4430
Ind. Eng. Chem. Res. 1997, 36, 4430-4433
CORRELATIONS Estimation of Infinite-Dilution Diffusion Coefficients in Supercritical Fluids Chao-Hong He* and Yong-Sheng Yu Department of Chemical Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China
Based on the hard-sphere theory and an extensive literature database, new correlations are developed to estimate the binary diffusion coefficients of liquid and solid solutes in supercritical solvents. The correlations can predict the behavior of the binary diffusion coefficients in a range of the reduced solvent density larger than 0.21, with an average absolute error of 7.8% for 107 systems and 1167 data points. The input data required for the correlations are the solute molecular weight, temperature, and density and the solvent basic properties (molecular weight, critical volume, and critical pressure). Introduction A knowledge of binary diffusion coefficients of solutes in supercritical solvents is required in the equipment design and process development involving supercritical fluids. It is important, therefore, to be able to estimate the diffusion coefficients at conditions and for solutes where no data are available. Though correlations for this purpose are available in many forms (Eaton and Akgerman, 1997; Liu et al., 1997; Akgerman et al., 1996; Catchpole and King, 1994; Funazukuri et al., 1991), most of them are either specific to the given supercritical solvent, cover a limited density range, or give poor prediction results. So it is the aim of this paper to propose correlations to predict the binary diffusion coefficients in various supercritical solvents with better results. Correlations According to the hard-sphere theory, Dymond (1972) gave a good correlation for predicting the self-diffusion coefficient given below: 2/3
D11 ) A′(V1
- B′)xT/M1
(1)
The success of eq 1 in correlating the self-diffusion coefficients leads one to expect, in analogy with eq 1, that the binary diffusion coefficients of solutes in supercritical solvents will obey an equation of the form
D21 ) A′′(V12/3 - B′′)xT/M2
(2)
where A′′ and B′′ are constants for the given solutesolvent system. Equation 2 was applied to correlate the infinitedilution diffusion coefficients of liquid and solid solutes in supercritical solvents, where the solvent molar volume V1 is obtained by 1000M1/F1 (if F1 is not available with the quoted data source, V1 is then calculated by * Author to whom correspondence is addressed. Telephone: 0086-571-7996837. Fax: 0086-571-7951358. E-mail: ceyuyu@ emb.zju.edu.cn. S0888-5885(97)00167-X CCC: $14.00
the BWR equation of state (Perry, 1984)). The correlation results showed that eq 2 gave good results if the reduced solvent density Fr (Fr ) F1/Fc1 ) Vc1/V1) was greater than 1 but gave poorer results if Fr was less than 1. Thus, eq 2 was modified as
D21 ) A × 10-9(V1k - B)xT/M2
(3)
k ) (2/3)[1 - 0.28 exp(-0.3xM1Fr)]
(4)
In order to make eq 3 predictive, the parameters A and B must be related to the solute-solvent properties. For this purpose, a large database has been used: 107 binary systems performing 1167 data points. Table 1 contains the systems studied, along with the data sources. It was found that the parameters A and B were approximately dependent on the properties of the solvent only. Further analysis shows that A and B can be approximated as the following equations:
A ) 0.56362 + 2.1417 exp(-0.95088xM1Vc1/Pc1),
Fr g 0.21 (5)
B ) 8.9061 + 0.93858xM1Vc1/Pc1,
Fr g 0.21
(6)
Results and Discussion Equations 3-6 have been used to predict the binary diffusion coefficients of all the solvent-solute systems given in Table 1. The total average absolute deviation (AAD %) is 7.8% for 1167 data points with the density range Fr g 0.21, and the maximum AAD % is 24.5%, where AAD % is
