Formation and dissociation of clathrate hydrate in stoichiometric tetrahydrofuran–water mixture subjected to one-dimensional cooling or heating

Chemical Engineering Science 56 (2001) 4747–4758

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Formation and dissociation of clathrate hydrate in stoichiometric tetrahydrofuran–water mixture subjected to one-dimensional cooling or heating Tomoyuki Iidaa; 1 , Hideaki Moria; 2 , Takaaki Mochizukib , Yasuhiko H. Moria;∗ a Department

of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan of Technology Education, Tokyo Gakugei University, Tokyo 184-8501, Japan

b Department

Received 25 September 2000; accepted 12 December 2000

Abstract Formation and growth, or melting, of a polycrystalline layer of tetrahydrofuran (THF) hydrate from, or into, a liquid solution having the same composition as that of the hydrate have been observed in a macroscopically one-dimensional heat-transfer system under atmospheric pressure. Experiments were performed with either the liquid THF–water solution or the THF hydrate initially 6lling a 260 cm3 cell which was sealed top and bottom by temperature-controlled copper plates and by glass plates on the side. In one group of experiments, the cell 6lled with the solution was initially adjusted at the hydrate–solution equilibrium temperature, ◦ Teq (4:4 C). The top copper plate was then cooled quasi-stepwise, while the temperature at the bottom copper plate was either unchanged or increased quasi-stepwise, resulting in the growth of a planar polycrystalline hydrate layer down from the surface of the top plate. In another group of experiments, the cell was initially 6lled with a polycrystalline THF-hydrate phase at a temperature slightly lower than Teq . Successively, the temperature at the bottom plate was increased quasi-stepwise to exceed Teq , resulting in the melting of the hydrate phase from the bottom. The behavior of such growth and melting of the hydrate layers observed in the experiments is in general agreement with that predicted by relevant theoretical=numerical analyses of transient conductive and=or free-convective heat transfer from=to the hydrate–solution interface, where the temperature is assumed to be 6xed at Teq . Also described in this paper is an unexpected 6nding in a particular experimental condition—the formation of column-like hydrate crystals extending almost across the 20-mm spacing between the top and bottom plates, which precedes the growth of a planar polycrystalline layer. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Clathrate hydrate; Crystallization; Melt growth; Heat transfer; Heat conduction; Convective transport

1. Introduction Among numerous compounds known as hydrate formers, tetrahydrofuran (abbreviated as THF hereafter) is unique in that (a) it is in the state of a liquid under atmospheric pressure, (b) it is unlimitedly soluble in liquid water, and (c) it forms a hydrate under atmospheric pressure. Owing to its unique nature mentioned above, THF has received the attention of hydrate researchers interested ∗ Corresponding author. Tel.: +81-45-566-1522; Fax: +81-45566-1495. E-mail address: [email protected] (Y. H. Mori). 1 Present address: Power Plant Division, Kawasaki Heavy Industries, Ltd., Tokyo 136-0072, Japan. 2 Present address: Hamamatsu Research Laboratory, Nichias Corporation, Hamamatsu, Shizuoka 431-2103, Japan.

in studying the physical properties (Gough & Davidson, 1971; Ross, Andersson, & BGackstGom, 1981; Ross & Andersson, 1982; Leaist, Murray, Post, & Davidson, 1982; Handa, Hawkins, & Murray, 1984; RueH & Sloan, 1985; White & MacLean, 1985; Ashworth, Johnson, & Lai, 1985; Tse & White, 1988; Andersson & Suga, 1995) and the crystal growth process (Pinder, 1965; Scanlon & Fennema, 1972; Makogon, Larsen, Knight, & Sloan, 1997; Larsen, Knight, & Sloan, 1998; Devarakonda, Groysman, & Myerson, 1999; Bollavaram, Devarakonda, Selim, & Sloan, 2000) of hydrates. In contrast to hydrophobic hydrate formers that allow hydrate crystals to form, in general, only at the interfaces where they meet a liquid-water phase, THF mixed with water into a solution allows hydrate crystals to form and grow anywhere within the con6nes of the solution. If

