(14) Hildebrand, J. H., Scott, R. L., “Solubility of Non-Electrolytes,” 3rd ed., Reinhold, New York, 1950. (15) Janz, G. J., “Estimation of Thermodynamic Properties of Orpanic ComDounds. “Academic Press. New York. 1958. (16) karasz, F.’E., Bair, H. E., O’Reilly, J. M., j. Phys. Chem. 69, 2657 (1965). (17) Lagemann, R. T., J . Chem. Phys. 17, 369 (1949). (18) Lewis, G. N., Randall, M., “Thermodynamics,” 2nd ed., rev. by K. S. Pitzer and Leo Brewer, McGraw-Hill, New York, 1961. (19) Lielmesz, J., Bondi, A., Chem. Eng. Sci.20, 706 (1965). (20) Moelwyn-Hughes, E. A,, “States of Matter,” Cambridge, 1961. (21) Pals, D. T. F., Delft, T. N. O., private communication, 1960. (22) Pessaglia, E., Kevorkian, R., J . Appl. Phys. 34, 90 (1963). (23) Pitzer, K. S., J . Am. Chem. Sac. 61, 331 (1939). (24) Pitzer,K. S., J . Chem. Phys. 8,711 (1940). (25) Pitzer, K. S., “Quantum Chemistry,” Prentice-Hall, New York. 1953. (26) PGzer, K. S., Gwinn, W. D., J . Chem. Phys. 10, 428 (1942); 16. 303 - ~ (1948). - .(27) Prigogine, I:, Raulier, S., Physica 9, 396 (1942). (28) Reid, R. C., Sherwood, T. K., “Physical Properties of Liquids and Gases,” McGraw-Hill, New York, 1958, 1965. (29) Riedel, L., Chem. Zng. Tech. 26,83, 259, 679 (1954). (30) Piedel, L., in “Kaltetechnologie,” by R. Planck, Vol. IV, Springer, Berlin. (31) Rihani, D. N., Doraiswamy, L. K., IND.ENG.CHEM.FUNDAMENTALS 4, 17 (1965). (32) Rowlinson, J. R., “Liquids and Liquid Mixtures,” Academic Press, New York, 1959. (33) Wilski, H., KunJtstofe 50, 281, 335 (1960). (34) Wood, L. A,, Martin, G. M., J. Res. Natl. Bur. Std. 68A, 259 (1964). RECEIVED for review June 6, 1966 ACCEPTED July 20, 1966
5cRT/Eo 5cRTp/Eo sound velocity, cm./sec. molal volume of liquid, cc./mole van der Waals volume ( 5 ) ,cc./mole height of potential energy barrier, cal./mole or units of R T concentration, mole fraction number of nearest neighbors per molecule cubic thermal expansion coefficient, K.-l deviation factor pair potenti,al ( L J D ) , ergs/molecule E0/5cR or L1,0/5c,R accentric factor (78) References
(1) Allen, G., Gee, G., Mangaraj, D., Sums, D., Wilson, G. J.. Polymer 1 , 467 (1960). (2) American Petroleurn Institute, Research Project 44, tables, Selected Properties of .Hydrocarbons. (3) Bondi, A., A.Z.Ch.E. J . 8, 610 (1962). (4) Bondi, A., J . Phys. Chem. 58, 929 (1954). (5) Ibid., 68, 441 (1964). ( 6 ) Zbid., 70, 530 (1966). (7) Bondi, A,, J . Polymer Sci. A2, 3159 (1964). (8) Bondi, A., Simkin, D. J., A.Z.Ch.E. J . 6, 191 (1960). (9) Brown, 0. L. I., J . A m . Chem. SOC.74, 6096 (1952). (10) Curl, R. C., Pitzer, K. S., Zbid., 77, 3433 (1955). (11) Halford, R. S., J . Chem. Phys. 8, 496 (1940). (12) Hellwege, K. H., Knappe, W., Lehmann, P., Kolloid Z. 183, No. 2, 110 (1962). (13) Hellwege, K. H., Knappe, W., Wetzel, W., Zbid., 180, 126 (1962).
I
\ - -
THERMODYNAMICS OF TERNARY, LIQUIDSUPERCRITICAL GAS SYSTEMS WITH APPLICATIONS FOR HIGH PRESSURE VAPOR EXTRACTION J.
R. BALDER AND J.M. PRAUSNITZ
Department of Chemical Engineering, University of California, Berkeley, Calif.
