Mathematical Modeling of High Pressure Tubular LDPE Copolymerization Reactors

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Mathematical Modeling ofHigh Pressure Tubular LDPECopolymerization ReactorsG. Verrosa, M. Papadakisa & C. Kiparissidesa

a Department of Chemical Engineering ChemicalProcess Engineering Research Institute AristotleUniversity of Thessaloniki P.O. Box 472, 54006,GreecePublished online: 04 Oct 2013.

To cite this article: G. Verros, M. Papadakis & C. Kiparissides (1993) MathematicalModeling of High Pressure Tubular LDPE Copolymerization Reactors, Polymer ReactionEngineering, 1:3, 427-460, DOI: 10.1080/10543414.1992.10744438

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POLYMER REACTION ENGINEERING, 1(3) 427-460 (1992-1993)

MATHEMATICAL MODELING OF HIGH PRESSURE TUBULAR LOPE

COPOLYMERIZATION REACTORS

G. Verros, M. Papadakis and C. Kiparissides*

Department of Chemical Engineering

Chemical Process Engineering Research Institute

Aristotle University of Thessaloniki

P.O. Box 472, 54006, Greece

* To whom all correspondence should be addressed.

ABSTRACT

In this study a comprehensive mathematical model for high pressure tubular ethylene/vinyl acetate copolymerization reactors is developed. A fairly general reaction mechanism is employed to describe the complex free radical kinetics of copolymerization. Using the method of moments, a system of differential equations is derived to describe the conservation of total mass, energy, momentum and the development of molecular weight and compositional changes in a two-zone jacketed tubular reactor . In addition, the model includes a number of correlations describing the variation of physical, thermodynamic and transport properties of the reaction mixture as a function of temperature, pressure, composition and molecular weight distribution of polymer. Numerical solution of the reactor model equations permits a realistic calculation of monomer and initiator concentrations, temperature and pressure profiles, number and weight average molecular weights, copolymer composition as well as the number of short and long chain branches per 1000 carbon atoms under typical industrial operating conditions. Simulation results are presented showing the effects of ethylene, vinyl acetate, initiator and chain transfer agent on the polymer quality and reactor operation. The results of this investigation show that, in principle, we can obtain a

427

Copyright© 1993 by Marcel Dekker, Inc.

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428 VERROS, PAPADAKIS, AND KIPARISSIDES

copolymer product of desired molecular weight and composition by controlling the process variables . The procedure developed in this work is general and can lead to a more systematic design, optimization and control of industrial high pressure ethylene copolymerization reactors.

INTRODUCTION

The high pressure free radical polymerization of ethylene in tubular and

vessel reactors is a process of considerable economic importance. The

polymerization is carried out at high temperatures (150-300°C} and

pressures (200-350 Mpa). Under these conditions, the ethylene behaves

as a supercritical fluid . Ethylene polymerization is a highly exothermic

process . Approximately one-half of the heat of reaction is removed

through the reactor wall by a heat transfer fluid which circulates through

the reactor jacket. This results in a nonisothermal reactor operation.

One of the most important problems encountered in operating a low

density polyethylene (LOPE) reactor is to select the optimum operating

conditions that maximize the reactor productivity at the desired product

quality. To accomplish optimal operation of an LOPE tubular reactor or

design a new one, it is necessary to know how the product quality

changes relative to the variations of the controlling process variables

(i.e. monomer concentration , initiator and chain transfer agent

concentrations, temperature , pressure, etc) . This information can be

obtained from a detailed reactor model describing the molecular weight

and compositional changes in terms of the process operating conditions.

Over the past twenty years several modeling studies on free radical

polymerization of ethylene have been reported (Chen et al., 1976 ; Thies,

1979 ; Lee and Marano, 1979 ; Goto et al., 1981 ; Mavridis and

Kiparissides, 1985 ; Shirodkar and Tsien, 1986 ; Brandolin et al., 1988;

Zabisky et al. , 1992). In the present work , a detailed mathematical model

is developed for a high pressure ethylene copolymerization tubular

reactor. The variation of the physical properties of the reaction mixture

with position is accounted for. The elements of the model are the

reaction mechanism, the mass, energy and momentum balances and the

moments of "live" and "dead" polymer distributions.

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TUBULAR LDPE COPOLYMERIZATION REACTORS 429

In what follows, the kinetic mechanism of ethylene copolymerization is

reviewed and the rate functions describing the net production rate of all

molecular species present in the reactor are derived. Subsequently,

general design equations describing the molecular weight and

compositional changes in a high-pressure tubular ethylene

copolymerization reactor are derived. Finally , simulation results are

presented illustrating the effect of process parameters (i .e. initiator

concentration, solvent) on product quality and temperature profile

along the reactor length.

