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A new predictive activity model for aqueous salt solutions
Fluid Phase Equilibria 181 (2001) 33–46
A new predictive activity model for aqueous salt solutions Asle Jøssang∗ , Ellen Stange Norsk Hydro ASA, 0246 Oslo, Norway Received 28 February 2000; accepted 28 November 2000
Abstract A general and predictive water activity coefficient model for brines has been developed. To predict the water activity of the brines, the model developed requires the knowledge of ionic radius, the charge on the ions, and the concentration of each salt. Mixing rules have been established for brines with different salts and brines containing alcohols. The model has proven to be consistent with experimental data. As a practical example, the model has been used to predict the inhibiting effect of salts on gas hydrate formation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Activity coefficient; Predictive model; Brine; Water activity; Gas hydrates
1. Introduction To predict water activities of sea-water and formation water, a model capable of predicting water activities of brines containing mixtures of salts is needed. To eliminate the need for experimental data or parameters for activity equations, an equation which do not need parameters for each salt present in the brine is desired. Different activity coefficient models have been extended to electrolytes and can be used to calculate the water activity of brines. Models that can be used are the Pitzer model [1,2] and the van Laar, Wilson, Redlich-Kister, UNIQUAC, and NRTL equation [3]. These models predict the activity with good accuracy, but can only be used if parameters are available or if parameters can be derived by regression of experimental data for each component. An interesting method has also been presented by Furst and Renon [4]. The article presents an equation of state (EOS) for electrolyte systems where some of the parameters are estimated from ionic radii. In this work, a predictive water activity model for brine is developed using water activity data. The model developed requires knowledge of the ionic radius, the charge of the ion and the concentration of each salt to predict the water activities of the brines. It is suitable for use in engineering applications and it illustrates which chemical properties of the salt affect the water activity coefficient. ∗ Corresponding author. Tel.: +47-22-53-81-00; fax: +47-22-53-81-76. E-mail address: [email protected] (A. Jøssang).
0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 5 1 5 - X
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2. Development of the model The following assumptions and simplifications are made for the model development. • Water is the continuous phase and the salt is dissolved as ions. • The salt solution is homogeneous with no concentration gradients. • The reference state is pure water at a pH of 7.0. The activity coefficient is per definition 1.00 in this state for all pressures and temperatures. • The water activity model developed has no pressure and temperature dependence. • The ionic radius of the salt crystal lattice is used as ionic radius in the water activity model [5]. 2.1. Conventions As a convention, the different salts will be referred to according to the charge on its cation (M+ ) and anion (X− ). Salts will then referred to as 1:1 salts, 1:2 salts, etc. As an example NaCl (Na+ , Cl− ) is a 1:1 salt and Na2 SO4 (2Na+ , SO4 2− ) is a 1:2 salt. 2.2. Evaluation of chemical and physical properties that affect the water activity When salts are dissolved in water, they dissociate into ions. This requires a negative change in Gibbs energy, which can be caused by a negative change in enthalpy or by a positive change in entropy. The change in interaction forces between ions and water causes the change in enthalpy. The change in entropy is caused by increased disorder in the water phase, and it is accounted for by the change in water activity. The water activity is defined by the activity coefficient of the water phase multiplied by the mole fraction of water. To evaluate which chemical and physical properties of the salts that affect the activity of the water, the activity coefficient of water containing LiCl, NaCl or KCl has been plotted at different concentrations [5–9]. This is shown in Fig. 1. Fig. 1 shows that the water activity coefficient increases with increased ionic radius. This change in water activity is general for all cations and is in accordance with Robinson and Stokes [10], Berecz and Balla-Achs [11] and Makogon [12]. The reason for the change in activity with ionic radius is assumed to be due to different disorder in the water phase caused by the different ions. This is a result of the difference in interaction forces between water molecules and ions. Small ions have a greater surface charge than larger ones. This results in stronger interactions between the ion and the water molecules and causes more disorder in the water phase. In Fig. 1, only 1:1 salts are described, but the correlation is seen for other types of salt. Fig. 2 shows that the water activity decreases with increasing charge on the cation. The effect of charge and ionic radius can be expressed through the surface charge, defined as the charge of the ion divided by the surface area of the ion. The ions are assumed to be hard spheres with a defined radius. The water activity coefficient decreases with increasing salt concentration. This is accounted for by the mole fraction of the ions. In general, the water activity coefficient is given by cation zion,i e γw ∝ 1 − (1) xion,i 2 4πrion,i anion
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Fig. 1. Activity coefficient of water in brines of LiCl, NaCl or KCl plotted vs. concentration at a temperature of 298.15 K.
Where γ w is the water activity coefficient in brines with only one salt present, cation anion all ions in the salt in the solution, zion,i the charge of ion i, e the elementary charge (Coulomb), rion,i the ionic radius of ion i (m), xion,i is the mole fraction of ion i. Fig. 3 shows a decrease in water activity with an increase in anionic radius. This is the opposite of what was found for cations. The effect of hydration of the anions with increased surface area, as Cl− , Br− and
Fig. 2. Activity coefficient of water in brines of NaCl, MgCl2 or AlCl3 plotted vs. concentration at a temperature of 298.15 K [5–9].
