Emulsion Polymers and Emulsion Polymerization - American

33

Downloaded by PENNSYLVANIA STATE UNIV on June 15, 2013 | http://pubs.acs.org Publication Date: October 7, 1981 | doi: 10.1021/bk-1981-0165.ch033

A Simulation Study on the Use of a Dead-Time Compensation Algorithm for Closed-Loop Conversion Control of Continuous Emulsion Polymerization Reactors KENNETH W. LEFFEW and PRADEEP B. DESHPANDE Department of Chemical and Environmental Engineering, University of Louisville, Louisville, KY 40292 Although continuous emulsion polymerization is an area of considerable industrial interest and importance, it has received very little attention in the literature when compared to the vast amount of published work available for the batch process. Largely in response to the growing interest expressed by industry, the amount of work on continuous emulsion polymerization appearing in the technical literature has been increasing in recent years. The majority of this work has dealt with the steady-state operation of a single continuous stirred tank reactor (CSTR). Few workers have dealt with the problems associated with operation of a train of reactors, which is the type of reaction system most commonly found in industry. Even less attention has been given to control strategies for these multi-reactor trains, in which the system dynamics are of concern. In the commercial manufacture of polymers by continuous emulsion polymerization, perhaps the primary concerns are maintenance of uniform product quality and avoidance of production of poor quality material resulting from swings away from the steady-state levels of the process variables. Therefore, a control system which provides tight regulatory control of the process is a need of the industry. Design of such a system for many continuous emulsion polymerized monomers is complicated by the occurrence of a steady-state limit cycle in the number of polymer particles produced and in the monomer conversion achieved in the reaction system. In this paper, control strategies are suggested, and demonstrated by simulation of the vinyl acetate system, which are designed to provide the required regulatory control of a series of continuous emulsion polymerization reactors. Background The two prominent variables to be controlled in a continuous polymerization system are the reaction temperature and the monomer conversion achieved in the reaction system. Final polymer properties are directly influenced by changes in these process 0097-6156/81/0165-0533$08.25/0 © 1981 American Chemical Society

In Emulsion Polymers and Emulsion Polymerization; Bassett, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.


534

EMULSION POLYMERS AND EMULSION POLYMERIZATION

variables. S e v e r a l c o n t r o l systems f o r c o n t i n u o u s e m u l s i o n p o l y m e r i z a t i o n have been s u g g e s t e d . Wismer and B r a n d (1) m a n i p u l a t e d r e a c t i o n temperature over a p o r t i o n o f a continuous r e a c t o r t r a i n t o c o n t r o l t h e monomer c o n v e r s i o n f r o m t h e f i n a l r e a c t o r i n a styrene-butadiene emulsion polymerization. Changes i n t e m p e r a t u r e w e r e b a s e d on a f e e d - f o r w a r d c o n t r o l scheme d e v e l o p e d f r o m a v e r y s i m p l e l i n e a r p r o c e s s m o d e l and t h e c o n t r o l a l g o r i t h m was i m p l e m e n t e d on a d i g i t a l c o m p u t e r . The a p p l i c a b i l i t y o f t h i s c o n t r o l s t r a t e g y i s l i m i t e d t o s y s t e m s w h i c h obey t h e s i m p l e p r o c e s s m o d e l and t o t h o s e where c h a n g e s i n r e a c t i o n t e m p e r a t u r e c a n be made w i t h o u t a f f e c t i n g p o l y m e r p r o p e r t i e s . F r a n c i s and S o n n t a g ( 2 ) s u g g e s t e d an a l t e r n a t e c o n t r o l s t r a t e g y i n w h i c h t h e residence time i n the l a s t of a t r a i n o f continuous s t i r r e d tank r e a c t o r s was m a n i p u l a t e d a s r e q u i r e d t o c o n t r o l t h e c o n v e r s i o n e x i t that reactor. T h i s t e c h n i q u e a l s o has l i m i t e d a p p l i c a b i l i t y due t o t h e r e l a t i v e l y s l o w r e s p o n s e o f t h e c o n t r o l l o o p m a i n t a i n i n g t h e l a s t r e a c t o r l e v e l and a l s o b e c a u s e t h e p o l y m e r i z a t i o n r a t e o f many s y s t e m s h a s b e e n f o u n d t o b e n o n l i n e a r l y a f f e c t e d by r e a c t o r r e s i d e n c e t i m e ( P o e h l e i n and D o u g h e r t y ( 3 ) ) . The most common c o n t i n u o u s e m u l s i o n p o l y m e r i z a t i o n s y s t e m s r e q u i r e i s o t h e r m a l r e a c t i o n c o n d i t i o n s and p r o v i d e f o r c o n v e r s i o n c o n t r o l through manipulation of i n i t i a t o r feed r a t e s . Typically, as shown i n F i g u r e 1, f l o w r a t e s o f monomer, w a t e r , and e m u l s i f i e r solutions into the f i r s t reactor of the series are controll e d a t l e v e l s p r e s c r i b e d b y t h e p a r t i c u l a r r e c i p e b e i n g made and r e a c t i o n temperature i s c o n t r o l l e d by changing t h e temperature o f the coolant i n the reactor j a c k e t . Manipulation of the i n i t i a t o r feed r a t e t o t h e r e a c t o r i s then used t o c o n t r o l r e a c t i o n r a t e a n d , s u b s e q u e n t l y , e x i t c o n v e r s i o n . An a s p e c t o f t h i s c o n t r o l s t r a t e g y which has n o t been c o n s i d e r e d i n t h e l i t e r a t u r e i s t h e c o m p l i c a t i o n p r e s e n t e d by t h e apparent dead-time w h i c h e x i s t s b e t w e e n t h e p o i n t o f a d d i t i o n o f i n i t i a t o r and t h e p o i n t w h e r e c o n v e r s i o n i s measured. I n many s y s t e m s t h i s d e a d - t i m e i s o f the o r d e r o f s e v e r a l h o u r s , p r e s e n t i n g a problem which convent i o n a l c o n t r o l systems a r e i n c a p a b l e o f s o l v i n g . This apparent dead-time o f t e n encountered i n i n i t i a t i o n o f p o l y m e r i z a t i o n . S e v e r a l c o n t r o l t e c h n i q u e s h a v e b e e n d e v e l o p e d t o compensate f o r l a r g e d e a d - t i m e s i n p r o c e s s e s and h a v e r e c e n t l y b e e n r e v i e w e d by G o p a l r a t n a m , e t a l . ( 4 ) . Among t h e most e f f e c t i v e o f t h e s e t e c h n i q u e s and t h e one w h i c h a p p e a r s t o be most r e a d i l y a p p l i c a b l e t o continuous emulsion p o l y m e r i z a t i o n i s the a n a l y t i c a l p r e d i c t o r method o f d e a d - t i m e c o m p e n s a t i o n (DTC) o r i g i n a l l y p r o p o s e d b y Moore ( 5 ) . The a n a l y t i c a l p r e d i c t o r h a s b e e n demons t r a t e d b y Doss and Moore ( 6 ) f o r a s t i r r e d t a n k h e a t i n g s y s t e m and b y M e y e r , e t a l . ( 7 ) f o r d i s t i l l a t i o n column c o n t r o l i n t h e only experimental a p p l i c a t i o n s p r e s e n t l y i n the l i t e r a t u r e . Implem e n t a t i o n o f t h e a n a l y t i c a l p r e d i c t o r method t o monomer c o n v e r s i o n c o n t r o l i n a t r a i n of continuous emulsion p o l y m e r i z a t i o n reactors i s t h e s u b j e c t o f t h i s paper.


