Fluid Phase Equilibria 506 (2020) 112376
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Fluid Phase Equilibria
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Density-based UNIFAC model for solubility prediction of solid solutes
in supercritical fluids
Yueqiang Zhao a, *, Weibing Wang b, Weiwei Liu a, Jing Zhu a, Xiaoqin Pei a
a
b
Department of Chemical Engineering, Jiangsu Ocean University(originally Named Huaihai Institute of Technology), Lianyungang, 222005, Jiangsu, China
Department of Applied Chemistry, Yuncheng University, Yuncheng, 044000, Shanxi, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 19 June 2019
Received in revised form
27 October 2019
Accepted 28 October 2019
Available online 31 October 2019
A new theoretical expression of group interaction parameter as a function of fluid density was put
forward in this work. The group interaction parameters between the supercritical solvents (carbon dioxide, ethane, fluoroform and chlorotrifluoromethane) and the functional groups of solutes at reference
fluid density were presented, which exhibit good compatibility with the UNIFAC Vapor-Liquid Equilibrium (VLE) group interaction parameters. For the solid Solute þ C2H4 systems, satisfactory solubility data
representation results can be obtained using the UNIFAC VLE group interaction parameters of “C]C”.
This density-based UNIFAC model was examined for the solubility of 20 solid solutes in 5 supercritical
solvents (51 systems, 1111 data points); the Average Relative Errors (ARE) in solubility prediction are
19.1%, 18.7%, 21.6%, 20.4% and 12.8% for CO2, C2H6, C2H4, CHF3 and CClF3 correspondingly. In the case of
each UNIFAC group being one molecule, the density dependent expression of group interaction
parameter derived in this model may be applicable to other activity coefficient models (UNIQUAC,
Wilson, NRTL) containing binary interaction parameters for solubility prediction of solids in supercritical
fluids.
© 2019 Elsevier B.V. All rights reserved.
Keywords:
Solid solutes
Supercritical fluid (SF)
Solubility prediction
1. Introduction
The Supercritical Fluid Extraction (SFE) technology has been
widely used to separate and purify the valuable components in food
and pharmaceutical processes [1]. Most of the valuable components in food and pharmaceutical processes are solids, such as
caffeine, theobromine, artemisinin, naproxen, benzoic acid, and
acridine; therefore the solubility of solid solutes in supercritical
solvents is essential for the design and optimization of industrial
SFE processes [2]. Until now, many methods have been developed
to correlate or predict SFE solubility efficiently, such as equation of
state (EOS) methods [3e8], solution models [9e15], and empirical
equations [16e25]. Quiram et al. [5] developed a model using the
density series virial EOS truncated at the third coefficient to predict
the SFE solubility, which yields a linear relationship between the
solid solute solubility and the supercritical fluid density. The EOS
methods [3e8] consider the supercritical fluid as a compressed gas;
in contrast, the solution models [9e15] look upon the supercritical
fluid as an expanded liquid, where the activity coefficient of solute
* Corresponding author.
E-mail address: zyqlyg@jou.edu.cn (Y. Zhao).
https://doi.org/10.1016/j.fluid.2019.112376
0378-3812/© 2019 Elsevier B.V. All rights reserved.
depends on the supercritical fluid density. The commonly used
activity coefficient models (UNIQUAC [26], UNIFAC [27,28], COSMORS [29], COSMO-SAC [30], NRTL-SAC [31], FH-HSP [32], etc) have
been widely used in Vapor-Liquid Equilibrium (VLE), Liquid-Liquid
Equilibrium (LLE) and Solid-liquid equilibrium (SLE) predictions.
Since the activity coefficient of solute for these activity coefficient
models [26e32] is a function of liquid mixture composition and
temperature, which is nearly irrelative with the density of fluid
mixtures; as a result, they can not be applied to SFE solubility
predictions. The COSMO-VAC model [14] utilizes solvent vacancy to
account for the variation of solute activity coefficient as a function
of fluid density; this model is capable of predicting the solid solute
solubility in liquid solvent and supercritical carbon dioxide at high
pressures. Our previous work [15] developed an empirical model
based on UNIFAC, in which the pressure influence on solute activity
coefficient is taken into account by a power exponent for the
reduced density; this model fails to predict the solubility minimum
and solubility maximum behavior for SFE(i.e., the naphthaleneethylene system [33]). Most of the activity coefficient models
(UNIQUAC [26], UNIFAC [27,28], COSMO-RS [29], COSMO-SAC [30],
NRTL-SAC [31], FH-HSP [32], etc) are incapable of SFE solubility
prediction, COSMO-VAC [14] and our previous model [15] fail to
2
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
predict the solubility extrema; therefore, there is a need for a new
density-based UNIFAC model to be developed.
In this work, we proposed a density-based UNIFAC model,
where the activity coefficient of solute is a function of fluid density,
composition and temperature. The model parameters were estimated, and the solubility prediction capability of this model was
verified with successful results.
