Water Content of Methane Gas in Equilibrium with Hydrates E. Dendy Sloan;
Fouad M. Khoury, and Riki Kobayashi
William Marsh Rice University, Houston, Texas 7700 1
Experimental measurements of water content of methane gas in equilibrium with hydrate are presented at 1000 and 1500 psia for temperatures greater than -10 O F . A new chromatographic technique for determining water concentration of gas in equilibrium with hydrate was used to eliminate some of the errors inherent in previous investigations. A method is suggested for calculatingthe water content of methane in equilibrium with its hydrate above 32 O F at any pressure up to 1500 psia. The experimental data indicate that methane should be significantly drier than indicated by existing water content charts in order to prevent hydrate formation. The data reported herein should be strictly interpreted as applying only to methane, rather than natural gases, due to the fact that methane forms a different hydrate structure from that normally formed by natural gases.
Introduction A natural gas hydrate is a solid, crystalline-like structure composed of cavities, formed by water molecules, which are stabilized by a natural gas molecule within each cavity. Hydrates may form from the water evolved at natural gas wellheads a t temperatures above 32 O F when the pressure is elevated and can hamper operation of heat exchangers, expanders, pipelines, etc. The discovery of natural gas in relatively cold regions has renewed interest in determining the water concentration in the gas which is necessary to form hydrates a t a specified temperature and pressure. There is a paucity of accurate equilibrium data in the gas-hydrate region due to the difficulty in analyzing low water concentrations in the gas and due to the unusual metastability of liquid water in the gas-hydrate region. The purpose of this paper is to present a method of obtaining data and a correlation for the water content of methane gas in equilibrium with hydrate above 32 O F as a first step toward specifying conditions for the elimination of hydrates from natural gas operations. Previous Work Nonpolar gas molecules which are smaller than pentane are required to stabilize the hydrate lattice(s) in either a body centered cubic lattice known as Structure I or a diamond lattice known as Structure 11. Molecular size is the factor which regulates the structure formed, with methane or ethane among those forming Structure I with water, and pure propane among those forming Structure 11. The work done to describe the water content of gas in equilibrium only with hydrates in the two-phase region has been limited in accuracy due to two factors: the fact that metastable liquid water may extend well into the gas-hydrate region and the experimental restraint that the existing analysis methods required large amounts of gas in equilibrium with hydrates. Kobayashi and Katz (1955) note that the data anomalies of both Hammerschmidt (1934) and Records and Seely (1951) may be explained by water metastability in the hydrate region. Laulhere and Briscoe (1939) used an adsorption train method to gravitrically determine the water content of natural gas at hydrate conditions. Deaton and Frost (1946), noted the inherent inaccuracies of using the gravimetric method for low water concentrations and resorted to the use of the Bureau of Mines ASTM dew point tester combined with the vapor pressure of liquid water, as did Skinner (1948) and Records and Seely (1951). Water content charts for gas in both the gas-liquid region and the gas-hydrate region have been To whom correspondence should be addressed at Department of Chemical and Petroleum-Refining Engineering, Colorado School of Mines, Golden, Colo. 80401. 318
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
presented by Deaton and Frost (1946),McCarthy et al. (1950), McKetta and Wehe (1958), and Laughlin (1969). Because the above charts are usually intepreted as applying to natural gases, which normally form a different hydrate structure than does methane, it was of interest to determine the water content of methane gas in equilibrium with its hydrate.
