Znd. Eng. Chem. Res. 1987,26, 1173-1179 Greek Symbols
p = parameter defined as a’6 6 = liquid film thickness 6 = dimensionless distance from gas-liquid interface, defined
as x / 6 q = liquid holdup = stoichiometric coefficient 4~ = enhancement factor for species E, defined as -(de/dt),,,, or RE/(kLa‘E*) pG = gas density pL = liquid density kL = viscosity of liquid Y
Superscript * = gas-liquid interface Subscripts
0 = liquid bulk
A = species A E = species E Literature Cited Calderbank, P. H.; Moo-Young, M. B. Chem. Eng. Sci. 1961,16,39. Chaudhari, R. V. In Frontiers in Chemical EngineeripgProceedings of the International Chemical Reaction Engineering Conference held in Poona; Doraiswamy, L. K., Mashelkar, R. A. Eds.; Wiley: Eastern, New Delhi, 1984;p 291. Chaudhari, R. V.; Deshpande, R. M. Progress in Catalysis and Chemical Engineering Proceedings of the 1st Indo-Soviet Con-
1173
gress on Catalysis, Novosibirsk, USSR, 1984,p 63. Chaudhari, R. V.; Doraiswamy, L. K. Chem. Eng. Sci. 1974,29,675. Dake, S. B.; Chaudhari, R. V. J. Chem. Eng. Data 1985,30,400. Deshpande, R.M.Ph.D. Thesis, University of Poona, India, 1986. Evans, D.; Osborn, J. A.; Wilkinson, G. J.Chem. SOC. A . 1968,3133. Falbe, J. Synthesis with Carbon Monoxide;Springer-Verlag: Berlin, 1980. Hikita, H.; Asai, S.; Ishikawa, H. Ind. Eng. Chem. Fundam. 1977,16, 215. Juvekar, V. A. Chem. Eng. Sci. 1974,29,1842. Luss, D.Chem. Eng. Sci. 1968,23,1249. Luss, D. Chem. Eng. Sci. 1971,26,1713. Luss, D.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1967,6,457. Natta, G.; Ercoli, R.; Castellano, S.; Berbieri, F. H. J . Am. Chem. SOC.1954, 76,4049. Ramachandran, P. A.; Sharma, M. M. Trans. Inst. Chem. Eng. 1971, 49,253. Roberts, G.W.; Satterfield, C. N. Ind. Eng. Chem. Fundam. 1966, 5,317. Roper, G.H.;Hatch, T. F.; Pigford, R. L. Ind. Eng. Chem. Fundam. 1962,1 , 144. Seidell, A. Solubilities of Inorganic and Metal Organic Compounds; D. Van Nostrand: New York, 1940;Vol. 1. Van Boven, M.; Alemdarogly, N. H.; Penninger, J. M. L. Ind. Eng. Chem. Prod. Res. Deu. 1975,14,259. Wilke, C. R.; Chang, P. AIChE J . 1955,1, 264. Yagi, H.; Yoshida, P. Ind. Eng. Chem. Process Des. Deu. 1975,14, 488. Zarzycki, R.; Ledakowicz, S.; Starzale, M. Chem. Eng. Sci. 1981,36, 105.
