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THERMODYNAMICS

JOURNAL REVIEW

Equations of State for the Calculation of Fluid-Phase Equilibria

Ya Song Wei and Richard J. Sadus Computer Simulation and Physical Applications Group, School of Information Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

Progress in de®eloping equations of state for the calculation of fluid-phase equilibria is re®iewed. There are many alternati®e equations of state capable of calculating the phase equilibria of a di®erse range of fluids. A wide range of equations of state from cubic equations for simple molecules to theoretically-based equations for molecular chains is considered. An o®er®iew is also gi®en of work on mixing rules that are used to apply equations of state to mixtures. Historically, the de®elopment of equations of state has been largely empirical. Howe®er, equations of state are being formulated increas- ingly with the benefit of greater theoretical insights. It is now quite common to use molecular simulation data to test the theoretical basis of equations of state. Many of these theoretically-based equations are capable of pro®iding reliable calculations, partic- ularly for large molecules.

Introduction Equations of state play an important role in chemical engi- 1979; Gubbins, 1983; Tsonopoulos and Heidman, 1985; Han neering design, and they have assumed an expanding role in et al., 1988; Anderko, 1990; Sandler, 1994; Economou and the study of the phase equilibria of fluids and fluid mixtures. Donohue, 1996. . Originally, equations of state were used mainly for pure com- The van der Waals was the first equation ponents. When first applied to mixtures, they were used only to predict vapor- coexistence. Later, the Redlich-Kwong for mixtures of nonpolarŽ Soave, 1972; Peng and Robinson, equation of stateŽ. Redlich and Kwong, 1949 improved the 1976.Ž and slightly polar compounds Huron et al., 1978; Asse- accuracy of the by introducing tem- lineau et al., 1978; Graboski and Daubert, 1978. . Subse- perature-dependence for the attractive term. SoaveŽ. 1972 quently, equations of state have developed rapidly for the and Peng and RobinsonŽ. 1976 proposed additional modifica- calculation of phase equilibria in nonpolar and polar mix- tions to more accurately predict the vapor , liquid tures. There are many advantages in using equations of state density, and equilibria ratios. Carnahan and StarlingŽ. 1969 , for phase equilibria calculations. Equations of state can be GuggenheimŽ. 1965 , and Boublik Ž. 1981 modified the repul- used typically over wide ranges of temperature and pressure, sive term of the van der Waals equation of state to obtain and they can be applied to mixtures of diverse components, accurate expressions for hard body repulsion. Other workers ranging from the light to heavy . They can be ŽChen and Kreglewski, 1977; Christoforakos and Franck, used to calculate vapor-liquid, liquid-liquid, and supercritical 1986; Heilig and Franck, 1989. modified both the attractive fluid-phase equilibria without any conceptual difficulties. and repulsive terms of the van der Waals equation of state. The calculation of phase equilibria has been discussed exten- In addition to modeling small molecules, considerable em- sively elsewhereŽ Sadus, 1992a, 1994; Sandler, 1994; Dohrn, phasis has been placed recently on modeling chain-like 1994; Raal and Muhlbauer,¨ 1998. and earlier reviews of vari- molecules. Based on the theories of PrigogineŽ. 1957 and ous aspects of equations of state are also availableŽ Martin, FloryŽ. 1965 , other workers Ž Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978. developed a perturbed hard- Correspondence concerning this article should be addressed to R. J. Sadus. chain theoryŽ. PHCT equation of state for chain molecules.

AIChE Journal January 2000 Vol. 46, No. 1 169 To overcome the mathematical complexity of the PHCT Table 1. Modifications to the Attractive Term of equation of state, Kim et al.Ž. 1986 developed a simplified van der Waals Equation versionŽ. SPHCT by replacing the complex attractive part of Equation Attractive Term Ž.yZatt a PHCT by a simpler expression. To take into account the a increase in attractions due to dipolar and quadrupolar , Redlich-KwongŽ.Ž. RK 1949 RT1.5 Ž. Vqb Vimalchand and DonohueŽ. 1985 obtained fairly accurate aTŽ. multipolar mixture calculations by using the perturbed SoaveŽ.Ž. SRK 1972 q anisotropic chain theoryŽ. PACT . Ikonomou and Donohue RTŽ. V b Ž.1986 extended PACT to obtain an equation of state which aTŽ. V Peng-RobinsonŽ.Ž. PR 1976 takes into account the existence of hydrogen bonding, namely, RTw VŽ.Ž. Vqb qbVyb x the associated perturbed anisotropic chain theoryŽ. APACT aTŽ. FullerŽ. 1976 equation of state. RTŽ. Vqcb Advances in statistical and an increase of com- aTŽ. V puter power have allowed the development of equations of HeyenŽ. 1980 RTw V 2qŽŽbT .qcV .ybT Ž . cx state based on molecular principles that are accurate for real Ž.Sandler, 1994 fluids and mixtures. Using Wertheim’s theoryŽ Wertheim, aTŽ. V Schmidt-WenzelŽ. 1980 1984a,b.Ž.Ž. , Chapman et al. 1990 and Huang and Radosz 1990 RTŽ. V 22qubVqwb developed the statistical associating fluid theoryŽ. SAFT aTŽ. V which is accurate for pure fluids and mixtures containing as- Harmens-KnappŽ. 1980 RTw V 22qVcbyŽ.cy1 b x sociating fluids. Recently, various modified versions, such as aTŽ. V LJ-SAFTŽ Banaszak et al., 1994; Kraska and Gubbins, Ž. Kubic 1982 2 1996a,b.Ž and VR-SAFT Gil-Villegas et al., 1997 . have been RTŽ. Vqc developed. A common feature of many newly developed aTŽ. V Patel-TejaŽ.Ž. PT 1982 equations of state is the increasing use of insights gained from RTw VŽ.Ž. Vqb qcVyb x molecular simulationŽ. Sadus, 1999a to improve the accuracy aTŽ. V Adachi et al.Ž. 1983 of the underlying model. RTwŽ.Ž. VybVqb x Our aim is to present a wide-ranging overview of recent 23 aTŽ. V progress in the development of equations of state encompass- Stryjek-VeraŽ.Ž SV 1986a . ing both simple empirical models and theoretically-based RTwŽ. V 22q2bVyb x equations. Many useful equations of state can be constructed aTŽ. V Yu and LuŽ. 1987 by combining different models of repulsive and attractive in- RTw VŽ.Ž. Vqc qb 3Vqc x termolecular interactions. The theoretical backbone of the aTŽ. V contribution from intermolecular repulsion is the concept of Trebble and BishnoiŽ.Ž. TB 1987 RTw V 22qŽ.ŽbqcVy bcqd .x hard-sphere repulsion, whereas attractive interactions are aTŽ. V modeled commonly using empirical or semi-empirical ap- Schwartzentruber and RenonŽ. 1989 w q q q x proach. Polymers and large chain-like molecules can be RTŽ.Ž. V cV2c b treated successfully by using a hard-sphere chain model and this aspect is discussed. Equations of state for associating molecules are examined in detail and comparisons between different equations of state are considered. Commonly, equa- the critical properties of the fluid. The van der Waals equa- q tions of state are developed initially for pure substances. Mix- tion can be regarded as a ‘‘hard-sphereŽ. repulsive ing rules are required to extend equations of state to mix- attractive’’ term equation of state composed from the contri- tures, and this topic is also addressed. bution of repulsive and attractive intermolecular interactions, respectively. It gives a qualitative description of the vapor and liquid phases and phase transitionsŽ Van Konynenburg and Equations of State for Simple Molecules Scott, 1980. , but it is rarely sufficiently accurate for critical The van der Waals equation of state, proposed in 1873 properties and phase equilibria calculations. A simple exam- Ž.Rowlinson, 1988 , was the first equation capable of repre- ple is that for all fluids, the critical factor pre- senting vapor-liquid coexistence dicted by Eq. 1 is 0.375, whereas the real value for different hydrocarbons varies from 0.24 to 0.29. The van der Waals equation has been superseded by a large number of other, Va Zsy Ž.1 more accurate equations of state. Many of these equations V y b RTV can be categorized in terms of modifications to the basic van der Waals model. where Z is the compressibility factor Ž.Zs pVrRT , T is tem- perature, V is , p is the pressure, and R is the molar © universal constant. The parameter a is a measure of the Modification of the attracti e term attractive forces between the molecules, and the parameter b Many modifications of the attractive term have been pro- is the covolume occupied by the moleculesŽ if the molecules posed. Some of these are summarized in Table 1. are represented by hard-spheres of diameter ␴ , then bs Benedict et al.Ž. 1940 suggested a multiparameter equation 2␲ N␴ 3r3. . The a and b parameters can be obtained from of state, known as the Benedict-Webb-RubinŽ. BWR equa-

170 January 2000 Vol. 46, No. 1 AIChE Journal tion showed that the Redlich-Kwong equation can be adapted to predict both vapor and liquid properties. Several other work- y y r 2 y ␣ ersŽ. Deiters and Schneider, 1976; Baker and Luks, 1980 ap- BRT000A C T bRT a a Zs1qqqplied the Redlich-Kwong equation to the critical properties RTV RTV25 RTV ž/ž/ and the high-pressure phase equilibria of binary mixtures. c ␥␥ For ternary mixtures, Spear et al.Ž. 1971 gave seven examples q 1q exp y Ž.2 ž/ž/ž/RT32 V V 2 V 2 of systems for which the vapor-liquid critical properties of hydrocarbon mixtures could be calculated by using the ␣ ␥ Redlich-Kwong equation of state. The results showed that the where A000, B , C , a, b, c, , are eight adjustable parame- ters. The BWR equation could treat supercritical compo- accuracy of the Redlich-Kwong equation of state calculations nents and was able to work in the critical area. However, the for ternary systems was only slightly less than that for the BWR equation suffers from three major disadvantages. First, constituent binaries. the parameters for each compound must be determined sep- The success of the Redlich-Kwong equation has been the arately by the reduction of plentiful, accurate pressure- impetus for many further empirical improvements. Soave Ž. r 1.5 volume-temperatureŽ. PVT and vapor-liquid-equilibrium 1972 suggested replacing the term a T with a more gen- Ž. Ž.VLE data. Secondly, the large number of adjustable param- eral temperature-dependent term aT , that is eters makes it difficult to extend to mixtures. Thirdly, its ana- lytical complexity results in a relatively long computing time. VaTŽ. Zsy Ž.5 Today, because of advances in computing capabilities, the V y bRTVŽ.q b latter disadvantage has lost much of its significance but the other disadvantages remain. where Perhaps, the most important model for the modification of the van der Waals equation of state is the Redlich-Kwong 2¶ equationŽ. Redlich and Kwong, 1949 . It retains the original RT2 c2 T 0.5 van der Waals hard-sphere term with the addition of a tem- aTŽ.s0.4274 1q m 1y pTccž/ perature-dependent attractive term ž/½5• 6 ms0.480q1.57␻ y0.176␻ 2 Ž. Va RT c Zsy Ž.3 bs0.08664 ß V y b RT1.5Ž. V q b pc

For pure substances, the equation parameters a and b are and ␻ is the acentric factor. To test the accuracy of Soave- usually expressed as Redlich-KwongŽ. SRK equation, the vapor of a number of hydrocarbons and several binary systems were cal- s 2 2.5r a 0.4278 R Tccp culated and compared with experimental dataŽ. Soave, 1972 . s r Ž.4 In contrast to the original Redlich-Kwong equation, Soave’s b 0.0867RTccp 5 modification fitted the experimental curve well and was able to predict the phase behavior of mixtures in the critical re- Ž. Carnahan and Starling 1972 used the Redlich-Kwong equa- gion. Elliott and DaubertŽ. 1985 reported accurate correla- tion of state to calculate the gas-phase enthalpies for a r tions of vapor-liquid equilibria with the Soave equation for variety of substances, many of which are polar and or not 95 binary systems containing hydrocarbon, hydrogen, nitro- spherically symmetric. Their results showed that the gen, hydrogen sulfide, carbon monoxide, and carbon dioxide. Redlich-Kwong equation is a significant improvement over Elliott and DaubertŽ. 1987 also showed that the Soave equa- Ž. the van der Waals equation. Abbott 1979 also concluded tion improved the accuracy of the calculated critical proper- that the Redlich-Kwong equation performed relatively well ties of these mixtures. Accurate resultsŽ. Han et al., 1988 were Ž for the simple fluids Ar, Kr, and Xe for which the acentric also obtained for calculations of the vapor-liquid equilibrium . factor is equal to zero , but it did not perform well for com- of symmetric mixtures and methane-containing mixtures. plex fluids with nonzero acentric factors. In 1976, Peng and RobinsonŽ. 1976 redefined aT Ž.as The Redlich-Kwong equation of state can be used for mix- tures by applying mixing rules to the equation of state param- 2¶ eters. Joffe and ZudkevitchŽ. 1966 showed that a substantial RT2 c2 T 0.5 improvement in the representation of fugacity of gas mix- aTŽ.s0.45724 1q k 1y pTccž/ tures could be obtained by treating interaction parameters as ž/½5• 7 empirical parameters. Calculations of the critical properties ks0.37464q1.5422␻ y0.26922␻ 2 Ž. of binary mixtures indicated that, for most binary mixtures, RT c the accuracy of the predicted critical properties could be im- bs0.07780 ß pc proved substantially by adjusting the value of the interaction parameter in the mixing rule for the a term. Spear et al. Ž.1969 also demonstrated that the Redlich-Kwong equation Recognizing that the critical compressibility factor of the s of state could be used to calculate the vapor-liquid critical Redlich-Kwong equation Ž.Zc 0.333 is overestimated, they properties of binary mixtures. Chueh and PrausnitzŽ. 1967a,b also proposed a different volume dependence

