6182
J. Phys. Chem. 1995, 99, 6182-6193
Water-Salt Phase Equilibria at Elevated Temperatures and Pressures: Model Development and Mixture Predictions Ioannis G. Economou,t Cor J. Peters,* and Jakob de Swaan Arons Del$ University of Technology, Faculty of Chemical Engineering and Materials Science, Laboratory of Applied Thermodynamics and Phase Equilibria, Julianalaan 136, 2628 BL Dew, The Netherlands Received: May 18, 1994; In Final Form: November 30, 1994@
A model is developed to predict the water-salt phase equilibria at elevated temperatures and pressures based on the associated-perturbed-anisotropic-chain theory (APACT). At high temperatures, salts in aqueous solutions exist in the form of ion pairs, unlike at ambient temperatures, where they strongly dissociate. Therefore, a molecular approach can be used to model their thermodynamic behavior. APACT accounts explicitly for the strong dipole-dipole interactions between water and salt molecules. Molecular salts have a dipole moment that is on average 5 times higher than the water dipole moment. In addition, a chemical equilibrium model is used to describe the hydration of salt molecules by water molecules and the water self-association. The salt parameters for APACT are obtained on the basis of molecular properties (crystal ionic radius, molecular polarizability, enthalpy and entropy of hydration of corresponding ions). The model is applied to predict the vapor-liquid equilibria of binary water-salt mixtures for 10 alkali halide salts with very good accuracy in the temperature range 150-500 "C. In addition, it predicts accurately the equilibrium pressure of the solidliquid-vapor equilibria of three binary water-salt mixtures in the temperature range 200-450 "C. In general, the model is shown to be most accurate in the temperature range 300-420 "C. Below 200 "C, when salt dissociation becomes dominant, the model seems to be less accurate.
Introduction Thermodynamic properties and phase equilibria of aqueous salt mixtures at high temperatures and pressures are crucial in many areas of science and technology such as geology and geochemistry, chemical engineering, and nuclear engineering. The experimental determination of these properties is quite cumbersome, and thus far only for a few salts is such information available. Therefore, there is a need for thermodynamic models able to predict the phase behavior of watersalt mixtures over a wide range of temperature, pressure, and composition. The phase behavior of aqueous salt mixtures is well-known at ambient conditions, and there is a large variety of equationsof-state and activity-coefficient models able to describe it accurately.'-5 However, as temperature increases, the accuracy of these models becomes doubtful, and for conditions close to and above the water critical point these models fail. At ambient conditions, salts strongly dissociate, and therefore only single ions are found in the mixture. The strong ionic forces are dominant and have to be taken into account accurately. As temperature increases, the dielectric constant of water decreases For example, at a constant pressure of about 250 bar, the dielectric constant of water decreases by almost 2 orders of magnitude from 25 to 500 "C. As a result, ions in aqueous environment tend to associate, and at temperatures above a few hundred degrees Celsius salts exist as ion pairs having a large dipole m ~ m e n t . ~Such . ~ a transition in the molecular behavior of salt mixtures affects the phase behavior considerably and should be accounted for properly by a thermodynamic model. The high-temperature and -pressure phase behavior of watersalt systems is less accurately known than the phase behavior
* To whom all correspondence
should be addressed. Exxon Research and Engineering Company, PO Box 101, Florham Park, NJ 07932. Abstract published in Advance ACS Abstracts, March 1, 1995.
' Currently: @
0022-365419512099-6182$09.0010
at ambient conditions. Most of the experimental work has been devoted to the water-NaC1 mixture whose phase behavior is known accurately up to 500 "C. Bischoff and Pitzer'O critically evaluated all the available experimental data for this mixture in the temperature range 300-500 "C. Much less is known for the water-KC1 mixture1'-I3 and even less for other watersalt mixtures. Considerable effort has been devoted by Russian scientists to understanding the global phase behavior of aqueous salt systems,I4-l6 and equilibrium pressure data for water-salt mixtures at high temperatures have been reported in the literature.' 1 , 1 7 - 2 2 On the other hand, molecular simulation has proven to be a useful tool in understanding the molecular structure of the fluid mixtures as a function of temperature and pressure, and it provides the basis for developing thermodynamic models. Kalinichev and H e i n ~ i n g e rpresented ~~ a critical review of the molecular simulation studies of aqueous salt systems at high temperatures and pressures. Recently, Cummings and coworkers reported molecular dynamics calculations of the local structure of a water-NaC1 mixture at supercritical condition^^^.^^ and Monte Carlo results for the VLE of the same mixture at 100 and 200 "C and of a water-methanol-NaC1 mixture between 70 and 90 0C.26 An accurate thermodynamic model (for example, an equation of state) is developed on the basis of the information of the microstructure of the fluid obtained from molecular simulation and the macroscopic thermodynamic properties measured experimentally. Pitzer and c o - w o r k e r ~ ~have ~ ~ ~developed ~,~* various models to describe the phase behavior and volumetric properties of the water-NaC1 mixture at temperatures above 300 "C. These models are accurate over the range of conditions for which they were developed. However, a large number of parameters need to be regressed from experimental mixture data, and as a result, these models are inappropriate for systems where limited experimental data are available. Similarly, the equation of state of Lvov and Wood29is very accurate for water-NaC1
0 1995 American Chemical Society
Water-Salt Phase Equilibria
J. Phys. Chem., Vol. 99, No. 16, 1995 6183 TABLE 1: Crystal Ionic Radii and Ionic Dispersion Energy Parameters* Li+ Na+
0.70 0.95 1.33 1.36 1.81 1.95 2.20
K+
F-
c1Br-
I-
(H20)i+ H20 (I-120)i+, c=)
(A'B-)
-
+ i(€-I2O)
i=1,2, ...
