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Modeling Polyethylene Fractionation Using the Perturbed-Chain Statistical Associating Fluid Theory Equation of State Eric L. Cheluget,*,† Costas P. Bokis,‡,§ Leigh Wardhaugh,† Chau-Chyun Chen,‡ and John Fisher† NOVA Chemicals Corporation, NOVA Research & Technology Center, 2928-16 Street N.E., Calgary, Alberta, Canada T2E 7K7, and Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141
This paper illustrates how polymer fractionation can be characterized using the perturbed chain version of the statistical associating fluid theory equation of state, a computationally efficient algorithm, and pseudocomponents. We reparametrize the PC-SAFT equation of state and propose an efficient method of generating pseudocomponents to describe polydisperse polymer molecular weight distributions from the results of size exclusion chromatography analysis by numerical integration. We perform rigorous phase equilibrium calculations to investigate the effects of temperature, pressure, and feed composition on polyethylene fractionation. In these calculations, the polyethylene molecular weight distribution is treated as an ensemble of pseudocomponents consisting of chemical homologues of varying size. The simulation results are compared with plant data from an industrial solution polymerization process using cyclohexane as the solvent to demonstrate the applicability of the method to an industrial situation. The results indicate versatile representation of the polymer polydispersivity as well as accurate prediction of the liquid-liquid, vapor-liquid, and fluid-liquid fractionation process. 1. Introduction In this investigation, we examine several factors influencing the phase behavior of polyethylene (PE) produced by the solution process. This process is mainly used to manufacture high-density and linear low-density polyethylene (HDPE and LLDPE, respectively). In polyolefin solution polymerization processes, ethylene is polymerized in a solvent medium such as cyclohexane.1,2 The monomer, comonomer, and catalyst are dissolved in solvent and simultaneously injected into the reactor, where polymerization takes place under adiabatic conditions. Downstream of the reactor, the process stream is heated and discharged through a pressure let-down valve into a vapor-liquid flash unit. Prior to the pressure let-down, the polymer-containing process stream temperature sometimes exceeds its lower critical solution temperature3 (LCST) value, resulting in the formation of two liquid-like phases, between which the polymer is fractionated. One of the phases is rich in polymer, while the other is mostly solvent. If the temperature is a few degrees above the critical temperature of the solvent, then the solvent-rich phase is supercritical, and the solubility of polymer in this phase is a function of pressure.4 This phase separation phenomenon might be undesirable because the phase containing most of the polymer usually has a high viscosity, which might cause difficulties in controlling the process. This type of LCST-driven liquid-liquid phase separation has been experimentally characterized for several solvents in the laboratory.5-7 The thermo* Author to whom correspondence should be addressed: E-mail
[email protected]. Tel.: (403) 250-4535. Fax: (403) 250-0621. † NOVA Chemicals Corporation. ‡ Aspen Technology, Inc. § Current address: Exxon Mobil Research and Engineering, 3225 Gallows Road 7A-1730, Fairfax, VA 22037-0001.
dynamic driving force for the phase separation in these systems is the difference between the thermal expansion of the solvent and polymer as the solvent approaches its critical point, the so-called free volume effect.8 In the vapor-liquid flash separator unit operating below the critical point of the solvent, most of the unreacted ethylene and comonomer, such as 1-butene or 1-octene, as well as a significant portion of the solvent, leave as vapor. The liquid stream leaving this unit is a concentrated solution of polymer in solvent, with a small amount of monomer and comonomer. Depending on the operating conditions of this unit, lowmolecular-weight oligomers of the polymer might leave with the vapor stream. This happens because, at intermediate and high pressures, polymer components at the low end of the molecular weight distribution (MWD) curve are appreciably soluble in the gaseous solvent and are extracted from the resin product in the liquid.1 This fractionation might or might not be desirable, depending on the product being produced; at any rate, it is advantageous for process engineers to identify the degree of fractionation at a given temperature and pressure. Whereas recent calculations utilizing the Sanchez-Lacombe equation of state9 have identified the existence and mapped the boundaries of a three-phase liquid-liquid-vapor (VLL) coexistence region in similar systems containing polydisperse polyethylene,10 solvent, and some residual monomer, this phenomenon, which is restricted to a fairly narrow pressure range, is not investigated in this study and will be the subject of future calculations. This is because the investigation of this important region of the phase diagram requires multiphase equilibrium calculations involving a minimum of three simultaneously present phases. A typical phase diagram for a quasibinary polydisperse polymersolvent system with a small amount of ethylene is shown in Figure 1. The locations of the LCST curve as well as the VLL, LL, and VL regions are shown in the
10.1021/ie010287o CCC: $22.00 © 2002 American Chemical Society Published on Web 09/22/2001
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Figure 1. Illustration of the phase behavior of a polymer solution displaying LCST behavior. This curve is typical of a 5-15 wt % polydisperse polyethylene solution with a small amount of ethylene present. The arrows represent the solution polymerization process trajectory.
