Measurement and Modeling of Hydrate Composition during Formation

Dec 15, 2010 - During hydrate formation, the mole fraction of methane in the gas phase ... A gas clathrate hydrate is a crystalline inclusion compound...
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Ind. Eng. Chem. Res. 2011, 50, 694–700

Measurement and Modeling of Hydrate Composition during Formation of Clathrate Hydrate from Gas Mixtures Thomas J. Hughes† and Kenneth N. Marsh* Department of Chemical and Process Engineering, UniVersity of Canterbury, Christchurch, New Zealand

In studying the relative dissociation times of structure I (sI) and structure II (sII) gas hydrates prepared from methane + ethane mixtures, X-ray diffraction analysis indicated the simultaneous presence of both hydrate structures at conditions where only one hydrate structure was expected. The aim of this work was to develop a model describing the formation of a gas hydrate from a mixture of methane + ethane in a closed system as the hydrate forms and to validate the model by experiment. Two methods of laboratory hydrate preparation from ice particles were studied: (1) a closed-system pressure-drop formation and (2) a closed-system constantpressure formation. Models were developed for both methods 1 and 2, and method 2 was also investigated experimentally using a mixture of mole fraction 0.897 methane to form the hydrate. Compositions of the gas were monitored periodically during the hydrate formation by gas chromatography (GC). The water-free hydratephase composition of methane and ethane was also measured by GC at the completion of hydrate formation. During hydrate formation, the mole fraction of methane in the gas phase increased. The model showed the same trend, although there was a discrepancy between the model and experiment that is most likely due to the assumption in the model that equilibrium between the phases exists at all times. 1. Introduction A gas clathrate hydrate is a crystalline inclusion compound with an icelike appearance. On a molecular scale, the hydrate consists of hydrogen-bonded water cages in which gas molecules are encaged or enclathrated. In typical natural gas systems, there are two structures of hydrate that can form depending on the gas composition, pressure, and temperature. These structures are designated structure I (sI) and structure II (sII).1 It has been known for 75 years that the composition of the guest molecules in a gas hydrate on a water-free basis can be considerably different from that in the gas phase that formed the hydrate. Hammerschmidt2 performed an analysis on a hydrate formed from a pipeline natural gas and found that the hydrate was significantly enriched in heavier components such as propane and depleted, relative to the pipeline gas, in methane. The distribution coefficient method of estimating the hydrate formation conditions of gas mixtures, first suggested by Wilcox et al.3 and fully developed by Carson and Katz4 in the 1940s, is based on the fact that, at a high gas-to-water mole ratio, the water-free hydrate-phase composition is different from that of the gas phase as a function of temperature and pressure. The vapor-to-solid distribution coefficient for gas hydrates, Kvsi, is defined as Kvsi ) yi /xsi

(1)

where yi is the gas-phase mole fraction of component i and xi is the hydrate-phase mole fraction of component i on a waterfree basis. From knowledge of Kvsi as a function of P and T, it is possible to estimate the hydrate formation conditions using a method analogous to the K-value method for dew-point calculations in a gas mixture. The success of phase boundary predictions using this method is based on being able to predict * To whom correspondence should be addressed. ken.marsh@ canterbury.ac.nz. † Current address: Centre for Energy, School of Mechanical Engineering M050, University of Western Australia, 35 Stirling Highway, Crawley 6009, Western Australia, Australia.

the hydrate-phase mole fractions on a water-free basis. As the values of Kvsi for individual components are functions of P and T, the composition of the hydrate will change with P and T in addition to the composition of the gas phase. The Kvsi method has been surpassed by methods based on the van der Waals and Platteeuw statistical hydrate-phase model.5 The model of van der Waals and Platteeuw calculates the composition of the hydrate based on fractional occupancies from Langmuir adsorption constants derived from statistical thermodynamics tempered by experimental data. This model provides a way of calculating the hydrate-phase composition from the pressure (P), temperature (T), and composition of the gas. The CSMGem (“Colorado School of Mines Gibbs Energy Minimization”) software developed by Ballard6 further allows flash calculations in the hydrate region at pressures higher than the incipient pressure or temperatures lower than the dissociation temperature and, hence, allows for the calculation of more than an infinitely small amount of the hydrate phase. This means that the effect of the ratio of the number of moles of water to gas can be investigated. Molecules that have a van der Waals diameter of approximately 6 Å are of a size close to the point at which they are too large to stabilize the smaller 51262 cage of sI and will instead stabilize the larger 51264 cage of sII.1 Two molecules of about this size, cyclopropane7 (5.9 Å) and trimethylene oxide8 (6.1 Å), have been shown to form both sI and sII hydrates depending on the P and T conditions. More unusually, however, certain binary mixtures of hydrate formers that, as single-guest components, form sI hydrates have been shown to form sII double-guest hydrates. This effect was first observed in the mid1950s by von Stackelberg and Jahn,9 who measured sII-sized lattice parameters by X-ray diffraction for binary mixtures of H2S with CH3Br, COS, and CHF2CH3, each of which forms an sI hydrate as single-guest components. Hendriks et al.10 demonstrated, with the use of a van der Waals and Platteeuw hydrate-phase model, that the available experimental P, T phase data for the system methane + ethane + water indicated that either sI or sII hydrates could form depending on the gas mixture

