a thermodynamic model for gas hydrates in the

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model of aqueous electrolyte solutions to predict hydrate formation conditions in .... NRTL electrolyte model (Aspen Plus, 1988) has been applied to correlate.
I108T001046 . 108 T001046d.108 Chem. Eng. Comm., 2000, Vol. 00, pp. 1 ± 18 Reprints available directly from the publisher Photocopying permitted by license only

# 2000 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. Printed in Malaysia.

A THERMODYNAMIC MODEL FOR GAS HYDRATES IN THE PRESENCE OF SALTS AND METHANOL JULIAN YOUXIANG ZUOa,*, DAN ZHANGa and ERLING H. STENBYb a

DB Robinson Research Ltd., 9419-20 Avenue, Edmonton, Alberta, Canada T6N IE5; b Engineering Research Center IVC-SEP, Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800, Lyngby, Denmark (Received 11 November 1999; In ®nal form 1 June 2000)

The equation of state (EOS) for aqueous electrolyte solutions developed by F urst and Renon (1993) has been extended to predict vapor ± liquid equilibria (VLE) of ternary water ± methanol ± salt systems. The model parameters have been determined by ®tting only binary data and related to the cationic Stokes diameters. The predictions of vapor ± liquid equilibria of ternary water ± methanol ± salt systems are in good agreement with experimental data. Then the extended EOS has been utilized to develop a predictive method for gas hydrate formation conditions in the presence of electrolytes and methanol. The new hydrate method employs the Barkan and Sheinin (1993) hydrate model for the description of the hydrate phase, the extended EOS for the vapor phase fugacity and for the activity of water in the aqueous phase. The agreement is good between the predicted hydrate formation pressures and experimental data. Keywords: Equation of state; Electrolyte; Mixed-solvent; Modeling; Gas hydrate; Inhibitor

INTRODUCTION The formation of gas hydrates can result in serious problems during the exploitation, transportation and processing of wet natural gas. For avoiding the formation of gas hydrates, inhibitors like methanol and electrolytes are usually used to lower the hydrate formation temperature. It is, therefore, of great importance to obtain gas hydrate formation conditions. *Corresponding author. Tel.: 780-463-8638, Fax: 780-450-1668, e-mail: [email protected] 1

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Holder et al. (1988) and Sloan (1990) have extensively compiled hydrate equilibrium data available in the literature for natural gas components in pure water and in the presence of inhibitors like alcohols. In the past few years, the measured hydrate formation conditions of natural gas components in aqueous single and mixed electrolyte solutions have been reported by several researchers (Dholabhai et al., 1991, 1993; Englezos and Bishnoi, 1991; Bishnoi and Dholabhai, 1993; Dholabhai and Bishnoi, 1994; Englezos and Hall, 1994; Mei et al., 1996). More recently, Dholabhai et al. (1996) measured the equilibrium data of the hydrate formation of carbon dioxide in mixed solvent (water and methanol) electrolyte solutions. Although the models for predicting hydrate formation conditions of natural gas in pure water have been available to the industry for a long time, it is only the past few years that the predictive models for aqueous systems containing methanol or electrolytes have been reported. Englezos and Bishnoi (1988) coupled the statistically thermodynamic hydrate model of van der Waals and Platteeuw (1959) with the available activity coecient model of aqueous electrolyte solutions to predict hydrate formation conditions in aqueous systems containing single or mixed electrolytes. Although this model produces excellent results for systems with substances sparingly soluble in water (e.g., light hydrocarbons and nitrogen), it is not suitable for gases like CO2 and H2S which have signi®cant solubilities in the aqueous phase. To solve this problem, Englezos (1992) used an electrolyte equation of state (EOS) to calculate the hydrate formation conditions of carbon dioxide only in aqueous NaCl solutions. Tse and Bishnoi (1994) proposed a method to predict the hydrate formation conditions of carbon dioxide in aqueous electrolyte solutions. The method utilizes the hydrate model of van der Waals and Platteeuw (1959) to describe the solid hydrate phase. Three di€erent models were examined for the representation of the liquid phase: Chen and Evans (1986); Zuo and Guo (1991) and AasbergPetersen et al. (1991). It was found that the model of Zuo and Guo (1991) gave the best results and the predictions agree well with experimental data. However, the methods of Englezos (1992) and of Tse and Bishnoi (1994) are complicated because the solubility of carbon dioxide in aqueous electrolyte solutions has to be calculated in each iterative step. Recently, Zuo et al. (1996) developed a generalized hydrate model to predict hydrate formation conditions in aqueous alcohol solutions or in aqueous electrolyte solutions. The model has a good accuracy in predicting hydrate formation conditions. However, all the models mentioned above are not suitable for the situations where another solvent like methanol is present in addition to electrolytes.

