Predictive Darken Equation for Maxwell-Stefan Diffusivities in

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Predictive Darken Equation for Maxwell-Stefan Diffusivities in Multicomponent Mixtures Xin Liu,†,‡ Thijs J.H. Vlugt,‡ and Andre Bardow*,†,‡ † ‡

Lehrstuhl f€ur Technische Thermodynamik, RWTH Aachen University, Schinkelstrasse 8, 52062 Aachen, Germany Process & Energy Laboratory, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands

bS Supporting Information ABSTRACT: This Article presents the derivation and validation of a rigorous model for the prediction of multicomponent Maxwell-Stefan (MS) diffusion coefficients. The MS theory provides a sound framework for modeling mass transport in gases and liquids. Unfortunately, MS diffusivities are concentration dependent, and this needs to be taken into account in practical applications. There is therefore a considerable interest in models describing the concentration dependence of MS diffusivities. While current practice employs empirical models for this purpose, recent work on molecular simulations favor the physically based Darken equation. The Darken equation, however, is limited to binary mixtures and is not predictive. In this study, a multicomponent Darken model for MS diffusivities is derived from linear response theory and the Onsager relations. In addition, a predictive model for the required self-diffusivities in the mixture is proposed, leading to the predictive Darken-LBV model. We compare our novel model to the existing generalized Vignes equation and the generalized Darken equation using molecular dynamics (MD) simulation. Two systems are considered: (1) ternary and quaternary systems in which particles are interacting using the Weeks-Chandler-Andersen (WCA) potential; (2) the ternary system n-hexane-cyclohexane-toluene. Our results show that, in all studied systems, the novel predictive Darken-LBV equation describes the concentration dependence better than the existing models. The physically based Darken-LBV model provides a sound and robust framework for prediction of MS diffusion coefficients in multicomponent mixtures.

1. INTRODUCTION For the description of multicomponent mass transport in gases and liquids, the Maxwell-Stefan (MS) theory is commonly used in chemical engineering as it provides a convenient description of diffusion in n-component systems.1 The key point of this approach is that the driving force for diffusion of component i (i.e., the gradient in chemical potential rμi) is balanced by a frictional force, resulting in the following transport equation 1 ∇μi ¼ RT

xj ðui uj Þ îj j ¼ 1, j6¼ i n

∑

x f 1 xj

îj ¼ ðîjxi f 1 Þxi ðîjj

ð1Þ

The frictional force between components i and j is proportional to the velocity difference (ui uj). R and T represent the gas constant and absolute temperature respectively. xj is the mole fraction of component j. The MS diffusivity îj acts as an inverse friction coefficient describing the magnitude of the friction between components i and j. It is usually difficult to obtain the MS diffusivities îj from experiments,24 and obtaining the MS diffusivities from molecular dynamics (MD) simulations requires extensive amounts of CPU time.5 Predictive models for MS diffusivities based on easily measurable quantities are therefore highly desired. Prediction of MS diffusivities usually follows a two-step approach: (1) prediction of MS diffusivities ^xijif0 in binary systems at infinite dilution; and (2) estimation of MS diffusion coefficients îj in concentrated mixtures using mixing rules. For the first step, a large number of models are available with the Wilke-Chang equation6 still being the most popular r 2011 American Chemical Society

(for a critical review we refer the reader to ref 7). In this work, we focus on the second step. For the estimation of MS diffusion coefficients of concentrated mixtures, empirical models are typically employed interpolating the diffusion coefficients at infinite dilution ^xijif0. The logarithmic interpolation suggested by Vignes8 is typically recommended.1,7 The Vignes equation has been generalized to multicomponent mixtures as follows9 Þ

n Y

ðîjxk f 1 Þxk

ð2Þ

k ¼ 1, k6¼ i, j

The generalized Vignes equation requires ^xijkf1 which describes the friction between components i and j when both are diluted in component k. This quantity is not directly accessible in experiments.4 Several predictive models have been proposed to estimate ^xijkf1 from binary diffusion data.913 However, all of these models are empirical and lacking a sound theoretical basis. We recently derived a physically based model for estimating this quantity using the self-diffusivities obtained from pure and binary mixtures at infinite dilution14 îjxk f 1