10.1021/ie101162z  2011 American Chemical Society Published on Web 12/15/2010

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composition, pressure, and temperature. (Note that methane has been shown to form both sII and sH hydrates as a singlecomponent hydrate but only sH at pressures in excess of 99 MPa.11,12) Hendriks et al.’s work was confirmed by Subramanian et al.,11-13 who measured the mole fractions of the hydratephase guests as a function of the gas mixture mole fractions. The compositions of each phase was calculated from 1H NMR or Raman spectrum peak areas. A step change in the hydratephase composition was observed for methane + ethane mixtures as the mole fraction of methane in the mixture was increased to between 0.7 and 0.8. This shift in the hydrate-phase composition was attributed to a change in hydrate structure from sI to sII. As the mole fraction of methane in the mixture was increased between 0.97 and 0.98, another step change in the hydrate composition was observed and attributed to a change in the hydrate structure from sII to sI. The upper transition was determined more accurately by Raman spectroscopy to occur between methane mole fractions in the vapor phase of 0.992 to 0.994, depending on the temperature. Processed natural gases or natural gases that are lean in components other than methane have been shown to form both sI and sII hydrates depending on the composition, pressure, and temperature by hydrate-phase prediction packages and from experimental measurements.14,15 Several authors have conducted experimental studies showing that the usual sI hydrate formers carbon dioxide16,17 and methane18 can transiently form metastable sII crystals. Schicks et al.19 also showed evidence of the formation of both sI and sII hydrates formed from a mixture of methane + ethane + propane. Hydrates were formed under flow conditions or conditions where there was large gas-to-water volume ratio. The authors showed that these conditions would result in no or limited changes in the gas composition, so that compositional changes were not the reason for the formation of different structures. Morphological changes were observed when crystals of hydrate formed just below the three-phase hydrate + aqueous + vapor phase (H + Aq + V) line were cooled isobarically to temperatures below the H + Aq + V line of the pure methane hydrate. Large euhedral crystals transitioned to a fine crystal foamy mass. A reversal in morphology could be achieved by reheating above the H + Aq + V line of pure methane hydrate. “Raman spectroscopy and X-ray diffraction data indicated that formation or decomposition of sI methane hydrates was part of this process”.19 In this work the equilibrium hydrate predicted by CSMGem was assumed to be formed at the given conditions. 2. Model Development The conditions in the laboratory for the formation of a gas hydrate from a mixture are often very different from the conditions within a gas pipeline. Typically, in the laboratory, only just enough gas is contacted with the water + ice to allow for complete conversion to hydrate. In a gas pipeline, there is typically a large excess of gas associated with a small quantity of water. Hydrate formation from gas mixtures under these various conditions can be considered by thought experiments. First, consider hydrate formation from a gas mixture under the conditions of excess water: A cell containing a piston maintained at constant temperature contains water and a gas mixture (containing only components that can be enclathrated in hydrate cages) at a ratio such that there is excess water to fully convert all of the gas mixture to hydrate. If the piston is compressed so that the conditions are within the hydrate formation P, T region, hydrate will begin to form until the pressure drops to the incipient pressure at the set temperature. However, if the mixture is continuously compressed by the piston to maintain the

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Figure 1. Thought experiment for hydrate formation in a closed system with excess water.

