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In this paper, how the performance of a PBMR for partial oxidations is influenced by the extent of intraparticle diffusion limitations has been invest...
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Ind. Eng. Chem. Res. 2007, 46, 7513-7523

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Effect of Mass-Transport Limitations on the Performance of a Packed Bed Membrane Reactor for Partial Oxidations. Intraparticle Mass Transport M. van Sint Annaland,* U. Ku1rten, and J. A. M. Kuipers Department of Science and Technology, UniVersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

For partial oxidation systems, where the reaction order in oxygen of the formation rate of the target product is smaller than the reaction order in oxygen of the consecutive reaction rate toward the waste product, a packed bed membrane reactor can be applied to distributively dose oxygen along the reactor in order to improve the selectivity toward the desired intermediate product. In this paper, how the performance of a PBMR for partial oxidations is influenced by the extent of intraparticle diffusion limitations has been investigated. First, the intrinsic effects for a single catalyst particle are discussed, and subsequently the integral effects over the entire reactor on the conversion and selectivity are studied with numerical models (considering different sets of reaction orders, different ratios of reaction rates of the primary and secondary reaction, and different stoichiometric coefficients). In Part 2, the effects of radial oxygen concentration gradients from the membrane wall to the center of the packed bed are investigated. The main objective of this work is to provide guidelines to assess whether mass-transfer limitations inside the particles (Part 1) and/or from the membrane wall to the center of the catalyst bed (Part 2) affect the reactor performance and to quantify these effects. 1. Introduction For partial oxidation systems, the formation rates for the target product and the waste product often differ in their dependency on the oxygen concentration. Thus, the oxygen concentration can be used to influence the ratio of these reaction rates, by which the product selectivity or yield can be improved. For example, for some partial oxidation processes, like the ethylene epoxidation1 or the selective oxidation of butane to maleic anhydride,2 the product formation shows a higher reaction order in oxygen than the sequential or parallel formation rate of the waste products and is thus optimally operated with an excess of oxygen. In the opposite case, where the oxygen dependency of the target product formation is less pronounced than the waste product formation rate, as observed, for example, for the oxidative dehydrogenation (ODH) of ethylbenzene to styrene3 or the ODH of methanol to formaldehyde,4,5 the process should be operated under oxygen deficiency for optimal product yield. For the case of a premixed feed, however, this results in a low single-pass conversion. By means of distributive addition of oxygen to the reaction mixture along the axial coordinate of the reactor, high reactant conversions can be obtained while operating at a low oxygen level. High reactant conversions combined with operation at a low oxygen concentration level can be accomplished in a packed bed membrane reactor (PBMR). Oxygen is fed distributively along the axial coordinate to the packed bed of the reactor via a (porous) membrane and is subsequently dispersed from the membrane wall to the center of the packed bed perpendicular to the main flow direction. The performance of the PBMR, in terms of reactant conversion and product selectivities, can be affected by mass transfer limitations inside the catalyst particles, as well as by mass transfer limitations from the membrane wall to the center of the packed bed. In this first part, we investigate how the performance of a PBMR for partial oxidations is influenced by the extent of intraparticle diffusion limitations * Corresponding author. Tel: 31-53-4894478, Fax: 31-53-4892881. E-mail: [email protected].

(i.e., concentration profiles inside the catalyst particles). First, the intrinsic effects for a single catalyst particle are discussed, and subsequently, the integral effects over the entire reactor on the conversion and selectivity are studied. In part II, the effects of radial oxygen concentration gradients from the membrane wall to the center of the packed bed, which prevail when the local chemical consumption rate exceeds the radial transport rate, are investigated. The main objective of this work is to provide guidelines to assess whether mass-transfer limitations inside the particles (part I) and/or from the membrane wall to the center of the catalyst bed (part II) affect the reactor performance and to quantify these effects. To investigate the effects of the oxygen concentration profiles due to mass-transfer limitations, we have selected the following general reaction scheme for the partial oxidation of a hydrocarbon with power law kinetics