AIChE Journal January 2000 Vol. 46, No. 1 171 VaTVŽ. The compressibility factor at the critical point is Zsy Ž.8 V y bRTVVwxŽ.Ž.q b q bVy b y ␤ q ␤ y q ␤ Ž.1 c Ž.Ž.2 ccc1 c cc Z Ž.␤ s Ž.12 The Peng-RobinsonŽ. PR equation of state slightly improves c q ␤ y ␤ 2 Ž.Ž.2 ccc1 c the prediction of liquid and predicts a critical com- s pressibility factor of Zc 0.307. Peng and RobinsonŽ. 1976 gave examples of the use of their equation for predicting the Fuller’s modification contains two useful features. First, the vapor pressure and volumetric behavior of single-component equation of state leads to a variable critical compressibility systems, and the phase behavior and volumetric behavior of factor, and secondly, a new universal temperature function is the binary, ternary, and multicomponent system and con- incorporated in the equation making both the a and b pa- cluded that Eq. 8 can be used to accurately predict the vapor rameters functions of temperature. Fuller’s equation can be pressures of pure substances and equilibrium ratios of mix- reduced to the Soave-Redlich-Kwong and van der Waals ␤ s s ⍀ s tures. The Peng-Robinson equation performed as well as or equations. If ca0.259921, then we have c 1, ⍀ s s better than the Soave-Redlich-Kwong equation. Han et al. 0.4274802, bc0.0866404, Z 0.333, and the Soave- ␤ Ž.1988 reported that the Peng-Robinson equation was supe- Redlich-Kwong equation is obtained. If c has a value of r s ⍀ s ⍀ s s rior for predicting vapor-liquid equilibrium in hydrogen and 1 3, then c 0, abc0.421875, 0.125, Z 0.375, and nitrogen containing mixtures. the van der Waals equation is obtained. The Peng-Robinson and Soave-Redlich-Kwong equations FullerŽ. 1976 reported that Eq. 9 could be used to corre- are used widely in industry. The advantages of these equa- late saturated liquid volumes to a root-mean-square devia- tions are that they can accurately and easily represent the tion of less than 5%. In the majority of cases it also improved relation among temperature, pressure, and phase composi- the vapor-pressure deviations of the original Soave-Redlich- tions in binary and multicomponent systems. They only re- Kwong equation. The results of calculations indicated that quire the critical properties and acentric factor for the gener- this equation is capable of describing even polar molecules alized parameters. Little computer time is required and good with reasonable accuracy. phase equilibrium correlations can be obtained. However, the In common with Fuller’s equation, Table 1 shows that a success of these modifications is restricted to the estimation feature of many of the empirical improvements is the addi- of vapor pressure. The calculated saturated liquid volumes tion of adjustable parameters. A disadvantage of three or are not improved and are invariably higher than experimen- more parameter equations of state is that the additional pa- tal measurements. rameters must be obtained from additional pure component FullerŽ. 1976 proposed a three parameter equation of state data. They almost invariably require oneŽ. or more additional which has the form mixing rules when the equation is extended to mixtures. The Peng-Robinson and Soave-Redlich-Kwong equations fulfill the requirements of both simplicity and accuracy since they VaTŽ. Zsy Ž.9 require little input information, except for the critical proper- V y bRTVŽ.qcb ties and acentric factor for the generalized parameters a and b. Consequently, although many equations of state have been The c parameter in Eq. 9 is defined in terms of the ratio of developed, the Peng-Robinson and Soave-Redlich-Kwong the covolume to the volume Ž.␤ s brV equations are widely used in industry, and often yield a more accurate representationŽ. Palenchar et al., 1986 than other alternatives. The application of cubic equations of state to 1133 cŽ.␤ syy Ž.10 the calculation of vapor-liquid equilibria has been discussed ␤␤ž/( 42 recently by Orbey and SandlerŽ. 1998 .

The other equation of state parameters are calculated from Modification of the repulsi©e term ¶ The other way to modify the van der Waals equation is to RTc examine the repulsive term of a hard-sphere fluid. Many ac- bs⍀ ␤ bŽ. curate representations have been developed for the repulsive pc interactions of hard spheres and incorporated into an equa- Ž.Ž.Ž.1y ␤ 2qc␤ y 1qc␤ ⍀ ␤ s ␤ tion of state. Some alternative hard-sphere terms are summa- bŽ. Ž.Ž.2qc␤ 1y ␤ 2 rized in Table 2. Perhaps the most widely used alternative to ⍀ ␤ 2 ␣ the van der Waals hard-sphere term is the equation proposed acŽ.RT Ž.T aTŽ.s • by Carnahan and StarlingŽ. 1969 who obtained an expression pc Ž.11 for the compressibility factor of hard-sphere fluids that com- Ž.Ž.1qc␤ 2 ⍀ ␤ pares very wellŽ. Figure 1 with molecular-dynamics data Ž Al- ⍀ ␤ s b der and Wainwright, 1960. . The form of the Carnahan-Star- aŽ. 2 ␤ Ž.Ž.1y ␤ 2qc␤ ling equation is ␣ 1r21s q ␤ y r2 Ž.T 1 q Ž .Ž.1 Tr 2 3 ␤ s ␤r 1r4 1q␩q␩ y␩ qŽ. Ž0.26 .m Z hss Ž.13 ß y␩ 3 ms0.480q1.5740␻ y0.176␻ 2 Ž.1

172 January 2000 Vol. 46, No. 1 AIChE Journal Table 2. Modifications to the Repulsive Term of the Žas0.4963RT2 c2rpccc, bs0.18727RT rp .Ž.. Sadus 1993 has van der Waals Equation demonstrated that Eq. 14 can be used to predict the Type III Equation Repulsive Term Ž.Z hs equilibria of nonpolar mixtures with considerable accuracy. The Guggenheim equationŽ. Guggenheim, 1965 is a simple 1q␩q␩ 2 Ž. alternative to the Carnahan-Starling equation. It also incor- Reiss et al. 1959 3 Ž.1y␩ porates an improved hard-sphere repulsion term in conjunc- 1q␩q␩ 2 tion with the simple van der Waals description of attractive Ž. Thiele 1963 3 interactions Ž.1y␩ 1 Ž. 1 a Guggenheim 1965 4 sy Ž.y␩ Z 4 Ž.15 1 Ž.1y␩ RTV 1q␩q␩ 23y␩ Ž. Carnahan-Starling 1969 cc Ž.1y␩ 3 The covolume Žb s 0.18284RT rp .Žand attractive as 2 c2r c RTŽ. V q b 0.49002 RT p . equation of state parameters are related Scott et al.Ž. 1971 to the critical properties. VVŽ.y b The Guggenheim equation of state has been used to pre- 1qŽ.Ž3␣ y2 ␩q 3␣ 2223y3␣ q1 .␩ y ␣␩ BoublikŽ. 1981 dict the critical properties of a diverse range of binary mix- Ž.1y␩ 3 turesŽ Hicks and Young, 1976; Hurle et al., 1977a,b; Hicks et al., 1977, 1978; Semmens et al., 1980; Sadus and Young, 1985a,b; Waterson and Young, 1978; Toczylkin and Young, 1977, 1980a,b,c; Sadus, 1992a, 1994; Wei and Sadus, 1994a, ␩s r where b 4V is the packing fraction defined in terms of 1999. . Despite the diversity of the systems studied, good re- Ž. the molecular covolume b . sults were reported consistently for the vapor-liquid critical To improve the accuracy of the van der Waals equation, locus. The critical liquid-liquid equilibria of Type II mixtures Ž. Carnahan and Starling 1972 substituted Eq. 13 for the tra- was also represented adequately. In contrast, calculations in- rŽ.y ditional term V V b in Eq. 1, resulting in the following volving Type III equilibria are typically only semiquantitative Ž. equation of state CSvdW Ž.Christou et al., 1986 because of the added difficulty of pre- dicting the transition between vapor-liquid and liquid-liquid 2 3 1q␩q␩ y␩ a behavior. The Guggenheim equation has also proved valu- Zsy Ž.14 Ž.1y␩ 3 RTV able in calculating both the vapor-liquid critical properties Ž.Sadus and Young, 1988 and general critical transitions of ternary mixturesŽ. Sadus, 1992a; Wei and Sadus, 1994b . Good Both a and b can be obtained by using critical properties results have also been reported for the vapor-liquid and liq- uid-liquid equilibria of binary mixtures containing either he- liumŽ.Ž. Wei and Sadus, 1996 or Wei et al., 1996 as one component. BoublikŽ. 1981 generalized the Carnahan-Starling hard- sphere term for molecules of arbitrary geometry via the intro- duction for a nonsphericity parameter Ž.␣ . Svejda and Kohler Ž.1983 employed the Boublik expression in conjunction with Kihara’sŽ. 1963 concept of a hard convex body Ž HCB . to ob- tain a generalized van der Waals equation of stateŽ. HCBvdW

1qŽ.Ž3␣ y2 ␩q 3␣ 2 y3␣ q1 .␩ 2 y ␣ 2␩ 3 a ZsyŽ.16 Ž.1y␩ 3 RTV

Sadus et al.Ž. 1988 and Christou et al. Ž. 1991 have used Eq. 16 for the calculation of the vapor-liquid critical properties of binary mixtures containing nonspherical molecules. The re- sults obtained were slightly better than could be obtained from similar calculations using the Guggenheim equation of state. SadusŽ. 1993 proposed an alternative procedure for ob- taining the equation of state parameters. Equation 16 in con- junction with this modified procedure can be used to predict Type III critical equilibria of nonpolar binary mixtures with a Figure 1. Hard-sphere compressibility factors from dif- good degree of accuracy. ferent equations of state with molecular simu- SadusŽ. 1994 compared the compressibility factors pre- lation data. dicted by the van der Waals, Guggenheim and Carnahan- ^, Alder and WainwrightŽ. 1960 ; `, Barker and Hender- Starling hard-sphere contributions with molecular simulation sonŽ. 1971 for hard spheres. dataŽ Alder and Wainwright, 1960; Barker and Henderson,

AIChE Journal January 2000 Vol. 46, No. 1 173 1971.Ž. for a one-component hard-sphere fluid Figure 1 . The Christoforakos and FranckŽ. 1986 proposed an equation comparison demonstrates that the Guggenheim and Carna- Ž.CF of state which used the Carnahan-Starling Ž 1969 . ex- han-Starling hard-sphere terms are of similar accuracy at low pression for the repulsive term and a square-well intermolec- to moderately high densities. It is at these densities that most ular potential for attractive intermolecular interactions. fluid-fluid transitions occur. The failure of the van der Waals hard-sphere term at mod- V 32qV ␤ qV␤ 23y ␤ 4␤⑀ erate to high densities has important consequences for the sy␭3y y Z 3 Ž.1 exp 1 prediction of phase equilibria at high pressure. For example, Ž.V y ␤ VRTž/ it has been demonstratedŽ. Sadus, 1992a that equations of Ž.19 state such as the Redlich-Kwong and Peng-Robinson equa- tions do not describe correctly the liquid-liquid critical behav- c 3rm ior of mixtures that can be classified as exhibiting Type III where ␤ s bTŽ rT . , m is typically assigned a value of 10, behavior in the classification scheme of van Konynenburg and and V denotes molar volume. The other equation of state ScottŽ. 1980 . In contrast, equations of state with more accu- parameters can be derived from the critical properties rate hard-sphere terms, such as the Carnahan-Starling or Guggenheim terms, can be often used to reproduce quantita- bs0.04682 RT crpc ¶ tively the features of Type III behaviorŽ Sadus, 1992a, 1993; ⑀ • c 3 Ž.20 Wei et al., 1996. . sT ln 1q2.65025rŽ.␭ y1 ß R © © Combining modification of both attracti e and repulsi e ⑀ terms The parameter reflects the depth of the square-well inter- molecular potential, and ␭ is the relative width of the well. Other equations of state have been formed by modifying This equation was applied successfully to the high tempera- both attractive and repulsive terms, or by combining an accu- ture and high-pressure phase behavior of some binary aque- rate hard-sphere model with an empirical temperature de- ous mixturesŽ. Christoforakos and Franck, 1986 . pendent attractive contribution. Heilig and FranckŽ. 1989, 1990 modified the Christo- Carnahan and StarlingŽ. 1972 combined the Redlich-Kwong forakos-Franck equation of state. They used a temperature- attractive term with their repulsive termŽ. CSRK dependent Carnahan-StarlingŽ. 1969 representation of repul- sive forces between hard spheres and an alternative square- 1q␩q␩ 2 y␩ 3 a well representation for attractive forces. The functional form Zsy Ž.17 of the Heilig-FranckŽ. HF equation of state is Ž.1y␩ 31.5RTŽ. V q b

