Ind. Eng. Chem. Res. 1997, 36, 923-931
923
Infinite-Dilution Diffusion Coefficients in Supercritical Fluids Anthony P. Eaton and Aydin Akgerman* Chemical Engineering Department, Texas A&M University, College Station, Texas 77843-3122
A predictive equation, based on the rough-hard-sphere theory, was developed for the determination of infinite-dilution molecular diffusion coefficients in supercritical fluids. The equation parameters were determined from molecular dynamics simulations and from experimental data on 1-octene diffusion in supercritical ethane, propane, and hexane over a wide range of temperatures and pressures for supercritical fluid densities in the range of 69.9-500.2 kg/m3. An extensive database on diffusion in supercritical fluids was collected through the literature in order to evaluate the proposed equation and to determine the accuracy of the predictions with no additional parameter adjustment. The predictions were very good, with an average absolute deviation of 15.08% over 1548 data points. For systems where actual data on the critical constants were available, the average error of the predictions reduces to 9.61% over 939 points. Introduction Supercritical fluids (SCFs) have been finding new and increased applications during the last decade. Their desirable properties of low viscosity, high diffusivity, and adjustable solvency power have led to numerous uses as solvents in chemical reactions, adsorption/ desorption processes, and extraction processes. Proper evaluation of the technical feasibility of using supercritical fluids in these processes necessitates the hydrodynamic and mass-transfer parameters, the estimation of which requires reliable values for the molecular diffusion coefficients of various solutes in the supercritical fluid of interest. Traditional predictive equations for diffusion in liquids or for diffusion in gases are not applicable in the supercritical region, giving estimates which may be off by an order of magnitude or more. Equations based on two theories, the hydrodynamic theory and the Enskog dense-gas theory, give acceptable predictions at different density ranges. Variations of the Stokes-Einstein equation, which is based on the hydrodynamic theory, yield acceptable predictions in the high-density region of supercritical fluids, probably due to the clustering phenomenon. The Enskog dense-gas theory, on the other hand, is accurate for predictions in the gaslike low-density region of supercritical fluids. The definition of high and low density of SCFs is somewhat arbitrary, and neither equation can be extrapolated throughout the whole range of densities. Since supercritical fluids cover a wide range of densities for a given solvent/solute interaction, a predictive equation which is applicable through a wide range of conditions would prove useful. The rough-hard-sphere theory has also been employed by a few investigators for modeling diffusion coefficients in supercritical fluids (Chen, 1983; Erkey et al., 1990; Akgerman et al., 1996). In this model the ratio of the smooth-hard-sphere diffusivity to Enskog diffusivity, which is calculated through molecular dynamics simulations, is needed. Unfortunately, molecular dynamics simulations have only been completed for a very narrow range of densities, molecular weight ratios, and molecular diameter ratios. Therefore, Chen (1983) attempted to use experimental data from Swaid and Schneider (1979) to improve molecular dynamics corrections, * Author to whom correspondence should be addressed. Telephone: 409-845-3375. Fax: 409-845-6446. Email: akg9742@ chennov2.tamu.edu. S0888-5885(96)00580-5 CCC: $14.00
which resulted in considerable improvement to the rough-hard-sphere prediction above the accuracy of the Enskog prediction used by Swaid and Schneider. Erkey et al. (1990) developed subsequent correlations based on both rough-hard-sphere and hydrodynamic theories to account for self-diffusion data of supercritical fluids, improving Chen’s approach, but did not account for binary diffusion coefficients. Akgerman et al. (1996) were able to adjust these correlations to account for binary diffusion of high-molecular-weight hydrocarbons in supercritical carbon dioxide. However, without extensive literature data, their correlation cannot be extended beyond this realm. Sun and Chen (1985a,b) have taken a different approach based on the work of Sung and Stell (1984). Their theory is based on an extensive derivation of the Stokes-Einstein equation. The correlation is extremely accurate in predicting selfdiffusion coefficients but fails in the prediction of infinite-dilution binary diffusivities. Diffusion coefficient measurements in the supercritical regime are somewhat limited. A recent review summarized some of the experimental techniques, data, and theoretical approaches (Liong et al., 1991a). There are approximately 95 solvent/solute pairs investigated in the literature. Of these, roughly 80% fall into two categories: diffusion of fatty acid ethyl and methyl esters in carbon dioxide (Funazukuri et al., 1989, 1991, 1992; Liong et al., 1991b, 1992) and diffusion of benzene and benzene derivatives in carbon dioxide and lowmolecular-weight alcohols (Bueno et al., 1993; Suarez et al., 1993; Chen, 1983; Swaid and Schneider, 1979). All diffusion measurements were made by the Taylor dispersion technique except for Catchpole and King (1994), who used the capillary evaporation technique. Theory Infinite-dilution molecular diffusion coefficients in liquids are readily correlated by the molecular dynamics approach employing the rough-hard-sphere theory of diffusion (Dymond, 1985; Dymond and Woolf, 1982; Easteal and Woolf, 1984a,b; Easteal et al., 1983). In molecular dynamics simulations, Newton’s laws of motion are used to calculate collisions of an ensemble of particles through computer simulations. From the time correlation of this motion, the transport properties of the fluid interaction can be determined. The calculations require the mass and diameters of the particles as well as the density of the solvent. The theory is © 1997 American Chemical Society
924 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Figure 1. Correlation of molecular dynamics simulation data given in Table 1.
