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«SUBMITTED TO Dr. Jayant Kumar Singh Associate Professor Department of Chemical Engineering Division Indian Institute of Technology, Kanpur Kanpur-208 ...»

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A Simplified SAFT Equation of State for Associating

And Non-Associating Compounds and Mixtures


Dr. Jayant Kumar Singh

Associate Professor

Department of Chemical Engineering Division

Indian Institute of Technology, Kanpur

Kanpur-208 016




Department of Chemical Engineering Indian Institute of Technology, Kanpur Kanpur-208 016


The development of accurate equations of state firmly based in statistical mechanics is one of the main goals in applied physical science since it allows for an accurate description of the

thermodynamic properties of real substances. Till now we have many equation of states like:

Ideal gas state: PV=nRT

Empirical Cubic equation of states:

RT a p 2

Vander wall state:

V b V BC Z  1 

Virial Equation of state:

V V2  RT a p  1/ 3


V  b T V (V  b)  RT a p 

Redlich-Kwong Soave:

V  b RT(V  b)  RT a p 

Peng- Robinson:

V  b RT(V  b)  b(V  b)  Statistical Associating Fluid Theory (SAFT) is a powerful equation of state model for thermodynamic property and phase equilibria calculations for fluid mixtures.

 Strong attractive forces, such as hydrogen bonding, affect the physical properties of associating compounds. For example, the boiling and critical points of such compounds are higher than those of similar size non- associating compounds. Also associating compounds may form highly non-ideal mixtures.

On the basis of thermodynamic perturbation theory, Wertheim developed a theory of associating fluids. In this theory, the molecules are treated as different species according to the number of bonded associated sites. The key result of Wertheim’s cluster expansion is written as a first-order perturbation theory (TPT1) that establishes a direct relation between the change in the residual Helmholtz energy due to association and the monomer density. This monomer is, in turn, related to a function characterizing the “association strength”.

Although Wertheim’s theory considers that the potential has a short-range highly directional component that is the cause of the formation of associated species, it does not specify any particular intermolecular potential for the reference fluid. It is necessary to select one in order to implement the theory.

In a first stage, the known hard-sphere model was used in order to study the influence of the molecular association on the phase coexistence properties of hard-sphere molecules with one or two bonding sites Wertheim and Chapman deduced that in the limit of infinite association (in an infinitesimal small volume), the system becomes a polymer. The hard-sphere model has accurate analytical expressions for its quation of state and pair distribution.

In Original SAFT EOS for chains of Lennard-Jones segments, a perturbation expansion is used to describe the monomer contribution and the hard-sphere radial distribution function is used to describe the chain contribution.


–  –  –


The SAFT EOS, 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 non-spherical molecules) and association, a res  a hs  a disp  a chain  a assoc A modified SAFT equation of state is developed by applying the perturbation theory of Barker and Henderson to a hard-chain reference fluid. With conventional one-fluid mixing  rules, the equation of state is applicable to mixtures of small spherical molecules such as gases, nonspherical solvents, and chainlike polymers. Depending upon the model being used we name SAFT Equation of state are as follows: LJ -SAFT, HS -SAFT, SW-SAFT etc.


Aref considers the residual Helmholtz free energy of nonassociated spherical segments, and it is not specified within SAFT. It can refer to atoms, functional groups or even a full molecule (methane, argon). Most SAFT equations differ in the reference term, keeping formally identical the chain and the association term, both obtained from Wertheim’s theory.

The original SAFT of Chapman and Huang and Radosz use a perturbation expansion using a hard-sphere fluid as a reference term and a dispersion term as a perturbation.

The square-well potential (SW), The SAFT equation with an intermolecular potential of variable range is known as SAFT-VR. More recently, SAFT-VR has been slightly modified extending the potential range to higher values.

The Lennard-Jones(LJ) potential, which accounts for both the repulsive and attractive interactions of the monomers in the same term. This potential has been used to develop different SAFT versions like the LJ-SAFT The Yukawa potential has also been used with variable range in SAFT-VR.

Summary of these potential is given as follows:


SAFT-HS is seen to work best in systems with strong association, where the dispersion forces can be adequately represented as a weak mean-field background interaction.

The SAFT expressions are continually being improved. An accurate representation of the monomer-monomer distribution function has been included to deal with chains of Lennard-Jones(LJ) segments, and the approach has been extended to different types of monomer segments such as square wells SW.

The effect of many-body interactions in the chain has also been included in dimer versions of the theory for chains formed from both LJ and SW chains.


