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P e r g a m o nComputers chem EngngVol. 20, No. 4, pp. 427-435, 1996Copyright 1996 Elsevier Science LtdPrinted in Great Britain. All rights reserved0098-1354 95)00032-1 0098-1354/96 15.00 + 0.00

DESIGN ISSUES REL TED TO THE CONTROL OFCONTINUOUS CRYST LLIZERS

R. A. EEK,:~ A. J. HOOGENBOEZEM and O. H. BOSGRAtMechanical Engineering Systems and Control Group, Mekeiweg 2, 2628 CD Delft, Delft Universityof Technology, The Netherlands

Abstract - -This paper addresses design and control issues related to a continuous crystallization process.Design addresses the configuration of actuators for this process. Results from both model studies andfrom experimental analysis show how different operation and design choices, related to the actuators,influence the process behaviour. The main conclusion is that classified removal of small crystals (fines)and large product crystals improves the shape of the crystal size distribution (CSD) at steady-state,however, at the expense of open-loop cycling of the CSD. A simple SISO feedback control strategy thatacts on the fines removal rate can be used to stabilize these cycles effectively. Fine particles are classifiedwith an annular zone. The size of the annular zone can be decreased without affecting the dynamiccontrollability, significantly. Classified removal of product crystals from the crystallizer is achieved witha vibrating screen. The aperture size of the screen and the flow feeding the screen are consideredimportant design parameters. Good product quality is obtained with a low feed flow and a large aperturesize. Controllability analysis further reveals that for both designs only two degrees of freedom exist, i.e.a quantity related to the small particles in the CSD and the total mass production rate per uni t of timecan be controlled independently.

1. INTRODUCTIONCrystallization from solution is a well-establishedand impo rtant industrial separation and purificationprocess. The produc t of a crystallization process ismainly characterized by a spectrum of differentlysized particles, called the crystal size distribution(CSD). The economic importance of crystallizationis enormous. World-wide production of basic mat-erials, such as adipic-acid, sucrose, ammonium sul-phate and sodium chloride exceed 106 T/yr.

The quality and qua ntity of crystals produced withcontinuous crystallizers is often limited due to insta-bilities or oscillatory behaviour of the CSD. Inaddit ion to this, the shape of the size distribut ion ofcrystals often does not meet requirements for opti-mal separation of the solid crystal phase from theproduct slurry in the crystallizer downstream.Impro-vement of the process performance can be achievedwith feedback control, proper selection of the ope-ration conditions and finally with design changes,like the impleme ntatio n of additional actuationmechanisms . For a large class of crystallizers, a gooddesign incl uding a feedback contro ller will stabilizeoscillatory behaviour, suppress disturbances andproduce coarse product particles, which enhancest To whom all correspondence should be addressed.~tCurrently wit h Bayer AG. ZF-TST. Leverkusen,Germany.

solid-liquid separation in the crystallizer down-stream. The median crystal size can be used as anindicator for the presence of coarse particles.

In general the design of a crystallization processinvolves decisions on several micro- and macro-sca leprocess conditions, of which the most impo rtan t are:1. Th e effective crystallizer volume ;

2. Suspension mixing, to ensur e good micro andmacro transport characteristics;

3. The supersaturation of mother liquor, as themain driving force for crystallization;

4. The effective nucleation (birth) or seedingrate;

5. And the re sidence time of crystals.Aspects concerning mixing are reviewed by Tavare(1986). In Hallas and Ha nn an (1990) a first attemptis described for modelling of flow patterns for acertain (idealized) crystallizer geometry. Nyvilt(1992) and Mersmann (1988) present several basicrules for the design and the selection of optimaloperation conditions. In Bennet (1992), heuristicdesign rules for the selection of a crystallizer type,the process scale, and for nominal process con-ditions like the pressure in the boiling zone, theslurry residence time and the heat input areprovided. In Randolph (1965), the mixed suspen-sion mixed product removal (MSMPR) concept forcrystallizer design is introd uced. A main assump tionfor this design is that all particles have the same

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4 2 8

p r o b a b i l i t y t o b e r e m o v e d f r o m t h e v o l u m e . M o r erea l is t ic mode ls desc r ibe th is p robab i l i ty a s a func-t io n o f t h e c rys t a l s i ze . In R a n d o lp h a n d L a r s o n(1 9 8 8 ), i d e a l i z e d m o d e l s a re u s e d to s tu d y th e e f f e c to f t h e r e s id e n c e t im e o f f in e a n d l a rg e p a r t i c l e s o nth e s t a t io n a ry a n d d y n a m ic p e r fo r m a n c e o f c ry s t al -l i ze r s . A re v ie w o n m o d e l l in g a n d c o n t ro l o f c ry s t al -l i ze r s is p ro v id e d b y R a w l in g s et a l . (1993).

D e c i s iv e a n s w e rs , h o w e v e r , a s t o w h ic h d e s ig na c h ie v e s th e b e s t s t a t io n a ry p ro d u c t q u a l i ty ( e . g .s h a p e o f t h e C S D ) a n d th e b e s t a b i li t y to s u p p re s sd i s tu rb a n c e s , a re n o t p ro v id e d in l i t e r a tu re d u e toth e l a c k o f g o o d m o d e l s a n d r e l i a b le e x p e r im e n ta lf a c i l i t i e s n e c e s s a ry fo r t h e e v a lu a t io n o f t h e s em o d e l s .

Th is con tr ibu t ion wil l be l imited to a s ing le s tagesuspens ion c rys ta l l ize r and the des ign aspec ts wi l l befo c u s e d p r im a r i ly o n th e s e l e c t io n o f a c tu a t io nm e c h a n i s m s fo r t h e c ry s t a l n u c le a t io n r a t e a n d th ere s id e n c e t im e . In p a r t i c u la r , f i n e s r e m o v a l w i l l b es t u d i e d a s a n a c t u a t i o n m e c h a n i s m f o r t h e n u m b e ro f s m a l l p a r t i c le s , t o im p r o v e th e s t a t io n a ry C S D , toa c h ie v e s t a b i l i t y a n d to r e j e c t d i s tu rb a n c e s in th ee ffec t ive nuc lea t ion ra te . Secondly , p roduc t c lass i f i -c a t io n w i l l b e s tu d ie d fo r i t s a b i l i t y to e n h a n c e th es t e a d y -s t a t e p ro d u c t q u a l i ty , e . g . a l a rg e r e l a t iv en u m b e r o f c o a r s e p a r t ic l e s , a n d i t s ab i l it y to r e j e c tp ro c e s s d i s t ru b a n c e s . Wi th r e s p e c t t o th e o th e rth re e a fo re m e n t io n e d d e s ig n i s s u e s , w e a s s u m e th a ta w e l l -m ix e d 1 m 3 c o n s ta n t e f f e c t iv e v o lu m e i s p re s -e n t , i n w h ic h a s u p e r s a tu ra t io n i s m a in ta in e d b ye v a p o ra t io n o f s o lv e n t .

A f i r s t p r in c ip l e s m o d e l fo r t h e C S D d y n a m ic sin c lu d in g a m o d e l fo r t h e C S D s e n s o r , a s d e s c r ib e db y E e k et a l . (1995) , i s used as a too l fo r p rocesss im u la t io n a n d a n a ly s i s . E x p e r im e n ta l e v a lu a t io n o fp l a n t d y n a m i c s a n d c o n t r o l l e r p e r f o r m a n c e i s d o n ew i th a 9 7 0 1 p i lo t e v a p o ra t iv e c o n t in u o u s c ry s t a ll i z e r ,w h ic h i s e q u ip p e d w i th a n o n - l in e C S D s e n s o r . T h ee f fe c t o f t h e tw o a c tu a t io n m e c h a n i s m s o n th ec lo s e d - lo o p p e r fo rm a n c e w i l l b e e v a lu a te d a n d th ein f lu e n ce o f s o m e d e s ig n p a ra m e te r s w i ll b e d i s -cussed .

