Parameter mixing rule

Due to restriction for space the results on modeling the high-pressure phase behaviour of the system carbon dioxide-water-1 -propanol are presented only briefly. The model used in this work was the Peng-Robinson EOS [8] with an temperature dependent attractive term due to Melhelm et al. [9], Although several mixing rules have been tested, the discussion will be restricted to the two-parameter mixing rule of Panagiotopoulos and Reid [10],... 

Wilson s expressions for the LCs (eqs 9 and 10) can also be used to generate mixing rules the combination of eqs 9 and 10 with eqs 17a and 18b leads to two-parameter mixing rules, with parameters I12 — An and A21 — A22- Three-parameter mixing rules with parameters A°2 — All, A21 — A22, and Agi — A°2 can be obtained by combining eqs 15 and 16 for the LCs and eqs 17a and 18b. This flexibility allows one to use two- or three-parameter mixing rules for systems for which the one-parameter mixing rules fail. 

A two-parameter mixing rule is used with several cubic equations of state and is shown to be relatively successful in correlating the phase equilibrium behavior of biomolecules that cannot be correctly represented by conventional one-parameter mixing rules. The modification is related to the idea of local composition, which has been shown to improve the representation of the phase equilibrium in asymmetric mixtures. However, further improvement is still needed. 

The mixture constants for the Patel and Teja equation of state were calculated using several mixing rules. Results of the calculations using classical one- and two-parameter van der Waals mixing rules are summarized in Table 1.9. The one-parameter mixing rule (denoted by vdWl in Table 1.9) contains one adjustable... 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

Several authors, notably Leland and co-workers (L2), have discussed vapor-liquid equilibrium calculations based on corresponding-states correlations. As mentioned in Section II, such calculations rest not only on the general assumptions of corresponding-states theory, but also on the additional assumption that the characterizing parameters for a mixture do not depend on temperature or density but are functions of composition only. Further, it is necessary clearly to specify these functions (commonly known as mixing rules), and experience has shown that if good results are to be obtained, these... 

In these equations, a and e are parameters in the Lennard-Jones potential function for interactions between unlike molecules, the customary mixing rules were used ... 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

In this section we consider typical examples. They cover all possible cases that could be encountered during the regression of binary VLE data. Illustration of the methods is done with the Trebble-Bishnoi (Trebble and Bishnoi, 1988) EoS with quadratic mixing rules and temperature-independent interaction parameters. It is noted, however, that the methods are not restricted to any particular EoS/mixing rule. 

Next, the VLE was calculated using these parameters and the results together with the experimental data are shown in Figure 14.4. The erroneous phase behavior has been suppressed. However, the deviations between the experimental data and the EoS-based calculated phase behavior are excessively large. In this case, the overall fit is judged to be unacceptable and one should proceed and search for more suitable mixing rules. Schwartzentruber et al. (1987) also modeled this system and encountered the same problem. 

It was shown by Englezos et al. (1998) that use of the entire database can be a stringent test of the correlational ability of the EoS and/or the mixing rules. An additional benefit of using all types of phase equilibrium data in the parameter estimation database is the fact that the statistical properties of the estimated parameter values are usually improved in terms of their standard deviation. 

A number of authors have suggested various mixing rules, according to which the quantity a could be calculated for a measured electrolyte in a mixture, starting from the known individual parameters of the single electrolytes and the known composition of the solution. However, none of the proposed mixing relationships has found broad application. Thus, the question about the dependence of the mean activity coefficients of the individual electrolytes on the relative contents of the various electrolytic components was solved in a different way. 

Ehase Inversion Temperatures It was possible to determine the Phase Inversion Temperature (PIT) for the system under study by reference to the conductivity/temperature profile obtained (Figure 2). Rapid declines were indicative of phase preference changes and mid-points were conveniently identified as the inversion point. The alkane series tended to yield PIT values within several degrees of each other but the estimation of the PIT for toluene occasionally proved difficult. Mole fraction mixing rules were employed to assist in the prediction of such PIT values. Toluene/decane blends were evaluated routinely for convenience, as shown in Figure 3. The construction of PIT/EACN profiles has yielded linear relationships, as did the mole fraction oil blends (Figures 4 and 5). The compilation and assessment of all experimental data enabled the significant parameters, attributable to such surfactant formulations, to be tabulated as in Table II. 

The calculated critical points of the binary pairs, particularly the critical pressures, are quite sensitive to the values used for the interaction parameters in the mixing rules for a and b in the equation of state. One problem in undertaking this study is that no data are available on the critical lines of any of the binary pairs except for CO2 - H2O. Even for C02 - H2O, two sets of critical data available (18, 19) are in poor quantitative agreement, though they present the same qualitative picture of the critical phenomena. 

The interaction parameters for binary systems containing water with methane, ethane, propane, n-butane, n-pentane, n-hexane, n-octane, and benzene have been determined using data from the literature. The phase behavior of the paraffin - water systems can be represented very well using the modified procedure. However, the aromatic - water system can not be correlated satisfactorily. Possibly a differetn type of mixing rule will be required for the aromatic - water systems, although this has not as yet been explored. 

Equation 9 was used to calculate H for steam + n-heptane as follows. The Peng-Robinson equation with parameters obtained from criticality conditions was used to calculate the residual enthalpy H of methyl fluoride. Peng-Robinson parameters for n-heptane were obtained by fitting to the residual enthalpy of the fluid at temperatures below the critical, and by using criticality conditions at higher temperatures. The mixing rules given in equation 4 with k.. = 1 were used to calculate H. As... 

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