Ind. Eng. Chem. Res. 2001, 40, 465-469
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RESEARCH NOTES Expression for the Viscosity and Diffusivity Product Applicable for Supercritical Fluids Geert F. Woerlee† Laboratory for Process Equipment, Delft University of Technology, Leeghwaterstraat 44, NL-2628 CA Delft, The Netherlands
This study presents an expression for the viscosity-diffusivity product, derived using the kinetic gas theory and an expansion term into supercritical densities by applying exponential Boltzmann factors. This leads to a transport relation, which describes the diffusivity and viscosity at any fluid density.The equation can be combined with an equation of state such as the Peng-Robinson equation of state (Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59-64). Using the transport relation and viscosity data, binary Maxwell-Stefan diffusion coefficients at supercritical conditions can be estimated. It is shown that, when increasing the density, the model predicts diffusion coefficients that subsequently follow the kinetic theory for gas densities, the hard sphere relation for dense fluids and the Stokes-Einstein equation for liquids. Data are applied to illustrate the predicting quality of the expression. It is shown that accuracies for diffusion coefficients of 25% is obtained. This is in most cases enough for engineering purposes. Introduction Separation design requires a substantial knowledge of the thermodynamic and physical properties of the components involved. The available data for the design will often not be complete, leaving the engineer with the option of either measuring the required properties or estimating them. Given the time and money restrictions, a balance of experiments and estimations has to be found. Although elaborate methods are available to estimate vapor-liquid equilibria,1-3 it is standard practice to determine the thermodynamic behavior experimentally and to estimate most of the transport properties, i.e., thermal conductivities, viscosities, and especially the diffusion coefficients. Several correlations for the diffusion coefficients in the liquid and gas phases have been suggested,4-6 but studies have shown that over a wide temperature and viscosity range simple empirical correlations for the estimation of the diffusion coefficient are still inadequate.7 Considerable progress has been made over the past few years for the estimation of diffusion coefficients in the vicinity of the critical point,8-11 but simple estimation methods for diffusivity covering the whole gas-liquid density range are still not available. However, estimation methods for the viscosity and diffusion coefficient are crucial in the design of separation processes, and a continuous improvement of the available estimation methods is required. Most of the current estimation methods for the gaseous transport properties are based on the kinetic gas theory,4 while the available relations for dense fluids or liquids use either a hard-sphere approach12-17 or the hydrodynamic Stokes-Einstein relation.18-20 † E-mail:
[email protected]. Tel: +31 20 4196050. Fax: +31 20 4196051.
In this paper the diffusivity and the viscosity are expressed using the kinetic gas theory and an activation energy theory using a Boltzmann factor. This activation energy theory is similar to the Eyring theory of absolute reaction rates.21 However, it is shown that the activation energies of viscosity and diffusivity cancel when multiplied. Although the remaining expression after multiplication is strongly determined by the kinetic gas theory only, it is shown that the derived expression is valid for any fluid density. When the new relation is applied at liquid densities, it resembles the StokesEinstein relation. This so-called modified Stokes-Einstein relation for a liquid was also empirically found.20 The Exponential Transport Property Description It is often suggested that the essential mechanism for molecular motion in a liquid or solid material is hopping from one vacancy to another place to the other.21-24 To enable a jump, a characteristic activation energy (v) is required. The probability of a displacement can be described using a Boltzmann factor, which is an exponential function of the required activation energy. Hence, the diffusion coefficient is proportional to the number of net displacements; the diffusivity of a component i can be described as25
^i ) ^i,0 exp
( ) -v kT
(1)
In this equation, ^i,0 is a reference value for the diffusivity in case no activation energy is required, k is Boltzmann’s constant, and T represents temperature. The viscosity is determined by the shear stress on the fluid. When no shear stress is applied on the fluid, an
10.1021/ie0002881 CCC: $20.00 © 2001 American Chemical Society Published on Web 12/08/2000
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equal number of molecules jump from one side of an imaginary plane to the other side. This is equivalent to the diffusion mechanism. However, when a shear stress is applied parallel to the imaginary plane, the activation energy perpendicular to the central plane, with activation energy s, is decreased by ∆s on the one side and increased by ∆s on the other side.25 When N0 symbolizes a constant of the jump rate, the net jump rate (N) over a molecular layer toward one side can be expressed,by application of a Taylor expansion, as
