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Ind. Eng. Chem. Res. 2007, 46, 5074-5079
Thermodynamic Modeling of Salt Precipitation and Gas Hydrate Inhibition Effect of Salt Aqueous Solution Amir H. Mohammadi and Dominique Richon* Centre Energe´ tique et Proce´ de´ s, Ecole Nationale Supe´ rieure des Mines de Paris, CEP/TEP, CNRS FRE 2861, 35 Rue Saint Honore´ , 77305 Fontainebleau, France
Thermodynamic modeling of salt precipitation/solubility and hydrate inhibition effects of salt aqueous solutions are presented in this communication. A modified version of the Patel-Teja equation of state and nondensitydependent mixing rules are used for modeling fluid phases. Salt is treated, in the equation of state, as a pseudo-component, similar to the other components in the system, by calculating its parameters from corresponding cation and anion parameters. A solid-liquid equilibrium theory is used for modeling equilibrium between precipitated salt and aqueous phase. The hydrate phase is modeled using the van der Waals-Platteeuw theory. The model predictions are compared with some selected experimental data reported in the literature on salt solubility and hydrate inhibition characteristics of salt aqueous solutions. The predictions through this model are found in acceptable agreement with the independent experimental data (not used in developing the model), demonstrating its convenient reliability. 1. Introduction Saline water is often present in the hydrocarbon reservoirs and transfer lines, as well as in the surface separation facilities. A major concern with the processes involving saline water is gas hydrate formation causing serious operational, safety, and economical problems.1 High water salinities can pose a serious problem because of the potential for salt deposition in the production, transportation, and processing facilities.2-6 The deposition of salt may result in flow restriction due to salt plug formation, as well as hydrate formation (as a result of neglecting the effect of salt deposition on reducing the overall hydrate prevention characteristics of the system).3-6 It is, therefore, vital that the salt and hydrate free regions of petroleum fluids can be accurately predicted.3-6 Prediction of salt precipitation and hydrate stability zones in the presence of saline water requires reliable thermodynamic model. Most of the models reported in the literature use the wellknown equations of Pitzer7 to describe the thermodynamic properties of the aqueous electrolyte solutions at 298.15 K over wide ranges of ionic strength. This model7 allows for extension of calculations to higher temperatures if some empirical functions for temperature are included.8,9 Other researchers have developed models based on the activity coefficient methods with a number of adjusted parameters, which are applicable at high temperatures and pressures.10,11 Other thermodynamic models have been developed to predict the properties of aqueous electrolyte solutions.3-6,12-26 AasbergPetersen et al.15 adopted a method that divides the fugacity coefficient of each component in the liquid water phase into two terms,
ln φi ) ln φiEoS + ln γiEL
(1)
where φ is the fugacity coefficient. Superscripts EoS and EL stand for equation-of-state and electrolyte terms, respectively, and subscript i represents the nonelectrolyte component in the aqueous phase. In eq 1, an equation-of-state term (short-range * Corresponding author. E-mail:
[email protected]. Tel.: +(33) 1 64 69 49 65. Fax: +(33) 1 64 69 49 68.
interactions), which is employed to calculate the effect of nonionic (molecular) species in the aqueous phase, and a Debye-Hu¨ckel electrostatic term (long-range interactions), which is used for calculating the effect of salts on the fugacity coefficients of molecular species in the solution, are found. It should be mentioned that, in eq 1, salt is not considered as a component in the equation of state (EoS), but the model takes into account its effect on the fugacity coefficient of other molecular species.3-6,15 There are some models that consider salt as a pseudocomponent in an equation of state by defining its critical properties.3-6,25,26 However, some of these models may have a deviation in their predictions from the literature data, and also their proposed critical properties for salts may be in discrepancy from the reported values in the literature.27-29 Only a few investigations have been carried out concerning salt precipitation.3-6,16-20 The gas hydrate inhibition effect of electrolyte solutions has been investigated and modeled by several workers.21-24 However, the majority of these models consider only the effect