Received for review August 12, 1985 Accepted February 2, 1987
Two-Phase Liquid Hydrocarbon-Hydrate Equilibrium for Ethane and Propane E. Dendy Sloan,* Kevin A. Sparks, and Jeffrey J. Johnson Chemical Engineering and Petroleum Refining Department, Colorado School of Mines, Golden. Colorado 80401
An experimental method was generated and proven for the measurement of the water content of a liquid hydrocarbon which is in equilibrium only with hydrates. The method was used to determine the water content of both liquid propane and liquid ethane when hydrates were present, without the presence of a free water phase. The dielectric constant apparatus enabled the water concentration measurement of the hydrocarbon liquid without sample withdrawal. The temperature range was 246-276 K with pressures of 0.77 MPa (propane) and 3.5 MPa (ethane), and the water concentrations in the hydrocarbon were from 16 to 176 ppm (mol). An a priori predictive method, based on parameters determined from three-phase equilibria, was determined to be accurate, with mean absolute errors of 2.5% and 5.8% for ethane and propane, respectively. Natural gas hydrates are solid enclosure compounds which form when water encages a hydrocarbon, such as methane, ethane, or propane. Hydrates, which are extensively reviewed by Davidson (1973) and Makogon (1981), are important industrially because they can plug transmission lines, foul heat exchangers, and erode expanders. The primary objective of this work was to measure the amount of water in a predominately hydrocarbon liquid phase in the region where hydrates exist. The two hydrocarbons studied were ethane and propane, as a function of temperature and pressure. Such measurements give the natural gas liquids processor an indication of how dry the liquid hydrocarbon must be in order to prevent hydrate formation. The work may best be understood by considering the qualitative isobaric propanewater phase diagram of Figure
1, taken from Kobayashi (1951); the lines of the diagram are not to scale. This diagram has four single-phase regions: a vapor area marked “A”, a liquid water area marked “B”, a liquid propane area marked “C”, and a hydrate line marked “H”. In our experiments, two-phase equilibriumexisted between lines H and L1. We measured the water concentration along line L1, as a function of temperature and pressure for both propane and ethane. In order to achieve this objective, we had to generate and verify an experimental method to measure low water concentrations in a hydrocarbon in equilibrium with.hydrates. The method was constructed to include the advantages of in situ measurement so that many of the problems accompanying the measurement of small water contents (typically less than 0.001 mol fraction) are not encountered. No other experimentaldata exist, which were
0888-5885/87/2626-1173$01.50/0 0 1987 American Chemical Society
1174 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 PROPANE - WATER
I12 PSlA
P
Vapor
1 A
A
INCREASING T DECREASING T
I
I
I68
3.4
I
I
I I
3.6
I
I
3.8
I
I
4.0
I
I
4.2
[I.O / T ( K ) ] XIOOO F i g u r e 2. Propane dielectric as a function of temperature. 0
MOLE 70 PROPANE
100 Micro -Coax Leads
F i g u r e 1. Propane-water phase diagram (after Kobayashi (1951)).
Filling Tube
explicitly planned to measure equilibrium in the liquid hydrocarbon-hydrate region. A secondary objective of the work was to generate a correlative or predictive method for such data and to compare that method, together with other correlation method(s) with the data obtained. Optimally, the method would have three characteristics: (1)it would be a priori predictive rather than correlative, (2) it would be based on molecular thermodynamic properties, and (3) the method would be compatible with computer calculations.
Thermometer Central Mandrel
Experimental Method The apparatus constructed for the determination of hydrate properties is based on oscilliometry, the study of dielectric properties. The dielectric constant of a material is defined as the capacitance of a material divided by the corresponding capacitance of the assembly under vacuum. Davidson (1962) gives the relation of the dielectric constant E to the polarizability cy and dipole moment M of a polar solute (2) in a nonpolar solvent (11, as F i g u r e 3. Capacitance cell.
The modified Clausius-Mosotti function, on the left in eq 1, is linearly related to the dielectric constant in our experimental range to within 0.35%. The fact that line L1 (Figure 1)has an almost infinite slope severely mitigates any change of N z with temperature in the liquid hydrocarbon-hydrate region. Thus, other parameters being approximately constant, the dielectric constant is linear with respect to reciprocal temperature. If the water concentration in the hydrocarbon were to change to another value, as a result of hydrate formation a t the phase boundary, a second straight line would result with reciprocal temperatures. As shown in Figure 2, which represents some of our experimental data, the intersection of both straight lines represents the temperature at which hydrates form for the initial water concentration and pressure of the system. Materials. Ultrahigh purity ethane was purchased from the Matheson Company bearing a stated purity of 99.99+ % . Research-grade propane donated by Phillips Petroleum Company with a stated purity of 99.94%, the only measurable impurity being trace ethane, was used along with degassed distilled water in all the experiments.
