A Simplified Approach to Modeling the Rate of Formation of Gas

Article pubs.acs.org/EF

A Simplified Approach to Modeling the Rate of Formation of Gas Hydrates Formed from Mixtures of Gases Carlos Giraldo,† Brij Maini,‡ Raj Bishnoi,‡ and Matthew Clarke*,‡ ‡

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

†

ABSTRACT: Experiments were conducted in a semibatch stirred tank reactor, equipped with an in situ particle size analyzer, to study the rate of formation of hydrates 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 in the current study. The experimental temperature ranged from 274 to 276 K and the experimental pressure ranged from 20 to 27 bar absolute. The range of temperatures was bounded by the freezing point of water and the temperature at which the CO2 vapor pressure curve intersects the methane hydrate curve. Initially, the results were analyzed using the kinetic model of Englezos et al. (Chem. Eng. Sci. 1987, 42, 2659−2666). After a careful reexamination of the model of Englezos et al., it was realized that the mathematical model for gas hydrate formation from gas mixtures could be simplified by directly incorporating the hydrate phase stoichiometry. The new approach has the added advantage that the intrinsic rate constant of gas hydrate formation is required for only a single component. Thus, a new approach was proposed to model the kinetics of gas hydrate formation from a mixture of gases, and the results from these predictions were compared to the results obtained using the model of Englezos et al. The root-mean squares of the relative errors between the experimental results and the predictions of the new model and the model of Englezos et al. were found to be 2.70% and 4.29%, respectively.

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INTRODUCTION On May 2, 2012, the U.S. Department of Energy (DOE) announced1 that it, along with several partners, had successfully completed a month-long “proof-of-concept” test in which a mixture of carbon dioxide and nitrogen was injected into a naturally occurring gas hydrate deposit to promote the production of methane. The naturally occurring gas hydrates that were encountered in this field test were icelike crystalline inclusion compounds of water and low-molecular-weight gases, such as methane and carbon dioxide, that form at appropriate conditions of low temperature and elevated pressure. Under such conditions, the crystalline structures of gas hydrates allow them to store extremely large amounts of gas; specifically, 1 m3 of gas hydrate can store up to 180 m3 of gas under standard conditions. Thus, the amount of methane that is sequestered worldwide is most likely far larger than the sum of all other conventional sources of methane; it has been estimated that the volume of methane locked in hydrate deposits in Canada2 alone is between 1012 and 1014 m3 under standard conditions. Although the tests conducted by the U.S. DOE reached the mainstream news only in 2012, the scientific community has long realized that naturally occurring gas hydrate deposits might also potentially provide a medium for CO2 storage. In this scheme, when CO2 contacts a methane hydrate, under certain conditions of temperature and pressure, a replacement reaction occurs in which the methane molecule is freed and CO2 is trapped in the hydrate structure.3 The methane that is displaced upon hydrate formation by the CO2 could then be recovered and utilized. This idea of combining methane recovery from naturally occurring hydrates with CO 2 sequestration3 offers several advantages: presenting a possible CO2 sink, reducing the amount of water produced during gas © 2013 American Chemical Society

production from naturally occurring gas hydrate deposits, and maintaining the geomechanical stability of the hydrate deposit. To be economically viable, it is likely that this alternative will need to be carried out in conjunction with another method, such as thermal stimulation or pressure reduction. However, to undertake any future computer simulations studies, it is necessary to examine the kinetics of hydrate formation and decomposition of a mixture of CO2/CH4 in the absence of any other additional process. Uddin and co-workers4,5 have remarked that, with today’s powerful petroleum reservoir simulation software, it is possible to couple the hydrate formation and decomposition kinetics, once they are known, with other phenomena such as mass and heat transfer and multiphase fluid flow. In previous studies, the kinetics of gas hydrate formation from pure gases was investigated for several systems,6−11 whereas the kinetics of gas hydrate formation has been studied for only a relatively small number of mixed-gas systems.6,10,12,13 The work of Englezos et al.6 was the first experimental study of gas hydrate formation kinetics from mixtures, as well as the first work that attempted to quantify the rate of formation of gas hydrates formed from gas mixtures. In their study, Englezos et al.6 proposed a predictive model for the rate of gas hydrate formation and used their model to match experimental data on the rate of gas hydrate formation from mixtures of methane and ethane. More recently, Al-Otaibi et al.11 applied the model of Englezos et al.6 to analyze the results of gas hydrate formation from mixtures of ethane and propane. Malegoankar12 made a Received: September 10, 2012 Revised: November 28, 2012 Published: March 1, 2013 1204

dx.doi.org/10.1021/ef301483h | Energy Fuels 2013, 27, 1204−1211

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Figure 1. Simplified schematic diagram of the experimental apparatus.

