Simulation of Flash Separation in Polyethylene Industrial Processing

Ind. Eng. Chem. Res. 2009, 48, 8613–8628

8613

Simulation of Flash Separation in Polyethylene Industrial Processing: Comparison of SRK and SL Equations of State G. M. N. Costa,*,† Y. Guerrieri,‡ S. Kislansky,† F. L. P. Pessoa,§ S. A. B. Vieira de Melo,† and M. Embiruçu† Programa de Engenharia Industrial, Escola Politećnica, UniVersidade Federal da Bahia, Rua Prof. Aristides NoVis, 2, Federaçaõ, 40210-630, SalVador, Bahia, Brazil, Braskem S. A., Rua Eteno, 1582, Po´lo Petroquı´mico, 42810-000, Camaçari, Bahia, Brazil, and Departamento de Engenharia Quı´mica, Escola de Quı´mica, UniVersidade Federal do Rio de Janeiro, Centro de Tecnologia, bloco E, S. 209, Ilha do Fundaõ, 21949-900, Rio de Janeiro-RJ, Brazil

The Soave-Redlich-Kwong (SRK) and the Sanchez and Lacombe (SL) equations of state are applied to the flash simulation of polyethylene industrial separators, specifically low-density polyethylene (eight resins) and linear low-density polyethylene (25 resins). Three mixing rules are used in the SRK equation: van der Waals (VDW) one-fluid, Wong-Sandler, and LCVM (linear combination of the Vidal and Michelsen mixing rules). The latter two mixing rules incorporate the Bogdanic and Vidal activity coefficient model. All these models involve two adjustable parameters. The results indicate that SL is the best model to simulate the flash separation of polyethylene under industrial conditions. 1. Introduction The thermodynamics of polymer solutions have been used in the design of many industrial processes such as polymerization, devolatilization, and separation of solvent from polymer solutions. The successful description of the vapor-liquid equilibrium (VLE) behavior in solvent-polymer systems is required in order to optimize the other two processes. Polymer systems are generally more complex than systems of low molecular weight substances due to the great difference in the molecular sizes of polymers and solvents, and the phase behavior of polymer systems often exhibits a pronounced density dependence at high temperatures. Analogous to the modeling of conventional phase equilibrium, there are two basic approaches available to describe phase equilibria of polymer-solvent mixtures: activity coefficient models and equations of state (EOS). There are several problems with the activity coefficient approach. For example, it is hard to define standard states (especially for supercritical components), the parameters of the activity coefficient models are very temperature dependent, and critical phenomena are not predicted because a different model is used for the vapor and liquid phases. Furthermore, other thermodynamic properties (densities, enthalpies, entropies, etc.) cannot usually be obtained from the same model because the excess Gibbs free energy is rarely known as a function of temperature and pressure. The use of equations of state in phase equilibrium modeling instead of activity coefficient models is mainly a result of the recent development of a class of mixing rules that enable the use of liquid activity coefficient models in the EOS formalism. The implication of this change is far-reaching as an EOS offers a unified approach in thermodynamic property modeling. With this approach, the applicability of simple cubic EOS has been extended to complex systems such as polymeric systems when coupled with the appropriate activity coefficient model. There* To whom correspondence should be addressed. Tel.: +55-7132839800. Fax: +55-71-32839801. E-mail: [email protected]. † Universidade Federal da Bahia. ‡ Braskem S. A. § Universidade Federal do Rio de Janeiro.

fore, there is much interest in mixture EOS models capable of describing higher degrees of nonideality than that possible with the van der Waals one-fluid model and its modifications. Some studies in this formalism with polymer solutions can be found in the literature.1-4 However, they focus on low-pressure ranges and laboratory experimental data, which poorly reflect the operational conditions found in industrial separation processes. Another approach for determining the phase equilibrium of polymer systems is based on the Sanchez-Lacombe equation of state (SL EOS).5,6 This is a lattice model EOS developed using statistical mechanics. It can be used to predict thermodynamic properties and the phase behavior of both polymeric and nonpolymeric systems. A large number of evaluations of the SL EOS are reported in the literature, chiefly regarding liquid-liquid equilibrium.7-17 For vapor-liquid equilibrium, there is a lack of references in the literature18,19 and none that use industrial plant data. Buchelli et al.20 investigated the performance of the PC-SAFT equation of state for modeling the high pressure separator (HPS) and low pressure separator (LPS) units downstream from a lowdensity polyethylene tubular reactor. Plant data were used to validate the equilibrium stage model prediction for the two gas-liquid flash separators; however, the pure component and binary interaction parameters of this model were obtained exclusively from experimental data published in the literature. The authors achieved a good agreement between the model and LPS plant data; however, the model’s predicted solubility was not in agreement with plant-measured values for the HPS. Other recent SAFT-type contributions specifically for the VLE of polyethylene + ethylene solutions are reported in refs 21-26, including another attempt to model industrial separations.26 This work presents the flash simulation of the polyethylene industrial process, more specifically low-density polyethylene (LDPE) and linear low-density polyethylene (LLDPE), with industrial plant operational data. The equilibrium behaviors from four different models based on the Sanchez-Lacombe (SL) equation of state5 and the Soave-Redlich-Kwong (SRK) cubic equation of state are compared.27 This last EOS is tested with three different mixing rules: the van der Waals given by Orbey and Sandler,28 LCVM (linear combination of the Vidal and

10.1021/ie801652q CCC: $40.75  2009 American Chemical Society Published on Web 08/20/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009

Figure 1. Simplified flow sheet of an LDPE plant.

Michelsen mixing rules),29 and the Wong-Sandler.30 Very recently, Costa et al.31 carried out a detailed evaluation of several activity coefficient models both at infinite dilution and at finite concentration, and the results indicated that the Bogdanic and Vidal activity coefficient model32 was the best one, with superior results in describing several polymer-solvent system liquidliquid equilibria. Based on this, we chose to incorporate this activity coefficient model in the mixing rules. 2. Process Description Despite the simple molecular structure of ethylene, polyethylene is a complex macromolecule that can be manufactured via many different types of processes. Today polyethylene production processes offer a versatile range of products. Highpressure processes can produce linear low-density polyethylene (LLDPE) in addition to the usual range of low-density polyethylene (LDPE) and ester copolymers. Some low-pressure processing plants also produce LLDPE and VLDPE (very low density polyethylene), as well as high-density polyethylene (HDPE), and in many cases, they compete with LDPE and the ester copolymers for the same market. 2.1. Low-Density Polyethylene (LDPE) Processing. Lowdensity polyethylene (LDPE) is industrially synthesized at relatively high temperatures (180-300 °C) and pressures (1000-3000 bar) by free-radical bulk polymerization in supercritical ethylene. In this process, ethylene acts as both reagent and solvent for the polymer. A schematic diagram is shown in Figure 1. In the LDPE process, an important step is the flash separation of monomers and other small molecules from the polymer produced. The process is carried out adiabatically in two stages. In the first step, the reactor effluent (F2) is depressurized through a pressure reduction valve down to 150-250 bar. This allows the separation of the polymeric product from the unreacted ethylene in a high-pressure separator (HPS). The overhead monomer-rich stream (F4) is cooled and recycled back to the reactor, whereas the bottom polymer-rich stream (F3) undergoes a second separation step at a nearatmospheric pressure in a low-pressure separator (LPS). The overhead stream of LPS (F6) is recycled back to the reactor, while the residual ethylene and comonomers dissolved in the molten polymer (F5) may be stripped under vacuum conditions in a devolatilizing extruder. There are two important issues in the design of these flash separators. The first is the prediction of the amount of monomer left in the final product. The presence of monomer in the polymer is undesirable for several reasons and should be minimized. The second issue is a consequence of the polydispersity of the polymer. The polydisperse nature of the polymer results in relatively smaller oligomer molecules (waxes) that remain in the product and can potentially go to the gas phase during the flash operation. When the gas phase is

