Impact of Initiation and Deactivation on Melting during Gas-Phase

We determine here the impact of accounting also for the polymerization initiation and deactivation reactions on the safe region of operation, i.e., on...
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Ind. Eng. Chem. Res. 2004, 43, 4789-4795

4789

Impact of Initiation and Deactivation on Melting during Gas-Phase Olefin Polymerization Hao Song and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4004

Olefin polymerization in fluidized-bed reactors may lead to particle overheating. The resulting softening or melting of polymer particles generates polymer sheets, requiring reactor shutdown. Previous criteria predicting the operating conditions leading to this overheating accounted only for the propagation reaction. We determine here the impact of accounting also for the polymerization initiation and deactivation reactions on the safe region of operation, i.e., one in which the maximum transient particle temperature does not reach the melting temperature. The polymeric-flow model predicts a slightly smaller region of safe operation than a lumped-thermal model and uniformly distributed catalytic sites. The reason is that the polymeric-flow model predicts a slightly higher maximum temperature under the same operating conditions. In general, accounting for the initiation and deactivation steps increases the region of safe operation. A decrease of either the initiation or propagation rate increases the size of the safe operating region, while decreasing the deactivation rate decreases its size. The deactivation impact is important mainly when its rate is not much slower than that of initiation. Parametric sensitivity does not occur before melting for typical gas-phase olefin polymerization operation. Thus, this potential problem need not be considered. Introduction Industrial experience revealed that gas-phase polymerization of olefins in fluidized-bed reactors can lead to local overheating of the formed polymer particles. The softening or melting of polymer particles causes them to stick to each other, forming polymer “sheets”, which require a shutdown of the reactor. To avoid this highly undesired event, it is important to be able to predict a priori the operating conditions for which this overheating does not occur, i.e., the region in which the operation is safe. Several investigations of overheating of growing particles in gas-phase olefin polymerization have been reported.1-6 Those studies accounted only for the polymer chain propagation, during which most of the heat is released, and not for the chain initiation and deactivation steps. This is equivalent to assuming that the chain initiation is instantaneous and the deactivation rate of the active sites is much slower than the rate of particle overheating. Under these assumptions, the total number of active sites (i.e., the total number of living polymer chains) within a growing particle is constant during its sojourn in the reactor. Assuming a uniform distribution of the active sites within a growing particle, the active sites concentration is C*(t) ) C*(0) Vp(0)/Vp(t) ) C*(0)/ gr3, where gr is the particle growth factor [gr ) R(t)/R(0)]. In addition, when a pseudo-steady-state assumption is made, it is possible to obtain simple bounds on the operation conditions for which melting does not occur.1 A simplified reaction network of olefin polymerization is

where C0 denotes the concentration of potential active sites, C* active sites with no attached polymer chain, C*(pj) active sites with an attached living polymer with chain length j. The total concentration of living polymer ∞ chains C* is ∑j)0 C*(pj). Values of the reaction rates and activation energies of the chain initiation, propagation, and deactivation have been reported in the literature.7-12 Neglecting the chain initiation and catalyst deactivation steps may lead to large errors in the predicted maximum temperature during the initial period of polymerization, during which this maximum is attained. The goal of this study is to examine the impact of accounting for the initiation and deactivation rate on the boundaries of safe operation. We determine the bounds on the operating conditions that do not lead to particle overheating during olefin polymerization, as predicted by both a lumped-thermal uniformly distributed model and the more realistic polymeric-flow model.

Lumped-Thermal Uniformly Distributed Catalyst Model We use first a lumped-thermal uniformly distributed catalyst model to predict the boundary of the safe operating conditions region. That model ignores any intraparticle temperature gradients, assumes that the active site concentration remains uniform in the growing polymer particle, and accounts for the transient reaction-diffusion interactions by an isothermal effectiveness factor. The model consists of the equations

dM ) kcav(Ma - M) - kp(Tp) C*Mη dt * To whom correspondence should be addressed. Tel.: (713) 743-4305. Fax: (713) 743-4323. E-mail: [email protected].