The phase behavior of a ternary system consisting of two liquids and a supercritical gas is predicted b y thermodynamic considerations employing only binary data. The thermodynamic properties of the liquid phases are described by two-suffix Margules equations with activity coefficients normalized by the unsymmetric convention. The vapor phase is treated as a nonideal gas consisting only of the supercritical component. An iterative numerical technique is used to solve the resulting equations of phase equilibria. This approach yields good qualitative results for a variety of ternary phase diagrams, some of which appear to have applicability for extractive separation processes wherein a supercritical gas is the selective solvent. Finally, criteria are developed for choosing a suitable supercritical gas for efficient separation.
use of a compressed gas above its critical temperature the separation of icomponents in a binary liquid solution has been suggested by several investigators and most notably by Elgin and Weinstoclc (7). Such a separation process is based on the ability of the gaseous component to introduce a n immiscible region in the ternary system; the major technical advantage of such a process is the ease with which the solvent may be recovered. The types of systems for which this process might find application are difficult to envision because of the HE
Tfor
limited number of experimental studies. Thermodynamic considerations, however, permit qualitative prediction of ternary phase behavior using only binary solution data. Experimental work can thus be reserved for those systems which, after preliminary thermodynamic analysis, look promising for industrial purposes. Figure 1 illustrates two types of phase diagrams being considered here. I n the following development, the qualitative features of such diagrams are discussed from a thermodynamic VOL. 5
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449
LIQUID I
Equations 2 and 3 are substituted into the stability criteria for a binary system (Equation I), it can be verified that a single phase will exist over the entire composition range if a12 < 2. The condition for the formation of a liquid-liquid miscibility gap, then, is ff12
>2
(5)
The effect of using additional terms in the Margules equations for the activity coefficients has been presented by Shain (8). For a ternary system, the stability of a single phase at constant temperature and pressure requires that LIQUID I
(*)
>o
(7)
n1,nt
viewpoint. The liquid phases are described by expressions for the activity coefficients containing binary Margales parameters. The vapor phase is treated as a nonideal gas consisting only of the light component (supercritical gas). The approximate behavior of the ternary system is then predicted using the equations of phase equilibria.
A two-phase region occurs if any one of these inequalities is not satisfied. Substitution of expressions for the activity coefficients, unfortunately, does not yield a simple statement of the ternary stability criteria in terms of the Margules constants analogous to Equatibn 5 . I t is important to note the distinction between stability criteria and two-phase equilibrium criteria. Stability criteria specify whether any region of immiscibility exists; however, they do not determine the compositions of the equilibrium liquid phases. This is illustrated in Figure 2. The equality of Equation 8 determifies the spinodal curve. The equilibrium liquid phases and the boundary for the two-phase region are given by the coexistence curve. For the construction of a ternary phase diagram, then, it is the coexistence curve, not the spinodal curve, which must be found.
Stability Criteria: Spinodal Curve
Equilibrium Criteria: Coexistence Curve
The conditions which determine the formation of the twophase liquid-liquid region are provided by the thermodynamic stability criteria. The derivation of these criteria may be found in various references such as Prigogine and Defay (7). For a binary system a t constant temperature and pressure, a single liquid phase can exist only if
In the two-phase region the two liquid phases must satisfy the conditions for thermodynamic equilibrium :
LlOUlD 3
GAS
2
Figure 1. Two types of ternary systems containing a supercritical gas
=
=
PP
+ R T In
ytxt
450
In yl =
(~12x2~
(3)
In yz =
( ~ 1 ~ x 1 ~
(4)
IbEC FUNDAMENTALS
(9)
= xfr'yr]'
(i
= 1, 2, 3)
If
(10)
T o extend this treatment to the ternary system in which one of the components exists as a gas as a pure component, the adjusted activity coefficients (6) are defined using the unsymmetric convention employed by O'Connell ( 5 ) . Subscripts 1
(2)
where pi0 is some reference chemical potential, yt is the activity coefficient, and x i is the mole fraction of the ith component, all a t temperature T . If the reference chemical potential is taken as that of the pure liquid, the activity coefficients may, to a first approximation, be expressed by Margules equations,
where a12 is the Margules constant for the 1-2 binary.