KINETIC MECHANISM AND POLYMERIZATION RATE FUNCTIONS

A fairly general kinetic mechanism describing the free-radical

copolymerization of ethylene-vinyl acetate in high pressure reactors is

considered (Ehrlich and Mortimer, 1970 ; Goto et al., 1981). The kinetic

mechanism includes the following elementary reactions :

1. Initiation (by Peroxides or Azo Compounds) :

2. Chain Initiation Reactions :

kl1 A"+ M1 -7 P1. 0, o

kl2

A"+ M2 -7 00,1, 0

3. Propagation Reactions :

kp11

Pp, q, r + M1 -7 p 1 P+ 'q, r

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kp12

Pp, q, r+M2 ~ ap,q+1 , r

kp21

ap, q, r + M1 ~ PP 1 q r + ' '

kp22

ap, q, r + M2 ~ ap, q+ 1, r

4. Chain Transfer to Monomer Reactions :

ktm11

~ p 1, 0, 0 + Dp, q, r

ktm12

P p ,q, r + M2 ~ Oo, 1, 0 + Dp, q, r

ktm21

~ P1 , 0, o+Dp,q,r

ktm22

ap, q, r + M2 ~ Oo, 1, o + Dp, q, r

5. Chain Transfer to Solvent (Chain Transfer Agent) Reactions :

k1s1

P + S ~ A" + Dp, q, r p, q, r

kts2

ap, q, r + S ~ R" + Dp, q, r

6. Chain Transfer to Polymer :

ktp11 P +D ~ P 1 +D p, q,r x,y, z x,y,z+ p, q,r

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TUBULAR LDPE COPOLYMERIZATION REACTORS

ktp12

P p, q, r + Ox, y, z ~

ktp21

op, q, r +ox, y, z ~

ktp22

0 1 +0 X, y, Z+ p, q, r

p 1 +0 X, y, Z+ p, q, r

Op, q,r+Mx,y,z ~ O x,y, z+1 +0p,q,r

7. Termination by Oisproportionation :

ktd11 p + p p,q,r x,y, z ~ 0 +0 p,q, r x, y, z

ktd12 0 + p p,q, r x,y, z ~ 0 +0 p,q,r x,y, z

ktd22

0 +0 p, q, r x,y, z ~ 0 +0 p, q,r x, y, z

8. Termination by Combination :

ktc 11 p + p ~ 0 p, q, r x, y,z p+x, q+y, r+z

ktc12

op, q, r + Px, y, z ~ OP+X ,q+y, r+Z

ktc22

0 +0 ~ 0 p, q,r x,y, z p+x, q+y, r+Z

9. Intramolecular Transfer (Short Chain Branching) :

kb1 p

p, q, r ~ p p, q, r or op, q, r

431

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kb2

op, q, r ~ op, q, r or p p, q, r

1 0. Overall 13-Scission of Radicals :

k[31

P p, q, r ~ D~. q, r + R"

kf32 Q ~ D= r + R" p, q, r p, q,

To identify a copolymer chain, we introduce a general notation Gp, q, b

which denotes the concentration of "live" or "dead" polymer molecules having p units of monomer 1 (M 1), q units of monomer 2 (M 2), and b long

chain branches (LCB) per polymer molecule. It should be noted that the ultimate monomer unit in a "live" copolymer chain can be either of M1 or

M 2 type. As a result, two different symbols, P and Q, are introduced to

identify the live copolymer chains ending in an M1 or an M2 monomer

unit, respectively .

By assuming (i) one phase flow (Chen et al., 1976); (ii) steady-state conditions (Thies, 1979); (iii) absence of axial and radial mixing (Donati

et al., 1981; Yoon and Rhee, 1985), one can derive the following

general steady-state molar balance differential equations to describe

the conservation of various species in a tubular reactor :

d(uG0 . q, r) r dx = Gp, q , r ( 1)

where r G denotes the polymerization rate of various species present in

the reaction mixture (i.e. initiator(s), monomer(s), solvent(s}, "live"

polymer chains of type "P" or "Q" and "dead" polymer chains, D). These

rate functions can be obtained by combining the rates of the various

elementary reactions describing the generation and consumption of

"live" and "dead" copolymer molecules based on the general kinetic

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mechanism of ethylene copolymerization described above. For

simplification, we choose to work with the bivariate number chain length

distributions (NCLDs) of the polymer chain populations , P(p , q), Q(p, q),

D(p, q) . Accord ingly , we write the following generalized expressions

describing the net rates of appearance/disappearance of individual

molecular species :

In itiator Consumption Rates :

Primary Radical Formation Rate :

rR. = 2fikdicdi + kts1CsPoo + kts2Cs0oo + k~1Poo + k~20oo

- (kp11 + kp12 )CR.Cm1- (kp22 + kp21 )CR.Cm2

Monomer(s) Consumption Rate (Propagation Rate) :

Solvent (CTA) Consumption Rate :

Net Formation Rate of "Live" Polymer Chains :

(2)

(3)

(4)

