A REPORT ON

A Simplified SAFT Equation of State for Associating

And Non-Associating Compounds and Mixtures

SUBMITTED TO Dr. Jayant Kumar Singh Associate Professor Department of Chemical Engineering Division Indian Institute of Technology, Kanpur Kanpur-208 016

SUBMITTED BY POOJA SAHU & UTSAV KUMAR Department of Chemical Engineering Indian Institute of Technology, Kanpur Kanpur-208 016

INTRODUCTION

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:

Vander wall state:

p RT

V ba

V 2

Virial Equation of state:

Z 1B

VC

V 2

Redlich-Kwong:

p RT

V b

a

T1/ 3V (V b)

Redlich-Kwong Soave:

p RT

V b

a

RT(V b)

Peng- Robinson:

p RT

V b

a

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 AND RELATED APPROACHES

Chapman constituted the first stone in the SAFT history. However, the pure

formalism of the SAFT equation was presented in the papers of Chapman et al.

(1989, 1990) and Huang and Radosz (1990, 1991). SAFT has been especially

successful in some engineering applications for which other classical EoSs failed.

The success of the equation in its different versions is proved by the amount of

published works since its development.

MODIFIED SAFT EQUATION OF STATE:

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,

ares ahs adisp achain aassoc

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.

THE REFERENCE TERM

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:

RELATION BETWEEN DIFFERENT SAFT EOS:

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.

ASSUMPTIONS INVOLVED IN SAFT EOS:

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.

SAFT PARAMETERS:

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:

i j (i j ) /2

ij i j

FOR NON-ASSOCIATING FLUIDS:

FOR ASSOCIATING FLUIDS:

CONTRIBUTION OF DIFFERENT TERMS FOR PURE FLUID:

ares ahs achain adisp aassoc

Z res 1 Z hs Z chain Z disp Z assoc

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

ahs ma0hs

where ahs

is the Helmholtz free energy for a hard sphere in a hard sphere fluid at the same

packing fraction as in the chain fluid.

a0hs

RT4 32

(1)2

s m

(Nav /6)sd3

d [1cexp(3u0 /kT)]

v 00 (Nav /6 ) 3

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

adisp ma0disp

a0disp

RT

u

kT

j

i

i

j

a0disp

RT Zm ln

vs

vs vY

Y expu

2kT

1

u u0[1 (e /kT)]

From fitting e/k=-10

Chain Term

For chain contribution to the Helmholtz free energy

achain

RT (1m)ln

1 (1/2)

(1)3

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.

SIMPLIFIED SAFT EOS (FOR PURE FLUIDS)

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

Z 1 Zhs Zchain Zdisp

P=

ZRT

Where:

Z hs

RT4 22

(1)3

Z chain

RT (1m)ln

(5 /2) 2

(1)(1 (1/2))

Z disp

RT Zm ln

vY

vs vY

B. FOR MIXTURES:

For mixtures, we use the same procedure to develop the equation of state as for a pure

component. The total residual Helmholtz energy in the SSAFT equation of state is again

given by

ares ahs adisp achain aassoc

For hard sphere mixtures, the Helmholtz energy is

a0hs

RT

6

Nav

(2)3 3123 312()

2

3(13)2

0 23

32

ln(13)

k 6

Navx i

i

mi(dii)k

Dispersion term for attractive force between segments, again assuming that attractive

potential is square well potential is-

m x imii

a0disp

RT Zm ln

vs

vs vY

dij dii d j j

2

uij (1 kij ) uiu j

vY

Nav x ix jmim j (dij /kT) 1j

i

x ix jmim j

j

i

Next chain molecules in the system are formed by the bonding of chain sites.

achain

RT x i

i

(1mi)ln(giihs(dii))

giihs(dii)

1

13 3

diid jj

dii d jj

2(13)

2 2

diid jj

dii d jj

2

22

(1 3)3

For segments of the same diameter, this equation becomes

giihs(dii)

1

1 3 3dii

2

2(1 3)

2 2

dii

2

222

(1 3)3

The final simplified form of SSAFT Equation of state for mixture:

Z 1 Zhs Zchain Zdisp

Where

Z hs 6

Nav

0313

12

(13)2

323

(13)3

233

(13)3

Z chain

RT x i

i

(1mi)

giihs(dii)

(giihs(dii))

Z disp

RT mZm ln

vY

vs vY

APPLICATION:

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

Entropy S

Internal energy U

U=A+TS

Enthalpy

H=U+PV

Gibbs energy G

G=H-TS

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:

Ex.Water-Ethylacetate-system

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 follows-

This 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.