V 3q ␤ V 2 q ␤ 2V y ␤ 3 B Their results demonstrated that this combination improved Zsq21 3 q r Ž. the prediction of hydrocarbon densities and supercritical Ž.V y ␤ V Ž.C B phase equilibria. De Santis et al.Ž. 1976 also tested Eq. 17 and concluded that it yielded good results for the case of The B and C terms in Eq. 21 represent the contributions pure components spanning ideal gases to saturated liquids. from the second and third virial coefficients, respectively, of When applied to mixtures for predicting vapor-liquid equilib- a hard-sphere fluid interacting via a square-well potential. ria, good accuracy in wide ranges of temperature and pres- This potential is characterized by three parameters reflecting sure can be obtained. intermolecular separation Ž.␴ , intermolecular attraction Chen and KreglewskiŽ. 1977 demonstrated that a good rep- Ž.⑀rRT , and the relative width of the well Ž.␭ . The following resentation of the phase behavior of simple fluids could be universal valuesŽ. Mather et al., 1993 were obtained by solv- obtained using an equation of state formed by substituting ing the critical conditions of a one-component fluid, ␭s the a term in Eq. 17 with the power series fit of simulation ␴ 3r c s 1.26684, NAAV 0.24912Ž where N denotes Avogadro’s data reported by Alder et al.Ž. 1972 for square-well fluids. constant. and ⑀rRT cs1.51147. Accurate calculations of the The functional form of this equation dubbed ‘BACK’ is critical properties of both binary and ternary mixturesŽ Heilig and Franck, 1989, 1990. have been reported. Shmonov et al. 1q␩q␩ 23y␩ u ij␩ Ž.1993 used Eq. 21 to predict high-pressure phase equilibria sy Z ÝÝjDij Ž.18 for the waterqmethane mixture and reported that the 1y␩ 3 kT ␶ Ž. ij Heilig-Franck equation of state was likely to be more accu- rate than other ‘‘hard-sphereqattractive term’’ equations of where Dij are universal constants which have been fitted to state for the calculation of phase equilibria involving a polar accurate pressure-volume-temperature, internal energy, and molecule. second virial coefficient data for argon by Chen and Kre- DeitersŽ. 1981 reported a further example of an equation glewskiŽ. 1977 , and urk is the temperature-dependent dis- of state, based on the Carnahan-Starling hard-sphere term persion energy of interaction between molecules and ␶ s plus an improved attractive term. A feature of Deiters’ equa- 0.7405. Historically, the complicated nature of the attractive tion of state is that the Carnahan-Starling term is adjusted to term has precluded it from routine phase equilibria calcula- more accurately reflect the repulsion of real fluids and the tions. However, as discussed below, the BACK equation has attractive term is obtained from an approximate formulation been the inspiration for further development. of square-well fluid interactions. The equation works well at

174 January 2000 Vol. 46, No. 1 AIChE Journal low pressures, but the critical properties at high pressures are for one-component hard-sphere fluid. Figure 1 shows that the not predicted accuratelyŽ. Mainwaring et al., 1988 . accuracy of hard-sphere term of Shah et al. equation is close Shah et al.Ž. 1994 developed an equation of state by defin- to the accuracy of the Carnahan-Starling hard-sphere term. ing empirical relationships for the repulsive ŽZ hs. and the att attractive ŽZ . contribution to the compressibility factor Equations of State for Chain Molecules Perturbed hard chain theory ␣ ¶ V kV1 Z hssq PrigogineŽ. 1957 introduced a theory to explain the proper- V y k ␣ y ␣ 2 Ž.0 Ž.V k0 ties of chain molecules which is based on the premise that 2 q ␣ some rotational and vibrational depend on density aV k0 cV Zatt sy and hence affect the equation of state and other configura- VVŽ.q eVŽ.y k ␣ RT 0 • tional properties. Based on Prigogine’s ideas, FloryŽ. 1965 T Ž.22 proposed a simple theory for polymer behavior. Flory’s work ␣ s0.165V exp y0.03125ln is similar to Prigogine’s theory except the expressions used to c T ½ ž/c account for intermolecular interactions are taken from free- 3 2 volume concepts instead of lattice theory. Limitations of both T y0.0054 ln Prigogine’s and Flory’s theories are that they can only be used T ß ½5ž/c 5 at high densities and are limited to calculations of liquid- phase properties. They give qualitatively incorrect results at low densities, because they do not approach the law In Eq. 22, ␣ represents the molar hard-sphere volume of the at zero density. s s fluid, k011.2864, k 2.8225, and e is a constant, and a and Based on perturbed hard-sphere theory for small molecules c are temperature-dependent parameters. A new, quartic Ž.valid at all densities and Prigogine’s theory for chain equation of state equation is formed by combining the above moleculesŽ. valid only at liquid-like densities , Beret and relationships PrausnitzŽ. 1975 developed the perturbed hard-chain theory Ž.PHCT equation of state for fluids containing very large ␣ q ␣ molecules, as well as simple molecules. The PHCT equation V kV10 aV k c Zsq y of state is valid for both gas- and liquid-phase properties. The V y k ␣ y ␣ 2 RT V q eVy k ␣ Ž.00Ž.V k0 Ž. Ž.PHCT equation differs from Prigogine’s and Flory’s equa- Ž.23 tions in two important aspectsŽ Vimalchand and Donohue, 1989. . First, to increase the applicability of the PHCT equa- tion to a wider range of density and temperature, more accu- Equation 23 only needs three properties of a fluid, namely, rate expressions in the PHCT equation are used for the the critical temperature, the critical volume, and acentric repulsive and attractive partition functions. Secondly, the factor to reproduce pressure-volume-temperature and ther- PHCT equation corrects the major deficiency in the theories modynamic properties accurately. Although it is a quartic of Prigogine and Flory by meeting the ideal gas limit at low equation and yields four roots, one root is always negative densities. and, hence, physically meaningless; therefore, it behaves like The PHCT equation of state is derived from the following a cubic equation. Shah et al.Ž. 1994 compared their quartic partition function Ž.Q equation with the Peng-RobinsonŽ. 1976 and Kubic Ž. 1982 equations of state and concluded that it was more accurate V N V NNy␾ than either the Peng-Robinson or the Kubic equation of state. f N Qs exp Ž.q ® Ž.24 Lin et al.Ž. 1996 extended the generalized quartic equation N !⌳3 N ž/V ž/2kT r, of stateŽ. Eq. 23 to polar fluids. When applied to polar fluids, the equation requires four characteristic properties of the where q is the contribution of rotational and vibrational pure component, namely: critical temperature, critical vol- r, ® of a molecule, N is the number of molecules, ⌳ is the ume, acentric factor, and dipole moment. They calculated thermal de Broglie wavelength and ␾ is the mean potential. thermodynamic properties of 30 polar compounds, and also The V term is the free-volume, which is defined as the vol- compared with experimental literature values and the Peng- f ume available to the center of mass of one molecule as it Robinson equation for seven polar compounds. Their results moves around the system holding the positions of all other showed that various thermodynamic properties predicted molecules fixed. The value of V can be calculated from the from the generalized quartic equation of state were in satis- f Carnahan and StarlingŽ. 1972 expression for hard-spheres factory agreement with the experimental data over a wide range of states and for a variety of thermodynamic proper- ties. The generalized quartic equation of state improves the Ž.Ž␶r®˜˜3␶r®y4 . s accuracy of calculations of enthalpy, second virial coeffi- Vf V exp2 Ž. 25 Ž.1y␶r®˜ cients, and the pressure-volume-temperature properties. In Figure 1, we compare the compressibility factor pre- s ® ®s r ®0 ®0 dicted by the Shah et al.Ž. 1994 hard-sphere term Žk0 1.2864, where ˜˜is a reduced volume defined as V nr Žwhere s s ␣r k1 2.8225, and y V . with molecular simulation data is the close-packed volume per mole and r is the number of Ž.Alder and Wainwright, 1960; Barker and Henderson, 1971 segments per molecule. .

AIChE Journal January 2000 Vol. 46, No. 1 175 In general, the PHCT equation can be written as equation with low values of the binary interaction parameter, and they extended the PHCT equation to predict the second a virial coefficient of both pure fluids and mixtures. Liu and Zs Z Ž.hard chain y Ž.26 RTV PrausnitzŽ. 1979a showed that the PHCT equation can be used to accurately predict the solubilities of gases in liquid When Eq. 25 is used to determine the hard chain compress- polymers, where the light component is supercritical, whereas ibility factor, and a is calculated from Alder’s polynomial the usual approach using excess functions is not useful in relationshipŽ. Alder et al., 1972 for square-well molecular dy- treating such supercritical gas-polymer systems. Liu and namics data, the functional form of the PHCT equation of PrausnitzŽ. 1979b, 1980 also applied the PHCT equation for state is phase equilibrium calculations in polymeric systemsŽ poly- mer-solvent, polymer-polymer, polymer-polymer-solvent. , taking into account the molecular distribution of poly- 4Ž.Ž.␶r®˜˜ y2 ␶r® 2 Zs1qc meric molecules. Ohzone et al.Ž. 1984 correlated the pure Ž.1y␶r®˜ 3 component parameters reported by Donohue and Prausnitz Ž.1978 and Kaul et al. Ž. 1980 with the group volumes of Bondi ⑀ 4 M qmAnm 1 Ž. Ž. q Ž.r®0 Ž.27 1968 . Chien et al. 1983 proposed a useful chain-of-rota- ÝÝ ®my1 ny1 ž/kTV s s ž/˜ ž/T˜ torsŽ. COR equation in the spirit of the BACK and PHCT n 1 m 1 equations. where c is 1r3 the number of external degrees of freedom, The PHCT equation is successful in calculating the proper- ⑀ q is characteristic energy per molecule, and k is Boltzmann’s ties of fluids, but a practical limitation is its mathematical complexity, as a result of the use of the Carnahan-Starling constant. The Anm terms are dimensionless constants and are independent of the nature of the molecules, and n, m are free-volume term and the Alder power series. Consequently, the index for the exponent in a Taylor series in a reciprocal calculations, especially for mixtures, are time consuming. In- reduced volume. In effect, the PHCT equation of state ex- spired by the proven ability of the PHCT equation in calcu- tends the BACK equation of stateŽ. Eq. 18 for chain lating the properties, modifications of the theory have been molecules. proposed with the aim of simplifying the equation of state. The adjustable parameters wr®0, Ž.⑀ qrk and cx of the PHCT equation of state can be obtained from pressure-volume-tem- Simplified perturbed hard chain theory perature data for gases and liquids and from vapor-pressure Kim et al.Ž. 1986 developed a simplified version of PHCT data. Beret and PrausnitzŽ. 1975 gave values for these param- equation by replacing the attractive term of the PHCT equa- eters for 22 pure fluids and compared theory with experiment tion with a theoretical, but simple, expression based on the for several fluids. The results indicated that the PHCT equa- local composition model of Lee et al.Ž. 1985 ; the equation is tion was applicable to a wide variety of fluids, from hydrogen called the simplified perturbed hard chain theoryŽ. SPHCT to eicosane to polyethylene. However, they also reported that the PHCT equation was not good in the critical region in U V Y common with other analytical equations of state. rep Zs1qcZ y Z U Ž.29 In re-deriving the PHCT equation to allow calculation of ž/m V qVY mixture properties, Donohue and PrausnitzŽ. 1978 slightly modified the perturbed hard-chain theory to yield better pure where c is 1r3 the number of external degrees of freedom component results and, more importantly, extended the per molecule, Zm is the maximum coordination number, U PHCT equation to multicomponent mixtures. The partition V represents the closed-packed molar volume given by function was given by ␴ 3r NsAAŽ.'2 , where N is Avogadro’s number, s is the num- ber of segments per molecule, and ␴ is the hard-core diame- N N VV® y␾ c ter of a segment. The remaining terms are calculated from Qs expŽ. 28 N !⌳3N ž/V 2ckT 4␩y2␩ 2 ¶ rep Equation 28 accounts for the effect of rotational and vibra- Z s Ž.1y␩ 3 tional degrees of freedom on both the repulsive and attrac- U tive forces. The pure component and binary parameters can T • Y sexp y1 Ž.30 be obtained by fitting experimental data. Donohue and ž/2T U PrausnitzŽ. 1978 reported that the PHCT equation of state ␩s␶ V rV can represent the properties of most mixtures commonly en- U ß T s⑀ qrck countered in petroleum refining and natural-gas processing even when the components differ greatly in size, shape, or potential energy. For pure substances, the SPHCT equation of state can be Subsequently, many workersŽ Kaul et al., 1980; Liu and written as Prausnitz, 1979a,b, 1980; Ohzone et al., 1984. have applied the PHCT equation to predict thermodynamic properties of ␩y ␩ 2 U 4 2 ZcVm Y numerous and varied types of systems of industrial interest. Zs1qc y U Ž.31 y␩ 3 q Kaul et al.Ž. 1980 predicted Henry’s constants using the PHCT Ž.1 V VY

176 January 2000 Vol. 46, No. 1 AIChE Journal U The SPHCT equation of state has three parameters Žc, T , real fluids. WertheimŽ. 1987 proposed a thermodynamic per- U and V . and these parameters can be evaluated by fitting turbation theoryŽ. TPT which accommodates hard-chain simultaneously both vapor pressure and liquid density data. molecules. As described below, TPT is the basis of many Kim et al.Ž. 1986 obtained these three parameters for several equations of state. The TPT concept has been extended to n-alkanes and multipolar fluids, and van Pelt et al.Ž. 1992 and include dimer propertiesŽ. Ghonasgi and Chapman, 1994 and Plackov et al.Ž. 1995 reported values for more than one hun- application to star-like moleculesŽ. Phan et al., 1993 and hard dred fluids. disksŽ. Zhou and Hall, 1995 have been reported. Work on To extend the SPHCT equation to mixtures, Kim et al. TPT models continues to be an active research areaŽ Stell et Ž.1986 proposed the following set of mixing and combining al., 1999a,b. . rules Chapman et al.Ž. 1988 generalized Wertheim’s TPT model to obtain the following equation of state for the compressibil- ity factor of a hard-chain of m segments s ␴ 3 ¶ UUs s iii ²:V ÝÝxViix i '2 i Ѩ ln g hsŽ.␴ hcs hsy y q␩ ²:c s xc Z mZ Ž.m 11 Ž.34 Ý ii ž/Ѩ␩ i ⑀ UUijq i • ® s ® y Ž.32 hs ²:c Y Ý xxcijiijexp 1 where g Ž.␴ is the hard-sphere site-site correlation function ž/2ckTi ij at contact, ␴ is the hard-sphere diameter, ␩s␲ m␳␴ 3r6is ␴ q␴ ␴ s ii jj the packing fraction, and ␳ is the number density. The com- ij 2 pressibility factor of hard sphere can be accurately deter- ⑀ s ⑀⑀ ß mined from the Carnahan-Starling equationŽ. Eq. 13 . For the ij' ii jj Carnahan-Starling equation, the site-site correlation function is The terms in angular brackets Ž²иии :. represent mixture y␩ properties. For mixtures, the SPHCT equation of state be- 2 g hsŽ.␴ s Ž.35 comes 21Ž.y␩ 3

