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Diffusion Model for Gas Replacement in Isostructural CH-CO Hydrate System Andrey N Salamatin, Andrzej Falenty, and Werner F. Kuhs J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b04391 • Publication Date (Web): 21 Jul 2017 Downloaded from http://pubs.acs.org on July 23, 2017

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The Journal of Physical Chemistry

Diffusion Model for Gas Replacement in Isostructural CH4-CO2 Hydrate System A.N. Salamatin 1, A. Falenty 2and W.F. Kuhs 2,* 1

Dept. of Applied Mathematics, Kazan (Volga Region) Federal University, 420008 Kazan, Russia

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GZG Abt. Kristallographie, Georg-August-Universität Göttingen, 37077 Göttingen, Germany

* Werner F. Kuhs, [email protected], tel. +49 551-39-3891, fax. +49 551-39-95-21

ABSTRACT. Guest exchange in clathrates is a complex activated phenomenon of the guest – host cage interaction on the molecular-scale level. To model this process, we develop a mathematical description for the non-equilibrium binary permeation of guest molecules during gas replacement based on the microscopic “hole-in-cage-wall” diffusive mechanism. The transport of gas molecules is envisaged as a series of jumps between occupied and empty neighboring cages without any significant lattice restructuring in the bulk. The gas exchange itself is seen as a two-stage swapping initiated by an almost instantaneous formation of a mixed hydrate layer on the hydrate surface followed by a much slower permeation-controlled process. The model is constrained by and validated with available time-resolved neutron diffraction data of the isostructural CH4-guest replacement by CO2 in methane hydrate, a process of possible ACS Paragon Plus Environment

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importance for the sequestration of CO2 with concomitant recovery of CH4 in marine gas hydrates.

Introduction The replacement of hydrocarbon-containing natural gas hydrates (GH) with CO2 or flue gas (CO2+N2) can be seen as an elegant way to combine resource recovery and sequestration of combustion products. From the thermodynamic standpoint the exchange between existing hydrates and injected fluids lowers the total free energy essentially without the involvement of latent heat thus is energetically more favorable than the gas recovery via decomposition. Indeed, the principal viability of this approach was repeatedly confirmed in laboratory experiments e.g. 15

and a field test 6, 7. Moreover, due to only marginal alteration of the existing microstructures 1, 8,

9

gas recovery based on this gas swapping method carries substantially lower hazard potential to

the mining operations and local environment. Moreover, the problematic water production accompanying any standard gas hydrate decomposition is considerably reduced in case of methane extraction via a gas exchange process

10

. In spite of these obvious advantages the

development of replacement technologies is hampered by the lack of a consistent picture of the process and the resulting inability to predict methane production rates needed for the calculation of the economic balance 1. With the recent progress in the understanding of kinetic factors and fundamental, molecular-scale mechanisms that allow relatively large gas molecules to move between hydrate cages it becomes now possible to develop a mathematical description of the exchange with predictive power. An important first milestone to achieve this goal has been set by the clear identification of two very distinct stages of the exchange: 1) The initial rapid, shortlasting surface reaction leading to a mixed hydrate layer formation followed by 2) a much slower replacement in the grains’ interior with clear indications for solid state diffusion. Convincing ACS Paragon Plus Environment

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experimental evidence

8, 11-15

reinforced by theoretical

8

and computational

13, 15

consideration

indicate that the rapid reaction in the first minutes may be actually not an inter-cage exchange but a full destruction and reconstruction of the interface. Melting of the contact surface is triggered by a sudden change in the chemical potentials and equilibration of the water vapor and guest components between the reservoir and injected dry fluid. The penetration depth in case of pure CO2 has been estimated about 10 µm or higher 1. Notably, less favorable GH-forming fluids like CO2+N2 are likely to increase this initial reaction volume, thus artificially improving the replacement efficiency via destruction/reformation processes on the gas hydrate surface. As soon as the affected outer film is reformed with a composition equilibrated to the local surrounding fluid, the remaining parent hydrate becomes protected from further decomposition; at that specific moment, the kinetic curves mark a clear change in the reaction rate as the further conversion is controlled by solid state diffusion

2, 4, 14, 16

. The nature of this slow process had

remained obscure for a long time and was limited to ad hoc assumptions of the permeation for the different gases. The first, thermodynamically consistent, approach

17

set the right track, but

had to neglect the coupled interference between in- and out-diffusion, assumed a locally equilibrated gas composition in smaller and larger cages and clearly demonstrated the need of a deeper understanding of guest transport mechanisms on a microscopic/molecular level. A further milestone, indispensable for a more physical modeling, has been reached with the recent progress in the understanding of the guest permeation in single guest hydrates. Both computational

