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Diffusion model of gas hydrate dissociation into ice and gas: Simulation of the self-preservation effect
International Journal of Heat and Mass Transfer 102 (2016) 631–636
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Diffusion model of gas hydrate dissociation into ice and gas: Simulation of the self-preservation effect Valeriy A. Vlasov Institute of the Earth Cryosphere, Siberian Branch of the Russian Academy of Sciences, P.O. Box 1230, 625000 Tyumen, Russian Federation
a r t i c l e
i n f o
Article history: Received 22 April 2016 Received in revised form 31 May 2016 Accepted 20 June 2016
Keywords: Gas hydrate Diffusion Chemical kinetics Kinetic model Self-preservation effect
a b s t r a c t The diffusion model of gas hydrate dissociation into ice and gas is presented. This model takes into account the possible porous structure of the formed layer of ice as well as the intrinsic kinetics of the process of gas hydrate dissociation into ice and gas. Specifically, the problem of the dissociation of a spherical gas hydrate particle into ice and gas was considered. In the framework of a quasi-stationary approximation, a simplified solution to this problem was obtained. From the comparison between the calculated data obtained in the framework of a quasi-stationary approximation and the available experimental data, the parameters of the developed model responsible for the kinetics of methane hydrate dissociation into ice and gas were estimated. In the framework of the developed diffusion model, an explanation for the anomalous preservation thermal regime of gas hydrates is given. From this explanation it follows that for some hydrate-forming gases such a regime may not occur. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the study of gas hydrates dissociation at temperatures below 273 K, their slow decomposition into ice and gas was detected [1,2]. This phenomenon is called the self-preservation effect. The self-preservation effect is attracting increasing interest because it can be the basis for the gas hydrate technologies for the longterm storage and transportation of various gases [3–5]. The self-preservation effect is being actively investigated [6–25]. This effect is observed during gas hydrates dissociation into ice and gas in the whole region of their thermodynamic instability at temperatures below the ice melting point. The self-preservation effect manifests itself not only in massive gas hydrates but also in disperse gas hydrates in some carrier medium [26–30]. The self-preservation effect is generally attributed to the formation of the ice layer on the gas hydrate surface. Depending on the temperature and pressure conditions, the ice layer may have a different degree of homogeneity. Herewith, the rate of gas hydrate dissociation into ice and gas is limited by the diffusion of the gas through this ever-increasing ice layer. It is obvious that the size effect will be the consequence of this dissociation mechanism: the larger the gas hydrate sample is, the longer it can exist in the thermodynamic instability region due to the self-preservation effect. The size effect is really manifested in experiments [31–33]. Moreover, the protective ice layer may appear after crystallisation of supercooled water,
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.06.057 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
which in turn may appear during the process of gas hydrate dissociation at temperatures below 273 K [34–38]. In works [39–42], methane hydrate dissociation into ice and gas at different fixed temperatures and a pressure of 0.1 MPa was investigated. These investigations revealed one feature: the rate of methane hydrate dissociation into ice and gas increases monotonically with increasing temperature in the range of 193–240 K. In the temperature range of 242–271 K, the anomalous preservation thermal regime (anomalous preservation effect) is observed: the rate of methane hydrate dissociation into ice and gas decreases noticeably and non-monotonically. According to the authors of works [39–41], for an explanation of the anomalous preservation thermal regime, it is not enough to assume that the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the ice layer. In this case, the molecular processes occurring at the ice–hydrate interface may be important [16,19]. Presently, there is no full kinetic model of gas hydrate dissociation into ice and gas. In this paper, an attempt is made to create such a model. The presented model is based on the use of the diffusion equation, and when describing the processes occurring at the ice–hydrate interface, the general theory of chemical kinetics is used. Previously, such an approach has been used to create a kinetic model of gas hydrate formation from ice [43,44]. 2. Theory Assume that gas hydrate dissociation into ice and gas proceeds by the diffusion mechanism: the gas molecules G diffuse through
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the formed ice layer from the ice–hydrate interface on which the chemical reaction occurs
G þ nH2 OðsolidÞ $ G nH2 O;
ð1Þ
where n is the hydration number. In the case under consideration, the rate of the forward reaction is lower than the rate of the reverse reaction. The thickness of the ice layer increases due to gas hydrate dissociation. Let us consider the situation in which the temperature at the front of the reaction (1) does not change during the process of gas hydrate dissociation. This situation is implemented under isothermal conditions in the case where the process of gas hydrate dissociation into ice and gas occurs rather slowly (even at the initial stage) or in the case where good thermal stabilisation exists. In both of these cases, the temperature at the front of the reaction (1) does not have time to change significantly, so we can assume that the process of gas hydrate dissociation into ice and gas occurs in the diffusion regime and should be considered only a diffusion problem. Consider the problem of the dissociation into ice and gas of a single spherical gas hydrate particle with an initial radius R0 (Fig. 1). During the process of gas hydrate dissociation, the radius of the gas hydrate core nðtÞ decreases. We assume that during the process of gas hydrate dissociation, the outside radius of the ice layer RðtÞ also decreases. Such a decrease can be due to the different densities of ice and the gas hydrate. Moreover, we assume that the particle under consideration is found in the atmosphere of the hydrate-forming gas at constant pressure p and constant temperature T. Herewith, the pressure of the hydrate-forming gas is less than the pressure of the ice–gas hydrate–gas equilibrium peq . Note that if gas hydrate dissociation into ice and gas occurs at a fixed temperature in the atmosphere of the non-hydrate-forming gas (as often occurs in real experiments), then it must be assumed that p ¼ 0. We formulate the problem in the spherical coordinate system by choosing its origin at the centre O of the particle under consideration. We take into account the possible presence of pores in ice by introducing the effective diffusion coefficient of the gas in ice:
Deff ¼ D þ Dpor ;
ð2Þ
where D is the diffusion coefficient of the gas in ice and Dpor is the coefficient characterising the diffusion of the gas through the system of pores in ice. In this case, the diffusion equation is written as
! @cðr; tÞ @ 2 cðr; tÞ 2 @cðr; tÞ ; þ ¼ Deff @t @r 2 r @r
t > 0;
where cðr; tÞ is the molar concentration of the gas in the ice layer, r is the radial coordinate, and t is the time. The initial condition is written in the form
0 6 r < R0 :
cðr; tÞjr¼RðtÞ ¼
p ; ZRT
t > 0;
ð5Þ
where Z is the compressibility factor of the gas under current thermobaric conditions and R is the gas constant. Gas molecules appear on the ice–hydrate interface due to reaction (1). Next, these molecules diffuse through the ice layer to the surface of contact of the particle under consideration with the gaseous phase. Thus, the internal boundary condition can be written as
rg ¼ jjr¼nðtÞ ;
t > 0;
ð6Þ
where rg is the rate of change in the moles of the gas on the ice–hydrate interface and j is the magnitude of the molar flux of the gas in the ice layer. In the framework of the general theory of chemical kinetics, it can be shown [45] that the following relation is satisfied for the quantity r g :
rg ¼ kdis v
! Z eq RT 1 cðr; tÞjr¼nðtÞ ; peq
t > 0;
ð7Þ
where kdis is the rate constant for the gas hydrate dissociation reaction at the ice–hydrate interface, v is the molar density of the gas hydrate, and Z eq is the compressibility factor of the gas under equilibrium conditions. The rate constant kdis obeys the Arrhenius equation and characterises the intensity of the gas hydrate dissociation reaction on the ice–hydrate interface. The molar flux of the gas in the ice layer is defined as
j ¼ Deff
@cðr; tÞ er ; @r
t > 0;
nðtÞ < r < RðtÞ;
ð8Þ
where er is the radial unit vector. Taking into account Eqs. (7) and (8) and given that r g > 0 and @cðr; tÞ=@r < 0, the internal boundary condition (6) is written in the form
kdis v 1
Z eq RT cðr; tÞjr¼nðtÞ peq
!