Experimental Section The experimental method combined a new analysis technique for measuring low water concentrations in gas, with a gas-hydrate equilibrium autoclave. The experimental apparatus size and accuracy limited the data range to temperatures above -10 O F and pressures of 500 psia or larger. Apparatus. The water analysis equipment, shown schematically in Figure 1,represents a modification of the method of Bloch and Lifland (1973) as detailed by Ertl et al. (1976). Briefly, the sample flows through sample absorption column where a stationary phase of glycerol absorbs essentially all of the water, and then into the calibrated volume sample loop. The switching valve B is then operated and dry helium elutes the water into two equal ffow rate streams. The two streams pass through two parallel columns which contain 5% (by mass) and 20% glycerol. Since glycerol has an extremely large affinity for water as compared to hydrocarbons, the thermal conductivity signals due to hydrocarbons cancel each other when fed into the balanced Wheatstone bridge. Water, on the other hand, is recorded because of the different retention times resulting from the different concentrations of glycerol on each analysis column. Since the peak areas are a measure of water content, the amount of water in the sample loop volume of gas is determined. The chromatograph was calibrated and checked at intermittent points by injecting water saturated ether at injection port I, as suggested by Bloch and Lifland (1973). A secondary calibration check was obtained by measuring the water concentration in a saturated air stream which was attached to the chromatograph at the point where the equilibrium sample was connected in normal operation. The equilibrium apparatus consisted of a hydrate formation autoclave combined with a ball mill to expose and convert the liquid water which may be occluded between the gas-solid interface and the autoclave wall. The apparatus schematic, shown in Figure 2, is essentially that detailed by Galloway et al. (1970), with the following modifications: the number and sizes of stainless steel balls in the autoclave were increased, the discharge line A was heated to prevent hydrate pluggage, and an inlet coil B was added so that sampling a t constant autoclave pressure could be achieved. The autoclave’s motor drive MD was designed with automatic reversing AR after
I N J E C T I ON
FLOWMETERS
ELUTION
SAMPLE VOLUME L O O P S
ABSO?PT'ON C H R W ATOGP A V
I
SAMPLE F53W
A'-T"CLA'IE
Figure 1. Analysis apparatus.
Figure 2.
approximately 2y4 turns in either direction. The cylindrical internal volume of the autoclave was 496.2 ml with rounded corners of 0.5 in. radius. The autoclave contained 67 stainless steel balls, 10 of Tl6-in. diameter, 10 of %-in. diameter, 1 2 of 7/G-in. diameter, 15 of %-in. diameter, and 20 of %-in.diameter, which rolled along the bottom, macroscopically grinding the hydrates on the autoclave walls. The sample discharge line was a coil of 18 f t of 0.019-in. i.d. tubing which, along with the three-way valve, was enclosed in an air chamber and heated to approximately 160 OF by three 400-W heaters in parallel; the heated chamber was isolated from the autoclave by a %-in. gap, filled with bath fluid. The lines from the autoclave three-way valve through the chromatograph sampling system were continuously eluted, when not sampling, with helium of 99.9999% (min.) purity. The inlet coil, which allowed the introduction of dry methane to the autoclave opposite the discharge port, was 24 f t of 0.055-in. i.d. tubing and was at the temperature of the bath. Distilled water was charged to the evacuated autoclave from valve V by a buret (not shown in Figure 2), graduated in 0.05-ml increments. The methane used had a minimum purity of 99.97 mol % with an initial cylinder pressure of approximately 2200 psig. All of the helium used in elution of the autoclave and chromatograph was of the above stated purity.
