Ind. Eng. Chem. Res. 2003, 42, 2409-2414
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Liquid Water-Hydrate Equilibrium Measurements and Unified Predictions of Hydrate-Containing Phase Equilibria for Methane, Ethane, Propane, and Their Mixtures Y. S. Kim, S. K. Ryu, S. O. Yang,† and C. S. Lee* Department of Chemical Engineering, Korea University, 1 Anamdong-5Ka, Sungbuk-ku, Seoul 136-701, Korea
With growing interest in gas hydrates as a possible energy resource, there have been necessities for more experimental data and prediction methods for properties of hydrate-containing systems. An experimental apparatus was developed for the determination of the equilibrium pressure of hydrates and solubilities of sparingly soluble gases in hydrate-liquid water equilibria. Methane and ethane solubilities in hydrate-liquid water equilibria were measured in the pressure range from 5 to 20 MPa and in temperature range of 276-282 K. A unified prediction method for the hydrate-containing phase equilibria for single- and mixed-guest species was developed. Hydratecontaining systems of methane, ethane, and propane were modeled using a unified approach. A lattice-fluid equation of state was used consistently for the fluid phases. Various hydratecontaining equilibria of simple and mixed hydrate were predicted and compared with literature data and the present experimental results. 1. Introduction With growing interest in methane hydrates as a possible energy resource, there have been necessities for more experimental data and prediction methods. Numerous studies have been devoted to three-phase equilibria, and several researches revealed the nature of hydrate-guest-rich fluid-phase equilibria. The present status of hydrate research is comprehensively reviewed in Sloan’s new edition.1 The review reveals that data for the water-rich phase in a two-phase equilibrium with hydrates are scarce and that comprehensive calculation methods for single- and mixed-guest systems still need development. Recently, the two-phase equilibrium data for carbon dioxide and methane systems were measured by the expansion of dissolved gas in a sampling valve and reported.2-4 A unified computation method was also proposed using a hydrogen-bonding lattice-fluid equation of state (EOS) and applied to various two- and three-phase equilibria for systems of water and a single guest species.2,3 The computation method was not extended to mixed-guest component systems. In this study two-phase equilibria of methane- and ethane-water systems with/without hydrate formation are experimentally studied, with particular attention given to the water-rich phase. Because the concentration of methane and ethane in the water-rich phase is expected to be very low and the previous determination of very low concentrations is subject to large errors, an indirect method of determining the equilibrium pressure at given temperatures and compositions is developed. The computation method is also extended to multiguest systems and applied to two- and three-phase equilibria. 2. Experiments The experimental apparatus of Yang et al.2,3 was modified to determine the concentration of sparingly * To whom correspondence should be addressed. Tel.: +822-3290-3293. Fax: +82-2-926-6102. E-mail:
[email protected]. † Current address: Center for Hydrate Research, Colorado School of Mines, Golden, CO 80401.
Figure 1. Experimental apparatus for measurements of equilibrium pressures and the solubilities of dissolved gases in hydratecontaining equilibria: (1) equilibrium cell; (2) density and temperature meter; (3) McHugh type variable volume view cell; (4) magnetic stirrer; (5) metering pump; (6) water bath; (7) syringe pump; (8) gas bomb; (9) line filter; (10) flask; (11) vacuum pump; (12) pressure generator; (13) water reservoir.
