1217
Ind. Eng. Chem. Res. 1991,30,1217-1222 Laddha, S. S.; Danckwerta, P. V. Reaction of COz with Ethanolamines: Kinetics from Gas-absorption. Chem.Eng. Sci. 1981,36, 479-482.
Perrin, D. D. Dissociation Constants of Organic Bases in Aqueous Solutions;Butterworths: London, 1965;No. 4099. Pinsent, B. R. W.; Pearson, L.; Roughton, F. W. J. The Kinetics of Combination of Carbon Dioxide with Hydroxide Ions. Trans. Faraday SOC. 1956,52,1512-1520. Roberta, B. E.; Mather, A. E. Solubility of COOand H2S in A Mixed Solvent. Chem. Eng. Commun. 1988,72,201-211. Sartori, G.; Savage, D. W. Sterically Hindered Amines for C02 Removal from Gases. Ind. Eng. Chem. Fundam. 1983,22,239-249. Sharma, M. M. Kinetics of Reactions of Carbonyl Sulphide and Carbon Dioxide with Amine and Catalysis by Bronsted Bases of
The Hydrolysis of COS. Trans. Faraday SOC. 1965,61,681-688. Tomcej, R. A., Otto,F. D. Absorption of C02and N20 into Aqueous Solutions of Methyldiethanolamine. AIChE J. 1989,35,861-864. Xu, S.;Wang, Y. W.; Otto, F. D.; Mather, A. E. Solubilities and Diffusivities of N20 and C02 in Aqueous Sulfolane Solutions. Submitted to J. Chem. Technol. Biotechnol. 1990. Xu, S.; Otto, F. D.; Mather, A. E. The Physical Properties of Aqueous AMP Solutions. J. Chem. Eng. Data 1991,36,71-75. Yih, S.M.; Shen, K. P. Kinetics of Carbon Dioxide Reaction with Sterically Hindered 2-Amino-2-methyl-l-propanol Aqueous Solution. Ind. Eng. Chem. Res. 1988,27,2237-2241. Received for reuiew December 20, 1990 Accepted January 9,1991
Estimation of Diffusion Coefficients in Multicomponent Liquid Systems Hendrik A. Kooijman and Ross Taylor* Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699
Simple formulas for estimating the Maxwell-Stefan diffusion coefficients in multicomponent liquid mixtures are described based on a geometrically consistent generalization of the Vignes equation for two-component systems. T h e new method requires only binary infinite dilution diffusion coefficients by way of basic data and is of comparable accuracy to other methods that require more basic data and/or fitted parameters.
Introduction Diffusion in multicomponent systems may be described by the Maxwell-Stefan (MS) equations n ( ~ i J-j ~jJi) di = C (1) j=1 c&j where Rj is the Maxwell-Stefan diffusivity and has the physical significance of an inverse drag coefficient. di, which can be considered to be a driving force, may be defined by
The subscripts TQ are used to emphasize that the gradient in eq 2 is to be calculated under constant-temperature and -pressure conditions (pressure gradients and external forces also contribute to di, but we shall ignore their influence here). The driving force, di, reduces to Vxi for ideal systems. Also, the sum of the n driving forces vanishes. The driving forces can be related to composition as follows
where the Dik are multicomponent Fick diffusion coefficients in the molar average reference velocity frame. At present, there is no adequate method for directly predicting the multicomponent Fick diffusion coefficients for liquid mixtures. Although some attempts to this end have been made, none has gained any measure of acceptance. Furthermore, we do not believe that any method of directly predicting the Fick diffusivities in multicomponent liquid mixtures is likely to be developed. Any really useful method of predicting the multicomponent Fick diffusion coefficients is likely to rely on the following relationship between the Fick and Maxwell-Stefan (MS) diffusivities (see the review by Krishna and Taylor (1986) for further background):
[D] = [B]-'[I']
(6)
where [B] is a square matrix (order n - 1)with elements given by
(7) Bij =-xi - - -
(3)
( A j
(4)
The matrix of thermodynamic factors has elements defined by eq 4. Thus, if we are to make any serious use of the generalized Fick or Maxwell-Stefan equations, we require methods of predicting the binary MS diffusivities in multicomponent liquid systems. The aim of this communication is to focus attention on some of the issues involved in developing simple methods for this purpose.
where the thermodynamic factors are defined by d In yi
rij= 6, + xi-
axj
...
