Measurement and prediction of phase equilibria for water+CO2 in hydrate forming conditions

Fluid Phase Equilibria 175 (2000) 75–89

Measurement and prediction of phase equilibria for water + CO2 in hydrate forming conditions S.O. Yang, I.M. Yang, Y.S. Kim, C.S. Lee∗ Department of Chemical Engineering, Korea University, 1 Anamdong-5ka, Sungbuk-ku, Seoul 136-701, South Korea Received 14 March 2000; accepted 8 August 2000

Abstract CO2 solubility in H–LW equilibria for which no reliable solubility data are available in the literature were measured together with H–LW –VCO2 and H–LW –LCO2 equilibria for water–CO2 –hydrate. The pressure range was from 2 to 20 MPa and the temperature range was from 273 to 285 K. Equations of state have been applied to three-phase equilibria and to gas-rich phase in two-phase equilibria by other investigators. However, they have not been applied to H–LW equilibria so far. In this study applicability of the lattice fluid equation of state was investigated for the unified description of various phase equilibria. With Langmuir constants in the Van der Waals and Platteeuw model for hydrates and hydrogen-bonding free energy of water fitted to data, the method was found to consistently describe various two- and three-phase equilibria of LW –VCO2 , LW –LCO2 , LW –VCO2 –LCO2 , H–LW –VCO2 /LCO2 , H–I–VCO2 , H–VCO2 –LCO2 , H–LW , and H–VCO2 /LCO2 . With a single binary interaction parameter, good agreements between observed and calculated results were obtained except for liquid composition in CO2 -rich phases for H–LCO2 and LW –VCO2 –LCO2 equilibria in the vicinity of CO2 vapor pressure. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Gas hydrate; Equation of state; Vapor–liquid equilibria; Experimental method; Water + CO2

1. Introduction Since the early 19th century, water molecules have been known to form clathrate hydrates when stabilized by guest molecules occupying cavities in the crystal even above the freezing point of water. After a period of scientific interest clathrate hydrates became a practical concern for the natural gas industry. Clathrate hydrate of CO2 attracted attention recently as a means of disposal in deep sea. The history and the recent status of clathrate hydrate research are well documented in Sloan’s book [1]. Hydrate–water–CO2 systems involve various phases; namely, hydrate (H), water-rich liquid (LW ), CO2 -rich vapor (VCO2 ), CO2 -rich liquid (LCO2 ), and ice (I). Experimental data for two- and three-phase equilibria have been reported. However, there are some ambiguities in available data on H–LW phase ∗ Corresponding author. Tel.: +82-2-3290-3290; fax: +82-2-926-6102. E-mail address: [email protected] (C.S. Lee).

0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 4 6 7 - 2

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equilibria. Teng and Yamasaki [2] reported CO2 solubility based on two liquid phases separated by thin hydrate film in a narrow tube. The apparently stable three-phase equilibria at two-phase equilibrium condition may be suspected if the state is truly in equilibrium. The other investigation on the solubility is based on the amount of CO2 supplied at the incipient crystallization condition [3]. At the condition, solutions are usually supersaturated and the solubility determined in this way may be suspected too high [4]. Two-phase equilibria are important for practical applications and developing thermodynamic models. Indeed, Wendland et al. [5] noted the importance of two-phase equilibria as well as of the calculation by equation of state approach. For general equilibrium calculations an equation of state approach is most preferable since the equilibria involving hydrates are usually in the high-pressure region. So far such an approach has been limited to three-phase equilibria [6–10] and to gas-rich phase [10–14] in two-phase equilibria. In the present study H–LW equilibria are experimentally studied for hydrate–water–CO2 systems. Since an equation of state and a set of parameters can describe all fluid phases, the equation of state approach is extended to liquid phases in this study. The lattice fluid equation of You et al. [15,16] and its extension to associating systems [17] is applied for a unified description of all these equilibria.

