582
Ind. Eng.
Chem. Process Des. Dev. 1983, 22, 582-588
t = top of column m = limiting value observed Literature Cited
Kolkna, E.; Unno, H.; Sato, Y.; Chkla. T.; Imai, H.; E&, K.; I ~ 1.; K&, Y13a d16 , J.; KaJi.H.; Nekanlshi, H.; Y a m a m , K. J . Cbem. Eng. 1960,
m.
_,
Tadaki. T.; Maeda, S. Kamku Kspaku 1963,27, 803. Takahashi, R.; Washimi, K. Pet. Dev. 11176,93. Towell. G. D.; Strand, C. P.; Ackerman, G. H. “Preprint of AIChE-Instn. Ch. E. Joint Meeting”; London, June 1965. Ueyama, K.; Miyauchi, T. Kagaku Kogaku Ronbunshu 1977. 3 , 19. Ueyama, K.; Miyauchi. T. A I C M J . 1979.25, 258. Ueyama. K.; Morooka, S.;Koide, K.; KaJi, H.; Miyauchi, T. Id. f n g . Chem. kocess Des. Dev. 1960. 19, 592. Vahtln, F. H. H. “Absorptbn in Qas-Liquid Dispersbns”; E. B F. N. Spon Ltd.: London, 1967; p 52.
Akita, K.; Yoshlda, F. Ind. Eng. Chem. Process D e s . Dev. 1973, 12, 76. Akita, K.; Yoshlda, F. Ind. fng. Chem. procesS Des. Dev. 1974, 13, 64. Calderbenk, D. H.; Moo-Young, M. B. Chem. Eng. &/. 1961, 16, 39. Fair, J. R. Chem. Eng. 1967, 74(14), 67. Higbie, R. Trans. A I C M 1935,3 1 , 365. Jackson, M. L.; Shen, C. C. A I C M J . 1978, 24, 63. Kataoka, H.; TakeucM, H.; Nakao, K.; Yagi, H.; Tadaki. T.; Otake, T.; Miyaw chi, T.; Washimi, K.; Watanabe, K.; Yoshkla, F. J . Chem. Eng. Jpn. 1979, 12, 105. Kato. Y.; Nishiwaki. A. Kaflku Kogaku 1971,35. 912. Kawagoe, M.; Nakao, K.; Otake, T. J . Chem. fng.Jpn. 197b,8 . 254. Kdde, K.; Morooka, S.; Ueyama, K.; Matswra. A.; Yamashlta. F.; Iwamoto, S.; Kato, Y.; Inoue, H.; Shigeta, M.; Suruki, S.; Akehata, T. J . Chem. Eng. Jpn. 1979, 12, 96.
Received for review December 29, 1981 Revised manuscript received February 4,1983 Accepted February 28, 1983
Correlatfon and Prediction of Solid-Supercritlcai Fluid Phase Equilibria David H. Zlger and Charles A. Eckert’ Department of Chemlal Englneedng, University of Illlnols, Urbana, Illinois 6 180 1
A semiempirical correlation has been developed to describe quantitatively the solubility of heavy nonpolar solis in slightly subcritical and supercritical solvents using only one temperatusindependent pure component parameter for each component. The method, based partly on regular sobtion theory and the van der Waats equatlon of state, correlates available solubility data for 24 binary solld-fluid systems within an average standard deviatlan of 25 %
with ethylene, ethane, and carbon dioxide as s o h ” . The range of solubility correlated was lo-’ to lo-* mole fraction, 20-80 “C,and 80-500 bar. Pure-component parameters for solutes are presented for naphthalene, anthracene, triphenylmethane, phenanthrene, fluorene, pyrene, 2,3-and 2,6dlmethylnaphthalene, biphenyl, hexamethylbenzene, and hexachloroethane along with solvent parameters for ethylene, ethane, and carbon dioxide.