T?,.rh,k#j==l n-1
where yi is the activity coefficient of species i in the mixture. An alternative method of describing diffusion in multicomponent systems is the generalized Fick's law, which takes the form (5)
i n )
Estimation of Maxwell-Stefan Diffusion Coefficients for Liquid Mixtures For two-component systems, eq 1 simplify to J~= -ctBlorvxl
0888-5885/91/ 2630-1217$02.50/0 0 1991 American Chemical Society
(9)
1218 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 Table I. Diffusion Coefficients in the System Acetone (1)-Benzene (t)-Carbon Tetrachloride (3) at 25 OC with Diffusivity Data from and Cited by Cullinan and Toor (1965) Fick Diffusion Coefficients (lo* ma 8-l) (in Molar Average Velocitv Reference Frame) X1
X2
~3
0.2989 0.1496 0.1497 0.6999
0.3490 0.1499 0.6984 0.1497
0.3521 0.7005 0.1519 0.1504
Dii
Diz
41
DB
1.8690 -0.2129 -0.0043 2.2730 1.5968 -0.0592 -0,0771 1.8132 1.9652 0.0137 -0.1522 1.9248 2.3041 -0.4019 0.1848 2.9969
Infinite Dilution Diffusion Coefficients W ; ;(lo* m2 8-l) W12 = 2.75 Wz1 = 4.15 W13 = 1.70 W31 = 3.57 W23 = 1.42 P 3 2 1.91 NRTL Parameters (Dimensionless, from Renon et al., 1971) r12= -0.46504; rzl = 0.74632;a12 = a21 = 0.2 7 1 3 = -0,42790;131 = 1.5931; a13 = a31 = 0.2 T23 = -0.51821;132 = 0.7338;aB = a32 = 0.2 Molar Volumes (10" m3/kmol) VI = 74.05 = 89.41 v 3 = 97.09
v2
?3
43 X
Figure 1. Two views of the Maxwell-Stefan diffusion coefficients calculated from eq 12 for the system acetone (1)-benzene (2)-carbon tetrachloride (3).
where the thermodynamic factor, r, is obtained from eq 4 as
I t is understood that the mole fractions xl and x 2 sum to unity when the partial derivative of In y1 is evaluated. The product %21' is the binary Fick diffusivity, D12. A number of methods for predicting 1312in concentrated solutions attempt to combine the infinite dilution coefficients, P 1 2 and P Z 1 in ,a simple function of composition. The Vignes (1966) equation, for example, €42 = ( ~ 1 2 ) x * ( ~ 2 1 ) x 1 (11) is recommended by Reid et al. (1987, p 614). Methods of estimating the infinite dilution diffusivities with a reasonable degree of accuracy are also discussed by Reid et al. (1987, pp 598-611). It has been suggested that, for multicomponent systems, methods of estimating binary diffusion coefficients (of which the Vignes equation is but one of many examples) be used with the actual mole fractions replaced by their equivalent binary compositions (Krishna, 1981, 1985) ej= ( e P i j ) X j i ( P j i ) X i j (12) where (13) X j j = X i / ( X i + x j ) ; x j i = 1 - x,
X
XI
Figure 2. Maxwell-Stefan diffusion coefficients as a function of mole fraction calculated from eq 15.
It should be pointed out that eq 12 possesses two different limits when both x i and x j vanish. The two limits are given by lim Bij = (Wji) xi = 0 Xl-0
In Figure 1 we plot the MS diffusivities as a function of composition, for the ternary system acetone-benzenecarbon tetrachloride, calculated by using eq 12. The infinite dilution diffusion coefficients for this system are given in Table I. The two distinct limits of B d can be seen in the corners of the B-x surfaces. If eq 12 does not provide a geometrically consistent method of estimating MS diffusivities in multicomponent systems, is there any alternative method of generalizing the Vignes equation? Wesselingh and Krishna (1990) have suggested the following generalization of the Vignes equation for ternary systems ej = ( ~ j ~ , - ~ ) x l ( ~ j ~ z - l ) x ~ ( ~ j , x ~ -i ~# ) jx ~= l , 2 , 3 (15) which is illustrated in Figure 2, where we see that the
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1219 limiting diffusivities, + , are the MS diffusion coefficients at the corners od tke diffusivity-composition surface. The multicomponent generalization of eq 15 is
Any model of the MS diffusivities based on eq 16 must obey some rules that place certain restrictions on the kind of model that can be used for the limiting diffusivities, First, the final expression for the MS diffusivities must be symmetric = +J.In addition, we must have
ejJh+.