33.

LEFFEW AND DESHPANDE

Emulsion

Polymerization

535

Reactors

The a n a l y t i c a l p r e d i c t o r , a s w e l l a s t h e o t h e r d e a d - t i m e compensation t e c h n i q u e s , r e q u i r e s a m a t h e m a t i c a l model o f t h e process f o r implementation. The b l o c k d i a g r a m o f t h e a n a l y t i c a l p r e d i c t o r c o n t r o l s t r a t e g y , a p p l i e d t o the problem o f conversion c o n t r o l i n an e m u l s i o n p o l y m e r i z a t i o n , i s i l l u s t r a t e d i n F i g u r e 2 ( a ) . I n t h i s a p p l i c a t i o n , t h e c u r r e n t measured v a l u e s o f monomer c o n v e r s i o n and i n i t i a t o r f e e d r a t e a r e i n p u t i n t o the m a t h e m a t i c a l model w h i c h then c a l c u l a t e s t h e v a l u e o f conversion u n i t s o f t i m e i n t h e f u t u r e a s s u m i n g no c h a n g e s i n i n i t i a t o r flow o r r e a c t o r c o n d i t i o n s occur during t h i s time. H e r e , T^ i s t h e sum o f t h e p r o c e s s d e a d - t i m e , 0 dt

t

}

-

^

= Θ

+ f (n,t)

k £(V (n-l,t)) j ^ l Θ Pj

dVpj(n,t) dt

=

(2)

η

V (n,t) P i

η

+

φ(η,^ (η,0Ν (η,0 ±

±

(3)

k £(Xj(n-l,t)) - X ( n , t ) j ^ l Θη ±

dXj(n,t) dt

Ap.(n,t)

=

+

^51

=

(36^

φ( ^)ς (η,ί)Ν (η^) η

l / 3

ί

ί

(Vp.(n,t))

2 / 3

Ni(n,t)

(4)

(5)

where Ap^(n,t), V ( n , t ) , X i ( n , t ) , and N-j_(n,t) are the t o t a l p a r t i c l e area, t o t a l volume, t o t a l conversion and t o t a l number of p a r t i c l e s i n r e a c t o r η of the i ^ p a r t i c l e generation. Also, P i

f c


33.


Emulsion

Polymerization

539

Reactors

k Y ^ | ( N j ( n - l , k ) ) i s t h e t o t a l number o f p a r t i c l e s c o m i n g i n t o =

r e a c t o r η f r o m t h e d i s c h a r g e o f r e a c t o r n - 1 , summing t h e number of p a r t i c l e s i n a l l k generations that occurred i n r e a c t o r n-1. I f more t h a n one p a r t i c l e g e n e r a t i o n o c c u r s i n a r e a c t o r , t h e n t h e t o t a l p r o p e r t i e s a t t h e e x i t o f t h e r e a c t o r w i l l b e t h e sum of t h e p r o p e r t i e s o f t h e i n d i v i d u a l generations: k £(Xi(n,t),A (n,t) i-1


X(n,t) =

p

k £v i=l

=

p

(n,t),N(n,t)

=

=

k ^Ap ( n , t ) , V ( n , t ) i = l p

k lN (n,t) i = l

(6)

±

The o t h e r v a r i a b l e s i n e q u a t i o n s (2)-(6) a r e d e f i n e d i n t h e n o m e n c l a t u r e . The s e r i e s o f d i f f e r e n t i a l e q u a t i o n s above a r e s o l v e d s i m u l t a n e o u s l y w i t h m a t e r i a l b a l a n c e s on t h e i n i t i a t o r and e m u l s i f i e r c o n c e n t r a t i o n s i n t h e r e a c t o r s :

1

ί

ί

^

-

|^((In-l)w " dn)w " kddn)w

(7)

T h i s p a r t i c u l a r model a l l o w s f o r p a r t i c l e n u c l e a t i o n t o o c c u r b y e i t h e r m i c e l l a r o r homogeneous n u c l e a t i o n m e c h a n i s m s . The d e t a i l s o f t h e m a t h e m a t i c a l d e v e l o p m e n t a r e a v a i l a b l e i n the paper by K i p a r i s s i d e s (9). S o l u t i o n o f t h e s e t o f d i f f e r e n t i a l e q u a t i o n s (2)-(8) r e q u i r e s t h e a d d i t i o n a l i t e r a t i o n o v e r t h e number o f r e a c t o r s i n t h e t r a i n . The c o n t r o l l e r i n F i g u r e 2 was c h o s e n a r b i t r a r i l y t o b e a p r o p o r t i o n a l + i n t e g r a l + d e r i v a t i v e t y p e , t h e d i s c r e t e form ( f o r i m p l e m e n t a t i o n on t h e d i g i t a l computer) o f w h i c h i s : i% - m _ n

1

=

Δπ^ =

+ g ( e

where K e T

c

n

s

= = = =

n

K ((e

- 2 e

c

n

_

1

n

+

- e _ ) + n

1

τι (9)

e _ )) n

2

c o n t r o l l e r output a t sample p e r i o d η c o n t r o l l e r gain error s i g n a l (setpoint-feedback value) period η sampling period

a t sample


540

EMULSION POLYMERS AND

EMULSION POLYMERIZATION

Tj = r e s e t time T-p = d e r i v a t i v e t i m e The f e e d b a c k v a l u e u s e d i n t h e c a l c u l a t i o n o f e r r o r i s t h e measured c o n v e r s i o n f o r t h e s t a n d a r d feedback l o o p , w h i l e i t i s the p r e d i c t e d v a l u e of f u t u r e c o n v e r s i o n f o r the a n a l y t i c a l p r e d i c t o r a l g o r i t h m , the p r e d i c t e d v a l u e being obtained from the model of K i p a r i s s i d e s .