0009-2509/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 1 2 8 - 2

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the solution is composed of THF and water at a molar ratio of 1 : 17, the ratio corresponding to the stoichiometric composition of THF hydrate of structure II, any mass transfer process can be eliminated from the process of crystal growth from the solution, thereby favoring researchers’ intention of studying hydrate-crystal growth in a most simpli6ed system. In such a stoichiometric binary system, we need to consider only two rate processes which possibly control the rate of crystal growth: (i) the transfer of heat generated at the surfaces of growing crystals to the surrounding solution, and (ii) the reaction at the surfaces. It seems reasonable to assume that, unless the heat transfer is strongly enhanced by some means, the growth rate of the crystals is mostly controlled by the heat transfer and their surfaces are held under nearly the hydrate–solution equilibrium condition (Larsen, 1997). However, we have little evidence of this notion. This is because hydrate-formation experiments using THF or another water-soluble hydrate former, ethylene oxide, have rarely been performed so far in such a way that the heat transfer from the crystal surfaces is accurately predictable. The only exception may be a very recent study by Bollavaram et al. (2000), who observed the growth of single, octahedral crystals each held stationary in a steady Low of temperature-controlled, stoichiometric THF solution. Comparing the observed crystal-growth rates with corresponding predictions of forced-convective heat transfer from the crystal surfaces, Bollavaram et al. suggested that the surface-reactive restraints may not be overlooked when the heat transfer coeMcient is much increased. No such experiments with polycrystalline hydrate samples have been reported. Besides the fundamental hydrate-kinetic issue described above, we 6nd an engineering need to study the growth and melting of polycrystalline THF hydrate from and into a stoichiometric THF solution. Owing to the restrictions on the use of chloroLuorocarbons and hydrochloroLuorocarbons some of which were considered to be favorable substances for hydrate cool storage systems for residential air conditioning, we need to select some alternative substances which are non-toxic and form hydrates under moderate pressures. THF is one of the candidates for the hydrate former for such cool storage systems (Akiya et al., 1997). Its solubility in water can be a favorable factor because any mechanical device to continuously mix the hydrate former and water is no longer necessary when a stoichiometric solution is employed as the cool storage medium. What we should know to have better prospects for THF-hydrate cool storage systems are the characteristics of the formation=growth and melting (crystal dissociation) as well as the morphological nature of polycrystalline THF hydrate phases. Considering both the scienti6c issue and the engineering need discussed above, we have performed

experiments involving macroscopically uni-directional growth and shrinkage (due to melting) of polycrystalline THF-hydrate layers. This paper shows both qualitative observations of the macroscopic morphologies of growing and shrinking hydrate layers and quantitative results of chronological changes in hydrate layer thickness. The quantitative results are compared with simulations of heat-transfer-controlled growth and shrinkage of hydrate layers, and we 6nd generally good agreement between them. 2. Experiments 2.1. Apparatus The experimental apparatus was so constructed as to enable a horizontal view of a planar, or nearly planar, hydrate layer thickening or thinning in the vertical direction in which a temperature gradient is held. Fig. 1 illustrates the test cell, the principal portion of the apparatus, in which a liquid solution of THF and water mixed at the molar ratio of 1 : 17 was converted into the THF hydrate and vice versa. The cell was made of a square borosilicate glass frame, 100 mm × 100 mm × 26 mm (height) in inside dimensions and 10 mm in thickness, and two thick, rectangular copper plates each machined

Fig. 1. Test cell holding THF–water mixture inside.

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2.2. Procedure

Fig. 2. Schematic of experimental setup.

to 140 mm × 170 mm × 30 mm. The glass frame was placed, together with two butyl rubber O-rings, between the copper plates so that a rectangular space, 100-mm square by 20 mm in height, was left inside. This space was closed from the outside except for three ports used as the inlet of the THF solution, an air vent and a drain. A 90-W plate-type Peltier cooler with a 200 mm × 118 mm cooling=heating surface (Model PW-246 manufactured by Netsu Denshi Kogyo Co., Tokyo) was pressed on the rear side of each copper plate to control its temperature independently of the temperature of the other plate. Three thermocouples were so inserted into each copper plate as to be aligned vertically with the intention of determining the temperature distribution inside the plate and thereby the heat Lux across the plate. However, this idea failed. We hardly detected any diHerence between the temperatures measured by the three thermocouples, and thus these thermocouples were actually used to determine only the temperature at the surface of the copper plate in contact with the THF–water solution or the THF hydrate. The entirety of the experimental setup is schematically illustrated in Fig. 2. The two Peltier coolers were separately controlled by two PID controllers (MT-702-2405 and MT-802-15P24, Netsu Denshi Kogyo Co., Tokyo). The heat released from the Peltier coolers was continuously removed by water Lowing through a loop connecting the Peltier coolers and a low-temperature water circulator (MUC-65, Tokyo Rikakikai Co., Tokyo). A stereomicroscope (Olympus SZ6045-TRCTV) and a video camera (Sony DXC-151A) connected to each other and held horizontally on a vertically movable stand were used to record the processes of formation and growth or dissociation of the THF hydrate inside the test cell with the aid of back lighting by a Lood lamp. The lighting was limited for only short periods of intermittent video-recording to minimize the eHect of radiative heating of the test cell by the lamp.