(A+B-)-(H20)i i=1,2, ...
Figure 1. Schematic picture of the molecular model used in APACT for the alkali halide salts and the chemical equilibria that describe water self-associationand salt hydration. from 273 K up to 973 K but requires 10 adjustable parameters. Harvey30presented a qualitative model for the critical behavior of water-salt mixtures based on the mean spherical approximation. In order to develop an accurate model for water-salt phase behavior, one has to account for all different molecular interactions exhibited between the constituent components. At high temperatures, salt molecules exist in the form of ion pairs having a large dipole moment. In addition, hydration of salt molecules by water molecules is quite strong and extended over many coordination shell^.^^.^^ Molecular simulation showed that water self-association might exist even in the supercritical r e g i ~ n , ~although ~ . ~ ' recent experimental data seem to disagree with this result.32 In this work, a model is developed based on the associated-perturbed-anisotropic-chaintheory (APACT) for the water-salt phase equilibria at elevated temperatures and pressures. APACT33*34 is an equation of state that accounts explicitly for dispersion, polar, and hydrogen-bonding interactions, based on statistical mechanics and physical considerations. It was proven to be accurate in predicting the phase behavior of water with aliphatic and aromatic hydrocarbons over a wide range of t e m p e r a t ~ r e . In ~ ~APACT, the salt molecule is considered as a molecular species having a strong dipole moment that can be up to 5 times larger than that of water depending on the particular salt. In addition, salt hydration and water self-association are modeled using the chemical equilibrium scheme of Campbell et al.35 The model is used to predict with very good accuracy the vapor-liquid equilibria of watersalt mixtures for 10 different alkali halides over the temperature range 150-500 "C. In addition, the equilibrium pressure of the solid-liquid-vapor equilibria for three water-salt mixtures is predicted accurately. Model Development APACT is an equation of state developed specifically for molecules that differ in size, shape, and energetic interactions. It is based on perturbation theory and accounts explicitly for repulsive, attractive, and specific interactions such as hydrogen bonding. The equation is written in terms of the compressibility factor 2 as
where zdssoc,ZeP, and Pttr are the contributions from the association, repulsions, and attractions, respectively. Equation 1 is applied to both pure components and multicomponent mixtures, and the various terms of the equation are given in
96.8 147.4 221.4 175.3 225.4 330.8 380.6
TABLE 2: APACT Molecular Parameters for the Salts salt r* (K) v* (cm3/mol) elk (K) KF 663.9 66.320 197.0 LiCl 434.2 147.7 53.878 NaCl 647.2 7 1.633 182.3 KCl 1025.4 105.481 223.4 LiBr 585.0 63.405 178.9 NaBr 865.5 83.096 220.8 KBr 1355.7 120.229 270.6 LiI 752.2 83.096 191.9 NaI 1094.5 106.493 236.9 KI 1686.6 149.869 290.3 TABLE 3: E~perimental~~ and APACT-Correlated Dipole Moment, Experimental Average Molecular Polarizability,"l and Enthalpy and Entropf2 of Hydration of the Salts KF LiCl NaCl KCl LiBr NaBr KBr LiI NaI KI
8.59 7.13 9.00 10.27 7.27 9.12 10.41 7.43 9.24 10.55
6.50 5.00 7.00 8.60 . 5.00 7.00 8.60 5.00 7.00 8.60
21.0 13.1 23.4 32.1 18.9 26.8 42.0 23.4 26.9 36.3
46.1 46.1 35.6 38.5 50.2 35.6 38.5 52.3 37.7 39.8
12.0 12.0 11.0 11.0 12.5 11.5 11.5 11.5 10.5 10.5
the Appendix for the case of a multicomponent mixture. The repulsive term of the equation is based on the hard-sphere equation of Carnahan and Starling36multiplied by the shape parameter c which accounts for the nonsphericity of the molecules. Attractive interactions are calculated as a perturbation over the reference fluid. Two perturbation schemes are used for the isotropic (Lennard-Jones) and anisotropic (polar) interactions. In the first case, perturbation is performed over the repulsive reference fluid, while in the second case the reference fluid is the Lennard-Jones fluid. Hydrogen-bonding interactions are calculated on the basis of chemical theory which assumes that oligomers are formed. The distribution of the resulting species is calculated as a function of temperature, density, and composition by solving the material balances. In the case of water, different equilibrium schemes have been proposed and used in APACT,34on the basis of the number of active hydrogen-bonding sites per molecule. Although the threesite model is physically more realistic than the two-site model, for practical calculations it is not clear which of the two models is more accurate. In this work, the two-site model is used for water self-associationand combined with a solvation model for salt hydration. The two-site model was chosen because the combined two-site self-association and hydration model gives a closed-form expression for the material balances that have to be solved. The threelsite model results in equations that are solved by an iterative procedure.37 The water self-association and salt hydration are expressed by the following chemical equilibria:
(H20)i
+ H 2 0 == (H20)i+,
i = 1,2, ...
