NUCLEATION IN CONTINUOUS AGITATED CRYSTALLIZERS
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NUCLEATION IN CONTINUOUS AGITATED CRYSTALLIZERS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. R. VAN NAUTA LEMKE, HOOGLERAAR
IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 23 MEI 1973
DES MIDDAGS OM 16.00 UUR DOOR
ERROLL PAUL KAREL OTTENS scheikundig ingenieur geboren te Willemstad, Curacao
r
Dit proefschrift is goedgekeurd door de p r o m o t o r e n PROF. IR. E.J. DE JONG EN PROF. DR. IR. H.A.C. THIJSSEN
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TABLE OF CONTENTS
page
SUMMARY 8 SAMENVATTING 10
CHAPTER 1 INTRODUCTION 12 CHAPTER 2 MECHANISMS OF NUCLEATION IN SOLUTION 14
2.1 Homogeneous nucleation 14 2.2 Primary heterogeneous nucleation 17
2.3 Secondary nucleation 18 2.4 Factors affecting collision breeding 19
2.4.1 Effect of the supersaturation 19 2.4.2 Effect of the impact energy 22 2.4.3 Effect of crystal size and concentration 25
CHAPTER 3 A MODEL FOR COLLISION BREEDING IN A STIRRED VESSEL 27 CRYSTALLIZER
3.1 Crystal^impeller collision breeding 28 3.2 Crystal-wall collision breeding 30 3.3 Crystal-crystal collision breeding 30 CHAPTER 4 THE APPARENT NUCLEATION RATE IN CONTINUOUS CRYSTALLIZERS 33
CHAPTER 5 EXPERIMENTAL AND DISCUSSION 44
5.1 Introduction 44 5.2 The sieve analysis 44
5.3 Supersaturation measurement 45 5.4 The experiments in the stirred vessel crystallizer 46
5.4.1 The apparatus 46 5.4.2 Results and discussion of the potassium alum-water system 47
5.4.3 Results and discussion of the magnesium sulphate-water
system 56 5.5 The experiments in the froth bed crystallizer 52
5.5.1 The apparatus 62 5.5.2 Results and discussion of the potassium alum-water system 63
CHAPTER 5 DESIGN CONSIDERATIONS OF STIRRED VESSEL COOLING
CRYSTALLIZERS 67 LISTS OF SYMBOLS 77 REFERENCES 80 APPENDIX I 83 APPENDIX II 86 APPENDIX III 87
SUMMARY
The aim of the present work is to investigate the relation between the nucleation rate and the supersaturation, crystal concentration, growth behaviour of the crystals and the hydrodynamical behaviour of the suspension in a continuous, agitated cooling crystallizer. Two crystallizing systems with different growth behaviour have been studied: the potassium alum-water system and the magnesium sulphate-water system. In order to examine the influence of the hydrodynamical behaviour two types of cooling crystallizers have been used: a stirred vessel crystallizer and a froth bed crystallizer, in which the crystals are suspended by bubble agitation.
The theory of crystallization is described in chapter 2 through 4 in so far as relevant to the interpretation of the experimental data (chapter 5). The differ-ent nucleation mechanisms, namely: homogeneous, primary heterogeneous and
secondary nucleation, are shortly reviewed in chapter 2. Because of the impor-tance of secondary nucleation in a suspension crystallization process this mechanism is described in more detail. The most acceptable secondary nucleation mechanism is collision breeding. Factors affecting the nucleation rate such as the supersaturation, crystal size, crystal concentration and the impact energy at the crystal surface are discussed.
A nucleation model for collision breeding in a stirred vessel crystallizer is derived in chapter 3. The nucleation model is based on the nucleation rate data for controlled impacts at crystal surfaces and a simplified model for collisions of crystals with the impeller, a vessel wall or other crystals. The nucleation model results in a relation between the nucleation rate and the supersaturation crystal mass concentration and the power dissipated by the impeller per unit mass of suspension.
The calculation of the nucleation rate and growth rate of crystals from a crystal size distribution (CSD) is described in chapter 4. The factors influencing the calculation of the nucleation rate such as size dependent growth rate and
classification of crystals near the discharge opening are discussed. Some experi-ments with glass spheres in order to determine the magnitude of classification near the discharge are described in appendix III.
The experimental results are discussed in chapter 5. For the experiments with potassium alum two vessels of different volume have been used. Potassium alum shows a regular growth behaviour at varying supersaturation. Variables of
particular interest were the impeller type, speed and location, the crystal mass concentration and the supersaturation. The nucleation model appears to be applic-able to the interpretation of the experimental results. It can be concluded that nucleation due to crystal-impeller collisions dominates in the overall nucleation process. The nucleation rate depends on the impeller type but not on the vessel volume. This enables one to design larger crystallizers of the same geometrical shape.
The irregular growth behaviour of magnesium sulphate at various supersaturations causes cyclic behaviour in the crystal size distribution as a function of time. This cyclic behaviour is ascribed to an abrupt transition from collision breeding at low supersaturation to needle breeding at higher supersaturation. A computer simulation of the periodically varying CSD agrees well with the experimental observations. A steady state crystallization process can be attained at low supersaturation where needle breeding is suppressed.
In view of the importance of crystal-impeller collisions in the nucleation
-process, another type of crystallizer has been studied, in which the crystals are suspended by bubble agitation. This froth bed cooling crystallizer is constructed as a sieve plate column. The nucleation rate gradually decreases with increasing superficial gas velocity. This phenomenon is explained from the flow behaviour in the froth bed.
The design procedure of stirred vessel cooling crystallizers is described in chapter 6. Three important factors are considered in this procedure: the product quality of the crystallizer, the possible formation of incrustations on the cool-ing wall and the attainment of a homogeneous suspension. The exponent of the crystal growth rate or supersaturation in the nucleation rate expression in-fluences the design procedure.
SAMENVATTING
Het doel van dit werk is na te gaan welke de relatie is tussen de kiemvormings-snelheid en de oververzadiging, kristalconcentratie, groeigedrag van kristallen en het hydrodynamische gedrag in de suspensie in een continue, geroerde kristal-lisator. Twee kristallisatiesystemen met verschillend groeigedrag zijn bestu-deerd: het kaliumaluin-water systeem en het magnesiumsulfaat-water systeem. De invloed van het hydrodynamisch gedrag is onderzocht in twee typen koelkristalli-satoren: een geroerd vat kristallisator en een schuimbed kristallisator, waarin de kristallen gesuspendeerd worden door de beweging van bellen.
De kristallisatie theorie wordt beschreven in hoofdstuk 2 tot en met 4, voor zo-ver deze relevant is voor de interpretatie van de experimentele gegevens (hoofd-stuk 5 ) . De verschillende kiemvormingsmechanlsmen, te weten: homogene, primaire heterogene en secondaire kiemvorming worden in het kort beschreven in hoofdstuk 2. Vanwege het belang van secondaire kiemvorming in een suspensie kristallisatie-proces zal dit mechanisme gedetailleerder worden behandeld. Het meest waarschijn-lijke secondaire kiemvormingsmechanisme is kiemvorming ten gevolge van botsingen van het kristal. De factoren die de kiemvormingssnelheid beïnvloeden, zoals de oververzadiging, de kristalgrootte, de kristalconcentratie en de botsingsenergie op het kristaloppervlak, zullen worden besproken.
