sII, Gas-hydrate Growth for a Methane

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Kinetics of sII, and Mixed sI/sII, Gas-hydrate Growth for a Methane/Propane Mixture Using Neutron Diffraction Alice Klapproth, Ross O Piltz, Shane Kennedy, and Karen A Kozielski J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 07 Jan 2019 Downloaded from http://pubs.acs.org on January 7, 2019

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The Journal of Physical Chemistry

Kinetics of sII, and Mixed sI/sII, Gas-hydrate Growth for a Methane/Propane Mixture Using Neutron Diffraction A. Klapproth1, R.O. Piltz1*, S.J. Kennedy12, and K.A. Kozielski3

1

Australian Centre for Neutron Scattering, Lucas Heights, Australia

2

European Spallation Source, Lund, Sweden

3

CSIRO Energy, Clayton North, Australia

ABSTRACT: Gas hydrates are solid solutions formed from water and enclathrated gas molecules. The hydrate types sI and sII are formed from pure methane or propane, while mixtures of these gases can form either type. In this study, eight experiments are presented of gas hydrates grown under semi-batch conditions using D2O ice powder and a feed gas of methane(90%)-propane(10%). In situ neutron scattering was used to measure the phase fractions of sI and sII hydrates, ice, and liquid-water components during hydrate growth and dissociation. The composition of the free gas and the methane/propane cage occupancies were derived from

* E-mail: [email protected]

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the observed phase fractions using mass-balance equations and Gibbs-energy calculations. In all cases the sII hydrate was formed before, or concurrent with, the sI phase. After an initial period of hydrate growth from ice, the temperature was increased to release liquid water by melting the ice particles. Different reaction mechanisms were observed; two occurred while the ice was melting and were heat-transfer limited, two occurred while the water was completely frozen or liquid and were mass-transfer limited. It is shown that the constantly changing gaseous-propane concentration is an important factor affecting sI growth and the stoichiometry and heterogeneity of the sII hydrates. It was observed that sI can grow more rapidly than sII during ice melting, despite the latter having a much higher driving force. A similar effect can be seen in other mixed-hydrate studies using stirred batch reactors; the similarities and differences to our own results are discussed. Complex equilibration processes occur involving the homogenization and stoichiometric change of the sII hydrates. For mixed sI/sII hydrates the equilibration also involves an interconversion between sI and sII. Within the energy sector, gas hydrates are of great importance, whether it is due to the safety and economic risks of gas-pipeline blockages; or the possible benefits of gas hydrates as an energy resource, a method of carbon-dioxide sequestration, or as a means for gas transport. In each case, this study offers important new information on the kinetics of gas-hydrate growth, and highlights the complications that must be considered when dealing with mixed gas and mixed hydrate types.

INTRODUCTION Gas hydrates are clathrate compounds consisting of a framework of hydrogen-bonded water molecules with different sized cavities (also known as cages) that can accommodate guest molecules, generally small gaseous hydrocarbons. For guest molecules of methane and propane two crystal structures are possible, sI and sII, which are stable at temperatures around 0° C and


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pressures of some tens of bar. The sI structure consists of 46 water molecules forming two 512 and six 51262 face-bonded polyhedrons, referred to respectively as sI small and large cages. For sII the corresponding values are 136 water molecules, 16 small 512 cages, and 8 large 51264 cages, where the large cage of sII is slightly larger than that of sI1-2. The occupation of the cages by guest molecules are non-stoichiometric and gas hydrates are described as solid-solutions3 where the occupations vary continuously with changes in temperature, pressure, and gas composition. Kinetic studies of gas-hydrate growth often use a semi-batch reactor vessel4, that is, a temperature-controlled pressure vessel where gas is supplied to the vessel to maintain the internal pressure. The vessel is commonly stirred to prevent a layer of hydrate forming at the gas-water surface and hindering further formation of hydrates. To estimate the observed growth rates, a number of models based on the transfer of gas molecules from the bulk gas through the liquid water to the water-hydrate interface have been proposed5. An important difference between these models is the significance, if any, given to crystal nucleation and growth mechanics6-8. Hydrate growth has also been studied for the case of a hydrate film formed at the interface of water droplets suspended in a gaseous or liquid form of the guest molecules. Kinetic models developed for these studies9-12 stress the importance of heat flow away from the growth site. The highly exothermic growth reaction can cause local temperature rises approaching the stability limit for hydrates, thus slowing or stopping further hydrate growth. An effect commonly referred to as heat-transfer limited, as against the mass-transfer limited commonly observed in semi-batch reactor vessels. - In the introduction, the authors should also note that the high degree of exothermicity during hydrate formation can also result in rapid expansion of the hydrate surface



area (dendritic growth); that is, the exothermic nature is only limiting to hydrate growth when local heat transport limitations exist (e.g. in quiescent systems). Hydrates can also be grown from powdered ice instead of liquid water13-16. This method has aspects of both the stirred liquid and water-droplet methods. The ice particles are in direct contact with the bulk gas, at least in the initial phase of the reaction. The heat of formation can then be removed by conduction into the ice phase, a more efficient process than conduction into the gas phase, though less efficient than into a liquid phase where large heat flows are possible via liquid convection and sample stirring. Several variants of the shrinking core model14-16 have been used to analyse the hydrate growth from ice. The base model is the thickening of a hydrate layer covering the ice particle thereby restricting the transport of gas molecules to the ice in the core of the particle. The variants are used to account for incomplete coverage of the ice particles in the early stages of hydrate growth13, and the influence of microscopic pores in the hydrate layer15. Powdered ice can also be used to study hydrate growth from liquid water released by the melting of the ice. Several studies have observed periods of rapid hydrate growth observed during the melting of the ice particles.17-19 This melting may occur deliberately by raising the reactor temperature, or inadvertently by adiabatic heating17 when feed gas is added to the reactor. The latent heat for hydrate formation is sufficient to melt enough ice to replenish the liquid water consumed by the hydrate growth20, thus allowing complete melting of the ice under adiabatic conditions once ice melting has begun. It also should be noted that this combination of hydrate growth and ice melting buffers the local temperature between the melting point of ice and the hydrate stability limit. Chen et al.17 used a variant of the shrinking-core model for hydrate formation on the ice particles, but then proposed that during the melting of the ice the hydrate