(5)

r p = (k11 R" Cm 1 )o(p-1, q) + kp11 (P(p-1 , q) -P(p, q))Cm1 - kp12P(p, q)Cm2

+ kp21Q(p-1' q) Cm1- AP(p, q) + pktp11D(p, q)Poo + pktp21D(p , q)Ooo

00 00

- k1P11 P(p .q) I I pD(p, q)- k1P12P(p,q) I I pD(p, q) P=1q=1 p=1q=1

r a= k12 R"Cm2°(P, q-1) + kp22(Q(p, q-1) -Q(p, q))Cm2- kp21 Q(p, q)Cm1

+ kp 12P(p , q-1) Cm2- BQ(p, q) + qk1P22D(p, q)000 + qk1p 12D(p, q)Po

00 00

- ktp22 O(p, q) I I q D(p,q) -ktp21Q(p.q) I I q D(p,q) p=1q=1 p=1q=1

(6)

(7)

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Net Formation Rate of "Dead" Polymer Chains :

ro =(A- ktc11 Poo-ktc120oo)P(p, q) + (B - ktc12Poo- ktc220oo)O(p, q)

- Pktp11D(p, q)Poo- Pktp21D(p, q)Ooo- qktp22D(p, q)Ooo

00

0000 1 0000

+ ktp220(p.q) I I qD(p,q) + 2 ktc11 I I P(x, y)P(p-x, q-y) p=1 Q=1 p>X Q>Y

1 00 00

+ 2 ktc12 I I {Q(x, y)P(p-x, q-y )+ P(x, y)Q(p-x, q-y)} P>X Q>Y

1 +2 ktc22 I I O(x, y)Q(p-x, q-y) P>X Q>Y

where

(8)

and P00 , 0 00 denote the concentrations of "live" polymer chains of type

"P" and "0", respectively :

P oo= I I P(p,q) p=1 q=1

00 00

Ooo= I I O(p,q) p=1 q=1

(11)

Based on the above definitions of rate functions and the general

reactor design equation (1 ), one can write an infinite set of differential

equations which must be solved numerically to obtain desired

information on molecular weight and compositional developments in a

high pressure copolymer reactor . However, for modeling purposes it is

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often impractical to solve the resulting infinite system of molar balance

differential equations. As a result, one has to resort to mathematical

techniques such as the method of moments (MM), the instantaneous

property method (IPM), and the property moment method (PMM) to

reduce the infinite system of molar balance equations into a lower order

system of differential equations.

Achilias and Kiparissides (1992) have shown that the method of

moments is the most general technique and can be applied to both

linear and branched copolymerizations either in the presence or

absence of diffusional limitations in the termination and propagation rate

constants . The method of moments is based on the statistical

representation of molecular properties (i.e. Mn, Mw) through the use of

leading moments of the distributions (i.e. molecular weight distribution

(MWD) , degree of branching distribution (DBD), cumulative copolymer

distribution (CCD)). The leading moments of the bivariate number chain

length distributions of "live" and "dead" macromolecules are defined

as (Arriola, 1989; Achilias and Kiparissides, 1992):

p 00 00

Amn = L L pmqn P(p, q) p=1 q=1

(12)

(13)

(14)

Accordingly, one can obtain the corresponding rate functions for the

moments of the bivariate number chain length distributions of "dead"

and "live" polymer chains by multiplying each term of equations (6-8) by

the term pmqn and summing the resulting expressions over the total

variation of p and q :

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436 VERROS, PAPADAKIS, AND K.IPARISSIDES

p p p p Q

- kp12A.mnCm2- AA.mn + ktp11(~m+1,nAoo- ~10A.mnl + ktp21~m+1,nA.oo p

- ktp12~01Amn (15)

Q • nno Q nnp rA = kl2 R Cm2 i5(m, n) + kp22(I. C) A.mi- A.mn)Cm+ kp12 .I G) A.miCm2

mn 1=0 1=0 Q Q Q Q p

- kp21AmnCm1 - BA.mn+ ktp22(~m.n+1Aoo- ~o1Amnl + ktp12~m.n+1AOO

Q - ktp21~10Amn (16)

1 mm n n P P 1 m m n n 0 0 rllmn =2ktc11i~O{j) k~O (k) AjkAm-j,n-k +2ktc22j~O {j) k~O (k) AjkAm-i,n-k

p Q p p Q Q -(A- ktc11Aoo - ktc12A.oo)A.mn- (B- ktc12Aoo - ktc22Aoo)A.mn

p p Q Q + ktp11 (~m+ 1 ,nAoo- ~10Amnl + ktp21 (~m+ 1 ,nAoo - ~10Amnl

p p Q Q + ktp12(~m.n+1A.oo - ~o1Amn) + ktp22(~m.n+1Aoo - ~o1Amn) ( 17)

To break down the dependence of the "dead" polymer moment

equations on higher order moments the mathematical simpification of

Lee and Marano (1979) was employed. By adding equations (15) and

(16) to the corresponding moment rate function of "dead" polymer chains, equation (17), we obtain the following expression for r" which .. mn

is independent of the higher order moments:

mm P p nn P P rllmn = kp11(I. (i ) \n- Amn)Cm1 + kp12CI. (i) Ami- Amn)Cm2

1=0 1=0

mmOO nno Q + kp21( i~O (i ) \n- Amn)Cm1 + kp22(i~ (i) Ami- Amn)Cm2

1 m m n n P P 1 m m n n PQ +2 ktc1\~o (j) k~O (k) AjkAm-i,n-k+2ktc12 i~O (j) k~O (k) {A.jkAm-i,n-k

