Slurry HDPE Process Technology

Ind. Eng. Chem. Res. 2002, 41 , 5601-5618

5601

Stead Ste adyy-St Stat ate e and Dyna Dynam mic Mo Mode deli ling ng of C ommerci rcial al Slurr Slurry y Hig Hi ghh-De Dens nsity ity Polye Polyethy thylen lene e (HDPE ) Pr Pro oce ces sses Nee Ne eraj P. Kha Khare re,, Ke Kevin vin C. Se Sea ave vey, y, and Y. A. Li u* H one oneywell ywell Center Center of Excell Excell ence in Comput er-Ai ded Design Design , Depar Depar tm ent of Chemi cal En gin eeri ng, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 

Sunda Sun daram ram Rama Ramana natha than, n, Simo Simon n L ing ingard, ard, and Chau-Chyun Chau-Chyun C he hen n Aspen Aspe n T echnology, I nc., 10 Cana l Par k, Cam bri dge, M ass assachuse achusetts tts 02141 

We present present t he develo development pment of both both stea dydy-sta sta te a nd dyna mic models models for for a slurry H DP E process process using f undamenta l chemical engineering principles a nd advanced software to ols, Polymers Plus and Aspen Dynamics. The discussion includes thermodynamic properties, phase equilibrium, reaction kinetics, polymer polymer pro prope perties, rties, a nd other m ode odeling ling issues. We cha cha ra cterize a Zi Ziegl egler erNatta catalyst by assuming the existence of multiple catalyst site types and deconvoluting data from gel permeation chromatography to determine the most probable chain-length distributions a nd relat ive amounts of polymer polymer produced produced at each site ty pe pe.. We We valida te th e model model using plant da ta fro from m t w o lar lar gege-scale scale comme commercial rcial slurry H DP E processes. processes. Signific Significaa ntly, t he model model conta conta ins a single set of kinetic and thermodynamic parameter s that accuratel y predicts the polymer production rate, molecular weight, polydispersity index, and composition for several product grades. We We illustra illustra te t he utility utility of the dynamic model model by simul simulat at ing a grade change. Finally, we pro propo pose se a pro proccess retrofit retrofit t hat pe permits rmits a n inc increase rease in t he HD P E pro produc ductio tion n rat e of up to 20%w 20 %w hil hilee ma intaining t he produc productt qua li lity. ty. 1. I ntro ntroduc ductio tion n 1.1. Slurry HDP HDPE E Proce Proces ss Techn Technolo olog gy. The slur ry polymerization of polymerization of HD PE is the oldest oldest a nd most w idel idelyy used method of productio production. n. Figure 1 pro provide videss a flo flowwchart for a typical slurry HDPE process. Slurry processess utilize either co esse continuous ntinuous stirredstirred-ta ta nk reacto reactors rs 1 (CSTRs), as in our case, or loop reactors. The monomers, cha cha inin-tra tra nsfer agent , solvent, solvent, and cata lyst species species enter t he reactors for polymeriza polymeriza tion. The The vaporiza tion of the solvent removes a large portion of the highly exotherm exo therm ic heat of polymeriza polymeriza tion. The The resulting slurry undergoes separation, removing unreacted monomer, solvent, and oligomeric species from the polymer. Solvent is separated from the oligomer and recycled to the reactor inlets, and the oligomer is processed and packaged. Meanw hile hile,, the po polymer lymer un dergo dergoes es mixing, pelletization, and packaging. The reactor temperature remains below the polymer melting point. The polymer crystallizes upon formation, creat cre at ing a slurry of solid solid par ticl ticles es in t he solvent. solvent.1 The introduction of a co comonomer monomer species species (typically prop propylylene, 1-butene, or 1-hexene) allows for the adjustment of the po polymer lymer pro propertie perties, s, bec because ause of the shortshort-chain chain branches resulting from the alkyl groups on the comonomer. mono mer. In creasing t he comonome comonomerr content dec decreases reases the crysta lli llinity nity of the polymer polymer product product an d increases the rate of ethylene polymerization.2 An increase in t he comonomer co monomer content content also decreases t he polymer polymer density 1 and melting point. The major advantages of a slurry process include mild operating conditions, high monomer conversion, ease of h ea ea t r em e m ov ov a l , a n d r el e l a t i ve ve e a s e of p ro r oce ss ss in in g . I t s * To whom correspondence should be addressed. Phone: (540) (5 40) 231231-7800. 7800. Fax : (5 (540) 40) 231231-5022. 5022. E-ma E-ma il: design @vt .edu.

disadvantages include long residence times (1-2.5 h per reactor), and limited production rates of polymers that ha ve rela tively low densities (lower (lower tha n 0.9 0.940 40 g/cm3) because of resin swelling.1 The Ziegler -Natta catalyst system involves a primary ca t a l y s t a n d a coca t a l y s t . Th Th e p r i m a r y c a t a l y s t i s a t r a n s i t i on on -m -m e t a l s a l t , w i t h a m e t a l f r o m g r o u ps ps I V t o VIII of the periodic ta ble. The The cocata cocata lyst is a ba sese-meta meta l 3 halide or or a lkyl, with a metal from groups groups I to III. Ou r modeled process uses titanium tetrachloride (TiCl4) a s the catalyst and triethyl aluminum [Al(C 2H 5)3] a s t h e cocatalyst. Ziegler -Natta catalysts produce polymers with broad molecular weight distributions because of the chemical properties of the catalyst. Two theories currently exist tha t ex explain plain t his heterogeneo heterogeneous us beha vio vior. r.2 The first is the existence of different site types within the catalyst, each with its own reactivity, caused by differences in the loc locaa l chemical compositions compositions of th e a ctive sites. The The s ec econ d i s t h e p r es es en en ce ce of t r a n s p or or t r e si si s t a n ce ce s t h a t affectt t he ra te a t w hic affec hich h monomer monomer species species travel to the active sites. However, under most polymerization conditions, the effect of different catalyst site types is the dominat domi nat ing fa cto ctor. r.4 We therefore incorporate this catalytic effec effectt by kinetic kinetically ally model modeling ing mult iple cat cat alyst site types. We discuss this approach in section 3.5. 1.2. Mode Modeled led Pr Proce ocess sses es.. We obtained process data for eight grades of HDPE produced in two large-scale slurr y polymeriza tion pla nts (144 00 000 0 an d 240 000 000 tons/ year). In this paper, we present modeling methodology and results of modeling these two commercial plants. We refer to these a s plant A and plant B , respectiv respectivel ely. y. Ea ch plant hous houses es two produc production tion trains. One tra in has a para lle llell reactor reactor configu configura ra tio tion, n, and the other other tr ain ha s t w o r e a c t or or s con n e ct ct e d i n s e r ie ie s o r t a n d e m . I n t h e

10.1021 10.1 021/ /ie02 ie02045 0451n 1n CCC : $22.0 $22.00 0 © 200 2002 2 American American C hemical Society P ubl ish ed on Web 10/22/2002

5602 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002

Figure 1. Flowchart of the slurry HDPE process.

Figure 2. Pr ocess flow diagra m for th e para llel reactor configuration.

following sections, we provide more details about each configuration. 1.2.1. Parallel Reactor C onfiguration. F i g u r e 2 shows the para llel arr an gement, in w hich tw o continuous stirred-ta nk reactors (CS TRs) produce the HDP E. The comonomer for the parallel process is propylene. The slurry streams leaving the two reactors are combined and enter a flash unit for the removal of light hydrocar bons. The va por streams leaving th e reactors contain hexane, monomer, and light gases present in the system. These streams ar e cooled a nd flashed into vapor an d liquid st reams, w hich recycle to the monomer and solvent feed streams, respectively. The vaporization of hexane is the prima ry mea ns for removal of the heat of polymerizat ion. The polymer slurr y leaving the fla sh unit enters a centrifugal separator that removes hexane from the polymer. This mother liquor returns to the reactor inlets, while the polymer strea m t ra vels to the processing and packaging phases of production. 1.2.2. Series Reactor Configuration. F i gu r e 3 illustrates the series layout, where raw materials feed to the first C STR a nd t he slurry product is t hen pumped to the second CS TR, w hich a lso receives fresh monomer, catalyst, and solvent. The comonomer for the series process is 1-butene, an d it ent ers only as a feed strea m to the second r eactor. The va por outlet fr om each rea ctor undergoes cooling a nd r ecycles to the rea ctor inlet. The slurry strea m leaving the second rea ctor enters a flash unit for removal of volat iles. The resulting st rea m enters a ce n t r if u ga l s ep a r a t o r , w h i ch r e m ov es a n d r e t u r n s hexane to th e reactor inlets. Although the temperatures of the tw o reactors in t he series configuration ar e the sa me, the hydrogen par tial pressures are different, permitting the production of polymers with different average molecular weights in

the two reactors. This results in a bimodal molecular weight distribution for the final polymer product. One can also vary the amount of comonomer fed to each reactor, providing a means of producing polymers with a variety of specific properties. 1 1.3. Modeling Technology. Our modeling incorporates fundamental chemical engineering principles and a d v a n ce d s of t w a r e t ool s f or b ot h s t ea d y -s t a t e a n d d y n a m i c p r oce ss s i m ul a t i on . We i n cl u de m a s s a n d energy balances, thermophysical properties, phase equilibrium, polymeriza tion kinetics, a nd rea ctor modeling. We use both P olymers P lus an d Aspen Dyna mics to s i m ul a t e t h e H D P E p r oce ss . P ol y m er s P l u s a p p li es process modeling t echnology t o a wide va riety of industrial polymerization processes. It considers the characterization of polymers and tra cking of their structural properties thr oughout t he flowsh eet, phase equilibrium for polymer systems, polymerizat ion kinetics, and reactor modeling. Polymers P lus uses a segment-based a pproach for computing the physical properties of polymer species. By considering a polymer chain as a series of segments whose structures are well-defined, Polymers Plus can model the polymer properties that commonly vary with time in a synt hesis process. This technique permits the modeling of properties such as molecular weight and copolymer composition and can account for the fact that most polymer products conta in a n ensemble of molecules ha ving a distribution of chain lengths. It facilita tes t he use of group-contr ibution methods for th e estima tion of properties such a s heat capacity, density, an d melt- an d gla ss-tra nsition t empera tures. One can a lso incorporat e subr outines for u ser-defined correlations of polymer properties such as density a nd melt index. Polymers Plus can interfa ce with Aspen Dyn am ics to create dynamic models of polymer processes. We incorporate control schemes and track changes in polymer attributes with modifications of process variables such as reactor conditions or component feed ra tes. This integrated software package provides powerful modeling and predictive capa bilities t o the process design engineer.

