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Measurement and calculation of multicomponent diffusion coefficients in liquids
Fluid Phase Equilibria 256 (2007) 99–104
Measurement and calculation of multicomponent diffusion coefficients in liquids Sebastian Rehfeldt ∗ , Johann Stichlmair Lehrstuhl fuer Fluidverfahrenstechnik, Technische Universitaet Muenchen, Boltzmannstrasse 15, 85747 Garching, Germany Received 30 June 2006; received in revised form 9 October 2006; accepted 9 October 2006 Available online 20 October 2006
Abstract In the literature, several prediction models for binary diffusion coefficients are found. For multicomponent systems however, only a few models exist. Due to lack of ternary diffusion data, these models have not been verified for real systems until now. To overcome this limitation, multicomponent diffusivities were measured within the whole concentration space of the ternary system acetone–1-butanol–1-propanol. Fick diffusivities were transformed to less concentration-dependent Maxwell–Stefan diffusivities, using the Wilson model for calculating the thermodynamic correction factor. Three suggested prediction models and one newly developed were tested by a comparison of predicted values to experimental data. The new prediction method shows the best results. All tested prediction models show larger deviations with increasing distance to the binary edge of the concentration triangle. © 2006 Elsevier B.V. All rights reserved. Keywords: Diffusion; Maxwell–Stefan diffusivities; Multicomponent systems
1. Introduction ji = − Diffusion is a physical phenomenon occurring in every natural or industrial process involving mass transfer. It can play an important role as the rate determining step. Therefore, the determination of diffusivities in liquids is of great interest for the calculation of mass transfer processes. Most studies about diffusion concentrate on measuring and predicting diffusivities in binary mixtures. Research on this topic advanced, also due to a fair database of experimentally determined binary diffusivities. However, real processes mostly deal with multicomponent mixtures. In this case, both the experimental and theoretical investigations of diffusion are much more complex. Therefore, few multicomponent diffusion data are found in the literature. 2. Theory There are two well-established theories describing diffusion. The more commonly used is Fick’s Law [1]: ∗
Corresponding author. Tel.: +49 89 289 16517; fax: +49 89 289 16510. E-mail address: [email protected] (S. Rehfeldt).
0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.10.008
n−1
Dij · ∇cj ,
i = 1, . . . , n − 1
(1)
j=1
The flow j is defined as the product of diffusion coefficients Dij and a driving force, i.e. the concentration gradient cj . In case of a binary mixture, only one diffusion coefficient is needed to describe all diffusional fluxes. However, in systems with more than two components, more coefficients are necessary and the structure of the flux equations is getting more complex. For example, four diffusivities Dij are required to describe mass flow rates in a ternary system: j1 D11 D12 ∇c1 =− · (2) j2 D21 D22 ∇c2 D11 and D22 are called the main coefficients. They show the dependency of the flow rate of a component on its own concentration gradient. The influence on the flow rate of the concentration gradient of the other component is taken into account by the cross coefficients D12 and D21 . However, only n − 1 flow rates are independent. In the ternary case, the third component is called the “solvent”, its flow rate can be determined with the flux balance condition. The four ternary diffusivities implicitly contain information on the flux of the solvent, so their values depend
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S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
