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Ind. Eng. Chem. Res. 2010, 49, 9542–9547

Correlation of Infinite-Dilution Diffusion Coefficients in Supercritical Fluids Yingzi Shi School of Education Science, Hangzhou Normal UniVersity, Hangzhou 310036, China

Jiangang Lu* State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang UniVersity, Hangzhou 310027, China

A correlation based on an extensive literature database was proposed to estimate the infinite-dilution diffusion coefficients in supercritical fluids. The input parameters required for the correlation are temperature, the solute molar mass, the solute density near ambient temperature, the solvent critical temperature, the solvent critical density, and the solvent reduced density. The total average absolute deviation (AAD%) of the prediction is 6.55% for 153 binary systems and 1968 data points over the solvent reduced temperature range of 1.00-1.78, the solvent reduced density range of 0.21-2.34, the solute molar mass range of 58.08-885.40 g/mol, and the solute ambient-temperature density range of 0.63-2.04 g/cm3. Introduction of solute density into this new correlation makes it possible to predict with reasonable accuracy the diffusivities in supercritical fluids over a wide solute density range. 1. Introduction Diffusion in supercritical fluids has long been a subject of intensive research efforts because of its vital importance in the development, design, and analysis of a process involving supercritical fluids.1-5 Although experimental and theoretical publications on diffusion in supercritical fluids abound, the quantitative knowledge of infinite-dilution diffusion coefficients in supercritical fluids and the ability to predict them is still in its relative infancy. During the past three decades, mainly through the Taylor-Aris chromatographic dispersion technique, thousands of data points of infinite-dilution diffusion coefficients in supercritical fluids have been measured and published.1-5 In general, such experimental apparatus is relatively expensive and the experimental procedure is always time-consuming and difficult to measure with high precision. Therefore, more than 20 famous equations have been proposed to estimate the infinite-dilution diffusion coefficients in supercritical fluids,6-9 which can be roughly classified into two groups: Stokes-Einstein type and rough-hard-sphere type. Among them, the He-Yu equation10 is highly recommended for its accuracy, generality, convenience, and simplicity.11 However, after detailed evaluation, the He-Yu equation may not hold for high-density solutes.11 The objective of this work is to collect the data available in the literature so far and to propose a new correlation equation, which takes the solute density into account, to estimate the infinite-dilution diffusion coefficients in supercritical fluids. This new equation has the advantage to predict with reasonable accuracy the diffusion coefficients of highdensity solutes in supercritical fluids; meanwhile, for other solutes, it maintains almost the same accuracy as the He-Yu equation. 2. Correlations The famous He-Yu equation for the infinite-dilution diffusion coefficients in supercritical fluids, which is related * To whom correspondence should be addressed. Tel.: + 86 571 87952829. Fax: + 86 571 87951879. E-mail: [email protected].

to rough-hard-sphere formula, is a free-volume-based equation:10

[

105DAB ) 14.882 + 5.9081

TcBVcB 3

10 MB

(

exp -

+ 2.0821

( )] TcBVcB

2

3

10 MB

)( )

0.3887VcB T VB - 0.23VcB MA

0.5

×

(1)

where DAB is the infinite-dilution molecular diffusion coefficient of solute A in supercritical solvent B, cm2/s; T is the system temperature, K; MA and MB are the molar mass of solute A and solvent B, g/mol; TcB is the critical temperature of solvent B, K; VcB is the critical molar volume of solvent B, cm3/mol; VB is the molar volume of supercritical solvent B at the system temperature T, cm3/mol; the term of (0.3887VcB) is related to the critical hole volume; and the term of (VB - 0.23VcB) is the average free volume. To illustrate solvent density dependence, eq 1 can be rearranged as follows: 105DAB ) [14.882 + 5.9081(10-3TcBFcB-1) + T 0.3887 2.0821(10-3TcBFcB-1)2]exp - -1 M FrB - 0.23 A

(

)( )

