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Crystallization kinetics of sodium sulphate from 9N sulphuric acid solution Nyakiamo, Anthony P. 1991

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C R Y S T A L L I Z A T I O N K I N E T I C S O F S O D I U M S U L P H A T E F R O M 9 N S U L P H U R I C A C I D S O L U T I O N by ANTHONY P. NYAKIAMO B.Sc. (UNIVERSITY OF NAIROBI) 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Chemical Engineering) We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January 1991 ©Anthony Nyakiamo, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CH/6MICAL £NG>/AI&^/ZlK]^ The University of British Columbia Vancouver, Canada Date "<54NJU MM 3& R 0 ! . DE-6 (2/88) A B S T R A C T The c r y s t a l l i z a t i o n k i n e t i c s of sodium sulphate from 9N s u l p h u r i c a c i d was s t u d i e d under c o o l i n g c o n d i t i o n s . The c r y s t a l growth and n u c l e a t i o n r a t e s were determined us i n g the P o p u l a t i o n Balance concept i n a continuous mixed-suspension mixed-product-removal (MSMPR) c r y s t a l l i z e r . The e f f e c t s of su p e r s a t u r a t i o n , c r y s t a l suspension d e n s i t y and temperature on the c r y s t a l l i z a t i o n k i n e t i c s were a l l i n v e s t i g a t e d . The study was conducted at c r y s t a l l i z e r temperatures of 45, 50, 55, and 60 °C. The c r y s t a l growth r a t e data were c o r r e l a t e d t o the su p e r s a t u r a t i o n with a power-law, G = KQS^. The c r y s t a l n u c l e a t i o n r a t e data were f i t t e d to both primary (B° = KgS^) and secondary (B° = K NM T^S U) n u c l e a t i o n models. Growth and n u c l e a t i o n r a t e data were c o r r e l a t e d according t o the primary (B° = Kj^G1) and secondary (B° = K nM T^G v) r e l a t i v e k i n e t i c models. The study determined that the growth r a t e data f i t the expression, (G = KQS w•° ), and tha t secondary n u c l e a t i o n was the dominant mode of n u c l e i generation (B° = K N M T ^ • • 2 7 ) _ The sodium sulphate c r y s t a l l i z e s from s o l u t i o n as the a c i d s a l t , sodium sesqui-sulphate (Na3HSC>4) . The r a t e constants, K G and K N, were both f u n c t i o n s of temperature and were f i t t e d t o Arrhenius type expressions : K G = 3.50xl0 8 exp(-46,700/RT) i i % = 4 . 7 1 x l 0 1 6 exp(-58,900/RT) where R : u n i v e r s a l gas constant (kJ/kmol.K) T : absolute temperature (K) +46,700 kJ/kmol i s the growth a c t i v a t i o n energy +58,900 kJ/kmol i s the n u c l e a t i o n a c t i v a t i o n energy The r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s can be expressed as B° = 1.14xl0 6M T°- 9 4G 1- 4 2 TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENTS x i i CHAPTER ONE : INTRODUCTION 1 CHAPTER TWO : LITERATURE REVIEW 4 2.1 : C r y s t a l l i z a t i o n K i n e t i c s 4 2.2 : Nu c l e a t i o n K i n e t i c s 5 2.2.1 : Primary N u c l e a t i o n 5 2.2.2 : Secondary Nu c l e a t i o n 10 2.2.3 : Factors a f f e c t i n g Secondary N u c l e a t i o n 14 2.2.4 : E m p i r i c a l Expressions For Nu c l e a t i o n 15 2.3 : Growth K i n e t i c s 18 2.4 : R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s 24 2.5 : The E f f e c t Of Hydrodynamics On The C r y s t a l l i z a t i o n K i n e t i c s 25 2.6 : The E f f e c t Of Temperature On The C r y s t a l l i z a t i o n K i n e t i c s 27 2.7 : The Pop u l a t i o n Balance Concept 28 2.8 : The Steady State Mixed-Suspension Mixed-Product-Removal (MSMPR) C r y s t a l l i z e r 30 2.8.1 : The C r y s t a l P o p u lation Balance 31 2.8.2 : D e r i v i n g Pure C r y s t a l l i z a t i o n K i n e t i c s For Size Independent C r y s t a l Growth 33 2.8.3 : The C o e f f i c i e n t Of V a r i a t i o n Of The C r y s t a l i v S i z e D i s t r i b u t i o n (CSD) 35 2.8.4 : Si z e Dependent C r y s t a l Growth 35 CHAPTER THREE : EXPERIMENTAL SECTION 38 3.1 : Experimental Set-up 38 3.2 : V e r i f i c a t i o n Of Uniform Mixing In The C r y s t a l l i z e r 40 3.3 : Determination Of Time To Steady State 42 3.4 : Determination Of C r y s t a l S ize D i s t r i b u t i o n (CSD) 44 3.5. : Determination Of The C r y s t a l Suspension Density 44 3.6 : Determination Of S o l u b i l i t y 45 3.7 : Determination of Supersaturation 47 3.8 : Determination of C r y s t a l Shape Factor 48 3.9 : V e r i f i c a t i o n Of The Type Of C r y s t a l Produced 50 CHAPTER FOUR : RESULTS AND ANALYSIS 51 4.1 : Determination Of C r y s t a l N u c l e a t i o n And Growth Rates 51 4.2 : M o d e l l i n g The Growth And Nu c l e a t i o n Rate Data 54 4.3 : D e r i v i n g The A c t i v a t i o n Energy Of C r y s t a l l i z a t i o n 58 4.4 : D e r i v a t i o n Using The Combined Data 61 4.5 : Res u l t s 63 4.5.1 : Growth K i n e t i c s 65 4.5.2 : N u c l e a t i o n K i n e t i c s 65 v 4.5.3 : R e l a t i v e N u c l e a t i o n K i n e t i c s 70 4.5.4 : A c t i v a t i o n Energy For Growth 71 4.5.5 : A c t i v a t i o n Energy For Nu c l e a t i o n 75 CHAPTER FIVE : DISCUSSION AND CONCLUSION 7 9 5.1 : E f f e c t Of Supersaturation 79 5.2 : E f f e c t Of Residence Time 82 5.3 : E f f e c t Of Suspension Density 86 5.4 : E f f e c t Of Temperature.... 87 5.5 : C o e f f i c i e n t Of V a r i a t i o n Of The CSD 90 5.6 : Accuracy Of The Study 90 5.7 : Conclusions 93 NOMENCLATURE 94 REFERENCES 99 APPENDICES 105 A : LOTUS P r i n t - o u t s For The Determination Of C r y s t a l Growth And N u c l e a t i o n Rates 105 B : CSD P l o t s For The Experimental Runs 127 C : MINITAB Regression P r i n t - o u t s For The Determination Of The Model Parameters 139 D : MINITAB Regression P r i n t - o u t s For The Determination Of The Rate Constants And The A c t i v a t i o n Energies 152 E : MINITAB Regression P r i n t - o u t For The C r y s t a l l i z a t i o n Models And A c t i v a t i o n Energies Using The Combined Data 155 F : D e r i v a t i o n s Of Expressions Used In Chapter F i v e 160 G : C r y s t a l l i z e r D e t a i l s 165 v i i L I S T O F T A B L E S 3.1 : V e r i f i c a t i o n Of Uniform Mixing In The C r y s t a l l i z e r 41 3.2 : S o l u b i l i t y Of Sodium Sulphate In 9N Sul p h u r i c A c i d 46 4.1 : T y p i c a l Lotus Regression P r i n t - o u t For C r y s t a l N u c l e a t i o n And Growth Rates 53 4.2 : T y p i c a l MINITAB Regression P r i n t - o u t For The C r y s t a l l i z a t i o n Model Parameters 55 4.3 : MINITAB Regression P r i n t - o u t For E f f e c t Of Temperature On The K i n e t i c Rate Constants 59 4.4 : Summary Of MINITAB Regression For The C r y s t a l l i z a t i o n Model Parameters And A c t i v a t i o n Energies Using The Combined Data 62 4.5 : Summary Of The Experimental Results 64 4.6 : Comparison Of I n d i v i d u a l And O v e r a l l Regression For Growth Rate 66 4.7 : Comparison Of Primary And Secondary N u c l e a t i o n Models ( I n d i v i d u a l Approach) 67 4.8 : Comparison Of I n d i v i d u a l And O v e r a l l Regression For Secondary N u c l e a t i o n 69 4.9 : Comparison Of Primary And Secondary R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s ( I n d i v i d u a l Approach) 72 4.10 : Comparison Of I n d i v i d u a l And O v e r a l l Regresssions For Secondary R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s 74 v i i i 4.11 : Comparison Of I n d i v i d u a l And O v e r a l l Regresssions For The E f f e c t Of Temperature On The Growth Rate K i n e t i c Constant 7 6 4.12 : Comparison Of I n d i v i d u a l And O v e r a l l Regresssions For The E f f e c t Of Temperature On The N u c l e a t i o n Rate K i n e t i c Constant 77 5.1 : Reported K i n e t i c Orders 80 5.2 : V a r i a t i o n Of C r y s t a l Median Size With Residence Time 81 5.3 : E f f e c t Of Residence Time On The Corrected N u c l e a t i o n Rate 85 5.4 : Reported A c t i v a t i o n Energies 8 9 L I S T O F F I G U R E S 2.1 : Nu c l e a t i o n Mechanisms 6 2.2 : Mier's Diagram 7 2.3 : Analogy Of The Various States Of S t a b i l i t y 9 2.4 : Free Energy Diagram For N u c l e a t i o n 9 2.5 : E f f e c t Of Supersaturation On C l a s s i c a l N u c l e a t i o n 16 2.6 : Concentration D r i v i n g Forces In C r y s t a l l i z a t i o n From S o l u t i o n 22 2.7 : CSD For Size Dependent C r y s t a l Growth Rate 37 3.1 : Schematic Of The Experimental Set-up 39 3.2 : Time To Steady-State In An MSMPR C r y s t a l l i z e r 43 3.3 : Sodium Sulphate S o l u b i l i t y In 9N Sul p h u r i c Acid...46 4.1 : A T y p i c a l CSD P l o t 52 4.2 : E f f e c t Of Supersaturation On The Growth Rate 66 4.3 : E f f e c t Of Supersaturation On The Corrected N u c l e a t i o n Rate 69 4.4 : R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s 74 4.5 : E f f e c t Of Temperature On The Growth Rate Constant 7 6 4.6 : E f f e c t Of Temperature On The Nu c l e a t i o n Rate Constant 77 5.1 : E f f e c t Of Residence Time On The Supersa t u r a t i o n . . . 83 5.2 : E f f e c t Of Residence Time On The Corrected N u c l e a t i o n Rate 85 5.3 : S c a t t e r P l o t Of Growth Rate Data About The P r e d i c t e d Values 91 x S c a t t e r P l o t Of N u c l e a t i o n Rate Data About The P r e d i c t e d Values A C K N O W L E D G E M E N T S I wish t o thank my research s u p e r v i s o r Dr. K.L. Pinder f o r h i s guidance and support throughout my course. I a l s o thank the Kenya goverment and the Canadian goverment (through C.I.D.A.) f o r t h e i r f i n a n c i a l support t h a t enabled me t o pursue t h i s degree course. I am deeply g r a t e f u l t o my wi f e , M i l l i c e n t , and daughter, S h e i l a , whose love and support c a r r i e d me through some very d i f f i c u l t times. x i i C H A P T E R O N E 1.1 INTRODUCTION C r y s t a l l i z a t i o n has been p r a c t i s e d by mankind si n c e the dawn of c i v i l i z a t i o n . For c e n t u r i e s , cooking s a l t has been produced by the evaporation of sea water. Today, few se c t i o n s of the chemical i n d u s t r y do not, at some stage, u t i l i z e c r y s t a l l i z a t i o n as a method of production, p u r i f i c a t i o n , or recovery of a s o l i d m a t e r i a l . Before c r y s t a l l i z a t i o n equipment can be designed and operated i n the most e f f i c i e n t manner, the p r i n c i p l e s that c o n t r o l the formation and growth of c r y s t a l s must be known. In the past, the n u c l e a t i o n and growth r a t e s of c r y s t a l l i z i n g systems were measured s e p a r a t e l y under d i f f e r e n t c o n d i t i o n s , and mainly using s i n g l e c r y s t a l s . The r e s u l t s were t h e r e f o r e of l i t t l e relevance t o i n d u s t r i a l c r y s t a l l i z a t i o n , where both n u c l e a t i o n and growth are o c c u r r i n g simultaneously i n a mass of c r y s t a l s . A l l t h i s changed however, with the development of the P o p u l a t i o n Balance concept (Randolph and Larson (1962)) and i t ' s use i n Mixed Suspension Mixed Product Removal (MSMPR) c r y s t a l l i z e r s . C r y s t a l l i z a t i o n k i n e t i c s data could now be exper i m e n t a l l y obtained i n the l a b o r a t o r y under c o n d i t i o n s c l o s e r t o those p r e v a i l i n g i n d u s t r i a l l y . 1.2 REASON FOR THE STUDY AND IT'S OBJECTIVES 1 There are over 200 i n s t a l l a t i o n s world-wide producing c h l o r i n e d i o x i d e , CIO2, f o r the pulp and paper i n d u s t r y . The processes commonly used are the Mathieson, Solvay, and R-8. The CIO2 i s produced by the red u c t i o n of sodium c h l o r a t e , NaClC>3, i n a strong (9N s u l p h u r i c a c i d , H2SO4) a c i d environment. These processes discharge an e f f l u e n t which i s e s s e n t i a l l y sodium sulphate (Na2SC>4) d i s s o l v e d i n f a i r l y concentrated s u l p h u r i c a c i d . The current p u l p - m i l l p r a c t i s e i s t o use the e f f l u e n t as s a l t - c a k e makeup by n e u t r a l i z i n g i t wi t h 'black l i q u o r ' p r i o r t o burning i n the recovery b o i l e r . However, the growing environmental and economic pressures are f o r c i n g m i l l s t o i n c r e a s i n g l y c l o s e t h e i r gas and l i q u o r emission c i r c u i t s . Reduction of s a l t - c a k e l o s s e s through the brownstock washing system, the i n c i n e r a t i o n and scrubbing of non-condensable gases to produce sodium sulphide, and more c h l o r i n e d i o x i d e used per ton of pulp a l l mean a re d u c t i o n i n s a l t - c a k e makeup requirments. The 'L-Process' o f f e r s a way t o r e c t i f y t h i s growing chemical imbalance. This process envisages the removal by c r y s t a l l i z a t i o n of sodium sesquisulphate from the CIO2 generator e f f l u e n t p r i o r t o the recovery and r e c y c l i n g of the a c i d . The L-Process can be used i n conjunction with the Mathieson or Solvay processes. There i s a need f o r more b a s i c i n f o r m a t i o n on the c r y s t a l l i z a t i o n k i n e t i c s of sodium sesquisulphate i n t h i s 2 k i n d of environment, and t h i s study attempts to shed some l i g h t i n t h i s regard. This work attempts t o obt a i n the growth and n u c l e a t i o n r a t e s of sodium sulphate from 9N s u l p h u r i c a c i d , as a f u n c t i o n of s u p e r s a t u r a t i o n , under c o o l i n g c r y s t a l l i z a t i o n . The k i n e t i c s of t h i s system w i l l be d e r i v e d using Randolph and Larson's (1962) P o p u l a t i o n Balance Concept i n a continuous mixed suspension mixed product removal c o o l i n g c r y s t a l l i z e r . The e f f e c t s of temperature, s u p e r s a t u r a t i o n , and suspension d e n s i t y w i l l be i n v e s t i g a t e d . The c r y s t a l l i z i n g system w i l l be modelled wi t h w e l l e s t a b l i s h e d e m p i r i c a l r e l a t i o n s . C H A P T E R T W O 2.1 CRYSTALLIZATION KINETICS C r y s t a l l i z a t i o n i s the process by which an ordered s o l i d phase i s generated from l i q u i d s o l u t i o n s . The s o l u t i o n i n question must be supersaturated before c r y s t a l l i z a t i o n can occur; however, s u p e r s a t u r a t i o n alone i s not a s u f f i c i e n t c o n d i t i o n f o r c r y s t a l l i z a t i o n . When the s o l u t e c o n c e n t r a t i o n exceeds i t ' s e q u i l i b r i u m s o l u b i l i t y (at a given temperature) i n the so l v e n t , the s o l u t i o n i s s a i d to be supersaturated. The c r y s t a l l i z a t i o n process comprises three b a s i c steps : - achievement of su p e r s a t u r a t i o n - formation of c r y s t a l n u c l e i - and growth of the c r y s t a l s Under most circumstances, formation and growth of the c r y s t a l s are i n t i m a t e l y connected. Supersaturation may be achieved by : - c o o l i n g a s o l u t i o n i n which the s o l u b i l i t y of the so l u t e increases with i n c r e a s i n g temperature - or, conversely, heating the s o l u t i o n i f s o l u b i l i t y decreases with i n c r e a s i n g temperature - evaporating o f f the solvent thereby r a i s i n g the so l u t e c o n c e n t r a t i o n - a d d i t i o n of a l e s s e f f i c i e n t solvent t h a t i s m i s c i b l e w i t h the o r i g i n a l solvent 4 - chemical r e a c t i o n i n s o l u t i o n l e a d i n g t o the formation of a c r y s t a l l i n e substance - s a l t i n g - o u t by the common ion e f f e c t The k i n e t i c s of c r y s t a l l i z a t i o n can be separated i n t o n u c l e a t i o n and growth k i n e t i c s . 2.2 NUCLEATION KINETICS N u c l e a t i o n may be defined as the process i n which the smallest aggregates of a c r y s t a l l i n e phase are formed i n a c r y s t a l l i z i n g system. According to the mechanism i n v o l v e d , n u c l e a t i o n can be sub-divided as shown i n f i g . (2.1). A b a s i c c r i t e r i o n , f o r t h i s breakdown i s the presence or absence of a s o l i d phase. 2.2.1 PRIMARY NUCLEATION This occurs i n the absence of s o l i d p a r t i c l e s of the c r y s t a l l i z e d substance. I t i s f u r t h e r sub-divided i n t o : homogeneous n u c l e a t i o n , which occurs w i t h i n homogeneous s o l u t i o n s where no f o r e i g n surface p r e v i o u s l y e x i s t e d ; heterogeneous n u c l e a t i o n , which i s c a t a l y t i c a l l y i n i t i a t e d by any f o r e i g n surface, e.g. v e s s e l w a l l s , dust, c o l l o i d s , e t c . As p r e v i o u s l y mentioned, s u p e r s a t u r a t i o n alone i s not a s u f f i c i e n t c o n d i t i o n f o r c r y s t a l l i z a t i o n . I t has been observed t h a t some supersaturated s o l u t i o n s may l a s t 5 NUCLEATION PRIMARY SECONDARY HOMOGENEOUS HETEROGENEOUS INITIAL BREEDING NEEDLE BREEDING COLLISION BREEDING FLUID SHEAR IMPURITY GRADIENT CONCENTRATION Figure (2.1) N u c l e a t i o n Mechanisms S u p e r s o l u b i l i t y curve L a b i l e s Temperature Figu r e (2.2) Mier's Diagram i n d e f i n i t e l y without ever forming c r y s t a l s . Such s o l u t i o n s are termed metastajble (Ostwald, 1897) . The metastable zone (see Mier's, diagram f i g . 2.2) l i e s between the s o l u b i l i t y and the s u p e r s o l u b i l i ' t y curves. I t i s a region of s u p e r s a t u r a t i o n where spontaneous n u c l e a t i o n i s improbable, though d e p o s i t i o n of c r y s t a l l i n e phase onto e x i s t i n g c r y s t a l surface ( c r y s t a l growth) i s p o s s i b l e . The region beyond the s u p e r s o l u b i l i t y curve i s the labile zone where spontaneous n u c l e a t i o n i s h i g h l y probable. The analogy of a b r i c k r e s t i n g on a f l a t surface ( f i g . 2.3) can be used to represent the energy l e v e l s of the v a r i o u s s t a t e s of s t a b i l i t y . The height of the b r i c k ' s centre of g r a v i t y above the a r b i t r a r y datum of the f l a t surface represents the p o t e n t i a l energy l e v e l of each s t a t e . P o s i t i o n s A and E represent the lowest energy s t a t e s (maximum s t a b i l i t y ) , analogous to the s a t u r a t e d s o l u t i o n . P o s i t i o n C i s analogous to the metastable region i n a supersaturated s o l u t i o n . The b r i c k , though somewhat s t a b l e , has a higher energy l e v e l than e i t h e r A or E, and can withstand only very small displacements before r e v e r t i n g to the more s t a b l e p o s i t i o n s A or E. P o s i t i o n s B and D have the highest energy l e v e l s of the system and are unstable. They represent the l a b i l e supersaturated s o l u t i o n t h a t would tend to nucleate spontaneously. Just how a c r y s t a l nucleus i s formed w i t h i n a homogeneous s o l u t i o n i s not known. The probable mechanism i n v o l v e s a 8 Metastable Labile \ \ \ B 3 "~~7 /—Labile / D 1 / \ TV t\ ^ y \ * \ 1 \ I  E Stable Stable F i g u r e (2.3) Analogy Of The Various States Of S t a b i l i t y Size of nucleus, r Figure (2.4) Free Energy Diagram For N u c l e a t i o n s e r i e s of bi m o l e c u l a r a d d i t i o n r e a c t i o n s u n t i l a c r i t i c a l c l u s t e r s i z e i s reached. This c r i t i c a l s i z e corresponds w i t h the maximum i n the f r e e energy curve (see f i g . 2.4). The fr e e energy changes a s s o c i a t e d with the process of homogeneous n u c l e a t i o n may be considered as f o l l o w s . The o v e r a l l excess fr e e energy, AG, between a small s o l i d p a r t i c l e of s o l u t e and the s o l u t e i n s o l u t i o n i s equal t o the sum of the surface excess fr e e energy, AG S, i . e . the excess f r e e energy between the surface of the p a r t i c l e and the bulk of the p a r t i c l e , and the volume excess f r e e energy, AG V, i . e . the excess fr e e energy between a very l a r g e p a r t i c l e (r=°°) and the s o l u t e i n s o l u t i o n . AG S i s p o s i t i v e and p r o p o r t i o n a l to r , and i n a supersaturated s o l u t i o n , AG V i s negative and p r o p o r t i o n a l to . Thus: AG = AG s + AG V (2.1) Since the r i g h t hand side terms are of opposite s i g n and depend d i f f e r e n t l y on r (the nucleus r a d i u s ) , the f r e e energy of formation, AG, passes through a maximum. This maximum value, A G c r ^ 4 - , corresponds to the c r i t i c a l s i z e , r c , which i s the minimum s i z e of a s t a b l e nucleus. N u c l e i smaller than r c can only decrease t h e i r f r e e energy by d i s s o l v i n g , whereas those l a r g e r than r c do so by growing. The term A G c r j _ t , i s the a c t i v a t i o n energy f o r n u c l e a t i o n , and both A G c r ^ t and r c decrease as s u p e r s a t u r a t i o n increases (McCabe 1946). 