Symposium on Size Reduction,” pp. 217-31, Verlag Chemie GmbH., Berlin, 1962. Goldfarb, D., Lapidus, L., IND.EXG.CHEM.FUNDAMENTALS 7, 142-51 (1968). Klein, M., Klimpel, R. R., J. Znd. Eng. 18, 90-5 (1967). Klimpel, R. R., Ph.D. thesis, Pennsylvania State Cniversity, 1964. Klimpel, R. R., Phillips, E., J . Chem. Eng. Data 13, 97-101 (1968).
Kunzi, H. P., Krelle, W., Oettli, W., “Nonlinear Programming,” Blaisdell Publishing Co., Waltham, Mass., 1966. Reid, K. J., Chem. Eng.Sci.20, 953-60 (1965). Rosen, J. B., J . SOC.Ind. A p p l . Math. 8 , 181-217 (1960). Wilde, D. J., Beightler, C. S., “Foundations of Optimization,” Prentice-Hall, Englewood Cliffs, N. J., 1967. RECEIVED for review February 27, 1969 ACCEPTED December 2, 1969
Experimental Measurement of Hydrate Numbers for Methane and Ethane and Comparison with Theoretical Values Travis J. Galloway,l Walter Ruska, Patsy S. Chappelear, and Riki Kobayashi William X a r s h Rice University, Houston, Tex.’77001
Hydrate numbers, n, were directly determined by two methods using a new apparatus, designed to alleviate the occlusion of liquid water in the hydrate crystal and to operate up to 15,000 psi in order to evaluate the pressure dependence of the hydrate composition. Experimental conditions for ethane were 1 18.0 psia at 40.0°F, 119.4 psia at 40.2”F, and 225 psia at 48.9”F. The experimental ethane hydrate numbers were from 7.90 to 8.46 with an average maximum relative uncertainty of &4.24/,. Four sets of experimental conditions were used for methane hydrate: 1030 psia at 5O.O0F, 1032 psia at 50.1 O F , 1901 psia at 59.9”F, and 1902 psia at 60.0”F. The experimental hydrate numbers were from 5.84 to 6.34 with an average maximum relative uncertainty of =t1 5.6y0. Predictions for n from the solid solution theory of van der Waals and Platteeuw are discussed.
GAS
HYDRATES, which are included in the broad classification of the clathrate compounds, are ice-like inclusion compounds formed from gases (or condensed gases) and water under suitable conditions of temperature and pressure. Previous efforts in direct experimental measurement of the equilibrium compositions of gas hydrates have been aggravated by two problems: occlusion of liquid water in the hydrate crystals which causes iiicomplete conversion of water and gas to hydrates, and marked dependence of the composition on pressure and temperature for the hydrates of pure gases. It was the purpose of this research to surmount these difficulties and develop a n experimental method which would permit accurate direct determination of hydrate numbers.
Previous Work
Early investigators (Deaton and Frost, 1946; de Forcrand, 1902; Hammerschmidt, 1934; Roberts et al., 1940) employed thermodynamical analysis to calculate hydrate numbers. The method used b y Deaton and Frost (1946) involved calculating the heats of formation per mole of combined gas of the hydrate from two initial states: liquid water plus gas and ice plus gas. The difference between these two heats Present address, Hurricane Creek Plant, Reynolds Metals Co., Bauxite, Ark. 72011
represented the heat of fusion of the water present in the hydrate with 1 mole of gas. Division of this heat by the heat of fusion of 1 mole of water gave the number of moles of water per mole of gas in the hydrate, which is defined as the hydrate number. The calculations b y Deaton and Frost gave a hydrate number of 7 for methane hydrate and 8 for ethane hydrate; however, the computation is applicable only in the vicinity of the ice point. Deaton and Frost also determined experimentally the hydrate numbers of several different gases, including ethanemethane mixtures, but did not attempt to study the dependence of hydrate composition on pressure and temperature. Deaton and Frost obtained hydrate numbers of 7.11 and 7.04 for methane hydrate, and 8.06, 8.18, 8.54, and 8.33 for ethane hydrate; but they did not report the pressures a t which the determinations were made or make an error analysis. One method used here is similar to the procedure of Deaton and Frost, but a more effective device for achieving complete conversion to hydrate was developed in the present research. Structure. T h e earliest work on t h e structure of gas hydrates came from the determinations of x-ray diffraction patterns for a large number of gas hydrates b y von Stackelberg and PIIuller (1951, 1954) a n d their hypothesis of a cubic structure. Claussen (1951) ivas t h e first to conceive the idea of t h e packing of pentagonal dodecahedra into a diamond lattice, now known as Structure 11. Muller a n d Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
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Claussen then independently deduced from the x-ray diffraction d a t a t h a t not all gas hydrates conformed to Structure I1 and formulated Structure I, the body-centered cubic lattict:. This lattice was verified by Pauling and Marsh (1952) from their studies on chlorine hydrate. With these known lattices and the molecular dimensions of water and various gasvs, it is possible to calculate the limiting hydrate number for :my gas from strictly geometrical considerations. Theoretical and Experimental Development. Equations relating t h e thermodynamic properties of clathrates were developed from classical statistical mechanics b y v a n der K a a l s and Platteeuw (1959). Marshall, Saito, and Kobayashi (1962, 1964) attempted to measure t h e hydrate numbers for methane hydrate b y the indirect method of observing the pressure drop in a closed system during hydrate formation. Method I of this paper is a modification of t h e indirect method used by Marshall et al. Narshall obtained hydrate conversions of 30 to 55% of the theoretical value and attributed the low conversions to occlusion of liquid due to rapid hydrate formation. His suggestion of using a method for macroscopically breaking the hydrate crystals gave impetus to this research. Theoretical Hydrate Numbers