(
)
(
) ( )
s - ∆s s + ∆s - N0 exp ) kT kT s 2∆s N0 exp (2) kT kT
N ) N0 exp -
The net jump rate N is equal (or proportional) to the velocity gradient perpendicular to the flow direction ∂u/ ∂x. Expression (2), therefore, can be used in the definition of viscosity, which is expressed as the force F per area A divided by the velocity gradient perpendicular to the flow direction. The expression for the dynamic viscosity, therefore, can be written as
F F s A A ) ) η0 exp η) ∂u N kT ∂x
( )
(3)
where η0 is a reference viscosity in case no energy is required to activate a jump. An exponential dependence of the viscosity with temperature is well-known for many fluids.7,25,26 This simple analysis intends to demonstrate that the basic mechanism for diffusivity and viscosity is identical. This means that the exponential activation energies involved for diffusion, eq 1, and viscosity, eq 3, are the same.
s ) v
(4)
When assumption (4) is correct, it becomes rewarding to multiply the expressions for viscosity and diffusivity. Hiss and Cussler27 also suggested this. When the energies are not identical, an exponential term would remain. This would lead to a rapid deviation in the description even if the remaining energy difference were small. If, however, the energies are identical, the type of energy is not essential, because the exponential term would always cancel. Assumption (4) leaves the need to estimate a viscosity η0 and a diffusivity ^i,0 for which no activation energy is required. When the molecules are far apart, they do not have a significant mutual interaction and activation energy is not required. This indicates that these coefficients (η0 and ^i,0) can be found by using an expression for gaseous densities. For engineering purposes, an adequate description of the gaseous transport properties is given by the kinetic theory. The Kinetic Description The momentum of a particle with mass m and average velocity cj is transported over the mean free path λ plus its hard-sphere diameter σ.27 The kinetic gas viscosity is written as
η0 )
(
)
mNA(λ + σ)cj mcj 3 b ) 1+ 2 3vm 2 vm 3πσ
(5)
Here the mean free path of a particle λ with a diameter σ is expressed as
λ ) vm/(πσ2NA)
(6)
where vm is the molar volume and NA is Avogadro’s number. The hard-sphere diameter σ of the molecules in eq 5 can be calculated using the excluded-volume parameter b ) 2/3πσ3NA of an equation of state. The diffusivity of a component i is expressed using the kinetic gas theory as well.28
^i,0 )
jc vm 3 πσ 2N i
(7) A
The transport of a molecule has a reference system according to its center of mass. The mean free path is therefore not increased with an additional hard-sphere diameter. The diffusivity is linearly dependent on the mean free path and the average velocity. Consequently, the diffusivity is linearly dependent on the molar volume. Viscosity-Diffusivity Transport Relation for Gas and Liquid Densities. Using eqs 5 and 7 in the multiplication of the exponential expressions (1) and (3) and the condition that the exponential term cancels (5), a new relation is obtained as
λcj mNA(λ + σ)cj ) ^iη ) ^i,0η0 ) 3 3vm
(
)
3 b 2 vm vm 9π2σ 2σ2 NA
2 1 +
i
(8) In eq 8 the ^ molecular kinetic energy 1/2mc2 term is written as an average molecular energy j. The kinetic energy also can be expressed in terms of temperature as 2j ) 3kT, so that relation (8) can be written as
(
^ iη )
)
3 b 2 vm vm NA 9π2σ 2σ2
3kT 1 +
(9)
i
In this equation the diffusion coefficient ^i and the diameter σi are related to the diffusing molecule, while all other properties are related to the mixture. Both the exponential description and the kinetic gas expressions for the diffusivity and viscosity are well established. However, the combination of both expressions leads to a relation that can become very useful. It is valid for any density and it relates the transport properties as a function of other properties that are known. When the molar volume and one of the transport properties is known, the other transport property is simply expressed with relation (9). Normally one would estimate the diffusivity, because, first, it is more difficult to measure diffusivity and, second, in a mixture with n components, we have n(n - 1)/2 diffusion coefficients, while there is only a single mixture viscosity. Effective Collision Diameter. Relation (9) is valid for gas densities, because it is directly related to the transport properties at gaseous densities. However, because of the attractive forces of real molecules, it is
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 467
σi )
[
( ) 3b 2πNA
1/3
+
(
)](x
v m vc 1 1 + tanh 2 2 b b
Figure 1. Attractive forces affecting the collision of particles. Situation b shows an unaffected particle track, which can be found at high temperatures, and situation c shows a collision caused by the path changes due to the attractive forces.
