Ind. Eng. Chem. Res. 2000, 39, 1111-1114
1111
Modeling of Structure H Hydrates Using a Langmuir Adsorption Model Jesper Madsen and Karen Schou Pedersen* Calsep A/S, Gl. Lundtoftevej 7, DK-2800 Lyngby, Denmark
Michael L. Michelsen Institut for Kemiteknik, Building 229, Technical University of Denmark, DK-2800 Lyngby
A Langmuir adsorption model is used to represent the conditions at which structure H hydrates may form. The two smaller cavities of structure H hydrates are of similar size and are modeled using the same Langmuir constants. Parameters in a simple two-parameter Langmuir expression have been estimated for methane and nitrogen as guest molecules of the smaller cavities and for 12 possible guest molecules of the large cavity. The latter ones are all hydrocarbons with from 5 to 8 carbon atoms. Experimental hydrate formation temperatures are correlated with an average absolute temperature deviation of 0.15 K. Hydrate formation data for two ternary mixtures not used in the parameter estimation are modeled with approximately the same accuracy. Introduction Until recently only two hydrate structures were known, structures I and II. These could only accommodate gas molecules of rather modest size, the largest guest molecule being isobutane. Gas hydrates of structures I and II can be modeled using Langmuir adsorption theory.1-3 In 1987 a third hydrate structure was discovered.4 It was named structure H and is a hexagonal structure with six cavities per unit cell, three smallsized ones, two medium-sized ones, and one large-sized one. The large cavity can accommodate hydrocarbon molecules with from 5 to 8 carbon atoms. The mediumand small-sized cavities are occupied by gas molecules such as methane or nitrogen. Structure H gas hydrates have been reviewed by Dendy Sloan5 and Englezos.6 Structure H Hydrates Some important properties of structure H hydrates are listed in Table 1. The notation used to describe the i polyhedra structure is nm i , where ni is the number of edges in face type i and mi the number of faces with ni edges. Structure H hydrates differ from structure I and II hydrates by accommodating three different cavity sizes, not just two. As may be seen from Table 1, the two smaller cavities of structure H are of very similar size. The largest cavity in structure H hydrates is considerably larger than the largest cavity of the two other structures and can therefore accommodate larger guest molecules. Experimental Structure H Data Tables 2 and 3 give an overview of the experimental structure H hydrate formation data used in the model work. Structure H data have also been measured for 2,2dimethylpentane.7,8 The hydrate formation tempera* To whom corresodence should be addressed. Phone: +45 45 87 66 46. Fax: +45 45 87 62 72. E-mail:
[email protected].
Table 1. Physical Properties for Structure H Gas Hydrates5 (X Refers to Large Cavities, Y to Small Cavities, and Z to Medium-Sized Cavities) crystal system cavities polyhedra structure no. of cavities per unit cell average cavity radius (Å) ideal unit cell formula
hexagonal small/medium/large 512/435663/51268 3/2/1 3.91/4.06/5.71 1X*3Y*2Z*34H2O
Table 2. Sources of Experimental Structure H Formation Data with Methane as a Guest Molecule of Smaller Cavities large cavity occupied by
T range (K)
P range (MPa)
reference
isopentane neohexane 2,3-dimethylbutane 2,2,3-trimethylbutane 3,3-dimethylpentane methylcyclopentane cycloheptane methylcyclohexane 1,1-dimethylcyclohexane cis-1,2-dimethylcyclohexane ethylcyclopentane cyclooctane
274-279 245-288 276-286 276-289 275-286 276-288 281-290 276-290 280-293 276-290 280-290 282-290
2.2-4.2 0.3-7.5 2.1-8.2 1.5-7.6 1.7-7.3 2.2-10.0 3.4-10.91 1.6-11.2 2.0-11.5 1.9-11.3 3.4-10.9 4.2-11.7
10, 12 8, 10, 18, 19 7, 8 7 7, 8 7, 8, 20 21 7, 8, 16, 22 21 7, 8 8, 21 21
Table 3. Sources of Experimental Structure H Formation Data with Nitrogen as a Guest Molecule of Smaller Cavities large cavity occupied by T range (K) P range (MPa) reference methylcyclopentane methylcyclohexane
274-286 276-284
7.8-30.7 8.8-20.4
20 17
tures reported in these two literature sources do however deviate as much as 7 K for the same pressure. For this reason this component has been left out of the model work. Modeling of Hydrates The phase transition from pure liquid water or ice to the hypothetical state of an empty hydrate lattice is associated with a change in chemical potential of water
10.1021/ie990677z CCC: $19.00 © 2000 American Chemical Society Published on Web 02/26/2000
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Langmuir Adsorption Constant
Table 4. Phase Transition Parameters from Water to Structure H Hydrate9 ∆µ0w (J/mol) ∆H (J/mol) ∆Cp (J/mol K) ∆V (cm3/mol)
Cki )
as3
which can be approximated
µ/w - µ0w ∆µ0w(T0, P0) ) RT RT0 T ∆H° + ∆Cp(T - T0)
∫T
0
Mehta and Sloan9,10 have modeled structure H hydrate formation using the following expression for the Langmuir constant,
