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Stoichiometric Approach toward Modeling the Decomposition Kinetics of Gas Hydrates Formed from Mixed Gases Carlos Giraldo† and Matthew Clarke*,‡ †

Three Streams Engineering, 401 1925 18th Avenue Northeast, Calgary, Alberta T2E 7T8, Canada Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada

‡

ABSTRACT: Following a recent study by Giraldo et al. (Giraldo, C.; Maini, B.; Bishnoi, P. R. A simplified approach to modeling the rate of formation of gas hydrates formed from mixtures of gases. Energy Fuels 2013, 27, 1204−1211), in which a new stoichiometry-based approach was taken to model the rate of gas hydrate formation for mixed gases, a new stoichiometric approach has been proposed to describe the kinetics of gas hydrate decomposition from mixed gases. Similar to the model of gas hydrate formation by Giraldo et al., the newly proposed model of gas hydrate decomposition has the advantage that the intrinsic rate constant of gas hydrate decomposition is only required for a single component. To test the model, gas hydrate decomposition experiments were conducted in a semi-batch stirred-tank reactor, equipped with an in situ particle size analyzer, to study the rate of decomposition of gas hydrates formed from mixtures of carbon dioxide and methane. Two gas mixtures of CO2 (1) + CH4 (2), one with x1 = 0.4 and the other with x1 = 0.6, were used for the current study. The experimental temperatures ranged from 274 to 276 K, and the experimental pressures ranged from 16 to 22 bar. The new approach was used to model the kinetics of gas hydrate decomposition, and the results from these predictions were compared to the results obtained using the model by Clarke and Bishnoi (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 and Clarke, M. A.; Bishnoi, P. R. Determination of the intrinsic rate constant and activation energy of CO2 gas hydrate decomposition using in-situ particle size analysis. Chem. Eng. Sci. 2004, 59, 2983−2993). The root-mean-square of the relative errors between the predictions of the new model and the model by Clarke and Bishnoi are 4.15 and 6.21%, respectively. Further validation of the new model was performed using it to fit the data by Clarke and Bishnoi on the decomposition of both sI and sII gas hydrates formed from mixtures of CH4 (1) + C2H6 (2). When applied to these data sets, the root-mean-square of the relative errors between the predictions of the new model and the model by Clarke and Bishnoi are 2.58 and 1.60%, respectively.

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INTRODUCTION When water and small gaseous molecules, including the likes of light hydrocarbons and CO2, are contacted at elevated pressure and low temperatures, ice-like solids can form in the solution. These crystalline ice-like solids, in which small molecules occupy almost spherical holes in ice-like lattices made up of hydrogenbonded water molecules, are known as gas hydrates, and they belong to a class of crystalline solids known as clathrates.4−7 What makes gas hydrates of particular interest to the petroleum industry and to academia is, first, the fact that their formation in pipelines and process equipment can completely block the flow of fluids and, second, the fact that naturally occurring deposits of gas hydrates underlying the ocean floor and permafrost are believed to be a vast storehouse for methane gas.8 One estimate8 places the inventory of methane hydrates in marine sediments to be in excess of 455 Gt. In particular, it is the ability of the gas hydrates to store up to 160 m3 of methane per 1 m3 of hydrate, at standard conditions, that makes them such a vast resource of methane. While it is generally believed that naturally occurring methane hydrates contain more methane than all other conventional sources combined,9 the extreme conditions under which they are found and the fact that they tend to be found in dispersed deposits have been some of the factors that has delayed the development of large-scale natural gas production from gas hydrates. In addition, the production of natural gas from hydrates © 2013 American Chemical Society

may also be complicated by the fact that gas hydrate deposits provide mechanical strength to the soil, 10 and hence, decomposing gas hydrates in situ may have the effect of weakening the surrounding soil matrix.10 One possible remedy10 to this dilemma may be to inject carbon dioxide into the naturally occurring methane hydrate deposits and, in doing so, cause a direct displacement of methane with carbon dioxide. Thus, the solid hydrate structure that provides strength for the soil is maintained, while at the same time, methane is produced and carbon dioxide is sequestered. In addition, as noted by Schicks et al.,11 this proposed process could provide a means for CO2 sequestration. The potential field-scale application of producing methane from naturally occurring gas hydrate deposits while injecting carbon dioxide will likely involve several mechanisms: heat transfer, flow through porous media, mass transfer, and the intrinsic kinetics of gas hydrate decomposition. While it is likely that, at the field scale, all of these mechanisms occur simultaneously and that one mechanism will be controlling, the purpose of the current work is to examine the kinetics of gas hydrate decomposition in isolation from the other mechanisms. Received: March 5, 2013 Revised: June 26, 2013 Published: July 5, 2013 4534

dx.doi.org/10.1021/ef400389s | Energy Fuels 2013, 27, 4534−4544

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existing data by Clarke and Bishnoi2 on the decomposition of gas hydrates formed from CH4 (1) + C2H6 (2).

