Fugacity computation

With v, taken as the molar volume of the saturated liquid, relative fugacities computed from Eq, (1.2-34) for subcooled liquid nitrogen at 100 K produce results nearly identical lo those given by the solid curve in Fig. 1.2-2. 

EXAMPLE 5.3 Phase equilibrium from fugacity computation... 

The calculation of vapor and liquid fugacities in multi-component systems has been implemented by a set of computer programs in the form of FORTRAN IV subroutines. These are applicable to systems of up to twenty components, and operate on a thermodynamic data base including parameters for 92 compounds. The set includes subroutines for evaluation of vapor-phase fugacity... 

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. 

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. 

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. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

Use the value of zv to compute the vapor phase fugacity for each species from Equation 14.9. 

Alternatively, one may use implicit LS estimation, e.g., minimize Equation 14.23 where liquid phase fugacities are computed by Equation 15.5 whereas vapor phase fugacities are computed by an EoS or any other available method (Prausnitz et al., 1986). 

A number of problems formulated with data from the literature are given next as exercises. In addition, to the objective function given by Equation 15.11 the reader who is familiar with thermodynamic computations may explore the use of implicit objective functions based on fugacity calculations. 

Whereas the fugacity approach was used by Mackay for the computation of mass flows and the concentration levels, the SimpleBox adopt the concentration-based piston velocity type mass transfer coefficients (ms-1). This is, mainly, because most scientific papers express the mass transfer in these terms, rather than in terms of the fugacity-based conductivity type coefficients (mol h 1 Pa-1). Furthermore, the transfer and transformation phenomena are treated as simple pseudo first-order processes, similar to Mackay models. 

Garrels and Thompson s calculation, computed by hand, is the basis for a class of geochemical models that predict species distributions, mineral saturation states, and gas fugacities from chemical analyses. This class of models stems from the distinction between a chemical analysis, which reflects a solution s bulk composition, and the actual distribution of species in a solution. Such equilibrium models have become widely applied, thanks in part to the dissemination of reliable computer programs such as SOLMNEQ (Kharaka and Barnes, 1973) and WATEQ (Truesdell and Jones, 1974). 

Fig. 23.3. Variation in C02 fugacity in a computer simulation of sampling, cooling, and then reheating a hypothetical geothermal fluid. Bold line shows path followed when system is held closed. Fine lines show effects of an open system in which fluid is allowed to degas C02 as it cools.

Computation of the heat of any process. To compute the heat of any process involving the disappearance of a substance or substances in the states given in the table and the appearance of other or the same substances in states given in the table Add together the heat of formation (values for Qf) of the products of the process in the final states and subtract therefrom the sum of the heats of formation (values for Qf) of the reactants in their initial states. The value so obtained represents the heat evolved when the given process takes place at a constant pressure (or a fugacity) of one atmosphere and at a temperature of 18°. The following are examples ... 

Since the equations for fugacity and activity coefficients are complex, solution of this kind of problem is feasible only by computer. Reference is made in Example 13.3 to such programs. There also are given the results of such a calculation which reveals the magnitude of deviations from ideality of a common organic system at moderate pressure. 

Table B.12. Parameters for Duan et al. (1992b) gas fugacity model (Eqs. 3.37-3.48). (Numbers are in computer scientific notation where e xx stands for 10 xx)...

Although fugacity is theoretically and computationally significant, it is the pressure that we need for practical applications. To find the relation between / and P, we write, from Eqs. (31) and (51),... 

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. 

Because of the complex functionality of the K-values, these calculations in general require iterative procedures suited only to computer solution. However, in the case of mixtures of light hydrocarbons, in which the molecular force fields are relatively weak and uncomplicated, we may assume as a reasonable approximation that both the liquid and the vapor phases are ideal solutions. By definition of the fugacity coefficient of a species in solution, =fffxtP. But by Eq. (11.61), f f = xj,. Therefore... 

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. 

Related Calculations. As computer-based computation has become routine, a growing trend in the determination of K values has been the use of of cubic equations of state, such as the Peng-Robinson, for calculating the fugacities of the components in each phase. Such calculations are mathematically complex and involve iteration. 

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