¼

Dxi,kselff 1 Dj,xkselff 1 f1 Dk,xkself

¼

îkxk f 1 ^jkxk f 1 f1 Dxk,k self

ð3Þ

Received: May 10, 2011 Accepted: August 2, 2011 Revised: July 28, 2011 Published: August 02, 2011 10350

dx.doi.org/10.1021/ie201008a | Ind. Eng. Chem. Res. 2011, 50, 10350–10358

Industrial & Engineering Chemistry Research

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In the remainder of this paper, we will refer to this as the LBV equation (taken from the author’s names). At infinite dilution, kf1 is equal to the MS diffusivity the self-diffusion coefficient Dxi,self ^xikkf1.7 The central assumption in eq 3 is that velocity correlations between different particles are neglected. Equation 3 leads to superior and more robust predictions compared to the previous empirical models.14 By inserting eq 3 into eq 2, we obtain the socalled Vignes-LBV equation !xk n Y Di,xkselff 1 Dj,xkselff 1 xj f 1 xj xi f 1 xi îj ¼ ðDj, self Þ ðDi, self Þ f1 Dxk,kself k ¼ 1, k6¼ i, j ð4Þ This physically based multicomponent form of the Vignes equakf1 tion requires the pure component self-diffusivities ^xk,self as additional information. For this property, there are also predictive models available.11 The Vignes-LBV equation thus allows the prediction of multicomponent MS diffusion coefficients based on pure component data and binary mixture data at infinite dilution. The popularity of the generalized Vignes equation (eq 2) has, however, recently been challenged by a number of groups working on diffusion using molecular simulation.11,1520 Their simulations results show that the Darken equation21 is superior to the Vignes equation. In the Darken equation, the MS diffusion coefficient îj is estimated from the self-diffusion coefficients of the two components Di,self and Dj,self in the binary mixture21 îj ¼ xi Dj, self þ xj Di, self

ð5Þ

While often regarded as an empirical mixing rule, the Darken equation can be rigorously derived as a limiting case from statistical-mechanical theory,22 see also section S1.1 of the Supporting Information for an alternative derivation. Despite this theoretical advantage and the support from simulations, the Darken equation is generally seen as “of little practical use due to the fact that it relies on the self-diffusion coefficients in the mixture, which are rarely available.”18 In addition, the Darken equation had been only formulated for binary mixtures. To overcome these limitations, Krishna and van Baten11 recently proposed a generalized Darken equation for describing the concentration dependence of MS diffusivities in multicomponent mixtures based on molecular simulations of liquid mixtures of linear alkanes, xj xi Dj, self þ Di, self ð6Þ îj ¼ xi þ xj xi þ xj The authors suggested to estimate the self-diffusivities Di,self and Dj,self in eq 6 as mass-weighted averages of the values at infinite dilution Di, self ¼

n

∑ wjDi,wselff 1 j¼1 j

ð7Þ

model for the self-diffusivity (eq 7) are empirical in nature. In particular, it is important to note that in systems with 3 or more components, the limit where both xi f 0 and xj f 0 is not welldefined by eq 6.4 Furthermore, eq 6 suggests that the MS diffusivities of an infinitely diluted component in a mixture should all become equal, i.e., ^xijif0 = ^xiki f0 for all j,k 6¼ i. This assumption seems unphysical and is in fact not supported by the authors’ own data for linear alkanes.11 In this work, we derive a sound extension of the Darken equation to multicomponent mixtures. In addition, a physically motivated model for estimation of the required self-diffusivities is provided. The resulting predictive Darken-LBV equation for multicomponent mixtures is validated using molecular simulations in ternary and quaternary mixtures. The remainder of this paper is organized as follows. In Section 2, the predictive Darken-LBV equation is derived on the basis of linear response theory and the Onsager relations. In Section 3, we introduce the details concerning the MD simulations. In Section 4, we test our model for ternary and quaternary systems in which purely repulsive pair interactions are used, i.e., particles interact using a Weeks-ChandlerAndersen (WCA) potential.23 We also test our model for the ternary system n-hexane-cyclohexane-toluene. Our conclusions are summarized in Section 5. It is shown that the predictive Darken-LBV model is clearly superior to the generalized Darken equation and it even outperforms the Vignes-LBV equation for the prediction of multicomponent MS diffusion coefficients.

