Evaluation and Analysis of a Multisite Kinetic Model for Polymerization

Chapter 31

Evaluation and Analysis of a Multisite Kinetic Model for Polymerization Initiated with Supported Ziegler—Natta Catalysts C. C . Hsu, J. J. A. Dusseault, and M . F. Cunningham

Downloaded by UNIV OF ROCHESTER on July 15, 2013 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0404.ch031

Department of Chemical Engineering, Queen's University, Kingston, Ontario K 7 L 3N6, Canada

A kinetic model which accounts for a multiplicity of active centres on supported catalysts has recently been developed. Computer simulations have been used to mechanistically validate the model and examine the effects on its parameters by varying the nature of the distribuitons, the order of deactivation, and the number of site types. The model adequately represents both first and second order deactivating polymerizations. Simulation results have been used to assist the interpretation of experimental results for the MgCl /EB/TiCl /TEA catalyst; suggesting that 2

4

at low TEA/Ti ( 10.8 the distribution can become more bimodal in nature and the deactivation kinetics change to second order. Heterogeneous Z i e g l e r - N a t t a c a t a l y s t s used t o p o l y m e r i z e o l e f i n s e x h i b i t phenomena c h a r a c t e r i s t i c o f a c t i v e s i t e h e t e r o g e n e i t y ( 1 _5). Complex k i n e t i c models which account f o r t h i s l i k e l i h o o d have been developed and used o n l y i n s i m u l a t i o n s t u d i e s (6-7)· R e c e n t l y Dumas and Hsu (1,8-9) have proposed a k i n e t i c model d e r i v e d on t h e assumption o f a m u l t i p l i c i t y o f s i t e s w h i c h has a much s i m p l e r form t h a n those g i v e n by p r e v i o u s workers ( 6 - 7 ) . They have shown i t t o f i t p r o p y l e n e p o l y m e r i z a t i o n r a t e d a t a w e l l , when u s i n g a MgCl^ s u p p o r t e d T i C l ^ / t r i e t h y l a l u m i n u m (TEA) c a t a l y s t . To a s s e s s t h e v a l i d i t y o f t h e model, computer s i m u l a t i o n s have been used t o g e n e r a t e r a t e c u r v e s from v a r i o u s d i s t r i b u t i o n s o f a c t i v e c e n t r e s (1,10) . F u r t h e r s i m u l a t i o n work was c a r r i e d out i n an attempt t o e s t a b l i s h a r e l a t i o n s h i p between e x p e r i m e n t a l l y determined model p a r a m e t e r s , and t h e s i t e d i s t r i b u t i o n and c a t a l y s t d e a c t i v a t i o n . 0097-6156/89A)404-0403$06.00A) o 1989 American Chemical Society

In Computer Applications in Applied Polymer Science II; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

404

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE II

The M u l t i p l e - S i t e Model The c a t a l y s t s u r f a c e i s assumed t o be comprised o f p r o p a g a t i n g c e n t r e s , whose v a r y i n g a c t i v i t i e s a r e r e p r e s e n t e d by a d i s t r i b u t i o n . For species i the rate of polymerization i s d e s c r i b e d by

V

=V

[

c

î

1

[

M

1

( 1 )

where [C*] i s t h e c o n c e n t r a t i o n o f s i t e type i and corresponding propagation rate constant.

i s the

[C*] i s assumed t o


f o l l o w a f i r s t order d e a c t i v a t i o n , d[C*]

where k ^ i s t h e d e a c t i v a t i o n r a t e c o n s t a n t .

