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J. Phys. Chem. B 2009, 113, 7558–7563
Hydrogen Storage in sH Clathrate Hydrates: Thermodynamic Model ´ ngel Martı´n† and Cor J. Peters*,†,‡ A Faculty of Mechanical, Maritime and Materials Engineering, Department of Process and Energy, Laboratory of Process Equipment, Delft UniVersity of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands, and The Petroleum Institute, Department of Chemical Engineering, P.O. Box 2533, Abu Dhabi, United Arab Emirates ReceiVed: August 20, 2008; ReVised Manuscript ReceiVed: April 8, 2009
A thermodynamic model of equilibrium states of structure sH clathrate hydrates of hydrogen with methyl tert-butyl ether, methyl cylohexane, or 1,1-dimethyl cyclohexane is presented. The model uses the van der Waals-Platteeuw statistical-thermodynamical model to represent the hydrates and the cubic-plus-association equation of state to represent the fluid phases. Good agreement between experimental and calculated phase equilibrium data has been obtained, with average absolute pressure deviations between experiments and calculations ranging from 1.2 to 2.1% depending on the promoter. The model has also been used to estimate the occupancy of the cavities of the hydrate by hydrogen and the promoter, as well as the hydrogen storage capacity of the hydrate. This capacity has been found to vary between 0.85 and 1.05% of hydrogen by weight at the conditions of formation of the sH hydrates (270-280 K and 60-100 MPa). Introduction Hydrogen has a great potential as energy source for a number of reasons: the energy produced by oxidation per unit of mass of hydrogen (142 MJ/kg) is at least three times larger than that of other chemical fuels; its oxidation product is water, and in addition hydrogen can be generated from renewable sources in a closed system. Because of this, technologies of using hydrogen as alternative fuel in automotive applications are under continuous development.1 One of the main technological challenges in the implementation of these technologies is the development of a safe and efficient way of storing hydrogen.2,3 Because of its low density, hydrogen occupies large volumes when stored as a gas. For example, in conventional high-pressure tanks (usually filled up to 200 bar), the volume required to store hydrogen is as high as 56 L/kg. This volume can be reduced if hydrogen is stored as a liquid, but in this case cryogenic temperatures are required (-253 °C at 1 bar). Moreover, the energy cost of compressing and cooling down the gas is very high, and thus this option becomes very expensive and it is not practical for everyday use. Other possibilities include adsorption of gaseous hydrogen on solid surfaces of carbon4 and the use of metal hydrides5 with conflicting results about the reversibility of the process, that is, the possibility of releasing all the hydrogen that was stored in the carbon or in the hydride, and the energy storage density. Clathrate (or gas) hydrates are a type of inclusion compounds characterized by formation of a regular crystal lattice consisting of water molecules in which cavities are created, which are occupied by guest molecules.6 Depending on the size and properties of the guest molecules, different hydrate crystalline structures can be formed, and up to now three different clathrate hydrate structures were known, two cubic structures known as structures I (sI) and II (sII) and a hexagonal structure known as structure H (sH). Each structure is characterized by the formation * To whom correspondence should be addressed. E-mail: C.J.Peters@ tudelft.nl. † Delft University of Technology. ‡ The Petroleum Institute.
TABLE 1: Hydrate Crystal Cell Data and Geometry of Cavities for the Three Major Clathrate Hydrate Structures, sI, sII and SH6 characteristic cavity type Rcell (Å)a Z N Nw crystal type ac (Å) cc (Å)
sI 12
5 3.95 20 2
sII 12 2
5 6 4.33 24 6
12
5 3.91 20 16
46 cubic 12.03
12 4
5 6 4.73 28 8 136 cubic 17.31
sH 12
5 3.91 20 3
3 6 3
456 4.06 20 2
51268 5.71 36 1
34 hexagonal 12.21 10.14
a The average cavity radius also entails the radius of water (at the cavity wall). To obtain the cavity radius that shows the viable occupation space for a guest molecule, the radius of the water molecules (1.44 Å) must be subtracted from this value.