2. Theory Various phase equilibria may be calculated once chemical potentials or fugacities of components for different phases are calculated. For fluid phases, they may be obtained using an appropriate equation of state (EOS). The EOS approach is advantageous for systems involving hydrates that are formed at high pressures. In a theory based on partition functions, chemical potential is conveniently evaluated and is used extensively instead of fugacity. We write the two-phase equilibrium relation involving hydrate phase. For water, H µ
W = µW

(1)

where
denotes liquid, vapor or gas. The chemical potential of a component i in a fluid phase (
) is readily written from the work of You et al. [15,16] and Yeom et al. [17] for associating as well as non-associating systems. The expression for chemical potential of water in hydrates is based on Van der Waals and Platteeuw model [18]:   X 1 − Ci,j fi
H EH µW = µW + RT νj ln (2) 1 + Ci,j fi
j where j denotes cavities which guest molecules can occupy,
denotes a fluid phase, µEH W is the chemical potential of empty hydrates, ν j is the number of cavities of type j per water molecule in the hydrate lattice (ν1 = 1/23 and ν2 = 3/23), and Ci ,j is the Langmuir constant for each kind of cavities. An intermolecular potential was used to calculate the Langmuir constants in many previous studies. However, this complex approach does not eliminate the necessity for adjustable intermolecular potential parameters. Nagata and Kobayashi [19] showed that Langmuir constants can be approximated by Ci,j = exp(Ai,j + Bi,j T )

(3)

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77

Langmuir constants can be regressed from three-phase equilibrium data. Spectroscopic studies showed that only gases of modest size and of appropriate geometry can occupy the cavities. The fugacity of component i in Eq. (2) is calculated from the chemical potential, 
 µi − µ
0 i
(4) fi = P0 exp RT where µ
0 i is the ideal gas fugacity of i component at P0 = 1 bar and at the system temperature. To equate chemical potentials for equilibrium calculation between a fluid and hydrate phase, it is convenient to represent the chemical potential of empty hydrate in reference to ideal gas fugacity at 1 bar and system temperature:
0 EH
0 sat EH sat EH sat EH µEH φW ] + VWsat EH [P − PW ] W = µW + RT ln(fpure W ) = µW + RT ln[PW

(5)

sat EH and VWsat EH . For the latter the This relation requires the vapor pressure of the empty hydrate, PW literature correlation of Avlontis [20] was used. The vapor pressure of hypothetical empty hydrate was calculated by Sloan et al. [12] from calculation of hydrate–fluid hydrocarbon phase equilibria below freezing temperature of water. For structure I, the vapor pressure (atm) can be represented by

 sat EH  6003.9 ln PW = 17.440 − T From Eqs. (2) and (5) we have for the residual chemical potential of water in hydrates,   X 1 − Ci,j fi
H
0 sat EH sat EH sat EH + RT ln[PW φW ] + VWsat EH [P − PW ] µW − µW = RT νj ln
1 + Ci,j fi j

(6)

(7)

For computation, µ
W is also expressed in reference to ideal gas fugacity at 1 bar and system temperature so that Eq. (1) is written as
0 H
0 µ
W − µW = µW − µW

(8)

The equality condition of fugacity of guest molecules is embedded in the relation. For fluid phase equilibria not involving hydrate phase, µ
(1) = µ
(2) i i

(9)

where
(1) and
(2) are two fluid phases. For phase equilibria involving ice,
0 I
0 µ
W − µW = µW − µW

(10)

where sat I sat I sat I sat I µIW − µ
0 W = RT ln[PW φW ] + VW [P − PW ]

(11)

A three-phase equilibrium involving hydrates is defined by the simultaneous solution of Eq. (8) with either Eq. (9) or (10). Working equations are collected and given in Appendix A.

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Fig. 1. Experimental apparatus: (1) equilibrium cell; (2) variable volume cell; (3) expansion cell; (4) sampling valve; (5) sampling loop; (6) precision metering pump; (7) gas bomb; (8) water bath; (9) pressure generator; (10) density transducer; (11) magnetic stirrer; (12) vacuum pump.

