Ind. Eng. Chem. Res. 1991,30, 1329-1335 Taylor, G. Conditions under Which Dispersion of a Solute in a Stream of Solvent Can Be Used to Measure Molecular Diffusion. h o c . R. SOC.London, Ser. A 1954b,225,413-411. Van Loef, J. J. The Corrected Enskog Theory and the Transport Properties of Molecular Liquids. Physica 1977, 87A, 258-212.
1329
Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solutions. AIChE J. 1955, I , 264-210.
Received for review August 13, 1990 Accepted December 11,1990
Diffusion Coefficients of Long-chain Esters in Supercritical Carbon Dioxide K. Keat Liong,
P.Anthony Wells, a n d Neil R.Foster*
School of Chemical Engineering and Indwtrial Chemistry, University of New South Wales, P.O.Box 1, Kensington 2033, Australia
Binary diffusion coefficients, Ol2,of CIBfi,Cm, and Cz. ethyl esters and Cm6 and C2M methyl esters were measured in supercritical carbon dioxide in the temperature range 308-318 K and a t pressures between 96.7 and 210.5 bar. Measurements were obtained using a capillary peak broadening technique. The experimentally determined diffusion coefficients were approximately 1 X lo4 cm2/s a t 313 K and 97 bar, decreasing to approximately 0.5 X lo4 cm2/s with an isothermal rise in pressure to 210 bar. The applicability of several correlations to the experimental diffusivity data was examined. In particular, the free-volume-type diffusion model was found to correlate the experimental data to within f3%. Introduction Supercritical fluid (SCF) extraction has received considerable attention in recent years as a technique for the separation of relatively nonvolatile materials. It is a novel separation technique that embodies several features of conventional solvent extraction and distillation in addition to several important, unique features such as liquidlike densities, gaslike viscosities, and diffusivities between typical gas and liquid values. The combination of liquidlike solvent power and gaslike transport properties is exploited in SCF extraction. Another feature of this type of extraction is that the overall properties of the fluid are very sensitive to changes in pressure and temperature in the vicinity of the critical point. This means that small changes in extraction conditions are often all that is required to effect complex separations. In addition, the relatively low operating temperatures often required to achieve criticality make the technique suitable for treating heat-sensitive substances that could not be extracted by conventional distillation. Despite the potential that SCFs can offer to extraction processes, the lack of fundamental thermodynamic data required for process design and scaleup has hindered development of the technology to a substantial commercial level. Hence, much research effort has been directed to obtaining procsas design parameters under SCF conditions. Diffusion is the dominant rate mechanism and is significant in equipment design for and process development of SCF extraction. There are currently very few sets of data for binary diffusivity in SCF systems (Groves and co-workers, 1984). Therefore the aim of this study was to obtain binary diffusion data for long-chain ethyl and methyl esters in supercritical carbon dioxide. These long-chain esters are typical of those found in esterified marine lipids. Clinical studies have shown that diets rich in these lipids can play an important role in regulating human metabolism. In particular, both cis-5,8,11,14,17eicosapentaenoic acid (EPA, C20:6w-3) and cis4,7,10,13,16,19-docosahexaenoicacid (DHA, C,#-3) have attracted considerable attention from researchers as having beneficial effects on human health (Dyerberg, 1986). As few binary diffusivity data for solutes in SCFs exist, there have been few attempts to model this property. With
the substantial data obtained in this study the merits of several currently available correlation models have been assessed. Experimental Section Binary diffusivities in supercritical carbon dioxide were measured by using the capillary peak broadening (CPB) technique. The principle of the CPB technique is based on the fundamental work of Taylor (1953, 1954), later extended by Aris (1956),which involved the dispersion of a solute in a laminar flow of mobile phase through a tube. The application of this technique to high temperature and pressure has been demonstrated by a number of researchers (Matthews and Akgerman (1987), Sassiat and co-workers (1987)). A comprehensive discussion of the basic theory has been given by Groves and co-workers (1984) and Alizadeh and co-workers (1980). Theory. Taylor (1953) showed that a narrow pulse of solute will broaden into a peak due to the combined action of convection along the axis of the tube and molecular diffusion in the radial direction. This concentration profile of the peak can be described mathematically by
or in terms of the theoretical plate height H
H = a2/L
(2)
where DI2 = diffusion coefficient, cm2/s; L = length of column, cm; U,, = linear velocity, cm/s; 2 = variance, cm2; H = theoretical plate height, cm; and ri = internal radius of tube, cm. For flow through a straight tube, the concentration profile becomes essentially Gaussian (Levenspiel and Smith, 1957) if D,/U& < 0.01 (3) where (4) DefI= D12+ r?Uo2/48Dl2 However, due to the lengths of the tubing required for diffusion experiments, the tubing generally has to be coiled
0888-5885/91/2630-1329$02.50/00 1991 American Chemical Society
1330 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 Table I. Properties of Solutes solute behenic acid ethyl ester, Caa DHA ethyl ester, Cas DHA methyl ester, Czlg EPA methyl ester, Cma stearic acid ethyl ester, Clea
mol w t a 368.7 356.5 342.5 316.5 312.5
VI: cm3/mol 552.6 508.2 484.4 447.4 463.8
I.