first attempt at examining the kinetics of gas hydrates formed from mixtures of CH4 and CO2. Malegoankar12 did not have access to a particle size analyzer but did observe that the results obtained for this system did not conform to the model of Englezos et al.6 Finally, Mork and Gudmundsson13 performed experiments to study the kinetics of gas hydrate formation for a natural gas mixture. In their approach, reaction rate constants were neglected, and the modeling was based solely on masstransfer considerations. The current study was inspired by the concept of underground replacement of enclathrated CH4 by CO2, and thus, it began as an investigation into the formation kinetics of gas hydrates formed from mixtures of CO2/CH4 in which the hydrates were formed in a semibatch stirred tank reactor under isothermal and isobaric conditions and the particle size distribution was measured in situ with a focused-beam reflectance method (FBRM) probe. However, after the experimental data were initially processed using the model of Englezos et al.,6 it was realized that the number of ordinary differential equations could be significantly reduced by directly including the hydrate phase stoichiometry, and thus, the study grew to include the development of a new model for the kinetics of gas hydrates that have been formed from a gas mixture.

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type T thermocouples, and the uncertainty in the temperature measurement was ±0.05 K. The reactor pressure was kept constant by controlling the flow rate from the supply reservoir through a control valve; as the pressure in the reactor fell, due to gas consumption, the control valve opened to allow additional gas to flow into the reactor. The two bias reservoirs were used for the differential pressure transducer cells to accurately measure the pressure in the reactor and supply reservoirs. The differential pressure transducer connected to the reactor (Alphaline pressure transmitter, Rosemount Instruments Ltd., Calgary, Alberta, Canada) had a span of 2 MPa, and the differential pressure transducer connected to the gas supply reservoir (also an Alphaline pressure transmitter) had a span of 1 MPa. Both differential pressure transducers had an uncertainty of 0.25% of the span. The voltage signals from the temperature and pressure measurements were fed into a data acquisition unit (National Instruments, FP-TC-120 for the thermocouples and FP-AI-110 for the pressure transducers) and then subsequently into a personal computer through a National Instruments FP-1601 ethernet interface. A focused-beam reflectance method (FBRM) probe (Lasentec model D600X) was directly installed in the reactor to provide in situ measurements of the chord length distribution of the hydrate particles. In addition, the system was equipped with an in situ particle size analyzer (Lasentec model D600), which provided the transient particle chord length distribution (0.5−1000 μm). Finally, an SRI-8640 gas chromatograph integrated with the Peak Simple II software was used to analyze the molar composition of the reactor gas phase during hydrate formation. The gas chromatograph was equipped with a capillary column (Porapak Q column) that separated 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 was helium (purity ≥ 99.999%), and it was purchased from Praxair Technology Inc. Two different gas mixtures of CO2 (1) + CH4 (2) were used: One mixture had x1 = 0.4, with a supplier’s reported uncertainty in composition of ±0.8 mol %, and the other had x1 = 0.60, with a supplier’s reported uncertainty in composition of ±1.0 mol %. Deionized water (Millipore Simplicity water purification system, which

EXPERIMENTAL APPARATUS, MATERIALS, AND PROCEDURE

Apparatus and Materials. The equipment used in this study was the same as that used by Al-Otaibi et al.11 Figure 1 shows a schematic of the apparatus, which consisted of an isothermal, isobaric stirred tank reactor (STR); two bias reservoirs (R2 and R5); and a gas supply reservoir (R1). The temperature was maintained at a near-constant value by immersing the reactor in a circulating glycol cooling bath whose volume was significantly larger than that of the reactor. The reactor temperature and gas supply temperature were measured with 1205


