Activity coefficient-models correlative liquid mixture

We focus on the thermodynamic models that deal with the liquid mixtures in this chapter. From the two categories of activity coefficient models, the correlative one is not very useful for solubility prediction and solvent screening purposes. The main reason for this is the lack of experimental data for the binary interaction parameters of the solute-solvent, solute-antisolvent, and solvent-antisolvent systems. As an example, the activity coefficient from... 

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. 

Several Correlative Liquid Mixture Activity Coefficient Models 429... 

SEVERAL CORRELATIVE LIQUID MIXTURE ACTIVITY COEFFICIENT MODELS... 

Several Correlative Liquid Mixture Activity Coefficient Models 431 The simplest polynomial representation of G satisfying these criteria is... 

Be able to correlate the low-pressure vapor-liquid equilibrium data for a nonideal liquid mixture (that is. to be able to compute the conditions of vapor-liquid equilibrium and develop. r-v, T-x y, and P-x-y diagrams for nonideal mixtures using activity coefficient models (the y -cj) method) (Sec. 10.2)... 

Figure 10.3-9 Vapor-liquid equilibria of the acetone -f water binary mixture correlated using the combination of the Peng-Robinson equation of state, the Wong-Sandler mixing rule, and the NRTL activity coefficient model. The three parameters in this model have been fit to data for each isotherm.

Liquid-liquid equilibrium occurs when the species in a mixture are dissimilar. The most common situation is the one in which the species are of a different chemical nature. Such mixtures are best described by activity coefficient models, and that is the case considered in Illustration 11.2-3. However, liquid-liquid equilibrium may-also occur when the two species are of similar chemical nature but differ greatly in size, as in the methane-/t-heptane system, or when the species differ in both size and chemical nature, as in the carbon dioxide-n-octane system shown in Fig. 11.2-36. Since both carbon dioxide and n-octane can be described by simple equations of state, liquid-liquid equilibrium in this system can be predicted or correlated using equations of state, though with some difficulty. 

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

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 most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are based on the concept of local composition. Within a liquid solution, local compositions, different from the overall mixture composition, are presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. The concept was introduced by G. M. Wilson in 1964 with the publication of a model of solution behavior since known as the Wilson equation. The success of this equation in the correlation of VLE data prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two Liquid) equation of Renon and Prausnitz and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz. A further significant development, based on the UNIQUAC equation, is the UNIFAC method,tt in which activity coefficients are calculated from contributions of the various groups making up the molecules of a solution. 

A modified local composition (LC) expression is suggested, which accounts for the recent finding that the LC in an ideal binary mixture should be equal to the bulk composition only when the molar volumes of the two pure components are equal. However, the expressions available in the literature for the LCs in binary mixtures do not satisfy this requirement. Some LCs are examined including the popular LC-based NRTL model, to show how the above inconsistency can be eliminated. Further, the emphasis is on the modified NRTL model. The newly derived activity coefficient expressions have three adjustable parameters as the NRTL equations do, but contain, in addition, the ratio of the molar volumes of the pure components, a quantity that is usually available. The correlation capability of the modified activity coefficients was compared to the traditional NRTL equations for 42 vapor—liquid equilibrium data sets from two different kinds of binary mixtures (i) highly nonideal alcohol/water mixtures (33 sets), and (ii) mixtures formed of weakly interacting components, such as benzene, hexafiuorobenzene, toluene, and cyclohexane (9 sets). The new equations provided better performances in correlating the vapor pressure than the NRTL for 36 data sets, less well for 4 data sets, and equal performances for 2 data sets. Similar modifications can be applied to any phase equilibrium model based on the LC concept. 

Liquid Mixtures Compositions at the liquid-vapor interface are not the same as in the bulk liquid, and so simple (bulk) composition-weighted averages of the pure-fluid values do not provide quantitative estimates of the surface tension at the vapor-liquid interface of a mixture. The behavior of aqueous mixtures is more difficult to correlate and estimate than that of nonpolar mixtures because small amounts of organic material can have a pronounced effect upon the surface concentrations and the resultant surface tension. These effects are usually modeled with thermodynamic methods that account for the activity coefficients. For example, a UNIFAC method [Suarez, J. T. C. Torres-Marchal, and P. Rasmussen, Chem. Eng. Set, 44 (1989) 782] is recommended and illustrated in PGL5. For nonaqueous systems the extension of the parachor method, used above for pure fluids, is a simple and reasonably effective method for estimating a for mixtures. 