U 4␩y2␩ 2 ZcVY² : Dimer properties have also been incorporated successfully s q²: y m Z 1 c 3 U Ž.33 Ž. Ž.1y␩ V q²:²:cV Y r c into the generalized Flory-type GF-D equations of state ŽHonnell and Hall, 1989; Yethiraj and Hall, 1991; Gulati and Hall, 1998. . The SPHCT equation of state retains the advantages of the Ghonasgi and ChapmanŽ. 1994 modified TPT for the PHCT equation, and it also can be used to predict the prop- hard-sphere chain by incorporating structural information for erties, at all densities, of fluids covering the range from argon the diatomic fluid. The compressibility factor of a hard chain and methane to polymer molecules. The SPHCT equation can be determined from the hard-sphere compressibility fac- predicts pure-component molar volumes and vapor pressures tor and the site-site correlation function at contact of both almost as accurately as the original PHCT equationŽ Beret hard spheres Ž g hs.Žand hard dimers g hd. and Prausnitz, 1975. . It can also predict mixture properties with reasonable accuracy using only pure-component proper- m Ѩ ln g hsŽ.␴ ties. This simpler equation has been used in a number of Zs mZhsy 1q␩ applications, including for mixtures of molecules which differ 2 ž/Ѩ␩ greatly in sizeŽ.Ž. Peters et al., 1988 . Van Pelt et al. 1991, 1992 applied this simplified version to binary critical equilibria. m Ѩ ln g hsŽ.␴ yy11q␩ Ž.36 Ponce-Ramirez et al.Ž. 1991 reported that the SPHCT equa- ž/2 ž/Ѩ␩ tion of state was capable of providing adequate phase equi- librium predictions in binary carbon dioxide-hydrocarbon systems over a range of temperature of interest to the oil ChiewŽ. 1991 obtained the site-site correlation results for hard industry. Plackov et al.Ž. 1995 showed that the SPHCT equa- dimers tion of state can be used to improve the quality of predicted vapor pressures of ‘‘chain-like’’ molecules over the entire 1q2␩ g hdŽ.␴ s Ž.37 range of vapor-liquid coexistence. Van Pelt et al.Ž. 1991 used 21Ž.y␩ 2 the SPHCT equation to calculate phase diagrams for binary mixtures and to classify these phase diagrams in accordance with the system of van Konynenburg and ScottŽ. 1980 . The addition of dimer properties into the hard-sphere chain equation of state is also a feature of the Generalized Flory- DimerŽ. GF-D equation of state Ž Honnell and Hall, 1989 . . Hard-sphere chain equations of state Chang and SandlerŽ. 1994 proposed two variants of a ther- The concept of a ‘‘hard-sphere chain’’ is the backbone of modynamic perturbation-dimerŽ. TPT-D equation. The TPT- many systematic attempts to improve equations of state for D1 equation can be expressed as

AIChE Journal January 2000 Vol. 46, No. 1 177 1q␩q␩ 2 y␩ 3 m ␩Ž.5y2␩ Ž.Vimalchand and Donohue, 1985; Vimalchand et al., 1986 s y q Z m 3 1 developed the perturbed anisotropic chain theoryŽ. PACT ž/Ž.1y␩ 21ž/Ž.Ž.y␩ 2y␩ which is applicable to simple as well as large polymeric m 2␩Ž.2q␩ molecules with or without anisotropic interactions. The PACT yy11q Ž.38 equation of state accounts for the effects of differences in ž/21Ž.Ž.y␩ 1q2␩ ž/molecular size, shape, and intermolecular forces including anisotropic dipolar and quadrupolar forces. In terms of the whereas the TPT-D2 can be represented as compressibility factor, the PACT equation of state can be represented as 1q␩q␩ 2 y␩ 3 m ␩Ž.5y2␩ Zs m y 1q 3 s q rep q iso q ani ž/Ž.1y␩ 21ž/Ž.Ž.y␩ 2y␩ Z 1 Z Z Z Ž.42 ␩ y ␩y ␩ 2 m Ž.3.498 0.24 0.414 The 1q Z rep term is evaluated in an identical way to the cor- yy11q Ž.39 ž/21ž/Ž.Ž.Žy␩ 2y␩ 0.534q0.414␩ . responding of the PHCT equation of stateŽ. see Eq. 27 . By extending the perturbation expansion of Barker and HendersonŽ. 1967 for spherical molecules to chainlike Using the TPT-D approachŽ. Eq. 36 , Sadus Ž. 1995 pro- molecules, the attractive Lennard-Jones isotropic interac- posed that, in general tions are calculated as g hds g hsŽ.␣␩qc Ž.40 LJ LJ 2 AA22 Z iso s Z LJq Z LJy2Z LJ 1y 43 where ␣ and c are the constants for a straight line and the 12 1LJ LJ Ž. ž/žAA11 / values can be obtained by fitting the molecular simulation data for g hs and g hd. Equation 40 can be used in conjunction with Eq. 36 to obtain a new equation of state, called the sim- where A is the Helmholtz function, LJ stands for Lennard- plified Thermodynamic Perturbation Theory-DimerŽ STPT- Jones, and iso denotes isotropic interactions. The contribu- D. equation of state. The general form of the STPT-D equa- tions to Eq. 43 are evaluated from tion of state for pure hard-sphere chains is LJ ¶ AcA11m Ѩ ln g hs ␣ 2y m ␩ s hs Ž. Ý ®m Zs1q mZŽ.Ž.y1 q 1y m ␩ q Ž.41 NkT T˜ m ˜ Ѩ␩ 2Ž.␣␩qc AcCCLJ C 21sqqm 2 m 3m 2 Ý ®m ®mq1 ®mq2 SadusŽ. 1995 applied the STPT-D equation to the prediction NkT T˜ m 2˜ ˜˜2 of both the compressibility factor of 4-, 8-, 16-, 51- and 201- cmA1m Z LJs mer hard-chains and the second virial coefficients of up to 1 Ý ®m T˜ m ˜ • 128-mer chains. Comparison with molecular simulation data 44 q q Ž. indicated that the STPT-D equation generally predicts both cmCm1m Ž.1 Cm2 m Ž.2 C3m Z LJsqq 2 2 Ý ®m ®mq1 ®mq2 the compressibility factor and the second virial coefficient T˜ m 2˜ ˜˜2 Ž more accurately than other equations of state Chiew equa- T ckT tion, GF-D, TPT-D1, TPT-D2. . T˜ss U ⑀ By using some elements of the one-fluid theory, Sadus T q Ž.1996 extended the STPT-D equation to hard-sphere chain ®®'2 ®s s mixtures without requiring additional equation of state pa- ˜ U ß ® Nr␴ 3 rameters. The compressibility factor predicted by the STPT-D A equation of state was compared with molecular simulation data for several hard-sphere chain mixtures containing com- sy sy s with the constants A118.538, A 125.276, A 13 3.73, ponents with either identical or dissimilar hard-sphere seg- sy s sy sy A147.54, A 1523.307, and A 1611.2, C 11 3.938, ments. Good agreement with simulation data was obtained sy sy s s s C123.193, C 134.93, C 1410.03, C 2111.703, C 22 when the ratio of hard-sphere segment diameters for the y s sy sy s 3.092, C234.01, C 2420.025, C 3137.02, C 32 26.93 component chains is less than 2. The accuracy of the STPT-D s and C33 26.673. equation compared favorably with the results obtained for The anisotropic multipolar interactions are calculated us- other hard-sphere chain equations of state. Recently, a sim- ing the perturbation expansion of Gubbins and TwuŽ. 1978 plification of the STPT-D equation has been proposedŽ Sadus, assuming the molecules to be effectively linear 1999b. . The word-sphere chain approach has also been ex- Ž. tended Sadus, 2000 to include HCB chains. 2 AAani ani anis aniq aniy ani 33y Z Z23Z 2Z 2ani1 ani Ž.45 Perturbed anisotropic chain theory ž/ž/AA22 By including the effects of anisotropic multipolar forces ex- plicitly into the PHCT equation of state, Vimalchand et al. where the superscript ani denotes the anisotropic interac-

178 January 2000 Vol. 46, No. 1 AIChE Journal tions and The Zassoc term for one and two bonding sites per molecule is evaluated from the material balance and expressions for assoc Ž ani Ž10. ¶ the chemical equilibria. The Z is given by Ikonomou and AcJ2 sy12.44 Donohue, 1986; Economou and Donohue, 1991, 1992. NkT 2 ® T˜Q ˜

ani Ž15. nT AcJcK3 assoc sy s2.611 q77.716 Z 148Ž. NkT 332® ® n0 T˜˜QQ˜˜T cJ Ž10.ŽѨ ln J 10. Zani sy12.44 1q ␳˜ where n is the true number of moles, and n is the number 2 ˜2 ® Ѩ␳ T 0 TQ ˜ ˜ of moles that would exist in the absence of association. The cJ Ž15.ŽѨ ln J 15. repulsive and the attractive terms in the APACT equation ani s q ␳ Z3 2.611 1 ˜ are association independent because of the assumptions made T˜3 ® Ѩ␳˜ • Q˜ 46 about the variation of the parameters of the associating Ž. rep attr cK Ѩ ln K species with the extent of association. The Z and Z are q77.716 2q ␳˜ given by Vimalchand et al.Ž. 1985, 1986 and Economou et al. ˜32® Ѩ␳ TQ˜ ˜ Ž.1995 ®®'2 ˜®sU s ® ␴ 3 2 ¶ NrA 4␩y2␩ rep s Z c 3 • T ckT 1y␩ 49 T˜ ss Ž. Ž. Q U ⑀ ß TQQq attrs LJq LJqиииq aniq ani qиии Z Z12Z Z 23Z r NQ5 3 2 ⑀ s As Q ß '2 as Economou and DonohueŽ. 1992 extended the APACT equation of state to compounds with three associating sites per molecule. The three-site APACT equation of state was where a is the surface area of a segment, ␴ is the soft-core s developed to allow calculation of vapor-liquid equilibria and diameter of a segment and the quadrupole moment, Q is re- 2 liquid-liquid equilibria of systems of water and hydrocarbons. lated to the quadrupolar interaction energy per segment, Qs U r Economou and DonohueŽ. 1992 tested the accuracy of the sQ2rrŽ® .5 3. In Eq. 46, the J terms are the integrals given three-site APACT equation over a large range of tempera- by Gubbins and TwuŽ. 1978 . tures and pressures for aqueous mixtures with polar and non- The PACT equation is valid for large and small molecules, polar hydrocarbons. They concluded that the three-site for nonpolar and polar molecules, and at all fluid densities. APACT equation was accurate in predicting phase equilibria The calculations of Vimalchand et al.Ž. 1986 show that the of different types for aqueous mixtures of nonpolar hydrocar- explicit inclusion of multipolar forces allows the properties of bons with no adjustable parameters. For aqueous mixtures highly nonideal mixtures to be predicted with reasonable ac- with polar hydrocarbons, the three-site APACT equation re- curacy without the use of a binary interaction parameter. quires a binary interaction parameter for the accurate esti- However, for pure fluids, the predictive behavior of the PACT mation of the phase equilibria. The comparisonŽ Economou equation of state is similar to other comparable equations of and Donohue, 1992. of the two-site and three-site APACT state. equations of state was also investigated for the prediction of the thermodynamic properties of pure water from the triple Equations of State for Associating Fluids point to the critical point. For most of the systems examined, Associated perturbed anisotropic chain theory the three-site APACT equation was in better agreement with the experimental data than the two-site APACT equation. Ikonomou and DonohueŽ. 1986 incorporated the infinite Smits et al.Ž. 1994 applied the APACT equation to the su- equilibrium model and monomer-dimer model into the PACT percritical region of pure water and showed that, over a large equation to derive the associated perturbed anisotropic chain pressure and temperature range that included the near-criti- theoryŽ. APACT equation of state. The APACT equation of cal region, the agreement between calculations and experi- state accounts for isotropic repulsive and attractive interac- mental data was good. They also reported that, although the tions, anisotropic interactions due to the dipole and difference between the APACT two-site and the APACT quadrupole moments of the molecules and hydrogen bond- three-site model appeared to be small, the accuracy of the ing, and is capable of predicting thermodynamic properties of three-site APACT equation of state for volumetric properties pure associating components as well as mixtures of more than was higher than that of the two-site model. Economou and one associating componentŽ. Ikonomou and Donohue, 1988 . PetersŽ. 1995 demonstrated that the APACT equation can be The APACT equation of state is written in terms of the com- applied to correlate the vapor pressure and the saturated liq- pressibility factor, as a sum of the contributions from these uid and saturated vapor densities of pure hydrogen fluoride particular interactions from the triple point up to the critical point with good accu- racy. Economou et al.Ž. 1995 applied the APACT equation to rep att assoc Zs1q Z q Z q Z Ž.47 water-salt phase equilibria and showed that the APACT

