EXPERIMENTAL AND THEORETICAL STUDY OF FINES DESTRUCTION...

Experimental and theoretical study of finesdestruction in a mixed suspension crystallizer

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Authors Kraljevich, Zlatica Idalia

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EXPERIMENTAL AND THEORETICAL STUDY OF FINES DESTRUCTION

IN A MIXED SUSPENSION CRYSTALLIZER

by

Z l a t i c a I dal l a K r a i j e v i c h

A Th es is Submi t ted t o t h e F a c u l t y o f the

DEPARTMENT OF CHEMICAL ENGINEERING

In P a r t i a l F u l f i l l m e n t o f the Requirements For t h e Degree o f

MASTER OF SCIENCE

In the Graduate C o l le g e

THE UNIVERSITY OF ARIZONAi

1 .9 7 7

STATEMENT BY AUTHOR

T h is t h e s i s has been s u b m i t te d in p a r t i a l f u l f i l l m e n t o f r e q u i r e ments f o r an advanced degree a t The U n i v e r s i t y o f A r i z o n a and is deposi t e d in the U n i v e r s i t y L i b r a r y t o be made a v a i l a b l e t o bo r ro w ers under r u l e s o f the L i b r a r y .

B r i e f q u o t a t i o n s f rom t h i s t h e s i s are a l l o w a b l e w i t h o u t s p e c i a l p e r m is s i o n , p ro v id e d t h a t a c c u r a te acknowledgment o f source i s made. Requests f o r p e r m is s io n f o r ex tended q u o t a t i o n f rom o r r e p r o d u c t i o n o f t h i s m a n u s c r ip t in whole o r in p a r t may be g ra n te d by th e head o f the ma jo r depar tment o r the Dean o f the Graduate C o l le g e when in h i s judgment the proposed use o f the m a t e r i a l i s in th e i n t e r e s t s o f s c h o l a r s h i p . In a l l o t h e r i n s t a n c e s , however, p e r m is s io n must be o b t a i n e d f rom the a u t h o r .

APPROVAL BY THESIS DIRECTOR

T h is t h e s i s has been approved on the da te shown below:

A ___________A. D. RANDOLPH

P r o fe s s o r o f Chemical E n g in e e r in g17 Date

Ded ic a ted t o

Werner and The Small Great Fam i ly

ACKNOWLEDGMENTS

The a u t h o r w ishes t o express h e r s i nee r e s t thanks t o Dr. A lan D.

Randolph f o r h i s encouragement, p a t i e n c e , and i n v a l u a b l e a s s i s t a n c e

d u r i n g t h i s p r o j e c t . She a l s o acknowledges th e Department o f Chemical

E n g in e e r in g f o r p r o v i d i n g a s s i s t a n c e and p h y s i c a l f a c i l i t i e s f o r th e

p r o j e c t .

The a u t h o r is g r a t e f u l t o the N a t io n a l Sc ience Founda t ion f o r

f i n a n c i a l s u p p o r t th roug h Grant ENG75-04348.

The a u t h o r a l s o thanks Dr. James R. Beckman, who f r e e l y shared

h i s e x p e r ie n c e on the e x p e r im e n ta l equ ipment .

The a u t h o r w ishes t o express h e r g r a t i t u d e t o h e r f a m i l y f o r i t s

en thus iasm ac ross th e d i s t a n c e . And l a s t bu t n o t l e a s t , she g i v e s

s p e c i a l thanks t o her husband, Werner, whose u n s e l f i s h a d v i c e , encourage

m e n t , and s u p p o r t , d e s p i t e h i s own needs, were w i t h her a l l the t im e .

TABLE OF CONTENTS

Page.

LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . v i ?

LIST OF TABLES . ................................. i x

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . V . . . x

INTRODUCTION . . . . . . . . . . . . . , . . . . . . . . . . . . . . 1

THEORY . . . . . . . ................... .... . . . ....................... 6

Ki n e t i c s . ........................................... .... ................................................... . 6S ize I m p r o v e m e n t .......................................... 12

EXPERIMENTAL EQUIPMENT

Feed Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Feed -L in e Hea te r . .. . . . . . . . . . . . . . . . . . . . . . 20C r y s t a l l i z e r and F ines Trap . . . . . . . ........................ 21Fines D e s t r u c t i o n System . . . . . . ....................... 23C r y s t a l l i z e r C o o l in g System . . . . . ................... .... . . . . . . 23A n a l y s i s Equipment . ....................... 23

EXPERIMENTAL PROCEDURE ....................... 25

S t a r t - Up ............................ 25O p e ra t io n . . . . . . . 26Shutdown . . . . . . . . . . . ............................ . . . . . . . . 28

RESULTS . . . . . . . ................... . ................................. . . . . . . .. 30

K i n e t i c s Model .......................................... 30S e t t l i n g V e l o c i t y E f f e c t . . . .................................................................... 50Design o f a Fines D e s t r u c t i o n System .......................................................... 55Inc rem en ta l O p e ra t in g Cost w i t h FDS ............................ 62Fines D e s t r u c t i o n by H ea t ing . . . . . . . . . . . . . . . . . 65Fines D e s t r u c t i o n by D i l u t i o n . . . . .................... . . . . . . . 66

Water in Feed S o l u t i o n . . . . . . . . . . . . . . . . . . . 68• Water Requi red t o D is s o l v e the Fines . . . . . . . . . . . . . 68

SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 12

v

T ABLE OF C O N T E N T S - “ C o n t I n u e d

Page

APPENDIX A: SIZE IMPROVEMENT WITH FDS . . . . . . . . . . . . . . . 75

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

REFERENCES ....................... 82

L IS T OF ILLUSTRATIONS

F i g u r e

1.

2 .

3.

4.

5.

6 .

7-

8 .

9.

10.

11.

12.

13.

14.

15.

1 6 .

17.

18.

Page

I n t e r r e l a t i o n s h i p o f C r y s t a l Growth Rate, N u c le a t i o n R a te ,and CSD . . . . . . . . ........................ . . . . . . . . . . . 2

S te a d y - S ta te Compar ison o f MSMPR and FDS C r y s t a l - S i z eD i s t r i b u t i o n . . . ........................................................ 9

Schemat ic o f C r y s t a l l i z e r w i t h Fines D e s t r u c t i o n System . . 19

Schemat ic o f F ines Trap Device . . . . . . . . . . . . . . . 22

V a r i a t i o n o f S o l i d s C o n c e n t r a t i o n w i t h Time .......................... 31

C r y s t a l - S i z e D i s t r i b u t i o n , MSMPR Run 62176 ...................................... 34

C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 70176 . . . . . . . . . . . 35

C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 80476 . . . . . . . . . . 36

C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 31677 Using F r a c t i o n a l .F ines Trap Area . . . . . . . . . . . . . . . . . . . . . 37


C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 111176 . ............................. 39


C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 20377 . . . • ......... 41

C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run .70676 . . . . . . . . . . 42

Compar ison o f C o u l t e r Counter Data Obta ined f rom D i f f e r e n tA p e r t u r e s , Run 70176 , ................... . . . . . . . . . . . . 45

C r y s t a l -S i ze Di s t r i b u t i o n , Run 70676 . .. . . . . . . . . . . . 46

Fines D i s t r i b u t i o n in C r y s t a l l i z e r P roduc t and Compar ison o f Computer S im u la t i o n and Expe r im en ta l R e s u l t s f o r Run 71576 . . . . . . . . . . . . . . . . . . ................... 48

F ines D i s t r i b u t i o n in C r y s t a l l i z e r P roduc t f rom S ieve andC o u l t e r Counter Techn iques ....................... 49

v i i

v i i i

L IST OF ILLUSTRATIONS — Cont inued .

F ig u re Page

19- N u c le a t ion -G row th Rate K i n e t i c s C o r r e l a t i o n f o r the KCTS y s t e m ....................... 51

20. T h e o r e t i c a l and Expe r im en ta l C o r r e l a t i o n s between th eC r i t i c a l S iz e , Lp, and the S e t t l i n g V e l o c i t y I n s i d ethe Fines Trap , . . . . . . . . . . . .• . . . . . . . 53

21. Design C o r r e l a t i o n s f o r S iz e Improvement w i t h FDS . . . . . 57

22. F r a c t i o n o f F ines D is s o lv e d versus. E x p o n e n t ia l DecayR a t i o , X . . . . . . . . . . . . . . . . . . . . . . . . 59

23. D e c is io n Flow Diagram f o r S ize Improvement A n a l y s i s . . . . 61

LIST OF TABLES

Tab le Page

1. Summary o f MSMPR and FDS E x pe r im en ta l R esu l ts ........................ J2

2. E xp e r im e n ta l and P r e d i c t e d S ize Improvement . . . . . . . . . 63

3- Cost E s t im a t i o n f o r F ines D e s t r u c t i o n by H e a t ing . . . . . . 67

4. Water Requirement f o r F ines D e s t r u c t i o n by D i l u t i o n . . . . . 69

5. F ines D e s t r u c t i o n by D i l u t i o n v e rsu s H ea t ing Techn ique . . . 71

ABSTRACT

A n u c l e a t i o n - g r o w th r a t e k i n e t i c s model f o r p o tass iu m c h l o r i d e in

t h e KCT-NaCi-hÔ system was e x p e r i m e n t a l l y d e te rm ined in a mixed suspen

s io n mixed p ro d u c t removal (MSMPR) c r y s t a l 1?z e r , equ ipped w i t h a f i n e s

d e s t r u c t i o n system (FDS) . Data o b t a in e d f rom both s t e a d y - s t a t e and

dynamic runs w e re ,u s e d . A c r y s t a l - s i z e d i s t r i b u t i o n (CSD) model was used

In the t r a d i t i o n o f Randolph and Larson and proved t o be adequate t o p r e

d i c t observed d a t a . P a r t i c l e s le s s than 150 pm in s i z e were removed f rom

t h e system u s ing a f i n e s t r a p l o c a te d i n s i d e t h e c r y s t a l l i z e r . D i f f e r e n t

c a l c u l a t e d t e r m in a l v e l o c i t i e s in t h e t r a p were c o r r e l a t e d w i t h the

l a r g e s t s i z e o f p a r t i c l e s b e in g d i s s o l ved as c a l c u l a t e d f rom semi lo g

p o p u l a t i o n d e n s i t y p l o t s . The a p p r o p r i a t e mass and p o p u l a t i o n b a la n c e s ,

t o g e t h e r w i t h the n u c l e a t ion k i ne t i cs o f the system, were combined t o

o b t a i n d es ign e q u a t i o n s and c o r r e l a t i o n s wh ich a c c u r a t e l y d e s c r i b e the

b e h a v io r o f the f i n e s d e s t r u c t i o n sys tem. The " p o i n t " f i n e s t r a p

a n a l y s i s ( i . e . , n e g l i g i b l e f i n e s mass) was extended t o t he genera l case

o f d i s s o l v i n g l a r g e r p a r t i c l e s and r e s u l t s were d is c u s s e d . C a l c u l a t i o n s

were p re s e n te d wh ich a l l o w e s t i m a t i o n o f t he in c re m e n ta l o p e r a t i n g c o s t

o f a c r y s t a l l i z e r t o produce c r y s t a l s o f an in c rease d s i z e u s in g an FDS.

IN T R OD UC T ION

Many o p e r a t i n g problems in i n d u s t r i a l c r y s t a l l i z a t i o n a re r e l a t e d

t o c r y s t a l - s i z e d i s t r i b u t i o n (CSD). For example , c r y s t a l h a b i t , p u r i t y ,

s a l t i n g ( f o u l i n g ) , c a p a c i t y , s c a l e - u p , and c r y s t a l 1 i z e r s t a b i 1 i t y a re

s t r o n g l y i n f l u e n c e d by t h e s i z e d i s t r i b u t i o n o f t he p r o d u c t be ing

o b t a i n e d . The main mechanism o f i n t e r a c t i o n between the se problems and

CSD is th ro u g h th e l e v e l o f d r i v i n g f o r c e s in t h e sys tem, I . e . , s u p e r -

s a t u r a t i o n . F i g . 1 (Randolph and La rson , 1971) shows how s u p e r s a t u r a t i o n

de te rm ine s t h e CSD and in t u r n is i n f l u e n c e d by th e CSD th r o u g h p rocess

feedback in t he system. T h us , r e f e r r i n g t o F i g . 1, t h e l e v e l o f supei—

s a t u r a t i o n is d e te rm in e d by the p r o d u c t i o n r a t e and t h e t o t a l c r y s t a l

s u r f a c e a rea a v a i l a b l e f o r d e p o s i t i o n . S u p e r s a t u r a t i o n , in t u r n , d e t e r

mines the g ro w th r a t e ( t h e r a t e a t wh ich t h e c r y s t a l s grow in t h e i r

l i n e a r d im ens ion ) and. t he n u c l e a t i o n r a t e ( th e r a t e a t wh ich new c r y s t a l s

a t a s i z e c l o s e t o ze ro a re f o r m e d ) . C r y s t a l g rowth r a t e i s o f t e n a

l i n e a r f u n c t i o n o f s u p e r s a t u r a t i o n , G( s ) , w h i l e n u c l e a t i o n r a t e t y p i c a l l y

has a powei— law dependency on S u p e r s a t u r a t i o n , B ° (s ) . Growth and

n u c l e a t i o n r a t e s , th rou g h th e p o p u l a t i o n b a la n c e , d e te rm in e the dynamic

CSD a t any g iv e n t im e . Since the t o t a l s u r f a c e a rea i s a f u n c t i o n o f t h e

d i s t r i b u t i o n , the loop i s c lo s e d . CSD then becomes a f u n c t i o n o f

n u c l e a t i o n - g r o w t h r a t e k i n e t i c s . For many c r y s t a l s y s te m s , r e l a t i v e

n u c l e a t i o n - g r o w th k i n e t i c s a re such t h a t , under o r d i n a r y p rocess c o n d i

t i o n s in an MSMPR c r y s t a l 1 i z e r , the p ro d u c t s i z e is l e s s than the

Rate o f ProductProduction CSD

Nucleation B° = B°( s )

Growth Rate

G = G(s)

Supersaturation s = s ( I /A t )

Crystal SurfaceArea.

At = f (n )

PopulationBalance

F ig . 1. I n t e r r e l a t i o n s h i p o f C r y s ta l Growth Rate, N u c le a t i o n Rate, and CSD.

d e s i r e d average s i z e . To i n c re a s e p a r t i c l e s i z e , d i f f e r e n t o p e r a t i n g

c o n d i t i o n s and s p e c i a l d es ign f e a t u r e s can be used . I n c r e a s in g the

s u p e r s a t u r a t i o n u s u a l l y in c rease s n u c l e a t i o n more than g rowth and , t h u s ,

decreases p a r t i c l e s i z e . Changes in t he mean r e t e n t i o n t im e have a .

l im i t e d e f f e c t on p a r t i c l e s i z e (Rando lph, 1965) , as shown by

E qua t ion ( 1 ) : - '

Ld = KTr - T / I+ 3 _ ; ( i )

where K i s a c o n s ta n t depending on feed c o n c e n t r a t i o n . F u r t h e r ,

i n c r e a s i n g r e t e n t i o n would in c re a s e c a p i t a l in ve s tm en t a n d / o r decrease

p r o d u c t i o n . A more p r a c t i c a l way t o improve t h e average p a r t i c l e s i z e ,

w i d e l y accep ted in i n d u s t r i a l c r y s t a l l i z a t i o n , -is t h e co n t in u o u s removal

o f the s m a l l e r c r y s t a l s o r f i n e s f rom the c r y s t a l magma, l e a v i n g the

l a r g e r c r y s t a l s t o grow t o l a r g e r average s i z e (Saeman, 1956; Larson and

Rando lph, 1969)• C l a s s i f i c a t i o n on the sma l l end o f t h e c r y s t a l spec

t rum is ach ieved w i t h a f i n e s t r a p d e v i c e t h a t can be l o c a te d o u t s i d e o r

i n s i d e the c r y s t a l l i z e r .

The upward v e l o c i t y th rou gh the t r a p 1s s 1ow and o n l y f i n e

p a r t i c l e s reach th e t o p . L a rge r p a r t i c l e s whose t e r m i n a l s e t t l i n g

v e l o c i t y exceeds the t r a p v e l o c i t y f a l l t o the bot tom and a re re tu r n e d t o

t he c i r c u l a t i n g magma. The s t ream o f f i n e p a r t i c l e s i s drawn o f f t o a

h e a t e r o r u n s a tu r a te d re g io n where the y are d i s s o l v e d and th e c r y s t a l -

f r e e s o l u t i o n is r e c y c le d t o the c r y s t a l 1 i z e r . S?nee Saeman1s o r i g i n a l

work in 1956 th e re has been g row ing i n t e r e s t in the s tu d y and m o d e l l i n g

o f the e f f e c t on CSD o f f i n e s d i s s o l v i n g systems and in f i n e s t r a p

• ' ■ • .. 4

des ign ( Larson and Randolph, 1969; L e i , S h in n a r , and Ka tz , 1971;

Rando lph, Beer, and Keener, 1973; Juzaszek and Larson , 1977) .