soluble gases by an indirect method as shown in Figure 1. By minimization of the volume of the vapor or hydrates in equilibrium with the water-rich liquid phase, equilibrium temperatures were measured at constant pressures with predetermined compositions. Two high-pressure view cells were used. A smaller cell was a variable-volume view cell for the adjustment of the system pressure. An insertion-type density transducer (model 7826) from Solatron was installed in the larger view cell to measure the temperature inside the larger cell. The accuracy in temperature was claimed to be less than (0.05 K. To enhance mixing, an external circulation loop was installed to circulate the aqueous phase by a high-precision metering pump. Cells and the circulation loop were immersed in a temperaturecontrolled bath. Temperature fluctuations of the bath were found within 0.1 K. A Sensys pressure gauge with a claimed accuracy of 0.06 MPa was used after calibra-
10.1021/ie0209374 CCC: $25.00 © 2003 American Chemical Society Published on Web 04/12/2003
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Table 1. Experimental Solubility of Methane and Ethane in Water at 298.15 K species
pressure [MPa]
measd xCH4 × 103
calcd xCH4 × 103
species
pressure [MPa]
measd xCH4 × 103
calcd xCH4 × 103
CH4
2.3 4.9 4.9 6.9 11.0 16.6
0.531 1.062 1.062 1.592 2.123 2.654
0.546 1.089 1.096 1.454 2.045 2.624
C2H6
1.4 1.6 2.0 2.0 2.1 3.0 3.5 3.5 3.9
0.385 0.476 0.552 0.543 0.570 0.756 0.901 0.891 1.125
0.409 0.479 0.559 0.561 0.585 0.771 0.860 0.869 0.923
Table 2. Experimental Dissociation Temperatures for H-LW Equilibria of Water-Methane and Water-Ethane Systems
species CH4
C2H6
pressure [MPa]
solute mole fraction in water
temp [K]
calcd mole fraction
5.0 5.1 5.1 5.1 10.1 10.2 10.2 10.2 12.7 12.7 12.7 12.7 14.3 14.4 14.3 14.3 10.1 15.1 20.1
0.001 33 0.001 59 0.001 75 0.001 86 0.001 33 0.001 59 0.001 75 0.001 86 0.001 33 0.001 59 0.001 75 0.001 86 0.001 33 0.001 59 0.001 75 0.001 88 0.000 437 0.000 437 0.000 437
276.2 277.9 279.9 280.5 276.4 278.7 280.1 280.7 276.7 278.9 280.4 281.1 276.9 279.2 280.6 281.7 277.3 277.8 278.5
0.001 39 0.001 54 0.001 75 0.001 82 0.001 36 0.001 58 0.001 72 0.001 78 0.001 37 0.001 57 0.001 72 0.001 80 0.001 38 0.001 58 0.001 74 0.001 84 0.000 438 0.000 437 0.000 440
tion against a Heise gauge. Triple distilled water was prepared using a distillator (Barnstead, MegaPure System MP-3A). A programmable syringe pump from Eldex Co. was used for introducing gas components and distilled water into the equilibrium cells. The pump has a syringe volume of 10 mL with an estimated accuracy of (0.5%. Methane-ethane gas used in this work was from Air Products, and its claimed purity was higher than 99.99%. Gas components were filled in the syringe, and the pressure was measured with a Heise gauge. The amount of a gas component introduced was determined using the PVT relation.5 The known amounts of distilled water were charged into the cell by the same syringe pump. The composition was taken as the water-rich liquidphase composition in equilibrium with hydrates when negligible amounts of hydrates were present in the system. After the gas phase completely disappeared by subcooling and pressurizing using a pressure generator (HIP 50-6-15), the system was maintained at the given temperature and pressure and was monitored for more than 1 day. The first subdigit of pressure measurements in megapascals was maintained constant throughout the experiment using the same pressure generator. Then the hydrates were allowed to dissociate by raising the temperature very slowly, typically at 0.1 K/h. By visual inspection, the dissociation temperature at a given pressure was determined with the estimated accuracy of 0.1 K. The accuracy in mole fractions using this procedure was estimated to be less than 3.1% and 5.3% for methane and ethane, respectively. This procedure is also used for equilibrium pressure determination in vapor-liquid equilibria above the hydrate-forming temperature. The results for vapor-liquid equilibria are
Table 3. Pure Parameters for the NLF-HB EOS species
a [K]
b
c
ra
rb × 103
rc
H2O CH4 C2H6 C3H8
142.172 53.542 76.378 84.730
0.216 0.016 0.000 0.006
-1.305 -0.103 -0.110 -0.052
1.817 3.911 5.221 6.840
-1.067 0.942 -0.009 0.000
-0.033 0.004 0.001 0.008
shown in Table 1 and those for hydrate-water-rich phase equilibria in Table 2. Calculated values by the method described in the next section are also given in both tables. 3. Model and Parameters The equality condition of the chemical potential for different phases is the basic relation for phase equilibrium calculations. A Helmholtz free-energy model applicable to fluid phases conveniently yields expressions for the volumetric EOS and chemical potential. A nonrandom lattice-fluid hydrogen-bonding model was used in the present study.6-10 Working equations for this model are summarized in work by Yang et al.2 The physical parameters are the segment number, ri, and the segment interaction energy, ij, represented as functions of temperature by fitting saturated liquid volume and vapor pressure
ii/k ) a + b(T - T0) + c[T ln(T0/T) + (T - T0)] (1) ri ) ra + rb(T - T0) + rc[T ln(T0/T) + (T - T0)]
(2)
where T0 is 273.15 K. The correlation parameters are presented in Table 3. Methane is a supercritical gas in the hydrate-forming temperature. Its pure-component parameters were fitted to PVT data and were slightly adjusted in the hydrate-forming conditions. For water below its normal melting point, subcooled water properties were obtained from Perry and Green.11 For interactions between segments of different species, a binary interaction parameter is required.
