1. J. of Supercritical Fluids 101 (2015) 87–94
Contents lists available at ScienceDirect
The Journal of Supercritical Fluids
journal homepage: www.elsevier.com/locate/supflu
Experimental determination and theoretical correlation for the
solubilities of dicarboxylic acid esters in supercritical carbon dioxide
Ram C. Narayan, Jacob Varun Dev, Giridhar Madras∗
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
a r t i c l e i n f o
Article history:
Received 1 November 2014
Received in revised form 31 January 2015
Accepted 17 March 2015
Available online 24 March 2015
Keywords:
Bis (2-ethylhexyl) sebacate
Dimethyl sebacate
Retrograde behavior
Solution model
Wilson activity coefficient model
a b s t r a c t
The synthesis of high molecular weight esters such as bis (2-ethylhexyl) sebacate is of significance for
its use as a lubricant. This ester is synthesized by the transesterification of dimethyl sebacate with
2-ethylhexanol. Therefore, the solubilities of bis (2-ethylhexyl) sebacate and dimethyl sebacate were
determined at 308–328 K at pressures of 10–18 MPa in supercritical carbon dioxide. The solubility of
dimethyl sebacate was always higher than bis (2-ethylhexyl) sebacate at a given temperature and
pressure. The Mendez-Teja model was used to verify the self-consistency of data. Further, a new semi-
empirical model with three parameters was developed using the solution theory coupled with Wilson
activity coefficient. This model was used to correlate the experimental data of this work and solubilities
of many high molecular weight esters reported in the literature.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Many synthetic esters possess the consistency of conventional
hydrocarbon oils. Therefore, they have been widely utilized as base
oils directly or as additives to lubricants. The chemical structures
of these esters resemble triglycerides that constitute the major
portion of vegetable oils and animal fats [1]. These esters are also
good substitutes for petroleum based lubricants as they are more
eco-friendly, highly biodegradable, have tunable viscosities and
medium range viscosity indices. The ester chemistry offers a wide
scope for producing nearly 90% of all the currently available lubri-
cants as rapidly biodegradable lubricants without compromising
the technical performance [2]. Further, these lubricants have very
low volatility, high flash and fire points. Due to the polar nature of
the ester group, they adsorb well on the metal surfaces and also
have good compatibility with various lubricant additives unlike
hydrocarbon based lubricating oils. In addition, these synthetic
esters have good thermo-oxidative stability as compared to veg-
etable oils [3].
Of particular interest is the high performance functional fluid,
bis (2-ethylhexyl) sebacate, a dicarboxylic acid ester of sebacic acid
with 2-ethyl hexanol. This synthetic diester has been in use as
lubricating oil in aircraft engines since the 1940s. In a very recent
work, the tribological and oxidation properties were investigated
∗ Corresponding author. Tel.: +91 80 22932321; fax: +91 80 23600683.
E-mail address: giridhar@chemeng.iisc.ernet.in (G. Madras).
for this compound. In addition to good low temperature behavior,
it has high viscosity index and is non-volatile [4,5]. Some recent
patents on lubricants have used bis (2-ethylhexyl) sebacate as base
oil in the formulations. It has also been used in the formulation
of various lubricants for different applications [6–8]. The blends
of this dicarboxylic ester with various vegetable oils have also
been studied in the context of biodegradable lubricants [9,10]. Bis
(2-ethylhexyl) sebacate is also used as a plasticizer [11–13]. It is
synthesized by simple transesterification of dimethyl sebacate with
2-ethylhexanol in the presence of a catalyst [1,14]. Dimethyl seba-
cate is synthesized from sebacic acid, obtained by alkali pyrolysis of
castor oil, a biodegradable lipid basestock [15]. Dimethyl sebacate
can also be used in the synthesis of several other sebacates, many
polymer syntheses and as a plasticizer apart from being investi-
gated as a lubricity additive [16–19]. One of the prime barriers to
wide applicability of these esters is the high cost of production [20].