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shells covering the ice particles were incomplete or ruptured by the melting of the ice core. As a result, liquid water is expelled from the core of the particles and coats the external surface of the hydrate shells. The thickness of the water layer over the hydrate shells is small, while the surface area of the gas-liquid interface is large, both of which contribute to the rapid transport of gaseous molecules from the gas-liquid to liquid-hydrate interface. The phenomenon of rapid hydrate growth during ice melting results from removing both the mass-transfer and heat-transfer limitations. Methane-ethane mixed hydrates are known to form sI and sII depending on the feed-gas composition21-22. Coexisting sI/sII mixtures have also been observed in this system, along with the interconversion between types apparently based on the thermodynamically preferred phases23-24. The coexistence of sI/sII has also been reported for methane hydrates at high pressure25-26, and for mixed hydrates of methane with cyclopropane27, nitrogen28, or carbon dioxide29. Of particular relevance to this article are the reported coexistence of sI and sII phases in gas mixtures containing methane and propane30-31 and the observation of large-molecule depletion in the feed gas of these systems30,

32-34.

In this article we present eight examples of

hydrate growth using powdered ice samples in an unstirred pressure cell acting in semi-batch mode. A methane/propane feed gas will be used at temperatures and pressures to ensure that pure sII phase, and mixed sI/sII phases are created and can be compared.

EXPERIMENTAL SECTION Sample Preparation A slurry of deuterated (D2O) ice and liquid nitrogen was ground in a mortar and pestle. Optical microscopy examination showed highly fractured ice particles with an average particle size of 30 micron, or less. The ground ice was loaded in aluminum pressure cells, the cell sealed and then



stored at liquid nitrogen temperatures for intervals up to several days, or at -80° C for more extended intervals. The pressure cells have a total internal volume of 4.0 ml when sealed; a tool was used to load the cell with a consistent powder volume of 2.8 ml; and the measured packing ratio of the ice powder was 0.58.

Experimental Method The loaded pressure cell was transferred to an aluminum cell holder designed with access ports suitable for a neutron powder diffraction experiment (Fig. 1). The pressure cell does not include a mechanism to stir the sample. The cell holder and pressure cell were designed to minimize temperature variations due to heat flow. Cooling and temperature control was provided by recirculating water/glycol from a Julabo LH45 water bath. Sample temperatures are based on the temperature measured at the top of the cell holder and corrected for the temperature differential of the cell measured previously for the same nominal temperature. During the experiments the temperature remained constant within 0.1 K. To prevent condensation, the cell and holder were surrounded by a second aluminum container that was filled with dry nitrogen gas. The feed gas for the experiment was 10% propane in methane (BOC Scientific Support Centre, gas purity 99.995%). Gas pressure was controlled manually using the mechanical regulator of the gas cylinder, while the pressure was electronically logged using a 70 MPa Quartzdyn pressure sensor. Feed gas was supplied to the pressure cell via a 6m length of stainless steel tubing with an ID of 0.9 mm, effectively preventing gas movement by diffusion from the pressure cell back to the gas supply.


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Figure 1. The experimental set-up of the pressure cell and liquid-cooled cell holder. In-situ neutron powder diffraction measurements were performed using the Wombat diffractometer.35 Quantitative Phase Analysis (QPA) was used to determine the relative amounts of crystalline phases from the Bragg peak intensities. It was found necessary to correct these intensities using a neutron attenuation factor that varied during the course of our experiments as the attenuation depends on the amount of hydrogen in the form of methane and propane incorporated into the hydrate phases. An analysis of the peak background was also used to obtain an approximate estimate of the liquid water component in the sample. Detailed information on the Wombat configuration and the data analysis method are given in the Supporting Information.

Experiment Sequence The sample temperature was allowed to equilibrate prior to the start of each experiment. Feed gas was added to the sample cell 5 to 20 minutes after the start of data collection. After an interval of 1.5 to 12 hours the temperature was raised above the melting point of D2O (+3.8° C) resulting in the melting of any remaining ice. Finally, the pressure was reduced below the



hydrate stability limit to allow the study of hydrate decomposition. CSMGem36 was used to calculate the stability limits for the sI and sII phases, and corrected for use of D2O by adding 2.0 K to the temperatures37. The experiments were performed in four sets of experimental time within a 12 month period. Individual experiments, and therefore samples, are denoted by a number to identify the set, and a letter to distinguish them within a set, e.g. 1a, 4b. Details of the pressures and temperatures for these samples are given in Figure 2, while the schedule of pressure and temperature changes is given in Table 1.

Figure 2. Pressure versus temperature for each sample (denoted by label) at Steps 1 to 5 of an experiment (denoted by symbol given in legend). The horizontal and vertical arrows illustrate the experimental sequence for Sample 1b, i.e. Steps 1 to 2 to 4 to 5. The upper and lower curves are the stability limits of sI and sII phases calculated for 10% propane (solid curves) and 5% propane (dashed curves) in methane. The dotted vertical line at 3.8° C marks the D2O melting point.


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Table 1. Times for the pressure and temperature changes, Steps 1 to 5, of Figure 2.1 Sample Step 1 Step 2 Step 3 Step 4 Step 5 (min) (min) (min) (min) (min) 1a

13

431

668

1b

22

742

798

2a

13

131

264

3a

21

108

189

3b

30

141

280

316

3c

23

140

201

311

4a

20

83

262

4b

17

318

893

933

745

RESULTS The results of QPA are shown in Figure 3. Here the phase fractions have been normalised against the amount of D2O contained in that phase, therefore the sum of the phase fractions should equal 1. That this is not true after ice melting is a result in part of the liquid water draining to the bottom of the cell out of the extent of the neutron beam. This effect and details on the analysis method are given in the Supporting Information.

1

At Steps 1, 4, and 5 the pressure varies but the temperature is constant. At Steps 2 and 3 the temperature varies but the pressure is constant. Prior to Step 1 the pressure is 1 bar and the temperature the same as for after Step 1. Blank entries in the table indicate where an optional Step 3 or 5 did not occur. The experiments were performed in four sets of experimental time within a 12 month period.



Figure 3. The sI, sII, and Ice phase fractions versus time for all eight samples. Solid blue circles are estimates of the phase fraction for liquid D2O water, the line marked as Total is the sum of all phase fractions. Due to instrument problems, a few gaps exist in the data for Samples 1a, 3a, and 3b, while the water component could not be calculated for Sample 3c.