1+
xTcTc,i T
)(
-1
In eq 11 the attracting forces on a molecule are an interaction of the molecule itself and the surrounding molecules. The attractive forces therefore are calculated by mixing the critical temperature of the component (Tc,i) and the critical temperature of the mixture (Tc). Figure 2 illustrates the behavior of the formulated collision near the critical point. In this case molar volume and diffusion data of Harris,29 in methane, are combined by the viscosity estimation methods of Lucas30 for low-density viscosity and of Jossi et al.31 for the excess viscosity. The molar volume is calculated using the Peng-Robinson equation of state as is the hardsphere diameter using the excluded volume. Similar results are obtained when using experimental data for the viscosity. Figure 3 also show a good agreement with the measurements over the whole density range. Although the description around the critical point is less accurate, it still has accuracies of better than 25%. Stokes-Einstein Relation Revisited. If one considers a pure liquid, all molecular diameters are identical (σ ) σi) and the molar volume (vm) can be taken as the excluded volume b. In that case eq 9 reduces to a form similar to the Stokes-Einstein relation
2 3 3kT πNAσi3 1 + 3 2 5 kT ) ^ iη g 9 πσi 9π2σi2σi2NA
(
Figure 2. Measured33 and estimated diffusion coefficient at infinite dilution of benzene and naphthalene in carbon dioxide as a function of the density near the critical point.
required to adjust the collision diameter. The forces effectively increase the collision diameter of the molecules in comparison with the hard-sphere diameter. The influence of the attractive forces on the collision diameter is illustrated in Figure 1. When the temperature increases, the attractive forces relatively decrease compared with the velocity of the molecules. The attractive forces of the molecules are related to the critical temperature Tc,i of the component. It is suggested28 that the collision diameter of a real molecule (σi) can be approximated as
σi )
( )x 3b 2πNA
1/3
Tc,i 1+ T
(10)
In eq 10 the hard-sphere diameter of the molecule is represented by the excluded-volume relation. The gaseous effective diameter of eq 10 is connected with the liquid collision diameter by application of a smooth transition function from gaseous to liquid densities. Given the nature of the relations, an exponential interpolation of the two extremes is used where the transition takes place at the critical volume (vc) of the total medium:
)
3b 1/3 2πNA (11)
)
(12)
which is valid for self-diffusion. The Stokes-Einstein relation has proven its applicability over many decades. The outcome, therefore, validates assumption (4). The coefficient in this study equals 5/9, whereas the value deduced in the hydrodynamic Stokes-Einstein analysis equals 3/9. The analysis of Rutten20 of collected data led to an average coefficient of 4.93/9 for self-diffusion in liquid systems. In a system containing different molecules, the translation energies of all molecules are identical, so that when considering a condensed fluid, one finds
2 3 3kT πNAσ3 1 + 3 2 5 kT σ ) ^ iη g 2 2 2 9 πσi σi 9π σi σ NA
(
)
(13)
In an overview 200 binary systems were examined. The molecular systems were categorized, and an optimum coefficient was determined for each category. Although optimum coefficients were found for different categories, all values are close to the value of 5/9 found in this study. The result of eqs 12 and 13 for over 200 measured binary diffusion coefficients is compared with the estimated diffusion coefficients applying measured viscosity data and the excluded-volume parameter of the PengRobinson equation of state. The results are shown in Figures 4 and 5. The parity plot of Figure 5 shows the improved predictive quality of the modified StokesEinstein relation (13). The hypothesis of the equal activation energy, therefore, first predicts the empirical correct coefficient of 5/9 in the Stokes-Einstein relation
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Figure 3. Measured29 and estimated self-diffusion coefficients for methane as a function of the specific volume for various temperatures. Figure 5. Comparison of the measured and estimated liquid diffusion coefficients using relation (13).