To overcome this shortcoming, several catalytic and enzymatic sys-
tems have been investigated [2].
This reaction could potentially be carried out using supercriti-
cal carbon dioxide as the reaction media using a suitable catalyst,
preferably an enzyme. Supercritical carbon dioxide is a popular
non-aqueous green reaction media with well recognized thermo-
physical properties. The pressure or temperature can be tuned to
manipulate the solvent capacity of a critical fluid. The low mass
transfer resistance ensures that high reaction rates are attained and
solvent removal is complete and straightforward [21,22]. The near
ambient critical temperature of carbon dioxide ensures processing
of higher boiling point compounds and heat labile substances
http://dx.doi.org/10.1016/j.supflu.2015.03.008
0896-8446/© 2015 Elsevier B.V. All rights reserved.

2. 88 R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94
[23,24]. Further, supercritical carbon dioxide is miscible with co-
solvents like methanol/ethanol and, therefore, used extensively in
lipid processing [25–29]. Solubility in supercritical fluids has been
thoroughly investigated for mono carboxylic esters, higher alco-
hols and carboxylic acids [30–36]. Dicarboxylic acid esters have
not been studied so far in the context of supercritical fluids (except
for one study at elevated pressures, but in sub-critical or slightly
supercritical state [37]).
In the context of this work, it is essential to determine
the solubilities of the reactant (dimethyl sebacate) and product
(bis (2-ethylhexyl) sebacate) in supercritical carbon dioxide in
order to design reaction conditions and consequent downstream
processing. The phase equilibria of carbon dioxide-2-ethyl hexanol
has been reported [38], but the solubilities of these two dicarboxylic
acid esters have not been reported in literature.
The modeling of the solubilities of solids in supercritical car-
bon dioxide is well studied. Fundamental correlations such as
Chrastil [39], Mendez-Teja [40] and polynomial correlations such
as Gordillo [41], Yu [42] and Jouyban [43] are available. Association
based model [44] coupled with Wilson activity coefficient model
[45] have been developed for solid solute in supercritical carbon
dioxide system. In addition to these, models for mixtures of solid
solutes in supercritical carbon dioxide have also been developed
[46–48]. However, models that correlate liquid phase solubilities in
supercritical fluid–liquid equilibria are scarce [49]. Further, models
based on solution theory have not been reported for liquid solubili-
ties in supercritical carbon dioxide. Thus, a new model is developed
in this work for correlating liquid solubilities in supercritical carbon
dioxide.
The objectives of this work were threefold: (a) determine the
solubilities of these dicarboxylic esters in supercritical carbon
dioxide at different temperatures and pressures (b) verify self-
consistency of the solubility data of these dicarboxylic acid esters
in supercritical carbon dioxide by Mendez Teja model and (c) cor-
relate the solubilites by a new semi-empirical expression based on
solution theory coupled with Wilson Activity coefficient model.
2. Materials
Carbon dioxide gas, with mass fraction of 0.99 was purchased
from Noble Gases (Bangalore, India). The purity was increased to
0.999 (by mass fraction) by passing the gas through a silica bed
column. Dimethyl sebacate (CAS no. 106-79-6) with purity >97%
(GC) and bis (2-ethylhexyl) sebacate (CAS no. 122-62-3) with purity
>96% (GC) were purchased from TCI Chemicals, Japan. The chemical
structures of these compounds are shown in Fig. 1.
3. Experimental
The solubilities of the two dicarboxylic esters in supercritical
carbon dioxide were determined by the flow saturation technique.