3. Experimental All the experiments were performed using the apparatus shown in Fig. 1. Two high-pressure view cells were used. A smaller cell was a variable volume view cell for adjustment of the system pressure. An insertion-type density transducer (model 7826) by Solatron was installed in the larger view cell. Temperature was measured inside the larger cell by the same transducer. Its accuracy was claimed to be less than ±1 kg/m3 in density and ±0.05 K in temperature. To enhance mixing, an external circulation loop was installed to circulate aqueous phase by a high-precision metering pump. The cells and the circulation loop were immersed in a temperature-controlled bath. Temperature of the bath was measured by Cole Palmer (model 8436-00) thermometer with 0.01 K resolution. Valcom (model VPRT-350K) pressure gauge with the claimed accuracy of 0.06 MPa was used after calibration against a Heise gauge. In measuring three-phase equilibrium, degassed water was charged into the cell. Then CO2 was introduced. The system was cooled to about 3–4 K below the anticipated hydrate forming temperature. Once the hydrates were formed, they were allowed to decompose by raising the temperature slowly. Near the complete decomposition by visual inspection, the system was maintained at the temperature for 8 h. When no pressure change was detected, the condition was taken as the equilibrium condition. Subsequently, a different pressure was selected, and the procedure was repeated to obtain the incipient hydrate formation temperature. In two-phase equilibrium measurements, solubility of CO2 in aqueous phase was analyzed by sampling the fluid in the sampling loop and by expansion into the pre-calibrated chamber. Degassed water was charged into the cell until most of the cell was filled with solution. After CO2 in the view cell was completely changed to hydrate by sub-cooling, the system was maintained at a given temperature and pressure and was monitored for more than a day. With no detectable pressure change, small amounts of aqueous liquid phase were introduced into the evacuated sampling loop with a calibrated volume. The dissolved CO2 molecules in the sample were expanded in the expansion chamber whose volume

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79

Fig. 2. Comparison of experimental and calculated dissociation pressure of CO2 hydrate in three-phase equilibria.

was determined precisely using methane PVT measurements. A pressure transducer (Valcom, 0–200 psi, 0.2%) was used to measure the pressure of expansion chamber. Amount of dissolved fluid was determined using the PVT relation. The accuracy in mole fraction measurements was estimated to be less than 7.7%. To test the procedure, the solubility of CO2 in water were determined and compared with literature values. 4. Results and discussion To test the reliability of the present experimental study, three-phase equilibria and solubility of CO2 in water was measured and compared with literature values in Figs. 2 and 3. AAD errors of the measured dissociation pressure with best-fitted literature values were 4.4% for H–LW –VCO2 and 13.9% for H–LW –LCO2 . These values were comparable with AAD fitting errors of literature values of 1.3 and 11.0%, respectively. For solubility of CO2 in water, AAD error of measured values with best-fitted value was 2.9% and that of literature value was 1.8%. Estimated errors were presented in Section 3. Present experimental results were found to be in good agreement with other experimental data within a maximum deviation of 0.003 in mole fraction. For the calculation of phase equilibria, the lattice fluid equation of state of You et al. [15,16] and Yeom et al. [17] was used extensively. This equation of state uses two temperature-dependent molecular

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Fig. 3. Comparison of experimental data from DECHEMA Series [34] and calculated solubility of CO2 in water and water in CO2 of LW –VCO2 or LW –LCO2 equilibria at 298.15 K.

parameters representing segment number and segment interaction energy for each pure species. They are determined from saturated liquid density and vapor pressure and tabulated in Table 1. Hydrogen-bonding energy and entropy for water was proposed by Luck [35], and adjusted to give better results. The adjusted hydrogen-bonding free energy was −8.75 kJ/mol for all hydrate-forming temperature. Below the freezing point of water, sub-cooled liquid vapor pressures in Perry’s handbook [36] were used for parameter determination. For ice the vapor pressure and density are needed. The vapor pressure of ice was from Perry’s handbook and the density was from Avlontis [20]. The binary interaction energy parameter was fitted to data and represented as a function of temperature as kij = −0.304 + 8.8 × 10−4 × T . Calculated results are compared with data and shown in Fig. 3. The Langmuir constants A and B in Eq. (3) were fitted to available three-phase equilibrium data. For CO2 they were 53.072 and −0.199 K for smaller cavities and 5.901 and −0.016 K for larger cavities, respectively. Table 1 Equation of state parameters for Eqs. (A.2) and (A.3)