on its choice. It is obvious that ternary Fick diffusivities are not related to the binary diffusion coefficients. This is a great disadvantage for prediction. However, Maxwell and Stefan developed a physically more consistent diffusion theory [2]. Based on the kinetic theory of ideal gases, the assumption of the Maxwell–Stefan-equations is an equilibrium of molecular friction and thermodynamic interaction. Any deviation from equilibrium leads to a diffusion flux. For liquid, non-electrolytic mixtures, the chemical potential gradient μi is used as the driving force d, which is counterbalanced by friction. The friction between two components is proportional to their velocity difference vi − vj and their amount of molar fractions x: di =
xi xj xi · ∇(T,p) μi = Γij · ∇xj = − (vi − vj ) RT Ðij n−1
n
j=1
j=1
(3)
∂ ln γi Γij = δij + xi ∂xj T,p for i = j and δij = 0
for i = j
(4)
In binary systems, there is again a single diffusion coefficient Ðij . In contrast to Fick’s law, for description of diffusion in a ternary mixture only three Maxwell–Stefan diffusivities Ðij are needed. Friction of each couple of components is taken into account by one coefficient. Thus, the choice of a solvent is dispensable. Therefore, multicomponent Maxwell–Stefan diffusivities can be related to the binary case. This should be advantageous for prediction. In the literature, most publications only contain Fick diffusivities, since they can be measured experimentally. Binary Maxwell–Stefan diffusivities can be calculated from experimental Fick diffusion coefficients and the thermodynamic correction factor: Ð=
D Γ
Fick diffusivities were experimentally measured by holographic laser-interferometry [4]. In this method, a measuring cell is filled with two mixtures with slightly different concentrations. Unsteady diffusion causes a changing concentration profile inside the diffusion cell, which causes an interference pattern. The cell is photographed in intervals with a digital camera. Analyzing the change of the interference pattern, Fick diffusion coefficients can be calculated [5]. For binary datapoints, only one experiment is necessary. Determination of ternary Fick diffusivities is much more complex. Several experiments with pairs of mixtures with different concentrations are necessary. 4. Results
Ðij denotes the Maxwell–Stefan diffusivities which can be interpreted as inverse drag coefficients of components i and j. The thermodynamic correction factor Γ ij can be calculated e.g. from the activity coefficients γ i :
with δij = 1
3. Experimental setup
Fick diffusion coefficients were measured over the whole concentration range of the ternary system acetone–1-butanol–1propanol and its three binary systems. Maxwell–Stefandiffusivities were also computed from the experimental Fick diffusivities, employing the Wilson excess Gibbs energy model for calculation of the thermodynamic correction factor [1]. Binary Fick diffusivities of acetone–1-propanol (Fig. 1) and acetone–1-butanol (Fig. 2) show a similar dependency on molar fraction of acetone. Values in the binary system acetone–1propanol are slightly higher. The Maxwell–Stefan diffusion coefficients in both systems are less dependent on concentration. In acetone–1-propanol, a linear relation of the Maxwell–Stefan diffusivity and concentration can be observed. Such functions of the binary diffusion coefficient were proposed by Darken [6]. Here, the binary diffusion coefficient is calculated as a sum of the binary diffusivities at infinite dilution multiplied with the molar fraction: x1 →1 x1 →0 Ð12 = D12 · x1 + D12 · x2
In the binary system of acetone–1-butanol, the graph of the Maxwell–Stefan diffusion coefficient is slightly curved. This
(5)
Transformation in binary systems is quite simple. Regarding ternary systems, the situation is more complex. Indeed, there is an algebraic relationship between Fick and Maxwell–Stefan diffusion coefficients [1]. Since there are four Fick coefficients, but only three Maxwell–Stefan coefficients, the transformation is only straightforward from Maxwell–Stefan to Fick. Transforming Fick diffusivities to Maxwell–Stefan, the system of equations is over-determined. Fick diffusion coefficients and thermodynamic factors both have experimental errors and are, in turn, usually inconsistent [3]. Therefore, a least-square fitting is required to compute Maxwell–Stefan diffusivities from Fick diffusivities.
(6)
Fig. 1. Binary diffusivities acetone–1-propanol.
S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
Fig. 3. Binary diffusivities 1-propanol–1-butanol.