The schematic of the apparatus used is shown in Fig. 2. The car-
bon dioxide (at ∼6 MPa) in the gas cylinder (1) is connected to a
pump (JASCO PU-1580-CO2) (3). Carbon dioxide from the cylinder
passes through a silica gel bed (2) to absorb traces of water prior
to entering the CO2 pump. The CO2 pump is equipped with a cool-
ing system that condenses carbon dioxide before being pumped to
the required pressure (within ±0.2 MPa). The system is operated in
two modes: offline and online mode. The flow path during offline
mode is indicated by a broken line, as shown in the figure. Carbon
dioxide from the CO2 pump passes through the accumulator (5)
(via the pump valve), encased within a thermostat (13) maintain-
ing the desired temperature (within ±0.1 K). Carbon dioxide attains
supercritical conditions in the accumulator, after which it flows
through the injector port (6) (not used in solubility experiments)
OO
CH3
O O
CH3
(a)
(b)
OO
O O
CH3
CH3
CH3
CH3
Fig. 1. Chemical structures of (a) dimethyl sebacate and (b) bis (2-ethylhexyl) seba-
cate.
and switching valve (7). The pressure is maintained by the back
pressure regulator (JASCO-PU-1580-BP) (8), by depressurizing at
regular intervals. The system is then allowed to stabilize at the
desired pressure, temperature and flow rate. The offline mode is
necessary to minimize the fluctuations in these parameters, before
the liquid solute of interest is saturated with carbon dioxide. In
the online mode, by using the switching valve (7), supercritical
carbon dioxide is made to pass through the liquid column (EV-
3-50-2 from JASCO) (10), preloaded with 30 mL of desired ester.
A period of 2 h is required for the carbon dioxide to equilibrate
with the liquid solute in the vessel. The saturated fluid stream
from the vapor phase passes through a static mixer (11). It then
flows through an entrainer column (12) that traps the entrained
liquid solute. The accumulator, liquid column, static mixer and
entrainer column are all housed in the thermostat. Completely sat-
urated carbon dioxide then passes through the switching valve and
then to back pressure regulator as shown in the schematic. The
binary mixture, carbon dioxide + dicarboxylic ester then depressur-
izes through the back pressure regulator. The gas (carbon dioxide)
escapes through the gas vent (14) leaving behind the solubilized
ester in the glass collection trap (9). A heating coil is wound around
all the tubing outside the thermostat, carrying supercritical car-
bon dioxide that maintains the desired temperature. The amount
of solute collected was determined gravimetrically. In order to
remove the traces of dicarboxylic esters present, n-hexane was
pumped through the flow lines after every experiment. To deter-
mine the flow rate beyond which solubility would not vary with
flow rate, experiments were performed at different flow rates
(0.05–0.3 mL min−1) keeping temperature and pressure constant.
No changes were observed when the flow rate was less than
0.2 mL min−1. All experiments were conducted at 0.1 mL min−1 to
ensure complete saturation. The solubility (y2) at every tempera-
ture and pressure is expressed in terms of mole fraction calculated
using the expression,
y2 = 1 +
Q × t ×
w2
−1
(1)
In Eq. (1), w2 represents the moles of liquid solute collected. Q
is the volumetric flow rate of carbon dioxide through the liquid
column, is the molar density of carbon dioxide and t is the sam-
ple collection time. Every experiment at a given temperature and
pressure was repeated in triplicate, and the standard deviation was
found to be lesser than 5%.

3. R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94 89
Fig. 2. Schematic representation of the flow saturation apparatus. (1) CO2 cylinder; (2) silica-gel bed; (3) CO2 pump; (4) valve; (5) accumulator; (6) injector valve; (7) six-port
switching valve; (8) back pressure regulator; (9) collection trap; (10) liquid column vessel; (11) static mixer; (12) entrainer column; (13) thermostat; (14) gas vent.
4. Models and correlations
4.1. Mendez-Teja Model
The Mendez-Teja (MT) model [40] is a popular three parameter
model that can be used in checking the consistency across various
isotherms for both liquid and solid solutes in supercritical carbon
dioxide. The model is given by
T ln(y2p) = A + B + C T (2)
Based on Eq. (2), a plot of T ln(y2p) − C T with mass density is
linear, with all isotherms collapsing onto a single straight line.