diagram. Note that, below the solvent’s critical temperature, the polymer-dilute phase in the LL region is a liquid, whereas slightly above the critical temperature, it is a fluid phase (FL). Hence, accurate modeling of these systems requires calculation of vapor-liquid equilibria (VLE), liquid-liquid equilibria (LLE), fluidliquid equilibria (FLE), and vapor-liquid-liquid equilibria (VLLE). Additionally, the precipitation temperature of polymer in solution is an important variable, and its determination requires solid-liquid equilibrium calculations (SLE). This study examines all but the latter two of these equilibria. Although polymers produced in the solution process are usually copolymers of ethylene with either 1-butene, 1-octene, or possibly 1-hexene, the present study models all polyethylene as linear polyethylene, and no account is taken of the degree of branching. The phase behavior of polyethylene is affected by the degree of branching11-14 although, for hydrocarbon solvents with six and more carbon atoms, this effect is of minor importance.15 In this study, we couple a rigorous thermodynamic model based on statistical mechanics with an efficient computer algorithm16 and the use of pseudocomponents to investigate liquid-liquid and vapor-liquid polymer fractionation in the polyethylene solution process. Initially, a brief description of the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EOS) is given. This is followed by a description of how the values of parameters in the PC-SAFT EOS were determined. The next section examines the use of pseudocomponents for characterizing the composition of polydisperse polymer. Results of calculations performed to investigate the effects of operating parameters such as temperature and pressure on the degree of fractionation in liquid-liquid and liquid-vapor systems are then reported. The final section of the paper is devoted to the application of the fractionation model to a vapor-liquid separator in an actual plant, with modeling results being compared to plant data.
2. The Perturbed-Chain Statistical Associating Fluid Theory Equation of State The PC-SAFT EOS17 is an improved form of the original SAFT EOS.18-20 Over the past 10 years, there have been many modifications of the now-ubiquitous SAFT EOS. Some of modifications have resulted in a reduction in the complexity of the equation21 with accompanying improvements in its accuracy. Others have improved the accuracy of the EOS in the nearcritical region.22-24 Yet others have generalized the basis of the model.25,26 In this study, we use the PC-SAFT modification, which distinguishes itself in that the dispersion term, described below, is based on the real properties of n-alkanes, allowing the equation to better represent data for hydrocarbon fluids. The original SAFT EOS is based on the perturbation theory of classical statistical mechanics27,28 where the unknown properties of a real fluid are obtained as a perturbation of the properties of a model system for which molecular properties can be calculated. In the SAFT EOS, the reference fluid is a chain of hard spheres. In this approach, the Helmholtz energy of the system, A, is given by
A ) Aid + Ahc + Apert
(1)
where Aid, Ahc, and Apert are the ideal gas, hard-sphere chain, and perturbation contributions to the Helmholtz energy, respectively. In the original SAFT EOS, the properties of real interacting chain fluids were obtained as sums of terms accounting for ideal gas behavior, Aid; the repulsive energy of noninteracting hard-spheres, the energy of forming chains from these spheres through covalent bonding, and the energy of forming clusters (associations) via, say, hydrogen bonding, with all three constituting Ahc; and the attractive (dispersion) energy due to interactions between isolated spheres (based on the real behavior of argon), Apert ) Adisp. The systems examined in this study do not exhibit association, and
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hence, this term is ignored. Details on the hard-sphere, chain connectivity, and association models used in SAFT can be found in papers by Chapman et al.18 and Huang and Radosz.19,20 These same models are used in PCSAFT to calculate the Ahc contribution. The difference between the two equations (SAFT and PC-SAFT) lies in the perturbation, or dispersion, term, which, instead of being based on the properties of argon,29 a fluid consisting of single spheres, is based on actual bonded-sphere “chain” fluid behavior. The perturbation contribution is obtained from Barker and Henderson’s second-order perturbation theory as a sum of first- (A1) and second- (A2) order terms