Saeman (1961) d iscu ssed the o p e r a t i n g c h a r a c t e r i s t i c s o f a n u c l e i

d i s s o l v i n g sys tem. C a ld w e l l (1961 ) , d i s c u s s i n g the d r a f t - t u b e - b a f f l e

c r y s t a l 1 i z e r o p e r a t i o n , emphasized th e s t r o n g i n f l u e n c e t h a t removal of.

excess f i n e s has on c r y s t a l - s i z e d i s t r i b u t i o n . Larson and Randolph

(1969) p re s e n te d a gen e ra l p o p u l a t i o n ba lance f o r p a r t i c l e s in an

a r b i t r a r y su spe n s io n . W i th p ro p e r assumpt ions, , these e q u a t i o n s may

d e s c r i b e the t r a n s i e n t response t o changes in n u c l e i d i s s o l v i n g r a t e .

S ince the n , s e ve ra l s t u d i e s o f c r y s t a l 1 i z e r s w i t h f i n e s d e s t r u c t i o n

systems have appeared in th e l i t e r a t u r e .

Nauman and Szabo (1971) c h a r a c t e r i z e d the s t e a d y - s t a t e b e h a v io r

o f c o n t in u o u s r e c y c l e c r y s t a l 1 i z e r s w i t h n o n - s e l e c t i v e f i n e s t r a p s and ,

Nauman (1971) a na lyze d th e s e l e c t i v e f i n e s t r a p s and the d ra m a t i c im prove

ment in c r y s t a l 1 i z e r pe r fo rm ance t h a t t h e y can cause. FInes t r a p s a l l o w

o p e r a t i o n a t h i g h e r s u p e r s a t u r a t i o n 1e v e l s than m igh t o th e r w i s e be

p o s s i b l e . 11 is usual 1y assumed t h a t th e t r a p behaves as a p o i n t f i n e s

t r a p , i . e . , f i n e s removal a t a s i z e n e g l i g i b l y smal l compared t o p r o d u c t -

s i z e c r y s t a l s .

Randolph and Larson (1971) r i g o r o u s 1y ana lyzed the e f f e c t o f

f i n e s d e s t r u c t i o n where the f i n e s are n o t n e g l i g i b l y sm a l l compa red t o

p r o d u c t - s i z e c r y s t a l s . F ines t r a p s no t o n l y a l l o w a d ju s tm e n t o f p a r t i c l e

s i z e t o a s i z e wh ich has a s p e c ia l i n t e r e s t f o r the i n d u s t r y pe r s e , b u t

the y a l s o have become an im p o r t a n t t o o l i n t h e c o n t r o l o f i n d u s t r i a l

V - ; 5

c r y s t a l 1 i z e r s s in c e p ro p e r m a n i p u l a t i o n o f t h e f i n e s d e s t r u c t i o n a l l o w s

s t a b i l i z i n g o f the o p e r a t i o n o f the c y r s t a l 1 i z e r and p r e v e n t i o n o f

c y c l i ng.

The im por tance o f CSD in any c r y s t a l l i z a t i o n p rocess due to i t s

i n t e r a c t i o n w i t h some o f t h e main p roblems u s u a l l y e nco un te re d in the

o p e r a t i o n o f i n d u s t r i a l c r y s t a l 1?z e rs has been emphas ized. I t was shown

how CSD is d e te rm ine d by t h e r e l a t i v e n u c l e a t i o n - g r o w th k i n e t i c s o f a

g iv e n sys tem and how th e o p e r a t i o n o f a co n t in u o u s c r y s t a l 1 i z e r equ ipped

w i t h a f i n e s d e s t r u c t i o n sys tem i n f l u e n c e s th e CSD be ing o b t a i n e d . I t i s

th e purpose o f t h i s s tud y to d e te rm in e a n u c 1e a t i o n - g r o w t h r a t e k i n e t i c s

model f o r the KC! sys tem, as w e l l as t o g i v e more i n s i g h t in t h e u nd e r

s ta n d in g o f the b e h a v io r o f f i n e s t r a p s . In p a r t i c u l a r , a d e s i g n - u s e f u l

model f o r t he p r e d i c t i o n o f c r y s t a l 1 i z e r per fo rm ance as a f u n c t i o n o f

f i n e s t r a p o p e r a t i o n w i l l be p re s e n te d . The e f f e c t o f t h e t e r m in a l

v e l o c i t y i n s i d e th e t r a p on the l a r g e s t s i z e o f p a r t i c l e s be ing d i s s o l v e d

w i l l be d e te rm in e d . The e f f e c t s o f a " p o i n t " f i n e s t r a p as w e l l as t h a t

o f a f i n e s t r a p where the mass o f f i n e s b e ing d e s t ro y e d i s no t n e g l i g i b l y

sma l l compared t o p r o d u c t - s i z e c r y s t a l s w i l l be d is c u s s e d . F i n a l l y , a

rough e s t i m a t i o n o f the o p e r a t i o n c o s t o f f i n e s d i s s o l v i n g w i l l be p r e

sen ted . A l t h o u g h the system used in t h i s s tu d y was KC1, t h e te c h n iq u e s

used to f i n d the k i n e t i c m ode l , and a l l t h e r e s u l t s c o n c e rn in g th e f i n e s

d e s t r u c t i o n loop shou ld be c o m p le te l y gene ra l and sh o u ld a p p ly t o any

o t h e r sys tem.

THEORY

Ki n e t i cs

C o n ven t io na l t h e o r e t i c a l c o n s i d e r a t i o n s a lo ne Have been demon

s t r a t e d t o be i n s u f f i c i e n t t o p r e d i c t CSD. However, i t has been shown

(Randolph and L a r s o n , 1.971) t h a t a p o p u l a t i o n ba lance o v e r a c r y s t a l l i z a

t i o n sys tem can r e l a t e t he c o n s t r a i n t s o f t h e system and the c r y s t a l l i z a

t i o n k i n e t i c s o f t he s o l i d be ing c r y s t a l l i z e d t o t he s i z e d i s t r i b u t i o n

o b t a i n e d . The concep t o f t h e p o p u l a t i o n ba lance and I t s u s e fu ln e s s in

the a n a l y s i s o f p a r t i c u l a t e systems i s p re s e n te d e x t e n s i v e l y by Randolph

and Larson (1 971 ) . Systems wh ich a re s u b s t a n t i a l l y w e l l mixed in th e

m a jo r p o r t i o n o f t h e i r c r y s t a l l i z a t i o n volume w i l l have a c r y s t a l s i z e

d i s t r i b u t i o n independent o f s p a t i a l l o c a t i o n in t h e c r y s t a l 1 i z e r . For

such a sys tem, t he b a s i c p o p u l a t i o n ba lance i s :

^ ; i ^ + n d ^ . B . D , 2)'

' . ' k

Th is e q u a t i o n is averaged in e x t e r n a l phase space and d i s t r i b u t e d in

i n t e r n a l phase space, n i s the p o p u l a t i o n d e n s i t y p e r u n i t volume o f

suspens ion a t s i z e L, G is the l i n e a r g rowth r a t e , V i s t h e suspens ion

volume, B and D r e p r e s e n t e m p i r i c a l b i r t h and death f u n c t i o n s a t any -

p o i n t in the phase space , and a re f l o w s go ing i n t o V ( n e g a t i v e ) o r o u t

o f V (pos i t i v e ) .

6

For a s i n g l e - s t a g e , mixed suspens ion mixed p r o d u c t removal c r y s

t a l l i z e r (MSMPR), E qua t ion (2) can be s i m p l i f i e d i f i t is assumed t h a t :

McCabe's AL law can be a p p l i e d t o t h e system. T h is r e q u i r e s G

no t t o be a f u n c t i o n o f L, i . e . , G = d L / d t t6 G ( L ) . Under most

i n d u s t r i a l c o n d i t i o n s , t h i s law may be assumed t o h o l d .

c r y s t a l l i z e r as the r a t i o o f suspens ion volume to v o l u m e t r i c f l o w r a te o f

the p ro d u c t Stream, r = V/Qp, the p o p u l a t i o n b a lan c e , Equ a t ion ( 2 ) ,

becomes:

The p o p u l a t i o n d e n s i t y , n , re p re s e n t s the number o f p a r t i c l e s

(AN) in a g iv e n s i z e range (AL) pe r u n i t vo lume:

The suspens ion volume i s h e ld c o n s t a n t i n t i m e , i . e . ,

d (1og V ) / d t = 0.

No a g g lo m e ra t i o n o r g ross c r y s t a l f r a c t u r e o c c u r s , i . e . .

B = D = 0.

A l l feeds t o t h e c r y s t a l l i z e r a re c r y s t a l - f r e e , i . e . , n. = 0

The p o p u l a t i o n d e n s i t y o f the d i s c h a r g e i s the same t h a t e x i s t s

in t he mixed s u spe n s io n , i . e . , n^ = n

Under these c o n d i t i o n s * and d e f i n i n g the re s id en c e t im e o f the

(3)

I f t h e c r y s t a l l i z e r o pe ra tes under s t e a d y - s t a t e c o n d i t i o n s , Equa

t i o n (3) can be d i r e c t l y i n t e g r a t e d t o o b t a i n the e x p o n e n t i a l

d i s t r i b u t i o n :

n = n° exp [~L/Gt ] (4)

In Equa t ion ( 4 ) , n° is c a l l e d the n u c l e i d e n s i t y and can be expressed as

L=0

n o(5)

E qua t ion (4) p l o t s as a s t r a i g h t l i n e on s e m i - l o g a r i t h m i c graph paper .

in F i g . 2. The s lo p e o f t h e l i n e i s equal t o - 1 / Gt . Then, f o r a g ive n

o f the c r y s t a l 1 i z e r p r o d u c t .

The f o r m a t i o n o f new c r y s t a l s can r e s u l t f rom d i f f e r e n t mecha

n ism s : . homogeneous n u c l e a t i o n , he te rogeneous n u c l e a t I o n , secondary

n u c l e a t i o n , and a t t r i t i o n . Homogeneous n u c l e a t ion is t h e f o r m a t i o n o f

new c r y s t a l s f ro m the l i q u i d phase as a r e s u l t o f s u p e r s a t u r a t i o n o n l y .

In he te rogeneous n u c l e a t i o n , c r y s t a l s a re formed due t o the presence o f

where n° r e p re s e n ts t he i n t e r c e p t o f t h i s l i n e a t z e ro s i z e , as shown

s t e a d y - s t a t e e x p e r i m e n t , CSD a n a l y s i s and t h e known v a lu e o f r a l l o w s t h e

d e t e r m i n a t i o h o f th e n u c l e i d e n s i t y , n , and t h e g rowth r a t e , G.

In a d d i t i o n t o t h e c o n s e r v a t i o n e q u a t i o n ( 4 ) , s u i t a b l e k i n e t i c

e q u a t i o n s a re needed. The n u c l e a t ion r a t e , B°, t h e r a t e a t wh ich new

c r y s t a l s a t s i z e near t o ze ro a re fo rmed, can be re p re s e n te d by

6° = d N / d t , when L -> 0 , and can be r e w r i t t e n in the fo rm :

(6 )L~0

T h us , the n u c l e a t ion r a t e can be e a s i l y de te rm ined f rom the CSD a n a l y s i s

PO

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LATI

ON

D

EN

SIT

Y,

n,

num

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mic

ron

9

n°2 o e-L/G|T, o < L < 00

o -L R /G .T - ■ ■V n2 e ' un /u2 1 -,0< L< L F

n2= A n2 e "L / G 2 ^ ; L p < L < 0 0 J9 = e -'h

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£n°MSMPR

FDS

LpCRYSTAL SIZE, L, MICRONS

F ig . 2. S te a d y -S ta te Compar ison o f MSMPR and FDS C r y s t a l - S i z e D i s t r i b u t i o n .

f o r e i g n p a r t i c l e s in t h e suspen s io n . Secondary nucTeat ion is a k i n d o f

he te rogeneous n u cT e a t ion where the f o r m a t i o n o f new c r y s t a l s i s induced

. by th e presence o f suspended c r y s t a l s in t h e s o l u t i o n . A t t r i t i o n r e f e r s

t o mechan ica l d e g r a d a t i o n o f suspended c r y s t a l s .

A number o f d i f f e r e n t n u c l e a t i o n models have been proposed s i n c e

the b e g in n in g o f t h e c e n t u r y . Vo 1 mer and Weber (1926) proposed an

A r r h e n i o u s - t y p e e x p r e s s io n f o r homogeneous n u c l e a t iO n :

B° = c exp ( -AG° /kT ) (7)

where c i s a p r o p o r t i o n a l i t y c o n s t a n t , AG° i s the f r e e energy o f f o rm a t

t i o n o f a n u c l e u s , k i s t h e B o l tz m an 's c o n s t a n t , and T i s t h e a b s o lu te

t e m p e ra tu r e . T h is e q u a t i o n can a l s o be expressed in terms o f the

geom et ry , s u r f a c e t e n s i o n , cr, f r e e e n e rg y , and s u p e r s a t u r a t i o n r a t i o , S:

B° = c exp ( -16 NmM^c^/3r^T^p^ log ^ S) (8)

The d i f f i c u l t y o f t h i s e x p r e s s io n is t h a t i t p r e d i c t s n u c l e a t i o n o n l y a t

e x t r e m e ly h ig h s u p e r s a t u r a t i o n , a phenomenon no t Observed in most

i n o r g a n i c c r y s t a l l i z a t i o n sys tems. A t p r e s e n t , i t i s accep ted t h a t t h e

p redom in an t n u c l e a t i o n mechanism in i n d u s t r i a l c r y s t a l l i z e r s is secondary

o r heterogeneous n u c l e a t i o n . A n o th e r n u c l e a t i o n model wh ich takes i n t o

accoun t he te rogeneous e f f e c t s was proposed by T u r n b a l l and F i s h e r ( 1 9 6 5 ) :

B° = B exp [ - ( l / k T ) l 6 7 r c 3v /3 k T l o g 2 S)b] (9)n

A g a in , t h i s model p r e d i c t s c r i t i c a l , s u p e r s a t u r a t i o n dependence, and o f t e n

f a i l s in p r e d i c t i n g e x p e r im e n ta l b e h a v i o r . Miens ' n u c l e a t i o n model is

. . . ; - ' ■ . V.

based on the concept o f a m e ta s ta b le , b u t s u p e r s a t u r a t e d , re g io n w i t h i n

wh ich n u c i e a t i o n does n o t o c c u r :

B" = k(C - C J 1 ; Cm > Cs ■ (10)

where C is the m e ta s ta b le t h r e s h o l d o f n u c l e a t i o n , C i s th e s o l u t e con-m

c e n t r a t io n , and C i s th e s a t u r a t e d so l u te c o n c e n t r a t i o n , k can be a

f u n c t i o n o f t e m p e r a t u r e , b u t i i s n o t . Randolph and. Larson (1971) have

c a r r i e d o u t e x p e r im e n ta l work t a k i n g C equal t o C w i t h c o n s i d e r a b l em s

success . In t h a t case, the model becomes the s im p le p o w e r - 1 aw f u n c t i o n :

B° = k(C - Cg) = ks (11)

N e v e r t h e le s s , t h i s s im p le powei— law model does no t c o n s i d e r secondary and

he te rogeneous e f f e c t s wh ich can be i d e n t i f i e d as sources o f n u c l e i

(C lo n t z and McCabe, 1971) . To be adequate as a gene ra l r e p r e s e n t a t i o n o f

n u c l e a t i o n r a t e , the model must a l s o c o n t a i n a dependence on such

phenomena. The n u c l e a t i o n p ow er - la w m o d e l :

B° = k s ' M j (12)

i s w i d e l y accep ted f o r i t s p r a c t i c a l i t y and u t i l i t y in d e s c r i b i n g

secondary n u c l e a t i o n . In Equa t ion ( 1 2 ) , My r e p re s e n ts t h e t o t a l mass o f

c r y s t a l s p e r u n i t volume o f s l u r r y . I f t he g rowth r a t e , G, i s assumed t o

be a l i n e a r f u n c t i o n o f the s u p e r s a t u r a t i o n :

G = kgtC - Cs ) (13)

12

then the p o w e r -1 aw model can be expressed in the more common form:

b ° = (14)

The r a t e c o n s t a n t , k ^ , i s l i k e l y t o depend on t e m p e r a tu r e , degree o f

a g i t a t i o n , and the presence o f i m p u r i t i e s .