Growth of sI and sII from Ice The plots of Figure 4 show the initial 2 hours of hydrate growth for our eight samples. Group 1 and 3 samples all show a near linear growth of sII in the first hour, and, in the cases when sI


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growth occurred, a delay of ~40 minutes between the onset of sI and sII growth. By contrast, Group 2 and 4 samples show a rapid growth of sII in the first few minutes followed by a significant reduction in growth rate. For the one case of sI growth, Sample 2a, the onset of sI and sII growth was simultaneous within our measurement time of 1 minute. Further discussion on the growth of the sI phase is postponed until later in this section.

Figure 4. Phase fractions of sII (continuous lines) and sI (dotted lines) versus time from the onset of sII growth. Plots are labelled with the sample identifier and grouped accordingly. Figure 5 is a plot of the sII growth from 5 minutes after growth onset up to the point of the first temperature change. Samples 1a and 1b both reach sII phase fractions above 0.4 after 6 hours, and show continued growth at their time limits of 6 and 12 hours. By contrast, the 4b sample reaches a maximum sII phase fraction of 0.08, with no significant growth after 1 hour. We have no data past 1 hour for Sample 4a, though this sample has a much lower sII growth rate at 1 hour than any other sample apart from Sample 4b. It is believed that these anomalously small phase fractions are due to the formation of a plug, or hydrate barrier, near the top surface of the



powdered ice. This and other anomalous behaviours for the Group 4 samples will be discussed later in this section.

Figure 5. The sII phase fraction starting from 5 minutes after the onset of hydrate growth. The upper graph is for mixed sI/sII samples; the lower for pure sII samples. All plots are shifted to start at the origin. Excluding Group 4 samples, the sII growth rates at the start of Figure 5 show a limited correlation with pressure; the highest rate being for the highest pressure, the lowest rate for the lowest pressure. However, Samples 1b, 2a, and 3b are all at similar pressures and temperatures yet their growth rates vary by a factor of 2, apparently due to extrinsic factors such as ice preparation or gas-dosing method.

Ice Melting Ice melting depends on the heat flow into the sample from the aluminum cell and the heat generated by the exothermic formation of hydrate phases. As expected, the samples at the highest cell temperature of 15° C, Samples 3a and 3b, have the highest ice-melting rates (see Figure 6). At lower temperatures, the heat of formation becomes more important and the relationship between cell temperature and melting rates is less clear. Though Samples 1a, 1b, and 2a have


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low cell temperatures of 4 to 6° C they all show rapid ice melting. This is a result of rapid hydrate growth resulting from their higher pressure, and hence higher driving force for hydrate formation. The two slowest melt rates are for Samples 4a and 4b, apparently due to a hydrate plug. Sample 3c shows a mixed behaviour; a fast melt up to 12 minutes from onset, followed by a much slower melt once sII growth has finished.

Figure 6. The ice phase fraction versus time after the onset of ice melting. The inset graph shows an enlarged view of the first 15 minutes.

Growth of pure sII Hydrate During ice melting, rapid sII growth is evident for the pure sII samples, with the exception of Sample 4b (see Figure 7). Excluding Sample 4b, the pure sII samples all show similar sII growth behaviour; constant growth in the first 20 – 30 minutes followed by a change to much slower growth. It will be shown in a later section that this behaviour is due to the depletion of gaseous propane and that the final sII phase fractions of Samples 3b and 3c are consistent with their respective pressures of 48 and 30 bar.



Figure 7. The phase fraction of sII versus time from the onset of ice melting for pure sII samples.

Growth of mixed sI/sII Hydrate During ice melting, the growth of sII for the mixed sI/sII samples is varied with no growth for Samples 1a and 1b, medium growth for 2a, and fast growth for 3a. With the exception of Sample 3a which is heated to 15 C, all samples show a later slow sII growth towards a phase fraction of ~0.6 (see Figure 8).

Figure 8. The phase fraction of sII versus time from the onset of ice melting for mixed sI/sII samples.


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The ratio of sI to sII phase fractions versus time (see Figure 9) has several interesting features. During ice melting this ratio increases for all sI/sII samples, the most dramatic example being Sample 2a where the increase is from 0.3 to 1.1. Once ice melting is complete, the ratio then decreases for three of the four samples, the exception is Sample 1b where it remains constant or slightly increases. For Samples 1a and 2a (see Figure 3) the ratio decreases not only by the growth of more sII phase, but also by a partial decomposition of the sI phase. Before ice melting the ratio appears to tend to values of 0.2 to 0.3, and to 0.3 to 0.4 after ice melting. The exception in this case is Sample 3a after ice melting where the high cell temperature causes rapid sI decomposition resulting in a ratio of zero.

Figure 9. The ratio of the phase fractions, RI/II = f(sI) / f(sII), for the four mixed sI/sII samples. Dashed vertical bars indicate the onset of ice melting. For clarity, an inset is used for Sample 3a. It appears that factors other than the minimization of Gibbs energy are driving the rapid increase in sI during ice melting. However, once ice melting has finished the amount of sI and sII



phases equilibrates towards the lowest Gibbs energy configuration. This can also be argued for the apparent equilibration occurring before the ice is melted.

Decomposition on Temperature Increase The unusual behaviour of Sample 3a observed in Figures 8 and 9 is due to the much higher cell temperature, 15° C as compared with 4 to 6° C for other samples. The higher temperature is well outside the stability limits for sI, and approaching the stability limits for sII. However, the temperature within the cell will be partially buffered during ice melting, lowering the local temperature so that growth of both sI and sII phases can occur. Once the ice has completely melted at 8 minutes, the sII phase starts a slow partial decomposition to a phase fraction of 0.2 after 70 minutes (see Figure 8). The sI phase is even closer to its stability temperature and begins decomposition after 3 minutes, even before the ice is completely melted, with complete decomposition occurring 3 minutes later (see Figure 3). To investigate further the slow partial decomposition of sII, an additional temperature increase at Step 3 was performed for Samples 3b and 3c. Figure 10 shows the slow decomposition of sII for these samples and for Sample 3a.


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Figure 10. The sII phase fractions for the three samples where the cell temperature exceeded 9° C and the sample contained no ice. The numbers following the sample identifiers are the cell temperatures (° C) and pressures (bar).

