Chemical Engineering Science 58 (2003) 1743 – 1749
www.elsevier.com/locate/ces
Extension of Valderrama–Patel–Teja equation of state to modelling
single and mixed electrolyte solutions
Rahim Masoudi, Mosayyeb Arjmandi, Bahman Tohidi∗
Centre for Gas Hydrate Research, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK
Received 1 April 2002; received in revised form 15 November 2002; accepted 18 November 2002
Abstract
The Valderrama modi4cation of Patel–Teja equation of state (VPT EoS) has been extended to predicting 5uid phase equilibria in the
presence of single and mixed electrolyte solutions at high-pressure conditions. Salts have been introduced as components in the EoS
by calculating their EoS parameters from corresponding cation and anion parameters. A non-density dependent mixing rule is used for
calculating a, b, and c parameters of the EoS. The inclusion of salts in the EoS resulted in the omission of the Debye–H9uckel electrostatic
contribution term in the fugacity coe;cient calculations. Water–salt binary interaction parameters (BIPs) are optimised using freezing
point depression and boiling point elevation data of aqueous electrolyte solutions. Gas solubility data in aqueous electrolyte solution are
used for optimising salt–gas BIPs. The predictions of the model have been compared with independent experimental data, demonstrating
the reliability of the approach.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Modelling; Phase equilibria; Equation of state; Solutions; Electrolyte; Gas solubility
1. Introduction
There are numerous calculations in chemical and
petroleum engineering where a reliable prediction of phase
behaviour of electrolyte solutions is required. Early models
for predicting vapour liquid equilibria in systems containing
aqueous electrolyte solutions were either based on activity
coe;cient approach (i.e., not suitable for high-pressure
conditions), or have not included non-condensable gases.
Aasberg-Petersen, Stenby, and Fredenslund (1991) adopted
a new approach. They divided the fugacity coe;cient of
each component in the liquid water phase into two terms:
Ln ’i = Ln ’EoS
+ Ln EL
i
i :
(1)
An equation of state term (short-range interactions),
which is employed to calculate the eEect of non-ionic
(molecular) species in the aqueous phase and a Debye–
H9uckel electrostatic term (long-range interactions), which
is used for calculating the eEect of salts on the fugacity
coe;cients of molecular species in the solution. It is worth
mentioning that they did not consider salt as a component
in the EoS, but took into account its eEect on the fugacity
coe;cient of other molecular species.
They used Adachi–Lu–Sugie equation of state (ALS EoS,
Adachi, Lu, & Sugie, 1983) with a volume-dependent mixing rule for the attraction parameter of the EoS, and the following Debye–H9uckel activity coe;cient expression for the
second term in the right-hand side of Eq. (1):
Ln DH
= 2A his Mm f(BI 1=2 )=B3 ;
i
(2)
where Mm is the salt-free mixture molecular weight, I is the
ionic strength, and his may be interpreted as an interaction
coe;cient between the dissolved salt and a non-electrolyte
component (gas and water), or a tuning parameter. They
assumed his to be independent of temperature, composition
and ionic strength of the solution. The function f(z) is obtained from:
f(z) = 1 + z − 1=(1 + z) − 2 ln(1 + z):
(3)
The parameters A and B are given by
∗
Corresponding author. Tel.: +44-131-451-3672;
fax: +44-131-451-3127.
E-mail address: bahman.tohidi@pet.hw.ac.uk (B. Tohidi).
0:5
=(m T )1:5 ;
A = 1:327757 × 105 dm
(4)
0:5
=(m T )0:5 ;
B = 6:359696 × dm
(5)
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0009-2509(03)00007-1
1744
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
where m is the salt free mixture dielectric constant and dm
is salt free mixture density (kg=m3 ).
Tohidi, Danesh, Burgass, & Todd (1995) modi4ed the
above model to predict the solubility of gases in aqueous electrolyte solutions at high- and low-pressure conditions. In their model, Valderrama–Patel–Teja (VPT) EoS
(Valderrama, 1990) with a non-density dependent (NDD)
mixing rule (Avlonitis, Danesh, & Todd, 1994) was employed for the 4rst term in the right-hand side of Eq. (1),
and Eq. (2) was used for the second term with some modi4cations. The main modi4cations were to express the his as a
function of salt concentration and temperature of the system.
