Ind. Eng. Chem. Res. 1993,32, 1442-1448
1442
Phase Equilibria of Binary and Ternary n-Alkane Solutions in Supercritical Ethylene, 1-Butene, and Ethylene 1-Butene: Transition from Type A through LCST to U-LCST Behavior Predicted and Confirmed Experimentally
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Christopher J. GreggJ Fred P. Stein,+Shen-jer Chen? and Maciej Radosz*** Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, and Exxon Research & Engineering Company, Annandale, New Jersey 08801 Binary phase equilibria of CZO+, c 3 0 +, and c40 + ethylene are correlated with three equations of state, the Redlich-Kwong-Soave (Soave), the generalized van der Waals-Prigogine (vdW-P), and the statistical associating fluid theory (SAFT), with special emphasis on the mixture critical region. The SAFT correlation is found to be the most reliable. SAFT predictive powers are tested on cZ6 ethylene and a new set of experimental data taken in this work for c36 1-butene and c36 ethylene 1-butene. T h e SAFT predictions are found to be in good agreement with the experimental data. Equally important, the SAFT predictions allow for understanding the continuous transition of phase behavior, from type A ( c 3 6 + 1-butene) through LCST (c36 ethylene 1-butene) to U-LCST (c36+ ethylene).
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Introduction Normal alkanes in the CZOto Cqg range form miscible solutions with olefins at elevated temperatures and pressures. Such solutions are prototypes of commercial monomer-comonomer solutions of small polyethylene or wax molecules. Hence, it is important to understand their phase behavior. Toward this end, de Loos et al. (1984, 1986) measured the phase equilibria of ethylene solutions of n-eicosane (CZO), n-hexacosane ( C d , n-triacontane (c30), and n-tetracontane (c40) and correlated the data with the Peng-Robinson equation of state. In spite of having to use two binary interaction parameters, they found that the Peng-Robinson correlation was still poor at pressures near the mixture critical region. The goal of this work is to understand the phase behavior of ternary systems containing one large n-alkane and two olefins in the mixture critical region. The approach is to take only a few phase equilibrium data and to use these data to test existing equations of state. For example, we test one typical cubic equation of state (Soave, 1972) and one cubic equation of state applicable to polyethylene systems (Sako et al., 1989), referred to as the van der Waals-Prigogine (vdW-P). Since we prefer to minimize the experimental effort, we also select a theoretically based equation of state, such as the statistical associating fluid theory (SAFT), that is likely to have better predictive powers. In addition, the SAFT pure component parameters are well behaved and easy to estimate for large molecules (Huang and Radosz, 1990). Experimental Section Phase boundaries of binary and ternary mixtures of n-hexatriacontane (C36) in ethylene and in 1-butene are measured up to 200 "C in a variable volume optical cell. The unit layout, apparatus, and experimental procedure are given by Chen et al. (1993) so only a simplified flow diagram (Figure 1) and an overview are provided here. The cell is equipped with a borescope for visual observation of phase transitions and phase disengagement,
* Author to whom correspondence should be addressed. + Lehigh
University.
* Exxon Research & Engineering Co.
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ll
. I
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'm-I
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r
1
I I Device 1 I I I I Sampling
-
-
_
Q Syrlnge
Figure 1. Simplified diagram of the batch cell showing the cell with a movable piston, boroecope, video camera, high-temperature oven, pressure transducer, and high-pressure syringe pump.
a movable piston for pressure control at constant feed composition, and a magnetic stirring bar (not shown in Figure 1) for enhancing mass transfer. In operation, the cell is loaded and the contents are pressurized into a onephase homogeneous solution by advancing the piston. Once equilibrated, the solution is slowly depressurized at constant temperature. The pressure at which a second phase appears is recorded as either a bubble- or a dewpoint pressure depending upon the formation of a bubble (bubble point) or dew (dew point). At the completion of the experiment, the cell contents are repressurized into the one-phase homogeneous region and vented through a low-volume chromatographic valve and a trap filled with glass wool. The heavy alkane component precipitates in the trap while the solvent gas is collected in a graduated flask. For the ternary system, the collected gases are subsequently analyzed by gas chromatography or mass spectrometry. The overall feed composition is determined from the material balance.