Anomalous Group 4 Samples Samples 4a and 4b show several extreme behaviours. They have the lowest sII growth rates from ice (see Figure 7), and the lowest ice melting rates (see Figure 6). Among the pure sII samples, they have much lower final phase fractions (see Figure 3). Sample 4b shows the more extreme behaviour of the two with sII growth stopping after the first 60 minutes, while Sample 4a shows only a very small growth rate after the first 5 minutes. The most plausible explanation for the Group 4 behaviour is the formation of a plug, or hydrate barrier, near the top surface of the powdered ice in the first few minutes of the experiment. This barrier significantly restricts gas diffusion and impedes further hydrate growth. The high initial cell temperatures only 1 to 3 K below the melting point for these two samples, compared to 8 to 15 K below for other samples, will greatly assist the local melting of ice and contribute to the formation of a hydrate barrier.

Estimating Depletion of Gaseous Propane The depletion of large-molecule (non-methane) gases has been reported in other isobaric semibatch experiments30,32-34. Even though the gas concentrations were not measured in our own experiments the change in gas mixture compositions can be calculated from mass balance and the measured phase fractions of the gas hydrate, albeit with the cage occupancies of the hydrates as unknown parameters. Details of the derivation are given in the Supplementary Information with only a summary given here.



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From the volumes of the sample cell and the ice sample we calculate the void space within the cell as 2.38 ml. Within this void the amounts of gaseous methane and propane are 𝑛G(CH4) = 0.106(mmol. bar ―1)

(

𝑛G(C3H8) = 0.106(mmol. bar ―1)

𝑃 (1 ― 𝑦C3) 𝑍 𝑇 273.2(K)

(

)

𝑃 𝑦C3 𝑍 𝑇 273.2(K)

)

(1a) (1b)

where 𝑦C3 is the unknown mole fraction of gaseous propane in the void space, and Z is the compressibility factor that we calculate using the Peng-Robinson38 equation-of-state which requires the values of P, T, and 𝑦C3. The amounts of methane and propane contained within the hydrates can be calculated from the sI and sII phase fractions, the number and type of cages in each hydrate structure, and the occupancies of these cages, the latter being unknown at this stage. 𝑛H(CH4) = ( 3.90(mmol) C1(sI,S) + 11.69(mmol) C1(sI,L)) 𝑓(sI) + ( 10.54(mmol) C1(sII,S) + 5.27(mmol) C1(sII,L)) 𝑓(sII) 𝑛H(C3H8) = 5.27(mmol) C3(sII,L) 𝑓(sII)

(2a) (2b)

where X(Y,Z) denotes the methane or propane occupancies (X=C1 or C3) of the small or large cages (Z=S or L) of the hydrate phase (Y=sI or sII). Due to its size, we assume that propane can only occupy the large sII cages. The sample cell is maintained at constant pressure by the inflow of feed gas (10% propane in methane) from the gas supply rig, so the total amount of methane in the sample cell must be 9 times that of propane, or nG(CH4) + nH(CH4) = 9 nG(C3H8) + 9 nH(C3H8). From this mass balance condition and eqs 1 and 2 we can solve for the unknown propane concentration.

{ ( 3.68(bar) C1(sI,S) + 11.03(bar) C1(sI,L)) 𝑓(sI) ― (44.75(bar) C3(sII,L)– 9.94(bar) C1(sII,S)– 4.97(bar) C1(sII,L)) 𝑓(sII) } (3)

𝑦C3 = 0.1 + (Z T/273.2 / P)


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A caveat exists that this equation is only valid for the flow of gas into the sample cell. During hydrate decomposition the gas flows out of the cell and, as the propane concentration of the outflowing gas is unknown, the mass balance condition shown above is not applicable. If we consider the case of full occupancy of the large sII cages by propane and the full occupancy of all other cages by methane, eq 3 becomes 𝑦C3 = 0.1 + (…) {14.71 f(sI) - 30.04 f(sII)}. Note the large numeric factor for the sII term, and also the fact that in all of our experiments the sI phase fraction only once exceeds the value for sII, and then only by ~5%. As such, the propane concentration must decrease below 0.1 as soon as any sII hydrate begins to grow. This result is not dependent on the full propane occupancy assumption. As we shall show in the section “Comparison to CSMGem Calculations” the estimated propane occupancy for propane concentrations of 0.1 is ~98% rather than the 50% or less that would be required for the propane concentration not to decrease. It should be noted that the depletion of the propane concentration is not due to sII consuming more propane than methane. As sII contains twice the number of small to large cages, and as only the large cages can contain propane, it is expected that sII will consume twice, or more, methane than propane. Instead it is the mismatch of this 1:2 propane to methane ratio required for sII growth compared to the 1:9 ratio in the feed gas that results in the propane depletion.

Estimated Propane Cage Occupancies For sI and sII mixtures, we observed in Figure 9 that RI/II = f(sI) / f(sII) tends towards values of 0.25 or 0.35. By inspecting eq 3, this is due to the phase fractions increasing with time, thus decreasing the importance of the initial gas inside the sample cell compared to the amount of gas consumed by the formation of hydrate. The limit is the asymptote of eq 3, given by RI II  (44.75(mmol) C3(sII,L) ― 9.94(mmol) C1(sII,S) ― 4.97(mmol) C1(sII,L))



/ (3.68(mmol) C1(sI,S) + 11.03(mmol) C1(sI,L))

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(4)

To obtain approximate estimates for the propane occupancy of the large sII cages, we must assume values for the unknown methane occupancies of eq 4. We will assume 100% methane occupancies for all cages except the large sII cage which is 100% occupied by an unknown mix of methane and propane. For RI/II asymptotes of 0.25 and 0.35, this predicts occupancies of 0.37 and 0.40, respectively. Other estimates of propane occupancies can be obtained from the pure sII samples 3b and 3c which have sII phase fractions that tend towards 0.60 and 0.40, respectively. If we assume negligible amounts of gaseous propane at equilibrium we can calculate the propane occupancy from the phase fractions and the total amount of propane in the system using eq 3. Using the same assumptions as above for the unknown occupancies, both Sample 3b and 3c estimate the same propane occupancy tends towards 0.43. A propane occupancy for the sII large cage of ~0.4 may appear unusual, though it must be remembered that the value depends not only on pressure and temperature but also the composition of the gas mixture. Due to the design of our experiments, the amount of gaseous propane is depleted with time and the final hydrate phases must be in equilibrium with a propane mole fraction much smaller than 0.1. Put simply, in our experiments too much sII phase is created from too little propane to allow full propane occupancy.

