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PREDICTION OF CRITICAL PROPERTIES OF 1-HEXENE/HEXENE ISOMERS/CARBON DIOXIDE MIXTURES WITH ACUBIC EOS: SENSITIVITY TO MIXTURE COMPOSITION ANDTO THE PATH OF APPROACH TO CRITICAL POINTSSAID SAIM a & BALA SUBRAMANIAM aa Department of Chemical and Petroleum Engineering, The University of Kansas Lawrence,KS, 66045-2223Published online: 29 Oct 2007.

To cite this article: SAID SAIM & BALA SUBRAMANIAM (1993) PREDICTION OF CRITICAL PROPERTIES OF 1-HEXENE/HEXENEISOMERS/CARBON DIOXIDE MIXTURES WITH A CUBIC EOS: SENSITIVITY TO MIXTURE COMPOSITION AND TO THE PATH OFAPPROACH TO CRITICAL POINTS, Chemical Engineering Communications, 125:1, 121-137, DOI: 10.1080/00986449308936198

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Chem. Eng. Comm. 1993, Vol. 125, pp, 121-137Reprints available directly from the publisher.Photocopying permitted by license only.© 1993 Gordon and Breach Science Publishers S.A.Printed in the United States of America

PREDICTION OF CRITICAL PROPERTIES OF 1­HEXENE/HEXENE ISOMERS/CARBON DIOXIDEMIXTURES WITH A CUBIC EOS: SENSITIVITY

TO MIXTURE COMPOSITION AND TO THEPATH OF APPROACH TO CRITICAL POINTS

SAID SA1M and BALA SUBRAMANIAMt

Department of Chemical and Petroleum EngineeringThe University of KansasLawrence, KS 66045-2223

(Received December 22, 1992; in final form June 18, 1993)

Critical loci of l-hexcne/Cfr, and (hexene isomer pseudo-ccmponenu/Co; binary mixtures, alongwith the critical properties of the ternary mixture of CO2/1-hexene/(hcxene isomer pseudo­component) were predicted using Gibbs' thermodynamic criteria for criticality in conjunction with thePeng-Robinson (P-R) EOS. Maximum deviation in the critical pressures of the l-hexene/Cfr, binarymixture is less than 2% from reported experimental values. The addition of hexenc isomer to the1-hexene/C02 binary and the path of approach to mixture critical points are shown to havesurprisingly strong effects on predicted mixture critical properties. For a given ternary mixture.multiple critical points are predicted depending upon the isopleth (either constant CO 2 mole fractionpath or constant l-hexene mole fraction path) along which the critical point is approached. Criticaldensities predicted along the CO 2 isopleth of approach to critical points are always smaller than theones predicted along the l-hexene isopleth of approach. Furthermore, upon hexene isomer addition tothe t-hexene/Ctr, binary, whereas the predicted critical property surfaces of the ternary system varymonotonically along the CO 2 isopleth of approach, extrema in these surfaces are predicted whenapproaching the mixture critical points along l-hexene isopleths. These anomalous predictions areattributed to the inability of the cubic EOS to accurately model the phase behavior near the criticalpoint.KEYWORDS 1-Hexene/(hexene isornersj/Cfr, systems Critical property prediction.

INTRODUCTION

In recent years, supercritical reaction media have been employed to extract heavyhydrocarbon or "coke" compounds in situ from porous catalysts (Saim andSubramaniam, 1991; Adschiri et aI., 1991). The increased desorption of heavyhydrocarbons in dense supercritical reaction mixtures (as compared to subcriticalgaseous reaction mixtures) coupled with enhanced transport rates of the coke­laden supercritical fluid phase (as compared to liquid mixtures) through the catalystpores have been exploited to control both catalyst activity and product selectivity.In such applications, knowledge of the phase behavior of the reaction mixture is

t Author to whom correspondence should be addressed.

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122 SAID SAIM AND BALA SUBRAMANIAM

essential to determine reactor operating conditions for dense supercriticaloperation and for reliable interpretation of kinetics data. Ehrlich and Mortimer(1970) emphasize this point by noting that some of the kinetic studies reported inthe literature on the high pressure polymerization of ethylene are of little valuesince the reaction rate data were analyzed assuming that the reaction wasoperated in a single phase when in fact two phases were present.

