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. 

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. 

The methane-methanol binary is another system where the EoS is also capable of matching the experimental data very well and hence, use of ML estimation to obtain the statistically best estimates of the parameters is justified. Data for this system are available from Hong et al. (1987). Using these data, the binary interaction parameters were estimated and together with their standard deviations are shown in Table 14.1. The values of the parameters not shown in the table (i.e., ka, kb, kc) are zero. 

Isomerization products and reactants have more or less identical physico-chemical properties. For example, critical properties of 1-hexene and its isomers are similar, especially among the isomers. For this reason, this critical property determination procedure was applied to only binary and ternary mixtures of 1-hexene and isomers (as a pseudo-component) in CO2. Binary interaction parameters were taken from Vera and Orbey (14). Figures 2-5 show critical property loci of binary mixtures of 1-hexene and C02 Maximum deviation from reported experimental critical pressures in Figure 1 ( is less than 2%. 

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. 

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. 

Table IV. Binary Interaction Parameters (Ay and /,) and Standard Deviations (s/cm3 mol 1) between Model Calculations and the...

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. 

In this study, a modified Simplex method was used to regress the binary interaction parameter, fcy, using a packaged algorithm, DBCPOL (13), The objective function minimized by the optimization routine was the percent absolute average relative deviation (%AARD)... 

In this work, the binary interaction parameter (k,j) was fitted to experimental vapor-hquid equilibrium data. The objective fimction used to optimize the kij parameter for any binary mixture was the minimization of the deviation between the model s prediction of the bubble pressure and the experimental value. The optimized ky values are suimnarized in Table 1. For comparison, optimum ky values are also shown for the PR EoS. A semi-predictive approach was used to determine the sensitivity of the results to the interaction parameter using the CO2 coupled binary interaction parameters. 

As expected, PC-SAFT performs better than PR EoS for both systems. A comparison between the predictions by PC-SAFT and PR and experimental data measured by Creton et al. [6] is seen in Figure 1. Furthermore, Table 3 shows the average absolute deviation for PR and PC-SAFT predictions for all three systems. The molecular simulation data for the C02-N2-Ar-02-S02 mixture lie far from the critical point so the density is more accurately predicted by both PC-SAFT and PR EoS resulting in a lower %AAD. These results show that PC-SAFT follows the trend of experimental data more closely and with higher accuracy than PR. The use of the optimized binary interaction parameters further improves the prediction of density by both PR and PC-SAFT. 

For fitting the binary interaction parameters nonlinear regression methods are applied, which allow adjusting the parameters in such a way that a minimum deviation of an arbitrary chosen objective function F is obtained. For this job, for example, the Simplex-Nelder-Mead method (21j can be applied successfully. The Simplex-Nelder-Mead method in contrast to many other methods [22] is a simple search routine, which does not need the first and the second derivate of the objective function with respect to the different variables. This has the great advantage that computational problems, such as "underflow or overflow with the arbitrarily chosen initial parameters can be avoided. 

Fig. 10.26 Schematic illustration of types of possible polymer blend phase diagrams, for binary blends where additional complications that can be introduced by competing processes (such as crystallization of a component) are absent. The coefficients di and d2 refer to a general functional form (as a function of temperature and component volume fractions) of the binary interaction parameter that quantifies deviations from ideal mixing (Courtesy Online resources)...

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 ... 

The characteristic parameters for calculating the density using the S-L EOS are given in Table 1[19]. The binary interaction parameter, ku, was determined in order to minimize the relative experimental deviation at each given temperature, and the determined binary interaction parameters at different temperatures are included in Table 2 [20,21],... 

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

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 Fig. 3.1, several ideal structures are also plotted with the + mark. All of these structures have no adjustable parameter and most of them lose some of the symmetry elements when they are distorted. As shown in the figure, most of the ideal structures have some deviation from the fitting curve. It may be related to the fact that some of these ideal structures are deformed in real binary compounds. 

Data for the carbon dioxide-methanol binary are available from Hong and Kobayashi (1988). The parameter values and their standard deviations estimated from the regression of these data are shown in Table 14.2. 

Monton and Llopis (1994) presented VLE data at 6.66 and 26.66 kPa for binary systems of ethylbenzene with m-xylene and o-xylene. The accuracy of the temperature measurement was 0.1 K and that of the pressure was 0.01 kPa. The standard deviations of the measured mole fractions were less than 0.001. The data at 26.66 for the ethylbenzene (1) - o-Xylene (2) are given in Table 15.8 and the objective is to estimate the NRTL and UNIQUAC parameters based on these data. 

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). 

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