Ind. Eng. Chem. Res. 1998, 37, 3793-3798
3793
New Equation for Infinite-Dilution Diffusion Coefficients in Supercritical and High-Temperature Liquid Solvents Chao-Hong He and Yong-Sheng Yu Department of Chemical Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China
On the basis of the idea that molecular diffusion occurs by the movement of molecules into voids, a new equation was developed for the determination of infinite-dilution molecular diffusion coefficients in supercritical and high-temperature liquid solvents (Tr g 0.7). An extensive database on diffusion was collected through the literature in order to evaluate the proposed equation and to determine the accuracy of the predictions with no additional parameter adjustment. The predictions were good, with an average absolute deviation of 8.2% for 113 binary systems and 1303 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 temperature, and critical volume). Introduction A knowledge of binary diffusion coefficients of solutes in liquid and supercritical solvents is required in equipment design and process development. It is important, therefore, to be able to estimate the diffusion coefficients at conditions and for solutes where no data are available. So far, many correlations have been proposed to estimate the infinite-dilution diffusion coefficients in liquid solvents (Reid et al., 1987). They often gave better prediction results at low temperatures such as 298 K but gave poor results at high temperatures (He and Yu, 1997a). The reason for this might be that the effect of solvent density was neglected. On the other hand, though the density effect was well considered in the correlations estimating diffusion coefficients in supercritical solvents (Eaton and Akgerman, 1997; Liu et al., 1997; Akgerman et al., 1996; Catchpole and King, 1994; Funazukuri et al., 1991; He and Yu, 1997b), few of them could simultaneously predict the behavior of diffusion in supercritical and high-temperature liquid solvents with satisfactory results. So it is the aim of this paper to propose correlations to estimate the infinite-dilution diffusion coefficients in supercritical and high-temperature solvents with better results. Model
total probability P(v*) of finding a hole of volume exceeding v* given by the solvent molecules. Approximately, it may be assumed that the diffusion rate of the solute is proportional to the average solute molecular velocity u2 [equal to (3RT/M2)0.5] and to P(v*), that is
D21 ) A′u2P(v*) ) A′(3RT/M2)0.5P(v*)
(1)
where A′ is a constant for the given solute-solvent system. According to the work of Cohen and Turnbull (1959)
P(v*) ) exp(-γ′v*/vf) ) exp(-γ′V*/Vf)
(2)
where γ′is a numerical factor to correct for overlap of free volume, vf is the average free volume per molecule, and V* ) NAv*, Vf ) NAvf. Inserting eq 2 into eq 1 gives
D21 ) A′(3RT/M2)0.5 exp(-γ′V*/Vf)
(3)
In eq 3, Vf can be expressed as (V1 - V0), where V0 is the close-packed molar volume. If designate A′(3R)0.5 ) A × 10-9, eq 3 can be rewritten as
x
D21 ) A × 10-9
(
)
T γ′V* exp M2 V1 - V0
(4)
Correlations
Let us first consider molecular diffusion in a dense fluid consisting of hard spheres. In such a fluid the potential energy of a molecule is constant except that it becomes infinite upon intermolecular contact, and molecules move with the gas kinetic velocity u but most of the time are confined to a cage bounded by their immediate neighbors (Cohen and Turnbull, 1959). According to the idea of Cohen and Turnbull (1959), molecular transport occurs by the movement of molecules into voids, with a size greater than some critical value v*, formed by redistribution of the free volume. Therefore, the diffusion of solute molecule in solvent is dependent on both the solute molecular velocity and the
Since the concentration of the solute is approximate to zero, the close-packed molar volume V0 depends only on the solvent properties, that is
V0 ) NAσ13/x2
(5)