Our research group has been investigating the coking and activity of highsurface area Pt/ y-A120 3 reforming catalysts in supercritical reaction mixtures.The isomerization of I-hexene, a reaction that is subject to deactivation bysimultaneous catalyst coking, was studied as the model reaction system. Ourreaction mixtures include a multitude of components such as 1-hexene and itsisomers, branched C6's, hexene oligomers and polynuclear aromatic (PNA)hydrocarbons. In some cases, cosolvents such as carbon dioxide and n-pentanewere also included. The hexene oligomers typically constituted (1-2) wt% (Saim,1990) while the PNAs were roughly up to 1000 ppm of the reaction mixture. Thus,the mixture physical properties are primarily dictated by 1-hexene, producthexene isomers and any added cosolvents.

Although there is much need for high pressure phase equilibrium data, there isrelatively little published literature for multicomponent mixtures. Needless to say,the complexity of the phase behavior of such mixtures renders any thoroughexperimental work very difficult even at low pressures. Most reported experimen­tal investigations deal with binary mixtures of some important solvents such asCO 2 with a variety of hydrocarbons and with polymers of importance to thepetroleum industry (see for example, McHugh and Krukonis, 1985).

Table [ summarizes physical properties of the primary components in ourreaction mixture. Because the physical properties of 1-hexene and its isomers aresimilar, especially among the isomers, the reaction mixture may be viewed aseither a pseudo-pure component or a pseudo-binary mixture without significantlycompromising on accuracy for predicting critical properties. In the presence of acosolvent such as CO2 , the reaction mixture may correspondingly be treated as apseudo-binary or a pseudo-ternary mixture.

Palenchar et al. (1986) compared the ability of five different cubic equations ofstate (Redlich and Kwong, 1949; Soave, 1972; Peng and Robinson, 1976; Teja

TABLE I

Physical property data for relevant mixture components]

Component T, (K) P, (bar) V, (cc/gmole) w PI

I-hexene 504.0 31.7 350.0 0.285 0.673Cis 3-hexene 517.0 32.8 350.0 0.225 0.680

Trans 3-hexcnc 5t9.9 32.5 350.0 0.227 0.677Trans 2-hexene 516.0 32.7 351.0 0.242 0.678

Cis 2-hexene 518.0 32.8 351.0 0.256 0.687CO 2 304.2 72.8 94.0 0.225 0.777

I Taken [rom Reid er al. (1977).

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PREDICTION OF CRITICAL PROPERTIES 123

and Patel, 1982; Adachi et al., 1983) to predict critical loci of binary mixtures.These authors report that only Type I critical temperature and pressure loci arepredicted reliably over the entire composition range. The Teja-Patel EOS gavebest predictions of Type I critical pressure and temperature. The Peng-Robinson(P-R) EOS was best in predicting Type I critical volumes and performed reliablyfor other critical properties as well. Hong et al. (1983) used the P-R EOS with asingle constant interaction parameter to predict the overall P-T -x diagrams offour binary mixtures exhibiting several types of phase behavior in the criticalregion. Predictions included two and three phase equilibria, azeotropic points,critical lines, critical end point, and spinodal curves. They concluded that the P-REOS predicts high pressure phase equilibria qualitatively over the entire fluidrange of non-polar substances, and were able to delineate regions that were notexperimentally explored. More recently, near-critical phase equilibria for theCH.-COz-HzS system were predicted using P-R EOS within 12% of experimen­tal values (Morris and Byers, 1991). In this paper, we employ the rigorousthermodynamic approach in conjunction with P-R EOS to predict criticalpressures and critical temperatures for pseudo-binary and pseudo-ternary 1­hexene/(hexene isomerj/Co, mixtures.

THEORETICAL DEVELOPMENT

The critical point of a mixture corresponds to the conditions under whichincipient separation into two phases occurs. Critical points can be described byconsidering the variation of Gibbs energy (G) with component mole fraction (x).For an n-component mixture, the criteria for a critical point require that thedeterminants of the following two Jacobian matrices be equal to zero (see forexample, Walas, 1985).

eo aZG aZG

axi aXI axz aXl aXn-laZG aZG aZG

u= axzax, ax~ axzaXn-l =0

aZG aZG aZG

aXn-l aXI aXn-l axz aX~_,

au au au

aXI axz aXn- 1

aZG aZG aZG

M= axzaXl axz axzaXn-l =0

aZG aZG aZG

aXn-l aXI aXn-1 axz aX~_,

(1)

(2)

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124 SAID SAIM AND BALA SUBRAMANIAM

where G = G(T, P, x) and the partial derivatives are numerically evaluated by themethod of central differences. As shown in the appendix, the P-R EOS (Pengand Robinson, 1976) is employed to express the Gibbs' energy terms in the U andM matrices in terms of T, V and x. For any given mixture of known composition,the system to be solved is then of the form:

V = V(T, V) = 0 (3)