To
Obtained from Sigma Chemicals. *Estimated by using Le Bas method (Reid and co-workers, 1989).
Ver
a
in order to be placed in a constant-temperature bath. In such a situation, for eq 1 to be valid (Moulijn and coworkers, 1977) De(Sc)o.6< 10 (5) where De and Sc are the Dean and Schmidt numbers respectively and are defined as De=- P U P ( 2Jbk s c = P/PD12
, I . Syringe Pump 2 . Solute Reservior
3. Injection Valve
4. Pressure Transducer 5 . Diffusion Column 6 . Water Bath 7 . UV Detector
9. 8. Regulating Integrator Valve 10. Sample Collector
Figure 1. Schematic of the experimental apparatus.
(6) (7)
where p = solvent density, g/cm3; p = solvent viscosity, poise; ddb, = id. of tube, cm; and dd = diameter of tubing coil, cm. If the constraints imposed by eqs 3 and 5 are fulfilled, the diffusion coefficient can then be determined by combining eqs 1 and 2 to give the following equation:
Ol2 = (U0/4)(H - (e - $/3)'/2) (8) The theoretical plate height is determined by first measuring the width of the peak at 0.607 times the peak height (this is in effect the half-height of the Gaussian peak), and then by using the equation H = Uo2W0.so72/4L (9) where W0.,, = width at 0.607 of peak height, s. Materials. The relevant properties of the solutes used in this study are presented in Table I. All solutes were obtained from Sigma Chemicals (9990 purity). As the ethyl ester of EPA (Cm& was not available, a comparative study was undertaken using the methyl and ethyl esters of DHA (C,) to determine the effects of the extra carbon in the chain. The purity of the liquid carbon dioxide used was 99.8%. Equipment and Procedure. A schematic of the experimental apparatus is shown in Figure 1. The syringe pumps (ISCO,Models No. LC-2600 and LC-5OOO), capable of delivery pressures up to 250 bar and delivery rates from 0.8 to 400 mL/h (on a liquid carbon dioxide basis), were used to bring the system to pressure. The temperature of the system was monitored by using a type K thermocouple, and the system pressure was monitored by pressure transducers (Druck). The overall error was k0.2 "C and f0.5 bar for temperature and pressure, respectively. The experimental apparatus consisted of two sections, namely, the solute stream section, and the solvent stream section. In the solute stream section, liquid carbon dioxide was contacted with solute in the solute reservoir. The concentration of solute in carbon dioxide was regulated by adjusting the valve on the bypass line. Solid solutes (for example, stearic acid ethyl ester), were packed into a 1.3-cm-0.d. tube, while liquid solutes were impregnated into filter paper that was cut up into small strips and packed into the tube. To prevent entrainment of the solutes, a filter (7 pm) was located immediately after the reservoir. The solute-laden stream was then passed from the reservoir into a heating coil located in a water bath (not
shown in Figure 1). A six-port injection valve (Rheodyne) with a 20-pL sample loop was used to inject the solute into the solvent stream. When injection of the solute into the solvent stream was not required, it was bypassed into a sample collector, and the carbon dioxide was vented. In the solvent stream section, carbon dioxide was passed through a heating coil located in the water bath (not shown in Figure 1)in order to reach the system temperature. At a particular solvent condition, three pulses of solute were injected into the solvent stream at 15-min intervals. This period was sufficient to prevent the peaks from overlapping during elution from the diffusion column. The solute injected into the solvent stream was passed into a diffusion column (i.d. = 0.1025 cm). Two diffusion columns of different lengths (1443.5 and 255.8 cm) were employed in determining the diffusion coefficients in order to overcome the effects of dead volume and initial dispersion. A subtraction procedure was used whereby the variance of the peak obtained from the short column was subtracted from that obtained with the long column. The resulting variance then corresponded to a peak that would have been produced in the same column without dead volumes and initial peak dispersion. On exiting the diffusion column, the solute was passed into a UV-vis variable-wavelength detector (ISCO V4). The tubing connecting the detector to the water bath was maintained at the system temperature by way of a furnace controller (Shinko). An integrator (Shimadzu C-RGA) was used to provide a hard copy of the peaks obtained. The experiments were conducted at temperatures in the range 308-318 K and pressures from 96.7 to 210 bar. A summary of the experimental conditions is provided in Table 11.
