Mathematical Modeling of Bulk and Solution Polymerization in a

Chapter 28

Mathematical Modeling of Bulk and Solution Polymerization in a Tubular Reactor Carl J . Stevens and W. Harmon Ray 1

Downloaded by EAST CAROLINA UNIV on November 13, 2016 | http://pubs.acs.org Publication Date: August 29, 1989 | doi: 10.1021/bk-1989-0404.ch028

Department of Chemical Engineering, University of Wisconsin, Madison, WI 53706

A detailed fundamental model for bulk and solution free radical polymerization in a tubular reactor is presented. An initial model including accurate viscosities and diffusivities was formulated for laminar flow. A generalized adaptive grid PDE solver using collocation with B-splines was developed and used to solve the model for the concentration, temperature and molecular weight profiles in the reactor. The effects of secondary flows from buoyant forces (generated by the density change on polymerization), and from flow in curved tubes were included in a more refined model. It was found that the secondary flows can be as large as the primary flow, and can cause convective mixing which increases the mass and heat transfer. The model predictions were compared to experimental data for the bulk polymerization of styrene and solution polymerization of vinyl acetate. Both of these models underpredict the conversion as a result of overpredicting the mass transfer limitations and degree of channeling. Turbulent or unsteady flows are possible due to the large magnitude of the secondary flows, and a model with r e a l i s t i c empirical turbulent d i f f u s i v i t i e s gives good agreement with the experimental data. 1

Current address: The Dow Chemical Company, Midland, MI 48674 0097-6156/89AM04-0337$06.75/0 c 1989 American Chemical Society

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

338

C O M P U T E R APPLICATIONS IN APPLIED P O L Y M E R

SCIENCE II


INTRODUCTION Tubular r e a c t o r s a r e used f o r p o l y m e r i z a t i o n because they a r e c o n t i n u o u s , u s e s i m p l e equipment, have good heat t r a n s f e r p r o p e r t i e s , and t h e i r behavior can approach that o f t h e b a t c h r e a c t o r . However, a n a l y s i s o f t h e s e r e a c t o r s i s complicated by t h e fact that t h e k i n e t i c s , t h e r h e o l o g y , mass t r a n s f e r , and heat t r a n s f e r a r e a l l important i n d e t e r m i n i n g t h e r e a c t o r b e h a v i o r . Under m i l d c o n d i t i o n s a t u b u l a r r e a c t o r w i l l behave l i k e a p l u g f l o w r e a c t o r . However, t h e optimum o b j e c t i v e i s t o b a l a n c e t h e b e n e f i t s o f increased r e a c t i o n rates o r tube s i z e s a g a i n s t t h e accompanying problems o f e v e n t u a l heat and/or mass t r a n s f e r l i m i t a t i o n s . Mass t r a n s f e r l i m i t a t i o n s w i l l l e a d t o higher polymer concentrations a t t h e tube w a l l (due t o l o n g e r residence time t h e r e ) , a broader m o l e c u l a r weight d i s t r i b u t i o n a n d reduced c o n v e r s i o n . The goal o f t h i s research i st o p r e d i c t t h ebehavior o f t u b u l a r r e a c t o r s under t h e c o n d i t i o n s where heat and/or mass t r a n s f e r l i m i t a t i o n s become i m p o r t a n t . The f o c u s i s on r e a c t o r s o p e r a t e d a t f l o w r a t e s b e l o w t h e t r a n s i t i o n t o t u r b u l e n t f l o w . T h e s e r e a c t o r s may b e o p e r a t e d a t l o w e r p r e s s u r e d r o p s , a n d a r e s h o r t e r t h a n some i n d u s t r i a l r e a c t o r s such as t h e h i g h p r e s s u r e p o l y e t h y l e n e r e a c t o r which c a n be a m i l e i n l e n g t h .

