The Solubility of Binary Mixed Gases by the Fluctuation Theory

On the basis of the Kirkwood-Buff theory of solution, two transcendental equations are derived that allow one to predict the solubility of binary mixe...
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Ind. Eng. Chem. Res. 2002, 41, 6279-6283

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The Solubility of Binary Mixed Gases by the Fluctuation Theory I. Shulgin† and E. Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260

This paper is devoted to the solubility of mixed gases in a liquid, the goal being to predict their solubilities from binary data. Only sparingly soluble and weakly interacting gases are considered. On the basis of the Kirkwood-Buff theory of solution, two transcendental equations are derived that allow one to predict the solubility of binary mixed gases from the solubilities of pure individual gases. The suggested method was tested for the solubilities of methane-ethane, methane-n-butane, and methane-carbon dioxide gas mixtures in water at high pressures. Good agreement between experiment and predictions was found. 1. Introduction The removal of acid gases from natural gas streams; the solubilities of hydrocarbons and natural-gas components such as CO2 and H2S in water under highpressure/high-temperature conditions; and the solubilities of air and other mixed gases in water, blood, seawater, rainwater, and many other aqueous solutions are a few examples for which information about the solubility of mixed gases in a solvent is needed. This topic has attracted the attention of both experimentalists and theoreticians.1-8 Whereas the solubilities of many individual gases in liquids have been precisely measured,9-11 those of mixed gases have rarely been determined; even complete information about the solubility of air in water in a wide range of pressures and temperatures is not available.9,10 So far, there is no rigorous method for predicting the solubilities of gaseous mixtures in liquids; only an empirical method for mixtures of hydrocarbons has been suggested.2 As mentioned in the literature,6 the usual methods for predicting vapor-liquid equilibrium, such as the Wilson, NRTL, and UNIQUAC approaches, cannot be straightforwardly extended to the solubility of mixtures of two supercritical gases. Cubic equations of state (EOS) such as the Peng-Robinson12 and the SoaveRedlich-Kwong13 EOS provide accurate descriptions for the solubility of single gases in liquids but can not be extended to the solubility of gaseous mixtures, especially when the solvent is polar,13,14 because of the empirical nature of the interaction parameter in the van der Waals mixing rule. Whereas the interaction parameter can be taken zero for multicomponent gaseous mixtures containing similar compounds,13 it cannot be predicted for unsymmetrical multicomponent mixtures, such as CH4 + C2H6 + polar solvent, from the interaction parameters for binary (individual gas-solvent) mixtures. However, the combination of one of the above EOS with modern mixing rules and group contribution methods14-16 seems to be promising in predicting the solubilities of gaseous mixtures in liquids. The aim of this paper is to propose a method for predicting mixedgas solubilities from the solubilities of the constituent gases in the same solvent, without using an EOS, and to compare the obtained results with available experi* Corresponding author. E-mail: [email protected]. Fax: (716) 645-3822. Phone: (716) 645-2911/ext. 2214. † E-mail address: [email protected].

mental data. The fluctuation theory of Kirkwood and Buff17 for ternary mixtures will be employed to develop the aforementioned method. 2. Theory 2.1. General Expressions for the Solubility of a Gas Mixture in a Single Solvent. Let us consider the solubility of a mixed gas (composed of two supercritical gases: component 2 with mole fraction y2 and component 3 with mole fraction y3) in a single solvent (component 1). At equilibrium, the fugacities of the components in the liquid and gaseous phases should be equal. Therefore, one can write

f G(t) ) f L(t) (i ) 2, 3) i i

(1)

where the superscripts G and L refer to the gaseous and liquid phases, respectively, and t indicates a ternary mixture. The Lewis-Randall rule18 for the fugacity of a species in a gas mixture will be adopted; hence, the fugacity of a component in a mixture is obtained by multiplying its fugacity as a pure gas with its mole fraction. In addition, for the sake of simplicity, the solubilities of both gases will be assumed small, and the concentration of the solvent in the gas phase will be neglected. Therefore, for the fugacities of the two species of the gas mixture, one can write

) f 0i (P,T)yi (i ) 2, 3) f G(t) i

(2)

where f0i (P,T) is the fugacity of the pure gas i at the pressure and temperature of the system. The fugacities of the components in the liquid phase can be expressed as18