Figure 2. Simplified flow sheet of an LLDPE plant.

recycled, the waxes must be removed; otherwise they accumulate and eventually precipitate in the recycling lines. 2.2. Linear Low-Density Polyethylene (LLDPE) Processing. Linear low-density polyethylene (LLDPE) is industrially produced at relatively high temperatures (200-300 °C) and moderate pressures (100-300 bar) through solution polymerization. A schematic diagram is shown in Figure 2. Some R-olefins such as 1-butene, 1-hexene, and 1-octene are usually used as comonomers with cyclohexane as the solvent. The polyethylene solution on leaving the reactor mixture (F8) passes through absorbers where the catalyst is removed. Two depressurization stages follow. The LLDPE is separated from the cyclohexane, the unreacted ethylene, and comonomers in the intermediate-pressure separator (IPS), where the pressure is reduced to 30 bar. At this point, most of the cyclohexane (and sometimes 1-octene) vaporizes and forms the IPS overhead gas stream (F10). The solution that leaves the IPS bottom (F9) contains about 40-60 wt % polyethylene and cyclohexane. The remaining solvent is removed in the low-pressure separator (LPS) through the overhead gas stream (F12), and the bottom stream feeds the pelletizing extruder. 3. Thermodynamic Models A brief description of each model follows. 3.1. Sanchez-Lacombe (SL). The Sanchez and Lacombe EOS5,6 is given by 1 F˜ 2 + P˜ + T˜ ln(1 - F˜ ) + 1 - F˜ ) 0 r

[

(

)]

(1)

where T T˜ ) T* T* )

ε* kB

P P˜ ) P* P* )

ε* V*

F˜ )

F F*

F* )

Mw rV*

(2)

(3)

where T is the absolute temperature, P is the pressure, F is the density, Mw is the weight-average molecular weight, kB is the Boltzmann constant, and r, ε*, and V* are pure component parameters related to the corresponding scale factors T*, P*, and F*, respectively. These scale factors are independent of the molecular size of the polymer. For mixtures the model


parameters become composition dependent through the following mixing rules:

∑ ∑ φ φ V*

V*mix )

i j ij

i

1 V*mix

ε*mix )

(4)

∑∑ i

(5)

j

φj

∑r

(6)

j

j

wi F*i V*i

∑ j

( )

1 V*ij ) [V*ii + V*jj ](1 - SL1ij) 2 ε*ij ) √ε*ii ε*jj (1 - SL2ij)

RT a(T) V-b V(V + b)

(9)

(10)

∑ ∑xxa

(11)

∑ ∑xxb

(12)

i j ij

j

i j ij

j

with the combining rules for the cross energy (aij) and covolume (bij) parameters given by aij ) (aiaj)0.5(1 - SR1ij) bij )

(

)

bi + b j (1 - SR2ij) 2

(13)

(14)

where SR1ij and SR2ij are the binary interaction parameters. 3.3.2. Wong-Sandler Mixing Rule. The Wong-Sandler (WS) mixing rule30 is based on the condition of infinite pressure and is given by a)b

(∑ i

i

where c is the numerical constant equal to -ln 2 for the SRK EOS and WSij is the interaction parameter. 3.3.3. LCVM Mixing Rule. Boukouvalas et al.29 proposed the LCVM mixing rule, a linear combination of the Vidal33 and MHV1 Michelsen34 mixing rules as follows: δ ) λδV + (1 - λ)δM

(8)

3.3. Mixing Rules. In order to extend the SRK EOS to polymer-solvent systems, the following mixing rules are considered. 3.3.1. van der Waals One-Fluid (VDW) Mixing Rule.28 The van der Waals one-fluid (VDW) mixing rule28 is given by

i

i

xiai GE + bi c

)

(18)

where δ ) a/(bRT) and δV and δM are given by expressions from the Vidal and Michelsen mixing rules, respectively. Their corresponding contributions to δ are weighed by the factor λ ) 0.36, as originally proposed by Boukouvalas et al.29 Accounting for δV and δM in eq 18, the expression for δ becomes

where SL1ij and SL2ij are binary interaction parameters. 3.2. Soave-Redlich-Kwong Equation of State. The Soave-Redlich-Kwong (SRK) equation of state27 has the following form:

b)

E

i

1 (b - a/RT)ij ) [(b - a/RT)i + (b - a/RT)j](1 - WSij) 2 (17)

δ)

i

(16)

∑ x (a /b RT) - (G /cRT) i

(7)

wj F*j V*j

The cross parameters are

a)

1-

ij

j

with φiφjε*ij V*ij

where the segment fraction of component i, φi, is calculated as a function of the weight fraction wi, given by

P)

i

j

1 ) rmix

φi )

∑ ∑ x x (b - a/RT) i j

b)

8615

(15)

∑xδ

i i

+

(

)

1-λ β 1 - λ GE + + AV AM RT AM

∑ x ln( bb ) i

i

(19)

where AV ) -ln 2 and AM ) -0.53 for the SRK equation of state. The covolume parameter mixing rule is given by b)

∑xb

i i

(20)

i

3.3.4. Excess Gibbs Free Energy (GE) Model. LCVM and Wong-Sandler mixing rules require a GE model, and the model proposed by Bogdanic and Vidal32 is in agreement with previous results presented by Costa et al.31 and was selected for this work. This was the first time that Bogdanic and Vidal’s GE model has been incorporated into a mixing rule, and the results are very good, even for high-pressure conditions, since both liquidand vapor-phase compositions predicted by the model compare very well with experimental ones. It is important to point out that vapor-liquid equilibrium calculation for polymer systems usually adopts GE models to describe liquid-phase behavior at low pressure. In this paper, Bogdanic and Vidal’s GE model is incorporated as an alternative in the SRK equation of state mixing rule to describe the vapor-liquid equilibrium at both low and high pressures for industrial flash separation. The Bogdanic and Vidal (BV) model is a segment-based thermodynamic model containing the combinatorial, freevolume, and energetic contributions to the excess Gibbs energy. The model is derived from the entropic free-volume model following the idea of associating the nonideality of polymersolvent mixture with polymer segment-solvent interaction parameters. The energetic contribution is based on interactions between individual segments (repeating units) of polymer or copolymer and solvent molecules. Segment activity coefficients are calculated by the UNIQUAC model.35 The activity coefficient is given by the following equation: ln γi ) ln γFV + ln γRES i i

(21)

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and the Nelder-Mead simplex method39 is applied to minimize the following objective function:

Table 1. Identification of the Models and Their Parameters model

parameters

Sanchez and Lacombe (SL) SRK with van der Waals mixing rules (SRK-VDW) SRK with LCVM mixing rules (SRK-LCVM) SRK with Wong-Sandler mixing rules (SRK-WS)

