Crystallite-pore network model of transport and reaction of multicomponent gas mixtures in polycrystalline microporous media

Accepted Manuscript Crystallite-pore network model of transport and reaction of multicomponent gas mixtures in polycrystalline microporous media Wenjin Ding, Hui Li, Peter Pfeifer, Roland Dittmeyer PII: DOI: Reference:

S1385-8947(14)00656-1 http://dx.doi.org/10.1016/j.cej.2014.05.081 CEJ 12175

To appear in:

Chemical Engineering Journal

Received Date: Revised Date: Accepted Date:

19 February 2014 17 May 2014 19 May 2014

Please cite this article as: W. Ding, H. Li, P. Pfeifer, R. Dittmeyer, Crystallite-pore network model of transport and reaction of multicomponent gas mixtures in polycrystalline microporous media, Chemical Engineering Journal (2014), doi: http://dx.doi.org/10.1016/j.cej.2014.05.081

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Crystallite-pore network model of transport and reaction of multicomponent gas mixtures in polycrystalline microporous media Wenjin Dinga, Hui Lia, Peter Pfeifera, Roland Dittmeyera,b,* a

b

Institute for Micro Process Engineering, Institute of Catalysis Research and Technology, Karlsruhe

Institute of Technology, Eggenstein-Leopoldshafen, D76344, Germany *Corresponding author. Tel.: +49-721-608-23114; Fax: +49-721-608-23186. E-mail address: [email protected] (R. Dittmeyer).

GRAPHICAL ABSTRACT Crystallite-pore network model representing polycrystalline microporous media (left), in which the crystallite orientation is described by two angles Ψ and Χ (right).

ABSTRACT A three-dimensional pore network model has been developed to simulate anisotropic multicomponent diffusion and reaction in polycrystalline microporous media with coexisting intracrystalline micropores and intercrystalline mesopores (i.e., defects). Transport in these pores is modeled with the generalized Maxwell-Stefan surface diffusion model proposed by Krishna and co-workers (1990) and the Knudsen diffusion model, respectively. A new feature highlight of this model is the representation of polycrystalline media with a crystallite-pore network model. In contrast to previous pore network models, the crystallite-pore network model has the novel aspect of modeling the anisotropic transport inside the crystallites forming a polycrystalline layer by assigning to every crystallite two parameters to describe its 1

orientation. The model was applied to simulate xylene isomerization in a polycrystalline ZSM5 zeolite membrane, which had been experimentally investigated in a Wicke-Kallenbach cell by Haag et al. (2006). First, their experimental data were used to estimate adsorption and diffusion parameters of the xylene isomers in the ZSM-5 membrane via fitting single-gas permeance data of the xylene isomers. Second, adopting these parameters, the experimental data for xylene isomerization were used to determine kinetic parameters for xylene isomerization in the ZSM-5 membrane. Finally, effects of selected structural parameters - concentration of defects, connectivity of defects, crystallite orientation, and crystallite size - were investigated using the obtained adsorption, diffusion, and reaction parameters. The simulation results show that high selectivity towards p-xylene requires a low concentration of defects in the polycrystalline layer and a low loading of xylene isomers in the membrane. The novel crystallite-pore network model is also applicable to many other reaction systems. Keywords: Pore network model; Surface diffusion; ZSM-5 membrane; Xylene isomerization.

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1. Introduction Rational catalyst design methodology, which combines computational and experimental approaches, is strongly demanded to reduce the costs of development and improvement of new catalysts. In the last three decades, much attention has been paid to the development of porous heterogeneous catalysts via computer-aided optimization of their pore system, since diffusion and reaction processes can be affected significantly by the catalyst pore system. Pore network models have proven to be a powerful tool to study the effect of the pore system on the catalytic and separation performance of porous media [1-4]. In 1997 Rieckmann and Keil [2] developed a three-dimensional cubic micro-macro pore network model to simulate transport and reaction in the bimodal pore system of pelletized catalysts. Multicomponent transport in a single pore of the pore network was modeled by the dusty-gas model, which combined the contributions of Knudsen diffusion, molecular diffusion and viscous fluxes. As an example, this model was applied in modeling the deactivation of a pelletized ZSM-5 catalyst due to coke formation. In 2008 Chen et al. [4] developed a three-dimensional pore network model to simulate transport and separation of binary gaseous mixtures in amorphous microporous membranes. Compared to Rieckmann and Keil’s model, the contributions of hindered and Knudsen diffusion as well as the viscous flux were considered in modeling multicomponent transport of gaseous mixtures in micropores. The results of that article showed that the model was able to predict the single-gas permeances and the ideal selectivity of a silicon-carbide membrane for a helium-argon system. In recent years, the interest in polycrystalline microporous media for catalysis and separation has grown due to their potential for application in a wide range of industrial processes [5-8]. For instance, due to excellent thermal and chemical stability, polycrystalline microporous zeolite membranes have been applied in fuel cells [7] and membrane reactors [6,8], including catalysis, high temperature separation, and combined catalysis and separation. In order to optimize the separation performance of the zeolite membranes, several one-dimensional models, for example reported in Refs. [9-10], have been used to simulate transport processes in them under the assumption that they are isotropic and homogeneous (pseudohomogeneous). For example, Van de Graaf et al. [9] used a one-dimensional model to simulate permeation of binary gas mixtures of light alkanes through a silicalite-1 membrane with randomly intergrown crystallites. In that model, multicomponent diffusion in the microporous membrane was simulated with the generalized Maxwell-Stefan Surface Diffusion Model (SDM) proposed by Krishna and co-workers [11]. The simulation results

3

show that the model is able to predict the permeances of methane and ethane in binary gas mixtures through the silicalite-1 membrane. In 2003 Lai et al. [12] synthesized a b-oriented ZSM-5 membrane and found that it had a superior performance for separation of organic mixtures, such as xylene isomers, compared to conventional membranes with randomly intergrown crystallites. This implies that the orientation of the crystallites can influence the transport in the polycrystalline membranes significantly. The three-dimensional pore network models for the amorphous media as well as the one-dimensional models for the polycrystalline microporous media fail to simulate such anisotropic polycrystalline membranes. In order to simulate the effects of the crystallite orientation and size on the separation and catalytic performance of the polycrystalline media, a more realistic description of the pore network is required. In this work, we propose a crystallite-pore network model to simulate the polycrystalline structure of polycrystalline microporous media. Compared to previous pore network models, this model is able to simulate anisotropy and heterogeneity of such media. Based on this pore network model, a simulation model has been developed to simulate transport and reaction of multicomponent gas mixtures in the pore space of polycrystalline microporous media. In this model, multicomponent diffusion in the intracrystalline micropores and intercrystalline mesopores (defects) is modeled with SDM and Knudsen Diffusion Model (KDM), respectively. Reaction represented by any type of kinetic expressions, e.g., a nonlinear Langmuir-Hinshelwood kinetics model, is allowed so that this model is applicable to a variety of practical problems. The simulation model was applied to simulate xylene isomerization in a ZSM-5 zeolite membrane, which had been investigated by Haag et al. [13] experimentally in a WickeKallenbach cell. Using the experimental data determined by Haag et al. [13], the diffusion and adsorption parameters of the xylene isomers in the ZSM-5 membrane were estimated via fitting the data of single-gas permeances. Based on the obtained parameters, the kinetic parameters of the xylene isomerization were estimated via fitting the xylene isomerization data. Finally, some selected structural parameters - concentration of defects, connectivity of defects, crystallite orientation, and crystallite size - were varied to study their effects on the separation and reaction performance of the ZSM-5 membrane.

