Behavior of Active Sites in a Changing, Supported Metallocene Catalyst Particle: Modeling Monomer...

Behavior of Active Sites in a Changing, Supported

Metallocene Catalyst Particle: Modeling Monomer

Transport and Kinetics

Mario G. Chiovetta,* Diana A. Estenoz

INTEC, UNL-CONICET, Guemes 3450, 3000 Santa Fe, ArgentinaFax: þ54 342 455 9185; E-mail: [email protected]

Received: March 16, 2004; Revised: July 10, 2004; Accepted: August 31, 2004; DOI: 10.1002/mame.200400069

Keywords: modeling; metallocene catalysts; morphology; polymer yield; supports

Introduction

In polyolefin polymerization, the analysis of the actual

participation of the metallocene active sites in the polymer

synthesis process as well as the degree of effectiveness of

these sites in the overall reactor productivity is of relevance.

Proper active site utilization is a key factor when conside-

ring the design and operating conditions required for an

efficient use of the reactor holding a specific catalyst-

support system.

The solid-within-a-solid[1] growth scenario derived from

the nature of the physical system involved is of significant

impact on the polymer manufacture process. The polymer, a

solid, is being synthesized inside a particle of the support,

another solid, producing a complex that, in turn, is affected

by the fluid environment in the polymerization reactor.

Metallocene active sites bonded to the porous surface of

the support structure are impregnated following several

techniques during catalyst preparation time.[2–6] These

procedures determine the specific rate of polymerization

depending on the chemical nature of the metallocene and

the co-catalyst, the sequence of impregnation (co-catalyst

placed before or after the metallocene),[6] and the degree

(density) of pore coverage with active sites. Additionally,

Summary: Support-catalyst-polymer particles composed ofmillions of microparticles arranged in cells and having silicanuclei covered with metallocene-methyl alumoxane (MAO)active sites are studied to analyze cell participation duringpolymerization. Main variables are the changing particlemorphology and the kinetic-diffusion effects determininglocal monomer availability during residence time. The pheno-mena were studied by means of a mathematical model usedto produce a set of predictions for particles polymerizingethylene in a toluene slurry continuous stirred tank reactor(CSTR) under various operating conditions. This informationis employed to predict the micro- and macroparticle behaviorin situations designed to explore catalyst activities, monomeravailabilities and reactor conditions. Kinetic constants andconcentrations range from reference values up to 6 timesthese figures, with reactor temperatures between 323 and353 K and particle Reynolds numbers on a 1 to 10 relativescale. Heat transfer and temperature elevation during poly-merization are predicted, with no relevant overheating ob-served. Morphology changes, in the form of density profilesinside the support-catalyst-polymer particle, are monitoredwith time, and their interaction with transport and reactionphenomena analyzed. Increasing catalyst activity alone maynot produce proportional raises in yield; it appears moreefficient to improve the monomer availability instead. High

catalyst activity may produce monomer depletion at innercells delaying their fragmentation and decreasing localpolymer-production.

Cell density vs. time for cells located at the exterior, at thecenter and at half the radius of the macroparticle.

Macromol. Mater. Eng. 2004, 289, 1012–1026 DOI: 10.1002/mame.200400069 � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1012 Full Paper

the support introduces elements such as its compactness,

density and microsphere size that also affect catalyst behav-

ior. These lists of features for the active sites and support

represent the basic set of variables catalyst manufacturers

can resort to tailor-make properties, such as the kinetic

constant, k, of the overall support-catalyst complex. Here,

the term, catalyst, includes the metallocene, the co-catalyst

and any other added chemical, if used. When the prepara-

tion process is finished, a particle several microns in

diameter, composed of millions of microspheres each one

covered by a given number of metallocenic active sites, is

ready to produce polymers, as described in the section

below.

When exposed to the monomer after the support-catalyst

particles are introduced into the reactor fluid medium, each

microsphere is designed to contribute to the polymer

production. The degree of participation of the microsphere

is essentially affected by a series of factors: a) the particle

changing morphology and b) the kinetic/diffusion effects

determining the local monomer availability. The magnitude

of these two major factors shows varying extension with

time during the particle residence in the reactor. These

factors also depend on the microsphere considered and

change significantly according to its position within the

overall particle. In the case of very active catalysts, which

are common in the industrial manufacture of polyolefins,

the period of time required for the particle morphology

changes forced to occur by polymerization may be several

hundred seconds, with the transition thus spanning a signif-

icant portion of the total particle residence time in the

reactor.

The analysis of the two factors mentioned in the preced-

ing paragraph is performed for the metallocenic complex on

a silica-support microsphere in the interior of a catalyst

particle during its residence time in a typical reactor en-

vironment. The changes are studied by performing poly-

merization calculations with the associated mass and heat

transfer processes with a mathematical model previously

validated through the available laboratory scale data.[6] The

latter is used to produce a set of simulations for particles

polymerizing ethylene in a toluene slurry continuous stirred

tank reactor (CSTR) with silica-supported metallocene

catalysts with a methyl alumoxane (MAO) co-catalyst and

for various operating conditions with a reference, base-case

set of parameters summarized in Table 1.

Mathematical predictions are produced for particle

conditions departing from those in the base-case to explore

several catalyst activities and monomer concentrations

as well as reactor conditions. When catalysts with higher

activities are studied, the results are presented by means of a

parameter, kf or kinetic factor, for simplicity, indicating the

ratio of the value of the particular kinetic constant analyzed

to that for the base-case. Similarly, to explore several

monomer concentrations in the fluid phase, a concentration

factor, Cf, is introduced to express the value in the reactor

fluid phase as times the value, MB, in the base-case. In an

analogous fashion, several mixing conditions are studied

via the introduction of an initial-particle Reynolds factor,

Ref, that reflects the ratio of Re0 for the condition under

analysis to the dimensionless number in the base-case.

Particle Dynamics

A single particle (a combination of the support, catalyst and

polyolefin, termed macroparticle) is taken as the unit where

the analysis is performed. The heterogeneous processes

taking place in each macroparticle residing in the reactor

fluid medium are modeled using the mathematical scheme

of Estenoz and Chiovetta, whose main features are sum-

marized in the Appendix, as previously mentioned.[6]

The sequence of changes suffered by the macroparticle

starts with the alterations produced by polymerization in

the morphology of the support-catalyst particle fed to the

reactor. Following Estenoz and Chiovetta,[6,7] this particle

is considered as a set of cells composed of nonporous silica

microspheres with radius RC initially arranged in a pattern

the shape of a cube. The cube-cell arrangement is repeated

within the whole particle to form a three-dimensional

network: the overall support-catalyst particle is thus consi-

dered as a set of cube-cells arranged in concentric layers.

Each cube has a characteristic edge dimension, lc¼ n(2RC),

in which n is the integer number of microspheres located

between two adjacent vertices of the cube determined by the

centers of the microspheres; each of them is in contact with

a number, b, of other microspheres.[8] For the selected

geometrical description, each concentric layer of cube-cells

in the initial support-catalyst particle has a thickness, lc.

Several high-porosity, silica-support particles can be repre-

sented using this scheme when a additional spheres are

added inside the cube-cell,[4,6] according to the description

Table 1. Parameters for the model in the Appendix, base-caseconditions.

Parameter Value Unit Ref.