2. Physical Properties 2.1. Introduction. Choosing appropriate property models for thermodynamic calculat ions can be a chall en g i n g e n d ea v o r . Th e p h a s e b eh a v i or a n d t h e r m ophysical properties of polymer systems are generally much more complicated than those for conventional m i xt u r e s . O n e ca n d e scr i b e t h e p h a s e b eh a v i or of polymer sy stems by using a ctivity-coefficient models a nd equations of state. The latter typically give pressure as

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Figure 3. Pr ocess flow diagra m for t he series reactor configuration. Table 1. Unit Operations for Which We Use the Sanchez-Lacombe and the Chao-Seader Property Methods polymer-containing units (Sanchez -Lacombe method)

nonpolymer units (Chao -Seader method)

polym er r ea ct or s polymer devolatilizers (flash units) poly mer r ecy cle pu mps

r a w -feed pum ps overhead compressors over hea d fla sh un it s

a function of tempera ture, molar volume, a nd composition, wh ile the former provide a correction to th e idealsolution assumption of Ra oult’s la w. 5 B ecause polymer equa tions of sta te do not normally perform as well as simple cubic equations of state for small components, 5 we use different property methods for the units and streams that contain polymer and those that do not. In the H DP E process, the polymer is present in the rea ctors a nd the subsequent separa tion units. The va por r ecycle conta ins only monomer, solvent, and other small-molecule components, because the polymer is nonvolatile. We use the Sanchez-Lacombe equa tion of sta te for the polymer-conta ining sections of t h e p l a n t a n d t h e C h a o-Seader method for the nonpolymer a rea s. Table 1 lists the portions of th e process m o d e l f o r w h i c h w e u s e t h e S a n c h e z -L a c o m b e a n d C h a o-S e a d e r m e t h od s . We d e scr i b e t h e s e m e t h od s next. 2.2. Sanchez-Lacombe Equation of State for Polymer Systems. We use an equation of stat e (EOS) developed by Sanchez and Lacombe6-8 for the polymercon t a i n i n g p or t i on s of t h e f low s h e et . I t i s b a s e d on lattice theory, which states that fluids are mixtures of molecules and holes that are confined to sites in a lattice. The Sanchez -La combe EOS provides a ccura te predictions of the phase behavior an d t hermodyna mic properties of th e specific components in our system. It is valid for polymer species as well as conventional components. These predictions include molar volume; fugacity coefficients; heat capacities; and depart ures for entha lpy, entropy, and G ibbs free energy. The model is given by

[

(

Fj2 + P  h + T  h ln(1 - Fj) + 1 -

1 r 

)Fj] ) 0

(1)

j , P  h , a n d T  h are the reduced density, pressure, where F and tempera ture, respectively. These qua ntities relat e to the density, pressure, and temperature via

Fj )

P  T  F , P  h ) , T  h) P * T * F*

(2)

w h e r e F*, P * , and T * a re scale factors tha t completely characterize each pure fluid. We typically determine values for these pa ra meters by regressing experimenta l data for each species (usually, vapor-pressure data for conventional components an d liquid-volume dat a for p ol y m er s p eci es ). Al t er n a t i v e ly , w e ca n u s e v a l u es published in the open literature, provided that the data used to obta in the para meter values were measured at or nea r the conditions of the modeled process. The Sanchez -Lacombe EOS a lso has tw o binary interaction para meters tha t one ca n determine by regressing binar y p h a s e d a t a f or t h e c om p on e n t s o f i n t e r es t . R e f er t o sections 2.4 an d 2.5 for the regressed para meter va lues and data sources we used. 2.3. Chao-Seader, Scatchard-Hildebrand, and Redlich-KwongModels for Conventional Species. The Ch ao -Seader correlation provides excellent predictions of the reference-state fugacity coefficients for pure liquid hydrocarbons under our system conditions.9 I t s form is (0) (1) ln (φliq  i  ) ) ln (νi  ) + ωi  ln (νi  )

(3)

w h e r e ν(0) a nd ν(1) ar e functions of the system t emperi  i  ature and pressure and the critical temperature and pressure for component i  a nd ωi  is the acentric factor for species i . Th e C h a o-S ea d er m et h od u ses t h e Redlich -Kw ong EOS for vapor-phase fugacities and t he S c a t c h a r d -Hildebrand model for liquid activity coefficients.10 2.4. Pure-Component Properties. Table 2 lists the primary chemical species that exist in the slurry HDPE process. Impurit ies can be incorporat ed if t heir concent r a t i on s a r e s i gn i f ica n t . Th e s e m i g h t i n cl u d e h y d r o ca r b o n s s u ch a s m e t h a n e a n d e t h a n e . The component list in cludes segments for th e monomer species. As described in section 1.3, Polymers Plus considers polymer species using a segment-based approach, where the ma cromolecules consist of chains conta ining segm ent versions of each monomer species. The polymerization reactions are written in terms of these segments as well. A comprehensive model for t he slurry HD P E process must provide an accurate description of the density, saturation pressure, heat capacity, and heat of vaporizat ion for each species. This is especially importa nt in the rea ctor beca use the kinetics calculat ions depend on a c cu r a t e p h a s e c on ce n t r a t i o ns , a n d t h e h e a t of p ol y merization is primarily removed t hrough the va porization of the solvent, hexane.

5604 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002 Table 2. Components Used in the Slurry HDPE Process Model species

fun ct ion

t it an ium t et ra ch lor ide t r iet h y l a lum in um et h ylen e et h ylen e segm en t pr opylen e pr opylen e segm en t 1-but en e 1-but en e segm en t h ig h-d en si t y p ol yet h y le ne oligom er h y dr ogen n -h exa n e n it r ogen m et h a n e et h a n e pr opa n e n -but a n e

ca ta lyst coca t a ly st m on om er m on om er segm en t com on om er com on om er segm en t com on om er com on om er segm en t p ol ym er w a x by pr oduct ch a in -t r a n sfer a gen t solven t pur ge ga s im pur it y bypr oduct im pur it y im pur it y

Figure 4. Saturation pressure of hexane. 11

Table 3. Pure-Component Parameters for the Sanchez-Lacombe EOSa  component

T * (K)

P * (bar)

F* (k g/m 3)

ca t a ly st coca t a lyst h exa n e et h y len e segm en t pr opy len e segm en t 1-but en e segm en t et h y len e pr opy len e 1-but en e h y dr ogen n it r ogen m et h a n e et h a n e pr opa n e but a n e

924.87 924.87 483.13 663.15 724.3 924.87 333 360.43 396.62 45.89 140.77 224 315 354.33 412.78

4000 4000 2900 4000 2800 4000 2400 3100 2900 1000 1786.17 2482 3273 2800 3257.9

866.97 866.97 786 896.6 938.87 866.97 631 670.83 671.5 142.66 922.5 500 640 615.91 755.68

Figure 5. Saturated liquid density of hexane. 11

a  T *, P * , a n d F* are the characteristic temperature, pressure, and density, respectively.

2.4.1. Sanchez-Lacombe Pure-Component Parameters. Ta ble 3 provides pure-component para meters for t he Sa nchez -La combe EOS for r elevant species. Note that, in Polymers Plus, one must enter the polymer unar y par am eters for th e individual segments tha t comprise it. The density a nd heat -capacity da ta for polyethylene, polypropylene, a nd poly(1-butene) m ust be regressed and the resulting para meters used for each respective segm ent species. We choose pa ra meters for the cat alyst a nd cocata lyst such tha t they remain in the liquid phase. Figure 4 compares model predictions with experimenta l data for the hexane satura tion pressure. Figures 5-7 compare model predictions with experimental data for the densities of hexane, hydrogen, and ethylene, respectively. F igure 8 compares model predictions w ith dat a for t he heat of vaporization of hexane. B oth the S an chezLacombe and Chao-Seader methods accurately describe these properties. 2.4.2. Heat Capacity. One ca n regress hea t-capacity data for pure species to determine parameters for the ideal-gas heat capacity model used in enthalpy predictions ig

C p

) C 1 + C 2T 

(4)

Table 4 gives t he par am eters for the ma jor species in the slurry HDPE process. Figures 9-11 compare model predictions with experim e n t a l d a t a f or t h e h e a t ca p a c i t ie s of h e xa n e , H D P E ,

Figure 6. Density of hydrogen vapor. 11 The predictions of the two property methods are almost identical.

and ethylene, respectively. Note that the EOS predictions tend to be less accurate near the critical re gion, producing sma ll deviations at higher tempera tures a nd pressures. 2.5. Mixture Properties. The vapor -liquid equil ib r i u m i n t h e s l ur r y p ol y m er i z a t i on p r oc es s i s i m portant because the solubilities of the monomer, comonomer, and hydrogen in the hexane directly affect the rates of reaction and the resulting polymer properties. Tables 5 and 6 give values for each of the binary interaction parameters used in the Sanchez-Lacombe m od e l. U n l ik e t h e c a s e f or t h e u n a r y p a r a m e t e r s , w e use the HD PE species when specifying binary int eraction parameters in Polymers Plus.

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002

Figure 7. Density of ethylene va por. 12

Figure 9. Heat capacity of liquid and vapor hexane. 11

Figure 8. Heat of vaporization of hexane. 11

Figure 10. H e a t c a p a c i t y o f H D P E . 13

5605

Table 4. Parameters for the Ideal-Gas Heat C apacity (Eq 4)a  component H D P E (R-C 2H 4) H DP E (R-C 3H 6) H DP E (R-C 4H 8) h exa n e et h ylen e pr opy len e 1-but ene h y dr ogen

C 1

C 2

10 4

3.51 × 4.3205 × 10 4 8.2932 × 10 4 1.6321 × 10 4 2.3194 × 10 4 1.0638 × 10 4 4.6593 × 10 4 2.8332 × 10 4

68.22 133.58 115.29 431.71 78.6581 178.06 154.94 1.96

a  Values were determined by regressing pure-component data. Un its for hea t ca pacity a re J /kmol ‚K .

