Results for Binary Systems

Sanchez and H. Lentz, High Temperature—High Pressure, 1974, 5, 689. 

Biichner, Systeme mit zwei flussigen Phasen , in ref. 16 (Part 2). 

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By adopting mixing rules similar to those given in Section II, Chueh showed that Eq. (55) can be used for calculating partial molar volumes in saturated liquid mixtures containing any number of components. Some results for binary systems are given in Figs. 7 and 8, which compare calculated partial molar volumes with those obtained from experimental data. 

The terms on the right-hand side of the first line represent the difference between the molar free enthalpies of the solid and liquid solutions at composition X Mb. In addition, a standard result for binary systems (e.g., Swalin, 1962) states for G and n the following relationship... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

Using a recent equation of state of the van der Waals type developed to describe non-polar components, a model is presented which considers water as a mixture of monomers and a limited number of polymers formed by association. The parameters of the model are determined so as to describe the pure-component properties (vapour pressure, saturated volumes of both phases) of water and the phase equilibria (vapour-liquid and/or liquid-liquid) for binary systems with water including selected hydrocarbons and inorganic gases. The results obtained are satisfactory for a considerable variety of different types of system over a wide range of pressure and temperature. 

In the previous sections the results of surfactant accumulation by foam or purification of a solution from surfactants were considered mostly for binary systems. The aim was to increase the accumulation ratio or the degree of extraction of one of the components with the... 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

Using our asymmetric combining rules, the data for the binary systems were combined and led to the results for ternary systems given in Fig. 3 this figure illustrates... 

Once the interaction energies were obtained, they were used to calculate the parameters in the UNIQUAC and Wilson models given by Eq. (24). To test the validity of the method, low-pressure vapor-liquid equilibrium (VLE) predictions were made for several binary aqueous systems. The calculations were done using the usual method assuming an ideal vapor phase (Sandler, 1999). Figures 7 and 8 show the low-pressure VLE diagrams for the binary aqueous mixtures of ethanol and acetone [see Sum and Sandler (1999a,b) for results for additional systems and values of the... 

The analyses in Section 9.4.1 for binary systems can be extended to multicomponent systems by using the Toor-Stewart-Prober approximation of constant [D]. We will not go through the details of the derivations here, our readers can verify these results for themselves. 

The material balance relations presented above are valid for any number of components. We shall discuss solutions to this system of equations for binary mixtures in the remainder of this section of Chapter 12 before moving on to obtain generalized results for multicomponent systems in Section 12.2. In the analyses that follow we shall ignore the effects of heat transfer between the vapor and liquid phases. 

Difficulties are also encountered when water and alcohol mixtures are considered. The correlation of the 2-propanol and water binary system at 353 K is shown in Figure 3,4.5. Here we see that at a temperature of 35 3 K, and also at lower temperatures, the IPVDW mixing rule gives a false liquid split and poorly represents the VLE data. At higher temperatures, the results for this system improve somewhat, as shown in Figure 3.4.6 for 523 K, but the correlation is still not acceptable for industrial design. 

With the treatment of gases as individual groups, some binary (or multicomponent) gas-liquid mixtures are reduced to mixtures of only two groups. For example, the carbon dioxide and methanol mixture considered at the conclusion of this section is actually a molecular mixture because both molecules are treated as groups by the UNIFAC approach, Similarly, mixtures of carbon dioxide with benzene or with paraffinic hydrocarbon liquids contain only two groups. The results for such systems are remarkably successful, as will be discussed in this section. The description of mixtures with more than two groups is possible for some of the present models, and the results look promising (Apostolou et al. 1995). 

The applicability of the multicomponent mass diffusion models to chemical reactor engineering is assessed in the following section. Emphasis is placed on the first principles in the derivation of the governing flux equations, the physical interpretations of the terms in the resulting models, the consistency with Pick s first law for binary systems, the relationships between the molar and mass based fluxes, and the consistent use of these multicomponent models describing non-ideal gas and liquid systems. 

Redden, G.D., Li, J., and Leckie, J., Adsorption of U and citric acid on goethite, gibbsite, and kaolinite. Comparising results for binary and ternary systems, in Adsorption of Metals by Geomedia, Jenne, E., ed., Academic Press, New York, 1998, p. 291. 

