Modelling of free radical polymerisation of ethylene using difunctional initiators

Chemical Engineering Science 57 (2002) 2735 – 2746

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Modelling of free radical polymerisation of ethylene using difunctional initiators R. Dhib ∗ , N. Al-Nidawy Department of Chemical Engineering, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3 Received 22 August 2001; received in revised form 18 January 2002; accepted 4 March 2002

Abstract Low-density polyethylene (LDPE), commonly produced in high-pressure free radical polymerisation processes, is very important for the manufacture of engineering and commodity plastics. However, the thermodynamic conditions of the process hinder ethylene from going to full conversion. Other than recycling the product, one way of improving the monomer conversion is to initiate the polymerisation with difunctional organic peroxides. But, the kinetic decomposition of multi-=difunctional peroxides is still a controversial issue. The present study proposes a kinetic model based on a postulated reaction mechanism for free radical ethylene polymerisation initiated by difunctional initiators. The model describes the rates of initiation, propagation and the population balance equations in an isothermal autoclave reactor ◦ operated at a constant pressure of 1700 bars and a temperature range of 110 –300 C. The simulation con9rmed the trends of experimental data collected from literature for one monofunctional and two difunctional initiators. Due to the dual functionality of difunctional peroxide, the ethylene conversion obtained was about twice as much as that obtained with monofunctional peroxide, for only a half amount of the initial initiator concentration. The model also gives good prediction of the number average molecular weights data of the polymer and predicts reasonable values of the polydispersity index. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Kinetics; Population Balance; Modelling; Simulation; Ethylene polymerisation; Difunctional peroxides

1. Introduction Low-density polyethylene (LDPE) constituting a large tonnage of plastics is commercially manufactured in high-pressure processes. The technique requires a highly puri9ed ethylene in the feed, and the process must be operated at an elevated pressure in the range 1000 –3000 atmo◦ spheres and a temperature range of 120 –300 C. Tempera◦ tures exceeding 300 C cause ethylene to decompose and are not recommended in practice. The high-pressure process is usually a bulk polymerisation initiated by organic peroxide used as an initiator. Over the last few decades, a lengthy list of academicians and industrialists incessantly attempted to establish a uni9ed tangible understanding of ethylene polymerisation in high-pressure autoclave reactors (Goto, Yamamoto, Furui, & Sugimoto, 1981; Feucht, Tilger, & Luft, 1985; Lorenzini, Pons, & Villermaux, 1992a, b; Chan, Gloor, & Hamielec, 1993) and tubular reactors (Chen, Vermeychuk, Howell, & Ehrlich, 1976; Shirodkar & Tsien, ∗

Corresponding author. E-mail address: [email protected] (R. Dhib).

1986; Brandolin, Capiati, Farber, & Valles, 1988; Baltsas, Papadopoulos, & Kiparissides, 1998). The eCorts and interests in the studies were propelled by the industrial and commercial importance of LDPE for the economical growth in contemporary society. In spite of the recent, visible success of metallocene and ziegler-Natta catalysts in polymerising ole9ns at relatively low pressure and temperature conditions (Soares, Beigzadeh, Duever, & da Silva Filho, 2000; Wang, Yan, Zhu, & Hamielec, 1999), the industrial demand and usage of organic peroxides in the production of LDPE at elevated pressure remain substantially signi9cant. Profound comprehension of the reaction mechanisms and kinetic analysis was the core of the aforementioned studies. For an autoclave reactor, Feucht et al. (1985) presented a mathematical model for high-pressure ethylene polymerisation to predict the weight and number averages of weight of the polymer. The performance of their model was con9rmed by a good agreement with experimental data. Goto et al. (1981) and Chan et al. (1993) included in their models the contribution of the -scissions. Furthermore, the work of Ham and Rhee (1996) incorporated predictions of the monomer conversion, reaction temperature, polydispersity and weight

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 5 6 - 2

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R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