independent of the fluid state and hence is applicable in the supercritical region as well. The rough-hard-sphere approach to diffusion results in the equation
D12,RHS )
[
]
kT (m1 + m2) 3 2 2π m1m2 8n2σ12
[
1/2
]
1 D12,SHS g(σ12) D12,E
)
D12,SHS D12,E
(
)F MD
A12 (1)
)
m1 σ1 V , , m 2 σ2 V 0
[
]
()
1 V )a V g(σ ) MD 0 12
(2)
R
-b
()
σ2 σ2 1 ) σ1 σ1 3
(3)
(4)
for a limited number of systems (naphthalene, phenanthrene, and hexachlorobenzene diffusion in supercritical carbon dioxide over a narrow temperature and pressure range). The same value of R was used in fitting eq 3 to the available molecular dynamics simulation data (Easteal and Woolf, 1990), and the results for the fitting are shown in Figure 1 for select cases. The linear fits are excellent, with regression coefficients better than 0.99 for most cases. Equation 3 can be rewritten as
[
]
V D12,SHS V0 D12,E
[( ) ]
1 V )a V g(σ ) MD 0 12
R
-
b a
2
0.8491
where a and b are constants relating to the slope and intercept of the fitted line. The value of R was determined by Akgerman et al. (1996) as
R)f
()
[
σ2 b ) -0.2440 a σ1
MD
Molecular dynamics simulations for (D12,SHS/D12,E)MD at different m1/m2, σ1/σ2, and V/V0 ratios (m2/m1 in the range 0.6-10, σ2/σ1 in the range 1.0-2.0, and V/V0 in the range 1.5-2.0) are provided by Easteal and Woolf (1990). The accuracy of these data has been tested against the molecular dynamics simulations of Chen (1983). The available data can be correlated by
V D12,SHS V0 D12,E
curves. These curves can be superimposed on each other and collapsed into a single relationship as shown in Figure 2 by adjusting the b/a ratio with the molecular weight ratio. A parabolic fit of the curve in Figure 2 yields the equation
b2 )
Through the molecular dynamics simulations it is shown that the ratio of the smooth-hard-sphere diffusivity to Enskog diffusivity is
(
Figure 2. Functionality for b2 (eq 6).
(5)
When the values of the intercept to slope ratio b/a are plotted versus σ2/σ1, they result in a set of parallel
+
()
]( )
σ2 m1 + 0.6001 σ1 m2
-0.03587
(6)
The equation enables prediction of (D12,SHS/D12,E)MD with an average absolute departure of less than (1.5%. Equation 4 reduces to the value of R ) 2/3 for selfdiffusion as reported by Erkey et al. (1990). The roughhard-sphere equation can now be represented by
D12 )
] [( ) ]
[
3aA12V0 kT (m1 + m2) 8(n2V)σ122 2π m1m2
1/2
V V0
R
- b2
(7)
where n2V and k are constants. Erkey and Akgerman (1989) showed that the translational rotational coupling parameter A12 is dependent only upon the ratio σ1/σ2, and we have shown that the constant a also depends only on the same ratio (Eaton, 1996); this function can be represented by
()
a or A12 ) δ
σ1 σ2
(8)
where δ and are constants but have different values for a and for A12. Substituting the correlation for a and A12 in eq 7 and combining the constants, eq 7 can then be reduced to
D12 ) βxT
( )[ σ1 σ2
γ
] ( )[( ) ]
(m1 + m2) m1m2
1/2
V0
σ12
2
V V0
R
- b2
(9)
Thus, in eq 9 there are two constants to be fitted from the data, β and γ, which are evaluated completely separately from R and b2 in eq 5. The values of β and γ are determined from experimental data, where R and b2 have already been determined separately from the molecular dynamics simulations. Although all the data in the literature can be correlated and these parameters fitted, β and γ would then be fitted parameters and it would be dangerous to extend the correlation to new systems. We postulate that β and γ are global parameters and, therefore, if they are evaluated from an accurate set of data at a wide range of conditions, then the same values should be applicable for different
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 925
systems at different conditions. We obtained data employing a noninteracting system at a wide range of conditions (temperature 314-563 K, pressure 44-124 bar, solvent density 70-500 kg/m3, and solvent molar volume 85-637 m3/mol) in a carefully designed Taylor dispersion apparatus and determined β and γ. In using eqs 4 and 9 with Figure 2 (or eq 6), the effective hard-sphere diameters of the solute and the solvent, σ1 and σ2, respectively, are needed. These values are determined from the Purkait and Majumdar (1981) semiempirical equation given by
σr ) 0.552803 - 0.0026776Tr
Table 1. Diffusion Coefficients of 1-Octene in Ethane
T/K
P/bar
322 340 357 378 414 503