The system pressure was sensed with a 2000 psia Heise, Bourdon tube gauge, graduated in 2-psia increments and calibrated against a certified precision dead weight gauge. The autoclave temperature was sensed with a 10-junction chromel-constantan thermopile T, calibrated against a platinum resistance thermometer. A millivolt potentiometer, PT, with a distilled water-ice reference junction was used. The pressure and temperature measurements were determined to be accurate to f l psia and f0.05 "C, respectively. The bath temperature was controlled to f 0 . 2 O F through the combined use of refrigeration unit, a baseload heater, and a fluid control heater which was regulated by a temperature feedback controller (not shown in Figure 2) with off-on control. The bath fluid was 66% (mass) ethylene glycol in water which was circulated by two impellers so that throughout the bath volume, the temperature did not vary more than 0.1 OF a t a given time. Method. Before the system was immersed in the constant temperature bath it was evacuated, pressurized to about 20 psia with methane, and re-evacuated; this was repeated twice. The water charging system was connected to valve V and purged of all air; then 12.5 ml of distilled water was charged to the autoclave. The water charging assembly was disconnected and the methane inlet coil was purged and connected Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
319
Table I. Data for Water Concentrations in Methane Gas in Equilibrium with Hydrate W ,l b m l lo6 ft3 at 14.7 psia
P, psia
T, OF
and 60 OF
P, psia
T, OF
and 60 OF
1010 1025 1017 1020 1001 1001
43.6 44.36 18.7 18.5 -9.7 -9.75
10.5 10.38 3.52 3.52 1.03 0.95
1494 1485 1478 1490 1484 1514
39.5 39.25 19.25 19.0 -10.25 -11.0
6.47 6.12 2.15 2.05 0.49 0.54
to valve V. Gas was charged to the autoclave until the desired pressure was obtained and the bath was raised with valve V open and the inlet flow shut off by valves. The initial temperature and pressure of the autoclave were such that the system was in the two-phase region of gas and water-rich liquid. Rotation of the autoclave was begun and the system cooled about 1 O F per hour along an isochor toward the gas-hydrate-liquid condition. As the three-phase conditions were obtained and metastable phase subcooling occurred, the cooling rate was reduced to about 0.5 "Flh. When hydrate formation occurred, the system pressure decreased rapidly at constant temperature, primarily due to the inclusion of methane molecules within the hydrate cavities. Cooling was continued until the pressure-temperature curve obtained approximated the same slope as the isochor in the gas-liquid region, or until no pressure drop was obtained at constant temperature within 12 h. The system was then heated to the three-phase P-T line a t about 0.25 "F/h. Since no metastability is encountered in hydrate dissociation, the point a t which the pressure begins to increase from the isochor on slow heating represents a true three-phase pressure-temperature condition. Heating was continued until all hydrates had dissociated and the pressure increased along the original gasliquid isochor. This cycle around the three-phase conditions was repeated a t least twice more to ensure that no further hydrates could be formed from the liquid phase, as indicated by a lower pressure in the gas-hydrate region. Then cooling was continued in the gas-hydrate region, with methane addition to the autoclave as necessary to obtain the desired pressure at the temperature of interest. Since all water concentrations in the gas phase were less than 2 X mole fraction, constant pressure and temperature, as measured with our instruments, were necessary but not sufficient to describe water concentration equilibrium in the gas phase. Equilibrium was determined by sampling for gas phase water concentration. After obtaining constant temperature and pressure, about 4 days were normally required to obtain equilibrium. For sampling the autoclave rotation was stopped. A sample loop of approximately 30,60, or 90 ml was selected based upon the approximate amount of water anticipated in the sample. The sample column was placed in the flow path from the autoclave by the eight port valve. The line from the autoclave through the sample column to needle valve 2 was pressurized with helium before a sample was withdrawn from the autoclave. At the same time, dry gas was added to the opposite end of the autoclave to maintain constant pressure. The effect of gas addition was determined to be negligible by varying the gas withdrawal rate over several experiments to find an optimum rate so that all of the gas withdrawn was at equilibrium with hydrates. After the sample was withdrawn, blocking valve 1 was closed, the autoclave discharge valve was shut, rotation restarted, and the sample was released to the atmospheric exhaust to depressurize the sample column. The eight port valve was operated to cause the helium elution of the sample column's mobile phase onto 320
W ,lbm/
lo6 ft3 at 14.7 psia
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
- 20
20
40
TEMPERATURE,
60
80
100
120
140
160
O F
Figure 3. Water content of methane.
the two analysis columns for water analysis as detailed by Ertl et al. (1976). When successivesamples spaced at least 4 h apart were determined to contain the same amount of water, equilibrium was assumed and the autoclave pressure-temperature conditions were changed to the next point desired. Experimental Results Experimental data were obtained for the gas-phase water composition of gas in equilibrium with hydrate at pressures of 1000 and 1500 psia and temperatures of 40, 20, and -10 OF.