SL1ij SR1ij UQij* UQij

SL2ij SR2ij λ WSij

As in the original model, the combinatorial and the freevolume contributions are combined in a single term, the socalled FV part, given by ln γFV ) ln i

( )

φFV φFV i i +1xi xi

(22)

and xi are the free-volume fraction and the molar where φFV i fraction, respectively. The free volume is defined as VFV,i ) Vi - VW,i

(23)

where Vi and VW,i are the liquid molar volume and the van der Waals volume, respectively. , the mixture For the calculation of the residual term, ln γRES i is considered a solution of segments, and the molar fraction of each segment is calculated as nc

∑xυ

(i) i k

Xk )

i

(24)

nc nseg

∑∑ j

xjυm(j)

m

where xi and xj are the mole fractions of the respective components i and j. The summations are extended to the total number of components, nc, and to the total number of segments, (j) nseg. Parameters υ(i) k and υm are the numbers of segments k in the component i, and of segments m in the component j, respectively. 4. Results and Discussion 4.1. General Observations. In order to perform calculations with the SRK equation of state for polymers, the pure parameters are not known from critical property data but are calculated following Kontogeorgis’s method.36 According to this method, polymer parameters a and b are fitted to two experimental volumetric data points at essentially zero pressure. Instead of using experimental volumetric data in this equation of state and in the BV model, these properties are obtained using the group contribution method GCVOL.37 Data for van der Waals volumes are taken from Bondi.38 The pure component parameters and segment parameters of the polymer for the SL EOS are taken from refs 6, 10, and 16. The original Sanchez and Lacombe6 and Koak et al.10 parameters were fit to LDPE density and the LDPE parameters from Xiong and Kiran16 were fit to PVT data for pure polymer. Generally, data from the literature are used to estimate binary interaction parameters. This is the case for the SRK model, since this is the first time it is used coupled with Bogdanic and Vidal (BV) model for both low and high pressures. However, at high pressure, experimental data from the literature are scarce and simulation results using only these data are less accurate. Thus, for both the SRK-BV and SL models plant data were used to estimate binary interaction parameters. Adjustable characteristic parameters of the various models, described in Table 1, are fitted to industrial operational data,

nc

nc

OF )

∑ (w

calc i

i)1

- wi exp)2 +

∑ (w

calc i

- wi exp)2

i)1

(25) where the superscripts “calc” and “exp” are related to the calculated and experimental values, respectively; w′i is the mass composition of component i in the heavy phase; wi′′ is the mass composition of component i in the light phase; and nc is the number of components. In Table 1, the UQij parameter is related to the UNIQUAC contribution34 for the BV model,31 and a distinction is made for the two mixing rules (LCVM and Wong-Sandler). As regards polymer phase equilibrium calculations, the component composition is usually presented as a mass fraction because the polymer has a much higher molecular weight than the solvent. The equation of state is defined in terms of molar fractions, which can lead to numerical problems for computing the solvent molar fraction as it becomes very small. A similar problem may occur when modeling polymer vapor-liquid equilibrium: in the gas phase, there is almost no polymer, but thermodynamically this value cannot be zero, even though it may be extremely low. This produces some difficulties for calculating phase equilibrium in flash separators. For collecting and treating the industrial data, the following steps were followed:40,41 1. The sampling time for data acquirement in the PIMS (plant information management system) was 1 min and data compression treatment was suppressed. 2. The collected raw data were analyzed to eliminate outliers and to identify near steady state operation regions; 10 steady state points were selected for each resin. 3. From the selected data, mass and energy balance calculations were made using the basic balance equations in order to reconcile the data. 4. To validate the thermodynamic model, an average of the 10 steady state points was used for each resin. 4.2. Assumptions Used To Obtain the Experimental Data in the Separation Stage of the Process in LDPE Plant. Figure 1 shows a schematic diagram of the LDPE flow sheet. Circles identify streams used in this study. Stream F1 contains ethylene and small amounts of propene and propane and feeds the reactor. The discharge stream (F2) containing ethylene, polymer, a small amount of ethane (formed in the reaction), propene, and propane, is sent to the HPS (highpressure separator). The HPS bottom stream (F3) is sent to the LPS (low-pressure separator). The LPS top (F6) returns to the process and its bottom (F5), containing essentially pure polymer, is sent to the extruder. The following experimental data are necessary to validate the HPS model: F2, THPS, PHPS, and zHPS (input data); F3, xHPS, F4, and yHPS (output data). zHPS, xHPS, and yHPS are the mass compositions of the feed, bottom, and top streams, respectively; F2, F3, and F4 are the total mass flow at the feed, bottom, and top, respectively; THPS and PHPS are the HPS operational temperature and pressure, respectively. Similarly, the following experimental data are needed to validate the LPS model: F3, TLPS, PLPS, and zLPS (equal to xHPS) (input data); F5, xLPS, F6, and yLPS (output data). zLPS, xLPS, and yLPS are the mass compositions of the feed, bottom, and top streams, respectively; F3, F5, and F6 are the total mass flow at the feed, bottom, and top, respectively; TLPS and PLPS are the LPS operational temperature and pressure, respectively.

Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009 Table 4. LLDPE Resins and IPS Operational Conditions

Table 2. LDPE Molecular Weight and Separators Operational Conditions resin

Mw (g/mol)

Mn (g/mol)

LDPE-1 LDPE-2 LDPE-3 LDPE-4 LDPE-5 LDPE-6 LDPE-7 LDPE-8

335 000 295 200 181 000 166 000 340 000 322 000 236 600 426 300

23 103 25 894 23 086 21 813 20 000 28 196 11 901 16 497

PDI

THPS (°C)

PHPS (bar)

TLPS (°C)

PLPS (bar)