Then, the isothermal compressibility data on other pure substances, generally for V/Vc < Vi, were used to obtain their T and V values. Table II reports the results for the systems here. More extensive tables are reported elsewhere (5,6). In general, the fitting is most sensitive to the value of V, so when few data are available, T can be estimated as the critical temperature and a one-parameter fit is used. A consequence of this is that the binary parameter, K12, will compensate for any erroneous... 

Much less information is available on the methods for the determination of the interphasial thickness in polymer blends, Al, than that of Vj - For binary systems, assuming that these two parameters are interrelated, one may estimate Al from Vj, the latter determined using one of the above-described methods. To determine the experimental value of Al in any system, diverse methods have been used, viz. electronic microscopy. X-ray scattering, eUipsometry [Inoue, 1991 Fayt, 1986], light scattering, etc. Results of these measurements are presented in the next section (see Table 4.6). 

Figure 1. Boiling-point diagram for the binary system -octane and n-dodecane at P = 20 kPa. Open and filled symbols are used for the TraPPE and SKS force fields, respectively. The dashed line represents the experimental data. The calculated boiling points for the pure substances are shown as circles. Simulation results for binary mixtures are depicted as diamonds, upward-pointing triangles, squares, and downward-pointing triangles for simulations containing total mole fractions of -octane of 0.1,0.25,0.5, and 0.75, respectively.

A theoretical basis for different shapes of microemulsions (even for small W/O or O/W volume fractions) has been established on the basis of the relationship between shape and interfacial curvature [350,351]. It is reasonable to expect that the relevant properties of the surfactant film are represented by a bending elasticity with a spontaneous curvature, Co (as was demonstrated for binary systems). If the elastic modulii k, ksT, the fluctuations in curvature of the film are very small, and the entropy associated with them can be neglected. The actual morphology is the result of the competition between the tendency to minimize the bending free energy (which prefers spheres of optimal radius of curvature, = l/c ) and the necessity to use up all of the water, oil, and surfactant... 

For binary systems Perez de Ortiz and Sawistowski (1973a) proposed stability criteria, which were obtained following a similar mathematical treatment to the one by Stemling and Scriven and applied them successfully to several binary systems Perez de Ortiz and Sawistowski (1973b). However, they also found some disagreements with experimental results. 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

Having obtained the necessary flux equations for binary systems, we can easily generalise the result to ternary systems. Similar to eq. (8.2-20) applicable for binary systems, we can write the flux equation for the species 1 in a ternary mixture as follows ... 

Fillers and additives may also interact with the solvent, with the polymer, or with both. The electrospinnability of tbe solution may change as a result of these interactions, but it may also remain imcbanged, regardless of tbe additive. It has, for example, been shown that conductive solutions can form different kinds of three-dimensional loose structures on the substrate, instead of a tbin coating layer attached to the substrate surface. Tbe form of these structures can vary from fibrils perpendicular to the substrate surface to a layer with a fluffy cotton-like structure. Fibrils can even extend to cover tbe entire electrospinning zone. If this kind of conductive fibrous structure connects tbe nozzle and collector plate, the electric field discharges and the process stops. The optimisation of the process parameters for ternary systems can be even more challenging than for binary systems composed only of polymer and solvent. 

In Ellegaard, Abildskov, and O Connell (2010), data were analyzed from nine pharmaceutical solutes in a total of 68 binary mixtures of 10 solvents, with some of the mixtures at different temperatures. The absolute average relative deviation of using parameters from binary data was 23% while that from correlation of ternary data was 11%. Additional study of this method is described in Ellegaard (2011) with more solutes and binary solvents, as well as in some ternary solvents. Fignre 9.10 shows excess solubility results for representative systems. 

An exhaustive study of high-pressure vapor-liquid equilibria has been repotted by Knapp et at., who give not only a comprehensive literanire survey but also compare calculated and observed results for many systems. In that stu, several popular equations of state were used to perform the calculations but no one equation of state emerged as markedly superior to the others. All the equations of state used gave reasonably g results provided care is exercised in choosing the all-important binary constant k,j. All the equations of state used gave poor results when mixtures were close to critical conditions. 

For several binary systems recommended model parameters for the -models Wilson, NRTL, and UNIQUAC are given in Tables 5.9-5.11. Typical results for the system acetone-water using the NRTL model are shown in Figure 5.34. It can be... 

Figure 5.41 VLE results for the system n-butane(l) C02(2) using the binary parameter ku = 0 and an adjusted binary parameter ( -270K A — 292.6 K — 325.01 K ...

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