H2 O CO2

εa

εb

εc

ra

rb

rc

258.549 88.809

−0.506 −0.140

−0.366 0.430

1.822 3.670

0.000 0.002

0.001 −0.068

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81

Table 2 Comparison of experimental and calculated three-phase equilibrium pressure in H2 O–CO2 system Phase

No. of Present points calculationa

H–I–VCO2 10 H–LW –VCO2 116 22 H–LW –LCO2 H–VCO2 –LCO2 37 LW –VCO2 –LCO2 8 a b

6.5 2.9 15.9 0.8 0.5

Calculated Correlation by T range by Sloan [1]a Wendland et al. [5]a (K)

P range (MPa)

5.8 2.5 NAb NAb NAb

0.55–1.05 1.05–4.50 4.50–88.1 2.18–4.70 4.28–6.41

0.20 1.85 NAb 0.12 0.01

256.8–271.2 273.2–283.3 282.9–289.2 258.8–285.0 281.2–298.2

[5,21] [5,21–27], present study [25–27], present study [5,21,27] [5]

AADP (%). Not available.

Calculated results of three-phase equilibria were graphically demonstrated in Fig. 2. For the calculation of H–LW –VCO2 equilibria, most other investigators used EOS approach for calculation of vapor phase fugacity. However, for chemical potential in liquid phase, pure water is assumed and the following relation was used:  Z T Z P H
 1HW 1VWH
H
H
dP (12) 1µW = 1µW (T0 , P0 ) + dT + RT2 RT T0 P0 Sloan [1] used Peng–Robinson or Soave–Redlich–Kwong EOS. Still this method was not applied to H–LW –LCO2 equilibria. Anderson and Prausnitz [7] represented LCO2 phase by UNIQUAC and vapor phase by an equation of state. Lundgaard and Mollerup [9] applied Soave–Redlich–Kwong EOS combined with the density-dependent local composition (DDLC) mixing model in H–LW –LCO2 and H–LW –VCO2 equilibria. Recently, Bakker et al. [8] applied Eq. (12) to H–LW –LCO2 equilibria by using different sets of Kihara parameters to obtain Langmuir constants in H–LW –LCO2 and H–LW –VCO2 equilibria. Calculated results by the present EOS for various three-phase equilibria are compared with data and summarized in Table 2. The H–LW –LCO2 branch is very steep and error in P appears large, but dissociation pressure itself has a large best-fitting AAD error of 11.0%. It is to be noted that H–I–VCO2 , H–VCO2 –LCO2 and Table 3 Comparison of experimental and calculated composition of two-phase equilibria Phase

No. of points

LW –VCO2

32

LW –LCO2

xAAD (%)

T range (K)

P range (MPa)

Reference

Data type

7.4

298.2

1.06–6.35

Mole fraction of CO2 in aqueous phase

7

14.5

298.2

6.35–20.00

LW –VCO2 LW –LCO2 H–LW H–VCO2 H–LCO2

23 16 32 15 10

12.5 17.3 3.0 7.9 17.7

285.2–298.2 286.3–302.7 277.8–281.0 251.8–294.3 256.2–276.2

0.69–6.42 6.69–20.00 4.99–14.20 0.69–3.45 8.28–13.79

[28–31], present study [28,31], present study [11,30–33] [11,31,33] Present study [11] [11]

H–LCO2

12

231.8

245.2–280.2

2.07–6.21

[11]

Mole fraction of CO2 in aqueous phase Mole fraction of H2 O in CO2 -rich phase Mole fraction of H2 O in CO2 -rich phase Mole fraction of CO2 in aqueous phase Mole fraction of H2 O in CO2 -rich phase Mole fraction of H2 O in CO2 -rich phase, far from CO2 vapor pressure Mole fraction of H2 O in CO2 -rich phase, around CO2 vapor pressure

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Table 4 Experimental mole fraction of CO2 in aqueous phase in H–LW equilibria Temperature (K)