(ej
-
-
to avoid discontinuities. Finally, eq 16 must reduce to the 0 (k binary Vignes equation when x i + x, 1 and x k i , j ) . Two of the limiting diffusivities in eq 16 may, therefore, be identified as the binary i-j infinite dilution diffusivities: #
Bij,x,+l
=p
Bij,x+l
= pj i
X
XI
Figure 3. Maxwell-Stefan diffusion coefficientscalculated from eq 22 for the system acetone (1)-benzene (2)-carbon tetrachloride (3).
i j
(18)
Substitution of these limiting values into eq 16 gives Bij =
( P i j P ( P j i P i
A
k=l
(ejsr'l)xh
?2
(19)
?3
k#ij
D13
Equation 19 is similar to an equation derived by Cullinan and Cusick (1967) as an extension of Cullinan's (1966) treatment of binary diffusion coefficients. His equation includes binary thermodynamic factors similar to I', but these are treated as adjustable parameters by Cullinan. It remains only to supply a model for the limiting difwhere k # i or j . We have tested several fusivities simple models of these limiting diffusivities that are functions of, at most, six infinite dilution diffusivities
e.,
&j4h-l
=f
X
( p i j pj i p i k pk i , pj k p k j )
Of the many models that we have evaluated, only two will be discussed in detail: the geometric average of the i-j and j-i infinite dilution diffusivities Bij,xh-l
- (PijWji)1/2
(20)
tentatively suggested by Wesselingh and Krishna (1990) and the geometric average of the i-k and j-k infinite dilution diffusivities (i # j # k ) . The limiting diffusivities obtained from eq 20 are the same for all possible species k ( k # i , j ) , whereas the limiting diffusivities obtained from eq 21 depend on the nature of component k . When we combine eqs 19 and 20, we obtain the following elegant expression for the binary MS diffusivity in a multicomponent system: &. V = ( p i j ) ( 1 + x , - x i ) / 2 ( p J.i . ) ( l + ~ x j ) / 2 (22) For systems where the i-j infinite dilution diffusivities are equal, eq 22 suggests that the MS diffusivity will be independent of the composition of the mixture. Substitution of eq 21 into eq 19 gives n
€)id =
(iPij)xj(@'ji)xi
Il k=l k#ij
(pikpjk)xh'/2
(23)
XI
Figure 4. Maxwell-Stefan diffusion coefficientscalculated from eq 23 for the system acetone (1)-benzene (2)-carbon tetrachloride (3).
The Qj obtained from eq 23 remain a function of composition unless all the limiting i-j diffusivities are equal to both the i-j infinite dilution diffusivities. To illustrate these two models, we have calculated the MS coefficients for the ternary system acetone-benzenecarbon tetrachloride, data for which is provided in Table I. The diffusivities computed by using eq 22 are shown in Figure 3. It can be seen that, as would be expected, eq 22 eliminates the discontinuities at the corners of the * x surfaces. Note, further, that the i3-x surfaces do not intersect, although such behavior is not precluded by eq 22. The MS diffusivities calculated with eq 23 are shown in Figure 4. All three B-x surfaces have the same general form, and they intersect each other as they must. The Fick diffusion coefficients calculated from eq 6 for the acetone-benzene-carbn tetrachloride system at 25 OC are shown in Figures 5-7. The thermodynamic factors, r,,,were calculated with the NRTL equation. The model parameters are listed in Table I. The MS diffusivitiw were calculated from eq 23. These figures clearly demonstrate the more complicated composition dependence of the Fick Coefficients when compared to the MS coefficients (at least when the models described here are used to calculate the MS diffusivities). Also demonstrated is the fact that multicomponent Fick diffusivity values depend on the order in which the components are numbered. Figure 6
1220 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
Dl I D*z
Table 11. Diffurion Coefficientr in the System Methanol (I)-Ilobutaaol (2)-l-Propaaol (3) at 30 O C with Diffusivity Data from Shuck and Toor (1963) Fick Diffusion Coefficients (lo4 m2s-l) (in Molar Average Velocity Reference Frame) XI 119 ~n Dii Dii Dm Dn 0.4633 0.2406 0.2961 1.0368 0.0071 -0.0535 0.8772 0.2531 0,1098 0.6371 0.9092 0.0149 -0.0212 0.7208 0.2781 0.5708 0.1511 0.7658 0.0125 -0.0932 0.6232 0.8283 0.0770 0.0947 1.5097 0.1043 -0.0138 1.3783 Infinite Dilution Diffusion Coefficients Pi, (lo* m2 8-l) 9 2 1 = 1.838 B O 1 2 = 0,587 9 3 1 = 1.966 P i 3 = 0.804 9 3 2 = 0.398 9 2 3 = 0.584
D*I D12
X
X
Figure 5. Multicomponent Fick diffusion coefficientsfor the system acetone (1)-benzene (2)-carbontetrachloride (3). Calculations were made with eqs 6 and 23.