Introduction Although supercritical fluids have been noted as excellent solvents of heavy hydrocarbon solids for nearly a century (Hannay and Hogarth, 1879, Biichner, 1906), a quantitative understanding of the underlying phenomena has eluded investigators. The obvious difficuIty researchers encounter when modeling two-component solid-fluid equilibrium is adequately describing complex, usually asymmetric, interactions present in the lighter phase. Applications taking advantage of solubility enhancements of 103-108 are numerous and varied because supercritical fluid extraction often has several advantages over more conventional liquid-liquid extraction. The most important of these is the relative ease in which a solute may be cleanly loaded and unloaded in a separation process when a supercritical fluid is used as the soIvent. This is possible since the solvating power of a supercritical fluid is directly related to its relative density. By using a fluid whose critical point is readily accessible, a lighter solvent phase may be heavily loaded as the fluid is in a very dense, supercritical state. By slightly manipulating process conditions downstream below the solvent’s critical point (e.g., lowering the pressure), the solvent’s density and solvating power are decreased dramatically, causing the vast bulk of solute to be precipitated and collected easily. If the chosen solvent’s critical point is near ambient temperature, large thermal savings are normally realized over conventional extraction methods since an energy-intensive 0198-4305/83/1122-0582$01.50/0
distillation operation is avoided. Final considerations are the low cost, low viscosity, and low toxicity of useful supercritical fluids compared to ordinary organic liquid solvents. The coal, food, and oil industries, in particular, have found supercritical fluid extraction either profitable or promising (Paulaitis et al., 1982). Perhaps the greatest liability limiting the use of supercritical fluids has been the lack of a reliable correlation for existing sold-fluid equilibrium data. Due to the complicated nature of the supercritical phase, proposed thermodynamic analyses usually involve the use of one or more temperature dependent solutesolvent cross parameters which are difficult to correlate generally or are applicable only to a few well-studied systems. In this paper, a quantitative semitheoretical corresponding states treatment will be discussed which involves only two correlated temperature-independentpure-component parameters. Theory 1. Background. In general, the light component is virtually insoluble in the solid phase, for which the fugacity of the heavy component is given by its pressure (except for solid COP,where 42sis needed) modified by a Poynting correction.
If the ratio between the observed solubilities to that pre0
1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
dicted by the ideal gas law is defined as the enhancement factor
Equation 1 can be further simplified as
by chromatography. Furthermore, Giddings observed that solubility of solids in dense gases should be related to a solute chemical nature, 62L and a relative solvent state, A. 2. Present Model. A modified version of eq 6 can be derived from direct application of van der Waals (vdW) equation of state to obtain an expression for In & for dilute mixtures. u b2 2(a,~~)l/~ In = In - --In z (7) V-b U-b uRT As noted by Giddings according to the vdW theory, 6; = ( ~ ; ) ' / ~ pFurthermore, ~. if the heavy component may be considered dilute, the solution properties can be found as those of the pure solvent. Inserting the vdW equation (in terms of solubility parameters) into eq 7, then assuming / ~ , following result is that b2 = upL and u12 = ( U , U ~ ) ~ the readily obtained
a2
One thermodynamic approach to the same problem has been to consider an equilibrium existing between a lighter fluid phase imagined as an expanded liquid solvent and a subcooled liquid solute. Introducing activity coefficients, y2 in the lighter phase, eq 1 can be rewritten as fiL=
(4)
Y2Y?f2m0
where f i L / f i m 0 is the ratio of fugacities for the heavy component in the condensed state and mixture standard state, respectively. Scatchard-Hildebrand regular solution theory (RST)has been widely respected and used for its ability to predict qualitatively and correlate various types of phase equilibrium. Besides predicting liquid-vapor equilibrium of nonpolar solvents, RST and related theories have been applied to polar systems, solid solutions, metallic solutions, solid-liquid equilibrium, and solubility of gases in liquids (Hildebrand and Scott, 1964). Although the basic assumptions of RST on a molecular level (e.g., size, orientation effects) are sometimes violated in the above theoretical extrapolations, semiempirical parameters are often incorporated in these treatments to account for the neglected effects. Since the sizes of these effects are often of the same relative magnitude, fewer adjustable parameters are normally needed to correlate phase equilibrium of similar systems than sometimes necessary with more involved theoretical treatments. Prausnitz (1965) first suggested applying RST to solidfluid equilibria almost 2 0 years ago by using eq 4 in conjunction with a solute activity coefficient for a dilute subcooled solute in the fluid phase, thereby obtaining fiL
