Energy Fuels 2010, 24, 5012–5022 Published on Web 08/20/2010
: DOI:10.1021/ef100560f
Formation Kinetics of Structure I Clathrates of Methane and Ethane Using an in Situ Particle Size Analyzer Faisal Al-Otaibi,† Matthew Clarke, Brij Maini, and P. R. Bishnoi* Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada. † Present address: Saudi Aramco, Post Office Box 5000, Dhahran 31311, Saudi Arabia. Received May 5, 2010. Revised Manuscript Received July 20, 2010
Experimental data on the kinetics of structure I (sI) clathrate formation were obtained for pure methane and pure ethane in a semi-batch stirred-tank reactor that is equipped with an in situ particle size analyzer. The particle size analyzer used is a Lasentec focused beam reflectance measurement (FBRM) probe capable of measuring particle chord lengths between 0.5 and 100 μm. Experiments were conducted at temperatures ranging from 274 to 282 K and with pressures ranging from 0.99 to 7.25 MPa. A novel procedure for cleaning the FBRM probe tip, which eliminated the need to remove the probe between runs, was developed. Experimental data on the pressure, temperature, and particle size distribution were analyzed using the approach of Clarke and Bishnoi (Clarke, M. A.; Bishnoi, P. R. Determination of the intrinsic kinetics of CO2 gas hydrate formation using in situ particle size analysis. Chem. Eng. Sci. 2005, 60, 695-709), with minor modifications, and intrinsic rate constants were obtained for methane and ethane in sI.
I (sI),5 structure II (sII),6,7 or potentially, structure H (sH).8 On their own, methane and ethane will form sI, which has a body-centered cubic structure with two cavities, a small cavity and a large cavity. Only methane is known to occupy both cavities, while the larger ethane molecule can only fit inside the larger cavity in a sI hydrate. Propane, being a larger molecule than ethane or methane, is not able to fit into either of the sI cavities, and thus, it forms a sII hydrate. sH hydrates,8 which have three cavities, can enclathrate bigger molecules still. However, to form sH, a helper gas (usually a small gas molecule, such as methane) is required in addition to the larger gas molecule. While the thermodynamics of gas-hydrate-forming systems has been studied extensively, it has only been in the last 2 decades that the kinetics of gas hydrate formation and decomposition has begun to receive comparable coverage in the literature. Vysniauskus and Bishnoi9,10 made the first measurements of the rate of methane hydrate formation, and in their study, the process was modeled as a homogeneous reaction. Following the work of Vysniauskas and Bishnoi,9,10 several researchers attempted to describe gas hydrate kinetics as a homogeneous reaction.11-14 The first work to recognize and quantitatively
Introduction Since their discovery by Sir Humphry Davy in the early 19th century, gas hydrates have gone from being a scientific curiosity to being of major importance to the energy industry. Gas hydrates, which belong to a class of non-stoichiometric crystalline inclusion compounds known as clathrates, are formed when water and certain gases are mixed at elevated pressures and low temperatures. Historically, the industry has been interested in devising ways to avoid the formation of gas hydrates in natural gas transmission lines. However, it has recently come to light that naturally occurring deposits of methane hydrates, which are found in the permafrost as well as under the ocean floor, may contain the world’s largest reserve of naturally occurring methane. The potential size of this naturally occurring methane reserve is made possible by the fact that gas hydrates have the ability to trap up to 160 m3 of methane (at standard conditions) in 1 m3 of gas hydrates. Being crystalline compounds, gas hydrates occur in regular crystal structures. Although several new structures for gas hydrates have been discovered in recent years,2-4 naturally occurring gas hydrates predominantly occur as either structure *To whom correspondence should be addressed. Telephone: 1-403220-6695. Fax: 1-403-284-4852. E-mail:
[email protected]. (1) Clarke, M. A.; Bishnoi, P. R. Determination of the intrinsic kinetics of CO2 gas hydrate formation using in situ particle size analysis. Chem. Eng. Sci. 2005, 60, 695–709. (2) Udachin, K. A.; Ripmeester, J. A. A complex clathrate hydrate structure showing bimodal guest hydration. Nature 1999, 397, 420–423. (3) Udachin, K. A.; Enright, G. D.; Ratcliffe, C. I.; Ripmeester, J. A. Structure, stoichiometry, and morphology of bromine hydrate. J. Am. Chem. Soc. 1997, 119, 11481–11486. (4) Udachin, K. A.; Ratcliffe, C. I.; Ripmeester, J. A. A dense and efficient clathrate hydrate structure with unusual cages. Angew. Chem., Int. Ed. 2001, 40, 1303–1305. (5) Claussen, W. F. Suggested structures of water in inert gas hydrates. J. Chem. Phys. 1951, 19, 259–261. (6) Jeffry, G. A.; McMullan, R. K. The clathrate hydrates. Prog. Inorg. Chem. 1967, 8, 43–108. (7) von Stackelberg, M.; M€ uller, H. R. Zur struktur der gashydrate. Naturwissenschaften 1951, 38, 456–461. r 2010 American Chemical Society