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produces ultrapure water with a resistivity of 18.2 MΩ·cm at 298 K) was used for all experiments. Experimental Procedure. The experimental procedure for the current study was identical to that followed by Al-Otaibi et al.11 Initially, the reactor was flushed several times with experimental gas and ultrapure water. After being flushed, the rector was filled with 280 mL of ultrapure water, and experimental gas was slowly introduced into the system until the desired pressures were reached. The pressure in the reactor was selected so as to be approximately 10% above the computed three-phase equilibrium pressure at the given temperature. Subsequently, stirring was started, and the control valve was set to automatic control to ensure that the pressure was kept constant. Measurements of pressure, temperature, and chord length distribution were recorded every 10 s. The reactor was monitored for the appearance of hydrates, as indicated by the solution becoming translucent and the slope of the total number of particles on the FBRM display changing sharply. Sampling of the gas phase was performed at only three times during each experiment, to avoid removing an appreciable amount of gas from the system: at the beginning of the run (t = 0), the start of hydrate formation, and the end of the hydrate formation process. The gas hydrates were allowed to form only until the total particle counts, as displayed by the FBRM probe, peaked. It is believed that this point corresponds to the onset of particle breakage.11

0.38:0.52 and not 1:1. Thus, to reformulate the original model of Englezos et al.,6 consider the following generalized hydrate ω1M1ω2 M 2ω3 M3 ··· ω NC MNCω W H 2O

where ωi (i = 1, ..., NC) represents the stoichiometric coefficients of the NC hydrate-forming compounds in the hydrate phase, Mi (i = 1, ..., NC) represents the hydrateforming compounds, and ωw is the stoichiometric coefficient of water in the hydrate. Let the component with subscript 1 be defined as the reference component. Thus, the rate of gas consumption of component i can be related to the rate of consumption of the reference component as ωi ⎛ dn ⎞ ⎛ dn ⎞ ⎜ ⎟ = ⎜ ⎟ ⎝ dt ⎠i ω1 ⎝ dt ⎠1

⎛ dn ⎞ ⎜ ⎟ = ⎝ dt ⎠ p

NC j=1

j=1

⎛

⎞

j=1

dt

j

⎛ ωj ⎞⎛ dn ⎞ = ∑ ⎜ ⎟⎜ ⎟ ⎝ ⎠ j = 1 ⎝ ω1 ⎠ dt j NC ⎛ ωj ⎞ = A p ∑ K1*⎜ ⎟(f ̂ − feq̂ )1 ⎝ ω1 ⎠ j=1

NC

⎛ dn ⎞ ⎟ = A ∑ K *(f ̂ − f ̂ ) p j eq j dt ⎠ j

NC

∑ ⎜⎝ dn ⎟⎠ NC

THEORY In the current study, the experimental results were initially correlated using the model of Englezos et al.6 In their model, Englezos et al.6 wrote that the total rate of consumption is the sum of the individual rates of consumption of all of the hydrateforming compounds

∑ ⎝⎜

(3)

Inserting eq 3 into eq 1 gives

■

⎛ dn ⎞ ⎜ ⎟ = ⎝ dt ⎠ p

(2)

(4)

⎞ ⎛ ω1 ω3 ⎛ dn ⎞ ω2 ⎜ ⎟ = A K *(f ̂ − f ̂ ) ⎜ + + + ··· ⎟ p 1 1 eq ⎝ dt ⎠ p ω1 ω1 ⎠ ⎝ ω1

(1)

= A pK1*(f ̂ − feq̂ )1ω

where NC is the number of hydrate-forming gases and K*j is the intrinsic rate constant of gas hydrate formation for component j. For the purpose of the following discussion, three points are important to note: First, eq 1 assumes that the overall rate of gas consumption during the formation of gas hydrates from a gas mixture is the unweighted sum of the rates of consumption of all species in the mixture. In other words, it assumes that the rate of diffusion and adsorption of a hydrateforming gas is unaffected by the presence of the other gases. As a justification for this assumption, Englezos et al.6 reasoned that, in the case of a mixture of methane and ethane, “there is no difference in the way the two gases interact with the water in the hydrate phase”.6 Second, solving eq 1 requires the solution of 2NC ordinary differential equations, which are usually fairly stiff. Finally, the solution of eq 1 also requires that the intrinsic rate constants of gas hydrate formation be known for all of the hydrate-forming compounds in a gas mixture. Simplified Approach to Modeling the Kinetics of Gas Hydrate Formation from Gas Mixtures. Upon reexamining the model of Englezos et al.6 in light of the three points highlighted in the previous paragraph, we decided to reformulate the original model of Englezos et al.6 Gas hydrates formed from a given gas mixture will have a well-defined stoichiometry, and thus, it can be safely assumed that the rate of consumption of each hydrate former does not occur in a 1:1 ratio. For example, if a gas hydrate particle that has been formed from a mixture of CH4 and CO2 has a stoichiometry of approximately 0.38CH4·0.52CO2·5.75H2O, then the ratio of the CH4 consumption rate to the CO2 consumption rate is