2. THE PREDICTIVE DARKEN-LBV EQUATION MS diffusivities îj can be obtained from linear response theory, i.e., from local particle fluctuations as obtained in equilibrium MD simulations.24,25 First, the Onsager coefficients Λij are calculated directly from the motion of the molecules,11 * + Z Nj Ni 1 ∞ Λij ¼ dt ∑ vl, i ð0Þ 3 ∑ vp, j ðtÞ ð8Þ 3N 0 p¼1 l¼1 In this equation, N is the total number of molecules in the simulation, Ni is the number of molecules of component i, and vl,i(t) is the velocity of lth molecule of component i at time t. Note that the matrix [Λ] is symmetric, that is Λij = Λji and that the Onsager coefficients are constraint by ΣiMiΛij = 0 in which Mi is the molar mass of component i.11 One can express îj in terms of the mole fractions x1, x2,...xn (n refers to the number of components) and the Onsager coefficients Λij. For a binary system, the resulting equation is rather simple, x2 x1 ^12 ¼ Λ11 þ Λ22 Λ12 Λ21 ð9Þ x1 x2 The explicit expressions for îj for ternary and quaternary systems are more complex and can be found in the Supporting Information. For the terms Λii, we can write14 1 Λii ¼ 3N

Here, wj is mass fraction of component j. The generalized Darken equation (eqs 6 and 7) thus allows one to predict MS diffusivities using the same information as for the Vignes-LBV equation: the diffusion coefficient in the binary mixtures at infinite dilution and the pure component self-diffusivities. Note that in the limit of if1 . infinite dilution, both are equal, i.e., ^xijif1 = Dxj,self While retaining the Darken equation (eq 5) as the binary limit, both the generalization of the Darken equation (eq 6) and the

¼

1 3N

1 þ 3N

Z

*

∞

dt 0

Z

*

∞

dt 0

Z

*

∞

dt 0

Ni

∑ vl, i ð0Þ 3 g∑¼ 1 vg, i ðtÞ

l¼1 Ni

∑

l¼1 Ni

+

vl, i ð0Þ 3 vl, i ðtÞ Ni

∑ ∑

l ¼ 1 g ¼ 1, g6¼ l

≈ xi Cii þ x2i NCf ii 10351

+

Ni

+ vl, i ð0Þ 3 vg, i ðtÞ

ð10Þ

dx.doi.org/10.1021/ie201008a |Ind. Eng. Chem. Res. 2011, 50, 10350–10358


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Figure 1. Computed and predicted self-diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent the computed self-diffusivities using MD. Solid lines represent the predictions using eq 16. Dashed lines represent the predictions using eq 7. (a) x1 = 0.2; (b) x1 = 0.4; (c) x1 = 0.6; (d) x1 = 0.8. Simulation details: F = 0.2; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400. f in which Cii and Cii account for self and cross correlations of the velocities of molecules of component i, respectively. We assumed here that Ni2 Ni ≈ Ni2. In general, the self-diffusivity of component i in a medium can be computed from a Green-Kubo relation24 Z 1 ∞ Di, self ¼ dtÆvi ð0Þ 3 vi ðtÞæ ð11Þ 3 0

in which Di,self is the self-diffusivity of component i, and vi(0) and vi(t) are the velocities of a molecule of component i at time 0 and time t, respectively. From eq 10 and eq 11, it follows that Cii ¼ Di, self

cross-correlation functions, Cij,i6¼j and Cf ii , are small compared to the terms involving autocorrelation functions (Cii).14 For idealdiffusing mixtures, we assume that this is also allowed at finite concentrations. As shown in the Supporting Information, this approach leads to a multicomponent Darken-like equation for an n-component system: îj ¼

∑

∑

It has been shown that, in systems in which particles are not or weakly associated (i.e., the system in which particles are interacting using a WCA potential or the n-hexane-cyclohexanetoluene system), the terms involving integrals of these velocity

ð14Þ

with Dmix defined as 1 ¼ Dmix

ð12Þ

For Λij with i 6¼ j, i.e., the correlations between unlike molecules, we can write14 * + Z Nj Ni 1 ∞ Λij ¼ dt vl, i ð0Þ 3 vk, j ðtÞ 3N 0 l¼1 k¼1 Z Ni Nj ∞ ≈ dtÆv1, i ð0Þ 3 v1, j ðtÞæ 3N 0 ð13Þ ¼ Nxi xj Cij

Di, self Dj, self Dmix n

∑ i i ¼ 1 Di, self x

ð15Þ

We will refer to eqs 14 and 15 as the multicomponent Darken equation. For binary systems (n = 2), the multicomponent Darken equation reduces to the well-known Darken equation (eq 5).11,20,26,27 To demonstrate this correspondence, the derivation of the binary Darken equation is presented using the same approach in the Supporting Information. It is important to note that, for a ternary system at infinite dilution, eqs 14 and 15 directly result in eq 3. As noted above, the applicability of the Darken equation has been suffering from the fact that it requires the self-diffusivities of all components in the mixtures. When eqs 3, 14, and 15 are compared, it suggests that the quantity Dmix describes the effective self-diffusivity of the medium. Its mixing rule (eq 15) motivates the following equation to estimate the self-diffusivity 10352



