The eutectic crystallization of NaCl.2H2O and ice

The eutectic crystallization of NaCl.2H2O and ice

Citation for published version (APA):

Swenne, D. A. (1983). The eutectic crystallization of NaCl.2H2O and ice. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR50645

DOI:

10.6100/IR50645

Document status and date: Published: 01/01/1983 Document Version:

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'l'BE EUTECTIC CRYSTALLIZATION OF

NaCl.2B

₂

0 AND ICE

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN A.AN DE TECHNISCHE HOOESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNlFICUS, PROF. DR. S. T. M. ACIERMANS, VOOR

EEN

COMMJSSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRJJDAG 11 NOVEMBER 1983 TE 14.00 UUR DOOR

DIRK ADRIA.AN SWENNE

GEBOREN TE EINDHOVEN

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN

PROF.DR.IR.D.THOENES PROF.DR.IR.H.A.C.THIJSSEN

Acknowledgements

I wish to express my thanks and appreciation to the students messrs. F. van Dijk, J. van den Heijkant, R. van Herten, E. Linthorst, C. Smits, R. Vervoort and T. de Weerd for their contribution to the experimental and theoretical studies. I would also like to thank messrs. W. Hiethaar, Th. van der Hoeven and B. Wienk of Akzo Zout Chemie Research for their valuable advice. I am indebted to the members of the Vakgroeo Fysische Technologie for their co-operation during this investigation. Thanks also to Anniek van Benrnelen for conscientiously typing the manuscript and to 11\Y wife, Ida Goede for skilfully preparing the figures and doing the scissors-and-paste work.

Contents

Acknowledgements Contents Summary Samenvatting 1. Introduction 1.1. Salt

1.1.1. Salt sources, methods of recovery, production and uses

1.1.2. The system NaCl-H2

o

1.1.3. The issue of energy 1.2. Freezing processes

1.2 .1. Overview

a. Eutectic freezing processes

a.a. Eutectic freezing processes with direct cooling

a.a.a. Direct contact cooling without evaporation

a.a.b. Direct contact cooling with evaporation

a.a.c. Direct contact cooling with hydrate formation

a.a.d. Direct cooling by evaporation of water

a.b. Eutectic freezing process with in-direct cooling

b. Noneutectic and partly eutectic freezing processes

1.2.2. State of the art

1.3. Objective of this study and set-up of thesis 2. Direct contact cooling ~ith evaporation

2.1. Introduction

2.2. The shape and size of a two-phase drop 2.2.1. Classification of shapes

2.2.2. The size of a two-phase drop in a stirring-tank iii iv ix xi 1 1 1 1 4 4 4 7 8 9 9 9 9 11

2.2.3. The shape of a two-phase drop as a function of the vapour mass fraction

2.2.4. The motion of a rising two-phase drop in a quiescent liquid

2.3. Heat transfer to two-phase drops

2.3.1. Heat transfer to a rising evaporating two-phase drop in a quiescent liquid

2.3.2. Heat transfer to a rising condensing two-phase drop in a quiescent liquid

2.3.3. Heat transfer between two invniscible liquid layers with simultaneous boiling and stirring 2.3.4. Heat transfer to evaporating drops in a

stirring-tank

2.3.5. Heat transfer to evaporating drops in a eutectic stirring-crystallizer

2.4. The particle size distributions of drops and of bubbles

2.5. Conclusions

3. Crystal nucleation and growth 3.1. Introduction

3.2. Crystal single growth 3.2.1. Interfacial energy 3.2.2. Homogeneous nucleation 3.2.3. Crystal growth rate 3.2.4. Growth classification 3.2.5. Heterogeneous nucleation 3.3. Ice single growth

3.3.1. Introduction 3.3.2. C-axis growth 3.3.3. A-axis growth

a. A-axis growth in pure water b. A-axis growth in NaCl solutions 3.4. NaCl.2H2

o

single growth

3.5. Heat and mass transfer 3.5.1. Introduction 3.5.2. Models 13 15 16 16 19 19 19 19 24 24 25 25 26 26 28 29 30 31 31 31 33 34 40 41 41 41

a. Isotropic turbulence model b. Slip velocity model

c. Penetration model

d. Transpiring and shrinking 3.5.3. Correlations 3.6. Secondary nucleations 3.6.1. Introduction 3.6.2. Sources of nuclei 3.6.3. Detachment mechanisms a. Spontaneous detachment b. Fluid shear c. Crystal-crystallizer collisions d. Crystal-crystal collisions 42 44 44 45 45

3.6.4. Relative importance of detachment mechanisms 46

3.6.5. Rate controlling stages 46

3.6.6. Correlations 47

3.6.7. Theoretical models 47

3.7. Ice growth from a NaCl solution in a stirring-tank 48 3.7.1. Ice growth in a batch crystallizer 48

a. Ice growth in a batch crystallizer with direct coo 1 i ng

b. Ice growth in a batch crystallizer with indirect cooling

3. 7 .2. Ice growth in a continuous crystallizer with 49 direct-contact cooling

3.8. Particle size distributions

3.8.1. Representation of distributions

50 50 3.8.2. The steady state continuous mixed suspension 54

mixed product removal crystallizer

3.9. Conclusions 57 4. Crystallization experiments 4.1. Introduction 59 59 60 60 60 61 63 64 4.2. Set-up and procedure

4.2.1. General description of set-up 4.2.2. Crystallizer

4.2.3. Brine circuit 4.2.4. Refrigerant circuit 4.2.5. Purity of substances

4.3. Measurement techniques 64 4.3.1. Subcooling 64 4.3.2. Temperatures 65 4.3.3. Pressure 65 4.3.4. Flow rates 66 4.3.5. Crystallizer volume 66 4.3.6. Stirring speed 66 4.3.7. Particle size 66 4.3.8. Crystal shape 68 4.3.9. Drop concentration 68

4.3.10.Crystal mass fraction 68

4.3.11.Residence time 69 4.3.12.Growth rate 69 4.3.13. Nucleation rate 69 4.3.14. Purity of crystals 69 4.4. Results 70 4.4.1. Qualitative observations 70

4.4.2. Unsteady state behaviour 71

4.4.3. Crystal size 74

4.4.4. Drops 85

4.4.5. Purity of crystals 87

4.5. Discussion 87

4.5.1. Crystal shape 87

4.5.2. Unsteady state behaviour and crystal size 87 distributions

4.5.3. The influence of pressure 89

a. Introduction

b. Determination of hd and he of a two-phase drop

c. Subcooling detennination d. Growth and nucleation 4.5.4. The influence of T, Na and b

4. 5 . 5 . Drops

4.5.6. Purity of crystals 4.6. Conclusions

5. Separation of two solids in a hydrocyclone 5.1. Introduction 5.2. Hydrocyclone properties 94 98 99 99 101 101 101

5.3. Experimental set-up and procedure 5.3.1. Set-up 5.3.2. Procedure 5.4. Results 5.5. Conclusions 6. Cone 1 us ions

6.1. Conclusions of this study 6.2. Scaling-up 103 103 106 107 107 110 110 111 6.3. Economical feasibility 112

A. Power consumption of freezing processes 114

A.1. Introduction 114

A.2. Eutectic freezing process with saturated feed 114 A.3. Eutectic freezing process with unsaturated feed 115 A.4. Noneutectic and partly eutectic freezing processes 118

with saturated feed

A.5. Process with incomplete ice-NaC1.2H20 separation 118 A.6, Process with incomplete crystal-brine separation 121 B. Calculation of the size of a two-phase drop in a 124

st i rri ng-tank Notation References Curriculum vitae 127 132 141

Summary

This thesis deals with the eutectic crystallization of NaCl.2H2

o

and ice from a NaCl solution, obtained by solution mining, with the object of separating NaCl from H20. The eutectic freezing process is an al-ternative for the evaporation of the NaCl solution. Calculations have shown that under certain conditions the freezing process may be more economical than the evaporation process.