Target product formation: A + ν1O2 f P r1 ) k1cAcn (1) Waste product formation: P + ν2O2 f W r2 ) k2cPcm (2) where A represents the hydrocarbon reactant, P the target product, W the waste product, νi the stoichiometric constants, and ki the reaction rate constants. The reaction rates of both reactions have been assumed to be first order in the hydrocarbons. The reaction orders of oxygen for the target product formation and for the waste product formation are indicated with n and m (m > n), respectively (note that c indicates the local concentration of O2). 2. Intrinsic Effects of Intraparticle Mass-Transport Limitations Two opposing effects determine the change in product selectivity that is due to intraparticle transport limitations. On the one hand, the concentration of reactant A decreases toward the particle center, whereas that of the intermediate product increases, as long as the product formation rate exceeds its consumption rate. This results in a selectivity loss. On the other

10.1021/ie070638k CCC: $37.00 © 2007 American Chemical Society Published on Web 08/15/2007

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hand, the decrease in the oxygen concentration toward the particle center favors the product selectivity. Which of these effects is dominating depends on the oxygen concentration gradients relative to the hydrocarbon concentration gradients. The effects of intraparticle mass-transfer limitations are quantified in terms of the particle effectiveness factor η and the product selectivity ratio F, defined, respectively, as

particle effectiveness: actual conversion rate of catalyst particle η) conversion rate corresponding to bulk concentrations (3) selectivity ratio: actual selectivity of catalyst particle F) (4) selectivity corresponding to bulk concentrations By numerical integration of the differential species mass conservation equations, the effects of intraparticle mass-transfer limitations on the particle effectiveness factor and the product selectivity ratio are quantified for the reaction system under consideration for different reaction orders and different ratios of the rate constants. First, the main assumptions of the particle model are briefly outlined. 2.1. Particle Model. The differential species mass conservation equations have been formulated with the following model assumptions: • The concentration of oxygen in the gas phase is small compared to the hydrocarbon concentrations of A and P (cb/ cAb ) cb/cPb ) 0.1). • The particle is isobaric and isothermal, i.e., the pressure and temperature inside the catalyst particle are uniform (which allows the use of concentration gradients in the description of the molecular transport inside the particle). • The effective diffusion coefficient is constant and assumed to be the same for all species (which allows a Fickian description of the molecular mass transport inside the particle). • External mass transfer limitations are ignored. The effects of external mass transfer limitations are discussed in section 3.6. As will be shown later, under these assumptions, the particle performance is dominated by the oxygen concentration profile. Therefore, a characteristic Thiele modulus referring to the conversion and mass-transfer rate of oxygen is defined. Because oxygen is consumed in parallel by both reactions, the modified Thiele modulus is defined as

R φ′ ) 3

x

m+1 n+1 ν k c cn-1 + ν2k2cPbcm-1 b 2 1 1 Ab b 2 D

(5)

Introducing the following dimensionless variables

c cb

(6)

r F) R

(7)

γ)

pnm )

k2cPb m-n rate of undesired reaction c ) k1cAb b rate of desired reaction

(8)

the differential mass conservations equations for the hydrocarbons A and P and oxygen can be written in dimensionless form as

(

)

cb 9 1 d F2 dγA ) γ Aγ n 2 dF 2 dF m + 1 n + 1 c F φ′ ν + ν2pnm Ab 2 1 2

(9)

( )

cb 1 d F2 dγP 9 )γ γn + 2 dF 2 dF n+1 cPb A m+1 F φ′ ν + ν2pnm 2 1 2 cb 9pnm γ γm (10) n+1 cPb P m+1 ν + ν2pnm 2 1 2

( )

9ν1 1 d F2 dγ ) γ Aγ n + 2 dF 2 dF m+1 n+1 F φ′ ν + ν2pnm 2 1 2 9pnmν2 γPγm (11) m+1 n+1 ν + ν2pnm 2 1 2 with the boundary conditions: γi|F)1 ) 1 and (∂γi/∂F)|F)0 ) 0. By numerical integration, the radial concentration profiles are obtained, from which the particle effectiveness factor for the consumption of A, P, and oxygen and the product selectivity can be computed. Subsequently, the effect of intraparticle concentration profiles on the particle effectiveness factor and the selectivity ratio is analyzed as a function of the modified particle Thiele modulus for different sets of reaction orders and stoichiometric coefficients. 2.2. Effect on Particle Effectiveness. For the partial oxidation reaction system, with a simultaneous consecutive and parallel reaction path, the standard effectiveness models cannot be employed to account for intraparticle mass-transfer limitations. The particle effectiveness factor for the consumption of A and the production of W are not just a function of a single Thiele modulus.6 Only the oxygen consumption rate can be described with a single parameter, in the case that the oxygen concentration gradients are dominating, i.e., when the oxygen concentration is small compared to the concentrations of the hydrocarbons A and P. The particle effectiveness factor for the consumption of oxygen is given by