ij ) (iijj)1/2(1 - kij)
(3)
The binary parameters for the interactions between gas molecules were assumed to be zero. The binary interaction parameters are assumed to be temperaturedependent as
kij ) A + B/T
(4)
The vapor-liquid equilibrium data from Knapp et al.12 and the present experimental results given in Tables 1 and 2 were used in the estimation of binary interaction parameters. Regressed binary parameters are listed in Table 4. The temperature-dependent hydrogen-bonding energy was regressed from the solubility of the guest species
Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2411 Table 4. Binary Interaction Energy Parameters between Water and Guest Molecules
Table 5. Optimized Kihara Potential Parameters for Structure I and II Hydrate Formers
species
CH4
C2H6
C3H8
species
/k [K]
σ [Å]
aa [Å]
A B
0.7610 -177.23
0.6147 -120.25
0.3333 -33.43
CH4 C2H6 C3H8
141.52 145.52 157.43
2.9488 3.2849 3.3371
0.3834 0.5651 0.6502
in water with two donors and two acceptors per molecule. The fitted hydrogen-bonding energy and entropy were -19.95 kJ/mol and -25.0 J/mol‚K for each bond in a temperature range of 240-295 K. Luck’s energy and entropy values13 for water are -15.5 kJ/mol and -16.6 J/mol‚K, respectively. In terms of free energy at 280 K, the Luck value corresponds to 84% of the present value. For solids such as ice, we can still write the chemical potential departure from the pure ideal gas at 1 bar if we know the saturation pressure and molar volume of ice satI satI satI µIW ) µ0W + RT ln(PsatI W φW ) + VW [P - PW ] (5)
where the fugacity coefficient of water, φsatI W , in the ice phase is assumed to be unity. The regressed vapor pressure of ice was regressed from literature data of Perry and Green11 and is represented by
ln[PsatI W /MPa] ) 15.6217 -
6415.37 T + 5.5171
(6)
The molar volume of ice was obtained from the same reference. For hydrate-containing phase equilibria, we need the chemical potential of water in hydrate phases that has been modeled by van der Waals and Platteeuw,14 in which the chemical potential of the hypothetical empty hydrates, µEH W , and the Langmuir constant of guest component j for the mth type of cavities, Cj,m, are needed. The empty hydrate is a hypothetical solid whose chemical potential is also represented as a departure function from ideal gas at 1 bar using eq 5.2,3 The , in the empty hyfugacity coefficient of water, φsatEH W drate phase is also assumed to be unity. Vapor pressures of empty hydrates of each structure were fitted to equilibrium pressures of multiguest and simple hydrates. The regressed vapor pressure are represented as
/MPa] ) a - b/T ln[PsatEH W
(7)
For structure I and II hydrates, a and b values were 15.107 and 6072.25, and for structure II, they were 15.212 and 6121.34. Sloan values are 15.137 and 6003.9 for structure I and 15.029 and 6017.6 for structure II, respectively in the same unit.1 For the molar volume of empty hydrates, the correlation regressed by Avlonitis16 was used. Structural information of hydrates is from Sloan.1 For the evaluation of Langmuir constants, an averaged cell potential, W(r), is needed. McKoy and Sinanoglu15 presented the cell potential for the Kihara potential with a spherical core. The parameters are , a, and σ that are unique to every guest molecule, and they remain unchanged in different cavity types. z is the coordination number of the cavity. In this work, these potential parameters were obtained from the threephase equilibrium pressure of methane, ethane, and propane. With spherical core radii (a) fixed at Sloan1
a
The radius of the spherical core (a) was from work by Sloan.1
