CRYSTALLIZATION OF AMMONIUM SULFATE

secondary nucleation and growth kinetics in suspension

P.J. Daudey

TR diss 1544

CRYSTALLIZATION OF AMMONIUM SULFATE - SECONDARY NUCLEATION AND GROWTH KINETICS IN SUSPENSION

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CRYSTALLIZATION OF AMMONIUM SULFATE - SECONDARY NUCLEATION AND GROWTH KINETICS IN SUSPENSION

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus,

Prof.dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan

van een commissie aangewezen door het College van Dekanen

op dinsdag 26 mei 19&7 te 14.00 uur

door

PIETER JOHANNES DAUDEY

geboren t e Den Haag, scheikundig i n g e n i e u r

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Dit proefschrift is goedgekeurd door de promotor

PROF.IR. E.J. DE JONG

PROF.DR.IR. G.M. VAN ROSMALEN heeft als begeleider in hoge mate bijgedragen aan het totstandkomen van het proefschrift. Het College van Dekanen heeft haar als zodanig aangewezen.

S t a b i e l e opera t ie van een kontinue suspensie k r i s t a l l i s a t o r i s mogelijk dankzij het bestaan van sekundaire kiemvorming.

Het "gemak" ( l age orde afhankeli jkheid in G of AC) waarmee sekundaire kiemvorming optreedt geeft op z ich a l aan d a t he t een groei-gebonden proces i s , waarb i j g roe i a l t i j d voorafgaat aan het "losraken" van een kiem.

Het visueel waarnemen van een "kiemexplosie" i n een aanvankelijk heldere oververzadigde oplossing i s geen bewijs voor de gedachte dat de t i j d e n s de kiemexplosie gevormde k r i s t a l l en primair van aard zi jn (zie hoofdstuk 2 ) .

Bi j de n o t a t i e van eenheden i s het wenselijk om ongelijke eenheden met ident ieke no ta t i e gescheiden t e houden. Tevens verdient het aanbevel ing om een symbool voor de eenheid van " a a n t a l " in t e voeren, en wel b i j voorkeur: "#". De eenheid van p o p u l a t i e d i c h t h e i d b i j voo rbee ld wo_rdt volgens deze aanbevelingen genoteerd a ls [#/mJ .m] in p laa t s van [m ] . Dit geeft duidel i jk aan dat het gaat om een a a n t a l (#) g e d i s t r i b u e e r d over twee ruimten, n l . de cartesiaanse (m3 ) en een l i nea i r e (m), in d i t geval een maat voor de k r i s t a lg roo t t e .

Het v e r d i e n t aanbeve l ing om binnen he t u n i v e r s i t a i r onderwijs in een vroeg stadium de student vertrouwd te maken met een a a n t a l management t e c h n i e k e n , d i e hem/haa r h e l p e n om de s t u d i e v o o r t g a n g en eigen p r e s t a t i e s op een ob jek t i eve wijze t e beoorde len . Hierdoor kan of e n e r z i j d s de s t u d i e op een e f f e k t i e v e r e wi j ze v o l t o o i d worden, of anderzi jds, wanneer nodig, vroegti jdig b e s l o t e n worden t o t een andere c a r r i è r e .

Publ ikat ies zouden aan waarde kunnen winnen wanneer he t v e r p l i c h t zou z i j n een o b j e k t i e f u i t t r e k s e l van de d i s k u s s i e t u s s e n r e f e r e e s en auteurs aan het betreffende a r t i k e l toe te voegen.

Als h e t onts taan van nieuwe ideeën a l leen mogelijk zou z i jn door middel van een mechanisme vergelijkbaar met homogene kiemvorming, zou het droef g e s t e l d z i j n met de wetenschap. Gelukkig echter bieden sekundaire en in uitzonderingsgevallen ook pr imai r heterogene kiemvormingsmechanismen h ier uitkomst.

Het i n d i t p r o e f s c h r i f t verworpen C a t a l y t i c B r e e d i n g mechanisme v e r k l a a r t hoe h e t m o g e l i j k i s p l a g i a a t t e plegen zonder van he t oorspronkelijke idee kennis te hebben.

Noch de samenhang tussen de soorten, noch de geologische ontwikkelings­geschiedenis van de aarde geven aanleiding to t de v e r o n d e r s t e l l i n g dat een scheppende God n i e t bestaat .

De kloof tussen kunstmatige i n t e l l i g e n t i e en de menselijke geest kan het s n e l s t v e r k l e i n d worden door aan b e i d e z i j d e n een verregaande s tandaard isa t ie door te voeren.

11. De term "Music for the Millions" is van bijzondere toepassing op het genre dat in warenhuizen ten gehore wordt gebracht.

12. Het gezegde "Een goed verstaander heeft slechts een half woord nodig" is onzinnig aangezien een slecht verstaander eveneens aan een half woord genoeg heeft.

3

Aan mijn ouders Dini Henriette Renier Andries

5

ACKNOWLEDGEMENT

The project is the result of a joint effort of DSM Research and the Technical University Delft. The author is grateful to all those within DSM and the Technical University who contributed to the present results. Especially he wishes to mention the pleasant working environment offered by the Laboratory for Process Equipment.

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SUMMARY

The exact knowledge and understanding of nucleation and growth kinetics are of fundamental importance to the design and operation of industrial suspen­sion crystallizers. The development of physical models for nucleation and growth in suspensions is hampered by problems concerning: > - the direct observation of nucleation at the (sub-) micron scale; - the formation of "structure" in solution near the surface of a crystal as a potential source of nuclei;

- the scale up of single crystal growth and secondary nucleation models to the mixed suspension level;

- the interpretation of empirical suspension crystallization kinetics in terms of basic mechanisms.

The objective of this thesis, chapter 1, is to study the phenomenon of nucleation in connection to the growth of crystals, and to arrive at a cor­rect physical model for the nucleation and growth kinetics in suspension. The theoretical part of this study, chapter 2, starts with a critical review of the existing nucleation models. It was concluded that: - secondary nucleation is the only important mechanism in the crystal­lization of moderate to high solubility systems;

- in secondary nucleation mechanisms three main categories c.q. mechanisms can be distinguished, namely "catalytic", "surface" and "mechanical" breeding;

- catalytic breeding, the induction of crystalline structure(s) in the su­persaturated solution near the crystal surface leading to nucleation, is not proven since experimental evidence is lacking and the theoretical models proposed are in conflict with the well-proven homogeneous and heterogeneous nucleation theories; therefore only two categories remain:

- surface breeding, encomprising all mechanisms in which the growth deter­mined micro-relief is involved, explains the fundamental relation between secondary nucleation and the growth rate and the habit of the parent crys­tals . The dislodgement of nuclei after their formation on the surface may be either spontaneous or by fluid shear or mechanical action;

- mechanical breeding, nucleation due to attrition of breakage of the parent crystals independent of their growth can be discriminated from surface breeding by the absence of the growth rate as determining factor;

- the outgrowth of secondary nuclei, and/or their "survival", depends on the size, shape, dislocation content of the nuclei and the bulk supersaturation. Extra dislocations kinetically may enhance the growth rate, but additionally increase the solubility of the nuclei, which reduces their growth rate;

- for the kinetic modelling of secondary nucleation it is advantageous to divide the nucleation process in three process steps: the formation of "proto-nuclei" , their removal from the surface and their survival/ out­growth in the solution.

The experimental verification of the kinetic model was done on the system ammonium sulfate-water in a small scale suspension crystallizer (10-20 1).

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Chapter 3 describes the population balance mathematics necessary for the analysis of nucleation and growth rates in a well mixed continuous crystal-lizer, including a derivation of the population balance for growth dispersion. In chapters h and 5 the suspension crystallization results are presented and compared with literature data. In chapter 4 pure ammonium sulfate was investigated. The nucleation kinetics were modelled and identified to belong to the group of surface breeding (secondary) nucleation mechanisms. The results obtained from "oxime-liquor", chapter 5, were completely different. Using the "discrete" growth dispersion model developed in chapter 3 it was proven that the crystal size distribution consisted of two types of secon­dary nuclei: "normal", good growing nuclei formed by surface breeding, and slowly growing nuclei formed by abrasion /mechanical breeding of existing crystals. Separate growth and nucleation experiments with ammonium sulfate in a fluid-bed are discussed in chapter 6. In chapter 7 the abrasion of ammonium sulfate in a suspension crystallizer is modelled and experimentally determined. The results of chapters 6 and 7 support the conclusions on the suspension crystallization kinetics, chapters 4 and 5-

Finally, in chapter 8 the implications and relevance of the results of this study for the understanding and modelling of industrial crystallization are discussed.

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SAMENVATTING

Een j u i s t begrip van de kiemvormings- en groei-kinet iek i s van fundamenteel be lang voor he t on twerpen en b e d r i j v e n van i n d u s t r i ë l e s u s p e n s i e -k r i s t a l l i s a t o r e n . De ontwikkeling van fysische modellen voor kiemvorming en groei in suspensie wordt e rns t ig belemmerd door problemen betreffende: - de d i rekte waarneming van kiemvorming op (sub-)mikron schaal; - de vorming van " s t r u k t u u r " i n de oplossing vlakbij een kristaloppervlak

a l s een mogelijke bron van kiemen; - he t opschalen van è è n - k r i s t a l g roe i en kiemvormingsmodellen naar het

niveau van een gemengde suspensie; - de i n t e r p r e t a t i e van een p r o e f o n d e r v i n d e l i j k v a s t g e s t e l d e suspensie-

k r i s t a l l i s a t i e k i n e t i e k in termen van de onderliggende mechanismen.

Het doel van d i t proefschr i f t , hoofdstuk 1, i s om het verschi jnsel van kiem­vorming te bestuderen in samenhang met de groei van k r i s t a l l e n , t ene inde t e komen t o t een k o r r e k t fysisch model voor de kiemvormings en groei-kinet iek in suspensie. Het t h e o r e t i s c h e dee l van d i t werk, hoofdstuk 2, begint met een k r i t i s ch review van de bestaande kiemvormingsmodellen. Di t l e i d d e t o t de volgende konklusies:

- sekundaire kiemvorming i s het enige belangri jke mechanisme b i j de k r i s t a l -l i s a t i e van systemen met een gemiddelde to t hoge oplosbaarheid;

- sekundaire kiemvorming kan het best worden opgedeeld in dr ie hoofdgroepen of mechanismen namel i jk " k a t a l y t i s c h e " , "oppervlakte" en "mechanische" kiemproduktie;

- k a t a l y t i s c h e kiemvorming, het induceren van k r i s t a l l i j n e s t ruktuur in de oververzadigde oplossing vlakbi j een k r i s t a l o p p e r v l a k , i s n i e t bewezen aangezien exper imenteel bewijs ontbreekt en de voorgestelde theoretische modellen in s t r i j d bl i jken te z i jn met de onomstotelijk vaststaande model­len van homogene en heterogene kiemvorming.

- oppervlakte kiemvorming, omvattend a l l e mechanismen waarin h e t door de g r o e i van h e t k r i s t a l bepaa lde o p p e r v l a k t e - r e l i e f een r o l s p e e l t , verk laar t de fundamentele samenhang tussen sekunda i r e kiemvorming ener­z i j d s en de g r o e i s n e l h e i d en habitus van de kiem genererende k r i s t a l l en anderzi jds . De verwijdering van de aan he t oppervlak gevormde kiempjes g e s c h i e d t öf spon taan , óf door v l o e i s t o f k r a c h t e n , óf door mechanische a k t i e .

- mechanische kiemvorming, tengevolge van s l i j t a g e of breuk van de reeds aanwezige k r i s t a l l e n , onafhankel i jk van h e t f e i t of deze a l of n i e t g roe i en , kan onderscheiden worden van o p p e r v l a k t e kiemvorming gebruik makend van de afwezigheid van de groeisnelheid a l s bepalende f ak to r in de kinet iek;

- het ui tgroeien van sekundaire kiemen, en/of hun "overleving", hangt af van de g r o o t t e , vorm, d i s loka t i e gehalte van de kiemen en de heersende over­verzadiging. Extra d is lokat ies kunnen, k ine t i sch gezien, de g roe i sne lhe id

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verhogen, echter zij zullen, als neveneffekt, de oplosbaarheid van de kiemen doen toenemen. Hierdoor kan de groeisnelheid afnemen;

- voor de kinetische modellering is het nuttig om het kiemvormingsproces onder te verdelen in drie processtappen: de vorming van "voor-kiemen", hun verwijdering van het oppervlak en hun overleving c.q. uitgroei in de oplossing.

De experimentele verifikatie van het kinetische model is gedaan in het sys­teem ammoniumsulfaat-water, in een kleine (10-20 1) suspensie kristallisator.

Hoofdstuk 3 beschrijft de "aantal-balans" wiskunde, nodig voor de experimen­tele bepaling van kiemvormings en groeisnelheden in een goed gemengde kontinu kristallisator, inklusief een afleiding van de aantal-balans voor groei-dispersie.

In de hoofdstukken k en 5 worden de resultaten van de suspensie kristal-lisatie experimenten gepresenteerd en vergeleken met gegevens uit de literatuur. Zuiver ammoniumsulfaat is onderzocht in hoofdstuk 4. De kiemvor-mingskinetiek is gemodelleerd en geïdentificeerd als behorend tot de groep van oppervlakte-kiemvormingsmechanismen. De resultaten verkregen in "oxim-loog", hoofdstuk 5. toonden een geheel ander beeld. Door gebruik te maken van het "diskrete groei-dispersie" model, ontwikkeld in hoofdstuk 3. kon aangetoond worden dat de kristalgrootteverdeling bestond uit twee typen sekundaire kiemen: "normale", goedgroeiende kiemen c.q. kris­tallen, afkomstig van oppervlakte-kiemvorming, en daarnaast langzaam groeiende kiemen, gevormd door slijtage/mechanische kiemvorming.

Afzonderlijke groei- en kiemvormingsexperimenten in een fluid bed worden be­schreven in hoofdstuk 6. In hoofdstuk 7 wordt de slijtage van ammoniumsulf aat gemodelleerd en ex­perimenteel bepaald. De resultaten van de hoofdstukken 6 en 7 zijn in overeenstemming met en geven een ondersteuning aan de konklusies betreffende de kiemvormings- en groeikinetiek in suspensie, hoofdstukken k en 5-

Tenslotte worden in hoofdstuk 8 de implikaties en het belang van de bereikte resultaten voor het begrip en de modellering van de industriële kristal-lisatie besproken.

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CONTENTS

Summary 7

Samenvatting 9

1. The importance of nucleation and growth kinetics in industrial crystallization from suspension 15 1.1 Objective 15 1.2 Modelling and experimental determination of nucleation and

growth kinetics 16 1.2.1 Nucleation and crystal growth 16 1.2.2 The key role of nucleation in the modelling of suspension

crystallization 17 1.2.3 The investigation of the nucleation kinetics 19

1.3 Scope of the thesis 22 2. Review and modelling of secondary nucleation mechanisms In suspension

crystallization 25 2.1 Introduction 25 2.2 The stability of supersaturated solutions 25

2.2.1 The meta-stable zone concept 25 2.2.2 Homogeneous nucleation 27 2.2.3 Heterogeneous nucleation 29 2.2.4 The effect of solution structure on nucleation 30 2.2.5 Discussion and conclusions 33

2.3 Secondary nucleation 34 2.3.1 Introduction 34 2.3.2 Secondary nucleation from growing single crystals 35

2.3.2-1 Outline 35 2.3.2-2 Nucleation by spontaneous dislodgement of nuclei 36 2.3.2-3 Nucleation by mechanical action 37 2.3.2-4 Nucleation by fluid-shear 48 2.3.2-5 The Catalytic Breeding hypothesis 51 2.3.2-6 Discussion of the single crystal experiments 54

2.3.3 Secondary nucleation from suspensions of growing crystals 55 2.3.3-1 Types of experiments 55 2.3.3-2 The value of nucleation experiments in suspension 66 2.3.3-3 Concluding remarks 69

2.3.4 Growth of secondary nuclei 70 2.3.5 Final conclusions on secondary nucleation mechanisms 78

2.4 Modelling of secondary nucleation in suspension 80 2.5 Conclusions 84

3. Derivation of the population balance mathematics for a CMSMPR crystal-lizer, including the phenomenon of growth dispersion 87 3.1 Introduction 87 3.2 The population balance 87

3.2.1 The crystal size distribution 87 3.2.2 Definition of the growth rate 90 3.2.3 Derivation of the population balance 90

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3«3 Determination of the average growth rate and the effective nucleation rate from steady state CSD's 93 3-3•! The steady state: general 93 3.3.2 The steady state: CMSMPR conditions 94 3.3.3 The steady state: CMSMPR conditions and size-

independent average growth rate 95 3.3.3-1 The "ideal" CSD 95 3.3.3-2 Determination of the average growth rate 96 3.3.3-3 Determination of the effective nucleation rate 96 3.3.3-4 Properties of the "ideal" CSD 97 3.3.3-5 Numerical procedure for the calculation of the

average growth rate and the effective nucleation rate 98

3.3.3-6 Determination of the empirical kinetics from steady state CMSMPR experiments 99

3.4 Determination of the actual nucleation rate from steady state CSD's 101 3.4.1 Growth models for the extrapolation in the sub-sieve range 102 3.4.2 True size-dependent growth 102 3.4.3 Growth dispersion 103

3.4.3-1 Introduction 103 3.4.3-2 Stochastic dispersion 103 3.4.3-3 Permanent dispersion 104 3.4.3-4 The population balance for permanent dispersion 105 3.4.3-5 The steady state solution for permanent disper­

sion assuming size independent growth 108 3.4.3-6 Determination of the growth rate distribution

and the nucleation rate 109 3-5 Discussion 111

Experimental determination of secondary nucleation and growth rate of pure Ammonium Sulfate in a CMSMPR crystallizer 113 4.1 Introduction 113 4.2 Review of CMSMPR kinetics 113

4.2.1 CMSMPR-experiments 113 4.2.2 Chambliss (1966) 114 4.2.3 Larson and Mullin (1973) 114 4.2.4 Larson and Klekar (1973) 115 4.2.5 Bourne and Faubel (1982) 115 4.2.6 Youngquist and Randolph (1972) 115 4.2.7 Comparison of the reported kinetics 118

4.3 Experimental 119 4.3.1 The pilot plant 119 4.3.2 Operation 120 4.3.3 Sample treatment 122

4.4 Results of the experiments 122 4.4.1 Classification of the experiments 122 4.4.2 Calculations o 123 4.4.3 Evaporative crystallization at 50°C 124 4.4.4 Evaporative experiments at 67°C 126

4.4.4-1 Results 126 4.4.4-2 Formation and survival limitation 128 4.4.4-3 Influence of the supersaturation profile on

the nucleation rate 128 4.4.5 Cooling experiments at 40°C 131

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4.5 Discussion 132 4.5.1 Comparison of cooling with evaporative crystallization

experiments 132 4.5.2 Comparison of present results and literature data 135

4.6 Conclusions 136 5. Experimental determination of secondary nucleation and growth rate of

ammonium sulfate from "oxime-liquor" in a CMSMPR crystallizer 137 5.1 Introduction 137 5.2 Experimental 138 5.3 Results 139

5-3»l Classification of the experiments 139 5.3.2 The total nucleation rate 140 5.3.3 The inverse-gamma model l4l 5-3-4 The continuous polynomal distribution 144 5-3-5 The discrete dispersion model 146

5.4 Discussion 154 5.4.1 Comparison of the dispersion models 154 5.4.2 Comparison with the pure system 154

6. The effect of the hydrodynamics on growth and nucleation 157 6.1 Introduction 157 6.2 Modelling of growth 157

6.2.1 Theoretical models 157 6.2.2 Relations for crystal growth 159

6.3 Description of the growth experiments 162 6.4 Results of the growth experiments 164 6.5 Nucleation experiments 167 6.6 Comparison of fluid bed nucleation kinetics with CMSMPR kinetics 170

7. Effect of mechanical abrasion on nucleation and growth 175 7.1 Introduction 175 7.2 Abrasion models 175 7.3 Abrasion experiments 178

7-3'l Experimental set-up 178 7.3-2 Analysis of the abrasion experiments 179 7.3-3 Results of the abrasion experiments 180

7.4 Conclusions 186 8. Conclusions 187

8.1 General remarks 187 8.2 Theoretical modelling 187 8.3 Experimental results 189 8.4 Final remarks 190

Appendices * 2A Homogeneous nucleation rate and size of the critical nucleus

for ammonium sulfate 191 3A Calculation of the growth rate from the crystal production

unit per unit of area 197 3B Properties of the steady state CMSMPR distribution for

diffusion limited growth 201 * Appendix numbering according to chapter numbering

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3C Evaluation of size-dependent growth rate 205 3D The effect of stochastic dispersion on steady state CMSMPR

distribution 206 3E The inverse-gamma distribution function 208 3F Derivation of the polynomial model 211 4 Results of the CMSMPR experiments with pure Ammonium Sulfate 213 5 Results of the CMSMPR experiments with "oxime liquor" 217 6A Physical properties of ammonium sulfate / water solutions 223 6B The convective-diffusive mass transfer in ammonium sulfate

solution 233 6C 237 7 Results of the abrasion experiments 24l

References 243

List of symbols 257

Curriculum vitae 261

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CHAPTER 1

THE IMPORTANCE OF NUCLEATION AND GROWTH KINETICS IN INDUSTRIAL CRYSTALLIZATION FROM SUSPENSION

1.1 Objective 1.2 Model l ing and exper imenta l de t e rmina t i on of n u c l e a t i o n and growth

k ine t i c s 1.3 Scope of the thes i s

1.1 Objective

The product qua l i ty of a c rys ta l l ine material produced in a suspension crys-t a l l i z e r i s determined by the growth and nucleation k ine t ics in cooperation with the hydrodynamic cond i t i ons in the c r y s t a l l i z e r . Genera l ly a l a r g e c r y s t a l s i z e i s s p e c i f i e d which i s benef ic ia l for the customer as well as for the f u r t h e r p roces s ing ( f i l t r a t i o n , d r y i n g , s t o r a g e ) and p r o d u c t handling.

A c r y s t a l l i z e r , designed for the production of coarse "granular" c rys ta l s i s for instance the OSLO or Krystal c r y s t a l l i z e r , which i s shown in figure 1.1.

Vapour

Crystallizer

Slurry withdrawal

Circulation pump Fig. 1.1 Evaporative suspension

c r y s t a l l i z e r of the OSLO type

C lea r , s a t u r a t e d mother l i q u o r i s c i rcu la ted through a heat exchanger and flashes in the boi l ing zone. Due to the evaporation of the solvent , which i s water i n most c a s e s , the s o l u t i o n becomes supersaturated, and enters the

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crystal slurry which is fluidized by the circulating solution, and the su­per saturation is consumed by crystal growth. Product slurry is continuously removed. New crystals are formed by nucleation.

Much effort has been invested during the last thirty years in the modelling of the nucleation and growth kinetics, but still no reliable design equa­tions for nucleation and growth are available. Due to the severe problems encountered in the observation and exact quantification of nucleation and growth the verification and improvement of theoretical models for these processes is seriously hindered. As a result of this the available models are to qualitative, and are not in agreement with each other.

The objective of this investigation is to study the phenomenon of nucleation in connection with crystal growth, and to arrive at a correct physical model serving as a basis of design equations for industrial suspension crystal­lization.

The theoretical part of the study is restricted to the crystallization of well soluble materials from aqueous solutions. The experiments were per­formed on the ammonium sulfate-water system. The results of this study will be used as a starting point for the dynamic modelling of industrial crystal-. lizers. Before defining the line of approach followed in this work, some specific problems encountered in the modelling and experimental determination of nucleation and growth kinetics, will be discussed.

1.2. Modelling and experimental determination of nucleation and growth kinetics

1.2.1 Nucleation and crystal growth

Traditionally, the nucleation mechanisms are divided in two classes, primary nucleation, nucleation from clear solution and secondary nucleation, nuclea­tion due to the presence of the crystalline material itself. Figure 1.2 lists the most important mechanisms within these two classes, which will be discussed in more detail in chapter 2.

NUCLEATION PRIMARY

SECONDARY

HOMOGENEOUS

HETEROGENEOUS

CATALYTIC

SURFACE

MECHANICAL

Fig. 1.2 The most important nucleation mechanisms

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Of the two primary nucleation mechanisms only the second one, heterogeneous nucleation, i.e. nucleation on heterogeneities in the solution, might become important in highly supersaturated zones in a crystallizer (e.g. a boiling zone). The parameters governing the heterogeneous nucleation rate are: - the supersaturation level - the presence of suitable heterogeneous substrate particles - the presence of impurities - the degree of mixing in the crystallizer (locally high supersaturations) The other mechanism, homogeneous nucleation, can occur at extremely high su­persaturations, which can be attained only in the precipitation of sparingly soluble substances. A proven example is the precipitation of barium sulfate where a 1000-fold supersaturation was needed to observe homogeneous nuclea­tion (Nielsen, 1964). Secondary nucleation is considerably more complicated due to the participa­tion of growing crystals in the nucleation process. In addition to the parameters listed already for primary heterogeneous nucleation one has: - growth rate of the crystals - surface structure and habit of the crystals - hydrodynamics of crystals and crystallizer - geometry and size of the crystallizer - geometry and speed of impellers or pumps - hardness of the crystals. Crystals grow in supersaturated solutions by the incorporation of ions or molecules. The growth mechanisms may be divided in: - continuous growth - two dimensional nucleation growth - spiral growth Continuous growth is not possible for facetted crystals at supersaturations encountered in industrial practice. The other two mechanisms encomprise the formation of steps on the surface either by two dimensional nucleation or by the presence of a screw dislocation, and layerwise growth by the lateral movement of these steps. The growth kinetics can be influenced by: - supersaturation - impurities - mixing of the suspension - the presence of dislocations in the crystals A number of the parameters listed for nucleation and growth is specific for the particular crystalline material. A second complication is the interrela­tion between growth, nucleation, supersaturation and state of mixing, which makes the independent variation of all parameters impossible. The conse­quences of this interdependence are discussed in the next section.

1.2.2 The key role of nucleation in the modelling of suspension crystal­lization

In suspension crystallization no independent variation of parameters is pos­sible without disturbing many other parameters. This will be explained using an interaction scheme, figure 1.3. It will be seen that nucleation plays a key role in determining the crystal size distribution (CSD). The CSD, characterized for instance by the average crystal size (e.g. Lj..), is governed directly by three processes: growth, shifting the size of tne crys­tals in time, nucleation, the production of new crystals and withdrawal of crystals. The effect of these parameters on the CSD is accounted for by the "population balance", (Randolph, Larson, 1971), which will be detailed in chapter 3- The linear growth rate, G, is proportional to the production of

18

crystalline material, P, per unit area of crystals, AT, provided that all crystals exhibit the same growth rate. At a constant value of the produc­tion, governed by the mass balance, the growth rate is inversely proportional to the crystal surface area. Since the effect of growth on the CSD is to increase the crystal size and thereby the crystal surface area, it is seen that growth counteracts itself. Another, also negative, feedback loop is formed by the fact that the growth rate has a dominant influence on the nucleation rate, either directly or via the supersaturation. An increase of growth rate therefore leads to an increase in number of crystals and thereby of the surface area. In case of secondary nucleation a third, posi­tive, feedback loop is formed by the effect of the CSD on nucleation and vice-versa.

slurry density MT=pk,/nL3dl |kg/ni3]

+

+

CD

r product withdrawal / np /x («/m^s j

< -0 p crystal size distribution

• /m*l n

t X — L[m]

* nucleation rate

B=B{G.AcB.N,MT);|«/m3s]

L

+

/

)

■ •

+

/ hydrodynamics / fH (stirrer -ipee «v

supersaturation

AcB=G/kg

to r

+ ^

I +

€£>

surface area

AT= k./nUdl [mVm3]

growt .e<= ti rate

G ~'P/AT |m/s] , +

)

/ production / / P (kg/ ,H\J

Fig. 1.3 Interrelation of parameters in a suspension crystallizer

If, for example the crystal mass is the property which controls the nuclea­tion rate, then it is seen that the total effect of the crystals of a given CSD on the nucleation rate is proportional to the slurry density, M ,. Apart from the mutually interacting, internal parameters, there are three independent, external parameters, namely: - the production, governed by the mass balance; - the hydrodynamics, including the geometry, and - the product withdrawal, mentioned before as one of the three parameters affecting the CSD directly.

From this description of the kinetics of a continuous suspension crystal­lizer it is evident that the quantification of the nucleation model will be complicated since no independent variation of parameters is possible. Further, the measurement of nucleation and growth rate bears a global character since no direct methods for measuring nucleation rates and growth rates in suspension are available.

19

The most common method employed is to determine the nucleation rate in­directly from the rate at which crystals enter the region of the CSD which can be analyzed, by e.g. sieving or other techniques. Thereby assumptions concerning the growth behaviour of the nuclei have to be made, which can not easily be verified. For the growth rate within the analyzed region only an average value of crystals of a given size can be obtained. The knowledge and understanding of the nucleation mechanism, including the growth behaviour of nuclei and crystals, is therefore essential in the modelling of suspension crystallization. The investigation of these kinetics 'will be discussed in the next section.

1.2.3 The investigation of the nucleation kinetics

Many attempts have been made to delineate the parameter dependence of secon­dary nucleation by performing nucleation experiments with "parent" crystals under well defined conditions. In this case independent variation of parameters is possible. The scale-up of these small scale experiments to suspension crystallization, however, is a difficult step. Some factors are: - the use of large seed crystals which are not formed under the same condi­tions as the experiments may induce significant errors;

- the method of stimulating the nucleation rate, e.g. by contacts, must be complemented by the quantification of the contacts in the crystallizer;

- the hydrodynamics in a suspension of crystals deviates from the hydrodynamics around a large single crystal.

Therefore, the results of small scale experiments can not be used quantita­tively in suspension crystallization, and consequently only semi-empirical equations are used in the modelling of nucleation leading to a general cor­relation of the form:

B = kN Nh G1 M^ (1.1)

where B is the nucleation rate, k . a constant, N the stirrer speed, G the growth rate and M, the slurry density. This so-called "power-law" kinetics can always be used to correlate the aforementioned parameters over a limited range. In that case the exponents need not have a clear physical meaning but should be considered simply as regression parameters. A justification of equation (1.1) is often given by assuming:

B ~ Acn (1.2)

and

G ~ AcS (1.3)

where Ac is the supersaturation. Since Ac cannot be measured easily in most cases it is eliminated, leading to:

B ~ G1 with i = n/g (1.4)

suggesting that the exponent i of the growth rate does reflect the ratio of two independent exponents. This, however, has never been proven, see chapter two. Equation (1.1) has been used widely to describe the nucleation kinetics in suspension crystallization experiments (Garside, Shah, 1980).

20

In order to simplify the mathematical analysis of the experiments, well mixed crystallizers featuring a non-classifying crystal withdrawal are generally employed. In that case the growth rate and the number flux of crystals along the size axis can be derived directly from a steady state CSD, see chapter 3. This type of experiments is generally referred to as Continuous Mixed Suspension Mixed Product Removal (CMSMPR) experiments. The application of eq. (1.1), quantified in lab-scale CMSMPR experiments on an industrial, size is hampered by its semi-empirical character. If such a large scale unit, see fig. 1.4, is modelled using e.g. a reactor engineering approach (Grootscholten, 1982) it is doubtful whether equation (1.1) can be applied to any of the reactor elements. Even a mixed part of such a crystal-lizer is essentially different from a small scale CMSMPR crystallizer. This is clearly shown in figure 1.5t where the periodic behaviour of the super-saturation in the circulating solution is sketched. The peak values of the supersaturation reached in large scale crystallizers can easily be two or­ders of magnitude higher than the average value. If the exact nucleation mechanism, leading to equation (1.1) is not known, extrapolation of these semi-empirical kinetics is not allowed and consequently additional experi­ments have to be performed on the large scale. Other factors connected to the large scale are: - classification of the crystals due to settling; - higher impeller tip speed than in small scale crystallizers; - less wall and baffle area per m3 compared to the small scale. In order to characterize a large scale crystallizer all these effects have to be modelled, whereby the understanding of the basic nucleation kinetics and mechanism in suspension is a prerogative.

1—». vapour

circulation pump

/ f > — steam v3*— heater

feed

product

vapour

clear liquid (feed)

PFR : Plug Flow Reactor FB : Fluid Bed

Fig. 1.4 Modelling of an evaporative Draft Tube Baffled (DTB) crystallizer

21

AC

flashing in boiling zone

,decrease due to crystal growth

injection of superheated liquid

slight dissolution of crystals

time

Fig. 1.5 Periodic behaviour of solution supersaturation in the mixed part of an evaporative DTB

Since experimentation on a large scale is to expensive, the scale of ex­perimentation should be chosen as small as possible without loosing relevance to the industrial scale. This means that in any case suspension crystallization at say 1 - 20 1. scale should be employed. Since at this scale the mixing and classification effects present at the large scale can­not be investigated, it is recommended to use the CMSMPR approach. Only when the basic kinetics is understood at this level, whereby single crystal and other experiments may be used additionally, scale-up to larger sizes can be successful. To illustrate this the various levels of experimentation are sketched in fig. 1.6. The approach followed in the study to investigate the nucleation kinetics is presented in the next section.

crystallizer

miier/ crystalüier fluid bed plug flow

reactor

seral-technlcal crystallizer

CMSMPR bitch

nudeatlon growth abrasion

full iialc 12-200 m>>

reactor sodding

Intermediate icale 10.1-2 ml)

s«all scale (1-20 I)

teed crystals In suspension

l i t single crystal single crystal experiments

Fig. 1.6 The v a r i o u s l e v e l s of exper imenta t ion and modelling

22

1.3 Scope of the thesis

In order to obtain a correct modelling of nucleation and growth, relevant to industrial crystallization in suspension, it is first of all necessary to remove the.discrepancies between the published nucleation models on the one hand, and the discrepancies between the basic, small scale nucleation c.q. growth experiments and the empirical crystallization kinetics on the other hand. This demands a study of the literature on nucleation and growth as well as experimental work. In chapter 2 the existing literature on both primary and secondary nuclea­tion is reviewed. A main controversial point is the hypothesis of an extensive molecular structuring in supersaturated solutions as a precursor of both nucleation and growth. It can be shown, however, that the theoreti­cal basis of this hypothesis is very weak and that no experimental proof has been given yet. The formation of clusters due to the ordering forces originating from the crystal surface will not lead to free clusters or nuclei in solution but, contrarily, to growth of the crystal. The existing models for secondary nucleation are subdivided in three main mechanisms, which number can be reduced if it is taken that the catalytic breeding mechanism, involving the free cluster hypothesis, can be rejected. In order to obtain a link with suspension crystallization, the main mechanisms are subdivided in process steps. This approach facilitates the derivation of kinetic equations in terms of measurable parameters in a well mixed suspension crystallizer, such as the stirrer speed, supersaturation, growth rate and the moments of the CSD. The process of crystal growth is not discussed in great length. Some causes for the observed low growth rates of nuclei will be considered. It is sug­gested that if nuclei are formed by fracture, the solubility of these nuclei will be slightly enlarged, leading to lower growth rates in low supersatura­tion systems. In the experimental part of the thesis, chapters 4 to 7. two types of ex­periments will be presented.

First, CMSMPR experiments in order to find the overall kinetics. The theoretical derivation of the population balance equations is presented in chapter 3. where also the effect of growth dispersion on the population balance is taken into account. Basically, the "permanent" dispersion model of Janse and de Jong (1976) is extended and further improved, allowing for a direct determination of the growth rate distribution at zero size from the steady state CSD of a CMSMPR crystallizer. Chapters 4 and 5 deal with CMSMPR experiments in the 10 - 20 1. scale crys­tallizer. The experiments were performed using evaporative crystallization. Pure ammonium sulphate is treated in chapter 4, experiments with "oxime-liquor", a side-product of the caprolactam sythesis, in chapter 5-

Secondly, in order to delineate the CMSMPR kinetics, separate nucleation and growth experiments were performed. Chapter 6 describes growth experiments on ammonium sulfate crystals in a small scale fluidized bed. In a number of experiments the nucleation rate was measured by sampling the solution from the fluid bed. The number of nuclei was obtained by counting after a period of growth. Chapter 7 describes a model abrasion experiment on ammonium sulfate crystals in a stirred vessel which was identical with the 10 - 20 1. crystallizer. This enabled the direct comparison of abrasion rates measured in the model experiments, and the nucleation rates from the CMSMPR experiments.

23

Finally, in chapter 8, the most important results are discussed, together with the implications for the analysis of the more complex kinetics of a large scale, industrial crystallizer.

24

25

CHAPTER 2

REVIEW AND MODELLING OF SECONDARY NUCLEATION MECHANISMS IN SUSPENSION CRYSTALLIZATION

2.1 Introduction 2.2 The stability of supersaturated solutions 2.3 Secondary nucleation 2.4 Modelling of secondary nucleation in suspension 2.5 Conclusions

2.1 Introduction

Nucleation, and in particular secondary nucleation has been, and still is a subject of many controversies. A major difficulty in its investigation is the final verification of a proposed mechanism e.g. by microscopical obser­vation of a nucleation event on micron or sub-micron scale in the supersaturated environment. As a result of this, for instance, it has not yet been resolved whether or not nuclei may be formed in the solution layer close to a seed crystal by the action of ordering forces orginating from the surface of this crystal.

The aim of this chapter is to review the proposed secondary nucleation mechanisms and their experimental evidence (section 2.3). Based on this review a simple classification of secondary nucleation mechanisms will be proposed (section 2.4). This classification, combined with a general modell­ing of secondary nucleation as 3_step process, offers a useful tool for the interpretation of the complex empirically determined relations for the kinetics of secondary nucleation (section 2.5). Prior to the review of secondary nucleation, the stability of crystal-free solutions will be discussed, since a number of secondary nucleation mechanisms refer to the supersaturated solution near the crystal as a source of secondary nuclei (section 2.2).

2.2 The stability of supersaturated solutions

2.2.1 The meta-stable zone concept

It is a general observation that solutions can be considerably super­saturated while no crystals are spontaneously formed. Those virtually stable solutions were named "meta-stable" by Ostwald (1897). According to his ob­servations only seeding with a crystal of the particular substance to be crystallized, or of a highly isomorphic substance can induce crystallization in these solutions. On the other hand, above a certain limit in supersatura-tion, called the "meta-stable limit", a solution will become unstable, or "labile". According to Ostwald (1897) such a labile solution will spon­taneously crystallize sooner or later. The idea of a sharp boundary between a meta-stable and a labile zone of su-persaturation was disputed by de Coppet (1872, 1875, 1907) who considered nucleation as a process of collisions of low energetic molecules, being of a statistical nature. The probability of nucleation would increase strongly with the subcooling. In fact, this model is a precursor of the "homogeneous

26

nucleation" theory (Volmer and Weber, 1926). From experiments with solutions supersaturated with respect to Na-SO^.yH-O and Na2S0^.10H_0 de Coppet con­cluded that no sharp meta-stable limit exists, but that at low levels of supersaturation the time needed for the formation of a nucleus becomes so long that these solutions are virtually stable.

o Contrarily to de Coppet's opinion Ostwald denied the possibility of spon­taneous formation of crystals within the meta-stable zone. By careful experimentation (*) he was able to show that all nucleation phenomena ob­served in his experiments were the result of extraneous seeding. Since no such extreme precautions were taken by de Coppet it is questionable whether the results of de Coppet apply to spontaneous nucleation.

The meta-stable zone concept of Ostwald found a wide acceptance. Miers (1906) used the idea to explain the observation of the stages of nucleation during continued cooling of a solution, see figure 2.1. At first small crys­tals made their appearance at the surface of the solution (figure 2.1, point B ) , but after some time, at continued cooling, suddenly a copious "crystallization" took place (C), not only at the surface and bottom of the solution, but as a cloud of small crystals throughout the solution. This second stage would correspond, according to Miers, to spontaneous nucleation in the labile zone, whereas the first stage was caused by extraneous seeding in the meta-stable zone.

"labile" zone

D/

LJK

T

/ B

"meta-stable" zone „

S

A /

undersaturation

temperature

Fig. 2.1 Crystallization in the meta-stable zone bounded by solubility curve S and supersolubility curve T

(*) Note: Ostwald for instance preferred to work with substances having a high v o l a t i l i t y a t room tempera ture thus ensuring the automatic and effect ive destruct ion of atmospheric nuclei which could pos s ib ly interfere with his nucleation experiments.

27

However, it is very unlikely that Miers did observe spontaneous nucleation at all, since a rigorous exclusion of foreign crystalline material was not undertaken. Moreover the copious number of nuclei formed at C can be easily induced by the crystals formed at B (Cayey, Estrin, 1967), i.e. by secondary nucleation.

Against this use of the meta-stable zone concept two objections can be made at this point:

a) It is difficult to prove that a solution is labile. According to the definition of Ostwald (1897) the seeding action of crystalline par­ticles has to be excluded. This can be performed by filtration or heat treatment, but the complete elimination of nucleation inducing particles can never be proven nor achieved (Nielsen, 1964).

b) It is not evident whether spontaneous nucleation is really impos­sible within a region of low supersaturations (de Coppet, 1907)•

This last point has been resolved by further theoretical developments on spontaneous, or homogeneous nucleation.

2.2.2 Homogeneous nucleation

In order to initiate crystallization in a supersaturated solution a seed or nucleus must be of a certain minimum size (Schweigger, 1813)• This early ob­servation was explained by Ostwald (1900) who showed that the solubility of a small particle increases very strongly with decreasing radius:

2 o V RT In - = (2.1)

c r where c i s the concentration of the solut ion in equi l ibr ium with a c r y s t a l of radius r , and c a i s the so lub i l i t y of the substance (r=«>). Equation (2.1) i s known as the Ostwald-Freundlich equa t ion (F reund l i ch , 1922). However, where Ostwald was only concerned with seeding and the ef fec t of seed par­t i c l e s i z e t he reon , the importance of e q u a t i o n ( 2 . 1 ) t o s p o n t a n e o u s n u c l e a t i o n from a homogeneous solut ion was recognized much l a t e r by Volmer, who derived a more q u a n t i t a t i v e model for spontaneous n u c l e a t i o n . This mechanism i s known as the homogeneous nucleation theory. (Volmer, Weber, 1926, Farkas 1927; Volmer Flood, 1934; Kaishew, S t r a n s k i , 1934; Becker, During, 1935; Kashchiev, 1984). In fact with equation (2.1) i t i s possible to ca lcu la te the minimum s ize of a nucleus which i s s t i l l s t ab l e in a super­s a t u r a t e d s o l u t i o n ( i . e . c > c ) . In the formation of such a " c r i t i c a l nucleus" by a se r ies of molecular addit ions to a c l u s t e r of molecules , an amount of work has to be performed by the system s e r v i n g as act ivat ion energy to the process. The ra te of nucleation J can be given by (Nie l sen , 1964): s

Jg= A exp - g - (2.2)

where AG* i s the i n c r e a s e in f r ee en tha lpy due t o the fqrjnation_iDf the c r i t i c a l nucleus. A i s a k ine t ic constant of the order of 10 cm s . The

28

increase in free enthalpy AG* and the size L* of thf! critical nucleus can be written as (see appendix 2):

AG* = 1/27. (a o)3/(Au)2 (2.3)

2 k V o

v a K

where Ay i s the s u p e r s a t u r a t i o n , expressed as a d i f f e r ence in chemical po ten t i a l , and a the i n t e r f ac i a l energy. Therefore:

4 (ao/kT)3

27(Au/kT)' J„ = A exp - " l " » / * " ^ (2. 5)

The nucleation rate is an extremely steep-function of the supersaturation, which can be approximated by:

Jg - Aun* (2.6)

where n # is the number of molecules in the critical nucleus, usually n* £ 100. As a consequence there will be a sharp transition between no nucleation and an extremely high nucleation rate. Usually the underlimit of the„homogeneous nucleation region, i.e. the labile region, is set to J = 1 cm ^s (Becker, Döring, 1935; Nielsen, 1964). Due to the sharpness of this transition, the meta-stable zone concept (Ostwald, 1897) is fully justified by the homogeneous nucleation theory.

Strong evidence for homogeneous nucleation is offered by Nielsen (1964) , Nielsen and Söhnel (1971) for the precipitation of a number of slightly soluble substances. A remarkable example is formed by BaSOj, where a 1000-fold supersaturation is needed in order to have homogeneous nucleation (Nielsen, 1964), whereas for silver chloride "only" a ten-fold supersatura­tion is needed. However, though the theory of homogeneous nucleation is well developed, it is not possible to predict the supersaturation where homogeneous nucleation becomes appreciable, i.e. the meta-stable limit for a given substance. The reason for this is that the interfacial energies of the crystals have to be known. In fact, only by measuring the homogeneous nucleation kinetics as function of the supersaturation it is possible to estimate an average inter­facial energy for a particular substance. (Nielsen, Söhnel, 1971; Söhnel, 1982). Direct measurement of this interfacial energy from the enhanced solubility of small crystals, see equation (2.1) is unreliable since it is almost impossible to obtain equally sized, neatly prepared small crystals. For instance, milling produces crystal fragments which show a more than ex­pected increase in solubility which is due to the introduction of dislocations in the crystal fragment during its plastic deformation in the process of milling (Harbury, 1946). Notwithstanding the lack of reliable values of the interfacial energies, calculations of the homogeneous nucleation rate using estimated values of a show that the meta-stable zone is much wider than assumed by Miers (1906).

29

Obviously Miers did not recognize the powerful nucleation capacity of the first crystals which he clearly observed prior to the following showers of nuclei. Calculations on ammonium sulfate, see appendix 2, show that no homogeneous nucleation is possible even if a hundred-fold supersaturation compared to the normal operating values is applied. Thereby an extremely low value of the interfacial energy was chosen. This conclusion is confirmed by Kubota and Fujisawa (1984). But even if all extraneous seeding material is excluded it is still possible that nucleation is initiated by foreign crys­talline particles. This mechanism, called heterogeneous nucleation will be discussed in the next paragraph. Further reading on the homogeneous nuclea­tion mechanism is provided in reviews by Bradley (1951). La Mer (1952), Turnbull (1956), van Hook (1961) , Nielsen (1964), Walton (1965), Packter (1982) and Kashchiev (1984). Recent developments in the field of simulation of nucleation by molecular dynamics are given by Mountain and Brown (1984).

2.2.3 Heterogeneous nucleation

Since homogeneous nucleation proceeds only at extremely high supersaturation ratios, a different explanation had to be provided for the occurrence of nucleation in clear, unseeded supersaturated solutions. By I85O it was al­ready quite generally accepted that the introduction of foreign bodies could initiate the relief of supersaturation. Gernez (1865 a, b) noted that a nucleus of an isomorphic crystalline substance was more effective than a nonisomorphic one, but required a higher degree of supersaturation than the salt itself. Ostwald (1897) believed that only strictly isomorphic crystals could induce the process of crystallization in the meta-stable zone, apart from crystals of the particular substance itself. A theoretical model for this so called "heterogeneous" nucleation mechanism was developed by Volmer and Weber (1926) and Volmer (1929). Essential in this model is that the nucleus is formed on the surface of a crystal of a different substance, which serves as substrate. The increase of surface energy of the nucleus is compensated partly by the fact that the in­terfacial energy between nucleus and substrate is lower than that of the nucleus-solution interface, and by the decrease of the interfacial area of the substrate in contact with the solution. Kinetically, the mechanism has been modelled analogously to homogeneous nucleation but with a lower activation energy (excess free energy of the critical nucleus). The rate and mode of nucleation is strongly dependent on the affinity be­tween the nucleus and the substrate. If, for instance, the substrate is the particular crystalline material itself, nucleation will proceed through a 2-dimensional (2D) nucleation mechanism, the size of the critical 2D nucleus being determined by the edge free energy of that nucleus. In this particular case one simply has crystal growth by 2D nucleation. At higher supersaturations oriented three-dimensional nucleation will prevail and at still higher supersaturations the orientation of the 3D nuclei will become random. (Volmer, Weber, 1926; Kaischew et al., 198l). For a heterogeneous substrate 2D nucleation, causing a regular overgrowth of the substrate, is only possible if the substrate and the nucleus are highly isomorphic, for instance potassium aluminium alum and potassium chromium alum. In other cases 3D nucleation will be predominant, whether or not oriented in a crystallographic direction of the substrate (Kaischew et al. , 1981). The

30

i n t e r f a c i a l energy between s u b s t r a t e and nucleus was modelled by Turnbull and Vonnegut (1952) in terms of chemical a f f in i ty and l a t t i c e matching. According to these authors the r e l a t i v e mismatch should not exceed 1.5?» in order t o o b t a i n s i g n i f i c a n t h e t e r o g e n e o u s n u c l e a t i o n a t m o d e r a t e s u p e r s a t u r a t i o n s . Si lver iodide, which i s able to provoke the nucleation of ice a t under-coolings of only 2.5 to 4°C (Vonnegut, 1947; Walton, 1965) was analyzed by Fletcher (1959) who showed tha t the nucleation was only possible on the prism faces of the hexagonal Agl c r y s t a l , though t h e r e was no d i f ­ference in l a t t i c e matching with ice between the basal plane and the prism faces. The explanation forwarded by him i s t ha t the b a s a l p lane tends to or ienta te the water dipoles p a r a l l e l to each other which i s entropical ly not favoured, and ra i ses the free energy of an i ce nucleus on t h i s p l a n e . The prism faces do not show th i s effect (see also Zettlemoyer e t a l . , 1963) and are therefore preferred for ep i t ax i a l nucleation. I t should be mentioned a t t h i s p o i n t t h a t s ince a d i r ec t bonding between substrate and nucleus i s e s sen t i a l , i t i s not probable that the nuc l e i w i l l be d i s lodged from t h i s s u b s t r a t e . Therefore t h e amount of new c rys ta l s formed w i l l depend d i r e c t l y on the amount of s u i t a b l e h e t e r o g e n e i t i e s present in the solut ion. This was confirmed experimentally by Nielsen (1964) who observed an only moderate increase in n u c l e a t i o n r a t e s for BaSOj, , the r a t e being c l e a r l y l imited by the number of heterogenei t ies present, up to log S = 3. the onset of homogeneous nucleation. E x c l u s i o n of h e t e r o g e n e i t i e s from t h e s o l u t i o n i n o r d e r t o avoid heterogeneous nucleation i s extremely d i f f i c u l t . Generally, f i l t r a t i o n , and a l so prolonged h e a t i n g of the s o l u t i o n above the sa tura t ion temperature, decrease the nucleation tendency (Kubota, Fujisawa 1984). Both effects point a t the r o l e of s o l i d p a r t i c l e s in n u c l e a t i o n (Melia, Mof f i t t , 1964b). Heating e i the r leads to the d issolut ion of pa r t i c l e s of a s l i g h t l y s o l u b l e m a t e r i a l (Nie l sen , 1964) o r to t h e i r i n a c t i v a t i o n by dissolving the own c rys ta l l ine material which may be present and s tab i l i zed in crevices i n the substrate p a r t i c l e (Volmer, 1929) •

This l a t t e r phenomenon can also explain why a so lu t ion , once c r y s t a l l i z e d , seems to c r y s t a l l i z e much eas ie r the next time. The subst ra te pa r t i c l e can hide minute amounts of the own c r y s t a l l i n e m a t e r i a l , seemingly g iv ing the solution a "memory" for nucleation. The same effect was shown for t races of grease present in the system, which can also preserve c rys t a l l ine material (Melia, Moffitt, 1964b). Another source of heterogeneous s u b s t r a t e p a r t i c l e s a re the wal l s of the c o n t a i n e r in contact with the supersaturated solut ion. Tapping against such a wall may free pa r t i c l e s inducing n u c l e a t i o n . Microscopica l evidence i s reported by Nielsen (1964).

2.2.4 The effect of solution s t ruc ture on nucleation

Recently the formation of molecular aggregates i n solutions near the sa tura­t ion concentration has been invest igated. Mullin and Leci (1969b) found t h a t s u p e r s a t u r a t e d aqueous solut ions of c i t r i c acid kept quiescent a t a constant temperature develop c o n c e n t r a t i o n g r a d i e n t s wi th the h i g h e s t c o n c e n t r a t i o n s in the lower regions. This be­haviour was cons idered t o be evidence for t h e e x i s t e n c e of m o l e c u l a r c l u s t e r s i n s o l u t i o n . Larson and Garside (1986a) repeated the experiments for aqueous solut ions of c i t r i c a c i d , u r e a , sodium n i t r a t e and potassium s u l f a t e . The e x i s t e n c e of concentra t ion gradients was confirmed. Assuming

31

the c lus t e r s to be s tab le and monodisperse, a q u a s i - s t a t i c thermodynamic model was developed l ead ing to estimates of c l u s t e r s izes in the order of 3-10 nm. Physically, the c lus te r concentration g r a d i e n t p r e d i c t e d by t h i s model i s a r e s u l t of s e t t l i n g of c l u s t e r s coun t e r ac t ed by diffusion of c l u s t e r s , according to the Stokes-Einstein diffusion c o e f f i c i e n t for small p a r t i c l e s . The time, however, needed to es tab l i sh any concentration g r a d i e n t over the l eng th L of a column, L = 0.3 D, can be estimated from the Fourier number, Fo, for diffusion:

Fo = 5 | > 0.1 (2.7) L

The minimum time follows ferom the largest possible diffusion coefficient in the system, say ID = 10 m /s, for diffusion of the solute, which leads to t > 10 days. Since the authors determined the concentration gradient after only 24 hours it is impossible that the gradient was formed by diffusion of the solute. Settling of the proposed 1Q Jim clusters according to Stokes law leads to settling velocities of 10 m/s. Consequently the build-up of a concentration gradient in 24 hours cannot be explained by settling of small clusters either. Particles of 1 um would be demanded to see an effect within 24 hours, having a settling velocity of approximately 10 cm/24 hours, but these particles are already much larger than the critical nucleus. It is therefore concluded that, based on the proposed model, no measurable concentration gradient will develop within the time scale of the experiment. Therefore the observed gradient must result from other effects and does not prove the formation of clusters in solution, Furthermore it should be noted that in the derivation of their model the authors make a fundamental mistake by stating that the equilibrium condition for the cluster distribution as a function of height is that in "a differen­tial element in the column the free energy change dG of the clusters must be zero" . Actually the Gibbs free energy change of the whole system should be zero, i.e. of the clusters plus the surrounding solution. Fortunately their final result is practically unaffected since two of three extra terms cancel against each other, and the third term, a mixing term of the solu­tion, can be neglected at cluster mole fractions much smaller than one.