4.2. Chrastil model
The Chrastil model, which hypothesizes the formation of solvato
complex at equilibrium [39], was one of the earliest correlations for
solubility in supercritical fluids. The expression is
ln(c) = Ä ln( ) +
a
T
+ b (3)
In Eq. (3), c (g/L) is the solubility of the liquid or solid solute in
supercritical carbon dioxide at mass density and temperature T
and Ä denoting the association number. Thus, a plot of ln(c) with
ln( ) is linear at a constant temperature.
4.3. New model
When the supercritical fluid is equilibrated with a liquid solute,
both the phases dissolve in each other to an extent, as determined
by the temperature and pressure of the system. At equilibrium, the
fugacity of solute in both the phases can be equated, as given in Eq.
(4),
ˆf
lp
2
= ˆf CO2
2 (4)
ˆf
lp
2
and ˆf CO2
2
represent the fugacity of the solute in the liquid
phase and the supercritical phase, carbon dioxide, respectively.
Expressing the fugacity of solute in supercritical fluid phase and
fugacity of solute in liquid phase in terms of their respective activity
coefficients,
ˆf CO2
2 = y2
CO2
2 f l
2 = ˆf
lp
2
= x2
l
2f
lp
2
(5)
In Eq. (5), y2, CO2
2
, f l
2
are the vapor/supercritical phase mole
fractions of solute, activity coefficient of solute in carbon dioxide
and fugacity of pure sub cooled liquid, respectively. x2, l
2
, f
lp
2
are the
liquid phase mole fractions of solute, activity coefficient of solute in
the liquid phase and fugacity of the pure liquid solute, respectively.
It should be noted here that l
2
is not close to one, due to the non-
ideality in the liquid phase as the mole fraction of carbon dioxide
is significant (x2 ≈ 0.8 − 0.9).
The ratio of pure fugacities of a component between a solid and
liquid phase is given by [50]
ln
f
sp
2
f l
2
=
Hf
2
R
1
T2,m
−
1
T
(6)
Adapting Eq. (6) for supercritical fluid–liquid equilibria, we
relate the logarithm of fugacity ratio as given by Eq. (7)
ln
ˆf
lp
2
f l
2
= A −
B
T
(7)
Using Eq. (5) in Eq. (7), the following expression can be written
ln
CO2
2
y2
l
2
x2
= A −
B
T
(8)

4. 90 R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94
The activity coefficient of the solute in Eq. (8) can be determined
using a suitable local composition model such as Wilson activity
coefficient model [51],
ln( i) = 1 − ln
⎛
⎝
j
xj ij
⎞
⎠ −
k
xk ki
j
xj kj
(9)
If the liquid solute forms a very dilute solution in supercritical
carbon dioxide phase, the activity coefficient of the liquid solute can
be approximated to the infinite dilution activity coefficient. For a
binary system, Eq. (9) simplifies to
ln( ∞
2 ) = 1 − ln( 21) − 12 (10)
In Eq. (10), 12 and 21 are the binary interaction parameters,
given by 12 = (V2/V1) exp(− a12/RT) and 21 = (V1/V2)exp(− a21/RT),
where V1 and V2 are pure component molar volumes of the super-
critical fluid and liquid solute, respectively. The expression for
infinite dilution activity coefficient in terms of density, = 1/V1 and
temperature, T
ln( ∞
2 ) = 1 + ln( V2) +
a21
RT
− V2 exp
−a12
RT
(11)
Substituting the above expression for infinite dilution coeffi-
cient in Eq. (8),
ln(y2) = A −
B
T
− 1 − ln( V2) −
a21
RT
+ V2 exp
−a12
RT
+ ln( l
2x2)
(12)
In case of solid solubilities, exp(− a12/RT) is normally expanded
in series. However, as −a12/RT is not close to zero for liquids, the
exponential cannot be expanded as a series. Further, for smaller
temperature ranges exp(− a12/RT) can be assumed to be a constant.
Thus, Eq. (12) can be written for correlating liquid solute solubility
in supercritical carbon dioxide in terms of and T as
ln( y2) = E1 + E2 +
E3
T
(13)
In Eq. (13), = V2, E1 = A − 1 + ln( l
2
x2), and E2 = exp(− a12/RT)
and E3 = − B − (a21/R).