Pressure (MPa)

xCO2 × 102

Temperature (K)

Pressure (MPa)

xCO2 × 102

280.32 280.59 280.72 280.74 279.98 278.71 278.99 279.01 279.12 279.32 279.45 279.49 279.58 279.94 280.43 280.98

4.99 6.08 6.08 6.08 6.09 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10

2.56 2.70 2.68 2.68 2.62 2.21 2.31 2.36 2.32 2.42 2.39 2.35 2.49 2.60 2.62 2.69

280.26 280.34 278.37 278.39 278.39 279.18 278.65 279.41 279.13 279.60 280.30 280.30 280.40 280.00 277.84 280.34

8.16 10.04 10.35 10.35 10.37 10.40 10.41 10.43 10.44 10.44 10.44 10.44 10.44 10.45 10.47 14.20

2.68 2.63 2.27 2.24 2.23 2.26 2.35 2.48 2.41 2.50 2.58 2.57 2.60 2.56 2.01 2.60

Fig. 4. Comparison of experimental and calculated solubility of CO2 in water in H–LW equilibria at 280.0 K.

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83

Fig. 5. Comparison of experimental and calculated solubility of water in CO2 -rich phases of H–LCO2 or H–VCO2 equilibria.

LW –VCO2 –LCO2 branches are also included in the present calculation. In this way the ice branch continuously extends to LCO2 branch as opposed to Sloan’s calculation [1]. Lundgaard and Mollerup [9] pointed out that the discontinuity in Sloan’s calculation is due to the quadruple point fixed at 273.15 K without doing equilibrium calculations. Calculated water solubility of CO2 for LW –VCO2 or LW –LCO2 equilibria are compared with other investigator’s data in Fig. 3 and errors are summarized in Table 3. Solubility errors of water in CO2 -rich phase are larger compared with those of CO2 in water-rich phase. It is interesting to note that the present method applies to both phases fairly accurately with a temperature-dependent binary parameter. Aasberg-Petersen et al. [37] and Mollerup and Clark [38] applied EOS approach to CO2 –water system with three temperature-independent binary parameters. Their temperature range is much higher than the present range of interest and no comparisons are made with their works. The solubility data of CO2 at 280 K in H–LW equilibria were obtained in this work and are listed in Table 4. They were compared with data of Teng and Yamasaki at 278 K [2] and the results of present calculation at 280 K in Fig. 4. Above the three-phase equilibrium pressure, present solubility slowly decreases with pressure. Handa [39] calculated the solubility of methane in H–LW equilibria and showed that the solubility decreases with pressure in this region. Solubilities described by Teng and Yamasaki increase with pressure. It is not obvious if Teng and Yamasaki measurements are in true equilibrium. Errors in calculation for present H–LW data are included in Table 3 together with those for H–LCO2 and

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Fig. 6. Isobaric phase diagram of water + CO2 system at 3.45 MPa.

H–VCO2 by other investigators. Except for water solubility in the pressure range from 2.07 to 6.21 MPa, calculated values agree with data fairly closely. Data on two-phase equilibria of H–LCO2 and H–VCO2 was known to be difficult to measure due to its metastability and low concentrations of water [1]. The results of present calculation for H–LCO2 and H–VCO2 equilibria were compared with data of Song and Kobayashi [11] and errors are also summarized in Table 3. The results for water concentration in CO2 -rich phase are good for H–VCO2 and relatively high-pressure region of H–LCO2 as shown by isobaric T–x curves for 0.69 and 10.34 MPa in Fig. 5. Although not shown in the figure, in the vicinity of vapor pressure of CO2 , however, large errors were observed. In this region, multiple roots of water solubility are found and the stable phase was determined by comparing molar Gibbs free energy. The isobaric phase diagram for 3.45 MPa was illustrated in Fig. 6. It is seen that calculated LCO2 composition is not very accurate. Difference between calculated and experimental water contents in LCO2 becomes smaller as temperature decreases from H–LCO2 –VCO2 equilibrium temperature. The vapor–liquid branch near pure CO2 was enlarged and shown in the inset. The H–LCO2 –VCO2 locus continuously extends to LW –VCO2 –LCO2 when temperature or pressure changes. Song and Kobayashi [11] reported water concentrations in LCO2 phase at LW –VCO2 –LCO2 equilibria that are shown in Fig. 7. We find again that LCO2 composition is not very accurate. Also shown in the figure are calculated lines and interpolated points to LW –VCO2 –LCO2 equilibria from other investigator’s [30–33] two-phase equilibrium data at 298.15 K represented in Fig. 3. Song and Kobayashi’s water contents in