18-20

and experimental

21-23

evidences strongly suggest that guest molecules can

hop between cages without significant overall changes to the host water network, even if their nominal size exceeds the opening in the hexagonal or pentagonal cage walls. This random walk of guest species is enabled by the presence of empty cages and is assisted by intrinsic and extrinsic water vacancies that, by migrating though the host lattice, can open and close cage walls, resulting in the so-called "hole-in-cage-wall" permeation scenario 21, 22 (Fig. 1). In case of ACS Paragon Plus Environment

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binary and more complex systems the migration appears to be guest specific and preferentially proceeds along interconnected larger cavities 1. Smaller cages of type sI (and, possibly, sII) structures are likely to not actively participate in the replacement process and serve primarily as sinks and sources for guest molecules. The rate-limiting process that controls the gas transport rates at high temperatures has been traced to the creation of water vacancy-interstitial pairs, which, in turn, determines the activation energy and temperature dependency of the permeation rates. It is noteworthy that – in contrast to the formation of GH from ice

21, 22

– the transport of

water molecules appears to be unimportant in the gas exchange process. A recent publication24, referring to the initial approach17, assumed, without any foundation, that the methane fugacity and carbon-dioxide cage-averaged occupancy in the mixed hydrate are linearly correlated with the mean CH4 content. Such an approximation, cannot properly account for changes in (p, T)-conditions and changing mixed hydrate composition (total occupancy). As a result, the driving force of diffusion was not correctly represented in the study. This and further questionable assumptions about the SSA of the used hydrate samples lead to unreasonably high activation energies and a wrong interpretation of the obtained data. In the following we take the basic concept of

17

as starting point and extend the theory of the

"hole-in-cage-wall" diffusion in single-gas clathrate hydrates

21

to develop a physical model for

the binary diffusion in gas hydrates of structure I that is validated by the kinetic exchange data in CH4-CO2 systems presented earlier on 1. Although the experiments cover only a timescale of about 30h, they run well into the diffusion-limited stage so that reliable information could be gained; there is no physical reason for any changes in this regime at much longer time-scales so that reliable extrapolations can be made eventually.

Experimental

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Two batches of deuterated CH4 hydrates used in the neutron diffraction experiments

1

were

grown at 276 K and 6 MPa (methane fugacity fA* = 5.09 MPa) from ice spheres formed by spraying D2O water into liquid nitrogen

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. Recovered CH4 clathrates were crushed and sieved

through a set of 200 and 300 µm meshes, being transformed to a polydisperse hydrate powder with the mean particle radius 〈r0〉 ≈ 141 µm and normalized standard deviation γ0 ≈ 0.42. Two representative samples in runs 1 and 2 considered below were preconditioned for a few hours in the pressure cell directly before the experiments to reform a minor fraction of GH clathrate surface dissociated during the crushing. The other two samples in runs 3 and 4 had been annealed for 10 days at the melting point of D2O ice to recover original GH surfaces. As a result, the surface re-growth of hydrate particles lead to development of large inter-grain contacts, clearly discerned in SEM images 1. This considerably reduced the samples' porosity and specific surface area exposed to environmental fluid. In-situ CH4 > CO2 exchange experiments performed were followed with the high-flux 2-axis neutron diffractometer D20 at the Institut Laue-Langevin (ILL), Grenoble, France. Quantitative structural information on the exchange kinetics was obtained with a full pattern Rietveld refinement and was already presented in 1. Time resolved information on the fluid composition in pore volume was retrieved in experimental runs 1 and 2 from the background levels of same neutron diffraction patterns by means of a newly developed method 1. However, the uncertainty in sample geometry and much lower exchange rates did not allow in case of run 3 and 4 for complete data refinement and straightforward interpretation. Further details on the experiment can be found in our previous publication 1.