¼ Deff
@cðr; tÞ ; @r r¼nðtÞ
t > 0:
ð9Þ
Appendix A shows that there is the following interrelation between the quantities RðtÞ and nðtÞ:
nðtÞ < r < RðtÞ; ð3Þ
cðr; tÞjt¼0 ¼ 0;
Taking into account the fact that the molar concentration of the gas outside the particle under consideration is determined from its thermal equation of state, the external boundary condition is written as follows:
ð4Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 vM h ð1 eh Þ Mg RðtÞ ¼ R30 n3 ðtÞ þ n3 ðtÞ; 1þ xMw ð1 ei Þ nMw
t P 0; ð10Þ
where x is the molar density of ice; Mw , M h and Mg are the molar masses of water, the gas hydrate and the gas, respectively; and ei and eh are the porosities of ice and the gas hydrate, respectively.
Fig. 1. Geometry of the problem at (a) the initial time and (b) later times.
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The equation of motion for the front of the reaction (1) can be obtained from the molar balance ratio for the gas hydrate, which has the form
From Eq. (3), it follows that, in the framework of a quasistationary approximation, the concentration profile cðr; tÞ should satisfy the following equation:
jr h j ¼ vð1 eh Þ
@ 2 cðr; tÞ 2 @cðr; tÞ ¼ 0; þ @r 2 r @r
dRðtÞ ; dt r¼nðtÞ
t > 0;
ð11Þ
where rh is the rate of change in the moles of the gas hydrate on the ice–hydrate interface and RðtÞ is the thickness of the ice layer (RðtÞ ¼ RðtÞ nðtÞ). Note that a similar molar balance ratio can also be written for the water phase. However, as the analysis shows, the choice of one of the two molar balance ratios is equivalent because this choice has little effect on the results of the simulation for the kinetics of gas hydrate dissociation into ice and gas in the framework developed by the model. The molar balance ratio (11) accounts for (by introducing the factor ð1 eh Þ) the possible pore structure of the dissociating gas hydrate. Given the geometry of the problem under consideration, Eq. (11) can be written as follows:
dnðtÞ jr h j ¼ ; dt vð1 eh Þ
t > 0:
ð12Þ
From work [45], it follows that the quantity rh is represented as
t > 0;
nðtÞ < r < RðtÞ:
It is easy to verify that the solution to Eq. (19) is the function of the form
cðr; tÞ ¼
A þ B; r
t > 0;
nðtÞ 6 r 6 RðtÞ;
ð20Þ
where the coefficients A and B do not depend on the radial coordinate r. Determining the coefficients A and B from the boundary conditions (5) and (9) and substituting them into Eq. (20), we obtain that
peq Deff nðtÞ p Zeq RT þ kdis vnðtÞ 1 r cðr; tÞ ¼ ZRT peq Deff þ k vnðtÞ 1 nðtÞ dis Z eq RT RðtÞ 1 kdis vn2 ðtÞ 1r RðtÞ peq ; t > 0; nðtÞ 6 r 6 RðtÞ: þ Z eq RT peq Deff þ k vnðtÞ 1 nðtÞ dis
Z eq RT
RðtÞ
ð21Þ
!
r h ¼ kdis v
Z eq RT cðr; tÞjr¼nðtÞ 1 ; peq
t > 0:
ð13Þ
Taking into account that, in the case of the problem under consideration, r h < 0, Eq. (12) can be written in the form
!
dnðtÞ kdis Z eq RT ¼ cðr; tÞjr¼nðtÞ ; 1 dt peq ð1 eh Þ
t > 0:
ð14Þ
Under the terms of the problem, there is no ice layer at the initial (zero) time. Thus, the initial condition for the quantity nðtÞ is as follows:
nðtÞjt¼0 ¼ R0 :
ð15Þ
Eqs. (3)-(5), (9), (10), (14) and (15) form a closed system of equations describing gas hydrate dissociation into ice and gas. This system of equations was obtained under the assumption that the outside radius of the ice layer changes during gas hydrate dissociation. The case in which this radius does not change is considered in Appendix B. 3. Quasi-stationary approximation
RðtÞ dRðtÞ=dt
ð16Þ
is much larger than the characteristic time for the diffusion of the gas through the ice layer
sdif ¼
R2 ðtÞ Deff
:
ð17Þ
Thus, if the condition
sth Deff ¼ >> 1; t > 0 sdif RðtÞdRðtÞ=dt
After substituting Eq. (21) into Eq. (14), we obtain the differential equation
dnðtÞ kdis Deff ddis h i ; ¼ peq Deff nðtÞ dt vð1 eh Þ Zeq RT v þ kdis nðtÞ 1 RðtÞ
t > 0;
ð22Þ
where
ddis ¼
peq p ; Z eq RT ZRT
ð23Þ
and the quantity RðtÞ is given by Eq. (10). Integration of the differential Eq. (22) with the initial condition (15) leads to the following transcendental equation:
peq 1 ðR0 nðtÞÞ þ ðR2 n2 ðtÞÞ 2Deff 0 Z eq RTkdis v h i2=3 1 R20 R30 k þ n3 ðtÞð1 kÞ 2Deff ð1 kÞ ddis t ; t P 0; ¼ vð1 eh Þ
ð24Þ
where
The presented system of equations can only be solved numerically. However, in the framework of a quasi-stationary approximation, this system of equations can be solved analytically. The quasi-stationary approximation can be used when the characteristic time for an increase in the ice layer thickness
sth ¼
ð19Þ
ð18Þ
is satisfied, then the concentration profile cðr; tÞ can be considered steady in the ice layer at each time. This approach allows us to neglect the partial derivative with respect to time in the diffusion Eq. (3). Further, we shall assume that the condition (18) holds.
