Ind. Eng. Chem. Fundam. 1982,27, 391-395
391
Measurement and Interpretation of the Water Content of a Methane-Propane Mixture in the Gaseous State in Equilibrium with Hydrate Kyoo Y. Song and Rlkl Kobayashl' Department of Chemical Engineering, Rice University, Houston, Texas 7700 1
Experimental measurements of water content in a gaseous phase of a fixed Composition (5.31mol percent propane in methane) in equilibrium with hydrate are reported for pressures of 300, 500, 1000, and 1500 psia and at temperatures ranging from 234 to 278 K. Water content in the gas mixture obtained in the present study shows much lower values than those obtained at the same condition for methane gas. Moreover, the discrepancies continue to increase as the temperature decreases. Predictions of water content in the gaseous phase have been made using the solid solution theory of van der Waals and Platteeuw and the Redlich-Kwong-Soave equation of state, assuming that only structure I1 crystals were formed at all of the experimental condition of this investigation. The predictions show close agreement with both the smoothed experimental data and the raw data, in view of lo-' mole fraction). the uncertainties involved in measuring small concentrations of water (as low as
Introduction Earlier efforts by Galloway et al. (1970); Sloan et al. (1976);Aoyagi et al. (1979) were successful in eliminating metastable liquid water in the measurement of the water content of methane gas in equilibrium with hydrates. They reported water concentrations as low as to lo* mole fraction in the gaseous phase. For a gas mixture of various hydrate forming components, it is necessary to successively enrich the overall mixture composition of a system with a component that preferentially enters the solid hydrate phase in order to maintain the gas phase concentration at a constant value. Furthermore, it is necessary to decompose and recrystallize the hydrate phase to ensure that each hydrate crystal is indeed in equilibrium with the gas phase. That this state of equilibrium can indeed be achieved cannot be shown experimentally, but it would be of some comfort if it were possible to analyze the hydrate phase as well as the gas phase. Nevertheless, earlier measurements show that the water content of a gas in equilibrium with hydrates is considerably lower than that of the same gas in equilibrium with metastable liquid water. Earlier measurements also indicate that the higher the initial hydrate formation conditions, the greater will be the distance between the stable and metastable equilibrium values at a given pressure and temperature as shown by Aoyagi et al. (1979). Earlier works above the initial hydrate formation temperatures for water-hydrocarbon systems reported by Olds et al. (1942), and McKetta and Katz (1948) indicate that the water content of the fluid phases in equilibrium with liquid water is strongly influenced by the fluid state. Due to the hydrogen bonding nature of water, the water content of the gaseous phase is considerably greater than that of the liquid phase, as shown in the paper by Kobayashi and Katz (1953). The establishment of gas-hydrate equilibrium constitutes the principal problem associated with properly conducting this work. The second serious problem is the sampling and analysis of the gas phase for the small equilibrium concentrations of water. The third difficult problem was to maintain a constant composition in the gas phase by successivelyadding propane, which was depleted 0196-4313/82/1021-039 1$01.25/0
preferentially during the course of hydrate formation. Experimental Method The experimental method used in this investigation has been discussed previously by Galloway et al. (1970),Sloan et al. (1976), Aoyagi et al. (1979). To improve the experimental methods, som e significant modifications to the apparatus were made: (1)for system temperatures below -20 "C, a Freon-502 Unit rather than a Freon-12 Unit was used; (2) a Spectra-Physics Autolab Minigrator was used to integrate the chromatographic peaks for analysis of methane-propane mixtures: (3) an improved absorption column was used for the determination of the very low water concentrations; (4) a Ruska hand pump was inserted into the system for all constant pressure sampling; (5) the heater from the sample discharge line was removed to prevent downstream hydrate blockage; (6) minicomputer with ca. 8K storage capacity was incorporated into the system to store and display the system pressure and temperature at desired time intervals; and (7) a tandem Ruska proportioning pump was operated at a controlled rate to circulate the gaseous mixture. Figure 1shows a schematic diagram of the equilibrium apparatus. A constant water-free gaseous composition was maintained by successively analyzing and adding propane or propane-rich gas to the system while operating the Ruska proportioning pump and simultaneously grinding the hydrates with a rotary motion of the auoclave for several hours. The system pressure and temperature were recorded in the storage area of a minicomputer, at desired time intervals, connected to a pressure transducer and a tenjunction chromel-constantan thermopile with an ice junction. In addition, a 2000-psia Heise gauge calibrated against a Ruska dead weight gauge was used to visually indicate the pressure. The temperature of bath was controlled to better than 0.1 "C and the pressure better than *l.O psi. . The calibration of the chromatographic peaks has been discussed for moisture measurements by Bloch and Lifland (1973) and Sloan et al. (1976). The gas mixture was prepared and certified by Matheson Gas Co., and the gas composition was maintained at 5.31 f 0.09% propane in 0 1982 American Chemical Society
302
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
Table I. Water Content of Methane-Propane Mixture mole fraction x
P, a t m
T, “ F
20.41
39.2 20.0 8.4 -6.6 -16.5 -38.1 34.7 14.0 -6.0 -16.5 -38.1 37.4 14.0 8.4 -6.0 -16.5 -3 8.1 40.0 20.0 8.4 -6.0 -16.5 -38.1
34.01
68.03c
102.O4lc
smoothed exptl 380.00 165.40 88.60 41.15 21.10 6.12 209.00 76.00 25.87 12.66 3.44 107.61 35.79 25.19 11.50 6.33 1.50 9 5.00 31.60 17.70 7.17 3.80 0.99
exptl 427.28 161.99 85.20 41.54 24.28 6.86 187.89 78.76 27.50 13.85 3.47 103.73 35.78 25.42 12.25 7.03 1.92 81.15 26.75 14.67 7.33 3.75 1.15
a Soave modified R-K equation of state (1972). carbon-water in R-K-S.
