Ind. Eng. Chem. Process Des. Dev. 1982, 21, 409-475
the University of Leeds, GRB 20 (1945). GermMan, J. E. W.D. Thesis, Massachusetts Insmute of Technology, Cambridge, MA, 1980. Heppner, D. B.; Halick, T. M.; Schubett, F. H. “Performance Characterlratbn of a B o b C02 Reduction Subsystem”: NASA CR-152342 (Feb 1980). Hofer, L. J. E.; Cohn, E. M.; Peebles, W. C. J . phvs. COlkM Chem. 1949, 53, 861-669. Hofw. L. J. E.; Cohn. E. M.; Peebies, W. C. J . phvs. Cdbkj Chem. 1950, 54. 1161-1169. Holmes, R. F.; Keller, E. E.; King, C. D. “A Carbon Dioxide Reduction Unit Usina Bosch Reactbn and Expendable Catalyst Cattridges”; NASA CR1682 (Nov 1970). Manning, M. P. Sc.D. Thesis. Massachusetts Instltute of Technology. Camb*. MA, 1976. Manning, M. P.; Reid, R. C. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 356. Nagakura, S. J . Wys. SOC.Jpn. 1957, 12(5),482-494. Nagakura, S. J . h y s . Soc.Jpn. 1901, 16(6), 1213-1219.
409
Renshaw, G. D.; Roscoe, K.; Walker, P. L. J . Catai. 1971, 22, 394. RostrupNielsen, J. J . Catai. 1972, 27. 343. RostrupNieisen, J.; Trimm, D. L. J . Catal. 1977, 48, 155-165. Sacco, A. Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1977. Sacco, A.; Reid, R. C. AIChE J . 1979, 25, 839. Schenck, R. 2.Anorg. Chsm. 1927, 164, 315-324. Trimm, D. L. Catai. Rev. 1077, 16. 155-189.
Received for review September 15, 1980 Revised manuscript received January 27, 1982 Accepted February 18, 1982 The authors gratefully acknowledge the support of the National Aeronautics and Space Administration under Grant No. NGR22-009-123.
Thermodynamic Model for Solvating Solutions with Physical Interactions Eugene E. Spala‘ and Ne11 L. Rlcker’ Depam“
of Chemical Engineering, University of Washington, Seattk, Washington 98 195
A method for the correlation of phase equllibrlum data for solvating, multicomponent liquid solutions is proposed. Chemical equilibrium constants are used to calculate the extent of formation of discrete solvation complexes, and the UNIFAC group-contrlbution theory is wed to predict the physical interactions between species in solution. The method is applied to example binary and multicomponent solvating systems including a quaternary trioctylaminelacetic acid/sohrent/water system from a developmental llquid extraction process that exhibits unusually complex phase equllibrla. The proposed method gives a much better representation of such systems than has been reported previously. Potential shortcomings of the approach are also discussed.
Introduction Many separation processes in the chemical industry exploit solvation effects, i.e., specific chemical interactions between two or more components in liquid solution. Common examples of such processes include extractive and azeotropic distillation, acid-gas absorption, and the separation of metal ions by liquid extraction. Recently, Ricker et al. (1980a,b) proposed a liquid extraction process for the recovery of carboxylic acids from aqueous wastes;the process was based on an extractant mixture of a long-chain tertiary aliphatic amine and an organic diluent. Wardell and King (1978) and Ricker et al. (1979) showed that under the optimal extraction conditions acetic acid and similar solutes were highly solvated in this extractant phase, resulting in favorable equilibrium distribution coefficients. They found, however, that the phase-equilibrium behavior was a complicated function of the amine structure, the type of diluent, the relative amounts of amine and diluent in solution, and the equilibrium concentration of the acidic solute(s). I t was clear that the optimization of the process would require either a great deal of experimentation or a calculational model that one could use to predict the equilibrium distributions of the various components. In the present paper, we develop such a model and present several example applications. Stauffer Chemical Co., Richmond, CA 94804. 0196-4305/82/1121-0409$01.25/0
The Calculational Model Traditional theories of liquid solutions interpret solution nonidealities exclusively in terms of either strong “chemical”intermolecular interactions or weak “physical” interactions (see, e.g., Prausnitz, 1969). These distinct viewpoints lead to relatively simple thermodynamic models that can be used to correlate data for solutions in which one type of interaction dominates over the other. Unfortunately, neither of these simple viewpoints adequately represents the equilibria of the acid/amine/diluent systems studied by Ricker et al. (1979). In these and in analogous systems, a representative model must allow for the simultaneous effects of chemical and physical interactions in solution. In other words, a unified treatment is required. Work along these lines has been performed by a number of previous authors, including, e.g., Harris and Prausnitz (1969),Renon and Prausnitz (1967),Nagata (1973,1977), Nagata and Kawamura (1979a,b), and Chen and Bagley (1978). The most common approach has been to express the excess Gibbs free energy as the s u m of a chemical and a physical contribution The chemical contribution results from the dependence of the “true” composition of the solution on the degree of solvation and/or association of the various species. Many of the past investigations have involved solutions in which one component, usually an alcohol, was assumed to self@ 1982 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
