Interaction obtained from binary data

The solubilities of ammonia, carbon dioxide, and hydrogen sulfide were obtained from binary data and expressed in terms of a Henry s constant for infinite dilution and an interaction parameter ... 

Although Procedure C is a good predictive method, it should not be used as a substitute to reducing good experimental data to obtain activity coefficients. In general, higher accuracy can be obtained from empirical models when these models are used with binary interaction parameters obtained from experimental data. 

Two adjustable parameters, K i and A21, need to be determined by fitting binary-solution data to Equation (4.380) and Equation (4.381). Parameters obtained from binary solutions are useful and sufficient in Equation (4.379) for multicomponent solutions, since no higher interaction parameters are required in the multicomponent equation. 

For the UNIQUAC equation, there are two adjustable equation parameters for each binary. For the binary that is partially miscible, the best way to determine the two binary parameters is to fit the mutual solubility data. For the completely miscible binaries, useful interaction parameters can be obtained from vie data. However, fitting vie data to within experimental accuracy does not uniquely determine the binary parameters. The choice of a particular set of parameters can have a significant effect on the representation of the ternary lie. For the ternary system of chloroform, water, and acetone at 333°K, for example, the two binary parameters are first determined from mutual solubility data for chloroform and water and then the other binary parameters for the two miscible binaries. Somewhat improved predictions occur by fitting binary parameters to the miscible binaries. Similar predictions have also been found for ternary systems of ethyl acetate, ethanol, and water. 

The application of the AEOS to hydrocarbon-water mixtures requires temperature-dependent binary interaction coefficients. These data can be obtained from binary water-hydrocarbon data. Once such data are available, then the AEOS can be used to predict the phase behavior of H2 0-crude oil systems. Figure 3.22 shows the binary interaction coefficients between Cj and heavier hydrocarbons from Shinta and Firoozabadi (1997). Binary interaction coefficients of... 

Water-Hydrocarbon Systems. The application of the PR equation to two and three-phase equilibrium calculations for systems containing water has recently been Illustrated by Peng and Robinson ( ). As in the case of other hydrocarbon-non-hydrocarbon mixtures, one fitted binary interaction parameter for water with each of the hydrocarbons is required. These parameters were obtained from experimental data available in the literature on each of the water-hydrocarbon binaries. 

Data at two temperatures were obtained from Zeck and Knapp (1986) for the nitrogen-ethane system. The implicit LS estimates of the binary interaction parameters are ka=0, kb=0, kc=0 and kd=0.0460. The standard deviation of kd was found to be equai to 0.0040. The vapor liquid phase equilibrium was computed and the fit was found to be excellent (Englezos et al. 1993). Subsequently, implicit ML calculations were performed and a parameter value of kd=0.0493 with a standard deviation equal to 0.0070 was computed. Figure 14.2 shows the experimental phase diagram as well as the calculated one using the implicit ML parameter estimate. 

Results for the various binary mixed surfactant systems are shown in figures 1-7. Here, experimental results for the surface tension at the cmc (points) for the mixtures are compared with calculated results from the nonideal mixed monolayer model (solid line) and results for the ideal model (dashed line). Calculations of the surface tension are based on equation 17 with unit activity coefficients for the ideal case and activity coefficients determined using the net interaction 3 (from the mixed micelle model) and (equations 12 and 13) in the nonideal case. In these calculations the area per mole at the surface for each pure component, tOj, is obtained directly from the slope of the linear region in experimental surface tension data below the cmc (via equation 5) and the maximum surface pressure, from the linear best fit of... 

It is important to note, that the interaction parameters between the components (two per binary) were estimated solely from binary phase equilibrium data, including low-pressure VLE data for the binary acetone - water no ternary data were used in the fitting. The values of the interaction parameters obtained are shown in Table III. 

The Patel-Teja equation of state Is able to correlate the data for the binary systems reasonably well provided a binary Interaction coefficient (kj.) is Included In the calculations. It Is Interesting to note that the binary Interaction coefficients obtained from correlation of data for the odd members of the series are an order of magnitude smaller than those obtained for the even members of the series and that they show regular behavior with carbon number. These differences are due to differences In... 