kv
=
lo6 R-K-S 414.79 162.56 89.04 39.10 21.99 5.70 204.76 73.87 25.22 13.84 3.67 107.17 33.85 25.23 11.55 6.33 1.66 69.36 29.44 17.43 8.85 5.07 1.39
0.45 for hydrocarbon-water in R-K-S.
44
r
dev, PPm 34.79 2.84 0.44 2.05 0.89 0.42 4.24 2.13 0.65 1.18 0.23 0.44 1.93 0.04 0.05 0.00 0.16 25.64 2.16 0.13 1.68 1.27 0.40 kij = 0.60 for hydro-
TEMPERATURE, 1 0 ~ 1 ~ 40 38 36 34
,
THERMOCOUPLE
I
HIGH PRESSURE TO ANALYSIS
PROPORTIONING PUMP
GAS CYLINDER
Figure 1. Schematics of equilibrium apparatus.
methane gas for the period of the experiment. The water-free gas composition was checked by taking small sample in 0.07 cm3 loop into a six-port Valco switching valve. Analysis of the sample was carried out by a Tracor ultrasonic detector. Normally, after allowing 3 to 4 days for equilibration (confirmed by reproducibility of measured data), 5 to 6 measurement of the moisture content were performed (over a 3 or 4 h interval for each measurement) to establish a single experimental data point. To follow the constraint that the hydrocarbon gas composition be constant in the gaseous phase along the hydrate formation conditions, the current investigation was rendered much more time-consuming compared to other studies of solid-gas equilibria, e.g., initial hydrate formation studies. Experimental Results The experimental data obtained in this study at 300, 500, 1000, and 1500 psia are presented in Table I and Figure 2. Table I and Figure 3 show the comparisons between the smoothed experimental data and the predicted values using the solid solution theory, RedlichKwong-Soave (1972) equation of state, and the hypothetical vapor pressure of water in the empty hydrate
1 I,
-
EXPERIMENTAL
,
, --*--DEW
0 01 -50
1
CH4-C3H8(5 31% mole). H20
-30
-IO
, AOYAGI et
01
(1179).
POINT LOCUS CALCULATED FOR CH4 - C3H8 I 5 31 %) 40
14
TEMPERATURE,
70
F
Figure 2. Water content in gas mixture of methane-propane (5.31 mol % ) in equilibrium with hydrate.
which was obtained through the courtesy of Sloan (1978) and included in the Appendix. The phase diagram of the water-free mixture was determined from the isochoric P-V-T data of the water-free mixture of Arai et al. (1980), and data processed by Magee (1981) has been superimposed on Figures 2 and 3 to show its relative position to the data obtained in this study. The water content values above the initial hydrate formation conditions were taken from the measurements of Olds et al. (1942) for the methane-water system. The study of McKetta and Katz (1948) for the methane-nbutane-water system indicates that the compositional
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 393
where fw and fwm are the fugactities of water in the filled and empty hydrate lattices, respectively. The effect of pressure on the empty hydrate lattice vapor pressure is given by
I O ~ / T , K-'
where PwMT is the vapor pressure of water in the empty hydrate lattice, and 4wmis the correction for the nonideal behavior of water at the saturation pressure. Since PwMT is of the order of to lo4 atm in the current study, 4wm was taken to be unity. The exponential of the integral is the so-called Poynting effect. The fugacity of water in the gaseous phase is given by fwB = Y W 4 W B P (7) where yw = mol fraction of water, and 4wg = fugacity coefficient of water in the gaseous phase to be determined by an appropriate equation of state. From the simultaneous solution of eq 1,5,6, and 7 one obtains the following expression for the concentration of water in the gaseous phase
0 SMOOTHED
5
f 0.011
EXPERIMENTAL 0 PREDICTED --a-- DEW POINT LOCUS CALCULATED FOR CH4-C3Hg (5.31 Z)
; ,'
4'
"
I
0
I
-10
14
40
I
I
-50
-30
I
70
I
TEMPERATURE, 'F
Figure 3. Smoothed experimental and predicted isobars of water content in gas mixture of methane-propane (5.31 mol % ) in equilibrium with hydrate.