M=M(T, V)=O (4)

The computer program used to solve these equations first reads guesses for thecritical temperature and critical volume of the given mixture along with thedesired temperature and volume increments to be employed for evaluating thepartial derivatives associated with matrices V and M. All variables are defined indouble precision. For the ternary mixtures, the triangular composition plane(Xl + X2 + X3 = 1) is spanned along either CO2 or l-hexene isopleths, whilevarying the remaining two mole fractions. As discussed later, critical propertypredictions for a given ternary mixture can be markedly different depending uponthe isopleth of approach to the critical point. For the chosen mixture composition,the program evaluates the partial derivatives and the values of V and M. Valuesof V and M were both less than 10-4 at convergence. For ternary mixtures,however, while V values were less than 10-" such M values were not achievableeven though initial values of M were reduced by factors of up to 109

• This isattributed to the stiffness of M (matrix of third derivatives which defines theuniqueness of the critical point). In this latter case, convergence was assumedwhen the pressure changes no more than 10-4 or 10-6 bar from one iteration tothe next. In all cases, the values of I~VI/V and I~TI/Twere less than or equal to10- 6 at convergence.

If the convergence criteria are not satisfied, improved guesses of criticaltemperature and volume are obtained using the following Newton's iterativescheme:

V(k+l) = V(k)

V(k)(~r)_M(k)(~(k)

JM(k)(~)(k) _V(k)(¥i)(k)

J

(5)

(6)

where k is the iteration level and

J = (:~rt~r)-(:~rt~r) (7)

After convergence, the program reads a new mixture composition that containsthe same mole CO 2 fraction and different mole fractions of l-hexene and hexeneisomer. For example, when xco

2= 0.2, the program successively solves for critical

point of mixtures containing Xl-hexene = 0.05, 0.1, 0.15 and so on (see Table II).Critical temperature and volume obtained for the previous mixture are usedas initial guesses for the new one. After the full range of ternary mixturecompositions at this CO2 mole fraction is covered, the program reads a new CO 2

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PREDICTION OF CRITICAL PROPERTIES 125

TABLE II

Converged M function values and critical properties for several l-hexene/thexene isomer)/COzmixtures: Comparison of predictions along CO, and I-hexene isopleths of approach to mixture critical

points

I Predictions along 'Predictions alongMole fractions CO, isopleth l-hexene isopleths

CO, 1- Hexene r, J; V< M p< T< V< Mhexene isomer (bar) (K) (m3/kmol) (bar) (K) (m3/kmol)

0.05 0.75 50.27 502.6 0.331 19.0 50.58 503.1 0.330 2.180.15 0.65 49.90 500.9 0.333 0.54 50.71 502.1 0.331 0.250.20 0.60 49.71 500.1 0.334 0.73 50.72 501.6 0.331 0.050.25 0.55 49.51 499.3 0.335 0.0005 50.70 501.0 0.331 0.130.30 0.50 49.30 498.4 0.337 0.42 50.66 500.3 0.331 0.180.35 0.45 49.10 497.6 0.338 0.22 50.59 499.6 0.331 0.210.40 0.40 48.88 496.8 0.339 0.66 50.50 498.9 0.331 0.73

0.2 0.45 0.35 48.66 495.9 0.341 0.13 50.38 498.0 0.331 0.770.50 0.30 48.42 495.1 0.342 0.41 50.24 497.1 0.331 0.070.55 0.25 48.19 494.2 0.344 0.054 50.06 496.0 0.331 0.800.60 0.20 47.95 493.4 0.346 0.14 49.84 495.3 0.331 0.510.65 0.15 47.70 492.5 0.347 1.8 49.59 494.2 0.331 1.220.70 0.10 47.43 491.6 0.349 32.0 49.28 493.1 0.331 0.900.75 0.05 47.17 490.7 0.351 16.0 48.90 491.9 0.332 10.6

0.05 0.55 73.43 480.6 0.260 0.01 76.05 482.9 0.255 0.470.10 0.50 72.53 479.7 0.264 0.02 77.27 483.8 0.254 0.070.15 0.45 71.60 478.8 0.267 0.12 78.07 484.3 0.254 0.150.20 0.40 70.61 477.7 0.271 0.007 78.52 484.3 0.254 0.030.25 0.35 69.59 476.6 0.275 0.08 78.68 483.9 0.254 0.01