AIChE Journal January 2000 Vol. 46, No. 1 179 equation accounted explicitly for the strong dipole-dipole in- from teractions between water and salt molecules. Other theoreti- cally based models of association are discussed elsewhere Achain 1y␩r2 Ž.Stell and Zhou, 1992; Zhou and Stell, 1992 . sŽ.1y m ln Ž. 55 NkT Ž.1y␩ 3 Statistical associating fluid theory assoc By extending Wertheim’sŽ. 1984a,b, 1986a,b,c, 1987 theory, The term A is the Helmholtz function change due to as- Chapman et al.Ž. 1988, 1990 proposed a general statistical sociation and for pure components it can be calculated from associating fluid theoryŽ. SAFT approach. Huang and Ra- assoc doszŽ. 1990 developed the SAFT equation of state. The SAFT AX␣ 1 equation of state accounts for hard-sphere repulsive forces, s ln X␣ yqM Ž.56 NkT Ý 22 dispersion forces, chain formationŽ for nonspherical ␣ molecules. and association, and it is presented as a sum of four Helmholtz function terms where M is the number of association sites on each molecule, X␣ is the mole fraction of molecules that are not bonded at ␣ AAideal Aseg Achain Aassoc site , and the summation is over all associating sites on the sqqq Ž.50 molecule. NkT NkT NkT NkT NkT Huang and RadoszŽ. 1990 also gave the SAFT equation in terms of the compressibility factor where A and Aideal are the total Helmholtz function and the ideal gas Helmholtz function at the same temperature and s q hsq disp q chain q assoc density. Z 1 Z Z Z Z Ž.57 The term Aseg represents segment-segment interactions and can be calculated from where

Aseg Aseg 0 4␩y2␩ 2 ¶ s m Ž.51 hs NkT NkT Z s m Ž.1y␩ 3 ij␩ seg Ž u where m is the number of segments per chain, and A0 per Zdisp s mjD mole of segments. is the residual Helmholtz function of the ÝÝ ij kT ␶ ij • nonassociated spherical segments. It has two contributions: Ž.58 Ž.5r2 ␩y␩ 2 the hard-sphere and dispersion Zchain sŽ.1y m Ž.Ž.1y␩ wx1y 1r2 ␩ Aseg Ahs Adisp 11Ѩ X A 000sq Zassoc s ␳ y Ž.52 Ý A Ѩ␳ ß NkT NkT NkT A X 2

The hard-sphere term can be calculated as proposed by Car- Ž. nahan and StarlingŽ. 1969 Chapman et al. 1990 reported that the agreement with molecular simulation data was good at all the stages of model development for associating spheres, mixtures of associating hs ␩y ␩ 2 A0 4 3 spheres, and nonassociating chains up to ms8. Huang and s Ž.53 NkT Ž.1y␩ 2 RadoszŽ. 1990, 1991 applied their SAFT equation to corre- late vapor-liquid equilibria of over 100 real fluids, and they also demonstrated that the SAFT equation was applicable to Ž. For the dispersion term, Huang and Radosz 1990 used a small, large, polydisperse, and associating molecules over the Ž. power series that was initially fitted by Alder et al. 1972 to whole density range. Huang and RadoszŽ. 1991 tested 60 molecular dynamics data for a square-well fluids phase equilibrium data sets for asymmetricŽ. smallqlarge and associating binary systems. They concluded that the mixing disp ij␩ Au0 rules for the hard sphere, chain, and associating terms were s D Ž.54 NkTÝÝ ij kT ␶ not required when using rigorous statistical mechanical ex- ij pressions. Only the dispersion term required mixing rules, and only one binary temperature-independent parameter was re- where Dij are universal constants which have been fitted to quired to represent the experimental data. Yu and Chen accurate pressure-volume-temperature, internal energy, and Ž.1994 also used the SAFT equation to examine the liquid- second viral coefficient data for argon by Chen and Kre- liquid phase equilibria for 41 binary mixtures and 8 ternary glewskiŽ. 1977 , and urk is the temperature-dependent dis- mixtures using many of the parameters of Huang and Radosz persion energy of interaction between segments. Ž.1990, 1991 . Economou and Tsonopoulos Ž. 1997 applied The term Achain is due to the presence of covalent chain- APACT and SAFT equations of state to predict the phase forming bonds among the segments and can be determined equilibrium of waterrhydrocarbon mixtures.

180 January 2000 Vol. 46, No. 1 AIChE Journal ␨ Simplified statistical associating fluid theory The reduced densities l for a binary mixture are defined as Fu and SandlerŽ. 1995 developed a simplified statistical as- ␲␳n ␲␳ sociating fluid theoryŽ. SSAFT equation by modifying the dis- ␨ s ␴ llls ␴ q ␴ liiiÝ xm Ž.xm111xm 2 22 Ž.63 persion term of the equation. The SSAFT equation is of the 66is1 same form as Eq. 57. It retains the original hard-sphere ŽZ hs., chain ŽZchain.Ž, and association Zassoc.Žterms see Eq. 58 . but ␨ 3 is the overall packing fraction of the mixture, mi is the the contribution of dispersion interactions is given by ␴ number, and i is the diameter of spherical segments of chain i. U V Y The contribution due to the dispersive attractive interac- Zdisp sy mZ 59 M q U Ž. tions is given at the mean-field level in terms of the van der ž/Vs VY Waals one-fluid theory. For binary mixtures Equation 59 is the attraction term for a square-well fluidŽ Lee Amf ␳ et al., 1985. and was used in the SPHCT equation of state. sy ␣ 22q ␣ q ␣ 22 Ž.11xm 1 12 12xxmm 1 2 1 2 22xm 2 2 Ž.64 For pure components, the parameters of the SSAFT equa- NkT kT tion of state were obtained by fitting vapor pressure and liq- uid density data and the resultsŽ. Fu and Sandler, 1995 where ␣ represents the integrated strength of segment-seg- showed that the SSAFT equation was generally similar to, or ment mean-field attraction, and m2 is the number of spheri- slightly more accurate than, the original SAFT equation. cal segments of chain 2. When the SSAFT equation applied to both self-associating By using the HS-SAFT equation, Galindo et al.Ž. 1996, 1997 and cross-associating binary mixtures, only one binary ad- performed the phase equilibria predictions with good agree- justable parameter was needed. The comparisonŽ Fu and ment with experimental results for binary mixtures of water Sandler, 1995. with the original SAFT equation for binary q n-alkanesŽ. Galindo et al., 1996 and waterqhydro fluoride mixtures demonstrated that the simplified SAFT equation of Ž.Galindo et al., 1997 . Garcia-Lisbona et al. Ž. 1998 used HS- state usually led to better correlated results than the original SAFT to describe the phase equilibria of aqueous solutions SAFT equation, and was simpler and easier to use. of alkyl polyoxyethylene mixtures and reported that the HS- SAFT equation was able to describe the phase behavior of these systems. The results showed the reasonable agreement Hard-sphere statistical associating fluid theory() HS-SAFT between the theoretical prediction and experimental results. The HS-SAFT equation of state is a simplified version of the SAFT equation of state, which treats molecules as chains Lennard-Jonesstatistical associating fluid theory() LJ-SAFT of hard-sphere segments with van der Waals interactions. In the HS-SAFTŽ. Chapman et al., 1988 equation, the Helmholtz Banaszak et al.Ž. 1994 incorporated Lennard-Jones interac- function A is separated into different contributions as tions in the context of thermodynamic perturbation theory to formulate an equation of state for Lennard-Jones chains. They demonstrated the feasibility of the LJ-SAFT formula- AAideal Ahs Amf Achain Assoc s qqq q Ž.60 tion, however, comparison of the calculated compressibility NkT NkT NkT NkT NkT NkT factor for 8-mer, 16-mer, and 32-mer chains did not match simulation data. hs where A is the contribution from hard spheresŽ. HS to the Later, Kraska and GubbinsŽ. 1996a,b also developed an mf Helmholtz function and A are the contributions from equation of state for Lennard-Jones chains by modifying the long-range meanfieldŽ. mf dispersion forces to the Helmholtz SAFT equation of state in two major ways. First, a Lennard- hs mf mono function. The sum of A and A is called A Jones equation of state was used for the segment contribu- tion; secondly, a term was added that accounts for the Amono Ahs Amf dipole-dipole interaction in substances like the 1-alkanols and sq Ž.61 water. In terms of the Helmholtz function the general expres- NkT NkT NkT sion for the LJ-SAFT equation of state is

that is, the monomer-monomer contribution to the Helmholtz ideal seg chain assoc dipole seg AA A A A A function which refers to the term A in the original SAFT sqqqq Ž.65 equationŽ. Chapman et al., 1990; Huang and Radosz, 1990 . NkT NkT NkT NkT NkT NkT Galindo et al.Ž. 1996, 1997 and Garcia-Lisbona et al. Ž. 1998 used the expression of BoublikŽ. 1970 for the hard-sphere where Adipole is a term for the effect of long-range dipolar contribution interaction. For the hard-sphere term

hs ␨ 33␨␨ ␨ Ahs 5 ␩Ž.34y33␩q4␩ 2 A 6 s 3 12 2 s y␩ q sy␨ lnŽ. 1y␨ qq lnŽ. 1 2 Ž.66 NkT ␲␳ ␨ 22031y␨ ␨ y␨ NkT 3 61Ž.y␩ ž/3 Ž.3 33Ž.1 62 Ž. For the Lennard-Jones segment term, Kraska and Gubbins Ž.1996a,b used the Kolafa and Nezbeda Ž. 1994 equation which where ␳ s NrV is the total number density of the mixture. covers a larger range of temperature and density and is more

AIChE Journal January 2000 Vol. 46, No. 1 181 reliable outside the region of fit. It can be expressed as Square-well statistical associating fluid theory() SW-SAFT The thermodynamic perturbation formalism was extended Aseg m s hsq y␥␳U2 ␳ ⌬ q ir2␳U j by Banaszak et al.Ž. 1993 to obtain an equation for chains of A expŽ.T B2, hBHÝ CT ij NkT NkT ž/Ž.square-well statistical associating fluid theoryŽ. SW-SAFT ij molecules Ž.67 Ѩ ln g sws Ž.␴ , ␩ Z swcs mZ sws qŽ.1y m 1q␩ Ž.71 where ref ž/Ѩ␩ 0 ¶ sws ⌬ s ir2 where g is the site-site correlation function at contact of B2, hBHÝ CT i sws isy7 square-well spheres, and Zref is the compressibility factor of ␳U s r square-well spheres which can be calculated from Barker- mb Vm ␲ • Henderson perturbation theory. Tavares et al.Ž. 1995 also ␩s ␳␴U 3 Ž.68 BH proposed equation of states for square-well chains using the 6 SAFT approach. In addition to properties of monomer seg- 1 r ments, they included dimer properties in the spirit of the ␴ s DTi 2 q D ln T BHÝ i ln ß TPT-D1 equationŽ. Eq. 38 . The concepts have been extended isy2 to hetero-segmented moleculesŽ Adidharma and Radosz, 1998. . Recently, six different SW-SAFT models have been where Ciiijand D , C and Dln are numerical constants and reviewedŽ. Adidharma and Radosz, 1999 . According to the adjustable parameters, hBH stands for hybrid Barker- comparison reported by Adidharma and RadoszŽ. 1999 , the Henderson, and details were given by Kolafa and Nezbeda addition of dimer structure does not improve the usefulness Ž. 1994 . of SW-SAFT equations for representing real fluids signifi- For the dipole-dipole term, the resulting Helmholtz func- cantly. Originally, the validity of SW-SAFT equations were tion is limited to a square well with a width of 1.5␴ . This limitation has been addressed by subsequent workŽ Chang and Sandler, dipole A A2 1 . s 69 1994; Gil-Villegas et al., 1997 . y r Ž. NkT T ž/1 Ž.A32A By providing an additional parameter which characterized the range of the attractive part of the monomer-monomer where potential, Gil-Villegas et al.Ž. 1997 proposed a general ver- sion of SAFT for chain molecules formed from hard-core UU4 monomers with an arbitrary potential of variable rangeŽ. VR . 2␲␳␮ ¶ sy Ž6. As discussed above, the general form of the SAFT equation A2 U J 3 T of state can be written as U U 32␲ 3 14␲␳␮2 6 s 333 A U K ideal mono chain assoc 3222135( 5 T 2 AA A A A • sq q q Ž.72 U TTkB Ž.70 NkT NkT NkT NkT NkT T ss T˜ ⑀ In VR-SAFTŽ. Gil-Villegas et al., 1997 , the ideal Helmholtz U ␮ s␮r'⑀ m␴ 3 function Aideal and the contribution to the Helmholtz func- U ␳ s ␳b tion due to interaction association Aassoc are the same as in ß Eq. 50. The contribution due to the monomer segments Amono bs N ␴ 3 L is given by

Ž6. 333 The coefficients J and K 222 are integrals over two-body Amono Am Ahs s s q ␤ q ␤ 2 and three-body correlation function for the Lennard-Jones m m A12A Ž.73 fluid and have been calculated by Twu and GubbinsŽ. 1978a,b . NkT NkTž/ NkT Kraska and GubbinsŽ. 1996a,b applied the LJ-SAFT equa- m tion of state to pure fluids and binary mixtures. The results where A is the Helmholtz function per monomer, m is the ␤ s r for pure fluids showed substantially better agreement with number of monomers per chain, 1 kT, and A12and A experiment than the original SAFT equation for the phase are the first two perturbation terms associated with the at- diagram the n-alkanes, 1-alkanols, and water. The LJ-SAFT tractive well. equation of state was also foundŽ. Kraska and Gubbins, 1996b The contribution to the Helmholtz function due to the for- chain to be more accurate in describing binary mixtures of n- mation of a chain of m monomers A is alkanern-alkane, 1-alkanolrn-alkane, and waterrn-alkane chain than the original SAFT equation. Recently, Chen et al.Ž. 1998 A syŽ.Ž.1y m ln y M ␴ Ž.74 reported an alternative LJ-SAFT equation. They reported NkT that the vapor pressures calculated from their equation can be fitted to experiment more accurately than the original where y MŽ.␴ is the monomer-monomer background correla- SAFT equation. tion function evaluated at hard-core contact.

