Prediction of Vapor-Liquid Equilibrium at High Pressure Using a New ExcessFree Energy Mixing Rule Coupled with the Original UNIFAC Method and theSRK Equation of State

Zhen-hua Chen, Zhen Yao, Yan Li, Kun Cao,* and Zhi-ming Huang

State Key Laboratory of Chemical Engineering (ZJU), Institute of Polymerization and Polymer Engineering,Department of Chemical and Biological Engineering, Zhejiang UniVersity, Hangzhou 310027, China

A new excess free energy mixing rule that employs a low-pressure reference state is proposed in this work.This new low-pressure mixing rule (LPMR), coupled with the original UNIFAC method and theSoave-Redlich-Kwong (SRK) equation of state (EoS), leads to an improved EoS/GE predictive model(LPMR-SRK model) that was applied to the high-pressure vapor-liquid phase equilibria of various systemsincluding symmetric and asymmetric systems of n-alkanes with different gases (C2H6, C3H8, CO2, H2) andpolar systems. The results are compared with those of other EoS/GE predictive models, such as the predictiveSoave-Redlich-Kwong (PSRK), Chen-modified PSRK (MPSRK), volume-translated Peng-Robinson groupcontribution equation of state (VTPR), and linear combination of the Vidal and Michelsen rules (LCVM)models. It is demonstrated that LPMR-SRK method gives similar accuracy as the PSRK approach forsymmetric or slightly asymmetric systems and gives better results than the MPSRK, VTPR, and LCVMmodels. For highly asymmetric nonpolar systems, the results indicate that the LPMR-SRK method has anaccuracy similar to or slightly better than that of the MPSRK and VTPR models and gives a much betterperformance than the PSRK model. Furthermore, the LPMR-SRK approach is competitive with or betterthan the LCVM approach, except for some CO2/heavy alkane systems. In addition, although it is slightlyinferior to the PSRK and LCVM approaches, the LPMR-SRK method can still provide satisfactory descriptionsof polar systems that are more accurate than those of the MPSRK and VTPR models. Moreover, three ternaryand one quaternary systems can also be predicted successfully by the LPMR-SRK method.

1. Introduction

Successful process design usually requires an accuratedescription of phase equilibrium. The cubic equation of state(CEoS) is widely utilized for this purpose because of its abilityto be applicable over an extended range of both temperatureand pressure for mixtures composed of various substances.When a CEoS is applied to phase equilibrium calculation, aconsistent description is obtained, and problems with standardstates do arise as in the case of the γ-� approach.

Mixing rules are key factors in the performance of a CEoS.Conventional van der Waals one-fluid (vdW1f) mixing rulesare limited to nonpolar fluids and are incapable of describingthe behavior of nonideal mixtures containing polar or associatingfluids. These shortcomings can be resolved with the equationof state (EoS)/excess Gibbs energy (GE) formalism, which wasoriginally developed by Huron and Vidal.1 Orbey and Sandler2

pointed out that EoS/GE mixing rules combine the successexpected from the γ-� and �-� methods. In general, thereare two approaches in developing EoS/GE mixing rules. Thefirst is to combine the CEoS with the GE mixing rule at infi-nite pressure as proposed by Huron and Vidal.1 The second isthe zero-pressure approach developed by Heidemann and Kokal3

and Michelsen.4,5 Furthermore, there are other EoS/GE mixingrules that were developed with excess Helmholtz energy (AE)function.2,6-9 In the past decades, several more or less successfulEoS/GE mixing rules such as the modified Huron-Vidalfirst-order (MHV1),5 modified Huron-Vidal second-order(MHV2),10 predictive Soave-Redlich-Kwong (PSRK),7

Wong-Sandler (WS),6 linear combination of the Vidal and

Michelsen rules (LCVM)11 models, among others, have beenproposed. Evaluations and comparisons of various mixing ruleshave been made by different authors.12-18 Actually, the activitycoefficient models coupled with the CEoS can be chosenarbitrarily in order to implement the EoS/GE mixing rules.However, when a group contribution approach such as theUNIFAC19 or ASOG20 method is selected, the resulting EoS/GE models are purely predictive tools. It should be pointed outthat most EoS/GE mixing rules (PSRK, MHV2, WS) providepoorperformanceswhenappliedtohighlyasymmetricsystems.16,21

