International Conference on Gas Hydrates (ICGH) (6th : 2008)

MODELING THE METHANE HYDRATE FORMATION IN AN AQUEOUS FILM SUBMITED TO STEADY COOLING Avendaño-Gómez, Juan Ramón; García-Sánchez, Fernando; Vázquez Gurrola, Dynora 2008-07

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MODELING THE METHANE HYDRATE FORMATION IN AN AQUEOUS FILM SUBMITED TO STEADY COOLING  Juan Ramón Avendaño-Gómez ∗  Laboratorio de Ingeniería Química Ambiental ESIQIE Instituto Politécnico Nacional Sección de Estudios de Posgrado UPALM ed. 8, México D. F. 07738 MEXICO  Fernando García-Sánchez Laboratorio de Termodinámica Instituto Mexicano del Petróleo Eje Central Lázaro Cárdenas, 172 México D. F., 07730 MEXICO  Dynora Vázquez Gurrola Laboratorio de Diseño de Plantas UPIBI Instituto Politécnico Nacional Acueducto s.n La Laguna Ticoman. México D. F., 07340 MEXICO  ABSTRACT The aim of this work is to model the thermal evolution inside a hydrate forming system which is submitted to an imposed steady cooling. The study system is a cylindrical thin film of aqueous solution at 19 Mpa, the methane is the hydrate forming molecule and it is assumed that methane is homogeneously dissolved in the aqueous phase. The model in this work takes into account two factors involved in the hydrate crystallization: 1) the stochastic nature of crystallization that causes sub-cooling and 2) the heat source term due to the exothermic enthalpy of hydrate formation. The model equation is based on the resolution of the continuity equation in terms of a heat balance. The crystallization of the methane hydrate occurs at supercooling conditions (Tcryst < TF), besides, the heat released during crystallization interferes with the imposed condition of steady decrease of temperature around the system. Thus, the inclusion of the heat source term has to be considered in order to take into account the influence of crystallization. The rate of heat released during the crystallization is governed by the probability of nucleation J(T ). The results provided by the model equation subjected to boundary conditions allow depict the evolution of temperature in the dispersed phase. The most singular point in the temperature–time curve is the onset time of hydrate crystallization. Three time intervals characterize the temperature evolution during the steady cooling: (1) linear cooling, (2) hydrate formation with a release of heat, (3) a last interval of steady cooling.   Keywords: gas hydrates, undercooling, stochastic nucleation.     ∗ Corresponding author E-mail: Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. NOMENCLATURE  A Coefficient in function J(T) B Coefficient in function J(T) Cps Specific heat of the dissolved methane at solid phase Cpl Specific heat of the dissolved methane at liquid phase. ∆G Gibbs free energy E Coefficient in function J(T) i  Radial position in discretized terms J Probability of nucleation k Boltzman constant k Global heat transfer coefficient L Apparent latent heat of the dissolved phase m Time in discretized terms M Final time of the cooling experiment in discretized terms N Total number of liquid micro-domains n Number of crystallized micro-domains • q  Rate of heat released by the crystallization of micro-domains r Radial position R Width of the annulus cylinder t Time tend  Final time of the cooling experiment TF Melting temperature of the dispersed phase Tend Lower limit of temperature of the cooling experiment β Cooling rate λs Thermal conductivity of the dissolved phase in liquid state λl Thermal conductivity of the dissolved phase in solid state φ Fraction of solidified micro-domains Φ Mass fraction of dissolved phase Θ Temperature in discretized terms µ Chemical potential ρ Density of the dissolved phase  INTRODUCTION Gas hydrates or clathrate hydrates are one class of inclusion compounds. The gas hydrate is an ice- like compound; it results from kinetic process of crystallization of an aqueous solution supersaturated with a dissolved gas. From the thermodynamics point of view, the prediction of hydrate formation is performed out knowing the gas and water composition and the temperature (or pressure) of the gas-water system. In this way, the series of equilibrium points allows to construct the phase diagram of the hydrate system. The state of the art in equilibrium thermodynamics of methane hydrates has evolved since the pioneering works of Parrish and Prausnitz (1972), Ng and Robinson (1976), Vyasnuskas and Bishnoi (1983), Englezos et al (1987). However, when the hydrate former solution is subject to dynamic conditions of pressure and temperature, the onset point of methane hydrate formation differs from that stipulated by equilibrium thermodynamics. Such a dynamic scenario characterized