The same EoS and mixing rules have been used in this
work. The VPT EoS has the following form:
P=
a
RT
−
;
v − b v2 + (b + c)v − bc
(6)
where a = ac . For pure component the four parameters of
Eq. (6) are obtained as follows:
ac = a
R2 TC2
;
PC
(7)
b = b
RTC
;
PC
(8)
c = c
RTC
;
PC
(9)
= (1 + m(1 −
Tr0:5 ))2 ;
(10)
where the parameters in the Eqs. (7)–(10) are calculated by
the following equations:
a = 0:66121 − 0:76105ZC ;
(11)
b = 0:02207 + 0:20868ZC ;
(12)
c = 0:57765 − 1:87080ZC ;
(13)
m = 0:462823 + 3:5830!ZC + 8:1817(!ZC )2 ;
(14)
where ! and ZC are acentric factor and critical compressibility factor, respectively.
The NDD mixing rule was developed for asymmetric mixtures to calculate the mixture parameters (a; b; c). In this
mixing rule, attraction parameter (a) has been separated into
two parts, classical mixing rule part (aC ) and asymmetric
contribution part (aA ), as follows:
a = aC + a A ;
C
a =
i
aA =
p
api =
√
j
xp2
(15)
0:5
xi xj (ai aj ) (1 − kij );
xi api lpi ;
(16)
(17)
i
ap ai ;
(18)
where kij is the classical binary interaction parameter, p
stands for polar component and lpi is the binary interaction
parameter between the polar component and the other components, which is a function of temperature, calculated by
the following expression:
lpi = l0pi − llpi (T − T0 );
(19)
where l0pi and llpi are binary interaction parameters and T0
is the ice point in K. The other parameters of EoS (b; c) are
calculated through the classical mixing rule as follows:
b=
xi bi ;
(20)
i
c=
x i ci ;
(21)
i
where xi is the mole fraction of component i.
Zuo and Guo (1991) presented their model for predicting gas solubility in saline solution under both high- and
low-pressure conditions. In their model ions/salts in solutions are considered as components in the EoS. They suggested the following equation for calculating fugacity coef4cient of each component:
Ln ’i = Ln ’Eos
+ Ln ’DH
i
i :
(22)
For the 4rst term in the above equation they employed Patel–
Teja EoS (Patel & Teja, 1982), and extended its application
to ions (cations and anions) in solution by de4nition of a,
b, and c parameters for them as follows:
a = 2:57012&'Na2 )3 f;
(23)
c = b = (2=3)&Na )3 ;
(24)
where Na is Avogadro’s number, ) is ionic diameter, f is
an empirical constant equal to 6, ' is ionic energy parameter
estimated from dispersion theory (Mavroyannis & Stephen,
1962):
' = 2:2789 × 10−8 0:5 *1:5 )−6 k;
(25)
where is number of electrons in an ion, * is polarizability
of an ion, and k is Boltzmann’s constant.
Kurihara, Tochigi, and Kojima (1987) mixing rule for
energy parameter a was used along with van der Waals
mixing rule. The second term of Eq. (22) is a Debye–H9uckel
expression using fully dissociated salt as standard state (Li
& Pitzer, 1986):
ln +DH
= −A
i
2 0:5 2
1 + BI 0:5
2Zi
I Zi − 2I 1:5
√
ln
×
;
+
B
1 + BI 0:5
1 + B= 2
(26)
where
I = 0:5,xi Zi2 ;
(27)
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
0:5 2 1:5
2&Na do
e
;
Ms
DkT
0:5
do
;
B = 2150
DT
A=
1
3
1745
(28)
in the original approach, has been studied and it appeared
to be negligible.