I n 1959 van der Waals and Platteeuw in their comprehensive treatment of clathrate solutions included a description of the nature and structure of gas hydrates as solid solutions in which gas molecules occupy cavities in a framework of tetrahedrally coordinated water molecules linked together by hydrogen bonds. Ethane and methane hydrates crystallize in Structure I, for which a unit cell contains 46 water molecules, which form two smaller, nearly spherical cavities of approximately 5.1 A diameter and six larger, slightly oblate cavities with a free diameter of approximately 5.8 A. Methane molecules with a molecular diameter of 4.1 X can occupy both large and small cavities, so the limiting hydrate number for methane hydrate is 46/73, or j 3 / 4 . Saito, Marshall, and Kobayashi (1964), who confirmed that the solid solution theory of van der Waals and Platteeuw could be applied above the ice point for gas hydrates, presented the basic equations of van der Waals and Platteeuw as follows :
(3) Equation 1 is a generalized form of Raoult’s law for a solvent with negligible solute-solute interaction. pzo and f i g represent the chemical potential of water in the hydrate and in the empty lattice structure modification, respectively. For gas hydrates of Structure I, u1 = 1/23and v2 = 3/23. Equation 2 is equivalent to a Langmuir isotherm for localized adsorption with no interaction between the adsorbed molecules. The term y K t represents the probability of finding a solute molecule, K , in a cavity of type i; and f K is the equilibrium fugacity of the solute K in the clathrate. The term C K $represents the Langmuir constant; and T and k are the temperature and Boltzmann’s constant, respectively. I n Equation 3, which is the defining equation for the Langmuir constant used in Equation 2, hK,(T,V) is the molecular partition function of a solute molecule of type K enclosed in a cavity of type i; and + K ( T )is the molecular partition function of a solute molecule of type K with the volume factor removed. 238
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Saito, Marshall, and Kobayashi (1964) used the LennardJones and Devonshire (1937, 1938) potential (as assumed by var. der Waals and Platteeuw) to evaluate the cell partition functions, h ~ %from , which the Langmuir constants, C K , , were calculated by Equation 3 for methane hydrate. The theoretical methane hydrate numbers (with occupation of both size cavities assumed possible) presented in Table I of Saito, Marshall, and Kobayashi were calculated from n = 23/[Culf~/(l
+
C M l f d
+ ~ C M Z ~ M+ /C( M~ Z ~ M (4)) ]
The theoretical hydrate numbers used in this paper were read from a plot of the Saito, Marshall, and Kobayashi values for methane hydrate from 382 to 15,000 psia. Nagata and Kobayashi (1966) have shown that the assumption that the ethane molecules with a molecular diameter of 5.5 A occupy only the larger cavities of Structure I meets the thermodynamic requirements for that system. The limiting hydrate number for ethane hydrate, according to this assumption, would be 46/6, or 72/3. Nagata and Kobayashi developed the equation
CEZ= exp (12.701 - 0.04373 T)
(5)
for C E l = 0 where T is the equilibrium temperature in OK. The Langmuir constants calculated from this equation were used to calculate the ethane hydrate numbers from the following equation, which assumes that only the larger cavities of Structure I are occupied: n = 23/[3CEzfE/(l f
CEZfE)]
(6)
Experimental System
Equipment. T h e experimental equipment as shown schematically in Figure 1 consisted of a n autoclave, ii, in which hydrate formation occurred, apparatus for charging water, B , and gas, C, t o the autoclave, a pressure-measuring system, P G , a temperature control and measuring system, T , P T , R , and a system for collecting and measuring the quantity of gas evolved during hydrate decomposition, D T , M , F . Shown as lines are the l/&inch stainless steel tubing used for interconnecting t h e equipment. illuminum tubing with Tygon connectors and vacuum hose are indicated b y heavier lines. The cylindrical autoclave had a bore of 25/8 inches, was j 3 / 4 inches long, and constructed of Type 410 stainless steel, heat-treated to approximately 260 Brinell. The end closures of the autoclave were machined with rounded corners of 5/16-inch radius, as showp in the detail of Figure 1.The vessel design pressure was in excess of 15,000 psi, but further modifications in the seals of the autoclave may be necessary to achieve higher pressure than those reported in this study. The Teflon seal, shown in the detail of Figure 1, failed when a final attempt was made to achieve a high pressure run a t 5000 psi. To release occluded water by macroscopically breaking u p the hydrate agglomerate, the autoclave was designed t o function also as a ball mill. It was suspended in its controlled temperature bath on journals so that it could rotate about a horizontal axis. Its cavity contained 25 stainless steel balls, 15 of 1/2-inch diameter and 10 of 9/16-inch diameter, which rolled along the bottom of the autoclave to produce a grinding or crushing action. The motor drive, X D , for the autoclave was arranged in such a way that the autoclave could cycle continuously approximately four turns clockwise, and by automatically reversing, AR, the motor, four turns counterclockwise, at a rate of about 16 seconds per cycle of rotations. The autoclave cavity was connected to the gas charging and collecting systems through a 1/16-inch tubing spiral, S , which allowed for its rotation. Distilled water which had been boiled to remove dissolved gases was charged to the evacuated autoclave from a 50-ml buret, B, graduated in 0.1 ml. The methane used had a minimum purity of 99.97 mole %, and the initial cylinder pressure vias approximately 2200 psig. The ethane had a 99.99
Figure 1 .
Diagram of experimental apparatus
mole yo purity, and the initial cylinder pressure was approximately 595 p i g . The system pressure was sensed with Heise Bourdon tube gages, PG, calibrated against a certified mecision dead weight gage. A 2000-psi gage-was used for