Fpcp

(1)

dTp ) hfav(Ta - Tp) + (-∆H)kp(Tp) C*Mη (2) dt

10.1021/ie049767g CCC: $27.50 © 2004 American Chemical Society Published on Web 07/08/2004

4790 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004

d(C*Vp) ) [ki(Tp) C0 - kd(Tp) C*]Vp dt

(3)

d(C0Vp) ) -ki(Tp) C0Vp dt

(4)

dVp Mw ) k (T ) C*MηVp dt Fp p p

(5)

3(ψ coth ψ - 1)

x

kp(Tp) C* De

[ (

∆Ej 1 1 kj(Tp) ) kj(Ta) exp R Ta Tp

)]

(7)

(j ) i, p, d)

(8)

M ) C* ) 0, Tp ) Ta, C0 ) C0(0), Vp ) Vp(0), t ) 0 (9) Introducing the dimensionless variables and parameters

M Ma

R gr ) R(0) x* ) ∆Ep RgTa

γp )

φp ) R(0)

C* C0(0) γi )

x

kp(Ta) C0(0) De

y)

Tp Ta

C0 C0(0)

∆Ei RgTa

γd )

φi ) R(0)

∆Ed RgTa

x

ki(Ta) De

φd ) R(0)

Bim )

kc(0) R(0) De

Le )

Fpcpkc(0) hf(0)

β)

x

kd(Ta) De

(-∆H)kc(0) Ma

p)

hf(0) Ta MwMa Fp

(10)

and using the fact that the particle Sh (Sh ) kcR/Db) and Nu numbers (Nu ) hfR/λf) are constant for small particle Re number,13 the following dimensionless equations are obtained:

dx 3Bim ) (1 - x) - φp2 exp[γp(1 - 1/y)]xx*η 2 dτ g r

dgr gr ) p φp2 exp[γp(1 - 1/y)]xx*η dτ 3

(15)

ψ ) φpgrxx* exp[γp(1 - 1/y)]

(16)

where

The corresponding initial conditions are

(17)

Typical ranges of the physical and dimensionless parameters encountered in gas-phase olefin polymerization are reported in Tables 1 and 2, respectively. We define the melting set (MS) as the set of parameters at which the maximum particle temperature is the melting point of the polyolefin, i.e.,

ymax ) ymelt

t τ) R(0)2/De x0 )

dx0 ) -φi2 exp[γi(1 - 1/y)]x0 - pφp2 exp[γp(1 dτ 1/y)]xx*x0η (14)

x ) x* ) 0, y ) 1, x0 ) 1, gr ) 1, τ ) 0

The corresponding initial conditions are

x)

dx* ) φi2 exp[γi(1 - 1/y)]x0 - φd2 exp[γd(1 - 1/y)]x* dτ pφp2 exp[γp(1 - 1/y)]xx*2η (13)

(6)

ψ2

ψ)R

dy 3Bim (1 - y) + βφp2 exp[γp(1 - 1/y)]xx*η (12) ) 2 dτ g r

where

η)

Le

(18)

The MS divides the plane of two parameters into two regions. In one, denoted as the safe operating region, the maximum particle temperature is lower than the polyolefin melting temperature. In the second, the maximum particle temperature exceeds the olefin melting temperature. The predictions of the model are not valid in that unsafe region because it accounts neither for the melting nor for the behavior of a molten polymer. To compute the MS, we first find by numerical iterations two close parameter values on the set. We then use linear extrapolation to estimate a new point. By small perturbations of the estimated parameter values and iterations, we determine the exact value within the desired accuracy. This numerical continuation is used to determine the MS within any desired parameters region. In addition to finding the MS, we also check whether the maximum particle temperature is sensitive to small perturbations in the system parameters, i.e., whether parametric sensitivity (PS) exists in the safe operating region. It is important to know whether PS occurs for temperatures lower or higher than the polymer melting point. Cleary, if PS occurs for temperatures exceeding the melting point (for which the model is invalid), it is not a practical constraint. Following Morbidelli et al.,14-19 we define the critical PS as the set of input parameters at which the objective sensitivity s(ymax,π)

s(ymax,π) ) ∂ymax/∂π

(19)

(11) attains its extremum value. We choose β as the bifurca-

Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4791 Table 1. Typical Ranges of Parameter Values Encountered in Gas-Phase Olefin Polymerization1-13 R(0) [µm] polymerization rate [kg of polymer/ (g of catalyst‚h)] kp(353 K) C0(0) [s-1] ki(353 K) [h-1] kd(353 K) [h-1] De [cm2/s] Ma [mol/cm3] Ta [K] (-∆H) [J/mol] ∆Ep, ∆Ei, and ∆Ed [kJ/mol] λf [J/(cm‚s‚K)] λe [J/(cm‚s‚K)] Fpcp [J/(cm3‚K)]

