rate activity distribution

SD Standard deviation Se Weight of solvent lost by evaporation, kg/kg original solvent Si Sine F ourler transform of response population density Sh Sherwood number S1 Separation intensity, kg/m3h STEPMX An estimate of Euclidean distance I Time, age, s tg Time for growth of the critical nucleus to detectable size, s tind Induction period, s In Time for formation of critical nucleus, s Ir Relaxation time, s T Temperature, DC, K T Dimensionless time (= tit) T Tank diameter, m aT Temperature difference (= T - Tw), DC Ta Taylor number, Ta= (27tR.jepN/~)(elRj)~ U Superficial solution velocity, mls U Dummy variable for crystal volume, m3 ut Particle terminal velocity, mls u(y-L) Step input function; = I ify>L; = 0 ifr, < L U Overall heat transfer coefficient, W1m K u Characteristic velocity, mls U Mean axial velocity, cmls Ui ith derivative oflogarithmic transfer function Uk kth moment of dimensionless population density with respect to dimensionless time Ur Ratio of heat transfer coefficient to heat capacity, m2/s V Velocity of crystal in re~on R V Crystal volume, m3, ~ , m3lkmol V Velocity, cm/s v(y-L) Delta Dirac input function; = I ify = L; = 0 ify:#; L V(T, x) Modified dimensionless population density function var Variance ve External (or spatial) crystal velocity VI Internal crystal velocity vm Molecular volume, Llmol Vz Velocity in axial direction, cmls va Velocity in angular direction, cmls V Volume of crystal suspension, volume of crystal bed, m3 V Working vessel volume, L; solvent capacity, kg V Volume of annular space, cm3 Vd Diluent addition, kg/kg original free solvent LIST OF SYMBOLS nUl

~ Solvent loss, kg/kg original free solvent w Valance of ions w Mass of one crystal, kg w Weight percent W(T, X) Modified dimensionless population density function W Weight of crystals, kg W Mass of seed crystals, total, kg or specific, kg/kg solvent AW Weight of crystals retained on a sieve, kg X Weight of solid, kg X Coefficients X Variable X Concentmtion of salt, kg/kg X Stoichiometric coefficient for silica or molar ratio of Si02 to Na20 (= 3.25) X Dimensionless crystal size (= LIGt, LIGt, (L -Lo)/Gt) X Population weighted mean (i.e. the mtio of first to zeroth moment) X,X Dimensionless mdial position for equal axial velocity X Set of independent variables X Vector coordinates of region R X Concentmtion of solids in suspension, % Y Coefficients Y Dimensionless population density, ninO Y Variable Y Crystal yield, kg Y Observable dependent variable z Dimensionless parameter in R-z crystallizer model equal to ratio of withdmwal of product at size L to MSMPR mte z Dimensionless residence time of crystals (tiT) z Valance of ions

GREEK SYMBOLS a. 3 pr!2F, kglm3 a. Degree of dissociation a. Solid solute deposition mte, kg/kg s a. Ratio of outside radius of inner cylinder to inside radius of outer cylinder (a. =R/R) a. Age a. Dimensionless constant in size dependent model xxiv LIST OF SYMBOLS a Model parameter in classification or size-dependent growth rate models Step length J(1+4IPe) Model parameter in classification or size-dependent growth rate models ~ Dimensionless temperature rise (Eq. (3.23» ~ Ratio of initial concentrations of reactants ~ Agglomeration kernel, Llno.s, kg/no.s W Agglomeration kernel at any time for vessel volwne, IIno.s ~I' ~2 Model parameters in classification functions ~ Parameters in model ~d Dimensionless temperature rise for diffusion step (Eq. (3.24» Y JI + (4IPe) Y Activity coefficient Y Relative desupersaturation (= Aco/Acp) f Dimensionless concentration driving force fD Capillary constant from Gibbs-Thomson equation, m 8 Delta Dirac function, m-I 8 Correction or improvement vector A Differential A Difference I; Power dissipation per unit mass, m2/ s3 I; Energy dissipation rate I; Molal ratio of diluent to initial solvent e Solid voidage or bed voidage I; Solid voidage &0 Arrhenius nwnber ~ jth central moment of the growth rate activity Effectiveness factor "9 Dimensionless residence time (9 =th) 9 Dummy variable Expected value ofY at x for given ~ "9 New time variable, s 9 Temperature,oC 9w Cooling water temperature, K K Reciprocal Debye-Hiickellength, m-I Kc Overall solution conductivity, Sim A Equivalent conductivity, S m2 equiv-I A Dimensionless residual life A Crystal size intensity function A. Dimensionless classification function LIST OF SYMBOLS xxv