Comparison to CSMGem Calculations The CSMGem software was also used to calculate the unknown cage occupancies of eq 3 using the Gibbs-energy minimization method. Occupancies were first calculated for a grid of pressures (30, 40, 50 bar), temperatures (260, 265, 270, 275, 280 K), and gaseous-propane mole fractions (0.001, 0.003, 0.01, 0.05, 0.1, 0.2). CSMGem predicted that the large sII cages were close to


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100% occupied by a propane or methane molecule, and that only these occupancies were strongly dependent on gaseous-propane concentrations. Empirical equations were fitted to the occupancy values calculated by CSMGem and interpolation functions were created with an average error of less than 0.01 for the grid values. The interpolation function and eq 3 were then iterated until the occupancies and propane mole fractions converged. Smoothing of the experimental phase fractions versus time was found necessary to prevent serious instabilities in this iterative process. The degree of smoothing was manually directed to ensure minimal smoothing during periods of non-linear growth, and stronger smoothing during periods of near linear growth. Figure 11 contains the results for Group 1, 2, and 3 samples, the plots being truncated at the point where eq 3 is no longer valid, that is when gas flows out of the sample cell.

Figure 11. The results of CSMGem calculations made self-consistent with the measured phase fractions and the conservation of gas molecules as expressed in eq 3. The solid curves marked by



sI and sII are the respective phase fractions, C3 the propane occupancy of the large sII cage, and 10 yC3 is 10 times the gaseous propane mole fraction. The dashed curves, for Samples 2a and 3c, are variants of C3 and 10 yC3 calculated using heterogeneous models as described in the Discussion section. For the two pure sII samples, Samples 3b and 3c, the propane occupancy of the large sII cage tends to a value of 0.41, in agreement with the earlier estimates of 0.37 to 0.43. For the mixed sI/sII samples the final calculated propane occupancies range from 0.49 to 0.60, significantly higher than the 0.41 value obtained for the pure sII samples. However, care must be taken in making the comparison as hydrate growth or dissociation is continuing in all of the mixed sI/sII samples. It is possible that the propane occupancy would eventually tend to the same value as for the pure sII samples. In Figure 11, the gaseous-propane mole fractions all tend to values of ~0.2%. However, this value is an estimate calculated by CSMGem, which may be inaccurate for such low mole fractions. For instance, CSMGem calculates the aqueous-vapor-sI-sII quadruple point at 277.6 K to have a propane mole fraction of 0.04% compared to the literature value1 of 0.06%. It is also useful to compare to a recent batch-reactor study39 which observed a depletion of propane in methane from an initial value of 13% to 1%, within an uncertainty of 1%. Within the limitation of our experiment it is not possible to determine the true amount of gaseous propane at equilibrium, though it does appear to be significantly less than 1%. Despite such inaccuracies the cage occupancy at equilibrium can still be estimated with some certainty as it principally relies on the measured amount of hydrate and the estimated amount of propane consumed. The difference between 0 and 1% propane in the free gas of our pressure cell is insignificant compared to total amount of propane stored in the hydrate. We therefore expect the minimum


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propane-cage occupancy values of Figure 11 to be accurate, though the corresponding propane mole fractions may not be.

DISCUSSION Hydrate Growth from Ice Several samples show a period of rapid hydrate growth lasting a few minutes after the initial introduction of the feed gas (see Figure 4). This effect is a result of ice melting, initially due the adiabatic heating of the feed gas,17 and then by the heat of hydrate formation1,10. To sustain this growth by successive ice melting and hydrate formation it is necessary for the reaction to occur in a localised region so as to reduce heat loss out of the region. However, as the reaction front expands this heat flow increases, eventually preventing the local melting of ice and ending the phase of rapid hydrate growth. Samples 4a and 4b are within 3 K of the melting point of D2O ice when the feed gas is introduced and, as such, these two samples are exemplars of heterogeneity where rapid growth affecting < 10% of the ice sample results in hydrate barriers created at the top of the sample. Excluding the rapid growth which may occur in the first few minutes, all samples show an early period where the hydrate growth rate is nearly constant and much smaller than during the rapid growth period (see Figure 5). Samples 1a and 1b were studied the longest and both show continuous, albeit slowing, growth up to the end of the study period. This behaviour is similar to the predictions of the published14-16 variants of the shrinking-core model (VSCM) where the initial growth rate is constant and then slowly decreases while the total amount of hydrate tends to an asymptote.



Figure 12. Phase fractions of sII (solid lines) for Samples 3a, 3b, and 3c versus time up to the first temperature increase. The dashed lines are linear fits to the initial sII growth rates. The second (and lower) solid line for Sample 3a is the phase fraction of sI. Alternatively, Figure 12 shows three examples where the hydrate growth rate increases with time, an impossible behaviour for a VSCM. For Sample 3a we observe an increase in sII growth rate approximately 5 minutes after the initiation of sI growth. This can be explained by the sI growth consuming methane gas which increases the flow of feed gas into the cell, and increases the amount of propane available for sII growth. Samples 3b and 3c also show an increase in growth rates, but as they are both pure sII samples the explanation used for Sample 3a is not applicable. The reason for the growth rate increases in Samples 3b and 3c is unknown. Though the rapid hydrate growth is necessarily a localized phenomenon, the slower growth does not have to be and can occur throughout the sample volume. It appears reasonable to assume the slow growth process occurs concurrently with the rapid growth process and that both can act relatively independently of each other. Unfortunately, our experiments were unable to distinguish between localized and delocalized growth.