(2)
Economou et al.
6184 J, Phys. Chem., Vol. 99, No. 16, 1995
230 I
I
I
I
lr-.----'-
1
- I
v 300oC
+ *
3100C 320oC 330oC A 340oC 0 350oC 360oC e.__ 3700~ __
21 0 190
-
+
170 150 130 110 90 70
50
0
I
1
1
I
10
20
30
40
1 50
wt 70NaCl .__.
-. ...
D 330oC
A
340OC
210
9 350oC
-I- 360oC 190 e 3700~
-;r 2 2
a
2
170 150 130 110
90 70 10-5
10-6
104
103
10.2
10-1
100
101
102
wt % NaCl Figure 2. Water-NaC1 VLE from 300 to 370 "C, experimental datal0 (various symbols) and APACT predictions (lines): (a) bubble pressure
curves; (b) bubble and dew pressure curves. (A+B-)
+ ~(H,o) == ( A + B - ~ ( H ~ o ) ~
i = i , 2 , ...
(3)
where (A+B-) denotes the salt molecule, with A+ being the cation and B- the anion. The equations that describe the water self-association and salt hydration35are given in the Appendix. The equilibrium constant for the chemical equilibrium is expressed according to
AH" AS"
hK=--+-
RT
R
(4)
where AH" is the standard enthalpy and AS" is the standard entropy of association. In APACT, the equilibrium constant is independent of the size of the water oligomers formed (eq 2) or the size of hydration shell (eq 3). This reduces the number of parameters to be regressed, without losing accuracy. APACT is a five-parameter equation of state. These param-
eters are the characteristic energy parameter P, the characteristic size parameter v*, the shape factor c, and the two variables for hydrogen bonding, AIP and AS". In general, these parameters are calculated by fittin.g the equation of state simultaneously to experimental liquid density and vapor pressure data for the pure component. Using this approach, APACT correlates subcritical liquid density and vapor pressure data within l%.34This work focuses mainly on the critical region of water, and therefore accurate representations of the water critical properties are necessary as well. As a result, a different procedure is used to calculate the APACT parameters for water. Initially, parameters are calculated by fitting subcritical data (liquid densities and vapor pressures). Later on, the energy and size parameters (P and v*) are readjusted in order to fit the water critical point accurately. This way, P becomes temperature dependent in order to correlate accurately subcritical and critical water
J. Phys. Chem., Vol. 99, No. 16, 1995 6185
Water-Salt Phase Equilibria
100
'
'
1
10-2
1
6
,
, 1 1 # / 1
10-1
2
2
I
f
1
I
,I
/ , I 1
100
2
I
101
3
2
, , , I ,
3
wt % NaCl Figure 3. Water-NaC1 VLE at 380, 400, 450, and 500 "C, experimental datal0 (various symbols) and APACT predictions (lines). 600
I
I
I
'
I
I
' -
e
500 h
tii
n
Y
$
400
u) u)
?! n
300
200 360
I
1
1
1
I
1
I
380
400
420
440
460
480
500
Temperature
(C)
Figure 4. Water-NaC1 critical curve from water critical point (374 "C) up to 500 "C, experimental datal0 (symbols) and APACT predictions (line). properties. Finally, the following parameters are obtained for water:
F = (200.527
+ 0.3157)T (3.6 x low4)?
(Pand T i n K)
v* = 14.83 cm3/mol, c = 1,
AlP = -20.79 kJ/mol, AS'IR = -10.54 This set of parameters together with the experimental dipole moment (1.85 D), average molecular polarizability (1.59 A3), and dispersion energy parameter (elk = 350 K) correlate the water critical temperature and pressure to better than 1% and the vapor pressure from the triple point up to the critical point to better than 0.2%. The experimental data for liquid density and vapor pressure of the salt are very limited and out of the range of interest. For example, the NaCl melting point is 1074 K,38 and the only
available experimental vapor pressures are above 1250 K.39For other salts, the available experimental data are even less. As a result, a different procedure must be used to calculate the APACT parameters for salts. In this work, the molecular parameters of salts are obtained based on the parameters for the constituent ions using proper combining rules, experimental data for ion hydration, and physically realistic assumptions. Jin and Donohue2 evaluated the parameters for several ions for the perturbed-anisotropic-chaintheory (PACT). These parameters form the basis for calculating the salt parameters. It is assumed that the salt molecule (A+B-) is a spherical molecule whose radius is the sum of the crystal radii of its constituent ions:
+
(5) rAB= rA+ r,In Figure 1, a schematic representation is shown of the molecular model used in this work for the salts. It is obvious that the size of the salt molecule is larger than the sum of the sizes of
6186 J. Phys. Chem., Vol. 99, No. 16, 1995
220 1
Economou et al. I
I
I
I
I
1
I
200 180 160 n L
m
140
0,
120
9 L
1 u) u)
100
n
80
2
60 SLE
40 20 0 ‘ 0
I
1
10
20
I
1
1
1
40
50
60
70
I
30
wt % KCI Figure 5. Water-KC1 VLE from 200 to 370 OC, experimental bubble pressure data (symbols at 200-350 OC are taken from Wood et al.;’* open squares at 370 “C are taken from Khaibullin and Borisov;” SLVE data are taken from Hovey et al.I3) and APACT predictions (lines).