Fig. 2. Binary diffusivities acetone–1-butanol.
result resembles the prediction model of Vignes [7]: Ð12 =
x1 →1 x1 (D12 )
x1 →0 x2 · (D12 )
101
(7)
1-propanol–1-butanol is a nearly thermodynamic ideal mixture. The Fick diffusion coefficient has nearly constant values (Fig. 3). Due to values of the thermodynamic correction factor near unity, there is only a small difference between Fick and Maxwell–Stefan diffusion coefficient. The experimental results shown prove that in these binary mixtures, Maxwell–Stefan diffusivities can be predicted quite accurate, if diffusivities at infinite dilution are known and an exact thermodynamic description of the system is possible. However, errors in the thermodynamic phase equilibrium data have a big influence on the quality of the Maxwell–Stefan diffusion coefficients. Since the thermodynamic factor involves the
first derivative of the activity coefficient with respect to composition, MS coefficients are quite sensitive to the parameters used to calculate the thermodynamic factor. These parameters are fitted against vapor–liquid equilibrium data. Even if the fitted parameters lead to good results for phase equilibrium, the first derivatives of the activity coefficients can show large deviations [1,3]. The ternary Fick diffusivities are displayed in Fig. 4. Each measured data point is plotted, a line indicates the position in the base triangle. A surface is fitted to the data points to illustrate the shape of the ternary diffusivities. They show stronger concentration dependence than the binary diffusivities. Especially the cross diffusion coefficients vary significantly with concentration. While values of the cross coefficients D12 are small, all values of D21 are negative.
Fig. 4. Ternary Fick diffusion coefficients of acetone–1-butanol–1-propanol.
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Fig. 5. Ternary Maxwell–Stefan diffusivities of acetone–1-butanol–1-propanol.
Obviously, there is no relation between binary and ternary Fick diffusion coefficients. The binary coefficient D12 has a completely different meaning as the ternary coefficient D12 . Contrary to the ternary Fick diffusivities, ternary Maxwell–Stefan diffusivities can be related to the relevant binary diffusivities. Regarding the binary edges of the ternary diffusion coefficients (Fig. 5), the shapes of the diffusion coefficient surfaces resemble their respective binary coefficients. Noteworthy, only ternary diffusion coefficients were used to fit the surface. Prediction of ternary Maxwell–Stefan diffusion coefficients usually starts with values of the binary diffusion coefficients at the binary edge. Therefore, behavior of the diffusion coefficient with increasing molar fraction of the respective third component is interesting. Here, values of all three Maxwell–Stefan diffusivities decrease while approaching the opposite corner of the triangle.
Prediction methods for ternary Maxwell–Stefan diffusivities start with a ternary extension of the binary Vignes-equation [8]: (8)
The first two diffusion coefficients at infinite dilution can be derived from binary experiments, values for the third diffusion coefficient at infinite dilution are unknown. Therefore, several authors tried to find an expression for this unknown coefficient. Wesselingh and Krishna developed this equation [8]: Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðxijj →1 · Ðxiji →1 )xk /2
xk →1 xk /2 ) Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðxikk →1 · Ðjk
(9)
Ðij at infinite dilution in component k is replaced by the geometric average of the other two diffusion coefficients at infinite dilution. Using this model, the ternary diffusion coefficient Ðij is completely independent of the nature of the third component k.
(10)
In this model, the ternary diffusion coefficient Ðij is also a function of the binary diffusivities Ðik and Ðjk . Rutten extended the model of Wesslingh and Krishna with a viscosity correction [10]. By this way, component k has an influence on the ternary diffusivity Ðij : √ ηi · ηj xk /2 Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · Ðxijj →1 · Ðxiji →1 · ηk (11) Additionally, a fourth model is proposed here: Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · xk →1 xk →1 xk /4 (Ðxijj →1 · Ðxiji →1 · Ðik · Ðjk )
5. Models
Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðijxk →1 )xk
Kooijman and Taylor use the geometric average of other diffusion coefficients at infinite dilution in their prediction model [9]:
(12)
Using the geometric average of all four diffusion coefficients at infinite dilution of the models developed by Wesselingh and Krishna and Kooijman and Taylor, this new model constitutes a combination of those approaches. The diffusion coefficient Ðij depends on component k, but comparing to the model of Kooijman and Taylor, the influence of the foreign binary diffusivities Ðik and Ðjk is reduced. 6. Discussion Of all introduced approaches, the model of Rutten shows the largest deviations from experimental data (Fig. 6). Comparing the results of the ternary system acetone–1-butanol–1-propanol, the viscosity correction has obviously too much weight. Although some points can be predicted quite accurate, the model of Kooijman and Taylor calculates several values with deviations of more than 50%. The model of Wesselingh and Krishna works
S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
103
relationship between binary and ternary diffusivities Ðij . All four introduced models show the largest deviations at high amounts of the third component k, with great distance to the binary edge. This indicates that presently no model is able to describe the diffusivity of i and j at infinite dilution in component k properly. 7. Conclusions
Fig. 6. Comparison of several prediction methods.
