Equation binary deviation parameters

Figure 1. Optimum binary deviation parameters (k and ji j) and confidence ellipses for the fit of the LHW equation of state to the Nz + Ar VE data of Ref. 11, pp. 166-168...

The remaining binary deviation parameter k 2 was determined by again fitting the 50MPa VE data of Reference 11. These ki2 values, the corresponding /i2 values from Equation 20, and the standard deviations for the VE data fit are listed in Table IV for the LHW (VEo° =0) model. By comparison with the Table III LHW values (also listed), only for N2 + CH4 and Ar -f- C2H6 are the standard deviations significantly increased for this method. 

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 method can be extended to include nonpherical, nonpolar species (such as the lower molecular weight alkanes) by introduction of a third parameter in the equation of state, namely the Prigogine factor for chain-type molecules (9). This modified hard-sphere equation of state accurately describes VE(T, x) for liquefied natural gas mixtures at low pressures. Ternary and higher mixture VE values are accurately predicted using only binary mixing rule deviation parameters. 

It is generally much better to use the Equation 25 approach than the mole-fraction average of the component values to obtain mixture isothermal compressibilities. Very little is gained by trying to optimize the deviation parameters to binary data. It would appear that = 0 is a satisfactory approximation in this method. 

An explicit account of hydrogen bonding in water by the equation of state results in substantial improvement of the correlation of (water + hydrocarbon) LEE. In Figure 4.6, LEE for (water + hexane) is shown. The CPA equation of state correlates the water solubility with an Absolute Average Deviation of 4.5 % and reasonable agreement is obtained between experiment and calculations for hexane solubility. Unfortunately, the minimum of the solubility of hydrocarbon cannot be captured with a single temperature-independent binary interaction parameter. 

Mixing rules with a binary interaction parameter and with an excess function were not studied by Centeno etal. (2011). To examine their capability for prediction of blend viscosity, datareported in the literature were taken as example (Table 1.7). The method developed by Ratcliff and Khan indicates that it is necessary to include an excess function to account for deviation from ideality by means of the following equation ... 

For gas-liquid solutions which are only moderately dilute, the equation of Krichevsky and Ilinskaya provides a significant improvement over the equation of Krichevsky and Kasarnovsky. It has been used for the reduction of high-pressure equilibrium data by various investigators, notably by Orentlicher (03), and in slightly modified form by Conolly (C6). For any binary system, its three parameters depend only on temperature. The parameter H (Henry s constant) is by far the most important, and in data reduction, care must be taken to obtain H as accurately as possible, even at the expense of lower accuracy for the remaining parameters. While H must be positive, A and vf may be positive or negative A is called the self-interaction parameter because it takes into account the deviations from infinite-dilution behavior that are caused by the interaction between solute molecules in the solvent matrix. 

In general, data are fit quite well with the model. For example, with only two binary parameters, the average standard deviation of calculated lny versus measured InY of the 50 uni-univalent aqueous single electrolyte systems listed in Table 1 is only 0.009. Although the fit is not as good as the Pitzer equation, which applies only to aqueous electrolyte systems, with two binary parameters and one ternary parameter (Pitzer, (5)), it is quite satisfactory and better than that of Bromley s equation (J). 

Equations 5.15 and 5.16 give only approximate values for the volumes of olivine compounds. More accurate study of binary mixtures outlines important deviations from ideal behavior, which result in slight curves in cell parameter vs. composition plots, as shown in figure 5.7. The greatest deviations regard cell edges and are particularly evident for the (Mg,Ni)2Si04 mixture. 

Predictions for Ternary Mixtures. Excess volumes were calculated from the equations of state for the nearly equimolar ternary mixtures of N2 - Ar —J— CH4 and Ar -f- CH4 -f- C2H6 for which data are given in Appendix B of Reference 11. The root mean square deviations between the experimental and calculated VE values are given in Table V. Only component and binary parameters from Tables I, II, and III were used in these calculations. 

