Competitive Diffusion of Gases in a Zeolite Bed ... - ACS Publications

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Competitive Diffusion of Gases in a Zeolite Bed: NMR and Slice Selection Procedure, Modeling, and Parameter Identification M. Petryk,† S. Leclerc,‡ D. Canet,§ I. Sergienko,∥ V. Deineka,∥ and J. Fraissard*,⊥,# †

Modélisation du Transfert de Masse, University Ivan Pul’uy, 46001 Ternopil, Ukraine LEMTA-UMR 7563 CNRS-Université de Lorraine, F-54504 Vandœuvre-lès-Nancy Cedex, France § Institut Jean Barriol (FR CNRS 2843), Université de Lorraine, F-54506 Vandœuvre-lès-Nancy Cedex, France ∥ Glushkov Institute of Cybernetics of NAS of Ukraine, 40 Glushkova av., 03187 Kyiv, Ukraine ⊥ Sorbonne Universités, UPMC Univ Paris 06, UFR Chimie, F-75005 Paris, France # ESPCI, UMR 8213, LPEM, 10 rue Vauquelin, F-75231 Paris Cedex 05, France ‡

S Supporting Information *

ABSTRACT: Taking into account the previous experimental results obtained by NMR, new procedures for identifying diffusion coefficients for codiffusing benzene and hexane in intra- and intercrystallite spaces of ZSM 5 zeolite were implemented, using high-speed gradient methods and mathematical diffusion models, as well as the NMR spectra of the adsorbed mass distribution of each component in the zeolite bed. These diffusion coefficients were obtained as a function of time for different positions along the bed. Benzene and hexane concentrations in the inter- and intracrystallite spaces were calculated for every position in the bed and for different adsorption times.

1. INTRODUCTION When a heterogeneous catalysis reaction is performed by flowing gases through a microporous catalyst bed it is necessary to determine at every moment the diffusion coefficient of each reactant in the presence of the others and its instantaneous distribution along the length of the catalyst bed. With conventional NMR, measurements characterize the entire sample at the equilibrium of adsorption and without spatial resolution. Several authors have shown that in MFI zeolite the self-diffusion of each component in the adsorbed mixture is influenced by the coadsorbed species;1 almost invariably the more mobile species is slowed down due to the presence of more sluggish partners.2−4 The use of classical magnetic resonance imaging (MRI) is complicated since the experiment must be performed several times under identical conditions and each time with only one incompletely deuterated gas.5 To remedy these drawbacks we have proposed a new NMR imaging technique6 that can be used to follow the diffusion and adsorption of a gas in a microporous bed7,8 and also to visualize directly and quantitatively the codiffusion of several gases. We have presented in a previous paper the experimental results of the codiffusion of benzene and hexane through a silicalite bed.9 We use these results here to calculate the variation of the diffusion and adsorption coefficients against time and the corresponding concentrations of each adsorbed gas along the bed. © 2015 American Chemical Society

2. EXPERIMENTAL SECTION The sample (length l = 15 mm) is displaced vertically by 0.5 mm steps in the magnet, relative to a very thin coil detector during the adsorption of the gas.6 The bed is assumed to consist of N very thin layers of solid (Figure 1). The region probed is limited to each layer, so that the variation of the concentration of each gas absorbed at the level of each layer is obtained as a function of time. Experimental NMR conditions, characteristics of the crystallite size and the description of the sample-holder bulb containing the liquid phase in equilibrium with the gas phase are given in ref 9. We have to mention that the instantaneous results for one component take into account the entire environment including the solid and the other adsorbate. The upper face of the bed of zeolite is exposed to a constant pressure of each gas (Figure 1). The diffusion of the two gases is axial in the macropores of the intercrystallite space (z direction) and radial in the micropores of the zeolite. According to the experimental conditions, the zeolite bed consists of a large number, N, of very thin layers of solid, of thickness Δlk = lk − lk−1, perpendicular to the z direction. The corresponding coefficients of inter- and intracrystallite space are Dinter,k and Dintra,k, respectively. Received: August 16, 2015 Revised: October 21, 2015 Published: October 23, 2015 26519

DOI: 10.1021/acs.jpcc.5b07974 J. Phys. Chem. C 2015, 119, 26519−26525

Article

The Journal of Physical Chemistry C

Figure 1. Distribution of the layers (left) and corresponding characterizing parameters (right).

