462

Leonard, G. L.; Mitchner, M.; Self, S. A. J. Fluid Mech. 1983,127, 123. ation, Montreal, Quebec, 1980.

Marietta, M. G.; Swan, G. W. Chemical Engineering Science; Per- gamon: Elmsford, NY, 1976; Vol. 31, p 795.

Melcher, J. R. Continuum Electromechanics; M.I.T. Press: Cam- bridge, MA, 1981; pp 5.2-5.16.

Pyle, B. E.; Pontius, D. H.; Snyder, T. R.; Sparks, L. E. Presented

Ind . Eng. Chem. Res. 1987,26, 462-469

at the 73rd Annual Meeting of the Air Pollution Control Associ-

Robinson, M., PhD Thesis, Cooper Union, NY, 1975. Taylor, G. I. Proc. London Math Soc., 1921, A20, 196.

Receiued for review June 3, 1985 Accepted July 17, 1986

Solid-Vapor Azeotropes in Hydrate-Forming Systems Jashwantsinh L. Thakore and Gerald D. Holder* Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261

Three-phase vapor (V), water-rich liquid (LJ, and hydrate (H) equilibrium conditions and solid hydrate azeotropic compositions were determined for various ternary mixtures in the temperature range from 274.15 to 281.15 K. The experimental measurements demonstrate that the systems, methane-propane-water and krypton-propane-water, form hydrate azeotropes in the temperature range 274.15-281.15 K. The systems, methane-cyclopropane-water and methane-isobutane-water, do not form hydrate azeotropes in this temperature range. Azeotropes occur as a result of the hydrate crystal structure and are not caused by hydrate-phase nonidealities.

Gas hydrates are crystalline compounds which form from mixtures of light, nonpolar gases and water. In particular, hydrate crystals can form from mixtures of natural gases (methane, ethane, and propane) and water. Hydrates can form in pipelines (Hammerschmidt, 1934) or in association with an underground hydrocarbon res- ervoir, where they can impede production by blockage of rservoir pores (Katz, 1971, 1972; Holder e t al., 1976). Because hydrates can form at temperatures well above the freezing point of ice, they have been considered as a me- thod of removing salt from sea water (Barduhn et al., 1962; Barduhn, 1969) and as a potential vehicle for the storage of natural gases (Parent, 1948). In any of these applica- tions, the conditions determining the stability of a hydrate are pressure (P), temperature (T) and equilibrium gas- phase composition (y). Any thorough evaluation of the conditions of hydrate stability requires knowledge of the extreme limits of such stability. This limit will be the maximum temperature at a given pressure or the minimum pressure at a given temperature for which hydrates are stable.

Most thermodynamic studies of gas hydrates have fo- cused on the conditions where a vapor (V), a water-rich liquid (L), and a solid hydrate (H) are all in equilibrium. The pressure and temperature conditions describing such three-phase equilibria are univariant-a line on the P-T coordinates as in Figure 8-for any binary gas-water mixture or for any multicomponent mixtures of fixed gas-phase composition. For a specified gas mixture, such equilibrium conditions determine the limits of stability (maximum temperature or minimum pressure) for the hydrate phase.

The limiting composition conditions for hydrate for- mation have not been thoroughly studied however. Thermodynamic studies have shown, in general, that larger hydrate-forming molecules, such as propane, form hydrates with relatively low equilibrium dissociation pressures, and smaller molecules, such as methane, form hydrates with relatively high dissociation pressures. Hydrates from mixtures of such larger and smaller molecules will have dissociation pressures nearer to the dissociation pressure of the larger molecules than might be expected. For ex- ample, a 99% methane-1% propane gas mixture, when contacted with water, will form hydrates with a dissociation

* Address to whom correspondence should be addressed.

0SSS-5SS5/S7/2626-0~62$01.50/0

pressure which is less than half the dissociation pressure of pure methane (Deaton and Frost, 1949).

A few studies (Snell e t al., 1961; Verma, 1974; van der Waals and Platteeuw, 1959; Holder and Grigoriou, 1980) have shown that some ternary (gas + gas + water) mixtures can, in fact, have dissociation pressures which are lower than those of either gas-water binary pair which comprise the ternary mixture. For example, at 278 K a methane- propane mixture containing 25 mol% propane in the gas phase forms hydrates a t a pressure which is about 10% lower than the hydrate-formation pressure of pure propane (Verma, 1974). For this ternary system, there will be some mixture which will have a lower dissociation pressure than any other mixture; this pressure will be lower than the dissociation pressure of both propane and methane hy- drate. Such a mixture will be an azeotropic mixture, but the azeotrope’s existence does not correspond to noni- dealities in the classical equilibrium model but rather to the nature and structure of crystalline hydrates in general.