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2. Model development 2.1. Crystallite-pore network model for polycrystalline microporous media A polycrystalline microporous medium such as a ZSM-5 zeolite membrane is represented schematically in Fig. 1. The microporous crystallites, in which the intracrystalline micropores exist, are surrounded by intercrystalline mesopores (i.e., defects such as gaps, pinholes and cracks). In order to obtain defect-free membranes with excellent separation performance, defects should be avoided or controlled at a low concentration in membrane preparation via special treatments such as filling the defects by a post-synthetic coking treatment [14].

Defects

Fig. 1. Schematic showing a polycrystalline microporous membrane.

Different from amorphous microporous materials such as silica membranes, many polycrystalline microporous membranes have an anisotropic pore system with micropores of different structure and/or size in different orientation, e.g., ZSM-5 crystallites have zig-zag and straight intracrystalline micropores in a- and b-direction, respectively, as shown in Fig. 2. As a result, there has been intensive interest by the research community to synthesize zeolite membranes of oriented structure in order to obtain better permeation and separation performance unavailable from the randomly oriented zeolite membranes, e.g., preparation of a b-oriented ZSM-5 zeolite membrane by Lai et al. [12]. In order to simulate the complex pore space in the polycrystalline membranes as shown in Fig. 1, we developed a crystallite-pore network model by including distributed nodes representing crystallites in the conventional cubic pore network model. As a sample, a crystallite-pore network with 8 crystallites (2×2×2,

) is illustrated in Fig. 3. Considering that

crystallites grow together to form the polycrystalline membrane, the adjacent nodes representing different crystallites are connected with each other via the interface nodes ( ),

5

b

a c

Fig. 2: Three-dimensional microstructure of a ZSM-5 crystallite, which has zig-zag and straight micropores in a- and b-direction, respectively (adopted from Koriabkina [15]).

as shown in the exemplary crystallite-pore network. These nodes are also connected with the nodes of the intercrystalline pores ( ). In this way, the interface nodes establish a connection between the intra- and intercrystalline pore networks. The connectivity of the intercrystalline pores can be determined experimentally, e.g., by the method of Seaton [16]. In our model, the connectivity of the intercrystalline pore network is adjusted via giving a connection probability P to the intercrystalline pores (P = 100% indicates all intercrystalline pores are connected; P = 0 means all of them are isolated). In order to simulate anisotropy of the crystallites in the polycrystalline layer, every crystallite in the pore network model is allowed to have two angles Ψ and Χ to describe its individual orientation, as shown in Fig. 5. In reality the crystallites forming a polycrystalline layer are inter-grown to reduce the defects between them, i.e., the intercrystalline pores. The crystallite-pore network model is fully applicable to such membranes. Via the six interface nodes situated at the faces of each crystallite node a connection to the neighbor crystallite nodes and at the same time to the intercrystalline pore network is made. For the transition from the micropore system of one crystallite into those of its neighbors, distinct transport resistances can be included reflecting pore blockage or surface barrier effects. The presence of isolated intercrystalline pores is taken care of via the connection probability of the intercrystalline pores. Transport through the crystallites is only via surface diffusion in the zeolitic micropores as implemented in the crystallite nodes. Consequently, in contrast to previous pore networks, the crystallite-pore network model is advantageous with regard to the following aspects: (1) The effect of the defects can be investigated via varying their concentration, connectivity, and pore size;

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(2) The effect of anisotropy of crystallites, e.g., the dependence of the trans-membrane flux on the orientation of the crystallites, can be studied by varying the orientation of the crystallites; (3) The effect of crystallite size can be simulated; (4) The effect of the interaction between intra- and intercrystalline pore systems can be simulated; (5) Blocking on the crystallite surface or in intracrystalline pores due to coking (catalyst deactivation) can also be simulated; (6) As an extension on the network model, secondary pore networks are allowed to replace the nodes representing the crystallites so that the concentration distribution of reactants and products within crystallites can be simulated.

Fig. 3: Crystallite-pore network model with the size 2×2×2 crystallite³. : Intracrystalline micropores : Intercrystalline mesopores (defects) : nodes representing crystallites; : interfaces nodes; : nodes of intercrystalline pores.

2.2. Mathematical model A Wicke-Kallenbach (WK) cell is a simple and common setup for studying transport and separation in porous membranes. Fig. 4 shows the geometry of the cell that was used by Haag et al. [13] as a catalytic membrane reactor to test the performance of a ZSM-5 membrane for xylene isomerization. The ZSM-5 membrane was prepared on a porous sinter metal support with an excellent mechanical strength but also a high porosity and a large pore size to ensure negligible transport resistance. Details on the experimental setup and the operating conditions can be found in the article by Haag et al. [13]. In this study, our

7

crystallite-pore network model was applied to simulate diffusion and reaction in the pore space of the ZSM-5 membrane. For the modeling both compartments were treated as being well mixed to avoid having to consider concentration gradients along the membrane surface. Such gradients would have to be determined by linking the crystallite-pore network model to computational fluid dynamics methods, which would dramatically increase the computational effort. Although in fact no information is available about the specific flow patterns in this cell, the assumption of uniform concentrations can be justified by the fact that only data from experiments at low conversion (< 5%) were used for the modeling. Hence variations of the concentration along the membrane surface should not exceed that level and may therefore safely be ignored for the purpose of this paper. The defects typically found in ZSM-5 membranes vary with the size of the crystallites and generally have a size of 2-20 nm. The Knudsen diffusion model (KDM) was used to simulate the diffusion in these defects in this work, since the flux by surface diffusion is assumed to be negligible compared to the flux by Knudsen diffusion given the expected low surface coverage in the defect pores, particularly at higher temperatures. The flow of species i through a single intercrystalline mesopore is calculated by:

J inter,i

ci 

 p y  = − Ainter DK ,i ∇ total i   RT 

with the cross-section area of an intercrystalline pore: Ainter = π

(1)

d 2p ,inter and the Knudsen 4

diffusion coefficient of species i :

DK , i =

d p ,inter 3

8RT . πM i

(2)

8

Fig. 4: Design and geometry of the Wicke-Kallenbach cell with installed zeolite membrane used by Haag et al. [13] (adapted from unpublished project report).

