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Ind. Eng. Chem. Res. 2002, 41, 3326-3328
CORRELATIONS A Modification of the Stokes-Einstein Equation for Diffusivities in Dilute Binary Mixtures Hendrik A. Kooijman† van Breestraat 131B, 1071 ZL Amsterdam, The Netherlands
An improved correlation is presented for the estimation of binary diffusion coefficients in dilute liquids. The correlation is based on the Stokes-Einstein equation with empirical corrections for the nonroundness and the size ratio of solute and solvent molecules. These corrections are computed from the UNIQUAC or UNIFAC group contribution R and Q parameters. The method works especially well for diffusivities involving water. b. From the molecule’s UNIQUAC Ri with
Introduction A knowledge of diffusion coefficients is important for the proper design of some chemical processes, for example,1,2 distillation, absorption, and chemical reactors, because these processes often are limited by a diffusional process. There are many methods available for estimating diffusion coefficients in dilute binary liquid mixtures (see, for example, refs 1 and 3). However, there is no method that works equally well for all compounds, and there also appears to be very little interest in developing improved correlations. This is somewhat surprising because while the methods presently available have an average relative error of around 12-30%, their maximum errors are much higher, ranging from 45 to 190%.3 Here a simple modification of the Stokes-Einstein (SE) equation for diffusion coefficients is proposed that appears to be an improvement over other methods and a step toward the development of more general methods. SE Equation The SE equation4 for the diffusion of a spherical solute molecule i in solvent j is
DSE ij ) kT/NSEπηjri
(1)
where NSE has the value 6. The SE equation was originally derived for the diffusion of large spherical molecules in solvents of low viscosity. Unfortunately, for solute molecules i that are similar in size to the solvent molecules j, the method predicts diffusivities that are too low. The SE equation often is used as a basis for developing improved correlations. Reid et al.3 provide an overview. ri in the SE equation is the solute molecular radius. It may be estimated in several different ways:
a. From the critical volume with ri ) (3πVci /4NA)1/3
ri ) 3.18 × 10-10Ri1/3
The Ri (and Qi) parameters in UNIQUAC are related to the (van der Waals) molecular volume and surface area.3 Using eqs 2 and 3 in a reverse manner to compute critical volumes gave good results except for water and some alcohols. Interestingly, computing diffusivities with radii from 3 gave good results!
c. Using (3) and the UNIFAC subgroups Rik contributions3 in the molecule Ri )
E-mail:
[email protected].
∑k νikRik
(4)
where νik is the number of occurrences of group k in molecule i. These three methods of estimating the solute radius give slightly different results, as we will see in the comparison with experimental data to be discussed below. New Correlation In the method proposed here, the SE diffusivity is corrected for the nonroundness and the size ratio of solute and solvent molecules. The nonroundness for a spherical molecule should be zero and a value larger than zero for any molecule that is not a sphere. The nonroundness of molecule i is assumed to be a function of the quotient of the UNIQUAC R (volume) and Q (surface area) values:
|
φi ) 1 -
Ri/Qi (R/Q)ref
|
(5)
where the reference UNIQUAC unit, the methylene (-CH2-) group, is used to normalize the correction; thus, (R/Q)ref ) 1.249. To correct for the size ratio of solute i and solvent j molecules, we multiply with the following correction factor:
(2) θij )
†
(3)
Ri1/3 + Rj1/3 Ri1/3
10.1021/ie010690v CCC: $22.00 © 2002 American Chemical Society Published on Web 05/30/2002
)1+
() Rj Ri
1/3
(6)
Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002 3327 Table 1. Constants for Equation 7 set
A0
Ai
Aj
Ar
set
A0
Ai
Aj
Ar
A B
2.50 3.53
0 1
0 1
0 0
C
1.58
1
1/3
1
where molecule radii are approximated with the third power of the UNIQUAC R parameters. For ri ) rj, this results in a correction factor of 2, while the correction becomes unity when ri . rj (see also ref 5 for size-ratio effects). Now we define the new diffusivity correlation as:
Dij ) A0(1 - φi)Ai(1 - φj)AjθAij r DSE ij
(7)
