Predicting Mixture Diffusion in Zeolites: Capturing Strong Correlations

Article pubs.acs.org/JPCC

Predicting Mixture Diffusion in Zeolites: Capturing Strong Correlations with a Simplified Theory and Examining the Role of Adsorption Thermodynamics Sanjeev M. Rao† and Marc-Olivier Coppens*,†,‡ †

Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, New York 12180, United States ‡ Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom ABSTRACT: The Maxwell−Stefan (MS) equations describing diffusion in zeolites are simplified to predict mixture diffusion behavior in the case of strong correlation effects. Intermolecular exchange coefficients are not required. No empirical relations are employed. This simplified model is used to predict single-file diffusion behavior in a binary system. The effect of using only the pure component thermodynamic correction factors in predicting mixture diffusion behavior is also examined with application to two systems of practical interest. In all cases, predictions from the simplified models are favorably compared to results from the more complex MS equations or experiments, indicating that the simplified models can serve as good starting approximations, and can also speed up mixture diffusion calculations in zeolites.

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Đi is the mobility of species i in the zeolite lattice (also referred to as the “corrected” diffusivity), and Đij is an intermolecular exchange coefficient. Molecule-framework interactions are captured by the Đi term, but, if correlation effects are important, they are also captured by the Đij term. Đij is interpreted as an inverse friction coefficient that accounts for the interactions between the molecules of species i and j. The mobility, Đi, can be determined from pure component uptake experiments,10 though it is often calculated from MD simulations.7,11 However, there is no straightforward way of calculating the Đij term. Often, an empirical equation such as the Vignes relation1,4,6 is used, given by

INTRODUCTION Adsorption and diffusion within zeolites are closely linked owing to the tightly confining pore space. Molecules that adsorb strongly on the adsorption sites within the zeolite spend a longer time in the zeolite as compared to a weakly adsorbing species. Therefore, in the same period of time, a weakly adsorbing molecule (in the absence of a strongly adsorbing species) will explore a larger fraction of the zeolite lattice, and, hence, will possess a much higher mobility. In mixtures, the difference in adsorption strengths and mobilities within the zeolite pore space is responsible for slowing down a weakly adsorbing species, without notably affecting the strongly adsorbing species.1 Therefore, the flux of the weakly adsorbing species is strongly correlated to the flux of the strongly adsorbing species. To describe adsorption and diffusion in zeolites, the Maxwell−Stefan (MS) theory, which is a phenomenological theory, has been adapted to zeolites and other microporous materials such as metal organic frameworks (MOFs).1−9 The MS theory describes the fluxes as being comprised of molecule−molecule interactions and molecule− framework interactions. The general form of the MS equations is given by7 −ρz qis

∇μi R gT

n

=

∑ j = 1, j ≠ i

qjsNi − qisNj qjs ,sat Đij

+

Ni Đi

(θ / θi + θj) (θj / θi + θj) Đ jj

Đij = Đii i

In eq 2, Đii represents the interactions between the different molecules of species i and is estimated as a function of Đi and the loading of species i, θi, by fitting to an empirical equation.12 In mixtures, the pure species loading is replaced by the total loading in the zeolite. For multicomponent mixtures, the Đii term has to be computed for each species, and it depends on the zeolite structures as well. These calculations can become a computational hurdle. The MS theory also requires the computation of mixture thermodynamic correction factors given by

(1)

In eq 1, ρz is the density of the zeolite, qis is the loading of the adsorbed species i in the zeolite, μi is the chemical potential of species i, Rg is the universal gas constant, and T is the temperature. On the right-hand side, Ni is the flux of species i, © 2012 American Chemical Society

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Received: September 19, 2012 Revised: November 19, 2012 Published: December 4, 2012 26816

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The Journal of Physical Chemistry C qis ⎛ ∂pis ⎞ ⎟ Γij = ⎜⎜ pis ⎝ ∂qjs ⎟⎠

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In eq 4, μi,0 is the reference chemical potential of species i. The gradient in chemical potential can be written as (3)

