Fugacity generalized method

The generalized methods developed in Sec. 3.6 for the eompressibility faetor Z and in Sec. 6.7 for the residual enthalpy and entropy of pure gases are applied here to tlie fugacity coefficient. Equation (11.34) is put into generalized fonn by substitution of the relations. 

Unfortunately, very few mixtures are ideal gas mixtures, so general methods must be developed for estimating the thermodynamic properties of real mixtures. In the dis-, cussion of phase equilibrium in a. pure fluid of Sec. 7.4, the fugacity function was especially useful the same is true for mixtures. Therefore, in an analogous fashion to the derivation in Sec. 7.4. we start from... 

Fugacity has proved useful in a number of ways. One way is to provide a relatively simple way to evaluate the integral / VdP. In 5.7.1 we saw one way to do this. That is, for solids, we often assume that the molar volume is constant, making the integration very simple. Another way, for gases, is to assume the ideal gas law (see below). This is actually a special case of the most general method, which is to develop an equation of state for the system (Chapter 13), from which you can generate all its thermodynamic properties. 

The experimental data at high pressures have been analyzed by different equations of state to calculate the fugacity coefficients in K. Gillespie [11] and Gillespie, Beattie [12] have used the Beattie-Bridgeman equation of state. The method by Newton [44] is a generalized method for calculation of fugacity coefficients. 

General Properties of Computerized Physical Property System. Flow-sheeting calculations tend to have voracious appetites for physical property estimations. To model a distillation column one may request estimates for chemical potential (or fugacity) and for enthalpies 10,000 or more times. Depending on the complexity of the property methods used, these calculations could represent 80% or more of the computer time requited to do a simulation. The design of the physical property estimation system must therefore be done with extreme care. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Use of generalized fugacity coefficients (e.g., see Example 1.18) eliminates some computational steps. However, the equation-of-state method used here is easier to program on a programmable calculator or computer. It is completely analytical, and use of an equation of state permits the computation of all the thermodynamic properties in a consistent manner. 

Recently, a method [5] for the prediction of the solubility of a solute in a SC fluid in the presence of an entrainer has been proposed. The method, based on the Kirkwood-Buff (KB) formalism, was however developed for cases in which the entrainer was in dilute amounts. The present paper is focused on the solubility of a solid in a non-dilute mixture of a SC fluid and an entrainer. The theoretical treatment, which is more complex than for the dilute case, is also based on the KB formalism. In this paper the following aspects will be addressed (1) general equations for the solubility in binary and ternary mixtures will be written for the cases involving a small amount of solute (2) the KB formalism will be used to obtain expressions for the derivatives of the fugacity coefficients in a ternary mixture with respect to mole fractions (3) these expressions will be employed to derive an equation for the solubility of a solute in a SC fluid containing an entrainer at any concentration (4) a predictive method for this solubility will be proposed in terms of the solubilities of the solute in the SC fluid and in the entrainer (5) the derived equation will be compared with experimental results from literature regarding the solubility of a solute in a mixture of two SC fluids. 

The set of 2C simultaneous equations is nonlinear and fairly complex since it involves calculating fugacities and enthalpies, themselves nonlinear functions of the temperature, pressure, and composition. The equations may be solved simultaneously or by some iterative method, tn general, the computational methods depend on which two variables are selected as the independent variables. Although in principle any two independent variables may be fixed, the problem complexity may vary from case to case. It is found, for instance, that a solution is more readily reached if P and Trather than P and Q are the independent variables. Since most of these calculations are carried out on computers, the solution methods should be designed for speed of convergence and reliability. Several methods have been proposed for handling the different types of flash calculations, some of which are discussed herewith. 

Calcnlation of vapor-liqnid eqnilibrinm states nsing K values is particularly convenient for ideal mixtures for which the K values are independent of composition, changing only with temperature or pressure or both. While convenient, the A -value method does satisfy all the equilibrium conditions the component fugacities are equal in both phases for all components in the mixture, and temperature and pressure are equal in both phases. It follows that the A-value method can be used in general for noifideal mixtures as well. The composition dependence of the A values of nonideal mixtures is addressed next. 

For normal fluids, an alternative to the nse of vapor pressnre to find the standard liqnid fugacity /l, is to use a generalized correlation snch as the correlation by the acentric factor method. 

Fugacity is a key concept in phase equilibria. The phase equilibrium condition consists of the equality of fugacities of a component among coexistent phases. The computation of fugacities implies two routes equation of states, for both pure components and mixtures, and liquid activity coefficients for non-ideal liquid mixtures. The methods based on equations of state are more general. 

Liquid-solid equilibria are attacked with the gamma-gamma method in the same general way as liquid-liquid systems however, the two applications differ in how the standard-state fugacities are treated. We still start from the equality of fugacities. 

In general, the activity of gas components will require the calculation of the fugacity coefficient using the methods discussed in Chapter lo. If the ideal-gas approximation is applicable, then we may set [Pg.507]

Fugacity coefficients and hence activity coefficients can be calculated with the help of appropriate equations of state (see Section IV). This is possible, however, only for the gas phase (van der Waals equation, Redlich-Kwong equation, virial equation) for condensed phases no useful general equations of state are available. Experimental determination of activity coefficients in condensed phases is based on the study of equilibria. There are numerous methods, but only typical examples will be given. 

The vapor pressure methods derive directly from the properties of the equilibrium between the condensed solution and the vapor - a consequence of the equality of the chemical potentials of the component in question in the solution and in the gaseous phase at equilibrium with it. This equality gives us a very general relation between fugacity coefficient of component i in the gaseous phase, the total pressure, the activity of the component in the solution and an equilibrium constant ... 

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