A1 A2 Adisp ) + RT RT RT
(2)
The Barker and Henderson theory was applied to a system of chain molecules interacting via a square-well potential using Chiew’s30 expression for the radial distribution function of interacting hard chains. The resulting two integrals of the radial distribution function in A1 and A2 were approximated using a temperatureindependent power series ranging to sixth order in reduced segment density and dependent on the number of segments in the molecule. The constants in this series were determined by fitting pure-component data for n-alkanes. First, the integrals were fully evaluated for Lennard-Jones chains, and n-alkane vapor pressure and density data were regressed to obtain the values of three pure-compound parameters m, σ, and /k (the segment number, segment diameter, and reduced interaction energy, respectively). Once this was done, the constants in the polynomial expressions for the integrals were evaluated for n-alkanes from methane to triacontane using vapor pressure and liquid, vapor, and supercritical density data. Hence, the PC-SAFT equation reproduces n-alkane data very accurately. For applications to other fluids, the constants in the expressions for the integrals are set to the values optimized for n-alkanes and, hence, are assumed to be universal for all compounds. The values of the other three pure-compound parameters, that is, the segment number m, the segment diameter σ, and the reduced interaction energy /k, are then fit to pure-compound data for the compound of interest. The EOS was extended to mixtures using the usual one-fluid mixing rules with a binary interaction parameter kij to correct the segment-segment interaction energies of unlike chains. Comparison to the original SAFT equation of state reveals improvement for both mixtures of small components and polymer solutions.17 The following section describes the approach taken in determining the values of the pure-compound parameters for the compounds of interest in this study. The PC-SAFT EOS was implemented in the Polymers Plus process simulation package. The fractionation calculations described in this study, some of which utilize hundreds of pseudocomponents were, however, performed using a versatile and efficient phase equilibrium computation code supplied by Aspen Technology. The code was originally developed at the Technical University of Berlin under the direction of Professors W. Arlt and G. Sadowski. The code accepts the composition in the form of discrete weight fractions corresponding to polymer pseudocomponents and conventional components.
Table 1. Pure-Component Parameter Values Used for Conventional Components in the PC-SAFT EOSa parameter ethylene 1-butene cyclohexane MW m σ /k a
28.05 1.5399 3.4470 180.68
56.11 2.3000 3.6392 221.27
84.16 2.4694 3.8679 282.29
polyethylene m/MW ) 2.63 × 10-2 4.0218 252.0
Polyethylene parameter values were determined by Gross.34
3. Determination of Values of Parameters in the PC-SAFT EOS Pure-Compound Parameters. Parametrization of the EOS is an important aspect of phase equilibria modeling, if accurate predictions are to be made. The compounds of interest here are ethylene, cyclohexane, 1-butene, and polyethylene. Although Gross and Sadowski17 have determined pure-compound parameters for many compounds, including those above, there are other compounds of interest that do not have listed parameters. As a result, a standardized method for determining the values of pure-compound parameters for conventional components with the PC-SAFT EOS was developed and used to determine new values of parameters as part of this study. For these conventional components, as in Gross and Sadowski,17 vapor pressures and saturated liquid density data were used to determine the values of parameters. To determine the values of parameters on an even basis, these data were used in the temperature range 0.5 < TR < 1.0, where TR is the reduced temperature, even for the