k¼
vMh ð1 eh Þ Mg 1þ xMw ð1 ei Þ nMw
1 :
ð25Þ
Using Eq. (24), we can determine the value of the quantity nðtÞ at each time t. 4. Comparison with experimental data The process of gas hydrate dissociation can be described by such a quantity as the degree of gas hydrate dissociation gdis ðtÞ. This quantity is the mass fraction of the dissociated gas hydrate:
gdis ðtÞ ¼
m0h ðtÞ ; mh0
t P 0;
ð26Þ
where m0h ðtÞ is the mass of the dissociated gas hydrate and mh0 is the initial mass of the gas hydrate. It is easy to show that in the case of the dissociation of a spherical gas hydrate particle into ice and gas, we have
gdis ðtÞ ¼ 1
n3 ðtÞ R30
;
t P 0:
ð27Þ
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Note that real experiments studied the dissociation of a powdery gas hydrate into ice and gas, i.e., the dissociation into ice and gas of a large set of gas hydrate particles whose shape is close to spherical. In this case, if the quantity R0 is the characteristic average radius of the gas hydrate particles forming the powder, then Eq. (27) can also be applied. This makes it possible to compare the results of numerical calculations and the results of experiments designed to study the kinetics of the dissociation of a powdery gas hydrate into ice and gas. In the diffusion model presented in this work, there are three main parameters that determine the kinetics of the process of gas hydrate dissociation into ice and gas: the extrinsic parameter ddis and the intrinsic parameters Deff and kdis . The parameter ddis determines the driving force of the process of gas hydrate dissociation into ice and gas, whereas the intrinsic parameters Deff and kdis determine the intensity of this process at the molecular level. Thus, in the framework of the developed model, on the kinetics of gas hydrate dissociation into ice and gas in the general case affects both the intensity of the diffusion process of the gas through the formed ice layer (the parameter Deff ) and the intensity of the gas hydrate dissociation reaction on the ice–hydrate interface (the parameter kdis ). In the particular case, as the analysis shows, one of the parameters Deff or kdis has considerably more influence on the kinetics of gas hydrate dissociation into ice and gas than the other. If the influence of the parameter Deff is much greater than the influence of the parameter kdis , then we can assume that the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the formed ice layer. If the influence of the parameter kdis greatly predominates over the influence of the parameter Deff , then we can assume that the rate of gas hydrate dissociation into ice and gas is limited only by the gas hydrate dissociation reaction on the ice–hydrate interface. In work [7], the dissociation of powdery methane hydrate samples into ice and gas at different temperatures and the Earth’s atmosphere (p ¼ 0) was studied using the energy-dispersive X-ray diffraction method. The characteristic radius of the methane hydrate particles in a single sample ranged from 10 to 25 lm, and these particles themselves were initially prepared from ice particles. The latter fact suggests that the methane hydrate in the experiments [7] had a porous structure [46,47]. Fig. 2 shows the experimental data [7]. Furthermore, Fig. 2 shows the calculated data obtained for each fixed temperature using Eqs. (24) and (27) by assuming that R0 ¼ 17:5 lm and eh ¼ ei ¼ 0:15. These calculated data, which are in good agreement with the experimental data [7], were obtained by varying the parameters Deff and kdis in
178 K
1.0
168 K
158 K
ηdis (t )
0.8 0.6 0.4 0.2 0
0
50
100
150
200
250
300
t, h Fig. 2. Experimental and calculated data for the dissociation of a powdery methane hydrate into ice and gas at atmospheric pressure (p ¼ 0). The symbols are experimental data [7] obtained at different temperatures (the characteristic radius of the methane hydrate particles in a single experiment ranged from 10 to 25 lm); the solid lines are calculated data obtained using Eqs. (24) and (27), where R0 ¼ 17:5 lm and eh ¼ ei ¼ 0:15. The values for the quantities Deff and kdis used in the calculations are presented in Table 1.