1187.33 846.57 -39.16 3.85
RT
2
dT +
∆V dP ∫PP RT h 0
(1)
where µ/w is the chemical potential of water in the empty hydrate lattice and µ0w the chemical potential of pure water. T0 is a reference temperature for which 273.15 K is a convenient choice. At this temperature the vapor pressure P0 of water is low and may be approximated to zero. T h ) (T + T0)/2 is the average 0 temperature. ∆µw is the difference at temperature T between the chemical potential of water in the empty hydrate lattice and in the form of pure liquid water or ice. ∆H° and ∆V stand for the enthalpy and volume differences between the same two states. ∆Cp is the difference between the heat capacity of water in the empty hydrate lattice and as either ice or pure liquid water. Mehta and Sloan9 have proposed the values given in Table 4 for the quantities ∆µ0w, ∆H°, ∆Cp and ∆V. ∆Cp is assumed to be independent of temperature and ∆V to be independent of pressure. An additional relation for the chemical potential difference in eq 1 may be derived from conventional thermodynamic relations and an expression suggested by van der Waals and Platteuw.1 This leads to the following relation,
∆µ0w RT0
-
∫TT
0
∆H° + ∆Cp(T - T0)
()
RT2 ln
fRw
dT +
∆V
∫0P RTh dP )
NCAV
-
f0w
∑ i)1
NC
νi ln(1 -
∑ nki)
(2)
k)1
where f0w is the fugacity of pure liquid water, fRw the fugacity of water in the nonhydrate phase R, νi the number of cavities of type i, NCAV the number of different cavities, NC the number of hydrate-forming compounds excluding water, and nki the probability of the cavity of type i being occupied by a hydrate-forming compound of type k. This probability can be expressed as
nki )
Ckifk NC
1+
(3)
Cjifj ∑ j)1
where fk is the fugacity of component k and Cki the Langmuir adsorption constant for component k in a cavity of type i. With expressions available for evaluating the fugacities f0w and fRw, eq 2 may be used to determine the temperature T at which an incipient hydrate phase may form.
4π kT
∫0R exp(-
)
ω(r) 2 r dr kT
(4)
where ω(r) is the potential function of guest k in cavity i in the radial distance r from the center of the molecule. R is the radius of the cell and k Boltzmann’s constant. The potential function was modeled using a Kihara spherical core pair potential. This is a function of r, R, and the three Kihara parameters specific for each guest component. The three latter parameters were in Sloan and Mehta’s work determined by a parameter fit to experimental structure H hydrate data. For structure I and II hydrates Parrish and Prausnitz2 have presented an alternative empirical twoparameter expression for the Langmuir constant
Cki )
( )
Aki Bki exp T T
(5)
which has further been used by Munck et al.3 Aki and Bki are parameters determined by fitting to experimental data. Simplified Structure H Model The two smaller cavities of structure H hydrates are of approximately the same size and will in this work be treated as one type of cavity. With this simplification NCAV ) 2, ν1 ) 5, and ν2 ) 1. The subindex 1 stands for the small- and medium-sized cavities lumped into one, and the subindex 2 stands for the large cavity. The expressions presented in the literature for the fugacity of the guest molecules in the structure I and II hydrate lattices make explicit use of the fact that there are only two cavity sizes.11-13 Assuming that structure H like structure I and II only comprises two different types of cavities, it is possible to use the same type of phase equilibrium calculation algorithms and hydrate fugacity expressions for structure H hydrates as those used for structures I and II. Equation 5 is used for the Langmuir constant. Because only one type of smaller cavities is considered, only one set of Langmuir parameters (A and B) is to be estimated for each guest component of the smaller cavities. With these simplifications it is straightforward to estimate model parameters for structure H hydrates and to extend existing phase equilibrium algorithms handling structure I and II hydrates to also handle structure H hydrates. The values in Table 49 are used for µ0w, ∆H°, ∆Cp, and ∆V. The fugacity of pure water is evaluated using the Soave-Redlich-Kwong equation.14 For aqueous solutions the mixing rule of Huron and Vidal15 is used with the interaction parameters proposed by Pedersen et al.16 The applied modification of the Huron and Vidal mixing rule is the one reducing to the classical mixing rule in the absence of components requiring a nonclassical mixing rule. Parameter Estimation In eq 5 the parameter Bki appears inside an exponential term and the parameter Aki does not. There is therefore a relatively large correlation between the
Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000 1113 Table 5. Estimated Langmuir Parameters for Structure H Hydrates small cavities / Aki
(K/Pa)
Bki (K)
large cavities / Aki