Een kiemvormingsmodel voor kiemvorming ten gevolge van botsingen van kristallen in een geroerd vat kristallisator wordt afgeleid in hoofdstuk 3. Het kiemvor-mingsmodel is gebaseerd op kiemvormingsgegevens voor gecontroleerde botsingen op kristalvlakken en een vereenvoudigd model voor botsingen van kristallen met het roerwerk, de vatwand of andere kristallen. Het kiemvormingsmodel resulteert in een relatie tussen de kiemvormingssnelheid en de oververzadiging, kristalmassa-concentratie en het gedissipeerde vermogen door het roerwerk per massa eenheid suspensie,
De berekening van de kiemvormingssnelheid en de groeisnelheid van kristallen uit een kristalgrootte verdeling wordt beschreven in hoofdstuk 4. De factoren die de berekeningswijze van de kiemvormingssnelheid beïnvloeden zoals grootte afhanke-lijke groeisnelheid en classificatie van kristallen bij de uitstroomopening wor-den besproken. Een aantal experimenten met glasbolletjes om de grootte orde van de classificatie te bepalen wordt beschreven in appendix III.
De experimentele resultaten worden besproken in hoofdstuk 5. Voor de experimenten met kalium aluin werden twee vaten met verschillende inhoud gebruikt. Kaliumaluin vertoont een regelmatig groeigedrag bij verschillende oververzadigingen. Belang-rijke variabelen in dit onderzoek waren het roerwerktype, toerental en de plaats van het roerwerk in het vat, de kristal massa concentratie en de oververzadiging. Het kiemvormingsmodel blijkt toepasbaar bij de interpretatie van de resultaten. De kiemvorming ten gevolge van kristal-roerder botsingen overheerst waarschijn-lijk in het totale kiemvormingsproces. De kiemvormingssnelheid hangt af van het roerdertype maar niet van de inhoud van het vat. Hierdoor is men in staat grote kristallisatoren met identieke geometrie te ontwerpen.
Het onregelmatige groeigedrag van magnesiumsulfaat bij verschillende overver-zadigingen veroorzaakt cyclisch gedrag in de kristalgrootte verdeling als functie van de tijd. Dit cyclisch gedrag wordt toegeschreven aan een plotselinge overgang van kiemvorming ten gevolge van botsingen bij lage oververzadiging naar kiemvor-ming door naaldgroei bij hogere oververzadiging. Computer berekeningen die dit cyclisch gedrag simuleren komen goed overeen met de experimenten. Het cyclisch gedrag treedt niet op bij lage oververzadiging waarbij de kiemvorming door naald-groei onderdrukt wordt.
-Gezien het belang van kristal-roerder botsingen in het gehele kiemvormingsproces is ook onderzoek verricht in een kristallisator waarin de kristallen gesuspen-deerd worden door de beweging van de bellen. Deze schuimbed-koelkristallisator is geconstrueerd als een zeefplatenkolom. Het blijkt dat de kiemvormingssnelheid afneemt met toenemende superficiële gassnelheid. Dit verschijnsel wordt ver-klaard uit het stromingsgedrag in het schuimbed.
De ontwerpprocedure van geroerd vat koelkristallisatoren wordt beschreven in hoofdstuk 6. Drie belangrijke factoren in deze procedure worden behandeld, name-lijk de productkwaliteit van de kristallisator, het mogename-lijk optreden van aan-korstingen op de koelwand en het verkrijgen van een homogene suspensie. Het blijkt dat de exponent van de kristal groeisnelheid of de oververzadiging in de kiemvormingsrelatie hierin een belangrijke rol speelt.
CHAPTER 1 INTRODUCTION
Crystallization from solution is a simultaneous process of the creation of a supersaturated solution, nucleation and growth of crystals and the withdrawal of the produced crystals from the crystallizer. The product quality of a continuous industrial crystallizer has to meet specifications as to the crystal habit, purity and crystal size distribution (CSD). Besides it is important to know how the product quality changes as a result of an increase in the production rate or an increase in the size of the crystallizer. An important factor in this problem is the supersaturation.
The supersaturation can affect the crystal habit and purity. The crystal habit is determined by the ratio of the growth rate of the different crystal faces, which may depend on the supersaturation. Irregular growth behaviour usually involves occlusions of mother liquor within the crystal thereby reducing the purity of the crystal. Irregular growth behaviour appears especially at higher supersaturation levels. The supersaturation maintains the nucleation process as well as the growth of crystals. The nucleation and growth rate and the residence time of the individual crystals in the crystallizer determine the CSD. The CSD determines the total available crystal surface area in the crystallizer, which in turn deter-mines the supersaturation level at a given production rate. By this phenomenon, the mutual dependency of the different quantities in a crystallization process is obvious. At present, methods to predict the CSD in an arbitrary crystallization process are not available, due to the lack of knowledge of the individual
processes which determine the CSD.
In this thesis the relation between the nucleation rate and the supersaturation, crystal concentration, growth behaviour of the crystals and the hydrodynamical behaviour of the suspension in the crystallizer is emphasized. Several mechanisms are known for nucleation in a supersaturated solution, namely: homogeneous,
primary heterogeneous and secondary heterogeneous or secondary nucleation.
Usually, homogeneous or primary heterogeneous nucleation can be ruled out during suspension crystallization because they occur at high relative supersaturation. Secondary nucleation takes place at low relative supersaturation in the presence of parent crystals. Much experimental work on secondary nucleation has been reported in the literature. However, no consistent physical theory is available as yet. Several mechanisms can be distinguished for secondary nucleation depend-ing on the supersaturation level, the growth behaviour of the crystals and the chance that collisions against the crystals may take place. It is likely that the nuclei are generated by the dislodgement of crystalline agglomerates from the crystal surface or the adsorption layer near the crystal. This can be achieved by an impact on a crystal surface. In an agitated suspension of crystals,
collisions may occur with the impeller, the vessel wall and other crystals. These collisions are greatly affected by the hydrodynamical behaviour of the suspension in the crystallizer.
In this work the effect of the hydrodynamical behaviour on the nucleation rate has been studied in two types of cooling crystallizers: a stirred vessel crystal-lizer and a froth bed crystalcrystal-lizer, in which the crystals are suspended by bubble agitation. Variables of particular interest in the stirred vessel crystallizer were the supersaturation, the crystal mass concentration, the size of the vessel and the impeller type, speed and position. In the froth bed crystallizer the superficial gas velocity, the supersaturation and the crystal mass concentration have been varied.
-The structure of the crystal surface affects the total number of nuclei, generat-ed by an impact on the crystal surface. The surface structure is relatgenerat-ed to the growth behaviour of the crystal, which, in turn, depends on the supersaturation level.
In order to investigate the influence of the growth behaviour on the nucleation behaviour, two crystallizing systems have been studied.
First, the potassium alum-water system which shows a regular growth behaviour at varying supersaturation and second, the magnesium sulphate-water system which shows an irregular growth behaviour.
CHAPTER 2
MECHANISMS OF NUCLEATION IN SOLUTION
Three stages can be distinguished in the formation of a solid phase from a solu-tion. First, the development of a supersaturated solution, which can be achieved by cooling or evaporation of the solution. Second, the generation of nuclei of the solid phase. Third, the growth of these nuclei to macroscopic sizes.
In practice, the development of a supersaturated solution depends on the solubility of the solute in the solvent as a function of temperature. If the solubility increases considerably with increasing temperature, the supersaturated solution can be achieved by cooling the solution. Evaporation of the solvent can be applied, when the solubility hardly depends on temperature. A combination of cooling and evaporation is also possible. If the solubility of the solute de-creases with increasing temperature, the solution must be heated to obtain a supersaturated solution.
Several mechanisms for the generation of nuclei are known. These are schematical-ly shown in figure 2.1. The mechanisms are described in the following sections.
