Mathematical Modeling of the Slurry Polymerization of Ethylene: Gas

2322

Ind. Eng. Chem. Res. 1994,33, 2322-2330

Mathematical Modeling of the Slurry Polymerization of Ethylene: Gas-Liquid Mass Transfer Limitations Mahalaxmi S. Bhagwat, Sunil S. Bhagwat, and Man Mohan Sharma' Department of Chemical Technology, University of Bombay, Matunga, Bombay, 400 019, India

A mathematical model for the isothermal, slurry polymerization of ethylene using solid ZieglerNatta catalysts is developed, accounting for the effect of gas-liquid mass transfer limitations on the overall rate and polymer properties, such as molecular weight and polydispersity index. The existence of micron-sized catalyst particles in the initial stages of polymerization influences the final polymer properties and also leads t o the diffusion and reaction steps being in parallel rather than in series. T h e gas-liquid interfacial contribution to the polymer formation is shown t o significantly enhance the value of the polydispersity index. Higbie's penetration theory has been satisfactorily employed t o model the monomer absorption in the presence of growing polymer macroparticles. High polydispersities are shown to arise in the presence of gas-liquid mass transfer limitations, even when only one type of active catalyst site is considered, and when the intra-macroparticular diffusional resistance is low. Introduction The heterogeneous polymerization of olefins (such as ethylene and propylene) by Ziegler-Natta catalysts is of enormous commercial significance and has attracted a considerable amount of research interest over the past few decades. However, some issues continue to remain unexplained, with only a partial understanding of the complex operating mechanisms. A major bone of contention has been a physical explanation for the extremely broad molecular weight distributions (MWDs) encountered in practice, especially in the case of polyethylene, since the polymer properties, both rheological and mechanical, depend critically on the same. The breadth of a MWD is most frequently characterized by its polydispersity (denoted by Q), which is a measure of the inhomogeneity of the distribution, and is defined as the ratio of the weight-average to the number-average molecular weight. The origin of the inordinately broad MWDs in polyethylene continues to be a source of considerable speculation, with two broad schools of thought emerging over the years. The observed high values of polydispersity (8) typically range between 5 and 30 in the case of polyethylene (Buls and Higgins, 1970a,b;Crabtree et al., 1973;Zucchini and Cecchin, 1983). Several authors have proposed models to describe the polymerization process and have attributed the broad distributions to physical factors (diffusional limitations) (Begley, 1966;Schmeal and Street, 1971;Singh and Merrill, 1971), as well as to chemical factors (such as site multiplicity, deactivation, etc.) (de Carvalho e t al., 1989; Villermaux et al., 1989; Lorenzini e t al., 1991). Ray and co-workers (Choi and Ray, 1985;Floyd e t al., 1986a,b, 1988;Hutchinson e t al., 1992) and, more recently, Gupta and co-workers (Sarkar and Gupta, 1991, 1992a,b; Sau and Gupta, 1993)have, in a series of publications, employed the multigrain approach, as first suggested by Yermakov et al. (1970) to model the polymerization process. Galvan and Tirrell(1986), for the first time, considered the possibility of diffusion resistance coupled with catalyst sites having differing activities, and showed that fairly high polydispersities (around 15) can be accounted for with two types of catalyst sites, even when intraparticular diffusional limitations are small, provided that the ratios

* Author to whom correspondence should be addressed.