Therefore, the following equation is applied in this model:
(29)
Ln ’i = Ln ’EoS
i ;
where D, e, Z, Ms and do denote dielectric constant, electronic charge, number of charges, molecular weight of solvent, and solvent density, respectively.
2. The modied model
The main objective of this work is to develop a model
capable of predicting the phase behaviour of aqueous electrolyte solutions as well as gas solubility in these systems
over a wide range of pressure, temperature and salt concentrations. An EoS based approach has been adopted for
modelling all 5uid phases, including the water-rich phase.
VPT EoS, together with a NDD mixing rule is used in the
modi4ed model. Salts are treated as components in the EoS,
along with other components in the system. Eqs. (23) and
(24) are used to calculate the a, b, and c parameters in the
EoS for individual ions. The following relations are suggested for calculating the a, b, and c for the resulting salts:
√
as = (aa ac );
(30)
bs = (ba + bc )=2;
(31)
cs = (ca + cc )=2;
(32)
where subscripts s; a, and c denote salts, anions, and cations,
respectively.
By including salts in the VPT EoS with the NDD mixing
rules, the Debye–H9uckel electrostatic contribution term was
eliminated from Eq. (22). This resulted in a simple and 5exible approach for modelling aqueous electrolyte solutions.
It should be noted that the eEect of Debye–H9uckel term,
(33)
where i could be each component in the system, including salt (i.e., fugacity of salt in the aqueous phase can
be calculated from the above equation). Binary interaction
parameters (BIPs) between salt and water are optimised using water freezing point depression and boiling point elevation. A similar approach is used to optimise salt–salt BIPs.
In addition, gas solubility data in aqueous electrolyte solutions have been used in the optimisation of gas–salt BIPs.
Table 1 shows all BIPs of water–salt, salt–salt, and gas–salt
for the NDD mixing rule obtained through the optimisation
process, using the VPT EoS.
3. Results and discussion
3.1. Phase behaviour of single and mixed electrolyte
aqueous solutions
The new simpli4ed approach has been applied to modelling NaCl and KCl aqueous solutions. The resulting model
is capable of predicting the phase behaviour of aqueous electrolyte solutions, as detailed below:
Initially water–salt binary interaction parameters were
optimised, using freezing point depression and boiling
point elevation of their aqueous solutions. This provided a reliable water–salt phase behaviour model over
a wide temperature range (i.e., −25◦ C to 125◦ C). Fig. 1
presents the experimental (CRC, 1989) and calculated
freezing point of NaCl aqueous solutions. Fig. 2 presents
the vapour pressure lowering of NaCl and KCl aqueous
solutions at 353.15, 373.15, and 273:15 K. It should be
noted that the vapour pressure lowering experimental data
(from International critical tables (Washburn, 1926–1930))
has not been used in optimising the BIPs, so they could be
Table 1
BIPs for the VPT EoS and NDD mixing rule
Kij
H2 O
H2 O
CH4
CO2
NaCl
KCl
—
0.5058
0.2659
0.3791
−0:2386
l0ij
H2 O
NaCl
KCl
—
2.62
0.4476
l1ij 1E + 4
H2 O
NaCl
KCl
—
135.615
21.679
CH4
CO2
0.5058
—
0
0.8738
1.482
1.818
103.438
146.357
49
14670.5
68393.4
0.2659
0
—
0.206
—
0.81386
−24:68
—
18.3228
−12:598
—
NaCl
KCl
0.3213
0.8738
0.206
—
−1:737
−0:2836
1.482
—
−1:377
—
1.951
—
−9:374
−0:0957
−4:278
—
−9:468
—
2493.2
3.951
−1117:1
—
1746
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
273.15
Freezing point temp./ K
EXP, CRC 1989
Prediction
268.15
263.15
258.15
253.15
248.15
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Mole fraction of NaCl
Fig. 1. Experimental and calculated freezing point of NaCl aqueous solutions (0 –24:3 mass% NaCl).
Fig. 2. Experimental and predicted vapour pressure lowering of NaCl and KCl aqueous solutions at various temperatures (up to 36 mass%).
regarded as independent data. Excellent agreement between
experimental data and model predictions are observed.