by low temperatures and high pressures is commonly met during the transportation of hydrocarbon fluid in submarine pipelines. First, the water content of the hydrocarbon fluid condensates and then high pressure enhances the solubility of methane gas in the liquid aqueous phase. The concentration of methane in the aqueous solution gradually increases until the system reaches the saturation concentration. From this moment or super saturation condition, the environment required to begin the nucleation and crystallization is fulfilled. Nucleation and growth of the crystal occur predominantly on the gas-water interface, where the super saturation is higher than in bulk water. Nevertheless, in several cases the solution reaches a uniform super saturation before the appearance of the nuclei, and as a result, the nucleation process occurs everywhere in the liquid phase.  Unfortunately, the actual temperature of hydrate formation in undersea pipelines usually differs from that stipulated by equilibrium considerations. Such a difference between theoretical equilibrium temperature Te, and the actual temperature of hydrate crystallization is due to the stochastic nature of nucleation. The difference ∆T = Te – T is also known as sub-cooling. In addition, the non- equilibrium occurrence of hydrate formation is emphasized by the amount of heat released during the crystallization. Most of the hydrate work focuses on the equilibrium conditions of hydrate formation [13–18]. It has been confirmed since the nineteenth century that hydrate formation follows a complicated process of crystallization. In this sense, there are still few works that deals with the kinetic aspects of hydrate formation. Two of the pioneering works that deals with the induction time concept are due to Englezos et al. [9] and the work of Natarajan et al. [19]. It is worthwhile to mention the experimental works of Jakobsen et al. [20], Lekvam and Ruoff [3],Munck et al. [21] and Zatsepina and Buffet [22]. The works aforementioned make evident the influence of the supercooling on hydrate formation kinetics. As those fine writers have shown, the gas hydrate is formed through a kinetic process which requires three necessary steps: (1) The first one is supercooling, (2) The second step is nucleation, (3) The third step is the growth of the crystal.  The objective of this work is to develop a simplified model to determine the onset time of hydrate nucleation within a hydrate forming system subject to an imposed steady cooling. The model includes a function that allows to consider the stochastic nature of the nucleation. The model in this work necessarily takes into account both factors stated before: 1) the stochastic nature of crystallization that causes sub- cooling and 2) the heat source term due to the exothermic enthalpy of crystallization. The steady temperature decrease β acting on the hydrate former system is a known parameter in the model. Thus, once the onset point of hydrate crystallization is determined, it is possible to transform the sub-cooling ∆T into an induction time.  STATEMENT OF THE PROBLEM  Methane hydrate is formed trough a kinetic process of nucleation and crystallization. The need to predict the onset time of nucleation with a model necessarily requires to consider the induction time. In this work, the model is based on the resolution of the continuity equation in terms of the heat balance which includes the source term and the probability of nucleation J(T). The boundary volume of the aqueous solution is a cylindrical annulus of a few µm of width as shown in figure 1. The internal radius R of the crucible is small in comparison to its length L (R <L). In order to simulate the steady cooling of the emulsion we imposed a constant cooling rate around the external surface of the boundary volume.    Figure 1. The study system is an annulus cylinder of width R. The system may be assimilated to an aqueous film at the saturation concentration in methane. The aqueous thin film is partitioned in a grid of 40 points.  The interest of this work to the real industrial needs lies in the fact that it enables to link the thermodynamic aspects of hydrate formation with the kinetic aspects through the function probability of nucleation J(T).  