NRTL Parameters (DECHEMA. cal/mol) = 835.3472; &2l = -548.5758; ( ~ 1 2 a21 = 0.3021 4 1 , = 767.4888; 4 3 1 -444.2736; a13 a31 = 0.3086 423 = -387.4575; ag,, = 508.6723; ( Y =~ ( Y =~ 0.2791 ~
4 1 2
Molar Volumes (lo4 m3/kmol) = 92.91 = 75.14
v3
vl = 40.73
2 becomes component 3, and component 3 becomes component 1. Figure 7 represents a second rotation of the component indexes. These figures were inspired by a similar set of drawings for a ternary ideal gas mixture made by Wesselingh (1985). Rotating the component indices of the MS diffusivities does not alter the B - x surfaces, so the only change in Figures 1-4 would be to see the same surfaces from a different angle.
D22 Dl I
D2 I
DI 2 X
X
Figure 6. Multicomponent Fick diffusioncoefficientsfor the system in Figure 5, except that the components have been renumbered so that component 1 becomes component 2, and so on.
x3
XI
Figure 7. Multicomponent Fick diffusion coefficientsfor the system in Figures 5 and 6. The components have been renumbered a second time.
has been obtained simply by rotating the component indices; component 1 becomes component 2, component
Comparison with Experimental Data It would be nice to be able to compare the predictions of the models discussed above against experimental data. Unfortunately, data on multicompomponent MS diffusivities are extremely rare. Even when available, such data can be quite sensitive to the choice of thermodynamic model used to obtain the coefficients ri.. We have tested these models in a more roundabout fashion by using the MS diffusion coefficients to predict multicomponent Fick diffusion coefficients. Cullinan and Toor (1965) have presented experimental data on the matrix of Fick diffusion coefficients for the system acetone (1)-benzene (2)-carbon tetrachloride (3) at a temperature of 25 "C. Fick diffusion coefficients for the ternary system methanol (l)-l-propanol(2)-isobutanol (3) at 30 "C have been reported by Shuck and Toor (1963). Alimadadian and Colver (1976) developed an experimental method for measuring Fick diffusivitiesand have provided data for the system acetone (1)-benzene (2)-methanol(3) at 25 "C. In all three cases the Fick diffusivities are provided in the volume average reference velocity frame. The data of these investigators, converted to the molar average reference velocity frame, are shown in Tables I-III (see Krishna and Taylor (1986, pp 280-281) for details of the transformation). The Fick coefficients predicted by using eq 6 with the three generalizations of the Vignes equations are summarized in Table IV. Space limitations limit the quantity of computed results included in this paper; more complete tabulations of our results are available in the supplementary material. The discrepancy between the measured and predicted Fick diffusivities is defined for each data point as tij
=
(Dij,calc
- Dij,expt)/Dii
where Dii is the diagonal element on the ith row of the matrix. The error is defined in this way to avoid placing
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1221 Table 111. Diffusion Coefficients in the System Acetone (1)-Benzene (2)-Methanol (3) at 25 OC with Diffusivity Data from Alimadadian and Colver (1976) Fick Diffusion Coefficienta (lo* mz8-l) (in Molar Average Velocity Reference Frame) x2
Xl
0.3500 0.7660 0.5530 0.4000 0.2990 0.2060 0.1020 0.1200 0.1500
33
Dii
Di2
0.3020 0.3480 3.6499 -0.0693 0.1140 0.1200 3.9119 0.3259 0.1900 0.2570 4.2273 0.2454 0.5OOo 0.1000 3.8649 0.8928 0.1500 0.5510 3.1311 0.0611 0.5480 0.2460 3.2822 0.2711 0.7950 0.1030 3.0454 0.5488 0.1320 0.7480 3.0828 0.0259 0.2980 0.5520 3.0017 0.0248