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85

Fig. 7. Comparison of calculated and experimental in CO2 -rich phases of LW –LCO2 –VCO2 equilibria.

LCO2 is seen as quite different from interpolated value that is close to the result of present calculation. A similar poor agreement is observed in Fig. 6 between calculated water contents in LCO2 and Song and Kobayashi’s data in H–LCO2 –VCO2 and H–LCO2 in the vicinity of vapor pressure. Further study is needed for the clarification.

5. Conclusion An experimental apparatus for the determination of the hydrate dissociation pressure of H–LW –VCO2 and H–LW –LCO2 as well as CO2 solubility in H–LW equilibria was built. The apparatus was used to obtain dissociation pressure of H–LW –VCO2 and H–LW –LCO2 equilibria and CO2 solubility in H–LW equilibria. Estimated experimental accuracy was 0.06 MPa in pressure, 0.05 K in temperature and 7.7% in composition. A variable volume view cell was used with a magnetic stirrer inside. Concentrations of CO2 in aqueous phase were determined by expanding dissolved gas from external sampling loop. The lattice fluid equation of state by present authors was used consistently for fluid phases in the calculation of two- and three-phase equilibria. The Langmuir constants in the Van der Waals and Platteeuw model for hydrates and hydrogen-bonding free energy of water were fitted to data. Various two- and three-phase equilibria of LW –VCO2 , LW –LCO2 , LW –VCO2 –LCO2 , H–LW –VCO2 /LCO2 , H–I–VCO2 , H–VCO2 –LCO2 , H–LW , H–VCO2 /LCO2 were predicted with a single binary interaction parameter. Good agreements between

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observed and calculated results were obtained except for water contents in LCO2 phases for H–LCO2 and LW –VCO2 –LCO2 equilibria in the vicinity of CO2 vapor pressure. List of symbols j al , dki number of acceptor type l in species j, or donor type k in species i, respectively A parameter for Langmuir constant Helmholtz free energy of k–l hydrogen bond formation AHB kl AAD absolute average deviation B parameter for Langmuir constant (K) C Langmuir constant (1 bar) H hydrate phase I ice phase k, l indices of donor and acceptor types L liquid phase Ndk , Nal number of donor type k, or acceptor of type l; defined by Eq. (A.12) defined by Eq. (A.13) Nr qi surface area parameter of component i segment number of component i ri SklHB entropy of k–l hydrogen bond formation UklHB energy of k–l hydrogen bond formation V vapor phase unit lattice volume VH z coordination number Greek letters β reciprocal temperature (1/kT) interaction energy between segments of components i and j ε ij µi chemical potential of component i the number of cavities of type j per water molecule νj ν HB volume fraction of hydrogen bond surface area fraction of component i; defined by Eq. (A.9) θi ρi reduced density of component i; defined by Eq. (A.8) Superscripts EH empty hydrate sat saturated
a fluid phase 0 pure component reference state at 1 bar Subscripts a, d acceptor or donor, respectively CO2 -rich phase CO2 W water-rich phase