of the presence of salt on the fugacity of water and gases and, therefore, are not able to predict salt precipitation. It is, therefore, evident that few works in the literature cover the simultaneous representation of salt precipitation and gas hydrate inhibition effects of saline solutions,3-6 because considering the effect of electrolytes on the fugacity of water, models are not able to predict potential salt precipitation.3-6,15 The aim of this work is to develop a thermodynamic model that is capable of predicting salt precipitation and the hydrate stability zones in the presence of salt aqueous solutions. This model should be particularly advantageous in that it will allow for the prediction of salt precipitation from saline water. The vapor and liquid phases are modeled using a modification of the Patel-Teja equation of state (PT-EoS)30 and nondensitydependent mixing rules (NDD),31 in which salt is treated as a component in the equation of state by calculating its parameters from corresponding cation and anion parameters.32 The theory of solid-liquid equilibria is applied to aqueous solution, in order to develop a model for predicting salt precipitation. The hydrate phase is modeled using the van der Waals-Platteeuw theory.33 The predictions of the model are finally compared with some selected data from the literature on salt precipitation/solubility
10.1021/ie061686s CCC: $37.00 © 2007 American Chemical Society Published on Web 06/15/2007
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5075
and hydrate phase equilibria in the presence of salt aqueous solutions. The results can provide a better understanding of salt precipitation and hydrate inhibition effects of salts associated with the petroleum industry. 2. Thermodynamic Model 2.1. Fluid-Phase Model. A modification of the Patel-Teja equation of state30 and nondensity-dependent mixing rules31 is used for modeling the fluid phase(s), in which salt is treated as a component in the equation of state, similar to other components in the system by calculating its parameters from corresponding cation and anion parameters.32 Using this EoS and the associated mixing rules (the details of the equation of state and mixing rules are given in the Appendix), the fugacity of each component (fi) in all fluid phases is calculated from
fi ) xiφiP
(9)
where fLi and fSi stand for the fugacity of species i in aqueous and solid phases, respectively. When the solvent does not enter the solid phase, the fugacity of the solid phase, salt, remains that of the pure solid. The ratio of fugacity of pure solid at subcooled conditions to that of solid at the same temperature, fLi /fSi , considering the effect of pressure and assuming that the volume change on melting is independent of pressure, is expressed as3-6,37
fLi
(
)
∆hfi T 1 1+ ln S ) RT T RT f m i
∆Cp dT + m T
∫TT ∆Cp dT - R1 ∫TT m
∆V h fi (P - PTm) (10) RT
(2)
where P is the pressure and xi is the mole fraction. The fugacity coefficient of component i, φi, is calculated using the EoS.1 2.1.1. Electrolyte Model. Zuo and Guo34 proposed the following equations for calculating a, b, and c parameters of PT-EoS35 for anions and cations,
a ) 2.57012πNav2σ3hf
(3)
c ) b ) (2/3)πNavσ3
(4)
where Nav is Avogadro’s number, σ is the ionic diameter, hf is an empirical constant set to 6 in the original paper, and is the ionic energy parameter estimated from dispersion theory,36 -8
fLi ) fSi
0.5 1.5 -6
) (2.2789 × 10 )η χ σ kh
(5)
where η is the number of electrons in the considered ion, χ is the polarizability of the considered ion, and kh is Boltzmann’s constant. Masoudi et al.32 later extended the earlier method to the Valderrama modification of the Patel-Teja equation of state (VPT-EoS)30 along with NDD mixing rules.31 Equations 3 and 4 were used to calculate the a, b, and c parameters in the EoS for individual ions. The following relations were suggested and used for calculating the a, b, and c for the resulting salts,32
asalt ) xaanaca
(6)
bsalt )
ban + bca 2
(7)
csalt )
can + cca 2
(8)
where subscripts salt, an, and ca stand for salt, anion, and cation, respectively. In the method of Masoudi et al.,32 binary interaction parameters (BIPs) between salt and water were optimized using water freezing point depression and boiling point elevation data. A similar method was used to optimize salt-salt BIPs.32 Gas solubility data in aqueous electrolyte solutions were used in the optimization of gas-salt BIPs.32 2.2. Salt Precipitation/Solubility Model. In order to predict salt precipitation/solubility from/in the aqueous phase, it is necessary to examine solid-liquid equilibria. In isothermal systems, the equality of fugacities can be expressed as3-6,37