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Fig. 3. Activity coefficient of water in brines of NaCl, NaBr and NaI [5–9] plotted vs. concentration at a temperature of 298.15 K.
I− , is larger than the effect of decreased surface charge. This results in a decrease in water activity with increasing ionic radius. Since the ionic radius of each of the anions Cl− , Br− and I− is much larger than for most cations, the change in surface charge between anions are less than between cations. Consequently, the effect of anionic charge on the water activity will be less than for cations. The degree of disorder in the water phase is a function of how much each water molecule is influenced by the ion and by the number of water molecules involved per ion. How much each water molecule is affected is a function of the interaction forces. The number of molecules affected is a function of the charge, the surface area of the ions and the distance from the surface of the ion to the different water molecules affect by the ion. In Eq. (2), this increase in hydration due to increased surface area, is accounted for by a function of the reciprocal of the ionic radius. This function is introduced in the equation as shown below. γw = 1 −
cation anion
f (ion, i)
zion,i e xion,i , 2 4πrion,i
(2)
where f (ion, i) is a function of the ion i (m2 /C). Figs. 1–3 show that the activity change with concentration is slightly curved. An exponent on the mole fraction of the ion accounts for this curvature. This exponent is given empirically and is introduced in Eq. (3) as k3 (zM+ ,i : zX− ,i ). 2.3. The water activity coefficient model for brines of one salt The activity coefficient can now be expressed by γw = 1 −
cation anion
f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i )
zion,i e k3 (zM+ ,i :zX− ,i ) xion,i , 2 4πrion,i
(3)
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where f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) represents a common function of the radius and charge to the anion–cation pair i. The function takes into account the increase in hydration of the ions with increased ionic radius and are expressed by f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) = k1 (zM+ ,i : zX− ,i ) −
k2 (zM+ ,i : zX− ,i ) rM+ ,i + rX− ,i
(4)
and k3 (zM+ ,i : zX− ,i ) is a empirical constant accounting for the interaction of the ions. The constants (k1 , k2 , k3 ) in these equations are given in Table 1 in Appendix A. 2.4. The model for water activity of mixtures of salt Formation and sea-water contain a mixture of different salts. To account for this the mixing rule developed by Patwardhan and Kumar [13] is used. ln aw =
m mi
mo i=1 i
o ln aw,i ,
(5)
where m is the number of components, i is the component index, mi is the concentration of the salt in question (mol/kg), moi is the concentration of component i which gives an ionic strength equal to the total o ionic strength of the brine (mol/kg), a w is the water activity of the brine, aw,i is the water activity if salt i was present alone at the same ionic strength as the total brine. The activity model developed for the water phase in brine can then be expressed as follows: m cation mi zion,i e k3 (zM+ ,i :zX− ,i ) ln aw = ln 1− f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) xion,tot xw,i (6) 2 mo 4πrion,i i=1 i anion 2.5. Mixing rule for brines containing alcohols The activity model can be used to calculate the water activity of an aqueous mixture of salts, methanol or ethylene glycol. The water activity of the mixture is then calculated by multiplying the water activities calculated as if salts, methanol and ethylene glycol were alone in the mixture. For salts, Eq. (6) are used. The basis for this equation is that the water activity is included in the EOS by addition of the natural logarithm of the water activity. When there are different components adding to the water activity, it can be assumed that the logarithm of water is added several times. A summation of the logarithms of water activity is equal to the logarithm of the multiple of the water activities. The water activity is then equal to the product of the water activities caused by the different components. For a mixture contains salts, methanol and ethylene glycol, the following expression for the mixing rule has been used: aw,total = aw,salt aw,MeOH aw,EG ,
(8)
whereaw,salt , aw,MeOH , and aw,EG are the activities of water if only salt, methanol and ethylene glycol, respectively, were present in the water phase.
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Fig. 4. Calculated water activities compared with experimental data and calculated values from other activity models given by ASPEN PLUS [15] at a temperature of 298.15 K.
3. Comparison of the new activity model with experiments 3.1. Solutions with single salts The calculated activities of water in brines are compared with experimental data and with results from established activity models. The water activity is calculated from experimental data for salt activity by the Gibbs–Duhems equation. The data are given by Patil et al. [14] and Robinson and Stokes [10]. The Pitzer, Bromley–Pitzer and NRTLelectrolyte models are used with the parameters given by ASPEN PLUS [15]. Fig. 4 shows calculated water activities in a KCl brine compared with experimental data and calculated results from other activity models. Fig. 10 in Appendix B shows calculated water activities compared with experimental data [5,9] for BaBr2 and with results from other activity models. Fig. 5 shows calculated water activities compared with experimental data [5–9] for brines of NaCl, CuCl2 and LaCl3 . Fig. 11 in Appendix B, calculated water activities are compared with experimental data [5,7,9] for brines of HNO3 and NaNO3 . In Fig. 12, calculated water activities are compared with experimental data [5,9] for brines of NH4 Cl and Na2 CO3 . In Fig. 6, calculated water activities are compared with experimental data [5,9] for brines of LiBr, NaBr, KBr, RbBr, and CsBr. In this article, six graphical comparisons of the water activity model to experimental data are given. Experimental data for 46 different salt solutions have been compared with calculated water activities with good results [16].