where σ1 can be determined from the Purkait and Majumdar (1981) semiempirical equation given by
σr ) σ/σc ) 0.552803 - 0.0026776Tr
(6)
σc ) [6Vc/(πNA)]1/3
(7)
S0888-5885(97)00898-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/14/1998
3794 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 Table 1. Systems Studieda and Data Sourcesb Solvent: Carbon Dioxide (Tr ) 0.95-1.13, 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, 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, limonene, indole) [6], (1-octene, dipentene, dichloromethane, chloroform) [5], hexachlorobenzene [9], (n-pentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane, n-undecane, n-dodecane, n-tetradecane) [3], 1,4-dioxane [5] Solvent: Ethylene (Tr ) 1.13, Fr ) 1.86-2.29, N ) 5) 1 solute: naphthalene-d8 [11] Solvent: Ethane (Tr ) 0.96-1.78, Fr ) 0.21-1.99, N ) 34) 2 solutes: 1-octene [20-22], 1-tetradecene [20] Solvent: Propane (Tr ) 0.79-1.47, Fr ) 0.32-2.03, N ) 50) 4 solutes: benzene [8], m-cresol [8], 1-octene [20, 22], 1-tetradecene [20] Solvent: n-Hexane (Tr ) 0.66-1.11, Fr ) 0.64-2.67, N ) 97) 7 solutes: (benzene, toluene, p-xylene, mesitylene, naphthalene, phenanthrene) [23], 1-octene [22] Solvent: 2,3-Dimethylbutane (Tr ) 1.05-1.10, Fr ) 1.43-1.91,N ) 41) 4 solutes: (benzene, toluene, naphthalene, phenanthrene) [24] Solvent: Methanol (Tr ) 0.73-0.92, Fr ) 2.06-2.62, N ) 15) 5 solutes: (benzene, toluene, mesitylene, naphthalene, phenanthrene) [25] Solvent: Ethanol (Tr ) 0.73-1.08, Fr ) 1.44-2.60, N ) 75) 5 solutes: (benzene, toluene, mesitylene, naphthalene, phenanthrene) [26] Solvent: 2-Propanol (Tr ) 0.73-1.05, Fr ) 1.53-2.59, N ) 72) 6 solutes: (benzene, toluene, naphthalene, phenanthrene, n-decane, n-tetradecane) [25] Solvent: Chlorotrifluoromethane (Tr ) 1.04-1.05, Fr ) 0.69-1.73, N ) 30) 3 solutes: (p-xylene, 1,3-dibromobenzene, acetone) [27] Solvent: Sulfur Hexafluoride (Tr ) 0.89-1.06, Fr ) 0.41-2.25, N ) 90) 5 solutes: (benzene, toluene, p-xylene, mesitylene, carbon tetrachloride) [27] total: 113 binary systems (Tr ) 0.66-1.78, Fr ) 0.22-2.62, ΣN ) 1303) 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, 1987, 26, Sun and Chen, 1986; 27, Kopner et al., 1987.
From eqs 5-7 and neglecting the effect of Tr on σr, one can obtain
V0 )
6 6 σr13Vc1 ≈ (0.552803)3Vc1 ) 0.23Vc1 x2π x2π
(8)
Thus, eq 4 becomes
x
D21 ) A × 10-9
(
T γ′V* exp M2 V1 - 0.23Vc1
)
solvent only. Further analysis showed that they could be approximated as the following equations:
γ′V* ) 0.3887Vc1
(10)
A ) 14.882 + 5.9081k + 2.0821k2
(11)
k) (9)
To make eq 9 practical, parameters A and γ′V* must be related to the solute-solvent properties. For this purpose, eq 9 was applied to correlate the infinitedilution diffusion coefficients in supercritical and hightemperature liquid solvents, where the solvent molar volume V1 was obtained by 1000M1/F1 [if F1 is not available with the quoted data source, V1 is then calculated by the BWR equation of state (Perry, 1984)]. The large database collected for correlation consists of 113 binary systems containing 1303 data points and is shown in Table 1 along with the data sources. It was found that the parameters A and γ′V* were approximately dependent on the properties of the
Tc1Vc1 1000M1
(12)
Results and Discussions Equations 9-12 have been used to estimate the infinite-dilution diffusion coefficients of all the solventsolute systems given in Table 1. The total average absolute deviation (AAD%) is 8.2% for 1303 data points in the range 0.22 e Fr e 2.62, 0.66 e Tr e 1.78, 58.1 e M2 e 885.4, and the maximum AAD % is 26.7%. Table 2 shows the typical results predicted by this work (eqs 9-12). Tables 3-5 and Figures 1-3 give the detailed prediction results for toluene in carbon dioxide, 1-octene in ethane, and naphthalene in ethanol. As can be seen in Tables 2-5 and Figures 1-3, this work usually gives good results for predicting the infinitedilution diffusion coefficients in supercritical and hightemperature liquid solvents (Tr g 0.66).