) xti γti f Li (T,P) (i ) 2, 3) f L(t) i

(3)

where xti and γti are the mole fraction and the activity coefficient, respectively, of component i in the liquid phase and f Li (T,P) is the fugacity of the pure component i in the (hypothetical) liquid state. Under the same conditions, for the solubilities of the pure gases in the same solvent (neglecting the concentration of the solvent in the gaseous phase), one can write

10.1021/ie020016t CCC: $22.00 © 2002 American Chemical Society Published on Web 05/04/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

f 0i ) xbi γbi f Li (T,P) (i ) 2,3)

(4)

where xbi and γbi represent the mole fraction and the activity coefficient, respectively, of component i in the liquid phase of the binary mixture 1-i, where i is 2 or 3. The combination of eqs 1-4 yields the relations

xt2 γt2 ) y2xb2 γb2

(5)

xt3 γt3 ) y3xb3 γb3

(6)

and

For dilute binary and ternary mixtures (all solute mole fractions are small), one can write19-22

for the binary mixtures 1-2 and 1-319 b ln γb2 ) ln γb,∞ 2 - k22x2

(7)

xt2 ) xb2y2 exp[k22(xt2 - xb2) + K23xt3]

(13)

xt3 ) xb3y3 exp[k33(xt3 - xb3) + K23xt2]

(14)

and

Equations 13 and 14 can be used to calculate the solubilities of mixed gases if the solubilities of the pure constituent gases in the same solvent and the values of k22, k33, and K23 are known. Whereas the values of k22 and k33 can be determined from the solubilities of the individual gases,19,24 an expression for K23 will be obtained below using the fluctuation theory of solution. 2.2. Expressions for the Derivative of the Activity Coefficient (D ln γt2/Dxt3)P,T,x2 in a Ternary Mixture through the Kirkwood-Buff Theory of Solution. General expressions for the derivatives of the activity coefficients in a ternary mixture with respect to the mole fractions were obtained in a previous paper23 in the form

(

)

∂ ln γ2,t

and b ln γb3 ) ln γb,∞ 3 - k33x3

(8)

where γb,∞ is the activity coefficient of component i (i ) i 2, 3) at infinite dilution of component i in the binary mixture 1-i and

( )

k22 ) and

∂ ln γb2 ∂xb2

( )

k33 ) -

(9)

-(c1 + c2 + c3)(c1[G11 + G23 - G12 - G13] + c3[-G12 - G33 + G13 + G23])/(c1 + c2 + c3 + c1c2∆12 + c1c3∆13 + c2c3∆23 + c1c2c3∆123) (15)

where ck is the bulk molecular concentration of component k in the ternary mixture 1-2-3 and GRβ is the Kirkwood-Buff integral given by

GRβ )

(10) P,T,x3f0

( ) ∂ ln γt2 ∂xt2

+ xt3

P,T,x3,0

( )

∆Rβ ) GRR + Gββ - 2GRβ, R * β

and t ln γt3 ) ln γt,∞ 3 + x2

( ) ∂ ln γt3 ∂xt2

P,T,x3,0

t t ≡ ln γt,∞ 3 - x2K32 - x3K33

2G12G23 + 2G13G23 - G122 - G132 - G232 2G11G23 - 2G22G13 - 2G33G12 (18)

(11)

+ xt3

One can show25 that the factors in the square brackets in the numerator of eq 15 and ∆123 can be expressed in terms of ∆Rβ as

( ) ∂ ln γt3 ∂xt3

(17)

∆123 ) G11G22 + G11G33 + G22G33 + 2G12G13 +

P,T,x2,0

t t ≡ ln γt,∞ 2 - x2K22 - x3K23

(16)

and

∂ ln γt2 ∂xt3

∫0∞ (gRβ - 1)4πr2 dr

In the above expressions, gRβ is the radial distribution function between species R and β, r is the distance between the centers of molecules R and β, and ∆Rβ and ∆123 are defined as

for the ternary mixture 1-2-3 at high dilutions of components 2 and 320-22 t ln γt2 ) ln γt,∞ 2 + x2

) T,P,xt2

P,T,x2f0

∂ ln γb3 ∂xb3

∂xt3

P,T,x2,0

G12 + G33 - G13 - G23 )

∆13 + ∆23 - ∆12 (19) 2

G11 + G23 - G12 - G13 )

∆12 + ∆13 - ∆23 (20) 2

(12)