Mixtures with nonsimilar molecular structures, and in particular mixtures containing water, generally exhibit nonideal behavior. When this situation prevails, recourse must be made to published experimental data or correlations of liquid phase activity coefQcients. With a minimum of experimental data, some very good models are available to provide activity coefficients over a broad range of conditions. If no experimental data are available, and there is no azeotrope, then approximate results can be obtained from the UNIFAC model. But it is better to make some experiments. 

The introductory discussion of models for liquid-phase activity coefficients, presented in Chapter 5, included a description of the Wilson equation, which is appropriate for many nonelectrolyte mixtures that exhibit large deviations from ideality. However, the Wilson model cannot correlate liquid-liquid equilibrium data, and therefore it cannot be used in LLE and VLLE calculations. To overcome this deficiency, Renon and Prausnitz [1] devised the NRTL model for (NonRandom, Two-Liquid). 

Whereas the models given above can be used to correlate solvent activities in polymer solutions, attempts also have been made in the literature to develop concepts to predict solvent activities. Based on the success of the UNIFAC concept for low-molecular liquid mixtures,Oishi and Prausnitz developed an analogous concept by combining the UNIFAC-model with the free-volume model of Flory, Orwoll and Vrij. The mass fraction based activity coefficient of a solvent in a polymer solution is given by ... 

Figure 6.10 shows activity coefficient derivatives over the whole composition range for experiment from three correlations and the Verlet method. A procedure for experimental data analysis was described by Wooley and O Connell (1991), in which one extracts the isothermal compressibility, partial molar volumes, and activity coefficient derivatives from experimental data. The activity coefficient derivatives are obtained by fitting mixture vapor-liquid equilibrium data to obtain parameters for at least two different models. Wooley and O Connell employed the Wilson, non-random, two liquid (NRTL) and modified Margules (mM) models. Partial molar volumes are obtained from correlations of mixture densities (Handa and Benson 1979). Isothermal compressibilities are either taken from measurements or estimated with... 

FIGURE 6.11 TCFIs for (a) water/water, (b) water/t-butanol, and (c) t-butanol/t-butanol obtained from simulation of water/t-butanol mixtures using the Verlet method (crosses) versus the water mole fraction Xj, compared with TCFIs obtained from experimental data nsing the Wooley/O Connell procedure, where either the Wilson (black line), NRTL (red Une), or mM (green line) models were employed for obtaining the activity coefficient derivatives. Note that the NRTL and mM model approaches infinity since they predict a phase split. (Calculated values from R. J. Wooley and J. P. O Connell, 1991, A Database of Flnctuation Thermodynamic Properties and Molecular Correlation-Function Integrals for a Variety of Binary Liquids, Fluid Phase Equilibria, 66, 233.) (See color insert.)... 

Nonelectrolyte G mcxlels only account for the short-range interaction among non-charged molecules (—One widely used G model is the Non-Random-Two-Liquid (NRTL) theory developed in 1968. To extend this to electrolyte solutions, it was combined with either the DH or the MSA theory to explicitly account for the Coulomb forces among the ions. Examples for electrolyte models are the electrolyte NRTL (eNRTL) [4] or the Pitzer model [5] which both include the Debye-Hiickel theory. Nasirzadeh et al. [6] used a MSA-NRTL model [7] (combination of NRTL with MSA) as well as an extended Pitzer model of Archer [8] which are excellent models for the description of activity coefficients in electrolyte solutions. Examples for electrolyte G models which were applied to solutions with more than one solvent or more than one solute are a modified Pitzer approach by Ye et al. [9] or the MSA-NRTL by Papaiconomou et al. [7]. However, both groups applied ternary mixture parameters to correlate activity coefficients. Salimi et al. [10] defined concentration-dependent and salt-dependent ion parameters which allows for correlations only but not for predictions or extrapolations. 

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