values, the remaining two parameters were fitted to available data at four different temperatures from the literature and intercorrelated. A set of values least sensitive to temperature variation was chosen and slightly adjusted to give better mixed-guest equilibria. The potential parameters are shown in Table 5. 4. Results and Discussion The validity and reliability of the present experimental apparatus were confirmed by comparing the present methane solubility data in water in Table 1 with those of Knapp et al.12 obtained by the usual bubble-point method. The average deviation between these data sets was 4.8% of AAD in mole fraction, while the deviation between different data sets in the DECHEMA Series was 5.5%. The three-phase equilibrium pressure (H-I-VC, H-LW-VC, and H-LW-LC) and the two-phase equilibrium composition (H-LW, H-VC, and H-LC) for pure and mixed guests are calculated in the present method. Yang et al. reported the results of the phase equilibria for carbon dioxide hydrates2 and methane hydrates3 by the present approach. This method is a typical one for calculating phase equilibria involving hydrates.1 However, it is to be noted in the present method that an association hydrogen-bonding model was used and that all equilibria including those with water-rich phases and mixed guests are calculated with a single model. Calculations for H-LW were not reported so far except for by Yang et al.2,3 Both LW and LC phases are very lean in solutes, and calculations using a single binary parameter may result in large errors in such cases. Three-phase equilibria are less sensitive to the LW model that is often assumed to be pure water in previous studies. Klauda and Sandler17 reported results by a similar approach. In their work, the guest-dependent vapor pressures of empty hydrates were used. They did not include LW in two-phase equilibria, and extensions to mixtures were not reported yet. Predicted equilibrium pressures for three-phase equilibria were compared with calculated results of Sloan,1 Barkan and Sheinin,18 and the present study in Table 6. Experimental data sets were from Sloan’s book. The Sloan results were obtained using the program given in his book.1 The Barkan results were reported for limited data points. For the methane equilibria, data for the equilibrium pressures up to 100 MPa were included for comparison. The AAD errors larger than 10% were found with the experimental data of Dyadin and Aladko19 for H-LW-V equilibria of methane. Some of their experimental data show large discrepancies with other experimental measurements. All three methods were generally acceptable for H-I-V and H-Lw-V, although the Sloan calculations were subject to somewhat larger errors. The calculated equilibrium pressures for H-LW-LC of ethane and propane by the present method show large deviations. Barkan and Sheinin did not report calculated results for H-LW-LC equilibria. The Sloan calculations were worse for H-LW-LC2H6 and
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Table 6. Comparisons of Experimental and Calculated Equilibrium Pressures for Three-Phase Equilibria Containing Simple Hydrates AAD [%] species
phase
no. of points
Barkan (1993)
Sloan (1998)
this work
T range [K]
P range [MPa]
methane
H-I-V H-LW-V H-I-V H-LW-V H-LW-Lc H-I-V H-LW-V H-LW-Lc
6 95 7 61 16 15 56 11
2.5 (5)a 2.7 (44)a 2.1 (4)a 2.5 (17)a
1.7 5.9 6.8 9.0 35.8 12.8 3.3
3.7 4.2 3.0 2.6 15.5 4.0 3.6 91.7
259.1-270.9 273.2-303.6 260.8-272.0 273.4-288.2 287.0-290.6 247.9-272.9 273.2-278.4 278.4-278.8
1.65-2.39 2.68-84.0 0.29-0.46 0.51-3.30 3.33-20.34 0.10-0.99 0.17-0.54 0.68-16.8
ethane propane
a
2.5 1.8 (24)a
Numbers in parentheses represent the number of data points used in the calculation.
Table 7. Comparisons of Experimental and Calculated Equilibrium Pressures for Three-Phase Equilibria Containing Mixed Hydrates AAD [%]
C1 + C2 C1 + C3
C2+C3 a
structure
data ref
phase
no. of points
I I I II II II II I II
20 21 22 20 22 23 24 25 25
H-LW-V H-LW-V H-LW-V H-LW-V H-LW-V H-LW-V H-LW-V H-LW-V/L H-LW-V
24 15 16 25 17 12 15 16 37
Barkan 2.1 9.7 25.1 (12)a 3.3 2.5 (28)a 2.5 (28)a
Sloan
this work
T range [K]
P range [MPa]
x1 range [mole fraction]
12.1 8.3 14.3 3.3 13.3 6.5 23.6 12.3 15.0
14.0 3.8 10.4 11.2 10.2 3.5 26.4 2.7 8.4
274.8-283.2 281.5-287.8 284.9-304.1 274.8-283.2 290.5-304.9 274.4-280.2 275.2-278.2 274.3-283.3 273.1-276.9
0.95-6.09 0.99-3.08 6.93-68.6 0.27-4.36 6.93-68.98 0.26-0.95 0.28-3.37 0.66-2.03 0.44-0.79
0.564-0.988 0.016-0.177 0.809-0.946 0.362-0.990 0.945-0.965 0.238-0.371 0.000-1.000 0.740-0.857 0.280-0.814
Numbers in parentheses represent the number of data points used in the Barkan and Sheinin calculation.