14.50 11.40 7.84 7.61 17.00 11.42 19.88 25.84

280.29 278.80 230.29 253.03 237.68 277.17 247.92 265.33

250.06 251.90 249.88 250.55 249.87 249.54 254.00 253.28

241.04 239.49 212.17 226.73 217.91 242.73 224.93 235.69

0.35 0.42 0.37 0.46 0.42 0.41 0.40 0.42

Table 3. LPDE Feed Range Composition in the HPS and LPS Separators HPS

ethylene ethane propene propane LDPE

8617

LPS

min

max

min

max

73.4 0.3 0.0 0.0 19.5

77.0 2.6 0.2 1.6 25.5

20.7 0.1 0.0 0.0 72.9

25.9 0.8 0.1 0.5 79.2

We selected eight resins to study the several aspects of the LDPE flash separation. Table 2 shows the HPS and LPS operational conditions and the characteristics of the resins, including the polydispersity index (PDI) and weight-average (Mw) and number-average (Mn) molecular weights. In order to present sufficient information to follow the simulation problem, Table 3 shows the feed range composition of the HPS and LPS separators. 4.3. Assumptions Used To Obtain Experimental Data in the Separation Stage of the LLDPE Processing Plant. Figure 2 shows a simplified flow sheet of the LLDPE industrial process. Circles identify the streams used in this study. Stream F7, containing ethylene, 1-butene, 1-octene, and cyclohexane, is fed to the reactor. Stream F8, containing polymer, ethylene, 1-butene, 1-octene, and cyclohexane, is discharged from the reactor to the IPS (intermediate-pressure separator). The IPS top stream (F10) is sent to a light separation unit. The IPS bottom stream (F9) is fed to the LPS, where the final separation between the polymer and the lights is carried out. In order to validate the model, the following experimental data are necessary: F8, T, P, and z (input data); F10, y, F9, and x (output data). z, y, and x are the mass compositions of the feed, top, and bottom streams, respectively; F8, F10, and F9 are the total mass flow at the feed, top, and bottom, respectively; T and P are the operational temperature and pressure, respectively. Table 4 presents the LLDPE resin types according to the following contents of the reactor feed: ethylene and cyclohexane (LLDPE-A); ethylene, 1-butene, and cyclohexane (LLDPE-B); ethylene, 1-octene, and cyclohexane (LLDPE-C); ethylene, 1-butene, 1-octene, and cyclohexane (LLDPE-D). Each type is numbered to indicate its pressure and temperature operational conditions. Table 5 shows the feed range composition information of the IPS, required to solve the simulation problem. The low-pressure separator (LPS) in the LLDPE plant was not examined in this study because it was not possible to estimate its process variable values since there were no available measurements of vapor properties, such as rates and compositions. 4.4. Simulation Results. The polymer is assumed to be monodisperse for all systems. As stated earlier, the pure component parameters and segment parameters of the polymer for the SL EOS are taken from refs 6, 10, and 16 and reported in Table 6. 4.4.1. Binary Interaction Parameters of the Models for Simulation of the LPS (LDPE Resin). The binary interaction

resin

Mw (g/mol)

Mn (g/mol)

PDI

T (°C)

P (bar)

LLDPE-A1 LLDPE-A2 LLDPE-A3 LLDPE-A4 LLDPE-A5 LLDPE-A6 LLDPE-A7 LLDPE-A8 LLDPE-B1 LLDPE-B2 LLDPE-B3 LLDPE-B4 LLDPE-B5 LLDPE-B6 LLDPE-B7 LLDPE-B8 LLDPE-B9 LLDPE-B10 LLDPE-B11 LLDPE-B12 LLDPE-C1 LLDPE-D1 LLDPE-D2 LLDPE-D3 LLDPE-D4

141 400 54 960 84 000 94 000 113 500 162 000 160 000 169 700 156 000 135 400 127 000 126 600 84 500 100 800 151 000 112 000 59 900 56 460 92 000 94 100 149 000 146 000 145 000 149 200 164 000

12 855 17 729 20 488 22 927 29 868 18 000 14 953 11 785 34 667 30 089 24 902 26 936 20 610 24 000 25 167 21 132 17 114 17 109 25 556 25 432 33 111 31 064 25 439 36 390 29 818

11 3.1 4.1 4.1 3.8 9 10.7 14.4 4.5 4.5 5.1 4.7 4.1 4.2 6 5.3 3.5 3.3 3.6 3.7 4.5 4.7 5.7 4.1 5.5

269.93 269.21 271.80 275.90 271.60 275.96 273.34 277.05 258.02 259.64 256.75 258.06 266.11 258.64 256.07 262.79 259.64 256.57 263.48 267.46 275.26 267.08 268.38 268.02 274.41

31.70 30.54 30.87 30.90 30.71 30.58 31.32 31.61 31.49 32.02 31.44 31.17 29.55 31.29 30.65 31.12 31.88 29.06 30.74 31.05 29.46 27.90 30.86 29.58 28.43

Table 5. LLDPE Feed Range Composition in the IPS Separator

ethylene 1-butene cyclohexane 1-octene LLDPE

LLDPE-A1 f LLDPE-A8

LLDPE-B1 f LLDPE-B12

LLDPE-D1 f LLDPE-D4

min

max

min

max

min

max

0.6 76.3 16.9

1.5 81.9 22.4

0.9 2.2 66.7 18.3

1.4 11.5 75.8 21.3

1.4 1.3 48.5 13.0 18.9

1.9 4.1 65.4 28.1 19.6

Table 6. Pure Component or Segment Parameters Used in Sanchez-Lacombe EOS parameter

ethane ethylene propane propene segment of LDPE 1-butene cyclohexane 1-octene

P* (atm)

T* (K)

F* (g/cm3)

3230 2000 3090 1913 3543 1967.88 3780.21 1882.32

315 327 371 385 650 418.35 497.00 521.12

0.64 0.515 0.69 0.555 0.895 0.592 0.902 0.656

parameters (LDPE-solvents) required for each model to fit the experimental data for each LDPE resin are shown in Table 7 at low pressure. Their identifications are displayed in Table 1. These binary interaction parameters are the same for LDPE + (any solvent, including monomers), and all other binary interaction parameters were set to zero. The fitted parameters SL1ij, SL2ij, SR1ij, and SR2ij presented in Table 7 show molecular weight independent values for SL EOS and SRK EOS with the VDW mixing rule. For the latter, the two interaction parameters connected to volume and force are equal (i.e., SR1ij ) SR2ij;). Parameter SR1 is related to the parameter a, and this result can be explained because the chemical structure of the compounds evolved to be the same. However, this result is unexpected for the parameter SR2 because the molecular volumes of the components are very different. Moreover, identical values of binary interaction parameters were obtained for different resins for both SL and SRK-VDW models. A possible explanation is that at low pressures these parameters are not very sensitive as they are

8618


Table 7. Binary Interaction Parameters of EOS Used for LPS Simulation (LDPE Resin with Any Solvent) resin

SL1ij

SL2ij

UQ*ij

WSij

UQij

λ

SR1ij

SR2ij


0.0098 0.0098 0.0098 0.0098 0.0098 0.0098 0.0098 0.0098

0.0101 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102

0.0101 0.0101 0.0101 0.0101 0.0439 0.0123 0.0108 0.0149

0.0101 0.0101 0.0101 0.0101 0.0197 0.0227 0.0136 0.0105

0.5420 0.6230 0.5700 0.1030 -0.5590 -0.5060 -0.5350 -0.3580

-0.6320 -0.7280 -0.6650 -0.6170 -0.6530 -0.5910 -0.6230 -0.4140

0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100

0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100

not capable of capturing the influence of the interaction between segments and the difference between the size of molecules in different systems, as these phenomena are less pronounced in these conditions. For LCVM and Wong-Sandler, as expected, the interaction UNIQUAC parameter values with the WS mixing rule UQ*ij are very different from those of the LCVM mixing rule UQij, due to the differences in the mixing rules. Since this parameter is related to forces of attraction and the molecular configurations of the systems are very similar, the low values of UQ*ij seem more realistic than those of UQij. Unlike the original paper,28 it is evident that the LCVM parameter λ should be fitted. Similar to SL and SRK-VDW models, we also observe identical SRKWS binary interaction parameters (UQ*ij and WSij) values for LDPE-1 to LDPE-4. Again, the low sensitivity of the parameters under these conditions is a possible explanation. Several attempts have been made to establish a connection between the model parameters λ (LCVM) and WSij (WongSandler) and the molecular weight of the resin in the LPS. The only reasonable correlation is obtained with the λ parameter. These results are shown in Figure 3 and the data points are regressed by the following equation: λ ) 0.051β2 - 0.2578β + 0.8272

(26)

with β ) Mw/105, where Mw is the molecular weight of the resin in units of grams per mole. It should be stressed that this is an empirically developed correlation, and there is no guarantee that it will work well beyond the molecular weight range where it was regressed. It is difficult to predict the range of extrapolation where the correlation will work well, but it is certain that greater deviations are expected as the correlation is used farther from its experimental data range. This is true not only because it is a general trend for empirical models, but also because the correlation predicts increasing dependence with increased molecular weight, which is not expected on a meaningful physical basis. Moreover, also considering a physical reasoning, a correlation capable of making great extrapolations should predict λ going to zero when the molecular weight goes to zero, which is also not observed.