0.5

(2)

where FcB ) MB/VcB, is the solvent critical density, g/cm3; and FrB ) VcB/VB, is the solvent reduced density. The He-Yu equation can reasonably estimate the infinitedilution diffusion coefficients in supercritical fluids with an average absolute deviation (AAD%) of 8.2% for 113 binary systems and 1303 data points.10 However, for high-density solutes, the He-Yu equation may not hold.11 For example, the prediction deviation for chloroform (density is about 1.5 g/cm3) in supercritical carbon dioxide is greater than -20%.11 Because the solute property considered in the He-Yu equation is the solute molar mass only, a new equation accounting for the effect of the solute density can be expected to improve its accuracy for systems where the solute is of high-density. To investigate the solute density dependence of diffusion coefficients, the solute density can be theoretically selected from

10.1021/ie101030p  2010 American Chemical Society Published on Web 09/01/2010

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

the following five options: the solute density at the system temperature (T), the solute density at the melting point (Tm), the solute density at the boiling point (Tb), the solute density at the critical temperature (Tc), and the solute density near ambient temperature. Generally speaking, for most organic compounds that are liquid or solid, the reliable values of their densities near ambient temperature (293-303 K) are always readily available with satisfactory accuracy in the literature, whereas the solute densities at (T, Tm, Tb, Tc) usually have to be estimated by a group contribution method or other estimation techniques which may lead to large errors. With these caveats, it is clearly seen that, the correlation involving the solute density near ambient temperature (293-303 K), which is herein simply denoted as FA,298K (g/cm3), will not generate any additional error caused by inaccuracy of the solute density value. We investigated the effect of solute density more extensively and proposed a new correlation equation, which accounts for solute density and is consistent with the rough-hard-sphere freevolume-diffusion theory of the He-Yu equation, to estimate the infinite-dilution diffusion coefficients of liquid and solid solutes in supercritical fluids:

10 DAB ) FA,298K(θ1 + 5

-θ2 FA,298K

-3

-1 θ3

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+ 10 TcBFcB ) [θ4 + θ5 ×

(

exp(-θ6FrB)] exp -

)

-θ8 - θ9 Tθ11 θ7FA,298K FrB-1 - θ10 MAθ12

(3)

where θ1-θ12 are empirical constants. The solvent reduced density FrB was obtained by VcB/VB, where the value of VB at the system temperature, in order to maintain consistency, was always calculated by the Lee-Kesler equation of state,12 even though in the quoted data source VB (or FB) may be obtained by other estimation techniques. As presented in Table 1, we collected the experimental data of infinite-dilution diffusion coefficients in supercritical fluids available in the literature, including 153 binary systems and 1968 data points, over the solvent reduced temperature range of 1.00-1.78, the solvent reduced density range of 0.21-2.34, the solute molar mass range of 58.08-885.40 g/mol, and the solute ambient-temperature density range of 0.63-2.04 g/cm3. Data measured not under supercritical conditions were excluded. As discussed in ref 5 and ref 10, the diffusivities in supercritical CO2 of ref 51 were omitted, as they were systematically about 50% lower than the data of ref 5; the data of ref 52 were also omitted, as they were systematically about 50% higher than the results reported by the same group in ref 31.