The greater stability (as evidenced by lower equilibrium pressures) of the hydrates formed from certain mixtures is due to competing effects. For example, in propane hydrates, propane molecules can enter only the large cavities of structure I1 hydrate, and as in all hydrate phases, the hydrate stability is derived from its cavities being occuppied. Methane, a relatively small molecule, normally forms structure I but can enter the small cavities of structure I1 and contribute to the stabilization of that structure in a way that propane cannot, since propane does not enter the small cavities and therefore can never sta- bilize them. The net effect of decreasing the mole fraction of propane in a propane-methane mixture is to increase the stability of the hydrate provided by methane and to decrease the stability provided by the propane. Under certain conditions, including the specific point mentioned above, the increase in stability by methane is greater than the decrease in stability by propane, and a lower pressure is required to form the hydrates. If propane could enter the small cavities in a crystal structure, the increase in methane mole fraction would result in a higher, not a lower, equilibrium pressure.

Model The stability of a crystalline hydrate phase depends on

the fraction of its cavities occupied by gas molecules. If an insufficient fraction is occupied for a given set of

0 1987 American Chemical Society

Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 463

pressure, temperature, and composition conditions, the hydrates become unstable and will dissociate into gas and water or gas and ice, depending on whether the tempera- ture is above or below 273 K.

By use of chemical potential as a measure of the hy- drate's stability, the effect of the fraction, 0, of the cavities which are occupied was developed by van der Waals and Platteeuw (1959) and is expressed as A p = pp - pH = -RT[vl In (1 - e l )] - RT[v2 In (1 - e,)]

(1) where pH is the chemical potential of the hydrate a t the specified temperature, T; pp is the chemical potential of the hypothetical empty (gas-free) hydrate lattice; O1 and O2 are the fractional occupancies of the small and large cavities, respectively; and v1 and v2 are constants equal to the numbers of small and large cavities, respectively, per molecule of water in the hydrate. The quantity, Ap, is the reduction in chemical potential of water in the hydrate phase caused by the occupancy of the cavities by the gas molecules.

The fractional occupancies are given by classical Lang- muir-type adsorption theory,

c C,f, 0, = ' j = l , 2

1 + zcjifi i

where fi is the gas-phase fugacity of the ith gas species. The summation extends over all hydrate-forming gas species. The Langmuir constant, Cji, applies to an i molecule in a j cavity. Equations 1 and 2 are combined to give

- v1 In [l + CClir#qyiP] = v2 In [I + CC2i,piuiP] (3) AP

To find the composition of the hydrate azeotrope, the derivatives with respect to the number of moles of each species, n, are taken (here y is the gas-phase mole fraction). Because an extremum is desired, derivatives of pressure are set to zero as are derivatives of the chemical potential difference, ApIRT, since this quantity varies monotonically with pressure at a fixed temperature.

_ - R T 1 i

= - - - 0 a[Acl/RTl ap

an, a n K

K = l , 2 , 3 ,..., N (4) As shown by Holder and Grigoriou (1980) for a system

containing two gases and water, the following is obtained for the azeotropic composition

(5) Y - 1 + ($22 - C12P

[Cl , - Cl2 + Y(C22 - C21)IP Y1 =

where

Y l c12 - c11

v2 c21 - c22 Y = -

Equations 5,6, and 3 can be solved simultaneously for the pressure and composition of a hydrate azeotropic point.

The development above shows that some mixtures will form hydrate azeotropes, but it does not indicate the properties of the gases which are most important for azeotropes to form. Equation 5 shows that the composition a t the hydrate azeotropic condition depends strongly on the Langmuir constants, which in turn depend upon the gas-water Kihara parameters and the cavity radii. To determine whether two gases will form an azeotrope, the

Table I. Kihara Parameters and Diameter Ratios for Selected Gasesa

aid, for species u, pm u / d l structure I structure I1

methane 350.1 0.452 0.422 0.372 ethane 403.6 0.521 0.486 0.429 ethylene 381.9 0.493 0.460 0.406 propane 439.9 0.567 0.530 0.468 cyclopropane 419.1 0.541 0.505 0.446 isobutane 483.8 0.624 0.583 0.514 krypton 353.1 0.456 0.425 0.375

Kihara Parameters are from John et al. (1985).

ratio of their gas Kihara diameters, u, to the cavity diam- eter, d , is examined. This ratio does not give the hydrate azeotropic condition but only indicates whether two gas species can be combined to form a azeotrope.