500
400 n L-
m
e
5
300
u) u)
2
n
200
100
1 0-1 100 wt % KCI
10-2
10-3
10’
102
Figure 6. Water-KC1 VLE at 380,410, and 450 “C. Experimental data (solid symbols are taken from Hovey et al.;I3 open squares are taken from Khailbullin and Borisovl’) and APACT predictions (lines).
the constituent ions. In Table 1, the crystal ionic radii for the salts examined in this work are given.2 The characteristic size parameter V*AB is calculated from the expression
where NA” is Avogadro’s number. The normalized extemal surface area per molecule, q A B , is calculated from qAB
= ‘AB2IrCH,
2
energy parameter for ions, elk, is calculated from the expression2
-‘ionk_ - 3
3/2 1/2 aim %,ion
5
6
7
‘ion
where ion can be either A+ or B-, aionis the average polarizability of the ion, ne,ion is the number of electrons per ion, and ‘ion is the crystal ionic radius. In Table 1, the elk parameter for the ions examined in this work is given together with the crystal ionic radius. The dispersion energy parameter, eABlk, of salts is calculated from the combining rule
(7)
where rCH2 = 1.465 8, is the radius of a -CH2- segment that ~ . ~dispersion ’ is assumed to have a q value equal to ~ n i t y . ~ The
(9) Finally, the characteristic energy parameter PAB is evaluated
J. Phys. Chem., Vol. 99,No. 16, 1995 6187
Water-Salt Phase Equilibria
60
L
40 20
00
I
300oC
80
60 40
20 0 0
10
20
30 40 wt % LiCl
50
60
70
Figure 7. Water-LiC1 VLE from 150 to 350 "C, experimental bubble pressure data (solid symbols, data taken from Wood et al.;Izopen symbols, data taken from Ravich and Yastrebova") and APACT predictions (lines).
Erom the expression
Results and Discussion
where c = 1 for all salts according to the spherical model assumed (Figure 1). In Table 2, the APACT molecular parameters for the salts examined in this work are summarized. APACT accounts explicitly for polar and induced polar interactions, and the corresponding expressions are given in the Appendix. These expressions are a function of the dipole moment and the average molecular polarizability of the components. For salts, the dipole moment in the gas phase is known e~perimentally.~~ In the present work, the salt dipole moment was adjusted in order to obtain best agreement between experimental data and model correlations. Vapor phase compositions are most sensitive to the salt dipole moment value. The correlated value is approximately equal to 3/4 of the experimental value. A similar approach was used by Anderko and Pitzer28for their water-NaC1 model. In this work, the same dipole moment is used for all salts containing the same cation. In Table 3, the experimental and calculated dipole moments for the different salts are given. For the induced polar interactions, the experimental average molecular polarizability for the salts is used (Table 3). Salt hydration by water molecules is calculated on the basis of eq 3. The equilibrium constant (eq 4) is a function of the standard enthalpy and entropy of hydration. Following Pitzer and Pabalan?? the entropy of hydration of salts was evaluated on the basis of the spectroscopically measured entropy of hydration of distinct ions.43 However, the enthalpy of hydration of salts is expected to be smaller than the enthalpy of hydration of ions, and this value is regressed from the experimental phase equilibria data examined in this work. In Table 3, the enthalpy and entropy of hydration for the various salts used in APACT are given. The W value for the NaCl hydration is close to the average AZP value of the successive hydration model of Pitzer and In addition, the equilibrium constant for salt hydration is higher than the equilibrium constant for water self-association, and therefore the local density enhancement around a salt molecule is higher than around a water molecule, which is in accordance with molecular simulation result^.^^-^^
The APACT model is applied to predict the fluid phase equilibria of several water-salt binary mixtures at temperatures above 150 "C and, in general, up to 500 "C. For the systems where experimental data are available comparison is made with the theoretical predictions. Water-NaC1 phase equilibria at high temperature and pressure are very important for geology, geochemistry, and chemical technology, and as a result, this is the most well documented system over a wide range of condition^.^-'^^^^^^-^^ A lot of attention has been devoted also to the critical behavior of this m i ~ t u r e . ~ Bischoff ~ , ~ ~ , and ~~ Pitzer'O critically evaluated all the available experimental data for the water-NaC1 mixture in the temperature range 300500 "C and provided detailed analysis of the vapor-liquid (VLE) and solid-liquid-vapor equilibria (SLVE) and critical behavior of the mixture. In this work, bubble pressure type of calculations were performed; that is, for a given temperature and composition of the dense phase, the pressure and composition of the light phase is calculated. In addition, at temperatures above the water critical point, flash calculations were performed (given temperature and pressure, composition of the two phases was calculated). APACT predictions are compared with the Bischoff and Pitzer data for both the VLE and the critical behavior. Comparisons are shown in Figures 2-4. In Figure 2a, bubble pressure isotherms in the temperature range 300-370 "C are presented as a function of the NaCl composition. APACT is in very good agreement with experimental data from pure water vapor pressure up to the halite-saturated line. The overall percentage average absolute deviation (%AAD)between the experimental data and APACT predictions for the bubble pressures is 1.19% (Table 4). In addition, the %AAD between APACT and experimental data for the equilibrium pressure of the SLVE is 1.90%. It should be pointed out here that, for the case of SLVE calculations, the solid phase is not accounted for explicitly in the calculations. Essentially, SLVE calculations reflect the boundary of the two-phase (LV) region. Both the liquid and vapor phase compositions at the same conditions are presented in Figure 2b. APACT is able to predict accurately the very low salt composition in the vapor phase that changes over almost 3 orders of magnitude from 300 "C up to the water
Economou et al.