Results To verify the accuracy of the measurements and the reliability of the technique, the diffusion coefficients of naphthalene at 308 and 318 K were determined for various pressures between 96 and 210 bar. The experimental data were then compared with those of Knaff and Schlunder (1986). The results obtained are presented graphically in Figures 2 and 3. The experimentally determined diffusion coefficienta for the esters in supercritical carbon dioxide are presented in Table 111. Each diffusivity data point represents an average of three measurements, the results of which differed by no more than 2.5% and were generally fl%. Preliminary studies revealed that the subtraction technique provided a correction of less than 1.5%. Consequently, the bulk of the data were determined by using the
Ind. Eng. Chem. Res., Vol. 30, No. 6,1991 1331 Table 11. Summary of Experimental Conditions temp, K press., bar viscosity," CP density," g/L 308 98.7 0.0566 700 112.0 0.0638 750 120.0 0.0666 768 138.5 0.0720 800 0.0771 160.1 828 0.0814 181.3 850 600 0.0450 313 96.7 103.0 0.0505 650 114.0 0.0567 700 132.7 0.0637 750 0.0720 163.2 800 0.0815 210.5 850 0.0452 318 109.4 600 117.9 0.0507 650 131.7 0.0569 700 153.7 750 0.0639 0.0721 188.0 800
Table 111. Experimental Binary Diffusion Coefficients for the Ester-Supercritical Carbon Dioxide Syrtems experimental binary diffusion coefficienta, 1@ cm2/s density, g/L Cna C m ethyl Cz+ methyl Cma C1&* T = 308 K 700 7.54 7.79 7.97 8.08 7.95 750 6.75 6.78 6.92 7.11 6.95 768 6.41 6.52 6.67 6.76 6.80 6.17 6.23 800 5.85 6.28 6.20 5.77 5.79 5.72 5.74 828 5.56 850 5.17 5.50 5.50 5.36 5.39
T = 313 K
Data from Angus and co-workers (1976).
600 650 700 750 800 850
9.91 8.75 7.79 6.78 6.11 5.31
600 650 700 750 800
10.1 8.80 7.95 6.90 6.13
1.4 -
cn
-5
T
N '
1.2
0
308 K
-
9.98 8.95 7.98 6.92 6.18 5.48
T = 318 K 10.2 8.73 8.00 7.01 6.26
10.1 8.86 7.96 7.11 6.27 5.57
10.5 9.21 8.07 7.19 6.36 5.62
10.1 9.12 8.10 7.24 6.29 5.60
10.3 9.00 8.17 7.18 6.41
10.6 9.41 8.18 7.27 6.41
10.5 9.4 8.18 7.34 6.40
1.4
1-
1.2
(u
0
-
313 K
A
co
0'
rii
0.8 -
'
0.0 50
I 100
150
200
d 0.8
250
Pressure (bar) Figure 2. Diffusion coefficienta of naphthalene in supercritical carbon dioxide as a function of pressure at 308 K (literature data from Knaff and Schlunder, 1987).
b 0.8
1.1
-
0.9
-
h
v)
' N
E
d
I
0.6 50
I 100
150
200
250
Pressure (bar) Figun, 3. Diffusion coefficients of naphthalene in supercritical carbbn dioxide as a function of pressure at 318 K (literature data from Knaff and Schlunder, 1987).