RESULTS AND DISCUSSION

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 segment-

segment 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:

For Hexane

m=7.5189

u0/k=37.2425

v00=0.0839

For Pentane

m =6.3018

u00/k =41.9472

v00=0.0556

For Heptane

m =7.5540

u00/k=35.2257

v00=0.1452

For mixture:

k11 = 1.8446

k12 = 2.4261

k22 = 2.8444

1. Propane (m1)

m=3.399 (m1)

u0/k=41.5961(u01)

v00=0.0672(v0011)

2.Benzene (m2)

m=4.3811

u0/k=69.7743

v00=0.0424

\

GRAPHS AND DISCUSSION FOR PURE COMPONENTS:

FOR HEXANE:

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

100 200 300 400 500

Temperature

Pressure vs Temperature

exp

saft

FOR PENTANE:

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01

Pressure vs Density

exp

saft

0

2

4

6

8

10

12

14

16

18

100 150 200 250 300 350 400 450

Pressure vs Temperature

exp

saft

FOR HEPTANE:

0

2

4

6

8

10

12

14

16

18

0 0.1 0.2 0.3 0.4 0.5 0.6

Pressure vs Density

exp

saft

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

100 200 300 400 500

Temperature

Pressure vs Temperature

exp

saft

Observations:

Pressure vs temperature Total pressure of the pure system increases with increase in temperature. Reason-

larger the temperature of a gas the faster the molecules will move (temperature is

proportional to the average kinetic energy of the particles) and the larger the force

they will exerted by molecules, which is responsible for the higher the pressure at

higher temperature.

As the chain length of n-alkanes increases, deviation of SAFT equation of

state from experimental values decreases.

Pressure vs density Total pressure of the pure system increases with increase in temperature. Reason-

As we increase the density of the system at same temperature, lesser volume

(area) is available to molecules for movement, which increases pressure (ie force

per unit area) of the system.

As the chain length of the pure fluid increases, a small change in density causes

higher change in pressure of system.

SAFT data matches well with experimental data for lower values of applicable

range for the system.

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01

Pressure vs Density

exp

saft

GRAPHS AND DISCUSSION FOR MIXTURE: (Benzene and Propane)

For non-associating binary mixtures, SAFT equation of state gives appropriate

results when compared to experimental values.

Overall Observations:

As with any Vander Waals equation of state the SAFT approach is inadequate

close to the critical point: the critical temperature and especially the critical

pressure are overestimated.

It has been observed that although the equation is able to capture the behavior of

these properties in all cases, the agreement with correlated experimental data

deteriorates as the chain length increases.

Good accuracy is obtained for the critical temperature and pressure values. while

the critical density is overestimated.

SSAFT is more accurate as compared to original equation.

Since dispersion term account for only weak attraction forces, thatsy segment

interaction energy for both associating and non-associating compounds is almost

same.

Molecule having high polarity shows higher interaction energy. For ex, Water has

higher interaction energy than any other compound.

Association energy depends on association sites available.

This is reason that acids have approximately twice association energy as

compared to alcohols.

-2

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

pre

ssu

re

mole fraction of propane (y/x)

Pressure vs Mole fraction

exp(p-x)

exp(p-y)

saft(p-x)

saft(p-y)

For alcohol –alkane mixture, both simplified and original SAFT equations

represents phase diagram quite well.

For acids original SAFT equation produces better results because original SAFT

equation of state results in less association for acids than with SSAFT equation.

If the both components of the mixture are associating types then cross association

occurs. Then both SAFT and SSAFT equation of state produce small error in

calculation of phase diagram.

In azeotropic region SSAFT equation of state results better than SAFT equation

of state.

CONCLUSION:

We have presented a version of the SAFT approach for chain molecules formed

from spherical segments with attractive potentials of variable range SAFT-VR.

The theory is based on a general treatment of the dispersion forces using a

compact expression for the mean-attractive energy with first order term within a

high-temperature perturbation expansion.

Standard perturbation theory can be used to describe the properties of the

monomeric segments, including the contact value of the cavity function, which is

used to evaluate the contribution to the free energy due to chain formation and

association. We have presented simple analytical expressions for the Helmholtz

free energy of chain molecules formed from square –well potential,

For pure compounds SSAFT equation of state correlate vapor pressure and liquid

density very well for both associating and non-associating compounds.

For self-associating mixtures, both SAFT and SSAFT equation of state, produce

small error in pressure and vapor phase mole fraction.

SSAFT leads better-correlated results than original SAFT equation. Also it is

easier to use.

For some mixtures like water-acetic acid, neither equation of state correlate better

results.

For high-pressure binary vapor-liquid equilibrium data we can use SSAFT

equation of state.

It has been observed that a linear relationship exists between binary interaction

parameter in SSAFT equation and temperature, which allows this model to be

used for extrapolation and prediction.

PC-SAFT equation of state seems to be more predictive for liquid- and vapor-

phase compositions as well as in the vicinity of the critical point.

Molecular pure-component parameters can be used in conjunction with these

binary parameters to predict the phase behavior of the binary mixtures at different

thermodynamic conditions.