390
Ind. Eng. Chem. Res. 1992,31, 390-399
Diffusion of Fatty Acid Esters in Supercritical Carbon Dioxide K. Keat Liong,P. Anthony Wells, and Neil R. Foster* School of Chemical Engineering and Industrial Chemistry, University of New South Wales, P.O.Box 1, Kensington 2033, Australia
The binary diffusion Coefficients of palmitic acid (C16:O) ethyl ester were measured in supercritical carbon dioxide using a capillary peak broadening technique. Measurements were obtained in the temperature range of 308-318 K and at pressures between 96.7 and 210.5 bar. In addition, the applicability of several correlations to predict the diffusion coefficients of fatty acid ester-carbon dioxide systems determined by other workers was assessed. The models based on molecular connectivity were found to be the most quantitative in prediction of diffusion coefficients and were also found not to be system specific.
Introduction Supercritical fluid (SCF) extraction is an area of rapid technological development. However, the lack of fundamental thermodynamic data that are required for process design and scale up has hindered development of the technology to a commercial level. The quantitative knowledge of rate mechanisms such as diffusion coefficients in SCFs, and the ability to predict them, is of considerable importance in the design and efficient operation of SCF processes. Specific mention is made of the lack of experimental diffusion data in numerous reviews of SCFs (Paulaitis et al., 1983; Liong et aL,1991a). ks a consequence of the high pressures involved, these systems are highly nonideal and are not readily described by predictive mathematical models. Thus, few theoretical and empirical models exist with which to predict diffusion coefficients. The primary objective of this study was, therefore, to present experimental binary diffusion coefficients of palmitic acid (C160) ethyl ester in supercritical carbon dioxide, as a supplement to the existing results obtained by Wells (1991) for butyric (C4:0), caprylic ((2801, capric (ClaO), and myristic (C140) acid ethyl eaters and by Liong et al. (1991b) for stearic (C18:0), behenic (C22:O) and docosahexaenoic (C22:6) acid ethyl esters, and eicosapentaenoic (C205) and C22:6 acid methyl esters. (It must be noted that, in this study, the stated carbon number refers to the long-chain fatty acid moiety and does not include the methyl or ethyl group in the ester linkage.) Another objective was to use the substantial data obtained by these workers to assess the applicability of several currently available models for the diffusion behavior of fatty acid esters in supercritical carbon dioxide. Experimental Section The experimental binary diffusion coefficients in supercritical carbon dioxide were measured using the capillary peak broadening (CPB) technique. The details of the experimental equipment and procedure have been presented previously (Liong et al., 1991b), and therefore will only be described briefly here. Material. The palmitic acid ethyl ester was obtained from Sigma Chemicals (99% purity). The purity of the liquid carbon dioxide used was 99.8%. Equipment and Procedure. A flow diagram of the experimental apparatus used is presented in Figure 1. The apparatus consisted of a length of tubing (the diffusion column: L = 1443.5 cm; i.d. = 0.1025 cm) mounted in a constant-temperature environment, a six-port injection valve (Rheodyne), a UV-visible variable-wavelength detector (ISCO V4), an integrator (Shimadzu C-RGA), and two syringe pumps (ISCO LC-2600 and LC-5OOO). The
temperature and pressure of the system were monitored by a type “K”thermocouple connected to a multimeter (Fluke 8060A) and pressure transducers (Druck), respectively. The experimental apparatus was made up of two sections: the solvent-stream section and the solute-stream section. Supercritical carbon dioxide was passed through the solvent-stream section under laminar flow conditions. In the solute-stream section, liquid carbon dioxide was contacted with solute in the solute reservoir. The solute-laden stream was then directed through a heating coil to a six-port injection valve (Rheodyne, BpL sample loop). At a particular system pressure and temperature, three pulses of solute were injected into the solvent stream at 15min intervals. This time lapse was to prevent the peaks from overlapping during elution from the diffusion column. The pulse of solute, which broadened as it traveled down the length of the column, was monitored using an UVvisible detector. The diffusion coefficient was then determined from the shape of the peak recorded, as described by Liong et al. (1991b). When injection of the solute into the solvent stream was not required, it was bypassed into a collection device, and the solvent vented. The experiments were conducted at pressures in the range of 96.7-210.5 bar for temperatures from 308 to 318
K. Results and Discussion The experimentally determined diffusion coefficients for the palmitic acid ethyl eater in supercritical carbon dioxide are presented in Table I and are shown graphically as a function of pressure in Figure 2. Each diffusivity data point represents an average of three measurements, the results of which were generally precise to f l % . In the following discussion, the diffusion coefficients measured in this study, as well as those determined in previous studiea, for fatty acid eaters in supercritical carbon dioxide are examined. The diffusion coefficientsmeasured by the above-mentioned workers (Liong et al., 1991b; Wells, 1991) are presented in Table 11. The results indicate that the diffusivities of the esters in supercritical carbon dioxide are approximately cmz/s. The influence of the various solvent properties on the diffusion coefficient of these esters is initially discussed, followed by the influence of the solute parameters. Influence of Solvent Parameters. Influence of Pressure at Constant Temperature. The binary diffusion coefficients of palmitic acid ethyl ester in supercritical carbon dioxide were determined as a function of pressure at 308,313, and 318 K. The influence of pressure on the diffusion coefficients was less significant at higher pressures, as is illustrated in Figure 2. A similar trend was
0888-5885/92/2631-0390$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 391 1.4
1.21
$ 1 *
7.
i
4
I I I I I I I I I I
I
I I I I
+
I
5.
I I
0,s
2 0.6
-
I
0.4
50
100
150
200
250
Pressure (bar) Figure 2. Diffusion coefficient of the C16:O ethyl ester in carbon dioxide as a function of pressure.