u2L
In E = In -- --a12(&f2mo RT
P + In -
PZS
(5)
Available data including the naphthalene-ethylene (Diepen and Scheffer, 1953) and solid carbon dioxide-air (Webster, 1952) systems were modeled by calculating 61 from pure solvent P-V-T data and by adjusting a2 to fit the observed data. The difficulty encountered with this approach was generalizing the pressure (or density) and temperature dependence of a2 used as an adjustable parameter. While investigating the advantages of supercritical fluid chromatography, Giddings et al. (1969) and Czubryt et al. (1970) proposed using RST to calculate the difference in solubilities at two states (A, B)along an isotherm. From eq 4, this is approximately given by Y2A Y2
In,=-
u2L(62L)2
RT
(AA - AB)(2 - AA - AB)
(6)
where AA = 61A/62L. Using eq 6 as a guide, saturation data for various heavy carbowaxes and alcohols dissolved in supercritical carbon dioxide at 40 O C were correlated. In addition, the threshold pressure of each species was estimated; that is, the minimum pressure for which the solubility of a particular species is large enough for detection
583
+
where e2* = (62)2 u , ~ / ~ . ~ RInserting T. this into eq 3 and assuming that the subcooled liquid volume is about that of the pure solid, leads to eq 9. log E = ez*A(2 - A) - log [l + (b12/P)]
(9) The above theory is consistent with the ideal regions of solubility, that is as P 0, log E 1. Since a1 ull/'pl, log E is predicted as exactly proportional to solvent density whenever 2 >> A and log [ l + (a12/P)] has a negligible contribution. This linear density dependence of the enhancement factor and closely related quantities (e.g. log E / z ) can be obtained from the virial equation and also has been observed for solutions of even moderate pressures (Ewald et al., 1953, Robin and Vodar, 1953; Johnston and Eckert, 1981). Both the vdW equation and RST involve many implicit assumptions for molecular behavior which are inexact in the supercritical regime. For example, the great asymmetry in size and intermolecular forces poses a severe restraint. Therefore we do not expect eq 9 to be exact; rather we use it as a qualitative guide to suggest a parameterization for a semiempirical correlation. Using eq 9 as a guide, we introduce two empirical parameters q and
-
-
-
V.
~2*A(2- A) - log 3. Results. Solubility data of dilute solid solutes in supercritical carbon dioxide, ethylene, and ethane were modeled since supercritical fluid extraction is most useful using solvents whose critical temperatures are near ambient conditions. A solubility matrix composed of available dilute solid-fluid data in these gases was constructed. Table I summarizes the systems and literature sources used. In carbon dioxide and ethane, solid-liquid isotherms at 30 OC were also modeled to determine whether the liquid state affected the applicability of the generalized form. Figures 1 and 2 illustrate the application of eq 10 (see Appendix for the source and evaluation of pure-component parameters); each system shows a linear behavior as plotted in this fashion while collapsing the temperature effect with a presumed average experimental error of 5% independent of whether the solvent was in a fluid or liquid state. Furthermore, the slopes are dependent only on the solvent gas to a good approximation while appearing to share common intercepts for each solute as suggested in
584
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983 -
ANTHRACENE 3
T.30
3
T.50 T.70
V
ETHYLENE FLUORENE
T.25 T:45 T.70
0 4
3
5W
-0m v 2 : 0,704
3-
1
C @
I
I
I
1
I
3 * A
5
7
9
Yl
Figure 1. Application of eq 11to solubility isotherms for the system ethane-anthracene.
I
(2-
E*-
I
a2
-log( I +
iI
Yl
Figure 2. Application of eq 11to solubility isotherms for the system ethylene-fluorene.
Table I. Systems Utilized for Modeling carbon ethylene dioxide naphthalene anthracene phenanthrene triphenylmethane fluorene pyrene hexamethylbenzene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene hexachloroethane biphenyl
a, b
C
b b d
d d d d d d
d d d e e
e e
-
f f
-
6
ethane d d d d -
W
z4
-
2 A
-
-
0
Johnston and Eckert Diepen and Scheffer (1953). (1981). Tsekhanskaya e t al. (1964). Johnston et al. (1982). e Kurnik et al. (1981). Van Leer and Paulaitis (1980).
;
(2-;)-lOg
(1
+
$)] +
5
10
Figure 3. Extrapolation to common intercept using solubility data of triphenylmethane dissolved in ethylene (a1 = 0.619), carbon dioxide (al = 0.497),and ethane (al = 0.581).