D21 Dn -0.3022 2.3021 -0.5119 3.1681 -0.2281 2.8137 -0.3052 2.2371 -0.1057 2.4289 -0.3408 2.1788 -0.8806 1.5806 -0.1560 2.2672 -0.1617 2.2983
Infinite Dilution Diffusion Coefficients 130, (lo4 m2 s-l) P i 2 = 3.0368 13021 4.2778 13013 = 2.6009 13031 4.7951 P a = 2.4159 9 3 2 = 3.6480
NRTL Parameters (DECHEMA, cal/mol) Ag12= -241.2745; Agzl = 816.6221; a12= azl = 0.2998 &I3 = 812.1131; &31 = -312.9786; a 1 3 = a31 = 0.2942 Agg23 = 1362.3125; &32 = 719.5238; = a32 = 0.5023
Molar Volumes (lo4 m3/kmol) = 89.41 = 40.73
9, = 74.05
v2
v3
Table IV. Summary of Fick Diffusion Coefficient Calculations discrepancies eq 12 eq 22 eq 23 System: Acetone (1)-Benzene (2)-Carbon Tetrachloride (3) overall average 0.1145 0.1125 0.0763 average diagonal 0.0799 0.0712 0.0916 average off-diagonal 0.1491 0.1538 0.0611 average maximum 0.1741 0.1828 0.1192 maximum diagonal 0.1950 0.2133 0.1593 maximum off-diagonal 0.2222 0.2484 0.1242 System: Methanol (1)-l-Propanol (2)-Isobutanol (3) overall average 0.1476 0.1413 0.0558 average diagonal 0.1685 0.1684 0.0748 average off-diagonal 0.1268 0.1143 0.0367 average maximum 0.2420 0.2418 0.0770 maximum diagonal 0.2852 0.2908 0.1352 maximum off-diagonal 0.3753 0.3770 0.0604 System: Acetone (1)-Benzene (2)-Methanol (3) overall average 0.2151 0.2199 0.1862 average diagonal 0.2207 0.2257 0.2062 average off-diagonal 0.2096 0.2141 0.1662 average maximum 0.4219 0.4346 0.3898 maximum diagonal 0.7285 0.7427 0.7051 maximum off-diagonal 0.4885 0.5144 0.3910
too great an emphasis on the errors between measured and predicted off-diagonal elements. Off-diagonal elements of the Fick matrix can be very sensitive to the values of the MS diffusion coefficients; a small change in one of the binary MS coefficients can change the sign of the calculated Fick diffusivity. The error defined above is equivalent to the error in the diffusion fluxes,Ji, computed from eq 5 for a mixture where all the driving forces are equal. Since we are ultimately interested in predicting masstransfer rates, it is the prediction of the flux that we consider to be more important than the prediction of the diffusivity. Equation 12 is, on average, the worst of the methods discussed above. However, despite its geometric consistency, eq 22 is not significantly better. The best performance comes from eq 23, with an average error of less than 0.08 for the system acetonebenzene-carbon tetrachloride and less than 0.06 for the methanol-l-propanol-isobutanol system. In view of the way that the discrepancy has been
Table V. Summary of Maxwell-Stefan Diffusion Coefficient Calculations discrepancies eq 12 eq 22 eq 23 System: Acetone (1)-Benzene (2)Xarbon Tetrachloride (3) overall average 0.3312 0.3240 0.1645 average maximum 0.6558 0.6554 0.2802 overall maximum 1.3120 1.3123 0.3420 ~~
System: Methanol (1)-l-Propanol overall average 0.2999 average maximum 0.6194 overall maximum 0.7376
(P)-Isobutanol (3) 0.2626 0.1333 0.5043 0.3175 0.5527 0.6976
System: Acetone (1)-Benzene (2)-Methanol (3) overall average 0.4320 0.4232 0.3473 average maximum 1.1944 1.2016 0.9569 overall maximum 1.5620 1.5777 1.1260