exP(
effects of the water content above the initial hydrate formation should be small. The experimental water content based on the calibration is estimated to be accurate to 5 to 6% for all experimental conditions. Analysis and Representation of Data Theoretical Relations. The solid solution theory of clathrates developed by van der Waals and Platteeuw (1959) expresses the differences of the chemical potentials of the filled (actually partially filled) hydrate and the metastable empty hydrate structures as pw - pwMT=
-RT& i
In (1
+ CC,if,) m
(1)
where C L = ~ chemical potential of water in the filled hydrate structure, km= chemical potential of water in the empty hydrate structure, f, = fugacity of solute gas m, vi = number of hydrate cavities of type i per water molecule, Cmi = Langmuir constant as determined by statistical mechanics for gas molecule m in a cavity of type i, R = gas constant, and T = system temperature in absolute units. The chemical potential difference in eq 1 is also expressed in terms of the probability of a gas molecule occupying a cavity of type i A p = RTCui In (1- Cymi) m
i
(2)
where (3) m
and the total moles of gas m per mole of water in the hydrate phase can be obtained by summing over m and i to give (4) or in terms of fugacities as pw
fw
- pwMT= R T In fwMT
(5)
pw
; F M T p W g (8) P
Equation 8 can be further simplified with reasonable assumptions being valid for the experimental conditions encountered in this work, such as unity of that 4wMT and the near constancy of the specific volume D of the hydrate crystal as determined by the X-ray studies of von Stackelberg and Miiller (1954). Data Reduction. The data reduction step involved the application of eq 8 to minimize the deviation between the measured and calculated water content values. In eq 8 reasonable assumptions were made for most of the parameters needed to calculate the composition of water in the gaseous phase at equilibrium with hydrate. Specifically, PwMT or the empty hydrate vapor pressures have been reported by Sloan (1978) for structure I and structure 11. The values of Pwmare reported as a function of temperature, and the expressions are given in the Appendix. The nonideality of water vapor pressure of the empty hydrate at saturation seems to be negligible due to the small quantity to atm for the current experimental condition). However, they have been corrected for pressure by Poynting effect. The values of D were reported by von Stackelberg and Muller (1954) from their X-ray crystallographic studies, and were assumed to be constant for various P and T. The fugacity or fugacity coefficient for water in the mixture was determined by applying the R-K-S (1972) equation of state. The Langmuir constants accounting for the interaction between the gas and water molecules in the cavities were reported by Parrish and Prausnitz (1972) for a range of temperatures. The integration procedure was followed in obtaining the Langmuir constants for lower temperatures using the Kihara potential function with a spherical core according to the study by McKoy and Sinanoglu (1963). The Langmuir constants are consistent with initial hydrate data. However, since the experimental conditions go far below the initial hydrate formation conditions, the assumptions that are evidently valid at the initial hydrate formation conditions may be invalid elsewhere. Never-
394
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
Table 11. Summary of Structure II/Coexistence I
T, K w Redlich-Kwong,Chueh eia1.(1967), k (CH4-C3He);0.02 K-x
6-W-R-S. Lin et
PI.