where ∆V h fi , PTm, ∆Cp, and ∆hfi are volume change on melting, pressure at the melting point, heat capacity change on melting, and enthalpy change of melting, respectively. R, Tm, and T represent universal gas constant, melting temperature, and temperature, respectively. In order to apply eq 10, the data on melting point temperature, along with volume, heat capacity, enthalpy change on melting for each salt, and fugacity of the pure subcooled liquid salt are required. The latter can be calculated using the EoS for the pure salt. The reported heat capacities of liquid and solid sodium chloride are the same up to 800 K, but they become different at higher temperatures. Therefore,3-6,38
∆Cp ) 0
J T e 800 K mol‚K
(11)
L S where ∆Cp ) Cp,NaCl - Cp,NaCl . L The heat capacity of liquid KCl, Cp,KCl , is reported to be S , is a constant, but for solid KCl, the heat capacity, Cp,KCl 3-6,38 39 function of temperature. Robie et al. proposed the L S following equation to calculate Cp,KCl - Cp,KCl ,3-6,39
∆Cp ) A + BT +
D J C h + xT T2 mol‚K
(12)
where A ) 98.113, B ) -4.852 × 10-2, C h ) -1.371 × 103, and D ) 1.605 × 106. Other model requirements, enthalpy change on melting and melting point temperature, as well as volume difference between liquid and crystal salt, are presented in Table 1 for NaCl and KCl.3-6,38,39 The fugacity of the salt in the solid phase is finally calculated from eq 10. Then, from the fugacity equality conditions, we get the equilibrium relations that equate the fugacity of the pure solid and the fugacity of the salt in the aqueous phase (which was calculated from the EoS). This enables the prediction of the potential salt precipitation/solubility conditions. 2.3. Gas Hydrate Model. The fugacity of water in the hydrate phase, fHw, is given by40
(
fHw ) fβw exp -
)
∆µβ-H w RT
(13)
where fβw is the fugacity of water in the empty hydrate lattice. is the chemical potential difference of water between ∆µβ-H w the empty hydrate lattice, µβw, and the hydrate phase, µHw, and is
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Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007
Figure 1. Boundary of NaCl solubility in water at various temperatures: (O), experimental solubility data;41,42 solid curve, model predictions.
Figure 2. Boundary of KCl solubility in water at various temperatures: (O), experimental solubility data;41 solid curve, model predictions.
Table 1. Enthalpy of Melting (∆hfi ) and Melting Point Temperature h fi ) of NaCl and (Tm) as Well as Volume Change on Melting (∆V KCl3-6,38,39 salt
∆hfi /(kJ/mol)
Tm/K
∆V h fi /(cm3/mol)
NaCl KCl
28.158 26.284
1074 1044
6.3 6.4
obtained from the van der Waals and Platteeuw theory.33 The fugacity of water in the empty hydrate lattice, fβw, is given by the method of Anderson and Prausnitz.40 3. Results and Discussions As mentioned earlier, Masoudi et al.32 used the electrolyte model introduced earlier to model freezing point depression and boiling point elevation as well as gas solubility over wide ranges of pressure, temperature, and salt concentrations. Systems including NaCl and KCl were studied. Water-salt interaction parameters were optimized, using freezing point depression and boiling point elevation data of the aqueous solutions, which provided a water-salt phase behavior model over a wide temperature range (i.e., 248-398 K).32 To model mixed electrolyte aqueous solutions, experimental data on boiling point elevation and freezing point depression of NaCl and KCl aqueous solutions were used to optimize the interaction parameters between NaCl and KCl.32 The gas solubility data in aqueous electrolyte solutions were used for optimizing gassalt interaction parameters.32 However, no attempt was made to evaluate the performance of the model for predicting salt precipitation/solubility and hydrate inhibition effects of salts. In order to extend this model for predicting salt precipitation/ solubility, all the electrolyte model parameters were set to those previously adjusted and reported by Masoudi et al.32 However, the interaction parameters between water and salt, which were already adjusted using freezing point depression and boiling point elevation data of aqueous electrolyte solutions,32 were slightly tuned to reduce the errors between experimental and predicted salt solubility data. Figures 1 and 2 show experimental and predicted boundaries of salt (NaCl and KCl, respectively) solubility in water at various temperatures. As can be seen, acceptable agreement is achieved, demonstrating the capability of the model for estimating salt precipitation/solubility from/in aqueous solutions. In Figures 3 and 4, the capability of the model is investigated for predicting hydrate phase boundaries of methane in the presence of NaCl and NaCl + KCl aqueous solutions, respectively. As can be observed, the model with no adjustable parameters (based on hydrate phase equilibrium data)