Table 1. Data Sources of Infinite-Dilution Diffusion Coefficients in Supercritical Fluids solvent (B): carbon dioxide (114 binary systems, 1575 data points, TrB ) 1.00-1.31, FrB ) 0.42-2.34) solutes (A) (MA ) 58.08-885.40 g/mol, FA,298K ) 0.63-2.04): acetone,13-15 acridine,16 allylbenzene,9 aniline,17 benzaldehyde,15 benzene,13-15,18-24 benzoic acid,5,16 benzyl acetate,25 benzyl ether,15 benzylacetone,26 2-bromoanisole,9 bromobenzene,2,6 2-butanone,15 (C4:0; C8:0; C10:0; C14:0; C16:0; C18:0; C22:0; C22:6a) ethyl ester,27 (C14:1; C18:2; C20:1; C20:5; C22:1; C22:6a) methyl ester,20,27,28 caffeine,29,30 chlorobenzene,2,6 chloroform,15 chrysene,13 cis-jasmone,20 cycloheptanone,31 cyclononanone,31 cyclopentanone,31 dibenzyl ether,25 1,2-dichlorobenzene,32 dichloromethane,15 1,2-diethylbenzene,33 1,4-diethylbenzene,33 diethyl ether,15,34 diisopropyl ether,34 2,3-dimethylaniline,35 2,6-dimethylaniline,35 1,4-dioxane,15 1,3-divinylbenzene,9 ethyl acetate,15,29 ethyl benzoate,26 ethylbenzene,15,19 2-ethyltoluene,36 3-ethyltoluene,36 4-ethyltoluene,36 eugenol,26 2-fluoroanisole,9 fluorobenzene,6 glycerol trioleate,5 3-heptanone,31 hexachlorobenzene,37 indole,20 iodobenzene,2,6 isopropylbenzene,15,19 limonene,15,20 m-cresol,22 m-xylene,2,15 mesitylene,2,13,15,17,18 2-methylanisole,35 4-methylanisole,35 n-butylbenzene,7 n-decane,14 n-dodecane,14 n-heptane,14 n-hexane,14 n-nonane,14 n-octane,14 n-pentane,14 n-pentylbenzene,7 n-propylbenzene,15,18,19 n-tetradecane,14 n-undecane,14 naphthalene,5,13,20,24,30,37-41 2-nitroanisole,32 nitrobenzene,2 3-nitrotoluene,35 5-nonanone,31 o-xylene,2,15 1-octene,15 oleic acid,5 p-xylene,2,15 3-pentanone,31 phenanthrene,13,16,37 phenol,29 2-phenyl-1-propanol,8 3-phenyl-1-propanol,8 phenylacetic acid,42 phenylacetylene,33 1-phenyldodecane,7 1-phenylethanol,8 2-phenylethanol,8 2-phenylethyl acetate,25 1-phenylhexane,7 phenylmethanol,8 1-phenyloctane,7 3-phenylpropyl acetate,25 pyrene,13 styrene,17 5-tert-butyl-m-xylene,33 tert-butylbenzene,32 tetrahydrofuran,34 toluene,15,19,23,29,43 vanillin,42 vitamin A acetate,20 vitamin E,5,20 vitamin K1,20 vitamin K320 solvent (B): chlorotrifluoromethane (3 binary systems, 30 data points, TrB ) 1.04-1.05, FrB ) 0.69-1.73) solutes (A) (MA ) 58.08-235.91 g/mol, FA,298K ) 0.79-1.95): acetone,44 1,3-dibromobenzene,44 p-xylene44 solvent (B): 2,3-dimethylbutane (4 binary systems, 41 data points, TrB ) 1.05-1.10, FrB ) 1.43-1.91) solutes (A) (MA ) 78.11-178.23 g/mol, FA,298K ) 0.86-1.15): benzene,45 naphthalene,45 phenanthrene,45 toluene45 solvent (B): ethane (2 binary systems, 27 data points, TrB ) 1.01-1.78, FrB ) 0.21-1.74) solutes (A) (MA ) 112.22-196.37 g/mol, FA,298K ) 0.71-0.78): 1-octene,4,46,47 1-tetradecene46 solvent (B): ethanol (5 binary systems, 55 data points, TrB ) 1.00-1.08, FrB ) 1.44-2.06) solutes (A) (MA ) 78.11-178.23 g/mol, FA,298K ) 0.84-1.15): benzene,48 mesitylene,48 naphthalene,48 phenanthrene,48 toluene48 Solvent (B): ethylene (2 binary systems, 20 data points, TrB ) 1.01-1.13, FrB ) 1.00-2.34) solutes (A) (MA ) 128.17-136.24 g/mol, FA,298K ) 1.15-1.22): naphthalene,40 naphthalene-d839 solvent (B): n-hexane (7 binary systems, 66 data points, TrB ) 1.00-1.11, FrB ) 0.64-1.96) solutes (A) (MA ) 78.11-178.23 g/mol, FA,298K ) 0.71-1.15): benzene,49 mesitylene,49 naphthalene,49 1-octene,4 p-xylene,49 phenanthrene,49 toluene49 solvent (B): propane (3 binary systems, 32 data points, TrB ) 1.04-1.47, FrB ) 0.31-2.00) solutes (A) (MA ) 78.11-112.22 g/mol, FA,298K ) 0.71-1.03): benzene,22 m-cresol,22 1-octene,4 solvent (B): 2-propanol (6 binary systems, 48 data points, TrB ) 1.00-1.05, FrB ) 1.53-2.08) solutes (A) (MA ) 78.11-198.39 g/mol, FA,298K ) 0.73-1.15): benzene,50 n-decane,50 n-tetradecane,50 naphthalene,50 phenanthrene,50 toluene50 solvent (B): sulfur hexafluoride (7 binary systems, 74 data points, TrB ) 1.00-1.06, FrB ) 0.68-1.99) solutes (A) (MA ) 78.11-153.82 g/mol, FA,298K ) 0.84-1.58): benzene,44 benzoic acid,51 mesitylene,44 naphthalene,51 p-xylene,44 tetrachloromethane,44 toluene44 total: 153 binary systems, 1968 data points, TrB ) 1.00-1.78, FrB ) 0.21-2.34, MA ) 58.08-885.40 g/mol, FA,298K ) 0.63-2.04 g/cm3 a