The THF solution used in the present experiments was prepared from a THF reagent of 99.9% certi6ed purity (Aldrich Chemical Co.) and water that was deionized and distilled in advance. After pouring the solution into the test cell, all visible gas bubbles trapped inside the cell were carefully removed by manipulating a syringe connected to a 6ne Lexible tube which was inserted into the cell through one of the two ports drilled in the upper copper plate. Every experimental run was commenced by simply changing both or either of the “upper-boundary temperature” (i.e., the temperature indicated by the thermocouples inserted into the upper copper plate) and the “lower-boundary temperature” (i.e., the temperature indicated by the thermocouples inserted into the lower copper plate) nearly stepwise from their initial levels, T1; i and T2; i , to certain prescribed levels, T1 and T2 , which were maintained afterwards. Depending on the changes imposed on the boundary temperatures, the experiments reported in this paper are classi6ed into four groups, I–IV, as summarized in Table 1. The procedures used in these groups are described below in order. 2.2.1. Groups I and II—growth of a hydrate layer down from the upper boundary In advance of each experimental run, we performed some preliminaries to make the THF solution 6lling the test cell experience the formation and dissociation of THF hydrate; this was to shorten the induction time for hydrate formation in the succeeding run. First, the upper-boundary temperature was decreased to, and then ◦ held constant at, −5 C to promote the formation of hydrate crystals in the THF solution adjacent to the upper boundary. At 5 min after we 6rst found hydrate crystals inside the test cell, the upper-boundary temperature was ◦ raised to 8 C to completely dissociate the hydrate crystals inside the cell. Both the upper- and lower-boundary ◦ temperatures were then decreased to 4:4 C, the hydrate– solution equilibrium temperature Teq , and held at this level for two hours or longer to make the THF solution ◦ in the test cell isothermal at 4:4 C. Subsequent to the above preliminaries, an experimental run was started by ◦ decreasing the upper-boundary temperature below 4:4 C ◦ and keeping the lower-boundary temperature at 4:4 C (in the case of a run in group I) or increasing it above ◦ 4:4 C (in the case of a run in group II), resulting in the growth of a planar (or nearly planar) hydrate layer down from the upper boundary. If we assume that the temperature at the interface between the hydrate layer and the ◦ residual liquid-solution layer was held at Teq , i.e., 4:4 C, it turns out that the solution layer was kept isothermal at ◦ 4:4 C until it vanished in every run in group I, while the solution layer was inevitably non-isothermal and a free

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Table 1 Temperature adjustment in the experimentsa Group of experiments

Upper-boundary temp. T1; i T1 → ◦ ◦ C C

Lower-boundary temp. T2 T2; i → ◦ ◦ C C

I II III IV

4:4 → 0:1; 1:0; 2:1; 3:3 4:4 → 0:0 ± 0:3 3:0 → 3:0 ¿ 4:4 → −3:0

4:4 → 4:4 4:4 → 4:7; 5:7; 6:5; 7:2; 8:6 3:0 → 5:0; 6:4; 7:4; 8:2 ¿ 4:4 → 4:4

a Note:

T1 and T2 were held within ±0:1 K of their prescribed levels listed in this table during each experimental run.

convection possibly occurred in the layer in each run in group II. 2.2.2. Group III—melting of a hydrate layer upward from the lower boundary The upper- and lower-boundary temperatures were 6rst ◦ ◦ adjusted to −5 C and 4:4 C, respectively, to make a planar hydrate layer grow down from the upper boundary. After the hydrate layer had grown to the lower boundary, taking the form of a monolithic brick, both the upper- and ◦ lower-boundary temperatures were changed to 3:0 C and then held constant for 2 h (or even longer) to make the ◦ hydrate layer isothermal at 3:0 C. An experimental run was then started by increasing the lower-boundary tem◦ perature above 4:4 C, while the upper-boundary temper◦ ature was being held constant at 3:0 C, thereby resulting in the melting of the hydrate layer upward from the lower boundary. 2.2.3. Group IV—growth of column-like hydrate crystals down from the upper boundary We unexpectedly found that when the test cell 6lled with a fresh THF solution having no prior experience of hydrate formation=dissociation was cooled down below ◦ ∼ 2 C, a small number of column-like crystals of THF hydrate sometimes grew down from the upper boundary, or grew upward from the lower boundary, much faster than a planar, polycrystalline hydrate layer. Thus, we added experiments to focus on observing such crystals. In these experiments, the upper- and lower-boundary temperatures were simply decreased from room temperature ◦ ◦ to −3:0 C and 4:4 C, respectively, and then held constant at these levels.