5. Results and discussion
The solubilities of bis (2-ethylhexyl) sebacate and dimethyl
sebacate were determined experimentally at 308, 313, 318, 323
and 328 K in the pressure range of 10–18 MPa, as given in Table 1.
The solubilities of the two dicarboxylic esters are plotted against
pressures at different temperatures, as shown in Fig. 3.
Fig. 3 shows that the solubility of bis (2-ethylhexyl) sebacate was
lower, almost by an order of magnitude, as compared to the solubil-
ity of dimethyl sebacate at a particular temperature and pressure.
This can be attributed to the higher molecular weight and lower
volatility of bis (2-ethylhexyl) sebacate. This is observed in other
homologous mono esters such as methyl and ethyl stearate [36,52].
In the perspective of supercritical fluid reactions, during the syn-
thesis of the lubricant bis (2-ethylhexyl) sebacate from dimethyl
sebacate and 2-ethylhexanol using supercritical carbon dioxide as
reaction medium, the product has much lower solubility than both
the reactants at a particular temperature and pressure. Across any
isotherm, the solubility increases with pressure. The retrograde
behavior, a phenomenon marked by the decrease of solubility with
increase in temperature was observed, in the entire pressure and
temperature range was observed for both the compounds.
In order to test the self-consistency of experimental data, the
solubilities were regressed with the Mendez-Teja model [40]. Based
on Eq. (2), T ln(y2p) − C T was plotted against , as shown in Fig. 4.
180160140120100
0
1
2
3
4
5
6
y2
x103
p(MPa)
(a)
180160140120100
4
6
8
10
12
14
y2
x103
p(MPa)
(b)
Fig. 3. Solubilities of (a) bis (2-ethylhexyl) sebacate and (b) dimethyl sebacate at
, 308 K; ᭹, 313 K; , 318 K; , 323 K; , 328 K in supercritical carbon dioxide. The
lines represent the model given by Eq. (13).
900800700600500400300
-4500
-4200
-3900
-3600
-3300
-3000
-2700
Tln(y2
p)-C'T
ρρ (kg.m-3
)
(a)
850800750700650600550500
-3400
-3300
-3200
-3100
-3000
-2900
-2800
-2700
-2600
Tln(y2
p)-C'T
ρρ (kg.m-3
)
(b)
Fig. 4. Solubilities of (a) bis (2-ethylhexyl) sebacate and (b) dimethyl sebacate at
, 308 K; ᭹, 313 K; , 318 K; , 323 K; , 328 K in supercritical carbon dioxide. The
lines represent the correlation based on the MT model (Eq. (2)).

5. R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94 91
Table 1
Experimental mole fraction solubilities (y2) of liquid solutes, bis (2-ethylhexyl) sebacate and dimethyl sebacate (2, vapor phase mole fraction of liquid solute) in supercritical
carbon dioxide (1) at temperature, T and pressure, p.a
T (K) p (MPa) Bis(2-ethylhexyl) sebacate
y2 × 103
(mol mol−1
)
Dimethyl sebacate
y2 × 103
(mol mol−1
)
308 10 1.1 10.1
12 1.9 10.7
14 3.2 11.6
18 5.5 11.8
313 10 0.5 8.0
12 1.1 10.4
13 1.3 10.5
14 2.4 11.5
16 2.6 11.8
18 4.4 11.8
318 10 0.3 6.9
12 0.6 9.1
13 1.2 10.3
14 1.4 10.6
16 2.1 11.6
18 3.4 11.8
323 10 0.1 2.0
12 0.4 8.3
13 0.8 10.2
14 1.1 10.4
16 1.6 11.1
18 2.3 11.2
328 10 0.1 1.7
12 0.4 4.5
13 0.5 6.5
14 0.6 10.2
16 1.1 10.5
18 1.6 11.0
a
Standard uncertainties u are u(T) = 0.1 K, u(p) = 0.2 MPa, and ur(y2) = 0.05.
