Modeling Polyolefin Deformation Resistance in a Growing Microparticle

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Modeling Polyolefin Deformation Resistance in a Growing Microparticle U. S. Agarwal* Polymer Technology Group, Faculty of Chemical Engineering, The Dutch Polymer Institute, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

When polyolefins are produced on heterogeneous catalysts, they encapsulate the catalyst fragments and present diffusional resistance to further monomer transport to the catalyst fragments. In addition, the deposited polymer layer brings in viscoelastic resistance, since it must be deformed to make space for the polymer being formed at the catalyst surface. In the present work, we analyze the influence of the growing polymer layer on the development of the stress and the concentration profiles and, hence, on the polymerization rate. Introduction Polyolefins are most often produced by heterogeneous catalysis. During polymerization, the heterogeneous catalyst undergoes fragmentation to diameters of 1 nm0.5 µm. These fragments get encapsulated by the polymer being produced, forming microparticles that continue to grow. The deposited polymer offers diffusional resistance to further monomer transport to the catalyst fragment surface.1-7 The multigrain model3 takes into account the concentration gradient-driven monomer transport not only through the microparticle but also through the macroparticle. In our recent modeling at the microparticle level,8,9 we considered the additional phenomenon that, as monomer molecules are added to a catalyst fragment surface, further availability of monomer requires space that can be provided only by continuous movement of the polymer mass away from the catalyst surface. Though the phenomenon involves a moving boundary (outer surface of the polymer layer), we made a quasi-steady-state approximation (QSSA) that the concentration profile reaches a steady state much faster than the particle growth. Here we eliminate this assumption. Instead, we carry out a coordinate transformation to freeze the outer boundary of the polymer layer. This allows us to analyze the influence of the growing polymer layer on the stress and the concentration profiles during particle growth. In Model for Polymerization in Microparticles, we present the unsteady-state model incorporating the concentration gradient-driven diffusion, the polymer deformation resistance, and the reaction at the catalyst fragment surface. The results are presented and discussed in Results and Discussion, and the conclusions are summarized.

Figure 1. A growing microparticle. The spherical catalyst fragment has a radius rc. Monomer is converted into polymer at this surface. This polymer, swollen with monomer, encapsulates the catalyst fragment, forming a microparticle of increasing radius R(t). The shown element (of entangled polymer molecules) is formed at the catalyst fragment at time to and deforms as it moves outward to make space for the polymer being formed at rc.

ing monomer volume fraction in the polymer mass at the outermost layers (r ) R) in the microparticle. If macroparticle diffusion were not rate limiting, φo would correspond to equilibrium with the surrounding fluid that provides the monomer. The governing equation for the monomer diffusion is10

* To whom correspondence should be addressed. Tel.: 31 40 247 3079. Fax: 31 40 243 6999. E-mail: [email protected].

(1)

φ ∂φe J ) -D φe ∂r

(2)

where

Model for Polymerization in Microparticles The growing microparticle and the catalyst fragment therein are assumed to be spherical, with radii R(t) and rc, respectively (Figure 1). Let φ(r) represent the volume fraction of the monomer penetrant in the monomer swollen polymer at the radial position r (rc < r < R) outside the catalyst fragment. Let φo be the correspond-

∂φ 1 ∂ ) - 2 (r2J) ∂t r ∂r

is the monomer flux, D being the diffusivity of the monomer through the polymer and φe being the local equilibrium value of φ. Since φ(r) e φe(r) e φo in the monomer swollen polymer mass, a swelling (osmotic) pressure comes into effect, and is given by11

P)

()

φe kT 1 (µ - µ) ) ln Ω e Ω φ

10.1021/ie034272x CCC: $27.50 © 2004 American Chemical Society Published on Web 07/09/2004

(3)

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where k is the Boltzmann constant, T is the absolute temperature, µ ) kT ln(γφ) is the chemical potential, γ is the activity coefficient, and Ω is the molecular volume of the monomer. P relaxes to zero as φ and µ approach the local equilibrium values φe and µe ) kT ln(γφe). Equation 3 was first derived by Thomas and Windle11 in connection with their description of the case II diffusion phenomena during swelling of polymers. Equations 2 and 3 can be combined to obtain12

J ) -D

∂P (∂φ∂r + Ωφ kT ∂r )