As

the freezing process had hardly been investigated, it was chosen as the subject of the present study.

In the framework of this study experiments with a continuous stirring-crystallizer containing 2 litres of suspension were conducted, the heat being removed by means of direct-contact cooling with evaporation of freon 114.

In the experiments NaCl.2H20 and ice crystals grew separately. The fol-lowing quantities were determined: the size, size distribution, shape, growth velocity, nucleation rate and purity of the ice and NaCl.2H20 crystals, the size, size distribution and concentration of the freon drops and the subcooling. These quantities were determined as functions of average residence time, freon pressure, stirring rate and crystal

' mass fraction ..

It appeared that for a residence time of 15 minutes and a crystal mass fraction of 5% the sizes of ice and NaCl.2H2

o

crystals were approximate-ly 100 µm. The size increased with increasing residence time T as

,o.

3 and with increasing crystal mass fraction b as b-0·2• The pressure and stirring rate had little or no influence on the crystal size. The size distribution of the NaCl.2H2

o

was exponential, that of the ice exhibited a maximum. The ice growth was heat or mass transfer limited, that of NaCl.2H2

o

was partly inbuilding limited. The nucleation of both kinds of crystal was slightly surface regeneration limited, and crystal-crystal collisions made a larger contribution to the total nucleation than crystal-crystallizer collisions.

The freon drops had sizes of approximately 50 µm. °the subcooling was approximately 0.1 K for a residence time of 15 minutes.

The separation of the crystals formed in the eutectic crystallization process was studied in a hydrocyclone of 1 cm in diameter with a model system at ambient temperature. The separation efficiencies of the light

and heavy particles were determined as functions of pressure drop, overflow/underflow ratio, and particle mass fraction. The fraction of light particles appearing in the overflow increased to approximately 98% with increasing overflow/underflow ratio, and the fraction of heavy particles appearing in the underflow was approximately 98% under all conditions.

The conclusion is that the eutectic freezing process for separating NaCl from H2

o

is technically feasible. It may be economically feasible for a grass root plant, if high purity salt is required or if the energy prices remain high.

Samenvattiog

Dit proefschrift behandelt de eutectische kristallisatie van NaC1.2H2

o

en ijs uit een door "solution mining" verkregen NaCl-oplossing, met als doel NaCl en H2

o

te scheiden. Het eutectisch vriesproces is een alter-natief voor het indampen van de NaCl-oplossing. Uit berekeningen is gebleken dat onder bepaalde ornstandigheden het vriesproces goedkoper kan zijn dan het indampproces. Aangezien het vriesproces nog nauwelijks onderzocht was, werd het gekozen als onderwerp van deze studie.

In het kader van dit onderzoek werden experirnenten uitgevoerd met een 2 liter kristallisator, waarbij de warmte werd onttrokken door middel van direct contact koeling met verdamping van freon 114.

In de experimenten bleken de NaCl .2H2

o-

en ijskristallen gescheiden van elkaar te ontstaan. De volgende grootheden werden bepaald: de grootte, grootteverdeling, vorm, groeisnelheid, nucleatiesnelheid en zuiverheid van de ijs- en NaC1.2H20-kristallen, de grootte, grootteverdeling en concentratie van de freondruppels, en de onderkoeling. Deze grootheden werden bepaald als functies van de gemiddelde verblijftijd, freondruk,

roersnelheid en kristalmassafractie.

Het bleek dat voor een verblijftijd van 15 minuten en een kristal-massafractie van 5% de ijs- en NaCl.2H20-kristallen ongeveer 100 µm groot waren. De afmeting nam toe met toenemende verb1ijftijd, als

,o.

3 en met toenemende kristalmassafractie b als b-0·2. De druk en de roer-snelheid hadden weinig of geen invloed op de. kristalgrootte. De

grootte-verdeling van het NaCl .2H2

o

was exponentieel. die van het ijs vertoonde een maximum. De ijsgroei was warmte- of massatransportgelimiteerd, die van het NaCl.2H2

o

was gedeeltelijk inbouwgelimiteerd. De nucleatie van beide kristalsoorten was enigszins "surface regeneration limited", en kristal-kristal botsingen droegen meer bij tot de totale nucleatie dan kristal-kristallisator botsingen.

De freondruppels hadden afmetingen van ongeveer 50 µm. De onderkoeling was ongeveer 0.1 K voor een verblijftijd van 15 minuten.

De scheiding van de in het eutectisch vriesproces gevormde kristallen werd bestudeerd bij omgevingstemperatuur met een modelsysteem in een hydrocycloon van 1 cm diameter. De scheidingsefficienties van de lichte en zware deeltjes werden bepaald als functies van drukval, overloop/ onderstroom-verhouding, en deeltjesmassafractie. De fractie lichte

deeltjes in de overloop nam toe met toenemende overloop/onderstroom-verhouding tot ongeveer 98%, en de fractie zware deeltjes in de onder-stroom was onder al le omstandigheden ongeveer 98%.

De conclusie is dat het eutectisch vriesproces om NaCl en H20 te scheiden technisch realiseerbaar is. Het is wellicht economisch haal-baar voor een "grass root plant" als zout met een hoge zuiverheid is vereist of als de energieprijzen hoog blijven.

1. Introduction

1.1. Salt

1.1.1. Salt Sources, Methods of Recovery, Production and Uses Salt {sodium chloride} is principally obtained from the following three sources:

{i} Rock salt: the salt is mined or quarried. This source yields 45 Mt/a of salt.

(ii} Sea salt: the salt is obtained from sea water, the water of which being removed by solar evaporation. This source yields 60 Mt/a of salt.

{iii) Salt obtained by solution mining. This source accounts for 65 Mt/a of salt. In this process, water is pumped into a brine well {bore hole}, the salt dissolves in the water, and sub-sequently the solution is pumped up. From this solution 20 Mt/a of salt is obtained by evaporation of water in closed vessels, mostly in multi-effect evaporators. This salt is called "vacuum salt". The rest of the solution is used directly in industry. The total annual world production thus amounts to 170 Mt.

About 50% of the quantity of salt is utilized for the industrial production of

c1

_{2, NaOH and Na2}

co

_3.

Further data can be found in [Ka 60].

1.1.2. The System NaCl-HzO

Figure 1.1 shows the phase diagram of the system NaCl-H2

o

between T

=

250 K and T

=

_{280 K. Two components are present: NaCl and H2}

o.

The phases that may exist are: vapour, solution, ice, solid NaCl and solid NaCl .2H2

o.

The degrees of freedom that are possible are: temperature (T}, pressure (p} and mass fraction NaCl (w}. According to Gibbs' phase rule, the sum of the number of degrees of freedom and the number of phases equals 4 for this system. Only 4 phases can thus exist simultaneously; temperature, pressure and composition are then fixed. This occurs at point E: the eutectic point {TE = 252.03 K, wE = 0.2331, pE = 92.6 Pa) and at point P: the peritectic point {Tp = 273.25 K, wp = 0.26285, Pp= 464 Pa).