$
'
$
0.7 -
0
""
550
1443.5-cm column. The results obtained indicate that the diffusivities of the esters in supercritical carbon dioxide are between 0.5 X lo-' cmz/s at 210 bar and 1X lo-' cm2/s at 97 bar. Influence of Pressure at Constant Temperature. The binary diffusion coefficienta of the esters in supercritical carbon dioxide were determined as a function of pressure and density at 308,313, and 318 K. In all cases, the influence of pressure on the diffusion coefficients was less significant at higher pressures, as is illustrated for the ClF0ethyl ester in Figure 4. The sharp change in diffusiwties at low pressures indicates that the solvent density and the solvent viscosity are important factors, as these
600
650
700
750
800
550
900
Density (g/l) Figure 5. Diffusion coefficients of stearic acid ethyl ester as a function of carbon dioxide density.
properties change more rapidly at lower pressures. The influence of the density and viscoeity (density and viscosity data obtained from Angus and co-workers (1976)) on the diffusion coefficients is shown in Figures 5 and 6, respectively. As the density of the solvent increases, the molar volume decreases. In such a situation, collision transfer, rather than molecular transfer, becomes the dominant transport mechanism, and this results in a sharp decline in diffusivity. Therefore the effect of pressure is
1332 Ind. Eng. Chem. Res., Vol. 30, No.6,1991
P
0 313 K
8O8
0
-
C I S 0 Ethyl Eater
> ZO8-
VI
* g040 L
B020.2
'
I
0
2
4
8
8
10
Viscosity lo2 (CP) Figure 6. Diffusion coefficients of stearic acid ethyl ester as a function of carbon dioxide viscosity.
0
N '
0
,
, 0.6
0.4
0.2
1
1.2
DT
_--co
1
o.71
I"
306
0.8
Measured Diffusivity lo4 (cm2/s) Figure 8. Comparison of the Wilkdhang equation with experimental data.
The original Stokes-Einstein equation, which is based on a model where a solute sphere is considered to move through a continuum of the solvent, is
760 QII
Y
f
E n
310
316
320
Temperature (K) Figure 7. Diffusion coefficients of stearic acid ethyl ester as a function of temperature.
probably due to the combination of changes in the density and viscosity of the solvent. Influence of Temperature at Constant Density. The effect of temperature at constant density is shown in Figure 7. It was observed that the diffusion coefficient changed by no greater than 10% over the 10 K temperature range investigated. As the degree of change in the diffusion coefficient was only marginally greater than the experimental errors incurred, it was difficult to quantitatively analyze the influence of temperature at constant density. However, this observation is consistent with the observations of Swaid and Schneider (1979), who reported that the influence of isopycnic changes in temperature was smaller than the experimental error. In a study of the diffusion behavior of linoleic acid methyl ester over a 20 K temperature range, Funazukuri and co-workers (1989) observed an increase in diffusion coefficient of approximately 12% at a constant density of 800 g/L.
Discussion The experimentally determined diffusion coefficients were used to test the performance of a Stokes-Einstein based equation, the Hayduk-Minhas correlation, and a generalized free-volume expression based on the rough hard sphere (RHS) theory of diffusion. As diffusion coefficients in SCF systems are similar to those found for liquid systems (in the range 0.5 X x 1od cm2/s), several authors have attempted to correlate diffusivities in SCF systems using the empirical relations developed for liquid systems (Saseiat and co-workers (1987), Funazukuri and co-workers (1989)). Some of these empirical equations were derived from the Stokes-Einstein expression.
