Vapor-liquid equilibrium fugacity coefficient

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... 

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. 

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. 

At system pressures up to several tens of MPa, the fugacity coefficients, < > and (j), and the Poynting factor, 7Zp are usually near unity. A simplified version of equation 19 can therefore be used for the majority of vapor—liquid equilibrium problems ... 

In terms of activity and the fugacity coefficients, the vapor-liquid equilibrium from Eq. (1.189) becomes... 

This equation, expressing equality of fugacity coefficients, is an equally valid criterion of vapor/liquid equilibrium for pure species. 

The mole fraction and the fugacity coefficient are y and total pressure is P. This is the so-called gamma/phi formulation of vapor-liquid equilibrium. ... 

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 fugacities f or activity coefficients of a liquid solution are measured in vapor-liquid equilibrium experiments. In commonly employed methods, the liquid solution is brought in contact and kept in contact with a vapor mixture of the same components until equilibrium is attained between the phases. A sample of the vapor is then withdrawn and analyzed to determine its mole fractions y, i = 1,2,. Similarly for the liquid sample, the mole fractions X = 1,2,... are determined. Together with the measured p, an experimental point of vapor-liquid equilibrium is given by... 

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. 

In a significant departure from conventional practice, Chueh and Prausnitz (11,12) proposed that the critical constraints on the RK equation be relaxed, and that parameters b and c be treated as empirical constants, determined separately for the liquid phase and for the vapor phase of a given substance. The conventional RK expression for (T) was retained the application was to vapor-liquid equilibrium calculations, in which the vapor-phase version of the equation was used for computation of vapor-phase fugacity coefficients, but in which the liquid-phase version was used only for Poynting corrections. Thus, they proposed that... 

In the analysis and correlation of vapor-liquid equilibrium (VLE) data it is essential, especially at superatmospheric pressures, to take into account the effect of vapor-phase nonideality. This is expressed by the fugacity coefficient which, as long as the density of the mixture is not greater than one fourth of its critical value, can be calculated reliably with the following equation (for a binary mixture) ... 

The vapor-liquid equilibrium K-ratio can be calculated via the fugacity coefficients from an equation of state as a function of temperature, pressure, and compositions of liquid and vapor mixtures. 

Despite widespread use of the ideal K-value concept in industrial calculations, particularly during years prior to digital computers, a sound thermodynamic basis does not exist for calculation of the fugacity coefficients for pure species as required by (4-85). Mehra, Brown, and Thodos discuss the fact that, for vapor-liquid equilibrium at given system temperature and pressure, at least one component of the mixture cannot exist as a pure vapor and at least one other component cannot exist as a pure liquid. For example, in Fig. 4.3, at a reduced pressure of 0.5 and a reduced temperature of 0.9, methane can exist only as a vapor and toluene can exist only as a liquid. It is possible to compute vl or f v for each species but not both, unless vl = vy, which corresponds to saturation conditions. An even more serious problem is posed by species whose critical temperatures are below the system temperature. Attempts to overcome these difficulties via development of pure species fugacity correlations for hypothetical states by extrapolation procedures are discussed by Prausnitz. ... 

Problem 11.16 At 80 "C, 1.32 bar the system methanol(i)/ethanol(2) is in vapor-liquid equilibrium. The composition of the two phases is Xi = 0.25, y, = 0.37. Calculate the fugacity, fugacity coefficient of each component in each phase. You may assume that the vapor phase is in the ideal-gas state. 

Experimental studies were carried out to derive correlations for mass-transfer coefficients, reaction kinetics, liquid holdup and pressure drop for the new catalytic packing MULTIPAK (see [9,10]). Suitable correlations for ROMBOPAK 6M were taken from [70] and [92], The vapor-liquid equilibrium is calculated using the modification of the Wilson method [9]. For the vapor phase, the dimerization of acetic acid is taken into account using the chemical theory to correct vapor-phase fugacity coefficients [93]. Binary diffusion coefficients for the vapor phase and for the liquid phase are estimated via the method purposed by Fuller et al. and Tyn and Calus, respectively (see [94]). Physical properties like densities, viscosities and thermal conductivities are calculated from the methods given in [94]. 

Equation (1.114) relates the vapor-liquid equilibrium ratio, Ki, to the ratio of fugacity coefficients. The fugacity coefficients can be obtained from the volumetric properties given by an EOS. However, as Eq. (1.109) demands, the volumetric data are required from zero pressure to pressure P of the system at constant temperature and composition. Therefore, the EOS should represent the volumetric behavior over the whole range. 

For calculation of the vapor-liquid equilibrium composition, the activity coefficients of each species in the liquid phase are necessary. They can directly be computed from g -models or from EOS with an intermediate step involving the fugacity coefficients as follows ... 

The condition for vapor-liquid equilibrium, Equation 1.19, may be stated in terms of fugacity coefficients ... 

Experimental vapor-liquid equilibrium data, namely equilibrium temperature, pressure, and vapor and liquid compositions, can be used to calculate the activity coefficients from Equation 1.29. The AT-values are calculated from the composition data, A", = lyx,. The fugadties and fugacity coefficients,/ f, are calculated from the compositions, temperature, and pressure, using, for instance, an equation of state and liquid density data as described earlier. The activity coefficients are then calculated by rearranging Equation 1.29 ... 

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. 

The properties of dissolution as gas solubility and enthalpy of solution can be derived from vapor liquid equilibrium models representative of (C02-H20-amine) systems. The developments of such models are based on a system of equations related to phase equilibria and chemical reactions electro-neutrality and mass balance. The non ideality of the system can be taken into account in liquid phase by the expressions of activity coefficients and by fugacity coefficients in vapor phase. Non ideality is represented in activity and fugacity coefficient models through empirical interaction parameters that have to be fitted to experimental data. Development of efficient models will then depend on the quality and diversity of the experimental data. 

The main quantity involved in low pressure vapor-liquid equilibrium is the activity coefficient, the measure of the nonideality in the liquid phase. (The same way that the fugacity coefficient reflects the nonideality in the vapor phase. Notice, however, that ideality refers to Raoult s law in the liquid phase and to ideal gas law, in the vapor phase.)... 

The methodology for solving the vapor-liquid equilibrium problem using an equation of state has been outlined in Section 13.4 and involves expressing the fugacity coefficients of the mixture components in the liquid (/) and vapor (v) phases through an equation of state (EoS). The equilibrium ratio Kj is then calculated from ... 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

The computer subroutines for calculation of vapor-phase and liquid-phase fugacity (activity) coefficients, reference fugac-ities, and molar enthalpies, as well as vapor-liquid and liquid-liquid equilibrium ratios, are described and listed in this Appendix. These are source routines written in American National Standard FORTRAN (FORTRAN IV), ANSI X3.9-1978, and, as such, should be compatible with most computer systems with FORTRAN IV compilers. Approximate storage requirements and CDC 6400 execution times for these subroutines are given in Appendix J. 

Thus, equilibrium is achieved when the escaping tendency from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, fj, can be defined by the expression ... 

There are several characteristics common to the describing equations of all types of multicomponent, vapor-liquid separation processes, both single- and multi-stage, that make it possible to exploit the inside-out concept in similar ways to solve them efficiently and reliably. The equations have as common members component and total mass balance, enthalpy balance, constitutive and phase equilibrium equations. In addition, all such processes require K-value or fugacity coefficient and vapor and liquid enthalpy models. In all cases the describing equations have similar forms, and depend on the primitive variables (temperature, pressure, phase rate and composition) in essentially the same ways. Before presenting the inside-out concept, it will be useful to identify two classes of conventional methods and discuss their main characteristics. 

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