The experimental study of gas hydrate decomposition kinetics is a relatively new area of research. The first attempt at quantifying the kinetics of gas hydrate decomposition was attempted by Kim et al.12 In their work, methane gas hydrates were decomposed at a constant temperature and pressure. Because Kim et al.12 did not have access to particle analysis technology, the initial particle size was estimated from settling times and Stokes law and it was assumed that all particles had the same diameter before decomposition. In the following decade, Clarke and Bishnoi2,3,14 studied the rate of decomposition of gas hydrates formed from methane, ethane, and their mixtures. A significant contribution of the aforementioned study was the inclusion of particle size analysis, albeit ex situ. Kawamura et al.15 measured the dissociation rates of hydrate pellets formed from mixtures of methane and ethane in both pure water and a synthetic drilling mud. Their results indicated that the free gas composition around the dissociation surface is determined by kinetics and not the equilibrium thermodynamics of dissociation and that the dissociation rates of gas hydrates in the synthetic drilling fluid are essentially proportional to the concentration of the fluid. Shortly thereafter, Clarke and Bishnoi3 modified their apparatus by adding an in situ particle size analyzer to study the decomposition kinetics of carbon dioxide gas hydrates in pure water. While it was still not possible, because of gas bubbles, to measure hydrate particle size distributions during decomposition, Clarke and Bishnoi3 were able to measure experimentally the particle size distribution in situ at the beginning of gas hydrate decomposition. In 2008, Giavarini et al.16 examined the influence of small amounts of tetrahydrofuran (THF) on the rate of decomposition of methane gas hydrates. Giavarini et al.16 found that the rate of gas hydrate decomposition is substantially lowered because of the addition of small amounts of THF. Nihous et al.17 examined the decomposition of relatively large, synthetic, gas hydrate plugs in the presence of methanol. The modeling of their experimental results suggested that, in their system, the inclusion of the intrinsic kinetics of gas hydrate decomposition might not be applicable to situations when hydrates are not decomposed by depressurization. Daraboina et al.18 examined the rate of gas hydrate decomposition, for hydrates formed from a mixture of light hydrocarbons, in the presence of several kinetic hydrate inhibitors. It was observed that the gas release curve that was obtained during decomposition varied noticeably depending upon which kinetic hydrate inhibitor was being used. As outlined in the previous paragraphs, only a small number of studies have made observation of the kinetic behavior of gas hydrate decomposition. However, at the time of writing, none of them had examined the decomposition kinetics of gas hydrates formed from a mixture of methane and carbon dioxide. In fact, as previously noted, the intrinsic rate constant for gas hydrate decomposition is currently only available for methane and ethane in both sI and sII2,14 and for carbon dioxide in sI.3 In the current work, which is a sister work to the study by Giraldo et al.1 on gas hydrate formation kinetics, a new model for the decomposition kinetics of gas hydrates formed from mixed gases is presented. Similar to the model by Giraldo et al.1 for gas hydrate formation kinetics, the newly proposed model for gas hydrate decomposition kinetics relies on the explicit inclusion of the gas hydrate stoichiometry to produce a model that is relatively simple yet produces accurate predictions. The model is subsequently tested against newly obtained data on the decomposition of gas hydrates formed from CO2 (1) + CH4 (2) as well as against the

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EXPERIMENTAL APPARATUS, MATERIALS, AND PROCEDURE

To validate the new gas hydrate decomposition model, which will be presented in a subsequent section, experimental gas hydrate decomposition kinetic data were obtained for two mixtures of CH4 (1) + CO2 (2). In addition, the data by Clarke and Bishnoi2 were also used for further validation of the new kinetic model. The apparatus and procedure for the two studies were almost identical, with two major exceptions: Clarke and Bishnoi2 used an externally connected particle size analyzer and a magnetic stirring rod for agitation, whereas the current study uses an in situ particle size analyzer and a magnetic agitator for stirring. The apparatus and procedure that will be outlined in the following section are valid for the newly obtained data. Readers that are interested in the apparatus, materials, and procedure used by Clarke and Bishnoi2 are referred to that paper.2 Apparatus and Materials. The experimental apparatus used in the current study was the same constant temperature/constant pressure semi-batch stirred-tank reactor that was used by Clarke and Bishnoi,3 and a detailed equipment diagram is available in that paper.3 The heart of the apparatus is a semi-batch stirred-tank reactor that can maintain a near-constant pressure and temperature and whose inner cavity volume is approximately 500 mL. In a typical experiment, the reactor is filled with 270 mL of water. The reactor pressure is maintained at a near constant value by controlling the flow rate of gas from the reactor to a gas collection reservoir, and the reactor temperature is maintained at a near constant value by immersing it in a large refrigerated glycol bath. The differential pressure transducer that is connected to the gas collection reservoir (Alphaline pressure transmitter, Rosemount Instruments, Ltd., Calgary, Alberta, Canada) has a span of 1 MPa, and the differential pressure transducer that is connected to the reactor (Alphaline pressure transmitter, Rosemount Instruments, Ltd., Calgary, Alberta, Canada) has a span of 2 MPa. The stated uncertainty in the pressure measurement by the manufacturer is ±0.25% of the full span. Type-T thermocouples, whose measurement uncertainty is ±0.05 K, are used to measure the gas-phase temperature in the reactor and in the gas collection reservoir. A data acquisition unit (National Instruments FPTC-120 for the thermocouple and FP-AI-110 for the pressure transducer) is used to transmit voltage signals from the differential pressure transducer and from the thermocouples to a personal computer, via a National Instruments FP-1601 Ethernet interface, where it is subsequently logged. The one difference between the apparatus as used in the current study versus that used by Clarke and Bishnoi3 is the stirring mechanism in the reactor; Clarke and Bishnoi3 agitated the reactor contents with a magnetic stir bar, whereas for the current study, the stir bar has been replaced by an Autoclave 075MagneDrive magnetic agitator. For the analysis of the reactor gas phase, an SRI-8640 gas chromatograph (GC) model integrated with the Peak Simple II software was used to analyze the molar composition of the reactor gas phase during the hydrate formation. The GC is equipped with a capillary column (Porapak Q column) that separates the injected gases. The column had a length of 15 cm and an inner diameter of 0.530 mm. The signals of the separated gases were measured by a thermal conductivity detector (TCD). The carrier gas is helium (purity ≥ 99.999%), and it was purchased from Praxair Technology, Inc. Finally, the particle size analysis was performed with an in situ focused beam reflectance method (FBRM) probe (Lasentec model D600), which provides a transient chord length distribution of particles. The FBRM probe can measure chord lengths between 0.5 and 1000 μm. The gas mixtures that were used for the current study were two different gas mixtures of CO2 (1) + CH4 (2); one mixture was x1 = 0.4, with a reported uncertainty in composition by the supplier of ±0.8 mol %, and the other was x1 = 0.60, with a reported uncertainty in composition by the supplier of ±1.0 mol %. In addition, deionized water (Millipore Simplicity water purification system, which produces 4535