QP 1 mmnnoO P QP + Aik Am-i,n-k} + 2ktc22i~O (j ) k~O {k) Aik Am-i,n-k- (ktc11 Aoo +ktc12A.oo)Amn

( 18)

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TUBULAR LOPE COPOLYMERIZATION REACTORS 437

The other reaction rates of interest are given by the following

equations :

Rate of Long Chain Branching :

Rate of Short Chain Branching

(20)

THE REACTOR MODEL EQUATIONS

In the present study a comprehensive reactor model is developed to

simulate the operation of a two-zone industrial high pressure tubular

ethylene copolymerization reactor. The reactor consists of two reaction

zones , Figure 1. At the exit of the first zone, the reaction mixture is

quenched by the addition of fresh ethylene and the combined stream is

fed to the second zone. The present model accounts for the presence

of multiple initiators and multiple solvents (i.e. chain transfer agents) .

To simplify the numerical solution of the reactor model equations we

introduce the following dimensionless quantities:

Dimensionless Length: z = x/ L

Dimensionless Temperature: 0 = (T-T )/T 0 0

Dimensionless Coolant Temperature : 0 = (T -T )/ T c c 0 0

Fractional Conversion : y. =(F. -F.)/ F. J JO J JO

(21)

(22)

(23)

(24)

where L is the total reactor length , T 0

the temperature of the reaction

mixture at the reactor inlet and F; (j : m, d, s, r) denotes the molar flow

rate of monomer, initiator, solvent and inhibitor, respectively. The molar

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438

CD

FIGURE 1:

VERROS, PAPADAKIS, AND KIPARISSIDES

lev @

G) ®

l L, b ZONE 1 ZONE 2

t ~ ~

@ 0 @

Schematic representation of a two-zone high pressure

tubular LOPE reactor. (1) Reactor Feed, (2) Quenching

Stream, (3) and (5) Coolant Inlet, (4) and (6) Coolant

Outlet, (7) Initiator Feed.

flow rate Fi can be expressed in terms of the flow velocity u and the

corresponding concentration of species j :

(25)

The concentration Ci (j: m, d, s, r) is related to the corresponding fractional conversion y. as in equation (26) :

J

(26)

Using the above dimensionless variables and the expressions for the

various rate functions, one can derive the following steady-state design

equations for a high pressure tubular ethylene copolymerization

reactor:

Continuity Equation :

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2 du/dz =- (u/p)[T0 (dp/dT)(d0/dz) +I (dp/dymi)(dYm/dz)]

1=1 (27)

Fractional Conversions :

i = 1, 2, ... Ni (28)

N

dR"/dz = (L[i 2fi kdi Cdi- k11 Cm 1(R"]- k12Cm2(R"]]- (R"~]/u (29) i=1

"Dead" Polymer Moments Equations :

p p Q

d~ooldz = {L[A.oo(ktm11Cm1 + ktm12Cm2 + ktd11A.oo + ktd12Aoo + ks1C s

Q p Q

+ k131) + "oo(ktm22Cm2 + ktm21Cm1 +ktd12A.oo + ktd22A.oo + ks2C s+k132)

P 2 0 2 P 0 du + 0.5ktc11(Aoo) + 0.5ktc22(Aoo) + ktc12A.oo"ool-lloodz}/u (32)

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(37)

"Live" Polymer Moment Equations :

p p Q dAoofdz = {L[k11[R"]Cm1- (kp12 + ktm12)Cm2Aoo + (kp21 + ktm21)Cm1Aoo

Q p p p Q Q + ktp21~10Aoo- ktp12~01Ao0 - Aoo (ktc11A0o + ktc12Aoo + ktd12 Aoo + ktd11

(38)

p p Q p p dA10/dz = {L[ki1[R"JCm1 + kp11Cm1Aoo + kp21Cm1 A1 o- kp12A10Cm2- AA1 o

P P P P du + ktp11 (~2oAoo- ~10A10l + ktp21 ~2o0oo- ktp12~o1A10] - A1 Odz }/u (39)

p Q p p p p dAo1/dz = {L[kp21Ao1Cm1- kp12A01Cm2- AAo1 +ktp11(~11Aoo- ~10Ao1l

a P P du + ktp21~11AOO- ktp12~01A01 ]- A01 dz }/u (40)

Q Q p dA00/dz = {L[k12[R"]Cm2- (kp21 + ktm21)Cm1Aoo + (kp12 + ktm12)Cm2Aoo

p Q Q Q p p +ktp12~o1Ao0- ktp21~1oAoo -A oo(ktc22Aoo + ktc12Aoo + ktd12Aoo

P a du + ktd22Aoo)] - Aoodz} /u (41)

Q Q p Q Q dAo1/dz = {L[ki2[R"JCm2 + kp22Cm2Aoo + kp12Cm1A01- kp21A01Cm1 - AAo1

a a P a a du + ktp22(~o2Aoo- ~o1Ao1) + ktp12~o2Aoo- ktp21~10Ao11- Ao1dz}/u (42)