124.1054 124.1054 124.1054 124.1054 124.1054 124.1054
(11)
σr ) σ/σc
(12)
σc ) (6Vc/πN)1/3
(13)
Experimental Section To determine the parameters β and γ, the infinitedilution diffusion coefficients of 1-octene in supercritical ethane, propane, and hexane were measured using the Taylor dispersion technique. The alkane system was chosen because it is noninteracting; the solute is an alkene since alkanes are UV invisible, whereas alkenes are visible. Ethane, propane, and hexane are structurally identical (except for chain length) but provide measurements at a wide range of temperatures and pressures, yielding a significant range of densities for the solvent. Previously, it was shown that mutual diffusion of homologous series components (alkane/ alkane, alkane/alcohol, and alcohol/alcohol) with chain lengths as high as C16 can be represented by the roughhard-sphere approach which assumes spherical solute and solvent molecules (Erkey and Akgerman, 1989; Erkey et al., 1991). The equipment was designed following the criteria of Erkey and Akgerman (1991), and preliminary experiments were performed to ensure the accuracy of the measurements. The supercritical fluid is pumped into a syringe pump and pressurized in a closed environment to the desired operating conditions. The fluid flows through stainless steel tubing to a preheater in the furnace with (1 °C temperature control. The supercritical fluid then passes through an injection valve and through a 15.84 m long Taylor dispersion coil, or through a bypass line. The switching between the diffusion tube, and the bypass was established by two zero-dead-volume connections. After the dispersion coil, the supercritical fluid passes through a cooling coil, in which it is cooled to ambient temperature, and the response is recorded by a high-pressure UV detector. The bypass was employed to measure the amount of dispersion in the system without the diffusion tube since there was significant dispersion due to the cooling. The system pressure is set by a series of two backpressure regulators, which also serve the function of reducing pulsations of flow through the system. The system pressure was determined by an internal pressure transducer controlled within (0.5 bar. The method of moments was used for parameter estimation (calculation of the diffusion coefficient from pulse response data) instead of the conventional curve fitting in the real-timedomain technique, since there was dispersion due to cooling before detection. The apparatus utilized and the
349.9 299.9 249.2 199.9 150.5 100.9
85.8703 100.1867 120.5698 150.3052 199.6412 297.7800
109D12/ (m2 s-1)
absolute error of prediction
16.26 ( 0.625 18.52 ( 0.940 22.67 ( 0.612 31.12 ( 0.346 37.61 ( 2.515 55.49 ( 4.476
0.704 5.84 5.67 4.63 1.98 1.60
Table 2. Diffusion Coefficients of 1-Octene in Propane
(10)
Tr ) T/Tc
molar F/ volume/ (kg m-3) (m3 mol-1)
T/K
P/bar
314 354 384 408 429 453 485 544 523 533 543
124.1054 124.1054 124.1054 124.1054 124.1054 124.1054 124.1054 124.1054 62.0527 62.0527 62.0527
molar F/ volume/ -3 (kg m ) (m3 mol-1) 500.7 450.7 401.1 350.2 300.6 249.6 200.6 150.4 73.64 71.31 69.18
88.0647 97.8345 109.9327 125.9109 146.6866 176.6587 219.8106 293.1782 598.7778 618.3425 637.3808
109D12/ (m2 s-1)
absolute error of prediction
11.81 ( 0.288 14.31 ( 0.393 16.29 ( 0.746 22.70 ( 0.547 26.88 ( 0.843 24.59 ( 0.664 44.88 ( 0.640 54.65 ( 0.395 86.89 ( 1.610 88.77 ( 3.551 90.61 ( 0.451
10.70 4.62 4.67 7.20 3.98 7.12 9.33 0.455 0.560 0.227 0.960
Table 3. Diffusion Coefficients of 1-Octene in Hexane
T/K
P/bar
483 124.1054 523 103.4212 523 82.7369 523 62.0527 523 44.8158 563 62.0527 563 44.8158
molar F/ volume/ -3 (kg m ) (m3 mol-1) 500.2 437.7 416.5 381.6 312.1 257.2 149.8
172.2711 196.8700 206.8908 225.8124 276.0974 335.0311 575.2336
109D12/ (m2 s-1)
absolute error of prediction
18.50 ( 0.325 22.50 ( 0.110 26.22 ( 0.715 29.70 ( 1.521 38.97 ( 2.336 53.00 ( 2.755 82.10 ( 3.145
4.79 2.54 1.74 1.82 5.52 5.08 4.53
data analysis technique are described in detail elsewhere (Eaton et al., 1995). Results and Discussion Data were collected for diffusion of 1-octene in supercritical ethane, propane, and hexane. These data are presented in Tables 1-3. The data were obtained over a wide temperature and pressure range, resulting in supercritical fluid densities in the range 43.57-500.70 kg/m3. Each entry in Tables 1-3 represents an average of at least six repetitive measurements. The uncertainties reported in these tables are for 3 standard deviations. All chemicals used had a purity of better than 98% and were used as received. The constants β and γ of equation 9 were then fit by a nonlinear regression technique and were determined as