NUCLEATION MECHANISMS HOMOGENEOUS NUCLEATION ^ HETEROGENEOUS NUCLEATION PRIMARY HETEROGENEOUS NUCLEATION '— SECONDARY NUCLEATION Figure 2.1 Mechanisms of nucleation in solution
2.1. Homogeneous nucleation
It is generally accepted that crystalline agglomerates are formed in solution by means of bimolecular reactions of solute molecules according to:
A t A -»• A o
AT + A -> A: (2-1)
An + A ^n+1
where A represents a single solute molecule. Agglomerate An+i contains (n+1) solute molecules. The formation of these agglomerates causes a change in Gibbs
free energy AG according to:
-AG $ + A- (2-2)
where n^ is the number of solute molecules in the agglomerate , <I> is the differ-ence in free energy between the solid and the solution per solute molecule, A^ is the surface area of the agglomerate and a^ is the surface free energy per unit surface area. The right hand side of equation (2-2) represents both a negative change in free energy due to the formation of the agglomerate (n^ $) and a positive change in free energy due to the formation of the agglomerate surface
It is assumed that macroscopic thermodynamic properties such as the surface free energy and volume free energy can be applied in the description of these very small agglomerates. These assumptions are rather questionable (1). Figure 2.2 illustrates the competition of the two factors affecting the change in total free energy:
volume free energy ^
Figure 2. 2 Change of Gibbs free energy as a function of the size of an agglomerate
A maximum in the curve occurs if:
d (AG) ^ , —^—- = $ + Oa ( dn = dA_ c dn- ) = 0 (2-3) I f we assume t h a t ; A^ = k v 2 / 3 = k ( n ^ V j n ) 2 / 2 ( 2 - 4 )
where k is the shape factor of the agglomerate, it follows from equation (2-3) that: -2 A3 vjn 3 V (2-5) Since: '^bT $ = £n ( ) (2-6) "'m ^b
one arrives at the following expression for the size of a spherical critical nucleus:
^ ^ «a
Lcr = (2-7) Pg R T £n ( — )
^s
where kb is the Boltzmann factor, T is the absolute temperature, ab, as is the electrolyte activity in the bulk solution and the saturated solution respective-ly, Mjjj is the molecular weight of the solid and p^ is the density of the solid. Replacing the activity by concentration, equation (2-7) is simplified to:
4 M a ^ '^m "a
Lcr = ~ (2-8)
Pg R T Jin ( )
^ ^s .
The critical nucleus is in metastable equilibrium with its surrounding super-saturated solution. Both the addition or removal of a solute molecule to a
critical nucleus results in a decrease of the total free energy. The addition of solute molecules to an agglomerate with a size smaller than Lcr is opposed by the increase of the total free energy. In a supersaturated solution an equilibrium distribution of crystalline agglomerates exists (2). The equilibrium concentra-tion of agglomerates decreases with increasing size and is in fact in its minimum at the size of the critical nucleus. For agglomerates larger than the critical nucleus the addition of solute molecules is favoured due to the continuous de-creasing total free energy. Consequently, these agglomerates have a high probability to grow to a macroscopic size.
The steady state nucleation rate can be calculated (2) by considering the rate of addition of a solute molecule to a critical nucleus and the rate of destruction of the newly formed agglomerate. Here, equilibrium thermodynamics have been modified to a kinetic situation. One arrives at an equation for the nucleation rate of the following form:
^ = '^n e^P ^ T i T ^ ^ ^2-9)
-where k^ is an empirical constant, which depends on the absolute temperature T, the surface tension aa and the energy barrier for diffusion of solute molecules from the bulk of solution to the agglomerate.
If the nucleation rate is expressed as a function of relative supersaturation, one obtains:
J = kjj exp (
Jin2 ( )
(2-10)
A schematic representation of equation (2-10) is shown in figure 2.3,
Figure 2.2 Rate of homogeneous
nucleation versus
relative supersaturation
-^ £b
Below a certain critical relative supersaturation, no measurable nucleation takes place. However, when this critical supersaturation is exceeded, an outburst of nuclei occurs. This critical supersaturation is referred to as the metastable limit for homogeneous nucleation. In general, the critical supersaturation is very high and is not reached in industrial crystallization processes.
2. 2. Primary heterogeneous nucleation
Although the critical supersaturation for homogeneous nucleation is very high, nucleation may still take place below this critical supersaturation due to the presence of solid substrates (1, 2, 3, 4, 5). This is caused by interactions between the molecules in the solid substrate and the solute molecules. These interactions depend on the similarity of the crystal lattice of both materials. When the two crystal lattices do match, an enhanced ordering process of solute molecules occurs at the interface of the solid substrate. This ordering process reduces the surface free energy term in equation (2-2) and as a result, the critical relative supersaturation, above which excessive nucleation occurs. The difference in the nucleation rate of homogeneous and primary heterogeneous nucleation as a function of the relative supersaturation is shown in figure 2.4. In practice, the solid substrates such as the atmospheric dust particles or the vessel wall of the crystallizer are always present in solution. Hence, it is difficult to exclude all solid substrates in crystallization studies. It is more likely that primary heterogeneous nucleation overrules the homogeneous nuclea-tion in practice.
primary heterogeneous nucleation homogeneous nucleation
Figure 2.4.
Rate of homogeneous
and primxry
hetero-geneous nucleation
versus relative
supersaturation
2.2. Secondary nucleation
csMacrosized crystals can generate new crystals at low supersaturation (5, 7). Several mechanisms can be distinguished as shown in figure 2.5. The mechanisms are discussed briefly.
SECONDARY NUCLEATION INITIAL BREEDING NEEDLE BREEDING POLYCRYSTALLINE BREEDING CATALYTIC BREEDING | COLLISION BREEDING | CRYSTAL BREAKAGE
Figure 2.5 Mechanisms of secondary nucleation
Initial breeding may occur when a seed crystal is added to a supersaturated solution (8). Small crystalline particles usually adhere to a dry crystal. These small crystalline particles detach in the solution of which the stable ones continue to grow. Initial breeding can be avoided by washing the seed crystals with undersaturated solution before it is added to the supersaturated solution. Needle breeding occurs when needle shaped crystalline structures, dendrites, growing from the parent crystal, are broken off by mechanical or fluid shear forces (9). These surface irregularities can occur at rather high supersaturation or due to the presence of habit modifying additives (10, 11, 12). Needle breeding is always accompanied by imperfect crystal growth and consequently, has to be avoided in industrial crystallization processes.
Poly crystalline breeding can occur when crystals grow as a polycrystalline mass (6). The crystal is an agglomerate of single crystals, which are loosely attached to each other. Due to mechanical impacts this agglomerate breaks up. Polycrystal-line crystal growth usually occurs at relatively high supersaturation. Because
the produced crystals are imperfect, polycrystalline breeding is unfavourable in industrial crystallization processes.
It has been suggested in the literature (13, 14, 15, 15), that preordered crys-talline agglomerates in the adsorption layer at the crystal surface can be torn away by fluid shear forces. This mechanism is known as catalytic breeding. How-ever, at present there is no experimental evidence for the existence of this mechanism. Clontz and McCabe (17), Lai et al. (8), Denk and Botsaris (18) have shown that fluid shear forces, below the limiting supersaturation for needle breeding, are ineffective for the production of nuclei. Also in crystal growth studies where a supersaturated solution flows past a crystal no nucleation has been observed.
A collision between a crystal and another object can result in the generation of new crystals (5). This phenomenon is known as collision breeding. Clontz and McCabe (17) suggest that preordered crystalline agglomerates are dislodged from the crystal adsorption layer. It is also possible that very small fragments are detached from the crystal (micro attrition). No visible damage of the crystal surface has been observed with a microscope of magnification 40x. Experiments with a controlled impact between a crystal and a rod have been carried out by Clontz and McCabe (17), Denk and Botsaris (18) and by Johnson et al. (19). These experiments are described in section 2.4.2.