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Figure 2. Computed and predicted self-diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent the computed self-diffusivities using MD. Solid lines represent the predictions using eq 16. Dashed lines represent the predictions using eq 7. (a) x1 = 0.2; (b) x1 = 0.4; (c) x1 = 0.6; (d) x1 = 0.8. Simulation details: F = 0.5; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400.

of component i in a multicomponent system 1 ¼ Di, self

n

x

j ∑ x f1 j¼1 D j

ð16Þ

i, self

We thus model the self-diffusivity of species i in the mixture as the inverse mole-fraction weighted sum of its pure component value and its values at infinite dilution. We refer to the combined models of the multicomponent Darken equation (eqs 14 and 15) and the predictive model for the self-diffusivities (eq 16) as the predictive Darken-LBV equation. MD simulations are used to generate benchmark values. Comparisons are made between our model and existing models.

3. MOLECULAR DYNAMICS SIMULATIONS The predictive models for îj and Di,self described in Sections 1 and 2 are tested for two systems: (1) systems in which only repulsive pair interactions are considered, i.e., particles interact using a WCA potential;23 (2) the ternary system n-hexane-cyclohexane-toluene in which united atoms interact using LennardJones (LJ) potentials. Errors in computed diffusivities are calculated by performing at least 10 independent MD simulations for each system and analyzing their differences. Using 10 simulations with different initial positions of the molecules allows us to provide a better estimate of the error in the average values of the diffusivities. 3.1. WCA System. Both ternary and quaternary mixtures are considered. All components only differ in their molar mass. Diffusivities were obtained from MD simulations in the

microcanonical (NVE) ensemble. Self and MS diffusivities are computed using eqs 8 and 11 and the equations provided in the Supporting Information. An order-n algorithm was used to compute correlation functions.24,28 All simulations were carried out in a system containing N = 400 particles. The WCA potential is constructed by truncating and shifting the LJ potential at 21/6σ. A linked-cell algorithm is applied to improve the efficiency.24 For convenience, we express all quantities in reduced units by setting the LJ parameters σ and ε as units for length and energy.24 The mass of the lightest component is set as unit of mass, i.e., M1 = 1. The equation of motion was integrated using a time step of Δt = 0.001 (in reduced units), and typically, 109 steps were needed. 3.2. n-Hexane-Cyclohexane-Toluene. The united-atom model is used in which CHx groups are considered as single interaction sites. For n-hexane and cyclohexane, the SHAKE algorithm is used to constrain the bond length.29 Bond bending and torsion potentials describe the interaction between three and four consecutive atoms, respectively. Toluene is modeled as a rigid molecule. The LJ potentials describe the nonbonded interactions which are truncated and shifted at 12 Å. The LorentzBerthelot mixing rules are applied to obtain the LJ parameters for the interaction of unlike atoms.30 The equations of motion were integrated using the velocity Verlet algorithm with a time step of 2 fs. The simulations were performed at a constant total energy and density corresponding to a pressure of 1 atm and a temperature of 298 K. All force field parameters for this system are taken from our previous work; see refs 5 and 14. Typical simulations to extract MS diffusivities took at least 50 ns. 10353



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Figure 3. Computed and predicted self-diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent the computed self-diffusivities using MD. Solid lines represent the predictions using eq 16. Dashed lines represent the predictions using eq 7. (a) x1 varies, x2/x3 = 1; (b) x2 varies, x1/x3 = 1; (c) x3 varies, x1/x2 = 1. Simulation details: F = 0.2; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400.