It can be clearly observed that the experimental data points fall
onto a single straight line, thus determining the consistency of
the experimental data. The density of carbon dioxide, used in the
above correlation and subsequent correlations, was estimated at
the operating temperature and pressure by using the 27 parameter
equation of state [53]. For correlating the equilibria data, multi-
ple linear regression was used. The parameters were determined,
by minimizing the objective function, namely the average absolute
relative deviation (AARD %), defined by
AARD(%) =
1
N
N
i=1
|ycalc
2
− y
exp
2
|
y
exp
2
× 100 (14)
In Eq. (14), N represents the number of data points and y2
represents the solubility of solute in terms of mole fraction. The
superscripts calc and exp represent the calculated and experimen-
tal values, respectively. All the multiple linear regressions were
done using the software Polymath 5.1®.
The experimental data for the two dicarboxylic esters investi-
gated in this work were also modeled by Chrastil equation (Eq.
(3)), represented in Fig. 5, which shows straight lines for differ-
ent isotherms for the variation of logarithm of mass solubility (c)
with the logarithm of mass density ( ). The isotherms for bis (2-
ethylhexyl) sebacate are quite close. The low vapor pressures for
this compound makes the solubility almost density dependent, in
the range of temperatures and pressures investigated.
Having correlated the solubilities by MT (Eq. (2)) and Chrastil
model (Eq. (3)), the experimental data was also correlated using
Eq. (13). Eq. (13) is rearranged in the following manner to yield Eq.
(15)
ln( y2) −
E3
T
= E1 + E2 (15)
7.06.86.66.46.26.05.8
-2
-1
0
1
2
3
4
5
ln(c)
ln(ρρ)
(a)
6.86.76.66.56.46.36.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
ln(c)
ln(ρρ)
(b)
Fig. 5. Solubilities of (a) bis (2-ethylhexyl) sebacate and (b) dimethyl sebacate at
, 308 K; ᭹, 313 K; , 318 K; , 323 K; , 328 K in supercritical carbon dioxide. The
lines represent the model represented by the Chrastil model (Eq. (3)).

6. 92 R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94
9876543
-4
-3
-2
-1
0
1
ln(ρρV2y2
)-E3
/T
ρρV2
(a)
5.04.54.03.53.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
ln(ρρV2y2
)-E3
/T
ρρV2
(b)
Fig. 6. Solubilities of (a) bis (2-ethylhexyl) sebacate and (b) dimethyl sebacate at
, 308 K; ᭹, 313 K; , 318 K; , 323 K; , 328 K in supercritical carbon dioxide. The
solid lines represent the correlation given by Eq. (15).
Thus, ln( y2) − (E3/T) varies linearly with and different
isotherms fall onto a single straight line, with slope E2 and inter-
cept E1. Fig. 6 shows this plot for both the dicarboxylic esters
investigated in this work. The line representing solubilities of bis
(2-ethylhexyl) sebacate has a higher slope than the line represent-
ing the solubilities of dimethyl sebacate, which is clearly due to the
E2values.
Eq. (13) was also used to correlate the experimental solubilities
of other esters reported in literature. Vapor phase compositions of
other supercritical fluid–liquid binary systems were collected for
higher molecular weight carboxylic esters reported in literature.
The database included 17 data sets, consisting fatty acid methyl
and ethyl esters with a total of about 340 data points. Eq. (13) is a
simple three parameter equation relating solubility of liquid solute
with and T. In Eq. (13), has been calculated as the product of
density of the supercritical fluid and the molar volume of the pure
liquid solute, i.e., the ratio of molar volumes of pure liquid solute
and supercritical carbon dioxide. The variation of molar volume of
liquid solute is less than 2% for a variation in temperature of 50 K.