0.4 0.30 0.30 68.52 475.5 0.280 0.18 78.57 483.1 0.254 0.050.35 0.25 67.43 474.2 0.284 0.09 78.19 481.9 0.254 0.160.40 0.20 66.30 472.9 0.289 0.03 77.53 480.3 0.254 0.270.45 0.15 65.14 471.5 0.294 0.02 76.53 478.4 0.254 0.290.50 0.10 63.96 470.1 0.299 1.0 75.15 476.0 0.255 0.010.55 0.05 62.76 468.5 0.304 20.0 73.28 473.2 0.256 0.31

0.05 0.35 101.2 444.3 0.192 0.74 113.4 448.4 0.178 0.150.15 0.25 94.61 442.5 0.206 0.17 123.3 452.4 0.172 0.15

0.60.20 0.20 91.02 441.0 0.214 0.09 123.9 451.9 0.171 1.930.25 0.15 87.31 439.1 0.223 0.06 121.6 449.6 0.172 0.090.30 0.10 83.54 436.9 0.232 0.25 116.6 445.9 0.174 0.230.35 0.05 79.75 434.2 0.242 4.0 108.9 441.2 0.179 0.21

0.05 0.25 112.3 416.8 0.161 0.82 135.0 417.3 0.138 0.04

0.70.15 0.15 101.2 415.4 0.180 0.93 153.6 414.9 0.128 0.110.20 0.10 95.19 413.4 0.192 1.0 145.0 413.8 0.124 0.230.25 0.05 89.14 410.5 0.204 2.0 129.2 412.3 0.137 2.21

1 Critical properties are evaluated by approaching the given ternary mixture critical point along theCO, isopleth corresponding to the CO, mole fraction noted in the first column. The remaining twomole fractions were varied as shown in the second and third columns.

'Critical properties are evaluated by approaching the given ternary mixture critical point along thel-hexene isopleth corresponding to the l-hexene mole fraction noted in the second column. Theremaining two mole fractions were varied such that the overall mole fraction i,. unity. The CO, andl-hexene isomer mole fractions shown in the first and third columns were chosen for purposes ofcomparison with predictions along the CO, isopleth of approach.'

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126 SAID SAIM AND BALA SUB RAMANlAM

mole fraction that is slightly greater than the previous one, and solves for criticalpoints of the range of ternary mixtures possible at this CO 2 mole fraction. Asimilar procedure is used when evaluating critical points along 1-hexeneisopleths. A listing of the computer program used to solve for the criticalproperties of binary and ternary mixtures by this method is provided elsewhere(Saim, 1990).

SIMULATION RESULTS

Using the procedure described in the previous section, critical loci of binarymixtures CO2/1-hexene, and CO2/(hexene isomer pseudo-component), alongwith the pseudo-ternary mixture of CO2/1-hexene/(hexene isomer pseudo­component) were predicted.

(a) CO2/I-hexene Mixtures

Figures 1a-1d represent predicted T-x, pox, V -x and P-T loci respectively forbinary mixtures of CO2 and 1-hexene. The binary interaction parameter forI-hexene/Co, mixtures was taken to be 0.095 as determined by Vera and Orbey(1984). As seen in Figures 1a-1d, the critical loci are continuous between thecritical points of the pure components (Type I behavior). Maximum deviationfrom experimental P-T critical locus data plotted in Figure 1 (Leder and Irani,1975) is less that 2%. Note from Figure (lb) that the inclusion of CO2 ascosolvent can increase the critical pressure of the reaction mixture by more thanthree-fold as compared to 1-hexene/(hexene isomer) reaction mixtures. Clearly,plots such as those in Figures 1a-1d aid in the determination of pressure andtemperature for dense supercritical operation with a given feed mixture.

(b) CO,/(Hexene Isomer Pseudo-Component) Mixtures

Leder and Irani (1975) note that the critical loci of binary mixtures of n-hexaneand 1-hexene in CO2, and of cyclohexane and benzene in CO 2 are similar, andthat size disparities and shape of the molecules are dominant factors indetermining critical loci. Hence, it is reasonable to expect the critical loci ofmixtures of hexene isomers and CO2 to be similar to those of l-bexene/Crj,mixtures. The critical properties of the hexene isomer pseudo-component aretaken as the arithmetic averages of the individual isomer properties listed inTable 1. Values of critical pressure, temperature and volume of the pseudo­isomer (viz., Pc = 32.7 bar, 'Fe = 517.7 K, Vc = 350.5 cc/gmole) differ from those of1-hexene by only 0.97bar (3%), 13.7K (3%), and O.5cc/gmole «0.1%),respectively. The binary interaction parameter for the hexene isomer/Co,mixture was assumed to be 0.095, the same as that for the t-hcxenc/Ctr, binary.Figure 2 compares predicted P-T-x critical loci of binary mixtures of CO2/1­

hexene and of CO2/(hexene isomer pseudo-component). As expected, the

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PREDICTION OF CRITICAL PROPERTIES 127

600(ta)

,-. 500::.d'-'a....:I- 400ell...a.Q,

ea.300Eo<

2000.0 0.2 0.4 0.6 0.8 1.0

CO2 Mole Fraction

ISO(lb)

.......'" 100...ell~'-'

a....:I

'"'" SOa....C.

