1-Butene Copolymerizations in Industrial Slurry

Ind. Eng. Chem. Res. 2005, 44, 2697-2715

2697

Modeling Ethylene/1-Butene Copolymerizations in Industrial Slurry Reactors Antoˆ nio G. Mattos Neto,† Marcelo F. Freitas,‡ Ma´ rcio Nele, and Jose´ Carlos Pinto* Programa de Engenharia Quı´mica/COPPE, Universidade Federal do Rio de Janeiro, Ciudade Universita´ ria, CP 68502, Rio de Janeiro 21945-970 RJ, Brazil

In this work, a comprehensive model is developed for the ethylene/1-butene copolymerization in an industrial slurry polymerization reactor for linear low-density polyethylene synthesis. The model is able to describe the dynamic evolution of the molecular weight averages, comonomer content, particle size averages, melt index, and density of the final polymer resin and extends modeling results available in the open literature. A new modeling approach is used to describe the evolution of particle sizes, which is based on the definition of a joint distribution of mass and catalyst concentration of solid polymer particles. It is shown that the model successfully describes the operation of an industrial slurry polymerization reactor. For this reason, the model is used to analyze how sensitive the final polymer properties are to variations of the feed conditions and to development of segregated mixing zones inside the reactor vessel. It is shown that the ethylene feed flow rate is the most influential process variable and that the existence of very small segregated reactor zones can lead to very serious operation problems, such as particle agglomeration. Introduction The polyolefin market is a commodity-based market with fierce competition among the main resin producers. New products are introduced at high rates with low profit margins, despite the steady sales volume growth observed in the past decades.1 Furthermore, this is a performance-driven market where buyers are interested in the final physical properties of the material rather than in its molecular structure or chemical composition. On the basis of this scenario, the importance of sound polymerization process engineering in the polyolefin business cannot be overestimated. A proper description of the polymerization process through mathematical models associated with the description of the polymer physical properties allows process engineers to improve the plant operation, increasing the overall profit margins, safety, and environmental cleanness of the production process. The use of process simulators in the polymer industry is not as widespread as it is in the petrochemical segment, even though there are a number of process simulators available for polymerization processes, such as Polyred (Hypro-tech), Polymers Plus (Aspen), Pro II (Simulation Science), and Simulpol2 (COPPE/Polibrasil). This is certainly due to the complexity of the polymerization processes, which may show different behaviors for different catalysts and/or operation conditions and due to the small number of well-understood standard unit operations in the industrial plant, when compared to a refinery for example. Even so, tailored reactor models are often used to describe a particular process for use at the industrial site. Ethylene polymerization in slurry reactors is still the most widely used process to produce polyethylene.3 * To whom correspondence should be addressed. Tel.: 5521-25628337. Fax: 55-21-25628300. E-mail: pinto@ peq.coppe.ufrj.br. † Works for Polibrasil Resinas SA, Saõ Paulo, Brazil. ‡ Works for BR Distribuidora SA, Rio de Janeiro, Brazil.

Nevertheless, in the past decades, most of the reactor modeling efforts presented in the literature have been focused on the development of mathematical models for the more modern gas-phase ethylene polymerization,4 while slurry models have normally been used to interpret polymerization kinetics from laboratory-scale reactors. Khare et al.3 used a polymerization process simulator to describe the steady-state operation of an industrial plant, analyzing production rates and certain key molecular properties of the final polymer material (average molecular weights and polydispersity index). The model was able to simulate the transient operation of the polymerization plant so that it became possible to suggest modifications of the operation conditions in order to increase the polymer production rate. Ha et al.5 investigated the operation of a semibatch slurry ethylene polymerization reactor. A model describing the gas phase, the liquid phase (including gas bubbles), and the solid polymer particles inside the reactor was developed to assess the effect of the hydrogen concentration and the initial size of the catalyst particles on the polymerization reaction. It was reported that the average molecular weight of the polymer material can be influenced by the initial size of the catalyst particles because of the existence of masstransfer limitations, an effect that may become very important when catalyst activities are high.6,7 Fontes and Mendes8 developed a similar model for a continuous slurry ethylene polymerization reactor that was able to predict the production rates and the number and weight molecular weight averages of the final polymer. In this work, a comprehensive model is developed for the ethylene/1-butene copolymerization in an industrial slurry polymerization reactor for linear low density polyethylene (LLDPE) synthesis. The model is able to describe the dynamic evolution of the molecular weight averages, comonomer content, particle size averages, melt index, and density of the final polymer resin and

10.1021/ie049588z CCC: $30.25 © 2005 American Chemical Society Published on Web 12/31/2004

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

perature profiles inside the polymer particles using wellestablished particle models at industrial operation conditions confirmed that mass- and heat-transfer resistances might be neglected. Therefore, it may be assumed that the gaseous constituents are readily solubilized in the reaction medium and that the chemical components present in the liquid, gas, and polymer phases are in thermodynamic equilibrium. Based on these assumptions, the mass balance for the total ethylene monomer (et) present in the reactor is given by eq 1, where mget is the mass of ethylene in the

d (mg + mlet + mset) ) dt et Fet - Ret Figure 1. Reactor scheme.

therefore extends the modeling results available in the open literature. Besides, a new modeling approach is used to describe the evolution of particle sizes, which is based on the definition of a joint distribution of mass and catalyst concentration of solid polymer particles. The model parameters are obtained from an analysis of laboratory-scale polymerization data, of offline polymer characterizations, and of industrial reactor operation conditions. It is shown that the model successfully describes the operation of an industrial slurry polymerization reactor. For this reason, the model is used to analyze how sensitive the final polymer properties are to variations of the feed conditions and to the development of segregated mixing zones inside the reactor vessel. Model Development Reactor Description. The industrial reactor studied here is presented in Figure 1. The reactor is a continuous stirred tank reactor (CSTR), cooled by a cooling jacket and two identical external shell-and-tube heat exchangers. The heat removal through the cooling jacket is controlled by manipulation of the water flow rate and is responsible for the fine-tuning of the reactor temperature. The circulation flow rate through the heat exchangers is varied only when a steep change in the reactor temperature is desired. Diluent and catalyst (prepolymerized Phillips catalyst) are fed into the reactor in the liquid phase. Gaseous ethylene, 1-butene, and hydrogen are sparged at the reactor bottom. The polymerization slurry is composed of liquid and solid phases. The liquid phase is composed of solubilized gaseous reagents, liquid catalyst components (such as alkylaluminum), diluent, and oligomers. The solid phase (polymer particles) is composed of solid catalyst, polymer and absorbed diluent, gaseous reagents, and alkylaluminum. The reactor product is then fed into a flash drum for removal of volatiles, and the polymer is finally separated from the solvent through centrifugation. Reactor Model. The mathematical model was developed based on available experimental results that showed that mass- and heat-transfer resistances in the reactor medium are negligible.9 Freitas9 showed that the catalyst activity was not sensitive to modification of the catalyst size distribution and of the agitation speed in experiments performed in a pilot plant. Besides, computation of monomer concentration and tem-

q (mlet + mset) (1) s V +V l

reactor head, mlet is the mass of ethylene dissolved in the liquid phase, mset is the mass of ethylene dissolved in the solid phase, Fet is the mass flow rate of the ethylene feed stream, Ret is the ethylene polymerization rate, q is the volumetric flow rate of the slurry output stream (liquid and solid phases), Vl is the volume of the liquid phase, and Vs is the volume of the solid phase. The mass balances for the remaining components [1-butene monomer, bt (eq 2); hydrogen, h (eq 3); n-hexane, nhx (eq 4); cocatalyst, cc (eq 5); catalyst, c (eq 6)] can then be written as

d (mg + mlbt + msbt) ) dt bt Fbt - Rbt -

q (mlbt + msbt) (2) Vl + Vs

d (mg + mlh + msh) ) dt h Fh - Rh -

q (mlh + msh) (3) Vl + V s

d (mg + mlnhx + msnhx) ) dt nhx Fnhx -

q (mlnhx + msnhx) (4) Vl + V s

q d (mscc) (ms ) ) Fcc - Rcc - l s dt cc V +V

(5)

d q (msc) (ms) ) Fc - l dt c V + Vs

(6)

where all variables are defined as described previously. The overall mass balance inside the reactor can be given by eq 7, where the summations are performed over

d dt

∑i mgi + ∑i mli + ∑i msi ) )

(

∑i Fi -

q l

s

V +V

(

∑i mli + ∑i msi )

(7)

all of the components present in the reactor medium. Equation 7 may be regarded as an algebraic constraint that allows for computation of q, the volumetric flow rate of the slurry output stream, as shown below. If the volume of gas bubbles in the polymerization mixture is negligible when compared to the volumes of