(4)

where the last term is the effect of osmotic pressure. When this last term is negligible, Fickian diffusion is observed. The monomer undergoes polymerization at the catalyst fragment surface, and the monomer flux is equal to the reaction rate

J ) -krφ

at

r ) rc

(5)

where kr (m s-1) is the rate constant for the surface reaction, assumed first order. φ(r) increases from φs at the catalyst surface (rc) and approaches φo (the saturation value) as the material element moves outward (Figure 1). With progress of polymerization, the spherical microparticle grows and R(t) increases. We assume that the densities of the polymer and the monomer are same and that there is no volume change on mixing. Thus, monomer and polymer fluxes are equal and opposite. Hence, accounting for the polymer volume fraction (1 - φ), J is related to the polymer radial velocity (v) as follows:

v ) -J/(1 - φ)

τθθ + τφφ ∂τrr 3 ∂P 1 ∂ ) - 2 (r2τrr) + )- τrr ∂r r ∂r r r ∂r

where the inertial effects have been neglected in comparison to the viscoelastic stress terms. In the earlier work,9 we employed a QSSA so that φ(r,t) could be treated as a function φ(r,R(t)). Further, we assumed that, for R(t) greater than some critical Rc, i.e., after the microparticle has grown sufficiently large, the penetrant flux at smaller r becomes constant, and we concentrated our attention on this steady state. We now wish to account for the unsteady-state microparticle growth phenomenon. This increasing R(t) introduces a moving boundary,

dR/dt ) v(R)

x ) (r - rc)/(R - rc)

xv(1) ∂φ ∂φ 1 ∂J 2 ) + J ∂t R - 1 ∂x R - 1 ∂x r J ) (1 - φ)v )

where η and G are the viscosity and the elastic modulus of the monomer swollen polymer and are here assumed to be constant throughout the microparticle. (d/dt) is the material derivative, indicating that the history of the polymeric material element determines its stress level. The diagonal components of the deformation rate tensor ˘ for the compressible polymer are given by14

˘ rr )

2 ∂v v ∂v 1 - (∇‚v) ) ∂r 3 3 ∂r r

(

)

(8)

and

1 ∂v v v 1 ˘ θθ ) ˘ φφ ) - (∇‚v) ) r 3 3 ∂r r

(

)

(9)

In an earlier work,9 we had made a simplifying assumption by neglecting the compressibility terms (containing(∇‚v)). Thus, only the balloonlike deformation of the polymer, and not volume change by swelling, was considered contributing to the deformation that determines the stress for the polymer. The equation of motion in spherical geometry14 describing the deformation of the polymer under the isotropic swelling pressure reduces to

(12)

which limits the swollen polymer between x ) 1 (at the outer surface of the deposited polymer) and x ) 0 (at the catalyst surface). This transformation is essentially similar in spirit to our earlier work.15,16 Further defining dimensionless variables rj ) r/rc, R h ) R/rc, J h ) -J/(krφo), ht ) tkrφo/rc, vj ) v/(krφo), P h ) PΩ/ (kT) ) ln(φe/φ), and θ ) τrr Ω/(kT), we arrive at the following dimensionless steady-state governing equations

The following constitutive equation is used to describe the viscoelastic behavior of the polymer13

(7)

(11)

We overcome the complications of the moving boundary during the computational analysis by carrying out the following coordinate transformation:

(6)

η dτ ) -2η˘ τ+ G dt

(10)

θ+

(13)

1 ∂P A ∂φ + Aφ R - 1 ∂x R - 1 ∂x

(

)

(14)

1 ∂P 1 ∂θ 3 )- θ R - 1 ∂x R - 1 ∂x r

(15)

r ) 1 + x(R - 1)

(16)

dR/dt ) v(1)

(17)

∂θ 1 ∂v v 1 Λ ∂θ + ux )A ∂t ∂x 3ADh R - 1 ∂x r

(

)

(

)

(18)

whereux ) (v(x) - xv(1))/(R - 1)) is the velocity of the polymeric material element in the x-frame, and it appears in the constitutive equation to track its stress deformation history. Here (and hereafter) the superscores in rj, ht, vj , R h, J h , and P h have been dropped for convenience, and A ) D/krφorc, Dh ) kTrc2/4DΩη, and Λ ) ηD/Grc2 are the dimensionless system parameters. The parameter A determines the ratio of reaction to diffusional resistance. Hence, highest and lowest values of A would correspond to reaction control and diffusional control regimes. The parameter Dh represents the ratio of diffusional to viscous resistance. Λ represents the elastic characteristics of the polymer.