The water concentrations in the gas phase as functions of temperature and pressure are presented in Table I and plotted in Figure 3. Also included in Figure 3 is one data point obtained in the gas-liquid region as a check on the method compared to the extrapolated data of the gas-liquid region obtained by Olds et al. (1942) and Dodson and Standing (1944). The three-phase (gas-liquid-hydrate) line was calculated from the Parrish and Prausnitz empirical relation (1972) and verified experimentally at lo00 and 1500 psia. The intersection of the three-phase line with the extrapolated data from the gas-liquid region yields the gas-phase water con-
centration in equilibrium with hydrates (and water-rich liquid) at temperatures above 32 O F . D a t a Analysis a n d Reduction Theoretical Analysis. The solid solution theory of van der Waals and Platteeuw (1959) described hydrates with a generalized form of Raoult’s law for a solvent when the solutesolute interactions are negligible, together with a Langmuir type isotherm for localized adsorption. For the binary component case of methane and water their relations reduce to fi,
- pwMT= -RT C v m In (1+ C m f C H 4 )
(1)
m
In eq 1fiwand wwMT represent the chemical potentials of water in the filled and empty hydrate, respectively, while fCH4 is the fugacity of methane in the gas phase and vrn is the number of cavities of type m per water molecule in a unit hydrate cell. The Langmuir constant, C,, for methane’s interaction with each type cavity has been determined as a function of temperature from statistical mechanics as well as from data at the three-phase line and are expressed in a convenient form by Parrish and Prausnitz (1972)
c,
=
3.7237 X T
exp
2.7088 X lo3
Table 11. Data Reduction Results and Comparison with Experiment PWMT,
T , “F/K
atm X lo2
351274.8 401271.6 451280.39 501282.75
1.385 1.758 2.210 2.653
yw at 1000 psia Calcd Exptl ( ~ m ~ / g - r n o l )X~ lo4 X lo4 CUT,,
3115 4980 4515 4625
1.45 1.79 2.28 -
1.56 1.897 2.34 -
hydrate partial molar volume equals the molar volume and is relatively is independent of pressure, and (2) that PwMT small (on the order of atm) so that 4wMT= 1.0. With the above assumptions we obtain from eq 7
n
(YwrnXwrn)Ym
rn
In eq 8 the fugacity coefficient is determined by a virial relation
(2a)
for pentagonal dodecahedra and
c,
=
1.8373 X
exp
T
2.7379 X lo3
(2b)
for tetrakaidecahedra, where T is temperature in Kelvin and C, has units of reciprocal atmospheres. We can define activity coefficients ywrnof water in the two types of cavities in the solid phase in an analogous manner to eq 1 using the empty hydrate lattice as a reference FW
- kwMT
= RT
C U r n In YwrnXwrn m
(3)
lim ywrn= 1.0 Xwm
-
1.0
Fugacity of water in the filled hydrate is related to chemical potential by llw
- llwMT
(4) RT where f , and fwMT are the fugacities of water in the filled and empty hydrate lattice, respectively. The fugacity of water in the empty lattice is expressed as fw
= f w M T exp
where PwMT is the vapor pressure of the empty hydrate lattice, dWMT is the correction for the deviation of the saturated vapor of the pure (hypothetical) lattice from ideal behavior, and the exponential term is a Poynting type correction. The fugacity of water in the gas phase is expressed by fwg
= YW 4 d P
(6)
where y w is the water gas phase composition and 4,g is the fugacity coefficient of water in the gas phase. Equating the fugacities of water in the gas phase and in the hydrate phase, a phase equilibrium relation is derived
Equation 7 may be simplified by two assumptions: (1)that the