quite well for the ternary system described here. Questionable is, of course, the independency of the diffusion coefficient Ðij on component k. This model predicts the same results for the ternary diffusion coefficient Ð of acetone and 1-propanol in this system as well as e.g. acetone–water–1-propanol. Further investigations on this topic have to be done. The best results are delivered by the new model introduced here. The average deviation is 23.9%, which seems quite good for prediction of ternary diffusivities in the whole concentration range of a system. In Fig. 7, all predicted values are displayed once more. In this figure, the color of the symbols indicates the distance to the binary edge of the predicted value. Points drawn as black symbols have low amounts of the third component k and are therefore close to the binary edge. Here, prediction works well and shows only small deviations, independent of the prediction model. Since ternary prediction starts with values of the binary diffusion coefficients at the binary edge, this demonstrates the
Binary and ternary Fick diffusion coefficients were measured by holographic interferometry in the system acetone–1-butanol–1-propanol. Maxwell–Stefan diffusion coefficients were derived from the experimental Fick diffusivities using the thermodynamic correction factor. Binary VLE-data was used with the Wilson excess-enthalpy model for calculation of the thermodynamic correction factor. Using an accurate description of the thermodynamic behavior of the system, the Maxwell–Stefan diffusivities show a lower concentration dependency than the Fick diffusivities. Several ternary prediction methods were tested against the experimental data. A new model, which combines the approaches of Wesslingh and Krishna and Kooijman and Taylor yields the best results for the ternary system considered. All models show good results for data points near the binary edge of the concentration triangle, but have higher uncertainties when calculating diffusion coefficients for concentrations with higher molar fractions of the third component k. For general evaluation of ternary prediction, the models have to be tested against other ternary systems. If diffusion coefficients of further systems with accurate description of their thermodynamic behavior can be investigated, dependencies of the diffusion coefficient of i and j at infinite dilution in k could be potentially identified. List of symbols c molar concentration d driving force D Fick diffusion coefficient Ð Maxwell–Stefan diffusion coefficient j molar flux n number of components R universal gas constant T absolute temperature v velocity x molar fraction Greek letters δij Kronecker delta γ activity coefficient Γ thermodynamic factor η viscosity μ chemical potential References
Fig. 7. Dependence of predicted values from distance to binary edge.
[1] R. Taylor, R. Krishna, Multicomponent Mass Transfer, Wiley & Sons Inc., New York, 1993. [2] J. Stefan, Sitzungsber. Akad. Wiss. Wien 63 (1871) 63–124, Abt. II.
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[3] A. Wild, Multicomponent diffusion in liquids, Ph.D. Thesis, TU Munich, VDI-Verlag, D¨usseldorf, 2003. [4] M. Pertler, Die Mehrkomponenten-Diffusion in nicht vollst¨andig mischbaren Fl¨ussigkeiten, Ph.D. Thesis, TU Munich, 1996. [5] D.G. Miller, J. Phys. Chem. 90 (1988) 1706–1717. [6] L.S. Darken, Am. Inst. Min. Metall. Pet. Eng. 175 (1948) 184–201.