2 PROCESS DESCRIPTION

A s c h e m a t i c d ra w in g o f t h e s in g le s t a g e s u s p e n -s ion c rys ta l l ize r is g iven in F ig . 1 . A de ta i le d desc r ip -t io n i s g iv e n in E e k et a l . (1995) . The p rocess isa s s u m e d t o b e i s o t h e r m a l w i t h a c o n s t a n t v o l u m e o f970 I.

T h e C S D i s d e s c r ib e d b y a p o p u la t io n d e n s i tyfu n c t io n , d e f in e d b y :

A A N ( x , t )n (x , t ) = l im a , s 0 A x

R. A. EEK et al.

vessel QP p. .l l l q . . . . l

P,~ Qi,T,Fig. 1. Schematic drawing of an evaporative crystallizer,equipped with a fines and product removal system.with N ( x , t ) t h e c u m u la t iv e n u m b e r fu n c t io n d e s -c r ib in g th e n u m b e r o f c ry s t a l s p e r u n i t o f v o lu m e ,w i th a s i z e e q u a l o r l o w e r th a n x . T h e s u b s t a n c es y s t e m w e u s e d i s a m m o n iu m s u lp h a te .2 . 1 . A c t u a t o r s

A s p ro c e s s a c tu a to r s , t h e f in e s r e m o v a l r a t e Q f ,th e to t a l h e a t i n p u t d e f in e d b y : P , o t = P ~ , + P e x , a n dth e p ro d u c t r e m o v a l r a t e Q o f , a re c o n s id e re d .S e le c tiv e r e m o v a l o f f in e a n d l a rg e p ro d u c t p a r t i c le sin f lue n c e s th e r e s id e n c e t im e s o f s m a l l a n d l a rg ep a r t i c l e s in th e c ry s t a l l i z e r v o lu m e . T h e h e a t i n p u tw i l l m a in ly a f f e c t t h e s t e a d y -s t a t e m a s s p ro d u c t io nra t e . In F ig . 2 , t h e e x p e r im e n ta l f i n e s a n d p ro d u c tc lass i f ica t ion sys tem a re desc r ibed . F ines a rere m o v e d s e l e c tiv e ly w i th a n a n n u la r s e t t l i n g z o n e , i nw h ic h a s lo w u p w a rd s lu r ry v e lo c i ty i s m a in ta in e d .F in e s a re a s s u m e d to b e d i s s o lv ed to t a l ly b y h e a t in g(Pex) the f ines f low up a fu r the r 10C, in an ex te rna lh e a t e x c h a n g e r . T h e v o lu m e o f th e a n n u la r z o n e i sa p p ro x . 6 0 0 1 , w h ic h i s l a rg e in c o m p a r i s o n to th ec ry s t a l l i z e r v o lu m e . T h e c ro s s s e c t io n a l a re a o f t h eannulu s is 0 .7461 m 2 . Be lo w w e wil l inves t iga te h owth e s i z e o f t h e a n n u la r z o n e a f f e c t s t h e c lo s e d - lo o pd y n a m ic s . T h e f in e s r e m o v a l f lo w ra t e c a n b e v a r i e din th e r a n g e o f 0 . 3 -3 . 5 l / s . P ro d u c t c l a ss i f ic a t io n i sa c h ie v e d w i th a v ib ra t in g s c re e n (F ig . 2 ) . T h e f e e d

Qv

QfP e x ~ Pin

CSD ensor

o

Fig. 2. S chematic draw ing o f pilot crystallizer equippedwith fines and product classifiers.

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Design issues related to the control of continuous crystallizers 429

:~ 0 . 81 t~ ' 'h f . . . .. . . .. . . . ]

f _ _ . 2 . '0.2 Y '0 xt 200 400 600 xp 800 1000 1200size in micronsFig. 3. Re mo val efficiency functions.

f l o w t o t h e s c r e e n c a n b e v a r i e d i n t h e r a n g e o f0 . 5 - 2 . 0 l / s . W h e n p r o d u c t i s r e m o v e d w i h o u t c la s s i-f i c a t i o n Q p f w i l l e q u a l Q p .

B o t h c l a s s if i e d f i ne s a n d p r o d u c t r e m o v a l a r ed e s c r i b e d b y e f f i c i e n c y f u n c t i o n s h f ( x , t ) a n dhp(x, t ) , r e s p e c t i v e l y , w h i c h d e s c r i b e t h e p r o b a b i l i t yt h a t a p a r t i c l e w i t h s i ze x is r e m o v e d f r o m t h e m a i nc r y s t a l l iz e r v o l u m e . T h e f u n c t i o n h f is g i v e n b y :

h f x , t ) [ x Ira2 1 )B e c a u s e t h e s o l i d c o n c e n t r a t i o n is l o w ( < 1 ) , t h ec u t - s iz e x f ( t) , d e f i n e d a s t h e c r y s t a l le n g t h w h e r e t h ep r o b a b i l i t y o f w i t h d r a w a l f r o m t h e c r y s t a l l i z e rv o l u m e i s 5 0 , c a n b e d e r i v e d f ro m S t o k e s ' r e l a t i o nf o r c a l c u l a t i n g s e t t l i n g v e l o c i t i e s :

x , ( t ) = ~ 6 ) P , , / A (2 )w h e r e P n i s a c o n s t a n t f a c t o r d e t e r m i n e d b y t h ep h y s i c a l p r o p e r t i e s o f t h e s u b s t a n c e s y s t e m a n d At h e c r o s s s e c t io n a l a r e a o f t h e a n n u l a r z o n e . T h ef u n c t i o n p i s g i v e n b y :

/ x \ ,p2P p3 + (1 - - 2 p p3 ) ~ Z )hp(x) - (3 )

1 + ( 1 - 2 p p3 ) ( x / p ~ 2\ P p I /T h i s f u n c t i o n c o n t a i n s t h r e e p a r a m e t e r s : t h e c u t - s i z eX p, p a r a m e t r i z e d b y a n u n k n o w n f a c t o r P p l, t h ei m p e r f e c t i o n Pp 2 a n d a n o f f s e t P p3 . T h e l a t t e r h a s al a r g e i n f lu e n c e o n t h e v o i d f r a c t i o n o f t h e p r o d u c t .

I n F i g . 3 , a n e x a m p l e o f t h e s e S - s h a p e d c u r v e s i sg i v e n . A d i s t i n c ti o n is m a d e b e t w e e n c l a ss i f ie d a n du n c l a s si f ie d r e m o v a l o f p r o d u c t s l u r r y . I n t h e l a t t e rc a s e h p ( x ) = 1 , O < x ~ x e , w i t h x , t h e l a r g e s t p a r ti c l es i z e a c c o u n t e d f o r i n t h e m o d e l .2 .2 . Sensors

O n - l i n e m e a s u r e m e n t o f t h e C S D i s n o t s t r a i g h t-f o r w a r d . I n t h i s w o r k , f o r w a r d l a s e r l i g h t s c a t t e r in gi s a p p l i e d . T h e s c a t t e r i n g o f a n i n c i d e n t l a s e r b e a mb y a p o p u l a t i o n o f d i f f e r e n t l y s i z e d p a r t i c l e s , s u s -p e n d e d i n a l i q u o r f l o w in g t h r o u g h a n o p t i c a l c e ll ,c a n b e d e s c r i b e d b y s e v e r a l s o - c a l le d d i f fr a c t i o n

m o d e l s ( H e c h t , 1 9 87 ) , w h ic h r e l a t e a n o b s e r v e dl i gh t d i f fr a c t i o n p a t t e r n t o t h e d i s t r i b u t i o n w h e r e i to r i g i n a t e s f r o m . A s c a t t e r m a t r i x is d e v e l o p e d b a s e do n F r a u n h o f e r d i f f r a c ti o n t h e o r y . D i s c r e t i z a t i o n o ft h i s m o d e l r e s u l ts i n t h e f o l l o w i n g l i n e a r o u t p u tm o d e l : L = G n , w i t h G t h e s e n s o r m o d e l m a t r i x , Lt h e s c a t t e r e d l i g h t e n e r g y v e c t o r m e a s u r e d o n 3 1c o n c e n t r i c a i d e t e c t o r r i n g s, a n d n a d i s c r e t e p o p u l a -t i o n d e n s i t y .

