Modeling of Olefin Gas-Phase Polymerization in a ... - ACS Publications

Ind. Eng. Chem. Res. 2006, 45, 3081-3094

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PROCESS DESIGN AND CONTROL Modeling of Olefin Gas-Phase Polymerization in a Multizone Circulating Reactor† Jose´ L. Santos,‡ Jose´ M. Asua, and Jose´ C. de la Cal* Institute for Polymer Materials (POLYMAT) and Grupo de Ingenierıá Quı´mica, Facultad de Ciencias Quı´micas, The UniVersity of the Basque Country, Apdo. 1072, 20080 Donostia-San Sebastia´ n, Spain

The multizone circulating reactor allows the production of intimate mixtures of polyolefins of widely different characteristics, enlarging the envelope for properties of these polymers. A mathematical model for the gasphase polymerization of olefin in a multizone circulating reactor (MZCR) is presented. The system is modeled as a series of two interconnected polymerization zones working with different gas-phase compositions. The model takes into account particle and reactor levels, as well as the particle population balance. Simulations show that a wide range of product characteristics can be achieved by varying the operation conditions in the reactor and that very active catalysts are required to fully exploit the advantages of this reactor. 1. Introduction Over the last 10-15 years, there have been remarkable developments in high performance catalysts for olefin polymerization both in terms of improved activity and stereospecificity. These developments have stimulated new reactor designs capable of achieving further improvements in process technology. These major scientific and technical breakthroughs have expanded the property envelope of polyolefins, increasing their market share. End-use properties of polymer materials are determined by the polymer microstructure and morphology. In catalytic polyolefins, the polymer microstructure is defined to a large extent by the molecular weight distribution (MWD), the copolymer composition, and the chemical composition distribution (CCD). In some cases, practical advantages can be obtained by broadening the MWD or the CCD. For example, polymers with broad or bimodal MWD present a better balance between rheological behavior (processability in melt state) and final mechanical properties than polymers with narrow MWDs. The production of polymers with broad MWDs has been achieved by using two reactors in series operating under different conditions1,2 and using mixed catalyst systems.3,4 However, a certain segregation between high and low molecular weight polymer inevitably occurs in these systems, which negatively affects properties. To obtain an intimate mixture of different polymers, Basell has developed a new reactor called the multizone circulating reactor (MZCR).5-8 A simplified scheme of this new reactor is presented in Figure 1. In the first polymerization zone (riser), the growing polymer particles flow upward under fast fluidization conditions and, then, they are separated from the transport gas and enter a second polymerization zone (downer) through which they flow downward in a densified form under the action of gravity. To establish the circulation of polymer around the two polymerization zones, the polymer is transferred from the †

This article is dedicated to Prof. Cecilia Sarasola. * To whom correspondence should be addressed. Telephone number: +34 43 01 53 31. Fax number: +34 43 01 52 70. E-mail address: [email protected]. ‡ Current address: Basell Poliolefinas Ibe´rica SA, Technical Centre, Apdo. 18, 43080 Tarragona, Spain.

Figure 1. Multizone circulating reactor scheme.

bottom of the downer through an “L” valve to the bottom of the riser. The main characteristic of the MZCR is the possibility of obtaining in-reactor intimate blends of polymers having different molecular weights and/or compositions. This is achieved by maintaining different gas compositions in the two polymerization zones. Low molecular weight polymer is produced in the riser by feeding a hydrogen rich reaction mixture through point A. The downer is maintained substantially free of molecular regulator by means of feed C (preferably located just below the solid level), which avoids the gas phase that comes from the riser to enter into the downer and also adjusts the gas composition in the downer. The prepolymerized catalyst is fed into the riser, the makeup monomer is continuously fed through point D, and the product is withdrawn from the bottom of the downer. The high recycling rate of the solids in the reactor allows it to obtain a high homogeneity of the product, i.e., to produce an intimate blend of polymer chains of different chain lengths and, hence, to minimize the segregation between low and high molecular weight polymer at the active center level (Figure 1). It is claimed9 that the MZCR is more flexible than the traditional multiple-step technologies as it has the possibility of producing a wide range of polymers, from polypropylenes with wide or narrow MWDs to polypropylene and ethylene/ propylene rubber blends. In this work, a mathematical model for the MZCR is presented. The model accounts for the polymerization at particle and reactor levels and considers the particle size distribution

10.1021/ie0500523 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/23/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006

by using the particle population balance. The model was applied to the isothermal gas-phase polymerization of propylene. First, the sensitivity of the model to the diffusional limitations in the polymer particle was assessed. Second, the influence of process conditions and catalyst characteristics on product properties, such as molecular weights and polydispersity, were studied. This model differs from the one existing in the literature10,11 in several ways. Thus, our model takes into account the particle size distribution and its evolution with time. The model was used to check the effect of the gas-solid relative velocities in the downer, the solid recycling flow rates, the location of the gas feed in the downer, and the catalyst characteristics on process productivity, the riser/downer polymer production ratio, and the polymer properties. 2. Model Description 2.1. Basic Assumptions. The mathematical model developed for the MZCR outlined in Figure 1 included the following assumptions: (1) The catalyst is a single site catalyst; (2) All catalytic sites were activated, and for the sake of simplicity, it was considered that the catalyst did not suffer deactivation. (3) The gas phase included monomer, hydrogen, and an inert gas. (4) Chain transfer to hydrogen was the sole chain stopping event. (5) For the sake of simplicity, the influence of the hydrogen on the reaction rate12 was not considered. (6) Polymerization in the polymer particles was described by means of the multigrain model13 (MGM). In this model, the polymer particles are called macroparticles, which in turn are formed by microparticles. In this paper, the names of particles and macroparticles are used indistinctly. (7) The flow of the gas phase and polymer particles in both the riser and the downer was described by the plugflow model. (8) The reactor temperature was perfectly controlled at the set-point. (9) The reactor was initially charged with polymer. (10) At time zero, the gas-phase compositions in the riser and downer were those of streams A and C. (11) The downer height was that corresponding to the particulate bed, with a fixed porosity. (12) The particle size distribution (PSD) fed to the MZCR was the one obtained at the exit of two continuous stirred tank prepolymerization reactors which operated under steady-state conditions (slurry polymerization, 6 min. residence time in each reactor, T ) 35 °C, and a monomer concentration of 3 × 10-3 mol/cm3 in the feed). (13) Monodispersed spherical catalyst particles of 50 µm in diameter were continuously fed into the prepolymerization reactors. (14) A key aspect of this reactor is avoiding solid particle fluidization in the downer in order to have a packed moving bed. So, the gas-solid relative velocity must be lower than the minimum fluidization one:

umf g upart - ug

(1)