SCF/Solute
0
313K
I.
Figure 1. Schematic of the capillary peak broadening apparatus: (1) syringe pump; (2) regulating valve; (3) injection valve; (4) solute pulse; (5) diffusion column; (6) constant-temperature bath; (7) detector; (8)integrator; (9) Gaussian profile. Table I. Experimental Diffusion Coefficients for the C160 Ethyl Ester-Carbon Dioxide System pressure/ density”/ viscosity”/ D,z x IO6/ bar k/L) CP (cm*/s) T = 308 K 8.42 96.7 700 0.0566 7.41 112.0 750 0.0638 7.15 120.0 768 0.0666 6.58 138.5 800 0.0720 6.04 160.1 828 0.0771 181.3 850 0.0814 5.74 ~~
i
0.4 50
100
150
200
Pressure (bar) Figure 3. Diffusion coefficient of the C16:O ethyl ester in carbon dioxide a function of pressure (ln-ln).
T = 313 K 96.7 103.0 114.0 132.7 163.2 210.5
600 650 700 750 800 850
109.4 117.9 131.7 153.7 188.0
600 650 700 750 800
0.0450 0.0505 0.0567 0.0637 0.0720 0.0815
10.8 9.61 8.47 7.53 6.75 5.89
0
n
T = 318 K 0.0452 0.0507 0.0569 0.0639 0.0721
11.0 9.72 8.49 7.56 6.74
“Data from Angus et al. (1976).
observed in all previously reported studies (Swaid and Schneider, 1979; Debenedetti and Reid, 1986). The sharp change in the diffusivities at low pressures indicates that the solvent density and/or solvent viscosity are important, as these properties change more rapidly at lower pressures. The influence of the density and viscosity on the diffusion coefficients will be discussed later. The effect of pressure on the diffusion coefficient also varies with temperature, as is illustrated on a ln-ln plot in Figure 3 for the C16:O ethyl ester. The results of a regression analysis of the experimental diffusivity data obtained in this study, as well as those obtained by Liong et al. (1991b) and Wells (1991),are presented in Table III. For the estexwarbon dioxide system at 308 K, the diffusion coefficient was observed to be inversely proportional to P0.6(Ho.02. This trend is consistent with the observations
306
310
315
320
Temperature (K) Figure 4. Influence of temperature on the diffusion coefficient of the C16:O ethyl ester at constant density.
made by Paulaitis et al. (1983), who reported that DI2a
PO.s. However at 318 K, the pressure exponent decreased to -0.71 i 0.03. As will be shown later, the influence of pressure is essentially the combination of changes in the density and viscosity of the fluid, and, as such, the effect is more pronounced at higher temperatures. Influence of Temperature. The influence of temperature on the diffusion coefficient will be discussed as a function of density and as a function of pressure. Influence of Temperature at Constant Density. The effect of temperature at constant density is shown in Figure 4. It was observed that the diffusion coefficient changed by no more 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
392 Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 Table 11. Experimental Binary Diffusion Coefficients for the Ester-Carbon Dioxide Systems DIz X 10'"/(cm2/s) Dlz x 106*/(cm2/s) preesure/bar C40 C80 C100 C140 C180 C220 C22:6 C205M' T = 308 K 1.35 1.09 1.00 0.91 7.95 7.54 7.79 96.7 8.08 1.18 0.98 0.93 0.80 6.95 6.75 6.78 112.0 7.11 120.0 1.14 0.94 0.87 6.80 0.78 6.41 6.52 6.76 1.07 0.80 0.70 6.20 0.87 5.85 138.5 6.17 6.28 0.80 0.74 0.64 160.1 0.99 5.79 5.56 5.72 5.77 0.76 0.71 0.61 181.3 0.94 5.50 5.17 5.36 5.50
C226M 7.97 6.92 6.67 6.23 5.74 5.39
T = 313 K 96.7 103.0 114.0 132.7 163.2 210.5
1.54 1.41 1.24 1.11 0.96
1.23 1.12 0.98 0.90 0.77
1.19 1.05 0.94 0.82 0.73
1.01 0.94 0.84 0.73 0.62
109.4 117.9 131.7 153.7 188.0
1.85 1.57 1.45 1.29 1.11
1.50 1.27 1.15 1.03 0.92
1.43 1.24 1.06 0.94 0.83
1.20 1.03 0.96 0.84 0.74
10.1 9.12 8.10 7.24 6.29 5.60
T = 318 K 10.5 9.40 8.18 7.34 6.40
9.91 8.75 7.79 6.78 6.11 5.31
9.98 8.95 7.98 6.92 6.18 5.48
10.5 9.21 8.07 7.19 6.36 5.62
10.1 8.86 7.96 7.11 6.27 5.57
10.1 8.80 7.95 6.90 6.13
10.2 8.73 8.00 7.01 6.26
10.6 9.41 8.18 7.27 6.41
10.3 9.00 8.17 7.18 6.41
"Data from Wells (1991). *Data from Liong et al. (1991b). CUnlessindicated by the letter 'M" (for methyl ester), the solute is an ethyl ester.
Table 111. Regression Analysis of the Diffusion Coefficient as a Function of Pressure slope of In DIz as a Function of In P ester 308 K 313 K 318 K -0.50 -0.61 c220 -0.72 C226 -0.48 -0.60 -0.72 -0.51 -0.58 -0.73 C22:6M -0.58 -0.68 -0.51 C205M -0.50 -0.60 -0.69 C18:O -0.50 -0.60 C160 -0.70 -0.48 -0.59 c40 -0.74 900 I
-10' 5.72
IO.9
0.8 0.7
I
500 50
0.6
100
150
0
DBIlDlfy
A
Vlmcomlty ( 3 U K )
200
5.74
5.75
5.76
5.77
In (Temperature, K) Figure 6. Influence of temperature on the diffusion coefficient of the C160 ethyl ester at constant pressure.