DISCUSSION As a s t a r t i n g p o i n t a m o d e l i s d e v e l o p e d f o r l a m i n a r a x i s y m m e t r i c f l o w i n a s t r a i g h t t u b e . The e f f e c t s o f secondary, t u r b u l e n t o r non-steady flows a r e considered l a t e r . The model d e v i a t e s from p r e v i o u s models i n t h e l i t e r a t u r e [1,2,3,4,5,6] i n t h a t a c c u r a t e d e s c r i p t i o n s , v a l i d over t h e e n t i r e o p e r a t i n g range, a r e used f o r t h e v i s c o s i t y a n d d i f f u s i o n . The c o r r e l a t i o n f o r t h e v i s c o s i t y i n t h e c o n c e n t r a t e d r e g i o n i s based on t h e f r e e volume/chain e n t a n g l e m e n t t h e o r y o f B e r r y a n d F o x [7] w i t h e x t e n s i o n s made b y R i c h a r d s [ 8 ] , a n d i n t h e d i l u t e r e g i o n t h e v i s c o s i t y i sbased on t h e M a r t i n e q u a t i o n [ 9 ] . The d i f f u s i o n o f t h e s p e c i e s i s b a s e d o n t h e a s s u m p t i o n t h a t a l l t h e low molecular weight species a r e e q u i v a l e n t . The d i f f u s i v i t i e s i n t h e c o n c e n t r a t e d r e g i o n a r e b a s e d o n t h e f r e e v o l u m e t h e o r y o f V r e n t a s a n d Duda [10,11,12], and t h e d i f f u s i v i t y o f t h e polymer i n t h e d i l u t e r e g i o n i s based on Kirkwood Reisman t h e o r y [12], w h i l e t h e d i f f u s i v i t i e s o f t h e low molecular weight species i n t h e d i l u t e r e g i o n a r e based on a c o r r e l a t i o n by D u l l i e n [13]. I t i s assumed t h a t t h e polymer c h a i n s a r e e n t a n g l e d a n d a l l d i f f u s e w i t h t h e same v e l o c i t y . I t i s a l s o assumed t h a t a x i a l d i f f u s i o n i s n e g l i g i b l e due t o t h e l a r g e l e n g t h t o r a d i u s r a t i o o f most r e a c t o r s . The e q u a t i o n s for t h e model a r e :


28.

339

Modeling of Bulk and Solution Pdymerization

STEVENS & RAY

Continuity f o r polymer: 3w

3w +

PVâf

-j. 9

v

P râT

3w

7â7P°

-

r

V>i

+

Ί7

( 1 )

Continuity f o r monomer: 0 P

.

+ V

z

P

dz

V

r

9r

1

i_

r

9

s

Ρ

r

P

^ dr

D

d

r (2)


W

P

Continuity f o r i n i t i a t o r : 9Υϊ μ

^Υτ ι 3 3y p(2> - 2> ) 3w 3y pv — - = — — o(D r — - + —3r r ar ^ s 1 - w ar 3 T

q

n

2

pv — + 3z z

P

r

P

T

R

r

9

r

r

p

1

+

1

P

Q

1 - wp

1

(3)

Energy balance: 3τ

ι 3

3τ

3T

V * 5 7 * " V r 5 7 - 7 57 "«57

r

/dp

>

2

^

+

3inp

dp

+ v„ — — 3lriT ρ z

d

+ ΔΗ P

z

Q l

-.R P

.

(4)

Q l

Continuity f o r polymer moments:

P V z

3z

R V r

.

3r

li. r 3r

o

!

P

n s

R

, pol

3r

r

^

i

P w p

^ ^ ^ 3r 3r

( ϊ I reaction" ^ ) μ

W

i ,.

p

Momentum Equation:

3 r

2

^

3

J a

r

r

2

η dz

w

D e f i n i t i o n of stream function: ν ζ

1 3ψ = - — — rp 3r

1 3ψ

,

ν r

= —

— rp 3z


(7)

340

C O M P U T E R APPLICATIONS IN

Boundary

conditions at

3r Ψ = \

R p 2

Boundary

f


t

3r

3r ( 8 )

the

tube

dy =

^

wall:

3y

M

dΤr

3τ —

3r

f

conditions at

r = R :

tube c e n t e r l i n e :

3r

1

6

6

(

Q

r

S c ) 0

.22

by

( 2 0 )

molecular The a x i s y m m e t r i c model and e f f e c t i v e d i f f u s i v i t y i n E q u a t i o n 20 p r e d i c t t h e same Sherwood Number as E q u a t i o n 19. (The c o e f f i c i e n t s 0.325 and 0.28 i n E q u a t i o n 19 were changed t o 0.166 and 0.22 i n E q u a t i o n 20 i n o r d e r t o account f o r d i f f e r e n c e s i n the d e f i n i t i o n of the Grashof Number u s e d i n R e f e r e n c e 16 and t h e d e f i n i t i o n u s e d i n t h i s work (see R e f e r e n c e 1 4 ) . The v a l u e o f 1 was added t o E q u a t i o n 20 t o g i v e t h e c o r r e c t l i m i t i n g b e h a v i o r f o r s m a l l v a l u e s o f G r S c ) . Cup a v e r a g e p r o p e r t i e s were u s e d t o d e t e r m i n e t h e v a l u e s o f Gr and S c . F o l l o w i n g common p r a c t i c e [17] i t was assumed t h a t a l l eddy d i f f u s i v i t i e s (heat, s e l f d i f f u s i o n o f low m o l e c u l a r w e i g h t s p e c i e s , and d i f f u s i o n between polymer and low m o l e c u l a r w e i g h t s p e c i e s ) a r e t h e same. The t h e r m a l c o n d u c t i v i t y t h e n becomes : «effective which