Simulation

Results

K i p a r i s s i d e s considered the continuous p o l y m e r i z a t i o n of v i n y l a c e t a t e i n a s i n g l e CSTR a t e m u l s i f i e r c o n c e n t r a t i o n s w h i c h r e s u l t e d i n e i t h e r s t e a d y - s t a t e o p e r a t i o n (0.06 m o l e s s u r f a c t a n t / l i t e r o f w a t e r ) o r s u s t a i n e d o s c i l l a t i o n s i n number o f p a r t i c l e s and c o n v e r s i o n (0.01 m o l / 1 ) . The o p e n - l o o p p e r f o r m a n c e o f a two e q u a l - s i z e d r e a c t o r t r a i n i n t e r m s o f c o n v e r s i o n ( F i g u r e 4 ) , number p o l y m e r p a r t i c l e s ( F i g u r e 5) and f r e e soap a r e a ( F i g u r e 6) f o r an i n i t i a t o r f e e d c o n c e n t r a t i o n o f 0.005 mol/1 f o r r e g i o n s o f s u s t a i n e d o s c i l l a t i o n s and s t e a d y - s t a t e o p e r a t i o n i n d i c a t e s t h a t the second r e a c t o r i s q u i t e s i m i l a r i n performance to the f i r s t . I t s h o u l d be o b s e r v e d , h o w e v e r , t h a t i n b o t h regions of s u r f a c t a n t concentration, p a r t i c l e formation occurs i n both r e a c t o r s . K i p a r i s s i d e s chose t o manipulate e m u l s i f i e r f e e d r a t e and i n i t i a t o r f e e d r a t e t o c o n t r o l r e a c t o r p e r f o r m a n c e . I n many c o m m e r c i a l p o l y m e r i z a t i o n s t h e e m u l s i f i e r c o n c e n t r a t i o n i n t h e r e a c t i o n s y s t e m i s f i x e d by t e c h n i c a l o r e c o n o m i c r e a s o n s and i s , t h e r e f o r e , n o t an a v a i l a b l e m a n i p u l a b l e v a r i a b l e . We, t h e r e f o r e , have c o n s i d e r e d c o n v e r s i o n c o n t r o l u t i l i z i n g a s i n g l e v a r i a b l e , e i t h e r i n i t i a t o r flow r a t e or r e a c t o r temperature, a t b o t h h i g h and l o w , f i x e d l e v e l s o f s u r f a c t a n t . Conversion C o n t r o l of the F i r s t Reactor. I n i t i a l l y , values f o r the t u n i n g constants, K , τι, and Tj) f o r t h e c o n t r o l l e r w e r e s e l e c t e d by a c r u d e t r i a l and e r r o r method and a r e g i v e n i n T a b l e I . L a t e r , " o p t i m u m " v a l u e s w e r e d e t e r m i n e d f r o m t h e IAE t u n i n g r e l a t i o n s f o r l o a d d i s t u r b a n c e s w h i c h w i l l be d i s c u s s e d b e l o w . A s e t o f c o n s t a n t s a r e shown i n T a b l e I f o r c o n v e r s i o n c o n t r o l b o t h by m a n i p u l a t i o n o f i n i t i a t o r f l o w r a t e and by manipulation of r e a c t o r temperature. For the a r b i t r a r i l y s e l e c t ed t u n i n g c o n s t a n t s , t h e s i m u l a t e d s y s t e m r e s p o n s e d u r i n g s t a r t - u p , w h i l e u n d e r c l o s e d - l o o p c o n t r o l , i s shown i n F i g u r e s 7 t h r o u g h 10. I n t h e s e s i m u l a t i o n s , t h e i n i t i a t o r f l o w r a t e was h e l d constant f o r three residence times of the r e a c t o r at which p o i n t t h e c o n t r o l a c t i o n was i n i t i a t e d . Figure 7 i l l u s t r a t e s the c o n v e r s i o n p r o f i l e i n the f i r s t r e a c t o r at the low e m u l s i f i e r l e v e l (0.01 m o l / 1 ) w i t h t h e c o n t r o l s y s t e m m a n i p u l a t i n g i n i t i a t o r f e e d r a t e . The c a l c u l a t e d c o n t r o l a c t i o n f o r b o t h s t a n d a r d f e e d b a c k and a n a l y t i c a l p r e d i c t o r a l g o r i t h m s i s shown i n F i g u r e 8. Despite p r e d i c t i o n of the f u t u r e occurrence of a p a r t i c l e genera t i o n , these c o n t r o l a l g o r i t h m s were not capable of p r e v e n t i n g o r c




Emulsion

Polymerization

Reactors

DIMENSIONLESS TIME

DIMENSIONLESS TIME Figure 4. Simulated open-loop conversion of vinyl acetate system at initiator concentration of 0.005 mol/L H 0 and 50°C: (a) S = 0.06 mol/L; (b) S = 0.01 mol/L 2


EMULSION POLYMERS AND EMULSION

POLYMERIZATION


542

DIMENSIONLESS TIME Figure 5. Number of particles under open-loop conditions at initiator concentration of 0.005 mol/L H 0 and 50°C: (a) S = 0.06 mol/L; (b) S = 0.01 mol/L 2




Emulsion Polymerization Reactors

543

DIMENSIONLESS TIME

Figure 6. Free soap area in the first reactor with no control action at initiator concentration of 0.005 mol/L and 50 C: (a) S = 0.06 mol/L; (b) S = 0.01 mol/L



544


0.5

POLYMERIZATION

H

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

DIMENSIONLESS TIME

Figures 7. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L Η,Ο) under closed-loop control with arbitrarily selected con troller tuning constants and manipulation of initiator flow rate at 50°C: conversion in Rl—STD feedback ( ) vs. DTC ( )

DIMENSIONLESS TIME

Figures 8. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L H.O) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of initiator flow rate at 50°C: Rl—no. of par ticles—STD feedback ( ; vs. DTC ( )



33.


Emulsion

Polymerization

Reactors

545

Figure 9. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L Η,Ο) under closed-loop control with arbitrarily selected con troller tuning constants and manipulation of initiator flow rate at 50°C: free soap area—STD feedback ( ) vs. DTC ( ;

DIMENSIONLESS TIME

Figure 10. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L Η,Ο) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of initiator flow rate at 50°C: catalyst feed concentration—STD feedback ( ) vs. DTC ( )



546



e v e n s e v e r e l y dampening p a r t i c l e n u c l e a t i o n a t t h e low e m u l s i f i e r c o n c e n t r a t i o n as shown i n F i g u r e 9. The same t y p e o f r e s p o n s e was s i m u l a t e d when t e m p e r a t u r e was t h e m a n i p u l a t e d v a r i a b l e ( F i g u r e s 1 1 - 1 3 ) . The w i d e e x c u r i s o n s i n t e m p e r a t u r e t h a t were s i m u l a t e d , o f c o u r s e , w o u l d n o t be p o s s i b l e i n p r a c t i c e , b u t were a l l o w e d i n t h e s i m u l a t i o n t o see i f t h e s p o r a d i c n u c l e a t i o n c o u l d be p r e c l u d e d . The a d v a n t a g e s o f d e a d - t i m e c o m p e n s a t i o n u n d e r these c o n d i t i o n s are f a s t e r r e t u r n to s e t p o i n t a f t e r n u c l e a t i o n s and t i g h t e r r e g u l a t o r y c o n t r o l b e t w e e n n u c l e a t i o n s . It is apparent from t h e s e s i m u l a t i o n s , however, t h a t r e a c t o r c o n t r o l i n t h e r e g i o n s o f low e m u l s i f i e r c o n c e n t r a t i o n i s v e r y p o o r , e v e n w i t h d e a d - t i m e c o m p e n s a t i o n . These r e s u l t s c l e a r l y d e m o n s t r a t e the d i f f i c u l t y of r e a c t o r c o n t r o l i n the presence of s p o r a d i c particle nucleations. P r e v e n t i o n o f f o r m a t i o n o f new p a r t i c l e s a p p e a r s t o be i m p o s s i b l e by means o f a c o n t r o l s t r a t e g y i f o n l y one v a r i a b l e i s m a n i p u l a t e d . The f r e e soap a r e a i n t h e f i r s t r e a c t o r i s shown as a f u n c t i o n o f t i m e i n F i g u r e 10 f o r c l o s e d l o o p c o n t r o l u t i l i z a i n g i n i t i a t o r f e e d r a t e . The s u c c e s s f u l c o n t r o l s t r a t e g y w o u l d p r o v i d e a c o n s t a n t f r e e soap a r e a and, hence, constant p a r t i c l e g e n e r a t i o n r a t e i n the r e a c t o r . Because t h i s r e a c t o r i s being run under a "soap-starved" c o n d i t i o n , the f r e e soap a r e a i s n o r m a l l y n e g a t i v e . I t m i g h t be p o s s i b l e t o s t a r t - u p the r e a c t o r under c o n d i t i o n s t h a t r e s u l t e d i n a constant p a r t i c l e g e n e r a t i o n r a t e and t h e n s l o w l y d e c r e a s e t h e f e e d soap c o n c e n t r a t i o n to the d e s i r e d l e v e l w i t h o u t i n i t i a t i n g the l i m i t c y c l e p e r f o r m a n c e . However, t h i s w o u l d n o t be an acceptable o p e r a t i n g c o n d i t i o n b e c a u s e t h e i n t r o d u c t i o n o f any disturbance i n t o t h e p r o c e s s w o u l d be l i k e l y t o send t h e s y s t e m i n t o t h e l i m i t c y c l e c o n d i t i o n . For t h i s operating r e g i o n , t h e r e f o r e , e l i m i n a t i o n of the unsteady-state c o n d i t i o n by means o f c o n t r o l s t r a t e g y a p p e a r s u n l i k e l y and s o l u t i o n o f t h e p r o b l e m r e q u i r e s a design m o d i f i c a t i o n that would r e s u l t i n a constant p a r t i c l e g e n e r a t i o n r a t e i n t h e s y s t e m . P o e h l e i n and D a u g h e r t y (3) s u g g e s t e d one p o s s i b l e means o f a c h i e v i n g t h i s g o a l ; by f e e d i n g a " s e e d e d " l a t e x i n t o t h e c o n t i n u o u s r e a c t o r s y s t e m where t h e seed i s formed e i t h e r i n a b a t c h r e a c t o r or i n a t u b u l a r r e a c t o r p l a c e d i n f r o n t o f t h e CSTR t r a i n . The c o n c e n t r a t i o n o f s e e d i n t h e f e e d l a t e x must be h i g h enough t o p r e c l u d e f u r t h e r p a r t i c l e n u c l e a t i o n i n the continuous r e a c t o r s . This type of d e s i g n m o d i f i c a t i o n a p p e a r s t o be n e c e s s a r y t o a c h i e v e s t a b l e operation of a t r a i n continuous emulsion polymerization reactors when i t i s n e c e s s a r y t o r u n a t l o w e m u l s i f i e r c o n c e n t r a t i o n s . At h i g h e m u l s i f i e r c o n c e n t r a t i o n , t h e u t i l i t y o f t h e a n a l y t i c a l p r e d i c t o r i s much more a p p a r e n t . U n d e r t h e s e o p e r a t i n g c o n d i t i o n s , t h e f i r s t two r e a c t o r s r e a c h a s t e a d y - s t a t e l e v e l o f p o l y m e r p a r t i c l e g e n e r a t i o n r a t e and monomer c o n v e r s i o n as was shown i n F i g u r e s 4-6. Again using the a r b i t r a t i l y s e l e c t e d tunging constants, the simulated system response during s t a r t - u p i s shown i n F i g u r s 14 t h r o u g h 17. F i g u r e 14 i l l u s t r a t e s t h e c o n v e r s i o n p r o f i l e f o r b o t h t h e s t a n d a r d f e e d b a c k and DTC