Most zeolites have intracrystalline pores with a diameter smaller than 1 nm, e.g., the intracrystalline pores in the b-direction of ZSM-5 crystallites have a diameter of 5.6 Å. Due to their narrow pore space, configurational diffusion such as surface diffusion and hindered diffusion is the dominant mechanism of transport in these micropores. It was reported by Van de Graaf et al. [9] that the generalized Maxwell-Stefan surface diffusion model (SDM) without considering hindered diffusion could successfully predict the binary permeation of ethane/methane and propane/methane through a silicalite-1 membrane (silicalite-1 has the same microstructure as ZSM-5). In our simulation model, SDM was therefore used to simulate diffusion in the intracrystalline micropores. Moreover, only the flux in the b-direction was considered, since diffusion measurements with pulsed field gradient-nuclear magnetic resonance (PFG-NMR) spectroscopy by Hong et al. [17] had indicated that the diffusivities of molecules in a- and c-direction are much lower than that in b-direction. SDM considers the chemical potential gradient as the driving force. Its mathematical expression for a multicomponent system is represented as follows:

−

θi RT

q j N iS − qi N Sj N iS + , S qsat ,i ⋅ ρ memb ⋅ DiS j =1 qsat , i ⋅ qsat , j ⋅ ρ memb ⋅ Di , j n

∇µi = ∑

(3)

where DiS, j and DiS are mutual Maxwell-Stefan surface diffusion coefficient and corrected surface diffusion coefficient, respectively. The mutual Maxwell-Stefan surface diffusion coefficient describes the friction between the adsorbed molecules, while the corrected surface diffusion coefficient represents the friction between the adsorbed molecules and the adsorption sites on the surface. The corrected surface diffusion coefficient can be 9

determined experimentally from singe-gas permeances [9]. The mutual Maxwell-Stefan surface diffusion coefficient cannot be determined directly from experiments but calculated by the following equations [11, 18]:

qi D Sj ,i = q j DiS, j = (q j DiS,i ) i q

(qi + q j )

q j qi + q j

θi

DiS,i =

1 S self , i

D

−

(

⋅ (qi D Sj , j )

1 DiS

)

,

(4)

(5)

,

S where DiS,i and Dself , i represent the Maxwell-Stefan self-exchange surface diffusivity and self-

diffusivity, respectively. Eq. (5) shows that the self-diffusivity must be known for the calculation of the mutual Maxwell-Stefan surface diffusivity. The self-diffusivity can be measured by PFG-NMR spectroscopy or quasi-elastic neutron scattering (QENS), or estimated by molecular dynamics simulations (MD). Unfortunately, the self-diffusivities of the xylene isomers in ZSM-5 are not available in the literature. Therefore, following the method proposed by Krishna [11], the mutual Maxwell-Stefan surface diffusivities were calculated replacing the Maxwell-Stefan self-exchange surface diffusivities in Eq. (4) with the corrected surface diffusivities:

qi D Sj ,i = q j DiS, j = (q j D0S,i ) i q

(qi + q j )

⋅ (qi D0S, j )

qj

(qi + q j )

.

(6)

The chemical potential gradient in Eq. (3) can be related to the surface coverage gradient by the thermodynamic factor Γ :

θi RT

n

∇µi = ∑ Γi , j∇θ i .

(7)

j =1

Substituting (7) into (3), the following expression in matrix notation is obtained to calculate the surface fluxes, if the species are assumed to have the same saturation loading qsat :

N S = − ρ memb qsat D S ∇θ .

(8)

In this work, linear concentration profiles in the crystallites were assumed for the calculation of the coverage gradients of the species. The matrix of the surface diffusivity can be calculated with the average coverage of species θ as follows: (9)

D S = B −1Γ , 10

with Bi , j

 θi  − DS , i, j  = 1 n θj  S +∑ S , j ≠ i Di , j  Di j =1 

for

i ≠ j;

for

i = j.

(10)

If the multicomponent adsorption isotherm can be modeled with the extended Langmuir equations, the thermodynamic factor between species i and j is rewritten with the following equations:

 θi n   1 − ∑θ j ∂ ln pi  j =1 Γi , j ≡ θi = ∂θ j 1 + θni  θj  1− ∑ j =1 

i≠ j

i, j = 1,2, n.

(11)

i= j

where n is the species number in the gas mixture. If the kinetic diameters of molecules are close to or slightly larger than the pore diameter, s

mutual passage of molecules is excluded (i.e. Di , j → 0 ), particularly when the loading on the surface inside the intracrystalline pores is high. This special transport mechanism is known as single-file diffusion (SFD). Detailed introduction about SFD can be found in the book of Kärger et al. [19]. Since p-, o- and m-xylenes have a diameter of 5.8, 6.8, and 6.8 Å, respectively, the effect of SFD was observed in the separation of xylene isomers with silicalite-1 membranes in experiments by Baertsch et al. [20] and Xomeritakis et al. [21]. They have used SFD to explain the absence of any separation on gas mixtures of xylene isomers with silicalite-1 membranes. The simulation results of Van den Broeke et al. [10] showed that SFD could predict the separation factor for binary mixtures of n-butane and i-butane through a Silicalite-1 membrane well. Thus, SFD was used to simulate transport of the xylene isomers in the intracrystalline micropores of ZSM-5 crystallites in this work. When

( )

the single-file surface diffusion occurs, the elements of the matrix B S

0 Bi−, 1j =  S  Di

for for

i ≠ j; i = j.

−1

are:

(12)

Finally, the flow of species i through one intracrystalline pore can be calculated from the flux of species i ( N iS ) with the following equation:

11

J intra,i = −αAintra N iS ,

(13)

where the cross-sectional area of the intracrystalline pore Aintra = π

d 2p ,intra and the factor α 4

describes the influence of the crystallite orientation on the transport in the crystallites. The angles Ψ and Χ shown in Fig. 5 represent the angle of the b-direction of a crystallite with the xy-plane and the angle of the projection line of b-direction on the xy-plane with the x-coordinate, respectively. Based on these two parameters, the factor α representing the distribution of the flow in different coordinates can be calculated as follows:

cos 2 ψ cos 2 χ  α =  cos 2 ψ sin 2 χ  sin 2 ψ 

x − coordinate , y − coordinate , z − coordinate .

(14)

The angles Ψ and Χ can be determined experimentally either from analyzing SEM micrographs or by using X-ray diffraction pole-figure texture analysis. The latter method has been adopted, e.g., by Lovallo et al. [22] to determine the crystal orientation of MFI-type membranes. In a practical membrane, in general not all crystallites will have the same orientation. Depending on the preparation method used random orientation or a preferred orientation, yet with some statistical variation may prevail. This can be treated in the model by introducing distributions for the angles Ψ and Χ which, however, must be known from membrane characterization.

Fig. 5: One Crystallite with angles Ψ and Χ surrounded by the intercrystalline mesopores.

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Rieckmann and Keil [2] pointed out that the porosity in a pore network without micro and macro scaling is orders of magnitude less than that of a real porous catalyst. Here, the porosity represents the ratio of the pore volume and the network volume. If the pore size and length do not vary, the porosity is identical to the pore concentration. In order to obtain realistic prediction results, the flows in the inter- and intracrystalline pores therefore were scaled by the factors ς inter and ς intra following Rieckmann and Keil’s method:

′ ,i = J inter,i J inter

Vinter, memb = J inter,i ⋅ ς inter , Vinter,net

′ ,i = J intra,i J intra

Vintra, memb = J intra,i ⋅ ς intra . Vintra,net

(15) (16)

Due to the high specific internal surface area of microporous crystallites, it can be assumed that most of the catalytic active sites are found within the crystallites. Thus, we can assume in the crystallite-pore network model that the xylene isomerization takes place only in the intracrystalline micropores within the ZSM-5 crystallites. According to the component material balance of species i for the node representing a crystallite, it holds:

∑ J′

ip intra,i

+ mcry (ν ⋅ r )i = 0

ip ip ∈ I

(17)

where I represents all the pores connected at the node representing the crystallite, and

(ν ⋅ r )i

is the reaction rate of species i .

Under the assumption of zero coverage gradients within the crystallites, the vector of the reaction rates

r in a crystallite can be calculated with the coverage of the species at the

node representing this crystallite:

r = r (θcry ,i , i = 1,, n ) .

