Equation of state mixing rules

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

Redlich-Kwong Equation of State — Peng-Robinson Equation of State—Mixing Rules... 

Shibata, S.K. and Sandler, S I. Critical Evaluation of Equation of State Mixing Rules for the Prediction of High-Pressure Phase Equilibria, Ind. Eng. Chem. Res. Vol 28, 1989, pp. 1893-1898. 

Orbey, H., and Sandler, S. 1., 1996b. A comparison of various cubic equations of state mixing rules for the simultaneous description of excess enthalpies and vapor liquid equilibria. Fluid Phase Eq., 121 67-83. 

Sandler, S. L, Lee, K.-H., and Kim, H., 1986. The generalized van der Waals partition function as a basis for equations of state Mixing rules and activity coefficient models in Equations... 

Shibata, S. K., and Sandler, S. L, 1989. Critical evaluation of equation of state mixing rules for the prediction of high pressure phase equilibria. Ind. Eng. Chem. Res., 28 1893-1898. 

Wong, D. S. H., Orbey, H., and Sandler, S. I., 1992. Equation of state mixing rule for nonideal mixtures using available activity coefficient model parameters and that allows extrapolation over large ranges of temperature and pressure. lEC Res., 31 2033-2039. 

The evolution of equation of state mixing rules has been reported by Copeman and Mathias and the mixing rules used in cubic equations of state have been discussed by Knapp et al However, except for the van der Waals equation, there is no direct relationship between the two parameters in the equation of state and those of the intermolecular potential. Also there is only a tenuous basis for similar mixing rules for other cubic equations, and little evidence that any are superior to the van der Waals one-fluid rules. 

Analysis of Equation of State Mixing and Combining Rules Using... 

ANALYSIS OF EQUATION OF STATE MIXING AND COMBINING RULES USING FLUCTUATION SOLUTION THEORY... 

Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

The mixture cohesive energy density, coh-m> was not to be obtained from some mixture equation of state but rather from the pure-component cohesive energy densities via appropriate mixing rules. Scatchard and Hildebrand chose a quadratic expression in volume fractions (rather than the usual mole fractions) for coh-m arid used the traditional geometric mean mixing rule for the cross constant ... 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

Schwartzentruber J., F. Galivel-Solastiuk and H. Renon, "Representation of the Vapor-Liquid Equilibrium of the Ternary System Carbon Dioxide-Propane-Methanol and its Binaries with a Cubic Equation of State. A new Mixing Rule", Fluid Phase Equilibria, 38,217-226 (1987). 

Modeling Vapor-Liquid Equilibria Cubic Equations of State and their Mixing Rules, Hasan Orbey and Stanley I. Sandler... 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

In the last 25 years, calculations of the detonation properties of condensed explosives from their chemical compositions and densities have been approached in various ways.2 All have used the necessary conservation conditions for steady flow with the detonation discontinuity satisfying the Chapman-Jouguet hypothesis (minimum detonation velocity compatible with the conservation conditions or sonic flow behind the discontinuity in a reference frame where the discontinuity is at rest). In order to describe the product state and the thermodynamic variables which fix its composition, an equation of state applicable to a very dense state is required. To apply this equation to a mixture of gaseous and solid products, a mixing rule is also needed and the temperature must be explicitly defined. Of the equations of state for high-density molecular states which have been proposed, only three or four have been adapted to the calculation of equilibrium-product compositions as well as detonation parameters. These are briefly reviewed in order to introduce the equation used for the ruby computer code and show its relation to the others. 

Mixing rules for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 0 for mixtures are usually given by the following, where, as for the second virial coefficient, 0 = 0ji. 

The greatest use of cubic equations of state is for phase equilibrium calculations involving mixtures. The assumption inherent in such calculations is that the same equation of state as is used for the pure fluids can be used for mixtures if we have a satisfactory way to obtain the mixtures parameters. This is most commonly done using the van der Waals one-fluid mixing rules,... 

Panagiotopoulos A.Z., Reid R.C., "A new mixing rule for cubic equations of state for highly polar, asymmetric systems", in... 

Van der Waals Mixing Rules for Cubic Equations of State... 

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