Prediction of Zeolites Diffusivities

Fundamentals of Adsorption Proc. IVth h i . Conf. on Fundamentals of Adsorption, Kyoto, May 17-22, 1992 Copyright Q 1993 International Adsorption Society

Prediction of Zeolites Diffusivities

Klaus Dahlke', Gerhard Emig' and Eberhard Aust' 'Institut fiir Chemische Technik, Universitat Karlsruhe Kaiserstrde 12, D-7500 Karlsruhe 1, F.R.G.

'AKZO Corporate Research Laboratories Obernburg D-8759 Obernburg, F.R.G

Abstract In this paper a possible method to predict diffusion coefficientsin zeolites is presented. This method consists mainly in a random-walk part and a part of rescaling the arbitrary units of the random-walk experiment into real units. The so obtained diffusion coefficients can be compared directly with experimental values. The only necessary input into the procedure is the knowledge of the activation energy for the diffusion process, which has to be obtained by experiments or other types of simulations.

Introduction Zeolitic catalysts and adsorbents play a major role in industrial applications and will gain increasing importance in the future [l]. The diffusion process in zeolites is a so called restricted or configurational one [2], due to a ratio of molecular and pore diameter close to unity. Therefore, the selectivity of catalytic reactions in zeolites leading to fine chemicals (i. e., high value products) will be strongly affected by the Wusivities of the reactands. In gas purification often adsorption is the only possible unit operation to clean off-gases in order to meet the limits for environmental protection, especially at high flow rates combined with low loadings of the pollutant. In both cases, catalysis and adsorption, the knowledge of the diffusion coefficient becomes important in the mathematical model for a optimal and efficient design of the process (31. Unlike in the bulk or Knudsen diffusion no simple rules for estimation of the diffusion coefficients in zeolite exist. This might be a reason for several research groups to focus on the estimation and calculation of the adsorption constants and the diffusion coefficients in zeolites. This task is done either by molecular dynamics [4-71 or Monte-Carlo methods [8, 91. Several other groups investigate the diffusion and adsorption process by randomwalk methods [lo-141.In our laboratory we are also working with a random-walk method

129

I30

K. Dahlke, G . Emig and E. Aust

to investigate the influence of several different microdynamic assumptions on the diffusion process [15, 161.

Model A detailed description of the used model is given in our two previous papers [15,161. We want to outline here only a few basic model assumptions. The zeolite is simulated as a two-dimensional array. Each array element is equivalent to a supercage (zeolite types A, X, Y) or a channel intersection (zeolite type ZSM-5). The cages are separated by windows in which no passing of diffusing molecules is allowed. A molecule being in a supercage can migrate to one of the four neighbouring cages with equal probability. In the case of investigations of transport diffusion a constant concentration gradient along

the x-direction exists and the upper and lower boundary of the array are connected. The net flux through the array is monitored and according to Fick’s first law the diffusion coefficient is calculated [15]. In the case of self-diffusion a tagged molecule is placed in the center of the array and the diffusion path of this molecule is monitored during a preset number of steps. Here, the upper and lower as well as the left and right boundary are connected. The self-diffusion coefficient is evaluated via the Einstein equation from the mean square displacemant of the tagged molecule and the number of steps. Each cage can contain at least one or more molecules, up to the maximum number of molecules allowed per cage (Nmam). The whole number of molecules in the array is denoted by the pore filling factor 0 = c/c-=. The cages are activated randomly once in a single simulation run and it is assumed, that a molecule inside a cage will try to leave the cage when the cage is activated. If the cage in the randomly chosen direction contains at least one empty adsorption site (i. e., the actual number of molecules in the cage is less than the maximum number of molecules allowed per cage) and no other molecule diffuses through the same window at the same time, the molecule performs a successful jump, leaves the old cage and is placed in the new cage. If the neighbouring cage is completely filled or another molecule migrates through the same window one may distinguish four different possibilities how the diffusion process of the observed molecule is affected by such an obstacle: 0

A counter-diffusing molecule stops the current diffusion process, a completely filled cage changes the direction of the diffusion process; -+ Model 2.0

-

0

The observed molecule shall try to leave the cage;

0

The observed molecule has only one chance to leave the cage; +.-

0

Model 2.1 Model 2.2

A completely filled cage stops the current diffusion process, a counter-diffusing molecule changes the direction of the diffusion of the observed molecule; -+ Model 2.3

Prediction of Zeolites Diffusivities

131

In Fig.1 the results of the different assumptions are shown for the transport diffusion coefficient D e f t . . . .