In order to make the existence of stable clusters of a certain size in su­persaturated solutions plausible Larson and Garside (1986 b) assumed that the interfacial energy of a crystalline cluster is a function of its radius:

o(r) = o(») exp (- 1.3 -] (2.8)

where 6 is the half-width of the interphase region of the cluster. The free enthalpy of formation of the cluster will show a local minimum at cluster radii somewhat below the value of 6, leading to the formation of stable clusters which are still sub-critical. The size distribution of these clusters is affected by the supersaturation in the solution. Since, however, the formation of these stable micro-clusters is favoured in this model, the initially clusterfree solution will "demix" in microclusters and solution of a lower concentration c.q. supersaturation. Eventually an equilibrium will be reached.

J

32

The consequence of this model for homogeneous and heterogeneous nucleation is less clear, since the effective supersaturation in the solution contain­ing the micro-clusters is decreased. It seems therefore that if this model is correct the thermodynamic modelling of the solution properties, sijch as the supersaturation (Au) for crystal growth and nucleation, has to be modified. It is expected that the activity coefficients of the solute will drop due to the formation of these stable microclusters. The authors use a tentative value of 2.5 nm for 6. It is questionable whether such a large value is justified in the formation of crystalline nuclei.

Since the radius of the proposed stable micro-clusters is calculated to bé smaller than 6, it: seems that the lattice parameters within this micro-cluster will deviate appreciably from those in a large crystal. This will lead to a higher solubility of the microcluster, which increase in AG for the micro-clusters is per definition to be assigned to the interface free energy o. Therefore it is also questionable whether the model (equation 2.8) is valid in the proposed microcluster region of 6 < r.

Support for the existence of clusters is given by Chang and Myerson (1984) who report a sharp drop in diffusion coefficients around the saturation point of aqueous solutions of KC1, NaCl and glycine, which was contributed to the formation of clusters, though these clusters need not be crystalline. Hussmann (et al., 1984) reported potential evidence for the existence of ag­gregates in concentrated sodium nitrate solutions using Raman spectroscopy. The relative intensity of the band attributed to these aggregated species, however, increased approximately linearly with the concentration of the solution over the full range of concentrations and showed even a slight decrease in the supersaturated region. Therefore no connection can be made with the formation of crystalline clusters, which is only possible in super­saturated solutions.

Mullin and Raven (1961, 1962) used the cluster hypothesis to explain some anomalous results detected in their nucleation experiments. With clear am-monnium dihydrogen phosphate solutions, cooled at a constant rate, they observed the critical supercooling, i.e. the supercooling where the first 'crystal was detected, to increase with the stirrer speed, and accordingly the nucleation tendency to decrease. The authors contributed this effect to the disruptive action of stirring on small clusters, breaking them into sub-critical cluster fragments. This explanation, however, seems very unlikely since the hydrodynamic forces due to the stirring can be fully neglected for small, sub-micron particles compared to the thermal motion of the molecules (Brownian motion). Normally, nucleation is found to increase with increasing stirrer speeds. This observation has been described in terms of pressure and density fluc­tuations, leading to the formation of clusters (Young, 1911; Mullin, 1972). However the effect of the pressure on the free enthalpy of formation of clusters can be fully neglected since solubility is hardly affected by pressure. Shmidt and Shmidt (1985) consider the effect of turbulent eddies on the spa­tial distribution of clusters and heterogeneous particles. They calculated the fraction of particles of a given size that is segregated on the boundary of an eddy by the cehtrifugal forces active in the rotating eddy, taking, into account the Brownian motion too. They assume that local enhancements of clusterconcentrations near the boundary of the eddy can significantly change the rate of homogeneous c.q. heterogeneous nucleation.

33

The formation of a critical nucleus by the collision of large, subcritical clusters, however, is statistically unimportant (Kashchiev, 198*0 • Furthermore, the average life-time of a single sub-critical cluster is ex­tremely short, and it is apt to disintegrate before it reaches the eddy "boundary".

So far no convincing fundamental, theoretical and experimental evidence has been brought forward for the existence of extensive cluster formation in homogeneous solutions at supersaturation levels where homogeneous nucleation can be neglected, i.e. within the meta-stable zone. Possible effects of the crystal-solution interface on the structure of the solution will be dis­cussed in relation to the mechanism of secondary nucleation (section 2.3).

The implications of the primary homogeneous and heterogeneous nucleation theory and the investigation of the proposed phenomenon of "crystalline" ag­gregate formation in supersaturated solutions for the stability of solutions in nucleation experiments in the range of supersaturations where the opera­tion of a suspension crystallizer is possible, will be shortly discussed in the next section.

2.2.5 Discussion and conclusions

Since homogeneous nucleation is only possible at extremely high supersatura­tions, which can only be reached in the rapid precipitation of certain slightly soluble substances, it is evident that most of the nucleation ex­periments reported in the literature have been performed within the meta-stable zone, according to Ostwald's (1897) definition. Modified definitions for the meta-stability of solutions as the region where no primary heterogeneous nucleation is possible (Nyvlt, et al., 1985) or the region where, in the presence of seeds, no secondary nuclei are produced (Mersmann, Förster, 1984) have a very limited use, since neither heterogeneous nor secondary nucleation predict a sharp boundary between nucleation and the absence of nucleation. The very use of the term "meta-stable limit" is misleading in connection to primary heterogeneous and secondary nucleation, since it should be used only for thermodynamic or kinetic properties of the homogeneous solution and not for the fortuitious presence or absence of seeds or heterogeneous particles. Moreover, qualitative models featuring limits between various types of nucleation do not provide the kinetic information needed for the design and operation of industrial crystallizers. It is also very improbable that the occurence of "massive" nucleation or an "explosive" increase in crystal num­bers , as frequently observed in the nucleation from clear solutions, is due to primary heterogeneous nucleation. Kubota and Fujisawa (1984) noted that after the appearence of the first crystal, which was proven to be a statis­tical event, rapid nucleation followed. Therefore it is ascertained that it is secondary nucleation which provides all of the nuclei after the formation of one single crystal by primary nucleation. Also at lower supersaturations than those used by Kubota and Fujisawa (1984) this remains true (Cayey, Estrin, I967). The similarity between spontaneous nucleation and nucleation induced by seeding, and the fact that nucleation of a substance which can give right and left hand forms, in most cases yields either the right hand form or the left hand form exclusively (Rogacheva, Belyustin, 1967) though on average the chance is 50/50, proves that the first crystal triggers a secondary nucleation process, yielding nuclei of the same form as the first

34

crystal. It has to be concluded, therefore, that secondary nucleation will be much more important than primary heterogeneous nucleation, since it is capable of producing large numbers of nuclei at low and medium levels of supersaturation. Since only the lower levels of supersaturations are impor­tant in industrial crystallization it is clear that secondary nucleation is the only important mechanism. Possible exceptions are precipitation reac­tions (Packter, 1982) of slightly soluble substances.

The modelling of nucleation from clear solutions should therefore not be based on primary nucleation, leading to expressions for the nucleation rate of the form B - ACn (Nyvlt, 1968; et al., 1970; Janse, de Jong, 1978) but on secondary nucleation, triggered by the statistical event of the formation of a primary nucleus. In this respect the momentum-balance method of analysis of batch crystallization (Kane, et al., 197^5 Randolph, Larson, 1971) is valuable, in combination with the waiting-timë distribution theory (Kubota, Fujisawa.1984). The formation of clusters, embryo's, molecular aggregates or other struc­tured species in solution, with exception of those subcritical clusters predicted by the primary nucleation models, is not considered to be relevant since up to now no convincing theoretical nor experimental evidence or necessity has been forwarded. Although a meta-stable supersaturated solution is not stable in a thermodynamic sense, physically it is a stable phase which will not "collapse" upon the introduction of a seed, but will show a controlled reaction at the phase boundary between solution and seed crystal.

2.3 Secondary nucleation

2.3»! Introduction

Secondary nucleation has been defined by Botsaris (1976) as nucleation which occurs only because of the presence of the material being crystallized. Though the phenomenon as such has been known for a long time (see e.g. de Coppet, 1872; Ostwald, 1897; and also the historical review by Van Hook, 1961), it has been the recognition of the fundamental importance of secon­dary nucleation in the field of large scale crystallization (Ting, Mc Cabe, 1934) which initiated the investigation of the subject. The aim of this section (2.3) is to review the proposed secondary nucleation mechanisms with the main emphasis on the mechanistic side. This implies that the experiments in stirred suspensions will be treated as far as they provide fundamental insight in the mechanism of nucleation and growth. Nucleation from growing single crystals is treated in section 2.3-2. Section 2.3.3 is devoted to the nucleation experiments in suspensions, including the results of CMSMPR experiments. A separate discussion on the growth of secon­dary nuclei is given in section 2.3.4. The review will serve as the basis for the classification and kinetic modelling of secondary nucleation, sec­tion 2.4, linking the basic level with the seeded suspension crystallization level and the CMSMPR level. This provides the essential tool for the inter­pretation of the information derived from seeded suspension and CMSMPR experiments. Earlier reviews on secondary nucleation are given by Botsaris and Denk (1972), Ottens (1973). Strickland-Constable (1973), Garside and Davey (1976), Estrin (1976), Botsaris (1976), de Jong (1979), Garside and Shah (1980), Larson (1981), Nyvlt et al. (1985) and Garside (1985).

35

The research into secondary nucleation can be divided into several levels, see also chapter one. First we have the basic level, where models are developed and verified by the observation of nucleation and growth in small scale experiments under well defined conditions, employing large, single seed crystals. Secondary nucleation is provoked by mechanical impacts, fluid shear, sliding, scratching or other stimuli. The resulting nuclei are analyzed either by direct microscopical observa­tion, or by (automatic) counting whether or not after a period of growth under defined conditions. Also the growth behaviour of nuclei is studied at this level. Secondly at the next higher level, the behaviour of suspensions of seed crystals is investigated. Complications in the interpretation of the experi­ments arise from the fact that i) no direct observation of the processes at the crystal surface of individual crystals is possible and ii) that several mechanisms can in principle operate at the same time making it difficult to identify which mechanism dominates. The advantage of seeded suspension experiments is that seed sizes of practi­cal interest can be used, and that the nucleation by free impacts of crystals against stirrers, walls and other crystals (Ting, McCabe, 193*0 can be studied. Moreover, the use of multiple crystals provides a good averaging of individual nucleation and growth rates. In the practice of suspension crystallization all crystal sizes are present in solution. No external seeding has to be provided, which rules out the potential dependence of measured nucleation and growth rates on the origine of the seeds. Independent variation of parameters, however, becomes very difficult as discussed in chapter one. Nevertheless, experimentation at this level is very useful. The solution of the population balance for a con­tinuous mixed suspension mixed product removal (CMSMPR) crystallizer is a powerful tool to measure growth and nucleation rates from the steady state crystal size distribution. Modelling of the nucleation kinetics of a stirred suspension at this level forms a sound basis for reactor engineering models for more complex crystallizers, which is beyond the direct scope of this thesis, since at these higher levels of operation ("high" meaning "more complex" or "closer to the industrial practice") additional factors such as the degree of mixing of the solution, internal and product classification, fines destruction, become important, which bear no direct relation to the investigation of the underlying growth and nucleation kinetics.

2.3-2 Secondary nucleation from growing single crystals

2.3.2-1 Outline

The experiments on growing s i n g l e c r y s t a l s may be d iv ided in t h r e e main groups according to the stimulus used in provoking the secondary nucleation. F i r s t , "spontaneous" secondary n u c l e a t i o n i s t r e a t e d , where t h e only s t i m u l u s i s the supersaturated environment in which the c rys ta l i s placed. Subsequently the effect of mechanical action on a growing seed c r y s t a l w i l l be d i s c u s s e d . F i n a l l y , f luid-shear as a means to obtain nucleation wi l l be considered. In the d i s c u s s i o n fol lowing t h e review of the experimental evidence, the source of the nuclei wi l l be discussed. Though d i r e c t observat ion of the event of nucleat ion i s almost impossible, i t w i l l be shown on theore t ica l grounds tha t c a t a l y t i c b reed ing , i . e . the

36

formation of clusters, or structure, in an aggregated lay(ir of solution ad­jacent to the crystal surface, is a very unlikely mechanism.

2.3.2-2 Nucleation by spontaneous dislodgement of nuclei

Parent crystals placed in a stagnant or mildly agitated supersaturated solu­tion are able to produce nuclei even when no external stimulus is applied. Two sources of nuclei have been suggested:

a) the spontaneous detachement of crystallites formed or present at the growing surface (Brewer, Palmer, 19^9)

b) the "inoculating" or "catalytic" effect of the crystal on the super­saturated solution (Ting, McCabe, 193*0-

Brewer and Palmer (19^9) visually observed the dislodgement of tiny dendrites from ice crystals growing in a supersaturated vapour. This effect was called "splintering". Gordon (1957) studied the growth of thin filamentary needles of various or­ganic and inorganic compounds. During the thickening of the initialy very thin (0.1-lum), curved filaments of hydroquinone, observed by microscopy, considerable stress would build up resulting in either straightening or breakage of the filaments. Also the thicker (30-50um), already straightened needles sometimes were observed to break in the absence of any bending strain. This latter effect could be explained by the stress present in polycrystalline needles formed by the intergrowth of individual filaments or needles.

Initial breeding

Mason (1962) and Mason and Strickland-Constable (1966) report observations on large growing single crystals of magnesium sulphate (MgSCL .7H-0, Epsom's salt). Various effects were noted here. Shortly after the introduction of a dry crystal in a supersaturated solution, nuclei could be detected. This phenomenon was termed "initial breeding", since after a first crop of crys­tals no further nucleation occurred. No direct visual observation of the actual formation of the nuclei could be made. The most plausible explanation is the spontaneous dislodgement of crystallites already present on the crystal before its introduction in the solution. This explanation was confirmed recently in a series of papers by Shimizu et al. (1984 a.b.c) and Shimizu, Sunagawa (1984) for K-alum. Whereas initial breeding was observed to be independent of the supersatura-tion, above a certain minimum value of the supersaturation two other effects were noted: needle and polycrystalline breeding.

Needle breeding

MgSOj-.yH-O single crystals were mounted on a rod and rotated in a super­saturated solution (Mason, 1962). Depending on the level of supersaturation a strong preference for growth in the c (long) axis direction was observed, the "end-growths" of the crystals being either hollow or in the form of bundles of fibres. In a stirred solution these outgrowths could break off and constitute fresh nuclei in the form of tiny needles.

37

The evidence for t h i s needle breeding mechanism i s provided by the one to one correspondence between the occurence of secondary n u c l e a t i o n y i e l d i n g n e e d l e s and t h e p r e s e n c e of n e e d l e - l i k e outgrowths on the c r y s t a l s . Additionally i t was noted tha t t h e phenomenon could a l s o be e f f ec t ed by touching a needle-l ike outgrowth with a rod causing breakage. In teres t ingly , in t h i s case not only the detached needles were observed but a l s o a l o t of small pa r t i c l e s which were suggested to be " sp l in te r s " (Mason, 1962).

Polycrys ta l l ine breeding

Growing KBr s ingle c rys ta l s in supersaturated solut ions (Mason, 1962; Mason, S t r i ck land-Cons tab le , 1966) r e s u l t e d in a p o l y c r y s t a l l i n e mass of KBr. Breeding of n u c l e i i n t h i s case consisted in the spontaneous breaking away of v i s i b l e pieces from the crys ta l mass.

Dendritic breeding

An observation simular to the needle-breeding mechanism was made by Melia and Moff i t t (1964a) . When observing growing c rys t a l s of NH^Cl and NHj.Br in stagnant so lu t ions , on several occasions p a r t i c l e s were seen to break away from the dendr i t i ca l ly growing c r y s t a l s , and to grow as they were carried up in the convection stream of exhausted s o l u t i o n r i s i n g from the c r y s t a l . S ince convect ion i n the absence of a seed c rys t a l fa i led to induce nuclea­t ion i t was concluded tha t the convection currents in the growth/nucleat ion exper iments were merely a consequence of c rys t a l growth and did not cause nucleat ion. Similar e f f e c t s were noted by Melia and Moff i t t for KC1 and CaS0..2H20. A d d i t i o n a l l y i t was found tha t the p resence of PbCl_ i n t h e s o l u t i o n inh ib i ted the formation of secondary nuclei for NH^Cl even though convection occurred, confirming t h a t no n u c l e i were produced wi th in the convect ion s t r e a m s . Though the authors do not offer an explanation, i t i s cer ta in that the growth morphology i s affected by the presence of PbCl_(Yamamoto, 1939) s u p p r e s s i n g the dendr i t ic habit . This i s in accordance with the correlat ion between n u c l e a t i o n and the f r i a b i l i t y of t he growth form, n e e d l e s or dendr i t es . Also the other 1-1 metal halogenides are affected by Pb , Kading (1932) ; Booth (1951) B ien fa i t e t a l . ( 1 9 6 5 ) ; Kleber , Schiemann (1966) , B o t s a r i s e t a l . ( I966)(1967); Malicskó, Jeszenszky (1970)(1972); Pilkington, Dunning, (1972); Stepien-Damm, Lukaszewicz (I98O), Hat tor i , (1984) .

2.3-2-3 Nucleation by mechanical action

Contact Breeding

The e f f e c t of mechanical a c t i o n on the secondary nucleat ion process was f i r s t reported by Ting and McCabe (193*0 for batch cool ing c r y s t a l l i z a t i o n expe r imen t s . In t h i s s e c t i o n only experiments with s ingle parent crysta ls wi l l be d i s cus sed ; in s e c t i o n 2 . 3 - 3 exper iments i n suspens ions w i l l be t r ea t ed .

The breakage of needle- l ike outgrowths by contacting a c rys t a l of MgSCv.7H20 was r epo r t ed by Mason (1962). Additionally i t was noted tha t also a t lower supersaturat ions in the "good growth region" due to c o l l i s i o n s of c r y s t a l s

38

with presumably a stirrer or other crystals new nuclei were formed. This ef­fect, called collision breeding, could not be directly related to the needle breeding mechanism, since no growth irregularities could be observed. The phenomenon of collision breeding was further elaborated by Lai (1966; Lai, et al.,1969) who employed three techniques to provoke nucleation from a seed crystal of MgS0^.7H20:

i) free movement of the crystal in a stirred solution ii) touching a crystal with a glass rod iii) sliding of a crystal in a slightly tilted tube.

In all three types of experiments it was noted that the number of nuclei produced was a strong function of the supersaturation, expressed as AT, see figure 2.2. Needle breeding was excluded by limiting the range of super-saturation 0 < AT < 4°C.

C F NUCLEI/MIN NUCLEI NUCLEI

Total nuclei against AT Series C : stirred solution

F : touching with glass rod E : tilted tube

Fig. 2.2 Results of Lai et al.(1969) en MgSCv.THpO

Nucleation in the solution layer between the crystal and the contacting sur­face can, according to the authors, not be explained by any molecular model. This item will be discussed in more detail in section 2.3.2-5- The other possibility is breakage (attrition) as already suggested by Ting and McCabe (1934). The strong dependence of the nucleation rate on the supersaturation, see figure 2.2, was believed not to be caused by an increase in the tendency for breakage. Consequently Lai et al. suggest a different explanation, namely the survival of nuclei. Assuming mechanical action to produce a wide range of particle sizes it is clear that particles falling below the "critical size", see section 2.2.2, will not survive but dissolve. Increasing the su­persaturation leads to a decrease of the critical size causing more crystals to survive. This theory has gained a wide acceptance as the "survival theory". Additional experiments to substantiate this theory are reported by Garabedian and Strickland-Constable (1972a). 'it was shown that the rate of

39

production of nuclei of NaClCL during the sliding process depended only on the supersaturation after the sliding process, which was varied independent of the supersaturation during the sliding. It should, however, be noted that this proof was not given for MgSOj, • 7H?0. The rejection of the idea of an in­crease of breaking tendency with the supersaturation was not substantiated.

1 temperature

saturation, Ts

Ts-1 -

le-L I J H

T ft l s - ö

Ts-16 -

growth

good growth

veiled growth

dendritic, spikewise brooming growth

nucleation absence of J presence of

crystal-solid | crystal-solid contact i contact

o

ro OJ CJ 3 C

O c:

splintering and

attrition from

splinters

r - " ~ J en best

^ operating i H region , i <-L"° L

ntac

ti i i 1 l ■1 i

o

splintering and

attrition of

colliding crystals

heterogeneous nucleation

Effect of supersaturation on crystal growth quality and type of nucleation for magnesium sulfate heptahydrate.

Fig. 2.3 Best operating region for contact experiments (Clontz, McCabe, 197D

The effect of mechanical impacts was investigated more quantitatively by Clontz (1970) and Clontz and McCabe (1971).

As test system also MgSO^^HpO was used. In order to avoid needle breeding the best operating region was aetermined first, see fig.2.3- In this region, according to Mason (1962) no "spontaneous" secondary nucleation by needle breeding is expected. The contact device is shown in figure 2.k. The energy transferred to the crystal by dropping a known weight on top of the contacting rod was correlated with the nucleation rate, calculated from the number of nuclei developing into crystals downstream of the contactor after having stopped the flow of solution.

40

thermistor probe Fig. 2.4 Contacting device employed by Clontz and McCabe (1971)

The following conclusions were reached: a) b)

c)

d)

The solution velocity has no effect in the range of 0,5 to 2,5 cm/sec. The nucleation rate is proportional to the relative supersaturation up to log. The nuc l ea t i on r a t e i s p ropor t iona l to the contact energy, up to 2600 ergs . No measurable threshold energy i s required. The number of n u c l e i i s p ropor t iona l to the square root of the impact area for c rys ta l - rod c o n t a c t s . A l i n e a r dependence was observed for c rys ta l -c rys ta l contacts .

diffusion O - MgSOi,,7H20 molecules

O - hydrated Mg2*

A - h y d r a t e d SO,,2" O

V, crystal surface

Fig. 2.5 P ic to r i a l view of the c rys ta l -so lu t ion in ter face , (Clontz, McCabe, 197D

Supersaturation dependence in contact breeding

The supersa tu ra t ion during the development of the nuclei was not varied in­dependently. Therefore no survival effect could be validated. The dependence of the n u c l e a t i o n r a t e on the supersaturation was not explained by an in­crease in breaking tendency nor by the survival theory, but by the existence

kl

of a "zone of partially organized solute clusters, microscopic dendrites, and unstable agglomerates, defying quantitative description", which is in line with the suggestions of Powers (1956, 1963) • The thickness of this "zone" is suggested to depend on the supersaturation, as is the rate at which nuclei are produced by this layer. The physical relevance of this sug­gestion, visualized in figure 2.5, does neither follow from their experiments, nor from Powers observations (1956) , and will be discussed from a theoretical point of view in section 2.3•2-5-

Evidence against the formation of nuclei in the liquid layer directly in contact with the. crystals is given by Denk and Botsaris (1972) who performed contact experiments on NaCIO- crystals. Because NaC10_ crystallizes in two easily distinguishable enantiomorphic forms, it was possible to determine that all of the nuclei came from the parent crystal. Additionally, the authors believe that two factors determine the dependence of secondary nucleation on the supersaturation, namely:

a) the number of fragments originally produced b) the number of fragments surviving.

The first point is in contrast with the opinion of Mason and Strickland-Constable (1966) and of Clontz and McCabe (1971) who do not consider the effect of supersaturation on the breaking tendency of the crystals to be relevant.

From these investigations it can be concluded at this point that three dif­ferent explanations have been suggested for the supersaturation dependence of the secondary nucleation rate by mechanical action (apart from the spon­taneous, growth related breeding mechanism given in section 2.3.2-2):

1. The effect of supersaturation on a hypothetical aggregated layer con­taining clusters c.q. embryomic species

2. The effect of supersaturation on the survival of fragments 3. The effect of supersaturation on the breaking tendency of the crystals

It should be realized that mechanisms 1 and 3 &re connected to the mechanism by which the crystal is growing, independent of the question whether clusters from the "aggregated" layer or microscopic growth details act as a source of nuclei assisted by mechanical action. The correspondence between the enantiomorphic form of the nucleus and that of the parent crystal for NaC10_ is more likely for the second and third mechanism. For MgSO^^HpO the same correspondence was noted by Rogacheva and Belyustin (1967). Therefore, apart from the disputable physical background for the "cluster" mechanism, there is no urgent need to invoke such a hypothetical mechanism to explain these contact breeding experiments.

Further evidence for the third mechanism, involving the breakage of micro­scopic details from the growing surface of a parent crystal is provided by Johnson et al.(1972) who extended the experiments of Clontz and McCabe. The (110) face of MgS0j,.7Hp0, which is microscopically rough, produces about 6 times as many nuclei as the (111) face, which is smooth, under the same con­ditions of supersaturation and impact energy level. Unexpectedly, the yield of nuclei for a given impact energy does not increase monotonously with the supersaturation but levels of at a relative supersaturation of 4.5%t see figure 2.6.

42

ra

» too-u fD

ë 320-01

E ^ 240-5 2 160-**-O

1 80-E 3 C

rod-crystal pres 0110 4111

^ i

face face

Oy

ï 1 i

sure contacts

O

^8

i i

- MgSt\,7H20

8 o o

i i

o

1 1 1 0.02 0.04 0.06 0.08 0.10 supersaturation (a-1)

Fig. 2.6 Supersaturation dependence of contact nucleation, MgSCK^H-O (Johnson et al.,1972)

This can mean two things: either the number of nuclei generated by the im­pact is constant but their survival is dependent on the supersaturation, where all nuclei which are removed survive for supersaturations greater than 4.5#. or the number of nuclei removed varies with the supersaturation. In the latter case, above 4.5# supersaturation the number of removable en­tities is limited, which would imply that the observed surface roughness increases up to an upper limit, or alternatively that there is only space for a limited number of removable entities on the surface of the crystal. This last point is in line with the observation of Johnson et al.(1972) that a further increase of the impact level did not increase the number of nuclei. Both explanations, however, seem to hold. Tai et al.(1975) performing con­tacts experiments with MgS0j,.7H_0, K-alum, K_S0j, and citric acid observed that the form of the supersaturation dependence of the nucleation rate was identical with that of the growth rate of a given substance. This strongly suggests a close relationship between growth, and in particular the growth related phenomenon of surface roughness, and the rate at which nuclei are observed. Additionally the survival effect was demonstrated by Rousseau et al.(1975) who used a growth chamber to collect the nuclei formed by the con­tacting of a K-alum crystal. The number of nuclei observed was shown to depend on the supersaturation in this growth chamber too. The authors con­clude that the supersaturation indeed seems to play a dual role in nucleation.

43

The time between contacts

When the c o n t a c t s a re made s h o r t l y a f t e r each other the number of nuclei formed per contact decreases. The minimum time between two c o n t a c t s below which t h i s e f f e c t becomes measurable was reported by Johnson e t al.(1972) for MgS0K.7H_0 t o be 15 s a t an undercool ing of 1 C and 18 s a t 0.5°C (supersaturat ions of ~ 2.2 and 1.1# respec t ive ly) . The explanation forwarded by the authors i s that the surface needs time to r e s to r e i t s o r i g i n a l t ex­t u r e c . q . m i c r o - r e l i e f . The r a t e of t h i s r e g e n e r a t i o n depends on the supersaturat ion, governing the growth r a t e . This observat ion supports the opinion tha t the number of removable e n t i t i e s on the surface i s a key factor in contact nucleation. On the other hand, the effect óf survival of nucle i i s l ike ly to play a role as w e l l . D i r e c t l y a f t e r a c o n t a c t t h e c o n t a c t e d a r e a w i l l s t a r t to regenerate i t s r e l i e f : the number, but also the dimensions of the "removable e n t i t i e s " wi l l s t a r t to increase . I f , s h o r t l y a f t e r the p rev ious con tac t t h e s e e n t i t i e s a re removed again, i t i s l ike ly that these nuclei w i l l have smaller dimensions and have less chance to survive than n u c l e i from a com­p le t e ly regenerated surface. The effect of the time between the contacts was inves t iga ted in more de t a i l by Bauer e t al . (1974b), Larson and Bendig (1976) and Khambaty and Larson (1978), a l l for MgSO^^HpO. The contacting was per­formed by a s p e c i a l l y designed r e p e t i t i v e c o n t a c t i n g d e v i c e . C o n t a c t f r equenc i e s up to 8 contacts per minute were used. The contactor was placed with the c rys ta l submerged below the solution leve l in a continuous s t i r r e d v e s s e l , where simultaneously growth and washout of the produced nuclei took place . The s ize d i s t r ibu t ion was analyzed by means of a Coul te r Counter, over a p a r t i c l e s i z e range of 8 to 70 um (Khambaty and Larson, 1978). The nucleation ra te B was obtained from the popu la t ion d e n s i t y d i s t r i b u t i o n u s ing the population balance for a MSMPR c r y s t a l l i z e r (Randolph and Larson, 1971)' The dependence of the nuc l ea t i on r a t e ob ta ined in t h i s way on the c o n t a c t frequency, f, was correlated by p lo t t ing B°/f, the number of nuclei produced per contact , versus the contact frequency, see figure 2 .7 .

1 70-

1 60-

£■ 50-

;o.

-

edge

- C L ^

B G s „ d =10.56 u

A = 5.65mm2

AT E =

u

S - * Q -

i!' . p u ­il 13 'S SM

. 3.3°C

939 ergs

I

experimental results

^ AT 939 ergs

m/min. \

\

\

1 1 1 6 7 8

f,frequency, contacts/mln

o o

51

0

" 1 1 1 6 7 e

Fig. 2.7 Number of n u c l e i per contact versus the con-t a c t f r e q u e n c y , Khambaty and L a r s o n (1978)

f,frequency, contacts/min

44

From the decrease in B°/f at a critical frequency of 6.3 contacts/min. a surface regeneration time of 9.5 s was calculated. At higher frequencies the regeneration becomes rate-limiting. This becomes even more evident when the value of B° is plotted versus the contact frequency, see figure 2.8. It ap­pears that the nucleation rate is almost independent of the contact frequency. At low contact frequencies below the critical frequency B /f will become constant, see figure 2.7, which implies that the rate B is propor­tional tö f, i.e. the removal of surface entities is limitative. The dependence of the surface regeneration time on the supersaturation, see fig. 2.9, reflects the rate at which surface is regenerated, or* alternately spoken, the rate at which removable entities are formed on the surface of a crystal, directly after the contact.

600

1.00

300

200

(.70

370

188 170

100 / / /

1 1

1 1

/

0 D A X

position ,

110 110 edge B edge A

G um/min

i.,0 5.0

10.6 6.7

. mm* 2 2 5.65 11.9

□ E

ergs 16697 16520

939 C218

0^—i J 5 1 f—6 5 5 f, frequency, [contacts/min.)

Fig. 2.8 Nucleation rate versus con­tact frequency, calculated from Khambaty and Larson (1978)

O JOHNSON A BENDIG □ BAUER O KHAMBATY

supersaturation, AT, C

Fig. 2.9 Surface regeneration time vs. supersaturation for MgSOi, (Khambaty, Larson, 1978 f

It can be concluded therefore that at elevated removal frequencies the rate of formation, or regeneration, of surface details becomes limitative to the process of nucleation. At low contact frequencies the removal frequency is the rate limiting factor.

Effect of the contact energy, area of contact and hardness of contactor

Johnson et al. (1972) and Bauer et al. (1974b) found that increasing the im­pact energy for MgS0j,.7H_0 leads to an increase in nucleation rate. This increase levelled off at more elevated values of the contact energy. A same levelling off was observed in pressure contacts (Johnson et al.,1972). It is noteworthy that the maximum levels reached were equal for impact and pressure contacts when the same rod was used for both types of contacts.

45

This again can be explained by assuming that a maximum number of removable entities is present on the surface of the crystal. At the lower energy levels the number of nuclei produced per mm2 of contact area was linearly dependent on the impact energy (Johnson et al.,1972; Clontz, McCabe, 1971; Kubota et al.,1986b).

At a given impact energy this number of nuclei produced per unit of contact area was reported to be inversely proportional to approximately the square root of the contact area (Clontz, McCabe, 1971; Johnson et al.,1972):

N ~ E = E r A JA A , J A

Apparently a large contact area is more effective than a smaller one. Bauer et al.,( 1974) observed that it was difficult to obtain full contact between the total area of the contacting rod and the crystal. This, however, does not lead to a clue to explain the observed area effect. Edge contacts, or contacts made at a slight angle produce more new crystals than normal contacts (Clontz, McCabe, 1971; Khambaty, Larson, 1978). Crystal-crystal contacts produce less nuclei per mm2 than crystal-rod con­tacts (Clontz, McCabe, 1971; Johnson et al.,1972). The hardness of the contactor plays a dominant role. A rubber tipped contacting rod yielded no nuclei at impact energies which were normally sufficient to shatter the crystal (Johnson et al.,1972).

The effect of dissolved impurities

The effect of ppm levels of Cr on the nucleation and growth of MgSCv . 7H-0 was investigated by Khambaty and Larson (1978). At a constant supersatura-tion the growth rate of both the nuclei and the seed crystal was reduced by the addition of up to 30 ppm Cr to the solution. The nucleation rate fell accordingly. Since the contact frequency remained low and no visible damage of the contacted area could be detected, the authors argue that the regeneration is still fast enough, and is not rate-limiting. If this is cor­rect, the decrease in nucleation rate could be caused by a change in surface micro-texture leading to a lower amount of removable entities. On the other hand, the time needed for regeneration might be expected to be inversely proportional to the growth rate, which dropped by a factor of four due to the addition of 30 ppm Cr . A reduction of the critical frequency of 6.3 in ,the pure experiments by this factor of four would yield a value of 1.6 which is much lower than 4.4 used in those experiments. Therefore it is not un­likely that the rate of regeneration of microrelief has become rate limiting caused by the reduction of the growth rate.

Microscopic observations of contact breeding

Microscopic observations of the crystal surface before and after contact and of the resulting nuclei are presented by Garside and Larson (1978). The sys­tems investigated were K-alum and MgSOjj^H-O. Two types of fragments or nuclei were observed for K-alum. First, micro particles, 1-4 um were formed in large number by the contacts independent of the supersaturation level. The source of these particles is identified to be the crystal itself since even at the slightest impact a damaged area was observed.

46

Additionally, larger particles up to 50 um were obtained at more elevated supersaturations, often of a plate-like shape. Many of the micro particles (< 4 vim) appear not to grow or to grow very slowly. Since their rate of formation was observed not to depend on the su-persaturation, and since they were even formed in undersaturated solutions, where they rapidly dissolved, it is evident that at least most of them are formed by mere attrition. The slow to zero growth rate of these smallest fragments considerably retards the outgrowth of these nuclei to visible crystals. Therefore the authors conclude that the survival theory is not necessary to explain why more nuclei are observed at increased supersatura­tions, but that, instead of survival, the growth behaviour of the nuclei is essential and should be investigated in more detail.

Size and growth rate of contact nuclei

In two complementary papers Garside et al.(1979) and Rusli et al.(1980) in­vestigated the initial size distribution and the growth rate of contact nuclei off K-alum by following the shift of the nuclei size distribution with time employing the Coulter Counter. The analyzed size range was 2.2 to 28 um. It was confirmed that also here two types of nuclei were involved namely:

1. Large numbers of small (2.2-4 um) nuclei which seemed to grow hardly at all. The total number of these small nuclei was almost independent of the supersaturation.

2. Larger crystals (4-28 um), with a negative exponential initial size distribution. The growth rate depended both on the size of the nuclei and on the supersaturation. The total number of these growing nuclei rose linearly with the supersaturation.

Clearly, contact nuclei were produced directly within an observable size range (2-28 um). These particles are so large that no survival effects based on the Ostwald-Freundlich equation may be expected. Since the number of growing nuclei (4-28 um) in the initial size distribution depended on the supersaturation it had to be concluded that the formation of these nuclei by the contacting depended on the supersaturation and therefore on the struc­ture of the growing surface.

Khambaty and Larson (1978) used a different method to measure the growth rate of contact nuclei, employing a small, well mixed continuous crystal-lizer (CMSMPR) with a repetitive contactor providing the nuclei. Short residence times were used to ensure rapid washout of the nuclei. Consequently the generation of second generation nuclei could be neglected. Essentially this technique is a modification of the seeded "washout" crys-tallizer (Randolph, Rajagopal, 1970) where the seeds were retained by a screen and the nuclei were washed out. From the slope of steady state crystal size distribution the growth rate of the contact nuclei of MgSO^^H-O could be determined. It appeared that the growth rate was independent of the size in the range of 14-70 um. However, below 14 um, a very steep increase in the slope of the CSD was noted. Three possible explanations were discussed: a) direct birth of nuclei into the 0-14 um size range. b) size-dependent growth: the growth rate increases with size c) the existence of two groups of nuclei, slow and fast growing ones.

47

Since the slope of the CSD changes in a discontinuous way at L=l4 um the size-dependent growth was not considered to be the predominant mechanism. It was concluded that a combination of direct birth into the size interval and slow growth of the nuclei was the most plausible explanation. Therefore also for MgSOj,.7H~0 the occurrence of two types of nuclei was confirmed.

The growth of contact nuclei as a function of their initial size has been studied microscopically for K-alum (Garside, 1979). citric acid (Berglund and Larson, 1982), potassium nitrate (Berglund et al.,1983) and sucrose (Shanks and Berglund, 1985)• Three observations could be made:

1. The growth rate of an individual crystal is constant with time. 2. Nuclei of the same initial size exhibit a wide spread in growth rate

(growth dispersion). 3. Notwithstanding this growth dispersion there is a definite trend of in­

creasing growth rate with increasing initial size.

Clearly the individual growth characteristics of a nucleus are determined at the instant of birth. It is noted that these microscopical observations are in accordance with the Coulter Counter measurements of Garside et al.(1979), Rusli et al.(1980) and Khambaty, Larson (1978) discussed above. The causes for the observed low growth rates will be discussed in section 2.3.4.

Discussion of the contact nucleation experiments

Mechanical action is capable of removing particles from the crystal. The source of these nuclei may be:

1. The crystal itself. 2. A spectre of roughnesses on the crystal surface caused by the growth

process. 3. A structured layer of solution surrounding the crystal.

The first source is evident since microscopical observations show that even the slightest impact causes damage. Additionally it was evidenced that many small fragments were formed due to impacts, and that the number of these small fragments was insensitive to the supersaturation. It should be noted, though, that these fragments exhibited a spectre of growth rates, which ex­plains the supersaturation dependence of the fraction of these fragments which is capable to grow to visible sizes. This provides an alternative to the survival theory (Garside, Larson, 1978). Survival effects are only to be expected if an appreciable part of the secondary nuclei is formed at near critical sizes. The second and third source cannot be conclusively distinguished from each other, based on the treated experiments only. Intuitively the second source is preferred over the third since there is a number of arguments in favour of the surface texture as a governing factor:

a) The direct dependence of contact nucleation on the microscopical rough­ness of the different faces (MgSCL .7H20, Johnson et al.,1972).

b) The suggestion that growth rate is the controlling factor in contact nucleation and not the supersaturation as such (K-SCv, K-alum, MgS0|,.7H?0, citric acid; Tai et al, 1975)- However, this argument is

48

also valid for attrition, with the crystal as source, followed by sur­vival modified according to Garside and Larson (1978), who point at the role of the low to zero growth rates among the small nuclei.

c) The regeneration of the surface as a limiting factor can be understood only if the "texture" of the crystal surface is taken into account, and is not expected in straightforward attrition. Therefore regeneration proves that the texture of the growing surface is decisive. On the other hand, the regeneration of a structured layer of solution is also a possibility, though intuitively much shorter regeneration times would be expected, with a strong supersaturation dependence.

d) The effect of crystal and contactor hardness is easily explained by the first two sources, but not by structured solution layer as a source.

e) The nucleation rate increases sharply near the threshold values ob­served for the needle breeding mechanism, AT > 4 C for MgS0i,.7H20 (Bauer et al., 197*0 , which also points at the surface relief of the growing crystal.

f) Experiments with crystalline materials exhibiting optical enantiomor-phism show that contact nuclei invariably have the same structure as the seed crystal (NaC10_, Denk and Botsaris, 1972; MgSO^^HpO, Rogacheva, Belyustin, I967)/

From these arguments it is concluded that only two sources are really proven in contact nucleation, namely the solid crystalline phase, yielding new nuclei by attrition, and the growing crystal surface, yielding nuclei depending on the rate of growth. The third source, a structured layer of solution from which nuclei are removed, was not evidenced, though its exist­ence was not fully disproven either. This point will be discussed further in the next section dealing with fluid shear as a means to stimulate secondary nucleation.

The behaviour of nuclei removed from their source seems to depend on this source: i) Small fragments formed by attrition appear to grow slowly or not

at all. ii) Larger fragments, in number depending on the growth rate, appear

to grow faster. This growth is either dependent on the size (K-alum, K-SOj.) or independent of size (MgSOn -7H20).

2.3-2-4 Nucleation by fluid-shear

Fluid-shear experiments

The effect of the shearing action of the solution on the generation of nuclei from a seed crystal was postulated by Powers (1956), though the solu­tion velocities employed by this author are not large (*• 4 mm/s). Moreover, in his experiments no precautions to avoid initial breeding seem to be taken. The action of a "sufficient violent shearing force" was to produce a cloud of nuclei from a "fluidized layer" of molecules "stockpiling" on the interface, acting as a "buffer reservoir layer" of fluidized material. An analogy was drawn with "the action of wind shearing off a spume or spray from the crest of waves" (Powers, 1963). Discussing the possible form óf these embryo crystals, the author reports microscopic observations of needles and monoclinic crystals as. a result of the secondary nucleation of sucrose at elevated values of the supersaturation,

49

Following Powers' suggestions, Clontz and McCabe (1971) suggest that fluid shear might play a role in the formation of nuclei from a "zone of partially organized solute clusters, microscopic dendrites, and unstable agglomerates" which is formed at high growth rates when ions or molecules strike the sur­face in numbers much too great to be organized into the lattice. The generation of nuclei from such a layer can be effected by either fluid shear or mechanical contacts. The term catalytic nucleation was introduced since this type of nucleation would have no noticable effect on the parent crystal, which simply keeps growing as usual. It should be noticed that Clontz and McCabe discern two essentially different mechanisms applying to the catalytic nucleation idea. One picture postulates the growth of microscopic dendrites, especially at corners and edges, which are weakly held and are broken off by the flowing solution. The other is the picture of a "fluidized" state which when dis­turbed by shear stresses, yields crystallites of the size of the critical nucleus or larger. In the following, however, the term Catalytic Breeding will be reserved for the latter mechanism only where the catalytic effect of the crystal on the solution leading to the formation of nuclei in solution is ment. The formation of dendrites is to be considered as crystal growth, leading to a more friable "texture" or "relief" of the crystal surface, resulting in the needle and dendritic breeding mechanisms as described in sections 2.3.2-2. Clontz and McCabe (1971) reported that no nucleation was observed when a MgS0j..7Hp0 crystal was placed in a supersaturated solution near an agitator. The supercooling was 4.5°C and the liquid velocity was estimated at 1 m/s. This observation was in accordance with the results of Mason (1962), Mason and Strickland-Constable (1966), Lal et al.(1969).

Sung et al.(1973) argued that the non-observation of fluid shear nuclei by both Lai et al.(1969) and Clontz and McCabe (1971) might be a result of the dissolution of these nuclei at the low values of the supersaturation, ac­cording to the survival theory (Garabedian, Strickland-Constable, 1972a; Strickland-Constable, 1973). In order to prove fluid-shear nucleation, a crystal of MgSO^^H-O was placed in a supersaturated solution near a cylindrical rotor, distance 1-4.6 mm; tangential speed 4.7-7.9 m/s. The saturation temperature was 40°C, the su­percooling around the seed crystal was varied from 0.5 to 3'0°C. The nuclei were allowed to grow in a cooler region of the vessel. Above a critical su­persaturation of ~ 2.5°C around the crystal, nucleation was always observed, provided that the undercooling in the cooler growth zone for the nuclei ex­ceeded - 8°C. The interpretation of the authors called upon the classical homogeneous or heterogeneous nucleation theory as a source of clusters. By fluid-shearing action these clusters would be transferred to the growth zone. A theoretical model was developed. The results could be fitted assuming a) heterogeneous nucleation on* the own crystallinematerial and b) extremely low values of the interfacial energy, 2.5 - 5 * 10 J/m* . The surviving nucleus was es­timated to contain 300 molecules, having a spherical diameter of 5«2 nm. It can be shown quite easily, however, that if such a nucleus originates as a spherical cap heterogeneous nucleus on a substrate, the removal of such a nucleus demands extreme shear stresses. It seems therefore highly unrealis­tic to assume that the removal of such small clusters from the surface by fluid-forces is possible. An alternative is the possibility that internal stresses build up when a 3-dimensional nucleus, which must have some mismatch with the seed's lattice

50

(section 2.2.3) grows to larger sizes, causing spontaneous fissure and dis-lodgement of a subcritical or supercritical cluster. Subcritical clusters, however, will rapidly dissolve unless they are very near the critical size. Dendritic or needle breeding was not considered as a possible source of nuclei, since needle breeding at AT < 4°C had not been found by Clontz and McCabe (1971). However, the original thesis of Mason (1962), see also Mason and Strickland-Constable (1966) shows a definite increase in needle breeding rate with the stirrer speed as well as an increase in growth rate, at a fixed- value of the supersaturation. It seems therefore that the high liquid velocities used by Sung et al (1973) are capable of inducing needle breeding. The needles need a high supersaturation to survive since the authors find no nuclei when the supersaturation during the development is low.

Youngquist et 31.(197*0 and Jagannathan et al. (I98O) used a liquid jet causing a liquid velocity of 6 m/s (MgSOj, .7HpO). The supersaturation around the growing crystal, AT and in a separate nuclei outgrowth-and observation-chamber, AT_, were well defined and could be varied independently. The following observations were made:

- No nuclei were observed for AT < 3°C even if AT > 22°C. - At increased values of AT lower values of AT_. were needed to develop the

1 j c B nuclei.

- Increasing the liquid velocity leads to a decrease in the supercooling ATR needed for the development.

The value of ATR can be considered as a measure for the instability c.q. solubility of the nuclei, connected to the size of the nuclei according to the survival theory. The authors therefore conclude that the size of the removed species from the seed surface increases with the supersaturation around the growing seed and with the shear rate. However, they had to admit that the shear rate also in­fluences the local values of the supersaturation near the crystal, which leads to higher local growth rates (Kumar, 1980; Kumar et al.,198l) and which could explain the larger size of the nuclei removed at higher shear rates. , It seems therefore that the local growth rate or supersaturation should be taken as parameter. This is confirmed by the critical behaviour at other temperatures: no nucleation was found at values of AT below 5t 3 and 1°C at saturation temperatures of 30, 40 and kl°C respectively. Since the dif-fusional mass transfer coefficient increases-with the temperature it is to be concluded that the growth rate at these critical supercoolings is ap­proximately a constant, independent of the saturation temperature. The same critical dependence of nucleation on the growth rate was found for needle breeding of MgS0j,.7H20 (Mason, 1962; Mason, Strickland Constable, 1966) , who found an additional relation between the occurence of "veiling" and growth rate. It seems therefore that fluid shear nucleation for MgS0j,.7H20 is completely in accordance with the evidence from spontaneous breeding ana contact breed­ing of MgS0|,.7Hp0, all referring to the texture c.q. relief of the surface as a nucleation rate determining factor.

No effect of jet velocity and no effect of changes in supersaturation was observed in thé nucleation of ice (Estrin et al.,1975)« The authors suggest

51

growth limited nucleation. The similarity in supersaturation dependence of nucleation and growth rate supports this suggestion. Wang et al.(198l) investigated the fluid shear nucleation of citric acid in a Couette flow crystallizer. The number of nuclei was seen to increase with the shear rate, initially, but levelled of at higher shear rates. Also here regeneration limited nucleation was suggested. K-alum was investigated by Jagannathan et al.(1980); Wang and Yang (198l) studied the behaviour of e-caprolactam in a Couette flow crystallizer from toluene solutions. Of the investigated crystal-solution systems only MgSO^.yH-O exhibits a critical supersaturation/growth rate for the production of nuclei by fluid shear. This might well be correlated with the needle forming capability of MgSOj..7H?0, which is also critically dependent on the supersaturation/growth rate.

Discussion of the fluid-shear experiments

Though initially the fluid-shear experiments were inent to prove the role of "embryonic species" in secondary nucleation, it now appears that the ex­perimental results may equally well or even better be interpreted in terms of "spontaneous" breeding c.q. needle breeding where the formation and removal of nuclei from the surface of the crystal is assisted by fluid-shear. A strong argument supporting this view is the correlation between growth rate and "fluid-shear" nucleation, and the suggested existence of a critical growth rate for this type of nucleation. It is concluded therefore that no need exists to invoke a mechanism based on the removal of "embryonic species" from the solution layer close to the crystal. On the other hand, the experiments do not prove that such a mechanism is impossible. It is therefore useful to inspect the physical basis of the embryonic-cluster model. The term "catalytic breeding" will be used for this type of secondary nucleation models.

2.3.2-5 The Catalytic Breeding hypothesis

The basic postulate of catalytic breeding is that the properties of the solution at the crystal-solution interface are modified in such a way that nuclei are formed. In order to separate catalytic breeding from the spon­taneous dislodgements of the surface relief, such as dendrites, needles etc. , as discussed in section 2.3.2-2, the mechanism will be restricted to the formation of new entities, clusters, within and surrounded by the liquid phase near the crystal surface, and not in immediate contact with the sur­face itself. Cluster formation on the surface, has to be regarded as growth by two or three-dimensional nucleation. In this sense heterogeneous nuclea­tion can be regarded as epitaxial growth on a foreign substrate. Therefore the following definition will be used:

Catalytic breeding is the enhancement of homogeneous or heterogeneous nucleation in the solution due to the interaction between the solution and the substrate.

The choice of this definition simplifies the treatment of the problem, since all kind of clusters formed on the surface and connected to the surface clearly attribute to crystal growth and not to catalytic breeding. Possible

52

roughening of the c rys ta l surface due to a pa r t i cu l a r mechanism of growth i s a growth-bound phenomenon contr ibuting to t he micro r e l i e f of the c r y s t a l s u r f a c e . Therefore the mechanism of the removal of 3~dimensional nuclei by f lu id forces as proposed by Sung e t a l . , c lear ly belongs to growth followed by removal of a p r o t r u d i n g p a r t , a t l e a s t in the way desc r ibed by the authors . Schemat ica l ly the p lace of ca t a ly t i c breeding amongst the other mechanisms involving c lus t e r formation i s shown in figure 2.10.

in solution V~* \3DJ *• homogeneous nucleation

dusters 4- (near substrate)-*(3D) »]~ catalytic breeding ~j

oh substrate xs> heterogeneous nucleation

growth (polycrystalline) twinning

<§> birth & spread growth

Fig. 2.10 The place of catalytic breeding

The mechanism of catalytic breeding was explained by Nyvlt et al.(1964). According to these authors the concentration of solute molecules is enhanced near the surface of a crystal due to attracting forces orginating from the crystal lattice. This enhanced concentration would lead to primary nuclea­tion on heterogeneities in the solution near the crystal, which is directly followed by secondary nucleation, triggered by the primary nuclei. The activation of heterogeities was also suggested by Belyustin and Rogacheva (1966), though the hypothesis was not referred to their next paper on the subject (Rogacheva, Belyustin, 1967). Denk and Botsaris (1972) suggest that local supersaturation may be enhanced due to the ordering of the solvent (not solute) at the crystal solution interface. Sung et al.(1973) suggest, apart from their 3-D clustering on the surface, that alternatively "clusters may be visualized as not being attached to the crystal surface but rather concentrated in the stagnant region of the crys­tal surface and under the influence of the structured force field of the surface". According to these authors in this way subcritical homogeneous clusters are formed. Subsequently they are removed by fluid shear to a region of high supersaturation where they might prove to be stable. Based on their model they estimated that the interfacial energy in case of "homogeneous" catalytic nucleation of MgSO^^HpO should be approximately 2.5 mj/m* .

53

I t i s e a s i l y v e r i f i e d , us ing t h e i r assumpt ions and t h e e q u a t i o n s of homogeneous nucleation that the width of the meta-stable zone would be only 3.5°C! Since supercoolings of 22.1°C could be e a s i l y reached the va lue of the i n t e r f a c i a l energy (of MgSC>K.7H„0) should be a t l ea s t 10 mJ/m* which in­va l ida tes t h e i r hypothes i s . Therefore , homogeneous c a t a l y t i c n u c l e a t i o n without energet ic in te rac t ion with the c rys ta l i s not f eas ib le . Also heterogeneous ca ta ly t i c nucleation as proposed by these authors i s not p o s s i b l e based on t h e i r assumptions: o = 5 mJ/m* and the energy of the heterogeneous c lus te r i s 10% of t ha t of a homogeneous c lus t e r containing the same number of molecules. Since o £ 10 mJ/m* i t can be calculated that the energy of the heterogeneous nucleus must be less than one hundredth of the c r i t i c a l nucleus energy for a homogeneously formed c lus t e r , which means that only a s l i g h t mismatch between c lus te r and s u b s t r a t e i s t o l e r a t e d . At the same time i t fol lows t h a t the re has t o be a s t r ong i n t e r a c t i o n between c lu s t e r and subs t ra te , which means that the c lu s t e r i s s trongly a t t a ched to the c ry s t a l surface, and i s in fact a pa r t of the c r y s t a l . If such a c luster ever comes loose i t must be e i ther due to the bui ld up of in te rna l s t r e s s e s a t t he s u b s t r a t e - c l u s t e r boundary, or to external mechanical act ion. I t i s therefore concluded that neither homogeneous nor heterogeneous nuc l ea t i on gives r i s e to ca t a ly t i c breeding. The argument of an increased supersaturation near the c rys ta l surface due to o r d e r i n g forces i s not correct e i t h e r . I f such a pos i t ive absorption occurs by a t t r a c t i n g forces in supersaturated s o l u t i o n s , i t w i l l occur a l s o in a s a t u r a t e d s o l u t i o n , l e a d i n g to a "supersaturated" layer . I t i s c lear that t h i s i s not the case. Even i f the concen t r a t i on i s l o c a l l y enhanced, the chemical p o t e n t i a l w i l l not show a maximum. In equilibrium u i s constant throughout the system; in supersaturated solutions there i s a g rad ien t from t h e s u p e r s a t u r a t e d s o l u t i o n towards the c r y s t a l su r f ace s e r v i n g as a "driving force" for diffusion. At the in terface the value of u drops t o the va lue of vi of the c rys ta l l a t t i c e , which i s mostly taken equal to the equi­librium value for an unstressed i n f i n i t e l y extended c r y s t a l l a t t i c e , u The d r i v i n g force for homogeneous and heterogeneous c lus t e r formation 3u equals u-u and has i t s maximum i n the bulk of the solut ion! eq I t can t h e r e f o r e be concluded tha t up to now no physically correct models have been proposed which explain c a t a l y t i c breeding in t he " t w i l i g h t zone" (Powers, 1963) between c r y s t a l and s o l u t i o n , n e i t h e r do the experiments demand such an e x p l a n a t i o n . Fur ther i t should be noted t h a t a r e a l ex­p e r i m e n t a l proof of c a t a l y t i c b r eed ing i s imposs ib le with the presen t observational techniques, because i t has to be ascertained that the nucleus i s formed in s o l u t i o n close to the c rys t a l , and not in d i r ec t contact with the c r y s t a l . The re fo re , Catalyt ic Breeding, defined as the enhancement of homogeneous or heterogeneous nucleat ion in the s o l u t i o n due t o i n t e r a c t i o n between the s o l u t i o n and a s u b s t r a t e , has t o be re jected. The main arguments are sum­marized below:

i ) T h e o r e t i c a l l y , the enhancement of c lus te r formation demands an ener­ge t i c in te rac t ion between c lus te r and subs t ra te , suf f ic ien t to provide a major pa^rt of the su r f ace energy of t h e c r i t i c a l n u c l e u s . This i n t e r a c t i o n u n a v o i d a b l y i m p l i e s a b o n d i n g f o r c e be tween t h e c l u s t e r / n u c l e u s and the s u b s t r a t e , which i s strong enough to prevent any spontaneous, f luid-shear or even mechanical removal of the c l u s t e r from the s u b s t r a t e , wi thout a f f l i c t i n g c o n s i d e r a b l e damage t o the

54

substrate. Clusters bonded to the surface of the crystal contribute to growth and fall outside the definition of catalytic breeding.

ii) Experimentally, no convincing evidence was found.

iii) Practically, it seems not possible to verify by in-situ observation whether a nucleus has originated at the crystal surface, or at a few molecular layers distance of the surface in the solution.

iv) Additionally, for primary heterogeneous nucleation it has been proven that direct contact (epitaxial) between nuclues/cluster and the heterogeneous substrate is absolutely necessary. Why should secondary nucleation behave differently?