Table 1 The values of the quantities Deff and kdis at which there is good agreement between the experimental [7] and calculated data for the dissociation of a powdery methane hydrate into ice and gas. T; K
Deff ; m2/s
kdis ; m/s
158
2:9 1014
> 2 109
168
2:8 10
14
> 3 109
178
2:5 1014
> 5 109
Eq. (24). At such variation was considered (following the authors of work [7]) that the rate of methane hydrate dissociation in the selected temperature range (158–178 K) is limited only by the diffusion of methane through the formed ice layer. Table 1 shows the values of the quantities Deff and kdis at which the calculated curves were constructed in Fig. 2. As mentioned above, when receiving the calculated data shown in Fig. 2, we assumed that the characteristic radius of the methane hydrate particles in the experiments [7] was 17.5 lm, whereas, in reality, this radius varied over a wide range (from 10 to 25 lm). As a result, the data presented in Table 1 should be regarded as estimates. In particular, this is evidenced by the fact that, according to Table 1, the value of the diffusion coefficient Deff slightly decreases with increasing temperature. However, this decrease may be attributed to the morphology of the formed ice layer. Note that the values of the diffusion coefficient Deff presented in Table 1 close in the order of magnitude to the values of the diffusion coefficient of methane in ice, which were calculated in the framework of the method of molecular dynamics [48]. Also note that, as the analysis shows, the ratio sth =sdif decreases with increasing gdis ðtÞ. Herewith, however, the condition (18) is not satisfied only for large values of the quantity gdis ðtÞ (at least this is true in the case of the experimental conditions [7] and for the values of the quantities Deff and kdis given in Table 1). Thus, we can often use Eq. (24) to simulate the kinetics of the dissociation of spherical gas hydrate particles into ice and gas, thereby avoiding the numerical solution of the system of Eqs. (3)-(5), (9), (10), (14) and (15). The values of the quantities Deff and kdis depend on the temperature. This fact allows us to explain the anomalous preservation thermal regime of methane hydrate, which was reported in works [39–42]. The monotonic increase of the rate of methane hydrate dissociation into ice and gas in the temperature range of 193–240 K observed in these works is due to the fact that the rate of methane hydrate dissociation into ice and gas in this temperature range is limited only by the diffusion of methane through the formed ice layer: the parameter Deff has a much greater influence on the kinetics of methane hydrate dissociation into ice and gas than the parameter kdis . In the temperature range of 242–271 K, the intensity of the methane hydrate dissociation reaction on the ice–hydrate interface significantly affects the kinetics of methane hydrate dissociation into ice and gas. In this temperature range, the rate of methane hydrate dissociation into ice and gas is limited both by the diffusion of methane through the formed ice layer and by the methane hydrate dissociation reaction on the ice–hydrate interface: the influence of the parameter Deff on the kinetics of methane hydrate dissociation into ice and gas is not overwhelming compared to the influence of the parameter kdis . A significant influence of the parameter kdis on the kinetics of methane hydrate dissociation into ice and gas leads to a decrease in the rate of methane hydrate dissociation into ice and gas in the temperature range of 242–271 K, which was observed in the works [39–42]. If we assume that the temperature of 241 K is the temperature above which the anomalous preservation thermal regime of methane hydrate begins to occur, then the parameter kdis can be explicitly calculated at this temperature in the framework of the developed diffusion model without an analysis of the
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specific kinetic curve of methane hydrate dissociation into ice and gas. Using in the calculations the value of the parameter Deff equal to the value of the molecular diffusion coefficient of methane in ice, taken from the work [48], we obtain by Eq. (24) that for methane hydrate kdis ¼ ð8 3Þ 108 m=s at a temperature of 241 K. In the framework of the developed diffusion model, the anomalous preservation thermal regime of gas hydrates may potentially be observed for all hydrate-forming gases, not only for methane. Herewith, the explanation for this regime will be similar to the above explanation for methane. However, in the framework of the developed diffusion model, the anomalous preservation thermal regime of gas hydrates may not occur for some hydrateforming gases because, for these gases at temperatures below the ice melting point, the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the formed ice layer.
where mi ðtÞ and mg ðtÞ are the masses of ice and the gas, respectively, and nw ðtÞ and ng ðtÞ are the moles of water and the gas, respectively. From Eqs. (A1)-(A3), it follows that
Mg ; m0h ðtÞ ¼ mi ðtÞ 1 þ nM w
ðA4Þ
Substituting Eq. (A4) into Eq. (26), we can obtain the relation
gdis ðtÞ ¼
mi ðtÞ Mg ; 1þ mh0 nM w
t P 0:
ðA5Þ
Taking into account the geometry of the problem under consideration (Fig. 1), the quantities mi ðtÞ and mh0 can be represented as follows:
mi ðtÞ ¼ qi ð1 ei Þ mh0 ¼ qh ð1 eh Þ
5. Conclusions In the formulation of the diffusion model of gas hydrate dissociation into ice and gas, it was not specified which kind of the hydrate-forming gas is considered. Thus, the diffusion model presented in this work is applicable for all hydrate-forming gases. In this work, the general theory of chemical kinetics was used in the development of the diffusion model of gas hydrate dissociation into ice and gas. This allowed to take into account the intrinsic kinetics of the process of gas hydrate dissociation into ice and gas. The developed diffusion model allows for simulation of the gas hydrates self-preservation effect. In the framework of this model, the anomalous preservation thermal regime of gas hydrates can be explained. It is important to note that the diffusion model presented in this work is universal, i.e., it can be generalised and applied to any geometry of the problem under consideration. Furthermore, the model can be generalised to the case in which the pressure of the hydrate-forming gas is not constant but increases as a result of the process of gas hydrate dissociation. When creating the diffusion model of gas hydrate dissociation into ice and gas, we assumed that the temperature at the front of the gas hydrate formation/dissociation reaction is unchanged. This assumption is not always valid when considering the initial stage of the process of gas hydrate dissociation into ice and gas at high temperatures (in this case, the dissociation occurs intensively). If the assumption regarding the constancy of the temperature at the front of the gas hydrate formation/dissociation reaction is not valid, then it is necessary to solve a diffusion problem involving the heat equation.
t P 0:
4 3 p R ðtÞ n3 ðtÞ ; 3
t P 0;
ðA6Þ
4 3 pR ; 3 0
ðA7Þ
where qi and qh are the mass densities of ice and the gas hydrate, respectively. From Eqs. (A5)–(A7), it follows that
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 q ð1 eh Þ Mg h RðtÞ ¼ R30 gdis ðtÞ þ n3 ðtÞ; 1þ qi ð1 ei Þ nM w
t P 0:
ðA8Þ
Taking into account Eq. (27) and given that qi ¼ xMw and
qh ¼ vMh , Eq. (A8) becomes Eq. (10).