The carbon di oxi de/lemon oil P-x behavior shown in Figures 4, 5, and 6 is typical of binary carbon dioxide hydrocarbon systems, such as those containing heptane (Im and Kurata, VO, decane (Kulkarni et al., 1 2), or benzene (Gupta et al., 1 3). Our lemon oil samples contained in excess of 64 mole % limonene so we modeled our data as a reduced binary of limonene and carbon dioxide. The Peng-Robinson (6) equation was used, with critical temperatures, critical pressures, and acentric factors obtained from Daubert and Danner (J 4), and Reid et al. (J 5). For carbon dioxide, u> - 0.225 for limonene, u - 0.327, Tc = 656.4 K, Pc = 2.75 MPa. It was necessary to vary the interaction parameter with temperature in order to correlate the data satisfactorily. The values of d 1 2 are 0.1135 at 303 K, 0.1129 at 308 K, and 0.1013 at 313 K. Comparisons of calculated and experimental results are given in Figures 4, 5, and 6. 

Although equations of state based on statistical mechanics, like the Perturbed Hard Chain and Chain of Rotators equations of state are good at predicting phase equilibria at conditions far from the critical point of mixtures, a critical evaluation of six of these type of equations of state showed that they are rather inaccurate in the mixture critical region[3]. Satisfactory correlation of the data is obtained with a Peng Robinson equation of state using two interaction parameters per binary as proposed by Shibata and Sandler[4], The correlations of Huang[5] were used for the pure component parameters. 

Initial values of the binary interaction parameters were obtained from vapor-liquid equilibrium data for ternary mixtures (9). These interaction parameters were then adjusted to minimize the difference between calculated and measured phase compositions for the three-phase equilibria measured at 50 C. 

The insertion of Eqs. (7) and (9) into Eq. (4) provides an expression which can be integrated analytically (Appendix B). The Flory-Huggins interaction parameter X can be used either as an adjustable parameter, or can be obtained from phase equilibrium data for the binary mixture polymer + water. 

Wilson s equation of state is found from Equations (14) and (15). It can be seen that for obtaining the activity coefficient of a component 1 in a pure solvent 2, we need four interaction parameters (A12, A21, An a A22, which are temperature dependent. It is evident that for calculating the value of the binary interaction parameters, additional experimental data, such as molar volume is needed. Other models which belong to the first category have the same limitations as Wilson s method. The Wilson model was used in the prediction of various hydrocarbons in water in pure form and mixed with other solvents by Matsuda et al. [11], In order to estimate the pure properties of the species, the Tassios method [12] with DECHEMA VLE handbook [13] were used. Matsuda et al. also took some assumptions in the estimation of binary interactions (because of the lack of data) that resulted in some deviations from the experimental data. 

The pure component parameters of the models can be retrieved by using the volumetric data above the glass transition temperature, for the polymers, and using volumetric data and/or vapor pressure data for the penetrants. The binary interaction parameters can be obtained from gas-polymer equilibrium data in the rubbery phase, when available. In the absence of any direct experimental information, the first-order approximation can be used or, alternatively, they can be treated as adjustable parameters. 

Values for critical temperature, pressure, and acentric factor for all five components participating in the system are given in Table 9.2. Values for the binary interaction parameters used in Equation 9.28c are given in Table 9.3. Note that the values found in Table 9.3 were obtained from different sources. Where the interaction parameters are unknown, the k, values are zero for the purposes of demonstration. More accurate predictions may be obtained by fitting the relevant vapor-liquid equilibrium data, if available. 

Based on the interaction parameters for the mixing rules obtained from the binary subsystems, the ternary phase behavior has been modeled by inter- and extrapolation of the interaction parameters. Because of lack of suflicient experimental data, extrapolation is considered to be the only option in some cases, although it obviously wiU introduce errors. Table 14.7 shows the interaction parameters of the various mixing rules used for the prediction of the ternary phase behavior, whereas the resulting partition coefficients per isothermal series... 

In principle, Rh thus determined should coincide with the value obtained from DLS. However, it should be noted that the figures derive from measurements performed in vastly different concentration regimes DLS refers to high dilution, whereas the rheological data are usually measured at volirme fractions where the binary interaction embodied in the quadratic term of eqn [4] can no longer be neglected. 

---