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The boundary conditions are

J ) φ/φo φ ) φo

x)0

at

(19)

x)1

at

(20)

and the initial condition is taken as

R ) Ro

t)0

at

(21)

where Ro is a parameter of the system and represents the initial radius of the microparticle. It is set to a value larger than (and not equal to) the catalyst radius 1 to enable numerical computations and, thus, represents an initial polymer layer of thickness (Ro - 1) around the catalyst. We select a value of Ro ) 2 such that a smaller chosen value does not affect the results in this work. Further, we consider the initial polymer shell of thickness (R - 1) to be stress free

θ)0

t)0

at

(22)

with the corresponding concentration profile φ (x, t)0) determined by steady-state Fickian diffusion along with the boundary conditions (eqs 19 and 20). The simultaneous governing differential eqs 13-18 are solved numerically by finite elements method (thirdorder orthogonal collocation) using gPROMS (Process System Enterprise, London, http://www.psenterprise.com/ gPROMS). The spatial domain x(0,1) is discretized into 200 elements, so chosen that increasing the number of elements did not influence the results. The commercial software adapts the time steps during the simulation to ensure convergence. Very sharp concentration, stress, and deformation gradients are set up immediately, making it difficult to achieve numerical convergence. This problem is alleviated by multiplying the right-hand side of eq 18 by a prefactor

f)

(t/tc)4

(23)

1 + (t/tc)4

that increases from 0 to o(1) in t ∼ tc and thus corresponds to a slower buildup of viscoelastic stress in the polymer up to (t ∼ 2tc) when (f ) 0.94). We selected (tc ) 5) and a smaller value did not affect the results in this work beyond (t ) 10). No Deformational Resistance. In the limit of η f 0 (Dh-1 f 0), the deposited polymer will offer no deformational resistance to the monomer influx, and Fickian diffusion will prevail throughout the microparticle. Assuming QSSA, the concentration profiles can be obtained as3

φ(r) ) φo φs )

φs 1 1 Aφo r R

(

φo 1 1 1+ 1Aφo R

(

)

dR φs 1 1 ) dt φo R2 1 - φo

)

(24)

(25)

and the microparticle growth rate can then be written as

(26)

Results and Discussions Parameter Estimation. As in our earlier work,9 we considered ethylene polymerization with parameter values as listed in Table 1 and catalytic activity of ∼105 Table 1. Parameter Values for Ethylene Polymerization on Highly Active Catalyst rc φo D h G T

10-25 × 10-9 m 0.05 10-11-10-12 m2 s-1 106-108 kg m-1 s-1 106 N m-2 60 °C

kg of monomer (kg of catalyst‚h)-1. The dimensionless parameter values are estimated as A ∼ 102-2 × 104, Dh ) 10-6-10-3, and Λ ) 103-107. Fickian Diffusion Case. For the Fickian diffusion case, eqs 25 and 26 are solved simultaneously, in Figure 2a we plot the development of φs for a set of values of the parameter A, and in Figure 2b we plot the corresponding growth of the microparticle dimension. Equation 19 implies that the monomer concentration φs ) φ(x)0) is proportional to and hence representative of the reaction rate for the first-order reaction. φs decreases with time corresponding to the increasing diffusion path length (R - 1) for the monomer. We recall that eqs 24 and 25 are based on the QSSA assumption. However, when the set of eqs 13-18 are solved (setting the terms containing stress θ to zero), we find the results to be indistinguishable from those plotted in Figure 2. This justifies the validity of the QSSA in the absence of viscoelastic resistance, at the parameter values (A . 1) considered here. Further, for the parameter values considered here (Aφo . 1, eq 25), the diffusional limitation in the microparticle is small (φs not much smaller than φo, Figure 2a) when the monomer diffusion is Fickian. We now examine whether the monomer transport to catalyst surface can become rate limiting as the deposited polymer offers resistance to deformation. With an objective of analyzing the sensitivity to the viscoelasticity parameters Dh and Λ, here we restrict our attention to A ) 102, above which the diffusion limitations play a diminishing role (Figure 2). Effect of Viscous Limitations. The case of purely viscous deformation of the deposited polymer in the absence of any elastic deformation (G f ∞) is represented by eqs 13-18 with Λ set to zero. The parameter Dh represents the ratio of diffusional to viscous resistance. Since the concentration and stress gradients are expected to be the largest near the catalyst surface, in Figure 3 we plot the influence of Dh on the stress and concentration at the catalyst surface (φs). For a given value of Dh, φs decreases with time (Figure 3a) from the initial value of 0.0454 (corresponding to Ro ) 2, eq 25) first rapidly (though slowed by the artificial prefactor, eq 23), as we begin to account for the deformational resistance, and then slowly corresponding to the increasing monomer diffusion path length and deforming polymer shell thickness (R - 1). Similarly, the stress at the catalyst surface increases with time (Figure 3b) corresponding to deformation of the shell of increasing thickness (R - 1). For Dh ) 0.01 and higher corresponding to low viscosity, though stress development occurs (Figure 3b) for the deformation to take place, it is not