T0¼ TB 323 K [5]P 1.7 atm [5]MB 155 mol/m3 [5]RM0 4.50� 10�5 m [5]RC 6.00� 10�9 m [6]N 1 250 [6]k 1.20� 10�6 m/s [5]E 13 500 cal/mol [7]DH 20 000 cal/mol [7]e0 0.825 [5]ea 0.409 [18]rs 2.27� 103 kg/m3 [19]rp 9.00� 102 kg/m3 [7]cpM 0.27 cal/g �K [19]kM 0.47 cal/m � s �K [19]DL/t 5.00� 10�9 m2/s [19],[20]DEP/tm 3.33� 10�11 m2/s [21],[22]c* 1.1 [12],[13],[20]

Behavior of Active Sites in a Changing, Supported Metallocene Catalyst Particle: . . . 1013

Macromol. Mater. Eng. 2004, 289, 1012–1026 www.mme-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

in Estenoz and Chiovetta summarized in Table 2.[6] The

support-catalyst complex analyzed in this work is in the first

line of Table 2.

Once the reaction begins, polymer accumulates around

these support microspheres; the support-catalyst nucleus

plus the polymer deposit constitutes a microparticle.

When the polymer thickness is sufficient, fragmentation

occurs:[12,13] forces arising between any two adjacent mi-

croparticles in the cell due to polymer accumulation break

the initial support’s rigid pattern. Microparticles now are

partially free to move and fill the holes of the initial cube-

cell structure, forcing porosity to decay. Polymer-molecule

entanglements between microparticles that permit some

displacements but prevent the complete disruption of

the microparticle aggregate restrict this process. Accom-

modation of microparticles proceeds until the cell attains a

fairly homogenous compactness in all directions, with poro-

sity being practically constant at a value, e¼ ea, hereafter.

Figure 1 schematically shows the interior of a macro-

particle at a time when fragmentation has already began and

a set of cube-cells is undergoing or has already finished the

process of rearrangement of its microparticles. The lower-

left portion of the picture shows the macroparticle with: a)

the nucleus, at which fragmentation has not yet occurred; b)

the zone undergoing microparticle rearrangement and

porosity decay; and c) the zone where changes in micro-

particle-morphology are finished. One of the cube-cells

located on the fragmentation-limiting surface of the

macroparticle is blown up in the central portion of the

picture. Here, a microparticle in the cell (upper-right

portion of Figure 1) is holding the amount on polymer

required to produce the local rupture of the initial network

of microspheres in the support.

The variable introduced to follow polymer accumulation

is the growth factor, cL, for a microparticle located at rM, the

radial position of the microparticle in layer, L, within the

macroparticle. The growth factor is defined as the ratio of

the microparticle radius, Rm, at time, t, to the support-

catalyst, nonporous microsphere radius, RC:

cL ¼ RmðrM; tÞRC

ð1Þ

The model uses two calculation levels, as indicated in the

Appendix, employing rigorous mathematical techniques to

follow the changes in the polymerizing particle along time

for: a) the processes taking place in the overall support-

catalyst-polymer complex or macroparticle and b) the local

phenomena of polymerization occurring at the active sites

in the microparticles.[6]

Particle Density Evolution

Changes in the particle morphology have an impact on the

effective use of the active sites and are one of the major

factors affecting the macroparticle performance in terms of

the actual exposure of microspheres to monomer. One of the

indicators of the macroparticle degree of change locally, for

each microparticle, is the density of the cells at any given

macroparticle radius and time. Density evolution is, thus, a

first indicator of the progress of the particle-arrangement

process.

At initial conditions, the local, cube-cell density can be

calculated as a function of the density of pure silica, rs, and

the initial porosity, e0. The mass ratio of catalyst to support

is usually below 1%.[5,6] Hence, the support-catalyst micro-

sphere density is assumed to be equal to that of silica.

rM0 ¼ rsð1 � e0Þ ð2Þ

With a density and initial porosity of 2 273 kg/m3 and 0.825,

respectively, corresponding to the support studied in this

Table 2. Representation of typical supports according to ref.[6]

Ref. Pore diameter Pore volume Specific surface area Porosity (experimental) n a Porosity (modeled) RC

nm m3/g m2/kg nm

[5] 40 2.20� 10�6 2.20� 105 0.83 3 2 0.83 6.00[9] 20 1.70� 10�6 3.00� 105 0.79 3 4 0.79 4.40

[10] 15 2.20� 10�6 3.00� 105 0.83 3 2 0.83 4.40[11] 20 1.70� 10�6 3.00� 105 0.79 3 4 0.79 4.40

Figure 1. Scheme of the macroparticle during olefin polymer-ization, with nucleus, region after fragmentation and exterior zonewith microparticle-rearrangement completed. Blow-up for cube-cell and microparticle at fragmentation point.

1014 M. G. Chiovetta, D. A. Estenoz


work (Table 1), a value of 397.8 kg/m3 is obtained for the

support-catalyst particle density. The initial conditions are

also instrumental to calculate the number of microspheres

present. At any given support-catalyst layer, the solid

volume, (Vsolid)L0, can be expressed as a function of the

number of microspheres, NL, in all cube-cells in layer, L,

and each microsphere volume:

ðVsolidÞL0 ¼ NLð4=3ÞpR3C ð3Þ

The volume of the layer is given by:

VL0 ¼ ð4=3Þpðr3MÞL � ð4=3Þpðr3

MÞL�1

¼ ð4=3Þpð2nRCÞ3ðL3 � ðL� 1Þ3Þ ð4Þ

in which the support-catalyst particle radius, rM, is

expressed as a function of the layer number, L, as:

rML ¼ nð2RCÞL ð5Þ

Then, the solid fraction at any layer can be expressed using

the porosity, e0, and Equation (3) and (4) above:

1 � e0 ¼ ðVsolidÞL0

VL0

¼ NLð4=3ÞpR3C

ð4=3Þpð2nRCÞ3ðL3 � ðL� 1Þ3Þð6Þ

Solving for NL:

NL ¼ ð1 � e0Þð2nÞ3ðL3 � ðL� 1Þ3Þ ð7Þ

As soon as the support-catalyst particle is placed in the

monomer-rich reactor environment, polymerization begins

and the polyolefin accumulates on a stratum located outside

the nonporous microsphere, as schematically shown in

Figure 1. During a certain amount of time, the initial cube-

cell geometry does not undergo any changes; porosity

decreases while cell density increases due to polymer filling

of the original voids in the structure. A simple way of

relating cell density with its morphology dynamics is via the

cell porosity at this stage. To do so, Equation (3) and (6) are

updated to take into consideration the polymer deposit.

First, the solid volume in a layer is calculated using Equa-

tion (3) modified to account for the microparticle growth:

ðVsolidÞL ¼ NLð4=3ÞpðcLRCÞ3 ð8Þ

The volume of the layer is still given by Equation (4), since

the cube-cell dimension, lc, spanning the layer thickness

is the same as initially. Similarly, the expression for NL in

Equation (7) still holds true, since the number of micro-

spheres in a layer is independent of time.