where ro = radius of spherical solute, cm; p = solvent viscosity, cP; T = temperature, K; and where D12has the units of cm2/s, as indicated in eq 2 (unless otherwise indicated, all notations have the same units as those mentioned previously). This relationship has since been modified by Wilke and Chang (1965) to provide a useful estimation procedure for liquid systems. The equation for unassociated liquids is 7.4 Dl2
=
X
lO+TG
pV10.6
where A4 = molecular weight and VI= molar volume of solute at its normal boiling point, cm3/mol. The ability of the Wilke-Changequation to predict the diffusion coefficients of the esters in supercritical carbon dioxide is shown in Figure 8. It was found that the Wilke-Chang equation consistently underpredicted the experimental data, the average deviation ranging from 14% for Cm5methyl ester to approximately 21 % for behenic acid (C22:o)ethyl ester. A similar trend was observed by Funazukuri and co-workers (1989) with C18 unsaturated fatty acid methyl esters. The esters studied in the present investigation were essentially long straight-chain molecules compared with the solutes upon which the derivation of the Wilke-Chang equation was based. The solutes studied by Wilke and Chang (1955) were more bulky in nature (for example, benzene, bromobenzene, carbon tetrachloride, and iodine, among others). Hence the underprediction of the diffusion coefficients by the Wilke-Chang equation is consistent with steric considerations and with the concept that in compressed gases a molecule diffuses along ita longest axis (Swaid and Schneider, 1979). One of the major weaknesses of the Wilke-Chang equation is the necessity to use the molar volume to characterize the solute size. As an example, n-propylbenzene, being a more stretched molecule, has been shown to diffuse faster than its structural isomer 1,3,5-trimethylbenzene (Swaid and Schneider, 1979) despite having similar molar volume. The Wilke-Chang equation does not account for this structural difference. As a result of ita simplicity, the Wilke-Chang equation cannot be used to predict diffusivities to a high degree of accuracy. It follows from eq 10 that a plot of DI2versus Tp-l for any given system exhibiting hydrodynamic behavior should
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1333
m
x
‘N
-E
08-
*
0 6 -
b
04-
C20:6 Methyl Eller
: 0
/’
02-
I
OL 0
1
2
3
TIP*
4
5
6
7
8
0
(KICP)
0.2
0.4
0.6
1
0.8
1.2
Measured Diffusivity * lo4 (cm2/s)
Figure 9. Diffusion coefficiente of stearic acid ethyl ester as a function of TIP.
Figure 10. Comparison of the Hayduk-Minhan equation with experimental data.
yield straight lines through the origin. Feist and Schneider (1982) examined the diffusion coefficients of benzene, phenol, naphthalene, and caffeine in supercritical carbon dioxide and concluded that the Stokes-Einstein equation did not apply since the intercepts from a plot of D12versus 2’p-l were nonzero. Debenedetti and Reid (1986) concluded that predictive correlations based on the StokesEinstein equation (such as Wilke and Chang (1955),Reddy and Doraiswamy (1967),and Luis and Ratcliff (1968))will overestimate diffusion coefficients in SCFs. This is not consistent with the data observed in this study. The binary diffusion coefficients for the CILo ethyl ester-carbon dioxide system plotted as a function of reciprocal carbon dioxide viscosity is shown in Figure 9. Within experimental error, the data yielded a straight line through the origin. Similar behavior was observed for all the esters studied. These observations are not consistent with those of other workers, as discussed above, and suggest that the structure of the molecule can have an important effect on its diffusion in the supercritical solvent. This could also explain why the W i l k d h a n g equation did not overestimate the experimental diffusion coefficients. The experimentally determined diffusion coefficients of the esters in supercriticalcarbon dioxide are also compared with the predictions from the Hayduk and Minhas (1982) equation. Hayduk and Minhas (1982) realized the need for a more specific correlation and, by regression analysis, proposed several correlations depending on the type of solumolvent system. For normal paraffin systems, the diffusion coefficient is given by
A model based on the RHS theory was also assessed. The basis of this model is that the dense fluid is considered to be an assembly of hard spheres with transport coefficients that are related directly to the low-density coefficients by the approximate Enskog formulation (Chapman and Cowling, 1970). In terms of the low-density B o l t z ” diffusion coefficient Do, the self diffusion coefficient DE at high density is DE = DO/g(‘J) (13)
D12= 13.3 X 10-sP*47pe/V~11
(12)
where e = (10.2/V1) - 0.791. In addition, Hayduk and Minhas (1982) took into account the interaction between the solvent viscosity and the solute molar volume, since the effect of the solvent viscosity on the rate of diffusion depends on the size of the diffusing molecules. The deviation of the experimental values from those predicted by the Hayduk-Minhas relation is shown in Figure 10. The average deviation was considerably less compared with the Wilke-Chang equation with deviations of 5% for Cm6methyl ester (cf. 14% with the Wilke-Chang equation) to about 16.4%for C22a ethyl ester (cf. 21%). However the model was developed from a regression of data from systems involving nparaffins. It would be expected that for systems where both the solute and solvent were dissimilar,the interactions between these solute-solvent molecules would be less significant, with consequently higher diffusion rates. The tendency of the Hayduk-Minhas equation to underpredict the ester-SCF system is consistent with this argument.
