Integrated Stefan–Maxwell, Mean Field, and Single-Event

Article pubs.acs.org/JPCC

Integrated Stefan−Maxwell, Mean Field, and Single-Event Microkinetic Methodology for Simultaneous Diffusion and Reaction inside Microporous Materials B. D. Vandegehuchte,† I. R. Choudhury,†,§ J. W. Thybaut,*,† J. A. Martens,‡ and G. B. Marin† †

Laboratory for Chemical Technology, Chemical Engineering Department, Ghent University, Technologiepark 914, B-9052 Ghent, Belgium ‡ Center for Surface Chemistry and Catalysis, KU Leuven, Kasteelpark Arenberg 23, B-3001 Heverlee, Belgium S Supporting Information *

ABSTRACT: The assessment of intrinsic reaction kinetics in the presence of diffusion limitations within a porous material remains one of the key challenges within the field of catalysis. The model-guided design of medium-pore zeolite catalysts which typically give rise to mass transport limitations would offer a feasible alternative to conventional trial-and-error procedures. Intracrystalline diffusion limitations during n-hexane hydroconversion on Pt/H-ZSM5 were assessed using an integrated Stefan− Maxwell, mean field, and Single-Event MicroKinetic (SEMK) methodology. The former theory quantifies multicomponent diffusion through a microporous substituent from pure component properties, while framework parameters inherent to the ZSM5 topology are incorporated via a mean field approximation. The complex chemistry involved in n-hexane hydroconversion was described by an SEMK model which is based upon the reaction family concept. Model regression against experimental data resulted in excellent agreement between the model and experiment. In addition, the estimated values for, among others, the component diffusion coefficients were physically meaningful. A sensitivity analysis of the catalyst descriptors demonstrated that especially the total acid site concentration and the crystallite geometry impact the catalyst activity and product distribution, establishing them as critical catalyst design parameters.

1. INTRODUCTION

adsorption and shape selectivity, among other phenomena, could be unambiguously separated from the overall reaction kinetics as demonstrated in the large body of literature available on alkane hydroconversion modeling using single-event microkinetics.3,5−9 Through a mere sensitivity analysis of the model’s catalyst descriptors, the identification of optimal catalysts comes within reach, offering an alternative to exhaustive trial-and-error procedures often encountered in industry.4,10,11 A major challenge in catalysis remains to adequately describe the reaction kinetics in the presence of pronounced mass transport limitations in the catalyst micropores. The diffusion through a porous substituent occurring simultaneously

Reactor design and process optimization are nowadays guided by versatile reactor and kinetic models enabling comprehensive process analyses in the desired operating range, as well as during start-up and shut-down.1,2 For complex reaction networks involving thousands of elementary reaction steps and intermediates, various lumping methodologies were developed, each aiming at reducing the number of adjustable model parameters. The Single-Event MicroKinetic (SEMK) methodology based upon the reaction family concept distinguishes itself from lumped methodologies and allows an accurate capture of the fundamental reaction kinetics.3 Even more for catalytic reactions, the SEMK methodology allows a straightforward interpretation of the catalyst role within the overall kinetics via so-called “catalyst descriptors”, i.e., model parameters which account for the catalyst used.4 As a result, physical © 2014 American Chemical Society

Received: June 18, 2014 Revised: August 21, 2014 Published: August 21, 2014 22053

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was not observed. A rather simple correlation between the Fick diffusion coefficient and the total site occupancy was applied to maintain the focus of that study on determining the dominant shape-selective effects occurring during reaction. The present work further elaborates on this matter, aiming at a more fundamental description of simultaneous diffusion and reaction inside a catalyst crystallite based upon eq 1. Therefore, an integrated Stefan−Maxwell, mean field, and SEMK methodology is developed which unequivocally quantifies the catalyst role in the overall diffusion and reaction mechanisms and, hence, offers a maximum level of detailed insight into both phenomena via explicit analytical equations. The model is validated using the n-hexane hydroconversion data reported earlier.34 A comprehensive analysis of the catalyst descriptors impacting the sorbate diffusion behavior and catalyst performance is performed afterward. The latter demonstrates the versatility of the presented methodology in the “model-guided” design of potentially any microporous catalyst giving rise to intracrystalline diffusion limitations.

with reaction is quantified by Fick’s second law and enters as follows into a component mass balance over a catalyst crystallite: ∂Ci 1 ∂ ⎛ s ∂Ci ⎞ = s ⎜y Di ⎟ + Ri ∂t ∂y ⎠ y ∂y ⎝

(1)

Herein, Ci represents the intracrystalline concentration of component i, Ri its net production rate, and s the crystallite shape factor.12 The latter equals 0, 1, and 2 for respectively slab, cylindrical, and spherical geometries. A comprehensive assessment of multicomponent diffusion via Fick’s law is hampered by the complex dependence exhibited by the Fick diffusion coefficient Di on the diffusing mixture composition.13 Especially for diffusion through microporous sorbents where van der Waals interactions usually dominate over conventional bulk and Knudsen diffusion,14 this regime being denoted as “configurational diffusion”, various methodologies were developed to physically interpret the peculiar trends in Fick diffusion coefficients with varying sorbate concentrations. Some approaches make use of two- and three-dimensional lattice models combined with kinetic Monte Carlo (kMC) simulations to determine the movements of interacting sorbate molecules per time unit,15−19 while others use approximate analytical and numerical methods for often simple lattice models aiming at a qualitative assessment.14,20−23 The Stefan−Maxwell formulation for multicomponent diffusion, on the other hand, is a phenomenological theory which enables the description of multicomponent diffusion in terms of pure component parameters. The theory was intensively elaborated by Krishna and co-workers and has been reported in a large body of literature.24−28 Trends in diffusion coefficients as determined via the Stefan−Maxwell theory were found to agree well with molecular dynamics simulation results in case topology effects, and correlations in molecular motion play a somewhat minor role.26−29 One of the main advantages of the Stefan−Maxwell theory is that the diffusion coefficient of each species within a multicomponent mixture can be expressed as a function of single-component diffusion coefficients, also taking the sorbent structure into account. The effect of the pore channel connectivity, physisorption site occupancy, and heterogeneity on the sorbate diffusion behaviors was thoroughly investigated by Coppens and coworkers,30−32 who considered diffusion as an activated hopping mechanism between different sites located in a well-defined unit cell model. Application of a mean field approximation neglecting any correlation between successive hops led to a relatively simple expression for the self-diffusion coefficient. The combination of this mean field approximation with the Stefan−Maxwell theory was found to agree rather well with KMC simulation results in the case of ZSM5 materials containing a low to moderate number of acid sites.33 The integration of both approaches into a single methodology would enable assessment of multicomponent diffusion via a limited set of analytical equations explicitly containing catalyst descriptors similarly to those introduced in SEMK models to describe reaction kinetics. In a preceding SEMK modeling study,34 it was found that n-hexane hydroconversion on a highly siliceous Pt/H-ZSM5 was primarily subject to intracrystalline mass transport limitations while other forms of shape selectivity could be discarded. n-Hexane was found to be an ideally suited model molecule to probe the shape selectivity exhibited by ZSM5, as pronounced cracking of the feed isomers for a molecule as small as n-hexane