defined above, the average error for the off-diagonal elements of [D] should be lower than the average error for the main diagonal elements. Only for eq 23 is this the case. None of the methods does as well for the system acetone-benzene-methanol. However, the acetone-methanol and benzene-methanol systems are associating binaries; the Vignes equation does not perform well for such systems (Reid et al., 1987, p 614). An alternative approach is needed to predict binary MS diffusivities for such systems (see, for example, McKeigue and Gulari, 1989). Several other simple models for the limiting diffusivities have also been tested as part of this investigation. The modifications included arithmetic-as opposed to geometric-averages of the infinite dilution diffusivities, or the appropriate root of the product of other combinations of the infinite dilution diffusivities, W i k . Also, methods of combining the binary infinite dilution coefficients in some way other than that provided by the Vignes (1966) equation were considered as starting points for the development of a generalized formula (a mole fraction weighted sum). Of the many alternatives that were considered, none proved to be any better than eq 23. We have also compared the MS diffusion coefficients predicted by the three models with the values of the that give the best fit to the measured Fick diffusivities. Detailed results are available in the supplementary material, and a summary is provided in Table V. The agreement between predicted and fitted MS diffusivities is not as good as might be hoped for. However, the results confirm our earlier observation that eq 23 appears to provide better predictions of the MS diffusivities than does eq 22, which, in turn, is marginally better than eq 12.
ej
Conclusion We have discussed a geometrically consistent generalization of the Vignes method for estimating MaxwellStefan diffusion coefficients in multicomponent liquid systems. The generalized method requires only binary infinite dilution diffusion coefficients and the limiting diffusivities, We have also discwed the evaluation of these limiting diffusivities from simple functions of infinite dilution diffusion coefficients. No adjustable parameters are required. Other published methods (Cullinan and Cusick, 1967; Bandrowski and Kubanka, 1982)require more basic data and employ fitted parameters. In the absence of additional data that can provide a more stringent test of these models, we recommend eq 23 for predicting the Maxwell-Stefan diffusivities in a multicomponent liquid mixtures. We hope that this paper will warn our readers of the problems of incorrectly generalizing methods of estimating diffusivities in binary systems and promote further work with the aim of developing new and
Ind. Eng. Chem. Res. 1991, 30, 1222-1231
1222
better methods of estimating these coefficients from fundamental models. Acknowledgment This paper grew out of work based on a project supported by the National Science Foundation under Grant No. CBT 88821005. Supplementary Material Available: Tables listing experimental and predicted diffusion coefficienta for the systems acetone (1)-benzene (2)-carbon tetrachloride (3),methanol (1)-1-propanol (2)-isobutanol(3), and acetone (1)-benzene (2)-methanol (3) (27 pages). Ordering information is given on any current masthead page. Literature Cited Alimadadian, A,; Colver, C. P. A New Technique for the Measurement of Ternary Diffusion Coefficients in Liquid Systems. Can. J. Chem. Eng. 1976,54, 208-213. Bandrowski, J.; Kubaczka, A. On the Prediction of Diffusivities in Multicomponent Liquid Systems. Chem. Eng. Sci. 1982, 37, 1309-1313. Cullinan, H. T. Concentration Dependence of the Binary Diffusion Coefficient. Ind. Eng. Chem. Fundam. 1966,5, 281-283. Cullinan, H. T.; Toor, H. L. Diffusion in the Three-Component Liquid System Acetone-Bemendarbon Tetrachloride. J. Phys. Chem. 1965, 69, 3941-3949.
