33
Ind. Eng. Chem. Fundam. 1980, 19, 33-36
A Method for Predicting the Equilibrium Gas Phase Water Content in Gas-H y drat e EquiIibrium H.-J. Ng and Donald 6. Robinson* Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
Recent experimental data on the equilibrium water content of the gas phase in gas-hydrate equilibrium have been used to develop expressions for the fugacity of water over the unfilled hydrate lattice as a function of pressure and temperature, for both Structure I and Structure I1 hydrates. The predicted water content of the gas phase using these equations agrees very well with the experimental values and for Structure I1 hydrates indicates that the parameters in the equations are independent of composition. The general applicability of the proposed equations to other mixtures can only be evaluated as additional data become available.
Introduction Interest in problems related to the formation of hydrates in natural gas pipelines and related processing equipment dates back to the mid 1930's. This interest continued for a period covering more than two decades. During this time a great deal of expermental information was accumulated on the formation of hydrates from natural gas components and their mixtures and the statistical mechanical theory of hydrate formation was developed. Recently, the discovery of natural gas in the Arctic and near-Arctic regions has renewed the interest in both the theoretical and experimental investigation of gas hydrates, particularly in the ice-gas-hydrate and gas-hydrate regions. Most of the earlier work was done on the hydrate-gas-liquid water, liquid hydrocarbon-liquid water-hydrate, and liquid hydrocarbon-liquid water-gas-hydrate regions. This work is extensively reported in the literature, notably in reviews by van der Waals and Platteeuw (1959), Byk and Fomina (1968), and Parrish and Prausnitz (1972), and more recently by Ng and Robinson (1976,1977). However, in spite of these comprehensive studies, information on the formation of hydrates in the gas-hydrate region is scanty. A knowledge of the water content of natural gas that is permissible without the formation of hydrates is of technical as well as economical importance in the transmission of the gas through cold regions and in low temperature gas processing. Recent experimental data on the methanehydrate system presented by Sloan et al. (1976) and by Aoyagi et al. (1979) showed that the permissible water content without hydrate formation in the gas-hydrate region was considerably lower than that indicated by an extrapolation of the data in the gas-liquid region as shown in the Gas Processors Association Engineering Data Book (1976). In the study undertaken in this work a procedure is proposed for estimating the water content of natural gas in equilibrium with hydrate in the gas-hydrate region. Thermodynamic Relationships. The general equation relating the properties of gas hydrates to their molecular parameters as developed by van der Waals and Platteuw (1959) may be expressed as follows ApwH = pwMT- pw =
RT
Cv, In
[1+ CC,,fl] (1) I
where pwm - pw is the difference in the chemical potential between the empty hydrate lattice and the filled gashydrate lattice, f j is the fugacity of any hydrating component j , Y, is the number of type m cavities per molecule of water, and C , is the Langmuir constant for solute j in 0019-7874/80/1019-0033$01.00/0
the cavity of type m. The Langmuir constant C,j, describing the hydrating gas interaction with each type of cavity, is expressed by
c,
-1-exp[-W(r)/kT]4rr2 dr
= 1
kT o
(2)
where W(r) is the spherically symmetric cell potential in the cavity of radius a, with r measured from the center. In treating the problem of the water content of methane gas in equilibrium with hydrates, Sloan et al. (1976) related the fugacity of water in the filled lattice, fw, to the fugacity of water in the empty lattice, fWMT, and the chemical potential difference by fw = fwMT expbw - wMT)/RT
(3)
They evidently visualized a process whereby the metastable unfilled hydrate lattice structure initially formed at PWMT, the vapor pressure of water in the empty lattice. This lattice then underwent an increase in pressure to the stable hydrate pressure, P. The fugacity of water in the empty lattice, fwm, was then expressed in terms of PWm, a Poynting type correction to get from PWMT to P, and a fugacity coefficient for the water vapor over the hydrate lattice a t PWMT. The fugacity of the water in the filled lattice thus obtained from eq 3 was then equated to the fugacity of water in the gas phase calculated from