2.2.2 SECONDARY NUCLEATION The presence of a c r y s t a l i n i t ' s supersaturated s o l u t i o n u s u a l l y induces the formation of f u r t h e r c r y s t a l s at l e v e l s of s u p e r s a t u r a t i o n too low f o r spontaneous, i . e . primary, n u c l e a t i o n to occur. Secondary n u c l e a t i o n then, i s n u c l e a t i o n c a t a l y z e d by the presence of p a r t i c l e s of the c r y s t a l l i z i n g substance. Cayey and E s t r i n (1967) s t u d i e d secondary n u c l e a t i o n of magnesium sulphate and found t h a t the i n t r o d u c t i o n of seed c r y s t a l s produced a shower of n u c l e i . Larson et a l . (1968), and Timm and Larson (1968) found t h a t the n u c l e a t i o n r a t e was d i r e c t l y p r o p o r t i o n a l to the c r y s t a l suspension d e n s i t y . The f i r s t order dependency of n u c l e a t i o n on suspension d e n s i t y was l a t e r confirmed by Youngquist and Randolph (1972). Mason and S t r i c k l a n d - C o n s t a b l e (1966) suggest three types of secondary n u c l e a t i o n . These are : i n i t i a l breeding, needle breeding, and c o l l i s i o n breeding. B o t s a r i s et a l . (1972) proposed a f o u r t h category, v i z . , i m p u r i t y c o n c e n t r a t i o n gradient n u c l e a t i o n . A d d i t i o n a l l y , secondary n u c l e a t i o n as a r e s u l t of f l u i d shear was put forward by Powers (1963). INITIAL BREEDING When dry c r y s t a l s are f i r s t introduced i n t o a supersaturated s o l u t i o n , they leave behind them a t r a i l of t i n y c r y s t a l s . These n u c l e i appear only when the seed c r y s t a l s are f i r s t i ntroduced, hence the term i n i t i a l breeding. I t i s b e l i e v e d t h a t the n u c l e i r e s u l t from s t r a y c r y s t a l l i t e s t h a t are present on the surface of the seed c r y s t a l s . NEEDLE BREEDING Once i n i t i a l breeding i s complete, the seed c r y s t a l s are s t i l l capable of generating more n u c l e i . In h i g h l y supersaturated s o l u t i o n s , S t r i c k l a n d - C o n s t a b l e and Mason (1966) found t h a t 'needles or d e n d r i t e s ' tended to grow out of the ends of the c r y s t a l s . Under a g i t a t e d c o n d i t i o n s these needles b r e a k - o f f and become new c r y s t a l s . COLLISION BREEDING L a i et a l . (1969) were perhaps the f i r s t t o demonstrate c o n c l u s i v e l y the exi s t e n c e of c o l l i s i o n breeding. They showed t h a t f l u i d shear alone does not give r i s e t o n u c l e a t i o n when seed c r y s t a l s were placed i n a supersaturated s o l u t i o n . C r y s t a l - s o l i d c ontacts, however, d i d produce n u c l e i . Large numbers of n u c l e i can be produced when the c r y s t a l s i n a suspension are f r e e to c o l l i d e . C o l l i s i o n breeding can be subdivided i n t o : Fragmentation, the means of n u c l e a t i o n here i s s e l f -evident, and occurs i n systems whose c r y s t a l s break e a s i l y . Examination of the parent c r y s t a l s w i l l r e v e a l v i s i b l e damage. N u c l e a t i o n by a t t r i t i o n i s simply fragmentation of a l e s s e r degree, and c r y s t a l damage w i l l be correspondingly l e s s . Contact n u c l e a t i o n , i s by f a r the most important source of n u c l e i i n mixed suspension c r y s t a l l i s a t i o n . I t stems from the c r y s t a l s c o n t a c t i n g the a g i t a t o r , v e s s e l w a l l s , and other c r y s t a l s . N u c l e i are produced by the displacement of the adsorbed l a y e r of s o l u t e t h a t has not yet become c r y s t a l l i n e , and leaves no v i s i b l e damage on the parent c r y s t a l . Clontz and McCabe (1971) confirmed t h i s mechanism. They showed that low energy contacts normal t o the surface of a growing c r y s t a l produced a number of new c r y s t a l s a f t e r a short growth-period. FLUID SHEAR I f the f l u i d v e l o c i t y r e l a t i v e to the c r y s t a l v e l o c i t y i s l a r g e enough, some of the 'adsorbed l a y e r ' or 'dendrites' (depending on the p a r t i c u l a r mechanism) are swept o f f to form new c r y s t a l s . Powers (1963) put forward two p o s s i b l e mechanisms. The f i r s t i s s i m i l a r to needle breeding i n t h a t i t assumes d e n d r i t i c c r y s t a l growth. The shearing a c t i o n of the f l u i d breaks o f f these dendrites producing n u c l e i . A l t e r n a t i v e l y , there i s an adsorbed l a y e r of s o l u t e molecules weakly attached t o the c r y s t a l s u r f a c e . Shearing o f f of a s u f f i c i e n t l y l a r g e c l u s t e r of these s o l u t e molecules can e v e n t u a l l y lead to new c r y s t a l s . Sung et a l . (1973) have shown that shear-produced n u c l e i normally need to be exposed to supersaturations much higher than those i n which they were produced i f they are to grow. Consequently, they do not represent an important n u c l e i source. IMPURITY CONCENTRATION GRADIENT NUCLEATION B o t s a r i s et a l . (1972) suggest t h a t there are i m p u r i t i e s which suppress spontaneous n u c l e a t i o n i n the bulk of a s o l u t i o n . According to them, i f t h i s i m p u r i t y i s inc o r p o r a t e d i n t o the l a t t i c e of a growing c r y s t a l , an imp u r i t y c o n c e n t r a t i o n gradient may a r i s e w i t h i n the boundary l a y e r . The imp u r i t y c o n c e n t r a t i o n i n the boundary l a y e r may then be reduced enough t o allow spontaneous n u c l e a t i o n to occur. 2.2.3 FACTORS AFFECTING SECONDARY NUCLEATION M e l i a and M o f f i t (1964) found that the number of n u c l e i grown to a dete c t a b l e s i z e increased w i t h the su p e r s a t u r a t i o n , r a t e of c o o l i n g , and the degree of a g i t a t i o n of the s o l u t i o n . Increasing the degree of a g i t a t i o n , increases the secondary n u c l e a t i o n r a t e . The increas e d s t i r r e r speeds lead t o more frequent and higher energy c o l l i s i o n s of the c r y s t a l s w i t h each other and the s t i r r e r , as w e l l as to higher l e v e l s of f l u i d shear. Ottens (1972) has shown that f o r a given i m p e l l e r , the n u c l e a t i o n r a t e i s p r o p o r t i o n a l to the power input d i s s i p a t e d by the i m p e l l e r per u n i t mass of s l u r r y . Clontz and McCabe (1971) showed the n u c l e a t i o n r a t e per u n i t c r y s t a l area t o be approximately l i n e a r with impact energy and wit h s u p e r s a t u r a t i o n . They a l s o showed th a t c r y s t a l / c r y s t a l contacts were twice as e f f i c i e n t i n generating secondary n u c l e i than c r y s t a l / r o d c o n t a c t s . The hardness of the i m p e l l e r was a l s o found to a f f e c t the n u c l e a t i o n r a t e . Randolph and Sikdar (1974,197 6) confirmed Johnson et a l . 1 s (1972) f i n d i n g t h a t s o f t - c o a t e d i m p e l l e r s produce fewer n u c l e i than s t e e l i m p e l l e r s . I m p u r i t i e s can have a marked e f f e c t on n u c l e a t i o n . Shor and Larson (1971) found t h a t heavy metal ions tended t o reduce secondary n u c l e a t i o n whereas surface a c t i v e agents appeared t o increase the n u c l e a t i o n of potassium n i t r a t e . 2.2.4 EMPIRICAL EXPRESSIONS FOR NUCLEATION C l a s s i c a l t h e o r i e s describe the phenomenon of n u c l e a t i o n according to Arrhenius type expressions such as : B° = A exp (-167ia 3v n 2/3k B 3T 3 (In S ) 2) (2.2)* where B° = c r y s t a l n u c l e a t i o n r a t e A = pre-exponential f a c t o r k B = Boltzmann constant T = absolute temperature G = i n t e r f a c i a l t e n s i o n v n = volume of a s p h e r i c a l nucleus S = su p e r s a t u r a t i o n The n u c l e a t i o n r a t e i s governed by three main v a r i a b l e s : temperature, s u p e r s a t u r a t i o n , and i n t e r f a c i a l t e n s i o n . The expression p r e d i c t s ( f i g . 2.5) a r a p i d increase i n the n u c l e a t i o n r a t e once a c r i t i c a l s u p e r s a t u r a t i o n i s exceeded. However, experimentation has shown tha t the ra t e of * M u l l i n (1972) 142 CD "5 C o o (J Theoretical \ Experimental \ \ Supersaturation, S F i g u r e (2.5) E f f e c t Of Supe r s a t u r a t i o n On C l a s s i c a l N u c l e a t i o n 16 n u c l e a t i o n peaks and may even d e c l i n e beyond a c e r t a i n l e v e l of s u p e r s a t u r a t i o n . Furthermore, the p r e d i c t i o n of n u c l e a t i o n only at very high supersaturations i s not borne out i n f a c t . C l a s s i c a l n u c l e a t i o n t h e o r i e s t h e r e f o r e f a i l t o adequately describe n u c l e a t i o n , and a l s o i n v o l v e parameters such as the i n t e r f a c i a l t e n s i o n and the nucleus volume th a t are unknown and cannot be determined. Miers and Isaac (1906) proposed a power-law expression f o r n u c l e a t i o n t h a t accounted f o r the f a c t that n u c l e a t i o n does not occur at very low s u p e r s a t u r a t i o n s . I t was based on the concept of m e t a s t a b i l i t y and i s given as : where c : s o l u t e c o n c e n t r a t i o n c m : a c o n c e n t r a t i o n greater than the s a t u r a t i o n c o n c e n t r a t i o n but below which n u c l e a t i o n does not occur c : s a t u r a t i o n c o n c e n t r a t i o n In most systems, c m i s very c l o s e to c*, and many workers have s u c c e s s f u l l y taken c m as equal to c . Thus, we have : B° = K ( c - c m ) b , c m > c (2.3) B° = K(c-c ) *xb _ = K BS b (2.4) f o r primary n u c l e a t i o n , and B° = K NM T^S (2.5) f o r secondary n u c l e a t i o n where S : su p e r s a t u r a t i o n M T : c r y s t a l suspension d e n s i t y K B, K N : n u c l e a t i o n r a t e constants f o r primary and secondary n u c l e a t i o n r e s p e c t i v e l y Expression (2.5) i s the more general i n nature, t a k i n g i n t o account the e f f e c t the presence of other c r y s t a l s may have on the n u c l e a t i o n process. In many systems where secondary n u c l e a t i o n i s a f a c t o r , j has been found to be c l o s e t o one. When secondary n u c l e a t i o n i s not s i g n i f i c a n t , j = 0, and the expression then r e v e r t s to the primary n u c l e a t i o n form. The n u c l e a t i o n r a t e constants, K B and K N, are f u n c t i o n s of temperature, hydrodynamics and imp u r i t y . Their temperature dependence i s modelled with an Arrhenius type expression ( s e c t i o n 2.6, eqn(2.19)). 2.3 GROWTH KINETICS The growth of c r y s t a l s from s o l u t i o n i n v o l v e s three steps: 1) the t r a n s p o r t of s o l u t e molecules from the bulk of the s o l u t i o n t o the c r y s t a l surface (volume d i f f u s i o n ) . 2) the adsorption and o r i e n t a t i o n of the s o l u t e molecules i n t o the c r y s t a l l a t t i c e (surface i n t e g r a t i o n ) . 3) the d i s s i p a t i o n of the heat of c r y s t a l l i z a t i o n l i b e r a t e d at the c r y s t a l surface, back i n t o the bulk of the f l u i d . This step i s u s u a l l y neglected as an o v e r a l l c r y s t a l l i z a t i o n r a t e c o n t r o l l i n g f a c t o r . I f c r y s t a l growth i s l i m i t e d by the r a t e of d i f f u s i o n through a laminar l a y e r of s o l u t i o n , then growth i s s a i d t o be ' d i f f u s i o n c o n t r o l l e d ' . In such cases, the c r y s t a l growth r a t e increases as the v e l o c i t y of the.supersaturated s o l u t i o n r e l a t i v e to the c r y s t a l surface i n c r e a s e s . At some p o i n t , f u r t h e r increase i n the s o l u t i o n v e l o c i t y does not a f f e c t the growth r a t e , and the surface i n t e g r a t i o n step becomes c o n t r o l l i n g . Noyes and Whitney (1897) considered c r y s t a l growth t o be e s s e n t i a l l y a d i f f u s i o n a l process. They assumed th a t c r y s t a l l i z a t i o n was the reverse of d i s s o l u t i o n and proposed the f o l l o w i n g equation f o r c r y s t a l l i z a t i o n : dm/dt = k m A c ( c - c * ) (2.7) where m : mass of s o l i d deposited on the c r y s t a l face i n time,t A c : surface area of c r y s t a l c : s o l u t e c o n c e n t r a t i o n i n the supersaturated s o l u t i o n c : e q u i l i b r i u m s a t u r a t i o n c o n c e n t r a t i o n k m : c o e f f i c i e n t of mass t r a n s f e r By assuming the presence of a t h i n stagnant f i l m of l i q u i d next t o the c r y s t a l face, Nernst (1904) modified t h i s expression t o : dm/dt = (D/8) A c(c-c*) (2.8) where D : c o e f f i c i e n t of d i f f u s i o n of the s o l u t e 6 : length of the d i f f u s i o n path A c: c r y s t a l surface area c : s o l u t e c o n c e n t r a t i o n i n the supersaturated s o l u t i o n c : e q u i l i b r i u m s a t u r a t i o n c o n c e n t r a t i o n Wulff (1901) and Marc (1908,1909,1910) found t h a t some c r y s t a l faces of a given c r y s t a l grow f a s t e r than others whereas they d i s s o l v e at the same r a t e ; as w e l l , i t was shown th a t some substances r e t a r d or i n h i b i t growth while having no e f f e c t on the d i s s o l u t i o n r a t e s . Therefore c r y s t a l growth was not simply the reverse of d i s s o l u t i o n . Marc a l s o found that i n s t r o n g l y s t i r r e d s o l u t i o n s , the f i l m t h i c k n e s s i s v i r t u a l l y zero. This would l e a d t o near i n f i n i t e growth r a t e s (eqn 2.8), so c l e a r l y , the concept of f i l m d i f f u s i o n by i t s e l f could not e x p l a i n c r y s t a l growth. Berthoud (1912) and Valeton (1923,1924) subsequently suggested t h a t there were two steps i n v o l v e d i n c r y s t a l growth. These were: a d i f f u s i o n process by which s o l u t e molecules are t r a n s p o r t e d from the bulk of the s o l u t i o n t o the c r y s t a l s u r f a c e ; then a ' f i r s t - o r d e r ' r e a c t i o n during which the s o l u t e molecules are arranged i n t o the c r y s t a l l a t t i c e . A p i c t o r i a l r e p r e s e n t a t i o n of the two processes i s given i n ( f i g . 2.6). These two steps occurred under d i f f e r e n t c o n c e n t r a t i o n d r i v i n g f o r c e s , and were represented as: 20 dm/dt = k d A c ( c - c i ) ( d i f f u s i o n ) (2.9) dm/dt = k rA c(c-L-c*) (reaction) (2.10) where k d : c o e f f i c i e n t of mass t r a n s f e r by d i f f u s i o n k r : r a t e constant f o r the surface r e a c t i o n c : s o l u t e c o n c e n t r a t i o n i n the supersaturated s o l u t i o n Cj_ : s o l u t e c o n c e n t r a t i o n i n the s o l u t i o n at the c r y s t a l - s o l u t i o n i n t e r f a c e c* : e q u i l i b r i u m s a t u r a t i o n c o n c e n t r a t i o n A c : c r y s t a l surface area Because the i n t e r f a c i a l c o n c e n t r a t i o n , c^, i s very d i f f i c u l t to measure, an ' o v e r a l l ' c o n c e n t r a t i o n d r i v i n g f o r c e i s used: dm/dt = K g A c ( c - c * ) b (2.11) where Kg : o v e r a l l c r y s t a l mass growth r a t e constant I f b=l, the i n t e r f a c i a l c o n c e n t r a t i o n , c^, can be e l i m i n a t e d from equations (2.9) and (2.10) to g i v e : dm A c(c-c*) = (2.12) dt l / k d + l / k r Equating equations (2.11) and (2.12) g i v e s : Adsorption layer Driving force for diffusion C: Driving force for reaction — C IpStagnant Bulk of solution llfilm Crystal: solution interface c o c a> o c o F i g u r e (2.6) Concentration D r i v i n g Forces In C r y s t a l l i z a t i o n From S o l u t i o n 22 For cases of r a p i d surface r e a c t i o n (large k r) Kg = and the c r y s t a l l i s a t i o n i s d i f f u s i o n c o n t r o l l e d . For r a p i d d i f f u s i o n (large k^) Kg * k r and surface i n t e g r a t i o n i s c o n t r o l l i n g . The mass r a t e expression can be converted to growth r a t e u s i n g volume and area shape f a c t o r s and the c r y s t a l d ensity. m = k v p L 3 ; dm = 3k vpL 2dL ; A c = k a L 2 where k a, k v are the c r y s t a l area and volume shape f a c t o r s r e s p e c t i v e l y p : c r y s t a l d e n s i t y L : c h a r a c t e r i s t i c l i n e a r dimension of the c r y s t a l This leads to a s i m i l a r expression: dL K g k a ( c - c * ) 9 G = = — (2.14) dt 3k vp or G = K G ( c - c * ) 9 = K GS9 (2.15) where K G : o v e r a l l c r y s t a l l i n e a r growth r a t e constant Many workers (for example, M u l l i n and Garside (1967); Bransom et a l . (1969); Timm and Cooper (1969); Desai et a l . (1974)) have s u c c e s f u l l y c o r r e l a t e d o v e r a l l c r y s t a l growth r a t e s w i t h the o v e r a l l d r i v i n g force through such a power law. I t should be pointed out th a t Garside and M u l l i n (1968), Garside et a l . (1974), and P h i l l i p s and E p s t e i n (1974) a l l show t h a t growth r a t e s are i n general c o n s i d e r a b l y s m a l l e r than the corresponding d i s s o l u t i o n r a t e s under i d e n t i c a l hydrodynamic c o n d i t i o n s . I t can t h e r e f o r e be concluded t h a t surface i n t e g r a t i o n o f f e r s greater r e s i s t a n c e t o growth than does volume d i f f u s i o n . However, the d i s s o l u t i o n r a t e s vary l i n e a r l y w i t h the c o n c e n t r a t i o n d r i v i n g f o r c e , as would be expected i f volume d i f f u s i o n were more important. As f o r the o v e r a l l c r y s t a l growth r a t e , i t does not vary l i n e a r l y with the c o n c e n t r a t i o n d r i v i n g f o r c e , suggesting t h a t the surface i n t e g r a t i o n k i n e t i c s are not f i r s t - o r d e r as put forward by Berthoud (1912) and Valeton (1923,1924). 2.4 RELATIVE CRYSTALLIZATION KINETICS As p r e v i o u s l y mentioned, growth G, and n u c l e a t i o n B°, r a t e s can be c o r r e l a t e d t o a power f u n c t i o n of the s u p e r s a t u r a t i o n S, according t o : G = K GS9 (2.15) and B° = KgS1-5 f o r primary n u c l e a t i o n , (2.4) or B° = K NM T3s u f o r secondary n u c l e a t i o n (2.5) I t i s o f t e n d i f f i c u l t t o a c c u r a t e l y measure the very low s u p e r s a t u r a t i o n e x i s t i n g i n MSMPR c r y s t a l l i z e r s . Many workers have consequently opted to e l i m i n a t e s u p e r s a t u r a t i o n from the c o r r e l a t i o n s by combining (2.15) wi t h (2.4) or (2.5) to g i v e : B" = K^G1 f o r primary n u c l e a t i o n , (2.16) and B° = K nM TJG v f o r secondary n u c l e a t i o n (2.17) Equations (2.16) and (2.17) represent the r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s f o r an MSMPR c r y s t a l l i z e r . They provide a r e l a t i o n s h i p between the competing k i n e t i c s of nu c l e a t i o n and growth, and set l i m i t s on the the a t t a i n a b l e c r y s t a l s i z e d i s t r i b u t i o n (CSD). Most a v a i l a b l e data i n d i c a t e t h a t i , j , and v are not fu n c t i o n s of temperature or a g i t a t i o n . and K n account f o r the e f f e c t s of temperature, a g i t a t i o n , hydrodynamics and imp u r i t y c o n c e n t r a t i o n . The r e l a t i v e k i n e t i c orders i and v are the r a t i o s of the n u c l e a t i o n and growth orders. For c o o l i n g and evaporative c r y s t a l l i z a t i o n , secondary n u c l e a t i o n i s dominant and the r e l a t i v e k i n e t i c order i s g e n e r a l l y between 1 and 2 (Garside and Shah (1980)). In p r e c i p i t a t i o n and s a l t i n g - o u t c r y s t a l l i z a t i o n , i t i s primary n u c l e a t i o n t h a t i s important. The higher r e l a t i v e k i n e t i c orders i n these systems are t o be expected due to the n o n l i n e a r i t y of primary n u c l e a t i o n with s u p e r s a t u r a t i o n . 2.5 THE EFFECT OF HYDRODYNAMICS ON THE CRYSTALLIZATION  KINETICS The hydrodynamics of a c r y s t a l l i z e r are f u n c t i o n s of the c r y s t a l l i z e r g e o m e t r y a n d d e g r e e o f a g i t a t i o n . A s p r e v i o u s l y m e n t i o n e d , c r y s t a l g r o w t h i n v o l v e s t h e t w o s t e p s o f d i f f u s i o n a n d s u r f a c e r e a c t i o n . Where t h e d i f f u s i o n s t e p i s i m p o r t a n t , i n c r e a s i n g t h e a g i t a t i o n r e d u c e s t h e t h i c k n e s s o f t h e s t a g n a n t l a y e r n e x t t o t h e c r y s t a l s u r f a c e ( d e c r e a s i n g d i f f u s i o n r e s i s t a n c e ) , r e s u l t i n g i n an i n c r e a s e i n t h e c r y s t a l g r o w t h r a t e . G a r s i d e (1984) p o i n t s o u t t h a t t h e g r o w t h r a t e w i l l c o n t i n u e t o i n c r e a s e w i t h i n c r e a s i n g a g i t a t i o n u n t i l some l i m i t i n g v a l u e i s r e a c h e d c o r r e s p o n d i n g t o t h e p u r e s u r f a c e r e a c t i o n s t e p . T h i s h a s b e e n e x p e r i m e n t a l l y o b s e r v e d b y s e v e r a l w o r k e r s , e . g . R o s e n a n d H u l b u r t ( 1971a ) u s i n g p o t a s s i u m s u l p h a t e a n d L i u e t a l . (1971) w i t h m a g n e s i u m s u l p h a t e . The n u c l e a t i o n r a t e i s a l s o a f f e c t e d b y t h e h y d r o d y n a m i c s . I n c r e a s e d a g i t a t a t i o n w i l l i m p r o v e t h e d e g r e e o f m i x i n g i n t h e c r y s t a l l i z e r t h e r e b y m i n i m i s i n g l o c a l a r e a s o f h i g h s u p e r s a t u r a t i o n a n d h a v e an e f f e c t on p r i m a r y n u c l e a t i o n . The h i g h e r i m p a c t e n e r g i e s o f c r y s t a l - c r y s t a l a n d c r y s t a l -c r y s t a l l i z e r c o l l i s i o n s a s s o c i a t e d w i t h i n c r e a s e d a g i t a t i o n w i l l l e a d t o i n c r e a s e d s e c o n d a r y n u c l e a t i o n . E m p i r i c a l c o r r e l a t i o n s b e t w e e n n u c l e a t i o n r a t e s a n d t h e h y d r o d y n a m i c s h a v e b e e n o f t h e f o r m : B ° oc w f ( 2 . 