Then the two mean intra-molecular potential energy terms are
nearly identical [38].
ðaÞ
ðbÞ
< uintra;i > z < uintra;i >
(5b)
Substituting eqs (4) and (5b) into eq (3), we obtain
ðbÞ
ðaÞ
U inter;i U inter;i V ðbÞ
Ki ¼ ðaÞ ¼ ðaÞ exp RT
V
x
ðbÞ
xi
2. Theory
(6)
i
2.1. Solid-liquid equilibrium equation
ðaÞ
The solubility of solid solute in supercritical fluid (expanded
liquid) is determined by the following equation [15,34].
lnðxi gi Þ ¼
ðP V S V L
i
i
RT
dp DHm
RT
1
P0
T
Tm
(1)
where DHm and Tm are enthalpy of fusion and the melting point of
the solute correspondingly; V Si and V Li are the molar volume of
solute in solid and liquid states correspondingly; T and P are system
temperature and pressure correspondingly; R is the gas constant; P0
is the reference pressure, here P0 is selected as P0 ¼ 1atm. xi is the
solute solubility (mole fraction); gi is the activity coefficient of solute in supercritical fluid (expanded liquid), which is calculated by
the density-based UNIFAC model.
Wang and Lin proposed to estimate the solid molar volume V Si
by the co-volume term of Peng-Robinson (PR) EOS [8].
V Si
.
¼ 0:0778RTc Pc
(2)
where Tc and Pc are the critical temperature and critical pressure,
respectively; and the liquid molar volume V Li is determined from PR
EOS.
2.2. Density-based UNIFAC model
The statistical mechanical expression of equilibrium ratio is
given as [35].
ðbÞ
ðaÞ
ðbÞ
U p;i U p;i x
V ðbÞ
Ki ¼ ði aÞ ¼ ðaÞ exp RT
V
xi
(3)
whereV(a)and V(b)are the molar volume of mixture in fluid phases a
ðaÞ
and b correspondingly; xi is the mole fraction of species i in fluid
ðaÞ
phase a; U p;i is the mean molar potential energy of component i in
ðaÞ
ðaÞ
ðaÞ
fluid mixture, U p;i ¼ NA < ui > , NA is Avogadro's constant, < ui >
is the ensemble average potential energy experienced by a molecule i in fluid mixture a.
The average potential energy for one molecule < ui> contains
mean intra-molecular potential energy and mean intermolecular
potential energy [36,37].
< ui> ¼ < uintra,i > þ < uinter,i>
ðVÞ
ðLÞ
ðaÞ
ðaÞ
ðbÞ
Eii ¼ Eii RT ln V i
Eii ðrÞ ¼ Eii ðr0 Þ RT lnðr0 = rÞ
(7)
(8a)
or
Eii ðPÞ ¼ Eii ðP0 Þ RT lnðr0 = rÞ
(8b)
where r0 is the reference density of pure liquid i at pressure P0, r is
the fluid density at system pressure P.
According to UNIFAC model [27], the group-interaction parameter is defined as
aij ¼ Uij Ujj
R
(9)
where Uij characterizes the interaction energy between groups i
and j.
Bruin and Prausnitz [40] suggested an estimation equation of Ujj
Ujj ¼ cEjj
(10)
where c is a pre-factor, Ejj is the total molar cohesion energy of
similar (like) groups i.
The combination of eqs (9) and (10) gives
aij ¼ Ejj Eij c R
(11a)
ðv;uÞ
.
ðu;uÞ
ðv;uÞ
c R
¼ Ejj Eij
(11b)
ðu;uÞ
.
ðu;uÞ
ðu;uÞ
c R
¼ Ell
Ekl
(11c)
ðu;vÞ
.
ðv;vÞ
ðu;vÞ
c R
¼ Eii Eji
(11d)
(4)
(5a)
ðbÞ
Vi
where Eii is the total molar cohesion energy of similar (like) molecules (or the molar internal energy change on vaporization for
ðoÞ
ðoÞ
ðoÞ
pure liquid i) [39], Eii ¼ U inter;i ¼ NA < uinter;i > , < uinter;i > is the
mean intermolecular potential energy deserved by one molecule of
ðoÞ
pure liquid i, U inter;i is the mean molar intermolecular potential
energy of pure liquid i at system temperature and pressure.
If fluid b is chosen as the reference state, then eq (7) becomes
aij
The Monte Carlo simulation results of Jorgensen et al. [36]
showed that the value of mean molar intra-molecular potential
energy for pure components (alkanes, alcohols, ethers, amides,
ketones, amines, etc) generally makes little difference in going from
liquid phase to vapor phase
U intra;i zU intra;i
where U inter;i is the mean molar intermolecular potential energy of
species i in liquid mixture system a.