The deviations for the ternary mixtures are on the same order as those for the constituent binaries. These simple equations of state can predict ternary VE values for mixtures of N2, Ar, CH4, and C2H6 without requiring any ternary parameters in the formulation. 

The two binary parameters were determined by least squares fit of the binary VE data (0-50 MPa) from Appendix B of Reference 11, exactly as was done for the equations of state discussed above. Binary parameters and standard deviations for the fits are compared with those from Table III for the LHW model in Table IV. MOL1 is the extended corresponding... 

Liquid Solution Behavior. The component activity coefficients in the liquid phase can be addressed separately from those in the solid solution by direct experimental determination or by analysis of the binary limits, since y p = 1. Because of the large amount of experimental effort required to study a ternary composition field and the high vapor pressures encountered in the arsenide and phosphide melts, a direct experimental determination of ternary activity coefficients has been reported only for the Ga-In-Sb system (26). Typically, the available binary liquidus data have been used to fix the adjustable parameters in a solution model with 0,p determined by Equation 7. The solution model expression for the activity coefficient has been used not only to represent the component activities along the liquidus curve, but also the stoichiometric liquid activities needed in Equation 7. The ternary melt solution behavior is then obtained by extending the binary models to describe a ternary mixture without additional adjustable parameters. In general, interactions between atoms in different groups exhibit negative deviations from ideal behavior... 

Many equations, either empirical or derived from models, are available to represent excess thermodynamic properties as a function of the liquid mole fraction and of a number of adjustable parameters for use in Equations 4.3 and 4.4 (Prausnitz, Lichtenthaler, and Gomes de Azevedo 1999). In the treatment of our experimental data, we have examined the capability of many of these equations to fit data with the smallest number of parameters and with the least standard deviation of the O.F. (Lepori et al. 1998). For the majority of systems, either binary or ternary, examined by us in the course of about two decades, the rational form (Myers and Scott 1963) of the Redlich-Kister (RK) expression (Redlich and Kister 1948), and the Wilson (Wilson 1964) equation in the extended form (Novtik, MatouS, and Pick 1987), have resulted in the most appropriate representations of and For ternary systems, the excess functions can be expressed as the sum of a contribution (subscript B), which depends only on the parameters of the three binary systems, and a ternary contribution (subscript T), which involves additional parameters. 

In practice almost exclusively VLE data are used to fit the required parameters. Since a distillation column works nearly at constant pressure, most chemical engineers prefer thermodynamically consistent isobaric VLE-data in contrast to isothermal VLE-data to fit the model parameters. But that can cause problems, in particular if the boiling points of the two compounds considered are very different [24], as for example, for the binary system ethanol-n-decane. The result of the Wilson equation after fitting temperature-independent binary parameters only to reliable isobaric data at 1 atm is shown in Figure 5.31 for the system ethanol-n-decane, where the sum of the relative deviations of the activity coefficients was used as objective function. 

Calculation of binary diffusion coefficients based on Eqs. (3.1.69),(3.1.71),(3.1.72) and (3.1.79) is limited because estimations of the collision cross-section of the molecules cr (and of the influence of the temperature on kinetic theory of gases. The so-called Hirschfelder equation is frequently presented in many textbooks and used in the literature, but values of parameters such as the collision diameters of the molecules and characteristic energies are needed. Instead, many authors have developed empirical relations. For non-polar gas pairs, D B.g is in good approximation (deviation <10%) given by the equation of Slattery and Bird (1958) ... 

All the experimental binary values and calculated deviations in liquid and vapor compositions and temperature, (calculated using the equation 4), are siunmarized in Table 1.2 where also the mean absolute deviations in the individual variables are reported. The resulting Wilson parameters optimized in this work for the two binaries are listed in Table 1.3. 

Equation (61) makes allowance for differences in the genesis of ternary contacts but it does not yet consider that the number of segments of the third component in the coordination sphere of a certain binary contact might deviate from that expected from the average composition due to very favorable or unfavorable interactiOTis (quasi chemical equilibria). One way to model such effects consists of the introduction of composition-dependent ternary interaction parameters, as formulated in the following equation ... 

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