(18)

It is assumed that no reaction takes place in the intercrystalline mesopores. According to the component material balances for nodes representing intercrystalline mesopores and interfaces, the flows of species entering the node must therefore be equal to the flows of species leaving the node. Thus, the following equation holds at each node representing an intercrystalline pore or an interface:

∑ J′

ip

i

= 0.

(19)

ip ip∈I

13

Due to the much larger pores in the support ( d p ≥ 250 nm), the transport resistance in the support was neglected in this work. As introduced earlier, concentration gradients in the permeate and retentate compartments are not considered in the model. However, the existence of a mass transfer resistance in the boundary layers between the membrane surface and the bulk gas phase was included. Consequently, the flow of species i from the permeate and retentate bulk gas phase through the boundary layers into the outside nodes of the network is calculated by: per  J inter,i ′ per = ς inter ⋅ Ainter ⋅ βiper (ciper − coutside ,i ) Permeate:  per , per per per ′ J = ς ⋅ A ⋅ β ( c − c ) intra,i intra intra i i outside , i 

(20)

ret  J inter,i ′ret = ς inter ⋅ Ainter ⋅ βiret (ciret − coutside ,i ) Retentate:  ret . ret ret ret ′ = ς intra ⋅ Aintra ⋅ βi (ci − coutside,i ) J intra,i

(22)

(21)

(23)

where β i is the mass transfer coefficient of species i . Given the low permeance of the zeolite membrane under study and the high gas flow velocity applied in both compartments, it was assumed that external mass transfer was not limiting the trans-membrane fluxes. The coefficient β i was therefore given a high value of 5 m/s for all species. Local blocking due to imperfect match of the pore systems of the zeolite and support layer is considered by introducing the proportion of blocked pores b in the above Equations: per per  J inter,i ′ per = (1 − binter ) ⋅ ς inter ⋅ Ainter ⋅ βiper (ciper − coutside ,i )

Permeate: 

per intra,i

J ′

per intra

= (1 − b

per

per i

) ⋅ ς intra ⋅ Aintra ⋅ βi (c

per outside,i

−c

)

ret ret  J inter,i ′ret = (1 − binter ) ⋅ ς inter ⋅ Ainter ⋅ βiret (ciret − coutside ,i )

Retentate: 

ret ret ′ret = (1 − bintra ) ⋅ ς intra ⋅ Aintra ⋅ βiret (ciret − coutside ,i ) J intra,i

,

(24) (25) (26)

.

(27)

A no-flux boundary condition (symmetry condition) was assumed for the other four surfaces of the network, namely,

J i′ = 0 .

(28)

According to the component material balances for the retentate and permeate compartments, the following equations apply:

′ yisweep − J ′per yiper − ΣJ i′ per = 0 , Permeate: J sweep

(29)

feed ′ yiret − ΣJ i′ret = 0 . Retentate: J ′feed yi − J ret

(30)

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Finally, assembling all the above equations for nodes in the network according to the component material balances, a large system of nonlinear equations is obtained:

F( c ) = 0 .

(31)

2.3. Implementation The simulation model reported in the last section was implemented in Matlab®. The large nonlinear equation system in (31) was solved with the function ‘fsolve’ from the Optimization Toolbox® in Matlab®. Its default algorithm – “trust-region dogleg” was chosen. In this algorithm, the trust region sub-problem, which decides whether the new result or worse than

ck +1

is better

c k , is solved via the dogleg approach. For a detailed mathematical description,

see Powell [23]. The implemented overall algorithm of the crystallite-pore network model is illustrated in Fig. 6.

Fig. 6: The overall algorithm of the crystallite-pore network model simulating diffusion and reaction in polycrystalline microporous media.

At first, the geometric parameters (e.g., network size, connection probability of intercrystalline pores, orientational angles) are read in to construct the crystallite-pore network. During the iteration process, the parameters for diffusion and reaction are read in to calculate the flows into the nodes of the network and the reactions in the nodes representing the crystallites. In the initial guess for the iteration process, we assume that the concentrations of the species in the permeate and retentate compartments are the same as in the sweep and feed gases, 15

respectively. In addition, the concentration distribution across the membrane is assumed to be linear. During each iteration step, the nonlinear equation system F( c ) was calculated using the diffusion and kinetic models introduced in the last sub-section. An improved guess was obtained by the trust region algorithm, while the trust region sub-problem was solved by the dogleg approach. This process stops when the component material balances at all nodes are fulfilled, i.e., the function value of

1 F(c ) 2

2

is smaller than the termination tolerance.

After that, in order to ensure the results are trustable, we check the component material balances at all nodes, i.e., whether each element in the final F (c ) is much smaller than its corresponding flows and/or reaction terms. Finally, using the obtained concentrations of species at every node, the fluxes of the species through the membrane can be calculated. If reactions take place in the membrane, conversion of feed gas and selectivities towards products can be predicted.

3. Application of the model to xylene isomerization over ZSM-5 membrane In order to verify the developed pore network model, this model was applied to simulate the xylene isomerization over a polycrystalline ZSM-5 membrane described by Haag et al. [13]. Their experimental data were used to estimate the diffusion and adsorption parameters of xylene isomers in the ZSM-5 membrane as well as the kinetic parameters of xylene isomerization. This system was selected because the conversion during the experiments was low and far from the equilibrium. Moreover, the xylene partial pressures in the feed were higher than in most other studies on xylene isomerization in membrane reactors.

3.1. Estimation of diffusion and adsorption parameters In the estimation of diffusion and adsorption parameters of xylene isomers, the geometric and operating parameters from the paper by Haag et al. [13] were used. The ZSM-5 membrane used by Haag et al. [13] has a surface area of 2.54 cm2 and a zeolite layer with thickness of 6 µm. The scanning electron microscope (SEM) image in that article showed that the ZSM-5 crystallites in the membrane had a size of ≈ 1×2×3 µm³. Thus, the unit pore length in the network was set to 1 µm. At a pressure of 1 bar, single gas permeation of xylene isomers was carried out by Haag et al. [13] at temperatures of 100 and 200 °C, respectively. The feed was supplied in liquid form at a flow rate of 0.0574 ml min-1 (20 °C). A nitrogen carrier gas flow of 50 ml min-1 (STP) was used to transport the evaporated feed to the retentate space. A nitrogen flow of 250 ml min-1 (STP) served as permeate sweep gas. 16

Besides these data, other parameters adopted from the paper by Haag et al. [13] are summarized in Table 1. The single-component adsorption isotherms for p- and o-xylene on a ZSM-5 membrane were carried out by Xomeritakis et al. [24] at a temperature of 100 °C. Via fitting their experimental data with Langmuir equations, we determined the adsorption parameters of p- and o-xylene in the ZSM-5 membrane at a temperature of 100 °C. The determined adsorption parameters are shown in Table 2 (for the fitting results, refer to Fig. A.1-2 in the appendix). Unfortunately, adsorption isotherms for m-xylene were not measured by Xomeritakis et al. [24]. Therefore, the adsorption constant of o-xylene at 100 °C was used as a first approximation also for mxylene in this work based on the similarity of their physical properties. For a unary system, only friction between molecules and the solid surface occurs. Eq. (8) is therefore simplified as follows:

N iS = − ρqsat

DiS ∇θi . 1 − θi

(32)

Table 1: Pore network properties used in the standard simulation.