Both hydrocarbons were determined to be within the stated purity, using a chromatograph in this laboratory, and were used as received. Apparatus. The capacitance cell used in this experiment is shown in detail in Figure 3. The ultrasonic oscillator at the bottom is designed to keep the contents of the cell well mixed, while also minimizing the degree of occlusion and metastability encountered a t hydrate formation conditions. The capacitor and capacitance bridge are identical with those described by Pan et al. (1975). The instrument is described in detail by Johnson (1981) and Sparks (1983). The low-pressure loading apparatus is shown in Figure 4. It is designed to load the capacitance cell with known amounts of water. Figure 5 shows the high-pressure experimental apparatus used: (1) to load the condensed hydrocarbon into the capacitance cell and (2) to measure the dielectric constant of the liquid phase. Procedure. Water is loaded into the evacuated capacitance cell as a vapor, using the low-pressure apparatus. The temperature and pressure are measured, and the cell is subsequently valved shut. An accurate determination of the amount of water loaded into the cell is made by
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1175
pJ /Temp1 F l Controller
Indicator
Vacuum
Pump
Table I. Ethane Liquid-Hydrate Equilibrium: System Pressure, 3.45 MPa concentration, ppm temp, K/OF exptl predicted 270.5127.1 86.2 f 5.4 82.3 270.0126.3 80.0 f 5.0 79.6 266.7120.4 62.7 f 3.9 61.9 264.9117.1 50.1 f 3.2 53.7 261.7111.4 41.6 f 2.6 42.1 259.116.8 34.8 f 2.2 34.2
A: Capacitance Cell 8: Water Flask C: Barocel Manometer D: M i x i n g Fan
E: Kovar Seal F: Union G: Temperature Sensor H: Heater Figure 4. Low-pressure apparatus for loading water.
WATER CONTENT OF
ETHANE
0
Thi8 Work ( 5 0 0 p r i a ) 0 Pollin (Phillips)
0
0.1
I
4.3
Figure 5. High-pressure apparatus for forming hydrates.
using the virial equation and the known cell volume. In order to prevent adsorption errors, loading pressures were limited to a maximum of 10% of the vapor pressure of water at the loading temperature. The capacitance cell, once loaded with the desired amount of water, is installed in the high-pressure system. The cell is cooled to about 278 K, and the hydrocarbon, at ambient temperature and its corresponding vapor pressure, is condensed directly into the cell. The pressure range is limited by the pressure of the hydrocarbon reservoir. Helium was used to pressurize the system when pressures higher than the ambient vapor pressure of the hydrocarbon were required. The basis of the data analysis employed in this study involves the fact that as the new hydrate phase forms, the concentration of water in the liquid phase changes. This translates to a change in slope of the dielectric constant plotted against inverse temperature, as shown in Figure 2. The point at which the change in slope occurs corresponds to the hydrate equilibria temperature for the specific concentration initially loaded into the equilibrium cell. Hysteresis does not appear to occur as shown by Figure 2. The determination of the "break" point involves the fitting of the data with two straight lines. The solution is obtained by considering every possible division of experimental points to the first and second lines, estimating the parameters by linear least squares and evaluating the residual sum of squares for each division. The division and set of parameters that give rise to the smallest of all the residual s u m of squares are then chosen as the appropriate description of the experimental data. The resulting two lines are set equal to each other, thus allowing the deter-
4.1
3.9 3.7 ( I I T K )x 103
35
Figure 6. Water content of liquid ethane.
1 WATER CONTENT -- This Work - A Kahre
100
= --
---
0 Pollin v Koboyashi"
*Extrapolated
5n d
IO
=-
Q
- IO
O V
a '
