Equation vapor fugacity predictions

By now we should be convinced that thermodynamics is a science of immense power. But it also has serious limitations. Our fifty million equations predict what — but they tell us nothing about why or how. For example, we can predict for water, the change in melting temperature with pressure, and the change of vapor fugacity with temperature or determine the point of equilibrium in a chemical reaction but we cannot use thermodynamic arguments to understand why we end up at a particular equilibrium condition. 

Table VII shows results for the first system, isopropanol -isopropyl ether - water - propylene, in which the experimental compositions in each of the three phases are compared with the values predicted by the method just described. A modified Redlich-Kwong equation of state for vapor fugacity, Chao-Seader equation with adjusted parameters for liquid fugacity, and the Wohl equation for the activity coefficients were used. The predictions were based only on data for binary systems.

If an activity coefficient model is to be used at high pressure (Equation 4.27), then the vapor-phase fugacity coefficient can be predicted from Equation 4.47. However,... 

Take a mixture of two or more chemicals in a temperature regime where both have a significant vapor pressure. The composition of the mixture in the vapor is different from that in the liquid. By harnessing this difference, you can separate two chemicals, which is the basis of distillation. To calculate this phenomenon, though, you need to predict thermodynamic quantities such as fugacity, and then perform mass and energy balances over the system. This chapter explains how to predict the thermodynamic properties and then how to solve equations for a phase separation. While phase separation is only one part of the distillation process, it is the basis for the entire process. In this chapter you will learn to solve vapor-liquid equilibrium problems, and these principles are employed in calculations for distillation towers in Chapters 6 and 7. Vapor-liquid equilibria problems are expressed as algebraic equations, and the methods used are the same ones as introduced in Chapter 2. 

Equilibrium compositions of liquid phases at equilibrium are calculated by equating the component fugacities, similar to vapor-liquid equilibrium calculations, described in more detail in Chapter 2. The activity coefficients may be calculated by equations presented in Section 1.3.3, in particular the UNIQUAC and NRTL equations. The composition dependence of these equations is developed to the point where the same equation with the same constants can predict activity coefficients over wide ranges of composition, thus allowing it to predict two immiscible liquid phases at equilibrium. 

The PR eos has been modified by Stryjek and Vera to extend to polar substances that do not follow the three-parameter principle of corresponding states. The modified eos is fitted to the vapor pressure of polar substances with additional substance-specific parameters. The PRSV equation has been described in Equation (4.163) et seq. The free-energy-matched mixture eos parameters are given in Equations (4.436) and (4.438) the fugacity coefficients are given in Equation (4.439). PRSV eos using the UNIEAC activity coefficient predicts the vie data for both ethanol/water mixtures at 423-623°K and acetone/water mixtures at 373-523°K from low to high pressure. 

Other approaches to the computation of solid-liquid equilibria are shown in Table 11.2-3. The Soave-Redlich-Kwong equation of state evaluates fugacities to calculate solid-liquid equilibria,7 while Wenzel and Schmidt developed a modified van der Waals equation of state forthe representation ofphase equilibria. The Wenzel-Scbmidt approach generates fugacities, from which the authors developed a trial-and-error approach to compute solid-liquid equilibrium. Unno et a .9 recently presented a simplification of the solution of groups model (ASOG) that allows prediction of solution equilibrium from limited vapor-liquid equilibrium data. 

In low- to moderate-density vapors, mixture nonidealities are not very large, and therefore equations of state of the type discussed in this text can generally be used for the prediction of vapor-phase fugacities of all species. [However, mixtures containing species that associate (i.e., form dimers, trimers, etc.) in the vapor phase, such as acetic acid, are generally described using the virial equation of state with experimentally determined virial coefficients.] The Lewis-Randall rule should be used only for approximate calculations it is best to use an equation of state to calculate the vapor-phase fugacity of vapor mixtures. 

As the first illustration of the use of these equations, consider vapor-liquid equilibrium in the hexane-triethylamine system at 60°C. These species form an essentially ideal mixture. The vapor pressure of hexane af this temperature is 0.7583 bar and that of triethylamine is 0.3843 bar these are so low that the fugacity coefficients at saturation and for the vapor phase can be neglected. Consequently, Eqs. 10.1-3 and 10.1-4 should be applicable to this system. The three solid lines in Fig. 10.1-1 represent the two species partial pressures and the total pressure, which were calculated using these equations and all are linear functions of the of liquid-phase mole fraction the points are the experimental results. The close agreement between the computations and the laboratory data indicates that the hexane-triethylamine mixture is ideal at these conditions. Note that this linear dependence of the partiaLand total pressures on mole fractions predicted by Eqs. 10.1-2 and 10.1-3 is trae only for ideal mixtures it is not true for nonideal mixtures, as we shall see in Sec. 10.2. 

Equation 25 was developed from an empirical representation of thg second virial coefficient correlation of Pitzer and Curl (I) parameter b was left unchanged at its classical value of 0.0866. Because of the substantial improvement in the prediction of and its temperature derivatives for nonsimple fluids, the Barner modification of the RK equation gave improved estimates of enthalpy deviations for nonpolar vapors and for vapor-phase mixtures of hydrocarbons. However, the new equation was unsuitable for fugacity calculations. 

For the SRK-EOS, the dimensionless constants and 0 arc 0.4274 and 0.0867, respectively. However, in the ZJKK-EOS, these two parameters are determined from saturated liquid density and the equality of the saturated liquid and vapor phase fugacities. At the critical temperature and above, these two parameters are assigned values of 0.4274, and 0.0867, respectively. In the PR and SRK equations, no parameter is adjusted for density. As a result, these two equations have a density-predict on deficiency. Figure 3.9 shows the deviation in liquid molar volume of selected substances at = 0.7 versus v). The SRK-EOS underestimates the liquid density of all substances that are shown in the figure. The PR-EOS overestimates the density to to — 0.35. and then underestimates the density of n-alkanes heavier than Q. This figure clearly shows that at 0.7, the SRK-EOS is best suited for density prediction of pure hydrocarbons with while the PR-... 

The vapor pressure data for pure components are used to obtain the y. parameter of the EOS the density is then predicted. For mixtures, the EOS can be used to calculate not only the mixture density but also the phase behavior. In a limited sense, the phase behavior means the compositions and amoimts of the equilibrium phases. The next chapter presents the equations for phase-behavior calculations. Essentially, phase-behavior calculations rely on the use of the expression for the fugacity of component i in the mixture given by Eq. (3.32) for the PR-EOS. 

The second reason for developing new equations of state concerns the exceptional power and utility of an equation of state. When combined with appropriate thermodynamic relations, a well-behaved equation can predict with high precision isothermal changes in heat capacity, enthalpy, entropy and fugacity, vapor pres-... 

Chao-Seader Correlation. Reference was made earlier to the well known and much used Chao-Seader correlation for the prediction of vapor-liquid equilibrium for principally hydrogen-hydrocarbon systems with small amounts of CO2, H2S, O2, N2, etc. The heart of the correlation consists of several equations to represent liquid fugacity. The other two constituents, the Scatchard-Hildebrand equation for activity coefficients and the Redllch-Kwong equation for the vapor-phase nonideality, were already well established. 

Equation-of-state methods appear to be the most likely candidates for reliable accurate data. They are capable of predicting enthalpy, entropy, density, fugacity, vapor-liquid, and liquid-liquid data from one equation in regions where both low and high densities are encountered. Rapid computation requires that the equation be simple yet accurate without computational difficulties in areas surrounding the critical point. 

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