β ) 4.486599 × 10-29
[x m s
]
g g mol K
(14)
γ ) 1.7538 The units of β give the units of the diffusion coefficient in m2/s; if other units are desired, then the value of β (as well as the units of other parameters in eq 9) should be adjusted accordingly. Figure 3 gives the predictions of the experimental data given in Tables 1-3 using the values of β and γ from eq 14 in eq 9 together with eq 4 and Figure 2. The predictions are excellent, with an average absolute error of 4.01%. The absolute errors of prediction are also included in Tables 1-3 for reference.
926 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Figure 3. Comparison of experimental and predicted diffusion coefficients for data presented in this study (Tables 2-4).
Figure 5. Parity plot for self-diffusion of supercritical carbon dioxide, ethylene, toluene, and chlorotrifluoromethane.
Figure 4. Comparison of experimental and predicted values of diffusion of coefficients of various solutes in alkanes.
Figure 6. Comparison of experimental and predicted diffusion coefficients of aromatics in carbon dioxide.
An accurate value of the density is needed for the determination of the solvent molar volume V in eq 9. Thus, a modified equation of state developed by Starling (1973) was used to determine the densities of alkane solvents at the experimental conditions for the diffusion data. The solution technique utilized a trial-and-error computer solution for the density. With accurately known pressures and temperatures, the equation of state is accurate to within a few percent. We evaluated the equation for 22 different systems (different alkanes, Tr in the range 0.8-1.1, Pr > 1.0) and found that the deviation was less than 3.5%. In evaluating the diffusion data reported in the literature, the measured values (temperature and pressure) reported were accepted as being exact; however, the density values (molar volume) reported at these conditions were in error of up to 15% for some cases. In comparing the model predictions to the literature data, the value of the molar volume at the experimental conditions, calculated from Starling’s equation, was used in eq 9 rather than the reported density values. Similarly, carbon dioxide densities were calculated from the IUPAC equation given by Angus et al. (1976), which is accepted as one of the most accurate equations of state for carbon dioxide. At a given temperature and pressure, if the densities reported in the literature were more than a few percent off from that calculated by the IUPAC equation, the calculated values are used in eq 9. Figure 4 presents data from the literature on diffusion in supercritical alkane solvents and the predictions using the same values of β and γ given in eq 14. Our data presented in Figure 3 are not included in Figure 4. The results indicate that the predictions are slightly higher than the experimental values but are still within the reported accuracy of the experimental data. We
Figure 7. Comparison of experimental and predicted diffusion coefficients of various solutes in carbon dioxide.
next attempted to use the same values of β and γ given above to predict self-diffusion coefficients and data on diffusion in supercritical carbon dioxide, the solvent that is used most extensively. Figure 5 shows our predictions of the data available on the self-diffusion coefficients of carbon dioxide (Chen, 1983; Takahashi and Hiroji, 1966), ethylene (Arends et al., 1981; Baker et al., 1984), toluene (Baker et al., 1985), and chlorotrifluoromethane (Harris, 1978). The predictions, with no parameters adjusted from the data, are excellent, with an average absolute error of 3.64%. Figures 6 and 7 are predictions of diffusion coefficients of various solutes in supercritical carbon dioxide. Some of the scatter in the predictions is due to the scatter in the data obtained by different investigators for the same system at the same conditions. Figures 8-10 show predictions of diffusion of various solutes in supercritical alcohols, 2,3dimethylbutane, and sulfur hexafluoride, respectively.
