Ind. Eng. Chem. Res. 2000, 39, 835-837
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RESEARCH NOTES On the Measurement of Anomalous Binary Diffusion Coefficients in the Near-Critical Region Toshitaka Funazukuri,*,† Chang Yi Kong,‡ and Seiichiro Kagei‡ Department of Applied Chemistry, Institute of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan, and Department of Information and Systems, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan
The anomaly in binary diffusion coefficients has been reported by Nishiumi and co-workers in the near-critical region of pure CO2 solvent with the Taylor dispersion technique when the relatively large amount of solute as a tracer was injected. We have demonstrated that the mixture of the solute and CO2 solvent does not attain to the supercritical state at most axial positions of the diffusion column under their measurement conditions showing the anomaly. Introduction Recently, Nishiumi et al.1 have reported from the Taylor dispersion measurements that binary diffusion coefficients of acetone in carbon dioxide at 314.25 K seem to become zero as the pressure approaches decreasingly to the critical pressure of carbon dioxide when the injected amount of the tracer solute species is relatively large (higher than 5 µL). In their subsequent studies2,3 the similar observation on binary diffusion coefficients of benzene in carbon dioxide at 313.15 K has also been made. They claimed that the phenomenon was ascribed to the critical anomaly in binary diffusion coefficients and explained this with irreversible thermodynamics. Nishiumi and co-workers regarded their measurement conditions as the supercritical state because the temperatures and the pressures were higher than the critical point properties of pure carbon dioxide. In the Taylor dispersion method, a small amount of tracer solute species is usually imposed in a diffusion column so that we can regard the concentration of a solute as an infinite dilution. When a larger amount of tracer is injected, however, the system is affected by the properties of the mixture, not those of the pure solvent. The critical temperature of the system depends on the compositions of a solute and a solvent, namely, on the axial distance from the point the tracer is injected in the Taylor dispersion. This implies that the critical temperature of the mixture changes with the composition at the axial position of the column. When a relatively large amount of the solute is loaded like the measurements of Nishiumi and co-workers,1-3 the critical temperature of the mixture of the solute and the solvent is sometimes much higher than that of the pure solvent. In this case the measurement temperature is * To whom correspondence should be addressed. † Chuo University. Tel.: +81-3-3817-1914. Fax: +81-3-38171895. E-mail:
[email protected]. ‡ Yokohama National University. Tel.: +81-45-339-4399. Fax: +81-45-331-1027. E-mail:
[email protected];
[email protected].
lower than the critical temperature of the mixture even though it is higher than that of the pure solvent. Recently, the critical point properties of carbon dioxide with some entrainers at various compositions are available.4 In this paper we estimate the concentration profile of the tracer species in the column and examine whether the mixtures at various axial distances in the column actually attain to supercritical condition using the data of Reaves et al.4 Estimation of the Critical Temperature for the Mixture in the Diffusion Column As shown in the literature,5,6 the cross-section averaged concentration at z ) L of a tracer solute injected at z ) 0 can be described in eq 1,
C)
[
]
-(L - ut)2 m exp 4Kt πR2(4πKt)1/2
(1)
where K ) D12 + u2R2/48D12, m is the amount of a solute injected, D12 is the binary diffusion coefficient, u is the average solvent velocity, and R is the column inner radius. By using eq 1 with the data of Nishiumi et al.,1 the solute concentration along the axial distance is estimated. The critical temperature, Tc, of the mixture of acetone and carbon dioxide in Kelvin can be expressed by eq 2,
Tc ) 304.21 + 341x1
for x1 < 0.06
(2)
obtained by the least-squares method from the data of Reaves et al.,4 as shown in Figure 1, where x1 is the acetone mole fraction. We use the critical temperature of 304.21 K for pure carbon dioxide.7 Instead of comparison between the critical temperature of the mixture and their measurement temperature of 314.25 K, the acetone concentrations are compared. Because the critical temperature of the mixture increases with increasing acetone mole fraction in the mixture, as shown in Figure 1, the higher acetone concentration leads to the higher critical temperature
10.1021/ie990275r CCC: $19.00 © 2000 American Chemical Society Published on Web 11/06/1999
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Figure 1. Critical temperature of the mixture vs acetone mole fraction for the data of Reaves et al.4