Cm:n ) number of carbons (m), number of double bonds (n) in fatty ester.

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

Figure 1. Comparisons between calculated and experimental values of DAB in supercritical fluids.

Figure 3. AAD of DAB for 153 binary systems as a function of solute density near ambient temperature: (b) this work; (O) He-Yu; (solid line) AAD of DAB by this work at every 0.25 interval of FA,298K; (dashed line) AAD of DAB by He-Yu at every 0.25 interval of FA,298K.

Figure 2. (DAB,cal - DAB,exp) as a function of solvent reduced density. Table 2. Comparison of AAD in Diffusivity Prediction between This Work and He-Yu Equation within Different Intervals of Solute Density AAD (%) FA,298K (g/cm3)

systems

N

eq 4

He-Yu

0.50-0.75 0.75-1.00 1.00-1.25 1.25-1.50 1.50-1.75 1.75-2.00 2.00-2.25 AAD (%)

15 81 45 7 2 2 1 153

102 1126 612 60 21 33 14 1968

11.36 5.69 7.22 7.63 5.63 8.88 2.54 6.55

12.87 5.86 7.89 12.90 13.99 17.41 26.74 7.50

In this extensive literature database, the smallest molecular weight solute is acetone (58.08 g/mol), the largest molecular weight solute is glycerol trioleate (885.40 g/mol), the lowest density solute is n-pentane (0.63 g/cm3), and the highest density solute is hexachlorobenzene (2.04 g/cm3). Then, the Marquardt-Levenberg nonlinear regression algorithm was used to seek the best values of the empirical constants (θ1-θ12) that minimize the sum of the squared deviations between the calculated values and the experimental data of DAB. It is interesting to find that θ2 ≈ θ3 ≈ 2.25; θ8 ≈ 0.1; θ9 ≈ θ10 ≈ 0.247; θ11 + θ12 ≈ 1.0. To fix these empirical constants and

Figure 4. Deviation of DAB for alkanes in supercritical CO2 as a function of alkane carbon number: (O) this work; (0) He-Yu.

run the nonlinear regression algorithm again, the following equation was obtained: 105DAB ) FA,298K(2.79 + FA,298K-9/4 + 10-3TcBFcB-1)9/4[0.4336 + 0.2955 exp(-2.342FrB)] × 0.547FA,298K-1/10 - 0.247 T0.52 exp (4) FrB-1 - 0.247 MA0.48

(

)

3. Results and Discussions A complete list of prediction results for infinite-dilution diffusion coefficients in supercritical fluids by eq 4 is provided as the Supporting Information. The total average absolute deviation (AAD%) is 6.55% for these 153 binary systems and 1968 data points. Comparisons between calculated and experimental values of DAB are shown in Figure 1. Among all these 1968 data points, most calculated values of DAB are located within the range of DAB,exp ( 10, whereas there are nine data points of (DAB,cal < DAB,exp - 10). Further analysis showed that the small solvent reduced density may result in the significant negative errors of these nine data points, as can be seen in Figure 2. This is because