2. MODEL DEVELOPMENT 2.1. Simultaneous Diffusion and Reaction in a Catalyst Crystallite. The impact of diffusion limitations on the apparent reaction rates is commonly assessed by means of the dimensionless Thiele modulus.35 A preceding study found that the Thiele moduli of the elementary isomerization steps in n-hexane hydroconversion on Pt/H-ZSM5 can significantly exceed 1 depending on the reaction type and reactant alkane.34 The n-hexane hydroconversion network is depicted in Figure 1. Prior to any reaction within the catalyst pores, alkanes are stabilized with respect to the bulk phase by van der Waals interactions with the catalyst framework. Migration to and chemisorption onto a platinum site are instantaneously followed by dehydrogenation toward an alkene. The latter is susceptible to protonation on the acid sites, yielding a carbenium ion which,36 in turn, can undergo branching over protonated cyclopropane (PCP) transition states, methyl shifts (MSs), and β-scission (β) depending on its structure.6 Note that not all of the hydrogenation and dehydrogenation reactions are depicted in Figure 1 to avoid overloading the picture. Hydride shifts are not shown either. Specifically for ZSM5 frameworks, the transport of both dimethylbutanes through the catalyst pores is strongly, if not fully, hindered, resulting in extremely low yields for these isomers compared to those obtained over non-shape-selective catalysts.34,37,38 The 2-methylpentane/3-methylpentane yield ratio is substantially higher than would be expected from thermodynamic equilibrium. The latter was indicative of stronger diffusion limitations for 3-methylpentane inside the ZSM5 framework, as evidenced by a lower diffusion coefficient for this species compared to 2-methylpentane.34,39,40 Intracrystalline diffusion limitations induce a nonuniform concentration profile over the catalyst crystallite. The net production rate of each sorbate species in the continuity equations constitutes an averaged value over the catalyst crystallite characteristic dimension: R̅ i =

s+1 Ls + 1

∫0

L

R i(y)y s dy

(2)

with R̅ i the average net production rate of component i and L the crystallite characteristic dimension, e.g., the crystallite radius in the case of a spherical geometry. The net production rate of 22054

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Figure 1. Detailed representation of the n-hexane hydroconversion network. Metallic platinum sites and acid sites are represented as dark and light gray areas, respectively.

component i at a specific location inside the crystallite was calculated from the steady-state site occupancies, which are, in turn, determined from integrating eq 1 up to the point that no significant differences were observed in time. Setting the accumulation term directly equal to zero was not opted for as the uniqueness of the solution could not be guaranteed. The integration routine is implemented in practice by introducing the dimensionless spatial coordinate ξ

ξ=

2y L

The concentration of any sorbate molecule inside the catalyst crystallite should initially equal zero: ⎪ θi = θisurf at ξ = 1⎫ ⎬ at t = 0 θi = 0 at ξ ≠ 1 ⎪ ⎭

The following sections elaborate on how the Fick diffusion coefficient in eq 4 is expressed as an analytical function of the site occupancy and the introduced catalyst descriptors. In case of an n-component system, eq 4 is often expressed in matrix form:

(3)

and by expressing eq 1 in terms of the pore occupancy θi Cis

∂θi 4C s ⎛ s ∂θ ∂Di ∂θi ∂ 2θ ⎞ = − 2i ⎜ Di i + + Di 2i ⎟ + R i ∂t ∂ξ ∂ξ L ⎝ ξ ∂ξ ∂ξ ⎠

Cs

with Ci Cis

(5)

A saturation concentration equal to zero would imply a complete restriction of the corresponding component from intracrystalline physisorption and, hence, a zero pore occupancy anywhere in the catalyst crystallite. The boundary conditions for eq 4 imply a symmetric concentration profile over the crystallite: θi = θisurf at ξ = 1⎫ ⎪ ⎬∀t δθi = 0 at ξ = 0 ⎪ δξ ⎭

4Cs ⎛ s ∂θ ∂θ ∂D ∂θ ∂ 2θ ⎞ =− 2 ⎜ D + + D 2⎟ + R ∂t ∂ξ ∂ξ L ⎝ ξ ∂ξ ∂ξ ⎠

(8)

with D an n × n nondiagonal matrix containing the Fick diffusion coefficients, Cs an n × n diagonal matrix with the physisorption saturation concentrations, and θ and R n × 1 matrices containing the site occupancies and net production rates, respectively. The determination of the net production rates following the SEMK methodology is concisely described in section 2.4. 2.2. Stefan−Maxwell Theory for Configurational Diffusion. In meso- and macroporous materials, diffusion is mostly controlled by molecule−molecule interactions and is commonly denoted as bulk diffusion.13 Molecule−wall collisions as in Knudsen diffusion typically prevail when pore dimensions are in the range of the mean free path. In the case of strong van der Waals interactions between the diffusing species and the sorbent, diffusion is governed by physisorption effects and can be considered as a series of activated hops

(4)

θi =

(7)

(6) 22055

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C jsDij̃ = (C jsDĩ )θi / θi + θj (CisDj̃ )θj / θi + θj

between regions of low potential energy representing the physisorption sites.41 The latter regime is usually denoted as configurational diffusion and is characterized by often strong concentration gradients.25 As commented in the Introduction, the diffusion coefficient introduced by Fick’s law is rather impractical in fundamentally describing the diffusion phenomenon in such a case, and hence, the Stefan−Maxwell theory is commonly applied. The theory is based upon a force balance on each diffusing species within the sorbate mixture and is elaborated in detail in the Supporting Information.25 In short, the driving force for mass transport is taken proportional to the chemical potential gradient and is balanced by friction with the pore walls and the other diffusing species in the mixture. On the basis of this force balance, the Fick diffusion coefficient matrix in eq 8 can be expressed as the product of the other two n × n matrixes: B, which contains the so-called Stefan−Maxwell diffusion coefficients related to the aforementioned friction effects, and the thermodynamic correction factor matrix Γ portraying the diffusing mixture nonideality:24 D = B−1Γ

Herein, possible differences in sorbate capacities are accounted for by incorporating the corresponding saturation concentrations in eq 13.47 Also, the so-called self-exchange coefficients D̃ ii and D̃ jj assessing the correlation between successful jumps of respectively species i and j were approximated by the corresponding corrected diffusion coefficients following the work of Paschek and Krishna.48 The corrected diffusion coefficients can be extracted from molecular dynamics computation, which is often integrated with the ideal adsorption solution theory (IAST) to describe multicomponent physisorption.49 Doing so, the main advantage of the Stefan−Maxwell theory to unravel the complex nature of the Fick diffusion coefficient is counterbalanced by time-consuming computations which severely extend as the feed carbon number increases. Vast model simulations are preferably avoided in modelguided process analysis and catalyst design routines, and an explicit, analytical expression for the dependency of the corrected diffusion coefficient on the site occupancy and the catalyst framework parameters is needed.50 The latter is accomplished by integrating a mean field approximation for diffusion through a catalyst unit cell as demonstrated in section 2.3. 2.3. Mean Field Approximation for Diffusion through a ZSM5 Unit Cell. The corrected diffusion coefficient can be related to the mean displacement distance λ and the jump frequency νi, which, in turn, depends on the total site occupancy. Assuming that a successful jump toward another physisorption site is only possible when the latter is vacant, a linear decrease of the corrected diffusion coefficient is often introduced:25

(9)

with Bii =

1 + Dĩ

n

∑ k=1 k≠i

θk Dik̃

Bij(i ≠ j) = −

θi Dij̃

i , j = 1, 2, ..., n (10)

Γij = θi

∂ ln fugi ∂θj

= δij +

θi θn + 1

i , j = 1, 2, ..., n (11)