Model Description Thermodynamic properties of mixed binary-gas hydrates


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Two basic crystallographic structures of gas hydrates (GH), the Stackelberg Structure I (sI) and Structure II (sII), with different ratios ν1 and ν2 of small (SC) and large (LC) cages per one water molecule are distinguished 25, 26, depending on the guest-gas nature. In this study, we concentrate on the sI-hydrates engaged in the CH4-CO2 gas-replacement experiments. The principal characteristics of sI-clathrate water framework can be found in 29

26-28

. Also, recent measurements

at temperatures of 270-280 K, typical for gas replacement conditions, predict the sI-hydrate

water frame density of ρw ≈ 44.43±0.11 kmol/m3 with an accuracy better than 0.25%. Theoretical considerations

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express partial fugacities fA and fB of the enclathrated A- and B-

gas solutes in thermodynamic equilibrium with the gaseous constituents in the ambient atmosphere via the respective guest-gas occupancies yAi and yBi in small and large (i = 1, 2) cavities, as

fJ =

yJ1 yJ 2 1 1 = , J = A, B, C J 1 1 − y A1 − y B1 C J 2 1 − y A 2 − y B 2

(1)

where CJi are the temperature dependent Langmuir constants of the J-th gas. Accordingly, chemical potentials µJ of the gas molecules encaged in SCs and LCs are expressed via their partial fugacities fJ or occupancies yJi,

 1 y Ji  C Ji 1 − y i

µ J = R g T ln f J = R g T ln

  , 

J = A, B;

i = 1, 2.

(2)

Here Rg is the gas constant, T is the Kelvin temperature, and yi = yAi + yBi is the total cage occupancy in the hydrate cage of the i-th type. It should be noted that the above approach does not consider all the complexity of mixing effects in details, e.g. the cage occupancies, cage diameters and related physical parameters like lattice constants are assumed to linearly depend on composition. Yet, there are indications that this approximation works reasonably well

30, 31

. On the other hand, the sometimes assumed full

occupancy of small cages 32 does neither find experimental 1 nor theoretical support 33, 34.


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Although theoretical concepts (1) and (2) are originally constrained to certain assumptions, they definitely capture the basic thermodynamic principles, at least locally for the given (p, T)conditions with properly tuned Langmuir constants.

General description of "hole-in-cage-wall" binary diffusion in sI-hydrate Based on analyses of natural phenomena and laboratory observations (e.g.

23, 28, 35

) as well as

molecular modeling 36 and experimental studies 37, 38, a principal conclusion was made22 that the gas diffusion in hydrates, most probably, is preferentially assisted by the presence of empty cages and water vacancies in the cage walls, i.e. is the so-called "hole-in-cage-wall" permeation confirmed later in 18, 20. Small cavities in sI-hydrate are not directly linked to each other, and the activation energy for guest molecules to hop from SC to LC was found

36

to be 1.5-2 times

higher than between LCs. Thus, we assume after 22 that the long-range transport and replacement of guest gases in sI-hydrate mainly proceed through LCs, while SCs work rather as sinks or sources of guest-gas molecules, facilitating the gas transport by evacuating, at least partly, guests from LCs, although without direct participation as the permeation paths in the mass transport process. The theory of the "hole-in-cage-wall" diffusion in single-gas clathrate hydrates has been recently developed and validated 21 in application to CO2-hydrate formation from ice powders. Here we extend this approach to consider the coupled "hole-in-cage-wall" migration of A- and B-gas molecules realized via water vacancies connecting neighboring LC cavities in the crystalline hydrate structure I. Following equilibrium thermodynamics

39, 40

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, in accordance with the general concepts of non-

and the Onsager Reciprocal Relations, for the coupled molar

mass fluxes qA and qB of the gas molecules through larger cages in sI-hydrate lattice, we write at a given (constant) temperature qA = −

LA L ∇µ A − AB ∇µ B , T T

qB = −

L AB L ∇µ A − B ∇µ B , T T


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where LA, LB, and LAB are the mass transport (phenomenological) coefficients and ∇ is the differential Nabla-operator. The LA- and LB-factors are conventionally (e.g. 39) assumed to be proportional to the respective molar concentrations, yA2, yB2, and potential individual mobilities, θA, θB, of the guest-gas molecules in LC cavities. At the same time, following

21

, they must be additionally reduced by

the probability for gas molecules to meet an empty large cage, i.e., L J = ρ wν 2 y J 2 (1 − y 2 )θ J ,

J = A, B .

As for the coupling effects of simultaneous jumps of A- and B-species, we formally assume L AB = ρ wν 2 y A 2 y B 2 (1 − y 2 )θ AB .