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The objective of the present work is to develop a new method for the prediction of hydrate formation conditions in water ± methanol (mixedsolvent) electrolyte solutions. To accomplish this objective, the aqueous electrolyte equation of state proposed by F urst and Renon (1993) has ®rst been extended to water ± methanol electrolyte systems. The model parameters have been determined by ®tting vapor ± liquid equilibrium (VLE) data of only binary systems. The predictions of VLE for ternary water ± methanol electrolyte systems agree very well with experimental data. Then, the extended EOS has been combined with the Barkan and Sheinin (1993) hydrate model in pure water to develop a predictive method for hydrate formation conditions. No adjustable parameters are required in addition to those for the hydrate model in pure water and for the extended electrolyte EOS. The new method has been applied to predict hydrate formation conditions of a great number of aqueous electrolyte systems and water ± methanol electrolyte systems with a good accuracy. EXTENDED ELECTROLYTE EQUATION OF STATE F urst and Renon (1993) derived an aqueous electrolyte equation of state based on an expression of the Helmholtz free energy. The molar Helmholtz energy of an electrolyte mixture consists of four contributions:           a a a a a ˆ ‡ ‡ ‡ …1† RT RT RF RT SR1 RT SR2 RT LR The ®rst term (RF) is relative to repulsive forces, the second one (SR1) represents attractive short-range interactions involving no ions. The two last terms are speci®c to ionic contributions, the ®rst one (SR2) represents solvation interactions and the second one (LR) is related to long-range interactions. This model may be considered as an extension of a classical cubic nonelectrolyte EOS and the RF and SR1 terms are similar to the corresponding terms of the SRK equation of state modi®ed by Schwartzentruber et al. (1989). The LR contribution is expressed using a simpli®ed version of the MSA model. The SR2 term is a speci®c one and involves symmetrical cation-solvent (Wcs) and cation-anion (Wca) interaction parameters. The other interactions involving ions (cation ± cation, anion ± anion or anionsolvent) are ignored due to charge repulsive e€ects and low solvation of

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anions (compared to cations). The others ionic adjustable parameters are the anionic (ba) and cationic (bc) covolumes. Hence the model contains up to four kinds of adjustable ionic parameters. However, it has been shown (F urst and Renon, 1993) that all these parameters could be related to Stokes diameters cS ( for cations) and Pauling diameters aP ( for anions): bc ˆ 1 …cS †3 ‡ 2

and ba ˆ 1 …aP †3 ‡ 2

…2†

and Wac ˆ 5 …cS ‡ aP †4 ‡ 6

…3†

And, for interaction parameters: Wcs ˆ 3 cS ‡ 4

The use of Pauling diameters in the case of anions is justi®ed by their lower solvation. Correlation coecients 1 ± 6 have been deduced from a data treatment of numerous experimental coecients for aqueous halide and non-halide systems (F urst and Renon, 1993). Table I gives the values of Stokes cationic diameters used in the calculations and 1 ± 6 of halide systems. To extend the EOS to water ± methanol electrolyte systems, the model parameters have to be determined from some experimental data. For pure molecular components, critical properties and acentric factors are taken from Reid et al. (1987). The pure component parameters are the volume translation parameters c and the three polar parameters p1, p2 and p3 used for the calculation of pure component energy parameters of molecular compounds aSR s : asSR ˆ

p …RTc †2 ‰1 ‡ m…!†…1 ÿ Tr † ÿ p1 …1 ÿ Tr †…1 ‡ p2 Tr ‡ p3 Tr2 †Š2 1=3 9…2 ÿ 1† Pc 1

…4†

TABLE I Cationic stokes diameters ( Aî ) in single solvents at 298.15 K and 1 ± 6 of halide systems ki,  106 Ion ‡

Li Na ‡ K‡ Cs ‡ Rb ‡ Ca2 ‡ * The ®tted value.