3. EXPERIMENTAL RESULTS: BENZENE AND HEXANE ADSORPTION CURVES The purely experimental results have been summarized in ref 9: the spectrum of each gas at every instant and every level of the solid, and the benzene and hexane concentrations along the sample, for each diffusion time. Here we shall only present the evolution, as a function of time, of the benzene and hexane concentrations at different levels of the sample,9 on which are based the calculations of the diffusion coefficients and the instantaneous inter- and intracrystallite concentrations (Figure 2).

restriction does not affect the calculation of the diffusion parameters. Figure 2 shows clearly that benzene hinders the diffusion of hexane, and this at every moment. Moreover, at equilibrium, the amount of adsorbed benzene is twice that of hexane, indicating quantitatively the relative affinity to the two adsorbates. These curves display modulations as a function of time, which must be averaged for all subsequent mathematical representations. These modulations are weak at the lower layers of the tube and can be due to errors in the measurement of small amounts. Those closer to the arrival of the gas are greater and are similar for the two gases. We suggest that these fluctuations may be due to the fact that intercrystallite adsorption at levels close to the gas phase is faster than liquid−gas equilibration, which is not as rapid for a mixture as for a single component. To each slight decrease in the gas pressure could correspond a small fast desorption.

4. MATHEMATICAL MODEL OF COMPETITIVE DIFFUSION IN MICROPOROUS SOLIDS The model presented is analogous to the biporous model.7−10 The diffusion process of the two diffusing components (benzene and hexane) involves two types of mass transfer: diffusion in the macropores (intercrystallite space) and diffusion in the micropores of crystallites (intracrystallite space). The following assumptions have been made for each component: (i) diffusion occurs in the Henry’s law region of the adsorption isotherm; (ii) the effect of heat is negligible; (iii) during the evolution of the system towards equilibrium there is a concentration gradient in the macropores and/or in the micropores; (iv) all zeolite crystallites are spherical and have the same radius R; (v) the crystallite bed is uniformly packed. Calculation of Diffusion Coefficients. The coefficients of competitive diffusion in intracrystallite space, Dintras, and intercrystallite space, Dinters, s = 1, 2 (s = 1 for benzene and s = 2 for hexane) being unknown, the mathematical model of gas diffusion kinetics in the zeolite bed is defined in domains: Ω kT = (0, T ) × Ωk ,

Figure 2. Evolution versus time of the benzene and hexane concentrations at different levels of the sample (continuous, experimental curves; dotted, their approximations used for simulation).

(Ωk = (Lk − 1 , Lk ),

k = 1, N ,

L0 = 0 < L1 < ... < LN + 1 = 1)

by the solutions of the system of differential equations (with dimensionless coordinates defined in the nomenclature):

We have considered neither the first diffusion time nor the first layers very close to the gas phase, because it takes about 5 min from the moment when the glass is broken (therefore the beginning of diffusion) to get the first spectrum.9 It is obvious that after such a time the first layers are saturated, but this

∂Csk(t , Z) ∂t 26520

=

Dinters ∂ 2Cs k k l

2

∂Z

2

− einterkK sk

Dintra s ⎛ ∂Q s ⎞ k k ⎟⎟ ⎜⎜ R2 ⎝ ∂X ⎠ X=1

(1)