between the activity of the two sites are substantially large. Floyd et al. (1988) have also demonstrated that, with two types of sites, the average molecular weights for each population must differ by at least an order of magnitude to obtain a high polydispersity and that the latter is maximum when both types of sites produce equal weights of the polymer. Most of the authors conclude that broad MWDs produced by heterogeneous Ziegler-Natta catalysts are most likely due to the presence of multiple active catalyst sites, although, in certain instances, substantially high levels of diffusional resistance could also play a role. However, while diffusional limitations have been considered only for the transfer of monomer through the polymer grain, the effect of mass transfer limitations at the gasliquid interface have been largely ignored. Floyd et al. (1986b) have addressed the question of gas-liquid mass transfer limitations arising in industrial slurry polymerization reactors, where the speed of agitation is restricted by limitations of the power consumption per unit volume. Reichert et al. (1986) have also considered the effect of mass transfer near the gas-liquid interface on the rate of polymerization. In all these cases, however, gas-liquid mass transfer has always been considered to be a step in series with the monomer uptake by the solid particles. However, in reality, the presence of growing polymer particles near the gas-liquid interface implies that monomer uptake should be a step in parallel with the gasliquid mass transfer step. The impact of the parallel nature of this diffusional limitation on the polymer particles has not been studied so far, although the treatment of the same for reacting systems by steady and unsteady state theories of mass transfer is well documented (Danckwerts, 1970;Doraiswamy and Sharma, 1984). In the present work, a model has been developed to describe the polymerization of ethylene in an isothermal, slurry reactor, and to rationalize the effects of gas-liquid mass transfer resistance, in the presence of micron-sized particles, on the polydispersity and other polymer properties.

Model Development In the typical slurry polymerization of ethylene catalyzed by solid Ziegler-Natta catalysts, one common observation is the disintegration or fracture of the initial catalyst particles into smaller fragments (referred to henceforth

0888-5885/94/2633-2322$04.50/00 1994 American Chemical Society

Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2323

i o 0

O

Crn

0

0

Scheme 1. Kinetic Scheme for Reaction of the Monomer at the Surface of a Catalyst Micrograin

C

0

0

0 ,Depth

of liquid element-

I I

! I

Pn

t

ki

PI

Inititation

Po t

s,

Termination by hydrogen

Po t

s,

Spontaneous termination

PI

i-

S,

Transfer to monomer

P,

t

s,

Transfer to dead polymer

ktB

Hz

, J

I

Figure 1. Slurry polymerization of olefins: physical picture.

as "micrograins"), during the very initial stages of polymerization (Hock, 1966). These catalyst micrograins constitute the loci of polymerization which proceeds by a coordination mechanism on the surface of these micrograins (Webb et al., 1991). Polymer growth occurs radially outward via encapsulation of these micrograins, each forming a polymer "microparticle". All microparticles originating from a single catalyst particle, however, continue to adhere loosely to each other, resulting in a single catalyst particle ultimately giving a single "macroparticle" or polymer bead at the end of the polymerization. This is the so-called phenomenon of replication, characteristic of solid-catalyzed Ziegler-Natta polymerization, both at the micro- and macroparticular levels. Another very significant observation reported is that the microparticular sizes throughout a single large macroparticle are uniform, even at the end of polymerization-as evident from transmission electron micrographs of polymer beads taken at different sections during various stages of polymerization (Kakugo et al., 1989). The gaseous monomer encounters four levels of diffusional resistance before finally reacting at an active site on the surface of a catalyst micrograin, resulting in the polymer. In the absence of gas-phase mass transfer limitations, these arise (1)at the gas-liquid interface, (2) through the liquid film surrounding each polymer macroparticle, (3) through the macroparticular interstices, and (4) finally through the polymeric shell encapsulating the individual micrograin. Each subsequent level acts as a "sink" for the previous levels, and may be in series or in parallel. For microparticular transport, the surface reaction is the "sink" term. Steps 1and 2 can be in series as well as in parallel; steps 2 and 3 are always in series, while 3 and 4 are always in parallel. Step 4 is in series with the step of reaction of the

>

kta

P,

POLYMER BEAD ( MACROPARTICLE)

I

+ M,

Mk

I

Depth of penetration

Po

Pn

t

M,

P,

t

s,

ktm

A

ktc

monomer a t the micrograin surface. The physical picture is schematically depicted in Figure 1. The multigrain model (which considers catalyst fragmentation as well as transport at micro- and macroparticular levels) as described by Ray and co-workers (Hutchinson et al., 1992)has been adopted for the polymer particle. A model which incorporates the presence of multiple sites is expected to predict higher polydispersities than a corresponding single site model. The presence of multiple active catalyst sites has not been considered in this paper since the focus has been to determine whether the presence of gas-liquid mass transfer limitations can result in high polydispersity values. Based on the heterogeneous coordination mechanism for Ziegler-Natta polymerization with the transition metal-carbon bonds as active sites, the kinetic scheme adopted for the polymerization reaction at a catalyst micrograin surface is given in Scheme 1. To characterize the molecular weight distribution, the method of momenta, as elaborated by Ray and Laurence (19771, has been employed. The ith moment may be defined for the live and dead polymer chains as m

p:

= xmiPm

ith moment for live polymer

m=l m

pp =Em%,

ith moment for dead polymer

m=2

The balances for the live and dead species as well as the equations for the dimensionless moments are given in Table 1. y1 is also a measure of the fractional occupancy of the catalyst sites. Balance for the Monomerin the Microparticle. The unsteady state balance for spherical geometry may be written as

The boundary conditions are at R = R,,, at R = Rsh, M = MIKdsh The initial condition is given as: at t = 0, M = Ms0 (initial concentration) (3) The quasi-steady-state assumption is made for microparticular diffusion since the associated time scale (Le.,

2324 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 Table 1

R,h2/Dmp)is much smaller than that for particle growth. The resultant ordinary differential equation can be solved analytically for M , in terms of M I and rsh,the microparticle growth factor (=R,h/R,). Concentrations are nondimensionalized with respect to Cmi, the interfacial monomer concentration, and are indicated as the correspondinglower case variable (e.g., m, for M,, ml for M I , etc.). Thus,

m, =

mlKdsh+ (1- 1/r& 1+ (1- 1/rsh)B

The consumption term on the right-hand side of eq 8,

rV,is a function of the radial location, and is given as

(4)

where

The q terms are akin to dimensionless Thiele moduli for the corresponding kinetic constants and are defined as

The porosity of the initial catalyst particle (e) and that of the polymer bead have been taken to be equal. Gas-Liquid Mass Transfer. The monomer diffusing in at the gas-liquid interface has a parallel mode of consumption through uptake by the macroparticles present in the liquid phase. The unsteady state balance for the monomer can be written as

The boundary conditions are

Macroparticular Monomer Balance (Interstitial). The governing unsteady state balance for interstitial transport of the monomer through the macroparticle can be written for spherical geometry as e

%

= $$[Def2

a,]- rv (for 0 Ir IRpb) (8)

-

03,

-aM1 - 0 ar

aMl/ax = o

and the initial condition is at t = 0,

(by symmetry)

(9)

at r = Rob,

The initial condition is given as at t = 0,

at x

aM1

The boundary conditions are at r = 0,

at x = 0, Ml = cmi (interfacial monomer concentration) (15)

Ml = M,, (the initial concentration) (11)

A t the outer bound of the macroparticle, the condition of no accumulation holds as given by the boundary condition (10).

M , = Mlb

for all x

(16)

The gas-liquid transfer can, in principle, be analyzed using the film theory (steady state) or the unsteady state theories (Higbie's penetration theory and Danckwerts' surface renewal theory) (Danckwerts, 1970; Doraiswamy and Sharma, 1984). As per the unsteady state theories, the particles from the bulk liquid phase would be exposed to the interface along with the liquid element for a short period, and would then be returned to the bulk liquid phase. Recently, the application of several standard mass transfer theories to such systems has been discussed, and a unified theory for transport in the presence of the socalled "microphase" has been presented (Mehra, 1988, 1990). The film theory approach has been adopted by several authors (Janakiraman and Sharma, 1985; Bhagwat and Sharma, 1988),and this corresponds to the "relative solubility controlled regime" of the unified theory.

Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2326 However, for exhaustible microphases, into which category the macroparticles (with limited capacity) fall, a more realistic conceptualization would be through the unsteady state theories, where the time of exposure of the macroparticles near the interface arises naturally through the contact time, t, for Higbie’s penetration theory (given as 40,,d[?rkGL2I), and the surface renewal frequency for Danckwerts’ surface renewal theory. On the basis of the relative rates of diffusion and chemical reaction, multiphase reaction systems may be conveniently classified into four major regimes by the theory of mass transfer with chemical reaction as elaborated by Doraiswamy and Sharma (1984). By the steady state film theory approach, for gas-liquid reactions in the slow reaction regime (where no reaction occurs in the diffusion film but the dissolved solute concentration is zero) and the regime between very slow and slow reactions (wherea part of the resistance is due to gas-liquid diffusion and the bulk concentration of solute is finite, but much lower than the interfacial concentration), the small contribution toward reaction in the interfacial region is neglected, and the entire reaction is presumed to occur at the bulk liquid phase concentration.

Solution Strategy In the case of the slurry polymerization of olefins, the aforementioned approach would not give grossly different values for the rate of monomer absorption. If the rate of diffusion is assumed to be much higher than the rate of monomer consumption (Floyd et al., 1986b; Reichert et al., 1986; Sau and Gupta, 1993), the gas-liquid mass transfer would be modeled as a step in series with monomer uptake. An attempt made on these lines, however, with gas-liquid mass transfer coefficients in the range of 0.0010.07 s-1, yielded polymer polydispersities of up to 8, even under conditions of severe intraparticular mass transfer limitations. Sau and Gupta (1993) have analyzed the transients arising due to the build-up of the gas phase pressure by the slow introduction of monomer into the reactor. The resultant transients in the degree of polymerization and polydispersity were observed to persist for longer durations than the corresponding monomer concentration transients. Another aspect that merits consideration is that, in gasliquid reaction systems generally encountered, the nature of the product formed is independent of the concentration of the reactant at which it was formed. In the case of polymerization reaction systems, the film treatment holds only in terms of the amount of polymer formed (or monomer consumed), but the polymer formed at different monomer concentrations differs in terms of the polymer chain lengths. Hence, it is preferable to use a theory that can be conveniently adapted to account for polymer formation at different concentrations, especially near the interface, where the concentration profiles are the steepest. The limited uptake capacity of the macroparticle also needs to be taken into account. Higbie’s penetration theory is more amenable to a numerical approach (owing to its constant uniform exposure time) and has therefore been employed to solve the governing equations of gas-liquid mass transfer in this Ziegler-Natta polymerization model. In a situation where the microphase is an emulsion droplet or a catalyst particle which merely adsorbs the substrate at its external surface, it is easy to visualize the “averaging”that would occur once the microphase particle returns from the interfacial region to the bulk. In the case of growing polymeric particles, the situation is complicated by the accumulation of the product (polymer)

within the particle itself. The macroparticles would grow at different rates depending on the distance from and the time spent near the interface. A more rigorous approach would involvepopulation balances at multiple levels. Using the simpler moment-based approach for the MWD, an “averaging”for the particle properties has been employed, as elaborated. Representative particles were tracked simultaneously-from the interfacial region along with the parent liquid element, as well as from the bulk. All particles were presumed to remain in their respective regions for the contact time, after which the particles from the penetration element near the interface would return to the bulk and lose their identity. The properties were then averaged for particles in the interfacial and bulk regions, and from this average bulk “pool”, a fresh batch of particles reentered the interfacial region, the whole cycle thus being repeated. Thisis not to imply a physical mixing of the polymer particles or a sharing of growth. Rather, in this manner, the aim was to average out the effect of the phenomenon that all macroparticles, at some time or other, would be exposed to the near-interface region, and this interfacial exposure (averaged over many such exposures) would be the same for all macroparticles. The six moment equations given in Table 1 (eqs v-x) and the balance for (PI/PT)(eq xi) as well as the macroparticular monomer balance (eq 8) are coupled and have to be solved simultaneously for each macroparticle. The macroparticle balance is a parabolic partial differential equation. The spatial derivative can be discretized (into divisionswhich are initially equal and become unequal as the polymerization proceeds) to obtain a set of ordinary differential equations, which include the boundary conditions at the center and at the outer surface of the macroparticle. The resultant stiff set of equations was solved employing the Gear’s algorithm (subroutine IVPAG from IMSL library). In addition, N,, different types of macroparticles were tracked in the penetration element, N,, being the number of divisions into which the penetration element was divided. The gas-liquid balance was solved numerically by employing an explicit method after the discretization of the gas-liquid monomer balance equation (14). Integration was carried out for Nar independent representative particles simultaneously for the penetration element and one representative particle for the bulk over the contact time, tc. Solving the moment and monomer balance equations was found to take inordinately long computation times. Hence, as suggested by Sarkar and Gupta (19911, the strategy of decoupling these two sets of equations over very short time periods was adopted, without jeopardizing the accuracy. The number-average and weight-average molecular weights, as well as the polydispersity, Q, were determined from the moments of live and dead polymer as