γt,∞ i

where is the activity coefficient of component i (i ) 2, 3) at infinite dilutions of components 2 and 3 in the ternary mixture 1-2-3 and the subscript 0 indicates that the derivatives should be calculated for xt2 f 0 and xt3 f 0. It should be noted that kii in eqs 7-10 refers to binary mixtures, whereas Kij, defined by eqs 11 and 12, refers to ternary mixtures. b,∞ t,∞ b,∞ Because22,23 γt,∞ 2 ) γ2 , γ3 ) γ3 , and K23 ) K32, the combination of eqs 5 and 6 with eqs 7 and 8 and eqs 11 and 12 yields

and

∆123 ) -[(∆12)2 + (∆13)2 + (∆23)2 - 2∆12∆13 2∆12∆23 - 2∆13∆23]/4 (21) The insertion of eqs 19-21 into eq 15 provides an expression for the derivatives (∂ ln γ2,t/∂xt3)T,P,x t in 2 terms of ∆Rβ and concentrations.

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It should be noted that, according to Ben-Naim,26 ∆Rβ is a measure of the nonideality of the binary mixture R-β because, for an ideal mixture, ∆Rβ ) 0. At infinite dilution of components 2 and 3, eqs 15 and 19-21 lead for K23 (defined by eq 11) to

(

)

lim ∂ ln γ2,t K23 ) -xt2f0 ∂xt3 xt3f0

Table 1. Dependence of the Parameter k22 ) -(D ln γ2/Dx2)P,T,x2f0 on Pressure at T ) 344.25 K for the Systems Investigated pressure (MPa)

k22

water (1)-methane (2)

system

100 75 50 20

-14.3 -14.4 -14.5 -14.6

water (1)-ethane (2)

100 75 50 20

-40.2 -40.3 -40.5 -40.7

water (1)-n-butane (2)

100 75 50 20

-77.6 -77.7 -77.8 -78.1

water (1)-carbon dioxide (2)

100 75 50 20

-33.7 -33.8 -34.0 -34.3

)

T,P,xt2

(

)

lim ∆12 + ∆13 - ∆23 c01 xt f0 (22) 2 2 t x3f0 where

lim c01 ) x2f0 c1 x3f0 When the pair 2-3 (pair of nonpolar gases) is ideal or its nonideality |∆23| is much smaller than that of the combined binary pairs 1-2 and 1-3 (solvent-gases) |∆12 + ∆13|, one can write

∆12 + ∆13 - ∆23 ≈ ∆12 + ∆13

(23)

Taking into account eq 23, eq 22 acquires the form

(

)

lim ∆12 + ∆13 K23 ) c01 x2f0 2 x3f0

(24)

Because23 ∆12 is the same for a binary mixture 1-2 in the limit xb2 f 0 and for a ternary mixture in the limit xt2 f 0 and xt3 f 0

lim k22 ) K22 ) c01 x2f0 ∆12 x3f0

(25)

lim k33 ) K33 ) c10 x2f0 ∆13 x3f0

(26)

and

Consequently, eqs 13 and 14 become

xt2 ) xb2y2 exp k22(xt2 - xb2) +

[

k22 + k33 t x3 2

]

(27)

[

k22 + k33 t x2 2

]

(28)

and

xt3 ) xb3y3 exp k33(xt3 - xb3) +

The system of transcendental eqs 27 and 28 can be used to predict the mixed-gas solubility from the solubilities of the individual gases. 3. Calculation Procedure Calculations were carried out for the solubilities of mixtures of hydrocarbons (methane-ethane and methane-n-butane) and for the mixture methane-carbon dioxide in water, because experimental data regarding the solubilities of binary gas mixtures and individual gases are available for these mixtures.7,8,27

Table 2. Comparison between Predicted and Experimental Solubilities of Methane-n-Butane Mixtures in Water at T ) 344.25 Ka experimental solubilities8 P (MPa)

y2

103xt,exp

100 100 100 75 75 75 50 50 50 20 20 20 20

0.043 0.230 0.455 0.043 0.230 0.455 0.043 0.230 0.455 0.043 0.230 0.455 0.830

0.286 1.148 2.329 0.233 1.118 2.118 0.198 1.003 1.884 0.127 0.799 1.441 1.886

2

predicted solubilities

103xt,exp

103xt,calc 2

103xt,calc 3

0.090 0.075 0.052 0.070 0.076 0.048 0.062 0.081 0.042 0.084 0.056 0.037 0.024

0.233 1.232 2.399 0.213 1.124 2.192 0.168 0.889 1.740 0.096 0.513 1.008 1.820

0.093 0.071 0.048 0.098 0.075 0.051 0.087 0.068 0.046 0.091 0.072 0.049 0.015

3

a xt,exp and xt,exp are experimental solubilities (mole fractions) 2 3 and xt,calc are their of methane and n-butane in water and xt,calc 2 3 solubilities (mole fractions) in water predicted by eqs 27 and 28.

