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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.