(18)

The cumulative number-average and weight-average degree of polymerization are calculated after summing up the moments over all the zones of the macroparticle as

2326 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994

The monomer gradient within the macroparticle, G,, has been defined as a measure of the intra-macroparticular diffusional limitation as the ratio of monomer concentration at the center of the particle to that at the surface.

- M,(innermost zone)

G~ - M,(outermost zone)

(22)

Table 2 lists the reference parameters used for the simulation and are typical values for the Stauffer AA catalyst system (the details of which are given by Jejelowo et al. (1991)),and lie within the ranges suggested by Floyd et al. (1986b).

Results and Discussion Several research workers have emphasized that gasliquid mass transfer becomes increasingly important as the activity of the catalyst increases (Bergerand Grieveson, 1965; Boor, 1979; Boocock and Haward, 1966). Floyd et al. (1986b) concluded that, in general, gas-liquid mass transfer resistance would be more severe for ethylene polymerization than for propylene polymerization, for the same temperature and pressure conditions. I t is significant that the polydispersity values for polyethylene are also reported to be much higher than those for polypropylene, for which the tacticity is a more important property. Sau and Gupta (1993) have observed that, under conditions of changing monomer concentration, higher polydispersities are obtained. It is reported that in industrial operations higher polydispersity is obtained when the polymerization is carried out in a series of stirred reactors instead of a single reactor (Zucchini and Cecchin, 1983). Nagel et al. (1980) have modeled olefin polymerization without considering the effect of gas-liquid mass transfer limitations. They have reported results which show a maximum value of polydispersity of 7 for a single type of catalyst site and a random distribution of catalyst micrograin sizes within a single macroparticle, even under severe intraparticular diffusional limitations. Thus, intraparticular diffusional resistance alone cannot account for high polydispersity values (often as high as 30) in the absence of catalyst sites of multiple activities. To obtain even reasonable values of polydispersity, the imposed intraparticular mass transfer limitations are so severe that the interior of the macroparticle would be starved of monomer. This would imply that microparticle growth factors would vary widely with radial location. This directly contradicts experimental observations from transmission electron micrographs taken at various radial locations as reported by Kakugo et al. (1989). Hence, including the gas-liquid mass transfer step in the analysis of Ziegler-Natta polymerization of ethylene is crucial to rationalize the observed effects. The values and trends of the polydispersity, 8, for the reference parameters adopted in this simulation match well with experimental values reported by Crabtree et al. (1973)and Zucchini and Cecchin (1983) for typical values. The microparticular growth factor, rsh, as a function of dimensionless macroparticular radial location, is shown in Figure 2, for the reference case. The variation has been

q

1.0 0.00

0.50

1.00

1.50

2.00

2. 0

Dimensionless Radial Location Figure 2. Intra-macroparticular variation of microparticlegrowth factor. (- -) Polymer yield = 4.9 kg/kg-cat; (- - -) polymer yield = 2.2 kg/kg-cat; (-) polymer yield = 1.2 kg/kg-cat. Table 2. Reference Values of Parameters Used in Simulation 0.4 2.8 0.5 kgm" 0.01 8-1 1 x 10-4ms-1 1 x 107 m3.km01-1-84 1 x 107 m3.km01-1.84 Dmp 1 x 10-lorn%-1 1.0 8-1 R, 2.5XlVm 0.0 rn%nol-W R, 5~10-8rn 0.0 rn*.kmol-l+ Calk 1 x 10-2kmol.site/ (m3of catalyst)