For the prediction of the mixed-gas solubilities from the solubilities of the pure individual gases, the pressure dependence of the binary parameters kii is needed. The Peng-Robinson12 EOS was used to determine the binary parameters kii. The binary interaction parameter q12 in the van der Waals mixing rule was taken from ref 28, where it was evaluated for the water-rich phases of water-hydrocarbon and water-carbon dioxide binary mixtures. The calculated binary parameters kii are listed in Table 1. One should note that, as expected for a liquid phase, the above parameters are almost independent of pressure, in contrast to their dependence on pressure in the gaseous phase near the critical point.19,24 4. Results and Discussion The results of the present calculations are compared with experiment in Table 2 and Figures 1 and 2, where y2 is the mole fraction of methane in the gas phase. One can see that there is good agreement between the two. The deviations at P ) 20 MPa for the methane-nbutane gas mixture are possibly caused by the experimental uncertainties regarding the solubility of the pure n-butane in water.8 Our calculations indicate that the solubility of methane-ethane gaseous mixture in water (Figure 1) ex-

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Figure 1. Solubility of the methane-ethane gas mixture in water at T ) 344.25 K and different pressures. The solubilities calculated from eqs 27 and 28 are represented by the solid lines [(1) xt2, (2) xt3, and (3) (xt2 + xt3)]. The experimental solubilities are taken from ref 8. (9) Mole fraction of methane, (b) mole fraction of ethane, (O) sum of the mole fractions of methane and ethane, and (0) solubilities of the pure gases.

|k22(xt2 - xb2) +

k22 + k33 t x3| , 1 2

(29)

|k33(xt3 - xb3) +

k22 + k33 t x2| , 1 2

(30)

and

5. Conclusion Figure 2. Solubility of the methane-carbon dioxide gas mixture in water at T ) 344.25 K and different pressures. The total solubilities (xt2 + xt3) calculated from eqs 27 and 28 are represented by the solid lines [(1) P ) 100 MPa, (2) P ) 75 MPa, (3) P ) 50 MPa, (4) P ) 20 MPa and (5) P ) 10 MPa]. The experimental solubilities are taken from ref 7. (×) 100, (O) 75, (b) 50, (9) 20, and (2) 10 MPa and (0) solubilities of the pure gases.

hibits almost linear behavior (this means that the solubility of each constituent of the gas mixture can be determined by multiplying the solubility of the pure component by its mole fraction in the gaseous mixture). This conclusion is in full agreement with the experimental results obtained in ref 8 but in disagreement with those of ref 2, where extrema in the dependence on composition of the solubilities of hydrocarbon mixtures in water (at P, T ) const) were found. Our calculations also show, in agreement with experiment,7,8 that the solubility of methane-n-butane gaseous mixtures (Table 2) exhibits a slight nonlinear behavior and that of methane-carbon dioxide mixtures (Figure 2), a nonlinear one. For ideal binary mixtures, k22 and k33 are equal to zero, and eqs 27 and 28 reduce to xt2 ) xb2y2 and xt3 ) xb3y3. Of course, linear behavior can be reached when either kii and/or the solubilities xb2 and xb3 are small enough for

The purpose of this paper was to propose a predictive method for the solubilities of binary mixed gases in a liquid in terms of the individual solubilities. For this aim, the derivatives of the activity coefficients in a ternary mixture with respect to the mole fractions were derived through the fluctuation theory of solutions and used to obtain expressions for the solubility at high dilutions of both gases. The suggested method was tested at 344.25 K and in the pressure range 20-100 MPa for the solubilities of methane-ethane, methane-n-butane, and methanecarbon dioxide in water. The predicted solubilities were compared with experimental data and good agreement was found. Literature Cited (1) McKetta, J. J.; Katz, D. L. Methane-n-butane-water system in two- and three-phase regions. Ind. Eng. Chem. 1948, 40, 853-863. (2) Amirijafari, B.; Campbell, J. Solubility of gaseous mixtures in water. Soc. Pet. Eng. J. 1972, 21-27. (3) Mathias, P. M.; O’Connell, J. P. Molecular thermodynamics of liquids containing supercritical compounds. Chem. Eng. Sci. 1981, 36, 1123-1132. (4) Campanella, E. A.; Mathias, P. M.; O’Connell, J. P. Equilibrium properties of liquids containing supercritical substances. AIChE J. 1987, 33, 2057-2066. (5) Gurikov, Yu. V. Solubility of a mixture of nonpolar gases in water. Zh. Strukt. Khim. 1969, 10, 583-588.

