Mathematical Model of Low Density Polyethylene Tubular Reactor

45 Mathematical Model of Low Density Polyethylene Tubular Reactor G. DONATI, L . MARINI, G. MARZIANO, C. MAZZAFERRI, and M . SPAMPINATO Istituto Guido Donegani S.p.A., Research Center, Novara, Italy

Downloaded by COLUMBIA UNIV on July 19, 2012 | http://pubs.acs.org Publication Date: September 16, 1982 | doi: 10.1021/bk-1982-0196.ch045

E . LANGIANNI Montepolimeri S.p.A., Research Center, Ferrara, Italy

A mathematical model was developed, able to predict monomer conversion and temperature profiles of indus trial tubular reactors for the production of low-den sity polyethylene, in different operating conditions. The usual limitations (isothermal wall, radicals qua si-steady state, constant pressure) found in the lit erature for similar models were released, and the im portance of correctly evaluating the propagation and termination rate constants, k and k , was shown. The model parameters were determined through fluid -dynamic experiments in a mock-up, and from the anal ysis of data obtained on an industrial reactor. p

t

Nearly a l l low d e n s i t y polyethylene (LDPE) i s produced at high pressure e i t h e r i n s t i r r e d autoclaves o r i n t u b u l a r react o r s . The high pressure polyethylene t u b u l a r r e a c t o r (Figure 1) i s c h a r a c t e r i z e d by a very high l e n g t h to diameter r a t i o , that ranges from 1000 t o 15000. Heat i s t r a n s f e r r e d from o r t o the r e a c t o r by means o f an o i l j a c k e t that surrounds i t . T h i s j a c k et i s subdivided i n t o s e v e r a l zones, as the o i l temperature must vary along the r e a c t o r due t o the d i f f e r e n t heat requirements o f the process: i n a f i r s t p a r t (heating zone) the c o l d feed must be heated to the r e a c t i o n s t a r t i n g temperature, while i n the subsequent zones the o i l has the duty o f removing the r e a c t i o n heat. Thus the r e a c t o r can be imagined as d i v i d e d i n t o as many zones as are the o i l input p o i n t s . In order t o prevent the build-up o f polymer deposits on the c o l d e r r e a c t o r w a l l s - that i n severe instances can lead to plugging or, i n l e s s severe ones, to a g r e a t e r production o f c r o s s - l i n k e d (gel) polymer o f lower q u a l i ty - the flow rate i n the tube i s kept as high as p o s s i b l e , and, i n most o f the r e a c t o r s p r e s e n t l y i n o p e r a t i o n , flow pulses are imposed to the r e a c t i o n mixture. These p u l s e s , the frequency o f which i s once every 2 t o 10 seconds, are b e l i e v e d to accomplish a r e g u l a r t e a r i n g away o f the accumulated polymer both by means 0097-6156/82/0196-0579$06.00/0 © 1982 American Chemical Society In Chemical Reaction Engineering—Boston; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


580

CHEMICAL REACTION ENGINEERING

+ Polymer Figure 1.

High pressure polyethylene tubular reactor.

In Chemical Reaction Engineering—Boston; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


45.

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Mathematical Model of Tubular Reactor