L E F F E W AND DESHPANDE

Emulsion

Polymerization

Reactors

547

DIMENSIONLESS TIME

Figure 11. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L Η,Ο) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of reactor temperature at initiator concentration of 0.005 mol/L: conversion in Rl—STD feedback ( ) vs. DTC ( )

R1-N0. OF PARTICLES-STD FEEDBACK

I

V S DTC

ι ι ι ι I ι ι ι

DIMENSIONLESS TIME

Figure 12. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L H O) under closed-loop control with arbitrarily selected con troller tuning constants and manipulation of reactor temperature at initiator concentration of 0.005 mol/L: Rl—No. of particles—STD feedback ( ) vs. DTC ( ) z

American Chemical Society Library 1151 16th St 1 . 1 . Washington, . C. 2I03SBassett, D., et al.; In Emulsion Polymers and Emulsioni Polymerization; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

EMULSION


548

POLYMERS

AND EMULSION

POLYMERIZATION

D I M E N S I O N L E S S TIME

Figure 13. Simulated start-up of vinyl acetate polymerization at low emulsifier level (0.01 mol/L H 0) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of reactor temperature at initiator concentration of 0.005 mol/L: reactor temperature—STD feedback ( ) vs. DTC ( ; 2

D I M E N S I O N L E S S TIME

Figure 14. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L H 0) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of initiator flow rate at 50°C: conversion in Rj—STD feedback ( ) vs. DTC ( ) 2



33.


Emulsion

Polymerization

549

Reactors

a l g o r i t h m s m a n i p u l a t i n g i n i t i a t o r f l o w r a t e . F i g u r e 15 i n d i c a t e s t h e d e g r e e o f c o n v e r s i o n c o n t r o l a c h i e v e d i n t h e downstream r e a c t o r s when t h e f r o n t r e a c t o r i s u n d e r c o n t r o l . A t the h i g h e r l e v e l o f e m u l s i f i e r , f r e e soap a r e a i s a v a i l a b l e a t s t e a d y - s t a t e r e s u l t i n g i n a constant p a r t i c l e s generation r a t e (Figure 16). The i n h e r e n t a d v a n t a g e s o f d e a d - t i m e c o m p e n s a t i o n u n d e r t h i s o p e r a t i n g c o n d i t i o n a r e f a s t e r a p p r o a c h t o s e t p o i n t and t i g h t e r regulatory control. As s e e n i n F i g u r e 17 t h e DTC a l g o r i t h m r e q u i r e s s m a l l e r a n d l e s s f r e q u e n t changes i n t h e m a n i p u l a t e d variable. The s y s t e m r e s p o n s e , u t i l i z i n g t e m p e r a t u r e a s m a n i p u l a t e d v a r i a b l e , i s s i m i l a r ( F i g u r e s 18 - 1 9 ) . The " l e a d i n g " t e n d e n c y o f t h e DTC a l g o r i t h m i s a p p a r e n t i n F i g u r e 19 where t h e temperature t r a c e i s presented. More t y p i c a l l y , i n s t e a d o f s e t p o i n t c h a n g e s , t h e r e g u l a t o r y problem o f responding t o a system d i s t u r b a n c e i s encountered i n c o m m e r c i a l r e a c t o r s . F o r t h i s r e a s o n , t h e optimum t u n i n g c o n s t a n t s f o r t h e P I D c o n t r o l l e r were d e v e l o p e d f r o m t h e I A E r a l a t i o n s for load disturbances. F i r s t , however, i t i s n e c e s s a r y t o o b t a i n a p r o c e s s m o d e l o f t h e s y s t e m . B r a n t l e y (10) h a s developed a process i d e n t i f i c a t i o n technique which f i t s process data t o the second order p l u s dead-time form: K e-0d

s

p

G (s) p

(10)

= (TIS+1)(T2S+1)

F o r t h e c o n t i n u o u s p o l y m e r i z a t i o n o f v i n y l a c e t a t e a t 0.06 mol/1 e m u l s i f i e r c o n c e n t r a t i o n , t h e m o d e l p r a m e t e r s a s d e t e r m i n e d by i n t r o d u c i n g s t e p changes i n t o the m e c h a n i s t i c model a r e shown i n T a b l e I I . G i v e n t h e f o r m o f e q u a t i o n ( 1 0 ) , t h e "optimum" c o n t r o l l e r t u n i n g c o n s t a n t s f o r a P I D c o n t r o l l e r w e r e d e t e r m i n e d b y t h e method o f G a l l i e r and O t t o ( 1 1 ) , w h i c h m i n i m i z e s the i n t e g r a l o f the a b s o l u t e e r r o r (IAE) t h a t r e s u l t s f o l l o w i n g a process disturbance. The e f f e c t i v e t i m e d e l a y , Τ', u s e d i n c a l c u l a t i n g t h e optimum t u n i n g c o n s t a n t s a s shown b y M e y e r , e t a l (7) i s 0 + T f o r t h e s t a n d a r d f e e d b a c k l o o p a n d T for the a n a l y t i c a l p r e d i c t o r loop. The d e g r e e o f c o n t r o l a c h i e v e d b y t h e s e a l g o r i t h m s i s m e a s u r e d b y t h e common i n t e g r a l criteria : 1. I n t e g r a l o f the square e r r o r ( I S E ) : d

s

OO

ISE 2.

f

ο 2

Q

(e(t)) dt

I n t e g r a l o f the absolute e r r o r ISE

3.

