Free-Radical Polymerization - American Chemical Society

10 Free-Radical Polymerization: Sensitivity of Conversion and Molecular Weights to Reactor Conditions KIU

H . LEE and J O H N P. M A R A N O , JR.1

Downloaded by TROY UNIV on October 14, 2014 | http://pubs.acs.org Publication Date: July 31, 1979 | doi: 10.1021/bk-1979-0104.ch010

Union Carbide Corporation, P. O . Box 8361, South Charleston, W V 25303

One o n - g o i n g objective in a c o m m e r c i a l polymerization reactor of a fixed size is to m a x i m i z e t h e reactor productivity at the desired p r o d u c t properties. Polymerization reactors a r e sensitive to c h a n g e s in operating p a r a m e t e r s b e c a u s e the reactors involve highly e x o t h e r m i c reactions. A relatively m i n o r fluctuation in t h e operating variables could c a u s e w i d e fluctuations in the reactor r e s p o n s e s . T h e r e f o r e , it is i m p o r t a n t to search o u t underlying relationships c o n c e r n i n g t h e reactor p e r f o r m a n c e and t h e modes of reactor operations. The conceptual t r e n d s obtained in this t y p e of investigation provide valuable information regarding t h e operating limits o f a given reactor a n d also aid in d e t e r m i n i n g t h e future actions a i m e d a t further i m p r o v i n g t h e limits o f the reactor p e r f o r m a n c e . The c o n c e p t u a l s t u d y may dictate c h a n g e s in the initiator s y s t e m , t h e solvent s y s t e m ( c h a i n transfer a g e n t s ) and the h e a t transfer s y s t e m for a reactor of fixed size to provide t h e maximum possible c o n v e r s i o n a t desired p r o d u c t properties. The s t u d y o f t h e peak t e m p e r a t u r e s e n s i t i v i t y t o t h e r e a c t o r o p e r a t i n g p a r a m e t e r s and t h e c o n s t r u c t i o n o f s e n s i t i v i t y b o u n d a r y c u r v e s f o r s t a b l e r e a c t o r o p e r a t i o n were p r e v i o u s l y r e p o r t e d ( l ) . T h i s p a p e r p r e s e n t s a computer s t u d y on c o n c e p t u a l r e l a t i o n s h i p s between t h e c o n v e r s i o n - p r o d u c t p r o p e r t i e s and t h e r e a c t o r o p e r a t i n g parameters i na p l u g f l o w t u b u l a r r e a c t o r of f r e e r a d i c a l polymerization. I n p a r t i c u l a r , a c o n t o u r map o f c o n v e r s i o n molecular weight r e l a t i o n s h i p s i n a r e a c t o r o f f i x e d s i z e i s presented and t h e s e n s i t i v i t y o f i t s r e l a t i o n s h i p t o t h e c h o i c e o f i n i t i a t o r system, s o l v e n t system and heat t r a n s f e r system a r e discussed. In t h e study, t h e k i n e t i c r a t e constants a p p l i c a b l e t o t h e p o l y m e r i z a t i o n o f e t h y l e n e (2_,3_) were u s e d w i t h a n assumed a c t i v a t i o n v o l u m e . These v a l u e s a p p e a r t o b e a r e a s o n a b l y c o n s i s t e n t s e t o f c o n s t a n t s f o r t h e p o l y m e r i z a t i o n o f e t h y l e n e and, a s shown Current

a d d r e s s : 'Now l o c a t e d a t M o b i l C h e m i c a l Company, E d i s o n , New J e r s e y .

0-8412-0506-x/79/47-104-221$07.75/0 © 1979 American Chemical Society In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

222

POLYMERIZATION REACTORS AND PROCESSES

i n F i g u r e 1 , g e n e r a t e a t e m p e r a t u r e p r o f i l e w h i c h a p p e a r s t o be r e a s o n a b l y t y p i c a l o f t h e one m e a s u r e d i n a h i g h - p r e s s u r e ethylene polymerization reactor. T h i s work p a r t i c u l a r l y emphasizes t h e importance o f s e l e c t i n g t h e i n i t i a t o r s y s t e m f o r optimum r e a c t o r o p e r a t i o n a n d r e v e a l s g e n e r a l concepts w h i c h s p e c i f y t h e d e s i r e d p r o p e r t i e s and o p e r a t i o n a l modes o f an optimum i n i t i a t o r s y s t e m . In addition,the e f f e c t s o f t h e s y s t e m h e a t t r a n s f e r a n d t h e CTA ( c h a i n t r a n s f e r a g e n t ) l e v e l on t h e c o n v e r s i o n - m o l e c u l a r w e i g h t s r e l a t i o n s h i p s are presented.


P o l y m e r i z a t i o n Tubular

Reactor

Model

The c o m p u t e r m o d e l u s e d f o r t h i s a n a l y s i s i s b a s e d on a p l u g f l o w t u b u l a r r e a c t o r o p e r a t i n g u n d e r r e s t r a i n t s o f t h e commonly a c c e p t e d k i n e t i c mechanism f o r p o l y m e r i z a t i o n r e a c t i o n s (3. h). The c o m p u t e r m o d e l c o n s i s t s o f t h e n u m e r i c a l i n t e g r a t i o n o f a s e t of d i f f e r e n t i a l equations which conceptualizes t h e high-pressure p o l y e t h y l e n e r e a c t o r . A Runge-Kutta technique i s used f o r i n t e g r a t i o n w i t h t h e use o f an a u t o m a t i c a l l y a d j u s t e d i n t e g r a t i o n step s i z e . The e q u a t i o n s u s e d f o r t h e c o m p u t e r m o d e l a r e shown i n A p p e n d i x A. The e l e m e n t s o f t h e m o d e l a r e t h e r e a c t i o n m e c h a n i s m , h e a t and mass b a l a n c e e q u a t i o n s , a n d t h e m o l e c u l a r w e i g h t moment e q u a t i o n s , w h i c h a r e n u m e r i c a l l y i n t e g r a t e d w i t h r e a c t o r l e n g t h . The m o l e c u l a r w e i g h t moment e q u a t i o n s w e r e d e r i v e d u s i n g moment g e n e r a t i n g f u n c t i o n s (5.). T h i s method o f d e r i v a t i o n a p p e a r s t o be t h e most r e l i a b l e t e c h n i q u e f o r d e r i v i n g m o l e c u l a r w e i g h t moments. The a s s u m p t i o n s u s e d i n t h e c o m p u t e r p r o g r a m a r e l i s t e d i n t h e Appendix. B u t u n l i k e many p o l y m e r i z a t i o n m o d e l s , no a s s u m p t i o n s a r e made c o n c e r n i n g t h e s t e a d y - s t a t e c o n c e n t r a t i o n o f r a d i c a l s s i n c e t h e r a d i c a l s w i l l n o t be a t s t e a d y - s t a t e under c o n d i t i o n s o f r a p i d l y changing temperature over t h e e n t i r e range o f r e a c t o r c o n d i t i o n s w h i c h must b e c o n s i d e r e d i n t h i s a n a l y s i s . The c o m p u t e r m o d e l u s e d i n t h i s a n a l y s i s was d i s c u s s e d p r e v i o u s l y (l 6) a n d a r e s i m i l a r , i n g e n e r a l c o n c e p t s , t o o t h e r models ( £ , 8 ) d i s c u s s e d i n t h e l i t e r a t u r e . The c o m p u t e r p r o g r a m was w r i t t e n f o r u s e on IBM 370/65 c o m p u t e r . 9