(i = 1, 2, 3)
where superscripts I and I1 denote the two liquid phases. If the same reference chemical potentials are used in phase I and phase 11, the set of Equations 9 may be rewritten with the aid of Equation 2 in the form xi'yi'
where is the chemical potential and nl is the number of moles of component 1. If this inequality is not satisfied, two liquid phases with dissimilar compositions form. T o express the above relationship in terms of the more useful parameters of temperature, pressure, and composition, it is necessary to employ a solution theory. From the definition of the activity coefficient,
Pp
P = PLAIT POINT COEXISTENCE S P = SPI NODAL
co =
32Figure 2.
CONSTANT
Stability (spinodal) curve and equilibrium (coexistence) curve in a ternary system
and 3 refer to heavy (liquid) components; subscript 2 refers to the light (gaseous) component.
where tjl. stands for the partial molar volume of the ith component. Equation 23 gives the effect of pressure on the two phases in equilibrium and hence the pressure dependence of the coexistence curve. For the present work, however, it is assumed that the partial molar volume is independent of composition, in which case &I =
&I1
and (24) [x,y*(P1S)]I
where fl denotes the fugacity and 81 the partial molar volume of component 1. The solution temperature is T and the total pressure is P. PlS represents the saturation pressure of pure component 1 a t temperature T . The reference fugacity, f 1 o ( P 1 ~ )is , the fugacity of pure liquid 1 a t temperature T and pressure PIs, andf30(PjS)is the fugacity of pure liquid 3 a t temperature T and pressure Pis. The reference fugacity, HzS,i,is Henry's constant for solute 2 in solvent 1 defined by
Hz*',l
lim X1-0
e)
(14)
= [xiyiP1s)]"
(i
=
1, 2, 3)
Under this particular assumption, then, the equilibrium conditions (Equation 24) are independent of the total pressure. The coexistence curve is now considered as the envelope of all the tie lines which connect two equilibrium phases. The six concentrations required for any tie line are calculated using the numerical iteration technique suggested by Hennico and Vermeulen ( 3 ) . The composition xll was chosen as the independent parameter. The set of Equations 24 together with the two material balance equations are solved for the remaining five compositions.
Simple expressions for the unsymmetric activity coefficients in the ternary system are developed by first assuming that the molar excess free energy in the symmetric convention is
where g123E(sym) is the molar excess free energy and the aij's are the Margules constants for the i-j binary referred to the reference pressure, P1'. The activity coefficients (symmetric convention) are related to the molar excess free energy by
where ni denotes the number of moles of component i and
nT
=
ni. I
I t has been shown ( 5 ) that the activity coefficients in the symmetric and unsymmetric conventions are related in the following way : y 1 (unsymmetric) = y1 (symmetric) y2* (unsymmetric) = yz (symmetric) exp (-aiz) y 3 (unsymmetric) =
y3
(symmetric)
(17) (18)
(19)
From Equations 15 through 19 the final forms for the unsymmetric activity coefficientsare obtained. In yl = In
y2*
In
7 3
alzxz2
+
- I)
=
a12(x12
=
(~1~x12
+
+
(Y13x3'
+
cY23x32
az:Ixz2
+
(a12
+ (a13
+
(a23
+
- (Y23)xZX3 a i 2 - (YlI)XlX3
a13
+
a23
-
ai2)xixz
(20) (21)
An arbitrary set of concentrations is assumed and used to evaluate the activity coefficientsfrom Equations 20, 21, and 22. Solving the set of Equations 25 yields a new set of concentrations which are used to calculate a new set of activity coefficients. Repeated iteration gives the five concentrations to any desired accuracy. By incrementing the value of x1I, the entire coexistence curve may be calculated. This procedure is well suited for programming on a digital computer. Vapor Phase: Solubility Curve
In the above discussion of the coexistence curve it has been tacitly assumed that liquid phases can exist for any given composition. However, since component 2 is a supercritical gas, the physically allowable liquid compositions are limited by the solubility of the gas in the binary solvent solution. As the pressure is increased the solubility of the gas increases, thus enlarging the region of allowable liquid phases. This is illustrated in Figure 3. The effect of this limited solubility on the coexistence curve will become evident when the construction of the final phase diagrams is discussed.