Parameters

Values

Source

Number of nodes

175 (5×5×7)

Zeolite thickness [13]

Number of nodes representing crystallites

12 (2×2×3)

Zeolite thickness [13]

Unit pore length

1 µm

Crystallite size [13]

Mean size of intercrystalline mesopores d p ,inter

12 nm

Assumed

0.56 nm

Size of intracrystalline micropores d p ,intra

[15]

Porosity of intercrystalline mesopores ξ inter

9×10-3

Estimated

Connection probability of intercrystalline pores P

100 %

Assumed

Porosity of intracrystalline micropores ξ intra

0.2

mass transfer coefficient β

[31]

5

Assumed

per Fraction of blocked intercrystalline mesopores binter

95 %

Estimated

per Fraction of blocked intracrystalline micropores bintra

50 %

Estimated

1990 kg/m³

Density of ZSM-5 membrane ρ Angle Ψ

45°

Angle Χ

Rand(1) * × 360°

[13] Randomly intergrown Crystallites [13]

*Rand(1): Matlab function, which returns a pseudo-random value drawn from the standard uniform distribution on the open interval (0, 1). Estimated: chosen based on plausibility of parameter values and simulation results for single-gas permeation.

17

Table 2: Adsorption constants of xylene isomers at 100 °C

Adsorption constants at 100 °C

p-xylene

m-xylene

o-xylene

-1 K i , 373 K Pa

(1.00±0.10)×10-3

2.25×10-4

(2.25±0.40)×10-4

0.693

0.693

0.693

qsat

mol/kg

Using the parameters from Table 1 and adsorption parameters from Table 2, the corrected surface diffusion coefficients of the xylene isomers DiS at 100 and 200 °C were determined via fitting the predicted single gas permeances to the experimental data of Haag et al. [13]. The obtained parameter estimates are summarized in Tables 3 and 4, whereas the predicted and experimental single gas permeances are shown in Fig. A.3 in the appendix. For comparison with literature results, Table 3 also provides corrected surface diffusion coefficients (i.e., based on the chemical potential gradient) derived from zero-length-column studies by Brandani et al. [25] as well as uncorrected diffusion coefficients (i.e, based on the loading gradient) from uptake measurements by Zheng et al. [26]. Table 3: Corrected surface diffusion coefficients DiS of the xylene isomers at 100 °C and 200 °C obtained via fitting single gas permeance data from Haag et al. [13]; comparison to literature values, i.e., corrected surface diffusion coefficients at 200°C [25] and uncorrected diffusion coefficients at 100°C [26].

Diffusivities

T (°C)

p-xylene

m-xylene

o-xylene

(m2/s)

D ps : Dms : Dos or D p : Dm : Do

100

(4.69±0.47)×10-10

(2.20±0.33)×10-10

(1.99±0.30)×10-10

2.36:1.10:1

200

3.26×10-9

1.41×10-9

4.94×10-10

6.59:2.85:1

DiS [25]

200

3.0×10-12

5.6×10-13

3.0×10-13

10:1.86:1

Di [26]

100

4.7×10-17

3.8×10-19

6.3×10-18

7.46:0.06:1

DiS (this work)

Table 4: Activation energies of DiS and K i determined via fitting predicted single gas permeations of xylene isomers at 200 °C with experimental data of Haag et al [13].

Activation energy

p-xylene

m-xylene

o-xylene

E a ( D iS ) kJ/mol

28.8±1.6

27.6±2.8

13.5±5.5

E a ( K i ) kJ/mol

62±2

51±3

58±6

18

It turns out that the corrected surface diffusion coefficients of the xylene isomers obtained here are rather close to each other whereas in literature in some cases up to 3 orders higher diffusion coefficients for p-xylene have been found compared to m-xylene and o-xylene. However, an interpretation of the differences has to consider the deviant definition of the two types of diffusion coefficients. The corrected surface diffusion coefficients DiS defined by (32) are based on the chemical potential gradient as the driving force for transport, whereas the uncorrected diffusion coefficients Di are based on the loading gradient and are defined as follows:

N iS = − ρ Di ∇qi = − ρqsat Di ∇θi .

(33)

Equating (33) and (32), and rearranging leads to:

Di = DiS Γ =

DiS 1 = DiS ⋅ = DiS ⋅ (1 + K i pi ) , K p 1 − θi i i 1− 1 + K i pi

(34)

where Γ is the thermodynamic factor introduced in (7). A pronounced difference between the adsorption constants of p-xylene and the other isomers, as given in Table 2, can therefore cause a marked difference of the uncorrected diffusion coefficients of the xylene isomers even if the corrected diffusion coefficients are close to each other. The ratio of the corrected surface diffusion coefficients D ps : Dms : Dos estimated from the data at 200 °C turned out as 6.59:2.85:1. This is not far from the ratio based on the results of zero-length-column studies by Brandani et al. [25], who found D ps : Dms : Dos = 10:1.86:1 for 200°C. On the other hand, comparing the absolute values shows that the corrected surface diffusion coefficients from this work are orders of magnitude larger than those from literature. This may be caused by deviations of the assumed structural parameters from the real values, by inaccuracies of the adsorption constants of the xylene isomers, which have been adopted in the estimation of the corrected surface diffusion coefficients (in particular for m-xylene), and by a different size of the crystallites. Moreover, the present model ignores gradients of the surface coverage inside the crystallites, which may also contribute to errors in estimating the corrected surface diffusion coefficients. The activation energies of DiS and K i were determined via fitting the single gas permeance data of the xylene isomers at 200 °C by Haag et al. [13]. The results are shown in Table 4. The activation energies of DiS for p-, m-, and o-xylene are 28.8±1.6, 27.6±2.8 and 13.5±5.5 19

kJ/mol, respectively. They agree reasonably well with the zero length column (ZLC) studies of Brandani et al. [25] and with the frequency response studies of Song et al. [27], which both led to activation energies for the corrected surface diffusion coefficients of the xylene isomers in silicalite-1 in the temperature range of 100-400 °C within 20-30 kJ/mol. The studies of Wu et al. [28] and Xomeritakis et al. [24] showed that the activation energies of the adsorption constants of xylene isomers in ZSM-5 varied from 50 to 70 kJ/mol. The activation energies of

K i obtained in this work are 62±2, 51±3 and 58±6 kJ/mol for p-, m-, and o-xylene, respectively. The agreement of the estimated diffusion and adsorption parameters with the data of previous researchers provides evidence that the model can be an alternative tool in the investigation of diffusion and adsorption in the polycrystalline microporous media.

3.2. Estimation of kinetic parameters of xylene isomerization Haag et al. [13] have performed the xylene isomerization over a ZSM-5 membrane at a pressure of 1 bar and a temperature of 300°C. The feed was supplied in liquid form at a flow rate of 0.0574 ml min-1 (20 °C). A nitrogen carrier gas flow of 50 ml min-1 (STP) was used to transport the evaporated feed to the retentate space. A nitrogen flow of 100 ml min-1 (STP) served as permeate sweep gas. The general triangular scheme shown in Fig. 7 was found by Haag et al. [13] to describe the reaction kinetics of xylene isomerization in the ZSM-5 membrane best. They expressed the reaction rates of xylene isomerization based on the partial pressures. As opposed to this, we introduce the reaction rates here based on coverage as the reactions occur on the surface: p-xylene: r1 = k 2θ 2 + k6θ 3 − (k1 + k5 )θ1 ,

(35)

m-xylene: r2 = k1θ1 + k 4θ 3 − (k 2 + k3 )θ 2 ,

(36)

o-xylene: r3 = k5θ1 + k3θ 2 − (k 4 + k6 )θ 3 .