η O2 )

ν1〈r1〉 + ν2〈r2〉 ) ν1r1b + ν2r2b

∫0R4πr2ν1k1cAcn dr + ∫0R4πr2ν2k2cPcm dr 4π 3 R (ν1k1cAbcnb + ν2k2cPbcmb ) 3 3

∫01F2γAγn dF + ν2pnm∫01F2γPγm dF

ν1

ν1 + ν2pnm

)

)

ν1〈γAγn〉 + ν2pnm〈γPγm〉 (12) ν1 + ν2pnm where 〈x〉 represents the spatial average of variable x. In Figure 1, the particle effectiveness factor for the consumption of oxygen is plotted as a function of the modified Thiele modulus for two sets of reaction orders (n ) 0.5, m ) 1 and n ) 1, m ) 2) and for different ratios of the reaction rates of the undesired and desired reactions (pnm) ranging from 0.01 to 2. The figure clearly demonstrates that the particle effectiveness factor ηO2 is virtually independent (for n,m g 0.5) of the ratio of the reaction rates of the primary and secondary reaction pnm

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Figure 1. Particle effectiveness factor for the consumption of oxygen (ηO2) as a function of the modified Thiele modulus (φ) for different values of pnm (cb/cAb ) cb/cPb ) 0.1). (a) n ) 0.5, m ) 1 and (b) n ) 1, m ) 2 (the dashed line indicates the analytical solution for high Thiele-moduli).

correlated to pnm and takes the value of zero, if the rates of the wanted and unwanted reactions are equal

σb )

k1cAbcnb - k2cPbcmb k1cAbcnb

) 1 - pnm

(14)

The actual product selectivity is given by

Figure 2. Normalized radial concentration profiles inside the catalyst particle for cb/cAb ) cb/cPb ) 0.1 (full lines) and cb/cAb ) cb/cPb ) 1 (dashed lines) (n ) 0.5, m ) 1, φ′)1, pnm)0.1).

∫0R4πr2k1cAcn dr - ∫0R4πr2k2cPcm dr 〈σ〉 ) ) ∫0R4πr2k1cAcn dr 1 - pnm

(e2), and that the particle effectiveness factor can very well be described by

(

3φ′ 1 -1 η O2 ) 3φ′2 tanh(3φ′)

)

(13)

which is the analytical solution for a single, first-order reaction in a spherical particle.7 Only for reaction orders close to n,m ) 0 is the numerically calculated particle effectiveness factor slightly higher than that predicted with the above equation at intermediate values of the Thiele modulus φ′ (e.g., φ′ ) 1 and n ) 0.1, ηO2 ) 0.74 instead of 0.67). In conclusion, with the modified Thiele modulus, a very good approximation of the particle effectiveness factor ηO2 can be found for all relevant reaction orders and ratios pnm of the reaction rates of the consecutive and primary reaction. 2.3. Effect on Product Selectivity. If the hydrocarbon concentrations in the bulk of the gas phase are on the same order of magnitude as the concentration of oxygen, intraparticle mass transport limitations result in a selectivity loss. This situation prevails only close to the reactor inlet, where the concentrations of the intermediate product are smaller than the oxygen concentration. However, in the most part of the PBMR, the oxygen concentration is small compared to those of the hydrocarbons. Therefore, the relative oxygen concentration gradients within the catalyst particles are much larger than the relative hydrocarbon concentration gradients (see full lines in Figure 2), and consequently have a larger effect on the local reaction rates. In the following, the effect of intraparticle masstransport limitations will be discussed for low oxygen concentrations in the gas phase compared to those of the hydrocarbons. First, the case where both A and P react with one oxygen molecule (i.e., ν1 ) ν2 ) 1) is considered and subsequently the influence of the ratio of stoichiometric coefficients is elucidated. 2.3.1. Reaction Stoichiometry: ν1 ) ν2 ) 1. The product selectivity with respect to the bulk concentration is directly