average of molecular weights. The reactor considered was divided into two compartments, each having two cells, to get simpli9ed equations for mass and energy balances. The eCects of various operation parameters on the reactor performance and the polymer properties were investigated systematically showing that the temperature distribution plays the central role in determining the properties of the polymer products. Villermaux and Blavier (1984) proposed a new method based on the concept of tendency model and the z-transform for modelling complex processes. Recently, Brandolin, Lacunza, Ugrin, and Capiati (1996) did an extensive review on the modelling of polyethylene processes and proposed a comprehensive model for an industrial tubular reactor. The model was proven useful in predicting reactor states and product properties over a wide range of operation conditions. In all these attempts, monofunctional initiators were used. However, the setback of obtaining only limited low conversion of ethylene in high-pressure polymerisation has been persistently an unpleasant and discouraging reality in spite of the classical approach to enhance the conversion upon recycling the product. Another route of improving ethylene conversion in this kind of polymerisation process is to investigate the eCectiveness of initiators. The chemistry and thermal behaviour of organic peroxides used as initiators in free radical polymerisation have been the focus of many researchers who were investigating potential routes to improve the eJciency of polymerisation reactors. Under thermal effects, organic peroxides exhibit diCerent rates of decomposition owing to their dissimilar half-life temperatures. For relatively low temperature polymerisation processes such as styrene and methyl methacrylates, a number of difunctional peroxides were tested and analysed, and consequently were proven to be eCective in increasing polymerisation rates (Kim & Choi, 1989; Villalobos, Hamielec, & Wood, 1991; Dhib, Gao, & Penlidis, 2000). Focusing on the chemistry and kinetics of certain organic peroxides, Luft and co-workers carried out a series of sound detailed studies on their kinetic decomposition, reaction rates, eCectiveness and eCects on polyethylene product (Luft, Bitsch, & Seidl, 1977; Seidl & Luft, 1981; Luft, Lim, Pavlaskis, & Seidl, 1985; Luft & Seidl, 1985; Luft & Dorn, 1988). The studies incorporated the setting-up of a database of the most common peroxide initiators in the production of polyethylene. Over 20 initiators have been used in experiments to produce LDPE polymers and the consumed amount of each initiator constituted the basic criterion of assessing the initiator eCectiveness. Continuing a previous experimental work (Luft & Seidl, 1985), Luft and Dorn (1988) studied the eCectiveness of four difunctional initiators: 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate, 2,5-dimethylhexane-2,5-bis-perpivalate, 2,2-bis(tert-butylperoxy)butane and 2,5-dimethylhexane-2, 5-di-t-butyl-peroxide, judged suitable for high-pressure polymerisation of ethylene at optimum temperatures of ◦ ◦ ◦ ◦ 273 C; 235 C; 268 C and 287 C, respectively. For in-

stance, Luft et al. (1985) studied thermal decomposition of the difunctional peroxide 2,2-bis(tert-butylperoxy)butane in a Low reactor and determined values for the rate of reaction for a pressure up to 2000 bar and a temperature range ◦ of 135 –212 C. The conversion rate, peroxide consumption, and product properties were analysed through experimental observations. A simpli9ed kinetic model was developed for a reaction mechanism based on monofunctional initiation, but the difunctionality of the initiators utilised was not accounted for. In comparison with monofunctional peroxides, difunctional initiators accelerate the polymerisation rate and produce polymers of higher molecular weight at high temperature. Besides, they can produce special polymers like star and hyperbranched polymers. Existing kinetic models for ethylene polymerisation with monofunctional initiators are not quite valid for data collected with di(multi)functional initiators. Consequently, the present study proposes a kinetic scheme and a comprehensive model for a thorough description of free radical polymerisation of ethylene catalysed by difunctional initiators. It is important to note that experimental data on polyethylene are limited. Nevertheless, enough data were collected from literature for testing the model performance. 2. Reaction mechanism Free radical ethylene polymerisation is carried out at high pressure and elevated temperature. The crucial thermodynamic setting in which the reaction should take place triggers a complicated chain of reactions. Even with a simple, monofunctional peroxide, literature shows a number of different kinetic models. For instance, the model of Brandolin et al. (1996) encompasses the most likely kinetic events in an industrial tubular reactor and takes into account the heat transfer and pulse valve eCects. For reaction temperatures ◦ ◦ between 150 C and 250 C, the eCects of thermal polymerisation become important and cannot be neglected. Transfer to non-saturated telogens such as propylene generates radicals having end double bonds, which favour long chain branching. However, under intense heat eCects, the growing polymer radicals may undergo -scission (thermal degradation) or breaking-up at the secondary or tertiary carbon atoms in the backbone making dead polymer chains with low molecular weights, thus aCecting the polymer properties. Besides, under these severe thermal conditions, organic peroxides—especially those with more than one oxygen– oxygen bond—may become unstable and undergo undesirable reactions, which are not easily predictable (Luft et al., 1985). Before presenting the overall kinetic mechanism, it would be helpful to review brieLy the decomposition of two types of peroxide-based initiators. 2.1. Monofunctional initiation Monofunctional peroxides such as benzoyl and dioctanoyl have one oxygen–oxygen bond separating two identical or

R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

diCerent organic groups. Under the eCect of heat, the oxygen bond breaks up giving rise to two possible free radicals: K

d 2R1 − O• ; R1 – O – O – R1 −→

(1)

where R1 is an organic ligand and Kd is the rate constant of the initiator decomposition. Initiators with this structure are widely used in practice. 2.2. Difunctional initiation

Kd

R1 – OO – R2 – OO – R1 2Kd

(3)

This decomposition alternative still depends on the stability of the initiator bridge. Now, postulating that the second radical is relatively stable enough for not undergoing further decomposition, scheme (3) is therefore generalised as 2Kd

•

• I −→1 Rin + R˜ in ;