Equa t ion (14) has been found t o be a s u i t a b l e c o r r e l a t i o n in many

systems (Randolph and C is e , 1972; Randolph and Youngqui s t , 1972; Larson ,

Timm, and W o l f f , 1968; Juzaszek and L a rs on , 1977).

S ize Improvement

The u s e fu ln e s s o f the. f i n e s d e s t r u c t i o n sys tem in im p rov ing th e

average s i z e o f the c r y s t a l l i z e d p r o d u c t i s l a r g e l y re c o g n iz e d . The

e x a c t b e h a v io r o f t h e f i n e s t r a p can be o b ta in e d by. s o l v i n g the appro

p r i a t e mass arid p o p u l a t i o n b a la n c e s , t o g e t h e r w i t h the n u c l e a t ion

k i n e t i c s o f th e sys tem. The f o l l o w i n g development o f des ign e q u a t io n s

f o r s i z e improvement i s based on t h e t h e o r y p re sen ted by Randolph and

Larson (1 9 71 ) .

By d e f i n i t i o n , ndL re p re s e n t s t h e number o f p a r t i c l e s per u n i t

volume o f s l u r r y w i t h s i z e s between L and L+dL, I f t h e w e ig h t o f each

c r y s t a l can. be r e l a t e d t o t h e cube o f i t s s i z e in t he fo rm :

mp = Pky1-3 0 5 )

where p i s th e c r y s t a l d e n s i t y and k i s a v o l u m e t r i c shape f a c t o r

r e l a t i n g p a r t i c l e volume f o s i z e cubed, then th e mass o f p a r t i c l e s w i t h

s i z e s in (L , L+dL) i s g ive n by :

dM = pk L^ndL v

13

( 16)

" 3 ' - -M = pk / L ndL ' (17)

0

r e p r e s e n t s t h e t o t a l mass o f c r y s t a l s p e r u n i t volume o f s l u r r y . The

shape f a c t o r i s . in de pen den t o f s i z e f o r g e o m e t r i c a l l y s i m i l a r p a r t i c l e s

and can be taken o u t o f t he i n t e g r a t i o n . For the e x p o n e n t i a l d i s t r i b u

t i o n g ive n by E qua t ion ( 4 ) , t h e s o l i d s c o n c e n t r a t i o n e x p r e s s io n becomes:

ML = pk / L^n° exp ( - L / G t ) d L (18)T : V 0 ;

o r

4My = 6pkv n°(Gt ) (19)

In a Glass I I c r y s t a l l i z e r ( lo w s u p e r s a t u r a t i o n , h ig h y i e l d ) w i t h

a p o i n t f i n e s t r a p , bo th s o l i d s c o n c e n t r a t i o n and r e t e n t i o n t im e remain

i n v a r i a n t when f i n e s d e s t r u c t i o n is imp lemented. I f s u b s c r i p t s 2 and 1

r e f e r t o th e cases w i t h and w i t h o u t f i n e s d e s t r u c t i o n , r e s p e c t i v e l y ,

the n : . ■

Mt = Mt (20 )2 1

o r

In th e r i g h t - h a n d s i d e o f Equa t ion (21) i t has been assumed t h a t t he mass

o f t h e f i n e s d e s t ro y e d i s n e g l i g i b l e compared t o t h e p r o d u c t , i . e . , a

" p o i n t " f i n e s t r a p . For t h e e x p o n e n t i a l d i s t r i b u t i o n , n(l_) i s g ive n by

Equa t ion ( 4 ) . When t h e c r y s t a l l i z e r i s equ ipped w i t h a f i n e s t r a p , two . - ■ - , -

d i f f e r e n t CSD's a re o b ta in e d as shown in F ig . 2 , one c o r r e s p o n d in g to the

f i n e s s t ream and the o t h e r t o t h e p r o d u c t s t re am . The s lo p e o f t h e f i n e s

CSD has t h e v a lu e o f -R/Gt , where R = 1 + Q^/Qp, Qp b e i n g th e f i n e s

removal r a t e . 6 re p r e s e n t s the f r a c t i o n o f c r y s t a l s n o t removed by th e

f i n e s t r a p , i . e . , t he f r a c t i o n s u r v i v i n g as p r o d u c t . Comb?n a t io n o f

Equa t ions (6) and (14) g i v e s :

n° = k NGM MTj (22)

B r i n g i n g (19) and (22) t o (21) y i e l d s an e x p r e s s io n f o r s i z e improvement

w i t h f i n e s d e s t r u c t i o n :

where L , i s

T / i + 3(23)

the dominant c r y s t a l s i z e (w e ig h t b as i s ) d e f i n e d as :

f o r an e x p o n e n t i a l d i s t r i b u t i o n . E qua t ion (23) i n d i c a t e s t h a t t h e e f f e c

t i v e n e s s o f f i n e s d e s t r u c t i o n in i n c r e a s i n g th e p a r t i c l e s i z e decreases

w i t h systems hav ing a h ig h s e n s i t i v i t y o f nuc i e a t ion t o g rowth r a t e ,

i n d i c a t e d by l a r g e v a lu e s o f th e pa ram ete r i . E qua t ion (23) can a l s o be

expressed a s :

I d - , - 1r ^ = = x / i + 3 (24)

d , ■

X i s c a l l e d the e x p o n e n t i a l decay r a t i o and i s g iv e n by

Lf (R - 1)A = ------------— - = - In 8 (25)

2

where Lp is t h e l a r g e s t s i z e o f p a r t i c l e s b e ing d e s t r o y e d , d e te rm in e d by

t h e i n t e r s e c t i o n o f t h e two d i f f e r e n t s i z e d i s t r i b u t i o n s , as shown in

F ig . 2. I t must be emphasized t h a t E qua t ion (23) ho lds f o r a p o i n t f i n e s

t r a p . A more r i g o r o u s e x p r e s s io n f o r s i z e improvement f o r th e case where

t h e mass o f f i n e s d e s t ro y e d is n o t n e g l i g i b l e can be o b t a i n e d by

e x te n d in g E qua t ion (21) t o :

|_. , F , ” n

/ n . I Z d L - / n_L dL + / n_L d l (26)0 1 . 0 2 LF 2

where , i n the case o f an e x p o n e n t i a l d i s t r i b u t i o n :

16

n 1 = n°^ exp ( - L / G ^ t )

n2 ~ " ° 2 exp ( “ l r / G2T^

r>2 = 6 n °2 exp ( - L / G - t )

0 < L < “

0 < L < L r

LF < L ‘<

( 27 )

M athem at ic a l m a n i p u l a t i o n o f E qua t ions ( 1 9 ) , ( 2 6 ) , and (27) leads t o a

more gene ra l e x p r e s s io n f o r s i z e improvement in t he fo rm :

14 W R = 1 L + e - > 1 7

R .

(0( X / R - l ) ]

l / i + 3

( 28)

where <w is the d? men si On less we igh t f r a c t i o n which can be expressed by an

incomplete gamma f u n c t i o n :

u) - y- / e P p^ dp • 0

(29)

S in ce Ly - 36%, c o m b in a t io n o f E qua t ions (23) and (25) a l l o w s e x p r e s

s io n o f the mass ba lan ce c o n s t r a i n t as:

G2 = G exp. X i+3

(30)

So, t o s a t i s f y the mass b a la n c e , t he decay r a t i o must s a t i s f y th e

e q u a t i o n :

lfV ■X exp ( X / i + 3 ) = G y - ■ = K ( 3 0

17

In o r d e r t o s o l v e Equa t ion ( 3 1 ) , a v a lu e f o r Lp must be e s t i m a te d f rom a

Stokes l a w - t y p e c o r r e l a t i o n o r o b ta in e d f rom e x p e r im e n ta l da ta ( p i l o t

p l a n t c r y s t a l l i z e r w i t h f i n e s d e s t r u c t i o n ) . S imple MSMPR e xpe r im en ts

p r o v id e t h e v a lu e o f and a good e s t i m a t i o n o f t h e pa ram ete r 1. When

th e v a lu e o f X i s found f rom E qua t ion ( 3 1 ) , t h e v a lu e o f / L ^ can be2 1

c a l c u l a t e d d i r e c t l y .

The t o t a l p r o d u c t i o n r a t e is g i v e n as :

P = Qppk / n l ^ d l (32)P V 0

The amount o f f i n e s d e s t ro y e d i s :

F .

PF = QpPk / nL dL (33)F R V 0

and the f r a c t i o n o f n e t p r o d u c t i o n wh ich is d i s s o l v e d and re c y c le d can be

c a l c u l a t e d f ro m :

0) ( X / R - l )a) ( X R / R - l )

D e t a i l s o f t h e deve lopment o f E qu a t ions (28) and (34) a r e p re sen ted in

the Append ix .

EXPERIMENTAL EQUIPMENT

The po tass ium c h l o r i d e c r y s t a l T i z e r used in t h i s e x p e r im e n ta l

s tu d y was composed o f the f o l l o w i n g e le m e n ts :

1. Feed t a n k .

2. F e e d - l i n e h e a t e r >

3* C r y s t a l I r z e r and f i n e s t r a p .

4. F ines d e s t r u c t i o n sys tem.

5* C r y s t a l l i z e r c o o l i n g system.

6 . A n a l y s i s equ ipm ent .

A schem at ic d ia g ram o f t h e system i s shown in F ig u re 3• The equ ipment

used in t h i s s tu d y is p a r t o f a more complex system used by James R.

Beckman in a p r e v io u s work and th e re a de r is r e f e r r e d t o Beckman (1976)

f o r a more d e t a i l e d d e s c r i p t i o n .

Feed Tank

The 2 0 0 - l i t e r t a n k was made from: epoxy f i b e r g l a s s and d i v i d e d

i n t o two co m par tm en ts , bo th b e ing w e l l - m i x e d by mar ine im p e l l e r s con

nec ted t o a common c e n t r a l s h a f t d r i v e n a t 85 rpm by a 1/15 HP c o n s t a n t -

speed Dayton g e a m o t o r . T h is he lped t o assu re a c o n s t a n t s a l t c o n c e n t r a

t i o n in t h e l i q u i d be ing fed t o t h e c r y s t a l l i z e r . Each compartment had

abou t a h a l f o f the t o t a l c a p a c i t y and bo th were connec ted by a d ou ghn u t

shaped b a f f l e . The low er compartment r e c e iv e d the p r o d u c t s o l i d s and t h e

o v e r f l o w st reams f rom th e c r y s t a l 1 i z e r . In o r d e r t o assu re l i q u i d con

t a i n e d in t h e ta n k was s a t u r a t e d w i t h s a l t , an excess o f p o tass ium

Feed.Q

ReturnSample

Level ControlTIC Fines

trapCWSteam

FromFeedTank

M Sample

ProductCrystallizer

Cond.

Heater Hold Cooler

Fines Destruction System

Crysta l l izerCoolingSystem

To Feed TankSystem

Fig . 3. Schemat ic o f C r y s t a l l i z e r w i t h Fines D e s t r u c t i o n System.

VO

2 0

c h l o r i d e c r y s t a l s was m a in ta in e d in t h i s compar tment . A c o n s ta n t s a l t

s a t u r a t i o n in the l i q u o r was assured by keep ing -a c o n s t a n t t e m p e ra tu r e in

t h e t a n k . T h is was done by a te m p e ra tu r e c o n t r o l l e r , a m o d i f i e d Dayton

t h e r m o s t a t wh ich r e g u la t e d t h e s u p p ly o f 150 p s ig steam t o the steam

h e a t in g c o i l s , a l s o l o c a te d in t he low er com par tm en t . The feed tank

t e m p e r a tu r e , u s u a l l y a t 70°C, was a l lo w e d t o change 1°C f rom t h e se t

p o i n t .

S a tu ra te d l i q u i d f rom th e lower compartment f l o w e d th rough th e

c e n t r a l h o le in th e b a f f l e i n t o t h e upper compartment f rom where i t was

fed t o the c r y s t a l l i z e r . A c o n t in u o u s i n - l i n e CUNO f i l t e r was i n s t a l l e d

in t h i s compar tment . The f i l t e r pump to o k abou t 5 l i t e r s p e r .m in u t e o f

c l e a r l i q u o r f rom th e upper compartment and d i s c h a rg e d the f i l t e r e d

l i q u o r t o t he lo w e r co m p ar tm en t . The f i l t e r was a b le t o r e t a i n f o r e i g n

p a r t i c l e s as s m a l1 as 5 pm. The f i l t r a t i o n sys tem, bes ides c l e a n in g t h e

feed s o l u t i o n o f i m p u r i t i e s , he lped t o reduce the t ime, r e q u i r e d t o reach

s t e a d y - s t a t e c o n c e n t r a t i o n in th e feed ta n k l i q u o r . T h is was an

im p o r t a n t a i d t o system s t a r t - u p .

Feed -L in e Hea te r

S in ce i t was necessa ry t o assu re t h a t the l i q u i d be ing pumped t o

the c r y s t a l l i z e r was p a r t i c l e - f r e e , the s o l u t i o n coming o u t o f the feed

tank f l o w e d th roug h 6 f e e t o f 3 / 8 - inch s t a i n l e s s s t e e l t u b i n g , wrapped

w i t h a h e a t i n g ta p e . The te m p e ra tu r e in t h i s l i n e was m a in ta in e d between

70 and 75°C.

. . . . 2 '

C r y s t a l l i z e r and Fines Trap

The epoxy f i b e r g l a s s c r y s t a l 1 i z e r had a c a p a c i t y o f 20 l i t e r s .

I n s i d e th e c r y s t a l l i z e r , t h e r e were two c o n c e n t r i c , t i g h t - w o u n d c o o l i n g

c o i l s . The o u t e r c o i l was a 3 / 8 - inch s t a i n l e s s s t e e l t ube w i t h 8 - inch

d ia m e te r w i n d i n g , wound around a d r a f t tube and f i x e d by t h r e e v e r t i c a l

b a f f l e s . The i n n e r c o i l was a 1 / 2 - i n c h s t a i n l e s s s t e e l tube w i t h 4 - i n c h

d ia m e te r w i n d i n g , a l s o f i x e d in p o s i t i o n by t h r e e v e r t i c a l b a f f l e s

a t t a c h e d t o t he d r a f t t u b e . Two s e ts o f removable v e r t i c a l c o o l i n g tubes

o f 3 / 8 - i n c h s t a i n l e s s s t e e l t u b i n g were added in o r d e r t o m in im iz e

2f o u l i n g . The e f f e c t i v e t o t a l c o o l i n g area was abou t 0 .6 m . The s l u r r y

in the c r y s t a l l i z e r was mixed by a mar in e i m p e l l e r wh ich fo r c e d i t t o

move down i n s i d e th e d r a f t tube a n d . up the o u t e r annul us formed by the

o u t e r c o o l i n g c o i l and t h e c r y s t a l 1 i z e r w a l l . The s h a f t o f t h e i m p e l l e r

was d r i v e n ,a t 500 rpm. '

The c r y s t a l ! i z e r was equ ipped w i t h a f i n e s t r a p made o f p l e x i

g la s s w i t h a t o t a l area o f abou t 45 cm^. I t o ccup ie d 1 /3 o f t h e a n n u la r

a rea formed between the i n n e r and t h e o u t e r c o o l i n g c o i l s . The t r a p

a l l o w e d s e p a r a t i o n and removal o f c r y s t a l s up t o 150 m ic ron s in s i z e f rom

the c r y s t a l l i z e r . An o v e r f l o w d e v i c e was I n s t a l l e d a t t he top o f t h e

t r a p f o r l e v e l c o n t r o l . The f i n e s t r a p d e v i c e is shown in F ig . 4. The

2te m p e ra tu r e o f t h e c r y s t a l l i z e r was c o n t r o l l e d by an I R c o n t r o l l e r w h ich

a d j u s te d t h e c o o l i n g w a te r b lend t e m p e ra tu r e in o r d e r t o keep the

c r y s t a l l i z e r te m p e ra tu r e a t 40°C. .

TOP VIEW5 cm

A

sampling tubeoverflow

FRONT VIEW

14 cm

optional slabs to reduce area

g. 4. Schemat ic o f F ines Trap Dev ice .

Fines D e s t r u c t i o n System

S o l u t i o n coming o u t o f the c r y s t a l 1 i z e r th rou g h t h e f i n e s t r a p

and c a r r y i n g c r y s t a l s o f up t o 150 m ic rons in s i z e was hea ted t o about

70°C in a steam h e a t e r . Hot l i q u o r l e a v in g the h e a t e r was s t o r e d in a

ho ld tan k f o r abou t h a l f a m in u te t o a l l o w t im e f o r the f i n e s t o d i s

s o l v e . A f t e r t h i s t im e , t h e l i q u o r was coo led as much as p o s s i b l e w i t h

o u t h av ing n u c l e i f o r m a t i o n b e f o r e be ing r e tu r n e d t o th e c r y s t a l 1 i z e r .