CO 2 Mole Fraction

FIGURE I Critical loci for CO,/l-hexene mixtures.

critical loci of both binary mixtures are very similar varying smoothly throughP-T-x space, and showing no discontinuity in the critical lines.

(c) COz/l-hexene/(Hexene lsomer Pseudo-Component) Mixtures

As a more reasonable approximation, the reaction mixture may be viewed as aternary mixture of CO,/1-hexene/(hexene isomer pseudo-component). Figures3a-3c depict the critical temperature, critical volume and critical pressure surfacesof the ternary system. In Figures 3a-3c, the critical properties were predicted byapproaching the mixture critical points along CO2 isopleths (i.e., constant CO 2

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128 SAID SAIM AND BALA SUBRAMANIAM

0.5r---------------..,(lc)

~ 0.4Q,>

"0e.:0: 0.3--..,e'-"Q,> 0.2e="0 0.1;>

0.2 0.4 0.6 0.8

CO 2 Mole Fraction

120(ld) - predicted

D Exptl.100

~

'".. 80co.c/'-"

Q,> CO 2I

60 I.. I= \/'"'"Q,>.. 40 IQ" I

II t-hexene /

I

\,//20 I

I,, ,, .-'

0200 300 400 500 600

Temperature (K)

FIGURE l-(contd.)

mole fractions) corresponding to the CO2 mole fraction in the ternary mixture.Given the nearly identical critical phase behavior of t-hexene/Cfr, and of(hexene isomer pseudo-componenn/Ctr, binary systems (see Figure 2), additionof hexene isomer pseudo-component to the l-hexene/Cr.r; binary would not beexpected to have any significant effects on the critical properties of the latter.However, as seen from Figures 3b and 3c, whereas predicted critical volumes andcritical pressures are virtually independent of the relative amounts of l-hexeneand hexene isomer for CO2 mole fractions less than 0.2, these predictions showunexpectedly appreciable changes with increasing isomer content at higher CO 2

mole fractions.Figures 4a-4c depict critical temperature, critical volume and critical pressure

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PREDICTION OF CRITICAL PROPERTIES

120 ~l.Hexene Isomer110

~ 0 L-Hexene100 ~ --J

90 <e80 ~70 ~60 ~...So .~40 ~

U

FIGURE 2 Critical P·T-x loci for COz/l-hexene and COz/(hexene isomer) mixtures.

129

surfaces that were predicted by approaching the critical points along constantl-hexene 'l-hexene isopleths. These critical points, similar to the ones in Figures3a-3c, also satisfy the following mechanical stability criterion.

(dP ) RTdV T.X = - (V - bf

__2_a-,--(V_+_b.:....)_...,..< (I

V(V+b)+b(V-b)(8)

Because critical points are unique in P-T -x space, critical properties predicted byapproaching the critical points along l-hexene isopleths would normally beexpected to be identical to the ones obtained by approaching the critical pointsalong CO 2 isopleths. However, for CO 2 mole fractions greater than 0.2, thecritical properties (especially the critical volume and critical pressure surfaces) inFigures 4b and 4c show an unexpected curvature exhibiting maxima that were notpredicted when approaching the critical points along CO 2 isopleths (see Figures3b and 3c).

The quantitative effects of hexene isomer addition and of the path of approachto mixture critical points on critical property predictions are seen more clearly inFigures Sa-Se. Figures Sa-Se compare Pc- 'Fe loci of several constant-Cess-mole­fraction ternary mixtures whose critical properties are predicted by approachingthe various ternary mixture critical points along either the CO 2 isoplethcorresponding to the CO2 mole fraction noted in the figure or along the l-hexeneisopleth corresponding to the l-hexene mole fraction in the given mixture.Predicted critical property values used for constructing these plots are tabulatedin Table 2 and are the same as those used in Figures 3 and 4. On each Pc - T; locusin Figures Sa-Se, the hexene isomer content in the ternary mixture increases fromleft to right. For a given ternary mixture, while P; and 'Fe predictions by both

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130 SAID SAIM AND BALA SUBRAMANIAM

CO2 mole fraction

s20s:o f< • 0.05

SILO ""'- 0 0.2480 ~

'" 0 0.3..46() e .a. 0.5..440 ~ 0.6

~ ...420 ...e ./ 0.7400 .~

"t + 0.8o 0.1

rJOJ -2

04 ...005 ,§"o 0.6 .;:I::-~o ., ....:

0.9 .ll

040CO2 mole fraction

Oi'Cl • 0.05

°Js i 0 0.2

OJo ::i 0 0.4~

" .a. 0.50·25 E:.E!