associate to form linear polymeric species. For such cases, it was assumed that the components formed a FloryHuggins solution; gEchem was then derived in terms of Kretschmer-Wiebe (1954)-type chemical equilibrium constants for the association reactions. The physical contribution to gErepresents the residual, relatively weak interactions between species. Several different expressions have been used for gEph,,including a one-parameter Scatchard-Hildebrand equation (Renon and Prausnitz, 1967; Nagata, 1973) and a two-parameter UNIQUAC equation (e.g., Nagata, 1977). Once the gEpFp and gEcham expressions have been chosen, one can easily obtain equations for the activity coefficients of the solution components (Prausnitz, 1969). Harris and Prausnitz (1969), on the other hand, were concerned only with solvation effects for nonpolymeric species, and they used a different expression for the equilibrium constant of their solvation reactions, as will be described below. The general strategy of the present work is similar to that of Harris and Prausnitz; the main distinction is that we have chosen a more general representation for the physical interactions in solvating solutions. Solutions of Two Solvating Components We shall first develop a model for a binary solution of hypothetical components A and B at a single temperature and pressure; the extension to more complex situations will then follow naturally. We assume that independent evidence, such as spectroscopic data, indicates that A and B are solvating components and that solvation occurs through the following reversible reaction A+B--,AB (1) The thermodynamic equilibrium constant corresponding to (1) is
where aA,aB, and am are the activities, YA,YB,and TAB are the activity coefficients, and ZA,ZB, and ZAB are the “true” mole fractions of A, B, and AB at equilibrium. Usually, the true mole fractions cannot be measured directly. Therefore, following Harris and Prausnitz (1969), we define “apparent” mole fractions of the solvating components, XAand XB, that can be measured and are related to the true mole fractions through stoichiometry
where 5, the normalized extent of complexing, is given by 1 - (1- 4[K/(K K,)]XAXB]~/~ (4) 5= 2 The above equation implies that the activity coefficients (or K,) must be known as a function of concentration in order to predict the extent of complexing, E, and the activities of the solvating species. At this point, Harris and Prausnitz introduce a van Laar expression for the activity coefficients in order to obtain equations of the form YA = ~ A ( Z A ZB, , ZAB,0) YB = ~ B ( Z A ZB, , ZAB,a ) (5) TAB = f A B ( Z A 1 ZB, ZAB, a ) where the f s are algebraic functions involving the true mole fractions and a single adjustable parameter, a. Given values for K and a , it is easy to solve eq 3, 4,and 5 nu-
merically for the activities of each of the species in solution as a function of X A and XB. In the present model, we retain the assumptions that lead to eq 3 and 4 but use the UNIFAC method (Fredenslund et al., 1975) for the algebraic relationships required for the calculation of YA, YB,and y u . UNIFAC is a group-contribution version of the UNIQUAC model developed by Abrams and Prausnitz (1975). In contrast to the van Laar expression, the UNIQUAC model was developed for use with mixtures containing both polar and nonpolar components; it can be applied to partially miscible, multicomponent systems. In UNIFAC, a molecular species, i, is considered to be made up of one or more “functional groups” (e.g., methyl and carbonyl groups). As in UNIQUAC, the activity coefficient of species i is considered to be the product of a combinatorial and a residual term, so that In yi = In yic
+ In yiR
(6)
Expressions for In 7; and In yiRare given by Fredenslund et al. (1975). Briefly, the combinatorial contribution, y;, is due mainly to differences in the size and shape of the molecules in solution. Its calculation requires only group-size parameters that are easily determined from pure-component data. Evaluation of the residual portion, yiR, requires two “group-interaction” parameters for each pair of functional groups in the solution. These parameters account for the weaker specific chemical interactions that are not included in the model as explicit solvation equilibria of the type shown in eq 1. In general, these adjustable parameters must be determined from binary and/or multicomponent equilibrium data through the use of a parameter-optimization scheme. Several such schemes are discussed in detail by Prausnitz et al. (1980). In our use of the UNIFAC equations, we define the solvating species A and B as functional groups. The complex AB is then simply a combination of these two groups and using UNIFAC we can calculate YA, yB, and TAB based on the two A-B group-interaction parameters, a A B and aBA. In other words, instead of (5), we have YA