2.3.2-6 Discussion of the single crystal experiments

If nuclei are generated in the neighbourhood of a parent crystal, there must exist a mechanism responsible for the removal of these nuclei from the parent crystal towards the bulk of the solution. There are three possible removal mechanisms; - spontaneous removal - fluid shear removal - mechanical removal

All three mechanisms have been confirmed experimentally. However, it is not easy to deduce the source of the secondary nuclei from the removal mechanism. In principle, for secondary nuclei, three sources can be discerned: - the crystal - the growth-determined texture or micro relief of the surface - thé supersaturated solution adjacent to the crystal

("catalytic breeding").

The first two sources have been proven, for the last source neither ex­perimental nor theoretical evidence has been found. Therefore only two sources remain. The first, the crystal, can give rise to new nuclei by breakage or attrition, mainly due to mechanical action as removal mechanism. The second source, the growth-determined surface relief, is per definition dependent on the growth process. This means that this surface relief and its ease of removal depend on: - the growth mechanism - the particular substance - the growth rate - the mass transfer conditions leading to supersaturation gradients and

growth instabilities - the presence of habit modifying impurities

Easily removable surface details are e.g. protruding parts such as needles, dendrites or polycrystallinities. Fluid shear can bè sufficient; even spon­taneous removal for instance due to internal stresses, may occur. On the other hand mechanical impact is demanded for the removal of nuclei from macro-steps, overhangs, weak spots due to subsurface inclusions and other microscopic details which are not affected by fluid forces.

55

The removal mechanism therefore depends on the mechanical strength and size of the removable entities on the crystal surface. The exact modelling of the nucleation rate in relation to the surface relief should follow from a detailed study on growing crystal surfaces. The single crystal experiments prove to be useful in the analysis of the type of nucleation operative under given conditions. However, in order to obtain quantitative information on the nucleation be­haviour in agitated suspensions, it is necessary to perform experiments on suspensions of growing crystals. Some results are discussed in the next sec­tion , 2.3 • 3 • The growth of contact nuclei showed a wide variability. It has been sug­gested that non-growth of nuclei is an alternative for the survival theory. In section 2.3.^ these phenomena will be discussed, taking into account the evidence from suspension crystallization experiments.

2.3.3 Secondary nucleation from suspensions of growing crystals

2.3.3-1 Types of experiments

Many experiments have been designed and performed using suspensions of crys­tals. The aim of this section (2.3-3) is to discuss the advantages and disadvantages of various types of experiments in the quantitative analysis of the nucleation rate, and in the determination of the nucleation mechanism. As discussed in chapter 1, in crystallization from suspensions there are ex­ternal parameters, which can be varied independently and internal parameters, which are dependent on the choice of the external parameters and which may generally vary with time. In order to completely analyze a par­ticular nucleation experiment one should be able to measure or to calculate the values of all relevant internal parameters in addition to the external ones. If this is not possible the effect of internal parameters has to be estimated either from basic experiments (for instance single crystal experiments) or from physical models. Table 2.1 shows a summary of external and internal parameters active in the nucleation mechanism. In a general approach of secondary nucleation it is possible to discern three consecutive process steps within the mechanism, see the foregoing dis­cussion on the single crystal experiments, namely the formation of proto-nuclei due to growth of the parent crystal, the removal, and the survival or outgrowth of nuclei. In table 2.1 these three process steps are used to il­lustrate where the external and internal parameters have their relevance in the nucleation mechanisms and in the experiments.

It is seen that important internal parameters are: - the growth mechanisms and rate; shape, size and perfection of both parent crystals and nuclei, which are liable to vary during the ex­periment , and,

- the surface relief of the parent crystals. All of which may depend on the origine of the seeds used as parent crystals. These parameters are very difficult to quantify in a suspension experiment. Therefore, if seed crystals are used it has proven to be advantageous to give them a pretreatment which eliminates their "history". Mostly partial dissolution followed by slow growth at the experimental conditions is thought to yield parent crystals of reproducible growth and nucleation characteristics.

56

Table 2.1 External and internal parameters in secondary nucleation from suspensions

"process" steps

1. growth of parent crystals

2. removal of nuclei

3. growth of nuclei (or survival)

external parameters

- Ac, hydrodynamics, T - shape, initial size, number and origin of seeds

- habit modifying impuri­ties

- hydrodynamics - vessel geometry - stirrer (size, shape, speed)

- initial size and shape of seeds

- Ac, hydrodynamics, T - impurities

internal parameters

- growth mechanism - shape, size, number and perfection of parent crystals

- growth rate and surface relief of parent crystals

- size and shape of crystals

- growth mechanism and growth rate

- shape, size and perfection of nuclei

Since secondary nucleation is a surface related phenomenon it is plausible to suppose that the nucleation rate should be proportional to the available surface area of the parent crystals. It is proposed to subdivide the stirred suspension experiments in two classes, namely seeded and autogenous nuclea­tion experiments: see table 2.2. In seeded nucleation experiments the surface area of the seeds is responsible for the nucleation, whereas in autogenous nucleation experiments the surface area of the seeds can be neglected compared to the area of the secondary nuclei which have grown to larger sizes. In the latter case the secondary nuclei, grown to larger sizes, generate most of the new nuclei. Typical examples for seeded nucleation experiments are short batch experi­ments, where enough seed surface area is used to keep the supersaturation at a low level and to prevent excessive nucleation. Autogenous experiments are either continuous crystallization or prolonged batch experiments using a minimum amount of seeds.

Table 2.2 Type of nucleation experiments from suspension

type

seeded

autogenous

mode of operation

a) batch b) semi-continuous

a) continuous b) batch (initiated)

configuration

- stirred vessel (•- stirred vessel l- fluid bed

- stirred vessel - stirred vessel

57

The use of seeded batch experiments has as a main advantage that the amount and type of seeds i s an external parameter. Drawbacks are that the amount of nuclei formed during the experiment has to be l i m i t e d and t h a t the seed mater ia l has to be well defined. A seed c rys ta l should preferably behave ex­ac t ly l i k e a c rys ta l grown in s i t u under the c o n d i t i o n s of the nuc l ea t i on experiment in order to make the outcome of the batch nucleation experiments relevant to continuous c rys ta l l i za t ion ( i . e . the autogenous c a s e ) . A d i s ­cuss ion on the growth of c rys ta l s of various or ig ines and growth h i s to r i e s wi l l be given in section 2.3-4. In autogenous batch or continuous experiments the influence of seed material i s to t r igger the secondary nucleation; the subsequent nucleation behaviour i s ful ly determined by the in s i t u generated nucle i a f t e r t he i r outgrowth to c r y s t a l s . An advantage of continuous over ba tch autogenous n u c l e a t i o n i s t h a t the c rys t a l s ize d i s t r ibu t ion of the steady s t a t e of a continuous well mixed c r y s t a l l i z e r (CMSMPR-type) can be used to derive d i r e c t l y the growth and nucleation ra t e (Randolph, Larson, 1971). see chapter 3-The analysis of an autogenous batch process i s compl ica ted (Kane, e t a l . , 197^; Evans e t a l . , 197*0. The main disadvantage of autogenous nucleation experiments i s that always a wide d i s t r i bu t ion of c r y s t a l s i s r e spons ib l e f o r t h e n u c l e a t i o n , which obscures the s ize-dependency of secondary nucleat ion. Based on the popula t ion balance analysis for a CMSMPR c r y s t a l l i z e r , an im­provement of the seeded batch experiments has been proposed (Randolph, Rajagopal , 1970): t he seeded semi-continuous c r y s t a l l i z e r , which i s con­tinuous with respect to the smallest c r y s t a l s / n u c l e i but r e t a i n s a l l seed c rys t a l s by means of a classifying device in the ou t l e t of the c ry s t a l l i z e r , for instance a wire mesh. From the CSD of the f i ne s leaving the out le t i t was possible to calculate t he i r nuc lea t ion r a t e and, i f t h e c o r r e c t assumptions were made, t h e i r growth r a t e . Since a l l nuclei are washed out i t i s possible to operate th is c r y s t a l l i z e r for a longer per iod than the seeded ba tch c r y s t a l l i z e r , moreover i t can be opera ted a t a cons tan t t empera ture , while the super-sa tura t ion var ies only slowly. Due to the extended operation time the seeds w i l l have more time to ad jus t the i r cha rac t e r i s t i c s to the process condi­t ions , which makes the experiment more r e l i a b l e . A drawback of the method i s that the CSD of the fines i s very steep due to the s h o r t r e t e n t i o n t imes needed to wash o u t t h e s e f i n e s or n u c l e i . Therefore the growth behaviour of the new born nuclei for larger s izes can­not be analyzed d i r ec t l y . A solution for t h i s problem could be to use the f i n e s conta in ing stream as feed to a second continuous c r y s t a l l i z e r operat­ing a t a la rger residence time. In fact t h i s would mean t h a t the secondary n u c l e i genera ted i n a p a r t i c u l a r experiment are used as seeds for a sub­sequent growth experiment. The advantage of t h i s proposal i s t h a t both the i n i t i a l s ize d i s t r ibu t ion of the nuclei as well as the resu l t ing CMSMPR d is ­t r ibu t ion can be obta ined. The procedure has been a p p l i e d in a s l i g h t l y d i f f e r e n t form by Toyokura et a l . (1977) who used a seeded fluid-bed crys­t a l l i z e r to produce secondary n u c l e i , and p e r f o r m e d a b a t c h growth experiment with the solut ion containing secondary nuclei from the fluid-bed. Table 2.3 l i s t s the various possible methods for the analys is of the i n t e r ­nal parameters. Table 2.4 through 2.6 give an overview of published nucleation exper iments , l i m i t e d to wel l so lub le substances in aqueous so lu t ions , and i c e . A second l imi ta t ion i s tha t only cooling and evaporative production of s u p e r s a t u r a -t i o n i s cons ide red , excluding r educ t ion of t h e s o l u b i l i t y by a mixable organic solvent, and mixing of reactants ( p r ec ip i t a t i on ) .

58

Also not included are experiments on abrasion of c r y s t a l s under cond i t i ons where growth does not take p l a c e , i . e . in s a t u r a t e d so lu t ions , or where growth and dissolut ion are prevented, for i n s t a n c e in non - so lven t s or in systems where both growth and dissolut ion are blocked by an inh ib i to r . These experiments wil l be discussed in chapter 7 which deals with the ab ra ­sion experiments of ammonium sul fa te in ethanol.

Table 2.3 The analysis of in ternal parameters in nucleation experiments

internal parameters

seeds - CSD

- growth ra te

- surface re l i e f & other character ic t ics

nuclei - nucleation ra te

- growth ra te

- shape

experimental determination

a) as a function of time: - photographs - sieving of samples - Coulter Counter

b) i n i t i a l and final CSD by sieve analysis - from CSD transients - from mass balance and c r y s t a l s u r f a c e

area - from the s t e a d y s t a t e CSD in CMSMPR

experiments - separate growth experiments

- microscopical inspection of c rys ta l samples

- v isua l detection (batch) - f ina l number of c rys ta ls (batch) - CSD transients - photographs

- Coulter Counter - Fraunhofer diffraction

- steady s t a t e CSD (CMSMPR) - r a t e of depletion of the supersaturation

(autogenous batch)

- CSD-transients - steady s t a t e CSD - s e p a r a t e growth experiments

- opt ical analysis of samples

The resu l t s of the nucleation experiments, the i r usefulness in the quant i ta­t i v e d e t e r m i n a t i o n of k i n e t i c d a t a and t h e i r c o n t r i b u t i o n to the understanding of the mechanisms of secondary nucleation in a d d i t i o n to the understanding obtained from the s ingle crys ta l experiments wi l l be discussed in the next paragraph (2 .3 .3-2) .

59

Table 2.k Seeded batch nucleation experiments

authors

Cayey, Estrin

Lal et al. 2) Garabedian,

Strickland-Constable 3) Shah et al. 4) Bauer et al.

Ness, White ^'

Rousseau et al. 7) Toyokura et al.

Q\ Kubota et al.

9) Yamamoto,Harano 10) Kubota, Kubota

Kubota, Kubota 8 , U )

Kubota et al. 8 , 1 2 )

13) Girolami, Rousseau

year

1967 1969

1972 b

1973

1974 a

1976

1976

1977

1978

1980

1982

1986 c

1986

1986

substance

MgSO^I^O

MgSOjj^O

Ice

MgS0v7H20

MgSOü.7H?0 K2S0^

MgSO^l^O

MgS0^.7H20

K-alum

K-alum

K-alum

MgS04.7H20

MgS0j,.7H2O

(NH,4)2S0(r7H20

K-alum

nucleation kinetics

-

-B - A T

-

-

B - N ^ 2 AC2'5 L*

B " N 4 ^ L 4- 6

B " N 3 G2 Kj,

B " H1-* AC1'"

B " AC1"8 - G AC0'7

B - ^ A c 1 - ^ 3

B " Ac3

B - N1-2 6 Ac1"9

-

Notes on seeded batch experiments

1) Minimum s i z e of seeds c r y s t a l s necessary for nucleation (~ 225 um). 2) S i n g l e s e e d c r y s t a l employed, s t i r r i n g for 5 to 10 s . Nucleation rate

proportional to supercooling AT up t o AT = 0 .3°C , Above AT = 0 .3°C a more than proportional increase was noted.

3) P o l y e t h y l e n e impel lers produced l e s s n u c l e i than s t a i n l e s s s t e e l im­p e l l e r s ( factor 4 ) .

k) The impact e f f i c i e n c y was shown to be a f u n c t i o n of the c r y s t a l s i z e . Crysta ls ranging from 1-5 mm were employed.

5) Poly-ethylene p r o p e l l e r s gave lower n u c l e a t i o n r a t e s than s t a i n l e s s s t e e l ones.

6) 2 . 5 - 6 . 5 mm c r y s t a l s used. Original f i t B - aL exp(bL). , - , , 7) Seeds generated i n a f lu id bed. Growth rate 10-100 p seeds " Ac D / 3 ; same

growth rate for larger c r y s t a l s in a fixed bed; 1000-2000 urn f l u i d i z e d seeds have a 60X higher growth rate .

8) The nuc leat ion probabi l i ty was derived from the w a i t i n g t ime d i s t r i b u ­t ion curve a t constant supercooling and a l s o from the observat ion of the f i r s t nucleus during l inear cool ing experiments.

9) G - A c 1 " 1 ' 4 ; B - N2 Ac 1 - 7 " 2 - 0 - N 2 G . A c 0 - 6 - 0 - 7 ; N i s the number of c r y s t a l s with radius > r . = 100-200 um.

10) Seed c r y s t a l s suspended on threads were brought i n contac t with the s t i r r e r .

11) Fluid shear experiments. 12) G = 2.57 * 10" 7(AT) 1* 3 3 j T = 35°C. Therefore B - QlA. 13) The growth rato of nuc le i produced in s i t u by i n i t i a l breeding has been

i n v e s t i g a t e d using the Coulter Counter to analyze the CSD.

60

Table 2.5 Seeded semi-continuous batch.experiments

a) s t i r r ed vessel

author(s)

Randolph, Rajagopal 2) Cise, Randolph

Youngquist, Randolph-"

Randolph, Cise

Randolph, Sikdair '

Sikdar, Randolph '

7) Randolph, S i k d a r "

year

1970

1972

1972

1972

197H

1976

1976

substance

K 2 S 0 4

K2S04

(NtygSO,,

K2é0H

K2S04

MgSO^^Ö

c i t r i c ac id

KjSO^

nucleation

-

■ -

N8

N6

-

N 2 - 6

kine t i c s

G 1 ^

AC0"6

G.

G.

A C 0 ' 7

Gseeds f™/»l

*T 1.1

Pi,

h?-1

Ö.8

.0 .5

-

-

i b - 28

-

-

26- 71

11 - 37

-

b) fluid bed

Mullin, Gaska9'

Toyokura e t a l .

Karpinski, Toyokura '

Karpinski-

•

12 \ Toyokura e t a l . '

Toyokura e t a l . J /

Budz e t a l .

Budz e t a l .

Budz e t a l .

1969

1976

1979

1980

1981

1982

1981

1982

1982

1984

1985

K2S0j,

K-alum

CuSOjj.SHgO

MgSO^^H^

MgS04.7H20

CuSOjj.Sf^O

MgSO^f^O

CuSOjj.5H20

K-alum .

K-alum

MgS0^.7H20

CuSOj,.5H20

K-alum

K-alum

K-aium

Na2S20-.5H20

K-alum

Na2S20-.5H20

3 -2 uJe uV2

(uV2)1

N3

Re*"8

R e \

Ré 1 ' 8

i e 9 - 0 4 - 5

Ac3 '3Re2-5

Ac1"8

Ac1""

G1-2

G1-1

Ac l '5

•** Ac1"5»

Ac 2 ' 6

i c 1 . 3 - 1 . 6

G 3- 2

G

G1-2

G1-1

G i . o

a1-* a1-* Q 1.8

*T

* j

f^-Q.f^

*r

AT

" m

AT

AT

AT'

AT

AT

Af

AT

AT

1.5r200

-

7- 80

40-375

6-300

4-200

6-3ÓO

4-200

-

-

180-400

46-320

-

-

-

-

-

-

61

Notes on the semi-continuous seeded batch experiment

1) Growth rates of 10-30 um crystals are linearly dependent on supersaturation. Value of the growth rate is lower than expected (3-25 nm/s). Incorrect interpretation of nucleation kinetics, see Cise, Randolph (1972).

2) Slowly growing rate nuclei: direct birth into observed size interval (3~ 7 um); strong stirrer speed dependence; incomplete suspension at lowest stirrer speeds.

3) Slowly growing nuclei; direct birth in (I.3-25 um) size interval. 4) Growth rate seeds - ic . Calculation of nucleation rate based on direct

nucleation into the observed size interval. 5) Effect of soft impeller coating. 6) Growth rate of crystals > 8 um independent of size. Nucleation rates

based on particle flux extrapolated to zero size (B _ _ ) . Below 8 um much lower average growth rates. Direct birth at sizes > 8 um. Strong stirrer-speed dependence below 8 um. No stirrer speed dependence and su­persaturation dependence in B „„ Same conclusions hold for botn s; systems.

7) Growth ra te of seeds - Ac .N . 8) Symbols : u = s u p e r f i c i a l s o l u t i o n v e l o c i t y , c = p o r o s i t y . Re i s

Reynolds- number based on par t i c le size and re la t ive veloci ty , u / e . 9) Rec rys t a l l i zed "commercial" pure-grade crys ta ls were used. Above 8% su­

persaturation opaque crys ta ls were grown. The 9th order i n c r e a s e of B s t a r t e d a t t h i s value of the s u p e r s a t u r a t i o n . Growth r a t e of seeds depends on crystal s ize and on the supersaturation squared. Nucleat ion ra te calculated from mass of f ines.

10) G . - A c 3 ' 3 ■> B - G ' . Outgrown nuc le i (100-200 pm) grow more slowly than the seeds.

11) G . » Ac1,2'* * B * a1'**.. seed seed

12) Small paddle type s t i r r e r s inserted in the fluid bed. For the smallest 1 6 stirrer B - Ac " . _ n 1 0 13) For potassium alum B - G ' for a fluid bed experiment; B - G when

the paddle type stirrer is used. The exponent of G decreases from 2 to 0.6 on an increase in seed size from 150 to 2000 pm (exponent van Ac: 3.2 + 1.0).

62

Table 2.6 Autogenous batch experiments

authors

Ting, McCabe1'

Mullin, Raven

Nyvlt2»

Nyvlt, et al.

Melia, Moffitt3» 4) Cayey, Estrin

Nyvlt5)

Nyvlt et al.5»

Mullin et al.

Mullin. Nyvlt

Mullin, Leci

Mullin, Osman

Jones, Mullin5*

Wey, Estrin '

Kane, et al. 7))

Evans, et al.

Mullin, Ang

Broul

Janse

9) Janse, -de Jong-"

Shestov et al.

Mullin, Jancic10»

Nyvlt11»

, Ishii, Fujita

Broul

Nyvlt12)

Mullin, Williams

Nyvlt et al.

year

1934

1961 1962

1963

1964

1964b

1967

1968

1970

197P

1971

1972

1973

1973

1974

1974

1976

1976

1977

1978

1979

1979 1980

1981

1982

1982

1984

1985

substance

MgS0j,.7H20

citric acid citric acid

succinic acid KDP

. (NH4)2SOjj

KC1 MgS04.7H20

KC1 (NH^SO,, 25 systems (NHjj)2S04

£ & » * Citric acid

NAS K2S0i,

Ice

Ice

Ni(NH4)2(S04)2.6H2O

(NHZ|)2SOl4

K-alum K 2Cr 20 7

K 2Cr 20 ?

Lactose

K-alum

K 2Cr 20 7

CuSO^HgO (NH^SC^ NH -alum K2S04 various systems

operation

linear cooling

ti

* «

it

it

linear cooling + desupersat.

linear cooling ft

tt

n

progr.cooling

desupersat.

lin. cooling

lin. cooling + desupersat.

desupersat.

desupersat. -

linear cooling

evaporation

linear cooling

ff

desupersat.

linear cooling

desupersat.

evaporation

linear cooling

linear cooling

linear cooling

CSD

sieve

photogr.

L50

photogr.

L50 sieve

supersaturation

AT, „ AT d e t max

AT, t det

AT, t det

AT., t det

AT, .. det AT. t det

Ac(t)

AT. t det AT, k

Ac(t)

ATdet iTdet

iTdet waiting time

iTdet Ac(t)

4Tdet ATdet ATdet ATdet

63

Notes

1) First seeded secondary nucleation experiments, employing characteristic values of the undercooling/supersaturation, expressed as AT, at the mo­ment of detection of the first crystals, AT, , and the maximum desupersaturation rate, AT , detected visually from the "explosive" increase in crystal numbers.

2) No seeds employed: primary nucleation (meta-stable zone approach) assumed. More likely secondary nucleation triggered by extraneous seeds or heterogeneous nucleation.

3) Few seeds used. Final number of crystals independent of number of seeds. The authors deduce the autocatalytic nature of secondary nucleation from this observation.

4) Initial breeding, depending on amount of seeds, followed by a sudden in­crease in nucleation rate, interpreted as secondary nucleation induced by the nuclei formed by initial breeding, after having reached a minimum size of 160-210 pm.

5) The method employed is known as the Nyvlt polythermal method. 6) Couette flow configuration; nucleation and growth calculated using the

! _., u ! D° «1-5. 1.8 0.2 population balance: B - N Ac u_ 7) Analytical solution of the momentum balances (Larson, Randolph, 1971)

for batch crystallization. Kinetic results of limited value since secon­dary nucleation was assumed to be independent of the growth rate.

8) Primary nucleation assumed as dominant mechanism. 9) Shows that there can exist an appreciable difference between the time of

birth and the moment of detection. 10) "Metastable zone" smaller when Coulter Counter is used for detection. 11) Growth of nuclei is accounted for, though still only primary nucleation

is considered: B - Ac . For K_Cr?0_ it was found that n = I.56. However, it had to be assumed that the growth rate was linear with the supersaturation. According to Janse and de Jong (1978) G - Acs; 1.2k < g < 1.63. This would lead to -0.33 < n <0.81». The nucleation order n can be calculated from the apparent nucleation order m (Nyvlt, 1968) with n = 4m = 3g - 4. The assumption that B - Ac will be correct only if a) seeding is per­formed in an exactly reproducible way and b) if it is ensured that autocatalytic breeding does not lead to a factor detection of nuclei.

12) The improved treatment, Nyvlt (I98O), has not been employed though the author does refer to this paper. Further discussion on the difference between true and apparent kinetic order is provided by Söhnel (1983) and Nyvlt et al.(1985).

64

Table 2.7 CMSMPH experiments '

author(s)

Chambliss

Bransom et al. 2) Timm, Cooper

Shor, Larson-" 4) Rosen, Hulburt ' ■5) Hargolis ét al.

Huige

Genck, Larson

7)' Larson, Mullin

Larson, Klekar

Ottens, de Jong Ottens

Ottens, de Jong 9) Bennett et al.

Desai et al.10'

Jancic11'

Janse 13) Randolph et al. J'

Juzaszek, Larson

Helt, Larson

Zabelka 15) Asselbergs

Garside, Jancicl6)

Shah1?» lfi) Rousseau, Woo

Bourne, Hungerbuehler

Hartel et al . 1 9 *

Epstein, Sowul '

Sowul, Epstein 21) Bourne, Paubel 22) Rousseau, Parks

Beer, Mersmann

Grootscholten Grootscholten et al.

2^) KuiJvenhoven -"

Ploss et al.

Chen, Larson

year

1966

1969

1971

1971

1971

1971 1972

1972

1973

1973 1973 1973

1971

1973

1974 1976

1977

1977

1977

1977

1978 1978

1979 1980

1980

1980

1980

1980

1981

1981

1981

1981

1982 1984 1985

1983 1985

1985

substance

(NV-jSO,,

NH^-alum

MgS0^.7.H20

K2Cr20?

KNO,

K2SO„

Ice Ice KC1 KNO,

(NH^SO,, (my-.so,, K-alum

K-alum

NaCl

K 2 C r 2 0 ?

K-alum

K 2 C r 2 0 ?

KC1 KNO,

KNO 3 )

K-alum NaCl K-alum

NAS K-alum

K-alum

Sucrose

Sucrose

(NHj^SO,,

MgSOj^hy) KC1

NaCl

Sucrose KC1 (NH^SO,,

Ca(NO,)2>4Ha

nucl.kinetics

N3

N3

N3

N* N3

(S-S*)1-7

N1-8 Ac1'8

N2

N1-8

Q 1 ^

G 2 - 0

(4C-4C0)

AC°-9-G°-5 Qi.38 Q-0.1

AT2"1

«T "T AT

«T *r *T

v0% G2-55.AC2.55

G 1 ' 3 ^ 1 - 3

01.03

G4-5

(G0)1-8

(0°-q) 00.98

Ac A C 2 - 5 2

G0.48

G4.99

G2.06

Wr-Q 1- 7

MT

"T M4 M0-6

"T MT

"r «?-14

"T ■

*r V N1-0 (O 0) 0' 9 ^

G2 (Qo,1.58

temperature dependence

N3

N1-"

N2

N2

N0.5 „1.8

„1.8

>°

(G 0) 1- 8

G1^ 0.0.51

G2-1

( Q O ) 0 . 5 I

G2-1

G2

(G')0-69

G2

G1.65

G2

"r *r

-

*r «T M,0-53

«T M0.68 "T M , 1 - 3 3

«T

M/- 5* M,0-75

"T *T

T[S]

15- 45

15- 45 -

10- 45

15- 45 19- 88 6- 13

10- 33 15- 45 15- 45 8- 20

9- 32 5- 50

5- 50 108-117

15- 60 10- 60

9- 30 -

10- 40

-

17- 37 10- 54

10- 60

-

10- 60

-27- 37

6- 33 15- 42

7.5- 92

24- 63

3-250

20-230

15- 45

G[nm/s]

43-122

35- 83 -

16- 81

48-133 12- 95

16- 82 117-207

43-157 150-230

97-210

10-150

10-150

6.3-12.7 90-122

9- 75 26-100

-77-132 40-110

14- 43 35-130

9- 75

8- 45

25- 62

12- 25

46-406

13- 55 -

17- 47 9-200

10-150

62-157

65

Notes to the CMSMPR kinetics. 1) For the absolute values of the nucleation rates the original references

should be inspected. 2) Supersaturation between 1.37 and 5.07X. 3) Effect of various additives investigated. *)) Evaporation, 30°C. Slow growth below 85 urn. Supersaturation between 2.15

and 5. *4 -5) Measured growth rates of the crystals decrease very strongly with size

(factor 30-100) from 150 um to 100 pm. 6) Nucleation rate decreased for KN0- with increasing temperature, but in­

creased for KC1 (at constant growth rate). 7) The effect of Cr-** was investigated. o 1 8 8) Original equation of the authors B°/(e.MT) - (G ) . 9) Evaporative crystallization. Growth rate dependence not reliable due to

the small range of T employed. Small size (< 100 um) and very large size crystals grow more slowly than-the average.

10) Growth rate proportional to Ac . Additional seeding employed. 11) Total nucleation rate from total number divided by residence time. 12) Effective nucleation and growth rates based on extrapolation of the

8l-l62 um size interval. 13) Solution nearly saturated with NaCl; impurity: 0.75 mg MgSO^/100 gH-0. 14) Nucleation decreases with increasing temperature. 15) Evaporative crystallization at pilot plant and at 55 1 scale. 16) Effective nucleation rates; lower absolute values than the total nuclea­

tion rate (Janci6 1976)> lower stirrer speed dependence. 17) Agglomeration of fines investigated. 18) Nucleation increases with temperature (at constant growth rate). 19) Size-independent average growth rate for crystals > 60 um. 20) Evaporative crystallization. 21) Rubber coated stirrer showed a negative stirrerspeed dependence (see

also chapter ** ) . 22) Average growth rate strongly size-dependent. 23) B*. G* numberflux and growth rate at L=100 um.

66

2.3-3-2 The value of nucleation experiments in suspension

The aim of this paragraph is to assess globally the value of the various types of suspension nucleation experiments in comparison with single crystal experiments, and to see which method leads to reliable quantitative data needed for the modelling of a suspension crystallizer. The kinetic equations for nucleation are generally in the form of a power law (see 1.2.3). Comparison of the results presented in tables 2.4 to 2.7 shows that one category of experiments, namely the autogenous batch experiments is par­ticularly unsuccesful in generating kinetic data. The main difficulty here is the need for analysing the CSD as a function of time, which was performed only in four out of the 30 references. In one of these four a kinetic equation was derived (Wey, Estrin, 197*0 • In order to circumvent the tedious process of CSD analysis Nyvlt developed a method to calculate the dependence of the nucleation rate on the super-saturation from cooling crystallization experiments (e.g. Nyvlt, 1968). The mathematical analysis, however, rests on these questionable assumptions:

The nucleation mechanism is primary homogeneous nucleation, B ~ Ac , in­dependent of the amount of crystals already formed. The nuclei are supposed to come into existence with a fixed non-zero size, whereby further growth is neglected, which leads to a crystal mass production proportional to the nucleation rate: dm/dt - B. There is a sharp transition supersaturation between no nucleation and "massive" nucleation. This transition was assumed to be justified from the meta-stable zone concept. When the solution supersaturation ap­proaches this critical supersaturation the "mass nucleation rate" dm/dt will almost instantaneously increase to a value which is more or less equal to the rate at which the supersaturation is produced, which is proportional to the cooling rate in cooling experiments.

Curiously enough the same analysis has been used under addition of seeds. Comments on this "Nyvlt-polythermal" method are given by Janse and de Jong (I978), Mullin and Jancii (1979) Nyvlt (1980); Söhnel (19.83) and Nyvlt et al. (1985). A physically correct and very elegant method is given by Kubota and Kubota (I986) , where both secondary nucleation and the stochastic character of the appearance of the first nucleus have been incorporated (see table 2.4,. seeded batch nucleation experiments). The autogenous character of secondary nucleation has been recognized and purposefully employed by Kane et al (1974). The analytical solution of the momentum-balances (Randolph, Larson, 1971) was used to derive the nucleation kinetics of ice from the desupersaturation curve. Unfortunally, in order to reach an analytical solution, the authors had to,. assume that the nucleation rate was independent of the supersaturation*::, which is not correct. • A second drawback of the momentum-balance method is that it assumes the growth rate to be independent of the crystal size. This condition is clearly not fulfilled in many systems (see section 2.3-4). In the crystallization of ice small crystals grow much faster than large ones (Margolis et al.,1971)-This effect was not taken into account by Kane et al. (197*0. It is concluded that in performing autogenous batch experiments it is neces­sary to follow the CSD and, if possible, the supersaturation as a function of time in order to be able to calculate the relevant quantities. The moment of seeding and the amount of seeds certainly do have an influence on the

67

subsequent nucleation and growth behaviour. Therefore autogenous batch ex­periments should perferably be performed using a well defined seeding technique in order to obtain reproducible experimental conditions. If no seeding is employed the nucleation process may proceed very rapidly in some cases, due to the high supersaturations reached without seeding, which may interfere with the time needed for the CSD-analysis.

More information is obtained from "fully" seeded batch experiment (table 2.4). Since the size and amount of seeds is easily varied it is possible to measure the dependence of the nucleation rate on the size of the crystals. The analysis of the nucleation rate can be performed by storing solution samples and counting the nuclei developed after some time. Mostly the batch experiments are used to determine the growth rate of the seeds, too, as a function of the supersaturation. One of the main disad­vantages in seeded batch experiments is the need for a proper pretreatment of the seeds ("curing"). The best procedure would be to take the seeds from a growing suspension without separating them from the solution and to perform a subsequent nucleation experiment under the same conditions of supersaturation, etc. Toyokura (et al.,1977) used the nuclei containing solution from a fluid bed of growing crystals as seeds for a subsequent nucleation experiment. A second possibility is to remove the first nuclei from the experiment and start the analysis of nucleation after some time. In this case, however, the removal of nuclei has to be rapid enough, and selective towards the nuclei. The third possibility is to remove the secondary nuclei continuously and to analyze them either directly or after sampling and "development". These "semi-continuous" batch experiments are listed in table 2.5- The stirred vessel experiments employed a screen in the outlet to retain the seeds crystals. The CSD of the nuclei was analyzed on-line in the small size range (~2-30 jam) using a Coulter Counter. As in the fully continuous experi­ments, the steady state solution of the population balance for continuous operation can be used to analyze the nucleation rate and the growth rate of the nuclei. The growth rate of the seeds was analyzed by sampling and weigh­ing, since the Coulter Counter technique is not suitable here. The analysis of the nuclei population densities produced by the growing seeds showed that initially a large number of nuclei is formed. These nuclei are washed out, and, after some time a stationary CSD of the nuclei is found. According to the population balance theory of a CMSMPR crystallizer (see chapter 3) a linear decrease of the logarithm of the nuclei population density is expected if a) the nuclei are formed at sizes smaller than the lower bound of the size

range covered by the CSD analysis, and, b) the growth rate of the nuclei is independent of their size.

The experimental CSD's, however, yielded curved In n versus L plots. For KpSOj, and (NH^J^SOj, this curvature was noticed over the full range of the CSD analysis, for citric acid and MgS0j,.7H_0 there was a distinct discon­tinuity at a size of about 10 um. Above 10 urn these CSD's yielded straight lines. The curved part of the distribution can be explained either by size-dependent growth, or by direct birth of slow growing or non growing nuclei in this size interval. Strong evidence for direct birth was found from the observation that the "curved" part of the CSD reacted instantaneously on an increase in stirrer speed; the population densities at all sizes reacted at the same moment. In the straight part of the distribution (MgS0j,.7H_0 and citric acid) the effect of stirrer speed variations became noticable after a

68

de lay time depending on the observed s i ze , which suggests tha t the nuclei need some time to grow in order to reach the par t icu lar s ize i n t e rva l . These conc lus ions a r e i n l i n e with the observations on K-alum (Garside e t a l . ,1979 and Rusli e t al . ,1900) discussed in sect ion 2.3.2-3 and with those on MgS0j,.7H~0 (Khambaty and Larson, 1978) in the same sect ion. In the l a t t e r paper a discontinuity a t ~ 14 um was concluded. Apparently the same s t i m u l u s , i n th i s case mechanical impacts, yields two kinds of nuclei : slow to non growing n u c l e i which c l e a r l y a r e broken off from t h e pa ren t c r y s t a l s , and n u c l e i growing a t a d i s t i n c t l y higher and measurable r a t e . Of the l a t t e r group the nucleation rate cor re la tes strongly with t h e growth r a t e of the parent c r y s t a l s , which suggest tha t the growth r a t e determined surface r e l i e f i s an important factor . This n u c l e a t i o n be­hav iour was q u a l i f i e d as " r e g e n e r a t i o n " l i m i t e d or "generation" limited nucleation, thé former qua l i f ica t ion r e f e r r i ng t ó the o b s e r v a t i o n t h a t i n c o n t a c t experiments the nucleation ra te becomes constant above a given con­t a c t frequency (see s e c t i o n 2 . 3 . 2 - 3 ) (Rando lph , S i k d a r , I 9 7 6 ) . The n u c l e a t i o n r a t e of the o the r k ind of n u c l e i , the slow growing ones; i s c l ea r ly less growth r a t e or supersaturation dependent, but i s very s t r o n g l y a f f ec t ed by the s t i r r e r speed. Here the q u a l i f i c a t i ö n "removal l imited" nucleation has been proposed. However the s ize dependence for both types óf nuc l ea t i on mechanisms i s almost iden t i ca l ( thi rd or fourth power, ref lected by M_ and ML respect ively, see table 2 .5a) . This suggest t h a t the removal mechanism i s i d e n t i c a l in both c a s e s . The d i f f e r ence between the two mechanisms seems to be the type of species removed. The non growing nucle i , formed almost independently of the supersaturat ion, are l ike ly to be abrasion f ragments , whereas t h e growing n u c l e i seem to or ig inate from the growth determined surface r e l i e f . The causes for the dif^ ferent growth behaviour of the two s o r t s of n u c l e i w i l l be d i scussed in sect ion 2 .3 .4 . An explanation for thé extremely strong s t i r r e r speed dependence in the for­m a t i o n of t h e s low growing f r a g m e n t s i s given by Ot tens (1973) who calculated tha t thé parent c r y s t a l s i n the experiments of Youngquist and Randolph (1972) were not completely suspended. This probably holds for a l l the experiments in table 2.5a) s i n c e they were performed in thé same ex­per imenta l se t up. Actually Cise and Randolph (1972) report tha t suspension was incomplete a t the lower s t i r r e r speeds. Comparison of the seeded semi-continuous nucleation experiments with CMSMPR ones shows tha t the t o t a l nucleation ra te measured in the semi-cont inuous case employing the Coul te r Counter i s orders of magnitude higher than the effect ive nucleation ra tes (see a lso chapter 4 on ammonium s u l f a t e ) . This e f f e c t i s p a r t l y exp la ined by the low t o zero growth r a t e s of the grea ter part of the nuclei preventing the i r outgrowth to product s i z e c r y s ­t a l s . The fluid-bed nucleation experiments (see table 2.5b) a re t o be cons idered as semi-cont inuous exper iments , too. The main difference with the s t i r r e d vessel semi-continuous experiments i s the short residence time of the s o l u ­t ion i n the f luid bed. Direct measurement of the CSD of the nuclei y ie lds a s ize d i s t r ibu t ion which i s , un l ike thé s t i r r e d vessel case, not affected by growth, i . e . i t i s pos­s ib le to measure the i n i t i a l s ize d i s t r ibu t ion of the nuc le i . However, t h i s means that information on the subsequent growth behaviour i s not a v a i l a b l e . Therefore a d d i t i o n a l growth exper iments a r e n e c e s s a r y . Nucleat ion r a t e s a re mostly determined by counting the numbers of c rys ta l s developing i n s o l u t i o n samples s t o r e d a t c o n s t a n t t e m p e r a t u r e c . q .

69

supersaturation. By this method non-growing nuclei are not observed. The ex­istence of such non-growing nuclei is confirmed by Toyokura et al. (1984). A drawback of fluid beds is that the effect of the velocity difference be­tween solution and parent crystals on the nucleation rate cannot be investigated independent of the crystal size, since it is directly connected to the free fall velocity which is a function of the crystal size.

The autogenous continuous nucleation experiments allow for a relatively simple and straightforward calculation of nucleation and growth rates from the steady state CSD when the crystallizer is operated as a well mixed non classifying continuous crystallizer ("CMSMPR" conditions), see chapter 3. The determination of the nucleation and growth rates from dynamic experi­ments is not so easy and demands considerably more experimental calculational efforts. The benefits of the CMSMPR steady state method are:

simple calculational procedures simple operation over long periods no seeding outgrowth of nuclei under constant conditions wide variation of parameters possible.

On the other hand there are some drawbacks: supersaturation is a dependent variable and should be measured separately only number averaged growth rates can be measured (see chapter 3) the moments of the CSD are interrelated (see 3-3-3~6) and consequently the size dependence of nucleation cannot be determined unambiguously.

The growth behaviour of the crystals is in accordance with the observations made in the semi-continuous seeded batch experiments. The curvature of the In n versus L plot, can, in some cases extend over the whole range of analysis (see section 2.3-4).

2-3-3~3 Concluding remarks

From the foregoing discussion it has become clear that the phenomenon of growth has everything to do with the determination of nucleation rates. First, the growth of the parent crystals may lead to surface conditions en­hancing the production of secondary nuclei; secondly the outgrowth of these nuclei is decisive for their observation at larger sizes, and for their im­portance as "consumers". Therefore the parent crystals should have a well defined growth history as in the steady state CMSMPR and the semi-continuous seeded batch experiments. Not recommended are seeded batch experiments (initial breeding effects) and autogeneous batch experiments (varying Ac; analysis and interpretation of CSD complicated). Since the growth of nuclei is so important in the process of secondary nucleation it will be discussed separately in the next paragraph (2.3-4). The growth of parent crystals will not be discussed in detail. The nucleation rate is most frequently expressed as an empirical power law in terms of stirrer speed, growth rate (supersaturation) and one of the mo­ments of the CSD. It will be shown in section 2.4 that the coefficients in this power law equation depend on the nucleation mechanism, which makes it possible to determine the nucleation mechanism from the values of the power law coefficients.

70

2.3.4 The growth of nuclei

The observed curvature in the steady state In n versus L plots of CMSMPR ex­periments can basically be explained in two ways, see chapter 3 section 3.4. First, true size dependent growth is possible, see table 2.8. Essential here is that all crystals of a given size have the same (time-averaged) growth rate. Secondly; growth dispersion may cause the observed curvature.

Table 2.8 True size-dependent growth models

authors

McCabe Bransom Canning,Randolph Abegg, et al. Estrin, et al. Randolph, Cise Wey, Terwilliger Garside. et al. White, et al. Garside, Janció

year

1929 I960 1967 1968 1969 1972 1974 1976 1976 1979

comments

size-independent growth ("AL-law")

ASL model (Abegg, Steve, Larson)

discussion review

In this case crystals of the same size exhibit different growth rates, see table 2.9. Two modes of growth dispersion have been proposed. Stochastic fluctuations of the growth rate around a given average value was suggested by White and Wright (1971). However, Randolph and White (1977) point out that this mode does not produce any "curvature". Janse and de Jong (1976) assumed that each crystal retains it own unique growth rate. Further, they modified the population density by distributing the crystals óf a given size over the growth rate. Thus, the growth rate distribution (GRD) of the nuclei leads to growth dispersion. For mathematical details see chapter 3-The model of Janse and de Jong will be termed Permanent (growth) dispersion in contrast with the Stochastic (growth) dispersion model of White and Wright. These terms are preferred over less descriptive or accurate ones such as: Growth Diffusivity or Random Fluctuation (RF) model, instead of Stochastic dispersion, and the Constant Crystal Growth (CCG) model instead of Permanent dispersion, see tabie 2.9.

Table 2.9 Growth dispersion models

authors

White, Wright Janse, de Jong Janse Randolph, White Tavare, Garside Ramanarayanan et al. Levenspiel, Fitzgerald Berglund, Larson Ramanarayanan et al. Larson et al. Zumstein, Rousseau

year

1971 1976 1977 1977 1982 1982 1983 1984 1984 1985 1986

comments

stochastic dispersion growth rate distribution

growth diffusivity

RF and CCG model dispersion versus convection CCG versus ASL model CCG model various growth rate distributions RF versus CCG model

71.

In the very small s ize range addit ional "curvature" may be caused by d i rec t "b i r th" of nucle i , or by agglomeration. Of these two addit ional e f f e c t s the c u r v a t u r e caused by d i rec t b i r th wil l be pos i t ive (posi t ive curvature mean­ing tha t the second order der ivat ive of In n towards L i s pos i t ive ) only i f the growth r a t e of these nuclei i s e i the r s trongly size-dependent, decreas­ing with increasing s i ze , or if permanent dispersion p reva i l s , where only a small fract ion of the nuclei has a measurable growth r a t e . Therefore a posi­t ive curva ture always proves the preva lence of e i t h e r permanent growth d i s p e r s i o n or t r u e s i z e dependent growth. Systems showing such a posi t ive curvature are summarized in table 2.10. Curvature over the whole s i z e range i s observed for K-alum, K-CrJ)™, MgS0K.7H_0 and for sucrose two out of three t imes. A d i s t i n c t discontinuity i s observed for K-SOj, a t about 85 um and once for sucrose a t about 60 um.

The ef fec t of agglomeration on t h e shape of t h e CSD i s l e s s c l e a r . Table 2 .11 l i s t s the va r ious re fe rences on agglomeration. The only quant i ta t ive measurements of the phenomenon a r e repor ted by Kubota and Mullin (1984) . Microscopically the phenomenon has been observed by Toyokura e t a l . (19-84).

Table 2 .10 Curved In n versus L p lo t s in CMSMPR experiments

authors

Rosen, Hulburt Ottens, de Jong Jancic Janse Zabelka Garside.Janció Bourne,Hungerbuehler Hartel et al. Epstein, Sowul Sowul, Epstein Rousseau, Parks Grootscholten Kuijvenhoven

year

1971 1973. 1974 1976 1977 1978 1979 1980 1980 1980 1981 1981 1982 1983

substance

K2S04 K-alum K-alum K Cr 0 K-alum' K-alum K-alum sucrose sucrose sucrose MgSOü.7H-0 NaCl4 * sucrose

remarks

below 85 um

growth dispersion ,

various models tested be3,ow 68 um

below - 100 um

Table 2.11 Agglomeration in wel l so luble systems

authors

Davey, RUtti Sarig et al. Shah Offermann,Ulrich Ulrich Kuyvenhoven Toyokura et al. Kubota, Mullin Ulrich et al

year

1976 1979 1980 1980 -i 1981 J 1983 1984 1984 1985

substance

HMT KC1 NAS NaCl sucrose K-alum K-alum K-alum

remarks

conglomerates observed

abrasion experiments conglomerates observed microscopic observation direct measurement

72

Table 2.12 Permanent dispersion of contact nuclei

authors

Lai et al. Rousseau et al. Khambarty, Larson Garside, Larson Garside et al. Garside Rusli et al. Larson Berglund, Larson Berglund, et al. Shanks, Berglund

year

1966 1975 1978 1978 1979 1979 1980

1981 1982, 1984 1983 1985

substance

MgSOjj.THgO K-alum MgS0^.7H20 K-alum K-alum

K-alum

citric acid citric acid KNO, sucrose

remarks

spread in outgrowth time spread in outgrowth time non growing nuclei < 14 pm microscopic observation Coulter counter

Coulter counter: no growth < size dependent > 4 pm microscopic observation microscopic observation microscopic observation microscopic observation

4 pm

Table 2.13 Permanent dispersion of collision nuclei

authors

Randolph, Rajagopal Cise, Randolph Youngquist,Randolph Randolph, Cise Randolph,Sikdar Sikdar, Randolph

Randolph, Sikdar Belyshev et al. Bujac van 't Land, Wienk Girolami, Rousseau

year

1970 1972 1972 1972 1974 1976

1976 1979 1976 1976 1985 1986

substance

K SO. K2S04

KpSOZ MgS0..7H20 citric acid

Penta-erythrol NaCl K-alum

remarks

negligible growth 2-40 pm negligible growth 2-40 pm negligible growth 2-40 pm negligible growth 2-40 pm negligible growth 2-40 pm no growth of nuclei < 8 pm no growth of nuclei < 8 pm negligible growth 2-40 pm dissolution of abrasion fragments slow growth of fragments slow growth of fragments growth dispersion of initial breeding nuclei

73

Table 2.1*1 Permanent dispersion in crystal growth experiments

authors

Kucharenko Sipyagin

Mussard, Goldstaub Natal'ina, Treivus Valcic van 't Land, Wienk Davey et al. Human

Garside, Ristid Garside et al. Offerman, Ulrich

Girolami, Rousseau Berglund, Murphy Bhat et al.

year

1927 1968

1972 1974 1975 1976 1979 1981

1983 1984 1984

1985 1986 1986

substance

sucrose f NaC10_ 1 KC10 5

NaClè KDP i

sucrose NaCl ADP K-alum

ADP KC1 NaCl MgS0a.7H_O ) K 2 S O ; K-alum -i NaCLO, ' K-alum" sucrose K-alum

remarks

+ fluctuations (long term)

♦ long term fluctuations

+ long term fluctuations damaged crystal

{111}: dispersion + decrease of average G in time

{110}: fluctuations (long term) {100}: strong fluctuations (long

term)

crushed crystals grow more slowly small fragments grow more slowly than "normal";

small fragments grow faster than "normal"

correlation growth rate with edge dislocation density

a)

b)

D i r e c t exper imenta l evidence for permanent d i s p e r s i o n h a s b e e n c o l l a t e d i n t a b l e s 2 . 1 2 , 2 .13 and 2 .14 . Various obse rva t i ons were made:

A spread in t ime i s needed for the deve lopment of c o n t a c t n u c l e i of K-alum and MgSCL^H^O. Extremely slow growth of con tac t and c o l l i s i o n n u c l e i below a given s i z e : K-alum < 4 um, K^SO^ < 85 um, (Nlfy) SCfy, MgSCy7H20 < 8 - lk um and c i t r i c ac id < 8 um. •

c) M i c r o s c o p i c o b s e r v a t i o n of t h e growth of c o n t a c t n u c l e i shows t h a t i n ­d i v i d u a l c r y s t a l s grow with a s i z e - i n d e p e n d e n t r a t e (K-alum, c i t r i c ac id , KN0_, s u c r o s e ) . A d d i t i o n a l l y t h e r e i s a c o r r e l a t i o n between t h e i n i t i a l s i z e of a nucleus and i t s growth r a t e . Abrasion fragments do n o t o r h a r d l y grow (NaCl, P e n t a - e r y t h r o l , KC1, d)

e)

f)

g)

MgS04.7H20) K 2 S C V , . Damaged l a r g e s i n g l e c r y s t a l s of NaCl do not r e p a i r , bu t t h e damaged s i t e i s overgrown (van ' t Land, Wienk, 1976) . F r a g m e n t s may d i s s o l v e in s l i g h t l y s u p e r s a t u r a t e d s o l u t i o n s (KC1, Belyshev e t a l . , 1979). Broadening of t he CSD i n seeded ba tch exper iments can o n l y be e x p l a i n e d by permanent d i s p e r s i o n (K-alum, Girolami and Rousseau, 1984, 1985)-Dispers ion i n growth r a t e of l a r g e and small s i n g l e c r y s t a l s i s e v i d e n t ( suc rose , NaClO-, KC10-, KDP, ADP, K-alum).

h) Long term f l u c t u a t i o n s 3 i n g rowth r a t e a r e o b s e r v e d ( s u c r o s e , NaCIO KC1CL, K-alum). 3'

74

The long term fluctuations in the growth rate of large single crystals are observed as sudden unpredictable changes in growth rate. Since the periods between these changes are in the order of hours to days (Human, 1981) they can be neglected for practical purposes. The growth rates of contact nuclei remained constant over periods of one hour. No evidence for true size-'dependent growth of single crystals was found in these experiments, except for a slow decrease of the average growth fate of large K-alum crystals, grown from seed plates, over a period of.3 to 5 days (Human, 1981). Actually, direct observations of true size-dependent growth are very scarce. One example is the growth of ice in suspension, where the growth rate of small (~ 150 um) crystals is reported to be much (30-100 times) higher than that of large (~ 1000 um) crystals. A decrease of growth rate with size in continuous crystallization can never be explained by growth dispersion. In the experiments óf Offermann and Ulrich (1984) the growth fates of 3"30 um small "hurt" crystals or fragments are compared with the growth rates of small crystals obtained by "homogeneous nucleation", presumably by super­saturating a clear solution. As pointed out before the most likely nucleation process from clear solutions is secondary nucleation triggered by either external, undeliberate seeding (crystalline dust), or by primary heterogeneous nucleation. Therefore the authors actually compared the growth behaviour of fragments with the growth behaviour of nuclei produced by some unspecified secondary nucleation mechanism. For both types of nuclei the growth rate as a function of size showed a decrease towards smaller sizes, which is most probably caused by growth dispersion, too. This implies that both types of nuclei are mixtures of slower and faster growing crystals. If slow growth of the "hurt" fragments is a result of the damage afflicted during the milling, it might well be true that the slow growing crystals present amongst the "unhurt" crystals are a consequence of mechanical damage during the secondary nucleation.

