Crystallization as a Separations Process - American Chemical Society

Chapter 11

Derivation of State Space Models of Crystallizers 1

2

1

2

Sjoerd de Wolf , Johan Jager , B. Visser , Herman J. M. Kramer , and Ο. H. Bosgra 1

Downloaded by UNIV OF ARIZONA on September 22, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch011

1

2

Laboratory for Measurement and Control and Laboratory for Process Equipment, Delft University of Technology, Mekelweg 2, Delft, Netherlands

For effective control of crystallizers, multivariable controllers are required. In order to design such con trollers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSMPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be ap plied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. C r y s t a l l i z a t i o n from s o l u t i o n i s a w i d e l y u t i l i z e d s e p a r a t i o n and p u r i f i c a t i o n technique i n chemical industry. I t i s characterized by the formation o f a spectrum o f d i f f e r e n t l y s i z e d c r y s t a l s . T h i s spectrum, c a l l e d the C r y s t a l Size D i s t r i b u t i o n or CSD, i s highly im p o r t a n t f o r the performance o f the c r y s t a l l i z e r , t h e c r y s t a l h a n d l i n g e q u i p m e n t l i k e c e n t r i f u g e s and d r y e r s , and t h e marketability o f the produced c r y s t a l s . However, i n many i n d u s t r i a l c r y s t a l l i z e r s , the observed CSD*s show large transients due to d i s turbances o r a r e u n s t a b l e b e c a u s e o f t h e i n t e r n a l f e e d b a c k mechanisms o f the c r y s t a l l i z a t i o n process (1). The main l i m i t a t i o n f o r e f f e c t i v e CSD c o n t r o l was the l a c k o f a good o n - l i n e CSD measurement device, but recent developments show that t h i s hurdle i s taken (2). With the o n - l i n e CSD-measurement equipment a v a i l a b l e , i t be comes relevant to investigate and design control schemes f o r CSD*s. A number o f schemes were i n v e s t i g a t e d by simulation i n the past. Most o f these control schemes use one p r o p o r t i o n a l s i n g l e - i n p u t s i n g l e - o u t p u t (SISO) c o n t r o l l e r w i t h the n u c l e a t i o n r a t e (or a

0097-6156/90/0438-0144$06.00A) © 1990 American Chemical Society

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

11.

DEWOLFETAL.

Derivation ofState Space Models of Crystallizers


v a r i a b l e proportional to the nucleation rate) as the v a r i a b l e to be c o n t r o l l e d , see for example (3..4) • The main drawback of these simple proportional c o n t r o l l e r s i s that transients and o f f s e t s o f the CSD are s t i l l p o s s i b l e . In o r d e r to improve the c o n t r o l performance, more process inputs and outputs can be used i n the c o n t r o l l e r and, to cope with unwanted interactions due to simultaneous v a r i a t i o n s of d i f f e r e n t process inputs by the c o n t r o l l e r , ' t h e more power f u l l mul t i v a r i a t e c o n t r o l l e r s are to be used. Most design methods f o r m u l t i v a r i a b l e c o n t r o l l e r s r e q u i r e a d y n a m i c model o f the p r o c e s s i n the l i n e a r s t a t e space r e p r e sentation. Such a model i s given by x(t) = A x(t) y_(t) = C x(t)

+ B u(t) + D u(t)

n K

, '

where x(t) u(t) y_(t) A B C D n m £

state vector dimension vector of process inputs dimension vector of process outputs dimension state matrix dimension input matrix dimension output matrix dimension d i r e c t transfer matrix dimension number of states (order of the state space number of process inputs number of process outputs

(nxl) (mxl) (£xl) (nxn) (nxm) (£xn) (£xm) model)

which i s a set of l i n e a r f i r s t - o r d e r ordinary d i f f e r e n t i a l equations i n m a t r i x - n o t a t i o n . As d i s c u s s e d i n (5. and De Wolf, S . ; P h . D . Thesis, Delft University of Technology, i n preparation) the physical model f o r the CSD i n evaporative and non-evaporative c r y s t a l l i z e r s i s given by the set of d i f f e r e n t i a l equations - population balance - concentration balance - energy balance (for non-evaporative systems) and

(2a)

the set of algebraic equations - algebraic r e l a t i o n for the vapour flow rate (for evaporative systems) - r e l a t i o n between the input and output flow rates - growth rate k i n e t i c s - nucleation k i n e t i c s .

(2b)

The problem i n obtaining a state space model for the dynamics of the CSD from t h i s p h y s i c a l model i s that the population balance i s a (nonlinear) f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l equation. Consequently, to o b t a i n a state space model the population balance must be trans formed into a set of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . A f t e r t h i s t r a n s f o r m a t i o n , the s t a t e space model i s e a s i l y obtained by sub s t i t u t i o n o f the a l g e b r a i c r e l a t i o n s and l i n e a r i z a t i o n o f the ordinary d i f f e r e n t i a l equations.