9

Use o f R e a c t o r M o d e l . I n order t o begin t h e study o f t h e s e n s i t i v i t y o f t h e r e a c t o r responses, t h e s i m p l e s t r e a c t o r conf i g u r a t i o n p o s s i b l e was c h o s e n . T h i s paper considers t h e case o f c o n s t a n t p r e s s u r e , c o n s t a n t heat t r a n s f e r c o e f f i c i e n t and constant j a c k e t o r w a l l temperature, w i t h i n i t i a t i o n o c c u r r i n g by a f r e e r a d i c a l g e n e r a t o r w h i c h decomposes b y a f i r s t - o r d e r r a t e process. The e f f i c i e n c y o f t h e i n i t i a t o r s c o n s i d e r e d i s assumed c o n s t a n t (=0.5) and, as w i t h t h e i n i t i a t o r e f f i c i e n c y , a l l o t h e r r a t e c o n s t a n t s a r e assumed i n d e p e n d e n t o f v i s c o s i t y . Following t h i s i n i t i a l i n v e s t i g a t i o n , an o p t i m i z a t i o n s t u d y o f r e a c t o r c o n f i g u r a t i o n s i s p l a n n e d , l i f t i n g most o f t h e s e i n i t i a l r e s t r i c t i o n s .

In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

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Free-Radical

I

o5

0

I

I

I

I

I

I

I

I

10

20

30

40

50

60

70

80

1 90

I 100

P e r Cent R e a c t o r L e n g t h Figure 1. Typical reactor temperature profile for continuous addition polymerization: a plug-flow tubular reactor. Kinetic parameters for the initiator: I = 10 ppm; E = 32.921 kcal/mol; In k ' = 26.492 In sec' ; f = 0.5. Reactor parameter: [(4hTp)/ (DpC )] = 5148.2. [(C ) = heat capacity of the reaction mixture; (p) = density of the reaction mixture; (h) = overall heat-transfer coefficient; (Tj) = reactor jacket temperature; (t) = reactor residence time; (D) — reactor diameter]. o

d

d

p

1

p




224

The f i x e d v a r i a b l e s u s e d i n t h e c o m p u t e r s i m u l a t i o n a r e shown i n Table 1 along w i t h the k i n e t i c r a t e constants f o r the polymerization reactions. Since i t i s a conceptual study employing a t h e o r e t i c a l r e a c t o r model, i t i s a l s o important t o appreciate the l i m i t s o f t h i s type of i n v e s t i g a t i o n . The a d v a n t a g e o f t h e c o m p u t e r i n v e s t i g a t i o n over a p i l o t or p r o d u c t i o n r e a c t o r i n v e s t i g a t i o n i s the o b v i o u s c o s t and t i m e s a v i n g o v e r t h e r e a l r e a c t o r e x p e r i m e n t . The c o m p u t e r i n v e s t i g a t i o n c a n a l s o y i e l d a more d e f i n a b l e r e l a t i o n s h i p w i t h f e w e r p a r a m e t e r e x c u r s i o n s s i n c e t h e o u t p u t w i l l be free of s c a t t e r . I n a d d i t i o n , excursions i n r e a c t o r parameters c a n be t a k e n w h i c h m i g h t be c o n s i d e r e d u n s a f e on o r b e y o n d t h e e q u i p m e n t l i m i t a t i o n s o f an e x i s t i n g r e a l r e a c t o r . The p i t f a l l s o f a c o m p u t e r m o d e l a r e o b v i o u s i n t h a t i t i s o n l y a c o n c e p t u a l r e p r e s e n t a t i o n o f t h e r e a c t o r and i n c l u d e s o n l y as many a s p e c t s o f t h e r e a l r e a c t o r as p r e s e n t k n o w l e d g e permits. I n a d d i t i o n , e v e n t h e most p e r f e c t l y c o n c e i v e d d e s c r i p t i o n w i l l s t i l l depend u p o n t h e a c c u r a c y o f t h e p h y s i c a l l y measu r e d c o n s t a n t s used i n t h e model f o r t h e q u a l i t y o f t h e p r o c e s s representation. The g o a l o f t h i s r e p o r t i s , h o w e v e r , o n l y t o show c o n c e p t u a l t r e n d s and t h e t e c h n o l o g i c a l b a s e i s d e v e l o p e d t o t h e e x t e n t t h a t t h e c o n c e p t u a l t r e n d s w i l l be c o r r e c t . In some r e s p e c t s t h e c o m p u t e r m o d e l i s a b e t t e r p r o c e s s d e v e l o p m e n t t o o l t h a n t h e p i l o t p l a n t u s e d f o r t h e LDPE p r o c e s s s i n c e t h e p i l o t r e a c t o r does n o t y i e l d d i r e c t l y s c a l e a b l e i n f o r m a t i o n . The reader should take care t o d i r e c t h i s a t t e n t i o n t o the t r e n d i n f o r m a t i o n and c o n c e p t u a l d i f f e r e n c e s d e v e l o p e d i n t h i s w o r k ; v e r y l i t t l e a t t e n t i o n s h o u l d be p a i d t o t h e a b s o l u t e v a l u e s o f t h e parameters given. Theoretical

Considerations

The o v e r a l l r e a c t i o n s i n v o l v e d i n a f r e e r a d i c a l p o l y m e r i z a t i o n are d e s c r i b e d i n the Appendix. I t i s i n t e r e s t i n g however, t o l o o k i n t o s e v e r a l r e a c t i o n s t e p s w h i c h c o n t a i n t h e key r e a c t i o n p a r a m e t e r s and c o n t r o l t h e r a t e o f p r o d u c t i o n and t h e m o l e c u l a r weights o f the polymer. F o r i l l u s t r a t i o n o f some s i m p l e h i g h l i g h t s , t h e r a t e o f p o l y m e r i z a t i o n i s g i v e n by t h e r e l a t i o n s h i p Rp

= k [M][R*]

(1)

p

t h e r a t e o f r a d i c a l g e n e r a t i o n i s g i v e n by t h e R

R

= R.