of cavities with different size and geometry, as presented in Table 1. The unit cell of sI hydrates consists of 46 water molecules forming two small cavities (with the shape of a pentagonal dodecahedron, 512) and six large cavities (a tetradecahedron with twelve pentagonal faces and two hexagonal faces, 51262). The unit cell of sII hydrates consists of 136 water molecules forming sixteen small 512 cages and eight large hexadecahedron 51264 faces. Each unit cell of structure sH requires 34 water molecules and contains three small cavities of type 512, two medium cavities of type 435663 (similar in size with the small cavities of sI and sII hydrates), and one very large cavity of type 51268 (Table 1). Because of the large differences between the sizes of small and large cavities of sH hydrates, to form these type of hydrates the presence of two types of guests of different molecular sizes is required. The discovery of the gas hydrate of hydrogen by Mao et al. in 20027 gave rise to the possibility of using these compounds for hydrogen storage. Mao et al. determined that the hydrogen hydrate was a sII hydrate. The estimated concentration of hydrogen in the hydrate with a high H2/H2O molar ratio of 0.45 ( 0.05, required double occupancy of the small cavities and quadruple occupancy of the large cavities. Later, Lokshin
10.1021/jp8074578 CCC: $40.75 2009 American Chemical Society Published on Web 05/06/2009
Hydrogen Storage in sH Clathrate Hydrates et al.8 determined by high pressure diffraction studies that the hydrogen occupancy in the large cavity of the sII structure is reversible between two and four molecules of hydrogen depending on the conditions, while the occupancy of the small cavity remains constant at one hydrogen molecule, thus reducing the maximum H2/H2O molar ratio to approximately 0.36 at elevated pressures. The main limitation for the practical application of the discovery of Mao et al. is the high pressure required to stabilize the sII hydrogen hydrates (220 MPa at 249 K). In 2004, Florusse et al.9 demonstrated that the formation pressure of hydrogen hydrates can be reduced to less than 10 MPa using tetrahydrofuran (THF) as a promoter. In this mixed sII hydrate, the promoter occupies the large cavities, stabilizing the hydrate at low pressures, while hydrogen occupies the small cavities. The drawback of this approach is the reduction in the storage capacity of the hydrate caused by the occupation of the large cavities by the promoter. Several experimental10 and theoretical11,12 studies have shown that only one hydrogen molecule is stored in the small cavities at the conditions of formation of the hydrate. Thus, the H2/(H2O+THF) molar ratio is reduced to about 0.11, which corresponds to a hydrogen storage capacity of about 1% by weight at moderate pressures. An alternative for increasing the storage capacity of the hydrate is to produce structure sH clathrate hydrates of hydrogen stabilized by a promoter. Therefore, if the large cages are occupied by the promoter and all the small and medium cavities are occupied by one molecule of hydrogen, the H2/(H2O + promoter) ratio in structure sH can increase to up to 0.14 (approximately 1.4% by weight of hydrogen), which is about 40% higher compared to the hydrogen storage capacity in structure II. Duarte et al.13 were able to stabilize sH hydrates of hydrogen at moderate pressures and temperatures (270-280 K and 60-100 MPa). Three different promoters were considered: methyl tert-butyl ether (MTBE), methyl cyclohexane (MCH), and 1,1-dimethyl cyclohexane (DMCH). In this work, a fugacity-based van der Waals-Platteeuw statistical thermodynamical model is used to correlate the experimental formation conditions of sH hydrogen hydrates with MTBE, MCH, and DMCH as promoters. This model is based on the work of Klauda and Sandler,14 and it has been presented in detail and compared to other models available in the literature in a previous work.15 In another previous work,11 this model was used to correlate the formation conditions of sII hydrogen hydrates with different water-soluble and water-insoluble promoters and for calculating the hydrogen storage capacity of these hydrates as well as the variation of this capacity with different parameters such as pressure or promoter concentration. The aim of this work is to obtain the parameters required by this previously developed model by correlation of experimentally obtained formation conditions of sH hydrogen hydrates with different promoters and then to apply the model to obtain derived properties such as the occupancy of the cavities and the hydrogen storage capacity of the hydrate at different conditions, which provide valuable information complementary to the experimental data available at the moment. Model Description. The proposed phase equilibrium model is based on solving the condition of equal fugacity of water in the hydrate and the remaining phases.14,15 For solving this condition, the fugacity of water in the fluid phases is calculated with an equation of state, while the fugacity of water in the hydrate is calculated with eq 1