Crystal breakage of a crystal in macrosized fragments occurs when mechanical forces acting on a crystal exceed a certain energy level (20, 21, 22, 23). How-ever, before this macro attrition process takes place, the nucleation rate due to collision breeding will be comparatively very high. For that reason the influence of crystal breakage on the total nucleation rate is relatively small. A suspension crystallization process is generally operated at such a low super-saturation level where primary heterogeneous or homogeneous nucleation can be ruled out, except in those cases where locally very high relative supersatura-tions occur in the crystallizer. This is, for instance, possible in a zone where the high concentrated feed solution is mixed up with the vessel contents (24). For cooling crystallization, high supersaturations may occur near the cooling wall and for crystallization by evaporating near the vapour bubble.
Collision breeding is the most acceptable source of secondary nuclei, when no irregular growth phenomena occur during crystallization. The total nucleation rate due to collision breeding depends on:
the supersaturation
the total number of crystals present in suspension and the impact energy at the crystal surface.
These factors are discussed in the following sections.
2.4. Factors affecting collision breeding
2.4.1. Effect of the supersaturation
After an impact at the crystal surface, crystalline agglomerates are dislodged from the crystal surface. These agglomerates are distributed over a certain size range, which is known as the birth distribution function. Let us consider a single crystal in a supersaturated solution, which has been brought into contact with another surface with a certain impact energy. An arbitrary birth distribu-tion funcdistribu-tion, generated by this impact, is shown in figure 2.6.
The quantity TIQ dL represents the number of agglomerates between size L and L + dL. The size of the critical nucleus, which is in metastable equilibrium
with the surrounding supersaturated solution, is given by equation (2-8). As pointed out in section 2.1., the agglomerates larger than the critical nucleus will have a high probability to grow to a macroscopic size.
no 1
Figure 2.6 An arbitrary birth
distribution function
-*-L
The agglomerates smaller than the critical nucleus will have a high probability to dissolve. This model is known as the "survival theory".
Let us assume that only agglomerates with size L > Lcr will grow to macroscopic sizes. Hence, the total number of crystals produced after one collision is:
j no dL (2-11)
•jcr
The quantity Ns is represented by the shaded area in figure 2.6.
If the same birth distribution function could be reproduced at a higher super-saturation level, the fraction of surviving agglomerates is increased and hence, the total nucleation. An increase in supersaturation will decrease the size of the critical nucleus. It is imaginable that a situation occurs where a further increase of the supersaturation does not influence the total number of surviving agglomerates anymore. Then, the number of produced crystals becomes constant. There is experimental evidence for the occurence of this phenomenon (19, 2 5 ) . An example is shown in figure 2.7., which represents the total number of produced crystals after sliding of a single sodiumchlorate crystal along a tilted tube versus supersaturation (25).
Lai (26) suggested a normal distribution for the birth distribution function. Consequently, we may write for no:
/2Tr a
exp
(L - L ) '
7a'
(2-12)
where Nt is total number of agglomerates produced after a collision, L is the average size of these agglomerates and a is the variance in the distribution. The total number of surviving agglomerates is given by:
^cr ^27r a exp (L - L)^ 2a' ) dL (2-13) 20
- 600- 500- AOO- 300-200 100-1.0 2.0 - ^ AC.10^ (kg/kg solution)
Figure 2. 7 Net produced crystals as a function of supersaturation
The i n t e g r a l cannot be i n t e g r a t e d . If we s u b s t i t u t e : L - L
a / 2
then equation (2-13) can be written as:
(2-14) Ng = / exp (- z^) dz /IT Z (2-15) or: N = — \ I exp (- 7?-) dz - .' exp ( A o 0 - .2
) dz}
(2-16)for z > o one arrives at: Ng = 0.5 N^jl - erf (z)} and for z < o:
Ng = 0.5 N.^{l - erf (I z |)}
(2-17)
(2-18) Equations (2-17) and (2-18) represent the number of produced crystals as a func-tion of the critical nucleus size. The dependence on relative supersaturafunc-tion is found by substituting equation (2-8) in (2-17) and (2-18).
Figure 2.8 shows an arbitrary normal distribution and two possible curves for the nucleation as a function of the relative supersaturation, which can be calculated from this birth distribution.
"o
I.
_ £b:
Figure 2.8 A normal birth distribution function and the fraction of surviving
nuclei as a function of supersaturation
Curve 1 in figure 2.8. represents the case that stable agglomerates are formed at low relative supersaturation while curve 2 indicates that stable agglomerates have been formed at higher relative supersaturation.
The parameters L, a and Nt in equation (2-12) and the physical quantities,
determining the size of the critical nucleus, can be adjusted' in such a way that experimental data can be described properly. However, in the above it was assumed that the birth distribution function does not vary with the supersaturation level viz. N-t, L and a are constants and independent of the supersaturation. This as-sumption is not experimentally verified at present but it is likely that these quantities vary with the supersaturation.
Other types of distributions functions could be used approximating a birth distribution function like the log-normal, the gamma or an exponential distribu-tion (27). As with the normal distribudistribu-tion, the parameters which determine the shape of the distribution curve can be adjusted by fitting them to experimental data.
In summary, it can be concluded that the survival theory provides a good under-standing in the experimental data of the nucleation as a function of the super-saturation. However, the use of empirical distribution functions to describe the functional relationship between the nucleation and the supersaturation is
questionable.
For a continuous crystallization process , not only the fraction of surviving agglomerates but also the growth rate of these agglomerates is important. For instance, if the growth rate of these agglomerates would be very small, these agglomerates can be withdrawn from the crystallizer before they reach a visible size. In that case, the apparent nucleation rate would be zero, while the actual or physical nucleation rate has a finite value. Therefore, nucleation rate data obtained in a batch system as described before are not completely comparable with continuous systems. The apparent nucleation rate in continuous crystallizers is described in more detail in chapter 4.
2.4.2. Effect of the impact energy
The effect of a controlled impact at a crystal surface on the net produced
-crystals have recently been reported for magnesium sulphate single -crystals (17, 19) and for sodium chlorate single crystals (18). The contact device in both studies were quite similar. The experimental results for magnesium sulphate are described in more detail. The contact device is schematically shown in figure 2.9. —contacting rod
A$^"""°"
rubber diaphragm supersaturated solution seed crystal thermister probe -OlOface. 4 : ^ .
}--Figure 2.9 A schematic diagram of the contact device
magnesium sulphate crystal
A magnesium sulphate crystal is fixed on a wire. A supersaturated solution flows past the crystal. An impact at the crystal surface can be effectuated with a rod, placed opposite the crystal. The surface of the rod is parallel to the crystal surface. The impact energy can be measured accurately. After an impact the flow of supersaturated solution is stopped. The stable produced nuclei are allowed to grow to visible sizes so as to be able to count them.
By varying the surface area of the rod the effect of the impact surface area was determined (19). The effect of impact energy and impact surface area for impacts on the (110) face of the magnesium sulphate crystal is shown in figure 2.10.
a 0) a Q. E •M E E u
n
E c 1/16 inchFigure 2.10 The effect of impact energy and rod diameter on the net produced nuclei
200 AOO 600 800 1000 1200
The data for crystal-rod impacts can best be represented by: Ns _ ^^ — = kn Ei Ai ' ( —— ) ^1 Cs for — < 0.025 Cs (2-19)
where A^ is the impact surface area, Ej_ is the impact energy and k^ is an empirical constant. Also crystal-crystal impacts have been studied by fixing a second crystal on the contacting rod. These data can best be represented by:
"s 4c
(2-20) Ac
for -^ < 0.025
Cs
The contact area was calculated as the sum of the area of both crystals in contact at the time of impact. A crystal-crystal contact yields less nuclei per unit area than a corresponding crystal-rod contact, approximately 1/15 as many. Microscopic observation of the crystal surface after an impact did not reveal any damage of the surface. Consequently, macroattrition can be excluded in these experiments.
