Group Contribution Equation of State

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

The procedure is based on the group contribution equation of state by F. Chen, Aa. Fredenslund, and P. Rasmussen, "A Group-Contribution Flory Equation of State for Vapor-Liquid Equilibria" Ind. Engr. Chem. Res., 29, 875 (1990). 

The procedure is based on the group contribution equation of state by M. S. High and R. P. Danner, "A Group Contribution Equation of State for Polymer Solutions," Fluid Phase Equilibria, 53, 323 (1989) and M. S. High Prediction of Polymer-Solvent Equilibria with a Group Contribution Lattice-Fluid Equation of State, Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1990. Additional and modified group values are from V. S. Parekh Correlation and Prediction of the PVT Behavior of Pure Polymer Liquids, M.S. Thesis, The Pennsylvania State University, University Park, PA, 1991. 

High, M. S. Danner, R. P., "A Group Contribution Equation of State for Polymer Solutions," Fluid Phase Equilibria, 53, 323 (1989). 

Holderbaum, T. Gmehling, J. PSRK A group contribution equation of state based on UNIFAC. Fluid Phase Equilib. 1991, 70, 251-265. 

Wang, L.S. Ahlers, J. Gmehling, J. Development of a universal group contribution equation of state. 4. Prediction of vapor-liquid equilibria of polymer solutions with the volume translated group contribution equation of state. Ind. Eng. Chem. Res. 2003, 42, 6205-6211. 

Gros HP, Bottini S, Brignole EA. A group contribution equation of state for associating mixtures. Fluid Phase Equilibria 1996 116 537-544. 

Bertucco, A. and Mio, C., Prediction of vapor-liqnid eqnUibria for polymers with a group contribution equation of state. Fluid Phase Equilibria, 117, 18-25, 1996. 

Li, j. D., Vanderbeken, L, Ye, S. Y., Carrier, H. Xans, P. 1997. Prediction of the solubility and gas-liquid equilibria for gas-water and light hydrocarbon-water systems at high temperatures and pressures with a group contribution equation of state. Fluid Phase Equilibria, 131(1/2), 107-118. 

Chapter 5 gives a comprehensive overview on the most important models and routes for phase equilibrium calculation, including sophisticated phenomena like the pressure dependence of liquid-liquid equilibria. The abilities and weaknesses of both models and equations of state are thoroughly discussed. A special focus is dedicated to the predictive methods for the calculation of phase equilibria, applying the UNIFAC group contribution method and its derivatives, that is, the Mod. UNIFAC method and the PSRK and VTPR group contribution equations of state. Furthermore, in Chapter 6 the calculation of caloric properties and the way they are treated in process simulation programs are explained. 

Instead of g -models also equations of state can be used for the determination of azeotropic behavior of binary or multicomponent systems. In Figure 5.54 the experimental and predicted azeotropic points using the group contribution equation of state VTPR (see Section 5.9.4) for the system ethane-C02 up to pressures of 80 bar are shown. 

If no experimental data are available gas solubilities can be predicted today with the help of group contribution equations of state, such as Predictive Soave-Redlich-Kwong (PSRK) [43] or VTPR [44]. These models are introduced in Sections 5.9.4 and 5.9.5. 

Figure 5.92 Current parameter matrix of the group contribution equation of state PSRK...

Figure 5.95 Experimental [65] and predicted /C-factors for a 12-component system using PSRK at a temperature of 322 K as a function of pressure and the required parameter matrix for the group contribution equation of state PSRK.

Table 5.20 Main differences between the new group contribution equation of state VTPR and the PSRK model.

In the case of the group contribution equation of state VTPR, instead of temperature-independent group interaction parameters from original UNIFAC, temperature-dependent group interaction parameters as in modified UNIFAC are used. As for modified UNIFAC, the required temperature-dependent group interaction parameters of VTPR are fitted simultaneously to a comprehensive data base. Besides VLE data for systems with sub and supercritical compounds, gas... 