= ~A(ZA ZB, , ZAB, AB,
~ B A )
YB
= ~B(ZA ZB, , ZAB, AB,
~ B A )
(7)
YAB = ~AB(ZA, ZB, ZAB,QAB, QBA) Note that for a binary solution, the use of eq 7 results in a total of three adjustable parameters, a A B , a B A , and K, whereas the Harris-Prausnitz model only requires two, In the UNIQUAC equation (Abrams and Prausnitz, 1975), however, it is possible to use only one parameter. By analogy, one may here replace a A B and a B A by a single adjustable parameter. Given eq 3,4, and 7, we can either determine the optimal values of K, am, and CWBAfor a particular set of equilibrium data, or we can use previously determined values of these parameters to predict phase equilibria. The numerical methods needed to solve the equations are straightforward. Details are given by Harris and Prausnitz (1969) and Spala (1980). The stoichiometric relationships in the present model are readily extended to systems in which the solvating components form more than one complex, e.g., A2B,A3B, AB2,etc. In general, an additional equilibrium constant, K , is required for each additional solvation reaction, but in some cases it may be physically reasonable to assume an arbitrary algebraic relationship between these constants in order to minimize the number of adjustable parameters. Harris and Prausnitz also discuss an extension to noni-
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 411 Table I. Comparison of Models for Prediction of Activity Coefficient of Acetylene in Three Solvents std % dev in acetylene activity coeff predictions solvent hexane bu tyrolac tone methyl pyrrolidone
present model ( 3 param) 1.8 2.7 2.7
present model ( 2 param) 3.2 2.6
Harris-Prausnitz ( 2 param) 5.0 2.6 1.1
UNIQUAC ( 2 param) 1.8 6.5 11.7
I
sothermal systems. The systems to be considered here are, however, assumed to be at a single temperature. Extension to Multicomponent Solutions We now consider a hypothetical ternary solution in which species A and B solvate according to eq 1, but the third molecular component, C, solvates neither A nor B. This situation is analogous to the acid/amine/ diluent systems studied by Ricker et al. (1979). For such a ternary solution, eq 3 and 4 still hold and the true mole fraction of the diluent component is given by
2x)
0 F R O M H A R R I S (19691
-PRESENT
v *C
2, = -
..-..H A R R I S
n
1-5
-- U N I 9 U A C
0
We again consider C to be made up of a single functional group. In order to calculate the required activity coefficients using UNIFAC, we now need to specify four additional group-interaction parameters: two for the A-C pair and two for the B-C pair. In principle, it is possible to determine these from independent binary equilibrium data, but this gives unsatisfactory results in certain cases, as will be discussed in a later section. The numerical solution of the multicomponent equations is completely analogous to the binary case (Spala, 1980). Illustrative Applications-Binary Systems Harris and Prausnitz (1969) used their calculational method to correlate the solubility of acetylene in a variety of liquid solvents. We have used the UNIFAC-based approach described above on three of the systems they considered acetylene dissolved in hexane, butyrolacetone, and N-methyl pyrrolidone. These cases were chosen because they exhibit a range of solvation, from essentially no solvation (acetylene/hexane) to stron hydrogenbonding (acetylene/N-methyl pyrrolidone). n each case, adjustable parameters were optimized by the Nelder-Mead (1965) simplex method to minimize the following objective function
4
where yA’(calcd) and yA’(exptl) are the calculated and experimental values of the apparent activity coefficient of acetylene. The yA’(expt1) values are calculated from the data as discussed by Harris and Prausnitz, whereas 7A’(calcd) is obtained via the expression (Prigogine and Defay, 1954)
The results are summarized in Table I, where the accuracy of the correlation is represented by the standard deviation of [yA’(calcd)- yA’(exptl)]aa a percentage of the mean value of yA’(exptl). The results for three other models are also shown for comparison: the two-parameter version of our model, i.e., one equilibrium constant and one binary interaction parameter, the Harris-Prausnitz two-parameter model, and the two parameter UNIQUAC model. The hexane system is a test of the ability of the models to represent physical interactions. In this case, the Har-
MODEL
MODEL
o’2!