(37)

20

Fig. 7: Triangular reaction scheme of xylene isomerization used by Haag et al. [13].

Thermodynamic equilibrium constants for xylene isomerization have been determined by Chirico and Steele [29] and are presented as follows:

K 1, eq = K 2,eq = K 3,eq =

po − xylene = exp( −791.8 / T + 1575 × 10 2 / T 2 − 1417 × 10 4 / T 3 ) , pm − xylene pm− xylene

= exp( 0.693 + 4.792 / T + 37070 / T 2 − 5099 × 103 / T 3 ) ,

(39)

= exp( −0.693 + 787.1 / T − 1946 × 102 / T 2 + 1927 ×104 / T 3 ) .

(40)

p p − xylene

p p − xylene po − xylene

(38)

These expressions again are based on the partial pressures of the xylene isomers. Hence, they should be modified using the Langmuir equations before used in our model. The partial pressures in (38) - (40) can be replaced by:

θi =

K i pi n

1+ ∑ K j pj

⇒ pi =

θi

1

Ki

n

. (41)

1 − ∑θ i

j =1

j =1

So we have

K1,eq

  po − xylene  θ 3 1 = = n  p m− xylene K  3 1 − ∑θi j =1 

     

    1 θ  2  = θ3 ⋅ K2 , n  K2  θ 2 K3 1 − ∑θi   j =1  

(42)

  1 θ = 2 n  K2 1 − ∑θi  j =1 

     

  1  θ1 n  K1 1 − ∑θi  j =1 

(43)

K 2 ,eq =

p m − xylene p p − xylene

21

   = θ 2 ⋅ K1 ,  θ1 K 2  

K 3,eq =

p p − xylene po − xylene

    θ1 1   = n  K1   1 − ∑θi  j =1  

    1  θ3  = θ1 ⋅ K 3 . n  K3  θ 3 K1 1 − ∑θ i   j =1  

(44)

Finally, we get the thermodynamic equilibrium constants for xylene isomerization based on the coverages of the xylene isomers:

θ3 k3 K = = K1, eq ⋅ 3 , θ 2 k4 K2

(45)

θ 2 k1 K = = K 2, eq ⋅ 2 , θ1 k 2 K1

(46)

θ1 k6 K = = K3, eq ⋅ 1 . θ 3 k5 K3

(47)

We chose k1 , k3 , and k5 as kinetic parameters for fitting, so the other kinetic parameters could be calculated from (48) - (50).

 K  k 4 = k 3  K1,eq ⋅ 3  , K2  

(48)

 K  k 2 = k1  K 2 , eq ⋅ 2  , K1  

(49)

 K  k6 =  K 3,eq ⋅ 1  ⋅ k5 . K3  

(50)

For estimation of the kinetic parameters of xylene isomerization, the adsorption and diffusion parameters previously obtained and shown in Tables 2-4 were used. Via fitting to the experimental data of Haag et al. [13] on xylene isomerization, the kinetic parameters at 300 °C were obtained and are shown in Table 5. The fitting results are given in Table A.1 in the appendix. Table 5: Kinetics parameters determined via fitting with experimental data of the xylene isomerization [13] at 300 °C.

i -1 -1 k i ,573 K mol kg s

1

2

3

4

5

6

18.63

6.58

3.80

9.03

4.44

3.72

22

3.3. Investigation of the effects of structural parameters

The effects of the structural parameters of the polycrystalline H-ZSM5 layer on its performance for xylene isomerization were studied with the help of the simulation model. Two different cases were investigated: 1) the situation at low conversion corresponding to the experiments performed by Haag et al. [13], and 2) the situation at higher conversion, i.e., reached via a 100-fold increase of the contact time (membrane area). The observed trends were generally independent of the conversion level. An example is given for the influence of crystal orientation on selectivity and conversion in section 3.3.2.

3.3.1. Effects of concentration and connectivity of defects As mentioned above, Knudsen diffusion is the dominant mechanism of transport in the defects (intercrystalline mesopores). The defects have no separation for the xylene isomers, since the isomers all have the same molecular molar mass. They can be isolated or connected with other defects forming an intercrystalline mesopore network. The defect pore network also connects to the intracrystalline pore system through the interface nodes, bringing about the possibility to access the micropores inside the crystallites via the intercrystalline mesopore network. Hence the reaction and separation performance of a polycrystalline layer will be affected not only by the number of defects per unit area, i.e., its concentration, but also by the connectivity of the defects. The concentration of the defects, i.e., the intercrystalline porosity ξ inter was varied in the range of 1×10-2 to 1×10-7, whereas the connectivity of the defects was varied between 0 and 4.41. The connectivity is defined as:

C=

total number of intercryst alline pores total number of nodes of interfaces and intercryst alline pores

(51)

It gives the average number of intercrystalline mesopores connected on each node of the network. Practically, the connectivity is obtained by specifying a connection probability P of the intercrystalline mesopores during generation of the network. P ranges from 0 to 100%, where 0 means all defects are isolated and 100% all defects are connected. For all values in between, the connections between the defect nodes are of statistical nature. The connectivity 23

generally depends on the network size as well. For small networks, like used here, the connectivity of the resulting intercrystalline pore network for a given value of P is subject to statistical variation. This can be seen in Fig. 8a showing a plot of the connectivity C from (51) as a function of the connection probability P for a network according to Table 1. The error bars indicate the variation among five consecutive calculations (standard errors) with the same value of P. The statistical variation is also visible from Fig. 8b giving an example of how the connectivity of the defect pores influence selectivity and conversion in xylene isomerization when staring from o-xylene. Again five consecutive calculations with random assignment of the connections among the intercrystalline mesopores according to the specified connection probability were performed. In this case selectivity is more or less independent of the connectivity of the defects whereas conversion increases with increasing connectivity of the intercrystalline mesopores. 90

5

4

3 2 1

Selectivity to p-xylene [%]

Connectivity C / -

C = 0.0441P

4

3 60 2 30

0

1

0 0

20 40 60 80 100 Connection probability P / %

Conversion of o-xylene [%]

b)

a)

0 0

20 40 60 80 100 Connection probability P [%]

Fig. 8: a) Connectivity of intercrystalline mesopores as a function of the connection probability for a network according to Table 1. b) Dependence of p-xylene selectivity (solid line) and o-xylene conversion (dashed line) -3 at 300 °C on the connection probability. Porosity of intercrystalline mesopores: 9×10 . Error bars give the standard errors for five consecutive calculations.

Fig. 9 shows the predicted selectivity towards p-xylene and conversion of m- or o-xylene at 300°C as a function of the intercrystalline porosity. The selectivity increases and the conversion decreases significantly with reduced intercrystalline porosity, when the intercrystalline porosity decreases from 1×10-2 to 1×10-4. This is because more molecules will diffuse through the membrane via the intracrystalline micropores when decreasing the number of defects. Lower permeance of the xylene isomers is obtained but higher selectivity towards p-xylene. When the intercrystalline porosity falls below 1×10-4, most of the xylene isomers diffuse through the membrane via the intracrystalline micropores. Thus, the selectivity towards p-xylene and the conversion of m- or o-xylene does no longer vary with decreasing concentration of the defects.