>Equations (3) and (6) are the statistical mechanical expression
ðaÞ
ðbÞ
of equilibrium ratio for fluid mixtures. For xi ¼ xi ¼ 1(pure
fluid), we can obtain the relationship between the total molar
cohesive energy of similar (like) molecules and the molar volume of
pure fluid from equation (6)
akl
aji
where the superscripts v and u represent solvent and solute
ðu;uÞ
correspondingly, Ejj
is the total molar cohesive energy of similar
ðv;uÞ
(like) groups j from solute molecule, Eij
represents the total
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
molar cohesion energy between group i from solvent molecule and
ðu;uÞ
group j from solute molecule, akl
is the interaction parameter
between dissimilar (unlike) groups k and l from solute molecule,
ðv;uÞ
aij denotes the interaction parameter between group i from solvent molecule and group j from solute molecule. Equations (8a) and
(8b) were developed for pure components, namely eqs (8a) and
(8b) can be applied to groups from the same type of molecules
ðv;vÞ ðv;vÞ
ðu;vÞ
(Ejj Ekl ). For groups form different type of molecules (Eji ), the
ðu;vÞ
relationship between Eji
and fluid density r is unavailable; we
ðu;vÞ
presume that Eji follows a similar law of fluid density as that of
ðv;vÞ
pure solvent groupsEii .
Considering the change in aij due to system pressure variation at
constant temperature, we obtain
3
ðv;uÞ
gases); for ideal gas (r ¼ 0), the value of DEij will be infinity, an
empirical parameter c is introduced to resolve this issue
ðs;tÞ
c ¼ 4ij
ðs;tÞ
½1 expð 60ar = r0 Þ
0
1
¼
4ij
.
ðb þ zÞ
(14a)
ð i ¼ j and s ¼ t Þ
ðnotð i ¼ j and s ¼ t ÞÞ
(14b)
ðv;uÞ
ðv;uÞ
ðu;uÞ
ðv;uÞ
c R
Daðv;uÞ
¼ aij ðPÞ aij ðP0 Þ ¼ DEjj
DEij
ij
(12a)
DEjjðu;uÞ ¼ Ejjðu;uÞ ðPÞ Ejjðu;uÞ ðP0 Þ
(12b)
where z is the compressibility factor of fluid, s and t denote solvent
molecule or solute molecule, a and b are adjustable parameters.
As shown in eqs (13c) and (14a), in the limiting case of low fluid
density (r/0), the pre-factor c approaches zero (c/0), therefore a
low density compensation term should be added to rectify this
residual error. The molecular dynamics simulation results of
Lennard-Jone fluid by Nicolas et al. [41] demonstrated that the
depth of the potential well is proportional to the critical temperature of component
DEijðv;uÞ ¼ Eijðv;uÞ ðPÞ Eijðv;uÞ ðP0 Þ
(12c)
Tc;i
εii
¼
kB 1:35
.
The system pressure has little influence on the density of solid
solute (hypothetical sub-cooled liquid), while it has a significant
impact on fluid density; as a result, combining eqs. (8b), (12a), (12b)
and (12c) yields
DEjjðu;uÞ z0
(13a)
DEðv;uÞ
z RT lnðr0 = rÞ
ij
(13b)
Daðv;uÞ
zcT lnðr0 = rÞ
ij
(13c)
The molar cohesive energy of like groups from solute molecules
ðu;uÞ
Ejj
is nearly independent of system pressure (eq (13a)), the molar
ðv;uÞ
cohesive energy of like groups from different moleculesEjj is
sensitive to fluid density (eq (13b)). As can be seen from eqs
(11a)e(11d) and eqs (13a)-(13c), for like groups from the same type
of molecules, the value of group-interaction parameter for this pair
of groups is zero
ðu;uÞ
aii
.
ðu;uÞ
ðu;uÞ
c R¼0
¼ Eii Eii
.
ðv;vÞ
ðv;vÞ
ðv;vÞ
ajj ¼ Ejj Ejj
c R¼0
(13d)
(13e)
And the change in group-interaction parameter due to solvent
density variance is also zero
Daiiðu;uÞ ¼ 0
(13f)
Daðv;vÞ
jj
(13g)
¼0
ðv;uÞ
.
ðu;uÞ
ðv;uÞ
c R
¼ Ejj Ejj
(13h)
And the change in group-interaction parameter due to solvent
density variance is also nonzero
Daðv;uÞ
zcT lnðr0 = rÞ
jj
where kB is Boltzmann's constant, then the energy parameter for
unlike molecules is
0:5
εij
Tc;i Tc;j
¼
kB
1:35
(15b)
The combination of eqs (15a) and (15b) yields
εij
.