=

s

=

/

(IAE):

(e(t))dt

I n t e g r a l o f time m u l t i p l i e d by the absolute value o f the e r r o r (ITAE):


550


TABLE I : CONTROLLER PARAMETERS FOR SIMULATION STUDY


l b / h r o r °C fraction conversion Manipu lated Variable

Trç, m i n

τ j , min

By Trial

IAE Optimum

By Trial

Std. Feedback

8.0

1.68

1.0

16.83

2.0

4.29

Analytical Predictor

8.0

37.21

1.0

3.79

2.0

0.11

Std. Feedback

40.0

173.8

1.0

15.73

2.0

3.42

Analytical Predictor

40.0

2750.

1.0

2.37

2.0

0.12

Control Scheme

I AE IAE Optimum

IAE Optimum

By Trial

Initiat o r Flow

Reactor Temperature

TABLE I I : PROCESS MODEL PARAMETERS

M o d e l Form:

Kpe

Gp(s)

(T S+1)(T S+1) 1

K l b / h r o r °C fraction conversion

2

p

Manipulated Variable

0^, m i n

, min

τ 2, m i n

Initiator Flow

30.0

0.2793

3.6826

0.0200

Temperature

30.0

0.002529

2.2540

0.0200



33.

L E F F E W A N D DESHPANDE


DIMENSIONLESS TIME

Figure 15. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L H>0) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of initiator flow rate at 50°C: Rl—no. of particles—STD feedback ( ) vs. DTC ( )

L-L 1

I I I

I

I I I I

I J

I / ! i»/

J

I

DIMENSIONLESS TIME

Figure 16. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L H>0) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of initiator flow rate at 50°C: free soap area—STD feedback ( ) vs. DTC ( )


551

552

EMULSION

l

ι t ι ι

1

• ι ι ι

I

ι

POLYMERS AND EMULSION

ι • ι I ι

ι ιι •ι I 1 ι• I ιI I

I

ι ι • ι

A ;

1

1

I I • •

r

\

-- I

-

A

A ^

\ 'W é\

:


ι I I I

/

~ -

ι -

:

V U :·

i -0 .j0 -

I

POLYMERIZATION

• r r

r 2r. 5r

r r -

, ii , ^5.0 ,

ιι

ι7.5 ι.

r.

7 r τ' ι τ ι ι τ r ι · ι ι ι ι ι ι ι ι ι · ι ι 10.0 15.0 17.5 20.0 22.5 τ r τ-τ 12.5

DIMENSIONLESS TIME

Figure 17. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L Η,Ο) under closed-loop control with arbitrarily selected con troller tuning constants and manipulation of initiator flow rate at 50°C: catalyst feed concentration—STD feedback ( ) vs. DTC ( )

22.5

DIMENSIONLESS TIME

Figure 18. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L Η,Ο) under closed-loop control with arbitrarily selected con troller tuning constants and manipulation of reactor temperature at initiator con centration of 0.005 mol/L: conversion in Rl—STD feedback ( ) vs. DTC ( ;



33.


Emulsion

Polymerization

Reactors

553

DIMENSIONLESS TIME

Figure 19. Simulated start-up of vinyl acetate polymerization at high emulsifier level (0.06 mol/L HO) under closed-loop control with arbitrarily selected controller tuning constants and manipulation of reactor temperature at initiator concentration of 0.005 mol/L: (a) Rl reactor temperature—STD feedback ( ) vs. DTC ( ); (b) conversion in Rl—STD feedback ( ) vs. DTC ( )



554


ITAE

=

/Q


t(e(t))dt

where e ( t ) i s t h e d i f f e r e n c e b e t w e e n t h e s e t p o i n t v a l u e and the a c t u a l v a l u e of c o n v e r s i o n . These " c o n t r o l l a b i l i t y " c r i t e r i a a r e used t o j u d g e t h e p e r f o r m a n c e o f t h e v a r i o u s c o n t r o l s y s t e m s d e s c r i b e d above i n T a b l e I I I . As s e e n i n T a b l e I I I , d e a d - t i m e c o m p e n s a t i o n m a r k e d l y improves system performance at the h i g h e m u l s i f i e r c o n c e n t r a t i o n . A s y s t e m l o a d d i s t u r b a n c e was s i m u l a t e d by i n t r o d u c i n g a s t e p change i n t h e p r o p a g a t i o n r a t e c o n s t a n t , k p , o f - 5 % a f t e r t h e s y s t e m was a t s t e a d y - s t a t e u n d e r c l o s e d - l o o p c o n t r o l . This type of d i s t u r b a n c e i n w h i c h the system r e a c t i o n r a t e changes s u d d e n l y , i n most c a s e s due t o t h e i n t r o d u c t i o n o f an unknown d i s t u r b a n c e , i s common i n c o m m e r c i a l p r o c e s s e s . The i m p r o v e d p e r f o r m a n c e o f t h e a n a l y t i c a l p r e d i c t o r w i t h "optimum" c o n t r o l l e r t u n i n g i s shown i n F i g u r e 20 f o r r e g u l a t o r y c o n t r o l . Notice that these c o n t r o l l e r s e t t i n g s r e s u l t i n " r i n g i n g " of the c o n t r o l v a l v e i n t h e a n a l y t i c a l p r e d i c t o r a l g o r i t h m t h a t does n o t o c c u r w i t h the a r b i t r a r i l y s e l e c t e d t u n i n g constants. These s i m u l a t e d r e s u l t s f o r the h i g h e m u l s i f i e r c o n c e n t r a t i o n o p e r a t i n g c o n d i t i o n d e m o n s t r a t e t h e u t i l i t y o f d e a d - t i m e compens a t i o n t o the c o n t r o l of c o n v e r s i o n from the f i r s t r e a c t o r i n a train. W i t h i m p l e m e n t a t i o n o f t h i s d e g r e e o f c o n t r o l on t h e f i r s t r e a c t o r , c o n t r o l schemes f o r d o w n s t r e a m r e a c t o r s can be s i m p l i f i e d as d i s c u s s e d i n t h e n e x t s e c t i o n . C o n v e r s i o n C o n t r o l o f Downstream R e a c t o r s . With the f i r s t r e a c t o r of the t r a i n under c l o s e d - l o o p c o n v e r s i o n c o n t r o l , r e s p o n s e o f d o w n s t r e a m r e a c t o r s t o c h a n g e s i n i n i t i a t o r f l o w can be a p p r o x i m a t e d v e r y c l o s e l y b y t h e s e c o n d o r d e r p l u s d e a d - t i m e model of e q u a t i o n ( 1 0 ) , p r o v i d e d t h a t p a r t i c l e n u c l e a t i o n i n the downstream r e a c t o r o c c u r s a t a c o n s t a n t r a t e o r i s t o t a l l y p r e c l u d e d b e c a u s e o f l o w f r e e soap c o n c e n t r a t i o n i n t h e f e e d t o the r e a c t o r . A danger of o p e r a t i n g these r e a c t o r t r a i n s at high e m u l s i f i e r concentration to provide s t a b i l i t y i s that, a l t h o u g h t h e f r o n t r e a c t o r s do r e a c h a s t e a d y - s t a t e l e v e l o f p a r t i c l e n u c l e a t i o n r a t e , t h e r e may e x i s t t h e c o n d i t i o n i n a d o w n s t r e a m r e a c t o r t h a t l e a d s t o s p o r a d i c n u c l e a t i o n s and, hence, o s c i l l a t i o n s i n conversion. The r e q u i r e d c o n d i t i o n f o r s t a b i l i t y , t h e n , w o u l d be t h a t t h e f e e d t o t h e l a s t r e a c t o r would a l l o w f o r continuous p a r t i c l e n u c l e a t i o n , i . e . , that p o s i t i v e f r e e s o a p a r e a be p r e s e n t i n t h e f i n i s h e d e m u l s i o n . For many s y s t e m s t h i s r e q u i r e m e n t f o r c e s a l e v e l o f e m u l s i f i e r i n the feed that r e s u l t s i n p r o h i b i t i v e l y h i g h emulsion v i s c o s i t i e s . The s o l u t i o n t o t h i s p r o b l e m a g a i n r e q u i r e s a r e a c t o r d e s i g n m o d i f i c a t i o n which p r o v i d e s s e p a r a t i o n of the regimes of p a r t i c l e n u c l e a t i o n and p a r t i c l e g r o w t h . The f o l l o w i n g d e v e l o p m e n t f o r c o n t r o l o f downstream r e a c t o r s i s a p p l i c a b l e , however, t o t h o s e r e a c t o r s i n which a constant or n e g l i g i b l e r a t e of p a r t i c l e nucleation occurs.