∇μi = R gT ∇ ln pis =

In eq 3, Γij is the mixture thermodynamic correction factor and pis is the partial pressure of species i inside the zeolite. The pore space of the zeolite is too small for a bulk fluid phase to exist, so the concept of pressure is hypothetical in this framework. The thermodynamic correction factors arise because the gradient in chemical potential is used as the driving force for diffusion in eq 1. Computation of these thermodynamic terms requires accurate pure component adsorption isotherm data and an additional theory such as the ideal adsorbed solution theory.13 For large multicomponent systems, these calculations can become computationally tedious, especially if the different species have vastly different adsorption strengths. Moreover, it is expected that, from a thermodynamic standpoint, the primary effects arising from adsorption should be seen in the pure component thermodynamic correction factors; mixture thermodynamic correction factors should represent secondary effects. Thus, while the major advantage of the MS theory is that it relies entirely on pure component adsorption and mobility data, the empirical estimation of the exchange coefficient and the large scale computations involved in both adsorption and diffusion calculations are a significant drawback. A simplified model, which relies only on pure component data and no empiricism, and which can predict mixture diffusion in zeolites with reasonable accuracy, is valuable from a practical perspective. Therefore, in light of the above problem, we pose the following questions: (i) is the full form of the MS equations always needed to describe diffusion in mixtures? (ii) can the MS theory be simplified to predict strong correlation effects in mixture diffusion without resorting to empiricism? (iii) is the full matrix of thermodynamic correction factors needed to predict diffusion in mixtures? To answer these questions, we proceed as follows: first, the MS equations are simplified to account for cases where correlations are so strong, that diffusion occurs essentially in a single-file manner. We do not consider the exchange coefficients and instead rely only on pure component mobilities. Next, we examine the matrix of thermodynamic correction factors and argue that, for several cases of practical interest, the off-diagonal (mixture terms) elements represent secondary effects and, hence, do not play a role in mixture diffusion. We use the simplified models to predict data for three case studies taken from the literature. The model predictions are compared to experimental data or predictions from the full MS theory. The novelty in the proposed approach is that simplified calculations suffice to obtain very good starting approximations to completely describe mixture diffusion in zeolites. The relevance of this approach stems from the need to reduce the significant computational requirements encountered during the solution of the flux equations, especially when dealing with vast, multicomponent systems and different zeolite structures.

pis

∇pis

(5)

Since pis = f(q1s,q2s,...), we can express the gradient in the partial pressure as ∇pis =

⎛ ∂p ⎞ is ⎟ ⎟∇qjs q ∂ ⎝ js ⎠

∑ ⎜⎜ j

(6)

Combining eqs 5 and 6, we can write qis

∇μi

=

R gT

∑ j

qis ⎛ ∂pis ⎞ ⎜ ⎟∇ q = pis ⎜⎝ ∂qjs ⎟⎠ js

∑ Γij∇qjs j

(7)

Combining eqs 1 and 7, we can express the MS equations in matrix form as (N ) = −ρz [B]−1 [Γ]∇(qs)

(8)

In eq 8, the elements of the B matrix are given by (see Hansen et al.,7 for example) Bii =

1 + Đi

Bij = −

n

∑ j = 1, j ≠ i

qjs qjs ,sat Đij

(9a)

qis qjs ,sat Đij

(9b)

Next, we proceed to simplify the MS equations to describe strong correlation effects in the zeolite. For the sake of simplicity, we demonstrate the approach by considering a binary mixture, with adsorption described by the multicomponent Langmuir isotherm. In this case, the elements of the B and Γ matrix are given by [B]−1 =

[Γ] =

1 Đ12 + θ2 Đ1 + θ1Đ2 ⎛ Đ1(θ1Đ2 + Đ12 ) ⎞ θ1Đ1Đ2 ⎜ ⎟ ⎜ ⎟ θ2 Đ1Đ2 Đ2 (θ2 Đ1 + Đ12 )⎠ ⎝

⎛1 − θ2 θ1 ⎞ 1 ⎜⎜ ⎟⎟ 1 − θ1 − θ2 ⎝ θ2 1 − θ1⎠

(10)

(11)

