Fugacity coefficient composition dependence

Foam fractionation Fractional extraction Fractionation, seeDistillation Free-volume theory of diffusion Freezing-point determination Fugacity of nitrogen standard state Fugacity coefficient composition dependence of acetic acid vapor... 

A component in a vapor mixture exhibits nonideal behavior as a result of molecular interactions only when these interactions are very wea)c or very infrequent is ideal behavior approached. The fugacity coefficient (fi is a measure of nonideality and a departure of < ) from unity is a measure of the extent to which a molecule i interacts with its neighbors. The fugacity coefficient depends on pressure, temperature, and vapor composition this dependence, in the moderate pressure region covered by the truncated virial equation, is usually as follows ... 

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. 

Fifth, calculate the composition dependent coefficients necessary for calculating fugacity coefficients for both liquid and gas. 

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. 

When the liquid phase is ideal, Ki depends only on the temperature, the pressure, and the vapor composition. The procedure for determining the dew point in such a case is to (1) guess a temperature (2) calculate the Kt, which equal f°/iP, where is the fugacity of pure liquid i at the system temperature and pressure, , is the fugacity coefficient of the i th species in the vapor phase, and P is the system pressure and (3) check if the preceding dew-point equation is satisfied. If it is not, repeat the procedure with a different guess. 

Unfortunately, K values are generally composition-dependent through the fugacity and activity coefficients. Only in ideal systems is the composition dependency removed ... 

Note that with this modification to the van der Waals one-fluid model, the fugac-ity coefficient expression for species i given in eqn. (3.3.9) will change because an additional compositional dependence has been introduced to the a term of the EOS. For the PR EOS, with the van der Waals one-fluid model, a more general form of the fugacity coefficient expression of species i in a mixture is... 

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. 

When the distribution coefficients are composition-dependent, the above method must be modified to account for the effect of composition. A search for the unknown bubble point or dew point temperature or pressure is started on the basis of some composition-independent relationship between the X-values and the temperature and pressure, such as Equations 2.20 and 2.21. Component fugacities are then calculated for the vapor phase and the liquid phase, and the /f-values are updated using Equation 2.15. The calculations are repeated until Equation 2.16 or 2.17, as well as Equation 2.12, are satisfied. The iterative scheme for the bubble point pressure calculation may proceed along the following steps ... 

FIGURE 1.3-1 Composition dependence of fugacity coefficient i>i of component 1 in a binary gas mixture ai 300 K aed I bsr, Curves correspoed to different values of iha interaction recond virial coefficient Bl2, (See text for discussion.)... 

FIGURE 1.3 5 Composition dependence of fugacity coefficients at 40 C aed 0.025 aim in binary gas mixture containing acetic acid aed an inert component. Curves are computed from chemical theory, assuming dimerization of acetic acid, with K = 380. 

The temperature dependence of the fugacity J-, (actually, the fugacity coefficient (p — f /x P) can be gotten by differentiating Eq. 9.2-3 with respect to temperature at constant pressure and composition. 

Figure 4.5 Typical composition dependence of fugacity coefficients in gas mixtures. Fugacity coefficients in carbon dioxide + propane mixtures at 100°F, 200 psia. These curves are corrected from results tabulated by Walas [9].

When intermolecular forces are independent of composition, each fugacity deviates from its ideal-gas value by an amount that is also independent of composition. This means each ideal-solution fugacity coefficient does not depend on composition. 

This shows that, although the ideal-solution fugacity coefficient is independent of composition, it does depend on the choice made for the standard state. Consequently, the ideal-solution fugacity coefficient is not the same as the standard-state fugacity coefficient unless we choose = P. That is, in general... 

Figure 5.6 Schematic of the composition dependence of the fugacity /j and activity coefficient Yj in a binary mixture at fixed T and P. This activity coefficient is based on the Lewis-Randail standard state (5.4.11) and therefore satisfies the pure-fluid (5.4.12) and dilute-solution (5.4.13) limits. Note that the fugacity of the ideal-solution (broken line) is linear in the mole fraction and that, in the Lewis-Randail standard state, f = j.

With the above parameters and assumptions, for each concentration the fugacity coefficients can be determined for the given pressure and temperature. The calculation is performed in the same way as shown in Chapter 5 for the system N2-CH4. However, it is simpler, since only the fugacity coefficients in the gas phase are required. Nevertheless, the calculation has to be performed iteratively, because the fugacity coefficients depend on composition. 

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