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3795 Table 2. Prediction Results for Infinite-Dilution Diffusion Coefficients in Supercritical and High-Temperature Solvents solute
N
Tr
Fr
benzene naphthalene phenanthrene C8:0 ethyl ester C22:6 methyl ester ethyl acetate n-hexane toluene ethylbenzene phenol caffeine acetone cyclononanone phenylacetic acid vanillin naphthalene-d8 1-octene 1-octene 1-octene 1-octene benzene p-xylene phenanthrene toluene phenanthrene mesitylene benzene naphthalene toluene phenanthrene dibromobenzene acetone toluene p-xylene
85 75 27 16 17 15 5 18 15 21 21 12 8 16 15 5 19 6 8 11 12 15 15 10 11 3 15 15 12 12 12 10 11 52
1.00-1.10 0.95-1.10 1.00-1.10 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 0.98-1.10 1.03 1.01-1.05 1.01-1.05 1.13 0.97-1.78 1.05-1.65 0.80-0.91 0.85-1.47 1.04-1.09 0.66-1.07 0.66-1.07 1.05-1.10 1.05-1.10 0.73-0.92 0.73-1.08 0.73-1.08 0.73-1.05 0.73-1.05 1.05 1.04 1.03 0.89-1.06
0.60-2.00 0.85-2.23 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 0.21-1.97 0.50-1.73 2.04-2.37 0.32-2.31 1.47-2.03 1.35-2.67 1.35-2.67 1.43-1.91 1.43-1.91 2.06-2.62 1.44-2.60 1.44-2.60 1.53-2.59 1.53-2.59 0.69-1.73 0.69-1.73 0.41-1.91 0.41-2.25
solvent carbon dioxide
ethylene ethane propane n-hexane 2,3-dimethylbutane methanol ethanol 2-propanol chlorotrifluoromethane sulfur hexafluoride
total
AAD % (this work)
620
max % (this work)
6.9 10.0 14.0 2.9 2.8 7.6 7.9 6.0 5.5 3.3 4.9 4.8 4.9 2.4 2.7 16.7 9.3 7.9 6.7 16.5 9.0 9.2 8.4 1.7 3.3 13.8 5.0 5.7 10.1 11.8 7.1 8.9 9.8 4.9
21.1 26.1 24.8 12.3 9.4 19.6 15.9 12.7 8.9 7.1 23.9 11.2 11.9 5.1 10.6 31.1 20.2 15.2 14.3 34.8 28.1 23.9 22.2 3.9 5.3 18.8 15.5 19.9 16.6 31.3 16.8 22.1 37.2 37.8
ADD %a eq 13
eq 20
14.2 18.2 27.6 20.2 20.3 19.4 26.2 20.7 16.7 10.9 11.0 2.6
6.0 12.4 14.9 10.6 15.8
11.4 7.2 21.8
5.8 9.1
1-8 1, 5, 6, 9-12 1, 9, 13 15 15 14 3 14 4 14 14 1, 3 18 19 19 11 20, 21 22 20 22 8 23 23 24 24 25 26 26 25 25 27 27 27 27
8.9 5.3
3.4
7.3%
refb
20.1 38.2 6.2 10.4 13.9 2.5 10.9
7.9c 5.2c 14.0 4.9
26.7 14.9 9.8
11.7d 15.9d 8.2e 6.4e 40.8 40.4 19.2 16.1
15.5%
11.7%
4.5
Results of eq 20 are quoted from Eaton and Akgerman (1997). References are the same as those in Table 1. c Tr ) 0.83-1.07. ) 0.92-1.08. e Tr ) 0.93-1.05. a
b
Table 3. Prediction Results for Diffusion Coefficients of Toluene in Carbon Dioxide Using Equations 9-12a T (K) F (kg/m3) 308.0 308.0 308.0 308.0 308.0 308.0 308.0 308.0 318.0 318.0 318.0 318.0 318.0 318.0 328.0 328.0 328.0 328.0
279 454 621 657 723 759 786 806 214 329 541 644 694 728 246 340 567 629
Fr 0.595 0.969 1.325 1.402 1.543 1.619 1.677 1.720 0.457 0.702 1.154 1.374 1.481 1.553 0.525 0.725 1.210 1.342
Dexp Dcal (10-9m2/s) (10-9m2/s) 28.1 23.5 18.3 16.7 15.5 13.8 13.7 13.0 33.2 28.9 19.3 16.1 15.4 16.3 31.1 24.9 19.7 18.2
27.4 22.1 17.1 16.0 14.1 13.1 12.4 11.9 29.9 26.3 19.8 16.7 15.2 14.2 29.3 26.4 19.3 17.4
D21 ) 2.116 × 10-3
(
deviation %b -2.5 -6.1 -6.7 -4.0 -8.8 -4.8 -9.5 -8.8 -10.1 -9.0 2.4 3.5 -1.3 -12.7 -5.8 5.8 -2.2 -4.5 AAD % ) 6.0%
a Data source: Lai and Tan, 1995. b Deviation % ) 100(D cal Dexp)/Dexp.