The LCVM mixing rule can overcome this limitation success-fully.11 However, the LCVM mixing rule involves an incon-sistency in reference pressure as it is a linear combination ofthe Huron-Vidal and MHV1 mixing rules. To improve thecapacity of the PSRK model for asymmetric systems, Li et al.22

recommended an empirical correction to the PSRK witheffective van der Waals volumes and surface-area parametersfor the subgroups CH3, CH2, CH, and C. Yang and Zhong23

modified the PSRK by introducing an asymmetry factor thatcan be calculated by a generalized expression. As suggestedby Mollerup,24 the expressions for mixing rules based on zeroor low reference pressure (MHV1, MHV2, PSRK) for energyparameters of the CEoS contain two Flory-Huggins-type termsand should be canceled in principle, and the second part of thecombinatorial term in the UNIFAC19 or UNIQUAC25 has littlesignificance to retain. This proposition was complemented andexplained in detail by Kontogeorgis and Vlamos.26 Accordingto this theory, some EoS/GE mixing rules have been developedto improve the predictive capacity for vapor-liquid equilibriaof highly asymmetric systems.27-30

In this study, we propose a new EoS/GE mixing rule for theCEoS and then obtain a new EoS/GE predictive model applicable

* To whom correspondence should be addressed. E-mail: kcao@che.zju.edu.cn.

Ind. Eng. Chem. Res. 2009, 48, 6836–68456836

10.1021/ie900111h CCC: $40.75 2009 American Chemical SocietyPublished on Web 06/12/2009

to symmetric and asymmetric systems. The predictions ofvapor-liquid equilibria of binary and multicomponent systemsare compared with those of the PSRK,7 Chen-modified PSRK(MPSRK),27 volume-translated Peng-Robinson group contribu-tion equation of state (VTPR),28 and LCVM11 models. It isdemonstrated that the proposed model is an alternative tool thatcan readily be used in vapor-liquid phase equilibrium predictions.

2. Theory

2.1. Cubic Equation of State (CEoS). The van der Waalsequation of state (vdW EoS) and its modifications are specialcases of a generic cubic equation, which can be expressed asfollows

where a, b, c, and d are constants or functions of fluidthermophysical properties (critical temperature, critical pressure,acentric factor, etc.) and temperature. The most popular formsof CEoS are the Soave-Redlich-Kwong (SRK) EoS31 and thePeng-Robinson (PR) EoS.32 In this work, the SRK EoS is usedto describe the new mixing rule. The SRK EoS can be expressedas follows

where

The function R(Tr,ω) is applicable to hydrocarbons and othernonpolar compounds. When it is used for polar fluids, thecalculated vapor pressures are not accurate. Other expressionshave been developed for polar substances, which usually containone or more parameters that are specific to the fluid of interest.33

In this work, we use the following temperature-dependent Rexpression proposed by Mathias and Copeman34

2.2. Derivation of GE Mixing Rule for the EnergyParameter a. According to the deduction process of the PSRKmixing rule reported by Fischer and Gmehling,35 the expressionfor AE obtained by SRK EoS is

It can be assumed that36

Then, eq 8 is simplified as follows

To obtain the PSRK mixing rule, Fischer and Gmehlingcalculated the saturation liquid molar volumes (VL) of differentpure components at normal pressure (1 atm) and the covolumeparamter (b) from critical properties. They found that the meanvalue for u ) VL/b was about 1.1, which was the value used inthe PSRK mixing rule. Thus, the reference pressure chosen forthe PSRK mixing rule was about 1 atm. Later, Noll andFischer37 applied the procedure of calculating quasi-experimentalu values mentioned above to different pressures, investigating458 components. The results showed that the optimum constantpacking fraction u was 1.189 for the SRK EoS and that thecorresponding reference pressure was about 2.3 bar. Thisoptimum value of u is used in this study, so that eq 10 becomes

At low pressures, the AE is related to GE by34

Substituting this expression into eq 11, one obtains the followingequation

In this work, the value of GE is obtained according to theoriginal UNIFAC approach as

where

P ) RTV - b

- aV(V + d) + c(V - d)

(1)

P ) RTV - b

-acR(Tr, ω)

V(V + b)(2)

ac ) 0.42747RTc

2.5

Pc(3)

b ) 0.08664RTc

Pc(4)

R(Tr, ω) ) [1 + (0.480 + 1.574ω - 0.176ω2)(1 - Tr0.5)]2

(5)

Ri(T) ) [1 + c1,i(1 - √Tr,i) + c2,i(1 - √Tr,i)2 +

c3,i(1 - √Tr,i)3]2 Tr e 1 (6)

Ri(T) ) [1 + c1,i(1 - √Tr,i)]2 Tr > 1 (7)

AE

RT) ln( u

u - 1) - abRT

ln(u + 1u ) - ∑ xi[ln( ui

ui - 1) -

ai

biRTln(ui + 1

ui)] (8)

u ) ui )VL

b)

ViL

bi(9)

abRT

) ∑ xi

ai

biRT+ AE/RT

ln( uu + 1)

(10)

abRT

) ∑ xi

ai

biRT+ AE/RT

-0.6103(11)

AE

RT- GE

RT) ∑ xi ln( b

bi) (12)

abRT

) ∑ xi

ai

biRT+

GE

RT+ ∑ xi ln( b

bi)

-0.6103(13)

Figure 1. Vapor-liquid equilibrium results for the C2H6/octane system.Experimental data from ref 52.