J. Phys. Chem. B, Vol. 113, No. 21, 2009 7559
f wH(T, P)
)
(
f wβ (T, P)exp
-∆µwH(T, P) RT
)
(1)
In eq 1, fwβ is the fugacity of the hypothetical, empty hydrate lattice. This fugacity depends on the guest(s) of the hydrate, which allows to take into account the different degree of lattice distortion caused by different guests.14,15 The chemical potential difference between the empty and filled hydrate ∆µH is calculated according to the van der Waal-Platteeuw (VdWP) statistical thermodynamics theory15
∆µwH(T, P) ) -RT
∑ Vm ln(1 - ∑ θmj) m
(2)
j
In this equation, Vm is the number of cages of type m per water molecule in the hydrate lattice, and θmj is the fraction of cages of type m that are occupied by the guest j. This fraction is calculated with a Langmuir adsorption relation. The form of the Langmuir relation depends on the number of molecules that occupy the cavity.17 In a previous work,11 it was shown that single occupancy of the small cavities of sII hydrogen hydrates was expected in the pressure range 10-100 MPa. The same conclusion has been recently obtained by several authors, both experimentally10 or by simulations of sII and sH hydrates.18,19 The results obtained for the small cavities of sII hydrates can be extrapolated to the small and medium cavities of sH hydrates, which are smaller and of the same size of the small cavities of sII hydrates, respectively (Table 1). Therefore, single occupancy of the small and medium cavities by hydrogen has been assumed, and thus the Langmuir relation takes the form presented in eq 3
θmj(T, P) )
Cmj(T )fj(T, P) 1+
(3)
∑ Cmi fi(T, P) i
The Langmuir constants Cmi required for application of eq 3 have been calculated assuming spherical symmetry, as presented in eq 4. Since the interaction between guests of adjacent cavities can represent a major contribution to the calculation of Langmiur constants, the contribution of the guest-guest interactions to the Langmuir constant have been explicitly taken into account in eq 4 by introducing the intermolecular potential between guests W gg17
Cmj )
gg gl Cmj · Cmj
( )
gg Wmj ) exp × kT 4π Rcell-a exp(-W(r)/kT)r2 dr (4) kT 0
∫
The fugacity of the reference empty lattice fwβ in eq 1 can be calculated referring it to the gas phase as shown in eq 5
(∫
f wβ (T, P) ) Pwsat,β(T)exp
)
V wβ dP Psat RT P
(5)
For the evaluation of the integral in eq 5, the molar volume of water in sH hydrates has been evaluated with eq 6, derived from the expressions of Tse for the crystal lattice parameter of structure sH hydrates as a function of temperature.20 In eq 6, the isothermal compressibility of the hydrate has been considered equal to 1.05 × 10-4 MPa-1, a value obtained by molecular simulations of empty structure sI hydrates by Docherty et al21
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Martı´n and Peters TABLE 4: Lennard-Jones 6-12 Potential Function Parameters
31.5 (12.268 + 0.697 × 10-3(T - 80) + 2 1.33 × 10-6(T - 80)2)2(9.997 - 0.332 × 10-3(T 10-30NA -5 2 80) + 4.30 × 10 (T - 80) ) × NHW
Vwβ,H )
exp(-1.05 × 10-10(P - 101325)) (6) Model Parameters and Numerical Methods. Equation of State. For the calculation of the fugacity of water in the fluid phases as well as the calculation of fluid phase equilibrium, the cubic-plus-association equation of state (CPA-EoS22,23) has been used. This equation was specifically developed for applications with mixtures of water and hydrocarbons or gases and therefore is especially suitable for the description of hydrate-forming systems, normally comprised by mixtures of these compounds. The application of this equation allows modeling systems either with water-miscible or with water-immiscible promoters. The CPA-EoS is constituted by a physical term and an association term. The physical term is the Soave Redlich Kwong cubic equation of state and the association term is taken from the statistical associating fluid theory equation of state (SAFT)22
P)
RT a RT + F V-b V(V + b) V
∑ A
[
]
1 1 ∂X A 2 ∂F XA
(7)
For mixtures with only one self-associating component, the extension of the association term of eq 7 is straightforward, while an interaction coefficient is required for the calculation of the a parameter of the physical term (eq 8)
a)
∑ ∑ xi xj√aiaj(1 - kij) i
(8)
j
The four-site 4C association scheme has been used for water,23 while the remaining components have been considered to be nonassociating. Pure component parameters are summarized in Table 2. The parameters of water have been obtained from a previous work,15 while the parameters of the remaining components have been estimated from their critical properties. The interaction coefficients kij of eq 8 presented in Table 3 have been obtained by correlation of experimental phase TABLE 2: Pure Component Parameters, CPA EoS substance