6188 J. Phys. Chem., Vol. 99, No. 16, 1995
I
180
I
I
I
I
I
1
I
i
- Water - LiCl
160
-. Water
140
Water
120 100
/I
- NaCl - KCI
-I
300 OC
80 60
250 OC
40 20
0
I
I
I
I
I
I
10
20
30
40
50
60
I 70
wt % salt
160
c
-
140 h
. I
0
e. L
Water -Water
120 -
1. 100
\
- NaCl
- KCI
8
0.
I
'
O!
8!
\
\\
\ \
I
1 VI v)
80
L
n
60 \
40 20
0 1 wt % salt Figure 8. Water-LiC1, water-NaC1, and water-KC1 VLE at 250, 300, and 350 OC, experimental data (symbols) and APACT predictions (lines). (a) Bubble pressures. Data sources: water-LiC1, solid triangles, data taken from Wood et a1.,I2 open triangles, data taken from Ravich and Yastrebova;" water-NaC1; solid squares at 300 and 350 "C, data taken from Bischoff and Pitzer,Io solid squares at 250 "C, data taken from Wood et al.;l2 water-KC1, filled dots, data taken from Wood et al.l2 (b) Bubble and dew pressures. Data sources are the same as in (a) except the KCl vapor phase compositions (solid triangles) and the NaCl phase compositions at 250 "C (open diamonds) are taken from Khaibullin and Borisov."
critical temperature. At temperatures above the water critical point, APACT is in good agreement with the experimental data f o r the VLE up to 420 "C (Figure 3). At higher temperatures deviations are found between APACT predictions and experimental values for the vapor phase composition at low salt concentrations. Nevertheless, the overall agreement is quite satisfactory, given the fact that no parameter adjustment has been made in order to correlate these data. The water-NaC1 mixture has a continuous critical line starting from the water critical point at T = 374 "C and terminating at the NaCl critical point estimated at T = 3600 OC.lo The mixture critical curve locus is well-known up to 500 "C. The APACT-predicted critical curve is in good agreement with the experimental critical curve as shown in Figure 4. Bubble pressure isotherms for the water-KC1 mixture were measured by Wood et a1.I2for the temperature range 200-350
"C, but they provided no information on the vapor phase composition or for higher temperatures. Khaibullin and Borisov" reported both vapor and liquid phase compositions at temperatures up to 440 "C. Their data, however, are not very a c ~ u r a t e , ' ~and . * ~ in this work these data are treated only as qualitative information. In Figure 5, experimental data and APACT predictions are shown for bubble pressure isotherms for the water-KC1 mixture at temperatures below the water critical point. The agreement is very good, and in fact the %AAD between experimental data and APACT is 0.60% in the temperature range 200-350 "C. Limited experimental information is available for the supercritical water-KC1 mixture. In Figure 6, APACT predictions are shown for three different temperatures above the water critical point. Experimental data1','3 at 380 and 410 "C are shown for comparison. The data from Hovey et aLi3are over a small concentration range,
6189
\
300 l-
\
v
h
t
e L 3
200
v) v)
?!
n 100
0 0.
20
10
30
50
40 Wt
60
70
70KF
Figure 9. Water-KF VLE from 200 to 450 "C, experimental bubble pressure curves1*(symbols) and APACT predictions (lines).
180 I
I
I
I
I
160 140
-
*
120 100
II-
+
250 OC
0
10
20
30
40
50
wt % NaBr Figure 10. Water-NaBr VLE from 150 to 350 "C, experimental bubble pressure curves19(symbols) and APACT predictions (lines).
whereas the data of Khaibullin and Borisov" have a lot of scatter. From Figure 6 it can be concluded that APACT predicts accurately the mixture critical point at 380 "C, whereas at 410 OC the predicted critical point is higher than the experimental one. In Figure 7, bubble pressure isotherms are shown for waterLiCl in the temperature range 150-350 "C. APACT predictions are in good agreement with the experimental data over the entire temperature and composition range. In Figure 8a,b, comparison is made of the VLE of water-salt mixtures at different temperatures for salts containing the same anion (C1-) but different cations (namely, Li+, Na+, and K+). In Figure 8a, only the bubble pressure curves are shown, while in Figure 8b both liquid and vapor compositions are presented. Several conclusions can be drawn from these figures. At constant temperature and salt composition, the bubble pressure increases with the size of the salt cation. In addition, at constant temperature and pressure, salt composition in the vapor phase
increases as the size of the salt cation decreases. In this case, at constant temperature and salt composition the water-LiC1 mixture has the lowest bubble pressure and water-KC1 the highest, whereas at constant temperature and pressure the water-LiC1 mixture has the highest salt vapor composition and the water-KC1 the lowest. These trends are observed experimentally and predicted quantitatively by APACT. In fact, APACT gives reliable results even for systems where limited or no experimental data are available as for the case of waterLiCl vapor phase composition. In Figure 9, bubble pressure isotherms are presented for the water-KF mixture both below and above the water critical point. The agreement between APACT predictions and experimental data is very good even at high salt concentrations. Only at the highest temperature (450 "C) are the APACT predictions lower than the experimental data. Similarly, in Figure 10 experimental data and APACT predictions are shown for the water-NaBr VLE from 150 to 350 "C. APACT predictions
Economou et al.