illustrated at three different yields. In all three cases, the radial variation of rsh is seen to be low, implying that microparticles at the center and near the surface of the macroparticle are comparable in size in keeping with experimental observations. Effects of VariousParameters. The effects of various physicochemical and operating parameters on the polymerization rate and properties of the polymer were assessed using the present model. 1. Effect of Catalyst Particle Size. The effect of the catalyst particle size was investigated by changingthe value of R,, in the range of 2 X 1o-S to 1 X 10-5 m and is shown in Figure 3. As the average catalyst particle size increases, intraparticular diffusional limitation increases, and this is reflected by a lowering of G,, the monomer gradient within the macroparticle. With an increase in the catalyst particle size, the interfacial contribution to the total polymer formation decreases (due to a decrease in the solid-liquid interfacial area), resulting in a lowering of rates. This results in a higher value of Mtb, Le., a lower difference in the monomer concentration experienced by the particles in the bulk liquid phase, and those in the interfacial region. This, in turn, results in a lower interfacial contribution to the total polymer relative to that from the bulk. The propagation rates are lower in comparison with the termination or the transfer processes, and the degree of polymerization therefore decreases with an increase in R!,. The decrease in G, contributes to an increase in the polydispersity index, Q, while the increase in Mtb contributes to a lowering of Q. The overall effect of both these competing factors is a net decrease in Q as R,, is increased.

Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2327

- -

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x 10-8,. I

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R, = 5.0 X 10-8 m; (-

2. Effect of Catalyst Micrograin Size. The catalyst micrograin size, R,,, was varied in the range of 2 X 10-8 to 1 X lo-’ m, and the polydispersity, Q, was seen to decrease with an increase in R,, as seen in Figure 4. An increase in RCI,implies a higher intra-microparticular diffusional resistance for the same yield. Since microparticular uptake acts as a “sink” for macroparticular diffusion, the monomer gradient in the macroparticle, G,, rises with increasing RPB‘ Since the overall rate is ultimately dependent on surface reaction, which in turn depends on microparticular uptake, the rate of polym,. This also leads erization decreases with an increase in R to an increased value of the bulk monomer concentration, Mlb, and, hence, a reduced difference between the interfacial and bulk monomer concentrations. This again implies a lower interfacial contribution to the total polymer with respect to the bulk. All these factors contribute to a decrease in the polydispersity, as discussed in the

a

1

10

100


Figure 5. Effect of monomer diffusivity through polymer. (---) D, = 3.0 X lo-” m2/s;(-) D,, = 1.0 X m2/s;(- -) D,, = 3.0X m2/s.

preceding section. The reduced monomer uptake by the microparticles also leads to a lower degree of polymerization. 3. Effect of Microparticular Diffusivity of the Monomer. When the diffusivity of the monomer through the polymer, Dmp,is varied in the range of 3 X W1to 3 X 10-lo m2/s, the asymptotic value of the polydispersity, Q, was seen to increase 3-fold from around 10, as seen in Figure 5. The increase in D m p implies a decrease in the intra-microparticular diffusional limitation and, hence, a higher uptake by the microparticles, all other factors being the same. This higher microparticular uptake leads to a steeper monomer gradient in the macroparticle, hence G, decreases. This explains the apparently anomalous effect of an increasing diffusivity causing increased diffusional limitations. The enhanced microparticular monomer uptake causes an increase in the propagation rates relative to the termination rates and, hence, a higher degree of polymerization, DP,, as well as polymerization rate. The latter implies a lower Mlb and, hence, a higher interfacial contribution to the polymer formed. The polydispersity, Q, therefore increases with an increase in D m p as a result of all these factors. 4. Effect of Volumetric Gas-Liquid Mass Transfer Coefficient. The effect of volumetric mass transfer coefficient, kGLuGL, was studied by varying the gas-liquid interfacial area from 50 to 200 m2/m3, holding the true mass transfer coefficient, kGL, constant a t 1 X 10-4 m/s (typical value for slurry polymerization systems in the range suggested by Floyd et ul. (198613)) and is illustrated in Figure 6. The intra-macroparticular phenomena, as represented by G,, are expectedly unaffected by this change, which is external to the particle. It is significant that, with intra-macroparticular conditions remaining unchanged, the polydispersity, Q, increases with gas-liquid diffusional resistance, Le., for lower values of koLaoL. This is due to a lower value of Mlb which enhances the interfacial contribution to the polymer formed and, hence, increases the value of Q at lower values of the volumetric gas-liquid mass transfer coefficient. The degree of polymerization increases marginally with an increase in kGLuGL, and the rate of monomer absorption also increases.