Thus, for small temperature ranges, the molar volume of pure liquid
solute can be assumed to be independent of temperature. It should
be noted here, that the molar volume of pure liquid solute is used,
which is unaffected by the presence of carbon dioxide. This is an
important motivation in using Wilson activity coefficient model for
the development of Eq. (13). The overall density of the liquid phase
in the supercritical fluid–liquid equilibria changes with respect to
operating pressure, due to dissolution of carbon dioxide in the liq-
uid phase, which is more significant in the sub-critical and critical
region and saturates at higher pressures [54]. Hence, the dependent
parameter can be considered a somewhat scaled dimensionless
form of , density of supercritical fluid. Nevertheless, the formula-
tion of Eq. (13) is a unique semi-empirical correlation, as it takes
into consideration the liquid solute density in the form of molar vol-
ume of pure liquid solute, V2 which is available for most chemical
compounds from literature.
The solubility of the liquid solute depends on its density. This
aspect is not captured by other popular semi-empirical models,
which relate solely to the density of supercritical fluid and tem-
perature alone. Although the liquid phase compositions do not
significantly vary in supercritical regions as vapor phase com-
positions, the model provides a scope for using liquid phase
compositions in the final expression which correlates vapor phase
compositions to density and temperature.
This simple model is most suitable for liquids that are spar-
ingly soluble in supercritical carbon dioxide. At high solubilities,
the assumption of infinite dilution activity coefficient, Eq. (10) is
no longer valid. The activity coefficient then becomes depend-
ent on the vapor phase composition in addition to the molar
volume ratio and temperature, Eq. (11), making the model more
complex. However, if the assumption of the infinite dilution is
relaxed, the regression is not improved. Many small molecular
Table 2
AARD (%) of the models for the liquid solubilities of various esters.
Compound T (K) p (MPa) N E1 E2 E3 R2
AARD (%) of
Eq. (13)
AARD (%) of
MT-model
Eq. (2)
AARD (%) of
Chrastil
model Eq. (3)
Ref
Bis(2-ethylhexyl)
Sebacate
303–328 10.0–18.0 28 −7.6 0.95 −1327.0 0.978 14.4 19.9 22.4 This work
Dimethyl Sebacate 308–328 10.0–18.0 26 −4.0 0.68 −648.6 0.900 7.8 6.4 6.5 This work
Butyl stearate 308–323 10.0–16.0 19 −6.5 1.08 −1333.7 0.987 9.9 9.8 11.1 [55]
Butyl laurate 313–328 10.0–16.0 20 −4.6 1.19 −1057.2 0.920 22.0 17.3 13.7 [55]
Methyl laurate 313–333 7.0–12.0 10 −0.6 2.58 −3361.7 0.950 24.3 8.8 9.6 [56]
Methyl myristate 313–333 7.9–16.0 10 −9.4 2.25 −547.5 0.984 14.9 16.0 16.8 [52]
Methyl palmitate 313–343 8.6–18.3 16 −8.3 1.95 −980.7 0.978 16.5 22.8 31.5 [52]
Methyl oleate 313–353 9.5–22.3 14 1.8 1.50 −4022.5 0.991 10.5 14.9 26.0 [57]
Methyl oleate 313–333 9.1–18.0 12 5.6 2.89 −7470.7 0.957 26.8 21.2 23.8 [58]
Methyl stearate 313–343 9.0–20.4 18 −4.1 1.97 −2894.7 0.968 15.1 19.2 17.4 [52]
Ethyl linoleate 313–333 9.0–17.0 16 −2.9 1.69 −2856.4 0.973 14.5 17.4 17.7 [36]
Ethyl oleate 313–333 9.1–18.6 21 −3.6 1.79 −2853.3 0.983 12.7 14.5 16.5 [36]
Ethyl stearate 313–333 8.9–18.3 19 −3.8 1.30 −2109.6 0.967 17.1 23.5 23.5 [36]
Ethyl EPA 313–333 9.0–21.1 24 −12.8 1.69 −114.0 0.987 13.0 14.6 16.9 [36]
Ethyl EPA 323–353 9.6–23.3 24 −5.9 1.05 −890.3 0.967 15.9 18.8 17.0 [59]
Ethyl EPA 313–333 11.3–18.0 12 22.3 3.21 −14,250.0 0.953 29.5 28.8 38.2 [60]
Ethyl DHA 313–333 9.0–21.1 26 −5.9 1.42 −2139.0 0.982 10.2 14.1 14.1 [36]
Ethyl DHA 313–353 8.1–23.1 34 −5.0 1.12 −1667.0 0.930 28.6 32.4 33.0 [59]
Ethyl DHA 313–333 9.7–23.5 17 −1.9 2.92 −6738.8 0.970 26.6 16.8 11.8 [60]