A 0.10.0
0.2 0.4 0.6 0.8 ACETYLENE MOLE FRACTION
1.0
Figure 1. Activity coefficient of acetylene in acetylene/N-methyl pyrrolidone binary.
ris-Prausnitz model with its single adjustable binary-interaction parameter is less successfd than those with two such parameters, as might be expected. For the N-methyl pyrrolidone (NMP) system, the Harris-Prausnitz model with K = 4.05, CY = 4.83 gives a better representation than our model with K = 4.25, am = 377, and CYBA= -7.76 (cal/mol). This is shown in Table I and in Figure 1. We note, however, that Harris and Prausnitz took advantage of additional data in their scheme to reduce the number of adjustable parameters. In particular, they assumed that the molar volume of the AB complex was given by Vm = 0.75vA + VB,where the factor 0.75 was derived from mixture volumetric data. In our UNIFAC-based model, we assumed simple additivity of the ri and q istructural parameters for A and B to give the corresponding AB values. Although Harris and Prausnitz claim that the coefficient 0.75 was not critical, we found that if a value of 1.0 was used instead, the optimum values of their adjustable parameters changed to K = 4.41, CY = 8.24 and the standard deviation of the fit increased to 2.7%, i.e., essentially the same as for our model. Since the van Laar equation is a subset of UNIQUAC, it is likely that with a simple adjustment of the rm and qm values we could have achieved the same degree of accuracy as Harris and Prausnitz. Prausnitz et al. (1980) have, in fact, advocated adjustments in qcvalues for certain alcohols and for water in order to improve the UNIQUAC representation for those compounds. In general, however, allowance for adjustment of structural parameters complicates the parameter optimization process and is rarely justified by the quantity and accuracy of the available data. As shown in Table I and Figure 1, UNIQUAC does well for the hexane system, but does progressively worse as the degree of solvation increases. It seems clear that any of the solvation models will yield better results than UNIQUAC for solvating systems in which one can identify the important solvation reactions.
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
Table 11. Binary Interaction Parameters Determined from Independent Binary Equilibrium Data interaction parameters, K binary pair Alamine 336/water TO A/ wat er DIBK/water 2-ethyl-1-hexanol/water chloroform/water acetic acid/water amine/diluent (all combinations)
O1 ij
Application to Multicomponent Systems Multicomponent solvating solutions are generally more difficult to model than the binary systems described above. In multicomponent systems, the relative importance of "chemical" and "physical" effects can change much more dramatically as the composition of the solution varies. Hence, to be useful for multicomponent systems, a model must be able to describe a wide range of solution behavior with a minimum number of adjustable parameters. We have used the UNIFAC-based model described previously to correlate quaternary LLE data for the distribution of acetic acid between an aqueous phase and an extractant phase containing a tertiary amine, which is a strong organic base, and an organic diluent. Experimental data for equilibria at 19 "C were taken from Ricker (1978), Ricker et al. (19791, and Spala (1980). Three different amine/diluent combinations were studied: Alamine 336/diisobutyl ketone (DIBK), Alamine 336/2-ethyl-1hexanol, and trioctylamine (TOA)/chloroform. Alamine 336 is an industrial grade liquid ion-exchange reagent manufactured by General Mills; it has been used extensively in the purification of metals by liquid extraction. It is >96 w t % mixed tertiary amines, &N, with an average molecular weight of 406. The R groups are predominantly straight chains in the C8 to Clo range (Ricker, 1978). Alamine 336 and TOA were found to be very similar in their solubility in water and their ability to extract acetic acid. Infrared spectra for such systems (Barrow and Yerger, 1954; Smith and Vitoria, 1968; De Tar and Novak, 1970) indicate that the acid and amine solvate to form not only a strong 1:i acid-base complex as in eq 1, but also a 2:l complex. It is postulated that the additional acid molecule in the 2:l complex is hydrogen-bonded to the carbonyl oxygen of the acid in the 1:l complex. There is also evidence that additional acid molecules may hydrogen bond to the 2:l complex to form 3:l and higher-order complexes. We therefore allowed for the formation of 1:1,2:1, and 3:l complexes in the organic phase. The applicable solvation equilibria are K
A+B&AB
K1 =
-.-YAB YAYB
A
(11)
2.413
ZAZB
+ AB 2A2B
(12)
Y A ~ B ZA~B K2 = -.YAYAB ZAZAB
A
K3
+ A2B + ASB
(13)
ZA~B YAYA,B ZAZA,B where A represents an acetic acid molecule and B represents an amine molecule. Since K 2 and K , both represent K, =
-.-Y A ~ B
data ref
O1 ji
761.7 761.7 803.4 306.9 813.0 434.1 0.0
--45.6 --45.6 54.1 191.8 360.5 -218.6 0.0
Ricker (1978) Ricker (1978) Stephen (1964) Mellan (1950) Conti e t al. (1960) Sebastiani and Lacquaniti (1967)
...