(5)

where NC

ω=

∑ j=1

ωj ω1

(6)

Thus, in the reformulated approach, the total rate of growth per particle is simply equal to the rate of consumption of the arbitrarily defined reference compound multiplied by ω. Following the same logic as Englezos et al.,6 it can be shown that (see Appendix A for the derivation), in the current study, the computation of the number of moles of a gas mixture consumed during hydrate formation consists of computing the number of moles of the reference component that have been consumed using eq 5 and then multiplying this result by the appropriate stoichiometric ratios to determine the numbers of moles of the other species that have been consumed during hydrate formation. Compared to the original model of Englezos et al.,6 this reduces the number of ordinary differential equations that must be solved from 2NC to 2, and these two equations are as follows ̂ ̂ ̂ ̂ dn1 ⎛ D1*γAg−l ⎞ (fg,1 − feq,1 ) cosh(γ ) − (fb,1 − feq,1 ) ⎟ = ⎜⎜ ⎟ dt sinh(γ ) ⎝ yL ⎠ (7) 1206


Energy & Fuels ̂ dfb,1 dt

=

Article

̂ )D *γ a (H1 − feq,1 1 1 H1c woyL sinh γ1 cosh γ1] −

time the number of particles began to decrease, likely as a result of the process of agglomeration. At each measurement, there was an associated chord length distribution that had to be transformed into a particle size distribution for use with the kinetic model of Englezos et al.6 Unlike in the work of AlOtaibi et al.,11 the current study used the non-negative minimization technique proposed by Li and Wilkinson15 for this transformation. A full explanation of the technique can be found elsewhere.15 Essentially, the method is an iterative nonnegative least-squares technique that has the advantage of being insensitive to measurement noise and does not presuppose that the particles are spherical. Once the chord length distribution had been converted into a particle size distribution, the second moment of the particle size distribution was found using the method outlined by Clarke and Bishnoi.10 Results Obtained Using the Model of Englezos et al.6 For the analysis, the intrinsic rate constant for methane hydrate formation was taken from Al-Otaibi et al.,11 and the intrinsic rate constant for CO2 hydrate formation was taken from Clarke and Bishnoi.10 Also, the binary diffusion coefficients were computed using the Wilke−Chang correlation,16 along with improved parameters given by Hayduk and Laudie.17 Thus, no adjustable parameters were required for modeling the formation kinetic behavior of gas hydrates formed from the two mixtures of CO2 (1) + CH4 (2). The results obtained using the model of Englezos et al.6 are shown as the dashed lines in Figures 2−7. It can be seen that the predictions of the model of

̂ − f ̂ ) − (f ̂ − f ̂ ) [(fg,1 eq,1 b,1 eq,1

̂ )2 πK1*μ2 (H1 − feq,1 H1c wo

̂ − f̂ ) (fb,1 eq,1 (8)

Once eqs 7 and 8 have been solved to estimate the number of moles of the reference component that have been consumed during hydrate formation, the numbers of moles of the other species that have been consumed during hydrate formation can be found by stoichiometry as ω ni = i n1 ω1 (9) It should also be noted that the new approach requires the intrinsic rate constant for gas hydrate formation for only one component in a gas mixture. This could be useful for modeling situations in which a gas mixture contains a hydrate-forming compound whose intrinsic rate constant of gas hydrate formation is not known. Finally, it should be noted that, in the limit as the number of hydrate-forming components in the gaseous mixtures goes to one, the proposed model reduces to the original model of Englezos et al.6 for gas hydrate formation from a pure gas.