Now, the expression equivalent to Equation (6) at this

stage of polymerization is obtained by combining Equation

(7) and (8):

1 � eðrM; tÞ ¼ ðVsolidÞL

VL

¼ ð1 � e0Þð2nÞ3ðL3 � ðL� 1Þ3Þð4=3ÞpR3Cc

3LðrM; tÞ

ð4=3ÞpR3Cð2nÞ

3ðL3 � ðL� 1Þ3Þð9Þ

Simplification renders:

eðrM; tÞ ¼ 1 � ð1 � e0ÞðcLðrM; tÞÞ3 ð10Þ

One particular value of the void fraction corresponds to the

time when the critical polymer thickness necessary for the

fragmentation of the microspheres is reached in the cubic

cell, namely c¼ c*.

e*ðrM; tÞ ¼ 1 � ð1 � e0Þðc*ðrM; tÞÞ3 ð11Þ

The cell density can be calculated at this point. The mass of

polymer on each microsphere is computed as a function of

the growth factor in the selected cell:

mpðrM; tÞ ¼ rp½ð4=3ÞpR3m � ð4=3ÞpR3

C�¼ rpð4=3ÞpR3

C½ðc*ðrM; tÞÞ3 � 1� ð12Þ

With the information that the number of microspheres in a

cube is (3n-2þa) and that the number of microspheres in an

edge is n,[6] the cell density becomes:

rðrM; tÞ ¼ mc þ mp

V¼ mc

Vþ mp

V

¼ rM0þrp

ð3n� 2 þ aÞð4=3ÞpR3C½ðc*ðrM; tÞÞ3�1�

ð2nRCÞ3

ð13Þ

The support/catalyst mass in a cell,mc, and the cell volume,

V, are introduced to calculate r. At this stage of the

polymerization process, the dimensions of the cell have not

changed and the ratio of support/catalyst mass to cell-

volume in Equation (13) (mc/V) can be replaced by the

initial value, rM0, found in Equation (2). Density increases

are related to the right-hand side term in the equation, with a

growing mass of accumulated polymer being added to that

of the support/catalyst initially present.

After fragmentation, Equation (10) still holds for the

calculation of the void fraction since, while microparticles

rearrange, cube-cell dimensions do not change. The density

can be calculated with Equation (13), by simply writing c

instead of c*, while the basic cube-cell-skeleton volume

remains constant.

Finally, when the porosity reaches the value, ea, corres-

ponding to the end of the microparticle arrangement

process,[6] cell density reaches its maximum; from here

on, porosity does not change and the arrangement of

microparticles will grow because of polymer accumulation

without change in cell morphology (constant ea). Conse-

quently, there will be a gradual decrease in cell density

because of polymer accumulation. The fraction of void

volume is maintained, but the microparticles forming the

solid phase turn into a lighter material with an ever-

increasing ratio of polymer-mass to support/catalyst-mass

as time passes. The equation for the cell density after

rearrangement is dominated by the constant-porosity ea

condition. The expressions for the whole cell volume,V, and



the support/catalyst and polymer volumes, Vc and Vp,

respectively, are:

V ¼ Vc þ Vp

1 � ea

ð14Þ

Vc ¼ ð3n� 2 þ aÞð4=3ÞpR3C ð15Þ

Vp ¼ ð3n� 2 þ aÞð4=3ÞpR3Cðc3 � 1Þ ð16Þ

The cell density at any given time, after rearrangement is

completed, can be calculated with the expressions in

Equation (14)–(16):

r ¼rsVc þ rpVp

Vð17Þ

r ¼ ð1 � eaÞ rs

1

c3þ rp 1 � 1

c3

� �� ð18Þ

The behavior of the macroparticle density at very long time

approaches a final, limiting value fixed by porosity and

polymer density:

limc!1

r ¼ ð1 � eaÞ½rp� ¼ rMf ð19Þ

The density evolution of any cubic-cell in the macroparticle

follows a typical hump-type plot, as predicted by the model.

The curves are shown in Figure 2, in which density for a cell

is plotted vs. time. Conditions are the same as those given in

Table 1, except for the polymerization rate used, with a

kinetic factor, kf¼ 3. The results for three cells at dimen-

sionless radial positions 0, 0.5 and 1 inside the macro-

particle are presented. Fragmentation points are indicated

using the c* symbol. In the portions of the evolution

immediately before and after the fragmentation points,

density increases with time in all three cells shown because

of pore-filling by the polymer, whereas the cell volume is

unchanged. Density reaches a maximum value when rear-

rangement is finished, with porosity reaching e¼ ea. After

this point, the cell density decreases with an increasing

polymer mass fraction. A flat, almost horizontal portion of

the curves indicates a final value if the time is sufficient. For

the innermost cells, a fraction of the residence time in the

reactor is spent prior to fragmentation, signaling a lower

contribution to polymer production during this period.

For the parameters given in Table 1 and the correspond-

ing equations, cell density values in Figure 2 are 449.8 kg/

m3 at c*¼ 1.1, 770.8 kg/m3 at ea¼ 0.409, and 531.9 kg/m3

for long periods of time.

The analysis of density evolution above can be applied

to the macroparticle in Figure 1 to stress the simultaneous

presence of cells with microparticles in various arrange-

ments at any given time during the particle residence in the

reactor fluid. Figure 3 depicts the three regions participating

to different degrees of activity in the overall particle produc-

tion. Through the cell density at each stage, the progress in

the morphology-modification sequence is depicted. The

pictures for three typical cells after 200 s in the reactor are

schematically shown in Figure 3, for the conditions used to

compute the plots in Figure 2.

The nonfragmented nucleus (a) in the macroparticle in

Figure 1 is related to the cell schematically shown in the left,

upper region of Figure 3; this cell has not reached fragmen-

tation. The cube-cell not only keeps its initial dimensions,

but also the original microparticle distribution. Although

polymerization takes place, the net polyolefin accumulation

is relatively low, and thus, cell fragmentation occurs at

approximately 500 to 1 000 s. This piece of information is

important since, according to model predictions, a portion

of the microspheres will participate actively only after seve-

ral minutes in the reactor. Region (b) in the macroparticle in

Figure 1 is represented with the scheme in the left, lower

portion of Figure 3. Here, the cell has passed the fragmen-

tation point and is undergoing microparticle rearrangement,

while the cube-cell keeps the original dimensions. Finally,

the bottom-right schemes in Figure 3 correspond to cells in

region (c) of the macroparticle. Microparticles have passed

the ea mark, showing constant porosity and falling density

during this stage. The cell is no longer tied to the initial lcvalue, since the microparticles grow and push the outer limit

of the macroparticle constantly.

The general distribution of cells according to their mor-

phology in Figure 3 is representative of any typical polyme-

rization. However, the duration of each stage may vary

substantially, being a strong function of catalyst activity and

monomer concentration. The scheme in Figure 3 is com-

mon to a variety of process situations in future sections of

this work, but with varying time scales.

Figure 2. Cell density vs. time for kf¼ 3; cells located at theexterior (rM/RM¼ 1), at the center (rM/RM¼ 0) and at half theradius (rM/RM¼ 0.5) of the macroparticle. For all other conditions,see Table 1.



Reactor Feasible Operation Domain

Polyolefin synthesis is an exothermic process: in most

industrial technologies, the heat balance is the determining

factor in the reactor design. Industrial syntheses are affected

by the manner of how both the particles and the reactor deal

with the heat of polymerization and convey it to the exterior.

The case under study in this work corresponds to one of the

most favorable situations with regard to heat management,

since polymerization is performed in a CSTR with a well-

stirred liquid slurry carrying the growing support-catalyst-

polymer micro-reactors.

To perform the analysis of the active-site performance in

the macroparticle, the first set of data obtained from the

model is devoted to finding the thermal conditions the

macroparticle must tolerate in the reactor. In this section,

heat transfer is studied for various fluid temperatures and

stirring conditions, to establish the parameter range in

which the macroparticle will polymerize efficiently.