581

of the p e r i o d i c a l l y i n c r e a s e d v e l o c i t y o f the r e a c t i o n mixture (that can be 2 to 5 times the average v a l u e ) and due to an expansion of the ethylene d i s s o l v e d i n the polymer as the pressure i s reduced. The tube s i z e s i n use today range from 1 to 2 inches i n t e r n a l diameter and from 500 to 1000 metres l e n g t h , w i t h p r e s sure drops between 100 and 700 atm. Reactor temperatures are comp r i s e d i n the 100-300°C i n t e r v a l : temperatures h i g h e r than 300°C are not used, p r i m a r i l y because decomposition of ethylene can occur above t h i s v a l u e . The p o l y m e r i z a t i o n r e a c t i o n i s known t o be of the r a d i c a l type, s i n c e i t i s i n i t i a t e d by s u i t a b l e compounds ( " i n i t i a t o r s " ) that, e i t h e r by r e a c t i o n w i t h the monomer or by seIf-decomposit i o n , give o r i g i n to primary r a d i c a l s : i n the p a s t , oxygen was used almost e x c l u s i v e l y (1), w h i l e the modern t r e n d i s to use organic i n i t i a t o r s such as peroxides, hydroperoxides and so on. As the temperature at which s i g n i f i c a n t i n i t i a t o r decomposition takes p l a c e depends on the i n i t i a t o r i t s e l f , a s u c c e s s f u l operat i o n of the r e a c t o r r e q u i r e s a proper choice of the i n i t i a t o r : i n many cases, s u i t a b l e mixtures of d i f f e r e n t i n i t i a t o r s are a l s o used. The r e a c t o r performances are o f t e n enhanced by a proper use of m u l t i p l e feed streams of c o l d ethylene and/or i n i tiator(s). In the l i t e r a t u r e many s t u d i e s on LDPE t u b u l a r r e a c t o r s are found ( 2 - 6 ) . A l l these s t u d i e s present models of the t u b u l a r r e a c tor, able to p r e d i c t the i n f l u e n c e , on monomer conversion and temperature p r o f i l e s , of s e l e c t e d v a r i a b l e s such as 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 temperature. With the e x c e p t i o n of the models of M u l l i k i n , that i s an analog computer model o f an i d e a l i z e d plug-flow r e a c t o r , and o f Schoenemann and T h i e s , f o r which i n s u f f i c i e n t d e t a i l s are g i v e n , a l l the other models developed so f a r appear to have some l i m i t a t i o n s e i t h e r i n the b a s i c hypotheses or i n the f i e l d s of a p p l i c a t i o n . A l l authors, f o r i n s t a n c e , c o n s i d e r the j a c k e t o i l at constant temperature. T h i s assumption, e q u i v a l e n t to that of i n f i n i t e o i l flow r a t e , makes i t impossible to c o r r e c t l y compute the o v e r a l l heat t r a n s f e r c o e f f i c i e n t and the thermal d r i v i n g f o r c e . Since heat exchange p l a y s an important r o l e i n the conduction of i n d u s t r i a l r e a c t o r s , where more than one t h i r d o f the p o l y m e r i z a t i o n heat i s removed through the e x t e r n a l c o o l i n g o i l (only very low conversion r e a c t o r s can be assumed a d i a b a t i c , as claimed by Chen et a l . ) , t h i s l i m i t a t i o n cannot be accepted. A second p o i n t regards the assumption, e i t h e r e x p l i c i t l y or i m p l i c i t l y made by a l l authors, o f r a d i c a l s pseudo-steady s t a t e along the r e a c t o r . T h i s assumption, t h a t i s adequate as long as the i n i t i a t o r i s not completely decomposed (by the way, t h i s i s the case i n the o p e r a t i n g c o n d i t i o n s considered by Agrawal and Han), does not allow to d e s c r i b e most i n d u s t r i a l r e a c t o r s , where there i s experimental evidence that some p o l y m e r i z a t i o n takes p l a c e a l s o a f t e r the temperature peak, i n a zone where the i n i t i a t o r i s completely decomposed. I t ' s worthy to p o i n t out that


582


even when the r a d i c a l s pseudo-steady s t a t e assumption i s not exp l i c i t l y made, p r a c t i c a l l y e q u i v a l e n t r e s u l t s can f o l l o w from an unappropriate choice of the values of the chain propagation and termination constants (k and k ) : t h i s i s a c t u a l l y the case w i t h the model of Chen et a l . None of the models p r e v i o u s l y mentioned, then, takes i n t o account the pressure v a r i a t i o n along the r e a c t o r . T h i s v a r i a t i o n i s not n e g l i g i b l e , i n view of the h i g h v e l o c i t i e s u s u a l l y imposed to the r e a c t i o n mixture; moreover, pressure i s known to p l a y a very important r o l e on the p o l y m e r i z a t i o n r a t e (4,7)· l e a s t a f i r s t order estimate of the pressure p r o f i l e along the r e a c t o r seems to be necessary. F i n a l l y , a proper i n v e s t i g a t i o n about the e f f e c t , on a x i a l mixing, pressure drop and heat t r a n s f e r c o e f f i c i e n t , of the end p u l s i n g v a l v e i s missing. In t h i s paper a computer model o f the LDPE t u b u l a r r e a c t o r i s presented, i n which the p r e v i o u s l y d i s c u s s e d l i m i t a t i o n s are avoided. fc

