Calculation of high-pressure phase equilibria

Calculation of High-Pressure Phase Equilibria Involving Light Gases... 

Bertucco et al. investigated the effect of SCCO2 on the hydrogenation of unsaturated ketones catalyzed by a supported Pd catalyst, by using a modified intemal-recycle Berty-type reactor [63]. A kinetic model was developed to interpret the experimental results. To apply this model to the multiphase reaction system, the calculation of high-pressure phase equilibria was required. A Peng-Robinson equation of state with mixture parameters tuned by experimental binary data provided a satisfactory interpretation of all binary and ternary vapor-liquid equilibrium data available and was extended to multicomponent... 

Beginning in the 1990s calculations of high-pressure phase equilibria of polydisperse polymer systems were performed. For example, Enders and de Loos calculated cloud-point and spinodal curves in the high-pressure range for methylcyclohexane + poly(ethenylbenzene) and compared their results with experimental data. Enders and de Loos ° used a Gibbs-energy model with pressure dependent parameters and models that include an equation of state, such as the lattice fluid model introduced by Hu et for the monodisperse and... 

WOH Wohlfarth, C. and Ratzsch, M.T., Calculation of high-pressure phase equilibria in mixtures of ethylene, vinyl acetate, and ethylene-vinyl acetate copol miers,Zcto Polym., 34,255,1983. 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

The purpose of phase equilibria calculations is to predict the thermodynamic properties of mixtures, avoiding direct experimental determinations, or to extrapolate the existing data to different temperatures and pressures. The basic requirements for performing any thermodynamic calculation are the choice of the appropriate thermodynamic model and knowledge of the parameters required by the model. In the case of high pressure phase equilibria, the thermodynamic model used is generally an equation of state which is able to describe the properties of both phases. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

The Sako-Wu-Prausnitz equation of state was also applied to high-pressure phase equilibria of polyolefin systems by Tork et alP The calculations were based on the pseudo-component method where the number of pseudo-components used were between 2 and 8. The small number of pseudo-components is a result of the very efficient estimation method used to adjust the pseudo-components to the moments of the distribution function (described in section 9.3.1). In so doing Tork et alP were able to provide a good description of the experimental data and show, perhaps not surprisingly, the agreement between calculated and experimental data improved with increasing number of pseudo-components. 

Eor the calculation of high-pressure vapor-liquid equilibria or liquid-liquid equilibria an equation of state is always used for both phases and the equilibrium condition used is given by Eq. (23). 

It is difficult to measure partial molar volumes, and, unfortunately, many experimental studies of high-pressure vapor-liquid equilibria report no volumetric data at all more often than not, experimental measurements are confined to total pressure, temperature, and phase compositions. Even in those cases where liquid densities are measured along the saturation curve, there is a fundamental difficulty in calculating partial molar volumes as indicated by... 

While the dilated van Laar model gives a reliable representation of constant-pressure activity coefficients for nonpolar systems, the good agreement between calculated and experimental high-pressure phase behavior shown in Fig. 14 is primarily a result of good representation of the partial molar volumes, as discussed in Section IV. The essential part of any thermodynamic description of high-pressure vapor-liquid equilibria must depend,... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Deiters, U.K. (1985) Calculation of equilibria between fluid and solid phases in binary mixtures at high pressures from equations of state. Fluid Phase Equilibria 20, 275-282. 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

It has been shown that ab initio total energy DFT approach is a suitable tool for studies of phase equilibria at low temperatures and high pressures even when small energy differences of the order of 0.01 eV/mol are involved. The constant pressure optimization algorithm that has been developed here allows for the calculation of the equation of state for complex structures and for the study of precursor effects related to phase transitions. 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

The conditions of metamorphism of BIF of different types are examined on the basis of thermodynamic calculations consistent with experimental investigations of phase equilibria at high temperatures and pressures. 

Browarzik, C. and Kowaleski, M., Calculation of the cloud-point and the spinodal curve for the system methylcyclohexane/polystyrene at high pressure. Fluid Phase Equilibria, 451, 194-197, 2002. 

Deiters, U., and G. M. Schneider. 1976. Fluid mixtures at high pressures Computer calculations of the phase equilibria and the critical phenomena in fluid binary mixtures from the Redlich-Kwong equation of state. Ber. Bunsenges. Phys. Chem. 80 1316. 

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