b 105
water 1.4515 hydrogen 1.8418 methyl tert-butyl ether 10.419 methyl cyclohexane 11.838 1,1-dimethyl cyclohexane 14.481
a0 0.12277 0.02492 2.1236 2.7767 3.5089
c1
εAB
β
0.67359 16655 0.0692 0.1426 0.8883 0.8599 0.8455
substance
ε/kb
σ (A)
reference
water hydrogen methyl tert-butyl ether methyl cyclohexane 1,1-dimethyl cyclohexane
102.134 37 394.74 454.30 469.31
3.564 2.93 5.53 5.80 6.20
ref 14 ref 31
a
a a a
Estimated with the method of Chung et al.29
equilibrium data, when available. The correlation has been performed by minimization of the average absolute deviation (AAD) between experimental and calculated data (eq 9). In the case of the water-hydrogen and water-promoter systems, the interaction parameters have been obtained by correlation of solubility data of the promoters and hydrogen in water. On the other hand, experimental solubility data of hydrogen in the promoters is not available. For this reason, the interaction coefficients of the binary systems hydrogen-promoter have been obtained by correlation of solubility data of hydrogen in the promoter estimated with the predictive soave redlich kwong equation of state (PSRK EoS24). Since model results are insensitive to the interaction parameters between gaseous and liquid compounds,25 the accuracy of the predictions of the model is not expected to be significantly affected by the lack of these experimental data
AAD% )
100 ndata
ndata
i |X iexp - X calc |
i)1
X iexp
∑
Lennard-Jones 6-12 Potential Function. Since neither the parameters of the Kihara potential function, nor the parameters of the Lennard-Jones 6-12 potential function were available for the promoters, these parameters were estimated. In this work, the Lennard-Jones parameters reported in Table 4, estimated from the critical properties of the promoters with the method of Chung et al.,29 were used. The estimation of Lennard-Jones parameters by the method of Chung et al. was preferred to the estimation of Kihara parameters by the method of Tee30 because the method of Chung et al. is based on a larger database and therefore it has been considered to be more reliable. Lennard-Jones parameters of water and hydrogen have been obtained from refs 13 and 30, respectively. It must be emphasized that the Lennard-Jones parameters have not been correlated to experimental hydrate phase equilibrium data. The common practice of correlating the parameters of the potential function has the disadvantage that with this method, possible inaccuracies of other parts of the model are compensated by adjusting the occupancies (which are determined by the potential function parameters), therefore rendering the calculation of occupancies unreliable.
TABLE 3: Interaction Coefficients kij of CPA-EoS
a
binary system
kij
temperature range (K)
H2O-H2 H2O-MTBE H2O-MCH H2O-DMCH H2-MTBE H2-MCH H2-DMCH
-1546.9/T + 5.7616 120.34/T - 0.6087 0.1041 0.0314 -94.663/T + 0.5461 -42.793/T + 0.2547 0.1045
292.65 K - 296.15 K 275.5 K - 298.15 K 275.5 K - 298.15 K 273.15 K - 313.15 K 273 K - 285 K 273 K - 285 K 273 K - 285 K
Correlated to data generated with the PSRK EoS.24
(9)
AAD and ref 2.5% 0.6% 8.0% 2.1% a a a
of of of of
pressure, ref 26 solubility, ref 27 solubility, ref 27 solubility, ref 28
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TABLE 5: Correlated Empty Lattice Vapour Pressure Pwsat,β Equations and Deviations between Model Results and Experimental Data for sH Hydrogen Hydrates with Different Promoters promoter
ln Pwsat,β
temp range (K)
R2
AAPD%
methyl tert-butyl ether methyl cyclohexane 1,1-dimethyl cyclohexane
-6096.17/T + 29.15 -6189.95/T + 29.52 -6217.36/T + 29.65
269.2-272.5 K 274.0-275.9 K 274.7-279.5 K
0.99993 0.9998 0.9998
1.2% 2.1% 2.0%
With the approach followed in this work, the compensation of inaccuracies and deficiencies of other parameters of the model is performed by the correlated empty lattice vapor pressures, Pwsat,b, and therefore the calculated occupancies are not affected by these possible inaccuracies, and depend only on the calculated Langmuir constants, and particularly on the intermolecular potentials used for this calculation. Numerical Methods. For the resolution of the condition of equality of the fugacity, a flash calculation algorithm based
on Gibbs energy minimization formalism has been implemented. This formalism allows simultaneously performing stability and multiphase equilibrium calculations.32,33 This method consists in minimizing the total Gibbs energy of the system as in eq 10
∂G ) λk ∂Rk
(10)
Where Rk is the mole phase fraction of phase k and λk are the Lagrange multipliers used for the minimization. It can be shown that in a system in equilibrium the relation presented in eq 11 must be fulfilled for all phases
Rkλk ) 0
(11)
That is, in a stable phase with Rk > 0, necessarily λk ) 0, and in an unstable phase with Rk ) 0, in general λk > 0. Thus λk provides an estimation of the stability of phase k.