3. Heat transfer analyses 3.1. Modeling and formulation The heat transfer from, or to, the hydrate–solution interface across the hydrate and solution layers was analyzed to simulate the growth or shrinkage of the hydrate layer in each of the experiments in groups I–III. The as-

Fig. 3. Planar hydrate layer growing down from the upper boundary (x = 0) toward the lower boundary (x = H ) — illustration of one-dimensional heat transfer model.

sumptions which underlie the present analyses are summarized below. (1) The heat Low across the hydrate and solution layers is one dimensional. (2) The hydrate–solution interface is always held at ◦ the two-phase equilibrium temperature, Teq (=4:4 C). (3) As a consequence of the above two assumptions, the thickness of the hydrate layer, , is uniform (see Fig. 3). (4) The mechanism of heat transfer through the solution layer is either transient or quasi-steady conduction as long as Ra is lower than 1708; the mechanism is replaced by quasi-steady free convection, if Ra exceeds 1708; here Ra is the Rayleigh number de6ned as Ra =

g L (T2 − Teq )(H − )3 ; L L

(1)

where g is the acceleration due to gravity, L the thermal expansion coeMcient of the solution, H the height of the space occupied by the THF–water mixture, the thickness of the hydrate layer, and L and L are the thermal diHusivity and the kinematic viscosity, respectively, of the solution. (5) Physical properties are constant throughout each of the hydrate and solution layers irrespective of the temperature distributions inside them, except for the temperature dependency of the solution density L assumed only implicitly in evaluating L .

T. Iida et al. / Chemical Engineering Science 56 (2001) 4747–4758

(6) The hydrate density H and the solution density L are identical. Thus, neither the formation nor the dissociation of hydrate crystals at the interface between the two layers causes any vertical displacement of either layer. The governing equation for the heat transfer in the hydrate layer is given by @2 T @T = H 2 ; (2) @t @x where T is the temperature as a variable, t time, H the thermal diHusivity of the hydrate, and x the vertical coordinate directed downward from the upper boundary of the space 6lled with the THF–water mixture. The heat transfer in the solution layer also needs to be formulated except for the case to simulate the experiments in group I, in which the solution layer is assumed to be isothermal, i.e., T = Teq (for 6 x 6 H ). This formulation is done in two alternative ways, depending on the value of Ra at each instant. When Ra ¡ 1708, the following equation applies: @T @T = L 2 : (3) @t @x When Ra ¿ 1708, the heat Lux across the solution layer, q˙L , is given as follows:
 T2 − Teq ; (4) q˙L = NukL H − where kL is the thermal conductivity of the solution, and Nu is the Nusselt number for the free-convection heat transfer in the solution layer. According to assumption (4), Nu is evaluated by an empirical correlation for steady free-convection heat transfer through an in6nite horizontal layer of a Luid heated from below (Churchill, 1983): 
1:5  51=15 Rac Raf(PrL ) ; + Nu = 1 + 1:446 1 − Ra 1420 (5) where






f(PrL ) = 1 +

0:5 PrL

9=16 −16=9

;

Rac denotes the value of Ra critical for the occurrence of free convection (=1708), and PrL is the Prandtl number of the solution. The boundary conditions are written as follows: at x = 0;

T = T1 ;

(6)

at x = H;

T = T2 ;

(7)

at x = ; T = Teq ; @T @ = q˙L + WhHL H ; kH @x x= −0 @t

(8) (9)

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where kH is the thermal conductivity of the hydrate, T= x|x= −0 is the hydrate-side temperature gradient at the hydrate–solution interface, and WhHL is the heat of hydrate formation per unit mass. In calculations using Eq. (3) instead of Eq. (4), Eq. (8) needs to be read as follows: @T @T @ (10) = k + WhHL H ; kH L @x x= −0 @x x= +0 @t where @T=@x|x= +0 is the solution-side temperature gradient at the hydrate–solution interface. The initial condition is given in two alternative forms. For calculations to simulate the experiments in groups I and II, it is given as follows: at t = 0;

T = Teq (0 6 x 6 H ):

