Finite Element Modeling of Polymer Flow and Heat Transfer in

Chapter 40

Finite Element Modeling of Polymer Flow and Heat Transfer in Processing Equipment

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C. G. Dumas and R. S. Dixit Engineering Research Laboratory, Central Research, The Dow Chemical Company, Midland, M I 48674

A mathematical model was developed based on a constitutive equation for polymer rheology and the fundamental equations for motion, momentum and energy. The power law model was extended to include higher order shear rate dependence. As a r e s u l t , p a r t i a l d i f f e r e n t i a l equations for momentum and energy are highly coupled. For a heated rectangular channel, the equations were solved in an i t e r a t i v e manner u n t i l convergence was achieved between velocity and temperature f i e l d s . The model was general, i n that the temperature p r o f i l e in the metal between the heat source and the polymer was also obtained. A marching technique was used to obtain the velocity, temperature and pressure p r o f i l e s in the flow d i r e c t i o n . The fundamental nature of the model made i t a useful predictive tool for design. Many p o l y m e r processing unit operations involve heat transfer t o polymer melt flowing through irregular g e o m e t r i e s . The r h e o l o g i c a l p r o p e r t i e s o f polymer m e l t s complicates t h e mathematical a n a l y s i s o f these systems. The v i s c o s i t y i s h i g h l y dependent on t e m p e r a t u r e and shear r a t e . I n t h i s p a r t i c u l a r case, t h e power l a w model was extended t o include higher order shear rate dependence. As a r e s u l t , p a r t i a l d i f f e r e n t i a l e q u a t i o n s for momentum and energy are highly coupled. A m a t h e m a t i c a l m o d e l was d e v e l o p e d b a s e d o n a c o n s t i t u t i v e equation f o r polymer rheology and t h e fundamental e q u a t i o n s f o r momentum a n d e n e r g y . Model

Formulation

The g e o m e t r y Polymer melt

f o rthis enters a

system i s shown i n F i g u r e 1 . rectangular metal channel with

0097-6156/89/0404-O521$06.00/0 ο 1989 American Chemical Society

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

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE Π


522


40. DUMAS & DIXIT

Modeling of Polymer Flow and Heat Transfer

523


h e i g h t D a n d w i d t h W. T h e w i d t h W i n c r e a s e s a l o n g t h e l e n g t h o f t h e c h a n n e l L. The m e t a l c h a n n e l i s i n s u l a t e d on t h e u p p e r a n d l o w e r s u r f a c e s , w h e r e a s t h e s i d e s a r e h e a t e d u s i n g Dowtherm. The f o l l o w i n g a s s u m p t i o n s a r e made i n d e v e l o p i n g t h e m a t h e m a t i c a l model:

1) P o l y m e r m e l t i s i n c o m p r e s s i b l e . 2) T h e v e l o c i t y p r o f i l e i s f u l l y d e v e l o p e d at the plate entrance. 3) T h e R e y n o l d s n u m b e r i s v e r y small ( i . e . creeping flow); inertial terms i n t h e equation of motion are neglected. 4) S t e a d y s t a t e f l o w o c c u r s ; t r a n s i e n t b e h a v i o r i s n o t analyzed. 5) A x i a l h e a t c o n d u c t i o n i n t h e p o l y m e r i s n e g l e c t e d d u e to i t s low thermal c o n d u c t i v i t y and i t s r e l a t i v e l y s m a l l magnitude, compared w i t h a x i a l t h e r m a l c o n v e c t i o n . 6) T h e p o l y m e r b e h a v e s a s a g e n e r a l i z e d N e w t o n i a n f l u i d , where v i s c o s i t y i s an a r b i t r a r y f u n c t i o n o f shear rate and temperature. 7) N o r m a l c o m p o n e n t s o f t h e s t r e s s t e n s o r a r e i g n o r e d . 8) T h e r e i s n o f l o w i n t h e ζ d i r e c t i o n . 9) F l o w i n t h e t r a n s v e r s e ( y ) d i r e c t i o n i s n e g l i g i b l e . Although flow i n the transverse (y) d i r e c t i o n will actually be non-zero, because the channel width i n c r e a s e s ; i t i s assumed t o be n e g l i g i b l e , r e l a t i v e t o v . A f t e r s o l v i n g t h e p r o b l e m b a s e d on Vy=0, an e s t i m a t e of V y can be o b t a i n e d by then s o l v i n g t h e c o n t i n u i t y equation. 10) H e a t c o n d u c t i o n t h r o u g h m e t a l , i n t h e χ d i r e c t i o n , i s n e g l i g i b l e because the metal temperature gradient, i n this direction, i s small. With these assumptions, the v e l o c i t y f i e l d i s g i v e n by, x

v = v (x,y,z) x

The

equation

x

(1)

of continuity i s ,

(2)