[7] A. Vignes, IEC Fund. 5 (2) (1966) 189–199. [8] J.A. Wesselingh, R. Krishna, Mass Transfer, Ellis Horwood, Chichester, England, 1990. [9] H.A. Kooijman, R. Taylor, Ind. Eng. Chem. Res. 30 (6) (1991) 1217– 1222. [10] Ph.W.M. Rutten, Diffusion in Liquids, Delft University Press, 1992.
Measurement and calculation of multicomponent diffusion coefficients in liquids Sebastian Rehfeldt ∗ , Johann Stichlmair Lehrstuhl fuer Fluidverfahrenstechnik, Technische Universitaet Muenchen, Boltzmannstrasse 15, 85747 Garching, Germany Received 30 June 2006; received in revised form 9 October 2006; accepted 9 October 2006 Available online 20 October 2006
Abstract In the literature, several prediction models for binary diffusion coefficients are found. For multicomponent systems however, only a few models exist. Due to lack of ternary diffusion data, these models have not been verified for real systems until now. To overcome this limitation, multicomponent diffusivities were measured within the whole concentration space of the ternary system acetone–1-butanol–1-propanol. Fick diffusivities were transformed to less concentration-dependent Maxwell–Stefan diffusivities, using the Wilson model for calculating the thermodynamic correction factor. Three suggested prediction models and one newly developed were tested by a comparison of predicted values to experimental data. The new prediction method shows the best results. All tested prediction models show larger deviations with increasing distance to the binary edge of the concentration triangle. © 2006 Elsevier B.V. All rights reserved. Keywords: Diffusion; Maxwell–Stefan diffusivities; Multicomponent systems
1. Introduction ji = − Diffusion is a physical phenomenon occurring in every natural or industrial process involving mass transfer. It can play an important role as the rate determining step. Therefore, the determination of diffusivities in liquids is of great interest for the calculation of mass transfer processes. Most studies about diffusion concentrate on measuring and predicting diffusivities in binary mixtures. Research on this topic advanced, also due to a fair database of experimentally determined binary diffusivities. However, real processes mostly deal with multicomponent mixtures. In this case, both the experimental and theoretical investigations of diffusion are much more complex. Therefore, few multicomponent diffusion data are found in the literature. 2. Theory There are two well-established theories describing diffusion. The more commonly used is Fick’s Law [1]: ∗
Corresponding author. Tel.: +49 89 289 16517; fax: +49 89 289 16510. E-mail address: [email protected] (S. Rehfeldt).
0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.10.008
n−1
Dij · ∇cj ,
i = 1, . . . , n − 1
(1)
j=1
The flow j is defined as the product of diffusion coefficients Dij and a driving force, i.e. the concentration gradient cj . In case of a binary mixture, only one diffusion coefficient is needed to describe all diffusional fluxes. However, in systems with more than two components, more coefficients are necessary and the structure of the flux equations is getting more complex. For example, four diffusivities Dij are required to describe mass flow rates in a ternary system: j1 D11 D12 ∇c1 =− · (2) j2 D21 D22 ∇c2 D11 and D22 are called the main coefficients. They show the dependency of the flow rate of a component on its own concentration gradient. The influence on the flow rate of the concentration gradient of the other component is taken into account by the cross coefficients D12 and D21 . However, only n − 1 flow rates are independent. In the ternary case, the third component is called the “solvent”, its flow rate can be determined with the flux balance condition. The four ternary diffusivities implicitly contain information on the flux of the solvent, so their values depend
100
S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