I

.

.

.

I

.

.

.

I

"

. . . b Model 4.0'

'

+-+ Model 2.1

3rmc Model 2.2 4-t Model 2.3

1 3

1 0%

10-17

L..'..)'.' ...I 104: 0.0

.

.

.

I

0.2

I

,

.

I

0.4

.

I

.

I

0.6

.

.

I

I

0.8

I

I

. , ' I 1 .o

pore filling factor 8 Figure 1: Concentration dependence of the effective diffusion coefficient for the different model assumptions, N,,, = 1 Model 2.0 shows a concentration dependence similiar to experimental results for the diffusion of para-Xylene in NaX [17]. To get realiable results for the models 2.2 and 2.3 the number of simulation runs had to be increased up to 90000. Using these two models the effective diffusion coefficient seems to be independent of concentration. This behaviour is reported for the Benzene/H-ZSM-5 system [18]. It seems that the diffusion mechanism inside the pore network might follow either a hopping process similiar to the assumptions of Model 2.0 or 2.2, depending on the observed real combination adsorbate/zeolite. We don't have any explanation for the sharp drop of D,ff of Model 2.1. It might be that at low loadings a disorded motion occurs and no net flux can be observed. At higher loadings more cages at the high concentration side are filled up to N,,, and the main flow will follow along the concentration gradient. But this behaviour might be a hint that diffusion in zeolites will not follow a jump mechanism according to Model 2.1. As mentioned above for the simulation of self-diffusion the diffusion path of a tagged molecule is monitored. By knowing the starting point, the final point and the number of steps, the sel-diffusion coefficient can be calculated similiar to the Einstein equation. Fig. 2 shows the results of the simulation, again for the four different model assumptions. The difference between the models is less pronounced as in the case of the transport diffusion. Experimental data show the same concentration dependence as shown here: a decrease of the self-diffsuion coefficient with increasing concentration. Setting N,,, greater than one [15] the decrease is slowed down and the sel-diffsuion coefficient stays constant up to 0 M 0.6 as observed by experiments [19, 201. Surprisingly, the model with the highest number of tries for leaving

132

K. Dahlke. G. Emig and E. Aust

a cage (Model 2.1) does not produce the highest values for the self-diffusion coefficient, whereas the lowest values are obtained by the model with only one try (Model 2.2).

+-+ Model 2.1

H H K Model 2.2 4-t

'c

Q) v)

n

Model 2.3

-

10-17

10-2,

0.0

*

-

'

I

0.2

"

'

I

0.4

'

'

'

I

0.6

'

'

'

I

0.8

=

'

"

1.o

-

pore filling factor 8 Figure 2: Concentration dependence of the self-diffusion coefficient for the different model assumptions, N,,, = 1

Rescaling of Units From the simulations described above one gets the diffusion coefficients in arbitrary units. For comparison with experimental data it is necessary to rescale these arbitrary units into real units. A simple method to do this is proposed by Karger and co-workers [19,201: A modification of the free-volume theory is applied, and from the volumes of the supercage as well as the molecule the mean square jump length can be estimated: OD

(A2) = ~ / ~ ~ ~ ~7"f e x p =( (5)2/3r - ~ ) d t ~ ~ "f 7 " f V.

As a rough approximation for the volume of the supercage one may take the volume of a sphere with the same diameter as the supercage and use the van der Wads radii to estimate the volume of the molecule [21]. For the mean life time of a molecule on an adsorption site one obtains: 1

Ea

-=6-~.e~p(--) T RT

(3)

Prediction of Zeolites Difisivities

133

Table 1: Comparison of experimental and simulated value for r and X

1

Benzene in

NaX

rnim

reap

1I

at 458K Xeap

0.13 50 I 950 0.46 2.8 10.2 11 51- 1I 825 I 0.35 II 2.2 1 0.33 I 50 678 10.24 1.8 I

I

I

1

Ethane in NaX at 253K

Xaim

I

I

I

0.73 I 22.2

I 7.35 I 0.44 I 1.3

I

I

Experimental data from [24]

where

Using the simple assumptions of an harmonic oscillator the values for T tend to be erroneous: [22, 241, but r decreases with higher pore filling factors (tab.1). To avoid the wrong dependence of r with pore filling factor 0,r was calculated at a low pore filling factor (0= 0.1) and held constant for all other pore filling factors of the array. Calculating (A') according to eqn.(l) and using the constant value for r we obtain simulated self-diffusion coefficients in the same magnitude as experimental diffusion coefficients (Fig.3). r should stay constant or increase with increasing pore filling factor

I

10.000 4 4.000

-...