(22)
I
I
The equilibrium criteria given by Equation 10 are valid at any solution pressure, whereas Equations 20, 21, and 22 have been referred to pressure PI'. Rewriting Equation 10 so that the activity coefficients are based on pressure P18,
(i
=
1, 2, 3)
Figure 3. solvent
Effect of pressure on gas solubility in a mixed
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The existence of a gas phase requires an additional equilibrium condition. If solvents 1 and 3 are relatively nonvolatile, the gas phase may be assumed to consist of only component 2. At equilibrium f z G = fz" (26) where f z G is the fugacity of component 2 in the gas phase and fzL is the fugacity in the liquid phase. Substituting Equation 12 into Equation 26
Assuming that the partial molar volume, pressure, Equation 27 becomes
02,
i
A
SOLUBILITY CURVE COEXISTENCE C U R V E
YIELDS
3
2
is independent of
3
2
YIELDS 3
By applying a numerical iteration technique (similar to the one described above) to Equation 28 and to the material balance equation, lines of constant fugacity of component 2 in the liquid phase may be calculated. These are the solubility curves of gaseous component 2 in the binary liquid solvent at various pressures. Parameters used to construct Figures 4, 5, and 6 are given in Table I.
Figure 4.
Phase diagram from computer results A. No two-liquid region B . Two-liquid region C. No vapor region
Construction of Phase Diagrams
Once the solubility and coexistence curves have been determined, the complete ternary phase diagram may be constructed. This procedure is illustrated by example in Figure 4. The following parameters have been used : a12 0.5 02 = 60 cc./gram mole 0.13 = 1.0 H;,, = 1500 p.s.i.a. a23 = 4.0 T = 15' C. f z G = P0.93 where P has units of p.s.i.a. Depending on the value of the total pressure, the following cases can arise : CASEI. At sufficiently low pressures the solubility curve does not intersect the coexistence curve. In this case the gas solubility is too low for immiscibility to occur, since the coexistence curve describes only liquid-phase behavior. Stated in another way, the points on the coexistence curve are not allowed because fugacity fzL on this curve exceeds the prescribed gas-phase value, f 2 O . The ternary phase diagram, then, consists only of the solubility curve as shown in Figure 4A. CASE11. Raising the pressure increases the solubility of the gas such that the solubility curve intersects a part of the coexistence curve. The stability criteria do not allow the existence of a single phase inside the coexistence curve; therefore, a liquid-liquid region and a vapor-liquid-liquid region are formed. This is illustrated in Figure 4B.
Table 1.
Parameters Used to Construct Figures 4, 5, and 9 82,
Hz,l, P, Figure a23 ff13 a12 P.S.I.A. P.S.I.A. 4A 4.00 1.00 0.50 1500 400 B 4.00 1.00 0.50 1500 700 C 4.00 1.00 0 . 5 0 1500 1900 715 5.20 1.80 0.90 5A 780 1.00 1850 5.20 0.30 B 515 2.25 0.70 1400 C 5.20 9 A 4.00 0.50 0.20 1500 1400 B 4.00 1.80 1.50 1500 350 a Pressure greater than critical solution pressure.
452
l&EC FUNDAMENTALS
T, ' C. 15 15 15 15 15 15
15 15
cc./ Gram Mole 60 60 60 65 65 65 60 60
CASE111. As the pressure is increased still further, the solubility curve intersects larger liquid-liquid regions until the critical solution pressure of the system has been reached. Above this critical pressure no vapor phase exists and the phase diagram consists of only the coexistence curve as shown in Figure 4C. The methods developed in this article are compared in Figure 5 with several ternary systems studied experimentally by Elgin and Weinstock (7). Quantitatively, the agreement is not good, but this is to be expected, since the simple expressions for the activity coefficients based on only a total of three binary Margules constants cannot accurately describe highly polar systems. More significant, however, is the excellent qualitative agreement with respect to the presence of multiple-phase regions and the proper slope of the tie lines. It is precisely this qualitative picture which is necessary to determine the possible industrial applicability of a particular system. Evaluation of Margules Constants
Certain experimental data are required to evaluate the binary Margules constants. The parameter al2 is obtained from solubility data of the gas in pure solvent 1 such as that presented in Figure 6. Todd and Elgin (70) and Chappelear and Elgin ( 2 ) have reported such data for ethylene and chlorotrifluoromethane in organic liquids. For the 1-2 binary, x3 = 0 and Equation 21 becomes In yz* =
a12(x12
- 1)
(29)
The activity coefficient, YZ*, is found from Equation 12 and from the solubility data. The value of cy12 is then chosen which best represents these data by means of Equation 29. The simplified one-parameter expression does not account for the binary liquid-liquid regions in the solubility data which have been observed in some cases by Todd and Elgin; therefore, only the vapor-liquid portions should be used in the calculations.