Materials Ethylene (polymer grade, minimum purity 99.9 % 1, 1-butene (CP grade, minimum purity 99.0%), and n-
0 1993 American Chemical Society 088S-5885/93/2632-1442$04.0~/~
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1443 Table I. SAFT Parameters Used in This Work molar uoo component maas (g/mol) rn (cms/(molseg)) 18.15 1.46 ethylene CzH4 28.05 13.15 3.16 butene C4H8 56.10 12.00 13.94 eicosane C f i ~ 282.55 12.00 366.63 17.94 hexacosane C&u 20.57 12.00 triacontane C&z 422.73 12.00 506.99 24.44 hexatriacontane C&I,d 12.00 tetracontane C&Iaz 562.97 26.84
uolk
(K)
212.06 202.49 211.25 211.25 211.25 211.25 211.25
hexatricontane (minimum purity 98 % ) were purchased from Matheson and Aldrich, respectively. The chemicals were used without further purification.
Equations of State The SAFT model is applicable to small and large, chain, and associating molecules. The SAFT fluid can be viewed as a collection of equisized hard spherical segments that not only are exposed to mean field (dispersion) forces but can also be covalently bonded to form chains and weakly bonded to form short-lived clusters. SAFT is developed in terms of the residual Helmholtz (area)energy = aref
ares
+ adisp
(1) where adiap accounts for the dispersion forces and the Helmholtz reference contribution (are?is the sum of hardand association contributions sphere (a”), chain (achain), (CPW):
pf=
+ achain + a a a a ~
(2) Since the molecules used in this work do not exhibit specific interactions that lead to association, a ~ sisWset equal to zero. Each nonassociating pure substance is characterized by three molecular parameters: urn, temperature-independent segment volume in cm3 per mole of segments; m segment number; and uO/k, segment energy (k E Boltzmann constant), which are determined by fitting pure component vapor pressures and liquid densities. The parameters and their correlations with respect to molecular weight are given by Huang and Radosz (1990). The values used in this work are given in Table I. SAFT is extended to mixtures using the volume fraction mixing rules, proposed by Huang and Radosz (19911, which improve the correlation in the mixture critical region. These mixing rules are given below,
5 kT =
77 g] f$j[
(3)
where f is the volume fraction defined as
ximiuoi fi
(4)
=
Cxjmjuoj I
uij = (1- kij)(UiiUjj)1/2
m=
T,?,
xixjmij
1 mij = s(mi + mj)
I
I
,--.
250
(5) (6) (7)
where x i is the mole fraction of component i, uoi is the temperature-dependent segment volume, and kij is the binary parameter characterizing interactions between species i and j . The Soave equation of state (Soave, 1972) is applicable to mixtures of small components. The Soave pure
50
1
,:;g;et;. ---
I
I
(19*4;.‘y
: VdW-P
3 4;
4
0 0
0.20
0.40 0.60 0.80 C,, Weight Fraction
1.00
Figure 2. Pressure-concentration diagram for Cm + ethylene at 100 O C : experimentaldata by de Loos et al. (1984);curves calculated from SAFT, Soave, and vdW-P. Table 11. Critical Properties and Acentric Factors Used for Estimating the Soave Parameters molar comDonent mass (dmol) T.(OC) P,(bar) w ethyleneCzH4 28.05 9.33 50.400 0.089 eicosane C&42 282.55 493.89 11.100 0.907 639.03 8.726 1.445 tetracontane CmH82 562.97 ~~
component parameters (a and b), derived from vapor pressure data only, are correlated to the critical temperature, critical pressure, and Pitzer’s acentric factor. Hence, Soave is off with respect to liquid densities, including the critical density, but accurately accounts for the critical temperature and pressure. The critical properties and acentric factors for (220, (236, and (240,used in this work to estimate the pure component parameters, given in Table 11, are based on Teja et al. (1990) and Lee et al. (1975). The mixing rules used are the common van der Waals one-fluid mixing rules. The vdW-P model is a cubic equation of state extended to fluid mixtures containing polymers by Sako et al. (1989). vdW-P invokes Prigogine’s (1957) concept of factoring the partition function into internal and external rotational and vibrational degrees of freedom. This concept requires a third parameter, c, in addition to the Soave-like a and b, that is related to the number of external degrees of freedom. The pressure explicit form of vdW-P is given below:
RT(u- b + bc) -- ~ ( 7 ‘ ) (8) u(u - b) u(u + b) If c is equal to 1,as it does for small molecules, vdW-P reduces to Soave. The difference between vdW-P and Soave is 2-fold. The pure component parameters for vdW-P are defined on a per segment basis, while those for Soave are defined on a per molecule basis. The pure component parameters a and b for vdW-P are estimated from London formula for dispersion forcesand from the van der Waals volume rather than from the critical properties that are not available for polymer molecules. Like Soave, the vdW-P pure component parameters are extended to mixtures using the van der Waals one-fluid mixing rules, with one adjustable binary interaction parameter (kij) for aij. CalculationalMethod. Regardless of the equation of state used, phase equilibria are calculated by equating the
P=
1444 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
is.
4
0
0.20
0.40 0.60 0.80 C, Weight Fraction
I
I
1.oo
Figure 3. Pressure-concentration diagram for Ca + ethylene at 175 OC: experimental data by de Loo8 et al. (1984);curves calculated from SAFT, Soave, and vdW-P.
chemical potentials or fugacity and adjusting only one binary interaction parameter (kij) to fit the experimental data. The kij is allowed to vary with temperature and molecular weight but not with concentration. Although we are concerned with the overall quality of fit, our focus is on the representation of the mixture ciritcal region. Bubble- and dew-point calculations are performed by iterating on pressure using the Richmond algorithm proposed by Chen et al. (1992).
Results and Discussion Correlating C20 +, Cao +, and C40 + EthyleneBinary Phase Equilibria. Although 12isothermal data sets, four isotherms for each n-alkane + ethylene, reported by de Loos et al. (1984, 1986) are correlated with the three equations of state, only two sample results are presented to illustrate the fitting quality. The experimental and calculated (from fitting) data are shown in the form of pressure-concentration diagrams; for CZO+ ethylene at 100 "C in Figure 2, and for C a + ethylene at 175 "C in Figure 3. Figure 2 shows that a t low pressures (below 50 bar) all three equations are in reasonable agreement with the experimental data. However, at higher pressures, vdW-P usually overpredicts the mixture critical region and dew pressures and underpredicta the bubble pressures. Soave and SAFT, on the other hand, are in better agreement with the experimental data. As the n-alkane size and the temperature increase, the quality of fit for Soave begins to deteriorate. This is illustrated in a pressure-concentration diagram, shown in Figure 3, for C a + ethylene at 175 "C; Soave retains the characteristic flatness in the mixture critical region. However,its dew- and bubble-point representation a t lower pressures is poor. As in Figure 2, vdW-P is off near and away from the mixture critical region. On the other hand, SAFT calculated pressures are in reasonable overall agreement with the experimental data. Since SAFT also
0
0.20
0.40 0.60 0.80 C, Welght Fraction
'
D
Figure 4. Pressureconcentration diagram for Cm + ethylene at 125and 150O C : experimental data by de Low et al. (1984);calculated curves from SAFT with kij = 0.085.