• • + R˜ in ; I −→ Rin 1 • + M −→ R1• ; Rin

• • K2 R˜ 1 ; R˜ in + M −→ Kd2 • Rin + Rr• P˜r −→

r ¿ 2;

• 2Kd2 • Rin + R˜ r P˜˜r −→

r ¿ 2:

Thermal initiation: K

th 3M −→ 2R1• :

Kp

(2)

However, there is no enough evidence to support this reaction scheme. Nevertheless, two investigations (Luft & Seidl, 1985; Villalobos et al., 1991) using independent approaches indicated that an eventual formation of diradicals • O–R 2 –O• is unlikely to occur. Therefore, the oxygen–oxygen bonds generate two diCerent free radical fragments:

−→1 R1 – O• + R1 – OO – R2 – O• :

Chemical initiation by difunctional peroxide: 2Kd1

Propagation:

R1 – OO – R2 – OO – R1 1 −→ 2R1 – O• + • O – R 2 – O• :

below, considering the assumptions: no diradical formation; chain transfer to telogens (propylene) is negligible; termination occurs by combination (coupling) only. Let M, I and P stand for the monomer (ethylene), difunctional initiator and the polymer (polyethylene), respectively.

K

The presence of two or more oxygen–oxygen bonds in multi-functional peroxides makes their kinetic decomposition into radical fragments quite particular. The splitting of oxygen bonds into free radicals may take several routes. For instance, for symmetrical peroxides in a high pressure and temperature process environment, Luft et al. (1985) investigated several schemes of decomposition into radical fragments producing hydrocarbons. According to Kim and Choi (1989) and Kim, Liang, and Choi (1989), there may be a formation of a diradical • O–R 2 –O• from (un-) symmetrical peroxides as

(4)

• • where Rin is a primary initiator radical, R˜ in is a second initiator radical with one undecomposed peroxide and Kd1 is the rate constant of initiator decomposition. Both initiator • • radicals Rin and R˜ in participate in the polymerisation process and are able of generating independent polymer chains. The polymer chain associated with the undecomposed peroxide • radical R˜ in behaves as a ‘macro’-peroxide radical and may subsequently contribute to the initiation step. Reaction (4) is selected in this study.

2.3. Reaction mechanism The reaction mechanism for bulk free radical polymerisation of ethylene using difunctional initiators is proposed

2737

• Rr• + M −→ Rr+1

r ¿ 1;

Kp • • R˜ r + M −→ R˜ r+1

r ¿ 1:

Termination: K

tc Rr• + Rs• −→ Pr+s

• K

r; s ¿ 1;

tc P˜r+s Rr• + R˜ s −→

r; s ¿ 1;

• • Ktc ˜ P˜r+s R˜ r + R˜ s −→

r; s ¿ 1:

Transfer to monomer: K

tfm Rr• + M −→ Pr + R1•

r ¿ 1;

• Ktfm P˜r + R1• R˜ r + M −→

r ¿ 1:

-scission to secondary radicals: K

• Rr+1 −→ Pr + R1•

r ¿ 1;

K • R˜ r+1 −→ P˜r + R1•

r ¿ 1:

Intermolecular transfer (Backbiting): K

b • Rr• −→ Rbr

r ¿ 1;

• Kb • R˜ br R˜ r −→

r ¿ 1:

Transfer to polymer: Kfp

Rr• + Ps −→ Pr + Rs•

r; s ¿ 1;

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R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

Rr• + P˜s −→ P˜r + Rs•

r; s ¿ 1;

where RI and R˜ I are the initiation rates de9ned as, respectively:

Kfp • • R˜ r + Ps −→ Pr + R˜ s

r; s ¿ 1;

RI = 2Kth [M]3 + 2f1 Kd1 [I] + f2 Kd2 (˜0 + 2˜˜0 );

(11)

Kfp • • R˜ r + P˜s −→ P˜r + R˜ s

r; s ¿ 1;

R˜ I = 2f1 Kd1 [I]:

(12)

Kfp

• where Rr• (r ¿ 1) is the regular radical, R˜ r (r ¿ 1) the macro-radical with one undecomposed peroxide, Pr the ˜ dead poly(R − R) dead polymer molecule, P˜r the (R − R) mer molecule with one undecomposed peroxide and P˜˜r the ˜ dead polymer molecule with two undecomposed (R˜ − R) peroxides. The remaining symbols are listed in the notation. Note that a factor 2 multiplies the decomposition rate constant Kd1 of the primary difunctional initiator I and Kd2 of macro-peroxide P˜˜r . This factor stands for the dual functionality of peroxides in I and P˜˜r .