.

I

,

I

I

I

~

I

I

I

~

I

I

I

1

<

n

N

1.000

E ol

0.400

0

0.100

I

7

Y

"0

0.040 _

_

.

V-V Experimental data, n nni , w w v~ iPropane in ZSM-5 at 300 K [2 i-k Simulated data, Benzene in NaX at 458 K 31c.31c Experimental data, Benzene in NaX at 458 K [24]

Y f

0.001

l

"

'

l

'

'

'

l

'

'

'

i

'

'

'

pore filling factor Q Figure 3: Comparison between experimental and simulated self-diffusion coefficients The excellent match for the Benzene/NaX-system is accidental, usually the deviations of the simulated values from experimental results are in the range of one order of magnitude. Most

134

K.Dahlke, G.Emig and E.Aust

of the times the simulated values are larger than the experimental values. The best results are obtained for parfines in Faujasite, whereas for Benzene in ZSM-5 [25] no coincidence could be achieved. The differences in the simulated self-diffusion coefficients of the four different model assumptions vanish when rescaling the simulated values, all values of the self-diffusion coefficient in the real units fall close together so that no difference depending on the model assumptions is apparent. The model we used for the simulation of the transport diffusion is a representation of the permeation through a zeolite membrane in a Wicke-Kallenbach type of experiment [26]. This time we rescaled the simulated flow of molecules per time step (from Model 2.0) into a real 00w of mol/s and converted the flow into the appropiate units to obtain the permeabilities of the membrane. The results are shown in Tab. 2. Table 2: Experimental and simulated permeabilities through a silicalite membrane Species

0

n

[part./step]

7-

[SI

n

[mol/s]

-

date

I

.

tzp

[moZ m / ( m 2 s Pa)]

Again, the simulated values are quite satisfactory, but unlike the simulation of the selfdiffusion the simulated values are about one order of magnitude lower than the experimentally observed permeabilities. In the case of transport diffusion the different model assumptions will have a much greater effect than in the case of the self-diffusion, due to the substantial difference of the concentration dependence. Because of the lower D e f t for Model 2.2 and 2.3 the apparent permeabilities will be lower than those obtained with Model 2.0 and the difference between experiment and simulation would be greater.

Conclusion In this paper we have shown a simple method to estimate diffusion coefficients in zeolites. Starting with the number of mean square displacements or aflow obtained by a random-walk experiment we are able to rescale those values into real units. The so predicted diffusion coefficients are in agreement with experimental data. The rescaling procedure requires easily accessible geometric data ( v , ~ ,v,d, r),the molecular weight of the diffusing particle and the activation energy for diffusion. A weakness of the method is that the activation energy has to be obtained by experiments or other estimation methods.

With random-walk methods we are able to achieve a closer look into the microdynamic behaviour of diffusion in zeolites and thus, we are able to give hints for the explanation of the different appearence of the concentration dependence of the diffusion coefficients.

prediction of Zeolites Diffusivities

I35

Acknowledgement K.D. and G.E. gratefully acknowledge financial support by AKZO Corporate Research, Obernburg, Germany.

Not ation

4(A2),

distance between adjacent sites M m concentration total number of molecules in the simulation array maximal number of molecules allowed in the array Nma,x number of cages effective diffusion coefficient, m2/s self-diffusion coefficient, m2/s activation energy, J/mol molecular weight, Icg/moZ actual number of molecules per supercage maximal number of molecules allowed per supercage gas constant J / ( m o l -K) temperature, K free volume, m3 mean free volume, m3 volume of the adsorbed molecule, m3 minimum free volume for diffusion M v,,I, m3 volume of a supercage, m9 overlapping factor incomplete gamma function,

m

r(m, x) = J tm-le-' D

dt

jump length, m mean square jump length, m2 jump frequency, 11s mean life time on an adsorption site, s pore filling factor permeability of the zeolite membrane, mol m/(ma s Pa)

.