1 1 Dĩ = λ 2νi(θtot) = λ 2νi(0)(1 − θtot) = Dĩ 0(1 − θtot) y y

Especially the possibly nonideal behavior of the mixture was found responsible for the often complex trend in Fick diffusion coefficient with increasing occupancy. The convenient expression for the thermodynamic correction factors follows directly from the single-site multicomponent Langmuir isotherm applied in this work:24 θi =

(14)

Equation 14 is approximative in nature, and a more complex function of the corrected diffusion coefficient in the site occupancy was suggested from KMC simulations.19 Owing to the relatively poor channel connectivity within the ZSM5 unit cell, the negative effect of occupied physisorption sites on the corrected diffusion coefficient was more pronounced than observed for, e.g., cubic lattices.30 Baur and Krishna51 empirically introduced an exponentially decreasing function into the site occupancy. Coppens and co-workers30−33,52 described diffusion through a ZSM5 unit cell by explicitly identifying the physisorption site locations within the unit cell and by accounting for possible physisorption site heterogeneity. The ZSM5 framework is schematically represented in Figure 2 along with its unit cell. The framework is typically built from perpendicularly intersecting straight (0.51 nm × 0.55 nm) and sinusoidal (0.53 nm × 0.56 nm) channels.53 Six distinct physisorption sites are identified within the unit cell, four of which reside in the pore channels (α). The sites located at the channel intersections (β) exhibit the highest accessibility as four neighboring α sites lie within reach for a successful hop. Framework acid sites generated from aluminum incorporation into the ZSM5 framework induce stronger sorbate stabilization by physisorption and could, therefore, slow the diffusion process considerably.54 The location of the acid sites is not entirely unraveled yet. Twenty-six different crystallographic positions can accommodate a bridging hydroxyl group with Si−O−Al angles ranging from 140° to 175°.55 However, experimental studies indicated a single proton affinity exhibited by acid ZSM5 zeolites and suggested that the acid sites most likely

K iLpi n

1 + ∑ j =par1 KjLpj

(13)

(12)

Herein, KLi constitutes the Langmuir physisorption coefficient of i. Even though equivalent in nature, eqs 10 and 11 are slightly different from those reported by Lettat et al.,42,43 who considered volume fractions instead of site occupations. In cases where physisorption is considered a form of friction against mass transport, the surface Stefan−Maxwell or corrected diffusion coefficient D̃ i in eq 10 is related to the inverse of a drag coefficient quantifying the drag force between sorbate species i and the sorbent only. D̃ ij in eq 10 is the Stefan− Maxwell interspecies or countersorption diffusion coefficient and is a measure of the frequency at which component j is replaced by component i at each physisorption site. Faster diffusing species are usually hindered by the slow components, while the latter experience a facilitated mass transport thanks to the “drag” induced by the faster species. Especially the diffusion rates of the faster species tend to be strongly affected by the presence of slower sorbate molecules.41,44 Other interspecies correlation effects might arise from the concerted motion of sorbate clusters.45 The countersorption diffusion coefficient at a given surface coverage is approximated via an extension of the Vignes expression for bulk liquid mixtures:46,47 22056

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Coppens and co-workers.31,33 The average physisorption times on the channel and weak intersection sites are considered identical. The average physisorption time equals the inverse of the desorption rate coefficient and follows an Arrhenius-type relationship with the standard physisorption enthalpy.57 φ can therefore be related to the ratio of the sorbate standard physisorption enthalpies evaluated on the weak and strong sites: °,phy

φi = e−ΔHs,i

°,phy −ΔH w, i / RT

(19)

where ΔH°s and represent the standard physisorption enthalpies with respect to the strong and weak physisorption sites, respectively. Application of the Stefan−Maxwell formulation to tracer diffusion enabled the self-diffusion coefficient of a component to be related to its corrected diffusion coefficient: ,phys

⎛ ⎜ θ 1 +i = ⎜ + i + ⎜ Dĩ Diĩ ⎜ ⎝

Figure 2. Simplified representation of the ZSM5 framework and indication of its unit cell. Physisorption sites in the pore channels are depicted as “α” and those at the channel intersections as “β”.

(16)

(1 − f )(1 − θ βw) + f (1 − θ βs)

K iL = 0.5(p°)−1 exp

4θα[(1 − f )(1 − θ βw) + fφi(1 − θ βs)] 1 − θα

j≠i

(20)

(21)

As a result, the corrected diffusion coefficient of a component through any zeolite framework could be analytically derived from the self-diffusion coefficient determined via a mean field approximation. Equation 9 enables the Fick diffusion coefficient to be related to the framework properties via eqs 15−17 including the catalyst descriptors s, L, f, and φ, which can be determined a priori from independent characterization. The single-component corrected diffusion coefficients at zero occupancy and zero strong physisorption site concentration should theoretically be applicable to any catalyst of the same framework type. Additional catalyst descriptors which are exclusively related to the reaction kinetics are introduced in section 2.4. 2.4. Single-Event Methodology for n-Hexane Hydrocracking. Kinetic model development for n-hexane hydroconversion is based upon the reaction network depicted in Figure 1 and described in section 2.1. Alkane physisorption is described via a single-site Langmuir isotherm (eq 12). The Langmuir physisorption coefficient comprises the standard physisorption enthalpy and entropy:

2(1 − θα)[(1 − f )θ βw + fθ βsφi−1]

+

j=1

+ iMFT ≈ +i(Diĩ , Dij̃ → ∞) = Dĩ

A = 2(1 − θα)[(1 − f )θ βw + fθ βsφi−1]

B=

n

∑

⎞−1 ⎟ θj ⎟ Dij̃ ⎟ ⎟ ⎠

The self-diffusion coefficient calculated using a mean field approximation takes no correlation whatsoever into account, hence reducing eq 20 to32

exclusively reside at the channel intersections.56 In the present methodology, the location of the acid sites is, hence, associated with strong β physisorption sites (βs), while the intersection sites which do not exhibit acid properties are denoted as weak β sites (βw). The mean field approximation implies no correlation between successive hops and is most applicable in the low occupancy range to describe self-diffusion through microporous substrates.30 In the present methodology, correlation effects are incorporated with the Stefan−Maxwell theory via eq 13. By accounting for the site-to-site flows between the different physisorption site types in the unit cell, the methodology developed by Coppens et al.31,33 allows expression of the self-diffusion coefficient as a function of the total site occupancy and the diffusion coefficient at zero occupancy and acid sites present: A + iMFT = D0,̃ 0 i (15) B with

+ 4θα[(1 − f )(1 − θ βw) + f (1 − θ βs)]

ΔH°w,phys

(17)

ΔSi°,phy −ΔHi°,phy exp R RT

(22)

A dual-site Langmuir isotherm was not adopted as n-hexane and its isomers exhibited a single-site physisorption behavior at the reaction conditions considered in this work.34 Alkane dehydrogenation toward alkene intermediates was assumed to be quasi-equilibrated corresponding to “ideal” hydroconversion conditions.58 In addition, protonation toward and deprotonation of carbenium ions are not rate determining compared to the acid catalysis involving isomerization and cracking. The corresponding reaction rates can consequently be expressed in terms of the alkane and hydrogen partial pressures, including the equilibrium coefficients for physisorption, dehydrogenation, and protonation. For example, for a reactant carbenium ion k

Herein, θα, θβw, and θβs represent the occupancies of the α, βw, and βs sites, respectively, f is the fraction of strong physisorption sites at the channel intersections, and φ is the ratio of average physisorption times on the strong and weak sites: τs, i φi = τw, i (18) A more detailed derivation of eqs 15−17 and the methodology followed to calculate the individual site occupancies can be found in the Supporting Information and in the publications of 22057