where g(a) is the radial distribution function at contact. However, in the original Enskog theory, no account was taken of successive hard-sphere collisions that led to a 30% underestimation of the actual value of the diffusion coefficient (or 40% overestimation at high densities). Dymond (1974) made a modification to the Enskog coefficient to account for its limitations by adding a correction term: DE = (DO/g(a))(DHS/DE) (14) where DHS/DE is the molecular dynamic ratio of hard sphere to Enskog diffusivity. Through computer calculations, Dymond (1974) was able to relate viscosity and diffusivity to the following:
2.306 D12 =
X
10-6(T/M)0*6 V00.67
(V2 -
1.384Vo)
(16)
where Vo = hard-sphere close packed molar volume, cm3/mol; V, = molar volume of the solvent, cm3/mol; and p = solvent viscosity, poise. Ignoring the PId” term, eq 15 is similar to the free-volume equation proposed by Batchinski (Hildebrand, 1971): l / P = a(V2 - VJ (17) where a and V, are constants characteristic of the fluid, and p has units similar those indicated in eq 15. In comparison to eq 16, the diffusion coefficient based on the free-volume equation will be of the form (Chen and coworkers, 1982) D12 = PT(V2 - VD) (18) where V, = minimum free volume for diffusion, cms/mol, and the coefficient ,8 is a function of the interaction between the solute and the solvent. An example of the plot of D12/Tas a function of V2 is presented in Figure 11, where a linear relationship was observed. The coefficient (3 was obtained from the slope
1334 Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991
, 3
C22 6 Ethyl Ester
A
C22.9 Methyl Ester
x
C20 6 Methyl Ester
0
C18 0 Ethyl Eeter
/.
/'
/"
/'' 0
02
04
06
08
1
12
Measured diffusivity lo4 (cm2/s) Figure 13. Comparieon of diffusion coefficients based on the Dymond plot with experimental data.
The mathematical expression for 8, eq 22, was then obtained by multiple regression of the slopes of the plots of D12/!P6as a function of molar volume for all the esters studied in supercritical carbon dioxide:
C22.8 Ethyl E i t e r
B = 5.89 x 10-3vD-2*72v14'26
0
0.4
B 0
p 0.2 t
0 0
/
///-. 0.4
02
06
08
1
I
12
Measured Diffusivity lo4 (cm2/s) Figure 12. Comparison of diffusion coefficients based on the freevolume expression with experimental data.
of the plot while VDrepresented the molar volume of the solvent at which the diffusivity approached zero. The binary diffusion coefficients were then determined from eq 18. The binary diffusivities obtained by using the free-volume relation were then compared to the experimental diffusivities. The average deviation obtained was less than 5% in all cases and is shown graphically in Figure 12. According to Chen and co-workers (1982), the results obtained by Dymond for hard spheres suggest that instead of eq 18, the equation should be expressed in the form 4
2
= j3P*6(VZ- V,)
The expression above was adopted by Matthews and Akgerman (1987) to describe mutual diffusion in n-alkane systems. The authors suggested simple relationships for both VD and j3 where
VD
0.308Vc
(20)
fl = 32.88M14's1V~-1.04 (21) Equations 19-21 enable one to calculate the binary diffusion coefficient given the critical volume of the solvent, V,, and the solute molecular weight, MI. However, instead of using eq 21, the coefficient j3 was calculated from the slope of the plot of D l 2 / P 5as a function of molar volume, for which the regression lines have been forced through 0.308Vc at D12equal zero. The average deviation between the experimental diffusivities and those correlated using the Dymond plot is shown graphically in Figure 13. An excellent correlation was observed for all the ester compounds studied with an average deviation of less than 2.6%.
