Computational methods Fugacity coefficient

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). 

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. 

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. 

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. 

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. 

Inner loop. The search begins by computing the liquid phase and vapor phase fugacity coefficients. The equations of state used in these problems are explicit in pressure, so the (ps are determined from (4.4.23), which involves an integration over the volume. This requires us to compute the molar volumes and from the equation of state. If the equation is cubic in v, then it should be solved analytically using Cardan s method (Appendix C). However, if the equation is fifth order or higher, then it will have to be solved by trial for v. With the qis known, we compute the K-factors from (11.1.2) and hence get calculated values for the vapor mole fractions xf). Typi-... 

Outer loop. When the sum S stops changing within the inner loop, we test whether that sum equals unity (conservation of mass). If it does not, we adjust the temperature and compute new values for the liquid-phase fugacity coefficients. This step closes the outer search over temperature. In many cases each K-factor responds to a change in temperature in a sufficiently well-behaved way that T can be adjusted by the simple secant method at the end of the iteration of the outer loop, the next guess k + 1) for T is taken to be... 

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 ... 

There is no direct way to measure fugacity, activity, or an activity coefficient. (If you can invent an instrument to do it, you will become rich and famous ) All values you will ever see have been calculated, either by the estimating methods shown in this and later chapters or computed from the things we can measure experimentally, such as temperature, pressure, density, and the concentrations (normally mol fractions) of the various species in the coexisting phases at equilibrium. The following example (which will be referred to and... 

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