6190 J. Phys. Chem., Vol. 99, No. Id, 1995 180 160 140 120 100 80 60 40 20 0 ' 0
I
I
1
I
1
1
I
10
20
30
40
50
60
70
wt % salt Figure 11. Water-KF, water-KC1, and water-KI VLE at 250,300, and 350 "C. The highest isotherm for water-KF is at 346.5 "C. Experimental bubble pressure curves (symbols) and APACT predictions (lines). Data sources: water-KF are from Urusova and Ravich; vater-KCl are from
Wood et al.;'* water-KI-are from Lvov et al.2i
160
I
1
I
I
I
0
140 120 h
Q
L
100
2
80
e 3
In
60
n 40
200
260
320
380
Temperature
440
500
(C)
Figure 12. Water-KF and water-KC1 SLVE, experimental data (Urusova and Ravich'* for KF; Hovey et al.I3 for KCl; symbols) and APACT
predictions (lines). are in best agreement with the data at low and intermediate salt compositions. In Figure 11, the VLE of water-salt mixtures for three salts having the same cation ( K') but different anions (F-, C1-, and I-) is shown at different temperatures. The effect of anion size in the equilibrium pressure is similar to the effect of cation size shown in Figure 8a. At constant temperature and salt composition, the mixture containing the salt with the largest anion (water-= in this case) exhibits the highest pressure. Again the quantitative agreement between experimental data and APACT predictions is very good. In Table 4,a summary of the water-salt mixtures examined in this work is shown together with a comparison with experimental data for the temperature range where such data are available in the literature. The %AAD between the experimental and APACT-predicted bubble pressures is reported. For the case of iodine salts (LiI, NaI, and KI), a small binary
parameter (eq A50) was used in the calculations with a constant value of kv = -0.08 for all cases. APACT calculations were most sensitive to a binary adjustable parameter for the mixtures where the salt dispersion energy parameter (elk) had a value close to the water value of 350 K. To this extent, water-IU mixture calculations showed the highest sensitivity to the introduction of a binary parameter. For mixtures for which the VLE is known well experimentally (NaCl and KC1 solutions) the agreement between experimental data and model predictions is excellent. For the mixtures where the %AAD is high (for example, higher than lo%), part of the deviation between experiments and theory should be attributed to the scatter or discrepancies of the experimental data themselves. For example, for the water-LiC1 mixture two different sets of data were examined (Figure 7). For the data of Wood et al.'* in the temperature range 150-350 "C, the %AAD is
J. Phys. Chem., Vol. 99, No. 16, 1995 6191
Water-Salt Phase Equilibria TABLE 4: Percentage Average Absolute Deviation (%AAD)between Experimental and APACT-Predicted Pressure for the VLE and SLVE of Water-Salt Mixture VLE T(OC)
%AAD
water-KF water-LiC1 water-NaC1 water-KC1 water-LiBr water-NaBr water-KBr water-LiI water-NaI water-KI
200-450 150-350 300-370 200-350 150-350 150-350 150-350 150-350 150-350 150-350
10.49 12.64 1.19 0.60 27.39 5.32 4.00 14.98 2.61 3.17
T(T)
%AAD
200-450
17.02
300-400 200-425
1.90 5.35
TABLE 5: Universal Constants for APACT AIm
-8.538 022 -5.276 135 3.730 389 -7.539 783 23.306949 -11.197068
APACT for multicomponent mixtures is given in terms of the compressibility factor 2 from the following e x p r e s ~ i o n : ~ ~ , ~ ~
SLVE
mixture
m 1 2 3 4 5 6
Appendix
CIm -3.938 1909 -3.192 6783 -4.929 9963 10.029 636
-
CZm
C3m
11.702 62 -3.091 5196 4.009 2149 -20.025 379
-37.018 96 26.927 112 26.672 754
0.0
10.35%, while for the data of Ravich and YastrebovaI7 in the temperature range 250-350 "C and %AAD is 18.07%. In addition, for a given mixture higher deviation between experiments and predictions is found at lower temperatures. For example, for the water-LiC1 and water-LiBr mixtures the %AAD at 150 "C is higher than the average value given in Table 4. This might be attributed to the fact that the model becomes less accurate as temperature decreases and ionic effects become important. At the same time, the salt vapor phase composition predicted from the model at low temperatures (in the region 150-250 "C depending on the salt) might be larger than the experimental composition, again due to the ionic effects. APACT was used to predict the equilibrium pressure for the SLVE of three different water-salt mixtures. In Table 4,the %AAD is presented between experimental data and APACT predictions, and in Figure 12 the SLVE pressure is given as a function of temperature. The agreement between APACT predictions and experimental data is very good in all cases.