(11)

For calculations relevant to group III, the initial condition is changed to the following: at t = 0;

T = T1 (0 6 x 6 H ):

(12)

3.2. Solution procedures The procedures to deduce − t relations for individual processes of growth or shrinkage of hydrate layers are outlined below. But for one exception, the procedures rely on numerical calculations with a computational algorithm into which the equations listed above are incorporated. The exception is the procedure relevant to the experiments of group I. The problem posed here is to solve Eq. (2) under the conditions speci6ed by Eqs. (6), (8), and (11). This is simply a problem known as the Stefan problem, and it can be analytically solved to give the following solution (Carslaw & Jaeger, 1959):   2kH (Teq − T1 )t 1=2 : (13) = WhHL H The computational algorithm used in simulating the experiments of group II includes an explicit 6nite-diHerence solution procedure applied to Eq. (2) and, as long as Ra ¡ 1708 at t = 0( = 0), Eq. (3). Fifty nodal points were laid uniformly on each of the hydrate and solution layers, and the interval between successive time steps is maintained below the limit de6ned by the stability criterion. (In advance of its use in simulating the experiments of group II, the above 6nite-diHerence solution procedure was tested for its accuracy in simulating the experiments of group I. The obtained − t solutions showed excellent agreement with corresponding predictions given by Eq. (13), thereby suggesting the solution procedure to be generally suitable for use in the present study.) If Ra ¿ 1708 at t =0 ( =0), the algorithm works with Eqs. (4) and (5), leaving Eq. (3) untouched. When t has advanced until Ra has decreased to 1708; Nu in Eq. (4) is

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Table 2 ◦ Physical properties of THF–water mixture of 1 : 17 molar ratio (evaluated at 4:4 C unless otherwise noted) Property −1

Thermal conductivity k(W m−1 K ) −1 Speci6c heat cp (kJ kg−1 K ) −3 Density (kg m ) Thermal diHusivity (m2 s−1 ) Kinematic viscosity L (m2 s−1 ) Thermal expansion coef. L (K −1 ) Prandtl number Pr (dimensionless) Heat of hydrate formation WhHL (kJ kg−1 )

Liquid solution

Hydrate

0.562a 4.21b 997d 0:134 × 10−6e 3:04 × 10−6f 3:91 × 10−4g 22.7h

0:525a 2:123c 997d e 0:248 × 10−6

260.0c

a Ross

and Andersson (1982). for pure water substituted for that of THF solution. c Leaist et al. (1982). d Gough and Davidson (1971). e Derived from k;  and c . p f Measured by the authors; see Appendix A. g Evaluated at 6:3◦ C; see Appendix A. h Derived from  and  . L L b Value

set at unity, causing only a minute stepping down in Nu from the value given by Eq. (5). Thereafter, Eq. (4) is used with Nu 6xed at unity, assuming quasi-steady conduction heat transfer through the residual solution layer. The algorithm used in simulating the experiments of group III is similar to that described above. It works with Eq. (3), together with Eq. (2), until Ra is increased to 1708. Thereafter, the algorithm substitutes Eqs. (4) and (5) for Eq. (3). The physical properties of the THF solution and THF hydrate used in deducing the − t solutions are summarized in Table 2. For simplicity, all of these properties ◦ are evaluated at 4:4 C (cf. assumption (5) in Section 3.1) ◦ except for L , which is evaluated at 6:3 C, the arithmetic ◦ ◦ mean of 4:4 C and 8:2 C, the highest value of T2 in the experiments of group III. Some details of evaluation of L and L are described in Appendix A.

Fig. 4. Snapshot of a planar hydrate layer growing down from the ◦ ◦ upper boundary. T1 = 3:3 C, T2 = 4:4 C, t = 270 min.

4. Results and discussion 4.1. Growth of hydrate layers (experiments in group I and relevant analysis) It was observed that, in each experimental run with T1 ◦ set at 2:1 C or lower, a hydrate phase 6rst appeared at one point on the surface of the upper copper plate and spread over the surface within several minutes in the form of a ◦ thin, apparently uniform layer. When T1 = 3:3 C, we 6rst observed several hydrate patches, which grew individually on the copper-plate surface until they coalesced into a single 6lm completely coating the surface. The hydrate layer thus spread over the copper-plate surface grew downward, increasing its thickness. The hydrate–solution interface was kept Lat and smooth, as shown in Fig. 4, until it arrived at the surface of the lower copper plate.

Fig. 5. Hydrate layer thickness versus time: comparison between observations in experiments (group I) and theoretical predictions given by Eq. (13).