T h is was necessary in o r d e r t o keep te m p e ra tu r e changes t o a minimum and

reduce f o u l i n g in t h e c o o l i n g c o i l s . In an i n d u s t r i a l s y s te m , f i n e s

wou ld n o r m a l l y be d e s t ro y e d by d i l u t i o n , r a t h e r than h e a t i n g . However,

no d i l u t i o n c o u ld be used in t h e p re s e n t s tu d y due t o th e t o t a l r e c y c l e

o f p r o d u c t t o t h e feed t a n k .

C r y s t a l 1 i z e r C o o l jn g System

The c o o l i n g system c o n s i s t e d o f t h r e e Eas te rn c e n t r i f u g a l pumps,

a b lend t a n k , and the c r y s t a l 1 i z e r c o o l i n g c o i l s . The warm w a te r c i r c u

l a t i n g w i t h i n the c o o l i n g c o i l s was b lended w i t h f r e s h coo l w a te r in th e

b lend t a n k . Use o f a tempered w a te r system r a t h e r than d i r e c t c o n t r o l o f

c o o l i n g w a te r f l o w r a t e ensured t h e lo w e s t p o s s i b l e AT d r i v i n g f o r c e

ac ross the c o o l i n g c o i l s and th e re b y m in im ize d f o u l i n g ,

• An a 1 y s i s Equipme n t -- ... - ...

Samples were taken f rom th e c r y s t a l l i z e r p r o d u c t s t ream and th e

f i n e s d e s t r u c t i o n loop s t re am . The c r y s t a l - s i z e d i s t r i b u t i o n o f the p r o

d uc t was o b t a i n e d by us ing an A11e n - B r a d le y Son ic S i f t e r Model L3P

equ ipped w i t h a 6 - t r a y s t a c k . The t r a y s i z e s ranged f rom 37 t o 2000 pm,

i n c r e a s i n g by a f a c t o r o f vT. The CSD o f t h e f i n e s s t ream was d e te rm ined

us ing a C o u l t e r Coun te r Model T equ ipped w i t h s e v e ra l probes w i t h d i f f e r

e n t a p e r t u r e s i z e s ra n g in g f rom 50 t o 400 pm.

EXPERIMENTAL PROCEDURE

Start-Up

Each e x pe r im en t was i n i t i a t e d by p r e p a r i n g t h e feed ta n k s o l u t i o n

f o r abou t 2 hours b e f o r e t h e s t a r t o f t h e ru n . The feed t a n k , f i l l e d

i n i t i a l l y w i t h 200 l i t e r s o f c o ld w a t e r , a p p r o x im a t e l y 40 k i l o g ra m s o f

s o l i d po ta ss iu m c h l o r i d e , and 48 k i l o g r a m s o f sodium c h l o r i d e , was warmed

up by t u r n i n g t h e steam on . The te m p e ra tu r e c o n t r o l l e r connec ted t o t h e

feed t a n k was a l s o t u r n e d on. The i m p e l l e r s were t u r n e d on a f t e r v e r i

f y i n g t h a t t h e v e r t i c a l s h a f t was f r e e o f c r y s t a l d e p o s i t i o n . Run

o p e r a t i n g te m p e ra tu r e o f between 68 and 70°C was a ch ieve d a f t e r abou t one

and o n e - h a l f h o u rs . Then, t h e CUNO f i l t e r connected t o t h e feed ta n k was -

s t a r t e d . S pe c ia l ca re was taken t o make su re t he feed t a n k c o n ta in e d

excess s o l i d - p h a s e s a l t on th e bo t tom . When the feed t a n k l i q u o r was a t

t h e r e q u i r e d t e m p e ra tu r e and c o n c e n t r a t i o n , t h e c r y s t a l l i z e r was f i l l e d .

F i r s t , i t was charged w i t h s a t u r a t e d l i q u o r r e s id u e saved f rom the end o f

t h e p r e v io u s run in a h o ld t a n k . T h is reduced i n i t i a l f o u l i n g . The

c r y s t a l l i z e r i m p e l l e r , t h e c o o l i n g w a te r sys tem, and t h e t e m p e ra tu r e con

t r o l l e r s were t u r n e d on a t t h i s t im e . The charge o f t he c r y s t a l l i z e r was

comple ted by pumping l i q u o r a t 70°C f rom t h e feed t a n k . Power was t u r n e d

on t o t h e f e e d - l i n e h e a t e r a f t e r t h e feed pump was s t a r t e d . When th e

c r y s t a l l i z e r was f i l l e d t o t he c o n t r o l l e v e l , t h e p r o d u c t pump was

s t a r t e d . The feed and p ro d u c t pumps were s e t a p p r o x im a t e l y t o the

d e s i r e d s e t t i n g s . Then th e f i n e s loop pump was t u r n e d on . When l i q u o r

25

. - ' 26

coming f rom th e c r y s t a l i i z e r s t a r t e d t o f i l l t he f i n e s h o l d . t a n k , steam

was tu r n e d on t o a c t i v a t e t h e h e a t e r . S pe c ia l ca re was taken in keep ing

th e f i n e s r e c y c l e s t ream warm enough t o a v o id n u c l e a t ion a n d , hence, l i n e

p l u g g in g a n d / o r system s e ed in g . A f t e r t h e f i n e s loop s t ream was f l o w i n g

c o n t i n u o u s l y f o r a moment, th e l e v e l in t h e c r y s t a l 1 i z e r was once aga in

ach ieve d and t h e e n t i r e process was on s t re am . The v o l u m e t r i c f l o w r a te s

in each pump were now a c c u r a t e l y s e t u s in g a g radu a ted c y l i n d e r and s to p

w a tch . The f l o w r a te s were p e r i o d i c a l l y checked , in t h i s way t o keep them

c o n s t a n t d u r i n g t h e e n t i r e r u n .

O p e ra t io n

In each r u n , t h e s l u r r y d e n s i t y in t h e c r y s t a l 1 i z e r , My, was

m o n i to r e d ve rsus t im e . To g e t t he v a lu e o f My, samples f rom the p r o d u c t

s t re am , wh ich is r e p r e s e n t a t i v e o f t h e c r y s t a l 1 i z e r c o n t e n t s , were taken

e v e r y h a l f hour by i n t e r r u p t i n g the p ro d u c t l i n e and c o l l e c t i n g the

s t ream in a 600-ml bea ke r . A P r e c i s i o n S c i e n t i f i c Lab t i m e r was used and

around 20 grams were c o l l e c t e d in a l - m l n sample. The sample was poured

i n t o a c l e a n , d ry 350-cc s i n t e r e d g la s s buchner f u n n e l . L i q u i d and s o l i d

phases were s e pa ra ted u s in g vacuum. The volume o f l i q u i d was measured in

a g ra dua te d c y l i n d e r and r e tu r n e d t o t h e c r y s t a l 1 i z e r . The s o l i d , p r e

v i o u s l y washed o f mother l i q u o r w i t h a c e t o n e , was vacuum d r i e d . T h is

wash was necessa ry t o p re v e n t p a r t i c l e a g g lo m e r a t i o n . A ru b be r po l iceman

was used t o c a r e f u l l y d i s t r i b u t e th e p a r t i c l e s d u r i n g t h e d r y i n g o p e ra

t i o n t o a v o id c r y s t a l a g g lo m e r a t i o n . The d r i e d c r y s t a l s were weighed in

a 100-gram M e t t i e r ba lance . . The w e ig h t o f s o l i d s was d i v i d e d by t h e

volume o f f i l t e r e d l i q u i d t o o b t a i n t h e s l u r r y d e n s i t y .

■■■ ' . ' 27

A p p r o x im a te l y t h r e e hours a f t e r t h e equ ipment was o p e r a t i n g w i t h

o u t i n t e r r u p t i o n , a s t e a d y - s t a t e c o n d i t i o n was re a ch e d . T h is was d e t e r

mined when no v a r i a t i o n in t h e magma d e n s i t y w i th , t im e was o b s e r v e d .

V a r i a t i o n s o f less than 5 p e r c e n t around a mean v a lu e o f MT were con-' : . I

s i d e r e d i n h e r e n t t o the e x p e r im e n ta l o p e r a t i o n . The t im e a t wh ich the

s teady s t a t e was reached was the " r e a l " s t a r t i n g p o i n t o f t h e run in

terms o f c o l l e c t i n g the d e s i r e d i n f o r m a t i o n on the system w o rk in g under

the chosen c o n d i t i o n s .

S t e a d y - s t a t e c o n d i t i o n s were s u s ta in e d f o r 5 -6 h o u rs . Dur ing

t h a t t im e , p r o d u c t and f i n e s st reams were sampled eve ry h a l f h o u r . The

p r o d u c t was sampled in t h e way d e s c r ib e d f o r v a lu e s . In a d d i t i o n ,

a f t e r w e ig h in g th e s o l i d s , th e y were s ie v e d t o d e te rm in e t h e c r y s t a l - s i z e

d i s t r i b u t i o n . A sma l l sample f rom th e f i n e s s t ream in t h e f i n e s d i s

s o l v i n g loop was o b t a i n e d by i n t e r r u p t i n g t h e l i n e a t a p o i n t l o c a te d

b e f o r e t h e h e a t e r . The sample was c o l l e c t e d in a 10 0 - c c b eake r . The

s o l u t i o n was q u i c k l y poured i n t o a 3007ml Pyrex fun ne l w i t h a p o r o m e t r i c

membrane (1 urn pore d ia m e te r ) where the c r y s t a l s were im m ed ia te l y

se p a ra te d f rom the l i q u o r , f l u s h e d w i t h methanol and a c e to n e . The who le

o p e r a t i o n t o o k o n l y a few se con d s . Rapid f i l t e r i n g was necessary t o

a v o id p r e c i p i t a t i o n o f new c r y s t a l s . The f i l t e r e d s o l u t i o n volume was

measured in a 10 0 -cc g ra d u a te d c y l i n d e r and re tu r n e d t o t h e c r y s t a l l i z e r .

The d r i e d c r y s t a l s were resuspended in e l e c t r o l y t i c i so p ro p a n o l s o l u t i o n

and coun ted w i t h an e l e c t r o n i c c o u n t e r . The Isop rop ano l s o l u t i o n con

s i s t e d o f i s o p r o p y l a l c o h o l s a t u r a t e d w i t h ammonium t h I o c y a n a t e and KC1

2 8

c r y s t a l s . The s o l u t i o n was c a r e f u l l y f i l t e r e d in a p o r o m e t r i c membrane

b e fo re use.

When u s ing the C o u l t e r Counte r Model T t o g e t t h e c r y s t a l - s i z e

d i s t r i b u t i o n o f the f i n e s s t re am , i t was found t h a t t h e 400 ym a p e r t u r e

was the most a p p r o p r i a t e f o r the s i z e range o f the c r y s t a l s being coun ted

(< 150 y m ) . i t was u s u a l l y n e cessa ry t o d i l u t e the resuspended c r y s t a l s

t o lower t h e p a r t i c l e c o n c e n t r a t i o n r e q u i r e d by t h e equ ipm en t .

Shutdown

The p ro ced u re d e s c r ib e d be low was f o l l o w e d in o r d e r t o e l i m i n a t e

any mechan ica l p roblems d u r i n g t h i s s t e p and in f u t u r e s t a r t - u p s . A l l

e l e c t r i c a l hea t arid steam was t u r n e d o f f t o cool th e s y s t e m . . The feed

pump was run in re v e rs e t o empty th e l i n e and then t u r n e d o f f . The feed

l i n e was d is c o n n e c te d f rom th e feed pump and f l u s h e d w i t h w a te r t o remove

a l l c r y s t a l s . T h is e l i m i n a t e d th e p o s s i b i l i t y o f h a v in g p lugs in the

l i n e w h ic h would cause d e la y in t he n e x t s t a r t - u p . The f i n e s pump was

a l s o re ve rsed and t h e f i n e s d i s s o l v i n g system l i n e was d r a i n e d back t o

t h e feed t a n k . The p ro d u c t pump was r e v e rs e d and s to p p e d , and the

c r y s t a l 1 i z e r c o n te n t s were d ra in e d p a r t i a l l y t o a h o ld t a n k , and then t o

t h e feed t a n k . The h o ld t a n k c o n te n t s wou ld be used t o charge th e

c r y s t a l 1 i z e r in t he n e x t r u n . When th e c r y s t a l 1 i z e r was empty, the

Impel l e t was t u r n e d o f f . The f i l t e r pump was a l s o s topped and th e f i l t e r

l i n e s were se pa ra ted f rom the feed t a n k . The e n t i r e system was then

f l u s h e d w i t h w a te r t o remove a l l c r y s t a l d e p o s i t i o n s . The f i l t e r i t s e l f

was washed w i t h h o t w a te r t o remove a l l accumula ted i m p u r i t i e s , so t h a t

i t was ready t o be used i n . a f u t u r e run . The feed ta n k was l e f t t o cool

29

w i t h t h e i m p e l l e r on t o a v o id a mass ive d e p o s i t i o n o f c r y s t a l s on the

v e r t i c a l s h a f t . I f t h i s p rocedu re was f o l l o w e d , the equ ipment would be

ready f o r t he nex t s t a r t - u p w i t h few problems and d e l a y s .

RESULTS

R e s u l t s a re p re sen ted in t h e o r d e r in wh ich t h e research program

was c a r r i e d o u t : n u c l e a t i o n - g r o w t h k i n e t i c s model , e f f e c t o f f i n e s t r a p

o p e r a t i o n on CSD, t e r m in a l v e l o c i t y measurements o f KC1 c r y s t a l s , des ign

e q u a t i o n s f o r m o d e l l i n g f i n e s t r a p o p e r a t i o n , and c o s t e s t i m a t i o n f o r

c r y s t a l s i z e im provem ent .

K i n e t i c s Model

In o r d e r t o d e te rm in e the pa ram ete rs o f E qu a t io n ( 1 4 ) , two d i f f e r

e n t t ypes o f e x pe r im e n ts were conduc ted under s t e a d y - s t a t e c o n d i t i o n s .

The c r y s t a l l i z e r was o p e ra te d as a s im p le MSMPR r e a c t o r and the c o r r e

spond ing va lues o f n 0 and G were o b t a i n e d f rom CSD a n a l y s i s . , Then a

f i n e s d e s t r u c t i o n system was implemented and CSD a n a l y s i s o f t h e p r o d u c t

and f i n e s st reams p r o v id e d th e c o r re s p o n d in g v a lues o f n ° , G, and Lp. In

each ru n , t h e d e n s i t y o f t he c r y s t a l 1 I z e r magma was reco rded versus t im e

and an average v a lu e o v e r t h e e n t i r e s t e a d y - s t a t e o p e r a t i o n was taken .

The v a r i a t i o n in so l ids c o n c e n t r a t i o n d u r i n g a t y p i c a l e xpe r im e n t is

shown in F ig . 5-

Each run was s u s ta i n e d as long as necessa ry t o reach s teady

s t a t e , u s u a l l y 6 -7 re s id e n c e t im e s . D u r ing t h i s p e r i o d , p ro d u c t and

f i n e s s t reams were sampled ev e ry h a l f h o u r . The gen e ra l o p e r a t i n g c o n d i

t i o n s were :

Feed r a t e , Qp - 400 cc /m in

P ro du c t r a t e , Qp = 400 c c /m in

30

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T IM E , HOURS

F ig . 5• V a r i a t i o n o f S o l i d s C o n c e n t ra t i o n w i t h Time.

32

C y r s t a l 1 i z e r I m p e l l e r = 500 rpm

C y r s t a l l i z e r t e m p e ra tu r e = 40°C

Feed s o l u t i o n t e m p e ra tu r e = 70°C

Residence t im e , t = 45 min

The feed and p ro d u c t r a t e were se t equal in o r d e r t o keep the o v e r f l o w

near z e ro because c l e a r l i q u o r advance ( o v e r f l o w ) a l s o i n f l u e n c e s CSD.

The r e s u l t s o f t h i s S e r ie s o f e x p e r im e n ts a re summarized in Tab le 1.

Tab le 1 . . Summary o f MSMPR and FDS E xpe r im en ta l R e s u l t s .