Figure 3. Regression of LCVM parameter λ with LDPE resin molecular weight (β ) Mw · 10-5) for LPS.

Table 8. Binary Interaction Parameters of EOS Used for HPS Simulation (LDPE Resin) resin

SL1ij

SL2ij

UQ*ij

WSij

UQij

λ


1.5975 1.5975 2.0199 2.0199 2.0199 1.5074 2.0000 1.5299

1.5075 1.5075 1.0099 1.0099 1.0099 1.5033 0.9900 1.4850

0.0010 0.0010 0.0010 0.0010 -0.0032 0.0068 -0.0297 0.0023

0.9440 0.9296 0.8237 0.9000 -0.4543 0.5873 -2.6891 0.0560

0.0010 0.0010 0.0013 0.0011 0.0028 0.0016 0.0025 0.0020

-0.3663 -0.2255 0.2235 -0.0703 3.6224 0.8922 3.0426 2.0432

4.4.2. Binary Interaction Parameters of the Models for Simulation of the HPS (LDPE Resin). The binary interaction parameters required for each model to fit the experimental data for each LDPE resin are shown in Table 8 at high-pressure conditions. Flash simulation of these conditions could not be done with the VDW mixing rules, as expected. The other models applied are identified in Table 1. As a basis for comparison, the SL parameters seem more meaningful than the other models. Table 8 shows the results given by SL EOS. They exhibit a slight relationship between the mass ratio of polyethylene/ethylene and the molecular weight of the resin. This behavior is illustrated in Figure 4 and is distinct from that for the LPS, shown in Table 7. SL EOS parameters assume different values depending on polymer molecular weight and on the weight ratio of polyethylene to ethylene, as given by the following equations: SL1ij ) SL2ij )

{ {

2.0150; 4.7 < R < 5 1.5593; 5 e R < 5.12

(27)

1.0050; 4.7 < R < 5 1.5008; 5 e R < 5.12

(28)

with R ) log[Mw(%Pol/%Et)], where %Pol is the polymer percentile weight and %Et is the ethylene percentile weight, both in the feed stream.

Figure 4. Sanchez-Lacombe parameters SL1ij and SL2ij as a function of LDPE resin weight-average molecular weight (Mw) and the mass ratio of polyethylene to ethylene (%Pol/%Et) for HPS, where R ) log [Mw(%Pol/ %Et)].


Plant data for monodisperse polymers are used to estimate the binary interaction parameters alone, and their values are correlated as a function of polymer molecular weight and feed composition. In order to show the predictive ability of SL EOS with these correlations, six (LDPE-1, -2, -3, -4, -7, and -8) of these eight resins were randomly chosen to estimate a new correlation for the binary interaction parameters and the other two resins (LDPE-5 and -6) were used to evaluate the prediction capability of this new correlation. In these figures, case a indicates a normal simulation, i.e., a simulation with the original correlation (eqs 27 and 28), where all resins were used for parameter estimation. Case b indicates a simulation with the new correlation, where resins LDPE-5 and -6 were not used for parameter estimation. As Figure 5 shows for LDPE-5, there is no difference between the compositions calculated with normal simulation and the new correlation. In Figure 6 (LDPE6) we can observe that there is a difference between the two approaches, although differences between polymer compositions at both top and bottom and ethylene compositions at top are negligible. For the other compositions, the results of the new correlation are even better, which demonstrates the prediction capability of our procedure. As can be seen above, we assume monodisperse polymers throughout this study because polydispersity data are available only for resins LDPE-1 and LDPE-6, each characterized by a histogram with five molecular weight ranges with their respective weight fractions from a chromatogram. In order to select a specific molecular weight value inside each of the five ranges, the following optimization problem was solved for each resin:

(

r

min ppc,i

∑ i)1

ppc,i - pmp,i pmp,i

s.t.

)

8619

2

(29)

i ) 1, ..., r

plb,i e ppc,i e pub,i r

∑wp

i pc,i

i)1

r

(

) Mw /MEt

r

∑ w p ∑ w /p i pc,i

i)1

i

i)1

)

pc,i

) PDI

where r is the number of molecular weight ranges (here, equal to 5), p is the polymer chain length, wi is the weight fraction of polymer range i, M is molecular weight, Et refers to ethylene, pc refers to the selected pseudocomponent in each histogram range, and mp, lb, and ub refer to their middle point, lower bound, and upper bound, respectively. This formulation reflects the desire to select pseudocomponents closest to the range middle points as possible, while ensuring polymer average molecular weight and polydispersity index, as well as satisfying range bounds. The equality constraints of eq 29 may be solved for two of the unknowns, reducing the problem to a (r - 2)order one, without equality constraints. Indeed, inequality constraints are not strictly necessary because if the problem solution does not match these restrictions some sort of inconsistency in the experimental data is expected. Therefore, the problem may be formulated in the following simpler form, which is an unconstrained multivariate problem and can be solved with standard optimization techniques, even analytical ones:

Figure 5. Modeling LDPE-5: (a, c) using the normal simulation (all eight resins used in correlation estimation); (b, d) using six resins in the estimation of the new correlation.

8620

min

ppc,i,i*s,t


[

r

∑

i)1,i*s,t

(

ppc,i - pmp,i pmp,i

) ( 2

+

ppc,s - pmp,s pmp,s

(

)

2

Table 9. Polymer Polydispersity Characterization

+

ppc,t - pmp,t pmp,t

)]

LDPE-1

where s and t refer to the two unknowns solved by the equality constraints and ppc,s and ppc,t are explained in eqs 31 and 32. The result of applying this procedure is shown in Table 9. ppc,t )

(

d ) wt

-e + √e2 - 4df 2d

(PDI)MEt Mw

wi p i)1,i*s,t pc,i

∑

(

r

e ) ws2 - wt2 - PDI +

∑

wippc,i

i)1,i*s,t r

(

)

(PDI)MEt Mw

)

Mw r wi wi + p M p Et i)1,i*s,t pc,i i)1,i*s,t pc,i

∑

f ) wt

(31)

r

Mw MEt

r

∑

i)1,i*s,t

wippc,i

∑

)

The monodisperse assumption is also due to the negligible

ppc,s )

Mw MEt

r

∑

i)1,i*s

ws

component

molecular weight

molecular weight

mass composition (feed)

pseudo-1 pseudo-2 pseudo-3 pseudo-4 pseudo-5

556 3 089 19 223 428 084 2 255 596

0.14 2.21 12.71 8.23 2.10

470 8 339 52 959 298 844 1 968 130

0.10 1.99 8.02 12.14 2.07

2

(30)

wippc,i (32)

effect of polydispersity on the overall model performance, previously investigated and shown in Table 10, where the