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ultrapure water with a resistivity of 18.2 MΩ at 298 K) was used for all experiments. Experimental Procedure. As previously mentioned, the current set of experiments was performed in conjunction with the work by Giraldo et al.,1 and details of the procedure that was used for forming the gas hydrates are given in that publication. Following the formation of gas hydrates, the decomposition of the hydrates proceeds in a two-step manner, identical to that used by Clarke and Bishnoi.2,3,14 In this procedure, the pressure in the reactor is first reduced to an intermediate pressure. The intermediate pressure, which is roughly 50 kPa above the three-phase equilibrium point, is called the “stabilization pressure”, and it allows for dissolved gas to be liberated from the solution prior to decomposing the hydrates. At a pressure that is only 50 kPa above the three-phase equilibrium value, it is expected that, because the driving force for hydrate particle growth is so low, the particle size distribution will essentially remain constant during this period. Thus, the stabilization period also allows for the measurement of the chord length distribution at the beginning of the decomposition. Also, during the stabilization period, the gas-phase composition is measured via gas chromatography. The stabilization period lasts until the number of particle counts, as displayed by the FBRM probe, remains constant. The constant value of particle counts is taken as an indication that gas is no longer being liberated from solution. Once the stabilization period has concluded, the pressure is further lowered to a value that is approximately 10% below the three-phase equilibrium pressure. It is important to note that the final pressure cannot be too far below the three-phase equilibrium pressure, otherwise Joule Thomson cooling of the gas that is escaping from the hydrate phase will induce the formation of ice,19 which would impede the decomposition of the hydrates. It should also be noted that, for experiments dealing with the decomposition kinetics of gas hydrates, the FBRM probe is unable to differentiate between gas hydrate particles and bubbles. Thus, it is not possible to monitor the particle size distribution once the gas hydrates have begun to decompose. The decomposition experiments typically last for 10 min or until no more hydrates are visible to the naked eye. At the end of the experiment, a final gas-phase composition measurement is taken.

In the current study, the logic that Giraldo et al.1 applied in modeling the formation of gas hydrate, from gas mixtures, will be extended to modeling the kinetics of gas hydrate decomposition. Giraldo et al.1 wrote the following generalized formula for a hydrate that has been formed from a gas mixture

THEORY In 2001, Clarke and Bishnoi2 extended the ideas of Kim et al.12 for use in modeling the decomposition kinetics of gas hydrates formed from mixed gases. In their model, Clarke and Bishnoi2 wrote that the total rate of gas liberation, during hydrate decomposition, is the direct sum of the rate of gas liberation of each hydrate-forming compound or

where

ω1M1ω2M 2ω3M3 , ..., ωNC MNCωw H 2O

where ωi values (i = 1, ..., NC) are the stoichiometric coefficients of each hydrate-forming compound, in the hydrate phase, Mi values (i = 1, ..., NC) are the hydrate-forming compounds, and ωw is the stoichiometric coefficient of water in the hydrate. Further, Giraldo et al.1 defined the component with the subscript 1 as the “reference component”. As will be shown, the introduction of a reference component will simplify the mathematics. Thus, the rate of gas liberation of component i may be related to the rate of consumption of the reference component as