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Figure 2. Reduction in reaction rate (∝ φs) and increase in particle radius (R) with polymerization time, for the Fickian diffusion case with various values of the parameter A (∝ kr-1). Note that the time axis has been scaled with respect to kr-1.

Figure 3. Effect of polymer viscosity (∝ Dh-1) on the reaction rate (∝ φs) and the deformation stress at catalyst surface (θ(x)0)) for the purely viscous polymer case, for A)100 and Dh values shown on the plot.

Figure 4. Effect of polymer elastic modulus (∝ Λ-1) on the reaction rate (∝ φs) for the viscoelastic polymer, for A ) 100, Dh ) 0.002, and Λ values shown in the figure. Also shown are the curves for the purely viscous case (Λ f 0) and the Fickian diffusion case.

sufficiently large to influence the overall mass-transfer and reaction rate (Figure 3a). With decrease in Dh to smaller values, the viscous stress resistance increases rapidly (Figure 3b), with the corresponding decrease in φs (Figure 3a). Viscoelastic Deformation Limitations. We now consider the situation where the polymer being produced behaves as a viscoelastic material as described by the Maxwell model (eqs 7 and 18). The influence of elasticity is quantified in the parameter Λ. For a given value of Dh, a decrease in Λ corresponds to an increase in G. We will here consider the effect of Λ for fixed values of the parameters A ) 100 and Dh ) 0.002. Figure 4 shows the influence of the parameter Λ on the overall reaction rate as represented by φs (eq 19). The uppermost and lowermost curves correspond to the Fickian diffusion and the purely viscous resistance (Λ f 0), respectively. For a sufficiently small value of Λ, G is so large that the polymer deformation is entirely by viscous mechanism. This results in φs values close

to that for the purely viscous case. At higher values of Λ, the deformation by elastic mechanism facilitates the overall deformation, resulting in higher φs profiles approaching that corresponding to the Fickian diffusion case. The highest Λ corresponds to such small values of G that deformation is dominantly by elastic mechanism, with the corresponding stress and deformation resistance being small. Let us now consider in detail the results for a small value of Λ. Figure 5 shows the corresponding concentration profiles at various times, extending to larger r (up to r ) R(t)) with increase in time. With increase in shell thickness (R - 1) with time, the concentration profiles become lower (Figure 5a), partly because of the increasing diffusion lengths and partly because of the deformation resistance (dP/dx, eq 14) to the monomer flux (Figure 5b). The value of dP/dx remains negative throughout the radial domain, indicating that the stress distribution resists the monomer flux toward the catalyst. Though the concentration gradients exist throughout the growing particle (Figure 5a), dP/dx nearly vanishes beyond r ) 1. 2, indicating that mass transfer at large r is essentially Fickian. This is related to the reducing velocity gradients (∼r-3 at large r) and hence viscous stresses. The results for Λ ) 0.1 (Figure 5) are close to the results of the corresponding calculations for purely viscous case (Λ f 0). This is because at this small value of Λ, the polymer deformation occurs primarily by the viscous mechanism. Increasing φs(t) with increasing Λ in Figure 4 illustrates that the elastic deformation of the polymer provides an additional mode of deformation, resulting in reduced resistance to monomer flux. We now consider the Λ ) 2.5 case in detail, with the corresponding concentration and dP/dx profiles at various times being shown in Figure 6. We find that the concentration profiles near the catalyst surface (r ) 1) deviate considerably from the nearly viscous case (Fig-