Figure 1. Experimental (symbols) and calculated (lines) pressuretemperature diagrams of sH hydrogen hydrates with different promoters.
4. Results and Discussion 4.1. Correlation of P-T Phase Equilibrium Diagrams. As discussed in the previous sections, the only parameter of the model that has to be correlated to experimental data is the vapor pressure of the empty β-lattice Pwsat,β (eq 9). This parameter depends on the degree of lattice distortion caused by the guest molecules, and therefore it depends on the promoter used to
Figure 2. (a) Cage occupacies and (b) hydrogen storage capacity of sH hydrogen hydrates with MTBE as promoter as a function of pressure. The temperature considered for the calculations is the temperature of formation of the sH H2 + MTBE hydrate at each pressure.
Figure 3. (a) Cage occupacies and (b) hydrogen storage capacity of sH hydrogen hydrates with MCH as promoter as a function of pressure. The temperature considered for the calculations is the temperature of formation of the sH H2 + MCH hydrate at each pressure.
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Martı´n and Peters
Figure 4. (a) Cage occupacies and (b) hydrogen storage capacity of sH hydrogen hydrates with DMCH as promoter as a function of pressure. The temperature considered for the calculations is the temperature of formation of the sH H2 + DMCH hydrate at each pressure.
stabilize the hydrates.14,15 Furthermore, the dependence of Pwsat,β with temperature can be described by a simple ClausiusClapeyron equation25
ln Pwsatat,β ) a + b/T
(12)
The equations obtained by correlation of the experimental data of sH hydrogen hydrates with MTBE, MCH, and DMCH as promoters are presented in Table 5. These equations have been obtained by first calculating, for each of the experimental points reported in ref 12 the Pwsat,β required for reproducing the experimental hydrate formation conditions with the model, and then correlating the dependence of the calculated Pwsat,β on temperature with eq 12. A good linear regression coefficient R2 has been obtained in the correlation of Pwsat,β versus T with this procedure. Thus it is expected that the model is able to extrapolate to other conditions of pressure and temperature with good accuracy. The average absolute pressure deviations (AAPD%, eq 9) between experimental and calculated formation pressures of the hydrates are reported in Table 5 as well. Figure 1 also presents a comparison between experimental and calculated data. It can be seen that the agreement between the model and the experiments is excellent with AAPD% ranging from 2.3 to 3.6% depending on the promoter used to stabilize the hydrate. 4.2. Calculation of Occupancies and Hydrogen Storage Capacity. The proposed model can also be used to estimate the occupancies of the cavities of the sH hydrate by the guests as well as the hydrogen storage capacity. The results are presented in Figures 2-4. In these Figures, the occupancies and storage capacity are displayed as a function of pressure for the three promoters (MTBE, MCH, and DMCH). At each pressure, the temperature considered for the calculations is the temperature of formation of the hydrate shown in Figure 1. It can be seen that the results obtained with the three promoters have similar characteristics: the promoter selectively occupies the large cavities with nearly complete occupation of these cavities, while the occupancy of small and medium cavities by hydrogen increases with pressure, being in the range 0.75-0.85. Therefore, the hydrogen storage capacity also increases with pressure. In the range of pressure and temperature of formation of hyrates considered in this work (270-280 K and 60-100 MPa), the storage capacity varies between 0.85 and 1.05% of hydrogen by weight. It is worth mentioning that since the occupation of large cavities by promoter is very close to unity, and the chemical potential of the hydrate varies with the logarithm of (1-θ) (eq 2), model results regarding the formation conditions of the hydrates are very sensitive to the calculations of this parameter, and therefore to the interaction
Figure 5. Occupancy of the medium cavity of the sH hydrates as a function of pressure at a fixed temperature of 273 K.