where either the UNIQUAC or UNIFAC R and Q parameters can be used. Model parameters A0, Ai, Aj, and Ar were determined through fitting the above equation to a database of experimental diffusivities. Table 1 lists the resulting sets of values for these parameters. Parameter set A is just the SE correlation with a different constant, namely, NSE ) 2.4 (this is close to the average NSE value in Table 5.1 of work by Rutten6). Parameter set B represents a modified SE correlation with corrections for the nonroundness of the molecules. Finally, parameter set C corrects for the nonroundness as well as the size ratio of the solute and the solvent. Note that including the size ratio reduces the optimal parameter value for the nonroundness of the solvent. It was observed that deviations were rather large for carbon tetrachloride. Therefore, a Q value of 2.4 was used for this compound (instead of 2.91). Similarly, a Q value of 0.938 was used for methane. Comparison with Other Estimation Methods Here we provide a comparison of the proposed method with the Wilke-Chang7 equation, the modified method of Tyn and Calus,8 and the modified Hayduk-Minhas9 correlations. The Wilke-Chang correlation is (in SI units)
(φjMj)T Dij ) 1.1728 × 10-16 ηj(θiVbi)0.6
(8)
where φ is a correction term for water (2.6), methanol (1.9), and ethanol (1.5) as the solvent (otherwise, φ ) 1). Normally, θi ) 1, and this method functions well, except for when water is the solute. Here, a modification of the Wilke-Chang correlation is proposed where we assume θH2O ) 4.5 and θi ) 1 for all other components. The Tyn-Calus correlation includes the parachor P as parameters:
Dij ) 28.25 × 10-16
( )() Vbi
Vbj
2
1/6
Pj Pi
0.6
T ηj
(9)
It is important to note that when water is the solute, the values of the molar volume and parachor are doubled, assuming water occurs as a dimer (without this modification, the predictions for water as the solute are much worse). The same is true for organic acid solutes in a solvent other than water, methanol, or butanol. Furthermore, for a nonpolar solute in monohydroxy alcohol solvents, the liquid volume and parachor are multiplied by factor 8 × 103ηj. The parachor for compound i can be computed from the liquid molar volume
Table 2. Average/Maximum Relative Errors: 100|Dcalc/ Dexp - 1| method
overall
alkanes
aqueous
7-A-a 7-B-a 7-C-a 7-C-b 7-C-c Wilke-Chang modified Wilke-Chang modified Tyn-Calus modified Hayduk-Minhas
21.1/71 14.1/54 10.6/50 9.9/48 10.5/55 22.6/167 14.9/114 11.4/45 12.9/82
19.7/58 21.3/48 13.4/42 13.7/42 13.7/42 8.8/28 8.8/28 16.7/45 9.2/20
31.3/71 12.3/38 8.7/22 6.7/22 8.9/55 30.7/167 8.7/41 11.4/41 17.7/82
at boiling point and the surface tension:
Pi ) Vbiσi1/4
(10)
The modified Tyn-Calus method uses the Brock and Bird method for estimating the surface tension:3
Tbr ) Tbi/Tci
(11)
pr ) p/101325
(12)
Rc ) 0.9076[1 + Tbr ln pr/(1 - Tbr)]
(13)
σ ) 10-3(0.132Rc - 0.278)Pr2/3 Tc1/3(1 - Tbr)11/9 (14) Thus, the modified Tyn-Calus method does not depend on the availability of the parachor but computes it from the normal boiling and critical temperatures of the component. The Hayduk-Minhas method consists of three different equations, one for water as the solvent
Dij ) 1.25 × 10-12[(103Vi)-0.19 - 0.292]T1.52(103ηj) (15) (with ) 9.58 × 10-3/Vi - 1.12), one for n-paraffins
Dij ) 13.3 × 10-12
T1.47(103ηj) (103Vi)0.71
(16)
(with ) 10.2 × 10-3/Vi - 0.791), and one for nonaqueous mixtures
Dij ) 1.55 × 10
-12
(
Pj0.5
)
T1.29 1.995 0.42 Pi (103ηj)0.92(103Vj)0.23 (17)
The modified Hayduk-Minhas method uses the computed parachors just like the modified Tyn-Calus method. The correlations were evaluated with diffusivity data from various sources that included organic-organic and organic-water systems (including associating systems;6 experimental diffusivities of alkanes-alkynes in water with extreme deviations were omitted). The purecomponent liquid viscosities that are required by all correlations were computed from the DIPPR correlation.10 Table 2 compares the average and maximum relative errors of the various methods. Listed here are the results for the modified Tyn-Calus and HaydukMinhas methods because not all of the parachors were available in our component database (with little or no impact on the performance; see ref 3). Surprisingly, the SE equation with a different value for NSE can give reasonable estimates for the diffusion
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Ind. Eng. Chem. Res., Vol. 41, No. 13, 2002
(see www.chemsep.org/downloads/#databases). I also thank Professor Ross Taylor (of Clarkson University) and the reviewers for helpful comments. Supporting Information Available: Table of values obtained for the comparisons of the various models. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature
Figure 1. Parity plot for the original Wilke-Chang correlation.