Article

The Journal of Physical Chemistry C ∂Q s (t , X , Z) k

∂t

⎞ Dintra s ⎛ ∂ 2Q s 2 ∂Q sk ⎟ k⎜ k = + X ∂X ⎟⎠ R2 ⎜⎝ ∂X2

each iteration, by minimizing the difference between the calculated values and the experimental data. The method proposed is a generalization of the calculation approaches presented in ref 8, 10, and 11; it allows to reduce the number of iterations to 2−3 orders of magnitude for each cycle. It can also be used to identify parameters for more complicated adsorption systems and to calculate three or more parameters simultaneously. The solution of the problem of calculating diffusion coefficients (eqs 1−7) is reduced to the problem of minimizing the functional of the difference (eq 9) between the model solution and the experimental data, the solution being refined incrementally by means of a special calculation procedure, which uses fast highperformance gradient methods.12,14,15 Gradient methods of calculating diffusion coefficients, based on a Lagrange functional of residuals (target, error, etc.), have been used by Lions14 (problems of mechanics and of thermoelasticity), later by Alifanov (calculation of the temperature of elements of planes),15 then by Sergienko, Deineka, Petryk, and Fraissard (problems of hydromechanics, filtration, adsorption, etc.).10−12 Gradient Method of Calculation. According to refs 11 and 12 and using the error minimization gradient method for the calculation of the competitive diffusion coefficients Dintrask and Dintersk of the sth diffusing component, we obtain the iteration expression for the n + 1-th calculation step:

(2)

with initial conditions: Csk(t = 0, Z) = 0; X ∈ (0, 1);

Q s (t = 0, X , Z) = 0; k

Z ∈ Ωk ;

k = 1, N

(3)

boundary and interface conditions for coordinate Z: Cs1(t , L1) = 1,

∂Cs1 ∂Z

(t , Z = 0) = 0,

t ∈ (0, T )

(4)

[Csk(t , Z) − Csk(t , Z)]Z = Lk = 0, ∂ [Dinters Csk −1(t , Z) − Dinters Csk(t , Z)]Z = Lk = 0, k−1 k ∂Z k = 1, N , t ∈ (0, T )

(5)

boundary conditions for coordinate X of the particle: ∂ Q (t , X = 0, Z) = 0(symmetry conditions) ∂X sk Q s (t , X = 1, Z) = Csk(t , Z)(equilibrium conditions), k

t ∈ (0, T ),

Z ∈ Ωk ,

k = 1, N − 1

(6)

n+1 n (t ) = Dintra (t ) − ∇JDn Dintra s s

additional condition (experimental data): [Csk(t , Z) + Q̅ s (t , Z)]|hk = M sk(t , Z)|hk , k hk ∈ Ωk ,

hk = (Lk − Lk − 1)/2,

×

s = 1, 2,

t ∈ (0, T )

intra s k

k

k

n [Csk(Dinter , sk

n ; Dintra sk

n n , Dintra ; t , γk) − M sk(t )]2 t , γk) + Q̅ s (Dinter s s k k

||∇JDn

intra s k

(7)

(t ) k

(t )||2 + ||∇JDn

(t )||2

inters k

,

t ∈ (0, T )

The problem of the calculations (eqs 1−7) is to find unknown functions Dintras ∈ ΩT, Dinters ∈ ΩT (Dintras > 0, Dinters > 0, s = 1, 2), when absorbed masses Csk(t,Z) + Q̅ sk(t,Z) satisfies conditions (7) for every point hk ⊂ Ωk of the kth layer.11,12 Here: εinterkcsk εinterk ; ≈ einterk = (1 − εinterk)Ksk εinterkcsk + (1 − εinterk)qsk qs ∞ eintrak = 1 − einterk , K sm = m c sm ∞

n+1 n (t ) = Dinter (t ) − ∇JDn Dinter s s k

×

inters k

m

(t )

n n n n [Csk(Dinter , Dintra ; t , γk) + Q̅ s (Dinter , Dintra ; t , γk) − M sk(t )]2 s s s s k k

k

k

||∇JDn

intra s k

k

(t )||2 + ||∇JDn

intersk

(t )||2

,

(8)

t ∈ (0, T )

where J(Dintersk, Dintrask) is the error functional, which describes the deviation of the model solution from the experimental data on hk ∈ Ωk, and is written as 1 2