M e a n i n g f u l q u a n t i t i e s , l i k e th e m e d i a n c r y s t a l s i ze( d e n o t e d a s X s0 ), a n d t h e s p r e a d o f t h e d i s t r i b u t i o n ,c a n b e e s t i m a t e d f r o m t h e d i f f r a c t io n p a t t e r n ( B o x -m a n et a l . , 1 9 9 1 ) w i t h t h e i n v e r s e d i f f r a c t i o n m o d e l .T h e s e q u a n t i t i e s a r e a d v a n t a g e o u s a s t h e y h a v e ad i r e c t p h y s i c a l i n t e r p r e t a t i o n . H o w e v e r , t h e i n v e r s ed i f f r a c ti o n m o d e l i s i l l- c o n d i t i o n e d ( s e e a l s oR a w l i n g s et a l . , 1 9 9 3 ), a n d i t i s n o t s u r e w h e t h e r t h eb e s t i n f o r m a t i o n p r e s e n t i n t h e o u t p u t s i s u s e d .T h e r e f o r e , i n a d d i t i o n t o t h is m e t h o d , w e d e c o m -p o s e d t h e s c a t t e r e d l i g h t e n e r g y v e c t o r t o i t s p ri n c i -p a l c o m p o n e n t s . T h e p r i n c i p a l c o m p o n e n t s a r e e s t i-m a t e d f r o m a n e i g e n v a l u e d e c o m p o s i t i o n o f t h ec o r r e l a t i o n m a t r i x e s t i m a t e d f r o m a l a r g e s e t o fs a m p l e s o f t h e s c a t t e r e d l i g h t e n e r g y v e c t o r sr e c o r d e d f r o m t h e s e n s o r o v e r a l a rg e t im e s p a n . F o rd e t a i l s o f t h is m e t h o d w e r e f e r t o F u k u n a g e ( 1 9 72 ) .T h e e i g e n v e c t o r s c o r r e s p o n d i n g t o t h e l a r g e s t e i g e n -v a l u e s a r e o r t h o g o n a l v e c t o r s . P r o j e c t i o n o f t h e 3 1m e a s u r e d e n e r g y v a l u e s o n t o t h e s e s i g n a l d ir e c t i o n s ,r e s u l t s in a r e d u c e d s e t o f o u t p u t s i g n a l s , w h i c h a r ed e n o t e d a s Y a , Y ,2 , I n F i g . 4 , t h e t h r e e p r i n c i p a le i g e n v e c t o r s o f s i x d i f f e r e n t c o r r e l a t i o n m a t r i c e s ,w h i c h w e r e c a l c u l a t e d f r o m s i x i n d e p e n d e n t h i s t o r i -c a l d a t a s e t s , a r e g i v e n . T h e c o r r e s p o n d i n g e i g e n v a -l u e s r e v e a l t h a t m o r e t h a n 9 5 o f t h e t o t a l s i g n a le n e r g y i s c a p t u r e d b y t h e s e t h r e e c o m p o n e n t s . T h e

0 . 5 1

o lU -_ 0 . 5 L _ ~ . i

0 . 5 1

5 10 15 20 25 30ring numberFig. 4. Principal compo nents of outpu t vector est imatedfrom eigh t different data sets .

C k C E Z O : 4 - 1

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430s ig na l Yr, i s c o r re l a t e d s t ro n g ly to th e n u m b e r o f f in epar t ic les . As wil l be i l lus t ra ted be low, th is s igna l hasexce l len t p roper t ie s to se rve as an inpu t to a s tab i l iz -ing f ines con tro l le r .

I n a d d i t i o n t o t h e C S D s e n s o r , a n i n d e p e n d e n ts e n s o r i s u s e d to me a s u re th e s o l id s f r a c t io n in th ep ro d u c t f lo w . T h e s o l id s f lo w in k g / s ) i s c a l c u la t e df r o m t h e t h i rd m o m e n t o f t h e C S D a n d t h e p r o d u c tf low:

M , = k v p c Q p f ] n x 3 d x ,w h e re k v i s a s h a p e f a c to r , a n d P c th e d e n s i ty o fc rys ta ls .

3 PROCESS DYNAMICS

Crys ta l l iza t ion p rocesses a re in p r inc ip le d is t r i -b u te d p a ra me te r p ro c e s s e s in w h ic h th e d y n a mic sa re ma in ly in f lu e n c e d b y c ry s t a l g ro w th , b i r th a n dc ry s t a l r e mo v a l . T h e c ry s t a l l i z e r c o n s id e re d h e re i sd e s c r ib e d b y a s u p e r s a tu ra t io n b a la n c e , fo r t h es u p e r s a tu r a t i o n o f m o t h e r l i q u o r a n d a p o p u l a t i o nb a l a n c e f o r t h e d y n a m i c s o f t h e C S D . T h e s u p e rs a -tu ra t io n i s a n o n l in e a r O D E d e s c r ib e d b y :

d A C t )d ~ - ~ fa c U , n , t ) , 4)with th e p rocess inpu ts : u = {Qf, Qp , eto t} . Th e sup er-s a t u r a t i o n m a i n l y d e t e r m i n e s t h e g r o w t h r a t e o fp a r t i cl e s . H o w e v e r , s u p e r s a tu ra t io n ma y a l s o a f f e c tth e p u r i ty o f cry s t a ls , t h e i r me c h a n ic a l p ro p e r t i e s ,a n d t he i r m o r p h o l o g y M e r s m a n n , 1 9 88 ). T h e r e f o r eth e l e v e l o f s u p e r s a tu ra t io n a t w h ic h th e c ry s t a l l iz e ro p e ra t e s i s t a k e n a s a n imp o r t a n t d e s ig n p a ra me te rfo r w h ic h o f t e n a n u p p e r l im i t e x i s t s . B e lo w th ere l a t iv e s u p e r s a tu ra t io n in p e rc e n ta g e o f t h e s a tu -ra t io n c o n c e n t ra t io n i s u s e d .