2.2. Model Equations. The model included different levels of description. The polymer particles were considered to be represented by the multigrain model. In this model, each particle was composed by a number of microparticles, each of them having a core formed by a catalyst fragment surrounded by a growing polymer shell. In this model, the mass transfer resistances occurred at two levels: mass transfer through the particle and mass transfer through the polymer shell of the microparticle. The reactor was modeled as a loop consisting of two plug flow sections with a recycling ratio R defined as the ratio between the recycling and outlet solid flows. The reactor model included the mass and momentum balances as well as the polymer particle population balances. The mass balances

allowed the calculation of the concentration of the reactants at any point of the reactor. The momentum balance gave the gas velocities in the various parts of the reactor. It is worth pointing out that the velocity of the solid in the riser was the gas velocity minus the terminal velocity of the particles in the gas phase. The polymer particle population balance gave the particle size distribution. 2.2.1. Balances at the Particle Level. The particle was considered to be represented by the multigrain model. The polymerization occurred at the surface of the catalyst fragments located within the microparticles. To reach the catalyst, the reactants should diffuse through the polymer shell. 2.2.1.1. Monomer and Hydrogen Mass Balances in the Microparticles. The non-steady-state mass balance for the species i was given by the classic nonreactive diffusion equation in a sphere

s

∂Fis ∂τ

)

(

)

∂Fis 1 ∂ 2 D r Si s ∂rs rs2 ∂rs

(2)

where DSi is the diffusion coefficient of species i in the polymer shell. The reaction rate appeared in the boundary condition at the surface of the catalyst fragment

rs ) rc w 4πrc2DSm rs ) rc w 4πrc2DSH2

∂Fms ∂rs

[C] ) kp F [Np] mc

∂FH2s

[C] ) kt F 1/2MWH21/2 ∂rs [Np] H2c

(3) (4)

where [Np] is the number of microparticles per unit volume of particle and [C] is the concentration of active centers per unit volume of particle (mol/L). The other variables are explained in the Nomenclature section. The boundary conditions at the surface of the microparticle were the following:

rs ) Rs w Fms ) η/mFmp

(5)

rs ) Rs w FH2s ) ηH/ 2FH2p

(6)

which indicates that the concentration at the surface of the microparticle was given by the equilibrium solubility of the reactant in the polymer. The characteristic time for diffusion is substantially shorter than that required for particle growth. Therefore, the pseudosteady state can be applied to eq 2. Under these circumstances, an analytical solution for eq 2 can be obtained.14 For the monomer, the solution is as follows

η/mFmp

F mc ) 1+

kp[C] 4π[Np]DSm

(

1 1 rc rc + e

)

(7)

τ ) 0 w Fmc ) Fmce For hydrogen (low values of kt and high value of DSH2), simple calculations show that the denominator of eq 7 is equal to one. Therefore, the concentration of hydrogen at the surface of the catalyst fragment is given by

FH2c ) ηH/ 2FH2p

(8)

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3083

Figure 2. Macroparticle monomer profile (a) for a particle size of 250 µm and (b) for a particle size of 2000 µm: (b) QSSA; (-) non steady-state condition.

2.2.1.2. Monomer and Hydrogen Mass Balances in the Macroparticles. The non-steady-state mass balance for species i is given by the classic diffusion-reaction equation in a sphere

p

∂Fip ∂τ

)

(

)

∂Fip 1 ∂ 2 D r - Rtransf.i Pi p ∂rp rp2 ∂rp

(9)

where Rtransf.i is the mass transfer rate to the microparticles, which under pseudo-steady-state conditions is equal to the rate of polymerization at the surface of the catalyst fragments

Rtransf.m ) kp[C]Fmc

(10)

Rtransf.H2 ) kt[C]FH2c1/2MWH21/2

(11)

Figure 3. Active centers profile (a) for a particle size of 250 µm and (b) for a particle size of 2000 µm: (b) [M]b ) 10-3 mol/cm3; (O) [M]b ) 5 × 10-5 mol/cm3; (-) neural network profile.

µm. It can be seen that QSSA can be safely used. This means that monomer diffusion is much faster than particle growth, and thus, the monomer profile along the particle is instantaneously adjusted to process conditions. The same conclusions applied for hydrogen which has a higher diffusion coefficient. Equations 7, 10, and 11 include the concentration of active sites ([C]), the number of microparticles per unit volume of particle ([Np]), and the thickness of the polymer shell in the microparticles (e). To calculate these variables, the corresponding balances should be developed. 2.2.1.3. Concentration of Active Centers in the Macroparticle. Because catalyst deactivation was not considered, the concentration of active centers depended only on particle growth.

∂[C] 1 ∂ ) - 2 (u[C]rp2) ∂τ r ∂rp

(14)

p

The boundary conditions for eq 9 are

rp ) 0 w rp ) Rp(V) w DPi

∂Fip ∂rp

∂Fmp ∂rp

rp ) 0 w )0

) kSi(Fi - Fip|rp)Rp(V))

(12) (13)

where kSi is the mass transfer coefficient at the particle surface. The complexity of the mathematical model is substantially reduced if the quasi-steady-state assumption (QSSA) could be applied to eq 9. To check this point, the growth of a single polymer particle was simulated for different conditions. Simulations were carried out for an initial particle size of 50 µm and a monomer diffusivity in the macroparticle equal to 1 × 10-3 cm2/s, varying both the propagation rate constant (kp ) 7.2 × 105, 2 × 106, and 3.2 × 106 cm3/(mol s)) and the bulk monomer concentration ([M]b ) 1 × 10-3, 6 × 10-4, 5 × 10-5, and 8 × 10-5 mol/cm3). Figure 2 compares the results obtained using QSSA with the solution obtained under non-steady-state conditions for the case of kp ) 2 × 106 cm3/(mol s) and [M]b ) 5 × 10-5 mol/cm3 when the particles reached a size of 250 and 2000

∂[C] )0 ∂rp

(15)

where u is the rate of particle growth (u ) ∂rp/∂t). Although all particles contained the same total amount of catalyst, each particle had a different history, and hence, the particle size and the distribution of active centers in the particle were in principle decoupled. To check the extent to which the history of the particle affected the distribution of active centers within the particle, the growth of two particles under widely different monomer concentrations was simulated. It was found that the activity profile did not depend on the history of the particle, but on the particle size (an example is presented in Figure 3 for two monomer concentrations and two particle sizes). This simplified tremendously the mathematical model because this relationship could be calculated independently, and in this work, it was represented by a neural network with two inputs (particle size and dimensionless position) and a hidden layer of six neurons, the concentration of active centers at the chosen dimensionless position being the output. The neural network was trained with results obtained integrating eq 14 under about 50 different conditions. Figure 3 shows that the neural network

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perfectly represented the dependence of the active distribution on the particle size. 2.2.1.4. Number of Microparticles per Unit Volume of Particle. Because no catalyst fragmentation after the prepolymerization stage was considered, the concentration of microparticles was ruled by particle growth.