3. Dynamic model of an autoclave CSTR reactor A dynamic model describing the kinetics of the polymerisation process was developed from molar balance of all the species involved in the mechanism; mainly the monomer, initiator, live and dead polymer radicals. 3.1. Live radical molar concentrations Performing radical molar balances on the growing tem• porary radicals Rr• and R˜ r leads to a set of diCerential equations for their moments i and ˜i (i =0; 1; 2) that are de9ned in Appendix A. In general, a steady-state hypothesis for i and ˜i is assumed yielding the set of algebraic equations: R˜ I + 2f2 Kd2 ˜˜0 ˜0 = ; (5) Ktfm [M] + K + Kb + Ktc to

An expression for a direct computation of the overall molar concentration of the live polymer radicals is shown as
RI + R˜ I + f2 Kd2 (˜0 + 2˜˜0 ) to = (13) Ktc which allows a straightforward analytical computation of the steady-state radical concentrations of live polymer chains • Rr• and R˜ r and the 9rst and second moments in the order written above. Initiator eJciencies f1 and f2 are introduced to account for the fraction of initiator amount eCectively contributing to the polymerisation. The overall live radical concentration to (13) determines the rate of polymerisation Rp (16) which is required in the reactor model (14) and (15), whereas the individual moments 0 (t); 1 (t); 2 (t); ˜0 (t); ˜1 (t) and ˜2 (t) are required in the polymer model (17) – (25). 3.2. Continuous reactor model As the reaction proceeds, the monomer and the initiator are continuously consumed: d([I]) = −2Kd1 [I] + q([I]f − [I])=V; dt

(14)

d([M]) = −Rp [M] + q([M]f − [M])=V; dt

(15)

where Rp is the rate of polymerisation Rp = Kp to :

It is assumed that the reactor volume, temperature and pressure remain invariant.

0 =

RI + (Ktfm [M] + K + Kb )˜0 + f2 Kd2 ˜0 ; Ktc to

1 =

RI + (Ktfm [M] + K + Kb )to + f2 Kd2 ˜1 + Kp [M]0 + Kfp (2 + ˜2 )0 ; Ktc to + Ktfm [M] + K + Kb + Kfp (1 + ˜1 )

R˜ I + 2f2 Kd2 ˜˜1 + Kp [M]˜0 + Kfp (2 + ˜2 )˜0 ˜1 = ; Ktc to + Ktfm [M] + K + Kb + Kfp (1 + ˜1 ) 2 =

(6)

(8)

RI + f2 Kd2 ˜2 + Kp [M](21 + 0 ) + Kfp (3 + ˜3 )0 ; Ktc to + Ktfm [M] + K + Kb + Kfp (1 + ˜1 ) (9)

R˜ I + 2f2 Kd2 ˜˜2 + Kp [M](2˜1 + ˜0 ) + Kfp (3 + ˜3 )˜0 ; ˜2 = Ktc to + Ktfm [M] + K + Kb + Kfp (1 + ˜1 ) (10)

(16)

(7)

3.3. Polymer model De9ning the moments of each polymer chain as in the appendix and writing molar balances for each polymeric ˜˜ a polymer model was species concentration P, P˜ and P, developed and it is written as follows: Dead polymer P: d(0 ) = Ktc 02 =2 + (Ktfm [M] + K + Kb )0 − q0 =V; dt (17)

R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

Conversion

d(1 ) = Ktc 0 1 + (Ktfm [M] + K + Kb )1 dt +Kfp ((1 + ˜1 )1 − to 2 ) − q1 =V;

(18)

(19)

˜ Temporary polymer P:

Number average molecular weight:

= (21)

+(Ktfm [M] + K + Kb )˜2 (22)

˜˜ Temporary polymer P: 2 d(˜˜0 ) = Ktc ˜0 − 2Kd2 ˜˜0 ; dt

(23)

d(˜˜1 ) = Ktc ˜0 ˜1 − 2Kd2 ˜˜1 ; dt

(24)

2 d(˜˜2 ) = Ktc (˜0 ˜2 + ˜1 ) − 2Kd2 ˜˜2 : dt

(25)

It is assumed that there is no outLow of polymer chains with two undecomposed peroxides. The expressions for the second moments of the live and dead polymers include a term of the third moment 3 making them open-ended. To close oC open-ended moment equations, Hulburt and Katz (1964) solved the problem of moment closure by approximating higher moments using Laguerre functions about -distributions and showed that: 2 3 = (20 2 − 12 ): (26) 0 1 Similarly, we deduce: ˜2 (2˜0 ˜2 − ˜21 ): ˜0 ˜1

1

1total (t) MU n (t) = Mwm 0total (t)

d(˜2 ) = Ktc (2 ˜0 + 21 ˜1 + 0 ˜2 ) − Kd2 ˜2 dt

˜3 =

Mwm {2 (t) + ˜2 (t) + ˜˜2 (t) + 2 (t) + ˜2 (t)} : 1 (t) + ˜ (t) + ˜˜ (t) + 1 (t) + ˜1 (t) (29)

(20)

+(Ktfm [M] + K + Kb )˜1

+Kfp ((2 + ˜2 )˜1 − to ˜3 ) − q˜2 =V:

(28)

2total (t) MU w (t) = Mwm 1total (t)