For comparison, the systems given in Table 2 were also predicted using the following two methods: (1) TLSM model by Liu et al. (1997), which is for diffusion coefficients of solutes in supercritical and liquid solvents (not including hydrogen-bonding solvents):
x
8314.4T V1 M12 NA σeff
exp -
12
1.2588 - F*1
Tr
2
( )
0.75F*1
d
-
)
0.27862 (13) T*12
where
M12 )
2M1M2 M1 + M2
σ12LJ ) (σ1LJ + σ2LJ)/2
(14) (15)
12LJ/kB ) x(σ1LJ)3(1LJ/kB)(σ2LJ)3(2LJ/kB)/(σ12LJ)3 (16) T*12 )
T 12LJ/kB
(17)
σ12eff ) 21/6σ12LJ[1 + (1.3229T*12)1/2]-1/6
(18)
F*1 ) NA(σ1LJ)3/V1
(19)
and σ1LJ, σ2LJ, 1LJ/kB, and 2LJ/kB are from Liu et al. (1997).
3796 Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 Table 4. Prediction Results for Diffusion Coefficients of 1-Octene in Ethane Using Equations 9-12a T (K) F (kg/m3) 296.2 296.2 296.2 311.2 317.1 322.2 322.0 340.0 357.0 378.0 414.0 503.0 523.0 523.0 523.0 523.0 523.0 533.0 533.0 533.0 533.0 533.0 543.0 543.0 543.0
399 389 376 350 330 308 350 300 249 200 150 101 46 62 78 95 111 45 60 76 92 108 44 59 74
Dexp Dcal (10-9m2/s) (10-9m2/s)
Fr 1.968 1.917 1.855 1.727 1.628 1.521 1.726 1.479 1.229 0.986 0.742 0.498 0.225 0.305 0.387 0.468 0.547 0.220 0.297 0.376 0.455 0.533 0.215 0.290 0.366
12.6 13.5 14.2 15.6 17.0 18.1 16.3 18.5 22.7 31.1 37.6 55.5 70.3 61.0 49.9 44.9 43.7 69.1 59.0 57.1 52.7 54.5 67.0 60.0 62.9
11.4 12.2 13.2 15.6 17.4 19.5 15.9 20.8 26.1 31.9 38.7 48.5 56.1 54.2 52.2 50.2 48.2 56.8 54.9 52.9 51.0 49.1 57.4 55.6 53.7
deviation% -9.2 -9.5 -7.3 0.0 2.4 7.5 -2.3 12.1 15.2 2.4 2.8 -12.6 -20.2 -11.2 4.6 11.8 10.4 -17.8 -6.9 -7.4 -3.2 -9.9 -14.3 -7.3 -14.6
Figure 1. 109D21/T1/2 vs Fr for the diffusion of toluene in carbon dioxide. (b) Lai and Tan, 1995; (s) this work (eqs 9-12); (- - -) TLSM model (eqs 13-19).