Figure 2. Thought experiment for hydrate formation with excess gas.

conditions within the hydrate formation region, then the gas will continue to combine with the remaining water until all of the gas is consumed. This thought experiment is illustrated in Figure 1. The hydrate composition on a water-free basis must, on average, be equal to the initial gas mixture composition (assuming negligible gas solubility in the remaining water phase). The same process of hydrate formation in a piston cell can be considered for a system containing a very small number of moles of water and a large excess of gas. If such a system is in the hydrate formation P, T region, then hydrate formation will occur to consume all of the aqueous phase. This thought experiment is illustrated in Figure 2. The hydrate composition under these conditions is defined by the measured Kvsi values of eq 1. The hydrate composition on a water-free basis can be considerably different from the composition of the gas mixture, as certain components of the gas phase will be more likely to enter the hydrate phase than others. This phenomenon is referred to as “preferential enclathration” in this work. Isothermal pressure versus mole fraction pseudo-binary phase diagrams for the system methane + ethane + water can be generated using hydrate-phase prediction packages based on the van der Waals and Platteeuw model by considering either water to be in excess or the gas mixture to be in excess. Figures 3 and 4 show the pseudo-binary phase diagrams for the methane (1) + ethane (2) + water system generated using CSMGem under the conditions of excess water and excess gas, respectively. In these figures, y1 represents the mole fraction of methane on a water-free basis. Varying the ratio of the number of moles of water to the number of moles of gas mixture can result in the formation of a different structure or multiple hydrate structures. To form hydrate in the laboratory, ice particles are typically melted just above the fusion temperature in a pressure cell in the presence of a hydrate-forming gas at a pressure above the hydrate formation incipient pressure. This method was developed by Stern et al.20 The amount of gas added to the known amount of ice + water in the cell is typically only in slight excess to that required for complete conversion of the gas to

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Figure 3. Isothermal (T ) 273.25 K) pressure versus composition phase diagram of methane (1) + ethane (2) + excess water, where y1 is the waterfree mole fraction of methane. Generated using CSMGEM. Phase labels: Aq ) aqueous phase, V ) vapor, sI ) structure I hydrate, sII ) structure II hydrate.

Figure 5. Constant-pressure syringe or piston pump formation of hydrate from ice particles.

Figure 4. Isothermal (T ) 273.25 K) pressure versus composition phase diagram of methane (1) + ethane (2) + water with excess gas, where y1 is the water-free mole fraction of methane. Generated using CSMGEM. Phase labels: Aq ) aqueous phase, V ) vapor, sI ) structure I hydrate, sII ) structure II hydrate.

hydrate. These conditions of formation lie between those of the two extreme conditions discussed above, so the hydrate composition formed will be between the two limits. In the method of Stern et al., hydrate formation proceeds with a pressure drop as the gas initially charged to the cell is depleted (method 1). An alternative method (method 2) is to connect the pressure cell to a gas-filled piston cell or syringe pump and maintain a constant pressure during the formation. A model of the conditions described by method 2 was considered, and experimental results for a mixture of methane + ethane in a system were compared to model predictions. A model for method 1 was also developed and is included in the Supporting Information (see part 1 therein). In the development of these models, the main objective was to develop a prediction tool to assess whether more than one structure of hydrate would form in the laboratory. As evidenced by Figure 4, the formation of more than one structure of hydrate in a flowing system is unlikely unless the water fraction is very high. 2.1. Constant-Pressure Preferential Enclathration Model for the Formation of Gas Hydrates from Ice Particles and Gas Mixtures (Method 2). A constant-pressure preferential enclathration model was formulated to predict changes in the gas phase and hydrate phase during hydrate formation in a constant-pressure closed system (method 2), such as that shown in Figure 5. The initial input values were the mass of ice, total system volume, pressure, temperature, and gas composition. The model algorithm is shown in Figure 6, and the model development is

Figure 6. Algorithm for gas-phase stripping and preferential enclathration for a constant-pressure syringe or piston pump formation of gas hydrate for a gas mixture. Note: Initial conditions must be calculated prior to using this algorithm. In step 8, NIST Reprop was used to calculate the gas denisty.21

described in detail in part 3 of the Supporting Information for this article. In brief, a step is taken in the conversion of the ice + water mixture to hydrate, the hydrate composition is calculated using CSMGem, the gas composition is recalculated, a small correction is made for the volume change of each phase, and the next step is taken until all of the ice is converted to hydrate. 3. Experimental Section To test the model of method 2, an experiment was devised to measure the gas-phase composition during hydrate formation from a mixture of methane + ethane + water. Samples of the gas mixture from the system were taken periodically and analyzed by gas chromatography (GC). After hydrate formation ended, a sample of hydrate was dissociated to measure the composition of the enclathrated gas.