To model mixed electrolyte aqueous solutions, experimental data on boiling point elevation (Washburn,
1926–1930) and freezing point depression (Hall, Sterner,
& Bodnar, 1988) of NaCl and KCl aqueous solutions have
been used to optimise the BIPs between NaCl and KCl. Fig.
3 shows the experimental and predicted water freezing point
temperatures in the presence of NaCl and KCl at 0.6 NaCl
mass fraction [NaCl=(NaCl + KCl)]. The predictions are in
excellent agreement with the experimental data, considering that only the freezing point depression data at 0.2 and
0.8 NaCl mass fractions have been used in the optimisation
process.
3.2. Gas solubility in single and mixed electrolyte
aqueous solutions
The gas solubility data in aqueous electrolyte solutions
has been used for optimising gas–salt binary interaction
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
1747
273.15
EXP, Hall et al., 1988
Prediction
271.15
269.15
T/K
267.15
265.15
263.15
261.15
NaCl / (NaCl+KCl)=0.6
259.15
257.15
255.15
0
5
10
15
20
25
(KCl+NaCl) / wt%
Fig. 3. Experimental and predicted water freezing point temperature in the presence of NaCl and KCl.
45
EXP, O'Sullivan & Smith 1970, 1M NaCl
Mole% of CH4 in Liq.*1E+4
40
EXP, 4 M NaCl
35
EXP, Pure water
30
Predictions
25
20
15
10
5
0
0
10
20
30
40
50
60
70
P / MPa
Fig. 4. Experimental and calculated methane solubility in pure water and NaCl aqueous solutions at 324:65 K.
parameters. Fig. 4 presents the experimental data
(Takenouchi & Kennedy, 1965) and the calculated methane
solubility in pure water and NaCl aqueous solutions.
Fig. 5 presents the experimental data (O’Sullivan & Smith,
1970) and the calculated carbon dioxide solubility in pure
water and NaCl aqueous solutions. The agreement between
experimental data and model predictions is very good.
Limited data is available on the phase behaviour of
mixed electrolyte systems. Independent gas solubility data
has been used in validating the developed thermodynamic
model. Table 2 shows the only available experimental data
(Byrne & Stoessell, 1982) on CH4 solubility in NaCl/KCl/
water ternary system. The thermodynamic model prediction
is also presented in the table, demonstrating the reliability
of the developed model.
4. Conclusions
The previous approaches (Zuo & Guo, 1991; Tohidi
et al., 1995) in modelling phase equilibria in the presence
of aqueous electrolyte solutions have been improved by
introducing salts as components in the equation of state
1748
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
Fig. 5. Experimental and calculated carbon dioxide solubility in pure water and NaCl aqueous solutions at 423:15 K.
Table 2
Experimental and predicted CH4 solubility in NaCl/KCl/water system
NaCl (mass%)
KCl (mass%)
T (K)
P (MPa)
Experimental solubility
(mole%)
Predicted solubility
(mole%)
Error
(%)
5.158
6.58
298.15
3.795
0.051
0.045
11.76
(EoS) and eliminating the Debye–H9uckel electrostatic contribution term in the fugacity coe;cient calculations. In this
work the Valderamma modi4cation of Patel and Teja EoS
with a non-density dependent (NDD) mixing rule has been
used. Water–salt and gas–salt binary interaction parameters
are calculated from literature data.
The approach, which could be applied to other EoS, provides a much simpler method for predicting phase equilibria
in the presence of aqueous electrolyte solutions, at a wide
temperature and pressure ranges. The predicted water freezing point depression, boiling point elevation, and gas solubility in aqueous electrolyte solutions are compared with
independent literature data, demonstrating the reliability of
the simpli4ed model.