Table 2.15 Proposed explanations for growth dispersion

authors

Valcic Strickland-Constable Rusli e t a l .

Human Garside e t a l .

Girolami, Rousseau

year

1975 1979 1980

1981 1984

1985

cause for dispers ion

a c t i v i t y and number for screw d i s l o c a t i o n s small nuclei do not contain screw d i s l o c a t i o n s a) a c t i v i t y screw d i s l o c a t i o n s b) disappearance of mobile d i s l o c a t i o n s c) increased s o l u b i l i t y of h i g h l y curved p a r t s

o f the c r y s t a l d) fragments grow f a s t e r due to t e n d e n c y t o

repair during growth the a c t i v i t y of s p i r a l s changes small nucle i produced by mechanical e f f e c t s must be deformed p l a s t i c a l l y and are p r e d i c t e d to grow fa s t er h i s tory of s e e d s : e f f e c t of d r y i n g and damage due to s i e v i n g .

75

The explanations given by various authors for the phenomenon of growth dis­persion are summarized in table 2.15.

Basically three effects were mentioned: a) The number or activity of screw-dislocations is responsible for the

variations in growth rate. Hereby it is assumed that the crystal grows according to a spiral growth mechanism.

b) Fragments have a higher specific surface area and will consequently grow faster.

c) Highly curved parts of the crystal will show an enhanced solubility, which lowers the growth rate.

Points b) and c) will not lead to permanent dispersion in growth rate. The crystal will very rapidly obtain its normal growth habit by filling up re­entrant corners and by dissolution and rounding of highly curved parts.

The variation of the number and activity of screw dislocations in small nuclei or crystals is generally held responsible for the variations in growth rate. For small nuclei two conflicting opinions have been presented: On the one hand, Strickland-Constable (1979) explains slow growth of nuclei by assuming that the process of breaking is such that the resulting "crystallites" contain no dislocations. Rusli et al. (1980) additionally suggest that mobile dislocations can leave the crystal. On the other hand, Garside et al. (1984) argue that fragments below a cer­tain critical size, termed the brittle/plastic transition size, would have to be plastically deformed. Therefore the smallest fragments were predicted to contain many dislocations and to have a high growth rate, combined with a negligible growth rate dispersion. Larger fragments formed by brittle frac­turing would contain less dislocations leading to lower growth rates and an appreciable growth rate dispersion. In order to verify their hypothesis the authors performed batch growth experiments with both crushed and uncrushed KCl crystals. The observed growth dispersion was characterized by evaluating the growth rate diffusivity from the increase in the variance of the seed CSD with time. It should be noted that this procedure is only allowed if the dispersion is caused by stochastic growth rate fluctuations. For permanent dispersion the increase in standard deviation of the CSD would be a more ap­propriate measure (Levenspiel, Fitzgerald, 1983; Girolami, Rousseau, 1985. 1986; Zumstein, Rousseau, 1986). Nevertheless it could be concluded that small fragments, 32-38 um and 106-125 um crushed crystals, show an appreciably lower growth rate diffusivity than 106-125 um uncrushed crystals. This observation was considered to sup­port the theoretical model. However, a closer inspection of the experimental results reveals that also the growth rates of the crushed crystals were very much lower than those of the uncrushed crystals, which is in conflict with the model. Moreover, the low growth rates clearly have to be considered as the prime source of the observed low values of the growth rate diffusivity. If it is accepted that: a) breakage introduces many dislocations of both screw and edge type which

are not mobile (standard theories on breakage and comminution) b) growth proceeds either by a spiral growth mechanism or by two-dimensional

nucleation, and, c) secondary nucleation always involves growth of an entity prior to dis-

lodgement of such an entity, then it follows that:

76

- a nucleus formed by breakage will always contain many dislocations, depending on the amount of energy expended (a),

- a nucleus formed by spontaneous dislodgement (e.q. by dissolution of the base of an dendrite) will not change its growth kinetics drastically (b and c).

Therefore it is concluded that there must be another effect explaining the slow growth of fragments. A simple explanation (Daudey, de Jong, 1984) is the increase in solubility due to the energy contained in plastically deformed regions of a crystal. Harbury (1946) has shown that the determina­tion of the interfacial free energy from the solubility of small crystals, see eq. 2.1, will give erroneous values if milled crystals are employed. The energy contained by screw and edge dislocations can be estimated from the elastic deformation of the lattice around the dislocation line (Cabrera, Levine, 1956; van der Hoek et al.,1982; van der Hoek, 1983; Meyer, 1968). Unfortunately, for small particles no values of the dislocation contents were found. Theories on comminution show that the energy expended in the process of breaking is used for approximately 1% in the production of new surface area. The remaining ~ 99% is used by the process of plastic deforma­tion where it is partly stored as elastic energy in the formation of dislocations and partly dissipated as thermal energy in the glide of the lattice along glide planes, containing the (edge-) dislocations. It is evi­dent that the energy stored in the lattice can easily outweigh the interfacial energy. Heavily deformed fragments will dissolve in slightly supersaturated solutions. The "non-growing" fragments observed in contact and collision breeding experiments are either in the process of dissolving, or they have been partly dissolved already which is likely if only the outer zone of the fragment had been deformed. In that case the dissolution will proceed until the deformed fragment is in equilibrium with the supersaturated solution.

Partly damaged fragments will continue to grow on the non damaged faces while the damaged zone is etched away yielding a non-growing "face". Therefore the overall growth rate will remain lower than that of an un­damaged crystal/nucleus, unless, after some time the damaged face is sufficiently overgrown.

The extent of deformation increases with decreasing fragment size (Garside et al., 1984). Therefore it is expected that dissolution of nuclei formed by breakage will be observed in any system below a given size. This size will depend on the energy expended in the comminution to that size and on the su-persaturation in terms of molar Gibbs free energy. According to Harbury (1946) it is possible to use a fictitious value of the interface free energy in the Ostwald-Freundlich equation (2.1), composed of the true interface free energy, and the deformation energy stored in the stress fields caused by the presence of dislocations, expressed per unit of surface area. It is clear that complete dissolution is enhanced by low supersaturations. In fig. 2.11 the growth rate of a small crystal is tentatively sketched as a function of the dislocation content, at a fixed supersaturation. It is seen that the growth rate increases first with increasing dislocation content, goes through a maximum and eventually drops to negative values.

77

growth

t

dissolution

I

^ " " ^

1 lm j

Bf

Ï

enhanced growth

t normal growth

1 reduced growth dislocation content [J/mole]

partial dissolution

t complete dissolution

Fig 2.11 Growth and dissolution of a small crystal as a function of the dis­location content (spiral growth model)

Legend: I: reduced growth due to enhanced perfection II: enhanced growth due to decreased perfection

III: - suppression of growth by increased solubility - non growing zones

IV: partial dissolution yielding non-growing nuclei V: complete dissolution

Depending on the extent of deformation and the supersaturation it is pos­sible to discern four types of fragments:

a) heavily deformed fragments which completely dissolve (V) b) less deformed fragments dissolving partly and yielding non-growing nuclei

(IV) c) partly deformed fragments growing at a lower rate (III) d) partly deformed fragments growing at an enhanced rate due to the intro­

duction of extra dislocations (II).

The last two categories are expected to regain their normal habit by over­growth of the damaged parts but due to the different dislocation structure a kinetic difference will remain. The former two categories will lead to the presence of many non growing c.q. dissolving particles. Therefore we have two types of permanent growth dispersion namely i) dispersion amongst the growing crystals due to the variations in spiral growth activity, and ii) dispersion due to the existence of non growing crystals. The first type of dispersion has been verified for contact nuclei of K-alum, citric acid, KNO-, sucrose, table 2.12 and for large single crys­tals of sucrose, NaC10„, -KC10_. KDP and K-alum table 2.14. Non-growth has been observed for the smaller contact nuclei of K-alum, KpSOj,, (NH^-SCL,

78

MgS0|,.7H_0 citric acid, and for abrasion fragments of NaCl, Penta-erythrol, and KC1.

Since K-alum, sucrose, NaCIO,, KC1CL, and KDP are known to grow according to the spiral growth mechanism, it is likely that the first type of dispersion indeed is a consequence of differences in screw dislocation activity and number. The occurrence of non growing nuclei appears to be a universal phenomenon, independent of the growth mechanism, but clearly connected to extensive mechanical damage.

The relation between the growth phenomena treated in this section and the secondary nucleation mechanisms from the previous sections will be discussed briefly in the next section.

2.3*5 Final conclusions on the secondary nucleation mechanisms

Secondary nucleation starts with a growing crystal, called the parent crys­tal. Depending on the level of supersaturation, type of substance, presence of impurities, hydrodynamics and mechanical damage the various faces of this parent crystal will attain a growth mechanism and rate resulting in a characteristic growth habit. Part of this growth habit is the surface relief. Examples:

- smooth faces - steps, kinks - overhangs, occlusions - dendrites, needles - polycrystalline outgrowth.

It is of course not suggested that the understanding of the growth phenomena is complete. From the surface relief of the parent crystal particles are removed,, .called secondary nuclei. The removal may be effected either by the growth process itself, i.e. spontaneous dislodgement, or by external action such as fluid shear and, more important, by mechanical forces arising from impacts against other solid objects. The growth rate of the resulting nuclei depends on the dislocation content after the removal, which will be considerably enhanced in case of plastic deformation. It appears that easily removable entities are removed without much damage, whereas on the other hand heavily damaged fragments are formed by abrasion of all types of crystals. These fragments may, depending on the level of supersaturation and their dislocation con­tent, dissolve or dissolverpartly yielding non or slowly growing nuclei, or heal to become neatly growing crystals. The differences in .growth rate lead to permanent growth dispersion.

Apart from the surface of the crystal, the solution adjacent to the crystal has been suggested >to yield nuclei through the formation of "clusters" or "embryos's" as a result of ordering forces from lattice of the parent crys­tal. No theoretical nor experimental evidence has been found for this mechanism. For this reason it is not justified to use Catalytic Breeding as a possible secondary nucleation mechanism.

79

Kinetically it is proposed to model secondary nucleation as a rate process consisting of three consecutive steps:

- the formation of the "growth-habit" of the parent crystals - the removal of parts of the parent crystal - the survival and outgrowth of these secondary nuclei in the solution

The proposed mechanisms for secondary nucleation, see sections 2.3.2 and 2.3.3 have been qualified either according to their formation step, examples: needle-, dendritic-, and polycrystalline breeding, or according to their removal step, examples: fluid shear breeding, attrition (micro-and macro-), contact breeding, collision breeding. The supersaturation dependence has been qualified either as survival limited or as (re)generation limited, the latter term being used frequently as the opposite of removal limited. It is proposed here that the prime classification should be done according to the formation step, i.e. the various growth phenomena leading to more fragile crystals. The rate of nucleation will depend on the rate of removal on the one hand and growth rate determined rate of formation of the "growth habit" on the other hand. To this view there is one exception namely the pure abrasion and fragmentation mechanism which will proceed without demanding growth of the parent crystals. It is suggested to use the term "mechanical breeding" for the latter class of mechanisms, whereas the growth determined ones are named "surface breeding". It is now possible to sketch the various possible com­binations between the formation, removal and survival/outgrowth characteristics, see table 2.16. The surface breeding mechanisms may if necessary be subdivided according to the ease of removal, which at the same time implies that the outgrowth characteristics might be different too: mechanical removal may introduce growth dispersion, whereas spontaneous dis-lodgement of needles or dendrites leads to nuclei (needles or dendrites) which are expected to retain the high growth rate which they possessed before their dislodgement. Mechanical removal seems to be necessary to remove less pronounced growth details. In the last column of table 2.16 the mechanisms as proposed in the literature are given.

Table 2.16 Classification of secondary nucleation mechanisms

Class

Surface breeding

Mechanical breeding

Formation

-growth parent of crystals ("growth habit")

-formation = removal

Removal

spontaneous & fluid shear

mechanical

mechanical (+ plastic deformation

Survival/Outgrowth

all survive & normal growth

growth dispersion

dissolution or non-growth or -growth dispersion

"traditional" mechanisms

needle breeding dendritic breeding polycrystalline breeding

contact-breeding collision-breeding

micro-attrition abrasion macro-attrition breakage

80

The classification scheme can now be used as a tool to determine what kind of nucleation mechanism is active. In the next section kinetic models will be developed as a guideline for the interpretation of the experiments. Finally it should be remarked that the present review and discussion of secondary nucleation mechanisms covers the various levels of investigation as discussed before ranging from nucleation in a stirred suspension down to the relation between nucleation and single crystal growth phenomena. The discussion on the secondary nucleation mechanisms did not answer the question why a given growth habit leading to a particular nucleation mechanism shows up:. Further studies on crystal growth, especially directed at the occurence of three-dimensional surface details and their ease of removal and subsequent growth behaviour are highly necessary in order to solve this fundamental question.

2.4 Modelling of secondary nucleation in suspension

In suspension crystallization experiments mostly a wide CSD. of parent crysr tals is responsible for the secondary nucleation. Consequently first the nucleation kinetics for a single parent crystal will be modelled, followed by an integration over all crystals in the CSD. The classification scheme derived in the previous section will be used to model the various depend­encies for the separate process steps within surface and mechanical breeding. The aim of the modelling is to obtain kinetic dependencies of the nucleation rate in suspension on measurable parameters, especially:

- growth .rate (G) - supersaturation (Ac) - s t i r r e r speed (N) - the moments of the CSD,

I t i s proposed to model the. consecut ive s t e p s i n the mechanisms s t a r t i n g with a formation s t e p where removable e n t i t i e s called "proto-nuclei" are genera ted on the p a r e n t c r y s t a l . The fo l lowing s t e p s , a removal and a su rv iva l / ou tg rowth s t e p , w i l l be cha rac t e r i zed by e f f ic ienc ies , since of course only a f r a c t i o n of the p r o t o - n u c l e i i s removed and s u r v i v e s . Mechanical b reed ing w i l l be t r ea t ed in a different way since in t h i s case the separate formation of "proto-nuclei" does hot occur: the fragments a re considered to be produced in the removal s t ep .

Table 2.17 Modelling of single parent crystal nucleation rates

surface breeding

mechanical breeding

formation

bf = F(G),.a

-

removal

b.c= cf.R(L,N).a

= V bf V br(L.N)

survival/outgrowth

: bs. - "s(ic>-br

.VVA-C,'br

81

Three rates are considered, b„,b and b which are the rate at which proto-nuclei are formed, b„, at which nuclei are removed, b and at which removed nuclei survive, b . The survival efficiency, n , is defined as the ratio be­tween b and b , fhe removal efficiency as the ratio between b and b„. Concerning the survival efficiency it should be remarked that if the sur­vived nuclei have a lower growth rate than the average of the CSD they will not be properly detected when the CSD analysis does not cover the size range in which these nuclei are produced. Therefore there might be a discrepancy between the effective nucleation rate and the rate at which surviving nuclei (b ) are formed. It is possible to account for this effect by modifying the definition of the survival efficiency through incorporation of an outgrowth effect, lowering the value of n . This procedure is not necessary if the growth rate including the effect of growth dispersion is properly modelled. In general the survival efficiency is a function of the supersaturation, Ac, in the bulk of the solution, and the extent of plastic deformation, which is connected to the size of the removed nucleus and its source (surface breed­ing or mechanical breeding). The removal function R depends on the crystal size and the stirrer speed N and some geometric factors, which are not considered here. See chapter 7 for a review of abrasion functions. What happens if a proto-nucleus happens to be not removed? There are two possibilities. Either it is overgrown by new layers of crystalline material, or, in case of needles or dendrites it becomes increasingly friable leading to removal sooner or later. Denoting the rate at which proto-nuclei are in­corporated as b. and the concentration per unit of surface area, a, of proto-nuclei as c„ [#/m*] then either b.=0, i.e. no incorporation, or

b±= cf.I(G).a, (2.9)

where 1(G) is the growth rate dependent incorporation frequency per proto-nucleus. Be F(G) the formation function of proto-nuclei then it follows that for non zero b. (equation 2.9 and the equations in table 2.17):

c = IM (2 10) Cf R(L,N) + 1(G) U , 1 U '

and for the removal efficiency i t i s derived t h a t :

V R-TÏ <2 'u> It is seen that for 1=0 the removal efficiency has its maximum value, n =1. In this case removal is not rate-limiting, and

b = n Uc).F(G).a (2.12) s s

Since in t h i s case the nuclei can not be damaged heavily (easy removal) i t i s expected tha t n « 1 , leading to :

bs= F(G).a (2.13)

for formation limited surface breeding.

82

The other extreme is given by R«I, i.e. the proto-nuclei are hard to remove and easily incorporated. It follows (2.11) that:

n = 7 « 1 (2.1*0 r 1

and (2.10) c - - (2 15) cf- I ( G ) U.15;

Consequently (table 2.17):

V 1(G)" * R<L'N>-a <2-l6> and:

bs= ns(Ac). Ulj- . R(L,N).a (2.17) It is reasonable to suppose that the incorporation frequency is proportional to the growth rate.

1(G) ~ G (2.18)

The formation frequency F(G), dimension [#/m2s], is at least proportional to G. Now if in the extreme case F(G) ~ G it is seen that b , equation (2.17) t becomes independent of G. From equation (2.15) it follows that c„ is con­stant in this case, and therefore the nucleation rate is not limited by the formation step:

b ~ n (Ac).R(L,N).a (2.19) s s

i.e. surface breeding limited by removal and survival. Conversely if F(G) is more than proportional with G, say

F(G) " G1+n (2.20)

then i t follows for su r f ace b reed ing l im i t ed by format ion , removal and survival :

b ~ TI (Ac).Gn.R(L,N).a (2.21) s s

For mechanical breeding it is derived that

bg= ns(Ac).br(L,N) (2.22)

The kinetic equations, derived for the various cases, are summarized in table 2.18.

83

Table 2.18 Secondary nucleation k inet ics with various ra te- l imi t ing steps

Mechanism

Surface Breeding

Mechanical breeding

limiting steps

formation

X

X

removal

X

X

X

survival

X

X

X

single crystal kinetics

b =F.a-Gn.a s

b -n ,Gn.R.a s s

b =n .R.a s s

b -n .b s s r

integrated kinetics

• B ~n .Qn (R.a.ndL s s J

o •

B =n R.a.ndL s s J o •

B -n b .n dL s s J r o

The l a s t two mechanisms, su r face breeding l imited by removal and survival y ie ld k ine t i c equations of s imilar shape, (2.19) and (2.22). I t w i l l t h e r e ­f o r e be d i f f i c u l t to d i s c r i m i n a t e between t h e two. Therefore sepa ra t e abrasion experiments in e .g. a non-solvent may be very use fu l (see chapter 7 ) . If a complete parent crys ta l CSD i s to be considered, in tegra t ion should be performed according to :

. - ƒ b .n.dL s (2.23)

where L i s the c rys ta l s i ze , see the l a s t column of t a b l e 2 . 1 8 . I f the ef­f e c t of c r y s t a l - c r y s t a l c o l l i s i o n s i s not important (see chapter 7) • the removal function R can always be wri t ten-as a polynomal in L. The a r e a , a, of a s ingle c rys ta l i s proportional to L , therefore:

R.a.n.dL ~ i 2 2 c..L .L .n.dL = 2 c l i j n'L 'dL = Z c , .u i ' H i + 2 (2.24)

It is seen that the total nucleation rate B can be written as proportional to a linear combination of the moments u.of the CSD. The same arguments can be applied to mechanical breeding.

For simple abrasion b is proportional to L , yielding a linear propor­tionality of B with the fourth to fifth moment of the CSD. In no case a power law dependence in one of the moments is expected.

8k

From table 2.18 and equation (2.24) a general kinetic equation can be ex­tracted which can be used for the evaluation of experimental nucleation kinetics:

General B - Ga. Acb.NC.u. (2.25) s j

where as ah approximation it is assumed that all dependencies can be written as power-law depéncies. If measurement of the supersaturatioh is not possible, it can be written as a function of the growth rate. diffusion limited growth: Ac = G/ka(N) ~ G.N~d (2.26)

G1/r reaction limited growth : Ac = - j — (2.27)

r For d i f fu s ion l i m i t e d growth k , i s known to depend on the s t i r r e r speed: This i n t roduces a nega t i ve s t i r r e r speed c o n t r i b u t i o n t o t h e g e n e r a l equation:

General B - Ga+bNc"du. (2.28) s * j

For reaction l imited growth

B - G a + r .NC .y. (2.29) s K j

Therefore finally one overall simplified equation is found:

Bg - l^.G1.^ (2.30)

It is advised to start the evaluation of the experimental kinetics with either equation (2.25) or, if the supersaturation is hot measured, with equation (2.30), (see chapter 4 and 5 for ammonium sulfate). From .the values of the exponents the type of nucleation mechanism should be judged. Thé cbB-clusive identification of a nucleation mechanism, however, is expected to demand separate experiments in order to unravel the individual process steps by for instance the characteristics of the growth behaviour of thé parent crystals (see chapter 6) and of the nuclei (growth dispersion, see chapter 3) and the abrasion rate in a non solvent (see chapter 7)-

2.5 Conclusions

1. The meta-stable zone

a) True meta-stability, according to the definition of Ostwald (1897),. of supersaturated solutions is due to the fact that the homogeneous nuciea-tion rate is zero over a wide range of supersaturatiohs.

b) The limit of the meta-stable zone can, in general, hot be determined, since primary heterogeneous and secondary nucleation wili prevent thé at­tainment of high supersaturations. An exception is thé precipitation of slighly soluble substances.

c) The pragmatical use óf the meta-stable zone concept to define an area where no nucleation occurs is misleading since this erroneously suggests

85

that outside this zone the solution becomes unstable and meaningful ex­periments are hardly possible.

d) In crystallization experiments with unseeded, clear solutions nucleation starts with one or two primary heterogeneous nuclei which induce the rapid formation of secondary nuclei. Since each new crystal will generate secondary nuclei, an "explosive", autocatalytic increase in crystal num­bers is observed, which should not be considered as primary nucleation, but as secondary nucleation, triggered by primary nucleation.

e) No conclusive evidence has been found for the extensive formation of ag­gregates of crystalline nature in supersaturated solutions.

2. Mechanisms of Secondary Nucleation

a) No evidence has been found for Catalytic Breeding, defined as the en­hancement of homogeneous or heterogeneous nucleation due to the influences of the crystalline phase on the supersaturated solution in contact with that phase.

b) Two sources for secondary nuclei may be discerned, namely, the crystal itself and the growth determined micro-relief of the crystal surface.

c) The removal of secondary nuclei from their source may be effectuated by three causes: spontaneous dislodgement, fluid-shear and mechanical action.

d) The properties of the secondary nuclei depend both on their source and on the removal process by which they are removed.

e) Plastically deformed fragments show a higher solubility than equally sized perfect crystals. In "low" supersaturation systems growth of these fragments will be blocked, in "high" supersaturation systems fragments might exhibit a higher growth rate than more perfect crystals.

3. The classification and modelling of secondary nucleation mechanisms

a) The secondary nucleation mechanisms as proposed in the literature may be subdivided in two groups/mechanisms. - Surface Breeding, source in the "growth-habit" of the surface. - Mechanical Breeding, abrasion of fragments from the "body" of the crys­tal.

b) Each mechanism can be considered as composed of three consecutive steps: Formation of proto-nuclei, in case of surface breeding. Removal from source to the solution. Survival in the solution, combined, if necessary, with their outgrowth.

c) The overall nucleation kinetics may be modelled mathematically by combin­ing the separate equations for the consecutive steps of a particular mechanism.

86

87

CHAPTER 3

DERIVATION OF THE POPULATION BALANCE MATHEMATICS FOR A CMSMPR CRYSTALLIZER, INCLUDING THE PHENOMENON OF GROWTH DISPERSION

3.1.Introduction 3.2 The population balance 3.3 Determination of the average growth rate and the effective nucleation

rate from steady state CSD's 3.4 Determination of the actual nucleation rate from steady state CSD's 3.5 Discussion

3.1 Introduction

The modelling of the nuc l ea t ion and growth k i n e t i c s has been in t roduced s chema t i ca l l y in chapter 1, and elaborated in chapter 2. The d i rec t deriva­t i o n of growth and nuc l ea t i on r a t e s e . g . by o p t i c a l measurements and observation of nucleation events for quant i t a t ive purposes in suspensions of c r y s t a l s i s not feas ib le . Therefore one has to r e l y on the measurement of average q u a n t i t i e s such as the average r a t e s of growth and the effective nucleat ion r a t e . I t w i l l be shown t h a t t he s e average q u a n t i t i e s can be d e r i v e d from the measurement of s teady s t a t e c rys ta l s ize d i s t r ibu t ions , provided tha t the sol ids c lass i f i ca t ion i s characterized independently. For a continuous well mixed c r y s t a l l i z e r (CMSMPR) an extremely simple solu­t ion i s reached (Randolph, Larson, 1971) (section 3-3-3) • The p r o p e r t i e s of t h i s " i d e a l " CSD can be used t o c a l c u l a t e n u c l e a t i o n and growth r a t e s d i r e c t l y from the raw s ieve d a t a wi thout u s i n g g r a p h i c a l methods. The k ine t i c re la t ions derived in chapter 2 can now be tes ted . When under CMSMPR conditions the r e s u l t i n g CSD i s curved when p l o t t e d on semi-logarithmic paper, the in te rpre ta t ion i s no longer straight-forward. In p r inc ip le two causes for th i s curvature are known, namely t rue s i z e depend­ent growth and permanent growth dispersion. The mathematical consequences of the l a t t e r cause are evaluated in order to be able to i n t e rp re t the e x p e r i ­ments with oxim-liquor described in chapter 5-The treatment in th i s chapter i s r e s t r i c t e d to t h e a n a l y s i s of the s teady s t a t e . This i s j u s t i f i e d s ince no dynamic experiments wi l l be t reated in t h i s t h e s i s . The main reason for not doing dynamic exper iments i s not the mathemat ical complexity but the n e c e s s i t y of having a r e l i a b l e on-line method of CSD analys is .

3-2 The population balance

3.2.1 The c rys t a l s ize d i s t r ibu t ion ,

Depending on n u c l e a t i o n r a t e , growth r a t e and c r y s t a l res idence time a suspension c r y s t a l l i z e r wi l l produce a c r y s t a l s i z e d i s t r i b u t i o n , as out ­l i n e d in sec t ion 1.2.2, from which valuable information on the k ine t ics and performance of the c rys t a l l i za t ion process can be extracted. By e . g . s i e v e a n a l y s i s the s i z e d i s t r i b u t i o n can be ob ta ined from the weights of c r y s t a l s , or iginat ing from a known amount of product s l u r r y , be­tween two succe s s ive s ieves . From these weights the number of c rys ta l s can be ca lcu la ted i f the c r y s t a l shape i s known. In o rde r t o normal ize the r e s u l t i n g numbers they a re p resen ted per u n i t amount of product slurry

88

(1 m') and per unit distance between two successive sieves (1 m). The resul­ting quantity, called the population density, n(L) [number/m*], tells us how the total number (concentration) U [#/m* ] is distributed over the crystal size L [m].

n(L) |*/ro*l -shaded area = / ndL = NT

- I [ml Fig. 3-1 Example of a crystal size distribution

Figure 3-1 shows a typical example of a CSD. The integral of n over the size L is the total number concentration:

N T = ƒ n(L)dL o

[#/">»] (3-D

If we define the (undersize) cumulative number as the number of crystals per m3 with sizes below a given value of L:

N (L) = ƒ n(L)dL cumx ' x o

[#/m'] (3-2)

it is easily seen that the population density:

3 N L) ,. . f cumv n(L) = (- 3L [#/m3.m] (3-3)

is the derivative of N (L) towards L, see figure 3-2. cum

89

[*/m3] N cum

a

— L [m]

Fig. 3-2 The p o p u l a t i o n d e n s i t y n i s t h e d e r i v a t i v e of the unders ize cumulative number, N

cum

Additional properties of the crystals are given in table 3-1- All properties are defined per m3 of slurry in order to simplify the calculations for con­tinuous crystallizers, having a constant slurry volume.

Table 3»! The most important distributive and cumulative properties of the crystals per m3 slurry

<property>

crystal number, '#

crystal length, L

2 crystal area, k L

crystal mass, pk L

<property> distribution

n = n

1 = nL

a = nk L a

m = np k L c v

[#/m'.m]

[m/m3.m]

[m*/m3.m]

[kg/m'.m]

cumulative <property>

L N = ƒ ndL cum

0 L

L = ƒ nLdL cum o L 2 A = ƒ n k L dL cum a o L 3 M = ƒ np k L/dL cum c v o

[#/m3]

[m/m3]

[m*/m3]

[kg/m3]

total <property>

N T [#/m']

LT [m/m' ]

A , [m* /m' ]

Hp [kg/m']

The crystal number distribution is mostly referred to as the Crystal Size Distribution (CSD) or the population density (distribution).

dL

90

3.2.2 Definition of the growth r a t e

The choice of the c rys ta l s i ze as parameter to character ize the c rys ta l s was not a r b i t r a r i l y made bu t i s a consequence of one of the most important propert ies of c rys t a l s : t h e i r ab i l i t y to grow when exposed to a supersatured s o l u t i o n . Since the s i e v i n g process i s determined by only one pa r t i cu la r dimension of the c r y s t a l s , t h i s p a r t i c u l a r dimension i s commonly used t o characterize both the c ry s t a l s i ze , L = L . , and the growth r a t e G:

sieve ° G = [m/s] (3-4)

as the increase of sieve-size with time.

3.2.3 Derivation of the population balance

A general derivation of the population balance will be given for a non-ideally mixed crystallizer. The principle of conservation of numbers will be applied on an interval between L and L + AL of the size axis. Since both the supersaturation and crystals are not well distributed, the values of the population density, n(L) and the growth rate G(L) locally deviate from their average values. Moreover, the local value of the growth rate may even vary from crystal to crystal. This phenomenon, known,as growth rate dispersion or growth dispersion, will be discussed in greater detail in section 3.4. Locally the growth rate of equally sized crystals will be averaged in such a way that the number flux F, caused by the growth along the size axis, is equal to:

F = n . G [#/m»s] (3-5)

It will be shown later that G is the number-averaged growth rate, where the "-" means number-averaging. The growth flux, F, equation (3-5), is a also local quantity. The total crystal number in the crystallizer between L and L + AL is found by integration:

total number {L, L+AL} : ƒ nAL dV (3-6) V

which is set equal to (n.AL.V). Consequently the average population density n, defined by:

ƒ n dV n = —v (3-7)

i s the volume-averaged population density. Integrat ion of the local growth flux, F, over the volume l e a d s to the t o t a l f l ux , which a l so equa l s the product of average flux and volume:

d p f -ƒ F.dV = ƒ n G dV === F.V (3-8)

V V It is seen that the average flux F,

F = J- / F dV (3-9) V

91

i s the volume-averaged f l ux . I t i s now demanded t h a t , analogous to eq. (3-5) t t h i s average flux i s the product of the volume-averaged popu la t ion d e n s i t y n and .an averaged value of G over the c r y s t a l l i z e r , which sha l l be designated by <G>:

F i iC n <G> (3-10)

Subst i tu t ion of equations (3-7) and (3-9) shows that <G>,

ƒ n.G dV

/ n dV <5> = I = V (3>11)

n V is obtained by averaging of G over all crystals in the crystallizer. A second interpretation of <G> is found by considering the average closed pathway of the crystals of size L, through the crystallizer. This closed pathway is chosen in such a way that the whole volume of the crystallizer is covered once, i.e. the pathway fills the whole crystallizer and never inter­sects with itself. All crystals of the considered size travel within this pathway. This implies that the numberflux through any cross section of this special average pathway is constant:

flux within pathway = v.. . x n x cross section = -pjr x n (3-12)

The average growth rate, equation (3.11) may be written now as follows:

ƒ C n G dt ƒ C G dt 1_ t . <5> = ~t ^ ~ = ~t = 'e o' C G d t <3'13)

, c dV ,. , c ,. ƒ n -j- dt I dt o dt o where t is the time needed to travel along the total pathway. Clearly, <G> can also be obtained by averaging G in time along a closed pathway covering the whole volume of the crystallizing slurry. For simple configurations the pathway may be a simple loop, but in general a rather complicated loop may be expected. The use of the time-averaged growth rate <G> is justified as long as the characteristic time t needed for the averaging is much smaller than other characteristic times, such as the mean residence time of the crystals in the crystallizer. The accumulation of crystals in the specified interval {L, L+AL} for the whole crystallizer is given by equation (3.6):

accumulation = (8 ' ° ^ V ' ) L = ( M J ^ ) L . AL (3-lt) There are two input terms, the flux of crystals growing into_the interval at the lower boundary L and a size-dependent "birth" function, b (L) , which is also averaged over the volume:

i n p u t = F(L) V . + b A L V (3-15)

92

Likewise the output has one term representing the flux of crystals growing out of the interval at L + AL; a size-dependent "death"-term, d(L), and ad­ditionally, a term giving the withdrawal of crystals as a product slurry:

output = F(L+AL) V + d AL V + n AL O (3-16) where 4> and n are the volumetric flowrate of, and the population density in the product slurry respectively. The population balance is found by equating the accumulation to the dif­ference between input and output see figure 3>3. and division by (ALV):

V l 3t'L n EM + K - F(L * AL) d - -£

AL D AL T ,3m - d In V - lim ,F(L + AL) - F(Lh ,r •=. WL + n "dT" = AL+O H AL ^ ♦ (b - d) - -

n _£.

tBTi\ idF\ /C j\ _P d In V y L

= (aT)t + (b _ d ) T ■ " " d t -

Introducing equation 3-10 finally leads to:

y L= - 1-aiHt + (b " d) " T " n "dT" {3'17)

In this general population balance the product withdrawal term can be sub­stituted after having defined a crystal mean residence time T (L) as:

s def total number between L and L + AL

s total number rate, between L and L + AL leaving the crystallizer

n AL V n ._ 1Q. T = —r-j—Z~ - ~ • T (3.18) s n AL * n \J i

P v p The ratio between the solids and slurry mean residence time is the clas­sification parameter A(L), as defined by Asselbergs (1978):

A(L) = T s ( L ) = "- (3.19) T p

Further it can be seen that: n n f-=— (3-20)

s In the next sections it will be shown how the growth rate and the number fluxes along the size-axis may be derived from steady state CSD's, employing the population balance, equation (3-17)•

93

FIU.V

accumulation <^-V), AL 1 öt L

growth flux in j . ( growth flux out

-j» - f»- F IUAU.V ■ total number |

I S.AL.V '

i ; i

i

F =fi.<G>

birth BAIV

, U A L

death dALV

product with drawal

np Aüfvp

Fig- 3-3 Total number fluxes in [s ] entering or leaving the considered size interval in a non ideally-mixed crystallizer

3-3 Determination of the average growth rate and the effective nucleation rate from steady state CSD's

3.3-1 The steady state; general

The general population balance for a poorly mixed crystallizer, equation (3-17) which incorporates both internal and external classification will be used to obtain nucleation and growth rate. First some assumptions will be made:

(A.l) : Birth and Death rates negligible

(A.2) : Constant slurry volume

Consequently, equations (3.17). (3.20):

13m /3n <G>% n l 3 t J L l 3L J t " T (3-21)

Most crystallization processes obtain a steady state after operation for 6 to 12 residence times without disturbancies. The properties of the steady state CSD can be derived from equation (3*21) assuming:

(A.3) : Steady State

94

We can write equation (3-21) as follows:

Steady State population balance;

d n <G> n_ ,_ „_. dL = " x \S'**)

s With equation (3-19) the steady state population balance can be written as:

d(A n <G>) n ,_ _oN P = - _P_ (3-23) dL T

Integration over L between the bounds L=L and L=<°, realizing that n =0 at L=«, gives:

• . ~ . • ov r d (A n <G>) „ - N°

-£ dL = - A n <G> = - - n dL = - -B-J dL p T J p T L L

i P <G> A = -* (3-24)

n In this way the product <G>A can be obtained directly from any steady state product CSD. In order to calculate the growth rate <G>, which of course may be a function of the crystal size, the classification parameter A(L) must be determined separately. This can be done by determining the CSD of the total contents of the crys-tallizer, see equations (3.18), (3.19). For those cases where the determination of N is not accurate or impos­sible, for instance if the CSD is seen to be not yet stable for the larger sizes, another equation was derived:

<G> A = J I (-l)j ^ - (3-25) j=l (dlnn p) J

which can be applied on the steady state part of the CSD. The derivation is given in appendix 3c. In order to obtain the nucleation rate, the size analysis has to be extended to very low sizes, which is not possible by sieving. The derivation of the nucleation rate is given in section 3.4 dealing with these problems.

3.3.2. The steady state: CMSMPR conditions

The separate determination of A (L) which demands an analysis of the total crystallizer contents can be avoided if the slurry is well mixed and the product population density equals the average population density in the crystallizer:

95

equation (3-19) : n = n = n } Ap = 1 > (A.4): CMSMPR T = T

A well mixed slurry can easily be obtained in small scale crystallizers, but special precautions must be taken to ensure that n = n (Becker, 1970; Janse, 1977). For large crystallizers the mixing will seldom be adequate. Consequently slurry samples from the total volume of the crystallizer should be taken. Crystallizers operating under assumption A.4 are mostly referred to as Continuous Mixed Suspension Mixed Product Removal (CMSMPR) crystallizers. With assumption (A.4) and equations (3-24) and (3• 25)( the size-dependent growth rate can be determined. The nucleation rate is treated in section (3.4). The time dependent population balance, equation (3-21), can be simplified by inserting assumption (A.4), to:

SHE* lg)L-- l ^ t - f O-*) which can be employed for the dynamics of a CMSMPR crystallizer.

3.3'3- Steady state; CMSMPR conditions and size-independent average growth

3-3.3-1 The "ideal" CSD

For the steady s t a t e , assumption (A.3), equation (3-26) can be further simplified to:

CMSMPR ^ 7 ^ ^ = " - (3-27) ::— dL T steady state The analytical solution of this equation may be obtained if <G(L)> is known. A very special case often applies:

(A.5): the growth rate is independent of size

Introduction of (A.5) in the CMSMPR - steady state population balance and integration from L=0 to L=L gives the steady state crystal size distribution:

In n = In n° - — — (3-28a) <G>T

n = n° exp ( — ) (3-28b) <G>x

A plot of the logarithm of the population density (equation 3.28a) versus the crystal size yields a straight line, see figure 3.4.

96

In n

1 y-\n n°

-1

^ * ^ ^ V <G>X N.

Fig. 3«^ Steady State CSD under CMSMPR conditions with size-independent average growth rate

3-3-3~2 Determination of the average growth rate

In many cases sieve analysis is used to determine the crystal size distribution. If, within the sieve range a straight line is obtained when the natural logarithm of the population density is plotted versus the crys­tal size, it can be concluded that the growth rate does not depend on size, at least for the observed size range. The slope of the straight line, given by equation (3.28a), can be used to evaluate the growth rate. If it is assumed that this same growth rate prevails in the sub-sieve range, the experimental straight line may be extrapolated to L=0, see figure 3-A, to obtain the population density at zero size, n . Strictly spoken, however, zero sized crystals do not exist and the size-independent crystal growth assumption will not be valid for crystals having a near critical size.

3-3-3-3 Determination of the effective nucleation rate

The numberflux n <G> of crystals of zero size along the size axis equals the number of crystals which enters the crystallizer per second_and per unit of volume. This numberflux will be called the nucleation rate, B :

B° = n° <G>° (#/m3s) (3-29) ,:vo Under the given assumptions <G>" equals <G>. Since in most cases the assump­

tion of size-independent growth in the subsieve range is not experimentally verified or, when tested, has been shown not'to be correct, the value of B obtained by extrapolation has to be considered as a practical or effective nucleation rate, B

R° Beff eff

eff" <G> (3-30)

97

The differences between the true nucleation rate given by equation (3«29) and the effective rate (3-30) must be accounted for in the nucleation model. This subject will further be discussed in section 3 «4.

3.3.3-4 Properties of the "ideal" CSD

The "ideal CSD", given by equation (3.28), has properties which can easily be calculated. One important group of properties is formed by the moments of the distribution:

j=0,1.2. u. = ƒ n IAL o

substitution of (3-28) and repeated partial integration leads to:

u. = j ! n° (Gx)j+1 J

(3-31)

(3-32) The slurry density VL,, defined as the mass of the crystals per unit volum^ of slurry, is obtained by integration of the mass of a single crystal, pk L~ over all crystals:

Up - J n (pkvL3)dL = pckvu3 o

With equation (3-32): M = 6p k n°(G-c) (3-33) The total number, length, area and the fourth moment of the CSD were ob­tained in this way.

Table 3.2 Moments of the distribution

Quantity

NT total number

LT total length

A, total area

MT total mass

Uj. fourth moment

moment

^0 ul V2 Pckvu3 u4

equation

NT= n"Gx

4= n°(Gx)2

/ y 2kan°(GT)3

M ^ 6cpkvn°(GT)4

u4= 24 n°(Gx)5

eq. (3.34) (3-35) (3.36) (3-33) (3.37)

Two other quantities which are frequently used to characterize the CSD are L,.-., the median size of the weight distribution, i.e. the crystal size below wnich 50% of the sample weight is found in the analysis,

L50 = 3'67 GT (3.38)

98

and C.V., the coefficient of variation, which has the value of 50% for the ideal CSD:

CV. =L84~ L16 x 100% (3-39) 2L50

Product specifications are most often expressed in terms of L_n, CV. and percentage fines or granular. -

3.3.3-5 Numerical procedure for the calculation of the average growth rate and the effective nucleation rate

Using _the properties in the previous section the size independent growth rate <G> can be calculated from:

- ka P <G> -öf-jT • r~ (3.40)

iPc v A ,

where P/A,-, represents the crystal production P per unit of surface area A„,.

In steady state the average production is given by: P=ÜTiP_ (3.41).

steady state A detailed derivation of equation (3-40) can be found in appendix 3a. The effective nucleation rate can be calculated after substitution of (3«30) in (3-33), since for the ideal CSD, equation (3.33), n°= n° f f,

R° eff 1 with: r ^ * i^-jr (3.42) T,p 6 k p <G> 3T H

V C The above set of equations facilitates the calculation of <G> and B _„ without graphical procedures or linear regression analysis. Equation (3-40) is valid also for non steady state conditions, including batch opera­tion, if only the crystallizer is well mixed and the growth rate is size-independent, however, in case of external classification the value of A_ from the crystallizer and not from the product should be employed. The production of crystals for non-stationary operation can be given by:

d "T *T n P = - d T + x (3.43)

where M_ is the product slurry density. In apperiaix 3a it is shown that if eq. (3.40) is evaluated for cases where the growth rate is not size-independent a value of <G> is obtained which can be considered as a crystal surface area averaged value. This average growth rate, G , may be combined with eq. (3-42) to give an approximate value of B __ for the steady state CSD. In this way a reliable method is obtained to calculate average growth and average effective nucleation rates for any steady state distribution. The

99

advantage for industrial practice is that overall empirical kinetic equa­tions can be derived for a crystallization process without need to model size-dependent growth rates or to measure the solids classification.

3.3.3-6 Determination of the empirical kinetics from steady state CMSMPR experiments

The determination of the nucleation kinetics can be done by regression analysis using the kinetic relations for nucleation as suggested in Chapter 2. Over a limited range of measured parameters a generalized power law will apply, see equation (2.30):

B eff k N*1 n G 1 ^ (3-44) From the values of the fitting parameters, h,i and j, the nucleation mechanism can be determined, see chapter 2. The value of j, however can not be determined from steady state experiments alone and this will hinder the direct evaluation of the other parameters in equation (3-44). If for instance the third moment, j=3, is used in equation (3.44) substitu­tion of equation (3.32) in equation (3.44) leads to:

and: Beff = n ° G = 6 k- ^G±n° (Q T ) 4

n T = (6 k N G v n

M i +3)i-F )"4

(3-45)

(3-46)

When equation (3-46) is introduced in equation (3.32) for other values of j, the following results are obtained:

■j - 2 B;

j = 4 B' eff o eff=

3 ^ 3/4 M3/4 „(31+D/4 N U-I k 5/4 N5/4L r(5i-D/4 4 N G ^4

(3-47) (3-48)

Therefore a linear dependence of B „» on one of the moments does, actually, not prove that the proper moment was chosen, since equations (3.44), (3-47) and (3.48) are equivalent in the "ideal" steady state. However, it is ob­served that the exponents of N and G vary with the value of j , see table 3-3. Table 3»3 The effect of the choice of the

power law .th moment on the generalized

moment

^2 u3 "4

nucleation constant

*T "N 4"

exponent of N

3/4.h h

5/4h

exponent of G

i-(i-D/4) i

i+U-D/4)

value of j

2 3 4

100

It is, as shown above, impossible to determine the correct values of the ex­ponents in equation (3-^) if j is unknown. In the steady state experiments the data will be plotted first using j=3, and if the nucleation mechanism can be inferred from this fit or from other considerations, it is possible to choose the correct value of j and to calculate the true exponents using table 3.3.

Since the knowledge of the proper value of j and the exponents of G and N is very important for both the interpretation and the dynamics of the nuclea­tion and crystallization kinetics, it is suggested to develop methods to analyse the nucleation and growth rate for non-steady state experiments.

For the evaluation of the exponents of equation 3-^t it is advantageous to use a log-log plot of B p^/M™, where VL, is essentially proportional to the third moment, versus the growth rate, see figure. 3«5- Equation (3.^2) yields lines with a slope of -3, for fixed values of the residence time, T. From this graph it is seen that, selecting a value of the growth rate and a residence time fixes the value of B __/MT. The nucleation kinetics for a given stirrer speed are represented by a straight line of slope + i. It is immediately seen that for the determination of this line it is necessary to vary the residence time. The stirrer speed dependence is evaluated by plot­ting log (B „„/Up) vs. log N at constant growth rate. Table 3^3 can be used to derive the equivalent equations (in steady state) for the other moments.

log Deff MT

log G

Fig. 3.5 Kinetic p lo t used in the determination of "power-law nucleation kinetics

101

3.4 Determination of the actual nucleation rate from steady state CSD's

3.^.1 Growth models for the extrapolation in the sub-sieve range

The determination of the actual nucleation rate from an "ideal" crystal size distribution obtained by sieve analysis would be very simple if the averaged growth rate of the crystals were size-independent for the sub-sieve range too. But more accurate measurements have shown that the "ideal" steady state CSD, plotted on semi-log paper, invariably shows a pronounced upward curva­ture at small sizes, which indicates lower growth rates for the smallest crystals (Randolph, Sikdar ,1976). This implies that the actual nucleation rate is higher than the effective one which is obtained from the straight semi-log CSD extrapolated to L=0. Although the use of this effective nuclea­tion rate, B „„, in a kinetic equation is pragmatically correct, it does not lead to a clear insight in the nucleation mechanism. The ratio between the actual and the effective nucleation rate remains a function of the unknown small-particle growth behaviour. A second drawback of the use of effective nucleation kinetics is that the incorrectly assumed size-independent growth in the sub-sieve range leads to deviations between the actual dynamic be­haviour and the behaviour based on the assumed size independency of the growth rate in the sub sieve range.

In n in n°?

size-dependent <G>

size-independent <G> Fig. 3.6 Behaviour of CSD

in the small-size range

The situation becomes even more complex when the semi-logarithmic CSD (see fig. 3-6) is _curved over the full size range. In this case the average growth rate <G> can be shown to depend on the crystal size over a major part of the sieve_range. Here a correct extrapolation to L=0 in order to derive B or even B _„ is not possible. Instead of B __ the use of the number flux at 100 jam in the nucleation kinetics has been suggested JJLanciè, 19-82) but the same objections can be used against the use of this B as against B __. When direct measurements in the small size range are impossible a correct physical model for the growth of the smallest crystals is needed to perform the extrapolation towards L=0.

Two growth models exist that explain the curvature observed in the semi-log plot of the CSD, see chapter 2:

102

a) True s ize dependent growth: The growth r a t e i s a function of s ize for a l l c ry s t a l s , t rue meaning t h a t e q u a l l y s i z ed c r y s t a l s e x h i b i t i d e n t i c a l growth r a t e s .

b) Growth dispersion: In t h i s case the c r y s t a l s of a given s i z e a l l grow with d i f f e r e n t r a t e s , which may lead to an apparent size-dependency in the average growth ra te <G(L)>.

The mathematical consequences of these models wi l l be t reated in the next sec t ions .

3.4.2 True size-dependent growth

Various empirical expressions have been published in o rde r to d e s c r i b e the s i z e dependent growth concluded from the curved semi-logarithmic steady s t a t e CSD's, see chapter 2, table 2 . 8 . However no sa t i s fac tory phys ica l ex­p l a n a t i o n fo r the i n c r e a s e of growth r a t e with s ize has been found. Since growth can be considered as a two s tep process ( see chap te r 6) e i t h e r the mass t r a n s f e r in the solut ion adjacent to the c rys ta l surface or the incor­poration react ion at the surface of the c r y s t a l s o r both have to be s i z e -dependent to explain t h i s phenomenon. The mass t ransfer c o e f f i c i e n t for the d i f f u s i o n in the s o l u t i o n , k , , i s modelled with a term a/L representing the s ta t ionary diffusion in a stagnant medium and a forced convection term b(N), due to the terminal s l i p v e l o c i t y enhanced by t h e l i q u i d t u r b u l e n c e , caused by the i m p e l l e r ( impe l l e r speed:N):

V f + b(N) (3.49) The slight size-dependence of b(N) for the largest crystal size has been neglected, see appendix 6b, eq.11.

From eq. (3-49) it is seen that if the growth is controlled by diffusion, the growth rate will increase towards smaller crystal sizes which is in con­flict with the observed dependence. This implies that the slow growth of the smallest crystals is not caused by diffusion in the solution, which means that the cause of the slow growth must lie in the crystalline phase itself. But even if the growth rate were completely diffusion controlled, it can be shown that the ratio between the actual nucleation rate B° and the effective one calculated with eq. (3.42), Beff' i s v i r t u a l l v constant, independent of the stirrerspeed, supersatura-tion and growth rate, see appendix 3b, which implies that diffusion limited growth will not affect the functional form of the nucleation kinetics derived from steady state kinetics. However, the dynamics of the crystal­lization may be affected since the nuclei grow out sooner in case of diffusion limited growth than when size-independent growth is assumed.

Apart from size-dependent growth caused by diffusion two other causes for size-dependent growth exist, see chapter two: a) change of the dislocation density with time, which is responsible for the formation of growth steps, and b) the Ostwald-Freundlich effect leading to an increased solubility of small particles. At this stage no further modelling of true size-dependent growth is pursued, since no significant improvement for the extrapolation to L*0 is expected.

103

3.4.3 Growth dispersion

3.4.3-1 Introduction Growth dispersion is the phenomenon that crystals of equal size exhibit dif­ferent growth rates. In the modelling of growth dispersion we have to consider two extremes. First, the dispersion may be caused by random fluc­tuations of the growth rate around a mean value. This is called the Random Fluctuation (RF) theory or Stochastic Dispersion. The other extreme is that the growth rate of individuals is constant. The dispersion in rates in­itially present amongst the nuclei will persist at larger sizes. This model will be called the Permanent Dispersion model. The differences between the two models are illustrated in fig. 3 «7.

True size-dependent G ö=eaii (D

Permanent dispersion 0=5 (L)

Stochastic dispersion G*G (L)

Fig. 3.7 Potential sources of size-dependent average growth rates

3.4.3-2 Stochastic dispersion

Stochastic dispersion has been modelled by Randolph and White (1977), for both batch and continuous operation, by introducing a growth diffusivity. The stochastic growth rate fluctuations are thought to have the same effect on the CSD as diffusion has on a concentration gradient. However, for steady state continuous operation they showed that stochastic fluctuations do not have any effect on the shape of the distribution. Therefore this effect cannot be evaluated from steady state CMSMPR distribu­tions, see appendix 3d. In batch operation, especially when narrow seed

Modelling of G

G(h o-

G(r 2 )

O *" O-

o—»- o-L=0

G(h) o • o-0 - 0 » o—»-—o—•» o-*----o-*-

G(t2)

L=0 ^ L G(f-|) G(t2

O* o--o—•-

o»-

L=0

104

fractions are employed, stochastic dispersion will cause peak broadening, but it has to be discriminated from permanent dispersion which will also cause peak broadening. It is expected that for stochastic dispersion the standard deviation of the distribution will increase with the square root of the size increase, whereas for permanent dispersion a linear increase is expected.' This last effect is analogeous to axial dispersion in liquid-liquid extraction columns where permanent dispersion is caused by the fact that larger drops have a higher velocity than small drops. Stochastic dis­persion in these columns can be caused by either backmixing or coalesence redispersion phenomena. In order to characterize the type of the dispersion Levenspiel and Fitzgerald (1983) pointed out that in doing pulse-response experiments the increase of the pulse-width must be determined as a function of the length of the column, which demands at least a three-point measurement. Since no such measurements on crystal growth were found in literature and since steady state CSD's are not influenced by stochastic dispersion the effect is not yet experimentally proven for suspension crys­tallisation. See futher the discussion in section 2.3-4.

3.4.3~3 Permanent dispersion

Permanent dispersion was first suggested by Janse and de Jong (1976). The underlying assumption of constant, size-independent individual growth rates was confirmed by the work of Berglund and Larson (1984) for nuclei formed by contacting parent crystals with a rod. Permanent dispersion leads to an increase in the average growth rate with size, see section 3>4.3~4, fig. 3.8, and never to a decrease. This is in accordance with the experimental observations, which indicate that permanent dispersion is far more important than true size-independent growth or stochastic dispersion. Two possible explanations for permanent growth dispersion were found, as discussed in chapter two:

a) a variation of growth spiral activity and number per crystal b) a variation of the solubility of equally sized crystals due to plas­

tic deformation (Daudey and de Jong, 1984a). In the first case no very large spread in growth rates is expected. In the second case it is probable that the crystal or fragment will heal after some time and consequently the growth rate will gradually increase upon outgrowth of the fragments. The term "permanent dispersion" allows for this non-stochastic variation in growth rate. In both cases it is evident that both the type and number of dislocations present in the crystal are important. This connects the growth behaviour of the nuclei to the mechanism by which the crystals or nuclei are formed, see chapter two. The analysis of the growth behaviour of the smallest crystal may therefore help in determining the nucleation mechanism(s) by which these crystals are formed.