Figure 12 compares model predictions with experimental data for the solubility of hydrogen in hexane. Hydrogen approaches th e supercritical sta te under t he con d i t ion s u s ed i n t h e H D P E p r oce ss . B e ca u s e t h e Sanchez -Lacombe EOS generally tends to overpredict the critical point, we do not use data near the critical region when determining pure-component parameters. F or t h is r ea s on , a s w e ll a s t h e i mpor t a n ce of t h e solubility prediction of hydrogen in the hexa ne solvent, we used th e solubility d at a in Figure 12 to regress purecomponent parameters for hydrogen. 2.6. Polymer Properties. 2.6.1. Heat of Polymerization. The heat of ethylene polymerization is the d i ff er e n ce b et w e e n t h e e n t h a l py of e t h y le n e a n d t h e enthalpy, per segment, of the polymer under t he sa me conditions. The reaction is15

C 2H 4(g)

f 

1

(C 2H 4)n (amorphous)

n 

∆H 

) - 24.3 kca l/mol (5)

Figure 11. Heat capacity of ethylene vapor. 12 Table 5. Values for Bi nary Interaction Parameters ηi j  for the Sanchez-Lacombe EOS component j  component i 

et h y len e

HDP E

h exa n e

hydrogen ethylene 1-butene h exa n e propylene

-0.0867 -

-0.1093 -

0.100 705 0.1476

0.1476

-

-

0.14

w h e r e ∆H  is the heat of ethylene polymerization. The difference between the entha lpies of ethylene an d H DP E u n d er t h e r e a ct o r con d i t ion s r e pr e se n t s t h e h e a t of p ol y m er i z a t i on i n t h e m od e l. Th e se e n t h a l pi es a r e computed using the Sanchez -Lacombe EOS. Table 7 gives model predictions for the heat of polymerization

5606 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002 Table 6. Values for Binary Interaction Parameters k  i j  for the Sanchez-Lacombe EOS component j  component i  et h y lene hexane m et h a n e et h a n e pr opylen e h ydr ogen butane

h exa n e

HDPE

0.0248

-0.14 -0.14 -

-

0.019 51 0.008 53 0.024 73 0.100 705 -0.002 286

Table 7. Computation of the Heat of Ethylene Polymerization Using the Sanchez-Lacombe EOSa  T 

P 

H ethylene

H  H D P E

∆H f

(°C)

(bar)

(kca l/mo l)

(kca l/mo l)

(kca l/mol )

70 75 80 85 70 75 80 85

5 5 5 5 6 6 6 6

12.2 12.2 12.3 12.3 12.2 12.2 12.3 12.3

-12.7 -12.6 -12.5 -12.4 -12.7 -12.6 -12.5 -12.4

-24.9 -24.8 -24.8 -24.7 -24.9 -24.8 -24.8 -24.7

Figure 13. C o m p a r i s o n o f t h e a c t u a l p h a s e b e h a v i o r i n t h e reactor with the modeling assumption. The actual situation has vapor a nd liquid pha ses, with solid polymer dispersed in the liquid phase. Our system considers the polymer as solubilized in the liquid phase. Table 8. Representative Species and Mass Fractions Used for Simulating the Phase Separation in a Slurry HDPE Reactor

a 

Results compare favorably with the literature value given in eq 5.

com pon en t

m a ss fr a ct ion

et h y lene pr opylen e h exa n e h ydr ogen HDPE

0.25 0.02 0.653 0.007 0.07

n  segments. The expression for bulk (live plus dead)

chains is ∞

 λi  )

∑ n i ([P n ] + [D n ])

(7)

n )1

w h e r e [D n ] i s t h e con ce n t r a t i on of d e a d (i n a ct i v e) polymer chains. The rate expressions involving polymer chains a re summed over all n , yielding a small number of closed expressions tha t a re functions of the moments. Typically, the zeroth, first, and second moments are s u ff ici en t f or t h e com p u t a t i on of com m on p ol y m er properties. Among them is the number-average molecu l a r w e ig h t (M n )

Figure 12. Solubility of hydr ogen in hexane.14

for representat ive r eactor conditions. These va lues compare fa vora bly with the literatur e value given in eq  5.

2.6.2. Molecular Weight from Method of Moments. To model a polymer reactor r igorously, one would need individual ra te expressions for polymer molecules of every chain length. B ecause polymers commonly contain distributions of chains consisting of up to hundreds of thousands of segments, this would lead to an impractical number of model equations. A useful technique for tracking the leading moments of the chain-length distribution of HDPE is the method of moments. 16 The moment s a re sum s of polymer concentra tions weighted by chain length. The moment expression for live polymer chains is ∞

 µi  )

∑ n  [P n ] i 

(6)

n )1

w h e r e µi  is t h e i t h m om en t f or l iv e c h a in s a n d [P n ] is the concentration of polymer chains containing

M  n

)

λ1

(8)

 λ0

The weight-average molecular weight (M w ) is M  w  )

λ2

(9)

 λ1

The polydispersity index (PD I) is

P DI )

M  w  M  n

)

λ2 λ0  λ12

(10)

P o l y m er s P l u s i m pl em e n t s t h e m e t h od of m om e n t s approach in tracking polymerization kinetics and polymer properties. 2.7. Reactor Phase E quilibrium. As mentioned previously, the polymer forms a crysta lline solid wit hin t h e l iq u i d p h a s e u pon f or m a t i o n . S ol id p ol y m er i s genera lly considered t o be inert an d not to part icipate i n p h a s e e q u il ib r i um .5 A r i g or ou s a p p r oa c h w o u ld consider only the solubilities of gases that can dissolve in th e solid polymer. H owever, most commercial pr ocess simulators do not ha ve reactor models tha t permit the

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Table 9. Comparison of the Liquid Compositions for Cases where the HDPE Has Its Own Liquid Phase (VLL E) and where It Is Dissolved in the Diluent (VLE) VL L E ca se liquid 1 (kg /h )

species h exa n e et h y len e pr opy len e h y dr ogen HDPE

350.805 5.520 1.248 0.007 1.200

606 1 240 829 687 94 724 188 71 × 10 -32

liquid 2 (kg /h )

total liquid (kg /h )

liquid (kg /h )

9.815 07 × 10 -5 0.351 363 342 6.896 22 × 10 -5 0.000 382 869 69.995 868 21

350.8057 5.871 604 1.248 757 0.008 107 69.995 87

367.9578 5.466 046 1.495 278 0.014 171 70

Table 10. Comparison of the Vapor Compositions for the VLE and VLLE Cases vapor mole fraction species h exa n e et h y len e pr opy lene h ydr ogen

VL E ca se 0.207 0.547 0.027 0.217

643 203 605 547

2 48 84 48

VL L E ca se 0.217 0.539 0.027 0.214

584 793 725 896

VL E ca se

7 44 36 5

existence of distinct species tha t a re th ermodynamically inert. Alternatively, one can model the polymer as being dissolved in t he liquid pha se. Figure 13 compares t hese physically different situations. In the remainder of this section, we demonstrate that we can make this assumption without undermining the robustness of the reactor model. We justify our simplification by comparing two approaches for m odeling t he slurry syst em. The first option is to treat the polymer as residing in a separate liquid phase (vapor -liquid -liquid equilibrium, VLLE ), an d in t h e s e con d op t i on , t h e p ol y m er i s t r e a t e d a s b ei n g dissolved in the solvent (vapor-liquid equilibrium, VLE). We model results for a flash vessel that simulates the phase separation for a mixture of components of mass fractions and conditions that are representative of an industria l slurry HD PE reactor. Table 8 gives the species and mass fractions that we use. For the case where HDPE is absent, we normalize the mass fractions of the remain ing species to unity. We flash t he mixtur e a t 75 ° C a n d 4 ba r . For the two-liquid case, we choose k i j  ) 0.5 and ηi j  ) 0 for a ll polymer/convent iona l-component pa irs t o force the polymer int o a separa te liquid phase. Ta ble 9 shows the r esulting a mounts of species in each liquid phase. The light components do not appreciably dissolve in the polymer phase. We also see that the differences in the total amounts of each species in the liquid phase for each case are relatively small. Table 10 compares the vapor-phase predictions for each case. The vapor compositions are approximately t h e s a m e . B e ca u s e w e u s e t h e s a m e p r e s su r e i n e a c h case, these vapor fractions also represent the relative magnitudes of partial pressures of components in the mixtures. I n t h e V L LE s it u a t i on , t h e r ea c t in g p ha s e i s t h e second liquid pha se, and in our VLE simplification, it is t he single liquid pha se. Although t he compositions of the reacting phases for these two approaches are different, the r esults a re sa tisfactory. We can r easonably m od e l t h e s l ur r y H D P E p r oce s s b y con s i de r in g t h e polymer a s dissolved in th e liquid phas e (VLE) without compromising the predicted phase behavior of the major components.

3. Polymerization Kinetics 3.1. Introduction. The reactions an d kinetics of Ziegler -Natta polymerization have been studied exten-

sively for different catalyst systems and processes.2,16-19 Further, it is generally accepted that Ziegler -N a t t a cata lysts produce polymers w ith w ide molecular weight distribut ions (MWD) beca use of the mult isite na tur e of the catalyst. It is believed that there are several site types on t he cata lyst, each with its own r eactivity. The composite polymer, defined as the sum of the polymer made from all of the catalyst sites, has a broad MWD even t hough the polymer ma de by each site ty pe has a narrow (most probable) MWD. In t his w ork, we develop a multisite kinetic model for t h e s l u r r y H D P E p r oce ss . We s e l ect a s u b se t of t h e Ziegler -Nat ta polymerizat ion reactions tha t a llows t he model to describe the observed kinetic phenomena a nd ma tch the production ra te, melt index (MI), an d density ta rgets for several gra des. Sections 3.2 a nd 3.3 describe the reactions for homopolymerization and copolymerizat ion kinetics, respectively. We include additional reactions, described in section 3.4, to account for the production of w a x (low-molecula r-weight polymer species) that dissolves in the hexane diluent. As there are ma ny reactions and kinetic para meters, we develop a detailed methodology t o fit t he kinetic parameters to data for several grades from both the parallel and series modes of operation. The methodology, described in section 3.5, involves a three-step p roce ss t o s im pl if y t h e t a s k of f it t i n g t h e k in et i c parameters. In the first step, we develop a single-site k in e t ic m od e l a n d f it t h e r a t e c on s t a n t s t o m a t ch t h e polymer production rate, comonomer conversion, and polymer M n for several par a llel an d series grades. Next, we deconvolute the measured polymer molecular weight distribut ion into a num ber of Flory distr ibutions. In the third step, we use the deconvolution results to expand the single-site kinetics t o multisite kinetics. We a djust the kinetic par am eters in the m ultisite model to fit the polymer production rate, comonomer conversion, M n , and PDI of the polymer for several parallel and series grades. In the available plant data, the polymerization react o r s a r e a t a p p r ox im a t e l y t h e s a m e t e m p er a t u r e . We therefore do not consider the effect of temperature on the polymerization kinetics. 3.2. Homopolymeri zation Ki netic Scheme. We develop a Ziegler -Natta reaction subset to describe the ob s er v ed p h en om e n a a n d m a t c h t a r g e t s f or s e ve r a l grades in the HDPE slurry process. Table 11 lists the homopolymerization r eactions tha t we consider. We describe these reactions in the following sections. 3.2.1. Catalyst Activation. In Ziegler -N a t t a s y s tems, an aluminum alkyl cocatalyst, such as triethyl aluminum, is typically used to a ctivat e the sites on the cata lyst. The cocata lyst is believed to form a complex w i t h t h e c a t a l y st s it e s t h a t m a k es t h em a c t iv e f or polymerization. Equation 11 shows the reaction for s it e a c t iv a t i on b y coca t a l y st . I n t h i s r ea c t ion , t h e t r a n s i t i on -m e t a l ca t a l y s t (C AT) r e a c t s w i t h t h e co -