LDPE-6

mass composition (feed)

polymer pseudocomponents are presented with increased molecular weight. The two adjusted parameters SL1 and SL2 and the vaporized fraction V/F are also presented for the monodisperse and the polydisperse polymers. We conclude that there is no significant difference between model predictions with plant data for both monodisperse and polydisperse polymers. The only advantage that we can obtain in considering a polydisperse instead of monodisperse polymer is an evaluation of the wax composition obtained in the top separator. The Wong-Sandler EOS fitted parameter values (WS) presented in Table 8 exhibit a pronounced dependence on the relative polymer and ethylene amounts and molecular weight of the resin. This behavior is illustrated in Figure 7. The UNIQUAC fitted parameter of the Wong-Sandler EOS (UQij*) for the high-pressure separator (HPS) shows irregular behavior for the LDPE-5 to LDPE-8 resins. However, for the LDPE-1 to LDPE-4 resins, this parameter is independent of pressure because its values remain the same for both the LPS and HPS, as shown in Table 7 (LPS) and Table 8 (HPS). It is important to point out that accounting for such behavior only on the basis of polymer molecular weight is not sufficient, as seen in Table 2.

Figure 6. Modeling LDPE-6: (a, c) using the normal simulation (all eight resins used in correlation estimation); (b, d) using six resins in the estimation of the new correlation.


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Table 10. Comparison between Monodisperse and Polydisperse Results Using Sanchez-Lacombe EOS LDPE-1 exp

LDPE-1 calc

LDPE-6 exp

LDPE-6 calc

monodisperse

polydisperse

monodisperse

polydisperse

SL1

1.597

1.716

1.507

1.469

SL2

1.507

1.716

1.503

1.509

top pseudo-1 pseudo-2 pseudo-3 monodisperse pseudo-4 pseudo-5 ethane ethylene propane propene V/F (kg/kg)

bottom

0.0 0.42 0.0 6.89 0.0 39.60 0.0 79.08 0.0 25.62 0.0 6.53 0.35 0.07 99.59 20.83 0.05 0.01 0.01 0.00 0.6790

top

bottom

0.00 79.76 0.26 0.26 99.69 19.94 0.04 0.04 0.01 0.01 0.6816

top

0.14 0.14 0.33 6.51 0.41 40.79 0.26 26.41 0.31 6.19 0.26 0.26 98.24 19.65 0.04 0.04 0.01 0.01 0.6954

The WS EOS parameter presents a quadratic dependence on polymer molecular weight and on the weight ratio of polyethylene to ethylene, as given by the following equation: WSij ) -143.25R2 + 1441.6R - 3626

(33)

with 4.85 < R < 5.15. Unlike the parameter value fitted at low pressure, the LCVM parameter (λ) at high pressure is not constant and is strongly dependent on the system properties. Figure 8 shows the parameter λ dependence on polymer molecular weight and on the weight ratio of polyethylene to ethylene. The parameter λ can have a linear or a nonlinear dependence on polymer molecular weight and on the weight ratio of polyethylene to ethylene, as given by the following equation: λ)

{

bottom

22.205R - 105.71; 4.7 < R < 5 2 422.89R - 4259.7R + 10727; 5 e R < 5.12 (34)

top

bottom

0.0 0.32 0.0 6.40 0.0 25.77 0.0 78.13 0.0 38.99 0.0 6.66 1.80 0.39 98.15 21.46 0.02 0.00 0.03 0.01 0.6886

top

bottom

0.0 76.25 0.069 4.11 99.93 19.51 0.00 0.063 0.00 0.062 0.6809

top

bottom

0.1 0.1 0.27 5.74 0.22 25.04 0.34 37.92 0.27 6.01 0.0 4.33 98.80 20.74 0.00 0.064 0.00 0.064 0.6859

both situations where good performance is a hard task for the model. In other words, we may say that the first linear correlation is physically meaningful, with good predictive capability, whereas the second quadratic correlation is physically meaningless, with poor predictive capability. The correlation coefficients reinforce this analysis. Simulation of the LPS and HPS behaviors for the LDPE resin and comparison of the equation of state performances suggest that the Sanchez-Lacombe EOS is the best choice. Furthermore, it is simpler to use and more accurately describes parameter dependence on polymer molecular weight. 4.4.3. Binary Interaction Parameters of the Models for Simulation of the IPS Separator (LLDPE Resin). Two of the four models are able to simulate the IPS in the LLDPE industrial plant: Sanchez-Lacombe and SRK-WS. Their results are shown in Table 11. A comparison, shown below, demonstrates that the best results are obtained with the SanchezLacombe EOS, irrespective of the specific component considered. We can observe some unusual great values, such as

This couple of correlations poses a nonmonotonic and nonsmooth behavior which is difficult to explain in physical terms, even if we consider it was empirically developed. Nevertheless, we can speculate that the abrupt change in the shape of λ dependence relative to R independent parameter should mean a bound value in R regarding its ability to describe the problem properly and therefore ultimately meaning the inadequacy of the model in coping with the description of the equilibrium phenomenon beyond this limiting R value. Indeed we think this make sense, since a greater value of R indicates greater values of polymer concentration and/or polymer length,

Figure 7. Wong-Sandler parameter WSij as a function of LDPE resin weight-average molecular weight (Mw) and the mass ratio of polyethylene to ethylene (%Pol/%Et) for HPS, where R ) log [Mw(%Pol/%Et)].

Figure 8. LCVM parameter λ as a function of LDPE resin weight-average molecular weight (Mw) and the mass ratio of polyethylene to ethylene (%Pol/ %Et) for HPS, where R ) log[Mw(%Pol//%Et)].

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Table 11. Binary Interaction Parameters of EOS used for IPS Simulation (LLDPE Resin) resin

SL1ij

SL2ij

UQij

WSij

LLDPE-A1 LLDPE-A2 LLDPE-A3 LLDPE-A4 LLDPE-A5 LLDPE-A6 LLDPE-A7 LLDPE-A8 LLDPE-B1 LLDPE-B2 LLDPE-B3 LLDPE-B4 LLDPE-B5 LLDPE-B6 LLDPE-B7 LLDPE-B8 LLDPE-B9 LLDPE-B10 LLDPE-B11 LLDPE-B12 LLDPE-C1 LLDPE-D1 LLDPE-D2 LLDPE-D3 LLDPE-D4

-0.6610 -0.7410 -0.6030 -0.5850 -0.6120 -0.6310 -0.6580 -10.1700 0.5670 0.3350 -0.6030 -0.6080 -0.5621 -0.6060 -0.6230 0.2190 -0.6660 -1.5760 -0.7400 -0.5583 -1.8930 -0.7890 -0.2030 -2.3580 -0.9860

-1.345 -1.875 -0.865 -1.000 -1.065 -1.187 -1.257 -1.020 -1.860 -1.550 -1.030 -1.030 -0.891 -0.979 -1.031 -1.410 -1.779 -0.736 -0.926 -0.865 -1.279 -0.952 -0.922 -0.927 -1.132

0.519 0.793 0.957 0.016 0.682 0.879 -0.004 -0.004 0.358 0.373 0.728 0.916 0.978 0.987 1.173 -3.840 -3.585 -0.157 -2.550 0.865 -0.020 0.422 0.693 0.468 0.159