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Onset of sI Growth from Ice Within our time resolution of 1 minute, we observed no appreciable delay between the onset of sII growth and the initial pressure increase. This was by design as a large driving force ensured minimal delays, and so optimal use of the neutron-scattering instrument. All of our samples are within the stability zone for sI, albeit for Sample 4a only after a significant depletion of propane (see Figure 2). Yet, we have 4 samples with no sI growth; three of these samples are at 30 bar and one at 48 bar. The lack of nucleation for sI in the three 30 bar samples can be explained by the low driving force, less than 10 bar as measured by the sample pressure minus the stability pressure for sI. Sample 3b cannot be explained in this way as the sI driving force is 29 bar, the third highest of all samples, yet no sI nucleation was observed. Comparing Samples 2a and 3b we find vastly different behaviours in terms of delay time between the onset of sII and sI, despite near identical pressures and temperatures. The former has a delay of 1 minute or less, while the latter has a delay of 2 hours or more. Despite the problems of determining probabilities from a sample set of two, it is highly improbable that the two samples have the same stochastic probability of sI nucleation. The difference in probabilities may result from differences in ice preparation. Hydrates are known to nucleate preferentially at high activity sites on the ice particles such as contact points between particles, along fractures, and certain crystallographic faces15. Our ice samples were created by grinding particles in a slurry of liquid nitrogen until less than 30 micron in size; a method that undoubtedly creates many sites of high activity. Annealing of the ice could have occurred while the ice samples were stored at -80 C, or possibly when the sample cell is transferred to the neutron instrument prior to an experiment. Differences in the amount of annealing may account for the large differences in sII to sI delay time. An alternative theory is that the sII quickly nucleates on high activity sites making them unavailable for the slower sI nucleation. This conjecture could be tested



experimentally by determining if the probability of sI formation is truly stochastic or if it varies with the degree of sII growth.

Hydrate Growth from Melted Ice All samples, excepting the anomalous Group 4 samples, show a rapid, though not necessarily large, growth of hydrates while the ice is melting due to the cell temperature increase. The reason is the same as for the initial rapid growth; a self-supporting solid structure, this time of hydrates rather than ice; liquid water from the melting ice covering this structure and making a large gasliquid interface; temperature regulation by melting ice absorbing the heat flow from the hydrate formation. However, in this case the cell temperature is raised above the melting point and the process can continue until all ice has melted, aided by the hydrate heat of formation1,10. Unexpectedly, for mixed sI/sII samples the rapid growth occurs for sI instead of sII, though once the ice has completely melted the ratio of sI and sII phase fractions slowly changes to a more energetically favourable value. Figure 13 shows the hydrate growth during ice melting for Samples 2a (mixed sI/sII) and 3b (pure sII). These samples are useful for comparison as they have similar pressures and though the cell temperatures during ice melting differ (4 C for 2a, 9 C for 3b) this will mainly affect the rate of melting while the local sample temperatures are buffered close to Tm(D2O) = 3.8 C. Sample 2a shows a remarkable growth of the sI phase during ice melting with a growth rate 6 times larger than for sII. By comparison, the sII growth rate for Sample 3b is 10 times larger than for Sample 2a. It should be noted that both samples have comparable total hydrate growth rates.


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Figure 13. Phase fractions for sII (solid lines), sI (dashed line), and ice (dotted lines) versus time from the onset of ice melting for Samples 2a and 3b. The rapid growth of sI, but not sII, during ice melting for Sample 2a is surprising as there are several factors favouring a higher sII growth rate compared to sI. The larger driving force for sII growth is one such factor. At the start of ice melting for Sample 2a, there is much more sII than sI phase which implies a greater availability of sII compatible growth sites, again favouring faster sII growth. Finally, at the start of ice melting the estimated propane concentration is 0.32% for Sample 2a, significantly larger than 0.19% for Sample 3b, yet the former has a sII growth rate that is 10 times smaller. Using an approach similar to Skovborg et al.7 we are able to perform a rough estimate of the propane diffusion through the water layer that covers the hydrate surface. From available propane data36, 40 the propane diffusion rate is calculated as ~0.2 mmol/sec for Sample 2a at the start of ice melting (see Supporting Information). From eq 2, and assuming 50% occupancy of the large cage, we calculate the f(sII) rate of increase as 8 x 10-2 s-1, some 800 times greater than the observed rate of 10-4 s-1 for Sample 2a. We conclude that the transport of propane molecules across the water layer is not the rate-limiting factor for sII growth, and does not directly account for the reduced sII growth in Sample 2a.



Due to the strong correlation observed between ice melting rate and the total hydrate growth rate, we conclude the rate-limiting factor for hydrate growth during ice melting is the availability of liquid water. Using this assumption the relative diffusion rates of methane and propane through the water layer can explain the unusual growth behaviour seen in Figure 13. For mixed sI/sII hydrates, both phases will compete for the limited supply of liquid water. The rapid growth of sI relative to sII results from the higher diffusion rate of methane compared to propane due to the much higher concentration of methane in the free gas. At the hydrate-water interface the scarcity of propane limits the growth of sII relative to sI resulting in most of the liquid water being consumed by sI growth and relatively little by sII growth. For single sII hydrates, there is no competition for the consumption of liquid water, and so the sII growth rate only depends on the rate that liquid water is supplied to the interface. The rate of ice melting is determined by heat flow, so the latter can be considered the ultimate rate-limiting process. As a result, the higher cell temperature of Sample 3b compared to Sample 2a paradoxically increases the total growth rate due to the faster production of liquid water. However, the higher temperature also increases the amount of pooled liquid water which does not contribute to hydrate growth but is necessary to regulate the sample temperature. Use of the ice-melting method for the production of gas hydrates will require further work to optimize the heat flows so that both hydrate yields and growth rates are optimal. Finally, once all ice has been melted the growth rate reduces to a value comparable with the slow growth from ice. For pure sII samples, the hydrate phase fraction tends to a value significantly less than 100% with significant amounts of liquid water still present in the sample. By contrast, the mixed sI/sII samples tend to the complete conversion of liquid water to hydrates.