Water

Methanol

i

4.76 3.68 2.50 2.36 2.38 6.50

7.56 6.54 5.56 4.80 5.64 7.50*

1 2 3 4 5 6

Water 0.10688 6.5665 35.090 6.0040 ÿ 0.4304 ÿ 27.510

Methanol 0.10688 6.5665 82 .398 ÿ 422.45 ÿ 0.007162 ÿ 402.90

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where m…!† ˆ 0:48508 ‡ 1:55171! ÿ 0:15613!2

…5†

The parameters are determined by ®tting experimental vapor pressure and density data of pure substances (Boublik et al., 1973; Reid et al., 1987), respectively. The values of these parameters are given in Table II. In the SR2 term of the model, we have also to introduce some values for the molecular diameters. In the case of water, the chosen value ( ˆ 2.5  10 ÿ 10 m) is small compared to the published estimations of water diameters. The reason is that this parameter has some in¯uence on the quality of the representation of excess properties. The smallest values are associated to the best results. In the present work, it is assumed that the same value could be given to the diameter of methanol. To extend the model more easily to mixed-solvent electrolyte systems, it is assumed that the values of cationic covolumes (bc) and size parameters (c) in methanol electrolyte solutions are the same as in aqueous electrolyte solutions. On the contrary, binary interaction parameters (Wij) are solvent dependent and estimated from Eq. (3). However, coecients 3 ± 6 in Eq. (3) are determined by ®tting the experimental vapor pressure data of binary salt ± methanol mixtures at 298.15 K. The optimized values (3 ± 6) are also given in Table I. Note that the Stokes diameter of calcium used in the calculation is less than experimental one. The ion pairing may be present in CaCl2 ± methanol mixtures. In this case, the solvation and the Stokes diameter of calcium decrease. If the ion pairing is taken into account, the association constant has to be included and to be adjusted. This makes the model more complicated. Therefore, the Stokes diameter of calcium is ®tted to experimental vapor pressure data. Table III summarizes the correlated vapor pressure results. The overall average relative deviation is 0.57%. To extend the model to mixed-solvent systems, the Wong-Sandler mixing rule (1992) with the UNIQUAC model has been introduced into the model. For binary water (1) and methanol (2) mixtures, adjustable parameters are k12 ˆ k21 (set to zero),  12 and  21, where  12 and  21 ( ij ˆ Uij/RT ) are the TABLE II Parameters (p1, p2, p3 and c) of pure solvents Solvent Methanol Water

p1 ÿ 0.023387 0.023175

p2 ÿ 15.960 4.6462

p3 21.509 ÿ 8.8079

c,  10 ÿ 5 m3/mol 1.29912 0.60473

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J. Y. ZUO et al.

TABLE III Average absolute deviations of the calculated vapor pressures for binary methanol-salt systems at 298.15 K Solvent ‡ Salt Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Overall

‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

Bu4NI Am4NBr Bu4NBr Et4NBr CaCl2 CuCl2 KI NaI NaBr LiCl LiBr CsI RbI KBr NaCl

No. of data points

AAD, %

43 29 35 30 16 11 55 66 37 18 11 41 46 23 33 494

0.47 0.83 0.69 0.35 1.23 0.93 0.24 0.84 0.36 2.33 1.99 0.21 0.20 0.19 0.17 0.57

Maximum molality

Source of data

0.907 1.560 1.651 1.874 2.635 3.396 1.122 4.520 1.556 5.355 4.345 0.130 0.436 0.134 0.219

b b b b c,e c a,c a,c,d a,c c,d f a a a a

Source of Data: (a) Barthel et al. (1986a); (b) Barthel et al. (1986b); (c) Bixon et al. (1979); (d ) Tomasula et al. (1987); (e) Hongo et al. (1990).

energy parameters in the UNIQUAC model. Two adjustable parameters ( 12 and  21) have been determined by ®tting the corresponding binary vapor ± liquid equilibrium data (Gmehling and Onken, 1977).  12 ˆ 0.3742 and  21 ˆ ÿ 0.1268. When the model is applied to ternary aqueous-methanol salt systems, the model parameters are estimated by Eqs. (2) and (3), which are determined from experimental data only in binary single (water or methanol) solvent electrolyte systems at 298.15 K and treated as temperature independent. The dielectric constant of the mixed solvent is evaluated by a linear volume fraction-mixing rule. The interaction parameters between cations and anions (Wca) in the mixed solvent are estimated by a linear mixing rule of solvent mole fractions because they are di€erent in single solvents (water or methanol): Wca ˆ