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2699 Table 1. Henry’s Constants for Gas-Liquid and Gas-Solid Equilibria component

n-hexane (kgf/cm2)

polymer (cm2/kgf)

ethylene 1-butene hydrogen n-hexane

Hlv et(T) ) 50.178 + 0.65506T 2 Hlv bt(T) ) 1.4344 + 0.036734T + 0.00091522T Hlv h (T) ) 1615.6 - 6.566T ln(Psat nhx) ) 9.2363 - 2697.547/(224.366 + T)

Hvs et ) 0.0012 - 0.000004T Hvs bt ) 0.028 - 0.00019T Hvs h ) exp[-12.323 - 567.44/(T + 273.15)] Hvs nhx ) exp[-10.591 + 2529.6/(T + 273.15)]

the liquid and solid phases,9 the total volume of the slurry may be given by the sum of the volumes of the liquid and solid phases. Furthermore, if the reactor level is controlled tightly, the volume of the slurry may be regarded as constant. In this case, eq 8 can be written as

Vl + V s )

mli

∑i F

i

+

msi

∑i F

) constant

(8)

i

where Vl + Vs is the volume occupied by the slurry and Fi is the density of the component i, assumed to be the same in both the liquid and solid phases. Equation 8 can be regarded as an additional algebraic constraint that allows for computation of the relative amounts of liquid and solid phases inside the reactor. If eq 8 is derived with respect to t, it is possible to write

dmspe dt

)

[

mspe dFpe Fpe dT

+

∑ i*pe

]

msi + mli Fpe dFi dT Fi

-

Fi dT dt Fpe d (msi + mli) (9) F dt i*pe i

∑

where pe stands for polyethylene. If eqs 1-6 and 9 are inserted into eq 7, then it becomes possible to calculate the reactor residence time (Θ), as presented in eq 10. It

1 Θ

)

q

)

Vl + Vs

( )

d

∑i Fi - ∑i dt (mgi ) - i*pe ∑ 1-

[

∑i mspe dFpe Fpe dT

Fpe d

(msi + mli)

Fi dt

-

(msi + mli)

+

∑ i*pe

]

(msi + mli) Fpe dFi dT Fi

Fi dT dt

∑i (msi + mli)

∑i ngi

(10)

R(T + 273.15) Vg

dT

∑i mticpi dt ) -∑i Ficpi(T - Tei) - Q˙ c - Q˙ t + (-∆Hpol)VsRpol (12) where ∆Hpol is the heat of polymerization, Q˙ c is the heat removed by the cooling jacket, Q˙ t is the heat removed by the external heat exchangers, Tei is the temperature of the feed of component i, and Rpol is the overall polymerization rate. Q˙ c and Q˙ t can be given by eqs 13 and 14 as

Q˙ c ) UAc(T - T0)

(13)

Q˙ t ) 2UAt(T - T0)

(14)

where UAc is the overall heat-transfer constant of the cooling jacket, UAt is the overall heat-transfer constant of the external heat exchangers, and T0 is the temperature of the cooling water feed stream. UAc and UAt can be calculated as functions of the thermal properties of the slurry, of the thermal properties of the refrigeration fluid (water), and of the flow rates of the cooling water feed and slurry recirculation streams. Detailed presentation of these equations is beyond the scope of this text. Detailed information about the computation of the heat-transfer constants can be found elsewhere.9,10 The time scale of the mass- and heat-transfer processes was found to be much shorter than the time scale of the reaction.9 Therefore, it sounds reasonable to assume that the distribution of chemical components among the distinct phases can be described by equilibrium relations. If equilibrium conditions are assumed to exist between the gas and liquid phases, then it is possible to write

yPi ) xiHlv i

is assumed that the gas phase occupies the reactor head only and that it follows the ideal gas law. Therefore, the reactor pressure (P) can be given by eq 11 as

P)

inside the reactor but neglects the shaft work.9 The energy balance can be written as

(11)

where T is the reactor temperature (in degrees Celsius), R is the gas constant, and ngi is the mass of component i in the gaseous phase (in moles). The energy balance, given by eq 12, takes into account the existence of multiple cooling systems, the enthalpies of the reactor streams, and the distinct components

(15)

where yPi is the partial pressure of component i in the gas phase, xi is the molar fraction of component i in the liquid phase, and Hlv i is Henry’s constant for component i in the polymerization medium. It is assumed that Hlv i is equal to the binary Henry’s constant for the pair diluent/gaseous component because the overall concentration of gaseous components in the liquid phase is normally very small. Table 1 presents binary Henry’s constants for gaseous components in the diluent phase. The liquid-vapor equilibrium of the solvent (n-hexane) was assumed to follow the Lewis-Randall rule, as described by eq 16. The solubility of liquids and gases

yPnhx ) xnhxPsat nhx

(16)

in polymers may also be described by Henry’s law11-13 when polymer swelling is not high. Therefore, the following gas-solid equilibrium equations

2700


Table 2. Densities of the Pure Components in the Liquid and Solid Phases

Table 4. Kinetic Mechanisms Initiation

component

density (kg/m3)

ethylene 1-butene hydrogen n-hexane polyethylene

Fet(T) ) 680.0635 - 0.7343T Fbt(T) ) 637.2882 - 1.6351T FH2(T) ) 678.6159 - 1.0421T Fnhx(T) ) 686.1365 - 1.0466T Fpe(T) ) 976 - 0.6T

by ethylene

S(i) + E98 P1,0(i)

by 1-butene

S(i) + B 98 Q0,1(i)

kiE(i)

kiB(i)

Propagation by ethylene

Table 3. Specific Heats of the Pure Components component

cp (kJ/kg/°C)

component

cp (kJ/kg/°C)

ethylene 1-butene hydrogen

1.87 1.41 14.4

n-hexane polyethylene water

2.50 2.25 4.20

kpEE(i)

n g 1; m g 0

kpBE(i)

n g 0; m g 1

kpEB(i)

n g 1; m g 0

kpBB(i)

n g 0; m g 1

Pn,m(i) + E 98 Pn+1,m(i) Qn,m(i) + E 98 Pn+1,m(i)

by 1-butene

Pn,m(i) + B 98 Qn,m+1(i) Qn,m(i) + B 98 Qn,m+1(i) Chain Transfer

can be written:

spontaneous s yPi Hvs i ) wi

(17)

where Hvs i stands for the Henry’s constant for component i in the polymer phase. As in the previous case, it is assumed that Hvs i is equal to the binary Henry’s constant for the pair polymer/component because the overall concentration of chemical components in the polymer phase is normally very small. Table 1 presents binary Henry’s constants for chemical components in the polymer phase. Henry’s constants for ethylene and hydrogen in nhexane were estimated from experimental data,9 while the Henry’s constant for 1-butene in hexane was evaluated using the Soave-Redlish-Kwong equation.14 For the polymer phase, Henry’s constants were estimated from experimental data provided by Yoon et al.15 For the sake of simplicity, it is assumed that the densities of the chemical components in the slurry are equal to the densities of the pure components so that the average density of the slurry can be given by eq 18.

1 Fj

)

wi

∑i F

(18)

i

The densities of the pure components in the liquid and solid phases are presented in Table 2. The densities of n-hexane and 1-butene were evaluated using the Rackett equation.14 Because ethylene and hydrogen are above their critical point, the Rackett equation should not be used. The densities of these components were evaluated by calculating the density of diluted solutions of n-hexane containing hydrogen and ethylene with the Rackett equation14 and evaluating the contribution of the gaseous components for the solution density using eq 18. It was found that the calculated densities were fairly constant and did not depend on the concentration of the gaseous components in the solution. The specific heats of the chemical constituents were assumed to be constant within the reactor operation range. The values of the specific heats presented in Table 3 were taken from the literature.16-18 Reaction Kinetics. In this section, a detailed kinetic scheme is developed to describe the kinetic events that take place at the catalyst sites. The derived rate equations are then coupled with the reactor model so that the properties of the produced polymer can be evaluated. The proposed kinetic scheme assumes that the catalyst contains distinct active sites and allows for prediction of the copolymer molecular weight distribution and composition. The scheme is composed of the

kte(i)

n g 1; m g 0

kte(i)

n g 0; m g 1

Pn,m(i) 98 S(i) + Dn,m(i) Qn,m(i) 98 S(i) + Dn,m(i)

by ethylene

ktmE(i)

n g 1; m g 0

ktmB(i)

n g 0; m g 1

kth(i)

n g 1; m g 0

kth(i)

n g 0; m g 1

Pn,m(i) + E 98 P1,0(i) + Dn,m(i) Qn,m(i) + E 98 P1,0(i) + Dn,m(i)

by hydrogen

Pn,m(i) + H2 98 S(i) + Dn,m(i) Qn,m(i) + H2 98 S(i) + Dn,m(i)

by cocatalyst (alkylaluminum)

kta(i)

n g 1; m g 0

kta(i)

n g 0; m g 1

Pn,m(i) + A 98 P1,0(i) + Dn,m(i) Qn,m(i) + A 98 P1,0(i) + Dn,m(i) Spontaneous Deactivation kd(i)