1

d(˜1 ) = Ktc (0 ˜1 + 1 ˜0 ) − Kd2 ˜1 dt +Kfp ((1 + ˜1 )˜1 − to ˜2 ) − q˜1 =V;

[M]0 − [M](t) ; [M]0

=

d(˜0 ) = Ktc 0 ˜0 − Kd2 ˜0 + (Ktfm [M] + K + Kb )˜0 dt −q˜0 =V;

X (t) =

where [M]0 is the monomer initial concentration. Weight average molecular weight:

d(2 ) = Ktc (0 2 + 12 ) + (Ktfm [M] + K + Kb )2 dt +Kfp ((2 + ˜2 )1 − to 3 ) − q2 =V:

2739

(27)

Integration of the kinetic model allows the computation of monomer conversion and molecular weight averages of the polymer as expressed below.

Mwm {1 (t) + ˜1 (t) + ˜˜1 (t) + 1 (t) + ˜1 (t)} ; 0 (t) + ˜0 (t) + ˜˜0 (t) + 0 (t) + ˜0 (t) (30)

where Mwm is the monomer molecular weight. The diCerential equations (20) – (25) describe the moments of the polymer chains grown out of the undecomposed peroxide and therefore are only speci9c to the difunctionality of the initiator. For ethylene polymerisation initiated by a monofunctional initiator, the kinetic model remains valid provided that f2 = 0; R˜ I = 0; ˜0 = 0 (i.e. 0 = to ) and Kd2 = 0, leading to retain only the 9rst three equations (17) – (19). In this case, the rate of initiator de-fragmentation is given as d([I]) = −Kd1 [I] + q([I]f − [I])=V: dt

(31)

Throughout the model development, it is assumed that the reactor is exceptionally maintained under a good control and therefore operates under isothermal and isobaric conditions. In regard to the controversial issue over the decomposition of organic initiators and the postulates of their kinetic schemes, the eCectiveness of initiators is still debatable. In particular, Luft et al. (1985) suggested few routes for the decomposition of the difunctional peroxide 2,2-bis(tert-butylperoxy)butane which is partially converted into few liquids (acetone, tert-butyl alcohol, tert-butyl methyl ether) and gases (methane, ethane, propane, ethylene). It is therefore understood that the decomposition of difunctional peroxide is not all dedicated to the initiation of ethylene polymerisation but also to the formation of other unwanted products. Thus, variable and probably low initiator eJciency is expected. For ethylene polymerisation, past experimental work showed that optimum monomer conversion was a function of initiator type. Therefore, different initiators, because of their dissimilarities in kinetic decomposition, are usually fed at diCerent points along a tubular reactor or between autoclave reactors in series for enhancing the reaction performance.

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R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

Table 1 Reaction rate constants in ethylene polymerisation

Rate constant Kth Kp Ktc Ktfm K Kfp Kb

Goto et al. (1981) K = A exp(−E=R1 T − VvP=R2 T )

Brandolin et al. (1996) K = A exp(−E=R1 T − VvP=R2 T )

A (1=s)

E (cal=mol)

Vv (cm3 = mol)

A (1=s)

E (cal=mol)

— 5.63e11 3.00e11 — 5.82e11 1.75e12 5.85e11

— 10,520.00 3000.00 — 15,760.00 14,080.00 13,030.00

— −17.91 12.61 — 21.86 −4.26 22.73

5.97e7 4.01e5 8.71e8 1.20e5 4.40e9 7.60e9 1.20e10

68,080.00 4210.00 3652.62 14,400.00 19,100.00 19,100.00 14,800.00

This study K = A exp(−E=R1 T − VvP=R2 T )

Vv (cm3 = mol) 0.00

−16.80

9.21 0.00 −9.90 −10.00 0.00

Vv (cm3 = mol)

A (1=s)

E (cal=mol)

6.04e3 5.45e5 4.35e8 1.20e5 1.40e9 1.80e8 3.250e5

72,945.00 4210.00 3650.00 14,400.00 19,300.00 9400.00 7474.00

0.00

−5.60

9.20

−20.00

9.90 0.00 0.00

R1 = 1:98 cal= mol K : R2 = 82 cm3 atm= mol K . Table 2 Decomposition rate constants of peroxide initiators

Initiator

Dioctanoyla peroxide 2,2-bis(tert-butylperoxy)-butane 2,5-dimethyl hexane-2-tbutylperoxy-5-perpivalate

This study Kd1 = A1 exp(−E1 =R1 T − Vv1 P=R2 T ) Kd2 = A2 exp(−E2 =R1 T − Vv2 P=R2 T )

Literature Kd1 = A1 exp(−E1 =R1 T − Vv1 P=R2 T ) Kd2 = A2 exp(−E2 =R1 T − Vv2 P=R2 T )