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RESULTS AND DISCUSSION Experimental results for the temperature, pressure, and chord length distributions were collected during gas hydrate formation from two different gas mixtures of CO2 (1) + CH4 (2), one with x1 = 0.4 and the other with x1 = 0.6. The range of temperatures over which experiments could be conducted was 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. However, to ensure that there was a finite driving force for hydrate formation, the upper temperature for experimentation was taken as 276 K, whereas the lower temperature limit for experimentation was 274 K. Based on the work of Al-Otaibi et al.11 with methane hydrate formation, the rate of stirring was taken as 700 rpm. The aforementioned study found that this rate of stirring was sufficient for eliminating heat- and mass-transfer resistances. The temperature, pressure, and gas-phase composition data were used, in conjunction with the Peng−Robinson equation of state,14 to compute the number of moles of the hydrate-forming gas present in the gas phase at any instant in time. A detailed explanation of the procedure for processing the experimental data is given by Clarke and Bishnoi10 and by Al-Otaibi et al.11 In the current study, however, an alternate approach was employed to convert the chord length distributions, which were measured by the FBRM probe, into particle size distributions. This approach is outlined in the next section. Conversion of Chord Length Distribution into Particle Size Distribution. The data obtained with the FBRM probe were the number of particle counts per second and chord length distribution at each time step. As seen by Clarke and Bishnoi10 and Al-Otaibi et al.,11 the particle count remained approximately constant prior to the formation of gas hydrates. However, when the systems reached the turbidity point, the total number of particles increased to a maximum, after which

Figure 2. Total number of moles of gas in the hydrate phase. Hydrates formed from a mixture of CO2 (1) + CH4 (2) with x1 = 0.40 at 274.2 K and 2.32 MPa. (■) Experimental data, (− − −) modeling obtained results using the model of Englezos et al.,6 and () modeling results obtained using the proposed model.

Englezos et al.6 in Figures 2−7 are generally good. The maximum relative deviation between the experimental data and the model predictions is only 9.35%, and the root-mean square of the relative error is 4.29%. As noted in the Introduction, a previous attempt was made to study the kinetic behavior of gas hydrates formed from mixtures of CO2 (1) + CH4 (2): Malegoankar12 attempted hydrate formation kinetics experiments with such mixtures, albeit without the aid of a particle size analyzer, and was unable to fit the experimental data with the model of Englezos et al.6 Although it remains unclear as to why the model did not fit the data, Malegoankar12 suggested modifying the model of Englezos et al.6 Although the results presented in Figures 1207


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Figure 3. Number of moles of gas in the hydrate phase. Hydrates formed from a mixture of CO2 (1) + CH4 (2) with x1 = 0.40 at 275.4 K and 2.58 MPa. (■) Experimental data, (− − −) modeling obtained results using the model of Englezos et al.,6 and () modeling results obtained using the proposed model.




2−7 are satisfactory, the aforementioned comments of Malegoankar12 provided some of the inspiration to re-examine the model of Englezos et al.6 Results Obtained Using the New Model. As seen in eq 8, the modeling of the experimental results with the new model requires the intrinsic rate constant of hydrate formation for only one component in the mixture, along with the appropriate stoichiometric constants. For the current study, methane was taken to be the reference component, and the intrinsic rate constants for methane hydrate formation, as measured by AlOtaibi et al.,11 were used. Although it should be possible to assign either component to be the reference component, methane was chosen because the rate constants for methane formation11 had a lower uncertainty than those for reported for carbon dioxide.10 The stoichiometric coefficients were computed using the thermodynamic model of van der Waals and Plaatteeuw,18 and as was done in the preceding section, the binary diffusion coefficients were computed using the Wilke− Chang correlation,16 with improved parameters given by Hayduk and Laudie.17 Additionally, as was done with the

modeling in the preceding section, the mass-transfer coefficients and Henry’s constants were obtained from solubility experiments, and the gas-phase fugacities of the hydrate-forming gases were computed using the Peng− Robinson equation of state.13 Thus, modeling of the formation of gas hydrates from mixtures of CO2 and CH4 with the proposed model required no adjustable parameters. The two ordinary differential equations (eqs 7 and 8) were solved simultaneously using MatLab’s ODE23S routine. The predictions from the new model for both gas mixtures are shown as the solid lines in Figures 2−7. As can be seen, the predictions with the new model match the data very well. Using the original model of Englezos et al.,6 the maximum relative percentage error in the predictions was 9.35%, whereas with the new model, the maximum relative percentage error in the predictions was 5.55%. Similarly, the root-mean square of the relative error for the predictions with the new model was 2.70%. Although a reduction of 3.80% in the maximum relative percentage error might not, in and of itself, be seen as a sufficient reason for adopting a new mathematical model for gas 1208


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If it is assumed that the fugacity difference and the intrinsic rate constant of the reference component are independent of particle size, eq A-1 becomes R y (t ) =

∫0

∞

A pK1*(f ̂ − feq̂ )1ωφ(L , t ) dL

= K1*(f ̂ − feq̂ )1ωπ

∫0

∞

L2φ(L , t ) dL

(A-2)