GE ) GFHE + GSG

E + GresE (14)

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 6837

In these equations, θi is the area fraction, φi is similar to avolume fraction and is called the segment fraction, and Γk isthe residual activity coefficient of group k. Details of theUNIFAC method are available in the original literature.19

As suggested by Mollerup24 and Kontogeorgis and Vlamos,26

the two Flory-Huggins-type terms, which are the Flory-Hugginspart [∑ixi ln(φi/xi] in the combinatorial term of the originalUNIFAC expression and the EoS-derived Flory-Huggins-typeGE combinatorial term [∑xi ln(b/bi)], should, in principle, cancel.Therefore, eq 13 becomes

where GSGE and Gres

E are the Staverman-Guggengeim part andthe residual part, respectively, in the UNIFAC expression.Usually, the contribution of the Staverman-Guggengeim partis small. However, large values of the Staverman-Guggengeimpart can be found for systems involving water or alcohols forwhich the values of θi and φi are empirical.38 Thus, the GSG

E

term in the GE expression was retained in this study.2.3. Mixing Rule for Covolume Parameter b. The mixing

rule for the covolume parameter b is the customary quadraticmixing rule

where

This combining rule for bij is especially flexible and encom-passes several rules appearing in the literature.39-41 In this work,the value of E was set to 2/3 as verified by preliminary

vapor-liquid equilibrium calculations for alkane/alkanesystems. The combining rule for bij becomes the Lee andSandler rule, which was developed on theoretical groundsfor molecules40

GFHE ) RT ∑

i

xi ln(φi

xi) (15)

GSGE ) 5RT ∑

i

xiqi ln(θi

φi) (16)

GresE ) RT ∑

i

xiVki (ln Γk - ln Γk

i ) (17)

Figure 2. Vapor-liquid equilibrium results for the C2H6/hexatriacontanesystem. Experimental data from ref 60.

abRT

) ∑ xi

ai

biRT+

(GSGE + Gres

E )/RT

-0.6103(18)

b ) ∑i

∑j

xixjbij (19)

bij ) (biE + bj

E

2 )1/E

(20)

Figure 3. Vapor-liquid equilibrium results for the C3H8/eicosane system.Experimental data from ref 63.

Figure 4. Vapor-liquid equilibrium results for the CO2/n-octane system.Experimental data from ref 73.

Figure 5. Vapor-liquid equilibrium results for the CO2/dotriacontanesystem. Experimental data from ref 82.

bij ) (bi2/3 + bj

2/3

2 )3/2

(21)

6838 Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Moreover, the same exponent is used for the relative vander Waals volume parameter ri in the combinatorial expres-sion of the modified UNIFAC approach.42

The new low-pressure mixing rule (LPMR) in eqs 18, 19,and 21 is coupled with the original UNIFAC method and theSRK EoS, which results in an improved EoS/GE predictivemodel (LPMR-SRK model).

3. Results and Discussion

3.1. Pure-Component and Group-Interaction Parameters.The critical properties, acentric factors, and Mathias-Copemanparameters for most components involved in this work weretaken from the Horstmann et al.43 Recently, Gao et al. examineda number of existing correlations for predicting n-paraffinphysical properties and successfully updated the asymptoticbehavior correlation (ABC) using the experimental criticalproperties of n-paraffins up to n-C36.

44 Thus, we chose the Gaoet al.’s ABC to predict the critical constants and acentric factorsof thermally unstable heavy alkanes that were not included inHorstmann et al.’s report. Moreover, the temperature-indepen-dent UNIFAC interaction parameters were taken from Hansenet al.,45 except those for the CO2/n-alkane and H2/n-alkanesystems, for which temperature-dependent interaction parameterscan be introduced into the UNIFAC model because the

experimental data cover a broad range of temperature, asfollows7

Table 1 lists the group-interaction parameters of the CO2/n-alkane and H2/n-alkane systems used in this new model. Theseinteraction parameters were obtained by fitting the vapor-liquidequilibrium data listed in Tables 4 and 5 below. The objectivefunction for this purpose was

where NP is the number of data points, P is the bubble-pointpressure, and y is the vapor-phase mole fraction.