Appendix B. Case in which RðtÞ ¼ const If we can assume that RðtÞ ¼ R0 ¼ const, then it is necessary to solve the system of Eqs. (3)-(5), (9), (14) and (15), where, in Eqs. (3) and (5), RðtÞ is replaced by R0 . Solving this system of equations in the framework of a quasi-stationary approximation, we obtain that
peq 1 2 ðR0 nðtÞÞ þ R n2 ðtÞ 2Deff 0 Z eq RTkdis v 1 3 R n3 ðtÞ 3Deff R0 0 ddis t ; t P 0: ¼ vð1 eh Þ
ðB1Þ
Solving Eq. (B1), it is possible to determine the value of the quantity nðtÞ at each time t . References
Acknowledgments The author is grateful to Dr. A.N. Nesterov for a useful discussion of the gas hydrates self-preservation effect. Appendix A. Derivation of Eq. (10) During the process of gas hydrate dissociation into ice and gas, the following relations are satisfied:
m0h ðtÞ ¼ mi ðtÞ þ mg ðtÞ; ng ðtÞ ¼
mg ðtÞ ; Mg
nw ðtÞ ¼ nng ðtÞ ¼
t P 0;
t P 0; mi ðtÞ ; Mw
ðA1Þ ðA2Þ
t P 0;
ðA3Þ
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Diffusion model of gas hydrate dissociation into ice and gas: Simulation of the self-preservation effect Valeriy A. Vlasov Institute of the Earth Cryosphere, Siberian Branch of the Russian Academy of Sciences, P.O. Box 1230, 625000 Tyumen, Russian Federation
a r t i c l e
i n f o
Article history: Received 22 April 2016 Received in revised form 31 May 2016 Accepted 20 June 2016
Keywords: Gas hydrate Diffusion Chemical kinetics Kinetic model Self-preservation effect
a b s t r a c t The diffusion model of gas hydrate dissociation into ice and gas is presented. This model takes into account the possible porous structure of the formed layer of ice as well as the intrinsic kinetics of the process of gas hydrate dissociation into ice and gas. Specifically, the problem of the dissociation of a spherical gas hydrate particle into ice and gas was considered. In the framework of a quasi-stationary approximation, a simplified solution to this problem was obtained. From the comparison between the calculated data obtained in the framework of a quasi-stationary approximation and the available experimental data, the parameters of the developed model responsible for the kinetics of methane hydrate dissociation into ice and gas were estimated. In the framework of the developed diffusion model, an explanation for the anomalous preservation thermal regime of gas hydrates is given. From this explanation it follows that for some hydrate-forming gases such a regime may not occur. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the study of gas hydrates dissociation at temperatures below 273 K, their slow decomposition into ice and gas was detected [1,2]. This phenomenon is called the self-preservation effect. The self-preservation effect is attracting increasing interest because it can be the basis for the gas hydrate technologies for the longterm storage and transportation of various gases [3–5]. The self-preservation effect is being actively investigated [6–25]. This effect is observed during gas hydrates dissociation into ice and gas in the whole region of their thermodynamic instability at temperatures below the ice melting point. The self-preservation effect manifests itself not only in massive gas hydrates but also in disperse gas hydrates in some carrier medium [26–30]. The self-preservation effect is generally attributed to the formation of the ice layer on the gas hydrate surface. Depending on the temperature and pressure conditions, the ice layer may have a different degree of homogeneity. Herewith, the rate of gas hydrate dissociation into ice and gas is limited by the diffusion of the gas through this ever-increasing ice layer. It is obvious that the size effect will be the consequence of this dissociation mechanism: the larger the gas hydrate sample is, the longer it can exist in the thermodynamic instability region due to the self-preservation effect. The size effect is really manifested in experiments [31–33]. Moreover, the protective ice layer may appear after crystallisation of supercooled water,
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.06.057 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
which in turn may appear during the process of gas hydrate dissociation at temperatures below 273 K [34–38]. In works [39–42], methane hydrate dissociation into ice and gas at different fixed temperatures and a pressure of 0.1 MPa was investigated. These investigations revealed one feature: the rate of methane hydrate dissociation into ice and gas increases monotonically with increasing temperature in the range of 193–240 K. In the temperature range of 242–271 K, the anomalous preservation thermal regime (anomalous preservation effect) is observed: the rate of methane hydrate dissociation into ice and gas decreases noticeably and non-monotonically. According to the authors of works [39–41], for an explanation of the anomalous preservation thermal regime, it is not enough to assume that the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the ice layer. In this case, the molecular processes occurring at the ice–hydrate interface may be important [16,19]. Presently, there is no full kinetic model of gas hydrate dissociation into ice and gas. In this paper, an attempt is made to create such a model. The presented model is based on the use of the diffusion equation, and when describing the processes occurring at the ice–hydrate interface, the general theory of chemical kinetics is used. Previously, such an approach has been used to create a kinetic model of gas hydrate formation from ice [43,44]. 2. Theory Assume that gas hydrate dissociation into ice and gas proceeds by the diffusion mechanism: the gas molecules G diffuse through
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the formed ice layer from the ice–hydrate interface on which the chemical reaction occurs
G þ nH2 OðsolidÞ $ G nH2 O;
ð1Þ
where n is the hydration number. In the case under consideration, the rate of the forward reaction is lower than the rate of the reverse reaction. The thickness of the ice layer increases due to gas hydrate dissociation. Let us consider the situation in which the temperature at the front of the reaction (1) does not change during the process of gas hydrate dissociation. This situation is implemented under isothermal conditions in the case where the process of gas hydrate dissociation into ice and gas occurs rather slowly (even at the initial stage) or in the case where good thermal stabilisation exists. In both of these cases, the temperature at the front of the reaction (1) does not have time to change significantly, so we can assume that the process of gas hydrate dissociation into ice and gas occurs in the diffusion regime and should be considered only a diffusion problem. Consider the problem of the dissociation into ice and gas of a single spherical gas hydrate particle with an initial radius R0 (Fig. 1). During the process of gas hydrate dissociation, the radius of the gas hydrate core nðtÞ decreases. We assume that during the process of gas hydrate dissociation, the outside radius of the ice layer RðtÞ also decreases. Such a decrease can be due to the different densities of ice and the gas hydrate. Moreover, we assume that the particle under consideration is found in the atmosphere of the hydrate-forming gas at constant pressure p and constant temperature T. Herewith, the pressure of the hydrate-forming gas is less than the pressure of the ice–gas hydrate–gas equilibrium peq . Note that if gas hydrate dissociation into ice and gas occurs at a fixed temperature in the atmosphere of the non-hydrate-forming gas (as often occurs in real experiments), then it must be assumed that p ¼ 0. We formulate the problem in the spherical coordinate system by choosing its origin at the centre O of the particle under consideration. We take into account the possible presence of pores in ice by introducing the effective diffusion coefficient of the gas in ice:
Deff ¼ D þ Dpor ;
ð2Þ
where D is the diffusion coefficient of the gas in ice and Dpor is the coefficient characterising the diffusion of the gas through the system of pores in ice. In this case, the diffusion equation is written as
! @cðr; tÞ @ 2 cðr; tÞ 2 @cðr; tÞ ; þ ¼ Deff @t @r 2 r @r
t > 0;
where cðr; tÞ is the molar concentration of the gas in the ice layer, r is the radial coordinate, and t is the time. The initial condition is written in the form
0 6 r < R0 :
cðr; tÞjr¼RðtÞ ¼
p ; ZRT
t > 0;
ð5Þ
where Z is the compressibility factor of the gas under current thermobaric conditions and R is the gas constant. Gas molecules appear on the ice–hydrate interface due to reaction (1). Next, these molecules diffuse through the ice layer to the surface of contact of the particle under consideration with the gaseous phase. Thus, the internal boundary condition can be written as
rg ¼ jjr¼nðtÞ ;
t > 0;
ð6Þ
where rg is the rate of change in the moles of the gas on the ice–hydrate interface and j is the magnitude of the molar flux of the gas in the ice layer. In the framework of the general theory of chemical kinetics, it can be shown [45] that the following relation is satisfied for the quantity r g :
rg ¼ kdis v
! Z eq RT 1 cðr; tÞjr¼nðtÞ ; peq
t > 0;
ð7Þ
where kdis is the rate constant for the gas hydrate dissociation reaction at the ice–hydrate interface, v is the molar density of the gas hydrate, and Z eq is the compressibility factor of the gas under equilibrium conditions. The rate constant kdis obeys the Arrhenius equation and characterises the intensity of the gas hydrate dissociation reaction on the ice–hydrate interface. The molar flux of the gas in the ice layer is defined as
j ¼ Deff
@cðr; tÞ er ; @r
t > 0;
nðtÞ < r < RðtÞ;
ð8Þ
where er is the radial unit vector. Taking into account Eqs. (7) and (8) and given that r g > 0 and @cðr; tÞ=@r < 0, the internal boundary condition (6) is written in the form
kdis v 1
Z eq RT cðr; tÞjr¼nðtÞ peq
!