The integral in eq A-2 is equal to the second moment of the particle size distribution. Therefore, this equation becomes


R y(t ) = πK1*(f ̂ − feq̂ )1ωμ2

(A-3)

R y(t ) = ωK (f ̂ − feq̂ )1

(A-4)

K = πK1*μ2

(A-5)

Note that special care must nonetheless be taken when interpreting eq A-4. The equilibrium fugacity that appears in eq A-4 must be that of the reference component in the mixture. The value of the other fugacity term will be clarified in a subsequent section. What is being implied in eq A-4 is that, because the ratios of the numbers of moles of the different hydrate formers consumed are constrained by the stoichiometry, it is necessary to solve only the differential equations (which will be presented shortly) for the reference component. Once it has been determined how many moles of that component have been consumed, it is possible to determine the numbers of moles of the other components that have been consumed by the stoichiometric ratios ω n1 = i n1 ω1 (9)

hydrate formation kinetics, the results with the new model show that the kinetics of gas hydrate formation from mixed gases can be accurately represented with only two ordinary differential equations and the intrinsic rate constant of gas hydrate formation for one of the hydrate-forming components. This is potentially advantageous when modeling the formation of gas hydrates in a natural gas pipeline, as some of the rate constants, such as those of H2S and C4H10 or CO2 in an sII hydrate, are not yet available. Finally, it must also be noted that, in the current study, the stoichiometric coefficients were computed from thermodynamics; we expect that the predictions of gas consumption during hydrate formation could be further improved if stoichiometric coefficients were computed from experimentally deduced fractional occupancies.

for i = 2, ..., NC.

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Two-Film Theory

CONCLUSIONS Experiments were conducted in an isothermal/isobaric semibatch stirred tank reactor to examine the kinetics of gas hydrate formation from two mixtures of CO2 (1) + CH4 (2), one with x1 = 0.4 and the other with x1 = 0.6. The experimental data were first analyzed with the model of Englezos et al.,6 which was found to fit the experimental data reasonably well. However, upon examination of the model of Englezos et al.,6 it was realized that the model could be simplified by explicitly accounting for the hydrate phase stoichiometry. The proposed model has no adjustable parameters, and it was found that the new model, when compared to the original model of Englezos et al.,6 was able to provide improved predictions for the rate of consumption of gas during gas hydrate formation from a gas mixture. Also, the proposed model reduces the number of ordinary differential equations while, at the same time, requiring the intrinsic rate constant of gas hydrate formation for only one of the components in a gaseous mixture.

Two-film theory was used to describe the absorption of gas at the gas−liquid interface. Assuming quasi-steady-state conditions, the mass balance for the reference component in a slice of thickness dy and unit cross-sectional area in the liquid film is given by D1,w

∫0

⎜

⎟

⎝ dt ⎠ p

= ωK (f ̂ − feq̂ )1

⎛ f̂ ⎞ eq,1 ⎟ c1 = c wo⎜ ⎜ H − f̂ ⎟ ⎝ 1 eq,1 ⎠

(A-6)

(A-7)

with the boundary conditions

APPENDIX A: DERIVATION OF EQS 7 AND 8 If eq 5 is integrated over all particle sizes, one obtains an expression for the global rate of reaction for all particles R y (t ) =

dy 2

Because the amount of water is in excess, the number of moles of water remains essentially constant, and the liquid solution is ideal, the concentration of the reference component in the aqueous phase can be estimated as

■

∞ ⎛ dn ⎞

d2c1

f ̂ = fĝ

y=0

(A-8)

f ̂ = fb̂

y = yL

(A-9)

To solve this boundary-value problem, eq A-6 is first converted to dimensionless form by the following changes of variables

φ (L , t ) d L (A-1) 1209


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y yL

χ=

yL =

(A-10)

(fĝ − feq̂ )1

dn1 = (J1)y = 0 Ag−l dt

(A-11)

Equation A-6 can now be rewritten as d2f * dχ

− γ 2f * = 0

2

γ = yL

πμ2 K1* D1*

χ=1

f* =

(fb̂ − feq̂ )1 (fĝ − feq̂ )1

The initial condition for this equation is the number of moles of the reference component that have been transported into the aqueous phase up to the turbidity point. This can be computed from the experimentally measured pressure and temperature values. To determine f b as a function of time, a mass balance in the bulk was performed. The mole balance of the reference component in the bulk liquid is