(8) Ripmeester, J. A.; Tse, J. S.; Ratcliffe, C. I. A new clathrate hydrate structure. Nature 1987, 325, 135–136. (9) Vysniauksas, A.; Bishnoi, P. R. A kinetic study of methane hydrate formation. Chem. Eng. Sci. 1983, 38, 1061–1072. (10) Vysniauksas, A.; Bishnoi, P. R. Kinetics of ethane hydrate formation. Chem. Eng. Sci. 1985, 40, 299–303. (11) Lekvam, K.; Ruoff, P. A reaction kinetics mechanism for methane hydrate formation in liquid water. J. Am. Chem. Soc. 1993, 115, 8565–8570. (12) Shindo, Y.; Lund, P. C.; Fujioka, Y.; Komiyama, H. Kinetics and mechanism of the formation of CO2 hydrate. Int. J. Chem. Kinet. 1993, 25, 777–782. (13) Teng, H.; Kinoshita, C. M.; Masutani, S. M. Hydrate formation on the surface of a CO2 droplet in high pressure, low temperature water. Chem. Eng. Sci. 1995, 50, 559–564. (14) Kvamme, B. A new theory for kinetics of hydrate formation. Proceedings of the 2nd International Conference on Natural Gas Hydrates; Toulouse, France, June 2-6, 1996; pp 139-146.
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not present a value for the rate constant for ethane hydrate formation. Bergeron and Servio23 revised the model of Englezos et al.15 in such a manner as to express the driving force in terms of interfacial mole fractions. Subsequently, they used their model to regress rate constants for propane hydrate,23 methane hydrate,24 and CO2 hydrate25 formation from experimental data that was obtained in their laboratory. The experimental apparatus of Bergeron and Servio23 used an ex situ particle size analyzer that measured particles in the range from 0.6 nm to 6 μm. An examination of the data of Bergeron et al.23 reveals that the number of counts, from the zetasizer, before hydrate nucleation is roughly the same magnitude as the number of counts after hydrate nucleation. Murshed and Kuhs26 used neutron powder diffraction to investigate the formation of hydrates formed from a mixture of methane and ethane at temperatures below the freezing point of water. It was found26 that, in their system, sI and sII phases formed at different rates. Oiliviera et al.27 performed a computational fluid dynamic study of hydrate formation in petroleum pipelines. Their study also uses the fugacity difference, as proposed by Englezos et al.,15 as the driving force for hydrate formation. Finally, Talaghat et al.28 compared the effect of using different equations of state on the driving force for hydrate formation that is computed from the model of Kashiev and Firoozabadi.17 The current work builds on the previous study of Englezos et al.,15 Sharma,20 and Clarke and Bishnoi.1 The kinetic studies on ethane and methane hydrates, which were originally performed by Englezos et al.1 without any measurement of the particle size distribution, have been performed using an in situ particle size analyzer to directly measure the hydrate particle size distribution during hydrate growth.
account for the fact that the growth of gas hydrate particles is a crystallization process was that by Englezos et al.,15,16 which was conducted in the same laboratory as the present study. Englezos et al.15,16 studied the formation kinetics of hydrates formed from methane, ethane, and their mixtures. In their model, the only adjustable parameter is the intrinsic rate constant of hydrate formation. In the work of Englezos et al.,15,16 however, it was not possible, because of equipment limitations, to measure the particle size distribution and, thus, assumptions had to be made regarding the particle size distribution during hydrate formation. Since the groundbreaking work of Englezos et al.,15 there has been a flurry of activity directed toward understanding the kinetics of gas hydrate formation, particularly in the last 10 years. In 2002, Kaschiev and Firoozabadi17 derived a model to quantify the driving force for gas hydrate formation from a pure gas. In their model, the driving force for hydrate formation was taken to be a chemical potential difference. Giavarini et al.18 used modulated differential scanning calorimetry to examine the rate of formation of propane hydrates, from both ice and liquid water, and they found that propane hydrates formed almost instantaneously with ice, whereas the hydrates formed at a much slower rate when water was present as a liquid. Lee et al.19 constructed a closed loop pipeline system to