Calculations (Holder and Grigoriou, 1980) indicate that each cavity has an optimal gas-water Kihara diameter associated with it, and the optimal diameters are such that the diameter-to-cavity-diameter ratio, old, is always near 0.44. In general, a ternary mixture will form hydrate azeotropes if any only if one of the gases has a diameter ratio, u/d2, for the large cavity very near to 0.44 and the other gas has a diameter ratio, a/dl, for the small cavity near 0.44.

This analysis allows selection of gases which will form hydrate azeotropes. Table I shows the Kihara diameters and diameter ratios for the large and small cavities of both structures. Because of the presence of larger molecules, structure I1 hydrate will, in general, be more stable than structure I, but structure I systems can form azeotropes under some conditions.

On the basis of these tables, the mixtures considered in this study were (1) a methane-propane-water system at 275.15 and 278.15 K, (2) a krypton-propane-water system at 276.15 K, (3) a methane-cyclopropane-water system at 277.15 and 281.15 K, and (4) a methane-isobutane-water system at 274.35 K. Because old2 = 0.514 for isobutane, this mixture is the worst candidate. Note that the u value of Parrish and Prausnitz (1972) indicates that this mixture is a good candidate as reported by Holder and Grigoriou (1980). If this mixture forms an azeotrope, the 1972 value may be best; otherwise, the value in Table I is probably more useful.

These mixtures are good candidates for the formation of hydrate azeotropes. The temperatures were selected so that a liquid hydrocarbon phase would not form; otherwise, they are arbitrary.

Experimental Section The (P , T , y) loci for three-phase (vapor, water-rich

liquid, and hydrate, VLIH) equilibria in the above ternary systems were determined between 274.15 and 281.15 K. The mole fraction of water in the vapor, V, was so small that it did not affect the reported gas-phase composition.

The experiments were carried out by using the appa- ratus shown in Figure 1. A high-pressure glass-windowed cell was immersed in a constant-temperature bath during the experiments. The bath, containing a 50-50 metha- nol-water mixture, was held a t constant temperature by using a Neslab refrigeration unit, two immersible heaters, and a microset thermoregulator which could control the bath temperature to 0.02 OF. The bath was insulated on all sides, and the bath fluid was circulated by using three submersible circulating pumps. Two Plexiglas windows were inserted into the front and back of the methanol- water bath to enable constant viewing of the contents of the high-pressure cell during the course of experiments.

464 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987

.!, NO TITLL NO TiTLE

Figure 1. Experimental apparatus.

To obtain mixing of its contents, the cell was continuously rocked during the experiments.

The pressure was measured by using two Heise gauges of a 0.1% accuracy-one with a range of 3446 kPa and one with a range of 345 kPa. The latter was used only for low-pressure readings.

The temperatures reported were those taken from the methanol-water bath within 3 cm of the experimental cell. Measurements made at several points within the bath indicated no temperature variation within the readability of the thermometer, which was a mercury-in-glass type with 0.1 "C increments. The thermometer was calibrated at the ice point of water and was also checked with a NBS traceable standard thermometer.

The gases used were obtained from Matheson. The mole-fraction purities of the major constituents were 0.999; these purities were verified by using a Perkin-Elmer Sig- ma-1B gas chromatograph.

Before each experimental run, the high-pressure cell was flushed several times and finally filled to three-quarters of its volume with doubly distilled degassed water. The gas in the cell was removed by evacuation, and then the gas volume was purged several times with pure gas of component 2 (the gas species with the lower vapor pres- sure). Because of the low temperature, the gas partially liquefied, as was determined visually. About 7-8 cm3 of condensed hydrocarbon liquid was used to form the hy- drates. However, no condensed liquid hydrocarbon re- mained in the cell during P-T-y measurements.