4. RESULT AND DISCUSSION 4.1. Ternary WCA Systems. Figures 1 to 4 show a comparison between computed and predicted self-diffusivities in ternary WCA systems. Two predictive models are considered: (1) the mass-weighted average of the values at infinite dilution (eq 7); and (2) eq 16 as presented in this study. In Figures 1 and 2, the mole fraction of component 1 is kept constant while the ratio x2/x3 varies. At F = 0.2, an excellent agreement between computed self-diffusivities and predictions using eq 16 was observed. Equation 7 consistently underestimates the self-diffusivities computed by MD. At a higher number density (F = 0.5), slightly larger deviations between the computed self-diffusivities and predictions are observed as shown in Figure 2. Here, both models underestimate the self-diffusivities. However, eq 16 leads to much better predictions than eq 7 with an average absolute deviation of 5% instead of 18%. Exactly the same trends were observed when keeping xi/xj constant and varying xk; see Figures 3 and 4. Tables S1 and S2 (Supporting Information) compare predicted MS diffusivities to computed MS diffusivities using MD. The multicomponent Darken equation (eqs 14 and 15) is parametrized using data from MD simulations, while the Darken-LBV (eqs 14 to 16), Vignes-LBV (eq 4), and generalized Darken equations (eqs 6 and 7) are parametrized using purecomponent and binary diffusion data which is listed in Tables S3

to S6 of the Supporting Information. Absolute differences between the predictive models and the MD simulations are calculated and normalized with the corresponding MD results. The averaged absolute difference is used to qualify various models. At lower number densities (F = 0.2), the predictive DarkenLBV equation performs equally well as the Vignes-LBV equation and leads to lower maximum deviations (Table S1, Supporting Information). This suggests that the Darken-LBV model is more robust. The performance of the generalized Darken equation is worse than the other predictive models with average errors larger by a factor of 2. The differences between multicomponent Darken and Darken-LBV are small. At higher number densities (F = 0.5), similar observations are found as seen for lower densities, see Table S2, Supporting Information. The maximum deviation from MD data is always obtained for the generalized Darken equation. In Figures 5 and 6, the comparison is visualized for mixtures in which x1 varies and x2/x3 = 1 (results for mixtures in which x2 is varied while x1/x3 = 1 as well as for mixtures in which x3 is varied while x1/x2 = 1 are shown in Figures S1, S2, S3, and S4 of the Supporting Information). The results shown in Tables S1 and S2 (Supporting Information) and Figures 5 and 6 can be summarized as follows: (1) Differences between the multicomponent Darken equation and the Darken-LBV equation are small, as eq 16 accurately 10354



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Figure 4. Computed and predicted self-diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent the computed self-diffusivities using MD. Solid lines represent the predictions using eq 16. Dashed lines represent the predictions using eq 7. (a) x1 varies, x2/x3 = 1; (b) x2 varies, x1/x3 = 1; (c) x3 varies, x1/x2 = 1. Simulation details: F = 0.5; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400.

predicts self-diffusivities of components in a mixture from pure component data; (2) the Darken-LBV equation performs even slightly better than the Vignes-LBV equation and much better than the generalized Darken equation. The out-performance of the Vignes-LBV model by the Darken-LBV model becomes especially clear from Figures 5 and 6, in particular Figures 5b and 6c which allow for the clearest discrimination. It is important to note that the accuracy of the Vignes-LBV equation (eq 4) relies on an accurate estimate of ^xijkf1. It was previously shown that eq 3 provides the best prediction for ^xijkf1.14 With this model, the Darken-LBV and the Vignes-LBV models are in fact identical in the corners of the ternary diagram, where the mole fraction of one of the components approaches unity. Other predictive models for ^xijkf1 introduce deviations ranging from 10 to 80%.5 Therefore, much larger errors are expected when combining the Vignes equation with other predictive models for ^xijkf1. The reason for the poor performance of the generalized Darken equation can be 2-fold: (1) the quality of the generalized Darken equation itself; and (2) the quality of the predictive model for self-diffusivities. We also used the computed selfdiffusivities from MD at the given compositions to parametrize the generalized Darken equation. In this way, deviations introduced by the predicted self-diffusivities are avoided. Our results show that this results in averaged absolute differences of 1420% (see Figures 5 and 6 and Tables S7 and S8 of the Supporting

Information). These deviations are similar to the deviations when eq 7 is used for predicting self-diffusivities in a mixture, suggesting that the differences are not due to the choice of model for self-diffusivities but due to the generalized Darken equation itself. As stated above, the model structure of eq 6 suffers from inconsistencies at infinite dilution. 4.2. Quaternary WCA Systems. In Tables S9 and S10 of the Supporting Information, it is shown that self-diffusivities in quaternary systems are very well predicted using eq 16, while the predictions using eq 7 show large deviations from MD results. Tables S11 and S12 (Supporting Information) compare the computed and predicted MS diffusivities in quaternary mixtures. The multicomponent Darken equation (eq 14) provides reasonable predictions, and again, the Darken-LBV equation, which requires only binary- and pure-component diffusivities, performs equally well. Also, here, the performance of the Vignes-LBV equation is quite well with the present choice for the terms ^xijkf1. These models retain their prediction quality from the ternary case. The performance of the generalized Darken equation is again significantly worse with average and maximum deviations larger by at least a factor of 2. 4.3. n-Hexane-Cyclohexane-Toluene. In Table S13 (Supporting Information), computed and predicted self-diffusivities in the n-hexane-cyclohexane-toluene system at pressure of 1 atm and a temperature of 298 K are listed. A detailed comparison between 10355