which assumes accurate representation by a series truncated after the third virial coefficient, CL,k. D a t a Reduction. The method of calculation of P U l t T , the vapor pressure of the water in Structure I hydrate lattice is as follows at a given temperature between 32 and 50 OF (e.g., 40 O F ) . The value of n r n ( y w r n ~ w m ) l ’ mwas determined at the three-phase line and other pressures of interest by eq 1 , 2 , and 3 at the given temperature. The molar volume given by von Stackelberg and Muller (1954) is used. Pure component and interaction second virial coefficients were obtained from Dymond (1964) and Rigby and Prausnitz (1968), respectively, and extrapolated slightly to lower temperatures, using suggested parameters by Coan and King (1971).Since y,, < 0.001 for all cases considered here, the only third virial coefficient to contribute appreciably t o 4,g in eq 9 is C,,,; however, this interaction coefficient has not been measured and must be determined. Equations 8 and 9 were combined and solved for the two unknowns, P w M T and C,,,, at the three-phase line and and C,,, were used at 1500 psia at 40 OF. The values of PwMT to calculate yw at 1000 psia for comparison with the experimental values. Results are presented at 5 O F intervals in Table 11. Discussion a n d E r r o r Analysis There is a paucity of data for the water content of methane in equilibrium with hydrates. While methane and natural gases contain approximately equal amounts of water in the gas-water rich liquid region, this is not true in the gas-hydrate region due to the fact that methane forms Structure I hydrate while differing amounts of Structures I and I1 form for different natural gases. Only natural gas data, however, are available for comparison. The smoothed data of the present work a t 1000 psia is compared to the data of other investigators for natural gas in the gas-hydrate region, along with an extrapolation from the gas-liquid region in Figure 4. The plot at 1500 psia is very similar. The water content charts of McKetta and Wehe (1958) and McCarthy et al. (1950) were derived in the gas-hydrate region mainly from the data of Skinner (1948), while the I.G.T. water content charts presented by Laughlin (1969) have results only slightly higher Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
321
t 0 c
d
E a
C 1
L
-10
0
10
TEVpE?ATjRE
20
30
40
+ c
'C
I 20
Figure 4. Water content comparison at 1000 psia.
. . 43
I
,
,
6C
,
,
80
.
,
,
.
,
.
.
$00 '22 14s
TEYFERA'USE
,
'6c
,
1
"E
Figure 5. Third virial interaction coefficients from gas-liquid data and from gas-hydrate data. than Skinner's data. The data of Skinner may be in error due to either of two reasons; the condensed phase of the ASTM dew point tester may have been metastable water or ice rather than hydrate, and there is a high experimental error for gravimetric determinations of water contents a t mole fractions less than as noted by Deaton and Frost. The lines drawn by Deaton and Frost a t 1000 and 1500 psi do not represent data, but extrapolations to include a Poynting effect. Deaton and Frost used the ASTM tester and water vapor pressure to determine their water contents; however, the normal angle formed by hydrogen atoms (with oxygen a t the vertex) in the pentagonal dodechedra is 108' as opposed to 1 0 4 . 5 O for liquid water or 109.5' for ice as noted by Claussen (1951) so that the vapor pressure of water in the hydrate phase should differ from that in the liquid or ice states. The photographs by Deaton and Frost of metastable liquid water on the mirror of the ASTM tester a t -4 OF demonstrate that this method of determining dewlfrostlhydrate points in the gas-hydrate region is questionable. Estimates of water content errors based upon calibration data are as follows: the data obtained a t -40 O F are accurate to within 5.09/0,data a t -20 O F are accurate to within 5.09/0,and data at -10 O F are accurate to within 8%. The assumptions of the data analysis allow the calculation of the hypothetical vapor pressure of pure water in the hydrate lattice and of the third virial interaction coefficient. The molar volume is a good approximation to the partial