In both eqs 10 and 11, θi = qis/qis,sat, where qis,sat is the saturation capacity of the adsorbed species i in the zeolite. It should be noted that different species can have different saturation capacities in the zeolite; hence, the sum of the loadings θi need not necessarily equal 1. Before considering the case of strong correlations, we examine the other extreme: weak or no correlations, where the molecules of different species move independently of each other in the zeolite lattice. In such a scenario, there is no friction between molecules, and Đij → ∞. Therefore, the matrix of mobilities should be a diagonal matrix. Thus, from eqs 8, 11, and 12, the matrix form of the MS equations is given by

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MATHEMATICAL MODEL First, we write the MS equations in terms of the concentrations of the adsorbed phase species in the zeolite. The chemical potential of species i is related to the partial pressure of species i in the mixture as10 μi = μi ,0 + R gT ln pis

R gT

(4) 26817

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The Journal of Physical Chemistry C ⎛ (1 ⎜ ⎛ N1 ⎞ ⎛ Đ1 0 ⎞⎜ (1 − ⎜ ⎟ = −ρz ⎜ ⎟⎜ ⎝ N2 ⎠ ⎝ 0 Đ2 ⎠⎜ ⎜ ⎝ (1 −

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⎞ − θ2) θ1 ⎟ θ1 − θ2) (1 − θ1 − θ2) ⎟ ⎟ θ2 (1 − θ1) ⎟ ⎟ θ1 − θ2) (1 − θ1 − θ2) ⎠

⎛ ∇q ⎞ ⎜ 1s ⎟ ⎜ ∇q ⎟ ⎝ 2s ⎠

qis

⎛ Đ2 [D] = ⎜ ⎝0

qis

qis ⎛ ∂pis ⎞ ⎜ ⎟∇ q pis ⎜⎝ ∂qns ⎟⎠ ns

(15)

∇μi R gT

≈

qis ⎛ ∂pis ⎞ ⎜ ⎟∇ q pis ⎜⎝ ∂qis ⎟⎠ is

(16)

The use of eq 16 simplifies the Γ matrix to a diagonal matrix containing only the pure component thermodynamic correction factors and represented by ⎛ Γ11 0 ⎞ ⎟⎟ [Γ] = ⎜⎜ ⎝ 0 Γ22 ⎠

(17)

We now apply the simplified form of the MS equations, as developed in this section, to predict diffusive fluxes for the separation of binary mixtures through zeolite membranes, and to predict concentration profiles in a diffusion-reaction system catalyzed by a zeolite. All models were simulated using the commercial finite element method package, COMSOL Multiphysics 3.5a, in combination with MATLAB 2009b, on a Dell Precision T3500 workstation with Intel Quad Core (2.66 GHz) processors and 4GB of RAM.

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RESULTS AND DISCUSSION Case Study 1: Permeation of CH4/C3H8 Mixture through a Silicalite Membrane. van de Graaf et al.1 studied the permeation of a binary mixture of CH4 and C3H8 through a silicalite membrane grown on a porous stainless steel support. They modeled the experimentally measured fluxes using the full form of the MS equations wherein loading independent mobilities and a multicomponent Langmuir adsorption isotherm were assumed. C3H8 has a much higher adsorption equilibrium constant and a much lower mobility, as compared to CH4, and hence, strong correlation effects are expected. This is experimentally confirmed in Figure 8a of their paper wherein marginally increasing the mole fraction of C3H8 in the mixture results in a drastic reduction in the CH4 flux. van de Graaf et al. modeled transport through the zeolite membrane as well as the stainless steel support and accounted for the material balance in the retentate and permeate side mixing compartments. We are interested primarily in the ability of our simplified model to predict mixture diffusion in the zeolite; therefore, we have neglected transport through the support layer, as well as the mixing volumes at the upstream and downstream sides of the membrane. Instead, we assume that, at the upstream side, the total pressure at the membrane face is equal to the feed pressure, and that the downstream face contains no permeating components, i.e., the sweep gas removes any component adsorbed on the downstream side. In Figure 1, we plot the flux of CH4 at the downstream side of the membrane as a function of the partial pressure of C3H8 on the upstream side, using (i) the full MS equations, (ii) the single file model (Dij → ∞) of van de Graaf et al., and (iii) the new single file model proposed