,O
0
700
J 5.73
l318K)
0.3 260
Prerrure (bar) Figure 5. Effect of pressure on the density and viscosity of carbon dioxide.
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 (constant density) changes in temperature was smaller than the experimental error. In a study of the -on behavior of solutes such as benzene, naphthalene, and phenanthrene in supercritical carbon dioxide over a 30 K temperature range, Sasaiat et al. (1987) observed an increase in diffusion coefficient of approximately 10% at a constant density of 800 g/L. The effect of pressure on the density and viscosity of supercritical carbon dioxide is shown in Figure 5. It was observed that,at constant density, the viscosity of the fluid is not a strong function of temperature. (For example, at
Table IV. Influence of Temperature on the Diffusion Coefficient at Constant Pressure ethyl ester pressure/bar slope of In D12vs h T c100 110 11.6 8.5 130 150 7.3 170 6.3 C160 110 11.6 130 7.8 150 6.5 170 6.1 C226 110 11.2 130 8.3 150 6.7 170 6.2
a constant density of 700 g/L, the viscosity of carbon dioxide varies less than 1%over a 10 K temperature range.) As a result, the variations in the diffusion coefficient are small. The small rise in the diffusion coefficient with increasing temperature could be attributed to the solute molecules becoming more energetic. Influence of Temperature at Constant Pressure. The effect of isobaric changes in temperature on the diffusion coefficient is shown in Figure 6 in a In-ln plot. It should be notad that the experiments were not conducted isobarically and that the data shown in Figure 6 were interpolated from experimental data. An increase in the
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 393
O.'t
.
Q
X
W
8
1
0
0.0
: a8 X
'
100
20
16
26
Figure 9. Relationship between the diffusion coefficient of the C16:O ethyl ester as a function of reciprocal viscosity. 0
508K
A
S1OK
Q
-9.5
B
"
I
0.6
10
1/viscosity (CP")
1.1 -
-B
6
1
-9.8
:t
8
-
-9.7
I 110
120
130
140
160
100
170
180
190
ZOO
VDensity x 10' ( V g )
-8.8
'
8
Figure 8. Relationship between the diffusion coefficient of the C160 ethyl ester as a function of reciprocal density.
diffusion coefficient was observed with an increase in temperature. In addition, the results indicate that the dependence of the diffusion coefficient on temperature decreased as the pressure was increased. This is clearly illustrated in Table IV, where the slope of the plot of diffusion coefficient as a function of temperature decreases with increasing pressure. The higher temperature dependence of the diffusion coefficient at lower pressures may be linked to the increasing temperature dependence of the fluid density as the pressure approaches the critical pressure. Influence of Solvent Density. The effect of solvent density on the diffusion coefficient of the C160 ethyl ester is shown in Figure 7. The decrease in the diffusion coefficient with increasing fluid density can be explained in terms of the path which the solute molecule takes. As the density of the solvent increases, the molar volume decreases, therefore the path taken by the molecule through the fluid becomes more hindered. In such a situation, collision transfer, rather than molecular transfer, becomes the dominant transport mechanism, and this results in a decline in diffusivity. The linear relationship between the diffusion coefficient of the C16:O ethyl ester and the inverse carbon dioxide density is illustrated in Figure 8. This relationship is consistent with the kinetic theory of dilute gases that states that diffusion is inversely proportional to density. Influence of Solvent Viscosity. The binary diffusion coefficients for the C160 ethyl estel-carbon dioxide system are plotted as a function of reciprocal carbon dioxide viscosity, as shown in Figure 9. Within experimental error, the data yielded a straight line through the origin. Similar behavior was observed for all the ester-carbon dioxide systems examined by Liong et al. (1991b) and Wells (1991).
I 8.06
6.1
8.16
8.2
8.26
8.3
6.36
In (Molar Volume, cma/mol)
Figure 10. Influence of molar volume on the diffusion coefficient of the ester compounds at 308 K.
-8.8' 8
I 8.06
8.1
8.16
8.2
6.26
8.3
8.36
In (Molar Volume, cma/mol)
Figure 11. Influence of molar volume on the diffusion coefficient of the ester compounds at 313 K.
These observations are not consistent with those of other workers (Feist and Schneider, 1982; Debenedetti and Reid, 1986) and suggest that the structure of the molecule can have an important effect on diffusion in supercritical solvents. The esters studied in this investigation, as well as in those of Liong et al. (1991b) and Wells (1991),were essentially long straight chain molecules compared to the solutes studied by other workers such as Feist and Schneider (1982) and Debenedetti and Reid (1986). The latter studies involved solutes which were generally more compact in nature. Effect of Solute Parameters. The size and shape of solute molecules have been shown by various workers (Sassiat et al., 1987; Debenedetti and Reid, 1986) to influence the rate of diffusion in a given solvent. The effect
394 Ind. Eng. Chem. Res., Vol. 31, No. 1,1992
1
-8.9
-141
I h
-9.7
c
A
I
A
I
A
-9.91 6
8.05
8.1
6.2
8.15
8.25
6.3
-
1 . 3 , I - T r I m e t h y l b e n ~ ~ S l o e e -0.02)
6.35
In (Molar Volume, cm'lrnol)
Figure 12. Influence of molar volume on the diffusion coefficient of the ester compounds at 318 K.