(cT*2v;)(v;)m-1
+
Vm
(cT*v;)(T*)L(v;)m
(m+
vm+l
m
+2 2
(crv;)(F)'2'(v;)m+1 3',
Vm+2
where Aim, Clm,C2,, and C3, are universal constants given in Table 5 .
Conclusions An equation of state was developed for water-salt phase equilibria based on APACT. For 10 alkali halide salts in the temperature range 150-200 "C the model has been shown to be accurate in predicting the bubble pressure, the liquid and vapor compositions, the critical curve in the case of the waterNaCl mixture, and the equilibrium pressure for the SLVE. The accuracy of the model decreases at temperatures below 200 "C because the salt dissociation becomes important. The accuracy of the model at these intermediatetemperatures can be improved by introducing a term to account specifically for the ionic interaction^.^,^ In addition, more experimental work is needed in order to verify some of the predictions obtained from APACT. Although in this work calculations were restricted to binary water-alkali halide mixtures, the proposed model should be applicable to mixtures containing more than one salt and to mixtures containing other types of salts. For example, 1-2, 2-1, and 2-2 salts can be treated as well by means of suitable combining rules for the evaluation of salt parameters. In addition, it should be possible to calculate the effect of inert gases, such as methane, on the phase behavior of water-salt mixtures on the basis of this model.
+
NkT
?2\ 2
Vm
+
(c(&,Ti$ bjT;,)v*)
pin* = -8.886
Tv
Economou et al.
For a binary mixture where both components self-associate and solvate, ndno is given from the expression35
where W1 and W2 are the mole fractions of the monomers of the two components (Wi = nil/%) calculated from the following set of equations:
where U I = KIRTQ,a2 = K~RTQ, a12= KIzRTQ,a21 = KzIRTQ, and K1,K2 are the equilibrium constants for the self-association of the two components, K I ~ , K zare I the equilibrium constants for the cross-association with K12 = K Z I ,and Q is the molar density. In this work, since salts do not self-associate, K2 = 0. In APACT, the following mixing rules are used:40,41$50
(A351
(c) = &Ci i
(cT;v*)
A;P
-= -2.962-j6'
NkT
(-437)
?V
42
AfQ -- - -12.44NkT
+ '2)'*)j8)
-A;Q - -4.443Q;'('( NkT
?V
(CT;(~)~*(~))
APPP
38 --
NkT - 43.596
1
d-
pv2
KPPP
CJ
1
+ 0.2977F
+ 0.33163?'+
A g Q - 77.716 -NkT AWQ 3B
(cT*(3)v*(2)) Q KQQQ IJV2
(cTr(Po(3)v*(2))
-= 38.68 NkT
IJV2
KPPQ
(A381
where CJ is the soft-core segmental diameter and has a universal value of 2.93 A, corresponding to the diameter of a -CH2segment,
" T T=-.
'ifli
(F) = c&j-
P)'
NkT
0.0010477?
i
j
cik
(A39)
Water-Salt Phase Equilibria
J. Phys. Chem., Vol. 99, No. 16, 1995 6193
(-442)
(P)“’ =1
(A43)
(-446) Similarly for (cTQ*%*)and ( C T Q * ( ~ ) V * ( ~ ) ) ,
+
dji dji d.. = [I 2
‘&ej=
6
+
a,, ajj a,.= 2
lJ
‘Qp,
(‘449)
6
=.
For the pure components P = cqlck, T,* = c,q/ck = 3.0848p’I CY*,TQ* = Qqlck = 0.017464Q2/cv*5~3, and a = (4.2589 x 10-4)a/v*, where v* is in Llmol, p is the dipole moment (in Debye), Q is the quadrupole moment (in esu x and a is the average molecular polarizability (in L x loz7). In this work, quadrupolar interactions are ignored since all of the components have insignificant quadrupole moments. Finally,
F + B,(&Q>~+ C,,(&Q) In ? -I- D,,(&Q) + E,, In ? + F,,
1nlJ’”’I = ~,(2/20>’ In
(A53)
and similarly for the K’s. A,, through F, are constants given by Gubbins and TWU.~’
References and Notes (1) Pitzer, K. S. In Activity CoefJicients in Electrolyte Solutions, Pytkowicz, R. M., EX.; CRC Press: Boca Raton, FL, 1979; Vol. 1, Chapter 7.
