Evaluation of the Multiphase Behavior in Binaries Using a New

Ind. Eng. Chem. Res. 2010, 49, 1905–1909

1905

Evaluation of the Multiphase Behavior in Binaries Using a New Technique for Describing Solid Phases Based on the Soave-Redlich-Kwong Equation of State Quan Yang,*,† Shen-Lin Zhu,‡ Tian-Sheng Zhao,† Yi-Gui Li,‡ Wei-Bin Cai,‡ and Li-Xin Yu‡ Key Laboratory of Energy Resources & Chemical Engineering, Ningxia UniVersity, Yinchuan, 750021 China, Department of Chemical Engineering, Tsinghua UniVersity, Beijing, 100084 China

In the petroleum industry, knowledge of phase behavior is essential to solve many problems, such as the design of the catalytic cracking of heavy oil. The catalyst tends to lose activity as a result of the formation of the solid phase, so the calculation of multiphase behavior of systems containing solid phases will serve to solve the problem. The Soave-Redlich-Kwong equation of state is widely employed to evaluate multiphase behavior, but cubic equations of state are incapable of calculating the properties of solid phases and thus cannot evaluate the multiphase behavior of systems containing solid phases. In this research, a new technique for describing the solid phase has been developed. The multiphase behavior of propane binaries with ployaromatic hydrocarbons was then explored. To evaluate the critical end point, which is difficult to calculate, an algorithm combining the method of Heidemann and Khalil (AIChE J. 1980, 26, 769-780) to compute the critical point and the tangent-plane criterion was developed previously (Yang et al. Ind. Eng. Chem. Res. 2009, 48, 6877-6881). Setting the initial guesses with the values at the obtained critical end points, the three-phase loci were then computed successfully. The calculation results show that the three-phase loci terminate at a lower critical end point (LCEP), which is wrong according to the experimental data. To correct this mistake, the newly developed technique was employed, and the quadruple points where the three-phase loci really end were evaluated successfully. 1. Introduction The Soave-Redlich-Kwong equation of state, a widely used cubic equation of state, is often employed to evaluate the phase behavior of mixtures. In 1980, van Konynenburg and Scott1 proposed a classification scheme to classify the phase behavior of binaries according to the characteristics of the phase behavior. The phase diagram corresponding to type V phase behavior in temperature-pressure space is presented in Figure 1. Peter et al.2 stated that, when the triple point of the less volatile component of a binary mixture is located in the nearcritical region of the more volatile component, a solid phase might be present, in which case the corresponding phase diagrams according to van Konynenburg and Scott’s classification scheme will be incomplete because the solid phase is not taken into consideration in the classification scheme. According to Peter et al.,2 the phase diagram in temperaturepressure space considering the presence of a solid phase should be as in Figure 2. From Figure 2, it is observed that the threephase locus l1l2g, which extends to a lower critical end point in Figure 1, terminates at a quadruple point, where four phases, a vapor phase, two liquid phases, and one solid phase, coexist. The cubic equations of state are extensively used to evaluate multiphase behavior, but because of the incapability of these equations to describe the solid phase,2 cubic equations of state cannot be employed to predict the quadruple points or other phase behavior of systems containing a solid phase, such as the system encountered in the catalytic cracking of heavy oil. Different equations3-6 of state have been proposed to calculate the solid phase. These equations are isothermal equations or have too many parameters, with different values * To whom correspondence should be addressed. Tel.: +86-951-2062328. Fax: +86-951-2062323. E-mail: [email protected]. † Key Laboratory of Energy Resources & Chemical Engineering, Ningxia University. ‡ Department of Chemical Engineering, Tsinghua University.

of the parameters required at different temperatures and for different components. Employing these equations is rather inconvenient. Baonza et al.4,7,8 employed their equation of state to compute the properties of both solid and liquid phases. Fang and Chen9 employed a universal equation of state that is widely used for describing the solid phase to evaluate the liquid phases and obtained good results. In this research, a new algorithm has been developed. The solid phase is computed according to the calculated results for the equilibrium liquid phase obtained using the Soave-Redlich-Kwong equation of state. The critical end points, three-phase loci, and quadruple points, the benchmarks of phase diagrams, were calculated to evaluate the multiphase behavior of selected binaries of propane and ployaromatic hydrocarbons. The technique proposed by Heide-

Figure 1. Type V phase behavior according to van Konynenburg and Scott:1 (b) critical end points, (9) pure-component critical points. l ) g and l ) l represent vapor-liquid and liquid-liquid critical loci, respectively; lg represents vapor-pressure loci of pure components; UCEP and LCEP represent upper critical end point and lower critical end point, respectively; and l1l2g represents the three-phase loci.