Heat Transfer from the Macroparticle

The Ranz-Marshall[14] and Nelson-Galloway[15] correla-

tions are used to model h, the heat transfer coefficient

between the particle and the surrounding fluid, as well as the

manner in which it varies with time as particle temperature

and dimensions change due to polymerization. Both corre-

lations are expressions for the Nusselt number as functions

of Prandtl and Reynolds numbers for a single sphere of

diameter, dp, into a fluid medium with density, rf, viscosity,

mf, heat capacity, Cpf, and thermal conductivity, ktf. The

main variable in the heat transfer analysis is h, whereas the

parameter considered is u, the particle velocity relative to

the fluid, as they appear in the Nu and Re numbers:

Nu ¼ hdp

ktf

ð20Þ

and

Re ¼ rfudp

mf

: ð21Þ

The Prandtl number, a relationship among the fluid

thermal properties and its viscosity is:

Pr ¼ mfCpf

ktf

: ð22Þ

The expression for the Ranz-Marshall’s correlation is:

Nu ¼ 2 þ 0:6Re1=2Pr1=3: ð23Þ

The Nelson-Galloway’s equation is:

Nu ¼2zþ 2z2ð1�cÞ1=3

1�ð1�cÞ1=3½ �2� 2

� �

z1�ð1�cÞ1=3 � tanh zh i ½ð1 � cÞ�1=3 � 1�; ð24Þ

where:

z ¼ ½ð1 � cÞ�1=3 � 1� b2Re1=2Pr1=3: ð25Þ

The equations above show functional relationships for

the Nu with the Re and Pr, the difference between the

expressions being the presence of a fluid volumetric

fraction, c, in the slurry, and of a bed-porosity related

functional z to include particle-to-particle effects in

Equation (24).

Figure 3. Microparticle arrangements after 200 s of polymerization; cube-cellslocated in the nucleus (top, left, rM< rM1), in the zone undergoing rearrangement(bottom, left, rM1< rM< rM2) and in the region with rearrangement completed(bottom, right, rM> rM2) of the macroparticle. Kinetic factor, kf¼ 3; for all otherconditions, see Table 1.



For the CSTR hosting the polymerization in a liquid

slurry under pseudo steady-sate conditions, a relatively

uniform fluid environment around a particle can be assum-

ed. From a global reactor perspective, the system is

homogeneous because the particle size distribution remains

virtually constant and is only affected by relatively neg-

ligible catalyst-input and product-output. However, the

conditions within each particle vary due to changes in

morphology and size during polymerization. Density chan-

ges may become relevant to the particle dynamics in the

fluid medium, but are of relative importance for the heat-

transfer correlations, affected mostly by the properties of

the liquid bulk phase. Conversely, the film heat-transfer

coefficient, h, is affected by macroparticle size.

To introduce several mixing conditions in a typical slurry

into the model, values of the macroparticle velocity, u, with

respect to the fluid are selected, ranging from 0.01 up to

0.1 m/s. For the fluid conditions under analysis, these

figures for u correspond to Reynolds numbers of 1.89 and

18.9, respectively, for a support-catalyst initial particle

diameter of 90 mm (Table 1) and TB¼ 323 K. These values

of Re0 are typical of laboratory slurries. A similar level of

mixing is also likely in commercial reactors, in which

values for the initial Reynolds number of the order of 1 to

10 times the base-case figures are, again, to be expected.

This is indicated in this work by means of factors, Ref,

ranging from 1, corresponding to Re0¼ 1.89 for the base-

case, to 10.

As it can be seen in Equation (23) and (25), the film

coefficient decreases with particle growth. However, and as

it was stated in ref.[12,13] in spite of the decay in h, the factor

effectively controlling the heat transfer is the product of

the film coefficient times the external area of the particle.

Because of the square-power dependence of hdp2 with the

particle size, the net heat transfer rate always increases with

polymerization, with a stronger dependence on dp for larger

radii.

Both the Ranz-Marshall and Nelson-Galloway correla-

tions are used in the model to predict temperature evolu-

tions, always choosing the most conservative result. The

goal is to detect the conditions for the polymerization such

that the particle will not be subject to temperatures above

383 K, as shown in what follows.

Maximum Temperatures

Maximum particle temperature is one of the parameters that

establishes the bounds of the practical operating-variable

domain in polyolefin reactors.[6,13,16] Although severe over-

heating is to be expected in industrial gas-phase polyolefin

synthesis,[17] heat transfer conditions are usually milder in

liquid-phase processes. If a thermal runway situation arises,

the macroparticle will not be able to evacuate the heat of

polymerization to the fluid at a rate high enough to keep the

temperature within controllable operational limits. In fact,

runaway conditions restrict the actual productivity of a

reactor, this being particularly important in gas-phase pro-

cesses, where, despite high-activity catalysts and significant

monomer pressures, the effective bound for the maximum

attainable yield is fixed by heat transfer conditions.

The model predictions for the thermal behavior of a

macroparticle in the case of the liquid slurry being studied,

under several operating conditions, are presented in

Figure 4.

When the system of equations [Equation (A-1)–(A-11),

Appendix] is simultaneously solved by the algorithm, the

value of T(rM,t) in the equations is predicted by the model.

The maximum value, Tmax, of the said variable, T(rM,t),

predicted at any point in the macroparticle during its resi-

dence time in the reactor is displayed in Figure 4. Maximum

temperatures reached in the particle are studied when

several values of the activity factor, kf, and the concentra-

tion factor, Cf, are employed. Simulations were performed

for the pairs, (kf,Cf)¼ (1,1), (2,2), (3,3), (4,4), (5,5) and

(6,6). Since kf¼Cf for all pairs, the plots are shown as

functions of (kf �Cf)1/2. The other independent variable in

the plot is the fluid temperature, TB, in the CSTR. This

variable stretches between 323 and 353 K, with the range

selected taking into consideration that fluid temperatures

below 323 K will diminish the reaction rate significantly,

whereas temperatures above 353 K will negatively affect

the heat transport from the particle to the fluid. The

parameter in the plot is the Reynolds factor, Ref.

In Figure 4, all maximum temperatures in the macro-

particle remain below 393 K, considered a theoretical upper

limit, since common practice is to run the reactors such that

Tmax remains below that value to prevent particle agglome-

ration and polymer degradation, and to avoid exposing the

macroparticle to conditions close to overheating. Accord-

ing to the practical limits usually accepted in polyolefin

polymerization for particle temperatures, it should be kept

below 383 K.

Results in Figure 4 indicate, for the set of catalyst and

reactor parameters chosen, the region of operation by

Figure 4. Maximum temperature reached in the macroparticlevs. liquid bulk phase temperature and the product (kf �Cf)

1/2 forRef¼ 10. For all other conditions, see Table 1.



means of a curved surface. Although it is not indicated in the

plots, Tmax develops at early stages during polymerization,

when the less favorable conditions related to the product,

hdp2 mentioned in the previous section are present. As

expected, macroparticle temperatures are higher for hotter

slurries as shown by the surface in the figure, with the

liquid-slurry temperature, TB, having a direct impact on the

particle temperature. Nonetheless, even considering DTmax

for the points in the curve with the highest value of TB

(TB¼ constant¼ 353 K), it is apparent from the figure that

the maximum temperature in the particle does not increase

substantially with (kf,Cf). This fact allows operation with

very active catalysts and high monomer concentrations in

the slurry.

Microparticle Effectiveness DuringPolymerization

Once no thermal limitations are found for the operating

conditions in the reactor, the analysis of the effectiveness of

the macroparticle as a micro-reactor is in order.