investigate the effect of flowing water on hydrate formation. Their study resulted in an empirical relationship that correlated the required formation temperature in terms of the system pressure and the velocity of the flowing fluid. The study of Sharma,20 which was conducted in the same laboratory as the present study, marked the first attempt to include particle size analysis in the study of gas hydrate formation kinetics. In his study, an external particle size analyzer was used to measure the particle size distribution during hydrate formation. After 9 years, Clarke and Bishnoi1 made the first attempt at incorporating an in situ particle size analyzer directly into a high-pressure reactor, from which the intrinsic rate constant of CO2 hydrate formation was regressed. Their particle size analyzer was capable of measuring chord lengths between 0.5 and 100 μm. In the work of both Sharma20 and Clarke and Bishnoi,1 the mathematical model of Englezos et al.15 was modified to allow for the inclusion of the measured particle size analysis data. Also, in 2005, Lee et al.21 measured the rate of sH hydrates that were formed from methane with either neohexane, tert-butyl methyl ester, or methylcyclohexane. Hussain et al.22 used a system with controlled cooling to study the rate of ethane hydrate formation. In their work, Hussain et al.22 acknowledge the equation of Englezos et al.15 as describing the process; however, they do
Experimental Section Apparatus. The apparatus used in the current study is essentially that which was used by Clarke and Bishnoi,1 with the addition of a new stirring system. A detailed description of that apparatus is given by Clarke and Bishnoi.1 Figure 1 shows a schematic of the apparatus, which consists of a semi-batch stirredtank reactor that is maintained at a constant temperature and pressure. The reactor is constructed from stainless-steel 316, and it is built to withstand a maximum operating pressure of 110 bar. The cavity volume of the reactor is 650 mL without the particle size probe and the stirrer, which are inserted in the reactor. The content of the reactor can be viewed from two sides via two Plexiglas windows held in place with Viton o-rings. The reactor is connected to four lines: a solution feed/flush or slurry withdrawal line at the bottom of the reactor, a slurry return line, and a flush/feed gas line connected to a three-way valve to allow sampling of the gas phase of the reactor when needed. The slurry
(15) Englezos, P.; Dholabhai, P.; Kalogerakis, N.; Bishnoi, P. R. Kinetics of formation of methane and ethane gas hydrates. Chem. Eng. Sci. 1987, 42, 2647–2658. (16) Englezos, P.; Dholabhai, P.; Kalogerakis, N.; Bishnoi, P. R. Kinetics of gas hydrate formation from mixtures of methane and ethane. Chem. Eng. Sci. 1987, 42, 2659–2666. (17) Kashchiev, D.; Firoozabadi, A. Nucleation of gas hydrates. J. Cryst. Growth 2002, 243, 476–489. (18) Giavarini, C.; Maccioni, F.; Santarelli, M. L. Formation kinetics of propane hydrate. Ind. Eng. Chem. Res. 2003, 42, 1517–1521. (19) Lee, J. H.; Baek, Y. S.; Sung, W. M. The kinetics on hydrate formation in pipelines. Energy Sources 2005, 27, 875–885. (20) Sharma, S. Gas hydrate particle size measurements. M.Sc. Thesis, University of Calgary, Calgary, Alberta, Canada, 1996. (21) Lee, J. D.; Susilo, R.; Englezos, P. Kinetics of structure H gas hydrates. Energy Fuels 2005, 19, 1008–1015. (22) Hussain, S. M. T.; Kumar, A.; Laik, S.; Mandal, A.; Ahmad, I. Study of the kinetics and morphology of gas hydrate formation. Chem. Eng. Technol. 2006, 29, 937–943.
(23) Bergeron, S.; Servio, P. Reaction rate constant of propane hydrate formation. Fluid Phase Equilib. 2008, 265, 30–36. (24) Bergeron, S.; Beltran, J. G.; Servio, P. Reaction rate constant of methane clathrate formation. Fuel 2010, 89, 294–301. (25) Bergeron, S.; Servio, P. Reaction rate constant of CO2 hydrate formation and verification of old premises pertaining to hydrate growth kinetics. AIChE J. 2008, 54, 2964–2970. (26) Murshed, M. M.; Kuhs, W. F. Kinetic studies of methane ethane mixed gas hydrates by neutron diffraction and raman spectroscopy. J. Phys. Chem. B 2009, 113, 5172–5180. (27) Oliviera, M. B.; de Castro, J. A.; da Silva, A. J. Study of hydrate formation kinetics in petroleum pipes by the phase field. Heat Transfer Eng. 2009, 30 (4), 309–315. (28) Talaghat, M. R.; Esmaeilzadeh, F.; Fathikalajahi, J. Effect of various types of equations of state on driving force calculation and gas consumption prediction in simple and double hydrate formation with or without presence of kinetic inhibitors in batch systems. Chem. Eng. Commun. 2010, 197, 584–605.
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Figure 1. Schematic of the experimental apparatus.