To form hydrates, the temperature of the bath was lowered below the ice point, so that ice crystals would form and serve as nucleation sites. Once nucleation sites had been obtained, large amounts of hydrates would form. The bath temperature was then raised to the desired value and held constant. The system pressure was recorded, and small amounts of gas were purged until the dissociation pressure of component 2 was reached. Since this pressure was below the vapor pressure of liquid component 2, only three phases (vapor + water-rich liquid + hydrate) were present at this point.

At this state, the cell was pressurized with component 1, and the system pressure and temperature were recorded as a function of time for the first few hours. As indicated by a large drop in the system pressure, more hydrates were forming. After 1 day, during which the cell was continu- ously rocked, the system pressure reached a steady value, indicating the equilibrium between the vapor phase, V, the

U L k

z

w [L 3 Lo Lo w n.

- 10'

n.

arm0 0

2n 1 o2

YOLF FRRCTION "FTPAYE

Figure 2. Methane-propane VLH equilibria at 275.15 K- equilibrium pressure vs. gas-phase composition.

water-rich liquid, L,, and the hydrate phase, H. It is assumed that the finely divided hydrate crystals were in equilibrium with the other phases. The nature of the experimental P-x diagrams indicates this assumption is reasonable.

Next, the system pressure was reduced by opening a valve to allow some of the gas to escape into a sampler cylinder. This reduction in pressure caused some of the remaining hydrates to dissociate and, hence, a new equi- librium condition to be obtained. The gas sample was analyzed on a Perkin-Elmer Sigma-10 chromatograph with a PORPAK-Q column. During the course of determining any (P-T-y) point, the cell pressure changed very little (ca. 10 kPa).

If only a small amount of the hydrate phase was left, the experiment was repeated again with the initial pressure equal to the pressure at which the previous experiment had been stopped. Repeating this procedure gave a complete locus of P-Y measurements at one temperature. The azeotropic condition is the P-Y value which gave the minimum hydrate dissociation pressure.

The azeotrope was confirmed by the observation that the pressure in the cell eventually retained its previous value upon allowing gas to escape. At the true equilibrium azeotropic condition, the dissociation pressure and com- position of the gas sample should not change until all hydrates have dissociated. The successive P-Y measure- ments in a single experiment cannot pass through the azeotrope, as in vapor-liquid azeotropes. Regardless of which side of the azeotropic composition the experiments started on, the P-Y measurements, as expected, always moved toward the azeotrope. The points reported here as azeotropes are the results of three or more consecutive measurements.

Experimental Results and Discussion The experimental and predicted hydrate azeotropic

conditions are shown in Table 11-VI11 and Figures 2-7. The solid curves (1 for the hydrate phase; 2 for the vapor phase) in these figures are obtained by the theoretical model (John et al., 1985) using the T-Y measurements obtained in this study as independent variables; P and the hydrate-phase composition are predicted. Table IX gives

Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 465

Table 11. Experimental and Predicted Hydrate Azeotropic Conditions for Methane-Propane-Water System at 275.15 K"

pressure, kPa

hydrate-phase composition vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction

exptl predicted structure methane propane methane propane 3370 3027 1 0.000 0.000 1.000 0.000 672 57 1 2 0.903 0.097 0.539 0.461 424 393 2 0.765 0.235 0.464 0.536 393 377 2 0.727 0.273 0.450 0.550 365 356 2 0.700 0.300 0.435 0.565 303 300 2 0.516 0.484 0.357 0.643 279 288 2 0.420 0.580 0.315 0.685 279 278 2 0.366 0.634 0.288 0.712 269 277 2 0.352 0.648 0.281 0.719 245b 271b 2b 0.190b 0.810 0.190 0.810 245 272 2 0.083 0.917 0.097 0.903 245 272 2 0.081 0.920 0.094 0.906 245 273 2 0.054 0.946 0.066 0.934 245 272 2 0.046 0.954 0.058 0.942 245 274 2 0.037 0.963 0.047 0.953 245 276 2 0.021 0.979 0.028 0.972 258 278 2 0.000 1.000 0.000 1.000

Average absolute error (AAE) in predicted pressure = 3.36%. Azeotrope (as determined multiple repetitions of pressure and compo- sition).