- 2k R*

=

(2)

2

t

and t h e r a t e o f i n i t i a t i o n i s g i v e n toy t h e %

relationship

relationship

2fk [l] d


(3)

10.

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Polymerization

225

TABLE I

REACTION AND REACTOR PARAMETERS USED I N THE COMPUTER SIMULATION

KINETIC RATE CONSTANTS


Log Base e Frequency F a c t o r ( l , mole, s e c ) Pll TC T CFM CFP CFS BETA*

IT.891 20.796 1.198 -U.616 -U.963 -0.577 27.0

A c t i v a t i o n Energy/R (°C) 3573.0 150.0 150.0 1988.k 30U.3 1U76.2 139^0.0

A c t i v a t i o n Volume/R (°C/atmo) -0.2800 -0.1710 -0.1100 0.0366 0.0366 0.0366 0.21+U

PHYSICAL PARAMETERS BMO = 1 7 . 1 7 6 PP = 33,000 CP = 16.2399 BM = 28.0 HEATRO = 22,300.00 TF = 5 8 . 0 DMO = 1 7 . 1 7 6 P l l = propagation rate TC = t e r m i n a t i o n b y c o m b i n a t i o n r a t e c o n s t a n t T = r a t i o o f t h e t e r m i n a t i o n r a t e constant f o r combination to t h e r a t e constant f o r d i s p r o p o r t i o n a t i o n CFM = r a t i o o f t h e r a t e c o n s t a n t f o r monomer t r a n s f e r t o t h e constant f o r propagation CFP = r a t i o o f t h e r a t e c o n s t a n t f o r p o l y m e r t r a n s f e r ( l o n g chain branching) t o t h e constant f o r propagation CFS = r a t i o o f t h e r a t e c o n s t a n t f o r s o l v e n t t r a n s f e r t o t h e constant f o r propagation BMO = i n l e t monomer ( e t h y l e n e ) c o n c e n t r a t i o n , m o l e / l PP = r e a c t o r p r e s s u r e , p s i a BM = monomer ( e t h y l e n e ) m o l e c u l a r -weight CP = h e a t c a p a c i t y a t c o n s t a n t p r e s s u r e o f t h e r e a c t i o n f l u i d , cal/mole-°C HEATRO = h e a t o f r e a c t i o n f o r t h e p o l y m e r i z a t i o n , c a l / m o l e TF = r e a c t o r i n l e t t e m p e r a t u r e , °C DMO = r e a c t o r f l u i d d e n s i t y , m o l e / 1 BETA = 3 - s c i s s i o n r e a c t i o n r a t e c o n s t a n t *Though t h i s r e a c t i o n i s i m p o r t a n t i n LDPE r e a c t o r s , i t was i g n o r e d i n t h e present s i m u l a t i o n because o f t h e u n c e r t a i n t y o f t h e r a t e c o n s t a n t v a l u e and f o r s i m p l i f i c a t i o n aimed a t r e p r e s e n t i n g t r e n d s .



226

F u r t h e r , the major m o l e c u l a r weight c o n t r o l s are accomplished i n t h e r a t e o f c h a i n t r a n s f e r "by s o l v e n t ( R t s ^ * " "ke o f r a d i c a l t e r m i n a t i o n s (R-t ) b y c o m b i n a t i o n and d i s p r o p o r t i o n a t i o n . a n c

tlie

ra

r

*ts R

t r

= k = k

t s

t r

W

[s][R*] [R*]

2

(5)


where kp = p r o p a g a t i o n r a t e "constant" k^ = t e r m i n a t i o n r a t e "constant" k^ = f i r s t o r d e r r a t e " c o n s t a n t " f o r t h e i n i t i a t o r breakdown [M] = monomer c o n c e n t r a t i o n [i] = initiator concentration [R*] = r a d i c a l c o n c e n t r a t i o n f = initiator efficiency •^ts chain t r a n s f e r rate "constant" ktr t e r m i n a t i o n (combination or d i s p r o p o r t i o n a t i o n ) rate constant [s] = solvent (chain t r a n s f e r agent) c o n c e n t r a t i o n =

=

E q u a t i o n ( l ) shows t h e r a t e o f p o l y m e r i z a t i o n i s c o n t r o l l e d by t h e r a d i c a l c o n c e n t r a t i o n and as d e s c r i b e d by E q u a t i o n (2) t h e r a t e o f g e n e r a t i o n o f f r e e r a d i c a l s i s c o n t r o l l e d by t h e i n i t i a tion rate. I n a d d i t i o n , E q u a t i o n (3) shows t h i s r a t e o f g e n e r a t i o n i s c o n t r o l l e d b y t h e i n i t i a t o r and i n i t i a t o r concentration. F u r t h e r , the r a t e of i n i t i a t i o n c o n t r o l s the r a t e of propagation which c o n t r o l s the rate of generation of heat. T h i s combined w i t h t h e h e a t t r a n s f e r c o n t r o l s t h e r e a c t i o n t e m p e r a t u r e and t h e v a l u e o f t h e v a r i o u s r e a c t i o n r a t e c o n s t a n t s o f t h e k i n e t i c mechanism. T h r o u g h t h e s e e v e n t s i t becomes o b v i o u s t h a t t h e i n i t i a t o r i s a prime c o n t r o l v a r i a b l e i n the t u b u l a r p o l y m e r i z a t i o n r e a c t i o n system. F u r t h e r , t h e r a t e c o n s t a n t s may be w r i t t e n i n i t s u s e f u l f o r m as k

= k"e-

E / R T

e"

A

V

P

/

R

T

(6)

where E R T P AV k"

= = = = = =

a c t i v a t i o n energy i d e a l gas c o n s t a n t absolute temperature pressure a c t i v a t i o n volume frequency f a c t o r

The v a l u e s o f k" and E a r e h i g h l y d e p e n d e n t on t h e i n i t i a t o r t y p e s and t h e i r e f f e c t s on t h e s o l v e n t t y p e s a r e l e s s o v e r w h e l m i n g . The t y p e s o f s o l v e n t u s e d as c h a i n t r a n s f e r a g e n t a r e u s u a l l y f i x e d


10.