T h e e v o lu t io n in t ime o f t h e C S D i s d e s c r ib e d b yth e p o p u la t io n b a la n c e :

O n x , t ) O G ~ x , t ) n x , t )- - d -Ot OxI-Qf t)yf Qpf t )+ t ) + V h p x ,~ h f x , t ) ]

x n x, t) = 0, 5)with the in i t ia l cond i t ion ns x , to ) , t h e b o u n d a r yc o n d i t io n : n x o , t) = B t ) / G k t ) and ) , f a mass sepa-ra t io n e f f i c i e n cy f a c to r . T h e b i r th a n d g ro w th r a t e o fc ry s t a l s a re b o th d e s c r ib e d b y e mp i r i c a l r e l a t io n s

E e k e t a l . , 1995) . In F ig . 5 , the p rocess dynamicsa re fu r th e r e x p la in e d . T h e s u p e r s a tu ra t io n , d e t e r -min e d f ro m m a s s a n d h e a t b a l a n c e s , a f f e c t s b o th th e

R A E E l < e t a l

removal

ACCSD

Fig. 5. Interconnection diagram for a s ingl e stage con-tinuous crystallizer from Randolph and Larson, 1988).

g ro w th r a t e o f c ry s ta l s s u s p e n d e d in th e s lu r ry a n dth e b i r th r a t e o f n e w c ry s t a ls . T h e b i r th r a t e ma in lyd e te rmin e s th e b o u n d a ry c o n d i t io n o f t h e c ry s t a lp o p u la t io n . T h e g ro w th r a t e , t o g e th e r w i th th ere mo v a l r a t e o f p a r t i c le s , d e t e rm in e s th e s lo p e o fth e p o p u la t io n d e n s i ty fu n c t io n n x , t ) . For a s ta -t i o n a ry c ry s t al l i z er , o p e ra t e d w i th a c o n s ta n t g ro w thra t e G e = g , w i th h f = 0 a n d hp = 1 , t h e p o p u la t io ndens i ty vs s ize desc r ibes a s t ra igh t l ine with s lope- Q p / g V o n a s e mi lo g y p lo t . F o r a n o n s ta t io n a ryc ry s t a l l i z e r c y c l in g o f t h e C S D o c c u r s d u e to th ep h y s ic a l f e e d b a c k lo o p s s e e F ig . 5) b e tw e e n th eC S D a n d t h e b i r t h r a t e , a n d b e t w e e n t h e C S D a n dth e s u p e r s a tu ra t io n , v i a th e to t a l s u r fa c e a re a o fp a r t i c le s S A ) a v a i l a b le fo r c o n s u m p t io n o f t h es u p e r s a tu ra t io n .

F o r s imu la t io n a n d a n a ly s i s p u rp o s e s , t h e p o p u la -t io n b a la n c e m o d e l i s l u mp e d w i th a f in i t e d i f f e re n c es c h e me a n d l in e a r i z e d a n a ly t i c a l ly b y a p p ly in g af i r s t -o rd er T a y lo r a p p ro x im a t io n d e W o l f , 1 99 0 ).T o p e r fo rm a n a n a ly t i c l i n e a r i z a t io n i s l a b o r io u s ,h o w e v e r , t h i s a p p ro a c h w a s n e c e s s a ry to a v o idn u me r ic a l p ro b le ms th a t c l e a r ly ex i s t w i th n u m e r ic a ll i n e a r i z a t io n s c h e me s . T h e r e s u l t in g mo d e l u s e d fo rprocess ana lys is is chose n as a 100-order l inea r s ta te -s p a c e mo d e l { A , B , C , D } . T h e s t a t e v e c to r c o n ta in s9 9 e l e me n t s fo r t h e d i s c re t e p o p u la t io n d e n s i ty a n do n e e l e me n t fo r th e s u p e r s a tu ra t io n . A n o n l in e a rlu mp e d p o p u la t io n b a la n c e mo d e l , i n c lu d in g 1 9 9d i s c re t e e l e me n t s fo r t h e C S D , i s u s e d fo r p l a n ts imu la t io n . We fo u n d th a t t h e l i n e a r i z e d mo d e la c c u ra t e ly p re d ic t s t h e s t a b i l i ty o f t h e p ro c e s s , a n dg iv e s a s u f f i c i e n t ly a c c u ra t e d e s c r ip t io n o f t h e p ro -c e s s i n p u t - o u t p u t b e h a v i o u r i n t h e f r e q u e n c y d o -ma in .

4 FINES REMO VAL

The f i rs t des ign cons ide red he re , i s a c rys ta l l ize re q u ip p e d w i th a f in e s r e mo v a l s y s t e m, a n d a nu n c la s s i f i e d p ro d u c t r e mo v a l s y s t e m. A s th e ma ind e s ig n p a ra me te r s , t h e c ro s s se c t io n a l a re a o f t h ef ines c lass if ie r F ig . 2 ) and the rem ova l f low a re

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Design issues related to the control of continuous crystallizersc o n s i d e r e d . E q u a t i o n 2 ) r e l a t e s t h e p r o d u c t f lo wa n d t h e c r o s s s e c t i o n a l a r e a t o t h e c u t - s i z e x f.4 .1 . O p e n - l o o p p l a n t b e h a v i o u r

I n F i g . 6 , t h e o p e n - l o o p s t a r t - u p r e s p o n s e o f t h ep r o c e s s i s g i v e n f o r t h r e e d i f f e r e n t f i n e s f lo w s a n d i nF i g . 7 t h e s l u r r y d e n s i t y c o r r e s p o n d i n g t o t h e s t a r t -u p w i t h 1 . 0 1 / s f in e s r e m o v a l , i s g i v e n . A l s o t h er e t e n t i o n t i m e o f f i n e p a r t i c le s d e f i n e d a s :

r f = V / Q f ,i s g i v e n . B e c a u s e p r o d u c t i s r e m o v e d u n c l a ss i f ie df r o m a w e l l - m i x e d v e s s e l w i t h a c o n s t a n t r a t e , a n dt h e h e a t i n p u t i s c o n s t a n t , t h e r e p s o n s e o f th e s l u r r yd e n s i t y f o l l o w s a f i r s t - o r d e r e x p o n e n t i a l r e s p o n s e ,t h a t r e a c h e s a s t a t i o n a r y v a l u e a f t e r s ix r e s id e n c et i m e s , w h i c h i s 8 h s e e F i g . 7 ) . T h e e x p e r i m e n t a lr e s u l t s r e v e a l t h a t i n c r e a s i n g t h e f i n e s r e m o v a l r a t es t r o n g l y p r o m o t e s c y c l i n g o f t h e C S D .4 . 2 . M o d e l r e su lt s

M o d e l s t u d i e s p e r f o r m e d w i t h t h e n o n l i n e a r p r o -c e s s m o d e l [ w h i c h i s f i t t e d t o t h e s e e x p e r i m e n t a ld a t a a s d e s c r i b e d i n E e k et al. 1 9 9 5 ) [ , r e v e a l t h a tt h e p r o d u c t q u a l i t y o f t h e s t a t i o n a r y c r y s t a l l i z e ri n c r e a s e s w i t h i n c r e a s i n g f i n e s f l o w s . I n T a b l e 1 , t h em e d i a n c r y s t a l s i z e , t h e r e l a t i v e n u m b e r o f f i n e s :d e f i n e d a s :

R = f l f n d x / f i n d xa n d t h e l e v e l o f s u p e r s a t u r a t i o n i n p e r c e n a g e o f th es a t u r a t i o n c o n c e n t r a t i o n , f o r t h r e e d i f f e r e n t fi n e sf l o w s a r e g i v e n . A s c a n b e s e e n , f i n e s r e m o v a li n c r e a s e s t h e m e d i a n c r y s t a l s i z e , h o w e v e r , a l s oi n c r e a s e s t h e r e l a t i v e a m o u n t o f f in e s a n d t h e l e v e l

E 100 _ Qf= l.0 l /s= .4O0200

1 0 0 0 ~ Q f = 2 . 2 / s' ~ 8 0 0 [ t / ~ r tf = 7 . 3m i n

600

i2 0 0 J8O06OO4OO2OO

/ A 3 4 ~ s / ~ 1/ ~ / / 4 7 r a in / / t5 10 15 20 25 30 35 40time [hours]

Fig. 6. Exp erime ntal s tar t-up responses o f median crystalsize for three different f ines flows from E ek et a l . 1995).