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originating from alkene j, in turn formed from alkane i, the reaction rate of an elementary isomerization and cracking step is calculated as follows:36 rk = kk

exhibited a near-spherical geometry with an average equivalent diameter of 0.5 μm. The majority of the catalyst descriptors introduced in section 2 were determined a priori. The physisorption saturation concentration of n-hexane was determined from the ratio of the sorbent micropore volume and the molar volume of the alkane at the reaction temperature considered. The latter was calculated via the Hankinson−Brobst−Thomson method.62 A typical value of 1.9 × 102 cm3 kg−1 was taken for the micropore volume of a ZSM5 framework.63 According to Zhu et al.,64 the physisorption saturation concentration of 2-methylpentane and 3-methylpentane should be similar to the saturation concentration of n-hexane. Conversely, a saturation concentration of half this value was reported for both dimethylbutanes. The Langmuir physisorption parameters in eq 22 were determined from physisorption data reported by Denayer et al.65 for n-hexane and its isomers on a ZSM5 with an identical Si/Al ratio. Propane physisorption was not accounted for. The physisorption saturation concentrations and the Langmuir physisorption coefficients evaluated at 503 K are reported in Table 1

−1 deh L C acidCisKjpro , k K i , j K i pp i H 2

1+

n ∑upar K Lp =1 u u

+

n n −1 ∑upar ∑vole C sK proK dehK iLpp =1 =1 i j,k i,j i H2

(23)

The summations in the denominator originate from site balances over the physisorption and protonation sites, respectively. The catalyst descriptors in eq 23 are the Langmuir physisorption coefficient KLi , the sorbate saturation concentration Csi , the total acid site concentration Cacid, and the protonation equilibrium coefficient Kpro j,k quantifying the carbenium ion stabilization upon alkene protonation at the acid sites. The dehydrogenation equilibrium coefficient Kdeh i,j is calculated from pure component thermodynamic data obtained from Thermodynamics Research Center (TRC) tables.59 The rate coefficient kk depends solely on the reaction kinetics and is therefore denoted as a “kinetic descriptor” in contrast to the previously mentioned parameters. A total of 21 rate coefficients are to be determined according to acid catalysis depicted in Figure 1. The SEMK methodology aims at significantly reducing the number of independent rate coefficients while maintaining the fundamental character of the model. This methodology introduces elementary reaction families based upon the elementary step type and the types of reactant and product ions involved in the reaction, i.e., secondary or tertiary. A unique “single-event” rate coefficient is defined per reaction family and differs from the actual rate coefficient of an elementary step belonging to this family by the so-called “number of single events”, ne, a factor assessing the number of structurally equivalent ways in which the elementary step can occur:

k = nek ̃

Table 1. Predetermined Catalyst Descriptor Values of Pt/H-ZSM5 (Si/Al = 137)a

(24)

For more details on the SEMK methodology, see a recent review by Thybaut and Marin.60 Specifically for the n-hexane hydroconversion network, only six distinct single-event rate coefficients have to be determined assuming that primary ion formation is not taking place. The activation energies and single-event standard activation entropies were determined earlier from SEMK modeling studies on n-alkane hydroconversion on Pt/H-USY catalysts.5,36 Similarly to the isomerization and cracking rate coefficients, the SEMK methodology enables the protonation equilibrium coefficient in eq 23 to be related to a single-event protonation equilibrium coefficient of a reference alkene. The standard protonation enthalpy involved depends strongly on the type of carbenium ion formed while only marginally on the reactant and product structures.6 This parameter was, hence, identified as an accurate descriptor of the catalyst average acid strength.61 While the single-event standard protonation entropy can be determined a priori from the standard translational and physisorption entropies,5 the latter is difficult to assess from conventional characterization techniques and is, therefore, commonly estimated via model regression.

catalyst descriptor

value

s L (m) CsnC6, Cs2MP, Cs3MP (mol kg−1)

2 5.0 × 10−7 7.3 × 10−1 b

Cs22DMB, Cs23DMB (mol kg−1) KLnC6 (Pa−1)

3.7 × 10−1 b 9.8 × 10−5 b

KL2MP (Pa−1) KL3MP (Pa−1) KL22DMB (Pa−1) KL23DMB (Pa−1) φnC6

7.2 × 7.1 × 6.4 × 9.0 × 4.2b

Cacid (mol kg−1) f

1.2 × 10−1 3.3 × 10−1

10−5 b 10−5 b 10−5 b 10−5 b

a

Hexane isomers are denoted as nC6 (n-hexane), 2MC5 (2-methylpentane), 3MC5 (3-methylpentane), 22DMC4 (2,2-dimethylbutane), and 23DMC4 (2,3-dimethylbutane). bEvaluated at 503 K.

along with the other catalyst descriptors. The average physisorption time ratio φ, see also eq 19, is approximated using the standard physisorption enthalpies reported by Arik et al.66 for the ZSM5 catalysts with the highest and lowest Si/Al ratios. Only the value obtained for n-hexane is reported in Table 1 as it is nearly identical for any of the hexane isomers. The total acid site concentration was determined at 1.2 × 10−1 mol kg−1 from the catalyst’s Si/Al ratio assuming that each Al atom generates an accessible acid site.55 The fraction of acid sites within the unit cell, f, was approximated by the ratio of the total acid concentration and the physisorption saturation concentration of n-hexane. As commented earlier, the alkene standard protonation enthalpy was determined via model regression against experimental data. The standard protonation enthalpy for secondary carbenium ion formation was estimated, while the corresponding enthalpy contribution for tertiary ion formation was set 30 kJ mol−1 more negative.36 The latter is not estimated because of the limited contribution of tertiary carbenium ions to the acid catalysis depicted in Figure 1. Owing to the large discrepancies between reported corrected diffusion coefficients

3. PROCEDURES 3.1. Catalyst Descriptors. The catalyst consisted of a commercial ZSM5 catalyst (Si/Al = 137) loaded with approximately 0.5 wt % Pt by means of incipient wetness impregnation with an aqueous Pt(NH4)3Cl2 solution.34 The catalyst crystallites 22058

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measured via different techniques,67−70 the corrected diffusion coefficients for 2-methylpentane, 3-methylpentane, and 2,2dimethylbutane were also estimated. The corrected diffusion coefficient of 2,3-dimethylbutane was chosen identical to the value for 2,2-dimethylbutane. A lower diffusion coefficient for 2,2-dimethylbutane was found from uptake measurements probably owing to a higher critical molecular diameter compared to that of 2,3-dimethylbutane.43,71 However, as will be commented further on, the n-hexane hydroconversion data indicated a negligible production of any of the dimethylbutanes, and hence, an average corrected diffusion coefficient value which is equivalent to the total immobilization of both species inside the catalyst crystallite should be sufficient to capture this effect. Considering the relatively low diffusional activation energies reported in the literature,69,72−74 and the limited reaction temperature range applied in this work, the corrected diffusion coefficients were assumed constant with respect to the reaction temperature. Parameter estimates and simulation results are explicitly compared with the results reported in an earlier work on this matter.34 Herein, a linearly decreasing function of the corrected diffusion coefficient with the total site occupancy was adopted as in eq 14, and interspecies correlations were not accounted for. 3.2. Experimental Data and Data Treatment. The n-hexane hydroconversion data acquired in a Berty-type continuous stirred-tank reactor (CSTR) are described elsewhere.34 A total of 4.83 × 10−3 kg of catalyst was loaded into the reactor. The reaction temperature ranged from 503 to 523 K, while the reactor pressure was varied from 1 to 3 MPa. The reactant space time range was 94−429 kg s−1 mol−1, and the inlet hydrogen-to-hexane molar ratio was situated between 50 and 100. The total n-hexane conversion is defined as X tot =