The flow rate of the supersaturated solution did not influence the total number of net produced crystals and hence, catalytic breeding can be excluded too. The direction of approach of the two colliding surfaces has a considerable effect on the total net produced crystals. It was suggested, that this is due to a scraping effect of the rod over the crystal surface. However, the effect may also be
ascribed to a higher impact energy density (Ei/Ai) at the impact location due to a relatively smaller impact surface area.
In addition, also experiments have been carried out on the (111) face of the magnesium sulphate single crystal (19). The results are shown in figure 2.11. Only the effect of the supersaturation has been reported. As can be seen from figure 2.11, the (110) face yields more nuclei than the (111) face. This is attributed to a difference in crystal surface roughness on microscale which was confirmed by electron microscope pictures of both faces. The (111) face appeared to be smoother and flatter than the (110) face. Because of the agreement between the nucleation behaviour and the surface roughness of both faces, they concluded that microattrition takes place during collision breeding. However, it is known that the (111) face grows two to four times faster than the (110) face (9, 28). This fact indicates that solute molecules are built in the crystal lattice of the
(111) face much faster than in the (110) face. This may result in another struc-ture of the adsorption layer near both faces. As a consequence, the dislodgement of preordered crystalline agglomerates may be more difficult from the adsorption layer at the (111) face than from the (110) face. No conclusions can be drawn from these experiments about the exact mechanism of collision breeding.
Johnson et al. (19) have also determined the effect of two successive impacts within a certain time interval. It appeared that only after approximately 14 seconds the second impact yields an equal number of crystals as the first one. In a smaller time interval the second impact yields less crystals. Obviously, the
crystal surface needs time to attain its original structure after a collision. Youngquist and Randolph (29) suggest that the surface healing process depends on the supersaturation. In a suspension crystallization process, impacts on crystals occur continuously. Nevertheless, the effect of the crystal surface healing
process on the nucleation behaviour will not be considered in the derivation of the model for collision breeding in a stirred vessel crystallizer as described in chapter 3. a u a a E ^E E E 3 1.05 ^s
Figure 2.11 Net produced crystals versus supersaturation for the two crystal
faces of magnesium sulphate
2.4.2. Effect of crystal size and crystal concentration
In principle, all crystals in a suspension can produce nuclei by collisions with other surfaces such as the impeller, the vessel wall or with other crystals. The chance that a collision takes place depends on the flow conditions near the target, the resultant of the forces acting on a crystal and the size of the target. Small crystals have a tendency to follow the trajectory of the fluid elements, while the larger crystals do not. Therefore, it can be expected that there is a minimum crystal size which contributes to the total net nucleation. Cayey and Estrin (30) have found a minimum crystal size of 180-220 ym in an agitated suspension of magnesium sulphate crystals. Besides, the effect of a collision of a small crystal on the nucleation rate is less than for a large crystal due to the low impact energy which is involved.
Because the contribution to the total nucleation rate is not the same for the different crystal sizes , the nucleation rate J at a given supersaturation can be written as:
ƒ no ^l F(L) dL (2-21)
where no dL is the number of crystals between size L and L + dL, w^ is the
collision frequency and F(L) is the contribution of a crystal with size L to the total nucleation rate at a given supersaturation.
As outlined above: •
F(L) = o L < Lmin (2-22) For crystals with size L > Lmin> F(L) depends on the impact energy Ej,, which
in-creases with increasing crystal size. The dependence of tog, on crystal size L can-not be predicted beforehand. Hence, F(L) and tOj^ are unknown and equation (2-21) cannot be further evaluated. This problem is simplified by relating the nuclea-tion rate J with the total crystal mass concentranuclea-tion M (29, 31, 32, 33) as well as the total crystal surface area A^ (34, 35) or the total number of crystals N-(-(36).
-CHAPTER 3
A MODEL FOR COLLISION BREEDING IN A STIRRED VESSEL CRYSTALLIZER
The empirical nucleation rate expressions for collision breeding obtained in an experimental apparatus, as described in section 2.4.2, cannot be used in stirred vessel crystallizers. The quantities impact energy, impact surface area and collision frequency of the crystals in a suspension are ill-defined. Some assump-tions are to be introduced in order to derive an expression for the nucleation rate in a stirred vessel crystallizer, based upon the empirical correlations given by Clontz and Mc Cabe (17) and Johnson et al (19):
- 1. The impact energy is proportional to the square of the relative velocity between the colliding crystal and the other surface. A certain part of this energy is used for the generation of nuclei, independent of the crystal size. - 2. The impact mostly happens on an edge or a corner of the crystals. This
impact surface has no relation to the total crystal surface area. Hence, the effect of the total crystal surface area on the nucleation rate is not
considered.
- 3. The effect of the direction of approach during a collision on the nucleation rate is ruled out by taking some statistical average.
- 4. The nucleation rate is proportional to the collision frequency of a crystal by assuming that the surface healing process of the crystal is infinitely fast.
Let us consider a number of crystals no dL with size between L and L + dL. These crystals collide with an arbitrary surface with a frequency co£ , where an impact energy E^ is involved. The contribution of these crystals to the total net
nucleation rate can be written as:
d J c - i = k i cojj, E^ ( — )P no dL . ( 3 - 1 )
Cs
where k^ is an empirical constant. The subscript c-i in dJc-i denotes a collision of a crystal,c with an arbitrary surface i. The exponent p in equation (3-1) can not be predicted beforehand and has to be determined experimentally. For iso-thermal crystallization processes, the saturation concentration Cg is a constant and equation (3-1) is rewritten as:
dJ _. = k2 i^Z Eg. (Ac)P no dL (3-2)
The total net nucleation rate of crystals in suspension is obtained, if equation (3-2) is integrated from a certain minimum crystal size Lmin to fhe maximum crystal size Lmax present in suspension, according to:
J^ax „
Jc-i = ƒ k2 üJg, Ej;_ (Ac)P no dL (3-3) '-'min
As mentioned in section 2.4.2, Lmin is the minimum crystal size which contributes to the total nucleation.
The nucleation rate expressions for crystal-impeller (Jc-ms)> crystal-wall (Jc-w) and crystal-crystal (Jc-c^ collision breeding are derived in the following
sections.
2.1. Crystal-impeller collision breeding
The impeller causes a circulation of the suspension through the vessel. The generalized flow patterns are shown in figure 3.1.