40~ 2o

0

431

5 10 15 20 25 30 35 40time [hours]Fig. 7. Experim ental response of mass pro duction ratecorresponding to s tar t-up experiment with 1.01/s f inesremoval.

o f s u p e r s a t u r a t i o n . T h i s i s f u r t h e r e x p l a i n e d i n F i g . 8w h e r e w e p l o t t e d t h r e e n o r m a l i z e d s t a t i o n a r y d i s t r i-b u t i o n s a f t e r p o s i t i v e s t e p c h a n g e s i n t h e f i n e s f l o w ,t h e p r o d u c t f l o w a n d t h e h e a t i n p u t . T h e s e d i s t r ib u -t i o n s a r e n o r m a l i z e d b y s u b d i v i d i n g a n o m i n a l d i s t ri -b u t i o n , r e p r e s e n t i n g a c r y s t a l l i z e r o p e r a t e d w i t h af i n e s f l o w o f 1 .0 l/ s , a p r o d u c t f l o w o f 0 .2 1 5 l / s a n d ah e a t i n p u t o f 1 2 0 k W . C l e a r l y , f i n es r e m o v a l c o u n -t e r a c t s i t s el f b y i n c r e a s i n g t h e n u m b e r o f f i ne s i n t h er e g i o n o f v e r y s m a l l p a r t i c l e s x ~< 5 0 ~ tm ) , f o r a ni n c r e a s e d r e m o v a l r a t e . T h i s i s e x p l a i n e d b y a d d i -t i o n a l n u c l e a t i o n w h i c h i s c a u s e d m a i n l y b y t h ei n c r e a s e i n s u p e r s a t u r a t i o n a n d t h e i n c r e a s e i n t h en u m b e r o f l a r g e m o t h e r c r y s t a l s . F i g u r e 8 f u r t h e rs h o w s t h a t t h e m a i n e f f e c t s o f f i n e s a n d p r o d u c tr e m o v a l a n d h e a t i n p u t v a r i a t i o n i s a r o t a t i o n a n dt r a n s l a t i o n o f t h e s t a t i o n a r y C S D . H e n c e , o n l y tw oi n d e p e n d e n t e f f ec t s a r e o b s e r v e d . A p p l i c a t i o n o f as i n g u l a r v a l u e d e c o m p o s i t i o n t o t h e d y n a m i c l i n e a r -i z e d p l a n t m o d e l w i t h t h e i n p u t s u = { Q f , Q p , P to t}a n d t h e o u t p u t s y = { y r l g s o M t } s h o w s t h a t t h ec o n d i t i o n n u m b e r i . e . t h e r a t i o o f t h e l a r g e s t t o t h es m a l l e s t s i n g u l a r v a l u e ) i s l a r g e r t h a n 1 0 0 o v e r t h ew h o l e f r e q u e n c y d o m a i n . T h i s p o i n t s a t a l a r g ed i r e c t i o n a l i t y o f th e p l a n t [ s e e S k o g e s t a d a n d M o r a r i1 9 8 8 ) f o r d e t a i l s o n t h e u s e o f t h e s i n g u l a r v a l u e

d e c o m p o s i t i o n f o r c o n t r o l l a b i l i ty a n a ly s i s ]. A s i m -p l i f ie d p l a n t is o b t a i n e d b y o m i t t i n g t h e m e d i a n X s0a s o u t p u t a n d t h e u n c l a s si f ie d p r o d u c t f l o w a s a c t u a -t o r , w h i c h i s d e s i r e d a s th i s i n p u t h a s a l a r g e i n t e r a c -t i o n w i t h t h e u p - a n d d o w n - s t r e a m p r o c e s s e s , p r e s -e n t i n t h e s a m e c a s c a d e . A t w o - i n p u t - t w o - o u t p u tp l a n t i s t h u s o b t a i n e d , w h i c h h a s a l i m i t e d i n t e r a c -t i o n a n d a c o n d i t i o n n u m b e r l o w e r t h a n 5 o v e r t h ew h o l e f r e q u e n c y d o m a i n o f in t e r e s t . H e n c e , a m u l t i -l o o p P I c o n t r o l c o n f i g u r a t i o n i s a s s u m e d t o b e s u ff i-c i e n t . T h e f i r s t l o o p , t h a t c o n t r o l s yf ~, i s e x p e c t e d t o

T a b l e 1 . S t a t i o n a r y C S D v a l u e s a t d i f f e r e n t f i n e s f l o w sQ f ( l / s ) X s o ( ~ m ) R f ( - - ) A C ( )

0 . 0 0 5 1 7 0 . 6 4 0 . 3 00 . 7 5 6 1 1 0 . 6 6 0 . 3 71 . 5 0 7 3 5 0 . 7 3 0 . 4 92 . 2 5 8 1 1 0 . 8 0 0 . 6 4

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432 R .A . EEK et al3 [ 1 . 5 * P t o t t

crysta l s ize [microns]Fig. 8. Sim ulated effect of 50 positive changes in processinput parameter values on normalized stat ionary crystalsize distr ibutions.

s t a b i l iz e t h e C S D a s d i s t u r b a n c e s o r c y c l in g b e h a v -i o u r i s i n i t i a t e d a s d i s t u r b a n c e s t o t h e C S D i n t h es m a l l p a r t ic l e s i z e r a n g e t h a t g r o w t o t h e r e g i o n o fl a r g e p a r t i c l e s . T h e e x p e r i m e n t a l e v a l u a t i o n o f t h i ss c h e m e i s d e s c r i b e d i n t h e n e x t s e c t i o n .

I n F i g . 9 , a G a i n p l o t o f t h e l i n e a r m o d e l , w i t hf i n e s r e m o v a l Q f a s i n p u t a n d t h e f i r s t r e d u c e do u t p u t o f t h e d i f f r a c t i o n m o d e l a s o u t p u t , i s c a l c u -l a t e d f o r d i f f e r e n t v a l u e s f o r t h e c r o s s s e c t i o n a l a r e aA o f t h e f in e s c l a s s i f i e r , a n d f o r d i f f e r e n t r e m o v a lf lo w s . F r o m t h e s e p l o t s , i t c a n b e c o n c l u d e d t h a tl o w e r i n g t h e c r o s s s e c t i o n a l a r e a o f t h e s e t t l in g z o n es l i g h tl y i n c r e a s e s t h e b a n d w i d t h o f t h e o p e n - l o o pp r o c e s s , w h i c h i s d e s i r e d , a l s o a s m a l l e f f e c t o n t h ep h a s e p l o t w a s o b s e r v e d . H o w e v e r , f r o m t h e s i m u l a -t i o n s w e a l s o f o u n d t h a t t h e s o l i d s c o n c e n t r a t i o n i nt h e f i n e s l o o p c o r r e s p o n d i n g t o t h e d e c r e a s i n gv a l u e s f o r A w e r e 5 2 , 1 3 0 a n d 3 0 0 g / m 3 , r e s p e c t i -v e ly . C o n s e q u e n t l y , t h e m i n i m u m v a l u e f or A ,s h o u l d b e b a s e d o n t h e m a x i m u m c a p a c i ty o f th ef i n es d i s so l v i n g s y s t e m , r a t h e r t h a n o n t h e b a s i s o fp r o c e s s d y n a m i c s . I n c r e a s i n g t h e r e m o v a l f l o wm a i n l y e n l a r g e s t h e b a n d w i d t h o f t h e p r o c e s s , a ss h o w n i n t h e l o w e r p l o t o f F i g . 9 . T h e s o l i d s f r a c t i o ni n t h e f i n e s l o o p w i l l, h o w e v e r , a l s o i n c r e a s e s f o r a ni n c r e a s i n g f lo w . T h e r e s p e c t i v e v a l u e s f o r t h e s o l i d sc o n c e n t r a t i o n w h e r e 6 1 , 1 30 a n d 1 89 g / m 3. H e n c e ,t h e n o m i n a l f i n e s f lo w s h o u l d b e s e l e c t e d o n t h eb a s i s o f b o t h t h e d e s i r e d f a s t e s t s p e e d o f r e s p o n s e