Figure 3. Hydrate phase boundary of methane in the presence of NaCl aqueous solutions; experimental data: (∆), 3 wt % NaCl;43 (O), 10 wt % NaCl;44 (]), 20 wt % NaCl;44 bold solid curves, predictions of this model; solid curves, predictions of the HWHYD model.45
Figure 4. Experimental43 and predicted hydrate phase boundary of methane in the presence of aqueous solutions composed of various concentrations of NaCl and KCl; experimental data:43 (∆), 3 wt % NaCl + 3 wt % KCl; (0), 5 wt % NaCl + 5 wt % KCl; (]), 5 wt % NaCl + 10 wt % KCl; (+), 5 wt % NaCl + 15 wt % KCl; (O), 10 wt % NaCl + 12 wt % KCl; (×), 15 wt % NaCl + 8 wt % KCl; bold solid curves, predictions of this model; solid curves, predictions of the HWHYD model.45
yields promising results. In general, acceptable agreements are achieved between the predictions of the model and experimental data reported in the literature. The deviations can be attributed to unreliability of some experimental data, as logarithm of hydrate dissociation pressure versus temperature of the system is approximately linear and, therefore, any deviation of experi-
Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007 5077
mental data from this behavior can indicate unreliability of the data. The model predictions for hydrate phase boundaries have also been compared with the predictions of a well-proven thermodynamic model, the HWHYD model (the Heriot-Watt University Hydrate model),45 which is capable of predicting different scenarios in hydrate phase equilibrium calculations. A detailed description of this model is given elsewhere.46,47 The model45 is briefly based on equality of fugacity concept, which uses the Valderrama modification of the Patel-Teja equation of state30 and nondensity-dependent mixing rules31 for modeling the fluid phases. The model uses the van der Waals and Platteeuw theory33 for modeling the hydrate phase. However, the HWHYD model45 cannot be used for predicting salt precipitation conditions, as this tool is based on the electrolyte model of Aasberg-Petersen et al.,15 in which salt is not considered as a component in the EoS, but the model takes into account its effect on the fugacity coefficient of other molecular species, as mentioned earlier. Good agreement between the results of two models in Figures 3 and 4 demonstrates the capability of the new model developed in this work for estimating hydrate inhibition effects of salt aqueous solutions.
The coefficients Ωaj, Ωbj, Ωcj, and F are given by
Ωaj ) 0.661 21 - 0.761 05Zc
(A.6)
Ωbh ) 0.022 07 + 0.208 68Zc
(A.7)
Ωcj ) 0.577 65 - 1.870 80Zc
(A.8)
F ) 0.462 86 + 3.582 30(ωZc) + 8.19417(ωZc)2
where Zc is the critical compressibility factor and ω is the acentric factor. Tohidi-Kalorazi47 relaxed the alpha function for water, Rw(Tr), using experimental water vapor pressure data in the range of 258.15-374.15 K, in order to improve the predicted water fugacity:
Rw(Tr) ) 2.4968 - 3.0661Tr + 2.7048Tr2 1.2219Tr3 (A.10) Nonpolar-nonpolar binary interactions in fluid mixtures are described by applying classical mixing rules as follows,31
∑i ∑j xixjaij
(A.11)
b)
∑i xibi
(A.12)
c)
∑i xici
(A.13)
a ) ajR )
4. Conclusions A previously reported equation of state based model, in which salt is treated as a pseoudo-component in the equation of state by calculating its parameters from corresponding cation and anion parameters, was combined with a solid-liquid equilibrium theory based model to predict precipitation/solubility of salt (NaCl and KCl) from/in aqueous solution. Promising results were obtanied when the model predictions were compared with some selected experimental data from the literature. The model was further developed for estimating hydrate inhibition effects of salt (NaCl and KCl) aqueous solutions by employing a van der Waals-Platteeuw theory based hydrate phase equilibrium model, and the validity of the model was successfully examined against some experimental data reported in the literature. Appendix The VPT EoS30 is given by
ajR(Tr) RT P) V h -b V h (V h + b) + c(V h - b) with
aj )
ΩajR2Tc2 Pc
(A.2)
ΩbhRTc Pc
(A.3)
b)
aij ) (1 - kij)xaiaj
ΩcjRTc c) Pc
(A.4)
R(Tr) ) [1 + F(1 - Trψ)]2
(A.5)
where P is the pressure, T represents the temperature, V h stands for the molar volume, R is the universal gas constant, and Ψ ) 1/ . The subscripts c and r denote critical and reduced properties, 2 respectively.