Ehmimed et al. [4] measured the influence of the global heat transfer coefficient k on the rate of thermal transfer between a wall and a dispersed system by differential scanning calorimetry. In that work [4] the authors concluded that thermal transfer is influenced by the heat transfer coefficient only for values of k lower than 100 W·m−2·K−1. The modeling conditions in this particular work are similar to that reported by Zeraouli [5]. The coefficient of heat transfer greater than 500W·m−2·K−1, so, it is possible to neglect the influence of the global heat transfer coefficient k on the rate of thermal transfer. In addition, the viscosity of the aqueous solution allows to consider the liquid as a Newtonian fluid. Therefore, we can assume that the thermal transfer between the cooling boundary and the solution is not ruled by the global heat transfer coefficient k, neither by the viscosity of the emulsion. In contrast, the thermal transfer is strongly influenced by an important amount of heat released during the formation of the solid hydrate. In consequence, the temperature distribution in the time interval of r = R r = 1 ∆r r = 2 r = 3 r = R-1 Width of the aqueous film Innermost position of the aqueous film hydrate formation in the emulsion is determined by two effects: (1) The external effect of the imposed steady cooling (Temperature decrease from the bulk liquid phase towards the boundary wall). (2) The internal effect due to the exothermic heat released by the crystallization of the droplets.  For long, thin cylinders having negligible axial variations in heat flux, the gradient of heat transfer in the radial direction dominates over the axial. Since the geometry of the boundary volume fulfills this condition of R > L we can focus the analysis on the heat transfer in radial direction. Additionally, since the methane is homogeneously dissolved in the solution volume, we can assume that the temperatures varies in the radial direction, such a variation of temperature is axisymmetrical [6,7]. The metastability of the solution is broken when the nucleation begins. In this manner, the problem is to determine the temperature of solution T (r, t). at any time, t during the steady cooling, and for all position r between the wall of the annulus cylinder and the radius R (interface of the aqueous film in contact with the methane gas). The initial and final temperatures of the cooling interval are free parameters that must be specified. In this work, the steady cooling begins at 298 K and the lower limit of temperature is 240 K. The cooling rate denoted as β is 10 K·hr−1.  Therefore, the numerical modeling of the thermal history is expressed as a heat transfer balance. A heat source term is taken into consideration within the energy balance of the dispersed phase. During the thermal history of the aqueous solution, the onset time of crystallization initiates the release of heat inside the aqueous solution. The source term depends on the probability of nucleation. Not all the aqueous volume crystallizes simultaneously due to the steady cooling. Actually, a progressive crystallization occurs and the rupture of the metastable state in the liquid phase begins when the rate of nucleation J(T ) > 0  [8].  Hereafter Φ shall denote the mass fraction of the disolved methane phase in the solution. This parameter is the ratio of the mass of methane to the total mass of aqueous solution. It is assumed that the thermal conductivity λ and the specific heat Cp do not vary in the range of cooling temperatures. Nevertheless, λ and Cp are considered dependent functions of the fraction of crystallized mass ϕ in the solution. The fraction of solid state within the solution at position r and time t shall be denoted by ϕ(r, t) and it varies between 0 < ϕ(r, t) < 1. Therefore, from the work of Fouconnier et al. [10] a linear law of variation allows to write the following equations:  λ(ϕ) = λl +(λs − λl)  (1) Cp(ϕ) = Cpl +(Cps −Cpl)   (2)  The thermal transfer during the cooling process of the aqueous solution is modeled with the continuity equation of heat conduction:  •       • +∇λ∇= ∂ ∂ρ qTC Tp t  (3)  The heat source term ˙q is different from zero at the onset time of nucleation. The occurrence of such event at undercooling conditions is governed by a probabilistic law of nucleation [9,11,12]. A fraction of the undercooled aqueous volume crystallizes ϕ(r, t) ≠  0  only when the probability of nucleation J(T) > 0 The latent heat of solidification of the fraction ϕ(r, t) represents the heat source term • q . The solution of equation (3) enables the determination of the space–time distribution of temperature across the radial position in the annulus cylinder T (r, t). Before enterprising the solution of equation (3) it is important to point out the connection between the heat source term • q  and the probability of nucleation J(T). In order to precise such a connection, it is convenient to remind the mechanisms of nucleation and crystallization that allow to derive an expression of the type • q  = J(T). Then, the continuity equation (3) can be expressed in terms of the dependent variable T (r, t) and known functions.   The probability of nucleation .  