3.2. Ethane/n-Alkane and Propane/n-Alkane Systems.Systems containing only alkanes can be considered athermalsolutions, and no interaction parameters are used for them inthe UNIFAC model, as Gres

E ) 0. The GSGE term usually has no

significance for alkane solutions and can also be omitted.Therefore, eq 18 is simplified to

Table 1. UNIFAC Group-Interaction Parameters for the LPMR-SRK Model

n m anm (K) bnm cnm × 103 (K-1) amn (K) bmn cmn × 103 (K-1)

CH2 CO2 919.8863 -3.8630 4.6658 -38.6782 0.7916 -1.6142CH2 H2 612.3832 -2.2713 1.6479 316.4417 -1.1079 1.2208

Table 2. Vapor-Liquid Equilibria of C2H6/n-Alkane Systems Using the PSRK, MPSRK, VTPR, LCVM, and LPMR-SRK Models

4P (%)a 4yb

n-alkanetemperaturerange (K)

pressurerange (bar) NP PSRK MPSRK VTPR LCVM

LPMR-SRK PSRK MPSRK VTPR LCVM

LPMR-SRK

datasource

C3H8 260–280 3.620–26.15 22 4.5 4.4 4.2 3.5 4.4 1.5 1.4 1.4 1.4 1.5 ref 48C4H10 260–280 1.613–25.01 24 9.6 9.2 8.8 7.7 9.4 0.7 0.7 0.6 0.5 0.7 ref 48C5H12 277.6–410.9 3.447–62.053 56 2.0 1.7 1.8 3.3 1.7 0.6 0.8 0.9 1.0 0.8 ref 49C6H14 298.2 5.078–35.494 7 4.1 2.7 2.1 3.4 3.1 0.5 0.5 0.4 0.4 0.5 ref 50C7H16 338.7–449.8 31.371–85.15 32 3.3 2.6 2.4 2.6 3.0 0.6 0.5 1.3 0.6 0.4 ref 51C8H18 313.2–373.2 4.053–52.69 46 1.7 2.8 4.1 9.2 1.7 0.8 0.8 0.7 0.7 0.8 ref 52C10H22 227.6–410.9 3.94–82.36 66 9.3 2.2 2.0 8.1 3.5 - - - - - ref 53C11H24 298.2–318.2 11.97–54.27 19 5.6 1.5 1.3 4.1 2.8 - - - - - ref 54C12H26 298.2–373.2 3.75–56.40 50 11.7 2.5 2.5 9.7 3.1 - - - - - ref 55C16H34 307.6–513.7 7.02–27.14 43 33.1 7.6 5.2 7.2 11.2 0.3 0.3 0.3 0.3 0.3 ref 56C20H42 264.9–451.8 2.34–168.47 355 14.8c 7.9 6.5 7.2 6.7 - - - - - ref 57C22H46 295.0–367.90 1.70–99.10 118 29.6 10.9 7.3 13.3 9.6 - - - - - ref 58C24H50 300.9–368.7 4.19–136.5 104 26.6 5.0 4.7 5.6 5.9 - - - - - ref 59C28H58 348.2–423.2 5.89–51.82 24 90.7 9.6 6.8 6.4 13.4 - - - - - ref 60C36H74 373.2–423.2 3.68–47.6 13 131.3 8.7 9.9 7.3 8.4 - - - - - ref 60C44H90 373.2–423.2 3.87–29.81 15 162.8 10.8 14.6 7.4 10.2 - - - - - ref 60

overall average 21.3 6.4 6.1 8.6 6.3

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆y ) (1/NP)∑i)1NP |yi

exp - yical| × 100. c Bubble-point pressure calculation is not possible for one

point.

Table 3. Vapor-Liquid Equilibria of C3H8/n-Alkane Systems Using the PSRK, MPSRK, VTPR, LCVM, and LPMR-SRK Models

4P (%)a 4yb

n-alkanetemperaturerange (K)

pressure range(bar) NP PSRK MPSRK VTPR LCVM

LPMR-SRK PSRK MPSRK VTPR LCVM

LPMR-SRK

datasource

C8H18 363.2–473.2 24.13–41.38 28 2.4 2.3 2.4 2.5 2.4 1.2 1.0 1.0 1.0 1.0 ref 61C10H22 277.6–510.9 1.72–68.70 42 5.5 3.2 2.5 3.7 3.7 0.4 0.6 0.2 0.3 0.6 ref 62C20H42 279.3–358.6 4.03–32.50 166 9.1 6.7 6.9 10.7 4.9 - - - - - ref 63C34H70 320.0–428.2 10.49–95.19 80 8.3 15.9 15.9 13.7 12.1 - - - - - ref 64C60H122 351.7–431.3 13.40–129.36 60 126.7c 61.1 61.0 48.2 58.5 - - - - - ref 65

overall average 26.8 16.6 16.6 15.9 14.7

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆y ) (1/NP)∑i)1NP |yi

exp - yical| × 100. c Bubble-point pressure calculation is not possible for six

points.