10.1021/ie9011003  2010 American Chemical Society Published on Web 01/11/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 4, 2010

2.2. Computing the Critical End Points. The algorithm of Heidemann and Khalil10 takes the temperature and the volume as independent variables. At a critical point, the following equation, obtained by expanding the Helmholtz free energy in a Taylor series, must be satisfied (∆A -

∑µ i

1 3!

i0∆ni0)T0,V0

1 2!

)

∑ ∑ (∂ A/∂n ∂n )∆n ∆n 2

j

j

∑ ∑ ∑ (∂ A/∂n ∂n ∂n )∆n ∆n ∆n i

j

j

i

j

+

+ O(∆n4) ) 0

3

i

i

i

k

i

j

k

(4)

k

The quadratic term and the cubic term must each equal zero

∑ ∑ (∂ A/∂n ∂n )∆n ∆n

Figure 2. Type V phase behavior considering the presence of a solid phase composed of the pure less-volatile component. The symbols also present in Figure 1 represent the same elements. (2) Triple point, (solid vertical bar) quadruple point. sB - l1 ) l2 represents the critical end point with a liquid-liquid critical phase and a solid equilibrium phase; sBl and sBg represent the melting locus and sublimation locus, respectively; and sBl1l2, sBl1g, and sBl2g represent the three-phase loci of the binaries.

mann and Khalil10 was employed to calculate the critical points in this work because their method is rather stable and converges rather rapidly. An algorithm based on the tangent-plane criterion11-13 was developed previously to determine the stability of the computed critical point and predict the possible equilibrium phase of the critical phase and the corresponding critical end point. The three-phase points were computed using the condition that the same component in different phases at equilibrium has equal fugacities. When the fugacity of the less volatile component of the three phases, one gas and two liquid phases, evaluated using the Soave-Redlich-Kwong equation of state, equals that of the pure solid component computed using the new algorithm for describing the solid phase, the quadruple point is determined. 2. Algorithm 2.1. Soave-Redlich-Kwong Equation of State. In this research, the Soave-Redlich-Kwong equation of state14,15 was employed. This equation can be presented in the form P)

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

(1)

where a and b are the energy parameter and the size parameter, respectively, of the Soave-Redlich-Kwong equation of state. The energy parameter and the size parameter for mixtures were evaluated using the van der Waals mixing rules a)

∑ ∑ x x (1 - K )a i j

i

ij

1/2 i

aj1/2

2

j

j

i

i

j

)0

∑ ∑ ∑ (∂ A/∂n ∂n ∂n )∆n ∆n ∆n 3

i

i

j

(5)

i

j

k

i

j

k

)0

(6)

k

With eqs 5 and 6, the critical points at different compositions for different mixtures can be evaluated using the Newton-Raphson method16 of solving equations. After a critical point was determined, the algorithm based on the tangent-plane criterion11,12 was developed to determine the equilibrium phase of the critical point and thus obtain the critical end point (Appendix A). 2.3. Calculating the Three-Phase Points and the Quadruple Points. The condition that the same component in different phases at equilibrium has equal fugacities was employed herein to evaluate the three-phase loci ln f1,1 ) ln f2,1 ) ln f3,1 ln f1,2 ) ln f2,2 ) ln f3,2

(7)

In fi,j, i represents the phase index, and j represents the component index. The initial guesses for the calculation of the three-phase loci were set according to the previously evaluated critical end points terminating the loci. Usually, j ) 2 represents the less volatile component of the binaries. When the fugacity of the less volatile component of three phases, one gas and two liquid phases, obtained using the Soave-Redlich-Kwong equation of state equals that of the solid phase computed using the new algorithm proposed herein, the quadruple point is obtained. Because 2 represents the less volatile component and the solid phase is composed of pure less volatile component,2 the equations to determine the quadruple point are ln f1,1 ) ln f2,1 ) ln f3,1 ln f1,2 ) ln f2,2 ) ln f3,2 ) ln fs,2

(2)

(8)

j

b)

∑

(3)

xibi

i

where Kij is the interaction parameter for components i and j. Usually, Kij ) Kji.

