Peng-Robinson Equation of State PR-EOS

For pure substances, the first and the second derivatives of pressure with respect to volume at the critical point are equal to zero. (The condi tion of criticality will be derived in the next chapter.) Note that these derivatives may not be zero for mixtures. Using the criteria of criticalitj 

It is well known that the critical compressibility factor depends on the substance. Table 3.1 gives the critical compressibility factors of n-alkanes and some nonhydrocarbons. This table shows a wide variation in of various substances. All the values listed in Table 3.1 are less than = 0.307 predicted from the PR-EOS. The SRK-EOS gives a critical compressibility factor of 0.333. On the basis of Z predictions, one expects the PR-EOS to predict pure component densities better than the SRK-EOS. Density predictions will be further discussed later. 

At temperatures other than the critical temperature, the parameter a in Eq. (3.6) is given by 

The dimensionless parameter a is a function of Tj. and the acentric factor, CO. Vapor pressure data are used to obtain a in the following manner. 

TABLE 3.1 Critical properties and normal boiling points of n-alkanes and some selected nonhydrocarbons  

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This work presents a temperature-dependent volume translated model for Peng-Robinson equation of state (PR EOS) for calculating liquid densities of pure compounds and mixtures in the saturated region. For pure compounds, the average absolute percent deviation (AAPD) were calculated in the reduced temperature range of (0.3-0.99). Similarly for mixtures, the (AAPD) of different binary, ternary and multicomponent mixtures were determined. The AAPD for 29 pure compounds and different mixtures(binary, ternary and multicomponents) were 1.29 and 1.35 respectively. The accuracy of this model was compared well with three well-known liquid density correlations and other earlier volume translated models. 

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. 

As discussed in the thermodynamics chapter (Chapter 4), an equation of state (EOS) can be used to calculate the fugacities of all components in a mixture. This approach hnds widespread use in the chemical and petroleum rehning industries cubic equations of state are used most often, particularly the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) equations. 

EOS is normally either the Soave-Redlich-Kwong (SRK) or the Peng-Robinson (PR). Both are cubic EOSs and hence derivations of the van der Waals EOS, and like most equations of state, they use three pure component parameters per substance and one BIP per binary pair. There are other more complex EOSs (see Table 8.4). EOS models are appropriate for modeling ideal and real gases (even in the supercritical region), hydrocarbon mixtures, and light-gas mixtures. However, they are less reliable when the sizes of the mixture components are significantly different or when the mixture comprises nonideal liquids, especially polar mixtures. 

Cubic Equations of State (EoS) are progressively becoming the main tool for phase equilibria calculations and, even though they are - so far -successful for nonpolar/weakly polar systems only, it will not be long before they can handle polar systems as well. The Soave-Redlich-Kwong (SRK, Soave, 1972) and the PR (Peng and Robinson, 1976) EoS - modifications of the first EoS proposed, that of van der Waals (vdW) - are the most commonly used among them. 

The pressure of a saturated Peng-Robinson liquid with k= 1.15 is between 0 and 2 atm for all the substances tested. Since Gibbs energy of a liquid is insensitive to pressure at low pressures, vT = 1.15 fc is adopted as the standard state in PR eos for pure liquids at low-pressure vapor-liquid equilibrium. Similarly, the volume of the liquid mixture is set to be v = 1.15 b. Substitution of Equation (4.434) into Equation (4.433) leads to... 

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