5608 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002 Table 11. Reaction Subset Used in the Ziegler-Natta Homopolymerization Kinetics r ea ct ion n um ber

descr ipt ion

1 2 3 4 5 6 7

ca t aly st sit e a ct iv at ion by coca t aly st ch a in in it ia t ion ch a in pr opa ga t ion ch a in t r a nsfer t o h y dr ogen ch a in t r a nsfer t o m on om er r ev er sible ca t a ly st sit e in hibit ion spon ta n eou s ca t aly st sit e d ea ct iva t ion

catalyst (COCAT) to form vacant sites (P 0, i ) of t ype i 

C ATi  + COCAT

k act, i 

98 P

0, i 

(11)

where k act,i  is the ra te consta nt for activat ion of site type i  by cocata lyst. The vaca nt sit es ar e ca pable of producing polymer chains by reacting with monomer during chain initiation and subsequent propagation. Typical ly, Ziegler-N a t t a ca t a l ys t s a c t iv a t e a l m os t completely in several min utes, a nd w e therefore choose a r el a t i ve ly h i gh r a t e con s t a n t f or s it e a c t iv a t i on . Alternat ively, we can determine the ra te consta nts for site activation and deactivation using data for cata lyst a ctivity profiles from experiments usin g labora tory-scale semibat ch rea ctors. As typical Ziegler -Nat ta cata lysts are heterogeneous in na tur e (a ctive meta l on a support), the model includes a para meter (ma x-sites) for th e concentra tion of cata lyst s i t es p er u n i t m a s s of c a t a l y s t . Ty p ica l v a l u es of t h e ma x-sites para meter ra nge from 1.0 × 10-5 to 1.0 × 10 -3 mol of sites per g of cat alyst. This pa ra meter controls the sensitivity of the polymer production rate to changes in cat alyst flow rat e. Changing the value of this para meter proportionally scales the effects of the site-based reactions (propa gat ion, chain tra nsfer, etc.). We can use it t o cha nge the ma gnitudes of these reactions without changing th eir values relative to each other. Hence, it changes t he polymer production ra te but does not a ffect the polymer molecular weight averages or copolymer composition. 3.2.2. Chain Initiation. A monomer molecule reacts w i t h a v a c a n t s i t e t o i n it i a t e c h a i n g r o w t h

P 0, i  + M

k ini, i 

98 P

1, i 

(12)

wh ere M is th e monomer (ethylene), P1,i  is a propagat ion site of type i  with an attached polymer chain containing one segment, and k ini, i  i s t h e r a t e c o n s t a n t f o r c h a i n initiation at site type i . It is not possible to determine the ra te consta nts for chain initiat ion a nd propaga tion separ at ely, because of the limited types of data measurements that can be ma de. Hence, we set th e rat e consta nt for ethylene cha in initiation equal t o the ra te consta nt for propaga tion of ethylene monomer on ethylene a ctive segments. S imilarly, we set the rate constants for comonomer chain initiation equal t o the ra te consta nts for homopropaga tion of these monomers. 3.2.3. Chain Propagation. The polymer chain grows ra pidly by t he successive ad dition of monomer molecules at the catalyst site

P n , i  + M

k p, i 

98 P

n +1, i 

(13)

where P n , i  a n d P n +1, i  a re polymer chains of length n  a nd n  + 1 segments, respectively, associated with cata lyst

s i t e t y p e i  a n d k p, i  i s t h e r a t e con s t a n t f or ch a i n propagation for site type i . In general, a linear increase in the rate constants for propagation yields a linear increase in m olecular weight. 3.2.4. Chain Transfer. Chain transfer occurs when a monomer or chain-tra nsfer agent disengages a polymer chain from the catalyst, rendering it inactive or dead, and initiates the growth of a new chain. Most slurry HD P E processes use hydrogen a s a chain-tra nsfer agent to control the molecular weight of the polymer product. For hydrogen, the reaction is

P n , i  + H 2

k th , i 

98 D

+ P 0, i 

n 

(14)

w h e r e D n  is a dead polymer chain of length n  a n d k t h, i  is the rate constant for chain transfer to hydrogen for site type i . The chain-transfer reaction to monomer is slightly different, as it produces an initiated chain instead of a vacant catalyst site

P n , i  + M

k tm , i 

98 D

n 

+ P 1, i 

(15)

w h e r e k tm , i  i s t h e r a t e c o n s t a n t f o r c h a i n t r a n s f e r t o monomer for site type i  a n d P 1, i  i s a n i n i t i a t e d c h a i n associated with site type i . We adjust the rate constants for chain transfer to h y d rog en a n d t o m on om er t o m a t c h t h e m ol ecu la r weight of the HD P E produced over several reactors a nd g r a d e s w i t h d i f f er e n t H 2/C 2H 4 r a t i o s i n t h e r e a c t or overheads. Note that the reaction with hydrogen produces a va cant cata lyst site, whereas t he reaction with monomer produces an initiated catalyst site. Adjusting the rate constant for chain transfer to monomer can disrupt the equilibrium number of inhibited catalyst sites, beca use the ra te of cha in initia tion competes wit h tha t of hydrogen inhibition. 3.2.5. Forward and ReverseCatalyst Inhibitions. Some species, such as hydrogen, a re known to cause a decrea se in th e ra te of polymeriza tion in some ZieglerNatta catalyst systems. This rate depression appears t o b e r e ve r s ib le a n d d i s a pp ea r s u p on r e m ov a l of t h e hydrogen. Slurry HD P E processes typically opera te w ith tw o reactors in series to ma ke a polymer product with a bimodal MWD. They do this by making close to 50% of t h e t o t a l p ol ym er i n t h e f i r st r ea c t or w i t h a l ow   avera ge molecular weight and 50% of the polymer in t h e s econ d r ea c t or w i t h a h i gh a v er a g e m ol ecu la r weight. The catalysts used in these processes exhib it the reaction for reversible site inhibition by hydrogen, an d incorporat ion of this effect is essentia l for modeling t h e p rod u ct i on r a t e s i n e a ch r ea c t or of t h e s er i es con f ig u r a t i on . We u s e f o r w a r d a n d r e ve r s e c a t a l y s t inhibitions by hy drogen to represent t his beha vior. The forward reaction is

C ATi  + x H 2

k finh, i 

98 

ICATi 

(16)

w h e r e k finh,i  is the rate constant for forward inhibition of cat alyst of site type i . The r everse rea ction is

ICATi 

k rinh, i 

98 

C ATi  + x H 2

(17)

w h e r e k rinh, i  is the r at e consta nt for r everse inhibition of site type i . We adjust these ra te consta nts t o match

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002

the production rate of HDPE in each reactor in the series configurations. 3.2.6. Spontaneous Catalyst Deactivation. The active sites on the catalyst can undergo spontaneous deactivation to form dead sites that are no longer active

P 0, i 

k d, i 

98  C AT

(18)

i 

w h e r e k d, i  is the ra te consta nt for sponta neous cata lyst deactivat ion for site type i . Increasing this rate constant decreases the production rate of HDPE. Also, if the chain-transfer rates are low, catalyst deactivation can affect the number-avera ge molecular weight. 3.3. Copolymerization Kinetic Scheme. Comonomers are commonly used to produce HDPE products of var ying densities. The int roduction of R-olefins, such a s propylene, 1-butene, and 1-hexene, creates short-chain branching along the polymer backbone, lowering the crystallinity of the polymer. We assum e tha t t he ra te of propaga tion of a m onomer (or comonomer) depends only on the a ctive segment (last m on om er a d d ed t o t h e ch a i n ) a n d t h e p rop a g a t in g monomer. This is commonly referred t o as t he term ina l model for copolymeriza tion kinetics. For a system w ith two monomers, we expand the propagation reactions as follows 11

1

k p, i 

P n , i  + M 1

98  P

P 1n , i  + M 2

98  P

1

(19)

2

(20)

1

(21)

2

(22)

n +1, i 

12

k p, i 

n +1, i 

21

2

P n , i  + M 1

k p, i 

98  P

n +1, i 

22

2

P n , i  + M 2

k p, i 

98  P

n +1, i 

w h e r e P  j n , i  is a polymer chain of length n , associated with site type i , that has an active segment corresponding to monomer of type j  a nd k  jp,k i  i s t h e r a t e c on s t a n t f or p r op a g a t i o n, a s s oci a t e d w i t h s i t e t y p e i , f o r a monomer of type k  a d d i n g t o a c h a i n w i t h a n a c t i v e segment of type j . For H DP E, t he concentra tion of comonomer segments in the polymer and the concentration of comonomer active segments (i.e., segments attached to an acti ve site) are small. Hence, the homopropagation reaction for ethylene is the primary factor responsible for ethylene conversion, whereas the propagation reaction for comonomer a dding to a chain ending with a n ethylene active segment dominates the consumption of comonomer. The concentration of ethylene active segments is very high relative to that of comonomer active segments. As a result, the propagation reactions involving comonomer active segments provide only minor contributions to the conversion of monomer and comonomer, as well a s t h e H D P E p r od u ct i on r a t e . We expand the reactions for chain initiation and chain transfer to both monomer and hydrogen in a similar fashion, to consider th e rea ction of these species w ith th e different m onomers/a ctive segment s on the polymer chains. For chain initia tion  j 

P 0, i  + M j 

k ini, i 

98 P

 j 

1, i 

(23)

5609

where M j  is a monomer of type j  a nd k  j ini,i  is the rate of chain initiation for monomer j  a t s i t e t y pe i . For chain transfer to hydrogen  j 

P

 j  n , i 

+ H2

k th , i 

98 D

n 

+ P 0, i 

(24)

w h e r e k  j t h, i  i s t h e r a t e c o n s t a n t f o r c h a i n t r a n s f e r t o hydrogen a ssociat ed with a chain ending with a monomer unit of type j  at site type i . S i m i la r l y , w e h a v e t h e copolymerization reactions for chain transfer to monom er  jk 