The supercooling phenomenon is characterized by the persistence in the liquid state of a material below its melting point TF. However, the rupture of the metastable equilibrium of the supercooled liquid occurs at the beginning time of hydrate formation. One of the main parameters that has influence on the supercooling is the volume of the liquid. For example, a bulk volume of water of a few cm 3  is characterized by a supercooling degree of ∆T = −14 K, while micro-droplets of water dispersed in a w/o emulsion (volumes of a fraction of µm3) the crystallization occurs at 233 K or lower temperatures [5, 6, 25].  The steady cooling of an aqueous solution shows the stochastic nature of crystallization [4].   During the induction time the nuclei growth in size, once the nuclei attain a stable size the metastable equilibrium of the aqueous solution no longer exists. The occurrence of such instant in time is a phenomenon of random and probabilistic nature if the conditions of supercooling, pressure and composition are fulfilled [11, 27]. At conditions of nucleation, it is not possible to determine the place where the nuclei are formed inside the aqueous volume. Nevertheless, a probability P = VJt of appearance of solid phase within the aqueous volume can be predicted because it is possible to calculate the nucleation rate J [5, 22].  The appearance of a nucleus is essentially a random phenomenon for which a probability can be predicted. If we consider a population of N aqueous micro-domains whose volumes are V , the nucleation will not occur in all those micro- domains at the same time as temperature decreases. Instead, the birth of nuclei occurs randomly. Let n(r, t) be a fraction of N that represents the number of solidified micro-domains at position r and time t , then the proportion of solidified micro-domains in the total population of N at position r and time t is:  N ,n ,           =ϕ tr tr  (4)  and the number of micro-domains remaining in liquid state at time t is N −n(r, t). Assuming that each solidified micro-domain of n corresponds to the appearance of a nucleus, there is an equal probability of forming a nucleus in each of them. The probability of nucleation in a micro-domain of volume V in a time increment dt is 1VJdt .  Let us consider not only one aqueous micro- domain but a large population composed of N micro-domains per unit volume of aqueous solution. The aqueous solution is submitted to a steady cooling from a starting temperature of 298 K until an imposed temperature lower than TF. At the beginning of the cooling all the aqueous micro- domains N are in liquid state, as the cooling progresses the metastable equilibrium will be broken at instant time t . In consequence, at time t > 0 there are n(r, t) micro-domains in solid state per unit volume aqueous solution.  The rate of crystallization is proportional to the number of micro-domains remaining in liquid state, N − n(r, t), and proportional also to the probability of crystallization by unit of time and volume J(T). The rate of crystallization of those remaining micro-domains in a time increment dt is expressed as: dn(r, t)                         ⋅−= TJ,nN d ,nd tr t tr   (5)  We can also express equation (5) in terms of the proportion of crystallized micro-domains ϕ(r, t), then:                        ⋅ϕ−=ϕ TJ,1 d ,d tr t tr   (6)  Equation (6) expresses the kinetics of crystallization of the dispersed micro-domains inside the total volume of aqueous solution subject to steady cooling.  4. Estimation of the rate of nucleation J(T )  In classical nucleation theories the rate of homogeneous nucleation J represents the number of nuclei formed per unit time and unit volume, the expression of J(T ) is given by:                ∆+−= ⋅ Tk GEexpATJ    (7)  J(T ) is a nonlinear function of temperature T. For temperatures above the melting temperature the function J(T ) is close to zero. On the contrary, as supercooling increases J(T) also increases exponentially and very quickly [8,28].  The coefficient E is a function of the viscosity of the liquid whereas ∆G is the energy barrier that nuclei must overcome in order to reach a stable critical size. The energy barrier is basically imposed by the interfacial tension between the nuclei and the liquid phase [12,29,30].  Moreover, if the viscosity of the liquid is too large in comparison to ∆G, then the value of E must be taken into consideration. A large value of E implies that the rate of nucleation is limited by viscosity. In this work, the viscosity of the aqueous solution behaves as a Newtonian liquid, hence, it has a weak influence on the crystallization behavior of the dispersed phase [1]. It is assumed the following form of Eq. (7):                ∆= ⋅ Tk GexpATJ    (8)   Quantifying the rate of hydrate nucleation is a difficult experimental problem. First, we must be able to detect the appearance of hydrate nuclei. Second, we need to observe a large number of nucleation events in order to obtain a meaningful statistical average for the nucleation rate.  