P


T l l u s

Fluiddynamic

a t

study

Due to the l a c k of p u b l i s h e d data on the s p e c i a l flow f i e l d generated i n the LDPE t u b u l a r r e a c t o r by the end p u l s i n g v a l v e , the development of the mathematical model was preceded by a f l u iddynamic study, w i t h the aim of e v i d e n c i n g the i n f l u e n c e , i f any, o f the p u l s e d motion on the a x i a l mixing, the heat t r a n s f e r c o e f f i c i e n t and the pressure drop i n the r e a c t o r . A f u l l s c a l e mock-up was b u i l t , c o n s i s t i n g of a 6 m l e n g t h , 32 mm i n t e r n a l diameter s t a i n l e s s s t e e l tube, f e d by a r e c i r c u l a t i n g pump. Since ethylene at process c o n d i t i o n s i s a very l i g h t l y comp r e s s i b l e gas with a d e n s i t y of 500 kg/m , water was used as a model f l u i d : the e f f e c t of v i s c o s i t y , then, was s t u d i e d by adding to i t small amounts of c a r b o s s y m e t h y l c e l l u l o s e . Because of the i n c o m p r e s s i b i l i t y of water, the flow pulses i n s i d e the model were obtained by f e e d i n g a constant water flow to both the model and a p a r a l l e l c i r c u i t at p e r i o d i c a l l y v a r y i n g p r o p o r t i o n s , through the use of a s u i t a b l e v a l v e . Thus an approximately s i n u s o i d a l flow i n s i d e the mock-up could be obtained, with p e r i o d i n the 2 to 10 sec range and amplitude comprised between .2 and .5 times the average value. T h i s average value, then, could be v a r i e d from 10 to 40 m^/h. The experimental r e s u l t s were r a t h e r s u r p r i s i n g . I t appeared that no s i g n i f i c a n t d i f f e r e n c e e x i s t e d between the average values of mixing e f f i c i e n c y , heat t r a n s f e r c o e f f i c i e n t and p r e s sure drop obtained i n a pulsed flow and those obtained i n a cons t a n t flow of the same mean r a t e , probably because of the very low p u l s a t i o n frequency, which i s superimposed to a t u r b u l e n t motion c h a r a c t e r i z e d by frequences three orders of magnitude h i g h e r . Thus, as f a r as only average values are concerned, the


45.

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AL.


583

p u l s e d motion i n s i d e the LDPE t u b u l a r r e a c t o r can be simulated by a constant flow of the same average r a t e . The s i g n i f i c a n c e of these r e s u l t s w i l l be d i s c u s s e d t h o r oughly i n a subsequent paper. K i n e t i c assumptions Many k i n e t i c s t u d i e s on the h i g h pressure ethylene polyme r i z a t i o n are found i n the l i t e r a t u r e (8-11). A l l authors agree on the f o l l o w i n g main r e a c t i o n steps: k


0

O2+M

i n i t i a t i o n by oxygen

• R-

(1)

» 2fR^

(2)

k j i n i t i a t i o n by peroxide

I

propagation

R +M

? » R

(3)

termination

R +R »Ρ (4) n m Since we are i n t e r e s t e d i n the computation of monomer con v e r s i o n and temperature p r o f i l e s , a l l r a d i c a l s of whatever c h a i n l e n g t h may be considered as a unique s p e c i e s , the c o n c e n t r a t i o n of which i s thus given by:

With t h i s p o s i t i o n , balances may be extended to only f o u r chemical s p e c i e s : oxygen C^, i n i t i a t o r I , monomer M and r a d i c a l s A . The r a t e s of appearance of these s p e c i e s are e a s i l y d e r i v e d from the k i n e t i c expressions (1) to (4). As an example, the r a t e of r a d i c a l appearance i s given by: q

R

A

q

- 2f k

x

[I ]+ k

Q

[M ] [0

2

] - k

t

[ A

q

]

2

(5)

The r a t e o f polymer p r o d u c t i o n , then, i s equal to the r a t e of monomer consumption. The r e a c t i o n r a t e constants are assumed to f o l l o w a m o d i f i e d Arrhenius law: r Ε- AV(p-p ) -ι k = A exp

L

o

J

In t h i s study the values l i s t e d i n Table I f o r the p a r a meters A, E, AV and ρ were used. As to the values of k and k , i t ' s w e l l known (9,10) t h a t from p i l o t experiments on i v e s s e l r e a c t o r these r a t e constants cannot be s e p a r a t e l y evaluated: only the value of the parameter k /\/k^ can be obtained. For t h i s reason, w h i l e most authors a^ree on the value of the above parameter, they s t r o n g l y d i s a g r e e on the separate values of the two r a t e constants. In a t u b u l a r r e a c t o r , however, the conversion can be shown to depend a l s o from the r a t i o k / k i f the r a d i c a l s quasi-steady s t a t e assumption i s r e l e a s e d . TSUS, i n a f i r s t approximation, both constants can be t>


584


evaluated from the a n a l y s i s of the performances of an i n d u s t r i a l t u b u l a r r e a c t o r . This was done i n the present work, and l e d to the values r e p o r t e d i n Table I . TABLE I - Rate constants used i n the

computation

\

P

(1/mol.s) k

k


15

31334.