% n° G mt l f

, Exp (c c /m in ) Mode (# /cc -ym ) (ym/min) (gm/T) (pm)

62176 0 MSMPR 32 2 .2 4070676 : 0 : MSMPR 24 2 .4 45 - -

70176 1520 FDS 71 3 .0 42 5572876 1515 FDS 42 3.1 64 3080476 1550 FDS 58 3 .2 64 5231677 1500 FDS 456 2 .5 55 90

111 176 2200 FDS 500 3.1 50 82

60776 2920 FDS 1160 4 .7 44 13520377 3000 FDS 1000 4.1 63 13071576 2990 FDS 600 4 .6 43 15070676 3540 FDS 600 5 .0 43 130

Tw o .expe r im en ts were conducted w i t h the MSMPR mode (0_R = 0) . Ri

'0676 was s t a r t e d under FDS <c o n d i t i o n s and conducted in t h i s mode a t

; teady s t a t e f o r 6 res Idence t im e s t o t a k e samples o f th e c r y s t a l l i z e r

p ro d u c t and f i n e s s t re a m s . A f t e r t h i s p e r i o d , t h e o p e r a t i n g c o n d i t i o n s

were changed t o MSMPR mode, s h u t t i n g down the f i n e s d e s t r u c t i o n lo o p . A

■ 33

s l i g h t l y d i f f e r e n t s teady s t a t e was reached a f t e r 2 re s id en c e t im es and

m a in ta in e d f o r 6% w h i l e sam p l ing the p r o d u c t .

A n a l y s i s o f the da ta shows t h a t the system can change e a s i l y f rom

one mode t o the o t h e r w i t h o u t becoming u n s t a b l e . F i g . 6 shows the

expec ted s e m i - l o g p o p u l a t i o n d e n s i t y o b t a i n e d f rom an MSMPR e x p e r i m e n t .

A s e r i e s o f n in e e x pe r im e n ts was conduc ted under FDS c o n d i t i o n s

f o r t h r e e d i f f e r e n t l e v e l s o f the f i n e s removal r a t e , CL : 1500, 2200,

and 3000 c c /m in . F ig s . 7 th rough 14 are th e p o p u l a t i o n d e n s i t y p l o t s f o r

t y p i c a l runs a t d i f f e r e n t s .

In these f i g u r e s , the s t r a i g h t l i n e s r e p r e s e n t i n g th e CSD were

de te rm in e d as f o l l o w s . From the s ie v e a n a l y s i s d a ta , the c u m u la t i v e

w e ig h t d i s t r i b u t i o n f u n c t i o n ,

W(L) = pt<v f p n (p) dp/M-j. ,0

was c a l c u l a t e d and p l o t t e d ve rsus th e c r y s t a l s i z e , L. Ta k in g d e r i v a

t i v e s o f t h i s c u r v e , the w e ig h t d i s t r i b u t i o n f u n c t i o n , w, was d e te rm ine d

and the p o p u l a t i o n d e n s i t y was then c a l c u l a t e d f rom th e r e l a t i o n - 3

n = wMy/pkyL , where L is t h e average s i z e between two s u c c es s i v e s ie v e

s c r e e n s , By u s ing l i n e a r r e g r e s s io n a n a l y s i s , the b e s t l i n e was f i t t o

these p o i n t s . The symbol A was used in a 1.1 the p o p u l a t i o n d e n s i t y p l o t s

t o r e p re s e n t a c tu a l da ta p o i n t s as th e y were d i r e c t l y o b t a i n e d f rom s ie v e

a n a l y s i s .

I t can be obse rved t h a t as Q, i n c re a se s h i g h e r v a lu es o f n° and

Lp were o b t a i n e d . As d e c r e a s e s , t he s lo pes o f the s t r a i g h t l i n e s

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RUN 6/21/76Q r = Odc/min (MSMPR) Qf = Qp= 4 0 0 cc/min

A Sieve Data

— Linear Regression

4 0 0 8 0 0200

CRYSTAL S IZ E , L, M IC RONS

F ig . 6, C r y s t a l - S i z e D i s t r i b u t i o n , MSMPR Run 62176.

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RUN 7 /0 1 /7 6 Qr = 1500 c c /m in

Coulter Counter Data

Sieve Data

L inear Regression

Stat ist ically Insignificant Data (less than 10 counts per channel)

10

I

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4 0 0 6 0 0200CRYSTAL S I Z E , L, MICRONS

F i g . 7- C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 70176.

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R U N 8 / 0 4 / 7 6

Qr = 1500 cc/m in

Qf = Qp = 4 0 0 cc /m ii)

o Coulter Counter Data

A Sieve Data

— L inear Regression

X Statistically Insignificant Data (less than 10 counts per channel)10

IQ-'

100 300 500 700 900

C RYSTA L S IZ E , L , M ICRONS

F ig . 8. C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 80476.

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RUN 3 / 1 6 / 7 7Q r = 1 5 0 0 cc/min Qp = Qp = 4 0 0 cc/min Fines Trap Area = 1/3

o Coulter Counter Data 6 Sieve Data

— Lin ear Regression x Sta t is t ica l ly Ins ignif icant Data

(less than 10 counts per channel)

, 0 - 4 # i i i________ I________ !_________I________ !________ I________ I________ !_________I________ I___

2 0 0 4 0 0 6 0 0 8 0 0 1000 1200CRYSTAL S I Z E , L, MICRONS

F ig . 9. C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 31677 Using F r a c t i o n a l F ines Trap Area.

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RUN 7 / 2 8 / 7 6 Q r = 1500 cc /m in

o Coulter Counter Data & Sieve Data — Linear Regression

x S t a t is t ic a l ly Ins ig n i f ican t Data (less than 10 counts per channel)

10

4 0 0 6 0 0200C R Y S T A L S IZ E , L, M IC R O N S

F i g . 1 0 . C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 7 2 8 7 6 .

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RUN I I / I 1 / 76

Qr = 2 2 0 0 cc /m in Qc = Qp = 4 0 0 c c / m in

o Coulter Counter Data A Sieve Data — Lin ear Regression

x Stat is t ica l ly In signif icant Data (less than 10 counts per channel)

10-

- 24 0 0 6 0 0200 8 0 0

CRYSTAL S IZ E , L, MICRONS

F i g . 1 1 . C r y s t a l - S i z e D i s t r i b u t i o n , FDS Run 1 1 1 1 7 6 .

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RUN 6 / 0 7 / 7 6Qr = 3 0 0 0 cc /m in Qp = Qp = 4 0 0 cc/min

o Coulter Counter Data A Sieve Data — Linear Regression x Sta t is t ica l ly Ins ig n i f ic ant Data


2 0 0 4 0 0CRYSTAL S IZ E , L, MICRONS

600


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R U N 2 / 0 3 / 7 7

Qr = 3 0 0 0 cc/min Qp = Qp = 4 0 0 cc/min

o Coulter Counter Data a Sieve Data — L i n e a r Regression x Sta t is t ica l ly Insigni ficant Data


10

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800 1000600200 400CRYSTAL S IZ E , L, MICRONS


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RUN 7 / 0 6 / 7 6Q r = 3 5 0 0 cc /m in Qp = Qp = 4 0 0 cc /m in

o Coulter Counter Data a Sieve Data

— L in e a r Regression x S t a t is t ic a l ly Insignificant Data


2 0 0 4 0 0 6 0 0 8 0 0

C R Y ST A L S I Z E , L, M IC RONS


. f - 43

re p r e s e n t i n g the p o p u l a t i o n d e n s i t y o f t he l a r g e and sma l l c r y s t a l s

become s t e e p e r , thus i n d i c a t i n g a d e c r e a s in g g rowth r a t e , G. The va lu es

o f the g rowth r a t e c a l c u l a t e d f rom the s lop es o f the p r o d u c t and f i n e s

CSD's sh ou ld be the same. . For a l l e x p e r im e n t s , these v a lu es o f g rowth

r a t e as measured f rom the two s t r a i g h t - l i n e segments o f the p o p u l a t i o n

d e n s i t y p l o t matched q u i t e W e l l , w i t h a maximum d i s c r e p a n c y o f 10%.

The l a r g e s t p a r t i c l e s i z e be ing d e s t ro y e d in t h e f i n e s lo o p , Lp,

was o b ta in e d e x p e r i m e n t a l l y f rom the i n t e r s e c t i o n between the f i n e s and

p ro d u c t CSD's. I t can be observed though t h a t v a lu e s o f L g r e a t e r than

Lp were a p p a r e n t l y measured by th e C o u l t e r Co un te r . T h is r e s u l t has been

r e p o r te d in t he l i t e r a t u r e a l t h o u g h n o t c o m p le te l y e x p l a i n e d . H e l t and

Larson (1976) a t t r i b u t e t h i s r e s u l t t o the e x i s t e n c e o f a p a r a b o l i c

v e l o c i t y p r o f i l e i n s i d e th e f i n e s t r a p due t o l a m in a r f l o w wh ich e n t r a i n s

some c r y s t a l s l a r g e r than expec ted based on the average f l o w r a t e .

A f l u i d v e l o c i t y h i g h e r than th e mean v e l o c i t y wou ld then e x i s t

a t t he c e n t e r o f the t r a p c a r r y i n g c r y s t a l s o f l a r g e r s i z e . However, i f

p a r t i c l e s a t s i z e s g r e a t e r than Lp were a c t u a l l y be ing removed f rom th e

t r a p , a n o t i c e a b l e e f f e c t on the p r o d u c t CSD shou ld be o bse rve d , i . e . ,

t h e l i n e r e p r e s e n t i n g t h i s d i s t r i b u t i o n would be s h i f t e d downwards.

Sieve a n a l y s i s o f the p r o d u c t d id n o t show such an e f f e c t , i n d i c a t i n g

t h a t f i n e s l a r g e r than Lp were n o t a c t u a l l y be ing removed.

T h is sugges ts t h a t these da ta p o i n t s are a p p a r e n t , p ro b a b ly due

t o c o in c id e n c e o r n o i s e when us ing the C o u l t e r C o u n te r . Data p o i n t s t h a t

a re c l o s e to the re a l d i s t r i b u t i o n , i . e . , a t s i z e s n o t much g r e a t e r than

Lp, m ig h t have been caused by c o in c id e n c e , i . e . , two p a r t i c l e s o f a smal l

4 4

s i z e a re counted a t the same t im e by th e c o u n t e r . Thus , th e equ ipment

d e te c t s a doub led volume and a s s o c ia te s i t t o a s i z e wh ich is l a r g e r than

the rea l s i z e . Th is e x p l a n a t i o n o f s p u r io u s coun ts i s borne o u t by the

f a c t t h a t s i m i l a r s i z e channe ls a re p o p u la te d re g a r d le s s o f the va lue

o f Lf .

Data p o i n t s p rocessed f rom less than 10 coun ts p e r channel were

c o n s id e re d t o be s t a t i s t i c a l l y i n s i g n i f i c a n t and are p re s e n te d in each

p l o t f o r i n f o r m a t i o n o n l y .

Two d i f f e r e n t a p e r t u r e s i z e s were used w i t h the C o u l t e r Counte r

in t h i s s t u d y : 280 and 400 m i c r o n s . I t i s recommended by th e manufac

t u r e r o f the C o u l t e r Counte r t o use an a p e r t u r e s i z e o f 2 .5 t imes the

maximum s i z e o f p a r t i c l e s t o be coun ted . Since the s i z e o f the p a r t i c l e s

p r e s e n t i n s i d e the f i n e s t r a p ranged between 10 and 130 m ic r o n s , a l a r g e

p o r t i o n o f t h e s i z e range was common to both a p e r t u r e s a nd , t h u s , a

d e t e r m i n a t i o n o f the b e s t o r i f i c e s i z e was necessary f rom o t h e r c o n s i d e r

a t i o n s , e . g . , p l u g g i n g . The 400 m ic ron a p e r t u r e would g i v e more i n s i g h t

in the l a r g e r s i z e r e g i o n , and t h e 280 m ic ron a p e r t u r e would be b e t t e r in

t h e s m a l l e r s i z e r e g io n . F ig s . 15 and 16 show a compar ison o f r e s u l t s

f o r two d i f f e r e n t e x p e r i m e n t s / W i th the 280 m ic ron a p e r t u r e , a d e p l e t i o n

in p o p u l a t i o n d e n s i t y a t sma l l s i z e s ( l e s s than 50 m ic ro n s ) was o bs e rv e d ;

however, a r e p r e s e n t a t i v e number o f p a r t i c l e s was observed in the

channe ls c o r re s p o n d in g t o the l a r g e s t s i z e s . Th is r e s u l t suggested t h a t

the l a r g e r a p e r t u r e shou ld be used . No a p p r e c i a b l e d i f f e r e n c e between

both a p e r t u r e s was found f o r coun ts a t s i z e s s m a l l e r than 10 m ic r o n s .

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10" '

4 0 0200100CRYSTAL S I Z E , L , MICRONS

F i g . 15. Compar ison o f C o u l t e r Counter Data Obta ined w i t h D i f f e r e n t A p e r t u r e s , Run 70176.

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C R YSTA L S I Z E , L , M IC RO NS

F i g . 16. C r y s t a l - S i z e D i s t r i b u t i o n , Run 70676.

4 7

T h u s , the 400 m ic ron a p e r t u r e appeared t o p r o p e r l y cove r the e n t i r e s i z e

range and was used t h r o u g h o u t the course o f t h i s r e s e a r c h .

F ig s . 17 and 18, c o r re s p o n d in g t o expe r im en ts 71576 (FDS) and

70676 (MSMPR), show the d i s t r i b u t i o n o f f i n e s found in the c r y s t a l l i z e r

p r o d u c t , compar ing s i e v e and C o u l t e r Counte r a n a l y s i s t e c h n iq u e s . These

data, were o b ta in e d by s a v in g the f i n e p a r t i c l e s a f t e r t he produce s ie v e

a n a l y s i s . These f i n e s were then resuspended in i s o p ro p a n o l s o l u t i o n and

counted u s in g the C o u l t e r Cou n te r . For t h e FDS e x p e r im e n t 71576, c l e a r l y

th e d i s t r i b u t i o n o f the f i n e s found in t h e p r o d u c t was the same as the

f i n e s i n s i d e the t r a p .

A d i f f e r e n t s e t o f data was c o l l e c t e d d u r i n g the dynamic runs

conduc ted and r e p o r te d by Beckman (1976 ) , The equ ipment used in t h a t

case was the s o - c a l l e d "comp lex c r y s t a l 1 i z e r " ; a c r y s t a l l i z e r whose con

f i g u r a t i o n in c lu d e d c l e a r l i q u o r o v e r f l o w , p ro d u c t c l a s s i f i c a t i o n , and a

f i n e s d e s t r u c t i o n system. Even though the c r y s t a l l i z e r was in a t r a n

s i e n t mode o f o p e r a t i o n , the CSD from th e f i n e s loop formed a q u a s i -

e q u i l i b r i u m e x p o n e n t i a l d i s t r i b u t i o n due t o the s m a l l r e t e n t i o n o f the

s i z e s a f f e c t e d by the d i s s o l v i n g system. The same da ta p ro c e s s in g t e c h

n iques used f o r MSMPR s t e a d y - s t a t e runs c ou ld then be used t o ana lyze

these d a ta .

The n u c l e a t i o n and g rowth r a t e v a lu e s o b ta in e d f ro m a n a l y s i s o f

f i n e s loop CSD's, under s t e a d y - s t a t e and t r a n s i e n t o p e r a t i o n , were

c o r r e l a t e d in a pow er - la w fo rm g iven by Equa t ion ( 1 4 ) :

B° = k^G 'Myj (14)

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RUN 7 / 1 5 / 7 6 Q r = 3 0 0 0 cc/min

o Coulter Counter Data

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x Statistically Insignificant Data (less than 10 counts per channel)

8 0 06 0 04 0 0200CRYSTAL S I Z E , L, MICRONS

F i g . 17. F ines D i s t r i b u t i o n in C r y s t a l 1 i z e r Produc t and Compar ison o f Computer S im u la t i o n and Expe r im en ta l R e su l t s f o r Run 71576.

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— L i n e a r Regression

-o

1000600200C R YSTA L S IZ E , L, MICRONS

F i g . 18. Fines D i s t r i b u t i o n in C r y s t a l I i z e r Produc t f rom Sieve and C o u l t e r Counte r Techn i ques .

■ - 50

The k i n e t i c param ete rs o f t h i s model were found t o be i = 4 . 9 9 , j = 0 .1 4 ,

and = 0 .657 by m u l t i p l e l i n e a r r e g r e s s io n o f the d a ta .

F ig . 19 shows the a c tu a l e x p e r im e n ta l da ta compared w i t h the

c o r r e l a t i o n . In t h i s f i g u r e , t h e r e a re two p o i n t s i n d i c a t e d by (0) w h ich

co r respond to expe r im en ts c a r r i e d o u t w i t h a d i f f e r e n t f eed s o l u t i o n co n

t a i n i n g no MgSQ^ i m p u r i t y . These two p o i n t s w i t h the d i f f e r e n t feed were

n o t c o n s id e re d in t h e m u l t i p l e r e g r e s s io n a n a l y s i s .