J(Dinters , Dintra s ) =

Q̅ s(t,Z) = ∫ 10Qs(t,X,Z)XdX, average concentration of adsorbed component s (s = 1, 2) in micropores; and Ms(t,Z)|hk, experimental distribution of absorbed mass in macro- and micropores at hk ⊂ Ωk (results of NMR data, Figure 2). Mathematical Justification of the Solvability of the Calculation (eqs 1−7). The calculation of the diffusion coefficients Dintrask and Dintersk is a complex mathematical problem. In general, it is not possible to obtain a correct formulation of the problem (eqs 1−7) and to construct a unique analytical solution because of the complexity of taking into account all the physical parameters (variation of temperature or pressure, crystallite structures, nonlinearity of Langmuir isotherms, etc.), as well as the insufficient number of reliable experimental data, measurement errors, and other factors. Therefore, according to the principle of Tikhonov,13 later developed by Lions14 and Sergienko and Deineka,12 the calculation of diffusion coefficients requires the use of the model for

k

k

̄

∫0

T

[Cs(τ , Z , Dinters , Dintra s ) k

k

+ Q s(t , Z , Dinters , Dintra s ) − M sk(t )]h2k dτ , k

hk ∈ Ωk ,

k

k = 1, N

(9)

∇JnDintra,sk(t), ∇JnDinter,sk(t) are the gradients of the error functional, J(Dintersk, Dintrask). ||∇JDn

intrask

||∇JDn

(t )||2 =

intersk

∫0

(t )||2 =

T

[∇JDn

∫0

intrask

T

[∇JDn

(t )]2 dt ,

intersk

(t )]2 dt

Supporting Information contains the description of the mathematical methodology for calculating diffusion coefficients and modeling of competitive diffusion in a microporous crystallite bed. 26521


Article


Figure 3. Variation of intracrystallite diffusion coefficients for benzene Dintra1,k (left) and hexane Dintra2,k (right) against time at different positions of the bed. dark red = 6 mm, dark blue = 8 mm, green = 10 mm, black = 12 mm, purple = 14 mm.

Figure 4. Variation of intercrystallite diffusion coefficients for benzene (left) and hexane (right), against time at different positions of the bed. dark red = 6 mm, dark blue = 8 mm, green = 10 mm, black = 12 mm, purple = 14 mm.

Figure 5. Variation of the intercrystallite concentration calculated for benzene (left) and hexane (right) against time and at different positions of the bed. dark red = 6 mm, dark blue = 8 mm, green = 10 mm, black = 12 mm, purple = 14 mm.

Figure 6. Total concentration of benzene (left) and hexane (right) versus time at different positions of the catalytic bed: dotted, experimental curves; continuous, model curves. 26522


Article


Figure 7. Distribution of the benzene (left) and hexane (right) concentrations in the intracrystallite space from the surface (abscissa 1) to the center (abscissa 0) of the crystallites, at different times. a-dark blue: t = 25 min.; b-green: t 50 min.; c-blue: t =100 min.; d-red: t =200 min.

5. NUMERICAL SIMULATION AND ANALYSIS: COEFFICIENTS OF COMPETITIVE DIFFUSION OF BENZENE AND HEXANE. CONCENTRATION PROFILES IN INTER- AND INTRACRYSTALLITE SPACES The benzene and hexane intracrystallite diffusion coefficients Dintra1,k and Dintra2,k are presented in Figure 3 as functions of time for the five coordinate positions: 6, 8, 10, 12, and 14 mm, defined now from the top of the bed. The curves for benzene Dintra1,k