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 3. Comparison of AAD in Diffusivity Prediction between This Work and He-Yu Equation within Different Intervals of Solute Molar Mass AAD (%) MA (g/mol)

systems

N

eq 4

He-Yu

0-100 100-200 200-300 300-400 400-500 885.40 AAD (%)

28 102 12 8 2 1 153

395 1333 141 91 6 2 1968

7.48 6.43 6.43 4.21 5.46 18.9 6.55

7.25 7.55 10.47 2.91 6.85 22.22 7.50

the supercritical fluid with smaller solvent reduced density is closer to the gas state and the solute particle can move more quickly than predicted. To partially reverse the trend of DAB,cal getting smaller than DAB,exp with decreasing FrB, we introduced the term of [0.4336 + 0.2955 exp(-2.342FrB)] into eq 4, which significantly reduced the data point number of (DAB,cal < DAB,exp - 10) but whereas produced two data points of (DAB,cal > DAB,exp + 10) as an attendant side effect, as shown in Figures 1 and 2. To illustrate solute density dependence of diffusivities, we divide the solute density range by 0.25 interval and compare

Figure 5. AAD of DAB for 153 binary systems as a function of solute molar mass: (b) this work; (O) He-Yu.

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the AADs in diffusivity prediction between this work and He-Yu equation within each interval. As can be seen in Table 2 and Figure 3, this work usually gives better prediction results within each solute density interval and covers a wider range of solute density more reasonably in comparison with the He-Yu equation. With increasing solute density, AAD by this work remains small; however, AAD by the He-Yu equation increases significantly. The total AAD is 6.55% for this work and 7.50% for He-Yu equation, respectively. Although the total AAD reduction of about 1% seems not very remarkable, it can be expected that the advantage of this work may become more significant when more data of high-density solute diffusivities are available. Meanwhile, it should be pointed out that the correlation proposed in this work is also less accurate within the lower solute density interval of 0.50-0.75 g/cm3. Among the 15 binary systems within this interval, the eight systems of alkanes (C5H12 to C12H26)14 in supercritical CO2 constitute the main source of prediction errors, as shown in Figure 4. It is interesting to find that with increasing carbon chain length, the alkanes diffuse more quickly than predicted, which implies that the chain-shaped solute molecules with the same density, molar mass, and molar volume as the sphere-shaped solute molecules can move more quickly than the latter. Introduction of more solute molecular properties (for example, solute molecular shape parameter) into the rough-hard-sphere free-volume-diffusion theory of this new correlation or the He-Yu equation can be expected to give better prediction results for the solutes which are very different from the sphere-shaped molecules. Table 3 and Figure 5 investigate solute molar mass dependence of diffusivities by comparing the AADs in diffusivity prediction between this work and He-Yu equation within different intervals of solute molar mass. It can be found that this work gives better prediction results within the solute molar mass intervals of 100-200, 200-300, and more than 400 g/mol; meanwhile, the He-Yu equation is somewhat better within the solute molar mass intervals of 300-400, and less than 100 g/mol. However, generally speaking, the improvement of this work to predict solute molar mass dependence of diffusivities is not significant. Table 4 and Figure 6 give the detailed prediction results for iodobenzene in supercritical carbon dioxide, where iodobenzene is a typical solute with a high density of 1.8229 g/cm3. This

Table 4. Prediction Results for Diffusion Coefficients of Iodobenzene in Supercritical Carbon Dioxidea solute (A) properties:53 MA ) 204.01 g/mol; FA,298K ) 1.8229 g/cm3 solvent (B) properties:11 MB ) 44.01 g/mol; TcB ) 304.12 K; PcB ) 73.74 bar; VcB ) 94.07 cm3/mol; ωB ) 0.225; FcB ) MB/VcB ) 0.4678 g/cm3 eq 4