A 1 ; A2 (1=s) 2.2925E14 0 1.8120E16 6.0400E13 1.8120E16 6.0400E17

A 1 ; A2 (1=s) 9.98E14b

E 1 ; E2 (cal=mol) 30,861.2b

Vv1 ; Vv2 (cm3 = mol) 5.9b

3.28E16c — — —

36,886c

22.3c — — —

E 1 ; E2 (cal=mol) 30,131.70 0 42,307.83 41,943.11 43,037.28 43,766.72

Vv1 ; Vv2 (cm3 = mol) 5.9 0 28.0 22.3 25.0 22.3

— —

R1 = 1:98 cal= mol K : R2 = 82 cm3 atm= mol K . a A = E = Vv = 0. 2 2 2 b Seidl and Luft (1981); for 110 –330◦ C and 1–3000 bar c Luft et al. (1985); for 135 –212◦ C and up to 2000 bar , no values for K . 2

4. Simulation results and discussions The proposed kinetic model for free radical ethylene polymerisation initiated by a difunctional peroxide in an isothermal autoclave reactor, regarded as perfectly mixed, describes the rates of initiation, propagation, and the moments of live and dead polymers. The model was integrated using Runge–Kutta–Fehlberg algorithm in Matlab environment. For using the data from the experimental setting by Seidl and Luft (1981), a small reactor of 15 mL operated at a 1700 bar pressure was considered in the simulation. Ethylene conversion data versus reaction temperature are available for a given residence time. For each simulation test, the reactor was assumed isothermal and maintained at constant pressure. As discussed in Brandolin et al. (1996), values of kinetic parameters vary signi9cantly in the literature. In this study, some of the rate constants of propagation, termination and transfer to monomer were re-adjusted and are tabulated in Table 1 with other values from literature. Values of the kinetic parameters for the initiators dioctanoyl and 2,2-bis(tert-butylperoxy)butane reported, respectively, in Seidl and Luft (1981) and Luft et al. (1985) were slightly modi9ed; whereas the kinetic parameters for 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate were estimated from the data (Table 2).

Fig. 1 shows the evolution of ethylene conversion versus time for diCerent temperatures. Conversion pro9les ◦ rise for increasing temperatures up to 150 C. For higher temperatures, the polymerisation rate is very steep at the beginning and then slows down to reach a 9nal conversion ◦ value lower than that obtained with 150 C. This trend is better displayed by plots of conversion versus temperature as shown in subsequent graphs. Fig. 2 shows ethylene conversion versus reaction temperature for a residence time of 65 s. Polymerisation was carried out at diCerent tempera◦ ◦ tures from 110 C to 220 C and triggered by dioctanoyl, a monofunctional initiator, with an initial concentration of 41 mol ppm. Ethylene conversion remains very low for ◦ moderate reaction temperature around 110 C, then accelerates quickly until a maximum value of 15% was reached ◦ corresponding to approximately a temperature of 165 C above which the conversion starts dropping gradually. High temperatures favour the side reactions, which lower the number of primary radicals, thus hindering the initiation of polymer chains. This trend was observed experimentally and con9rmed by the model in this study. The model follows adequately the data trend but predicts an optimum ◦ conversion at a temperature slightly lower than 160 C, then it follows the data very closely for higher temperatures after the peak. For difunctional initiators used in the produc-

R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

2741

16.0 T=125 C T=150 C T=175 C T=200 C T=225 C

14.0

Conversion (%)

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0

5

10

15

20

25

30

35 40 Time (s)

45

50

55

60

65

70

Fig. 1. Predictions of conversion versus time for diCerent temperatures. Ethylene bulk polymerisation using Dioctanoyl with initial concentration of 41 mol ppm; P = 1700 bar; !r = 65 s.

16

Ethylene Conversion (%)

14 12 10 8 6 4 2

Model Predictions Experimental Data

0 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 Temperature (˚C) Fig. 2. Predictions and data conversion versus temperature. Ethylene bulk polymerisation using Dioctanoyl with initial concentration of 41 mol ppm; P = 1700 bar; !r = 65 s, (Data: Luft et al., 1977).

tion of polyethylene in high-pressure processes, the only experimental work existing in the literature was that reported by Luft and Seidl (1985) and Luft and Dorn (1988). Data were collected for validating the model developed in this study. For instance, Fig. 3 shows an ethylene polymerisation run carried out at 1700 bar over a temperature ◦ ◦ range from 220 C to 295 C using the difunctional initiator 2,2-bis(tert-butylperoxy)butane with a concentration of 10 mol ppm. Simulated monomer conversion reaches a ◦ maximum of 16% at 265 C and then drops gradually for higher temperatures. The model gives a satisfactory pre◦ diction of the data except for temperature less than 250 C where it visibly drifts away. In comparison with dioctanoyl peroxide, only a quarter of 2,2-bis(tert-butylperoxy)butane

concentration is required to get similar maximum ethylene conversion, but at the expense of higher temperature. For similar conditions, Luft and Seidl (1985) repeated the experiment using a 20 mol ppm concentration of 2,2-bis(tert-butylperoxy)butane for 40 s residence time. As in Fig. 4, the model follows closely the conversion data ◦ trend as it rises to a maximum value of nearly 20% at 265 C and then falls down. Luft and Dorn (1988) conducted few more experimental runs on ethylene polymerisation in a continuous autoclave reactor at a constant pressure of 1700 bar and collected ethylene conversion data for a residence time of 60 s. The reaction was accomplished with another difunctional initiator [2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate] of