ωi ⎛ dn ⎞ ⎛ dn ⎞ ⎜ ⎟ = ⎜ ⎟ ⎝ dt ⎠ ω1 ⎝ dt ⎠1

NC

⎛ dn ⎞ −⎜ ⎟ = ⎝ dt ⎠ p

NC

⎛

j=1

dt

⎛ ωj ⎞⎛ dn ⎞ ⎟⎜ ⎟ = A p ⎝ ⎠ j = 1 ⎝ ω1 ⎠ dt 1 NC

⎞

∑ ⎜⎝ dn ⎟⎠

= j

∑⎜

⎛ ωj ⎞ ⎟(f ̂ − fĝ )1 ⎝ ω1 ⎠ eq

NC

∑ Kd,1⎜ j=1

(4)

Expanding the summation ⎤ ⎡ω ω ⎛ dn ⎞ ω −⎜ ⎟ = A pKd,1(feq̂ − fĝ )1⎢ 1 + 2 + 3 + ...⎥ ⎝ dt ⎠ p ω1 ω1 ⎦ ⎣ ω1 = A pKd,1(feq̂ − fĝ )1ω

NC

ω=

∑ j=1

(5)

ωj ω1

(6)

For this work, the stoichiometric coefficients will be assumed to remain constant throughout the decomposition process. The main reason for invoking this assumption was due to the fact that it was not possible to determine these values experimentally with the current laboratory equipment. This assumption may be revisited in a future study. Integrating eq 5 over all particle sizes yields

NC

⎛ dn ⎞ ̂ ⎟ = A ∑ K (f ̂ p d, j eq − fg )j ⎠j d t j=1 j=1

∑ ⎜⎝

(3)

Inserting eq 3 into eq 1 gives

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⎛ dn ⎞ −⎜ ⎟ = ⎝ dt ⎠ p

(2)

(1)

where NC is the number of hydrate-forming gases and Kd,j is the intrinsic rate constant of gas hydrate decomposition for component j. A detailed explanation of the mechanism involved in the development of eq 1 is given by Clarke and Bishnoi.13 In addition to using an unweighted summation, the solution of eq 1 also requires that the intrinsic rate constants of gas hydrate decomposition be known for all of the hydrate-forming compounds in a gas mixture. Recently, Giraldo et al.1 observed that the modeling of gas hydrate formation from a gas mixture could be simplified by explicitly accounting for the hydrate stoichiometry. This technique slightly improved the accuracy of the predictions of the kinetics of gas hydrate formation. However, this technique also removed the need to know the intrinsic rate constant for gas hydrate formation for each hydrate-forming component in a gashydrate-forming mixture.

R y (t ) =

∫0

R y (t ) = −

∞ ⎛ dn ⎞

∫0

⎜

⎟

⎝ dt ⎠ p ∞

φ(L , t )dL (7)

A pKd,1(feq̂ − fĝ )1ωφ(L , t )dL

= −Kd,1(feq̂ − fĝ )1ω

π ψ

∫0

∞

L2φ(L , t )dL

(8)

The quantity within the integral is the second moment of the particle size analysis. Thus, eq 8 can be rewritten as π R y(t ) = − Kd,1ωμ2 (t )(feq̂ − fĝ )1 ψ (9) Equation 9 is the global rate of reaction, and if it is assumed that the volume of the reacting mass remains constant, it can be 4536


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proposed model was performed using it to fit the experimental gas hydrate decomposition data of Clarke and Bishnoi,2 which includes both sI and sII hydrates. Results from a Typical Experiment. During the experiments, the temperature and pressure were recorded and the number of moles of hydrate-forming gas in the gas phase was computed using the Peng−Robinson equation of state. The chord length distribution of the hydrate particles was measured in situ, just prior to the onset of gas hydrate decomposition. Finally, the gas-phase composition was analyzed at the onset of decomposition as well as at the end of the experiments. As was the case in the formation kinetics study by Giraldo et al.,1 the range of temperatures over which experiments could be conducted was only relatively small; the lower boundary was the freezing point of water (the apparatus is not equipped to operate below 273 K), and the upper boundary was approximately the intersection of the CO2 vapor pressure curve with the pure methane hydrate curve or 277 K. For each experiment, the pressure in the reactor was kept at a value that is approximately only 50 kPa below the three-phase equilibrium pressure, at the given temperature. This was chosen to avoid the gas hydrate selfpreservation effect.19 Table 1 summarizes the experimental

related to the overall rate of gas liberation, during decomposition as

R y (t ) =

1 ⎛⎜ dn ⎞⎟ V ⎝ dt ⎠ H

(10)

or ⎛ dn ⎞ π ⎜ ⎟ = − Kd,1ωVμ2 (t )(feq̂ (t ) − fĝ (t ))1 ⎝ dt ⎠ H ψ

(11)

In eq 11, the fugacity difference has been explicitly written as a function of time because both the temperature and the gas-phase composition can vary with time. However, because of the relatively short duration of the experiment, it is usually only possible to obtain gas-phase composition measurements at the onset of decomposition as well as at the end of the experiments. In this case, a time-averaged gas-phase composition can be used when computing the fugacity difference. Also, strictly speaking, the hydrate-phase composition and, thus, the stoichiometric coefficient, ω, may also be time-dependent. However, with the current apparatus, it is not possible to perform in situ Raman spectroscopy measurements, and thus, the stoichiometric coefficient will be assumed to be constant. Finally, the second moment of the particle size distribution is also a function of time. However, because the FBRM probe is unable to differentiate between hydrate particles and the gas bubbles that are liberated during hydrate decomposition, the technique outlined by Clarke and Bishnoi,2 which is given below in eqs 12−15, is used to estimate the second moment as a function of time.