ψnm ) exp(-anm + bnmT + cnmT2

T ) (22)

obj ) ∑i)1

NP (|Piexp - Pi

calc

Piexp | × 100 + |yi

exp - yicalc| × 100)

(23)

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 6839

This mixing rule was developed by Zhong and Masuoka tomodel vapor-liquid equilibria of polymer solutions and gassolubilities in molten polymers.46,47 Thus, the LPMR-SRKmodel can be extended to the calculation of polymer solutions,as will be reported in a future work.

The calculated results for the C2H6/octane and C2H6/n-hexatriacontane systems are presented in Figures 1 and 2,respectively. In the case of the C2H6/octane system, the resultsobtained by the LPMR-SRK and PSRK models are better thanthe predictions based on the MPSRK, VTPR, and LCVMapproaches. However, the PSRK method fails to predict thephase equilibria of the more asymmetric system (i.e., C2H6/n-hexatriacontane). As shown in Figure 2, the LPMR-SRK modelis more accurate in the high-pressure region but less satisfactoryin the lower-pressure region when compared with the MPSRK,VTPR, and LCVM approaches. Figure 3 shows the predictionsfor the C3H8/eicosane system, for which the LPMR-SRK model provides a good description, especially as the pressure of the

system increases.

Vapor-liquid equilibrium predictions for C2H6/n-alkane andC3H8/n-alkane binary mixtures are presented in Tables 2 and 3,respectively. From these two tables, one can see that theLPMR-SRK model is similar to or slightly better than theMPSRK and VTPR approaches. Compared with the PSRKmethod, the above three models can obtain improved predictionsof the bubble pressure, especially for asymmetric systems. TheLCVM model is also satisfactory for asymmetric systems butwith an ambiguous reference pressure.

3.3. CO2/n-Alkane and H2/n-Alkane Systems. Figures 4and 5, respectively, present the results for moderately (CO2/n-octane) and strongly (CO2/n-dotriacontane) asymmetric systems.As shown in Figure 4, the LPMR-SRK, MPSRK, and VTPRmodels are more accurate than the PSRK and LCVM ap-proaches. On the other hand, the LPMR-SRK model has lesssatisfactory performance than the LCVM approach for CO2/n-dotriacontane, as shown in Figure 5. The results of theLPMR-SRK, MPSRK, and VTPR approaches are similar.

Figure 6. Vapor-liquid equilibrium results for the H2/hexadecane system.Experimental data from ref 89.

abRT

) ∑ xi

ai

biRT(24)

Table 4. Vapor-Liquid Equilibria of CO2/n-Alkane Systems Using the PSRK, MPSRK, VTPR, LCVM, and LPMR-SRK Models

4P (%)a 4yb

n-alkanetemperaturerange (K)

pressure range(bar) NP PSRK MPSRK VTPR LCVM

LPMR-SRK PSRK MPSRK VTPR LCVM

LPMR-SRK

datasource

C2H6 207–270 3.72–33.44 73 0.4 0.5 1.9 5.5 1.9 0.6 0.6 1.2 1.6 0.9 ref 66C3H8 230–361.2 2.65–49.75 140 2.7 2.0 2.5 3.5 1.5 1.5 1.2 1.5 1.8 0.9 refs 67 and 68C4H10 277.9–418.5 3.45–80.67 109 3.0 2.2 1.9 1.3 2.0 1.9 1.9 1.9 1.8 1.6 ref 69C5H12 310.2–363.2 5.66–96.71 41 7.3 5.2 5.8 5.4 4.5 2.1 1.6 1.6 1.2 1.4 ref 70C6H14 298.2–313.2 4.44–76.57 20 4.9 7.4 5.7 4.6 7.9 0.5 0.6 0.5 0.5 0.6 ref 71C7H16 310.7–477.2 1.86–133.14 64 3.5 5.4 4.7 4.4 6.8 1.1 1.0 1.1 1.3 1.0 ref 72C8H18 313.2–348.2 15.0–113.5 20 4.3 2.7 3.8 4.8 2.3 1.5 0.2 0.1 0.1 0.2 ref 73C9H20 343.25 37.3–118.04 6 2.2 2.7 2.9 2.1 3.9 1.6 0.3 0.2 0.6 0.3 ref 74C10H22 277.6–510.9 3.45–172.38 88 2.2 2.0 2.2 5.2 2.0 0.4 0.4 0.4 0.6 0.3 refs 74 and 75C14H30 344.3 110.3–163.7 25 27.9 0.6 0.7 6.1 0.8 3.1 5.7 0.6 1.2 0.8 ref 76C15H32 313.2 17.03–64.15 8 8.1 4.0 4.9 10.5 3.2 – – – – – ref 77C16H34 463.1–663.8 20.06–50.76 16 4.0 11.7 16.3 8.6 6.8 6.2 7.1 1.4 2.6 5.9 ref 78C19H40 313.2–333.2 9.36–79.58 35 26.2 3.9 5.7 5.8 3.5 – – – – – ref 79C20H42 310.2–373.2 5.07–75.99 83 28.0c 5.8 6.9 4.9 5.4 – – – – – refs 80 and 81C22H46 323.2–373.2 9.62–71.78 44 36.8 5.6 5.6 4.3 5.6 – – – – – ref 82C28H58 348.2–423.2 8.07–96.04 23 73.2c 9.7 9.5 4.3 10.3 – – – – – ref 81C32H66 348.2–398.2 9.46–72.29 37 88.0 9.3 9.0 5.5 9.2 – – – – – ref 82C36H74 373.2–423.2 5.24–86.32 18 107.7 34.7 37.1 5.8 35.7 – – – – – ref 81C44H90 373.2–423.2 5.79–70.81 14 155.2c 14.4 13.1 9.4 14.3 – – – – – ref 81