A ) correlation constants D ) diffusivity (m2/s) k ) Boltzman constant ) 1.3807 J/K NA ) Avogadro’s constant ) 6.022 × 1026 kmol-1 NSE ) constant in the Stokes-Einstein equation (6) p ) pressure (Pa) P ) parachor (m3 kg1/4 s-1/2 kmol-1) Q ) UNIQUAC molecular surface area Qk ) UNIFAC molecular surface area of group k R ) UNIQUAC molecular volume Rk ) UNIFAC molecular volume of group k T ) temperature (K) Tb ) normal boiling point temperature (K) Tc ) critical temperature (K) Vc ) molar volume at the critical point (m3/kmol) Vb ) liquid molar volume at the normal boiling point (m3/ kmol) R ) Riedel parameter at the critical point η ) viscosity (Pa‚s) φ ) nonroundness correction σ ) surface tension (N‚m) θij ) solute-solvent size-ratio correction, eq 6 θi ) correction term for the modified Wilke-Chang correlation, eq 8
Figure 2. Parity plot for the correlation 7-C-b.
Literature Cited
coefficient. It performed similarly to the original WilkeChang method, which exhibits large deviations when water is the solute. The modified Wilke-Chang method proposed here drastically improves the performance for water as the solute. Next in line are the modified Hayduk-Minhas and Tyn-Calus methods. However, the new correlation (7) using parameter set C performed better than all of the others that were tested. A slight improvement was obtained when solute radii were computed from UNIQUAC R volumes (b) instead of the critical volumes (a). Parity plots (Figures 1 and 2) for the original Wilke-Chang and the new correlation show the performance of the proposed correlation.
(1) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; Wiley: New York, 1993. (2) Westerterp, K. R.; van Swaaij, W. P. M.; Beenackers, A. A. C. M. Chemical Reactor Design and Operation; Wiley: New York, 1984. (3) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (4) Einstein, A. U ¨ ber die von der molekularkinetischen Theorie der Wa¨rme geforderte Bewegung von in ruhenden Flu¨ssigkeiten suspendierten Teilchen. Ann. Phys. 1905, 17, 549. (5) Bosma, J. C.; Wesselingh, J. A. Estimation of Diffusion Coefficients in Dilute Liquid Mixtures. Trans. Inst. Chem. Eng. 1999, 77, 325. (6) Rutten, Ph. W. M. Diffusion in Liquids. Ph.D. Thesis, University of Delft, Delft, The Netherlands, 1992. (7) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264. (8) Tyn, M. T.; Calus, W. F. Diffusion coefficients in dilute binary liquid mixtures. J. Chem. Eng. Data 1975, 20, 106. (9) Hayduk, W.; Minhas, B. S. Correlations for prediction of molecular diffusivities in liquids. Can. J. Chem. Eng. 1982, 60, 295. (10) Daubert, T. E.; Danner, R. P.; Sibul, H. M.; Stebbing, C. C. Physical and Thermodynamic Properties of Pure Chemicals; Data Compilation, Taylor and Francis Publishing: New York, 1998.
Conclusions A modification of the SE equation has been put forward for the estimation of diffusion coefficients in dilute liquid mixtures. The proposed correlation has an overall average error of 10% for an experimental data set of 245 points. The advantage of the new correlation is that by using group contributions it becomes predictive and eliminates the need for critical and normal boiling point temperatures as well as liquid molar volumes. Acknowledgment I thank Dennis Bosse at the (University of Kaiserslautern) for making his diffusivity database available
Received for review August 20, 2001 Revised manuscript received March 12, 2002 Accepted April 5, 2002 IE010690V