AAD % )
100 N N
|(Dexp - Dpred)/Dexp)| ∑ i)1
(7)
For comparison, the systems given in Table 1 have also been predicted using the method proposed by Catchpole and King (1994), which is expressed by © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4431 Table 1. Systems Studieda and Data Sourcesb Carbon Dioxide (Tr ) 1.00-1.13, Fr ) 0.43-2.23, N ) 794) 71 solutes: benzene [1-8], naphthalene [1, 5, 6, 9-12], phenanthrene [1,9,13], toluene [4, 5, 14], ethylbenzene [4, 5], (o-, m-, p-)xylene [5], n-propylbenzene [2, 4, 5], isopropylbenzene [4, 5], mesitylene [1, 2, 5], pyrene [1], chrysene [1], ethyl acetate [5, 14], (C4:0, C8:0, C10:0, C14:0, C16:0, C18:0, C22:0, C22:6) ethyl ester [15], (C20:5, C22:6, C18:2, C14:1, C20:1, C22:1) methyl ester [6, 15, 16], glycerol trioleate [17], vitamin A acetate [6], acetone [1, 3, 5], 2-butanone [5], (cyclopentanone, 3-pentanone, cycloheptanone, 3-heptanone, cyclononanone) [18], 5-nonanone [18], cis-jasmone [6], phenylacetic acid [19], benzoic acid [13, 17], oleic acid [17], vanillin [19], benzaldehyde [5], phenol [14], m-cresol [8], benzyl ether [5], diethyl ether [5], caffeine [12, 14], acridine [13], R-tocopherol [17], (vitamin E, vitamin K1, vitamin K3) [6], limonene [6], indole [6], 1-octene [5], (dipentene, dichloromethane, chloroform) [5], hexachlorobenzene [9], (n-pentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane) [3], n-undecane [3], n-dodecane [3], n-tetradecane [3], 1,4-dioxane [5] Ethylene (Tr ) 1.13, Fr ) 1.86-2.29, N ) 5) 1 solute: naphthalene-d8 [11] Ethane (Tr ) 1.01-1.78, Fr ) 0.21-1.96, N ) 28) 2 solutes: 1-octene [20-22], 1-tetradecene [20] Propane (Tr ) 1.04-1.47, Fr ) 0.32-2.03, N ) 32) 3 solutes: benzene [8], m-cresol [8], 1-octene [22] n-Hexane (Tr ) 1.00-1.11, Fr ) 0.64-1.97, N ) 66) 7 solutes: (benzene, toluene, p-xylene, mesitylene, naphthalene, phenanthrene) [23], 1-octene [22] 2,3-Dimethylbutane (Tr ) 1.05-1.10, Fr ) 1.43-1.91, N ) 41) 4 solutes: (benzene, toluene, naphthalene, phenanthrene) [24] Ethanol (Tr ) 1.00-1.07, Fr ) 1.44-2.06, N) 55) 5 solutes: (benzene, toluene, mesitylene, naphthalene, phenanthrene) [25] 2-Propanol (Tr ) 1.00-1.05, Fr ) 1.53-2.08, N ) 48) 6 solutes: (benzene, toluene, naphthalene, phenanthrene, n-decane, n-tetradecane) [26] Chlorotrifluoromethane (Tr ) 1.04-1.05, Fr ) 0.69-1.72, N ) 30) 3 solutes: (p-xylene, 1,3-dibromobenzene, acetone) [27] Sulfur Hexafluoride (Tr ) 1.01-1.06, Fr ) 0.41-1.91, N ) 68) 5 solutes: (benzene, toluene, p-xylene, mesitylene, carbon tetrachloride) [27] total: 107 binary systems (Tr ) 1.00-1.78, Fr ) 0.21-2.29, ∑N ) 1167) a The data of Debenedetti and Reid (1986) and Dahmen et al. (1990b) are omitted, as they are systematically low or high (Catchpole and King, 1994). Cno.:no. ) number of carbons, number of double bonds in fatty ester. b References: 1, Sassiat et al., 1987; 2, Swaid and Schneider, 1979; 3, Umezawa and Nagashima, 1992; 4, Suarez et al., 1993; 5, Funazukuri, 1996; 6, Funazukuri et al., 1992; 7, Mei et al., 1995; 8, Olesik and Woodruff, 1991; 9, Akgerman et al., 1996; 10, Lauer, 1983; 11, Lamb et al., 1989; 12, Knaff and Schlu¨nder, 1987; 13, Shenai et al., 1993; 14, Lai and Tan, 1995; 15, Liong et al., 1992; 16, Funazukuri et al., 1991; 17, Catchpole and King, 1994; 18, Dahmen et al., 1990a; 19, Wells et al., 1992; 20, Noel et al., 1994; 21, Eaton et al., 1995; 22, Eaton and Akgerman, 1997; 23, Sun and Chen, 1985a; 24, Sun and Chen, 1985b; 25, Sun and Chen, 1986; 26, Sun and Chen, 1987; 27, Kopner et al., 1987.