hydrogen bonding to a carboxyl group, we assumed K z = K3 in order to reduce the number of adjustable parameters. We should note that while acetic acid is likely to dimerize in nonpolar solvents, the formation of dimers is less likely in the systems described here and would, in any case, be difficult to distinguish from the formation of 2:l and 3:l complexes. For a system with four functional groups (water, acid, amine, and diluent), UNIFAC requires a total of 12 group-interaction parameters to describe the physical interactions in solution. To the greatest extent possible, these were determined from independent measurements on the relevant binary systems; the results of this are shown in Table 11. The water/amine and water/diluent parameters were determined from mutual-solubility data. The water/acid parameters were based on the extensive VLE data of Sebastiani and Lacquaniti (1967). The parameters for the amine/diluent interaction were set to zero because data were lacking ahd this interaction was expected to be less important than the others. In our initial work, we also fixed the acid-diluent parameters based on binary data; this was found to cause problems in some cases, as discussed below. Two equilibrium constants, K1 and K2, and the two group-interaction parameters for the acid/amine binary were determined from the quaternary extraction data. These parameters were optimized as for the acetylene systems described previously, except that in this case the objective function was C[xA"(CalCd) - X~"(eXptl)],~ 1
where XAorefers to the apparent mole fraction of acid in the organic phase at equilibrium. We could have included the distributions of other components, (e.g., water) in the objective function, but chose the simpler form shown above because it was more sensitive to the parameters we were attempting to optimize. Calculational details are given by Spala (1980). Each of the three quaternary systems was studied individually; i.e., one set of optimized parameters was determined for each system. Typical results for the DIBK diluent are shown in Figure 2. The three curves in the figure correspond to model predictions for different amine/diluent ratios in the extractant phase. In all cases, the maximum acid concentrations correspond to a large excess of acid based on a 1:l acid-base stoichiometry. Agreement between the model and the experimental results is quite good over a wide range of amine/diluent ratios and acid concentrations, especially when one considers that the results are sensitive to the distribution of water, which the model predicts with parameters based on binary data alone. The optimized parameters and the resulting percent standard deviation in the organic phase acid mole fraction are shown in Table 111. Additional calculations were made to determine the importance of the 2:l and 3:l complexes. We found that these could be neglected at low acid:base ratios, but were
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 413
Table 111. Results of Parameter Optimization for Acid/Amine/Diluent Systemsa equilib const interaction param, K
a
~
sM % dev
K,
K2
"AB
OLBA
"AD
CVDA
Alamine 336/DIBK Alamine 336/2-ethyl-l-hexanol
5.9 34.6
24.5 1.9
184.8
146.4
310.7
311.5
TOA/chloroform
153.5 84.5
36.0
-32.2 -59.0 -82.5
in
xA'
0.9 1.1 0.9
1.6 360.2 199.4 92.5 Subscripts A, B, and D refer to acid, amine, and diluent, respectively. Equilibrium constants are defined in eq 11-13. 3.01
I
4 J 2.5
1
' 0
.25
.5
.75
1.0
1.25
MOLE X ACID IN AQUEOUS
1.5
PHS.
Figure 2. Equilibrium extraction of acetic acid from water for Alamine-336/DIBK extractant mixtures.