280 T (K)

t

270 260 250 slope: ~ = 14 kK

dw NaCl & saturated solution

Ti = 273.16 K P; = 611.2 Pa _P wh = 0.61863 1---~---i Wp = 0.26285 Pp = 464 Pa Tp = 273.25 K unsaturated ice & unsaturated solution E NaC1.2H₂0 & saturated so 1 uti on

NaCl & NaCl • 2Hz0

1 - - - L - - - 1 T E = 252.03 K WE = 0.2331 PE = 92.6 Pa 0.0 0.2 0.4 0.6 0.8 - w (1) 1.0

Fig!AX'e 1.1. Phase diagram of NaCl-H₂

o.

In each point of the diagram the vapour is in equiZibrium u>ith the other phases.

The phase diagram is 3-dimensional, with pressure as the third variable, but because the pressure has a negligible influence on the equilibrium temperature or on the equilibrium composition, the vapour is omitted in this diagram. In each point of the diagram the vapour volume is assumed to be negligible compared to the volume of the other phases.

If the pressure of the system is increased, the vapour is no longer in equilibrium with the other phases. Then the melting temperature of ice and the eutectic temperature will decrease and the peritectic tempera-ture and composition will increase.

In the eutectic point, vapour, solution. ice and NaCl.2H20 are in equi-librium and in the peritectic point, vapour, solution. NaCl and

NaCl.2H2

o

are in equilibrium. The point where vapour, solution. ice and NaCl are in equilibrium is the metastable eutectic point. It was found

at T = 250.8 K,

w

=

0.23 [Me 05). The intersection of the extended ice and NaCl solubility curves, however, is T

=

249.6 K,

w

=

0.26. This discrepancy has not been explained so far.

In addition

t?

_{NaCl.2H20 another NaCl hydrate is known: NaCl.H20, but}

only as a molecule, not as a crystal lattice [Au 78).

NaCl has the mineralogical name of "halite" and it occurs in 2 crystal fonns [Ba 68). The low pressure {p < 20 GPa) structure is cubic. NaCl .2H2

o

has the mineralogical name of "hydrohalite" and has a mono-clinic lattice [Kl 74).

Ice occurs in 11 crystal fonns [Mi 80]. The stable low pressure {p < 100 MPa, T < 100 K) crystal structure is hexagonal, and is called ice Ih.

Only the low pressure structures of NaCl and ice are relevant to this study.

All three kinds of crystals {NaCl, NaCl.2H2

o

ice Ih) are colourless.

100 slope: if! = 50 kJ/g dw

-

unsaturated solution l;T> ;:;- 0

-

_~ ... :::i.: -100 ~ ~ _m

'

_T. :5-200 ~ l c <11

L30o

-400 0.0

ice & NaCl .2H 2o

0.2 0.4

Figu:t.>e 1. 2. Enthalpy diagram of NaCZ-H₂0.

NaCl & saturated solution

0.6 Tp = 273.25 K TE = 252.03 K T; = 273.16 K 0.8 - w (1) 1.0

H = 0 for "liquid water and solid NaCZ at T = 273.15 K. In each point of the diagram the vapour is in equiZibriwn with the other phases.

Figure 1.2 shows the enthalpy diagram for the system NaCl-H2

o

between T

=

250 K and T = 290 K; in this diagram H = 0 is taken for liquid water and solid NaCl at T = 273.15 K [Fa 56, Ku 57]. From this diagram specific heats and heats of melting can be read. The enthalpy H is also a function of pressure, but in this temperature range the pressure has only little influence on H. In the diagram the vapour volume is assumed to be negligible compared to the volume of the other phases.

1.1.3. The Issue of Energy

As has been mentioned in section 1.1.1, multi-effect evaporation is a widely applied method of separating NaCl form H2

o.

The energy costs of this process depend on several factors and constitute a considerable part of the total production costs. In view of the rising energy prices in the past it became of interest to consider less-energy consuming alternatives. Freezing processes appear to be promising from an energy point of view. Therefore, this type of processes was chosen as a sub-ject for this study.

1.2. Freezing Processes 1.2.1. Overview

Freezing processes applied to produce NaCl from brine can be divided into two categories: eutectic and noneutectic processes. Both catego-ries can be subdivided into two subcategocatego-ries: processes with direct cooling and processes with indirect cooling.

Direct cooling can be effected by means of a nonevaporating nonsoluble refrigerant, an evaporating nonsoluble refrigerant, an evaporating hydrate former or by evaporation of water.

Power consumptions of the freezing processes are calculated in appen-dix A.

a. Eutectic Freezing Processes ,

Figure 1.3 shows the principle of the eutectic freezing process. The solution is cooled to T

=

252 K (the eutectic temperature}. At the trajectory T = 273 K + T = 252 K some NaCl.2H2

o

crystallizes. At T

=

273 K the solution is (partly} frozen in the crystallizer. Because this freezing occurs at the eutectic temperature, ice as well as

NaCl.2H2

o

crystal.lizes. These two kinds of crystals are then fed to a separator. This device separates the ice-NaCl.2H20-brine slurry into an ice-brine and a NaCl.2H20-brine slurry. Separation can be effected on the ground of the density difference of the two substances: ice is lighter than the brine, NaCl.2H 20 is heavier than the brine. The NaCl.2H2

o

crystals are heated and dissociated at T

=

273 K into NaCl and saturated solution. The solution is recirculated to the crystal-lizer. The NaCl is the end product. The ice crystals are heated, washed and melted at T

=

273 K. The resulting water can be used to dissolve new salt.

If flow Qhl' 0_{11 , Qh or Qi is partly recirculated to the crystallizer,} the ice or NaCl.2H2

o

crystal concentration in the crystallizer can be varied independently of residence time or total production.

a.a. Eutectic Freezing Processes with Direct Cooling a.a.a. Direct Contact Cooling without Evaporation

An immiscible nonevaporating liquid of temperature lower than TE is suspended in the slurry and separated from the slurry afterwards. The density difference between the slurry and the refrigerant is utilized to separate them. This method is not economically feasible, because the power consumption and the size of the compressors are prohibitively large.

a.a.b. Direct Contact Cooling with Evaporation

An immiscible liquid is suspended in the slurry and evaporated by lowering the pressure. Table 1.1 lists possible refrigerants. Some literature is available [Ba 75, Ba 77, Ba 78, Fl 75, Fl 79, St 73a, St 73b]. This process may be economically feasible.

a.a.c. Direct Contact Cooling with Hydrate Fonnation

An immiscible liquid is suspended in the slurry. The heat is removed by evaporating part of the refrigerant. The remaining refrigerant forms a hydrate (solid) which decomposes at a higher temperature into water and vapour. This process may be economically feasible.

a.a.d. Direct Cooling by Evaporation of Water

In this process the heat is removed by evaporation of water. This pro-cess is not economically feasible because the compressors are

prohibit-Nae 1. 2H₂o & ice & solution NaCl solution ~~~~~____,.~~~~-+-~ crys ta 11 i zer solution product water washing water

ice & solution Qil

solution

Figure 1.3. Principle of eutectic fPeezing pPOcess (with diPect cooling with evapoPation). NaCl .2H₂0 & solution crys ta 11 i zer solution NaCl solution NaCl solution

Figia>e 1.4. Principle of noneuteatic fPeezing pPOaess (with diPect cooling with evapoPation).