10-31 0 . . A = 0 . 3 7 5 m ^ 2 ~lOI .-

I t' ~ 1 0 1 ~ . .. .. q f = 2. 0 l / s ' ~- - q f = l . 0 I /s . . . . ~11~: . . . . . . : .: i . q f= 0.5 I, . . . . ---- ,--- . ~

10 ~ 10 4 10-3 10-2f r e q u en c y ] l / s ]Fig. 9. Bod e plots for different cross sect ional area s of theannulus (upper) , and different values of the stat ionaryfines removal flow (lower).

0-1-2-38 0 ( 3 - -,...y

70c~ 6~

I 3 4 0~ '~ 320~ 1 3 0 0

5 1 0 1 5 2 0 2 5 3 0Time [hrs]Fig. 10. Experimental open-loop response on a blockedproduct removal f low at A and C; a control ler is usedbe tween B and C and a f te r D to s tab i l i ze the open- loopresponse.

a n d t h e m a x i m u m c a p a c i t y o f t h e f i n e s d i s s o l v in gs y s t e m .4 3 E x p e r i m e n t a l c o n t r o l l e r e v a l u a t i o n

T o s t a b i l i z e t h e C S D a n d t o r e j e c t C S D d i s t u r -b a n c e s , a s i n g l e - l o o p P I c o n t r o l l e r u s i n g Yr~ a s i n p u tp a r a m e t e r a n d Q f a s o u t p u t p a r a m e t e r i s e v a l u a t e dw i t h s i m u l a t i o n . T h e r e s u l t s r e v e a l e d t h a t s t a b i l i t yc a n b e a c h i e v e d e a s i l y w i th t h i s c o n t r o l l e r . T h ec o n t r o l l e r i s t e s t e d f o r i t s a b i l i t y t o r e j e c t a d i s t u r -b a n c e c a u s e d b y b l o c k a g e o f th e p r o d u c t d i s c h a r g es y s t e m f o r I h . D u r i n g t h i s d i s t u r b a n c e t h e p r o c e s s i so p e r a t e d b a t c h w i s e . I n F i g . 1 0 , t h e o p e n - l o o p r e s p -s o n s e o f t h e p r o c e s s i s g i v e n f o r t w o r e p e a t e d d i s t u r -b a n c e s . A s c a n b e s e e n t h e s l u r r y d e n s i t y i n c r e a s e s

8 0 ) . . . . . . . .

6 0 0

1 3 4 0 A B~ 1 3 2 0

[ , ,0 2 4 6 8 10 12 14 16 18T i m e ] h r s ]Fig. 11. Expe rimental closed-loop process response onblocked pro duct f low for 1 h at A and B. Yr~ is control led ona se tpo in t (dashed) o f -0 .5 .

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Design issues relat ed to the con trol of continuous crystallizers 433

Time [hrs]Fig. 12. Experim ental man ipulated variable response(f ines f low) on a blocked pro duct f low during 1 h at A a ndB; dashed-clotted line is constraint.

l i n e a r l y a f t e r c l o s i n g t h e p r o d u c t d i s c h a r g e a n d ab a d l y d a m p e n e d c y c le o f th e C S D o c c u rs .

F o r c o n t r o l a S I S O M P C c o n t r o ll e r is u se d ( E e k ,1 9 9 5) . T h e r e s u l t s o f a c l o s e d - l o o p e x p e r i m e n t w i t ht h i s c o n t r o l l e r a r e d e s c r i b e d i n F i g s 1 1 a n d 1 2 .A n a l y s i s p e r f o r m e d a f t er t h e e x p e r i m e n t r e v e a l e dt h a t t h e p e r f o r m a n c e o f t h e S I S O M P C c o n t r o l le r iss t r o n g l y c o m p a r a b l e t o th e S I S O P I c o n t r o l l e r .T h e r e f o r e t h e l a t t e r i s r e c o m m e n d e d f o r i n d u s t r i a la p p l i c a t i o n . A s c a n b e c o n c l u d e d f r o m t h i s r e s u l t ac r y s t a l l i z e r e q u i p p e d w i t h a f i n e s r e m o v a l s y s t e m ,o p e r a t e d i n a c l o s e d - l o o p , im p r o v e s b o t h t h e s t a -t i o n a r y a n d d y n a m i c p r o c e s s b e h a v i o u r .

5 . C L A S S I F I E D P R O D U C T R E M O V A L

T h e s e c o n d d e s i g n i n c l u d e s b o t h a f in e s r e m o v a la n d a c l a s s i f ie d p r o d u c t r e m o v a l s y s t e m . T h ec r y s t a l s l e a v i n g t h e d e c k o f t h e v i b r a t i n g s c r e e n ( s e eF i g . 2 ) a r e n o w c o n s i d e r e d a s p r o d u c t c r y s t a l s . T h ep r o d u c t i s a v i s c o u s c a k e w h i c h d o e s n o t d i s p e r s e .T h e s o l i d s f r a c t i o n i n t h i s c a k e i s a p p r o x . 5 5( m e a s u r e d b y h a n d ) , a n d f a i r l y c o n s t a n t o v e r t i m e .T h e c r y s t a l s a n d t h e m o t h e r l i q u o r p a s s i n g t h es c r e e n a p e r t u r e s a r e r e t u r n e d t o t h e c r y s t a l l i z e r .5.1. Open loop plant behaviour

T w o s e r i e s o f o p e n - l o o p e x p e r i m e n t s a r e p e r -f o r m e d , u s i n g a s c r e e e n w i t h a n a p e r t u r e s i z e o f 80 0

~ 7 0 ob

s o o t ::~ o or _- 0 ~ , ~ ~ _ _ Y . . . . . 5 t4 0 0 [ . . . . . . . . . . 9 p f = 0 5 0 /s j8 0 0 . . . . . . . . - -- - [ O p f = ] . 5 0 l / s

,o o t b - - -o o ~ J5 10 15 20 25 30 35

Time [hours]Fig. 14. Experimental response of median crystal s ize ofcrystals in the crystallizer vessel for two experiments atd i f feren t f eed f lows Qpf ( - - - ) ; small f eed f lows (upper)large feed f lo ws ( lower); the screen is switched on atT = 8 h .