potentials between promoter and hydrate lattice, which are the main parameter that determines this occupancy. The proposed model can also be used to extrapolate experimental data in order to obtain the pressure required to achieve nearly complete occupation of the small and medium cavities by hydrogen and therefore the maximum storage capacity of the hydrates. The results obtained are presented in Figure 5, which presents the occupancy of the medium cavity of sH hydrates with the three promoters as a function of pressure. It can be seen that very similar results are obtained with the three promoters, and that very high pressures of about 300 MPa are required in order to achieve occupations of about 0.99. Nearly identical results are obtained when the occupancy of the small cavities is examined. Conclusions A fugacity-based van der Waals-Platteeuw statisticalthermodynamical model of equilibrium states of structure sH hydrogen hydrates with three different promoters (methyl tertbutyl ether, methyl cyclohexane and 1,1-dimethyl cyclohexane) has been presented. The model has been used to correlate experimental phase equilibrium data of sH hydrogen hydrates with excellent results, with average absolute pressure deviations between experiments and calculations ranging from 1.2 to 2.1% depending on the promoter. The model has also been used to calculate the cage occupancies as well as the hydrogen storage capacity of the hydrate. It has been found that the promoter occupies selectively and completely the large cavities, while hydrogen occupies the small and medium cavities with fractional occupancies ranging from 0.75 to 0.85 at the conditions of formation of the hydrates (270-280 K and 60-100 MPa). At these conditions, the storage capacity of the hydrate varies between 0.85 and 1.05% of hydrogen by weight.
Hydrogen Storage in sH Clathrate Hydrates
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Acknowledgment. The authors gratefully acknowledge the financial support of the Stichting Technische Wetenschappen (STW) in The Netherlands.
L V β
Nomenclature
References and Notes
a ac a0 b c1 cc f k kij n NA NW P Psat r R Rcell T V V W(r) x XA z
(1) Dresselhaus, M. S.; Thomas, I. L. Nature 2001, 414, 332–337. (2) Schlapbach, L.; Zuttel, A. Nature 2001, 414, 353–358. (3) Zuttel, A. Naturwissenschaften 2004, 91, 157–172. (4) Dillon, A. C.; Heben, M. J. Appl. Phys. A 2001, 72, 133–142. (5) Hydrogen in intermetallic compounds. In Surface and dynamic properties, applications; Schlapbach, L. Ed.; Springer: Berlin 1992; Vol 2. (6) Sloan, E. D. Clathrate Hydrates of Natural Gas, 2nd ed.; Marcel Dekker: New York, 1998. (7) Mao, W. L.; Mao, H.-K.; Goncharov, A. F.; Struzhkin, V. V.; Guo, Q.; Hu, J.; Shu, J.; Hemeley, R. J.; Somayazulu, M.; Zhao, Y. Science 2002, 297, 2247. (8) Lokshin, K. A.; Zhao, Y.; He, D.; Mao, W. L.; Mao, H.