Fugacity coefficients, Peng-Robinson

Fugacity coefficient Peng-Robinson equation of state... 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

Example 6.5 Repeat the calculations from Example 6.4 taking into account vapor-phase nonideality. Fugacity coefficients can be calculated from the Peng-Robinson Equation of State (see Poling, Prausnitz and O Connell6 and Chapter 4). 

Combining the Peng-Robinson equation of state and Equation 15-22 results in an equation for fugacity coefficient of each component.3... 

The expression for the fugacity coefficient 4> depends on the equation of state that is used and is the same for the vapor and liquid phases. In calculating the mixture properties with the Peng-Robinson equation of state we have used the following combining rule ... 

The fugacity coefficient < >2 is calculated by using a thermodynamic model. In this work the SRK (Soave) and Peng Robinson (PR) Equations of State were considered. 

The fugacity coefficient is usually obtained by solving an equation of state (e.g., Peng-Robinson Redlich-Kwong). The activity coefficient is obtained from a liquid phase activity model such as Wilson or NRTL (see Walas, 1985). 

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. 

The Soave/Redlich/Kwong (SRK) and the Peng/Robinson (PR) equations of state, both expressed by Eqs. (4-230) and (4-231), were developed specifically for VLE calculations. The fugacity coefficients implicit in these equations are given by Eq. (4-232). When combined... 

The graphs are based on the Peng-Robinson equation of state (1) as improved by Stryjek and Vera (2, 3). The equations for thermodynamic properties using the Peng-Robinson equation of state are given in the appendix for volume, compressibility factor, fugacity coefficient, residual enthalpy, and residual entropy. Critical constants and ideal gas heat capacities for use in the equations are from the data compilations of DIPPR (8) and Yaws (28, 29, 30). 

Further, the fugacity coefficient was calculated as a function of X2, and the value of K2 was obtained from the slope of the curve In 02 against X2 (for additional details see Appendix 2). The calculated values of K2 for the CO2 + naphthalene and CO2 + pyrene systems are plotted in Figure 1. Similar calculations were carried out using the Peng—Robinson (PR) EOS. Good agreement was found between the values of K2 obtained from the two equations of state. 

Compare these experimental results with the predicted fugacity coefficients for hydrogen in these mixtures calculated using the Peng-Robinson equation of state. Determine the sensitivity of the predictions to the value of the binary interaction parameter. 

The difficulty in solving this equation is that the fugacity coefficients, and therefore Ky (computed using, for example, the Peng-Robinson equation of state), depend upon the mole fractions that must be obtained from the equilibrium relation. Consequently, the calculation is a complicated, iterative one. The results obtained using a specially prepared MATHCAD worksheet are shown in. Figures 1 and 2. 

Table 3. Fugacity coefficients for the reagents and product calculated from the Peng-Robinson equation of state.

Therefore, the calculation of the fugacity coefficient by a cubic equation of state does not require much in terms of additional computations, since the required expressions for the residual enthalpy and entropy have been obtained already. In particular, the result for the Soave-Redlich-Kwong and the Peng-Robinson equation are... 

If an equation of state is to be used for the calculation, it is often better to introduce the fugacity coefficient This is particularly important if there is no expression for the Gibbs energy as in the case of empirical cubic equations of state like Soave-Redlich-Kwong and Peng-Robinson. In usual thermodynamics the fugacity coefficient may be calculated from... 

The same equation of state is then employed to get both the numerator and denominator in this expression using standard thermodynamic relationships. The work of Benedict, Webb and Rubin (.6), of Starling (.2) > and the series of Exxon papers (Lin et al., ( ) Lin and Hopke,( ))—all on various forms-of the BWR, Soave, and Peng-Robinson equations of state are examples of the use of one equation of state to perform the whole calculation. It should be added that the original developments in this area treated the fL /x term in the numerator as a separate entity and multiplied the final answer by l/ir for consistency. Most contemporary approaches relate both numerator and denominator to equations of state via the fugacity coefficient route, the only difference in liquid and vapor being the density and the equation constants obtained from the respective mixing rules. 

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