24

60

2 30

0 1E-08

Selectivity to p-xylene [%]

3

Conversion of m-xylene [%]

Selectivity to p-xylene [%]

a)

b) 3

30

2

15

1

0 1E-08

1E-05 1E-02 Intercrystalline porosity [-]

Conversion of o-xylene [%]

45

90

1 1E-05 1E-02 Intercrystalline porosity [-]

Fig. 9: Dependence of p-xylene selectivity (solid line) and m- or o-xylene conversion (dashed line) at 300 °C on the intercrystalline porosity: a) m-xylene as Feed; b) o-xylene as Feed. Connection probability of intercrystalline mesopores P: 100 %, i.e. C = 4.41.

Consequently, the simulation results suggest a careful control of the concentration of defects by appropriate actions during membrane preparation or a post synthetic treatment to reduce it afterwards in order to improve the selectivity towards the product p-xylene.

3.3.2. Effect of crystallite orientation Due to the anisotropic pore system, crystallite orientation is thought to play an important role on the catalytic performance of polycrystalline membranes, at least for high-quality almost defect-free layers, in which the reactants diffuse mainly through the intracrystalline micropores. In the investigation of the effects of crystallite orientation on xylene isomerization, the intercrystalline porosity was set to a low level of 1×10-7 to ensure that this condition is fulfilled. Random variation of the orientation angle of the crystallites was allowed for a more realistic representation of a polycrystalline layer. Fig. 10 summarizes the effects of crystal orientation on selectivity and conversion for isomerization of m-xylene and o-xylene at low and high conversion levels for a reaction temperature of 300°C. In each case a normal distribution of the orientation angle Ψ (see Fig. 5) with a standard deviation of 5° was assumed and five consecutive simulations were performed. The error bars indicate the randomness effect. Note that Ψ = 0° means that the b-direction of the ZSM-5 crystallites is parallel to the membrane surface whereas for Ψ = 90° it is perpendicular to the membrane surface. Both for low and high conversion levels, the selectivity to p-xylene decreases and the conversion of m- and o-xylene increases with increasing angle Ψ. The increase of

25

conversion is caused by faster reactant supply into the catalytic membrane at higher angle Ψ. For Ψ = 90° the trans-membrane flux is maximized as the straight pores in the crystallites, which offer high diffusivity, run parallel to the membrane thickness. Hence, with increasing angle Ψ the concentration of xylene isomers inside the membrane rises. At low angle, high pxylene selectivity is obtained, as the reaction is limited by reactant supply so that the loading remains below the limiting value where single file diffusion occurs. In this regime, p-xylene formed by the isomerization reaction quickly diffuses out of the membrane, giving rise to high p-xylene selectivity. At higher angle Ψ, the diffusive flux into the membrane is increased. In turn, the concentration of the xylene isomers in the membrane gets larger and single-file diffusion gains control. In this regime, the faster moving p-xylene is slowed down by the slower moving o- and m-xylene and is increasingly converted back to m-xylene. This is indicated in Table 5 showing that the constant k1 describing the conversion of p-xylene to mxylene is much larger than all other rate constants. 100

4

45

4

3

60 2 40 1

20 0

Selectivity to p-xylene [%]

80

3 30 2 15

0 0

1

0

0

30 60 90 Crystal orientation Ψ [degree]

0

80

30 60 90 Crystal orientation Ψ [degree]

60

d) Selectivity to p-xylene or Conversion of o-xylene [%]

c) Selectivity to p-xylene or conversion of m-xylene [%]

Conversion of o-xylene [%]

b) Conversion of m-xylene [%]

Selectivity to p-xylene [%]

a)

60

40

20

0

40

20

0 0

30 60 90 Crystal orientation Ψ [degree]

0

30 60 Crystal orientation Ψ [degree]

90

Fig. 9: Dependence of the selectivity to p-xylene (solid line) and the conversion (dashed line) at 300 °C on the angle Ψ describing the crystallite orientation: a) m-xylene as Feed, low conversion; b) o-xylene as Feed, low conversion; c) m-xylene as Feed, high conversion; d) o-xylene as Feed, high conversion.

26

These results indicate that high selectivity to p-xylene can only be obtained when the loading of the xylene isomers in the membrane is low. Additional calculations were performed to study whether a layer with randomly oriented crystallites (i.e., Ψ = Rand(1)×90°) can be represented by a layer with a uniform orientation angle of Ψ = 45 °. Five consecutive calculations were carried out. The predicted conversions and selectivities as well as their standard errors are summarized in Table 6. For a uniform orientation angle of 45° slightly higher conversions and lower p-xylene selectivity are detected than for random orientation of the crystallites. However, differences are very moderate, also when compared to the standard errors. Hence, a uniform orientation angle of 45° appears to be valid as a first approximation of a layer with random orientation of the crystallites.

Table 6: Predicted xylene conversions and p-xylene selectivities for two cases: 1) all crystallites aligned at Ψ = 45° and 2) crystallites randomly orientated, i.e., Ψ = Rand(1)×90°.

Feed

Conversion (%)

m-xylene

o-xylene

S p− xylene %

S m− xylene %

S o−xylene %

Aligned

Random

Aligned

Random

Aligned

Random

Aligned

Random

1.82

1.67±0.09

72.5

73.3±1.1

-

-

27.5

26.7±1.1

31.9

31.4±0.2

47.4

49.3±0.6

-

-

52.6

50.7±0.6

1.55

1.40±0.12

31.1

31.4±0.4

68.9

68.6±0.4

-

-

51.3

49.4±1.3

23.2

23.6±0.1

76.8

76.4±0.1

-

-

3.3.3. Effect of crystallite size Besides crystallite orientation, the effect of the crystallite size on the catalytic performance of the ZSM-5 membrane was investigated. The intercrystalline porosity and the angle Ψ were set to 1×10-7 and 45°, respectively. The crystallite size was varied between 0.75 and 3 µm while maintaining the same thickness of the polycrystalline layer, which means the number of intracrystalline nodes on top of each other was varied accordingly. Interfacial resistances between the intergrown crystallites, e.g., due to pore blockage or surface barrier effects were neglected at this stage. Fig. 11 shows the selectivity to p-xylene and the conversions of m- and o-xylene at 300 °C as a function of the crystallite size. Selectivity to p-xylene decreases slightly and conversion increases with decreasing crystallite size. This can be explained by the fact that the concentration inside the crystallites is taken as uniform in the present model. Therefore, like 27

in a cascade of stirred tank reactors, conversion is increased with increasing number of units because on average the concentration becomes larger. Due to the reaction kinetics of xylene isomerization, the selectivity to p-xylene shows the opposite trend. In reality, however, for bulky molecules like xylene isomers interfacial effects are expected to become more important for the whole intracrystalline transport when the crystallites are smaller. In addition to pore blockage the interfacial effects also include a barrier of entrance into the intracrystalline pores and the existence of a physisorbed precursor state on the outside of the zeolite crystallite. For example, the experimental results of Zheng et al. [26] show that the diffusion coefficients of xylene isomers in smaller ZSM-5 crystallites are lower than those in large crystallites, in particular for p-xylene. This implies that the interfacial effects are more significant for smaller crystallites. For the calculations shown in Fig. 11, interfacial effects were ignored because no quantified information on the magnitude of the interfacial resistance is available. However, taking these effects into account could well lead to a reversed trend, i.e., higher conversion for larger crystals, because then fewer interfaces between the crystallites exist. To investigate this more in detail experimental data from polycrystalline layers with distinctly different size of the crystallites would be required as well as a more accurate model taking into account the interfacial effects. 90

40

2

30

1

0

0 0

1 2 3 Crystallite size [µm]

3

2 30 1

20

4

Conversion of o-xylene [%]

60

Selectivity to p-xylene [%]

b) 3

Conversion of m-xylene [%]

Selectivity to p-xylene [%]

a)

0 0

1 2 3 Crystallite size [µm]

4

Fig. 10: Dependence of p-xylene selectivity (solid line) and conversion (dashed line) at 300 °C on the crystallite size: a) m-xylene as Feed, low conversion; b) o-xylene as Feed, low conversion.