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87

Acknowledgements The authors acknowledge the support by the Korea Science and Engineering Foundation under Contract nos. 98-0502-04-01 and 94-0562-06-01. Appendix A. Working equations for non-random lattice fluid hydrogen bonding theory You et al. [15,16] proposed an explicit model based on non-random lattice fluid model. Yeom et al. [17] extended the model to associating systems based on Veytsman statistics [40]. Lee et al. [41] proposed a normalization of the Veytsman statistics. Lattice-hole theories are derived from the lattice model by replacing one of the components in the mixture by holes. In a c component mixture, Ni is the number of molecules of i component that are assumed to occupy ri sites of the unit cell size VH . Molecules of component i interact with surface area qi with neighboring segment of r-mers in a three-dimensional lattice of the coordination number z. N0 is the number of holes. ri values for holes are assumed to be unity. VH and z are set to 9.75 cm3 /mol and 10, respectively. For linear or branched chains, the following relation between ri and qi forms the basis for the derivation: zqi = (z − 2)ri + 2

(A.1)

They are fitted to saturated liquid volume and vapor pressure. They are conveniently represented as functions of temperature:     273.15 εii = εa + εb (T − 273.15) + εc T ln + (T − 273.15) (A.2) k T     273.15 + (T − 273.15) (A.3) ri = ra + rb (T − 273.15) + rc T ln T For interactions with a segment of different species, a binary interaction parameter is required: εij = (εii εjj )1/2 (1 − kij )

(A.4)

Expression for pressure is            z 2 εM qM z 1 − 1 ρ − ln(1 − ρ) − νHB ρ − ln 1 + θ P = βVH 2 rM 2 VH where



εM = rM =

   1 XX β XXXX θi θj εij + θi θj θk θl εij (εij + εkl − εik − εjk ) θ2 2θ 2

c X

xi ri ,

qM =

i=1

ρ=

c X i=1

ρi ,

c X

xi qi

(A.5)

(A.6)

(A.7)

i=1

ρi =

N0 +

Ni ri Pc

i=1 Ni ri

(A.8)

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θ=

c X

θi ,

θi =

i=1

N0 +

Ni qi Pc

(A.9)

i=1 Ni qi

PK PL HB0 HB l=1 Nkl − k=1 l=1 Nkl Pc i=1 Ni ri

PK PL k=1

νHB =

(A.10)

NklHB is the solution of a set of simultaneous equations: ! ! L K X X HB k HB l HB Nkl Nr = Nd − Nkm Nnl exp(−βAHB Na − kl ), m=1

n=1

k = 1, 2, . . . , K, l = 1, 2, . . . , L

(A.11)

HB0 HB HB where AHB is the solution of the same equations with AHB kl = Ukl − TSkl · Nkl kl = 0. k and l are indices of donor types and acceptor types. The number of donor type k, Ndk , and that of acceptor type l, Nal , are j defined in terms of the number of donor type k in species i, dki , and that of acceptor type l in species j, al , as

Ndk =

c X dki Ni , i=1

Nal =

c X

j

al Nj

(A.12)

j =1

The set of Eq. (A.11) yields an explicit solution when only one type of hydrogen bonds are present in a mixture. The volume is defined by the relation V = V H N r , where X Nr = N0 + Ni r r (A.13) and the chemical potential is          
0 µ
VH zβqi εM θ 2 qM θi i − µi = −ln + ri ln 1 + + − 1 ρ − ri ln(1 − ρ) + ln RT RT rM qi 2 P PPP   2 θk εik + β θj θk θl εij (εij + 2εkl − 2εjk − εik ) ri × 1− − qi εM θ 2 m n HB HB X X Nk0 N0k + aki ln HB0 (A.14) + dki ln HB0 N N k0 0k k=1 k=1 References [1] [2] [3] [4] [5] [6] [7]

E.D. Sloan, Clathrate Hydrates of Natural Gases, Marcel Dekker, New York, 1990. H. Teng, A. Yamasaki, J. Chem. Eng. Data 43 (1998) 2–5. H. Kimuro, T. Kusayanagi, F. Yamaguchi, K. Ohtsubo, IEEE Trans. Energy Convers. 9 (1994) 732–735. V. Natarajan, P.R. Bishinoi, N. Kalogerakis, Chem. Eng. Sci. 49 (1994) 2075–2087. M. Wendland, H. Hasse, G. Maurer, J. Chem. Eng. Data 44 (1999) 901–906. E.S. Barkan, D.A. Sheinin, Fluid Phase Equilib. 86 (1993) 111–136. F.E. Anderson, J.M. Prausnitz, AIChE J. 32 (1986) 1321–1333.