dμ0 dt dμ1 dt dμ2 dt

=0

(12)

= G(t )μ0

(13)

= 2G(t )μ1

(14)

G (t ) = −

Table 1. Experimental Conditions for the Formation and Decomposition of Gas Hydrates Formed from CO2 (1) + CH4 (2)

M φs Kd,1ω(feq̂ (t ) − fĝ (t ))1 3ρ φv

x1

formation temperature (K)

formation pressure (MPa)

decomposition temperature (K)

decomposition pressure (MPa)

0.4 0.4 0.4 0.6 0.6 0.6

274.2 275.4 276.2 274.4 275.4 276.4

2.32 2.58 2.73 2.02 2.17 2.38

274.4 275.4 276.3 274.3 275.3 276.1

1.6 1.8 1.9 2.0 2.1 2.2

conditions for gas hydrate decomposition as well as the conditions at which the gas hydrates were formed. As noted earlier, the gas hydrate formation experiments are summarized by Giraldo et al.1 Figures 1 and 2 show an example of the moles versus time and chord length distribution data that are obtained for a typical experiment. In this case, the data are for the gas mixture of CH4 (1) + CO2 (2) with x1 = 0.4 and pressure and temperature of 1.6 MPa and 274.4 K, respectively. As would be expected, because the apparatus in the current study is almost identical to that used by Clarke and Bishnoi,3 the results depicted in Figures 2 and 3 are qualitatively identical to those obtained by Clarke and Bishnoi.3 The chord length distribution that was measured at the onset of decomposition has a shape that is approximately log-normal, with the largest number of counts being at approximately 8 μm. The presence of a small number of particles with large chord lengths is likely an indication that there was agglomeration occurring during the formation of the hydrates. Modeling Results for the CH4 (1) + CO2 (2) Hydrates. The results were analyzed using both the proposed model as well as the previous model by Clarke and Bishnoi.2,3 In the current study, the chord length distribution is converted to a particle size distribution using the non-negative minimization technique proposed by Li and Wilkinson15 for this transformation, whereas Clarke and Bishnoi3 presented their own technique for this task. A full explanation of the technique is given by Li and Wilkinson.20 For both cases, once the chord length distribution has been

(15)

Equations 12−14 are derived from a population balance, in which it is assumed that there is no breakage or agglomeration during the decomposition.2 The initial values of the zeroth, first, and second moments of the particle size distribution are inferred from the experimental measurement of the chord length distribution, which occurs during the stabilization period. The important difference between eqs 11−15 and the equations given by Clarke and Bishnoi2 is that eq 11 only requires the intrinsic rate constant for a single component. This is advantageous if one wishes to model the gas hydrate decomposition kinetics for a gas mixture where not all of the intrinsic rate constants are known. As previously mentioned, at the time of writing, the intrinsic rate constants for gas hydrate decomposition were only available for methane and ethane in both sI and sII hydrates2,14 and for CO2 in sI hydrates.3

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RESULTS AND DISCUSSION Experiments were conducted to study the decomposition kinetics of gas hydrates that were formed from two different mixtures of methane and carbon dioxide. The experiments were performed as follow-on experiments to the gas hydrate formation study of Giraldo et al.1 In addition, further validation of the 4537


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Figure 1. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of gas hydrates formed from a mixture of CO2 (1) + CH4 (2), with x1 = 0.4, T = 274.4 K, and P = 1.6 MPa.

Figure 2. Measured chord length distribution (normalized) at the onset of the decomposition of gas hydrates formed from a mixture of CO2 (1) + CH4 (2), with x1 = 0.4, T = 274.4 K, and P = 1.6 MPa.

given in eq 16. The constants that Clarke and Bishnoi2,3 determined for the Arrhenius equation are given in Table 2.

converted to a particle size distribution, the second moment of the particle size distribution is found using the method outlined by Clarke and Bishnoi.3 The experimentally measured temperature, pressure, and gasphase composition values were used to compute the number of moles of hydrate-forming gas in the hydrate phase at any time during the decomposition process, using the Peng−Robinson equation of state.21 The number of moles of the hydrate-forming gas that remain in the hydrate phase during decomposition is simply the value of the moles of hydrate-forming gas in the gas phase at time t minus the value of the moles of hydrate former in the gas phase at time t = 0. For the modeling, the intrinsic rate constants for gas hydrate decomposition by Clarke and Bishnoi2,3 were used. Thus, no adjustable parameters were required for the current study. The temperature dependence of the rate constants was found by Clarke and Bishnoi2,3 to follow the Arrhenius equation, which is

⎛ ΔE ⎞ ⎟ Kd = Kdo exp⎜ − ⎝ RT ⎠

(16)