T. Iida et al. / Chemical Engineering Science 56 (2001) 4747–4758

◦

4753

◦

Fig. 6. Sequence of growth of a hydrate layer down from the upper boundary. T1 = −0:1 C, T2 = 5:7 C. Note the wavy pattern growing at the hydrate–solution interface, which is ascribable to free convection in the solution layer.

In the course of its growth in thickness, the hydrate layer occasionally suHered crack propagation and, at the same time, underwent a change in transparency. The crack propagation occurred more frequently at lower lev◦ els of T1 . It hardly occurred when T1 = 3:3 C. In contrast to the crack propagation being erratic in nature, the change in transparency progressed rather regularly. At early stages, the hydrate layer growing at relatively high rates was substantially opaque. With the lapse of time, and with a decrease in the thickening rate, the front portion of the layer became increasingly transparent. Also, the base portion solidi6ed at early stages gradually became transparent. These observations are presumably ascribable to some change in polycrystalline structure in the hydrate layer with the lapse of time and with a decrease in the rate of hydrate-crystal formation. Fig. 5 compares the observations of the hydrate-layer growth in thickness and theoretical predictions given by Eq. (13). Despite the variable appearance of each hydrate layer discussed above, the observed behavior of hydrate-layer growth is well approximated by the analytic solution, Eq. (13), which assumes hydrate layers to be homogeneous and invariable with respect to their physical properties. Some irregularity in the data points at the early stages (t . 20 min) is ascribable to an uncertainty in de6ning the origin of time, t = 0, in each experimental run. (For each run, the instant when we 6rst visually con6rmed hydrate formation was de6ned as the origin of time, t = 0.) 4.2. Growth of hydrate layers (experiments in group II and relevant analysis) The qualitative observation of hydrate layers in the experiments of group II diHers from that described in Section 4.1 only in that the hydrate–solution interfaces were not always Lat but sometimes appreciably wavy as demonstrated in Fig. 6. In other words, the thickness of each hydrate layer at each instant could be spatially variable. In Fig. 7, the amplitude of observed at each instant was indicated by a vertical bar piercing a

Fig. 7. Hydrate layer thickness versus time: comparison between observations in experiments (group II) and numerical solutions. Each of the vertical bars laid on some data points indicates the range of spatial variation in thickness of a hydrate layer with a wavy interface.

data-point symbol indicating the spatial average of . It is reasonable to assume that the wavy pattern observed at a hydrate–solution interface was caused by the free convection induced in the solution layer. However, the relation between the amplitude of and the strength of free convection measured by, for example, Ra is not straightforward. As observed in Fig. 7, the amplitude was the ◦ largest when T2 was not the highest (8:6 C) but was in◦ termediate (5:7 C). No clear explanation of this fact can be provided at present. The wavy pattern induced at a hydrate–solution interface could aHect the heat transfer from the solution layer to the interface. Fig. 7 shows, however, that the general behavior of the growth of hydrate layers with such wavy interfaces is still predicted with reasonable accuracy by the numerical solutions not taking into account such wavy interfaces.

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◦

◦

◦

◦

Fig. 8. Two sequences of melting of a hydrate layer. (a) T1 = 2:6 C, T2 = 5:0 C. (b) T1 = 2:5 C, T2 = 6:4 C. Note the wavy pattern growing at the hydrate–solution interface in (b), which is ascribable to free convection developing in the solution layer. Several gas bubbles are observed in (b).

4.3. Melting of hydrate layers (experiments in group III and relevant analysis) ◦

The hydrate layer held at 3:0 C in advance of each melting experiment was highly transparent. As soon as the ◦ lower-boundary temperature was increased above 4:4 C, thereby causing hydrate dissociation at the bottom of the layer, we sometimes noted many streaks extending from the hydrate–solution interface into the hydrate layer to a depth of a few millimeters. At the same time, the layer began to become opaque gradually, indicating a change in the polycrystalline structure of the layer reversed to that referred to in Section 4.1. The hydrate–solution interface is kept almost Lat, as shown in Fig. 8(a), when the ◦ excess of T2 over Teq ; 4:4 C, is relatively small. The interface became increasingly wavy with the thickening of the solution layer, as shown in Fig. 8(b), when the excess of T2 over Teq was so large as to induce free convection in the solution layer. We also found that gas bubbles formed and grew at, or near, the interface. In general, the rate of increase in the total volume of such gas bubbles tended to increase with the progress of the melting of the hydrate layer in each experimental run. This fact indicates that the hydrate crystals formed earlier and hence located at an upper position in the test cell released a larger amount of gases when they dissociated than did the crystals formed later and located at a lower position. Because we did not perform any chemical analysis of the gases composing the bubbles, we cannot de6nitely identify their chemical species. By consulting a discussion on air inclusions in

Fig. 9. Hydrate layer thickness versus time: comparison between observations in experiments (group III) and numerical solutions. Each of the vertical bars laid on some data points indicates the range of spatial variation in thickness of a hydrate layer with wavy interface.