E qu a t ion (.14) f i t s t h e e x p e r im e n ta l nucl e a t ion da ta w i t h an

average d e v i a t i o n o f 7.8%.

The r e s u l t s o f the computer s i m u l a t i o n o f t h e s t e a d y - s t a t e run

71576 u s in g th e k i n e t i c pa ram ete rs found in t h i s s tu d y (Beckman, 1976)

a re a l s o shown in F ig . 17,

S e t t ! i n g V e l o c i t y E f f e c t

The l a r g e s t p a r t i c l e s i z e be ing d i s s o l v e d in the f i n e s lo o p , Lp,

i s one o f the pa ram ete rs t h a t has t o be co ns id e re d in t he des ign o f a

f i n e s d e s t r u c t i o n sys tem. Lp i s de te rm ined f rom p o p u l a t i o n d e n s i t y p l o t s

as p r e v i o u s l y d e s c r i b e d . I t s v a lu e depends upon the s e t t l i n g v e l o c i t y

wh ich e x i s t s i n s i d e t h e f i n e s t r a p . The S e t t l i n g v e l o c i t y i s g iven by

the r a t i o between the v o l u m e t r i c r a t e a t wh ich the f i n e s a re removed,

Qn , and the c r o s s - s e c t i o n a l area o f t h e t r a p .

I f the dependency o f Lp on the v e l o c i t y , v , f o r a g iven system is

w e l l - k n o w n * then Lp can be p r e d i c t e d and used in the p r o p e r des ign

c o r r e l a t i o n s . I f t h i s r e l a t i o n s h i p i s unknown, Lp wou ld have t o be

o b t a i n e d e x p e r im e n ta l 1y In a p i l o t p l a n t c r y s t a l 1 i z e r w i t h f i n e s loop in

o r d e r t o des ign the FDS.

BO

/M

51

FINES REMOVAL RUNS

DYNAMIC RUNS F IN E S REMOVAL WITH

D I F F E R E N T FEED SOLN.

O

©

kN = 0 . 6 5 7

I = 4 . 9 9

4 . 9 9

I0"1 I 10

GROWTH RATE G, M IC R O N S / M I N

F ig . 19. N u c le a t i o n -G ro w th Rate K i n e t i c s C o r r e l a t i o n f o r the KC1 System.

' ■' ■

An e x p e r im e n ta l c o r r e l a t i o n between Lp and th e v e l o c i t y , v , was

found and t e s t e d u s in g two e q u i v a l e n t t e c h n iq u e s : k e ep ing the f i n e s loop

area c o n s t a n t and chang ing the f i n e s removal r a t e , CL; and f i x i n g Q andK K

v a r y i n g the a re a . .

The f i r s t approach was used to d e te rm ine the e m p i r i c a l c o r r e l a

t i o n between Lp and v f o r the KC1 sys tem. , The second approach was used

to check these r e s u l t s , as shown in F ig . 20. The s o l i d l i n e co r responds

to the t h e o r e t i c a l r e l a t i o n s h i p between, the s i z e o f a s p h e r i c a l p a r t i c l e

moving th roug h a f l u i d by e f f e c t o f g r a v i t y and i t s s e t t l i n g v e l o c i t y , v:

v =4 g L p ( p - p s )

3ps CD

1 /2(35)

Under the o p e r a t i n g c o n d i t i o n s in t h e f i n e s t r a p , t he Reynolds\

number v a r i e d between 0 .5 and 2 . 0 , and the drag c o e f f i c i e n t was con

s id e r e d t o be g iven by the i n t e r m e d ia t e law:

' < # >

i f was found t h a t , f o r the KOI sys tem, the t h e o r e t i c a l dependence

o f Lp upon v , com b in ing Equa t ions (35) and (3 6 ) , i s exp ressed by:

L f = 1 6 8 v 0 ; 875 (37)

By m u l t i p l e l i n e a r r e g r e s s io n o f d a ta , i t was found t h a t the e x p e r im e n ta l

c o r r e l a t i o n o f t h e Lp va lues measured f rom CSD p l o t s was:

MIC

RO

NS

5 3

o Fines Trap Data o Experimental Settl ing Data

— Theory, Stokes Intermediate Region

u__i

1.0

S E T T L IN G VELOCITY v , cm/sec

F ig . 20. T h e o r e t i c a l and Expe r im en ta l C o r r e l a t i o n s between the C r i t i c a l S iz e , Lp, and the S e t t l i n g V e l o c i t y I n s i d e the F ines T rap .

54

Lp = 96 v 1 *276 (38)

In F ig . 20, the data p o i n t i n d i c a t e d by A co r re sp on d s t o e x p e r i

ment •31677 wh ich was c a r r i e d o u t w i t h a f i nes t r a p area equal t o one-

t h i r d o f t h e t o t a l t r a p a re a . As e x p e c t e d , t h i s d a ta c o r r e l a t e d w e l l

w i t h o t h e r e x p e r im e n ts w i t h the same t e r m i n a l v e l o c i t y . F ig . 20 a l s o

shows the r e s u l t s o f t he e x p e r im e n ta l d e t e r m i n a t i o n o f the s e t t l i n g

v e l o c i t y f o r s i n g l e KC1 c r y s t a l s . These measurements were done u s in g a

1000 -cc g ra du a ted c y l i n d e r f i l l e d w i t h s a tu r a t e d KC1 s o l u t i o n and

immersed in a t h e r m o s t a t i c bath whose tem p e ra tu re was 40°C , the same as

t he c r y s t a l l i z e r t e m p e r a t u r e .

KC1 c r y s t a l s a t d i f f e r e n t known s i z e s were dropped i n t o the s o l u

t i o n and the t im e r e q u i r e d f o r the p a r t i c l e s t o f a l l a p re d e te rm in e d

d i s t a n c e was measured w i t h a t i m e r .

A l t h o u g h the v a lu es o f Lp o b ta in e d f rom the' e x p e r im e n ts were

found t o in c re a s e when t h e removal v e l o c i t y i n s i d e th e t r a p in c re a s e d , as

e x p e c t e d , they were s t i l l lo w e r than the v a lues p r e d i c t e d by s e t t l i n g

t h e o r y . T h is i s p r o b a b ly due t o t h e c r o s s - s e c t i o n a l a rea o f the removal

tube r e l a t i v e t o t h a t o f t he t r a p . I t i s b e l i e v e d t h a t i f t he r a t i o

between t h e removal tube area and the t r a p was Inc reased then h ig h e r

va lues o f Lp wou ld be o b t a i n e d . At th e p re s e n t c o n d i t i o n s , t h i s r a t i o i s

v e ry sma l l and near the top o f the t r a p the f l o w near t o the w a l l s has t o

d e c e l e r a t e in the a x i a l d l r e c t i o n and a c c e l e r a t e in th e h o r i z o n t a l d i r e c

t i on t o reach the removal tub e . Since t h i s is a g ra dua l p ro c e s s , the

l a r g e s t p a r t i c l e s in t he s t ream q lo s e t o the w a l 1s are g r a d u a l l y

55

d e c e le r a t e d and s t a r t t o f a l l b e fo re th e y reach the to p o f th e t r a p , thus

d e c r e a s in g th e observed v a lu e o f Lp. Of c o u r s e , t h i s c o n j e c t u r e a ls o

must i n c l u d e a c i r c u l a t i n g f l o w a t some p o i n t w i t h i n the t r a p t h a t wou ld .

u l t i m a t e l y a l l o w the l a r g e r p a r t i c l e s t o e x i t back t o th e magma.

Design o f a Fines D e s t r u c t i o n System

The e q u a t i o n s necessa ry t o des ign a f i n e s d e s t r u c t i o n system have

been p r e v i o u s l y d e r i v e d f o r t h e s p e c i f i c case o f a " p o i n t " f i n e s t r a p ,

i . e . , n e g l i g i b l e f i n e s mass d i s s o l v e d , and f o r a more gen e ra l case o f

d i s s o l v i n g l a r g e r p a r t i c l e s .

I t has been shown t h a t , under " p o i n t " f i n e s t r a p c o n d i t i o n s , the

p ro d u c t s i z e improvement w i t h f i n e s d e s t r u c t i o n has the s im p le form:

However, when t h e f i n e s a re n o t n e g l i g i b l y sma l l compared t o

p r o d u c t - s i z e c r y s t a l s and r e p r e s e n t an a p p r e c ia b l e s o l u t e re c y c le s t ream ,

the s i z e improvement w i l l be:

%1

l / i + 3

(28)

The f i n e s - t o - p r o d u c t mass r a t i o (wh ich l i m i t s t h e p ro p e r use o f

the p o i n t f i n e s t r a p a p p r o x i m a t i o n ) , E qua t ion (2 4 ) , has n o t been s p e c i f ! '

c a l l y i n v e s t i g a t e d . However, i t i s g e n e r a l l y accep ted t h a t t h i s r a t i o

shou ld be le ss than 5% f o r t h e p o i n t f i n e s t rap, assumpt ion t o h o l d .

'■ : 56 Larson and Gars ide (1973) assume th e concept o f t h e p o i n t f i n e s t r a p i s

v a l i d i f t he d e s t ro y e d f i n e s a re less than 10 ym in s i z e .

C a l c u l a t i o n s o f the s i z e improvement f o r e x p e r im e n ts c a r r i e d o u t

d u r i n g t h i s s tu d y have shown t h a t the s i m p l e r a p p r o x im a t io n (Eq ua t ion 24)

ho lds f o r f i n e s - t o - p r o d u c t r a t i o s as h ig h as 35% w i t h o n l y 1% e r r o r .

In o r d e r t o e s t i m a te th e s i z e improvement w i t h f i n e s d e s t r u c t i o n

e i t h e r f rom E qu a t ion (24) o r Equa t ion ( 2 8 ) , the v a lu e o f the e x p o n e n t i a l

decay r a t i o . A, must bq d e te rm in e d f rom the mass ba lance c o n s t r a i n t .

Equation (31):

Lf (LA exp ( A / i+ 3 ) = q—7 - = K (31)

1

In o r d e r t o s o lv e Equat ion ( 3 1 ) , t he r a t i o K and the paramete r i

must be p r e v i o u s l y d e te rm in e d . I f t h e r e l a t i v e k i n e t i c o r d e r o f n u c l e a -

t i o n t o g ro w th , i , is unknown, i t can be e s t i m a te d w i t h a good app rox im a

t i o n f rom two MSMPR runs whose da ta have been p rocessed in the manner

d e s c r ib e d a t t he b e g in n in g o f t h i s c h a p t e r . The same MSMPR e xpe r im en ts

w i l l p r o v i d e the v a lu e o f the g rowth r a t e , G j . The f i n e s removal r a t e ,

is s e t by c h o i c e , w h i l e Lp can be e s t i m a te d f rom s e t t l i n g c o r r e l a

t i o n s o r e x p e r i m e n t a l l y o b t a i n e d . E qua t ion (31) was s o lv e d n u m e r i c a l l y

u s in g Newton's method and the r e s u l t s a re shown in F ig . 21a f o r d i f f e r e n t

va lues o f t h e k i n e t i c pa ram ete r I .

W i th i and A known, the s i z e improvement w i t h f i n e s d e s t r u c t i o n

can be d i r e c t l y o b t a i n e d f rom Equa t ion ( 2 4 ) . In o r d e r t o make use o f

Equa t ion ( 2 8 ) , the Incom p le te t h i r d - o r d e r gamma f u n c t i o n must be o b ta in e d

57

10.0

8.0

6.0

0.0 20 30

5.0

4.0

3.0

2.0

3020

Fig. 21. Design Correlat ions for Size Improvement with FDS.

58

f o r the c o r re s p o n d in g va lues o f A and R = 1 + QR/Qp . Tab les f o r the

in c o m p le te t h i r d - o r d e r gamma f u n c t i o n a re found in Randolph and

Larson (1971) .

S ize improvement c a l c u l a t e d f rom Equat ion (24) has been p l o t t e d

versus K as shown in F ig . 21b. I f t he d es ign e n g in e e r i s i n t e r e s t e d in

d e t e r m i n in g s i z e improvement o n l y , then the i n t e r m e d i a t e s tep o f d e t e r

m in in g A. is no t necessa ry and F ig . 21 b can be used d i r e c t l y .

The f r a c t i o n o f n e t p r o d u c t i o n wh ich is d i s s o l v e d and re c y c le d

can be c a l c u l a t e d f rom E qua t ion ( 3 4 ) :

: - •• — ----------- — --------- — :— r (34)1 + rV x

1 - a) ( X / R - l ) co (AR/R-1 )

The d i s s o l v e d f r a c t i o n , p , i s p l o t t e d ve rsus A in F ig . 22 f o r d i f f e r e n t

v a lu es o f R. T h is f i g u r e has c o n s i d e r a b l e i n t e r e s t f rom the des ign

v i e w p o i n t because i t shows t h a t , f o r a g iv e n v a lu e o f R, t h e r e e x i s t s a

l i m i t i n g v a lu e o f A o v e r wh ich the d i s s o l v e d f r a c t i o n no l o n g e r in c re a s e s

The a s y m p to t i c v a lu e f o r each cu rve co r responds t o ( R - l ) . The f r a c t i o n

d i s s o l v e d and r e c y c le d can be expressed as :

* = / = (39)

where MR would be t h e f i n e s s l u r r y d e n s i t y . As 0 R = (R - l )Q.p, then

E qua t ion (39) becomes:

FRA

CTI

ON

D

ISS

OLV

ED

, <j)

= P

F/P

59

20

10

I

PF Qr Mr ( R - | ) ^ M

A = (R-1)

F ig . 22. F r a c t i o n o f F ines D is s o lv e d ve rsus E x p o n e n t ia l Decay R a t i o , X .

60

which means t h a t , when <j> = R-i , t h e d i s t r i b u t i o n o f t h e f i n e s co r responds .

t o t he e n t i r e d i s t r i b u t i o n and the p ro d u c t and f i n e s d i s s o l v e d are in the

same r a t i o as t h e i r r e s p e c t i v e f l o w s .

There e x i s t s , t h e n , a p h y s i c a l l i m i t a t i o n f o r t h e v a lues o f A.

For any r e a l i s t i c d e s ig n , the f i n e s t r a p o p e r a t i n g c o n d i t i o n s must be

such t h a t t h e c r i t i c a l v a lu e o f A w i l l no t be re a ched . D i f f e r e n t combina

t i o n s o f R and Lp r e s u l t in e q u i v a l e n t v a lu e s o f the p a ra m e te r A. Which

v a lu es a re chosen i s an economic d e c i s i o n . Large v a lues o f R mean l a r g e r

f l o w r a t e s wh ich makes t h e o p e r a t i o n more e x p e n s iv e . On the o t h e r hand,

la r g e v a lues o f Lp a re I m p r a c t i c a l because o f the d i f f i c u l t y in d i s

s o l v i n g such l a r g e p a r t i c l e s . F ig . 23 shows th e d e c i s i o n f l o w d iagram

f o r such a s i z e improvement a n a l y s i s . To u t i l i z e t h e p rocedu res o f

F ig . 23, the f o l l o w i n g i n f o r m a t i o n a n d / o r a c t i o n s are r e q u i r e d :

A s im p le MSMPR run p r o v id e s the v a lu e o f Gp

• Lp i s e s t im a te d f rom a s e t t l i n g c o r r e l a t i o n o r e x p e r im e n ta l 1y

o b t a i n e d by the method o f s e m i - l o g p l o t i n t e r s e c t i o n s .

» Qp is s e t by c h o i c e .

The volume o f the r e a c t o r , V, i s known.

K is c a l c u l a t e d f rom K = LpQ^/G^V.

An a d d i t i o n a l MSMPR run a l l o w s e s t i m a t i o n o f th e n u c l e a t i o n

pa ram ete r I .

W i th i known, A can be g r a p h i c a l l y o b t a i n e d ( F i g . 21 a) f o r any

v a lu e o f K.

61

- Settling correlation

-P i lo t Plant FDS

f M S M P R A V Run # r 2 J

A / i + 3

A / i +3FDS MSMPR

Fraction DissolvedSize Improvement

F ig . 23. D e c is io n Flow Diagram f o r Size Improvement A n a l y s i s .

' ' ' - . 62

* W i th i known, the s i z e improvement r a t i o can be g r a p h i c a l l y

de te rm ined " ( F i g . 21 b.) f o r any v a lu e o f K.

F i n a l l y , f o r a g iv e n v a lu e o f R, the f r a c t i o n d i s s o l v e d and

r e c y c le d can be g r a p h i c a l l y de te rm in e d ( F ig . 22) f o r any v a lu e

o f X.