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Fig. 5. Calculated water activities compared with experimental data at a temperature of 298.15 K.
In Table 2 in Appendix A, the range of validity for the model is given based on a maximum deviation from experimental data of ±0.01. 3.2. Prediction of gas hydrate equilibrium The hydrate model of van der Waals and Platteeuw [17] modified by McKoy and Sinanoglu [18] is used to calculate the chemical potential of the water phase in the hydrate lattice. The model by Parrish
Fig. 6. Calculated water activities compared with experimental data at a temperature of 298.15 K.
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Fig. 7. Calculated gas hydrate temperature compared with experimental measurement when KCl is in the water phase.
and Prausnitz [19] is used to calculate the chemical potential of the liquid water phase in equilibrium with the hydrate lattice. This model uses the water activity to predict the equilibrium. The water activity model is used in the calculation of equilibrium temperature of gas hydrate where there are salts in the water phase. In Figs. 7–9, the calculated values are compared to experimental hydrate equilibrium data [20,21].
Fig. 8. Calculated gas hydrate temperature compared with experimental measurement for 9.98 wt.% NaCl, 9.99 wt.% methanol (65), and 20 wt.% ethylene glycol 5.02 wt.% CaCl2 (75).
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Fig. 9. Calculated gas hydrate temperature compared with experimental measurement for NaCl, KCl brines or mixture of 5.00 wt.% NaCl and 10.00 wt.% KCl (32) or 10.17 wt.% NaCl and 5.08 wt.% CaCl2 (33).
In Fig. 7, calculated equilibrium hydrate temperatures for natural gas hydrates 1 in KCl brines are compared with experimental data. In Figs. 8 and 9, calculated hydrate temperatures for hydrates of a gas containing ∼80 mol% CH4 and 20% CO2 are compared to experimental data. Calculations for pure water are included to indicate the deviation from experimental data resulting from using the hydrate model without the water activity model. The model has previously been compared to other experimental data with good results in [16]. 4. Discussion 4.1. General The model can be used to predict the activity of water in brines without relying on experimental data. The empirical constants in the model were derived from experimental data for some of the salts. Calculations showed that the predicted water activity also compared well with experimental data for other salts. An example of this is the NH4 Cl brine, which is shown in Fig. 12 in Appendix B. The model is suitable to evaluate the change in water activity caused by various salts and it indicates which chemical properties that affect the water activity. The model only needs the ionic radius, the charge of the ions and the concentration of each salt to predict the effect of the salts on the activity. The development of the model is based on surface charge of the ion and the assumption that the ion–water distance is equal for all ions when the ion is given a defined radius as a hard sphere. The strength of the interaction forces are then represented by the surface charge of the ion since the charge of 1 The natural gas composition: 0.10 N2 , 1.76 CO2 , 80.53 CH4 , 10.30 C2 H6 , 4.99 C3 H8 , 0.72 i-C4 H10 and 1.60 n-C4 H10 (all in mol%).
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the water molecules are the same. The calculation of the actual water–ion interactions are not necessary when using the model. The water activity data are obtained from data of salt activity in water converted to water activities with the Gibbs–Duhems equation. A consistency test between the converted data and available experimental data for water activity [10,14] has been done with good consistency [16]. The Gibbs–Duhems equation can also be used to calculate the mean ionic activity coefficient from the water activities given by the equation developed in this paper. The activity equation developed in this paper will presents the conditions at infinite dilution in accordance with the Debye–Huchel limiting law. The ion radius of an ion in a salt crystal lattice is used as the ionic radius of the ions in the water phase. It is assumed that this will not affect the results. For ions consisting of more than one atom, a radius is predicted from the covalent and ionic radius of the different atoms. This method to estimate the ionic radius gives results in accordance with experimental data [16]. 4.2. Limitations of the model Going from pure water to pure salt there will be different concentration zones where the mixture has different chemical and physical state. To simplify this, the salt solution is considered to be in three different states. The first state are dissolved ions in water continuous phase, the second one is the intermediate state where there are both water–water and ion–ion clusters but the solution is still unsaturated. The third state is saturated solution with solid salt present. Many activity models try to describe all three states. Because of the different chemical properties in these three states, this model considers only the first state. This gives the model a limitation in concentration. The model is valid from pure water up to between 10 and 50 wt.% dependent on the salt (see Table 2 in Appendix A). The calculation of water activity when salt, methanol and glycol are present does not account for the interaction effect between the different components. Due to this, the calculated water activity is higher than indicated by experimental measurements for a given composition. The model will, therefore, always be conservative when mixtures of components are present in the water phase. 4.3. Improvements of the model The model does not account for changes in activity caused by pressure. In a further development of the model, the pressure effect should be evaluated. The pressure effect can be the reason why the model gives a larger deviation from the experimental data at higher pressures as shown in Figs. 7–9 than for the calculations at ambience pressure. To simplify the development of the model, the increase in hydration caused by increased ionic radius has been expressed by a linear function of the reciprocal ion radius. A further improvement of the equation will be to express this increase in hydration by a correlation found in chemical properties for example using hydration numbers. This could eliminate nearly all use of parameters derived from equilibrium data. 5. Conclusions A general predictive water activity model has been developed. Three general parameters have to be fitted to experimental data for each salt group (1:1, 1:2, etc). After this regression, it is possible to calculate the activity of water containing other salts, without any regression for this particular salt.