(2) Jim, G.; Donohue, M. D. Ind. Eng. Chem. Res. 1988, 27, 1073. (3) Jin, G.; Donohue, M. D. Ind. Eng. Chem. Res. 1991, 30, 240. (4) Fiirst, W.; Renon, H. AIChE J. 1993, 39, 335. ( 5 ) Lu, X.; Maurer, G. AIChE J. 1993, 39, 1527. (6) Quist, A. S.; MarshalL, W. L. J. Phys. Chem. 1968, 72, 684. (7) Helgeson, H. C. In Chemistry and Geochemistry of Solutions at High Temperatures and Pressures; Physics and Chemistry of the Earth, Vol. 13 & 14; Rickard, D. T., Wickman, F. E., Eds.; Pergamon Press: Oxford, 1981; p 133. (8) Pitzer, K. S. Proc. Natl. Acad. Sci. U S A . 1983, 80, 4575. (9) Pitzer, K. S. J. Chem. Thermodyn. 1993, 25, 7. (10) Bischoff, J. L.; Pitzer, K. S. Am. J. Sci. 1989, 289, 217. (11) Khaibullin, I. Kh.; Borisov, N. M. Teplojiz. Vys. Temp. 1966, 4, 518. (12) Wood, S. A.; Crerar, D. A.; Brantley, S. L.: Borcsik, M. Am. J. Sci. 1984, 284, 668. (13) Hovey, J. K.; Pitzer, K. S.; Tanger, J. C.; Bischoff, J. L.; Rosenbauer, R. J. J . Phys. Chem. 1990, 94, 1175. (14) Valyashko, V. M. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 388. (15) Valyashko, V. M. Pure Appl. Chem. 1990, 62, 2129. (16) Valyashko, V. M. Phase Equilibria and Properries of Hydrothermal Systems; Nauka: Moscow, 1990 (in Russian). (17) Ravich, M. I.; Yastrebova, L. F. Russ. J . Inorg. Chem. 1963, 8, 102. (18) Urusova, M. A.; Ravich, M. I. Russ. J. Inorg. Chem. 1966, 11, 353. (19) Mashovets, V. P.: Zarembo, V. I.; Fedorov, M. K. Zh. Priks. Khim. 1973, 46, 650. (20) Fedorov, M. K.; Antonov, N. A.; Lvov. S. V. Zh. Priks. Khim. 1976, 49, 1226. (21) Lvov, S. N.; Antonov, N. A,; Fedorov, M. K. Zh. Priks. Khim. 1976, 49, 1048. (22) Zarembo, V. I.; Antonov, N. A,; Gilyarov, V. N.; Fedorov, M. K. Zh. Priks. Khim. 1976, 49, 1221. (23) Kalinichev, A. G.; Heinzinger, K. In Thermodynamic Data: Systematics and Estimation, Advances in Physical Geochemistry, Vol. 10; Saxena, S. K., Ed.; Springer-Verlag: New York, 1992; Chapter 1. (24) Cummings, P. T.; Cochran, H. D.; Simonson, J. M.; Mesmer, R. E.; Karabomi, S. J . Chem. Phys. 1991, 94, 5606. (25) Cochran, H. D.; Cummings, P. T.; Karabomi, S. Fluid Phase Equilib. 1992, 71, 1. (26) Strauch, H. J.; Cummings, P. T. Fluid Phase Equilib. 1993, 86, 147. (27) Pitzer, K. S.; Pabalan, R. T. Geochim. Cosmochim. Acta 1986,50, 1445. (28) Anderko, A.; Pitzer, K. S. Geochim. Cosmochim. Acta 1993, 57. 1657. (29) Lvov, S. N.; Wood, R. H. Fluid Phase Equilib. 1990. 60, 273. (30) Harvey, A. H. J. Chem. Phys. 1991, 95, 479. (31) Mountain, R. D. J. Chem. Phys. 1989, 90, 1866. (32) Postarino, P.; Tromp, R. H.; Ricci, M.-A,; Soper, A. K.; Neilson, G. W. Nature 1993, 366, 668. (33) Ikonomou, G. D.; Donohue, M. D. AlChE J . 1986, 32, 17i6. (34) Economou, I. G.; Donohue, M. D. Ind. Eng. Chem. Res. 1992,31, 2388. (35) Campbell, S. W.; Economou, I. G.; Donohue, M. D. AlChE J. 1992, 38, 611. (36) Camahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (37) Economou, I. G.; Donohue, M. D. AIChE J. 1991, 37, 1875. (38) Knacke, O., Kubaschewski, O., Hesselmann, K., Eds. Thermochemical Properties of Inorganic Substances; Springer-Verlag: Dusseldorf, 1991. (39) Barton, J. L.; Bloom, H. J. Phys. Chem. 1956, 60, 1413. (40) Vimalchand, P.; Donohue, M. D. Ind. Eng. Chem. Fundam. 1985, 24, 246. (41) Vimalchand, P.; Donohue. M. D.; Celmins, I. ACS Symps. Ser. 1986, 300, 297. (42) CRC Handbook of Physics and Chemistry, 73rd ed.; CRC Press: Boca Raton, FL, 1992; p 9-42, 10-200. (43) Kebarle, P. Ann. Rev. Phys. Chem. 1977, 28, 445. (44) Sourirajan, S.; Kennedy, G. C. Am. J. Sci. 1962, 260, 115. (45) Bischoff, J . L.; Rosenbauer, R. J.; Pitzer, K. S . Geochim. Cosmochim, Acta 1986, 50, 1437. (46) Rosenbauer, R. J.; Bischoff, J. L. Geochim. Cosmochim. Acta 1987, 51, 2349. (47) Bischoff, J. L.; Rosenbauer, R. J. Geochim. Cosmochim. Acta 1988, 52, 2121. (48) Armellini, F. J.; Tester, J. W. Fluid Phase Equilib. 1993, 84, 123. (49) Pitzer, K. S. J. Phys. Chem. 1986, 90, 1502. (50) Donohue, M. D.: Prausnitz, J. M. AIChE J . 1978, 24, 849. (51) Gubbins, K. E.; Twu, C. H. Chem. Eng. Sci. 1978, 33, 863.
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