Data c o l l e c t e d f o r t h r e e FDS e x p e r im e n ts in t h i s s tu d y were used

t o d e te rm in e the s i z e improvement by u s ing E qua t ions (24) and (2 8 ) , a lo ng

w i t h F ig s . 21a, 21b, and 22, u s in g the sys tem k i n e t i c s and o p e r a t i n g

parameters as e x p e r im e n t a l1 y d e t e r m i n e d . The r e s u l t s a re summarized in

Tabl 'e 2. I t can be observed t h a t , f o r a f r a c t i o n o f f i n e s d i s s o l v e d as

h ig h as 35%, E qua t ion (24) i s s t i l l an e x c e l l e n t a p p r o x im a t i o n . The

e x p e r im e n ta l va lu es o f s i z e improvement a re g iven by th e r a t i o between

the g rowth r a t e a t FDS c o n d i t i o n s , G^, and the growth r a t e a t MSMPR

c o n d i t i o n s , , f o r the same re s id e n c e t im e . The v a lu e s o f G^ were

o b ta in e d f rom Tab le 1 and G was c o n s id e re d to be 2 .3 mi c ro n s /m i n . The

e x p e r im e n ta l and p r e d i c t e d v a lu e s o f s i z e improvement a re seen t o be in

e x c e l l e n t agreement. The lower than expec ted v a lu e c o r re s p o n d in g t o =

2200 c c /m in is due t o t h e low v a lu e o f G o b ta in e d in e x p e r im e n t 111176.

Inc rem en ta l O p e ra t in g Cost w i t h FDS

Fines d i s s o l v i n g can be accom p l i shed us ing h e a t i n g a n d / o r d i l u

t i o n . On a l a b o r a t o r y s c a l e , the f i n e p a r t i c l e s can e a s i l y be d e s t ro y e d

by h e a t i n g , as was done in the p r e s e n t s t u d y . The l a b o r a t o r y f i n e s

d e s t r u c t i o n system c o n s i s t e d o f a steam h e a te r u n i t , a h o l d i n g t a n k , and

a c o o l e r u n i t . A te m p e ra tu r e r i s e th ro u g h the h e a te r o f about 10°C was

Tab le 2. Expe r im en ta l and P r e d i c te d S iz e Improvement.

G = 2 .3 ym/min i = 5

V = 18,000 cc K = L fQr /G V

S ize Improvement, L, / L .2 a l

R%

( c c /m in )

Lp,

Exp. (ym) K X

<!>(%)

Equa t ion (2 4 ) , Approx im ate Equat ion (2 8 ) ,

Exact (a)

Exper imen ta l S ize

1mprovement(a) (b) (c)

4 .8 1500 55 2 .0 2 .0 1.3 1.28 1.40 1.20 . . 1.30 1.30

6 .5 2200 82 4 .4 3.3 4 ,5 1.51 1.75 1.42 1.53 1.35

8 .5 3000 150 11 .0 5 .7 35.0 2.04 2.20 1.80 2.03 2.00

(a) Using t h e . e x p e r im e n t a l s e m i - l o g i n t e r s e c t i o n f o r Lp.(b) Using the t h e o r e t i c a l s e t t l i n g c o r r e l a t i o n f o r Lp, Equa t ion ( 3 7 ) .(c) Using the e x p e r im e n ta l s e t t l i n g c o r r e l a t i o n f o r Lp, Equat ion (38)

64

r e q u i r e d to t o t a l l y d i s s o l v e the f i n e s u t i l i z i n g a re s id e n c e t im e o f

c a . 1 m in u te . S ince t h i s is ah e xpe n s ive sys tem, f i n e s would be

d e s t ro y e d by d i l u t i o n r a t h e r than h e a t i n g on an i n d u s t r i a l s c a le .

Potass ium c h l o r i d e is produced i n d u s t r i a l l y in a f l a s h c o o l i n g

c r y s t a l l i z e r where the p r e c i p i t a t i o n o f the s a l t is induced by c o o l i n g

produced by f l a s h e v a p o r a t i o n . About 8-10% o f the w a te r c o n te n t in the

feed s o l u t i o n i s f l a s h e d . S ince th e magma a l s o c o n t a i n s N a d , wh ich i s .

p r e c i p i t a t e d by e v a p o r a t i o n , 50% o f the f l a s h e d w a te r is r e c y c le d t o the

c r y s t a l l i z e r t o m a in t a i n KC1 p u r i t y . I f a f i n e s d e s t r u c t i o n system were

t o be implemented on such a c r y s t a l l i z e r , a l o g i c a l and in e x p e n s iv e p r o

cedure would be t o make use o f the w a t e r f l a s h e d o u t t o d i s s o l v e the

n u c l e i . T h is t e c h n iq u e a vo ids inc re ased o p e r a t i n g c o s ts f o r f i n e s

h e a t i n g and p r o v id e s the w a te r necessary t o m a in t a i n the re q u i r e d p u r i t y

l e v e l .

A l o g i c a l l o c a t i o n o f the f i n e s d e s t r u c t i o n d e v i c e i s i n s i d e the

c r y s t a l l i z e r . T h is l o c a t i o n m in im iz e s r e q u i r e d p i p e s , th e need f o r l e a k -

p r o o f c o n s t r u c t i o n , hea t lo ss p ro b lem s , and e x t r a mechan ica l d r i v e s . The

f i n e s s t ream coming o u t o f the t r a p i s mixed w i t h d i l u t i o n w a te r in an

a u x i l l i a r y t a n k and then r e c y c le d t o the c r y s t a l l i z e r .

Some sample c a l c u l a t i o n s a re p re se n ted here t o a l l o w e s t i m a t i o n

o f th e in c re m e n ta l o p e r a t i n g c o s t f o r a c r y s t a l l i z e r w i t h a f i n e s t r a p ,

c o n s i d e r i n g both h e a t i n g and d i l u t i o n te c h n iq u e s f o r f i n e s d e s t r u c t i o n .

65

Fines D e s t r u c t i o n by Hea t ing

T h is c a l c u l a t i o n was done f o r t he l a b o r a t o r y c r y s t a l l i z e r o f

18 l i t e r s c a p a c i t y and c o n s id e rs a re s id e n c e t im e o f a p p r o x im a t e l y 45 min

and a s l u r r y d e n s i t y o f 50 g / 1 .

The volume and the re s id e n c e t im e de te rm ine the p ro d u c t r a t e ,

Op = V /t = 400 c c / m in , and the p r o d u c t r a t e i s then P - OpM-p = 28.8x10 ^

t o n s / d a y . The f ines removal r a t e . Op,, was d e te rm ined as a f u n c t i o n o f R

f rom 0R = (R - l ) O p .

I t i s necessa ry t o e s t i m a te the d e n s i t y and hea t c a p a c i t y o f the

KC1 s o l u t i o n :

1. D e n s i t y o f a KC1 aqueous s o l u t i o n ( P e r r y , 1973) :

P40°C = 1•030 g / c c f o r a 4-8% KOI s o l u t i o n .

2. Heat c a p a c i t y ( P e r r y , 1.973) : a t 40°C, the s o l u b i l i t y i s

19.5 g KCl /100 g s o l u t i o n , wh ich co r responds t o 0 .2 5 moles o f KC.l

and 4 .47 moles o f HÔ. Then the m o la r f r a c t i o n o f KC1 in the

s o l u t i o n i s :

0 .26 . .

4 .47 + 0 . 2 6 ° ' 5

The c o r re s p o n d in g v a lu e o f th e hea t c a p a c i t y was e s t im a te d a t

Cp = 0 .775 c a 1 /g - ° G .

3. I t was c o n s id e re d t h a t an i n c re a s e in t e m p e ra tu r e o f 10°C is / '•

enough t o d i s s o l v e t h e p a r t i c l e s . Th is assumpt ion c o n s id e rs both

s o l u b i l i t y changes and d i s s o l v i n g k i n e t i c s , and i s c o n s i s t e n t

w i t h l a b o r a t o r y e x p e r ie n c e w i t h the KC1 system.

66

Then th e steam r e q u i r e d t o hea t the f i n e s s t ream and d i s s o l v e the

n u c l e i Is g i v e n by:

Qr P Cp AT steam = —— g /m in

where 4H - ^ „ a t e r - (1190 - 330 .5 ) B t u / l b .

The amount o f t h e steam r e q u i r e d depends on the v a lu e o f R. The

c o s t o f t h e steam was co n s id e re d t o be $1 .75 /1000 l b .

The w a t e r necessa ry t o cool the s o l u t i o n r e c y c le d t o the c r y s t a l -

l i z e r must a l s o be c a l c u l a t e d as a f u n c t i o n o f R. The amount o f w a te r

r e q u i r e d i s :

% p" Cp AT

w a te r ' p Cp AT •'w Pw w

A t 50°C, p . = 0 .990 g / c c and CD = 1 c a l / g - ° C ; AT was taken asw a te r ■ Pwater w

10°C. The c o s t o f w a te r Was c o n s id e re d t o be $ 0 .26 /1000 gal .

F i n a l l y , a rough e s t i m a t i o n o f th e pumping c o s t (pump + .

c i r c u l a t i n g ) was done c o n s i d e r i n g a d e p r e c i a t i o n o f 10 y e a r s . Pump c o s t

da ta were o b ta in e d f rom P e r ry (1973, p. 6 -6 ) f o r s t a i n l e s s s t e e l pumps.

The r e s u l t s a re summarized in Tab le 3•

F ines D e s t r u c t i o n by D i l u t i o n

C a l c u l a t i o n s were made based on the e x p e r im e n ta l c o n d i t i o n s p r e s

en t in t h i s s t u d y . However, the c o n c l u s i o n s a p p ly t o an i n d u s t r i a l

s e a le as w e l l .

T a b l e 3 . C o s t E s t i m a t i o n f o r F i n e s D e s t r u c t i o n b y H e a t i n g .

R%

( cc /m in )

Steam ( $ / t o n o f p ro d u c t )

Water ( $ / t o n o f p ro d u c t )

Pump + Power ( $ / t o n o f p r o d u c t )

T o ta l Cost L , / L , , ( $ / ton o f 2 1 p r o d u c t ) Expected

3 800 2 .5 8 2.21 .0 .1 9 3 4.99 1.08

5 1600 5.16 4.41 0.392 9.97 1.25

7 2400 7.75 6 .62 0 .598 14.97 1.50

10 3600 11.68 9 .93 0.886 22.50 2 .16

15 5600 18.18 15.45 1.389 35.03 3.40

ON —4

68

C a l c u l a t i o n s compared th e d i l u t i o n w a te r a v a i l a b l e f rom f l a s h i n g

th e feed s o l u t i o n and th e w a te r r e q u i r e d t o d i s s o l v e the f i n e s l e a v in g

th e t r a p .

Water in Feed S o l u t i o n

The te m p e ra tu re o f t h e feed s o l u t i o n was 70°C. From the mutual

s o l u b i l i t y cu rve o f KC1 -NaCl in w a t e r , a t t h i s te m p e ra tu re in 100 grams

o f s o l u t i o n t h e r e were 27 .3 grams o f KC1, i . e . , 72 .7 grams o f HÔ. The

d e n s i t y o f t he s o l u t i o n a t t h i s t e m p e ra tu r e was e s t i m a te d a t 1.015 g / c c

( P e r r y , 1973) . The feed r a t e t o t h e c r y s t a l l i z e r was 400 c c / m i n . T h us :

W = 72- 7 9 H2° . n11- g so l u t l o n r nn cc so l u t i o n1 100 g s o l u t i o n 1 cc s o l u t i o n m inu te

g H O= 295 — - T - m inu te

is the t o t a l w a te r in th e feed s o l u t i o n .

Water Requi red t o D i s s o l v e th e Fines

The w a te r r e q u i r e d w i l l depend on the f i n e s removal r a t e , ,

i . e . , on th e v a lu e o f R as w e l l as f i n e s s i z e , Lp. T a b le 4 summarizes

t h e w a te r re q u i rem en t as a f u n c t i o n o f R. The c o r r e s p o n d in g va lu es of- Lp

were o b ta in e d f rom the e m p i r i c a l c o r r e l a t i o n (E qua t ion 3 8 ) . K was c a l

c u l a t e d from. Equa t ion (31) and the v a lu e s o f X and <f> were g r a p h i c a l l y

d e te rm in e d . The v a lu es o f Pp were c a l c u l a t e d c o n s i d e r i n g t h a t Pp - <j>-P =

<j>QpMr f

The d i l u t i o n w a te r r e q u i r e d co r respon ds t o the w a t e r necessary t o

fo rm a s a t u r a t e d s o l u t i o n w i t h the f i n e s coming o u t o f the f i n e s t r a p .

T a b l e 4 . W a t e r R e q u i r e m e n t f o r F i n e s D e s t r u c t i o n by D i l u t i o n .

Rqr

( c c /m in )V

(cm/sec)l f

(vim) K X 4>PF

(g /m in )

WaterRequired(g /m in )

wExpected

3- 800 0 .30 21 0 .4 2 0 .7 0.0012 0.024 0.121 1.08

5 1600 0 .60 50 2 .02 1.8 0.0072 0.144 0.594 1.25

• 7 2400 0 .90 84 5 .0 9 3.6 0.07 1.40 5.78 1.50

.8 2800 1.05 102 7.21 4 .4 0.14 2 .8 11.56 1.70

. 9 3200 1 .20 121 9 .8 0 5.3 0.23 4 .6 18.99 1.92

10 3600 1.35 141 12.82 6 .2 0 .42 8 .4 34.68 2.16

15 5600 2 .10 247 33.4 9 .8 4 .7 94 .0 388.1 3.40

cr\VO

7 0

At 40°C, the s o l u b i l i t y o f KC1 in t h e KC1-NaCl-wate r sys tem i s 19-5 9 KC1

per 100 g o f s o l u t i o n ; t h u s , the r e q u i r e d w a te r can be c a l c u l a t e d f rom

th e s im p le r e l a t i o n s h i p :

PF = 19.5 PF + X 100

where X is the r e q u i r e d w a t e r .

I f 8% o f the w a te r c o n t e n t in t h e feed s o l u t i o n is f l a s h e d t o

produce the r e q u i r e d c o o l i n g , then t h e w a t e r a v a i l a b l e i s 23 .6 g HgO/min

and p r o v id e s the r e q u i r e d w a te r f o r a v a lu e o f R up t o S. For va lu es o f

R h ig h e r than 9, the d i f f e r e n c e between the a v a i l a b l e and the r e q u i r e d

w a te r has t o be s u p p l l e d . A c o s t e s t i m a t i o n f o r f i n e s d e s t r u c t i o n by

d i l u t i o n was done c o n s i d e r i n g pumps, p i p i n g , and power f o r pumping c o s ts

as w e l l as the r e s u l t i n g water , c o s t f o r R >_ 10. These c a l c u l a t i o n s

i n v o l v e o n l y i n t e n s i v e v a r i a b l e s a n d / o r r a t i o s and w o u ld , t h e r e f o r e , h o ld

f o r f u l l - s c a l e equ ip ment .

Ta b le 5 shows a compar ison between t o t a l c o s ts o f f i n e s d e s t r u c

t i o n by d i l u t i o n and h e a t i n g . I t appears obv ious t h a t , f o r a d e s i r e d

s i z e improvement , f i n e s d e s t r u c t i o n by h e a t i n g is too e xp e n s ive and

d e s t r u c t i o n o f n u c l e i by d i l u t i o n is more a t t r a c t i v e a t an i n d u s t r i a l

s c a le .

71

Tab le 5- F ines D e s t r u c t i o n by D i l u t i o n ve rsus H e a t in g Techn ique.

R

D i l u t i o n T o ta l Cost

( $ / ton o f p r o d u c t )

H e a t ing T o ta l c o s t

( $ / t o n o f p r o d u c t )

Expected S ize Improvement

‘ W

3 0 .109 4 .9 9 1.08

5 0 .224 9 .97 1.25

7 0 .329 14.97 1.50

10 0 .532 22.50 2.16

15 2.050 35.03 3.40

SUMMARY AND CONCLUSIONS

• A nucTeat i o n - g r o w th r a t e k i n e t i c s model wh ich a d e q u a te l y d e s c r i b e s

the po tass iu m c h l o r i d e sys tem can be expressed as:

B° = O.657 M ^ ' ^ n o / c c - m in

These k i n e t i c s were de te rm ine d a t f i x e d c o n d i t i o n s o f re s id en c e

t im e , a g i t a t i o n , c r y s t a l l i z a t i o n t e m p e r a tu r e , and feed s a t u r a t i o n tem pe ra

t u r e in t h e normal o p e r a t i n g range o f these . v a r i a b l e s . I t was no t the

purpose o f t h i s s tu d y t o de te rm ine t h e i n d i v i d u a l e f f e c t s o f these v a r i

ab le s on the sys tem , b u t t o p r o v id e a re ason ab le k i n e t i c s model t o a l l o w

computer s i m u l a t i o n and s t a b i l i t y c o n t r o l s tu d y o f t h e KC1 system. The

same p o w e r - 1 aw model f i t s bo th MSMPR and FDS d a ta , in bo th s t e a d y - s t a t e

and dynamic o p e r a t i n g c o n d i t i o n s .