33.



555

Γ— CM m m

ON

r- m Γ- m

CM m

CT. CM m ο ο r H CM m O mN oo cr r- 00 ο σ>
rH ON Ο CM ο r H I CM cd cd

φ Ο ΡΜ Η 2 w ρ Η

§•

φ Η


Η

556



X 10 1.3 -

CATALYST FEED CONC-STD FEEDBACK VS DTC I I I

I

I I I I

I

I I I I

I

I I I I

I

ι ι ι ι

I

ι ι ι ι

I

I

' ' ' ι

Ο 2

1.0 •

Ο ζ ο ο

(Λ >-

< Ο

10.0

Ι

ι ι ι ι Ι ι ι ι ' Ι ι ' ' ' I ' ' ι ' I '

12.5

15.0

17.5

20.0

22.5

'' 'I' ' 1

25.0

1

1

1

27.5

1

'

1

I

1

30.0

1

' ' .

32.5

DIMENSIONLESS TIME Figure 20. Simulated conversion response of continuous polymerization system to a load disturbance under closed-loop control with IAE optimum controller tuning constants and manipulation of initiator flow rate at 0.06 mol/L H O sur factant and 50°C: catalyst feed concentration—STD feedback ( ) vs. DTC (—; s

point

T

s

ΤΓ-

Figure 21.

Block diagram of digital control loop for downstream reactors


33.


Emulsion

Polymerization

557

Reactors

F o r t h e v i n y l a c e t a t e s y s t e m a t 0.06 m o l / 1 o f e m u l s i f i e r i n the f e e d , the process i d e n t i f i c a t i o n technique o f B r a n t l e y (10) was u s e d t o f i n d t h e c o n v e r s i o n r e s p o n s e o f t h e t h i r d e q u a l - s i z e d r e a c t o r i n t h e t r a i n t o changes i n f e e d r a t e o f i n i t i a t o r t o that reactor:

Gp(s)

=

3 0

Xo(s)

r — 5

F

e

m (s) 3


0 S

O.O525 - " (19.864s+l)(0.020s+l)

=

(ID

B r a n t l e y ' s method u s e s a m u l t i v a r i a b l e s e a r c h t e c h n i q u e t o determine t h e model parameters o f a second o r d e r p l u s dead-time model t h a t m i n i m i z e s t h e square o f t h e e r r o r between the a c t u a l o u t p u t and t h e m o d e l - p r e d i c t e d o u t p u t f o r a g i v e n s e t o f i n p u t values. The e m p i r i c a l m o d e l o f e q u a t i o n (11) p r e d i c t e d t h e r e s p o n s e o f t h e m e c h a n i s t i c m o d e l t o a s t e p change i n i n i t i a t o r f l o w v e r y c l o s e l y ( t h e average a b s o l u t e d e v i a t i o n between t h e e m p i r i c a l m o d e l and m e c h a n i s t i c m o d e l was 0.8% o f t h e r e s p o n s e ) . Three a l g o r i t h m s have been c o n s i d e r e d f o r c o n t r o l o f t h e downstream r e a c t o r modeled by e q u a t i o n ( 1 1 ) . 1. Optimum P I D C o n t r o l l e r : T h i s t e c h n i q u e f o l l o w s t h e work o f G a l l i e r and O t t o (11) a n d i s d e s c r i b e d i n t h e previous section. 2. Z - T r a n s f o r m - D e s i g n e d A l g o r i t h m : T h i s t e c h n i q u e i n v o l v e s s p e c i f y i n g the d e s i r e d response t o a load o r s e t p o i n t change a n d u s i n g t h a t r e s p o n s e t o d e s i g n t h e c o n t r o l l e r . The d i g i t a l p r o c e s s c o n t r o l l o o p f o r downstream r e a c t o r c o n t r o l i s shown i n F i g u r e 2 1 . The c o n t r o l l e r t r a n s f e r f u n c t i o n , D ( z ) , can be o b t a i n e d from t h i s loop a s : D ( Z )

1

" HG(zT

.

C(z)/R(z)

.

n

F~C(z)/R(z)

where HG(z) i s t h e Z - t r a n s f o r m o f t h e z e r o - o r d e r times the process t r a n s f e r f u n c t i o n , o r

(

1

2

)

holds

-sT HG(z) = ( —

· G(s))

(13)

s T y p i c a l l y , one s p e c i f i e s t h e d e s i r e d r e s p o n s e , C ( z ) / R ( z ) , w h i c h y i e l d s f r o m e q u a t i o n (12) t h e r e q u i r e d d e s i g n o f the c o n t r o l l e r , D(z). I n p r a c t i c e , however, t h i s d e s i g n technique r e s u l t s i n a c o n t r o l l e r which r e q u i r e s excess i v e v a l v e movement, an u n d e s i r a b l e s i t u a t i o n . Conseq u e n t l y , Kalman (12) d e v e l o p e d a Z - t r a n s f o r m a l g o r i t h m w h i c h s p e c i f i e s t h e d e s i r e d o u t p u t , C ( z ) , and t h e d e s i r e d v a l v e t r a v e l , M(z) f o r a s e t p o i n t change. The d e s i r e d r e s p o n s e a n d v a l v e t r a v e l f o r a u n i t s t e p change i n s e t p o i n t i s shown i n F i g u r e 22. T h e s y s t e m r e s p o n s e ,


558


then,

POLYMERIZATION

follows the expression:

C(z) = z'^qz^+zVV*")

(14)

where Ν i s t h e i n t e g e r number o f s a m p l i n g p e r i o d s process dead-time. L i k e w i s e , t h e v a l v e t r a v e l c a n be d e s c r i b e d a s :


M(z)

= M

+

0

MT^Z"

+

1

M z"" f

2

+ M z"

3

+

f

...

i nthe

(15)

The t r a n s f e r f u n c t i o n s r e l a t i n g r e s p o n s e and v a l v e _ t r a v e l t o a u n i t change i n s e t p o i n t ( R ( z ) = 1 / ( 1 - z ) ) are then: C(z) -1 χ -N,_ -1, -2, -3, ν ^ = ( 1 - z ) z ( C ^ z +z +z +···) β