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(43)

Molecular Property Equations :

p p Q Q dCLcsldz = L[(ktp11"-ool-l10 + ktp12"-ool-l10 + ktp21"-ool-lo1 + ktp22"-ool-lo1)

du - CLCBdz]/u (44)

(45)

Energy equations :

Pressure drop :

dP/dz = - 2f(L!D)p u2 (48)

On the assumption that (i) the long chain hypothesis (LCH) and the

quasi-steady state approximation (OSSA) can be applied to the "live"

radical moments (Mavridis and Kiparissides, 1985 ; Yoon and Rhee,

1985 ; Shirodkar and Tsien, 1986) and (ii) the velocity gradient is

negligible, one can replace the "live" moment differential equations by

the following algebraic equations:

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(49)

Q p p [k,2[R"1Cm2 - (kp21 + ktm21lCm1Aoo + (kp12 + ktm12lCm2Aoo +ktp12l-io1Aoo

Q Q Q p p p - ktp21J..110Aoo- "oo(ktc22Aoo + ktc12"oo+ ktd12"oo + ktd22Aooll =0 (50)

p Q p p [k,1[R"]Cm1 + kp11Cm1Aoo + kp21Cm1A10- kp12A10Cm2- AA.1 o

p p p + ktp11(1-12o"oo- J..110A10) + ktp211-i2oOoo- ktp12l-lo1A.10] =0 (51)

Q p p p p [kp21A.01cm1- kp12A.o1Cm2- AA.o1 + ktp11(1-i11Aoo- J..110Ao1l

Q p + ktp21I-I11Aoo - ktp12l-lo1A.o1l =0 (52)

Q p Q

[k,2[R" ]Cm2 + kp22Cm2Aoo + kp12Cm1A01- kp21A.01Cm1

Q Q Q p Q

- AA.o1+ktp22(1-1o2"oo-l-lo1A.o1l + ktp12l-lo2"oo- ktp21l-l1o"o1l =0 (53)

p Q Q Q Q

[kp12A.10Cm2- kp21A.10Cm1 - BA.o1 + ktp22(l-l11"oo -1-1o1A.1ol

(54)

By solving the above system of linear algebraic equations (49-54) with

respect to the moments of the "live" polymer distribution and

substituting the resulting expressions into the reactor design equations

(28)-(37), (44)-(46), it can be shown that the final model equations depend only on the values of the ratios of the kinetic parameters, "8ii",

and not on the absolute values of kinetic rate constants :

(55)

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i,i=1' 2 (56)

£ .. = k ../k .. QIJ QIJ pll

g=p, ts, tm, tp i,i=1' 2 (57)

i,i=1' 2 (58)

This important observation explains the fact that one cannot obtain

independent estimates of the individual rate constants by fitting model

predictions to kinetic data from a tubular reactor.

The molecular weight averages , the SCB and LCB per 1000 carbon

atoms as well as the cumulative copolymer composition can be

expressed in terms of the moments of the bivariate NCLD of "dead"

polymer chains :

Number Average Molecular Weight :

(59)

Weight Average Molecular Weight :

(60)

Long Chain Branches per 1000 Carbon Atoms :

(61)

Short Chain Branches per 1000 Carbon Atoms :

(62)

Cumulative Copolymer Composition :

(63)

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Balances at the Quenching Points

At the quenching point, the reaction mixture is quenched by the

addition of fresh ethylene (Figure 1 ). At this point, the following mass,

molar end energy balances must be established in order to calculate

the initial conditions at the inlet of the second reaction zone :

Total Mass Balance :

(64)

where m is the mass flow rate ' Q is the volumetric flow rate and p is the

density of the corresponding stream.

Component Mass Balances :

(65)

where wi is the weight fraction of the "i" component and Ci is the

concentration of "i" species (Cmi• csi• cdi• ~mn• Cscs· CLcsl·

Energy Balance :

(66)

where H is the specific enthalpy associated with a stream. Finally, the cumulative monomer(s) conversion, Ycum is calculated by the

expression :

(67)

where F mo, 1, F mo,2 are the molar flows of monomer at the inlet of the

first and second reaction zone respectively and ymi, 1, ymi,2 are the

corresponding fractional monomer(s) conversions .

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KINETIC RATE CONSTANTS

One of the most difficult problems in simulating the operation of high

pressure LOPE reactors is the selection of appropriate values of the

kinetic rate constants. Detailed information on the kinetic rate constants

for the high pressure ethylene polymerization can be found in the

articles of Ehrlich and Mortimer (1970), and Goto et al. (1981 ). It is

interesting to note that although these investigators agree on the values of £: .. (= k.J k .) parameters, the reported values for the individual rate

IJ IJ pi

kinetic constants show a large discrepancy.

The most complete set of kinetic data for high pressure ethylene

polymerization, including the effect of pressure through activation

volumes, has been reported by Goto et al. (1981).The reported values

were estimated from experimental measurements on monomer

conversion, number and weight average molecular weights, amount of

unsaturated double bonds and total methyl content per 1 o3 carbon

atoms. The measurements were obtained from a continuous stirred

autoclave operated under typical industrial conditions.