P a r t i c l e s l a r g e r than th e c u t s i z e , Lp, were a p p a r e n t l y measured

in t he f i n e s t r a p s t ream . These l a r g e p a r t i c l e coun ts a re a t t r i b u t e d t o

c o in c id e n c e e f f e c t s a n d / o r i n s t r u m e n t n o i s e . These p a r t i c l e counts a re

c o n s id e re d s p u r i o u s ; I f f i n e s were Indeed removed a t these l a r g e r s i z e s ,

the p o p u l a t i o n d e n s i t y in t he p ro d u c t wou ld be much lo w e r than obse rved .

The c r i t i c a l s i z e Lp o b ta in e d f rom p o p u l a t i o n p l o t s was c o r r e

l a t e d w i t h the upward v e l o c i t y i n s i d e t h e f i n e s t r a p assuming p lug f l o w .

The va lues o f Lp p r e d i c t e d by s e t t l i n g t h e o r y a re g r e a t e r than those

e x p e r i m e n t a l l y o b t a i n e d . The assumpt ion o f p lug f l o w in the t r a p has

been d is c u s s e d by Juzaszek and Larson (1977) and compared w i t h the

. 72

■ 7 3

assumpt ion o f a l a m in a r f l o w . T h e i r r e s u l t s do n o t c o n c l u s i v e l y d i s

c r i m i n a t e between models and a re a l s o s u b j e c t t o the c r i t i c i s m t h a t lo w e r

p ro d u c t p o p u l a t i o n d e n s i t i e s wou ld have been observed had these l a r g e r

f i n e s been removed a t t h e ra te s i n d i c a t e d .

Design e q u a t i o n s were deve loped t o p r e d i c t t h e b e h a v io r o f a

f i n e s d e s t r u c t i o n sys tem. These e q u a t i o n s were p l o t t e d as f u n c t i o n s o f

t he im p o r t a n t v a r i a b l e g ro u p in g s o v e r a r e a l i s t i c range . These des ign

c h a r t s a l l o w q u i c k p r e d i c t i o n ( w i t h o u t u s in g a computer) o f the s i z e

improvement expec ted by im p lem en t ing a FDS. C a l c u l a t i o n s u s ing these

c h a r t s i n d i c a t e d the " p o i n t " f i n e s t r a p i d e a l i z a t i o n was a c c u r a te f o r

amounts o f d i s s o l v i n g up to a t l e a s t 35% o f n e t p r o d u c t i o n .

A p h y s i c a l l i m i t a t i o n was found t o e x i s t f o r t h e f r a c t i o n o f

f i n e s d i s s o l v e d . T h us , t h e r a t i o o f d i s s o l v e d f i n e s t o n e t p r o d u c t can

o n l y a s y m p t o t i c a l l y approach th e v a lu e ( R - 1) no m a t t e r how h igh t h e d i s

s o l v i n g pa ram ete r X. When th e o p e r a t i n g c o n d i t i o n s in t he f i n e s t r a p a re

such t h a t t h e f r a c t i o n o f f i n e s d e s t ro y e d and r e c y c le d approaches th e

v a lu e ( R - l ) , then th e d i s t r i b u t i o n o f f i n e s co r responds t o t he e n t i r e.1

d i s t r i b u t i o n in the c r y s t a l l i z e r , i . e . , t h e r e a re no p a r t i c l e s a t s i z e s

l a r g e r than Lp p r e s e n t in the system and th e f i n e s sys tem i s o b v i o u s l y

i n e f f e c t i v e . Such ovei—d i s s o l v i n g o f t he f i n e s f r a c t i o n (such t h a t t h e r e

a re no s u r v i v i n g n u c l e i t o p o p u la te the p ro d u c t s i z e ranges) has been

obse rved i n d u s t r i a l l y and t y p i c a l l y r e s u l t s in w i l d o s c i l l a t i o n s o f the

CSD.. T h us , c h o ic e o f R and Lp must be based on f e a s i b i l i t y as w e l l as

economic c o n s i d e r a t i o n s .

7 4

A rough c a l c u l a t i o n o f FDS c o s t sugges ts t h a t t he im p le m e n ta t io n

o f f i n e s d i s s o l v i n g produces a lm o s t no a d d i t i o n a l o p e r a t i n g expenses i f

t h e d e s t r u c t i o n o f n u c l e i i s done by d i l u t i o n . T h is r e s u l t , o f c o u rse ,

assumes t h a t such amounts o f d i l u t i o n w a t e r a re a v a i l a b l e , e . g . , f rom

f l a s h c o o l i n g . F ines d e s t r u c t i o n by h e a t i n g is e x p e n s iv e and would n o t

compete e c o n o m ic a l l y w i t h d e s t r u c t i o n by d i l u t i o n in an i n d u s t r i a l u n i t .

In f a c t , i n d u s t r i a l . c r y s t a l ! i z e r s seldom use h e a t i n g as a method o f f i n e s

d e s t r u c t i o n .

\

APPENDIX A

SIZE IMPROVEMENT WITH FDS

E qua t ion (24) was deve loped f o r th e a pp ro x im a te ease o f a " p o i n t

f i n e s t r a p . A more r i g o r o u s a n a l y s i s o f s i z e improvement c o n s id e rs the

gene ra l case where th e mass o f p a r t i c l e s d i s s o l v e d i s n o t n e g l i g i b l e

compared t o the mass o f p r o d u c t . What f o l l o w s i s the development o f

E qua t ion (28) f o r t h i s gen e ra l case.

The t o t a l mass o f c r y s t a l s per u n i t volume o f s l u r r y i s :

00 •?Mt = p kv / L ndL (17)

An e q u i v a l e n t e x p r e s s io n f o r an e x p o n e n t i a l d i s t r i b u t i o n i s :

M, = 6pk n ° (Gt ) ^ (19)I V

The s o l i d s c o n c e n t r a t i o n w i t h o r w i t h o u t f i n e s d e s t r u c t i o n is

g ive n by E qua t ion (17) upon s u b s t i t u t i o n o f the p ro p e r f u n c t i o n f o r popu

1 a t ion d e n s i t y . When f i n e s d e s t r u c t i o n i s implemented and the mass o f

f i n e s d i s s o l v e d is n o t n e g l i g i b l e , the s l u r r y d e n s i t y , ML , can be2

expressed by Equa t ion (17) in the fo rm :

IF . » .

M = pk [ / n„L d l + / n l ^ d l ] ( A . l )2 0 L,

: 75 '

For a Class I I sys tem:

o r o of n 1L dL = / n „L dL + / n . L ^ d l0 0 L_ 2F

( A . 2 )

where n^ and n^ a re t h e p o p u l a t i o n d e n s i t i e s w i t h o u t and w i t h f i n e s

d i s s o l v i n g . For an e x p o n e n t i a l d i s t r i b u t i o n :

n 1 = n ° 1 exp ( - L / G ^ r )

n2 = n ° 2 exp ( -LR/G2t )

n2 = ^n °2 exp

0 < L < 00

0 < L < Lr

Lp < L <

(A. 3)

S u b s t i t u t i n g ( A . 3) i n t o ( A . 2) and making uSe o f E qu a t ion (19)

x r

n 0. (G.t ) ^ / e Xx -d x = n° (G9t )^ / e Rxx^dX 0 0

» CO

+ gn°2 (G2t ) f e Xx dx XF

(A . 4)

where x = L/Gt . L e t t i n g p = Rx, the f i r s t i n t e g r a l on the r i g h t - h a n d

s i d e can be r e w r i t t e n a s : ' .

Xp KXp

f e Rxx ^d x = —r ’ f e Pp^dp0 R 0

R e c a l l i n g t h a t t h e i n c o m p l e t e gamma f u n c t i o n i s :

i t ?i ( t ) = t " / e "p dp

■ 0( 2 9 )

then Equa t ion ( A . 4) becomes:

. n° (G t ) . h

n ° 1 ( ^ 1 T ) = If---------u (RXp) + Sn02 (G2r ) [1 - w (x p ) ]R

Rea r r a n g i n g :

(A .5)

vG1t

4 n°. to(RXp)7j— + B [1 - w(xp) ] (A. 6)

S ince B° = n°G = k^G 'M j "* , then n° = k^G* ^My"*. For a c o n s t a n t s l u r r y

d e n s i t y and r e t e n t i o n t im e :

G „ t V I { d „ V 1

Gr(A .7)

'1

Combining ( A . 7) w i t h ( A . 6 ) , the e x p r e s s io n f o r s i z e improvement i s

o b ta in e d as:

’ d2 i i+3

1 ,co (Rxp) ( A . 8 )

+ 3 [1 - o)(xp) ]

S u b s t i t u t i n g 6 = e and Xp = Lp/Gr = X/ ( R - i ) , the f i n a l e x p r e s s io n f o r

s i z e improvement , Equ a t ion (2 8 ) , i s o b t a i n e d :

Ld l / i + 32 (28)Ld1 R

[ i - t o ( X / R - l ) ]

The f r a c t i o n o f ne t p r o d u c t i o n wh ich i s d i s s o l v e d is expressed by

E qua t ion (3 4 ) . The d e t a i l e d deve lopment o f t h i s e q u a t i o n is p resen ted

he re .

The t o t a l p r o d u c t i o n r a t e , P = Q.pf' j j can be e xpressed a c c o r d in g

t o E qua t ion (17) as:

S i m i l a r l y , t he r a t e o f f i n e s d e s t ro y e d i s :

PF = QRp kv / nL3d l0

( 3 3 )

T h us , the f r a c t i o n wh ich i s d e s t r o y e d , <f>, can be expressed as:

79

For an e x p o n e n t i a l d i s t r i b u t i o n :

x p

QRn°(Gt )^ / e x^dx

"

apn- (GT) ' ,.[ / F = " Rx x 3dx + / e" x F<R- , ) e - x x 3dx]0 x F

(A.TO)

- X p t f T - l ) _ x

where e = e = g . S ince = (R-l)Q.p and Rx = p, t h i s e q u a t i o n

becomes:

Rx_

/ e Pp 3dp

♦ R- ----------° ----------------------------------------- (A. I I ), F - P 3 - x f ( R - D — „ x ,—r- / e Pp dp + e / e x dxR 0 x F

Using t h e d e f i n i t i o n g iv e n by E qua t ion (29) f o r the gamma d i s t r i b u t i o n :

R- li (Rx f )

* = — " - X p ( R - l )—r- w (Rxp) + e [ i - co(xF) ]R

Exp ress ing ^ as a f u n c t i o n o f X:

R-l

1 + r " e ' 1 [ 1 m0) (RA/R-T)

NOMENCLATURE

A- t o t a l s u r f a c e area p e r u n i t volume o f su spe n s io n .1 y . ■ /. . -. ■ ■ ; ■■ ‘

B p a r t i c l e b i r t h f u n c t i o n .

B ° n u c l e a t ion r a t e .

C s o l u t e c o n c e n t r a t i o n .

C : s a t u r a t e d s o l i d c o n c e n t r a t i o n .: y - ; -vv ' - ■ . • ■■ :D p a r t i c l e dea th f u n c t i o n .

G growth r a t e .

i o r d e r o f n u c l e a t i o n ( s u p e r s a t u r a t i o n ) .

j o r d e r o f n u c l e a t i o n (su sp ens io n ) . .

k c o n s t a n t in g rowth r a t e k i n e t i c s e q u a t i o n ..9

c o n s t a n t in n ucT e a t io n k i n e t i c s e q u a t i o n .

k v o l u m e t r i c shape f a c t o r .

L p a r t i c l e s i z e .

Ly dom inan t p a r t i c l e s i z e .

Lp upper l i m i t i n f i n e s d e s t r u c t i o n .

m t o t a l mass o f i n d i v i d u a l p a r t i c l e .P .• ■■ '

Mp s l u r r y d e n s i t y in f i n e s t r a p .

M y s l u r r y d e n s i t y in c r y s t a l 11 z e r .

n p o p u l a t i o n d e n s i t y a t s i z e L.

n . i n l e t p o p u l a t i o n d e n s i t y ,

n o u t l e t p o p u l a t i o n d e n s i t y .

n° n u c l e i p o p u l a t i o n d e n s i t y .

80

81

N c u m u la t i v e members o f a p o p u l a t i o n d i s t r i b u t i o n .

P p r o d u c t i o n r a t e .

Pp f i n e s p r o d u c t i o n r a t e .

Q.p feed v o l u m e t r i c f l o w r a t e .

Qp p r o d u c t v o l u m e t r i c f l o w r a t e .

Qp f i n e s v o l u m e t r i c f l o w r a t e .

R ' f i n e s d e s t r u c t i o n r a t e , (Qp + Q p) /Q p .

s s u p e r s a t u r a t i o n as C - C^.

t t im e .

v s e t t l i n g v e l o c i t y .

V suspens io n vo lume,

x dimens ion le s s c r y s t a l s i z e .

3 f r a c t i o n o f c r y s t a l s s u r v i v i n g the f i n e s t r a p .

(f> f r a c t i o n o f n e t p r o d u c t i o n .

X e x p o n e n t i a l decay r a t i o ,

p c r y s t a l d e n s i t y .

p s o l u t i o n d e n s i t y .s . ■

t mean res idence , t im e ,

w w e ig h t d i s t r i b u t i o n f u n c t i o n .

S u b s c r i p t s

1 w i t h o u t f i n e s d i s s o l v i n g system.

2' w i t h f i n e s d i s s o l v i n g sys tem.

REFERENCES

Beckman, J . R . , Ph.D. D i s s e r t a t i o n , Dept , o f Chemical E n g in e e r in g , U n i v e r s i t y o f A r i z o n a , Tucson (1976) .

C a l d w e l l , H. B . , Ind . Eng. Chem. , 2_, 115 (1961) .

C l o n t z , N. A . , and McCabe, W. L . , Chem. Ena. P ro g r . Symp. S e r . , 67, 6 (1971).

H e l t , J . E . , and La rson , M. A . , p re s en ted a t th e Wor ld Congress on Chem. E n g r . , Amsterdam, The N e th e r la n d s , June 30, 1976.

Ju z a s z e k , P . , and Larson , M. A . , AlChE J . ( i n p r e s s ) , 1977.

Larson , M. A . , and Gars i d e , J . , Chem. E n g r . , Lond . , No. 261, 318 (1973 ) .

Larson , M. A . , and Rando lph, A. D . , Chem. Engr. P r o a r . Symp. S e r . , 6 5 , 1 (1 969 ) .

La rson , M. A . , Timm, D. C . , and W o l f f , P. R. , AlChE J . , J_4, 448 (1968 ) .

L e i , S. L . , S h in n a r , R . , and Ka tz , S . , AlChE J . , 1 7 , 6 (19 71 ) .

Nauman, E. B . , Chem. Engr. P ro g r . Symp. S e r . , No. 110, 6 7 , 116 (1971) .

Nauman, E. B . , and Szabo, T. T . , Chem. Engr. P ro g r . Symp. S e r . , No. 110, 67, 108 ( 1 9 7 1 ) . -

P e r r y , R. H. ( e d . ) , Chemical E n g in e e r ' s Handbook, 5 th e d . , McGraw-H i l l , I n c . , New York (1973) .

Randolph, A. D. , AlChE J . , ] ± , 424 (1 965 ) .

Randolph, A. D. , Beer, G. L. , and Keener, J . P . , AlChE J . , 9_, 1140(1973) .

Randolph, A. D . , and C ise , M. D . , AlChE J . , J8_, 798 (1972) .

Rando lph, A. D . , and L a r s o n , M. A . , Theory o f P a r t i c u l a t e P rocesses ,Academic P ress , New York (1971 ) . ~ "

Randolph, A. D . , and You r tgqu is t , G. R . , AlChE J . , 18 , 421 (1972).

Saeman, W. C. , AlChE J . , 2, 107 (1 956 ) .

8 2 - ■

Saeman, W. C. , Ind. and E n g r . Chem., 5 3 , 612 (19 6 1).

T u r n b a l l , D . , and F i s h e r , J . C . , J . Chem. Rhys . , 17, 71 (1965 ) .

Volmer, M . , and Weber, A . , Z. Rhys. Chem. ( L e i p z i g ) , 119, 277 (1926).

. ; h

B:

m a m ® .

S j

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