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227

Polymerization

f o r a g i v e n r e a c t o r and o n l y t h e c o n c e n t r a t i o n o f t h e s o l v e n t s are v a r i e d t o c o n t r o l t h e molecular weights. The i n i t i a t o r t y p e s , h o w e v e r , a r e c h a r a c t e r i z e d b y t h e s e p a r a m e t e r s , a n d s i n c e t h e e f f e c t o f p r e s s u r e i s s m a l l (l>9) a n d the t u b u l a r polymerization o f ethylene i sundertaken w i t h i n a n a r r o w r a n g e o f p r e s s u r e , t h e d e s c r i p t i v e c o n s t a n t becomes -E, /RT (T)

d

k' e


d

Any i n i t i a t o r w h i c h decomposes b y a f i r s t - o r d e r r a t e p r o c e s s c a n , t h e r e f o r e , b e c h a r a c t e r i z e d b y t h e t w o p a r a m e t e r s k*£ a n d E^. S u c h i m p o r t a n t m a t e r i a l s a s o r g a n i c p e r o x i d e s , a z o compounds, a s w e l l a s many o t h e r t y p e s o f m a t e r i a l s , a r e d e s c r i b e d b y t h e f i r s t order p r o c e s s and as such f o l l o w t h e g e n e r a l development g i v e n i n t h i s work. The e f f i c i e n c y w i l l b e assumed c o n s t a n t a n d t h e same f o r a l l i n i t i a t o r s w i t h 0.5. I n a case o f where t h e r a d i c a l s t e a d y - s t a t e a s s u m p t i o n c a n be made, t h e r e a c t o r h e a t b a l a n c e c a n b e w r i t t e n i n d i m e n s i o n l e s s f o r m a t c o n s t a n t p r e s s u r e (l_) T

?

d T / d Z = 6X ( l - Y ) " a Q ( T - 1 / T

f

!

) - y(T -l)

(8)

r

where

AHk

f

fk

2

M I e^T p o o L

Q a i T

and

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a/T, Ep + ( E - E / 2 ) / R T/T, L/L temperature residence time monomer i n l e t c o n c e n t r a t i o n i n i t i a t o r i n l e t concentration initiator efficiency density heat c a p a c i t y j a c k e t o r w a l l t e m p e r a t u r e (The j a c k e t t e m p e r a t u r e when h i s d e f i n e d a s a n o v e r a l l heat-transfer coefficient; the inside wall t e m p e r a t u r e when h i s d e f i n e d a s a n h e a t transfer coefficient.) heat-transfer coefficient reactor diameter d

t

J

0



228

and

X Y AH R L L k p ,Ep

monomer c o n v e r s i o n i n i t i a t o r conversion heat of r e a c t i o n i d e a l gas c o n s t a n t reactor length t o t a l reactor length f r e q u e n c y f a c t o r and a c t i v a t i o n for chain propagation k , E = f r e q u e n c y f a c t o r and a c t i v a t i o n f o r i n i t i a t o r breakdown k ^.,E^. = f r e q u e n c y f a c t o r and a c t i v a t i o n for radical termination Q

T

f

d


f

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energy energy energy

E q u a t i o n (8) p r o v i d e s a g e n e r a l r e l a t i o n s h i p b e t w e e n t h e r e a c t o r t e m p e r a t u r e p r o f i l e and t h e o p e r a t i n g p a r a m e t e r s . In r e l a t i n g the system heat t r a n s f e r t o t h e c o n v e r s i o n - m o l e c u l a r weights r e l a t i o n s h i p f o r a r e a c t o r o f f i x e d s i z e , t h e heat t r a n s f e r c o e f f i c i e n t emerges as t h e c o r r e l a t i n g p a r a m e t e r . E f f e c t s o f I n i t i a t o r C o n c e n t r a t i o n and J a c k e t T e m p e r a t u r e . The a b i l i t y t o m a n i p u l a t e r e a c t o r t e m p e r a t u r e p r o f i l e i n t h e polymerization tubular reactor i s very important since i t d i r e c t l y r e l a t e s t o c o n v e r s i o n and r e s i n p r o d u c t p r o p e r t i e s . This i s often done b y u s i n g d i f f e r e n t i n i t i a t o r s a t v a r i o u s c o n c e n t r a t i o n s and at d i f f e r e n t r e a c t o r jacket temperature. The r e a c t o r t e m p e r a t u r e response i n terms o f t h e d i f f e r e n c e between t h e j a c k e t temperature and t h e p e a k t e m p e r a t u r e (6=Tp-Tj) i s p l o t t e d i n F i g u r e 2 as a f u n c t i o n of the j a c k e t temperature f o r v a r i o u s i n l e t i n i t i a t o r concentrations. The t e m p e r a t u r e r e s p o n s e n o t o n l y depends on t h e j a c k e t temperature but a l s o , f o r c e r t a i n combinations o f the v a r i a b l e s , i t i s very s e n s i t i v e t o the jacket temperature. The c o n v e r s i o n r e f l e c t s t h e t e m p e r a t u r e r e s p o n s e r e a l i z e d i n t h e r e a c t o r and t h e t e m p e r a t u r e r e s p o n s e shown i n F i g u r e 2 c a n be r e p l o t t e d i n t e r m s o f c o n v e r s i o n r e s p o n s e s and t h e y a r e shown i n F i g u r e 3. The f i g u r e c l e a r l y shows t h a t t h e c o n v e r s i o n i n a r e a c t o r o f f i x e d s i z e depends on b o t h t h e i n l e t i n i t i a t o r c o n c e n t r a t i o n and t h e j a c k e t t e m p e r a t u r e . T h e r e e x i s t optimum o p e r a t i n g c o n d i t i o n s t o maximize the c o n v e r s i o n i n a r e a c t o r o f f i x e d s i z e . The d a s h e d l i n e s i n F i g u r e s 2 and 3 n o t o n l y i n d i c a t e t h e optimum o p e r a t i n g c o n d i t i o n b u t a l s o show t h e l i m i t s o f s t a b l e r e a c t o r o p e r a t i o n f o r a g i v e n i n i t i a t o r system i n a f i x e d r e a c t o r . In a d d i t i o n , the average polymer m o l e c u l a r weights t h a t are produced i n t h e r e a c t o r depend on t h e t e m p e r a t u r e r e s p o n s e and t h u s , a r e r e l a t e d t o t h e c o n v e r s i o n . An example o f t h i s r e l a t i o n s h i p i s shown i n F i g u r e h. Optimum O p e r a t i n g L i n e . The r e l a t i o n s h i p s b e t w e e n t h e c o n v e r s i o n and t h e a v e r a g e m o l e c u l a r w e i g h t c a n be p l o t t e d as a f u n c t i o n o f i n i t i a t o r c o n c e n t r a t i o n w h i l e v a r y i n g t h e j a c k e t temperat u r e t o o p t i m i z e t h e c o n v e r s i o n . The r e l a t i o n s h i p s a r e shown i n