∂[Np] 1 ∂ ) - 2 (u[Np]rp2) ∂τ r ∂rp

the convective flux in the reactor and the mass transfer to the solid particles.

r

∂Fi ∂Fiug ) -r - kSiav(Fi - Fip|rp)Rp(V)) ∂t ∂z

(16)

p

rp ) 0 w

∂[Np] )0 ∂rp

(17)

The reasons given to calculate the concentration of active centers by means of a neural network also apply to the concentration of microparticles. In addition, as eq 16 is very similar to eq 14, [Np] was calculated by means of the following equation:

[Np] ) [Np]inic([C]/[C]inic) [Np]inic ) 3(1 - p)/(4πrc3) (18)

x 3

3(1 - p) 4π[Np]

- rc

τ)0 w e)0

(19) (20)

Equation 19 considers that the particle porosity remains constant during the process. 2.2.2. Balances at Reactor Level. The reactor was modeled as a loop consisting of two plug flow sections with a recycling ratio R defined as the ratio between the recycling and outlet solid flow rates. Mass and polymer particle population balances for each reaction zone and for the connections between zones were developed. 2.2.2.1. Overall mass Balance in the Gas Phase. In each reaction zone, the gas-phase density was affected by the convective flux in the reactor and by the monomer and hydrogen transfer to the solid particles.

r

∂Fug ∂F ) -r - kSmav(Fm - Fmp|rp)Rp(V)) - kSH2av(FH2 ∂t ∂z FH2p|rp)Rp(V)) (21) t ) 0 w F ) Fe

(22)

z ) 0 w F ) Fe

(23)

where r is the reactor porosity, F is the gas-phase density, ug is the gas velocity, kSm and kSH2 are the monomer and hydrogen mass transfer coefficients, respectively, av is the surface area of the particles per unit volume of the reactor, Fm and FH2 are the densities of the monomer and hydrogen in the gas phase, respectively, Fmp|rp)Rp(V) and FH2p|rp)Rp(V) are the densities of the monomer and hydrogen at the particle surface, respectively, and Fe is the gas-phase density at the entry of the particular reaction zone. 2.2.2.2. Mass Balance for Monomer, Hydrogen, and Inert Species in the Gas Phase. In each reaction zone, the concentration of monomer and hydrogen in the gas phase depends on

t ) 0 w Fi ) Fie

(25)

z ) 0 w Fi ) Fie

(26)

where Fie is the density of the monomer (i ) m) and hydrogen (i ) H2) at the entry of the particular reaction zone. The porosity of the riser was calculated as shown by eq I.5 (Appendix I), and that of the downer was taken as that to a close packing of spheres (dr ) 0.35). Because the inert did not react in the polymer particles, the mass balance for the inert in the gas phase in each reaction zone is as follows:

r

Equation 18 implicitly considers that the number of active sites per catalyst fragment remains constant. 2.2.1.5. Polymer Layer Thickness in the Microparticle. The thickness of the polymer layer is the radius of the microparticle minus the radius of the core (catalyst fragment):

e)

(24)

∂Fine ∂Fineug ) -r ∂t ∂z

(27)

z ) 0 w Fine ) Finee

(28)

t ) 0 w Fine ) Finee

(29)

where Fine is the inert density and Finee the inert density at the entry. 2.2.2.3. Momentum Balance. The momentum balance for the gas in each reaction zone is as follows:

r

∂Fug ∂Fug2 ∂P ) -r - F - r - rFg ∂t ∂z ∂z

(30)

t ) 0 w ug ) ugs

(31)

z ) Lr w ug ) ugs

(32)

where P is the pressure, g is the acceleration of gravity (which is positive in the riser and negative in the downer), Lr is the length of the reaction zone, ugs is the gas-phase velocity at the outlet of the reaction zone, and F accounts for the friction losses. This term is very different in each reaction zone because in the riser the particles are transported by the gas, whereas in the downer the gas flows through a moving packed bed. The equations used to calculate the friction losses in each reaction zone are summarized in Appendix I. 2.2.2.4. Polymer Particle Population Balance. As shown in the balances at the particle level, both the monomer concentration and the active site concentration profiles depend on the size of the polymer particle. Even though a monodispersed catalyst was considered, the residence time distribution led to a particle size distribution that was calculated using a particle population balance. In each reaction zone, the polymer particle population balance is given by the following equation:

∂q(V)upart ∂rv(V)q(V) ∂q(V) )∂t ∂z ∂V

(33)

where the first term on the right-hand side accounts for the convective flux of particles in the reactor, the second accounts for the particle growth by polymerization, q(V) is the number


density of particles of volume V, upart is the velocity of the solid particles, and rv(V) the rate of volumetric growth of particles of volume V given by:

rV(V) )

∫0VF

kp[C]Fmc pol(1

- p)

dV

(34)

where kp is the propagation rate constant, [C] is the concentration of active centers, Fmc is the density of the monomer in the active centers, Fpol is the polymer density, and p the particle porosity. The following boundary and initials conditions were considered:

t ) 0 w q(V) ) q(V)inic

(35)

z ) 0 w q(V) ) q(V)e

(36)

V ) V0 w q(V) ) 0

(37)

∂q(V) )0 ∂V

V ) V* w

(1 -

dr )

q(V)ds +

Qpartcat

q(V)cat )

(1 - cat)

Qpartrec + Qpartcat q(V)re (1 - rre) (39)

To solve this equation, the reactor porosity at the riser entry, rre, must be calculated taking into account the solid flow and the solid particle velocity as follows:

Qpartrec + Qpartcat ) (ugr e - ut)(1 - rre)Srr

(40)

The particle size distribution at the entry of the downer, q(Vde ), was calculated from that at the exit of the riser as follows:

q(V)de (1 - dr )

)

q(V)rs (1 - rrs)

(41)

It was considered that the concentration of the reactants within the particles at the entry of each reactor zone was the same as that at the exit of the other zone. The gas density, composition, and velocity at the entry of each of the reactor zones were those of the feeds. The total number of polymer particles per unit volume of the reactor is

[Npart] )

∫VV*q(V) dV

parameter

value

(42)

0

2.2.2.5. Polymer Production. Polymer was withdrawn from the exit of the downer. The polymer production was calculated as the difference between the flow rate of the solids at the end

parameter

value

3.01 × 10-2 Fme (g/cm3) (downer) 4.4 × 10-2 Fe (g/cm3) (downer) 200 uparts - ugs (cm/s) (downer) µ (g/(cm s)) (riser) 1.02 × 10-4 µ (g/(cm s)) (downer) Lr (cm) (riser) 300 Lr (cm) (downer) Dr (cm) (riser) 24 Dr (cm) (downer) P (atm) 30 T (K) 0.65 r (downer) cat 3 -4 [C]inic (mol/cm ) 10 kp (cm3/(mol s)) DPm (cm2/s) 10-3 DSm (cm2/s) Fpol (g/cm3) 0.9 Fpart (g/cm3) p 0.3 ut (cm/s) Rcat (cm) 2.5 × 10-3 KSm (cm/s) / 1.72 ηm (Henry law) ηH/ 2 (Henry law) Qpartrec (cm3/s) 2205 Qpartcat (cm3/s) Fme (g/cm3) (riser) Fe (g/cm3) (riser) ugs (cm/s) (riser)

