Experimental and theoretical study of finesdestruction in a mixed suspension crystallizer
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Authors Kraljevich, Zlatica Idalia
Publisher The University of Arizona.
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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
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 72876 . . . . . . . . . . 38
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 60776 . . . . . . . . . . 40
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-h^O 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
PU
LATI
ON
D
EN
SIT
Y,
n,
num
ber/
cc
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
on
-R
£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
SOLI
DS
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4 5 6 7
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
io-'
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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3 7
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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, n
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m
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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 .
POP
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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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4 0
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
(less than 10 counts per channel)
2 0 0 4 0 0CRYSTAL S IZ E , L, MICRONS
600
F i g . 1 2 . 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 6 0 7 7 6 .
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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
(less than 10 counts per channel)
10
10*
800 1000600200 400CRYSTAL S IZ E , L, MICRONS
F i g . 1 3 . 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 2 0 3 7 7 .
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4 2
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
(less than 10 counts per channel)
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 i g . 1 4 . 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 0 6 7 6 .
. 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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Coulter Counter Data
A 2 8 0 micron aperture
o 4 0 0 micron aperture
x S ta t is t ic a l ly Insignif icant Data (less than 10 counts per channel)
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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RUN 7 / 0 6 / 7 6
Coulter Counter Data A 2 8 0 micron ap er tu re
o 4 0 0 micron aper ture x S ta t is t ica l ly Insignif icant
Data (less than 10 counts per channel)
10
104 0 0200100
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
A Sieve Data
— Computer Simulation (after Beckman, 1976)
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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Y,
n,
num
ber/
cc
mic
ron
4 9
10'
RUN 7 / 0 6 / 7 6Q r = 0 cc /m in ( M S M P R ) Q r = Qp = 4 0 0 c c /m i n
o Coulter Counter Data a Sieve Data
— 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 H^O. 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 H^O. 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.
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