case of ethylene, which is supercritical under normal industrial conditions. Initial studies with a few compounds involved multiproperty regression in which saturated liquid, and in some cases supercritical, heat capacities were used to determine the values of the parameters. This had only a small impact on the accuracy and was not deemed necessary. The vapor pressure “data” were actually the predictions of an accurate extended Antoine equation available in Polymers Plus, whereas the saturated liquid density data were obtained from the Design Institute for Physical Property Data (DIPPR) compilation.31 The optimal values of the parameters determined by this method are listed in Table 1. Figure 2 compares the PC-SAFT vapor pressure correlation with experimental data for 1-butene. As reported by Gross and Sadowski,17 the PC-SAFT EOS is able to correlate vapor pressures accurately. The trend is similar for other compounds. Figure 3 compares the predicted and experimental saturated liquid densities for cyclohexane. Values are also calculated at higher pressures and compared with the Aalto and Keskinen32 model, which predicts liquid densities to 8000 bar and TR ) 0.995. Accurate liquid and vapor densities are often required for the conversion of flow rates from a volumetric to a mass basis in operating plants. Values of parameters determined using saturation data are good for predicting compressed liquid densities to about 300 bar; above this pressure, the accuracy begins to deteriorate. Improvements in the accuracy of the highpressure data could probably be made by including these data in the parameter regression. As a matter of fact, if liquid density data for a compound are scarce, the COSTALD liquid density model16 could be used within a simulator to generate data, as with the extended Antoine equation. A similar trend was observed for the other compounds, including ethylene. Because the critical temperature of ethylene is only 9.2 °C, the prediction
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Figure 2. Comparison between PC-SAFT EOS vapor pressure prediction and experimental data.
Figure 3. Comparison of PC-SAFT saturated and compressed liquid densities with experimental data and the predictions of the Aalto and Keskinen32 model for cyclohexane.
of supercritical densities based on parameter values determined from subcritical data was examined by comparing model predictions to the data of Sychev et al.,33 which are based on the predictions of a complex multiparameter EOS developed specifically for ethylene. The comparison is made in Figure 4 for the temperature range of 25-300 °C and pressure range of 60-300 bar. The results indicate the accurate prediction of these densities. The ability of the model to predict calorimetric properties based on parameter values determined using pressure-volume-temperature (PVT) information was also tested. The prediction of liquid heat capacities is examined in Figure 5, where predicted values for saturated liquid are compared with experimental values for cyclohexane. Except for the critical region, the results show an accurate prediction, which was also
observed for all conventional compounds. Moreover, a comparison with the Peng and Robinson35 EOS shows that PC-SAFT is more accurate, particularly at lower temperatures. The heat capacity predictions for supercritical ethylene based on the parameter values obtained under subcritical conditions are compared with Sychev et al.’s33 data in Figure 6. The results indicate that the model predicts heat capacities for ethylene quite accurately in the temperature range 25-300 °C and pressure range 60-300 bar, although the peak at ∼100 bar and 50 °C is slightly underestimated. Finally, the predictions of heat of vaporization by the PC-SAFT and Peng-Robinson EOS are compared to experimental data, and the case of cyclohexane is displayed in Figure 7. For the PC-SAFT EOS, the data are predicted accurately at all temperatures except for values near the critical point. The Peng-Robinson EOS
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Figure 4. Comparison of PC-SAFT predictions and Sychev et al.33 supercritical ethylene density data.
Figure 5. Comparison of experimental data and PC-SAFT predictions of the saturated liquid heat capacity for cyclohexane.