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Figure 2. Effect of jacket temperature on the maximum temperature in a plugflow reactor (f = 0.5; In k ' = 43.2261 In sec ; E,/R = 23481.06 °C; r/ C 0.67898 sec-ft?-°C/BTU; T = 58°C) d

1

P

0


p

=



230

A i r = 40 ppm

1.7

1.8

1.9

2.0

2.1

2.2

2.3

Jacket Temp/Ref., Temp., Dimensionless Figure

3.

Conversion-jacket temperature relation transfer coefficient: 75 cal/m 2

(computer sec °C

simulation).


Heat

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50,000


40

1 •H

—

30

40,000 0

o •H

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o o

20

30,000

10

20,000

Jacket

2.8

2.7

2.6

2.5

I

Temp/T f, re

Dimensionless Figure 4. Operation of a plug-flow tubular addition polymerization reactor fixed size using a specified free-radical initiator (initiator kinetic parameters: E 32.921 Kcal/mol; In k ' = 26.492 In sec' ; f = 0.5; 10 ppm initiation, 1.0 mol solvent)

d

d

1


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232


F i g u r e 5 and i t p r o v i d e s a b a s i s f o r c o n s t r u c t i n g t h e optimum operating l i n e . The f i g u r e shows t h a t t h e r e i s a maximum c o n v e r s i o n t h a t c a n be a c h i e v e d a t a g i v e n i n i t i a t o r c o n c e n t r a t i o n . In t e r m s o f m o l e c u l a r w e i g h t , no o p t i m i z a t i o n e x i s t s . H o w e v e r , an o p e r a t i o n a l l i m i t a t i o n f o r c e s an o p t i m i z a t i o n s i n c e t h e o p e r a t i o n o f t h e r e a c t o r a t m o l e c u l a r w e i g h t s w h i c h a r e h i g h e r t h a n c a n be o b t a i n e d a t optimum c o n v e r s i o n c o u l d r e s u l t i n an e x c e s s i v e i n i t i a t o r c o n c e n t r a t i o n a t t h e r e a c t o r e x i t and an e x c e s s i v e number o f f r e e r a d i c a l s i n the polymer r e c o v e r y system. T h i s o p e r a t i o n a l l i m i t a t i o n c o u p l e d w i t h t h e maximum c o n v e r s i o n s p e c i f i c a t i o n r e s u l t s i n a d e f i n e d o p t i m a l o p e r a t i o n a l mode which i s unique t o a given i n i t i a t o r type. The optimum o p e r a t i n g l i n e i s i l l u s t r a t e d i n F i g u r e 5 as t h e d a s h e d l i n e . T h i s optimum o p e r a t i n g l i n e c a n now be p l o t t e d f o r d i f f e r e n t i n i t i a t o r t y p e s a t v a r i o u s s o l v e n t c o n c e n t r a t i o n s and h e a t t r a n s f e r c o n d i t i o n s t o compare i n i t i a t o r t y p e s and t o s t u d y t h e r e a c t o r r e s p o n s e s t o t h e operating parameters. The o p e r a t i n g l i n e shows t h a t f o r a g i v e n i n i t i a t o r t y p e t h e r e i s a maximum m o l e c u l a r w e i g h t and maximum c o n v e r s i o n w h i c h c a n be p r o d u c e d i n a r e a c t o r o f f i x e d s i z e . The o p e r a t i n g l i n e s e r v e s as a s o u n d b a s i s f o r c o m p a r i n g t h e p e r f o r mance o f t h e r e a c t o r as t h e v a r i o u s i n i t i a t o r s a r e u s e d and f u r t h e r p r o v i d e s t h e d i r e c t i o n o f s e a r c h f o r optimum i n i t i a t o r system f o r a given product i n a r e a c t o r of f i x e d s i z e . E f f e c t of Solvent Concentration. The optimum c o n v e r s i o n m o l e c u l a r w e i g h t c u r v e c a n be d i v i d e d i n t o two z o n e s : one c h a i n t r a n s f e r c o n t r o l l e d and t h e o t h e r i n i t i a t o r c o n t r o l l e d . I n t h e upper m o l e c u l a r w e i g h t or c h a i n t r a n s f e r c o n t r o l l e d r e g i o n changes i n t h e i n i t i a t o r c o n c e n t r a t i o n s i g n i f i c a n t l y change t h e r e a c t o r c o n v e r s i o n b u t h a v e l i t t l e e f f e c t on t h e m o l e c u l a r w e i g h t . As s e e n i n F i g u r e 6, h o w e v e r , i t i s i n t h i s r e g i o n t h a t t h e c h a i n t r a n s f e r a g e n t has i t s l a r g e s t e f f e c t . I n the low molecular w e i g h t or i n i t i a t o r c o n t r o l l e d r e g i o n changes i n t h e i n i t i a t o r c o n c e n t r a t i o n a l t e r t h e m o l e c u l a r weight but have l i t t l e e f f e c t on t h e r e a c t o r c o n v e r s i o n . A f u r t h e r examination of a s i n g l e operating l i n e i n d i c a t e s t h a t t h e r e a c t o r i s m o l e c u l a r w e i g h t l i m i t e d b e c a u s e o f an i n v e r s e r e l a t i o n s h i p o f t h e i n i t i a t o r c o n c e n t r a t i o n and t h e j a c k e t t e m p e r ature. The l i m i t i n g m o l e c u l a r w e i g h t i s a p p r o a c h e d as t h e i n l e t i n i t i a t o r c o n c e n t r a t i o n a p p r o a c h e s z e r o and t h e j a c k e t t e m p e r a t u r e approaches a l i m i t i n g v a l u e d i c t a t e d by the i n i t i a t o r t y p e . The m o l e c u l a r w e i g h t at t h a t p o i n t i s g i v e n s i m p l y by t h e r a t i o o f t h e p r o p a g a t i o n t o monomer and s o l v e n t c h a i n - t r a n s f e r r a t e c o n s t a n t s evaluated at the l i m i t i n g r e a c t i o n temperature. On t h e o t h e r h a n d , t h e l i m i t i n g c o n v e r s i o n i n a r e a c t o r o f f i x e d s i z e i s d e p e n d e n t on t h e t e m p e r a t u r e and t h e r a d i c a l c o n c e n t r a t i o n i n t h e r e a c t o r and r e s u l t s f r o m a p r e d o m i n a t i n g radicalr a d i c a l i n t e r a c t i o n p r e c i p i t a t e d b y an i n c r e a s e d i n i t i a t o r c o n c e n t r a t i o n and t h e a c c o m p a n y i n g t e m p e r a t u r e e x c u r s i o n . At t h i s p o i n t t h e s o l v e n t c o n c e n t r a t i o n s h a v e l i t t l e e f f e c t on t h e m o l e c u l a r