m i c r o n s . I n a f i r s t e x p e r i m e n t t h e f e e d f l o w ( Q p f )w a s v a r i e d s t e p - w i s e . T o s t a b i l iz e t h e o p e n - l o o ps t a r t - u p r e s p o n s e , t h e c r y s t a l li z e r i s s t a r te d w i t h o u tp r o d u c t c l a s si f ic a t io n i n c l o s e d - l o o p w i t h t h e S I S OP I c o n t r o l l e r t h a t i s d e s c r i b e d i n t h e p r e v i o u ss e c t i o n . A f t e r r e a c h i n g s u f f i c ie n t s t a b i li t y o f t h eC S D ( 8 h i n t h e p l o t s ) , t h e c o n t r o l l e r i s s w i t c h e d o f fa n d t h e p r o d u c t c l a s s i f i e r i s s t a r t e d w i t h a f e e d f l o wo f 0 .7 5 l/ s . A f t e r 1 2 h t h e f e e d f l o w i s c h a n g e d s t e p -w i s e f r o m 0 .7 5 t o 0 .5 0 l / s . T h e f i n e s fl o w i s k e p tc o n s t a n t a t 1 . 4 1 / s , F i g u r e 1 3 a l s o s h o w s t h a t t h em a s s f l ow o f p r o d u c t a n d t h e d e n s i t y i n t h e c r y s t a l -l i z e r v e s s e l , c o r r e s p o n d i n g t o t h i s f ir s t e x p e r i m e n t ,e x h i b i t o s c i l l a to r y b e h a v i o u r w i t h s i m i l a r d y n a m i c s .T h u s p r o d u c t c l a s s i fi c a t io n i n t r o d u c e s a d d i t i o n a li n t e r a c t i o n b e t w e e n C S D d y n a m i c s a n d t h e m a s sp r o d u c t i o n r a t e , w h i c h w a s n o t t h e c a s e w i t h n o n -c l a s s i f i e d p r o d u c t r e m o v a l ( c f . F i g . 7 ) . A s e c o n de x p e r i m e n t i s p e r f o r m e d w i t h t h e s a m e s t a r t - u pp r o c e d u r e , n o w t h e s c r e e n i s s w i t c h e d o n w i t h a f e e df l o w o f i l / s . A f t e r 1 7 h a s t e p c h a n g e t o 1 .5 l / s i sm a d e , a n d a f t e r 8 h t h e f lo w is s w i t c h e d b a c k t o1 .0 i/ s ( s e e F i g . 1 4 ) . D u r i n g t h i s e x p e r i m e n t t h e f i n e s

2Ig o o -

7

0 5 10 15 20 25 30 35t ime in [hours]

Fig. 13. E xperim ental response of product mass f low (up-per) and the densi ty of s lurry in the crystal l izer ( lower) ,correspo nding to the f irs t experimen t with prod uct classif i-cat ion.

Median crys ta l s ize in vesse l8 0 0 r ]

4 0 0 [ : Q p f = 0 . 7 5 I / . . . . . . O p [ - - 0: 5 0 i / s IQ p f = l . 5 0 l / s

7 0 0 . . . . . . . .6 0 0 . . . . . . . . . . . . . . . . . . . i . . . .5 0 0 1 O p f = l . 0 0 l / s4001 . . . . . .1 0 1 5 2 0 2 5 3 0 3 5

Time [hours]Fig. 15. Simulated behav iour of open -loop process behav-iour using product classif icat ion with vibrat ing screen,correspond ing to the two op en-loo p experiments .

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434Table 2. Stationa ryproduc t CSD values at different classifier eedflowsQpf (J/s) xso (gm) Rf (-- ) AC ( )

0.50 672 0.451.00 620 0.421.50 597 0.41

R. A. EEK et als t a t i o n a r y s i m u l a t e d v a l u e s r e l a t e d t o t h e p r o d u c tC S D , a n d t h e l e v e l o f s u p e r s a t u r a t i o n i n t h e c r y s t a l -l i z e r a s a p e r c e n t a g e o f t h e s a t u r a t i o n c o n c e n t r a t i o n

0 .2 0 a r e g i v e n f o r d i f f e r e n t f e e d f l o w s t o t h e s c r e e n a n d a0 .2 6 c o n s t a n t f i n e s f l o w o f 1 .0 1 / s . A s c a n b e s e e n a0 . 3 2 f u r t h e r i m p r o v e m e n t o f t h e p r o d u c t q u a l i t y a t s t a -

t i o n a r y c o n d i t i o n s i s o b t a i n e d i n c o m p a r i s o n w i t hf i n e s c l a s s i f i c a t i o n . T h e r e l a t i v e n u m b e r o f f in e s i sl o w e r e d , t h e m e d i a n i s i n c r e a s e d a n d t h e s u p e r s a t u -r a t i o n i n t h e c r y s t a l l i z e r i s l o w e r e d .

I n t h e f i r s t i n s t a n c e , i t i s d e s i r a b l e t o c h o o s e t h ea p e r t u r e s i z e o f t h e s c r e e n a s l a r g e a s p o s s i b l e , t op r o d u c e a c o a r s e p r o d u c t . H o w e v e r , w e f o u n d t h a ti n c r e a s i n g X p b y 1 0 0 ~ tm , r o u g h l y i n c r e a s e s t h e s o l i d sf r a c t i o n i n t h e c r y s t a l li z e r b y 1 0 . T h e r e f o r e , t h eu p p e r l i m i t f o r X p s h o u l d b e d e t e r m i n e d f r o m t h em a x i m u m a l l o w e d s o l i d s f r a c t io n i n t h e c r y s t a ll i z e r .

F i n a l l y t h e s t e a d y - s t a t e a n d d y n a m i c c o n t r o l l a b i -l i t y w e r e s t u d i e d u s i n g a m o d e l w i t h u = { Q f , Q p f ,Pto t} as inp u ts and y = {Yrb Xs0 , Mr} as ou tpu ts . A lsof o r t h is c a s e w e f o u n d t h a t v a r y i n g t h e p r o c e s s i n p u tv a l u e s o n l y t w o m a i n e f f ec t s a r e o b s e r v e d i n t h e s i zed i s t r i b u t i o n : a r o t a t i o n a n d a t r a n s l a t i o n . T h e c o n -d i t io n n u m b e r o f th e t h r e e - i n p u t - t h r e e - o u t p u ts y s t e m i s l a r g e r t h a n 1 00 o v e r t h e w h o l e f r e q u e n c yd o m a i n . T h e r e f o r e , w e p r o p o s e t o o p e r a t e t h es y s t e m w i t h Q p f o n a c o n s t a n t a n d l o w v a l u e a n d t ou s e a m u l t i l o o p c o n t r o l l e r w i t h t h e f i n e s f l o w c o u -p l e d t o Y r~ a n d t h e h e a t i n p u t t o m 3 . T h e c o n d i t i o nn u m b e r o f t h i s s y s t e m i s f o u n d t o b e l o w e r t h a n 1 0o v e r t h e w h o l e f r e q u e n c y d o m a i n , i n d i c a t i n g e a s yc o n t r o l l a b i l i ty . T h e i n t e r a c t i o n b e t w e e n t h e s e p a r -a t e c o n t r o l l o o p s is fo u n d t o b e s m a l l .5 3 E x p e r i m e n t a l c o n t r o l l e r e v a l u a t i o n

M o d e l s i m u l a t i o n s t u d i e s r e v e a l t h a t f o r t h is c a s ea l s o , t h e c r y s t a l li z e r c a n b e s t a b i l i z e d w i t h t h e s i m -p l e S I S O P I c o n t r o l l e r , w h i c h w a s a l s o u s e d t oa c h i e v e s t a b i l i t y i n t h e c a s e o f u n c l a s s i f i e d p r o d u c tr e m o v a l . I n F i g . 1 6, t h e r e s u l t s o f a n e x p e r i m e n t a r eg i v e n w h e r e w e a p p l i e d t h i s c o n t r o l l e r a f t e r s w i t c h -i n g o n t h e s c r e e n , 6 h a f t e r a s t a r t - u p w i t h o u t p r o -d u c t c l a s s i f i c a t i o n . T h e r e s u l t s s h o w t h a t s t a b i l i t y i sa c h i e v e d . B e c a u s e t h e d e n s i t y in t h e c r y s t a l l i z e r ,a n d t h e r e b y t h e m a s s p r o d u c t i o n r a t e , s t il l s h o w ss o m e d e v i a t i o n s , a m u l t i l o o p c o n t r o l l e r i s e x p e c t e dt o f u r t h e r e n h a n c e t h e p r o c e s s b e h a v i o u r .