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Figure 7. Experimental apparatus for preferential enclathration formation tests at constant pressure (maintained by an ISCO syringe pump). PRT refers to platinum resistance thermometer.

3.1. Materials. Methane was supplied by Southern Gas Services from Linde Gas, U.K. Ltd., with a mole fraction purity of 99.975 %. Ethane was obtained from BOC Gases Ltd. and had a mole fraction purity of 99 %. These gases were used to prepare gas mixtures without further purification. Deionized water was used to prepare ice. A gas mixture was prepared in situ in an ISCO syringe pump (model 260D, which has a maximum volume of 266 mL, maximum pressure of 51.4 MPa, relative pressure uncertainty of (2 %, and relative flow rate uncertainty of (0.5 %). Initially, a mixture with a mole fraction of methane of 0.532 ( 0.003 that had previously been prepared gravimetrically was attached as the cylinder shown in Figure 7, and all valves were shut. Valves 8 to 12 and the cylinder valve were then opened and, the regulator was adjusted to the desired pressure. The empty ISCO pump was then set to refill mode, and the cylinder was reversed to a calculated volume such that the desired gas composition was yielded upon filling the ISCO pump to its maximum volume (266 mL) with methane. For the experiment, a mixture of mole fraction of 0.90 methane was used. The final pressure of this mixture was approximately 7.2 MPa for the full syringe. Gas chromatography measurements (described in more detail below) indicated a mole fraction of methane of 0.897 ( 0.003. 3.2. Hydrate Preparation. To form hydrate, sieved ice particles were loaded into a pressure cell, which was then filled with gas from the syringe pump in constant-pressure mode. The custom stainless steel 316 cell was designed with an internal diameter of 25.4 mm, an external diameter of 144.0 mm, and a length of 100 mm. The total cell volume was 51 cm3, with a pressure rating of 35 MPa. A schematic of the experimental apparatus is shown in Figure 7. The total system volume bound by valves 3, 13, and 21 and the cylinder regulator was approximately 330 cm3, estimated from the volume of the lines, fittings, and cell. The cell was cooled with liquid nitrogen and loaded with 19.3 g of ice particles prepared by crushing ice cubes in a food processor and sieving under liquid nitrogen to a particle size range of (250 to 850) µm. The cell was then placed in a thermostatted ethylene glycol + water bath (Poly-

science model 9601) preset to T ) 258 K. The cell was then connected to the system at valve 15. Platinum resistance thermometers (PRTs) (Omega 100 Ω PR-11-2-100-1/16-12-E, (0.04 K) were then placed in the thermowells of the pressure cell and in the bath. The cell and system were then evacuated and purged with the gas mixture at low pressure (below the hydrate formation pressure) for three cycles to ensure the removal of air. Pressurization of the cell was then begun by slowly opening valve 8 to bleed gas into the cell. This was done to ensure that the gas cooled sufficiently as it entered the cell so that the temperature in the cell did not rise more than 1 K above the bath temperature of 258 K. When the pressure between the cell and the ISCO pump had equalized, valve 8 was then fully opened. The pressure selected for the experiment was 6 MPa, and the ISCO pump was set to slowly pump gas (at 0.5 mL · min-1) into the system until the pressure reached 6 MPa, where it was maintained using its constant-pressure mode. The temperature of the bath was then raised to 273.25 K to initiate hydrate formation. Samples of the gas phase were analyzed by gas chromatography during hydrate formation as described in the following section. 3.3. Sampling and Gas Chromatography Measurements during Hydrate Formation. Before taking a gas-phase sample, the cell was cooled to 258 K, and a mixing process was initiated. The ISCO pump was set to pressurize to 8 MPa and depressurize to 4 MPa for four or more cycles in order to obtain a homogeneous gas sample. (For some of the earlier mixing processes, the depressurization was limited to about 5 MPa because the piston reached maximum extension.) The reason for cooling the cell to 258 K was to ensure that the hydrate would not dissociate because of the temperature rise during compression. The temperature in the cell never exceeded 278.5 K, approximately 3 K below the hydrate dissociation temperature at 4 MPa. After the mixing process had been completed, valves 3 and 7 were partially opened, and the back-pressure regulator (abbreviated BPR, Tescom model 26-1700, maximum pressure of 41.4 MPa) was adjusted so that gas just started to flow. A