¼ Deff
@cðr; tÞ ; @r r¼nðtÞ
t > 0:
ð9Þ
Appendix A shows that there is the following interrelation between the quantities RðtÞ and nðtÞ:
nðtÞ < r < RðtÞ; ð3Þ
cðr; tÞjt¼0 ¼ 0;
Taking into account the fact that the molar concentration of the gas outside the particle under consideration is determined from its thermal equation of state, the external boundary condition is written as follows:
ð4Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 vM h ð1 eh Þ Mg RðtÞ ¼ R30 n3 ðtÞ þ n3 ðtÞ; 1þ xMw ð1 ei Þ nMw
t P 0; ð10Þ
where x is the molar density of ice; Mw , M h and Mg are the molar masses of water, the gas hydrate and the gas, respectively; and ei and eh are the porosities of ice and the gas hydrate, respectively.
Fig. 1. Geometry of the problem at (a) the initial time and (b) later times.
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The equation of motion for the front of the reaction (1) can be obtained from the molar balance ratio for the gas hydrate, which has the form
From Eq. (3), it follows that, in the framework of a quasistationary approximation, the concentration profile cðr; tÞ should satisfy the following equation:
jr h j ¼ vð1 eh Þ
@ 2 cðr; tÞ 2 @cðr; tÞ ¼ 0; þ @r 2 r @r
dRðtÞ ; dt r¼nðtÞ
t > 0;
ð11Þ
where rh is the rate of change in the moles of the gas hydrate on the ice–hydrate interface and RðtÞ is the thickness of the ice layer (RðtÞ ¼ RðtÞ nðtÞ). Note that a similar molar balance ratio can also be written for the water phase. However, as the analysis shows, the choice of one of the two molar balance ratios is equivalent because this choice has little effect on the results of the simulation for the kinetics of gas hydrate dissociation into ice and gas in the framework developed by the model. The molar balance ratio (11) accounts for (by introducing the factor ð1 eh Þ) the possible pore structure of the dissociating gas hydrate. Given the geometry of the problem under consideration, Eq. (11) can be written as follows:
dnðtÞ jr h j ¼ ; dt vð1 eh Þ
t > 0:
ð12Þ
From work [45], it follows that the quantity rh is represented as
t > 0;
nðtÞ < r < RðtÞ:
It is easy to verify that the solution to Eq. (19) is the function of the form
cðr; tÞ ¼
A þ B; r
t > 0;
nðtÞ 6 r 6 RðtÞ;
ð20Þ
where the coefficients A and B do not depend on the radial coordinate r. Determining the coefficients A and B from the boundary conditions (5) and (9) and substituting them into Eq. (20), we obtain that
peq Deff nðtÞ p Zeq RT þ kdis vnðtÞ 1 r cðr; tÞ ¼ ZRT peq Deff þ k vnðtÞ 1 nðtÞ dis Z eq RT RðtÞ 1 kdis vn2 ðtÞ 1r RðtÞ peq ; t > 0; nðtÞ 6 r 6 RðtÞ: þ Z eq RT peq Deff þ k vnðtÞ 1 nðtÞ dis
Z eq RT
RðtÞ
ð21Þ
!
r h ¼ kdis v
Z eq RT cðr; tÞjr¼nðtÞ 1 ; peq
t > 0:
ð13Þ
Taking into account that, in the case of the problem under consideration, r h < 0, Eq. (12) can be written in the form
!
dnðtÞ kdis Z eq RT ¼ cðr; tÞjr¼nðtÞ ; 1 dt peq ð1 eh Þ
t > 0:
ð14Þ
Under the terms of the problem, there is no ice layer at the initial (zero) time. Thus, the initial condition for the quantity nðtÞ is as follows:
nðtÞjt¼0 ¼ R0 :
ð15Þ
Eqs. (3)-(5), (9), (10), (14) and (15) form a closed system of equations describing gas hydrate dissociation into ice and gas. This system of equations was obtained under the assumption that the outside radius of the ice layer changes during gas hydrate dissociation. The case in which this radius does not change is considered in Appendix B. 3. Quasi-stationary approximation
RðtÞ dRðtÞ=dt
ð16Þ
is much larger than the characteristic time for the diffusion of the gas through the ice layer
sdif ¼
R2 ðtÞ Deff
:
ð17Þ
Thus, if the condition
sth Deff ¼ >> 1; t > 0 sdif RðtÞdRðtÞ=dt
After substituting Eq. (21) into Eq. (14), we obtain the differential equation
dnðtÞ kdis Deff ddis h i ; ¼ peq Deff nðtÞ dt vð1 eh Þ Zeq RT v þ kdis nðtÞ 1 RðtÞ
t > 0;
ð22Þ
where
ddis ¼
peq p ; Z eq RT ZRT
ð23Þ
and the quantity RðtÞ is given by Eq. (10). Integration of the differential Eq. (22) with the initial condition (15) leads to the following transcendental equation:
peq 1 ðR0 nðtÞÞ þ ðR2 n2 ðtÞÞ 2Deff 0 Z eq RTkdis v h i2=3 1 R20 R30 k þ n3 ðtÞð1 kÞ 2Deff ð1 kÞ ddis t ; t P 0; ¼ vð1 eh Þ
ð24Þ
where
The presented system of equations can only be solved numerically. However, in the framework of a quasi-stationary approximation, this system of equations can be solved analytically. The quasi-stationary approximation can be used when the characteristic time for an increase in the ice layer thickness
sth ¼
ð19Þ
ð18Þ
is satisfied, then the concentration profile cðr; tÞ can be considered steady in the ice layer at each time. This approach allows us to neglect the partial derivative with respect to time in the diffusion Eq. (3). Further, we shall assume that the condition (18) holds.