has easier to estimate pure component parameters, we select SAFT as our working model. Our approach to generating phase diagrams for the CN and CB systems is, first, to develop an empirical kij correlation for CZO+, ($0 +, and CN + ethylene solutions and, next, to use this correlation to predicting the phase boundaries for cZ6 and c36 solutions with no additional fitting. Figures 4-6 illustrate additional SAFT correlated pressure-concentration diagrams for C20, C ~ Oand , CO that are used to fit the SAFT kij. These figures emphasize the mixture critical region, rather than the low-pressureregion where agreement between literature data and SAFT is excellent; the low-pressure predictions are typically less sensitive to kij. For clarity and breavity, only one lowand one high-temperature isothermal data set are plotted. In general, SAFT quantitatively captures the concentration effect as temperature and chain length increase, with an average absolute deviation from the experimental data below 7%. For CZO+ ethylene, the binary interation parameter is found to be constant, kij = 0.085. For Cm + and CN+ ethylene, on the other hand, the binary interation parameters are found to linearly depend on temperature; kij = 0.0791 + 3.60ebT and kij = 0.0368 + 1.63e4T (temperature in kelvin), respectively. To highlight the temperature and molecular weight effect, Figure 7 presents the experimental points and SAFT curves for CZO+, C30 +, and C a + ethylene solutions in pressure-temperature coordinates a t a constant feed concentration of 22.8wt % n-alkane. In general, the phase boundary pressure increases with increasing temperatures and molecular weight. Although some discrepancy exists at low temperatures and for high molecular weights, SAFT quantitatively captures these trends. For example, the largest deviation is observed for C a + ethylene a t 100 "C.
Ind. Eng. Chem. Res., Vol. 32,No. 7,1993 1445 I
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40
30C
!! g!
a !Pi 200
o : de Loos et ai. (1984, 1986)
100
-
o A : de Loos et ai. (1986)
-:
SAFF
0 I50
I
0
:SAn
0.20
I 0.40
I
I
0.80 0.80 CU Woight Fraction
I
1.1
Figure 5. Preseure-concentration diagram for Cw + ethylene at lOOand175OC experimentaldatabydeLoosetal.(1986);calculated curves from SAFT with k" = 0.0791 + 3.60e-V (K). I
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Temperatun,("C) Figure 7. Preseure-temperature diagrams of CW,(kij = 0.065), Cm and Cu,(kij 0.0368 + l.63e4T (K)) (kij 0.0791 + 3.60e4T (K)), + ethylene at a fixed concentration of 22.8 wt % n-alkane: experimental data by de Looe et al. (1984,1986); calculated curve^ from SAFT.
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400
350
-5
n
t
f
n
300
100
00
250
-
1
: de Loor et ai. (1984) : SAFF 0 I
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Figure 6. Pressure-concentration diagram for Ca + ethylene at 125and 175O C experimental databy de Loos et al. (19EM);calculated curves from SAFT with kij 0.0368 + 1.63e4T (K).
Weight Fraction Figure 8. Pressure-concentration diagram of C s + ethylene at 175 "C with a pressuretemperature inset at a fixed concentration of 22.8 wt % n-alkane: experimental data by de Looe (1986);predicted curves from SAFTusing the interpolated kij = 0.0681 + 5.33e4T (K).
In this case, the SAFT dew point is at 345.1 bar, which is 25.2 bar (6.6 % ) below the experimental dew point. Predicting (3% + and Cw + Ethylene Binary Phase Equilibria. The empirical kij equations obtained from the Cm, (230, and CMdata, in turn, are used for estimating
kij for C26 + and CM+ ethylene systems. Examples of the predicted pressuretemperature and pressure-concentration phase boundaries are shown in Figure 8 for a solution of c26 in ethylene a t 175 OC. As shown, SAFT predictions are in agreement with experimental data over
0.20
0.40 0.80 0.80 Cu, Weight Fractlon
1.(
C,,
1446 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 I
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400
-
Temperature ("c)
300
-2 -s
350
300
0
o :This work
(22.8 wt% alkane : Chou and Chao /l989]
250
0 0
0.20
0.40
0.60
0.80
1.00
C,, Weight Fraction
Figure 9. Pressure-concentration diagram of C s + ethylene at 100 "C with a pressure-temperature inset at a fixed concentration of 22.8 wt % n-alkane: experimental data by Chou and Chao (1989) and this work (circles). The predicted curves are from SAFT using the interpolated kjj = 0.0573 + 1.04e4T (K).