2 X iˆ1

…i† x0i Wca

…6†

where x0i is the salt-free mole fraction of solvent i and W (i) is the parameter Wca in the solvent i. Vapor ± liquid equilibria for some ternary systems containing water, methanol and a salt are predicted by use of the new model. In addition, the NRTL electrolyte model (Aspen Plus, 1988) has been applied to correlate

333.15 K 1.013 bar 1.013 bar 1.013 bar 1.013 bar

1.013 bar 298.15 K 1.013 bar 1.013 bar 1.013 bar

P

T, or 0.05245 0.04641 0.03455 0.06497 0.09195 0.13224 0.12300 0.03348 0.06564 0.09195 0.07101

xsalt

Max. 12 40 32 36 26 8 25 16 23 19 33 270

Np

0.57

0.61 0.86

1.63

0.56 0.37 0.34

T, K

This work 0.01364 0.01376 0.01922 0.02270 0.02071 0.01536 0.01150 0.01285 0.01916 0.01458 0.01479 0.01669

y1

This work 0.01142 0.01182 0.01731 0.01221 0.01748 0.01456 0.01245 0.01443 0.01705 0.00887 0.01011 0.01337

y1

NRTL model f g h i i j k h h i i

Source of data

Source of Data: (f). Nishi (1975); (g). Kumagae et al. (1992); (h) Boone et al. (1976); (i). Morrison et al. (1990); (j) Raastchen et al. (1987); (k) Hala (1983).

CaCl2 CaCI2 KCI KCl KBr LiBr LiCI LiCl NaBr NaBr NaCl

Water Water Water Water Water Water Water Water Water Water Water

Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Overall

‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

Salt

Solvent

System

TABLE IV Average deviations of the predicted VLE for salt ± methanol ± water mixtures by use of the extended FR EOS and the electrolyte NRTL model

I108T001046 . 108 T001046d.108

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J. Y. ZUO et al.

FIGURE 1 Vapor ± liquid equilibria for LiCl ± methanoal ± water mixtures at 101.32 kPa.

the same data sets. In the NRTL electrolyte model, two or three parameters are usually required to be adjusted for each system. In the present work, the methanol ± salt nonrandomness factor is set equal to the best value obtained by Mock et al. (1986). Two interaction parameters between methanol and a salt are determined by ®tting VLE data of each ternary system and di€erent parameters are used for isobaric and isothermal VLE data for the same ternary system, respectively. The calculated results by application of both models are summarized in Table IV. The NRTL electrolyte model produces slightly better results than the new model. By considering that all the parameters of the extended EOS are determined only from binary data instead of ternary data, the predictions of the new model are satisfactory. Figure 1 typically shows VLE of LiCl ± methanol ± water mixtures at 101.32 kPa. The predictions are in good agreement with experimental data.

HYDRATE THERMODYNAMIC MODEL For a system in which gas, solid hydrate and aqueous liquid phases coexist at equilibrium, the following condition must be satis®ed: H ˆ W

…7†

where H stands for the di€erence between the chemical potential of water in the empty hydrate lattice and in the hydrate, and W denotes di€erence between chemical potential of water in the empty hydrate lattice and in the aqueous solution.

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W is evaluated by the following equation: W 0 ˆ ‡ RT RT

Z 0

P

vW dp ÿ RT

Z

T

T0

 hW dT ÿ ln aW RT 2

…8†

where T0 ˆ 273.15 K;  hW and vW are the di€erences of water molar enthalpies and of water volumes between the empty hydrate lattice and ice (or pure water), respectively; aW denotes the water activity in the aqueous phase (calculated by the extended EOS). H is determined by the van der Waals and Platteeuw (1959) equation:   2 X H X ˆ j ln 1 ‡ Cij fi RT i jˆ1