Substance Structural Fonnula Tb (p= 101 kPa) Ref. Fll13 _CF2

=

CFCl 244.8 Ba 60 isobutene _CH2 = _{C - (CH3)2} 266.3 Br 62 1-butene CH

=

_{CH - CH2 - CH3} 266.9 Ba 60 FC318

_Cf

_{z -}

(CF2)z - 9F2 267.6 Ba 60 1,3-butadiene _CH2

=

CH - CH

=

_CH2 268.8 Ba 60 n-butane _{CH3 - (CH2)2 - CH3} 272.7 Ba 78 2-transbutene} CH3 - CH

=

CH - CH 3 274.0 Br 62 2-ci sb utene 276.9 Br 62 Fl14 _{CC1F2 - CC1F2} 277.0 Br 62 Fl33a _{CF3 - CH2Cl} 280.1 Ba 60 1-butyne _{CH :: C - CH2 - CH3} 281.3 Br 62 neopentane _{C - (CH3)4} 288.8 Ba 60 Fl112a _{CC1 2} = _CF2 293.6 Br 62 Fl42 _{CH2Cl - CHF2} 309 .2 Br 62

Tab le 1. 1. Compounds that form no hyd:l'ate and are in.so Zub Ze in ?Jater.

ively large.

a.b. Eutectic Freezing Process with Indirect Cooling

The heat is removed by cooling the wall of the vessel. Indirect cooling is not economically feasible. Some literature is available [Ne 54, Th 56. BJI 79,

no

101 •

b. Noneutectic and Partly Eutectic Freezing Processes

Figure 1.4 shows the principle of the noneutectic freezing process. In this process no ice is formed. The product solution can be used to dis-solve new salt. Processes intermediate between eutectic and noneutectic freezing processes are also conceivable. These processes are not econ-omically feasible because of the high capital investments of the entire equipment.

1.2.2. State of the Art

thermo-dynamic properties was available. The crystal size and crystal size distribution as functions of conditions were unknown. The size of the separation and of the washing equipment and consequently the capital costs depend on the crystal size. Also unknown was the performance of equipment that separates two kinds of crystals, one heavier and one lighter than the liquid.

It appeared that due to lack of data it could not be calculated whether freezing fs a cheaper method of salt recovery than evaporation.

1.3. Objective of this Study and Set-up of Thesis

The objective of this study was to produce the data that were still unknown. As has been mentioned in section 1.2.1, only alternatives a.a.b and a.a.c are perhaps economically feasible. Only direct contact cooling with evaporation has been proved to be possible [St 73b]; so it was decided to study this process. In addition, the separation of two solids was studied, in the device that has proved to function well

[St 73b]: a hydrocyclone.

Chapter 2 presents a theoretical model of direct contact evaporation. In chapter 3 the theory and the literature on crystallization is reviewed.

Chapter 4 deals with the experimental data on eutectic crystallization of NaC1.2H2

o

and ice.

The data on the separation experiments in a hydrocyclone are presented in chapter 5.

2. Direct Contact Cooling with Evaporation

2.1. Introduction

In this chapter the theory of direct contact cooling with evaporation is presented. In this process a liquid refrigerant is injected into a stirring-tank filled with suspension. The refrigerant must be only slightly miscible with the bulk liquid, and the bulk liquid must be much less volatile than the refrigerant. The refrigerant is evaporated by maintaining a low pressure by means of a compressor. In this way, heat is withdrawn from the bulk liquid.

The size, shape and motion of evaporating refrigerant drops is studied in section 2.2. The heat transfer characteristics of evaporating refrig-erant drops are treated in section 2.3. In section 2.4 the particle size distributions of drops and of bubbles are mentioned. The main con-clusions of this chapter are given in section 2.5.

2.2. The Shape and Size of a Two-phase Drop 2.2.1. Classification of Shapes

In this section the shape of a drop. consisting of a liquid and its vapour ("two-p~ase drop"), in another. inmiscible liquid will be de-scribed.

First. consider a liquid 1 upon which a drop of liquid 2 is dripped {see figure 2.1).

vapour (1&2) vapour (1&2)

liquid 2 _{liquid 2} iquid l

Liquids mutually saturated Liquids mutually saturated.

Fi(JUN 2.1. Fi(JUN 2. 2.

Initially this drop will take the shape of a lens. Next, either of two things will happen: the drop spreads over liquid 1 or it remains a lens.

The lens will spread if cr1 ~ _cr2 + _{cr12, where cr1 is the jnterfacial}

tension between liquid 1 and the vapour, cr2 the interfacial tension between liquid 2 and the vapour, and cr12 the interfacial tension between

liquid 1 _{and liquid 2. If the lens does not spread and p2} > p

1, then the

drop will sink to the bottom.

Next, consider a layer of liquid 2 upon which a drop of liquid 1 is dripped (see figure 2.2). Now the lens will spread if cr2 ~ cr₁ + _{cr12 .}

If cr 12 > icr 1 - cr2

!.

the lens will spread in neither case.

In these experiments the drop must of course be prevented from evapo-rating.

From these experiments the shape of a vapour bubble at the interface of the two liquids can be deduced (see figure 2.3).

liquid 2 (saturated with liquid 1)

vapour (1&2) vapour (1&2) vapour (1&2)

~

t)

b

f)

0_{2 :} 0_{1 +} 0₁₂ 0_{12 :lcr1-} 0₂₁ 0_{1 :} 0_{2 +} 0₁₂ 1 iqui d 1 (saturated with liquid 2)

Figure 2.3

If liquid 2 spreads over liquid 1, one could say that liquid 1 and the vapour "don• t want to see each other". It appears that the vapour bubble withdraws into liquid 2.

The cr's can also be measured by a surface tension measuring instrument, but the spreading experiment is a convenient way to determine which of the three cases is at hand.

Next consider an evaporating drop having surface tension crd imnersed in a liquid having surface tension crc. Figures 2.4, 2.5 and 2.6 show the shapes of the evaporating drops.

~

0

0

~=10-3 ~= 0.01 ~=0.03 0_{c ~} 0_{d +} 0_cd

0

0

~=0.01 ~=0.03

ocd >loc - odl

0 0 0 0 0

®

FigUPe 2.4 FigUPe 2.5 FigUPe 2.6

~

: evaporating liquid

D

: vapour

FigUPes 2.4, 2.5, 2.6. Evaporating drop immersed in a nonvoZatiZe Ziquid.

In figures 2.4 and 2.5 is denoted the evaporated fraction t of the total drop mass.

The shapes of two-phase drops were determined photographically, those of figure 2.4 by Sideman & Taitel [Si 64], Simpson et al. [Si 73b] and Pinder [Pi 80]. those of figure 2.5 by Tochitani et al. [To 77a] (furan-glycerol), and those of figure 2.6 by Iida

&

Takashima [Ii 80] {aceton

&

methanol-silicone oil).

To prevent explosive evaporation. the nucleation (creation of vapour bubble) must be induced. In a stirred suspension the nucleation is in-duced by collisions.

Homogeneous nucleation has been described byJarviset al. [Ja 75] and Avedisian

&

Andres [Av 78].

2.2.2. The Size of a Two-phase Drop in a Stirring-tank

In figure 2.7 the vapour bubble is apt to rise under the influence of the buoyancy force (also see Selecki

&

Gradon [Se 72]}.