Cullinan, H. T.; Cusick, M. R. Predictive Theory for Multicomponent Diffusion Coefficients. Ind. Eng. Chem. Fundam. 1967,6, 72-77. Krishna, R. Ternary Mass Transfer in a Wetted Wall Column. Significance of Diffusional Interactions. Part 11: Equimolar Diffusion. Trans. Inst. Chem. Eng. 1981, 59,44-53. Krishna, R. Model for Prediction of Multicomponent Distillation Efficiencies. Chem. Eng. Res. Des. 1985, 63, 312-322. Krishna, R.; Taylor, R. Multicomponent Mass Transfer-Theory and Applications. In Handbook of Heat and Mass Transfer; Cheremisinoff, N. C., Ed.; Gulf Publishing Co.: Houston, TX, 1986; Vol. 2,Chapter 7, pp 259-432. McKeigue, K.; Gulari, E. Affect of Molecular Association on Diffusion in Binary Liquid Mixtures. AZChE J. 1989, 35, 3W310. Reid, R.C.; Prausnitz, J. M.; Poling, B. Diffusion Coefficienta. The Properites of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987;Chapter 11, pp 577-631. Renon, H.; Asselineau, L.; Cohen, G.; Raimbault, C. CaZcuZ sur Ordinateur des Equilibres Liquide- Vapeur et -Liquide-Ziquide;Editions Technip, 1971. Shuck, F. 0.; Toor, H. L. Diffusion in the Three-Component Liquid System Methyl Alcohol-n-Propyl Alcohol-Isobutyl Alcohol. J. Phys. Chem. 1963,67,540-545. Vignes, A. Diffusion in Binary Solutions. Ind. Eng. Chem. Fundam. 1966,5, 189-199. Wesselingh, J. A. Is Fick Fout. Procestechnologie 1985, No. 2,3943. Wesselingh, J. A.; Krishna, R. Elements of Mass Transfer; Ellis Horwood: Chichester, U.K., 1990. Received for review October 9, 1990 Revised manuscript received January 18, 1991 Accepted February 4, 1991
Chemical Complexing Agents for Enhanced Solubilities in Supercritical Fluid Carbon Dioxide Richard M.Lemert and Keith P.J o h n s t o n * Department of Chemical Engineering, The University of Texas a t Austin, Austin, Texas 78712
The effect of the strong Lewis base tri-n-butyl phosphate (TBP) on the solubility of benzoic acid and hydroquinone (HQ) in supercritical fluid carbon dioxide is reported. TBP is shown to be a much stronger cosolvent for these solutes than methanol. For example, 2% TBP increases hydroquinone’s solubility by a factor of 250. The principles of chemical reaction equilibria are combined with the Peng-Robinson equation of state in order to model these results. The behavior of the hydroquinone-carbon dioxide-TBP system is shown to be attributable to the formation of an HQ-TBPZ complex having an enthalpy of formation of -18.9 kcal/mol. The performance of this chemical model is compared to that of a recently developed density-dependent local composition (DDLC)model. Introduction Ten years ago, studies of supercritical fluid extraction were usually either empirical investigations into the fractionation of complex naturally occurring matrices or rigorous investigations into the behavior of much simpler model systems (see, for example, Johnston, 1984, Paulaitis et al., 1983). Since then, this gap has been bridged to a significant extent as discussed in several reviews (Brennecke and Eckert, 1989a; Brunner, 1988; Johnston et al., 1989~).In some cases, it is now possible to predict the phase behavior of these systems as a function of the molecular structure of the components (Johnston et al., 1989d), something that is particularly useful in process design and evaluation. This early work showed fairly quickly that pure supercritical fluid solvents have some significant limitations for certain separations. Fluids with convenient critical temperatures (defined here as being between 0 and 100 OC), for example, are usually nonpolar.
* T o whom correspondence should be addressed. 0888-5885/91/2630-1222$02.50/O
As a result they are good solvents for solutes such as naphthalene and anthracene (Kurnik et al., 1981; McHugh and Paulaitis, 1980), but are unsuitable for amino acids or sterols (Krukonis and Kurnik, 1985; Wong and Johnston, 1986). In addition, solubilities and selectivities (defined as the ratio of solubilities) are determined primarily by the solute’s vapor pressure (Dobbs and Johnston, 1987; Kurnik and Reid; 1982; Johnston et al., 1989d). Much of the recent research in this area has been designed to enhance the solubilities of polar solutes in these solvents, with the greatest success coming from the use of cosolvents. Nonpolar cosolventa were the first to be studied and were shown to be useful in increasing solute solubilities (Brunner, 1983; Dobbs et al., 1986). This results from their greater polarizability compared to the smaller primary solvent molecules, which leads to increased dispersion forces in the solution. Because these forces are nonselective, however, there is usually no improvement in the selectivity of the process. Polar cosolvents usually produce much larger solubility enhancements (Dobbs et al., 1987; Schmitt and Reid, 1986; van Alsten, 1986), and 0 1991 American Chemical Society