(13)

Equation 13 should be used to calculate the fluxes as (N ) = −ρz [D]∇(qs)

q ⎛ ∂p ⎞ qis ⎛ ∂pis ⎞ ⎜⎜ ⎟⎟∇q + is ⎜⎜ is ⎟⎟∇q pis ⎝ ∂q2s ⎠ 2s pis ⎝ ∂q1s ⎠ 1s

We expect that only the pure component thermodynamic correction factors are important because they represent primary effects associated with the adsorption of a species from the vapor phase to the solid phase. Therefore, we approximate the gradient in chemical potential as

(12)

⎞ − θ2) θ1 ⎟ θ1 − θ2) (1 − θ1 − θ2) ⎟ ⎟ θ2 (1 − θ1) ⎟ ⎟ θ1 − θ2) (1 − θ1 − θ2) ⎠

R gT

=

+ ....

It should be noted that eq 12 has been referred to as the “single file diffusion” model or the GMS (Đij → ∞) model in the literature.1,14 However, single file diffusion is expected to arise only in situations where the difference in adsorption strengths between different species is so large, that a single strongly adsorbed species effectively forces the other comparatively weakly adsorbed species to move in a single file manner from site to site. If correlation effects are weak, one would not expect to see such single file effects because each species retains its pure component mobility. Therefore, the use of the “single file” terminology needs to be re-examined. For species with vastly different adsorption strengths, both species will have the same mobility, equal to that of the more strongly adsorbing species in the zeolite. There is no need for an exchange term, because in the narrow pore space of the zeolite the molecules cannot pass each other. In other words, the friction between the molecules tends to infinity, and Đij is zero. Within the framework of the MS equations, setting Đij → 0 while maintaining the individual pure component mobilities results in a singular [B]−1 matrix. However, if all species are assigned the same mobility as that of the slowest species, then we effectively model a system wherein correlation effects as captured by the Đij term are zero, i.e., Đij → ∞. So, we can once again reduce the mobility matrix to a diagonal one as (assuming species 2 is strongly adsorbing) ⎛ (1 ⎜ 0 ⎞⎜ (1 − ⎟ Đ2 ⎠⎜⎜ ⎜ ⎝ (1 −

∇μi

(14)

Thus, for strong correlation effects, which can lead to single file diffusion, the matrix of mobilities is simplified to contain the same diagonal terms, which are dictated by the mobility of the slowest species in the system. To the best of our knowledge, such an approach to describe strong correlation effects has not been attempted before. The set of eqs 13 and 14 constitute the proposed single file model in this work. Next, we examine the simplification of the Γ matrix. This is probably the first report that systematically examines the effect of neglecting the off-diagonal elements of the matrix of thermodynamic correction factors while modeling mixture diffusion in zeolites. As shown in eq 7, the gradient in chemical potential of species i is expressed as 26818

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Figure 2. separation from the adsorption Figure 1. Flux of CH4 in a binary CH4/C3H8 mixture permeating through a silicalite-1 membrane as a function of the partial pressure of C3H8 on the feed side, using the various models discussed in this work.

Comparison of N2/CH4 permeation selectivities for of a N2/CH4 mixture through a SAPO-34 membrane model neglecting correlations and with a simplified matrix. Comparison with the experimental data of Li et al.

the MS model neglecting correlations in diffusion, and with a simplified adsorption matrix. The pure component mobilities and adsorption isotherms were taken from Li et al. The IAST was used to compute the adsorbed phase concentrations at the upstream side of the membrane. From the pure component adsorption isotherms, the thermodynamic correction factor was fitted to a cubic spline function of the pure species loading using a cubic spline interpolation function available in MATLAB 2009b. The spline fit was required because Li et al. used a statistical isotherm to describe pure component adsorption. This spline function was then used in the mixture diffusion calculations to determine the pure component thermodynamic correction factors. The selectivity is defined as