the entire temperature and concentration range, away from and near the critical region. Similarly, Figure 9 illustrates the predicted pressuretemperature (at 22.8 wt % n-alkane) and pressureconcentration (at 100 "C) phase boundaries for c36 + ethylene. While there is some discrepancy at low temperatures, SAFT predictions are in agreement with the high-pressure data collected in this work (tabulated in Table 111) and especially with the low pressure data of Chou and Chao (1989). The agreement can improved by readjusting the ki, to the low-temperature high-pressure data, but this is not our goal. The c36 ethylene phase equilibria measured in this work are compared for consistency with those of de Loos (1984,1986)in a pressure-carbon number diagram. Figure 10illustrates this comparison at constant CNconcentration of 22.8 wt 5% and temperatures of 100, 125, and 175 "C. As it turns out, the phase boundary pressures are linear
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Carbon Number Figure 10. Pressure-carbon number diagram for n-alkanes + ethylene solutions at a fixed concentration of 22.8 w t % n-alkane.
with respect to chain length. The data measured in this work (squares) are in line with those measured by de Loos et al. Correlating CSS+ 1-ButeneBinary Phase Equilibria. In predicting the phase behavior of the ternary CM + ethylene 1-butene system, the binary interaction parameter for c36 + 1-butene is needed. This parameter is determined by correlating the experimental data collected in this work (Table 111). The experimental data and SAFT phase boundaries are shown in Figure 11 in pressure-temperature coordinates for a 14.2 wt % CN + 1-butene system. We note that CM+ 1-butene solutions exhibit vapor-liquid type A behavior, typical of systems having a low degree of asymmetry (Radosz, 1987). Characteristic of type A behavior, qualitatively shown in the inset of Figure 11, the critical mixture curve (dashed line) runs continuously from the critical point of the heavy component (c36)to that of the lighter component (1butene) forming a 2-phase closed loop. At temperatures = 146.5 lower than the critical temperature of 1-butene (T, "C, P, = 39.67 bar), the measured bubble points (solid symbols)essentially coincidewith the vapor pressure curve
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Table 111. EsDerimentally Measured Phase Boundaries for Ethylene + 1-Butene + C ~ H U ethylene 1-butene phase boundary ((2%free basis) CssH74 temperat UTe (CMfree basis) (*0.3 bar) (wt % ) (fO.l OC) (wt %) (wt % ) 14.2 80.0 10.9 0.0 100.0 14.2 100.2 16.5 0.0 100.0 39.6 0.0 100.0 14.2 150.0 55.3 0.0 100.0 14.2 169.6 80.3 0.0 100.0 14.2 200.0 57.5 35.0 65.0 10.0 80.0 10.0 100.2 77.2 35.0 65.0 133.7 35.0 65.0 10.0 150.0 165.5 35.0 65.0 10.0 200.0 8.5 80.0 83.4 44.0 56.0 8.5 100.2 118.1 44.0 56.0 8.5 150.0 164.1 44.0 56.0 194.0 44.0 56.0 8.5 200.0 22.8 100.2 344.2 100.0 0.0 22.8 150.0 364.1 100.0 0.0 375.0 100.0 0.0 22.8 200.0 ~~
a Key:
BP
bubble point transition; DP
= dew point transition.
transition typea BP BP BP BP DP BP DP DP DP DP DP DP DP DP DP DP
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1447 200,
150
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o : This work (14.2 wt% n-Alkane) : Pure Critical Point v : Mixture Critical Point (Calculated)
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1
o : Thts work 6
v
: Solvent Critical Point : Mixture Critical Point (Calculated)
I
0-100
1
WO Ethylene I
*)e0 0
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50
I00
150
200
Temperature ("C)
Figure 11. Pressure-temperature diagram for CB + 1-butenewith an inset showing a general type A behavior. Experimental data
/
0 wt% Ethylene
lo
(circles) from this work and calculated curve from SAFT with a temperature dependent (in kelvin) kij: kij = O.OO0 (2' < 419.65 K); kij = -0.247 0.00062' (419.65 K IT < 443.15 K);and kij = -0.512 + 0.00122' (443.15 K IT I473.15 K).