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Figure 5. Computed and predicted MS diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent computed MS diffusivities using MD. Blue lines represent predictions using the Vignes-LBV equation. Green lines represent predictions using the generalized Darken equation. Green symbols represent predictions using the generalized Darken equation with the self-diffusivities obtained from MD simulation. Red lines represent predictions using the Darken-LBV equation. Red symbols represent predictions using the multicomponent Darken equation with the self-diffusivities obtained from MD simulation. (a) ^12; (b) ^13; (c) ^23. Simulation details: F = 0.2; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400, x2/x3 = 1.

Figure 6. Computed and predicted MS diffusivities in ternary systems in which particles interact using a WCA potential. Open symbols represent computed MS diffusivities using MD. Blue lines represent predictions using the Vignes-LBV equation. Green lines represent predictions using the generalized Darken equation. Green symbols represent predictions using the generalized Darken equation with the self-diffusivities obtained from MD simulation. Red lines represent predictions using the Darken-LBV equation. Red symbols represent predictions using the multicomponent Darken equation with the self-diffusivities obtained from MD simulation. (a) ^12; (b) ^13; (c) ^23. Simulation details: F = 0.5; M1 = 1; M2 = 5; M3 = 10; T = 2; N = 400, x2/x3 = 1.

simulations and experiments for this mixture is given in ref 5 showing that the simulations correctly capture the experimental behavior. In this ternary system, the self-diffusivities are again much

better predicted by eq 16 than eq 7. Table S14 (Supporting Information) compares the predicted MS diffusivities to the computed MS diffusivities at the same conditions. The pure and 10356


Industrial & Engineering Chemistry Research binary diffusion coefficients were taken from ref 5. We observe that using the multicomponent Darken equation results in most accurate predictions of the MS diffusivities (average deviations of 7%), requiring self-diffusivities at a given composition. It is therefore very encouraging that the Darken-LBV performs equally well (average deviations of 9%), while the performance of the VignesLBV is similar but slightly worse (average deviations of 11%). The performance of the generalized Darken equation is significantly worse (average deviations of 15%). This is especially reflected by the large maximum deviations.

5. CONCLUSION In this work, we derived a predictive Darken-LBV model for MS diffusivities in multicomponent systems based on the linear response theory and the Onsager relations. MS diffusivities can be expressed as functions of easily obtainable self-diffusivities and the integrals of velocity correlations between different molecules. Neglecting the latter terms results in the multicomponent Darken equation (eqs 14 and 15). By combining the multicomponent Darken equation with a new predictive model for self-diffusivities in a mixture (eq 16), we obtain the so-called Darken-LBV model (eqs 14 to 16). We compared our DarkenLBV equation to the existing models, i.e., the Vignes-LBV model (eq 4) and the generalized Darken model (eqs 6 and 7) for WCA systems and the ternary n-hexane-cyclohexane-toluene system. Our results show that, in studied systems, the Darken-LBV equation provides very good predictions for MS diffusivities and is superior compared to the other models. Since the DarkenLBV model is rigorously derived, it should also provide the preferred framework to extensions for the prediction of diffusion in complex mixtures. ’ ASSOCIATED CONTENT

bS

Supporting Information. Details of how Maxwell-Stefan diffusivities in binary, ternary, and quaternary systems follow from the Onsager coefficients Λij and how the multicomponent Darken equation (eqs 14 and 15) can be derived from this; tabulated values for self and MS diffusivities for the investigated systems. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was performed as part of the Cluster of Excellence “Tailor-Made Fuels from Biomass”, which is funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities. ’ REFERENCES (1) Taylor, R.; Krishna, R. Multicomponent mass transfer; Wiley: New York, NY, 1993. (2) Medvedev, O. O.; Shapiro, A. A. Verifying reciprocal relations for experimental diffusion coefficients in multicomponent mixtures. Fluid Phase Equilib. 2003, 208, 291–301.

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