at NETLIB.75 Herein, the Fick diffusion coefficient matrix was expressed as the product of a matrix containing the thermodynamic correction factors and a matrix with the corrected diffusion coefficients; see eq 9. The dependencies of the corrected diffusion coefficients on the site occupancies were subsequently incorporated via eqs 15−17. The derivatives of the site occupancies to the dimensionless spatial coordinate ξ in eq 8 were calculated using the central difference method. The presence of hydrogen in the reaction mixture was not accounted for in the Stefan−Maxwell equations following experimental evidence that the transport of both hydrogen and hydrocarbons such as n-butane through silicalite remains close to unaffected at reaction conditions similar to the ones applied in this work.76 3.4. Objective Function Minimization. A consecutive Rosenbrock and Levenberg−Marquardt algorithm was applied for parameter estimation. The former routine was incorporated via an in-house-written FORTRAN code and is more robust against divergence77 than the Levenberg−Marquardt method, which, on the other hand, allows quadratic convergence around the optimum parameter values.78 The latter was included using ODRPACK v2.01 available online at NETLIB.75 The latter subroutine was complemented with some additional code to retrieve the F value for global significance and the variance− covariance matrix between the parameter estimates. The objective function to be minimized consists of the weighted sum of squared differences (SSQ) between the experimental and modeled outlet flow rates: nobs nresp

SSQ =

The weighting factors wi are calculated from the diagonal elements of the inverse of the variance−covariance of the experimental errors on the responses. No replicate experiments were available, and hence, the weighting factors were determined as follows:

(25)

The yield of an isomer product i is defined as Xi =

Fi 0 FnC 6

n

wi =

(26)

The experimentally observed n-hexane conversion ranged from 10% to 60%. The product mixture was primarily composed of 2-methylpentane and 3-methylpentane, while dimethylbutanes and propane could barely be detected. 3.3. Reactor Mass Balance and Solution Strategy. Assuming an ideal CSTR operation, the outlet flow rate of each reaction product is determined from the corresponding continuity equation, e.g., for reaction product i Fi ̂ − Fi0 − R̅ i(Fi )̂ W = 0

(28)

j=1 i=1

0 − FnC6 FnC 6 0 FnC 6

b

∑ ∑ wi(Fi ,j − Fi ,̂ j)2 → min

(∑kobs F )−1 =1 i,k n

n

∑ j =resp1 (∑kobs F )−1 =1 j,k

(29)

Both dimethylbutane outlet flow rates were lumped into a single response, resulting in a total of four responses in the SSQ expression.

4. RESULTS AND DISCUSSION 4.1. Model Regression. The F value for the global significance of the regression amounted to 8500, which significantly exceeds the tabulated value. An excellent agreement between simulated and experimental n-hexane conversions was obtained at different space times and reaction temperatures and pressures as demonstrated in Figure 3. The catalytic activity clearly increased with the reaction temperature and the reactant space time. The n-hexane conversion decreased with the reaction pressure and the inlet hydrogen-to-hydrocarbon molar ratio, which is indicative of quasi-equilibration between the sorbate alkanes and alkene intermediates.58 The alkene standard protonation enthalpy for secondary carbenium ion formation was estimated at −69.9 kJ mol−1; vide Table 2. The latter value is situated well within the range of reported alkene standard protonation enthalpies obtained on other Pt-loaded zeolite catalysts.9,36,61

(27)

The outlet flow rates of n-hexane and hydrogen were determined a posteriori from the atomic carbon and hydrogen balance, respectively. The set of algebraic equations was solved with the DNSQE routine available online at NETLIB.75 The average net production rate in eq 27 is calculated by discretization of eq 2 over a specific number of intracrystalline grid points distributed equidistantly over half the crystallite characteristic dimension. Symmetry around the crystallite center was implicitly assumed via the boundary conditions (eq 6). At each grid point, eq 8 with boundary and initial conditions in eqs 6 and 7, respectively, was integrated with time for each sorbate alkane using the DVODE integration routine available 22059

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for both methodologies were identical, this improved agreement and superior global model significance were achieved by the higher degree of complexity attained by the present model. Section 4.3.1 elaborates further on whether the implementation of interspecies correlations portrayed by the countersorption diffusion coefficients and/or of the channel connectivity within the ZSM5 unit cell was crucial in obtaining such an improvement. A peculiar feature in alkane hydroconversion over a ZSM5 catalyst is that the selectivity toward the 2-methyl-branched alkane within the feed isomers exceeds the value expected from thermodynamic equilibrium.79−81 This is demonstrated in Figure 5, showing the experimentally obtained 2-methylpentane/ Figure 3. Experimental (symbols) and simulated (lines) total n-hexane conversions (eq 25) as a function of the reactant space time at 523 K and 1 MPa (◆), 503 K and 1 MPa (■), 503 K and 2 MPa (▲), and 503 K and 3 MPa (●) and an inlet H2/nC6 molar ratio of 50. Simulated total n-hexane conversions were obtained using the methodology described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

Table 2. Estimated Catalyst Descriptors Using the Methodology Described in Section 2 and with the Catalyst Descriptors Reported in Table 1 parameter −ΔH°ref,pro (kJ D̃ 00,i (m2 s−1) i i i i a

= = = =

−1

mol )

nC6 2MC5 3MC5 DMC4

estimated value

Figure 5. Experimental (full symbols) and simulated (open symbols) 2-methylpentane/3-methylpentane yield ratio (eq 26) as a function of the total n-hexane conversion (eq 25) at 503 K (◆) and 523 K (■). Simulated 2MC5/3MC5 yield ratios were obtained using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2. Experimentally obtained 2MC5/3MC5 conversion ratios on a Pt/ H-USY catalyst (Si/Al = 30) (●) from data reported by Thybaut et al.6 are also depicted along with the thermodynamic equilibrium value evaluated at 503 K (dashed line).

69.9 ± 0.1a (1.8 (1.1 (2.5 (3.2

± ± ± ±

0.3) 0.1) 0.2) 0.4)

× × × ×

10−14 10−14 10−15 10−18

Ninety-five percent confidence interval.

3-methylpentane yield ratios along with the thermodynamic equilibrium value. Additionally, n-hexane hydroconversion data obtained on Pt/H-USY (Si/Al = 30) reported by Thybaut et al.6 are depicted and are reminiscent of the fast establishment of thermodynamic equilibrium between 2-methylpentane and 3-methylpentane on such a non-shape-selective catalyst. Moreover, on such catalysts, this equilibrium is approached from the other side, i.e., from lower 2- to 3-methylpentane yield ratios, as determined by the kinetically controlled formation of methylbranched isomers from n-hexane. The equilibration follows directly from fast intramolecular methyl shifts occurring between these two isomers, compared to the slower (de)branching and cracking reactions.36 Vandegehuchte et al.34 demonstrated that intracrystalline diffusion of 3-methylpentane is hindered to a higher extent than that of 2-methylpentane, resulting in a higher apparent reactivity of the former species and, consequently, in a higher 2-methylpentane yield. The corrected diffusion coefficient of 3-methylpentane was estimated at a nearly 4-fold lower value than that of 2-methylpentane; vide Table 2. A 3−4-fold difference in corrected diffusion coefficients was confirmed by the sparse comparative studies in the literature.82,83 A 1.6-fold higher diffusion coefficient of n-hexane compared to 2-methylpentane was estimated by the model and also agrees well with earlier reported differences.82,84,85 The corrected diffusion coefficients of both dimethylbutanes were estimated at a value which is 3 orders of magnitude lower than the coefficient obtained for 3-methylpentane. Such low diffusion

Figure 4. Experimental (symbols) and simulated (lines) 2-methylpentane (■) and 3-methylpentane (●) yields (eq 26) as a function of the total n-hexane conversion (eq 25). Simulated isomer yields were obtained using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2 (full lines). The model regression results from Vandegehuchte et al.34 are also shown in which a linearly decreasing corrected diffusion coefficient with the total pore occupancy was assumed (eq 14) (dashed lines).

