Computational methods Fugacity

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. 

The fugacity coefficient J may be calculated by use of an equation of state or other methods for computing vapor fugacities as described above. (Note, in this case, however, f jp is replaced by /,%,/P,-.) If the liquid state exists at all pressures between P, and the total pressure P of the mixture, the fugacity of the liquid is found by integrating Eq. (14-101) which gives... 

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. 

Equation (7-106) is the Lewis and Randall rule for computing the fugacity of a pure real gas. (Further methods of calculation are discussed in Gilbert Newton Lewis and Merle Randall, Thermodynamics and the Free Energy of Chemical Substances, chap. 17, McGraw-Hill Book Company, Inc., New York, 1923.)... 

The central portion of the algorithm in Figure 11.6 exactly parallels the standard Rachford-Rice procedure. First, we use (11.1.27)-(11.1.29) to compute the mole fractions for all phases, then we compute all fugacity coefficients and all activity coefficients. With those quantities we can obtain new estimates for the Cs and Ks from the phase-equilibrium relations (11.1.15) and (11.1.24). Now we use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and F2, and test for convergence. If our convergence criteria are not met at iteration k, then we use the Newton-Raphson method to estimate the unknown L and V at the next iteration (fc + 1). 

In principle, we could compute the fugacity of pure liquids and solids the same way we computed that of gases. This is impractical, however, so we use other methods. The reason is that the molar volume v of liquids and solids is so small that the volume residual a is practically the same as the ideal gas volume. 

There are many types of EOS with a wide range of complexity. The Redlich-Kwong (RK) EOS is a popular EOS that relies only on critical temperatures and critical pressures of all components to compute equilibrium properties for both liquid and vapor phases. However, the RK EOS does not represent liquid phases accurately and is not widely used, except as a method to compute vapor fugacity coefficients in activity-coefficient approaches. On the other hand, the Benedict-Webb-Rubin-Starling (BWRS) EOS [6] has up to sixteen constants specific for a given component This EOS is quite complex and is generally not used to predict properties of mixture with more than few components. 

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. 

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). 

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. 

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. 

Finally, we must select appropriate methods of estimating thermodynamic properties. lime (op. cit.) used the SRK equation of state to model this column, whereas Klemola and lime (op. cit.) had earlier used the UNIFAC model for liquid-phase activity coefficients, the Antoine equation for vapor pressures, and the SRK equation for vapor-phase fugacities only. For this exercise we used the Peng-Robinson equation of state. Computed product compositions and flow rates are shown in the table below. 

Due to lack of space, the few results presented here are primarily intended to demonstrate the validity of the proposed method. The pore space of the adsorbent is assumed to consist of slit-shaped pores of width 15 A, with parameters chosen to model activated carbon. The porosity values are fixed at q = 0.45 and qp = 0.6. The feed stream is atemary gas mixture of H2/CH4/C2H6. The vtqx>r-phase fugacities were computed from the virial equation to second order, using coefficients taken from Reid et al ... 

We now consider two cases. The first is a bulk liquid at a temperature T subject to an external pressure of P. The fugacity of this liquid, (7, P), can be computed by any of the methods described earlier in this chapter. The second case is that of a small droplet of the same liquid at the same temperature and external pressure. The fugacity of this droplet is... 

Although only compressibility factor calculations are used as an example in the explanation of the method, other properties can be predicted equally well. Because of the temperature and density dependence of the diameters and shape factors needed to relate them to critical constants it is best to determine separate values of them for each component. Three basic dimensionless properties should be determined. These are the ones best suited to the use of the HSE method with an equation of state in terms of temperature and density. These are the compressibility factor, z the internal energy deviation (U — V)/RT and a dimensionless fugacity ratio, ln(f/pRT). All other desired properties can be obtained from them. The ln(f/pRT) and z are calculated similarly. The computation scheme is outlined as shown in Table III. 

From the outset the relationships between the fugacity and the state variables are highly nonlinear. To determine the composition of each phase for a SLV system such that Equations 1 and 2 are satisfied requires that an iterative method be used. Because of the constraints imposed on the system by the phase rule somewhat different procedures were used in this study to compute the SLV equilibrium condition for multicomponent systems and for binary systems, respectively. Both procedures calculate the fluid-phase compositions of a given mixture at the incipient solid-formation condition. 

Conceptually, the simplest method for solving phase-equilibrium problems is the phi-phi method, but computationally it is usually more complicated than other methods. The conceptual simplifications arise in part because no decisions need to be made about reference states the reference state is the ideal gas and the choice of the ideal-gas reference is implicit in choosing to work with fugacity coefficients. Usually, the same pressure-explicit equation of state is used for all components in all phases, for this produces consistency in the results and helps in organizing the calculations. (The same calculations are to be done for all components in all phases, and therefore computer programs can be structured in obvious modular forms.) However, this need not be done different equations of states can be used for different phases. 

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