5-50 4-100 60-1500 5-50 0.5-20 8 × 10-5-5 × 10-4 1.2 × 10-3-4 × 10-4 343-363 ∼108 000 (polyethylene) ∼104 000 (polypropylene) 12-65 1.2-3 × 10-4 0.8-2 × 10-3 ∼1.26

Table 2. Typical Ranges of Dimensionless Parameters Encountered in Gas-Phase Olefin Polymerization γp, γi, and γd φp φi φd β

4-20 0.5-5 0.004-0.04 0.0005-0.02 2-6.5

Bim Bih Le p

10-60 0.1-0.3 5-30 0.01-0.03

Figure 1. Dependence of the general objective sensitivity s(ymax,β) on β and the corresponding transient maximum particle temperature. The critical objective sensitivity attains its maximum at βc.

tion parameter π and compute s(ymax,β) by the finite difference18,19

s(ymax,β) )

ymax(β+∆β) - ymax(β) ∆β

(20)

Figure 1 illustrates the dependence of the objective sensitivity on the input parameter β and the corresponding maximum particle temperatures. We denote by βc the β value at which the general objective sensitivity attains its maximum value. For β < βc, the maximum particle temperature is relatively low, while for β > βc, the maximum particle temperature is very high, indicating a runaway operation. The sensitivity of the maximum particle temperature to changes in β increases at large values of γp. Figure 2 describes the MS and PS in the (β, γp) plane. The maximum transient catalyst temperature is below the melting temperature for all of the operating conditions in the region below the MS. We refer to that set of parameters as the safe region. To illustrate the impact of the initiation and deactivation rates on the MS, we show also the MS for a case that accounts only for the propagation rate, which is equivalent to assuming an infinite rate of initiation and no deactivation. As expected, the finite rate of initiation and deactivation decreases the maximum temperature, and this increases the size of the region of safe operation. The locus of the PS boundary is above the MS, i.e., in the unsafe region in which the maximum temperature exceeds melting. This implies that PS does not occur in the safe operating region. The reason that PS was always above the MS is that for typical polyolefin polymerization when s(ymax,β) attains its maximum, the maximum particle temperature exceeds 200 °C. The melting temperatures of most polyolefins are well below that value. Figure 3 illustrates this argument, showing that the maximum particle temperature on the PS locus is much higher than the melting temperatures of polyolefins. Under normal gas-phase olefin polymerization conditions, PS is always encountered at temperatures exceeding the polyolefin melting point even under extreme ratios of the initiation to the deactivation rate of either 5:1 or 1:5. Thus, one need not be concerned in these cases with the possible occurrence of PS.

Figure 2. Loci of the MS and PS in the β-γp plane: (s) accounting for propagation, initiation, and deactivation; (- -) accounting only for propagation. φp ) 2.5, Bim ) 20, Le ) 25, and p ) 0.019.

Figure 3. PS and the corresponding maximum particle temperature Tmax. φp ) 2.5, φi ) φd ) φp/x18000, γi ) γd ) 10, Bim ) 20, Le ) 25, and p ) 0.019.

The simulations indicate that the PS curve asymptotically approaches the MS from above for large values of γp. For the case shown in Figure 2, the ∆β difference between PS and the MS is on the order of 10-5 when γp > 15, so that the MS and PS are practically the same in the β-γp plane for large γp. For small γp, the sensitivity of ymax with respect to β is small and just a few iteration steps are needed to determine the MS. However, at high γp values, many more iterations are required to determine the MS within the specified

4792 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004

Figure 5. Impact of ki(Ta) (or equivalently of φi) on the MS. γi ) 10, γd ) 10, φp ) 2.5, Bim ) 20, Le ) 25, and p ) 0.019 when (a) ki(Ta)/kd(Ta) ) 1 and (b) ki(Ta)/kd(Ta) ) 10.