εjj ¼ Tc;i Tc;j
0:5
(15c)
Our previous work [38] illustrated that the energy parameter of
dispersive interactions for like molecules is proportional to the total
cavity surface area of molecule from conductor-like screening
model
εii
fA0:5
COSMO;i
kB
(15d)
Similarly, the total molar cohesive energy of like groups (k) from
the same type of molecules (solvent) is proportional to the surface
area parameter of group
ðv;vÞ ðvÞ
Ekk f Q k
0:5
(15e)
Then the expression of the total molar cohesive energy between
group k from solvent molecule and group l from solute molecule
can be obtained
ðv;uÞ
Ekl
ðvÞ ðuÞ
f Qk Ql
0:25
(15f)
and
For like groups from different type of molecules, the value of
group-interaction parameter for this pair of groups is not zero
ajj
(15a)
(13i)
Equation (13b) is valid for dense fluids (liquids and compressed
ðv;uÞ
ðEkl
.
ðu;uÞ
Ell
ðvÞ
ðuÞ
f Q k =Q l
0:25
(15g)
From eqs (15c) and (15g), we have
ðv;uÞ
ðEkl
.
ðu;uÞ
Ell
ðv;vÞ
ðu;uÞ
f Tc
Tc
0:5 ðvÞ
ðuÞ 0:25
Q k =Q l
(15h)
As can be seen from eq (11b), as the fluid density approaches
zero (r/0), the fluid mixture becomes ideal gas, and the total
cohesion energy between group i from solvent molecule and group
ðv;uÞ
j from solute molecule will close in on zero (Eij /0); whereas the
total cohesion energy of pure solid solute (hypothetical sub-cooled
liquid) makes little change with the alteration of fluid density (eq
4
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
ðv;vÞ
Solvent
Cosolvent
a
CO2
C2H6
C2H4
CHF3
CClF3
acetone
ethyl acetate
ethanol
1-propanol
2-propanol
methanol
Tc (K)
Pc (bar)
u
r0a (kg/m3)
Ref.
304.25
305.35
282.40
299.25
301.95
508.2
523.2
513.9
536.8
508.3
512.6
73.8
48.8
50.3
49.5
39.2
46.6
38.3
61.4
51.7
47.6
80.9
0.225
0.098
0.089
0.272
0.172
0.318
0.362
0.635
0.623
0.665
0.556
771.20
380.36
440.84
819.00
924.00
790.55
900.63
789.30
803.60
786.30
791.30
[15,50]
[15,50]
[15,50]
[50,57]
[50,57]
[15]
[15]
[15]
[15]
[15]
[15]
ðv;uÞ
For r ¼ 0 and aij
ðv;uÞ
aij
ðr ¼ 0Þ
ðv;uÞ
aij ðr
¼ r0 Þ
ðv;uÞ
ðv;uÞ
aij
Rk
Qk
Ref.
CO2
C2H6
C2H4
CHF3
CClF3
C5HNa
caffeine
theobromine
1.0020
1.8022
0.9946b
1.6360
2.1721
2.3347
3.2873
2.9840
0.8800
1.6960
1.3448b
1.6060
2.1000
0.9930
2.6331
2.4396
[15]
[15]
[15]
Sub-group of “Pyridine”.
Estimated from experimental solubility data via eq (20).
ðv;uÞ
aij
r¼0
(16c)
ðr0 Þ > 0, eq (16a) can be rewritten as
ðv;vÞ ðu;uÞ
Tc
¼ 1 þ Tc
For r ¼ 0 and aij
group
a
8
< 1 ðx > 0Þ
SgnðxÞ ¼ 0 ðx ¼ 0Þ
:
1 ðx < 0Þ
Reference density of supercritical solvent at T ¼ 20 C and P ¼ 1 atm.
Table 1b
Group Volume and Surface area parameters.
b
ðu;uÞ
molecule; T c andT c are the critical temperature of solvent and
solute correspondingly; tanh(x) is the hyperbolic tangent function,
tanh(x)¼ (exe-x)/(ex þ e-x); and Sgn(x) is the sign function
Table 1a
Physical properties of supercritical solvents.
ðvÞ
Qi
.
ðuÞ 0:25
Qj
ðu;uÞ
¼ 1 T ðv;vÞ
Tc
c
0:5 ðvÞ
Qi
.
ðuÞ 0:25
Qj
As shown from eqs (16d) and (16e), Sgn(x) function ensures that
ðv;uÞ
ðr ¼ 0Þ > aij ðr ¼ r0 Þ.
The attenuation factors of fluid density in eq (16a), expð xpfÞand tanh(x), ensure that as the fluid density increases the
density compensation term (the right side of Eq (16a)) decreases
rapidly.