0.

0.

1.608-10ll

30670.

0.6

1.

6164.

0.6

1.

0.

0.

1.000·10

i

o k ρ t k

Mathematical

o (atm)

(cal/mol-atm)

(cal/mol)

3.100-10

4

4.000·10

4

750.

model

Each zone of the t u b u l a r r e a c t o r i s simulated as a sequence of N p e r f e c t l y mixed elementary volumes, as shown i n F i g u r e 2. Each volume can r e c e i v e a feed side-stream, and exchanges heat with a corresponding volume i n the o i l j a c k e t . For volume i and chemical species j ( e i t h e r i n i t i a t o r , oxygen, r a d i c a l s or monomer) the mass balance i s w r i t t e n as: v

F F Qfpf+Qi-i^-i Q?C. .+Q. C . . - ( ) C . ,+R. .V.=0 i j,i i-1 j,i-l j , i j,i ι 1

x

H

1

1

1

1

1

(9)

n

P i

where s u p e r s c r i p t F designates the feed side-stream, i f present, and R. . i s the production r a t e of species j i n the elementary v o l u m e * i t s e l f , as d e f i n e d by the above k i n e t i c equations. I f T ^ i n d i c a t e s the j a c k e t o i l temperature corresponding to volume i , ' U the o v e r a l l heat t r a n s f e r c o e f f i c i e n t and S the exchange area, the enthalpy balance f o r the r e a c t i o n mixture i s : Q

Qf^cF T^._ , _ i

i

i

l C p i

J F , _ T _ -(Q , Q _ p _ )c 1

i

1

i

i +

i

1

i

1

p i

T

i

+

+ X , r . .41LV.+US(T .-T.)=0 k k,i κ ι ο,ι ι

(10)

F i n a l l y , the thermal balance f o r the j a c k e t o i l reads: F

F

Q P (c« . ,T . -c .T .)-US(T .-T.)=0 o o ^ o , i + l o , i + l P o , i ο,ί' ο,ι ι' x

r

n

v

(11)

As i t can be seen, allowance i s made f o r v a r i a t i o n s i n the p h y s i c a l p r o p e r t i e s of the r e a c t i o n mixture and the j a c k e t o i l . For some p r o p e r t i e s i n the r e a c t i o n mixture, as w e l l as the k i n e t i c constants, depend on pressure, t h i s too i s to be comput ed, u s i n g the c l a s s i c a l equation:

P

i

P

- i-1 -

4 f

(u

/2

ÏÏ < > i i 8 >


( 1 2 >

45.

The

585


DONATI ET AL.

g l o b a l heat t r a n s f e r c o e f f i c i e n t i s given by:

U

D-h 2k h 1 o w m i n which k i s the w a l l c o n d u c t i v i t y , D and D^ a r e the i n n e r and outer reaclfor diameters, and h and h a r e the heat t r a n s f e r coeff i c i e n t s i n the j a c k e t and i n 2he r e a c t o r , computed through the c l a s s i c a l Nusselt equation: e8


Nu=.028(Re) (Pr)*

4

(14)