Figure 3 for triphenylmethane in all three solvents. Finally, substituting A/yl for A in eq 10 seemed to better collapse isotherms for more soluble solutes such as naphthalene and fluorene. All these observations combined suggest the following semiempirical correlation logE = %[ t2*
Cnbon Dioxide
8
6
s 8 4
W
-0"
v2
(11)
Fourteen parameters (a slope, ql, for each gas and an intercept, v2, for each of the 11solutes) were simultaneously fitted to the observed data according to eq 11 by using a generalized linear least-squares approach (Russel, 1970). Parameters calculated in this manner are summarized in Table I1 along with correlation coefficients describing deviations of proposed fits from experimental enhancement factors. In general, experimental enhancement
0
2
V
I/, ,
1
5
OO
1
1
10 l q ( I + +a I2
+(2-%YI
Phenonlhrene Triphenylmelhane
;I 15
YI
Figure 4. Solubility data for various solutes dissolved in ethylene.
factors were fitted within an average standard deviation of 2.1% in log E for 24 systems using a common slope for
Table 11. Parameters and Enhancement Factor Correlations ethylene 0.619
q, =
solute naphthalene anthracene phenanthrene triphenylmethane hexamethylbenzene fluorene pyrene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene hexachloroethane biphenyl
"2
0.724 0.704 0.682 0.390 1.10 0.923 0.892 1.14
1.09 0.54 2 1.77
0.013 0.016 0.012 0.016 0.040 0.016 0.017 0.032 0.030
-
carbon dioxide ethane q l = 0.497 q , = 0.581 average standard deviation 0.020 0.027 0.036 0.020 0.033 0.012 0.024 0.017 0.014 0.029 0.026
0.012 0.013 0.016 0.007
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983 585
Pressure (Bar I
Figure 6. Solubility data for various solutes dissolved in carbon dioxide.
Figure 6. P-y isotherms for phenanthrene dissolved in ethylene.
Table 111. P-y Calculations Average Standard Deviation
aaV
carbon ethylene dioxide ethane naphthalene anthracene phenanthrene triphenylmethane hexamethylbenzene fluorene PYrene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene hexachloroethane biphenyl
0.13 0.18 0.17 0.26
0.39 0.06 0.30 0.35 0.30
-
0.24 0.22 0.43 0.19 0.63 0.11 0.43 0.19 0.15 0.25 0.25
0.19 0.21 0.30 0.12
-
-
each solvent and a common intercept for each solute. The lines in Figures 1-3 visually illustrate typical fits obtained using parameters in Table 11. Figures 4 and 5 illustrate the collapsing of solubility data over six orders of solubility magnitude and two of solvent density for four solutes in ethylene and carbon dioxide when log E - v2 is plotted against e2*(A/yl) (2 - (A/yl) - log (1 + (S12/P). Obtaining solubility isotherms as functions of pressure is probably the most useful application of solid-fluid correlations. Since eq 11 is transcendental in nature, a simple iteration procedure is required. For solubilities less than 0.01 mole fraction, only two iterations by substitution are generally required and rarely are more than 5 ever necessary for more concentrated solutions. However, due to the assumptions in ita derivation, the correlation has a practical limitation to solubilities below 0.04, above which errors are often 3O-70%. At these concentrations, which correspond to about 15% by weight solute solutions, the assumptions that went into eq 11are obviously in much more error and the correlation is not expected to be as quantitative. Table I11 summarizes the average errors of calculated isotherms for each of the 24 systems below 0.04 mol fraction in terms of the average standard deviation N
a, = [ ( Z C Y 2 d C d i=l
- Y2)i2)/(N - 1)1"2/92
In general, the deviations were quite low considering the extreme range of data being fitted and the number of parameters used. The average standard deviation was less than 25% with the best fit systems less than 10% in error and the poorest about 60%. Typical calculated isotherms for various solutes are illustrated in Figures 6 and 7. Discussion 1. General Form of Correlation. The generalized from of eq 11should be acknowledged since it fails to obey
ln
0.004
0
0
200
400
600
Pressure (Bar1
Figure 7. P-y isotherms for fluorene dissolved in carbon dioxide.