2.4. New Technique for Describing the Solid Phase. Because the solid phase contains only pure less volatile component,2,17 the new technique employs the liquid phase composed of pure less volatile component in equilibrium with the solid phase as the reference to compute the solid phase. The liquid phase is computed using the Soave-Redlich-Kwong

Table 1. Experimental Results for the Propane Binaries with Fluorene, Phenanthrene, and Triphenylmethane2 LCEP

UCEP

Q-point

system

k1

T (K)

P (MPa)

T (K)

P (MPa)

T (K)

P (MPa)

propane (1)-fluorene (2) propane (1)-phenanthrene (2) propane (1)-triphenylmethane (2)

0.96 0.89 0.87

-a -a -a

-a -a -a

385.5 377.3 378.8

5.110 4.672 4.760

361.2 351.2 335.3

3.343 2.930 2.110

a

No corresponding experimental data available.


equation of state, and the solid phase is calculated with the equation (Appendix B) Vs ) Vl[k1 + k2(T - Tm)]

(9)

where Vs represents the molar volume of the solid phase composed of pure less volatile component, Vl represents the molar volume of the liquid phase composed of pure less volatile component in equilibrium with the solid phase, Tm represents the melting temperature, k1 represents the ratio of the molar volume between the solid phase and the corresponding liquid phase in equilibrium with the solid phase at Tm, and k2 is a constant that shows the difference in the thermal coefficients of expansion between the solid phase and the liquid phase in equilibrium composed of pure less volatile component. At the melting point, eq 9 is necessarily satisfied. At other temperatures, the coefficient k2 serves to correct the errors caused by the differences in the thermal coefficient of expansion between the solid phase and the liquid phase in equilibrium with the solid phase. For a binary for which the triple point of the less volatile component and the quadruple point lie in the nearcritical region of the more volatile component, the difference in temperature and pressure between the melting point of the less volatile component and the quadruple point is not too large. Therefore, the effect of the pressure change can be neglected, and the error resulting from using a constant coefficient k2 is small, as confirmed by the calculation results of the quadruple points. 2.5. Calculating the Fugacity of the Pure Solid Component. For pure components of vapor or liquid state, the fugacity18 is computed using the following equation based on the Soave-Redlich-Kwong equation of state ln(f/P) )

√2a z + B(1 - √2) ln - ln(z - B) + z - 1 4bRT z + B(1 + √2) (10)

where B)

bP RT

(11)

z)

PV RT

(12)

Nevertheless, for pure components in the solid state, the corresponding fugacity is calculated using the following algorithm. At the melting point of a pure component corresponding to pressure PI, the fugacity of the solid phase equals that of the liquid phase fs,I ) fl,I

(13)

where I represents the conditions at the melting point. The fugacity at temperature TII and pressure PII can be evaluated using the equation

ln

fs,II fs,I - ln ) PII PI

∫

PII

PI

(

1907

)

Vs 1 dp RT P

(14)

The final derived equation to compute the fugacity of the pure solid component is then ln

fs,II fl,II ) (k1 - k2Tm) ln - (k1 - k2Tm - 1) × PII PII fl,I PII k2 ln - ln + Ω PI PI R

(

)

(15)

where

(

Ω ) zII - zI - ln

)

(

VII - b VII VI + b a RT + ln VI - b b VI VII + b

)

(16)

zI )

PIVI RTI

(17)

zII )

PIIVII RTII

(18)

In these equations, II represents the conditions at temperature TII and pressure PII. 3. Results and Discussion In this work, the multiphase behavior of the binaries of propane with some ployaromatic hydrocarbons were explored. The three polyaromatic hydrocarbons, fluorene, phenanthrene, and triphenylmethane, were selected because the quadruple points of the corresponding binaries with propane have been experimentally determined.2 The experimental data for the three binaries are presented in Table 1. k1 represents the ratio of the molar volumes between the solid and liquid phases at the melting point.19,20 Correlated values of the molar volumes can be used to calculate k1 if no corresponding experimental data are available, because the calculation results show that the calculated values of the quadruple points are not sensitive to k1. Using the Soave-Redlich-Kwong equation of state, the phase behaviors of the three binaries of propane and the selected ployaromatic hydrocarbons were evaluated first. The Table 3. Calculation Results for the Quadruple Points of the Propane-Triphenylmethane Binary Corresponding to Different Values of k2 k2

y1,g

y1,l1

y1,l2

T (K)

P (MPa)

-0.0050 -0.0049 -0.0048 -0.0047 -0.0046 -0.0045

0.999995 0.999996 0.999998 0.999999 0.999999 -a

0.9348 0.9297 0.9246 0.9197 0.9147 -a

0.8663 0.8726 0.8786 0.8843 0.8899 -a

336.8 335.9 335.1 334.4 334.0 -a

2.235 2.190 2.153 2.125 2.105 -a

a

No corresponding computed values obtained.