Table 111. ExDerimental and Predicted Hydrate AzeotroDic Conditions for Methane-ProDane-Water System at 278.15 K" hydrate-phase composition

vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction pressure, kPa

exptl predicted structure methane propane methane propane 4495 4001 1 1.000 0.000 1.000 0.000 1306 1144 848 630 496 489 479 475 479 458 458b 479 480. 509

1123 1050 793 587 472 468 466 460 454 453 452b 524 526 548

2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.956 0.947 0.894 0.768 0.530 0.510 0.502 0.468 0.412 0.394 0.390b 0.030 0.026 0.000

0.044 0.053 0.106 0.232 0.470 0.490 0.498 0.532 0.588 0.606 0.610 0.970 0.974 1.000

0.610 0.600 0.559 0.502 0.416 0.408 0.405 0.392 0.369 0.361 0.359 0.064 0.057 0.000

0.390 0.400 0.441 0.498 0.584 0.592 0.595 0.608 0.631 0.639 0.641 0.936 0.943 1.000

"AAE = 6.4%. bAAzeotrope.

1 I

0.0 0.2 0.4 0.6 0.8 1 .o 3 ~ 1 0 '

MOLE FRRCTION METHRNE

Figure 3. Methane-propane VLH equilibria at 278.15 K- equilibrium pressure vs. gas-phase composition.

10'

D a Y

2

w 3 Lo L1 w w a

.L

Figure 4. Krypton-propane VLH equilibria at 276.15 K- equilibrium pressure vs. gas-phase composition.

466 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987

Table IV. Experimental and Predicted Hydrate Azeotropic Conditions for Krypton-Propane-Water System at 276.15 K" hydrate-phase composition

vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction pressure, kPa

exptl predicted structure propane krypton propane krvpton 1749 985 737 317 310 262 265 255 245 238 234 233 233 231b 239 251 262 255 309

1711 916 996 427 427 364 356 330 322 312 291 292 289 28gb 292 299 299 300 346

"AAE = 23.7%. bAzeotrope.

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0.000 0.020 0.016 0.161 0.161 0.250 0.267 0.339 0.365 0.412 0.575 0.613 0.654 0.669 0.761 0.833 0.835 0.844 1.000

t ? I

13 e 10'

f W iT 3 v) (0 W CI c

/'

,/'

0.: 0.2 0.4 0.6 0.8 ! .C I Y ~

POLE FRRCTICN tlETHRNE

Figure 5. equilibrium pressure vs. gas-phase composition.

the data obtained for the binary systems at the various temperatures. These binary data are in good agreement with the results obtained in other studies (Verma, 1974; Holder and Grigoriou, 1980; Hafeman and Miller, 1969; Holder and Godbole, 1982; Holder et al., 1982).

Figure 2 is useful in graphically demonstrating the limit of accuracy in the results, but more importantly, it clearly shows that a methane-propane-water system forms a hydrate azeotrope a t 275.15 K because the methane [ 19%]-propane [81%] hydrates have a lower disociation pressure than any other methane-propane mixture. This hydrate dissociation pressure is called the azeotropic point. At this point, the water-free vapor-hydrate phase must have the same gas composition as the gas phase. The gas composition in the hydrate phase is, in general, difficult to determine experimentally, but a t an azeotropic point it is identical with the vapor-phase composition. The hydrate and gas-phase compositions are both 19% meth- ane and 81% propane a t 221 kPa and 275.15 K. Hence,

Methane-isobutane VLH equilibria a t 274.35 K-

1.000 0.980 0.984 0.839 0.839 0.732 0.733 0.661 0.635 0.588 0.425 0.317 0.346 0.331 0.239 0.167 0.165 0.156 0.000

0.000 0.318 0.294 0.464 0.464 0.502 0.509 0.535 0.544 0.562 0.625 0.647 0.661 0.669 0.721 0.773 0.774 0.782 1.000

1.000 0.682 0.706 0.535 0.536 0.498 0.491 0.465 0.466 0.438 0.375 0.353 0.339 0.331 0.279 0.227 0.226 0.218 0.000

CRLCULRTED LFG% HYDRRTE PHRSE C f l L C U L f l ~ O P FOR YflPOR P H f l X ,

0 EXPERIWNIRL ORTR

i

2 8 ^ " " C c 2 : 4

"C-1 FR3C-';I J M I T H ' I U C

Figure 6. Methane-cyclopropane VLH equilibria a t 277.15 K- equilibrium pressure vs. gas-phase composition.

azeotropic measurements allow one of the many possible hydrate-phase compositions to be determined.