Average deviation 17 17.7 19.3

7. R.C. Narayan et al. / J. of Supercritical Fluids 101 (2015) 87–94 93
weight compounds like lower alcohols, ketones, carboxylic acids
and hydrocarbons completely dissolve completely near or just
above critical conditions. In these cases, the Wilson activity coeffi-
cient model, Eq. (10) and further Eq. (11) should include the mole
fraction of liquid solute and supercritical carbon dioxide in the
vapor phase. The model developed in this study can be used to
correlate the vapor phase compositions in case of higher molecu-
lar weight compounds, which are sparingly soluble. Instances for
the application of this model have been shown for carboxylic esters
(fatty acid methyl and ethyl esters) and dicarboxylic esters in this
work as they are more relevant in lipid processing.
The AARD (%) values obtained by Eq. (14) using the model equa-
tion represented by Eq. (13) were found to be around 8% and 14%
respectively for the dicarboxylic acid esters, dimethyl sebacate and
bis (2-ethylhexyl) sebacate. The deviation of the experimental solu-
bilities at 308 K and 328 K isotherms for bis (2-ethylhexyl) sebacate
and dimethyl sebacate (from the model given by Eq. (13)) are higher
compared to the other isotherms investigated in the work. The
regression results for Eq. (13) are represented in Table 2. The table
shows corresponding pressure and temperature range, number of
data points considered, AARD (%) and the correlation coefficients.
The correlation coefficients are above 0.95 for most compounds.
The value of E2 is always positive ranging from 0.6 to 3.0. As den-
sity increases along an isotherm, increases. It can be seen that
the derivative of solubility, y2 with respect to always increases
for the binary equilibria datasets included in the study. Therefore,
at constant temperature, the increase in density is always accompa-
nied with an increase in the mole fraction or solubility of the solute
in supercritical fluid. The value of E3 is always negative, indicat-
ing the decrease in solubility at constant density conditions with
increasing temperature. The effects of density and temperature on
solubility are thus suitably explained by the model.
The three parameter model given by Eq. (13) was compared with
the other three parameter models in literature such as MT model,
Eq. (2) and Chrastil model using Eq. (3) respectively. The AARD (%)
of these models is represented in Table 2. In general, the average
deviation of the model developed in this study was found to be
lesser than that of Chrastil and MT model. Thus, this model is supe-
rior to existing models and also provides physical insights into the
constants obtained by regression.
6. Conclusions
The solubilities of the two dicarboxylic acid esters, bis (2-
ethylhexyl) sebacate and dimethyl sebacate were determined
experimentally at 308, 313, 318, 323 and 328 K between 10 and
18 MPa. At constant temperature and pressure, the solubility of bis
(2-ethylhexyl) sebacate was less than dimethyl sebacate. A new
semi-empirical equation was formulated for correlating the solu-
bility of the liquid solute with the pure component molar volume
ratio and temperature and was used to correlate successfully vapor
phase compositions of binary systems of carbon dioxide + liquid
solute systems, with an average deviation of 17%. The new model
performed better at correlating than the existing three parameter
models, Chrastil and MT-model.
Acknowledgements
The authors thank the Council of Scientific and Industrial
Research (CSIR India) for financial support. The authors thank Ms.
Neha Lamba for help in some experiments.
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