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Heterogeneous Occupancy Models Bouillot has proposed a model41 for use with mixed-gas experiments39, 42 in a stirred batchreactor. It is assumed that particle growth occurs layer by layer and that each layer has a stoichiometry in equilibrium with the reactor conditions when the layer was formed. The result is a heterogeneous particle where the cage occupancy varies with distance from the particle center. It is possible that these heterogeneous particles will equilibriate with time to a more constant stoichiometry. As these experiments differ significantly from our own, further discussion of them will be left until later in this section. We will analyse our data under similar assumptions that the hydrate stoichiometry depends on the current reactor conditions and, to begin with, we will assume the stoichiometry does not equilibrate with time. For our experiments the reaction cell is a narrow cylinder, has no sample agitation, and the feed gas is introduced from a single fixed point at the top. For low hydrategrowth rates the propane concentration will tend to be uniform due to gas diffusion, but for high growth rates the bulk flow of gas due to internal consumption will tend to create variations in the gas concentration surrounding the ice and hydrate particles. At the top of the sample we would expect a gas concentration similar to the feed gas, while further into the sample more significant variations can occur. Though we can speculate on this spatial component of heterogeneity as against the temporal component (due to the layer by layer growth in a stirred reactor) our experiments give us little quantitative information to distinguish the two components. To introduce heterogeneity into our models we will modify eqs 2 and 3 by replacing the X (Y,Z) 𝑓(Y) terms by the integrals∫𝑑𝑡X(Y,Z) 𝑓′(Y), where 𝑓′(Y) is the time derivative of the phase fraction 𝑓(Y). Sample 3c was reanalysed using this method, and the results are shown as dashed lines in Figure 11. The calculated cage occupancies and the gaseous mole fractions of propane deviate strongly from the homogeneous occupancy model after 30 minutes. Despite the



occupancy dropping to ~0.35, complete depletion of propane is predicted at 45 minutes, less than half of the observed period of near constant sII growth. For such a model there is simply not enough propane in the gas phase to create the large f(sII) values obtained in our experiments. Furthermore, significant changes in propane cage-occupancies must occur within the first 45 minutes to prevent Sample 3c prematurely depleting propane from the gas phase. Sample 2a shows a dramatic increase in sI growth during ice melting. During this period the propane cage occupancies may be too slow to respond to this rapid change. To explore this further we use a hybrid model where the hydrate grown prior to ice melting has its stoichiometry fixed at 120 minutes, while hydrate grown after 120 minutes is assumed to be in equilibrium with the free gas. The results are shown in Figure 11 as dashed lines for Sample 2a. Equation 3 was used for time up to 120 minutes, and eq S3 (see Supporting Information) for time after 120 minutes. Using this method, a much larger increase in propane concentration was predicted compared to the homogenous model. As to the relative merit of the homogeneous and hybrid models, it appears reasonable that the actual gaseous propane concentration and hydrate occupancies are somewhere between the two models though quantitative values are not possibly without more knowledge of the timescale over which sample homogenization occurs. Excluding the above two examples, our calculation method for Figure 11 plots assumes a system in quasi-static equilibrium where all phases are homogeneous at any one time. We expect that this will be the case if sufficient time is given for the hydrate phases and the gas composition to equilibrate and homogenize. Without direct observations of occupancies or gas concentrations our experiments can only make limited estimates of what sufficient time actually means. It should be noted that the Samples 4a and 4b exhibit obvious spatial heterogeneity albeit of an extreme nature, and in the form of hydrate plugs. The validity of our heterogeneous models


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cannot be adequately checked given the limited direct evidence available from our experiments. Future experiments are required to measure the rate and degree of spatial/temporal heterogeneity. We will now return to the stirred batch-reactor experiments39, 42 and the authors’ observation that hydrates grown at fast crystallization rates contain a lower proportion of heavy to light hydrocarbons as compared to hydrates grown at a slow rate. As the slow experiments gave results in agreement with thermodynamics, the results of the fast experiments appear to be related to limitations in the transport of the gas molecules41. This effect has similarities to our observed increase in sI growth compared to sII during ice melting, and both can be explained by the slower transfer of the heavier hydrocarbon to the water-hydrate interface under conditions where the driving force is not a limiting factor. It must be stressed that the slower transfer is not due to the mass or size of the hydrocarbons, but results from the low gaseous concentration of heavy hydrocarbons in these particular experiments. At the start of the fast reactor experiments the driving force is very large, allowing the formation of hydrates with a higher Gibbs energy, i.e. a stoichiometry with fewer heavy hydrocarbons. In our case, sI was formed despite being much higher in Gibbs energy than the sII phase. For the slow reactor experiments the limiting factor is the small driving force, so the hydrate in the growth layer has a stoichiometry with the lowest Gibbs energy given the reactor conditions. As the fast reactor experiments approach equilibrium, the limiting factor will switch from mass transfer to driving force and the growth layers will become richer in heavy hydrocarbons because this stoichiometry has the lowest Gibbs energy. The difference between the fast and slow reactor experiments appears to agree with our own observations, albeit with a large difference in timescales for hydrate homogenization. This may be due to how the hydrates were grown. Hydrate particles commonly grow to several hundred



microns in stirred batch-reactors

43,

while our own self-supporting matrix of hydrate was made

from ice particles less than 40 micron in size. As well, hydrates grown from ice are known to be porous15, further enhancing the gas transfer. The difference in timescales may be due to the more rapid transport of gaseous molecules for our experiment which then facilitates a faster rate of homogenization.

Ultimate Hydrate Yields Our experimental results when combined with mass balance equations show that the propane occupancy of the large sII cage asymptotes to a value in the 40 - 60% range. The use of CSMGem calculations allowed some refinement of these values, though it appears that different asymptotes, 40% or 50%, may exist for pure sII or mixed sI/sII hydrates, respectively. For pure sII samples, the amount of hydrate tends to the point where propane has been almost completely removed from the free gas, and where the propane occupancy has reached the asymptote. For mixed sI/sII samples, the hydrate growth continues to total transformation of the available ice and liquid water to sI or sII hydrates. As for pure sII samples, virtually all propane in the free gas is consumed to grow sII hydrate with a propane occupancy equal to the asymptote. At the same time sI continues to grow, consuming methane gas, and drawing in new feed gas containing propane. The growth of sI can continue without propane, so complete transformation to hydrate phases can occur. The final ratio of sI to sII is determined by the minimization of the Gibbs energy of the system. If excess sI is formed, such as occurs for Samples 1a and 2a during ice melting, it will partially dissociate over a period of several hours releasing water molecules that can form the more energetically favourable sII phase, provided available propane gas is present.