Figure 2.1 Generalized flow patterns
in a stirred vessel
propeller turbine impeller
The volumetric flow rate of solution through the impeller area, $yp, is given by (37):
"vp = K^ n, (3-4)
where Kp is the impeller discharge coefficient, n^. the impeller speed and D-p the impeller diameter. The value of Kp depends on the impeller type and geometry. The average time interval between two subsequent passages of a volume element through the impeller area, tc > is given by:
'to ^c - ^
vp
(3-5)
where V.(-(-, is the total volume of the vessel. The circulation frequency of a volume element through the impeller area, to, is:
1__ J ^ ^c" Vtc
(3-5)
The circulation frequency of a crystal,cog, can be calculated in an analogous way. If the relative velocity of the crystal with respect to the surrounding solution is neglectable compared to the absolute velocity of the solution, then tO£ is of the order of magnitude of to, and becomes independent of the crystal size:
co£ - CO (3-7)
Every time a crystal passes through the impeller area a collision with one of the impeller blades may take place. The impact energy Ej^ is related to the mass of
the crystal m^ and the square of the tipspeed v^ of the impeller, according to:
Eg = k3 mg v | (3-8)
-The approach velocity of the crystal to the impeller is neglected compared to the tipspeed. The tipspeed v.^^ is given by:
v-j- = IT np Dp (3-9)
By substituting equations (3-5) to (3-8) in equation (3-2), one obtains:
"vp ''t ^
<iJ._^. = k4 ( -T, ) (Ac)P n^ m^ dL . (3-10)
'to
c-ms -'^ "• V^„ -' ^"^^ '^o '"g
With equations (3-4) and (3-9) this leads to:
"r ^r
'iVms = ^5 ( ^ - ^ ) (^c)P n^ m^^ dL (3-11) X c
n^ D^ "r "r
For a stirred vessel, the group ( —r ) can be correlated with the dissipated ^tc
power by the impeller per unit mass of suspension, E , according to: n3 D |
e = P Q ( ) (3-12) "tc
where P Q is the impeller power number (38). At sufficiently high Reynolds number (Re > lO'*) the impeller power number P^ is a constant (38). Substituting equation (3-12) in (3-11) yields:
<iJc-ms = 5^5 ( J^ ) (Ac)P no m^^ dL (3-13)
After integration of equation (3-13) one obtains:
^max F r^ F
Jc-ms = J ks ( - ^ ) (Ac)P no mji dL = ks ( - | - ) M^^ (Ac)P ( 3 - 1 4 ) T . ^O ^O
^ m m or:
Jc-ms = K ^ ^i ('^c)? (3-15)
for constant Po»
It should be noted that Mx represents the total mass of crystals above size Lmin-When Lmin is relatively small then the mass of crystals below size Lmin is small compared to the total mass of crystals in suspension. Hence, the real crystal mass concentration M can be used in equation (3-15).
2. 2. Crystal^wall collision breeding
Collisions of crystals with the vessel wall occur preferentially at places where the suspension flowing from the impeller has to change from direction. These places are shown in figure 3.2.
KJ
I.
J
L
C
CH-DFigure 2,2. Preferent places for
crystal-^wall collisions
propeller turbine impeller
After a collision with the wall the crystal participates again in the circulation pattern of the suspension in the vessel. In fact, this situation is comparable with that of crystal-impeller collisions. The velocity of the solution near the wall is low compared to the tipspeed of the impeller. The impact energy will be less and, consequently, the nucleation rate for crystal-wall collisions.
Collisions can also take place against the cooling coil. Hence, the collision frequency cojj^ and the impact energy Ej^ for each collision of a crystal during one circulation through the vessel cannot be estimated beforehand. However, it is assumed that the nucleation rate expression for crystal-wall collision breeding is analogous to that for crystal-impeller collision breeding, or:
c-w kn e Mx (Ac)P (3-15)
2.2. Cry stal-cry stal collision breeding
Crystal-crystal collisions can occur in the whole vessel volume. The suspended crystals are transported by turbulent eddies and describe complicated traject-ories. Due to a density difference between the crystal and the solution, the crystal moves with a relative velocity with respect to the surrounding solution, The average relative velocity, Vg, of a particle in a turbulent liquid has been calculated by Levich (39). Some assumptions had to be made by him. First, the turbulence in the vessel is considered to be homogeneous and isotropic through-out the vessel contents. Second, the turbulence is not affected by the presence of the particles. Third, the particle size is large compared to the energy dis-sipating eddy. By means of a force balance over the particle , Levich arrives at the following general expression for Vg:
Ps-Pg, )0.5 ( ^ ) 0.33 rO.33
-W •0. 33 ^0.33 (3-17)
where Ps, pg, is the density of the solid and liquid respectively, L is the par-ticle size, Cvf is the drag coefficient and e the power dissipated per unit mass of liquid.
It is extremely difficult to derive a general collision model for a poly-disperse particle system in a turbulent liquid. Hinze (40) has derived equations for the interaction between poly-disperse particles in a turbulent fluid. A collision model for unidisperse droplets in a turbulent liquid has been derived by Van
Heuven (41 ) and for raindroplets of two different sizes in turbulent clouds by Saffman and Turner ( 42 ). In order to derive an approximative nucleation rate
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expression for crystal-crystal collisions in a crystallizer, we introduce two, rather severe, simplifications.
First, all crystals above a certain minimum crystal size Ljuin ^^^s spherical with an average size L defined by:
N x i ^ i^' Ps = "x • (3-18)
where N^ is the total number of crystals in suspension above size L^^-^, according to:
^max
Njj = ƒ no dL (3-19) Lmin
Mx is the total mass of crystals above size Lmin*
Second, the local flow field around the crystal is not disturbed during a collision and the crystal retains its original relative velocity immediately after the collision. The nucleation rate expression now becomes:
Jo-c = kg Nj^ Ü Ê (Ac)P (3-20)
where kg is an empirical constant, co is the_mean collision frequency and E is the mean impact energy for a crystal with size L. An estimate of the collision fre-quency of a crystal can be obtained as follows. The crystal moves across a distance Vg per second and passes through a volume v, given by:
V = 0.25 TT £2 Vg (3-21)
In this volume N^, v crystals are present. Consequently, the collision frequency Ü) becomes:
ü = Nj^ V (3-22)
The impact energy depends on the direction of approach of both particles. The maximum impact energy is obtained when both crystals move in opposite direction along the connection of the mass centers. Any other direction of approach yields a lower value. The average value of the impact energy is taken as:
Ê = ky Pg £3 v| (3-23)
After substituting equations (3-17), (3-18) and (3-21) to (3-23) in equation (3-20), one obtains:
Jc-c = kn e M^ (Ac)P (3-24)
-Comparing the nucleation rate expressions for the three different types of crystal collisions, namely:
crystal-impeller collision breeding:
J = k e M (Ac)P (3-15) c-ms n X
crystal-wall collision breeding:
Jc-w = l<n ^ "x (^<^^P (3-15)
crystal-crystal collision breeding:
JQ-C = kn E M2 (AC)P (3-24) a simple relationship is obtained between the nucleation rate J and the
quantities e, Mx and Ac. It should be noted that the values of kn as well as Mx in these equations are different among themselves. The value of the minimum crystal size Lmin and, consequently, the value of Mx is unknown for the three types of collisions. Experimental determination of Lmin is difficult because the various types of collision breeding act simultaneously during crystallization.
CHAPTER 4
THE APPARENT NUCLEATION RATE IN CONTINUOUS CRYSTALLIZERS
During a crystallization process nucleation and growth of crystals occur simulta-neously. In a continuous crystallizer, operating in the steady state, the
nucleation rate must be equal to the rate of withdrawal of crystals from the crystallizer. The withdrawal of crystals from the crystallizer is a statistical process and determines the residence time of each crystal in the crystallizer. Consequently, the crystals in the outlet of the crystallizer are distributed over a certain size range, generally known as the crystal size distribution or CSD. The CSD of a crystallizer product is not only determined by the residence time distribution of the crystals. Two other quantities, the nucleation rate and growth rate of the crystals, have a direct influence on the CSD as shown in figure 4.1. FEED RATE OF SOLUTION Figure 4.1. EVAPORATING COOLING RESIDENCE TIME OF THE INDIVIDUAL CRYSTALS iPRODUCTJO^ 'RATE TOTAL CRYSTAL SURFACE AREA IN THE CRYSTALLIZER
T
Influence of the different quantities on the CSD of a continuous
crystallization process.
The CSD is of paramount importance for the design of crystallizers. Consequently, the nucleation and growth rate of the crystals as a function of supersaturation, crystal mass concentration and hydrodynamical behaviour of the suspension should be known.