R

η

= C^z

"N-i/n

η \ "Ν-2

(l-C^)z

„

= Ρχζ

-N-i

,

Ώ

+ P2Z

-N-2

P(z)

(16)

and,

(1-z

R(z)

MQ

= q

)(Mo+M

+

Q

(M -M )z 1

+ ς ζ χ

+Mfz

1Z

1

0

-ι 1

+ q z 2

+

+Mfz

(M -M )z F

Z

+·

2

x

= Q(z)

(17)

I t c a n b e s e e n f r o m F i g u r e 21 t h a t t h e r a t i o o f C ( z ) t o M(z) i s t h e process p u l s e t r a n s f e r f u n c t i o n s , HG(z):

H

G

(

Z

)

~

M(z)

"

I t f o l l o w s from t h e c o n t r o l l e r design that:

DM

=

(

Q(z) equation

1

8

)

(12)

^

(19)

T h e r e f o r e , t h e Kalman d e s i g n e d a l g o r i t h m i s s p e c i f i e d by o b t a i n i n g the modified Z-transform of the process pulse t r a n s f e r f u n c t i o n o t g e t P ( z ) and Q ( z ) and t h e n s u b s t i t u t i n g these values i n t o equation ( 1 9 ) . Self-Tuning Algorithm: This algorithm incorporates the method d e s c r i b e d j u s t above w i t h t h e added f e a t u r e o f



33.

LEFFEW AND

DESHPANDE

559


o n - l i n e t u n i n g of the c o n t r o l l e r . Here, process data are compared e a c h s a m p l i n g p e r i o d w i t h t h e m o d e l p r e d i c t i o n o b t a i n e d f r o m e q u a t i o n ( 1 1 ) . When s i g n i f i c a n t e r r o r d e v e l o p s , t y p i c a l l y as a r e s u l t o f t h e i n t r o d u c t i o n o f a process d i s t u r b a n c e , the process i d e n t i f i c a t i o n technique o f B r a n t l e y (10) i s u s e d t o c a l c u l a t e new p a r a m e t e r s o f t h e p r o c e s s m o d e l . These new p a r a m e t e r s r e s u l t i n a new c o n t r o l l e r a l g o r i t h m as o b t a i n e d by t h e Kalman method d e s c r i b e d above. T h i s c o n t r o l l o o p i s shown s c h e m a t i c a l l y i n F i g u r e 23. The p e r f o r m a n c e o f t h e a l g o r i t h m s d e s c r i b e d above i n c o n t r o l l i n g downstream r e a c t o r c o n v e r s i o n i s i l l u s t r a t e d f o r a s t e p change i n s e t p o i n t o f t h e t h i r d r e a c t o r i n t h e v i n y l a c e t a t e p o l y m e r i z a t i o n m o d e l i n F i g u r e 24 and f o r t h e i n t r o d u c t i o n o f a d i s t u r b a n c e ( 1 0 % d e c r e a s e i n kp) i n F i g u r e 25. The c o n t r o l c r i t e r i a r e s u l t s f o r these a l g o r i t h m s a r e g i v e n i n T a b l e IV. M a n i p u l a t i o n o f i n i t i a t o r f l o w r a t e t o t h e t h i r d r e a c t o r i s shown f o r e a c h c o n t r o l l e r f o r a s t e p d i s t u r b a n c e i n F i g u r e 26. The s e l f - t u n i n g a l g o r i t h m shows no improvement i n c o n t r o l o v e r t h e Z-transform technique over the p e r i o d of operation i l l u s t r a t e d i n F i g u r e 25. However, t h e p r o c e s s m o d e l had b e e n u p d a t e d t o a c c o u n t f o r t h e r e d u c t i o n i n kp by t h e p r o c e s s i d e n t i f i c a t i o n technique to the form: r b

(«\ P

i

s

;

"

3 0 S

0.00745e" (0.024s+l)(0.02s+l)

( t > 1

'

^ }

Consequently, the s e l f - t u n i n g a l g o r i t h m would respond to the i n t r o d u c t i o n o f a s e t p o i n t change o r a n o t h e r d i s t u r b a n c e f a s t e r than the o r i g i n a l model, i l l u s t r a t i n g the value of the s e l f - t u n i n g algorithm. The f r e e soap a r e a o f t h e t h i r d r e a c t o r o f t h e v i n y l a c e t a t e p o l y m e r i z a t i o n i s shown i n F i g u r e 27. N o t i c e that f o r the r e a c t o r c o n d i t i o n s chosen f o r t h i s s i m u l a t i o n t h a t p a r t i c l e f o r m a t i o n o c c u r s c o n t i n u o u s l y and a p p r o a c h e s a s t e a d y - s t a t e l e v e l i n t h e t h i r d r e a c t o r . F o r t h e c a s e o f v i n y l a c e t a t e , t h e homogeneous n u c l e a t i o n r a t e i s s u f f i c i e n t a t low f r e e soap c o n c e n t r a t i o n t o p r o v i d e a constant r a t e of p a r t i c l e g e n e r a t i o n i n the absence of m i c e l l e s and, h e n c e , the f r e e - s o a p a r e a i s a c t u a l l y d e p l e t e d i n t h e t h i r d r e a c t o r b u t r e a c h e s a s t e a d y - s t a t e l e v e l so t h a t m i c e l l a r n u c l e a t i o n never occurs. As d i s c u s s e d a b o v e , t h i s c o n d i t i o n does n o t a l w a y s e x i s t and t h e p o t e n t i a l f o r i n i t i a t i o n of s u s t a i n e d o s c i l l a t i o n s e x i s t s f o r downstream r e a c t o r s where t h e f r e e soap a r e a i s n e g a t i v e and t h e homogeneous n u c l e a t i o n r a t e i s not s i g n i f i c a n t compared w i t h t h e m i c e l l a r n u c l e a t i o n r a t e . T h i s s i t u a t i o n must be c o n s i d e r e d i n r e a c t o r d e s i g n and i n s e l e c t i o n o f o p e r a t i n g c o n d i t i o n s .



560


DESIRED SYSTEM

Figure 22.

RESPONSE

Desired system response C(z) and valve travel M(z) for a unit step change in setpoint according to Kalman's approach

Algorithm Adapter

Model Identificati*

Parameters

Algorithm Parameters Set-

Secondary Controller

Control Algorithm

point

IL j]

Ms)

ι Figure 23.

ξ

Block diagram of self-tuning control loop for downstream reactors


33.


Emulsion

Polymerization

Reactors

561


DOWNSTREAM REACTOR C O N T R O L - C O N V E R S I O N 0.5825 - f - ^ l x x . , L ^ ^ L ^ ^

DIMENSIONLESS TIME

Figure 24. Simulated response of third reactor of a continuous vinyl acetate polymerization to a step change in setpoint at high emulsifier feed concentration (0.06 mol/L ΗΌ) and manipulation of initiator flow rate to the third reactor at 50°C (( j optimum PID); ( ) Ζ transform)

0.5, W

0.56

L

W

I

DOWNSTREAM REACTOR CONTROL-CONVERSION I ^ X X W ^ L O ^ L I •• •

-J

0.55

L

0.54

L

0.53

1

0 52 -

-

0.51

-

0.50 4-τϊ-τ-Γ-Γ-r- r~-f~ τ j ι r~ τ τ j τ t r τ- -j τ ι ι ι j ι ι r ι | τ~~τ ι ι | ι ι ι ι '5.0

17.5

20.0

22.5

25.0

27.5

30.0

32.5

35.0

DIMENSIONLESS TIME

Figure 25. Simulated response of third reactor of a continuous vinyl acetate polymerization to a step disturbance at high emulsifier feed concentration (0.06 mol/L Η,Ο) and manipulation of initiator flow rate to the third reactor at 50°C (( ) optimum PID); ( ) Ζ transform; (XXX) self-tuning algorithm)


562

EMULSION POLYMERS

AND EMULSION POLYMERIZATION


DOWNSTRCAU REACTOR CONTROL-INITIATOR FLOW MANIPULA

DIMENSIONLESS TIME

Figure 26. Simulated initiator flow rate manipulation for closed-loop control of the third reactor in response to a step disturbance at high emulsifier feed concen tration (0.06 mol/L H>0) and 50°C (( ; optimum FID; ( ) Ζ transform; (XXX) self-tuningalgorithm)

DIMENSIONLESS TIME

Figure 27. Free soap area of emulsion in third reactor of a continuous vinyl acetate polymerization under closed-loop control at high emulsifier feed concen tration (0.010 mol/L H 0) and 50°C (( ; optimum PID; ( ) Ζ transform) 2



33.