Table 2. Fitted Interaction Parameters and the Corresponding Calculation Results for the Phase Behaviors of the Propane Binaries with Fluorene, Phenanthrene, and Triphenylmethane LCEP

UCEP

Q-point

system

Kij

T (K)

P (MPa)

T (K)

P (MPa)

T (K)

P (MPa)


-0.035 0.000 0.037

330.3 347.2 333.6

1.926 2.730 2.088

374.7 373.4 373.1

4.515 4.452 4.437

-

-

1908


Table 4. Calculation Results for the Propane Binaries with Fluorene, Phenanthrene, and Triphenylmethane Using the New Technique for Describing the Solid Phase UCEP a

system

a

k1

k2

Kij

LCEP

T (K)

P (MPa)

T (K)

P (MPa)

0.96 0.89 0.87

-0.0130 -0.0080 -0.0046

-0.035 0.000 0.037

-b -b

374.7 373.4 373.1

4.513 4.452 4.437

357.2 350.5 334.0

3.301 2.922 2.105

∆P/P (%)


1.20 0.27 0.24

Q-point

b

∆P is the difference in Q-point pressure: ∆P ) |Pcal - Pex|. b No corresponding computed values obtained.

calculation results varied corresponding to different interaction parameters. According to the experimental and calculated results for the upper critical end point (UCEP), eq 19 was used as the objective function to fit the interaction parameter,13,21 Kij, of the Soave-Redlich-Kwong equation of state diff )

(

Tcal - Tex Tex

) ( 2

+

Pcal - Pex Pex

)

2

(19)

Because, experimentally, the LCEP is obscured by a quadruple point and the quadruple point lies above the LCEP (Figure 2), not only should eq 19 be considered, but also the calculated LECP should have lower temperature and pressure values than the corresponding quadruple points. The fitted interaction parameters and the corresponding calculation results satisfying the above conditions are presented in Table 2. To obtain the experimentally determined quadruple points as shown in Table 1, the proposed new technique for describing the solid phase was employed to compute the multiphase behavior of these three binaries. The calculation results for the quadruple points of the propane-triphenylmethane binary corresponding to different values of k2 are presented in Table 3. y1,g, y1,l1, and y1,l2 represent the propane compositions of the vapor phase and the two liquid phases at the quadruple points, respectively. The solid phase is composed of pure less volatile component, triphenylmethane. From Table 3, it is observed that the calculation results for the quadruple points are sensitive to the values of k2. According to the experimental and calculated results for the quadruple points, the k2 values were fitted using eq 19 as the objective function. The fitted k2 values and the computation results for the propane-triphenylmethane binary and the other investigated binaries are listed in Table 4. It is observed from Table 4 that the quadruple points have been evaluated successfully and the largest error in the computed pressure at the quadruple points is 1.20%. With the new technique for describing the solid phase, not only were the quadruple points (which the Soave-Redlich-Kwong equation of state is unable to evaluate) obtained, but also the mistake of obtaining an LECP instead of a quadruple point when using the Soave-Redlich-Kwong equation of state to calculate multiphase behavior was corrected. Experimentally, the threephase locus extending from a UCEP does not end at an LCEP, but terminates at a quadruple point as determined with the new technique for describing the solid phase. 4. Conclusions Because of the incapability of the Soave-Redlich-Kwong equation of state to evaluate the properties of the solid phase, the Soave-Redlich-Kwong equation of state is unable to predict the multiphase behavior of the systems containing a solid phase. In this research, a new technique for describing the solid phase has been developed. The multiphase behavior of propane binaries with selected ployaromatic hydrocarbon was explored.