molar volume of the hydrate since the phase is solid. The nonideality of water under its own vapor pressure in the hypothetical empty hydrate is very small a t the temperature of interest. The fugacity of water in dense methane gas, f w g , is difficult to calculate, one reason being that the dense gas tends to break hydrogen bonds between water molecules, causing a change in intermolecular potential. Almost all of the available equations of state for f w g are for higher temperatures and lower pressures than those encountered here. Figure 5 shows the third virial interaction coefficient, C,,, calculated in this work along with those determined from the vapor data of Olds et al. (1942) combined with the liquid data of Culberson and McKetta (1951) as a function of temperature. While the gasliquid third virial coefficients do not appear to be independent of pressure, perhaps indicating premature virial truncation, they do yield values in the same range as those for the gashydrate region when extrapolated with reciprocal absolute temperature. Data reduction was not done below 32 O F due to large uncertainties in second virial coefficient extrapolations. More work is needed to describe gas phase nonidealities 322 Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
for water and hydrocarbons a t low temperatures. At present and to a lesser extent PwMT should be conthe values of ,,C, sidered parameters available to calculate gas phase water concentration in equilibrium with methane hydrate between the three phase pressure and 1500 psia a t temperatures from 32 to 50 O F . Conclusion Measurements have been made for water concentrations in methane gas in equilibrium with hydrates between -10 O F and the three-phase line a t 1000 and 1500 psia. Values obtained were significantly drier than previously available data for natural gases; however, it was noted that these water contents are not directly comparable since natural gases normally form a different hydrate structure than does methane. A method was proposed and checked for determining gas phase water concentration between the threephase pressure and 1500 psia and for temperatures between 32 and 50 O F . Acknowledgments Mr. Raymond J. Martin, Scientific Instrument Maker, constructed the apparatus and provided invaluable design and maintenance assistance. Nomenclature B,, = second virial coefficient for molecules i and j , cm3/gmol CiJh = third virial coefficient for molecules i, 1, and K (cm3/ g-moW C = Langmuir adsorption type constant, atm-' f = fugacity, atm P = pressure, atm R = universal gas constant T = temperature in K except as noted u = molar volume of hydrate, cm3/g-mol u g = molar volume of gas, cm3/g-mol u = partial molar volume of water in hydrate, cm3/g-mol x,, = mole fraction of water in cavity type m y L = mole fraction of component i in gas phase
Greek Letters activity coefficient of water cavity type m k , w W M T = chemical potential of water in the filled and empty hydrate, respectively, cal/mol u, = number of cavities of type m per water molecule in a unit ywm =
hydrate cell, Y23 for pentagonal dodecahedra and tetrakaidecahedra & = fugacity coefficient
3/23
for
Supercripts g = gas phase MT = empty hydrate lattice Subscripts c = methane i,j,k = component types m = cavity type m w = water
Literature Cited
'
Bloch, M. G.,Lifland, P. P., Chem. Eng. Prog., 69 (9), 49 (1973). Claussen, W. F. J. Chem. Phys., 19, 259 (1951). Coan, C. R., King, A. D.. Jr., J. . Chem. Soc., 93, 1857 (1971). Culberson, 0. L., McKetta, J. J.. Petrol. Trans. AlME, 193, 223 (1951). Deaton, W. H., Frost, E. M., U S . Department of the Interior, Bureau of Mines, Monograph 8, 1946. Dodson, G. R.. Standing, M. B., Proc. A.P.I. Drill. Prod. Pract., 173 (1944). Dymond, J. H.. "Compilation of Second and Third Virial Coefficients," Physical Chemistry Laboratory, Oxford University, 1964.