0,
*O LL 0
0 O.
X
0
*:
0 McKetta*
-
c
OF PROPANE
--I
e
0
- 0.1
O F -.L
-30-20-10 0 IO 20 32 I J I' 'I I I 1 1 ' 4.3 4.1 3.9 3.7
/ 'I
E
\
a
50 I
I, 3.5
1176 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
ethane-water system (3.45 MPa)
1000
c
I ETHANE LIQUID-HYDRATE EQUILIBRIUM
log x (ppm) = 10.8943 - 2426.9/T temperature range: 271-259 K
(24 0 500 psla dot0
propane-water system (0.772 MPa) log x (ppm) = 10.4573 - 2276.296/T temperature range: 277-246 K
loo
E
-This Work -Ng Robinron (modified)
-
(2b)
An elevated pressure data point a t 1.81 MPa was measured for the propane-water system, but it was not statistically different from that at low pressure. Two attempts were made a t measuring the pressure effect on the ethane-water system, specifically at 5.52 MPa, but dielectric constant measurement problems were encountered. The predicted pressure effect for the ethane-water system is negligible, similar to that measured and predicted for propane. Discussion of Results. Water solubility data exist which inadvertently extend into the two-phase, liquid hydrocarbon-hydrate region. The data are shown in Figures 6 and 7. The data of Kahre (1964) and Parrish et al. (1982), both of Phillips Petroleum Co., were taken without explicit planning for the possible existence of hydrate equilibrium, especially with regard to possible metastability and occlusion. Kobayashi and Katz (1955) indicate that without careful planning for hydrate equilibrium, experimental error can occur. We have, however, no conclusive reason why the data of Phillips Petroleum Co. investigators fall below our data, by a maximum of 50 ppm. Calculations indicate that both the data of the Phillips investigators and the data of the present work have lower concentrations than those calculated for metastable equilibrium between ice and liquid ethane or propane. Phase rule analysis indicates that the water content of liquid hydrocarbon in equilibrium with hydrate should have a slope greater than that for liquid hydrocarbon in equilibrium with an aqueous liquid, when plotted as in Figures 6 and 7. For propane, this relation holds for the indicated extrapolated data of Hoot et al. (1957) and the extrapolated data of Poettmann and Dean (19461, which are not indicated for clarity purposes. Also, the straight line of our propane data in the liquid hydrate region intersects the straight line extrapolation through the data of Hoot et al. (1957) and that of Poettmann and Dean (1946) a t the quadrapole point temperature, 42 OF. The Phillips Petroleum Co. data lie below the McKetta data, the Poettmann data, and our data. In Figure 7, the single point of Kobayashi was extrapolated from a concentration 82 ppm higher. Accuracy Estimation. The dielectric constant of pure liquid propane was measured at five temperatures between 243 and 283 K a t 1.8 MPa and compared against values from the National Bureau of Standards measured by Haynes (1983). In all cases, the measured values differed by less than 0.09%. The dielectric constant of methanol in liquid propane was measured at 228.4 K and 0.0898 MPa and compared to the values of Luo (1979) to within 0.02%. Such accuracies are common for dielectric constant experiments and are a justification for the choice of the experimental method. This study employed two methods for the estimation of the uncertainty of the experimental water concentration a t hydrate formation conditions. Firstly, the traditional propagation of error was used along with required partial derivatives which were numerically approximated to estimate the variance of the experimental dependent variable. Secondly, a Monte-Carlo-type error analysis was used
10
-
c
E
n
I
0 Y
X
IO
- / I J
- 0.1
O F -c
0 10 20 32
/30-20-10
; ; 'I
I
II $ 1
PROPANE LIQUID
50 I
- HYDRATE EQUlLlBRiUM
10
Predict ions Method for f --Ng-Robinson (modified)
E /
c
/
-
/
i I
0
I
I
X
I / - / I
- 0.1
OF-
/-30-20-10 0 10 20 32
50
J
1
I
I
/
I
I
I
"
,
( I / T K ) X 103
Figure 9. Propane water content data and predictions. Table 11. Liquid Propane-Hydrate Equilibrium: System Pressure, 0.772 MPa concentration, ppm temp, K/OF exptl oredicted 276.5/37.9 176.1 f 10.6 168.3 273.9133.4 136.9 f 8.2 142.2 273.oj31.6 125.8 f 7.5 133.0 123.9 f 7.5 272.2130.2 125.9 269.9j26.i 102.5 f 6.3 107.5 266.0/ 19.1 87.6 f 5.4 81.8 264.3/ 16.1 65.7 f 4.0 72.5 259.9/8.1 50.5 f 3.1 52.4 246.71-15.7 16.9 f 1.8 18.7
due to the ability to generate information regarding the probability distribution of the dependent variable. The two methods give identical results in regard to the variance of the dependent variable. The reported error limits in Tables I and I1 correspond to a confidence interval of 99.9%. A detailed description of the Monte Carlo error analysis is given by Sparks (1983).