Figure 2. Three-dimensional response curve reproduced under conditions listed in Table 1 when the amount of 13.1 µL is injected and the CO2 solvent flows at the lowest possible velocity (0.01 m s-1). Table 1. Parameters for the Estimation of the Critical Temperatures of the Mixtures Shown in Figures 2-4, for the D12 Data in the Paper of Nishiumi et al.1 temperature, K pressure, MPa solvent solvent velocity, m s-1 solute injected solute amount, m3 column radius, m column length, m binary diffusion coefficient, m2 s-1
314.25 11.87 CO2 0.01 and 0.025 acetone 0.7 to 13.1 × 10-9 0.44 × 10-3 10.2 1.98 × 10-8
of the mixture. The measurement temperature of 314.25 K is equivalent to the critical temperature of the mixture at the acetone mole fraction of 2.944 × 10-2, obtained from eq 2. Thus, we can regard the mixture as a supercritical state when the acetone concentration is lower than the equivalent value of 2.944 × 10-2. Results and Discussion Figure 2 shows a three-dimensional response curve reproduced by using the diffusivity data and the parameters reported in the paper of Nishiumi et al.1 These values are listed in Table 1, the injected tracer amount is 13.1 µL, and the CO2 flow velocity is the lowest possible value (0.01 m s-1). The L axis is the axial distance of the column. The t - Lu-1 axis is time difference from the mean residence time; i.e., t - Lu-1 ) 0 corresponds to the peak of the response curve measured at the axial distance at z ) L. The Tc axis is the tracer concentration in the mixture of acetone and
Figure 3. Regions showing supercritical and subcritical temperatures of the mixtures of acetone and carbon dioxide at 314.25 K and 11.87 MPa with 13.1 µL of acetone injection: (a) at u ) 0.01 m s-1 (the lowest possible solvent velocity); (b) at u ) 0.025 m s-1 (the highest possible solvent velocity) under the experimental conditions of Nishiumi et al.1 The broken lines are the peak widths at 10% of the peak heights of the response curves measured at L ) 10.2 m. The conditions employed in part a are the same as those in Figure 2.
CO2 in terms of the critical temperature of the mixture. The plane at Tc ) 304.21 K is equivalent to the tracer concentration of zero. The plane at Tc ) 304.33 K corresponds to the tracer concentration at 10% of the peak height of the response curve at column exit L ) 10.2 m. The response curve, at L ) 10.2 m, higher than the plane at Tc ) 304.33 K is employed for the fitting in the determination of the diffusion coefficient. The upper portion cut with the plane at Tc ) 314.25 K is the region where the critical temperature of the mixture is higher than the measurement temperature of 314.25 K, that is, in the subcritical state. Figure 3 shows the region where the mixture is in the subcritical or supercritical state on the L-(t - Lu-1) plane of the three-dimensional response curve shown in Figure 2. Unfortunately, the value of the solvent velocity is not available in their paper, and only we know the range of the values. Parts a and b of Figure 3 correspond to the lowest possible and the highest possible values, 0.01 and 0.025 m s-1, respectively, and the actual value at which Nishiumi et al. carried out is between them. The dotted regions designate that the acetone concentrations of the response curve are higher than those equivalent to the critical temperature of 314.25 K. Thus, the regions are under subcritical condition. The broken lines in parts a and b of Figure 3 show the peak widths at 10% of the peak height of the response curve measured at L ) 10.2 m, namely, the cross-sectional intercept lines obtained by cutting the three-dimensional response curve with the plane at Tc ) 304.33 K (see Figure 2). The area ratio of the dotted region showing the subcritical state to the inner region surrounded with the broken line affects the accuracy in the determination of the diffusion coefficients. When
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µL. Therefore, the binary diffusion coefficients, actually obtained under subcritical conditions, apparently lead to the anomaly. On the contrary, the anomaly does not appear when the injected amount is lower (0.7 and 1.1 µL). In this case the critical temperatures of the solute and the solvent mixture are lower than the measurement temperature in the nearly entire diffusion column. Conclusions The measurement conditions of the studies of Nishiumi and co-workers, showing the anomaly in binary diffusion coefficients under supercritical conditions of pure carbon dioxide, are found not to be actually in the supercritical state for the mixture when the relatively large amount of the tracer species is injected. Nomenclature C ) cross-section averaged concentration D12 ) binary diffusion coefficient L ) column length m ) injected amount of tracer R ) tube radius Figure 4. Effect of the amount of acetone injected on the critical temperatures of the mixtures of acetone and carbon dioxide at compositions of the top of the response curves (t ) Lu-1): (a) at u ) 0.01 m s-1; (b) at u ) 0.025 m s-1, under the same conditions as those shown in parts a and b of Figure 3, respectively; -‚‚-, 13.1 µL; -‚-, 5.7 µL, - - -, 4.3 µL; ‚‚‚, 1.0 µL; s, 0.7 µL.