1 8 ) w h e r e W : i m p e l l e r s p e e d ( p e r s e c o n d ) B o u r n e a n d H u n g e r b u e h l e r ( 1 9 8 0 ) f o u n d f = 3 f o r p o t a s h a l u m , Bourne and Faubel (1981) found f=1.5 f o r ammonium sulphate, and Grootscholten et a l . (1981), Scrutton (1979), and Asselbergs (1978) a l l determined t h a t f=2 f o r sodium c h l o r i d e . 2.6 THE EFFECT OF TEMPERATURE ON CRYSTALLIZATION  KINETICS Genck and Larson (1972) s t u d i e d the i n f l u e n c e of temperature on n u c l e a t i o n and growth r a t e s i n a continuous MSMPR c r y s t a l l i z e r . They observed t h a t under constant c r y s t a l suspension d e n s i t y , temperature could have a d i f f e r e n t e f f e c t on these r a t e s depending on the choice of s o l u t e . For example, w i t h potassium n i t r a t e , growth r a t e i n c r e a s e s while n u c l e a t i o n r a t e decreases as the temperature i s r a i s e d . For potassium c h l o r i d e , however, growth r a t e d e c l i n e s and n u c l e a t i o n r a t e increases with i n c r e a s i n g temperature. Working with potassium alum, Rousseau and Woo (1978) observed t h a t f o r a given s u p s e r s a t u r a t i o n , temperature i n f l u e n c e d the n u c l e a t i o n r a t e more s t r o n g l y than the growth r a t e . Several workers have sought to q u a n t i f y the e f f e c t of temperature on the c r y s t a l l i z a t i o n k i n e t i c s using an Arrhenius type expression : K = A exp(-E/RT) (2.19) where A : pre-exponential f a c t o r E : a c t i v a t i o n energy f o r c r y s t a l growth or n u c l e a t i o n R : gas constant T : absolute temperature The a c t i v a t i o n energy f o r c r y s t a l growth has g e n e r a l l y been reported t o have p o s i t i v e values of about 12,000-88,000 kJ/kmol. Budz et a l . (1985) determined a growth a c t i v a t i o n energy of +12,000 kJ/kmol f o r sodium t h i o s u l p h a t e pentahydrate c r y s t a l s . They concluded from t h i s low value and the l i n e a r dependence of the growth r a t e on sup e r s a t u r a t i o n , that the d i f f u s i o n step was c o n t r o l l i n g . Working w i t h potassium sulphate, Jones et a l . (1986) concluded t h a t the r e l a t i v e l y high growth a c t i v a t i o n energy of +40,400 kJ/kmol i n d i c a t e d surface r e a c t i o n c o n t r o l of the growth process. Both p o s i t i v e and negative values have been reported f o r the n u c l e a t i o n a c t i v a t i o n energy. For example, Helt and Larson (1977) found n u c l e a t i o n a c t i v a t i o n energy of -108,000 kJ/kmol f o r potassium n i t r a t e , and f o r potassium alum, Rousseau and Woo (1978) reported an a c t i v a t i o n energy of +100,000 kJ/kmol. 2.7 THE POPULATION BALANCE CONCEPT C r y s t a l l i z a t i o n from s o l u t i o n i s c h a r a c t e r i z e d by the formation of a spectrum of d i f f e r e n t l y s i z e d c r y s t a l s . This c r y s t a l s i z e d i s t r i b u t i o n (CSD) r e s u l t s from the continuous i n t e r a c t i o n between the n u c l e a t i o n and growth r a t e s of the c r y s t a l s , as w e l l as the c o n s t r a i n t s and geometry of the c r y s t a l l i z i n g system. In a c r y s t a l l i z e r , the law of conservation of p o p u l a t i o n may be expressed f o r the whole system as : N i n ~ N o u t = A c c u m u l a t i o n (2.20) where N : i s the t o t a l number of c r y s t a l s Where the c r y s t a l s are growing, the o v e r a l l p o p u l a t i o n balance f o r c r y s t a l s of s i z e ,L, w i l l be : N l i n " N l o u t = N I accumulation (2.21) Since the number of c r y s t a l s may change from p o i n t t o p o i n t w i t h i n the system, the c h a r a c t e r i s t i c property t o be considered i s the frequency p o p u l a t i o n d e n s i t y : n = n (x, y, z, L, t) (2 .22) where n i s the number of c r y s t a l s of s i z e ,L, at the p o s i t i o n (x,y,z) at time, t . Defined i n t h i s way, n i s r e f e r r e d t o as the po i n t p o p u l a t i o n d e n s i t y and has dimensions of c r y s t a l s per u n i t time per u n i t volume and u n i t l e n g t h . At steady s t a t e then, the t o t a l number of c r y s t a l s , N T , i n a c r y s t a l l i z i n g system i s : C O o o o o o o NT = f ,J J I t A n ( x ; y , z , L ) dx dy dz dL (2.23) 1 x=0 y=0 z=0 L=0 Expressing the po p u l a t i o n d e n s i t y f o r the c r y s t a l l i z i n g system as a whole : C O C O o o N(L) = J J ft I n(x,y,z,L) dx dy dz (2.24) x=0 y=0 z=0 where N(L) i s the number of c r y s t a l s of s i z e L at steady s t a t e . N(L) i s defined as the t o t a l p o p u l a t i o n d e n s i t y and has dimensions, number of c r y s t a l s per u n i t l e n g t h . Assuming tha t the v a r i a t i o n of pop u l a t i o n d e n s i t y from p o i n t t o po i n t i s n e g l i g i b l e , N(L) can be averaged over the volume, V, of the c r y s t a l l i z e r suspension g i v i n g : n (L) = N(L)/V (2.25) where n(L) i s defined as the average p o p u l a t i o n d e n s i t y , and has dimensions of number of c r y s t a l s per u n i t length and u n i t volume of the c r y s t a l l i z e r suspension. Thus, the number of c r y s t a l s present i n a small s i z e range, AL, and per u n i t volume of c r y s t a l l i z e r suspension i s given by : L+AL AN(L)/V = £ n(L) AL (2.26) 2.8 THE STEADY STATE MIXED-SUSPENSION MIXED-PRODUCT  REMOVAL (MSMPR) CRYSTALLIZER The MSMPR c r y s t a l l i z e r i s a concept according to which there i s no s p a t i a l v a r i a t i o n of i n t e n s i v e v a r i a b l e s , and the e x i t stream from the c r y s t a l l i z e r i s i d e n t i c a l i n a l l respects 30 w i t h the c r y s t a l l i z e r contents. The d e f i n i t i o n of the system and i t ' s c o n s t r a i n t s are : - the c r y s t a l l i z e r volume i s well-mixed, - the withdrawal stream i s u n c l a s s i f i e d and re p r e s e n t a t i v e of the c r y s t a l l i z e r contents, - there i s n e g l i g i b l e c r y s t a l breakage or a t t r i t i o n , - there i s a uniform c r y s t a l shape f a c t o r . 2.8.1 THE CRYSTAL POPULATION BALANCE At steady s t a t e , the number rat e of c r y s t a l s e n t e r i n g a s i z e range must equal the number rat e l e a v i n g . Consider an a r b i t r a r y s i z e range L^ t o L2, i n a volume, V, having p o p u l a t i o n d e n s i t i e s n^ and n2 r e s p e c t i v e l y . Let the growth r a t e of c r y s t a l s of s i z e range L^ be G^, and tha t of s i z e L2 be G2• For an increment of time 8t, the number of c r y s t a l s e n t e r i n g t h i s range by growth i s : I f the feed contains c r y s t a l s i n t h i s range, then the input to the d i s t r i b u t i o n i n the volume V i s : Vn-^St (2.27) and the number l e a v i n g by growth i s Vn 2G 25t (2.28) Q in i8L8t (2.29) where Qj_ volu m e t r i c flow r a t e n^ : average po p u l a t i o n d e n s i t y i n the range L-j_-L2 i n the feed 8L = L2-L-^ The number removal by bulk flow i n t h i s range i s : Qn8L8t (2.30) ( s u b s c r i p t s can be dropped due to assumption of mixed product removal). Since input must equal output f o r the s i z e range, we have C^n^LSt + VG 1n 18t = Qn8L8t + VG 2n 28t (2.31) and rearranging : V ( G 2 n 2 - G ^ ) = (Qini - Qn) 8L (2.32) As 8L tends to zero, the average values of n become p o i n t values and the equation takes the form : V d(Gn) = Qini - Qn (2.33) dL which f o r an unseeded system (n^ = 0) becomes : V d(Gn) + n = 0 (2.34) Q dL For systems t h a t obey McCabe 1s AL law, (that i s , c r y s t a l growth i s not a f u n c t i o n of c r y s t a l s i z e ) the expression becomes GT(dn) + n = 0 (2.35) dL where residence time, T = V/Q D e f i n i n g n° as the po p u l a t i o n d e n s i t y of c r y s t a l n u c l e i , then, as L tends to zero : n L J o dn/n = -J dL/ (Gx) (2.36) n 0 This i n t e g r a t e s to : n = n° exp(-L/GX) (2.37) or e q u i v a l e n t l y : In n = In n° - L/GX (2.38) which i s the po p u l a t i o n balance equation f o r s i z e -independent c r y s t a l growth, and describes the po p u l a t i o n d e n s i t y d i s t r i b u t i o n to be expected under the p r e v i o u s l y mentioned c o n s t r a i n t s . 2.8.2 DERIVING PURE CRYSTALLIZATION KINETICS FOR SIZE  INDEPENDENT GROWTH Equation (2.37) enables d e r i v a t i o n of the c r y s t a l l i z a t i o n k i n e t i c s from a s i z e a n a l y s i s performed on a r e p r e s e n t a t i v e sample of the c r y s t a l l i z e r contents at steady s t a t e . P l o t t i n g In n against L r e s u l t s i n a s t r a i g h t l i n e whose slope i s equal t o the negative r e c i p r o c a l of the product GT. Since the residence time X i s known from the operation of the c r y s t a l l i z e r , the growth r a t e G of the c r y s t a l s can be c a l c u l a t e d . The s t r a i g h t l i n e i n t e r c e p t gives the value f o r n°, the n u c l e i p o p u l a t i o n d e n s i t y . This parameter i s r e l a t e d t o the n u c l e a t i o n k i n e t i c s according to : the n u c l e a t i o n r a t e B° can be represented as B° = dN dt L=0 dN' dt (2.39) and the growth r a t e , G, may be w r i t t e n as dL/dt, t h e r e f o r e dN dt L=0 dL dt dN dL L=0 (2.40) r e c a l l i n g that dN" dL dN dL L=0 n' (2.41) leads to B° = n°G° - n°G (2.42) sin c e G° = G f o r s i z e independent growth. G° : growth ra t e of 'zero s i z e ' n u c l e i G : growth r a t e of c r y s t a l s 2.8.3 COEFFICIENT OF VARIATION OF THE CSD The c r y s t a l s i z e d i s t r i b u t i o n can be def i n e d by two parameters, namely, the median s i z e and the c o e f f i c i e n t of v a r i a t i o n . The c o e f f i c i e n t of v a r i a t i o n (cv) of a d i s t r i b u t i o n of c r y s t a l s from a steady s t a t e MSMPR c r y s t a l l i z e r i s given by : cv % = 1 0 0 ( a 1 6 - a 8 4 ) / 2 ( a 5 0 ) (2.43) where a^g and a g 4 are the aperture s i z e s of the siev e s t h a t r e t a i n 16 and 84% of the c r y s t a l s r e s p e c t i v e l y . The c r y s t a l median s i z e i s d e f i n e d as the aperture s i z e (a^g) of the sieve t h a t r e t a i n s 50% of the d i s t r i b u t i o n . The CSD generated from an i d e a l steady s t a t e MSMPR c r y s t a l l i z e r has a cv of 52%, and a median s i z e equal to 3.67GT. 2.8.4 SIZE-DEPENDENT GROWTH As has been p r e v i o u s l y mentioned, c r y s t a l growth depends on two processes, d i f f u s i o n and surface r e a c t i o n . In most c r y s t a l l i z i n g systems, the d i f f u s i o n r e s i s t a n c e i s much l e s s than the r e s i s t a n c e due to the surface r e a c t i o n , and the growth r a t e i s r e a c t i o n c o n t r o l l e d . Such systems obey McCabe 1s AL law, and have c r y s t a l growth r a t e s t h a t are independent of the c r y s t a l s i z e . Other systems, however, have greater d i f f u s i o n r e s i s t a n c e and e x h i b i t growth r a t e s t h a t are a f u n c t i o n of c r y s t a l s i z e . Examples of such systems are potash-alum, and potassium sulphate. Under these circumstances, expressions (2.37, 2.38) do not hold. Indeed, a p l o t of In n against L, shows a d i s t i n c t upward curvature ( f i g . 2.7) i n s t e a d of the expected s t r a i g h t l i n e . Various e m p i r i c a l models have been proposed t o account f o r t h i s s i z e dependence of growth r a t e , e.g. Bransom (1960) : Eqn (2.44) however, p r e d i c t s very la r g e growth r a t e s f o r l a r g e c r y s t a l s and zero growth f o r n u c l e i . An improved model was proposed by Canning and Randolph (1967) as : where G° i s the growth r a t e at zero s i z e and i s a f u n c t i o n of s u p e r s a t u r a t i o n . a^ i s an e m p i r i c a l l y determined constant. This expression was f u r t h e r r e f i n e d by Abegg et a l . (1968) to : G = k S a L d (2.44) G = G° (1+a-LL) (2.45) G = G°(l+a 2L) d, d < 1 (2.46) where a 2 and d are two a d j u s t a b l e e m p i r i c a l constants d e f i n i n g the s i z e dependence of the growth r a t e . Crystal siz«. L F i g u r e (2.7) C r y s t a l Size D i s t r i b u t i o n (CSD) For Size Dependent C r y s t a l Growth Rate 37 C H A P T E R T H R E E 3.1 EXPERIMENTAL SET-UP The experimental equipment ( f i g . 3.1) c o n s i s t e d of two g l a s s d i s s o l v i n g tanks, a g l a s s c r y s t a l l i z e r , and a p e r i s t a l t i c pump. The c r y s t a l l i z e r was jacketed, and had a d r a f t tube which served the dual purposes of enhancing mixing as w e l l as being a c o o l i n g surface. Pyrex g l a s s and C - f l e x t u b i n g were the m a t e r i a l s of choice because of the h i g h l y c o r r o s i v e nature of 9N s u l p h u r i c a c i d . C o o l i n g of the c r y s t a l l i z e r (working volume 1.45L) was achieved by passing c o o l i n g water from a water-bath through the d r a f t - t u b e and j a c k e t . This arrangement allowed f o r accurate c r y s t a l l i z e r temperature c o n t r o l to w i t h i n 0.5 C, as measured by a mercury thermometer i n the c r y s t a l l i z e r . Two d i s s o l v i n g tanks were r e q u i r e d to ensure t h a t a l l the c r y s t a l s were d i s s o l v e d p r i o r to the s o l u t i o n r e t u r n i n g to the c r y s t a l l i z e r . Tank A had a ca p a c i t y of 1.95L and was maintained at above 95 C. Tank B had a working volume of 17L and was maintained at about 85 C. The two tanks were heated using g l a s s - e n c l o s e d e l e c t r i c heating c o i l s . The residence time was c o n t r o l l e d by a d j u s t i n g the pumping ra t e with the pump speed c o n t r o l l e r . The flow would be set and measured at the beginning of the run, then re-measured at the end of the run. The mean value would be taken and used t o c a l c u l a t e the residence time f o r t h a t run. Sampling Heating c o i l s Heating c o i l s D i s s o l v i n g Tank B Pump Figur e (3.1) Schematic Of The Experimental Set-up 39 Rotameters were not used t o continuously monitor the flow during the run because they i n v a r i a b l y caused premature c r y s t a l l i z a t i o n and subsequent blockage of the system. The supersaturated s o l u t i o n was pumped from tank B t o the c r y s t a l l i z e r , from where i t flowed by g r a v i t y t o tank A and back t o tank B. Samples were obtained by i n t e r c e p t i n g the flow from the c r y s t a l l i z e r t o tank A. The c r y s t a l l i z e r and tanks were s t i r r e d by l J / H diameter polyethylene i m p e l l e r s connected to v a r i a b l e speed motors. 3.2 VERIFICATION OF UNIFORM MIXING IN THE CRYSTALLIZER One of the assumptions made i n the MSMPR c r y s t a l l i z e r i s that the c r y s t a l l i z e r contents are well-mixed and t h a t the e x i t stream i s r e p r e s e n t a t i v e of the c r y s t a l l i z e r contents. P r e l i m i n a r y work was done to confirm that t h i s was the case. The c r y s t a l l i z e r was operated at the lowest temperature and the longest residence time t h a t would be used during the main body of experimentation. This would give the greatest suspension d e n s i t y l i k e l y to be encountered. I f uniform mixing i s e s t a b l i s h e d under these c o n d i t i o n s , then under l e s s extreme c o n d i t i o n s the MSMPR assumption should be v a l i d . The i m p e l l e r rpm was adjusted u n t i l no c r y s t a l s e t t l i n g was d i s c e r n i b l e . Two samples each were then taken from the c r y s t a l l i z e r e x i t stream, and from w i t h i n the c r y s t a l l i z e r at the top, middle and lower l e v e l s of the s l u r r y . The suspension d e n s i t y and the c r y s t a l s i z e d i s t r i b u t i o n were then determined as des c r i b e d elsewhere SAMPLE ONE SAMPLE TWO AVERAGE CRYSTALLIZER REGION M T LM M T LM M T TOP 0.2627 0.323 0.2445 0.314 0.2536 0.318 MIDDLE 0.2502 0.321 0.2436 0.265 0.2469 0.324 BOTTOM 0.2377 0.323 0.2652 0.327 0.2514 0.325 EXIT STREAM 0.2417 0.309 0.2384 0.314 0.2401 0.312 M T : Suspension density (g c r y s t a l s / m l of suspension) L M : Median s i z e of c r y s t a l s (mm) Table (3.1) V e r i f i c a t i o n Of Uniform M i x i n g In The C r y s t a l l i z e r (sect.3.4 & 3.5). For an i m p e l l e r rpm of 1300, i t was found (table 3.1) tha t the s l u r r y of average d e n s i t y 0.2480 g/ml was unif o r m l y d i s t r i b u t e d and discharged. A l l the experimental work was t h e r e f o r e c a r r i e d out at t h i s i m p e l l e r rpm. 3.3 DETERMINATION OF TIME TO STEADY-STATE The c r y s t a l l i z e r must be at steady-state f o r the a n a l y s i s methods t o be v a l i d . I t i s t h e r e f o r e necessary to e s t a b l i s h the number of residence times that must elapse before steady-state i s reached and sampling may begin. Samples of the c r y s t a l l i z e r e x i t stream were taken at i n t e r v a l s of one residence time beginning at time zero (when the c r y s t a l l i z e r had s t a b i l i z e d at the operati n g temperature), f o r a t o t a l of eight residence times. The t e s t runs were c a r r i e d out at 45 C f o r the longest residence time (15 min.), and at 60 C f o r the sho r t e s t residence time (7 min.). Two samples were taken per run. The c r y s t a l s i z e d i s t r i b u t i o n was obtained by s i e v i n g , then the c r y s t a l growth r a t e , G, was c a l c u l a t e d according to eqn(4.2). P l o t s of G against residence time u n i t s show steady-state t o be e s t a b l i s h e d w i t h i n f i v e residence times. A l l f u t u r e experiments were allowed to run f o r s i x residence times before sampling was begun. G r a p h i c a l r e s u l t s are given i n f i g . (3.2). E E LU DC o DC CD 0.8 h 0.6 0.4 0.2 0 o 0 • 0 • 9 0 • 0 o 4 5 C • 0 0 • + + • • + • + + 4-a 6 0 C + a 1 i 2 4 6 8 R E S I D E N C E TIME? UNITS 10 * two samples/run * Fig u r e (3.2) Time To Steady-State In An MSMPR C r y s t a l l i z e r 43 3.4 DETERMINATION OF CRYSTAL S I Z E D ISTR IBUT ION (CSD) A f t e r the c r y s t a l l i z e r reaches steady s t a t e , samples of the c r y s t a l suspension are taken from the e x i t stream. The stream i s d i r e c t e d to a f i l t e r i n g funnel and about 120 ml of the suspension i s c o l l e c t e d . The f i l t e r i n g funnel w i l l have been pre-heated i n the oven to approximately the c r y s t a l l i z e r temperature. This prevents c r y s t a l l i z a t i o n t a k i n g place a f t e r sampling. Upon f i l t r a t i o n , the f i l t r a t e i s d i s c a r d e d and the r e t a i n e d c r y s t a l s are washed s e v e r a l times using denatured a l c o h o l . This removes any adhering a c i d and ensures that agglomeration of the c r y s t a l s does not occur. The washed c r y s t a l s are c a r e f u l l y t r a n s f e r r e d onto a watch-glass and oven-dried at about 190 C f o r 12 hours. The dry c r y s t a l s are then c l a s s i f i e d u sing a nest of sieves i n a mechanical shaker. The d u r a t i o n of the shaking i s 60 minutes, and the d i f f e r e n c e between the s t a r t i n g and ending weights of the i n d i v i d u a l sieves i s the mass of c r y s t a l s on each s i e v e . The c l a s s i f i e d c r y s t a l s are kept f o r l a t e r determination of the volume shape f a c t o r . There were 10 sieves used, v i z . , 355, 300, 250, 212, 180, 150, 125, 106, 90, and 75 micrometres. 3.5 DETERMINATION OF CRYSTAL SUSPENSION DENS ITY , Mrp About 120 ml of the e x i t stream i s c o l l e c t e d onto a f i l t e r i n g f u n n e l . The f i l t r a t e i s suctioned o f f and i t ' s volume noted. The c o l l e c t e d c r y s t a l s are t r a n s f e r r e d t o a watch-glass and oven-dried f o r 12 hours at about 190 C. The suspension d e n s i t y i s obtained from the oven-dry weight of the c r y s t a l s and the f i l t r a t e volume. 3.6 DETERMINATION OF SOLUB I L ITY Three stoppered f l a s k s c o n t a i n i n g the solvent (9N H2SO4) and an excess of the s o l u t e (Na2S04) are placed i n a water-bath and l e f t (with o c c a s s i o n a l shaking) f o r 72 hours t o reach e q u i l i b r i u m . A f t e r t h i s p e r i o d , each f l a s k ' s s a t u r a t e d s o l u t i o n i s c a r e f u l l y decanted i n t o two pre-weighed f l a s k s . The masses, s, of the s o l u t i o n s are noted and they are then t i t r a t e d t o the end-point with IN NaOH s o l u t i o n . The i n d i c a t o r used i s Bromothymol Blue, and the volume of c a u s t i c consumed i s noted. The f l a s k s are then oven-dried f o r 12 hours at about 190 C. They are then weighed and the weight d i f f e r e n c e , mc, i s the mass of the t o t a l Na2S04 present. That i s , i t i s the sum of the s a l t o r i g i n a l l y present i n the sample, plus the s a l t produced during the n e u t r a l i z a t i o n of the a c i d i n the sample. Knowing the volume of c a u s t i c consumed, v t , the mass of s a l t a r i s i n g from the ' n e u t r a l i z a t i o n can be c a l c u l a t e d and the s a l t / a c i d c o n c e n t r a t i o n can then be e a s i l y determined. The s o l u b i l i t y i s taken to be the mean value of the concentrations obtained. S o l u b i l i t y , c* = mc - 0.071v t s - (mc - 0.071v t) (3.1) TEMPERATURE °C SOLUBILITY, c* (g s a l t / g acid) Sample 1 Sample 2 Sample 3 Mean 30 0.462 0.468 0.465 40 0.482 0.486 0.484 50 0.522 0.509 0.518 0.516 60 0.549 0.549 0.543 0.547 70 0.569 - 0.580 0.583 0.577 80 0. 609 0.610 0.602 0. 