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Figure 5. Concentration and pressure gradient profiles in the growing microparticle considering the polymer to be viscoelastic (A ) 100, Dh ) 0.002, Λ ) 0.1). The profiles at t ) 5, 10, 50, 200, and 500 are shown, with lower curves for increasing time.

Figure 6. Concentration and pressure gradient profiles in the growing microparticle, considering the polymer to be viscoelastic (A ) 100, Dh ) 0.002, Λ ) 2.5). The profiles at t ) 5, 10, 50, 200, and 500 are shown, with the arrows pointing to increasing time.

ure 5). This is because, for the viscoelastic polymer, elastic deformation is the prominent mode of deformation during early deformation of the polymer material element (Deborah number De ) Λ/At > 1)13 that is gradually moving away from the catalyst surface. Alternatively stated, the required deformation rate is highest near the catalyst surface and hence viscous deformation is critically difficult. As a polymer material element moves away from catalyst surface, viscous deformation becomes the prominent mode of deformation, and the effect of Λ decreases. dP/dx is negative near the catalyst surface (r < 1.15), indicating that the stress distribution resists the monomer flux (eq 14). However, at the larger values of r, dP/dx is positive, indicating that the stress distribution assists the monomer flux. We recall that an intermediate maximum in pressure was also predicted during swelling of polymer sheets dipped in solvents.11 We also find a maximum in the monomer concentration at an interior point (r ) 1.15). At the largest values of r, the velocity gradients become small, the deformation becomes primarily viscous, and the corresponding stresses become vanishingly small. It is interesting to evaluate the impact of the assumptions in this work, namely, the zero initial stress, and the consequent need for employing delayed stress growth prefactor f (eq 23). Figure 7 shows the change in reaction rate (∝ φs) for the viscoelastic polymer with A ) 100, Dh ) 0.002, and Λ ) 0.1. The curve a therein is the same as the Λ ) 0.1 curve in Figure 4, i.e., the dynamic case with the general assumptions of this paper, namely, zero initial stress (eq 22), Ro ) 2, and the prefactor f (eq 23) with tc ) 5. The curve b corresponds to a similar dynamic simulation, but with the parameter tc in eq 23 set to 2.5, i.e., with a reduced delay in the initial stress growth. The curve c also corresponds to a dynamic simulation similar to curve a, but with a smaller initial particle size Ro ) 1.1. Curve d differs from curve a only in the respect that it replaces

Figure 7. Effect of initial conditions, QSSA, and stress contributions on the calculated reaction rate (∝ φs) for the viscoelastic polymer with A ) 100, Dh ) 0.002, and Λ ) 0.1. (a) Dynamic case with zero initial stress (eq 22), tc ) 5, Ro ) 2 (as in Figure 4). (b) Dynamic case with zero initial stress (eq 22), tc ) 2.5, Ro ) 2. (c) Dynamic case with zero initial stress (eq 22), tc ) 5, Ro ) 1.1. (d) Dynamic case with initial θ determined by steady state (eq 18 with ∂/∂t term set to zero at t ) 0), Ro ) 2, and prefactor f ) 1 (eq 23). (e) QSSA (φ and θ profiles determined by ∂/∂t terms in eqs 13 and 18 set to zero for all t), Ro ) 2. (f) QSSA (φ and θ profiles determined by ∂/∂t terms in eqs 13 and 18 set to zero for all t), Ro ) 2, while neglecting the polymer swelling contribution to the viscoelastic stresses.

the zero initial stress condition (eq 22) with a corresponding “steady”-state profile (eq 18 with ∂/∂t term set to zero at t ) 0), allowing this simulation to be carried out without delaying the stress growth (i.e., f )1, tc ) 0). A comparison of the curves a-d in Figure 7 indicates that the reaction rate results at t >10 are somewhat insensitive to the choice of the initial stress conditions and the initial particle size. This insensitivity can be considered as a justification of the selection of Ro ) 2, θ(t ) 0) ) 0, and tc ) 5. In our earlier work,8,9 we had proposed a QSSA approach, and here we examine the validity of that approach. Curve e of Figure 7 employs the same initial