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three-way stopcock was attached to a 25 mL syringe. A septum was attached to the second branch of the stopcock, and a rubber hose was connected to the third. To collect samples, the rubber hose was slipped over a barbed tube fitting leading from the BPR. The syringe was thrice flushed with gas before a sample was taken. Once a sample had been taken, valve 7 was closed, and the BPR was adjusted until all of the gas was bled from the line to prevent contamination in the next sample. Valve 3 was then also closed. Samples for GC of approximately 0.3 mL were taken from the three-way stopcock septum of the 25 mL syringe. A 1 mL syringe was flushed approximately 10 times with the gas before a sample was injected into the GC (Shimadzu GC-R1A). A Porapak Q packed column (1.83 m, 6.35 mm o.d.) was used with a thermal conductivity detector. The injection and detection temperatures were 423 K, the column temperature was 318 K, and the helium flow rate was 25 mL · min-1. It was not possible to remove all of the air from the samples, and a small amount was always present, with the oxygen and nitrogen peaks being indistinguishable and forming a single air peak. If there was significant air contamination, a fresh sample was used. Typical elution times for air, methane, and ethane were about 1.2, 1.75, and 8.9 min, respectively. For each 25 mL syringe of gas, 6 to 10 injections were made. 3.4. Hydrate-Phase Measurements and Structural Identification. When the consumption of gas had ended, as indicated by no volume change in the ISCO pump, the hydrate formation was considered complete, and valves 15 and 17 of Figure 7 were closed. The gas in the system (excluding the cell) was then vented to atmospheric pressure, and the cell was disconnected from the system at valve 15 and placed in liquid nitrogen. When the temperature inside the cell had dropped to below 180 K, valve 17 was opened to depressurize the cell (at 180 K, the hydrate was stable at atmospheric pressure). Once the cell had completely depressurized, it was tipped to one end, and the screws and lid of the other end were removed. The hydrate inside the cell was then quickly scraped out with a liquid-nitrogen-cooled spoon and placed into a Dewar flask also containing liquid nitrogen. Lumps of about (1 to 2) cm3 of hydrate were quickly removed from the liquid nitrogen Dewar flask and placed in the 25 mL syringe that had its plunger removed. The 25 mL syringe, the same syringe that was used in sampling, had a three-way stopcock attached with a septum for taking smaller syringe samples for GC analysis. The syringe plunger was inserted as far as possible to break apart the lumps of hydrate, and as the hydrate dissociated, the plunger became extended. When the plunger reached full extension, the gas was flushed out of the syringe. This process was repeated until dissociation was near-complete and the plunger was no longer being forced back by the dissociating hydrate gas. Gas samples were then taken for analysis by GC. The chromatographic method used for these samples was the same as previously described. Samples of the hydrate were also analyzed by powder X-ray diffraction. A sample lump of hydrate from the Dewar was placed, with liquid-nitrogen-cooled tweezers, in a shallow polystyrene dish filled with several centimeters of liquid nitrogen. One end of a glass capillary (internal diameter ) 0.3 mm) held with tweezers was inserted repeatedly into the sample until it was covered with hydrate powder. The capillary was then mounted, with the help of some petroleum jelly, onto the goniometer head of an X-ray diffractometer in place of a mounting pin. The goniometer head was cooled with a stream of nitrogen gas at 93 K to prevent hydrate dissociation. A

Figure 8. ([) Volume of the ISCO syringe pump and (0) temperature of the cell as a function of time.