0.014 0.005 0.130 0.606 0.018 0.005 -0.010 -0.010 0.100 0.501 0.524 -0.001 0.581 18.440 -3.840 1.170 0.529 -0.003 -8.192 0.022 0.001 0.012 0.018 0.013 0.006

-10.1700 for LLDPE-A8 with SL, 18.440 for LLDPE-B6 with SRK-WS, and -8.192 for LLDPE-B11 also for this last model. Regarding the SRK-WS model, these values reveal a certain insensitivity of the parameter relative to molecular weight variations. In the case of the SL model, although a similar interpretation may be given, we note that system LLDPE-A8 presents the greatest molecular weight and the greatest PDI, suggesting that these extreme property conditions may be influencing the unusual value obtained. This observation reinforces the superiority of SL behavior when comparing with SRK-WS. The low pressure separator (LPS) in the LLDPE plant was not evaluated in this study because it was not possible to measure or estimate its process variable values. 4.4.4. Comparison of the LPS Overhead Composition for LDPE Resin. Figures 9-12 show the results for the LPS overhead composition for LDPE resin obtained with SL, SRKLCVM, SRK-WS, and SRK-VDW (as stated earlier, all LDPE-solvent binary interaction parameters are the same for each model). According to these figures, the SL and SRK-VDW EOS perform well in the simulations. However, as previously discussed, the SL EOS has the advantage of constant parameters (see Table 7) for the LPS separator. It gives the best description of the LPS operational conditions and system properties. Despite its simplicity, the SRK-VDW EOS is able to simulate quite well with the same level of precision as the more elaborated SL EOS, if given an adequate pure parameter description. The results presented in Figure 10 reveal that the SRK-LCVM overpredicts ethylene composition for all eight resins tested. On the other hand, prediction of the ethane composition is in good agreement with experimental data. However, since the ethylene mass concentration is usually around 95%, we conclude that SRK-LCVM is not adequate for LDPE simulation. The simulation results with SRK-WS are shown in Figure 11. It can be seen that this model is completely inadequate for the prediction of the ethylene overhead composition of LDPE. The same can be said about the prediction of ethane composition, with large errors not graphically represented. 4.4.5. Comparison of the HPS Overhead and Bottom Compositions for LDPE Resin. As previously described, for the HPS it is important to evaluate both overhead and bottom

Figure 9. LPS overhead composition for LDPE resin: experimental and predicted data by SL EOS.

Figure 10. LPS overhead composition for LDPE resin: experimental and predicted data by SRK-LCVM EOS.

streams. The amount of ethylene in the overhead is about 99% on a mass basis. The bottom stream usually contains about 20% ethylene and 80% LDPE fractions. As mentioned in section 4.4.2, it was not possible to apply SRK-VDW to simulate the HPS. Figures 13-16 show the results for the HPS overhead composition of LDPE resin obtained with SL, SRK-LCVM, and SRK-WS (in these figures resins are identified by their numbers). The results presented in Figures 13 and 14 indicate that the


8623

Figure 11. LPS overhead composition for LDPE resin: experimental and predicted data by SRK-WS EOS.

Figure 13. HPS overhead compositions for LDPE resin: experimental and predicted data by SL EOS.

Figure 12. LPS overhead composition for LDPE resin: experimental and predicted data by SRK-VDW EOS.

best predictions of overhead and bottom compositions for seven of the tested resins were made using the SL EOS. The worst results for the overhead were for LDPE-6 (Figure 13), and the worst results for the bottom stream were for LDPE-5 (Figure 14). Thus, the SL EOS is also the best model to describe the HPS operational conditions and system properties. A comparison between Figure 9 and Figure 13 and Figure 14 describes the pressure influence on the simulation results. It is clearly shown in Figure 13 and Figure 14 that the experimental data at high pressure are far from the diagonal. It is important to say that the overhead composition depends on the polymer molecular weight, ethylene weight fraction and polymer weight fraction, expressed in terms of R in eqs 27 and 28 in the HPS feed stream. The results of the simulation with SRK-LCVM and SRKWS for HPS indicate that the overhead stream is made up of pure ethylene. As a consequence, only the composition of the bottom stream can be discussed with these models. Figure 15 shows the results of simulations with SRK-LCVM for the HPS. In order to evaluate the pressure influence, Figures 10 and 15 are compared. The results reveal that SRK-LCVM has increased

Figure 14. HPS bottom compositions for LDPE resin: experimental and predicted data by SL EOS.

the difficulty in describing the overhead stream composition as pressure increases. Figure 16 shows the results of simulations with SRK-WS for the HPS, and it can be compared to Figure 11 to evaluate the influence of pressure. The results indicate that SRK-WS gives a poorer performance at low pressure. This behavior is to

8624


Figure 15. HPS bottom composition for LDPE resin: experimental and predicted data by SRK-LCVM EOS.

Figure 16. HPS bottom composition for LDPE resin: experimental and predicted data by SRK-WS EOS.

be expected due to the intrinsic concept behind the WongSandler mixing rules that take infinite pressure as a reference. 4.4.6. Comparison of IPS Overhead and Bottom Stream Compositions for LLDPE Resin. Figures 17-20 show the results for IPS overhead and bottom stream compositions for LLDPE resins obtained with the SL and SRK-WS, respectively.

As mentioned in section 4.4.3, only SL and SRK-WS are able to simulate IPS with LLDPE resins. Figures 17-20 show the results for IPS overhead and bottom stream compositions for 25 LLDPE resins (see Table 4) obtained with SL and SRKWS, respectively. Figure 17 presents the overhead composition for ethylene, 1-butene, 1-octene, and cyclohexane predicted by SL. In these cases, the most important component to be monitored is cyclohexane because it is the solvent and therefore is present in all resins at the highest concentration. The results indicate a very good simulation not only for cyclohexane but also for the other components. Figure 18 shows the bottom stream composition for 1-butene, 1-octene, cyclohexane, and LLDPE for the same conditions. It the bottom stream, cyclohexane and LLDPE are the key components as they are at the highest concentrations. It can be seen that the SL EOS offers good accuracy for simulation in this pressure range. Figures 19 and 20 present the IPS overhead and bottom compositions for ethylene, 1-butene, 1-octene, cyclohexane, and LLDPE predicted by the SRK-WS EOS. A comparison between Figures 17 and 18 and Figures 19 and 20 reveals that SL EOS gives the best results for all components considered. Note that the cyclohexane composition in the overhead stream is overpredicted with SRK-WS and that, as a consequence, its composition is underpredicted with this same EOS in the bottom stream. Furthermore, LLDPE composition is underpredicted by this model for most resins. 4.4.7. Comparison of HPS Overhead and Bottom Compositions for LLDPE Resin Using Parameter Values Estimated for LPS. To investigate the influence of EOS fitted parameters at different simulation conditions, the HPS overhead and bottom compositions are predicted with the parameter values obtained for SL, SRK-WS, and SRK-LCVM at low pressures. Since the main difference between the two separators is the pressure, it is possible to evaluate the pressure influence on the parameters using this procedure. Figures 21-23 show the results for HPS composition for LDPE resin obtained with SRK-LCVM, SRK-WS, and SL, with parameter values estimated at high and low pressures. In Figure 21, the half under the diagonal repeats the same values plotted for LDPE composition in Figure 15. In a fixed LDPE composition, the distance between the low and high pressure predicted points indicates the degree of pressure dependence of the parameters. The points in the upper portion of Figure 21 are closer together, suggesting that they are more independent of pressure. The results presented in Figure 21 show that SRKLCVM gives an overprediction of the HPS bottom stream composition if low-pressure parameters are used to estimate high-pressure behavior. This same procedure is applied to SRK-WS, and the results are shown in Figure 22. The half under the diagonal repeats the same values plotted for the LDPE composition in Figure 16. A comparison of the results from Figures 21 and 22 shows that the SRK-WS parameters are more pressure dependent than the SRK-LCVM parameters. This same procedure was applied to the SL EOS, and the results are shown in Figure 23. Its half under the diagonal repeats the same values plotted for LDPE composition in Figure 11. A comparison of the results from Figures 21-23 indicates that the SL equation is the most pressure-dependent model. Binary interaction parameters of equations of state are usually estimated considering only temperature dependence. For polymer systems, parameter dependence on mixture composition cannot be neglected and should take into account the segment interactions and their molecular displacements. However, a