〈γPγm〉 〈γAγn〉

(15)

The effect of intraparticle mass-transfer limitations is indicated by differences between σb and 〈σ〉. In analogy to the effectiveness factor, a ratio Fσ of selectivity with and without transport limitations could be defined as

Fσ )

(

)

〈γPγm〉 〈σ〉 ) 1 - pnm /(1 - pnm) σb 〈γ γn〉 A

(16)

However, this ratio of the product selectivities is not very illustrative and is impractical if pnm approaches 1. Furthermore, for pnm values above unity, Fσ takes values below 1, because 〈σ〉 is less negative than σb ) 1 - pnm < 0, as illustrated in Figure 3. Therefore, the following definition is chosen to elucidate the effect of intraparticle diffusion limitations on the product selectivity

F1-σ )

m 1 - 〈σ〉 〈γPγ 〉 ) 1 - σb 〈γ γn〉 A

Note that F1-σ < 1 indicates a positive effect on the average product selectivity 〈σ〉. The effect of intraparticle mass-transfer limitations on the product selectivity is shown in Figure 4 as a function of the modified Thiele modulus for two sets of reaction orders. The figure clearly shows that the selectivity calculated with the bulk concentrations underpredicts the actual selectivity that accounts for the concentration profile inside the particle. Because of the higher oxygen dependency of the consecutive reaction, the oxygen concentration profiles decrease the average reaction rate of this undesired reaction more pronouncedly than the main reaction, which results in an improved product selectivity. As can be concluded from Figure 4, the selectivity gain caused by the oxygen profile inside the particle is larger for

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Figure 3. Influence of intraparticle transport limitations on the average product selectivity as a function of the modified Thiele modulus for different values of pnm (cb/cAb ) 0.1 and cb/cPb ) 0.1). (a) n ) 0.5, m ) 1 and (b) n ) 1, m ) 2.

Figure 4. Influence of intraparticle transport limitations on the average product selectivity as function of the modified Thiele modulus for different values of pnm (cb/cAb ) 0.1 and cb/cPb ) 0.1). (a) n ) 0.5, m ) 1 and (b) n ) 1, m ) 2.

Figure 5. Effect of radial oxygen profile inside the particle on the product selectivity as a function of the selectivity at bulk concentrations and the modified Thiele modulus. Comparison of cb/cAb ) cb/cPb ) 0.1 (full lines) and cb/cAb ) cb/cPb ) 1 (dashed lines). (a) n ) 0.5, m ) 1 and (b) n ) 1, m ) 2.

higher values of φ′ and pnm, i.e., at higher oxygen concentration gradients and at relatively higher reaction rates of the consecutive reaction. Thus, an increase of the catalyst particle size (with a corresponding increase of φ′) results in an improvement of the product selectivity as long as the oxygen concentration in the gas phase is small compared to the concentrations of A (cb < ν1cAb) and P. This effect has been reported in the literature for the oxidative coupling of methane both as a result of model calculations8 (referred to as intrapellet oxygen sieving) and as an experimental result.9 Reyes et al.8 concluded that low oxygen concentrations permit the use of larger pellets. They also indicated that the oxygen concentration has to be low compared to the hydrocarbon concentrations. Follmer et al.9 found in a packed bed reactor for the same oxygen conversion higher product selectivities when using larger catalyst pellets. Their reaction scheme has a parallel route of coupling (n ) 0.8) and combustion (m ) 1.6). In the case of a consecutive reaction scheme, however, the concentration of the intermediate product is low near the inlet of the packed bed reactor, and thus the intraparticle mass-transport limitations have a negative influence