(Figure 3, left) are pseudo exponentials. Dintra1,k decreases from 9.0 × 10−13 to about 1.0 × 10−14 (equilibrium) depending on the position of the crystallite and the time, as well as on the amount of adsorbed gas. The shapes of the variations of Dintra2,k for hexane are roughly the same, but the diffusion coefficients are higher, from about 9.0 × 10−12 to 2.0 × 10−12 (Figure 3, right). Figure 4 presents the variation of the benzene and hexane diffusion coefficients in intercrystallite space Dinter1,k and Dinter2,k against time, for coordinate positions from 6 to 14 mm. These 26523


Article

The Journal of Physical Chemistry C coefficients decrease with time from 6.0 × 10−6 to 1.0 × 10−6 (equilibrium) for benzene and from 2.7 × 10−6 to 1.0 × 10−5 for hexane, depending on the position in the bed and the increase in the adsorbed concentrations. Figure 5 shows the variation against time of the concentrations calculated for benzene and hexane in the intercrystallite space. As can be seen, the calculated concentration approaches equilibrium values for a diffusion time around 250 min. However, the variations of the concentrations with time are rather different for the two gases. Figure 6 compares the calculated and experimental variations against time of the total amount of adsorbed benzene and hexane in the catalytic bed. As can be seen in Figure 6, left, the distributions of adsorbed benzene are in good agreement with the experimental ones for each measurement position. The maximum deviation is generally less than 5%. A similar pattern is observed for hexane (Figure 6, right). The greatest differences (6−7%) are for the curves corresponding to positions 6 and 8 mm. Figure 7a,b shows the variation of the concentrations Qt,x,z of adsorbed benzene and hexane in the micropores of intracrystallite space from the surface-1 to the center-0 of the crystallites located between 6 to 14 mm from the top of the bed, and after 25 to 200 min of diffusion (a, b, c, and d, respectively). The gradients increase, and the mean concentrations decrease with increasing distance of the particles from the arrival of the gases. The particles at 6 and 8 mm are saturated with benzene after 100 min, but not yet with hexane.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b07974. Description of the mathematical methodology for parameter identification and modeling competitive diffusion in a microporous crystallite bed (PDF)

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REFERENCES

(1) Krishna, R.; Van Baten, J. M. Diffusion of Alkane Mixtures in Zeolites: Validating the Maxwell−Stefan Formulation Using MD Simulations. J. Phys. Chem. B 2005, 109, 6386−6396. (2) Förste, C.; Germanus, A.; Kärger, J.; Pfeifer, H.; Caro, J.; Pilz, W.; Zikanova, A. Molecular Mobility of Methane Adsorbed on ZSM-5 Containing Co-Adsorbed Benzene, and the Location of the Benzene Molecules. J. Chem. Soc., Faraday Trans. 1 1987, 83, 2301−2309. (3) Fernandez, M.; Kärger, J.; Freude, D.; Pampel, A.; Van Baten, J. M.; Krishna, R. Mixture Diffusion in Zeolites Studied by MAS PFG NMR and Molecular Simulation. Microporous Mesoporous Mater. 2007, 105, 124−131. (4) Krishna, R.; Van Baten, J. M. Diffusion of Hydrocarbon Mixtures in MFI Zeolite: Influence of Intersection Blocking. Chem. Eng. J. 2008, 140, 614−620. (5) N’Gokoli-Kekele, P.; Springuel, M. A.; Bonardet, J.-J.; Dereppe, J.M.; Fraissard, J. 129Xe NMR of Adsorbed Xenon and 1H NMR Imaging: New Methods to Study the Diffusion of Gaseous Hydrocarbons in a Fixed Bed of Zeolite. Stud. Surf. Sci. Catal. 2001, 133, 375−382. (6) Leclerc, S.; Trausch, G.; Cordier, B.; Grandclaude, D.; Retournard, A.; Fraissard, J.; Canet, D. Chemical Shift Imaging (CSI) by Precise Object Displacement. Magn. Reson. Chem. 2006, 44, 311−317. (7) Petryk, M.; Leclerc, S.; Canet, D.; Fraissard, J. Mathematical Modelling and Visualization of Gas Transport in a Zeolite Bed Using a Slice Selection Procedure. Diff. Fundamentals. 2007, 4, 11.1. (8) Petryk, M.; Leclerc, S.; Canet, D.; Fraissard, J. Modelling of Gas Transport in a Microporous Solid Using a Slice Selection Procedure: Application to the Diffusion of Benzene in ZSM5. Catal. Today 2008, 139, 234−240. (9) Leclerc, S.; Petryk, M.; Canet, D.; Fraissard, J. Competitive Diffusion of Gases in a Zeolite Using Proton NMR and a Slice Selection Procedure. Catal. Today 2012, 187, 104−107. (10) Deineka, V.; Petryk, M.; Fraissard, J. Identifying Kinetic Parameters of Mass Transfer in Components of Multicomponent Heterogeneous Nanoporous Media of a Competitive Diffusion System. Cybernetics and System Analysis 2011, 47, 705−723. (11) Petryk, M.; Fraissard, J. Mathematical Modelling of Nonlinear Competitive Two-Component Diffusion in Media of Nanoporous Particles. J. of Automation and Information Sciences. 2009, 41, 37−55.