Figure 4. Comparison of the measured and estimated diffusion coefficients in liquids using relation (12).
and second leads to a correction term also found empirically for liquid diffusion. The comparison considers 250 measured data points. The average deviation of measured and estimated values of eq 12 is equal to 14.7%, while the absolute average deviation equals 24.0%. In comparison, the average deviation of measured and estimated values of eq 13 equals 5.5%, while the absolute average deviation equals 15.0%. Concluding Remarks The viscosity-diffusivity transport equation (8) relates the diffusion of a component i into a viscous system, with mixture properties vm and η. In principle, no limitations are set to the types of systems. The system can, therefore, also be a mixture of components. However, the obtained diffusivity can only be used directly as a binary diffusion coefficient when the system is infinitely diluted; ^i ) ^ij. Direct estimation of the binary diffusion coefficients for other concentrations
requires an interpolation formula, such as, for instance, the Vignes32 interpolation. The presented transport relation connects the diffusivity and viscosity as a molar volume expansion (vm + 3/2b). Although this expansion is connected with the description of the collision diameter, it can be seen as an equation of state for transport properties. The fact that the equation is valid for any density is the main improvement of this equation compared with other estimation methods. Next to this, the relation can be connected directly with an equation of state. This connection is demonstrated by applying the PengRobinson equation, but by application of a more specific estimation method and equation of state, the predicting ability can be improved. In the paper of Liu and Ruckenstein11 describing the diffusion in supercritical fluids, their estimation method for this region has a better accuracy. However, their method is developed on trace diffusion in carbon dioxide and is only valid in the near-critical region. Currently the expression is extended to a general method for estimating nonideal diffusion coefficients. To enable this, an exponential description for mixture viscosity is required. Literature Cited (1) Peng, D. Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. (2) Thomas, E. R.; Eckert, C. A. Prediction of limiting activity coefficients by a modified separation of cohesive energy density model and UNIFAC. Ind. Eng. Chem., Process Des. Dev. 1984, 23, 194-209. (3) Fredenslund, A.; Gmehling, J.; Rasmusen, P. Vapor-Liquid Equilibria using UNIFAC; Elsevier: Amsterdam, The Netherlands, 1977. (4) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264. (5) Takahashi, S.; Hongo, M. Diffusion coefficients of gases at high pressures in CO2-C2H4 system. J. Chem. Eng. Jpn. 1982, 15, 57. (6) Oosting, E. M.; Gray, J. I.; Grulke, E. A. Correlating Diffusion Coefficients in concentrated carbohydrate solutions. AIChE J. 1985, 31, 773. (7) Reid, C. R.; Prausnitz, J. M.; Poling, E. P. Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1988.
Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 469 (8) Funazukuri, T.; Ishiwata, Y.; Wakao, N. Predictive Correlation for binary diffusion coefficients in dense carbon dioxide. AIChE J. 1992, 38, 1761-1768. (9) Debenedetti, P. G.; Reid, R. C. Binary diffusion in supercritical fluids. In Supercritical Fluid Technology; Penninger, J. M. L., Randosz, M., McHugh, M. A., Krukonis, V. J., Eds.; Elsevier Science Publishers B.V.: Amsterdam, The Netherlands, 1985. (10) Eaton, A. P.; Akgerman, A. Infinite-dilution diffusion coefficients in supercritical fluids. Ind. Eng. Chem. Res. 1997, 36, 923-931. (11) Liu, L.; Ruckenstein, E. Predicting diffusion coefficients in supercritical fluids. Ind. Eng. Chem. Res. 1997, 36, 888-895. (12) Dymond, J. H. Corrected Enskog theory and the transport coefficients of liquids. J. Chem. Phys. 1974, 37, 1643. (13) Sun, C. K. J.; Chen, S. H. Tracer diffusion in dense ethanol: a generalized correlation for nonpolar and hydrogenbonded solvents. AIChE J. 1986, 32, 1367. (14) Funazukuri, T.; Hachisu, S.; Wakao, N. Measurements of diffusion coefficients of C18 unsaturated fatty acid methyl esthers in supercritical carbon dioxide. Ind. Eng. Chem. Res. 1991, 30, 1323. (15) Matthew, M. A.; Akgerman, A. Hard-Sphere theory for correlation of tracer diffusion of gases and liquids in alkanes. J. Chem. Phys. 1987, 87, 2285. (16) Erkey, C.; Rodden, J. B.; Akgerman, A. A correlation for predicting diffusion coefficients in alkanes. Can. J. Chem. Eng. 1990, 68, 661. (17) Liong, K. K.; Wells, P. A.; Foster, N. R. Diffusion coefficients of long-chain esters in supercritical carbon dioxide. Ind. Eng. Chem. Res. 1991, 30, 1329. (18) Swaid, I.; Schneider, G. M. Determination of binary diffusion coefficients of benzene and some alkylbenzene in supercritical CO2 between 308 and 328K in the pressure range 80 to 160 bar with supercritical fluid chromatography (SFC). Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 969. (19) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (20) Rutten, P. W. M. Diffusion in Liquids. Ph.D. Dissertation, Technical University of Delft, Delft, The Netherlands, 1992. (21) Glasstone, S. N.; Laidler, K.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941.
(22) Kittel, C. Introduction to Solid State Physics, 6th ed.; John Wiley & Sons Inc.: New York, 1986. (23) Eyring, H. Viscosity, plasticity and diffusion as examples of absolute reaction rates. J. Chem. Phys. 1936, 4, 283-291. (24) Macedo, P. B.; Litovitz, T. A. On the relative roles of free volume and activation energy in the viscosity of liquids. J. Chem. Phys. 1965, 42, 245. (25) Walton, A. J. Three Phases of Matter, 2nd ed.; Oxford University Press: Oxford, U.K., 1983. (26) Andrade, E. N. C. The viscosity of liquids. Nature 1930, 125, 309-310. (27) Hiss, T. G.; Cussler, E. L. Diffusion in high viscosity liquids. AIChE J. 1973, 19, 698. (28) Tabor, D. Gases, Liquids and Solids and other states of matter, 3rd ed.; Cambridge University Press: Cambridge, U.K., 1991. (29) Harris, K. R. The density dependence of the self-diffusion coefficient of methane at -50°, 25° and 20 °C. Physica 1978, 94A, 448-464. (30) Lucas, K. Phase Equilibria and Fluid Properties in the Chemical Industry; Dechema: Frankfurt, Germany, 1980; p 573. (31) Jossi, J. A.; Stiel, L. I.; Thodos, G. The viscosity of pure substances in the gaseous and liquid phases. AIChE J. 1962, 8, 59. (32) Vignes, A. Diffusion in binary solutions. Ind. Eng. Chem. Fundam. 1966, 5, 189-199. (33) Feist, R.; Schneider, G. M. Determination of binary diffusion coefficients of benzene, phenol, naphthalene and caffeine in supercritical carbon dioxide between 308 and 333K in the pressure range of 80 to 160 bar with supercritical fluid chromatography (SFC). Sep. Sci. Technol. 1982, 17, 261.
Received for review March 2, 2000 Revised manuscript received October 17, 2000 Accepted October 20, 2000 IE0002881