P

 j  n , i 

+ M k 

k tm , i 

98 D

n 

+ P k 1, i 

(25)

k  w h e r e k  jtm , i  is the ra te consta nt for chain tra nsfer for a monomer of type k  reacting with a growing cha in ending in a monomer unit of type j  a t s i t e t y pe i . 3.4. Oligomer Production. Slurry H DP E processes produce oligomer, which is a low-molecular-weight polymer species that dissolves in the hexane diluent. U s in g pl a nt d a t a f or t h e m ol ecu la r w e ig ht of t h e oligomer, we m odel its production by rea cting st oichiometric amounts of ethylene and hydrogen

x C 2H 4

+ H2

f 

oligom er

(26)

w h e r e x  represents the number of ethylene segments i n t h e ol ig om er . B e ca u s e w e k n ow t h e a m ou n t of oligomer produced, we a djust th e extent of this reaction in the model to match the oligomer production rate. Specifically, the extent of reaction represents the changes in the number of moles of ethylene due to reaction divided by the stoichiometric coefficient x . 3.5. Determination of Kinetic Parameters. 3.5.1. Introduction. Here, w e provide a general m ethodology for simultaneously fitting the kinetic parameters to plant da ta for multiple product gra des. The fine-tunin g of kinetic parameters to match plant data can be a difficult task. Adjustment of the rate constant for each reaction can affect several simula tion varia bles simultaneously. The methodology assumes no information about the kinetic activity of the cata lyst or the number of catalyst sites per mass of catalyst. As mentioned previously, we do not consider temperature effects on the polymerization kinetics, as all of the reactors in the plant were operated at about the same temperature. Because the polymerization reactions are highly coupled, the determination of temperature dependence for each individual reaction would require extensive experimenta tion. Moreover, few da ta ar e ava ilable in th e open literatur e for th e tempera ture dependence of the reactions for Ziegler -N a t t a s y s t em s . We divide the procedure into tw o parts. In the first part, w e assume that t he cat alyst conta ins a single site type. This simplifica tion permits u s t o model a ccura tely o n l y t h e M n , n o t t h e M w  o r t h e P D I . W e a d j u s t t h e kinetic par ameters to mat ch the HDP E production rate in each reactor and the conversions of monomer and comonomer, in a ddition t o the HD P E M n . The second part of the procedure involves the introduction of multiple cata lyst site t ypes. We deconvolute the MWD for th e polymer product int o distribut ions for e a ch s i t e t y p e. Th i s p r oce d ur e g i ve s t h e m i n im u m number of site types allowing for the a ccura te computa tion of the polymer MWD, as well as the relative rate of propagation and the polymer M n produced by each

5610 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002 Table 12. Simulation Targets for the Models for Catalysts with Single and Multiple Site Types m odel for sin gle-sit e ca ta ly st

m odel for m ult iple-sit e ca t a ly st

H DP E pr oduct ion in ea ch r ea ct or m on om er a nd com on om er con ver sion s H D P E M  n

H DP E pr oduct ion in ea ch r ea ct or m on om er a n d com on om er con ver sion s H D P E M n relative production of HDP E by ea ch site type M n of HDPE at each site type fraction of inhibited cata lyst sites HDPE PDI

site type. The simulat ion ta rgets include those for the s i n gl e-s i t e m o d el , a s w e l l a s t h e r e la t i v e a m ou n t of polymer produced a t each site t ype, the M n f o r H D P E for ea ch site type, the fraction of inhibited cata lyst sites as determined in the single-site model, and the P DI of the H DP E product. Table 12 summarizes th e simulation ta rgets for th e single-site a nd multisite models. S e ct i on 3 .5 .2 d e scr i b es t h e d e t er m i n a t i on of t h e kinetic para meters for t he single-site model. We ma nually iterate between each of the polymer grades until we obtain a set of kinetic par am eters that sat isfies the simulation targets for each one (Table 12). Section 3.5.3 explains the procedure for deconvoluting the MWD of the HDPE to determine the number of catalyst site types a nd t he kinetic beha vior for each site. S ection 3.5.4 details t he methodology for using th e multisite m odel t o a d ju s t t h e k i ne t ic p a r a m e t er s s i m ul t a n e ou s ly t o match all of the simulation targets given in Table 12. 3.5.2. Single-Site Kinetic Model. We b eg i n b y modeling the catalyst as containing a single site t ype. Assuming a single-site catalyst, we can accurately model all of the reaction phenomena except the polymer PDI. Th e m o t i va t i o n f o r u s i n g t h i s a p pr oa c h i s t h a t t h e consideration of a multisite cat alyst significant ly increases the number of kinetic para meters tha t w e must determine, because of the increase in the number of reactions involving catalyst sites. The number of reactions can exceed 60, depending on the number of site t y p es u s ed . Th e p a r a m e t e r d e t er m i n a t i on i s m u ch simpler using a two-step method than trying to establish all of the parameter values at once. We establish a base set of kinetic parameters using sources in the open litera ture. Table 13 gives their values. We also use an initial concentration of active catalyst sites on the catalyst species of 0.0002 mol of si t es/m ol of Ti. 20 We use these numbers as initial va lues in the model and then apply an iterative methodology to adjust them to match model predictions with plant data. Figure 14 shows the methodology we use to determine the kinetic para meters for the single-site model to ma tch plant da ta for multiple product gra des (both pa ra llel an d series configurations). We step through the algorithm manually. Using the base set of kinetic parameters in Table 13, we first adjust the concentration of active sites on the cata lyst species. Only a fra ction of the tra nsitionm e t a l s i t es of t h e c a t a l y s t a r e a v a i l a b le f o r p ol y m er i zat ion. I ncreasing the active-site concentra tion on the cata lyst increases the ra tes of reaction involving cata l y st , s u ch a s a c t i va t i o n , c ha i n i n it i a t i on , a n d ch a i n propagation, while maintaining the relative ratios between them. Thus, we can increase the production rate of the HD P E while ma intaining its comonomer composition an d n umber-avera ge molecular weight. We adjust the rate constants for chain propagation to match the conversions of monomer and comonomer. The prima ry r eactions a ffecting monomer an d comonomer conversions a re t hose for m onomer-monomer a nd

Table 13. Base Set of Kinetic Parameters for the Single-Site Modela  r ea ct ion ca t -a ct ch ain -in i ch ain -in i ch ain -in i pr opa ga t ion pr opa g a tion pr opa ga tion pr opa g a tion pr opa g a tion pr opa g a tion pr opa g a tion ch at -a gen t ch a t-a g en t ch a t-a g en t ch at -m on ch a t-m on ch a t-m on ch a t -m on ch at -m on ch at -m on ch at -m on fw d sit e in h r ev sit e in h spon -dea ct

r ea ct a n t 1 r ea ct a n t 2 ca t a ly st ca ta ly st ca ta ly st ca ta ly st et hy len e et h ylen e pr opy len e pr opy len e et h ylen e 1-b ut en e 1-b ut en e et hy len e pr opy len e 1-b ut en e et hy len e et h ylen e pr opy len e p rop yl en e et hy len e 1-bu ten e 1-bu ten e ca ta ly st ca t a ly st ca t a ly st

coca t a lyst et hylen e pr opy len e 1-but en e et hy len e pr opy len e et hylen e pr opy len e 1-bu ten e et h yl en e 1-bu ten e h yd rog en h yd rog en h yd rog en et hy len e pr opy len e et h yl en e p ropy len e 1-but en e et hy len e 1-bu ten e h ydr ogen

k b

1 14.6 9.8 9.8 14.6 0.81 41 9.8 0.81 41 9.8 0.088 0.088 0.088 0.0021 0.006 0.0021 0. 006 0.006 0.0021 0.006 2000 0.0003 0.0001

r e f c om m e n t s 17 c  c  d 

18 18 18 18 d  d  d 

17 17 d 

17 e  e  e 

17 17 17 17 17 17

a 

We do not consider tempera tur e effects on the polymerizat ion k i n et i cs , a n d w e t h e r ef or e d o n o t t a k e a c t i va t i o n e n er g i es i n t o account. b  G eneral un its a re L/mol‚s. c  Assumed to be equal to tha t for homopropagation. d  Assumed to be equal t o the ana logous rate constant involving propylene. e  Assumed to be equal t o the a nalogous rat e constant involving 1-butene.

monomer -comonomer propagations, respectively, due to the high monomer concentration relative to that of comonomer. Chain propagation also affects the molecu la r w e ig h t . Ad ju s t in g t h e r a t e con s t a n t f or ch a i n transfer to monomer affects the number-average molecular weight of the HDPE, especially when the concentra tion of chain-tra nsfer a gent is low. F or t h e s e r ie s g r a d e s , w e a d ju s t t h e r a t e c on s t a n t s for forward and reverse catalyst inhibitions by hydrogen. Note that the series reactors operate at the same temperature. In general, increasing the rate constant for forw ard site inhibition decreases t he polymer prod u ct i on r a t e i n t h e f i r s t r e a c t or w h i l e i n cr e a s i n g t h e production rate in the second reactor. Increasing the ra te consta nt for reverse inhibition a ffects the r elative production rates in the two reactors simultaneously, so w e c a n u s e i t t o a d j u st t h e H D P E p r od u ct i on r a t e s i n the two reactors while maintaining the same ratio of production in each one. We iterate between all of the product grades until we obta in a set of single-site kinetic para meters tha t a llows the model to match the plant dat a (except the P DI) for each gr ade. The next section describes the use of gel p er m e a t i on ch r om a t o g r a p h y (G P C ) f o r ob t a i n i n g t h e M WD of t h e p ol y me r a n d a q u a n t i fi ca t i on of t h e multisite behavior of the catalyst. 3.5.3. Deconvolution of Molecular Weight Distribution Data. We can apply a statistical algorithm

5611

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002

Figure 15. GPC deconvolution results for a representative HDPE sample from the para llel reactor configuration. Five catalyst site t y p es a c cu r a t e l y d e sc r ib e t h e e xp er i m e n t a l m o le cu l a r w e i g h t distribution. Table 14. Deconvolution Results for a Representative Sample of HDPE Produced in the Parallel Process site type

HDPE weight fraction, m i 

τ i  (M n -1)

1 2 3 4 5

0.0529 0.295 38 0.269 79 0.256 85 0.125 04

1.76 × 10 -4 5.75 × 10 -5 2.85 × 10 -5 1.41 × 10 -5 4.76 × 10 -6

τ i -1 (M n ) 5.69 1.74 3.50 7.08 2.10

10 3 10 4 4 × 10 4 × 10 5 × 10 × ×

produced at each site t ype. The expression for t he tota l polymer is  j 

W (n )