Figure 2. Reactor flange showing the mixer and FBRM probe.
withdrawal and return lines connect the apparatus to a Raman spectrometer, which was not used in studying the kinetics of pure methane and ethane hydrate formation. Two type-T thermocouples are used to measure the gas phase and the reactor solution temperatures. Since Clarke and Bishnoi’s work of 2005, the stirring system of the reactor has been changed from a magnetic stirring bar to an Autoclave Engineers Magna Drive mechanical mixer, which is inserted through the flange at the top of the reactor, as shown in Figure 2. In addition to the bottom shaft impeller, which agitates the solution, an additional impeller is added midway from the upper part of the shaft to agitate the gas phase of the reactor. Each impeller consists of six blades, with dimensions of 13.5 9 1 mm. Heating of the stirrer is prevented by flowing cooled glycol through the provided cooling jacket, and the agitation rate is controlled via a magnetic hall effect sensor by an electric-driven motor capable of more than 1000 rpm. Establishment of an Optimum Focal Length for the Focused Beam Reflectance Measurement (FBRM) Probe. Prior to beginning the experimental study, it was necessary to first optimize the FBRM probe for use with gas hydrates. The position of the focal point of the laser can be manually adjusted relative to the probe window. Hence, the viewing region can be moved further into the solution (þ) or behind the probe window (-). The focal
Figure 3. Large particle counts and the total number of particles of hydrates at the þ10 μm focal point position.
point can be set anywhere between -100 and þ300 μm. The default manufacturer setting is at -20 μm. Greaves et al.29 conducted a detailed investigation of the FBRM probe, from which they concluded that it is necessary to calibrate the FBRM probe if it is to be used for quantitative measurements. According to the operation manual of the FBRM probe, the probe window is considered clean when it measures less than 300 counts/s submerged in distilled water, but for applications that generate less than 1000 counts/s, much lower counts per second should be obtained for the probe window to be considered clean.30 The presence of one particle in the viewing range of the probe results in 75 counts/s at the laser scanning speed of 2 m/s. (29) Greaves, D.; Boxall, J.; Mulligan, J.; Montesi, A.; Creek, J.; Sloan, E. D.; Koh, C. A. Measuring the particle size of a known distribution using the focused beam reflectance measurement technique. Chem. Eng. Sci. 2008, 63, 5410–5419. (30) Hukkanen, E. J.; Braatz, R. D. Measurement of particle size distribution in suspension polymerization using in situ laser backscattering. Sens. Actuators 2003, 96, 451–459.
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Figure 4. Small particle counts and the total number of particles of hydrates at the þ10 μm focal point position.
optimal performance was achieved when the focal point is set closer to the window. Larger particles may possess some difficulty entering the viewing region as the focal point is brought back toward or inside the window. On the other hand, a particle will appear larger when it is far from the focal point33 because of the broadening effect of the laser beam. In view of the above, the optimum focal point position is somewhere closer to the probe window yet external, where hydrate particles are in the viewing region of the probe. Thus, the focal point is set at þ10 μm for all of the hydrate experiments carried out in this study. Finally, once the new focal length had been selected, the reproducibility of the FBRM probe was checked by measuring the chord length distribution of a sample of standard latex microspheres, whose chord length distribution was known. The FBRM was found to yield reliable and reproducible results with the new focal length. Procedure. The procedure of Clarke and Bishnoi1 was followed in forming the hydrates. However, significant revisions were made to the procedure for cleaning the FBRM probe. Upon examination of the work of Clarke and Bishnoi,1 it was recognized that there was a need to achieve a “quieter” initial reading on the particle size analyzer, and thus, a new procedure was developed to clean the FBRM probe without requiring that it be removed from the flange between every experiment. Upon examining the previous results, it was concluded that the high number of counts per second that appear as background noise in a clean water solution between hydrate formation experiments is not solely an instrument noise; rather, it is a combination of that and a remaining fine layer of very small gas bubbles left because of hydrate decomposition from a previous run. The probe can be removed from the reactor and cleaned as recommended by the manufacturer. Nonetheless, frequent removal and reinstallation of the probe are not desirable because of the high-pressure operating environment. The high number of counts after hydrate formation and decomposition experiments and the decrease in that number at the onset of a new hydrate growth experiment in the instances of high background noise triggered the idea that the hydrate formation experiment can be set in a way to clean the probe window without the need to physically remove it. The principle here is simple; the way hydrates are decomposed in the solution affects the cleanliness
The growth of hydrate particles during kinetics experiments is indicated by a sudden rise in the number of counts of particles per second. Although the instant of hydrate growth can be detected easily by use of the FBRM probe, the increase in the number of counts per second is not high enough to eliminate the background noise effect (the initial counts of particles per second) on the total counts of particles. Subtraction of the background noise can be challenging because it is rarely exactly constant. Thus, it is recommended to start with the minimum possible number of counts per second. During the initial experimental stage, it had been noticed that, after any hydrate formation experiment, the background noise (the number of counts of particles per second in distilled water solution) differs from one run to another and rises to a high number in the majority of cases. Adjustment of the focal point position of the FBRM probe was found to have a significant effect on the number of counts of particles per second. The effect of focal point positioning on the number of counts per second during a hydrate formation experiment was also investigated. If the focal length is too far away from the window, the signal-to-noise ratio is too low, and if the focal length is too close to the window, the boundary layer hydrodynamics around the probe tip may prevent the laser from seeing a representative sample. It is expected that, during hydrate growth, the number of counts per second of smaller particles increases sharply at the early stage and then decreases, while the number of counts per second of larger particles increases at a later stage. Figures 3-6 show the effect of two different focal positions (þ10 and þ300 μm) on the number of counts per second of different particle size ranges. The spikes in the readings in Figure 3 are likely due to particle breakage and agglomeration. The conclusion drawn here is that a closer position to the window provides the expected trend of hydrate particle counts. This is in total agreement with what has been reported in the literature.31 Both groups observed substantial changes on the number of counts per second with focal position, suggesting that setting the position further into the solution (in the range of 0.8-2 μm) gave better results for larger particles. However, Heath et al.32 suggested that the (31) Monnier, O.; Fevotte, G.; Hoff, C.; Klein, J. P. Model identification of batch cooling crystallizations through calorimetry and image analysis. Chem. Eng. Sci. 1997, 52, 1125–1139. (32) Heath, A. R.; Fawell, P. D.; Bahri, P. A.; Swift, J. D. Estimating average particle size by focused beam reflectance measurement (FBRM). Part. Part. Syst. Charact. 2002, 19, 84–95.