This will happen if

{2.1)

where Fis the upward force {=buoyancy force minus weight). his the distance over which the bubble penetrates liquid 2, A2v is the area of

F liquid 2 h liquid 1 vapour FiguPe 2. 7.

the part of the sphere, wetted by liquid 2, and A12 is the liquid to liquid interfacial area.

A force is the derivate of an energy, hence an energy balance is equiv-alent to a force balance as far as the possibility of motion is

con-cerned. ·

The upward force equals:

The differentials of the areas are: dA2v = _{2lrrvdh and dA12} = 2lr(h -rv)dh~ Expression 2.1 must hold for all h (O ~ h ~ 2rv) so

This is true for a 11 r v :: 0 i f _{cr 1:: cr2+cr12 • If cr2'.:cr1+a 12 or} 0₁₂>

I

cr 1 - cr2

I

only bubbles with radius

can rise and leave the interface.

The same principle can be applied to a two-phase drop in a continuous liquid. This calculation is presented in appendix B. The result is shown in figure 2.8. For a given acceleration of the two-phase drop and for a given vapour mass fraction, this figure shows the size rv* of the vapour bubble for which the bubble detaches from the liquid drop. It is shown that this critical radius decreases if the acceleration increases.

9 8 7 6

-...

₅ I Ci) N ... ("I') ₄ E ("I') I 0 ... ₃ ~ i< > ₂ !...

t

1 0 Figure 0.0 2. 8. 0.2 0.4 0.6 0.8 1.0 -- c; (1)

r v*: bubble radius for 1.1Jhich bubble d.etaches from droplet at given acceleration a.

The critical radius for n-butane is larger than the critical radius for Fl14, because for n-butane the buoyancy force and the inertial force have the same direction but for F114 these forces have opposite direct-ions.

2.2.3. The Shape of a Two-phase Drop as a Function of the Vapour Mass Fraction

Tochitani et al. [To 77a] present a model in which ~(B) is given, where

s

is the half opening-angle. The total two-phase drop is assumed to be spherical (figure 2.9). It can be shown that

FiguP8 2.9. d

Figur>e 2. 10.

Pv(2 - 3cose + cos 3e}

~

=~---pd(2 + 3coss - cos3s} + Pv(2- 3cose + cos3a}

Pinder [Pi 80] found the following correlation by measurements:

p p p

a

=

~

f 1

~

3 tan 2

(~

arctan l} 2 dollc g

---z-z·

crd Pc (2.2} -1

P1 = _{-7.91 m d0 - 0.1565 (where d0 is the initial drop diameter)} P2 = 6.205

P3 = -6.032.

Pinder fixed ~ by using a mixture of a volatile and a nonvolatile com-ponent as the dispersed phase. When the drop size does not increase further.

a

is detennined.

The areas of the liquid to liquid interface (Acd>• of the vapour to con-tinuous liquid interface (Acv>• of the vapour to dispersed liquid inter-face (Adv> and of the total two-phase drop (A} can be determined as follows. Assume the vapour bubble to be spherical (figure 2.10}. The equivalent drop diameter is defined as de = ( d₁2d_{2} 1/3 where d1 is the} largest dimension and d2 is the smallest dimension of the total two-phase drop.

. Then A = Trd/

Acv= 2Trr

/(1-

cosB)

2

Adv= 2Trrv (1 + COSB) Acd= A - Acv

de== (1-i:; + (pd/pv}F;}l/3do

Figures 2.11 to 2.16 show B. AcvlArJ. A~vlArJ. _{Acd/Ao• de/d0 as}

of i:;, _{both axis linear, where Ao == Trd0 •}

1T 40 B

t

0.8 Acv _Adv

AO

_AO

t

t

0 1 _{0 1} -t;. -i:; (2.3} functions 1 - t;

Figure 2.11. Figure 2.12. Figure 2.13.

5 4 1.2 1.0 A de Acd

~

ao

AO

t

_t

1

t

0 1 l 0 -t;. 1 -t;. _-E:.

Figure 2.14. Figure 2. 16. Figure 2. 18.

2.2.4. The Motion of a Rising Two-phase Drop in a Quiescent Liquid This subject is described by Tochitani et al. [To 75], Selecki & Gradon

[Se 76] and Mokhtarzadeh

&

El-Shirbini [Mo 79a]. A general theoretical treatment of two-phase drops is given by Mokhtarzadeh

&

El-Shirbini [Mo 79b].

2.3. Heat Transfer to Two-phase Drops

2.3.1. Heat Transfer to a Rising, Evaporating Two-phase Drop in a Quiescent Liquid

Two evaporation regimes can be distinguished: one with ad ~ ac + acd and one with ac ~ ad + acdor acd > lac - crdl (see figures 2.4, 2.5 and

2.6). .

Selecki

&

Gradon [Se 72] incorrectly distinguished the regimes

ac < crd and crc > crd. This error has gone unnoticed thus far because nobody has studied the case ac < crd and crcd > lcrc - ad!'

The subcooling of a two-phase drop, AT, is defined as: the temperature of the bulk liquid T_{1 ,} minus the equilibrium temperature corresponding to the boiling pressure of the dispersed liquid at the given depth Tb. (The drops are so large that the curvature effect (see section 3.2.2) plays no role.) The area that is used to define the heat transfer coef-ficient h~ is the total two-phase drop area A.

The case crd ::-_ crc + crcd is treated by Selecki & Gradon [Se 72] and Gradon & Selecki [Gr 77]. No heat transfer coefficient is calculated or measured, however. The case ac ~ crd + crcdorcrcd > lac - crdl has been studied by several authors: In their experiments they allow evaporating drops to rise in a warm liquid; the size as a function of time is then determined. From this function de(t), the overall heat transfer coef-ficient h

0 has been determined. The experiments were conducted at atmo-spheric pressure.

Sideman & Taitel [Si 64] 'developed a model of the heat transfer to an evaporating drop and found:

h = kc ( 3coss - cos 3s + 2)!

Pee~

c

a;-

1T (2.4)

Tochitani et al. [To 77b] developed a model resulting in:

k . 2/3 1/3

he = o.463

a;

(1T -

s

+

~)

Pee · (2.5)

12 k a116 h - d d - 2

173

d 1/2 (2.6) \Id 0 and 25 70 (de/ d0) 116 ho = _l _+_o_.-2-06_(_de_/_do_).,..5""T'/ 1 ... 2 (2.7)

Sideman

&

Taitel [Si 64] found from experiments that for butane in sea water h0 = 1500 .:!:. 700 W/(m2K). with a slight maximum at t; = 0.03 and a decrease of h0 at larger ~T (4K < ~T < 13K) (figure 2.17E).

Tochitani et al. [To 77b] found for pentane in glycerol qualitatively the same relation (slight maximum, ah/a~T < _{0), but their h0 is one} order of magnitude smaller than the h0 of Sideman

&

Taitel for butane in sea water.

Simpson et al. [Si 73b] report that for butane in a NaCl-solution the h

0 = 2500.:!:_500W/(m

2

K). In their experiments h0 appeared to be independ-ent of ~T (2K < ~T < SK), however, and to increase monotonously as a function oft; (ah /at;> 0) (figure 2.18F). Furthermore, they found for de/do > 2 (t; _{> 0.03) that h0 is independent of the NaCl concentration} (0 < w < 0.08).

Prakash & Pinder [Pr 67], Sideman & Isenberg [Si 67] and Raina & Grover [Ra 82] give some more results.