Figure 4 zooms in on the agreement between simulated and observed methylpentane yields. An improved agreement was obtained compared to the results presented in earlier work where a linearly decreasing corrected diffusion coefficient with the site occupancy was assumed, see eq 14, without any interspecies drag.34 As the numbers of adjustable model parameters 22060

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Figure 6. Simulated steady-state site occupancies of n-hexane (◇), 2-methylpentane (□), 3-methylpentane (Δ), and dimethylbutanes (o) as a function of the dimensionless spatial coordinate at 503 K and 3 MPa (a) and 523 K and 2 MPa (b), an inlet H2/nC6 molar ratio of 50, and a space time of 100 kg s mol−1. Site occupancies were determined using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

4.3. Impact of the Catalyst Descriptors on the nHexane Hydroconversion Performance of ZSM5. 4.3.1. Channel Connectivity and Interspecies Correlations. Considering the improvement in agreement between the model and experiment compared to the results from previous research, vide Figure 4, the peculiar ZSM5 channel connectivity and/or interspecies correlations during diffusion seem to impact considerably the pure component diffusion and should be taken into account to attain a better approximation of the real corrected diffusion coefficients. Figure 7 illustrates the effect of the channel connectivity on the corrected diffusion coefficient, the latter represented as a function of the site occupancy (Figure 7a), and on the catalyst performance in n-hexane hydroconversion (Figure 7b). The corrected diffusion coefficient is related to the corrected diffusion coefficient at zero site occupancy via31

coefficients were also reported before84 and are mathematically equivalent to the former species being completely immobile upon formation. As a result, they tend to react further instead of diffusing out of the crystallite, leading to negligible dimethylbutane yields. Reaction of dimethylbutanes toward more mobile species exclusively involves debranching toward a 2-methylpentane; vide Figure 1. Of course, in the case of heavier feeds, cracking of the dibranched isomers will also become significant. The corrected diffusion coefficient estimates, D̃ i, are situated around the lower limit of the reported range of diffusion coefficients obtained from macroscopic measurements.42,72,86 The discrepancy between results from macroscopic and microscopic measurement techniques has been the subject of much research67−69 and is attributed to the experimental technique itself and not to the methodology used to extract the corrected diffusion coefficients from the obtained data. This was demonstrated by Jobic et al.87 and Millot et al.,73 who applied the Stefan−Maxwell formulation to determine the corrected diffusion coefficient of various alkanes from respectively quasi-elastic neutron scattering (QENS) and permeation measurements, the former being a microscopic technique while the latter is classified as a macroscopic technique. A recent study by Hansen et al.,88 who modeled benzene alkylation with ethylene, demonstrated that the application of corrected diffusion coefficients situated within the reported range of diffusion coefficients obtained from macroscopic measurements was vital to accurately describe the experimentally obtained product yields. 4.2. Steady-State Concentration Profiles. Figure 6 shows the steady-state concentration profiles of n-hexane, 2-methylpentane, and 3-methylpentane at distinctly different reaction temperatures and pressures, i.e., at 503 K and 3 MPa (Figure 6a) and at 523 K and 1 MPa (Figure 6b). In both cases, the relatively high diffusion coefficients of n-hexane and both monomethyl isomers result in rather moderate concentration profile gradients. Especially at higher conversions, such as in Figure 6b, an apparent discontinuity in the first-order derivative emerges at the internal grid point closest to the external surface boundary and results from the unconstrained reaction at the external surface boundary and the reaction inside the catalyst crystallite, i.e., in the presence of hindered mass transport. This effect is much more pronounced for the dimethylbutanes, where the negligible occupancy at the external surface is in clear contrast with the relatively high intracrystalline concentrations at steady state.

Dĩ =

(φi − 1)f + 2 A 0 (φi − 1)f + 2 0 Dĩ = D0,̃ i 2 B 2

(30)

with A and B determined from eqs 16 and 17, respectively. The additional factor in eq 30 accounts for the increase in average physisorption time owing to the presence of acid sites. As described in section 2.3 for a ZSM5 framework, Coppens et al.31 similarly determined a mean field expression for the selfdiffusion coefficient in a cubic lattice as a function of the total site occupancy and acid site concentration: Dĩ =

[1 + f (φi − 1)](1 − θtot)2

[1 + (φi − 1)f ][f (1 − θs)(φi − 1) + (1 − θtot)] 1 = Dĩ 0 1 + (φi − 1)f

D0,̃ 0 i

(31)

with θs the occupancy of the strong physisorption sites. Figure 7a additionally depicts a linearly decreasing function of the corrected diffusion coefficient with the total site occupancy as in eq 14. Pronounced differences in corrected diffusion coefficient trends were simulated between ZSM5 and a cubic lattice. A generally higher corrected diffusion coefficient is observed for the latter lattice type, which follows directly from the higher accessibility of the different physisorption site locations compared to those in the ZSM5 framework.30 Especially in the higher site occupancy range, the presence of vacant physisorption sites surrounding the sorbate position in the unit cell is more limited for a ZSM5 framework, resulting in a rapid decrease in diffusion coefficient. The higher diffusion coefficient 22061

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Figure 7. (a) Simulated corrected diffusion coefficient normalized to the corrected diffusion coefficient at zero occupancy as a function of the total site occupancy assuming a linearly decreasing corrected diffusion coefficient (eq 14) (full line) for a ZSM5 framework (eqs 16, 17, and 30) (longdashed line) and a cubic lattice (eq 31) (short-dashed line). (b) Simulated n-hexane conversion (eq 1) (black line, left axis) and 2MC5/3MC5 yield ratio (eq 2) (gray line, right axis) as a function of the space time at 503 K, 1 MPa, and an inlet H2/nC6 molar ratio of 50 assuming a linearly decreasing corrected diffusion coefficient with the site occupancy (full line) for a ZSM5 framework (long-dashed line) and a cubic lattice (shortdashed line). Conversions and yield ratios were determined using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

Figure 8. (a) Simulated steady-state site occupancies of n-hexane (full line), methylpentanes (long-dashed line), and dimethylbutanes (short-dashed lines) as a function of the dimensionless spatial coordinate at 503 K, 1 MPa, an inlet H2/nC6 molar ratio of 50, and zero n-hexane conversion. (b) Simulated n-hexane conversion (eq 1) (black line, left axis) and 2MC5/3MC5 yield ratio (eq 2) (gray line, right axis) as a function of the space time at 503 K, 1 MPa, and an inlet H2/nC6 molar ratio of 50. Site occupancies, conversions, and yield ratios were determined using the methodologies described in section 2, with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2 (black lines, a; full lines, b), and by additionally neglecting interspecies correlation effects (gray lines, a; dashed lines, b).