f w g = Yw4 w g p
(4)
The gas phase water fugacity coefficient 4Wg was obtained from a virial relation and hence it was possible to determine the equilibrium gas phase concentration, yw. The method of calculation by Sloan et al. (1976) was limited to estimating the gas phase water content over Structure I hydrates where methane was the only hydrating component. The use of the virial relation for 4Wp was awkward in that it required a knowledge or determination of at least three interaction virial coefficients Bm, Cand CWMM even for the pure methane system. No mdication is given on how the calculations could be extended to other components or mixtures. The method of calculation proposed in this work is applicable to the gas phase in equilibrium with both Structure I and Structure I1 hydrates for pure components and for mixtures, it uses an equation of state for which the parameters are already known to calculate the gas phase fugacity coefficient, and it eliminates the need for speculation regarding the physical process whereby the unstable
0 1980 American Chemical Society
34
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
empty lattice structure forms a t one pressure and then undergoes a change in pressure to the hydrating pressure. As in the case of Sloan's work, the proposed method makes use of the basic equations given in eq 1,3, and 4. Equation 1 is modified according to the work of Ng and Robinson (1976), and becomes ApwH = RT[nI(l I
+ 3(a,
-
l)y;
-
10-2
10'3
2(aj - l)yfj] X
[ C u m In (1 + CCc,jfj)I (5) rn
I
In this equation, yj is the mole fraction of component j in the gas mixture and ai is an interaction parameter between the least volatile and each of the other hydrating components in the gas mixture. Values of these parameters have been presented by Ng and Robison (1976). In the case of gas hydrates of Structure I, v l = 1/23 an: v2 = 3/23, and in and v 2 = lll7 the case of Structure I1 hydrates, v1 = (van der Waals and Platteeuw, 1959). The Langmuir constant, Cmj,was determined from eq 2 using the Kihara spherical core cell model as,explained in detail by Parrish and Prausnitz (1972). The Kihara parameters were taken from an earlier paper by Ng and Robinson (1977). The fugacity of water over the filled hydrate lattice is expressed by eq 3 slightly rearranged as follows
t
fw = fWMT exp(-AwH)/RT (34 The phase equilibrium relation for water in the gashydrate region is obtained by equating the fugacities of water in the gas phase and in the hydrate phase from eq 3a and 4 as follows YW$
wgP= fWMTeXp(-ApwH/RT)
(6)
which may be rearranged to give 10'~
-
In eq 6a, -ApWH/RTcan be calculated from eq 1 and 2 using the Kihara potential with a spherical core, and the fugacity coefficients of water and the gas may be calculated from an equation of state, or from a generalized correlation. Experimental data on the water content of the gas phase in a gas-hydrate equilibrium mixture can be used to estimate fWMTby rearranging eq 6a as follows
Data Reduction. Data on hydrate formation in the hydrate-gas region for two different gas mixtures have recently been made available by Aoyagi and Kobayashi (1978). These data cover a temperature range from 233 to 267 K and a pressure from 27 to 119 atm. Both these multicomponent systems form hydrates of Structure 11. The procedure for using these data for calculating fwm, the fugacity of the water in the unfilled hydrate lattice, was as follows. At each available experimental data point, it is possible to determine the fWMT by evaluating the right-hand side of eq 6b. In this equation, yw and P are known from the data, ApwH is obtained from eq 1, and 4Wp may be calculated from an equation of state. In this case the Peng-Robinson (1976) two-constant equation was used. The values of In fwm obtained in this manner were then plotted against pressure at each of eight constant temperatures as shown in Figure 1. These relationships were clearly linear, although the slope was a function of temperature. The values of fwom obtained in this manner were then plotted as In fw,om vs. 1/T. Figure 2 shows the resulting
-
I
"
220
240
I
I
I
2bO
280
300
Temperature, K
Figure 3. Effect of temperature on the rate of change of the fugacity of water with pressure over the unfilled Structure I1 lattice.