Experiments with various systems (no butane, no NaCl solution, up to high µc (10 mPas), Pc> pd) conducted by Adams

&

Pinder [Ad 72] re-vealed that

-h- = 7550 kd Pr -0. 75 ( µc )4.3 Bo0.33'

oO

CIQ

c µc + µd

(Pc - Pd)g do 2

where the Bond number Bo= a (also called the Eotvos number Eo) and ho0 is the overall cd heat transfer coefficient based on Ao and averaged over t;.

From figure 2.17 it is seen that h0(t;) is approximately constant. There-fore h00 A0 = h0 f A(t;) c!E;. _{Because A/Ao = (de/d0)}2 this yields with equation 2.3: O

h l} oO

- c; (1)

(2.8)

Figure 2.17. Heat transfer ooefj'iaient h as a funotion of vapoUP mass fro.otion t; for n-butane in a IV 0.04 NaCl so"lution.

Data f'l'Om various authors (see text seotion 2.2.1).

Figure 2.17 shows the various results of h(t;). To compare these results. for the refrigerant, NaCl mass fraction, and initial drop diameter the following choise had to be made: n-butane, w = _{0.04 and d0} = 3.7 mm respect i ve ly •

Curve A represents equation 2.4 with the rise velocity according to [Si 64}: figure 14 and S(t;) according to equation 2.2 with d0 = 3.7 mm.

Curve B shows equation 2.5 with rise velocity according to [Si 73bJ: equation 3 and B(~) according to equation 2.2.

Curve C illustrates equation 2.6 with d0 = 3.75 mm. Curve D represents equation 2.7 with d0

=

3.75 mm.

Curves El,2 are [Si 641: figure 17, butane - sea water (d0 = 3.6 mm) with El : lff

=

4K, E2 : AT

=

7K.

Curve Fis [Si 73J: figure 8 (d0

=

3.75 mm). Curve G represents equation 2.8 with d0

=

3.7 mm.

2.3.2. Heat Transfer to a Rising Condensing Two-phase Drop in a Quiescent Liquid

This case is described by Sideman

&

Hirsch [Si 65bJ and by Isenberg

&

Sideman [Is 70]. This phenomenon does not occur in the type of process-es we studied; it is therefore not treated here.

2.3.3. Heat Transfer Between Two Immiscible Liquid Layers with Simul-taneous Boiling and Stirring

This case is treated by Fortuna

&

Sideman [Fo 68J. It occurs at very low stirring rates, which are not applied in this study, for which reason this case is not treated here.

2.3.4. Heat Transfer to Evaporating Drops in a Stirring-tank

Sideman

&

Barsky [Si 65aJ conducted experiments in a stirring-tank with high dispersed phase concentration {Em= 0.03). Heat was supplied by a heating plate and the bulk subcooling was measured. The results can be represented by (Ptv_1/ 1 "'AT Ec2, where P is the heating power supplied by the heating plate and E: the agitation power per liquid mass. The exponents c1 and c2 depend on Rea and are approximately unity. 2.3.5. Heat Transfer to Evaporating Drops in a Eutectic Stirring

Crysta 11 i zer

In a eutectic crystallizer the bulk temperature T1 ·is fixed. The heat of crystallization is removed by evaporating a refrigerant by means of a compressor. If the refrigerant is com~letely suspended, the pressure is a variable. If the refrigerant is (partly) present as a layer on the bottom of the tank, the pressure is fixed. If the refrigerant floats

upon the suspension, then Ps = Pe(T₁); if the suspension floats upon the refrigerant, then Ps

=

_{Pe(T1) - Ph• where Ps =surface pressure,}

Pe

=

equilibrium pressure and ph = hydrostatic pressure across the sus-pension. It is assumed that both layers have the temperature r_1. If the drops are fully suspended, the following model is applicable: Consider a horizontal slice with thickness dz in the crystallizer. The slice generates a refrigerant vapour mass flow density

l

drJ> _ h0 AT b. T ( z) dz

Of

~

0 y '

(2.9)

with zb the 11

boilinq depth": the depth where Ph= Pe(T1), and Ar is the drop area density (m-1) (total drop interfacial area divided by bulk suspension volume, see section 3.8.1).

From section 2.2.l it is clear that for conditions that have not been experimentally studied, h₀ is best predicted by equation 2 .• 8. The re-sults of section 4.4.4 show that the maximum drop size d"" 150 µm; therefore, do is taken to be 150 µm. Then for a eutectic NaCl solution with n-butane as a refrigerant equation 2.8 yields h₀

=

1300 W/m2K, and with Fll4 as a refrigerant ho

=

310 W/m2K. This difference is

principal-ly caused by the difference in thermal conductivity of n-butane and Fll4. Equation 2.8 is applicable only if the velocity of the drop relative to the surrounding liquid equals the rise velocity in a gravitation field. In a stirring-crystallizer this velocity is larger. The above h₀ 's are therefore lower limits. In the calcu·lation of h

0 of Fll4, (Pc - pd) is

replaced by (Pd - Pc).

Two cases can now be distinguished: zb < _{z1 and zb} > _z1, w~ere _{z1 is the}

suspension depth. In the first case there is a zone where no evaporation takes place, in the second case evaporation takes place everywhere in the suspension.

If zb < _{z1• equation 2.9 can be integrated to give}

(2.10)

3 zb (m)

~Ts

(K)

1

30 zb I:~j AT(z)dz (Km) 0

t

2 1 0 0 10 20 30 40 - Ps (kPa)

Figure 2.18. Boiling depth zb, ~UPfaae subaooling AT

8(= Tl - Tb,s)

and I as a funation of BUPfaae pressu!'e p . _s

If zb > _{z1, equation 2.9 can be integrated to give:}

20

10

(2.11)

The function J(z 1,ps) for F114 is shown in figure 2.19.

It is now possible to calculate

Ar

by means of equation 2.10 or equa-tion 2.11. The area of a two-phase drop can be determined from the size of a two-phase drop (see section 2.2.2) and from this the refrig-erant volume fraction Ev can be calculated. If (part'of) the suspen-sion is heated all the refrigerant will evaporate. The refrigerant vol-ume fraction can then be measured and the refrigerant mass fraction Em calculated. The vapour mass fraction of a two-phase drop averaged over

7 1.2

~

6 1.0 0.8 5 0.6

-

4 _{parameter z} 1 (m)

J2

_0.4 N ₃ "1:1

-

N

-... <J ₂ 0.2 ,.... N"-~O II "'?

t

1 0 0 10 20 30 -Ps (kPa)

Figw:oe 2 .19. J as a funation of eurfaae pressure with parameter the liquid d.epth

zz.

all drops can now be written as

EV pd - 1

Em pl

~ = • with pl = the suspension mass density • pd - 1

Py

The time derivate of the vapour volume is given by

dVv h₀ AT A

at= PVLV and because

40

the growth velocity of the drops dde/dt equals 2h₀AT/pvlv (0 = 111111/s). where O denotes order of magnitude.

For the evaporation time 'v of a drop, this result yields:

p L d_ p l T :: V V \) {(_!!)! _ 1}

Integration of this equation gives the evaporation time averaged over all drops: d p L p

~

T = _J! V V {(~)"' v z₁ ~ pv zl dz If Ps

f:

0.6

Pt>

=

0 .6 Pe and zb > z₁ the function f: = f 1iT can be approximated by linearization of p(T) to give O

where c = 8,0 K/m for Fl14.