k¼
vMh ð1 eh Þ Mg 1þ xMw ð1 ei Þ nMw
1 :
ð25Þ
Using Eq. (24), we can determine the value of the quantity nðtÞ at each time t. 4. Comparison with experimental data The process of gas hydrate dissociation can be described by such a quantity as the degree of gas hydrate dissociation gdis ðtÞ. This quantity is the mass fraction of the dissociated gas hydrate:
gdis ðtÞ ¼
m0h ðtÞ ; mh0
t P 0;
ð26Þ
where m0h ðtÞ is the mass of the dissociated gas hydrate and mh0 is the initial mass of the gas hydrate. It is easy to show that in the case of the dissociation of a spherical gas hydrate particle into ice and gas, we have
gdis ðtÞ ¼ 1
n3 ðtÞ R30
;
t P 0:
ð27Þ
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Note that real experiments studied the dissociation of a powdery gas hydrate into ice and gas, i.e., the dissociation into ice and gas of a large set of gas hydrate particles whose shape is close to spherical. In this case, if the quantity R0 is the characteristic average radius of the gas hydrate particles forming the powder, then Eq. (27) can also be applied. This makes it possible to compare the results of numerical calculations and the results of experiments designed to study the kinetics of the dissociation of a powdery gas hydrate into ice and gas. In the diffusion model presented in this work, there are three main parameters that determine the kinetics of the process of gas hydrate dissociation into ice and gas: the extrinsic parameter ddis and the intrinsic parameters Deff and kdis . The parameter ddis determines the driving force of the process of gas hydrate dissociation into ice and gas, whereas the intrinsic parameters Deff and kdis determine the intensity of this process at the molecular level. Thus, in the framework of the developed model, on the kinetics of gas hydrate dissociation into ice and gas in the general case affects both the intensity of the diffusion process of the gas through the formed ice layer (the parameter Deff ) and the intensity of the gas hydrate dissociation reaction on the ice–hydrate interface (the parameter kdis ). In the particular case, as the analysis shows, one of the parameters Deff or kdis has considerably more influence on the kinetics of gas hydrate dissociation into ice and gas than the other. If the influence of the parameter Deff is much greater than the influence of the parameter kdis , then we can assume that the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the formed ice layer. If the influence of the parameter kdis greatly predominates over the influence of the parameter Deff , then we can assume that the rate of gas hydrate dissociation into ice and gas is limited only by the gas hydrate dissociation reaction on the ice–hydrate interface. In work [7], the dissociation of powdery methane hydrate samples into ice and gas at different temperatures and the Earth’s atmosphere (p ¼ 0) was studied using the energy-dispersive X-ray diffraction method. The characteristic radius of the methane hydrate particles in a single sample ranged from 10 to 25 lm, and these particles themselves were initially prepared from ice particles. The latter fact suggests that the methane hydrate in the experiments [7] had a porous structure [46,47]. Fig. 2 shows the experimental data [7]. Furthermore, Fig. 2 shows the calculated data obtained for each fixed temperature using Eqs. (24) and (27) by assuming that R0 ¼ 17:5 lm and eh ¼ ei ¼ 0:15. These calculated data, which are in good agreement with the experimental data [7], were obtained by varying the parameters Deff and kdis in
178 K
1.0
168 K
158 K
ηdis (t )
0.8 0.6 0.4 0.2 0
0
50
100
150
200
250
300
t, h Fig. 2. Experimental and calculated data for the dissociation of a powdery methane hydrate into ice and gas at atmospheric pressure (p ¼ 0). The symbols are experimental data [7] obtained at different temperatures (the characteristic radius of the methane hydrate particles in a single experiment ranged from 10 to 25 lm); the solid lines are calculated data obtained using Eqs. (24) and (27), where R0 ¼ 17:5 lm and eh ¼ ei ¼ 0:15. The values for the quantities Deff and kdis used in the calculations are presented in Table 1.