In Figure 3, the predicted hydrate azeotropic-phase composition was 36% methane and 64% propane. This figure is useful in demonstrating that the composition of the vapor is very sensitive to the dissociation pressure. Slight changes in the system pressure make a large change in the vapor-phase composition. Comparing experimental azeotropic conditions for this system at 275.15 and 278.15 K, it can be seen that the azeotropic hydrate phase be- comes richer in methane as the temperature increases.

Table V and Figure 5 show that the methane-iso- butane-water system does not form hydrate azeotropes at 274.35 K. The methane + cyclopropane + water system also does not form an azeotrope, probably because it forms structure I. I t is very interesting to note that the iso- butanes, a ld , ratio is apparently not close enough to 0.44, using the Kihara parameters of John et al. (1985). If the Kihara parameters of Parrish and Prausnitz (1972) are

Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 467

Table V. ExDerimental and Predicted Hydrate Azeotropic Conditions for Methane-Isobutane-Water System at 274.35 K hydrate-phase composition

vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction pressure, kPa

exptl predicted structure methane isobutane methane isobutane 3099 2811 1 1.000 0.000 1.000 0.000 841 553 2 0.949 0.051 0.547 0.453 461 446 2 0.919 0.081 0.514 0.486 268 286 2 0.792 0.208 0.426 0.574 234 250 2 0.725 0.275 0.390 0.610 210 218 2 0.632 0.368 0.344 0.656 180 189 2 0.500 0.500 0.282 0.718 156 165 2 0.313 0.687 0.191 0.809 136 154 2 0.172 0.828 0.114 0.886 134 153 2 0.150 0.850 0.101 0.899 133 151 2 0.124 0.876 0.085 0.915 132 149 2 0.091 0.909 0.064 0.936 131 149 2 0.086 0.914 0.061 0.939 131 148 2 0.076 0.924 0.054 0.946 131 148 2 0.073 0.927 0.052 0.948 129 148 2 0.066 0.934 0.047 0.953 129 148 2 0.056 0.944 0.041 0.959 129 147 2 0.051 0.949 0.037 0.963 127 147 2 0.048 0.952 0.035 0.965 129 147 2 0.036 0.964 0.027 0.973 128 145 2 0.000 1.000 0.000 1.000

Table VI. Experimental and Predicted Hydrate Azeotropic Conditions for Methane-Cyclopropane-Water System at 277.15 K" hydrate-phase composition

vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction

exptl predicted structure methane cyclopropane methane cyclopropane pressure, kPa

4102 3644 1 1.000 0.000 1.000 0.000 1337 2136 1 0.981 0.019 0.629 0.371 665 1045 1 0.931 0.069 0.363 0.637 513 699 1 0.889 0.118 0.272 0.728 444 601 1 0.856 0.144 0.244 0.756 382 531 1 0.831 0.169 0.222 0.778 324 429 1 0.779 0.221 0.186 0.814 337 411 1 0.776 0.224 0.180 0.820 304 363 1 0.729 0.271 0.160 0.840 278 333 1 0.698 0.302 0.147 0.853 262 310 1 0.671 0.329 0.136 0.864 236 272 1 0.616 0.384 0.117 0.883 218 239 1 0.553 0.447 0.098 0.902 207 223 1 0.515 0.485 0.088 0.912 185 204 1 0.463 0.537 0.076 0.924 172 182 1 0.388 0.612 0.060 0.940 154 172 1 0.345 0.655 0.051 0.949 151 162 1 0.302 0.698 0.044 0.956 137 146 1 0.216 0.784 0.030 0.970 127 142 1 0.191 0.809 0.026 0.974 124 136 1 0.150 0.850 0.020 0.980 111 117 1 0.000 1.000 0.000 1.000

"AAE = 19.0%.

Table VII. Experimental and Predicted Hydrate Azeotropic Conditions for Methane-Cyclopropane-Water System at 281.15 K"

hydrate-phase composition vapor-phase composition (exptl) (predicted) mol fraction mol fraction mol fraction mol fraction

exptl predicted structure methane cyclopropane methane cyclopropane pressure, kPa

6101 410 348 330 262 233 200 194 192 191 184

"AAE = 7.9%.