'). Marquardt parameter '). Dimensionless distance at which the momentum flux is zero '). Particle size at time e =0, 11m '). Latent heat of vaporization, kJ/kg solvent ').(L) Number flux caused by fines removal, m-4 s-I 11 Solution viscosity, kglm s, N s/m2, mPa s 11 Chemical potential 11 Mean Ilj jth moment of population density with respect to time about the origin Ilvj jth moment with respect to crystal volume, no.m3j/L V Molecular volume, Llmol V Kinematic viscosity of the solution, m2/s ~ Constant ~ Dimensionless residual time P Dimensionless growth rate P Density, kglm3, kgIL Pc Crystal density, kglm3 Pp Density of precipitated silica, kglL 3 Ps Solution density, kglm PsI Density of slurry, kglm3 cr Surface energy or interfacial tension, J/m2 cr Relative supersaturation (~c/c*) cr Width parameter in normal distribution cr Standard deviation cr' Width parameter in log-normal distribution, geometric standard deviation Variance Constant, kmol/kg Run time, s Slurry residence time, overall drawn-down time, crystal residence time, s ,- Mean residence time of crystals, s 'c(L) Size-dependent residence time of crystals, s 'I Liquid residence time, s L\"C Difference between two batch times, s U Kinematic viscosity, m2/s Uj jth central moment of population density, no mi/kg solvent ~ Mass fraction of the distribution around the mean ~ Volume fraction of solids ~(1) Modified dimensionless nuclei population density function ~(cr,n Dimensionless supersaturation function in the growth rate model \jI Dimensionless age uvi LIST OF SYMBOLS

n Ratio of growth rates 0> Frequency, lis, 11m 0> Fourier transform variables, 11m, lis 0> Angular velocity, radls m Ratio of molecular weights of hydrate to anhydrous salt

SUBSCRIPTS

a Addition, agitator aq Aqueous A Acid, component A b Bed, bend, bound, bulk, nucleation batch Value evaluated for batch case B Alkali (sodium silicate solution), feed brine c Downcomer, clean, contraction, critical, crystal, crystallizer, crystal size distribution cone Conical cryst Crystallization cs Complete segregation cw Cooling water C Component C, threshold for classification d Diffusion dil Dilution D Design, diluent, dirt, dissolution e Expansion E End f Feed,final,fluid,firee F Fines F,FI Threshold for fines removal g Growth G Growth, distribution with growth rate i Initial, inlet, inside, interfacial j Index variable js Just suspension k Index variable I Liquor 1m Log mean L At size L, distribution with size, liquid m External loop, mass density, maximum, threshold time at which appreciable solid formation starts, smallest crystals in the product, upper LIST OF SYMBOLS uvil max Maximum mf Minimum fluidization min Minimum mm Maximum mixedness n Nucleation, nucleus N Based on number, hydrotrope, newly generated crystals, NaS Sodium salicylate (total or ionic) NH3 Ammonia o Outside p Product, population, precipitate pw Process water P Point, clump PF Plug flow section r Reaction, relaxation, surface integration, s Solid, solution, solute, saturation, steam sl Slurry sol Sol phase soln Solution sus Suspension S Seed, solute, silicate SA Salicylic acid (total) SH Salicylic acid (molecular) t At time t, total, terminal settling velocity, distribution with time T Total v Volume coordinate, vapour, vaporizer w Weight distribution wm Weight mean o Bulk, feed, initial, lower, seed, zero ionic strength, reference, smallest crystals in the fluidized bed 1 Specific with respect to solvent 1,2 Addition stages I,II,III Models I, II and III respectively

SUPERSCRIYfS o Ion pair, nuclei, initial Average or transformed quantities * Equilibrium, critical condition, threshold Derivative with respect to a variable A Quantities based on total solvent capacity nvHi LIST OF SYMBOLS

Dummy variable Noncentral moments