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First, the propane occupancy of the large sII cages tends to a value of ~40% for pure sII samples, and ~50% for mixed sI/sII samples. Second, sII hydrate continues to grow until the gaseous propane is near depletion. Third, if sI hydrate is present it tends to the minimum amount of sI consistent with the maximum amount of sII. We use these rules with CSMGem calculations to determine the cage occupancies for propane concentrations that correspond to the asymptote value. Equation 3 is then solved for the phase fractions with the constraints of 𝑓(sI) = 0 for pure sII, and 𝑓(sI) + 𝑓(sII) = 1 for mixed sI/sII. A comparison of calculated and observed ultimate phase fractions is shown in Table 2, where the calculated values assume an asymptote of 40%, except for the values in brackets where an asymptote of 50% has been used for the mixed sI/sII samples. Table 2. Observed and calculated ultimate values of sI and sII phase.2 Sample

1a

Time (min.) 667

1b

2a

3b

3c

4a

4b

797

263

279

200

261

892

Calc. 𝑓(sII) 0.88(0.70) 0.92(0.74) 0.92(0.72) 0.67 0.42 0.42 0.42 Obs. 𝑓(sII)

0.63

Calc. 𝑓(sI)

0.12(0.30) 0.08(0.26) 0.08(0.28)

Obs. 𝑓(sI)

0.22

0.58

0.21

0.59

0.62 0.39 0.26 0.12

0.29

The calculated ultimate yields for the pure sII samples is good, once it is accepted that Samples 4a and 4b have hydrate barriers preventing full transformation. For the mixed sI/sII samples an asymptote of 40% significantly overestimates the amount of sII phase compared to sI. Increasing

2

The direction and number of arrows indicates that the observed value was not constant but appeared to be moving in the direction indicated. Sample 3a was not included as both the sI and sII hydrates are decomposing, that is, gas is being expelled from the cell which renders eq 3 invalid.



the asymptote to 50% gives much better values for the mixed hydrates. However, it must be kept in mind that the observed phase fractions for mixed sI/sII samples have not stabilized. Furthermore, it is known that significant amounts of liquid water pools at the bottom of the cell. Pooled water, as against the water coating the ice/hydrate structure, has a small surface area and makes no significant contribution to total hydrate growth. This limitation could be overcome by agitation of the sample, such as occurs in a stirred reactor. Though the three rules presented above are empirical it would be expected that they approximate the conditions that minimize the Gibbs energy of the system. This is true for Rule 2 and 3 as they reflect the lower energy of sII compared to sI except for very low concentrations of gaseous propane. It is then expected that Rule 1 should reflect the concentration of gaseous propane where sII is no longer is more stable than sI. CSMGem was used to calculate the sI-sIIwater-vapour quadruple point for methane/propane/water system, in this case for hydrogenous water rather than deuterated. At temperatures of 274.0 and 277.6 K the respective values are: pressures of 28.2 and 40.6 bar; molar propane concentrations of 0.030 and 0.040%; sII largecage propane occupancies of 13 and 14%. These propane occupancies are far less than our estimates of the asymptote. The discrepancy may be due to the limitation of our experimental method, or possibly inaccuracies in the CSMGem calculations for the case of very low propane concentrations. However, an alternative is that the homogenization of relative cage occupancy is incomplete and that some sII hydrate continues to contain more propane than the average value, or at least continues to do so over the timescale of our experiments. Using this assumption we require an average propane occupancy of 40-50% and a 13-14% occupancy on the outer growth layers so that they are in equilibrium with the free gas. The result would be a large variation in the relative cage occupancy stoichiometry of sII located in the inner and outer hydrate layers.


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Our experiments also provide information on the timescale of equilibration. For pure sII samples a significant change to propane cage occupancies must occur over a period of 45 minutes. By comparison, for mixed sI/sII samples the ratio of sI and sII phases is still changing significantly 2 hours after the end of the ice melting. The difference in speeds of the equilibration processes may have several reasons. Rule 3 does not apply for pure sII samples and the equilibrium is reached between the propane cage occupancy and gaseous-propane concentrations. For mixed sI/sII samples both Rules 1 and 3 apply, the equilibrium is between the sII propane cage occupancy, gaseous propane concentration, and phase fractions of the sI and sII phases. The more complex mechanism for mixed sI/sII samples will necessarily be slower than the simple equilibration of pure sII samples. However, it may also indicate that the change in propane cage occupancy for the pure sII phase is an intrinsically faster process than one involving conversion between sI and sII phases.

CONCLUSIONS We have argued that a similar effect to the rapid growth we observed during ice melting can explain why sII hydrate grown under high driving force in batch reactor experiments39,

42

contains less of the minority gas molecules than predicted by the thermodynamic model. For these cases it may be possible to incorporate this behavior in the flash-calculation method41 by changing the assumptions of the model. The layer by layer particle growth would be retained, as would the stoichiometry of each layer being in equilibrium with the reactor conditions when the layer was formed. However, it now becomes a quasi-static equilibrium with conditions at the liquid-hydrate interface rather than in the free gas. With this work we demonstrate the importance of the constantly changing gaseous concentration on the complex growth of mixed gas hydrates.



Though the variants of the shrinking core model have successfully described the growth rate of hydrates from ice particles14-16 we have found several difficulties with the model. In particular, the model in its current form does not include the possibility of heterogeneity and the complex interactions between multiple growth processes that can occur in mix gas hydrates. The limitations of our experiments suggest many improvements in the design of future experiments. Particularly, the ability to distinguish localized and delocalized growth and so determine the extent of spatial heterogeneity. The ability to measure directly the gaseous propane concentrations and/or the cage occupancies would also allow a more detailed analysis of the equilibration of propane denuding, temporal heterogeneity, departures from the shrinking-core models, and the physical reason behind the cessation of sII growth.

ACKNOWLEDGMENTS We would like to especially thank Dr Vanessa Peterson for her assistance in performing the Wombat experiments, and Prof. Garry McIntyre for his general advice, careful reading, and his hyphenations. We also thank the sample environment team (ACNS, ANSTO) for help with the design, approval and manufacturing of the pressure vessel and liquid coolant device. This project was jointly funded by CSIRO and ANSTO. Supporting Information. Experimental Configuration; Data Analysis; Estimate of f(sII) growth rate; Derivation of eqs 1-3; Equation S3. (PDF)

REFERENCES 1. Sloan, E. D.; Koh, C. Clathrate Hydrates of Natural Gases Third Edition, CRC Press: Boca Raton, FL, 2008. 2. Koh, C. A.; Sloan, E. D.; Sum, A. K.; Wu, D. T. Fundamentals and Applications of Gas Hydrates. Annu. Rev. Chem.Biomol. Eng. 2011, 2, 237-257. 3. Van der Waals, J.; Platteeuw, J. Clathrate Solutions. Adv. Chem. Phy. 2007, 2, 1-57.


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sII, Gas-hydrate Growth for a Methane

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