THF hydrate crystals given by Larsen (1997), however, we assume the gases to be nitrogen and oxygen that were 6rst dissolved in water in preparing the THF solution and then 6xed into the hydrate crystals of structure II as the guest molecules occupying a small proportion of the 512 cavities in the crystals.

T. Iida et al. / Chemical Engineering Science 56 (2001) 4747–4758

◦

4755

◦

Fig. 10. Sequence of growth of column-like crystals of hydrate. T1 = −3:0 C, T2 = 4:4 C. Note that their axial growth is faster than their growth in thickness and also much faster than the growth of a planar polycrystalline layer.

Fig. 9 shows the experimental data and relevant numerical predictions of the thinning of hydrate layers. The agreement between the experimental data and the predictions is rather poor at intermediate levels ◦ ◦ of T2 ; 6:4 C and 7:4 C. Except for the lowest level ◦ of T2 ; 5:0 C, the aligned data points for each experimental run exhibit an inLection point at ≈ 15 mm. This point presumably corresponds to the inception of the free-convection-dominated heat transfer in the growing liquid-solution layer. The prediction curve relevant to the experimental run also shows an inLection point, which corresponds to the transition from the transient-conduction regime to the quasi-steady free-convection regime at Ra = Rac = 1708 in the computation algorithm. However, the inLection point on the prediction curve always precedes the corresponding point shown by the experimental data, indicating an appreciable delay in the actual development of the free convection in the solution layer from the instantaneous development of the convection assumed in the algorithm. This is presumed to be the primary cause of a substantial discrepancy between the observed − t behavior and the predicted behavior. To obtain more accurate predictions, it is necessary to replace Eqs. (4) and (5) incorporated into the present algorithm by a numerical calculation scheme for transient free-convection heat transfer. 4.4. Growth of column-like hydrate crystals (experiments in group IV) Fig. 10 shows the sequence of the growth of columnlike crystals. These crystals were rectangular in cross section. Their axial growth was much faster than the planar growth of a polycrystalline hydrate layer. The 6rst picture in Fig. 10 demonstrates that the column-like crystals had grown to a few millimeters above the surface of ◦ the lower copper plate, which was held at 4:4 C, within 3 min, while the polycrystalline layer had not yet grown to 1 mm in thickness. The column-like crystals hardly elongated further but gradually thickened. In this course, they were getting deep into the growing polycrystalline layer from their root side.

The orientation of such column-like crystals could not be well de6ned. Their axes were scattered within ◦ some 30 C about the vertical, i.e., the normal to the isotherms in the test cell. It seems that the presence of a temperature gradient in the THF solution caused the formation of such column-like crystals, which are greatly diHerent in appearance from the regular octahedral hydrate crystals grown from an isothermal, stoichiometric THF solution (Makogon et al., 1997; Larsen et al., 1998). The reason why the column-like crystals grew only from a fresh THF solution having no prior experience of hydrate formation=dissociation is unclear. When a solution having prior experience of hydrate formation=dissociation was used, hydrate nucleation generally occurred before the upper copper plate was cooled enough to establish a steep temperature gradient in the solution, resulting in rapid formation of the polycrystalline layer covering the surface of the copper plate. This situation may have prevented some of the nuclei from excessively growing into column-like crystals without being hindered by neighboring crystals. Our observation of column-like hydrate crystals remains to be interpreted more convincingly.

5. Concluding remarks The phase change between a hydrate and liquid solution of a THF–water mixture having the same composition as that of the hydrate has been observed and examined by means of heat transfer analyses. The growth and shrinkage of a polycrystalline hydrate layer in response to a one-dimensional temperature distribution imposed upon the mixture was found to be predicted with reasonable accuracy by a conventional analytic means taking into account the transient conduction and free convection in the hydrate and liquid-solution layers and by assuming a hydrate–solution equilibrium being held at the interface between the two layers. This fact indicates the prospect of accurate control of an energy storage system using a stoichiometric solution of a water-soluble hydrate former,

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such as THF, with the aid of a rather simple heat-transfer calculation scheme. We unexpectedly found the formation of column-like hydrate crystals in the THF solution subjected to a temperature gradient. The growth of these crystals cannot be dealt with by the framework of one-dimensional heat-transfer analyses presented in this paper; it remains to be studied from the aspects of crystallography and heat transfer. If our interest is limited to the gross rate of growth or melting of a hydrate phase in, for example, an energy storage system, the eHect of possible formation of such column-like crystals will be safely neglected because of their small volumetric proportion in the system.