not available for H-LW-LC3H8. The large deviations in H-LW-LC equilibria are due to the very steep nature of the equilibrium line. A small error in the temperature can cause a large effect on the equilibrium pressure. Calculations for the three-phase equilibrium pressures of systems with mixed guests to form the same type of hydrate structure are relatively simple. Deviations between predicted and experimental three-phase equilibrium pressures of mixed hydrate are shown in Table 7. The binary systems of guest species forming the structure I hydrate are composed of methane and ethane. The present calculation shows good results in the binary mixtures of the structure I hydrate-forming species except for the data by Deaton and Frost20 for the mixture of methane + ethane hydrates. This is probably due to a phase transition that Subramanian et al.26 reported. Barkan and Sheinin did not report calculated results. Calculations of the hydrate-containing phase equilibria are complex for systems of mixed guests to form mixed hydrates of structures I and II. The calculation is very sensitive to the Kihara parameters and the vapor pressure of the empty hydrate. The stable structure type of the hydrate is subject to the system condition. Prior to the calculation of the equilibrium pressure, the stable structure type was determined. Applying the Clapeyron equation, Holder and Hand25 suggested a method to determine the proper structure type by their slope. In this work, the stable structures of hydrates were identified by the comparison of the total Gibbs free energy. The predictions for phase equilibria for water + methane + propane and water + ethane + propane are also compared with experimental data in Table 7. The largest error was observed in the water + methane + propane system by Thakore and Holder.24 Calculated results of the aqueous phase are compared in Table 8 for H-LW equilibria. The present prediction is good except for a data set of methane from Yang et
Table 8. Comparisons of Calculated and Experimental Solubilities of Hydrate Former in H-LW Equilibria species CH4 C2H6
data ref
no. of points
AAD [%]
T range [K]
P range [MPa]
3 4 this work this work
20 2 16 3
42.2 6.24 2.14 0.40
273.1-278.2 274.15-277.35 277.9-281.7 277.3-278.5
5.1-19.4 5.0 5.1-14.4 5.1-20.1
Table 9. Comparisons of Calculated and Experimental Water Contents for H-V and H-LC Equilibria Containing Simple Hydrates AAD [%] data no. of this species phase ref points Sloan work CH4 C2H6 C3H8
H-V H-V H-LC H-LC H-LC H-LC
28 29 29 30 29 30
12 3 4 6 5 9
23.3 69.3 14.0 16.6 49.3 7.9
20.9 19.8 12.0 23.6 8.7 39.9
T range [K]
P range [MPa]
240.0-270.0 276.2-283.7 260.1-281.2 259.10-270.5 255.6-276.2 246.7-276.4
3.5-10.3 2.48 3.45 3.45 1.10 0.77
al.3 in which concentrations were measured by the expansion of dissolved gas in the sampling valve. The error probably reflects the inaccuracy in the concentrations. It is interesting to note that present measurements show close agreements with Handa predictions27 that were linearly regressed with pressure and then interpolated to the same pressure experimented with in these experiments. The comparisons are graphically shown in Figure 2. Most available data for H-LC equilibria are from the laboratory of Kobayashi.28,29 Calculated results were compared with data in Table 9. For methane, the data of Aoyagi et al.28 were used. Two different sets of data were reported for water + ethane and water + propane by Song and Kobayashi29 and Sloan et al.,30 respectively. The comparisons for ethane were shown graphically in Figure 3. Discontinuities in the calculated values indicate the phase transitions of the ethane-rich phase from
Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2413 Table 10. Comparisons of Calculated and Experimental Water Contents for H-V and H-LC Equilibria Containing Mixed Hydrates species
phase
data ref
no. of points
Sloan
CH4 + C3H8 C2H6 + C3H8
H-V H-LC H-LC
31 29 30
23 14 6
91.9 14.1 24.6
a
AAD [%] this work 32.3 11.8 35.7
T range [K]
P range [MPa]
x2a
234.2-277.6 255.9-277.9 263.5-276.2
2.07-10.34 3.447 3.50
0.0531 0.105-0.500 0.085-0.898
Mole fraction of propane in the fluid phase.
Figure 2. Comparisons of calculated methane solubility in fluid phases with the present experimental data for H-LW equilibria of water + methane.