8625

Figure 17. IPS overhead compositions for LLDPE resin: experimental and predicted data by SL EOS.

Figure 18. IPS bottom compositions for LLDPE resin: experimental and predicted data by SL EOS.

typical problem that appears is the choice of reference composition (feed, top, or bottom). As the feed composition of the separator changes less than that of top and bottom, it is more consistent to assume that the parameters depend on feed composition. Wolfarth and Ra¨tzsch42 present a detailed discussion of the dependence of interaction parameter on temperature, polymer concentration, and molecular weight distribution. Buchelli et al.20 obtained the pure component and binary interaction parameters of the PC-SAFT model based exclusively on experimental data published in the open literature. However, a better alternative is to use plant data instead of literature data, and the results are quite good. To illustrate this, we suggest a

comparison of Figure 14 in this paper with Figure 8 from Buchelli et al.20 Note that our study describes HPS behavior more accurately. Buchelli et al.20 perhaps chose the nonequilibrium method due to the poor results they obtained using binary interaction parameters from literature data. 5. Conclusions To simulate the flash HPS, LPS, and IPS separators in the industrial polyethylene process, low-density polyethylene (LDPE) and linear low-density polyethylene (LLDPE) resin industrial plant data are used to test the Sanchez-Lacombe (SL) and

8626


Figure 19. IPS overhead compositions for LLDPE resin: experimental and predicted data by SRK-WS EOS.

Figure 20. IPS bottom compositions for LLDPE resin: experimental and predicted data by SRK-WS EOS.

Soave-Redlich-Kwong (SRK) EOS performance. For the SRK EOS, three different mixing rules are compared: the van der Waals (VDW) one-fluid, the LCVM, and the Wong-Sandler (WS). The Bogdanic and Vidal activity coefficient model is used coupled with LCVM and Wong-Sandler mixing rules. To estimate the pure polymer parameters in the SRK equation of state, the Kontogeorgis procedure is used coupled with the GCVOL method. The results for the LPS separator indicate that SL and SRKVDW are the best models. This conclusion is based on the comparison of predicted and experimental data for mass composition, and also on the constant values estimated for the

interaction parameters of these equations. It is important to point out that SRK-LCVM results are based on a realistic description of mass composition with a reasonable correlation for parameter λ. The results for the HPS separator indicate that the SL EOS is the best model. Similarly, this conclusion is based on the comparison of predicted and experimental data for mass composition as well as the constant values estimated for the interaction parameters of this equation. SRK-LCVM and SRKWS are unable to describe the HPS overhead composition because the results indicate the presence of only ethylene in this stream. Regarding the LDPE composition in the bottom


Figure 21. HPS bottom composition for LDPE resin: experimental and predicted data by SRK-LCVM EOS with parameter λ estimated from low and high pressure plant data.

Figure 22. HPS bottom composition for LDPE resin: experimental and predicted data by SRK-WS EOS with parameter WSij estimated from low and high pressure plant data.

8627

F ) mass flow rate GCVOL ) group contribution volume model HDPE ) high-density polyethylene HPS ) high-pressure separator IPS ) intermediate-pressure separator kB ) Boltzmann constant LCVM ) linear combination of the Vidal and Michelsen mixing rule LDPE ) low-density polyethylene LLDPE ) linear low-density polyethylene LPS ) low-pressure separator M ) molecular weight Mw ) average molecular weight (mass) Mn ) average molecular weight (number) nseg ) total number of segments (eq 24) nc ) number of components OF ) objective function p ) polymer chain length P ) pressure PDI ) polydispersity index R ) gas constant SL1 ) Sanchez-Lacombe binary interaction parameter (eq 8) SL2 ) Sanchez-Lacombe binary interaction parameter (eq 9) SR1 ) binary interaction parameter with van der Waals mixing rule (eq 13) SR2 ) binary interaction parameter with van der Waals mixing rule (eq 14) T ) temperature UQ* ) UNIQUAC interaction parameter in WS mixing rule (Table 7) UQ ) UNIQUAC interaction parameter in LCVM mixing rule (Table 7) V ) molar volume VW ) van der Waals volume VLDPE ) very low density polyethylene w ) weight fraction WS ) binary interaction parameter with Wong-Sandler mixing rule (eq 17) x, y, z ) molar fractions Greek Letters

Figure 23. HPS bottom composition for LDPE resin: experimental and predicted data by SL EOS with parameters estimated from low and high pressure plant data.

stream, SRK-LCVM and SRK-WS give similar results but are not as accurate as the SL EOS. Acknowledgment The financial support of Fundaçaõ de Amparo a` Pesquisa do Estado da Bahia (FAPESB) and Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) is gratefully acknowledged.

R ) correlation parameter (Figures 4, 7, and 8) β ) correlation parameter (Figure 3) δ ) LCVM mixing rule parameter (eq 18) ε ) pure component parameter Sanchez-Lacombe equation λ ) LCVM mixing rule parameter (eq 18) F ) density υ ) number of segments in the component φ ) segment fraction of component (eq 7) Superscripts calc ) calculated ∼ ) reduced properties ′ ) mass composition in the heavy phase ′′ ) mass composition in the light phase exp ) experimental FV ) free volume RES ) residual

List of Symbols a ) Soave-Redlich-Kwong attraction parameter (eq 10) b ) Soave-Redlich-Kwong covolume parameter (eq 10) BV ) Bogdanic and Vidal activity coefficient method c ) constant with Wong-Sandler mixing rule (eq 15) d, e, f ) constants of eq 31

Subscripts Et ) ethylene HPS ) HPS i ) component i j ) component j

8628


k, m ) segments lb ) lower bound LPS ) LPS M ) Michelsen mixing rule (eq 18) mp ) middle point pc ) pseudocomponent r ) number of molecular weight ranges s, t ) unknowns solved by the equality constraints (eq 30) ub ) upper bound V ) Vidal mixing rule (eq 18)

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ReceiVed for reView October 30, 2008 ReVised manuscript receiVed July 23, 2009 Accepted July 29, 2009 IE801652Q

Simulation of Flash Separation in Polyethylene Industrial Processing

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