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The activity model has been compared to experimental data and shows good agreement with the data available. The model estimates water activities with an accuracy of 0.01 within the concentration range as given in Table 2 in Appendix A. List of symbols aw activity of water aw,i activity to water in presence of component i o aw,i water activity if salt i was present alone at the same ionic strength as the total brine e elementary charge (C) k1 constant (m2 /C) k2 constant (m/C) k3 constant m number of components mi molar concentration of component i (mol/kg) moi molar concentration of component i which gives an ionic strength equal to the total ionic strength of the brine (mol/kg) rion,i radius of ion i rM+ radius of the cation rX− radius of the anion xi mole fraction of component i xion,i mole fraction of ion i xw mole fraction of water zion,i the charge on the ion zM+ charge on cation zX − charge on anion γw water activity coefficient Acknowledgements This work was supported by Hydro Technology and Projects, Norsk Hydro ASA. Appendix A Empirical constants developed for the water activity model for the different salt groups and range of concentration where the activity model has proven to be valid are shown in Tables 1 and 2. Table 1 Empirical constants in the water activity model Ion pair (zM+ ,i : zX− ,i )
k1 (zM+ ,i : zX− ,i ) (m2 /C)
k2 (zM+ ,i : zX− ,i ) (m/C)
k3 (zM+ ,i : zX− ,i )
1:1 2:1 3:1 1:2 1:NO3 −
4.9204 12.747 5.6604 0.40 0.40
9.3753E−10 2.9313E−9 1.2042E−9 0.00 0.00
1.27 1.30 1.10 1.00 1.00
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Table 2 The highest concentrations where the activity model has proved to be consistent with experimental data Salt
Highest mole fraction
Highest weight (%)
KF NaF HCl LiCl NaCl KCl RbCl CsCl HBr LiBr NaBr KBr RbBr CsBr HI LiI NaI KI RbI CsI HNO3 NaNO3 KNO3 AgNO3 NH4 NO3 NH4 Cl Na2 SO4 K2 SO4 Na2 CO3 MgCl2 CaCl2 SrCl2 BaCl2 FeCl2 CuCl2 ZnCl2 MgBr2 CaBr2 SrBr2 BaBr2 MgI2 CaI2 SrI2 BaI2 AlCl3 LaCl3
0.06 0.08+ 0.05 0.065 0.08 0.08+ 0.08 0.08+ 0.05 0.055 0.067 0.08+ 0.08+ 0.08 0.02 0.07 0.06 0.08+ 0.08 0.07 0.08+ 0.08+ 0.07 0.07 0.08+ 0.08+ 0.06 0.05 0.08+ 0.02 0.035 0.04 0.03 0.02 0.08+ 0.02 0.05 0.035 0.035 0.05 0,01 0.05 0.035 0.035 0.06 0.08
17.1 16.9 9.6 14.1 22.0 26.5 36.9 44.9 19.1 21.9 29.1 36.5 44.4 50.7 12.7 35.9 34.7 44.5 50.6 52.1 23.3 29.1 29.7 41.5 27.9 20.5 33.5 33.8 33.9 9.7 18.3 26.8 26.4 12.6 39.4 13.4 35.0 28.7 33.3 46.5 13.5 46.2 40.8 44.1 32.1 54.2
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Appendix B Calculated water activities compared with experimental data are shown in Figs. 10–12.
Fig. 10. Calculated water activities for brines compared with experimental data and calculated values from activity models given by ASPEN PLUS [14] at a temperature of 298.15 K.
Fig. 11. Calculated water activities for brines compared with experimental data at a temperature of 298.15 K.
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Fig. 12. Calculated water activities for brines compared with experimental data at a temperature of 298.15 K.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
K.S. Pitzer, J. Phys. Chem. 77 (2) (1973) 268–277. K.S. Pitzer, G. Mayorga, J. Phys. Chem. 77 (19) (1973) 2300–2308. R.C. Reid, J.M. Prausnitz, B.E. Poling, Properties of Liquids and Gases, 4th Edition, McGraw-Hill, New York, USA, 1987. W. Furst, H. Renon, AIChE J. 39 (1993) 335–343. D.R. Lide, CRC Handbook of Chemistry and Physics, 71st Edition, CRC Press, Boston, USA, 1990–1991. B.E. Conway, Electrochemical data, 2nd Edition, Greenwwood Press, London, UK, 1969. G.W.C Kage, T.H. Laby, Tables of Physical and Chemical Constants, 16th Edition, Longman, Harlow, UK, 1995. G. Borge, R. Castaño, M.P. Carril, M.S. Corbillón, J.M. Madariaga, Fluid Phase Eq. 121 (1996) 85–98. D. Dobos, Electrochemical Data, Elsevier, Amsterdam, The Netherlands, 1975. R.A. Robinson, R.H. Stokes, Electrolyte Solutions, Butterworth, London, 1970. E. Berecz, M. Balla-Achs, Studies in Inorganic Chemistry 4. Gas Hydrates, Elsevier, Amsterdam, The Netherlands, 1983. Y.F. Makogon, Hydrates of Natural Gas, Penn Well Publishing Company, Tuls, Oklahoma, 1981. V.S. Patwardhan, A. Kumar, AIChE J. 32 (9) (1986) 1419–1428. K.R. Patil, A.D. Tripathi, G. Pathak, S.S. Katti, J. Chem. Eng. Data 36 (1991) 225–230. Aspen Technology Inc., ASPEN PLUS, User Guide, Release 9, USA, 1994. A. Jøssang, Calculation of the Activity of Aqueous Salt Solution for Application in Gas Hydrate Formation Modelling, NTNU, 1996. J.H. van der Waals, J.C. Platteeuw, Adv. Chem. Phys. 2 (1959) 1. V. McKoy, O. Sinanoglu, J. Chem. Phys. 38 (1963) 2946. W.R. Parrish, J.M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev. 11 (1) (1972) 26–35. DB Robinson Research Ltd., Hydrate formation of a typical north sea gas mixture in the presence of selected aqueous solution, Alberta, Canada, 1991. P.R. Bishnoi, P.D. Dholabhai, K.N. Mahadev, Hydrate phase equilibria in inhibited and brine system, GPA, RR-156, Canada, 1996.