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1177 Table 111. Liquid Hydrocarbon-Hydrate Prediction Algorithm 1. Select hydrocarbon liquid-hydrate system. a. Liquid ethane-hydrate equilibrium (Structure I) b. Liquid propane-hydrate equilibrium (Structure 11) 2. Select temperature and pressure. a. Must be in liquid region of hydrocarbon 3. Calculate Langmuir constants. a. Kihara potential (see Parrish and Prausnitz (1972)) or b. Equation 5 4. Calculate fugacity of water in the empty hydrate, fgT. a. Method of this work (eq 7, 8a or 8b, and 9) or b. Method of Ng and Robinson (eq 10, l l a or l l b , and 12a or 12b) 5. Estimate mole fraction of water in liquid phase. 6. Calculate fugacity coefficients of components in liquid phase. a. Peng-Robinson equation of state b. Soave-Redlich-Kwong equation of state 7. Calculate Bij the fraction of type i cavities filled by guest j . a. Equation 4 8. Calculate the chemical potential difference of water between the empty and filled hydrate. a. Equation 3 9. Calculate fugacity of water in hydrate, by eq 6. 10. Calculate "newn mole fraction of water in liquid phase. a. Equation 13 11. Compare "newn mole fraction of water with "guessed" mole fraction. a. If within specific tolerance, stop, otherwise b. Use "new" mole fraction as the next estimate. 12. Go to step 6.
Table IV. Langmuir Constant Parameters and (A) Kihara Potential and (B) Cij = ( A i j / T )exp(Bij/T) between 260 and 300 K
io-%, cm-'
component ethane propane
0, A a,8, 3.2444 0.5651 3.3111 0.6502 large cavities
177.01 108.45 small cavities
component
103Aij
ethane propane
0.0 0.0
Bij 10*Aij Structure I 0.0 0.4071 0.0 0.0
ethane propane
0.0 0.0
Structure I1 0.0 2.9157 0.0 1.3212
1
Bij 3820.7119 0.0
3277.9254 4506.9810
0 Methene 0 Ethane A cyclo Propane 0 Oxygen 0 Hydiogen- Sulfide
-
-3.9 -3'1
6,
E
1
where Bij is the fraction of type i cavities filled by guest
i. Bij = Cijfj/(l
+ Cijfj)
(4)
The Langmuir constants, Cij, are only functions of temperature. These constants may be calculated via a Kihara potential using computerized numerical integration, as indicated by Parrish and Prausnitz (1972), or they may be calculated by hand, using the empirical expansion of Parrish and Prausnitz: Aij Bij cij= exp(5) T T with Cij in reciprocal atmospheres and T in Kelvins. Both the Langmuir-Kihara parameters and the constants in the above equation are given in Table IV. The Kihara parameters were generated by Dharmawardhana (1980) from the three-phase (V-L-H) data. The number of type i cavities per water molecule, vi, is equal to 1/17 for propane (Structure 11) and 3/23 for ethane (Structure I). The fugacity of the liquid hydrocarbon, f j , may be determined through an equation of state such as the PengRobinson or the Soave-Redlich-Kwong.
80
-6.3 -7.1
Prediction and Correlation Method An outline of the prediction method based upon an iteration for water content of the hydrocarbon is presented in Table 111. A detailed explanation of the method is given below. The general equation relating the thermodynamic properties of hydrates to their molecular parameters was derived by van der Waals and Platteeuw (1959) after the molecular structure of hydrates was determined. The model was extended by Parrish and Prausnitz (1972) to gas mixtures. The equation for the chemical potential difference of water between the empty and filled hydrate, for a single hydrate former which fills only one type of hydrate cavity, is Ap@ = RTvi In (1- Bij) (3)
S O *O
A
-
A
'
A A
366
376
386
396
406
416
426
1 0 5 / ~ ,K-I
Figure 10. Empty hydrate vapor pressure of Structure I hydrate.