S(i) 98 Y(i) kd(i)

n g 1; m g 0

kd(i)

n g 0; m g 1

Pn,m(i) 98 Y(i) + Dn,m(i) Qn,m(i) 98 Y(i) + Dn,m(i)

basic steps normally used to describe olefin polymerizations with Ziegler-Natta and Phillips catalysts, such as site initiation, chain propagation, chain transfer to chain-transfer agents, and site deactivation. The developed kinetic scheme is presented in Table 4 and is based on the mechanism presented originally by De Carvalho et al., which has been used successfully to describe the behavior of industrial19 and laboratory-scale20 reactors. In Table 4, E is an ethylene monomer unit; B is a 1-butene monomer unit; S(i) is an empty catalyst site; Pn,m(i) is a living polymer chain in catalyst site i with ethylene as the last inserted monomer unit, containing n inserted ethylene units and m inserted 1-butene units; Qn,m(i) is a living polymer chain in catalyst site i with 1-butene as the last inserted monomer unit, containing n inserted ethylene units and m inserted 1-butene units; Dn,m(i) is a dead polymer chain formed by catalyst site i, containing n inserted ethylene units and m inserted 1-butene units; and Y(i) is a dead catalyst site. The kinetic scheme does not consider the catalyst activation step, caused by the interaction between the heterogeneous catalyst compound and the alkylaluminum. In the process described herein, the catalyst is fed into the reactor after a short prepolymerization period, which means that the catalyst enters the reactor in its active form. Therefore, it is not necessary to take the activation step into account in the polymerization scheme. If the prepolymerization reactor had been included in the model, this step would have to be considered. The propagation reaction was assumed to be of firstorder in relation to the monomer concentration. Chaintransfer reactions limit the polymer chain length and

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2701

can occur spontaneously (β-elimination) or through transfer to monomer, to alkylaluminum, or to hydrogen.21 The relative importance of these reaction steps depends on the particular catalyst used in the industrial process. For conventional Ziegler-Natta catalysts, chain transfer to hydrogen is the most important process, while chain transfer to monomer is the most important chain transfer step for Phillips catalysts. It has been shown, though, that chain transfer to 1-butene is negligible during the copolymerization of ethylene and 1-butene, while the chain transfer of living 1-buteneterminated polymer chains to ethylene may be very important.22 Therefore, chain transfer to 1-butene is neglected in the kinetic scheme presented in Table 4. A spontaneous deactivation step was also included in the kinetic mechanism. This reaction exerts little influence on the polymer molecular weight under normal operation conditions but may exert a strong influence in the reactor productivity. It must be pointed out, however, that this reaction is not regarded to be very important for Phillips catalysts.23,24 Dead polymer chains formed after spontaneous chain transfer or transfer to monomer can present terminal double bonds and may be reincorporated into growing polymer chains, giving rise to long-chain branches. This phenomenon is known to occur in metallocene25 and Phillips catalysts,26 but the mechanism is not well understood yet.27 Reincorporation of dead chains is neglected here because available experimental data indicate that polymer chains do not present long-chain branches.9 The mass balance equations for the species defined in the kinetic mechanism, the reaction rate expressions, and the calculation of the average molecular weights and of monomer conversions using the standard method of moments are presented in the Appendix. Modeling of the Particle Size Distribution (PSD). The control of the PSD is very important for handling and processing of the final polymer product. Fine particles (1000 µm) may cause melting problems during extrusion and may lead to fish-eye formation. Despite the importance of the PSD of the final polymer, only recently has the modeling of the PSD received considerable attention in the literature. Talbot28 developed a model for the PSD in a gas-phase ethylene polymerization reactor and observed that the PSD of the polymer is broader than the PSD of the catalyst. Soares and Hamielec29 pointed out that the residence time distribution exerts an important effect on the PSD of polymer particles. Zacca et al.30 investigated the effect of the residence time distribution upon the final polymer properties, including the PSD. Zacca and Debling31 developed a comprehensive model to study the effect of the PSD on particle overheating during olefin polymerizations for different reactor configurations. Mattos and Pinto32 described the PSD of polypropylene in a slurry reactor and included it in a process simulator. Zoellner and Reichert33 were able to describe the experimental PSD of polybutadiene produced in a gas-phase laboratory-scale reactor using a population balance model coupled with a kinetic scheme. In the absence of breakage and coalescence, the reactor may be regarded as a mixing vessel for the number of particles. Therefore, the total number of particles inside the reactor can be described by the

following population balance equation:

dNp Np ) N˙ p,e dt Θ

(19)

where Np is the number of particles inside the reactor and N˙ p,e is the rate of catalyst particles fed into the reactor. When breakage and coalescence are absent, the particles change size because of the increase of mass due to polymerization. This depends on the number of active sites inside the particle and on the polymerization conditions. Therefore, the natural internal variables required to describe the PSD are the mass (directly associated with the particle diameter), the number of 1-butene- and ethylene-terminated living polymer chains of type j, and the number of empty active sites inside the polymer particle. The population balance equation that describes the number of particles as a function of the internal variables and of the reaction time can be given by

∂ ∂t

[Npf(t,m,S)] ) N˙ p,e fe -

Npf Θ

∂

-

∂m

[m ˘ Npf] ∂

∑i ∑j ∂S [S˙ ijNpf]

(20)

ij

where f(t,m,S) is the number density of polymer particles, fe is the number density of the catalyst particles, m is the particle mass and S is the matrix whose element Sij is the number of active sites of type ij in the polymer particle (site type i is empty if j ) 0, is occupied by a living ethylene-terminated chain if j ) 1, and is occupied by a living 1-butene-terminated chain if j ) 2). m ˘ and S˙ ij represent the rates of change of m and Sij.34 It is assumed that the polymer PSD is known at a given instant (initial condition), that the mass and the elements of the matrix S are nonnegative (boundary condition), and that the polymer PSD is limited (boundary condition), as defined in eq 21. Equations 19 and

∫m∫Sf(m,S) dm dS ) 1

(21)

20 can be combined as

Np

∂ ∂t

f(t,m,S) ) N˙ p,e[fe - f(t,m,S)] -

{

Np

∂

[m ˘ f(t,m,S)] +

∂m

∂

∑i ∑j ∂S

[S˙ ijf(t,m,S)] ij

}

(22)

The rate of mass increase within a polymer particle is given by the summation of the polymerization rates for each site type (ij), which is independent of the particle mass. The number of catalyst sites changes because of the kinetic events, in accordance with the kinetic scheme presented in Table 4. Thus, it is possible to write

m ˘ )

∑i ∑j φijSij

(23)

∑k ζijkSik

(24)

S˙ ij )

where φij is related to the mass increase due to the propagation and initiation steps, while ζijk is related to

2702


Table 5. Meaning of the Terms in the Population Balance term

origin

φi0 ) MEkiE(i) [E] + MBkiB(i) [B] φi1 ) ME[E][kpEE(i) + ktE(i)] + MBkpEB(i) [B]

particle mass increase due to initiation of an empty site particle mass increase due to propagation or chain transfer of an ethylene-terminated living polymer chain particle mass increase due to propagation or chain transfer of a 1-butene-terminated living polymer chain site transformation due to initiation site transformation from an ethylene-terminated living polymer chain to empty sites site transformation from a 1-butene-terminated living polymer chain to empty sites site transformation from an empty active site to an ethylene-terminated living polymer chain site transformation from living ethylene-terminated polymer chains to 1-butene-terminated living chains or dead sites site transformation from 1-butene-terminated living polymer chains to ethylene-terminated living polymer chains site transformation due to initiation by 1-butene site transformation from ethylene-terminated living polymer chains to 1-butene-terminated living polymer chains site transformation from living 1-butene-terminated polymer chains to ethylene-terminated living chains or dead sites

φi2 ) ME[kpBE(i) + ktB(i)][E] + MBkpBB(i) [B] ξi00 ) -kiE(i) [E] - kiB(i) [B] - kd(i) ξi01 ) kte(i) + kth(i) [H] ξi02 ) kte(i) + kth(i) [H] ξi10 ) kiE(i) [E] ξi11 ) -kte(i) - kth(i) [H2] - kpEB(i) [B] - kd(i) ξi12 ) ktB(i) [E] + ktA(i) [A] + kpBE(i) [E] ξi20 ) kiB(i) [B] ξi21 ) kpEB(i) [B] ξi22 ) -kte(i) - kth(i) [H2] - kpBE(i) [E] - kd(i)

the kinetic events that change the nature of the active sites, such as initiation, cross-propagation, and chain transfer. The expressions for φij and ζijk, consistent with the kinetic scheme proposed in Table 4, are presented in Table 5. The population balance described by eq 22 then becomes

Np

∂

Np

f ) N˙ p,e(fe - f) -

(∑ ∑

∂t

i

j

φij

∂ ∂m

[Sijf] +

∂

∑i ∑j ∑k ζijk ∂S

)