(22)
where VD is similar to that proposed by Matthews and Akgerman (1987). In this case, the molar volume of the solute at ita normal boiling point was used to characterize ita size instead of the molecular weight. The average deviation of 2.5% between experimental and predicted diffusivities obtained using j3 values calculated from eq 22 was not significantly different from the previous method used. It must be added that as VDis a constant (only one solvent was studied), it can be raised to any power and the regression would fit just as well, provided that the appropriate adjustment to the coefficient (5.89 X lo5) is made. On the basis of eq 21, with VD-l.O1,the coefficient was found to be 2.09 X (see eq 23):
0 = 2.09 x 10-6vD-1'04v1-o*26
(23)
It is clear that the free-volume expressions provided a more successful means of correlating the experimental data than the Wilke-Chang and Hayduk-Minhas equations. While the correlation was satisfactory for linear compounds, ita performance with aromatic or cyclic compounds needs to be investigated. One drawback of using these free-volume expressions is the need to evaluate j3 and VDfor different solute-olvent systems. As with the previously discussed models, the inability to account for the structure of the diffusing solute remains a major hindrance to successful modeling of diffusivity data. Conclusion The binary diffusion coefficienta of behenic acid ethyl ester (Czz:o),DHA ethyl and methyl esters (C2?.), EPA methyl ester (CmS),and stearic acid ethyl ester (Cleo) in supercritical carbon dioxide were determined a t 308, 313, and 318 K for pressures ranging from 96.7 to 210.5 bar. The experimental diffusivities of these esters ranged from 1 X lo-' cm2/s a t the lower pressures to 0.5 X lo4 cm2/s at the higher pressures. The reliability of several correlations for diffusion coefficients, namely, the Wilke-Chang equation, the Hayduk-Minhas equation, and the freevolume expressions based on the rough hard sphere theory, has been assessed. In particular, the free-volume diffusion model was the moat successful at correlating the experimental data. It is clear that the structure of the molecule can have an important effect on diffusion in a supercritical solvent.
1335
Ind. Eng. Chem. Res. 1991,30, 1335-1342
The use of the molar volume (Wilke-Chang and Hayduk-Minhas) or molecular weight (Matthews-Akgerman) of the solute to characterize the solute size has ita limitations, for it has been shown that solutes of similar molar volumes or molecular weight can diffuse at different rates. None of the three correlations studied were able to account for the effects of the structure of the solute molecule on diffusion.
Acknowledgment Financial support of this work was provided by TECHn-Nu P/L and the Australian Government under the auspices of the Australian Research Council, Grant No. A88930202. bgirtry No. COS,124-38-9;Ca+ ethyl ester, 5908-87-2;Cm ethyl ether, 81926-94-5;CW methyl ester, 2566-90-7;Cm5 methyl ester, 2734-47-6;CIS* ethyl ester, 111-61-5.
Literature Cited Alizadeh, A,; Nieto de Castro, C. A,; Wakeman, W. A. The Theory of the Taylor Dispersion Technique for Liquid Diffusivity Measurements. Znt. J. Thermophys. 1980,l, 243. Angus, S.;Armstrong, B.; de Reuck, K. M. Carbon Dioxide: Znternational Thermodynamic Tables of the Fluid State; Pergamon Press: Oxford, 1976;Vol. 3. Aria, R. On the Dispersion of a Solute in a Fluid Flowing Through a Tube. Proc. R. SOC.London 1956,235,67. Chapman, S.; Cowling, T. G. The Mathematical Theory of NonUniform Gases; Cambridge University Press: Cambridge, 1970. Chen, S. H.;Davis, H. T.; Evans, D. F. Tracer Diffusion in Polyatomic Liquids. 111. J. Chem. Phys. 1982,77,2540. Debenedetti, P. G.; Reid, R. C. Diffusion and Mass Transfer in Supercritical Fluids. AZChE J . 1986,32,2034. Dyerberg, J. Linolenate Derived Polyunsaturated Fatty Acids and Prevention of Atherosclerosis. Nutr. Rev. 1986,44,125. Dymond, J. H. Corrected Enskog Theory and the Transport Coefficients of Liquids. J. Chem. Phys. 1974,60, 969. Funazukuri, T.; Hachisu, S.; Wakao, N. Measurement of Diffusion coefficients of C18 Unsaturated Fatty Acid Methyl Esters, Naphthalene, and Benzene in Supercritical Carbon Dioxide by a Tracer Response Technique. Anal. Chem. 1989,61,118.