5316 532 366 356 286 238 212 209 207 206 196

1 1 1 1 1 1 1 1 1 1 1

1.000 0.693 0.517 0.500 0.353 0.197 0.107 0.071 0.058 0.053 0.000

0.000 0.307 0.483 0.500 0.647 0.803 0.893 0.929 0.942 0.947 1.000

1.000 0.169 0.109 0.104 0.068 0.035 0.018 0.012 0.010 0.009 0.000

0.000 0.831 0.891 0.896 0.932 0.965 0.982 0.988 0.990 0.991 1.000

468 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987

Table VIII. Summary of Hydrate Azeotropes hydrate-phase

pressure, kP a composition gas (1)-species (2) temp, K exptl predicted exptl predicted methane-propane 275.15 245.00 271.00 0.19 0.20 methane-propane 278.15 458.00 krypton-propane 276.15 231.00

Table IX. Experimental and Predicted VLIH Dissociation Pressures for Binary Gas-Water Hydrates above the Ice Point

dissociation pressure, kPa

temp, K structure exDtl Dredicted

275.15 276.15 277.15 278.15

276.15 277.15 278.15 279.15 281.15

274.15 275.15 276.15 279.15 278.15

275.35 275.15 277.15 278.15 279.15 281.15

274.35 274.65

Krypton 2 1620 2 1750 2 2130 2 2340

Cyclopropane 1 104 1 111 1 135 1 152 1 184

Propane 2 217 2 248 2 310 2 450 2 510

Methane 1 2870 1 3374 1 3900 1 4495 1 4900 1 6102

Isobutane 2 128 2 155

1594 1750 1921 2109

104 118 134 153 197

223 278 347 435 548

2812 3027 3643 4001 4394 5317

146 157

I I

O ? 3 2 C I 0 6 2 8 c “OLE FRRCTION UETHPLE

Figure 7. Methane-cyclopropane VLH equilibria at 281.15 K- equilibrium pressure vs. gas-phase composition.

used, one would expect methane-isobutane systems to form hydrates since u/dz = 0.44. This is evidence that this latter u value is incorrect. The value of u for cyclopropane

452.80 0.39 0.36 289.00 0.33 0.33

Figure 8. Methane-propane hydrate P-T-Y azeotropic locus.

given by John et al. (1985) may also be incorrect. Figure 8 gives a three-dimensional representation of the

P-T-y locus of krypton-propane hydrates, and Table VI11 gives a summary of the experimental hydrate azeotropes obtained in this study. The ability to predict the hydrate azeotropic composition is a strong factor in supporting the predictive model since these azeotropes are predicted in- dependently of any of the experimental data obtained in this study.

Acknowledgment

under Grant CPE81-15639.

Nomenclature C,, = Langmuir constant for species i in cavity j , kPa-’ d = cavity diameter, pm f, = gas-phase fugacity of species i, kPa n, = number of moles of species i in the gas or hydrate phase P = pressure, kPa R = gas constant, J mol-’ K-’ T = temperature, K y, = gas-phase mole fraction of species i Greek Symbols y = quantity defined by eq 6 0, = fraction of j cavities containing a gas molecule p = chemical potential, J/mol of H20 u, = number of cavities of type j per water molecule in the

u = Kihara diameter, pm Subscripts fl = empty (metastable) hydrate phase H = occupied hydrate phase

Literature Cited Barduhn, A. J. Chem. Eng. Prog. 1969, 63, 98. Barduhn, A. J.; Towlson, H.; Yee, C. H. AZChE J. 1962, 18, 176. Deaton, W. M.; Frost, E. M. US Bureau of Mines Monograph 8,

Hafeman, D. R.; Miller, S. L. J. Phys. Chem. 1969, 73, 1392.

We thank the National Science Foundation for support

hydrate phase

1949; US Bureau of Mines, Washington, DC.

Ind . Eng. Chem. Res. 1987, 26, 469-474 469

Hammerschmidt, E. G. Ind. Ing. Chem. 1934,28,851. Holder, G. D.; Corbin, G.; Papadopoulos, K. D. Znd. Eng. Chem.

Holder, G. D.; Godbole, S. P. AIChE J . 1982, 28 930. Holder, G. D.; Katz, D. L.; Verma, V. K. AAPG Bull. 1976,60,981. Holder, G. D.; Grigoriou, G. J. Chem. Thermodyn. 1980,12, 1093. John, V. T.; Papadopoulos, K. D.; Holder, G. D. AIChE J . 1985,31,

Katz, D. L. J. Pet. Technol. 1971, 23, 419. Katz, D. L. J. Pet. Technol. 1972, 24, 557.