(33) Pons, M.; Miferstedt, K.; Morgenroth, E. Modeling of chord length distributions. Chem. Eng. Sci. 2006, 61, 3962–3973.
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Figure 5. Large particle counts and the total number of particles of hydrates at the þ300 μm focal point position.
Figure 6. Small particle counts and the total number of particles of hydrates at the þ300 μm focal point position.
data acquisition program are started and the temperatures are closely monitored. The data acquisition program reads temperatures and pressures once every 30 s and calculates the moles of free gas in the system, and the FBRM reads chord length distributions every 10 s. Once the temperatures have stabilized, the stirrer is started and the reactor is put on automatic control. During the entire experiment, cooled glycol flows through the cooling jacket of the stirrer to ensure that the stirrer is not adding heat to the solution in the reactor. As the gas in the reactor is consumed, isobaric conditions are maintained by feeding fresh gas from the supply reservoir. The reactor is monitored for the appearance of hydrates, which are marked by the solution becoming turbid and by a sharp change in the slope of the total number of particles on the FBRM display. The experiments are allowed to proceed for roughly 20 min or until agglomeration
of the probe window. Moreover, it was found that decomposing the hydrate at a very slow rate by decreasing the pressure of the system very slowly while stirring until a clear solution (no bubbles or hydrate particles) is noticed reduces the background noise. Thus, after the necessary data of hydrate growth are obtained, the pressure is very slowly lowered until the solution is clear. Once the solution is clear, the pressure is further lowered slowly to release the soluble gas from the solution. At this point, the water is flushed out and fresh water is introduced to the system. Once the reactor has been prepared, using the procedure described above, the reactor and supply cell bias reservoir (R2) are charged to a pressure roughly equal to the experimental pressure and the reactor bias reservoir (R5) and the gas supply cell (R1) are charged to a slightly higher pressure still. After the reservoirs and the reactor have been filled, the particle size analyzer and the 5016
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A detailed derivation of the above equations can be found elsewhere.15 In solving the above pair of differential equations, Clarke and Bishnoi1 used the measured particle size analysis data to obtain a value of the second moment, μ2. In the present work, an alternate approach for computing the second moment from the particle size analysis data, which is presented below, is used. Because the FBRM probe only detects particles immediately inside its measuring area, it is necessary to extrapolate that which is measured by the FBRM probe to the whole reactor volume. The sample that the probe can see is assumed to be a representative sample of the hydrate particles. From the FBRM data alone, it is not possible to directly compute the moments because the probe does not extrapolate its readings to the entire reactor volume. Thus, it is necessary to combine the readings of the FBRM probe with the temperature/ pressure readings to arrive at a value of the second moment. One can, however, compute what the authors refer to as “pseudo-moments”. The zeroth pseudo-moment is the total number of particles detected by the probe in a given reading and is found from the FBRM readings by
starts to occur. Agglomeration can be seen on the display of the FBRM probe as a sudden drop in the total number of particles. Preceding each hydrate formation experiment, solubility runs were also conducted using the same procedure as above but at a pressure slightly lower than the three-phase equilibrium pressure. By measuring the rate at which gas is absorbed, one is able to determine the value of the liquid side mass-transfer coefficient times the gas-liquid interfacial area per unit volume (this product will be referred to as the apparent dissolution rate constant from here on) and the concentration of the dissolved gas in equilibrium with the gas at the gas-liquid interface.