Carbon dioxide was taken to be the reference component, and the intrinsic rate constants for carbon dioxide hydrate decomposition, as measured by Clarke and Bishnoi,3 were used. The reason for assigning carbon dioxide to the reference component was because the previous experiments that examined carbon dioxide hydrate decomposition used the same FBRM probe in situ particle size measurements, whereas the experiments by Clarke and Bishnoi14 that examined methane hydrate decomposition used a different apparatus for particle size measurement. The stoichiometric coefficients were computed using the thermodynamic model by van der Waals and Plaatteeuw.22 4538


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Figure 3. Sensitivity study of moles of hydrate-forming gas remaining in the hydrate phase during decomposition of gas hydrates formed from a mixture CO2 (1) + CH4 (2), with x1 = 0.4, T = 274.4 K, and P = 1.6 MPa. (●) Experimental data, (- - -) predictions using a 50% increase in the ratio of stoichiometric coefficients (ωCH4/ωCO2), () predictions using a 25% increase in the ratio of stoichiometric coefficients (ωCH4/ωCO2). (···) predictions using a 25% decrease in the ratio of stoichiometric coefficients (ωCH4/ωCO2), and (-·-) predictions using a 50% decrease in the ratio of stoichiometric coefficients (ωCH4/ωCO2).

Table 2. Constants for Use in the Arrhenius Equation as Applied to the Decomposition Kinetics of Gas Hydrates species

ΔE (kJ/mol)

Kod (mol m−2 Pa−1 s−1)

source

CH4 in sI CH4 in sII C2H6 in sI and sII CO2

81.0 77.3 104.0 102.9

3.60 × 104 8.06 × 103 2.56 × 108 1.83 × 108

Clarke and Bishnoi14 Clarke and Bishoi2 Clarke and Bishnoi2,13 Clarke and Bishnoi3

Figure 4. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of gas hydrates formed from a mixture of CO2 (1) + CH4 (2), with x1 = 0.4, T = 274.4 K, and P = 1.6 MPa. (●) Experimental data, (- - -) predictions using the model by Clarke and Bishnoi,13 and () predictions with the new model.

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Figure 6. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of gas hydrates formed from a mixture CO2 (1) + CH4 (2), with x1 = 0.4, T = 276.3 K, and P = 1.9 MPa. (●) Experimental data, (- - -) predictions using the model by Clarke and Bishnoi,13 and () predictions with the new model.

As noted previously, it was not possible to measure the stoichiometric coefficients as a function of time. Even though the authors did not initially feel that this would be a significant disadvantage when modeling gas hydrate decomposition using a macroscopic approach, a sensitivity study was performed, in which the value of the ratio of the stoichiometric coefficients

(ωCH4/ωCO2) was varied. The results of the sensitivity study are shown in Figure 3. Corresponding to the curves in Figure 3, the average percent deviations between the predictions and the data range from 0.69%, which corresponds to the stoichiometric ratio being 25% less than that which thermodynamics predicts, all of the way up to 3.80%, which corresponds to the ratio of 4540


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Figure 7. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of gas hydrates formed from a mixture CO2 (1) + CH4 (2), with x1 = 0.6, T = 274.3 K, and P = 2.0 MPa. (●) Experimental data, (- - -) predictions using the model by Clarke and Bishnoi,13 and () predictions with the new model.


that were obtained using both models, the fit is generally good. In all six cases, the results obtained with the new model seem to follow the experimental data more closely than the results obtained using the model by Clarke and Bishnoi.2,3 In all but one case (Figure 5), the model by Clarke and Bishnoi2,3 overpredicts the data. At this point in time, the cause of this discrepancy is not certain. However, the authors suspect that this is likely due to the fact that the particle size analyzer that was available to Clarke and Bishnoi14 assumed that the particles were perfectly spherical,

stoichiometric coefficients being 50% above that predicted by thermodynamics. As the aforementioned exercise shows, the modeling results are only mildly sensitive to perturbations in the ratio of the stoichiometric coefficients. Thus, for the subsequent predictions, the stoichiometric coefficient as predicted by thermodynamics will be used. The computed results, using both the model by Clarke and Bishnoi2 as well as the newly proposed model, are shown in Figures 4−9. For the analysis of the data, there were no adjustable parameters required for either model. For the modeling results 4541


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Figure 10. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of sI gas hydrates formed from a mixture of CH4 (1) + C2H6 (2). (◆) Experimental data2 with x1 = 0.25, T = 274.65 K, and P = 0.64 MPa and () predictions with the new model with x1 = 0.65, T = 274.65 K, and P = 0.64 MPa. (●) Experimental data2 with x1 = 0.65, T = 274.15 K, and P = 0.94 MPa and (- - -) predictions with the new model with x1 = 0.65, T = 274.15 K, and P = 0.94 MPa.

only 7.72% and the root-mean-square of the relative error is 3.64%. Modeling Results for the CH4 (1) + C2H6 (2) Hydrates. To further test the predictive capability of the new approach, the proposed model was used to fit the gas hydrate decomposition kinetic data that Clarke and Bishnoi2 obtained for sI and sII hydrates formed from mixtures of methane and ethane. As previously noted, the experimental results obtained by Clarke

whereas the FBRM probe used in the current study does not make this assumption. For the results obtained with the model by Clarke and Bishnoi,2,3 the maximum relative deviation, between the experimental data and the model predictions, is only 9.69% and the root-mean-square of the relative error is 5.47%. When the proposed model is used, the maximum relative deviation is 4542