Figure 4. Comparisons of calculated water contents in fluid phases with isobaric data for H-V equilibria of water + methane + propane. The mole fraction of propane in the vapor phase is 0.0531.
quantitatively correct. We also have the problem of two different sets for ethane-propane mixed systems and observed better agreements with Song and Kobayashi sets. It should be noted that both water contents in Tables 9 and 10 for LC phases and hydrocarbon contents in Table 8 for LW phases are very small and that the present model accurately predicted them with a single binary parameter. 5. Conclusions
Figure 3. Comparisons of calculated water contents in fluid phases with literature data for H-V and H-LC equilibria of water + ethane.
the liquid to vapor phase. We used Song and Kobayashi sets in determining the parameters, and agreements with their sets are better than those with the Sloan sets. The water contents of the hydrocarbon-rich phase in equilibrium with hydrates were also investigated for mixed guests, and the results are summarized in Table 10. The results for water + methane + propane systems were shown graphically in Figure 4. Although the error in the table appears to be large, the same system shown graphically indicates that the present prediction is
The H-LW equilibria for the methane and ethane + water system were experimentally investigated. An apparatus using a syringe pump was developed for the indirect determination of the water contents. A variable volume view cell was used with a magnetic stirrer inside. With known liquid compositions, the dissociation temperature of the hydrate phase was determined by visual inspection. The estimated experimental accuracy was 0.1 MPa in pressure, 0.1 K in temperature, and 3.1% and 5.3% in composition for methane and ethane, respectively. Various hydrate-containing two- and three-phase equilibria of simple and mixed hydrates were predicted and compared with the experimental results from the literature and the present experimental works. A hydrogen-bonding lattice-fluid EOS and the van der Waals and Platteeuw model were used in the prediction. In most hydrate-containing three-phase equilibria, good agreements were obtained between experimental and calculated pressures. The agreements were also good for solute contents in H-LC and H-LW equilibria using
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a single binary parameter. No other investigators reported calculations on H-LW equilibria. The results were generally similar to Sloan’s calculations expect for some cases where the present results were much better. Systems with large deviations between calculations and experiments were discussed. Acknowledgment The authors acknowledge support by the Korea Science and Engineering Foundation under Contract No. 98-0502-04-01 and by the Ministry of Education in the Brain Korea 21 project. Literature Cited (1) Sloan, E. D. Clathrate Hydrates of Natural Gases, 2nd ed.; Dekker: New York, 1998. (2) Yang, S. O.; Yang, I. M.; Kim, Y. S.; Lee, C. S. Measurement and Prediction of Phase Equilibria for Water + CO2 in Hydrate Forming Conditions. Fluid Phase Equilib. 2000, 175, 79. (3) Yang, S. O.; Cho, S. H.; Lee, C. S. Measurement and Prediction of Phase Equilibria for Water + Methane in Hydrate Forming Conditions. Fluid Phase Equilib. 2001, 185, 53. (4) Servio, P.; Englezos, P. Measurement of Dissolved Methane in Water in Equilibrium with Its Hydrate. J. Chem. Eng. Data 2002, 47, 87. (5) AllPROPS Property Package; http://www.uidaho.edu/∼cats/ software.htm. (6) You, S. S.; Yoo, K. P.; Lee, C. S. An Approximate Nonrandom Lattice Theory of Fluids. General Derivation and Application to Pure Fluids. Fluid Phase Equilib. 1994, 93, 193. (7) You, S. S.; Yoo, K. P.; Lee, C. S. An Approximate Nonrandom Lattice Theory of Fluids. Mixtures. Fluid Phase Equilib. 1994, 93, 215. (8) Yeom, M. S.; Park, B. H.; Lee, C. S. A Nonrandom Lattice Fluid Hydrogen Bonding Theory for Phase Equilibria of Associating Systems. Fluid Phase Equilib. 1999, 158-160, 143. (9) Lee, C. S.; Yoo, K.-P.; Park, B. H.; Kang, J. W. On the Veytsman Statistics as Applied to Non-random Lattice Fluid Equations of State. Fluid Phase Equilib. 2001, 187-188, 433. (10) Park, B. H.; Kang, J. W.; Yoo, K.-P.; Lee, C. S. An Explicit Hydrogen-Bonding Non-random Lattice-Fluid Equation of State and its Application. Fluid Phase Equilib. 2001, 183-184, 111. (11) Perry, R. H.; Green, D. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1989. (12) Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M. Vapor-Liquid Equilibria for Low Boiling Substances; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1982. (13) Luck, W. P. Angew. Chem., Int. Ed. Engl. 1980, 19, 28. (14) van der Waals, J. H.; Platteeuw, J. C. Clathrate Solutions. Adv. Chem. Phys. 1959, 2, 1.
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Received for review November 22, 2002 Revised manuscript received March 12, 2003 Accepted March 18, 2003 IE0209374