The polymer yield for the conditions used to explore the

domain in Figure 4 is considered in the first place. Results

are presented in Table 3 for Ref¼ 10. Yields in kg of poly-

mer per kg of the support-catalyst complex fed to the reactor

for a residence time of 8 000 s are displayed for slurry

temperatures, TB, between 323 and 353 K, and for the pairs,

(kf,Cf)¼ (2,2), (3,3), (4,5), (5,5) and (6,6). To stress the fact

that both the reaction rate and the monomer concentration

are increased at each point, Table 3 displays in the left-most

column the product, kf �Cf.

From the analysis of the figures in the table, it can be

observed that the larger kf �Cf, the higher the yield, as one

would expect. However, the growth of the latter is essen-

tially linear when the former is increased following a square

power. Additionally, raising the liquid temperature in the

slurry produces an increase in yield as expected, according

to the range for TB in the previous section. If percentage

variations are considered for yields at any given kf �Cf, it can

also be seen that the incremental rate decreases indicating

that, when higher kinetics are proposed, the raise in produc-

tivity exists, but at an ever-decaying incremental rate.

To explore the reasons for this behavior, a closer look at

the evolution of the microparticles inside the macroparticle

is performed. Two important variables are followed in time:

a) the monomer concentration available at the active sites

rendered dimensionless with the bulk liquid phase value

and b) the microparticle growth factor, cL, as defined in

Equation (1).

Since a typical macroparticle may have more than 1 000

cube-cell layers in its radius, it is of practical interest to

choose a few layers to follow the variables in the paragraph

above in a manner representative of their distribution in the

macroparticle. Typically, the outer- and inner-most layers

are chosen. In Figure 2, a third layer located at half the

macroparticle radius was chosen to complete the study at

three representative depths inside the macroparticle. At this

particular stage in the analysis, it is better to choose as the

interior representative layer that corresponding to the

surface that divides the sphere in two portions of equal

volume.With thisapproachandgiventhespherical symmetry

of the macroparticle, this layer will be a better representation

of an interior microsphere since, at initial conditions, the

number of active sites located towards the interior and the

exterior of the layer is approximately the same.

Considering the support-catalyst, initial particle as the

reference state to compute the radius at half-the-volume, the

value for such radius termed RHV is:

VM

2¼ 1

2ð4=3pR3

M0Þ ¼ ð4=3pR3HVÞ ) RHV ¼ RM0=ð2Þ1=3

¼ 0:794RM0 ð26Þ

With these considerations, the monomer concentrations are

explored in the case of a system with Ref¼ 10, TB¼ 323 K,

and with kf �Cf¼ 1 and 25, for microparticles located in

layers at the center, the outside and limiting the half-volume

of a macroparticle, respectively. For this system, the dimen-

sionless monomer concentrations available in the cell-pores

for the microparticles at said locations are plotted as a

function of time in Figure 5. The full lines correspond to the

base case scenario with kf �Cf¼ 1, whereas the dotted

lines show the concentration evolution when a very active

catalyst is placed in a concentrated monomer environment

(kf �Cf¼ 25).

Table 3. Polymer yield at 8 000 s for several activities andtemperatures (Ref¼ 10).

kf �Cf TB Polymer yield Maximum temperature

K kg/kg K

4 323 8 603 324.25333 11 992 334.84343 13 344 345.61353 14 179 356.67

9 323 17 793 325.49333 22 159 336.66343 23 742 348.25353 24 911 360.36

16 323 29 154 327.13333 33 602 339.16343 35 559 352.02353 37 173 366.03

25 323 42 016 329.27333 46 199 342.60343 48 588 357.66353 49 560 371,12

36 323 55 852 332.11333 59 802 347.65343 62 792 361.30353 64 048 382.00

a)All other conditions as given in Table 1.



When the curves for the outermost microparticles are

considered, it is apparent that in a few seconds both systems

reach a value close to 1 and remain in an almost constant

fashion all the way up to the 8 000 s mark adopted in this

plot. Although the kf �Cf¼ 25 plot shows a somewhat lower

concentration, the difference between the curves is

minimal.

At the half-volume radius, the situation is different. For

the kf �Cf¼ 1 condition, the microparticle at this location

will be exposed to a monomer concentration at least 20%

lower than that feeding the outermost microparticles until

almost one hour into the process, to settle at a 15% diffe-

rence after two hours. The variation is stronger for the

kf �Cf¼ 25 case, where these percentages are approxi-

mately 40 and 30%, respectively.

The microparticles located at the macroparticle center in

the kf �Cf¼ 1 case exhibit a behavior intensifying the trend

observed in the previous paragraph: at 4 000 and 8 000 s, the

concentration is 50 and 40% lower than in the cells located

in the exterior surface of the macroparticle, respectively.

When kf �Cf¼ 25, a cell in the macroparticle center will

remain virtually without significant monomer concentra-

tion for more than 3 000 s. After 8 000 s, the dimensionless

concentration will be only 0.3. Taking into account that

common residence times in a slurry are in the 4-h range, if

the center of the macroparticle is low in monomer for

almost one half that period its participation in polymer

production will be relatively low.

Combining the information in the profiles described in

the paragraph above, it is clear that when both the catalytic

activity and the monomer concentration in the liquid phase

in the slurry are increased the net effect is such that a portion

of the macroparticle will not have enough monomer as to

effectively contribute to the overall yield. This fact could

explain the linear increase in yield when rising kf �Cf in a

square-power fashion as mentioned earlier when discussing

Table 3. Another side effect of the mentioned increase

predicted by the model is on the macroparticle morphology.

Low concentrations as those shown for cells in the center of

the macroparticle for high kf �Cf are associated with a slow

accumulation of polymer on the nonporous nucleus of the

microspheres. This, in turn, could delay the cell fragmenta-

tion and the subsequent microparticle rearrangement, with a

postponement of the accessibility increase that these two

processes cause.

Growth-Factors inside the Macroparticle

To measure microparticle contribution to macroparticle

yield at various positions, the evolution of the growth factor

in Equation (1) is displayed in Figure 6 as a function of the

activity product, kf �Cf. A three-dimensional surface is used

to show the distribution of the relative size of the micropar-

ticles in cells located at the macroparticle center (Figure 6b)

and at the external portion of the macroparticle (Figure 6e).

The surface is presented for the pairs (kf,Cf)¼ (1,1), (2,2),

(3,3), (4,4), (5,5) and (6,6), with kf¼Cf in all pairs, and

displayed for the same time range used in Table 3 and

Figure 5.

The followings are observations from Figure 6b at the

macroparticle center: 1) The line for the kf¼Cf¼ 1 case,

placed on the vertical back-plane of the plot, shows a steady

increase with time from the very beginning of polymeriza-

tion. A final value on the order of 16 is reached after 8 000 s

in the reactor. 2) The curve for kf¼Cf¼ 6 shows no

significant polymer accumulation on the microspheres for

the first 5 000 s. Later, the growth factor increases up to a

value of approximately 26 after 8 000 s of residence time.

The final 16 and 26 values are of the same order of magni-

tude, but corresponded to activities kf �Cf¼ 1 and 36, res-

pectively, that were significantly different. 3) Lines for

intermediate values of (kf,Cf) show behaviors that depict a

transition between the lines at 1 and 36. An almost diagonal

line can be drawn on the inferior coordinate plane between

the point at [(kf,Cf)¼ (6,6), time¼ 5 000 s] and at [(kf,Cf)¼(1,1), time¼ 0 s] to show the moment the microparticle at

the center will start growing for each (kf,Cf) pair in the

(2,2)–(5,5) range. The maximum values for the growth

factor, reached at 8 000 s for all of these lines, show a peak

of 28 for the pair (kf,Cf)¼ (4,4).