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Figure 11. Moles of hydrate-forming gas remaining in the hydrate phase during decomposition of sI gas hydrates formed from a mixture of CH4 (1) + C2H6 (2). (◆) Experimental data2 with x1 = 0.72, T = 278.15 K, and P = 1.49 MPa and () predictions with the new model with x1 = 0.72, T = 278.15 K, and P = 1.49 MPa. (●) Experimental data2 with x1 = 0.75, T = 277.15 K, and P = 1.34 MPa and (- - -) predictions with the new model with x1 = 0.75, T = 277.15 K, and P = 1.34 MPa.

and Bishnoi2 were obtained in a previous study, which employed an external particle size analyzer. Aside from this item, the experimental apparatus used by Clarke and Bishnoi2 is virtually identical to that used in the current study. When applying the new model to the data by Clarke and Bishnoi,2 methane was taken to be the reference component. Figures 10 and 11 show the modeling results obtained when the newly proposed model is applied to the sI and sII decomposition data obtained by Clarke and Bishnoi.2 In the case of the two gases that are known to form sI hydrates, Figure 10 shows that the newly proposed model performs very well in fitting the experimental data. The root-mean-square of the relative error between the predictions and the data is 2.59% when the original approach by Clarke and Bishnoi2 is used and 1.76% when the new approach is used. In Figure 11, two gas mixtures, which were identified by Subramanian et al.23 as sII hydrate-forming gases, were used. Similar to the results shown in Figure 10, the results obtained with the new model are also slightly improved over those obtained by Clarke and Bishnoi.2 In the case of the sII hydrates, the root-mean-square of the error between the data and the predictions is 2.58% using the model by Clarke and Bishnoi2 and 1.45% using the newly proposed approach. When the entire set of gas hydrate decomposition data generated by Clarke and Bishnoi2 is considered as a whole, the root-mean-square of the relative errors between the predictions of the new model and the model by Clarke and Bishnoi2 are 2.58 and 1.60%, respectively. As was the case in the gas hydrate formation work by Giraldo et al.,1 the main contribution of the simplified approach is not a drastic increase in the numerical accuracy; rather, it is a mathematical model for gas hydrate decomposition from a gas mixture that does not require the intrinsic rate constants to be known for each component in a gas mixture. This could be advantageous, for example, if one wishes to include the decomposition kinetics when simulating the decomposition of

a gas hydrate plug in a natural gas pipeline because the intrinsic rate constants of gas hydrate decomposition are currently only known for methane, ethane, and carbon dioxide.

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CONCLUSION A new approach to modeling the decomposition kinetic behavior of gas hydrates formed from mixed gases has been proposed. In light of recent work by Giraldo et al.,1 the new modeling approach explicitly includes the stoichiometry of the gas hydrates formed from gas mixtures. This, in turn, reduces the number of required intrinsic rate constants to just one. To test the newly proposed model, experiments were conducted in an isothermal/ isobaric, semi-batch stirred-tank reactor to examine the kinetics of gas hydrate decomposition from two mixtures of CO2 (1) + CH4 (2): one with x1 = 0.4 and the other with x1 = 0.6. When modeling the gas hydrate decomposition kinetic behavior of these two gases, the proposed model required no adjustment; the intrinsic rate constant for methane hydrate decomposition that was used was that measured by Clarke and Bishnoi.14 A further test of the predictive capabilities of the model was performed using it to fit previously obtained experimental data2 on the decomposition kinetics of sI and sII hydrates formed from mixtures of CH4 (1) + C2H6 (2). For both sets of data, the experimental data were first analyzed with both the model by Clarke and Bishnoi2,3 and the newly proposed model. The new model was found to fit the experimental data well for both data sets. While the numerical accuracy of the predcitions that were obtained with the new model was only a modest improvement over those obtained using the model by Clarke and Bishnoi,2,3 the significant contribution of the current work was to present a means for modeling the decomposition kinetics of gas hydrate decomposition, from a mixed gas, with only a single intrinsic rate constant. This is significant because the intrinsic rate constants for gas hydrate decomposition have only been reported for methane,14 ethane,2 and carbon dioxide.3 4543