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 927
Figure 8. Comparison of experimental and predicted diffusion coefficients of various solutes in alcohols.
Figure 9. Comparison of experimental and predicted diffusion coefficients of various solutes in 2,3-dimethylbutane.
Figure 10. Comparison of experimental and predicted diffusion coefficients of various solutes in sulfur hexafluoride.
In general, the predictions are somewhat high for diffusion in 2,3-dimethylbutane and good for diffusion in alcohols and sulfur hexafluoride. In all of these cases (Figures 4-10) the predictions are within the range of errors of these sets of experimental data. However, for some systems, the predictions fail by more than 35%. The available literature data on diffusion in supercritical fluids, and very high density, near-critical fluids, and their predictions using eqs 4, 9, and 14 with Figure 2 (eq 6) are presented in Tables 4 and 5 again without any data reduction. These tables contain, to the best of our knowledge, all the published data in the literature on infinite-dilution diffusion coefficient measurements in supercritical and high-density near-critical fluids. It should be noted that we have not performed any data reduction and evaluated all the data in the literature as measured accurately and reported correctly. This was done on purpose in order to correctly show the
range of applicability of the proposed equation. Therefore, caution should be taken when attempting to use the data in these tables. The high-density near-critical fluids (Tr < 1.0) are also included to display the predictions of the proposed equation in that range as well. The tables are broken up into two classifications. It seemed that there was a definite differentiation between predictions of diffusivities of compounds with known critical constants (Tc and Vc) and predictions of diffusivities of compounds with estimated critical constants. Therefore, the two tables represent data for systems for which the critical constants are available from the literature (experimentally determined) and those for which these constants have to be predicted by group contribution techniques. The uncertainties involved in evaluating the critical constants introduce significant error in the prediction of the diffusion coefficient due to the fact that they are vital in the determination of the molecular diameters. The critical constants (Tc and Vc in eqs 11 and 13, respectively), if they are not available from the literature data, were determined by the group contribution methods of Joback, Ambrose, and Fedor (Reid et al., 1987). To determine the critical temperature of compounds for which the boiling point was unavailable or uncertain, the method of Fedor was used. For compounds for which the boiling point was known and accurate, the method of Joback was used. Reid et al. (1987) state that this is the most accurate way of determining the critical temperatures with known values of the boiling point. The critical volumes, in all cases, were also determined by the method of Joback, stated as the most accurate correlation by Reid et al. (1987). The method of Ambrose was used as a scale to determine if there were any predictions that could be considered unrealistic. Ambrose’s method predicted the values of the critical constants within 30% of the other methods utilized. Since this is the typical accuracy of the group contribution methods for a compound of this structure in determining the critical constants, the values were accepted and predictions were completed on these points. The average absolute departure for all points in Table 5 is 22.6%. In order to demonstrate the effect of the uncertainties in critical properties on the prediction of diffusion, we introduced 10-30% error (increase) to the critical temperature and volume of 1-octene (the solute) and calculated the diffusion coefficients with these new hypothetical critical constants compared to our experimental data. Table 6 shows the systematic increase in the error in predictions of diffusion coefficients due to this increase in critical constants. The first row is identical to the last column in Table 1 (except that in Table 1 the errors are reported as absolute errors); each row thereafter shows the prediction error due to error introduced to the critical constants shown in column 1 of Table 