Fundam. 1982,28,930.

251.

Parent, J. D. Inst. Gas Technol. Bull. 1948, 1 , 1. Parrish, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev.

Snell, L. E.; Otto, F. D.; Robinson, D. B. AIChE J . 1961, 7, 482. van der Waals, J. H.; Platteeuw, J. C. Adv. Chem. Phys. 1959,2, 1. Verma, V. K. Ph.D. Dissertation, University of Michigan, Ann Arbor,

1972, 11, 26.

1974.

Received for review June 3, 1985 Accepted August 4, 1986

Heat Transfer from a Particle to a Surrounding Bed of Particles. Effect of Size and Conductivity Ratios

S. Mohan Raot and H. L. Toor* Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

Heat transfer from a test particle to a particle bed has been measured as functions of the ratio of test-particle to bed-particle radius and of the ratio of the thermal conductivity of the discontinuous to that of the continuous medium. Continuum behavior occurs above a radius ratio which increases with conductivity ratio and is approached a t any radius ratio as the conductivity ratio decreases. The nonclassical behavior appears as an additional resistance near the test particle caused by the nonuniform distribution of the heat flux and the localized nature of the conduction paths. A discrete model which accounts for this behavior satisfactorily predicts both steady-state and transient data.

Heat transfer which takes place in a packed bed of particles is classically described by an effective conduc- tivity, a bed continuum property, predicted well by several models (Schumann and Voss, 1934; Deissler and Boegli, 1958; Schotte, 1960; Kunii and Smith, 1960; Crane et al., 1977). However, Rao and Toor (1984) showed that heat transfer from a hot particle to a surrounding bed of rapidly conducting particles is satisfactorily described by the classical continuum assumption only if the hot particle is much larger than the bed particles. When the hot particle was comparable in size to the bed particles, the heat transfer rate was less than half the continuum value. An elementary discrete model fitted this behavior.

The above work was limited to rapidly conducting particles. Clearly, if the thermal conductivity of the bed particles equals the thermal conductivity of the continuous medium (say, air), the discrete effects should vanish (ig- noring second-order effects such as natural convection and radiation) and the bed should behave as a continuum. Thus, the description of heat transfer from particles to a bed should include the effect of the ratio of thermal con- ductivity of the continuous and discontinuous phases. Hence, the need for this study.

Here we present results of studies of heat transfer from hot particles to bed particles of differing conductivities as well as the results for different ratios of test- to bed-particle sizes. A simple generalized discrete model that satisfac- torily predicts experimental data is also presented.

Experimental Section The experimental setup for steady-state and transient

measurements was the same as that described in the earlier paper (Rao and Toor, 1984). I t consisted of electrically heated "test spheres" (made of aluminum of different

* Author to whom correspondence should be addressed. 'Textile Fibers Department, E. I. du Pont de Nemours & Co.,

Seaford, DE 19973.

0888-5885/87/2626-0469$01.50/0

sizes), each suspended a t the center of a 16-cm-diameter spherical glass vessel filled with particles. A heater of known electrical resistance and a thermocouple were im- bedded in the test sphere. The spherical flask was im- mersed in a water bath, effectively at a constant temper- ature.

The packing of particles was done by slowly pouring particles through the necks of the glass vessel. A test particle was allowed to lie on the half-filled bed, and the rest of the flask was filled with bed particles. For steady-state measurements, a suitable current value was set, and the system was allowed to attain steady state, which took about 4 h. The heat-transfer coefficient was calculated from

(1)

where q is evaluated by knowing the current and the re- sistance of the heater. Each steady-state experiment was repeated by repacking the bed several times, using the same procedure. Due to the statistical nature of random packing, the heat-transfer coefficients were found to differ for every packing state. The averages of the heat-transfer coefficients over several packing states, as well as the standard deviation as a percentage of the mean value, are all reported in Table I.

The experiments were all conducted with the tempera- ture of the test sphere at values between 45 and 50 "C and boundary temperatures between 20 and 24 "C. The ef- fective conductivities of all beds used in the study were measured in a parallel-plate apparatus by using a com- parative method (Rao, 1984), and these values were pre- dicted satisfactorily by the method of Kunii and Smith (1960). The Nusselt numbers based on effective conduc- tivity were obtained from

4 4*r2(To - TN)

h =

2roh NU = -

Re

0 1987 American Chemical Society