Ertl, H.. Khoury, F., Slm, D., Kobayashi, R., Chem. hg. Techn. (1976) (in press). The article is available in English from this source. Galloway, T. J., Ruska, W.. Chappelear, P. S., Kobayashi, R., Ind. Eng. Chem., Fundam., 9, 237 (1.970). Hammerschmidt, E. G., Ind. Eng. Chem., 26, 851 (1934). Kobayashi, R., Katz, D. L., Trans. A I M , 204, 51 (Aug 1955). Laulhere. B. M., Briscoe, C. F., Gas, 15 (9), 21 (1939). Laughlin, A. R., "Water Content in Natural Gas," presented at the Gas Conditioning Conference, Norman, Okla., 1969. McCarthy, E. L., Boyd, W. L., and Reid, L. S., Trans. AIM€, 189, 241 (1950). McKetta, J. J.. Wehe. A. H.. Pet. Refiner, 37, 153 (Aug 1958). Olds, R. H., Sage, B. H., Lacey, W. N.,Ind. Eng. Chem., 34, 1223 (1942). Parrish, W. R., Prausnitz, J. M., Ind. Eng. Chem., Process Des. Dev., 11, 26 (1972). Records, L. R.. Seely, D. H., Trans. AIM€, 192, 61 (1951). Rigby. M., Prausnitz, J. M., J. Phys. Chem., 72, 330 (1968). Skinner, W., M. Ch.E. Thesis, University of Oklahoma, 1948. Stackelberg, M.. von Muller, H. R.. Z.Elektrochem., 58, 25 (1954). Walls, J. H. van der, Platteeuw, J. C., Adv. Chem. Phys., 2, 1 (1959).
Received for reuieu; February 6, 1976 Accepted July 22, 1976 Work reported here was supported in part by Northern Engineering
Services Co., Calgary, Canada, and by Columbia Gas System Service Corporation. Their support is gratefully acknowledged.
Empirical Description of the Liquid-Vapor Critical Region Based upon Coexistence Data Kenneth R. Hall' and Philip T. Eubank Chemical Engineering Department, Texas A&M University, College Station, Texas 77843
Empirical relationships from coexistence data form the basis for a description of the vapor-liquid critical region for pure fluids which is independent of but generally consistent with the scaling hypothesis. These relationships are: rectilinearity with temperature for mean density and for mean isochoric slope (dPld issuing from the coexistence curve; and power law behavior for the vapor-liquid differences of density, enthalpy, and (df / d 7&. The present description displays excellent agreement with data mapping divergences for various thermodynamic properties at the critical point. With one notable exception, this description also agrees with the theoretical predictions of the scaling hypothesis. The exception is: the present description produces 8 = 1 - 2 p as a lower bound which is somewhat larger than the scaling hypothesis assertion that 8 = a. Unfortunately, the data cannot distinguish between these results. Another interesting result is that the present description correctly predicts maxima with temperature for both mean enthalpy and mean entropy near the critical point.
nP.
Introduction A vast literature has developed during the past two decades concerning the vapor-liquid critical region of pure compounds; the recent work of Rowlinson (1972), Gielen et al. (1973), Levelt Sengers e t al. (1976), and Stanley (1971) exemplify these studies. The powerful scaling hypotheses proposed by Widom (1965) and Griffiths (1967) now form the basis for investigating critical phenomena. Essentially, the scaling hypothesis provides critical exponent equations t o describe the divergences of various thermodynamic properties along specific paths leading to the critical point (CP). In general, these relationships agree with the experimental data available, but in some instances agreement is uncertain causing controversy between experimentalists and theoreticians. Our purpose with this paper is to provide additional insight into the problem with a unique approach. Our description of the critical region derives solely from experimental evidence concerning the variation with temperature of coexisting densities (Figure l),isochoric slopes
issuing from the vapor pressure curve (Figure 2), and heat of vaporization. Although based in the two-phase region, this description correctly predicts critical exponents for the divergence of properties along paths in the single-phase region. We do not consider this description as a rival for the scaling hypothesis, but rather an inductive approach complementing the deductive theoretical model. In fact, only one serious disagreement appears to exist between the two approaches: the scaling hypothesis asserts that the exponent describing the curvature of the vapor pressure is 8 = LY (Vicentini-Missoni et al. (1969), Widom and Rowlinson (1970),and Green et al. (1971)),while the present description predicts 8 = 1 - 2 p as a lower bound. Direct experimental measurements of the slope of the vapor pressure curve are sparse and do not clearly indicate which value of 8 is correct; analysis of these older data as well as recent, yet unpublished data will be the subject of a subsequent paper. Ind. Eng. Chern., Fundarn., Vol. 15,No. 4 , 1976
323