Table 2. Prediction Results for Infinite-Dilution Diffusion Coefficients of Liquid and Solid Solutes in Supercritical Solvents eqs 3-6 solvent carbon dioxide
ethylene ethane propane n-hexane 2,3-dimethylbutane ethanol 2-propanol chlorotrilfluoromethane sulfur hexafluoride
eqs 8-12
solute
N
Tr
Fr
AAD %
max %
AAD %
max %
refb
benzene benzene naphthalene naphthalene phenanthrene C14:0 ethyl ester C22:0 ethyl ester ethyl acetate n-heptane toluene ethylbenzene phenol caffeine acetone cyclononanone phenylacetic acid vanillin naphthalene-d8 1-tetradecene 1-octene benzene mesitylene phenanthrene toluene naphthalene benzene naphthalene toluene n-decane dibromobenzene acetone p-xylene CCl4
11 22 11 19 14 16 17 15 5 18 15 21 21 7 8 16 15 5 6 13 12 10 10 10 9 11 11 8 8 12 10 30 7
1.00-1.10 1.01-1.08 1.00-1.10 1.01-1.08 1.01-1.08 1.01-1.05 1.01-1.05 1.01-1.08 0.98-1.01 1.01-1.08 1.03-1.10 1.01-1.08 1.01-1.08 1.00-1.10 1.03 1.01-1.05 1.01-1.05 1.13 1.01-1.06 1.71-1.78 1.04-1.09 1.00-1.07 1.00-1.07 1.05-1.10 1.05-1.10 1.00-1.07 1.00-1.07 1.01-1.05 1.01-1.05 1.05 1.04 1.01-1.06 1.03
1.09-1.88 0.60-1.71 1.09-1.88 1.50-2.23 0.85-1.92 1.28-1.82 1.28-1.82 0.45-1.72 1.53-1.65 0.46-1.72 1.30-2.00 0.76-1.72 0.91-1.72 1.09-1.88 1.27-1.73 1.50-1.82 1.50-1.82 1.86-2.29 1.52-1.77 0.21-0.55 1.47-2.03 1.35-1.97 1.35-1.97 1.43-1.91 1.43-1.91 1.44-2.06 1.44-2.06 1.53-2.08 1.53-2.08 0.69-1.72 0.69-1.65 0.41-1.91 0.41-1.91
8.8 11.8 11.9 5.6 12.2 2.7 2.4 10.4 12.8 8.5 5.4 6.2 7.7 3.8 6.4 2.5 3.3 3.2 8.4 13.3 13.1 3.0 7.3 4.9 5.0 3.4 2.0 5.3 3.4 6.0 7.0 6.2 13.9
19.5 29.6 19.7 13.4 24.7 6.2 7.1 29.5 18.9 18.5 8.3 20.4 27.3 7.2 12.2 5.5 11.2 6.4 19.2 22.0 28.7 8.0 10.7 10.5 8.5 7.3 4.9 10.5 8.8 9.6 13.7 16.1 29.3
14.1 14a 14.7 7.2 14a 3.3 5.5 32a 18.3 21a 11.2 15a 19a 6.4 3.4 2.0 10.6 7.6
24.1 56a 19.4 20.8 26a 6.1 7.8 89a 23.0 69a 15.2 61a 65a 10.9 7.3 5.6 19.1 10.5
6.6 9.5 10.2 5.9 5.2 6.4 6.9 7.4 9.1 15a 5.6a 6.6a 11a
25.0 20.0 17.0 9.7 8.7 12.5 17.3 12.5 16.7 38a 12a 32a 38a
1 2 1 11 9 15 15 14 3 14 4 14 14 1 18 19 19 11 20 21 8 23 23 24 24 25 25 26 26 27 27 27 27
total 423 6.9 Equations 8-12 are extended to the range that the reduced density is less than 1 for part of the data points. same as those in Table 1. a
b
References are the
4432 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997
D21 ) 5.152DcTr(Fr-2/3 - 0.4510)R/X,
1 < Fr < 2.5 (8)
where
∑ν12/3Fc)
Dc ) 4.300 × 10-7M11/2Tc10.75/(
X ) [1 + (Vc2/Vc1)1/3]2/(1 + M1/M2)1/2
(9) (10)
and R is calculated from eq 11 or eq 12 according to the solute-solvent system (Catchpole and King, 1994)
R ) 1.0 ( 0.1, R ) 0.664X0.17 ( 0.1,
2