N -PROPYL A LCO H 0 L,
N - PROPYL ALCOHOL
where 61 designates the solubility parameter and V I the liquid molar volume of component 1. T h e solubility parameters and “liquid” molar volumes for several common gases have been given by Shair ( 9 ) .
ACETIC ACID
Effect of Margules Constants on Phase Behavior
-
Since industrial separation processes operate in the L1 LZ region, it is important to determine how the Margules parameters affect the shape of the coexistence curve and the slope of the tie lines. I n order for any liquid-liquid region to exist, a t least one of the binary Margules constants must have a value greater than 2 (only positive values are being considered here) ; this is a consequence of the stability criteria given by Equation 5. Assume for illustrative purposes that the 2-3 binary satisfies this condition-that is,
METHYL ETHYL KETONE
METHYL ETH’YL KETONE
C
- LfV
,-
L LfV
ff23
H20
‘ZH4
Figure 5. Comparison of computed (left) and experimental (right) phase diagrams Experimental data from ( 7 )
‘..--
II
t
/SLOPE
FUGACITY
COMPONENT OF 2
-- H ~ , ,
CURVATURE
$
I x2 Figure 6. Binary solubility curve for Henry’s and Margules Iconstants
01
As shown by O’Connell, CY23 is given by
>2
(33)
The effect of increasing the value of 0123 is to increase the area enclosed by the liquid-liquid region, thereby providing greater separation a t a given over-all composition. This is demonstrated by Figure 7 in which, for illustrative purposes, the Margules constants for the 1-2 and 1-3 binaries have been set equal to zero. If the 1-2 and 1-3 Margules parameters are less than 2, their primary effect is to determine the slope of the tie lines. Three possible cases can occur. CASEI. CY13 = 0112 (Figure 8 A ) . The tie lines are very nearly parallel to the 2-3 binary base line; they are not exactly parallel owing to computational approximations in specifying the activity coefficients. CASE 11. 0113 > a12 (Figure 8B). The tie lines slope toward the 1-3 binary line. This could have been intuitively predicted by considering the limiting case of a n immiscibility band across the phase diagram as shown in Figure 8, C. Of necessity the tie lines become parallel to either the 1-3 or the 2-3 binary lines in the limit of pure 1-3 binary or pure 2-3 binary, respectively. CASE111. 0113 < cy12 (Figure 80,. Following consistently the explanation of Case 11, the tie lines slope toward the 1-2 binary line. I n general, then, the effect of the relative values of the 1-2 and 1-3 Margules parameters is to cause the tie lines to slope toward the least ideal of these two binaries. The absolute value of a12 affects the total pressure required to achieve any specified degree of separation. This is illustrated
where H2,3(PlS)is the Henry’s constant for component 2 in pure solvent 3 a t the saturation pressure of component 1. Thus it is necessary to have complete solubility data-Le., for solute concentrations exceeding the Henry’s law region-for only one of the gas-liquid pairs. The parameter 0113 is found by standard methods from vaporliquid equilibrium data for the binary solution of solvents 1 and 3. For this case Equations 20 and 22 reduce to In y1 =
ff13%32
(204
In
(Y13%1’
(224
73
=
If gas solubility or binary solvent data are not available, the Margules parameters at2 and ff13 may be estimated for nonpolar systems from solubility parameter theory. Using a simplified form of Hildelbrand’s equation (4,
Figure 7. Effect of 2-3 Margules constant on coexistence curve Systems 1-2 and 1-3 ideal System 2-3 highly nonideal VOL. 5
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453
and if the gas is readily soluble. In terms of the Margules parameters and Henry’s constant, these favorable criteria are: 1.
T o form an immiscible region : ff23
>2
2. T o obtain the proper slope of the tie lines, the 1-3 binary should be more nonideal than the 1-2 binary. Cy13
I
1
> ff12
3. T o minimize the pressure requirement: H2,1 is small (gas 2 is readily soluble in liquid 1). a12 is large and positive (1-2 binary is highly nonideal with positive deviations from Raoult‘s law).
The n-propyl alcohol-water-ethylene Figure 5A, satisfies these criteria.
system shown in
Conclusions Figure 8. Effect of 1-2 and 1-3 Margules constants on phase behavior A. 6.
0112 Dl12
c.
0112
D.
ai2
1.0, 0113 1.0, 0123 = 5.0 = 0.5, 0113 = 1.5, 0123 = 5.0 2, 0113 > 2, 0123 > 2 = 1.5, ai3 = 0.5, 0123 = 5.0