An algorithm combining the method of Heidemann and Khalil and the tangent-plane criterion was employed to calculate the critical end points. Setting the initial guesses to the values at the obtained critical end points, the three-phase loci were then computed. According to the calculation results obtained using just the Soave-Redlich-Kwong equation of state, the threephase loci end at an LCEP, which is wrong according to the experimental data. The quadruple points experimentally observed were evaluated successfully with the proposed new technique for describing the solid phase. The quadruple points also helped fit the interaction parameter of the Soave-RedlichKwong equation of state. The largest error in the calculated pressure at the quadruple points was 1.2%. Nomenclature a ) attraction parameter, Pa · m6/mol2 A ) Helmholtz free energy, J b ) co-volume (size) parameter, m3/mol fi ) fugacity of component i, Pa k1 ) ratio of the molar volume between the solid phase and the corresponding equilibrium liquid phase at the melting point k2) difference in the thermal coefficients of expansion of between the solid phase and the equilibrium liquid phase Kij ) interaction parameter LCEP ) lower critical end point n ) number of moles, mol P ) pressure, MPa Q-point ) quadruple point R ) universal gas constant, 8.314 J/(mol · K) T ) temperature, K UCEP ) upper critical end point V ) molar volume, m3/mol x ) mole fraction y ) mole fraction Greek Symbols ∆P ) |Pcal - Pex| Ω ) variables for calculating fugacity Subscripts I ) temperature and pressure condition at the melting point II ) condition of at certain temperature and pressure A ) more volatile component B ) less volatile component cal ) calculation results ex ) experimental values i, j ) component or phase indices m ) melting point s ) solid phase

Appendix A: Algorithm for Computing the Critical End Points13 After a critical point was determined, the tangent-plane criterion was employed to determine the equilibrium phase in the research. The criterion was first used to decide if a phase was


stable. At a given temperature and pressure, the component mole fractions of a M-component mixture are z1,z2, ...,zM. If this mixture is separated into two phases, the mole numbers of the second phase is ε, ε being infinitesimal, and the mole fractions of different components in the second phase are y1,y2, ...,yM. Because ε is infinitesimal, the compositions of the first phase remain as z1,z2, ...,zM. The variation of Gibbs energy for the process is ∆G )

∑ εy µ (yj) - ∑ εy µ (zj) ) ε ∑ y [µ (yj) - µ (zj)] 0 i i

i i

i

i

i

i

0 i

i

(a.1) To ensure stability of the system, the variation of the Gibbs energy for the above process should be larger than zero. If at composition jye, ∆G equals zero and at other trial composition jy, ∆G is always equal to or larger than zero, the phase with composition jye is then the equilibrium phase for the original mixture. That is to say if at the stationary point of the ∆G function ∆Gequals zero, the stationary point represents the equilibrium phase. According to the necessary condition of a stationary point, the following equation is obtained from eq a.1: µi(yj) - µ0i (zj) ) 0,

i ) 1, 2, ..., M

(a.2)

To calculate the composition of the equilibrium phase, assume µi(yj) - µ0i (zj) ) K,

i ) 1, 2, ..., M

(a.3)

The expression for the chemical potential is µi(yj) - µ0i (zj) ) RT[ln yi + ln φi(yj) - ln zi - ln φi(zj)] (a.4) With eqs a.3 and a.4, assuming hi ) ln zi + ln φi(zj), θ ) K/RT, and Yi ) exp (- θ)yi, eq a.5 is obtained: ln Yi + ln φi(yj) - hi ) 0,

i ) 1, 2, ..., M

(a.5)

The successive substitution method was employed to solve eq a.5. After obtaining the Yi for the previous iteration, the composition yi for the next iteration was calculated using the following equation: yi ) Yi /

∑Y

j

(a.6)

j

If the correct calculation results satisfy the condition ∑jYj > 1(θ < 0, K < 0, ∆G ) εK < 0), the original mixture is unstable and it has a potential to separate into two different phases from the original phase. If ∑jYj < 1, the mixture is stable. In the above two cases, no equilibrium phases exist for the original phase. If ∑jYj ) 1, the variation of the Gibbs energy equals zero and the corresponding compositions calculated are the compositions of the found equilibrium phase. Appendix B: Calculating the Volume of the Solid Phase Using the Value of the Volume of the Corresponding Liquid Phase At the melting point, assume Vs,m ) k1Vl,m

(b.1)

At other temperature, if assume the thermal coefficient of expansion of both phases are constant and assume Vs - Vs,m ) k1k2′(Vl - Vl,m)

(b.2)

1909

then Vs ) k1Vl,m + k1k2′(Vl - Vl,m)

(b.3)