(A-14)

(A-15)

accumulation = (number of moles in by diffusion)

= τ1

− (number of moles out by reaction)

(A-16)

(A-25)

The general solution to eq A-12 is f * = K1 exp(γχ ) + K 2 exp( −γχ )

The accumulation of the reference component in the bulk phase can be written as

(A-17)

⎛ df ̂ ⎞ ⎛ dn ⎞ ⎛ dc ⎞ V c b,1 ⎟ accumulation = ⎜ 1 ⎟ = Vbulk ⎜ 1 ⎟ = bulk wo ⎜⎜ ⎝ dt ⎠ ⎝ dt ⎠ H1 ⎝ dt ⎟⎠

where K1 and K2 are the constants of integration, which are determined from the boundary conditions to be

K2 =

τ1 − exp( −γ ) 2 sinh(γ )

(A-18)

exp( −γ ) − τ1 2 sinh(γ )

(A-19)

(A-26)

The value of the Henry’s constant, H1, used in eq A-26 was determined from a solubility experiment.6,10 The term that represents the number of moles of reference component that enters the bulk liquid through diffusion is

When the constants of integration are inserted into eq A-17, the result can be simplified to f* =

(A-24)

(7)

The diffusion coefficient that should be used in eq A-14 is the diffusion coefficient of the reference component in water, in the presence of the other compounds. The original boundary conditions can be transformed to f* = 1

y=0

̂ ̂ ̂ ̂ dn1 ⎛ D1*γAg−l ⎞ (fg,1 − feq,1 ) cosh(γ ) − (fb,1 − feq,1 ) ⎟ = ⎜⎜ ⎟ dt sinh(γ ) ⎝ yL ⎠

(A-13)

χ=0

(A-23)

⎛ df ̂ ⎞ ⎛ dc ⎞ (J1)y = 0 = −D1,w ⎜ 1 ⎟ = −D1*⎜⎜ 1 ⎟⎟ ⎝ dy ⎠ y = 0 ⎝ dy ⎠

(A-12)

⎛c + c ⎞ wo eq,1 ⎟ * D1 = D1,w ⎜ ⎜ H − f̂ ⎟ ⎝ 1 eq,1 ⎠

K1 =

(A-22)

The rate at which gas is being consumed to form hydrates can be obtained from

(f ̂ − feq̂ )1

f* =

Da kLa

τ1 sinh(γχ ) + sinh[γ(1 − χ )] sinh(γ )

number of moles in by diffusion ⎛ df ̂ ⎞ b,1 ⎟ * = −D1 Abulk−film ⎜⎜ ⎟ ⎝ dt ⎠ y = y

(A-20)

L

Upon reintroducing the original variables, eq A-15 becomes the result that was obtained by Englezos et al.6

The term that represents the number of moles of the reference component that has gone into forming hydrates is

⎧ ⎡ ⎛ ⎞⎤ ⎪ ̂ + ⎨(f ̂ − f ̂ ) sinh⎢γ ⎜1 − y ⎟⎥ f1̂ = feq,1 eq,1 ⎪ g,1 ⎢⎣ ⎜⎝ yL ⎟⎠⎥⎦ ⎩ ⎛ ⎞⎫ ⎤ ⎪⎡ ̂ − f ̂ ) sinh⎜γ y ⎟⎬⎢ 1 ⎥ + (fg,1 ⎜ y ⎟⎪⎣ sinh(γ ) ⎦ eq,1 ⎝ L ⎠⎭

(A-27)

number of moles out by reaction = VbulkπK1*μ2 (fb̂ − feq̂ )1 (A-28)

Combining eqs A-16 and A-26−A-28 gives ̂ dfb,1

(A-21)

dt

This equation represents the reference component’s fugacity profile in the liquid film that separates the gas phase from the bulk liquid at any given time. Finally, it should be note that the film thickness, yL, can be calculated from data obtained in a solubility experiment6,10 as

=

̂ )2 D *γ a (H1 − feq,1 1 1 H1c woyL sinh γ1 cosh γ1] −

̂ − f ̂ ) − (f ̂ − f ̂ ) [(fg,1 eq,1 b,1 eq,1

̂ )2 πK1*μ2 (H1 − feq,1 H1c wo

̂ − f̂ ) (fb,1 eq,1 (8)