overall average 19.0 4.6 5.1 4.5 4.6

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆y ) (1/NP)∑i)1NP |yi

exp - yical| × 100. c Bubble-point pressure calculation is not possible for one

point.

Figure 7. Vapor-liquid equilibrium results for the H2/octacosane system.Experimental data from ref 88.

6840 Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Moreover, the PSRK approach should be not used for the CO2/n-dotriacontane system.

The simulation results for the vapor-liquid equilibria of theH2/n-hexadecane and H2/n-octacosane systems are shown inFigures 6 and 7, respectively.

Because the interaction parameters of H2/CH2 are not reportedfor the MPSRK and LCVM approaches, the results of theLPMR-SRK model can be compared only with the results ofthe PSRK and VTPR models. From Figures 6 and 7, it is veryclear that the LPMR-SRK model provides a more satisfactorydescription of these highly asymmetric systems than the VTPRapproach. On the other hand, the PSRK model completely failsto describe the two systems.

Other results for CO2/n-alkane and H2/n-alkane systems arelisted in Tables 4 and 5, respectively.

As shown in Table 4, the performances of the LPMR-SRKmodel are analogous to or slightly better than those of theMPSRK and VTPR models for the CO2/alkane systems. Whencompared with the PSRK approach, the LPMR-SRK modelgives a similar accuracy for symmetric and slightly asymmetricsystems but much more accurate descriptions of highly asym-metric systems. Furthermore, the LPMR-SRK model obtainsbetter results than the LCVM approach for most of the systemsin which the number of carbon atoms in the n-alkane is lessthan 20 but shows less satisfactory performance for the moreasymmetric systems. However, it is worth pointing out againthat the LCVM approach is a purely empirical artifice that lacks

Table 5. Vapor-Liquid Equilibria of H2/n-Alkane Systems Using the PSRK, VTPR, and LPMR-SRK Models

4P (%)a 4yb

n-alkane temperature range (K) pressure range (bar) NP PSRK VTPR this work PSRK VTPR this work data source

C4H10 327.7–394.3 27.78–168.76 60 10.2 9.5 14.6 2.1 2.1 2.3 ref 83C5H12 323.2–373.2 6.93–275.90 20 11.3 12.7 16.5 1.5 1.5 1.6 ref 84C6H14 298.2–373.2 13.80–98.10 24 6.3 9.2 11.1 – – – ref 85C8H18 463.2–543.2 10.10–150.4 50 4.0 3.6 9.2 2.7 1.6 3.7 ref 86C10H22 462.5–583.5 19.26–255.24 26 7.6 3.0 8.8 2.3 1.4 1.8 ref 87C10H22 283.2–449.6 12.38–142.16 105 16.1 3.7 3.0 – – – ref 88C16H34 461.7–664.1 19.95–253.82 29 33.6 8.9 4.1 1.2 0.3 0.2 ref 89C16H34 298.1–448.2 12.16–151.34 114 59.3 14.9 8.6 – – – ref 88C20H42 373.4–573.3 9.94–50.81 15 65.2 18.7 10.2 0.2 0.1 0.1 ref 90C28H58 342.6–447.3 14.56–140.01 56 174.1 20.3 12.3 – – – ref 88C36H74 357.5–447.4 13.75–143.41 41 248.7 54.3 35.2 – – – ref 88C46H94 372.5–447.5 22.93–159.70 36 846.6c 17.5 28.4 – – – ref 88

overall average 97.6 13.8 11.9

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆y ) (1/NP)∑i)1NP |yi

exp - yical| × 100. c Bubble-point pressure calculation is not possible for eight

points.