4.5 × 10-3 4.5 × 10-2 0 9.55 × 10-5 150 24 353 0.35 2 × 106 10-6 0.63 100 Ranz-Marshall 0.12 7 × 10-2

of the downer and the solid recycling flow rate (input of the model). d Qparts ) Qpart - Qpartrec s

(43)

t ) 0 w Qparts ) 0

(44)

(38)

where q(V)inic is the initial number density of particles in the reactor, z is the axial coordinate, q(V)e is the number density of particles at the reactor entry, V0 is the minimum particle volume of the particle size distribution, and V* is the maximum particle volume of the particle size distribution. The particle size distribution at the riser entry, q(V)re was calculated summing the contributions of the particles recycled from the downer and those of the catalyst feed.

Qpartrec

Table 1. Simulation Parameters in the Comparison between the Complete Model and the Pseudohomogeneous Model

The solids flow rate at the exit of the downer was calculated as follows: d

1 dQpart ) Sdr dz

∫V

Vd* d 0

q(V)

d

d kp[C]dFm c

∫0 F V

pol(1

- p)

dV dVd

(45)

t ) 0 w Qdpart ) Qpartrec

(46)

d z ) 0 w Qdpart ) Qpart e

(47)

where Qpart is the solid flow, Sr is the reactor section, and the superscript d denotes the downer. 2.3. Model Resolution. The mathematical model is a set of nonlinear partial differential equations (PDEs) and algebraic equations. These PDEs were reduced to a set of ordinary differential equations (ODEs) using orthogonal collocation.15-18 The integration of the final system was performed by means of a routine for differential-algebraic equations which uses the Petzold-Gear backward differentiation formulas method19 (code DDASPG, IMSL library). It was found that, because of the large number of equations of the model, the integration was time-consuming. In addition, simulations yielded almost flat concentration profiles for monomer and hydrogen in the macroparticles. Because the computation time could be substantially reduced if the resistances to mass transfer around and inside the macroparticle were negligible, simulations were carried out using both the whole model and a reduced model in which flat concentration profiles in the polymer particles were considered. The values of the process variables and the parameters used are presented in Table 1. These process variables were chosen in order to have similar polymer productions in each section of the reactor, a global particle residence time of 1.5 h, and a polymer output of about 20 kgpol/h. Since the main claimed characteristic of the MZCR is the possibility of obtaining in-reactor intimate blends of polymers having different molecular weights, the recycling ratio should be high. For that reason, a value of 5000 kgpol/h was chosen for the solid recycling flow rate, which gave a recycling ratio of 250.

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where [Npart]inic and [Npart]e are the initial and entry number of particles per unit volume of the reactor, respectively. 2.4. Molecular Weights. The moments of the MWD were calculated for each reactor zone.

FH21/2 ∂[c0] Fm / ∂[c0]upart η [c ] + kt ηH/ 1/2µL0 )- kp ∂t ∂z MWm m 0 MWH21/2 2 (54)

Figure 4. Final particle size distribution: (-) complete model; (- -) reduced model.

The reactor is shown in Figure 1, where stream C is located at the top of the downer and it was considered that the amount of gas exiting the downer with the product and the recycling polymer corresponds to the gas filling the interparticle space, namely, the relative velocity of the gas with respect to the polymer particles is zero in the exit lines in the downer. Also, for the sake of simplicity, it was considered that the gas phase only contained monomer and inert gas. Figure 4 presents the PSD of the final product for the complete and reduced models. It can be seen that both PSDs are nearly identical, showing that the simplified model was accurate enough. Therefore, the reduced model was used in the simulations carried out in the study of the influence of process conditions and catalyst characteristics on polymer productivity and product properties. For this model, the polymerization rate within the particles corresponded to the mass transfer rate of monomer and hydrogen through the external surface of the particle.

FH21/2 Fm / ∂µL0 upart ∂µL0 η [c ] - kt ηH/ 1/2µL0 )+ kp ∂t ∂z MWm m 0 MWH21/2 2 (55) ∂µL1 upart ∂µL1 Fm / )+ kp η ([c ] + µL0 ) ∂t ∂z MWm m 0 FH21/2

ηH/ 1/2µL1 (56) kt MWH21/2 2 Fm / ∂µL2 upart ∂µL2 η ([c ] + µL0 + 2µL1 ) )+ kp ∂t ∂z MWm m 0 FH21/2 η/ 1/2µL2 (57) kt 1/2 H2 MWH2 FH21/2 ∂µD0 ∂µD0 upart ηH/ 1/2µL0 )+ kt ∂t ∂z MW 1/2 2

(58)

FH21/2 ∂µD1 ∂µD1 upart )+ kt ηH/ 1/2µL1 ∂t ∂z MW 1/2 2

(59)

FH21/2 ∂µD2 upart ∂µD2 )+ kt ηH/ 1/2µL2 ∂t ∂z MW 1/2 2

(60)

H2

kSmav(Fm - Fmp|rp)Rp(V)) ) kp[C]inicVcatη/mFm[Npart] (48) kSH2av(FH2 - FH2p|rp)Rp(V)) ) ktµL0 ηH/ 21/2FH21/2MWH21/2

(49)

H2

Because the monomer concentration is the same throughout the particle, the rate of volumetric growth of a particle depended only on the total amount of active sites in the particle, namely, the amount of active sites in the original catalyst particle:

∫0V[C] dV ) [C]inicVcat rV(V) )

∫0VF

kp[C]Fmc pol(1

- p )

dV )

kp[C]inicVcatη/mFm Fpol(1 - p)

(50)

Orthogonal collocation may lead to numerical instabilities when sharp peaks appear in the distributions. Nevertheless, in many cases, this does not affect the overall properties (e.g., total number of particles and total polymer produced). The accuracy of the particle size distribution was tested by comparison of the [Npart] obtained in eq 42 with the overall polymer particle population balance:

∂[Npart]upart ∂[Npart] )∂t ∂z

H2

The number and weight average molecular weights could be obtained from these MWD moments along the reactor. Thus, it was possible to calculate the average molecular weights of the polymer produced in each section of the reactor as well as those of the final product. Riser r

r Qpart µD s 1s

M hn)

(1 - rrs) r Qpart µDr s 0s

(1 - rrs)

r

-

r Qpart µD e 1e

(1 - rre)