Notation
aA
aC
ac
a; b; c
dm
contribution to the attractive term using
asymmetric mixing rules (Eqs. (15) and
(17))
contribution to the attractive term using classical mixing rules (Eqs. (15) and (16))
EoS attractive term at critical point
parameters of EoS
salt free mixture density (kg=m3 )
do
D
e
f
I
k
Mm
Ms
Na
Tr
Z
solvent density
dielectric constant
electronic charge
empirical constant equal to 6
ionic strength
Boltzmann constant
salt-free mixture molecular weights
molecular weight of solvent
Avogadro number
reduced temperature
number of charges (Eqs. (26) and (27))
Greek letters
i
'
m
)
’i
*
activity coe;cient of component i
ionic energy parameter
number of electrons in an ion (Eq. (25))
salt free mixture dielectric constant
ionic diameter
fugacity coe;cient of component i
polarizability of an ion
Acknowledgements
Mr Rahim Masoudi wishes to thank the National Iranian
Oil Company (NIOC) for the 4nancial support.
R. Masoudi et al. / Chemical Engineering Science 58 (2003) 1743 – 1749
References
Aasberg-Petersen, K., Stenby, E., & Fredenslund, A. (1991).
Prediction of high-pressure gas solubilities in aqueous mixtures of
electrolytes. Industrial & Engineering Chemistry Research, 30(9),
2180–2185.
Adachi, Y., Lu, B. C.-Y., & Sugie, H. (1983). A four-parameter equation
of state. Fluid Phase Equilibria, 11, 29.
Avlonitis, D., Danesh, A., & Todd, A. C. (1994). Prediction of VL and
VLL equilibria of mixtures containing petroleum reservoir 5uids and
methanol with a cubic EoS. Fluid Phase Equilibria, 94, 181–216.
Byrne, P. A., & Stoessell, R. K. (1982). Methane solubilities in multi-salt
solutions. Geochimica et Cosmochimica Acta, 46, 2395–2397.
CRC Hand Book of Chemistry and Physics (1989). Boca Raton, FL:
CRC Press Inc.
Hall, D. L., Sterner, M., & Bodnar, R. J. (1988). Freezing point depression
of NaCl–KCl–H2 O solutions. Economic Geology, 83, 197–202.
Kurihara, K., Tochigi, K., & Kojima, K. (1987). Mixing rule containing
regular solution and residual excess free energy. Journal of Chemical
Engineering of Japan, 20, 227–231.
Li, Y.-G., & Pitzer, K. S. (1986). Thermodynamics of aqueous
sodium chloride solutions at high temperatures and pressures (I):
thermodynamic properties over 373–573 K and 0.1–100 MPa. Journal
of Chemical Industry and Engineering (China), 1, 40–50.
1749
Mavroyannis, C., & Stephen, M. J. (1962). Dispersion forces. Molecular
Physics, 5, 629–638.
O’Sullivan, T. D., & Smith, N. O. (1970). The solubility of partial molar
volume of nitrogen and methane in water and in aqueous sodium
chloride from 50 to 150◦ C and 100 to 600 atm. Journal of Physical
Chemistry, 74(7), 1460–1466.
Patel, N. C., & Teja, A. S. (1982). A new cubic equation of state for
5uids and 5uid mixtures. Chemical Engineering Science, 46, 463–473.
Takenouchi, S., & Kennedy, G. (1965). The solubility of carbon dioxide in
NaCl solutions at high temperatures and pressures. American Journal
of Science, 263, 445–454.
Tohidi, B., Danesh, A., Burgass, R. W., & Todd, A. C. (1995).
Gas solubility in saline water and its eEect on hydrate equilibria.
Proceedings of the ;fth international o<shore and polar engineering
conference. The Hague, The Netherlands.
Valderrama, J. O. (1990). A generalized Patel–Teja equation of state for
polar and nonpolar 5uids and their mixtures. Journal of Chemical
Engineering of Japan, 23(1), 87–91.
Washburn (1926 –1930). International critical tables (ICT ) of numerical
data, physics, chemistry and technology. National Research
council.
Zuo, Y., & Guo, T. (1991). Extension of the Patel–Teja equation of state
to the prediction of the solubility of the natural gas in formation water.
Chemical Engineering Science, 46(12), 3251–3258.