T h i s assumption i s

j u s t i f i e d under c e r t a i n c o n d i t i o n s . I t s g e n e r a l a p p l i c a b i l i t y w i l l be d i s c u s s e d l a t e r . The o v e r a l l p o l y m e r i z a t i o n and decay r a t e s a r e then t h e sum of E q u a t i o n s 1 and 2 r e s p e c t i v e l y , over a l l a c t i v e s p e c i e s . R Ρ

- Σ (k [C*])[M] Pi ι

(3)

d2 [C*]

By m u l t i p l y i n g E q u a t i o n 2 w i t h k . and summing as above, E q u a t i o n 5 i s obtained. p

dZ (k

[C*]) Σ

αΤ^

(

k

k

pi di

[

C

*

]

)

(

5

)

An analogous s i t u a t i o n o c c u r s i n t h e c a t a l y t i c c r a c k i n g o f mixed f e e d gas o i l s , where c e r t a i n components o f t h e f e e d a r e more d i f f i c u l t t o c r a c k ( l e s s r e a c t i v e o r more r e f r a c t o r y ) t h a n t h e o t h e r s . The h e t e r o g e n e i t y i n r e a c t i v i t i e s ( i n t h e form o f E q u a t i o n s 3 and 5) makes k i n e t i c m o d e l l i n g d i f f i c u l t . However, Kemp and W o j c i e c h o w s k i (11) d e s c r i b e a t e c h n i q u e w h i c h lumps t h e r a t e c o n s t a n t s and c o n c e n t r a t i o n s i n t o o v e r a l l q u a n t i t i e s and t h e n , because o f t h e e f f e c t s o f h e t e r o g e n e i t y , account f o r t h e changes o f these q u a n t i t i e s w i t h t i m e , o r e x t e n t o f r e a c t i o n . F i r s t a f r a c t i o n a l a c t i v i t y i s d e f i n e d as x - Σ (k [ C * ] ) / Z (k [C*] ) pi i pi iο where t h e s u b s c r i p t ο corresponds t o i n i t i a l c o n d i t i o n s . 5 i s then r e - a r r a n g e d i n t o t h e form,

f

= -

θ

2

(6) Equation

F(x) χ

where


(7)

31. HSU ET AL.

405

Kinetic Model for Polymerization

and Σ (k F

U

)

k

[C*])/Z (k .k.. [C*l )

Σ (k [C*])/Z ( k [ C J ] ) pi

pl

( 9 )

o

Kemp and Wojciechowski referred to F(x) as the refractoriness function and i t describes the change in the specific overall rate constant (i.e., θ F(x)) as a function of extent of reaction (11).


2

The functional form of F(x) is determined by the proper choice of the distribution of the reaction rate constants. In the context of this study, the extent of reaction refers to the conversion of sites from active to inactive, and is given by Equation 6 (i.e., χ = 1 for no conversion, χ - 0 for total conversion). For a single site mechanism i t can be shown easily that F(x) reduces to 1.0. Solution of Equation 7 and substitution into Equation 3 yields the expected result: R = k C* β χ ρ ( - ^ t) [M] ρ ρ ο α

(10)

For multiple sites F(x) is obtained by assuming a distribution of rate constants. The summations in Equation 9 are then approximated by definite integrals evaluated for k^ from 0 to infinity. Thus a gamma distribution of rate constants will result in F(x) having the form in Equation 11, where ω is the reciprocal of the shape parameter of the gamma distribution (19) or is the square of the coefficient of variation of the distribution (12) (i.e., ( σ / μ ) , where σ is the standard deviation and μ is the mean)· 2

F(x) = χ

(11)

ω

Substituting for F(x) in Equation 7 results in a differential equation which can be solved for x, and can then be used in the overall polymerization rate (Equation 3), yielding θ, [M] R

P

=

ΪΤθΓ (1 + 8 9 t) 1

2

( 1 2 )

3

where Θ. - Σ (k [CJ] ) 1 pi I ο and

(13)

has been previously defined by Equation 8.

is the i n i t i a l activity of the catalyst;

Mechanistically

is the average

i n i t i a l first order deactivation rate constant, weighted by the i n i t i a l activity of each species; and (= ω), as discussed above, is a dispersion parameter, characterizing to some extent, the nature of the distribution of active sites. More complicated forms of F(x), derived from the binomial and Poisson distributions, are respectively:


406

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE Π F ( x ) = β - ( 0 - 1) x ~

V

(14)

F(x) = 1 - l n ( x ) / l n ( z )