(K/Pa) Bki (K)
methane 6.782 × 10-4 3390 nitrogen 1.425 × 10-4 3795 isopentane 8.242 × 10 neohexane 1.794 × 103 2,3-dimethylbutane 9.400 × 102 2,2,3-trimethylbutane 6.901 × 103 3,3-dimethylpentane 3.208 × 103 methylcyclopentane 1.584 × 103 cycloheptane 4.184 × 104 methylcyclohexane 9.681 × 103 1,1-dimethylcyclohexane 7.433 × 104 cis-1,2-dimethylcyclohexane 2.389 × 104 ethylcyclopentane 6.425 × 103 cyclooctane 6.101 × 104
1699 3175 3608 -39 3183 4024 5050 3604 4089 4114 4207 4135
Table 6. Compositions of Test Mixtures in mol % component
mixture 1
mixture 2
nitrogen methane methylcyclohexane methylcyclopentane water
23.17 16.05 3.79
12.38 9.65
56.99
Figure 1. Measured17 and calculated hydrate formation temperatures for mixture 1 in Table 6.
7.21 70.76
parameters Aki and Bki. A small adjustment of Bki will result in a large adjustment of Aki. An expression of this type is inconvenient for the purpose of fitting to experimental data. For this reason eq 5 was rewritten to
Cki )
(
)
( (
))
Aki A/ki Bki Bki Bki 1 1 exp exp Bki + ) T T T0 T0 T T T0 (6)
where
A/ki ) Aki exp
( ) Bki T0
(7)
T0 is a reference temperature taken to be 273.15 K. Instead of estimating Aki and Bki, A/ki and Bki are chosen as the two parameters to be estimated. They are less correlated than Aki and Bki, which makes the parameter estimation more reliable. A/ki provides a measure of the probability of component k to remain as the guest molecule in cavity i. Bki determines the temperature dependence of this probability. The fitted values of A/ki and Bki are listed in Table 5. Using these parameters, the average absolute deviation between experimental and estimated hydrate formation temperatures was found to be 0.15 K and to never exceed 0.53 K. A positive value of Bki indicates a decrease with temperature in the probability that the guest molecule will remain inside the hydrate lattice. This is the temperature dependence to be expected from basic physical considerations. As is seen from Table 5, the optimum B parameter is positive for all guest molecules of the structure H except for 2,2,3-trimethylbutane. Because only one literature source with experimental structure H data for this component exists, it is impossible to tell whether 2,2,3-trimethylbutane really has this unusual temperature behavior or the negative value of B is an artifact arising from inaccurate experimental data.
Figure 2. Measured17 and calculated hydrate formation temperatures for mixture 2 in Table 6.
Example Calculations Structure H test calculations have been made for two mixtures, each containing three possible guest molecules of structure H. The compositions of the two mixtures are shown in Table 6. Figures 1 and 2 show experimental17 and calculated hydrate formation conditions for each mixture. Conclusion The formation conditions for structure H hydrates can be modeled using a Langmuir adsorption model. A simple two-parameter expression for the Langmuir constant is sufficient to correlate experimental structure H data. The two smaller cavities of structure H hydrates may be regarded as only one cavity. This simplifies not only parameter estimation but also phase equilibrium calculations considering structure H hydrates. Nomenclature A ) empirical parameter entering the expression for the Langmuir constant
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Ind. Eng. Chem. Res., Vol. 39, No. 4, 2000
A* ) defined in eq 7 B ) empirical parameter entering the expression for the Langmuir constant C ) Langmuir constant Cp ) heat capacity f ) fugacity H ) enthalpy of water k ) Boltzmann’s constant m ) number of faces of hydrate crystal n ) probability of cavity being occupied or number of edges in hydrate crystal NC ) number of components NCAV ) number of cavities P ) pressure P0 ) vapor pressure R ) gas constant or radius of cell r ) distance from center of molecule T ) temperature T0 ) reference temperature T h ) average temperature V ) volume X ) large cavities Y ) small cavities Z ) medium cavities Greek Letters µ ) chemical potential ν ) number of cavities of given type Subscripts and Superscripts i ) cavity type k ) component index w ) water * ) empty hydrate lattice 0 ) reference (pure water) state R ) water phase
Literature Cited (1) van der Waals, J. H.; Platteuw, J. C. Clathrate Solutions. Adv. Chem. Phys. 1959, 2, 1. (2) Parrish, W. R.; Prausnitz, J. M. Dissociation Pressure of Gas Hydrates formed by Gas Mixtures. Ind. Eng. Chem. Des. Dev. 1972, 11, 26. (3) Munck, J.; Skjold-Jørgensen, S.; Rasmussen, P. Computations of the Formations of Gas Hydrates. Chem. Eng. Sci. 1988, 43, 2661. (4) Ripmeister, J. A.; Tse, J. S.; Ratcliffe; C. I.; Powell, B. M. A New Clathrate Hydrate Structure. Nature 1987, 325, 135. (5) Sloan, E. D. Clathrate Hydrates of Natural Gases, 2nd ed.; Marcel Dekker: New York, 1997. (6) Englezos, P. Clathrate Hydrates. Ind. Eng. Chem. Res. 1993, 32, 1251.