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(2) 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. (3) Clarke, M. A.; Bishnoi, P. R. Determination of the intrinsic rate constant and activation energy of CO2 gas hydrate decomposition using in-situ particle size analysis. Chem. Eng. Sci. 2004, 59, 2983−2993. (4) Claussen, W. F. Suggested structures of water in inert gas hydrates. J. Chem. Phys. 1951, 19, 259−261. (5) Jeffry, G. A.; McMullan, R. K. The clathrate hydrates. Prog. Inorg. Chem. 1967, 8, 43−108. (6) von Stackelberg, M.; Müller, H. R. Zur struktur der gashydrate. Naturwissenschaften 1951, 38, 456−461. (7) Ripmeester, J. A.; Tse, J. S.; Ratcliffe, C. I. A new clathrate hydrate structure. Nature 1987, 325, 135−136. (8) Wallmann, K.; Pinero, E.; Burwicz, E.; Haeckel, M.; Hensen, C.; Dale, A.; Ruepke, L. The global inventory of methane hydrate in marine sediments: A theoretical approach. Energies 2012, 57, 2449−2498. (9) Collette, T. S.; Lewis, R.; Uchida, T. Growing interest in gas hydrates. Oilfield Rev. 2000, Summer, 42−57. (10) Nixon, M. F.; Grozic, J. L. H. Submarine slope failure due to gas hydrate dissociation: A preliminary quantification. Can. Geotech. J. 2007, 44, 314−325. (11) Schicks, J.; Luzi, M.; Beeskow-Strauch, B. The conversion process of hydrocarbon hydrates into CO2 hydrates and vice versa: Thermodynamic considerations. J. Phys. Chem. A 2011, 115, 13324− 13331. (12) Kim, H. C.; Bishnoi, P. R.; Heidemann, R. A.; Rizvi, S. S. H. Kinetics of methane hydrate decomposition. Chem. Eng. Sci. 1987, 42, 1645−1653. (13) Clarke, M. A.; Bishnoi, P. R. Determination of the intrinsic rate of ethane gas hydrate decomposition. Chem. Eng. Sci. 2000, 55, 4869− 4883. (14) Clarke, M. A.; Bishnoi, P. R. Determination of the activation energy and intrinsic rate constant of methane gas hydrate decomposition. Can. J. Chem. Eng. 2001, 79, 143−147. (15) Kawamura, T.; Ohga, K.; Higuchi, K.; Yoon, J. H.; Yamamoto, Y.; Komai, T.; Haneda, H. Dissociation behavior of pellet-shaped methane−ethane mixed gas hydrate samples. Energy Fuels 2003, 17, 614−618. (16) Giavarini, C.; MacCioni, F.; Santarelli, L. Dissociation rate of THF−methane hydrates. Pet. Sci. Technol. 2008, 26, 2147−2156. (17) Nihous, G. C.; Kuroda, K.; Lobos-González, J. R.; Kurasaki, R. J.; Matsutani, S. An analysis of gas hydrate dissociation in the presence of thermodynamic inhibitors. Chem. Eng. Sci. 2010, 65, 1748−1765. (18) Daraboina, N.; Linga, P.; Ripmeester, J.; Walker, V.; Englezos, P. Natural gas hydrate formation and decomposition in the presence of kinetic inhibitors. 2. Stirred reactor experiments. Energy Fuels 2011, 25, 4384−4391. (19) Kuhs, W. F.; Genov, G.; Staykova, D. K.; Hansen, T. Ice perfection and onset of anomalous preservation of gas hydrates. J. Phys. Chem. 2004, 6, 1−6. (20) Li, M.; Wilkinson, D. Determination of non-spherical particle size distribution from chord length measurements. Part 1: Theoretical analysis. Chem. Eng. Sci. 2005, 60, 3251−3265. (21) Peng, D.; Robinson, D. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (22) Van der Waals, J. H.; Platteeuw, J. C. Clathrate solutions. Adv. Chem. Phys. 1959, 2 (1), 1−57. (23) Subramanian, S.; Kini, R. A.; Dec, S. F.; Sloan, E. D. Evidence of structure II hydrate formation from methane + ethane mixtures. Chem. Eng. Sci. 2000, 55, 1981−1999.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Faisal Al-Otaibi and Dr. Amit Majumdar for their help. Likewise, the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Ursula and Herbert Zandmer Graduate Scholarship, and the Department of Chemical and Petroleum Engineering, University of Calgary, is highly appreciated.

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NOMENCLATURE Ap = surface area per unit volume of a gas hydrate particle (m2/ m3) ΔE = activation energy in the Arrhenius equation (J/mol) fg,ĵ = fugacity of hydrate former j in a mixture, in the vapor phase (MPa) ̂ = fugacity of hydrate former j in a mixture, at the hydrate feq,j equilibrium point (MPa) G = linear growth rate (m/s) Kd,j = rate constant of gas hydrate decomposition for component j (mol m−2 Pa−1 s−1) Kod = pre-exponential term in the Arrhenius equation (mol m−2 Pa−1 s−1) L = characteristic length (m) M = molecular mass (kg/mol) n = number of moles (mol) NC = number of hydrate-forming components R = universal gas constant (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 reacting mass (m3)

Greek Letters

γ = Hatta number μn = nth moment of the particle size distribution (mn/m3) φs = surface shape factor φ = particle density function (m−4) φv = volume shape factor ψ = sphericity ρ = density of the hydrate (kg/m3) ωj = stoichiometric coefficient of component j, in the hydrate phase Subscripts and Superscripts

b = in the bulk liquid phase eq = at the three-phase (gas−liquid−hydrate) equilibrium conditions g = in the gas phase g−l = at the gas−liquid interface H = all hydrate particles p = individual hydrate particle wo = water

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REFERENCES

(1) Giraldo, C.; Maini, B.; Bishnoi, P. R. A simplified approach to modeling the rate of formation of gas hydrates formed from mixtures of gases. Energy Fuels 2013, 27, 1204−1211. 4544


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