MWm

r

-

r Qpart µD e 0e

(1 - rre) r

r Qpart µD s 2s

(51)

t ) 0 w [Npart] ) [Npart]inic

(52)

z ) 0 w [Npart] ) [Npart]e

(53)

M hw)

(1 - rrs)

r

-

r

r Qpart µD s 1s

(1 - rrs)

r µD Qpart e 2e

(1 - rre)

MWm (61)

r

-

r µD Qpart e 1e

(1 - rre)


Downer d

d Qpart µD s 1s

M hn)

(1 - dr )

d

-

d

d Qpart µD s 0s

(1 - dr )

d Qpart µD e 1e

(1 - dr )

MWm

d

-

d Qpart µD e 0e

(1 - dr ) d

d Qpart µD s 2s

M hw)

(1 - dr )

d

d Qpart µD e 2e

-

(1 - dr )

d

d Qpart µD s 1s

(1 - dr )

MWm (62)

d

d Qpart µD e 1e

-

Figure 5. Monomer concentration profile along the downer in the steady state: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s.

(1 - dr )

Product d

M hn)

µ1Ds

d

MWm M hw)

d

µ0Ds

µ2Ds

MWm

(63)

d

µ1Ds

Notice that the product was withdrawn from the reactor. 3. Reactor Behavior The effect of different operational variables, such as the gas velocity at the exit of the downer, the recycling flow rate of the solids, the location of stream C, and the catalyst characteristics on the product characteristics and reactor perfomance was studied in the case of obtaining broad MWD polypropylene. 3.1. Gas Velocity at the Exit of the Downer. For the reactor already shown in Figure 1, the parameters used are presented in Table 2.13,20-25 Two gas velocities at the exit of the downer were considered: ugs - uparts ) 0 cm/s and ugs - uparts ) 5 cm/s. There is no solid particle fluidization in the downer because the downward solid velocity is lower than the minimum fluidization one.

udpart )

Qpartrec Sdr (1

-

dr )

)

Table 2. Process Conditions in the Study of the Influence of the Gas-Solid Relative Velocity at the Exit of the Downer Fme (riser) FH2,e (g/cm3) (riser) Fe (g/cm3) (riser) µ (g/(cm s)) (riser) kp (cm3/(mol s)) Qpartrec (cm3/s) (g/cm3)

Figure 7. Evolution of the productivity: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s.

2205 ) 7.5 cm/s (64) π × 122 × 0.65

The minimum fluidization velocity for the average particle size in the reactor at steady state is 23 cm/s.26 It was found that the gas velocity in the downer did not affect the performance of the riser. Figures 5 and 6 present the effect of the gas velocity in the downer on the monomer and hydrogen concentration. It can be seen that both monomer and hydrogen concentration decreased along the downer. The decrease was more pronounced for the lower gas velocity. Nevertheless, polymer productivity (Figure 7) and the particle size distribution (Figure 8) were not significantly affected by the flow rate in the downer.

parameter

Figure 6. Hydrogen concentration profile along the downer in the steady state: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s.

value 10-2

3.01 × 4.2 × 10-4 3.5 × 10-2 1.02 × 10-4 2 × 106 2205

parameter

value

Fme (downer) FH2,e (g/cm3) (downer) Fe (g/cm3) (downer) µ (g/(cm s)) (downer) kt (cm3/2/(mol1/2 s)) ugs (cm/s) (riser)

4.5 × 10-3 4.14 × 10-8 4.5 × 10-2 9.55 × 10-5 3.2 × 102 200

(g/cm3)

Figure 8. Final particle size distribution: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s.

Figures 9 and 10 present the effect of the gas velocity at the exit of the downer on the evolution of the product weight average molecular weight and polydispersity index. It can be seen that, the lower the gas velocity in the downer, the lower the product molecular weight (Figure 9). Surprisingly, the polydispersity index (PI) was smaller than expected considering that different monomer/hydrogen ratios were used in each reaction zone. Table 3 shows that the PI calculated for the downer was lower than 2, which is the minimum value expected

3088


Figure 9. Evolution of the product weight average molecular weight: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s.

Figure 11. Gas velocity profile along the riser in the steady state, ugs uparts ) 0 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

Figure 10. Evolution of the product polydispersity index: (-) ugs - uparts ) 0 cm/s; (b) ugs - uparts ) 5 cm/s. Table 3. Effect of the Gas Velocity at the Exit of the Downer on the Product Propertiesa parameter

ugs - uparts ) 0 cm/s


M h w (g/mol) (riser) PI (riser) M h w (g/mol) (downer) PI (downer) wp (%) (riser) M h w (g/mol) (outlet) PI (outlet)

260 200 2.56 232 800 1.24 56.4 258 600 2.44

279 000 2.63 243 900 1.25 53.9 275 600 2.49

a

Qpartrec ) 2205 cm3/s.

Table 4. Values of the Average Chain Lifetime and Particle Residence Time in Each Section of the MZCR ht (s) τ (s)

riser

downer

0.6 3

63 20

Figure 12. Monomer concentration profile along the downer in the steady state where (a) ugs - uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

in polymerizations terminated by chain transfer. Table 4 compares the residence time of the particles in each section of the reactor “τ” with the average chain lifetime “th” calculated as

X hn )

kp[Mc] kt[H2]1/2 ht )

M hn)X h nMWm X hn kp[Mc]

(65)

(66)

It can be seen that residence time of the particles in the downer was shorter than the average chain lifetime. Therefore, under the conditions used in the downer, the residence time was too short to produce long polymer chains. Consequently, smaller recycling ratios should be used to produce polymers with wide MWDs. 3.2. Recycling Flow Rate of the Solids. Three recycling flow rates of the solids were studied, all of them being lower than the one chosen previously, Qpartrec ) 551 cm3/s, Qpartrec ) 368 cm3/s, and Qpartrec ) 184 cm3/s.

Figure 11 presents the effect of the solid recycling ratio on the gas velocity profiles along the riser. For the sake of simplicity, only the case ugs - uparts ) 0 cm/s is shown, since it was shown before that the profiles were not significantly affected by the gas velocity at the exit of the downer. It can be seen that the gas velocity in the riser increased with the solid recycling ratio because solid particles should flow faster but that it was almost constant along the riser. On the other hand, because of the high gas velocity, the monomer concentration was not affected by the solid recycling flow rate and remained constant along the riser. Figures 12-14 show the effect of the solid recycling flow rate on the monomer and hydrogen concentration and on the gas velocity profiles in the downer, respectively. It can be seen that the decrease was more pronounced for ugs - uparts ) 0 cm/s where also the differences between the solid recycling flow rate were higher because the gas velocity was limited by the solid velocity. Thus, Figure 12 shows that for the smallest solids recycling flow rate and for ugs - uparts ) 0 cm/s the monomer was almost completely depleted at about the middle of the downer. This means that at such a low solid recycling flow rate,


Figure 13. Hydrogen concentration profile along the downer in the steady state where (a) ugs - uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

Figure 15. Evolution of the productivity when (a) ugs - uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

Figure 16. Final particle size distribution: Qpartrec ) 551 cm3/s; ugs - uparts ) 0 cm/s.