TABLE I V :

Emulsion

Polymerization

Reactors

CONTROL SYSTEM PERFORMANCE - DOWNSTREAM REACTOR

Controller Algorithm

ISE

IAE

ITAE

S t e p Change i n S e t p o i n t 1.

Optimum P I D

0.0611

2.234

2.

Z-Transform

0.0612

2.219

94.17 94.43

S t e p Change i n D i s t u r b a n c e ( 1 0 % d e c r e a s e i n k p ) 1.

Optimum P I D

0.0140

1.339

2.

Z-Transform

0.0147

1.370

104.0

3.

Self-tuning

0.0147

1.370

104.0

Process Model:

ggl

-

89.29

( 1 9 . 8 6 4 s S ) (0.020s+l)


563

564

EMULSION

POLYMERS

AND

EMULSION

POLYMERIZATION


Conclusions The u t i l i t y o f t h e a n a l y t i c a l p r e d i c t o r method o f d e a d - t i m e compensation t o c o n t r o l o f conversion i n a t r a i n o f continuous e m u l s i o n p o l y m e r i z e r s has been demonstrated by s i m u l a t i o n o f t h e v i n y l a c e t a t e s y s t e m . The s i m u l a t e d r e s u l t s c l e a r l y show t h e extreme d i f f i c u l t y o f c o n t r o l l i n g t h e c o n v e r s i o n i n systems which are operated at "soap-starved" c o n d i t i o n s . The a n a l y t i c a l p r e d i c t o r was shown, h o w e v e r , t o p r o v i d e s i g n i f i c a n t l y i m p r o v e d c o n t r o l of conversion, i n presence of e i t h e r setpoint o r load c h a n g e s , a s compared t o s t a n d a r d f e e d b a c k s y s t e m s i n o p e r a t i n g r e g i o n s t h a t promote c o n t i n u o u s p a r t i c l e f o r m a t i o n . These s i m u l a t i o n s s u g g e s t t h e a n a l y t i c a l p r e d i c t o r t e c h n i q u e t o be t h e p r e f e r r e d method o f c o n t r o l when i t i s d e s i r e d t h a t o n l y one v a r i a b l e ( p r e f e r a b l y i n i t i a t o r f e e d r a t e ) be m a n i p u l a t e d . T h r e e a l g o r i t h m s were a l s o s u g g e s t e d f o r c o n t r o l o f c o n v e r s i o n f r o m d o w n s t r e a m r e a c t o r s when t h e f i r s t r e a c t o r o f t h e t r a i n i s o p e r a t i n g a t s t e a d y - s t a t e under c l o s e d - l o o p c o n t r o l : a c o n v e n t i o n a l P I D c o n t r o l l e r w i t h optimum I A E t u n i n g c o n s t a n t s , a Z - t r a n s f o r m d e s i g n e d a l g o r i t h m b a s e d on an a p p r o x i m a t e l i n e a r p r o c e s s m o d e l , and a s e l f - r e g u l a t i n g s y s t e m w h i c h u p d a t e s t h e c o n t r o l l e r a l g o r i t h m b a s e d on c h a n g e s i n t h e p r o c e s s m o d e l a c h i e v e d by o n - l i n e p r o c e s s i d e n t i f i c a t i o n . A l l three of these c o n t r o l l e r s were shown t o p r o v i d e e x c e l l e n t c o n t r o l o f d o w n s t r e a m r e a c t o r s i n which continuous polymer p a r t i c l e generation occurred, f o r b o t h s e t p o i n t and l o a d c h a n g e s . Nomenclature A C(z) dm dp D(z) e f(n,t) Gp (Dw HG(z) Μχ p

= = = = = = = = = =

MW M(z) ,m = = 3 = Ν q^(n,t)=

m

R(z) S Τ

= = = =

t o t a l s u r f a c e area o f polymer p a r t i c l e s s y s t e m o u t p u t o r r e s p o n s e i n z-domain monomer d e n s i t y polymer d e n s i t y controller transfer function system e r r o r s i g n a l net r a t e o f p a r t i c l e n u c l e a t i o n i n r e a c t o r η a t time t system open-loop t r a n s f e r f u n c t i o n i n i t i a t o r concentration process pulse t r a n s f e r function t o t a l monomer c o n c e n t r a t i o n i n t h e e m u l s i o n m o l e c u l a r w e i g h t o f monomer value of manipulated v a r i a b l e i n i t i a t o r flow rate to t h i r d reactor i n series t o t a l number o f p a r t i c l e s a v e r a g e number o f r a d i c a l s p e r p a r t i c l e i n r e a c t o r η at time t setpoint value emulsifier concentration sampling period sampling period


33. LEFFEW AND DESHPANDE

X Xo


total volume of polymer particles total monomer conversion monomer conversion from third reactor in series

Subscripts i = having to do with the i generation of particles within a given reactor η = reactor number or sample number t

n


Greek Symbols Φ = monomer volume fraction in a particle θ = mean residence time of a reactor °d - process dead-time time constant τ References 1. Wismer, D. Α., Brand, W., Joint Automatic Control Conference, Stanford, California, (1964), 147-154. 2. Francis, D. H. and H. R. Sontag, Chemical Engineering Progress, 1949, 45, (6). 3. Poehlein, G. W. and D. J. Dougherty, Rubber Chemistry and Technology, 1977, 50, (3). 4. Gopalratnam, P. C., P. B. Deshpande, and R. H. Ash, paper presented at ISA National Conference, Chicago, Illinois, paper No. C.I. 79-619 (1979). 5. Moore, C. F., Selected Problems in the Design and Implementa tion of Direct Digital Control, Ph.D. Thesis, Louisiana State University (1969). 6. Doss, J. E. and C. F. Moore, 74th National AICHE Meeting, New Orleans, Louisiana (1973). 7. Meyer, C., D. E. Seborg and R. K. Wood, Ind. Eng. Chem. Process Des. Dev., 1978, 17, (1). 8. Kiparissides, C., J. F. McGregor and A. E. Hamielec, Journal of Applied Polymer Science, 1979, 23, 401-418. 9. Kiparissides, C., Continuous Latex Reactor Modelling and and Experimental Studies, Ph.D. Thesis, McMaster University (1978). 10. Brantley, R. O., M.S. Thesis, University of Louisville, Louisville, Kentucky (1981). 11. Gallier, P. W. and R. E. Otto, Instrumentation Technology, 1968, 15, (2), 65-70. 12. Kalman, R. Ε., Trans. AIEE, 1954, 236-247. RECEIVED April 6, 1981.


565

Emulsion Polymers and Emulsion Polymerization - American

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