-
the ideal gas law as P 0 and appears to differ from vdW theory by about a factor of 2. First, it is obvious that the ideal gas law would be followed if v2 = 0 for each system. The empirical observation that each solute requires a nonzero intercept to correlate data at higher densities according to eq 11 would seem to imply that a specific solute-solvent interaction might be occurring in the dense fluid that is not possible with a low density gas. Therefore, the following modification of eq 11would seem appropriate
- -
(12)
with the understanding that as P 0,f b ) 0. An attempt was made to correlate f b )to ( p / a p C ) " ) where a and n were assumed universal constants, but in each case the observed fits were poorer than leaving f ( p ) = 1over the entire high density range. In retrospect, the exact form of f b )seemed secondary since the goal was to correlate data in the dense regime only. On the other hand, if the above expression were to be extrapolated to low solvent density, it would seem logical to set fbl = 0. For correlating v2, it was observed that the vapor pressure and subcooled liquid heat of vaporization dominate the relative solubility of a particular solute in a given solvent. Since the heat of sublimation Pub is proportional to the slope of a vapor pressure curve and is related to H E P , it is reasonable to expect that v2, can be correlated to &:/H&?', where H ? is the enthalpy change for a liquid-vapor phase transition. Figure 8 illustrates that v2, can in fact, be represented in this manner within an acceptable scatter. Finally, note that an error in v2 as large as h0.5 (roughly half the spread of the 24 v2 parameters) would still allow P-y isotherms to be estimated within half an order of magnitude.
586
lnd. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
Table IV. Solvent Effect Estimation. Data of Eisenbeiss (1964) ethylenel
P, bar ethane q , = 0.58
carbon dioxide q 1 = 0.50
methane
: )
275 413 551 275 413 551 275 413 551
q , = 0.4ga
ethylenel
(exptl)y,?O (eq 1 4 ) 0.97 1.1 1.2
0.90 1.2 2.0
2 3 2.5
2.0 2.0
3.0
84 70 46
157 88 72
~ C H ,was estimated using Figure 9.
Table V. Minimal Error Analysis 0,5
0.7
0,6 Solute
$1
0,8
t/i
0.03
p:
10 50 5
solvent 61
1
i
combined contributions of v z and log (1 + d12/P)are neglected then
i
If the ratio ~ ~ : ! is 4 ~evaluated ~ in these same units, the ratio of solubilities of a particular solute in two solvents (a, b) at the samestemperature and pressure is approximately given by
-i
0,5
0,4
% estimated error
HK~
Figure 8. Correlation of v2 with H&w/Pl): (1) biphenyl; (2) 2,3dimethylnaphthalene; (3) 2,6-dimethylnaphthalene; (4) hexamethylbenzene; (5) fluorene; (6) pyrene; (7) naphthalene; (8) anthracene; (9) phenanthrene; (10)hexachloroethane;(11) triphenylmethane.
4
solute properties
Ethylene Ethane Corbon Dioxide
0 04
0.05
QlIbl
Figure 9. correlation of q 1 with q/bl.
An explanation of 7, is difficult beyond the empirical observation that it is nearly the proportionality for each solvent between the experimental enhancement and the approximations that are included in eq 8. Correlation of 7, was based partly on the empirical order of the parameter, i.e., C2H4> CzH6 > C02. Increasing v1 increases solubility sharply, and therefore 7, should reflect the ability of a particular solvent to interact with a comparatively larger solute molecule. Since dispersion forces dominate nonpolar solute-solvent interactions, 7, was correlated to q / b l (Figure 9). This correlation reasonably implies that 17, extrapolated to such gases as H2,N2, and CHI should be substantially less than C2H6or C02, and thus, solute solubilities in the former should be dramatically different even at similar molecular densities. Solubility data for these same solutes in other solvents a t lower pressures (Najour and King, 1966, 1970; Bradley and King, 1970) seem to justify this presumption. 