Theory The data obtained from the experiments were analyzed using the approaches developed by Englezos et al.15 and Clarke and Bishnoi,1 with a minor modification to the way that the particle size analysis data is incorporated into the model. It should be noted that the theoretical approach of Clarke and Bishnoi1 was itself a modification of the theoretical approach of Sharma.20 In their approach, Clarke and Bishnoi1 modified the equations derived by Englezos et al.15 to incorporate measured particle size analysis data. Englezos et al.15 presented a system of five coupled differential equations; two describe the surface reaction and diffusion of the hydrate-forming gas, and the remaining three predicted the second moment of the particle size distribution. Clarke and Bishnoi1 were able to remove the last three differential equations and, instead, used the experimental particle size analysis to determine the second moment. Englezos et al.1 established that the rate of gas hydrate formation is given by dnH ¼ K Ap ð f - feq Þ ð1Þ dt p
μ0 0 ¼
NC X
C i ðLi Þ0
ð7Þ
i¼1
where NC is the number of channels, C i is the average number of particle counts per unit time in the ith channel, and Li is the chord length or the upper boundary of a channel. The first pseudo-moment is the total length of all particles detected by the probe in a given reading and is found from the FBRM readings by μ1 0 ¼
where
NC X
C i ðLi Þ1
ð8Þ
i¼1
1 1 1 ¼ þ K kr kd
ð2Þ
The second pseudo-moment is the total surface area of all particles detected by the probe in a given reading and is found from the FBRM readings by
If the experiments are conducted at conditions such that the diffusion rate constant approaches infinity, then the measured rate constant is equal to the intrinsic rate constant. Englezos et al.1 subsequently invoked the two-film theory to describe the absorption of gas at the gas-liquid interface, and the following two differential equations were obtained:1 D γAðg- lÞ fðfg - feq Þcosh γ - ðfb - feq Þg dnH ð3Þ ¼ sinh γ dt yL
μ2 0 ¼
NC X
C i ðLi Þ2
ð9Þ
i¼1
The third pseudo-moment is the total volume of all particles detected by the probe in a given reading and is found from the FBRM readings by μ3 0 ¼
dfb ðH - feq Þ Dγa ¼ fðfg - feq Þ - ðfb - feq Þcosh γg Hcwo yL sinh γ dt 2
NC X
C i ðLi Þ3
ð10Þ
i¼1
ð6Þ
Because the instrument gives a particle chord length distribution, which is a function of the true particle diameter distribution, a conversion method is needed to obtain the actual particle size. Many researchers have proposed methods to convert the chord length distribution to the actual particle size distribution.30,32,34 For simplicity, it is assumed that the hydrate particles are spherical and the simple conversion method in which the chord length can be divided by a factor of 0.791 to estimate the diameter is used.33 To calculate the volume of particles in a given reading, it is assumed that (i) the particles are spherical, (ii) the probe sees all of the particles in the
The mass-transfer coefficient and Henry’s constant in eqs 5 and 6 are obtained from solubility experiments, and the details of such an experiment can be found in the study by Englezos et al.15
(34) Wynn, E. J. W. Relationship between particle-size and chordlength distributions in focused beam reflectance measurement: Stability of direct inversion and weighting. Powder Technol. 2003, 133, 125–133.
πK μ2 ðH - feq Þ2 ðfb - feq Þ Hcwo where D γ ¼ kL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s πK μ2 D
Dcwo D ¼ H
ð4Þ
ð5Þ
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Figure 8. Number of moles of ethane in the hydrate phase during hydrate formation at 274 K and 0.99 MPa.
Figure 7. Effect of the stirring rate on the combined rate constant, K*, of ethane hydrate formation at 276 K and 1.18 MPa.
sampling volume, and (iii) the probe does not detect any gas bubbles. At a given time, the volume of particles is simply the sum of the particle volumes. Thus, the volume of the hydrate particles in the viewing region of the probe can be calculated from π 0 1 3 ð11Þ VH, FBRM ¼ μ3 6 0:791 For the sake of notational simplicity, let B be defined as the second pseudo-moment divided by the hydrate particle volume computed in the above equation or B ¼
μ2 0
3 π 0 1 μ3 6 0:791
ð12Þ Figure 9. Number of moles of ethane in the hydrate phase during hydrate formation at 276 K and 1.08 MPa.
The total volume of hydrate particles in the total reacting mass is computed from the temperature/pressure measurements as VH ¼ nH
MH FH
ð13Þ
The molecular mass and density are computed by the method described by Clarke and Bishnoi.35 By combining the previous two equations, the actual second moment, which is the total surface area of particles per unit volume of slurry, is written as μ2 ¼
BVH Vslurry
ð14Þ
Results and Discussion Elimination of Heat- and Mass-Transfer Resistances. The data obtained from a typical experiment consists of the pressure and temperature measurements obtained from the data acquisition unit plus the data from the particle size analyzer. From the temperature/pressure data, the number of moles in the gas phase at time t is computed using the Trebble-Bishnoi equation of state (TB EOS).36 At any time
Figure 10. Number of moles of ethane in the hydrate phase during hydrate formation at 279 K and 1.34 MPa.
t, the moles of gas that have been consumed is the difference between the moles of gas that were initially present minus the moles of gas present at time t or nH ðtÞ þ nL ðtÞ ¼ n0 - nG ðtÞ
(35) Clarke, M. A.; Bishnoi, P. R. Measuring and modelling the rate of decomposition of gas hydrates formed from mixtures of methane and ethane. Chem. Eng. Sci. 2001, 56, 4715–4724. (36) Trebble, M.; Bishnoi, P. R. Development of a new four-parameter equation of state. Fluid Phase Equilib. 1987, 35, 1–17.