182 January 2000 Vol. 46, No. 1 AIChE Journal Gil-Villegas et al.Ž. 1997 gave the analytical expressions of for a range of well widthsŽ. 1F ␭F2 , and it leads to good F ␭F the A12and A for square-well fluids ranges 1.1 1.8 predictions of the compressibility factor of 4-mer, 8-mer, and 16-mer square-well chains. A completely analytical equation sws ®dW hs ␩ is desirable for phase equilibria calculations, but it comes at A1A 1g Ž.1: eff Ž.75 the cost of a considerable increase in mathematical complex- Ѩ sw 1 A1 ity. Asws ⑀K hs␩ Ž.76 2 2 Ѩ␩ Cubic plus association equation of state ⑀ ␭ where and are the depth and the range parameter of the Kontogeorgis et al.Ž. 1996 presented an equation of state attractive well, respectively, and suitable for describing associating fluids. The equation com- bines the simplicity of a cubic equation of stateŽ the Soave- ®dWsy ␩⑀ ␭3 y ¶ . A1 4 Ž.1 Redlich-Kwong and the theoretical background of the y␩ r perturbation theory employed for the association part. The 1 eff 2 g hsŽ.1;␩ s • resulting equation, called cubic plus associationŽ. CPA equa- eff y␩ 3 Ž.77 Ž.1 eff tion of state, was given by ␩ s ␩q ␩ 23q ␩ ß effc 1c 2c 3 Va 11Ѩ X␣ Zsy q␳ y 80 y y Ý Ѩ␳ Ž. The coefficients c were given by the matrix V bRTVŽ.bX␣ ž/␣ 2 n

y where the physical term is that of the Soave-Redlich-Kwong c1 2.25855 1.50349 0.249434 1 equation of state and the associating term is taken from SAFT c s y0.669270 1.40049 y0.827739 ␭ 2 equationŽ. Huang and Radosz, 1990 . Kontogeorgis et al. c 10.1576 y15.0427 5.30827 ␭2 03 00 Ž.1996 applied this new equation of state to pure components Ž.78 and obtained good correlations of both vapor pressures and saturated liquid volumes for primary-alcohols, phenol, tert- butyl alcohol, triethylene glycol, and water. hs In the second-order termŽ. Eq. 76 , K is the isothermal Voutsas et al.Ž. 1997 applied the CPA equation of state to compressibility of the hard-sphere fluid and was given by liquid-liquid equilibrium calculations in alcoholqhydro- carbon mixtures. They used the conventional van der Waals Ž.1y␩ 4 one-fluid mixing rules for the attractive parameter a and the K hss Ž.79 1q4␩q4␩ 2 co-volume parameter b. Satisfactory results were obtained in all cases using only a single temperature-independent binary interaction parameter. They also compared the performance The analytical expressions of the A12and A for Sutherland of CPA equation of state with that of the SRK and SAFT fluids and Yukawa fluids over variable range were also given equations of state and concluded that the CPA equation pro- Ž. by Gil-Villegas et al. 1997 . vided an improvement over the SRK equation and performed The VR-SAFT equation broadens the scope of the original similar to the SAFT equation of state. SAFT equationŽ. Huang and Radosz, 1990 and improves the chain contribution and the mean-field van der Waals descrip- tion for the dispersion forces of the HS-SAFT treatment. Heteronuclear chains Gil-Villegas et al.Ž. 1997 demonstrated the adequacy of the The preceeding equations of state can only be applied to VR-SAFT equation in describing the phase equilibria of chain homogeneous chains, that is, chains composed of identical molecules such as the n-alkanes and n-perfluoroalkanes. segments. Instead, many real chain-like fluids are heteroge- Davies et al.Ž. 1998 showed the VR-SAFT provided a simple neous being composed of different alternating segments. and compact equation of state for Lennard-Jones chains, and Furthermore, we are often interested in studying mixtures of it is valid for ranges of density and temperature of practical different types of chains. Several equations of state have been interest. McCabe et al.Ž. 1998 used the VR-SAFT equation proposed relatively recently that tackle the problem of het- to predict the high-pressure fluid-phase equilibrium of binary eronuclear chains. mixtures of n-alkanes and obtained good agreement with ex- Amos and JacksonŽ. 1992 used a bonded hard-sphere ap- periment. Galindo et al.Ž. 1998 provided a detailed analysis proach to devise an equation of state for the compressibility of the VR-SAFT approach for mixtures of factor of hard spheres of different size. The compressibility fluids with nonconformal intermolecular potentials, that is, factor for a fluid mixture of heteronuclear hard chain which have attractive interactions of variable range. They ex- molecules is amined the adequacy of the monomer contribution to the Helmholtz function by comparing with computer simulation r s hsy q data. The results showed that the VR-SAFT equation of state Z Ž.Z 1 Ý xmii 1 s represented well the vapor-liquid and liquid-liquid phase i 1 y equilibria of mixtures containing square-well molecules. r m i 1 Ѩ hs ␴ q ln gj, jq1Ž.j, j 1 Ž. y x ␩ 81 Tavares et al. 1997 reported a completely analytical equa- ÝÝi Ѩ␩ Ž. tion of state for square-well chains. Their equation is valid is1 js1 ž/

AIChE Journal January 2000 Vol. 46, No. 1 183 where xiiis the mole fraction of chains of component i, m is Recently, Blas and VegaŽ. 1998 applied their modified ver- ␴ the number of segments in a chain of component i, j is the sion of the SAFT equation of stateŽ. Blas and Vega, 1997 to ␴ diameter of segment j on a chain of component i, and j, jq1 predict thermodynamic properties, as well as liquid-vapor s ␴ q␴ r Ž.jjq1 2 is the bond length between adjacent seg- equilibria, of binary and ternary mixtures of hydrocarbons. ments. Amos and JacksonŽ. 1992 reported good agreement Adidharma and RadoszŽ. 1998 have introduced hetero- between simulation data and the compressibility factor pre- bonding into the SW-SAFT equation of state. The resulting dicted by Eq. 81 for mixtures containing diatomic and tri- equation can provide reasonable predictions of the phase be- atomic entities. Malakhov and BrunŽ. 1992 also generalized havior of mixtures of large homo- and heterosegmented thermodynamic perturbation theory in a similar way to for- molecules. Kalyuzhnyi et al.Ž. 1988 and Lin et al. Ž. 1988 have mulate an equation of state for heteronuclear chains. also used the concept of heteronuclear chains to model poly- Song et al.Ž. 1994a developed a perturbed hard-sphere mers. chainŽ. PHSC equation of state that can be applied to het- erogeneous mixtures. In general terms, the PHSC equation Comparing Equations of State of state for m-component mixtures containing r segments can Interrelationships between different equations of state be represented asŽ. Song et al., 1994b It is evident from the preceding descriptions that there is a substantial inter-relationship between various different equa- 2␲␳ m s q ␴ ␴ tions of state. It is very rare for an equation of state to be Z 1 Ý xxrri j i j ijg ijŽ. ij 3 ij developed entirely from scratch. Typically, new equations of state are proposed as modifications of existing ones, or suc- mm␳ y y ␴ y y cessful components of one or more equations of state are ÝÝxriŽ. i1 g ijŽ. ij1 xxrra i j i j ij Ž.82 iijkT reused to form a new equation. This component reuse is common to both empirical and theoretical equations of state. Invariably, equations of state are formed by combining sep- This equation has been used successfully to predict the liq- arate contributions from repulsive and attractive interactions. uid-liquid equilibria for binary polymer solutionsŽ Song et al., If we consider the repulsion term as forming the underlying 1994b.Ž. . Comparison Sadus, 1995 of the hard-sphere contri- basis of the equation of state, the interrelationships between butions of Eq. 82 with molecular simulation data indicates various equations of state can be summarized conveniently by that this part of the equation is less accurate than other an equation of state tree, as illustrated in Figure 2. In Figure hard-sphere chain alternatives. However, an advantage of Eq. 2, the tree grows from our knowledge of intermolecular inter- 82 is that it can be applied easily to mixtures of heteroge- actions. The main branches of the tree represent different neous chains by simply using a different value of r for the ways of representing intermolecular repulsion. We can iden- different chains. An equation for heteronuclear chains has tify branches representing the van der Waals, Carnahan-Star- also been reportedŽ. Hu et al., 1996 using a cavity-correlation ling, HCB, PHCT, and TPT terms. The addition of a differ- function approach. ent attraction term to these branches results in a different Banaszak et al.Ž. 1996 extended the SAFT equation of state equation of state capable of predicting phase equilibria. Fig- to apply to copolymers. The copolymer-SAFT equation in- ure 2 illustrates the dichotomy between empirical and theo- cludes contributions from chain heterogeneity and mi- retical equations of state. The empirical equations of state crostructure. Banaszak et al.Ž. 1996 demonstrated that the stem almost exclusively from the van der Waals repulsive copolymer-SAFT equation of state reproduces the qualitative branch. In contrast, theoretical equations of state are formed cloud point behavior of propaneqpolyŽ. ethylene-co-butene . from different branches that represent alternative ways of ac- More recently, Han et al.Ž. 1998 reported that the copoly- counting for the repulsion of nonspherical bodies. These dif- mer-SAFT equation of state can be used to correlate the soft ferent repulsion branches converge to a common point, which chain branching effect on the cloud point pressures of represents the limiting case of hard-sphere repulsion as de- copolymers of ethylene with propylene, butene, hexene, and scribed by the Carnahan-Starling equation. octene in propane. The copolymer-SAFT equation can also The tree diagram illustrates the importance of the Red- be used to predict phase transitions in hydrocarbonqchain lich-Kwong equation as the precursor for the development of solutionsŽ. Pan and Radosz, 1998 and -liquid equilibria empirical attraction terms. In contrast, the PHCT and SAFT Ž.Pan and Radosz, 1999 . equations of state are the precursors of many theoretical at- Shukla and ChapmanŽ. 1997 extended the SAFT equation traction terms. of state for fluid mixtures consisting of heteronuclear hard chain molecules, and they formulated expressions for the Comparison with experiment compressibility factor of pure block, alternate, and random copolymer systems. For the relatively simple case of pure al- Experimental data provides the ultimate test of the accu-

ternate copolymers consisting of mab-mers of type a and m - racy of an equation of state. However, there are several fac- mers of type b, the compressibility factor is tors that make it difficult to make absolute quantitative judg- ments about the relative merits of competing equations of Ѩ hs ␴ state from a comparison with experimental data. The distinc- ln gabŽ. ab Zs Z hsy1 mq1y my1 ␩ 83 tion must also be made between correlation and genuine Ž. Ž.Ѩ␩ Ž. prediction. Nevertheless, we can make some useful general comments about the accuracy of different categories of equa- s q s where m mabm and m abm . tions of state. However, before doing so, it is instructive to

184 January 2000 Vol. 46, No. 1 AIChE Journal Figure 2. Equation-of-state tree showing the interrelationship between various equations of state. identify some of the reasons why an absolute quantitative the evaluation of alternatives, particularly if the alternatives judgment is difficult: are more complicated. Ž.1 The very large number of equations of state Ž Sandler, Partly because of the above factors, there are relatively few 1994. is an obvious factor that makes the assessment of dif- reports in the literatureŽ Spear et al., 1971; Soave, 1972; Car- ferent equations of state difficult. In this context, it should be nahan and Starling, 1972; Beret and Prausnitz, 1975; Peng noted that the problem is compounded by the practice of and Robinson, 1976; Abbott, 1979; Martin, 1979; Elliott and equation of state developers testing their equation of state Daubert, 1985; Kim et al., 1986; Han et al., 1988; Mainwaring against experimental data, but not offering an identical com- et al., 1988; Sadus, 1992; Plackov et al., 1995; Plohl et al., parison with other equations of state. Nonetheless, the close 1998. comparing the predictive properties of two or more inter-relationship between the different equations of state al- equations of state over the same range of physical conditions lows some scope for a category by category evaluation. and experimental data. Ž.2 The accuracy of equations of state is often dependent Equations of state are used frequently to correlate experi- on highly optimized equation of state parameters. These mental data rather than to provide genuine predictions. It is equations of state parameters are typically tuned to a particu- well known that the van der Waals equation cannot be used lar region of interest and breakdown outside this region. For to correlate accurately the vapor-liquid coexistence of pure example, an equation of state tuned for atmospheric pres- fluids. In contrast, the addition of further adjustable parame- sures is unlikely to predict high-pressure phenomena with ters and temperature dependence to the attractive term used equal accuracyŽ. Sadus, 1992a . in the Redlich-Kwong, Soave-Redlich-Kwong and Peng- Ž.3 Comparison of the effectiveness of the equation of state Robinson equations of state result in accurate correlations for mixtures is hampered by the additional uncertainties in- Ž.Abbott, 1979; Han et al., 1988 of the vapor pressure. How- troduced by different mixture prescriptions, combining rules ever, a large improvement in the prediction of vapor pres- and unlike interaction parameters. These topics are dis- sures can be obtainedŽ. Plackov et al., 1995 by using either cussed in detail in the section discussing mixing rules. the Guggenheim or Carnahan-Starling van der Waals Ž.4 Users of equations of states often adopt a favorite Ž.CSvdW equations of state. The predictions of the Guggen- equation of state with which they become expert in using. heim and CSvdW equations cannot compete with the accu- This can result in considerable inertia to change that hinders racy obtained from empirical correlations with cubic equa-