- 14.9
- 15.7
I
-3.5
I -3.4
-3.3
-3.2
-3.1
-3
-2.9
-2.8
-2.7
-2.8
-2.5
In (Viscosity, cP)
Figure 13. In (D,,/T) as a function of Ln fi for the ester compounds.
of the molar volume at the normal b o i i point of the ester compounds studied at 308,313, and 318 K at a constant density of 750 g/L is illustrated in Figures 10-12. A linear relationship was observed between ln D12and In Vl for the long-chain esters studied, with a slope of -0.32 f 0.05. This finding is in contrast to the results of Sassiat et al. (1987), who observed that the diffusion coefficient is proportional to V1-0.6f0.05. This discrepancy may be explained in terms of the different shapes of the molecules studied by Sassiat et al. (1987) compared to those studied in this investigation. The solutes studied by Sassiat et al. (1987) were more bulky (and hence voluminous) in shape compared with the esters, which are long straight chain molecules. The effect of the structure of a molecule on the rate of diffusion can also be observed from a plot of In (D12/T) as a function of In p (see Figure 13). The data of Sassiat et al. (1987) were used to highlight the effect of the structure of the molecule on the diffusion coefficient, as is illustrated in Figure 14. It was found through linear for the ester-carbon regression that D12/T 0: pF(.-1.05f0.03 dioxide systems. while the ring and substituted aromatic compounds exhibited a lowerpower dependence, D12j T oc U-0.69f0.04.
Data Correlation. The diffusivity data for the fatty acid esters were used to test the performance of several Stokes-Einstein-based equations, the Hayduk-Minhas correlation, a generalizedfree volume expression based on the rough hard sphere (RHS) theory of diffusion, and models based on molecular connectivity. Correlation with the Stokes-Einstein-Based Equations. 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 D12 = R T / 6 ~ p r
(1)
where 4 is an association factor. Scheibel equation
(
Dl2 = 8.2pVl1I3 x 10-8T( + 3:)2J3)
(3)
Reddy-Doraiswamy equation D12 = 52
TMz1J2 (4)
pV11J3V21/3
where 52 is a constant which is dependent on the relative molar volumes of the solvent and solute. For V2/Vl I1.5, 52 = 10 X otherwise 52 = 8.5 X Lusis-Ratcliff equation Dl2 = 8.52
X
(
10-'T( 1.40 ;)ll3
pV21J3
+
(:)-
(5)
where D12 is the diffusion coefficient (cm2/s), M is the molecular weight (g/mol), V is the molar volume at its normal boiling point (cm3/mol),p is the viscosity (cP),and T is the temperature (K). The applicability of these correlations to predict diffusion coefficients of the esters in supercriticalcarbon dioxide was studied. In all cases, the molar volumes of the solutes were estimated from the group contribution method of Le Bas (1915). The resulta of the average ratios of predicted diffusion coefficients to experimental diffusion coefficients are summarized in Table V. It was found that all the above-mentioned equations consistently overpredicted the experimental data with the single exception of the Wilke-Chang equation. This is in accordance with the trends observed by Debenedetti and Reid (1986) and Feist and Schneider (1982) that deviations from hydrodynamic behavior would lead
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 395 Table V. Average Ratios" of Predicted to Experimental Diffusion Coefficients for the Various Eauations Wilke- LusisReddy-Dorai- Haydukester Chann Ratcliff Scheibel swamv Minhas " 1.27 1.04 1.28 1.22 c40 0.95 1.34 0.99 1.17 0.88 1.21 C80 1.36 0.96 1.13 0.85 1.19 c100 1.42 0.92 1.20 0.81 1.20 c140 1.51 0.95 1.15 0.85 1.28 C160 1.50 0.91 1.11 0.84 1.25 C180 1.48 0.84 1.21 1.05 0.79 c22:o 0.88 1.49 1.23 1.08 0.82 C226 0.93 1.50 1.26 1.13 0.86 C205M 0.90 1.49 1.24 1.10 0.83 C226M "This average ratio represents the mean of all the ratios of the predicted and the experimental diffusion Coefficients obtained for each solute regardless of the pressures and temperatures at which they were obtained.
to overestimation of diffusion coefficients in SCFs by equations based on the Stokes-Einstein relationship. It must be noted that, for these equations, the molar volumes of the solutes have been evaluated at their normal boiling points according to the group contribution method of Le Bas and the molar volumes of carbon dioxide (the SCF) were evaluated by way of the empirical equation developed by Pitzer and Schreiber (1988), which depends on the system temperature and pressure. The latter approach was necessary because carbon dioxide does not have a normal boiling point. As such, the equations of Reddy and Doraiswamy (19671, Lusis and Ratcliff (19681, and Scheibel (1954) may not be applicable to systems involving carbon dioxide. The equation of Wilke and Chang does not involve the molar volume of the solvent but uses the molecular weight to characterize the solvent. In general, it was observed that the Wilke-Chang equation provided a good fit of the experimental diffusivity data, with an average ratio of predicted to experimental diffusion coefficients of 0.95 for the C40 ethyl ester to approximately 0.79 for the C22:O ethyl ester. The esters studied in the present investigation were essentially long straight chain molecules in comparison to 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, carbon tetrachloride, benzene, bromobenzene, 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 its longest axis (Swaid and Schneider, 1979). The experimentally determined diffusion coefficients of the esters in supercritical carbon dioxide were also compared with the predictions from the Hayduk and Minhas (1982) equation. This model was developed from a regression of data from systems involving binary mixtures of n-paraffins, and it was thought that it may well model the diffusion behavior of the structurally similar ester compounds. In contrast to the Stokes-Einstein-based equations, the Hayduk-Minhas correlation has a more flexible viscosity term. It is also interesting to note that the viscosity term is raised to a power that is dependent on the solute being considered. The average deviation was considerably less than that obtained with the Wilke-Chang equation. The ratios of predicted to experimental diffusivities ranged from 1.04 for the C4:O ethyl ester (cf. 0.95 with the Wilke-Chang equation) to about 0.84 for the C220 ethyl ester (cf. 0.79). As mentioned previously, the Hayduk-Minhas equation