Groves, F. R.; Brady, B.; Knopf, F. C. State of the Art on the Supercritical Extraction of Organics from Hazardous Wastes. CRC Crit. Rev. Environ. Control 1984, 15, 237. Hayduk, W.; Minhas, B. S. Correlations for Prediction of Molecular Diffusivities in Liquids. Can. J. Chem. Eng. 1982,60, 295. Hildebrand, J. H.Motions of Molecules in Liquids: Viscosity and Diffusivity. Science 1971,174,490. Knaff, G.; Schlunder, E. U. Diffusion Coefficients of Naphthalene and Caffeine in Supercritical Carbon Dioxide. Chem. Eng. Proc. 1987,21,101. Levenspiel, 0.; Smith, W. K. Notes on the Diffusion-Type Model for the Longitudinal Mixing of Fluids in Flow. Chem. Eng. Sci. 1957, 6,227. Lusis, M. A,; Ratcliff, G. A. Diffusion in Binary Liquid Mixtures at Infinite Dilution. Can. J. Chem. Eng. 1968,46,385. Matthews, M. A; Akgerman, A. Diffusion Coefficients for Binary Alkane Mixtures to 573 K and 3.5 MPa. AZChE J. 1987,33,881. Moulijn, J. A.; Spijker, R.; Kolk, J. F. M. Axial Dispersion of Gases Flowing Through Coiled Columns. J. Chromatogr. 1977,142,155. Reddy, K. A.; Doraiswamy, L. K. Estimating Liquid Diffusivity. Znd. Eng. Chem. Res. 1967,6,77. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Property of Gases and Liquids, 4th ed.; McGraw-HIlk Singapore, 1988. Saseiat, P. R.; Mourier, P.; Caude, M. H.; Roaset, R. H. Measurement of Diffusion Coefficienta in Supercritical Carbon Dioxide and Correlation with the Equation of Wilke and Chang. Anal. Chem. 1987,59,1164. Swaid, I.; Schneider, G. M. Determination of Binary Diffusion Coefficients of Benzene and Some Alkylbenzenes in Supercritical COz between 308 and 328 K in the Pressure Range 80 to 160 bar with Supercritical Fluid Chromatography (SFC). Ber. BunsenGes. Phys. Chem. 1979,83,969. Taylor, G. Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube. Proc. R. SOC.London 1953,219,186. Taylor, G. Conditions Under Which Dispersion of a Solute in a Stream of Solvent can be Used to Measure Molecular Diffusion. Proc. R. SOC.London 1954,225,413. Wells, P. A.; Foster, N.R.; Chaplin, R. P. Diffusion in Supercritical Fluids. CHEMECA '88, Z.E. Aust. Prepr. Papers, Sydney 1988, 898. Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solutions. AZChE J. 1955,1, 264.
Received for review August 13, 1990 Accepted December 10,1990
Extraction of Carboxylic Acids with Tertiary and Quaternary Amines: Effect of pH Shang-Tian Yang,* Scott A. White, and Sheng-Tsiung Hsu Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210
Extractions of carboxylic acids by tertiary and quaternary amines were studied under various equilibrium pHs ranging from 2.0 to 8.5. The quaternary amine Aliquat 336 extracted both dissociated and undissociated forms of acids, whereas the tertiary amine Alamine 336 extracted only the undissociated acid. Pure Aliquat 336 also had higher distribution coefficients, KD,than Alamine 336 at all pHs. In the intermediate pH range, KDdecreased with increasing the equilibrium pH of the aqueous phase. However, in the extremely high and low pH ranges, KD remained unchanged with pH. This pH dependency of KDcan be modeled by using a three-parameter equation derived from general chemistry principles. Extractions were also conducted with two diluents, kerosene and 2-octanol. In general, the polar diluent, 2-octanol, increased the extracting power of Alamine 336 by providing more solvating capacity for the nonpolar amine. In contrast, neither the polar nor the nonpolar diluent was active when used with Aliquat 336.
Introduction Recently, extractive recovery of carboxylic acids from dilute, aqueous solutions, such as fermentation broth and wastewater, which have acid concentrations lower than 10% (w/w), has received increasing attention (Helsel, 1977; Jagirdar and Shanna, 1980, Busche et al., 1982; Kertes and King, 1986; Tamada et al., 1990). Organic solvents used
for extraction can be categorized into three major types: (1)conventional oxygen-bearing and hydrocarbon extractanta, (2) phosphorus-bonded oxygen-bearing extractanta, and (3)high molecular weight aliphatic amines (Kertes and King,1986). Solvent extraction with conventional solvents such as alcohols, ketones, ethers, and aliphatic hydrocarbons is not efficient when applied to dilute, carboxylic
0888-5885/91/2630-1335$02.50/00 1991 American Chemical Society