on its choice. It is obvious that ternary Fick diffusivities are not related to the binary diffusion coefficients. This is a great disadvantage for prediction. However, Maxwell and Stefan developed a physically more consistent diffusion theory [2]. Based on the kinetic theory of ideal gases, the assumption of the Maxwell–Stefan-equations is an equilibrium of molecular friction and thermodynamic interaction. Any deviation from equilibrium leads to a diffusion flux. For liquid, non-electrolytic mixtures, the chemical potential gradient μi is used as the driving force d, which is counterbalanced by friction. The friction between two components is proportional to their velocity difference vi − vj and their amount of molar fractions x: di =
xi xj xi · ∇(T,p) μi = Γij · ∇xj = − (vi − vj ) RT Ðij n−1
n
j=1
j=1
(3)
∂ ln γi Γij = δij + xi ∂xj T,p for i = j and δij = 0
for i = j
(4)
In binary systems, there is again a single diffusion coefficient Ðij . In contrast to Fick’s law, for description of diffusion in a ternary mixture only three Maxwell–Stefan diffusivities Ðij are needed. Friction of each couple of components is taken into account by one coefficient. Thus, the choice of a solvent is dispensable. Therefore, multicomponent Maxwell–Stefan diffusivities can be related to the binary case. This should be advantageous for prediction. In the literature, most publications only contain Fick diffusivities, since they can be measured experimentally. Binary Maxwell–Stefan diffusivities can be calculated from experimental Fick diffusion coefficients and the thermodynamic correction factor: Ð=
D Γ
Fick diffusivities were experimentally measured by holographic laser-interferometry [4]. In this method, a measuring cell is filled with two mixtures with slightly different concentrations. Unsteady diffusion causes a changing concentration profile inside the diffusion cell, which causes an interference pattern. The cell is photographed in intervals with a digital camera. Analyzing the change of the interference pattern, Fick diffusion coefficients can be calculated [5]. For binary datapoints, only one experiment is necessary. Determination of ternary Fick diffusivities is much more complex. Several experiments with pairs of mixtures with different concentrations are necessary. 4. Results
Ðij denotes the Maxwell–Stefan diffusivities which can be interpreted as inverse drag coefficients of components i and j. The thermodynamic correction factor Γ ij can be calculated e.g. from the activity coefficients γ i :
with δij = 1
3. Experimental setup
Fick diffusion coefficients were measured over the whole concentration range of the ternary system acetone–1-butanol–1propanol and its three binary systems. Maxwell–Stefandiffusivities were also computed from the experimental Fick diffusivities, employing the Wilson excess Gibbs energy model for calculation of the thermodynamic correction factor [1]. Binary Fick diffusivities of acetone–1-propanol (Fig. 1) and acetone–1-butanol (Fig. 2) show a similar dependency on molar fraction of acetone. Values in the binary system acetone–1propanol are slightly higher. The Maxwell–Stefan diffusion coefficients in both systems are less dependent on concentration. In acetone–1-propanol, a linear relation of the Maxwell–Stefan diffusivity and concentration can be observed. Such functions of the binary diffusion coefficient were proposed by Darken [6]. Here, the binary diffusion coefficient is calculated as a sum of the binary diffusivities at infinite dilution multiplied with the molar fraction: x1 →1 x1 →0 Ð12 = D12 · x1 + D12 · x2
In the binary system of acetone–1-butanol, the graph of the Maxwell–Stefan diffusion coefficient is slightly curved. This
(5)
Transformation in binary systems is quite simple. Regarding ternary systems, the situation is more complex. Indeed, there is an algebraic relationship between Fick and Maxwell–Stefan diffusion coefficients [1]. Since there are four Fick coefficients, but only three Maxwell–Stefan coefficients, the transformation is only straightforward from Maxwell–Stefan to Fick. Transforming Fick diffusivities to Maxwell–Stefan, the system of equations is over-determined. Fick diffusion coefficients and thermodynamic factors both have experimental errors and are, in turn, usually inconsistent [3]. Therefore, a least-square fitting is required to compute Maxwell–Stefan diffusivities from Fick diffusivities.
(6)
Fig. 1. Binary diffusivities acetone–1-propanol.
S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
Fig. 3. Binary diffusivities 1-propanol–1-butanol.