607 c = 0.372 + 0.00291T" c* : s o l u b i l i t y (g s a l t / g acid) T' : temperature (°C) R-square = 99.5% Table (3.2) S o l u b i l i t y of Sodium Sulphate In 9N Sulp h u r i c A c i d -3 09 20 4 0 6 0 Temperature C 8 0 100 F i g u r e (3.3) Sodium Sulphate S o l u b i l i t y In 9N Sulph where 0.071v t : mass of s a l t from the n e u t r a l i z a t i o n (g) mc - 0.071v t : mass of d i s s o l v e d s a l t i n the s a t u r a t e d s o l u t i o n (g) s : mass of the sat u r a t e d s o l u t i o n (g) v t : volume (ml) of IN NaOH consumed mc : t o t a l mass of s a l t c r y s t a l s ( d i s s o l v e d + ne u t r a l i z e d ) (g) 0.071 : f a c t o r converting.the volume of c a u s t i c consumed to the mass of s a l t produced by n e u t r a l i z a t i o n The r e s u l t i n g s o l u b i l i t y data (table 3.2) were regressed t o obt a i n an expression r e l a t i n g the s o l u b i l i t y t o the temperature. These c a l c u l a t e d values were the ones used as the s a t u r a t i o n concentrations, c , when determining s u p e r s a t u r a t i o n s (sect. 3.7) i n the main body of experiments. Figure 3.3 shows the s o l u b i l i t y of sodium sulphate i n 9N s u l p h u r i c a c i d . 3.7 DETERMINATION OF SUPERSATURATION A sample of the c r y s t a l l i z e r e x i t stream i s c o l l e c t e d i n t o a f i l t e r i n g f u n n e l . The f i l t r a t e i s suctioned o f f i n t o a pre-weighed 125 ml f l a s k . The f l a s k i s re-weighed and the mass of f i l t r a t e noted. The f l a s k i s stoppered and placed i n a water-bath whose thermostat i s set t o the temperature p r e v a i l i n g i n the c r y s t a l l i z e r at the time of sampling. The f l a s k remains there f o r 72 hours w i t h p e r i o d i c shaking, so as t o reach e q u i l i b r i u m . At the end of t h i s time, the f l a s k contents are emptied i n t o a 60 ml coarse f r i t t e d g l a s s f i l t e r i n g funnel of known weight. The f i l t e r i n g funnel plus contents i s oven-dried f o r 12 hours at about 190 C, and i t ' s weight i s then noted. The d i f f e r e n c e between the s t a r t and end weights of the f i l t e r i n g funnel i s the mass of excess s a l t t h a t was d i s s o l v e d i n the o r i g i n a l f i l t r a t e . Knowing the s t a r t i n g weight of the f i l t r a t e , the excess s a l t present i n i t , and i t ' s f i n a l s a t u r a t i o n c o n c e n t r a t i o n (from the s o l u b i l i t y t e s t s ) , the su p e r s a t u r a t i o n can be determined according t o : S = x = x ( l + c*)/(m f - x) (3.2) (mf - x)*1/(1 + c ) where x : mass of excess s a l t (g) nif - x : mass of satu r a t e d s o l u t i o n (g) 1/(1 + c ) : mass of solvent (g)/unit mass of sa t u r a t e d s o l u t i o n (g) mf : mass of f i l t r a t e (g) : s a t u r a t i o n c o n c e n t r a t i o n (g s a l t / g acid) 3.8 DETERMINATION OF CRYSTAL SHAPE FACTOR For p a r t i c l e s below about 500 micrometres, determination of shape f a c t o r s i s very d i f f i c u l t . M u l l i n (1972) has come up wit h a s i m p l i f i e d procedure t h a t gives a good estimate. A f t e r the CSD samples have been shaken f o r 60 minutes, and the si e v e weights noted, each sieve (beginning w i t h the coarsest) i s emptied, cleaned and weighed again. Then, a piece of adhesive tape of known area i s a f f i x e d onto the underside of the empty s i e v e . The sieve (and tape) i s reweighed and the mass noted. Now, the c r y s t a l s from the next si e v e are poured onto t h i s f i r s t s ieve and i t i s manually shaken ( M u l l i n , 1972). A f t e r a few minutes, the adhesive tape i s removed and weighed. The tape now has a matrix of c r y s t a l s stuck to i t ' s surface, approximately one per si e v e aperture. Knowing the mesh number of the s i e v e , the area of the tape, and the d e n s i t y of the c r y s t a l s , the shape f a c t o r can be determined according t o : k v = m x/ypL 3 (3.3) where k v : shape f a c t o r y : number of c r y s t a l s on tape = NSA^.(1 in/25.4 mm) 2 mx : mass of c r y s t a l s on tape (g) p : c r y s t a l d e n s i t y = 2.27 g/cm3 * L : mean s i z e of c r y s t a l s , i . e . mean of the two sieve aperture s i z e s i n v o l v e d (mm) N s : si e v e mesh number, i . e . the number of apertures per sq. i n . A t : area of adhesive tape (sq.mm) The procedure i s repeated through to the f i n e s t s i e v e . This was done f o r a t o t a l of eighteen CSD samples, and an average * the c r y s t a l d e n s i t y was experimentally determined using a pycnometer value f o r k v obtained. The shape f a c t o r was found t o be 0.4 9 ( 0.06). Bourne and Faubel (1981) determined the volume shape f a c t o r f o r ammonium sulphate t o be 0.64, using a counting method.. Their r e s u l t i s an i n d i c a t o r t h a t the k v value found here i s reasonable. Both ammonium sulphate and sodium sesquisulphate form orthorhombic c r y s t a l s , though the l a t t e r has narrower c r y s t a l s . I t f o l l o w s t h a t the sodium sesquisulphate should have a lower shape f a c t o r . 3.9 VERIFICATION OF THE TYPE OF CRYSTAL PRODUCED Under the c o n d i t i o n s of experimentation, the sodium sulphate c r y s t a l l i z e s as the a c i d s a l t , sodium sesquisulphate (Na3HS0 4). The sesqui s a l t i s 18% by weight a c i d . T i t r a t i o n of the c r y s t a l l i z e r samples, a f t e r the CSD and shape f a c t o r analyses, found a l l the samples t o have a c i d contents between 17-20%, which i s c o n s i s t e n t w i t h the type of c r y s t a l form expected, i . e . the a c i d s a l t (sodium s e s q u i s u l p h a t e ) . C H A P T E R F O U R 4 .1 DETERMINATION OF NUCLEATION AND GROWTH RATES F o r a n MSMPR c r y s t a l l i z e r o p e r a t i n g a t s t e a d y - s t a t e , i t ' s n u c l e i p o p u l a t i o n d e n s i t y , n , i s g i v e n b y : n = n ° exp (-L/GT ) ( 2 . 3 7 ) F o r e a c h c r y s t a l s i z e f r a c t i o n r e t a i n e d b e t w e e n t w o a d j a c e n t s i e v e s o f a p e r t u r e s i z e s L-j_ a n d L 2 , (L2>L-^) , t h e n u c l e i p o p u l a t i o n d e n s i t y , n L , o f s i z e L, i s g i v e n b y : n L = w M T / p k v ( 8 L ) L 3 ( 4 . 1 ) w h e r e w : w e i g h t f r a c t i o n o f c r y s t a l s on t h e s i e v e , L^ M T : s u s p e n s i o n d e n s i t y p : c r y s t a l d e n s i t y k v : c r y s t a l v o l u m e s h a p e f a c t o r 8L : L 2 - L ] _ , b e i n g t h e d i f f e r e n c e s i n a p e r t u r e s i z e s o f a d j a c e n t s i e v e s L : ( L 2 + L 1 ) / 2 , t a k e n t o r e p r e s e n t t h e c r y s t a l s i z e F r o m e q n ( 2 . 3 7 ) ( s u b s t i t u t i n g n L f o r n ) , p l o t t i n g I n n a g a i n s t L, g i v e s a s t r a i g h t l i n e whose s l o p e i s e q u a l t o -1/Gx a n d whose i n t e r c e p t i s e q u a l t o I n n ° . T h u s , f o r G r o w t h r a t e : G = - 1 / ( s l o p e ) T ( 4 . 2 ) a n d f o r N u c l e a t i o n r a t e : ! N U C L E I D E N S I T Y , n (no. /mm.mm ) RUN #01 CRYSTAL DENSITY, p TEMPERATURE C SLURRY DENSITY Mt RESIDENCE t min 2.27 g/cm~3 50 0.2186 g/cmA3 7.19 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 5 . 7400 0 .871 0.300 0 . 1516 0 .023 2. 33E+00 0 . 845 0.250 0 .1713 0 .026 4. 89E+00 1 .587 0. 212 0 . 1450 0 .022 9. 18E-MD0 2 . 217 0.180 0 .1054 0 .016 1. 30E+01 2 . 564 0. 150 0 .0857 0 .013 1. 89E+01 2 . 937 0.125 0 .0791 0 .012 3. 61E+01 3 . 586 0. 106 0 .0330 0 .005 3. 34E+01 3 . 508 0.090 0 .0264 0 .004 5. 19E+01 3 . 950 0.075 0 .0198 0 .003 6. 96E+01 4 . 243 PAN 0 .0330 0 .005 TOTAL 6 . 5901 1 .000 SIEVE (mm) 0. 355 0. 300 0. 250 0.212 0. 180 0. 150 0. 125 0. 106 0.090 0.075 PAN dL (mm) 0.055 0.050 0.0.38 0.032 0.0.30 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7828 95.41081 96.1503 95.2151 91.5905 91.5093 86.5137 91.3118 87.2897 91.5688 86.8026 SIEVE + CRYSTALS (g) 100.5228 95.5597 96.3216 95.3601 91.6959 91.5950 86.5928 91.3448 87.3161 91.5886 86.8356 Regression Output: Constant 5.258 Std E r r of Y Est 0.118 R Squared 0.990 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -13.44 Std E r r of Coef. 0.5000 GROWTH RATE NUCLEI DENSITY NUCLEATION RATE SUPERSATURATION 0.621 mm/hr 192.1 number/mm.mm~3 119.3 number/mm^3 hr 0.040. g s a l t / g a c i d Table (4.1) T y p i c a l Lotus Regression P r i n t - o u t For C r y s t a l N u c l e a t i o n and Growth Rates B° = Gn° = G exp ( i n t e r c e p t ) (4.3) The least-squares r e g r e s s i o n f u n c t i o n of the LOTUS 123 spreadsheet program was used t o ob t a i n the c o r r e l a t i o n between In n and L. A sample p r i n t - o u t i s given i n t a b l e 4.1, and f i g u r e 4.1 gives the corresponding CSD p l o t . P r i n t -outs and p l o t s f o r a l l the runs are presented i n appendices A and B r e s p e c t i v e l y . 4.2 MODELLING THE GROWTH AND NUCLEATION RATE DATA The experiments were c a r r i e d out at four temperatures, 45, 50, 55, and 60 C. At each temperature, runs were conducted at three residence times, of approximately 7, 11, and 15 minutes. With the exception of the runs at 45 C, a l l experimental c o n d i t i o n s were r e p l i c a t e d t wice, and c a r r i e d out i n random order. Conditions at 45 C were r e p l i c a t e d only once, though two samples were taken of runs 19 and 21. The growth and n u c l e a t i o n r a t e data were c o r r e l a t e d at each temperature according to the models : G = K GS9 (2.15) B° = K B S b (2.4) B° = Kj^G1 (2.16) B° = K NM T^S U (2.5) B° = K nM TJG v (2.17) MTB > # TEMPERATURE 45 C MTB > PRINT CI2 C1-C4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 19 0 .037 0.459 190. 2 0.2720 2 19 0.037 0.442 184. 2 0.2720 3 20 0.055 0. 561 212. 1 0.2224 4 21 0.082 0.886 338. 1 0.2359 5 21 0.082 0.881 417.3 0.2359 MTB > # GROWTH RATE r e g r e s s i o n MTB > regr c6 1 c5 The r e g r e s s i o n equation i s Ln G = 1.97 + 0.847 Ln S P r e d i c t o r Constant Ln S Coef 1.9709 0.84679 Stdev 0.2273 0.07781 t - r a t i o 8.67 10.88 P 0.003 0.002 s = 0.06192 R-sq = 97.5% A n a l y s i s of Variance SOURCE DF SS Regression 1 0.45412 E r r o r 3 0.01150 T o t a l 4 0.46562 R-sq(adj) = 96.7% MS* 0.45412 0:00383 F 118.43 P 0.002 MTB > # PRIMARY NUCLEATION model MTB > regr c7 1 c5 The r e g r e s s i o n equation i s Ln B = 8.07 + 0.876 Ln S P r e d i c t o r Constant Ln S Coef 8.0744 0.8757 Stdev 0.5280 0.1808 t - r a t i o 15. 29 4. 84 P 0.001 0.017 s = 0.1438 R-sq A n a l y s i s of Variance 88. 7% R-sq(adj) = 84.9% SOURCE DF SS MS Regression 1 0.48561 0.48561 E r r o r . 3 0.06207 0.02069 T o t a l 4 0.54768 F 23. 47 P 0.017 Table (4.2a) T y p i c a l MINITAB Regression P r i n t - o u t For The C r y s t a l l i z a t i o n Model Parameters MTB > # PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 6.05 + 1.05 Ln G P r e d i c t o r Coef Stdev t - r a t i o p Constant 6.04542 0.08604 70.26 0.000 Ln G 1.0528 0.1503 7.00 0.006 s = 0.1026 R-sq = 94.2% R-sq(adj) = 92.3% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 1 0.51611 0.51611 49.04 0.006 E r r o r 3 .0.03157 0.01052 T o t a l 4 0.54768 MTB > # SECONDARY NUCLEATION model MTB > regr c7 2 c8 c5 < The r e g r e s s i o n equation i s Ln B = 23.1 + 1.70 Ln Mt + 1.18 Ln S P r e d i c t o r Coef Stdev t - r a t i o Constant 23 i.096 8 .064 2.86 0. 103 Ln Mt , 1. 7022 0. 9127 1.87 0.203 Ln S 1. 1798 0. 2109 5.59 0.031 s = 0.1064 R-sq = 95.9% R-sq(adj) = : 91.7% A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 0 . 52502 0.26251 23.17 0.041 E r r o r 2 0 .02266 0.01133 T o t a l 4 0 .54768 SOURCE DF SEQ SS Ln Mt 1 0 . 17050 Ln S 1 0 .35452 Table (4.2b) T y p i c a l MINITAB Regression P r i n t - o u t For The C r y s t a l l i z a t i o n Model Parameters 56 Unusual Observations Obs. Ln Mt Ln B 3 -8.41 5.3571 F i t S t d e v . F i t Residual St.Resid 5.3571 0.1064 0.0000 * X X denotes an obs. whose X value gives i t l a r g e i n f l u e n c e . MTB > # SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i s Ln B = 11.5 + 0.650 Ln Mt + 1.17 Ln G P r e d i c t o r Constant Ln Mt Ln G s = 0.1087 Coef 11.502 0.6501 1.1697 Stdev 6.662 0.7935 0.2139 t - r a t i o 1.73 0.82 5.47 P 0.226 0.499 0.032 R-sq = 95.7% R-sq(adj) = 91.4% A n a l y s i s of Variance SOURCE DF SS MS F Regression 2 0.52404 0.26202 22. 17 E r r o r 2 0.02364 0.01182 T o t a l 4 0.54768 SOURCE DF SEQ SS Ln Mt 1 0.17050 Ln G 1 0.35354 Unusual Observations Obs. Ln Mt Ln B F i t S t d e v . F i t R e s i d u a l 3 -8.41 5. .3571 5. 3582 0.1086 -0.0011 p 0.043 St.Resid -0.29 X X denotes an obs. whose X value gives i t l a r g e i n f l u e n c e . MTB > noou Table (4.2c) T y p i c a l MINITAB Regression P r i n t - o u t For The C r y s t a l l i z a t i o n Model Parameters 57 E x p r e s s i o n (2.15) r e l a t e s t h e g r o w t h r a t e t o t h e s u p e r s a t u r a t i o n , w h i l e e x p r e s s i o n (2.4) r e l a t e s t h e r a t e o f n u c l e a t i o n t o t h e s u p e r s a t u r a t i o n f o r t h e c a s e w h e r e s e c o n d a r y n u c l e a t i o n i s n o t a f a c t o r . I f s e c o n d a r y n u c l e a t i o n i s s i g n i f i c a n t t h e n e x p r e s s i o n (2.5) a p p l i e s . E x p r e s s i o n s (2.16) a n d (2.17) g i v e t h e r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s f o r t h e n o n - s e c o n d a r y a n d s e c o n d a r y c a s e s r e s p e c t i v e l y . T h e s e e x p r e s s i o n s w e r e l i n e a r i z e d b y c o n v e r t i n g t h e m t o t h e i r n a t u r a l l o g e q u i v a l e n t s . The r a t e c o n s t a n t s ( i n l o g f o r m ) a n d k i n e t i c o r d e r s g , b , i , u , j , a n d v w e r e t h e n o b t a i n e d t h r o u g h t h e l e a s t - s q u a r e s r e g r e s s i o n f u n c t i o n o f t h e M I N I T A B s t u d e n t ' s e d i t i o n , a s t a t i s t i c a l p r o g r a m . P r i n t -o u t s o f t h e r e g r e s s i o n r e s u l t s f o r 45 °C a r e g i v e n i n t a b l e s 4.2 ( a ) , ( b ) , a n d ( c ) ; t h e c o m p l e t e s e t i s g i v e n i n a p p e n d i x C . 4.3 DERIVING THE ACTIVATION ENERGIES OF CRYSTALLIZATION The r a t e c o n s t a n t s a r e f u n c t i o n s o f a g i t a t i o n , t e m p e r a t u r e , a n d i m p u r i t y . Among t h e s e , t e m p e r a t u r e was t h e o n l y v a r i a b l e t e s t e d i n t h i s s t u d y , a n d t h i s d e p e n d e n c y was f i t t e d t o an A r r h e n i u s t y p e e x p r e s s i o n : K = A e x p ( - E / R T ) (2.19) w h e r e E a c t i v a t i o n e n e r g y f o r n u c l e a t i o n o r g r o w t h ( k J / k m o l ) R u n i v e r s a l g a s c o n s t a n t , 8.315 k J / k m o l . K 58 MTB > # TEMPERATURE DEPENDENCE OF RATE CONSTANTS MTB > p r i n t c5 c l - c 3 ROW Temp. C 1/T (/K) LnKg LnKn 1 45 0.00314 1. 971 23.096 2 50 0.00310 2.085 18.245 3 55 0.00305 2. 693 17.883 4 60 0.00300 2.796 17.892 MTB > # GROWTH RATE CONSTANT r e g r e s s i o n MTB > regr c2 1 c l The r e g r e s s i o n equation i s LnKg = 22.6 - 6584 1/T (/K) P r e d i c t o r Coef Stdev t - r a t i o p Constant 22.617 4.381 5.16 0.036 1/T (/K) -6584 1426 -4.62 0.044 s = 0.1500 R-sq = 91.4% R-sq(adj) = 87.1% A n a l y s i s of Variance SOURCE DF SS MS • F p Regression 1 0.48015 0.48015 21.33 0.044 E r r o r 2 0.04502 - 0.02251 T o t a l 3 0.52517 MTB > # NUCLEATION RATE CONSTANT r e g r e s s i o n (45 C included) MTB > regr c3 1 c l The r e g r e s s i o n equation i s LnKn = - 80.9 + 32612 1/T (/K) P r e d i c t o r Coef Stdev t - r a t i o p Constant -80.92 57.41 -1.41 0.294 1/T (/K) 32612 18684 1.75 0.223 s = 1.966 R-sq = 60.4% R-sq(adj) = 40.6% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 1 11.779 11.779 3.05 0.223 E r r o r 2 7.732 3.866 T o t a l 3 19.511 Table (4.3a) MINITAB Regression P r i n t - o u t For The E f f e c t Of Temperature On The K i n e t i c Rate Constants MTB > It NUCLEATION RATE CONSTANT r e g r e s s i o n (45 C excluded) MTB > l e t c 3 ( l ) = ' * ' MTB > regr c3 1 c l The r e g r e s s i o n equation i s LnKn•= 7.24 + 3530 1/T (/K) 3 cases used 1 cases contain m i s s i n g values P r e d i c t o r Coef Stdev t - r a t i o p Constant 7.240 6.534 1.11 0.467 1/T (/K) 3530 2142 1.65 0.347 s = 0.1515 R-sq = 73.1% R-sq(adj) = 46.2% A n a l y s i s of Variance SOURCE DF SS MS F p Regi-ession 1 0.06230 0.06230 - 2.72 0.347 E r r o r 1 0.02294 0.02294 T o t a l 2 0.08525 Unusual Observations Obs.1/T (/K) LnKn F i t S t d e v . F i t Residual St.Resid 1 0.00314 * 18.3244 0.2117 * * X X denotes an obs. whose X value gives i t l a r g e i n f l u e n c e . MTB > noou Table (4.3b) MINITAB Regression P r i n t - o u t For The E f f e c t Of Temperature On The K i n e t i c Rate Constants 60 T : absolute temperature (K) A : pre-exponential f a c t o r A p l o t of l n K against T - 1 gives a slope equal t o -E/R, and an i n t e r c e p t equal to l n A. Thus : E = - R (slope) (4.5) Again the r e g r e s s i o n was performed using the MINITAB s t a t i s t i c a l program, and the p r i n t - o u t i s given i n t a b l e s 4.3 (a) and (b), and appendix D. 4.4 DERIVATION USING THE COMBINED DATA An o v e r a l l approach was a l s o used to determine the a c t i v a t i o n energies and the 'orders'. In t h i s case, the data are not d i v i d e d according t o temperature. Rather, a l l a v a i l a b l e data are used i n the reg r e s s i o n s , and the increas e d degrees of freedom should improve the accuracy of the c o r r e l a t i o n s . Expressions (2.16, 2.17) remain unchanged because they are independent of temperature. However, f o r expressions (2.4, 2.5, and 2.15) the r a t e constant, K, must be replaced by i t ' s Arrhenius form, thus : G = A G e x p ( - E G / R T ) ( 4 . 6 ) and B° = A B exp(-E B/RT)S b (4.7) B" = A N exp(-E N/RT)M T^S U (4.8) The MINITAB r e g r e s s i o n p r i n t - o u t f o r t h i s o v e r a l l approach GROWTH RATE model : G = A^ exp(-E n/RT) l n G = 19.674 - 5615/T + 0.8681n S R-square = 94 .1% PRIMARY NUCLEATION model : B = A R exp(-E p/RT)S b l n B = 14.872 -1656/T +1.4091n S R-square = 81 .7% PRIMARY RELATIVE KINETICS model : B = K b G i l n B = 5.883 +1.6961n G R-square = 50 .1% SECONDARY NUCLEATION model : B = A M exp(-E N/RT)M T3< 3 U . l n B = 38.392 - 7090/T + 0.8361n M T + l_.2661n S R-sguare = 90 .3% SECONDARY RELATIVE KINETICS model : B = K nM T^S u l n B = 13.950 + 0.9381n M T + 1.4181n G R-square = 91 • 5 % Table (4.4) Summary Of M i n i t a b Regressions For The C r y s t a l l i z a t i o n Parameters And A c t i v a t i o n Energies Using The Combined Data i s given i n appendix E, and t a b l e 4.4 i s a summary of the r e s u l t i n g c o r r e l a t i o n s . 4.5 RESULTS A summary of the r e s u l t s i s given i n t a b l e (4.5) . The complete r e s u l t s are a v a i l a b l e i n appendices A, B, C, D, and E . To choose between the var i o u s models (whether primary or secondary; i n d i v i d u a l temperatures or o v e r a l l combined approach), the R-square values, t - r a t i o s , 95% confidence l i m i t s , and the standard d e v i a t i o n s were compared. The R-square value i n d i c a t e s how w e l l the data c o r r e l a t e w i t h the model. A value of 0% means there i s no c o r r e l a t i o n , w hile 100% means e x c e l l e n t c o r r e l a t i o n . The t - r a t i o i s an i n d i c a t o r of the amount of s c a t t e r of the parameters about the p r e d i c t e d value. The b e t t e r the f i t (smaller d e v i a t i o n ) , the l a r g e r the absolute value of the t - r a t i o . The 95% confidence i n t e r v a l i s the region about the p r e d i c t e d (by the model) p o i n t w i t h i n which the tr u e value of the parameter can be expected t o occur with 95% p r o b a b i l i t y . The narrower t h i s i n t e r v a l , the b e t t e r the f i t . The confidence i n t e r v a l i s c l o s e l y r e l a t e d to the t - r a t i o ; but u n l i k e the t - r a t i o , i t i s dependent on the magnitude of the parameter. For i n s t a n c e , two parameters of magnitude 10 and 100 can have the same confidence i n t e r v a l of 5 u n i t s . C l e a r l y , i n r e l a t i v e terms a d e v i a t i o n of 5 u n i t s i s l e s s s e r i o u s f o r 100 than i t i s f o r 10. The parameter 10 would have a t - r a t i o s e v e r a l times smaller than the parameter 100. The standard RUN # TEMPERATURE RESIDENCE GROWTH NUCLEI NUCLEATION SUSPENSION SUPERSATURATION C.V. MEDIAN TIME, T RATE, G DENSITY, n° RATE, B° (no./mm3•hr) DENSITY, M T (g/ml) SIZE (°C). (min) (mm/hr) (no./mm«mm3 ) (g/g a c id) (%) (mm) 1 50 7.19 0.621 192.1 119.3 0.2186 0.040 61.8 1.06 2 60 7.19 0.686 214.0 146.9 0.0998 0.025 49.5 0.41 3 60 11.79 0.512 180.6 92.5 0.1036 0.016 46.0 0.34 4 55 14.87 0.345 125.3 43.2 0.1000 0.016 50.6 0.52 5 60 15.03 0.413 113.4 46.9 0.0992 0.016 45.0 0.39 6 60 7.18 1.004 208.1 208.9 0.1193 0.038 33.3 0.28 7 50 10.98 0.559 258.2 144.4 0.1836 0.036 42.4 0.36 8 50 14.30 0.439 279.3 122.5 0.2141 0.026 38.1 0.34 9 60 11.01 0.603 124.0 74.7 0.0871 0.023 40.8 0.33 10 55 7.03 0.857 301.1 262.7 0.1834 0.039 42.7 0.35 11 50 10.45 0.500 355.1 177.6 0.2070 0.037 44.2 0.41 12 55 6.95 0.710 339.6 132.1 0.1002 0.036 56.6 0.48 13 • 55 10.51 0.527 229.8 128.9 0.1312 0.030 42.1 0.35 14 55 10.59 0.538 261.3 140.6 0.1382 0.025 43.1 0.35 15 60 14.52 0.373 59.1 22.0 0.0428 0.012 52.0 0.48 16 50 7.26 0.714 511.7 365.4 0.2320 0.051 42.4 0.35 17 55 14.23 0.470 220.3 103.5 0.1828 0.021 39.4 0.35 18 50 13.62 0.402 287.7 115.7 0.2563 0.027 52.1 0.53 19b 45 14.73 0.459 414.4 190.2 0.2720 0.037 * * 19 45 14.73 0.442 417.1 184.2 0.2720 0.037 36.7 0.33 20 45 10.91 0.561 378.1 212.1 0.2240 0.055 * * 21 45 6.98 0.886 381.6 338.1 0.2359 0.082 40.3 0.34 21b 45 6.98 0.881 473.7 417.3 0.2359 0.082 * * * data l o s t due to computer f a u l t Table (4.5) Summary of Experimental Results d e v i a t i o n represents the s c a t t e r of the response v a r i a b l e from the value the model p r e d i c t s using the c a l c u l a t e d parameters. 