Figure 14. Gas velocity profile along the downer in the steady state where (a) ugs - uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

half of the downer did not contribute to polymer formation. The productivity decreased when the solid recycling ratio and gas velocity in the downer decreased (Figure 15). Figures 16-18 show the particle size distribution of the final product for all solids recycling flow rates. For the sake of simplicity, only the case ugs - uparts ) 0 cm/s is shown. It can be seen that as the solid recycling flow rate increased the obtained distribution was smoother. For high solid recycling flow rates, the system behaves like a continuous stirred tank reactor (CSTR), whereas for low flow rates some peaks appeared due to the plug-flow behavior (the first peak showed the particles


that exited the reactor in the first cycle, the second peak showed the particles in the second cycle, and so on). This behavior fully agrees with the residence time distribution in loop reactors.27 The average particle size was about 1.4 mm for all cases. Figures 19 and 20 present the effect of both the solid recycling flow rate and the gas velocity on the evolution of the product weight average molecular weight and polydispersity index. It can be seen that both properties increased with the gas velocity, because there was more monomer in the downer. On the other hand, molecular weight and PI increased as the solids recycling

3090



Figure 20. Evolution of the product polydispersity index when (a) ugs uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s. Table 5. Values of the Average Chain Lifetime and Particle Residence Time in Each Section of the MZCR ht (s) τ (s)

Qpartrec ) 551 cm3/s Qpartrec ) 368 cm3/s Qpartrec ) 184 cm3/s

riser

downer

0.6 12 18 36

63 80 120 240

Table 6. Effect of the Gas Velocity at the Exit of the Downer and the Solid Recycling Flow Rate on the Product Propertiesa Figure 19. Evolution of the product weight average molecular weight when (a) ugs - uparts ) 0 cm/s and (b) ugs - uparts ) 5 cm/s: (-) Qpartrec ) 551 cm3/s; (- -) Qpartrec ) 368 cm3/s; (-‚-) Qpartrec ) 184 cm3/s.

flow rate decreased, because the polymer chains in the downer had more time to grow (Table 5). It is worth pointing out that a decreasing recycling flow rate led to bigger polymer domains, namely, to a less homogeneous final product. Therefore, a compromise between the width of the MWD and the final product homogeneity should be reached. Besides, for ugs - uparts ) 0 cm/s, relatively narrow MWDs were still obtained under steady-state conditions for all solid recycling flow rates. This fact could be explained with two reasons. First, most of the polymer was produced in the riser (Tables 6-8). Second, the difference between the molecular weight obtained in both sections of the MZCR decreased with time because the molecular weight produced in the downer decreased due to the higher reactivity of the monomer in comparison with the hydrogen. Concerning the obtained results in each section of the MZCR, the molecular weight and polydispersity index obtained in the downer increased as the solid recycling flow rate decreased and the gas velocity at the exit of the downer increased because the monomer concentration was higher and the residence time of the solid particles was longer. Thus, it was possible to produce long molecular chains in a significant amount. For the smallest solid recycling flow rate and ugs - uparts ) 0 cm/s, the polydispersity index was higher than 2 (characteristic of polymerizations terminated by chain transfer at constant mono-

parameter




200 500 2.73 381 200 1.27 72.2 229 000 2.75

360 300 4.14 571 200 1.34 57.1 404 400 3.87

a


Table 7. Effect of the Gas Velocity at the Exit of the Downer and the Solid Recycling Flow Rate on the Product Propertiesa parameter




164 100 2.42 408 900 1.29 78.8 200 900 2.62

357 700 4.40 723 200 1.48 57.4 455 800 4.39

a


mer concentration) because the monomer concentration profile in this section of the reactor led to a certain heterogeneity in the obtained polymer. In the riser, the results were influenced by the living polymer chains that came from the downer because they died quickly due to the higher hydrogen/monomer ratio in the riser. Therefore, as the solid recycling flow rate decreased, there were two opposite effects. On one hand, the molecular weight of the living

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3091 Table 8. Effect of the Gas Velocity at the Exit of the Downer and the Solid Recycling Flow Rate on the Product Propertiesa parameter



128 100 2.03 472 500 2.46 86.4 173 800 2.51

a

ugs - uparts ) 5 cm/s 320 100 4.36 1 044 600 1.90 57.6 563 400 5.44

Qpartrec ) 184 cm3/s. Figure 22. Hydrogen concentration profile along the downer in the steady state: (b) Qpartrec ) 2205 cm3/s; (9) Qpartrec ) 551 cm3/s; (1) Qpartrec ) 368 cm3/s; (2) Qpartrec ) 184 cm3/s.

Figure 21. Monomer concentration profile along the downer in the steady state: (b) Qpartrec ) 2205 cm3/s; (9) Qpartrec ) 551 cm3/s; (1) Qpartrec ) 368 cm3/s; (2) Qpartrec ) 184 cm3/s.

polymer chains from the downer was higher, increasing the molecular weight and polydispersity index in the riser. On the other hand, these living polymer chains from the downer were relatively less important because more polymer was produced per cycle due to the higher residence time. 3.3. Feeding the Monomer at the Bottom of the Downer. When the monomer was fed at the bottom of the downer, the gas-solid relative velocity was (upart + ug) and the following condition had to be satisfied in order to avoid solid particle fluidization:

umf g |upart + ug|

Figure 23. Evolution of the productivity: (b) Qpartrec ) 2205 cm3/s; (9) Qpartrec ) 551 cm3/s; (1) Qpartrec ) 368 cm3/s; (2) Qpartrec ) 184 cm3/s.

(67)

with umf ) 23 cm/s. The most critical situation corresponded to the highest solid recycling flow rate (i.e., the highest solid velocity, upart ) 7.5 cm/s). Therefore, the gas velocity could not be higher than 15.5 cm/s. A gas velocity equal to 12 cm/s at the bottom of the downer was chosen for the simulations presented here. The process conditions are those in Table 2 with the only change being that the gas density at the bottom of the downer was higher, Fe ) 0.0451 g/cm3, because the pressure at the entry of the downer must be higher than at the exit of the riser (it must be taken into account that the circulation in the loop was defined by the balance of pressures between the two polymerization zones). Figures 21-23 show that neither monomer and hydrogen concentrations nor the polymer productivity were affected by the solid recycling flow rate. On the other hand, the molecular weight and polydispersity index of the final product increased as the solid recycling rate decreased (Figures 24 and 25). 3.4. Catalyst Characteristics. The MZCR allows the production of polymer particles containing intimate mixtures of polymers of widely different MWDs. It was discussed above that increasing the solid recycling flow rate led to better particle homogeneity but to a narrower overall MWD. Therefore, a compromise between the width of the MWD and the final product homogeneity should be achieved. This optimal solid

Figure 24. Evolution of the product weight average molecular weight: (b) Qpartrec ) 2205 cm3/s; (9) Qpartrec ) 551 cm3/s; (1) Qpartrec ) 368 cm3/s; (2) Qpartrec ) 184 cm3/s.