ð15Þ
where nH(t) is the moles of gas in the hydrate phase at time t, n0 is the moles of gas initially present in the gas phase, nG(t) is 5018
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the moles of methane in the gas phase at time t, and nL(t) is the moles of gas absorbed into the liquid phase. Prior to the turbidity time, nH(t) is 0 and all of the methane that is consumed are in the liquid phase. After the turbidity time, the moles of hydrate former that have gone into the liquid phase, nL(t), are found using the measured value of Henry’s constant and the three-phase equilibrium fugacity, calculated from the TB EOS. As noted elsewhere,1,15 the constant K* is obtained from the experimental data by minimizing the sum of the square of the errors between the measured num-
ber of moles consumed and the computed moles of gas consumed during an experiment. As noted in the Theory, the rate constant in eq 1 is an overall rate constant that contains contributions from both mass-transfer and kinetic resistances. Under conditions of increased agitation, the mass-transfer rate constant, kr, approaches infinity, and thus, the overall reaction rate constant is equal to the intrinsic rate constant. To find the optimum value of the agitation rate, experiments were conducted at varying rates of agitation and the overall rate constant is computed at each rate. Englezos et al.15 concluded that a plateau in the graph of K* versus revolutions per minute (rpm) is indicative of the mass-transfer resistance having been eliminated, and in Figure 7, it is seen that the rate constant sharply increases from 500 to 700 rpm and then levels off between 700 and 800 rpm. Gas entrainment was likely not an issue at 700 rpm because there were very few bubbles that were observed and because it was observed that the FBRM counts prior to hydrate formation are roughly the same before and after the stirrer has started. For subsequent experiments, 700 rpm was chosen as the stirring rate. On the basis of previous work by the likes of Englezos et al.,15 it was expected that K* versus rpm would be qualitatively similar to that of ethane. Thus, it was felt that additional experiments were not required. Results from Experiments with Ethane. Experiments for ethane gas hydrate formation were carried out at temperatures ranging from 274 to 282 K and pressures ranging from 0.99 to 1.6 MPa to determine the intrinsic rate constant of ethane hydrate formation at each isotherm. The results are shown in Figures 8-11. In these figures, the points are the experimental data points and the solid lines are the computed values from the model. The computed values by the model fit the experimental data well. The regressed intrinsic rate constants for each isotherm are tabulated in Table 1. These values are 4 orders of magnitude higher than those reported by Englezos et al.,15 without removing the inconsistency resulting from representing the particle size by its radius rather than its diameter. This implies that the measured interfacial area of all hydrate particles is less than what was estimated by
Figure 11. Number of moles of ethane in the hydrate phase during hydrate formation 282 K and 1.59 MPa. Table 1. Intrinsic Rate Constant of Ethane Hydrate Formation in sI Determined in This Work Using FBRM
temperature (K)
K* = kr (mol m-2 MPa-1 s-1) present study
K* = kr (mol m-2 MPa-1 s-1) Sharma20
274 276 279 282
0.9987 1.071 0.7944 0.9467
0.21 0.17 0.24
Figure 12. Measure of particle counts during ethane hydrate formation at 274 K and 0.99 MPa.
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Figure 13. Number of moles of methane in the hydrate phase during hydrate formation at 276 K and 4.65 MPa.
Figure 14. Number of moles of methane in the hydrate phase during hydrate formation at 279 K and 5.16 MPa.
Englezos et al.15 In Figure 12, the particle counts, as a function of time, are presented for ethane hydrates at 274 K and 0.99 MPa. It can be seen that, once turbidity has occurred, there is a sharp increase in the particle counts, which is believed to correspond to the initial primary nucleation. Following this brief phase, the slope changes and the number of hydrate particles increases linearly. The spikes in Figure 12 are believed to correspond to hydrate particle breakage. For the sake of comparison, one run of ethane gas hydrate formation at 274 K (from the present set of experiments) was analyzed following the same procedure that was used by Englezos et al.,15 without incorporating the actual particle size measurements carried out herein. The intrinsic rate constant thus obtained is identical to the value obtained by Englezos et al.,15 which was 0.12 10-4 mol m-2 MPa-1 s-1. It should be stressed that, in the decades that have elapsed since the work of Englezos et al.,15 the hydrodynamics of the reactor have been greatly changed because of the addition of the mechanical stirrer and the FBRM probe and the removal
of the baffles. Thus, the observation that the regressed rate constants of Englezos et al.15 are so close to the rate constant that is obtained from treating the current experimental data as was done by Englezos et al.,15 despite the vastly different hydrodynamics in the two studies, leads the authors to conclude that the rate constants obtained in the present study are the true intrinsic rate constants, which are independent of the hydrodynamics and geometry of the reactor. It is worth noting that the intrinsic rate constants of ethane hydrate formation determined herein have the same order of magnitude as those determined in the work of Sharma,20 which are also given in Table 1. The intrinsic rate constants determined in the current study are, however, slightly higher than those reported by Sharma.20 The measured surface area by Sharma20 is likely larger because the particle size measurements by Sharma20 were conducted outside the reactor in a sampling loop, where breakage of particles could possibly occur. Hence, the higher values of the intrinsic rate constants obtained in the present study are justifiable. 5020
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Figure 15. Number of moles of methane in the hydrate phase during hydrate formation at 282 K and 7.25 MPa.