The main approximations are:

1.Only three-like structures are permitted in theory, neglecting more complex structures like the ring bonding.

2.Only one single bond is allowed at each associating site.

It implies that:

 Two bonded associating sites (each one from a different molecule) prevent a third core of another molecule to bond to any of the occupied sites.

 Two associating sites of the same molecule cannot bond at the same time to another site of a different molecule.

 Double bonding between two molecules is not allowed.

3. The activity in each site is not affected by the activity in other sites of the same molecule. It means that the possible repulsion interactions of two molecules trying to join at two sites of a third molecule are neglected.

4. The first order approximation does not make any difference among the actual positions of the sites. As a consequence, the angles among the bonding sites are not specified and the properties are evaluated independently of the angle between the sites.


Pure-component parameters for molecules (non-polar, non-associating, uncharged):

• Segment diameter σ

• Segment number m

• Dispersion energy ε Mixtures: One-fluid theory m=∑ximi

mean segment number Berthelot-Lorenz combining rules between components i and j:

–  –  –

  Figure 1. Procedure to form a molecule in the SAFT model. (a) The proposed molecule.(b)Initially the fluid is a hard sphere fluid. (c) Attractive forces are added. (d) Chain sites are added and chain molecules appear. (e) Association sites are added and molecules form association complexes through association sites.

Hard sphere term Each compound is assumed to be a chain with m segments. The hard sphere Helmholtz energy is given by

–  –  –

  where s= molar density of hard spheres d = effective hard sphere diameter of a segment.

 = The molar density of hard spheres c = 0.333  uo= is the temperature independent interaction energy between segments.

V00 = temperature independent segment molar volume in a closed-packed arrangement  = 0.740 48.

 Dispersion Term  For dispersion contribution to the Helmholtz free energy a disp  ma0 disp

–  –  –

Association Term  The association Helmholtz energy due to hydrogen bonding was also estimated from Wertheim’s association theory. Here we are intended to non-associating compounds only. So we can neglect contribution of association term.


The simplified SAFT equation of state for pure fluids obtained from the volume derivative of the Helmholtz free energy is a sum of the compressibility factors from each of the contributions above and is given by

–  –  –

 Dispersion term for attractive force between segments, again assuming that attractive potential is square well potential is

–  –  –



A. SAFT Helmholtz Energy A may be used for calculation of various thermodynamics properties, such as Pressure p and compressibility factor Z  Density ρ by iteration  Chemical potential μ  Fugacity coefficient φ

–  –  –

B. Complete thermodynamic description of a system C. SAFT equation can be used to determine vapor-liquid, liquid-liquid and solid-liquid equilibria.

Vapor-liquid Equilibria:

Ex. Heptane-Ethanol system

Liquid-Liquid Equilibria:


Solid-Liquid Equilibria:

Ex. Amino Acids in water

ADVANTAGE OF SAFT EOS (Equation of state):

Advantages of SAFT compared to other EOS and activity-coefficient models are as followsThis is physically based model, which accounts for size and shape of molecules suitable also for complex and large molecules This equation of state account for the density (pressure) dependence also.

It is reliable for extrapolation to other conditions (T, p, concentration) to multi-component systems (binary, ternary,...) All thermodynamic properties can be derived from Helmholtz energy function SAFT and PC-SAFT EoS are used to predict the volumetric properties of fluid, which can be used for its transportation. This equation of state is much useful in Carbon capture and sequestration (CCS) technology. EoS predictions are in good agreement with experimental data, with the exception of the critical region, where higher deviations are observed.

It is also used for study of non-electrolyte solutions.


Pure component parameters were to experimental vapor pressure and saturated liquid density data taken from NIST The SSFT EOS developed for chains formed from square-well segments, are used in this demonstration although the Suther- land or Yukawa potentials could also have been used. The parameters m, u0/k, and v00 of the square-well chain model are optimized by fitting the calculated vapor-pressure curve and saturated liquid densities to the experimental data.

The parameters show the rough tendency to increase with increasing number of carbon atoms C, but whilst the range appears to continue increasing, the size and energy of the segmentsegment interaction appear to tend to a limiting value for the longer chains. As expected, the diameters and the well-depth energy of the segments are larger for the heavier n-alkanes.

It is important to note that the segments of our chain molecules are united atom models so that the number of segments in the chain does not represent the number of carbon atoms. Instead, the parameter m provides an indication of the non-sphericity aspect ratio of the non-sphericity of the molecule.

Parameter values (obtained by fitting) for our system are as follows:

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