Combining the latter relations with Eqs. (2), one finally arrives at the generalized Fick's law for the binary "hole-in-cage-wall" diffusion in sI-hydrate q A = − ρ wν 2 ( D A ∇y A2 + D AB ∇y B 2 ) ,

q B = − ρ wν 2 ( D BA ∇y A 2 + D B ∇y B 2 ) ,

(3)

where, by definition, the diffusion coefficients are given by D A = R g [(1 − y B 2 )θ A + y A 2 y B 2θ AB ] , D BA = R g y B 2 [θ B + (1 − y B 2 )θ AB ] ,

D AB = R g y A 2 [θ A + (1 − y A 2 )θ AB ] ; D B = R g [(1 − y A2 )θ B + y A 2 y B 2θ AB ]

and reveal strong non-linear dependence of the two-component diffusive transport on distribution of A- and B-species in LCs, i.e. on their cage occupancies, yA2 and yB2. Accordingly, it becomes possible to describe the A/B-guest replacement in the mixed hydrate structure I in terms of the single-gas diffusion coefficients, D J0 = R g θ J , J = A, B , e.g. inferred from A- or B-hydrate formation kinetics in ice powders 21-23, D A = (1 − y B 2 ) D A0 + y A2 y B 2ξ AB ,

D AB = y A 2 [ D A0 + (1 − y A2 )ξ AB ] ;

D BA = y B 2 [ D B0 + (1 − y B 2 )ξ AB ] ,

D B = (1 − y A 2 ) D B0 + y A2 y B 2ξ AB ,

(4)

with only one new (tuning) parameter ξAB = RgθAB, which controls the diffusive coupling of the guest gases and must be additionally specified in experiments. ACS Paragon Plus Environment

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Next, it should be emphasized that the diffusive gas-mass transfer through the non-equilibrium mixed (A + B)-gas hydrate bulk is a relaxation phenomenon which reveals itself as coupled Aand B-guest exchange between smaller and larger cavities, driven by the local differences between corresponding chemical potentials of the guest molecules. Hence, following

21

, we

introduce the mass fluxes σA and σB of A- and B-gas molecules from LCs to SCs in a unit of GH (water framework) volume per a unit of time, and, again, based on the conceptual considerations of non-equilibrium thermodynamics, with the use of Eqs. (2), we finally arrive (see Appendix A, Supporting Information) at the generalized analogues of the constitutive relations for σA and σB:

σ A = ρ wν 1ν 2 χ A0 [C A1 y A 2 (1 − y1 ) − C A 2 y A1 (1 − y 2 )] + + ρ wν 1ν 2ζ AB C A1 y A 2 [C B1 y B 2 (1 − y1 ) − C B 2 y B1 (1 − y 2 )] , (5)

σ B = ρ wν 1ν 2ζ AB C B1 y B 2 [C A1 y A 2 (1 − y1 ) − C A 2 y A1 (1 − y 2 )] + + ρ wν 1ν 2 χ B0 [C B1 y B 2 (1 − y1 ) − C B 2 y B1 (1 − y 2 )] , where χA0 and χB0 are the respective guest exchange coefficients between smaller and larger cages in single A- or B-gas clathrates, while ζAB is the mass exchange coupling parameter; C J i = C J i /(C J 1 + C J 2 ) , J = A, B, i = 1, 2, are the normalized Langmuir constants.

Reduced mass balance equations of binary diffusion in sI-hydrate Let us formulate the mass conservation laws for A- and B-guest molecules in small and large cages. For SCs, only gas exchange with neighboring LCs can influence their composition, and for a unit of GH volume per a unit of time t, we write

ρ wν 1

∂y A1 =σA , ∂t

ρ wν 1

∂y B1 =σB . ∂t


(6)

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The guest occupancies in LCs, being additionally affected by the diffusive molar mass fluxes qA and qB of gas molecules through larger cages, are governed by the overall mass balance

equations for each of A- and B-guests in hydrate:

ρ wν 1

∂y A1 ∂y + ρ wν 2 A 2 + ∇ ⋅ q A = 0 , ∂t ∂t

ρ wν 1

∂y B1 ∂y + ρ wν 2 B 2 + ∇ ⋅ q B = 0 . ∂t ∂t

Available quantitative data on the gas replacement

1, 9

(7)

point at possibly rather slow

readjustment of guest composition in SCs. Hence, in general, transient SC-occupancies of A- and

B-guests in Eqs. (6) and (7) cannot be directly identified with their equilibrium values corresponding to LC-filling,

y J1 = y J 2

CJ1 CJ 2

  C A1   C   − y B 2 1 − B1  , 1 − y A 2 1 −  C A2   C B 2  

J = A, B.