4 . 5 . 1 GROWTH K INET ICS Where the c r y s t a l l i z a t i o n k i n e t i c s were determined w i t h i n temperature groups, the growth r a t e k i n e t i c order r e l a t i v e t o the su p e r s a t u r a t i o n v a r i e d between 0.813 to 0.907. The k i n e t i c order obtained from the o v e r a l l r e g r e s s i o n gave a value of 0.868. The R-square value f o r the combined data at 94.1% was not d r a m a t i c a l l y d i f f e r e n t or b e t t e r than the i n d i v i d u a l ones which were between 88.7-96.7%. However, when the t - r a t i o s are considered, the combined data (18.68) i s c l e a r l y b e t t e r than the i n d i v i d u a l approach (6.34-10.88). The o v e r a l l r e g r e s s i o n , t h e r e f o r e , gives the best f i t t o the growth r a t e data. Table 4.6 compares the two approaches, and f i g u r e 4.2 shows the r e l a t i o n s h i p of growth ra t e to the s u p e r s a t u r a t i o n . The f i t t e d l i n e s are from the o v e r a l l approach. 4 . 5 . 2 NUCLEATION K INET ICS The n u c l e a t i o n r a t e k i n e t i c order r e l a t i v e to the sup e r s a t u r a t i o n v a r i e d between 0.876 and 1.770, f o r the primary n u c l e a t i o n model, and 1.168 to 1.670 f o r the secondary n u c l e a t i o n model. The o v e r a l l r e g r e s s i o n gave Model : G = K G S^ C r y s t a l Growth Rate Temperature CC) 45 50 55 60 O v e r a l l Ln K G 95% C.I. ± t - r a t i o 1.971 0.631 8.67 2.085 1.106 4.85 2.693 1.194 5.80 2.796 0.910 7.90 g 95% C.I. t - r a t i o 0.847 0.216 10.88 0.813 0.330 6.34 0.907 0.328 7.11 0.862 0.232 9.55 0.868 0.096 18.68 s t d dev. R-square % 0.062 96.7 0.072 88.7 0.096 90.8 0.083 94.7 0.072 :94.1 Table (4.6) Comparison Of I n d i v i d u a l And O v e r a l l Regressions For C r y s t a l Growth Rate Figu r e (4.2) E f f e c t of Supersaturation On The Growth Rate Primary : B° = K Bsk a and Secondary : B = % M T 3 s U Temperature 45 50 CC) Model Primary Secondary Primary Secondary Ln K 8.074 23.096 9.666 18.245 95% C.I. ± 1.466 22.386 4.597 5.507 t - r a t i o 15.29 2.86 5.41 8.52 • 1.702 0.888 95% C . I . I 2.534 0.532 t - r a t i o 1.87 4.29 b or u 0.876 1.180 1.375: 1.670 95% C.I. 0.502 0.585 1.371 0.618 t - r a t i o 4.84 5.59 2.58 6.95 s t d . dev. 0.144 0.106 0.301 0.130 R-square % 84.9 91.7 53.1 91.2 Table (4.7a) Comparison Of ( I n d i v i d u a l ) Primary And Secondary Nucleation Models Primary : = K B S b and Secondary : B° = K NM T3S U Temperature 55 60 C C ) Model Primary Secondary Primary Secondary Ln K 10.384 17.883 11.264 17.892 95% C.I. t 3.692 5.745 4.345 10.443 t - r a t i o 7.23 8.00 6.66 4.40 J 0.908 0.961 95% C.I. ± 0. 659 1.425 t - r a t i o 3.55 1.73 b or u 1.546 1.382 1.770 1.168 95% C.I. 1.014 0.528 1.108 1.270 t - r a t i o 3.92 6.73 4.11 2.36 s t d . dev. 0.298 0.151 0.395 0.332 R-square % 74.2 93.4 76.0 84.1 Table (4.7b) Comparison Of ( I n d i v i d u a l ) Primary And Secondary Nucleation Model : B° = K NM T^S U Secondary N u c l e a t i o n Temperature CC) 45 50 55 60 O v e r a l l Ln K N 95% C.I. t t - r a t i o 23.096 22.386 2.86 18.245 5.507 8.52 17.883 5.749 8.00 17.892 10.443 4.40 j 95% C.I. t t - r a t i o 1.702 2.534 1.87 0.888 0.532 4.29 0.908 0.659 3.55 0.961 1.425 1.73 0.836 0.398 4.35 u 95% C.I. ± t - r a t i o 1.180 0.585 5.59 1.670 0. 618 6.95 1.382 0.528 6.73 1.168 1.270 2.36 1.266 0,-2 96 8.87 s t d dev. R-square % 0.106 91.7 0.130 91.2 0.151 93.4 0.322 84.1 0.215 90.3 Table (4.8) Comparison Of I n d i v i d u a l And O v e r a l l Regression For Secondary Nucleation F i g u r e (4.3) E f f e c t of Supersaturation On The Corrected N u c l e a t i o n Rate 69 values of 1.403 and 1.266 f o r the primary and secondary n u c l e a t i o n models r e s p e c t i v e l y . The order, j , f o r the suspension d e n s i t y was 0.836 f o r the o v e r a l l r e g r e s s i o n , and between 0.888 and 1.702 based on the regre s s i o n s by temperature. Tables 4.7 (a) and (b) compare the primary and secondary n u c l e a t i o n models. An examination of the R-square, t - r a t i o and standard d e v i a t i o n f i g u r e s i n d i c a t e s t h a t the secondary model gives a b e t t e r f i t to the data. Table 4.8 compares the i n d i v i d u a l and combined approaches f o r secondary n u c l e a t i o n . The combined approach has b e t t e r t - r a t i o , and 95% confidence l i m i t s , but a lower R-square value. An examination of the f i t of the data as l a i d out i n t a b l e s (4.7) and (4.8) leads t o the choice of the parameters obtained using the o v e r a l l secondary n u c l e a t i o n model. Figure (4.3) shows the r e l a t i o n s h i p between the c o r r e c t e d n u c l e a t i o n r a t e and the su p e r s a t u r a t i o n . The c o r r e c t e d n u c l e a t i o n r a t e i s equal t o B°/MT^ and i t allows f o r comparison of the n u c l e a t i o n data at the same c r y s t a l suspension d e n s i t y . 4.5.3 R E L A T I V E NUCLEATION K INET ICS The r e l a t i v e k i n e t i c order i ( i n d i v i d u a l approach), f o r the primary n u c l e a t i o n model f e l l w i t h i n the range 1.053 t o 2.075, whereas v ( r e l a t i v e k i n e t i c order f o r secondary nucleation) was between 1.170 and 2.080. From t a b l e s (4.9 (a) and (b)), i t i s d i f f i c u l t to choose between the primary and secondary r e l a t i v e n u c l e a t i o n models. The secondary model tends t o have b e t t e r R-square values, poorer r a t e constant t - r a t i o s , and comparable growth order ( i or v) t -r a t i o s . However, since the primary n u c l e a t i o n model has already been r e j e c t e d e a r l i e r (sect.4.5.2) i t i s only l o g i c a l t h a t the primary r e l a t i v e n u c l e a t i o n model can not be acceptable here. The o v e r a l l secondary r e l a t i v e n u c l e a t i o n model gave the best f i t (table 4.10) to the data, and gave r e l a t i v e k i n e t i c order v = 1.418, with a suspension d e n s i t y dependence r a i s e d to the power j = 0.938. Figure 4.4 shows the secondary r e l a t i v e k i n e t i c s u s i n g the co r r e c t e d n u c l e a t i o n r a t e (B°/MT^) p l o t t e d against the growth r a t e . 4.5.4 A C T I V A T I O N ENERGY FOR GROWTH The i n d i v i d u a l temperature regressions r e s u l t i n an a c t i v a t i o n energy f o r growth of +54,746 kJ/kmol and a pre-exponential f a c t o r of 6.64x10^. The o v e r a l l r e g r e s s i o n gives the a c t i v a t i o n energy f o r growth as +46,693 kJ/kmol and the pre-exponential f a c t o r as 3.50xl0 8. Primary 0 : B = I^G 1 and Secondary : B° = K nM T3G v Temperature 45 50 C C ) Model Primary Secondary Primary Secondary Ln K 6.045 11.502 5.966 16.102 95% C.I. + 0.239 18.494 1.261 5.021 t - r a t i o 70.26 1.73 12.16 8.25 j 0. 650 1.136 95% C.I. ± 2.203 0.560 t - r a t i o 0.82 5.21 i or v 1.053 1.170 1.413 2.080 95% C.I. + 0.417 0.594 1.891 0.763 t - r a t i o 7.00 5.47 1.92 7.01 s t d . dev 0.103 0.109 0.354 0.129 R-square % 92.3 91.4 35.0 91.4 Table (4.9a) Comparison Of ( I n d i v i d u a l ) Primary And Secondary R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s Primary : B° = KhGL and Secondary : B* = K nM T3G v Temperature 55 60 (°C) Model Primary Secondary Primary Secondary Ln K 5.796 11.631 5.534 13.761 95% C.I. + 0.587 7.142 0.704 8.291 t - r a t i o 25.40 4.19 20.21 4.27 j 0. 668 0.919 95% C.I.± 0.817 0.924 t - r a t i o 2.10 2.55 i or v 1.709 1.507 2.075 1.450 95% C.I.± 0.885 0.694 1.069 0.936 ' t - r a t i o 4.97 5.58 4.99 3.98 s t d . dev. 0.245 0.180 0.336 0.218 R-square % 82.6 90.6 82.7 92.7 Table (4.9b) Comparison Of ( I n d i v i d u a l ) Primary And Secondary R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s Model : B° = K nM T3G v Secondary R e l a t i v e N u c l e a t i o n Temperature C O 45 50 55 60 O v e r a l l Ln K n 95% C.I. ± t - r a t i o 11.502 18.494 1.73 16.102 5.021 .8.25 11.631 7.142 4.19 13.761 8.291 4.27 13.950 1.660 17.43 j 95% C.I. + t - r a t i o 0.650 2.203 0.82 1.136 0.560 5.21 0.668 0.817 2.10 0.919 0.924 2.55 0.938 0.192 10.15 V 95% C.I. + t - r a t i o 1.170 0.594 5.47 2.080 0.763 7.01 1.507 0.694 5.58 1.450 0.936 3.98 1.418 O-i-308 9.56 s t d dev. R-square % 0.109 91.4 0.129 91.4 0.180 90. 6 0.218 92.7 0.202 91.5 Table (4.10) Comparison Of I n d i v i d u a l And O v e r a l l Regressions For Secondary R e l a t i v e C r y s t a l l i z a t i o n K i n e t i c s The i n d i v i d u a l l y o b t a i n e d g r o w t h r a t e c o n s t a n t s a r e g i v e n i n f i g u r e 4.5 t o g e t h e r w i t h t h e l i n e r e p r e s e n t i n g t h e r a t e c o n s t a n t s p r e d i c t e d u s i n g a l l t h e g r o w t h r a t e d a t a . T h e r e i s g o o d a g r e e m e n t b e t w e e n t h e t w o a p p r o a c h e s . T a b l e 4 . 1 1 c o m p a r e s t h e i n d i v i d u a l a n d o v e r a l l a p p r o a c h t o o b t a i n i n g t h e p a r a m e t e r s i n t h e A r r h e n i u s e q u a t i o n . 4.5.5 ACTIVATION ENERGY FOR NUCLEATION The r e g r e s s i o n o f t h e i n d i v i d u a l l y o b t a i n e d r a t e c o n s t a n t s ( R - s q u a r e = 4 0 . 6 % ) r e s u l t s i n an a c t i v a t i o n e n e r g y f o r n u c l e a t i o n o f - 2 7 1 , 1 6 9 k J / k m o l a n d a p r e - e x p o n e n t i a l f a c t o r o f 7 . 1 9 x l 0 ~ 3 ^ , when a l l t h e f o u r t e m p e r a t u r e v a l u e s a r e i n c l u d e d . When t h e r a t e c o n s t a n t v a l u e a t 45 °C i s e x c l u d e d , t h e R - s q u a r e v a l u e i m p r o v e s s l i g h t l y t o 4 6 . 2 % . I n t h i s c a s e , t h e a c t i v a t i o n e n e r g y i s - 2 9 , 3 5 2 k J / k m o l w i t h a p r e - e x p o n e n t i a l f a c t o r o f 1 , 3 9 4 . The c a s e e x c l u d i n g 45 °C g i v e s a m o r e r e a s o n a b l e v a l u e t h a n t h e p r e v i o u s o n e , e s p e c i a l l y c o n s i d e r i n g t h a t o t h e r w o r k e r s h a v e n o t r e p o r t e d a c t i v a t i o n e n e r g i e s f a r b e y o n d 1 0 0 , 0 0 0 k J / k m o l . The o v e r a l l r e g r e s s i o n g i v e s a c t i v a t i o n e n e r g y f o r n u c l e a t i o n o f +58,953 k J / k m o l a n d p r e - e x p o n e n t i a l f a c t o r o f 4 . 7 1 x l 0 1 ^ . T h i s i s i n m a r k e d c o n t r a s t t o t h e v a l u e s f r o m t h e i n d i v i d u a l a p p r o a c h . H o w e v e r , t h e o v e r a l l a p p r o a c h i s c l e a r l y b e t t e r a b l e t o f i t t h e d a t a a s c a n be s e e n ( t a b l e 4 . 1 2 ) b y c o m p a r i n g t h e R - s q u a r e v a l u e s , t - r a t i o s , a n d 95% c o n f i d e n c e l i m i t s . The Model : KG = AQ exp(-E G/RT) APPROACH INDIVIDUAL OVERALL Ln AQ 9 5 % C.I. t - r a t i o ± 2 2 . 6 1 7 1 3 . 9 4 0 5.16 1 9 . 6 7 4 3.01 8 1 3 . 5 2 - (EQ/R) 9 5 % C.I. t - r a t i o ± -6584 4 5 3 7 -4 . 6 2 - 5 6 1 5 900 - 1 2 . 9 5 s t d dev. R-square 0.150 87.1 0.072 94.1 Table ( 4 . 1 1 ) Comparison Of I n d i v i d u a l And O v e r a l l Regressions For The E f f e c t Of Temperature On The Growth Rate K i n e t i c Constant Figure (4.5) E f f e c t Of Temperature On The Growth Rate Constant Model : K N = A N exp(--EN/RT) APPROACH INDIVIDUAL OVERALL 45 °C inclu d e d excluded Ln A N 95% C.I. t - r a t i o + -80.92 182.7 -1.41 7.240 28.116 1.11 38.392 29.850 5.53 -<EN/R) 95% C.I. t - r a t i o ± 32,612 59,452 1.75 3,530 9,217 1.65 -7,090 3,733 -3.94 s t d dev. R-square o, "o 1.966 40.6 0.151 46.2 0.215 90.3 Table (4.12) Comparison Of I n d i v i d u a l And O v e r a l l Regressions For The E f f e c t Of Temperature On The Nu c l e a t i o n Rate K i n e t i c Constant 50 -40 30 20 10 -Individual } 5 R — , __ • • i i Overatiy I 1 1 1 2.98 3.02 3.04 3.06 3.08 3.1 3.12 I N V E R S E T E M P E R A T U R E (1 /K )x1000 3.14 3.16 Figure (4.6) E f f e c t Of Temperature On The Nu c l e a t i o n Rate Constant o v e r a l l approach i s t h e r e f o r e chosen as the most app r o p r i a t e . F i g u r e 4.6 shows the i n d i v i d u a l l y obtained n u c l e a t i o n r a t e constants and the s t r a i g h t l i n e r e p r e s e n t i n g the r a t e constants p r e d i c t e d when a l l the data are used. I t should be po i n t e d out here that the o v e r a l l r e g r e s s i o n l i n e i n f i g u r e 4.6 has nothing to do whatsoever wi t h the i n d i v i d u a l r a t e constant values a l s o presented i n the f i g u r e . I t i s there p r i m a r i l y to c o n t r a s t the i n d i v i d u a l r e g r e s s i o n l i n e t h a t a r i s e s from these i n d i v i d u a l r a t e constants. U n l i k e i n the case of growth r a t e , the two r e g r e s s i o n l i n e s c l e a r l y do not agree. C H A P T E R F I V E DISCUSSION AND CONCLUSIONS 5.1 EFFECT OF SUPERSATURATION The d r i v i n g f o r c e i n any c r y s t a l l i z a t i o n i s the su p e r s a t u r a t i o n . The temperature and residence time w i t h i n the c r y s t a l l i z e r can be c o n t r o l l e d d i r e c t l y . Both of these parameters a f f e c t the su p e r s a t u r a t i o n , but the a c t u a l l e v e l of s u p e r s a t u r a t i o n t h a t w i l l e x i s t i n the c r y s t a l l i z e r w i l l a l s o be a f u n c t i o n of the competing k i n e t i c s of c r y s t a l growth and n u c l e a t i o n . Consequently, the s u p e r s a t u r a t i o n can not be a r b i t r a r i l y f i x e d at a given l e v e l . As shown i n t a b l e 5.1, the r e l a t i v e k i n e t i c orders i , and v, are g e n e r a l l y greater than one. This means tha t the nu c l e a t i o n r a t e k i n e t i c orders b or u, are gre a t e r than the growth r a t e k i n e t i c order g. Supersaturation changes t h e r e f o r e have a greater impact on the n u c l e a t i o n r a t e than on the growth r a t e . I t f o l l o w s that f o r constant suspension d e n s i t y , a decrease i n the su p e r s a t u r a t i o n (longer residence time) w i l l cause a greater r e l a t i v e drop i n the n u c l e a t i o n r a t e than i n the growth r a t e . This should r e s u l t i n l a r g e r c r y s t a l s at longer residence times. In t h i s study, the r e l a t i v e k i n e t i c order (v=1.418), i s greater than one which i s i n agreement wi t h t y p i c a l values. The growth ra t e (g=0.868) and the secondary n u c l e a t i o n r a t e (u=1.266) k i n e t i c orders are a l s o w i t h i n the t y p i c a l range Models : G = K GS9 ; B = K B S b ; B = KhGl ; B = K nM T^G v g b i system authors 1.1 1. 6 1 .5 ammonium sulphate Larson & K l e k a r 1973 0. 65 0. 54 0 .83 c i t r i c a c i d S i kdar & Randolph 1976 2.29 2. 59 1 .13 magnesium sulphate Sikdar & Randolph 1976 1.33 2. 10 1 .58 potash alum Garside & J a n c i c 1979 1.7 0. 9 0 .5 potassium dichromate Timm & Cooper 1971 1.0 1. 6-1 .9 1.6 -1.9 potassium n i t r a t e H e l t & Larson 1977 1.29 0. 67 0 .52 potassium sulphate Randolph & Sikdar 1976 j V system authors 1. 0 2 .0 ammonium alum Chambliss 1966 0. 98 1 .22 ammonium sulphate Youngquist & Randolph 1972 1. 0 1 .8 potash alum Ottens & de Jong 1973 0. 6 0 .5 potassium dichromate Desai et a l 1973 1. 07 -3 .54 urea Lodaya et a l 1977 0. 84 0 .98 sodium c h l o r i d e Bennet et a l 1973 1. 21 2 .78 potassium c h l o r i d e Qian et a l 1987 Table (5.1) Reported K i n e t i c Orders RESIDENCE TIME min. TEMPERATURE °C 7 11 15 45 0.336 0.326 50 1.060 0.347 0.359 0.410 0.342 0.526 55 0.349 0.484 0.353 0.353 0.517 0.354 60 0.412 0.278 0.341 0.333 0.394 0.479 Table (5.2) V a r i a t i o n Of C r y s t a l Median Size ( W m ) With Residence Time found i n the l i t e r a t u r e . Examination of the c r y s t a l median s i z e data (table 5.2) does not c l e a r l y show the expected increase i n s i z e w i t h i n c r e a s i n g residence time. This can be a t t r i b u t e d t o an obscuring e f f e c t by the experimental e r r o r as w e l l as t o the v a r i a t i o n of the c r y s t a l suspension d e n s i t y . The dependency of growth r a t e on the s u p e r s a t u r a t i o n i s given by : G = KQ S 0 - 8 6 8 (5.1) where K G = 3.50xl0 8 exp(-46,689/RT) (5.2) while f o r n u c l e a t i o n , the expression i s : B° = K N M T 0 ' 8 3 6 s 1 ' 2 6 6 (5.3) where % = 4 . 7 1 x l 0 1 6 exp(-58,953/RT) (5.4) 5.2 EFFECT OF RESIDENCE TIME In an MSMPR c r y s t a l l i z e r , s u p e r s a t u r a t i o n and residence time cannot be separated. There i s no d i r e c t c o n t r o l over the su p e r s a t u r a t i o n , but changes t o i t can be e f f e c t e d by changing the residence time. In general, the longer the residence time, the lower the su p e r s a t u r a t i o n p r e v a i l i n g i n the c r y s t a l l i z e r . That was a l s o the case i n t h i s study ( f i g u r e 5.1). For constant MT, the r e l a t i o n s h i p has been put i n formal terms by Garside and Shah (1980) as : S <*= x -4/g(i+3) ( 5 > 5 ) 3 3 0.1 0.081 3 0 .06 5 0.041 < 0.021 OC U l ft 3 0 ' • 46 C + 60C • o 66 C • 60 C • ' • I L. + 0 + a 0 I I I I I 1 L 0 11 13 16 RESIDENCE TIME (min.) 17 Figure (5.1) E f f e c t Of Residence Time On The Supersaturation Since G v a r i e s as S^, s u b s t i t u t i n g f o r S i n eqn (5.5) gives : G - < r - 4 / < i + 3> (5.6) which shows the e f f e c t the residence time would have on the growth r a t e under c o n d i t i o n s of constant suspension d e n s i t y . Randolph and Larson (1971), and J a n c i c and Grootscholten (1984) have shown th a t the various average c r y s t a l s i z e s f o r a CSD are m u l t i p l e s of the product Gt. For example : mode s i z e , LQ = 3GT (5.7) median s i z e , L M = 3.67GT (5.8) mean s i z e , L = 4GT (5.9) Thus, combining eqn (5.6) wit h any of eqns (5.7, 5.8, or 5.9) r e s u l t s i n : average s i z e T ( i - l ) / d + 3 ) (5.10) We a l s o have th a t (Appendix F.l) : n o _ x - 4 ( i - l ) / ( i + 3) ( 5 > 1 1 ) From eqn (5.10), f o r i = l , the average c r y s t a l s i z e i s unaf f e c t e d by changes i n the residence time. For i < l , the average c r y s t a l s i z e decreases as residence time i n c r e a s e s , and f o r i > l , i t increases with i n c r e a s i n g residence time. And by eqn (5.11), f o r i = l , n° i s independent of residence time and t h e r e f o r e a l s o of s u p e r s a t u r a t i o n . When i < l , n° RESIDENCE TIME min. TEMPERATURE •c 7 11 15 45 1.130 0.745 0.565 50 0.425 1.239 0.596 0. 663 0.444 0.361 55 1.085 0.904 0.704 0.869 0.296 0.428 60 1.009 1.235 0. 616 0.575 0.324 0.306 co r r e c t e d n u c l e a t i o n r a t e = B/MT3 x 10 Table (5.3) E f f e c t Of Residence Time On The Corrected N u c l e a t i o n Rate j _ 7 11 * RESIDENCE TIME (min.) F i g u r e (5.2) E f f e c t Of Residence Time On The Corrected N u c l e a t i o n Rate 85 CD 4-) cd C<x> O I -H O 4J <H fd CD X. O - r - i S S TJ • <D CQ 4-> O CD U 5H O O 1.6 0.0 46 C a o + 50 C o 66C 60C t • increases as the residence time inc r e a s e s , while f o r i > l , n° decreases with i n c r e a s i n g residence time. In t h i s i n v e s t i g a t i o n , i=1.418 and we should expect t h a t the average c r y s t a l s i z e (and growth rate) increases w i t h i n c r e a s i n g residence time. The n u c l e i p o p u l a t i o n d e n s i t y and th e r e f o r e the n u c l e a t i o n r a t e (B°=n°G) should d e c l i n e as residence times i n c r e a s e . An examination of the r e s u l t s , c o r r e c t e d f o r suspension d e n s i t y (table (5.3), and f i g . (5.2)), shows tha t t h i s i s the case f o r n u c l e a t i o n . The average of the values i n t a b l e (5.3) were used t o generate f i g u r e (5.2). The tr e n d i n c r y s t a l s i z e (table 5.2), however, i s obscured by the e r r o r as expl a i n e d i n the previous s e c t i o n . 5.3 EFFECT OF SUSPENSION DENSITY Larson et a l . (1968) have shown that f o r the s i t u a t i o n where secondary n u c l e a t i o n i s s i g n i f i c a n t , the r e l a t i o n s h i p between the suspension d e n s i t y and the growth r a t e i s : G oc M TU-J) / ( i + 3) (5.12) The r e l a t i o n s h i p of the suspension d e n s i t y and the n u c l e i p o p u l a t i o n d e n s i t y i s given by : n . a M T ( i + 4 j - l ) / ( i + 3) ( 5 > 1 3 ) The complete d e r i v a t i o n of eqns (5.12, 5.13) i s given i n Appendix (F.2). 86 By eqn (5.12), f o r the case where j = l , growth r a t e (and the average c r y s t a l s i z e ) i s independent of the suspension d e n s i t y . For j greater than 1, the growth r a t e and the average c r y s t a l s i z e w i l l d e c l i n e as the suspension d e n s i t y i n c r e a s e s . I f j < l , growth r a t e and the average c r y s t a l s i z e w i l l i ncrease w i t h i n c r e a s i n g suspension d e n s i t y . Eqn (5.13) p r e d i c t s t h a t f o r j 0, as the suspension d e n s i t y i n c r e a s e s , so does the n u c l e i p o p u l a t i o n d e n s i t y . In t h i s study, j=0.836 or j=0.938 depending on whether the secondary model i n v o l v e s s u p e r s a t u r a t i o n or not. I t f o l l o w s then (since j < l ) , that the average c r y s t a l s i z e and growth r a t e should increase as the suspension d e n s i t y i n c r e a s e s . The n u c l e i p o p u l a t i o n d e n s i t y should a l s o increase w i t h g r e a t e r suspension d e n s i t y , r e s u l t i n g i n an increa s e d n u c l e a t i o n r a t e by eqn ((2.40), i . e . B°=n°G). The r e s u l t s do f o l l o w these trends q u a l i t a t i v e l y . I t can be concluded (Ottens and de Jong (1973)) th a t the predominant source of n u c l e i i n t h i s study has been c r y s t a l -i m p e l l e r and c r y s t a l - w a l l contacts (j approximately = 1) rat h e r than c r y s t a l - c r y s t a l contacts (j about equal t o 2). 