ðv;uÞ
The resulting equation of aij is given below
ðv;uÞ
aij
h
ðv;uÞ
ðv;uÞ
ðrÞ ¼ aij ðr0 Þ 1 þ Sgn aij ðr0 Þ expð xpfÞ
ðv;vÞ ðu;uÞ 0:5 ðvÞ . ðuÞ 0:25 i
Qi
Qj
Tc
Tc
n
.
o
ðs;tÞ
þT lnðr0 =rÞ 4ij ½1 expð 60ar=r0 Þ ðb þ zÞ
(17a)
ðv;uÞ
ðv;uÞ
ðu;uÞ
ðv;uÞ
ðr/0Þ
ðv;uÞ
(16e)
aij
(13a)). As a result, the value of aij ðr /0Þ will become more
ðv;uÞ
positive, and the low density compensation term of aij
can be
approximated semi-empirically
aij
(16d)
ðr0 Þ < 0, eq (16a) becomes
r ¼ r0
0:5 ðv;vÞ ðu;uÞ
ðv;uÞ
Tc
1 ¼ Sgn aij ðr0 Þ expð xpfÞ T c
Similarity, we can obtain the expressions of akl
ðu;vÞ
and aji
0:5
ðu;uÞ
aij ðr ¼ r0 Þ
.
ðvÞ
ðuÞ 0:25
Qi
Qj
(16a)
f ¼ tanhðpr = r0 Þ
(16b)
akl
ðvÞ
where Q i is the surface area parameter for group i from solvent
ðu;vÞ
aji
ðu;uÞ
ðPÞzakl
ðu;vÞ
ðrÞ ¼ aji
ðP0 Þ
(17b)
ðr0 Þ½1 expð 60ar = r0 Þ
(17c)
where the interaction parameter of dissimilar (unlike) groups k and
ðu;uÞ
l from solute moleculeakl is nearly independent of system pressure P. In the limiting case of low fluid density (r/0), the
Table 2
Physical properties of solid solutes.
Solute
M (g/mol)
Tm (K)
DHm (kJ/mol)
Tc (K)
Pc (bar)
u
Ref.
naphthalene
anthracene
phenanthrene
2,6-Dimethylnaphthalene
2,3-Dimethylnaphthalene
benzoic acid
hexachloroethane
biphenyl
phenol
p-chlorophenol
2,4-dichlorophenol
fluorene
triphenylmethane
naproxen
2-naphthol
2-aminofluorene
acridine
pyrene
caffeine
theobromine
128.174
178.234
178.234
156.227
156.227
122.124
236.740
154.210
94.113
128.550
163.000
166.220
244.330
230.263
144.170
181.230
179.220
202.260
194.200
180.18
353.35
489.65
369.45
383.29
377.47
395.50
458.00
342.08
314.04
316.00
318.00
387.92
365.45
426.15
395.15
404.15
384.00
423.15
511.15
627.15
18.875
28.972
18.715
25.060
19.353
18.020
9.750
18.570
11.510
14.070
20.100
19.580
21.500
26.023\
17.510
22.327
19.700
17.110
21.118
41.112
748.15
869.30
882.55
777.00
785.00
751.00
714.60
773.00
694.15
738.00
710.61
826.00
863.00
889.69
812.13
912.70
883.15
936.00
706.57
817.83
40.500
31.200
31.700
32.200
32.200
44.700
34.500
33.800
61.300
53.200
50.800
30.000
22.400
26.680
47.366
35.473
31.900
26.100
42.700
40.900
0.302000
0.353100
0.329900
0.420100
0.424000
0.602790
0.163000
0.402870
0.440000
0.484882
0.539112
0.404160
0.576000
0.966890
0.576533
0.665725
0.498000
0.507416
0.948000
0.725250
[15,51]
[25,51]
[15,51]
[25,51]
[25,51]
[25,51]
[15,51]
[15,51]
[25,51]
[52]
[52]
[51,52]
[51,52]
[15,52]
[25]
[53]
[25]
[52]
[38,59]
[50,59]
1729.64
2719.65
714.41
113.72
2798.38
472.10
2157.25
ACH
CH
ACOH
ACNH2
C5HN
COOH
0.00
113.00
526.46
185.42
234.25
385.57
289.03
452.81
743.64
5075.79
2827.77
1657.66
1843.19
aij (K)
CHF3
ACH
COOH
ACNH2
C5HN
ACOH
CHF3
ACH
COOH
ACNH2
C5HN
ACOH
0.00
238.30
361.87
391.18
445.23
314.89
802.50
1915.59
2387.41
1510.64
2398.71
Table 6
Group Interaction Parameters at reference fluid density for Chlorotrifluoromethane.
1623.72
564.05
63.39
theobromine
CH3CO
ACCl
C2H6
C2H6
ACH
CH
ACOH
ACNH2
C5HN
COOH
aij (K)
CClF3
ACH
COOH
ACOH
ACNH2
C5HN
CClF3
ACH
COOH
ACOH
ACNH2
C5HN
0.00
287.39
424.53
365.19
471.85
533.32
893.96
2385.94
3195.4
3611.81
2193.17
interaction parameter between group j from solute molecule and
ðu;vÞ
group i from solvent molecule approaches zero (aji /0).