Figure 25. Evolution of the product polydispersity index: (b) Qpartrec ) 2205 cm3/s; (9) Qpartrec ) 551 cm3/s; (1) Qpartrec ) 368 cm3/s; (2) Qpartrec ) 184 cm3/s.

recycling flow rate depends also on the catalyst characteristics since the origin of the problem was that the residence time in the downer was not enough to produce long molecular chains due to the average chain lifetime. The effect of the intrinsic catalyst activity was simulated by considering a catalyst for which the kinetic constants (kp and kt) were five times greater than those reported in Table 2. To maintain the same productiv-

3092


Table 9. Results for the Catalyst with a Hydrogen Transfer Rate Constant Five Times Highera

a

parameter

kp ) 2 × 106 cm3/(mol s) kt ) 3.2 × 102 cm3/2/(mol1/2 s)

kp ) 1 × 107 cm3/(mol s) kt )1.6 × 103 cm3/2/(mol1/2 s)


260 200 2.56 232 800 1.24 56.4 258 600 2.44

383 100 4.50 678 100 1.35 56.4 453 500 4.28


ity, the number of active centers were reduced by a factor of 5. Table 9 presents a comparaison of the characteristics of the polymers produced in both cases for the highest solid recycling flow rate Qpartrec ) 2205 cm3/s, gas feed at the top of the downer, and ugs - uparts ) 0 cm/s. It can be seen that the molecular weight and the polydispersity increased with catalyst activity. The reason was that the average chain lifetime decreased and longer chains could be produced in the downer. This result shows that very active catalysts are needed to fully exploit the advantages of the MZCR. 4. Conclusions A mathematical model for the gas-phase polymerization of olefins in a multizone circulating reactor (MZCR) is presented. The MZCR is intent on obtaining intimate mixtures of polymers of different characteristics. In this article, polymers with widely different molecular weights were considered. The outputs of the model include polymer production, particle size distribution, molecular weight, and polydispersity index. It was found that the particle size distribution was mainly determined by the residence time distribution. In addition, the polydispersity index of the final product (which is a measure of the success of the operation) increased as the solid recycling flow rate decreased. However, as the solid recycling flow rate decreased, the polymer domains were bigger and, therefore, the final product was less homogeneous. Therefore, a compromise between the width of the MWD and the final product homogeneity should be reached. It was found that the optimal solid recycling flow rate depended on the catalyst characteristics. Both the molecular weights and the polydispersity increased with catalyst activity. Very active catalysts are needed to fully exploit the advantages of the MZCR. Acknowledgment J.L.S. acknowledges the scholarship from the UPV/EHU. The financial support from the MEC (PB98-0234) and from the EC (POLYPROP, GRD2-2000-30189) is gratefully appreciated.

following equation28 2

F2 )

F1 )

2 2fgFrrrurg Drr

Drr

)

(

3FrDrrC urg - urpart 2Fpartdhp ur part

4fpart )

2

(I.3)

where Fpart and upart are the density and velocity of the particles, respectively, and dhp is the average particle size. (3) The net weight of the bed

F3 ) (1 - rr)(Fpart - Fr)g

(I.4)

The porosity in the riser can be calculated from the polymer particle population as follows:

rr ) 1 -

∫VV* q(V)rVr dVr r

r 0

t ) 0 w rr ) rrinic

(I.5) (I.6)

The velocity of the solids in the riser is related to the gas velocity by the solid terminal velocity (ut).

urpart ) urg - ut

(I.7)

where the solid terminal velocity was considered to be that corresponding to the average particle size, dhp.26

ut )

(

(I.1) CD )

)

4gdhp(Fpart - Fr) 3FrCD

1/2

(I.8)

24 Rep < 0.4 Rep

(I.9)

10 0.4 < Rep < 500 Rep1/2

(I.10)

CD )

where Drr is the diameter of the riser and fg is the Fanning friction factor. (2) Combined friction between particles and the reactor wall, between gas and particles, and between particles. It was assumed that this friction could be expressed by the

(I.2)

where the friction factor fpart was related to the particle drag coefficient (C) by a force balance on the particles in the reactor as follows:

Appendix I: Friction Losses A.1. Riser. Friction losses were considered macroscopically in the “F” factor as a sum of the following effects:28 (1) Friction between the gas and the reactor wall

2fpartFpart(1 - rr)urpart

CD ) 0.43 500 < Rep < 200 000 Rep )

dhpFrut µr

(I.11) (I.12)


A.2. Downer. The friction losses in the downer were calculated by means of the Ergun equation:28

F ) 150

(1 -

dr )2 d2

Fdr

(1 µd d (u - udpart) + 1.75 d (udg - udpart)2 2 g dhp r dhp (I.13) dr )

where dhp is the average particle size, µd is the viscosity of the gas phase in the downer, and udpart was the velocity of the solids in the downer, which was calculated from the solid flow rate

udpart )

Qdpart Sdr (1 - dr )

(I.14)

Nomenclature aV ) solid superficial area per reactor volume unit (cm2/cm3) [C] ) active centers concentration (mol/cm3) [c0] ) empty active centers concentration (mol/cm3) CCD ) chemical composition distribution dhp ) average particle size (cm) DPi ) diffusivity of component i in particle pores (cm2/s) Dr ) reactor diameter (cm) DSi ) diffusivity of component i in the polymer (cm2/s) e ) microparticle polymer layer thickness (cm) fg ) Fanning friction factor g ) gravity acceleration (cm2/s) [H2] ) hydrogen concentration (mol/cm3) kp ) propagation rate constant (cm3/mol s) kSi ) mass transfer coefficient for component i in the boundary layer (cm/s) kt ) hydrogen transfer rate constant(cm3/2/(mol1/2 s)) Lr ) reactor length (cm) MW ) molecular weight (g/mol) M h n ) average number molecular weight (g/mol) M h w ) average weight molecular weight (g/mol) MWD ) molecular weight distribution [Np] ) number of microparticles per unit volume of particles (part/cm3) [Npart] ) particle number per reactor volume unit (part/cm3) P ) pressure (atm) PI ) average polydispersity index q(V) ) number of particles with volume V per particle and reactor volume unit (part/cm6) Qg ) gas flow rate (cm3/s) Qpart ) solid flow rate (cm3/s) rc ) catalyst fragment radius in the microparticles (cm) Rcat ) catalyst particle radius (cm) rp ) macroparticle radial coordinate (cm) Rp ) macroparticle radius (cm) rV ) volumetric growth velocity (cm3/s) Sr ) reactor section (cm2) t ) time (s) ht ) average chain lifetime (s) T ) temperature (°C) umf ) minimum fluidization velocity (cm/s) ug ) gas-phase velocity (cm/s) upart ) solid particle velocity (cm/s) ur ) gas-solid relative velocity (cm/s) ut ) solid terminal velocity (cm/s) V ) particle volume (cm3) wp ) weight fraction of polymer produced (%) X h n ) average number polymerization degree z ) reactor axial coordinate (cm)