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20 18

Ethylene Conversion (%)

16 14 12 10 8

Experimental Data Model Predictions

6 4 2 0 200

210

220

230

240 250 260 Temperature (˚C)

270

280

290

300

Fig. 3. Predictions and data conversion versus temperature. Ethylene bulk polymerisation using 2,2-bis (tert-butylperoxy)-butane with initial concentration of 10 mol ppm; P = 1700 bar; !r = 40 s (Data: Luft & Seidl, 1985).

25

Ethylene Conversion (%)

20

15

10

Experimental Data 5

Model Predictions

0 200

210

220

230

240 250 260 Temperature (˚C)

270

280

290

300

Fig. 4. Predictions and data conversion versus temperature. Ethylene bulk polymerisation using 2,2-bis (tert-butylperoxy)-butane with initial concentration of 20 mol ppm; P = 1700 bar; !r = 40 s, (Data: Luft & Seidl, 1985).

concentrations of 25, 10 and 5 mol ppm. With the initial concentration of 25 mol ppm, a maximum ethylene conversion ◦ around 30% was obtained at just below 270 C as in Fig. 5. In this case, the kinetic model gives a good prediction of the data. Fig. 6 shows two more runs with 10 and 5 mol ppm. The model performs quite well for the higher initial initiator concentration, but does not agree with low temperature data obtained with the 5 mol ppm concentration. Again, in comparison with dioctanoyl peroxide, only a quarter of 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate concentration is required to yield twice the maximum ethylene conversion.

This study con9rms the experimental observations in the literature and shows that for similar experimental conditions, the latter difunctional initiator gave higher conversion than the former difunctional initiator. The initiator eJciency is a very signi9cant factor inLuencing the results of the simulation and was estimated in each experimental run. In Fig. 7, the study shows that the initiator eJciency decrease linearly with initial initiator concentration regardless of the initiator type. Kim et al. (1989) reported similar observation. For further testing of the model, the average molecular weights MU w and MU n were predicted and are plotted in Fig. 8 along with MU n data obtained with 20 mol ppm for each difunc-

R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

2743

35

Ethylene Conversion (%)

30 25 20 15

Experimental Data 10

Model Predictions

5 0 200

210

220

230

240 250 260 Temperature (˚C)

270

280

290

300

Fig. 5. Predictions and data conversion versus temperature. Ethylene bulk polymerisation using 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate with initial concentration of 25 mol ppm; P = 1700 bar; !r = 60 s, (Data: Luft & Dorn, 1988).

35

Ethylene Conversion (%)

30 25 20 15 Experimental Data, [I]o = 10 ppm

10

Model Predictions, [I]o = 10 ppm Experimental Data, [I]o = 5 ppm

5

Model Predictions, [I]o = 5 ppm 0 200

210

220

230

240

250

260

270

280

290

300

Temperature (˚C) Fig. 6. Predictions and data conversion versus temperature. Ethylene bulk polymerisation using 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate with initial concentration of 10 and 5 mol ppm; P = 1700 bar; !r = 60 s, (Data: Luft & Dorn, 1988).

tional initiator. For the same simulation run, the polydispersity index PDI being the ratio MU w = MU n is plotted in Fig. 9 displaying the variation of PDI with temperature. 5. Concluding remarks A mathematical model for free radical ethylene polymerisation initiated by a difunctional initiator in an autoclave reactor, regarded as perfectly mixed and maintained at constant pressure and temperature, was formulated to describe the rates of initiation, propagation, and the moments of live and dead polymers. The model was tested against conver-

sion data of ethylene collected from literature for one monofunctional initiator and two difunctional initiators. Due to the dual functionality of difunctional peroxide, the ethylene conversion obtained was about twice as much as that obtained with monofunctional peroxide, for only a half amount of the initial initiator concentration. Peroxides are quite expensive and it is more economical to use difunctional initiators than monofunctional ones. In spite of the fact that the decomposition of difunctional peroxide cannot be predicted easily due to the severe thermal conditions of the process, the mechanism adopted herein led to a model that succeeded well in predicting experimental data trend and estimated a polydipersity of 2-3 for polyethylene.

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1 Difunctional # 2 Difunctional # 1 Monofunctional Linear Regression Model

0.9 0.8 Initiator Efficiency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15

20

25

30

35

40

45

50

Initiator Concentration (Mol ppm) Fig. 7. Initiator eJciency (f1 = f2 ) versus initial initiator concentration.