For mixed solvent (supercritical solvent þ cosolvent) systems,
the reference density of fluid mixture r0 is given by
ðM xs þ Mc xc Þ
Ms xs
Mc xc
þ
r
r
r0 ¼ s
s0
816.40
CH3O
a
ACOH
aij (K)
Table 5
Group Interaction Parameters at reference fluid density for Fluoroform.
a
220.20
213.70
0.00
800.10a
1599.20a
16.05a
1835.00a
3266.43
200.70a
72.68
179.68
952.46
253.56
827.25
821.16
592.70
468.65
CO2
ACH
CH2
COOH
CH3O
ACOH
ACCl
ACNH2
C5HN
CH3COO
CH3CO
CH3OH
OH
theobromine
caffeine
UNIFAC group-interaction parameters from Ref. [15].
1149.10
a
COOH
a
CH2
a
ACH
CO2
aij (K)
Table 3
Group Interaction Parameters at reference fluid density for Carbon Dioxide.
5
Table 4
Group Interaction Parameters at reference fluid density for Ethane.
a
ACNH2
C5HN
CH3COO
CH3OH
OH
caffeine
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
(18)
c0
where xs and xc are the mole fraction (solute free) of solvent and
cosolvent correspondingly, Ms and Mc are the molecular weight of
solvent and cosolvent correspondingly, rs0 and rc0 are the reference
density (at T ¼ 20 C and P ¼ 1 atm) of solvent and cosolvent
correspondingly.
The group parameters Rk and Qk of supercritical solvents are
obtained from the van der Waals group volume and surface areas
Vwk and Awk given by Bondi [42]:
Rk ¼ Vwk/15.17
(19a)
Qk ¼ Awk /(2.5 109)
(19b)
ðv;uÞ
The model parameters (aij ðr0 Þ, a, b, x) are estimated by
minimizing the following objective function
f
ðv;uÞ
aij ðr0 Þ; a; b; x
100%
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u n .
2
u1 X
¼ ARE ¼ t
ðxcalc
xexpt
1
i
i
n i¼1
(20)
where xexpt
is the experimental value of solute solubility;xcalc
is the
i
i
calculated value of solute solubility via eq (1), where the activity
6
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
Table 7a
Summary of the estimated model Parameters and Accuracy for solids solubility prediction in pure supercritical solvents a.
Solvent
No. of Solutes
Nd
a
x
b
ARE
Ref.
CO2
C2H6
C2H4
CHF3
CClF3
18
8
6
7
6
453
237
143
72
68
0.638839
0.146707
0.344120
0.156616
0.902956
4.76109
1.51234
1.95355
3.83015
4.65904
1.023980
0.996624
0.600463
0.956236
1.746680
19.1%
18.7%
21.6%
20.4%
12.8%
[7,49,54e57,59]
[7,57]
[7,55,58]
[57]
[57]
a
Detailed information of estimated model parameters and accuracy in solubility prediction for solid solutes in various supercritical solvents refers to the Supporting
Material.
Table 7b
Solubility calculation results for Solid Solute in Supercritical Carbon Dioxide with Cosolvent.
Solid Solute
Solvent
naproxen
CO2
CO2
CO2
CO2
CO2
CO2
þ
þ
þ
þ
þ
þ
Ethyl Acetate
Acetone
Methanol
Ethanol
1-Propanol
2-Propanol
a
x
b
Nd
ARE
Ref.
0.638839
4.761090
1.023983
18
33
26
24
15
22
6.87%
9.75%
8.95%
7.97%
12.11%
7.01%
[49]
[49]
[49]
[49]
[49]
[49]
Table 8
Comparison of solubility prediction accuracy between this model and PR EOS method.
T ( C)
System
Naphthalene þ CO2
2,3-Dimethylnaphthalene
þ CO2
Naphthalene þ C2H6
Phenanthrene þ C2H4
a
b
35
55
35
45
55
20
25
35
35
45
55
45
55
65
Nd
9
16
5
5
5
7
7
6
15
12
14
5
5
5
This model
PR EOS
parameters
ARE
k12
a ¼ 0.638839
5.66%
19.81%
9.93%
6.53%
17.53%
26.00%
20.35%
23.67%
20.29%
18.90%
30.55%
14.78%
9.37%
13.66%
0.0980
0.0992
0.0989
0.1042
0.1078
0.0427
0.0457
0.0386
0.0448
0.0446
0.0448
0.0459
0.0355
0.0318
x ¼ 4.761090 b ¼ 1.023983
a ¼ 0.146707
x ¼ 1.512340 b ¼ 0.996624
a ¼ 0.344120
x ¼ 1.953550 b ¼ 0.600463
Ref.
ARE
a
a
a
a
a
a
a
a
a
a
a
b
b
b
13.48%
28.46%
18.50%
30.46%
22.57%
27.77%
21.31%
36.60%
37.21%
38.54%
44.07%
55.85%
49.12%
42.77%
[54]
[54]
[55]
[55]
[55]
[7]
[7]
[7]
[57]
[57]
[57]
[55]
[55]
[55]
Values of k12 from Ref. [51].