The surface in Figure 6e for the cells at the exterior of the

macroparticle shows that: 1) The line for the kf¼Cf¼ 1

case (vertical back-plane) shows the same steady-increase

trend from the beginning as in Figure 6b. The final value is

on the order of 20, close to that for the macroparticle center.

2) For kf¼Cf¼ 6, the line is similar in shape to that for

Figure 5. Dimensionless monomer concentration vs. time incells located at the exterior (rM/RM¼ 1), at the center (rM/RM¼ 0)and at the layer limiting half the volume (rM/RM¼ 0.794) of themacroparticle. Activity factors kf �Cf¼ 1 (filled line) and 25(dotted line); Ref¼ 10. For all other conditions, see Table 1.



(kf,Cf)¼ (1,1), with larger polymer accumulation for all

times. A final value of 65 is observed. 3) Intermediate lines

for (kf,Cf) ranging from (2,2) to (5,5) show analogous

behaviors, with maximum values for the growth factor, all at

8 000 s, monotonously increasing from the already men-

tioned 20 to 65.

The rate of increase of the growth factor decreases with

time in both Figure 6b and e regardless of kf �Cf. This is a

consequence of the plots showing cL, a measure of the

growth of the microparticle radius, Rm, as a function of

time. The phenomenon driving the growth is polymer-mass

accumulation; assuming constant polymer-stratum density

in the microparticle, the determining force is polymer-

volume accumulation which is a cubic function of Rm and,

hence, cL.

If model predictions in parts (b) and (e) are compared the

first observation is consistent with the findings in Figure 5

for monomer concentration in the cells. The growth of a

microparticle is driven by polymer accumulation on the

microsphere nonporous nucleus. For a microparticle, the

catalytic activity for any chosen kinetic factor, kf, is cons-

tant with time; hence, the variation in monomer availability

is the deciding factor in terms of the extent of polymeriza-

tion at this particular microparticle. A better look at cell

concentrations is in order, since it appears as being the

dominating variable in the process.

ConcentrationDistributionsAcross theMacroparticle

Monomer concentration evolution at the macroparticle

locations already employed in Figure 5 were obtained for

the set of simulation-runs corresponding to Table 3 and

Figure 6; results are shown in Figure 7. Curves for the

dimensionless monomer concentration in cells at the center,

the layer dividing the macroparticle volume in half, and the

exterior layer are plotted as a function of time, with the

product, kf �Cf, as the parameter. The effect of increasing

either kf or Cf was found to show no symmetry. Hence, to

maintain the comparison basis kf �Cf is set equal to 1, 4, 9,

16, 25 and 36 in all cases, but three columns are presented

in Figure 7. They show for each of these products the effect

of achieving them with values of kf<Cf (left column),

with kf¼Cf (center column), and with kf>Cf (right

column). The exact values employed for kf and Cf are

given in Table 4.

From the curves corresponding to kf¼Cf, it is possible to

observe the trend depicted in Figure 5, now for the various

pairs (kf,Cf) employed: 1) External cells rapidly enjoy and

steadily maintain high values of the monomer concentra-

tion for all (kf,Cf) cases, with maximum differences below

3% between the curves corresponding to the limiting case

for (1,1) and (6,6) presented. 2) At the half-volume position,

cells show a decay in concentration, more pronounced for

the higher values of (kf,Cf). Curves also indicate through the

shift of the shapes to the right that longer times are required

to reach a stable, practically flat profile. 3) For the innermost

cells, the situation is similar to that in the layer at half-

volume for (kf,Cf)¼ (1,1) and (2,2). However, a depletion in

monomer availability is observed for (kf,Cf)¼ (3,3) to (6,6).

As seen before, the period of time elapsed with the cell

practically not performing any polymerization due to lack

of monomer is longer for higher (kf,Cf). For (5,5) and (6,6),

the inner cells remain inactive for almost one half of the

macroparticle overall residence time in the reactor.

Figure 6. Microparticle growth factor vs. time and the product (kf �Cf)1/2, in cells located at

the center [(a), (b) and (c)] and at the exterior [(d), (e) and (f)] of the macroparticle, for kf<Cf

(left column), kf¼Cf (center column) and kf>Cf (right column); Ref¼ 10. For all otherconditions, see Table 1.



A different situation is predicted if the values of kf and Cf

are varied according to Table 4 to explore the effect of

relatively increasing/decreasing concentration vs. kinetics,

keeping the product kf,Cf constant for comparison purposes.

For the outermost cells in the upper rectangles in Figure 7,

there is no significant modification in the profiles when kf is

relatively increased or decreased with regard to Cf. Curves

are essentially those observed in the situation with kf¼Cf.

Nonetheless, observing the plots in detail, it can be noted

that when kf<Cf the line for the evolution with kf �Cf¼ 36

is closer to that for kf �Cf¼ 1 than it is when kf¼Cf.

Conversely, if kf>Cf plots for kf �Cf¼ 36 and 1 are more

separated if, again, compared with those in the kf¼Cf case.

Please note that for kf �Cf¼ 1 the lines depicting the

concentration evolution are exactly the same for the plots in

all three vertical columns following the definitions in

Table 4 (those for kf<Cf, kf¼Cf and kf>Cf, respectively).

The rectangles containing the plots located in the middle

horizontal section of Figure 7 contain the monomer concen-

tration evolution for the cells locates in the layer separating

the macroparticle into two equal volume domains. Here, the

effect described in the precedent paragraph is repeated.

Again, all curves for kf �Cf¼ 1 are identical in the left,

central and right half-volume individual plots and serve as a

reference. Increasing kf vs. Cf reduces the monomer

availability at all values of kf �Cf, the higher the product,

the stronger the effect. In practical terms, if one takesM/MB

for kf �Cf¼ 36 at a macroparticle residence time of 4 000 s,

half-way to the exit, when kf<Cf monomer concentration

is 50% higher than for kf>Cf. (M/MB¼ 0.6 and 0.4,

respectively). And additional fact related to the increasing

dominance of kf can be seen from the plots: a) For kf>Cf the

plots separate rapidly from the kf �Cf¼ 1 reference line. For

kf �Cf¼ 4, the departure is already noticeable. b) When

kf<Cf, it can be observed that the plot for kf �Cf¼ 4, with

kf¼ 1.333 and Cf¼ 3, crosses above the line for kf �Cf¼ 1,

showing for the first time a sequence inversion considering

the order when increasing kf �Cf that, up to this point, shows

a monotonous behavior.

As expected, for the three plots corresponding to the cells

in the macroparticle center, at the bottom horizontal band in

Figure 7, the plots describe a behavior conceptually similar

to that for the half-volume cells but stronger.

For kf �Cf¼ 36 and kf>Cf, the innermost cells do not

contribute substantially to the polymerization process since

Figure 7. Dimensionless monomer concentration vs. time for activity factors (kf �Cf)¼ 1,4, 9, 16, 25, and 36, in cells located at the exterior (rM/RM¼ 1), at the center (rM/RM¼ 0) andat the layer limiting half the volume (rM/RM¼ 0.794) of the macroparticle, for kf<Cf (leftcolumn), kf¼Cf (center column) and kf>Cf (right column); Ref¼ 10. For all otherconditions, see Table 1.