necessary for a good representation for acid:base ratios above 1.0. The same procedure was attempted for the other two systems, but the agreement between the model and the data was much less satisfying. It was eventually determined that the calculations for these systems were very sensitive to the two diluentlacid interaction parameters and that reasonable results could only be obtained when these were adjusted along with the other four parameters. Once this was done, the agreement was comparable to that obtained for the DIBK diluent. Overall results for all three systems are summarized in Table 111. Again, the standard deviations between the model predictions and the measured organic-phase acid mole fractions are about l %. Our problems with the use of acid/diluent parameters derived from binary data were perhaps to be expected. As noted by many authors including Prausnitz et al. (1980), binary parameters cannot be determined uniquely and those derived from VLE data can have large uncertainties. If such parameters are used in a situation where the calculations are more sensitive to the parameter values, as in many LLE systems, the results may be unsatisfactory. Prausnitz et al. (1980) suggest that ternary (or quaternary) LLE data be used in the determination of binary parameters for use in LLE. Our experience supports this advice. Prediction and Correlation of Distribution Coefficients In the design of an extraction process, the distribution coefficient, KD, is a primary concern. The acetic acid distribution coefficient data for the Alamine 336/DIBK system are shown in Figure 3 as a function of the concentrations of amine in the extractant phase and acid in the aqueous phase at equilibrium. Figure 4 is a similar plot for the Alamine 336/2-ethyl-l-hexanol system. The KD values shown in Figures 3 and 4 are defined as follows. KD = [equiv of acetic acid/kg of extractant phase] / [equiv of acetic acid/kg of aqueous phase]
'.%
20 40 60 80 100 VOL. % AMINE IN ORGANIC PHASE
Figure 3. Acetic acid distribution coefficient for Alamine-336/ DIBK extractant mixtures.
.01 WT.
.I Yo
I
IO
ACETIC ACID IN AQ. PHASE
Figure 4. Acetic acid distribution coefficient for Alamine-336/2ethyl-1-hexanol extractant mixtures.
The acid in the aqueous phase is assumed to exist entirely as "free" or uncomplexed acid because of the negligible solubility of the amine in the aqueous phase. In the extractant phase, however, we assume that the total equivalents of acid, which are determined by titration (Ricker et al., 1979),are attributable to acid in the various forms shown in eq 11-13. For a given acid activity in the aqueous phase, we will have at equilibrium the same activity of free acid in the extractant phase; the activities of the complex species will be such that the remaining equilibrium and material-balance constraints are satisfied. In general, KDwill increase with the degree of complex formation. Figures 3 and 4 illustrate some interesting features of the equilibria for acid/amine/diluent systems. For example, for a given activity of "free" acid (i.e., ZArA in the
414
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982
aqueous and organic phases), the KD value goes through a maximum with respect to amine concentration due to the combined effects of chemical and physical interactions in solution, as discussed by Ricker et al. (1979) and Spala (1980). Briefly, at low amine concentrations in a polar diluent, ym is relatively low and chemical effects dominate. Increasing the amine concentration shifts the equilibrium toward additional complex formation, hence increasing KD. As the concentration of amine continues to rise, however, ym begins to increase because the amine is nonpolar, and this eventually has an offsetting effect on KD. A related phenomenon is the effect of the free acid activity. In the DIBK diluent, the KD increases with increasing acid activity, while in the other two diluents the trend is reversed. A simple chemical theory of solution would suggest that for a given amine activity at low total-acid concentrations, the acid in the amine phase will be almost entirely complexed, i.e., ZAB >> Z A , yielding relatively high KD values. As the amount of acid in the amine phase increases, the ratio Z A : Z m increases, resulting in lower KDvalues. This is in agreement with the results for the 2-ethyl-1-hexanol and chloroform diluents. We postulate that the contrasting behavior for the DIBK diluent is due to its lack of a hydrogen-bonding capability, which makes it a less effective solvent for the AB complex than either chloroform or 2-ethyl-1-hexanol. Thus, in solvents like DIBK, the available uncomplexed acid may initially have a solvation effect on the complex, lowering the apparent value of ym and increasing the KD value. The above discussion may account for the large differences in the optimized equilibrium constants shown in Table I11 for the three systems. If the calculational model were an accurate representation of the phenomena occurring in solution, we would expect the thermodynamic equilibrium constants to be independent of the