Sloan et al. (1976) showed that the chemical potential difference of water in the filled and empty hydrate structure can be related to the fugacity of water in the hydrate phase by
f i = fMT
exp(&w/RTl (6) where fET is an experimentally determined parameter related to the (hypothetical) vapor pressure of the empty hydrate. Determining fET in the Present Work. As shown in detail in the Appendix, Dharmawardhana (1980) determined the empty vapor pressure for both hydrate structures which relates to the empty hydrate fugacity as
In the above equation, the empty hydrate vapor pressure is shown in Figures 10 and 11, as determined from three-phase (vapor-ice-hydrate) data for many components. The vapor pressures are fitted to equations in temperature as In P p = 17.440 - 6003.925/T
(84
for Structure I and In P p = 17.332 - 6017.635/T
(8b)
"' in atmospheres and T in Kelfor Structure 11, with p vins. The empty hydrate fugacity coefficient may be determined using the second virial coefficient by = exp(BwPW /R
Tl
(9) In most cases, $ET is within 1% of unity. Values for the &T
1178 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
- 4.2
Table VI. Eauation of State Interaction Parameters kij in equation of state svstem Penp-Robinson SRK ethane-water 0.50 0.516 propane-water 0.48 0.496
180 butane
0
0 Sulpha hexafluoride A Cyclo propane 0 Propane
E
+
a
c
-5.4 -5.8
1I
-6.2 360
0 OA A
I 365
I
I
1
1
370
375
380
385
I 390
Therefore, eq 10-12 represent a correlative method, rather than a predictive method. Final Prediction of Water Content Using Either of the Above Methods. The prediction of hydrate equilibria, specifically in the two-phase, liquid-hydrate region can be reduced to the equating of the fugacity of the water in the liquid phase to the fugacity of the water in the hydrate phase. Thus, the resulting equation becomes
I O ~ / T ,K - I
Figure 11. Empty hydrate vapor pressure of Structure I1 hydrate. Table V. Poynting Correction Factor temp, K/"F 273.15/32
253.15/-4.0
pressure, MPa/psia 101.33/14.7 689.47/100 3447.4/500 6894.7/1000 34474.0/5000 101.33/14.7 689.47/100 3448.4/500 6894.7/1000 34474.0/5000
Structure I
PCF Structure I1
1.001 000 1.006 890 1.034 960 1.071 124 1.408 445 1.001086 1.007 425 1.037 679 1.076750 1.445692
1.001 020 1.007013 1.035590 1.072 427 1.417 025 1.001 106 1.007 558 1.038 360 1.078 162 1.455 195
exponential term (the Poynting correction factor) are presented in Table V, for various pressures with both hydrate structures. Determining fgTby the Method of Ng and Robinson. An alternate method for the determination of the fugacity of water in the empty hydrate, was developed by Ng and Robinson (1980) who report expressions for fZT obtained by fitting the vapor-hydrate data of Prof. KObayashi's laboratory (Aoyagi et al., 1979; Aoyagi and KObayashi, 1978). The expression for the fugacity of water in the empty hydrate is
fgT,
The expressions for each structure are In = 14.269 - 5393/T
fw$
d In f#?r = 0.00036T - 0.1025 dP for Structure I and In f#: = 18.062 - 6512/T d ln fW3
dP
= 0.0001109T - 0.03192
(1la) (12a)
(llb) (12b)
for Structure 11. While the fgT values of Ng and Robinson were derived for an vapor-hydrate data, they should be valid for liquid-hydrate predictions, if fgT are only hydrate parameters; thus the method may be called a modified Ng-Robinson method. It should be noted that eq 10 represents an empirical fit to the two-phase (vaporhydrate) data; in particular, the pressure dependence of fgT is negative below 285 and 287 K in both eq 12a and 12b, respectively. Thermodynamics indicates the pressure dependence of fZT should always be positive, as given by the Poynting correction (exponential) factor in eq 7 .