…9†

where  j is a crystal chemical constant characterizing a ratio of the number of cavities of the jth type to the number of water molecules in the elementary hydrate cell; fi denotes the fugacity of hydrate former i (calculated by the extended EOS); Cij stands for the Langmuir constant of hydrate former i in the jth type cavity, which is calculated by the Barkan and Sheinin (1993) expression. Barkan and Sheinin (1993) proposed an improved general technique for calculating gas hydrate formation conditions in pure water, based on the statistical theory of gas hydrates. The Langmuir constant of gas i in the jth type cavity is expressed as:   Z ÿWij …r† 2 4 Rj Cij ˆ r dr exp …10† kT kT 0 where Rj is the radius of the jth type cavity. W(r) denotes the smoothed cell potential function: Wij …r† ˆ W1ij …r† ‡ W2ij …r† ‡ W3ij …r†

…11†

where W1, W2 and W3 are the contributions of the ®rst, the second and the third shells of water molecules. The thermodynamic properties and model parameters speci®c to gas hydrates in pure water are taken from Barkan and Sheinin (1993). It should be noted that gas solubilities in the aqueous phase are not required to be taken into account in the Barkan and Sheinin hydrate model. Compared to the van der Waals-Platteeuw hydrate model, this is an advantage and makes the calculation simpler when hydrate formation

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conditions of more soluble gases like carbon dioxide or hydrogen sul®de are predicted.

RESULTS AND DISCUSSION Gas Hydrates in Aqueous Electrolyte Solutions The new method has been extensively tested for predicting hydrate formation conditions in aqueous single and mixed electrolyte solutions. Since no parameters of bicarbonate are available in the original EOS, it is assumed that the Pauling diameter of bicarbonate is 3.9 A. î As the solubility of a bicarbonate system is small compared to a halide system and the activity coecient of water is very close to unity, this assumption has a slight in¯uence on the prediction of hydrate formation conditions. The model parameters related to bicarbonate can be estimated by Eqs. (2) and (3). The average absolute deviation (AAD) of the predicted hydrate formation pressures is 4.3% for 95 systems with 521 data points. The predictions by use of the new method match experimental data very well. Gas Hydrates in Water ± Methanol Electrolyte Solutions When the extended EOS is directly applied to predict hydrate formation conditions in water ± methanol electrolyte solutions, the e€ect of a salt on the water activity is over-predicted. The predicted hydrate formation pressures are always higher than experimental data. As we know, in the previous section, the cationic Stokes diameters in aqueous solutions have been treated as temperature independent and good predictions of water activity (hydrate formation conditions) have been obtained. This means that the cationic Stokes diameters in aqueous solutions are slightly dependent on temperature. In fact, Robinson and Stokes (1970) reported experimental ionic limiting conductance data at 273.15 ± 373.15 K, which can be converted into the Stokes diameters by the Stokes law. The temperature dependence is slight and can be ignored. However, the temperature dependence of the cationic Stokes diameters in methanol solutions can not be ignored. Barthel et al. (1986c) measured conductance data of some 1:1 electrolytes in methanol at temperatures less than 298.15 K. These data have been used to estimate the temperature dependence of the cationic Stokes diameters in

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methanol with the help of the Stokes law. The temperature dependence of the cationic Stokes diameters in methanol is expressed as:   1 1 s cs ˆ c;298:15 ‡ A…T ÿ 298:15† ‡ B ÿ …12† T 298:15 where c,298.15 is the cationic Stokes diameters at 298.15 K. A and B are two adjustable parameters. For simplicity, it is assumed that A and B are universal constants for all the cations in methanol, which are determined by ®tting the experimental Stokes diameters, A ˆ 0.075818 and B ˆ 8601.3. The interaction parameters between cations and methanol and between cations and anions in methanol are estimated by means of Eqs. (3) and (12). Figure 2 compares the measured and the predicted hydrate formation conditions of carbon dioxide in aqueous-methanol solutions and in aqueous-methanol KCl solutions. The average deviation of the predicted pressures is 3.0% and the maximum deviation is 11.7%. The measured and the predicted hydrate formation conditions of carbon dioxide in aqueous-methanol NaCl solutions are compared in Figure 3. The average deviation of the predicted pressures is 2.5% and the maximum deviation is 6.1%. A comparison of the measured and the predicted hydrate formation conditions of carbon dioxide in aqueous-methanol CaCl2 solutions is shown in Figure 4. The average deviation of the predicted pressures is 4.9% and the maximum deviation is 12.9%. The inhibiting e€ect of Me5Ca10 (5 mass % methanol and 10 mass % CaCl2) and Me5Ca15 (5 mass % methanol and 15 mass % CaCl2) is somewhat over-predicted. As can be seen from Figures 2 ± 4, the predictions by use of the new method are in good agreement with experimental data. We have also tried to combine the other activity coecient models for mixed-solvent electrolyte systems (Mock et al., 1986; Sander et al., 1986 and Macedo et al., 1990) with the Barkan-Sheinin hydrate model to predict hydrate formation conditions of carbon dioxide in aqueous-methanol salt solutions. It is obviously unsuccessful because these models could not produce accurate activity of water in the aqueous phase at low temperature although they can represent VLE well at temperatures higher than 298.15 K. Dholabhai et al. (1996) made some observations regarding relative inhibition strengths of NaCl and CaCl2 from their experimental data. However, they did not explain these observations. We have tried to explain these observations with the help of the new method. The profound e€ect of inhibitors is seen mathematically from Eq. (8) where a nonlinear term, lnaW, is added to the equation describing the