COMPUTER APPLICATIONS

524 The r a t e

IN APPLIED POLYMER SCIENCE Π

of deformation tensor i s , 3vx

0


Δ =

dz 0

0

ay

dz

The s h e a r r a t e

i s then g i v e n by, 1/2

γ =

* 3y

The c o m p o n e n t s o f s h e a r

x

τ

ap ax

{

stress 3v

(3)

a r e g i v e n by, v

= η 3y

x y

= η-

χ ζ

The e q u a t i o n o f m o t i o n

dz'

;

(4)

az

(5)

c a n now b e w r i t t e n a s , avv

ay

v

1

ay

av, y

az

v

1

az'

(6)

The temperature field f o r the polymer melt can be d e t e r m i n e d u s i n g an e n e r g y b a l a n c e w h i c h i n c l u d e s terms f o r c o n v e c t i o n , c o n d u c t i o n and v i s c o u s d i s s i p a t i o n .

Energy (i)

Equation

Fluid

A similar energy balance conduction terms only.

for

the

metal

includes


Modeling ofPolymer Flow and Heat Transfer

40. DUMAS & DIXIT

(ii)

Metal

0

The b o u n d a r y


v T

^[ "-df]

=

k

conditions

)

+

for this

( 8 )

system a r e ,

= 0 at the walls

x

ρ

D

T

525

= T

° 0

-W

=

m

-W W -D D toi I

Q - Qactual

End of channel

YES

< toi

END

NO

Compute the temperature field at χ •+· A Χ using PDE/PROTRAN

Figure

4

Numerical solution

procedure.



40. DUMAS & DIXIT

Modeling ofPolymer Flow and Heat Transfer

529

s e l e c t e d . The s e c a n t method i s a p p l i e d f o r s e l e c t i n g a new v a l u e . A f t e r c o n v e r g e n c e i s o b t a i n e d on t h e f l o w r a t e , t h e t e m p e r a t u r e p r o f i l e a t an i n c r e m e n t a l d i s t a n c e down the channel i s c o m p u t e d u s i n g PDE/PROTRAN. However, PDE/PROTRAN c a n n o t s o l v e p r o b l e m s i n t h r e e dimensions. T h e r e f o r e , because the system i s at s t e a d y - s t a t e , the t i m e v a r i a b l e i n PDE/PROTRAN i s r e p l a c e d by d i s t a n c e x. In t h i s manner, t h e p r o g r a m i s a b l e t o march a l o n g t h e χ d i r e c t i o n computing the temperature p r o f i l e i n t h e y-z plane at each c r o s s - s e c t i o n . A f t e r marching to the d i s t a n c e χ + Δ χ , t h e t e m p e r a t u r e p r o f i l e i s s t o r e d and t h e v e l o c i t y p r o f i l e i s computed a t t h i s new c r o s s - s e c t i o n . T h i s m a r c h i n g p r o c e d u r e c o n t i n u e s u n t i l t h e end o f t h e channel i s reached. At v a r i o u s c r o s s - s e c t i o n s along the flow d i r e c t i o n , (i) the polymer velocity profile, ( i i ) polymer t e m p e r a t u r e p r o f i l e , ( i i i ) m e t a l t e m p e r a t u r e p r o f i l e , and ( i v ) p r e s s u r e d r o p a r e a l l computed. B e c a u s e t h e p r e s s u r e a t t h e e x i t i s known, t h e a b s o l u t e p r e s s u r e p r o f i l e a l o n g t h e f l o w d i r e c t i o n can a l s o be d e t e r m i n e d . R e s u l t s and D i s c u s s i o n As a r e s u l t o f symmetry, i t was o n l y n e c e s s a r y t o model one q u a r t e r o f t h e f l o w c h a n n e l . A s e r i e s o f s i m u l a t i o n s were performed using a set of nominal operating c o n d i t i o n s . These c o n d i t i o n s a r e summarized i n T a b l e I . Table

I

Operating conditions

I n i t i a l width of f l u i d channel F i n a l width of f l u i d channel Height of f l u i d channel Metal p l a t e thickness M e t a l t h i c k n e s s a t t h e edges Channel l e n g t h V o l u m e t r i c polymer f l o w r a t e Polymer i n l e t temperature Temperature o f Dowtherm Polymer d e n s i t y Polymer h e a t c a p a c i t y Polymer t h e r m a l c o n d u c t i v i t y Metal thermal c o n d u c t i v i t y

1 .905 cm (3/4 i n ) 3 .175 cm (1.25 i n ) 0 .3175 cm (1/8 i n ) 0 .3175 cm (1/8 i n ) 0 .635 cm (1/4 i n ) 15.240 cm (6.0 i n ) 2 .0 c c /s 2 00°C 2 40°C 726 g/cc 5 c a l7g°

Finite Element Modeling of Polymer Flow and Heat Transfer in

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