6. Therefore, the prediction technique presented here necessitates accurate values for the critical temperature and the volume of the solvent and solute or accurate values for the effective hard-sphere diameters. The overall average absolute deviation of the predictions for both tables (including 101 systems and over 1548 data points) is 15.08%, and the predictions are excellent to acceptable for most solvent/solute pairs, including the polar compounds. The average absolute error for the systems in Table 4 is 9.62%. This average does not include four sets of data on diffusion of ketones in carbon dioxide, for which the average absolute deviation was 37.82%. Since the predictions on these
928 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 Table 4. Comparison of Model to Literature Data for Systems with Known Critical Constants (Tc and Vc) solvent sulfur hexafluoride
ethylene chlorotrifluoromethane toluene 2,3-dimethylbutane
carbon dioxide
ethane n-hexane
ethanol
2-propanol
carbon dioxide
solute
reduced temp range (T/Tc)
density range (kg/m3)
average absolute percent error
naphthalene 1,3,5-trimethylbenzene 1,4-dimethylbenzene benzene methylbenzene tetrachloromethane carbon dioxide ethylene chlorotrifluoromethane toluene benzene naphthalene phenanthrene toluene acetone naphthalene phenanthrene 1,3,5-trimethylbenzene benzene ethylbenzene hexachlorobenzene i-propylbenzene m-xylene mestiylene n-propylbenzene o-xylene p-xylene pentachlorophenol toluene ethylene carbon dioxide 1-octene 1-tetradecene benzene mesitylene naphthalene p-xylene phenanthrene toluene benzene mesitylene naphthalene phenanthrene toluene benzene n-decane n-tetradecane naphthalene phenanthrene toluene
0.998-1.030 1.029 0.889-1.061 1.029 1.029 1.029 1.056-1.233 1.056-1.410 1.004-1.153 0.969-1.222 1.047-1.097 1.047-1.097 1.047-1.097 1.047-1.097 0.997-1.095 0.948-1.155 0.997-1.095 1.013-1.095 0.997-1.095 1.030-1.095 0.980-1.078 1.030-1.095 1.030 1.030 1.013-1.095 1.030 1.030 1.012-1.078 1.007-1.095 0.980-1.145 1.013-1.227 0.970-1.055 0.960-1.055 0.933-1.051 0.834-1.051 0.834-1.051 0.834-1.051 0.834-1.051 0.834-1.051 0.917-1.073 0.917-1.073 0.917-1.073 0.917-1.073 0.917-1.073 0.931-1.054 0.931-1.054 0.931-1.054 0.931-1.054 0.931-1.054 0.931-1.054
2-butanone 2-pentanone 2-propanone 3-pentanone
1.034 1.034 1.034 1.034
1108.5-1489.71 300-1400 300-1650 300-1400 300-1400 300-1400 11.04-342.16 125-580.1 351.03-1807.94 200.49-717.26 392.72-523.35 392.72-523.35 392.72-523.35 392.72-523.35 510-880 325.6-1112.28 325.61-900 510-800 280-800 607.1-911.2 400-900 607.1-936.1 795.57 795.57 607.1-800 795.57 795.57 400-900 607.1-936.1 92.22-789.44 158-757 308.4-399.1 308-404 346-677 346-622 346-622 346-622 346-622 346-622 398-772 398-567 398-567 398-567 398-557 417-569 417-569 417-569 417-569 417-569 417-569 average: 531-812 531-812 619-792 586-812 average:
13.74 18.70 16.07 24.32 19.21 28.80 7.87 3.76 3.11 0.45 21.12 7.52 4.85 14.00 5.28 12.42 14.87 5.66 6.04 8.91 10.71 8.44 10.73 8.40 10.09 11.58 13.12 3.34 6.45 22.62 5.29 1.64 7.66 19.42 10.22 6.12 7.90 5.15 12.11 11.71 12.92 15.93 15.10 14.14 7.35 10.48 6.19 8.46 6.43 8.21 9.61 35.81 37.22 26.01 43.02 37.82
citations y k k k k k cc ee, ff hh gg t t t t r b, g-j, l, o, q, r, z o, q, r r, bb a, d, g, h, l, r, s, bb, ii a, s o, q a, s a a a, s, bb a a o, q a, s, ii cc d, dd p p c, v v v v v v u u u u u aa aa aa aa aa aa e e e e, x
a Bueno et al, 1993. b Catchpole and King, 1994. c Chen et al., 1985. d Chen, 1983. e Dahmen, 1990a. f Funazukuri et al., 1991. Funazukuri et al., 1992. h Funazukuri et al., 1989. i Iomtev and Tsekhanskaya, 1964. j Knaff and Schlunder, 1987. k Kopner et al., 1987. l Lauer et al., 1983. m Liong et al., 1991b. n Liong et al., 1992. o Madras et al., 1994. p Noel et al., 1994. q Orejula, 1994. r Sassiat et al., 1987. s Suarez et al., 1993. t Sun and Chen, 1985a. u Sun and Chen, 1986. v Sun and Chen, 1985b. w Wells et al., 1992. x Dahmen et al., 1990b. y Debenedetti and Reid, 1986. z Lamb et al., 1989. aa Sun and Chen, 1987. bb Swaid and Schneider, 1979. cc Takahashi and Hongo, 1982. dd Takahashi and Hiroji, 1966. ee Arends et al., 1981. ff Baker et al., 1984. gg Baker et al., 1985. hh Harris, 1978. ii Levelt-Sengers et al., 1993. g
