Statistical Associating Fluid Theory SAFT Equation Of State EOS

Here is a review of Equations of States (EOS) that was discussed in class:

And in equation form:

Statistical Associating Fluid Theory (SAFT) EOS development
These four empirical equations are mentioned because they show the major contributions to EOS development and more specifically, they focus on attractive term advancements.

Developments to the repulsive term by Carnahan and Starling (CS)  coupled with attractive term improvement (R-K) are shown as the following:

(1)
\begin{align} \frac{P\nu} {RT} =\frac{1+y+y^2-y^3} {(1-y)^3} - \frac{a} {RT(\nu+b)} \end{align}

Chain molecules

Perturbed Hard Chain Theory (PHCT) 
PHCT is derived from a molecular vibrational and rotational partition function using free-volume concepts:

(2)
\begin{align} Z = Z(hard chain) - \frac{a} {RTV} \end{align}

Perturbed Anisotropic Chain Theory (PACT) [1985-86]
PACT includes anisotropic multipolar forces in the PHCT equation and accounts for size, shape, and intermolecular effects:

(3)
\begin{equation} Z = 1+Z^{rep}+Z^{iso}+Z^{ani} \end{equation}

Hard-Sphere Chain Concept

• The backbone of many systematic improvements to EOS
• Thermodynamic perturbation theory (TPT), proposed by Wertheim , which accommodates hard-chain molecules
• Chapman generalized it, Compressibility factor of hard-chain of m segments,
(4)
\begin{align} Z^{hc} = mZ^{hs} - (m-1) (1+\eta \frac{\partial ln g^{hs} \sigma} {\partial\eta}) \end{align}

Associating Fluids
The associated perturbed anisotropic chain theory (APACT)  is derived from the infinite equilibrium model and monomer-dimer model into the PACT and accounts for isotropic repulsive and attractive interactions, anisotropic molecular interactions, and is capable of predicting thermodynamic properties of pure associating components as well as mixtures,

(5)
\begin{equation} Z = 1+Z^{rep}+Z^{att}+Z^{assoc} \end{equation}

The SAFT EOS is developed from Wertheim's theory of Helmholtz energy expansion and is expressed as residual Helmholtz energy and it describes hard-sphere repulsive forces, chain formation (for nonspherical molecules) and association,

(6)
\begin{equation} a^{res} = a^{seg}+a^{chain}+Z^{assoc} \end{equation}

The following diagram shows the interrelationship of EOS 