Figure 4. Impact of γi/γd on the MS. γi ) 10, φp ) 2.5, φi ) φp/ x18000, Bim ) 20, Le ) 25, and p ) 0.019 when (a) ki(Ta)/kd(Ta) ) 1 and (b) ki(Ta)/kd(Ta) ) 10.

accuracy. The calculation of the locus of the PS at large γp is much easier than that of the MS. Thus, at high γp, the PS may be used as a very close upper bound on the MS. The development of a catalyst requires understanding and an ability to predict the impact of the three rate constants (ki, kp, and kd) and the corresponding activation energies (∆Ei, ∆Ep, and ∆Ed) on the maximum particle temperatures and the region of safe operation. To gain this insight, we constructed the MS in the β-γp plane. Figure 4 shows the impact of γd on the MS when it is either larger, equal, or smaller than γi for a fixed ratio of ki(Ta)80 °C)/kd(Ta). Clearly, when γd > γi, the heat release by the reaction and the resulting temperature rise increase the rate of deactivation more than that of the initiation. The inverse effect occurs when γd < γi. The calculations show that increasing γd enlarges the safe region because this increases the impact of deactivation [at a fixed kd(Ta)], which, in turn, decreases the maximum particle temperature. A comparison between parts a [for which ki(Ta)/kd(Ta) ) 1] and b [for which ki(Ta)/kd(Ta) ) 10] of Figure 4 shows that the impact of γd on the MS decreases as the ratio ki(Ta)/kd(Ta) becomes larger, i.e., when the initiation step is much faster than deactivation. The ratio of γi/γd has a noticeable impact on the MS only if kd(Ta) is of the same order of magnitude as ki(Ta). The reason is that when kd(Ta) is much smaller than ki(Ta), it has a small impact on the initial reaction period, during which the particle attains its maximum temperature. The fact that the safe region in Figure 4b is smaller than that in Figure 4a indicates that an increase in the deactivation rate constant increases the size of the safe operating region because it decreases the maximum particle temperature. Computation of the MS for different ratios of ki(Ta)/ kd(Ta), shown in Figure 5, indicates that for a fixed φp decreasing the initiation rate constant ki(Ta) (or equivalently of φi) increases the size of the region of safe operation. The reason is that a decrease of ki(Ta)

Figure 6. Impact of kp(Ta) (or equivalently of φp) on the MS. γi ) 10, γd ) 10, φi ) 2.5/x18000, Bim ) 20, Le ) 25, and p ) 0.019 when (a) ki(Ta)/kd(Ta) ) 1 and (b) ki(Ta)/kd(Ta) ) 10.

increases the time at which the catalyst attains its maximum activity and decreases the maximum activity. Calculations, shown in Figure 6, indicate that an increase in kp(Ta) (or equivalently of φp) for a fixed φi decreases the region of safe operation. This is due to the increased reaction rate and corresponding rate of heat release, which, in turn, increases the maximum particle temperature. A comparison of the cases shown in parts a and b of Figure 6 shows that the region of safe operation is larger for higher values of the deactivation rate constant. The reason is that a higher deactivation rate decreases the maximum particle temperature. Predictions by the Polymeric-Flow Model The polymeric-flow model is a distributed parameter model. Unlike the lumped-thermal model, it accounts for the special distribution of the active sites and temperature within the particle. To check the sensitivity of the predicted boundary of the safe operating conditions region on the assumptions of the lumped-thermal uniformly distributed model, we computed these MSs also by the more realistic polymeric-flow model. That model assumes that the local expansion of the growing particle pushes the incompressible polymer and active sites outward at a velocity u.20-25 The polymeric-flow model is described by the set of equations

∂M 1 ∂ 2 ∂M ) De 2 r - kp(Tp) MC* ∂t ∂r r ∂r Fpcp

(

(

)

)

(21)

∂Tp 1 ∂ 2 ∂Tp ) λe 2 r + (-∆H)kp(Tp) MC* (22) ∂t ∂r r ∂r

Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4793

∂C* 1 ∂ ) - 2 (r2uC*) + ki(Tp) C0 - kd(Tp) C* (23) ∂t r ∂r ∂C0 1 ∂ ) - 2 (r2uC0) - ki(Tp) C0 ∂t r ∂r

(24)

Mw 1 ∂ 2 (r u) ) k (T ) MC* 2 ∂r Fp p p r

(25)

Introducing the following additional dimensionless quantities:

ξ)

r R(0)

β* )

v)

uR(0) De

(-∆H)DeMa λeTa

Le* )

Bih )

FpcpDe λe

hf(0)R(0) λe

(26)

Figure 7. Comparison of the MSs predicted by the lumpedthermal and polymeric-flow models. φp ) 2.5, Bim ) 20, Bih ) 0.2, Le ) 25, p ) 0.019, and Tmelt ) 120 °C.

the dimensionless model equations are

∂x 1 ∂ 2 ∂x ) ξ - φp2 exp[γp(1 - 1/y)]xx* (27) ∂τ ξ2 ∂ξ ∂ξ

( )