2. Further Applications. The most obvious application of eq 11is its ability to generate P-y isotherms using only pure component parameters. The general form should also be useful to extrapolate any known data for a particular system to conditions remote from experimental observation, provided that both states are relatively dilute in solute concentration. Equation 11also allows the effect of various solvents to be discerned semiquantitatively independently of a solute. If eq 11 at infinite dilution is rewritten by using vdW solubility parameter extrapolation (6, ~ , ' / ~ p ,and ) the
-
1
(alb)1/2plb (cal/cc)1/2 10
(14)
where 10 (cal/cc)1/2 has been used as an average value. Data of Eisenbeiss (1964) for phenanthrene in ethylene, ethane, carbon dioxide, and methane at 40 O C were used to check the validity of eq 14. All solvent parameters were evaluated from normal sources (Weast et al., 1980-1981) and results are summarized in Table IV using ethylene as an absolute basis for comparison. The agreement is quite good considering the numerous approximations that went into eq 14. Therefore this method should provide a rough criterion for comparing the solvating power of various supercritical fluids independent of the solute in question. Finally, the general methods suggested by eq 11may be appropriate to polar solid-fluid equilibria with slight modifications. Correlation was attempted on three polar-fluid systems available in the literature, benzoic acid in ethylene and in carbon dioxide and phenol in carbon dioxide. All three could be correlated well with the same 7, parameter derived from nonpolar solutes but for benzoic acid a slightly different v2 parameter could be utilized to better represent data in carbon dioxide and ethylene (Figure 10). If the qualitative interpretation of v2 is accepted, then it might be postulated that specific chemical
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
587
Table VI. Solute Parameters l o g P S = A - B / ( C + 2') naphthalene anthracene phenanthrene
US*
HgDW
(C," - CpL)
A
B
C
128.6
7.2144
2926.6 (d 4397.6
-35.8
142.6
50.75 (b) 62.18
0.056
(a
(a )
(e 1
167.6
152.7
70.76 (h1 80.62 (est) 55.48
(i 1
tilS
(a)
triphenylmethane
240.9
(a
hexamethylbenzene fluorene
139.3
pyrene 2,3-dimethylnaphthalene
(i )
159.1
62.17 (h 77.13
(i )
(m
154.7
(OlS
(0
0
3855 (k) 4561.8 (h+l) 4904 (h) 4302.5
9.2046 8.3946
(m)'
113.2
5228
(9
8.1336
0.097 (g1 0.076
0
(e
9,7858
ti)'
0
cn 4873.4
9.6310
0.122 (est)* 0.020
(a1
154.7
7.1464
k)'
0.050 @) 0.050 (g 0.053
(a)
hexachloroethane
0.089 (est)* 0.080
53.40 (h1 52.31 (h1 40.23
(a )
2,6-dimethylnaphthalene
IS
(c
9.0646
-21
0 0 0
(h
9.4286
0
4419.5 (h 2603
5.6356
0
(n
0.039 9.4068 4262 0 (est) ( 11 Data supplied from Technical Services Dept., Aldrich Chemical Milwaukee, WI (1980). Ambrose e t al. (1975). Sinke (1974). Fowler e t al. (1968). e Kudchadker e t al. (1979). f Wiedeman and Vaughan (1970). Finke et al. (1977). Osborn and Douslin (1975). Weast et al. (1980-81). MacDougall and Smith (1930). Ambrose et al. (1976). Bradley and Cleasby (1953). Smith et al. (1980). " Jordon (1954). O Majer e t al. (1980 . For all compounds, solid thermal expansivities were assumed to be 0. Based on naphthalene (Var aftik, 1975). a$ = 0.0007 g/ (cm3 K) and P , = ~ ~ 1 1 . 0 2a. Estimated from phenanthrene. Estimated from d?$"=/dT. Estimated from HLvap= 2.3R[d(log Pzs)di)l/T)] where P z s is liquid vapor pressure. biphenyl
131.0
4 5.
(a1
(V
*
I
,
/
I
,
71
#
I
BENZOIC ACID
I
I
ETHYLENE HEXAMETHYLEENZENE
r/
5 w
z4 3
'
OO
5
E'P
YI
15
IO
C2-P)-log YI
2
I
I
5
3
c;~(z-%-Iog(
I+$1
YI
7 I
+ +rJ2 I
YI
Figure 10. Application of eq 10 for solubility data of a polar compound in ethylene and carbon dioxide.
Figure 11. Error analysis showing effects of uncertainties in thermodynamic parameters toward the application of eq 10.