(15)

These forms r e s u l t i n t h e f o l l o w i n g r a t e e x p r e s s i o n s : R

= θ^ίΒ - 1)/B + 1/B e x p ( - B v 6 ) )

p

R


1 / v

(16)

2

= e, ( l - e x p ( e t / l n Z ) ) z

2

The parameter Ζ i n E q u a t i o n 17 i s e q u i v a l e n t t o t h e f r a c t i o n a l a c t i v i t y a t an i n f i n i t e p o l y m e r i z a t i o n t i m e . 3 and ν can a l s o be i n t e r p r e t e d i n a s i m i l a r way through E q u a t i o n 18, where α has t h e same meaning as Ζ. α = (1 - 1 / β )

1 / ν

(18)

Even though t h e model was d e r i v e d based on f i r s t o r d e r d e a c t i v a t i o n o f a c t i v e c e n t r e s , i t was found t h a t t h e model i s e q u a l l y capable o f f i t t i n g d a t a generated from a d i s t r i b u t i o n o f a c t i v e s i t e s undergoing second o r d e r decay. For second o r d e r decay, the c o n c e n t r a t i o n o f s p e c i e s i i s g i v e n by d[C*] k

- i ^

[C

d i

2

(

V

1

9

)

As w r i t t e n , E q u a t i o n 19 i m p l i e s a s i m u l t a n e o u s l o s s o f two s i t e s of t h e same t y p e . On a heterogeneous c a t a l y s t t h i s i s o n l y r e a l i s t i c f o r a d j a c e n t s i t e s , as has r e c e n t l y been suggested by Chien ( 1 5 ) . E q u a t i o n 19 assumes a d j a c e n t s i t e s a r e the same s p e c i e s , which appears c o n s i s t e n t w i t h a c t i v e s i t e s t r u c t u r a l models a p p e a r i n g i n t h e l i t e r a t u r e ( 1 7 - 1 8 ) . P e r f o r m i n g t h e same m u l t i p l i c a t i o n (by k ^ ) d summation d e s c r i b e d e a r l i e r y i e l d s a

n

p

dZ(k [ C * ] ) — S H

1

z ( k

- -

k

[c

pi di

2)

(

v

2

0

)

As b e f o r e , E q u a t i o n 20 c a n be r e - a r r a n g e d i n t o t h e form, |£

= -

Q' F ( x ) χ

(21)

2

where θ

2 =

Σ

ν ^ 1

[

0

1

]

ο

/

Σ

ν

[

Ε

Ϊ

]

w i t h the meanings o f F ( x ) and χ (and c o n s e q u e n t l y intact. later.

(

ο

2

2

Q^) r e m a i n i n g

The e v a l u a t i o n o f F ( x ) f o r t h i s case w i l l be d e a l t w i t h


)

31.

HSU ET AL.

407

Kinetic Model for Polymerization

Computer S i m u l a t i o n s t o A s s e s s Model V a l i d i t y The i n i t i a l s e t of s i m u l a t i o n s were used t o m e c h a n i s t i c a l l y v a l i d a t e the k i n e t i c model so i t c o u l d be used i n m e a n i n g f u l k i n e t i c i n v e s t i g a t i o n s . By p r e - d e t e r m i n i n g the d i s t r i b u t i o n of a c t i v e s i t e s , a c t u a l ( t h e o r e t i c a l ) v a l u e s of and θ can be 2

c a l c u l a t e d a p r i o r i u s i n g E q u a t i o n s 13 and 8, r e s p e c t i v e l y , and then compared t o the e s t i m a t e s o b t a i n e d from model ( E q u a t i o n 12) fitting. I n t h i s way s i m u l a t i o n s were performed assuming one hundred n o r m a l l y ( G a u s s i a n ) d i s t r i b u t e d s i t e s of d i f f e r e n t a c t i v i t i e s . The k ' s ranged from 0 t o 2000, and k ' s ( f i r s t