Le*

∂y 1 ∂ 2 ∂y ξ + β*φp2 exp[γp(1 - 1/y)]xx* ) ∂τ ξ2 ∂ξ ∂ξ (28)

( )

∂x* 1 ∂ 2 )- 2 (ξ vx*) + φi2 exp[γi(1 - 1/y)]x0 ∂τ ξ ∂ξ φd2 exp[γd(1 - 1/y)]x* (29) ∂x0 1 ∂ 2 )- 2 (ξ vx0) - φi2 exp[γi(1 - 1/y)]x0 ∂τ ξ ∂ξ 1 ∂ 2 (ξ v) ) pφp2 exp[γp(1 - 1/y)]xx* ξ2 ∂ξ

Figure 8. MS for a base case (solid line) and following a 50% increase of either ki, kp, and kd. For the base case, kd(Ta)/ki(Ta) ) 1, Bim ) 20, Bih ) 0.2, Le ) 25, and p ) 0.019.

(30) (31)

The corresponding initial and boundary conditions are

x ) x* ) v ) 0, y ) x0 ) gr ) 1, τ ) 0

(32)

∂x ∂y ) ) v ) 0, ξ ) 0 ∂ξ ∂ξ

(33)

∂x* ∂x0 ∂y Bih ∂x Bim (1 - x), (1 - y), ) ) ) ) 0, ∂ξ gr ∂ξ gr ∂ξ ∂ξ ξ ) gr(τ) (34) We used a finite-difference scheme with a nonuniform grid size to describe the spatial derivatives and integrated the resulting set of algebraic-differential equations by the LIMEX solver.26 The MS was computed by numerical continuation. The numerical calculations indicate that the maximum transient temperature predicted by the polymericflow model, which accounts for the intraparticle temperature gradient, is somewhat higher than that predicted by the lumped-thermal model. Consequently, the size of the safe operation region predicted by the polymeric-flow model is somewhat smaller than that predicted by the lumped-thermal model. This difference is clearly illustrated by the difference in the MS for the two models shown in Figure 7. The figure also shows the MS computed by a polymeric-flow model that

accounts only for the propagation step. It shows that also for this model accounting for the initiation and deactivation increases the size of the safe region because of the associated decrease in the maximum temperature. The same qualitative trends are predicted by both the polymeric-flow and lumped-thermal models. Specifically, increasing the value of either the propagation rate constant kp or the initiation rate constant ki decreases the size of the safe region because of the faster initial rate of heat release that leads to a higher maximum transient particle temperature. On the other hand, a larger deactivation rate constant kd increases the size of the safe operation region because it decreases the rate of heat release and hence lowers the maximum transient temperature. These qualitative features are illustrated in Figures 8-10, which show the impact of increasing one of the three rate constants by 50%. The difference between the three cases is that kd in Figure 8 is 5 times larger than that in Figure 9. This difference in the value of the deactivation rate constant causes the safe region of the base case in Figure 9 to be smaller than that in Figure 8. Moreover, it decreases the magnitude of the shift in the MS upon a 50% change in any one of the three rate constants. On the other hand, the value of kd in Figure 8 is 5 times smaller than that in Figure 10. Consequently, the safe region of the base case in Figure 10 is much larger than that in Figure 8. Moreover, the larger kd base case magnifies the shift in the MS upon a 50% change in any one of the three rate constants.

4794 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004

Figure 9. MS for a base case (solid line) and following a 50% increase of either ki, kp, and kd. For the base case, ki(Ta)/kd(Ta) ) 5, Bim ) 20, Bih ) 0.2, Le ) 25, and p ) 0.019.

constants, the higher their sensitivity to the temperature rise caused by polymerization and the larger is their impact. The impact of changes in the initiation and deactivation rate constants depends strongly on their relative values and on their sensitivity to changes in temperature, i.e., to the corresponding activation energies. Changes in the deactivation rate are much more pronounced when its rate is comparable or larger than that of initiation, so that it has an impact on the time at which the maximum polymerization activity is attained. Increasing the value of the propagation rate constant increases the rate of heat release and hence the maximum temperature. Thus, increasing its value will always decrease the size of the safe operation region. The fact that PS does not occur before melting is important because it implies that there is no need to address this problem in this class of reactions. Acknowledgment We are most thankful to Michael E. Muhle for very useful discussions and suggestions and to the NSF for financial support of the research. Notation

Figure 10. MS for a base case (solid line) and following a 50% increase of either ki, kp, and kd. For the base case, ki(Ta)/kd(Ta) ) 0.2, and under the condition Bim ) 20, Bih ) 0.2, Le ) 25, and p ) 0.019.