0·20 ~ ... 0.6...~

0.7.~

./.-;:C

FIGURE 3 Critical property predictions for l-hcxene/Ihexcnc isomcr)!CO, mixtures along CO,isoplcths of approach to critical points.

approaches to the critical points are almost identical for a CO 2 mole fraction of0.05 (Figure Sa), such predictions are significantly different for larger CO 2 molefractions. Critical pressure and temperature predicted by approaching a giventernary mixture critical point along the l-hexene isopleth are always greater thanthe ones predicted by approaching the critical point along the CO 2 isopleth.Furthermore, with hexene isomer addition, whereas the Pc'Tc locus predicted bythe CO2 isopleth of approach changes monotonically, a maximum in the P/Tc

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PREDICTION OF CRITICAL PROPERTIES 131

FIGURE 3-(contd.)

CO 2 mole fraction

• 0.05

0 0.1

• 0.2

.... 0.3

'" 0.4

./ 0.5

+ 0.6

• 0.7

., 0.8

locus is predicted by the 1-hexene isopleths of approach. As seen clearly in Figure5e, while no maximum in the Pc'Tc locus is predicted by approaching the criticalpoints along the CO2 isopleth, a maximum critical pressure (approximately154 bar at a molar composition of roughly 15% 1-hexene, 15% hexene isomer,and 70% CO2) is predicted by approaching the mixture critical points alongI-hexene isopleths.

Regardless of the path of approach to the critical points, the addition of hexeneisomer affects predicted critical properties, especially at higher CO2 molefractions. For the 0.2 and 0.7 CO2 mole fraction ternary mixtures (Figures 5b and5e), critical pressures predicted by approaching the critical points along 1-hexeneisopleths show maximum increases of roughly 2.3 bars (5%) and 35 bars (29%)respectively; corresponding increases along CO2 isopleths of approach areroughly 3.1 bars (7%) and 28 bars (31%) respectively. Predicted increases inmixture critical temperatures are relatively less sensitive to isomer addition, beingroughly 12 K and 6 K for the 0.2 and 0.7 CO 2 mole fraction ternary mixturesalong either isopleth of approach to the critical points.

Table II compares predicted Pc, Tc and Vc values along with values of M atconvergence for both approaches at the four CO 2 mole fractions considered inFigures 5b-5e. Because values of M in the first iteration range from 109 to 1017

,

the values of M at convergence (mostly between 0.01 and 1) shown in Table II aredeemed acceptable. Furthermore, at convergence, ~T :s 10-4 K, and the values ofI~VI/V and I~TI/T were less than 10-6

• Thus, although the liquid-vapor criticalpoint of a mixture of given composition is unique in the P-T-x space, multiplecritical points (all satisfying the stability criterion given by Eq. 8) are predicteddepending upon the isopleth along which the mixture critical point is approached.Using the Soave-Redlich-Kwong EOS, Heidemann and Khalil (1980) also

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132 SAID SAIM AND BALA SUBRAMANIAM

COz mole fraction

~ • 0.05

Q 0 0.2

~ • 0.4e::i • 0.5

~ " 0.6III 0.7e ~

~

~...~;:1::U

OJo

0·25

0·20

COz mole fraction

520 • 0.05

~ 0 0.2500 ....

e • 0.4

480~ • 0.5l!46Q III " 0.6

~ ~ 0.7440~

1Q420 ~:;::

a

0-4G

(b)

(a)

0.15

~~0~0'1~~ O~ -.-s.-~ OJ 0.' ., 0..\ 04 ~<:-

es>-'>. II'"' 0.6 04 ..;1:--es>.-:s, 0.1 o.s 0 0.7 -:

C>"-">es>_ oJ., 0.,"FIGURE 4 Critical property predictions for I-hexene/(hexene isomer)/COz mixtures along 1­hexene isopleths of approach to critical points.

predicted multiple critical points for multicomponent COz/hydrocarbon mixtures.Heidemann and Khalil attribute the lowest density critical points to vapor-liquidtransition and the high density critical points to possible liquid-liquid transitions.Given that I-hexene/COz binary mixtures do not exhibit LLV or LLG transitions(Leder and Irani, 1975), it appears highly unlikely that either the (hexeneisomer)/COz binary or the l-hexene/fhexene isomer)/COz ternary systems willexhibit such LLV or LLG transitions. Hence, the anomalous behavior reported

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PREDICTION OF CRITICAL PROPERTIES 133

FIGURE 4-(contd.)