In this study, the fugacity coefficient of water in the liquid, &, was determined by using both the Peng-Robinson equation of state and the Soave-Redlich-Kwong equation of state. The binary interaction parameter, kL,, defined by a , = ( ~ p , ) ~- /kL,) ~ (and l used in the determination of the fugacity coefficient, was taken from vapor-liquid equilibria data, specifically that reported by Peng and Robinson (1976). The interaction parameters used with the Soave-Redlich-Kwong equation of state were optimized to reproduce the vapor-liquid equilibria results predicted by using the Peng-Robinson equation of state. The interaction parameters used in the prediction are shown in Table VI. Figures 8 and 9 show the results using both methods of obtaining f Z T compared to the data of the present work. With our predictive method, the maximum error and mean absolute error are 5.6% and 2.5% for ethane, respectively, while corresponding values for propane are 10.7% and 5.8%, respectively. The results using the method of NgRobinson (1980) to determine fgT predict values systematically more conservative than our data.
Conclusions Experimental measurements were made to determine the water content of liquid ethane and liquid propane in the two-phase, liquid hydrocarbon-hydrate region. An a priori predictive technique, based on parameters obtained only from three-phase data, was shown to predict the two-phase data to within 10% maximum error in concentration, or to within 3 K in temperature. Work is currently under way to extent these results to the region of hydrocarbon mixture-hydrate equilibria. Acknowledgment The two-phase, liquid hydrocarbon-hydrate studies were conducted under the sponsorship of Gas Processors Association Project 775-B. We thank Dr. R. C. Miller, currently Dean and Professor of Chemical Engineering at Washington State University, for loaning us the original capacitance assembly and refrigeration equipment. We also thank Dr. W. R. Parrish of Phillips Petroleum Co. for his interest and help in developing the model used in this study. The research-grade propane used in this study was generously donated by Phillips Petroleum Co. Finally, M. S. Bourrie duplicated some of this data in his M.S. work.
Nomenclature a = force parameter in equation of state A = parameter in equation for Langmuir constants B = parameter in equation for Langmuir constant B , = virial coefficient of water C = Langmuir constant
Ind. Eng. Chem. Res. 1987,26, 1179-1184
1179
found to be a single function of temperature. Figure 10 shows the empty hydrate vapor pressure for Structure I hydrate formers. A similar plot for Structure I1 hydrate formers is shown in Figure 11. Registry No. C3H8, 74-98-6; C2H6,74-84-0; H20, 7732-18-5;
f = fugacity k = Boltzmann constant M = dipole moment N = number of molecules per unit volume P = pressure T = temperature x = mole fraction water in liquid phase V = molar volume
CH4,74-82-8 02,7782-44-7; H,S, 7783-06-4; (H,C)ZCHCHB, 7528-5; SF,, 2551-62-4; cyclopropane, 75-19-4.
Literature Cited
Greek Symbols a = polarizability c = dielectric 6 = fractional occupation of a cavity I.L = chemical potential Y = number of cavities per water molecule in a crystal unit C$ = fugacity coefficient
Aoyagi, K.; Kobayashi, R. Presented a t the Proceedings of the 57th Annual Convention of Gas Processing Association, New Orleans, LA, 1978; p 3. Aoyagi, K.; Song, K. Y.; Sloan, E. D.; Dharmawardhana, P. B.; Kobayashi, R. Presented at the Proceedings of the 58th Annual Convention of Gas Processing Association, Denver, CO, 1979; p 25.