The rate of nucleation can be determined by means of the differential scanning calorimetry technique (DSC) [6, 31]. An experimental function described in the form of equation (9) can be fitted from measurements by means of differential scanning calorimetry [5, 8].                              −⋅ −= ⋅ 2 FTTT BexpATJ   (9)  To carry out the experimental procedure may turn most difficult than the determination of A and B. Several temperatures must be recorded at different positions inside the sample container, this implies the use of thermocouple in contact with the liquid. In this work such method was not followed since it requires very coercive experimental conditions inside the crucible. A complete description of the method may be found in references [5,27,32,33]. The main condition to validate the quantification of  J(T ) by means of the cooling-heating thermogram is to use a cooling rate lower than 15 K·hr −1 . In this manner we can assume that the same temperature is maintained between the first layer of liquid inside the crucible and the surrounding resistance of the calorimeter [5].  In this work the coefficients A and B of equation (9) are fitted from kinetic data of a previous experimental work developed by Sloan et al [1]. The fitted values are A = 1.6755 × 1034 nuclei·m −3 ·s −1 and B = 4×106 K3.   MATHEMATICAL MODEL AND BOUNDARY CONDITIONS  Assumptions and derivation of the model.  The nonlinear heat conduction of a cylindrical annulus of aqueous solution is defined by the model equation (3). Equation (3) can be expressed in terms of the dependent variable T (r,t), the rate of crystallization J(T) and the fraction of crystallized micro-domains ϕ(r, t). The precedent assumption of R < L enables to consider a one- dimensional heat conduction in radial direction [34]. The analog form of equation (3) in cylindrical coordinates is:       •                            + ∂ ∂ λ ∂ ∂= ∂ ∂ ⋅ρ ⋅⋅⋅⋅ tr r tr trr rrt tr ,q ,T ,1 ,T Cp (10)  The thermal properties of the dissolved methane in Eqs. (1) and (2) are linear functions of ϕ(r, t). In this sense, during the steady cooling ϕ(r, t) = 0 for the time interval where the dissolved methane in the form of subcooled liquid. Nevertheless, from the moment of hydrate formation ϕ(r, t) > 0 and increases until ϕ(r, t) = 1. In consequence the thermal properties λ and Cp change in time. In this sense, the rupture of the metastable equilibrium of the liquid micro-domains points out the beginning of hydrate formation.  The source term is no longer • q = 0 in reason of the exothermic character of the phase transition. The heat flow per unit volume of dispersed phase (in J·m −3 ·s −1 ) is proportional to the kinetics of crystallization. The relation between • q (r, t) and the kinetics of crystallization can be expressed as follows:  t tr d ,d Lq     • ϕ ρ= ⋅    (11)  where L is the apparent latent heat of the dissolved methane.     The boundary conditions.  The temperature of the cooling fluid decreases at a constant rate, at t = 0, T∞(0) > 0 where TF is the melting temperature of the dissolved methane. At any time t > 0 the external surface of the cylindrical annulus is always in contact with the cooling fluid at T∞(t), therefore:  T (r, t) = T∞(t)  at      r = R and t > 0  The condition of axisymmetrical temperatures in relation to r imposes:  0 ,T = ∂ ∂      r tr   at      r = 0 and t > 0  The initial conditions  T (r, t) = T∞(0) > TF         for       0 < r <R, t = 0 ϕ(r, t) =0                    for       0 < r <R, t = 0  Numerical solution of the model equation  The equation (10) of heat conduction can be solved by means of a numerical algorithm for T (r, t). Here we choose an explicit forward finite difference method for the solution of the partial differential equation (10). In order to determine T (r, t) in the radial direction during the cooling, the position coordinate r and the time t are represented by discrete set points m and i respectively,  i  = 1, 2, 3,... I        for t1 = 0,  t2 = t1 + ∆t,  t3 = t1 + 2 ∆t , ... tend m = 1, 2, 3,... M     for r1 = 1,  r2 = r1+∆r,  r3 = r1 + 2 ∆r , ... R   Thus, the coordinate r in radial direction is represented by a dimensional grid shown in Fig. 1. The innermost position of the aqueous film corresponds to the radius r = 1 is assigned with the node m = 1, each node spans the radius with a fixed radial length ∆r. At radius R the corresponding node is M. The duration of time of