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icr


—

1

ppm

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2

io S^" *^^^^^ \

\ \

10

q

1

I . I . I 10

20

30

40

Conversion, % Figure 5. Molecular weight-conversion contour map for various concentrations of a free-radical initiator operating in a tubular-addition polymerization reactor of fixed size. Curves were constructed using varying jacket temperatures (kinetic parameters for the initiator: E = 32.921 Kcal/mol; In k ' = 26.494 In sec' ; f = 0.5; ( ) optimum operating line) d

d


1



234

Conversion, % Figure 6. Effect of solvent concentration on the molecular weight-conversion relationships of a tubular-addition polymerization reactor of fixed size using a specified initiator type. Each point along the curves represents an optimum initiator feed concentration—reactor jacket temperature combination, (kinetic parameters of the initiator: E = 24.948 Kcal/mol; In k ' = 26.494 In sec' ; f = 0.5) d

d

1


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weight. The l i m i t i n g c o n d i t i o n i s r e a c h e d when t h e c h a i n r a d i c a l s which a r e formed i m m e d i a t e l y t e r m i n a t e . T h i s c o n d i t i o n i s depend e n t on t h e t y p e o f i n i t i a t o r u s e d i n t h e r e a c t o r . E f f e c t s o f I n i t i a t o r Parameters. I n i t i a t o r t y p e s can b e s t be c h a r a c t e r i z e d b y t h e f r e q u e n c y f a c t o r ( k ^ ) and t h e a c t i v a t i o n e n e r g y ( E ^ ) , a n d t h e e f f e c t o f t h e s e p a r a m e t e r s on t h e m o l e c u l a r w e i g h t - c o n v e r s i o n r e l a t i o n s h i p i s shown i n F i g u r e s 7 a n d 8 . The c u r v e s shown a r e t h e r e s u l t o f c h o o s i n g t h e j a c k e t t e m p e r a t u r e i n l e t i n i t i a t o r c o n c e n t r a t i o n combination which maximizes t h e r e a c t o r c o n v e r s i o n f o r each i n i t i a t o r t y p e i n v e s t i g a t e d . F i g u r e 7 shows t h e l i m i t i n g maximum m o l e c u l a r w e i g h t o f p r o ducts from a r e a c t o r o f f i x e d s i z e v a r i e s d i r e c t l y w i t h t h e f r e quency f a c t o r o f t h e i n i t i a t o r a t a f i x e d a c t i v a t i o n energy, w h i l e the l i m i t i n g conversion v a r i e s i n v e r s e l y w i t h t h e frequency f a c t o r . I n a d d i t i o n , t h e l e n g t h o f t h e c h a i n - t r a n s f e r c o n t r o l l e d zone i s increased i n v e r s e l y w i t h t h e frequency f a c t o r . F i g u r e 8 shows t h e l i m i t i n g maximum m o l e c u l a r w e i g h t o f p r o ducts produced i n a r e a c t o r o f f i x e d s i z e v a r i e s i n v e r s e l y w i t h t h e a c t i v a t i o n energy o f t h e i n i t i a t o r a t a f i x e d frequency f a c t o r , while the l i m i t i n g conversion varies d i r e c t l y with the activation energy. In addition, the length of the chain-transfer controlled zone i n c r e a s e s d i r e c t l y w i t h t h e a c t i v a t i o n e n e r g y . T h e o r e t i c a l l y , as t h e i n i t i a t o r a c t i v a t i o n energy approaches z e r o , a v e r y h i g h m o l e c u l a r weight m a t e r i a l w i l l be produced a t a v e r y s m a l l c o n v e r s i o n and as t h e i n i t i a t o r a c t i v a t i o n approaches i n f i n i t y , a v e r y l o w m o l e c u l a r weight m a t e r i a l w i l l be produced a t v e r y h i g h c o n v e r s i o n . T h i s i m p l i e s t h a t an optimum c o m b i n a t i o n o f Ea a n d k ^ w h i c h p r o d u c e s a n i n f i n i t e r a n g e o f m o l e c u l a r w e i g h t s does n o t e x i s t . T h e r e i s , h o w e v e r , a n optimum c o m b i n a t i o n o f E and k*a f o r a g i v e n p r o d u c t ( g i v e n m o l e c u l a r w e i g h t ) p r o d u c e d i n a given reactor. I n a d d i t i o n t o t h e number a v e r a g e m o l e c u l a r w e i g h t o f t h e produced polymer, t h e breadth o f t h e molecular weight d i s t r i b u t i o n has i m p o r t a n t e f f e c t s on t h e p r o d u c t p r o p e r t i e s . F i g u r e s 9 a n d 10 show t h e e f f e c t o f i n i t i a t o r t y p e on t h e m o l e c u l a r w e i g h t d i s t r i b u t i o n o f t h e r e s i n as d e f i n e d by t h e r a t i o o f t h e weight t o number a v e r a g e m o l e c u l a r w e i g h t . The f i g u r e s show t h a t t h e b r e a d t h of molecular weight d i s t r i b u t i o n v a r i e s i n v e r s e l y w i t h t h e a c t i v a t i o n e n e r g y o f t h e i n i t i a t o r a t a n y g i v e n c o n v e r s i o n f o r an i n i t i a t o r o f s p e c i f i e d f r e q u e n c y f a c t o r and v a r i e s d i r e c t l y w i t h t h e f r e q u e n c y f a c t o r f o r an i n i t i a t o r o f s p e c i f i e d a c t i v a t i o n e n e r g y . This suggests t h a t t h e molecular weight d i s t r i b u t i o n o f a r e s i n c a n b e made t o assume a n y d e s i r e d v a l u e b y a p r o p e r c h o i c e o f t h e initiator. The i n i t i a t o r u s a g e c a n p l a y a r o l e i n t h e e c o n o m i c s o f r e s i n p r o d u c t i o n . The c o m p u t e r s i m u l a t i o n s show t h e u s a g e t o b e d e p e n d e n t on t h e i n i t i a t o r t y p e . The e f f e c t o f t h e i n i t i a t o r t y p e on t h e amount o f i n i t i a t o r r e q u i r e d t o p r o d u c e a g i v e n q u a n t i t y o f r e s i n a t optimum r e a c t o r c o n d i t i o n s i s shown i n F i g u r e s 11 a n d 1 2 .