In Table 1, the values of kinetic parameters for the ethylene-vinyl

acetate copolymerization are reported. For simplicity, we assume that etmij=etmii· etpij=etpii and the eij parameters of 13-scission and backbiting

reactions for vinyl acetate reactions are equal to the respective values for ethylene. The ratios of k12 I k11 and k1c11 / k1d 11 are assumed to be

equal to one.

To account for the effects of MWD, temperature, and polymer

concentration on the thermodynamic, physical and transport properties

of reaction mixture the correlations reported by Kiparissides et al. (1993)

were used.

SIMULATION RESULTS

For our simulation studies the reactor geometry shown in Figure 1 was

employed. The reactor consists of two reaction zones. At the inlet of

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TABLE 1 : Numerical Values of the "eij" Parameters a

Ko llE llV Ref. {J/ ~:~mol) {m3/ gmol)

(kp11/ktc11 O.S) 2500 31642 -26.2 10-6 Present work

(kp22/ktd22 o.s, 426 4900 Hamer, 1983

(kp12/ kp11l 1.07 Ehrlich and Mortimer, 1970

(kp21/ kp22) 1.09 Ehrlich and Mortimer, 1970

(ktm11/ kp11l 1Q-4 Odian, 1981

(ktm22/ kp22l 0.0142 11286 Hamer, 1983

(ktp11/kp11) 3.11 14952 24.1 10•6 Goto et al.,1981

(kts1/ kp11 l 0.218 9650 0.2 10-6 ))

(ki3'1/ kp11) 0.169 16842 0 ))

(kw-1/ kp11) 1.034 21840 2.9 1Q•6 ))

(kb1/ kp11) 8 10500 3.8 10·6 ))

a £=Ko e -(t:.E+Pt:. V) I RT

each zone, fresh ethylene and initiators are fed into the reactor . Each

zone, from a reaction point of view, can be divided into two sections,

namely, a reaction section and a cooling one. After the injection of

initiators (points 7 in Figure 1), the reaction proceeds rapidly, in an almost

adiabatic way, to a maximum (peak) temperature which corresponds to

the end of the reaction section and the depletion of all initiators. In the

cooling section, no reaction takes place and the reaction mixture is

cooled by the coolant flowing through the reactor jacket. At the end of

the first zone, the reaction mixture is quenched by the addition of fresh

monomer and the new stream is fed to the second zone. Each zone has

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0 0 ~

Q) ... :II

iii ... Q)

c. E Q)

1-

TABLE 2: Initiator Decomposition Kinetic Rate Constants (Halle , 1980)

Ko ~E ~v

(s-1) (J/gmol} (m3/gmol}

TBPV 7.95 1013 116630 3.4 10"6

DTBP 3.69 1014 146847 13.010"6

350 - Cdo = 2.E-5 Kgmol /m3

---o- Cdo = 4.E-5

300

250

200

150+---~--~--~--~--~--~--~--~--~--~~

0.0

FIGURE 2:

0.2 0.4 0.6 0.8 1.0

Dimensionless Axial Length

Effect of initial initiator concentration on reactor

temperature profile.

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448

c 0 Ill ... CD > c 0

(,)

... CD E 0 c 0

::E CD > iii :=o E :=o

(,)

VERROS, PAPADAKIS , AND KIPARISSIDES

0.2~--------------------------------------~ - Cdo = 2.E·5 Kgmol /m3

---o- Cdo = 4.E·S Kgmol /m3

0.1

0.0~--~~r-~--~--~--~--~--~--~--r-~

0 .0

FIGURE 3:

0.2 0.4 0.6 0.8 1.0


Effect of initial initiator concentration on ethylene

cumulative conversion .

a total length of 150m and the internal tube diameter is 5.0 em. The

mass flow rate of ethylene at the inlet of the first zone and at the side

quenching point is 10.0 and 4.0 Kg/s, respectively. The initiator mixture

consists of 80% per weight of t-butyl peroxypivalate (TBPV) and 20%

per weight di-t-butyl peroxide (DTBP). Typical values of initiator

decomposition rate constants are given in Table 2.

The TBPV nominal inlet concentration was 4.0E-5 gmol/1 in both

reaction zones. In all simulations runs, the nominal reactor inlet pressure

was 270 Mpa, while the temperature of the reaction mixture at the inlet

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60000~---------------------------------------,

-.c Cl ·a; :t ... 50000 Cll

::I u Cll 0 ::E Cll Cl Cll

Cii ~ 40000

- Cdo = 2.0E-5 Kgmol/m3

--o- Cdo = 4.0E-5

30000+---~~r-~--~--~--or--~--~--~-.--~

0.0

FIGURE 4:

0.2 0.4 0.6 0.8 1.0


Effect of initial initiator concentration on number

average molecular weight.

of each reaction zone was assumed to be equal to 190°C. A fouling

factor of 1000 W/m21K was assumed for both reaction zones while the

coolant flow was considered to be 5 kg/s and its outlet temperature was

set at 180°C.