was developed from a regression of data from systems involving n-paraffins. However, where dissimilar solutesolvent systems are being investigated, the interactions between these solute-solvent molecules would be less significant. This would consequently lead to higher than predicted diffusion rates. Thus the tendency of the Hayduk-Minhas equation to underpredict the estel-SCF system is consistent with this argument. One of the major weaknesses of the Stokes-Einsteinbased equations and the Hayduk-Minhas 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 Stokes-Einstein-based equations and the Hayduk-Minhas equations do not account for this structural difference. As a result of their simplicity, these correlations cannot be used to predict diffusivities to a high degree of accuracy. Correlation with a Generalized Free Volume Expression. 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 lowdensity diffusion coefficients by the approximate Enskog formulation (Chapman and Cowling, 1970). The results obtained by Dymond (1974) for hard spheres suggest that the free volume equation could be expressed in the form Dl2 = J[JF"(v2-
vd)
(6)
where Vd is a constant characteristic of the solvent and J[J is a constant characteristic of the solute and the solvent. 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 V, and 8, where vd
= 0.308Vc
at DI2= 0 cm2/s
(7)
0 = 32.88M14'61Vd-1'04 (8) Equations 6-8 enable one to calculate the binary diffusion coefficient given the critical volume of the solvent, Vc,and the solute molecular weight, Ml. In order to test the proposed model on the diffusion data obtained in this study, the coefficient B was calculated from the slope of the plot of DIz/ as a function of the molar volume of the solvent, for which the regression lines have been forced through 0.308Vc at DI2 = 0, rather than using eq 8. The mathematical expression for 8, eq 9, was then obtained by a regression of the slopes of the plots of D12/P5 as a function of molar volume for all the esters studied in supercritical carbon dioxide
8 = 9.50
X
10-5M1-'".53Vd-1.04
(9)
where v d (cm3/mol) is similar to that proposed by Matthews and Akgerman (1987). An excellent correlation was observed for all the ester compounds studied with an average deviation of less than 3%. It is clear that the free volume expressions provide a more successful means of correlating the experimental data than the Wilke-Chang and Hayduk-Minhas equations. While the correlation is satisfactory for linear compounds, the model does not appear to hold for aromatic or cyclic compounds. The free volume plot of the diffusion of such solutes as benzene and n-propylbenzene in supercritical carbon dioxide obtained by Swaid and Schneider (1979) is illustrated in Figure 15. At molar volumes below 80 cm3/mol, a linear relationship was observed between
396 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
0
20
40
60
60
100
120
140
-18.6' 4
180
Molar Volume (cm3/mol) Figure. 15. Free volume plot of the diffusion coefficienta of benzene and n-propylbenzene (data from Swaid and Schneider, 1979).
D12/!P5and V,. However, at molar volumes greater than 80 cm3/mol, there is considerable deviation between the experimental diffusivity measurementa and those predicted by the free volume expression (eq 6). Further, the x-axis intercepts were significantly lower than the value of 0.308 Vc as suggested by Matthews and Akgerman (1987). The intercepts varied between 10 and 15 cm3/mol for these compounds, and there appeared to be no correlation with either the location of the intercept or the slope of the lines. Therefore, the main drawback of using these free volume expressions is that they are system specific, in that @ and Vd must be evaluated for each solutesolvent system. The free volume expression for the diffusion behavior of fatty acid esters in supercritical carbon dioxide was found to be quantitatively described by eqs 6, 7, and 9. Correlation with the Models Based on Molecular Connectivity. All previous models discussed thus far suffer from an inability to account for the effects of molecular structure on diffusion. The shape and size of a solute have been shown by other workers, such as Swaid and Schneider (1979), to affect diffusion. In addition, the models discussed have been developed for describing diffusion in liquid systems, and not specifically for SCF systems. Wells (1991) has developed several models based on molecular connectivity specifically for SCF systems. It was clearly shown that the molecular connectivity term completely characterized the role of the shape and size of a solute in the diffusion process. Some background on the parameters describing molecular shape is provided in the Appendix. In adopting the Stokes-Einstein relationship as a starting point for analysis, Wells (1991)regressed the group (D1*/7') against Ox, OxM1, and 2 ~ M 1The . data of Swaid and Schneider (1979), Sassiat et al. (1987), and Wells (1991) at approximately 313 K and 160 bar were chosen as the basis for the regression. The details of this regression can be found elsewhere (Wells, 1991). The correlated data indicated the following relationships for the carbon dioxide solvent D l z p / T = 6.18 X 10-8(0x)4.50 (10)
D,+/T = 1.27 x 10-7(oX~,)4~25
(11)
D,,p/T = 1.07 X 10-7(2~M1)4~25
(12)
where Ox and 2x are the zero- and second-order molecular connectivity indexes, respectively. The average errors observed when comparison is made of the predictions obtained using eqs 10-12 with experimental diffusivity data were 2.7, 1.9, and 2.5%, respectively. However, it must be noted that eq 10 does not
I 4.6
5
5.5
6.5
6
7
In (Molar Volume) Figure 16. Plot of In (D1#/T) ae a function of the molar volume of the solute (1) acetone; (2) benzene; (3) C40; (4) n-propylbanzene; (5) 1,3,5-trimethylbenzene (Sassiat et al., 1987);(6) l,3,5-trimethylbenzene (Swaid and Schneider, 1979); (7) phenylacetic acid; (8) vanillan; (9) c80; (10) c100; (11)(2140; (12) C16:0; (13) C205M; (14) Cl80; (15) C226M; (16) C226; (17) C22:O. -17
1
-
t 3.