Acknowledgements This study has been supported in part by the Grant-in-Aid for Scienti6c Research from the Japan Society for the Promotion of Science (grant no. 10450088). We thank Hirofumi Migita, student in the Department of Mechanical Engineering, Keio University, for his help in preparing some of the drawings used in this paper.

Fig. 11. Kinematic viscosity of THF–water solution with a molar ratio of 1:17. Each data point represents the arithmetic mean of the data obtained by three independent measurements.

Appendix A. Kinematic viscosity, density and thermal expansion coe&cient of stoichiometric THF solution The kinematic viscosity of a THF–water solution adjusted to a molar ratio of 1 : 17 was measured, using an Ubbelohde viscometer immersed in a thermostated water bath. The obtained data are plotted in Fig. 11. The solid curve laid on the data points represents the following correlation prepared by means of a regression analysis of the data: T L = 3:695 − 1:697 × 10−1 ◦ [mm2 s−1 ] [ C]
2 T : +4:58 × 10−3 ◦ [ C]

Fig. 12. Mass density of THF–water solution with a molar ratio of 1:17.

(A.1)

The L value shown in Table 2 is given by the above correlation. A curve-6tting correlation for the density of the solution, L , was also prepared by a regression analysis of the data given by Gough and Davidson (1971) and by Kiyohara, D’Arcy, and Benson (1978). Kiyohara et al. (1978) measured the density, varying the THF-to-water molar ratio at each of the 6ve diHerent temperatures. The L value at a 1 : 17 molar ratio at each temperature was estimated, by a linear interpolation, from the original L data obtained at two diHerent molar ratios adjacent to

1 : 17. Five L values thus deduced from the original data of Kiyohara et al. (1978) were used, together with the four data points of Gough and Davidson (1971), in the regression analysis and are plotted in Fig. 12. The solid curve drawn in Fig. 12 represents the quadratic correlation obtained by the regression analysis, which is T L = 847:03 + 1:4585 −3 [K] [kg m ]
2 T −3 −3:3057 × 10 : [K]

(A.2)

T. Iida et al. / Chemical Engineering Science 56 (2001) 4747–4758

The expansion coeMcient L is deduced from Eq. (A.2) as follows:

L [K −1 ]

=

−1:4585 + 6:6114 × 10−3 (T=[K]) : 847:03 + 1:4585(T=[K]) − 3:3057 × 10−3 (T=[K])2

(A.3)

The L value shown in Table 1 is given by Eq. (A.3) at ◦ T = 279:45 K(6:3 C), which is the arithmetic mean of ◦ ◦ 4:4 C and 8:2 C, the highest T2 value in the experiments of group III. Notation cp g H WhHL k Nu Pr q˙L Ra T Teq T1 T1; i T2 T2; i t x

speci6c heat capacity at constant pressure, J kg−1 K −1 acceleration due to gravity, m s−2 height of the space 6lled with THF–water mixture, m heat of hydrate formation, J kg−1 thermal conductivity, W m−1 K −1 Nusselt number Prandtl number heat Lux on the THF solution side of the hydrate–solution interface, W m−2 Rayleigh number ◦ temperature, K or C hydrate–solution equilibrium temperature, K or ◦ C upper-boundary temperature maintained in ◦ each experimental run, K or C upper-boundary temperature maintained in ad◦ vance of each experimental run, K or C lower-boundary temperature maintained in ◦ each experimental run, K or C lower-boundary temperature maintained in ad◦ vance of each experimental run, K or C time, s coordinate directed downward from the upper boundary of the space 6lled with THF–water mixture, m

Greek letters 

L L 

Subscripts c H L

1 @L ≡− L @T

thermal diHusivity, m2 s−1 thermal expansion coeMcient of THF solution, K −1 thickness of hydrate layer, m kinematic viscosity of THF solution, m2 s−1 density, kg m−3

4757

critical condition for the occurrence of free convection in THF solution layer hydrate aqueous THF solution

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