as the flow i s t u r b u l e n t a l l over the r e a c t o r . For each elementary volume, then, f o u r equations l i k e (9) p l u s the three equations (10),(11) and (12) are w r i t t e n . The r e a c tor zones can be s e p a r a t e l y computed i n sequence: f o r each, 7N n o n l i n e a r a l g e b r a i c equations a r e t o be simultaneously s o l v e d . T h i s i s performed w i t h the a i d o f a general program f o r the s o l u t i o n o f l a r g e , sparse matrix, n o n l i n e a r equations systems, a l ready employed (12) f o r the s i m u l a t i o n o f the LDPE v e s s e l reactor. More d e t a i l s on the program are given elsewhere (13). Results The p r e v i o u s l y discussed model was a p p l i e d t o simulate an i n d u s t r i a l r e a c t o r , d i v i d e d i n t o 15 zones; f o r each zone, t e n elementary volumes were considered i n the computation. The r e s u l t s are reported i n Figure 3, where the corresponding e x p e r i mental data are a l s o shown: a f a i r l y good agreement between computed and measured temperatures i n the r e a c t o r i s apparent. The conversion p r o f i l e , whose f i n a l value i s very c l o s e t o the t o t a l monomer conversion i n the i n d u s t r i a l r e a c t o r , appears t o be q u i t e d i f f e r e n t from those p r e v i o u s l y r e p o r t e d i n the l i t e r a t u r e : the r e a c t i o n proceeds very q u i c k l y a f t e r the mixture has reached a c e r t a i n " s t a r t i n g temperature", but the temperature peak i s r a t h e r smooth and a f t e r i t the conversion i s s t i l l i n c r e a s i n g . T h i s behaviour i s e x p l a i n e d by the computed p r o f i l e s o f oxygen, o r ganic peroxide and r a d i c a l s c o n c e n t r a t i o n s , r e p o r t e d i n the same Figure 3: the organic i n i t i a t o r decomposition s t a r t s a t the end of the f o u r t h zone, and i s p r a c t i c a l l y complete when the temperature reaches 180-200°C; only a t t h i s p o i n t oxygen begins t o r e act very q u i c k l y , g i v i n g o r i g i n t o the w e l l known temperature peak. The r a d i c a l s c o n c e n t r a t i o n , that i n c r e a s e s u n t i l the temperature peak i s reached, begins t o decrease, s t i l l a l l o w i n g a f u r t h e r s i g n i f i c a n t p o l y m e r i z a t i o n i n the l a s t p a r t o f the reactor. T h i s r e s u l t , from a computational p o i n t o f view, depends on the choice o f the k i n e t i c parameters k and k : i n F i g u r e 4a are presented the temperature and conversion p r o f i l e s computed, f o r the same o p e r a t i n g c o n d i t i o n s o f F i g u r e 3, with values o f k and k such t h a t the r a t i o k //k~ i s the same but the r a t i o k /fe i s 100 times lower. While nB s i g n i f i c a n t v a r i a t i o n can be noled i n fc

t



( in//>in)


Q

— — * Vi-.

i-1

Î4l

T Figure 2.

1 Oil

Schematization of a section of the tubular reactor for computation.

1 2 3 4

5

6

7

8

9

10 11 12 13 14 Section number

15

Figure 3. Results of computation. Key: ° ° °, experimental values of reactor temperature; , computed reactor temperature; , computed conversion profile; - · -, computed initiator concentration; , computed oxygen concentration; and , computed radicals concentration.



45.

DONATi ET A L .


1 2 3 4

5

6

7

8

S e c t i o n

9 10 11 12 13 14

587

15

n u m b e r

Figure 4. Effect of variations of model parameters and operating condi tions. Key: a, Temperature and conversion profiles obtained with a 100 fold reduced value of ratio k /k (unchanged value of the ratio k /yjk ); b, Effect of +20% ( ) or —2Οψο (· · -) variation of oilflowrate in startup sections. p

t

v

t




588

1 2 3 4

5

6 7

β

S e c t i o n

9 10 11 12 13 U

15

n u m b e r

Figure 4c and 4d. Effect of variations of model parameters and operating condi tions, c, Effect of +20% ( ) or —20% ( · · -) variation of feed oxygen con centration, d, Temperature and conversion profiles for a split-feed condition.


45.

DONATI ET

AL.


the f i r s t part of the r e a c t o r , the p r o f i l e s a f t e r the temperature peak appear to be q u i t e d i f f e r e n t . The model was then employed to evaluate the e f f e c t , on the r e a c t o r behaviour, of small v a r i a t i o n s of the o p e r a t i n g c o n d i t i o n s . As an example i n F i g u r e 4bthe e f f e c t of a ^20% v a r i a t i o n i n the o i l flow r a t e i n the s t a r t - u p ( 3 * and 4 ) s e c t i o n s i s presented, while Figure 4c shows the e f f e c t of a +20% v a r i a t i o n i n the feed oxygen c o n c e n t r a t i o n . In a l l cases the computed p r o f i l e s appear to be c o n s i s t e n t with t h e o r e t i c a l expectations and i n d u s t r i a l experience. As p r e v i o u s l y discussed, one p o s s i b i l i t y of i n c r e a s i n g the r e a c t o r performance r e s i d e s i n the use of m u l t i p l e feeds. In F i g ure 4dan example i s presented of a " s p l i t - f e e d " c o n d i t i o n ; the same amount of monomer and i n i t i a t o r s considered i n the example of Figure 3 was supposed to be f e d p a r t at the r e a c t o r i n l e t , p a r t i n a r e a c t o r zone immediately a f t e r the temperature peak: as shown by the computed temperature and conversion p r o f i l e s , a small i n c r e a s e i n the monomer conversion can be obtained. r

Mathematical Model of Low Density Polyethylene Tubular Reactor

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