dense phase interactions for benzoic acid in ethylene and in carbon dioxide are of slightly different magnitudes. 3. Effect of Errors of Properties on Analysis. As noted earlier, application of eq 11 requires that several thermodynamic properties be evaluated for both the solute and solvent. The reader is directed to the Appendix for further discussion along these lines. In any event, due to the exponential nature of eq 11, small errors in properties are expected to have a large effect on the reliability of the generalized correlation and usefulness of the general parameters that might be presented by different investigators. A minimal error analysis was calculated according to standard methods in order to estimate the effect that even modest errors (Table V) contributed toward fitting the parameters ql and v2. Undoubtedly the single largest estimated contribution was the uncertainty in p:, since this quantity was reported for only naphthalene and required extrapolaration for the remaining solutes. The second ? though the ablargest was attributed to calculating H solute error is significantly lower than pf;lp. Therefore a
Table VII. Solvent Thermophysical Properties, 61(T P) and '1IT. Pb ethylene Dick and Hedley (1962) Newitt e t al. (1962) carbon dioxide ethane Goodwin et al. (1976)
minimum 7% property error propagated throughout the correlation. Figure 11pictures the result of detailed error calculations for hexamethylbenzene-carbon dioxide. In general, errors in p:, and H'GP affected v2 more severely than q1 since v2 is an extrapolated parameter and neither the linearity nor temperature independence of log E vs. c2*(A/y1)(2- (A/yl)) - log (1+ (6,2/P))would be greatly affected. Errors in the thermal coefficients and vapor pressure would cause a plot of log E vs. e2*(A/y1)(2(A/yl)) - log (1 (S12/P)) to appear more scattered, affecting both q1 and v2. Acknowledgment This project has been financed in part with Federal funds as part of the program of the Advanced Environ-
+
588
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
mental Control Technology Research Center, University of Illinois at U-C, which is supported under Cooperative Agreement CR 806819 with the Environmental Protection Agency. The contents do not necessarily reflect the views and policies of the Environmental Protection Agency, nor does the mention of trade names or commercial products constitute endorsement or recommendation for use. Additional funding was provided by Phillips Petroleum Co., Arco Oil and Gas Co., and Exxon Research and Engineering Co. Appendix Actual application of eq 11 required the evaluation of several pure component thermodynamic parameters for both solute and solvent. Since fluid phase interactions were considered liquid-like in nature, all solute parameters for log (i.e. H P p , p i ) were evaluated from extrapolating liquid properties from the melting point using thermal coefficients
a2
HPP z H
e
+ (C,L - (2,")
( T m p- T )
(A.l)
Whenever possible, thermodynamic parameters were taken directly from literature sources or evaluated from related properties. Thermal coefficients, if necessary, were estimated for some compounds on a per weight basis from similar molecules for which values were published (e.g., a: = 0.0007 g/cm3 K based on naphthalene (Vargaftik, 1975). Tables VI and VI1 summarize thermal properties for all solutes and solvents. Nomenclature a = energy parameter in vdW equation b = hard-sphere volume parameter in vdW equation = heat capacity = solubility enhancement factor (= y 2 P / P 2 s ) f = component fugacity H = enthalpy heat MW = molecular weight P = pressure, bar T = temperature, K y = mole fraction u = molar volume, mol/cm3 z = compressibility (= PujRT) Greek Symbols a = polarizability at = thermal expansion coefficient e* = dimensionless energy parameter (= HLvap/2.3RT) 6 = RST solubility parameter y = activity coefficient A = ratio of solubility parameters (= &/ij2) 6 = fugacity coefficient 7 = slope parameter in eq 11 v = intercept parameter in eq 11 CP = volume fraction p = density Subscripts 1 = light solvent component 2 = heavy solute component av = average c = critical point ref = reference