I
+
of 1-butene. At even lower temperatures (T 50 OC), the solid-liquid (SL) and solid-liquid-vapor (SLV)transitions are observed but not accurately measured. Upon increasing the temperature, at the mixture critical point, one crosses from the bubble-point (filled circle at T = 172 "C) to the dew-point (open symbol at T = 200 "C) branch of the phase boundary. A temperature-dependent kij gives SAFT the adequate flexibility to correlate this transition. Below the critical temperature of 1-butene, kij is set equal to zero. However, above the critical point of 1-butene, kij is found to be linearly dependent on temperature: kij = -0.247 + 0.0006T (K) (419.6 K IT < 443.2 K); kij = -0.512 0.00122' (K) (443.2 K 5 T < 473.2 K). For the record, SAFT predicts a mixture critical point at 185 "C and 71 bar for this binary system of 14.2 wt % c36. Predicting C38 + Ethylene + 1-Butene Ternary Phase Equilibria. A different type of phase behavior is observed for ethylene solutions of C36. Figure 12 shows the experimental data and SAFT predicted (no fitting) phase boundaries for the ternary C a + ethylene + 1-butene system at various ethylene concentrations (Table 111).The pressure-temperature diagram shown in Figure 12 is at 0 wt 7% (pure 1-butene), 35 wt %, 44 wt %, and 100 wt % ethylene on a (&-free basis. In addition to the ternary predictions, the figure shows the predicted binary vaporliquid equilibria for ethylene 1-butene. The critical points for ethylene and 1-butene are shown as diamonds, and their critical locus is shown as a dashed curve. Consistent with Figure 11,the c 3 6 + 1-butene vaporliquid curve is of type A. However, as ethylene is added and its content increased to 35 wt % ,the phase boundary shifts to higher pressures. More important, the phase behavior of such a ternary becomes vapor-liquid-liquid (type B). The corresponding phase boundaries that go through the open circles are referred to as LCST. SAFT quantitatively predicts the LCST boundaries and their end points (LCEP). This behavior is similar to that found for many polymer + solvent systems (e.g. Chen et al. 1992) where, below the LCEP, the bubble point (or vapor pressure) curve the solvent coincideswith the vapor-liquid boundary of the polymer solution. For example, such a region is experimentally verified and predicted with SAFT at 80 "C and 35 wt % ethylene (filled circles). If we increase the ethylene content from 35 to 44 w t % , we increase the phase boundary pressures by -25 bar and decrease the LCEP from 85 to 64 "C. Although not shown for better
+
+
-7-7 80
/ Butene I
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200
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Temperature ("C) Figure 12. Pressuretemperature diagram for CB + ethylene + 1-butene. Experimental data are from this work (Table 111) and predicted curves from SAFT using interpolated kij's. The curves corresponding to the open circles are referred to as LCST, and their low-temperature end points are referred to as LCEP.
clarity, a further increase from 44 to 55 wt % ethylene increases the phase boundary pressure by an additional -30 bar and reduces the LCEP to 38 "C. Interestingly, the experimental data for c36 + ethylene (0 w t % 1-butene) suggest a flat LCST curve. This is reminiscent of similar curves for polymer + solvent systems. Chen et al. (1992) found that an increase in the degree of asymmetry between the polymer and the solvent usually causes a flatness, or even a minimum in the LCST curve, that leads to an upper-lower critical solution temperature behavior (U-LCST). Furthermore, Gregg et al. (1993) demonstrated that for amorphous poly(ethylene-propylene) (PEP molecular weight of 790 g/mol) solutions in ethylene, U-LCST behavior was observed with a flat LCST branch and that SAFT quantitatively reproduced this trend. Although the c36 + ethylene phase behavior is terminated by the solid-liquid transition, the SAFT curve shows some inflection in the 50 to 60 "C range. This usually indicates proximity to a minimum in the U-LCST curve. These results obtained for c36 + ethylene + 1-butene aid our understanding of the more complex behavior of the commercially important monomer + comonomer polyethylene systems .