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6283 (6) Myers, A. K.; Myers, A. L. Prediction of mixed gas solubility at high pressure. Fluid Phase Equilib. 1988, 44, 125-144. (7) Dhima, A.; de Hemptinne, J.-C.; Moracchini, G. Solubility of hydrocarbons and CO2 mixtures in water under high pressure. Fluid Phase Equilib. 1999, 38, 129-150. (8) Dhima, A.; de Hemptinne, J.-C.; Jose, J. Solubility of light hydrocarbons and their mixtures in pure water under high pressure. Ind. Eng. Chem. Res. 1998, 145, 3144-3161. (9) Wilhelm, E.; Battino, R.; Wilcock, R. J. Low-pressure solubility of gases in liquid water. Chem. Rev. 1977, 77, 219-262. (10) Solubility Data Series; Pergamon Press: Elmsford, NY, 1982; Vol. 10. (11) Solubility Data Series; Pergamon Press: Elmsford, NY, 1981; Vol. 8. (12) Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. (13) Soave, G. Equilibrium constants from a modified RedlichKwong equation of state. Chem. Eng. Sci. 1972, 27, 1197-1203. (14) Orbey, H.; Sandler, S. I. Modeling Vapor-Liquid Equilibria. Cubic Equations of State and Their Mixing Rules; Cambridge University Press: New York, 1998. (15) Holderbaum, T.; Gmehling, J. PSRK: A group contribution equation of state based on UNIFAC. Fluid Phase Equilib. 1991, 70, 251-265. (16) Fischer, K.; Gmehling, J. Further development, status and results of the PSRK method for the prediction of vapor-liquid equilibria and gas solubilities. Fluid Phase Equilib. 1996, 121, 185-206. (17) Kirkwood, J. G.; Buff, F. P. Statistical mechanical theory of solutions. I. J. Chem. Phys. 1951, 19, 774-782. (18) Sandler, S. I. Chemical and Engineering Thermodynamics, 3rd ed.; Wiley: 1999. (19) Debenedetti, P. G.; Kumar, S. K. Infinite dilution fugacity coefficients and the general behavior of dilute binary systems. AIChE J. 1986, 32, 1253-1262.

(20) Munoz, F.; Li, T. W.; Chimowitz, E. H. Henry’s law and synergism in dilute near-critical solutionssTheory and simulation. AIChE J. 1995, 41, 389-401. (21) Chialvo, A. J. Solute solute and solute solvent correlations in dilute near-critical ternary mixturessMixed-solute and entrainer effects. J. Phys. Chem. 1993, 97, 2740-2744. (22) Jonah, D. A.; Cochran, H. D. Chemical potentials in dilute, multicomponent solutions. Fluid Phase Equilib. 1994, 92, 107137. (23) Ruckenstein, E.; Shulgin, I. Entrainer effect in supercritical mixtures. Fluid Phase Equilib. 2001, 180, 345-359. (24) Ruckenstein, E.; Shulgin, I. On density microheterogeneities in dilute supercritical solutions. J. Phys. Chem. B 2000, 104, 2540-2545. (25) Ruckenstein, E.; Shulgin, I. The solubility of solids in mixtures containing a supercritical fluid and an entrainer. Fluid Phase Equilib., 2002, in press. (26) Ben-Naim, A. Inversion of the Kirkwood-Buff theory of solutions: Application to the water-ethanol system. J. Chem. Phys. 1977, 67, 4884-4890. (27) Culberson, O. L.; McKetta, J. J. Phase equilibria in hydrocarbon-water systems. III. The solubility of methane in water at pressures to 10,000 psia. Pet. Technol. 1951, 3, 223226. (28) Daridon, J. L.; Lagourette, B.; Saint-Guirons, H.; Xans, P. A cubic equation of state model for phase equilibrium calculation of alkane + carbon dioxide + water using a group contribution kij. Fluid Phase Equilib. 1993, 91, 31-54.

Received for review January 8, 2002 Revised manuscript received March 18, 2002 Accepted March 18, 2002 IE020016T