(7) Mehta, A. P.; Sloan, E. D. Structure H Hydrate Phase Equilibria of Paraffins, Naphthenes and Olefins with Methane. J. Chem. Eng. Data 1994, 39, 887. (8) Thomas, M.; Behar E. Structure H Hydrate Equilibria of Methane and Intermediate Hydrocarbon Molecules. Proceedings of the 73rd Gas Processors Association Convention, New Orleans, LA, March 7-9, 1994. (9) Mehta, P. A.; Sloan, E. D. Improved Thermodynamic Parameters for Predictions of Structure H Hydrate Equilibria. AIChE J. 1996, 42, 2036. (10) Mehta, P. A.; Sloan, E. D. A Thermodynamic Model for Structure-H Hydrates. AIChE J. 1994, 40, 312. (11) Cole, W. A.; Goodwin, S. P. Flash Calculations for Gas Hydrates: A Rigorous Approach. Chem. Eng. Sci. 1990, 45, 569. (12) Bishnoi, P. R.; Gupta, A. K.; Englezos; P.; Kalogerakis, N. Multiphase Equilibrium Flash Calculations for Systems Containing Gas Hydrates. Fluid Phase Equilibr. 1989, 53, 97. (13) Michelsen, M. L. Calculation of Hydrate Fugacities. Chem. Eng. Sci. 1991, 46, 1192. (14) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (15) Huron, M. J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibria of Strongly Non Ideal Mixtures. Fluid Phase Equilibr. 1979, 3, 255. (16) Pedersen, K. S.; Michelsen; M. L.; Fredheim, A. O. Phase Equilibrium Calculations for Unprocessed Well Streams Containing Hydrate Inhibitors. Fluid Phase Equilibr. 1996, 126, 13. (17) Tohidi, B.; Danesh, A.; Burgass, R. W.; Todd, A. C. Hydrate Equilibrium Data and Thermodynamic Modeling of Methylcyclopentane and Methylcyclohexane. Proceedings of the 2nd International Conference on Natural Gas Hydrates, Toulouse, France, 1996. (18) Hu¨tz, U.; Englezos, P. Measurements of Structure H Hydrate Phase Equilibrium and the Effect of Electrolytes. Fluid Phase Equilibr. 1996, 117, 178. (19) Makogan, T. Y.; Mehta; P. A.; Sloan, E. D. Structure H and Structure I Hydrate Equilibrium Data for 2,2-Dimethylbutane with Methane and Xenon. J. Chem. Eng. Data 1996, 41, 315. (20) Danesh, A.; Tohidi, B.; Burgess, R. W.; Todd, A. C. Hydrate Equilibrium Data of Methylcyclopentane with Methane or Nitrogen. Trans. Inst. Chem. Eng. 1994, A72, 197. (21) Thomas, M.; Behar, F. Modeling of Structure H Hydrate Equilibria for Methane, Intermediate Hydrocarbon Molecules and Water Systems. Proceedings of the 75th Gas Processors Association Convention, 1995. (22) Becke, E.; Kessel; D.; Rahimian, I. Influence of Liquid Hydrocarbons on Gas Hydrate Equilibrium. Proceedings of the European Petroleum Conference, Cannes, France, Nov 16-18, 1992; SPE 25032.
Received for review September 13, 1999 Revised manuscript received January 26, 2000 Accepted January 27, 2000 IE990677Z