Table 4. Values of kf and Cf used in Figure 7.

kf �Cf kf>Cf kf¼Cf kf<Cf

kf Cf kf Cf kf Cf

1 1.000 1.000 1 1 1.000 1.0004 3.000 1.333 2 2 1.333 3.0009 4.000 2.250 3 3 2.250 4.000

16 5.000 3.200 4 4 3.200 5.00025 6.000 4.166 5 5 4.166 6.00036 7.000 5.143 6 6 5.143 7.000



little monomer is available for the reaction in the active sites

at this location. Once more, the separation between the plots

for kf �Cf¼ 1 and 4 is more pronounced than in the half-

volume cells in Figure 5.

Conversely, if kf<Cf, monomer concentration for

kf �Cf¼ 36 starts to grow at around 2 700 s and reaches a

value close to 0.4 at 8 000 s, this implying a considerable

improvement with regard to the kf>Cf condition. Again, an

inversion is observed when kf<Cf in the sequence for

the curves: concentrations for the curve at kf �Cf¼ 4 run

above those for kf �Cf¼ 1, as it was the case at the half-

volume radius.

It is now convenient to mention that these concentration

distributions correlate with the growth factors at the macro-

particle center in the sections termed (a) and (c) in Figure 6:

for the kf>Cf case, the surface in the macroparticle center

is mostly flat, indicating very small microparticles in the

cells; when kf<Cf, the whole surface is mostly separated

from the zero-value bottom coordinate plane, indicating

microparticles contribute to yield even deep in the

macroparticle.

As shown in the upper plots in Figure 7, the outermost

microparticles behave similarly regardless of the relative

values of kf and Cf. Consequently, three-dimensional plots

(d), (e) and (f) in Figure 6 are almost identical.

Exploring Kinetics vs. Concentration Effects

The analysis of the data in the previous section indicates

that the overall efficiency of the polyolefin production will

be affected dissimilarly if, to increase overall productivity,

the catalyst activity and the monomer concentration are

raised in a different manner. Figure 8 is devoted to analy-

zing the practical impact of the relative values of kf and Cf

upon the macroparticle behavior, and, hence, the polymer-

ization yield. First, a dimensionless relative polymer yield

is introduced as the ratio of the yield attained with a given

pair (kf,Cf) to that at kf �Cf¼ 1. Then, in Figure 8, this

relative yield is plotted as a function of kf �Cf for three

situations, namely at kf>Cf, kf¼Cf and kf<Cf. The

relative polymer yield is used as a measure of the reactor

efficiency when kinetics and concentration effects are

studied. A 458 dotted line in the plot is placed to make the

deviations from the behavior at kf �Cf¼ 1 visible. The

specific yields corresponding to the products 1, 4, 9, 16, 25

and 36 are shown in Figure 8.

First, it is predicted that higher yields are obtained when

the catalyst is designed and the reactor conditions selected

in a manner such that concentrations are predominant over

kinetic constants. The curve for relative yields correspond-

ing to kf<Cf is always above that for the inverse relation-

ship, with the line for kf¼Cf placed in between them. For

kf �Cf¼ 25, yields are approximately 40% higher if Cf> kf

when compared with case at Cf< kf (relative yields of 21

and 15, respectively).

As hinted to in Figure 7 by the sequence inversion

observed in the half-volume and center-cell plots when

kf<Cf, the corresponding relative polymer yield curve lies

above the 458 reference line, indicating, for low kf �Cf, an

efficiency higher than that in the reference (1,1) case.

Another general observation from the modeled results in

Figure 8 is that related to the fact that in all cases, save the

domain kf �Cf¼ [1–9], higher activities are followed by a

decrease in relative yield regardless of the relative monomer

concentration value. The plots in the indicated range bend

downwards for increasing activities, moving away from the

458 reference line.

As a general note, model predictions presented in

Figure 5–8 suggest a shift in the relative role played by

transport and kinetics in the highly nonlinear transport and

reaction processes taking place inside the macroparticle

concurrently with morphology changes. More active cat-

alysts are in principle desired, as well as higher monomer

concentrations in the liquid bulk-phase of the slurry.

However, the higher reaction rates associated with this

raise in kf �Cf could also produce a more rapid decay of the

reactant locally, in each of the cells. Additionally, cells

inside the particle are depending on the monomer amounts

existing in cells exteriorly located. If the latter are consum-

ing the monomer at a faster rate, less will be available for

active sites located to the macroparticle interior. This

situation is compounded with the fact that if monomer

access is more time-consuming and, consequently, polymer

accumulation slower, fragmentation, needed to increase

accessibility, will be further delayed.

Since fragmentation is strongly dependent on the struc-

tural properties of the support, the picture presented here

Figure 8. Relative macroparticle polymer yield vs. activityfactor kf �Cf, for kf>Cf, kf¼Cf and kf<Cf. Ref¼ 10. For allother conditions, see Table 1.



can be intensified or minimized according to the higher or

lesser breaking resistance of the silica matrix. It becomes

apparent at this point that the model can be used to predict

these trends provided the proper parameters, in particular,

the critical accumulation, c*, corresponding to the specific

support being studied is included in the model.

Conclusion

The cube-cell model and a set of parameters were used to

predict the behavior of the support-catalyst-polymer parti-

cle in the reactor. With this mathematical tool, it is possible

to build a realistic representation of the morphologies for

some typical silica supports employed in high-activity

metallocene-catalyst preparation into the model. The mo-

del, in turn, allows the coupling of the polymerization/heat-

and mass-transfer analysis with the transient in the

macroparticle structure occurring during particle growth.

In this work, the main effort is focused on modeling the

overall picture (transportþ polymerizationþmorphology

changes) despite complications derived from not assuming

instant fragmentation.

The model has shown its ability to predict the porosity

and density changes undergone by the macroparticle and

their interaction with the transport/polymerization phe-

nomena determining the said changes. The evolution of

these variables is followed by the model concurrently with

the changes in chemical and transport forces producing the

structural changes, thus recognizing their interactive nature.

Scouting simulations were performed with the model

under various operating conditions to assess the relative

significance of several variables of the process for a set of

typical silica-supported metallocene-based catalysts. Para-

meters used in the simulations to tailor-make polymeriza-

tion conditions are the kinetic constant, through the number

and chemical nature of the active sites deposited on the

microparticle nonporous nucleus, and the amount of mono-

mer available in the reactor fluid phase. According to

simulations, for the parameters used the catalyst activity

and the monomer concentration have the expected impact

on polymer production, although not in a similar manner

when each of them is varied.

Modeling the several stages in the morphology-changes

translates into density variations for cells at different

locations inside the macroparticle. This density distribution

at any given time is the first indicator of the ability of the

microparticles to effectively contribute to the macroparticle

yield. The cells near the macroparticle center are always in a

less convenient position when compared with those located

closer to the liquid bulk-phase, in terms of reaching the

final, accessible microparticle configuration that emerges

after fragmentation and subsequent rearrangement. The

relative extent of the time needed to attain this final stage,

characterized by an almost constant density, when com-

pared with macroparticle residence time in the reactor indi-

cates the actual participation of a cell at any given location

within the support-catalyst-complex.

Heat transfer and temperature elevation during poly-

merization do not pose practical limitations to the process.

For the domain studied, no overheating was predicted by the

model.

