Vapor-liquid equilibrium coefficients

Next, Equations 5.3 and 5.4 are combined into one equation, and the system of Equations 5.1 through 5.5 is reduced to two equations. Each of Equations 5.3 or 5.4, aided by a vapor-liquid equilibrium coefficient prediction method such as Equations... 

According to the rules proposed by Seader and Westerberg (1977), the components in the feed are arranged by decreasing volatility and two sets of parameters are checked the relative volatilities of adjacent pairs and the molar percentages of the components in the feed. The relative volatility, (x,y, of component i with respect to component j is used as an indicator of how readily they could be separated and is defined as the ratio of their vapor-liquid equilibrium coefficients ... 

The computations in three-phase distillation involve two sets of vapor-liquid equilibrium coefficients, or A -values, derived from the activity coefficients of each component in each liquid phase (Section 2.3.3). The calculations may be simplified in hydrocarbon systems if the second liquid phase is mostly water. In these situations it is possible to assume the aqueous phase to be pure water and account only for water dissolved in the organic phase. The description of rigorous solution methods for three-phase distillation is deferred to Chapter 13. The objective at this point is to consider tlie effect a liquid phase split can have on distillation. 

Step 1. Assume an average column temperature, and calculate the reference component vapor-liquid equilibrium coefficient at average conditions, using some A -valuc correlation of the form Also, at the same... 

The shortcut model is developed in terms of reduced parameters that are not strongly dependent on stream compositions, temperature, and pressure. The shortcut model, represented by Equations 12.33 through 12.37b, is solved in conjunction with reduced equations for calculating enthalpies, vapor-liquid equilibrium coefficients, and effective stripping factors based on the rigorous base case. 

The vapor-liquid equilibrium coefficients are expressed in terms of the relative volatility, oc, and a reference A -valuc at the top and bottom stages ... 

The other assumption in the model relates to the vapor-liquid equilibrium coefficients, or K-values. The -values at a given pressure are assumed to be a function of temperature only, and not of composition. It is further assumed that the temperature dependence of the -values for the different components is similar, i.e., the ratio of the 7 -values of any pair of components is independent of temperature. Thus, the relative volatilities, defined as the ratios of -values of any two components, are assumed constant throughout the column. 

K Vapor-liquid equilibrium coefficient V Vapor molar flow rate in the column... 

At this condition, tabulate the vapor-liquid equilibrium coefficient, the K value, for each component in the system. Then, calculate their volatilities relative to the heavy key component. 

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... 

Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

The analogy between equations derived from the fundamental residual- and excess-propeily relations is apparent. Whereas the fundamental lesidanl-pL-opeRy relation derives its usefulness from its direct relation to equations of state, the ci cc.s.s-property formulation is useful because V, and y are all experimentally accessible. Activity coefficients are found from vapor/liquid equilibrium data, and and values come from mixing experiments. 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

Gmehhng and Onken (Vapor-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) have reported a large collection of vapor-liqnid equilibrium data along with correlations of the resulting activity coefficients. This can be used to predict liqnid-hqnid equilibrium partition ratios as shown in Example 1. 

Gmehhng and Onken (op. cit.) give the activity coefficient of acetone in water at infinite dilution as 6.74 at 25 C, depending on which set of vapor-liquid equilibrium data is correlated. From Eqs. (15-1) and (15-7) the partition ratio at infinite dilution of solute can he calculated as follows ... 

Both liquid and vapor phases are totally miscible. Conventional vapor/liquid equilibrium. Neither phase is pure. Separation factors are moderate and decrease as purity increases. Ultrahigh purity is difficult to achieve. No theoretical limit on recovery. Liquid phases are totally miscible solid phases are not. Eutectic system. Solid phase is pure, except at eutectic point. Partition coefficients are very high (theoretically, they can be infinite). Ultrahigh purity is easy to achieve. Recovery is limited by eutectic composition. 

Several activity coefficient models are available for industrial use. They are presented extensively in the thermodynamics literature (Prausnitz et al., 1986). Here we will give the equations for the activity coefficients of each component in a binary mixture. These equations can be used to regress binary parameters from binary experimental vapor-liquid equilibrium data. 