1210


Energy & Fuels

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Article

(4) Uddin, M.; Coombe, D.; Law, D.; Gunter, B. Numerical studies of gas hydrate formation and decomposition in a geological reservoir. J. Energy Resour. Technol. 2008, 130, 032501-1−032501-14. (5) Uddin, M.; Coombe, D.; Wright, F.. Modeling of CO2-hydrate formation in geological reservoirs by injection of CO2 gas. J. Energy Resour. Technol. 2008, 130, 032502-1−032502-11. (6) 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. (7) 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. (8) Malegaonkar, M.; Dholabhai, P.; Bishnoi, P. R. Kinetics of carbon dioxide and methane hydrate formation. Can. J. Chem. Eng. 1997, 75, 1090−1098. (9) Herri, J.; Gruy, F.; Cournil, M. Methane hydrate crystallization mechanism from in-situ particle sizing. AIChE J. 1999, 45, 590−602. (10) 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. (11) Al-Otaibi, F.; Clarke, M.; Maini, B.; Bishnoi, P. R. Formation kinetics of structure I clathrates of methane and ethane using an in situ particle size analyzer. Energy Fuels 2010, 24, 5012−5022. (12) Malegaonkar, M. Kinetics of gas hydrate formation from methane, carbon dioxide and their mixtures. MSc. Thesis, University of Calgary, Calgary, Alberta, Canada, 1996. (13) Mork, M.; Gudmundsson, J. S. Hydrate Formation Rate in a Continuous Stirred Tank Reactor: Experimental Results and Bubbleto-Crystal Model. Presented at the Fourth International Conference on Gas Hydrates, Yokohama, Japan, May 19−23, 2002; http://www. ipt.ntnu.no/∼jsg/publikasjoner/papers2002/YokohamaReactor.pdf (accessed Feb 25, 2013). (14) Peng, D.; Robinson, D. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (15) 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. (16) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264−270. (17) Hayduk, W.; Laudie, H. Prediction of diffusion coefficients for nonelectrolytes in dilute aqueous solutions. AIChE J. 1974, 20, 611− 615. (18) van der Waals, J. H.; Platteeuw, J. C. Clathrate solutions. Adv. Chem. Phys. 1959, 2 (1), 1−57.

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 NSERC, a Ursula and Herbert Zandmer Graduate Scholarship, and the Department of Chemical and Petroleum Engineering is highly appreciated.

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NOMENCLATURE

a = gas−liquid interfacial area per unit volume (m2/m3) Ag−l = gas−liquid interfacial area (m2) Ap = surface area per unit volume of a gas hydrate particle (m2/m3) c = concentration (mol/m 3) Dj = diffusivity of component j in the gas phase (m2/s) f = fugacity (MPa) f b = fugacity of the hydrate former in the bulk liquid phase (MPa) feq = equilibrium fugacity (MPa) fg = fugacity of the hydrate former in the vapor phase (MPa) Hj = Henry’s constant of component j (MPa) K*j = combined rate parameter for component j [mol/ (m2·MPa·s)] L = characteristic length (m) n = number of moles (mol) NC = number of hydrate-forming components Ry(t) = global rate of reactions [mol/(m3·s)] t = time (s) V = volume of reacting mass (m3) yL = thickness of the gas−liquid interface boundary layer (m)

Greek Letters

γ = Hatta number μ2 = second moment of the particle size distribution (mn/ m3) φ = particle density function (m−4) ωj = stoichiometric coefficient of component j in the hydrate phase Subscripts and Superscripts

b = bulk liquid phase eq = three-phase (gas−liquid-hydrate) equilibrium conditions g = gas phase g−l = gas−liquid interface p = hydrate particle wo = water

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REFERENCES

(1) U.S. and Japan Complete Successful Field Trial of Methane Hydrate Production Technologies; U.S. Department of Energy: Washington, DC, 2012; available at http://www.netl.doe.gov/publications/press/2012/ 120502_us_and_japan.html. (2) Majorowicz, J. A.; Osadetz, K. G. Gas hydrate distribution and volume in Canada. Am. Assoc. Pet. Geol. Bull. 2001, 85, 1211−1230. (3) Oghaki, K.; Takano, K.; Sangawa, H.; Matsubara, T.; Nakano, S. Methane exploitation by carbon dioxide from gas hydratesPhase equilibria for CO2−CH4 mixed hydrate system. J. Chem. Eng. Jpn. 1996, 29, 478−483. 1211


A Simplified Approach to Modeling the Rate of Formation of Gas

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