p i

d i

o r d e r ) ranged l i n e a r l y from 0 t o 0.14, such t h a t the most a c t i v e s i t e s decay the f a s t e s t . The v a l u e s of the k's were chosen t o generate r e a l i s t i c p o l y m e r i z a t i o n r a t e c u r v e s , a g e n e r a l p h i l o s o p h y used throughout t h i s work. F i g u r e 1 shows a c t u a l r a t e d a t a w i t h the E q u a t i o n 12 p r e d i c t i o n s superimposed. The a c c u r a c y of the parameter e s t i m a t e s was a s s e s s e d by c a l c u l a t i n g the p e r c e n t d e v i a t i o n from the known v a l u e s ( i . e . , ( ( e s t i m a t e d - calculated 6_.)/calculated θ^)·100).

I t was

found t h a t the

and

θ

2

e s t i m a t e s were a c c u r a t e t o w i t h i n 1.5 % and 8 % of t h e i r known values, respectively. cannot be c a l c u l a t e d a p r i o r i , so comparison t o a known v a l u e i s n ' t p o s s i b l e . Some i n d i c a t i o n of i t s b e h a v i o r can be determined by c o r r e l a t i n g estimates to changes i n the a c t i v e s i t e d i s t r i b u t i o n .

decreased

( F i g u r e 2)

as the d i s t r i b u t i o n of a c t i v i t i e s and/or d e a c t i v a t i o n r a t e s , f o r a f i x e d number of s i t e s , became narrower; and as the number of s i t e s w i t h d i f f e r e n t a c t i v i t i e s and/or d e a c t i v a t i o n r a t e s , f o r a g i v e n d i s t r i b u t i o n , was lowered (becoming more homogeneous). T h i s s h o u l d be expected s i n c e Crickmore (12) has demonstrated t h a t when t h e r e i s a f i r s t o r d e r d e a c t i v a t i o n f o r any a r b i t r a r y distribution, with 2* 3 < 3 ( σ / μ ) . As the s t a n d a r d d e v i a t i o n ( σ) i n c r e a s e s , so s h o u l d . As shown i n F i g u r e 3, θ

θ

# 1 :

t

h

e

n

θ

s

2

ramains f a i r l y c o n s t a n t over a wide range of d i f f e r e n t numbers o f s i t e s , u n t i l t h i s number becomes s m a l l , and then decreases s h a r p l y . When t h e r e i s o n l y one s i t e F ( x ) = x° = 1 ) .

θ~ approaches z e r o

(i.e.,

0

F u r t h e r I n t e r p r e t a t i o n of Model Parameters F o r the s i m u l a t i o n s c a r r i e d out under the c o n d i t i o n s d e s c r i b e d p r e v i o u s l y , the 0^ v a l u e s never exceeded 0.38, i n c o n t r a s t t o the e x p e r i m e n t a l v a l u e s which v a r y from 0.02 t o 1.8, o b t a i n e d by Dumas (V) and Cunningham ( 1 0 ) . Thus the type of d i s t r i b u t i o n used i n t h e s i m u l a t i o n s d i d not a d e q u a t e l y r e p r e s e n t the e n t i r e s i t u a t i o n on the s u r f a c e of c a t a l y s t . For t h a t r e a s o n f u r t h e r s i m u l a t i o n s , employing d i f f e r e n t d i s t r i b u t i o n s , were conducted t o f u r t h e r study the e f f e c t s of s i t e d i s t r i b u t i o n on θ . 0


408

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE II


1000

< 01 CL

if) 20

30

TIME

40

50

60

(MINUTES)

F i g u r e 1,

S p e c i f i c R a t e = R /[M]/mol T i . T = 50 °C. A l / T i P 4. PPoo i rn t s a r e e x p e r i m e n t a l r e s u l t s ( 1 ) . L i n e i s t h e l e a s t 84· quares e s t i m a t i o n o f E q u a t i o n 12 w i t h θ = 1391, θ = 0.78, = 1.27. 1 Λ -J 1 Ζ ?

rO0.18

Evaluation and Analysis of a Multisite Kinetic Model for Polymerization

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