145

146

CRYSTALLIZATION AS A SEPARATIONS PROCESS


In t h i s paper, t h r e e methods to transform t h e p o p u l a t i o n b a l a n c e i n t o a s e t o f ordinary d i f f e r e n t i a l equations w i l l be d i s c u s s e d . Two o f t h e s e methods were r e p o r t e d e a r l i e r i n t h e c r y s t a l l i z e r l i t e r a t u r e . However, these methods have l i m i t a t i o n s i n t h e i r a p p l i c a b i l t y to c r y s t a l l i z e r s with f i n e s removal, product c l a s s i f i c a t i o n and size-dependent c r y s t a l growth, l i m i t a t i o n s i n the choice of the elements of the p r o c e s s output v e c t o r y_(t) t h a t i s used by the c o n t r o l l e r or r e s u l t i n high orders of the state space model which causes severe problems i n the c o n t r o l system d e s i g n . T h e r e f o r e another approach i s suggested. T h i s approach i s demonstrated and compared with the other methods i n an example. Transformation methods The discussion of the transformation methods w i l l be based on the population balance for a c r y s t a l l i z e r with fines removal and product classification: 3n(L,t)

a[G(L,t)n(L,t)] v

at +

3L

Q (t)h (L,t)n(L,t) • Q (t)h (L,t)n(L,t) - 0 p

p

f

f

(3)

with boundary condition n ( L , t ) = B ( t ) / G ( L , t ) = Kg G ^ t ) Q

Q

where n ( L , t ) « population density V « c r y s t a l l i z e r volume G(L,t) » c r y s t a l growth rate Q(t) = flow rate h ( L , t ) ' = c l a s s i f i c a t i o n function B(t) a nucleation rate m (t) = xth moment of n ( L , t ) indices p = product flow f = fines removal flow.

1

'

1

m (t)

J

(4)

x

[#/(m.m*)] [m ] [m/s] [m /s] [-] [#/(m s)] [m /m ] 3

3

3

3

Size-dependent c r y s t a l growth i s i n c l u d e d i n the model because i t can be important to describe d i f f u s i o n l i m i t e d growth rates or c r y s t a l a t t r i t i o n . As d i s c u s s e d i n ( 6 , £ ) , t h e s i z e r e d u c t i o n by a t t r i t i o n can be modelled by an e f f e c t i v e growth rate G ( L , t ) which i s the difference between the k i n e t i c growth rate G , ( L , t and an at t r i t i o n rate G ( L , t ) : e

&

G (L,t) = G (L,t) - G (L,t) e

k

a

(5)

I n c l u s i o n o f a t t r i t i o n can be important because the e f f e c t o f process inputs, as for example the fines flow rate and the residence time on f o r example the average c r y s t a l s i z e , i s strongly affected by a t t r i t i o n .


11.

147 Derivation of State Space Modds of Crystallizers

DEWOLFETAL.

Method of Moments. The moments of the population density can be used to c h a r a c t e r i z e the CSD. The f i r s t f o u r moments have p h y s i c a l i n t e r p r e t a t i o n s and the mean c r y s t a l s i z e (L^Q) and the Coefficient of V a r i a t i o n (CV) based on the mass d i s t r i b u t i o n are f u n c t i o n s o f the moments: L

50

CV

s

m

m

m

4^ 3

'• -'

= (m m /mj - l ) 5

1 / 2

3

.

(6)

[-]

When c o n t r o l i s r e q u i r e d f o r moments, L - or CV only, a model for the dynamics of the moments i s s u f f i c i e n t . S u c h a model can be ob tained by multiplying the population balance with L and integration over L :


Q

J

dm.(t) B(t)L

dt

J

+ j |

0

GfL.tJnfL.tjL^dL

L

0

Q (t> ;

o (t)

D

-

V ~

:

f

Jh

L

J

p

(L,t)n(L,t)L dL "

JV '

V

0

t ) n ( L

'

t ) L dL

*

(7)

0

This r e s u l t s i n a set of f i r s t - o r d e r ordinary d i f f e r e n t i a l equations for the dynamics of the moments. However, the population b a l a n c e i s s t i l l r e q u i r e d i n the model to determine the three i n t e g r a l s and no state space r e p r e s e n t a t i o n can be formed. Only f o r simple MSMPR (Mixed Suspension Mixed Product Removal) c r y s t a l l i z e r s with simple c r y s t a l growth behaviour, the population balance i s redundant i n the model. For MSMPR c r y s t a l l i z e r s , Q =0 and h ( L ) = l , thus: f

dm (t)

p

*

—^—

= B(t)L

J

* 3 J

Q

L

Q (t) G(L,t)n(L,t)L

J

dL -

m^t).

(8)

0

The f i n a l i n t e g r a l disappears when the size-dependent growth rate written as a polynomial of the c r y s t a l s i z e : G(L,t) = G ( t ) G ( L ) = G ( t ) [ a a

L

o

0

+ a^ + a L 2

2

is

* ...]