Table 6. Vapor-Liquid Equilibria of Binary Polar Systems Using the PSRK, MPSRK, VTPR, LCVM, and LPMR-SRK Models

4P (%)a 4yb

systemtemperaturerange (K)

pressurerange (bar) NP PSRK MPSRK VTPR LCVM

thiswork PSRK MPSRK VTPR LCVM

thiswork

datasource

methanol/water 363.2–442.2 1.17–20.93 25 3.2 15.1 10.2 3.8 4.6 – – – – – ref 91ethanol/water 423.2–623.2 5.6–189.7 77 2.1 8.3 7.9 1.4 4.4 0.8 4.5 6.4 0.7 1.6 ref 922–propanol/water 423.2–548.2 5.2–93.1 66 2.2 12.6 7.0 3.3 7.5 1.1 8.5 9.0 2.3 2.2 ref 92propane/ethanol 313.6–349.8 3.70–28.30 22 9.4 12.1 7.2 7.2 5.9 0.6 0.8 0.5 0.5 0.5 ref 93ethane/1-decanol 448.2 5.38–143.2 14 8.0 9.4 10.1 26.6 4.6 0.2 0.2 0.2 0.6 0.1 ref 94n-pentane/acetone 373.2–422.6 4.78–17.99 27 1.1 1.4 4.0 1.6 3.3 0.9 1.0 1.9 1.1 1.6 ref 95dimethyl ether/ethanol 332.2–373.6 1.91–31.60 43 9.8 9.8 9.3 8.1 11.1 1.2 1.2 1.2 1.3 1.2 ref 96dimethyl ether/2-propanol 323.2–373.7 1.29–26.21 29 11.1 12.4 11.7 9.0 13.2 1.4 1.5 1.3 1.2 1.5 ref 97

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆y ) (1/NP)∑i)1NP |yi

exp - yical| × 100.

Figure 8. Vapor-liquid equilibrium results for the ethanol/water system.Experimental data from ref 92.

Figure 9. Vapor-liquid equilibrium results for the ethane/1-decanol system.Experimental data from ref 94.

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 6841

a theoretical basis when compared with the LPMR-SRK model.As listed in Table 5, when compared with the PSRK and VTPRapproaches, the LPMR-SRK model gives much better resultsfor highly asymmetric H2/n-alkane systems (CnH2n+2, n g 10)at the cost of less accurate descriptions of low and moderatelyasymmetric systems.

3.4. Polar Systems. Predictions of vapor-liquid equilibriafor polar systems are presented in Table 6, and P-x-y phasediagrams for the ethanol/water and ethane/1-decanol systemsare shown in Figures 8 and 9, respectively. Figure 8 indicatesthat the LPMR-SRK model is less satisfactory than the PSRKand LCVM models but more accurate than the MPSRK andVTPR models for the ethanol/water system. The LPMR-SRKmodel provides the best results for the ethane/1-decanol system,as shown in Figure 9. Although the LPMR-SRK modelprovides less accurate predictions than the PSRK and LCVMapproaches for most of the polar systems, it is evident that theLPMR-SRK model can predict vapor-liquid equilibria of polarsystems with reasonable results and provide more satisfactoryperformance than the MPSRK and VTPR approaches, assuggested by the summary results reported in Table 6.

3.5. Ternary and Quaternary Systems. Predicted resultsfor three ternary systems and one quaternary system arepresented in Table 7. Accurate results are obtained using theLPMR-SRK model for these nonpolar and polar systems, andthe LPMR-SRK model predicts the most accurate bubble-pointpressures when compared with the other four models, exceptfor those of the propane/2-butanol/2-propanol system.

4. Conclusions

A new GE mixing rule for the CEoS is proposed in this study.According to the suggestions of Noll and Fischer, the referencepressure is about 2.3 bar, and the corresponding constant packingfraction u is 1.189 for the SRK EoS, which is used to implementthe new mixing rule based on a low-pressure reference state.The Staverman-Guggenheim term of the combinatorial partand the residual part of the UNIFAC model are included in the