AAD % ) 9.0% a
Data sources: Noel et al., 1994; Eaton and Akgerman, 1997; Eaton et al., 1995. b Deviation % ) 100(Dcal - Dexp)/Dexp. Table 5. Prediction Results for Diffusion Coefficients of Naphthalene in Ethanol Using Equations 9-12a T (K) F (kg/m3) 373.2 423.6 473.4 500.3 516.2 516.2 530.2 530.2 530.2 542.8 542.8 542.8 554.0 554.0 554.0
716 649 557 467 567 469 418 476 544 411 449 517 398 433 491
Dexp Dcal (10-9m2/s) (10-9m2/s)
Fr 2.597 2.354 2.020 1.694 2.057 1.701 1.516 1.727 1.973 1.491 1.629 1.875 1.444 1.571 1.781
4.44 8.17 14.2 21.9 18.2 23.5 26.6 23.3 19.1 28.3 25.8 21.5 29.6 27.1 24.2
4.6 8.2 14.7 22.3 14.6 22.4 27.3 22.1 16.5 28.2 24.8 18.9 29.7 26.5 21.3
deviation 3.6 0.4 3.5 1.8 -19.8 -4.7 2.6 -5.2 -13.6 -0.4 -3.9 -12.1 0.3 -2.2 -12.0 AAD % ) 5.7%
source: Sun and Chen, 1986. Deviation % ) 100(Dcal - Dexp)/Dexp. aData
b
(2) The Eaton and Akgerman method (Eaton and Akgerman,1997), which is expressed as
[
D21 ) βxT(σ2/σ1)γ
]
M1 + M2 M1M2
1/2
(V0/σ221)[(V1/V0)R - b2] (20)
where β is a constant related to its unit (Eaton and Akgerman, 1997) and γ ) 1.7538 and
R ) σ1/σ2 - 1/3
Figure 2. 109D21/T1/2 vs Fr for the diffusion of 1-octene in ethane. (b) Noel et al., 1994; (0) Eaton and Akgerman, 1997; (O) Eaton et al., 1995; (s) this work (eqs 9-12); (- - -) TLSM model (eqs 1319).
(21)
b2 ) [-0.2440(σ1/σ2)2 + 0.8491(σ1/σ2) + 0.6001] × (M2/M1)-0.03587 (22)
Figure 3. 109D21/T1/2 vs Fr for the diffusion of naphthalene in ethanol. (b) Sun and Chen, 1986; (s) this work (eqs 9-12).
As can be seen in Table 2, this work usually gives better prediction results and covers a wider range of density and temperature in comparison with the above two methods. The Eaton and Akgerman method is generally good whenTc2 and Vc2 are known, but it may lead to large errors when Tc2 and Vc2 are not available and are estimated by a group contribution method, such as dibromobenzene in chlorotrifluoromethane. As for the TLSM model by Liu et al. (1997), it has the merit of covering a wide range of density and temperature, but the prediction results are not very satisfactory. Conclusions New correlations, eqs 9-12, are proposed for predicting the infinite-dilution diffusion coefficients in supercritical and high-temperature liquid solvents. The
Ind. Eng. Chem. Res., Vol. 37, No. 9, 1998 3797
correlations can estimate the diffusion coefficients with satisfactory results under the condition 0.22 e Fr e 2.62, 0.70 e Tr e 1.78, 58.1 e M2 e 885.4. Nomenclature A ) parameter in eqs 4 and 9 and calculated by eqs 11 and 12 D ) infinite-dilution molecular diffusion coefficient, m2/s k ) parameter in eq 11 and calculated by eq 12 M ) molecular mass, kg/kmol N ) number of data points NA ) the Avogadro number P ) probability R ) the gas constant T ) temperature, K u ) molecular velocity V ) molar volume ()1000M/F), cm3/mol V* ) critical hole volume calculated by eq 10, cm3/mol V0 ) the close-packed molar volume, cm3/mol Greek letters γ′ ) numerical factor to correct for overlap of free volume /kB ) energy parameter, K F ) density, kg/m3 σ ) molecule diameter, cm Subscripts 1 ) solvent 2 ) solute c ) critical r ) reduced Superscripts eff ) effective hard-sphere diameter LJ ) Lennard-Jones
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Received for review December 8, 1997 Revised manuscript received April 29, 1998 Accepted June 23, 1998 IE970898+