References [I] W. Hpelderich, in Zeolites: Facts, Figures, Future (Eds. P.A. Jacobs and R.A van Santen),

Elsevier, Amsterdam, 1989, p.69

"4

Ch.N. Satterfield, Hetemgeneous Catalysis an Practice, McGraw-Hill, New York, 1980, p.171

[3] W. Schirmer,I(.Fiedler, € Stach, I. M. Suckow, in Zeolites as Catulgsts, Sorbents and Detergent Builders (Eds. H.G. Karge and J. Weitksmp), Elsevier, Amsterdam, 1989, p.439 [4]

F.R. l'rouw, L.E. Iton, paper 142 in 'Zeolites for the nineties' presented during 8th IZC Amsterdam 1989

136

K.Dahlke, G.Emig and E. Aust

[5] K.-P. Schroder, J. Sauer, M. Biilow, paper 144 in ’Zeolites for the nineties’ presented during 8th IZC Amsterdam 1989 [6] E. Cohen de Lara, R.Kahn, A.M. Goulay, M.Lebars, in Zeolites: Facts, Figures, Future (Eds. P.A. Jacobs and R.A van Santen), Elsevier, Amsterdam, 1989, p.753 [7] S . Yashonath, J.Phys.Chem., 03 (1991) 5877 [8] S.D. Pickett, A.K. Nowak, A.K. Cheetham, J.M. Thomas, in Recent Aduawes i n Zeolite Science (Eds. J. Klinowski and P.J. Bmie), Elsevier, Amsterdam, 1989, p.253 [9] B. Grauert, K. Fiedler, H. Stach, J. Jhchen, in Zeolites: Facts, Figures, Future (Eds. P.A. Jacobs and R.A van Santen), Elsevier, Amsterdam, 1989, p.711 (101 D.M. Ruthven, Canad. J. Chem., 52 (1974) 3523 [ll]D. Theodorou, J. Wei, J.Catal., 83 (1983) 205

[I21 M.G. Palekar, R.A Rajadhyaksha, Chem.Eng. Sci., 40, (1985) 1085 [13] Chr.Forste, A. Germanus, J. Kiirger, H. Pfeifer, J. Caro, W. Pilz, A. Zikanova, J.C.S. Faraday Trans. I, 83(8) (1987) 2301 [14] J. Kiirger, M. Petzold, H. Pfeifer, J. Weitkamp, St. Emst, t o be published in J. Catal. [15] E. Aust, K. Dahlke, G. Emig, J. Catal, 116 (1989) 179 [16] K. Dahlke, G. Emig, Catalysis Today 8 (1991) 439 [17] D.M. Ruthven, M. Eic, E. Richards, Zeolites 11 (1991) 647 [18] K. Beschmann, G.T. Kokotailo, L. Riekert, Chem. Eng. Process. 22 (1987) 223 [19] J, Kiirger, H. Pfeifer, M. Rauscher, A. Walter, J.C.S. Faraday Trans. I, 76 (1980) 717 [20] J. Caro, M. Biilow, W. Schirmer, J. Kiirger, W. Heink, H. Pfeifer, J.C.S. Faraday Trans. I, 81 (1985) 2541 [21] D.M. Ruthven; Principles of Adsorption and Adsorption Processes, John Wiley & Sons,New York, 1984, p.46 [22] H. Pfeifer, D. Fkeude, J. Kiirger, in Catalysis and Adsorption by Zeolites (Eds. G. OhImam, H. Pfeifer and R.Fricke), Elsevier, Amsterdam, 1991, p.89 [23] J. Kiirger, H. Pfeifer, D. F’reude, J. Caro, M. Biilow, G. Ohlmann, in New Developments in Zeolite Science and Tec~nology(Eds. Y. Murakami, A.Iijima and J.W. Ward), Elsevier, Amsterdam, 1987, p.633 Pfeifer, I. in Catalysis and Adsorption by [24] H. Jobic, M. Bke, J. Caro, M. Biilow, J. Kiirger, € Zeolites (Eda. G. Ohlmann, H. Pfeifer and R. Fricke), Elsevier, Amsterdam, 1991, p.445 [25] Chr. Forste,J. Kiirger, H. Pfeifer, L. Riekert, M. Biilow, A. Zikanova, J.C.S. Faraday Trans. I, 8615) (1990) 881 [26] E.R.Geus, A.E. Jansen, J.C. Jansen, J. Schoomann, H. van Bekkum, in Catalysis and Adsorption by Zeolites (Eds. G. Ohlmann, H. Pfeifer and R.Fricke), Elsevier, Amsterdam, 1991, p.457