1 ----_-___

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2328 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 ,6000

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O,-'

10

100



1

10

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Figure 8. Effect of termination rate constant. (---) kM, = 0.1 5-1; (-) kb, = 1.0s-1; (- -) kh, = 20 5-1; (- -) kb, = 200 5-1.

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Figure 7. Effect of propagation rate Constant. (---) k, = 3 X 106 m3/(kmol.s); (-) k, = 1 X lo7 m3/(kmol.s); (- -) k, = 3 X lo7 m3/ (kmo1.s).

5. Effect of Propagation Rate Constant. The effect of a variation in k,, the propagation rate constant, on the simulation results was investigated in the range of 3 X 106 to 3 X 107 m3/(kmol*s),holding other kinetic parameters (initiation, termination, and transfer reactions) constant, and this is illustrated in Figure 7. This would be akin to an increase in temperature, assuming that the activation energy is much higher for propagation relative to the other kinetic steps. The degree of polymerization,DP,, increases as a natural consequence of the increased propagation rates. A higher microparticular uptake leads to sharper macroparticular gradients, as reflected in lower values of G,. The rate of polymerization increases, leading to a lower bulk monomer concentration, Mlb, and a higher interfacial concentration-leading to a higher polydispersity. 6. Effect of Termination Rate Constant. The effect of change in the termination rate constant (kkm = kto k t ~ [ H d ~was . ~ )studied in the range of 0.1-200 s-l, and

+

corresponds to an increase in the concentration of the chain transfer agent, hydrogen. The monomer gradient in the macroparticle, G,, and the overall rate of polymerization were affected only at higher termination rates (kkm > 20 s-l), as shown in Figure 8. The asymptotic value of polydispersity displays a maximum with variation in kkm, as shown in Figure 9. Actual experimental observations have been reported to exhibit varying trends in the effect of hydrogen concentration on the polydispersity, such as an increase with increase in hydrogen concentration or no change in the polydispersity value (Zucchini and Cecchin, 1983). Hydrogen as a chain transfer agent decreases the degree of polymerization, DP,, over the entire range, but its effect on Q is thus not monotonic.

Conclusions A mathematical model for the batch Zieglel-Natta slurry polymerization of ethylene has been developed, taking into account the effect of gas-liquid mass transfer limitations, in the presence of micron-sized catalyst particles. The gas-liquid interfacial contribution to the polymer formation is seen to be crucial in deciding the molecular

Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2329 weight distribution of the polymer formed. Higbie’s penetration theory was satisfactorily employed to model the monomer absorption in the presence of growing polymeric particles, which can cause significant effects, especially in the early stages of polymerization. It is seen that slurry polymerizations can generate broad molecular weight distributions even when only one type of active catalytic site is present. The dispersity in polymer chain lengths, which has often been attributed to catalyst sites of differing activity, can also be explained on the basis of multiple levels of mass transfer resistance. Uptake by the polymer particles affords a mode of consumption that is in parallel with gas-liquid mass transfer rather than in series. The gas-liquid mass transfer limitations can account for broad distributions (Q 2 30) even while the interstitial macroparticular diffusional limitations may be only moderate. Macroparticular diffusional limitations alone cannot give rise to very broad molecular weight distributions.