in this work. For the full MS equations, the intermolecular exchange coefficient was calculated using the Vignes relation, as was done by van de Graaf et al. From Figure 8a of van de Graaf et al., it was observed that the full MS equations give a very good match to the experimentally measured flux of CH4. Hence, the simplified models should be compared with the full MS equations. In Figure 1, we see that the new single file model gives the closest match to the predictions from the full MS model over almost the entire range of C3H8 partial pressures studied. Please note that we did not simulate the entire range of C3H8 partial pressures, because, at higher partial pressures, the CH4 flux is almost zero. At low partial pressures of C3H8, the new single file model underpredicts the flux of CH4, because there is insufficient C3H8 adsorbed to induce single file diffusion; however, at intermediate to higher partial pressures, the strongly adsorbed C3H8 slows down the weakly adsorbing CH4 significantly, forcing the CH4 to move in a single file fashion. In comparison, the single file model of van de Graaf et al. overestimates the CH4 flux significantly because the pure component mobility of each species is used in the diffusivity matrix (eq 12). Thus, we are able to predict the flux of the weakly adsorbing species fairly accurately without using any empiricism; the only inputs to the diffusivity calculations are the pure component mobilities and the adsorption equilibrium constants. Next, we examine the effect of using a simplified adsorption matrix in the prediction of mixture diffusion fluxes. The following two case studies will demonstrate that the offdiagonal elements of the matrix of thermodynamic correction factors are not always important in flux predictions. Case Study 2: Permeation of N2/CH4 Mixture through a SAPO-34 Membrane. Li et al.5 studied the permeation of a N2/CH4 mixture through a SAPO-34 membrane and predicted the experimentally determined permeation selectivity using the MS equations. Li et al. report that correlation effects are not important in SAPO-34 for this system; therefore, the mobility matrix becomes diagonal with the pure component mobilities occupying the diagonal positions. We also simplified the adsorption matrix to a diagonal matrix. Figure 2 compares the experimental permeation selectivity with the predictions from

αperm =

N1/N2 f1,up /f2,up

(18)

In eq 18, f i,up is the upstream fugacity of component i in the mixture. For equimolar feed mixtures, the selectivity is simply the ratio of permeate fluxes. Figure 2 shows that the model predictions match the experimental data very well over the entire range of experiments. The maximum deviation of 15% between the model and experiments is observed at approximately 1 MPa total upstream pressure. This deviation could arise because of the simplification of the adsorption matrix. Case Study 3: Alkylation of C6H6 by C2H4 over H-ZSM5. Hansen et al.7 studied the diffusion limited alkylation of benzene by ethylene over H-ZSM-5 to produce ethylbenzene. They modeled the diffusion of the reactant and product species using the full MS equations, and with the MS model neglecting correlations in diffusion. They showed that the two models produce very similar effectiveness factors. We predicted the concentration profiles for this system using the simplified adsorption matrix and neglecting correlation effects. In Figure 3, the concentration profiles of the three species predicted from the simplified model are plotted as a function of the dimensionless radial coordinate in the zeolite crystal. For comparison, the concentration profiles of benzene and ethylbenzene as obtained by Hansen et al. assuming no correlations but with a full adsorption matrix, are plotted as symbols. For the simulations, the pure component mobilities and adsorption isotherms were taken from Hansen et al. The adsorbed phase loadings at the external surface of the zeolite crystal were calculated using the IAST. 26819