It is clear that permanent dispersion is not caused by the diffusion process, but either by dispersion in the incorporation process (or surface reaction) or dispersion in solubility of individual nuclei or fragments. Both causes for permanent dispersion can be modelled by assuming dispersion in k , the mass transfer coefficient of the incorporation reaction. The dis­persion in growth rates can be calculated from k and k,, eq. (3.49), leading to an overall mass transfer coefficient k . The dispersion in k is bounded by a maximum value given by k,. A consequence is that the increase in average growth rate with size resulting from permanent dispersion in k will level off at larger sizes, where the growth process is diffusion con­trolled. Here the dispersion in k is masked by the dominance of k,. Since

105

k, i s only weakly size-dependent for the larger c r y s t a l s , diffusion l imi t ed growth r e s u l t s i n s i ze - independen t growth r a t e s . Therefore the so-called "AL-law" (McCabe, 1929) s ta t ing that the growth r a t e i s independent of the c r y s t a l s i z e i s , fo r d i f f u s i o n - l i m i t e d growth, a r e s u l t of the s i z e -independence of k , for larger c rys ta l s (say > lOOu). Comparing permanent d i s p e r s i o n with s t o c h a s t i c dispersion and true s ize-dependent growth l e a d s to the conc lus ion t h a t a l l t h r e e e f f e c t s can in p r i n c i p l e occur , and t h a t n e i t h e r of them w i l l be encountered in a pure form. For the analysis of steady s t a t e CSD's s tochas t ic dispersion cannot be used since i t does not influence the " ideal" CSD. Consequently the size-dependent average growth r a t e s wi l l have to be i n t e r ­preted in terms of permanent dispersion and size-dependent growth. Therefore in the next sect ion the population ba lance w i l l be de r ived for permanent d i s p e r s i o n , i n c l u d i n g the effect of size-dependent growth of the c rys ta l s . In the analysis of "curved" semi- logar i tmic s t eady s t a t e CSD's,permanent d i s p e r s i o n and " t r u e " size-dependency wi l l have to be tes ted separately in order to see which model gives the best r e s u l t s . I t i s expected that separa­t i o n of the t h r e e growth phenomena i s only possible when dynamic, i . e . non steady s t a t e , experiments are performed (Girolami, Rousseau, 1985; 1986). In the next section permanent dispersion wi l l be modelled.

3 .^ .3"^ The population balance for permanent dispersion

Permanent dispersion i s most eas i ly visual ized by dividing the nucle i , with a population density n , in m f rac t ions . An. , each fract ion having i t s own growth r a t e G.:

n°= I An,° (3.50) i = l 1

Since G. is constant in the permanent dispersion model if the diffusion is not rate-limiting, the "ideal" CSD is found for each fraction:

An±= An±° exp(- ^ ) (3-5D i

The total population density at any size is obtained by summing An. over all fractions:

n n = X An (3.52)

i=l I t i s c l e a r tha t the dispersion can be described by a l i s t of corresponding (G.and An. values) . Fig. 3-8 shows the steady s t a t e d i s t r ibu t ions for th ree growth f r a c t i o n s accord ing to eq . 3-51 and t h e r e s u l t i n g , curved CSD on semi-log scale (eq.3-52).

This "d iscre te" dispersion approach can be replaced by a continuous model by dividing An. by AG. and taking the l i m i t for AG. ■» 0 (Janse and de Jong, 1976): x X x

f(L.Q) - (fg) (3-53)

. 106

In n

Fig. 3.8 Curved steady state CMSMPR CSD modelled with discrete dispersion using three growth fractions

The function f was called the "modified population density" (Janse, 1977) since it is a bivariate distribution of the crystal number over the crystal s i z e and over the growth rate. In the following the term Growth Rate Distribution (GRD) will be employed. The population density is given by:

■ - J f dG (3-54)

and the number flux by: + 00

F = I f G dG I (3-55)

The integration limits {-•, +»} were taken instead of {0, +»}, (Janse, 1977) in order to accompdate negative growth rates, i.e. dissolution, too. Introduction of (3-5*0 and (3-55) in (3-5) shows that:

G =

+m ƒ f G dG -a + » ƒ f dG

(3.56)

The average growth rate G is clearly number averaged. In order to derive a general population balance incorporating growth disper­sion an averaged value of f is used given by:

107

la<G>JL,

leading to :

(3.57)

= ■ ƒ f d<G> (3-58)

where <G> i s the time-averaged growth ra te of a c ry s t a l . AL

birth accumulation ÓÏALA<G>V

(_ÖL_) ALA<G>V d<G»l,t

di 'Mi .

A<G>

withdrawal (fpALA<G>S>¥)

death

dAG 'L.t Fig. 3.9 The derivation of the population balance for growth dispersion

Now the average flux F can be expressed as:

I f <G> d<G> (3-59)

From eq. (3-10) the average growth rate is derived:

<G> = ƒ f <G> d<G> — <D

+ 00

ƒ f d<G> (3-60)

which is seen to be the number averaged value of <G> , which was already time-averaged. The population balance can be derived by considering a region in the L,<G> space bounded by (L, L + AL) and (<G>, <G>+A<G>), see fig. 3.9. The various terms are:

accumulation = (^ (f AL A<G> V ) ) L | < Q > - AL A<G> ( f f \ < Q > (3.61)

108

ab input = (flux 1)A<G> V +(flux 3)AL V + (ffö>)L fcAL A<G> V (3-62)

output = (flux 2)A<G> V +(flux 4)AL V + (ffö>)L fcA<G> V +.f AL AG ♦ (3-63)

Equating the accumulation to input minus output and division by (AL A<G> V) yields:

f if. 1 ? d In V ,flux 1-flux 2, .flux 3-flux 4 . f3(b-dh p_ l3t JL,<G>+ dt =( AL ;+l A<G> ' + l 3<G> JL,t" T

(3-64) Due to the size dependency of the growth rate the fluxes in the L and G direction are coupled by (3<G>/3L) . Consequently:

[»<a>i - - f fÈL\ f3f<G>x f3f<G> l 3L ' ] filbzd]_1 _p_ T. d In V l3t'L,<G>~ " l 3L J<G>,t " l 3<G> J + l 3<G> J L , t T d t

(3.65)

After evaluation of the pa r t i a l differentials and some rearrangements the final number balance is obtained:

(f|)L.<G>= -<°> lf&<G>.t " tlife)t.t<a> * « I*^t ■ ■ ■ - ' <°> fe I ^ A . t * l ^ ë V t - i - ? 4 ^ <3.66,

T This form of the number balance allows for:

- maldistribution of solids and supersaturation - permanent growth dispersion - birth of nuclei having a specified size and growth rate. - classification - negative growth rates, i.e. dissolution - volume variations.

The only assumption which had to be made to derive (3-65) and (3.61) is that all crystals at a specified size and growth rate have the same value of (3<G>/3L) , i.e. they travel along the same trajectory in the L-G space. This assumption is fulfilled if there is only one single property which causes equally sized crystals to grow at different rates, for example the activity of a spiral growth mechanism or the shape of the crystal, or the bulk dislocation density due to plastic deformation.

3»4.3~5 Thé steady state solution for permanent dispersion assuming size-independent growth

The consequences of this permanent dispersion model will be analyzed under Steady State CMSMPR conditions, assuming no volume changes and no size-dependent birth and death rates (assumptions A.l-A.4). Further the growth rate is assumed to be independent of size for the separate growth fractions, assumption A.6:

109

(A.6): Growth rate independent of size for separate growth fractions

Under these assumptions eq. (3-66) simplifies to:

<G> (|f)<G>= with as solution:

f_ T

f = f° exp (- J-)

(3-67)

(3-68)

If f , the initial growth rate distribution is known, the steady state dis­tribution can be calculated with eq. (3.68). This is illustrated by fig.(3>10) where In f is plotted as a function of <G> and L. It is seen that the initial growth rate distribution is modified com­pletely due to growth and withdrawal, the maximum and also the average growth rate shifting to higher values of <G>.

initial growth rate distribution. In f

Fig, 3.10 Change of the growth rate distribution with crystal size

3-^.3~6 Determination of the growth rate distribution and nucleation rate

The nucleation rate equals the flux at zero size:

= l f ° < B~ = I f~<G>d<G> o

(3.69)

no

From the above_equations it is clear that the CSD in steady state can be calculated if f°is known and assumption A6 holds.. Eq. (3«69) Implies that a nucleation mechanism may produce nuclei with, a broad spectre of growth rates.

In the analysis of the initial growth rate distribution from steady state CSD's three approaches are possible. First we may use the discrete model eqs. (3-50)-(3.52) to fit the CSD. It can be expected that only a limited number of growth classes can be used since each extra class brings in one extra parameter assuming that the growth rates have fixed values.

The second approach is to test specific continuous dispersion functions. An example is the "inverse gamma function" (Janse and de Jong, 1976):

o_r-l f°= V(r-1) G"rexp (-Q/G) (3.70)

The d i s t r ibu t ion function goes to zero for both G=0 and G=«» and has the ad­v a n t a g e t h a t t h e i n t e g r a l s (3-5*0 and (3-55) and the moments may be eva lua ted a n a l y t i c a l l y . The d isadvantage i s t h a t i t h a s no p h y s i c a l background. The sharp i n c r e a s e of the curvature in the CSD at small s izes demands a d i s t r ibu t ion function which has i t s maximum a t zero growth r a t e , unlike eq. (3.70).

The average growth ra te i s evaluated in appendix 3e, t o g e t h e r with the mo­ments, nucleation r a t e , CSD e t c :

G(L) = ^ § " 2 (1 + ±~) (3.7D

It is seen that G has a finite value at L=0 and increases linearly with size. The parameters Q and r may be obtained by fitting <G(L)>, eq. (3-25) . versus L, for A=l.

The third approach is to employ general continuous dispersion functions. As an example a polynomial in (G /G - 1) is suggested:

fo ip " (%-!)* (3. 7 2, u G i=l G

where G is the maximum growth rate which can be obtained by a crystal if it is fully diffusion limited. The advantages of this function are:

a) when evaluated analytically a polynomial result for the population density n which can be fitted to the experimental CSD by standard polynomial procedures;

b) it assumes dispersion in k (first order growth kinetics), and at the same time it allows for an upperlimit in k due to mass transfer limitation in the solution. .

Details of the derivation are given in appendix 3f.

From the initial growth rate distributions the nucleation rate can be calculated. The determination of the nucleation mechanisms can now be based on the growth behaviour, represented by f , and the experimental nucleation

Ill

3.5 Discussion

The shape of the steady state CSD under CMSMPR conditions was shown to be influenced by the average growth rate as a function of the crystal size. This size-dependency could be caused by either true size-dependent growth, or permanent growth rate dispersion. In order to measure the actual nuclea-tion rate it is mostly necessary to extrapolate the CSD towards near zero size. This demands knowledge in advance of the growth mechanism. The description of the permanent growth dispersion given in this chapter is an extension of the approach of Janse and de Jong (1976). An extended population balance has been derived, eq.(3.66), accounting for permanent growth dispersion, but not for stochastic growth dispersion. With the presented mathematics it is possible to calculate the steady state CSD once the growth rate distribution at zero size is known, accounting also for additional true size-dependent effects, such as mass transfer resistance due to diffusion. The reverse is also possible, i.e. the growth rate distribu­tion at zero size can be evaluated from the steady state CSD, but then a high accuracy in the analysis of the CSD is demanded. It is clear that the slowest growing nuclei can only be analysed from the lower size range of the CSD. A simpler method, which will be used in chapter 5 in the analysis of the oxim liquor experiments, is the use of a discrete growth rate distribution with a limited number of growth classes. With the mathematical approach of this chapter it is possible to analyse the CMSMPR experiments in the next two chapters, 4 and 5- The theory for the analysis of the growth , nucleation and abrasion experiments will be presented separately in chapters 6 and 7-

112

113

CHAPTER 4

EXPERIMENTAL DETERMINATION OF SECONDARY NUCLEATION AND GROWTH RATE OF PURE AMMONIUM SULFATE IN A CMSMPR CRYSTALLIZER

4.1 Introduction 4.2 Review of CMSMPR kinetics 4.3 Experimental 4.4 Results of the experiments 4.5 Discussion 4.6 Conclusions

4.1 Introduction

CMSMPR experiments a re very use fu l i n the d e r i v a t i o n of the n u c l e a t i o n k i n e t i c s s i n c e the e f f e c t s of c l a s s i f i c a t i o n and n o n - i d e a l mixing are s t r o n g l y suppressed . In t h i s chap t e r the r e s u l t s of bo th c o o l i n g and e v a p o r a t i v e experiments on pure ammonium su l fa te solut ions employing the steady s t a t e analysis technique a r e p re sen ted and compared with repor ted nucleat ion k ine t ics (only cooling) . Comparable experiments on an indus t r i a l , impure solut ion wil l be given in chapter 5» For the i n t e r p r e t a t i o n of the n u c l e a t i o n mechanism the c la s s i f i ca t ion of secondary nucleation mechanisms as proposed in chapter 2 i s used. I t i s however not p o s s i b l e to d e r i v e the proper c o r r e l a t i n g moment in t h e n u c l e a t i o n k i n e t i c s from steady s ta te experiments alone, see section 3•3♦3 -6, which means tha t in addi t ion to the experiments described in t h i s thes i s dynamic experiment have to be performed which wi l l demand d i r ec t , on-line analysis of the CSD, chapter 8. Apart from the exper iments a shor t review of published CMSMPR experiments on ammonium su l f a t e wi l l be given in the next sect ion. Since an extended s e t of coo l ing experiments was found (Bourne & Faubel, 1982) the main emphasis of the experiments reported in t h i s chapter wi l l l i e on evaporative c ry s t a l l i z a t i on .

4.2 Review of CMSMPR kinetics

4.2 .1 CMSMPR-experiments

Five s e t s of CMSMPR experiments on ammonium su l fa te were found. Four of them employed coo l ing c r y s t a l l i z a t i o n and one used s a l t i n g out with methanol (Timm and Larson, 1968). These l a t t e r experiments wil l not be reviewed here s i n c e t he n u c l e a t i o n r a t e i s two o rde r s of magnitude higher than in the cooling exper iments , probably due to the e f f e c t of the methanol on the c r y s t a l l i z a t i o n . The experimental k ine t ics of the cooling experiments are summarized in table 4 . 1 . As a gene ra l cor re la t ion a power-law was used, see chapter 3 . section 3 • 3 • 3 - 6 , employ ing t h e t h i r d moment of t h e d i s t r i b u t i o n which i s proport ional to the s lur ry density M_:

Beff / MT = kN t " ^ 1 [ # / k g s ] { 4 > 1 )

S i n c e a l l exper imenta l growth r a t e s l i e i n t h e range of 0.4 * 10 to 3 * 10 , the growth r a t e was d iv ided by 10 'm/s which, makes a d i r e c t

114

comparison of the kJ, values obtained in this way, possible as the nucleation rate per kgof crystals [#/kg s] present in the crystallizer at a growth rate of 10 m/s. For the stirrer speed-dependence of k/. also a power law has been employed:

k^ ~ N11 (4.2)

4.2.2 Chambliss (1966) Continuous cooling experiments were performed in a 10.5 1 stainless steel crystallizer at 22°C. The suspension was stirred with a 75 mm propeller at 1725 rpm. The crystal size distributions gave straight lines when plotted on semi-logarithmic paper, which proved that the product of growth rate, G and classification factor, A was independent of size in the size range covered by the sieve analysis, see equation (3-25). Since this observation was made in all his experiments and since the agigation level, estimated with:

c = 0.25 £-f^~ [W/kg] (4.3) yielded a high value of 1.7 W/kg, it can be concluded that both the classification factor and the growth rate are size-independent individually. The values of the effective nucleation rates B „„ were recalculated (see appendix 4, table 4), using the volume shape factor k of Bourne and Faubel (1982) see section 4.4.2, and correlated with the growth rate according to equation (4.1):

1-37 Beff/MT = °'99 * 1Q5 (G/10"7) ' < s e e f iS- i4'10)

The linear dependence of B __ on M„ was separately verified proving secondary nucleation to prevail. Since the stirrer speed was not varied, no information on the removal step could be obtained and consequently the choice between mechanical breeding and surface nucleation cannot be made. .

4.2.3 Larson and Mullin (1973) The crystallizer consisted of a 1 litre beaker (working volume 800 ml) equipped with a cooling coil and a propeller of unknown diameter, with a speed of 500 rpm. The feed solution was saturated at 40°C, the operation temperature was 18°C. At the end of the run, which lasted 6 to 8 retention times, the entire contents of the crystallizer were filtered and the crystals were analyzed. Only three runs with pure, solution were reported, together with the effect of the addition of Cr^ ions. The values of B /JL and L,.,. have been recalculated, see appendix 4 table 5- The following correlation was obtained, see figure 4.10:

* c 1-°3 B^/M, = 0.82 *10 5(^)

Not a l l d i s t r i b u t i o n s , however, were s t r a i g h t when p lo t ted on semi-logarithmic paper, due to disturbancies in the crystallization process, see figure 4 in the original paper.

115

4.2.4 Larson and Klekar (1973)

The data of Larson and Klekar were taken from a review paper of Garside and Shah [l7]« The scale of operation was not given there, the operation temperature was 15°C. Cooling crystallization was applied, In addition to the common parameters, the supersaturation was determined: AC [10 ^kg/kg sol] : O.92-I.9O. The resulting kinetic equation, see figure 4.10 is given by:

B ^ / M , . 0.15 ♦ 105 l ^ ] 1 ' 5

From the range of AC and of G the value of k = G/AC can be calculated, assuming a linear dependence of G on AC: g

k = 10.8 * 10~5[-] g sJ

4.2.5 Bourne and Faubel (1982)

Contoured-base tanks of 1.7 and 42 1 working volume, fitted with a pitch 1.5 propeller and a draft tube (109 nun i.d.) were employed. Three impellers were used in the smaller tank, a steel propellor, the same, coated with rubber, and a teflon impeller. A pitch 1.5 steel propellor of unknown diameter was used in the 42 1 tank.

The experimental conditions are given in table 4.1. The resulting growth and nucleation rates were replotted and the exponents of the growth rates and stirrer speeds were determined. Within each series of experiments a considerable scatter was observed. The average lines are presented in figure 4.10 and the calculated exponents in table 4.1. The largest crystal size is obtained in the 42 1 crystallizer. Generally the nucleation rate increases with the stirrer speed, h ranging from 0.7 to 1.4, except for the experiments with the rubber coated stirrer where a decrease was observed, h = -1.0. It is also observed that the teflon impeller gives less nucleation than the steel one which is in agreement with the positive h-values. The negative value for h and the high value of the nucleation rate in the rubber-coated impeller experiments are not in line with the results for the teflon and steel impeller, since a lower nucleation rate should result from the soft coating. One possible explanation could be that due to the coating the clearance between the impeller tip and the drafttube was reduced leading to an increased removal level. For the discussion of the negative value of it is suggested that the removal efficiency n , is high, say unity and that further increase of stirrer speed reduces the bulk supersaturation and the survival efficiency, c.f. section 4.4.4.

4.2.6 Youngquist and Randolph (1972)

The experiments of Youngquist and Randolph cannot directly be compared with the previously described experiments since instead of the effective nucleation, obtained by sieve analysis, the total nucleation was measured with an on-line Coulter Counter method. The crystallizer, a thermostated vessel of 800 ml working volume was operated at 34°C, but only the produced nuclei were withdrawn, the larger

116

crys ta l s being retained by a stainless steel screen. Three different sizes of seed-crystals were used and three s t i r r e r speeds. The growth ra te of the seed-crystals was determined from photomicrographs and from a sieve analysis at the end of a run. A 70 um aperture Coulter tube enabled size analysis öf the nuclei from 1.26 up to 25 um. The authors observed that i) the d i s t r ibu t ion in th i s range i s created by d i r e c t b i r t h in the

observed size intervals and not by growth from smaller sizes and i i ) the growth of the nuclei is almost negligible. From the dependence of the nucleation rate with the s t i r r e r speed and the photomicrographs of the growing seed-crystals, which show that the corners of the c rys ta l s are damaged at the higher s t i r re r speeds, they concluded that the observed nuclei were formed by mic ro -a t t r i t i on . The following correlation was reported by the authors:

BT - 6.14 * IQ"8 N?'84 G1-22 M ^ 8

The high value of the exponent of the stirrer speed might be an artefact caused by incomplete suspension of the seed crystals (Ottens, 1972) , or it is due to the fact that a minimum value of the tipspeed is needed in the removal step, see also the abrasion experiments in chapter 7-From the classification of secondary nucleation mechanisms proposed in

chapter 2, sections 2.3-5 and 2.4, the following potential explanations can be derived:

i) the nucleation is caused by mechanical breeding and consequently the exponent of the growthrate reflects the effect of survival limitatipn_or

ii) surface breeding prevails with a formation step proportional to G It should be noticed, however, that B measured with the Coulter Counter is not an effective nucleation rate, but the rate at which the nuclei are removed from the surface or corners of thearent crystals, before growing out to larger sizes. Unfortunately the effective nucleation rate could not be measured. In order to compare the value of the nucleation rate, the original correlation is evaluated at the highest stirrer speed used by the authors:

B 1 22 N = 675 rpm : r£ = 26.5 * 105 ( =-^ ) ' [#/kg s]

™T 10"' [m/s]

From table 4.1 it is seen that e.g. the experiments of Bourne and Faubel (1982), performed with the steel propeller yield on average: k' =0.45 * 10 [#/kg s] which is a 60 times lower value. Although Youngquist and Randolph could not measure the effective nucleation rate, it is clear that this rate will be much lower than the observed rate due to the very low growth rate of the nuclei compared to the parent (seed) crystals, which would bring their measurements in line with those of Bourne and Faubel (1982).

Table 4.1 Comparison of cooling experiments of various authors, see fig. 7.10 B°/MT = *N fe^ '-f*

authors

Chambliss,1966

Larson and Mullin.1973

Larson and Klekar.1973

Bourne and Faubel.1982

Youngguist and Randolph,1972

temp [•c]

22

18

15

15

15 15

15

34

V [1]

10.5

0.80

1.7

1.7 1.7 42

1.0

stirrer

propeller

steel propeller

rubber coated teflon

steel propeller

-

diam [mm]

75

99

>99 92

-20

N [rpm]

1725 500

265-335

265-335 265-^50

65-125

545-675

v. . tip [m/s]

6.8

1-7

1.7 2.2

0.7

T

[s]

900-2700

510- 756

540- 720

360-2000

980-2000 360-2000

980-2000

300- 960

"T [kg/m3 ]

26-74 25-41

25-41

14-17

18-19 15-19

15-19

15-50

G [l0"8m/s]

4.2-12.5

15.2-22.5

9.7-21

7-32

5-10 8-32

9-19

-2-10

i

1.4 •1.0

1.5

2.1

1.6 2.1

2.1

-

h

1.4

-1.0 0.7 1.4

-

k' [l0?#/kgs]

0.99 0.82

0.15

0.45

2.05 0.24

0.11

5-27

AC [I0_3kg/kg]

0.92-1.90

-

#expts.

11

3

12

11 19 20

-

118

4.2.7 Comparison of the reported kinetics

From the published data on the crystallization of ammoniümsulfate the following conclusions can bë drawn:

a) The values of the exponent i of the growth rate lie between 1.4 and 2.1 except for the value published by Larson and Mullin (1973) which, however, was based on three points only.

b) The nucleation rate is removal limited. Both the positive h-values and the trend to lower nucleation rates when going from a small vessel with steel impeller to a teflon impeller, and to a larger vessel, point at this fact.

c) The low values of the exponent of the stirrer speed, 0.7 ^ h £ 1.4 and the relatively high values of the exponent of the growth rate, 1.4 £ i £ 2.1 indicate that the nucleation mechanism is surface breeding and not mechanical breeding.

d) The growth rate dependence of the nucleation is due to both the formation step of proto-nuclei and the outgrowth of the nuclei. The existence of an outgrowth step can be inferred from the comparison of the "total" nucleation kinetics measured with the Coulter Counter (Youngquist and Randolph, 1972), having a high nucleation rate, and a low growth rate dependence, with the effective nucleation kinetics obtained by e.g. Bourne and Faubel, showing a much lower rate and a much stronger growth rate dependence. Clearly this last point reflects the supersaturation dependence of the survival or outgrowth step.

Several questions, however, are not yet answered. First, though all authors use the third moment in their correlation, no direct evidence for this assumption was found. For a surface breeding mechanism a moment higher than, or equal to the second moment is expected, see chapter 2. Separate non steady state experiments should be performed in order to ascertain the correct moment, see section 3-3«3-6. Secondly, in industrial practice, evaporative crystallizers are employed having a highly turbulent boiling zone and much higher slurry densities than obtainable in cooling crystallization. No small scale CMSMPR evaporative experiments on ammonium sulfate were found. Thirdly, the effect of impurities is not yet understood, though it is clear that they may have a tremendous effect (Larson & Mullin, 1973).

In order to solve these questions both pure and impure solutions will be investigated, employing both cooling crystallization and evaporative crystallization. The steady state experiments of the pure system are presented in this chapter whereas the results obtained with oxime liquor, an industrial motherliquor, are given in chapter 5- Dynamic experiments as suggested to reveal the correct correlating moment were not undertaken since no reliable on-line CSD measurement method was available.

119

4.3 Experimental

4 .3 .1 The p i l o t plant

The c r y s t a l l i z e r was constructed from two QVF-glass p ipe segments, length 50 cm, inne r d iameter 22.5 cm on top of each other , r es t ing on a s ta in less s t e e l bottom par t , see figure 4 . 1 . T o g e t h e r w i t h a s t a i n l e s s s t e e l top f l a n g e , QVF g l a s s vapour l i n e s , condensors and condensate receiver vessels a u n i t was ob ta ined capable of o p e r a t i n g unde r f u l l vacuum in the evapora t ive mode and ye t easy to disassembly. A s t a i n l e s s s t e e l 140 mm diameter pitched blade impeller, shown in figure 4 . 1 , c i r cu la t e s the c r y s t a l s l u r r y downwards through the d r a f t tube and o u t s i d e the d r a f t tube up to t h e b o i l i n g zone. The draf t tube i s double-walled for operation as a heat exchanger for bo th coo l ing and evapora t ive c r y s t a l l i z a t i o n . The feed so lu t ion flows from a constant-head vesse l , see f i g u r e 4 . 2 , through an e l e c t r i c a l l y t r aced feed l i n e and e n t e r s the c r y s t a l l i z e r a t about 5 cm above the bottom f lange in the annular space between the d r a f t t u b e and the c r y s t a l l i z e r w a l l . In t h e e v a p o r a t i v e e x p e r i m e n t s a t 50°C t h e feed was s l i g h t l y s u p e r h e a t e d and f lashed consequently upon introduction, which resul ted in a good mixing of the feed with the s l u r r y in the b o i l i n g zone. To improve the mixing, in case of cooling c rys t a l l i z a t i on and in case of non-flashing feed, the i n l e t pipe was extended with a p i ece of f l e x i b l e P.V.C. tubing, bent over the top of the draf t tube to d i r ec t ly above the s t i r r e r . The feed flow i s c o n t r o l l e d manually by means of two need le va lves of d i f ferent bore placed para l l e l to each other i n the feed l i n e . Fresh feed solut ion i s c i rcula ted from a 500 1 storage vessel which i s kept at constant temperature, through f i l t e r s (Millipore, "Lifeguard", ~ 1 urn pore s i z e ) , to the constant head vesse l . The excess flows back to the storage tank. The volume of s lurry in the c r y s t a l l i z e r was I approximately 12 1, and was c o n t r o l l e d by a level electrode, see figure 4 . 1 , and a conductivity switch, which act ivated a s ingle stroke piston pump. The p i s t o n speed i s manual ly ad jus ted in o rder to ob t a in i s o k i n e t i c withdrawal of the c rys ta l s at the withdrawal p o i n t . The c r y s t a l s l u r r y i s col lected in a 500 1 receiver vesse l . In case of evaporation the necessary heat i s supplied by hot water (100 to l40°C) c i r c u l a t i n g in a closed loop from an e l e c t r i c a l l y heated (max. 9 kW) buffer vessel through the draft tube heat exchanger. The heat input i s kept c o n s t a n t i n o rde r to reach s teady s t a t e operating condit ions. To achieve t h i s , the flow of the hot water to the heat exchanger and the temperature drop were measured. From these va lues the hea t i npu t was automatically derived and used to control the hot water c i rcu la t ion pump. The temperature of t h e c r y s t a l l i z e r was c o n t r o l l e d by r e g u l a t i n g the vacuum in the condensors and by control l ing the evaporation r a t e with the control valve in the vapour l i ne to the condensors. In case of cooling operation the contents of the buffer v e s s e l were cooled with cold water , and the c i rcula t ion ra te to the draf t tube heat exchanger was manually s e t in order to maintain the desired operating temperature.

120

aechanlcat se «I

Fig. 4.1 Lower part of the 12-20 1 CMSMPR crystallizer, suitable for both evaporation and cooling operation

4.3.2 Operation

For the experiments solut ions prepared from municipal water and t e c h n i c a l l y pure ammonium s u l f a t e (DSM) were used. The so lub i l i t y of ammonium sulfa te increases l i n e a r l y with the temperature from 4 l . l 2 # a t 0°C to 50.3% a t 100°C, see appendix 6a. The concentration of the solution was determined with an Anton Paar DMA 602 p r e c i s i o n d e n s i t y meter , which was c a l i b r a t e d with s o l u t i o n s of known composition and with solut ions of known s a t u r a t i o n t empera tu re , see a l s o appendix 6a. The saturation temperature of the feed was kept at least 10 degrees below the actual temperature to prevent crystallization in the feed lines, while the actual temperature of the feed solution was at maximum 2 degrees above the crystallizer temperature in case of evaporative operation to prevent excessive flashing of the feed in the crystallizer. The volume of solution with constant composition in the feed supply vessel was sufficient for two runs of 8 to 10 retention times. During a run the temperature in the crystallizer, the flow and temperature of the feed and the heat input were recorded and automatically controlled. In case of evaporation the condensate flux was checked periodically. After 8 to 10 residence times the crystallizer was assumed to be in a steady state and samples were taken. The crystallizer can be operated under various conditions (see table 4.2).

r-oi VAT SOOltr , <wrw.S.8kW

5EW RFtDDTK «•rm. 0.55 kW

T-02 VOORRAADVAT SOOltr RVS304 w r w . 2 3 k W motwSEWRF*ODT»0 mot of « r m . 0 SS kW

T-03 MONSTEROPVANG co ï Itr Olas

T-04 HEETWVAT 60 I t r W S 3 0 J ko*lïpirOOt

nutorvcrm 0.71 kW

T-05 KONST.N1VEAU VAT

6 ( t r

T-06 KONDENS OPV.VAT

10 I t r

vacuum

T-07 KONDENS OPV. VAT

20 11 r.

vacuum

T-08 BUFFERVAT l O l t r . RVS 304 vacuum

R-01 KRISTALLISATOR

P-01 STUAHT16 RVS

30 itr. glo» vecuurr cap. motor RD1SS 1HDI motor v i r m . 0 SkW

q w h vtrm

, • „ , / . , . >.., 0.1 « .

P-02 JABSCOUIC-nORVS

cttp

v . r m

1 1 , , . , . , .

»..,. ..„.»

P-05 ECO C l RVS

v«rm

I I » . . , . .

0 . S . W

P-06 STUART I t RVS

cttp. 11 I t r / r i n lwtt 0.11 kW

P-07 STUART I S O RVS

cap

« r m .

M l t r / . i » 11 Btr. O l T k W

P-08 LEVBOLO S6

top

0 .3 Ï kW

C- 01 C0NOENSOR (1>1 qwrkf i l H[ I

A-01 VM-01

| Totaal v«rmogtn 9.94 kW

NR K I K f K ) Kt KS K* KT KR

KtO K i l K i l K i l

KIS KIK K IT K i l KW

K i l KT1 K M

K ï f

TYPE AFSLUITER MEMBRAAN RVS I t t -MEMBflAANRVS I A ' MEMBRAAN RVS 1 / f BOOEHAfSL RVS toorHat p K VERVALLEN ««VALLEN KOGEL RVS Jft" MCMBSAAN RVS l A * BODEMAFSL. RVS taflMl i n H » E l AFSL. RVS 1 -K0SEL APSL. RVS I A -SCHUIF RVS 3 / * -KOGEL RVS 1 M " K0G(L RVS 1/4" PLUG GLAS OM riant » ■ ■ NAALD RVS NAALD RVS HAAL0 RVS TEGENBRUKVENTIEL RVS KLIP 1 / l -RVS BODEMAFSL. RVS MEMBRAAN RVS I A " SCHWP RVS NAALD RVS NAALD «VS MEMBRAAN RVS 1/4"

NR KÏT

K IS K30 Ki t K M K33 K M

K M KIT K M K M K W KJ1 KJ1 KJ3 K44 K I S

K47 K4R K i l K M

K M

TVPE AFSLUITER MEMBRAAN RVS JA" MEMBRAAN RVS I f t " MEMBRAAN RVS 3 ' V VEILIGHEIDSKLCP BRONS VERVALUN RECELKLEP GLAS i o r t a a t » ) ) ] MEMBRAAN RVS 1/4"

VRYSTR0OMKIEP MET AFTAP AFTAP VAN K i t APTAP VAN KIT VRTSTROONKLEP HET AFTAP

RECELKLEP GIETTZCR H I P BRONS 1 ' MEMBRAAN RVS 1/7" NAALD MEMBRAAN 1/1" RVS MEMBRAAN 1/1" RVS PLUGKRAAH GLAS MEMBRAAN 1/l 'OVS MEMBRAAN 1ft*RVS MEMBRAAN l / l " RVS NAALO RÏGELKLEP KLEP BRONS 1 ' KLEP BRONS 1 ' MEMBRAAN RVS lil'

Fig. k.2 Flowsheet of the p i l o t plant FLOWSHEET 12 Itr KRISTALLISATOR

TECHNISCHE HOGESCHOOL DELFT

ARX WUKTUIGSOUWftUNM

A i 31317

122

Table 4.2 Operating conditions

Feed temperature

Crystallizer temperature

Retention time

Crystallizer volume

Evaporative duty

Slurry density

Stirrer speed

20 - 90°C

30-40°C - cooling mode 45-80°C - evaporative mode

600-7200 s

12-18 1

0-7 kW

0-300 kg/m'

0-1400 rpm

4.3»3 Sample treatment

A s l u r r y sample of about 1 1. was co l l ec t ed i n a thermostatted measuring vessel which was cal ibrated with known volumes of water. After f i l t r a t i o n over a P-4 g l a s s - f i l t e r the wet c rys ta l s were washed on the f i l t e r successively with a 40-60# methanol-water mixture s a t u r a t e d a t 25°C w i t h ammoniumsulfate, with pure methanol and with d i - e t h y l e t h e r . The c rys ta l s were dried under a IR-lamp and stored a t 50°C. Methanol was chosen as t h e only so lven t mixable with saturated ammoniumsulfate solut ions , see appendix 6a. The s i eve a n a l y s i s was performed u s i n g 120 mm d i a m e t e r e lec t roformed p r e c i s i o n sieves (Veep) with square holes ranging from 44 to H96 um with a progression factor of J 2. During s i ev ing the lower s i e v e s were kept f ree from mois tu re by a heated airstream from a blower. In t h i s way agglomerat ion of s p e c i f i c a l l y the f i n e r c r y s t a l s owing t o t h e i r hygroscopic character , i s avoided.

4.4 Results of the experiments

4 .4 .1 Classif icat ion of the experiments

In order to t e s t the k ine t i c equation see section 3-3• 3.-6 t he exper imenta l parameters were varied over a wide range, see table 4 . 3 . The residence time was varied from 600 - 3600 sec, s l u r r y densi t ies range from 10 to 300 kg/m3. Four types of s t i r r e r s were used in the 50°C evaporative experiments and the s t i r r e r speed was varied in most cases from 300 to 600 rpm. In a d d i t i o n to the evaporated experiments a t 50°C and 67°C, a limited number of cooling experiments was done to check t h e agreement wi th the pub l i shed da ta on cooling c rys t a l l i za t ion of ammonium su l f a t e .

123

Table 4.3 Type of experiments and range of parameters

system

pure

pure pure

mode

evap.

evap. cooling

stirrers

1 2 3 4

T [°c] 50

67 40

stirrer type

1 2 3 4 4 4

N [rpm]

150-600 300

300,450 300,600 300,700 300,600

[s] 580-3700 1160-3140 1550-2360 1460-3020 620-2860 1200-2650

type

pitched blade pitched blade/silicon rubber pitched blade propeller/pitch=l

[kgV ] 18-246 95-234 121-155 64-153 22-172 26- 83

diameter [mml

140 140 100 100

c!a 239-337 256-359 294-308 266-350 220-430 360-580

no of expt.pts

26 3 2 4 14 5

4.4.2 Calculations

Slurry densities were calculated from the weight of a dried crystal sample and the initial slurry volume of the sample. Residence times were calculated from the mass balance since due to the construction of the product withdrawal unit no accurate flow measurements were possible. For cooling crystallization:

V vi ( M i - i ) - i )

and for evaporative crystallization: v P c ÏL VI r l 1 rC

«T pici }

(4.4)

(4.5)

where 4> . , p. and c. are the volumetric flow rate, the solution density and the mass Traction of1solute of the feed to the crystallizer. Population densities were evaluated from sieve analysis using equation 16, appendix 3a-After having checked that the In n vs L plot gave a straight line, see equation (3.28), the growth rate was evaluated, using equation 17 in appendix 3a. assuming no external classification:

T = T, i.e. s (4.6) Nuclea t ion r a t e s were calcula ted using equation (3.42) as effect ive nucleation rates, assuming no classification:

Y P = M T (4.7)

124

The nucleation rate divided by the slurry density was correlated with the growth rate in a double logarithmic graph, equation (3-44) with j = 3:

B° —^£ " N V (4.8) **T

For constant values of the stirrer speed N straight lines should be obtained having as slope i, see figure (3«5)« I n this plot also equation (3-44) is shown for fixed values of the residence time T yielding straight lines with a slope of -3. It is clearly seen that in order to obtain the nucleation kinetics, equations (4.8) or (3-42) the residence time must be varied. The mean size of the crystals, represented by L_0, the median of the weight distribution, was evaluated directly from tne sieve analysis. The shape factors given by Bourne and Faubel (1982):

k = 5.4 and }

k = 0.64 (4.9) v

were used throughout the experiments, see appendix 6C.

4.4.3 Evaporative crystallization at 50°C

The results of the steady state evaporative crystallization experiments, performed at 50°C, are given in table 1 of appendix 4a. The semi-logarithmic crystal size distributions were straight proving size-independent growth. All three impellers were used. At the minimum impeller speed used still a good suspension of the crystals was obtained; the maximum speed was limited by the carry under of vapour bubbles. The nucleation rates were correlated with the growth rates using equation (4.8), see figure 4.3. The result is:

.B°ff- 6.42 • 105(-S-y) • Hj, (4.10)

The linear dependence of B __ on MT was verified by plotting B ff/G versus M_, figure (4.4), proving that the nucleation was secondary. No effect 01 the stirrer speed was observed within the experimental accuracy. the scatter amongst the experimental points, see figure (4.3) and (4.4) is probably due to the difficulties of operating under vacuum conditions. The absence of any stirrer speed dependence proves that the removal step is not rate limiting. This directly proves that the secondary nucleation is not governed by mechanical breeding. In chapter 7 it is calculated that if mechanical breeding were the only source of nuclei and if no survival effects prevail the dash-dotted line is obtained for N = 600 rpm and the l40 mm pitched blade impeller. Moreover a higher than fourth order stirrer speed dependence is predicted which is clearly not observed in the present experiments. This non observance of the expected contribution of mechanical breeding to the nucleation indicates that an additional survival limitation is active preventing the outgrowth of the fragments to larger sizes. The necessity of such a survival or outgrowth limitation under the conditions of the experiments will be evidenced later. As a result of the absence of mechanical breeding it can be concluded that surface breeding prevails. Furthermore, the removal efficiency is almost

125

unity explaining the non-sensitivity of the nucleation to stirrer speed variations. Consequently only the formation of proto-nuclei, proportional to a power of the growth rate and the parent crystal surface area must be rate-limiting, while an additional survival limitation is possible, see section 2.4. This means that it is theoretically expected that the second moment should be used in the correlation of the nucleation rate.

The following correlation is obtained:

B eff 5-59 * 10"1 G 10 '

2.13 A . [#/m*s]

It is seen that the power of G has been decreased from i A_, is used instead of M,_, see also section 3«3-3"6.

(4.11)

2.51 to 2.13 when

\ a \

T=600S

pitched blade 100mm

pitched blade 11.0mm

propellor 3 blade,100mm

symbl.

X

e <D O ® •

A

N[rpm] 300 (.50

150 300/sil

300 J.50 600

300 600

* 1 1

1 3

19 2 1.

3 1

1..50J

-7.0

Fig. 4 .3 Pure solution-evaporation at 50°C

126

». M T |kg/m3

Fig. 4.4 First order dependence of the nucleation rate on M,

4.4.4 Evaporative experiments at 67°C

4.4.4-1 Results

The r e s u l t s of the steady s t a t e experiments with the pure solution at 67°C a r e shown in f i gu re 4 . 5 and g i v e n i n a p p e n d i x 4a , t a b l e 2 . In al l-exper iments the 100 mm, pi tch = 1, three-bladed propel ler was used. Also in t h i s case the growth r a t e d id no t depend on t h e c r y s t a l s i z e . The r e p r o d u c i b i l i t y of t he exper iments was b e t t e r than the 50°C experiments, probably caused by the greater ease of opera t ion a t a h ighe r p r e s s u r e , i n cont ras t , however, with the 50°C experiments a strong negative dependence of the nucleation ra te on the s t i r r e r speed i s noticed, see figure 4 .5 :

„ 2.80 N = 300 rpm Beff /MT 6.180 * 10- (G/10 - 7 )

and: N = 700 rpm 6 ° ^ / ^ = 2.76I

Assuming a power-law dependence of B

,5

10- (G/10"7) 2.80

B eff /»T 6.180 * 10-

o eff -O.95I

(N/300)

on N yields an exponent h 7 2.80

(G/10~')

(4.11)

(4.12)

-O.95I:

(4.13) _ p 0

The plot of B __/G ' versus M T gave a straight line through the origine, which, as in the 50"C experiments proves secondary nucleation. The high exponent of the growth rate combined with the even negative value of the exponent of the stirrer speed prove, according to chapter 2, that surface breeding is the dominant mechanism, and not mechanical breeding.

127

These k ine t i c s can be explained in two ways:

a. both formation and survival are ra te l imi t ing , the negative s t i r r e r speed dependence ref lec t ing the decrease in survival efficiency a t high s t i r r e r speeds,

b . i t can be shown t h a t a s u p e r s a t u r a t i o n p r o f i l e e x i s t s i n t h e c r y s t a l l i z e r . Assuming t h a t the nucleation i s purely formation limited and t h a t t h i s n u c l e a t i o n responds i n s t a n t a n e o u s l y t o t h e p e r i o d i c s u p e r s a t u r a t i o n l e v e l encountered by a p a r e n t c r y s t a l in c i rcula t ion leads to an increase of the nucleation ra t e a t s t ronger s u p e r s a t u r a t i o n p r o f i l e s . The i n c r e a s e in s t i r r e r speed y i e l d s a r e d u c t i o n of t h i s p ro f i l e and consequently to a reduction in nucleation r a t e .

Both p o s s i b i l i t i e s will be discussed in the next sec t ions .

Fig. 4.5 P u r e s o l u t i o n , e v a p o r a t i o n a t 67°C

128

4.4.4-2 Formation and survival limitation A negative exponent of the stirrer speed is predicted by equation (2.28) for surface breeding if the removal is not rate limiting (c = 0, j = 2):

Beff " N'd ' Q ( a + b ) *2 C*-14) Replacing the t h i r d moment in equa t ion (4.13) by the second, see section 3»3«3_6 leads to :

Beff " N"Q'71 G 2 " 3 5 *T ^•15) Using d * 0.6, see chapter 6, and elimination of N with equation (2.26) yields:

a = 2^5 - 1.18 = 1.17 ^ (4,16)

which may be used in equation (2.25) with c=0 and j=2, to yield: B°ff " AC1'18 G1'17 ^ (4.17)

In deriving equations (4.14) and (4.17) from (2.28) and 12.25) the true rate at which stable nuclei are formed, B , was replaced by B _„. The consequence

s et L .. o of t h i s i s t h a t the bulk s u p e r s a t u r a t i o n dependence AC ' i n equa t ion (4 .17) can be caused by e i t h e r s u r v i v a l or outgrowth e f fec t s . The growth r a t e dependence c l e a r l y r e f l e c t s the format ion s t e p . Evidence for t h e assumption t h a t the growth of t h e c r y s t a l s i s diffusion limited wi l l be given in chapter 6. Up u n t i l now the c r y s t a l l i z e r was cons idered to be wel l mixed. Th i s , however, i s not t rue as far as the supersaturation is concerned, as w i l l be shown in the next sect ion.

4 .4 .4-3 Influence of the supersaturat ion prof i le on the nucleation ra te

I n t h i s s e c t i o n i t w i l l be shown t h a t even in a small c r y s t a l l i z e r a c o n c e n t r a t i o n p r o f i l e may e x i s t , and t h a t t h i s p r o f i l e might cause an enhanced n u c l e a t i o n r a t e i f c e r t a i n assumptions are fu l f i l l ed . Increasing the s t i r r e r speed would l e v e l t h e p r o f i l e and t h e r e f o r e r e d u c e t h e nucleation r a t e . I f we assume that for the 67 °C exper iments the n u c l e a t i o n r a t e i s pu re ly formation l imited then, according to equation (2.25) with b , c=0 and j=2:

Bg ~ Ga • u2 (4.18)

If it is assumed that the nucleation rate varies instantaneously, i.e. without induction time, with the local value of the growth rate, the local values of the nucleation rates have to be time averaged for the parent crystals:

<fi0> = <B°loca1> " <G*> (4'19)

129

It is easily shown that in general for i > 1:

<G1> > K O 1 (4.20)

C o n s e q u e n t l y t h e n u c l e a t i o n r a t e w i l l be h i g h e r when a s t r o n g supersaturat ion prof i le exis ts and i f the assumption of immediate respons i s f u l f i l l e d . F igure 4 .6 shows the concen t r a t i on p r o f i l e c a l c u l a t e d for experiment JH-281 using the following assumptions:

- the suspension i s r ad ia l ly well mixed - t h e suspension f l a she s a t t=0 in t he b o i l i n g zone where the maximum

supersaturat ion i s instantaneously reached - the undersaturated feed has the temperature of the c r y s t a l l i z e r and enters

above the s t i r r e r a t t = 1/4 t , where t i s the c i rcu la t ion time c c

- in order to supply the heat of evaporation the suspension is heated; the resulting temperature increase AT rises linearly from zero at t = 1/4 t up to AT just before the boiling zone, at t=t , where it drops to zero

- both growth and dissolution are considered. .10.0.

ho"Vkg| AC " 1 t '

.5.0.

0 0

-3.0-

I

N.

N

t s 0 : boiling zone : *tire. *6.ts , N«300rpm : fcj rc s2.7s . N*700rpm

X. 1 1 I I

\ I \ \ i

i 1 1 1 V 1

N \ \ N N

N

Fig. 4.6 Supersaturation profile, exp. JH-281, evaporation at 67°C

In the calculation the average value of AC over the profile must equal the average growth rate divided by the mass transfer coefficient k . It is seen in figure 4.6 that due to the temperature increase the solubility rises and consequently the supersaturation drops to negative values. It has to be mentioned that experiment JH-281 is one of the "worst" examples. Increasing the stirrer speed has several effects (see figure 4.6) a) reduction of the circulation time b) decrease of the maximum superheating before flashing c) decrease of the maximum supersaturation jump d) increase of the k -value

g This last point was not considered in the calculations since the value of k under these hydrodynamic conditions was not known. Instead of theunknown k the value for the fluid-bed experiments was taken (k = 24.5 * 10 m/s). Thi

s

130

supersaturation half-life times t_ ,_ are calculated from this k value i see table 2, appendix 4a. -5 g

From the calculated concentration profiles the deviation of B from the "ideally mixed" case is estimated from:

no ƒ C i r C G1 dt B o B° <G>i * t . id circ

(4.22)

In the evaluation of the integral only positive values of G have been considered. The values of B /B. , calculated for i = 2.35, given in the last

Bid column of table 2 of appendix 4, were employed to calculate j — from the o T

experimental values B /A_. In a plot of log B/A versus log G the corrected points (B. ,/A.) for both stirrer speeds fall on one straight line given by:

B7, ü r 3-00 id = 1.359 * 104 (-9-=) *T

(4.23) 10

In figure 4.7 the correlation is shown together with the original and the corrected points.

4.50-

Bid/AT

4.00-

3.50-

3.00-

2.50-

N (rpm) original

"corrected

700 O

•

300 o ■

^T -7.0 -6.5 —«-log (G/|m/s|)

Fig. 4.7 The " i d e a l l y mixed" n u c l e a t i o n r a t e , evaporation a t 67°C

131

Several arguments, however, plead against the effect of the supe r sa tu ra t i on p ro f i l e on the nucleation r a t e . - The s c a t t e r amongst the co r rec t ed p o i n t s i s l a r g e r than the o r i g i n a l

s c a t t e r . - The calculated slope i = 3-00 of the corrected points i s l a r g e r than the

o r ig ina l slopes ( i = 2.35) and i s not explained. - The l e v e l l i n g e f f e c t of i n c r e a s i n g t h e s t i r r e r s p e e d on t h e

s u p e r s a t u r a t i o n p r o f i l e i s counteracted by the simultaneous increase of k , s ince the growth i s diffusion l imited. g

- The model i s based on two assumptions which both are e s sen t i a l : F i r s t , the assumption tha t the nucleation i s almost completely formation c o n t r o l l e d and secondly , the assumption t h a t the nucleat ion reacts immediately on var ia t ions in the supersaturat ion.

- The f i r s t a ssumpt ion impl ies t h a t the format ion of p r o t o - n u c l e i i s 2 S5 p r o p o r t i o n a l to G which i s a r a t h e r h i g h o r d e r . A p o t e n t i a l

e x p l a n a t i o n might be a change-over in growth mechanism above a minimum value of the growth r a t e , leading to enhanced surface breeding.

- The second assumption demands i nduc t ion t imes for surface breeding of l e s s than about 0.1 s , since the c i r cu l a t ion time v a r i e s from 2 to 6 s . This means t h a t a s t a b l e nucleus can be formed within 0.1 s . Estimating the order of s ize of such a nucleus from the growth r a t e of the c r y s t a l s

-7 -ft of about 10 m/s , i t i s found that nuclei of approximately 10 m would have to be s t ab l e . This i s below the s ize of the c r i t i c a l n u c l e u s , being 2 . 9 * 10 m, b a s e d on an e s t i m a t e d v a l u e of t h e su r f ace energy a = 12 * 10 J/m2 and an average supersaturat ion.

I t was t h e r e f o r e decided t h a t the f i r s t explanation, see equation (4.15) offers the most reasonable explanation for negative s t i r r e r speed dependency for the 67°C experiments.

4 .4 .5 Cooling experiments a t 40°C

In o r d e r t o t e s t t h e r e l i a b i l i t y of t h e p u b l i s h e d da t a on cool ing c r y s t a l l i z a t i o n of ammoniumsulfate a few exper iments were performed. The r e s u l t s of these experiments a t 40°C are p r e s e n t e d in figure 4.8 and in tab le 3 of appendix 4 . The 100 mm t h r e e - b l a d e d p r o p e l l e r was used . The c r y s t a l s i z e d i s t r i b u t i o n s gave s t r a i g h t l i n e s when p l o t t e d on semi-logarithmic paper, proving size independent growth in the sieve range. The s t r a i g h t l i ne in figure 4.9 has been corre la ted by:

R 7 2 ' 8 8

Beff/MT = °'681 * 10 ( G / 1 ° " } (4,24)

Due to the limited amount of data the accuracy of the value of the exponent of G is low. The effect of the stirrer speed on the nucleation rate is positive pointing to a removal limitation. Due to this additional removal limitation it is not clear which moment should be used in the correlation, see section 2.4.

132

5.0

log B e f f /H T

|«/kg s|

<..5

«..OH

3.9

temp, «.0 C sHrrer type: propellor diameter 100mm O 600 rpm O 300 rpm

"fe- n ' r~ÏÏ r ■** ,09 ITI lm/s|

Fig. 4.8 Cooling crystallization at 40°C

4.5 Discussion

4.5-1 Comparison of cooling with evaporative crystallization experiments

In table 4.4 the results of the cooling and evaporative experiments are summarized. The nucleation mechanism of the evaporative experiments undoubtedly is identical with surface breeding having a low order dependence on the stirrer speed and a high order dependence on the growth rate. The value of the exponent of the growth rate i, using A„ as the correlating moment is on average 2.3« The 67°C experiments are not removal limited (h < 0) while the 40°C experiments possibly are (h > 0 ) ; at 50°C the behaviour is not yet clear. The negative value of h in the 67°C experiments can be explained by assuming a survival limitation in combination with a bulk diffusion controlled growth rate. That this same effect is not noticed at 50°C could be due apart from the scatter to a stronger decrease with the temperature of the reaction transport coefficient compared to the diffusion transport coefficient, which makes the growth at 50°C less diffusion controlled.

133

Table 4.4 Results of cooling and evaporative experiments

temp. [°C]

50 67 67 40

parameters

mode

evaporation evaporation evaporation cooling

N [rpm]

150-600 300 700

300-600

G [10"8m/s]

2.5-11.2 4.0-12.6 4.0-12.6 5.0-10.7

6 ° / ^ " G 1 ^

i h k^ [lO^/kg s]

2.5 » 0 6.4

2.8 ; 1 , u 2.8 2.9 > 0 0.68

B0/^ - G V

i h

2.1 * 0 2 4 2!Ï )_0-75 2.5 > 0

A second explanation of the negative value of h involved the action of the supersaturation profile on the nucleation rate. The underlying assumption of direct response of the nucleation on local supersaturation values is contradicted by the idea that a "proto-nucleus" needs time to reach the critical size for survival (or even for removal). If the value of h for the cooling experiments is greater than zero, meaning a removal limitation, this would be in accordance with the lower nucleation rate found in these experiments. This lower nucleation rate is found at stirrer speeds comparable with the evaporative experiments. This implies that in the evaporative experiments an additional removal mechanism must be active compared to the cooling experiments. Hereby it is assumed that the formation step remains unchanged, which is not unreasonable. The vigourous boiling and the flashing of superheated feed in case of the 50°C experiments most probably provide the additional mechanical action to enhance the removal present under non-boiling conditions, up to a removal efficiency n » 1. Apart from the presence of a boiling-zone there are three other differences between the cooling and the evaporative experiments. In the cooling experiments: i) no strong temperature fluctuations and ii) no undersaturated zones exist, iii) the fluctuations in supersaturation are smaller compared to the

evaporative crystallization. This last point is caused by the low slurry densities resulting in a longer decay time of the supersaturation, tabulated in appendix 4 as a half-life time (t„ ,_) of the supersaturation (Garside, 1985) • No effect of these differences between cooling and evaporation is expected. In the next section the results of this study will be compared with the published kinetics on ammonium sulphate, as treated in section 4.2.