particle size analyzer, to determine the intrinsic rate constants for sI gas hydrates formed from pure methane and pure ethane. The focal length of the FBRM probe was optimized for operations with high-pressure gas hydrate systems, and a novel procedure for cleaning the FBRM probe, which ensures a near zero baseline, was formulated. From the experimental results, a slightly modified version of the approach of Clarke and Bishnoi1 was used to extract the intrinsic rate constants for ethane and methane gas hydrate formation and it was found that the regressed results compare favorably to the results of Sharma.20 Finally, when the data from the present study were analyzed with the approach of Englezos et al.,15 it was concluded that the rate constants obtained in the present study are the true intrinsic rate constants.
Table 2. Intrinsic Rate Constant of Methane Hydrate Formation in sI Determined in This Work Using FBRM
temperature (K) 274 276 278 279 282
K* = kr (mol m-2 MPa-1 s-1) present study
K* = kr (mol m-2 MPa-1 s-1) Sharma20
0.0654
0.06 0.10 0.18
0.0812 0.0603
The intrinsic rate constants determined in this work using a particle size analyzer do not show a strong dependence upon the temperature within the temperature range studied. Results from Experiments with Methane. Ultra-pure methane gas (99.97 mol %) was used to perform hydrate formation experiments at temperatures ranging from 276 to 282 K and pressures ranging from 3.2 to 7.3 MPa to determine the intrinsic rate constant of methane hydrate formation at each temperature. The computed values by the model matched the experimental data well, as illustrated in Figures 13-15. The regressed intrinsic rate constants for each temperature are given in Table 2, along with those of Sharma.20 As was the case with ethane, the rate constants obtained in the present study are roughly 4 orders of magnitude higher than those obtained by Englezos et al.,15 but the determined rate constants from this study compare favorably to those reported by Sharma.20 Again, one methane hydrate run was analyzed following the same procedure that was used by Englezos et al.,15 without incorporating the actual particle size measurements carried out herein, and the same intrinsic rate constant, 0.65 10-5 mol m-2 MPa-1 s-1 at 274 K, was reproduced. Thus, the rate constants obtained in the present study are true intrinsic rate constants, which are independent of the hydrodynamics and geometry of the equipment.
Acknowledgment. The authors acknowledge the generous funding from Saudi Aramco, the COURSE program of the Alberta Energy Research Institute (AERI), and the Natural Sciences and Engineering Research Council of Canada (NSERC).
Nomenclature a = gas-liquid interfacial area per unit volume (m2/m3) A(g-l) = gas-liquid interfacial area (m2) Ap = surface area of a particle (m2) cwo = initial concentration of water molecules (mol/m3) D = diffusivity of the gas (m2/s) f = fugacity (MPa) fb = fugacity of the hydrate former in the bulk liquid phase (MPa) feq = equilibrim fugacity (MPa) fg = fugacity of the hydrate former in the vapor phase (MPa) H = Henry’s constant (MPa) K* = combined rate parameter (mol m-2 MPa-1 s-1) kd = mass-transfer coefficient for the liquid phase (mol m-2 MPa-1 s-1) kL = liquid side mass-transfer coefficient at the gas-liquid interface (mol m-2 MPa-1 s-1). kr = reaction rate constant (mol m-2 MPa-1 s-1) L = characteristic length (m) MH = mass (kg) of hydrate per mole of gas
Conclusion Experiments were conducted in an isothermal/isobaric semi-batch stirred-tank reactor, equipped with an in situ 5021
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nH(t) = number of moles of gas in the hydrate phase P = pressure (MPa) rpm = revolutions per minute R = universal gas constant (8.314 J mol-1 K-1) Ry(t) = global rate of reactions (mol m-3 s-1) T = temperature, K. t = time (s) V = volume of the reaction mass (m3)
yL = thickness of the gas-liquid interface (m) z = compressibility factor Greek Letters μn = nth moment of the particle size distribution (mn/m3) μn0 = nth pseudo-moment of the particle size distribution (mn/s) F = density (kg of hydrate/m3 of hydrate)
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