in the 0−0.3 site occupancy range for ZSM5 compared to the cubic lattice could only be attributed to faster sorption on the strong physisorption sites, which, owing to the higher physisorption time on these sites, reduce the overall corrected diffusion coefficient when unoccupied.31,32 Figure 7b shows the impact of the channel connectivity on the overall catalyst performance in n-hexane hydroconversion. Owing to a more efficient diffusion throughout a cubic lattice, a higher feed conversion is to be expected. In addition, the 2-methylpentane/3-methylpentane yield ratio evolves more quickly to the thermodynamic equilibrium value. A good agreement was obtained between the simulated n-hexane conversions considering a ZSM5 framework and by assuming a linearly decreasing corrected diffusion coefficient with the total site occupancy as assumed in previous work.34 As was indicated in Figure 7a, both methodologies approach each other rather well in the higher site occupancy range from 0.6 on. Minor discrepancies in product selectivities are, however, observed in the lower conversion range. This implies that accounting for the exact geometry of the catalyst framework contributed rather moderately to the distinct improvement in agreement

between the experimental data and regression results presented in Figure 4. The impact of interspecies correlations on the intracrystalline concentration profiles of n-hexane, methylpentanes, and dimethylbutanes is illustrated in Figure 8a. In the case in which no interspecies correlations are accounted for, i.e., for infinitely large countersorption diffusion coefficients, only the singlecomponent corrected diffusion coefficients and the net production rates determine the steady-state concentration profiles. As for n-hexane exhibiting the highest diffusion coefficient, its site occupancy varies slightly according to the dimensionless spatial coordinate, and consequently, a substantially higher n-hexane conversion was simulated; vide Figure 8b. Discrepancies in product selectivities emerge in the higher conversion range, and the model fails at accurately describing the experimental data. The latter was earlier illustrated in Figure 4 showing the SEMK modeling results from previous research.34 Interspecies correlations, hence, considerably impact the diffusion of hexane isomers through a ZSM5 crystallite at the reaction conditions considered in this work and become more pronounced as the catalyst micropores become saturated with 22062

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Figure 9. (a) Simulated corrected diffusion coefficient (eqs 15−17) normalized to the corrected diffusion coefficient at zero occupancy and zero acid site concentration as a function of the total site occupancy at total acid concentrations of 2.6 × 10−1 (full line), 1.3 × 10−1 (long-dashed line), and 0.5 × 10−1 (short-dashed line) mol kg−1. (b) Simulated n-hexane conversion (eq 1) (black line, left axis) and 2MC5/3MC5 yield ratio (eq 2) (gray line, right axis) as a function of the space time at 503 K, 1 MPa, and an inlet H2/nC6 molar ratio of 50 at total acid concentrations of 2.6 × 10−1 (full line), 1.3 × 10−1 (long-dashed line), and 0.5 × 10−1 (short-dashed line) mol kg−1. Conversions and yield ratios were determined using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

sorbate species.89 Krishna and Van Baten90 found that the interspecies correlations were difficult to assess in case of pronounced intersection blocking by immobile species. To this purpose, Beyne and Froment91 developed an effective medium approximation (EMA) theory for site blockage, more specifically for coking in ZSM5, which was further elaborated by Coppens and co-workers.32,52 However, in this work, dimethylbutanes as the immobile species in the reaction network tend to react further to more agile species, hence eliminating any permanent obstruction inside the catalyst crystallite. The impact of permanent intersection blocking on the component mass transport was therefore not accounted for. 4.3.2. Total Acid Site Concentration. While the physisorption saturation concentration, the Langmuir physisorption coefficient, and the alkene standard protonation enthalpy affect the sorbate net production rates only, vide eq 23, the total acid site concentration also directly impacts the sorbate diffusion coefficient following eqs 15−17. Figure 9a shows a lower corrected diffusion coefficient normalized to the diffusion coefficient at zero occupancy and zero acid site concentration for the catalyst containing the highest acid site concentration. As mentioned earlier in section 4.3, strong physisorption sites are first occupied by sorbate species, resulting in a nonlinear decrease of the corrected diffusion coefficient with the site occupancy.31 This nonlinearity becomes more pronounced with higher acid site concentrations as shown in Figure 9a. Note that no model simulations were carried out with acid site concentrations exceeding 2.6 × 10−1 mol kg−1, which would correspond to a strong physisorption site fraction of about 0.5. The Stefan−Maxwell theory was found to agree rather well with KMC simulation results up to such intermediate values, after which the former theory tends to fail to even qualitatively predict the dependence of the corrected diffusion coefficient on the total site occupancy.33 Figure 9b shows an increasing catalytic activity with the total acid site concentration. The latter follows directly from the rate expression in eq 23 in which the acid site concentration acts as a mere multiplicator of the isomerization and cracking rate coefficients. The total acid site concentration exhibits a negative effect on the sorbate corrected diffusion coefficients, and consequently, the product distribution should be affected by the aluminum content of the catalyst. Indeed, at an identical

feed conversion, a higher selectivity toward 2-methylpentane was simulated for the catalyst containing the highest acid concentration; vide Figure 9b. The selectivity toward the more agile 2-methylpentane increases when mass transport is generally hindered to a higher extent. Under such conditions, the global catalyst activity is also reduced. However, the latter is overcompensated by an increase in the number of active sites as evident from Figure 9b. This assessment demonstrates that the catalyst acid site concentration constitutes a design parameter which not only allows the overall catalyst activity to be boosted, but also opens up the route to tailor the product distribution to the design specifications. 4.3.3. Crystallite Geometry and Dimensions. Both the crystallite shape and dimensions are important parameters in eqs 2 and 4 in the determination of the average net production rates and the intracrystalline concentration profiles, respectively. The most commonly investigated crystallite shapes are the slab, cylindrical, and spherical geometries, which correspond to shape factors amounting to 0, 1, and 2, respectively. In the absence of reaction, analytical expressions for the steadystate intracrystalline concentration profiles for all three geometries indicate more pronounced concentration gradients over the crystallite characteristic dimension with decreasing shape factors.92 Simulation results adopting any of the above three crystallite geometries are shown in Figure 10a and indeed attribute the least uniform concentration profile to the slab geometry. As a result, the activity of a slab-shaped catalyst becomes inferior to that of a spherically shaped catalyst of the same characteristic dimensions as found earlier when comparing the corresponding effectiveness factors for a first-order reaction; vide Figure 10b.35 The slab-shaped catalyst exhibits a substantially higher selectivity toward 2-methylpentane, illustrating that the catalyst geometry could considerably affect the product distribution.93 A cylindrical shape gave rise to a steady-state concentration profile and n-hexane hydroconversion pattern intermediate between those obtained with slab and spherical geometries. Extension to irregularly shaped crystallites was elaborated by Burghardt and Kubaczka,92 who developed a general expression for the shape factor. Mass transport limitations become increasingly more important as the diffusion path inside the catalyst crystallite becomes longer. The impact of the diameter of a spherical crystallite on 22063

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Figure 10. (a) Simulated steady-state site occupancies of n-hexane (black lines) and methylpentanes (gray lines) as a function of the dimensionless spatial coordinate at 503 K, 1 MPa, an inlet H2/nC6 molar ratio of 50, and a space time of 100 kg s mol−1 for spherical (full line), cylindrical (longdashed line), and slab (short-dashed line) crystallite geometries. (b) Simulated n-hexane conversion (eq 1) (black line, left axis) and 2MC5/3MC5 yield ratio (eq 2) (gray line, right axis) as a function of the space time at 503 K, 1 MPa, and an inlet H2/nC6 molar ratio of 50 for spherical (full line), cylindrical (long-dashed line), and slab (short-dashed line) crystallite geometries. Site occupancies, conversions, and yield ratios were determined using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