[Sikf] (25) ij

Equation 25 explicitly couples the PSD, the kinetics of polymerization, the multisite nature of the catalyst, and the reactor conditions. The method of moments can be used to provide the main averages of the PSD, as presented in detail in the Appendix. On the basis of the moments, it is possible to define the average particle growth (DPC), which is the ratio between the polymer average particle size and the catalyst average particle size as 3

DPC ) xµm/µem

(26)

and the polydispersity index (Qp) of the PSD as

mw µm,m ) Qp ) mn (µ )2

(27)

m

where mw and mn are the weight- and number-average particle masses, respectively, and µ represents the moments of the distribution, as defined in the Appendix. Polymer Properties. As a first approximation, the physical properties of a polymer can be defined as functions of its molecular structure, e.g., molecular weight distribution, composition distribution, polymer architecture, etc. It is normally aimed to produce polymers with specified physical properties; however, these properties cannot be directly related to the reactor conditions through first-principle models. For this reason, they are often defined empirically as functions of the molecular structure of the final polymer.35 Empirical models for the melt index (MI), melt flow ratio (FR), and density (F) are presented below and used to transform

the obtained molecular properties into end-use properties that are used for process monitoring, process control, and grade design. The density of the LLDPE was assumed to be a function of the weight-average molecular weight (Mw) and of the 1-butene molar fraction (χB) of the final polymer material. The obtained correlation can be described as

F ) (1 - 0.009165χB0.148895) [1.137247 - 0.014314 ln(Mw)] (28) The fit of the empirical model to available plant data is presented in Figure 2. Results can be regarded as very good. The MI of polymer resins obtained with a single catalyst system can be correlated to the weight-average molecular weight for a broad range of molecular weights, even when the molecular weight distributions are broad.36 Bremner and Rudin37 reported that the MI of polyolefins can be correlated to the weight-average molecular weight with a simple power-law function such as

MI ) aMw-x

(29)

A power-law correlation was successfully used to describe the MI of LLDPE19 and was also found to describe available plant data very well, as shown in Figure 3. The obtained empirical correlation was

MI ) 901(Mw/105)-5.14

(30)

A simple linear relationship was used to describe how FR depends on the polydispersity index of the molecular weight distribution (Q), as previously reported by Spitz.36 The good adherence of the experimental data to the empirical model (eq 31) can be observed in Figure 4.

FR ) 1.11Q + 11.57

(31)

Estimation of Kinetic Parameters. The proper estimation of kinetic parameters is a fundamental step in the modeling. The kinetic parameters were estimated using literature and experimental data obtained from


Figure 5. Typical reaction rate profile for semibatch slurry polymerizations performed in the laboratory-scale reactor. Figure 2. Experimental and calculated polymer densities for different polymer grades.

Figure 3. Experimental and calculated melt flow indices for different polymer grades.

Figure 4. Experimental and calculated melt flow ratios for different polymer grades.

polymerization runs performed in a laboratory-scale reactor. Literature data provided important information that allowed for the reduction of the set of model parameters and suitable initial guesses for the estimation of the remaining unknown parameters. A typical polymerization rate profile that can be obtained in a semibatch laboratory-scale reactor, when the reaction is performed at conditions that are similar

Table 6. Kinetic Parameters and Hypothesis Taken from Literature Data kinetic constant

hypothesis

ethylene propagation22,41 ethylene cross-propagation40 1-butene propagation39,40 activation energy for the propagation reactions18 hydrogen chain-transfer rate constant24,26 spontaneous chain-transfer rate constant22,40 activation energy for chain transfer40 deactivation rate constant23,24

kpEE ≈ kpBE kpEB ≈ 1.42kpEE kpBB ) 0 Ep ) 41 kJ/mol kth ≈ 0 kte ≈ 0 Et ) 83 kJ/mol kd ) 0

to the industrial conditions, is shown in Figure 5. Figure 5 shows that catalyst deactivation is negligible during the experimental run. This is in accordance with literature data. Krauss24 observed that Phillips catalysts can remain active for weeks, although an excess of alkylaluminums frequently used to reduce the induction period may promote the catalyst deactivation.18 Phillips catalysts are very active for ethylene polymerization; however, the catalyst activity reduces sharply for longer olefins.38 Therefore, the homopropagation of R-olefin (1-butene) is not favored (although the comonomer incorporation in ethylene/R-olefin polymerizations is normally higher than that obtained with conventional Ziegler-Natta catalysts).22 The 1-butene reactivity ratio (r2 ) kpBB/kpBE) in ethylene/1-butene copolymerizations with CrO3/SiO2 in a slurry reactor was found to be negligible,39 while different values have been reported for the ethylene reactivity ratio (r1 ) kpEE/kpEB), ranging from 1.3-1.440,41 to 27.3. For the present work, the value of r1 ) 1.42 was used, in accordance with composition data obtained in laboratory-scale reactors.9 The main chain-transfer step in olefin polymerizations with Phillips catalysts is chain transfer to ethylene. Therefore, the reaction temperature is the most important variable for control of average molecular weights in the polymerization reactor.38,42,43 It is important to notice that the chain-transfer rates of a 1-butene-terminated living polymer chain to ethylene are about 11 times larger than the chain-transfer rates of an ethylene-terminated living polymer chain.22 Besides, the effect of hydrogen on the molecular weight, very important for conventional Ziegler-Natta catalysts, can be neglected42 for Phillips catalysts. The main assumptions used for parameter estimation are presented in Table 6. Parameter estimation was performed using polymerization rate profiles and deconvolution of molecular weight distributions of polymer samples obtained in different experimental runs, as discussed elsewhere.44 Figure 5 illustrates the fit of

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Figure 6. Deconvolution of the molecular weight distribution of a polymer grade for one, two, and three catalyst sites. Table 7. Kinetic Parameters Estimated from MWD Deconvolution

Table 8. Kinetic Parameters Used To Perform Simulations of Actual Industrial Operation

constant

site 1

site 2

site 3

constant

site 1

site 2

site 3

molar fraction kiE (L/mol/min) kpEE (L/mol/min) kpEB (L/mol/min) kpBE (L/mol/min) Ep (kJ/mol) ktmE (L/mol/min) ktmB (L/mol/min) Et (kJ/mol)

0.33 80000 80000 56000 80000 41 44.8 0.8 83

0.4 80000 80000 56000 80000 41 9.6 0.8 83

0.27 80000 80000 56000 80000 41 2.04 80 83

molar fraction kiE (L/mol/min) kpEE (L/mol/min) kpEB (L/mol/min) kpBE (L/mol/min) Ep (kJ/mol) ktmE (L/mol/min) ktmB (L/mol/min) Et (kJ/mol)

0.4 80000 80000 56000 80000 41 80 8.0 83

0.5 80000 80000 56000 80000 41 26.4 8.0 83

0.1 80000 80000 56000 80000 41 2.84 80 83

polymerization rate profiles, while Figure 6 illustrates the deconvolution of molecular weight distributions. Figure 6 shows that three distinct catalyst sites are required to describe the molecular weight distribution of the polymer samples. The estimated model parameters are presented in Table 7. Parameters presented in Table 7 were used for the fitting of actual industrial data, as shown below. Results Industrial Reactor Behavior. The set of model parameters presented in Tables 6 and 7 was used as

initial guesses for the fitting of the transient behavior of the industrial reactor. Simulation results were compared to experimental data (catalyst activity, molecular weight distribution, and comonomer composition of polymer samples collected during production of different grades), and some of the model parameters were adjusted heuristically in order to allow for a fair description of the reactor behavior. The final set of kinetic constants is presented in Table 8. As one may see, parameters presented in Table 8 are essentially the same parameters as those presented in Table 7, which gives independent support to the preliminary analysis performed in the laboratory-scale reactor. The main


Figure 8. Experimental and simulated melt flow indices and densities of the final polymer produced by the industrial reactor during the campaign.