the cooling experiment is also discretized by constant time steps ∆t. The cooling experiments begin at t = 0 that corresponds in the time grid to i = 1, the final instant of the experiment tend in the finite difference representation corresponds to i = I. Hereafter we denote the temperature and the crystallized fraction in node m and time i as Θim and ϕim respectively. The explicit forward scheme means that Θi+1 m for each m can be calculated explicitly from the quantities that are already known. The requirement of stability to maintain convergence is:  ( )               λλ ⋅ρ ≤ ∆ ∆ s,l4 psC,plC 2 max min r t       (12) The solution of the nonlinear second order partial differential equation (10) is carried out by means of the explicit finite difference approach. The resultant equations are as follows:  1M2 R2 − =∆ ⋅ ⋅ r     (13)  1I T − =∆t   (14)                               Θ−Θ λ ∆⋅ρ ∆ +Θϕ−∆+Θ=Θ ⋅⋅⋅⋅⋅⋅+ i1 i 2i 1 i 1 211i 1 i 1 1i m Cp 4iJi1 Cp 1L r t t  (15)                               Θ+Θ−Θ λ ∆⋅ρ ∆ +Θϕ−∆+Θ=Θ −+ + ⋅⋅⋅⋅⋅ i 1m i m i 1mi m i m 211i 1 i m 1i m 2 Cp iJi1 Cp 1L r t t                 Θ−Θ λ ⋅ −⋅∆⋅ρ ∆ + −+ ⋅⋅ i 1m i 1mi m i m 2 Cp2m2 1 r t (16)  Θ=Θ + ∞+ 1ii 1M   (17)  t∆Θ⋅Θ−+Θ=Θ ⋅               +∞ i 1M i m i m J1i  (18)    The numerical calculations were performed using a Fortran programming. The step size in time is ∆t = 0.0125 s and 40 radial positions are considered for ∆r = 0.0005125 m.   RESULTS OF THE SIMULATION  The evolution of the temperature T (r, t) of the cylindrilcal annulus of aqueous solution is obtained from the numerical solution of equations (15) and (16). The parameters used in the modeling are listed in Table 1. The thermo- physical properties of the dissolved methane are listed in Table 2. The aqueous solution contains Φ = 0.00305 mol fraction of dissolved methane (P = 19.0 MPa). The melting temperature of the aqueous hydrate solution is Tm < 293 K.      Symbol Value Units β 10 K / hr T∞(0) 298 K Tend 236 K ∆t 0.0125 s ∆r 0.0005125 m M 40 Number of radial positions R 0.0205 m   Table 1. Parameters used in the modeling of the cooling of the aqueous film of methane solution.         Symbol Value Units Φ 62.4 wt  % ρ 194.6748 Kg / m 3  A 1.6755×1034 s-1 B 4×10 6   K 3  λs 2.6973 W / m K λl 0.46022 W / m K Cps 1.6628 KJ / Kg K Cpl 4.029855 KJ / Kg K L 129.32 KJ / Kg TF 293 K  Table2. Thermochemical properties of the dissolved methane in the aqueous solution.   Before starting the cooling experiments, the temperature of the sample is homogeneous inside the boundary volume and equal to T∞(0) = 298 K. Then, from t = 0 a constant steady cooling process is imposed on the external surface of the boundary volume. The temperature of the cooling fluid decreases linearly at a constant cooling rate of β = 10 K·hr −1 . The lower limit of temperature reached in the cooling simulation is Tend = T (r, t) = 236 K.  The result of a simulation of the cooling process for the radial position r = 1 is shown in Fig. 2. 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Time h T em p er a tu re  K Induction time Crystallization beginsSubcooling degree = 12K Crystallization ends Cooling temperature step β = 55.55 x10 -5   K/h  Figure 2. Thermal evolution of the innermost position of the aqueous film.  The evolution of the curve T (r, t) reveals two different behaviours. The first one in the time interval 0 < t < 8 hr presents the profile of a continuous cooling. During this first stage the fraction of solidified droplets is zero, ϕ(r, t) = 0. The second time interval between 8 hr < t < 14 hr presents a different slope of the curve T (r, t). The sudden change of slope at t = 8.5 hr indicates the onset time of hydrate crystallization. From the beginning of the second stage t = 8 hr the fraction of solidified micro-domains starts to increase ϕ(r, t) > 0. This implies that formation of solid hydrate is occurring in the aqueous solution. The fraction ϕ(r, t) shall vary until it reaches the magnitude ϕ = 1. Correspondingly, an amount of heat is released as hydrate crystallization progresses in the large population of liquid micro- domains. The heat balance between the imposed steady cooling and the exothermic heat of crystallization results in the change of slope of T (r, t) which is almost horizontal.  From Fig. 2 it can be seen that for the innermost position r = 1 the stabilizing effect disappears at t > 14.5 hr. From this moment the cooling tendency becomes once again more important than the heat released. The behaviour of T (r, t) for the external position r = R is nearly constant due to the proximity with the cooling fluid. We can also identify the onset time of hydrate crystallization at t = 8.5 hr. 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