Figure 5.102 Experimental and predicted SLE data for the system ethane (1)-C02 (2) experimental [3, 68] — group contribution equation of state VTPR.

Figure 5.103 Experimental and predicted phase equilibrium data and excess enthalpies for alkanes with ketones predicted using modified UNIFAC respectively the group contribution equation of state VTPR modified UNIFAC, — group contribution equation of state VTPR.

At the same time the group contribution equation of state VTPR in contrast to modified UN I FAC can be applied for the prediction of phase equilibria including compounds not covered by modified UNIFAC, for example, the various gases. The predicted LLE results for the ternary system nitrogen-C02-methane at 122 K and the binary system nitrogen-C02 as a function of temperature are shown in Figure 5.104 together with the experimental data. 

A group contribution equation of state shows in particular great advantages compared to the usual equation of state approach in the case of multicomponent mixtures, when the multicomponent mixture consists of gases and various alkanes, alcohols, alkenes, and so on. The reason is that the same parameters can be used for all alkanes, alcohols, alkenes, so that the size of the parameter matrix is small in comparison to tlie typical equation of state approach. The results of VTPR for a 12 component system consisting of nitrogen-methane-C02-alkanes are shown in Figure 5.105. As can be seen, excellent results are obtained with the six required parameters (66 binary parameters would be required for the classical equation of state approach). 

An overview about the development of group contribution methods and group contribution equations of state for the prediction of phase equilibria and other thermophysical properties can be found in [69]. 

Compare the experimental data for the system ethanol-water measured at 70 ""C (see Figure 5.30 resp. Table 5.2) with the results of the group contribution method modified UNIFAC and the group contribution equation of state VTPR. 

P5.12 Predict the solubility of methane, carbon dioxide, and hydrogen sulfide in methanol at a temperature of —30 °C for partial pressures of 5 bar, 10 bar, and 20 bar using the PSRK and VTPR group contribution equations of state. Compare the results with the solubilities obtained using Henry s law and the Henry constants predicted in problem PS.11. 

In the case of nonideal systems, the real behavior has to be taken into account using activity coefficients obtained from g -models, e.g. group contribution methods like modified UNIFAC. The required activity coefficients can of course also be calculated using an equation of state or group contribution equation of state. 

Calculate the solubility of solid carbon dioxide in propane with the help of the group contribution equations of state PSRK and VTPR assuming simple eutectic behavior. Compare the results with the results assuming ideal behavior and the experimental data that can be downloaded from the textbook page on ivww.ddbst.com. All required parameters can be found in Appendix A. 

If no binary experimental data are available, powerful predictive models (group contribution methods and group contribution equations of state) can be applied today to reliably predict the missing pure component properties (see Chapter 3) and the phase equilibrium behavior (see Chapters 5 and 7). These predicted mixture data can be used, for example, to fit the missing binary parameters for a multicomponent system. 

With the help of the group contribution equation of state VTPR, it should be checked whether the quaternary system carbon dioxide (1)-ethane (2)-hydrogen sulfide (3)-propane (4) shows binary, ternary, or quaternary azeotropes at 266.5 K. 

Stored in the DDB. For three binary systems, azeotropic behavior is calculated. No ternary or quaternary azeotropic point is found using the group contribution equation of state VTPR. A comparison with the experimental data stored in the DDB shows that this is in agreement with the experimental findings. 

The results of Examples 11.2 and 11.3 show that today even predictive models can be applied successfully to find the binary and higher azeotropes of a multicomponent system. With the development of the group contribution equations of state like PSRK and VTPR, the range of applicability was extended to compounds which are not covered by group contributions methods such as UNI FAC or modified UNIFAC... 

Instead of a -model or a group contribution method like modified UN I FAC also an equation of state or a group contribution equation of state can be used for the calculation of residue curves and distillation boundaries. In Figure 11.15, the results are shown for the ternary system carbon dioxide-hydrogen sulfide-ethane at 266.5 K using VTPR. As can be seen, two binary azeotropes and one distillation boundary is observed. 

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