Using equations of state method

HIGH-PRESSURE VAPOR-LIQUID EQUILIBRIA USING EQUATIONS OF STATE METHOD)... 

High-Pressure Vapor-Liquid Equilibria Using Equations of State Method) 559... 

Quantitative solubility calculations are usually performed using equation of state methods (10). Kim, et al., (29), discuss solubility behavior in the immediate vicinity of the critical point using macroscopic thermodynamic properties. 

In Figure 10 are shown comparisons of the equation of state methods with the experimental data. The Lee-Kesler methods represent the data the best. Again, if the water acentric factor determined to best represent the pure steam enthalpy data is applied to the mixtures, further improvement is noted for the predictions by the Lee-Kesler method. Use of interaction constants within the Lee-Kesler, or other models, would undoubtedly provide even better representation of the data. 

The equation-of-state method, on the other hand, uses typically three parameters p, T andft/for each pure component and one binary interactioncparameter k,, which can often be taken as constant over a relatively wide temperature range. It represents the pure-component vapour pressure curve over a wider temperature range, includes the critical data p and T, and besides predicting the phase equilibrium also describes volume, enthalpy and entropy, thus enabling the heat of mixing, Joule-Thompson effect, adiabatic compressibility in the two-phase region etc. to be calculated. 

Critical reviews of existing methods to model phase behaviour at high pressure using equations of state have been made in recent years, and we refer to these papers for further details [24, 25]. The general conclusion is that modelling is still case-specific. When the critical point is approached, predictions and even correlations of critical curves and solubilities are extremely difficult because of the nonclassical behaviour in this region. 

We have applied a global optimization technique, based on interval analysis, to the high-pressure phase equilibrium problem (INTFLASH). It does not require any initial guesses and is guaranteed, both mathematically and computationally, to converge to the correct solution. The interval analysis method and its application to phase equilibria using equation-of-state... 

Many computational studies of the permeation of small gas molecules through polymers have appeared, which were designed to analyze, on an atomic scale, diffusion mechanisms or to calculate the diffusion coefficient and the solubility parameters. Most of these studies have dealt with flexible polymer chains of relatively simple structure such as polyethylene, polypropylene, and poly-(isobutylene) [49,50,51,52,53], There are, however, a few reports on polymers consisting of stiff chains. For example, Mooney and MacElroy [54] studied the diffusion of small molecules in semicrystalline aromatic polymers and Cuthbert et al. [55] have calculated the Henry s law constant for a number of small molecules in polystyrene and studied the effect of box size on the calculated Henry s law constants. Most of these reports are limited to the calculation of solubility coefficients at a single temperature and in the zero-pressure limit. However, there are few reports on the calculation of solubilities at higher pressures, for example the reports by de Pablo et al. [56] on the calculation of solubilities of alkanes in polyethylene, by Abu-Shargh [53] on the calculation of solubility of propene in polypropylene, and by Lim et al. [47] on the sorption of methane and carbon dioxide in amorphous polyetherimide. In the former two cases, the authors have used Gibbs ensemble Monte Carlo method [41,57] to do the calculations, and in the latter case, the authors have used an equation-of-state method to describe the gas phase. 

Use of generalized fugacity coefficients (e.g., see Example 1.18) eliminates some computational steps. However, the equation-of-state method used here is easier to program on a programmable calculator or computer. It is completely analytical, and use of an equation of state permits the computation of all the thermodynamic properties in a consistent manner. 

For the analytical equations, there are two methods to compute the vapour-liquid equilibrium for systems. The equation of state method (also known as the direct or phi-phi method) uses an equation of state to describe both the liquid and vapour phase properties, whereas the activity coefficient method (also known as the gamma-phi approach) describes the liquid phase via an activity coefficient model and the vapour phase via an equation of state. Recently, there have also been modified equation of state methods that have an activity coefficient model built into the mixing mles. These methods can be both correlative and predictive. The predictive methods rely on the use of group contribution methods for the activity coefficient models such as UNIFAC and ASOG. Recently, there have also been attempts to develop group contribution methods for the equation of state method, e.g. PRSK. " For a detailed history on the development of equations of state and their applications, as well as activity coefficient models, refer to Wei and Sadus, Sandler and Walas. ... 

High-Pressure Vapor-Liquid Equilibria Using Equations of State 4>-(j) Method) 563 and the summation conditions (Eqs. 10.1 -5 and 10.1 -6)... 

While Figs. 1.7-1 and 1.7-2 are for binaiy mixtures, the equation-of-state method for calculating vapor-liquid equilibria can be applied to mixtures with any number of components. When the quadratic mixing rules [Eqs. (1.3-32) and (1.3-33)] are used, only pure-component and binary constants are required these mixing rules therefore provide a powerful tool for scale-up" in the sense that only binary mixture data are needed to calculate equilibria for a mixture containing more than two components. For example, in the ternary mixture containing components 1, 2 and 3. only binary constants k,j, k2i (and perhaps c,. 

There are two methods for representing the fugacities in terms of the measurable state variables the equation-of-state method and the activity-coefficient method. The usual representation in the vapor phase is to use the equation-of-state method. 

Although experimental transport properties are measured at different temperatures and pressures, it is the density, or molar volume, which is the theoretically important variable. So, for the prediction of transport properties, it is necessary to convert data at a given temperature and pressure to the corresponding temperature and density, or vice versa, by use of a reliable equation of state. Accordingly, an account is given in this volume of the most useful equations of state to express these relationships for gases and liquids. For dense fluids, it is possible to calculate transport properties directly by molecular simulation techniques under specified conditions when the molecular interactions can be adequately represented. A description is included in this book of these methods, which are significant also for the results which have aided the development of transport theory. 

Up to this point, we have used activity coefficients and models for to describe the nonideality in the liquid phase. In this section, we learn another approach to solve VLE problems, where we describe both the liquid and vapor phases using the fugacity coefficients. We term this approach the equation of state method. 

When the system temperature is above one of the species critical temperature, it can be problematic to use activity coefficient methods, and the equation of state method is preferred. The presence of such a lighter component also leads to high system pressures at equilibrium, as shown in Figure 8.11, where the pressure of the methane—n-pentane system can exceed 150 bar. 

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