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conditions as curve d; i.e., φ and θ profiles at t ) 0 are determined by the corresponding steady-state profiles. However, instead of the dynamic simulation followed in the case of curve d, curve e is obtained by employing a QSSA for determination of φ and θ profiles, ignoring the time derivative terms in eqs 13 and 18 at all times. The complete overlap of curve e with curve d, but considerable deviation from curves a-c suggests that the QSSA employed for curves e is a good assumption beyond the initial stress growth (i.e., at t > 10 in the present case). However, an analysis of the stress growth at shorter times necessarily demands a dynamic analysis, and the solution employed here is successful only partly due to the need for artificially slowing down the initial stress growth. A dynamic description of the coupled diffusion-deformation reaction phenomena during early stages of particle growth5,17,18 is needed for an analysis of the physical/hydraulic forces responsible for the particle morphology development by the catalyst fragmentation process. Finally, following eq 9, we mentioned that the present work does away with the simplifying assumption made in our earlier work8,9 that only the balloonlike deformation of polymer determines the stress development; i.e., the contribution of the polymer swelling (∇‚v term in eq 18) to the viscoelastic stress is negligible. The impact of relaxing this assumption in the present work is evaluated by comparing the present QSSA result (curve e) with the curve f obtained by a similar QSSA, but by replacing9 the right-hand side of eq 18 with (1/ADh)(v/r) corresponding to the balloonlike deformation determining the stress development in the polymer. A higher φs of curve f suggests that the masstransfer limitation is underpredicted when the polymer swelling contribution to viscoelastic stress is ignored. Therefore, as practiced in the present paper, it is important to account for the contribution of polymer swellinig to the viscoelastic stresses. Conclusions We have examined the effect of the depositing polymer in a growing microparticle during olefin polymerization on heterogeneous catalyst. By modeling the monomer transport through this deforming polymer, we have evaluated rate limitations presented by this deformation resistance for some sets of parameter values. Our earlier model based on QSSA has been extended to permit the transient analysis and thus follow particle growth. This has been achieved using a coordinate transformation to freeze the moving boundaries. Incorporation of viscous resistance results in reduced reaction flux. When elastic deformation provides an additional mode of deformation (large Λ), the reaction flux increases, approaching the reaction rate for the Fickian case at sufficiently large Λ. The concentration profile at large Λ shows a maximum at an intermediate r such that pressure profile resists the monomer transport at smaller r and assists the monomer transport at higher r. The influence of relaxing the QSSA is seen to be limited to the early stage of particle growth. While our earlier work considered the viscoelastic stresses in the polymer to result only from the balloonlike deformation of polymer, we have now also accounted for the contribution of polymer swelling. A higher viscoelastic resistance to the reaction flux is thus predicted.

Notation A ) dimensionless parameter defining ratio of reaction to diffusional resistance Dh ) dimensionless parameter defining ratio of diffusional to viscous resistance D ) monomer diffusion coefficient, m2 s-1 De ) Deborah number G ) polymer elastic modulus, Pa J ) monomer diffusion flux at radial position r k ) Boltzmann constant kr ) intrinsic surface reaction rate constant, m s-1 P ) swelling pressure in the polymer, Pa r ) radial position variable, m rc ) catalyst fragment radius, m R ) microparticle outer radius, m Rc ) radial position in microparticle beyond which diffusion is Fickian T ) absolute temperature, K t ) time variable, s v ) radial velocity of the polymer (m s-1) Greek Letters Λ ) dimensionless parameter defining the polymer relaxation time Ω ) molecular volume of monomer, m3 φs ) monomer volume fraction at the catalyst fragment surface φ ) monomer volume fraction φo ) saturation monomer volume fraction in the polymer η ) viscosity of the polymer, Pa s θ ) dimensionless stress tensor ˘ ) strain rate tensor τ ) deformation stress tensor

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Received for review November 26, 2003 Revised manuscript received June 4, 2004 Accepted June 7, 2004 IE034272X