In the research, because the difference in the pressure and temperature between the melting points of less volatile component and the quadruple points is not large, the effect of pressure change can be neglected. If assume Vl - Vl,m ) k′′2 · Vl · (T - Tm), the following equation would be obtained: Vs ) k1Vl + (k1k2′′ + k1k2′k2′′)Vl(T - Tm)

(b.4)

If assume k2 ) k1 · k2′′ + k1 · k2′ · k2′′, then Vs ) Vl[k1 + k2(T - Tm)]

(b.5)

Literature Cited (1) van Konynenburg, P. H.; Scott, R. L. Critical lines and phase equilibriu in binary van der Waals mixtures. Philos. Trans. R. Soc. London, A 1980, 298, 495–540. (2) Peters, C. J.; Rijkers, M. P. W. M.; Roo, D. J. L.; Arons Swaan, J. D. Phase Equilibria in Binary Mixtures of Near Critical Propane and Poly-Aromatic Hydrocarbons. Fluid Phase Equilib. 1989, 52, 373–387. (3) Murnaghan, F. D. Compressibility of Media under Extreme Pressures. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244–247. (4) Baonza, V. G.; Caćeres, M.; Nuń˜ez, J. Universal Compressibility Behavior of Dense Phases. Phys. ReV. B. 1995, 51, 28–32. (5) Ferrante, J.; Smith, J. R. Theory of the Bimetallic Interface. Phys. ReV. B. 1985, 31, 3427–3434. (6) Vinet, P.; Smith, J. R.; Ferrante, J.; Rose, J. H. Temperature Effects on the Universal Equation of State of Solids. Phys. ReV. B. 1987, 35, 1945– 1952. (7) Baonza, V. G.; Caćeres, M.; Nuń˜ez, J. Extended Analytical Equation of State for Liquids from Expansivity Data Analysis. J. Phys. Chem. 1993, 97, 10813–10817. (8) Baonza, V. G.; Caćeres, M.; Nuń˜ez, J. Universal Behavior of Compressed Liquids. J. Phys. Chem. 1994, 98, 4955–4959. (9) Fang, Z. H.; Chen, L. R. The test and comparison of two isothermal equations of state for 32 liquids. J. Anhui Norm. UniV. 1994, 17, 39–43. (10) Heidemann, R. A.; Khalil, A. M. The calculation of critical points. AIChE J. 1980, 26, 769–780. (11) Michelsen, M. L. The isothermal flash problem. Part I: stability. Fluid Phase Equilib. 1982, 9, 1–19. (12) Michelsen, M. L. The Isothermal Flash Problem. Part II: Phase split calculation. Fluid Phase Equilib. 1982, 9, 21–40. (13) Yang, Q.; Li, Y. G.; Zhu, S. L. Tuning the interaction parameter for the Soave-Redlich-Kwong equation of state using the critical endpoints. Ind. Eng. Chem. Res. 2009, 48, 6877–6881. (14) Soave, G. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, 1197–1203. (15) Luo, M. J.; Hu, B.; Jiang, T.; Xia, S. Q.; Ma, P. S. Analysis and Comparison of the Alpha Functions of SRK Equation of State. Chin. J. Chem. Eng. 2008, 16, 766–771. (16) Plybon, B. F. An Introduction to Applied Numerical Analysis; PWSKENT Publishing Company: Boston, MA, 1992. (17) Lam, D. H.; Jangkamolkulchai, A.; Luks, K. D. Liquid-LiquidVapor Phase Equilibrium Behavior of Certain Binary Ethane + n-Alkanol Mixtures. Fluid Phase Equilib. 1990, 59, 263–277. (18) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids; McGraw-Hill Book Company: New York, 1987. (19) Dean, J. A.; Gokel, G. W. Handbook of Organic Chemistry; McGraw-Hill Book Company: New York, 1987. (20) Ali, M. F.; El Ali, B. M.; Speight, J. G. Handbook of Industrial Chemistry: Organic Chemicals; McGraw-Hill Book Company: New York, 2005. (21) Zhu, H. B.; Gong, M. Q.; Zhang, Y.; Wu, J. F. Review of the research on interaction coefficients used in two cubic equations of state for phase equilibria predictions. Cryogenics 2005, 5, 7–11.

ReceiVed for reView July 8, 2009 ReVised manuscript receiVed November 14, 2009 Accepted December 7, 2009 IE9011003

Evaluation of the Multiphase Behavior in Binaries Using a New

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