Average Molecular Weights: Mn and Mw

1.8E+05

Data of Mn, Dif Init #1 Data of Mn, Dif Init #2 Predictions of Mw, Dif Init #1 Predictions of Mw, Dif Init #2 Predictions of Mn, Dif Init #1 Predictions of Mn, Dif Init #2

1.6E+05 1.4E+05 1.2E+05 1.0E+05 8.0E+04 6.0E+04 4.0E+04 2.0E+04 0.0E+00 200

220

240

260 Temperature (˚C)

280

300

320

Fig. 8. MU w and MU n predictions and MU n data versus temperature for ethylene bulk polymerisation. For initiators 2,2-bis (tert-butylperoxy)- butane (Init #1) and with 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate (Init #2) each with [I]0 = 20 mol ppm; P = 1700 bar; !r = 40 s (Data: Luft & Seidl, 1985).

Notation f1 f2 I• [I] Kb K Kd Kd 1 Kd 2

eJciency of original initiator eJciency of initiator with undecomposed peroxide initiator radical initiator concentration, mol=L rate constant for backbiting, s−1 rate constant for -scission, s−1 decomposition rate constant of monofunctional peroxide, s−1 decomposition rate constant of original difunctional peroxide, s−1 decomposition rate constant of peroxide with undecomposed radical, s−1

Kp Kfp Ktc Kth Ktfm [M] Mw MU w (t) MU n (t) [Pr ] [P˜r ] [P˜˜r ]

rate constant for propagation, L=mol s rate constant transfer to polymer, L=mol s rate constant for termination, L=mol s rate constant for thermal initiation, L2 =mol2 s rate constant for transfer to monomer, L=mol s monomer concentration, mol=L molecular weight weight average molecular weight number average molecular weight dead polymer concentration, mol=L dead polymer concentration with one undecomposed peroxide, mol=L dead polymer concentration with two undecomposed peroxides, mol=L

R. Dhib, N. Al-Nidawy / Chemical Engineering Science 57 (2002) 2735–2746

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3.4 Predictions, Dif Init #1 Polydispersity Index (PDI)

3.2

Predictions, Dif Init #2

3.0 2.8 2.6 2.4 2.2 2.0 210

220

230

240

250 260 Temperature (˚C)

270

280

290

300

Fig. 9. Predictions of polydispersity index of LDPE versus temperature. For initiators 2,2-bis (tert-butylperoxy)-butane (Init #1) and with 2,5-dimethylhexane-2-t-butylperoxy-5-perpivalate (Init #2) each with [I]0 = 20 mol ppm; P = 1700 bar; !r = 40 s.

q RI R˜ I • Rin • ˜ R in Rr• Rs• • R˜ r • R˜ s

Rp t T V X

total Low rate of reactants in a CSTR, L=s rate of initiation without undecomposed peroxide rate of initiation with undecomposed peroxide primary initiator radical fragment initiator radical fragment with one undecomposed peroxide radical of chain length r radical of chain length s macroradical fragment of chain length r, with one undecomposed peroxide macroradical fragment of chain length s, with one undecomposed peroxide rate of polymerisation, mol=s L time of integration, s reaction temperature, K reactor volume, L monomer conversion

Subscripts  f m p r

-scission feed monomer polymer chain length

Greek letters i ˜i

moments of live temporary polymer radical (i = 0; 1; 2) moments of live temporary polymer radical with one undecomposed peroxide (i = 0; 1; 2)

[to ] i ˜i ˜˜i !r

total concentration of radicals, mol=L moments of P (i = 0; 1; 2; 3) moments of P˜ (i = 0; 1; 2; 3) moments of P˜˜ (i = 0; 1; 2) residence time, s

Acknowledgements The authors would like to thank the Natural Science and Engineering Research Council of Canada for the 9nancial support. Appendix A. Moments of polymer species Polymer species consist of the growing radical chains, dead polymer chains, and two diCerent polymer chains with one and two undecomposed peroxides. Concentration of the temporary polymer radicals: ∞  i = r i [Rr• ]; (A.1) r=1

˜i =

∞ 

•

r i [R˜ r ];

(A.2)

r=1

where the superscript i is 0; 1 or 2 for zeroth, 9rst and second moment, respectively. The total concentration of radical species includes radicals with the undecomposed peroxide group and radicals without any undecomposed peroxide group. ∞ ∞   • [R•r ] + [R˜ r ]: (A.3) to = 0 + ˜0 ≡ r=1

r=1

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Moments of dead polymers: ∞  i = r i [Pr ]:

(A.4)

r=1

Moments of dead polymers related to the undecomposed ˜ and P˜˜r : (R˜ − R), ˜ respectively: peroxide i.e. P˜r : (R − R) ∞  ˜i = r i [P˜r ]; (A.5) r=1

˜˜i =

∞ 

r i [P˜˜r ]:

(A.6)

r=1

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