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FIGURE 2 Hydrate formation conditions of carbon dioxide in aqueous-methanol solutions and in aqueous-methanol KCl solutions.

FIGURE 3 Hydrate formation conditions of carbon dioxide in aqueous-methanol NaCl solutions.

FIGURE 4 Hydrate formation conditions of carbon dioxide in aqeuous-methanol CaCl2 solutions.

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formation from pure water. The pressure/temperature predictions are sensitive to the calculated values of water activities. The water activity is equal to the mole fraction of water multiplied by the activity coecient of water (i.e., aW ˆ XW W ). The smaller the activity of water, the stronger the inhibiting e€ect. OBSERVATION 1 The inhibiting e€ects of NaCl and CaCl2 on a mass basis are close in aqueous solutions. Although the mole fractions of water in aqueous NaCl solutions are always lower than those in aqueous CaCl2 solutions on the same mass basis, the activity coecients of water in aqueous NaCl solutions are higher than those in aqueous CaCl2 solutions. Figure 5 shows the negative logarithm of the activity of water as a function of weight (mass) percents of salts for KCl, NaCl and CaCl2 at 298.15 K and 273.15 K. Experimental data are converted from experimental osmotic coecient data of Robinson and Stokes (1970) at 298.15 K and the curves are calculated by the new model. It can be seen that the values of the activities of water are close in aqueous NaCl and CaCl2 solutions. Therefore, the inhibiting e€ects of NaCl and CaCl2 on a mass basis are close in aqueous solutions. OBSERVATION 2 The inhibiting e€ect in CaCl2 solutions is lower than that in NaCl solutions in the presence of methanol when the compared pairs of solutions containing the same mass percent of methanol. The mole fractions of water in NaCl solutions are obviously lower than those in CaCl2 solutions because the molecular weight of NaCl is lower than that of CaCl2. For example, for Me5Na15 (5 mass % methanol and 15 mass

FIGURE 5 Activity of water in aqueous salt solutions at 298.15 K.

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FIGURE 6 Activity of water in 10 mass % salt ± methanol ± water solutions at 273.15 K.

% NaCl) and Me5Ca15 (5 mass % methanol and 15 mass % CaCl2) solutions, the mole fractions of water are 0.8689 and 0.8878, respectively. Figure 6 shows the activity coecients of water (calculated by use of the new model) in 10 mass % salt-methanol-water solutions at 273.15 K as a function of the salt-free mole fractions of water. The activity coecients of water in NaCl solutions are very close to (or slightly lower than) those in CaCl2 solutions when the salt-free mole percents of water are higher than 80%. Hence the activity of water (aW ˆ XW W ) in NaCl solutions is lower than that in CaCl2 solutions. Consequently, CaCl2 is a weaker inhibitor on the same mass basis than NaCl in the presence of methanol.