diluent. The optimized constants are very similar fot the two hydrogen-bonding diluents; the variation shown in Table I11 can be attributed to differences in the structures of Alamine 336 and TOA. Note also that K, >> K2 for these two systems, which seem reasonable since eq 11is a strong acid-base reaction whereas eq 12 and 13 represent weaker hydrogen-bonding associations. The picture for the DIBK diluent is, however, much different. As shown in Table 111, we need K2 > K1 to get a good representation of the data. Moreover, the results are insensitive to the values of the diluent/acid group-interaction parameters-again in contrast to the other two systems. These inconsistencies can be explained by the way the model handles the “physical” interactions between the acid and the AB complex. Let us reexamine the chemical equilibrium represented by eq 1 and 2 in a binary solution of A and B. The UNIFAC expressions for the activity coefficients are
+ In TAR In YB = In y B c + In yBR In ym = In ?ABC + In y A B R In
YA
= In
(14)
yAc
(15) (16)
If A and B are d e f i e d as functional groups, Spala (1980) shows that the UNIFAC expression for y A B R reduces to
In ymR= ln
y~~
+ In y~~ - In y
~ (- In ~ y)
~
(
~
(17) ~ )
where the last two terms are the activity coefficients of A and B at kfinite dilution in AB. We now rewrite the expressioii for K, in eq 2 as In K , = In TAB - In YA - In YB (18) combining eq 14-18 gives
In K, = In
yABC
+ In y A c + In yBc - lnyA(AB)- In yB(AB) (19)
The last two terms in (19) are functions of the group-interaction parameters but are independent of concentration. The combinatorial terms are independent of the groupinteraction parameters and are functions of concentration. We have found, however, that for the acetic acid/amine systems the combinatorial terms are only weak functions of concentration so that the predicted values of K, are essentially independent of acid concentration until the acid concentration reaches about 20 mol 9%. Therefore, if there is an important “physical” interaction between the A and A 3 species, and if the importance of this interaction varies with the concentration of A, the simple model involving the single solvation reaction (1)will give a poor representation of the data. In such cases, we need to allow explicitly for additional specific chemical interactions, as in reactions 12 and 13. If, on the other hand, the important physical interaction is between the diluent, C, and the AB complexes, this can be reflected through the A-C group-interaction parameters, since these exhibit much stronger concentration-dependent effects on K,. The unfortunate conclusion is that the optimized equilibrium constants obtained from the present model will be sensitive to the character of the interactions in solution and to whether one chooses to model a particular interaction as a “chemical” type using equilibrium constants, or as a “physical”type using empirical interaction parameters. Unless one can actually measure the concentration of each species in solution, it is impossible to tell with certainty whether a particular representation is thermodynamically correct. Still, the calculational method presented here provides a useful framework for the correlation of data for industrial applications. In that regard, it is a significant improvement over models that have been proposed in the past for use with acid/amine/diluent systems. Sakai et al. (1969), for example, proposed a “chemical” model involving five reaction equilibria and six adjustable parameters. The model was successful in the limited range of their experiments, but it was incapable of predicting the more complicated phenomena shown in Figures 3 and 4 of the present work.
Conclusions The example applications described above show that the present calculational model can represent a wide range of phase-equilibrium behavior. In solutions where nonidealities are mainly due to nonspecific “physical” effects (e.g., hexane/acetylene), the UNIQUAC/UNIFAC equations ensure a good representation. Moreover, we can provide for simultaneous solvation effects, where necessary. The calculational framework is flexible and can be extended easily to multicomponent solutions with an arbitrary number of solvation equilibria although one should try to minimize the number of adjustable parameters. For most binary systems, the two-parameter version of the present model performed at least as well as that of Harris and Prausnitz (1969). As discussed previously, Harris and Prausnitz minimized the number of adjustable parameters in their model by using data that might not be available in general. Moreover, our three-parameter model would appear to offer signficant advantages for systems in which “physical”effects dominate. Finally, the UNIFAC-based approach used here lends itself naturally to the study of homologous series. Once tbe solvation equilibrium constants have been determined for given A and B species, it should be possible to predict the equilibria
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 3, 1982 415