A new predictive activity model for aqueous salt solutions Asle Jøssang∗ , Ellen Stange Norsk Hydro ASA, 0246 Oslo, Norway Received 28 February 2000; accepted 28 November 2000
Abstract A general and predictive water activity coefficient model for brines has been developed. To predict the water activity of the brines, the model developed requires the knowledge of ionic radius, the charge on the ions, and the concentration of each salt. Mixing rules have been established for brines with different salts and brines containing alcohols. The model has proven to be consistent with experimental data. As a practical example, the model has been used to predict the inhibiting effect of salts on gas hydrate formation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Activity coefficient; Predictive model; Brine; Water activity; Gas hydrates
1. Introduction To predict water activities of sea-water and formation water, a model capable of predicting water activities of brines containing mixtures of salts is needed. To eliminate the need for experimental data or parameters for activity equations, an equation which do not need parameters for each salt present in the brine is desired. Different activity coefficient models have been extended to electrolytes and can be used to calculate the water activity of brines. Models that can be used are the Pitzer model [1,2] and the van Laar, Wilson, Redlich-Kister, UNIQUAC, and NRTL equation [3]. These models predict the activity with good accuracy, but can only be used if parameters are available or if parameters can be derived by regression of experimental data for each component. An interesting method has also been presented by Furst and Renon [4]. The article presents an equation of state (EOS) for electrolyte systems where some of the parameters are estimated from ionic radii. In this work, a predictive water activity model for brine is developed using water activity data. The model developed requires knowledge of the ionic radius, the charge of the ion and the concentration of each salt to predict the water activities of the brines. It is suitable for use in engineering applications and it illustrates which chemical properties of the salt affect the water activity coefficient. ∗ Corresponding author. Tel.: +47-22-53-81-00; fax: +47-22-53-81-76. E-mail address: [email protected] (A. Jøssang).
0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 5 1 5 - X
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2. Development of the model The following assumptions and simplifications are made for the model development. • Water is the continuous phase and the salt is dissolved as ions. • The salt solution is homogeneous with no concentration gradients. • The reference state is pure water at a pH of 7.0. The activity coefficient is per definition 1.00 in this state for all pressures and temperatures. • The water activity model developed has no pressure and temperature dependence. • The ionic radius of the salt crystal lattice is used as ionic radius in the water activity model [5]. 2.1. Conventions As a convention, the different salts will be referred to according to the charge on its cation (M+ ) and anion (X− ). Salts will then referred to as 1:1 salts, 1:2 salts, etc. As an example NaCl (Na+ , Cl− ) is a 1:1 salt and Na2 SO4 (2Na+ , SO4 2− ) is a 1:2 salt. 2.2. Evaluation of chemical and physical properties that affect the water activity When salts are dissolved in water, they dissociate into ions. This requires a negative change in Gibbs energy, which can be caused by a negative change in enthalpy or by a positive change in entropy. The change in interaction forces between ions and water causes the change in enthalpy. The change in entropy is caused by increased disorder in the water phase, and it is accounted for by the change in water activity. The water activity is defined by the activity coefficient of the water phase multiplied by the mole fraction of water. To evaluate which chemical and physical properties of the salts that affect the activity of the water, the activity coefficient of water containing LiCl, NaCl or KCl has been plotted at different concentrations [5–9]. This is shown in Fig. 1. Fig. 1 shows that the water activity coefficient increases with increased ionic radius. This change in water activity is general for all cations and is in accordance with Robinson and Stokes [10], Berecz and Balla-Achs [11] and Makogon [12]. The reason for the change in activity with ionic radius is assumed to be due to different disorder in the water phase caused by the different ions. This is a result of the difference in interaction forces between water molecules and ions. Small ions have a greater surface charge than larger ones. This results in stronger interactions between the ion and the water molecules and causes more disorder in the water phase. In Fig. 1, only 1:1 salts are described, but the correlation is seen for other types of salt. Fig. 2 shows that the water activity decreases with increasing charge on the cation. The effect of charge and ionic radius can be expressed through the surface charge, defined as the charge of the ion divided by the surface area of the ion. The ions are assumed to be hard spheres with a defined radius. The water activity coefficient decreases with increasing salt concentration. This is accounted for by the mole fraction of the ions. In general, the water activity coefficient is given by cation zion,i e γw ∝ 1 − (1) xion,i 2 4πrion,i anion
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Fig. 1. Activity coefficient of water in brines of LiCl, NaCl or KCl plotted vs. concentration at a temperature of 298.15 K.