) ∑m i w i (n )

(28)

i )1

Figure 14. Methodology for simulta neously determining the kinetic parameters for a single-site catalyst to match plant data for multiple HDPE grades.

to deconvolute the polymer MWD t o determine a mostprobable chain-length distribution for each of a determined number of catalyst site types. The methodology p re se nt e d b y S oa r e s a n d H a m i e lec4 a l l ow s on e t o determine the minimum number of cat alyst site types tha t gives a n a ccurat e representat ion of the molecular weight distributions generated by Ziegler-N a t t a c a t a l ys t s, a s w e ll a s t h e w ei gh t f ra c ti on a n d n u mb er average molecular weight of polymer produced by each site type. The consideration of these site types, each with its respective reactivity, enables us to model the broad molecular weight distribution of the HDPE accurately. Soares and Hamielec use the following expression to represent the most-probable weight chain-length distribution produced by each site type w i (n )

) τ i 2n  exp(-τ i n )

(27)

Here, w i (n ) is the weight fraction of polymer of chain length n  produced by site type i . τ i  is a fitting parameter for site type i  and represents the inverse of the M n of polymer produced a t t ha t site. The w eight chain -length distribution of the entire polymer is a weighted sum of t h e d i st r i b u t ion s p r od u ce d b y e a ch s i t e t y p e. Th e w e ig h t in g f a ct or i s t h e m a s s f r a ct i on of p ol y me r

w h e r e W (n ) is the w eight fra ction of polymer of chain length n , m i  is the mass fraction of polymer produced a t s i t e t y pe i , a n d j  is the total number of site types. We u s e s o ft w a r e p r od u ce d b y P o l y t h in k I n c.21 t o deconvolute the GP C da ta for HD P E. It incorpora tes the methodology presented by Soares and Hamielec. The program determ ines the minimum num ber of site types that accurately describes the MWD, the weight fraction of polymer a nd corresponding M n produced at each site t y p e, a n d t h e p r ed i ct e d M n a n d M w  f or t h e e n t ir e distribution. Ta ble 14 shows a representa tive set of deconvolution results for t he pa ra llel r ector configuration. Figure 15 illustrates the MWD predicted for each site type, as well as a comparison of the prediction of the overall MWD with the experimental curve. The results indicat e tha t a five-site m odel can describe th e m olecular weight d is t r ib u t ion of t h i s p a r t icu la r s a m p le . I n t h e n ex t section, we show how to use these results to determine kinetic para meters for a reaction set considering m ultiple cat alyst site types. 3.5.4. Multisite Kinetic Model. Once w e establish a set of single-site kinetic parameters that allows the model to match the simulation targets for each grade (see Table 12), we introduce multiple site types, the e xa c t n u m be r o f w h ich i s d e t er m in ed b y t h e G P C deconvolution met hod described in t he previous section. It is importa nt t o note tha t t he MWD from the second reactor in the series process is not useful for determining kinetic parameters for the site types. The polymer exiting this reactor results from reaction phenomena occurring in the t wo rea ctors, a nd t here is no reasonable wa y to decouple these events for the purpose of esta blishing kinetic para meters for individual reactions.

5612 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002

Figure 16. Methodology for simultaneously determining the kinetic parameters for a catalyst with multiple site types t o match plant d a t a f or s e v er a l H D P E g r a d e s.

The transition from the single-site to multisite catalyst introduces three new criteria for determining the kinetic para meters. These are t he w eight fraction a nd M  n of p ol ym er p rod u ce d a t e a ch s it e t y pe a n d t h e fraction of inhibited catalyst sites. The first two targets result from the MWD deconvolution described in the previous section. The fra ction of inhibited s ites a ffects the production rate of polymer, and one must maintain the fra ction tha t results from t he single-site modeling step to preserve the correct relative amounts of polymer produced in each reactor for the series grades. We multiply the rate constant for propagation determined in t he single-site model by the w eight fra ction of polymer produced at site type i , m i , (see Table 14) to

ob t a i n t h e i n it i a l v a l ue s f or e a ch p rop a g a t ion r a t e constant i j 

i j 

k p, i  ) n st k p m i 

(29)

w h e r e n st is the number of site types considered in the model. We must multiply the rate constants by n st (5, in our case) because the concentra tion of total sites is n st t i m e s t h a t o f t h e i n d i v i d u a l s i t e s . N o t e t h a t w e assume the cata lyst conta ins an equal number of moles of ea ch site type. We adjust t he ra te consta nts for chain transfer to hydrogen and to monomer to match the number-a verage molecular weight produced at each site type.

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002

5613

Table 15. Simulation Targets for E ach Polymer Grade and the Corresponding Adjusted Model Parameters s im ula t ion t a rget s monomer conversions (HDPE production rate) H D PE P DI

mas s fra ction of polymer produced a t each site ty pe M n produced by each site type

Figure 17. C o m pa r i s on o f M WD s f or t w o d i ff er e n t g r a d e s o f HD P E. We hypothesize that t he Ziegler-Nat ta site type producing high-molecular -weight polymer is insensit ive to h ydrogen concentration.

The single-site model yields an equilibrium mole fra ction of inhibited cata lyst sit es. We denote th is value a s C I S F R AC ss , w h e r e

CISFRAC ss ) number of moles of inhibited cat a lyst sites (30) tota l number of moles of cata lyst sit es One must maintain this total fraction in the multisite model to preserve the polymer properties computed in the single-site m odel. We represent the corresponding t a r g et s i n t h e m u lt i si t e m od el , C I S F R AC i , a s t h e quotient of the single-site value and the number of site types

CISFRAC i  ) number of moles of inhibited sites of type i  ) tota l number of moles of a ll site types CISFRAC ss (31) n st

Figure 16 shows the itera tive scheme for determining kinetic parameters in the multisite model. It is more complex than that for the single-site model because of t h e a d d i t ion a l s i m ul a t i on t a r g e t s . A s w i t h t h e s i n g l es i t e m od e l, w e s t e p t h r o u g h t h e a l g or i t h m m a n u a l ly . Beginning iterations with a parallel model, we adjust the rate constants for forward site inhibition to match the inhibited fraction for each site type. If the mass fraction of polymer produced at each site does not match the target values determined from deconvolution, we a d j u st t h e r a t e con s t a n t s f or p r op a g a t i o n. We t h e n adjust the ra te consta nts for chain tra nsfer to hydrogen to mat ch the number-avera ge molecular w eights at each site type. If th e number-avera ge molecular weight of the total polymer is not correct, w e adjust th e ra te consta nts for chain tr an sfer to hydrogen for ea ch cat alyst site type while maintaining their relative values. For the series gra des, we check the PD I produced in each reactor. If, at this point, the kinetic parameters d o n ot g i ve a m a t c h f o r t h e P D I s i n t h e f i r s t r e a c t or s f or a l l g r a d e s, w e h y p ot h e s iz e t h a t a t l ea s t on e o f t h e site types is insensitive to a reactant involving c hain transfer, such as hydrogen. We illustrate this approach u s in g M WD s f or r ep re se nt a t i v e H D P E . F i gu r e 17 compares the MWDs for HDPE produced in reactors t h a t h a v e d i ff er e n t h y d r og e n c on ce n t r a t i on s . I n t h i s case, the series reactor has a higher hydrogen concen-

a dju st ed pa ra met er s propagat ion rat e constan ts for monomer and comonomer rela tive propa ga tion ra te con st a nt for each site t ype, determined using deconvolution of MWD propagat ion rat e constan ts for individual site types chain-tra nsfer rate constants for individual site types

tration than the parallel reactor. Because hydrogen limits the molecular weight of the polymer, we expect to see the entire molecular weight distribution shi ft horizontally to the left as the hydrogen concentration increases. However, if a site type is insensitive t o the hydrogen concentration, a shift does not occur. This is consistent w ith t he behavior appearing in Figure 17. In a ccordan ce wit h this observat ion, we include steps in the methodology to allow the kinetic model to match the P DI for gra des of var ying hydrogen concentra tion. We return t o the previous fitted gra de in the a lgorithm, prefera bly a para llel model. We reduce the ra te of cha in tra nsfer to hydrogen by 10% for the insensitive site while increasing th e rat e of chain tr an sfer t o monomer to refit the num ber-a verage molecular weight produced a t t h a t s it e . We t h en p la c e t h e s e a d ju s t ed k in et i c parameters back into the model for the current grade and check the P DI. We repeat this loop until w e ma tch the PDI in the first reactor for each grade. We then check the PDI for the second series reactors and return to the beginning of the algorithm for tha t gra de if it does not match. We repeat this process until we match th e data for each reactor in each grade. Table 15 summa rizes the model para meters tha t w e adjust to match the primary targets for the plant data. In the next section, we compare model predictions to data for eight HDPE grades.

4. Simulation Results 4.1. Steady-State Model Validation. We va lidate the model using plan t da ta from four para llel and four series grades of HDPE, produced in two large-scale commercial plant s (144 000 a nd 240 000 tons/year ). Differences betw een the processes for ea ch gra de include reactor configurations (par allel a nd series), feed ra tes for raw materials, and unit-operation conditions. Because we used these same data when developing the kinetic parameters, we can expect that accurate prediction by our m odel is genera lly limited t o similar process con d i t ion s . Th e u t i l i z a t i on of p la n t d a t a a t v a r y i n g reactor temperatures, for example, would permit the con s i d er a t i on of a c t i v a t i on e n er g i es i n t h e k in e t i c expressions, resulting in a n expansion of the predictive capabilities of the model. Figures 18-20 compare the model predictions with d a t a f r om p la n t A f or t h e H D P E p rod u ce d i n e a ch reactor. The model accurately predicts the production r a t e , M n , and P DI for each gra de. The PD I predictions reflect the va lidity of the kinetic modeling and para meter determination for the multisite catalyst. Figures 21 and 22 show model validations for the v a p or f low i n t h e r e a c t or ov er h e a d a n d t h e r e a c t or residence time, respectively, for plant A. The process m od e l p r ov i de s g ood a g r e e m en t w i t h e a ch of t h e s e process variables. Accurate prediction of the volumetric

5614 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002

Figure 18. Comparison of model predictions with plant data for HDPE production rate.

Figure 19. Comparison of model predictions with plant data for the number-average m olecular weight of HD PE for each reactor.

Figure 21. Comparison of model predictions with plant data for the vapor flow rate in the reactor overhead.