predicts accurate values in the critical region and less accurate values at low temperatures. This behavior was also observed for other conventional compounds. The accuracy of the heat of vaporization and heat capacity at high temperature is intimately tied to the location of the vapor-liquid critical point predicted by the equation of state (at which point the heat of vaporization vanishes and the heat capacity diverges). This temperature is often overpredicted by both the original SAFT EOS and the PC-SAFT EOS (and all other unrescaled mean-field EOSs) when parameterized in the manner described here.22 The discrepancy between the PC-SAFT EOS-predicted critical temperature and the true value is, however, smaller than typical of the original SAFT EOS. This error could be eliminated by rescaling the equation to better predict the true critical point.24,36
In terms of the pure-compound parameters for polyethylene, experience with the original SAFT EOS indicates that the EOS can represent the PVT data of molten polymers accurately. As with other EOSs, such as the original SAFT EOS and the Sanchez-Lacombe EOS,9 however, the use of polymer pure-compound parameter values determined from PVT data often results in poor representations of binary or quasibinary phase equilibria between the polymer and low-molecular-weight solutes. For example, Koak et al.37 using the original SAFT EOS found that they had to adjust the value of the energy parameter to obtain a reasonable fit to binary polyethylene-ethylene FLE cloud-point data. Calculations performed by Gross34 indicate that the PC-SAFT EOS suffers from the same deficiency and that the use of values of parameters determined from
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Figure 6. Comparison of PC-SAFT predictions and Sychev et al.33 supercritical ethylene heat capacity data.
Figure 7. Comparison of PC-SAFT and Peng-Robinson EOS predictions of the heat of vaporization and experimental data for cyclohexane.
molten polymer PVT data alone does not result in good representations of binary phase equilibrium data for polyethylene-ethylene. Gross obtained a set of purecompound parameters for polyethylene by adjusting the value of the energy parameter and the binary interaction parameter to simultaneously fit molten polymer PVT data38 and cloud-point data collected in the polydisperse polyethylene-ethylene FLE study of de Loos et al.39 The density values for molten polymer predicted by the PC-SAFT EOS using Gross’s parameter values are compared with experimental data for LDPE in Figure 8. The fit is worse when compared for HDPE40 or LLDPE.41 The data are not well represented by the parameter values at high pressure. Because the molten polymer is relatively incompressible, however, the overall error is relatively small and introduces little error
in the solution for low (2, whereas PE7 is a narrowly distributed polymer with M h W ≈ 8.8 × 103 and a polydispersity index of 1.16. These pseudocomponents were used, along with the independently determined temperature-dependent kij of Table 2, to calculate isotherms for the two polymers with ethylene at 130 and 170 °C, the lowest and highest temperatures for which experimental data exist. The results of these calculations for the PE1 and PE7 systems are compared with data in Figures 18 and 19, respectively. Both the cloudand shadow-point curves were calculated for the systems. The intersection of the calculated cloud- and shadow-point curves is the location of the predicted critical point. For definitions of these terms and other details of polydisperse polymer LLE and FLE behavior, see Casassa71 or Folie and Radosz.3 As expected, the cloud-point curve for PE1, with the higher molecular weights and broader distribution, is located at higher pressures, is more skewed, and has the critical point located on the right-hand branch. The predictions of the PC-SAFT model with an independently determined value of kij are reasonably accurate when the absolute magnitude of the cloud-point pressure is taken into account. The EOS predictions are,
Figure 17. Weight fraction and molecular weights of pseudocomponents used to represent the de Loos et al.39 polymers.
however, more accurate at high temperature than at low temperature, with the discrepancy at 130 °C being larger than 100 bar for low polymer concentrations. In addition, the cloud-point pressures at concentrations below the critical point are underestimated, whereas cloud-point pressures at concentrations above the critical point are overpredicted. Similar phenomena can be seen in Figure 19, where the isotherms are at lower pressures, broader, and less skewed, as expected for a polymer of lower molecular weight and narrower distribution. The fit at low polymer concentrations is much better in this case. In general, however, the fits are inferior to modeling efforts in which both the polymer pure-component data and temperature-dependent kij are simultaneously adjusted.9,37 With respect to PC-SAFT, additional accuracy can be obtained by further adjusting the values of the temperature dependence of kij and the pure-compound energy parameter of polyethylene to simultaneously match PVT data for the type of resin of interest (HDPE, LLDPE), the binary monodisperse FLE data of Chun Chan et al.,14 and the polydisperse FLE data of de Loos et al.39 Polyethylene-Cyclohexane. Isotherms were also calculated for the polydisperse polyethylene-cyclohexane quasibinary system to examine the nature of the two phases formed in this system. The polymer molecular weight distribution was specified as the PE1 distribution. Two sets of calculations were performed. The first isotherm was calculated at a temperature of 270 °C, at which pure cyclohexane is subcritical, as is evident in Figure 15. In this case, there is a region of VLE at low pressures (Figure 1) and a LLE region at higher pressures. The solvent-rich phase formed under these conditions is a liquid. The second set of calculations was performed at 300 °C, which is above the critical temperature of cyclohexane. Under these conditions, an almost pure solvent phase is supercritical, and
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Figure 18. Comparison between experimental (de Loos et al.39) and PC-SAFT calculated isotherms for the ethylene-polyethylene (PE1) system at 130 and 170 °C.