linear relationship. Finally, the slopes of In fWMTfrom Figure 1 were plotted as a function of temperatures,
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 35
Table I. Experimental and Predicted Gas Phase Water Content for Two Gas Mixtures YH,O
Table 11. Experimental and Predicted Gas Phase Water Content for Methane
x 10-5
Y H , x~
TI K Platm exptl pred Composition: 75.02% CH,, 7.95% C,H,, 3.99% C,H,, 13.04% CO, 44.4 9.89 57.8 8.71 119.1 6.30 44.0 5.88 5.67 57.6 4.16 118.9 57.8 2.52 44.4 2.06 1.84 119.2 44.2 1.05 57.7 1.03 1.05 118.9 119.3 0.452 0.250 119.1
267.1 260.9 261.2 260.9 251.8 249.0 249.8 243.2 243.7 243.2 237.2 233.9
9.45 8.04 6.15 5.59 4.86 3.68 2.06 1.85 1.44 1.04 0.953 0.774 0.446 0.329
102.1 34.0 102.1 34.0 102.1 34.0
260.9 249.8
11:8 25.2 2.84 6.30 1.00 1.97
where
In
fWMT
dp
>,
= 0.0001109T - 0.03192
(7) (8)
The experimental values of the water content of the two multicomponent systems and the predicted values obtained by using eq 6b, 7, and 8 are presented in Table I. The average absolute difference between the experimental and predicted results is 9.1 '70.If three experimental points which depart significantly from a smooth curve are removed from the total of 20 points, the average difference drops to 6.0%. It was reported that the reliability of the experimental results is thought to be about 5%. The results indicate that although fWMT is a function of pressure and temperature, it is apparently independent of the system composition for Structure I1 hydrates. Experimental data on the saturated water content of the equilibrium gas phase in the methane gas-hydrate region were reported by Sloan et al. (1976) and Aoyagi et al. (1979). The revised data were used for obtaining values of fwMTfor hydrates of Structure I in a manner similar to that used for Structure 11. In fW,oMT = 14.269 - 5393/T
( 1;2MT)T
= 0.00036T - 0.1025
240
34.04 68.07 102.1 34.04 68.07 102.1 34.04 68.07 102.1 34.04 68.07 102.1
1,226 0.559 0.271 3,209 1,541 0.844 7.804 3.946 2.417 17.762 9.419 6.405
270
yielding the linear relationship shown in Figure 3. Thus for Structure I1 hydrates, the fugacity of water in the hydrate lattice may be expressed as
(
exptl
260
12.1 25.5 2.82 6.24 0.992 2.22
In fW,oMT = 18.062 - 6512/T
Plat m
250
Composition: 87.06% CH,, 7.96% C,H,, 3.88% C,H,, 1.10% CO, 277.6
TIK
(9)
10-5 pred 1.345 0.542 0.282 3.367 1.483 0.840 7.923 3.8 51 2.395 17.689 9.528 6.548
predicted values for this system is 2.9%. This is well within the reported experimental accuracy. Discussion The foregoing analysis of experimental data on the equilibrium water content of the gas phase in the gashydrate region has made it possible to express the fugacity of water over the unfilled hydrate lattice as a function of temperature and pressure. The relationships depend on hydrate structure but are independent of composition for the mixtures studied. Although the development is based on rather limited data, it is felt that the method is sound. It is believed that only minor modifications, if any, may be required in the numerical coefficients in order to use the equations for predicting the equilibrium gas-phase water content of any gas mixture in the gas-hydrate region. Nomenclature BWM= interaction second virial coefficient for water and methane CWWM, CWMM = interaction third virial coefficient for water and methane C , = Langmuir constant for component j in hydrate cavity f j = fugacity of component j in gas phase k = Boltzmann's constant P = pressure r = radial coordinate R = gas constant 5" = absolute temperature W ( r )= spherically symmetric cell potential y, = mole fraction of component j in gas phase Greek Letters CY, = binary interaction parameter AwwH .= chemical potential of water in the unfilled hydrate lattice minus the chemical potential of water in the filled
hydrate lattice 9, = fugacity coefficient for component j in the vapor phase u, = number of cavities of type m per water molecule in the hydrate Subscripts 0 = denotes zero pressure value
W = denotes water
Superscripts g = denotes gas phase MT = refers to unfilled hydrate lattice
Literature Cited (10)
The experimental and predicted results obtained for the Structure I system are presented in Table 11. The average absolute difference between the twelve experimental and
Aoyagi, K., Kobayashi, R., Proceedings 57th Annual Conventlon, Gas Processors Association, New Orleans, La., 1978. Aoyagi, K., Song, K. Y., Sloan, E. D., Dharmawardhana, P. E., Kobayashi, R., Proceedings 58th Annual Convention, Gas Processors Association, Denver, Colo, 1979. Byk, S. S.,Fomina, V. I., Russ. Chern. Rev., 37, 469 (1968). Engineering Data Book, Gas Processors Suppliers Association, Tulsa, Okla, 1977.
36
Ind. Eng. Chem. Fundam. 1980, 19, 36-39
McKoy, V., Sinanoglu, O.,J. Chem. Phys., 38, 2946 (1963). Ng, H.J., Robinson, D. B., Ind. €ng. Chem. Fundam., 15, 293 (1976). Ng, H.J., Robinson, D. B., AIChEJ., 23,477 (1977). Parrish, W. R., Prausnitz, J. M., Ind. fng. Chem. Process Des. Dev., 11, 26 (1972). Peng, D.-Y., Robinson, D. B., Ind. Eng. Chem. Fundam., 15, 59 (1976). Sloan, E. D., Khoury, F. M., Kobayashi, R., Ind. Eng. c&m. Fundam., 15,318 (1976).
van der Waals, J. H., Platteeuw, J. C., Adv. Chem. Phys., 2, 1 (1959).
