# Prediction of Phase Behavior by Statistical Associating Fluid Theory Models

EoS play an important role in chemical and biochemical engineering process design. They have assumed an important place in the prediction of phase equilibria of fluids and fluid mixtures. Originally, EoS were used mainly for pure components. Later on it was applied to mixtures of nonpolar components and subsequently extended for the prediction of phase equilibria of polar mixtures. The history of EoS goes back to the year 1873 with the development of van der Waals EoS. Based on van der Waals' EoS, researchers have developed various EoS such as Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson, Guggenheim and Carnahan-Starling (Wei & Sadus, 2000) to name a few. Advances in statistical mechanics allowed the development of EoS based on molecular principles that are able to describe thermodynamic properties of real fluids accurately. Using Wertheim's first-order thermodynamic perturbation theory (TPT-1; Wertheim, 1984a, 1984b, 1986a, 1986b, 1986c, 1987), Chapman, Gubbins, Jackson and Radosz (1989, 1990) and Huang and Radosz (1990, 1991) developed the SAFT which is an accurate EoS for pure fluids and mixtures containing associating fluids.

In the SAFT model, molecules are considered to be composed of equal-size, spherical segments. Numbers of segments and segment diameters vary with the molecule size and shape. For a pure component, the fluid is first assumed to consist of equal-sized hard spheres. Next, the segment-segment attractive forces are added to each sphere. Thereafter, the chain sites are added to each sphere and chain molecules are formed by the bonding of chain sites. Finally, specific association sites are added at the same position through attractive interactions and contribution to Helmholtz free energy for each step is evaluated. Therefore, in the SAFT model, the residual Helmholtz energy per molecule for a pure component has a hard sphere (я^{hs}), dispersion (я^{dlsp}), chain (я^{chain}) and an association (я ) contribution and is given by

*
*

where я^{res} is the residual Helmholtz free energy of the system (я^{res} = я^{total} - я^{ldeal}) and я=Л/Nk T.

A variant, namely the PC-SAFT EoS was developed in 2001 by Gross and Sadowski (2001, 2002). Using hard-chain reference fluid and applying a perturbation theory for chain molecules, Gross and Sadowski (2001, 2002) derived a dispersion expression for chain molecules. PC-SAFT uses the hard-chain fluid as the reference system, whereas in SAFT, a hard- sphere fluid is considered as a reference system. In this EoS, molecules are assumed to be chains composed of hard spheres, which repel each other. The attractive forces among molecules are accounted by adding perturbation terms to the reference system which includes dispersive forces and specific associative interactions (Figure 2.17). The residual Helmholtz energy can now be calculated as the Helmholtz energy of the reference

FIGURE 2.17

Procedure to form a molecule in the PC-SAFT model.

hc

system *(a* ) superposed with the Helmholtz energy of perturbation, that is, dispersion *(a*^{dlsp}) and association *(a*^{assoc}):

*
*

*Hard-Chain Contribution*

The Helmholtz energy of the hard-chain reference term is given as

*
*

where *x _{{}* is the mole fraction of chains of component i, is the number

of segments in a chain and the mean segment number in the mixture is defined as

*
*

The Helmholtz energy for the hard-sphere segments is given on a per-segment basis as

*
*

where *N _{s}* is related to the number of hard spheres.

The radial pair distribution function for the hard-sphere fluid is given by

*
*

and is defined as

*
*

The temperature-dependent segment diameter is obtained as *
* where:

*a,* is the temperature-independent segment diameter *ejk* is the depth of the pair-potential

*Dispersion Contribution*

The dispersion contribution to the Helmholtz energy is given by *
* with

*
*

Power series *I _{1}* and

*I*depend only on density and segment number according to the following equations:

_{2}*
*

where the coefficients *a(m)* and b,(m) are functions of the segment number: *
* *Association Contribution*

The association contribution to the Helmholtz energy is given as *
*

where *X ^{Al}* is the fraction of the free molecules

*i*that are not bonded at the association site A:

*
*

with

*
*

In terms of the compressibility factor *Z*, the EoS is given as the sum of the ideal gas contribution (Z^{id} = 1), the hard-chain contribution (Z^{hc}), the dispersion (attractive) contribution (*Z*^{disp}) and the contribution due to associating interactions (Z^{assoc}). Thus,

*
*

The expressions for all the terms are given in Appendix A. Pressure can be calculated by applying the following relation:

*
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