(9)

where G (t) and G (L) are the supersaturation dependent part and the s i z e - d e p e n d e n t p a r t o f the c r y s t a l growth rate respectively. This r e s u l t s for Equation 8: L

dm (t) -

d

T

q '

^

V

+

J Q

a

( t )

J

Q (t) a

0

m

± i j-l +

( t )

" " V —

m

j

(

t

)

(

1

0

)

and the p o p u l a t i o n balance i s no l o n g e r r e q u i r e d i n the model. However, there i s another problem. As already discussed i n (8), for a ^ 0 for i = 2 , 3 t . . . the equation for dm.(t)/dt c o n t a i n s h i g h e r mo ments than m.(t) and the set of equations i s not a closed model set. In order to solve t h i s problem, an approximation o f the p o p u l a t i o n density by orthogonal Laguerre polynomials , i . e . 1

American Chemical Society Library 1155 15th St., N.W. In Crystallization as a Separations Process; Myerson, A., et al.; Washington, O.C. 20036 ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

CRYSTALLIZATION AS A SEPARATIONS PROCESS

148 P

n(L,t) *

I b,(t) k=0

Lag, (L)

(11)


was investigated i n (10). However, t h i s r e s u l t s i n u n s t a b l e ap p r o x i m a t i o n s because i t i s not based on the minimization of the error involved i n Equation 11 (11_). In c o n c l u s i o n , the method of moments can be used to obtain a state space model for the dynamics o f the moments o f the CSD. The method i s l i m i t e d to MSMPR c r y s t a l l i z e r s w i t h s i z e - i n d e p e n d e n t growth or size-dependent growth described by G(L,t) - G ( t ) G ( L ) = G ( t ) [ a a

L

a

Q

+ a^]

(12)

and the r e s u l t i n g moment equations are given by: dm (t) % t

Q(t) • B(t)L

J Q

+ jG (t)[a m _ (t)+a m (t)] a

0

j

1

1

j

m^.(t). (13)

In a p p l y i n g the r e s u l t i n g s t a t e space model f o r c o n t r o l system d e s i g n , the order of the state space model i s important. This order i s d i r e c t l y affected by the number o f o r d i n a r y d i f f e r e n t i a l equa t i o n s (moment e q u a t i o n s ) r e q u i r e d to d e s c r i b e the p o p u l a t i o n balance. From the structure of the moment equations, i t follows that the dynamics of m.(t) i s described by the moment equations for m (t) to m.(t). Because the concentration balance contains e(t)=l-k m~Yt), at l e a s t the f i r s t four moments equations are required to close o f f the o v e r a l l model. The f i n a l number of equations i s determined by the moment m (t) i n the equation for the nucleation rate (usually mj(t)) and the highest moment to be c o n t r o l l e d . Q

J

Method o f L i n e s . The method of l i n e s i s used to solve p a r t i a l d i f f e r e n t i a l equations (12) and was a l r e a d y used by Cooper (13.) and T s u r u o k a (lj_) i n the d e r i v a t i o n of s t a t e space models f o r the dynamics of p a r t i c u l a t e processes. In the method, the s i z e - a x i s i s d i s c r e t i z e d and the p a r t i a l d i f f e r e n t i a l a [ G ( L , t ) n ( L , t ) ] / 3 L i s ap proximated by a f i n i t e difference. Several choices are p o s s i b l e f o r t h e a c c u r a c y o f t h e f i n i t e d i f f e r e n c e . The method w i l l be demonstrated for a fourth-order central d i f f e r e n c e and an e q u i d i s tant g r i d . For n o n - e q u i d i s t a n t g r i d s , the Lagrange i n t e r p o l a t i o n formulaes as described i n (15.) are to be used. U s i n g an e q u i d i s t a n t d i s c r e t i z a t i o n i n K+l p o i n t s (LQ, L ^ , L 2 , . . . L ^ _ ^ , L^) and the fourth-order central difference

A d i f f e r e n t approach i n the use o f o r t h o g o n a l p o l y n o m i a l s as a t r a n s f o r m a t i o n method for the population balance i s discussed i n (8,2.). Here the error i n Equation 11 i s minimized by the Method of Weighted Residuals. This approach releases the r e s t r i c t i o n s on the growth rate and MSMPR operation, however, at the c o s t o f the i n t r o d u c t i o n of numerical integration of the i n t e g r a l s involved, which makes the method c o m p u t a t i o n a l l y u n a t t r a c t i v e . The ap p l i c a b i l i t y i n d e t e r m i n i n g s t a t e space models i s p r e s e n t l y investigated and results w i l l be published elsewere.


11.

Derivation ofState Space Models of Crystallizers 149

DEWOLFETAL.

^ U

±

f

=

8f

+8f

i k ( i-2- i-i i r +

f

)•

i+2

Crystallization as a Separations Process - American Chemical Society

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