Received f o r review January 8, 1979 Accepted October 24, 1979 The financid support received from the Alberta Research council for this work is sincerely appreciated.
Kinetics of Absorption of Oxygen in Aqueous Solutions of Ammonium Sulfite K. Neelakantan and J. K. Gehlawat" Department of Chemical Engineering, Indian Institute of Technology, Kanpur, Kanpur-2080 16, India
The reaction between oxygen and ammonium sulfite is industrially important. The kinetics of absorption of oxygen in aqueous solutions of ammonium sulfite was studied in stirred cells. Cobaltous sulfate was used as the soluble catalyst. The absorption of oxygen in ammonium sulfite solutions was found to conform to the fast pseudenth-order mechanism. In the range of the reactant concentrations of 0.045to 0.45g-mol/L the reaction was found to be first order with respect to oxygen and second order with respect to ammonium sulfite. The third-order reaction rate constant at 30 OC was found to be 2.70 X IO4 [L/g-molI2 s-' and the energy of activation was found to be 14.5 kcal/g-mol.
Introduction Aqueous solutions of ammonia are used to absorb the lean mixtures of waste sulfur dioxide to control atmospheric pollution in several fertilizer plants producing SO2 and ammonia. Ammonium sulfite is thus obtained as a byproduct. It can be easily oxidized to ammonium sulfate, which is used as a fertilizer. In the chemical engineering literature the problem of oxidation of aqueous sodium sulfite has been studied exhaustively. On the other hand, the oxidation of ammonium sulfite has received very little attention. Some preliminary studies have been reported by Young (19021, Vorlander and Lainau (1929), and Hori (1937). Recently Grigorayan (1968) investigated the oxidation of ammonium sulfite by atmospheric oxygen in the presence of nitrogen oxides. Matsuura et al. (1969) studied this reaction in a batch reactor without catalysts. Mishra and Srivastava (1975, 1976) conducted a study for the homogeneous and heterogeneous liquid phase oxidation of ammonium sulfite by the Hatridge and Roughton method of rapid mixing. A mechanism of the reaction was proposed. It may be noted that detailed information on the kinetics of the heterogeneous reaction between oxygen and ammonium sulfite under conditions of industrial importance is not available in the literature. According to the theory of absorption with chemical reaction, this system is likely to conform to the fast reaction regime. It may be erroneous to infer anything about the kinetics of absorption of oxygen in ammonium sulfite solutions based on the controversial information available on the sodium sulfite-oxygen system. An independent study is needed. The present work was therefore undertaken to make a systematic study of the kinetics of reaction between dissolved oxygen and ammonium sulfite. An apparatus 00 19-78741801 1019-0036$0 1.OO/O
of well-defined interface geometry has been used. The theory of absorption accompanied by chemical reaction has been used to interpret the results obtained. Experimental Section Absorption experiments were carried out in stirred cells of various dimensions. The design features of the apparatus were similar to those employed by Gehlawat and Sharma (1968). Figure 1shows the schematic diagram for the experimental setup. A known amount of solution was added to the stirred cell which was installed in a constant-temperature bath. The absorption of oxygen was measured by the volumetric uptake method similar to that employed by Gehlawat and Sharma (1968) and the analytical method described by Jhaveri and Sharma (1967). In the volumetric uptake method, pure oxygen was placed in a balloon which was connected to the stirred cell. The volumetric uptake of oxygen was measured by a soap-film meter. In a few experiments the partial pressure of oxygen was varied from 12.5% to 99% by using nitrogen as the diluent. The mixture of oxygen and nitrogen in the desired proportion was passed through the apparatus for sufficient time so that the partial pressure of oxygen in the apparatus was the same as that in the incoming stream. The gas phase in the stirred cell was also agitated by another stirrer kept a t about 0.5 cm above the gas-liquid interface. A known amount of solution of known concentration was then introduced in the cell, the gas flow was stopped, and the unit was connected to a balloon containing pure oxygen at essentially atmospheric pressure. After several minutes the volumetric uptake of oxygen was noted. In the analytical technique oxygen or mixtures of oxygen and nitrogen were passed through the apparatus containing a known amount of solution of prefixed concentration for 0
1980
American Chemical Society