Figure 7. Normalized feed flow rate profiles for the industrial reactor during a production campaign.

difference regards the chain transfer to ethylene and 1-butene, which seems to be larger in the industrial environment than in the laboratory-scale reactor, leading to lower molecular weight averages during plant operation. This is probably related to recirculation of contaminants at the plant site. At the plant site, the solvent is purified and recirculated, which certainly allows for recirculation of contaminants, a relative decrease of the polymerization rates, and a relative increase of chain-transfer reactions. Figure 7 shows typical feed flow rate profiles during a production campaign at the plant site. Feed flow rates are normalized in order to maintain the confidentiality of the industrial data. However, it is possible to note that the feed flow rates of n-hexane (Fnhx) are fairly constant and that the feed flow rates of 1-butene are very noisy and follow the feed flow rates of ethylene for most of the campaign (in order to keep the polymer composition at the desired levels). Catalyst feed rates are periodically increased to about 10 times their nominal values and then decreased to the nominal values again. This is related to control of the slurry concentration. The periodic shots of catalysts allow for a sudden increase of the polymer concentration in the slurry and are introduced when the slurry concentration falls below a minimum desired value. As might be expected, this operation strategy exerts a strong impact on the number of particles inside the reactor and on the PSD of the polymer produced. Figure 8 presents model predictions and actual plant values for the polymer melt index and density during the campaign described previously. The model fit to the experimental data may be regarded as very good, and it seems that the model can be used with confidence to simulate the plant behavior and to monitor the polymer properties. Figure 9 presents model predictions for the variables that are directly related to the MI and F of the final polymer: the weight-average molecular weight (Mw), the polydispersity index (Q), and the 1-butene fraction. It is possible to observe that the catalyst feed strategy exerts a small impact on these variables and

Figure 9. Simulated weight-average molecular weight, polydispersity index and comonomer content of the final polymer produced by the industrial reactor during the campaign.

that the high-frequency perturbation of the 1-butene feed flow rates does not cause significant variations of the product quality. Both effects are filtered by the mixing nature of the CSTR. Figure 10 shows dynamic profiles for the number of particles (Np), polydispersity index of the PSD (Qp), and particle growth factor (DPC) during the production campaign described previously, as predicted by the model. It can be seen in Figure 10 that the reactor operation procedure does lead to significant perturbations of the variables related to the polymer PSD. Fluctuations of the feed flow rates, as shown in Figure 8, exert a significant impact on the number of polymer particles, which increases sharply after catalyst shots. The polydispersity index of the PSD also increases during catalyst shots because the number of small particles increases fast after catalyst shots. On the other hand, the particle growth factor decreases significantly after catalyst shots because a large population of very small particles is fed into the reactor. This indicates that catalyst shots may cause operation problems related to the production of fines at the plant site and that the strategy for control of the slurry concentration should be improved.

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Figure 12. Open-loop response of the polymer melt index and density to disturbances in the ethylene feed flow rate.

Figure 10. Simulated PSD (particle growth factor and polydispersity) of the final polymer produced by the industrial reactor during the campaign.

Figure 13. Open-loop response of the weight-average molecular weight, polydispersity index, reactor temperature, and comonomer fraction to disturbances in the 1-butene feed flow rate. Figure 11. Open-loop response of the weight-average molecular weight, polydispersity index, and reactor temperature to disturbances in the ethylene feed flow rate.

Open-Loop Responses to Feed Disturbances. Disturbances of the feed flow rates are frequently introduced at the plant site for a number of reasons, such as the modification of the production schedule, elimination of contaminants, and grade transition. For this reason, the model was used to analyze the openloop responses of the molecular properties, the physical properties, and the PSD of the final polymer to disturbances of the feed flow rates of ethylene, 1-butene, n-hexane, and catalyst. In all simulations, disturbances of (10% and (30% of the nominal value were applied, assuming that the reactor was initially operating at stable steady-state nominal conditions. Disturbances of this magnitude are not sufficient to lead the reactor to unstable operation because this reactor was found to present complex dynamic behavior in certain circumstances.10 Figures 11 and 12 present the effect of the ethylene feed flow rate perturbation on the reactor temperature and polymer molecular properties. The reactor temperature responds almost instantaneously to changes of the feed flow rate. The increase of the feed rate of ethylene increases the polymerization rate, causing the increase of the heat released by the polymerization and of the reactor temperature. As mentioned previously, this causes the decrease of the average molecular weight of the produced polymer. For this reason, the tight control

of the reactor temperature is of fundamental importance for this technology. However, because of the mixing characteristics of the CSTR and the significant average residence time, the polymer average molecular weight responds slowly to the change of the reactor temperature. When the reactor temperature changes, the properties of the produced polymer are modified suddenly, causing the polydispersity index to present a more complex response to feed perturbations. When perturbations are large, overshoots and nonlinear response gains can be observed for Q. The responses of the polymer melt index and density to modifications of the ethylene feed flow rates are shown in Figure 12 and essentially follow the observed responses of Mw. Figures 13 and 14 present the effect of the 1-butene feed flow rate perturbation on the reactor temperature and polymer molecular properties. Disturbances of the 1-butene feed flow rates are much less important for the reactor temperature and polymer properties than those in the previous case. First, 1-butene feed flow rates are much lower than ethylene feed flow rates. Second, 1-butene is much less reactive than ethylene, causing less important modifications of the reactor temperature and polymer properties. This explains why the variations of the 1-butene feed flow rates presented in Figure 7 do not cause a major impact on the process operation. As observed in Figures 13 and 14, only the polymer composition and density respond significantly to changes of the 1-butene feed flow rate. Even in these cases, though, the impact is less important than those observed in Figure 12, which means that density control


Figure 14. Open-loop response of the polymer melt index and density to disturbances in the 1-butene feed flow rate. Figure 17. Open-loop response of the weight-average molecular weight, polydispersity index, and reactor temperature to disturbances in the solvent flow rate.

Figure 18. Open-loop response of the polymer melt index and density to disturbances in the solvent feed flow rate. Figure 15. Open-loop response of the weight-average molecular weight, polydispersity index, and reactor temperature to disturbances in the catalyst feed flow rate.

Table 9. Experimental PSD Parameters for Catalyst and Polymer Particles mn (kg) DPC Qp

Figure 16. Open-loop response of the polymer melt index and density to disturbances in the catalyst feed flow rate.

might also be performed through the simultaneous manipulation of the ethylene feed flow rate and reactor temperature. Figures 15 and 16 present the effect of the catalyst feed flow rate perturbation on the reactor temperature and polymer molecular properties. The effect of the disturbance on the reactor temperature is small. Because most of the ethylene feed is consumed at normal operation conditions, the consumption of the remaining monomer does not exert any significant impact on the rate of polymerization and, consequently, on the final polymer properties. However, as discussed elsewhere,10 variations of the catalyst feed flow rates may lead to complex unstable operation. For similar reasons, modification of the feed flow rates of the diluent does not cause any significant impact on the reactor operation either, as shown in Figures 17 and 18. In this case, the most important modification is the reactor residence time. However, polymerization rates do not change

catalyst

polymer

7.2 × 10-13 1 425

2.34 × 10-9 16.7 950

significantly with the residence time because most of the ethylene feed is consumed at normal operation conditions. Table 9 shows the experimental number-average mass, polydispersity index, and growth factor for the PSDs of the catalyst and polymer particles at nominal operation conditions. The effect of the polymerization reaction on the average size of the particles is evident. It is also interesting to note the increase of the polidispersity of the PSD due to the mixing characteristics of the stirred tank reactor. The effect of the feed disturbances on the PSD can normally be related to the observed changes of the average residence time. For a constant reactor level, disturbances that cause a decrease (increase) of the overall feed flow rates while simultaneously causing an increase (decrease) of the reactor residence time and of the number of particles in the reactor. For this reason, the effect of the feed disturbances on the reactor residence time will be presented with the observed variations of the characteristic parameters of the PSD. Figures 19-21 present the effects of the perturbations of ethylene, catalyst, and diluent feed flow rates on the PSD of the final polymer. It is possible to note that disturbances that lead to a decrease (increase) of the polymerization rate through a decrease (increase) of the ethylene concentration also lead to a decrease (increase) of the average particle growth factor. It is also interesting to note that the sudden modification of the average

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Figure 19. Open-loop response of the PSD (particle growth factor and polydispersion) to disturbances in the ethylene feed flow rate.

Figure 21. Open-loop response of the PSD (particle growth factor and polydispersion) to disturbances in the solvent feed flow rate.

Figure 20. Open-loop response of the PSD (particle growth factor and polydispersion) to disturbances in the catalyst feed flow rate.

residence time causes the temporary modification of the polydispersity index of the PSD, although this effect is less important when diluent feed rates are perturbed and monomer conversions are high. This may be used to design operation policies aimed at the production of polymer grades with narrower and/or broader PSDs, especially when the remaining polymer properties are not affected significantly by the disturbance (as in the case of the diluent and catalyst feed flow rates). Perturbations of the catalyst feed flow rates exert no observable effect on the reactor residence time; however, disturbances of the catalyst feed flow rates may exert a significant impact on the number of particles inside the reactor, changing the average particle growth factor and the polydispersity index of the PSD. Reactor Multiple Steady States. It was shown previously, using a much simpler model, that the reactor analyzed here may present complex dynamic behavior, including multiple steady states and sustained oscillatory behavior.10 This more complete model confirms the results reported previously and shows that multiple steady states can indeed occur for conditions of low catalyst concentration. These conditions can be achieved,

Figure 22. Multiple steady-state responses.

in practice, during start-up or grade transitions or when the catalyst is partially deactivated by feed impurities. Figure 22 illustrates the existence of steady-state multiplicity. The presentation of a detailed bifurcation analysis is beyond the scope of this paper. The reactor is initially at stable steady-state conditions, with 25% of the ethylene nominal feed flow rate, 30% of the catalyst nominal feed flow rate, and 75% of the solvent feed flow rates (low polymer production condition). Even though the catalyst and ethylene are fed continuously into the reactor, the ethylene conversion (ξE ) 0) and the particle growth factor (DPC ) 1) are negligible. After 5 h of operation (t ) 5 h), the ethylene feed flow rate is increased to 50% of its nominal value and the reaction ignites, increasing the reactor temperature by 60 °C in only 9 min. The reactor then reaches a new stable steady state. After 20 h of operation (t ) 20 h), the nominal ethylene feed flow rate is decreased to the initial value of 25% of the nominal feed flow rate, and


