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Figure 1 shows second virial coefficients for four pure fluids as a function of temperature. Second virial coefficients for typical fluids are negative and increasingly so as the temperature falls only at the Boyle point, when the temperature is about 2.5 times the critical, does the second virial coefficient become positive. At a given temperature below the Boyle point, the magnitude of the second virial coefficient increases with... 

Figures 3 and 4 show fugacity coefficients for two binary systems calculated with Equation (10b). Although the pressure is not large, deviations from ideality and from the Lewis rule are not negligible.

Two additional illustrations are given in Figures 6 and 7 which show fugacity coefficients for two binary systems along the vapor-liquid saturation curve at a total pressure of 1 atm. These results are based on the chemical theory of vapor-phase imperfection and on experimental vapor-liquid equilibrium data for the binary systems. In the system formic acid (1) - acetic acid (2), <() (for y = 1) is lower than formic acid at 100.5°C has a stronger tendency to dimerize than does acetic acid at 118.2°C. Since strong dimerization occurs between all three possible pairs, (fij and not... 

Figure 3-7. Fugacity coefficients for a saturated mixture of propionic acid (1) and raethylisobutylketone (2). Calculations based on chemical method show large variations from ideal behavior.

Figure 1 compares data reduction using the modified UNIQUAC equation with that using the original UNIQUAC equation. The data are those of Boublikova and Lu (1969) for ethanol and n-octane. The dashed line indicates results obtained with the original equation (q = q for ethanol) and the continuous line shows results obtained with the modified equation. The original equation predicts a liquid-liquid miscibility gap, contrary to experiment. The modified UNIQUAC equation, however, represents the alcohol/n-octane system with good accuracy. 

Figure 4-4. Representation of vapor-liquid equilibria for a binary system showing moderate positive deviations from Raoult s law.

Figure 4 shows experimental and predicted phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate positive deviations from Raoult s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation. 

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

Figure 7 shows a fit of the UNIQUAC equation to the iso-baric data of Nakanishi et al. (1967) for the methanol-diethyl-amine system this system also exhibits strong negative deviations from Raoult s law. The UNIQUAC equation correctly re-... 

Figure 4-7. Vapor-liquid equilibria and activity coefficients in a binary system showing a weak minimum in the activity coefficient of methanol.

As discussed in Chapter 3, at moderate pressures, vapor-phase nonideality is usually small in comparison to liquid-phase nonideality. However, when associating carboxylic acids are present, vapor-phase nonideality may dominate. These acids dimerize appreciably in the vapor phase even at low pressures fugacity coefficients are well removed from unity. To illustrate. Figures 8 and 9 show observed and calculated vapor-liquid equilibria for two systems containing an associating component. 

Figure 4-9. Vapor-liquid equilibria for a binary system where one component dimerizes in the vapor phase. Activity coefficients show only small deviations from liquid-phase ideality.

Moderate errors in the total pressure calculations occur for the systems chloroform-ethanol-n-heptane and chloroform-acetone-methanol. Here strong hydrogen bonding between chloroform and alcohol creates unusual deviations from ideality for both alcohol-chloroform systems, the activity coefficients show... 

Predictions for the other isobaric systems (experimental data of Sinor, Steinhauser, and Nagata) show good agreement. Excellent agreement is obtained for the system carbon tetrachlor-ide-methanol-benzene, where the binary data are of superior quality. 

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Figure 4-14. Predicted liquid-liquid equilibria for a typical type-II system shows good agreement with experimental data, using parameters estimated from binary data alone.

Figure 15 shows results for a difficult type I system methanol-n-heptane-benzene. In this example, the two-phase region is extremely small. The dashed line (a) shows predictions using the original UNIQUAC equation with q = q. This form of the UNIQUAC equation does not adequately fit the binary vapor-liquid equilibrium data for the methanol-benzene system and therefore the ternary predictions are grossly in error. The ternary prediction is much improved with the modified UNIQUAC equation (b) since this equation fits the methanol-benzene system much better. Further improvement (c) is obtained when a few ternary data are used to fix the binary parameters. 

Figure 16 shows observed and calculated VLE and LLE for the system benzene-water-ethanol. In this unusually fortunate case, predictions based on the binary data alone (dashed line) are in good agreement with the experimental ternary data. Several factors contribute to this good agreement VLE data for the mis-... 

The continuous line in Figure 16 shows results from fitting a single tie line in addition to the binary data. Only slight improvement is obtained in prediction of the two-phase region more important, however, prediction of solute distribution is improved. Incorporation of the single ternary tie line into the method of data reduction produces only a small loss of accuracy in the representation of VLE for the two binary systems. 

Figure 4-16. Representation of ternary liquid-liquid equilibria using the UNIQUAC equation is improved by incorporating ternary tie-line data into binary-parameter estimation. Representation of binary VLB shows small loss of accuracy. ---- Binary...

Figure 17 shows results for the acetonitrile-n-heptane-benzene system. Here, however, the two-phase region is somewhat smaller ternary equilibrium calculations using binary data alone considerably overestimate the two-phase region. Upon including a single ternary tie line, satisfactory ternary representation is obtained. Unfortunately, there is some loss of accuracy in the representation of the binary VLB (particularly for the acetonitrile-benzene system where the shift of the aceotrope is evident) but the loss is not severe. 

Figure 4-18. Calculated distribution for ternary liquid-liquid systems show good agreement with experiment except very near the plait point.

Figure 4-19. Calculated selectivities in two ternary systems show large improvements when tie-line data are used to supplement binary VLB data for estimating binary parameters.

We consider three types of m-component liquid-liquid systems. Each system requires slightly different data reduction and different quantities of ternary data. Figure 20 shows quarternary examples of each type. 

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

Figure 2 shows similar results for ethanol(1)-n-hexane(2) at 1 atm. The liquid-phase enthalpy of mixing was again estimated from UNIQUAC using temperature-independent parameters. 

Finally, Table 2 shows enthalpy calculations for the system nitrogen-water at 100 atm. in the range 313.5-584.7°K. [See also Figure (4-13).] The mole fraction of nitrogen in the liquid phase is small throughout, but that in the vapor phase varies from essentially unity at the low-temperature end to zero at the high-temperature end. In the liquid phase, the enthalpy is determined primarily by the temperature, but in the vapor phase it is determined by both temperature and composition. 

Table 1 gives the measured data, estimates of the true values corresponding to the measurements, and deviations of the measured values from model predictions. Figure 1 shows the phase diagram corresponding to these parameters, together with the measured data. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

Cases 3 and 4 show strong vapor-phase nonidealities as well. 

Execution times for the higher level subroutines FLASH and ELIPS will be highly dependent on the problems involved. The times required per iteration can be estimated from times for lower level subroutines and the descriptions given for FLASH and ELIPS. Computation times for two specific cases calculated with FLASH and one case claculated with ELIPS are included in Table J-1 to show approximate magnitudes required. 

Figure 1.46 shows a flowsheet without any heat integration for the different reactor and separation system. As before, this is probably too inefficient in the use of energy, and heat integration schemes can be explored. Figure 1.5 shows two of the many possible flowsheets. 

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