(1973),
277.15 266.45 260.01
k (CHs-C3Hg)=0.001
pure )EcH,, atm
partial ~ C H 1, P , atm
35.91 19.67
18.60/20.408 18.52/20.408 18.47/20.408 120.408
15.87
lower T 274.65 263.15
28.23 17.62
lower T 276.20
lower T
1
1
I
io01
I
I
20
40
I
60 80 P, atm
I
100
I
120
1
140
Figure 4. Water vapor fugacity in hydrates of structures Iand 11.
theless, until experimental occupation numbers as a function of P-T, X etc. are obtained, there can be no independent check of this assumption as discussed by Davidson (1973). In the application of the R-K-S equation of state for partial fugacity of each component, the values of kij between methane and propane were determined by predicting the compressibility factor with an average accuracy of 0.5% for the experimental values of compressibility factor measured by Arai et al. (1980) and correlated by Magee (1981). The values of kij between hydrocarbon and water were determined by minimizing the discrepancies between the predictions and the experimental and the smoothed experimental values of water content assuming the sole presence of structure I1 crystals in the hydrate phase. The calculated partial fugacities of methane in the gaseous mixture at various temperatures via equations of state using the kij and compressibility factors were compared with the experimental fugacities for initial hydrate formation of structure I in the methane-water system at each temperature. Two different analytical expressions were generated to calculate the initial hydrate formation methane fugacities for the methane-water system above and below the ice point based on the P-T hydrate data reported by Deaton and Frost (1946). The expressions predicted the experimental values accurately to 0.51% . In case the partial fugacity of methane from the two equations of state exceeded the initial hydrate formation fugacity of methane predicted from the two different analytical expressions, it was thought that structure I would form along with structure 11. The stability conditions are presented in the Appendix. However, water vapor fugacity calculations in the hydrate phase using various equations of state show that the water vapor fugacities in structures I and I1 would not become identical for any temperature at the pressures covered in our experiment and/or calculations. An example calculation at 260 K is shown in Figure 4. Accordingly, it has been assumed that only structure I1 crystals exist in the hydrate phase for the predictions of water content in the gaseous phase in
I1 I1 II/I
1111 II/I 1111
II/I
32.77
56.48168.03 168.03
II/I II/I
38.98
80.68/102.041 /102.041
II/I
lower T 278.0
30.16134.014 29.91 /34.014 /34.014
structure(s)
II/I
equilibrium with hydrate at all of the current experimental conditions. Predictions of the moisture content agreed very well with the experimental values except 1500 psia isobar as shown in Table I. The corresponding interaction parameters between hydrocarbon and water were found by trial to be 0.45 for methane-water and for propanewater at 300 and 500 psia, and 0.60 for the same systems at 1000 and 1500 psia to produce the best prediction. It is interesting to note that Erbar et al. (1980) found the same parameters to be 0.52 and 0.53 for methane-water and propane-water, respectively, for the best fits of V-L-E data for the binary hydrocarbon-water systems at higher temperatures using the same equation of state as that used in this study.
Conclusions The water content in the gaseous phase containing 5.31 mol % propane in methane, in general, shows much lower values than that in the pure methane gas. The authors believe that the reasons for this behavior are the shift in the initial hydrate formation condition and differences in the hydrate crystal filling characteristics of the two molecules in the two structures. The possibility of the coexistence of structures I and I1 under the high pressures and low temperatures must be verified later by an independent method, e.g., composition analyses of hydrate and gaseous phases, etc. An attempt has been made to predict the water content, producing good agreement except 1500 psia isobar with experimental results by using the solid solution theory of van der Waals and Platteeuw, the R-K-S equation of state, and the hypothetical water vapor pressure of the empty hydrate lattice, and by assuming that only structure I1 exists at temperatures and pressures of the experimental condition of the present study. Some of the larger deviations in the predictions for lowest temperatures near the two-phase envelope are suspected to be due to the combined effects from the extrapolation of the water vapor pressure in the empty hydrate lattices to lower temperatures, from the equation of state for the partial fugacity of water far below the ice point, and from the Langmuir constants. However, the effect of the two-phase envelope near the low temperature data does not seem to justify the differences of as much as 20 or 30% between the experimental and predicted water contents considering the internal consistency in the experimental data. Thus, the authors recommend the use of experimental measurements at the lowest temperatures rather than the predicted values for process design purpose. Acknowledgment The authors acknowledge the support by the National Science Foundation under grant number ENG 78-20400,