Figure 19. Experimental (de Loos et al.39) and PC-SAFT calculated isotherms for the ethylene-polyethylene (PE7) system at 130 and 170 °C.
hence, this situation corresponds to FLE. Unfortunately, there are no cyclohexane data with which to compare the results of the calculations. The results of both sets of calculations are displayed in Figure 20, and they indicate that, once the pressure falls below the cloud-point pressure, the incipient solvent-rich phase contains virtually no polymer, regardless of the pressure and temperature and regardles of whether the solvent-rich phase is supercritical or subcritical. For the 270 °C calculation, the light-phase polymer concentration does not change much as the second phase is converted from liquid to vapor with a lowering of pressure. The line separating the LLE and VLE regions is drawn as an invariant pressure line, which is only the case for true binary systems composed of mondisperse polymer and solvent (because of Gibbs
phase rule); in reality, there is a (very) small threephase region here for broadly distributed polymer. If ethylene, 1-butene, or 1-octene is present, then this three-phase region is large, sometimes greater than 10 bar.10 For subcritical conditions (270 °C), there is a sharp kink separating the VLE and LLE regions, and for the case of FLE at 300 °C, there is still a kink, although diffuse. Compared to the ethylene-polyethylene case, the differences between the cloud- and shadow-point curves are not as pronounced, although the precipitation threshold (maximum in the cloud-point curve) is located at very low concentrations, below 1% for PE1, and the critical point is located on the righthand branch at about 3 wt % or so. The shapes of the calculated cloud-point curves are very similar to those measured for LLDPE in n-hexane by de Loos et al.15
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Figure 20. PC-SAFT calculated isotherms for PE-cyclohexane at sub- and supercritical temperatures (for pure cyclohexane).
Figure 21. PC-SAFT calculated isotherms for PE-1-butene at sub- and supercritical temperatures (for pure 1-butene).
The next section examines polyethylene MWD curves in equilibrated liquid and vapor phases. Polyethylene-1-Butene. A similar set of calculations was performed for the 1-butene-polyethylene binary system, with isotherms being obtained at suband supercritical conditions relative to the critical point of 1-butene. The calculations were performed at 130 and 200 °C (see Figure 16). As with cyclohexane, PE1 was chosen, because it has a high molecular weight and broad distribution, to emphasize the effects of polydispersity. The results of the calculations are displayed in Figure 21. Because the 1-butene molecule is smaller, the isotherms are located at higher pressures as a result of greater molecular size asymmetry in the system. Otherwise, the trends are very similar to those in the cyclohexane case, and the general conclusions remain
the same. The LLE and FLE regions are larger, however, because of the increased density difference between 1-butene and polyethylene. Fractionation in an Industrial Process. This section describes the modeling of polyethylene fractionation in the flash unit of a plant producing polymer using the SCLAIRTECH process.2 The unit effects vapor-liquid separation of subcritical solvent and residual monomer and comonomer from polymer. The initial case modeled involves operation of this unit at 267 °C and 33 bar. The feed contains 16.32 wt % polyethylene, 1.66 wt % ethylene, and 5.7 wt % 1-butene. Samples of polymer leaving in the liquid stream from this unit and of “grease”, consisting of low-molecularweight oligomeric material dissolved in the vapor, were collected at a plant during normal operation. These
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Figure 22. Results of plant flash simulation with the PC-SAFT calculated MWD curve of the liquid and vapor phases compared to plant data in terms of pseudocomponents. Table 3. Overall Results of the Flash Calculation liquid composition (wt %) compound
plant
PC-SAFT calculated
polyethylene ethylene 1-butene cyclohexane
44.62 0.74 4.48 50.15
50.8 0.21 1.44 47.6
vapor composition (wt %)
Table 4. Polymer Molecular Weight Averages in Equilibrated Liquid and Vapor Phases liquid phase
vapor phase
plant
PC-SAFT calculated