f

T

d




236

Conversion, % Figure 7. Tubular plug-flow addition polymer reactor: effect of the frequency factor (k ) of the initiator on the molecular weight-conversion relationship at constant activation energy (E ). Each point along the curves represents an optimum initiator feed concentration-reactor jacket temperature combination and their values are all different. (E = 32.921 Kcal/mol; In k = 35.000 In sec' ; 0.0 mol % solvent) d

d

d

d


1

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10

20

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Conversion, % Figure 8. Tubular plug-flow addition polymer reactor: effect of the activation energy (E) of the initiator on the molecular weight-conversion relationship at constant frequency factor (k ). Each point along the curves represents an optimum initiator feed concentration-reactor jacket temperature combination and their values are all different. (In k' = 26.494 In sec' ; 0.0 mol % solvent) f

1


In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979. 1

Conversion,

%

d

Figure 9. Effect of the initiator activation energy on the molecular weight distribution of an addition polymer produced in a tubular reactor: constant frequency factor and at widely different values of initiator-jacket temperature combination (the conversion is optimized: In k ' = 26.492 In sec' ; 0.0 mol % solvent)

20


In Polymerization Reactors and Processes; Henderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979. %

d

Figure 10. Effect of the initiator frequency factor on the molecular weight distribution addition polymer produced in a tubular reactor: constant activation energy and at widely ent values of initiator-jacket temperature combination (the conversion is optimized: E = kcal/mol; 0.0 mol % solvent)

Conversion,


of an differ32.921

CO



240

Figure 11. Effect of the initiator frequency factor on the initiator usage in an addition polymerization reactor: constant activation energy (the conversion is optimized; E = 32.921 kcal/mol) d


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60

Conversion, %

10

20

30

40

Square Root of the I n l e t I n i t i a t o r Concentration Figure 12. Effect of the initiator activation energy on the initiator usage in a tubular-addition polymerization reactor: constant frequency factor (the conversion is optimized; In k' = 26.492 In sec' ) 1



242

As s e e n i n t h e f i g u r e s , t h e q u a n t i t y o f i n i t i a t o r r e q u i r e d t o y i e l d a given conversion v a r i e s d i r e c t l y w i t h the frequency f a c t o r f o r an i n i t i a t o r w i t h a s p e c i f i e d a c t i v a t i o n e n e r g y and i n v e r s e l y w i t h t h e a c t i v a t i o n e n e r g y f o r an i n i t i a t o r o f s p e c i f i e d f r e q u e n c y factor. The r e l a t i o n s h i p s do n o t show a l i n e a r p r o p o r t i o n a l i t y b e t w e e n t h e r e a c t o r c o n v e r s i o n and s q u a r e r o o t o f t h e i n l e t i n i t i a t o r concentration. E f f e c t o f I n i t i a t o r Change on C o n v e r s i o n Improvement. Based on t h e s e d i s c u s s i o n s , i t i s a p p a r e n t t h a t a s e l e c t e d i n i t i a t o r c a n a l l o w c o n v e r s i o n improvements f o r a s p e c i f i e d m o l e c u l a r w e i g h t . T h i s c a n be i l l u s t r a t e d i n F i g u r e 13. I f product A i s using i n i t i a t o r A, c o n v e r s i o n A w o u l d r e s u l t . A switch to i n i t i a t o r B w o u l d c a u s e t h e p r o d u c t D t o be p r o d u c e d . The i n i t i a t o r c c n c e n t r a t i o n c o u l d t h e n be i n c r e a s e d a l o n g c u r v e B t o p r o d u c t A at c o n v e r s i o n A". This simple increasing of the i n i t i a t o r concentrat i o n a n d , t h e r e f o r e , t h e c o n v e r s i o n c o u l d n o t h a v e b e e n done w i t h the o r i g i n a l i n i t i a t o r s i n c e a decrease i n t h e m o l e c u l a r weight w o u l d have o c c u r r e d . I n o t h e r w o r d s a 0.1 m e l t i n d e x m a t e r i a l c a n be p r o d u c e d a t t h e same r a t e as a 10 m e l t i n d e x m a t e r i a l b y u s i n g an i n i t i a t o r o f t h e p r o p e r d e s i g n . The c o n v e r s i o n i m p r o v e ment f o r C p r o d u c t f r o m C c o n v e r s i o n t o C" c o n v e r s i o n i s now done b y t h e r e v e r s i n g o f t h e i n i t i a t o r t y p e s and shows t h e s e n s i t i v i t y o f t h e p r o d u c t p r o p e r t i e s on c o n v e r s i o n improvement w i t h i n i t i a t o r changes. The LDPE r e a c t o r i s sometimes t e r m e d h e a t t r a n s f e r l i m i t e d i n conversion. While t h i s i s t r u e , the molecular weight (or melt i n d e x ) — c o n v e r s i o n r e l a t i o n s h i p i s n o t s i n c e t h i s w o r k shows t h a t a s e l e c t e d i n i t i a t o r c a n a l l o w c o n v e r s i o n i m p r o v e m e n t s t o be made under a d i a b a t i c c o n d i t i o n s f o r a s p e c i f i e d m o l e c u l a r weight. The a c t u a l l i m i t a t i o n t o conversion i s the decomposition temperature o f t h e e t h y l e n e and g i v e n t h a t t e m p e r a t u r e as a maximum l i m i t a t i o n , an i n i t i a t o r ( n o t n e c e s s a r i l y c o m m e r c i a l o r even known w i t h p r e s e n t i n i t i a t o r t e c h n o l o g y ) c a n be f o u n d w h i c h w i l l a l l o w any p r o d u c t t o be made a t t h e r a t e d i c t a t e d b y t h i s t e m p e r a t u r e . Conc e p t u a l l y , t h i s i s a c o n s t a n t (maximum) c o n v e r s i o n r e a c t o r , r u n n i n g a t c o n s t a n t o p e r a t i n g c o n d i t i o n s where t h e p r o d u c t p r o d u c e d d i c t a t e s t h e i n i t i a t o r t o be u s e d . 1