Figure 11. (a) Simulated steady-state site occupancies of n-hexane (black lines) and methylpentanes (gray lines) as a function of the dimensionless spatial coordinate at 503 K, 1 MPa, an inlet H2/nC6 molar ratio of 50, and a space time of 100 kg s mol−1 for crystallite diameters of 5 μm (full line), 1 μm (long-dashed line), 0.5 μm (short-dashed line), and 0.1 μm (dotted line). (b) Simulated n-hexane conversion (eq 1) (black line, left axis) and 2MC5/3MC5 yield ratio (eq 2) (gray line, right axis) as a function of the space time at 503 K, 1 MPa, and an inlet H2/nC6 molar ratio of 50 for crystallite diameters of 5 μm (full line), 1 μm (long-dashed line), 0.5 μm (short-dashed line), and 0.1 μm (dotted line). Site occupancies, conversions, and yield ratios were determined using the methodologies described in section 2 and with the catalyst descriptors reported in Table 1 and the parameter estimates reported in Table 2.

the steady-state concentration profiles is demonstrated in Figure 11a and that on the catalyst performance in n-hexane hydroconversion in Figure 11b. No gradient in site occupancy is observed for n-hexane and both methylpentanes at the lowest crystallite diameter. In this case, the intracrystalline diffusion path is sufficiently short that the reaction remains kinetically limited. As a result, a relatively high n-hexane conversion was simulated and, additionally, thermodynamic equilibrium between 2- and 3-methylpentane from low conversions on. The synthesis of ZSM5 particles in the nanometer range constitutes a wellestablished method to attenuate the shape-selective properties exhibited by ZSM5-based catalysts.94,95 The gradient in steadystate site occupancies emerges from higher crystallite diameters on and becomes more significant up to the point where full micropore saturation is reached. Only the crystallites having the largest diameters were subjected to such saturation effects, while the total site occupancy ranged from 0.6 to 0.85 in the case of the other crystallite dimensions depicted in Figure 11. Under saturation conditions, Figures 7 and 9 imply a completely hindered diffusion for each sorbate species. As a result, the catalyst activity drops drastically with increasing crystallite dimensions, and in

addition, the product distribution is dominated by the fastest sorbate species in the diffusing mixture. Both features are demonstrated in Figure 11 and indicate that, similarly to the crystallite shape factor, the crystallite characteristic dimension constitutes one of the most important catalyst descriptors in optimizing the catalyst performance, particularly for a diffusioncontrolled reaction.

5. CONCLUSIONS Intracrystalline diffusion limitations during n-hexane hydroconversion on Pt/H-ZSM5 were described via Fick’s second law for mass transport, wherein the Fick diffusion coefficient was quantified via the Stefan−Maxwell theory for multicomponent diffusion. The effect of the framework geometry and the physisorption site heterogeneity on the pure component diffusion coefficients was assessed via a mean field approximation. An SEMK model was applied to simulate the reaction kinetics in a fundamental manner while retaining the number of adjustable parameters at an acceptable level. Integration of all three approaches into a single, comprehensive methodology enabled accurate description of experimentally observed n-hexane 22064

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Fick diffusion coefficient (m2 s−1) n × n matrix containing Fick diffusion coefficients corrected diffusion coefficient of component i (m2 s−1) self-exchange coefficient of component i (m2 s−1) countersorption diffusion coefficient of component i with respect to component j (m2 s−1) self-diffusion coefficient (m2 s−1) + f ug fugacity (Pa) f fraction of strong physisorption sites Fi experimental flow rate of component i (mol s−1) F̂i calculated flow rate of component i (mol s−1) ΔH° standard enthalpy (J mol−1) Kdeh dehydrogenation equilibrium coefficient (MPa) K̃ iso single-event isomerization equilibrium coefficient KL Langmuir physisorption coefficient (MPa−1) Kpro protonation equilibrium coefficient (kg mol−1) k rate coefficient (mol kg−1 s−1) k̃ single-event rate coefficient (mol kg−1 s−1) L crystallite dimension (m) n number of sorbate species ne number of single events nobs number of observations nole number of alkenes npar number of alkanes nresp number of responses p partial pressure (MPa) p° atmospheric pressure (Pa) R net rate of production (mol kg−1 s−1) R vector containing net production rates R̅ average net rate of production (mol kg−1 s−1) r reaction rate (mol kg−1 s−1) s crystallite shape factor ΔS° standard entropy (J mol−1 K−1) SSQ sum of squares T temperature (K) t time (s) W catalyst mass (kg) w weighting factor X yield Xtot conversion y spatial coordinate (m) D D D̃ i D̃ ii D̃ ij

hydroconversion activities and product selectivities via model regression. The set of model parameter estimates, including the alkane corrected diffusion coefficients, correspond well to reported values in the literature. The identification of catalyst descriptors in the model equations aided in rationalizing the impact of the catalyst properties on the diffusion and reaction phenomena occurring on the microscopic scale and on the individual product yields observed on the macroscopic scale. As the extent of diffusion limitations becomes higher, the resulting product distribution tends to shift toward the more agile species in the product mixture. Specifically for n-hexane hydroconversion, the selectivity toward 2-methylpentane increased with the total number of strong physisorption sites and the crystallite diameter at the expense of the more sluggish 3-methylpentane, and decreased with the channel connectivity and crystallite shape factor. Especially the crystallite dimension exhibited a significant impact on the catalyst performance in n-hexane hydroconversion. A sensitivity analysis of the catalyst descriptors, hence, brings a comprehensive catalyst design strategy within reach which is readily extendable to other reactions and catalytic materials, giving rise to mass transport limitations.

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ASSOCIATED CONTENT

S Supporting Information *

Elaboration of the Stefan−Maxwell methodology for multicomponent diffusion through a microporous material and derivation of eqs 15−17 following the mean field approximation to self-diffusion through a ZSM5 unit cell. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: + 32 9 331 17 52. E-mail: [email protected]. Present Address

§ I.R.C.: Indian Oil R&D Centre, Sector 13, Faridabad-121007, Haryana, India.

Notes

The authors declare no competing financial interest.

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Greek Symbols

ACKNOWLEDGMENTS This work was supported by the Research Board of Ghent University (BOF), by the Interuniversity Attraction Poles ProgrammeBelgian StateBelgian Science Policy, by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement No. 615456, and by Long Term Structural Methusalem Funding by the Flemish Government. Model regression was carried out using the Stevin Supercomputer Infrastructure at Ghent University, funded by Ghent University, the Hercules Foundation, and the Flemish GovernmentDepartment of Economy, Science and Innovation (EWI).

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Γ Γ θ θ λ ν ξ τ φ

Superscripts

0 acid MFT phy pro s surf

NOMENCLATURE

Roman Symbols

b B C Cs

thermodynamic correction factor matrix containing thermodynamic correction factors site occupancy vector containing site occupancies displacement distance (m) jump frequency (s−1) dimensionless spatial coordinate average physisorption time (s) ratio of average physisorption times on strong and weak sites

model parameter vector matrix containing corrected diffusion coefficients concentration (mol kg−1) n × n diagonal matrix containing saturation concentrations

initial, also at zero site occupancy acid sites calculated via mean field approximation physisorption protonation saturation at the external surface

Subscripts

0 22065

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The Journal of Physical Chemistry C 2MC5 3MC5 22DMC4 23DMC4 α β i, j, k, u, v nC6 s tot w

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2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane channel physisorption site intersection physisorption site component indices n-hexane strong physisorption site total weak physisorption site

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