Application of Activity Coefficient Models

The equation of state approach is very attractive for the calculation of VLE But it requires an equation of state and reliable mixing rules, which are able to describe the PvTbehavior not only of the vapor but also of the liquid phase with the required accuracy. In spite of the progress achieved in the last 20 years, up to now there is no universal equation of state and mixing rule which can be successfully applied to all kind of systems in a wide temperature and pressure range for pure compounds and mixtures. 

For the calculation of VLE with Approach B often the simplified Eq. (5.16) is applied. Then besides the activity coefficients as a function of composition and temperature only the vapor pressures of the components are required for the calculation. 

Using Eq. (5.16) the required activity coefficients and the excess Gibbs energies can directly be derived from complete experimental VLE data. This is shown in Example 5.3 for the binary system ethanol-water measured at 70 C. 

Calculate the activity coefficients and the excess Gibbs energies for the system ethanol (l)-water (2) at 70 C as a function of composition using Eq. (5.16) and Table 5.2. 

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The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

The non-random two-liquid segment activity coefficient model is a recent development of Chen and Song at Aspen Technology, Inc., [1], It is derived from the polymer NRTL model of Chen [26], which in turn is developed from the original NRTL model of Renon and Prausznitz [27]. The NRTL-SAC model is proposed in support of pharmaceutical and fine chemicals process and product design, for the qualitative tasks of solvent selection and the first approximation of phase equilibrium behavior in vapour liquid and liquid systems, where dissolved or solid phase pharmaceutical solutes are present. The application of NRTL-SAC is demonstrated here with a case study on the active pharmaceutical intermediate Cimetidine, and the design of a suitable crystallization process. 

Two activity coefficient models have been developed for vapor-liquid equilibrium of electrolyte systems. The first model is an extension of the Pitzer equation and is applicable to aqueous electrolyte systems containing any number of molecular and ionic solutes. The validity of the model has been shown by data correlation studies on three aqueous electrolyte systems of industrial interest. The second model is based on the local composition concept and is designed to be applicable to all kinds of electrolyte systems. Preliminary data correlation results on many binary and ternary electrolyte systems suggest the validity of the local composition model. 

Numerous studies on the thermodynamics of calcium chloride solutions were published in the 1980s. Many of these were oriented toward verifying and expanding the Pitzer equations for determination of activity coefficients and other parameters in electrolyte solutions of high ionic strength. A review article covering much of this work is available (7). Application of Pitzer equations to the modeling of brine density as a function of composition, temperature, and pressure has been successfully carried out (8). 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

We have seen many successful industrial applications of applied electrolyte thermodynamics models. In particular, the electrolyte NRTL activity coefficient model of Chen and Evans has proved to be the model of choice for various electrolyte systems, aqueous and mixed-solvent. However, there are unmet needs that require further development. 

Second, we need an equation-of-state for electrolyte solutions. Equations-of-state are needed for modeling high-pressure applications with electrolyte solutions. Significant advances are being made in this area. Given that the electrolyte NRTL model has been widely applied for low-pressure applications, we are hopeful that, some day, there will be an equation-of-state for electrolytes that is compatible with the electrolyte NRTL activity coefficient model. 

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. ... 

The calculations reported in this paper and a related series of publications indicate that it is now quite feasible to obtain reasonably accurate results for phase equilibria in simple fluid mixtures directly from molecular simulation. What is the possible value of such results Clearly, because of the lack of accurate intermolecular potentials optimized for phase equilibrium calculations for most systems of practical interest, the immediate application of molecular simulation techniques as a replacement of the established modelling methods is not possible (or even desirable). For obtaining accurate results, the intermolecular potential parameters must be fitted to experimental results, in much the same way as parameters for equation-of-state or activity coefficient models. This conclusion is supported by other molecular-simulation based predictions of phase equilibria in similar systems (6). However, there is an important difference between the potential parameters in molecular simulation methods and fitted parameters of thermodynamic models. Molecular simulation calculations, such as the ones reported here, involve no approximations beyond those inherent in the potential models. The calculated behavior of a system with assumed intermolecular potentials is exact for any conditions of pressure, temperature or composition. Thus, if a good potential model for a component can be developed, it can be reliably used for predictions in the absence of experimental information. 

Figure 4.5 Schematic plot showing the general applicability of different activity coefficient models as a function of ionic strength for a divalent cation. The dashed tangent to the curve at its origin is a plot of the Debye-Hiickel limiting law for the ion.

In usual applications, the parameters of the UNIQUAC and Wilson activity coefficient models are fitted to experimental phase equilibria data. However, in the development of these models, the adjustable parameters correspond to the difference of interaction energies between the like and the unlike species. 

This mixing rule has been successful in several ways. First, when combined with any cubic EOS that gives the correct vapor pressure and an appropriate activity coefficient model for the term, it has been shown to lead to very good correlations of vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria, indeed generally comparable to those obtained when the same activity coefficient models are used directly in the y-

mixing rule extends the range of application of equations of state to mixtures that previously could be correlated only with activity coefficient models. 

The y-O approach is applicable only at low pressures. Like equations of state, activity coefficient models are also mathematical functions, however, of only temperature and concentration. Because in many applications related to polymer solutions, e.g., in paints and coatings industry, low pressures are involved. Equation 16.18 is a useful approximation. If, furthermore, it is assumed that the vapor phase, which typically contains only the volatile components (solvents), is ideal, then Equation 16.18 can be further simplihed ... 

Following the successes of the common cubic equations of state (van der Waals, SRK, and PR), the Sako equation of state has been reevaluated from some researchers, as shown by some recent applications. " " In some of these recent efforts, an activity coefficient model suitable for polymers is employed in the mixing rule for the energy parameter. 

The EoS/G method combines the best features of cubic equations of state and classical activity coefficient models. This is because, via this technique, at low pressures the behavior of the activity coefficient model is recovered. However, the model is also applicable at high pressures in a predictive way. It is important to note that existing parameter tables from, e.g., UNIFAC, UNIQUAC, etc. can be used. From the above it is understood that it is essential that the activity coefficient employed in the mixing rule should be as accurate as possible (at the low pressure Umit). 

Simple cubic equations of state can correlate both VLE and LLE for polymer solutions and blends with a single interaction parameters. They can be combined with an activity coefficient model for predictive calculations using the so-called EoS/G mixing rules. Applications of cubic equations of state to high pressures are so far limited to those shown for the Sako et al. cubic equation of state. 

The modeling of activity coefficients in multicomponent systems and the application of the general model [Equation (12.21)] are discussed in detail by Walas and Reid et al. ... 

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