1

f


f

f

1

E f f e c t o f Heat T r a n s f e r . Because t h e r e a c t o r i s heat t r a n s f e r l i m i t e d , e f f o r t s a r e o f t e n made t o i m p r o v e t h e h e a t t r a n s f e r and c o n v e r s i o n . However, f o r a g i v e n i n i t i a t o r s y s t e m i n a s p e c i f i e d r e a c t o r , t h e r e are a l s o unique conversion-molecular weightheat t r a n s f e r r e l a t i o n s h i p s . F i g u r e ih shows a r e l a t i o n s h i p b e t w e e n t h e a v e r a g e m o l e c u l a r w e i g h t and t h e c o n v e r s i o n w i t h h e a t t r a n s f e r c o e f f i c i e n t as a parameter. The c u r v e i s b a s e d on o p t i mized conversion-jacket temperature r e l a t i o n s h i p s f o r d i f f e r e n t number a v e r a g e m o l e c u l a r w e i g h t s . The shape o f t h e c u r v e i m p l i e s a g i v e n i n i t i a t o r s y s t e m and r e a c t o r c o n f i g u r a t i o n . The shape o f c u r v e may change w i t h d i f f e r e n t r e a c t o r s y s t e m s , b u t i t does show


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si •H

Q)

A'

A" B' C

C"

Reactor Conversion Figure

13.

Effect of an initiator change on the conversion improvement tubular-addition polymerization reactor


in

the


244

5


10


10.

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a general trend of what can be expected of molecular weightconversion-heat transfer coefficient relationship. The figure shows that the conversion does increase with increasing heat transfer but the degree of increase depends on the average molecular weights of polymer being produced. At high average molecular weights (lower melt index), the lines representing different heat transfer levels converge. This implies that the conversion improvement due to heat transfer is small at high molecular weight. At lower molecular weights (higher melt index), the degree of conversion improvement is much larger. In order to utilize the improvement in heat transfer for a given product, therefore, an initiator system must be selected to provide a maximum conversion spread with increasing heat transfer.

Concluding Remarks The computer simulation study of the operation of the tubular free radical polymerization reactor has shown that the conversion and the product properties are sensitive to the operating parameters such as initiator type, jacket temperature, and heat transfer for a reactor of fixed size. The molecular weight-conversion contour map is particularly significant and it is used in this paper as a basis for a comparison of the reactor performances. The type of initiator used affects the molecular weight and conversion limits in a reactor of fixed size and the molecular weight distribution of the material produced at a given conversion level. The initiator type also dictates the amount of initiator which is necessary to yield a given conversion to polymer, the operating temperature range of the reactor and the sensitivity of the reactor to an unstable condition. Clearly, the initiator is the most important reaction parameter in the polymer process. The full utilization of improved heat transfer in a given reactor can only be made when the molecular weight-conversion relationships are carefully studied with various initiator types at different heat transfer levels. Then a particular initiator system must be selected for a maximum conversion improvement for a specified product. A study of this kind can be further extended to develop optimum reactor configurations which are needed to produce given products at the highest possible conversion.

Abstract A theoretical polymerization tubular reactor model was used to study the effects of reactor operating parameters on conversion


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APPENDIX A


EQUATIONS FOR A PLUG FLOW POLYMER TUBULAR REACTOR WITH BRANCHING KINETICS


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PROCESSES

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and average molecular weight. In particular, the kinetic rate constants specific to high pressure ethylene polymerization were used in the computer-study. There exists an optimum jacket temperature for maximizing conversion at a given average molecular weight product. The study further suggests that an unstable operating region exists where wide conversion fluctuations result from attempts to increase the reactor conversion by minor adjustments in initiator amount or jacket temperature. The initiator is the most important reactor parameter in the polymer process. The initiator type affects the molecular weight and conversion limits in a reactor of fixed size and the molecular weight distribution of the material at a given conversion level. The initiator type dictates the initiator amount for a given conversion, the operating temperature range and sensitivity of the reactor to an unstable condition. Optimized molecular weight-conversion relationship is related to the system heat transfer coefficient. The degree of conversion improvement from improved heat transfer depends on the average molecular weights of polymer being produced for a given initiator system. Acknowledgements The authors gratefully acknowledge Union Carbide Corporation for permission to publish this paper.

Literature Cited 1

Lee, K. H. and Marano, J . P., J r . , Paper No. 67d, AIChE Annual Meeting, November, 1977, New York.

2

Szabo, J., Luft, G. and Steiner, R., Chemie-Ing.-Techn., 41, 1007 (1969).

3

Ehrlich, P. and Mortimer, G. A . , Adv. Polymer S c i . , 1, 386 (1970).

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Graessley, W. W., AIChE-I. Chem. E. Symposium Series, No. 3, London, 1965.

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Ray, W. H., Canadian Journal of Chemical Engineering, 45, 356 (1967).

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Marano, J . P., J r . , A Seminar Presented at Northwestern University, Evanston, Illinois (February, 1974).

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B a m f o r d , C. H., et al., The Kinetics o f Vinyl Polymerization "by Radical M e c h a n i s m s , B u t t e r w o r t h s Scientific P u b . , London (1958).

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Free-Radical Polymerization - American Chemical Society

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