Equation of state methods

In a series of papers by Leung and coworkers (AlChE J., 32, 1743-1746 [1986] 33, 524-527 [1987] 34, 688-691 [1988] J. Loss Prevention Proc. Ind., 2[2], 78-86 [April 1989] 3(1), 27-32 [Januaiy 1990] Trans. ASME J. Heat Transfer, 112, 524-528, 528-530 [1990] 113, 269-272 [1991]) approximate techni ques have been developed for homogeneous equilibrium calculations based on pseudo-equation of state methods for flashing mixtures. 

Example 4.3 A mixture of ethane, propane, n-butane, n -pentane and n-hexane is given in the Table 4.3. For this calculation, it can be assumed that the K-values are ideal. For the mixture in Table 4.3, an equation of state method might have been a more appropriate choice. Flowever, this makes the calculation of the K-values much more complex. The ideal K-values for the mixture can be expressed in terms of the Antoine Equation as ... 

Oellrich L, Plocker U, Prausnitz J.M and Knapp H (1981) Equation-of-state Methods for Computing Phase Equilibria and Enthalpies, Int Chem Eng, 21(1) 1. 

Figure 8 depicts how the three popular equation-of-state methods cited previously perform on pure steam. From a theoretical viewpoint, none of the methods has the foundation to handle mixtures of polar/non-polar components. Although the agreement with experimental data is not very satisfactory for any of the methods, the Lee-Kesler equation-of-state does best. It was also found that by slightly adjusting the acentric factor of water, improvement in the representation of the enthalpy of steam can be obtained by this method at 598 K, the conditions of the experimental mixture data, and at other temperatures as well. 

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. 

Figure 9. Comparisons of predicted and experimental enthalpy departures for an equimolar steam-methane mixture at 598 K (equation-of-state methods)...

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. 

The object of this work was to extend the field of application of the equation-of-state method. The method was applied to aqueous systems in conjunction with a model that treats water as a mixture of a limited number of polymers, an approach similar to that previously adopted for the carboxylic acids (2). Association is calculated by the law of mass action corrections for non-ideal behaviour are made by means the equation of state. A major problem of the method is the large number of parameters needed to describe the properties and concentrations of the polymers together with their interaction with molecules of other substances. The Mecke-Kemptner model (15) (also known as the Kretschmer-Wiebe model (16) and experimental values for hydrogen-bond energies were usecT for guidance in fixing these parameters. 

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. 

There has been no shortage of attempts to estimate surface (Helmholtz-) energies from contact angles, by invoking some model. A controversial issue is Neumann s equation of state method ) which is based on the assumed validity of a second relationship between interfacial tensions, so that and y can be individually estimated. Another route starts by assuming [5.7.5] to be valid. For an apolar liquid on a solid S (y = y ), combination with Young s law gives... 

Values of the critical temperature and pressure are needed for prediction methods that correlate physical properties with the reduced conditions. It is also important to know the critical conditions when applying equation of state methods, as some of the equation of state models are unreliable close to the critical point. Experimental values for many substances can be found in various handbooks and in Appendix C. Critical reviews of the literature on critical constants and summaries of selected values have been published by Kudchadker et al. (1968), for organic compounds, and by Mathews (1972), for inorganic compounds. An earlier review was published by Kobe and Lynn (1953). 

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 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. 

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,. 

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