Where γ w is the water activity coefficient in brines with only one salt present, cation anion all ions in the salt in the solution, zion,i the charge of ion i, e the elementary charge (Coulomb), rion,i the ionic radius of ion i (m), xion,i is the mole fraction of ion i. Fig. 3 shows a decrease in water activity with an increase in anionic radius. This is the opposite of what was found for cations. The effect of hydration of the anions with increased surface area, as Cl− , Br− and
Fig. 2. Activity coefficient of water in brines of NaCl, MgCl2 or AlCl3 plotted vs. concentration at a temperature of 298.15 K [5–9].
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Fig. 3. Activity coefficient of water in brines of NaCl, NaBr and NaI [5–9] plotted vs. concentration at a temperature of 298.15 K.
I− , is larger than the effect of decreased surface charge. This results in a decrease in water activity with increasing ionic radius. Since the ionic radius of each of the anions Cl− , Br− and I− is much larger than for most cations, the change in surface charge between anions are less than between cations. Consequently, the effect of anionic charge on the water activity will be less than for cations. The degree of disorder in the water phase is a function of how much each water molecule is influenced by the ion and by the number of water molecules involved per ion. How much each water molecule is affected is a function of the interaction forces. The number of molecules affected is a function of the charge, the surface area of the ions and the distance from the surface of the ion to the different water molecules affect by the ion. In Eq. (2), this increase in hydration due to increased surface area, is accounted for by a function of the reciprocal of the ionic radius. This function is introduced in the equation as shown below. γw = 1 −
cation anion
f (ion, i)
zion,i e xion,i , 2 4πrion,i
(2)
where f (ion, i) is a function of the ion i (m2 /C). Figs. 1–3 show that the activity change with concentration is slightly curved. An exponent on the mole fraction of the ion accounts for this curvature. This exponent is given empirically and is introduced in Eq. (3) as k3 (zM+ ,i : zX− ,i ). 2.3. The water activity coefficient model for brines of one salt The activity coefficient can now be expressed by γw = 1 −
cation anion
f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i )
zion,i e k3 (zM+ ,i :zX− ,i ) xion,i , 2 4πrion,i
(3)
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37
where f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) represents a common function of the radius and charge to the anion–cation pair i. The function takes into account the increase in hydration of the ions with increased ionic radius and are expressed by f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) = k1 (zM+ ,i : zX− ,i ) −
k2 (zM+ ,i : zX− ,i ) rM+ ,i + rX− ,i
(4)
and k3 (zM+ ,i : zX− ,i ) is a empirical constant accounting for the interaction of the ions. The constants (k1 , k2 , k3 ) in these equations are given in Table 1 in Appendix A. 2.4. The model for water activity of mixtures of salt Formation and sea-water contain a mixture of different salts. To account for this the mixing rule developed by Patwardhan and Kumar [13] is used. ln aw =
m mi
mo i=1 i
o ln aw,i ,
(5)
where m is the number of components, i is the component index, mi is the concentration of the salt in question (mol/kg), moi is the concentration of component i which gives an ionic strength equal to the total o ionic strength of the brine (mol/kg), a w is the water activity of the brine, aw,i is the water activity if salt i was present alone at the same ionic strength as the total brine. The activity model developed for the water phase in brine can then be expressed as follows: m cation mi zion,i e k3 (zM+ ,i :zX− ,i ) ln aw = ln 1− f (rM+ ,i + rX− ,i , zM+ ,i : zX− ,i ) xion,tot xw,i (6) 2 mo 4πrion,i i=1 i anion 2.5. Mixing rule for brines containing alcohols The activity model can be used to calculate the water activity of an aqueous mixture of salts, methanol or ethylene glycol. The water activity of the mixture is then calculated by multiplying the water activities calculated as if salts, methanol and ethylene glycol were alone in the mixture. For salts, Eq. (6) are used. The basis for this equation is that the water activity is included in the EOS by addition of the natural logarithm of the water activity. When there are different components adding to the water activity, it can be assumed that the logarithm of water is added several times. A summation of the logarithms of water activity is equal to the logarithm of the multiple of the water activities. The water activity is then equal to the product of the water activities caused by the different components. For a mixture contains salts, methanol and ethylene glycol, the following expression for the mixing rule has been used: aw,total = aw,salt aw,MeOH aw,EG ,
(8)
whereaw,salt , aw,MeOH , and aw,EG are the activities of water if only salt, methanol and ethylene glycol, respectively, were present in the water phase.
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Fig. 4. Calculated water activities compared with experimental data and calculated values from other activity models given by ASPEN PLUS [15] at a temperature of 298.15 K.