The reactor model equations coupled with the appropriate algebraic

equations, describing the variation of thermophysical and transport

properties of the reaction mixture, were numerically integrated to

calculate the monomer and initiator concentrations, temperature and

pressure profiles as well as the variation of molecular properties of LOPE (i.e. Mn, Mw, LCB, SCB, CC) along the reactor length. A modified

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500000~--------------------------------------~

~ 400000 1:11 ·; 3: ... IV

:::J 300000 u 41

0 ::::E

41

:: 200000 ... 41 > ~ .. .c 1:11 ·; 100000 3:

- Cdo = 2.0E-5 Kgmol/m3

--o- Cdo = 4.0E-5 Kgmol/m3

o+---~--~--~~r-~--~--~---r--~--.-~

0.0

FIGURE 5:

0.2 0.4 0.6 0.8 1.0


Effect of initial initiator concentration on weight average

molecular weight.

Adams-Moulton routine was utilized for the numerical integration of the

model's stiff differential equations.

Figures 2-11 illustrate some representative results obtained for the

two-zone reactor of Figure 1. Note that an increase in the initial initiator

concentration results in an increase of the peak temperature (Figure 2), monomer conversion (Figure 3), weight-average molecular weight (Mw)

(Figure 5), LCB and SCB (Figures 6 and 7). The number-average molecular weight (Mn) decreases with increasing initial initiator

concentration (Figure 4) due to the larger number of initiated polymer

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TUBULAR LDPE COPOLYMERIZATION REACTORS

2 -II)

E 0 --o-..

<C c:: 0 .a ... ca 0 0 0 0 ,.... ... G) 1 c. II) G)

.c u c:: ca ...

11:1

c:: "iii .c 0

Cl c:: 0 ..J

0 0.0

FIGURE 6:

Cdo = 2.0E-5 Kgmol/m3

Cdo = 4.0E-5

0.2 0.4 0.6 0.8 1.0


Effect of initial initiator concentration on long chain

branching.

451

radicals. On the other hand , Mw increases with initiator concentration

due to the higher monomer conversion and the increased rate of long

chain branching reaction.

Chain transfer agents (CTA) as n-hexane and comonomers such as

propylene or vinyl acetate added in small amounts to the reaction

mixture are used to control the product quality. Note that chain transfer

agents or comonomers added in small amounts do not affect the

temperature profile (Figure 8) and overall ethylene conversion in the

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28~--------------------------------------~

II)

E 0

< c 26 0 .a ... cu 0 0 0 0 24 ... G) Q.

II) G)

.1:. u c cu ...

CXI

c "iii .1:. 0 -... 0 .1:. en

22

20 Cdo = 2.0E-5 Kgmol/m3

Cdo = 4.0E-5

18+---~--~--~--r---~~r-~--~--~--~--~

0.0

FIGURE 7:

0.2 0.4 0.6 0.8 1.0


Effect of initial initiator concentration on short chain

branching.

reactor. On the other hand, as the concentration of CTA increases both Mn (Figure 9) and Mw (Figure 10) significantly decrease. Notice that

although the values of Mn and Mw vary with the concentration of CT A,

the observed variations in SCB and LCB are not significant.

In Figure 11, the cumulative copolymer composition for the ethylene -

vinyl acetate system is plotted with respect to the reactor length for

three different simulation runs. In the first simulation run, the vinyl

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0 ~

CD ... ~ -ftl ... CD Q,

E CD 1-

350.-----------------------------------------, - So = 0% (w/W) n·Hexane

~ So:S%

300

250

200

150+---~-.,-~---.--~--.---~--.---~-.--~

0.0

FIGURE 8:

0.2 0.4 0.6 0.8


Effect of initial solvent concentration (S0 ) on

temperature profile.

1.0

acetate concentration in the feed and the quenching stream is 5% w/w

and 0%, respectively. In the second run, the vinyl acetate

concentration in both streams remains constant at 5% w/w. Finally, in

the third case the VA concentration changes from 5% w/w in the feed

stream to 10% w/w in the quenching stream. It should be pointed out

that the cumulative copolymer composition remains constant in the first

zone and is independent of the temperature profile. This behaviour can

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50000~--------------------------------------~

Q)

1:111 ca ..

40000

Q)

~ 30000 .. Q)

..a E ~

z

20000 0.0

FIGURE 9:

- So = 0% (w/W) n-Hexane

--o- So= 5%

0.2 0.4 0.6 0.8 1.0


Effect of in itial solvent concentration (S0 ) on number


be explained by the fact that the values of the reactiv ity ratios (kp 12/

kp 11 and kp 21 1 kp 22) are approximately equal to one. The observed

copolymer composition changes in the second zone are due to the step changes (i. e. ±5%) in the VA concentration at the quenching point.

These results clearly demonstrate that one can control the final product

copolymer compos it ion by appropriate sel ection of the VA

concentration of the side streams.