134

6.00-

log B ef f / M T

[»/kg s)

5.50

5.00-

4.50-

4.00-

evaporation tooling

-7.50 -7.00 — » - tog _G_

[■/si -6.50

Fig. 4-9 Comparison of present results and literature data

Legend to figure 4.9

1 2 3 4

Chambliss, 1966 ; cooling, Larson, Mullin, 1973; cooling,

Klekar, 1973; cooling, Faubel, 1982; cooling,

Larson, Bourne, a) 1. b) 1, c) 1.

22°C 18°C 15°C 15°C

.7 1, stainless steel impeller

.7 It rubber coated impeller

.7 1, teflon impeller d) 42 1, stainless steel impeller Present results

h h h h

1.4 -1.0 0.7 1.4

a) evaporation, 50 C, b) evaporation, 67°C, c) evaporation, 67°C, d) cooling, 40°C

all impeller speeds 300 rpm 700 rpm

135

4.5.2 Comparison of present results and literature data

The results of the present series of experiments (table 4.3) and the literature (table 4.1) are compared in figure 4.9- As the basis for the comparison the empirical kinetic equation (4.1) has been used, though it was concluded in chapter 2 that the use of the slurry density is not justified a priori. Furthermore it appeared that the present evaporative experiments are to be described by using the total crystal area as the correlating moment, see equation (4.17). The following observations were made:

1 - The 40°C cooling experiments are roughly in accordance with the experiments reported in literature, performed at lower temperatures (15-22°C). It appears that the nucleation rate increases with the stirrer speed pointing at a removal limitation, n < 1. This is confirmed by the effect of the hardness of the impeller and the scale of operation (Bourne, Faubel, 1982), though an exception has to be made for their rubber-coated impeller experiments.

2 - The evaporative experiments show significantly higher rates of nucleation than the cooling experiments though the overall growth rate dependence is comparable.

3 - Additionally the exponent h of the stirrer speed is considerably lower than in the cooling experiments and even reaches negative values (67°C, h = -1, see table 4.3). This latter point was interpreted as a combination of two effects, the absence of removal limitation, n * 1, and the lowering of the supersaturation (at constant G) with an increase of the stirrer speed, assuming the mass transfer process to be fully diffusion controlled, and assuming a supersaturation dependent survival/ outgrowth step in the nucleation mechanism.

4 - The high value of the removal efficiency (n * 1) might be caused by the more vigourous conditions in the evaporative crystallization. At the same time the high removal efficiency explains the high level of nucleation rate compared to the cooling experiments.

5 - The latter two observations, n * 1 and h = -1 also apply to the rubber-coated impeller experiments \cooling operation, Bourne, Faubel, 1982), lying in the same range as the evaporative experiments. An additional removal process (reduced clearance between impeller tip and draft tube, Grootscholten, 1982) would explain this behaviour.

6 - The explanation of the negative stirrer speed dependence as a decrease in average level of supersaturation, see points 3 and 5i which is observed even in cooling crystallization, rules out the effect of the rapid supersaturation fluctuations on the nucleation rate.

7 - The positive exponent of the stirrer speed, h B 1.4 (Bourne, Faubel, 1982) is composed of two contributions, a negative one, see points 3 and 5, and a positive one representing the true removal effect. The exact values of h and i, however, depend on the choice of the proper correlating moment, see chapter 3» section 3•3«3-6. The use of for instance uj, yields h = 1.75 and i = 2.38. Analogeous to the treatment in section 4.4.4.2 one could arrive at:

136

B°eff ~ N2'5 AC1'2 Q1'2 y4 {4,25>

0r: B°eff " N3'2 L°2A P 4 {k'26)

I t is clear that no definite choice between the various possibilities can be made at this point since the proper correlating moment can not be derived from steady state experiments alone. Theoretically, however, the second moment is to be preferred.

8 - The resul t s of the semi-continuous seeded batch experiments of Youngquist and Randolph (1972) show a growth rate dependence of i = 1.22. Since no growth of the nuclei could be observed and since direct birth in the observed size interval (2-25 um) was concluded, i t is most probable that i = 1.22 is a result of the formation step. This supports the interpretation of the 67°C evaporative experiments, see 4.4.4-2, where i t was found that (equation 4.17)

B°eff " A c l ' 1 8 Q 1 ' 1 7 *T { 4 ' 1 7 )

This would imply at the same time that the AC dependence in equation (4.17) reflects the outgrowth of the nuclei, since i t is absent in the semi-continuous batch experiments. The outgrowth efficiency is very low.

9 - Finally i t should be remarked that the spread in absolute values of the nucleation rate is s t i l l very high, about a factor 70 at G = 10 m/s, see figure 4.9. Therefore i t is necessary to investigate more parameters such as - geometry of stirrer and vessel

- scale of operation - effect of impurities on nucleation and growth - effect of temperature.

4.6 Conclusions

- The most likely interpretation of the nucleation kinetics of the pure ammonium sulfate system is secondary nucleation caused by surface breeding.

- The present results and the reported empirical kinetics can be explained by separating the overall growth rate dependence of. the nucleation rate into contributions _from a formation step (~ G * ) and a survival/ outgrowth step (~ Ac ). Comparison of total nucleation kinetics with effective nucleation kinetics indicates that i t is outgrowth rather than survival. The outgrowth efficiency is very low.

- The removal efficiency is high (n B 1) for the evaporative crystallization experiments, and considerably lower for the cooling crystallization experiments, the most likely kinetic expressions are:

Evaporation B°ff = Ac1'2 G1,2 A , (4.17)

Cooling B°ff = kjj N2'5 Ac1,2 G1'2 (4.25)

The value of lc. may depend on geometries, scale, temperature and presence of impurities.

137

CHAPTER 5

EXPERIMENTAL DETERMINATION OF SECONDARY NUCLEATION AND GROWTH RATE OF AMMONIUM SULFATE FROM "OXIME-LIQUOR" IN A CMSMPR CRYSTALLIZER

5.1 Introduction 5.2 Experimental 5-3 Results 5.4 Discussion

5.1 Introduction

In the previous chapter the k ine t ics of the pure ammonium sulfate-water sys­tem have been d i s cus sed . I t was concluded t h e r e t h a t the growth of the c rys t a l s was independent of their s i ze in the observed s i z e i n t e r v a l , and tha t the nuclei were formed according to the surface breeding mechanism, one of the two secondary nucleation mechanisms proposed from the review of the l i t e r a t u r e i n chap te r 2 . The e f f e c t i v e n u c l e a t i o n r a t e appeared to be proport ional to the surface area of the p a r e n t c r y s t a l s and t h e i r growth r a t e , r e f l ec t ing the formation of proto-nuclei on the c rys ta l surface. The removal of these proto-nuclei from the surface was not r a t e - l i m i t i n g in the e v a p o r a t i v e exper iments , s i n c e i n c r e a s i n g the s t i r r e r speed did not r e s u l t i n higher effect ive nucleation r a t e s . Cont rar i ly , a dec rease in ef­f e c t i v e nucleat ion ra te was observed which could be in terpre ted by assuming a bulk supersaturat ion dependence in the n u c l e a t i o n k i n e t i c s , see s e c t i o n 4 .4 .4 -2 .

The second of the two important nucleation mechanisms, mechanical b reed ing , which should be highly dependent on the s t i r r e r speed, see chapter 7. could not be detected though the number r a t e a t which fragments were formed in abrasion experiments under exactly the same conditions was of the same order as the observed nucleation rates i n the c r y s t a l l i z a t i o n experiments. This non-observation of mechanical breeding was t en ta t ive ly explained by the increased so lub i l i t y of the abrasion fragments which severely reduced t h e i r s u r v i v a l e f f i c i e n c y a t the low s u p e r s a t u r a t i o n s in t he c r y s t a l l i z a t i o n experiments. The experiments with the pure ammonium sulfa te-water system, however, cannot be applied d i rec t ly in the design of a large scale indus t r i a l c r y s t a l l i z e r . Apart of the problems encountered in s c a l i n g up which have to be inves­t iga ted a t the intermediate scale, i n 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 t he feed s o l u t i o n most ly contains trace amounts of chemical compounds which can in­fluence both the growth and nucleation r a t e . The adverse e f f e c t of so lub l e t r i v a l e n t metal ions present in the old sa tu ra to r process (Klempt, 1953) and the ro le of organic material from the caprolactam synthesis (Bennett , 1965) i l l u s t r a t e t h i s point . In general i t can be expected tha t the mass t ransfer coeff ic ient of the incorporation react ion of ammonium sul fa te in to the crys­t a l l a t t i c e wi l l be lowered i f other substances are preferent ly adsorbed to the c r y s t a l s u r f a c e . But not only the growth r a t e , b u t a l s o the growth mechanism w i l l be affected. An example of t h i s i s the growth of KCl, which from pure solut ions usually proceeds in a po lycrys ta l l ine fashion r e s u l t i n g in cub ic c r y s t a l s consis t ing of many small, s l i g h t l y d isor iented cubes. The addit ion of Pb ions d ras t i ca l ly changes t h i s pa t t e rn : both t h e p o l y c r y s -t a l l i n i t y and the growth r a t e a re suppressed and large s ingle c rys t a l l i ne

138

cubes are obtained with flat faces showing large circular growth spirals (Pilkington and Dunning, 1972). Since polycrystallinity and surface breeding are interrelated phenomena it is clear that this nucleation mechanism will be affected by substances which influence the growth of the crystals. The aim of this chapter, therefore, is to investigate the effect of impurities present in an industrial solution on the nucleation and growth kinetics. The experiments were performed with one type of solution, spent mother liquor ("oxime-liquor") supplied by DSM. Apart from a high nitrate level this solution contained a number of organic compounds from the cyclohexanone oxime synthesis, which is an intermediate in the route to e-caprolactam. No further impure solutions were tested since it was decided that in that case a more systematic approach would be demanded, including the modelling of the growth process influenced by various impurities, which is beyond the scope of this thesis.

5-2 Experimental

The experiments were performed in the pilot plant described in detail in section 4.3.1. The level control contact electrode in the crystallizer was replaced by an E&H "Liquiphant" level detector, mounted in a glass T-piece connected to the crystallizer as shown in figure 5«1«

--E&H Liquiphant

Fig. 5.1 Positioning of the new level detector

In this way a highly accurate and reliable level control was obtained which was insensitive to the height of the boiling.zone in the crystallizer.

The oxime-liquor, obtained from DSM was filtered before introduction in the pilot plant. Since it contained a high level of ammonium nitrate and organic "impurities" the density measurement could not be used. Prior to an experi­ment the solution was saturated at about the desired saturation temperature by adding just enough water to dissolve the excess of crystals present in

139

the feed supply vessel. During the run the temperature of the feed solution was kept at least 10 degrees above the saturation temperature. At the end of a run the condensate could be drained to remove water from the system. In this way the ratio nitrate/sulfate could be kept constant without analyzing the solution. The sample treatment used in the pure ammonium sulfate-water system, section '4.3»3» was used without difficulty. Though not fully correct, the same shape factors were used, which is practically justified as long as the value of the error introduced is small compared to the difference in behaviour be­tween the pure and the "impure" system. If necessary, the shape factors can be determined for each sieve-size by microscopic characterization of the crystals.

5-3 Results

5-3-1 Classification of the experiments

Three series of experiments were performed: one with cooling (40°C) and the other two with evaporative crystallization (50, 67°C). In all experiments the three blade, pitch = 1, diameter = 100 mm marine type propeller was used. The experimental conditions and the calculated nucleation and growth rates are listed in Appendix 5-Since, in constrast with the pure ammonium sulfate experiments, the semi-log population density plots yielded curved lines, the analysis of nucleation and growth rates was done according to the permanent dispersion model (see 2.3-4 and 3-4.3). The growth rate distributions at zero size were tested (3-^-3-6): 1. The inverse-gamma function 2. The continuous polynomal function 3- Discrete dispersion.

True size-dependent growth was not considered since all evidence (see 2.3-4) on the growth of nuclei/small crystals points at permanent dispersion. Stochastic fluctuations in "growth rate leading to a growth dif fusivity do not lead to the observed curvature.

mode

cooling evaporative evaporative

T[°C)

40 50 67

N[rpm]

300-700 300-700 300-700

T[S]

1100-3600 770-3600 650-3050

M.jlkg/m3 ]

6- 33 35-300 17-108

L50[um]

470-830 504-676 503-880

no of experi­mental points

16 8 8

Apart from the dispersion models it is possible to calculate the total nucleation rate in a direct way. This will be done first, in the next sec­tion (5-3-2). Thereafter the three growth rate distribution functions will be discussed.

140

5.3.2 The total nucleation rate The total number balance over the crystallizer is obtained when the popula­tion balance, equation (3-17) is integrated over the crystal size from L = 0 to L = •:

o

dïL " N, T ro ,;^o dt n° <G>° * ƒ (b-d)dL - - ^ - N T ^ V (5>1)

The total nucleation rate is the sum of the first two terms on the right hand side:

f l m g _ TJD _ fi dlnV ( } dt bT T NT dt ^* '

Assuming steady state and constant slurry volume leads to:

B T = - ^ (5-3) which says that the nucleation rate in steady state equals the number of crystals leaving the vessel per second (#/m3s]. Using the CMSMPR assumption we have:

N T = N T = N T > p (5.4)

The experimentally determined values are given in table 1 of Appendix 5-

Figure 5.2 shows this total nucleation rate divided by the total area A_ for the 40°C cooling experiments, plotted versus the average growth rate G defined by equation (3-40) see section 3.3-3~5. No satisfactory correlation is obtained. It is further noticed that:

a. The total nucleation rate increases with the stirrer speed, b. The slopes of the lines for constant stirrer speed also increase

with the stirrer speed. This shows that the effect of the stirrer speed on the nucleation rate is more pronounced at higher growth rates. The observed scatter is mainly caused by the inaccuracy of the sieve analysis at sizes below 100 um. The correlations to be obtained with the dispersion models should give a better result than this.

141

..9 ( i l^ l) |»/m2s|

3.50.

3.00-1

2.70 -7.50

N[rpm)

300 500 700

• O A O

slope 0.37 1.17 2.00

-7.00 .log (Gsv /{m/sll

Fig. 5.2 The total nucleation rate, cooling at 40°C

5.3»3 The inverse-gamma model The inverse-gamma growth rate distribution of the zero-sized crystals is given by equation (3-70). In Appendix 3f it has been shown that the popula­tion density (CMSMPR; steady state) can be expressed as:

n = n (1 1-r

(5-5) Since regression analysis of this non-linear equation is somewhat involved it was decided to use equation (3.71). the size-dependent average growth rate of the steady state CSD resulting from the inverse-gamma growth disper­sion model, to fit the experimental values of the average growth rate as a function of size, which were calculated from the steady state CSD using equation (3.24) with A = 1, i.e. no classification:

N ov <G(L)> n T P

Equation (3-71): G(L) = Q M L . F=2 (1 + Qï>

(5-6)

(3.7D

142

From the straight line obtained by fitting equation (3-71) the parameters Q and r of the inverse-gamma distribution are easily obtained. The growth rate at zero size is obtained by setting L=0:

G° r-2 (5-7). The nucleation rate is determined from the product of n G , where n is solved from the analytical solution of the slurry density, see Appendix 3f'•

Kj, (r-5)(r-4)(r-3) B° =

6p k (G°)V c v (r-2)-(5-8)

The results of the calculations are given in table 2 of Appendix 5-

• — O experimental inverse gamma approximation

exp DSM 18

"* L 'Mm'

Fig. 5«3 Size-dependent growth rate, 40°C cooling

Figure 5-3 shows the growth rate as a function of size, together with the linear approximation, equation (3-71)• It is seen that the experimental growth rates do not depend linearly on the crystal size, as predicted by the inverse-gamma distribution. Clearly, the inverse-gamma distribution underestimates the amount of slow growing crys­tals in the small size range. Figure 5.4 shows the CSD obtained by sieve analysis and the approximation using the inverse-gamma model. Also here the failure of the inverse-gamma function below 150 um is clearly shown. This means that growth rate distribution functions that are less skewed towards L=0 than the inverse-gamma function, such as the log-normal distribution and the normal distribution will prove to be even less successful. The nuclea­tion rate, equation (5>8) was plotted versus the average growth rate G which was calculated to be (see Appendix 3f): av

G av r-5 (5-9)

143

The result, shown in figure 5-5 is not encouraging: the correlation is even worse than the previous one, obtained from the total nucleation rate. It is therefore concluded that the inverse-gamma growth rate distribution function does not represent the true distribution of growth rates at zero size. A second discrepancy is noticed in figure 5-3: whereas the inverse-gamma ap­proach predicts a continuous increase in growth rate with size it appears that in reality the growth rate levels off and reaches a maximum value. This might be in accordance with the diffusion controlled growth rates measured in fluid bed experiments, see chapter 6, which were also performed on large crystals 800-1000 urn. It is therefore suggested that the largest crystals grow diffusion controlled. Before proceeding to the next model it is noticed that qualitatively the same stirrer speed dependence is obtained with the inverse-gamma approach compared with the direct calculation of the total nucleation rate.

30.00-

ln n [•/■si

25.00-

O sieve data

inverse gamma approximation

Gav = US nm/s T = 3624 s C.V. = 54 % N = 500 rpm

500 I M 1000

Fig. 5^4 Example of a crystal size distribution; upward curvature below - 500 u

144

log B°/AT

|-/m2sl

3.5-

3.0-

2.7-

• o A O

N |rpm] 300 500 700

slope

0.50 0.65 1.20

—i r -7.50 log

RA| -7.00

Fig. 5>5 Results of the inverse-gamma function, 40°C

5.3-4 The continuous polynomal distribution

The continuous polynomal distribution function, equation (3-72) leads to an analytical solution of the population density, see Appendix ?>f:

n =

b

b =

exp(-b) b

i s given L

G T max

z i=o

by

a . i ! l V (5.10)

(5.11)

The maximum growth r a t e i s the fully diffusion controlled growth rate at large particle sizes, and acts as a f i t t i n g parameter. The degree of the polynomal can be chosen freely, however, no inflection points in the popula­tion density are allowed. A good f i t was obtained for m=4, see figure 5.6. From the experimental values of a., equation (5.10), the growth rate dis­t r ibu t ion at zero s i z e , equation "\3.72) , was obtained d i r e c t l y . The

145

nucleation rate _was obtained using equation (3-69) • Since the flux term in equation (3-69). f <G>, becomes infinite for zero growth rates, see equation (3-72), a lower limit of integration was arbitrarily chosen at 5 um. The results are shown in figure 5-7. and in table 3 of Appendix 5. Comparison with the results booked by the previous methods shows that no im­provement is obtained. In fact the nucleation rate obtained in this way should have approximately the same value as the total nucleation rate, see equation (5-3) and figure 5-2, provided the fit of equation (5.10) is ac­ceptable. Also here the increasing effect of the stirrer speed at elevated growth rates is noticed. This is confirmed by inspection of the individual CSD's of the various experiments: the curvature caused by growth dispersion becomes more pronounced at elevated stirrer speeds and growth rates. Up to now the continuous functions have been considered. The use of these functions is practically limited to those which can be integrated analyti­cally to yield an expression for the population density. A second drawback of the continuous dispersion functions is that they have no clear physical background. Moreover it is difficult to obtain a function which can be in­tegrated analytically and which yields finite moments. No such problems are encountered in the discrete model, which will be discussed in the next section.

[•/ml

30.00-

20.00-

EXP. DSM 18

G a ï = 1.8 lnm/s) T » 3624 |s| C.V. = 54 \%\

-1 1 1 1 — 1000

~l 1 1 r 500 L [Mm|

Fig. 5«6 Experimental CSD approximated by the continuous polynomal function, 40°C

146

4.00-

log B°/A I |»/m2s|

3.50.

3.00.

-7.40

• o A O

N [rpm]

300 500 700

slope

0.8 1.5 2.2

— I 1 r -7.00 r.

* - log m/s)

—r~ -6.60

Fig. 5>7 Nucleation kinetics calculated from the polynomal approach

5.3-5 The discrete dispersion model

Since all distributions were straight at sizes over 500 um and some even over 200 um it was decided to treat the largest crystals as a single group having a unique growth rate. Both the growth and nucleation rate of this group were calculated from the slope of the straight part of the semi-log CSD and the intercept at L=0 respectively, according to the conventional size-independent growth approach (see equations (3-50-3«52)). The distribution thus obtained was substracted from the total distribution to yield a slower growing group of crystals at small sizes. Growth and nucleation rates were calculated as for the first group. Since the resulting distribution of slow growing crystals contained not many points it was decided not to split this distribution any further, so only two growth rates, G1 and G? are used, yielding two nucleation rates. The first group of crystals will be referred to as "fast growers" and the second as "slow growers". The results of the calculations are listed in table 4 of Appendix 5-In figure 5«8 two experimental CSD's are shown, the only difference being the stirrer speed. The curvature at the highest stirrer speed can be clearly seen. After extrapolation and subtraction of the straight part of the CSD,

147

l i ne I , the resu l t ing d i s t r i b u t i o n , p l o t t e d a l s o on semi- log paper , see f i g u r e 5 . 9 , y ie lds a s t r a igh t l i n e ( l ine I I in f igure 5 .8) . The 300 rpm CSD could not be s p l i t in two CSD's. The best corre la t ion of the nucleation rate B« of the fast growers was obtained with the area as correlating moment

o 1-08 B„ 'Ap = 1.146 x 10J (—5=) (5.12)

10 see f igure 5.10. No effect of the s t i r r e r speed was detected, though a l l ex­periments were included. From the modelling of the nucleation mechanisms in c h a p t e r 2 i t can be inferred tha t the absence of any s t i r r e r speed depend­ence means tha t the removal step i s not r a te con t ro l l i ng , and consequent ly equation (5.12) cannot be caused by mechanical breeding. I t i s concluded tha t surface breeding p r e v a i l s . The low va lue of t he ex­ponent of G sugges t s t h a t only the formation s tep and no survival s tep i s involved. The growth r a t e of the slow growers was much lower than tha t of the fast growers, see figure 5 - H . with on the average:

G2/Q1 * 0.27 (5-13)

0 X

exp.

OSM 7 DSM 12

N [rpm]

700 300

X

Is] 1121 1121

M T [kg/m3]

74.9 19.8

1-50 |pm|

U 9 615

25.00-

L [pm]

Fig. 5-8 E f f e c t of s t i r r e r d i s t r i bu t ion

s p e e d on t h e s h a p e of t h e c r y s t a l s i z e

148

30.00-

Inln/h/m''])

I ; 25.00-

20.00-

A

\° O V j

O V^

1 r ■ i

\°

' i -

Experiment : DSM 7

In n\= 30.85

^ r = 65 tim L50 . 240 MH,

G2 = 5.80 «. 10 [m/s] MT,2 = 303 [kg/m3]

o \

\°

— i 1 1 1 —

L IMI»] 500

Fig . 5.9 The popu la t ion d e n s i t y of "slow growers" . Expt. DSM 7

149

3.5-^

o .og(_L_L)

|*/nrs]

3.<M

3001 500 700

40

0 & O

50

®

a

67

•

■

Fig. 5.10 Nucleation rate of fast growing crystals

150

il t 0.50H

• # o

o

tO°C

o V & 0

50 °C

®

8

67 °C

•

■

N (rpml

300 100 SOO 700

O

"ö!T i 1.0 * 3 - AC [10 kg/kg]

2.0

Fig. 5'H Growth rate of slow growers (G?) compared with fast growers (G..)

Plotting the nucleation rate B_ divided by M_ versus the growth rate of the fast growers, figure 5-12, shows, notwithstanding the large scatter which is due to the subtraction process and the limited accuracy of the sieve analysis at small sizes, that a) the dependence on the growth rate is not

clear, b) the stirrer speed dependence is very strong

From these observations it was tentatively concluded that mechanical breed­ing is the most likely nucleation mechanism. Anyhow it is clear that the mechanism by which the slow growers are formed is totaly different from the one yielding fast growers, which justifies the use of two groups in the dis­crete dispersion model. In order to test this hypothesis the results of abrasion rate measurement on ammoniumsulfate in dry ethanol, see chapter 7. were applied on the present experiments. The abrasion rate was calculated using:

B mech 1.281 ■ 1016 ( G ^ ) 2 ' 8 2 2 V (5.14)

This equation was established with a 14 cm diameter pitched blade impeller rotating at 600 rpm, while a fourth power dependence on the stirrer speed was found. This fourth power is due to

a) circulation frequency (~ <t> vc vtiP " N> b) impact velocity squared (- v . )

c) impact efficiency <- W

151

The abrasion and crystallization experiments were performed in vessels with the same geometry and volume. Therefore

B ~ ♦ . v... 3 - K ND3(ND)3 mech vc tip p

KNV P

(5-15) A c o r r e c t i o n f a c t o r due to t h e d i a m e t e r r e s u l t s of app rox ima te ly (10/14) = 0.133 assuming ident ica l pump cons tan t K . Combining t h i s with

■equation (5.14) and equation (5.15) leads to :

B . * 0.133 » 1-281 • 10 mech 1 6 2.822 N 4

' ( G 1 T ) * {W5] ' *T.; (5.16)

tog B2 /MT

5.00-

4.50.

4.00-

3.50-

-7.50

N [rpm]

300 400 500 700

#

O

A O

-7.00 r -6.50 »- log .—L

(m/s)

Fig. 5.12 Nucleation r a t e of "slow growers", 40°C, cooling

152

-0.50-

-1.00-

-1.50-

/ -B2 =Bmech

N [rpm] 300 UO 500 700

#

O

0

-1.0 -0.5 log AC r e (

TT

Fig. 5.13 C o r r e l a t i o n between abras ion and t h e n u c l e a t i o n r a t e of slow growers, 40°C

Various corre la t ions were t e s t e d . The be s t r e s u l t was ob ta ined when the r a t i o of the experimental value of B? and the predicted mechanical abrasion r a t e , B . (equation 5-16) was p lo t ted versus the re la t ive supersaturat ion. Since the a b s o l u t e va lue of t h e supe r sa tu ra t ion could not be measured, a r e l a t ive supersaturat ion has been defined as the r a t io of the ac tua l super -s a t u r a t i o n and a reference supersaturat ion, at a r b i t r a r i l y chosen reference condit ions, G = 10 m/s and N = 300 [rpm]. For d i f f u s i o n l i m i t e d growth, which i s a good assumption for the^-öfast-growers", the mass t ransfer coeffi­c i e n t i s p r o p o r t i o n a l t o N ' , s e e A p p e n d i x 6 b . The r e l a t i v e supersaturat ion can be expressed as :

Ac rel = I — . -) (" N -) -O.58

(5.17) 10 '[m/s] 300 [rpm]

The resulting correlation is given for the 40°C cooling experiments, see figure 5.13• It is seen that a single line is obtained for all stirrer speeds, which reaches a level B_° = B . for the highest values of the

F 2. mecn supersaturation.

153

I t i s concluded that the nucleation of the "slow growers" can be descr ibed s a t i s f a c t o r i l y by a mechanical breeding mechanism, with a supersaturation dependent survival/outgrowth efficiency. SEM pictures of the c r y s t a l s show damage a t the corners of large c rys t a l s , especial ly at the highest s t i r r e r speed, while a large proportion of i r regu la r ly shaped c rys ta l s i s present in the smallest s ize f ract ions , see figure 5 . l4 .

a) Exp DSM 15, 1185 pm fraction, 40x b) Exp. DSM 16, 1185 pm fraction, 40x

c) Exp. DSM 15, 52 pm fraction, 500x d) Exp. DSM 16, 52 pm fraction, 200x Fig. 5.14 SEM pic tures of damaged large c rys ta l s and i r r e g u l a r small crys­t a l s

The slow growth of these small c r y s t a l s i s most l i k e l y caused by t h e breakage process , see chapter 2: p l a s t i c deformation r e su l t s in a high d i s ­locat ion densi ty . The e l a s t i c energy stored in the c r y s t a l l a t t i c e around t he se d i s l o c a t i o n s causes an i n c r e a s e in so lub i l i t y and, consequently, a decrease of the growth ra te and may even lead to ( local) d isso lu t ion . The cha rac t e r i s t i c s of the two groups of nuclei are summarized in table 5.2:

154

Table 5»2 Characteristics of the two groups of nuclei

fast growers slow growers

1 1 U 0 Pi

nucleation rate B ~B. A_ B ~ Ac N (L_„) ' VL,

growth diffusion limited Gp = 0.27 G1

stirrer speed no effect strong effect

nucleation type formation limited removal & survival limited

mechanism surface breeding mechanical breeding (abrasion)

5.4 Discussion

5.4.1 Comparison of the dispersion models

From the discrete dispersion approach it was concluded that two different nucleation mechanisms are operative producing two types of nuclei, each having their own growth rate. No attempts were made to see whether amongst the slow growers additional growth dispersion was operative, since the limited accuracy of the sieve date in the low size range did not permit this. The strongly bimodal growth rate distribution explains why the con­tinuous dispersion functions failed in the regression analysis of the data. Furthermore the continuous growth dispersion functions result in "total" nucleation rates where no discrimination between abrasion fragments and sur­face nuclei can be made. It seems that the continuous growth dispersion functions should only be used when the nuclei result from a single nuclea­tion mechanism. In the present case two totally different mechanisms were found yielding comparable amounts of nuclei with different growth charac­teristics, favouring the use of the discrete approach. Considering the separate groups of nuclei, it is expected that dispersion amongst the fast growers is limited due to the fact that their growth rate is most probably limited by volume diffusion. On the other hand growth dispersion is expected for the slow growers, where even negative growth rates are possible.

5.4.2 Comparison with the pure system

The differences between the pure and the oxime system are

- In the pure system no mechanical breeding is observed. - In the oxime-system the surface breeding rate is lower, resulting in higher growth rates, supersaturations and values of L ... The lower surface breeding rate is clearly a result of the presence of impurities which af­fect the growth mechanism and the habit of the crystals.

- In the pure system the surface breeding rate is additionally survival - or outgrowth - limited, n < 1:

s

155

B eff " ACB Gav • EÖS-n o x, 1.1 A B. - G . A_ oxime 1 av T

whereas for oxime liqour TI = 1 . This might be caused either by the lower supersaturations in the pure solution or by differences in growth charac­teristics of the nuclei.

- The shape of the crystals is different: "orthorhombic" plates in the pure system vs. large, hexagonally shaped prisms in the oxime system, caused by twinning along the pseudo-hexagonal c-axis.

The reason why no mechanical breeding is observed in the pure system might be:

a) the supersaturation is lower, decreasing the survival efficiency of the fragments,

b) the "parent crystals" are smaller leading to lower abrasion rates (B/Hj. - NHL *)

c) the surface breeding rate is higher, preventing the observation of the fragments.

The existence of non growing crystals in pure solutions in the 2-25 um range was evidenced by the semi-continuous seeded batch experiments of Youngquist and Randolph (1972) on ammonium sulfate, see chapter 4, section 4.2.6. Their stirrer speed and supersaturation dependence is in full agreement with the present results.

156

i

i

157

CHAPTER 6

THE EFFECT OF THE HYDRODYNAMICS ON GROWTH AND NUCLEATTON

6.1 Introduction 6.2 Modelling of growth 6.3 Description of the growth experiments 6.4 Results of the growth experiments 6.5 Nucleation experiments 6.6 Comparison of fluid bed nucleation kinetics with CMSMPR kinetics

6.1 Introduction

The CMSMPR experiments described in the previous chapters conclusively showed that two nucleation mechanisms can be operative, depending on the level of supersaturation. Thereby it was assumed that the growth rate is a linear function of the supersaturation. However, in the evaporative CMSMPR experiments it was not possible to measure the supersaturation, since the half-life times of the supersaturation in the growing suspension are much shorter than the time needed to obtain a crystal-free sample of the solution. In order to test the assumption of a first order dependence of the growth rate on the supersaturation, growth experiments have been performed. A small, conical fluidized bed was used in which seed crystals were gently fluidized with the supersaturated solution. The supersaturation and the growth rate could be freely varied since the nuclei produced by the seeds were removed by the upflowing solution preventing a rapid build up of mass transfer area in the fluid bed. In addition to the growth experiments it was tried to quantify the effective nucleation rate caused by the growing seed crystals. Samples of the solution which were taken after the passage through the fluid bed were stored at the same temperature at which the seeds were growing. By this procedure only those nuclei are expected to develop into nuclei which would be viable in a real crystallization experiment at the same, constant supersaturation. The modelling of the growth kinetics in the next section will be done in an empirical way since in fluid bed experiments the growth rates of the in­dividual faces cannot be measured.

6.2 The modelling of growth

6.2.1 Theoretical models

Crystal growth proceeds by the incorporation of "growth units", which may be ions, molecules, or molecular complexes, in the surface of the crystal. The essential role of the crystal-solution interface makes crystal growth very sensitive to. impurities, analogeous to other surface bound phenomena like heterogeneous catalysis or emulsification etc., which may be influenced by p.p.m. levels of specific "poisons".

On a planar crystal surface, which is perfectly flat on the molecular scale, a single growth unit will not attach since it has not enough bonds with neighbouring growth units.

158

solution

"step"

crystal surface

Fig. 6.1 Planar crystal growth by the incorporation of growth units in kinks of a step

Fig. 6.2 The spiral growth mechanism

a) Step caused by a screw dislocation. The dislocation line is in­dicated by line a.

b) The formation of a growth spiral from an initial straight step, starting at the dislocation line.

c) Scetch of a spiral hillock. The steps are polygonized according to the six-fold symmetry assumed in this example.

159

In order to explain growth at low supersaturations the presence of "steps" in the surface of the crystal is necessary, see figure 6.1. The incorpora­tion of a growth unit in the "kink" position is energetically "neutral" since it is seen that no extra surface area is formed. Two models describe the formation of steps on the crystal surface (Bennema, 1976) . From the energy per unit of step length (line tension) it can be cal­culated that a minimum amount of growth units must be adsorbed next to each other in order to form a stable, "2-dimensional cluster" or nucleus. The size of this 2-dimensional "critical" nucleus depends on the supersaturation. The growth rate of the crystal depends on the frequency at which these clusters are formed and on the step velocity. This growth model is known as the "Birth and Spread" model (B&S). It can be derived from this theory that in general the growth rate becomes appreciable only at more elevated supersaturations. In order to explain the growth at low super-saturations, Burton, Cabrera and Frank (1951) showed that the presence of screw dislocations in the crystal induces steps on the surface. In figure 6.2-a such a step is shown. Due to the growth of the step it winds itself up around the point at which the "dislocation line" intersects the crystal sur­face, figure 6.2-b. In this way a "spiral hillocks" are formed covering the crystal surface with steps. More details can be found in (Mullin, 1972).

6.2.2 Relations for crystal growth

From the models given in the previous section the supersaturation dependence of the growth rate can be derived. For the spiral growth model the following equation is given:

G = C.a2/a1 tanh [c^/a] (6.1)

where G is the growth rate [m/s] and o the relative supersaturation. For low supersaturations a << a. the growth rate is approximated by G ~ 0 ; at high supersaturations, a >> a., a linear law applies, G - a. For the B&S model the equations are more complex (Bennema, 1976). For the modelling of crystal growth at low supersaturations the spiral growth model is to be preferred. The flux R„ of growth units being incor­porated in the surface by this mechanism can be given by:

RG = kr . Ac..r [kg/m*s] (6.2) E •• 1 £ r £ 2 (spiral growth)

where R„ is the mass flux due to growth. The two-dimensional nucleation growth models can be approximated by a power-law dependence on the supersaturation too. In that case the value of r lies between 2 and 5 (Bennema, 1976). In series with the incorporation reaction the dif fusive/convective mass transfer process is operative in the solution:

RQ = kd. (AcB - Ac..) (6.3)

where k, is the mass transfer coefficient for diffusion in the solution (see figure 0.3). The overall mass transfer coefficient for growth, defined by

k = T 5- (6.4) * ACB

160

can be related to k and k,, assuming r 1_ k g

1_ k (6.5)

The case r = 2 is more complicated, see Karpinski and Koch (1979)- It 'is clear from equation (6.5) that the smallest of the two mass transfer coeffi­cients, k^ and k^, will have the largest effect on k . For instance, growth is diffusion limi ted i i f k, << k .

d r g

incorporation*»'AC|r

solution

interfaced

convection/diffusion — (ACB- AC|) CB__

w«q

fc_: AC0

Fig. 6.3 The mass transfer to a growing crystal

In order to separate the incorporation and diffusion step the hydrodynamics around the growing crystal must be varied. The effect of the hydrodynamics on the diffusion step is thought to be composed of three terms:

I) diffusion in stagnant solution II) convection due to the slip velocity of a crystal settling in a stagnant

solution III) extra convection due to the turbulence in a flowing solution.

In Appendix 6b these three terms are evaluated. The results for ammonium sulfate are presented in figures 6.4 and 6.5- The physical properties of the ammonium sulfate solution used in the calculations are given in Appendix 6a. The diffusion coefficient was estimated from literature values at 25°C near saturation, using Einsteins relation

ID. n = constant (6.6)

Since ammonium s u l f a t e c r y s t a l l i z e s from fa i r ly concentrated solut ions an appreciable "d r i f t flux" s e t s up towards the surface of the c r y s t a l enhanc­i n g t h e mass t r a n s f e r . In o rder to c o r r e c t for t h i s e f f e c t , known as

161

the Stephan-effect (Westphal and Rosenberger, 1978), an effect ive d i f fus ion coeff ic ient was used defined by:

ID eff JD_ 1-w (6.7)

where w is the mass fraction (Appendix 6a-5)

Figure 6.4 shows the contribution of stagnant diffusion (I), slip-velocity (II) and turbulence (Ilia and Illb) to the total value of k, as a function of size. The stagnant diffusion becomes important at small sizes only. For large crystals the value of k, is virtually independent of the crystal size.

| 1 0 - W S ]

k*

ï t 50 .

:i _ N»-58L-°14,U„= 0.1m/s _ N«-58L-o«,U„= 0.3m/s

• U„= 0.1m/s total mass transfer coefficients + U„= 0.3m/s

O stagnant diffusion O slip velocity effect A turbulence v turbulence

— L fuml

Fig. 6.4 Mass transfer coefficients in ammonium sulfate solutions at 70°C

Figure 6.5 shows the effect of the temperature and the turbulence on k,. The turbulence is given by the velocity of the liquid (cm/s) in a tube with a hydraulic diameter of 10 cm. The temperature dependence of k, is ap­proximated using the Arrhenius equation

kd,. 6 X P " RT (6.8)

where E. is the activation energy. It is seen that E. values of about 12 kJ/mole result. From the calculation of k, it should be noted that the tem­perature dependence of k, is rather complicated and that equation (6.8) is therefore only of empirical use. The theoretical k, values will be compared with the experimental ones in section 6.4.

162

-8.5

In k„

-9.0 -

-9.5

-10.0-

-10.5

\ CK N X N

V X

[ra/s| 0.0 0.1 0.3

EA [kj/mole |

12.0 12.0 12.1

\ , X

L = 1000 pm 0 M r . = 0.1m

\ N X x

V X X V \

\

V X \

V

X \

\ V \ N X

\ >

X V X

cl x 80 70 60

■J , , _ 50

_ 1 _

X

i.0 30 25 20 _ i i i l

10 i

2.80 3.00 3.50 - * • ( — » 10M

Fig. 6.5 Arrhenius plot of calculated mass transfer coefficients in saturated ammonium sulfate solutions

6.3 Description of the growth experiments

The growth experiments were performed with seed crystals in a fluid bed, see figure 6.6. The supersaturation was controlled by the saturation temperature of the feed, which was constant during the experiment, and thé temperature in the constant temperature bath in which a cooler made of stainless steel and the fluid-bed; made of perspex were submerged. The capacity of the sup­ply vessel was 6 1, its temperature was kept - 5°C above the saturation temperature in order to dissolve the nuclei produced by the fluid bed. The fluid bed was machined from perspex. The conical bore was chosen since the operation of straight bore fluid beds is difficult at the low void frac­tions employed: small crystals tend to be carried away by the upflowing liquid while the largest ones settled to the bottom of thé fluid bed.

The feed solution was prepared using reagent grade ammonium sulfaté and de-ionized water. The solubility of ammonium sulfate is given in Appendix 6a-1. Prior to an experiment the concentration of the solution was checked by measuring the density of the solution. A second check on the saturation tem­perature is obtained from a complete series of growth and dissolution

163

experiments done shortly after one another, by extrapolation of the growth and dissolution rates to find the temperature at which both are zero.

rheraoatter hort

o-rlng

overflow

derail: the conical fluid bed

i 10 Ld., feed

Fig. 6.6 The fluid bed growth set up

A known amount of seed crystals (1-1.5 g) was introduced. The average size was 800 um (738 urn < L < 837 um, Veco electroformed precision sieves with square openings). The standard growth, respectively dissolution time was 20 min. except for very high rates. Rinsing of the crystals prior to the ex­periment in order to remove excess fines from the surface was tested but discarded since it had a negative effect on the experimental accuracy. For most of the experiments a flow rate of 11 ml/s was used. The temperature in the fluid bed was determined with a thermometer with an accuracy of 0.1°C. After the growth experiments the crystals were filtered, rinsed with a methanol-ammonium sulfate-water mixture as described in chapter 4 and dried. From the weight increase the growth rate is calculated:

m 1/3 L 1 lm ' J At (6.9)

m , m are the initial and final weight, L is the initial size and At is tfie growth time. The calculation of k-values is given in Appendix 6c since due to the non-spherical shape it is not possible to compare k-values for a single crystal face directly with the overall k-value.

164

6.4 Results of the growth experiments The results of the growth experiments are given in table 6.1 and figure 6.7 together with reported values of k given by Parkash et al (1968), Mullin et al (1970) and Tengler and Mersminn (1983). Inspection of figure 6.7 shows that the present growth and dissolution experiments performed in the fluid bed (black dots and circles) lie close to each other over the whole tempera­ture range. The activation energy was calculated to be 20.4 kJ/mole for growth and 19.0 kJ/mole for dissolution. The fluid bed growth experiments of Parkash et al (1968) lie on average 25# below the present results. The ac­tivation energy compares well, E. = 23.8 kJ/mole. The fluid-bed experiments by Tengler and Mersmann at 20°C are exactly in line with the present results. The small arrows indicate the upper and lower limits of k values from single crystal growth experiments by Tengler and Mersmann. These over­all values are estimated in Appendix 6c from the measured growth rate of the (100) face. Tengler and Mersmann report that the growth rate of these (100) faces is second order dependent on Ac which is in contrast with Mullin (1970) who reports a linear dependence for the (100) face but a second order dependence for the (001) face. This controversy is not yet resolved but might be caused by the use of different conventions for the crystallographic axes.

■9.00-

In 'S-. [ra/s]

-10.00

9 d • O ▲ A ■ e ▼ V X

experiment

fluid bed fluid bed fluid bed mixed suspension single crystal single crystal

source

present results Tengler,Mersmann Parkash et al present results Tengler,Mersmann Mullin et al (1970)

-11.00

-—Jft l 7 0 . 60 5,0 UP 3,0 , 2,0 . 10 5

0.0028 0.0030 0.0032 0.0034 0.0036 — ~ 1 / T [ l f 1 |

Fig. 6.7 Growth and dissolution of ammonium sulfate

165

Table 6 . 1 Growth exper iments with ammonium s u l f a t e

# T ,. sat [*C]

5 24

29

'12

60

25

urel [cm/s]

8.1 1.9 1.0 5-0 10.2

1.9 7.4 10.8

2.0 4.4 6.4 11.3

2.3 6.8 11.9

0.6-2.7

face

[hkl]

(100)

(100) (100) (100) (100)

(100) (100) (100)

(100) (100) (100) (100)

(100) (100) (100)

(100)

k w #

g [10_5m/s]

8.2 4.9 6.3 7-8 12.5 6.4 8.9 10.6

7-1 10.2 11.8 17.8

12.0 17.8 24.4

3.47

w* k

[10~5m/s]

15.8

4.2 8.9 13-5 16.6

9-0 18.5 21.3

H.9 16.4 19-7 22.4

12.7 17.9 24.8

-

k e [10_5m/s]

6.9 4.1 5.3 6.6 11.5

5-1» 7.5 8.9 6.0 8.6 9-9

15.0

10.1 15.0 20.5

2.9

kd [10"5m/s]

13.3

3-5 7-5 11.3 13.9 7.6 15.5 17.9 10.0 13.8 16.5 18.8

10.7 15.0 20.8

-

Reference

Tengler & Mersmann (1983)

,

Mullin et al(1970)

20

25 35 45 55 65 75

35 36 43 34 32 51 72

YY 36 37

6-8 -----

---6 4 4 4

20 20

fluid bed

fluid bed fluid bed fluid bed fluid bed fluid bed fluid bed

fluid bed fluid bed fluid bed fluid bed fluid bed fluid bed fluid bed

1-1 batch 1-1 batch

k w g

8.4 4.66 8.30 11.3 14.9 19.7 23.9

7.65 8.11 12.0 10.9 10.7 17.4 26.5

13.3 16.1

C d 9-3 -----

-17.4 -

10.9 13.2 16.8 21.1

k g

4.22

2.34 4.17 5.68 7.49 9.91 12.02

3.84 4.08 5.18 5.47 5.37 8.75 13.3

6.68 8.08

k, d 4.68

-----

-8.73 -

5.47 6.62 8.43 13.9

Tengler, Mersmann (1983) Parkash et al(1968)

Present work

Hartgerink Hartgerink Hartgerink Winter Knaap Knaap Knaap

Yap

166

From the comparison of growth and dissolution rates it is concluded that the growth process in the fluid bed experiments is almost completely diffusion limited. The growth and dissolution rate of the (100) face are different, according to Tengler and Mersmann, which suggest that the growth of the (100) face is only partly diffusion limited. The observation of a second order dependence of the growth rate of the (100) face on the supersaturation is in line with the suggested reaction limited behaviour. It must be mentioned here that the value of the k values ex­tracted from the (100) growth data is only indicative since it was determined assuming a linear dependence of the growth rate on the supersaturation.

The value of the experimental mass transfer coefficient for convection/ dif­fusion is compared with the estimates given in section 6.2, see figure 6.8. It is seen that the predicted activation energies differ considerably from the experimental ones. This is most probably caused by the approximative character of the Einstein relation, equation (6.6) in predicting the tem­perature dependence of the diffusion coefficient. The actual difference between the experiments and prediction using the slip-velocity theory is 30$ at 25°C. At this temperature the diffusion coefficient was known and equa­tion (6.6) was applied using this value as starting point. In view of the experimental accuracy and the many parameters involved in the theoretical approach, a discrepancy of 30% is not a bad result. The larger differences at high temperatures can be eliminated by adjusting the estimate of D. It is therefore concluded that the mass transfer in a fluid bed under conditions of high void fractions as employed in the experiments can be described by the slip-velocity theory. In order to match theory and experiments an inde­pendent determination of the diffusion coefficient is needed at elevated temperature and concentration.

-9.0

In J<d_ |m/s|

t

-10.0-

-11.0.

N L n 1000 M") ^ N predicted, E» = 12kJ/mole

vx experimental. EA = 19kJ/mole

^ \ ♦ U„ = 0.3m/s.

slip-velocity

0.0028 0.0030 0.0032 0.003». 0.0036 — T IK-'1

Fig. 6.8 Comparison of calculated and experimental mass transfer coefficients

167

The effect of the stirrer speed viz. degree of turbulence on the mass trans­fer has not yet been verified. Preliminary experiments using a seeded continuous crystallizer with a classifying outlet designed to retain the seeds instead of a fluid bed were inconclusive since the amount of seeds employed was too large, which resulted in a rapid and substantial decrease of the supersaturation to near zero values, which fact was detected from mass balance considerations. Additionally, at more elevated supersatura-tions, extra surface area was produced due to nucleation. It is suggested that further growth experiments should be performed in a stirred vessel with smaller amounts of seed crystals while a shorter residence time for the smallest crystals/nuclei should be used.

6.5 Nucleation experiments Nucleation experiments were performed in the fluid bed growth experiments in order to compare the nucleation rates in a non stirred configuration with those in a stirred suspension. The procedure was as follows: Before the experiment the experimental setup was kept for at least 1 hour in an undersaturated state, AT = 2°C, in order to dissolve all traces of crys­talline material which could serve as a source of nuclei. For the detection of nuclei two 150 ml glass bottles with perforated rubber stoppers were placed in the overflow line of the fluid bed, and submerged in the same thermostatic batch as the fluid bed. The solution containing nuclei from the fluid bed flowed through these bottles, continuously replacing their contents. The nucleation rate at a given instant was determined by coupling off one of the glass sampling bottles and allowing the nuclei present in the solution to grow for three hours at the same temperature. By this procedure it was assured that the same supersaturation as in the fluid bed was main­tained during the growth of nuclei. The amount of nuclei was calculated from their weight, after filtering, rinsing and drying, and their size was deter­mined microscopically. The nucleation rate was obtained from

N.4> B = -y* (6.10)

where N is the number of nuclei formed in the volume V of the sampling bottle, and <> is the volumetric flow rate. In order to verify that all nuclei came from the fluidized crystals, prior to each growth experiment a blank sample of the supersaturated solution was taken. From the blank samples it was concluded that very careful experimentation is required in order to avoid nuclei from other sources than the fluid bed. Especially the cooling down of the undersaturated solution in the fluid bed should be done slowly. In order to avoid misinterpretation of the nucleation rates due to initial breeding, the second sample was taken at the end of the experiment. It is reasonable to suppose that the influence of initial breeding is negligible after 30 minutes of growth. The results of the experiments are presented in figure 6.9. From figure 6.9 it is seen that the nucleation starts above a minimum value of the growth rate and increases linearly with the growth rate. This observation was correlated with the qualitative observation that the seed crystals grew with smooth surfaces at low growth rates while a definite macro-roughness or polycrystallinity became apparent at higher growth rates.

168

6000

- |#/m2s|

1.000

2 0 0 0 -

"smooth zone of increasing j "rough' "roughness"

• O ^ I

i.ö T T -^ G llO-'m/sl

Fig. 6.9 Secondary nucleation induced by seeds growing in a fluid bed

This fact is shown in a series of SEM pictures of the crystals from normal growth experiments, see figure 6.10, taken at increasing growth rates. It is seen that the seed crystals (DSM, technical grade ammonium sulfate) grow out to poly-crystalline "buildings". At the highest growth rates it is observed that the largest protruding parts can break loose. From the magnification factor is is concluded that their size lies between 250 and 400 um. It is probable that crystals smaller than this size will be carried away by the upflowing solution. The nucleation experiments however were carried out for growth rates below 1.7 * 10 m/s. Therefore the breaking loose of large parts, of the order of 200 vim, was not considered as the source of nuclei. Moreover, crystals of these size would be immediately observed in the sampling bottles. The observation that nucleation and enhanced polycrystallinity or macro-roughness occur simultaneously is significant but is not sufficient justification to assume a direct causal relationship, since the size of the nuclei probably is much smaller than the observed surface details and since the removal of nuclei from the crystal surface at such a small size cannot be observed directly. In order to get an impression of the number of nuclei generated by one seedz-crystal it was calculated that an average crystal of 1 mm size has 5-4 * 10 m* surface area. The highest rate measured in the fluid bed experiments is 4500 nuclei/m2s. Consequently the rate per crystal is one nucleus in 41 s, which means that each seed generates about 40 nuclei

10 m/s. At lower a growth rate, in 1800 s, at_a growth rate of 1.7 * G = 0.7 * 10 m/s, B/A = 1000 nuclei/m*s, on average one nucleus is produced per 185 sec. per crystal.

169

a) Seeds, 768-837 pm b) G = 0.18*10 m/s c) G = 0.68*10 m/s

Fig. 6.10 The polycrystalline outgrowth of seeds in the growth experiments Magnification 28x

The lower limit of growth rate below which no nucleation is observed may be explained by three mechanisms:

a) the nucleation mechanism is pure mechanical breeding; a minimum super-saturation is required for the survival of nuclei. In this case the size of the nuclei will be in the order of the critical nucleus,

b) the nucleation mechanism is surface breeding, limited by formation and survival. The proto-nuclei are removed at small sizes, and the lower limit of nucleation is caused by the non survival of the nuclei below this limit,

c) the nucleation mechanism is surface breeding and is limited by formation. The macro-roughness acts as a source of nuclei, and increases strongly above the minimum growth rate for nucleation.

170

Strong evidence for the breaking loose of protruding parts of the polycrys-talline seeds was inferred from closer inspection of the SEM pictures, see figure 6.10, pictures c and d, where clearly depressions are seen, sometimes in the middle of a face. These depressions cannot be caused by simple attri­tion since this is highly improbable in the center of a face when the edges remain undamaged. It is therefore concluded that these depressions are caused by the removal of large protruding crystallites. Pictures e and f show that at high growth rates these crystallites can reach a large size before they are removed. The large size of these crystallites is not a result of growth after their dislodgement since a) small crystals are washed out of the fluid bed very easily and b) it can be seen that many of the crystallites in picture f are partly

damaged. The other two explanations cannot be ruled out, based on the fluid bed ex­periments only. Mechanical breeding seems less likely than surface nucleation since in the CMSMPR experiments with the pure system mechanical breeding is not important.

It can therefore be concluded that the nucleation is surface breeding. It is proven that at least a part of the nuclei is broken away from the crystals at larger sizes. But does this mean that the mechanism of the CMSMPR experi­ments described in chapter k is also governed by the formation of macro-roughness? This suggestion will be discussed in the next section.