5.4 E F F E C T OF TEMPERATURE The temperature dependency of the c r y s t a l l i z a t i o n k i n e t i c s i s contained i n the k i n e t i c r a t e constant of eqns (2.4, 2.5, and 2.15). This dependency of the ra t e constant on temperature i s i n t u r n modelled on an Arrhenius form : K = A exp(-E/RT) (4.4) As discussed p r e v i o u s l y , c r y s t a l growth i n v o l v e s two processes, d i f f u s i o n and surface r e a c t i o n . Where d i f f u s i o n i s dominant, lower growth a c t i v a t i o n energies are expected. I f the surface r e a c t i o n step i s more important, then higher a c t i v a t i o n energies are the norm. An a c t i v a t i o n energy of +12,000 kJ/kmol, as reported by Budz et a l (1985) f o r sodium t h i o s u l p h a t e , i s at the lower end of the s c a l e , while a value of +71,900 kJ/kmol ( I s h i i (1965) f o r potassium sulphate) i s on the high end. This study gives an a c t i v a t i o n energy f o r growth of +46,700 kJ/kmol which i s a r e l a t i v e l y high value i n d i c a t i n g the importance of surface r e a c t i o n . P r e v i o u s l y (sect. 2.5), the impact of a g i t a t i o n on the d i f f u s i o n r e s i s t a n c e was discussed. Increased a g i t a t i o n reduces the' d i f f u s i o n r e s i s t a n c e , lowering the importance of the d i f f u s i o n step i n the o v e r a l l growth k i n e t i c s . In the course of t h i s study, i t was necessary to maintain a high l e v e l of a g i t a t i o n t o ensure adequate mixing (sect. 3.2). This r a t h e r high a g i t a t i o n r a t e , would have minimised the d i f f u s i o n e f f e c t s and t h i s i s borne out by the high growth r a t e a c t i v a t i o n energy obtained. The a c t i v a t i o n energy f o r n u c l e a t i o n was e s t a b l i s h e d t o be +58,900 kJ/kmol. o A c t i v a t ] (kJ/ EG Lon Energy 'kmol) % System Authors 29,800 -39,750 c i t r i c a c i d Sikdar & Randolph 1976 12,000 sodium t h i o s u l p h a t e Budz et a l 1985 32,000 potassium n i t r a t e Genck 1969 31,000 -108,000 potassium n i t r a t e H e l t & Larson 1977 .40, 400 62,500 potassium sulphate Jones et a l 1986 18,000 89,000 45,100 potassium sulphate potassium sulphate M u l l i n & Gaska Randolph & C i s e 1969 1972 71,900 potassium sulphate I s h i i 1965 39,600 potassium sulphate Jones & M u l l i n 1973 38,400 64,600 potassium sulphate Jones & Mydlarz 1985 25,000 50,200 potassium alum Rousseau & McCabe 1976 31,800' 100,300 potassium alum Rousseau & Woo 1978 Table (5.4) Reported A c t i v a t i o n Energies Table 5.4 gives some reported a c t i v a t i o n energy values f o r v a r i o u s systems. No data were a v a i l a b l e f o r sodium sulphate. Even so, the values obtained i n t h i s study are comparable with those given i n the t a b l e f o r other systems. 5.5 C O E F F I C I E N T OF VAR IAT ION OF THE CSD The CSD c o e f f i c i e n t of v a r i a t i o n ranged between two extreme values of 33.3 and 61.8%, wi t h most of the runs f a l l i n g w i t h i n 40-56%. This i s reasonably c l o s e to the i d e a l 52% f o r MSMPR c r y s t a l l i z e r s . The median s i z e ranged from 0.28-0.53 mm, wit h the exception of one run tha t had a median s i z e of 1.06 mm. This i s the same run tha t had a cv of 61.8%. 5.6 ACCURACY OF THE STUDY An estimate of the accuracy of the data can be obtained by comparing the observed n u c l e a t i o n and growth r a t e s w i t h those t h a t are p r e d i c t e d by the models at the observed s u p e r s a t u r a t i o n s . The p r e d i c t i o n s are based on the o v e r a l l c r y s t a l growth and secondary n u c l e a t i o n models. Figures 5.3 and 5.4 show the s c a t t e r of the observed growth r a t e and c o r r e c t e d n u c l e a t i o n r a t e values about the p r e d i c t e d values. The s t r a i g h t l i n e shown represents the t h e o r e t i c a l l y expected r a t e s . The growth r a t e data f a l l w i t h i n 15% of the s t r a i g h t l i n e , whereas the n u c l e a t i o n r a t e data are more s c a t t e r e d , up to 30%. The'nucleation r a t e i s obtained by e x t r a p o l a t i o n of the CSD r e g r e s s i o n l i n e to 0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.9 1 11 12 1 3 14 PREDICTED GROWTH RATE (mm/hr) F i g u r e (5.3) S c a t t e r P l o t Of Growth Rate Data About The P r e d i c t e d Values i n t e r c e p t the n u c l e i d e n s i t y a x i s . Any s c a t t e r i n the data has a much more pronounced e f f e c t on t h i s i n t e r c e p t value, than on the growth r a t e value which i s obtained from the slope of the l i n e . 5.7 CONCLUSIONS The conclusions t h a t can be drawn from t h i s study are : - the d e f i n i n g k i n e t i c expressions are : G = 3.50xl0 8 exp(-46,700/RT) s 0 - 8 7 B° = 4 . 7 1 x l 0 1 6 exp(-58,900/RT) M T 0- 8 4 S 1 ' 2 7 B° = 1.14xl0 6 M T 0' 9 4 G 1' 4 2 - the growth r a t e a c t i v a t i o n energy i s +46,700 kJ/kmol; t h i s r e l a t i v e l y high a c t i v a t i o n energy i n d i c a t e s t h a t surface r e a c t i o n i s dominant. - the n u c l e a t i o n r a t e a c t i v a t i o n energy i s +58,900 kJ/kmol. - the n u c l e a t i o n r a t e i s dependent on the suspension d e n s i t y to the power of 0.84, and t h e r e f o r e the n u c l e i are predominantly generated by secondary n u c l e a t i o n mechanisms. - c r y s t a l - i m p e l l e r and c r y s t a l - w a l l contacts are more important than c r y s t a l - c r y s t a l contacts. - the c r y s t a l growth r a t e i s independent of the c r y s t a l s i z e . - the c r y s t a l s have a volume shape f a c t o r of 0.49 - the system produces the sesqui form of sodium sulphate. N O M E N C L A T U R E a, a^, a 2 : e m p i r i c a l l y determined constants i n e m p i r i c a l size-dependent c r y s t a l growth models a16' a50' a84 ' aperture s i z e s of sieves that r e t a i n 16, 50, and 84% r e s p e c t i v e l y of the c r y s t a l sample (by weight) A : pre-exponential f a c t o r i n the Arrhenius equation A B : pre-exponential f a c t o r i n the Arrhenius equation f o r primary c r y s t a l n u c l e a t i o n A c : c r y s t a l surface area AQ : pre-exponential f a c t o r i n the Arrhenius equation f o r c r y s t a l growth A N : pre-exponential f a c t o r i n the Arrhenius equation f o r secondary c r y s t a l n u c l e a t i o n A t : area of adhesive tape i n the determination of the c r y s t a l volume shape f a c t o r (sq.mm) b : primary n u c l e a t i o n r a t e k i n e t i c order B° : c r y s t a l n u c l e a t i o n r a t e (number/mm3 hr) c : s o l u t e c o n c e n t r a t i o n i n the supersaturated s o l u t i o n (g s o l u t e / g solvent) c* : s a t u r a t i o n (equilibrium) s o l u t e c o n c e n t r a t i o n (g s o l u t e / g solvent) c^ : s o l u t e c o n c e n t r a t i o n i n the s o l u t i o n at the c r y s t a l -s o l u t i o n i n t e r f a c e (g s o l u t e / g solvent) c m : s o l u t e c o n c e n t r a t i o n below which spontaneous n u c l e a t i o n does not take place (c>c m>c*) (g s o l u t e / g solvent) cv : c o e f f i c i e n t of v a r i a t i o n of the CSD CSD : c r y s t a l s i z e d i s t r i b u t i o n d : e m p i r i c a l l y determined constant i n e m p i r i c a l s i z e -dependent models D : c o e f f i c i e n t of d i f f u s i o n of the s o l u t e i n a supersaturated s o l u t i o n E : a c t i v a t i o n energy i n the Arrhenius equation (kJ/kmol) E B : a c t i v a t i o n energy f o r primary c r y s t a l n u c l e a t i o n i n the 94 Arrhenius equation (kJ/kmol) EQ : a c t i v a t i o n energy f o r c r y s t a l growth i n the Arrhenius equation (kJ/kmol) E N : a c t i v a t i o n energy f o r secondary c r y s t a l n u c l e a t i o n i n the Arrhenius equation (kJ/kmol) f : exponent on i m p e l l e r r o t a t i o n a l speed i n the expression r e l a t i n g c r y s t a l n u c l e a t i o n to the a g i t a t i o n g : growth r a t e k i n e t i c order i n the e m p i r i c a l c r y s t a l growth model G : c r y s t a l l i n e a r growth r a t e (mm/hr) G° : l i n e a r growth r a t e of c r y s t a l n u c l e i (mm/hr) AG : o v e r a l l excess fr e e energy of c r y s t a l AG S : surface excess fr e e energy of c r y s t a l AG V : volume excess f r e e energy of c r y s t a l i : n u c l e a t i o n r a t e k i n e t i c order i n e m p i r i c a l primary r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s model j : suspension d e n s i t y order i n e m p i r i c a l secondary n u c l e a t i o n and secondary r e l a t i v e c r y s t a l l i z a t i o n models k : r a t e constant f o r e m p i r i c a l size-dependent c r y s t a l growth model k a : c r y s t a l area shape f a c t o r k B : Boltzmann constant k^ : c o e f f i c i e n t of mass t r a n s f e r by d i f f u s i o n k m : c o e f f i c i e n t of mass t r a n s f e r k r : r a t e constant f o r surface r e a c t i o n k v : c r y s t a l volume shape f a c t o r K : r a t e constant K£ : p r o p o r t i o n a l i t y constant f o r e m p i r i c a l primary r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s Kg : primary n u c l e a t i o n r a t e constant Kg : o v e r a l l c r y s t a l mass growth r a t e constant 95 KQ : o v e r a l l c r y s t a l l i n e a r growth r a t e constant K N : p r o p o r t i o n a l i t y constant f o r e m p i r i c a l secondary r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s K N : secondary n u c l e a t i o n r a t e constant L : c r y s t a l s i z e measured along i t ' s c h a r a c t e r i s t i c l e n g t h (mm) L : c r y s t a l mean s i z e (mm) L D : c r y s t a l mode s i z e (mm) L M : c r y s t a l median s i z e (mm) L]_, L 2 : sieve aperture s i z e s (mm) of consecutive sieves 8A : trje Sicj^ epevxe iv oietue anepTDpe ai£ea; A2~A^ AL : an increment i n c r y s t a l s i z e (eqn. 2.26) m : mass of s o l i d deposited on the c r y s t a l face i n time,t mc : t o t a l oven-dry mass of c r y s t a l ( d i s s o l v e d + ne u t r a l i z e d ) i n the determination of s o l u b i l i t y mf : mass of f i l t r a t e i n the determination of sup e r s a t u r a t i o n mx : mass of c r y s t a l s deposited on the adhesive tape i n the determination of c r y s t a l volume shape f a c t o r M T : c r y s t a l suspension d e n s i t y (g crystals/mm 3 of suspension) n or n(L) : average c r y s t a l p o p u l a t i o n d e n s i t y (number/mm.mm3) n° : average c r y s t a l n u c l e i p o p u l a t i o n d e n s i t y (number/mm.mm3) n : c r y s t a l n u c l e i p o i n t p o p u l a t i o n d e n s i t y (number/mm.mm3.hr) n L : average c r y s t a l p o p u l a t i o n d e n s i t y f o r c r y s t a l s of s i z e L (number/mm.mm3) AN : number of c r y s t a l s i n a small s i z e range AL (eqn.2.26) N or N(L) : t o t a l c r y s t a l p o p u l a t i o n d e n s i t y (number/mm) N g : siev e mesh number i n the determination of c r y s t a l volume shape f a c t o r (number of apertures/sq. inch) N.p : t o t a l number of c r y s t a l s i n the c r y s t a l l i z i n g system Q : vol u m e t r i c f l o w r a t e : v o l u m e t r i c f l o w r a t e i n t o c r y s t a l l i z e r r : ra d i u s of a s p h e r i c a l nucleus r c : minimum rad i u s of a s t a b l e nucleus R : u n i v e r s a l gas constant (8.315 kJ/kmol.K) s : mass of sat u r a t e d s o l u t i o n i n determination of s o l u b i l i t y (g) S : s u p e r s a t u r a t i o n (g s o l u t e / g solvent) t : time (min) T : absolute temperature (K) T" : temperature C O u : secondary n u c l e a t i o n r a t e k i n e t i c order v : n u c l e a t i o n r a t e k i n e t i c order i n the e m p i r i c a l secondary r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s model v n : volume of a s p h e r i c a l nucleus v t : volume of IN NaOH consumed during t i t r a t i o n i n the determination of s o l u b i l i t y (ml) V : volume of suspension i n the c r y s t a l l i z e r w : weight f r a c t i o n of c r y s t a l s r e t a i n e d on the sieve during the determination of CSD W : i m p e l l e r r o t a t i o n a l speed ( s - 1 ) x : mass of d i s s o l v e d s o l u t e i n excess of the e q u i l i b r i u m mass i n the determination of s u p e r s a t u r a t i o n (g) y : number of c r y s t a l s deposited on the adhesive tape i n the determination of c r y s t a l volume shape f a c t o r GREEK LETTERS 5 : leng t h of d i f f u s i o n path a : i n t e r f a c i a l t e n s i o n p : c r y s t a l d e n s i t y (g/cmJ) X : residence time (min.) 98 R E F E R E N C E S ABEGG, C.F., J.D. STEVENS and M.A. LARSON AIChE J . . 14 (1) (1968) 118 ASSELBERGS, C.J. Ph.D t h e s i s , D e l f t Univ. of Tech. 1978 BENNET, R.A., H. FIEDELMAN, and A.D. RANDOLPH Chem. Eng. Prog. 69 (7) (1973) 86 BERTHOUD, A. J . Chem. Phys. 10 (1912) 624 BOTSARIS, G.D., E.G. DENK and J.O. CHUA CEP Symp. Ser. 118 (1972) BOURNE, J.R. and A. FAUBEL Proc. 8 t h Symp. on Ind. C r y s t . , Budapest (1981) 79 BOURNE, J.R. and K. HUNGERBUEHLER Trans. I n s t . Chem. Eng. 58 (1980) 51 BRANSOM, S.H. B r i t . Chem. Eng. 5 (1960) 838 BRANSOM, S.H., D.E. BROWN and G.P. HEELEY Symposium on I n d u s t r i a l C r y s t a l l i z a t i o n , 1969, London I n s t n . Chem. Engrs. BUDZ, J . , P.H. KARPINSKI and Z. NURUC AIChE J . 31 (1985) 259 CANNING, T.F. and A.D. RANDOLPH AIChE J . 1 (1967) 5 CAYEY, N.W. and J . ESTRIN Ind. Eng. Chem. Fund. 6 (1) (1967) 13 CHAMBLISS, C.W. Ph.D t h e s i s Iowa State Univ. (1966) 99 CLONTZ, N.A. and W.L.McCABE CEP Symp. Ser. 110 (67) (1971) 6 DESAI, R.M., J.W. RACHOW and D.C. TIMM AIChE J . 29 (1974) 43 GARSIDE, J . AIChE Symp. Ser. 240 (80) (1984) 23 GARSIDE, J . , and S.J. JANCIC AIChE J . 25 (1979) 948 GARSIDE, J . and J.W. MULLIN Trans. I n s t n . Chem. Engrs. 46 (1968) T i l GARSIDE, J . and M.B. SHAH Ind. Eng. Chem. Proc. Des. Dev. 19 (1980) 509 GARSIDE, J . , J.W. MULLIN and S.N. DAS Ind. Eng. Chem. Fund. 13 (1974) 299 GENCK, W.J. Ph.D t h e s i s Iowa State Univ. (1969) GENCK, W.J., and M.A. LARSON AIChE Symp. Ser. 121 (68) (1972) 57 GROOTSCHOLTEN, P.A.M., C.J. ASSELBERGS, and E.J. de JONG Proc. 8 t h Symp. Ind. C r y s t . Budapest, Hungary (1981) 61 HELT, J.E., and M.A. LARSON AIChE J . 23 (6) (1977) 822 ISHII, T B u l l . Tokyo I n s t . Tech. #67 (1965) JANCIC, S.J. and P.A.M. GROOTSCHOLTEN I n d u s t r i a l C r y s t a l l i z a t i o n , D e l f t U n i v e r s i t y Press (1984) JOHNSON, R.T., R.W. ROUSSEAU and W.L. McCABE AIChE Symp. 68 (121) (1972) 31 JONES, A.G., J . BUDZ and J.W. MULLIN AIChE J . 32 (12) (1986) 2002 JONES, A.G., and J . MYDLARZ Chem. Eng. Res. Des. 67 (5) (1989) 283 JONES, A.G., and J.W. MULLIN Trans. I n s t . Chem. Engrs. 51 (1973) 302 LAL, M.A., R.E.A. MASON and R.F. STRICKLAND-CONSTABLE J . C r y s t a l Growth 5 (1969) 1 LARSON, M.A., and S.A. KLEKAR AIChE 6 6 t h Annual Meeting, P h i l a d e l p h i a (1973) LARSON, M.A., D.C. TIMM and P.R. WOLFF AIChE J . 14 (3) (1968) 452 LODAYA,K.D., L.E. LAHTI, and M.L. JONES Ind. Eng. Chem. Proc. Des. Dev. 16 (7) (1973) 86 LIU, C.Y., H.S. TSUEI, and G.R. YOUNQUIST Chem. Eng. Prog. Symp. Ser. 110 (67) (1971) 43 MARC, R Z. Phys. Chem. 61 (1908) 385 I b i d . 67 (1909) 470 I b i d . 73 (1910) 685 MASON, R.E. and R.F. STRICKLAND-CONSTABLE Trans Faraday Soc i e t y 62 (1966) 455 McCABE, W.L. Ind. Eng. Chem. 38 (1946) 18 MELIA, T.P. and W.D. MOFFIT Ind. Eng. Chem. Fund. 3 (1964) 313 MIERS, H. A. and F. ISAAC J . Chem. Soc. London 89 (1906) 413 MULLIN, J.W. C r y s t a l l i z a t i o n (CRC Press, 1972) 398 I b i d . 382 MULLIN, J.W. and J . GARSIDE Trans. I n s t n . Chem. Engrs. 45 (1967) T285 MULLIN, J.W., and C. GASKA Canadian J . Chem. Eng. 47 (1969) 483 NERNST, W. Z. Phys. Chem. 47 (1904) 52 NOYES, A.A. and W.R. WHITNEY J. Am. Chem. Soc. 930 (1897) OSTWALD, W. Z. Phys. Chem. 22 (1897) 289 OTTENS, E.P.K. and E.J. de JONG Ind. Eng. Chem. Fund. 12 (1973) 179 OTTENS, E.P.K., A.H. JANSE and E.J. de JONG J . C r y s t . Growth 13/14 (1972) 500 PHILLIPS, V.R. and N. EPSTEIN AIChE J . 20 (1974) 678 POWERS, H.H.C. Ind. Chemist. 39 (1963) 351 QIAN, R.-Y., Z.-D. CHEN, H.-G. NI, Z.-Z. FAN, and F.-D. CAI AIChE J . 33 (10) (1987) 1690 RANDOLPH, A.D. and M.D. CISE AIChE J . 18 (4) (1972) 798 102 RANDOLPH, A.D. and M.A. LARSON AIChE J . 5 (1962) 639 RANDOLPH, A.D. and M.A. LARSON Theory of P a r t i c u l a t e Processes, Academic Press (1971) RANDOLPH, A.D and S.K. SIKDAR AIChE J . 20 (2) (1974) 410 RANDOLPH, A.D and S.K. SIKDAR Ind. Eng. Chem. Fund. 15 (1) (1976) 64 ROSEN, H.N., and H.M. HULBURT Chem. Eng. Prog. Symp. Ser. 110 (67) (1971a) 18 ROUSSEAU, R.W., andW.L. McCABE World Congress on Chemical Engineering, Amsterdam (1976) ROUSSEAU, R.W., and R. WOO AIChE 8 4 t h N a t i o n a l Meeting, A t l a n t a (1978) SCRUTTON, A. Proc. 5 t n Symp. on S a l t Hamburg, Germany 197 9 SHOR, S.M. and M.A. LARSON Chem. Eng. Prog. Symp. Ser. 110 (67) (1971) 32 SIKDAR, S.K., and A.D. RANDOLPH AIChE J . 22 (1) (1976) 110 SUNG, C.V., J . ESTRIN and G.R.YOUNGQUIST AIChE J . 19 (5) (1973) 957 TIMM,- D.C. and M.A. LARSON AIChE J . 14 (3) (1968) 452 TIMM, D.C. and T.R. COOPER AIChE J . 17 (1971) 285 YOUNGQUIST, G.R. and A.D. RANDOLPH AIChE J . 18 (2) (1972) 421 VALETON, J.J.P. Z. K r i s t a l l o g r . 59 (1923) 135, 335 I b i d . 60 (1924) 104 A P P E N D I X A 1 0 5 RUN it 01 CRYSTAL DENSITY, p 2.27 g/cm~3 TEMPERATURE C 50 SLURRY DENSITY Mt 0.2186 g/cm"3 RESIDENCE t min 7.19 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 5 . 7400 0. 871 0. 300 0 . 1516 0. 023 2. 33E+00 0 . 845 0.250 0 . 1713 0. 026 4. 89E+00 1 .587 0. 212 0 . 1450 0. 022 9. 18E+00 2 . 217 0.180 0 . 1054 0. 016 1. 30E+01 2 . 564 0. 150 0 .0857 0. 013 1. 89E+01 2 . 937 0.125 0 .0791 0. 012 3. 61E+01 3 . 586 0. 106 0 .0330 0. 005 3. 34E+01 3 . 508 0.090 0 .0264 0. 004 5. 19E+01 3 . 950 0.075 0 .0198 0. 003 6. 96E+01 4 . 243 PAN •0 .0330 0. 005 TOTAL 6 . 5901 1. 000 SIEVE SIEVE + IEVE dL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7828 100 . 5228 0. 300 0. 055 0 . 3275 95.4081 95 . 5597 0. 250 0. 050 0 . 2750 96.1503 96 . 3216 0. 212 0. 038 0 . 2310 95.2151 95 . 3601 0. 180 0. 032 0 . 1960 91.5905 91 . 6959 0. 150 0. 030 0 . 1650 91.5093 91 . 5950 0. 125 0 . 025 0 . 1375 86.5137 86 . 5928 0. 106 0. 019 0 .1155 91.3118 91 . 3448 0. 090 0. 016 0 .0980 87.2897 87 . 3161 0. 075 0. 015 0 .0825 91.5688 91 . 5886 PAN 86.8026 86 . 8356 Regression Output: Constant ' 5.258 Std E r r of Y Est 0. 118 R Squared 0.990 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -13.44 Std E r r of Coef. 0.5000 GROWTH RATE 0.621 mm/hr NUCLEI DENSITY 192.1 number/mm.mmA3 NUCLEATION RATE 119.3 number/mm"3 hr SUPERSATURATION 0.040 g s a l t / g a c i d RUN n 02 CRYSTAL DENSITY TEMPERATURE C SLURRY DENSITY Mt RESI D E N C E t min 2.27 g/cnT3 60 0.0998 g/cnT3 7.19 S I E V E (mm) 0.355 0. 300 0. 250 0. 212 0.180 0. 150 0.125 0. 106 0.090 0.075 PAN TOTAL CRYSTALS (g) 2.8620 0.4851 0.4123 0.2571 0.2377 0.2183 0.1892 0.0679 0.0728 0.0291 0.0194 4.8508 w n Ln n 0 . 590 0 . 100 4. 62E+00 1. 531 0 .085 7. 30E+00 1. 987 0 .053 1. 01E+01 2. 313 0 .049 1. 82E+01 2. 899 0 .045 2. 98E+01 3. 395 0 .039 5. 36E+01 3. 981 0 .014 4. 27E+01 3. 754 0 .015 8. 89E+01 4. 488 0 .006 6. 36E+01 4. 152 0 .004 1 .000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7941 97.6561 0. 300 0. 055 0. 3275 95.4201 95.9052 0. 250 0. 050 o: 2750 96.1588 96.5711 0. 212 0. 038 0. 2310 95.2164 95.4735 0. 180 0. 032 0. 1960 91.5951 91.8328 0. 150 0. 030 0. 1650 91.5066 91.7249 0. 125 0. 025 0. 1375 86.5157 86.7049 0. 106 0. 019 0. 1155 91.3123 91.3802 0. 090 0. 016 0 . 0980 87.2918 87.3646 0. 075 0. 015 0. 0825 91.5579 91.5870 PAN 86.8054 86.8248 Regression Output: Constant 5.366 Std E r r of Y Est 0.226 R Squared 0.959 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -12.16 Std E r r of Coef. 0.9561 GROWTH RATE 0.686 mm/hr NUCLEI DENSITY 214.0 number/mm^ NUCLEATION RATE 146.9 number/mmA3 hr SUPERSATURATION 0.025 g s a l t / g a c i d RUN # 03 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 60 SLURRY DENSITY Mt 0.1036 g/cnT3 RESIDENCE t min 11.79 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 2 .3634 0 .432 0.300 0 .8316 0 . 152 7. 29E+00 1 . 987 0.250 0 .5580 0 . 102 9. 09E+00 2 .207 0. 212 0 .6127 0 .112 2. 22E+01 3 .098 0.180 0 .3994 0 .073 2. 81E+01 3 . 335 0. 150 0 . 3228 0 .059 4. 06E+01 3 . 703 0.125 0 .1422 0 .026 3. 71E+01 3 .613 0. 106 0 . 1094 0 .020 6. 33E+01 4 . 148 0.090 0 .0438 0 .008 4. 92E+01 3 .896 0.075 0 .0492 0 .009 9. 90E+01 4 . 595 PAN 0 .0383 0 .007 TOTAL 5 .4708 1 .000 SIEVE SIEVE + 1IEVE DL L empty CRYSTALS mm) (mm) (mm) (g) (g) 0. 355 94.7835 97.1469 0. 300 0. 055 0. 3275 95.4060 96.2376 0. 250 0. 050 0. 2750 96.1495 96.7075 0. 212 0. 038 0. 2310 95.2135 95.8262 0. 180 0. 032 0. 1960 91.5895 91.9889 0. 150 0. 030 0. 1650 91.5036 91.8264 0. 125 0. 025 0. 1375 86.5131 86.6553 0. 106 0. 019 0. 1155 91.3106 91.4200 0. 090 0. 016 0. 0980 87.2905 87.3343 0. 075 0. 015 0. 0825 91.5525 91.6017 PAN 86.8031 86.8414 Regression Output: Constant 5. 197 Std E r r of Y Est 0. 223 R Squared 0. . 