CO2 mole fraction

160 " 0.05

140 ~ 0 0.2L.;

120Oil II 0.4~

100 ~ A. 0.5

'"~ " 0.680

I\: " 0.760 ...

~'~

40 .~

\J

in this paper is attributed to the inability of the P-R EOS (and cubic equations ofstate in general) to accurately model the region near the critical point. Theseequations of state predict parabolic coexistence curves, whereas real fluidcoexistence curves are almost cubic. Consequently, there appears to be regionsnear the true critical point in which Gibbs' criteria for criticality are satisfied.

The question arises therefore as to which one of the predicted critical points isacceptable for describing liquid-vapor transitions. In our case, the criticalproperties predicted by approaching the critical point along CO;, isopleths yieldedsmaller mixture densities as compared to those predicted along either I-hexcneisopleths (summarized in Table II) or hexene isomer isopleths (values notprovided for conciseness sake). Furthermore, along the COz isopleth, unexpectedextrema in critical property surfaces are not predicted upon addition of hexeneisomer to the I-hexene/COz binary. For these reasons, we consider the criticalproperties predicted along the COz isopleth of approach to mixture critical pointsto be more reliable for describing vapor-liquid transitions.

SUMMARY

Critical loci of l-hcxene/Ctr, and (hexene isomer pseudo-componentj/Ctj,binary mixtures, along with the critical properties of the ternary mixture ofCOz/l-hexene/(hexene isomer pseudo-component) were predicted using Gibbs'thermodynamic criteria for criticality in conjunction with the P-R EOS. Maxi­mum deviation in the critical pressures of the l-hexene/Cf), binary mixture is lessthan 2% from reported experimental values. The addition of hexene isomer tothe COz/l-hexene binary and the isopleth (either constant COz mole fraction or

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constant I-hexene mole fraction or constant hexene isomer mole fraction path)along which the ternary mixture critical point is approached are shown to havesurprisingly strong effects on critical property predictions. Upon hexene isomeraddition, whereas the ternary critical property surfaces predicted along the CO2

isopleth of approach vary monotonically, extrema in these surfaces are predictedwhen approaching the mixture critical points along l-hexene isopleths. For agiven ternary mixture, critical densities predicted along the CO2 isopleth ofapproach are smaller than the ones predicted along either the l-hexene isoplethor hexene isomer isopleth of approach to mixture critical points. Theseanomalous predictions are attributed to the inability of the cubic EOS toaccurately model the region near the critical point.

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PREDICTION OF CRITICAL PROPERTIES 135

(c)80 (5c) CO2 Mole Fraction =0.4

66~6 6

75

70

65

cn

n

nc

rPn

cn

cc

Predictions Along

c CO2 Isopleth

6 I-Hexene Isopleths

490485475 480

T (K)c

47060 L...~.L....-~"""'::==::::;::==::::;:::::::::::"J465

(d) 130

120

(5d) CO2

Mole Fraction =0.6

6 lt16

Predictions Along

c CO 2 Isopleth

6 I-Hexene Isopleths

8c

c c-r- ~

cc

c

90

80

100

lIO

70 L...............~...................;:::::;:::;::::::::=:i:::::::;::::;:::;:::::::==J430 435 440 445 450 455 4.60

Tc

(K)

(e)160 (5e) CO2 Mole Fraction =0.7

6

140

'I:''"e 120

g.u100

80

c

cc

Predictions Along

o CO 2Isopleth

6 I-Hexene Isopleths

415T (K)c

FIGURE 5-(conrd.)

4.Z0

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136

REFERENCES

SAID SAIM AND BALA SUBRAMANIAM

Adachi, Y., Lu, B.C-Y. and Sugie, H., "Three Parameter Equations of State," Fluid PhaseEquilibria, 13,133 (1983).

Adschiri, T., Suzuki, T., and Arai, K., "Catalytic Reforming of Coal Tar Pitch in Supercritical Fluid,"Fuel, 70, 1483 (1991).

Ehrlich, P., and Graham, F.B., "Solubility of Polymers in Compressed Gases," J. Polym. Sci., 45,246(1960).