AIChE Journal January 2000 Vol. 46, No. 1 185 tions of state. However, their improved accuracy alone der Waals, and other empirical equations, can be used to at demonstrates clearly that the success of the cubic equations least qualitatively predict these five types. However, since the of state is due largely to the ability of the empirical improve- work of van Konynenburg and Scott, many more phase types ments to compensate for the inadequacy of the van der Waals have been documented experimentally including a sixth type repulsion term. of behaviorŽ. Schneider, 1978 associated commonly with Theoretical equations of state can also be used to correlate aqueous mixtures involving liquid-liquid immiscibility at high experimental data. For example, van Pelt et al.Ž. 1992 corre- pressures. This so-called Type VI behavior can be predicted lated the SPHCT equation of state with experimental data by the SPHCT equation of stateŽ. Van Pelt et al., 1991 . More for the vapor-liquid coexistence of polyatomic molecules and recently, the existence of Type VI behavior has been pre- n-alkanes. The results indicate that the SPHCT provided bet- dicted using the CSvdWŽ. Yelash and Kraska, 1998 and ter correlations than cubic equations of state. This is a com- GuggenheimŽ. Wang et al., 2000 equations of state. In con- mon finding for other multiparameter theoretical equations trast, Type VI behavior cannot be calculated by using an em- of state. However, the benefit of using a theoretical equation pirical equation of state. for correlation is moot because they are considerably more Historically, both the empirical and theoretical ‘‘hard- complicated than cubic equations. The true value in using a sphereqattractive term’’ have been applied successfully be- theoretical equation is their improved ability to predict phase yond the natural range of validity of the hard-sphere concept. equilibria rather than merely correlate data. This success can be attributed in part to clever correlation At low pressures, either empirical or theoretical equations rather than the capabilities of the equation of state. How- of state can be used to at least qualitatively predict phase ever, it is evident that the quest to genuinely predict the phase equilibria. In this region, the highly optimized parameters of equilibria of large molecules will rely increasing on theoreti- empirical parameters generally enable empirical equations to cally-based models that account for the complexity of molec- provide superior accuracy than otherwise can be obtained ular interaction. Currently, either the SAFT or PHCT ap- from theoretical equations of state. However, empirical equa- proaches appear to be promising approaches towards this tions will almost invariably fail to predict the high-pressure goal. Some improvements have been reported recentlyŽ Chiew phase behavior of multicomponent mixturesŽ. Sadus, 1992a . et al., 1999; Kiselev and Ely, 1999; Feng and Wang, 1999. . For example, the results obtainedŽ. Sadus, 1992a, 1994 for the high-pressure equilibria of binary mixtures predicted by either the Redlich-Kwong and Peng-Robinson equations of Comparison with molecular simulation data state are not even qualitatively reliable. This failure can be Molecular simulationŽ. Sadus, 1999 is playing an increas- attributed unequivocally to the breakdown of the van der ingly valuable role in validating the accuracy of the underly- Waals repulsion termŽ. Figure 1 at the moderate to high den- ing theoretical basis of equations of state. Molecular simula- sities encountered at high pressures. In contrast, theoretical tion provides exact data to test the accuracy of theory. For equations using either the Carnahan-Starling or Guggenheim example, an equation of state for the compressibility factor of model of repulsion can be used to obtain quantitatively accu- Lennard-Jones atoms must reproduce the compressibility fac- rate predictions at high pressures. A curious exception to this tors for Lennard-Jones atoms obtained from molecular simu- rule is the high-pressure phase behavior of the heliumqwater lation. Discrepancies between theory and simulation can be binary mixture which can be represented accurately attributed unambiguously to the failure of theory to repre- Ž.Sretenskaja et al., 1995 by the simple van der Waals equa- sent adequately the underlying model. In contrast, direct tion of state. comparison of a theoretical model with experiment does not Another weakness of empirical equations of state, which is provide useful information regarding the validity of the the- also related to the inadequacy of the van der Waals repulsion ory, because experimental data does not represent the ideal- term, is that they cannot be used to calculate full range of ized behavior of the theoretical model. However, comparison phase equilibria of mixtures. Van Konynenburg and Scott of the results of a simulation-verified model with experiment Ž.1980 classified the phase behavior of binary mixtures into does indicate the strength or weakness of theory to represent five main types based on different critical behavior. The van experiment adequately. Probably, the best known example of the role of molecular simulation is the testing of hard-sphere repulsion terms against simulation data for hard spheresŽ. Figure 1 . The re- Table 3. Average Absolute Deviation of the Calculated sult of this comparison indicates that the Carnahan-Starling Compressibility Factor of m-Hard-Sphere Chains and Guggenheim terms are accurate theoretical models, Compared with Molecular Simulation Data whereas the van der Waals term fails at moderate to high AADŽ. % densities. UU U UUUMore recently, molecular simulation data have proved use- m GF-D TPT-D1 TPT-D2 Virial-PROSA STPT-D ful in determining the accuracy of theoretical models for 4 1.72 1.04 0.76 0.85 hard-sphere chains. A comparison with simulation data for 8 3.58 1.88 1.78 1.28 16 6.64 2.63 3.74 1.67 the compressibility factor predicted by several hard-sphere 32 10.44 4.27 5.75 1.08 chain equations of state is summarized in Table 3. Generally, 51 5.37 2.72 3.26 6.25 3.97 the data in Table 3 indicate that the STPT-D equation is 201 9.35 5.89 6.51 8.98 3.92 considerably more accurate than the GF-D, TPT-D1, TPT- U Ž. D2, and PROSA equations. The comparison of theory with UUFrom Sadus 1995 . From Stell et al.Ž. 1999b . simulation does not guarantee that using any of these equa-

186 January 2000 Vol. 46, No. 1 AIChE Journal tions will result in an accurate description of real chain where aiiand b ii are the constants of the equation for pure / molecules. However, using a simulation-verified model does component i, and cross parameters aijand bi ij Ž.j are mean that any discrepancies of theory with experiment are determined by an appropriate combining rule with or without not merely due to the failure of theory to represent ade- binary parameters. quately the underlying model. Instead, the discrepancies can Equations 86 and 87 are based on the implicit assumption be attributed unambiguously to the limitations of theory per that the radial distribution function of the component se to model real molecules. molecules are identical, and they both explicitly contain a contribution from interactions between dissimilar molecules. Mixing Rules A comparisonŽ. Harismiadis et al., 1991 with computer simu- lation has concluded that the van der Waals mixing rules are The great utility of equations of state is for phase equilib- reliable for mixtures exhibiting up to an eight-fold difference ria calculations involving mixtures. The assumption inherent in the size of the component molecules. The performance of in such a calculation is that the same equation of state used the van der Waals mixing rules has also been tested thor- for pure fluids can be used for mixtures if we have a satisfac- oughly for several equations of state by Han et al.Ž. 1988 . tory way to obtain the mixture parameters. This is achieved They used the van der Waals mixing ruleŽ. Eq. 86 to obtain commonly by using mixing rules and combining rules, which parameter a and the linear mixing ruleŽ. Eq. 85 to obtain related the properties of the pure components to that of the parameter b. Their results showed that most of equations of mixture. The discussion will be limited to the extension of a state with the van der Waals mixing rules were capable of and b parameters. These two parameters have a real physical representing vapor-liquid equilibria with only one binary ad- significance and are common to many realistic equations of justable parameter for obtaining aij. state. Equations 86 and 87 are also adequate for calculating the The simplest possible mixing rule is a linear average of the phase behavior of mixtures of nonpolar and slightly polar equation of state parameters compoundsŽ. Peng and Robinson, 1976; Han et al., 1988 . Voros and TassiosŽ. 1993 compared six mixing rules Ž the one- s a Ý xaii Ž.84 and two parameters van der Waals mixing rules; the pres- i sure- and density-dependent mixing rules; two based on ex- s . b Ý xbii Ž.85 cess Gibbs energy models: MHV2 and Wong-Sandler and i concluded that the van der Waals mixing rules give the best results for nonpolar systems. For the systems that contained Equation 85 is sometimes employedŽ. Han et al., 1988 be- strongly polar substances such as alcohols, water and ace- cause of its simplicity, but Eq. 84 is rarely used because it tone, the van der Waals mixing rule did not yield reasonable does not account for the important role of unlike interaction vapor-liquid equilibrium results. AnderkoŽ. 1990 gave some in binary fluids. Consequently, employing both Eqs. 84 and examples of the failure of the van der Waals mixing rules for 85 would lead to the poor agreement of theory with experi- strongly nonideal mixtures. ment. © © The ©an der Waals mixing rules Impro ed an der Waals mixing rules Ž The most widely used mixing rules are the van der Waals Many workers Adachi and Sugie, 1986; Panagiotopoulos one-fluid prescriptions and Reid, 1986; Stryjec and Vera, 1986a,b; Schwartzentruber et al., 1987; Sandoval et al., 1989. have proposed modifica- s tions for the van der Waals prescriptions. A common ap- a ÝÝxxaijij Ž.86 ij proach is to include composition-dependent binary interac- tion parameters to the a parameter in the van der Waals s b ÝÝxxbijij Ž.87 mixing rule and leave the b parameter rule unchanged. Some ij of examples are summarized in Table 4.

Table 4. Composition-Dependent Mixing Rules

Reference aij Term in Eq. 86 1r2w y q y x Adachi and SugieŽ. 1986 Žaaii jj .1 l ijmx ijŽ. ix j 1r2w y q y x Panagiotopoulos and ReidŽ. 1986 Žaaii jj .1 k ijŽ.k ijkx ji i 1r2 y y Stryjek and VeraŽ. 1986b Ž.Žaaii jj1 xk i ijxk j ji . Ž.Margules-type

kkij ji Stryjek and VeraŽ. 1986b Ž.aa 1r2 1y ii jj xk q xk Ž.Van Laar-type iij jji y mxij imx ji j Schwartzentruber et al.Ž. 1987 Žaa .1r2 1y k ylx Žq x . ii jj ij ijq i j mxij imx ji j s sy s y s s k jik ij; l jil ij; m ji1 m ij; k11l 11 0 1r2w y q y q y y x Sandoval et al.Ž. 1989 Žaaii jj .1 Ž.Ž.Ž.kx ij ikx ji j0.5 k ijk ji1 x ix j

AIChE Journal January 2000 Vol. 46, No. 1 187 Adachi and SugieŽ. 1986 kept the functional form of the linear mixing ruleŽ. Eq. 85 for the volume parameter b, the van der Waals mixing rule, left the b parameter unchanged, resulting expression for parameter a is and added an additional composition dependence and pa- rameters to the a parameter in the van der Waals one-fluid n agE s ii y ϱ mixing rules a bxÝ i Ž.91 is1 bii ln 2 s a ÝÝxxaijij Ž.88 E where gϱ is the value of the excess Gibbs energy at infinite 1r2 pressure and can be calculated fromŽ Renon and Prausnitz, a sŽ.aa 1yl y mx Žy x . Ž.89 ij ii jj ij ij i j 1968.

Adachi and SugieŽ. 1986 showed that their mixing rule can be n applied to the binary and ternary systems containing strongly Ý xGCjjiji n s polar substances. Es j 1 gϱ Ý xi n Ž.92 Ž. Ž s Sadus 1989 used conformal solution theory Sadus, 1992a, i 1 xG . Ý kki 1994 to derive an alternative to the conventional procedure 0k s1 for obtaining parameter a of equation of state. Instead of proposing an average of the pure component parameter data, with the a parameter for the mixture is calculated directly. Conse- quently, a is a function of composition only via the conformal s y ¶ Cjig jig ii parameters Ž.f, h and the contribution from the combinato- • C 93 rial entropy of mixing. The a parameter is obtained by taking s y ␣ ji Ž. Gjib jexp ji ß the positive root of the following quadratic equation ž/RT

YX22 Y XXX a26␪ y2 f rf qŽ.f rf q2h rhy2 fhrfhq Ž.hrh where gjiand g ii are the interaction energies between unlike ␣ Ž.gjiand like Ž.g iimolecules; ji is a nonrandomness param- q ␾␪2 y YYr y r eter. The Huron-Vidal mixing rule for a is deduced by apply- aRTV B Ä4h h f f ing Eq. 91 YXXX q2␾␪3 h rhq fhrfhyŽ.hrh 2 A Ä4 n

YX2 xGC q2␪ 6 y h rhq hrh q1rx 1y x n Ý jjiji Ä4Ž. Ž . aii 1 js1 as bx y Ž.94 Ý i b ln 2 n y 222␾ 2 XXr y r is1 ii Ž.RTV A Ä4 Ž.Ž.f f h h Ý xGkki k s1 q␾␪2 YXr y r 2y r y y␾␾ Yr ␪ s BABÄ4h h Ž.h h 1 x Ž1 x . h h 0 ␣ with ji, C ijand C ji as the three adjustable parameters. When Ž.90 ␣ s ji 0, the Huron-Vidal mixing rule reduces to the van der Waals mixing rule. Huron and VidalŽ. 1979 showed that their X Y where superscripts and denote successive differentiation of mixing rule yields good results for nonideal mixtures. Soave the conformal parameters, and ␪ and ␾ are characteristic of Ž.1984 found that the Huron-Vidal mixing rule represented the equation of state. The main advantage of Eq. 90 is that an improvement over the classical quadratic mixing rules and the a parameter can be calculated directly from the critical made it possible to correlate vapor-liquid equilibria for highly properties of pure components without using combining rules nonideal systems with good accuracy. The Huron-Vidal mix- for the contribution of unlike interactions. SadusŽ. 1992a,b ing rule has also been applied to a variety of polar and asym- has applied the above equation to the calculation of the va- metric systemsŽ Adachi and Sugie, 1985; Gupte and Daubert, por-liquid critical properties of a wide range of binary mix- 1986; Heidemann and Rizvi, 1986. . tures. The agreement was generally very good in view of the A weakness of the Huron-Vidal mixing rule is that the fact that no adjustable parameters were used to arbitrarily equation of state excess Gibbs energy at near atmospheric optimize the agreement between theory and experiment. pressure differs from that at infinite pressure. Therefore, the Huron-Vidal mixing rule has difficulty in dealing with low- pressure data. Several proposalsŽ Lermite and Vidal, 1992; Mixing rules from excess Gibbs energy models Soave et al., 1994. have been reported to overcome this diffi- Huron and VidalŽ. 1979 suggested a method for deriving culty. mixing rules for equations of state from excess Gibbs energy MollerupŽ. 1986 modified Eq. 94 by retaining that the ex- models. Their method relies on three assumptions. First, the cess volume is zero but evaluating the mixture parameter a excess Gibbs energy calculated from an equation of state at directly from the zero pressure excess free energy expression. infinite pressure equals an excess Gibbs energy calculated The modified mixing rule has the form from a liquid-phase activity coefficient model. Secondly, the covolume parameter b equals the volume V at infinite pres- a a f GE RT b s iiyq i sure. Third, the excess volume is zero. By using the Soave- ÝÝxiicx ln f Ž.95 Redlich-Kwong equationŽ. Eq. 5 and applying the common bbfffiiž/i ž/ ž /ž/ b