Finally, it should be mentioned that the calculated values of Mn

(22x 1 Q3-35x 1 Q3) and Mw (20x1 o4-30x 1 Q4), LCB(=1.0/1 Q3 C) and

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Ql Cl :! G)

> c(

500000~--------------------------------------~

- So = 0% (w/W) n-Hexane

--o- So= 5%

400000

300000

200000

100000

0+---~--.-~--~--~--~--~--~-----,--~

0.0 0.2 0.4 0.6 0 . 8 1. 0


FIGURE 10: Effect of initial solvent concentration (S0 ) on weight


SCB(,25/1 Q3 C) are in good agreement with experimental results

reported by Shirodkar and Tsien (1986), for a two-zone industrial

reactor.

CONCLUSIONS

This study presents a general mathematical framework for modeling

high pressure industrial tubular ethylene copolymerization reactors. A

fairly general kinetic mechanism is employed to describe the complex

kinetics of high pressure copolymerization of ethylene. To determine the

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456

< > c

c 0 -"iii 0 c. E 0 ()

... G)

E > 0 c. 0 () -

VERROS, PAPADAKIS, AND KIPARISSIDES

0.07 -r------------------------,

--o- 5% VA Feed Stream, 0% Quenching Stream

- 5% VA Feed Stream, 5% Quenching Stream

- 5% VA Feed Stream, 10% Quenching Stream

0.06

"Et 0.05 ·a; ~ G) Cll «< -c G) u ... G)

c. 0.04 +--.......---~-.......--~-...--..,..----..---T""""-..---T""""--1 0.0

FIGURE 11 :

0.2 0.4 0.6 0.8


Effect of initial vinyl acetate (VA) comonomer

concentration on copolymer composition.

1.0

variation of molecular properties along the reactor length the method

of moments is applied. Application of the OSSA and LCH to the moment

equations of "live" polymer chains shows that ethylene copolymerization kinetics will depend on the values of £ ( = k./ kpi) parameters . This

IJ lj

means that the absolute values of the kinetic rate constants cannot be

estimated by fitting model predictions to kinetic data obtained from an

industrial tubular reactor .

The present simulation results are in good agreement with available

experimental data on LOPE for typical industrial operating conditions. As

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TUBULAR LOPE CO POLYMERIZATION REACTORS 457

a result , the present modeling approach can be used in the design,

optimization and control of industrial high pressure tubular ethylene

copolymerization reactors .

NOTATION

C : concentration, kgmol/m3

D : inside tube diameter, m Dp,q,r : "dead" polymer , consisting of p ethylene units,

q comonomer units and r long chain branches Fi : molar flow of the j component, kgmol/s

f :friction factor , dimensionless fi : initiator(s) efficiency

kb : rate constant for intramolecular transfer, s·1

k~ : rate constant for overall3-scission of radicals, k~=k~,+kW" s·1

kW : rate constant for 13-scission of sec-radicals , s·1

kW' : rate constant for 13-scission of tert- radicals , s·1

kd : rate constant for initiator decomposition , s·1

k : propagation rate constant, m3fkgmol/s p

ktc : termination by combination rate constant, m3/kgmol/s

ktd : termination by disproportionation rate constant , m3/kgmol/s

ktm : rate constant for transfer to monomer, m3/kgmol/s

ktp : rate constant for transfer to polymer, m3/kgmol/s

kts : rate constant for transfer to solvent, m3fkgmol/s

L : reactor length, m

m : mass flow' kg/s

P : reactor operating pressure , MPa P :"live" polymer, consisting of p ethylene units , p,q,r

Qi Q p,q,r

q comonomer units , r long chain branches, ending

in an ethylene monomer unit :volumetric flow rate of the i stream, m3/s

: "live" polymer, consisting of p ethylene units,

q comonomer units, r long chain branches, ending

in a comonomer monomer unit

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R : universal gas constant, (J/gmoi/K)

R · : primary radicals T c : coolant temperature , K

T 0 : reactor feed temperature , K

u :fluid velocity, m/sec

w : weight fraction Y cum : cumulative monomer conversion .

yi : fractional conversion of the j component

z :dimensionless axial distance.

Greek letters

~E :activation energy, J/gmol (-~H)r: propagation reaction enthalpy , J/ kgmol

~ V : activation volume, m3/ gmol

o : Dirac impulse function

e : dimensionless temperature. Amn : moment of "live" polymer chains, kgmol/m3.

IJmn : moment of "dead" polymer chains, kgmol/m3.

p : reaction mixture density, kg/m3.

Subscripts

o : conditions at the inlet of the reaction zone c :coolant

di : "i " initiator

mi : " i " monomer si : "i " solvent

Superscripts

P : "live" polymer chains ending in an ethylene unit

Q : "live" polymer chains ending in a comonomer unit

: terminal double bond

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TUBULAR LDPE CO POLYMERIZATION REACTORS 459

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Date Rece i ved : Da t e Accep t ed :

Au g u s t 6, 1 99 2 February 23 , 1 993 D

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Mathematical Modeling of High Pressure Tubular LDPE Copolymerization Reactors

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