C
-18.6' 4
5
4.5
5.5
5
In (Molecular Weight)
Figure 17. Plot of In (Dl+/T) as a function of the molecular weight of the solute. (The numbers correspond to those given in the caption of Figure 16.)
account for the presence of double bonds in the esters. For example, the molecular connectivity indexes for both C22:O ethyl ester and C22:6 ethyl ester are identical. By taking into account the molecular weighta of the solutes, as in eq 11and 12, the molecular indexes for these compounds can be distinguished. Consequently, a reduction in the average errors was observed. It was evident from the results obtained that the molecular indexes have correlated well with the observed diffusion coefficients of the linear compounds. In particular, it is interesting to note that the lowest order molecular connectivityindex, Ox, correlated very well with the experimental diffusion data. In order to compare the data obtained to the models developed by Wells (19911, the group D1*/T was regressed against the molar volume of the solutes at the normal boiling point, and the molecular weight of the solutes. A similar regression procedure to that performed by Wells (1991)was used. The data obtained by Sassiat et al. (1987), Swaid and Schneider (1979), Liong et al. (1991b), and Wells (1991),as well as those obtained in this study, at 313 K and 160 bar were chosen as the basis for the regression. The resulta of the two regressions are shown in Figures 16 and 17, with regression coefficients of 0.976 and 0.984, respectively. The correlated data yielded the following relationships:
D 1 2 p / T = 2.10
10-7V1-0.43
(13)
D l 2 p / T = 2.42 x 10-7M14.48
(14)
X
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 397
4
4.6
6
5.6
e
8.5
7
7.5
8.5
8
o
0.5
ie
In (2x M1) Figure 18. Plot of In (D12c(/T)as a function of In (OxM1). (The numbers correspond to those given in the caption of Figure 16.) -17
t a
-17.6 -
I
0,
E
-18 -
-18.5' 4
'
'
'
'
'
'
'
4.6
6
6.6
8
8.5
7
7.5
' 8
'
8.5
' S
0.5
10
In ( *X x MJ Figure 19. Plot of In (D12c(/T)aa a function of ln(2xM1). (The numbers correspond to those given in the caption of Figure 16.)
The average errors obtained using eq 13and 14 to compare with experimental data were 4 and 5 % , respectively. In contrast, for the same compounds, the regression of In (D1+/T) as a function of ln (OxM1)and ln (2xMl)yielded regression coefficients of 0.993 and 0.994, respectively (see Figures 18 and 19). This regression analysis clearly highlights the superiority of the use of shape factors (molecular indexes) over the use of the molecular weight or the molar volume to characterize the solute size and shape. Compared to the predictions obtained using the free volume expressions, the correlations developed by Wells (1991) are not system specific and thus have a wide range of applicability. As a result of the limited amount of published diffusivity data in SCFs, a more detailed analysis of these correlations could not be undertaken. However, the excellent correlation of the predicted diffusivity data (using eqs 11and 12) against the experimental diffusion coefficients obtained in this study clearly shows the ability of the molecular indexes to characterize the shape and size of the solute in the diffusion process.
Conclusion The binary diffusion coefficients of palmitic acid ethyl ester (C160) in supercritical carbon dioxide were determined at 308,313, and 318 K for pressures ranging from 96.7 to 210.5 bar. The experimental diffusivities of the C16:O ethyl ester ranged from 1 X lo4 cmz/s at the lower pressures to 0.5 X lo4 cm2/s at the higher pressures. The diffusion coeffcients obtained in this study supplemented the existing data determined by Liong et al. (1991b) and Wells (1991) for fatty acid esters with carbon numbers ranging from 4 to 22 in supercritical carbon dioxide. Some
general trends have been established from the substantial data obtained. The binary diffusion coefficients were found to decrease with an isothermal increase in pressure, but increase with an isobaric increase in temperature-consistent with trends observed in all previously reported studies (Swaid and Schneider (1979) and Sassiat et al. (1987)). The influence of temperature on the diffusion coefficient at constant density was very small (approximately 10% change over a 10 K temperature range). It was observed that the viscosity of carbon dioxide is not a strong function of temperature at constant density. This led to the conclusion that changes in the diffusion coefficients at isothermal and isobaric conditions were a result of the variations in the density and viscosity of the fluid. The binary diffusion coefficients for the esters, plotted as a function of reciprocal carbon dioxide viscosity, yielded a straight line through the origin within experimental error. This observation is in agreement with the assumption made in the Stokes-Einstein relationship, that the diffusion coefficient is inversely proportional to viscosity. However, it is not consistent with the observations made by other workers, such as Swaid and Schneider (1979) and Debenedetti and Reid (1986). The solutes studied by these workers were more bulky in nature (for example, benzene, phenanthrene, and chrysene, among others), compared to the essentiallylong straight chain molecules studied in this present investigation. This suggests that the structure of the molecule has an important effect on diffusion in the supercritical solvent. The reliability of several correlations for diffusion coefficients, namely, the Stokes-Einstein-based equations, the Haydul-Minhas equation, the free volume expression based on the rough hard sphere theory, and the models based on molecular connectivity, has been assessed. In particular, the free volume diffusion model and the models based on molecular connectivity were most successful in correlating the experimental data. Correlation of the experimentally determined diffusion coefficients with the Stokes-Einstein-based equations resulted in an overestimation of the experimental data (up to 50%) with the single exception of the Wilke-Chang equation. The failure of these correlations to model the role of the size and shape of the solute and the fluid viscosity in the diffusion process is thought to be the main cause of the observed errors. The free volume relationship derived from rough hard sphere theory provided a good estimate of the experimental diffusion coefficients. However, it was found that the equations proposed by Matthews and Akgerman (1987), relating the regressed slopes and intercepts of the plot of D 1 2 / P 5versus solute molar volume, were not applicable to the systems studied in this investigation. This was because the regressed slopes and intercepts were characteristic of each solute. Thus the main disadvantage of the free volume equation is that it is system specific. As a consequence, each system needs to be assessed individudY. It is clear that the structure of the molecule can have an important effect on diffusion in a supercritical solvent. The use of the molar volume (Stokes-Einstein-based equations and that of Hayduk-Minhas) or molecular weight (Matthews-Akgerman equation) of the solute to characterize the solute size has its limitations, for it has been shown that solutes of similar molar volumes or molecular weight can diffuse at different rates (Swaid and Schneider, 1979). None of these three types of correlations were able to account for the effects of the structure of the
398 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 ~~
~~~
~
I
A
Valencies:
Pair of Edger
Edae
Vert..