(A.14)
where a, b, and c represent equation of state parameters and xi stands for mole fraction of component i. kij is the binary interaction parameter. For polar-nonpolar interaction, however, the classical mixing rules are not satisfactory and more complicated mixing rules are necessary. In this work, the NDD mixing rules developed by Avlonitis et al.31 are applied to describe mixing in the a-parameter,
a ) a C + aA (A.1)
(A.9)
(A.15)
where aC is given by the classical quadractic mixing rules (eqs A.11 and A.14). The term aA corrects for asymmetric interaction, which cannot be efficiently accounted for by classical mixing rules,31
∑P x2P ∑i xiaPilPi
(A.16)
aPi ) xaPai
(A.17)
0 1 lPi ) lPi - lPi (T - T0)
(A.18)
aA )
0 1 where lPi and lPi stand for binary interaction parameters in NDD mixing rules and P is the index of polar components. The BIPs of the NDD mixing rules used in this work are given elsewhere.3-6,47
Nomenclature BIP ) binary interaction parameter
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Ind. Eng. Chem. Res., Vol. 46, No. 14, 2007
EoS ) equation of state NDD ) nondensity-dependent mixing rules PT ) Patel-Teja VPT ) Valderrama-Patel-Teja A ) constant B ) constant C ) heat capacity C h ) constant D ) constant F ) parameter in equation of state N ) Avogadro’s number P ) pressure R ) universal gas constant T ) temperature V h ) molar volume Z ) compressibility factor a ) parameter in equation of state aj ) parameter in equation of state b ) parameter in equation of state c ) parameter in equation of state f ) fugacity hf ) empirical constant set to 6 h ) molar enthalpy k ) binary interaction parameter in classical mixing rules kh ) Boltzmann’s constant l ) binary interaction parameter in NDD mixing rules x ) mole fraction Subscripts P ) polar component Tm ) melting point an ) anion av ) Avogadro ca ) cation salt ) salt aj ) parameter in equation of state bh ) parameter in equation of state cj ) parameter in equation of state c ) critical property i ) component i j ) component j m ) melting point p ) pressure r ) reduced property w ) water 0 ) reference value Superscripts EL ) electrostatic contribution EoS ) equation-of-state contribution A ) asymmetric term C ) classical term H ) hydrate L ) liquid S ) solid f ) fusion β ) empty hydrate lattice Ψ ) parameter in equation of state 0 ) binary interaction parameter in NDD mixing rules 1 ) binary interaction parameter in NDD mixing rules Greek ∆ ) difference Ω ) parameter in equation of state R ) temperature-dependent parameter in equation of state
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(35) Patel, N. C.; Teja, A. S. A New Cubic Equation of State for Fluids and Fluid Mixtures. Chem. Eng. Sci. 1982, 37 (3), 463-473. (36) Mavroyannis, C.; Stephen, M. J. Dispersion forces. Mol. Phys. 1962, 5, 629-638. (37) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1998. (38) Chase, M. W.; Davies, C. A.; Downey, J. R.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF (Joint Army, Navy, Air Force) Thermochemical Tables. J. Phys. Chem. Ref. Data 1985, 14 (Suppl. 1), 756-770. (39) Robie, R. A.; Hemingway, B. S.; Fisher, J. R. Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 pascals) pressure and at higher temperatures. USGS Bull. 1978, 1452, 456. (40) Anderson, F. E.; Prausnitz, J. M. Inhibition of Gas Hydrates by Methanol. AIChE J. 1986, 32 (8), 1321-1333. (41) Stephen, H.; Stephen, T. Solubilities of Inorganic and Organic Compounds; Pergamon Press: Oxford/London/New York/Paris, 1963. (42) Breton, A. M. Postgraduate Thesis in Physical Sciences, University of Pau, France, June 6, 1967 (Quoted in ref 3-6). (43) Dholabhai, P. D.; Englezos, P.; Kalogerakis, N.; Bishnoi, P. R. Equilibrium Conditions for Methane Hydrate Formation in Aqueous Mixed Electrolyte Solutions. Can. J. Chem. Eng. 1991, 69, 800-805. (44) Kobayashi, R.; Withrow, H. J.; Williams, G. B.; Katz, D. L. Gas Hydrate Formation with Brine and Ethanol Solutions. Proc. 30th Ann. ConV. Nat. Gasoline Assoc. Am. 1951, 27-31. (45) Heriot-Watt University Hydrate model: http://www.pet.hw.ac.uk/ research/hydrate/ (accessed Dec 2006). (46) Avlonitis, D. Thermodynamics of Gas Hydrate Equilibria. Ph.D. Thesis, Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, U.K., 1992. (47) Tohidi-Kalorazi, B. Gas Hydrate Equilibria in the Presence of Electrolyte Solutions. Ph.D. Thesis, Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, U.K., 1995.
ReceiVed for reView December 29, 2006 ReVised manuscript receiVed April 23, 2007 Accepted May 7, 2007 IE061686S