3. Comparison of the new activity model with experiments 3.1. Solutions with single salts The calculated activities of water in brines are compared with experimental data and with results from established activity models. The water activity is calculated from experimental data for salt activity by the Gibbs–Duhems equation. The data are given by Patil et al. [14] and Robinson and Stokes [10]. The Pitzer, Bromley–Pitzer and NRTLelectrolyte models are used with the parameters given by ASPEN PLUS [15]. Fig. 4 shows calculated water activities in a KCl brine compared with experimental data and calculated results from other activity models. Fig. 10 in Appendix B shows calculated water activities compared with experimental data [5,9] for BaBr2 and with results from other activity models. Fig. 5 shows calculated water activities compared with experimental data [5–9] for brines of NaCl, CuCl2 and LaCl3 . Fig. 11 in Appendix B, calculated water activities are compared with experimental data [5,7,9] for brines of HNO3 and NaNO3 . In Fig. 12, calculated water activities are compared with experimental data [5,9] for brines of NH4 Cl and Na2 CO3 . In Fig. 6, calculated water activities are compared with experimental data [5,9] for brines of LiBr, NaBr, KBr, RbBr, and CsBr. In this article, six graphical comparisons of the water activity model to experimental data are given. Experimental data for 46 different salt solutions have been compared with calculated water activities with good results [16].
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Fig. 5. Calculated water activities compared with experimental data at a temperature of 298.15 K.
In Table 2 in Appendix A, the range of validity for the model is given based on a maximum deviation from experimental data of ±0.01. 3.2. Prediction of gas hydrate equilibrium The hydrate model of van der Waals and Platteeuw [17] modified by McKoy and Sinanoglu [18] is used to calculate the chemical potential of the water phase in the hydrate lattice. The model by Parrish
Fig. 6. Calculated water activities compared with experimental data at a temperature of 298.15 K.
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Fig. 7. Calculated gas hydrate temperature compared with experimental measurement when KCl is in the water phase.
and Prausnitz [19] is used to calculate the chemical potential of the liquid water phase in equilibrium with the hydrate lattice. This model uses the water activity to predict the equilibrium. The water activity model is used in the calculation of equilibrium temperature of gas hydrate where there are salts in the water phase. In Figs. 7–9, the calculated values are compared to experimental hydrate equilibrium data [20,21].
Fig. 8. Calculated gas hydrate temperature compared with experimental measurement for 9.98 wt.% NaCl, 9.99 wt.% methanol (65), and 20 wt.% ethylene glycol 5.02 wt.% CaCl2 (75).
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41
Fig. 9. Calculated gas hydrate temperature compared with experimental measurement for NaCl, KCl brines or mixture of 5.00 wt.% NaCl and 10.00 wt.% KCl (32) or 10.17 wt.% NaCl and 5.08 wt.% CaCl2 (33).
In Fig. 7, calculated equilibrium hydrate temperatures for natural gas hydrates 1 in KCl brines are compared with experimental data. In Figs. 8 and 9, calculated hydrate temperatures for hydrates of a gas containing ∼80 mol% CH4 and 20% CO2 are compared to experimental data. Calculations for pure water are included to indicate the deviation from experimental data resulting from using the hydrate model without the water activity model. The model has previously been compared to other experimental data with good results in [16]. 4. Discussion 4.1. General The model can be used to predict the activity of water in brines without relying on experimental data. The empirical constants in the model were derived from experimental data for some of the salts. Calculations showed that the predicted water activity also compared well with experimental data for other salts. An example of this is the NH4 Cl brine, which is shown in Fig. 12 in Appendix B. The model is suitable to evaluate the change in water activity caused by various salts and it indicates which chemical properties that affect the water activity. The model only needs the ionic radius, the charge of the ions and the concentration of each salt to predict the effect of the salts on the activity. The development of the model is based on surface charge of the ion and the assumption that the ion–water distance is equal for all ions when the ion is given a defined radius as a hard sphere. The strength of the interaction forces are then represented by the surface charge of the ion since the charge of 1 The natural gas composition: 0.10 N2 , 1.76 CO2 , 80.53 CH4 , 10.30 C2 H6 , 4.99 C3 H8 , 0.72 i-C4 H10 and 1.60 n-C4 H10 (all in mol%).