941 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.94 Std E r r of Coef. 0.9400 GROWTH RATE 0.512 mm/hr NUCLEI DENSITY 180.6 number/mm^ NUCLEATION RATE 92.5 number/mmA3 hr SUPERSATURATION 0.016 g s a l t / g a c i d RUN # 04 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1000 g/cm~3 RESIDENCE t min 14.87 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 3. 4758 0. 730 0. 300 0. 2476 0. 052 2. 41E+00 0. 879 0. 250 0. 3095 0. 065 5. 59E+00 1. 721 0. 212 0. 2000 0. 042 8. 02E+00 2. 082 0. 180 0. 1857 0. 039 1. 45E+01 2. 673 0. 150 0. 1381 0. 029 1. 92E+01 2. 957 0. 125 0. 0714 0. 015 2. 06E+01 3. 027 0. 106 0. 0571 0. 012 3. 67E+01 3. 602 0. 090 0. 0333 0. 007 4. 16E+01 3. 728 0. 075 0. 0190 0. 004 4. 25E+01 3. 749 PAN 0. 0238 0. 005 TOTAL 4. 7614 1. 000 SIEVE (mm) 0. 355 0.300 0. 250 0. 212 0.180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7887 95.4022 96.1502 95.2147 91.5917 91.4922 86.5138 91.3085 87.2933 91.5059 86.8077 SIEVE + CRYSTALS (g) 98.2645 95.6498 96.4597 95.4147 91.7774 91.6303 86.5852 91.3656 87.3266 91.5249 86.8315 Regression Output: Constant 4. . 831 Std E r r of Y Est 0 . . 130 R Squared 0. . 985 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -11.71 Std E r r of Coef. 0.5485 GROWTH RATE 0.345 mm/hr NUCLEI DENSITY 125.3 number/mm^ NUCLEATION RATE 43.2 number/mm^ hr SUPERSATURATION 0.016 g s a l t / g a c i d RUN # 05 CRYSTAL DENSITY 2.27 g/cmA3 TEMPERATURE C 60 SLURRY DENSITY Mt 0.0992 g/cm~3 RESIDENCE t min 15.03 SIEVE CRYSTALS w n Ln n (mm) (g) 0. .355 4. 2868 0. 597 0. . 300 0. 5888 0. 082 3. 77E+00 1. 326 0. . 250 0. 6822 0. 095 8. 11E+00 2. 093 0. . 212 0. 5744 0. 080 1. 52E+01 2. 718 0. , 180 0. 3088 0. 043 1. 58E+01 2. 762 0. . 150 0. 3303 0. 046 3. 03E+01 3. 411 0. 125 0. 2298 0. 032 4. 37E+01 3. 777 0. . 106 0. 0646 0. 009 2. 73E+01 3. 306 0. ,090 0. 0503 0. 007 4. 12E+01 3. 720 0. ,075 0. 0287 0. 004 4. 21E+01 3. 741 PAN 0. 0359 0. 005 TOTAL 7. 1806 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7658 99.0526 0. 300 0. 055 0. 3275 95. 3773 95.9661 0. 250 0. 050 0. 2750 96.1301 96.8123 0. 212 0. 038 0. 2310 95.1960 95.7704 0. 180 0. 032 0. 1960 91.5840 91.8928 0. 150 0. 030 0. 1650 91.4771 91.8074 0. 125 0. 025 0. 1375 86.5097 86.7395 0. 106 0. 019 0. 1155 91.2998 91.3644 0. 090 0. 016 0. 0980 87.2885 87.3388 0. 075 0. 015 0 . 0825 91.4717 91.5004 PAN 86.8057 86.8416 Regression Output: Constant 4. 731 Std E r r of Y Est 0. 258 R Squared 0. 918 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.66 Std E r r of Coef. 1.0877 GROWTH RATE 0.413 mm/hr NUCLEI DENSITY 113.4 number/mm"4 NUCLEATION RATE 46.9 number/mm"3 hr SUPERSATURATION 0.016 g s a l t / g a c i d RUN # 06 CRYSTAL DENSITY 2.27 g/cnTS TEMPERATURE C 60 SLURRY DENSITY Mt 0.1193 g/cm"3 RESIDENCE t min 7.18 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 0. 9830 0. 158 0. 300 1. 5056 0. 242 1. 34E+01 2. 593 0. 250 1. 3936 0. 224 2. 30E+01 3. 135 0. 212 0. 6719 0. 108 2. 46E+01 3. 203 0. 180 0. 4293 0. 069 3. 06E+01 3. 420 0. 150 0. 6408 0. 103 8. 16E+01 4. 401 0. 125 0. 2862 0. 046 7. 55E+01 4. 325 0. 106 0. 1929 0. 031 1. 13E+02 4. 727 0. 090 0. 0684 0. O i l 7. 79E+01 4. 356 0. 075 0. 0373 0. 006 7. 60E+01 4. 331 PAN 0. 0124 0. 002 TOTAL 6. 2215 1. 000 SIEVE (mm) 0. 355 0.300 0. 250 0.212 0. 180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7729 95.3787 96.1329 95.1971 91.5871 91.4703 86.5118 92.0153 87.2904 92.0961 86.8079 SIEVE + CRYSTALS (g) 95.7559 96.8843 97.5265 95.8690 92.0164 92.1111 86.7980 92.2082 87.3588 92.1334 86.8203 Regression Output: Constant 5. . 338 Std E r r of Y Est 0. .292 R Squared 0. . 867 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -8.32 Std E r r of Coef. 1.2328 GROWTH RATE 1.004 mm/hr NUCLEI DENSITY 208.1 number/mm~4 NUCLEATION RATE 208.9 number/mm"3 hr SUPERSATURATION 0.038 g s a l t / g a c i d RUN # 07 CRYSTAL DENSITY 2.27 g/cm"3 TEMPERATURE C 50 SLURRY DENSITY Mt 0.1836 g/cm~3 RESIDENCE t min . 10.98 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 4. 5754 0. 523 0. 300 0. 9536 0. 109 9. 27E+00 2. 226 0. 250 0. 9711 0. 111 1. 75E+01 2. 864 0. 212 0. 7873 0. 090 3. 16E+01 3. 452 0. 180 0. 4462 0. 051 3. 48E+01 3. 548 0. 150 0. 5074 0. 058 7. 07E+01 4. 258 0. 125 0. 2012 0. 023 5. 81E+01 4. 062 0. 106 0. 1050 0. 012 6. 73E+01 4. 209 0. 090 0. 1312 0. 015 1. 64E+02 5. 097 0. 075 0. 0350 0. 004 7. 80E+01 4. 357 PAN 0. 0350 0. 004 TOTAL 8. 7483 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS mm) (mm) (mm) (g) (g) 0. 355 94 . 7751 99 . 3505 0. 300 0. 055 0. 3275 95 . 3765 96 . 3301 0. 250 0. 050 0. 2750 96 . 1310 97 . 1021 0. 212 0. 038 0. 2310 95 . 1950 95 .9823 0. 180 0. 032 0. 1960 91 . 5874 92 .0336 0. 150 0. 030 0. 1650 91 .4628 91 . 9702 0. 125 0. 025 0. 1375 86 . 5130 86 . 7142 0. 106 0. 019 0. 1155 92 .0183 92 . 1233 0. 090 0. 016 0. 0980 87 . 2916 87 . 4228 0. 075 0. 015 0. 0825 92 . 1018 92 . 1368 PAN 86 .8073 86 . 8423 Regression Output: Constant 5.554 Std E r r of Y Est 0.298 R Squared 0.896 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.77 Std E r r of Coef. 1.2582 GROWTH RATE 0.559 mm/hr NUCLEI DENSITY 258.2 number/mm^ NUCLEATION RATE 144.4 number/mm^ hr SUPERSATURATION 0.036 g s a l t / g a c i d RUN # 08 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 50 SLURRY DENSITY Mt 0.2141 g/cm"3 RESIDENCE t rain 14.30 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 6 .3475 0. 523 0. 300 1 . 1408 0. 094 9. 32E+00 2 . 232 0.250 1 .6749 0. 138 2. 54E+01 3 .235 0. 212 0 .8496 0. 070 2. 86E+01 3 .354 0.180 0 .8253 0. 068 5. 40E+01 3 . 990 0. 150 0 . 5462 0. 045 6. 39E+01 4 . 158 0.125 0 .2427 0. 020 5. 89E+01 4 .076 0. 106 0 .2063 0. 017 1. 11E+02 4 . 711 0.090 0 . 1214 0. 010 1. 27E+02 4 .846 0.075 0 .0485 0. 004 9. 09E+01 4 . 510 PAN 0 . 1335 0. O i l TOTAL 12 . 1367 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. . 355 94.7857 101.1332 0. . 300 0. 055 0. 3275 95.3929 96.5337 0. . 250 0. 050 0. 2750 96.1445 97.8194 0. . 212 0. 038 0. 2310 95.2006 96.0502 0. . 180 0. 032 0. 1960 91.6047 92.4300 0. . 150 0. 030 0. 1650 91.4600 92.0062 0. . 125 0. 025 0. 1375 86.5213 86.7640 0. . 106 0. 019 0. 1155 92.0240 92.2303 0. .090 0. 016 0. 0980 87.2931 87.4145 0. .075 0. 015 0. 0825 92.1075 92.1560 PAN 86.7033 86.8368 Regression Constant Std E r r of Y Est R Squared No. of Observations Degrees of Freedom X C o e f f i c i e n t ( s ) Std E r r of Coef. Output: 5. . 632 0. . 245 0. . 924 9 7 -9. 57 1.0358 GROWTH RATE 0.439 mm/hr NUCLEI DENSITY 279.3 number/mm"4 NUCLEATION RATE 122.5 number/mm~3 hr SUPERSATURATION 0.026 g s a l t / g a c i d RUN # 09 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 60 SLURRY DENSITY Mt 0.0871 g/cnT3 RESIDENCE t min 11.01 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 4 .3749 0 .426 0. 300 1 .6740 0 . 163 6. 57E+00 1 .883 0.250 1 .3659 0 . 133 9. 96E+00 2 .299 0. 212 0 .8216 0 .080 1. 33E+01 2 . 588 0.180 0 . 5443 0 .053 1. 71E+01 2 . 841 0. 150 0 . 7291 0 .071 4. 10E+01 3 .715 0.125 0 .3492 0 .034 4. 08E+01 3 .708 0. 106 0 . 2259 0 .022 5. 85E+01 4 .070 0.090 0 .0822 . 0 .008 4. 14E+01 3 .723 0.075 0 .0513 0 .005 4. 62E+01 3 .834 PAN 0 .0513 0 .005 TOTAL 10 .2697 1 .000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7655 99.1404 0. 300 0. 055 0. 3275 95.3688 97.0428 0. 250 0. 050 0. 2750 96.1216 97.4875 0. 212 0. 038 0. 2310 95.1841 96.0057 0. 180 0. 032 0. 1960 91.5824 92.1267 0. 150 0. 030 0. 1650 91.4455 92.1746 0. 125 0. 025 0. 1375 86.5135 86.8627 0. 106 0. 019 0. 1155 92.0176 92.2435 0. 090 0. 016 0. 0980 87.2911 87.3733 0. 075 0. 015 0. 0825 92.1074 92.1587 PAN 86.8028 86.8541 Regression Output: Constant 4. . 820 Std E r r of Y Est 0 , . 245 R Squared 0. .916 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.04 Std E r r of Coef. 1.0348 GROWTH RATE 0.603 mm/hr NUCLEI DENSITY 124.0 number/mm~4 NUCLEATION RATE 74.7 number/mmA3 hr SUPERSATURATION 0.023 g s a l t / g a c i d RUN # 10 CRYSTAL DENSITY 2.27 g/cnT3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1834 g/cm~3 RESIDENCE t min 7.0 3 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 3. 4730 0. 607 0. 300 0. 5550 0. 097 8. 24E+00 2. 109 0. 250 0. 4634 0. 081 1. 28E+01 2. 548 0. 212 0. 3891 0. 068 2. 38E+01 3. 170 0. 180 0. 2460 0. 043 2. 93E+01 3. 377 0. 150 0. 2117 0. 037 4. 50E+01 3. 808 0. 125 0. 2117 0. 037 9. 34E+01 4. 537 0. 106 0. 0687 0. 012 6. 72E+01 4. 208 0. 090 0. 0458 0. 008 8. 71E+01 4. 468 0. 075 0. 0401 0. 007 1. 36E+02 4. 915 PAN 0. 0172 0 . 003 TOTAL 5. 7216 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94 . 7720 98. . 2450 0. 300 0. .055 0. 3275 95. 3676 95. 9226 0. 250 0 .050 0 . 2750 96. 1182 96. . 5816 0. 212 0 .038 0 . 2310 95. 1826 95. 5717 0. 180 0 .032 0. 1960 91. 5834 91. . 8294 0. 150 0 .030 0. 1650 91. 4390 91. . 6507 0. 125 0 .025 0. 1375 86. 5129 86. . 7246 0. 106 0 .019 0. 1155 92. 0187 92. 0874 0. 090 0 .016 0. 0980 87. 2900 87. . 3358 0. 075 0 .015 0. 0825 92. 1044 92. . 1445 PAN 86. 8018 86. . 8190 Regression Output: Constant 5.708 Std E r r of Y Est 0.189 R Squared 0.966 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -11.20 Std E r r of Coef. 0.7997 GROWTH RATE 0.762 mm/hr NUCLEI DENSITY 301.1 number/mm'4 NUCLEATION RATE 229.5 number/mm's3 hr SUPERSATURATION 0.039 g s a l t / g a c i d RUN #11 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 50 SLURRY DENSITY Mt 0.2070 g/cnT3 RESIDENCE t min 10.45 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 11. 2153 0. 616 0. 300 1. 6932 0. 093 8. 91E+00 2. 188 0. 250 1. 7296 0. 095 1. 69E+01 2. 828 0. 212 1. 0560 0. 058 2. 29E+01 3. 132 0. 180 0. 7465 0. 041 3. 15E+01 3. 450 0. 150 0. 6190 0. 034 4. 67E+01 3. 844 0. 125 0. 5280 0. 029 8. 26E+01 4. 414 0. 106 0. 2731 0. 015 9. 49E+01 4. 553 0. 090 0. 1457 0. 008 9. 84E+01 4. 589 0. 075 0. 1457 0. 008 1. 76E+02 5. 170 PAN 0. 0546 0. 003 TOTAL 18. 2066 1. 000 SIEVE (mm) 0. 355 0.300 0. 250 0.212 0. 180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7798 95.3697 96.1144 95.1833 91.5856 91.4315 86.5126 92.0166 87.2890 92.1028 86.8020 SIEVE + CRYSTALS . (g) 105.9951 97.0629 97.8440 96.2393 92.3321 92.0505 87.0406 92.2897 87.4347 92.2485 86.8566 Regression Output: Constant 5. .872 Std E r r of Y Est 0 . . 157 R Squared 0. . 977 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -11.48 Std E r r of Coef. 0.6643 GROWTH RATE 0.500 mm/hr NUCLEI DENSITY 355.1 number/mm^ NUCLEATION RATE 177.6 number/mm"3 hr SUPERSATURATION 0.037 g s a l t / g a c i d RUN # 12 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1002 g/cm~3 RESIDENCE t min 6.95 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 4. 5316 0. 642 0. 300 0. 5788 0. 082 3. 80E+00 1. 336 0. 250 0. 4306 0. 061 5. 26E+00 1. 660 0. 212 0. 4165 0. 059 1. 13E+01 2. 424 0. 180 0. 3882 0. 055 2. 05E+01 3. 018 0. 150 0. 2894 0. 041 2. 73E+01 3. 306 0. 125 0. 1835 0. 026 3. 59E+01 3. 580 0. 106 0. 0847 0. 012 3. 67E+01 3. 604 0. 090 0. 0565 0. 008 4. 76E+01 3. 863 0. 075 0. 0565 0. 008 8. 51E+01 4. 444 PAN 0. 0424 0. 006 TOTAL 7. 0585 1. 000 SIEVE (mm) 0. 355 0.300 0. 250 0. 212 0. 180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7383 95.3432 96.0839 95.1686 91.5568 91.4132 86.4881 92.0103 87.2770 92.1047 86.8022 SIEVE + CRYSTALS (g) 99.2699 95.9220 96.5145 95.5851 91.9450 91.7026 86.6716 92.0950 87.3335 92.1612 86.8446 Regression Output: Constant 5. .226 Std E r r of Y Est 0. . 178 R Squared 0. , 974 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -12.16 Std E r r of Coef. 0.7533 GROWTH RATE 0.710 mm/hr NUCLEI DENSITY 186.1 number/mm~4 NUCLEATION RATE 132.1 number/mm"3 hr SUPERSATURATION 0.036 g s a l t / g a c i d RUN it 13 CRYSTAL DENSITY 2.27 g/cm"3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1312 g/cnT3 RESIDENCE t min 10.51 SIEVE (mm) 0.355 0. 300 0.250 0. 212 0.180 0. 150 0.125 0. 106 0.090 0.075 PAN TOTAL CRYSTALS (g) 4.7283 1.0961 1.1054 0.5759 0.7060 0.3530 0.3623 0.1579 0.0836 0.0743 0.0464 9.2894 w n Ln n 0. 509 0. 118 7. 17E+00 1. 970 0. 119 1. 34E+01 2. 598 0. 062 1. 55E+01 2. 743 0. 076 3. 70E+01 3. 611 0. 038 3. 31E+01 3. 499 0. 039 7. 04E+01 4. 255 0. 017 6. 81E+01 4. 222 0. 009 7. 01E+01 4. 250 0. 008 1. 11E+02 4. 714 0. 005 1. 000 SIEVE (mm) 0. 355 0. 300 0. 250 0.212 0. 180 0. 150 0. 125 0. 106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7542 95.3549 96.0896 95.1722 91.5622 91.4100 86.4933 92.0166 87.2801 92.0805 86.8040 SIEVE + CRYSTALS (g) 99.4825 96.4510 97.1950 95.7481 92.2682 91.7630 86.8556 92.1745 87.3637 92.1548 86.8504 Regression Output: Constant 5. .499 Std E r r of Y Est 0. . 200 R Squared 0. , 959 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -10.83 Std E r r of Coef. 0.8428 GROWTH RATE 0.527 mm/hr NUCLEI DENSITY 244.5 number/mm"4 NUCLEATION RATE 128.9 number/mmA3 hr SUPERSATURATION 0.030 g s a l t / g a c i d RUN # 14 CRYSTAL DENSITY 2.27 g/cm"3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1382 g/cm"3 RESIDENCE t min 10.59 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 5 .6134 0. 460 0. 300 1 .9891 0. 163 1. .04E+01 2 . 345 0.250 1 .4278 0. 117 1. .39E+01 2 .633 0. 212 0 .8176 0. 067 1. . 77E+01 2 .873 0.180 0 .7078 0. 058 2. . 98E+01 3 . 393 0. 150 0 . 8420 0. 069 6. .33E+01 4 . 148 0.125 0 .2441 0. 020 3. . 80E+01 3 .639 0. 106 0 . 2685 0. 022 9. .29E+01 4 . 531 0.090 0 . 1098 0. 009 7. .39E+01 4 . 302 0.075 0 . 1342 0. O i l 1. .61E+02 5 .084 PAN 0 .0488 0. 004 TOTAL 12 . 2030 1. 000 SIEVE (mm) 0. 355 0. 300 0. 250 0. 212 0. 180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7451 95.3424 96.0788 95.1674 91.5594 91.4027 86.4920 92.0159 87.2804 92.0790 86.8012 SIEVE + CRYSTALS (g) 100.3585 97.3315 97.5066 95.9850 92.2672 92.2447 86.7361 92.2844 87.3902 92.2132 86.8500 Regression Output: Constant 5. . 566 Std E r r of Y Est 0. .316 R Squared 0. . 899 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -10.53 Std E r r of Coef. 1.3368 GROWTH RATE 0.538 mm/hr NUCLEI DENSITY 261.3 number/mnT4 NUCLEATION RATE 140.6 number/mm"3 hr SUPERSATURATION 0.025 g s a l t / g a c i d RUN it 15 CRYSTAL DENSITY TEMPERATURE C SLURRY DENSITY Mt RESIDENCE t min 2.27 g/cm"-3 60 0.0428 g/cm~3 14. 52 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 6 .9449 0. 661 0. 300 0 . 8615 0. 082 1. 62E+00 0 .485 0. 250 0 .7144 0. 068 2. 50E+00 0 .918 0. 212 0 . 5884 0. 056 4. 58E+00 1 . 521 0. 180 0 . 5043 0. 048 7. 63E+00 2 .032 0. 150 0 . 3677 0. 035 9. 94E+00 2 .297 0.125 0 .2311 0. 022 1. 30E+01 2 .562 0. 106 0 . 1156 0. O i l 1. 44E+01 2 . 666 0.090 0 .0735 0. 007 1. 78E+01 2 .879 0.075 0 .0630 0. 006 2. 73E+01 3 . 306 PAN 0 .0420 0. 004 TOTAL 10 . 5066 1. 000 SIEVE (mm) 0.355 0.300 0. 250 0. 212 0.180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7287 95.3120 96.0659 95.1596 91.5514 91.3894 86.4864 92.6661 87.2702 92.7314 86.7976 SIEVE + CRYSTALS (g) 101.6736 96.1735 96.7803 95.7480 92.0557 91.7571 86.7175 92.7817 87.3437 92.7944 86.8396 Regression Output: Constant 4.079 Std E r r of Y Est 0.109 R Squared 0.988 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -11.09 Std E r r of Coef. 0.4586 GROWTH RATE 0.373 mm/hr NUCLEI DENSITY 59.1 number/mm^ NUCLEATION RATE 22.0 number/mm^ hr SUPERSATURATION 0.012 g s a l t / g a c i d RUN # 16 CRYSTAL DENSITY 2.27 g/cm~3 TEMPERATURE C 50 SLURRY DENSITY Mt 0.2320 g/cm"3 RESIDENCE t min 7. 26 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 8.0318 0. 510 0. 300 1.6379 0. 104 1. 12E+01 2 .413 0.250 1.9056 0. 121 2. 41E+01 3 . 184 0. 212 1.0394 0. 066 2. 92E+01 3 . 376 0.180 1.0394 0. 066 5. 68E+01 4 .040 0. 150 0.6929 0. 044 6. 78E+01 4 . 216 0.125 0.5827 0 . 037 1. 18E+02 4 .772 0. 106 0.4567 0. 029 2. 06E+02 5 . 326 0.090 0.1102 0. 007 9. 65E+01 4 . 569 0.075 0. 1417 0. 009 2. 22E+02 5 . 402 PAN 0.1102 0. 007 TOTAL 15.7487 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7397 102.7715 0. 300 0. 055 0. 3275 95.3133 96.9512 0. 250 0. 050 0. 2750 96.0698 97.9754 0. 212 0. 038 0. 2310 95.1625 96.2019 0. 180 0. 032 0. 1960 91.5566 92.5960 0. 150 0. 030 0. 1650 91.3895 92.0824 0. 125 0. 025 0. 1375 86.4914 87.0741 0. 106 0. 019 0. 1155 92.6773 93.1340 0. 090 0. 016 0. 0980 87.2769 87.3871 0. 075 0. 015 0. 0825 92.7500 92.8917 PAN 86.8002 86.9104 Regression Output: Constant 6.238 Std E r r of Y Est 0.285 R Squared 0.930 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -11.57 Std E r r of Coef. 1.2020 GROWTH RATE 0.714 mm/hr NUCLEI DENSITY 511.7 number/mm"4 NUCLEATION RATE 365.4 number/mm^ hr SUPERSATURATION 0.051 g s a l t / g a c i d RUN 17 CRYSTAL DENSITY 2.27 g/cmA3 TEMPERATURE C 55 SLURRY DENSITY Mt 0.1828 g/cm"3 RESIDENCE t min 14.23 SIEVE CRYSTALS w n Ln n (mm) (g) 0.355 5 .5094 0. 504 0. 300 1 .4757 0. 135 1. 14E+01 2 .436 0. 250 1 .4102 0. 129 2. 03E+01 3 .010 0.212 0 . 7105 0. 065 2. 27E+01 3 . 122 0.180 0 .4591 0. 042 2. 85E+01 3 .350 0. 150 0 .6887 0. 063 7. 64E+01 4 . 336 0.125 0 .3170 0. 029 7. 30E+01 4 .290 0. 106 0 . 2186 0. 020 1. 12E+02 4 .716 0.090 0 .0765 0. 007 7. 60E+01 4 .331 0.075 0 .0437 0. 004 7. 77E+01 4 . 352 PAN 0 .0219 0. 002 TOTAL 10 . 9314 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) (g) (g) 0. 355 94.7436 100.2530 0. 300 0. 055 0. 3275 95.3133 96.7890 0. 250 0. 050 0. 2750 96.0698 97.4800 0. 212 0. 038 0. 2310 95.1598 95.8703 0. 180 0. 032 0. 1960 91.5578 92.0169 0. 150 0. 030 0. 1650 91.3864 92.0751 0. 125 0. 025 0. 1375 86.4916 86.8086 0. 106 0. 019 0. 1155 92.6829 92.9015 0. 090 0. 016 0. 0980 87.2774 87.3539 0. 075 0. 015 0. 0825 92.7578 92.8015 PAN 86.8006 86.8225 Regression Output: Constant 5.395 Std E r r of Y Est 0.287 R Squared 0.887 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -8.98 Std E r r of Coef. 1.2130 GROWTH RATE 0.470 mm/hr NUCLEI DENSITY 220.3 number/mm~4 NUCLEATION RATE 103.5 number/mm^ hr SUPERSATURATION 0.021 g s a l t / g a c i d RUN # 18 CRYSTAL DENSITY 2.27 g/cm'-S TEMPERATURE C 50 SLURRY DENSITY Mt 0.2563 g/cm"3 RESIDENCE t min 13.62 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 12. 3667 0. 711 0. 300 1. 2001 0. 069 8. 19E+00 2, , 103 0. 250 1. 0784 0. 062 1. 37E+01 2. , 615 0. 212 0. 7479 0. 043 2. 10E+01 3, .047 0. 180 0. 4870 0. 028 2. 66E+01 3. .282 0. 150 0. 6088 0. 035 5. 95E+01 4. .087 0. 125 0. 3653 0. 021 7. 41E+01 4. . 305 0. 106 0. 2609 0. 015 1. 17E+02 4. . 766 0. 090 0. 1044 0. 006 9. 13E+01 4. .515 0. 075 0. 0522 0. 003 8. 17E+01 4. .403 PAN 0. 1218 0. 007 TOTAL 17. 3934 1. 000 SIEVE (mm) 0. 355 0. 300 0. 250 0.212 0. 180 0.150 0. 125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7590 95.3372 96.0805 95.1686 91.5615 91.3996 86.4940 92.6785 87.2806 92.7450 86.8007 SIEVE + CRYSTALS (g) 107.1257 96.5373 97.1589 95.9165 92.0485 92.0084 86.8593 92.9394 87.3850 92.7972 86.9225 Regression Output: Constant 5. . 662 Std E r r of Y Est 0. . 241 R Squared 0. . 943 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -10.96 Std E r r of Coef. 1.0195 GROWTH RATE 0.402 mm/hr NUCLEI DENSITY 287.7 number/mraA4 NUCLEATION RATE 115.7 number/mnT 3 hr SUPERSATURATION 0.027 g s a l t / g a c i d RUN # 19 CRYSTAL DENSITY TEMPERATURE C SLURRY DENSITY Mt RESIDENCE t min 2.27 g/cmA3 45 0.2720 g/cm-3 14.73 SIEVE CRYSTALS w n Ln n (mm) (g) 0. 355 6 . 2842 0. 412 0. 300 2 . 5015 0. 164 2. 07E+01 3 .028 0.250 2 . 2422 0. 147 3. 44E+01 3 .538 0. 212 1 . 4338 0. 094 4. 88E+01 3 . 888 0.180 1 . 1440 0. 075 7. 57E+01 4 . 327 0. 150 0 . 5949 0. 039 7. 04E+01 4 . 254 0.125 0 .4881 0. 032 1. 20E+02 4 .786 0. 106 0 . 2440 0. 016 1. 33E+02 4 . 890 0.090 0 . 1830 0. 012 1. 94E+02 5 .267 0.075 0 . 1068 0. 007 2. 02E+02 5 . 309 PAN 0 .0305 0. 002 ,TOTAL 15 . 2528 1. 000 SIEVE SIEVE + SIEVE DL L empty CRYSTALS (mm) (mm) (mm) ( g) (g) 0. , 355 94 . 7629 101 .0471 0. . 300 0 . 055 0 . 3275 95. 3129 97 . 8144 0. . 250 0. 050 0 . 2750 96. 0711 98 . 3133 0. . 212 0. 038 0 .2310 95. 1608 96 . 5946 0. . 180 0. 032 0 . 1960 91. 5598 92 . 7038 0. . 150 0. 030 0 . 1650 91. 3790 91 . 9739 0. . 125 0. 025 0 . 1375 86. 4916 86 . 9797 0. . 106 0 . 019 0 .1155 92. 6786 92 . 9226 0. .090 0. 016 0 .0980 87. 2752 87 . 4582 0. .075 0. 015 0 .0825 92. 7565 92 . 8633 PAN 86. 7995 86 . 8300 Regression Output: Constant 6.033 Std E r r of Y Est 0 . 123 R Squared 0.978 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.22 Std E r r of Coef. 0.5183 GROWTH RATE NUCLEI DENSITY NUCLEATION RATE SUPERSATURATION 0.442 mm/hr 417.1 number/mm'M 184.2 number/mraA3 hr 0.037 g s a l t / g a c i d *Run #20 l o s t due t o computer f a u l t * 125 RUN # 21 CRYSTAL DENSITY 2.27 g/cnT3 TEMPERATURE C 45 SLURRY DENSITY Mt 0. 2359 g/cm~3 RESIDENCE t min 6. 98 SIEVE CRYSTALS DW n Ln n (mm) (g) 0. 355 7.1962 0 .450 0. 300 2.2708 0 . 142 1.55E+01 2 . 741 0. 250 2.0149 0 .126 2.56E+01 3 .241 0. 212 1.4872 0 .093 4.19E+01 3 . 735 0.180 1.2473 0 .078 6.83E+01 4 .224 0. 150 0.6237 0 .039 6.11E+01 4 . 112 0.125 0.5437 0 .034 1.10E+02 4 .704 0. 106 0.2719 0 .017 1.23E+02 4 . 808 0.090 0.1919 0 .012 1.68E+02 5 .125 0.075 0.0959 0 .006 1.50E+02 5 .013 PAN 0.0480 0 .003 TOTAL 15.9915 1 .000 SIEVE (mm) 0. 355 0. 300 0. 250 0.212 0. 180 0.150 0.125 0.106 0.090 0.075 PAN DL (mm) 0.055 0.050 0.038 0.032 0.030 0.025 0.019 0.016 0.015 L (mm) 0.3275 0.2750 0.2310 0.1960 0.1650 0.1375 0.1155 0.0980 0.0825 SIEVE empty (g) 94.7423 95.3084 96.0733 95.1645 91.5660 91.3767 86.5004 92.6940 87.2785 92.7697 86.7976 SIEVE + CRYSTALS (g) 101.9385 97.5792 98.0882 96.6517 92.8133 92.0004 87.0441 92.9659 87.4704 92.8656 86.8456 Regression Output: Constant 5.944 Std E r r of Y Est 0.138 R Squared 0.975 No. of Observations 9 Degrees of Freedom 7 X C o e f f i c i e n t ( s ) -9.70 Std E r r of Coef. 0.5819 GROWTH RATE NUCLEI. DENSITY NUCLEATION RATE SUPERSATURATION 0.886 mm/hr 381.6 number/mm~4 338.1 number/mm~3 hr 0.082 g s a l t / g a c i d A P P E N D I X B 127 CRYSTAL 8IZE, L (mm) RUN #2 CRYSTAL SIZE, L (mm) run 3 129 CRYSTAL SIZE, L (mm) run 6 I 1 1 1 i i L 0 0 .05 0.1 0.16 0.2 0 .26 0.6 0.86 CRYSTAL 8IZE, L (mm) run 6 130 CRYSTAL 8IZE, L (mm) ran 7 131 C R Y S T A L S IZE , L (mm) run 10 132 run ii 1 0 0 .09 0.1 0 .