2
TP = property or state evaluated at temperature, T, and pressure, P mp = melting point temperature
Superscripts ID = ideal gas state L = liquid, real fluid for T > Tmphypothetical subcooled state; extrapolated from melting point for T Tmp Lvap = vaporization from liquid state mo = standard state of component in mixture phase = property evaluated in mixture phase S = solid phase s = saturated vapor sub = sublimation v = vapor state Registry No. Ethylene, 74-85-1;ethane, 74-84-0; carbon dioxide, 124-38-9; naphthalene, 91-20-3; anthracene, 120-12-7; triphenylmethane, 519-73-3; phenanthrene, 85-01-8; fluorene, 86-73-7;pyrene, 129-00-0; 2,3-dimethylnaphthalene,581-40-8; 2,6-dimethylnaphthalene,581-42-0; biphenyl, 92-52-4; hexamethylbenzene, 87-85-4; hexachloroethane, 67-72-1. Literature Cited Ambrose, D.; Lawrenson, I.J.; Sprake, C. H. S.J. Chem. Thermdyn. 1975, 7 . 173. Ambrose, D.; Lawrenson, I.J.; Sprake, C. H. S. J. Chem. Therm. 1978, 8 , 503. Bradley, H.. Jr.; King, A. D.. Jr. J. Chem. Phys. 1970, 52, 285. Bradley, R. S.; Cleasby, T. G. J. Chem. SOC. 1953, 1690. BLichner, E. H. 2.fhys. Chem. 1008, 5 4 , 665. Czubryt, J. J.; Myers, M. N.; W k g s , J. C. J. Phys. Chem. 1970, 74, 4260. Dlepen, E. A.; Scheffer, F. E. C. J. Phys. Chem. 1953, 5 7 , 575. Dick, W.; Hedley, A. M. I n "Thermodynamic Functions of Gases", 1st ed., Din. F., Ed.; Butterworths: London, 1962 Voi. 11, p 86. Eisenbeiss, J. "A Basic Study of the SolubiHty of Sollds in Gases at High Pressures", Final Report, Contract No. DAW-108-AMC-244 Southwest Research Institute, San Antonio, TX, 1964. Ewald, A. H.; Jepson. W. B.; Rowiinson, J. S. Dlscuss. Faraday SOC. 1053-1054, 238. Finke, H. L.; Messeriy, J. F.: Lee, S. H.; Osborn, A. G.; Douiin. D. R. J . Chem. Thermdyn. 1077, 9 , 937. Fowler, L.; Trump, W. N.; Vogler, C. E. J. Chem. Eng. Data 1988, 73,209. Giddtngs, J. C.; Myers, M. N.; King, J. W. J. Chromatcgr. Sci. 1089, 7 , 276. Goodwin, R. D.; Roder, H. M.; Straty, G. C. National Bureau of Standards, Technical Note 684, 1976. Hannay, J. B.; Hogarth, J. Proc. R . SOC. London, Ser. A 1879, 29, 324. Hildebrand, J. H.; Scott, R. L. "The Solubility of Nonelectrolytes"; Dover Publications: New York, 3rd ed., 1964. Johnston, K. P.; Eckert, C. A. AIChE J. 1081, 2 7 , 733. Johnston, K. P.; Zlger, D. H.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1982, 27, 191. Jordon. T. E. "Vapor Pressure of Organic Compounds", Interscience: New YOrk, 1954. Kudchadker, S. A.; Kudchadker, A. P.; Zwolinski, B. J. J . Chem. Thermodyn. 1979, 7 7 , 1051. Kurnik, R. T.; Holla, S. J.; Reid, R. C. J. Chem. Eng. Data 1981, 2 6 , 47. MacDougali, F. H.; Smith, L. 1. J. Am. Chem. SOC. 1930. 52, 1998. Majer, V.: Svab, L.; Svoboda, V. J. Chem. Thermodyn. 1980, 12, 843. Najour, G. C.; King, A. D., Jr. J. Chem. Phys. 1988, 45, 1915. Najour, G. C.; King, A. D., Jr. J. Chem. Phys. 1070, 5 2 , 5206. Newitt, D. M.; Pai, M. V.; Kuloor. N. R.; Hugglli, J. A. I n "Thermodynamic Functions of Gases", 1st ed., Din, F., Ed.; Butterworths: London, 1962; p 102. Osborn, A. G.; Douslin, D. R. J. Chem. Eng. Data 1975, 2 0 , 229. Paulattis, M. E.; Krokonis, V. J.; Kurnik, R. T.; Reid, R. C. Rev. Chem. Eng. in press, 1963. Prausnttz, J. M. National Bureau of Standards, Tech Note 316, 1965. Robin, S.; Vodar, B. Discuss. Faraday Soc. 1953-1954, 233. Russel, D. "Optimization Theory"; W. A. Benjamin: New York, 1970; pp 131-135. Sinke, G. C. J. Chem . Thermodyn . 1974, 6 , 3 11. Smlh, N. K.; Osborn, A. G.; Scott, D. W. J. Chem. Thermodyn. 1980, T2, 919. Tsekhanskaya, Y. V.; Iomtev, M. 8.; Mushkina, E. V. Russ. J. Phys. Chem. 1964, 9, 1173. Van Leer, R. A.; Paulaltis, M. E. J. Chem. Eng. Data 1980, 25, 257. Vargaftik, N. B. "Tables on the Thermophysical Properties of Liquids and Gases"; Hemisphere Publications: New York, 1975. Weast, R. C.; Astie. M. J., Ed. "CRC Handbook of Chemistry and Physics", 61st ed.,CRC Press: Boca Raton, FL, 1980-1981; p D-194. Webster, J. J. Proc. R . SOC. London, Ser. A 1952, 214. Wiedemann, H. G.; Vaughan. H. P. Roc. Toronto Symp. Therm. Anal. 1970, 233.
Received f o r review August 24, 1981 Revised manuscript received November 15, 1982 Accepted December 21, 1982