Nomenclature = specific gas-liquid interfacial area, m2.m-3 ap = specific interfacial area of macroparticles, m2.m-3 B = dimensionless constant defined by eq 5 Csite = concentration of active sites, kmo1.m-3 of catalyst particle D,f = effective diffusivity of monomer through macroparticular interstices, m2.s-1 D,, = diffusivity of monomer through polymeric shell of microparticle, m2.s-1 Dd = diffusivity of monomer in liquid phase, m2.s-l DP, = number-average degree of polymerization, defined by eq 20 DP, = weight average degree of polymerization, defined by eq 21 G , = monomer gradient in macroparticle, defined by eq 22 K h h = distribution coefficient for the monomer between microparticular polymer shell and macroparticular interstices, dimensionless K d p b = distribution coefficient for the monomer between macroparticular interstices and solvent, dimensionless kGL = true gas-liquid mass transfer coefficient, ms-1 ki = rate constant for initiation, m3.kmol-14 k , = rate constant for propagation, m3.kmol-1-s-1 kSL = true solid-liquid mass transfer coefficient, m-s-1 kte, = rate constant for termination (spontaneous + hydrogen), s-1 ktH = rate constant for transfer to hydrogen, (m3.kmol-1)0%-1 kk = rate constant for transfer to polymer chain, m2-kmol-14 kt, = rate constant for transfer to monomer, m3.kmol-1-s-1 km = rate constant for spontaneous termination, s-1 M = volumetric monomer concentration variable in microparticle, kmo1.m-3 Ml = volumetricliquid phase monomer concentrationvariable, kmol-m-3 Mlb = volumetric liquid phase concentration of monomer in bulk liquid phase, kmol.m-3 Mh = volumetric liquid phase concentration of monomer at macroparticular surface, kmo1.m-3 M, = number-averagemolecular weight of polymer, kgkmol-1 M,o = initial volumetric concentration of monomer in microparticle, kmo1.m-3 M, = weight-averagemolecular weight of polymer, kekmol-1 ma = dimensionless value of Ma Ma = volumetric concentration of monomer at micrograin surface, kmol.m-3 ml = dimensionless value of M1 MI = volumetricconcentrationof monomer in macroparticular interstices, kmo1.m-3 A410 = initial value of MI, kmol.m-3 QGL

Nc = number of catalyst particles per unit volume of the liquid phase, m-3 N(rl = number of microparticles per unit volume of macroparticle at radial location r, m-3 N,, = number of computational divisions of the penetration element NNw = number of catalyst micrograins per catalyst particle, dimensionless P = pressure, kPa Pi = surface concentration of live (or growing)polymer chains of length i, kmo1.m-2 PT = active site concentration on micrograin surface,kmo1.m-2 Q = polydispersity index, defined by eq 19, dimensionless R = variable radius within microparticle, m R , = initial radius of catalyst particle, m I* = radial variable in macroparticle, m r s h = microparticular growth factor, dimensionless R,h = radius of the growing microparticle, m Rrg = radius of catalyst micrograin, m Sh = Sherwood number for the macroparticle, dimensionless Si = surface concentration of dead (or terminated) polymer chains of length i, kmol-m-2 t = time, s t, = contact time of penetration element by Higbie’s penetration theory, s w = molecular weight of monomer, kgkmol-1 x = distance variable in penetration element, m yi = for i = 1-6, dimensionless moments y7 = dimensionless surface concentration of live polymer of length n z = dimensionless constant defined by eq 6 Greek Symbols e = porosity of catalyst particle, dimensionless t i = dimensionless Thiele modulus for initiation 7, = dimensionless Thiele modulus for propagation tk = dimensionless Thiele modulus for chain transfer to polymer chain tterm = dimensionless Thiele modulus for termination ttm = dimensionlessThiele modulus for transfer to monomer rs= rate of consumption of monomer at micrograin surface, kmol-m-2.s-1 rv= volumetric rate of monomer uptake by microparticles per unit volume of macroparticle, kmol.m-3.s-1 4 = fractional volumetric hold-up of macroparticles, dimensionless pc = density of catalyst particle, kg.m-3 pp = density of polymer, kg.m-3

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* Abstract published in Advance ACS Abstracts, August 15, 1994.

Mathematical Modeling of the Slurry Polymerization of Ethylene: Gas

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