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literature, wherein the two species attain nearly the same selfdiffusivity, equal to that of the slowest moving species in the mixture. However, the self- and corrected diffusivities coincide only under the limit of low adsorbed species concentration.11 Therefore, further investigation is needed to clarify how single file diffusion affects the diffusivities of the different species in the mixture. From a practical standpoint, the calculation of mixture thermodynamic correction factors can become tedious, especially if the functional form of the pure component adsorption isotherms is complex. If experimental data on mixture behavior is available, for example, in the form of flux measurements, then the first effort at modeling can focus on using only the pure component thermodynamic correction factors and mobilities to estimate fluxes. We propose that the methodology introduced in this work, while not entirely rigorous from a fundamental point of view (but it should also be noted that the MS equations for zeolites are purely phenomenological), provides useful engineering approximations that can complement existing methods of predicting the behavior of mixtures, while simplifying and accelerating calculations. Finally, the models presented in this paper hold strictly for all-silica zeolites or zeolites with very low static heterogeneity (low Al content, for example) because the MS theory does not properly account for high static heterogeneity.15,16 Similarly, the calculation of adsorption thermodynamics using the IAST is questionable when there is variable energetic heterogeneity in the zeolite lattice induced by significant framework substitutions, for example.17 When there is large static heterogeneity (low Si/Al ratio), different adsorption sites can have vastly different adsorption strengths, and percolation effects may come into play. Alternate approaches such as an effective medium approximation (EMA)-based mean field theory (MFT)15,18 may be more applicable. Rare event simulations may also be a useful tool to calculate diffusivities in zeolites, as an alternative to the MS formulation.19

Figure 3. Predicted concentration profiles from the model assuming no correlations, and with a simplified adsorption matrix, as a function of the radius of a ZSM-5 crystal for the alkylation of benzene with ethylene. Symbols represent calculations of Hansen et al.

For ethylene, the concentration profile was not easily discernible from the data of Hansen et al.; hence, it is not plotted. However, they do report that the ethylene concentration at the center is approximately 61% of the value at the external surface. The corresponding fractional concentration from the simplified model is approximately 63%, indicating good agreement. The predicted profiles show fairly good agreement with the data of Hansen et al. The effectiveness factor from the weak correlations model of Hansen et al. is 52% compared to 58% from the simplified model. The simplified model using only pure component thermodynamic correction factors overpredicts the effectiveness factor by approximately 11% when compared to the model neglecting correlations alone. This is within the typical accuracy of experiments as well. The full MS equations predict an effectiveness factor of 47%. Therefore, from a computational standpoint, the simplified model can be used as a good starting point to estimate effectiveness factors, without calculating mixture thermodynamic correction factors.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support for this work from Synfuels, China. S.M.R. thanks Jeffrey Marquis of Rensselaer Polytechnic Institute and Dr. Niels Hansen and Prof. Frerich Keil of the Hamburg University of Technology, Germany, for helpful discussions.

CONCLUSIONS For very strong correlations, the proposed single file model provides a very simple way to estimate mixture behavior, provided accurate data on the pure component mobilities and adsorption isotherms are available, either from simulations or experiments. No empiricism is needed, because the intermolecular exchange coefficients do not play a significant role. Correlation effects are captured instead, by assigning equal mobilities to all species in the mixture, equal to that of the slowest species in the system. It must be reiterated that, while our work and previous studies have both used the limit of Đij → ∞ to describe single file diffusion, we do not retain pure component mobilities for each species, as explained above. Moreover, recent efforts at capturing strong correlation effects have emphasized the importance of correctly estimating the Đij term,9 which, as we show, is not required in the simplified model. Interestingly, van de Graaf et al.1 point to several PFGNMR studies of binary mixture diffusion in the published

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[B] Đi Đii Đij f i,up n Ni 26820

LIST OF SYMBOLS matrix of MS mobilities and loadings MS (or corrected) diffusivity or mobility of species i in the zeolite (m2/s) intermolecular exchange coefficient capturing species i−i interactions (m2/s) intermolecular exchange coefficient capturing species i−j interactions (m2/s) upstream fugacity of species i (Pa) number of species in the system flux of species i in the zeolite (mol/m2/s) dx.doi.org/10.1021/jp309315z | J. Phys. Chem. C 2012, 116, 26816−26821

The Journal of Physical Chemistry C pis qis Rg T

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partial pressure of species i inside the zeolite (Pa) molar loading of species i inside the zeolite (mol/kgcat) universal gas constant (J/mol/K) absolute temperature (K)

Greek

αperm Γij μi ρz θi

permeation selectivity mixture thermodynamic correction factor chemical potential of species i (J/mol) density of the zeolite (kgcat/m3) normalized molar loading of species i inside the zeolite

Subscripts

0 reference value sat saturation capacity

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REFERENCES

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