for chemically similar species with a minimum of additional experimentation. This might be advantageous, for example, in the selection of the optimal absorbent for an absorption/stripping process. For the quaternary acetic acid/amine/diluent/water systems, the present model gives a much better representation of the experimental data over a wide range of composition than the six-parameter model of Sakai et al. (1969), which fails to represent several important qualitative features of these systems. We have found, however, that the sometimes arbitrary distinction the present model makes between "chemical" and "physical" nonidealities limits the general utility of the method. Specifically, we found that the optimized equilibrium constants obtained from the quaternary extraction data varied dramatically with the chemical nature of the diluent and hence were not the true thermodynamic values. This implies that the optimized equilibrium constants obtained for a certain acid/amine pair dissolved in a particular diluent would not, in general, represent the data for the same acidlamine pair dissolved in another diluent, especially if the two diluents were chemically dissimilar. The value of K determined for the N-methyl pyrrolidone/acetylene system was also sensitive to small changes in pure-component volumetric and structural parameters. Nevertheless, the present model is a powerful tool for the correlation of data for a particular system as long as the appropriate solvation reactions are included explicitly. The chemical association model typified by the work of Nagata (1977) could also be used to correlate the data for the quaternary systems studied in the present work. We see no a priori reason for the use of the association model in preference to the model presented here, but we have not made a quantitative comparison of the two models. Nomenclature uL= activity of molecular species i g = molal Gibbs free energy g:&m = "chemical" contribution to g ph = "physical" contribution to g K , = thermodynamic equilibrium constants for solvation and association K , = ratio of activity coefficients, defined in eq 2 Xi= apparent mole fraction of molecular species i ; related to the true mole fraction by eq 3
2,
P
Xio = apparent mole fraction of i in the organic phase Zi = true mole fraction of molecular species i CY = adjustable interaction parameter in Harris-Prausnitz (1969) model aij, aji = UNIFAC groupinteractionparameters for the binary pair i-j (note, a i j # CY$ y i = true activity coefficient of molecular species i ti= normalized extent of solvation, defined in eq 4 y/ = apparent activity coefficient for molecular species i; related to true activity coefficient by eq 10 Literature Cited Abrams, D. S.; Prausnitz, J. M. AICHf J., 1975, 2 1 , 62. Barrow, G. M.; Yerger, E. A. J. Am. Chem. SOC., 1954, 76, 5211. Chen, S A . ; Bagley, E. B. Chem. f n g . Sci., 1978, 3 3 , 161. Conti, J. J.; Othmer, D. F.; Giimont, R. J. Chem. Eng. Dafa 1960, 5(3), 301. DeTar, D. F.; Novak, R. W. J. Am. Chem. SOC. 1970, 9 2 , 1361. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. A I C M J. 1975, 27, 1066. Harris, H. G.; Prausnitz, J. M. Ind. fng. Chem. Fundam. 1969, 8 , 180. Kretschmer, C. B.; Wiebe, R. J. Chem. phvs. 1954, 2 2 , 1697. Meiian, I. "Industrial Solvents", 2nd ed.; Rienhoid: New York, 1950. Nagata, I. 2.phvs. Chem. (Le/&) 1973, 252, 305. Nagata, I . FIuM Phase Equilib. 1977, 1 , 93. Nagata, I.; Kawamura, Y. FluidPhase fquilib. 1979a, 3 , 1. Nagata, I.; Kawamura, Y. Chem. f n g . Sci. 1979b, 3 4 , 601. Nelder, J. A.; Mead, R. Comput. J. 1965, 7 , 308. Prausnitz, J. M. "Molecular Thermodynamics of Fluid-Phase Equilibria"; Prentice-Hail: Engelwood Cliffs. NJ, 1969; Chapter 7. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A,; Hsieh, R.; 0'Conneli, J. P. "Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Eguiiibrla", Prentice-Hail, Engelwood Cliffs, NJ, 1980; Chapters 4, 6; Prigogine, I.; Defay, R. "Chemical Thermodynamics"; Wiiey: New York, 1954; p 410. Renon, H.; Prausnitz, J. M. Chem. €ng. Sci. 1987, 2 2 , 299, 1891. Ricker, N. L. Ph.D. Dissertation, University of California, Berkeley. 1978. Ricker, N. L.; King, C. J. "Solvent Extraction of Wastewaters from Acetic Acid Manufacture". EPA Report 600/2-80-064, April 1980a. Ricker, N. L.; Michaels, J. N.; King, C. J. J. Sep. Process Techno/. 1979, 7 , 36. Ricker, N. L.; Pittman, E. F.; King, C. J. J. Sep. Process Techno/. l98Ob 1(2), 23. Sakai, W.; Nakashio, F.; Inoue, K. Kegaku Kogeku 1969, 3 3 , 1243. Sebastiani, E.; Lacquaniti, L. Chem. f n g . Sci. 1967, 2 2 , 1155. Smith, J. W.; Vitoria, M. C. J. Chem. SOC.A 1968. 2468. Spaia, E. E. M.S. Thesis, University of Washington, Seattle, 1980. Stephen, H.; Stephen, T., Ed. "Solubilities of Organic and Inorganic Compounds"; Pergamon Press: New York, 1964. Wardeii, J. M.; King, C. F. J. Chem. Eng. Dafa 1978, 2 3 , 146.
Received for review January 19, 1981 Reuised manuscript receiued December 14, 1981 Accepted January 14, 1981
The authors are grateful for the support of the Weyerhauser Company and the Graduate School Research Fund and Academic Computer Center of the University of Washington.