B
C
D
E
F
C
6~ = 6 1 = 6 L = 1 6, = = dD = = dr = dc 6n = 3
H
=
J
K
L
All Vertices Represent Carbon Atoms with the Exception Of Atoms I and J which are Oxygen
& J = bK = 2
I
Order of Index
Paths
E. C. D
Zeroth
A.
Pint
A-E, E-C. C-D
ox =
etc.
etc.
1x
=
2x
=
(No. of Edge# = 1 1 )
Second
A-E-C, E-C-D. etc. (No. of Palm of Edger = IO)
Figure 20. Method for determining low-order molecular connectivity indexes for the C80 ethyl ester.
solute molecule on diffusion. However, it was found that the models based on molecular connectivity were able to describe the role of the shape and size of the solute in the diffusion process. The average percentage errors obtained from predictions using the molecular connectivity models with observed diffusion coefficients were significantly lower than those obtained using the Stokes-Einstein-based relationships and the Hayduk-Minhas relationship. In comparison to the free volume equation, the models based on molecular connectivity had a wide range of applicability. In summary, the transition of SCF extraction technology from laboratory-scale experiments to a pilot plant will depend on the progress that is made in understanding the fundamentals of the extraction process. As diffusion is often the dominant rate mechanism, the knowledge of, and the ability to predict, transport properties such as diffusion coefficients in SCFs is of considerable importance to the design of equipment for SCF extraction. Although there is still a lack of diffusion data available for SCF systems, it is possible to discern some trends in the relationships between the diffusion coefficient and the various solute and solvent properties. There are also indications that the size and shape of the solvent molecule may play a part in the diffusion process, in addition to the role of the solute (Wells, 1991). An investigation of the diffusion of a single solute in various SCFs would aid in a greater understanding of the role played by the solvent molecule. The development of reliable models for estimating the diffusion coefficient is still in the early stages. The models based on molecular connectivitydeveloped by Wells (1991) hold much promise. Appendix ParametersDescribing Molecular Shape. There are numerous examples of physicochemical properties that correlate well with molecular size and, as a consequence, lend themselves to estimation of properties by way of group contribution methods. One such example is that of Schroeder (Partington, 1949),who suggested a novel and simple additive method for the estimation of molar vol-
umes at the normal boiling point. The Schroeder method involves counting the number of atoms of carbon, hydrogen, oxygen, and nitrogen, adding 1 for each double bond, and multiplying the sum by 7. This rule is suprisingly good, in that results can be obtained to within 3 or 4% of the experimental values, except for highly associated liquids (Reid et al., 1987). However, there are other properties in which the structure of a molecule is also important. As an example, the boiling points of alcohols have been observed to be fundions of both shape and size. For example, the boiling point of 1-propanol is 370 K, while that of 2-propanol is 355 K. A greater degree of branching in alcohols has been postulated to lead to a reduction in the surface area available for interaction between the liquid-phase molecules. This resultant decrease in attraction results in a lowering of boiling points. It is, however, difficult to characterize the degree of branching. "Should 3-methylhexane, for instance, be considered more branched than 3-ethylpentane?" (Randic, 1975). The field of chemical topography is thus developed in order to answer this problem. In undertaking a topographical analysis of a molecule, its three-dimensional shape is ignored and the hydrogen atoms are stripped from the carbon backbone. The resulting skeleton structure is then "laid flat", and only the number of atoms and the way in which they are connected are considered. The lengths and the angles of the bonds are also ignored. Weiner (1948)proposed the first topographical index, in which the carbon atoms are represented by points, known as vertices. The carbon-carbon bonds are termed edges. The Weiner number is equal to the sum of the edges between all pairs of vertices. Therefore molecules that contain more vertices would result in a larger Weiner number. For example, the Weiner number for butane is 10,while that of 2-methylpropane is 9. A refinement of this topographical index, developed by Randic (1975),is more sensitive to the degree of branching. The manner in which the Randic number is determined is similar to that employed in the calculation of the Weiner number, in that the molecule of interest is stripped of its hydrogen atoms and double bonds are treated as if they
Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 399 Table VI. Low-Order Molecular Connectivity Indexes the Ester Compounds Studied molecular weight solute OX 6.406 116.2 c40 9.234 172.3 C80 10.648 200.3 c100 13.471 256.4 c140 14.891 284.5 C160 16.305 312.5 C180 19.134 368.7 c220 19.134 356.5 C22:6 17.012 316.5 C205M 18.427 342.5 C22:6M
for 2X
2.683 4.097 4.804 6.218 6.925 7.632 9.047 9.047 7.959 8.666
were single bonds. In molecular connectivity calculations the carbon atom is used as the “standard” atom. Heteroatoms are regarded as the equivalent of carbon atoms. Two methods can be employed in order to determine the molecular connectivity of a molecule. However, only the method employed in the determination of molecular connectivities in this study is discussed. The reader is referred to the paper by Randic (1975) for more information. The method involves assigning to each vertex a value equal to the number of carbon atoms to which it is directly connected. This value is known as the valence of the vertex (hi). In order to calculate the zero order of the Randic number or molecular connectivity index, O x , the reciprocals of the square roots of the valences are summed “ 1 0x = C (AI) i=l g . W where n is the number of vertices in a molecule. The zero order is the simplest order of the molecular connectivity index and conveys the least information about bonding in a molecule. The first-order molecular connectivity, ‘x, is calculated by assigning a value of (6ibj)4.5 to each edge, where 6;and Si are the valences of the atoms at each end of the edge. 4 1 (A21 1 x = Clj2 m=l
E,
where E, is the value assigned to edge m (=bibi), q is the number of edges possessed by the molecule, and 6i and bj are the valences of the vertex at each end of the edge. The second-order index, 2x, is calculated by assigning a valence to each pair of edges attached to a common vertex.