systems were not good and the literature data seemed to be taken in a well-defined manner, we have repeated the ketone experiments, using representative areas of the data including both end points and two additional data points in the middle, and measured diffusion coefficients of ketones (3-pentanone, 3-heptanone, and 2-heptanone) in supercritical carbon dioxide to compare our findings to the data in the literature. We obtained experimental values within (10% of the literature data. However, we observed that the Taylor dispersion peaks obtained were both non-Gaussian (exhibiting wider dispersion) and skewed with a tail, indicating that these
compounds do adsorb on the 316 stainless steel diffusion column (or somewhere else in the system). Figure 11 is a representative data set comparing the experimental data with the predicted response curve. The experimental pulse-response data will always result in a value for the diffusion coefficient, but it is imperative to make sure that the response curve is Gaussian. A nonGaussion curve with a tail implies adsorption, and the Aris/Taylor analysis of Taylor dispersion is invalidated (Alizadeh et al., 1980). Therefore, we believe that the data reported on these compounds are possibly in error as well since the authors also used a 316 stainless steel
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 929 Table 5. Comparison of Model to Literature Data for Systems for Which the Critical Constants Are Predicted by Group Contribution solute
reduced temp range (T/Tc)
density range (kg/m3)
average absolute percent error
citations
benzoic acid 2,2,4,4-tetramethyl-3-pentanone 2,4-dimethyl-3-pentanone 2-heptanone 2-nonanone 4-heptanone 5-nonanone 6-undecanone 3-heptanone behenic acid ethyl ester butyric acid ethyl ester capric acid ethyl ester caprylic acid ethyl ester cis-11-eicosenoic acid methyl ester docosahexenoic acid ethyl ester docosahexenoic acid methyl ester eicosapentenoic acid ethyl ester eicosapentenoic acid methyl ester erucic acid methyl ester linoleic acid methyl ester myristic acid ethyl ester myristoleic acid methyl ester nervonic acid methyl ester oleic acid methyl ester palmitoleic acid ethyl ester stearic acid ethyl ester benzoic acid caffeine chrysene d1-limonene glycerol trioleate indole linoleic acid oleic acid phenylacetic acid pyrene squalene stearic acid vanillin vitamin A acetate vitamin E vitamin K1 vitamin K3 1,3-dibromobenzene 2-propanone p-xylene
1.030-1.061 1.031 1.031 1.034 1.034 1.034 1.034 1.034 1.032 1.012-1.045 1.012-1.045 1.012-1.045 1.012-1.045 1.029 1.012-1.045 1.012-1.045 1.012-1.045 1.012-1.045 1.029 1.013-1.079 1.012-1.045 1.029 1.030 1.030 1.012-1.045 1.012-1.045 0.982-1.046 1.030-1.095 0.997-1.095 1.029 1.013 1.029 1.031 1.013-1.031 1.013-1.030 1.013-1.095 1.031 1.031 1.013-1.046 1.029 1.013-1.029 1.029 1.029 1.054 1.037 1.054
917.2-1400.6 619-792 619-792 531-812 586-812 619-792 586-812 586-812 596-812 600-850 600-850 600-850 600-850 795.57-879.97 600-850 600-850 600-850 600-850 795.57-879.97 739-929 600-850 795.57-879.97 792.3 792.3 600-850 600-850 537.2-858.7 550.1-795.31 800-880 795.57-879.97 867.3-902.5 795.57-879.97 703-812 657-902.5 600-850 650-880 703-812 739-792 600-850 795.57-879.97 795.57-902.5 795.57-879.97 795.57-879.97 400-1000 398-956 400-1000 average:
46.23 39.60 39.60 39.07 37.18 40.91 45.03 35.95 55.12 20.37 11.00 10.57 10.03 30.40 15.82 15.47 15.28 15.65 34.02 21.28 11.67 19.13 32.37 20.07 13.66 15.01 13.10 17.49 6.87 6.39 97.00 0.75 7.34 38.38 5.78 12.50 5.25 33.29 9.14 32.31 40.31 53.07 5.71 40.75 40.37 29.17 22.60
y e e e e e e, x e x m, n n n n g m, n f, m, n f, m n g f n g f f f, n m, n b, y j, l r g b g e b, e w r e e w g g g g g k k k
solvent sulfur hexafluoride carbon dioxide
chlorotrifluoromethane
a Bueno et al., 1993. b Catchpole and King, 1994. c Chen et al., 1985. d Chen, 1983. e Dahmen, 1990a. f Funazukuri et al., 1991. Funazukuri et al., 1992. h Funazukuri et al., 1989. i Iomtev and Tsekhanskaya, 1964. j Knaff and Schlunder, 1987. k Kopner et al., 1987. l Lauer et al., 1983. m Liong et al., 1991b. n Liong et al., 1992. o Madras et al., 1994. p Noel et al., 1994. q Orejula, 1994. r Sassiat et al., 1987. s Suarez et al., 1993. t Sun and Chen, 1985a. u Sun and Chen, 1986. v Sun and Chen, 1985b. w Wells et al., 1992. x Dahmen et al., 1990b. y Debenedetti and Reid, 1986. z Lamb et al., 1989. aa Sun and Chen, 1987. bb Swaid and Schneider, 1979. cc Takahashi and Hongo, 1982. dd Takahashi and Hiroji, 1966. ee Arends et al., 1981. ff Baker et al., 1984. gg Baker et al., 1985. g