4. CONCLUSION Using the experimental NMR data and models of competitive diffusion, new procedures for the calculation of the diffusion coefficients for both components in the intra- and intercrystallite spaces were developed. The diffusion coefficients were obtained as a function of time for different positions along the catalyst bed. Using these diffusion coefficients, the concentrations of benzene and hexane in the inter- and intracrystallite spaces were calculated for every time and every position in the bed.

■

Dintra: micropore diffusion coefficient, m2/s K: adsorption equilibrium constant l: bed length, mm Δl = lk − lk−1 ; k = 1, N : layer thickness (all layers have the same thickness) L: dimensionless bed length (L = 1) M: total uptake at time t MT: total uptake at equilibrium q: adsorbate concentration in micropores q∞: equilibrium adsorbate concentration in micropores Q = q/q∞: dimensionless adsorbate concentration in micropores x: distance from crystallite center, mm R: mean crystallite radius, mm (we assume that the crystallites are spherical) X = x/R: dimensionless distance from crystallite center z: distance from the bottom of the bed for mathematical simulation, mm Z = z/l: dimensionless distance from the bottom of the bed Lk: dimensionless position of the kth layer hk: (Lk −Lk‑1)/2. einter: porosity εinter: bed porosity T: total duration of diffusion, min n: iteration number of identification

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 3360-987-7865. Notes

The authors declare no competing financial interest.

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NOMENCLATURE k = 1, N : layer number. Subscript k will be added to all the following symbols to specify that they are characteristic of the kth layer s: symbol of the adsorbate, 1 for benzene and 2 for hexane c: adsorbate concentration in macropores c∞: adsorbate equilibrium concentration in macropores C = c/c∞: dimensionless adsorbate concentration in macropores Dinter: macropore diffusion coefficient, m2/s 26524


Article

The Journal of Physical Chemistry C (12) Sergienko, I. V.; Deineka, V. S. Optimal Control of Distributed Systems with Conjugation Conditions; Nonconvex Optimization and Its Applications Series; Kluwer Academic Publishers: New York, 2005. (13) Tikhonov, A. N.; Arsenin. V. Y.; Solutions of Ill-Posed Problems; V. H. Winston Wiley: New York, 1977. (14) Lions, J. L. Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal; Lecture Notes in Math Series; Springer: New York. 2008. (15) Alifanov, O. M. Inverse Problems of Heat Exchange; International Series in Heat and Mass Transfer; Springer Verlag: Berlin, 1994.

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Competitive Diffusion of Gases in a Zeolite Bed ... - ACS Publications

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