« 0.2 0 .25 0 .3 0 .30 CRYSTAL SIZE, L (mm) CRYSTAL SIZE, L (mm) ran 12 133 CRYSTAL SIZE, L (mm) ran 18 0 0 .00 0.1 0.18 0 .2 0 .20 OA 0 .00 CRYSTAL 8IZE, L (mm) ran 14 134 100 IS 0.1 0.10 0 .2 0 .80 CRYSTAL 8IZE, L (mm) 0.30 ran 18 0.00 0.1 0.10 0 .2 0 .20 CRYSTAL 8IZE, L (mm) ran 16 nm 18 CRYSTAL SIZE, L W mm m *Runs # 19b, 20, and 21b l o s t due to a computer f a u l t * A P P E N D I X C 139 MTB > it TEMPERATURE 45 C MTB > PRINT C12 C1-C4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 19 0.037 0.459 190. 2 0.2720 2 19 0.037 0.442 184.2 0.2720 3 20 0.055 0. 561 212. 1 0.2224 4 21 0.082 0.886 338. 1 0.2359 5 21 0.082 0.881 417. 3 0.2359 MTB > tt GROWTH RATE r e g r e s s i o n MTB > regr c6 1 c5 The re g r e s s i o n equation i s Ln G = 1.97 + 0.847 Ln S P r e d i c t o r Constant Ln S Coef 1.9709 0.84679 Stdev 0.2273 0.07781 t - r a t i o 8.67 10.88 P 0.003 0.002 s = 0.06192 R-sq An a l y s i s of Variance 97. 5% R-sq(adj) = 96.7% SOURCE DF Regression 1 E r r o r 3 Tot a l 4 SS 0.45412 0.01150 0.46562 MS 0 . 45412 0.00383 F 118.43 • P 0.002 MTB > it PRIMARY NUCLEATION model MTB > regr c7 1 c5 The r e g r e s s i o n equation i s Ln B : 8.07 + 0.876 Ln S P r e d i c t o r Constant Ln S Coef 8.0744 0.8757 Stdev 0.5280 0.1808 t - r a t i o 15. 29 4. 84 P 0.001 0.017 s = 0.1438 _ R-sq An a l y s i s of Variance 88. 7% R-sq(adj) = 84.9% SOURCE DF SS MS Regression 1 0.48561 0.48561 E r r o r 3 0.06207 0.02069 To t a l 4 0.54768 F 23. 47 P 0.017 140 MTB > # PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 6.05 + 1.05 Ln G P r e d i c t o r Constant Ln G Coef 6.04542 1.0528 s = 0.1026 R-sq A n a l y s i s of Variance Stdev 0.08604 0.1503 t - r a t i o 70.26 7.00 P 0.000 0.006 R-sq(adj) = 92.3% SOURCE DF SS MS Regression 1 0.51611 0.51611 E r r o r 3 0.03157 0.01052 To t a l 4 0.54768 F 49.04 P 0.006 MTB > SECONDARY NUCLEATION model MTB > regr c7 2 c8 c5 The r e g r e s s i o n equation i s Ln B = 23.1 + 1.70 Ln Mt + 1.18 Ln S P r e d i c t o r Constant Ln Mt Ln S Coef 23.096 1.7022 1.1798 Stdev 8.064 0.9127 0.2109 t - r a t i o 2. 86 1. 87 5. 59 P 0. 103 0.203 0.031 0.1064 R-sq = 95.9% R-sq(adj) = 91.7% A n a l y s i s of Variance SOURCE DF SS ' MS Regression 2 0.52502 0. 26251 E r r o r 2 0.02266 0. 01133 T o t a l 4 0.54768 SOURCE DF SEQ SS Ln Mt 1 0.17050 Ln S 1 0.35452 F 23. 17 P 0.041 141 Unusual Observations Obs. Ln Mt Ln B 3 -8.41 5.3571 F i t S t d e v . F i t R e s i d u a l 5.3571 0.1064 0.0000 St.Resid * X X denotes an obs. whose X value gives i t l a r g e i n f l u e n c e . MTB > » SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i s Ln B = 11.5 + 0.650 Ln Mt + 1.17 Ln G -P r e d i c t o r Coef Stdev t - r a t i o P Constant 11 . 502 6 . 662 1. 73 0.226 Ln Mt 0. 6501 0. 7935 0 . 82 0 . 499 Ln G 1. 1697 0. 2139 5.47 0.0 32 s = 0.1087 R-sq = 95.7% R-sq(adj) = : 91.4% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 2 0 .52404 0 .26202 22.17 0.043 E r r o r 2 0 .02364 0 .01182 T o t a l 4 0 .54768 SOURCE DF SEQ SS Ln Mt 1 0 .17050 Ln G 1 0 .35354 Unusual Observations Obs. Ln Mt Ln B F i t S t d e v . F i t Residual St.Resid 3 -8.41 5.3571 5.3582 0.1086 -0.0011 -0.29 X X denotes an obs. whose X value gives i t lar g e i n f l u e n c e . •MTB > noou 142 MTB > # TEMPERATURE 50 C MTB > p r i n t c l 2 c l - c 4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 1 0.040 0. 621 119.3 0.1118 2 16 0.051 0.714 365. 4 0.2320 3 7 0.036 0 . 559 144.4 0.1836 4 11 0.037 0 . 500 177. 6 0. 2070-5 8 0.026 0.439 122. 5 0 . 2141 6 18 0.027 0.402 115.7 0.2563 MTB > H GROWTH RATE r e g r e s s i o n MTB > regr c6 1 c5 The r e g r e s s i o n equation i s Ln G - 2.08 + 0.813 Ln S P r e d i c t o r Constant Ln S Coef 2.0849 0.8134 Stdev 0.4302 0.1283 t - r a t i o 4.85 6. 34 P 0.008 0 .003 s = 0.07243 R-sq = 91.0% R-sq(adj) 88. 7% An a l y s i s of Variance SOURCE DF SS Regression 1 0.21103 E r r o r 4 0.02099 To t a l 5 0.23202 MS 0.21103 F 40. 22 P 0.003 MTB > # PRIMARY NUCLEATION model MTB > regr c7 1 c5 The r e g r e s s i o n equation i s Ln B = 9.67 + 1.37 Ln S P r e d i c t o r Constant Ln S Coef 9. 666 1.3749 Stdev 1. 788 0 . 5331 t - r a t i o 5.41 2. 58 s = 0.3011 R-sq = 62.4% A n a l y s i s of Variance SOURCE " DF SS MS Regression 1 0.60285 0.60285 E r r o r 4 0.36257 0.09064 To t a l 5 0.96542 P 0.006 0.061 R-sq(adj) = 53.1% 6. P 0.061 143 MTB > it PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 5.97 + 1.41 Ln G P r e d i c t o r Constant Ln G Coef 5.9659 1.41-30 Stdev 0.4905 0.7356 t - r a t i o 12. 16 1.92 s = 0.3543 R-sq = 48.0% A n a l y s i s of Variance SOURCE DF Regression 1 E r r o r 4 Tot a l 5 0. 4632 0.5022 0.9654 MS 0.4632 0.1255 P 0.000 0 . 127 R-sq(adj) = 35.0% F 3. 69 P 0. 127 MTB > # SECONDARY NUCLEATION model MTB > regr c7 2 c8 c5 The r e g r e s s i o n equation i s Ln B = 18.2 + 0.888 Ln Mt + 1.67 Ln S P r e d i c t o r Constant Ln Mt Ln S Coef 18.245 0.8884 1.6698 Stdev 2. 142 0.2069 0.2403 t - r a t i o 8. 52 4.29 6. 95 P 0.003 0.023 0.006 s = 0.1300 R-sq = 94.7% R-sq(adj) = 91.2% A n a l y s i s of Variance SOURCE DF SS MS Regression 2 0.91468 0. 45734 E r r o r 3 0.05074 0. 01691 T o t a l 5 0.96542 SOURCE DF SEQ SS Ln Mt 1 0.09811 Ln S 1 0.81657 F 27.04 P 0.012 144 MTB > # SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i s Ln B = 16.1 + 1.14 Ln Mt + 2.08 Ln G P r e d i c t o r Coef Stdev t - r a t i o P Constant 16 .102 1.953 8. 25 0.004 Ln Mt 1. 1364 0.2180 5. 21 0 .014 Ln G 2. 0804 0.2969 7 .01 0.006 s = 0.1290 R-sq = 94.8% R-sq(adj) = : 91.4% A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 0 .91548 0 .45774 27.50 0. .012 E r r o r 3 0 .04994 0 .01665 T o t a l 5 0 .96542 SOURCE DF SEQ SS Ln Mt 1 0 .09811 Ln G 1 0 .81737 MTB > noou 145 MTB # TEMPERATURE 55 C MTB > .print c l 2 c l - c 4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 10 0 .039 0 . 857 262. 7 0.1834 2 12 0.036 0.710 132. 1 0.1002 3 13 0.030 0. 527 128. 9 0.1312 4 14 0.025 0. 538 140.6 0.1382 5 4 0.016 0. 345 43. 2 0.1000 6 17 0.021 0.470 103. 5 0.1828 MTB > # GROWTH RATE r e g r e s s i o n MTB > regr c6 1 c5 The r e g r e s s i o n equation i s Ln G = 2.69 + 0.907 Ln S P r e d i c t o r Constant Ln S Coef 2.6932 0.9069 Stdev 0.4645 0.1276 t - r a t i o 5.80 7,11 P 0.004 0.002 s = 0.09639 R-sq A n a l y s i s of Variance' 92. 7% R-sq(adj) 90.8% SOURCE DF SS MS Regression 1 0.46925 0.46925 E r r o r 4 0.03717 0.00929 To t a l 5 0.50642 F 50. 50 P 0. 002 MTB > # PRIMARY NUCLEATION model MTB > regr c7 1 c5 The r e g r e s s i o n equation i s Ln B = 10.4 + 1.55 Ln S P r e d i c t o r Constant Ln S Coef 10.384 1.5457 Stdev 1.436 0 . 3945 t - r a t i o 7. 23 3. 92 P 0.002 0.017 s = 0.2980 R-sq An a l y s i s of Variance SOURCE " DF Regression 1 E r r o r 4 To t a l 5 79. 3% 1.3632 0.3552 1.7184 R-sq(adj) = 74.2% MS 1.3632 0.0888 F 15. 35 P 0.017 146 MTB > # PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 5.80 + 1.71 Ln G P r e d i c t o r Coef Stdev t - r a t i o p Constant 5.7960 0.2282 25.40 0.000 Ln G 1.7087 0.3441 4.97 0.008 s = 0.2449 R-sq - 86.0% R-sq(adj) = 82.6% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 1 1.4786 1.4786 24.66 0.008 E r r o r 4 0.2398 0.0600 To t a l 5 1.7184 MTB > # SECONDARY NUCLEATION model MTB > regr c7 2 c8 c5 The r e g r e s s i o n equation i s Ln B = 17.9 + 0.909 Ln Mt + 1.38 Ln S P r e d i c t o r Coef Stdev t - r a t i o " P Constant 17 .883 2.236 8.00 0.004 Ln Mt 0. 9085 0.2562 3.55 0.038 Ln S 1. 3816 0.2052 6.73 0 .007 s = 0.1510 R-sq = 96.0% R-sq(adj) = : 93.4% A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 1 .65000 0.82500 36.18 0. .008 E r r o r 3 0 .06841 0.02280 To t a l 5 1 .71840 SOURCE DF SEQ SS Ln Mt 1 0 .61619 Ln S 1 1 .03381 147 MTB > # SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i s Ln B = 11.6 + 0.668 Ln Mt + 1.51 Ln G P r e d i c t o r Constant Ln Mt Ln G Coef 11.631 0.6684 1.5070 Stdev 2. 778 0.3177 0.2701 t - r a t i o 4.19 2. 10 5. 58 P 0.025 0 . 126 0.011 s = 0.1797 R-sq = 94.4% R-sq(adj) : : 90.6% A n a l y s i s of Variance SOURCE Regression E r r o r T o t a l DF 2 1 3 0 5 1 SS . 62152 .09688 .71840 0 0 MS .81076 .03229 F 25.11 0. P .013 SOURCE Ln Mt Ln G DF 1 0 1 1 SEQ SS .61619 .00534 MTB > noou 148 MTB > it TEMPERATURE 60 C MTB > p r i n t c l 2 c l - c 4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 2 0.025 0. 686 146. 9 0 .0998 2 6 0.038 1.004 208. 9 0.1193 3 3 0.016 0. 512 92. 5 0.1036 4 9 0.023 0.603 74. 7 0.0871 5 5 0.016 0 .413 46. 9 0.0992 6 15 0.012 0.373 22.0 0.0428 MTB > it GROWTH RATE r e g r e s s i o n MTB > regr c6 1 c5 The r e g r e s s i o n equation i s Ln G = 2.80 + 0.862 Ln S P r e d i c t o r Constant Ln S Coef 2.7958 0.86206 Stdev 0.3541 0.09027 t - r a t i o 7.90 9. 55 P 0.001 0.001 s = 0.08281 R-sq = 95.8% A n a l y s i s of Variance SOURCE DF SS Regression 1 0.62533 E r r o r 4 0.02743 T o t a l 5 0.65276 R-sq(adj) = 94.7% M b 0.62533 0.00686 F 91. 19 P 0.001 MTB > it PRIMARY NUCLEATION model MTB > regr c7 1 c5 The re g r e s s i o n equation i s q P r e d i c t o r Constant Ln S Coef 11.264 1. 7705 s = 0.3953 R-sq An a l y s i s of Variance Stdev 1.690 0.4310 80. 8% t - r a t i o 6. 66 4.11 R-sq(adj) : P 0.003 0.015 76.0% SOURCE DF Regression 1 E r r o r 4 To t a l 5 SS 2.6376 0.6251 3.2627 MS 2.6376 0.1563 F 16. 88 P 0.015 149 MTB > # PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 5.53 + 2.08 Ln G P r e d i c t o r Constant Ln G Coef 5.5343 2.0752 Stdev 0.2738 0.4159 t - r a t i o 20.21 4.99 P 0.000 0.008 s = 0.3360 R-sq = 86.2% A n a l y s i s of Variance R-sq(adj) = 82.7% SOURCE DF Regression 1 E r r o r 4 To t a l 5 SS 2.8112 0.4516 3.2627 MS 2.8112 0.1129 F 24.90 P 0.008 MTB > # SECONDARY NUCLEATION model MTB > regr c7 2 c8 c5 The r e g r e s s i o n equation i s Ln B r 17.9 + 0.961 Ln Mt + 1.17 Ln S P r e d i c t o r Coef Stdev t - r a t i o P Constant 17.892 4.062 4.40 0.022 Ln Mt 0.9613 0.5541 1. 73 0 .181 Ln S 1.1682 0.4941 2. 36 , 0.099 s = 0.3225 R-sq = 90.4% R-sq(adj) = : 84.1% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 2 2.9506 1.4753 14.18 0.030 E r r o r 3 0.3121 0.1040 T o t a l 5 3.2627 SOURCE DF SEQ SS Ln Mt 1 2.3691 Ln S 1 0.5815 150 MTB > # SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i s Ln B = 1 3 . 8 + 0 . 9 1 9 Ln Mt + 1 . 4 5 Ln G . P r e d i c t o r Coef Stdev t - r a t i o P Constant 1 3 . 7 6 1 3 . 2 2 5 4 . 2 7 0 . 0 2 4 Ln Mt 0 . 9 1 8 8 0 . 3 5 9 6 2 . 5 5 0 . 0 8 4 Ln G 1 . 4 4 9 7 0 . 3 6 4 1 3 . 9 8 0 . 0 2 8 s = 0 . 2 1 7 7 R-sq = 9 5 . 6 % R-sq(adj) r : 9 2 . 7 % A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 3 . 1 2 0 6 1. 5 6 0 3 3 2 . 9 2 0 . 0 0 9 E r r o r 3 0 . 1 4 2 2 0 . 0 4 7 4 T o t a l 5 3 . 2 6 2 7 SOURCE DF SEQ SS Ln Mt 1 2 . 3 6 9 1 Ln G 1 0 . 7 5 1 4 MTB > noou 1 5 1 A P P E N D I X D 152 MTB > # TEMPERATURE DEPENDENCE OF RATE CONSTANTS MTB > p r i n t c5 c l - c 3 ROW Temp. C 1/T (/K) LnKg LnKn 1 45 0.00314 1. 971 23.096 2 50 0.00310 2.085 18.245 3 55 0.00305 2. 693 17.883 4 60 0.00300 2. 796 17.892 MTB > # GROWTH RATE CONSTANT r e g r e s s i o n MTB > regr c2 1 c l The r e g r e s s i o n equation i s LnKg = 22.6 - 6584 1/T (/K) P r e d i c t o r Constant 1/T (/K) Coef 22.617 -6584 Stdev 4. 381 1426 t - r a t i o 5. 16 -4.62 P 0.036 0.044 s = 0 . 1500 R-sq = 91.4% A n a l y s i s of Variance R-sq(adj) ='87.1% SOURCE DF SS MS Regression 1 0.48015 0.48015 E r r o r 2 0.04502 0.02251 T o t a l 3 0.52517 F 21 . 33 P 0.044 MTB > # NUCLEATION RATE CONSTANT r e g r e s s i o n (45 C included) MTB > regr c3 1 c l The r e g r e s s i o n equation i s LnKn = - 80.9 + 32612 1/T (/K) P r e d i c t o r Coef Stdev t - r a t i o p Constant -80.92 57.41 -1.41 0.294 1/T (/K) 32612 18684 1.75 0.223 s = 1.966 R-sq = 60.4% R-sq(adj) = 40.6% A n a l y s i s of Variance SOURCE DF SS MS F p Regression 1 11.779 11.779 3.05 0.223 E r r o r ' 2 7.732 3.866 To t a l 3 19.511 MTB > # NUCLEATION RATE CONSTANT r e g r e s s i o n (45 C excluded) MTB > l e t c 3 ( l ) = ' * ' MTB > regr c3 1 c l The r e g r e s s i o n equation i s LnKn = 7.24 + 3530 1/T (/K) 3 cases used 1 cases c o n t a i n missing values P r e d i c t o r Constant 1/T (/K) Coef 7. 240 3530 Stdev 6. 534 2142 t - r a t i o 1. 11 1.65 P 0.467 0 . 347 s = 0.1515 R-sq = 73.1% A n a l y s i s of Variance SOURCE DF SS Regression 1 0.06230 E r r o r 1 0.02294 T o t a l 2 0.08525 Unusual Observations Obs.1/T (/K) LnKn 1 0.00314 * R-sq(adj) = 46.2% MS 0.06230 0.02294 F 2.72 P 0. 347 F i t S t d e v . F i t Residual 18.3244 0.2117 X denotes an obs. whose X value gives i t lar g e i n f l u e n c e . MTB > noou St.Resid * X 154 A P P E N D I X E 155 MTB > it ALL DATA COMBINED FOR REGRESSIONS MTB > p r i n t c l 2 c l - c 4 ROW RUN no. SUPSAT GROWTH NUCL. MAGMA 1 19 0. 037 0. 459 190. . 2 0. . 2720 2 19 0 . 037 0. 442 184. .2 0 . 2720 3 20 0. 055 0 . 561 212. . 1 0. . 2224 4 21 0. 082 0 . 886 338. 1 0. . 2359 5 21 0. 082 0. 881 417. . 3 0 . . 2359 6 1 0. 040 0. 621 119. 3 0. . 2186 7 16 0 . 051 0. 714 365. . 4 0. . 2320 8 7 0. 036 0 . 559 144. . 4 0. . 1836 9 11 0. 037 0 . 500 177. . 6 0. . 2070 10 8 0. 026 0 . 439 122. 5 0. . 2141 11 18 0. 027 0. 402 115. . 7 0 . 2563 12 10 0. 039 0. 857 262. . 7 0 . 1834 13 12 0. 036 0 . 710 132. . 1 0 , . 1002 14 13 0 . 030 0 . 527 128. . 9 0. . 1312 15 14 0. 025 0. 538 140. . 6 0. . 1382 16 4 0 . 016 0 . 345 43. . 2 0. . 1000 17 17 0. 021 0. 470 103. . 5 0 , . 1828 18 2 0 . 025 0. 686 146. . 9 0 . 0998 19 6 0 . 038 1. 004 208. . 9 0. . 1193 20 3 0. 016 0. 512 92. . 5 0 . . 1036 21 9 0. 023 0. 603 74. . 7 0 . .0871 22 5 0. 016 0. 413 46. . 9 0. .0992 23 15 0. 012 0. 373 22. .0 0. .0428 MTB > it GROWTH RATE re g r e s s i o n MTB > regr c6 2 clO c5 The r e g r e s s i o n equation i s Ln G '= 19.7 - 5615 1/T + 0.868 Ln S P r e d i c t o r Coef Stdev t - r a t i o P Constant 19.674 1. 455 13. 52 0.000 1/T -5615.5 433.8 -12. 95 0.000 Ln S 0.86786 0.04647 18 . 68 0.000 s = 0.07204 R-sq = 94.6% R-sq(adj) = 94.1% A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 1 .82022 0 .91011 175.38 0. .000 E r r o r 20 0 .10379 0 .00519 'Total • 22 1 .92401 SOURCE DF SEQ SS 1/T 1 0 .01000 Ln S 1 1 .81023 156 Unusual Observations Obs. 1/T Ln G F i t S t d e v . F i t Residual St.Resid 14 0.00305 -0.6406 -0.4901 0.0170 -0.1505 -2.15R R denotes an obs. w i t h a l a r g e s t - r e s i d . MTB > it PRIMARY NUCLEATION model MTB > regr c7 2 clO c5 The r e g r e s s i o n equation i s Ln B = 14.9 - 1656 1/T + 1.41 Ln S P r e d i c t o r Constant 1/T Ln S Coef Stdev 14.872 5.995 -1656 1787 1.4079 0.1914 t - r a t i o 2.48 -0. 93 7.36 P 0.022 0 . 365 0.000 s = 0.2967 R-sq = 83.3% R--sq(adj) = : 81.7% A n a l y s i s of Variance SOURCE Regression E r r o r T o t a l DF SS 2 8.8101 20 1.7607 22 10.5708 4 . 0 . MS 4050 0880 F 50.04 0 . P . 000 SOURCE 1/T Ln S DF SEQ SS 1 4.0459 1 4.7641 Unusual Observations Obs. 1/T Ln B F i t 23 0.00300 3.0910 3.6728 S t d e v . F i t 0.1380 Residual -0.5818 St.: R denotes an obs. w i t h a l a r g e s t . r e s i d . MTB > # PRIMARY RELATIVE KINETICS model MTB > regr c7 1 c6 The r e g r e s s i o n equation i s Ln B = 5.88 + 1.70 Ln G P r e d i c t o r Constant Ln G Coef Stdev 5.8827 0.2274 1.6961 0.3530 t - r a t i o 25.87 4 . 80 P 0.000 0.000 s = 0.4897 R-sq = 52.4% ' R--sq(adj) : - 50.1% A n a l y s i s .of Variance SOURCE Regression E r r o r T o t a l DF SS 1 5.5349 21 5.0359 22 10.5708 5 0 MS . 5349 . 2398 F 23.08 0 p .000 157 Unusual Observations Obs. Ln G Ln B 23 -0.99 3.091 F i t S t d e v . F i t 4.210 0.177 Residual -1.119 St.Resid -2.45R R denotes an obs. w i t h a l a r g e s t . r e s i d . MTB > # SECONDARY NUCLEATION model MTB > regr c7 3 clO c8 c5 The r e g r e s s i o n equation i s Ln B = 38.4 - 7090 1/T + 0.836 Ln Mt + 1.27 Ln S P r e d i c t o r Constant 1/T Ln Mt Ln S Coef 38.392 -7090 0.8362 1.2656 Stdev 6.937 1800 0.1921 0.1427 t - r a t i o 5.53 -3.94 4.35 8.87 P 0.000 0.001 0.000 0 .000 s = 0.2154 R-sq A n a l y s i s of Variance 91. 7% R-sq(adj) = 90.3% SOURCE DF SS Regression 3 9. 6894 E r r o r 19 0 . 8814 T o t a l 22 10. , 5708 SOURCE DF SEQ SS 1/T 1 4. .0459 Ln Mt 1 1. 9956 Ln S 1 3. . 6479 Unusual Observations Obs. 1/T Ln B 6 0.00310 4 . . 7816 5 23 0.00300 3. .0910 3 MS 3.2298 0.0464 F i t S t d e v . F i t 3204 0.0567 0923 0.1668 F P 69.62 0.000 Residual -0.5388 -0.0013 St.Resid -2.59R -0.01 X R denotes an obs. wit h a larg e s t . r e s i d . X denotes an obs. whose X value gives i t lar g e i n f l u e n c e . 158 MTB > .# SECONDARY RELATIVE KINETICS model MTB > regr c7 2 c8 c6 The r e g r e s s i o n equation i i s Ln B = 14.0 + 0.938 Ln Mt +1.42 Ln G P r e d i c t o r Coef Stdev t - r a t i o P Constant 13.9504 0.8005 17.43 0.000 ' Ln Mt 0.93848 0.09248 10. 15 0.000 Ln G 1.4183 0.1484 9. 56 0.000 s = 0.2024 R-sq = 92.3% R-sq(adj) = 91. 5% A n a l y s i s of Variance SOURCE DF SS MS F P Regression 2 9. 7519 4. 8759 11 9.08 0. .000 E r r o r 20 0. 8190 0. 0409 T o t a l 22 10. 5708 SOURCE DF SEQ SS Ln Mt 1 6. 0134 Ln G 1 3. 7385 Unusual Observations Obs. Ln Mt Ln B F i t S t d e v . F i t Residual St.Resid 6 -8.4 4.7816 5.3649 0.0530 -0.5832 -2.99R 23 -10.1 3.0910 3.1115 0.1307 -0.0205 -0.13 X R denotes an obs. wit h a la r g e s t . r e s i d . X denotes an obs. whose X value gives i t l a r g e i n f l u e n c e . MTB > noou 159 A P P E N D I X F 160 APPENDIX F F.l EFFECT OF SUPERSATURATION (OR RESIDENCE TIME) ON  CRYSTAL SIZE DISTRIBUTION Consider a h y p o t h e t i c a l c r y s t a l l i z e r o p e r a t i n g at steady-s t a t e , i n which the suspension d e n s i t y can be f i x e d at a given l e v e l r e g a r d l e s s of the residence time. The r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s i s given by : B = K^G1 (2.16) Combining eqns (2.16, 2.40) gives : n° = K b G i _ 1 (F.l) For size-independent growth, the s i z e d i s t r i b u t i o n i n terms of the po p u l a t i o n d e n s i t y i s : n = n° exp(-L/GT) (2.37) and the suspension d e n s i t y expressed i n terms of the d i s t r i b u t i o n i s : M T = 6kvpn° (GX) 4 (F.2) To compare the e f f e c t s of su p e r s a t u r a t i o n on CSD, consider two c r y s t a l l i z a t i o n s c a r r i e d out at the same temperature but at d i f f e r e n t s u p e r s a t u r a t i o n s . This can be achieved by operatin g the c r y s t a l l i z e r at d i f f e r e n t h o l d i n g times while m a i n t a i n i n g the same suspension d e n s i t y . For c r y s t a l l i z a t i o n s 1 and 2, we have 6k vpn° (G^T^) 4 = Miji = 6k vpn° ( G 2 T 2 ) 4 (F.3) or n 0 2 / n 0 ! = ( G 1 T 1 / G 2 T 2 ) 4 (F.4) Growth r a t e , G, can be e l i m i n a t e d u s i n g eqn (F.l) t o give : and e l i m i n a t i n g n° by combining ( F . l and F.4) gives : For %1 > t 2 , i t f o l l o w s from eqns (2.37, F.5, F.6) tha t : a) The CSD i s independent of su p e r s a t u r a t i o n when i = l ; n l ° = n 2 ° ' a n c i G l / t l = G 2 x 2 ' however, growth r a t e s i n c r e a s e i n pr o p o r t i o n t o (Ti/x 2) • b) Smaller c r y s t a l s are obtained at higher s u p e r s a t u r a t i o n s (shorter residence times) when i > 1. c) l a r g e r c r y s t a l s are obtained at higher s u p e r s a t u r a t i o n s when i < 1. Thus, the observations can be summarized as : f o r systems having i > l , higher supersaturations l e a d t o smaller c r y s t a l s i z e , and the higher the order of n u c l e a t i o n , the more d i f f i c u l t i t i s to produce c r y s t a l s of la r g e s i z e . F.2 E F F E C T OF SUSPENSION DENSITY ON CRYSTAL S I Z E n°2 / n°l = ( T l / T 2 ) 4 ( i - l ) / ( i + 3) (F.5) G 2/ G l = < T 1 / T 2 ) 4 / ( i + 3 ) (F.6) DISTR IBUT ION F.2.1 PRIMARY NUCLEATION Consider c r y s t a l l i z a t i o n s 1 and 2, operated at the same residence time and temperature but at d i f f e r e n t suspension d e n s i t i e s . From eqn (5.2) we have : M ^ / e ^ p n 0 - ^ 4 = T 4 = M T 2/6k vpn 0 2G2 4 (F.7) or M T 1 / n ° 1 G 1 4 = M T 2/n 0 2G2 4 (F.8) Rearranging and using eqns ( F . l , F.8) gives : G 2/G 1 = ( M T 2 / M T 1 ) 1 1 ( i + 3 ) (F.9) or n° 2/ n°l = ( M T 2 / M T 1 ) 1 ( i + 3 ) (F.10) The dominant c r y s t a l s i z e , L D, i s given by : L D = 3GX ( F . l l ) and i t i s r e l a t e d to the suspension d e n s i t y according t o : LD2 / LD1 = ( M T 2 / M T 1 ) 1 / < i + 3> (F.12) For non-secondary n u c l e a t i o n then, i t f o l l o w s t h a t i n c r e a s e d suspension d e n s i t y leads to l a r g e r c r y s t a l s i z e . The absolute d i f f e r e n c e , however i s s m a l l . F.2.2 SECONDARY NUCLEATION The expression f o r the r e l a t i v e c r y s t a l l i z a t i o n k i n e t i c s , where secondary n u c l e a t i o n i s a f a c t o r i s : B° = K nM TJG v (2.17) or i n terms of the po p u l a t i o n d e n s i t y : n° = K n M T J G v - 1 (F.13) Once again c o n s i d e r i n g two c r y s t a l l i z a t i o n s 1 and 2, having the same residence time and temperature but d i f f e r e n t suspension d e n s i t y gives as before : M T 1 / n ° 1 G 1 4 = M T 2/n° 2G2 4 (F.8) Combining eqn(F.8) wi t h (2.17) gives : n 2°/n 1° = (M T 2/M T 1) ( i + 4 j - D / ( i + 3) (F.14) or G2/G1 = (M T 2/M T 1) /<i + 3) (F.15) For secondary n u c l e a t i o n , j i s normally equal t o one. From eqns (F.14, F.15) i t f o l l o w s then t h a t : G 2/G 1 = 1 (F.16) and n 2°/n 1° = M T 2/M T 1 (F.17) Thus by eqn (F.16), the growth ra t e i s independent of the suspension d e n s i t y . Eqn (F.17) shows that the n u c l e i p o p u l a t i o n d e n s i t y , n°, changes i n p r o p o r t i o n w i t h the suspension d e n s i t y , whatever the su p e r s a t u r a t i o n k i n e t i c order. A P P E N D I X G 165 S I D E V I E W ; A l l dimensions i n mm. 

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