+
Conclusions Statistical associating fluid theory (SAFT) is found to correlate the phase behavior of CZO+, C30 +, and Cm + ethylene binaries away from and near the mixture critical region using pure component and binary parameters that are well behaved and easy to estimate. More important, SAFT is found to predict the phase behavior of (226 + ethylene and C s + ethylene, and C s + 1-butene binaries, and C36 + ethylene + 1-butene ternary. SAFT also captures the change in phase behavior type from type A for (236 + 1-butene, through LCST for C a + ethylene + 1-butene, to U-LCST for c36 + ethylene. This is because,
1448 Ind. Eng. Chem. Res., Vol. 32,No. 7, 1993
while far from perfect, SAFT can reasonable correlate and predict the mixture critical region.
Acknowledgment The authors wish to thank Professor John Prausnitz of University of California, Berkeley, for providing us with the Soave and vdW-P software. Literature Cited Chen, S.-j.; Economou, I. G.; Radosz, M. Density-tuned Polyolefin Phase Equilibria: 11. Multicomponent Solutions of Alternating Poly(ethylenepropy1ene)in Supercriticaland SubcriticalOlefiis. Experimental and SAFT Model. Macromolecules 1992,25,3089. Chen, S.-j.; Randelman, R. E.; Seldomridge,R. L.; Radosz, M. Mass Spectrometer Composition Probe for Batch Cell Studies of Supercritical Fluid Equilibria. J. Chem. Eng. Data 1993,38,211. Chou, J. S.; Chao, K. C. Solubility of Ethylene in n-Eicosane, n-Octacosane,and n-Hexatriacontane. J.Chem. Eng. Data 1989, 34,68. Gregg, C. J.; Chen, S. J.; Stein, F. P.; Radosz,M. Phase Behavior of Binary Ethylene-Propylene Copolymer Solutions in Sub- and Supercritical Ethylene and Propylene. Fluid Phase Equilib., in press. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Znd. Eng. Chem. Res. 1990,29,2284.
Huang, S. H.; Radoez, M. Equation of State for Small, Large, Polydisperse, and hociating Molecules: Extension to Fluid Mixtures. Znd. Eng. Chem. Res. 1991,30,1994. Lee, B. I.; Keeler, M. A Generalized Thermodynamic Correlation Basedon ThreaParameter Correaponding States. AZChE J.197S, 21,510. Loos,Th. W. de; Poot, W.; Lichtenthaler,R. N. FluidPhaseEquilibria in Binary Ethylene + n-AlkaneSvetems. Ber. Bunsen-Ges.Phvs. Chem. i984,i 8 , 855. Loos,Th. W. de;Poot, W.;SwaanArons,J. de. FluidPhaseEquilibria in the Binary Syetems Ethylene + n-Triacontane and Ethylene + Squalane.- Fiuid Phase Equilib. 1986,29,505. Prigogine, I. The Molecular Theory of Solutions; North-Holland Amsterdam 1967. Radosz, M. M u l t i p h e Behavior of Supercritical Fluid Systems: Oil Solutions in Light Hydrocarbon Solvents. Znd. Eng. Chem. Res. 1987,26,2134. Sako, T.; Wu, A. H.; Prauenitz, J. M. A Cubic Equation of State for High-pressurePhase Equilibria of Mixtures Containing Polymers and Volatile Fluids. J. Appl. Polym. Sei. 1989,38,1839. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sei. 1972,27,1197. Teja, A. S.;Lee, R. J.; Roeenthal, D.; Aneelme, M. Correlation of the Critical Properties of Alkanes and Alkanols. Fluid Phase Equilib. 1990,56,153. Received for review Auguet 21, 1992 Revised manuscript received March 1, 1993 Accepted March 17,1993