mixing rule for the cohesion parameter of the CEoS. For thecovolume parameter, a nonlinear combining rule for the crosscovolume parameter is used with the conventional quadraticconcentration-dependent mixing rule. This mixing rule coupledwith the SRK EoS and the original UNIFAC model leads to anLPMR-SRK model that was used to simulate high-pressurevapor-liquid phase equilibria of symmetric and asymmetricsystems. The results obtained by the LPMR-SRK model werecompared with those from the PSRK, MPSRK, VTPR, andLCVM models for binary and multicomponent systems. It wasdemonstrated that the LPMR-SRK model gives similar ac-curacy as the PSRK model for symmetric or slightly asymmetricsystems and provides better results than the predictions basedon the MPSRK, VTPR, and LCVM approaches. At the sametime, it provides much better descriptions of highly asymmetricsystems that are comparable to those obtained with the LCVMapproach. When compared with the MPSRK and VTPRmethods, the LPMR-SRK model is similar or slightly moreaccurate for nonpolar systems and achieves a significantimprovement for polar systems. Moreover, it is clarified thatthe LPMR-SRK model represents a simple and reliable toolfor vapor-liquid phase equilibria of systems whose componentscan involve various degrees of nonideality and asymmetry.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China through NSFC Major Project 50390097and NSFC Projects 20576115 and 50773069.

Nomenclature

a ) cohesive energy parameter of the CEoSAE ) excess Helmholtz energyanm ) temperature-independent group interaction parameterb ) volumetric parameter of the CEoSbnm ) linear temperature-dependent group interaction parameterc, d ) other two parameters of the CEoS

Table 7. Vapor-Liquid Equilibria of Multicomponent Systems Using the PSRK, MPSRK, VTPR, LCVM, and LPMR-SRK Models

temperature range (K) pressure range (bar) NP model 4P (%)a 4y1b 4y2

b 4y3b data source

CO2 (1)/n-C15H32 (2)/n-C16H34 (3)

313.15 17.14–64.05 8 PSRK 9.8 - - - ref 98MPSRK 3.4 - - -VTPR 5.4 - - -LCVM 11.6 - - -this work 0.8 - - -

CO2 (1)/C3H8 (2)/n-C5H12 (3)/n-C8H18 (4)

310.93-394.26 30.80–63.70 15 PSRK 8.6 2.4 1.4 1.4 ref 99MPSRK 7.4 5.9 1.6 3.0VTPR 7.4 2.5 1.3 1.3LCVM 5.6 2.9 1.4 1.4this work 4.1 4.7 1.6 2.3

Propane (1)/2-Butanol (2)/2-Propanol (3)

328.10-368.10 15.85–36.77 9 PSRK 1.9 0.2 0.2 - ref 100MPSRK 1.3 0.2 0.3 -VTPR 3.4 0.3 0.3 -LCVM 2.0 0.3 0.2 -this work 3.1 0.2 2.0 -

n-Butane (1)/Ethanol (2)/Water (3)

353.2 6.77–10.69 4 PSRK 8.0 2.2 2.5 - ref 101MPSRK 11.5 2.0 2.7 -VTPR 7.2 2.6 2.5 -LCVM 6.3 2.0 2.3 -this work 5.8 2.3 2.4 -

a ∆P (%) ) (1/NP)∑i)1NP [(|Pi

exp - Pical|)/Pi

exp] × 100. b ∆yj )(1/NP)∑i)1NP |yj,i

exp - yj,ical| × 100.

6842 Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

cnm ) quadratic temperature-dependent group interaction parameterc1,i, c2,i, c3,i ) specific constants of pure compound i in the Mathias

and Copeman expressionGE ) excess Gibbs energyNP ) number of experimental data pointsobj ) objective functionP ) pressure (bar)R ) general gas constant (J/mol ·K)T ) absolute temperature (K)u ) packing fractionV ) molar volume (m3/mol)x ) liquid mole fractiony ) vapor mole fraction

Greek Letters

R(T) ) temperature-dependent function of aΓ ) residual activity coefficient in UNIFACθ ) area fracion in UNIFACφ ) segment fraction in UNIFACψ ) temperature-dependent function in the residual part of UNIFACω ) acentric factor

Subscripts

c ) critical stateFH ) Flory-Huggins part of UNIFACi ) component ik ) group in component im, n ) main group m, nr ) reducedres ) residual part of UNIFACSG ) Staverman-Guggengeim part of UNIFAC

Superscripts

cal ) calculatedexp ) experimentalL ) saturation liquid

Acronyms

CEoS ) cubic equation of stateEoS )equation of stateLCVM ) linear combination of Vidal and MichelsenMPSRK ) Chen-modified PSRKPR ) Peng-RobinsonPSRK ) predictive Soave-Redlich-KwongSRK ) Soave-Redlich-KwongVTPR ) volume-translated Peng-Robinson group contribution

equation of statevdW EoS ) van der Waals equation of statevdW1f ) van der Waals one-fluid

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ReceiVed for reView January 21, 2009ReVised manuscript receiVed May 15, 2009

Accepted May 27, 2009

IE900111H

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