One of the conclusions of the analysis is that increasing

the kinetic constant should not necessarily produce a

proportional increase in polymer production. Moreover,

predicted results show that it is more efficient to raise the

monomer availability in the bulk phase. The highly non-

linear coupling of: a) particle morphology changes at all

levels, b) monomer transport, and c) polymerization creates

complex situations reflecting the interactive nature of these

phenomena. In the particular case of the macroparticle

yield, simulations suggest as practical to increase monomer

concentrations in the liquid reactor-phase feeding the active

sites above any increase in catalyst activity. It is clear that

the results may vary for other support-catalyst systems and

their corresponding parameter sets. The relevant point is

that the model may prove useful to analyze prospective

reactor conditions for each support-catalyst system and its

data, to prevent potential problems in their usage and to help

in the optimization of the economics of the process.

Appendix

The macroparticle is modeled as a pseudo-homogeneous

medium containing a solid phase composed of cells with

microparticles (nonporous, support-catalyst solid micro-

spheres covered by a growing polymer stratum) and a

porous region through which the monomer gains access to

the active sites located in the microparticles.[6] Spherical

symmetry is assumed for microparticles and the overall

macroparticle. In turn, the polymer stratum on the micro-

spheres is treated as pseudo-homogeneous. Both the com-

position and the temperature in the liquid bulk phase are

uniform and constant. External mass transfer limitations are

neglected, as well as temperature gradients across the

polymer stratum covering the microspheres.

The equations for mass and energy balances within the

macroparticle are:

@MðrM; tÞ@ t

¼ 1

r2M

@

@rM

DL

tr2

M

@MðrM; tÞ@ rM

� �� RðrM; tÞ

eðrM; tÞðA-1Þ

@TðrM; tÞ@t

¼ 1

r2M

@

@ rM

kM

rMcpM

r2M

@MðrM; tÞ@ rM

� �

� DHrMcpM

RðrM; tÞ ðA-2Þ



@M

@rM

¼ 0;@T

@rM

¼ 0 rM ¼ 0; t ¼ t ðA-3Þ

M ¼ MB;@T

@rM

¼ h

kM

ðT � TBÞ rM ¼ RMðtÞ; t ¼ t

ðA-4Þ

M ¼ 0; T ¼ T0 rM ¼ rM; t ¼ 0 ðA-5Þ

The pseudo-homogenous rate of polymerization in Equa-

tion (A-1) and (A-2) is modeled as a continuous sink term

R(rM,t):

RðrM; tÞ ¼ ð2nÞ3ð1 � e0Þ½L3 � ðL� 1Þ3�4p3½ðRLðtÞÞ3 � ðRL�1ðtÞÞ3�

� 4pR2CkðTÞMCðrM; tÞ ðA-6Þ

with k, the kinetic constant per unit area, being a function of

temperature:

kðrM; tÞ ¼ kðT0Þ expE

Rg

1

T0

� 1

TðrM; tÞ

� �� ðA-7Þ

At the microparticle level, the mathematical model includes

monomer transport across the polymer layer, and the

chemical reaction at the catalyst active sites is located on the

microsphere external surface boundary. The monomer first

dissolves into the polymer layer, and then moves across its

amorphous region, neglecting transport through the crystal-

line region. Temperature gradients in the polymer layer are

considered negligible. Expressions are:

a@M0ðrm; tÞ

@t¼ 1

r2m

@

@r

aDEP

tm

r2m

@M0ðrm; tÞ@rm

� �ðA-8Þ

aDEP

tm

@M0

@rm

¼ kM0 rm ¼ RC; t ¼ t ðA-9Þ

M0 ¼ ZMðrM; tÞ rm ¼ Rm; t ¼ t ðA-10Þ

@Rm

@ t¼ Mh i kR

2C

rp

M0ðRC; tÞR2

m

ðA-11Þ

The system of equations from Equation (A-1)–(A-11) are

solved simultaneously[6] using a numeric scheme.

Table 1 gives the parameters and constants employed in

the base, reference case used in the analysis.

Nomenclature

a number of additional microspheres added to the

basic cube-cell

b Iler’s coordination number

c dimensionless growth-factor for a microparticle in

a cube-cell

cpm macroparticle heat capacity (cal/kg �K)

c* growth factor at the fragmentation point

Cf concentration factor, ratio of MB to the base-case

monomer concentration in Table 1

Cpf slurry fluid-phase heat capacity (cal/kg �K)

dp macroparticle diameter (m)

DEP monomer diffusion coefficient in amorphous

polymer (m2/s)

DL monomer diffusion coefficient in the fluid medium

(m2/s)

E activation energy (cal/g �mol)

h macroparticle external heat-transfer coefficient

(cal/m2 � s �K)

k superficial kinetic constant (m/s)

kf kinetic factor, ratio of k to the base-case kinetic

constant in Table 1

kM macroparticle thermal conductivity (cal/m � s �K)

ktf slurry fluid-phase thermal conductivity

(cal/m � s �K)

lc characteristic edge dimension of a support cube-

cell (nm)

L macroparticle-layer number, with L¼ 1 at the

particle center

m mass (kg)

mc mass of supportþ catalyst in a cell (kg)

mp mass of polymer in a cell (kg)

M monomer concentration (mol/m3)

MB monomer concentration in the slurry fluid-phase

(mol/m3)

MC monomer concentration at the active sites (mol/m3)

M0 monomer concentration in the amorphous region of

the polymer layer(mol/m3)

hMi monomer molecular weight (kg/kg �mol)

n number of microspheres located between two

adjacent vertices of a support cube-cell

N overall number of layers in the macroparticle

NL number of microspheres in layer, L

Nu Nusselt number, Equation (20)

P pressure (atm)

Pr Prandtl number, Equation (22)

rm microparticle radial coordinate (nm)

rM macroparticle radial coordinate (mm)

R monomer consumption rate (mol/m3 � s)

RC radius of non-porous support microspheres in cube-

cell microparticles (nm)

Re Reynolds number, Equation (21)

Ref Reynolds number factor, ratio of Re0 to the base-

case initial Reynolds number of 1.89

Rg universal gas constant (cal/g �mol �K)

RHV radius of macroparticle layer, L, dividing the

macroparticle volume in two halves

RL external radius of macroparticle layer, L

Rm radius of microparticle (nm)

RM radius of macroparticle (mm)

RM0 radius of initial support-catalyst particle (mm)

t time (s)



T temperature (K)

T0 initial temperature of the particle (K)

TB slurry fluid-phase temperature (K)

u macroparticle velocity relative to the slurry fluid-

phase (m/s)

V cell volume (m3)

Vc support-catalyst volume in cell (m3)

Vp polymer volume in cell (m3)

Vsolid solid volume (m3)

VL layer-L volume (m3)

VM macroparticle volume (m3)

Greek Letters

a amorphous-phase volume fraction of the polymer

layer

b parameter in Equation (25), dimensionless value¼0.543

DH heat of polymerization (cal/g �mol)

e porosity

e0 initial porosity

ea final-product porosity, after the rearrangement pro-

cess is completed

e* cell porosity at fragmentation time

Z monomer-in-polymer solubility

mf slurry fluid-phase viscosity (kg/m � s)

r cell density (kg/m3)

rf slurry fluid-phase density (kg/m3)

rM macroparticle density (kg/m3)

rM0 initial support-catalyst particle density (kg/m3)

rMf final-product macroparticle density (kg/m3)

rp polymer density (kg/m3)

rs silica density (kg/m3)

t tortuosity in the macroparticle

tm chain immobilization factor

c slurry fluid-phase volumetric fraction

z Equation (25), functional in Nu expression in

Equation (24)

Subscripts

L corresponding to layer L

0 initially

Acknowledgements: The authors are deeply grateful to theUniversidadNacional del Litoral (CAIþD 4/23) and toCONICETand SETCIP for their financial support.

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