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## Comparisons of State for the Computation of Liquid Stage Equilibria

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**Equations of State for the Calculation of Fluid-Phase**Equilibria Y.S. Wei and R.J.Sadus, AIChE J., 46, 169-191, 2000 Kim, Yong-Soo Thermodynamics & Properties Lab. Korea University**Introduction**• Advantages of using Equation of State(EOS) • Wide ranges of temperature and pressure • Application of mixtures of diverse components • Various phase equilibria without any conceptual difficulties • This work • An overview of recent progress in EOS • Simple empirical EOS • Theoretically-based EOS • Relationships between different EOS • Role of molecular simulation data**EOS for simple molecules**• van der Waals EOS (vdW) (1873) • Hard-sphere (repulsive) + Attractive intermolecular interactions • A qualitative description of phase transitions • Inadequate to critical properties and phase equilibria • Requirement of modifications of attractive and repulsive terms**EOS for simple moleculesModification of Attractive term**• Benedict-Webb-Rubbin EOS (1940) • Disadvantage • Requirement of plentiful, accurate PVT and VLE data for parmamter estimation • Difficulty of extension to mixtures**EOS for simple moleculesModification of Attractive term**• Redlich-Kwong EOS (1949) • Significant improvement over the vdW EOS • The impetus for many further empirical EOS**EOS for simple moleculesModification of Attractive term**• SRK (1972) • Prediction of phase behavior of mixtures in the critical region and improvement of accuracy of critical properties**EOS for simple moleculesModification of Attractive term**• Peng-Robinson (1976) • Slight improvement of the predictions of liquid volumes • Superior to the VLE in hydrogen and nitrogen containing mixtures (Han et al., 1988)**EOS for simple moleculesModification of Attractive term**• The advantages of SRK and PR EOSs • Easy representation of the relation among temperature, pressure, and phase compositions in multicomponent systems • Only requirement of the critical properties and acentric factor • Little computing time • Overestimation of saturated liquid volumes.**EOS for simple moleculesModification of Repulsive term**• Reproducibility of complex phase transitions such as LLV equilibria.**EOS for simple moleculesModification of Repulsive term**• Hard-sphere compressibility factors from different EOS with molecular simulation data**EOS for simple moleculesCombining modification of both**attractive and repulsive terms • Carnahan and Starling (1972) • The prediction of hydrocarbon densities and supercritical phase equilibria. • Chen and Kreglewski (1977) • The substitution of attractive term with the power series fit of MC data by Alder et al. (1972) • This attractive term is the inspiration for further development.**EOS for simple moleculesCombining modification of both**attractive and repulsive terms • Shah et al. (1994) • Requirement of 3 properties of fluids : Tc, Vc, and acentric factor • Quartic equation, but it behaves like cubic equation. • Lin et al. (1996) • Extension to polar fluids. • Need of dipole moment Repulsive Attractive**EOS for Chain MoleculesPerturbed hard chain theory**• Prigogine (1957) • Rotational and vibrational motions are depend on density => EOS and configurational properties are affected.**EOS for Chain MoleculesPerturbed hard chain theory**• Beret and Prausnitz (1975) • Development of PHCT EOS • More accurate expressions for repulsive and attractive partition functions • Meeting the ideal gas law at low densities • Deficiency in Prigogine’s theory**EOS for Chain MoleculesPerturbed hard chain theory**• Equation of State • Parameters : • Sucessful in calculating the various properties of fluids and phase equilibria • A practical limitations as a result of the use of Carnahan-Starling free-volume term and the Alder power series • Simplifying the PHCT EOS**EOS for Chain MoleculesSimplified perturbed hard chain**theory • Kim et al. (1986) • Parameters : • The SPHCT EOS retains the advantages of the PHCT EOS.**EOS for Chain MoleculesHard-sphere chain theory**• Wertheim’s thermodynamic perturbation theory (TPT) • The association site are replaced by covalent, chain-forming bonds. • Chapman et al. (1988) : Generalization of TPT • Zhs is Carnahan-Starling equation.**EOS for Associating FluidsStatistical associating fluid**theory (SAFT) • Chapman et al. (1988, 1990)**EOS for Associating FluidsStatistical associating fluid**theory (SAFT) • Huang and Radosz (1990)**EOS for Associating FluidsStatistical associating fluid**theory (SAFT) • Development of variable SAFT model • Simplified SAFT : Fu and Sandler (1995) • Galindo et al. (1996) : The expression of Boublik for the hard-sphere contribution • LJ-SAFT (Banaszak et al., 1994), VR-SAFT (Gil-Villegas et al., 1997), and so on.**Comparing EOSInterrelationships between different EOS**• New EOS • Modification of existing ones • Reuse of successful EOS to form a new EOS • The branches in next figure show different ways of representing intermolecular repulsion. • van der Waals, Carnahan-Starling, HCB, PHCT, and TPT • The precursor for the development of EOS • SRK in empirical EOS’s • PHCT and SAFT in theoretical EOS’s**Comparing EOSComparison with experiment**• Experimental data • The ultimate test of the accuracy of an EOS • No absolute quantitative judgments about the relative merits of competing EOS • Why is an absolute quantitative judgments difficult? • EOS developers test their EOS against experimental data, but not offer an identical comparison with other EOS. • The accuracy of EOS is often dependent on highly optimized EOS parameters • EOS users adopt a favorite EOS with which they become expert in using.**Comparing EOSComparison with experiment**• The true value in using a theoretical EOS • Their improved ability to predict phase equilibria rather than merely correlate data. • Correlation of experimental data with PR/SRK at low pressure => Failure of the prediction of phase equilibria at high pressure • Breakdown of vdW repulsion term => Using Carnahan-Starling or Guggenheim repulsion term • Ability of calculating full range of phase equilibria of mixtures. • Theoretical EOS’s, such as SAFT and PHCT are promising approaches.**Comparing EOSComparison with molecular simulation data**• Molecular simulation • Provision of exact data to test the accuracy of theory • Discrepancies between theory and MC =>Failure of theory to represent the underlying model • Direct comparison of a theoretical model with experiment => No useful information • Direct comparison of a simulation-verified model with experiment =>To indicate the strength or weakness of theory • Example => Show figure • Carnahan-Starling and Guggenheim equation is accurate !**Comparing EOSComparison with molecular simulation data**• Failure of accuracy in comparison of EOS with MC • Not merely due to the failure of theory to represent adequately the underlying model • Because of the limitations of theory to model the real molecules**Conclusion**• To meet the challenge posed by large and complicated molecules, EOS are being developed with an improved theoretical basis. • These new EOS are playing an expanding role in the calculation of various phase equilibria. • Molecular simulation have an ongoing and crucial role in the improvement of the accuracy of EOS