The residence time averaged over all drops in the crystallizer is

with m₁ = suspension mass and er> = 4>At• the refrigerant mass flow, where

At

=

cross-sectional area of the tank.

Because 'f">> :rv• the refrigerant will be present mostly as a vapour. In this section it is assumed that AT is a function of the depth z. It could be, however, that as a result of strong agitation the drops have about the same temperature everywhere in the tank. The following argumentation shows that the assumption is correct:

The rate at which heat is supplied to the drop is P1

=

h₀

~

de2AT (0 = 100 µW), The minimum rate at which heat should be supplied to sat-isfy the requirement is P2 = vz Pp cp

~

d/

~T,

in which vz is the vertical component of the drop velocity and Pp and cp: the density and specific heat of the total two-phase drop respectively (Pp cp de 3 does not vary much as a function of~). _{It appears that P2 is 0} = 1 µW, so the drops are heated fast enough to reach the AT which corresponds to the given depth.

2.4. The Particle Size Distributions of Drops and of Bubbles

In a continuous stirring-tank reactor (CSTR) drops and bubbles have a particle size distribution (PSD) which is approximately normal. Under certain conditions the tail is longer than the tail of a normal distri-bution [ Ze 72].

2. 5. Con cl us ions

(i) An evaporating refrigerant drop consists of a liquid and a vapour part, which stick together ("two-phase drop") by interfacial tension.

(ii) If in a stirring-tank the vapour bubble grows too large, it is torn from the liquid drop.

(iii) The heat trans fer coefficient of an evaporating two-phase drop in a still liquid is of the order of magnitude of 1 kW/m2K.

(iv} In a stirring-tank, the growth velocity of a two-phase drop is of the order of magnitude of 1 mm/s, the evaporation time is of the order of magnitude of 100 ms.

(v) The subcooling of a two-phase drop is a function of the depth. The temperature of the drop corresponds to the local boiling pressure.

3. Crystal Nucleation and Growth

3.1. Introduction

In this chapter crystal nucleation and growth are described for the follCMing two conditions:

{i) nucleation and growth of a single crystal under a well controlled flow regime;

(ii) nucleation and grCMth of crystals in a stirring-tank.·

Case (i) is treated in sections 3.2 to 3.4; case {ii) is treated in section 3.5 to 3.8.

Nucleation is the creation of a new crystal. Two kinds of nucleation can be distinguished: primary nucleation and secondary nucleation. Primary nucleation is nucleation that occurs in the absence of parent crystals of the crystallizing material. Secondary nucleation is nuclea-tion due to the presence of parent crystals of the crystallizing mate-rial.

Primary nucleation can be subdivided into: homogeneous and heterogeneous nucleation. Homogeneous nucleation is nucleation that occurs in the absence of any solid material. Heterogeneous nucleation is nucleation due to the presence of solid material other than the crystallizing material.

By convention, the growth velocity of single crystals is denoted by v, the growth velocity of bulk crystals is denoted by G.

In section 3.2 is treated the theory of single crystal nucleation and grCMth. More details about this subject can be found in [De 66, Tr 75, Wo 73].

Section 3.3 gives the literature experimental data on ice single growth. The literature on nucleation and growth of NaCl.2H20 crystals is treated {very briefly) in section 3.4.

Section 3.5 describes models and correlations of heat and mass transfer in a stirring-tank, and section 3.6 deals with models and correlations of secondary nucleation.

Experimental data on ice nucleation and grCMth in a stirring-tank as presented in the literature are given in section 3.7.

Section 3.8 deals with the theory of particle size distributions. The main conclusions of this chapter are given in section 3.9.

3.2. Crystal Single Growth 3.2.1. Interfacial Energy

The particle, liquid and interface free energies are denoted by FP, F1 and F1 respectively, and the particle, liquid and interface free en-thalpies by GP, G1 and Gi respectively. The quantity F; is called the interfacial free energy. The total free energy F of the system equals FP + F1 + F1, the total free enthalpy G of the system equals Gp+ G1 +

G;.

The interfacial energy y is defined by y: = -b;cCF;-Gi)lT,V,µ· [J/m2l [Tr 75, p.561. It can be shown that the reversible work requir~d to increase the area A of the particle equals

oW

= ydA. Note that the interface is an open thermodynamical system.

Interfacial tension is defined by cr:

=

f-

[N/m]: the force required to increase the area divided by the length ~ver which the force acts (see figure 3.1).

A

L

Figu:J?e 3.1. Interfaae.

For fluids the interfacial tension satisfies the relation [De 66, p.61):

where nj,i [moll: amount of substance of component j adsorbed at the interface (surface excess).

If y is isotropic, the relation between cr and y is given by [Tr 75,

For fluids the interfacial energy is not dependent on the strain

(oy/alj = 0) so a= y, In general, for crystals a~ y because a crystal

can sustain stresses.

In the literature the quantities cr, y, Fi and F;/A are often confused. The interfacial energy of crystals can be anisotropic. It can be repre-sented in a 3-dimensional polar diagram, which is called a Wulff dia-gram. An example of a Wulff diagram is shown in figure 3.2.

Pi(JUI'e 3. 2. Two-dimensional cross-secticm of Wulff diagr>am (bent lines) and equilibl'ium shape of crystal (stroight lines).

The shape of a crystal in equilibrium with its vapour or liquid (or bubble or droplet in a crystal, or a bubble in a liquid or a droplet in a vapour} is given by the Wulff theorem: Draw planes perpendicular to the radius vectors that intersect the points where 3y/38 and 3y/3$ are discontinuous. The crystal has the shape composed by these planes. The equilibrium shape satisfies the conditions [De 66, p.299]: d ~ crij

=

0 and {dF}T,V =

O.

If Fis minimum, the equilibrium is stable.JFor a bubble in a liquid or a droplet in a vapour y is isotropic, so the equilibrium shape is a sphere.

The determination of y is not simple. The results of various methods can differ by a factor 2. For an overview of methods see e.g. [Wo 73, ch .2].

The interfacial energy may depend on the growth velocity of the crystal or on the bulk concentration of a foreign component (see section 3.3.3.b}.

3.2.2. Homogeneous Nucleation

Consider a spherical nucleus. The driving force for crystallization equals [Wo 73, p.23]: 8G

=

4nr28G; + jnr38Gp• where 8Gi

=

y and 8GP = - ppLnfiT/Tm • with 8T ~ 0 and Tm: equilibrium temperature for r = ro (see figure 3.3).

t

- - r

Figw:>e 3. 3. Free enthalpy difference foi' cPystaZ.

From ~r

8G

= 0 follows the critical radius

2T

* -

Y m

r -

µ-e;r ,

p m

(3 .1)

an unstable equilibrium. Crystals with radius r > r* will grow until all liquid has solidified {if the process is isothermal) or until sub-cooling has vanished (8T = 0) (if the process is adiabatic) with a probability > 0.5; crystals with r < r* will dissolve (or melt) with a probability> 0.5. Therefore, the melting point of a spherical crystal with radius r equals

2y Tm

T

=

T -_{m pplmr} (3.2)

At a concave part of a surface the melting point is locally higher than Tm.

This result also applies for liquid-vapour and solid-vapour equilib-riums, if Lm is replaced by Lv or Ls respectively.

As a result of statistical fluctuations crystals with r > r* can dis-solve (or melt) too. So in principle a suspension is only stable if

only one crystal is present. This means that in a suspension the mean crystal size steadily increases owing to the growth of the larger crystals at the cost of the smaller ones. This process is called ripening.