Subscripts 1 = hydrocarbon molecule 2 = water molecule
i = cavity type j = guest molecule type W = water
H = hydrate Superscripts
H = hydrate M T = empty
Appendix Method Employed i n This Study for the Determination of the Fugacity of Water i n t h e Empty Hydrate. By equating the fugacity of hydrate to ice for three-phase (V-I-H) data, Dharmawardhana (1980) showed that fKT can be expressed as an empty hydrate times a correction for nonideality, va or pressure, $#, as
GT,
In the above equation, all of the ice properties are well-known; the APE is obtained from three-phase (V-LH) data. The only unknown is PKTwhich was fit to a number of hydrate's three-phase data below 273 K and
Davidson, D. W. In Water: A Comprehensive Treatise; Franks, F., Ed.; Wiley: New York, 1973; Vol. 2, pp 115-234. Davidson. N. Statistical Mechanics: McGraw-Hill: New York, 1962: pp 402-418. Dharmawardhana, P. B. Ph.D. Thesis, Colorado School of Mines, Golden, 1980. Haynes, W. M. J. Chem. Thermodyn. 1983, 15, 419. Hoot, W. F.; Azarnoosh, T.; McKetta, J. J. Pet. Refin. 1957, May, 255. Johnson, J. J. M.S. Thesis, Colorado School of Mines, 1981. Kahre, L. C. Solubility of Water in Propane, Internal Report, Phillips Petroleum Co., 1964. Kobayashi, R. Ph.D. Thesis, University of Michigan, 1951. Kobayashi, R.; Katz, D. L. J. Pet. Technol. 1955, 7(8), 51. Luo, C. C. MSc. Thesis, University of Wyoming, Laramie, 1979. Makogon, Y. F. Hydrates of Natural Gas; Cieslewicoz, J., Transl.; PennWell Books: Tulsa, OK, 1981. Ng, H.; Robinson, D. B. Ind. Eng. Fundam. 1980, 19, 33. Pan, W. P.; Mady, M. H.; Miller, R. C. AIChE J. 1975, 21, 283. Parrish, W. R.; Pollin, A. G.; Schmidt, T. W. "Properties of Ethane-Propane Mixes, Water Solubility and Liuqid Densities", Proceedings of the 61st Annual Convention, Dallas, TX, 1982. Parrish, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 26. Peng, D. Y.; Robinson, D. B. Can. J. Chem. Eng. 1976, 54, 318. Poettman, F. H.; Dean, M. R. Pet. Refin. 1946, 25, 125. Sloan, E. D.; Khoury, F. M.; Kobayashi, R. Ind. Eng. Chem. Fundam. 1976,15, 318. Sparks, K. A. M.S. Thesis, Colorado School of Mines, 1983. van der Waals, J. H.; Platteeuw, J. C. Ado. Chem. Phys. 1959, 2, 1-59.
Received for review August 14, 1985 Revised manuscript received November 24, 1986 Accepted February 28, 1987
Intraparticle Ion-Exchange Mass Transfer in Ternary System H i r o y u k i Yoshida* and Takeshi K a t a o k a Department of Chemical Engineering, university of Osaka Prefecture, Sakai, Osaka 591, Japan
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Intraparticle ion-exchange mass transfer in ternary systems, [R-A] (B C) (adsorption process) and [R-B R C ] A (desorption process), has been analyzed according to the Nernst-Planck equation. The mean concentrations in the resin phase with time in a ternary system (H+,Na+, and Zn2+)were measured under the condition of intraparticle diffusion controlling. [R.H+] + (NaNO, Zn(N03)2),[RoNa'] (HNO, Zn(N03)2),and [R.Zn2+] (NaNO, HN03)as adsorption processes and [R.Na+ R.Zn2+] + HNO, and [R-H+ + R-Zn2+]+ NaN0, as desorption processes were investigated. The ways in which combinations of the ions affected the behavior of each ion were discussed. Ion exchangers were used with the degrees of cross-linking 870, 1070,and 16%. The experimental results are compared with the theoretical results and are discussed in relation to the theory.
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1. Introduction
In the theoretical analysis of intraparticle mass transfer in ion exchange, various complicated factors must be 0888-5885/87/2626-1179$01.50/0
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considered to estimate the ion-exchange rate accurately. There are three typical factors in the intraparticle ionexchange mass transfer. The first is the effect of the 0 1987 American Chemical Society