a liquid solute is directly injected into the column, this subcritical region is inevitable in the vicinity of the injection port. If the tracer in the smaller amount is injected into the longer diffusion column, this effect is smaller. Otherwise, when the solute dissolving in the supercritical solvent is injected, the effect is avoidable. As depicted, the mixture fluid at positions more than at least a half of the entire length of the diffusion column is found to be in the subcritical state. According to the paper of Nishiumi et al.,1 the diffusivities at pressures lower than ca. 11 MPa show the anomaly. At lower pressures the concentration of the solute is higher because the constant amount of the solute is injected into the lower density solvent. Consequently, the critical temperatures of the mixtures at pressures lower than 11.87 MPa are higher than those at this pressure. Although the details on their measurement conditions are not described in their paper, it is evident that the estimated critical temperatures of the mixtures are mostly higher than the measurement temperature for the diffusivity data showing the anomaly at pressures lower than 11.87 MPa. Note that eq 1 cannot represent the response curves precisely at shorter axial distances because of the initial transient effect of the column. Parts a and b of Figure 4 show the effect of the injected solute amount on the critical temperature of the mixture corresponding to the composition at the top of the response curve; the parameter values employed are the same as those in parts a and b of Figure 3, respectively. When the injected amounts are large (5.7 and 13.1 µL), the critical temperatures of the mixtures at upstream positions are higher than the measurement temperature, in particular, at most positions for 13.1
Tc ) critical temperature of the mixture t ) time u ) average velocity of the mixture x1 ) acetone mole fraction z ) axial distance
Literature Cited (1) Nishiumi, H.; Fujita, M.; Agou, K. Diffusion of Acetone in Supercritical Carbon Dioxide. Fluid Phase Equilib. 1996, 117, 356. (2) Ago, K.; Fujita, M.; Konda, A.; Asakura, T.; Nishiumi, H. Maximum Points of Diffusion Coefficients in Supercritical Carbon Dioxide. Proceedings of the International Symposium on Molecular Thermodynamics and Molecular Simulation, Jan 12-15, 1997, Tokyo; Hosei University: Tokyo, 1997; p 219. (3) Ago, K.; Nishiumi, H. Calculation of Mutual Diffusion Coefficients near the Critical Region from the Peng-Robinson Equation of State. Ind. Eng. Chem. Res. 1998, 37, 1692. (4) Reaves, J. T.; Griffith, A. T.; Roberts, C. B. Critical Properties of Dilute Carbon Dioxide + Entrainer and Ethane + Entrainer Mixtures. J. Chem. Eng. Data 1998, 43, 683. (5) Taylor, G. Dispersion of Soluble Matter in Solvent Flowing Slowly through a Tube. Proc. R. Soc. London 1953, A219, 186. (6) Aris, R. On the Dispersion of a Solute in a Fluid Flowing through a Tube. Proc. R. Soc. London 1956, A235, 67. (7) Angus, S.; Armstrong, B.; de Reuck, K. M. IUPAC International Thermodynamic Tables of the Fluid State, Carbon Dioxide; Pergamon Press: Oxford, U.K., 1976.
Received for review April 15, 1999 Revised manuscript received September 28, 1999 Accepted October 6, 1999 IE990275R