= «molecular

+

°·

1

6

6

lgç\4gr

( 2 4 )

-^s-molecular

A comparison between t h emodel p r e d i c t i o n s a n d t h e experimental data f o r v i n y l acetate and styrene homopolymerization i s g i v e n i n F i g u r e s 12 t h r o u g h 1 4 . F o r the v i n y l a c e t a t e case t h e d i f f u s i v i t i e s a r e i n c r e a s e d by a factor o f 5 and the v a r i a t i o n i nt h e logarithm o f v i s c o s i t y w i t h p o l y m e r c o n c e n t r a t i o n was r e d u c e d b y 1 0 % (these changes a r e c o n s i d e r e d t o be w i t h i n t h e u n c e r t a i n t y i n t h e s e p h y s i c a l p a r a m e t e r s [ 1 4 ] ) . The f i g u r e s show t h a t t h e e x p e r i m e n t a l d a t a a r e e x p l a i n e d w e l l by t h e axisymmetric model w i t h e f f e c t i v e t r a n s p o r t p r o p e r t i e s g i v e n b y E q u a t i o n s 20 t h r o u g h 2 4 .

CONCLUSIONS A d e t a i l e d a x i s y m m e t r i c model has been developed t o describe tubular bulk and solution polymerization r e a c t o r s . Based on t h i s model t h e r e g i o n i n parameter s p a c e t h a t g i v e s s i g n i f i c a n t mass t r a n s f e r l i m i t a t i o n s and c h a n n e l i n g h a s been d e t e r m i n e d . T h i s model o v e r p r e d i c t s t h e mass t r a n s f e r l i m i t a t i o n s a n d c h a n n e l i n g e x i s t i n g i n styrene and v i n y l acetate polymerization experiments, presumably because o f t h e s i g n i f i c a n t secondary flow e f f e c t s from n a t u r a l convection o r h e l i c a l f l o w . Steady l a m i n a r secondary f l o w s have been modeled. For n a t u r a l c o n v e c t i o n these flows can be l a r g e r than t h e p r i m a r y f l o w , a n d c a n c o m p l e t e l y change t h e shape o f t h e c o n c e n t r a t i o n p r o f i l e s . However, t h e i n c r e a s e d mass t r a n s f e r p r e d i c t e d f o r steady laminar secondary flows i s not s u f f i c i e n t t o e x p l a i n t h e e x p e r i m e n t a l d a t a . Based on e x p e r i m e n t a l work o f o t h e r r e s e a r c h e r s on s i m i l a r systems, i t i shypothesized that t h e secondary flows c o u l d be non-steady. E f f e c t i v e t u r b u l e n t d i f f u s i v i t i e s have been u s e d w h i c h do e x p l a i n t h e e x p e r i m e n t a l d a t a f o r v i n y l acetate and styrene polymerization. Flow and c o n c e n t r a t i o n v i s u a l i z a t i o n experiments would be u s e f u l i n determining t h eexact nature o f t h e secondary flows, and t h e s e r e s u l t s c o u l d be used t o g u i d e a n d c o r r o b o r a t e a d d i t i o n a l modeling o f these r e a c t o r s .


COMPUTER

APPLICATIONS IN

APPLIED POLYMER

SCIENCE


356

F i g u r e 12. C o m p a r i s o n b e t w e e n cup a v e r a g e c o n v e r s i o n p r e d i c t e d by a x i s y m m e t r i c model w i t h e f f e c t i v e t r a n s p o r t p r o p e r t i e s and e x p e r i m e n t a l l y measured v a l u e s f o r s t y r e n e [5] a n d v i n y l a c e t a t e [ 2 ] .


II

STEVENS & RAY

Modeling ofBulk and Solution Polymerization

200000

] a Vinyl Acetate '. Δ S t y r e n e

Is


1

100000

/ m

100000

200000

Measured M , F i g u r e 13. C o m p a r i s o n between number a v e r a g e m o l e c u l a r weight p r e d i c t e d by axisymmetric model w i t h e f f e c t i v e transport p r o p e r t i e s and experimentally m e a s u r e d v a l u e s f o r s t y r e n e [5] a n d v i n y l a c e t a t e [2] .