Concluding Remarks The rate of heat release during the polymerization is most strongly affected by the propagation rate constant. Hence, any increase in its sensitivity to temperature changes, i.e., an increase in its activation energy, increases the maximum transient particle temperature and decreases the maximum exothermicity (β values) of the reaction (monomer concentration) for which the operation is safe. This trend is observed in all of the figures of the MS. Accounting for the finite rate of catalyst initiation and for its deactivation decreases the maximum temperature from that predicted by models that account only for the propagation rate constant. Thus, the more detailed model leads to a larger region of safe operation than that predicted by the models that do not account for the initiation and deactivation of the catalytic sites. This point is illustrated by Figures 2 and 7 for the two models. The maximum transient temperature rise of a polymer particle is obtained during its initial growth. Increasing the values of the initiation and deactivation rate constants leads to two conflicting effects. The initiation rate constant affects mainly the period needed for the particle to reach its maximum polymerization activity and to some extent this maximum activity. The faster the initiation step is, the higher is the maximum temperature and the smaller is the size of the safe operation region. The faster the deactivation is, the lower is the maximum polymerization rate and the shorter is the time at which this activity is attained. Thus, higher deactivation rate tends to decrease the maximum transient temperature and increase the size of the safe operation region. The higher the activation energy of either the initiation or deactivation rate

av ) ratio of the particle external surface area to the volume, 1/cm Bih ) Biot number for heat transfer, defined by eq 26 Bim ) Biot number for mass transfer, defined by eq 10 cp ) polymer particle heat capacity, J/(g‚K) C* ) concentration of all active sites, mol of sites/cm3 C*(pj) ) active sites with an attached living polymer with chain length j C0 ) concentration of the potential active sites, mol of sites/ cm3 C0(0) ) initial concentration of the potential active sites, mol of sites/cm3 De ) monomer effective diffusivity, cm2/s ∆E ) activation energy, J/mol gr ) dimensionless particle radius, defined by eq 10 hf ) heat-transfer coefficient, J/(cm2‚s‚K) -∆H ) heat of polymerization, J/mol kc ) mass-transfer coefficient, cm/s kd ) deactivation rate constant, 1/s ki ) initiation rate constant, 1/s kp ) propagation rate constant, cm3/(mol of sites‚s) Le ) Lewis number for the lumped-thermal model, defined by eq 10 Le* ) Lewis number for the polymeric-flow model, defined by eq 26 M ) monomer concentration, mol/cm3 Mw ) molecular weight of the monomer, g/mol p ) dimensionless parameter, defined by eq 10 r ) radial position R ) particle radius, cm Rg ) gas constant, J/(mol‚K) s ) objective sensitivity, defined by eq 19 t ) time, s T ) temperature, K Vp ) particle volume, cm3 u ) velocity of moving outward active sites, cm/s v ) dimensionless velocity, defined by eq 26 x ) dimensionless monomer concentration, defined by eq 10 x0 ) dimensionless deposited catalytic sites concentration, defined by eq 10 x* ) dimensionless total active sites concentration, defined by eq 10

Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4795 y ) dimensionless temperature, defined by eq 10 ymelt ) dimensionless polymer melting temperature Greek Symbols β ) dimensionless temperature rise, defined by eq 10 β* ) dimensionless temperature rise, defined by eq 26 γ ) dimensionless activation energy, defined by eq 10 η ) isothermal effectiveness factor, defined by eq 6 λe ) thermal conductivity of the particle, J/(cm‚s‚K) λf ) thermal conductivity of the monomer, J/(cm‚s‚K) ξ ) dimensionless particle radial position, defined by eq 26 Fp ) particle density, g/cm3 τ ) dimensionless time, defined by eq 10 ψ ) nonisothermal Thiele modulus, defined by eq 7 φ ) Thiele modulus based on ambient conditions and the initial particle size, defined by eq 10 Superscript max ) maximum Subscripts a ) ambient d ) deactivation i ) initiation p ) propagation, particle

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Received for review March 24, 2004 Revised manuscript received May 14, 2004 Accepted May 21, 2004 IE049767G