Fig. 2. Binary diffusivities acetone–1-butanol.
result resembles the prediction model of Vignes [7]: Ð12 =
x1 →1 x1 (D12 )
x1 →0 x2 · (D12 )
101
(7)
1-propanol–1-butanol is a nearly thermodynamic ideal mixture. The Fick diffusion coefficient has nearly constant values (Fig. 3). Due to values of the thermodynamic correction factor near unity, there is only a small difference between Fick and Maxwell–Stefan diffusion coefficient. The experimental results shown prove that in these binary mixtures, Maxwell–Stefan diffusivities can be predicted quite accurate, if diffusivities at infinite dilution are known and an exact thermodynamic description of the system is possible. However, errors in the thermodynamic phase equilibrium data have a big influence on the quality of the Maxwell–Stefan diffusion coefficients. Since the thermodynamic factor involves the
first derivative of the activity coefficient with respect to composition, MS coefficients are quite sensitive to the parameters used to calculate the thermodynamic factor. These parameters are fitted against vapor–liquid equilibrium data. Even if the fitted parameters lead to good results for phase equilibrium, the first derivatives of the activity coefficients can show large deviations [1,3]. The ternary Fick diffusivities are displayed in Fig. 4. Each measured data point is plotted, a line indicates the position in the base triangle. A surface is fitted to the data points to illustrate the shape of the ternary diffusivities. They show stronger concentration dependence than the binary diffusivities. Especially the cross diffusion coefficients vary significantly with concentration. While values of the cross coefficients D12 are small, all values of D21 are negative.
Fig. 4. Ternary Fick diffusion coefficients of acetone–1-butanol–1-propanol.
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Fig. 5. Ternary Maxwell–Stefan diffusivities of acetone–1-butanol–1-propanol.
Obviously, there is no relation between binary and ternary Fick diffusion coefficients. The binary coefficient D12 has a completely different meaning as the ternary coefficient D12 . Contrary to the ternary Fick diffusivities, ternary Maxwell–Stefan diffusivities can be related to the relevant binary diffusivities. Regarding the binary edges of the ternary diffusion coefficients (Fig. 5), the shapes of the diffusion coefficient surfaces resemble their respective binary coefficients. Noteworthy, only ternary diffusion coefficients were used to fit the surface. Prediction of ternary Maxwell–Stefan diffusion coefficients usually starts with values of the binary diffusion coefficients at the binary edge. Therefore, behavior of the diffusion coefficient with increasing molar fraction of the respective third component is interesting. Here, values of all three Maxwell–Stefan diffusivities decrease while approaching the opposite corner of the triangle.
Prediction methods for ternary Maxwell–Stefan diffusivities start with a ternary extension of the binary Vignes-equation [8]: (8)
The first two diffusion coefficients at infinite dilution can be derived from binary experiments, values for the third diffusion coefficient at infinite dilution are unknown. Therefore, several authors tried to find an expression for this unknown coefficient. Wesselingh and Krishna developed this equation [8]: Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðxijj →1 · Ðxiji →1 )xk /2
xk →1 xk /2 ) Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðxikk →1 · Ðjk
(9)
Ðij at infinite dilution in component k is replaced by the geometric average of the other two diffusion coefficients at infinite dilution. Using this model, the ternary diffusion coefficient Ðij is completely independent of the nature of the third component k.