188 January 2000 Vol. 46, No. 1 AIChE Journal where tion to develop mixing rules to satisfy the second virial condi- ¶ tion. For the mixture parameters of an equation of state, a bi and b are f s i ® i E aAi ϱ b as bxq Ž.100 f s • Ý i ® Ž.96 bCi ®® a syi y y fc 1 1 ß ÝÝxxij b bbž/ ž/RT ij ž/i bs Ž.101 AaE qyϱ i 1 Ý xi The mixing ruleŽ. Eq. 95 depends on the liquid-phase volume RTž/ bi RT of the mixture and individual components, and is a less re- strictive assumption than the Huran-Vidal mixing rule. where C is a constant dependent on the equation of state This basic conceptŽ. Mollerup, 1986 was implemented by selectedŽŽ. for example, C is equal to 1r''2ln 2y1 for the E MichelsenŽ. 1990 . Using a reference pressure of zero and the Peng-Robinson equation of state. and Aϱ is the excess Soave-Redlich-Kwong equation of stateŽ. Eq. 5 , Michelsen Helmholtz function at infinite pressure, and Ž.Michelsen, 1990; Dahl and Michelsen, 1990 repeated the matching procedure of Huron and Vidal resulting in the fol- aa1y ka ysŽ.ijyqi y j lowing mixing rule called the modified Huron-Vidal first or- b bijb Ž.102 ž/RT ij 2 ž/RTž/ RT derŽ. MHV1 where k is a binary interaction parameter. nnE ij 1 Gb Ž. Ž ␣ s x ␣ qqx lnŽ. 97 Wong and Sandler 1992 tested the mixing rules Eq. 100 ÝÝiii i . is1 qRT1 is1 ž/ bii and Eq. 101 , and concluded that they were reasonably accu- rate in describing both simple and complex phase behavior of where binary and ternary systems for the diverse systems they con- sidered. Wong et al.Ž. 1992 demonstrated that the Wong-San- ␣ s arbRT ¶ dler mixing rules can be used for highly nonideal mixtures. n The mixing rules can be applied at temperatures and pres- s • b Ý xbiii Ž.98 sures that greatly exceed the experimental data used to ob- is1 tain the parameters. Huang and SandlerŽ. 1993 compared the ␣ s r ß iia iibRT ii Wong-Sandler and MHV2 mixing rules for nine binary and two ternary systems. They showed that either the MHV2 or sy Wong-Sandler mixing rules can be used to make reasonable with the recommended value of q1 0.593. In addition, Dahl and MichelsenŽ. 1990 derived an alternative mixing rule high-pressure vapor-liquid equilibrium predictions from low- referred to as the modified Huron-Vidal second-order pressure data. They used the Peng-Robinson and Soave-Re- Ž.MHV2 dlich-Kwong equations of state and found that the errors in the predicted pressure with the Wong-Sandler mixing rule were, on the average, about half or less of those obtained nnGbE n ␣ y ␣ q ␣ 22y ␣ sq when using the MHV2 mixing rule. Orbey and SandlerŽ. 1994 q1 ÝÝÝxiii q2 xiiix iln ž/ž/is1 is1 RTis1 ž/ bii used the Wong-Sandler mixing rule to correlate the vapor- q q Ž.99 liquid equilibria of various polymer solvent and solvent long chain hydrocarbon mixtures. They concluded that the Wong-Sandler mixing rule can correlate the solvent partial sy sy with suggested values of q120.478 and q 0.0047. pressure in concentrated polymer solutions with high accu- Ž. Dahl and Michelsen 1990 investigated the ability of racy over a range of temperatures and pressures with temper- MHV2 to predict high-pressure vapor-liquid equilibrium ature-independent parameters. when used in combination with the parameter table of modi- To go smoothly from activity coefficient-like behavior to Ž. fied UNIFAC Larsen et al., 1987 . They concluded that sat- the classical van der Waals one fluid mixing rule, Orbey and isfactory results were obtained for the mixtures investigated. SandlerŽ. 1995 slightly reformulated the Wong-Sandler mix- Ž. Dahl et al. 1991 demonstrated that MHV2 was also able to ing rules by rewriting the cross second virial term given in correlate and predict vapor-liquid equilibria of gas-solvent bi- Eq. 102 as nary systems and to predict vapor-liquid equilibria for multi- component mixtures using the new parameters for gas-solvent q aa 1y k interactions together with the modified UNIFAC parameter a Ž.bijb ' ijŽ. ij bys y Ž.103 table of Larsen et al.Ž. 1987 . ž/RT ij 2 RT Generally, the use of the infinite pressure or zero pressure standard states for mixing in the equation of state will lead to while retaining the basic equations, Eq. 100 and Eq. 101. Or- inconsistencies with the statistical mechanical result that the bey and SandlerŽ. 1995 tested five binary systems and one second virial coefficient must be a quadratic function of com- ternary mixture and showed that this new mixing rule was position. Wong and SandlerŽ. 1992 used the Helmholtz func- capable of both correlating and predicting the vapor-liquid

AIChE Journal January 2000 Vol. 46, No. 1 189 equilibrium of various complex binary mixtures accurately In terms of the equation of state parameters, the equiva- over wide ranges of temperature and pressure and that it can lent combining rules to Eq. 104 are be useful for accurate predictions of multicomponent vapor- liquid equilibria. ¶ aaii jj Castier and SandlerŽ. 1997a,b performed critical point cal- a s␰ b ij ij ij bb culations in binary systems utilizing cubic equations of state ( ii jj • Ž.105 combined with the Wong-Sandler mixing rules. To investigate 3 b1r31q b r3 the influence of the mixing rules on the shape of the calcu- s␨ Ž.ii jj ß bij ij lated critical phase diagrams, the van der Waals equation of 8 state was combined with the Wong-Sandler mixing rules Ž.Castier and Sandler, 1997a . The results showed that many where the rule for bij is referred to as the Lorentz ruleŽ Hicks different types of critical phase diagrams can be obtained and Young, 1975; Sadus, 1992a. . More commonly, the equa- from this combination. When the Wong-Sandler mixing rules tion of state parameters are obtained from are combined with the Stryjek and VeraŽ. 1986a version of the Peng-RobinsonŽ. 1976 equation of state Ž Castier and San- a s␰ aa ¶ dler, 1997b. , it was able to predict quantitatively the critical ij ij' ii jj • behavior of some highly nonideal systems involving com- q Ž.106 Ž.biib jj pounds such as water, acetone, and alkanols. For some highly b s␨ ß ij ij 2 asymmetric and nonideal mixtures, such as waterq n- dodecane, only qualitatively correct critical behavior could be predicted. where the combining rule for aij is referred to as the van der Several other mixing rules have been proposedŽ Heide- Waals combining rule. At this point, it should be noted that mann and Kokal, 1990; Soave, 1992; Holderbaum and there is a common misconception in the literature that Eq. Gmehling, 1991; Boukouvalas et al., 1994; Tochigi, 1995; 106 and not Eq. 105 is the equivalent of Eq. 104. This error is Novenario et al., 1996; Twu and Coon, 1996; Twu et al., 1998. understandable in view of the functional similarity of Eqs. based on excess free energy expressions. Comparison and 106 and 104. However, it should also be observed that the a evaluation for various mixing rules can be found in the works and b parameters have dimensions of energy=volume and of Knudsen et al.Ž. 1996 , Voros and Tassios Ž. 1993 , Michelsen volume, respectively, compared with a dimension of energy and HeidemannŽ. 1996 , Wang et al. Ž. 1996 , Abdel-Ghani and for ⑀ and a dimension of distance for ␴ . Another combining HeidemannŽ. 1996 , Orbey and Sandler Ž. 1996 , Heidemann rule for bij is the geometric mean rule proposed by Good Ž.1996 , and Twu et al. Ž. 1998 . and HopeŽ. 1970

s␨ Combining rules bij ij'bb ii jj Ž.107 As noted above, any mixing rule will invariably contain a contribution from interactions between unlike molecules. In SadusŽ. 1993 compared the accuracy of the Lorentz Ž Eq. / 105.Ž.Ž. , arithmetic Eq. 106 , and geometric Eq. 107 rules for other words, the cross terms aijand bi ij Ž.j must be evalu- ated. They can be determined by an appropriate combining bij when used in the prediction of Type III phenomena. For rule. The contribution from unlike interaction to the inter- molecules of similar size, all three combining rules give al- molecular parameters representing energy Ž.⑀ and hard- most identical results, but the discrepancy increases substan- sphere diameter Ž.␴ can be obtained from tially for mixtures of molecules of very dissimilar size. Sadus Ž.1993 proposed an alternative combining rule by taking a 2:1 geometric average of the Lorentz and arithmetic rules with- ⑀ s␰⑀⑀ ¶ out the ␨ parameter, that is, ij ij' ii jj • ij ␴ q␴ Ž.104 Ž.ii jj r ␴ s␨ ß s r 1r31r31q r3 2 q 1 3 ij ij 2 bijÄ41 42Ž.Ž.b iibb jjŽ. iib jj Ž.108

SadusŽ. 1993 reported that the new combining rule Ž Eq. 108 . ␰ Ž.y In Eq. 104, the ijalso commonly defined as 1 k ij and is generally more accurate that either the Lorentz, arith- ␨ Ž.y ijalso commonly defined as 1 l ij terms are adjustable metic, or geometric combining rules. parameters which are used to optimize agreement between ␨ theory and experiment. The ij term does not significantly improve the analysis of high-pressure equilibria, and it can be Conclusions ␨ s ␰ usually omitted Ž.ij1 . The ij term is required, because it Considerable progress has been achieved in the develop- can be interpreted as reflecting the strength of unlike inter- ment of equations of state. Many highly successful empirical action except the simple mixtures of molecules of similar size. equations of state have been proposed that can be used to ␰ This interpretation is supported by the fact that values of ij calculate the phase behavior of simple fluids. However, a obtained from the analysis of the critical properties of many more sophisticated approach is required for complicated binary mixtures consistently decline with increasing size dif- molecules. To meet the challenge posed by large and compli- ference between the component molecules as detailed else- cated molecules, equations of state are being developed in- whereŽ. Sadus, 1992a, 1994 . creasingly with an improved theoretical basis. These new

190 January 2000 Vol. 46, No. 1 AIChE Journal equations are playing an expanding role in the accurate cal- ␰sunlike interaction parameter ␲s culation of fluid-phase equilibria. Equation of state develop- 3.14159 ␶snumerical constant ment has been aided greatly by new insights into the nature ␻sacentric factor; orientation of molecule of intermolecular interaction and molecular simulation data. ⌺ssummation In particular, molecular simulation is likely to have an ongo- ing and crucial role in the improvement of the accuracy of Subscripts and superscripts equations of state. A continuing challenge is to improve the 0sdenotes component; reference system prediction of the phase behavior of mixtures. The main im- 1sdenotes component; first-order term of perturbation pediment to the prediction of mixture phenomena is our un- theory derstanding of interactions between dissimilar molecules. This 2sdenotes component; second-order term of perturba- tion theory is also an area that is likely to benefit from the input of U s configurational property; perfect gas contribution molecular simulation data. assocsassociation attsattractive Acknowledgments BHsBarker-Henderson cscritical property Y.S.W. thanks the Australian government for an Australian Post- chainschain term graduate Award. dipolesdipole-dipole term dispsdispersion s Notation Dij universal constants hcshard chain BWRsBenedict-Webb-Rubin hcbshard convex body CPAscubic plus association hBHshybrid Barker-Henderson CFsChristoforakos-Franck hdshard-dimer GF-Dsgeneralized Flory-dimer isith component HCBshard convex body js jth component HCBvdWshard convex body van der Waals LJsLennard-Jones HFsHeilig-Franck msmixture HS-SAFTshard sphere statistical associating fluid theory mfsmean field LJ-SAFTsLennard-Jones statistical associating fluid theory monosmonomer-monomer term MHV1smodified Huron-Vidal first order psperturbation MHV2smodified Huron-Vidal second order rsrotational motion of a molecule; number of segment PROSAsproduct-reactant Ornstein-Zernike approach per molecule RKsRedlich-Kwong repsrepulsive SRKsSoave-Redlich-Kwong ressresidual SSAFTssimplified statistical Associating fluid theory swssquare well SW-TPT-Dssquare-well thermodynamic perturbation theory-di- ®svibrational motion of a molecule mer TBsTrebble and Bishnoi TPT-D1sthermodynamic perturbation theory-dimer 1 Literature Cited TPT-D2sthermodynamic perturbation theory-dimer 2 ⌬ TPT1sfirst-order thermodynamic perturbation theory Abdel-Ghani, R. 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196 January 2000 Vol. 46, No. 1 AIChE Journal