Figure 23. Schematic representation of the segregated zone reactor model.

a new steady state, which is different from the initial one (observed at t ) 5 h), is reached. Therefore, the reactor presents at least two stable steady states (and at least one unstable steady state) for the starting conditions of 25% of the ethylene nominal feed flow rate, 30% of the catalyst nominal feed flow rate, and 75% of the solvent feed flow rates. Besides, the operation of the reactor at conditions of low reactor load may lead to reaction ignition after modification of the catalyst and/ or ethylene feed flow rates. So, operation at conditions of low polymer production should be carefully analyzed at the plant site. (This becomes even more important when one considers that usual plant knowledge assumes that the operation risks are much smaller when polymerization rates are low.) Analysis of Segregation. Slurry polymerization reactors are usually very large, and reactor volumes may reach a few hundred cubic meters. For this reason, mixing conditions may not be uniform and homogeneous inside the reactor, even when the reactor contains a single mixing zone. Different reaction zones with different reaction conditions can be formed in the reactor vessel because of nonideal mixing, leading to the formation of reagent-rich zones in reactor regions that are close to the feed points. These reagent-rich zones are characterized by higher polymerization rates and higher reactor temperatures45 and produce polymer chains with different properties, when compared to the reactor volume that surrounds these zones. Figure 23 shows the schematic representation of the polymerization reactor divided into two zones. It is assumed that higher reagent concentrations can be found in the region that surrounds the feed point. As a first simple approach, the two-zone reactor can be described as a train of two CSTR reactors, where the first reactor of the series receives the feed flows and operates adiabatically. Figure 24 shows steady-state responses (ethylene conversion, reactor temperature, polymer molecular weight, and particle growth factor) to a continuous increase of the segregated reactor volume. It is possible to observe that if the segregated zone is smaller that 0.5% of the reactor total volume, the monomer conversion in this zone is negligible and the impact on the reactor operation can be discarded. However, as the relative volume of this zone increases to 2%, which is a very small volume and might indeed represent the actual industrial operation, the temperature in this segregated zone may increase up to 200 °C, leading to polymer softening and particle agglomeration. Furthermore, in this zone the polymer molecular weight would be 5-10 times smaller than that in the nonadiabatic reactor zone, causing a significant decrease of the overall

Figure 24. Effect of the volume of the segregated zone on the reactor state and polymer properties.

polymer molecular weight and an increase of the polydispersity index. Therefore, the existence of very small segregated zones inside the reactor can exert a very significant impact on the reactor behavior, explaining abnormal formation of agglomerates and unusually broader molecular weight distributions. Conclusions A comprehensive model was developed for the ethylene/1-butene copolymerization in an industrial slurry polymerization reactor for LLDPE synthesis. The model is able to describe the dynamic evolution of the molecular weight averages, comonomer content, particle size averages, melt index, and density of the final polymer resin and extends modeling results available in the open literature. Besides, a new modeling approach was used to describe the evolution of particle sizes, which is based on the definition of a joint distribution of mass and catalyst concentration of solid polymer particles. The model parameters were obtained from the analysis of laboratory-scale polymerization data, of offline polymer characterizations, and of industrial reactor operation conditions. It was shown that the model successfully describes the operation of an industrial slurry polymerization reactor. For this reason, the model was used to analyze how sensitive the final polymer properties are to variations of the feed conditions and to the development of segregated mixing zones inside the reactor vessel. It was shown that the most influential process variable is the ethylene feed flow rate and that perturbation of this variable may cause significant modifications of reactor operation and final polymer properties. It was also shown that sudden modification of the average residence time (through manipulation of the diluent feed flow rate, for instance) may allow for the production of polymer grades with narrower and/or broader PSDs. Besides, the model indicates that multiple steady states and complex dynamic behavior may exist when the

2710


reactor operates at conditions of low polymer production rates. Finally, it was shown that the existence of very small segregated zones inside the reactor (about 2% of the overall reactor volume) can exert a very significant impact on the reactor behavior, explaining the abnormal formation of agglomerates and unusually broader molecular weight distributions. Acknowledgment The authors thank CNPq (Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico) and CAPES (Conselho para Aperfeiç oamento de Pessoal de Nı´vel Superior) for providing scholarships and supporting this project. The authors also thank Polialden SA, Camaçari BA, for supporting this project both technically and financially.

]

kpBE(i) [E] Q0,1(i) + kiB(i) [B]S(i) - kpBB(i) [B]Q0,1(i)

ktcB(i) ) ktA(i) [A] + ktmB(i) [B]

(A.7)

(v) Living polymer chains with m 1-butene units and no ethylene:

[ ]

q d + kd(i) + ktc(i) + ktcB(i) + Q (i) ) - s dt 0,m V + Vl kpBE(i) [E] Q0,m(i) - kpBB(i) [B][Q0,m(i) - Q0,m-1(i)]

Evaluation of the Moments of the Molecular Weight Distribution. The mass balance equations for the chemical species of the kinetic scheme presented in Table 4 are given below. (i) Living polymer chains with one ethylene unit and no 1-butene:

dt

(A.6)

where the overall chain-transfer constants for 1-buteneterminated living polymer chains are defined as

Appendix

d

[

d q Q (i) ) - s + kd(i) + ktc(i) + ktcB(i) + dt 0,1 V + Vl

[

P1,0(i) ) -

q

Vs + Vl

]

+ kd(i) + ktc(i) + ktcE(i) +

(vi) Living polymer chains terminated in 1-butene, with n ethylene and m 1-butene units (n + m > 1 and n > 0):

[

q d + kd(i) + ktc(i) + ktcB(i) + Q (i) ) - s dt n,m V + Vl

[ktcE(i) Pn,m(i) + ∑n ∑ m ktcE(i) Qn,m(i)] (A.1)

where the overall chain-transfer constants for ethyleneterminated living polymer chains are defined as

ktc(i) ) kte(i) + kth(i) [H2]

(A.2)

ktcE(i) ) kta(i) [A] + ktmE(i) [E]

(A.3)

(ii) Living polymer chains with n ethylene units and no 1-butene:

[

q d P (i) ) - s + kd(i) + ktc(i) + ktcE(i) + dt n,0 V + Vl

]

kpEB(i) [B] Pn,0(i) - kpEE(i) [E][Pn,0(i) - Pn-1,0(i)] (A.4)

(iii) Living polymer chains terminated in ethylene, with n ethylene and m 1-butene units (n + m > 1 and m > 0):

[

q d P (i) ) - s + kd(i) + ktc(i) + ktcE(i) + dt n,m V + Vl

]

kpEB(i) [B] Pn,m(i) + kpBE(i) [E]Qn-1,m(i) kpEE(i) [E][Pn,m(i) - Pn-1,m(i)] (A.5)

]

kpBE[E] (i) Qn,m(i) + kpEB(i) [B]Pn,m-1(i) -

kpEB(i) [B] P1,0(i) + kiE(i) [E]S(i) kpEE(i) [E]P1,0(i) +

(A.8)

kpBB(i) [B][Qn,m(i) - Qn,m-1(i)] (A.9)

(vii) Dead polymer chains formed by chain transfers and catalyst deactivation with n ethylene and m 1-butene units:

d D (i) ) [kd(i) + ktc(i) + ktcE(i)]Pn,m(i) + dt n,m q [kd(i) + ktc(i) + ktcB(i)]Qn,m(i) - Dn,m(i) s V + Vl (A.10) (viii) Empty catalyst sites chain-transfer reactions of the catalyst feed to the reactor:

q S(i) ) FS(i) - S(i) dt Vs + Vl [kiE(i) [E] + kiB(i) [B] + kd(i)]S(i) + d

∞

ktc(i) [

∞

∑∑

∞

Pn,m(i) +

n)1 m)0

∑ ∑ Qn,m(i)]

(A.11)

n)0 m)1

Only a fraction of the total number of chromium atoms in the catalyst is active for polymerization. Hence, the effective site i feed flow rate may be given by

FS(i) ) F iFcat.