Table 6. Average Errors for Prediction of 1-Octene Diffusion in Ethane with Error Introduced to Tc and Vc of 1-Octene average known values of Tc and Vc +10% error in Tc and Vc +20% error in Tc and Vc +30% error in Tc and Vc +10% error in Vc +20% error in Vc +30% error in Vc +10% error in Tc +20% error in Tc +30% error in Tc
0.704 3.64 8.90 15.43 4.10 7.04 9.62 0.986 1.22 1.42
5.84 11.23 17.75 25.83 1.64 2.00 5.21 5.48 5.17 4.91
diffusion column. In addition, although our predictions of the diffusion coefficients of acid esters in carbon dioxide were quite good, a closer analysis indicated that the trend in the data and predictions were skewed. Similarly, we measured diffusion coefficients of butyric acid ethyl ester, caprylic acid ethyl ester, maristic acid ethyl ester, and stearic acid ethyl ester in supercritical
5.67 11.82 19.26 28.52 0.900 3.24 6.88 5.23 4.87 4.56
-4.63 1.65 9.28 18.83 9.47 13.70 17.30 5.11 5.50 5.85
1.98 9.63 19.01 30.86 3.83 8.84 13.20 1.35 0.834 0.400
1.60 7.11 17.96 31.93 8.00 13.50 18.30 2.45 3.14 3.73
3.40 7.51 15.36 25.23 4.66 8.05 11.75 3.43 3.46 3.48
carbon dioxide at conditions reported in the literature. Our experimental data again matched the literature within (10%. However, we again observed that the Taylor dispersion peaks were non-Gaussian, more like a square peak (a peak with a flattened top portion and larger dispersion) with a tail. Figure 12 is a plot of a typical response, with the fitted theoretical curve which
930 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
A.P.E. was also supported by the Phillips Fellowship. The contribution of both is gratefully acknowledged. Notation Aij ) translational rotational coupling parameter a ) fitting parameter for self-diffusion b ) fitting parameter for self-diffusion D ) diffusion coefficient Dij ) mutual diffusion coefficient g(σij) ) radial distribution function k ) Boltzmann constant m ) mass of single molecule n ) number density T ) absolute temperature V ) molar volume VD ) molar volume scaling V0 ) close-packed hard-sphere volume Figure 11. Comparison of experimental response and the theoretical Gaussian response for diffusion of 3-pentanone in supercritical carbon dioxide.
Subscripts 1 ) solute 2 ) solvent E ) Enskog HSG ) hard-sphere gas MD ) molecular dynamics RHS ) rough hard spheres SHS ) soft hard spheres Greek Letters R ) exponential parameter, defined by eq 8 β ) slope parameter, defined by eq 11 γ ) V/V0 relationship parameter, defined by eq 11 σ ) diameter σij ) average diameter
Figure 12. Comparison of experimental response and the theoretical Gaussian response for diffusion of myristic acid ethyl ester in supercritical carbon dioxide.
lies within the experimental peak matching the maximum but neither the beginning and the end nor the steepness. This also indicates adsorption or other phenomena that invalidates Aris/Taylor analysis. Therefore, we believe that the data on the diffusion of acid esters should also be used with caution as well. Conclusions The proposed predictive equation based on the roughhard-sphere theory has been shown to successfully predict the range of binary infinite-dilution diffusion in supercritical fluid data available in the literature. The average absolute error of the predictions for 101 systems and over 1500 data points is 15.08%. This result is well within the experimental uncertainty of the binary systems studied. The equation developed involves two empirical global constants, β and γ, which are fitted from limited data at a wide range of conditions and fitted representation of the molecular dynamics simulations that can be duplicated by eq 5 with an accuracy better than 1.5%. The functionality of R has been shown to be a constant for all systems, including self-diffusion (σ1 ) σ2 and m1 ) m2). For systems where the predictions fail, it is shown that the data may be in error due to adsorption in the diffusion coil, invalidating the Aris/Taylor mathematical analysis of Taylor dispersion. Acknowledgment This work was supported by a grant from the Department of Energy (Grant No. DE-FG22-92PC92545), and
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Received for review September 23, 1996 Revised manuscript received December 13, 1996 Accepted December 16, 1996X IE9605802
X Abstract published in Advance ACS Abstracts, February 1, 1997.