Figure 22. Comparison of model predictions with plant data for reactor residence time. Table 16. Comparison of Model Predictions with Data for a Parallel Grade for Plant B

H D P E p r od u ct i on r a t e (k g /h ) et hy len e con ver sion (%) pr opy len e con ver sion (%) H D P E M n HDPE PDI r ea ct or r esiden ce t im e (h) H 2/C 2H 4 m o la r r a t i o i n o ve r h ea d

plant data

model prediction

50 96-5148 98-99 72 14 000-16 000 9-10 2.2 0 .6 5-0.68

5187 99.7 75.3 15 975 9.63 1.98 0.643

Table 17. Comparison of Model Predictions with Data for a Series Grade for Plant B

Figure 20. Comparison of model predictions with plant data for the HDPE polydispersity index.

f low i n t h e v a p or ov er h e a d i s i m p or t a n t b eca u s e of limitations in equipment capacity. The reactor residence t i m e m u s t b e a ccu r a t e b eca u s e i t a f f ect s a l l of t h e polymer propert ies. Tables 16 and 17 compare model predictions with data from plant B .22 The model provides good agreement with polymer properties and process variables in both the pa rallel a nd series configurations. Here, we h ave presented results for a single reactor section for each of the parallel processes (refer to Figure 2). Although each parallel process essentially contains two identical process sections that receive the sam e fresh feeds a nd opera te a t t he sam e conditions, identical

H D P E p r od u ct i on r a t e (k g/h ) et h yl en e con ver sion (%) 1-but en e con ver sion (%) H D P E M  n HDP E P DI r ea ct or r esid en ce t im e (h ) H 2/C 2H 4 m ola r r a tio in over h ea d

t ot a l t ot a l t ot a l reactor r ea ct or r ea ct or r e a ct o r r ea ct or r ea ct or r ea ct or r ea ct or

1 2 1 2 1 2 1 2

plant data

model prediction

9 212-9306 98-99 89

9151 97.4 74.8 3336 6537 10 38.9 2.68 1.17 7.25 0.084

-

7000

-

3 1-35 2.5 1.13 6.8-7.2 0.08-0.09

i nd u st r i a l s y s t em s r a r e ly b eh a v e i n t h e s a m e w a y . Therefore, when constructing a robust process model, one should consider both sections on an individual basis. 4.2. Dynamic Modeling. 4.2.1. Introduction. Whereas a steady-state model does not consider changes

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002

Figure 23. S i m p l i f i e d d i a g r a m o f t h e c o n t r o l s c h e m e f o r t h e para llel r eactor configuration.

5615

Figure 24. C h a n g e i n t h e H 2/C 2H 4 o v e r h e a d r a t i o d u r i n g t h e grade change in the parallel configuration.

Table 18. Controlled and Manipulated Variables for the Slurry HDPE Process con t rol led v a ri ab les

m a ni pu la t ed v a ria b les

r e a ct or t e mp er a t u r e overhead H 2/C 2H 4 molar ratio l i q ui d l ev e l i n f l a s h t a n k

ov er h ea d v a p or r e cy cl e t o r ea c t or hydrogen feed rate a nd overhead flare l i q u id r ec yc le r a t e t o r e a ct o r

Table 19. Specifications for the Grade Change in the Parallel Configuration et h ylen e feed r a t e (kg/h ) pr opylen e feed r a te (kg/h ) overhead H 2/C 2H 4 m ola r r a t io ca ta ly st feed r a te (kg/h ) coca ta lyst feed r at e (kg/h )

gr a de 1

gr a de 2

5949 81.7 0.56 0.175 0.658

5999 0 1.54 0.386 0.800

in the process with time, a dyna mic model permits t he varia tion of the feed ra tes or vessel conditions a nd t he t r a c ki n g of t h e r es u lt i n g p roce ss ch a n g es a s t h ey p r op a g a t e t h r o ug h t h e s y s t em . A d y n a m i c m o d el ca n a s s i s t i n op t i m iz i n g t h e t i m e r e q u i r ed t o c a r r y ou t a g r a d e ch a n g e , f a ci li t a t e t h e d e ve lop m en t of a n e w   polymer grade, or reveal the time-dependent effects of changing a process variable. Our goal here is to demonstrate the capability and utility of a dynamic model for an industrial slurry HDPE process. We cr e a t e t h e d y n a m i c p r oce ss m od e l i n As p en Dynamics by importing the corresponding steady-state model developed in Aspen Polymers Plus. We include vessel dimensions and geometries to account for the liquid levels in each unit properly. In the following sections, we illustrat e the ut ility of dynamic modeling by simulating a grade change for the parallel reactor configuration. Section 4.2.2 describes the contr ol scheme for t he process. S ection 4.2.3 gives t h e f ee d r a t e s a n d p r oces s v a r ia b l es f or t h e g r a d e change, as well as dynamic results. 4.2.2. Control Scheme. Figure 23 presents a simplified diagram illustrating the control scheme for the par a llel configura tion. The process includes (1) a rea ctor t e mp er a t u r e con t r ol le r t h a t a d ju s t s t h e a m o un t of recycle gas returned to the reactor inlet, (2) a compositional controller that adjusts the hydrogen feed rate and the ra te of recycle-gas flare to ma intain a consta nt ra tio of hydr ogen to ethylene in th e recycle-ga s strea m, an d (3) a level controller in th e overhead fla sh unit tha t a d j u st s t h e a m o u n t of l iq u i d r e cy cl ed t o t h e r e a c t or . Table 18 summa rizes the controlled and ma nipulat ed variables.

Figure 25. Effect of th e grade chan ge on the production ra te of HDPE in the parallel configuration.

4.2.3. Dynamic Grade Change. Ta ble 19 gives the key process modifica tions for a typical gr a de change for the pa ra llel configura tion. The unit -opera tion condit ions r e m a i n t h e s a m e b e t w e en t h e t w o g r a d e s . We assume an inst an ta neous change in the feed rat es in Table 19 to t he new values. We a lso cha nge the set point for t he H 2/C 2H 4 cont roller to the n ew va lue of 1.54. Figure 24 shows the dyna mic cha nge in the H 2/C 2H 4 ratio in the reactor overhead. The controller initi ally increases the hydrogen feed rate to achieve the new set p oi n t . Th e r a t e a t w h i c h t h e s y s t em r e a ch e s t h e n e w   s et p oi n t i s m a i n ly a f un ct i on of t h e v a l u e f or t h e proportional gain used for the controller. Figure 25 illustrat es the effect of the gra de change on the production rate of HDPE. The sharp increase is due to the increase in the cat alyst feed ra te. The production ra te then decreases to a value below tha t for the first gra de, beca use of the large increase in hydrogen concentra tion that results from the higher set point for the H 2/C 2H 4 ra tio in the overhead. The hydrogen inhibits t he cata lyst, effectively reducing t he concentra tion of active sites in t he rea ction mixture. Figure 26 shows the effect of the grade change on the HDP E M n . The increase in the hydrogen a nd cat alyst feed ra tes decreases the product M  n through chain transfer. 4.3. Process Retrofit. The purpose of this process retrofit is to increase the production ra te of HDP E w hile m a i n t a i n i n g t h e s a m e p r od u ct q u a l i t y (i .e ., p ol y m er a tt ributes). An increase in production requires a corresponding increase in heat removal due to polymeriza-

5616 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002

crease. The recycle gas flow rate increases, as does the heat duty of the slurry cooler. The slurry recycle flow  rate also increases.

5. Conclusions

Figure 26. Effect of the grade change on the number-average molecular weight of HDPE.

Figure 27. Flowsheet for a process retrofit a pplied to one reactor i n t h e p a r a l l el c on f i gu r a t i o n . I n c r e a s in g t h e p r od u ct i on r a t e increases the demands for heat removal, thereby necessitating the addition of an other pump an d cooler to recycle liquid to the rea ctor. Table 20. Comparison of HDPE Flow Rate and Attributes for Varying Increases in Feed R ates of Raw Material for the Process Retrofit %increase in ethylene, propylene, HDPE hexane, and production mother-liquid rate feed rates (kg /h ) 0a  10 15 20 a 

5 83 4 6426 6717 7017

M n

2 1 14 8 21 093 21 037 20 990

recycle heat duty slurry gas flow  of slurry recycle rate cooler flow ra te P D I (kg /h ) (kca l/h) (kg /h) 4 .5 8 4.58 4.58 4.58

24 8 09 25 622 562 108 23 244 28 804 807 327 33 382 28 982 1 075 000 44 459

Original model.

tion. If the overhead units a re a lready opera ting close t o ca p a ci t y , o n e ca n n ot i n cr ea s e t h e r a t e of v a p or r e m ov ed f r om t h e r e a ct o r s i gn i fi ca n t l y t o m e et t h e additional heat-transfer requirements. We propose the addition of a pump and a heat exchanger to enable the cooling a nd r ecycling of a portion of th e slurry product stream to the reactor. We use a single reactor from the para llel configurat ion to model the retrofit. F i g u r e 2 7 s h ow s a f low s h e et f or t h e r e t r of i t . We increase the feed rates of the monomer, comonomer, and solvent species by the same r elative rat ios to ma inta in the same relative concentra tions of these components in the reaction mixture. This allows the properties of the polymer product to be preserved. Table 20 summa rizes the retrofit results, indicat ing t h a t w e c a n i n cr e a s e t h e p r od u ct i on r a t e o f H D P E b y up to 20%if the overhea d unit s can h a ndle up to a 17% increase in the flow rate of recycle gas. As we increase the feed ra te, the HD PE production ra te increases, while the molecular weight undergoes an insignificant de-

We have developed both steady-state and dynamic m od e ls f o r a s l ur r y H D P E p r oce ss w i t h t w o r e a c t or con f ig u r a t i on s . We h a v e v a l i da t e d t h e m od e l u s in g design and operating data from two large-scale commercial HD PE processes. The model uses a single set of k in e t ic a n d t h e r m od y n a m i c p a r a m e t er s t o p r ed i ct a c cu r a t e l y t h e p r od u ct i on r a t e , M n , P D I , m on om er conversion, a nd comonomer composition, as well as trends in these polymer properties with changes in key process variables, for both reactor configurations. We illustrat e the utility of the dyna mic model by simulat ing a grade change for the parallel reactor configuration. The Sanchez -Lacombe E OS provides good predictions of th e pha se behavior of polymer mixtures. The C h a o-Seader property method gives accurate descriptions of the phase behavior of mixtures of light hydrocarbons. A GPC deconvolution algorithm developed by Soares a n d H a m i e l e c4 a n d i m p l e m e n t e d b y P o l y t h i n k I n c . 21 a c cu r a t e l y d e scr i b es t h e p ol y d is p er s e n a t u r e of t h e HDP E produced using a multiple-site-type ZieglerN a t t a ca t a l y st . We d e scr i be a u n i m od a l m ol ecu l a r weight distribution produced in a parallel reactor and a bimodal molecular weight distribution produced in the first reactor of a series configuration by a ssuming tha t the site t ype producing the longest chains is insensitive to t he h ydrogen concentra tion. We propose a process retrofit that permits an increase in HD PE production rat e of up to 20%while mainta ining the same product quality, provided that the overhead units can handle up to a 17%increase in capacity, on the ba sis of simulation results.

