Component fugacity coefficients, calculation

If equations of state are available for both the vapor and liquid phases, the above equations may be used to calculate the component fugacity coefficients in both phases by Equation 1.23, and the /f-values by Equation 1.25. Alternatively, the fugacity coefficient of a component in solution may be derived from the total fugacity coefficient expression (Equation 1.21) via the definition of partial properties ... 

Derive equations to calculate component fugacity coefficients in a binary mixture using the virial equation of state truncated after the second virial coefficient. The mixture second virial coefficient is given as... 

Once procedures for calculating pure-component parameters and mixing rules are established, the calculation of component fugacity coefficients 4>i for both vapor and liquid phases follows standard procedures (see e.g. (4)). For VLE calculations, the distribution of components between phases is expressed generally as the K-value—the vapor mole fraction divided by the liquid mole fraction—related to fugacity coefficients for each component by ... 

With the critical data, acentric factors, and the binary parameters, the pure component and mixture parameters have to be calculated for a temperature of 723.15 K. As initial composition, the mole fractions determined assuming ideal gas behavior are used and the parameters required for the calculation of the fugacity coefficients calculated. 

If equations of state are available for both the vapor and liquid phases, the above equations may be used to calculate the component fugacity coefficients in both phases by Equation 1.23, and the... 

Hint The component fugacity coefficient in a mixture is a partial property, and is calculated as the partial derivative of the corresponding total property with respect to the number of moles of the component in question in the mixture. 

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 fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure... 

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. 

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. 

Figure 13 presents results for a binary where one of the components is a supercritical, noncondensable component. Vapor-phase fugacity coefficients were calculated with the virial... 

CALCULATE FUGACITY COEFFICIENTS FOR NQN-ASSOCI ATING COMPONENTS ... 

CALCULATE FUGACITY COEFFICIENTS FOR ASSOCIATING COMPONENTS WITH CHEMICAL THEORY. FIRST CALCULATE THE EOUILIBRIUM CONSTANTS. 

PHIS calculates vapor-phase fugacity coefficients, PHI, for each component in a mixture of N components (N 5. 20) at specified temperature, pressure, and vapor composition. 

The fugacity coefficient of component i at saturation is obtained after the calculation of the vapor fugacity at saturation, by the relation ... 

The fugacity coefficients in Equation (7.29) can be calculated from pressure-volume-temperature data for the mixture or from generahzed correlations. It is frequently possible to assume ideal gas behavior so that = 1 for each component. Then Equation (7.29) becomes... 

The fugacity coefficients fa can be calculated from the Peng-Robinson Equation of State. The values of fa are functions of temperature, pressure and composition, and the calculations are complex (see Pohling, Prausnitz and O Connell6 and Chapter 4). Interaction parameters between components are here assumed to be zero. The results showing the effect of nonideality are given in Table 6.9 ... 

An initial guess for the pressure is assumed and the fugacity coefficient of each component in the liquid phase ( ) can be calculated. An initial guess is also assumed for the fugacity coefficient of each component in the vapour phase ( v), and consequently a first estimate of the vapour composition is evaluated. With this value of y, the fugacity coefficients in the vapour phase are recalculated using the equation of state and a second estimate for y,- is evaluated. This iterative procedure is continued until the difference between two successive values of the composition are below a predetermined error. At this point, the sum of y, is checked if the sum is different from unity a new value of the pressure is assumed for a new iteration. The iterative procedure ends when the y, differs from unity by less then a given value. 

Sixth, calculate the fugacity coefficients of the components of liquid and gas. ... 

The physical property monitors of ASPEN provide very complete flexibility in computing physical properties. Quite often a user may need to compute a property in one area of a process with high accuracy, which is expensive in computer time, and then compromise the accuracy in another area, in order to save computer time. In ASPEN, the user can do this by specifying the method or "property route", as it is called. The property route is the detailed specification of how to calculate one of the ten major properties for a given vapor, liquid, or solid phase of a pure component or mixture. Properties that can be calculated are enthalpy, entropy, free energy, molar volume, equilibrium ratio, fugacity coefficient, viscosity, thermal conductivity, diffusion coefficient, and thermal conductivity. 

Related Calculations. This procedure is valid only for those components whose critical temperature is above the system temperature. When the system temperature is instead above the critical temperature, generalized fugacity-coefficient graphs can be used. However, such an approach introduces the concept of hypothetical liquids. When accurate results are needed, experimental measurements should be made. The Henry constant, which can be experimentally determined, is simply y00/0, where y°° is the activity coefficient at infinite dilution (see Example 3.8). 

Two methods can be used to calculate the fugacities of a component in equilibrium. The first method requires an equation of state, which can be used with the following expression to calculate the fugacity coefficient. 

Here fugacity coefficient of component i, P is the pressure, and S is the mole fraction of component i. Fugacity coefficients are usually used only for the vapor phase, so yj is usually meant to represent the mole fraction of component i in the vapor phase and Xj is usually reserved to represent the mole fraction in the liquid phase. Equation (2B-6) can be used with any equation of state to calculate the fugacity of the components in the mixture in any phase as long as the equation of state is accurate for the conditions and phases of interest. An equation of state that is explicit in pressure is required to use Equation (2B-6). 

From Eqs. (9) and (10) one can see, that the calculation of the solubilities of solids in a SCF in the presence of an entrainer (cosolute or cosolvent) requires information about the properties of the pure components, the fugacity coefficients at infinite dilution and the values of K p. 

The value of the parameter 2 13 in a gas mixture can he calculated from PVT data using any traditional EOS. Eor the mixtures that obey the Lewis-Randall rule [16] (the fugacity of a species in a gaseous mixture is the product of its mole fraction and the fugacity of the pure gaseous component at the same temperature and pressure), the fugacity coefficients of the components of the mixture are independent of composition. In such cases, the KB equation [13] for the binary mixtures 1-3 ... 

In eqs 2 and 3, V is the molar volume, z is the compressibility factor, n is the total number of moles in the system, and ni is the number of moles of component i. Equations 2 and 3 show that the fugacity coefficient their derivatives with respect to the number of moles of solute are known. While near the critical point the fluctuations are important and an EOS involving them should be used, we neglect for the time being their effect. 

Equation (7) is obtained from Equation (2) by noting that the solid phase is pure, and therefore the mole fraction and activity coefficient in the solid phase are both unity. The ratio of pure-component fugacities can be obtained from any one of Equations ((3) to (6)), and the activity coefficient in the liquid, Ji, must be estimated. The composition-temperature behavior along the liquidus curves may then be calculated. The eutectic point is found from the intersection of the two liquidus curves. 

It is then possible to calculate the fugacity coefficient from an equation of state applied to Eq. (74). In this equation, n, is the moles of component i and ht is the total number of... 

---