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the water molecules are the same. The calculation of the actual water–ion interactions are not necessary when using the model. The water activity data are obtained from data of salt activity in water converted to water activities with the Gibbs–Duhems equation. A consistency test between the converted data and available experimental data for water activity [10,14] has been done with good consistency [16]. The Gibbs–Duhems equation can also be used to calculate the mean ionic activity coefficient from the water activities given by the equation developed in this paper. The activity equation developed in this paper will presents the conditions at infinite dilution in accordance with the Debye–Huchel limiting law. The ion radius of an ion in a salt crystal lattice is used as the ionic radius of the ions in the water phase. It is assumed that this will not affect the results. For ions consisting of more than one atom, a radius is predicted from the covalent and ionic radius of the different atoms. This method to estimate the ionic radius gives results in accordance with experimental data [16]. 4.2. Limitations of the model Going from pure water to pure salt there will be different concentration zones where the mixture has different chemical and physical state. To simplify this, the salt solution is considered to be in three different states. The first state are dissolved ions in water continuous phase, the second one is the intermediate state where there are both water–water and ion–ion clusters but the solution is still unsaturated. The third state is saturated solution with solid salt present. Many activity models try to describe all three states. Because of the different chemical properties in these three states, this model considers only the first state. This gives the model a limitation in concentration. The model is valid from pure water up to between 10 and 50 wt.% dependent on the salt (see Table 2 in Appendix A). The calculation of water activity when salt, methanol and glycol are present does not account for the interaction effect between the different components. Due to this, the calculated water activity is higher than indicated by experimental measurements for a given composition. The model will, therefore, always be conservative when mixtures of components are present in the water phase. 4.3. Improvements of the model The model does not account for changes in activity caused by pressure. In a further development of the model, the pressure effect should be evaluated. The pressure effect can be the reason why the model gives a larger deviation from the experimental data at higher pressures as shown in Figs. 7–9 than for the calculations at ambience pressure. To simplify the development of the model, the increase in hydration caused by increased ionic radius has been expressed by a linear function of the reciprocal ion radius. A further improvement of the equation will be to express this increase in hydration by a correlation found in chemical properties for example using hydration numbers. This could eliminate nearly all use of parameters derived from equilibrium data. 5. Conclusions A general predictive water activity model has been developed. Three general parameters have to be fitted to experimental data for each salt group (1:1, 1:2, etc). After this regression, it is possible to calculate the activity of water containing other salts, without any regression for this particular salt.
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The activity model has been compared to experimental data and shows good agreement with the data available. The model estimates water activities with an accuracy of 0.01 within the concentration range as given in Table 2 in Appendix A. List of symbols aw activity of water aw,i activity to water in presence of component i o aw,i water activity if salt i was present alone at the same ionic strength as the total brine e elementary charge (C) k1 constant (m2 /C) k2 constant (m/C) k3 constant m number of components mi molar concentration of component i (mol/kg) moi molar concentration of component i which gives an ionic strength equal to the total ionic strength of the brine (mol/kg) rion,i radius of ion i rM+ radius of the cation rX− radius of the anion xi mole fraction of component i xion,i mole fraction of ion i xw mole fraction of water zion,i the charge on the ion zM+ charge on cation zX − charge on anion γw water activity coefficient Acknowledgements This work was supported by Hydro Technology and Projects, Norsk Hydro ASA. Appendix A Empirical constants developed for the water activity model for the different salt groups and range of concentration where the activity model has proven to be valid are shown in Tables 1 and 2. Table 1 Empirical constants in the water activity model Ion pair (zM+ ,i : zX− ,i )
k1 (zM+ ,i : zX− ,i ) (m2 /C)
k2 (zM+ ,i : zX− ,i ) (m/C)
k3 (zM+ ,i : zX− ,i )
1:1 2:1 3:1 1:2 1:NO3 −
4.9204 12.747 5.6604 0.40 0.40
9.3753E−10 2.9313E−9 1.2042E−9 0.00 0.00
1.27 1.30 1.10 1.00 1.00
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Table 2 The highest concentrations where the activity model has proved to be consistent with experimental data Salt
Highest mole fraction
Highest weight (%)
KF NaF HCl LiCl NaCl KCl RbCl CsCl HBr LiBr NaBr KBr RbBr CsBr HI LiI NaI KI RbI CsI HNO3 NaNO3 KNO3 AgNO3 NH4 NO3 NH4 Cl Na2 SO4 K2 SO4 Na2 CO3 MgCl2 CaCl2 SrCl2 BaCl2 FeCl2 CuCl2 ZnCl2 MgBr2 CaBr2 SrBr2 BaBr2 MgI2 CaI2 SrI2 BaI2 AlCl3 LaCl3
0.06 0.08+ 0.05 0.065 0.08 0.08+ 0.08 0.08+ 0.05 0.055 0.067 0.08+ 0.08+ 0.08 0.02 0.07 0.06 0.08+ 0.08 0.07 0.08+ 0.08+ 0.07 0.07 0.08+ 0.08+ 0.06 0.05 0.08+ 0.02 0.035 0.04 0.03 0.02 0.08+ 0.02 0.05 0.035 0.035 0.05 0,01 0.05 0.035 0.035 0.06 0.08
17.1 16.9 9.6 14.1 22.0 26.5 36.9 44.9 19.1 21.9 29.1 36.5 44.4 50.7 12.7 35.9 34.7 44.5 50.6 52.1 23.3 29.1 29.7 41.5 27.9 20.5 33.5 33.8 33.9 9.7 18.3 26.8 26.4 12.6 39.4 13.4 35.0 28.7 33.3 46.5 13.5 46.2 40.8 44.1 32.1 54.2
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Appendix B Calculated water activities compared with experimental data are shown in Figs. 10–12.
Fig. 10. Calculated water activities for brines compared with experimental data and calculated values from activity models given by ASPEN PLUS [14] at a temperature of 298.15 K.
Fig. 11. Calculated water activities for brines compared with experimental data at a temperature of 298.15 K.
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Fig. 12. Calculated water activities for brines compared with experimental data at a temperature of 298.15 K.
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