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
and the Gas Processors Association under Project Number 775. The authors thankfully acknowledge the frequent valuable discussions and suggestions from Professor E. D. Sloan and Dr. P. B. Dharmawardhana of the Colorado School of Mines. Special thanks are due to Mr. K. H. Kilgren of Chevron Research Inc., for his interest in the project. In is acknowledged that Dr. John P. Schroeter assembled the computer-storage system and Mr. Joe Magee calculated the dew point-bubble point locus for the gas mixture used in this work. Mr. Raymod J. Martin’s effort in the maintenance of the experimental apparatus is acknowledged. Appendix 1. Using the Soave modification (1972) of the RedlichKwong equation of state, R-K-S, to predict the methane fugacities and the hydrate-liquid water-gas locus determined by Deaton and Frost (1946, Table 11),the expression for the methane fugacity along the three phase line can be represented by In ( f C H , ) = 0.0963T - 23.1084 where f C H , = initial hydrate formation fugacity, atm, and T = temperature, K. Below the ice point, using the hydrateice-gas locus,also determined by Deaton and Frost (1946, Table V), the expression for the initial hydrate formation fugacity for methane is represented by In ( ~ c H ,=) 0.03333T - 5.9017 The first expression predicted the initial hydrate formation fugacities with an average deviation of 0.675% while the second expression represented the fugacities with an average deviation of 0.341%. The R-K-S equation of state was also used to compute the partial fugacities of methane in the water-free gas phase for the methane-5.31 mol % propane mixture. The coexistence of structure I and structure I1 hydrate would be possible when the partial fugacity of methane in the mixture exceeded the fugacity stability condition for methane determined in the methane-water system. Otherwise, only structure I1 would form. Using the criteria stated above the predicted structures are summarized in Table 11. 2. Meanwhile, water vapor fugacities in the filled hydrates of structure I and I1 from the combined eq 1,5, and 6 using a various set of equations of state and the hypothetical empty hydrate vapor pressure of structure I and I1 reported by Sloan et al. (1978), were calculated, and they are shown in Figure 4. 3. The empty hydrate vapor pressures reported by Sloan (1978) are given as: Pwm(atrn) = exp(17.440 - 6003.93/2‘) for structure I, and Pwm(atrn) = exp(17.332 - 6017.64/2‘) for structure 11.
395
Nomenclature C = Langmuir constant, atm-’ exp = exponential f = fugacity of water, atm I, I1 = structure type K = interaction parameter P = system pressure, atm P, = water vapor pressure, atm R = gas constant T = temperature, K V = molar volume of water, cm3/g-mol Y = mole fraction of water in gaseous phase y = probability of filling a cavity Greek Letters = chemical potential
p
v = number of cavities per mole of water 4 = fugacity coefficient
Subscripts i = type of cavity i j = between component i and j mi = type of cavity i with molecule m w = water Superscripts
MT = empty lattice g = gaseous phase _ -- partial molar quantity
Literature Cited Aoyagi, K.; Song. K. Y.; Sloan E. D.; Dharmawardhana, P. B.; Kobayashi, R. Proceedings of the 58th Annual Gas Processors Association Convention, Denver, CO, March 1979. Arai, K.; Kobayashl, R. Adv. Cryog. Eng. 1980, 25, 640. Bloch, M. ‘3.; Lifiand, P. P. Chem. Eng. h o g . 1973, 69(9), 49. Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1967, 6 , 492. Davldson. D. W. “Water: A Comprehensive Treatise”, Frank, F., Ed.; Plenum Press: New York, 1973. Deaton, W. H.; Frost, E. M. US. Department of the Interior, Bureau of Mines, Monograph 8, 1946. Erbar, J. H.; Jagota, A. K.; Muthswamy, S.; Moshfeghian, M. GPA Research Report-42, Oklahoma State University, Stillwater, OK, Aug 1980. Galloway, T. J.; Ruska, W.; Chappeiear, P. S.; Kobayashl, R. Ind. Eng. Chem. Fundem. 1970, 9. 237. Lin, C. J.; Hopke, S. W. 75th National Meeting of the American Institute of Chemical Engineers, New Orleans, LA, March 1973. McKetta, J. J.; Katz, D. L. Ind. Eng. Chem. 1948, 40(5), 853. Magee, J. M.S. Thesis, Rice University, Houston, Tx, 1981. McKoy, V.; Sinanoglu, 0. J. Chem. fhys. 1969, 38, 2946. Olds, R. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1942, 3 4 , 1223. Parrish, W. R.; Prausnh, J. M. Ind. Eng. Chem. Process Des. D e w . 1972, 11, 26. Peng. D. Y.; Robinson, D. 8. Ind. Eng. Chem. Fundam. 1976, 15, 59. Sloan, E. D.; Khoury, F. M.; Kobayashi, R. Ind. Eng. Chem. Fundam. 1976, 15, 318. Sican, E. D. Private Communication, Colorado School of Mines, Golden, CO, Sept 1978. Soave, G. Chem. Eng. Sci. 1972, 2 7 , 1197. Stackelberg, M. von; Mulier, H. I?. 2.Elektrochem. 1954, 58, 25. van der Waais, J. H.; Platteeuw, J. C. A&. Chem. fhys. 1959, 2 , 1.
Received for review June 26, Revised manuscript received April 26, Accepted May 27,
1981 1982 1982