(A.12)

where F i is the concentration of the catalyst site i in the catalyst (in mol/kg). The moments of the molecular weight distribution of the ethylene-terminated living polymer chains are given by ∞

(iv) Living polymer chains with one 1-butene unit and no ethylene:

∞

λjk E (i) )

∞

∑ ∑ njmkPn,m(i) n)1 m)0

(A.13)


The mass balance equations for the ethylene-terminated polymer species can be summed up for m and n, and the first moments of these species are given by

[

d 00 q + kd(i) + ktc(i) + ktcE(i) + λ (i) ) - s dt E V + Vl

]

00 kpEB(i) [B] λ00 E (i) + kiE(i) [E]S(i) + kpBE(i)[E]λB (i) + 00 ktcE(i) λ00 E (i) + ktcB(i) λB (i) (A.14)

[

q d 10 + kd(i) + ktc(i) + ktcE(i) + λ (i) ) - s dt E V + Vl

]

kpEB(i) [B]

λ10 E (i)

kpBE(i)

+ kiE(i) [E]S(i) + kpEE(i)

M1[λ10 B (i)

[

+

λ00 B (i)]

+

[E]λ00 E (i)

+

ktcE(i) λ00 E (i) + ktcB(i) λ00 B (i) (A.15)

d 01 q λ (i) ) - s + kd(i) + ktc(i) + ktcE(i) + dt E V + Vl

]

01 kpEB(i) [B] λ01 E (i) + kpBE(i) [E]λB (i) (A.16)

[


]

kpEB(i) [B]

λ11 E (i)

[

+ kpEE(i)

MEλ01 E (i)

]

]

10 kpBE[E](i) λ10 B (i) + kpEB(i) [B]λE (i) (A.23)

[

d 11 q + kd(i) + ktc(i) + ktcB(i) + λ (i) ) - s dt B V + Vl

]

10 kpBE[E](i) λ11 B (i) + kpBB(i) [B]λB (i) +

[


]

02 kpEB(i) [B] λ02 E (i) + kpBE(i) [E]λB (i) (A.19)

The moments of the molecular weight distribution of the 1-butene-terminated living polymer chains are given by ∞

λjk B (i) )

∞

∑ ∑ njmkQn,m(i)

(A.20)

n)0 m)1

The mass balance equations for the 1-butene-terminated polymer species can be summed up for m and n, as performed previously, and the first moments of these species are given by

q d 00 λ (i) ) - s + kd(i) + ktc(i) + ktcB(i) + dt B V + Vl + kiB(i) [B]S(i) + kpEB(i)

10 kpEB(i) [B][λ11 E (i) + λE (i)] (A.24)

[

q d 02 λ (i) ) - s + kd(i) + ktc(i) + ktcB(i) + dt B V + Vl

]

01 kpBE[E](i) λ02 B (i) + kiB(i) [B]S(i) + kpBB(i) [B][2λB (i) +

λ00 B (i)]

01 00 + kpEB(i) [B][λ02 E (i) + 2λE (i) + λE (i)] (A.25)

[


]

The moments of the molecular weight distribution of the dead polymer chains are given by ∞

Λjk(i) )

10 00 00 kpBE(i) [E][λ20 B (i) + 2λB (i) + λB (i)] + ktcB(i) λB (i) (A.18)

]

[


01 kpBE(i) [E][λ11 B (i) + λB (i)] (A.17)

00 00 kpEE(i) [E][2λ10 E (i) + λE (i)] + ktcE(i) λE (i) +

kpBE[E](i)

00 kpEB(i) [B][λ01 E (i) + λE (i)] (A.22)

20 kpBE[E](i) λ20 B (i) + kpEB(i) [B]λE (i) (A.26)

kpEB(i) [B] λ20 E (i) + kiE(i) [E]S(i) +

λ00 B (i)

]

00 kpBE[E](i) λ01 B (i) + kiB(i) [B]S(i) + kpBB(i) [B]λ2 (i) +

+


[

[

d 01 q λ (i) ) - s + kd(i) + ktc(i) + ktcB(i) + dt B V + Vl

[B]λ00 E (i) (A.21)

∞

njmkDn,m(i), ∑ ∑ m)0 n)0

m+n>0

(A.27)

The mass balance equations for the dead polymer species can be summed up for m and n, as performed previously, and the first moments of these species are given by

d 00 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ00 E (i) + [kts(i) + dt q Λ00(i) (A.28) ktc(i) + ktcB(i)]λ00 B (i) - s l V +V d 10 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ10 E (i) + [kd(i) + dt q ktc(i) + ktcB(i)]λ10 Λ10(i) (A.29) B (i) - s V + Vl d 01 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ01 E (i) + [kd(i) + dt q Λ01(i) (A.30) ktc(i) + ktcB(i)]λ01 B (i) - s V + Vl d 11 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ11 E (i) + [kd(i) + dt q Λ11(i) (A.31) ktc(i) + ktcB(i)]λ11 B (i) - s V + Vl

2712


d 20 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ20 E (i) + [kd(i) + dt q Λ20(i) (A.32) ktc(i) + ktcB(i)]λ20 B (i) - s l V +V d 02 Λ (i) ) [kd(i) + ktc(i) + ktcE(i)]λ02 E (i) + [kd(i) + dt q ktc(i) + ktcE(i)]λ02 Λ02(i) (A.33) B (i) - s l V +V The mass balance equation for the empty active sites can be given by

[

d S(i) ) FS(i) - kiE(i) [E] + kiB(i) [B] + kd(i) + dt q 00 S(i) + ktc(i)[λ00 E (i) + λB (i)] (A.34) s l V +V

]

(χB) for the total polymer are given by Mn ) ns

∑{λ

10 E (i)

ME

i)1

ns

∑[λ

00 E (i)

00 + λ00 B (i) + Λ (i)]

i)1

(A.39) Mw ) ns

∑[λ ME

ns

20 E (i)

20 + λ20 B (i) + Λ (i)] + 2r

i)1

∑[λ

11 E (i)

11 + λ11 B (i) + Λ (i)]

j)1

+

ns

∑

{λ10 E (i)

+

λ10 B (i)

10

+ Λ (i) +

r[λ01 E (i)

+

λ01 B (i)

+ Λ (i)]}

ns

∑[λ

r2 ME

02 E (i)

02 + λ02 B (i) + Λ (i)]

j)1

ns

∑{λ

10 E (i)

01 01 10 01 + λ10 B (i) + Λ (i) + r[λE (i) + λB (i) + Λ (i)]}

i)1

(A.40) Q ) Mw/Mn

ns

λ10 E (i)

+

λ10 B (i)

10

+Λ

λ00 E (i)

01 (i) + r[λ01 E (i) + λB (i) 00 + λ00 B (i) + Λ (i)

01 01 [λ01 ∑ E (i) + λB (i) + Λ (i)] j)1

01

+ Λ (i)] (A.35)

ns

10 01 10 01 01 [λ10 ∑ E (i) + λB (i) + Λ (i) + λE (i) + λB (i) + Λ (i)] i)1

(A.42)

Mw(i) ) 20 11 11 20 11 λ20 E (i) + λB (i) + Λ (i) + 2r[λE (i) + λB (i) + Λ (i)]

λ10 E (i)

(A.41)

χB )

Mn(i) )

ME

01

i)1

From the leading moments of the molecular weight distribution of the living and dead polymer in the reactor, it is possible to evaluate the number-average molecular weight [Mn(i)], weight-average molecular weight [Mw(i)], polydispersity index [Q(i)], and 1-butene incorporation [χB(i)] for the polymer produced by each catalyst site i as

ME

10 01 01 01 + λ10 B (i) + Λ (i) + r[λE (i) + λB (i) + Λ (i)]}

+

λ10 B (i)

10

+ Λ (i) +

r[λ01 E (i)

+

λ01 B (i)

01

+ Λ (i)]

+

02 02 r2[λ02 E (i) + λB (i) + Λ (i)] ME 10 01 01 10 01 λE (i) + λ10 B (i) + Λ (i) + r[λE (i) + λB (i) + Λ (i)]

(A.36)

Q(i) ) Mw(i)/Mn(i)

(A.37)

χB(i) )

Evaluation of the Moments of the PSD. The averages of the PSD were evaluated using the method of moments for continuous variables,34 allowing for estimation of the average particle size and the polydispersity index of the PSD. To simplify the notation, it is convenient to define the mean value operator

〈P〉 )

∫m∫SP(m,S) dm dS

Then, the first moment for a variable x can be defined as

µx ) 〈x〉 ) 01 01 λ01 E (i) + λB (i) + Λ (i) 10 01 10 01 01 λ10 E (i) + λB (i) + Λ (i) + λE (i) + λB (i) + Λ (i)

(A.38) where ME is the ethylene molecular weight and r is the ratio between the molecular weights of ethylene and 1-butene. The final polymer properties are functions of the polymer produced by all catalyst sites. Therefore, for ns catalyst sites, the number-average molecular weight (Mn), the weight-average molecular weight (Mw), the polydispersity index (Q), and the 1-butene incorporation

(A.43)

∫m∫Sxf(m,S) dm dS

(A.44)

and the second-order moment, first order in relation to x and y, can be given by

µx,y ) 〈xy〉 )

∫m∫Sxyf(m,S) dm dS

(A.45)

In this work, x and y can be particle internal variables such as the particle mass (m) or the number of type i active sites that are occupied by living chains (j * 0) or are empty (j ) 0). These active sites are represented by ij, as discussed in the derivation of the population balance. Using the moment definitions in the population balance previously derived