If a liquid is cooled below a certain temperature, the (homogeneous) nucleation suddenly increases very much. At and below this temperature equation 3.2 does not apply [Wo 73, p.26].

3.2.3.Crystal Growth Rate

Figure 3.4 shows the temperatures appearing at a growing crystal in a solution. liquid crystal Tm AW00 melting point depression T _e

(r=oo) 2yT (r=oo) _e curvature

h.T pp[mr effect Te growth- A(W· - W ) ₁ 00 mass transfer driving force oT i nbui ldi ng heat transfer

Figure 3. 4. Temperatures at arystal interface.

The interface temperature is given by 2yT₀₀

Ti = T - -::"""'!"-: - AW • - oT , m ppi.mr i

where wi: the concentration of the solute at the interface, 1 a known constant and oT the driving force for inbuilding. The bulk subcooling is given by AT

=

Te(r=<>} - T00•

Several heat and mass transport models for crystal growth can be pre-sented. These models give v(vf,AT,w~,r}: the growth velocity of the ctystal tip as a function of flow velocity vf• of subcooling AT and of concentration of the solute w~ in the liquid, and of the radius of the (cylindrical) tip (see, for example, [Si 75]}.

Experimentaliy, however, it has been observed that v is detennined unam-biguously by vf' AT and w00 • Thus it follows that av/ar

=

0.

Obviously, an additional condition is satisfied. It is assumed that this additional condition is the "maximum velocity criterion": Most models predict a maximum in v(r}. It is now assumed that the radius r of the growing tip will adjust itself in such a way that v is maximum. Thus the additional condition is represented by av;ar = 0, which_ gives v(vf,AT,w00 ) . This criterion can be justified by a stability argument: if an area does not grow with the maximum velocity, it is overgrown by areas where v is maximum.

3.2.4. Growth Classification

At least three kinds of crystal growth can be distinguished: two-dimen-sional nucleation growth, screw dislocation growth and continuous (or nonnal) growth. In the two-dimensional nucleation growth process a crystal grows monolayer after monolayer on a smooth surface; in the screw dislocation growth process the crystal grows on screw dislocations; in the continuous (or normal) growth process the roughness of the surface is such that all places of the surface are equally suitable for growth. The three processes have the following growth velocities v as functions of the inbuilding subcooling OT [Wo 73, ch.SJ:

two-dimensional nucleation growth: v ~exp(- 1/oT); screw dislocation growth : v ~ (oT} 2; continuous (or normal} growth : v ~ oT.

The bulk subcooling can be determined experimentally. By means of heat conduction and diffusion models the quantities Ti - T₀₀ and wi can be calculated as functions of vf.

If y and r are known or if the maximum velocity criterion is applied, the inbuilding subcooling oT can be calculated.

If in an experiment several bulk subcoolings ~Tare applied, the growth velocity as a function of inbuilding subcooling v(oT) can be calculated subsequently, and from this the growth mechanism can be derived. If, however, (Ti - T00 ) + A(wi - w00) >> oT, the growth mechanism cannot be

determined in this way.

3.2.5. Heterogeneous Nucleation

Heterogeneous nucleation is the nucleation upon a foreign particle. Because the critical radius r* can be attained (locally) with fewer molecules than needed for a spherical particle, the subcooling for heterogeneous nucleation is smaller than that for homogeneous nucleation. For a plane substrate the following relations can be derived (see figure 3.5): crystal (2)

~liquid

(1) substrate (foreign particle) (3)

FigUPe 3.5. Nucleation on foreign particle.

If cr₁₂ > icr_{13 -} cr₂₃

1.

a lens-shaped crystal forms; if cr₁₃ > cr₁₂ + cr₂₃, the crystal covers the substrate entirely; i f cr₂₃ > cr₁₂ + cr

13, no crystal arises on the substrate.

3.3. Ice Single Growth 3.3.1. Introduction

Ice is known to have at least 11 crystal lattices. The crystal structure under normal conditions (P < 100 MPa, 100 K < T < 273 K) is called ice Ih and has a hexagonal lattice (P6 3/11111c). The unit cell is shown in figure 3.6.

730 pm I I I C I I

r--- --

--l

450 pm

Figu~e 3.6. Unit cell of ice Ih.

Several measurements on ice growth have been carried out: in still and in flowing water {figure 3.7), in pure water and in NaCl solutions up tow~

=

0.06, in the a-axis direction and in the c-axis direction and with different subcoolings.

liquid

tiT

Figure 3. 7. G!'ObJing ceystal in flObling liquid.

Growth velocities of crystals have been detennined in capillaries and in free conditions. The two methods generally give different results. Because in a stirring-crystallizer only free-growing crystals are pres-ent, only the literature on free-growing crystals will be treated in this chapter.

The a-axis growth. rate is two orders of magnitude faster than the c-axis growth rate. Therefore, ice crystals grow into flat plates.

A literature directory on ice growth is to be found in [Ke 79].

3.3.2. C-axis Growth

The c-axis growth of free-growing ice crystals is described by Simpson et al. 1 [Si 73al and Simpson et al. 2 [Si 76]. From comparison with measurements carried out in capillaries it appears that a free-growing crystal grows an order of magnitude more slowly than a crystal in a capillary.

Simpson et al. 1 [Si 73a] found that Ve= 1.734 10-6 exp(- 0.234/8T) in pure water. The flow velocity vf has no influence on vc.

Simpson et al. 2 [Si 76] correlated their data by

vc

=

2.5 10-6 exp(-0.320/8T) for pure water. The ·flow velocity has a slight effect on vc: for 8T ~ 200 mK the growth velocity increases with increasing flow velocity (avc/avf > O}. For NaCl solutions the growth rate decreases with increasing salt concentration (avc/aw00 < 0) (see figure 3.8}.

The method described in section 3.2.4 can be applied to calculate the inbuilding subcooling 8T. For pure water this results in

a 1 a: Yf=l25 11111/S Wcc=O b: vf=50 mm/s w.,=O c: w.,=0.0l d: w.,=0.02 e: w.,=0.03 f: w.,=0.04 0.1..._~~~~--''--->-~~~--' 10 20

vc

=

2.5 10-6 exp(- 0.159/oT).

An exponential correlationship has also been found for NaCl solutions. The c-axis growth is thus a two-dimensional nucleation growth (see section 3.2.4).

3.3.3. A-axis Growth

a. A-axis growth in Pure Water

The experimental results of various authors for the a-axis growth rate in quiescent pure water are shown in figure 3.9.

104 ~----,---, va (µm)

t

s 103 10 1 0.1 1 10 -AT (K) 1: Hu 69b:va=3.0 10·4.n2•22 2: Ry 69: va=9.0 10·5ar2•5 3: Si 73a:va=2.0 10·4ar3 4: Gi 76: v =1.0 10-4llT3 5: Ka 77: /=1.18 l0-_a 4llT2•17

FigUPe 3.9. A-axis gT'ObJth Pate of ice in quiescent

pupe 1'1ateP.

The growth rate in flowing pure water has been studied by several authors:

Fernandez & Barduhn [Fe 671 found va = 4.66 10-3 v/12

t.r

312• These authors developed a boundary layer model in which they assumed that the growth rate is limited by heat transport, that the ice conducts no heat, and that va is negligible compared to vf. In addition, they applied the maximum velocity criterion (see section 3.2.3}. This model results in