CONCLUSION The equation of state for aqueous electrolyte solutions proposed by F urst and Renon has ®rst been extended to predict vapor ± liquid equilibria of ternary water ± methanol ± salt systems. The model parameters are ®tted to only binary data and related to the cationic Stokes diameters. The predictions of vapor ± liquid equilibria of ternary water ± methanol ± salt systems match experimental data very well. Then a predictive method is presented to predict hydrate incipient equilibrium conditions in aqueous solutions containing electrolytes and containing methanol and electrolytes simultaneously. The new method employs the Barkan and Sheinin hydrate model for the description of the hydrate phase, the extended EOS for the vapor phase fugacities and for the activity of water in the aqueous phase. The extensive test results indicate that the new method is capable of

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accurately predicting the hydrate formation conditions in aqueous solutions containing electrolytes and containing methanol and electrolytes simultaneously. The proposed methodology provides a valuable tool for the investigation of the gas hydrate formation conditions in industries.

NOMENCLATURE a AAD,% b c C f k Np P r Rj T v W Wij(r) x

molar Helmholtz free energy (100/Np) j j1 ÿ Pcal/Pexpj j covolume volume translation parameter Langmuir constant fugacity in the vapor phase Boltzmann constant number of data points pressure radial distance of gas molecule from center of a hydrate cavity radius of the jth type cavity temperature molar volume interaction parameter spherically symmetrical cell potential mole fraction

Greek Letters    

property di€erence parameters de®ned in Eqs. (2) and (3) chemical potential diameter

Superscripts H P S W

hydrate Pauling diameter Stokes diameter water

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J. Y. ZUO et al.

Subscripts a c cal exp i j LR RF SR1 SR2 s 0

anionic properties cationic properties calculated experimental species type of cavities long-range term repulsive forces nonelectrolyte short-range term ionic short-range term solvent properties standard properties

References Aasberg-Petersen, K., Stenby, E. and Fredenslund, Aa. (1991). Predictions of High-Pressure Gas Solubilities in Aqueous Mixtures of Electrolytes. Ind. Eng. Chem. Res., 30, 2180 ± 2185. Aspen Technology Inc., ASPEN PLUS Electrolytes Manual, Cambridge, MA, 1988. Barkan, E. S. and Sheinin, D. A. (1993). A General Technique for the Calculation of Formation Conditions of Natural Gas Hydrates. Fluid Phase Equilibria, 86, 111 ± 136. Barthel, J., Neueder, R. and Lauermann, G. (1986a). Vapor Pressures of Non-Aqueous Electrolyte Solutions. Part 1: Alkali Metal Salts in Methanol. J. Solution Chem., 14, 621 ± 631. Barthel, J., Lauermann, G. and Neueder, R. (1986b). Vapor Pressure Measurements on NonAqueous Electrolyte Solutions. Part 2: Tetraalkylammonium Salts in Methanol. Activity Coecients of Various 1-1 Electrolytes at High Concentrations. J. Solution Chem., 15, 851 ± 867. Barthel, J., Krell, M., Iberl, L. and Feuerlein, J. (1986c). Electroanal. Chem., 214, 485 ± 492. Bishnoi, P. R. and Dholabhai, P. D. (1993). Experimental Study on Propane Hydrate Equilibrium Conditions in Aqueous Electrolyte Solutions. Fluid Phase Equilibria, 83, 455 ± 462. Bixon, E., Guerry, R. and Tassios, D. (1979). Salt E€ect on Vapor Pressure of Pure Solvents: Methanol with Seven Salts at 24.9 C. J. Chem. Eng. Data, 24, 9 ± 11. Boone, J. E., Rousseau, R. W. and Schoenborn, E. M. (1976). The Correlation of Vapor ± Liquid Equilibrium Data for Salt-Containing Systems. Present in Furter, W. F., Thermodynamic Behavior of Electrolytes in Mixed Solvents. Adv. Chem. Ser., 155, 36 ± 52. Boublik, T., Fried, V. and Hala, E., The Vapor Pressures of Pure Substances. Elsevier Sci. Pub. Co., Amsterdam, 1973. Chen, C.-C. and Evans, L. B. (1986). A Local Composition Model for the Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE J., 32, 444 ± 454. Dholabhai, P. D. and Bishnoi, P. R. (1994). Hydrate Equilibrium Conditions in Aqueous Electrolyte Solutions: Mixtures of Methane and Carbon Dioxide. J. Chem. Eng. Data, 39, 191 ± 194. Dholabhai, P. D., Englezos, P., Kalogerakis, N. and Bishnoi, P. R. (1991). Equilibrium Conditions for Methane Hydrate Formation in Aqueous Electrolyte Solutions. Can. J. Chem. Eng., 69, 800 ± 805.

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