Greek Letters p ) particle porosity r ) reactor porosity µ ) gas-phase viscosity (cp) µLk ) kth moment of the MWD of growing chains (mol/cm3) µDk ) kth moment of the MWD of dead chains (mol/cm3) ηH/ 2 ) hydrogen solubility coefficient in polymer η/m ) monomer solubility coefficient in polymer F ) gas-phase density (g/cm3) Fpol ) polymer density (g/cm3) τ ) residence time (s) Subscripts c ) active center surface cat ) catalyst e ) entry H2 ) hydrogen ine ) inert inic ) initial m ) monomer p ) macroparticle rec ) recycling s ) outlet Superscripts d ) downer r ) riser Literature Cited (1) Ahvenainen, A.; Sarantila, K.; Andtsjo, H.; Takakarhu, J.; Palmroos, A. Multistage process for producing polyethylene. US Patent 5,326,835, 1994. (2) Avela, A.; Karling, R.; Takakarhu, J. The Enhanced Bimodal Polyethylene Technology. DECHEMA Monogr. 1998, 134, 3. (3) Razavi, A.; Debras, G. L. G. Manufacture of polyolefins with multimodal molecular weight distribution. US Patent 5,719,241, 1998. (4) Pettijohn, T. M. Compositions useful for olefin polymerization and processes therefor and therewith. US Patent 5,622,906, 1997. (5) Govoni, G.; Rinaldi, R.; Covezzi, M.; Galli, P. Process and Apparatus for the Gas-Phase Polymerization of R-Olefins. US Patent 5,698,642, 1997. (6) Govoni, G.; Covezzi, M. Process and Apparatus for the Gas-Phase Polymerisation. PCT Int. Appl. WO 00/02929, 2000. (7) Covezzi, M.; Mei, G. The Multizone Circulating Reactor Technology. Chem. Eng. Sci. 2001, 56, 4059. (8) de Vries, A.; Izzo-Iammarrone, N. The MultiZone Circulating Reactor Technology. DECHEMA Monogr. 2001, 137, 43. (9) Galli, P.; Chadwick, J.; Mei, G.; Vecellio, G. Multizone circulating reactor: The novel frontier of the polyolefins technology. Presented at the Ninth Joint Business Forum on Specialty Polyolefins (SPO 99), Houston, TX, October 12-13, 1999. (10) Fernandes, F. A. N.; Lona, L. M. F. Multizone Circulating Reactor Modeling for Gas-Phase Polymerization. I. Reactor Modeling. J. App. Polym. Sci. 2004, 93, 1042. (11) Fernandes, F. A. N.; Lona, L. M. F. Multizone Circulating Reactor Modeling for Gas-Phase Polymerization. II. Reactor Operating with Gas Barrier in the Downer Section. J. App. Polym. Sci. 2004, 93, 1053. (12) Parasu Veera, U.; Weickert, G. Single Particle Modelling of GasPhase Propylene Polymerization: Viscous Modulus and Time Scale Analysis. Polym. React. Eng. 2003, 11, 33. (13) Nagel, E. J.; Kirilov, V. A.; Ray, W. H. Prediction of Molecular Weight Distributions for High-Density Polyolefins. Ind. Eng. Chem. Process Des. DeV. 1980, 19, 372. (14) Santos, J. L. Modelado de la polimerizacioń de olefinas con catalizadores soportados. Ph.D. Thesis, The University of the Basque Country, Donostia-San Sebastiań, Spain, 2002. (15) Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Pretice-Hall: Englewood Cliffs, NJ, 1978. (16) Villadsen, J.; Stewart, W. E. Solution of Boundary-Value Problems by Orthogonal Collocation. Chem. Eng. Sci. 1967, 22, 1483.

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(17) Finlayson, B. A. Orthogonal Collocation in Chemical Reaction Engineering. Catal. ReV. 1974, 10, 69. (18) Stewart, W. E. Simulation and Estimation by Orthogonal Collocation. Chem. Eng. Educ. 1984, 204. (19) Gear, C. W.; Petzold, L. R. ODE Methods for the Solution of Differential Algebraic Equations. SIAM J. Numer. Anal. 1984, 21, 716. (20) Hutchinson, R. A.; Chen, C. M.; Ray, W. H. Polymerization of olefins through heterogeneous catalysis. X. Modeling of particle growth and morphology. J. Appl. Polym. Sci. 1992, 44, 1389. (21) Floyd, S.; Choi, K. Y.; Taylor, T. W.; Ray, W. H. Polymerization of olefins through heterogeneous catalysis. III. Polymer particle modeling with an analysis of intraparticle heat and mass transfer effects. J. Appl. Polym. Sci. 1986, 32, 2935. (22) Galvan, R.; Tirrell, M. Orthogonal collocation applied to analysis of heterogeneous Ziegler-Natta polymerization. Comput. Chem. Eng. 1986, 10, 77. (23) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977.

(24) Floyd, S.; Heiskanen, T.; Taylor, T. W.; Mann, G. E.; Ray, W. H. Polymerization of olefins through heterogeneous catalysis. VI. Effect of particles heat and mass transfer on polymerization behavior and polymer properties. J. Appl. Polym. Sci. 1987, 33, 1021. (25) Boehm, L. L. Homo and copolymerization with a highly active Ziegler-Natta catalyst. J. Appl. Polym. Sci. 1984, 29, 279. (26) Kunii, D.; Levenspiel, O. Fluidization Engineering. In New Fluidization Engineering; Krieger, R. E., Ed.; Huntington: New York, 1977. (27) Warnecke, H. J.; Pru¨ss, J.; Langemann, H. On a mathematical model for loop reactors. I. Residence time distributions, moments and eigenvalues. Chem. Eng. Sci. 1985, 40, 2321. (28) Perry, R. H., Green, D. W., Eds. Chemical Engineer’s Handbook, 6th ed.; McGraw-Hill: New York, 1984.

ReceiVed for reView January 14, 2005 ReVised manuscript receiVed October 17, 2005 Accepted February 27, 2006 IE0500523

Modeling of Olefin Gas-Phase Polymerization in a ... - ACS Publications

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