f lo w is k e p t c o n s t a n t a t 0 . 8 i /s . A s c a n b e c o n c l u d e d ,a l o w e r f e e d f l o w Q p f , i n c o n j u n c t i o n w i t h a l a r g e rf i ne s fl o w , p r o m o t e c y c l i n g o f b o t h t h e C S D a n d t h em a s s p r o d u c t i o n . I n c r e a s i n g t h e f e e d f l ow , g i v e s ar e s p o n s e t h a t d a m p e n s a f t e r a p p r o x . 3 c y c l e s .B e s i d e s , s t e p - w i s e c h a n g e s i n t h e f e e d f l o w s e e m t oh a v e a l i m i t e d e f fe c t o n t h e C S D i n b o t h e x p e r i -m e n t s . H o w e v e r , p r o p e r j u d g e m e n t i s d i ff i c ul t i nt h i s c a s e , d u e t o t h e s t r o n g c y c l i n g b e h a v i o u r .5 2 M o d e l r e s u l ts

T h e e m p i r i c a l p a r a m e t e r s i n t h e p r o c e s s m o d e lr e l a t e d t o c r y s t a l l i z a t i o n k i n e t i c s a n d f i n e s r e m o v a l[ e q u a t i o n ( 1 )] w e r e e s t i m a t e d o n t h e b a s i s o f a s e ri e so f s t a r t - u p e x p e r i m e n t s , t h e r e s u l t s o f w h i c h a r ep r e s e n t e d b y E e k et al ( 1 9 9 5 ) . T h e p a r a m e t e r s p p~a n d P p3 r e l a t e d t o t h e p r o d u c t c l a s s i f i c a t i o ne f f ic i e n c y [ e q u a t i o n ( 3 )] c o u l d b e d e t e r m i n e d ap r i o r i f r o m t h e a p e r t u r e s i ze o f t h e s c r e e n a n d t h em e a s u r e d s o l i d s f r a ct i on o f t h e p r o d u c t ( = 5 5 ) .T h e e f f ic i e n c y f a c t o r Po2 w a s u s e d a s a t u n i n g f a c t o rt o m a t c h t h e s i m u l a t i o n r e s u l t s t o t h e e x p e r i m e n t a lr e s u l ts . T h e p r o d u c t c l a s s i f ic a t i o n p a r a m e t e r s t h u so b t a i n e d w e r e p p ~ = 8 0 0 ~ t m , p p 2 = 5 . 0 a n d pp 3 =0 . 0 29 . I n F i g . 1 5 , t h e m o d e l o u t p u t r e l a t e d t o t h es c r e e n e x p e r i m e n t s i s g i v e n . A s c a n b e s e e n , t h em o d e l g i v e s a r e a s o n a b l e d e s c r i p t i o n . I n T a b l e 2 ,

1 . . . . .0.5 -

0

-0.5

1 3 2 [t 3 1 ~~ 1 2 9 0 ~

1280 I0 5 1 0 15 2 0 25 3 0 35Time [hours1

Fig. 16. Compar i son o f exper imenta l open- loop andc losed- loop response o f y . , the m edian c rys ta l s ize and thedensi ty in the crystal l izer vessel af ter switching on thescreen at T = 8 h (A).

6 . C O N C L U S I O N SB o t h c l a ss i f ie d f i n es a n d p r o d u c t r e m o v a l e n h a n c e

t h e s t a t i o n a r y p e r f o r m a n c e o f a c r y s t a l li z e r , h o w -e v e r , a t t h e e x p e n s e o f o p e n - l o o p c y c l i n g o f t h eC S D . W i t h p r o d u c t c l a s s if i c a ti o n a d d i t i o n a l i n t e r a c -t i on b e t w e e n t h e C S D a n d t h e m a s s p r o d u c t i o n r a t e

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Design issues related to the control of continuous crystallizersand the density in the crystallizer is observed.Closed-loop stability is obtaine d for bot h cases, witha simple closed-loop feedback co ntroller which usesfines removal as its actuator and a linear combi-natio n of diffracted light energy values observed ona CSD detector as its input.

For both config urations only a translation androtatio n of the CSD could be achieved with changesin the process inputs. This obser vation is confirmedby dynamic con trollability analysis which shows thatonly a two-input-two-output system is easily con-trollable. For both configurations, therefore, a mul-tiloop control structure is proposed using the finesflow and the heat input as actuators and using theweighted number of fines and the slurry density asoutputs.

It is found that the feed flow to the screen can beoperated at a low value, while the maximum aper-ture size of the screen should be determined fromthe maximu m allowable solids fraction in the crystal-lizer. Large fines flows are preferable; however,they also increase the supersaturation level, whichmight be undesired. The size of the annular zonedoes not affect strongly the dynamics of thecontrolled process, however this parameter deter-mines strongly the amo unt of solids to be dissolved.A c k n o w l e d g e m e n t s - - T h e authors are indebted to HansGerla, Sven Rusticus and Jaap Both, who carried out partof the experimental work and to the Dutch Foundation ofTechnology STW), AKZO, ICI, DOW Chemicals, DSM,E.I. Dupont de Nemours, and Eastman ChemicalCompany for their financial support of the UNIAKproject.

R E F E R E N E S

Bennett R. C., Crystallizer selection and design.Handbook of Industrial Crystallization

435Butterworth-Heinemann Series in Chem. Engn g Chap.5, pp. 103-129 1992).Boxman A., H. G. Merkus, P. J. T. Verheijen and B.Scarlett, Deconvolution of light-scattering patterns byobserving intensity fluctuations. AppL Optics 30, 4818-4822 1991).Eek R. A., Ph.D. Thesis, Mechanical EngineeringSystems and Control Group, Delft University ofTechnology 1995).Eek R, A., Sj. Dijkstra and G. M. van Rosmalen,Dynamic modeling of suspension crystallizers, usingexperimental process data. A I C h E J l March issue1995).Fukunage K., Introduction to Stat is t ica l PatternRecognition Chap. 8, pp. 225-257. Academic Press,New York 1972).Hallas N. J. and M. L. Hannan, Measurement and compu-tation of hydrodynamics for a draft tube baffled crystal-lizer. Proc. 1 1th Syr up. Industrial Crystallization A.Mersmann, Ed.), pp. 107-112 1990).Hecht E., Optics 2nd Edn, Chap. 10. Addison-Wesley,Reading, MA 1987).Mersmann A., Design of crystallizers. Chem. EngngProcess. 11,213-228 1988).Nyvlt J., Design of Crystallizers. CRC Press, Boca Raton,FL 1992).Randolph A. D., The mixed suspension, mixed productremoval crystallizer as a concept in crystallizer design.A1ChE J I 11,424-430 1965).Randolph A. D. and M. A. Larson, Theory of ParticulateProcesses 2nd Edn. Academic Press, New York 1988).Rawlings J. B., S. M. Miller and W. B. Witkowski, Modelidentification and control of solution crystallization pro-cesses: a review. Ind. Engng Chem. Res. 32, 1275-12901993).Skogestad S. and M. Morari, Understanding the dynamicbehaviour of distillation columns. Ind. Engng Chem.Res. 27, 1848-1862 1988).Tavare N. S., Mixing in continuous crystallizers.A I C h E J l32, 705-732 1986).Wolf S. de, Modelling, system identification and control ofan evaporative continuous crystallizer. Ph.D. Thesis,Delft University of Technology, The Netherlands1990).