Values of k12 from Ref. [55].
coefficient of solid solute in supercritical fluid is computed by the
original UNIFAC model equations [27] plus the density dependent
expressions of group-interaction parameter (eq (17a), (17b) and
(17c)). The fitting quality of this model to the experimental solubility data is measured in terms of the average relative error (ARE)
defined in eq (20).
3. Results and discussions
Fig. 1. Solubility of naphthalene in carbon dioxide at T ¼ 328.15K
The reported density calculation accuracy of Hang EOS [43] for
carbon dioxide is within 1% around the critical point and within
0.1%e0.2% outside the critical region; and the Hang EOS [43] is
more reliable than other equations of state (BWR [44], Martin-Hou
[45] and Bender [46]) in the PeV-T property predictions of carbon
dioxide. The work of Peng and Robinson [47] showed that the PR
EOS [48] is reliable in predicting the critical properties of pure
substances and multi-component systems. Based on the reasons
mentioned above, we used the Hang EOS [43] to calculate the fluid
density of pure carbon dioxide, and utilized the PR EOS to compute
the fluid density of other pure supercritical solvents. The density of
mixed solvents (supercritical solvent þ cosolvent) was calculated
by PR EOS [48] with the binary interaction parameters from
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
7
Fig. 2. Solubility of 2,3-dimethylnaphthalene in carbon dioxide.
literature [49]. The physical properties of the supercritical solvents
(carbon dioxide, ethane, ethylene, fluoroform and chlorotrifluoromethane) and cosolvents (ethyl acetate, acetone, methanol,
ethanol, 1-propanol, 2-propanol) are shown in Table 1a, where the
reference density r0 is chosen as the density of saturated liquid at
normal temperatures (20 C); the group volume and surface area
parameters of the supercritical solvents and some groups are listed
in Table 1b; the physical properties of solid solutes [49e59] are
given in Table 2; the regressed group interaction parameters at
ðv;uÞ
reference density aij ðr0 Þfor supercritical solvents (CO2, C2H6,
CHF3 and CClF3) are given in Tables 3e6. The estimated model
parameter (a, b, x) and the average relative error (ARE) in solubility
calculation are illustrated in Table 7a, 7b and 8 (Tables S1eS5 in
Supporting Material), and some solubility calculation results
compared with the experimental solubility data [49,54,55,59] are
depicted in Figs. 1e7; from which we can observe that this model
describes the solubility behavior of solid solute in pure supercritical
solvents and mixed solvents (supercritical solvent þ cosolvent)
successfully. For system temperature T ¼ 333.1 K and pressure
P ¼ 179.3 bar with cosolvent composition xs ¼ 1.75% (solute free
Fig. 3. Solubility of 2,6-dimethylnaphthalene in ethylene.
mole fraction), the solubilities of naproxen (mole fraction naproxen 105) in supercritical carbon dioxide with cosolvents are
6.09, 5.26, 9.53, 9.55, 11.20 and 10.84 for acetone, ethyl acetate,
methanol, ethanol, 1-propanol and 2-propanol correspondingly
[49]; which shows that the best cosolvent for naproxen is 1propanol.
We can see from the accuracy of solubility calculation (Table 7a)
that the estimated group interaction parameters at reference density for supercritical solvents (CO2, C2H6, CHF3 and CClF3) exhibit
good compatibility with the UNIFAC VLE group interaction parameters. For the solid SoluteeC2H4 systems, where the entire
ethylene molecule was treated as one functional group and the
UNIFAC VLE group interaction parameters of “C]C” [27,28] were
used here, satisfactory solubility prediction results was obtained for
these systems (Table 7a, Table S3 in Supporting Material). This
density based UNIFAC model has been examined for the solubility
of 20 solid solutes in 5 supercritical solvents (CO2, C2H6, C2H4, CHF3
and CClF3) over a pressure range of 42.3e490.4 bar, a temperature
range of 20e70 C and a wide solubility range of 106101 (51
systems, 1111 data points), the average relative errors in predicted
solubility are 19.1%, 18.7%, 21.6%, 20.4% and 12.8% for CO2, C2H6,
C2H4, CHF3 and CClF3 correspondingly; which demonstrates that
this density-based UNIFAC model is simple and reliable for practical
application. For CaffeineCO2 and TheobromineCO2 systems,
both caffeine and theobromine molecules contain two adjacent
rings [59], and the structures of each ring for caffeine and theobromine are different form that of cyclohexane, benzene and pyridine; as a result, the whole molecule is treated as one UNIFAC
group for the two solid solutes; here, the UNIFAC model is reduced
to the UNIQUAC method; this means that the similar forms of
equations (17a), (17b) and (17c) may be applied to other activity
coefficient models (UNIQUAC [26], Wilson [60], NRTL [61]) containing binary interaction parameters for SFE solubility predictions.