He-Yu c

T (K) 313 313 313 313 313 323 323 323 323 323 333 333 333 333 333 AAD% a

P deviation (bar) VBb (cm3/mol) TrB ()T/TcB) FrB ()VcB/VB) DAB, exp (10-5cm2/s) DAB, cal (10-5 cm2/s) (%) 150 200 250 300 350 150 200 250 300 350 150 200 250 300 350

55.98 52.15 49.94 48.39 47.20 62.89 55.84 52.58 50.50 48.99 74.03 60.75 55.78 52.93 50.98

1.03 1.03 1.03 1.03 1.03 1.06 1.06 1.06 1.06 1.06 1.09 1.09 1.09 1.09 1.09

1.680 1.804 1.884 1.944 1.993 1.496 1.685 1.789 1.863 1.920 1.271 1.548 1.686 1.777 1.845

10.66 9.61 9.17 8.25 7.98 12.30 10.60 9.88 9.07 8.96 13.81 11.91 11.47 10.29 9.72

10.88 9.78 9.09 8.57 8.14 12.73 11.01 10.08 9.42 8.92 15.07 12.45 11.18 10.35 9.73

2.06 1.77 -0.87 3.88 2.01 3.50 3.87 2.02 3.86 -0.45 9.12 4.53 -2.53 0.58 0.10 2.74

DAB, cal (10-5 cm2/s) 8.38 7.32 6.67 6.19 5.81 10.16 8.47 7.57 6.95 6.48 12.46 9.84 8.59 7.79 7.21

Data source: ref 6. b VB was calculated by the Lee-Kesler equation of state (ref 12). c Deviation (%) ) 100 (DAB,cal - DAB,exp)/DAB,exp.

deviationc (%) -21.39 -23.83 -27.26 -24.97 -27.19 -17.40 -20.09 -23.38 -23.37 -27.68 -9.78 -17.38 -25.11 -24.30 -25.82 22.60

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 V ) molar volume, cm3/mol Greek Letters θ ) empirical constant of eq 3 F ) density, g/cm3 ω ) acentric factor Subscript A ) solute B ) solvent c ) critical r ) reduced

Literature Cited

Figure 6. Experimental and correlated diffusivities of iodobenzene in supercritical CO2: (b) at 313; (9) at 323; (2) at 333 K; The symbols are the experimental values, the solid lines are the calculated values by this work, and the dashed lines are the calculated values by He-Yu.

work gives perfect prediction with an AAD of 2.74%, which is much better than the He-Yu equation whose AAD is 22.60% for this binary system. 4. Conclusions A new correlation equation, eq 4, is proposed for predicting the infinite-dilution diffusion coefficients in supercritical fluids. The input parameters required for this correlation are temperature, the solute molar mass, the solute density near ambient temperature, the solvent critical temperature, the solvent critical density, and the solvent reduced density. The total average absolute deviation (AAD%) of the prediction is 6.55% for 153 binary systems and 1968 data points over the solvent reduced temperature range of 1.00-1.78, the solvent reduced density range of 0.21-2.34, the solute molar mass range of 58.08-885.40 g/mol, and the solute ambient-temperature density range of 0.63-2.04 g/cm3. Furthermore, this correlation has the advantage to estimate the diffusion coefficients of high-density solutes in supercritical fluids with satisfactory accuracy. Acknowledgment This work was partially supported by the National Natural Science Foundation of China (NSFC) (No. 21076179), the National High-Tech Research and Development Program of China (863 Program) (No. 2006AA04Z184), the Key Technologies R&D Program of Zhejiang Province (No. 2006C31051), and the Natural Science Foundation of Zhejiang Province (No. Y4080339). Supporting Information Available: Prediction results for infinite-dilution diffusion coefficients in supercritical fluids, involving 153 systems and 1968 data points. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature AAD ) average absolute deviation ) 100/N ∑Ni)1|(DAB,cal - DAB,exp)/ DAB,exp|, % D ) infinite-dilution molecular diffusion coefficient, cm2/s M ) molar mass, g/mol N ) number of data points T ) temperature, K

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ReceiVed for reView May 5, 2010 ReVised manuscript receiVed July 11, 2010 Accepted August 19, 2010 IE101030P