Deviation 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. 

These equations may be generalized to multicomponent systems with a linear mixing rule as in Eq. 1. It is worth remarking that the goodness of the fit tends to improve when the number of components of the mixture increases. For instance, the deviation shown in Fig. 5 (right) for the mixture of C9 and C15 ortho-xylene sulfonates, disappears if 20% of C12 ortho-xylene sulfonate is added. It is conjectured that the presence of intermediate species improves the collective behavior. 

It has been suggested that this deviation to Hnear mixing rules comes from the fact that the nonionic species partitions to a much larger extent than the ionic one [51,70]. 

Nishikido (21) has done a systematic study o-f mixed sur-factant solubilization. In that study, solubilization in mixed systems was compared to that predicted by application o-f a linear mixing rule to the solubilizations in the pure surfactant component micelles. For example, in this "ideal case, a micelle composed of a 50/50 molar mixture of two surfactants would have a solubilization capacity which is an average of that of the two pure surfactants involved. A system showing negative deviation from ideality would have less solubilization than this ideal system a system having positive deviation from ideality would have more. 

Zsg = correction for deviation from a geometric mean mixing rule. 

Polymer gel Polymer cohesive energy density P (cal cm ) Deviation from geometric mean mixing rule parameter z Non-Gaussian elasticity parameter, N Curve fit crosslink density (10s mol cm-3) Experimental crosslink density (10s mol cm-3)... 

The good agreement between experimental and calculated values is pointed out by the average absolute deviation, AAD. The results demonstrate the possibility to compute the solubility as well as the partition coefficients with the Peng-Robinson-equation of state and a mixing rule with two interaction parameters. 

Soave equation gives rather deviations in mixture density with all mixing rules ... 

To verify, among the employed mixing rules, the one or ones that best represent the equilibrium data, the C02(l)-Diphenyl(2) at 343.2 K and C02(l)-n-Propylbenzene(2) at 353.2 K systems were used. The data and their conditions are presented in the works of Zang et al. [6] and Maurer et al. [7], respectively. The pure properties of the components are given in Reid et al. [8], The standard deviation of the measured output variable is ap=0.05MPa and cp=0.07MPa, respectively. 

The most sensitive mixing rule is Equation 5. A characteristic constant for each binary, kijy expresses the deviation from the geometric mean for which applies (that is, kij = 0) only when i and / are very similar in size and chemical nature. Otherwise, in the absence of any strong specific chemical interaction between i and /, fcy should be positive and thus TC. would be less than the geometric mean. 

Excess volume predictions are extremely sensitive to the value used for the unlike-molecule size parameter. Small deviations must be allowed from the commonly used arithmetic mean mixing rule for size parameters if quantitative results are to be obtained for VE simultaneously with other measurable excess thermodynamic functions (GE and HE). It should be noted that similar conclusions have been drawn for mixtures of globular molecules (8). 

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. 

There are two mixing rule deviation parameters (fey and /y) which must be evaluated for each pair of species in a mixture. In the present investigation, only binary mixture VE data were used in the evaluation of these parameters. 

Using Equation 17 instead of Equation 16 in the mixing rules for the GIB model did not greatly change the ability of the equation to fit the VE data. There were only slight changes in deviation parameters and standard deviations for the N2 -f CH4 and Ar -f C2H6 systems. 

MSE/MST and MSEE/MST blends were miscible by DSC measurements. Their density vs. composition curves showed positive deviations from the linear additivity rule volume contraction upon mixing two miscible polymer pairs was confirmed. Permeability vs. composition plot also showed large negative deviations from the semilogarithmic mixing rule. The permeability coefficient of the 50/50 MSE/MST blend was the lowest of all the materials tested it is comparable to EVAL-F and much lower than any other commercial barrier polymers. The volume shrinkage in the compatible blends show that the free volume is reduced, thus reducing the permeability. 

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