A comparison of solubility prediction accuracy between the
original UNIFAC, density based UNIFAC and PR EOS method is given
in Fig. 1 and Table 8; which shows that both the proposed model
and PR EOS can describe solubility extrema (solubility minimum
and solubility maximum), and the original UNIFAC model gives far
worse solubility prediction results than density based UNIFAC and
PR EOS. In addition, the binary interaction parameter k12 for EOS
methods is temperature and solute-solvent system dependent;
8
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
Fig. 4. Solubility of Naproxen in supercritical Carbon Dioxide. with Ethyl Acetate Cosolvent at T ¼ 333.1K.
Fig. 5. Solubility of Naproxen in supercritical Carbon Dioxide. with Ethanol Cosolvent at T ¼ 333.1K.
while the parameters of this model (a, b, x) are irrelative to system
temperature, this may be due to that these parameters (a, b, x) were
estimated via experimental data from different system temperatures. For some solute-solvent systems (Table 7a and 7b,
Tables S1eS5 in Supporting Material), the values of model parameters (a, b, x) seem to be universal constants. The origin of the
prediction error of this model may come from experimental error in
solubility data or from the uncertainty in solute property values
(DHm, Tc, Pc, u) [5]. For example, the uncertainty (or reproducibility)
of the experimental solubility data from the same literature was
reported [48] within ±5%; the agreement of solubility results for
the same solutesolvent system between different reference
sources was reported [57] within ±10%.
4. Conclusions
A new density-based UNIFAC model for SFE Solubility prediction
was presented in this work, in which the group interaction
parameter is a function of supercritical fluid density. The group
interaction parameters for supercritical solvents (carbon dioxide,
ethane, fluoroform and chlorotrifluoromethane) at reference fluid
density were estimated, which manifest good compatibility with
the UNIFAC VLE group interaction parameters. For the solid SoluteeC2H4 systems, satisfactory solubility prediction results can be
obtained using the UNIFAC VLE group interaction parameters of
“C]C”. This new model has been verified for the solubility of 20
solid solutes in five superficial solvents (CO2, C2H6, C2H4, CHF3 and
CClF3) over a pressure range of 42.3e490.4 bar, a temperature range
of 20e70 C and a wide solubility range of 106101 (51 systems,
1111 data points) with successful results; which indicates that this
model is simple and reliable for practical application. This densitybased UNIFAC model is comparable to the PR EOS method with
regard to the accuracy of SFE solubility prediction. For some
solutesolvent systems, the values of model parameters (a, b, x)
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
9
Fig. 6. Solubility of Caffeine in supercritical Carbon Dioxide at T ¼ 333.15 K.
Fig. 7. Solubility of Theobromine in supercritical Carbon Dioxide.
seem to be universal constants. When each UNIFAC group is one
molecule, the density dependent expression of group interaction
parameter derived in this model may be applicable to other activity
coefficient models (UNIQUAC, Wilson, NRTL) containing binary
interaction parameters, which can expand the application range of
these activity coefficient models from liquid phase to supercritical
regime. This model is reliable for SFE solubility prediction of solid
solute in mixed solvents.
Acknowledgment
We gratefully acknowledge financial Support by a Project Funded by the Priority Academic Program Development of Jiangsu
Higher Education Institutions, the Public science and technology
research funds projects of ocean (201505023).
List of symbols
Declaration of competingf interest
Does the appearance of financial or other “material” interests by
authors of scientific papers present a more serious bias than other
competing interests so as to require a signed Conflict of Interest
(COI) statement?
NO, this paper does not require a signed Conflict of Interest (COI)
statement.
ACH
ðv;uÞ
aij
ARE
Eij
aromatic carbon group
interaction parameter between group i from solvent
molecule and group j from solute molecule, K
average relative error in solubility prediction defined via
eq(20)
total molar cohesive energy of dissimilar (unlike) groups
i and j, J/mol
10
Ejj
DHm
kij
M
Nd
P
P0
Pc
Qk
R
Rk
T
Tc
Tm
ðoÞ
U inter;i
V
z
Y. Zhao et al. / Fluid Phase Equilibria 506 (2020) 112376
total molar cohesive energy of similar (like) groups i, J/
mol
enthalpy of fusion for solid solute, kJ/mol
binary interaction parameter for equation of state
molecular weight, g/mol
number of experimental data points
pressure, bar
reference pressure, bar
critical pressure, bar
surface area parameter for group k
gas constant
volume parameter for group k
system temperature, K
critical temperature, K
melting point of solid solute, K
mean molar intermolecular potential energy of pure
liquid i at system temperature and pressure, J/mol
molar volume, m3/mol
compressibility factor
Greek Letters
g
activity coefficient
r0
reference density of supercritical solvent, or the density
of corresponding pure liquid solvent at normal
temperatures, kg/m3
r
the fluid density at system temperature and pressure,
kg/m3
u
acentric factor
Subscripts
i, j, k, l
group index
Superscripts
Calc
calculated
expt
experimental
u
solute
v
solvent
Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.fluid.2019.112376.
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