where E,, is the value assigned to a pair of edges p (‘bib,&), z is the number of pair of edges possessed by a molecule,
bi and bk are the valences of the vertex at each end of the edge, and Si is the valence of the vertex common to two edge pairs. According to Randic (1975),the number of orders of the molecular connectivity index can be extended to take into account a path length of any value. The formula for determining r orders of molecular connectivity is given in eq A4, where r is the order of the molecular connectivity index, f is the order of possible paths in a molecule.
An example of the methods for determining the first three orders of the molecular connectivity index for the C8:O ethyl ester is illustrated in Figure 20. An example
of the zero and second orders of molecular connectivities calculated for several fatty acid esters is presented in Table VI. It must be noted that the presence of heteroatoms such as chlorine or oxygen, which differ in mass from that of a carbon atom, may af€ectthe diffusion behavior of a solute and thus must be taken into account. One method of accomplishing this is to group the molecular connectivity index with the molecular weight of the solute in study. &&try NO. C22:0, 5908-87-2; C226, 73310-11-9; C22:6M, 2806146-3; C20:5M, 28061-45-2; C180,111-61-5; C160,628-97-7; C40,105-54-4; C100,110-38-3; C8:0, 106-32-1; C140, 124-06-1; CO, 124-38-9.
Literature Cited Angus, S.; Armstrong, B.; de Reuek, K. M. Carbon Dioxide: International Thermodynamic Tables of the Fluid State-9; Pergmon Press: Oxford, U.K., 1976. Chapman, S.; Cowling, T. G. The Mathematical Theory of NonUniform Gases; Cambridge University Press: Cambridge, U.K., 1970. Debenedetti, P. G.; Reid, R. C. Binary Diffusion in Supercritical Fluids. AIChE J. 1986, 32, 2034. Dymond, J. H. Corrected Enskog Theory and the Transport Coefficients of Liquids. J. Chem. Phys. 1974, 60, 969. Feist, R.; Schneider, G. M. Determination of Binary Diffusion Coefficients of Benzene, Phenol, Naphthalene and Caffeine in Supercritical Carbon Dioxide Between 308 and 333K in the Pressure Range 80 to 160 Bar with Supercritical Fluid Chromatography (SFC). Sep. Sci. Technol. 1982,17, 261. Hayduk, W.; Minhas, B. S. Correlations for Prediction of Molecular Diffusivities in Liquids. Can. J. Chem. Eng. 1982,60,295. Le Bas, G. The Moleculur Volumes of Liquid Chemical Compounds; Longmans: New York, 1915. Liong, K. K.; Wells, P. A.; Foster, N. R. Diffusion in Supercritical Fluids. J. Supercrit. Fluids 1991a 4,41. Liong, K. K.; Wells, P. A.; Foster, N. R. Diffusion Coefficients of Long Chain Esters in Supercritical Carbon Dioxide. Ind. Eng. Chem. Res. 1991b, 30, 1329. 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 573K and 3.5 MPa. AIChE J. 1987,3,881. Partington, J. An Advanced Treatise on Physical Chemistry, Fundamental Principles: The Properties of Gases; Longmans: New York, 1949; Vol. 1. Paulaitis, M. E.; Krukonis, V. J.; Kurnik, R. T.; Reid, R. C. Supercritical Fluid Extraction. Rev. Chem. Eng. 1983, 1, 179. Pitzer, K. S.; Schreiber,D. R. Improving Equation of State Accuracy in the Critical Region; Equations for Carbon Dioxide and Neopentane as Examples. Fluid Phase Equilib. 1988,41, 1. Randic, M. On Characterization of Molecular Bonding. J. Am. Oil Chem. SOC.1975,97,6609. Reddy, K. A.; Doraiawamy, L. K. Estimating Liquid Diffusivity. Id. Eng. Chem. Fundam. 1967,6, 77. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hik New York, 1987. Sassiat,P. R.;Mourier, P.; Caude, M. H.; Rasset, R. H. Measurement of Diffusion Coefficients in Supercritical Carbon Dioxide and Correlation with the Equation of Wilke and Chang. Anal. Chem. 1987,59, 1164. Scheibel, E. G. Liquid Diffusivities. Ind. Eng. Chem. 1954,46,2007. Swaid, I.; Schneider, G. M. Determination of Binary Diffusion Coefficients of Benzene and Some Alkylbenzenes in Supercritical C02between 308 and 328K in the Pressure Range 80 to 160 bar with Supercritical Fluid Chromatography (SFC). Ber. BunsenGes. Phys. Chem. 1979,83,969. Weiner, H. Vapour PreasureTemperature RelationshipsAmong the Branched Paraffin Hydrocarbons. J. Phys. Chem. 1948,52,425. Wells, P. A. Diffusion in Supercritical Fluids. Ph.D. Dissertation, The University of New South Wales, Kensington, Australia, 1991. Wilke, C. R.; Chang, P. Correlation of Diffusion coefficients in Dilute Solutions. AIChE J. 1955, 1 , 264.
Received for review April 15, 1991 Revised manuscript received August 19, 1991 Accepted August 27, 1991