molecular weight average
plant dataa
PC-SAFT calculated
plant datab
PC-SAFT calculated
0.25 2.18 6.41 91.39
0.05 2.34 7.72 89.9
MN MW MZ polydispersity index
13 200 126 000 695 800 9.6
11 323 126 592 710 230 11.2
140 170 220 1.21
164 190 222 1.16
samples were analyzed using SEC to determine MWD curves. Using the results of the SEC analysis and a material balance, the MWD curve of material entering the flash unit was reconstructed. This reconstruction involved some assumptions regarding the material balance, particularly the flow rates of the components, because of lack of plant measurements and difficulties with sampling and analytical techniques. Nonetheless, it represents the first attempt at experimentally characterizing fractionation in an operating plant. Material down to very low molecular weight, on the order of the solvent’s, was also included to investigate the effects of fractionation at low molecular weight. This distribution was discretized to yield ∼250 pseudocomponents using the techniques described previously and treated as the polymer feed in fixed-temperature and -pressure flash calculations. Table 3 compares the results of the flash calculations with plant data. Considering the uncertainty in accurately measuring mass flow rates at the plant, it is apparent that the PC-SAFT modeling provides a reasonable prediction of the data. The predicted vapor fraction is 68 wt %. Figure 22 shows a comparison between calculated and measured MWD curves for the grease and resin. There is some scatter in the reconstructed original feed distribution, and this is reflected in the vapor phase distribution. The plant data represent pseudocomponents generated from SEC data using numerical integration techniques. The PC-SAFT calculations provide an accurate characterization of the fractionation process, although there is some error in the middle molecular weight range, as is evident from a comparison of the calculated and measured average
a
Based on resin SEC analysis. b Based on grease SEC analysis.
molecular weights listed in Table 4. The main discrepancy is in the number-average molecular weight, which is sensitive to the low-molecular-weight end of the SEC data. Additional calculations were performed with this system to examine the effect of pressure and temperature on the molecular weight distribution of the polymer in the light phase. First, the LL boundary for the feed system going into the flash unit was calculated, as displayed in Figure 23. Also shown is the LL boundary for a quasibinary solution of polymer in cyclohexane at the same polymer concentration. The effect of the presence of 1-butene and ethylene is to increase the size of the LLE region (by ∼40 bar) and to raise the bubble point, or vapor pressure line. A flash calculation was performed at 84 bar, less than 1 bar into the LLE region, and the results of the overall LLE splits were very similar to the VLE case at 33 bar, with the only significant change being that the light liquid phase now contains 0.77 wt % polyethylene (versus 0.05 wt % for VLE at 33 bar, Table 4). In this region, the solubility of the polymer is increased by the higher pressure. The molecular weight distributions in the feed, liquid, and vapor phases are shown in Figure 24. The increased pressure results in the dissolution of higher-molecular-weight material in the light phase, so that the number- and weight-average molecular weights increase to 786 and 2320, respectively. Additional calculations indicate that the nature of the light phase does not change significantly as it changes from a liquid
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Figure 23. Effect of ethylene and 1-butene on LL phase boundary of polydisperse polymer in cyclohexane.
Figure 24. MWD curves in equilibrated LL phase at 267 °C and 84 bar.
to a vapor phase, which occurs at a pressure of approximately 55 bar for this mixture. Similarly, the difference in going from the LLE region to the FLE region is not dramatic. A set of flash calculations was performed with the feed described above to examine in detail the behavior of the equilibrated phases as the pressure is lowered from the cloud-point value of ∼84.5 bar to pressures as low as 1 bar in the VL region. As the pressure falls below the cloud-point value, the new liquid phase generated contains very little polymer,