∂

Np

∂t

(

f ) N˙ p,e(fe - f) - Np

∂

[m ˘ f] +

∂m

∂

∑i ∑j ∂S

[S˙ ijf]

ij

)

(A.46)

σm2 ) )

∫m∫Sm2f dm dS - 2mj ∫m∫Smf dm dS + m j 2∫m∫Sf dm dS

)

∫m∫Sm2f dm dS - 2mj 2 + mj 2

and the auxiliary equation

∫

∞ n ξ 0

n

∂ [φ(ξ) f(ξ)] dξ ) ξnφ(ξ) f(ξ)|∞0 ∂ξ

∫

∞ n-1 ξ φ(ξ) 0

f(ξ) dξ ) -n

∫

∞ n-1 ξ φ(ξ) 0

f(ξ) dξ (A.47)

which is valid if f(ξ) has exponential decay, it is possible to obtain the population balance equation in terms of the moments of the PSD

d

〈moSijpSklq〉 )

dt o

N˙ p,e (〈moSijpSklq〉e - 〈moSijpSklq〉) + Np

∑r ∑s φrs〈mo-1SijpSklqSrs〉 + p∑r ζijr〈moSijp-1SklqSir〉 + q∑ζklr〈moSijpSklq-1Skr〉 (A.48) r

When the values of p and q are conveniently applied in eq A.48, it is possible to obtain the leading moments of the PSD. The first-order moments are given by

d

µm )

dt

N˙ p,e (µem - µm) + Np

∑r ∑s φrsµrs

N˙ p,e (µeij - µij) + µij ) dt Np d

∑r ζijrµir

(A.49)

(A.50)

dt d dt

µm,m )

µm,ij )

dt

dt

∑r ∑s φrsµm,rs

N˙ p,e e (µm,ij - µm,ij) + Np

d

d

N˙ p,e e (µm,m - µm,m) + 2 Np

µij,kl )

µij,ij )

(A.51)

∑r ∑s φrsµij,rs + ∑r ζijrµm,ir

(A.52)

N˙ p,e e (µij,ij - µij,ij) + 2 Np

∑r ζijrµij,ir

N˙ p,e e (µij,kl - µij,kl) + Np

(A.53)

∑r ζijrµir,kl + ∑r ζklrµij,kr

(A.54)

After calculationg of the moments, it is possible to evaluate the particle average mass as

m j )

∫m∫Smf dm dS ) µm

) µm,m - µm2

(A.56)

The variance is an absolute measure of the breadth of the PSD so that it is convenient to define a relative measure such as the polydispersity index of the molecular weight distribution. The polydispersity index of the PSD is given by

Qp )

mw µm,m ) mn µ 2

(A.55)

and the variance of the particle mass distribution

(A.57)

m

where mw is the weight-average particle mass

∫m∫Sm2f dm dS µm,m ) mw ) ∫m∫Smf dm dS µm

(A.58)

and mn is the number-average particle mass

mn )

∫m∫Smf dm dS ) µm

(A.59)

It is important to note that the polydispersity index is related to the variance of the PSD as

j n) 2 Qp ) 1 + (σm/m

while the second-order moments are given by equations

d

∫m∫S(m - mj )2f dm dS

(A.60)

To compare the sizes of the polymer particle and of the initial catalyst particle, it is convenient to define the growth factor (DPC) as the ratio between the polymer average particle size and catalyst average particle size. Assuming that both catalyst and polymer particles are spherical and have similar densities 3

DPC ) xµm/µem

(A.61)

The moments of the PSD depend on the catalyst PSD. The PSD of the catalyst particles was obtained from light scattering (LLALS) measurements and is assumed to be known. In the feed, all catalyst sites were assumed to be empty so that only the moments related to the e e e e , µi0 , µi0,i0 , µi0,k0 , catalyst mass and empty sites (µem, µm,m e and µm,i0) were nontrivial. It was also assumed that the active sites are homogeneously distributed over the catalyst particle so that the initial number of active sites is proportional to the particle mass. Therefore e ) F iµem µi0

(A.62)

e e ) F iµm,m µm,i0

(A.63)

e e ) F i2µm,m µi0,i0

(A.64)

2714

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 e e µi0,k0 ) F iF kµm,m

(A.65)

where F i and F k are the concentrations of active sites of types i and k in the catalyst at the feed stream. The feed rate of catalyst particles is given by the ratio between the catalyst mass feed rate and the numberaverage catalyst mass as

N˙ p,e ) Fcat./µem

(A.66)

With these pieces of information, it is possible to evaluate the polymer PSD under process conditions. List of Symbols [A] ) cocatalyst concentration in the solid phase (mol/L) [B] ) 1-butene concentration in the solid phase (mol/L) [E] ) ethylene concentration in the solid phase (mol/L) [H2] ) hydrogen concentration in the solid phase (mol/L) cp ) specific heat (kJ/kg/°C) Dn,m(i) ) dead polymer chain produced by a catalyst site i containing n ethylene monomer units and m 1-butene monomer units (mol) dp ) particle diameter (µm) DPC ) particle growth factor Ep ) activation energy of the propagation reactions (kJ/ mol) Et ) activation energy of the transfer reactions (kJ/mol) f ) number density of the particle size distribution Fi ) mass feed rate of component i (kg/min) FR ) melt flow ratio HM ) Henry’s constant (kgf/cm2) kd ) deactivation rate constant (min-1) kiB ) initiation by the 1-butene rate constant (L/mol/min) kiE ) initiation by the ethylene rate constant (L/mol/min) kpBB ) 1-butene homopropagation rate constant (L/mol/ min) kpBE ) ethylene cross-propagation rate constant (L/mol/ min) kpEB ) 1-butene cross-propagation rate constant (L/mol/ min) kpEE ) ethylene homopropagation rate constant (L/mol/min) kta ) chain transfer to cocatalyst rate constant (L/mol/min) kte ) spontaneous chain-transfer rate constant (L/mol/min) kth ) chain transfer to hydrogen rate constant (L/mol/min) ktmB ) 1-butene-terminated living polymer chain transfer to ethylene rate constant (L/mol/min) ktmE ) ethylene-terminated living polymer chain transfer to ethylene rate constant (L/mol/min) mgi ) mass of component i in the gas phase (kg/m3) mli ) mass of component i in the liquid phase (kg/m3) msi ) mass of component i in the solid phase (kg/m3) m ) particle mass (kg) MB ) 1-butene molecular weight (g/mol) ME ) ethylene molecular weight (g/mol) MI ) melt index (g/10 min) mn ) number-average particle mass (kg) Mn ) number-average polymer molecular weight (g/mol) mw ) weight-average particle mass (kg) Mw ) weight-average polymer molecular weight (g/mol) Np ) number of particles in the reactor N˙ p,e ) number of catalyst particles feed rate 2 Psat nhx ) n-hexane vapor pressure (kgf/cm ) P ) reactor pressure (kgf/cm2) Pn,m(i) ) ethylene-terminated living polymer chain in a catalyst site i containing n ethylene and m 1-butene incorporated units (mol) Q(i) ) polydispersity index of the molecular weight distribution of the polymer produced by active site i

Q ) polydispersity index of the molecular weight distribution of the whole polymer in the reactor Qn,m(i) ) 1-butene-terminated living polymer chain in a catalyst site i containing n ethylene and m 1-butene incorporated units (mol) Qp ) polydispersity index of the particle size distribution of the particles in the reactor Q˙ c ) heat removed by the cooling jacket (W) Q˙ t ) heat removed by the external heat exchangers (W) Rpol ) polymerization rate (kg/L/min) S(i) ) empty active site (mol) Sij ) number of active sites of type i, empty (j ) 0), bearing an ethylene-terminated polymer chain (j ) 1) or bearing a 1-butene-terminated polymer chain (j ) 1) (mol) T ) reactor temperature (°C) T0 ) makeup water temperature (°C) Tei ) temperature of the component i at the feed (°C) UAc ) heat-transfer coefficient of the jacket (W/°C) UAt ) heat-transfer coefficient of the external heat exchangers (W/°C) Vl + Vs ) slurry volume (m3) wi ) mass fraction of the component i in the solid phase xi ) molar fraction of the component i in the liquid phase Y(i) ) dead active site (mol) yi ) molar fraction of the component i in the liquid phase Greek Letters χB ) 1-butene fraction in the polymer ∆Hpol ) heat of polymerization (kJ/kg) λB ) moments of the living 1-butene-terminated polymer chain λE ) moments of the living ethylene-terminated polymer chain µ ) moments of the particle size distribution Θ ) reactor residence time (min) F ) density (kg/m3) Λ ) moments of the dead polymer chains

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Received for review May 16, 2004 Revised manuscript received September 29, 2004 Accepted September 30, 2004 IE049588Z

1-Butene Copolymerizations in Industrial Slurry

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