S.O. Yang et al. / Fluid Phase Equilibria 175 (2000) 75–89 [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

89

R.J. Bakker, J. Dubessy, M. Cathelineau, Geochim. Cosmochim. Acta 60 (1996) 1657–1681. L. Lundgaard, J.M. Mollerup, Fluid Phase Equilib. 70 (1991) 199–213. J. Munck, S. Skjold-Jørgensen, M. Rasmussen, Chem. Eng. Sci. 43 (1988) 2661–2672. K.Y. Song, R. Kobayashi, SPE Form. Eval. Dec (1987) 500–507. E.D. Sloan, K.A. Sparks, J.J. Johnson, Ind. Eng. Chem. Res. 26 (1987) 1173–1179. K.Y. Song, R. Kobayashi, Ind. Eng. Chem. Res. 21 (1982) 391–395. H.J. Ng, D.B. Robinson, Ind. Eng. Chem. Res. 19 (1980) 33. S.S. You, K.P. Yoo, C.S. Lee, Fluid Phase Equilib. 93 (1994) 193–213. S.S. You, K.P. Yoo, C.S. Lee, Fluid Phase Equilib. 93 (1994) 215–232. M.S. Yeom, B.H. Park, C.S. Lee, Fluid Phase Equilib. 158–160 (1999) 143–149. J.H. Van der Waals, J.C. Platteeuw, Adv. Chem. Phys. 2 (1959) 1–57. I. Nagata, R. Kobayashi, Ind. Eng. Chem. Fundam. 5 (1966) 466. D. Avlontis, Chem. Eng. Sci. 49 (1994) 1161–1173. S.D. Larson, Phase studies of the two-component carbon dioxide–water system, involving the carbon dioxide hydrate, University of Illinois, 1955. C.H. Unruh, D.L. Katz, Trans. AIME 186 (1949) 83. W.M. Deaton, E.M. Frost Jr., Gas Hydrates and Their Relation to Operation of Natural-Gas Pipe Lines, US Bureau Mines Monogr. 8 (1946). D.B. Robinson, B.R. Mehta, J. Can. Petr. Tech. 10 (1971) 33. H.J. Ng, D.B. Robinson, Fluid Phase Equilib. 21 (1985) 145. S. Takenouchi, G.C. Kennedy, J. Geol. 73 (1965) 383–390. S. Fan, T. Guo, J. Chem. Eng. Data 44 (1999) 829–832. R. Wiebe, V.L. Gaddy, J. Am. Chem. Soc. 62 (1940) 815. Y.D. Zelvensky, Zh. Khim. Prom. 14 (1937) 1250. T. Nakayama, H. Sagara, K. Arai, S. Saito, Fluid Phase Equilib. 38 (1987) 109. P.C. Gillespie, G.M. Wilson, Res. Rep. GPA (1982) 48. C.R. Coan, A.D. King, J. Am. Chem. Soc. 93 (1971) 1857. R. Wiebe, V.L. Gaddy, J. Am. Chem. Soc. 63 (1941) 475. H. Knapp, R. Doring, L. Oellrich, U. Plocker, J.M. Prausnitz, Vapor–Liquid Equilibria for Low Boiling Substances, DECHEMA Chemistry Data Series, Frankfurt, 1982. W.P. Luck, Angew. Chem. Int. Ed. Eng. 19 (1980) 28. R.H. Perry, Perry’s Chemical Engineers’ Handbook, 6th Edition, McGraw-Hill, New York, 1989. K. Aaesberg-Petersen, E. Stenby, A. Fredenslund, Ind. Eng. Chem. Res. 30 (1991) 2180–2185. J.M. Mollerup, W.M. Clark, Fluid Phase Equilib. 51 (1989) 257–268. Y.P. Handa, J. Phys. Chem. 94 (1990) 2652–2657. B.A. Veytsman, J. Phys. Chem. 94 (1990) 8499–8500. C.S. Lee, K.P. Yoo, B.H. Park, J.W. Kang, Submitted to Fluid Phase Equil.