Heidemann, R.A., and Khalil, A.M., "The Calculation of Critical Points," AIChE J., 26,769 (1980).Hong, G.T., Modell, M., and Tester, J.W., "Binary Phase Diagrams from a Cubic Equation of State,"

in Chemical Engineering at Supercritical Fluid Conditions, Paulaitis, M.E.; Penninger, J.M.L.;Gray, R.D.Jr.; Davidson, P., Eds., Ann Arbor Science, Ann Arbor, MI, 1985, p. 263.

Leder, F., and Irani, CA., "Upper Critical Solution Temperature in Carbon Dioxide HydrocarbonSystems," J. Chem. & Eng. Data, 20,323 (1975).

McHugh, M.A., and Krukonis. V.J., Supercritical Fluid Extraction: Principles and Practice,Butterworths, Stoneham, MA, 1985.

Morris, J.S., and Byers, CH., "Near-critical Equilibria of the CH.-COz-HzS System," Fluid PhaseEquilibria, 66,291 (1991).

Palenchar, R. M., Erickson, D.O., and Leland, T.W., "Prediction of Binary Critical Loci by CubicEquations of State," in Equations of State; Theories and Applications, Chao, K.C; Robinson,R.L., Eds., ACS Symposium Series 300, 1986, p. 132.

Peng, 0.-Y., and Robinson, D.B., "A New Two-Constant Equation of State," Ind. Eng. Chem.Fundam., 15,59 (1976).

Peng, D.-Y., and Robinson, 0.8., "A Rigorous Method for Predicting the Critical Properties ofMulticomponent Systems from an Equation of State," AIChE J., 23, 137 (1977).

Reid, R.C, Prausnitz, J.M., and Sherwood, T.K., The Properties of Gases and Liquids, McGraw Hill,New York, 1977, p. 629.

Redlich, 0., and Kwong, J.N.S., "On the Thermodynamics of Solutions. V. An Equation of State.Fugacities of Gaseous Solutions," Chem. Rev., 44,233 (1949).

Saim, S., "Isomerization of l-hexene over Pt/y-AlzO, Catalyst at Subcritical and SupercriticalReaction Conditions: Temperature, Pressure and Solvent Effects on Catalyst Coking," Ph.D.Dissertation, U. of Kansas, Chap. 3, 1990. .

Saim, S., and Subramaniarn, B., "Isomerization of l-Hexene over Pt!y-Alz03 Catalyst: ReactionMixture Density and Temperature Effects on Catalyst Effectiveness Factor, Coke Laydown andCatalyst Micromeritics,' J. Catal., 131,445 (1991).

Soave, G., "Equilibrium Constants from a Modified Redlich-Kwong Equation of State," Chem. Eng.Sci., 27, 1197 (1972).

Teja, A.S., and Patel, N.C, "A Cubic Equation of State for Fluids and Fluid Mixtures," Chem. Eng.Sci., 37,463 (1982).

Vera, J.H., and Orbey, H., "Binary Vapor-Liquid Equilibria of Carbon Dioxide with 2-methyl-l­pentene, l-hcxcnc, I-heptene and m-xylene at 303.1, 323.15 and 343.15 K," J. Chem. & Eng.Data, 29,269 (1984).

Walas, S.M., "Phase Equilibria in Chemical Engineering," Butterworth, Stoneham, 1985.

APPENDIX

For evaluating the elements of the Jacobian matrices (U and M), the Gibbs freeenergy (G) is expressed in terms of the Helmholtz free energy (A) using the P-REOS (Peng and Robinson, 1977). The Helmholtz free energy is given by

~ (XiRT) a (V + 2A14b)A = RT 6 Xi Ln V _ b - 2.828b Ln V - OAl4b

iTn iT n dT+ LXiC;idT-T LXiCpi-- R T

70 1=1 To l=l T

where C; is the ideal gas heat capacity at constant pressure,

(AI)

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PREDICTION OF CRITICAL PROPERTIES 137

The second partial derivatives of G in terms of those of A and P are given by:

(A2)

The third partial derivatives of G in terms of those of A and P are given by:

ap a2p ap a2p ap a2p---+---+----

(a3G ) = a3A + ax,ax! aXk ax!ax, aXk aXk ax,aXj

ax, aXj aXk ax, aXj aXk apav

apap a2p ap ap a2p ap ap a2p----+----+-----aX, aXj aXk av aXjaXkaX, av aX,aXkaXjaV

(:~r

(A3)

where the subscripts have been dropped for convenience. The expressions forthese partial derivatives in terms of T, V and x are provided elsewhere (Saim,1990). Heidemann and Khalil (1980) employ equivalent expressions for predictingmixture critical points.

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