(10)
In this model, the ternary diffusion coefficient Ðij is also a function of the binary diffusivities Ðik and Ðjk . Rutten extended the model of Wesslingh and Krishna with a viscosity correction [10]. By this way, component k has an influence on the ternary diffusivity Ðij : √ ηi · ηj xk /2 Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · Ðxijj →1 · Ðxiji →1 · ηk (11) Additionally, a fourth model is proposed here: Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · xk →1 xk →1 xk /4 (Ðxijj →1 · Ðxiji →1 · Ðik · Ðjk )
5. Models
Ðij = (Ðxijj →1 )xj · (Ðxiji →1 )xi · (Ðijxk →1 )xk
Kooijman and Taylor use the geometric average of other diffusion coefficients at infinite dilution in their prediction model [9]:
(12)
Using the geometric average of all four diffusion coefficients at infinite dilution of the models developed by Wesselingh and Krishna and Kooijman and Taylor, this new model constitutes a combination of those approaches. The diffusion coefficient Ðij depends on component k, but comparing to the model of Kooijman and Taylor, the influence of the foreign binary diffusivities Ðik and Ðjk is reduced. 6. Discussion Of all introduced approaches, the model of Rutten shows the largest deviations from experimental data (Fig. 6). Comparing the results of the ternary system acetone–1-butanol–1-propanol, the viscosity correction has obviously too much weight. Although some points can be predicted quite accurate, the model of Kooijman and Taylor calculates several values with deviations of more than 50%. The model of Wesselingh and Krishna works
S. Rehfeldt, J. Stichlmair / Fluid Phase Equilibria 256 (2007) 99–104
103
relationship between binary and ternary diffusivities Ðij . All four introduced models show the largest deviations at high amounts of the third component k, with great distance to the binary edge. This indicates that presently no model is able to describe the diffusivity of i and j at infinite dilution in component k properly. 7. Conclusions
Fig. 6. Comparison of several prediction methods.
quite well for the ternary system described here. Questionable is, of course, the independency of the diffusion coefficient Ðij on component k. This model predicts the same results for the ternary diffusion coefficient Ð of acetone and 1-propanol in this system as well as e.g. acetone–water–1-propanol. Further investigations on this topic have to be done. The best results are delivered by the new model introduced here. The average deviation is 23.9%, which seems quite good for prediction of ternary diffusivities in the whole concentration range of a system. In Fig. 7, all predicted values are displayed once more. In this figure, the color of the symbols indicates the distance to the binary edge of the predicted value. Points drawn as black symbols have low amounts of the third component k and are therefore close to the binary edge. Here, prediction works well and shows only small deviations, independent of the prediction model. Since ternary prediction starts with values of the binary diffusion coefficients at the binary edge, this demonstrates the
Binary and ternary Fick diffusion coefficients were measured by holographic interferometry in the system acetone–1-butanol–1-propanol. Maxwell–Stefan diffusion coefficients were derived from the experimental Fick diffusivities using the thermodynamic correction factor. Binary VLE-data was used with the Wilson excess-enthalpy model for calculation of the thermodynamic correction factor. Using an accurate description of the thermodynamic behavior of the system, the Maxwell–Stefan diffusivities show a lower concentration dependency than the Fick diffusivities. Several ternary prediction methods were tested against the experimental data. A new model, which combines the approaches of Wesslingh and Krishna and Kooijman and Taylor yields the best results for the ternary system considered. All models show good results for data points near the binary edge of the concentration triangle, but have higher uncertainties when calculating diffusion coefficients for concentrations with higher molar fractions of the third component k. For general evaluation of ternary prediction, the models have to be tested against other ternary systems. If diffusion coefficients of further systems with accurate description of their thermodynamic behavior can be investigated, dependencies of the diffusion coefficient of i and j at infinite dilution in k could be potentially identified. List of symbols c molar concentration d driving force D Fick diffusion coefficient Ð Maxwell–Stefan diffusion coefficient j molar flux n number of components R universal gas constant T absolute temperature v velocity x molar fraction Greek letters δij Kronecker delta γ activity coefficient Γ thermodynamic factor η viscosity μ chemical potential References
Fig. 7. Dependence of predicted values from distance to binary edge.
[1] R. Taylor, R. Krishna, Multicomponent Mass Transfer, Wiley & Sons Inc., New York, 1993. [2] J. Stefan, Sitzungsber. Akad. Wiss. Wien 63 (1871) 63–124, Abt. II.
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