6.6 Comparison of fluid bed nucleation kinetics with CMSMPR kinetics

In order to answer the question whether the formation and subsequent breakage of macro-roughness might be important in the CMSMPR experiments too, the kinetics of both types of experiments are compared in figure 6.11 where log B / ^ is plotted for both evaporative and cooling CMSMPR experi­ments (see chapter l\) and the fluid bed experiments described in the previous section (6.5). It is seen that both the slope of the line and the position are in perfect accordance with other cooling crystallization results at 40°C. Inspection of SEM pictures of crystals obtained from the 40°C cooling crystallization ex­periments (chapter 4) reveals both a pronounced polycrystallinity and traces of breakage of protruding parts (see figure 6.12). Moreover it seems that the polycrystallinity may be caused by agglomeration of small crystals. The same observations can be made for the evaporative experiments at 50°C, see figure 6.13. This confirms the suggestion that macroscopic breakage of polycrystalline conglomerates plays a role in the CMSMPR experiments of am­monium sulfate. The consequences of this fact can be far reaching. Mason (I960) already observed that macroscopic breakage is accompagnied by what he called "splinter-breeding", the formation of a large number of fast growing crystals directly after the dislodement of a needle (MgS0,,.7H_0).

171

4.50 J

4.00

3.50

3.00 i

log

2.50

B°/AT

[#/m2s)

,300 rpm j 6?<>c e ¥ i p o r l h o n 700 rpm ' K

Bourne & Faubel.1.7l cooling crystallization,(,0°C

fluid bed, 40"C U i

Bourne & Faubel.421. cooling crystallization,40 C

cooling exp.,40t,section .4.4.5 I =500 rpm, 600 rpm)

log (G(m/s])

-7.5 i -7.0 -6.5

Fig. 6.11 Comparison CMSMPR and fluid bed nucleation kinetics

This offers the possibility for the following series of mechanisms: a) nuclei or small crystals are attached to each other or to a larger crys­

tal, b) consequently polycrystalline conglomerates are formed, c) depending on the size and structure of the conglomerates and the stirrer

speed loosely attached parts may break loose, d) due to this breakage additional small nuclei are formed. Therefore it is concluded that ammonium sulfate has a tendency to agglomerate. The resulting conglomerates are friable and enhance the "surface nucleation" by "polycrystalline breeding" (Mason, I960) and pos­sibly "splinter breeding" (Mason, i960).

It will be clear that agglomeration and breakage will be a function of the crystal and nucleus size and of the strength of the polycrystalline conglomerates. The reduction of the nucleation rate at lower stirrer speeds in the cooling crystallization experiments may be explained by the reduced fragmentation tendency. It is clear that agglomeration, if actually opera­tive, leads to a decrease of the effective nucleation rate.

172

c) 704 pm fraction, 18x d) 60x detail of C Fig. 6.12 SEM pictures of sieve fractions of product crystals obtained from

CMSMPR cooling experiments at 40°C. Arrows: traces of breakage, circles: adhering crystals

In the evaporative experiments agglomeration may be counteracted by the tur­bulence in the boiling zone or by increased stirrer speeds. On the other hand increasing the turbulence increases the mass transfer coefficient. This might explain why in the cooling experiments, figures 6.10 and 6.12 many deep crevices are seen: at low stirring speeds the boundary layer for diffu­sion is thicker leading to enhanced mass transfer rates at the corners of a crystal and at protruding parts, in comparison with the center of a flat face. Increasing the stirrer speed distributes the local mass transfer rates more evenly over the crystal. The uneven growth at the corners and edges (figures 6.10 and 6.12) suggests the operation of the "Birth and Spread" growth mechanism. The same conclusion was reached also by v. Enckevort et al (1983) who found a negative correlation between spiral dislocation density and growth rate for ammonium sulfate.

173

a) 30x exp. 7H-51 b) 120x c) 120x

Fig. 6.13 SEM p i c t u r e s of the 209 urn f r a c t i o n of product c r y s t a l s from CMSMPR evaporative experiments a t 50°C. G = 1.26 * 10 'm/s

The l a s t p o i n t to be discussed i s the lower l imi t of the growth ra te below which no nucleation i s observed in the f luid bed experiments . Two explana­t ions were considered: a) below t h i s l imi t the c rys ta l grows in a smooth way b) the seeds were produced in an environment were p o l y c r y s t a l l i n i t y i s

suppressed. Consequently i t takes some time to develop protruding p a r t s , and a t low growth r a t e s the du ra t i on of the experiment was not long enough.

To t e s t whether a prolongued experiment at low growth ra tes could induce macro-roughness a se r ies of experiments was performed a t cons t an t growth r a t e but increased growth times, 1200, 2400, 36OO and 7200 s . The resu l t i s shown in figure 6.14 where i t i s seen that both effects occur: the polycrys­t a l l i n i t y , which i s p r e s e n t in the seed, i s developing in time but the r e su l t i ng c rys ta l a t 7200 s has a much smoother appearance , in comparison with f igure 6.10 e. I t i s therefore concluded that i t i s not cer ta in whether the nucleation ra te i s n e g l i g i b l e below the observed lower l imi t a t longer growth times. On the other hand, the s i m i l a r i t y between the k i n e t i c s when p l o t t e d on double l o g a r i t h m i c s c a l e s i s s t r ik ing , which suggests tha t possibly in the CMSMPR experiments the nucleation ra te f a l l s to zero at low growth r a t e s . If t h i s i s t r u e the resu l t ing nucleation k ine t ics of the pure system might have the same form as the f luid bed nucleat ion k i n e t i c s , see f i g u r e 6 .9 , in which case i t i s seen t h a t the use of the double logarithmic p lo t i s misleading since i t does not suggest such a behaviour.

174

SW-51 G = 0.58*10" m/s A t = 1200 s magnification 21x

SW-52 7 G = 0.53*10 m/s A t = 3600 s magnification 16x

SW-54 ? G = 0.51*10 m/s A t = 7200 s magnification 16x

Fig. 6.14 Fluid bed growth experiments. The effect of the growth time on the po lyc rys ta l l in i ty

175

CHAPTER 7

EFFECT OF MECHANICAL ABRASION ON NUCLEATION AND GROWTH

7.1 Introduction 7.2 Abrasion models 7-3 Abrasion experiments

7.3.1 Experimental set-up 7-3.2 The analysis of abrasion experiments 7.3.3 Results of the abrasion experiments

7.4 Conclusions

7.1 Introduction

Mechanical breeding, the formation of fragments by pure mechanical action was introduced in chapter 2 as one of the two important groups of nucleation mechanisms. In order to discriminate this mechanism from surface breeding it was shown that in pure mechanical breeding the growth rate of the parent crystals does not influence the rate of nucleation directly, though it is likely that the survival of the fragments is a function of the bulk super-saturation and thus indirectly a function of the growth rate. Since no growth is needed to obtain a correct "removal" rate, see chapter 2, it is possible to analyze the rate at which fragments are produced in a non-solvent were growth and dissolution are absent. In this chapter the results of such abrasion experiments with ammonium sulfate are reported. The results of the experiments have been applied already in chapter 5 to model the nucleation ratge of slow growers. Before presenting the results in the next section a short summary of theoretical models for abrasion in suspension will be given.

7.2 Abrasion models

Abrasion of crystals in a stirred suspension may take place by collisions against the impeller, against the walls of the crystallizer and against other crystals. Generally the nucleation rate B [#/m3s] due to these impacts is assumed to be proportional to the energyE [J] transferred during an im­pact, the impact frequency per crystal, co [s ] and the concentration of the crystals, c [#/m3]:

B - Ê = E.co.c (7.1)

T h e o r e t i c a l r e l a t i o n s h i p s for the dependence of E, <o and B on the s t i r r e r speed, N, the c rys ta l s i ze , L, and the c rys ta l concent ra t ion , c , a re s#hown in tab le 7-1 for the various modes of impact. The energy t ransfer r a te E in case of a wide c r y s t a l s i z e d i s t r i b u t i o n i s expressed as (Ottens e t a l , 1972):

Ê t = ƒ E.co.n dL (7-2)

176

Table 7-1 Dependence of various impact models for abrasion on stirrer speed, crystal size and impact energy

mode of impact

crystal-impeller

crystal-wall crystal-crystal

authors

Ottens et al (1972) , Ottens, de Jong (1973) Evans et al (1974) Nienow (1976)

Garside, Janció (1979)

Kuboi et al (1984) Èvans et al (1974) Ottens et al (1972) Ottens, de Jong (1973) Evans et al (1974) { Nienow, Conti (1978) Conti, Nienow (1980) ' Kuboi et al (1984)

frequency

(i>

~N -NL: 1 ' 2

-N

N2L -N2L

-CNL2 !/3

. **

impact energy,E

V2

-N2

<

-N2

-N2

~N2

-NV'3

m

-L3

-L3 -L 3

-L3

-L3 turbulence gravity

Energy * Et

-c N3

~c N

-c N -c N -c N!| ~c N5

~c2N3 2 3

~c

~c N -c N

transfer rate

L3

L3

!>

1? J IO.42

8J.4

Generally only this energy transfer rate is modelled, assuming a propor­tionality factor between B and E depending on the supersaturation and other system parameters (Evans et al, 197*0. The nucleation rate is assumed to be proportional to this energy transfer rate, modified by a supersaturation de­pendent survival or outgrowth effect. The modelling of this energy transfer rate comes down to finding expressions for E and for o>. The energy per impact E is assumed to be proportional to the kinetic energy of the crystal where for crystal-object impacts the relative velocity, v, between crystal and object at the moment of impact is used. For a stirred vessel all velocities are assumed to be proportional to the stirrer speed, except for the gravity driven collisions (Evans et al, 1974). For crystal-crystal collisions additionally a slight effect of the crystal size on the velocity is noted.

The impact frequency to for crystal-impeller collisions is dominated by the circulation frequency of the suspension (Ottens et al, 1973):

(i> vc vc

* K N D J -XS.B _£ N V V c c

(7-3)

However not all crystals that pass the impeller zone are actually hit by the impeller. Bauer et al (1974) propose a target efficiency and show that this efficiency drops to zero below a minimum crystal size. This was verified by numerical calculations of particle trajectories around an impeller blade (Ramshaw 1974). A better applicable result was reached by Nienow (1976) who showed that the impact (or target) efficiency is inversely proportional to a parameter X given by:

_Dg_ v.ut (7-4)

177

where u is the terminal velocity of the crystals and D the diameter of the impeller. This factor X is the inverse of the separation number N used by Bauer et al (197*0. P

Consequently the target efficiency, n. is proportional to nt " X " V'Ut ~ N'Ut (7>5)

In the Stokes regime: U t ~ L

Consequently: n - N.L (7-6)

A slight modification, given by Garside and Jancic (1979) is needed for the intermediate flow regime:

u t - L1'14 (7-7) The impact frequency is obtained by combining (7-3) and (7.6):

u = TK . w - N2L (7.8) t vc

In all cases complete suspension is assumed. At low stirrer speeds, however, the largest crystals will have a lower circulation frequency than the liq­uid. To a first order approximation this effect can be modelled assuming a neat flow configuration with an impeller in a draft tube, designed to yield a uniform velocity throughout the vessel. In the upflowing liquid the velocity of the crystals will be lower than that of the liquid, while in the down flow region a higher velocity is found. The total circulation time may be expressed as:

t -_U2±_ + 1/2_L_ c v -u v +u,_ ax t ax t

where L is the length of the average path of a c rys ta l or f luid element around the draf t tube, and v is the linear velocity of the liquid in and outside the draft tube. The circulation time t of a liquid element i s re­lated to v by: v c

ax t = — (7.10)

V C V v i /

ax Combining (7.9) and (7-10) yields a circulation efficiency

t u 2 * = r ^ = i - (—) (7.i i) c t v c ax

n è 0 c

The modelling of crystal-crystal collisions is more complicated. Since the total slurry density can be written as:

1^ = c pckv L 3 (7.12)

178

The energy transfer rate for crystal-crystal collisions is proportional to MT (Ottens et al, 1973)• This proportionality, however, only holds for monosized crystals (Evans et al, 1974; Kuboi, 1984). The importance of crystal-impeller and crystal-wall interactions decreases when the system is scaled up. For large crystallizers it is expected that crystal-crystal impacts can become predominant.

It should be noted that none of the above mentioned models quantifies the effect of the habit and surface structure of the crystals on abrasion: under growing conditions phenomena such as step-bunching, roughness of crystal faces and polycrystallinity, all of which are governed by the supersatura-tion, hydrodynamics and impurities, are able to promote the fracturing and abrasion of crystals. In the classification of nucleation mechanisms developed in chapter two the enhancement of abrasion due to crystal growth is considered as surface breeding. This type of growth-related phemomena cannot be studied in an inert medium. At most an enhanced initial abrasion rate is to be expected. Furthermore, the shape, size and dislocation structure of the fragments has not been considered in the aforementioned models since the direct relation between energy transfer and breakage is difficult to model in general terms. As discussed in chapter two the plastic deformation of the fragments due to the fragmentation process may severely reduce the growth rate of the frag­ments, or even lead to dissolution. This effect is excluded in inert media. Summarizingly it can be stated that the models available at present incor­porate the most important parameters but do by no means allow for quantitative predictions. The "missing links" are a) the crystal description of the flow pattern and crystal trajectories up

to the point of impact b) the effect of material properties such as the hardness of crystals and

object on the fragmentation/abrasion c) the influence of the shape and surface structure of the crystal in the

process of fragmentation

In order to model the contribution of fragmentation of the parent crystals to the secondary nucleation rate abrasion experiments were performed in an abrasion vessel of identical dimensions with the CMSMPR crystallizer, as used to determine the crystallization kinetics of ammonium sulfate. The ex­perimental set-up, the method of analysis and the results of the abrasion experiments will be discussed in the next section (7.3).

7.3 Abrasion experiments

7.3.1 Experimental set up

Sieved fractions of good quality technical grade ammonium sulfate crystals were abraded in a 20 1 stirred vessel (figure 7-1) of the same dimensions as the crystallizer.

179

Fig. 7-1 Abrasion vessel

sample valve

At the beginning of the experiment 250 grams sieved c r y s t a l s were added to 13 1 of f i l t e r e d e t h a n o l sa tu ra ted with ammonium su l f a t e . The s t i r r e r was s t a r t ed and a t fixed times 200 ml samples were withdrawn through the sam­p l i n g v a l v e . With t h e a i d of an 80 pm s i eve the l a r g e c r y s t a l s were separated from the abrasion fragments. 100 ml of sample c o n t a i n i n g abrasion fragments was mixed with 200 ml f i l ­tered ethanol , sa tura ted with ammonium s u l f a t e , c o n t a i n i n g 2 g /1 LiJ to improve the c o n d u c t i v i t y of the s o l u t i o n . A good a l t e r n a t i v e i s sodium iodide which i s much cheaper. A Coulter Counter (model ZR) equipped with a 200 um-orifice tube was used in combination with a 1024 channels Puls Height Analyser (Northern Electronics TN1705) to obtain the p a r t i c l e s ize d i s t r ibu­t i o n of the f ragments . C a l i b r a t i o n was achieved with s t anda rd l a t e x e s ( p a r t i c l e s ize 8.79 um and 44.6 um) . Measurements were performed in the range between 8 and 35 um.

7-3.2 Analysis of the abrasion experiments

From t h e experiments fragment production ra tes are computed as the increase of the fragment population densi t ies n„ with time for one c r y s t a l s i z e L ( = L s t a r t ) :

180

dn_ bf = d T T <7'13)

st This fragmentation rate b_ is a function of the size L_ of the fragments. Now B(L .) is defined crystals of a single size L as:

OB

B = B(Lgt) = J bfdLf (7-14) Lf=0

the total fragment-production rate. Table 7-1 indicates that a crystal-impeller collision mechanism, which is expected to be predominant in small scale abrasion experiments, is propor­tional to the concentration of the monosized crystals and the mass of an individual crystal. Since:

*T " c p c k v Ls"t { 7 - l 2 )

B/MT was expected to be merely dependent on the stirrer speed. Therefore J the fragment production rate per kg of parent crystals, has been defined as:

B ( L s t } r

1 f I f Jf."Up dt""| =M^ (7-l6)

st The total fragment production rate in comparable crystallization experiments is obtained by multiplying J(L ) by the weight distribution function g(L<31.) followed by integration over L :

St

mech = | J(Lét).g(Lst).dLst (7.17a) o

which is equivalent to n(L J

mech r v st'

-J B ( L s t> - i r^ d L s t <7-17b> o

where n is the population density.

7.3.3 Results of the abrasion experiments

Four series of experiments were performed as specified in table 1-2. Size distributions of the fragments as measured with the Coulter Counter delivered straight plots of the logarithm of the fragment population density n„ versus the size of the fragments L_. SEM pictures indicate that the frag­ments were heavily deformed, see rigure 7.2. The abrasion rates were

181

c o n s t a n t for a given experiment a f t e r an i n i t i a l l y h ighe r v a l u e . This i n i t i a l value decreased within - 60 s to the f i n a l value but was not very r e p r o d u c i b l e , whereas the f i na l value was reproducible. This lead to the conclusion that the i n i t i a l l y higher value i s caused by " c r y s t a l l i n e dus t" on the su r face of the c r y s t a l s , which a l so i s held r e spons ib l e for the " in i t i a l -b reed ing" nucleation mechanism, of by "growth-details", see chapter 2.

Table 1.2 Abrasion experiments; ser ies 1-4

experiments

1 - 7 5, 8 - 11 5.12 - 15 16 - 19

varied parameter

crystal size stirrer speed viscosity slurry density

range

300 - 2000 um 300 - 600 rpm 1.2 - 2.5 cP 10 - 30 kg/m>

Fig . 7 .2 SEM p i c t u r e s of the heav i ly deformed fragments

182

Variation of parent crystal length L

The abrasion rate j„ as defined in section 7-3«2 is a function of both the parent-crystal size L and the fragment size L„ see figure 7-3- The slope of the straight lines is the same for all experiments and has a value of

4 _i 4 - 1 -8.56 * 10 m with a standard deviation of 0.95 * 10 m , so

log j f = log jf(Lf=0) -8.56 * 10U Lf (7.18)

From this value of the slope the average size of the crystals, L,-,-., is cal­culated to be 43 pm, using L,.Q = 3-67/slope. An overall fit describing tne dependence of j„ on both L„ and L is ob­tained by plotting the (extrapolated) value of log j„ at L„ = 0 versus log L . The best straight line through the datapoints has a slope of 2.822, so: s t>

log j f = 19.467 - 8.56 * 104 * Lf + 2.822 log Lgt (7.19)

Equation. (7-19) can easily be integrated over L„ to obtain the total frag­ment production rate, J(L ). In first instance the lower limit of integration if chosen to be 8 um since below this value the linearity of log j_ vs L_ could not be checked. In figure JA this dependence is illustrated together with values of J obtained by directly integrating equation (7-18) within the same limits. It is seen that a straight line can be drawn through the data points, intersecting the L -axis at L = 450 um. This means that J is linearly dependent on L . In section J .2 it Was concluded, that for impacts with walls or stirrer B - cL , causing J to be independent of L . It now appears that B has a fourth power dependence on L above a certain minimum size. This can be explained by the "collision-corridor" model of Ramshaw (1974), who demonstrated a) the existence of a minimum size for im­pact and b) the decrease of impact velocity on a decrease of particle size. It would be interesting to investigate whether an increase of the viscosity does result in a higher value of this minimum size for abrasion. Though cor­relation 7.19 does not provide the best description of the measured abrasion rates it has the advantage that it gives a better fit in the region below 500 um and enables us to do some predictions of the abrasion rates which have to be operative in the crystallization process.

183

11.00 ■

10.00

9.00

8.00-

7.00.

exp.no symbol • 0 A O O X +

L.,(M-> 293 418 648 917 1297 1600 2000

Fig. 7^3 Variation of abrasion rate j„ with the fragment size L_ and the crystal s ize L . s t

S.0

4.0

3.0

2.0

1.0

0

JIL.,1

[lOWkgs)

|

3 /

>.yj£'

18,. /

A/ / /

/ /

/

i 1 1 1 /

/ /

/ / /

z _ «q.15 Lf > 8pm

— . Lit IMIDI

70

500 1000 1500 2000

Fig. 7.4 Variation of total abrasion rate J with L ^ at 600 rpm st

Variation of stirrer speed

At L = 1300 um a good correlation is obtained plotting J(L ) versus N (see figure 7-5) • The straight line intersects the N axis at N - 280 rpm. From table 1 it follows that a fourth power dependence can be expected when the effect of a "collision-corridor" is taken into account. The sharp decrease of the attrition rates below - 250 um is probably due to in-homogenity of the suspension. To verify this explanation measurements at smaller particle sizes are to be performed.

Variation of viscosity and slurry density

At 600 rpm and with a L of 1300 um the viscosity was varied between 1.20 and 2.50 cP, covering the operating range of saturated ammonium sulfate solutions. No effect of viscosity could be detected. It can be expected that at the lower particle sizes the viscosity will influence the impact efficiency. The slurry density was varied at 600 rpm and a L of 917 um from 12 to 30 kg/m3 . The fragment production rate B was found to have a linear dependence on MT< This proves that the measured abrasion rates result from impacts against the stirrer, baffles and/or vessel walls.

184

5.50O-

4.000-

Icg -B- . log ■

-7.500

ibraston fragnents > Bpo ( effective miclejtion rite SO evaporation abrasion frigaents > Ojm

•log O

Fig. 7-5 Variation of j, J(L J with NH

st' Fig. 7«6 Comparison between nucleation and

abrasion for crystallization experiments

Comparison between nucleation and abrasion

Integration of equation (7.17a) using equations {7.13) to (7-16) and using for the weight-distribution function g(L):

3 g(L) = n p k L" o\ / rC V

(7.2Q)

with n given by equation (3«28) and inserting VL, from equation (7-12) allows us to derive the total fragment production rate by abrasion for the crystal­lization experiments as:

, Bmech " kf MT <GT) 2.822 (7.21)

This shows that abrasion in a mixed suspension crystallizer although first order dependent on i/L, has an additional strong dependence on GT and thus on L,_0(= 3.67 G T ) . Assuming that the only source of nucleation in the crystal­lizer is B . (equation 7-21) leads to a final result:

B mech _ ,, G0.4l4 *T

(7.22) 7 with k' = 2.07 * 10 presented by line I in figure 7-6 together with the

measured crystallization data, line II for the 50°C evaporative experiments. A simple extrapolation to L. = 0, shifting the lower integration limit for the integration of equation (7-19) to Lf = 0, gives k' = 6.73 * 10 result­ing in line III. As the abrasion experiments were done with good quality crystals under non-growing conditions the line I and III represent the minimum values of nucleation caused by abrasion.

185

Overall correlation 4 4

Since both the plots of J versus L and J versus N intersect the L and N axis respectively it was attempted to find an overall correlation incor­porating both effects. The best result was obtained when B divided by 3 3 (n .c.NJL ) is plotted versus (N.L). The circulation efficiency n , equation (7.11) , typically has values between 0.90 and 0.99 except for She 300 rpm experiments, see Appendix 7.1. In the calculation an impeller discharge coefficient k = 0.35 was used, determined by using a neutral buoyancy flow follower. The terminal velocity of the ammonium sulfate crystals was deter­mined in ethanol, see Appendix 7.1. Figure 7.7 shows that a straight line is obtained intersecting the (N.L) axis at 4.25 * 10 [m/s]:

B(L) = 4.472 * 107.n .CN3L3 (NL - 4.25 * 10~3) (7-23)

The parameter N.L was shown before to be proportional to the separation num­ber, N , see section 7-2. The section that a minimum value of N or L is

S 6 ID necessarjr f or an effective impact can also be inferred from the results of numerical simulations by Ramshaw (1974). The minimum value for N.L, 4.25 * 10 [m/s] below which abrasion is negligible, is valid only for am­monium sulfate and the particular impeller employed. The viscosity does not seem to affect the correlation within the tested range of 1.2 - 2.5 cP. The effect of the density of the liquid should be investigated, especially since the density of ammonium sulfate solutions is higher than that of ethanol.

[I0s#s3/m3]

5.0

H t .C.N3L3

0.0-J ** ca_

© impeller diamefer = 9.14m

— - N.L (10-3 m/s]

10.0 20.0

Fig. 7.7 Overall corre la t ion of abrasion data

186

7.4 Conclusions

From the abrasion experiments the following conclusions may be drawn:

1. The abrasion of ammonium sulfate crystals is, under the conditions employed in the experiments, determined solely by impacts against the im­peller and baffles of the vessel, and not notably by crystal-crystal impacts.

2. The abrasion rate B due to crystal-impeller collisions is proportional to the circulation frequency of the crystals, an impact efficiency, the con­centration and the kinetic energy of the crystal before the impact.

3. The circulation frequency , IÜ of the crystals is lower than the liquid circulation frequency u>. :

co = «_ [1 - (-H ] c L L lv ' J ax 4. The impact e f f i c i e n c y n. i s p r o p o r t i o n a l t o t h e p r o d u c t (N.L) minus a

minimum v a l u e :

TI - (NL) - (NL) . t min 5. For the abrasion experiments of ammonium sulfate in 13 1 ethanol using a

stainless steel 140 mm diameter four blade pitched blade impeller the following overall equation was obtained:

B = 4.472 * 107 . n .cN3L3 (NL - 4.25 * 10~3)

Further the use of ethanol as inert medium combined with Coulter Counter analysis of the fragment population density as a function of time proved to be an excellent method of measuring abrasion rates.

187

CHAPTER 8

CONCLUSIONS

8.1 General remarks

The field of industrial crystallization from suspensions can be divided into two research trajects, chapter 1: i) a reactor engineering traject, dealing with down-scaling, mixing, clas­

sification and reactor element modelling and ii) a basic growth and nucleation traject, dealing with the fundamental

growth and nucleation phenomena.

The present study has focussed mainly on the second traject. As a link between the two trajects the CMSMPR steady state was chosen. This common point is a very useful starting point of research in either of the two directions. For instance, Asselbergs (1978) and Grootscholten (1982) in­vestigated the effect of internal and external classification on the CSD for NaCl in a 2 m3 scale forced circulation evaporator, using 10, 55 and 90 1 evaporative CMSMPR crystallizers as reference point. The aim of this study is to describe the nucleation and growth kinetics, measured at the CMSMPR level, in terms of "more fundamental" physical processes. Both the theoreti­cal modelling as well as the experimental approach rise the CMSMPR as a reference level. In the following the results of the theoretical modelling and of the ex­perimental approach will be shortly summarized.

8.2 Theoretical modelling

The review of theories and experiments in the field of primary and secondary nucleation, chapter 2, has lead to the following conclusions:

1. Primary nucleation does not play a role of any significance suspension crystallization of well soluble substances. At the relatively low super-saturations encountered in these systems primary homogeneous nucleation is not possible, whereas primary heterogeneous nucleation at most is able to generate some nuclei which serve as a starting point for an overwhelm­ing "avalanche" of secondary nuclei. In this respect, the use of the meta-stable zone concept (Ostwald, 1897) to explain this "explosion" of secondary nuclei in terms of primary homogeneous or heterogeneous nuclea­tion seems not to be justified.

2. The secondary nucleation mechanisms as proposed in the literature can be classified using only three categories, namely catalytic breeding, sur­face breeding and mechanical breeding. Each categorie, or mechanism, can be modelled by considering a formation step, a removal step and a survival/outgrowth step. As a criterion for the classification the origin of the secondary nuclei has been employed.

k. Catalytic breeding, defined as the enhancement of homogeneous or heterogeneous nucleation in the solution due to interaction between the solution and a substrate, was rejected for the following reasons:

188

Theoretically, the enhancement of cluster formation demands an ener -getic interaction between cluster and substrate sufficient to provide a major part of the surface energy of the critical nucleus. This interaction unavoidably implies a bonding force between nucleus and substrate which is strong enough to prevent any spontaneous or fluid-shear driven removal of the cluster from the substrate. Experimentally, no convincing evidence was found, Practically, it seems not possible to observe in situ whether a nucleus has originated at the crystal surface or at a few molecular layers distance of the surface in the solution. For nucleation enhanced by a foreign substrate, i.e. primary heterogeneous nucleation, it has been proven that a nucleus can only be formed on the substrate. Why should secondary nucleation behave differently?

4. Surface breeding encomprises all secondary nucleation mechanisms in which the growth of the crystal plays a role in determining the surface relief or growth habit of the crystal, producing protruding parts or weak spots which are prone to be removed. This removal may proceed "spontaneously" by stresses building up in the crystal due to the growth process, or by external means such as fluid shearing action or mechanical action. Since growth plays a dominant role, the basic understanding of surface breeding will have to result from a detailed study on growth phenomena leading to three-dimensional growth structures. Some complicating factors in this study will be the effects of impurities, the origine of the seeds employed, the effect of attrition and agglomeration upon polycrystal-linity, and the dependence of the phenomena on the particular system under study.

5. Mechanical breeding, is pure abrasion, attrition or breakage of crystals producing fragments independent of the growth rate of the crystals. Consequently no formation of new entities prior to their removal is in­volved in this mechanism.

6. Apart from formation and removal a third process step is to be considered namely the survival or outgrowth step. In this study it is suggested that the slow growth of secondary nuclei can in many cases be caused by the plastic deformation of the nuclei due to the mechanical removal step. This may even result in dissolution or partial dissolution yielding non or slowly growing, stressed nuclei. The growth behaviour of the nuclei, therefore, can be indicative for the removal mechanism.

7. Kinetically, the secondary nucleation mechanisms are modelled using the three process steps mentioned above. The use of efficiencies for the removal and survival step facilitates the analysis of possible raté-limiting steps. The equations produced in this modelling can be described by a general power law with exponents depending on which nucleation mechanism prevails.

8. An evaluation of the experimental methods showed that CMSMPR experiments produce the most relevant information,.followed by the seeded semi-continuous batch experiments. The latter experiments, however, involve tedious evaluation of the fine particle CSD by e.g. Coulter Counter analysis.

i )

ü ) iii)

iv )

189

Based on the review of theories and experiments it was decided that the most promising experimental strategy was to perform CMSMPR experiments and to design some auxilliary experiments, preferently with suspensions, to study some of the phenomena, such as growth, nucleation and abrasion, separately.

In order to sustain the analysis of the CMSMPR experiments, especially in respect to the phenomenon of growth dispersion, an extended population balance has been derived, chapter 3, accounting for: - permanent growth dispersion - maldistribution of solids and growth rate - classification - negative growth rates, i.e. dissolution - direct birth of nuclei of a given size and growth rate.

8.3 Experimental results

For a detailed discussion of the various experiments using the ammonium sulf ate-water system the specific chapters should be consulted. Here only the most interesting results are presented:

1) In the CMSMPR-experiments with pure ammonium sulfate, chapter 4, the nucleation mechanism is surface breeding limited by a formation step proportional to the growth rate, and a survival step proportional to the supersaturation. The stirrer speed had a negative effect on the nuclea­tion rate, and is therefore certainly not rate-limiting.

2) The evaporative experiments show higher nucleation rates than comparable cooling crystallization experiments, while the overall growth rate de­pendencies are well comparable.

3) The negative stirrer speed dependence is a result of the decreased mass transfer resistance (diffusional) leading to lower supersaturations at a given value of the growth rate.

4) The CMSMPR nucleation kinetics are strongly influenced by impurities. Oxime liquor, chapter 5. an industrial motherliquor, gave considerable lower nucleation rates leading to larger crystals at a fixed residence time. The crystals had a pseudo-hexagonally twinned structure with well developed {001} faces, which is in contrast with the single, orthorombic crystals of the pure system showing mainly the {010}, {Oil}, {110} and {111} faces.

5) In order to model the size-dependent average growth rate found in the im­pure system it was necessary to split the CSD into two groups of crystals, each having its own growth rate. A separate regression analysis for the two groups showed that the nucleation kinetics of the "fast growers" was linearly dependent on the growth rate and the surface area of the crystals, but no stirrer speed dependence was found. The nuclea­tion kinetics of the "slow growers" was strongly stirrer speed dependent. The best fit was obtained by using the results of the abrasion experi­ments (chapter 7). Additionally a supersaturation dependence was noted, which might be explained as a survival limitation. It was concluded that the two groups of crystals actually resulted from the simultaneous opera­tion of two secondary nucleation mechanisms. The first is surface

190

breeding, limited only by the formation step, the second is mechanical breeding, i.e. abrasion, limited by removal and survival and/or outgrowth. These results confirm the slow growth of mechanically damaged fragments, as discussed in chapter 2.

In order to facilitate the evaluation of the nucleation kinetics in the CMSMPR experiments, a number of additional experiments have been designed, namely, growth and nucleation rate measurements using a small seeded fluid bed growth cell, and abrasion experiments with ammonium sulfate in ethanol:

6) Growth of pure ammonium sulfate seed crystals is diffusion limited. The nucleation found in the fluid-bed was predominantly caused by the polycrystalline outgrowth of the seeds. Therefore no direct comparison with the CMSMPR kinetics can be made. It was hypothesised that agglomera­tion might play a role in the polycrystalline outgrowth of crystals.

7) The abrasion of ammonium sulfate was caused by predominantly crystal-impeller impacts. The fragments had a negative exponential size distribution with a L,_n of 43 um. Their rate of production was propor­tional to roughly the fourth power of parent crystal size and the fourth power of the stirrer speed.

8.^ Final remarks

The main result of this thesis is that the wide variety of secondary nuclea­tion models has been reduced to two main mechanisms, surface breeding and mechanical breeding. The growth process of the parent crystals is the most important factor to be understood for surface breeding. A study of growth, directed at the understanding of the three-dimensional relief of these parent crystals, will have to be performed as a logical extension of the present modelling. Further it is recommended that additional methods are developed to charac­terize the growth behaviour of secondary nuclei in relation to the mechanism by which they are formed.

191

Appendix 2A

Homogeneous nucleation rate and size of the critical nucleus for ammonium sulfate

1) Size of the critical nucleus

The Gibbs free energy of formation of a cluster containing n molecules is given by:

AG = -n Au + aa n '^ (1)

where Au is the difference in chemical potential per molecule [j/#] between a supersaturated solution and a solution in equilibrium with large, un­stressed crystals, having a crystal-solution interfacial energy a [j/m2]. The shape factor, a, can be expressed as:

° = "a <inr)2 / 3 <2> a v

where V is the molar volume of the crystalline phase, N the Avogadro num­ber, [ff/mole], and k , k shape factors relating the area and volume of a crystal or cluster to a characteristic dimension of the crystal, L:

V™ (3)

k = ^ e {4) v L3

D i f f e r e n t i a t i o n of equa t ion (1) and demanding equ i l i b r ium between the c lu s t e r and the supersaturat ion,

Iff) - ° (5) for n=n , the so cal led " c r i t i c a l nucleus", y ie lds

»*■ !f-g)3 <6> and with equation (1):

AG*= ^ -feel- = 1/2 n\u (7) ' UP)

Introducing the volume of the critical nucleus in equation (4) leads with equations (6) and (2) to:

n V M ,„ „ V T * = f 3 , 1 / 3 . figoi f j n a / 3

lN k ' ~ l3AuJ lN kJ

a v r a v 2 k V a

v a K

192

2) Homogeneous nucleation rate The stationary homogeneous nucleation rate is given by:

Js= A exp (£§-] (9) where the k i n e t i c , pre-exponential. factor, A, i s estimated to be (Nielsen, 1966):

A * 1 0 3 2 - 5 [#/cm3s] = 1 0 3 8 , 5 [#/m3s] (10)

The underlimit of homogeneous nucleation i s generally se t a t J = 1 [#/cm 3s] which leads to a meta-stable zone l im i t , eqs. (9) and (10):

(11)

(12)

(11)

(12)

(13)

AGMET = 7 5 k T tj/#l , (17): AuMET 2 ,0013/2 kT 45 lkTJ

, (7): * _ f r kT,3 nMET " I15 ao]

, (8): LMET = 15 ^ k 0 ^ a

(13)

(14)

The "driving force" Ap for nucleation and crystal growth may be related to the supersaturation by:

^ = l n S (15) where S is the supersaturation ratio. This expression is valid if the ac­tivity coefficient of the supersaturated solution equals that of the saturated solution.

3) Calculation of physical constants for ammonium sulfate Estimate for o Söhnel (1982) presented a correlation between the interface free energies of crystalline substances and their solubilities, expressed in mol/1.

d° = - 17.8 dlog c eq From the values for K-SOj.: log c = - 0.21 and a = 2 7 mJ/m2 we can es­

timate a for (NHJJ-SOJ, using c - 45 weight percenx * 4,2 mol/1:

193

log c =0.62 eq o = 27 - 17.8 (0.62 + 0.21) = 12 mJ/m2

This is, however, only a crude estimate. Close inspection of Söhnel's fig. 1 shows that the group of potassium salts has on average the same range of o-values as the ammonium salts, o = 30 +. 20 mJ/m*, the latter ones having a higher solubility. Therefore ?Ke value of o for K_S0j. is the most probable one for ammonium sulfate.

a » 27 + 10 mJ/m*

It is however not clear whether the value for KpSOj, is reliable, since no technique exists to quantify exactly the interfacial energy of a crystalline solid, except perhaps the measurement of homogeneous nucleation rates. However, homogeneous nucleation only occurs at extremely high supersatura-tions (Nielsen, 1964; Kubota, Fujisawa, 1984).

Calculation of a

V = 7.47 * 10"5 [m'/mole]

N = 6.024 * 1023 [#/mole]

k = 5.4 [m'/m'] _ l 8 2 / -a ) see ch. 4 -» eq.(2) a = 1.81 * 11 [m' / iT 7 3 ]

k = 0.64 [m3/m3]

4) The width of the meta-stable zone

The temperature i s 50°C ■> T = 323 [K]

The gasconstant k = 1.28 * 10~23 [ j /# ]

eq. (11): AG*MET = 3-34 * 10"1 9 [ j ]

eq. (12): ^ p . 363.7 * a 3 / 2 [-] # eq. (13): nMT?T = 0.412 * o " ^ [#] 3/2 MET

eq. (14): L #M E T = 4.31 * 10_1° * a"1/2 [m]

eq. (15): S M E T = exp [——) [-J

For ammonium sul fate at 50 °C the solubility is 45-8$ and the slope of the solubility curve, which is constant, is 1% in 9.12°C. Consequently the width of the meta-stable zone, expressed as ATMp equals:

AT - f MET i rvi MET " l0.00239J l J

194

For the calculations 4 values of a were selected, .see table 1. The vaLue of 6 mJ/m* is the lowest value quoted by Söhnel for a crystalline substance.

Table 1 The meta-stable zone width for ammonium sulfate

o [mJ/mJ]

37 27 17 6

APMET kT ["] 2.588 1.614 0.806 0.169

» n MET [•] 58 93 186 886

* LMET [nm]

2.24 2.62 3-31 5.56

SMET [-] 13.3 5.0 2.24 1.18

ATMET [•c]

5150 1682 518 77

Homogeneous nucleation under normal operation conditions ,-7 m/s i s r e p r e s e n t a t i v e f o r most of t h e

e Ac leadd 0.059?». The driving force i s calculated with:

An average growth r a t e of 10 exper iments . At 50 C the average Ac l e a d ing to t h i s growth r a t e equals

kT In (l Ac \ + LH eq

Ac 0.059 c = 45.8 eq

1.29 * 10 ,-3

Table 2 Homogeneous nucleation for 0 = 6 mJ/m'

normal 10 fold 100 fold

Au/kT [-3

1.29*10"| 1.29*10, 1.29*10"

G [m/s]

10~6 10-5 10 p

AT [°c]

0.54 5.4 51»

• n

2.00*10? 2.00*10, 2.00*10;J

• L [nm]

3382 338 33-8

J [#/m's]

0 0 ft 4.0*10 °

195

5) Calculation of the width of the meta-stable zone for MgSOu^H^O

Assumptions of Sung et al. (1973)

o = 2.5 * 10~3 [J/m2]; T = 311 K; V /N = 2.4 * 10 [m3/molecule] ID £1

the shape factors for a sphere: k = n, k = n/6 lead to (equation 2 ) :

a = 1.87 * 10"8 [m*/#2 / 3] and jg- = 2.178 ApMET From eq . (12) : -^— = 0.071 and from

eq . ( 1 5 ) : SMET = 1.074 -> ^ * 100% = 7A% eq

From Clontz, McCabe (1971): AT = (— * 100%) y ' eq

Conversely if ATMET > 22.1°C ■»(££)> 0.381

and ~ > 4.19 or 0 > 9-62 * 10~3J/m*.

} ATMET= 3.5°C.

196

197

Appendix 3A Calculation of the growth rate from the crystal production per unit of area The production of crystalline material, P in [kg/m Js], is defined here as the mass formation rate per unit of volume. In general this production is divided over the net-production of crystals leaving the crystallizer per unit volume and the increase of the crystal mass in the crystallizer, per unit volume:

_ "T.p , ^ , dlnV dt dt (1)

Where VL, is the product slurry density and VL, is the average slurry den­sity in crystallizer. Since:

Mr = | n kvPcL3 dL, (2)

equation (1) becomes; under assumption of constant p k : 0»

o the limits of integration being constant in time (Rule of Leibnitz) . Using the general population balance and assuming that size dependent birth and death are absent, equation (3-17) + assumption (A.l), leads to:

f3n\ f3n<G>^ p - dlnV ,,..

leads to: <D CD (0

5 ^ . P T f 3n<G>, . . 3,. f % , t 3JT f - dlnV . T 3 „ s dlnV P = — t r ~ \—ri I . k p LJdL - -*1 k p LJdL - n ,. k p LJdL + M„ —r—-T J l 3L ' t v c J T v*c J dt vKc T dt o o o

(il£G>.) L 3 d L = _[ - < g > k L3]L=,° + [ " <G>k p 3L2dL J l 3L 't v c l vKc JL=o J v cJ

- \ 3 <G>p k L 2 dL (5)

c v KJ'

Since no infinite crystals exist at normal slurry densities, n(L=») = 0. The definition of the average total crystal area,

*T F n kQL2 dL (6)

198

with the area distribution " " 1 T 2 a = n k L a

shows that:

P = J3 <G> P k Kc v k a

(7)

(8)

If <G> and k are independent of the size it is seen from equations (6)-(8) that:

P = 3 P C ^ <G> ^ (9) a

Consequently the size-independent growth rate <G> can be calculated from the averag_e crystal mass production per unit volume, P, and the average crystal area A^ per unit volume in thé-crystallizer. Of course, under CMSMPR condi­tions, 7L, equals the area per m in the product slurry. This analysis also holds for batch operation, x being infinite. In both con­tinuous and batch operation the crystal mass production P must be measured or calculated from the mass balance.

The area averaged growth rate G a -. Q v

If the growth rate is not independent of size, evaluation of G from equation (9) leads to an averaged value of G:

Gav = , / - <10> 3pc -*- * AT c k L

a which after inserting (6), (7) and (8) can be written as:

CO

ƒ <G> a dL • G - £ av

= I G (r)dL ' (11) ƒ a dL o ^T

o This shows that G is the area averaged value of the growth rate, since the normalized area distribution a/A , is used as weight factor.

Propagation of errors in <G> and B in steady state The production rate P is calculated from the net-production in steady state:

5L P P = x'^ (12^ x steady state v '

199

The total area of the crystals in the crystallizer A_, [-7-] is related to the area in the product slurry by:

V _s_ T VP" A • V. (13)

where A is the classification coefficient (assumed independent of size). The size-independent growth rate G follows from equations (9) or (10):

<G> = JWl k T (it)

The area /L, can be evaluated from the sieve analysis of the product:

A_, = I k n. LV T,p £ a 1 1 AL, (15)

.th where L. is the average size of the crystals on the i sieve, AL. the dif­ference in sieve size between sieve (i+1) and sieve i. The population density follows from:

w. 1 5EJ

p k L.3 A L. I w. v i i i (16)

,th w. being the weight of the crystals on the i " sieve. Substitution of equations (16) in (15) and (15) in (14) gives:

<G> = Z«±

3 x 1 (w./L.) s 1' 1 (17)

It should be noted that the shape factors k and k do not affect <G>! The relative error in <G> can be expressed as:

A<G> <G>

AT s

T S

w AL

5 Irf (r^l ♦ 1 1 — ♦ s

1

A w. J

Z w i

A w. ,1

L. . J

1

(18)

The relative error is seen to be directly affected by the relative error in the solids residence time. Further the relative errors in the sieve-size, given by the second term on the r.h.s. of eq. 18, are weighed by the frac­tion of the total crystal area on that sieve. If we denote this relative area by a. ,, it is seen that: j, rel

w w j. rel (19)

L. L. J 1 It follows that, equations (17)-(19)

AT A<G> <G>

s + I a j, rel AL. H H ♦ I a L. J

j, rel h. 3<G>T

Aw. w (20)

200

The third term shows the influence of the relative errors in the weights of the crystal fractions, which are weighed by the relative area and a factor depending on the relative position of the sieve^ L., compared to the mode of the CMSMPR steady state weight distribution, 3<G>T^

Nucleation rate The effective nucleation rate is given by equation (3.42) The relative error in B pp/^iy and in < G V T H (see eq. f7). ' P The relative error in B pp/^iy is the sum of the relative errors in the k

i 7 W <G>3x 4 ~ T . U T H 3 (21)

s s lI w./L. h

A(<G>x ) AT . - AT s s _ fA<G> s i ,__>

^ T ö if" = " + 3 (— — ) (22) <G>D T s <G> s s

Consequently (3.42),(22):

AB° /M_ Ak ,.:. AT

«eff^.p v <G>

The relative error in B „„: erf

AB° AM_ Ak . - AT -eff = _JLB. V + 3 A < G > . . 2 _ S ( 2M Beff «T.p V <G>

is seen to be higher than the relative error in B „„/Mm err T,p

201

o r :

w i th

G =

G =

: G CD

£♦ c2Na)

* 1)

= c2NQAC

AC

Appendix 3B Properties of the steady state CMSMPR distribution for diffusion limited growth From the simplified mass transfer dependence equation (3«^9) the size de­pendent growth rate is expressed as:

(1)

(2)

(3) and: L 2 = / ( c ^ A C ) (4) It is seen from equation 2 that at L=L_ we have G = 2G B. Since at L=0 the growth rate becomes infinite, we will solve the population balance for the product n.G, the number flux, which for L=0 equals the nucleation rate:

dnG n nG (5) dL = " T = " Gx

. L L ƒ din (nG) = - ƒ | - dL o o

o o L L2 i /i L x

■ " G T + ÖT ln (1 + L > «o 2.

■* n = Q (1 + - ) . exp (- Q-^-) (6) 2 •

Using: x = — (7) L2 and: a = ^ (8)

we obtain for n, equations (6) and (7),:

B x i. x ia-1 , . /r>« n = Q - - (1 + - ] .exp (-x) (9)

o The r e su l t i ng CSD's are given in f ig . 1 in dimensionless form with parameter a = L 2 / ( G < D . T ) . The popu la t ion d e n s i t y s t a r t s with n=0 fo r L=0 and goes through a maximum at x = Ja . ° max v

202

a=0.01

- i 1 1 1 T -0 1 2 3 4 5

8°eff B°

t

2.00-

1.50-

1 00- 0-""" 1

5 - 4

. 0 B eff F(3) B° "*6 Hla)3

/ *

-3 -2 -1 Ó 1 2 3

-1.00

0.50

o B e f f ,

<WF (=x> - » In 9

Figure 1 Figure 2

The moments of the d i s t r ibu t ion are evaluated using:

u . = | n L'dL o

/r>\ B lr, v j+1 P 1 ,x ^\a-l ,. x j+lj (9)-» ]i. = Q- (G„T) J J - .(- + 1) exp (-x).xJ dx

(10)

u = B°x (G„T)J. F(j)

where F(j) = \ \ (f + l ) a _ 1 exp (-x).xj+1dx

(11)

(12)

is evaluated for j= 0,1,2,3 and several values of a, see table 1. The average growth rate of the CSD is calculated from appendix 3a. equation (10), (12):

G av 3 T u, (13)

203

Table 1 Numerical factors in equations (11) - (16)

a

10 1 0 . 5 0 .1 0 .01 0

F(0)

1 1 1 1 1 1

(Fl)

4.658 2.000 1.657 1.229 1.042 1

(F2)

29.264 6.000 4 .311 2.656 2.103 2

F(3)

226.547 24.000 15.417 8.338 6.341 6

H(a)

2.592 1.333 1.292 1.046 1.005 1

F(3) 6 H2(a)

2 .17 1.69 1.52 1.21 1.04 1.00

d In V d In a

0.834 0.859 0.921 1.00 1.00 1.00

Relating G to G^ with H(a) defined by:

H(a) = av (14)

and inserting eq. (13) and (11) for j=2 and j=3 leads to:

H(a, = IEÜI n[a) 3 F(2) (15) From table 1 it is seen that H(a) increases by 33$ going from a=0 to a=l, which means that the average growth rate is slightly enhanced by the in­crease of mass transfer at small sizes. The effective nucleation rate is calculated with equation (3.42) from G using VL, = pk u~, equation (14) and equation (11): av

B B°T (G OT) 3F(3) F(3)

eff" 6 G V - 6H3(a)GT< av v ' a. 6 H(a); . B (16)

The effective nucleation rate is seen to be higher than the true nucleation rate (table 1), ~ 70# for a=l. This is caused by the fact that the growth rate of small crystals is higher than the average value G and consequently the crystals need less time to reach average sizes than the time calculated by assuming a growth rate equal to G for all crystals. Therefore less crystals are withdrawn, increasing the effective nucleation rate over the true rate.

Effect of stirrer speed

The parameter a is dependent on the stirrer speed. From equations (1) and (2) we see that:

L2 = c2N a (17)

We now choose a simple nucleation rate equation:

1 av ^2 (18)

204

and substitute u_. After some rearrangements using eq. 15 we solve for T:

T = JL_ f__2—]i/3 (iq) T G l K 1 F ( 3 ) J U y ;

av 1 v J' The s t i r r e r speed dependence of a fol lows from:

L ( c J c J N " a = ÖT = " ? ^ T7T < 2 0 )

U » T H^(a) [ 3 / ( k F ( 3 ) ) ] V : S

c k a .H 2 (a) [ F ( 3 ) ] ' 1 / 3 = Z(a) = - 1 N~a ( - | ) 1 / 3 (21)

C 2 3

-» d In Z .„_ . , d In a d In Z ,_„ . X^;—ÏT = ~a (22) , and ^-^ rr = -a -r—: (23) d In N \ / • d In N d In a

The r a t i o between B „„ and B , t h e nuc l ea t i on enhancement f a c t o r i s c a l c u ­l a t e d from e q u a t i o n (16) and i s p l o t t e d v e r s u s a on d o u b l e l o g a r i t h m i c paper , f i g u r e 2 :

d l n (Beff /B°> d l n ^ f f / ^ d i n a S i n c e g - ï _ s = 5 - I i r - . ^ - j - j j . (24)

the maximum e f f e c t of the s t i r r e r speed occurs f o r a * 1, see f i g u r e 2 . For a - v a l u e s be tween 0 . 5 and 1 . 0 , ( d l n Z / d l n a ) * 0 . 8 8 , t a b l e 1 and wi th x = 0 .6 we f ind dlna/dlnN = - 0 . 6 * 0.88 = - 0 . 5 3 , e q u a t i o n ( 2 3 ) . From e q u a ­t i o n (24) and t h e maximum va lue of d In (B f f / B ) / d l n a * 0.145 from f i g u r e 2 , we o b t a i n :

d In (B° /B°) d l ^ N * - ° ' ° 7 7 (25)

An increase in stirrer speed of 100$ reduces B ff./B by 5.2$ which is prac­tically negligible in comparison with the direct stirrer speed dependence of the nucleation rate.

y = x =

dL

A <Ü>T

- In n P dx

y dL = -1 dL x dy_

* y = dx-* dt differentiation:

205

Appendix 3C Evaluation of size-dependent growth rate From equation (3-23) it follows that:

d (A n <G>T) §ï = - n (1) dL p v ' d (A <G>T) , - d n p ,_. -» n — i — T - J + A <G>x ,_r = - n (2) p dL dL p v '

Use y = A <G>T (3)

(4)

(5)

(6)

(7)

(8)

to:

(9)

(10)

(11)

(12)

(13)

dx '

d 2y dx 2

dn":

dx11'

= 2 + 2 dx dx

d 3L 4 d3y dx3 dx3

L ,n. ,n y d L + d y -1 , n , n dx dx

and subsequent substitution

' y =

with

•» y =

dL d2L d x + . 2 +

dx

R = d-^ n , n dx R = 0 for n = » n

j=l dxJ This finally results in:

SO*i l . 1 r , J <0>A - L \-s.j

j=l . (d

in (5) leai

+ + R dx n n

djL lnn)j

P

206

Appendix 3D

The effect of stochastic dispersion on steady state CMSMPR distribution

The dispersion in growth rates of crystals of a given size is described by a random movement of the size of the crystals superimposed on a constant group velocityt G . This model is analogeous to one-dimensional diffusion superimposed upon con­vection (Randolph and White, 1976). The number flux will therefore contain two terms, a convective and a dif­fusive one:

n G = n G * + D Q(-f ) t (1)

# This means that G is always larger than the group-velocity G for any value of L, and consequently G should be lower than the smallest average velocity measured at small particle sizes. The option that G could depend on the crystal size will not be considered here since this would imply true size dependent growth. Equation (1) can be introduced in the steady-state popula­tion balance assuming CMSMPR-conditions:

(2)

*«- v Gl dL' ; T

d (nG) n dL T

which leads to: * d nG d nG

dL dL

2 * , d n G dn

d + dL

n V (3)

assuming G and Dp to be independent of size.

This equation is of the form

(D-a)(D-b) y = 0 (4)

where D is the differential operator — , a and b are constants and y is the function to be differentiated. The solution of equation (4) is given by:

Y = C. exp ax + C_ exp bx (5)

Comparing equation (3) and (k) shows that:

n = Y; x = L (6) S-= a + b (7) G

207

The constants a and b can be solved from equations (7) and (8): 1 G 1 ffG i2 4 q/2 ...

a = 2 D G ^ n D G ) * D ^ <9)

b = 2D^ + 2 U D Q ) + D ~ ^ {10> Inspection of equations (5) (9) and (10) learns that the sign of a in equa­tion (9) may be arbitrarily chosen, whereby of course the opposite sign must be used in equation (10) for b. Introducing the minus sign in equation (9) and the plus in equation (10) yields values of a < 0 and b > 0. The term c_ exp (bL) in equation (5) after substitution of equation (6) clearly becomes infinite for large L-values c_= 0. Consequently:

n = t|) exp (a.L), a < 0 (11) At L = 0 eg. (11) reduces to:

n°= cx (12) *

Since both G and DQ were assumed to be constant equation (9) shows that a is a constant. Therefore the logarithmic of n varies linearly with size, equations (11), (12), according to:

In n = In n + a L a < 0 a = constant (13)

This solution is identical to the solution for size-independent average growth rate, where a equals (-1/Gx):

which can *

G G

GT

be

1 -

2 D G reduced

DG G2.x

2

t o : 'G "Q'

(It)

(15)

We have to conclude that, starting the analysis with the assumption of con­stant, size-independent group velocity, G , and constant growth diffusivity, D_, a size-independent average growth rate, G, is obtained. Therefore:

i) stochastic growtjh dispersion, modelled with the assumptions of con­stant D , and G does not lead to size-dependent average growth rates in a CMSMPR crystallizer, and

ii) the stochastic fluctuations can not be detected from the CMSMPR steady state crystal size distributions.