Experimental values for the activity coefficients for components 1 and 2 are obtained from the vapor-liquid equilibrium data. During an experiment, the following information is obtained Pressure (P), temperature (T), liquid phase mole fraction (x, and x2=l-X ) and vapor phase mole fraction (yi and y2=l—yi). 

Table 15.6 Vapor-Liquid Equilibrium Data and A ctivity Coefficients for Benzene(I)-i-Propyl Alcohol at 760 mmHg...

Table 15.7 Vapor-Liquid Equilibrium Data and Activity Coefficients for...

This expression provides the basis for vapor-liquid equilibrium calculations on the basis of liquid-phase activity coefficient models. In Equation 4.27, thermodynamic models are required for cf>y (from an equation of state) and y, from a liquid-phase activity coefficient model. Some examples will be given later. At moderate pressures, the vapor phase becomes ideal, as discussed previously, and fj = 1. For... 

VAPOR-LIQUID EQUILIBRIUM BASED ON ACTIVITY COEFFICIENT MODELS... 

These models are semiempirical and are based on the concept that intermolecular forces will cause nonrandom arrangement of molecules in the mixture. The models account for the arrangement of molecules of different sizes and the preferred orientation of molecules. In each case, the models are fitted to experimental binary vapor-liquid equilibrium data. This gives binary interaction parameters that can be used to predict multicomponent vapor-liquid equilibrium. In the case of the UNIQUAC equation, if experimentally determined vapor-liquid equilibrium data are not available, the Universal Quasi-chemical Functional Group Activity Coefficients (UNIFAC) method can be used to estimate UNIQUAC parameters from the molecular structures of the components in the mixture3. 

Before an equation of state can be applied to calculate vapor-liquid equilibrium, the fugacity coefficient < />, for each phase needs to be determined. The relationship between the fugacity coefficient and the volumetric properties can be written as ... 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

Example 4.5 2-Propanol (isopropanol) and water form an azeotropic mixture at a particular liquid composition that results in the vapor and liquid compositions being equal. Vapor-liquid equilibrium for 2-propanol-water mixtures can be predicted by the Wilson equation. Vapor pressure coefficients in bar with temperature in Kelvin for the Antoine equation are given in Table 4.113. Data for the Wilson equation are given in Table 4.126. Assume the gas constant R = 8.3145 kJ-kmol 1-K 1. Determine the azeotropic composition at 1 atm. 

A model is needed to calculate liquid-liquid equilibrium for the activity coefficient from Equation 4.67. Both the NRTL and UNIQUAC equations can be used to predict liquid-liquid equilibrium. Note that the Wilson equation is not applicable to liquid-liquid equilibrium and, therefore, also not applicable to vapor-liquid-liquid equilibrium. Parameters from the NRTL and UNIQUAC equations can be correlated from vapor-liquid equilibrium data6 or liquid-liquid equilibrium data9,10. The UNIFAC method can be used to predict liquid-liquid equilibrium from the molecular structures of the components in the mixture3. 

Care should be exercised in using the coefficients from Table 4.14 to predict two-liquid phase behavior under subcooled conditions. The coefficients in Table 4.14 were determined from vapor-liquid equilibrium data at saturated conditions. 

Although the methods developed here can be used to predict liquid-liquid equilibrium, the predictions will only be as good as the coefficients used in the activity coefficient model. Such predictions can be critical when designing liquid-liquid separation systems. When predicting liquid-liquid equilibrium, it is always better to use coefficients correlated from liquid-liquid equilibrium data, rather than coefficients based on the correlation of vapor-liquid equilibrium data. Equally well, when predicting vapor-liquid equilibrium, it is always better to use coefficients correlated to vapor-liquid equilibrium data, rather than coefficients based on the correlation of liquid-liquid equilibrium data. Also, when calculating liquid-liquid equilibrium with multicomponent systems, it is better to use multicomponent experimental data, rather than binary data. 

Prediction of liquid-liquid equilibrium also requires an activity coefficient model. The choice of models of liquid-liquid equilibrium is more restricted than that for vapor-liquid equilibrium, and predictions are particularly sensitive to the model parameters used. 

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