(8)

Preliminary estimates based on CSMGem software predictions 41 show that, at least in case of methane replacement by carbon dioxide from CH4-hydrate, the total LC filling y2 remains at a highest level of ~95-98% and vary slowly by not more than 1-2% with changes in guest composition, i.e. ∂y2/∂t → 0. Furthermore, at y2 ≈ 1, the total equilibrium SC-occupancy y 1 = y A1 + y B1 determined by Eqs. (8) must be also fairly high and little dependent on A- and Bgas exchange. CSMGem calculations confirm that the equilibrium SC-filling in mixed CH4-CO2 hydrate remains within ~75-85%-range up to 80%-level of CH4 extraction. Thus, especially in case of slow CO2-content readjustment in SCs, it may be assumed that y1 also does not vary much during an active phase of gas replacement, at the minimum rates ∂y1/∂t → 0. Consequently, at ν1 CO2 Replacement Used in Long-Term Modelling


32

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Fig. 1. Schematic drawing of the migration paths of water vacancies (white spheres and thin broken arrows) and CH4- and CO2-guest atoms (thick dark green and grey arrows) in a single crystalline sI hydrate lattice represented by polyhedral water cages. The cage color stands for the type of guest atom: CH4 in light green (large cages) and dark green (small cages), CO2 in light grey (large cages) and dark grey (small cages). A water vacancy is needed for a successful jump of a guest molecule into a neighboring (empty) cage. The guest migration paths are following mostly connected large cages. 43x29mm (300 x 300 DPI)


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Fig.2 Comparison of the gas-replacement kinetic data (red dots) and the best-fit model predictions (solid lines) in case of run 1: (a) volume fraction of mixed GH phase, (b) CO2-occupancy in LC’s in mixed GH phase, (c) CO2-molar fraction in environmental fluid, and (d) overall CO2-cage-weighted occupancy in GH versus time. Dashed and dotted lines are the sensitivity tests (see text) 288x201mm (300 x 300 DPI)



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Fig. 3 Comparison of the gas-replacement kinetic data (black dots) and the best-fit model predictions (solid lines) in case of run 2: (a) volume fraction of mixed GH phase, (b) CO2-occupancy in LC’s in mixed GH phase, (c) CO2-molar fraction in environmental fluid, and (d) overall CO2-cage-weighted occupancy in GH versus time. Dashed lines are the sensitivity tests (see text) 288x201mm (300 x 300 DPI)



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Fig. 4 Terminal simulated CH4- and CO2-occupancy profiles in large and small cages in (a) run 1 and (b) run 2 conditions for the fitted data 296x209mm (300 x 300 DPI)



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Fig. 4 Terminal simulated CH4- and CO2-occupancy profiles in large and small cages in (a) run 1 and (b) run 2 conditions for the fitted data 296x209mm (300 x 300 DPI)


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Fig. 5 CH4-extraction degree (a) and mixed GH volume fraction (b) in GH-particles in runs 1 and 2, in gaseous and liquid CO2-environment (red and black curves, respectively) extrapolated over 1000h. 288x201mm (300 x 300 DPI)



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Fig. 5 CH4-extraction degree (a) and mixed GH volume fraction (b) in GH-particles in runs 1 and 2, in gaseous and liquid CO2-environment (red and black curves, respectively) extrapolated over 1000h. 288x201mm (300 x 300 DPI)


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Fig. 6 CH4- and CO2-occupancy profiles in large and small cages predicted in (a) run 1 and (b) run 2 conditions at the end of 1000 h-replacement 288x201mm (300 x 300 DPI)



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Fig. 6 CH4- and CO2-occupancy profiles in large and small cages predicted in (a) run 1 and (b) run 2 conditions at the end of 1000 h-replacement 288x201mm (300 x 300 DPI)


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Fig. 7 CH4-extraction degree growth (a), apparent relative radius (b) of vanishing un-reacted parent hydrate core in particles versus dimensionless time Fo at 10 % CH4-concentration in the fluid (277 K, 6 MPa), and the cage occupancy profiles (c) in a hydrate particle versus relative radius at the end of the mixed-hydrate layer development, Fo ~ 10, at about 70% of extraction degree; sensitivity of extraction degree rates (d) to particle size (r0 = 69, 242, and 830 µm) versus real time. 288x201mm (300 x 300 DPI)



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Fig. 8 Long-term CH4-extraction degree versus time in a particle of size r0 = 242 µm at 6 MPa for (a) different CH4-concentrations in the fluid at 277 K and (b) different temperatures at 10 % CH4-concentration in the fluid. 288x201mm (300 x 300 DPI)



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Fig. 8 Long-term CH4-extraction degree versus time in a particle of size r0 = 242 µm at 6 MPa for (a) different CH4-concentrations in the fluid at 277 K and (b) different temperatures at 10 % CH4-concentration in the fluid. 288x201mm (300 x 300 DPI)


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Diffusion Model for Gas Replacement in an ... - ACS Publications

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