powder diffraction pattern was recorded with a Bruker Smart CCD area-detector diffractometer using the 360° phi drive and scan function (see Figure S3 in Supporting Information). 3.5. Modeling the Experiments. Some corrections were made to the model for method 2 to account for the nonuniform system temperature (the ISCO pump was at room temperature, and the cell was in the temperature-controlled bath) and for the steps where samples were removed. These changes are detailed in part 4 of the Supporting Information for this article. 4. Results and Discussion 4.1. Hydrate Formation. Figure 8 shows the pump volumes, Vpump, as well as the cell temperature, Tcell, as a function of time. The vertical spikes in the pump volume are due to the mixing process as described previously. The sudden drops in pump volume indicate the removal of samples for GC analysis. The flat period of both temperature and pump volume between about day 4 and day 11 was due to a small leak in the back-pressure regulator (indicated by the larger-than-normal sample drop in pump volume at about day 4). The leak meant that no samples could be taken until it was repaired, so the cell was maintained at 258 K for the time that the back-pressure regulator was out of service. (Note that the leak was treated like a sample removal for modeling purposes.) It was noted that, over this time, the ISCO pump volume did not change significantly, indicating the assumption that hydrate did not form directly from ice was correct. (Note that the sharp deviation at the end of the flat region was the BPR being tested.) After about day 15, no further changes in the methane mole fraction were observed by gas chromatography, within the uncertainty limits estimated by a 95 % student-t confidence interval (error bars in Figure 8), and hydrate formation was considered to be complete. (Note that the sharp spike in temperature and pump volume just before day 20 occurred after the cell had been disconnected and chilled in liquid nitrogen to extract the hydrate samples.) The desired pressure for the formation of the hydrate was 6 MPa. The average pressure over the time of hydrate formation was 6.03 MPa, and this value was used in the model. The room temperature (used in calculations of the overall gas density) was an average value of 294.7 K over the time of formation; the maximum variation from this temperature was (2.8 K. 4.2. Gas Chromatography Results and Comparison to Modeling. Figure 9 shows the mole fraction of methane in the gas phase as a function of the volume pumped from both the measurements and calculated from the constant-pressure model.

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Figure 9. Comparison of experimental and model preferential enclathration stripping of the gas phase for CH4 (1) + C2H6 (2): (O) experimental GC data, (×) preferential enclathration model.

The flat regions of constant composition represent the points when samples were removed, and the change in the pump volume with no change in the methane mole fraction represents sample removal (or back-pressure regulator leak for the widest flat region). The experimental and model data both display the same trends. The methane mole fraction in the gas phase increases with the conversion of ice to hydrate, indicating that ethane was being preferentially enclathrated in the hydrate phase. The slope of the methane mole fraction with pump volume is on average steeper from the model prediction than from the experimental data. In the model a step size in the conversion of ice to hydrate, f, of 0.02 was used. The final mole fraction of methane measured was 0.925 ( 0.002, whereas the model predicted a value of 0.943; the relative difference of the model value from the GC measurement is thus 0.018. The total volume of gas pumped was 183.68 mL, which was slightly larger than the model prediction of 183.12 mL. The above results suggest that the model is validated, and although the predictions are not perfect, the model could prove to be a useful tool for establishing the direction and likely extent of stripping of the gas phase. The assumption that the equilibrium hydrate-phase prediction can be used to determine the composition of the hydrate phase is probably the greatest source of uncertainty between the model and experimental results. The system is not at equilibrium during any of the steps in conversion, because, once an initial layer of hydrate has grown on the outer surface of the ice particles, the gases will begin to diffuse through fissures and cracks in the hydrate phase. Diffusive separation of the gases is not likely to occur by mechanisms such as Knudsen diffusion because, at elevated pressures, the mean free paths of the molecules are reduced. However, because of the smaller size of the methane molecule, it is most likely that it will diffuse through the hydrate layer more easily than ethane. Ethane might have to be orientated longitudinally to pass through the hydrate layer. It might be expected for this reason that the hydrate phase would form with a higher proportion of methane than would be predicted by equilibrium thermodynamics. This could explain the discrepancy between the model predictions and experimental measurements observed in Figure 9. A small discrepancy could also result from heat effects associated with the hydrate enthalpy of formation. In addition discrepancies could result from inaccuracies in CSMGem, but Ballard6 has shown that it predicts the phase boundary for methane + ethane better than other hydrate packages. The GC-measured water-free mole fraction of methane in the gas of the hydrate phase was 0.785 ( 0.007, compared to a