208

Appendix 3E

The inverse-gamma distribution function

The size of an inverse-gamma function to describe the distribution of zero-size nuclei over the growth rate at zero size (Janse and de Jong, 1976):

f° = Cx . G~r exp. (-Q/G) ; G è o (l)

i s mathematically very elegant s ince: i ) the funct ion f i s pos i t i ve over the en t i re pos i t ive range of G-

values. i i ) In the l i m i t s for G = 0 and for G = », f takes the value zero.

Consequently the i n t e g r a t i o n s over t he growth r a t e w i l l y i e l d f i n i t e values.

i i i ) The model has only two independent parameters de te rmin ing the shape (G and r ) .

Integrat ion of f over the growth r a t e , see eq. (3-5^) a t s i z e L = 0 g ives the population density a t zero s i z e :

o n = ƒ f° dG G=0

CD

= C1 ƒ G"re: o

resulting finally in

o n - c i ; r , , ê f~2 exp (-S- ) d(£ ) (2) Qr X Q/G=o G G G

The gamma function r(m) is defined as: CD

r(m) = ƒ tm_1' exp (-t) dt (3) o

C Thus: n° = —±7- r(r-l) (4)

Q oQr-l

and: fG = "r(-i) G~r exp (_Q/G) (5)

The population density at an arbitrary size L is calculated using equations (3«54) and (3.68), the latter incorporating the CMSMPR-conditions and the assumption of size-independent growth rates of the individual crystals:

OB OD

n = ƒ f dG = ƒ f° exp [-^) dG (6) o o

Inserting equation (1) leads to:

n = C1 ƒ G"r exp (-§ -£ ) dG o

209

This integral is analogeous to the one encountered in the derivation of equation (2) with (Q + L/T) instead of Q and therefore:

Ct r(r-l) n = — ^l (7)

(Q + W T ) r X

Combining equations (4) and (7) results in: n = n ° ( l ^ ) 1 _ r (8)

The average growth rate is calculated with equations (3-56), (3-68) and (1): o ƒ f GdG c G - °- — - 7 G-rexP ( 4 ) exp (-§ ) G dG

o Cl m

f _l-r ,0 + L/T» .„ = — ƒ G exp. -( Q-1—} dG o

which analogeous to the previous derivation leads to: 5 ■ • r' r' 2' 2 <S>

Inse r t ing equation (7) and using the recurs ive property of the gamma function:

r(m) = (m-1) r(m-l) (10) " ^ (1 ♦ -1 r-2 [X QT' yields: G = - ^ (l + ) (11)

o v

In order to be able to calculate n from experimentally determined values of Q, r and one of the moments of the crystal size distribution, an analytical expression for the_moments is derived. Take for instance the slurry density M„ (kg crystals/nr suspension) which is proportional to the third moment of the CSD:

CO

MT = pckv ƒ n.L3dL (12) o

The population density is given by equation (8):

MT = pckv ƒ n° (1 + -jj-J^lAtt. (13) o

The solution of the integral is reached by repeated partial integration: 4 ,, , o 6 (Qt) ..... M_ = p k n . -. =-r-7 .,0 7 '. , rr- (14) T c v (r-5)(r-4)(r-3)(r-2) x '

Inspection of this result shows that the value of r should be grea ter than five in order to obtain a f in i t e positive value of the slurry density. In general, the value of r should be grea ter than the order j of the moment plus two:

210

r > i * 2 (15)

The nucleation rate at zero size can be calculated from the growth rate and the population density at zero size:

„O O zO O Q IAC\ B = n G = n —r- (16) r-2 With n from equation (14) the following result.is found:

£ , (r-5)(r-4)(r-3) {ly) V 6Pk oA4 c v

which is analogeous to the equation reached for constant growth rate. With equation (11) for L=0, Q can be replaced:

B° 1 (r-5)(r-4)(r-3) (Q)

« r ' ö p k (G°) V • (r-2)3 ( Ö) c v

The dominant growth rate G_. at zero size is calculated by differentiation of equation (1) with respect to G and equating the result to zero:

«g ■ F <W Another useful quantity is the area averaged growth rate:

ka P av 3Pcky ^

where P is the production rate:

steady state (21)

The t o t a l a rea i s ob ta ined from the second moment of the d i s t r i b u t i o n analogeous to the derivat ion of the s lur ry densi ty . Combining equations (20) and (21) and elimination of VL, ( equa t ion 14) and AT (equa t ion not given) g i v e s :

G = -£=■ (22) av r~5

211

Appendix 3P Derivation of the polynomial model The distribution of zero sized nuclei over the growth rate at L = 0 is modelled with a polynomial in z,

m f° = I a. z 1 (1) 2 i-0 X

k, G • i-u d max .. ,_» with: z = k~" = — G ^ '

r

—= * f ° M - (fr)L.t (3)

The minus sign is used here since n is a cumulative property and is chosen to be increasing with increasing growth rate. Consequently an increase in z, means a decrease in G and a decrease in n. In order to keep f greater than zero a minus is added in the definition. The steady state crystal size distribution follows from equations (3-5^) and (3-68):

a>

n = ƒ f° exp - jj- dG (4) o

From equation (2) dz/dG is derived: , G dz max dG = - Q2

Consequently:

' ï - - l ï r l - - l £ ) t £ i - § L ' S . <6> max

(5)

equations (4)-(6):

G=0 G max z=0 P T

From equation (2): G = zm^]- * = i1—^-) (8) ma

CD

equations (7),(8): n = f f° exp (-b-bz)dz (9) o

with: b = L T (10)

max Substitute the polynomal equation (1) in equation (9):

«o m n = / £ a.z exp (-b-bz)dz

o i=o

212

= ï [ 1 e^|~b)-) . / (bz)iexp(-bz) d bz (11) i=o b o

The integral in equation (11) is recognized as the gamma function r(i + l) which reduces to i! for integer values of i:

nB«5El=bI l it (i)i (12) i=o

a polynomial in (r-) which can be fitted to the experimental data using stan­dard least squares curve fitting techniques.

213

Appendix 4

Results of the CMSMPR experiments with pure Ammonium Sulfate

Table 1 Evaporation a t 50°C - pure solution (see figure 4.3)

#

MB2 MB3 MB4 MB6 MB7* MB9 MB10* MB11 HV14 HV15 HV1 HV4 HV7 HV5 HV8 HV12 HV13 JH24 JH31 JH4l JH51 JH61 JH71 JH81 JH91 JH101 JH111 JH121 JH131 JHl4l JH151 JH161 JH171 JH181 JH191 JH201

symbol

0 0 0 0 0 0 0 0 0 0 Ü 8 e e e X + 0 0 0 0 • 0 0 0 0 • 0 0 <D (D (D A A A A

N

[rpm]

300 300 300 300 300 300 300 300 300 300 450 450 150 150 150 300 450 300 300 300 300 600 600 300 300 300 600 600 300 300 300 300 300 600 300 300

X

[s]

1211 II94 2456 2467 1192 3974 2433 3697 1237 2409 1484 1214 1211 2587 2484 2357 1547 1165 1081 1618 615 582 1580 1770 1801 1155 IO87 3309 1578 3142 1158 2000 2734 1468 1464 3017

"r [kg/m' ]

85.6 144.5 82.7 113.2 80.0 96.4 85.3 182.7 90 142.6 114.7 150.3 134.8 200.4 184.9 121.4 154.7 102.2 69.0 77.6 73-8 66.2 17.8 121.2 145.4 86.5 60.8

245.5 101.7 233.6

95.1 195.5 132.2 129.0 63.7

153.2

\

[m2/m3]

1650 3090 1370 1790 1610 1500 1500 2940 1750 2540 2100 3130 2860 3340 3470 2140 2600 1893 1365 1347 1361 1280

390 2161 2781 1625 1177 3721 1659 3271 1855 3166 2285 2447 1195 2229

G

l 0 g [m/s]

-7.I67 -7.206 -7.398 -7.39O -7.I79 -7.59O -7.43O -7.574 -7.I65 -7.417 -7.218 -7.I86 -7.I94 -7.420 -7.426 -7.403 -7.401 -7.132 -7.129 -7-248 -6.854 -6.849 -7-337 -7-397 -7.337 -7.134 -7.122 -7.500 -7.212 -7.441 -7.154 -7.308 -7.474 -7.244 -7.237 -7.441

— —

. Beff/MT l o g [#/kg s]

5.335 5.477 4.841 4.769 5.399 4.541 4.913 4.615 5.311 4.907 5.149 5.409 5.434 4.789 4.881 4.903 4.830 5.299 5.419 5.075 5.573 5.655 5-386 5.070 5.154 5.319 5-389 4.587 4.999 4.504 5.374 4.889 4.841 5.231 5.216 4.572

fc0.5 [s]

4.7 2.4 5-7 4.3 5.6 5.1 6.0 2.5 4.4 3.0 3.6 2.4 2.6 2.2 2.1 3.5 2.9 3-9 5.6 5-6 5-6 6.0 20.9 3-3 2.5 4.6 6.6 1.6 4.4 1.9 4.0 2.0 3.1 2.9 6.4 3.1

L50 [um]

274 286 308 326 254 332 299 337 262 294 301 256 243 311 304 294 308 281 259 296 274 259 239 291 276 273 266 335 318 359 256 316 296 266 276 350

* experiments a t 43 °C ** the symbols are explained in the corresponding figures

214

Table 2 Evaporation at 67°C - pure solution (see figures 4.5-4-7)

# JH-

251 261 271 281 291 301 311 341 351 371 381 421 441 451

symbol

0 0 D D 0 a a 0 0 D 0 0 0 D

N [rpm]

700 300 300 300 700 300 300 700 700 300 700 700 700 300

T

[s]

1100 1859 2836 693 1203 627 706 1670 2857 966 1179 736 1866 1686

MT [kg/m3 ]

21.6 95.8 123.6 40.8 85.7 45.5 37.8 79-7 133.6 85.6 89.7 61.3

171.8 150.4

AT [m* /m3 ]

354 1608 1841 942 1474 1062 869 1136 1821 1723 1500 1168 2329 2379

1 U S [m/s]

-7.O58 -7.294 -7-425 -7.OO5 -7.114 -6.963 -7.OO9 -7.I77 -7-389 -7.O88 -7.O92 -6.945 -7.2OI -7.224

, Beff/MT lufe [#/kg s]

5.205 4.970 4.632 5.819 5.191 5.867 5.802 4.807 4.509 5.491 5.157 5-535 4.689 4.933

fc0.5 [s]

15.8 3.2 2.7 5.8 3.5 5.1 6.3 4.6 2.7 3.0 3.4 4.6 2.0 2.0

L50 [um]

315 308 350 222 306 219 221 395 428 257 318 283 399 326

B/B. , ' id [-]

1.01 2.21 3.02 2.39 1.26 1.76 1.88 1.09 1.48 2.69 1.22 1.09 1.37 7.23

Table 3 Cooling at 40°C - pure solution (see figure 4.8)

JH-

211 221 231 24la 24lb

symbol

0 a 0 0 D

N [rpm]

300 600 300 410 600

X

[s]

1204 2645 1808 2532 2532

«T [kg/m3]

26.0 44.5 82.9 33-7 37-2

A™

[m* /m3 ]

364 511 859 285 372

G

l 0 S [m/s]

-7.026 -7.282 -7.072 -7.I3O -7.226

, Bïff/«r l u g [#/kg s]

4.923 4.323 4.356 3.945 4.213

fc0.5 [s]

28.0 19.6 24.9 35-3 27.1

So [um]

362 435 514 579 505

et CD

CD

o o c M Cu

D O et O* CD

CO H-N CD & o, e CD

et O et 3* CD

CD

M (B

O

<<! co et

ON ON U I U I U I "^J ON U I U I O O H H P U I O S H -P--P-O o o ONONO

l\J j r w ( j j (JO U ) UO LO U I p» 4=- Oo 0 0 0 0 0 0 0 0

o o o o o o o o

1 1 1 1 1 1 1 ON ON QN ON ON ON ON

* ON - P - U I U I O O - J ON ^ - H \ D h> H H U I t

- t u i U i U i v£> ~ J Oo

4 r j r j r - p - U l U I U I

i ON oo oo oo p* P* ro -~] M O 0 O 3 P - C - 0 O O W W W P U I I>J

ON U I U I U I -P- -P" -P-1 P» V£) - J —J M K) l\J

H \ O W l\) P O O P

M P H O O O U I U I O O O

1—1

co I 1

1—1

*• <rc B

1 1

l - l 5 * ^ CO l—J

p—ï

=**: w

CXJ

co 1 1

1—1

1 L_J

T3 •o 5

^

^

M O <w o

M o aq tn

CD O •-b •-b

^

r U I o ^^ o ^ <J0

+

H O*

UI O 3

50 I n

CD co c M et CO

O •-b

r f» ►3 en O 3

5 a 3 C F-3

CO 3 *

CD

I\J rv> i-» w P I\J P1

- J -~] O O V O V O - J 0 0 V O - ~ 4 0 0 VD o o o o o o o o o o o o o o o o o o o o o o

—J —J —1 ON U I -P- J i -UO L O U M 4=- .p- -p -v j l v l W O ^ W P U I

W O N O O O O O U J P ' M O N O N U I

1 l 1 l l l 1 1 1 1 1 - J ~ J — J O N O N - J ~ J O N ~ J —J ON

( j O U J P V D ^ U J P>vX>O0 P» U 3 - j ONOOU) p u i v o w u i OOUJ O l \ ) W ^ J W O W U 1 C D 0 O M

-p- -p- - p - U I U I -P- - P - U I -P" - P - U I

U I U I O \ P O - P ^ P U I - 4 P U I I U V D O N V O V D W U I P P -C-p» - J - p - U J 0 0 K) l\J U I 0 0 M \£>

-p- -p- -P"U0 -p- -p- -p"OJ -p- -P"U0 i\j<joooooo-p-r<o0o<-orv)0o W P W W -P-UJ U I -P-UI 00 ON

1—1 CO

1 1

1—1

oq s w

1 I

l - l 5 CO

l _ I

1—1

w co

1 1

1—1 ■c B

i i

H

3

O

cm o

O

w tn

CD O •-b •-b

U I O

216

Appendix 5

Results of the CMSMPR experiments with "oxime liquor"

Table 1 Cooling experiments at 40°C, see figure 5 «2, total nucleation rate

#

DSM-

5* 7 8 9 10 11 12 13 14* 15 16 17 18 19 20 21* 22*

symbol

a a a 0 0 0 0 0 a a 0 A A A A A V

N [rpm]

700 700 700 300 300 300 300 300 700 700 300 500 500 500 500 500 400

T

[s]

2295 1121 2745 1481 2134 1241 1121 2745 1086 3624 3624 2664 3624 1241 1709 1678 1241

*T [m2 /m3 ]

258 187 55-7 77.0 246 149 170 257 420 124 205 308 185 312 133 320 343

NT [kg/m' ]

25.4 14.9 5-7 9.1

28.3 17.1 19.8 29.7 36.3 12.8 27.2 32.3 20.1 27.0 14.0 38.5 33.0

L50 [um]

517 449 620 661 615 618 615 676 470 568 831 620 664 477 576 621 512

c.v. m 70 65 67 51 52 44 42 53 44 63 >50 56 54 54 50 42 47

G av [nm/s]

68 113 59 126 86 147 165 67 126

45 58 63 48 ill 98 114 124

** B T / A T

log [#/m*s]

3.064 3-728 3-097 3.173 2.987 3.174 3-213 3.173 3-620 3.005 3.081 3.261 3.099 3.622 3.422 3.014 3.350

*) not fully stationary **) total nucleation rate

218

Table 2 Cooling experiments at 40°C. The inverse-gamma function

# DSM-

5* 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21* 22*

lo« G° l 0 g [m/s]

-7-256 -7.007 -7.481 -7-008 -7.089 -6.838

--7-370 -6.959 -7-539 -7.561 -7.375 -7.541 -7.030 -7.087

--6.933

G av l 0 g [m/s]

-7.066 -6.951 -7.097 -6.873 -7.051 -6.827

--7.131 -6.860 -7-276 -7.066 -7.180 -7-276 -6.943 -6.979

--6.899

8 ° / ^ l o g [#/m2 s]

3.109 3-704 3.202 3-338 3.113 3.358

-3-166 3-629 3.089 2.862 3-264 3.126 3.590 3.350

-3.510

Q [10"6m/s]

0.204 0.727 0.104 1.041 1.681 16.28

-0.343 1.19 0.182 0.163 0.376 0.222 0.588 2.613 -4.67

r [-]

7.38 11.52 6.30 12.75 23.92 114.1 <5 9.62 13-63 8.43 6.89 10.74 9.18 10.18 29.86 -42.05

219

Table 3 Cooling experiments at 40°C. The polynomal model (see figure 5.7)

#

DSM-

7 8 9 10 11 12 13 15 16 17 18 19 20

N [rpm]

700 700 300 300 300 300 300 700 300 500 500 500 500

G av [nm/s]

113 59 126 86 147 I65 67 45 58 63 48 111 98

Gl [nm/s]

150 62 142 87 161 190 72 47 94 77 58 135 124

G * max [nm/s]

186 93 201 114 279 269 110 56 169 93 93 133 130

B° 6 [10 #/m2s]

1.89 0.175 0.206 0.363 0.453 0.409 1.12 0.141 0.303 0.877 0.883 1.34 0.266

log [#/m*s]

4.005 3.497 3.427 3.169 3.483 3-381 3.639 3.056 3.170 3.454 3.680 3-633 3.301

G ** max [nm/s]

243 87 210 103 187 248 101 40 190 113 76 253 95

* A polynomal of degree m = 2 was used: together with G a 4 parameter model. Sieve fractions having less than 0.1$ of the total weight were not used. Also the two largest sieves were skipped

** Values obtained with m = 4

* *

•• 03

l\J O

^ iD CO r t H-13 f» r t CD O.

*

• • O

l\>

CO CA r t H-B P r t a> O, f» r t

U I

W h O H H I - ' l - ' P P P P K H i\) O U ) 0 0 - J ON U I J="U) N P O U ) O O - J ON

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ON ON ON —J —J -~J ~-J ON-~J ON ON—J ON—J ON ON

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223

Appendix 6A

Physical properties of ammonium sulfate / water solutions

1. Solubility

The s o l u b i l i t y measurements of Ishikawa e t a l (1929) form the most extended s e t of d a t a , 17 p o i n t s from -18 .50°C ( e u t e c t i c p o i n t ) t o 108.5°C (atmospheric boi l ing po in t ) . A s t ra igh t l i n e describes these data very well:

w = 0.41179 + 9-121 » 10_Zf T

- 6.55°C £ T £ 90°C ) [kg/kg soln] (1)

Benrath et al (1937) give the solubility for temperatures above the atmos­pheric boiling point:

w = 0.41302 + 9-310 « 10-/|

100 ^ T ^ 400°C ) (2)

Both s e t s may be combined to one single s e t

) w = 0.41182 + 9.344 « 10"4 T (3)

10 S T S 400°C

2. Density of the solution

The d e n s i t y , p, of the solut ion i s given by Synowietz (1977), see table 1. The dens i t i e s were redetermined in a l imited reg ion around the s o l u b i l i t y l i n e u s i n g an Anton P a a r DMA 602 high p r e c i s i o n d e n s i t y measurement apparatus. The tempera ture of the measuring compartment was s t a b i l i z e d within 0.005°C and recorded from a cal ibrated platinum wire r e s i s t o r . In order to obtain long term s t a b i l i t y t h e counterweight s u s t a i n i n g the r e s o n a t i n g p a r t s had t o be thermostated too. Addit ional ly, the osc i l l a to r e lec t ron ics were removed from th i s thermostated counterweight and mounted in a separate casing. A s t ee l cover protect ing the glass resonator was replaced by an aluminium one and the two small magnets tha t held t h e s t e e l cover in i t s place were removed from the neighbourhood of the resonator , since i t was observed tha t changes in the magnetic f ie ld influenced the c a l i b r a t i o n . Air and b o i l e d - o u t de-ionized water were used for c a l i b r a t i on . The dens i t ies of water and a i r were taken from the Handbook of Chemistry and Phys i c s , 58 ed . , page F l l .

224

Table 1 The d e n s i t y of ammonium s u l f a t e s o l u t i o n s

(NH4)2S04 M=132,14

Gew.-%

0°C 20°C 40°C 60°C 80°C 100 °C

Gew.-#

0°C 20°C 40°C 60°C 80°C 100 °C

2

1.0124 1.0102 1.0039 0.9948 0.9836 0.9705

40

1.2350 1.2277 1.2196 1.2105 1.2011 1.1910

5

1.0310 1.0279 1.0213 1.0122 1.0012 0.9886

45

1.2626 1.2552 1.2471 1.2384 1.2290 1.2189

10

1.0618 1.0574 1.0503 1.0412 1.0304 1.OI85

50

1.2899 1.2825 1.2745 1.2657 1.2564 1.2466

15

1.0921 1.0866 1.0792 1.0700 1.0595 1.0480

20 .

1.1215 1.1154 1.1077 1.0986 1.0883 1.0772

25

1.1506 1.1440 1.1360 l.1270 1.1168 1.1061

30

1.1791 1.1721 1.1640 1.1548 1.1451 1.1346

35

1.2072 1.2000 1.1919 l.1829 1.1731 1.1629

Chertkov, B.A., Pekareva, I . I . : J . a n g e w . Chem.UdSSR 3jL ( 1 9 6 1 ) 1 4 3 / 5 0 ; J .appl .Chem. UdSSR 3JL (1961) 135/41

Simons, E .L . , R i c c i , I . E . : J.Am.Chem.Soc. 68 (1946) 2194/2202 Gibson, R.E. : J.Am.Chem.Soc. £6_ (1934) 4/14 Hantzsch, A. , DUrigen, F . : Z.physik.Chem. 136. (1928) 1/17 Gruner t : Z.anorg.Chem.151 (1926) 310 B e l l , I .M. , Taber, W.C.: J .physic .Chem. 10.(1905/06) 119/22 Tammann, Hasse lb l ad t u . Lerche: Z.physik.Chem. 1£ (1895) 620/36 Bodlander.G. : Z.physik.Chem. 7_ (1891) 308/22 L. Borns t . Neue S e r i e IV (1) p . 7 6 , Spr inger Verlag B e r l i n 1977

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U I ON ON ON —i —J —J — J 0 0 0 0 0 0 N O

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1 1

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226

Six s e r i e s of measurements were done using three standard solutions with concentrations 0.448, 0.457 and 0.467 kg/kg solution in order to ob ta in the t empera ture dependency of the d e n s i t y near s a t u r a t i o n , see t ab le 2. An average value of

(ff) = - 0.3840 ± 0.012 (4) w

was obtained. Additionally at seven temperatures a saturated solution was prepared by gentle agitation of large crystals in a 250 ml solution for at least 12 hours. The density of the solution was analyzed from a sample taken 5 minutes after the stirrer had stopped. The density of the saturated solu­tion is given by:

p » (1248.915 ± 0.010) + (0.1349 ± 0.0014) (T-50) (5) eq

The solubility line, equation (1) can be written as w = (0.45740 ± 5 * 10~4) + (9.121 ± 0.1) * 10-Z* (T -50) (6)

eq Combining (4), (5) and (6) yields:

p = 1007.900 - 0.384 T + 568.907 w (7)

The absolute error in the density is estimated to be AP = 0.010 + 0.012 lT-50 - < w - ° - ^ o ) | + 0.0014 |"-o-^o ■

9.121»10 9.121*10 = 0.010 + 0.012 IT - T I■ + 0.0014 IT -50I (8)

1 eq' ■ eq ' v ' In figure 1 the density given by equation (7) is compared with the values of Synowitz (1977) see table 1. It is seen that the values of Synowietz deviate from the present results for w > 0.40 and, at w = 0.40 for T > 40°C. The reason for this is most probably that the high temperature and high con­centration data of Synowietz have been obtained by extrapolation.

227

1300.0-

Synowieri (19771 ■new correlation

r Ik9/m3l

1250.0-

1200.0-

1900.0-80 90 „100 110

— T|°C]

Fig. 1 Density of ammonium sulfate solution near saturation

3. Vapour pressure

The vapour pressure over ammonium sulfate solutions is given by Tans (1958) as:

log p = 8.0635 - T?233 63 " O-OO11^ W (9)

with p in [mm of Hg], T in [°C] and w' in [g/100 g H_0]. Equation (9) holds for 10 < T < 110°C and 0 S w' £ 100 g/100 g L O .

228

4. The dynamic v iscos i ty of ammonium sul fa te solutions

Dynamic v iscos i ty data from Pulvermacher (1920), Kanitz (1897) and Grunert (1925) were used see table 4.

Table 4 Dynamic v iscos i ty of ammonium sul fa te solutions

T[°C]

0 10 20 25 30 40 50 60 70 80

w=w eq

-1.54 -1-32 1.166 1.072 1.013 0.892 O.78O 0.687 0.607 0.544

0.00

0.581* 0.268 0.002 -0.116 -0.226 -0.426 -O.609 -O.762 -O.906 -I.O36

w [kg/kg solution'

0.02

0.599 0.293 0.037 -O.O83 -O.I86 -O.387 -O.545 -O.717 -0.844 -O.983

0.10

O.678 0.405 0.162 0.057 -0.041 -0.238 -0.400 -0.553 -0.693 -0.799

1 0.20

0.850 0.593 0.360 0.261 0.166 -0.021 -0.186 -O.329 -O.478 -O.574

0.25

0.971 0.703 0.479 0.377 0.293 0.104 -0.062 -0.198 -0.342 -0.449

0.30

0.622 0.516 0.425 0.241 0.086 -O.O58 -O.I98 -O.3II

0.38

O.874 0.754 0.668 0.494 0.329 O.182' 0.049 -O.O67

T[°C]

25

0.42

0.858

* the va lues of In ( t i / [ c P ] ) have been tabulated

The dynamic v i s c o s i t y of the sa tura ted solution was extrapolated from the values a t w = O.38 [kg/kg] using (8 In n/3w)T = 5.833 see f i g u r e 2 . These s a t u r a t i o n v a l u e s w i l l be used in the e x t r a p o l a t i o n of t h e d i f f u s i o n coeff icient .

229

80 70 60 SO 40 30 25 20 10 O j 1 1 , 1 1 — i — i | _ i u

300*10-' J_ iK.,| 350»10-»

Fig. 2 Dynamic viscosity of ammonium sulfate solutions

5. The effective diffusion coefficient and the Schmidt number

The true diffusion coefficient ID was measured by Wishaw and Stokes (195*0 on a de Golly diffusiometer at 25°C, see table 5

Table 5 Diffusion coefficient at 25°C

c [moles/1]

0.0525 0.1002 0.2050 0.3824 0.5621 1.155 2.393 3.594

Ac

0.0419 0.0503 0.0660 0.0711 0.1227 0.2002 0.1735 0.2032

ID [10~9 m2/s]

0.803 0.825 0.869 0.916 0.950 1.026 1.084 1.125

230

from 4 .0 to 4 .6 h a s t h e v a l u e

(1.145 ± 0.005) * 10"3 [ m l / s ] . The values of ID a t other temperatures were estimated using Einsteins r e l a ­t ion , see table 6.

The saturat ion concent ra t ions of ammonium s u l f a t e ra m o l e s / 1 f o r 40 C T < 8 0 ° C . I n t h i s r a n g e ID

constant (10)

Due to the advective flux towards a growing crystal the mass transfer is larger than predicted if a stagnant liquid is assumed. To correct for this effect an effective diffusion coefficient must be used given by (Westphal and Rosenberger, 1978):

ID ID eff 1-w (ID

where w is the mass fraction. The value of Schmidt's number Sc, is also cal­culated with this effective diffusion coefficient:

Sc eff eff (12)

where v is the kinematic viscosity.

Table 6 Diffusion coefficients and effective Schmidt number at saturation

T

[°C]

0 10 20 25 30 40 50 60 70 80

ID

[10~9 m*/s]

0.650 0.841 1.018 1.140 1.235 1.436 1.660 1.884 2.103 2.310

meff

[10~9 m' /s]

1.105 1.453 1.785 2.016 2.202 2.603 3.060 3.531 4.011 4.483

<*w1/3

15.03 12.75 11.30 10.52 10.01 9.09 8.30 7.66 7.15 6.74

6. The conductivity of ammonium sulfate solutions

The conductivity of ammonium sulfate solutions near the saturation con­centration was redetermined in order to be able to perform some calculations for the optimization of the Coulter Counter and to see whether the conduc­tivity can be used to determine the concentration. The measurements were performed using a Wayne Kerr B642 universal bridge and a Philips PR9512 conductivity electrode. The resistance of the leads to the cell was measured by short circuiting the electrode dipping it in mercury.

231

The cell constant supplied by Philips, 0.746276 cm was used, since a check on a saturated NaCl solution showed the deviation to be less than 1%. The conductivities, K , of five solutions are given in table 7. including the saturated NaCl solution. The conductivities at 18°C are compared with values of Kohlrausch, and Klein (Abegg, 1907), see table 8 and figure 3- It is seen that the conductivity has a maximum value at *• 32#, and decreases with in­creasing concentration in the near saturated region. The concentration dependence of K at 18°C is estimated to be

f—1 = - 209 19 IsS/SSl <n) l3wJT ^09'19 [kg/kg] ( l i )

This value is used to test the consistency of the temperature dependencies of K, given in table 7-

Table 7 Conductivities, K, of ammonium sulfate solutions

w [kg/kg]

0.4172 «xO.4273 ;0.4379

saturated saturated

solute

(NH.) SO. (NH£);SO7 (NHJ)^SOJ (NHJ^SOJ NaCl

temp. range

25-49°C 27-49 29-49 19-49 9-29

K [mS/cm]

156.726 + 4.2211 » T 154.973 + 4.2105 • T 152.862 + 4.0864 • T 157-684 + 3.8229 * T 134.218 + 4.6776 * T s a t

concentration suspect

Table 8 Conductivities, K, compared with literature data, T = 18°C

w

0.032 0.063 0.092 0.122 0.151 0.180

<[mS/cm]

37-8 68.1 94.1 „. . 120.1 K l e i n

141.4 163.O

w

0.05 0.10 0.20 0.30 0.31 0.4172 0.4173 *0.4379 0.4483

K[mS/cm]

55-2 101.0 229 2 Kohlrausch 232.1

221.55 219 37 21230 Present determination 215.03

ft

* concentration erroneous?

For the variation of K with the saturation temperature the following equa­tion holds:

232

d K sat ■3KI d w sat (£5.1 fÓK) d T . l3TJ + lawJ„ " d T _ sat w T sat

= 4.215 - 209.19 • 9.121 ■ ÏO"4

= 4.024

which is a 5% higher value than the experimental one.

(14)

200

X[aS/.ca|

I HO­

MO

SO-

« Ikj/kgl —

200

—1 r- 1 1 O.tO 0.20 0J0 0.40 0.S0

ConducHvttfX.of ■■■oaiuB julflle solutions «I 18*C.

Fig. 3 Conductivity K, of ammonium sul fa te solut ions at 18°C

Comparing the temperature dependence, 4.215 mS/cm °C with the concen t r a t ion dependence 209.19 mS/cm.kg/kg shows tha t 1°C has the same effect on the con­duct ivi ty as a concent ra t ion change of 4 .215/209.19 * 0.020 kg/kg, or a change in sa tura t ion temperature of 22.1°C. Therefore when th i s technique i s used the measurement and s t a b i l i t y of the tempera ture a re of the utmost impor tance . In o rde r to determine the sa tura t ion temperature within 0.1°C the temperature must be held within 0.1/22.1 = 4.5 * 10 C. The accuracy of the conductivity should be be t t é r than 4.5 » 10"^°C • 4.215 mS/cnTc =0 .019 ms/cm a t a c o n d u c t i v i t y of about 400 mS/cm which i s about 5 » 10 . Therefore th is technique does not offer an accurate way to determine supersaturat ions in ammonium su l fa te solut ions .

233

Appendix 6B

The convective-diffusive mass transfer in ammonium sulfate solution

1. General

The mass transfer from the solution towards a crystal is governed by diffu­sion and convection. Diffusion to a sphere of size L in a stagnant medium is given by:

k .L Sh = "V = 2 (1)

I t i s assumed tha t d i l u t e solutions are used. In more concentrated solutions the d i f f u s i o n process i s enhanced by the veloci ty of the solut ion to the c rys t a l - so lu t ion in ter face , which can no longer be neglected compared to the average v e l o c i t y of the d i f f u s i n g s o l u t e . I t can be shown (Westphal and Rosenberger, 1978) that th i s effect can be accounted for by using an e f f e c ­t ive diffusion coeff ic ient :

m PP = ^T~ (2) eff 1-w

For the d e s c r i p t i o n of the mass t ransfer enhancement due to convection two models are currently used: the f i r s t based on the s l i p veloci ty of the pa r ­t i c l e s due to gravi ty , and the second based on the turbulence of the l iqu id . The turbulence i s described using Kolmogoroff' s theory r e l a t i n g the i n t e n ­s i t y of t h e tu rbu lence to the speci f ic power input . The r e su l t s are mostly presented in the form of empirical r e l a t i ons :

Sh = 2 + 0.72 R e . 1 / 2 S c 1 / 3 (3)

d e s c r i b i n g the mass t r a n s f e r accord ing to the t e rmina l v e l o c i t y - s l i p v e l o c i t y approach (Nienow, 1975) where Re i s the p a r t i c l e Reynolds number, and:

l / 3 T V 3 a , „ Sh = 2 + K [- f ] . Sc / : J (4)

following from the Kolmogoroff theory. The lower-limit of Sh = 2 was chosen in order to account for the stagnant diffusion, see equation 1. Both equa­tions, 3 and 4 have their shortcomings: the slip-velocity approach does not account for extra turbulence and will give the best results at minimum suspension conditions; the Kolmogoroff model holds strictly only for non-settling particles at appreciable power input. It seems therefore justified to combine the two approaches in order to estimate the effect of the hydrodynamics and crystal size on the liquid side mass transfer coefficient. However, Nienow (1975) seriously objects against the use of the Kolmogoroff theory in the form of equation 4 for overall correlations of Sh numbers, since the right and left hand side of equation 4 are dominated by the size L of the particles which has no great influence on the mass transfer: taking a = 0.63 (Ohashi et al, 1979) and solving for the size dependence of k yields:

kd - L-°-16 ,5)

234

In mass t r a n s f e r exper iments t h e s i z e of the p a r t i c l e s i s the parameter which i s most eas i ly varied over a wide range, about 1000-fold, whereas the v a r i a t i o n of the o t h e r parameters i s r e s t r i c t e d . P lo t t i ng equation 4 on double logarithmic paper i s equ iva len t t h e r e f o r e to p l o t t i n g k i .L ve r sus L . I t i s not su rp r i s i ng therefore, tha t due to- the dominating effect of the unimportant, but widely varied parameter L, the c o e f f i c i e n t a tends t o 3/4.

The insufficiency of equation 4 i n c o r r e l a t i n g mass t r a n s f e r da t a can be demonstrated by comparing the b a s i c r e s u l t s of Ohashi e t a l (1979) with t he i r f inal corre la t ion based on equation 4. The au thors i n v e s t i g a t e d the mass t ransfer towards spherical ion-exchanger p a r t i c l e s of 300, 600 and 800 um in v e r t i c a l tube-flow. The average l i q u i d v e l o c i t y U va r i ed between 0.47 and 2.52 m/s and two tube diameters, 2.0 and 3«0 cm were used. Their basic data can be correlated by:

k„ - U ° -5 8 . D - ° -5* . L-°'30 (6) d av T 1/3 4/3 For the definition of e in the specific power group (e L ' / v), also

referred to as the turbulent Reynolds number, they use:

e = 2 f U3 D "1 (7) av T w '

where f, the friction factor is given for a smooth tube by: f = 0.082 (ReT)"1/i4 (8)

where- Re,p and D , a r e the Reynolds number and diameter of the tube. By cor-<d r e l a t i ng k , with c a t constant p a r t i c l e s i ze they found:

kd - c (9)

The overall correlation, using equation 4, described the data with a stan­dard deviation of 10.7#:

Sh = 2 + 0.44 (e1/3 L 4 / 3 / v ) 0 - 6 3 Sc1/3 (10)

By inserting equation (7) in (10) and solving for k, it is found that: " TT 0.58 n -0.26 .-0.16 ,11N kd " Uav DT L (11)

It is seen that the exponents of D_ and L are halved compared to the straightforward analysis of the data, see equation 6. This means that the use of the specific power group from the Kolmogoroff approach is not justified.

235

2. Mass transfer in a crystallizer

The mass transfer in a stirred vessel is assumed to be an addition of stag­nant diffusion, terminal velocity and turbulence effects. The mass transfer at the terminal velocity is evaluated using the velocity needed for just complete particle suspension (Nienow, 1975)ï

(R APJ 0- 3 6 L1.07

(Sh-2) = 0.28 — 2 — - . Sc1/3 (12) v

Adding the effects given by equations 1, 6 and 10 and expressing k, as a function of L and the stirrer speed N which is proportional to U , Pleads to:

k„ - X cP L"0^0 N 0'* ♦ c, L°'°7 (13) a L £ 3

Equation 11, valid for dilute suspensions, shows that the mass transfer is strongly size-dependent for small particles only. The stirrer speed depend­ence will manifest i t se l f most clearly at the larger sizes, c,/L*0, and neglecting the other size dependencies, though the overall value or the ex­ponent of N will be lower than O.58 due to the constant slip velocity contribution. In the next section some calculations will be made for the am­monium sulfate-water system.

3. Mass transfer in an ammonium sulfate solution

The mass transfer in saturated ammonium sulfate solutions is calculated as the sum of three terms, diffusion (I) , slip velocity (II) and turbulence (III), using equations 1, 2, 10 and 12:

I 2ffleff

L

H = o.28 (S-^)0 '3 6 . L°'°7 S c ^ / 3 v"°-72 ^ . a . L0.07

ill - 0.30 U °'58 L"0-16 D - ° - 2 6 v-°'58 S c 1/3 D b u 0.58 L-0.16 av T eff eff av

The following values are used in the calculation of a and b:

g = 9-81

^ = 0.421 P 1/3 Sc __ J and ffl __ given in Appendix 6a.

v is calculated from n and p, see Appendix 6a.

H» 0 0 — J ON VJI . p - O ) IV) IV) H 1

O O O O O O O U I O O O

o o o o o o o o o o o \j\ j = - . * r - p - . p - . p - - p - 4 = - . p - - p - - p -i-» oo—i: ONVJI 4=-OJOJOJ iv> i-1

h j j r v j l ON—J OOVO . p - O O t-1

i-» oo ONVJI -P-OO rv) ON o vo oo

1—»■ 1—»■ 1—* 1—»• 1—»■■ 1—»■ I-* I—* H-* •—»• H^ I V > I V ) t V > I V > I V ) I V ) ! V > I V > t V > t V > I V ) VJI VJI VJI VJI -c- -P" -P- -P- -P- -er -P-—J IV) H» O 0 0 — ] ON U I -P-OO IV)

o vo ON tv) vo VJI iv> VJI OOVJI H» H» ON H ' ON IV) —J IV) -P"-—J | \ ) —J

h - ' l - ' l - » I V ) I V ) N ) I V ) < _ O U J - p -

—j oovo t-1 -P-—a vo iv) —J ON IV) OJ VO 0 0 -p-VJl IV) t-» -P- ON

i - ' i - 'H»i- ' i - ' iv) iv)rv)(_ouo UJ -P-VJI -J vo tv) OJ vn o —J —J ONVO VJI ONH> -p- CO ►-» VJI

U I 4=- -P"-OJ O ) . IV) IV) IV) H» H» I-»

VO -p- O VJI O ON IV) O —3 -P - H-» O O H W 0 \ O O W V O U 1 O

H 1 t -1

OJ O V O 0 0 — J ONVJI j r j r - u o IV)

1-» M O ) IV) IV) M OJ VO OJ VJI —J ON i-*- oo —J VJI rv) i-'—a co-p-

vo—i ONONVJI j r j r u o u o rv) iv> VJI VJI vo MVJ IOOI - 'VOVJ IVOOJ

■—1 OJ O ) -p- -p- v o O M VJl O J

o

o

1—1

l—J

1 - 1

B. It»

1 1

1 - 1

o 1

2 ! ca s

IS»

1 1

■1—1 B-

f**

co 1 1

1—1

o 1

VO B

N

ca 1 1'

1—1

o VJI

1—1

1—1

o VJI

1 1

H

s

T>

3

<;-

03

o*

IV) OJ ON

237

Appendix 6C 1. Shape of ammonium sulfate crystals From the work of Bourne & Faubel the value of k was obtained, together with the ratio of the dimensions in c and a direction:

k = 0.64 (1) v c/a * 1.2 (2)

Sieves with square holes were employed. Since the dimension in the b direc­tion is much smaller than in the c or a direction, the sieving classifies the crystals according to their a-dimension. Assume the a-dimension to be 85# of the diagonal of the sieve hole:

a * 0.85 L J 2 = 1.20 L (3) Approximation of the crystal shape by rectangular blocks with dimensions a, b and c leads to:

a b c (eq,3) b c kv = Ê^f = 1.73 (|) (f) (4) Lt

and: k = 2 (ab * be ♦ ac) = 2 m (c { 1 +b } + bj (5)

a j c. a a a Inserting equations (1) and (2) in (4) results in:

b/a = 0.31 (6) From equation (5) the value of k is calculated to be:

el ka = 5-4 (7)

The linear growth rate in the [100] direction (a-axis) can be related to the growth rate G defined as the increase of sieve dimension of the crystal:

i d a eq.(3) dL v [100] = I jj! = 0.60 ^r = 0.60 G (8a) + G = 1.67 v [100] (8b)

238

2. Conversion of concentrations and supersaturations

Three definitions of the concentration can be used:

a. The mass fraction , w , in kg/kg solution. b. The mass ratio , c', in kg/kg solvent. c. The mass concentration, c , in kg/m3 solution.

Conversion formulas for concentrations and supersaturations:

w = l/(l/c' + 1) = c/p (9)

c' = l/(l/w - 1) = l/(p/c - 1) (10)

c = pw = p/(l/c'+ 1) (11)

Aw = Ac'/U + c')« = Ac/p (12)

Ac'= Aw/(1 - w)' = (Ac/p)/(l-c/p)* (13)

Ac = pAw = pAc7(C + I)2 (14)

Additionally the supersaturation can be expressed as an undercooling AT:

AT = Aw/(d weq/dT) (15) For ammonium sulfate:

AT = 1096.4 Aw (16)

3. Definition and conversion of mass transfer coefficients

Two types of definitions are used:

a. the usual definition k = mass flux (R) ... , , }

concentration difference and, b.

linear growth rate concentration difference (18)

Definitions:

[(kg/m's) / (kg/m3)] (19)

[(m/s) / (kg/m>)] (20) [m/s °C] (21) [(m/s) / (kg/kg)] (22)

(23) (24) (25)

The overall growth rate is related to the mass flux R by

R = 3 Pc (V ka ) G (26)

In this work Aw is used preferentially. With equations (26), (25), (19) (12) the relation between k and k is found:

k = 3 <PC/P) (V ka ) k" (27)

Also, from equations (22) and (25) it is seen that:

kw = (G / v[hkl]) kw* (28)

For the [100] face of ammonium sulfate it is calculated with equation ( that:

kw = 1.67 kW [100] (29)

and k = 0.503 kW = 0.84 kW [100]

usual

single face growth rate

overall growth rate

: k = — Ac : k£ = v[hkl]

k* = v[hkl] k = v[hkl]

: k£ = G / Ac k = G / AT kw = G / Aw

/ / /

Ac AT Aw

N ) f O | y K H h * P M h » ( - * H M H I V J I - > O V 0 0 0 - J 0 N U l . t r U > t o t - » O V 0 0 3 - - J O N U l - J ï - t j J t o H » B B B 0 0 0 0 0 0 0 0 ® ® 0 o - O O O O O O O

1—» I— t—' t—*j l -» t -»H>H* H» (-» H» »-» I-» N> t-» U I U 1 W O O O O O O O O U l f l v > 1 0 0 0 0 0 0 0 0 0 o o o o o o o o o b o o M u i ^ J o o b b b b b O O O O U I O U 1

fO t~* t—* h-' H* H* I™* l~* f"* H* ÏSJ H* h- Q t o u i v o v o v o v o t o t o t o t o t o t o t o t o O O N r v j v o o N - e - t o O - J - & - J - J ~4 - J -»1 ~g - J - 4 - - J - 4 - 4 - J O O - - ] - J O O O O l j O

I-» I-» to to I-* >-> »-» M H » VO VO O M » U I !-• VO VOVO to M i O O U J U l l y toto (_>J OJ W 00 00 4=-U) UJVAJ

to NVJ1 ONUOOJ M Kj o o o c a o o w o o

O O O H P H H ^ U U ^ O O O W -trtjo N H O O O M O O O O V O J r » . P ^ 1 0 » 0 0 -P-OOWVOUI ON -Er vo O O toVOLOONto-JOOUJUIVOVOOOI-'Ulto4=-ON~} O - fc-J W ■ t r - J V O - J t o t o U I J r O t o O O I - ' O N t o - J t o v O U l v O U J O N O N JrvO O ~J U I 00 O M J r to

p o o - J to as

I - « I - » t-> j r i-» M (-» H» H» i-» H O O H O O O W O J r ^ - ^ O H W V O V O V O W a i J S O O O - I T H O x O O3 00V*>.to H ' O N . C ' O N V O W U I V A I O J - - I V O W O — l u i u i o o t o U J i o — l v * > v o - C ' t o t o u i o ' N O \ o \ O N . e - . t r u i vo O N O O O O NJ o o o \ v o to v o v o v o

-~1U1 I-» - J J r o \

M to ►-* a \ o O H ' K M H N v i ~ j v i o N i w o \ * i - ' u i i-* o o o O \ C 0 O 0 3 C 0 « 0 0 I U M O C » - I W H H ~ 1 0 U W * 0 0 ! - > V D O I - » O N l j i > < j J U l . C - U J O - t = - 0 ' N O l - ' O N t o l - » O N U l ( - » 0 O O h»U0—4 O O W to toUl M 00 00OJ L O O O I V I U I t o

00 »— 03 M U I - J 00ON

to totototo i - ' i - ' O o - c - t o O U 1 - J - J S ^ H K H P U l O N O l O O P ^ I U ' O - t r O - J — 1 - J - l 00 CD 00 00 -Er U I to-J O V O C O - 4 - J ~ J to O U 1 0 0 H P H P N W W M U I M ^ O O O \ M H M W U 1

u i O to o

V \S \S V

o o o o o o o o o o o o o o o o o o o o ON oovo vo vO VO V0 vo VO 00 00 VO vo VO vO VO vo vo vo vo >-»OJVO00 0 » O N O N O N Ö N O O V O to 4r O t - u u i o o v o v o v o U I 0 0 O O O O O O N O O - 4 -C-COVO O O ON OO

O) >-» u> o O U I U I 00

O t o t o l - » I V ) - - J J r * - 4 = - O t o t o J r O \ 4 * V J 0 l-> ►-> O

IS8SS8ggSS5ffiSSS&SIS o o LD O '

H» H» H» M I-» H» to H» I-» o o \ t - ' v o v o v D v o t o t o t o t o o \ o o v o H ' 0 0 \ t o v o o \ - c t o 0 - C O M - » H » > - » t - > V D V O V O V O . e - | - » — I W O C H O H - t H U ) ovoto—j—m- j - j> j~ i \OHwuioo-J~JOOOOV*>

r—i n

1 h-»

l—l

r—i

o 1

ON B

1 1

r—i ST w B w

1 1

I—1 o l_J

(—1

o U I

(XI (0

U f

c—i

o u i % --, *• m

(0 1 1

1—1 co •- B

l - J

r—i

o 1

VO B^

Dl

l—l

ss

r ca f t

3 co

3

00

I

B^

3 o o z to r

co V 03

* t o e*

3 o

1—1

o U I %

UI w

- I_J

1—1

o B

co U.J

3 o

o SS

o 00

i

?

s^uamxasdxa uotseaqB aqq. JO s^xnsen

/, xTpuaddv

0\\Z

241

Terminal velocity of ammonium sulfate crystals in ethanol

L Cum]

2200-2400 1180-1410 1000-1180 416- 496

L Cum]

2300 1295 1090 456

vt [m/S]

0.152 0.097 0.090 0.040

mass (mg)

7.86 1.86 1.38 0.13

eq. bol diam. [mm]

2.97 1.9^ 1.62 0.67

/t [m/s]

0.10 -

VT * 75 L [m/sj

Tb 25" Fig. 1 Terminal settling velocity

2k2

243

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257

List of symbols

1. Abbreviations

ADP ammonium dihydrogen phosphate ASL Abegg, Stevens and Larson B&S Birth & Spread CCG constant crystal growth CMSMPR continuous mixed suspension mixed product removal CSD crystal size distribution DTB draft tube baffled GRD growth rate distribution HMT hexa methylene tetramine KDP potassium dihydrogen phosphate NAS nickel ammonium sulfate RF random fluctuation 2D, 3D 2-, 3-dimensional

2. Variables

a a a a. A1

cum A

Ac

Ac .. . cv?1 d D Dr mG E E. f F F Fo S G AG* I

a-axis, [100] direction; dimension crystal area distribution area per crystal (chapter two) coefficients of polynomal eq.(5-10) pre exponential factor, eq.(2.2) cumulative, total area b-axis, [010] direction; dimension nucleation rate per crystal birth function; -of fragments dimensionless size, eq.(5«10) secondary nucleation rate c-axis, [001] direction; dimension concentration, supersaturation, Ch.1-5

Ch.6 surface concentration of proto nuclei dimensionless supersaturation, eq.(5«17) coefficient of variation death function diameter of stirrer growth diffusivity diffusion coefficient impact energy activation energy energy transfer rate modified population density, eq.(3-53) number flux of growing crystals formation frequency Fourier number constant of gravitation linear growth rate, eq.(3-4) formation free enthalpy of the critical nucleus incorporation frequency birth function of fragments total fragment formation rate

m2/m3.m] m2] #/m3.m] # / m 3 . s ] m* /m3 ]

# / s ] # /m 3 . s .m] - ] # / m 3 . s ]

kg/kg s o l n . ] kg/m3] # /m J ] - ] - ] # /m 3 . s .m] m] m 2 / s ] m ' / s ] J / i m p a c t ] kJ /mole] J / m 3 . s ] # / (m 3 .m . (m/s ) ) ] # /m 3s] #/m*s] - 3 N/kg] m/s]

s ] #/kg.s.m] #/kg.s]

258

k

kr

KV

1P

L L L-.E' 16 LSum m M

• cri'L' 84

cum n

n.n"

| M T

N

N Ncum psep Q r r , R R

R£ S Sc Sh t t Tl AT Ut ir av v V V m w. ï w

.N„

;o.5

stationary homogeneous nucleation rate gas constant area shape factor mass transfer coefficient for diffusion overall mass transfer coefficient for growth nucleation constant nucleation constant growth constant volume shape factor volumetric discharge coefficient crystal length distribution crystal size size below which 16, 50 or 84% of total mass cumulative, total crystal length size of the critical nucleus crystal mass distribution cumulative, total crystal mass fourth moment of the CSD crystal number distribution, or population density, or crystal size distribution number of molecules in a nucleus, critical nucleus stirrer speed Avogadro number cumulative, total crystal number separation number production rate of crystalline material coefficient, eq.(3.70) radius of spherical nucleus/crystal coefficient, eq.(3.70) removal frequency per proto-nucleus gas constant mass flux during growth Reynolds number (particle) concentration ratio c/c Schmidt number Sherwood number (particle) time; time of penetration, eq.(2.7) half-life time of supersaturation temperature undercooling terminal velocity average liquid velocity velocity volume of crystallizer molar volume, weight of i sieve fraction concentration: mass fraction inverse of separation number

[#/m3s] [J/K] [m* /m* ] [m/s] [m/s]

[#/kg s]

[m3 /m3 ] [-] [m/m3.m] [m] [m] [m/m3 ] [m] [kg/m3.m] [kg/m3] [m* /m3 ]

[#/m3.m]

[#] [rpm]

[#/m3] [-] [kg/m3s] [m/s] [m] [#/#s] [J/mole.K] [kg/m2 s] [-] [-] [-] [-] [s] [s] [K] [K] [m/s] [m/s] [m/s] [m3] [m3/mole] [g] [kg/m3] [-]

a r <5

shape factor, see app.2A gamma function, see app.3E, eq.(3) half width of cluster interphase region, eq.(2.8) specific power imput

[m] [W/kg]

259

efficiency, target efficiency [-] dynamic viscosity [mPas] conductivity [mS/cm] classification parameter [-] j moment of the CSD [nT/m* ]

n,nt n K K u . - ■ Ap supersaturation [J/mole]

kinematic viscosity [m2/s] liquid density [kg/m*] crystal density [kg/m3] interfacial energy [J/m2]

a relative supersaturation [-] a1 parameter, eq.(6.1) [-] x,T mean residence time of slurry and solids

s respectively [s] $ volumetric flow rate [i'_/s] a impact frequency per crystal [s ]

3. Subscripts

v P pc a

av ax B c cum d det eff eq f f g i i i id max P r r s sat St T V

average axial bulk circulation cumulative diffusion detection effective equilibrium formation fragment (Ch.7) growth incorporation (Ch.2) in feed (Ch.4) interface (Ch.6) ideally mixed maximum product removal reaction (Ch.6) survival saturation start (Ch.7) total volumetric

4. Superscripts o at zero size ov oversize ~, - averaging

260

261

Curriculum v i t a e

De a u t e u r , P i e t e r J o h a n n e s Daudey, i s geboren op 15 maart 1955 te ' s -Gravenhage. Na b e ë i n d i g i n g van z i j n m i d d e l b a r e s c h o o l o p l e i d i n g (Gymnasium-p) in j u n i 1972, begon h i j z i jn studie scheikundige technologie aan de Technische Hogeschool t e De l f t . In maart 1980 behaalde h i j z i j n i n g e n i e u r s d i p l o m a , waarna h i j s t a r t t e met z i j n promotie-onderzoek in opdracht van DSM. Sinds 1 maart 1986 i s h i j werkzaam a l s research technoloog b i j Akzo Research te Arnhem.