0

100000

200000

Measured M

300000

w

F i g u r e 14. Comparison between w e i g h t a v e r a g e m o l e c u l a r weight p r e d i c t e d by axisymmetric model w i t h e f f e c t i v e t r a n s p o r t p r o p e r t i e s and experimentally m e a s u r e d v a l u e s f o r s t y r e n e [5] a n d v i n y l a c e t a t e [2] .


357

358

C O M P U T E R APPLICATIONS IN APPLIED P O L Y M E R


LEGENP Q£

SYMBOLS

Cp

Heat

D

D i f f u s i v i t y between low m o l e c u l a r w e i g h t species

S

capacity

( w e i q h t

D

D i f f u s i v i t y between polymer and low mole c u l a r weight species

g

gravitational acceleration

Gr

Grashof

h

external transfer

number heat coefficient

k

Thermal

Ρ Pr

pressure Prandtl

number

r

Radial

coordinate

r*

r/R

Ri

Mass r a t e o f initiation

R

Se

Schmidt

Sh

Sherwood

Τ

Temperature

T v

Radial

r

v

ν

Angular

r

θ

r

ν *

ν

v

Axial

θ

z

v *

v

z

θ

z

number temperature

ρ R /

z

f

(Gr η) velocity

w

monomer) p

Weight f r a c t i o n w

Φ2

/

p

polymer

Length along

tube

Z

α

Viscosity

α

Thermal

coefficient

diffusivity,

k/(pC ) p

β

Density

ζ

Dimensionless along tube,

η

Viscosity

%

Viscosity

coefficient length

coefficient

n

Density

Pf D e n s i t y

of

feed

Ρο

Density

Φ

Theile

Ψ

A x i a l flow function

ψ*

Dimensionless secondary flow stream function

p R / ( G r η)

coefficient modulus stream

ω* D i m e n s i o n l e s s

velocity

/ z

Average i n l e t

-

2

V

p

ζ* D i m e n s i o n l e s s l e n g t h along tube. z* = z 2)/(R < V > )

Ρ

velocity

v *

1

t

number

Coolant

c

W p *

i n i t i a t o r )

w

f r a c t i o n

M

Wp

-

i molecular weight moment d i v i d e d b y t h e f i r s t moment

Mass r a t e o f polymerization

pol

1 ( w e i q h t

y

conductivity

Effective f i r s t order polymerization rate constant

pol

f r a c t i o n

Yi

ζ

DHpol Heat o f p o l y m e r i z a t i o n (mass b a s i s )

k

SCIENCE

axial

2

ι

a

r*

3r*

a

γ~ ——

3r*

vorticity -}-

a2

3Θ

velocity


2

II

28.

STEVENS & RAY


LITERATURE

Modeling of Bulk and Solution Polymerization

359

CITED

1. Vrentas, J.; Huang, W. Chem. Eng. Sci. 1986, 41, 2041. 2. Hamer, J . ; Ray, W. Chem. Εng. S c i . 1986, 41, 3083, 3093. 3. McLaughlin, H.; Mallikarjun, R.; Nauman, E. AIChE. J. 1986, 32, 419. 4. Gosh, M.; Foster, D.; Lenczyk, J . ; Forsyth, T. AIChE Symp. Ser. 160 1976, 72, 102. 5. Wallis, J.; Ritter, R.; Andre, H. AIChE J. 1975, 2 1 , 686, 691. 6. Lynn, S.; Huff, J . AIChE J. 1971, 17, 475. 7. Berry, G.; Fox, T. Adv. Polym. S c i . 1968, 5, 261. 8. Richards, W., Ph.D. Thesis, Princeton University, Princeton, 1983. 9. Martin, Α., A.C.S. Meeting, Memphis, A p r i l 1942. 10. Vrentas J . ; Duda, J. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 417. 11. Duda, J.; Vrentas, J . ; Ju, S.; L i u , H. AIChE J . 1982, 28, 279. 12. Vrentas J.; Duda, J . AIChE J.1979, 25, 1. 13. Dullien, F., AIChE J.1972, 18, 62. 14. Stevens, C., Ph.D. Thesis, University of Wisconsin, Madison, 1988. 15. Deaver, F.; Eckert, E. i n Heat Transfer; G r i g u l l , U.; Hahne, E., Eds.; Elsevier: Amsterdam, NC 1.1, 1970; Vol. IV. 16. Sedahmed, G.; Shemilt, L. Chem. Eng. Commun.1983, 23, 1. 17. Knudsen, J.; Katz, D. F l u i d Dynamics and Heat Transfer; McGraw Hill: New York, 1958. RECEIVED March 27, 1989


Mathematical Modeling of Bulk and Solution Polymerization in a

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