Acknowledgment We gra tefully a cknow ledge Alliant Techsyst ems, Aspen Technology (part icula rly J ila Ma ha lec, D irector of Wor l d w i d e U n i v er s i t y P r o g r a m , a n d J os ep h B o s t on , Senior Corporate Advisor and past President), China Petroleum and Chemical Corporation, China National Petroleum Corporation, Honeywell Specialty Materials, an d Honeywell Interna tional Foundat ion for supporting t h e com p u t er -a i d e d d e si g n e d uca t i o n a l p r og r a m a t Virginia Tech. We also thank Costas P. Bokis for his assistance in this work.

Symbols E n g l i sh S y m b ol s   C i 

) constant for ideal-gas heat capacity correlation ( i  )

1,2) ) idea l-ga s hea t capa city (kJ /kmol ‚K ) C 2H 4 ) ethylene C ATi  ) inactive cata lyst CISFRAC i  ) fraction of cat alyst site type i  tha t is inhibited CISFRAC ss ) fraction of catalyst sites that are inhibited in t he single-site m odel COCAT ) cocatalyst CSTR ) continuous stirred-tank reactor D n  ) inactive polymer chain containing n  segments ig

C p

Ind. Eng . C hem. R es., Vol. 41, No. 23, 2002 [D n ] ) concentrat ion of ina ctive polymer cha ins containing n  segm ent s (mol/L) E OS ) equation of state G P C ) gel permeation chromatography H 2 ) hydrogen H D P E ) high-density polyethylene ICATi  ) inhibited cata lyst of site type i  k  ) ra te consta nt (L/mol ‚s) k 0 ) preexponential factor for the rate constant k act, i  ) rate constant for activation of catalyst site type i  k finh, i  ) rate constant for forward hydrogen inhibition of catalyst site type i  k i j  ) binary intera ction par ameter for Sa nchez -Lacombe equation of state k ini, i  ) rate constant for chain initiation for catalyst sit e type i   j  k ini, i  ) rate constant for chain initiation of catalyst site type i  by monomer j  k p ) rate constant for chain propagation for the singlesite model k p, i  ) rat e constant for chain propaga tion for cata lyst site type i   j k  k p, i  ) rat e constant for chain propaga tion for monomer j  adding to segment type k  at catalyst site type i  k rinh, i  ) rate constant for reverse hydrogen inhibition of catalyst site type i  k t h, i  ) r a t e c o n s t a n t f o r c h a i n t r a n s f e r t o h y d r o g e n f o r catalyst site type i   j  k t h, i  ) r a t e c on s t a n t f or c ha i n t r a n s f er t o h y d r o ge n f or chain ending in segment ty pe j  on cata lyst site type i  k t m, i  ) rate constant for chain transfer to monomer for catalyst site type i   j k  k t m, i  ) rate constant for chain transfer to monomer j  for chain ending in segment type k  on catalyst site type i  M ) monomer species m i  ) ma ss fraction of polymer produced at cata lyst site type i 

M i  ) monomer species of type i  M n ) number-average molecular weight M w  ) weight-average molecular weight MWD ) molecular weight distribution n  ) number of monomer segments in a polymer chain n st ) number of cata lyst site types P  ) pressure (bar) P  h ) reduced pressure in th e S anchez -Lacombe equation of state P * ) p r es s ur e s ca l e f a ct o r i n t h e S a n c h ez -Lacombe equation of state (bar) P 0, i  ) activated cata lyst site of type i  P 1, i  ) initiated cata lyst site of type i  P D I ) polydispersity index, M  w /M  n P n  ) live polymer chain containing n  segments [P n ] ) concentration of live polymer chains containing n  segm ent s (mol/L) P n , i  ) live polymer chain containing n  segments attached to catalyst site type i   j  P n , i  ) live polymer chain containing n  segments, ending in segment type j , attached to catalyst site type i  r  ) molecular pa rameter in the S anchez -Lacombe equa tion of stat e T  ) temperat ure (° C) T  h ) reduced temperature in t he Sa nchez -Lacombe equa tion of stat e T * ) temperature scale factor in the Sanchez -Lacombe equation of state (K) VLE ) vapor -liquid equilibrium VLLE ) vapor -liquid -liquid equilibrium w i (n ) ) weight fraction of chains of length n  produced at catalyst site type i 

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) total weight fraction of chains containing n  segments x  ) stoichiometric coefficient ∆H  ) molar enthalpy change W (n )

Greek Symbols 

ηij  ) binary intera ction para meter for S anchez -Lacombe equation of state  λi  ) i th moment for bulk (live a nd dead) polymer chains  µi  ) i th moment for live polymer chains ν(0) i  ) parameter in Chao -Seader fugacity coefficient model ν(1) i  ) parameter in Chao -Seader fugacity coefficient model F ) den sit y (kg/m 3) Fj ) reduced density in the Sanchez -Lacombe equation of state F* ) density scale factor in the Sanchez -Lacombe equat ion of st a te (kg/m 3) τ i  ) adjustable par ameter for chain-length distribution function liq  φi  ) liquid fugacity coefficient of species i  ωi  ) acentric fa ctor for species i 

Li terature Cited (1) Scheirs, J .; Evens, G. Polyethylene (High-Density Preparation). In Polymeri c M ateri als En cyclopedi a ; Sa lamone, J . C., Ed.; CRC Pr ess: Boca Ra ton, FL, 1996; p 5965. (2) Xie, T.; McAuley, K. B.; Hsu, J . C. C.; Bacon, D. W. GasPh ase Ethy lene P olymerization: P roduction Processes, Polymer Properties, and Reactor Modeling. In d. E ng. Chem. Res. 1994, 33 , 449. (3) Peacock, A. J . H and book of Polyethyl ene: Str uctur es, Pr op-  erties, and Applications ; Ma rcel Dekker: New York, 2000. (4) S oares, J . B . P .; Ha mielec, A. E . Deconvolution of Cha inLength D istributions of Linear P olymers Ma de by Multiple-SiteType Catalysts. Polymer  1995, 36 , 2257. (5) Bokis, C. P .; Orbey, H.; Chen, C.-C. P roperly Model Polymer Processes. Chem. Eng. Prog. 1999, 95 , 39. (6) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80 , 2352. (7) Lacombe, R. H.; Sanchez, I. C. Sta tistical Thermodynamics of Fluid Mixtures. J. Phys. Chem. 1976, 80 , 2568. (8) Sa nchez, I. C.; Lacombe, R. H. Sta tistical Thermodynamics of P olymer Solutions. M acromolecul es  1978, 11 , 1145. (9) Cha o, K. C.; Seader, J . D. A General C orrelat ion of VaporLiquid Eq uilibria in Hydrocarbon Mixtures. A I C h E J . 1961, 7 , 598. (10) Reid, R. C.; Pra usnitz, J . M.; Poling, B. E . The Propert ies  of Gases and Liquids ; McGra w-Hill: New York, 1987. (11) Beat on, C. F.; Hewitt, G. F. Physical Pr opert y Data for t he  Design En gin eer ; Hemisphere Publishin g Corp.: New York, 1989. (12) Sychev, V. V.; Vasserma n, A. A.; Golovsky, E. A.; Kozlov, A. D.; Spiridonov, G . A.; Tsyma rny, V. A. Thermodynami c Proper-  ties of E thyl ene ; Hemisphere Publishing Corp.: New York, 1987. (13) Ga ur, U.; Wunderlich, B. H eat C apa city a nd Other Thermodynam ic Properties of Linear Ma cromolecules. II. P olyethylene. J. Phys. Chem. Ref. Data  1981, 10 , 119. (14) Kna pp, H.; D o¨r ing, R.; Oellrich, L.; P lo¨cker, U .; P ra usnitz, J . M . Vapor -L i q u i d E q u i l i br i a f or M i x t u r es of L o w B oi l i n g   Substances ; C h e m is t r y D a t a S e r ie s; D E C H E M A: F r a n k fu r t , Germany 1982; Vol VI, Part 1. (15) Leonard, J . Heats a nd E ntropies of Polymerizat ion, Ceiling Temperat ures, Equilibrium Monomer Concentrations, a nd Polymerizability of H eterocyclic Compounds. In Polymer H andbook ; B r a n d r u p , J . , I m m e r g u t , E . H . , G r u l k e, E . A. , E d s . ; Wi l ey & Sons: New York, 1999; p II /363. (16) Arriola, D . J . Modeling of Addition P olymerization Systems. P h.D. D isserta tion, Un iversity of Wisconsin, Madison, WI, 1989. (17) McAuley, K. B .; MacG regor, J . F.; Ha mielec, A. E. A Kinetic Model for Industria l G as-P hase E thylene Copolymerization. A I C h E   J . 1990, 36 , 837. (18) Cansell, F.; Siove, A.; Fontanille, M. Ethylene -Propylene Copolymerization In itiated with Solubilized Ziegler-N a t t a M a c -

5618 Ind. En g. C hem. R es., Vol. 41, No. 23, 2002 romolecular Complexes. I. Determinat ion of Kinetic P ara meters. J. Polym. Sci. A: Polym. Chem. 1987, 25 , 675. (19) Kissin, Y. V. Isospecific Polymerizati on of Olefins w ith  H eterogeneous Zi egler  -Natta Catalysts ; Springer-Verlag: New  York, 1985. (2 0) N a g e l, E . J . ; K i r i ll ov , V . A .; R a y , W. H . P r e d i ct i on o f Molecular Weight D istributions for H igh-Density P olyolefins. I n d . Eng. Chem. Prod. Res. Dev. 1980, 19 , 372. (21) Polythink Inc., 1005 Sydenham Rd., S.W., Calgary, Alberta, Ca nada T2T 0T3. http://ww w.polythink.com (accessed August 2002).

(22) Yan, R .; Xu, X.; Khar e, J .; Liu, Y. A.; Chen, C.-C. Modeling o f a C o m m er c ia l S l u r r y H D P E P r o c es s U s i n g P o ly m e r s P l u s . I n Proceedings of AspenWorld China 2000 ; Aspen Technology: Ca mbridge, MA, 2000; p 79. Receiv ed for r evi ew  J une 18, 2002 Revised ma nu scri pt received  September 3, 2002 Accepted  September 5, 2002

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