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Equations of State for the Calculation of Fluid-Phase Equilibria

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Equations of State for the Calculation of Fluid-Phase Equilibria 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 equation of state was the first equation ponents. When first applied to mixtures, they were used only to predict vapor-liquid 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 van der Waals equation 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 pressure, 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 gases to heavy liquids. 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,È 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 forces, 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 mechanics 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 compressibility 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 volume, p is the pressure, and R is the molar © universal gas 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 s , then bs Benedict et al.Ž. 1940 suggested a multiparameter equation 2p Ns 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 a 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 gg 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 a g Redlich-Kwong equation of state. The results showed that the where A000, B , C , a, b, c, , are eight adjustable parame- ters.
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