4. Conclusions The application of a new crystallite-pore network model for multicomponent diffusion and reaction in microporous media to xylene isomerization in a polycrystalline ZSM-5 zeolite membrane was demonstrated. The crystallite-pore network model was confirmed to be suitable to describe reaction and transport in the pore space of such polycrystalline

28

microporous media. Moreover, the model can be used for estimation of adsorption and diffusion parameters of xylene isomers and kinetic parameters of xylene isomerization in ZSM-5 membranes, which are difficult to determine experimentally. Effects of some structural parameters, such as the concentration and connectivity of intercrystalline defects and the crystallite orientation and size were investigated. The simulation results show that the intercrystalline pores and the crystallite orientation in ZSM-5 membranes may affect the catalytic performance significantly. Therefore, the concentration of defects in the ZSM-5 membrane must be controlled at a low level during membrane preparation or reduced afterwards in order to achieve high selectivity to p-xylene. Increasing the amount of b-oriented crystallites allows for an increased conversion of m- and o-xylene to p-xylene, but may also reduce the selectivity towards p-xylene if single-file diffusion control. If interfacial effects are ignored, the crystallite size has a minor effect when compared to intercrystalline defects and crystallite orientation. Future extension of the model will focus on overcoming the limitations of the current version, i.e., that interfacial effects are neglected and concentration gradients inside the crystallites cannot be resolved. This may lead to an over-prediction of the diffusion coefficients. A possible approach is using a secondary cubic pore network with several nodes (e.g., 5×5×5) to represent every crystallite. A further improvement on the modeling of transport in intracrystalline pores is the application of the ideal adsorbed solution theory (IAST) proposed by Myers and Prausnitz [30] for the modeling of multicomponent adsorption in order to simulate the coverage of components in a mixture more accurately, which is important for the SDM to simulate the multicomponent diffusion in the micropores. Moreover, a more accurate approach than Knudsen diffusion for modeling the transport through small mesoporous defects could be adopted by connecting to recent work by Bonilla and Bhatia [31] who developed an oscillator model from statistical-mechanical principles, which can predict diffusivities in nanopores more accurately than the Knudsen diffusion model.

Acknowledgments This research has been performed within the framework of the priority program 1570 of Deutsche Forschungsgemeinschaft (DFG): porous media with defined pore system in process engineering - modeling, application, synthesis. Wenjin Ding gratefully acknowledges Seungcheol Lee in Karlsruhe Institute of Technology, Michael Klumpp, Stephanie Reuß, and Wilhelm Schwieger in Friedrich-Alexander-University Erlangen-Nürnberg for their support and discussion. 29

Nomenclature A = area, m² b = fraction of blocked pores, c = concentration, mol/m³

c = vector of concentration C = connectivity of intercrystalline pores, d = diameter, m

D = diffusion coefficient, m²/s D = matrix of diffusion coefficients, m2/s

Ea = activation energy, kJ/mol J = molar flow rate, mol/s

J ′ = molar flow rate after scaling, mol/s

k = reaction rate constant, m3/kg/s

K = adsorption constant, 1/Pa m = mass, kg M = molar mass, kg/mol n = number of components, N = molar flux, mol/m²/s p = pressure, bar P = connection probability of intercrystalline pores, % q = loading, mol/kg

r = vector of reaction rates, mol/m3/s R = universal gas constant, J/mol/K S = selectivity, T = temperature, K y = mole fraction, -

30

Subscripts cry = crystallite eq = equilibrium feed = feed gas inter = intercrystalline intra = intracrystalline i, j = component number K = Knudsen diffusion memb = membrane net = network outside = outside surface of the network p = pore sat = saturation self = self-diffusion sweep = sweep gas Superscripts feed = feed gas ip = pores connected at the node representing a crystallite per = permeate side ret = retentate side sweep = sweep gas S = surface diffusion Greek letters

α = factor of flux distribution in x-, y- and z-coordinate, -

β = mass transfer coefficient, m/s

θ = coverage, µ = chemical potential, J/mol 31

ν = matrix of stoichiometric coefficients, ρ = density, kg/m³

ς = scaling of pore number, ψ = angle between the b-direction of crystal and xy-flat, ° χ = angle of the projection line of b-direction on the xy-flat with x-coordinate, ° Γ = matrix of thermodynamic factors

32

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34

Appendix

0.8 0.7 Loading q / mol/kg

0.6 0.5 0.4 0.3 0.2 0.1 0 0

5000

10000

15000

20000

25000

30000

pressure of p-xylene /Pa

Fig. A.1: Single-component adsorption isotherm of p-xylene in a ZSM-5 membrane at 100 °C. Symbols: experimental data of Xomeritakis et al. [22]; Line: Fitted data with the Langmuir equation.

0.7

Loading q / mol/kg

0.6 0.5 0.4 0.3 0.2 0.1 0 0

5000

10000

15000

20000

25000

30000

pressure of o-xylene /Pa Fig. A.2: Single-component adsorption isotherms of o-xylene in a ZSM-5 membrane at 100 °C. Symbols: experimental data of Xomeritakis et al. [22]; Line: Fitted data with the Langmuir equation.

35

8E-07 exp. 200 °C Sim. 200 °C Exp. 100 °C Sim. 100 °C

permeances / mol/(m²sPa)

7E-07 6E-07 5E-07 4E-07 3E-07 2E-07 1E-07 0E+00

p-xylene

m-xylene

o-xylene

Fig. A.3: Predicted and experimentally determined single gas permeances of xylene isomers at 100 and 200 °C. The experimental permeances have the measure errors of 6 %.

Table A.1: Comparison of predicted values with experimental data of the xylene isomerization in the ZSM5 membrane reactor at 300 °C. Error = (Sim. Value – Exp. Value)/ Exp. Value×100%.

Feed

Conversion (%)

S p−xylene %

S m −xylene %

S o−xylene %

Exp.

Sim.

Error

Exp.

Sim.

Error

Exp.

Sim.

Error

Exp.

Sim.

Error

p-xylene

5.6

5.9

+5%

-

-

-

82.6

81.2

-2%

17.4

18.8

+8%

m-xylene

2.2

2.0

-9%

63.6

64.4

+1%

-

-

-

36.4

35.6

-2%

o-xylene

2.6

2.5

-4%

30.8

29.5

-4%

69.2

70.5

+2%

-

-

-

36

HIGHLIGHTS •

Crystallite-pore network model is proposed to represent polycrystalline media.

•

Maxwell-Stefan surface diffusion model is used to simulate transport in micropores.

•

Reaction represented by any type of kinetic expressions is allowed in this model.

•

Structural effects of polycrystalline media on transport and catalysis are studied.

37