Table 1 The values of the quantities Deff and kdis at which there is good agreement between the experimental [7] and calculated data for the dissociation of a powdery methane hydrate into ice and gas. T; K
Deff ; m2/s
kdis ; m/s
158
2:9 1014
> 2 109
168
2:8 10
14
> 3 109
178
2:5 1014
> 5 109
Eq. (24). At such variation was considered (following the authors of work [7]) that the rate of methane hydrate dissociation in the selected temperature range (158–178 K) is limited only by the diffusion of methane through the formed ice layer. Table 1 shows the values of the quantities Deff and kdis at which the calculated curves were constructed in Fig. 2. As mentioned above, when receiving the calculated data shown in Fig. 2, we assumed that the characteristic radius of the methane hydrate particles in the experiments [7] was 17.5 lm, whereas, in reality, this radius varied over a wide range (from 10 to 25 lm). As a result, the data presented in Table 1 should be regarded as estimates. In particular, this is evidenced by the fact that, according to Table 1, the value of the diffusion coefficient Deff slightly decreases with increasing temperature. However, this decrease may be attributed to the morphology of the formed ice layer. Note that the values of the diffusion coefficient Deff presented in Table 1 close in the order of magnitude to the values of the diffusion coefficient of methane in ice, which were calculated in the framework of the method of molecular dynamics [48]. Also note that, as the analysis shows, the ratio sth =sdif decreases with increasing gdis ðtÞ. Herewith, however, the condition (18) is not satisfied only for large values of the quantity gdis ðtÞ (at least this is true in the case of the experimental conditions [7] and for the values of the quantities Deff and kdis given in Table 1). Thus, we can often use Eq. (24) to simulate the kinetics of the dissociation of spherical gas hydrate particles into ice and gas, thereby avoiding the numerical solution of the system of Eqs. (3)-(5), (9), (10), (14) and (15). The values of the quantities Deff and kdis depend on the temperature. This fact allows us to explain the anomalous preservation thermal regime of methane hydrate, which was reported in works [39–42]. The monotonic increase of the rate of methane hydrate dissociation into ice and gas in the temperature range of 193–240 K observed in these works is due to the fact that the rate of methane hydrate dissociation into ice and gas in this temperature range is limited only by the diffusion of methane through the formed ice layer: the parameter Deff has a much greater influence on the kinetics of methane hydrate dissociation into ice and gas than the parameter kdis . In the temperature range of 242–271 K, the intensity of the methane hydrate dissociation reaction on the ice–hydrate interface significantly affects the kinetics of methane hydrate dissociation into ice and gas. In this temperature range, the rate of methane hydrate dissociation into ice and gas is limited both by the diffusion of methane through the formed ice layer and by the methane hydrate dissociation reaction on the ice–hydrate interface: the influence of the parameter Deff on the kinetics of methane hydrate dissociation into ice and gas is not overwhelming compared to the influence of the parameter kdis . A significant influence of the parameter kdis on the kinetics of methane hydrate dissociation into ice and gas leads to a decrease in the rate of methane hydrate dissociation into ice and gas in the temperature range of 242–271 K, which was observed in the works [39–42]. If we assume that the temperature of 241 K is the temperature above which the anomalous preservation thermal regime of methane hydrate begins to occur, then the parameter kdis can be explicitly calculated at this temperature in the framework of the developed diffusion model without an analysis of the
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specific kinetic curve of methane hydrate dissociation into ice and gas. Using in the calculations the value of the parameter Deff equal to the value of the molecular diffusion coefficient of methane in ice, taken from the work [48], we obtain by Eq. (24) that for methane hydrate kdis ¼ ð8 3Þ 108 m=s at a temperature of 241 K. In the framework of the developed diffusion model, the anomalous preservation thermal regime of gas hydrates may potentially be observed for all hydrate-forming gases, not only for methane. Herewith, the explanation for this regime will be similar to the above explanation for methane. However, in the framework of the developed diffusion model, the anomalous preservation thermal regime of gas hydrates may not occur for some hydrateforming gases because, for these gases at temperatures below the ice melting point, the rate of gas hydrate dissociation into ice and gas is limited only by the diffusion of the gas through the formed ice layer.
where mi ðtÞ and mg ðtÞ are the masses of ice and the gas, respectively, and nw ðtÞ and ng ðtÞ are the moles of water and the gas, respectively. From Eqs. (A1)-(A3), it follows that
Mg ; m0h ðtÞ ¼ mi ðtÞ 1 þ nM w
ðA4Þ
Substituting Eq. (A4) into Eq. (26), we can obtain the relation
gdis ðtÞ ¼
mi ðtÞ Mg ; 1þ mh0 nM w
t P 0:
ðA5Þ
Taking into account the geometry of the problem under consideration (Fig. 1), the quantities mi ðtÞ and mh0 can be represented as follows:
mi ðtÞ ¼ qi ð1 ei Þ mh0 ¼ qh ð1 eh Þ
5. Conclusions In the formulation of the diffusion model of gas hydrate dissociation into ice and gas, it was not specified which kind of the hydrate-forming gas is considered. Thus, the diffusion model presented in this work is applicable for all hydrate-forming gases. In this work, the general theory of chemical kinetics was used in the development of the diffusion model of gas hydrate dissociation into ice and gas. This allowed to take into account the intrinsic kinetics of the process of gas hydrate dissociation into ice and gas. The developed diffusion model allows for simulation of the gas hydrates self-preservation effect. In the framework of this model, the anomalous preservation thermal regime of gas hydrates can be explained. It is important to note that the diffusion model presented in this work is universal, i.e., it can be generalised and applied to any geometry of the problem under consideration. Furthermore, the model can be generalised to the case in which the pressure of the hydrate-forming gas is not constant but increases as a result of the process of gas hydrate dissociation. When creating the diffusion model of gas hydrate dissociation into ice and gas, we assumed that the temperature at the front of the gas hydrate formation/dissociation reaction is unchanged. This assumption is not always valid when considering the initial stage of the process of gas hydrate dissociation into ice and gas at high temperatures (in this case, the dissociation occurs intensively). If the assumption regarding the constancy of the temperature at the front of the gas hydrate formation/dissociation reaction is not valid, then it is necessary to solve a diffusion problem involving the heat equation.
t P 0:
4 3 p R ðtÞ n3 ðtÞ ; 3
t P 0;
ðA6Þ
4 3 pR ; 3 0
ðA7Þ
where qi and qh are the mass densities of ice and the gas hydrate, respectively. From Eqs. (A5)–(A7), it follows that
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 q ð1 eh Þ Mg h RðtÞ ¼ R30 gdis ðtÞ þ n3 ðtÞ; 1þ qi ð1 ei Þ nM w
t P 0:
ðA8Þ
Taking into account Eq. (27) and given that qi ¼ xMw and
qh ¼ vMh , Eq. (A8) becomes Eq. (10).
Appendix B. Case in which RðtÞ ¼ const If we can assume that RðtÞ ¼ R0 ¼ const, then it is necessary to solve the system of Eqs. (3)-(5), (9), (14) and (15), where, in Eqs. (3) and (5), RðtÞ is replaced by R0 . Solving this system of equations in the framework of a quasi-stationary approximation, we obtain that
peq 1 2 ðR0 nðtÞÞ þ R n2 ðtÞ 2Deff 0 Z eq RTkdis v 1 3 R n3 ðtÞ 3Deff R0 0 ddis t ; t P 0: ¼ vð1 eh Þ
ðB1Þ
Solving Eq. (B1), it is possible to determine the value of the quantity nðtÞ at each time t . References
Acknowledgments The author is grateful to Dr. A.N. Nesterov for a useful discussion of the gas hydrates self-preservation effect. Appendix A. Derivation of Eq. (10) During the process of gas hydrate dissociation into ice and gas, the following relations are satisfied:
m0h ðtÞ ¼ mi ðtÞ þ mg ðtÞ; ng ðtÞ ¼
mg ðtÞ ; Mg
nw ðtÞ ¼ nng ðtÞ ¼
t P 0;
t P 0; mi ðtÞ ; Mw
ðA1Þ ðA2Þ
t P 0;
ðA3Þ
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