-K.; Hemley, R. J.; Lovanov, M. V.; Greenblatt, M. Phys. ReV. Lett. 2004, 93, 125503. (9) Florusse, L. J.; Peters, C. J.; Schoonman, J.; Hester, K. C.; Koh, C. A.; Dec, S. F.; Marsch, K. M.; Sloan, E. D. Science 2004, 306, 469– 471. (10) Strobel, T. A.; Taylor, C. J.; Hester, K. C.; Dec, S. F.; Koh, C. A.; Miller, K. T.; Sloan, E. D., Jr. J. Phys. Chem. B 2006, 110, 17121–17125. (11) Martín, A.; Peters, C. J.Thermodynamic modeling of structure II clathrate hydrates of hydrogen. J. Phys. Chem. C, DOI: 10.1021/jp807367j. (12) Papadimitriou, N. I.; Tsimpanogiannis, I. N.; Papaioannou, A. Th.; Stubos, A. K. J. Phys. Chem. B 2008, 112, 10294–10302. (13) Duarte, A. R. C.; Shariati, A.; Rovetto, L. J.; Peters, C. J. J. Phys. Chem. B 2008, 112 (7), 1888–1889. (14) Klauda, J. B.; Sandler, S. I. Ind. Eng. Chem. Res. 2000, 39, 3377– 3386. (15) Martín, A.; Peters, C. J.J. Phys. Chem. C 2009, 113 (1), 422-430. (16) van der Waals, J. H.; Platteeuw, J. C. AdV. Chem. Phys. 1959, 2, 1–57. (17) Klauda, J. B.; Sandler, S. I. Chem. Eng. Sci. 2003, 58 (1), 127–41. (18) Alavi, S.; Ripmeester, J. A.; Klug, D. D. J. Chem. Phys. 2005, 123, 024507. (19) Alavi, S.; Ripmeester, J. A.; Klug, D. D. J. Chem. Phys. 2006, 124, 204707. (20) Tse, J. S, J. Inclus. Phenom. Mol. 1990, 8, 25. (21) Docherty, H.; Galindo, A.; Vega, C.; Sanz, E. J. Chem. Phys. 2006, 125, 074510-1-074510-9. (22) Kontogeorgis, G. K.; Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. Ind. Eng. Chem. Res. 1996, 35, 4310–4318. (23) Kontogeorgis, G. K.; Michelsen, M. L.; Folas, G. K.; Derawi, S.; von Solms, N.; Stenby, E. H. Ind. Eng. Chem. Res. 2006, 45, 4855–4868. (24) Holderbaum, T.; Gmehling, J. Fluid Phase Eq. 1991, 70, 251– 265. (25) Zhang, Y.; Debenedetti, P. G.; Prud’homme, R. K.; Pethica, B. A. J. Pet. Sci. Eng. 2006, 51, 45. (26) Perry’s Chemical Engineers’ Handbook, 7th ed.; Perry, R. H., Green, D. W. Eds.; McGraw-Hill, New York, 1997. (27) Susilo, R.; Lee, J. D.; Englezos, P. Fluid Phase Equilib. 2005, 231, 20–26. (28) Dohanyosova, P.; Sarraute, S.; Dohnal, V.; Majer, V.; Costa Gomas, M. Ind. Eng. Chem. Res. 2004, 43, 2805–2815. (29) Poling, B. E., Prausnitz, J., O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; Mc Graw-Hill, New York, 2001. (30) Tee, L. S.; Gotoh, S.; Stewart, W. E. Ind. Eng. Chem. Fundam. 1966, 5 (3), 356–363. (31) McQuarrie, D. A., Simon, J D. Molecular Thermodynamics; University Science Books, Sausalito, CA, 1999. (32) Gupta, A. K.; Bishnoi, P. R.; Kalogerakis, N. Fluid Phase Equilib. 1991, 63, 65–89. (33) Ballard, A. L.; Sloan, E. D. Fluid Phase Equilib. 2004, 218, 15–31.
spherical hard-core radius, Kihara parameter (m) crystal lattice dimension (Å) CPA-EoS pure component parameter CPA-EoS pure component parameter CPA-EoS pure component parameter crystal lattice dimension (Å) fugacity (Pa) Boltzmann constant (J/K) interaction coefficient, CPA-EoS number of cavities per unit cell Avogadro number number of H2O molecules per unit cell pressure (Pa) vapor pressure (Pa) radius (m) gas constant (J/mol K) cavity radius (m) temperature (K) number of cavities per H2O molecule molar volume (m3/mol) intermolecular potential molar fraction fraction of occupied association sites, CPA-EoS coordination number of cavity
Acronyms AAD% MTBE MCH DMCH
average absolute deviation (eq 9) methyl tert-butyl ether methyl cyclohexane 1,1-dimethyl cyclohexane
Greek Symbols β ε/k εΑΒ µ θ F σ
CPA-EoS pure component parameter maximum attractive potential energy, Kihara parameter (K) CPA-EoS pure component parameter chemical potential (J/mol) cavity occupancy density (mol/m3) zero potential energy difference, Kihara parameter (m)
Subscripts and Superscripts calc H exp i
calculated result clathrate hydrate experimental data component i in mixture
JP8074578
liquid vapor empty β-lattice