An ideal crystallizer to investigate these phenomena is a continuous, homogeneous mixed crystallizer operated in the steady state. This type of crystallizer is known as the continuous , mixed suspension mixed product removal crystallizer or the CMSMPR-crystallizer (43). The simplest geometry is a stirred vessel. The advantage of the CMSMPR-crystallizer is that nucleation and growth occur under the same conditions as supersaturation, crystal mass concentration, hydrodynamic-al behaviour in the crysthydrodynamic-allizer and temperature. Second, the residence time distribution of the crystals in the crystallizer is equal to that of the mother liquor. From the CSD, both the nucleation rate and the growth rate of the crys-tals can be determined, which is discussed below.
The CSD can be represented by a number population density distribution versus particle size (27, 44). The number population density, no, is defined as the number of crystals, dN, in a sufficiently small size interval dL between L and L + dL:
AN _ dN lo = lim ^ = ^
Ag->o
(4-1)
The dimension of n^ is usually given as a number per ym per m^ mother liquor. The total number of crystals between size Lj and L2 is given by:
L2
Nt - j ^o '^^ Ll
(4-2)
Let us consider a steady state population balance in an arbitrary size range from Li to L2 during a time interval At in the CMSMPR-crystallizer. It is further assumed that no crystal breakage occurs.
g r o w t h into size i n t e r v a l L 1 L g r o w t h out of "" size interval 2 crystal size
Figure 4.2 Population balance over a size range L2-L2
removal from crystallizer
The total number of crystals growing into the size range at L^ (see figure 4.2) is equal to the sum of the total number of crystals growing out of the size range at L2 and the number of crystals in the size range L1-L2 removed from the crys-tallizer in the time interval At. Hence:
Vc ^1 "oi At "2 no2 At + Ov no AL At (4-3) where V^ is the total volume of mother liquor in the crystallizer, r^ and r2 are the growth rate of crystals with size L^ and L2 respectively, n^i and nQ2 are the population densities of crystals at size L^ and L2, $y is the volumetric flow rate of mother liquor in the outlet of the crystallizer and HQ is the mean popu-lation density of crystals between size L^ and L2. Rearranging equation (4-3) and taking the limit for AL^o one obtains:
d (n^ r)
dL ^ " o (4-4)
With:
where x is the mean residence time of the mother liquor, equation (4-4) can be rewritten as:
^"o dr ""o
^ dT ^ ^° dL ^ - = °
(4-6)If the growth rate is not size dependent, equation (4-6) reduces to: dn„ n^ o o
r -77- + — = o
dL T (4-7)
Integration of equation of (4-7) results in: -L
no = ng exp ( — ) (4-8)
where n§ is the crystal number population density at L = o. A population density distribution according to equation (4-8) is shown in figure 4.3.
In n
Figure 4.2 A population density
distribution
The growth rate of the crystals can be calculated from the slope in the (Jln nQ,L) plot. The apparent nucleation rate J is calculated according to:
J = dN dt £ = o dN dL £=o dL dt nX r £=o
The different moments of the CSD, u^^, are defined according to: oo Ltj= j no dL = Nt = ng r T o m = ƒ no L dL = L.^: = "o (^ T ) 2 o 00 kg U2 = ƒ HQ k L2 dL = A^ = 2 kg ng (r T ) 3 (4-9) (4-10) (4-11) (4-12) 35
-k,. y V ^^3 no k^ L 3 dL = V-t- = 6 k^ ng ( r x ) ' ( 4 - 1 3 )
k^ pg V3 = .1 riQ k^ Pg L 3 dL = M = 6 k^ Pg ng ( r r ) ' o
(4-14)
where N-^ is the total number of crystals, L-^ the total length of the crystals, At the total crystal surface area, V^ the total volume of the crystals, M the total mass of crystals, kg is the surface shape factor and k^ the volume shape factor of the crystals.
A population density distribution is usually obtained from a sieve analysis of the crystallizer product (45). The procedure is described in chapter 5.
The sieve analysis can be carried out with an acceptable accuracy for crystals larger than 50-100 ym. The value of ng is obtained by extrapolation of the
population density distribution to L = o. This extrapolation is questionable for several reasons. This is discussed below.
The generated nuclei are distributed over a certain size interval, as has been described in chapter 2. These nuclei are very small. For the very small size ranges the number population balance must be adjusted in the following way:
production of stable nuclei growth into size interval Ll growth out of
"size interval Figure 4.4 Population balance over a size range L2-L2
L2 crystal size removal from
crystallizer
For an arbitrary size range L^ < L < L2 with L^ > L^^ we can write, in analogy to equation (4-3):
Vc ^l " o l At + VG B ( L ) AL At = V^ r a no2 At + <i>v " o AL At (4-15)
where B(L) is the production of stable nuclei per unit of length, time and volume of mother liquor. After rearranging equation (4-15) and taking the limit for AL->-o , one obtains:
d (rno) ~"dL B(L) (4-15) or: dr dUr ^o dL ^ ^ dL" + B(L) T (4-17)
Equation (4-17) can be solved if r = f(L) and B(L) are known. The physical nucleation rate is given by:
J = ƒ B(L) dL L.
•"cr
(4-18)
Youngquist and Randolph (29) have determined the CSD in the size range between 1.25 ym to 25.4 ym for the continuous crystallization of ammonium sulphate with a Coulter Counter. An example of one of these CSD is shown in figure 4.5.
Figure 4.5 A typical population density distribution for small crystal sizes
(from (29))
They considered two extreme situations in order to solve equation (4-17): dr
-1) r = o, ^ = o, B(L) ^ o (4-19)
-2) B(L) = o, B(Lo) = J, r = f(L) (4-20)
Both assumptions are unrealistic. The first assumption, expressed by equation (4-19), implies that the generated nuclei donot grow and, therefore, are washed out from the crystallizer before populating the larger size ranges. In fact, no visible crystallization takes place. The values obtained for the nucleation rate according to equation (4-18) are larger compared with the situation that the growth rate of these nuclei would have a finite value.
The second assumption, expressed by equation (4-20), implies that all nuclei are generated at a certain size LQ. In order to describe the CSD, as shown in figure 4.5, the growth rate of the smallest nuclei is low and gradually increases with
-crystal size until a maximum growth rate is obtained. This growth phenomenon may be explained by the effect of the curvature of the nucleus on the driving force for growth (46, 47). The equilibrium solute concentration at the surface of a spherical nucleus of size L ,Cgj^ ,in a supersaturated solution is given by the Gibbs-Thomson equation:
Csl ^ Mm Pa
£n ( — — ) = - — - ^ ^ (4-21) Cs Ps R T L
For ( ) ;; 1, equation (4-21) can be rewritten as: ^s
c
s£ - S
"^
Mm °a
Pg R T L (4-22)
The driving force for growth of a nucleus of size L can now be written as:
^ M^ °a 1 1
^b - cs£ = -s(p-^TT-) ^ T ; , - L > (^-23)
Taking the limit for L-^-Lcr, the driving force cb - cs£ "*• o and for L •> <=°, Cb ~ Cs£ -»• cj2 - Cg. If we assume that the critical nucleus size Lcr is of the order of magnitude of 0.5 ym, it can be deduced from equation (4-23) that the influence of the curvature can be neglected for crystal sizes up to 10 ym. However, it has been shown experimentally (45) that the curvature effect by itself cannot be responsible for the extremely low growth rate of these nuclei. Strickland-Constable (48) suggests, that the generated nuclei are in a perfect state without defects in the crystal lattice. Because of this, the growth rate is very low, but as they grow they acquire an increasing number of dislocations which give rise to an increased growth rate. This phenomenon has been confirmed experimentally by Morris et al. (49).