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prediction of the model of 0.733. The difference of 0.052 is larger than the gas-phase difference of 0.018. A possible cause is that the sampling method for the GC measurement was for a sample that was nonrepresentative of the overall hydrate. In retrospect, it might have been better to powder the entire hydrate sample and mix it under liquid nitrogen in a mortar and pestle. On the other hand, the model predicts greater stripping of ethane from the gas phase than the experimental results, and this results in a lower methane mole fraction in the hydrate. The overall mole balance for methane was calculated by comparing the initial number of moles of methane in the gas feed to the cell and the final gas mole fractions after hydrate formation, as well as the hydrate methane mole fraction on a water-free basis. There was a 1.4 % normalized difference in the number of moles of methane calculated from the two methods. The X-ray diffraction of the hydrate indexed to an sII hydrate as expected. The gas-phase mole fractions did not change enough to result in the formation of an sI hydrate. If the amount of ice loaded initially into the cell were larger or the amount of initial gas were less, then this might have been a possibility, as both the experimental results and the model predicted that the gas-phase methane mole fraction increased with the conversion of ice to hydrate, becoming closer to the structural transition mole fraction of methane of approximately 0.993 (as measured by Subramanian et al.11). 4.3. Stepwise Models for Hydrate Formation. The models for stepwise hydrate formation could potentially be used for other hydrate formation methods. In the method of Stern et al.,20 for example, the formation of hydrate involves significant heating and cooling, and the model could be applied to these conditions. The constant-volume model presented in the Supporting Information could be reformulated in terms of conversion of a small fraction of ice to make it more like the constant-pressure model. The current model is very slow to implement because of the two internal interacting loops, first to calculate the mole ratio of ice consumed per step to gas and second to calculate the molar density of the gas at the next step. The models developed could potentially be applied to situations other than laboratory hydrate formation. One example is the modeling of a batch separation process for a gas mixture using gas hydrates. Hydrate could be formed from the mixture from an initial pressurization in a closed vessel. After sufficient conversion, the system could then be cooled to a lower temperature to stabilize the hydrate, and most of the gas phase could then be removed by depressurization. The hydrate in the vessel would then be dissociated. Similarly, the constantpressure model could be used if the pressure were maintained constant during the hydrate formation in such a vessel by a syringe or piston pump (without flow of fresh gas in to the vessel). Another potentially useful model would be one derived for an open system at constant pressure where the gas pressure is regulated in the system by flow from a supply source. Application of the model to two different gas mixtures to prevent changes in the gas-phase composition during hydrate formation as described below might enable the prevention of changes in the gas-phase and hydrate-phase composition formed. 4.4. Method to Avoid Changes in the Gas-Phase Composition During Hydrate Formation. To avoid changes in the gasphase composition during hydrate formation from melting ice particles would require two gas supplies (see Figure 10). Initially, a cell containing ice with the gas mixture of interest would be filled, and then the gas supply would be switched to

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(part 5). This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited

Figure 10. Laboratory experimental setup for hydrate formation from melting ice particles to avoid gas-phase composition changes by the use of two gas supplies.

a mixture that has compositions equal to the predicted waterfree hydrate-phase mole fractions for the initial gas. The second gas supply would then be regulated to the cell to maintain constant pressure during hydrate formation. As hydrate formation began, the components that were stripped out of the gas phase would be replaced by gas from the second gas supply, and thus, the gas would have the initial composition. This should also result in a uniform single hydrate composition and structure. This method needs to be tested experimentally. 5. Conclusions This work indicates that the composition of both the free gas phase and the enclathrated gas phase can be successfully modeled for the typical laboratory formation conditions for gas hydrates where only just enough gas is supplied to convert all of the water to hydrate. Some discrepancy between experiment and model was observed that was probably due to the assumption of equilibrium during the steps. The phenomenological explanation for the discrepancy is that methane can diffuse more quickly through a layer of hydrate than ethane because of its smaller size. It was concluded therefore that it might be expected that the hydrate would form with a higher mole fraction of methane than predicted by equilibrium thermodynamics. These results support the possibility of forming a hydrate containing both sI and sII hydrates if typical laboratory formation conditions are used, because of preferential enclathration and the accompanying stripping of components from the gas phase. The formation of both sI and sII hydrates is more likely to occur if the initial gas composition is close to an sI-sII boundary. In the experiments in this work, the initial feed gas composition did not have a high enough methane mole fraction to result in the formation of both sI and sII hydrates. Similar conclusions apply to multicomponent gas mixtures and natural gases. Acknowledgment The authors appreciate the financial support for this work from the Gas Processors Association. Supporting Information Available: Model for gas hydrate formation from a mixture using method 1 (part 1), detailed derivations of the constant-volume and constant-pressure preferential enclathration models (parts 2 and 3, respectively), details of slight modifications in the constant-pressure model used when modeling the experiments (part 4), and modeling nomenclature

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ReceiVed for reView May 26, 2010 ReVised manuscript receiVed November 16, 2010 Accepted November 18, 2010 IE101162Z