From the above it is obvious that the physical nucleation rate is not equal to the apparent nucleation rate obtained from a sieve analysis.
A curvature in the (log no» L) plot in the larger size ranges may also lead to misinterpretation of the CSD of the product crystals. The curvature can be caused by classification of crystals near the discharge opening or by a size dependent crystal growth phenomenon. Crystal breakage or inhomogeneous mixing of the crys-tals in the vessel can also result in a curvature, but are not considered here. Classification of crystals near the discharge opening causes a difference between the CSD, and its related moments, of the product crystals and the crystals in the vessel. Consequently, data obtained from a sieve analysis are not representative for the vessel contents. A size dependent growth rate of the crystals may cause an erroneous extrapolation in the (log n^, L) plot to L = o. Both factors are discussed below.
If the suspension is homogeneously mixed the outlet of the crystallizer can be located anywhere in the crystallizer. However, the presence of the discharge may affect the local flow pattern of the suspension. A change in the direction and absolute value of the velocity of the main stream can cause classification of crystals. The magnitude of this classification effect is determined by the ability of crystals to follow the fluid streamlines. Very small crystals follow the fluid streamlines while the larger crystals do not.
Let us consider a steady state population balance over an arbitrary size range from Li to L2 during a time interval At in the crystallizer.
growth into size interval L 1 L growth cut of *"size interval 2 crystal size
Figure 4,6 Population balance over a size range i^^-^S
removal from crystallizer
The number balance (see also figure 4.5) can be written as:
Vc ^1 noi At = V^ r2 no2 At + \ n^ AL At (4-24)
where no is the mean population density of crystals between size L^ and L2 in the product stream. Assuming:
no = A(L) n^ (4-25)
and taking the limit for AL-HD , one obtains: d (r no)
- 1
no A(L) X dL (4-25)
where A(L) is the separation coefficient, dependent on the crystal size. If the growth rate of the crystals is independent of size, equation (4-25) can be re-written as:
d n,
no X(L) rx dL (4-27)
Usually, X(L) is unknown. Hence, equation (4-27) cannot be further evaluated. Two possible cases are shown in figure 4.7.
For the smallest crystals, it is assumed that X(L) ->• 1 for L -»• o.
It is obvious from figure 4.7, that the CSD in the vessel and the product stream may differ considerably. Consequently, one has to be careful in correlating the nucleation rate obtained from a sieve analysis with one of the moments of the CSD of the product crystals if classification occurs due to the presence of the discharge. In order to get an impression of the measure of classification in the apparatuses, used in our investigations, experiments have been carried out with glass spheres in water. These experiments are described in appendix III.
Another reason for the curvature in the (log no, L) plot may be caused by the size dependent growth of the crystals. The growth of a crystal in a
super-saturated solution is a result of two mass transfer processes in series. First,
-the diffusional transport of solute molecules from -the bulk of -the solution to the crystal interface. Second, the integration of these solute molecules in the
crystal lattice. The concentration profile of solute molecules around a growing crystal is schematically shown in figure 4.8.
X(L)>1
-vessel contents
o<X(L)<1
product
-^L
Figure 4, 7 Comparison between the CSD of the product and the vessel contents
during classification
The mass flux, <I>m", for both transfer processes can be written as:
the diffusional transport: O^," = k^ (G]^ - c^) (4-28) and the rate of integration in the crystal lattice;
t'm " = kr (ci - Cs)" (4-29)
where k^ is the mass transfer coefficient, kr is a reaction constant for the surface integration, cb is the bulk concentration of solute, ci is the interface concentration and Cs is the saturation concentration. In this case, Cs is
considered to be independent of the crystal size. Equation (4-29) is an empirical approximation of the theoretical relation between the growth rate and the super-saturation, which is given by:
V " = kr ( — )
( 4 - 3 0 )interface
crystal X surface y^^,
- C K
Figure 4.8 Concentration profile
around a growing
crystal
concentration i
If ci Z Cb, the growth rate of the crystal is governed by the rate of the integration of solute molecules in the crystal lattice. For this mechanism of crystal growth Burton et al. (50) have derived equations for the growth rate as a function of the relative supersaturation. According to this theoretical model, the exponent n varies between one and two, dependent on the level of
super-saturation. The value of n decreases with increasing supersuper-saturation. The
theoretical model does not predict an influence of the crystal size on the growth rate.
If Ci ; Cs, the crystal growth rate is governed by the diffusional transfer rate of solute molecules from the bulk to the crystal interface which mainly depends on the value of k^. The mass transfer coefficient k^ for suspended particles has been subjected to many theoretical and experimental studies, of which an excel-lent review is given by Huige (51). He has suggested the following empirical correlations valid for suspensions which contain less than 5 % solids by volume:
kd L Sh = = 2 t 1.3 ( -'m zL' )0.17 SoO-25 for ( % ) < 10^ (4-31) and: Sh = kd L eL^ = 2 t 0.4 ( ^ ) ,0.243 Sc 0.25 for ) > 10' (4-32)
where Sh is the Sherwood number, Djj, is the diffusion coefficient, v is the
kinematic viscosity, e is the power dissipated per unit mass of suspension, Sc
is the Schmidt number, and zL^/v^ is the specific power group.
(m/sec) ii _i I u -J 1 u 200 ^00 600 BOO ^ L ( ^ m ) 1000
Figure 4.9 The mass transfer coefficient as a function of particle size
From equations (4-31) and (4-32) it is obvious that the relation between kd and L is rather complex. For an arbitrary inorganic solid kn has been calculated as a funcion of L for the following conditions: D^ = 5.10"^, c = 2, v = I C ^ and T = 30 °C. The result is shown in figure 4.9.
The value of k^ is approximately constant above a size of 300 ym, and increases considerably with decreasing particle size. If the effect of the curvature of the crystal on the saturation concentration is ignored, the growth rate of a very small crystal is relatively high and decreases gradually with increasing size until a constant growth rate is attained. This growth phenomenon causes a convex upward shape in the (log no,L) plot, which can be deduced from equation (4-5):
<i r^o dr ""o
^ "dT ^ ^° dL ^ ~ = ° ^^-^^
or: d no = - no ( - ^ + è J7 ) (4-33) dL ^ ' r X r dL d HoA maximum in the (log n o , L) plot occurs for - — j — = o or:
dr 1 /, ^, X
-rr = - — (4-34)
dL X
Generally, this maximum falls in the smaller size ranges (52). In this size range the sieve analysis becomes less accurate, which may easily lead to a misinter-pretation of the sieve analysis data. An example is shown in figure 4.10.
Obviously, the apparent nucleation rate is calculated for a hypothetical crystal-lization process which results in a population density distribution given by the dotted line.
In crystallization studies in stirred vessel crystallizers a convex downward (log no, L) plot have also been found (53, 54, 55). In principle, this phenomenon cannot be explained beforehand considering the growth behaviour of the crystals. Nevertheless, several 'growth models' have been proposed (55, 57, 5 8 ) :
r = c L^ (Ac)ni (4-35) r = ro (1 + a L) (4-35) r = ro (1 + - ^ ) ^ (4-37)
ro X
where c, b , m and a are empirical constants and rQ is the growth rate for L = o.
The influence of the supersaturation on the growth rate r is incorporated in VQ,
The above mentioned growth models are not based on a consistent physical theory. They have some practical value in facilitating computer curve fitting of the (log no, L) plot in order to calculate the apparent nucleation rate. However, there are some mathematical restrictions. First, the growth rate of the very small crystals must have a finite value, which in case of equation (4-35) is not
satisfied if b / o. Second, the different moments of the CSD must have a finite