on the product selectivity. In section 3, therefore, the effect of the particle size on the product selectivity will be investigated as an integral result over the entire reactor length. If the oxygen concentration is on the same order of magnitude as the hydrocarbon concentrations, mass-transport limitations inside the particle result in a decreased concentration of A and an increased concentration of P (see Figure 2), which decreases the product selectivity, especially at high selectivities. This is illustrated in Figure 5 for two different cases with different oxygen/hydrocarbon concentration ratios: cb/cAb ) 0.1 (full lines) and cb/cAb ) 1 (dashed lines). 2.3.2. Influence of the Stoichiometric Coefficients. If the oxygen concentration is small compared to the hydrocarbon concentrations, the particle efficiency for oxygen is independent of the stoichiometric coefficients ν1,2 * 1. Moreover, the effect on the product selectivity is the same as for the case ν1 ) ν2 shown in Figure 4, which was confirmed by calculations for ν1,2 ) 0.5 (cb/cAb ) cb/cPb ) 0.05) and ν1,2 ) 2 (cb/cAb ) cb/cPb ) 0.2) for n ) 0.5, m ) 1 and n ) 1, m ) 2 (not shown). In the case of an oxidative dehydrogenation reaction system (ν1 ) 0.5), the molar conversion rate of oxygen in the primary

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Figure 6. Influence of the stoichiometric coefficients. Full lines ν1 ) ν2 ) 1 and cb/cAb ) cb/cPb ) 0.1. Dashed lines (a) ν1 ) 0.5, ν2 ) 1, cb/cAb ) cb/cPb ) 0.05 and (b) ν1 ) 0.5, ν2 ) 10, cb/cAb ) cb/cPb ) 0.05.

Figure 7. Influence of the stoichiometric coefficients. Full lines: ν1 ) ν2 ) 1 and cb/cAb ) cb/cPb ) 0.1. Dashed lines: (a) ν1 ) 1, ν2 ) 2, cb/cAb ) cb/cPb ) 0.1 and (b) ν1 ) 0.5, ν2 ) 10, cb/cAb ) cb/cPb ) 0.05.

reaction is only half that in the consecutive reaction. Therefore, the cb/cAb ratio decreases to values below where the effect of the oxygen gradients dominates the effect of the hydrocarbon concentration gradients, and the following correlation may serve as a guideline: max{cb/(ν1cAb), cb/(ν1cPb), cb/(ν2cPb)} e 0.1. Especially for modified Thiele moduli above unity, the effect of intraparticle mass-transfer limitations on the product selectivity is influenced by the stoichiometric coefficients. This is shown for the reaction orders n ) 0.5, m ) 1 in Figure 6 and for n ) 1, m ) 2 in Figure 7. For smaller values of ν1 (Figure 6a) as well as higher values of ν2 (Figure 7a), the effect of intraparticle mass-transfer limitations on the product selectivity is more pronounced compared to the case of ν1 ) ν2 ) 1, especially for higher values of pnm. Cases with ν1 ) 0.5, ν2 ) 1 and ν1 ) 1, ν2 ) 2 (same ratio of ν1/ν2) will result in the same curves.6 In a PBMR, an axial concentration profile of the target product P will prevail. Near the reactor entrance, the concentration of the target product is very low, and large concentration gradients inside the particle are formed with negative consequences for the intrinsic selectivity of a catalyst particle. A reduction of the oxygen concentration helps to reduce this selectivity loss. Consequently, it should be advised to add all oxygen distributively to the packed bed to keep the zone where the oxygen concentration exceeds that of the hydrocarbons (A or P) as small as possible. However, the effect of the initial drop of the intrinsic selectivity near the reactor inlet on the integral reactor selectivity has to be evaluated in order to find the optimal inlet oxygen concentration. The low oxygen level in the PBMR requires an increased reactor volume (for n > 0) compared to a fixed bed reactor with premixed feed (FBR) and possibly an increase in the size of the catalyst particles because of pressure drop considerations. Yet, in contrast to the FBR, an increase in the particle diameter will not result in a selectivity loss in the PBMR.

Figure 8. Effect of particle size on the axial concentration profiles of A and P at a constant and uniform oxygen concentration (c0/cA0 ) 0.05).

3. Integral Effects of Intraparticle Transport Limitations In the previous section, it was shown that the concentration gradients inside the particles induced by mass-transport limitations can result in an improvement of the intrinsic product selectivity (m > n), provided that the oxygen concentration in the gas phase is small compared to the concentrations of the hydrocarbons. However, this condition is definitely not fulfilled in the initial zone of the packed bed, where the first intermediate product is formed, and eventually, if a high degree of conversion for reactant A is realized, near the end of the packed bed. Moreover, the axial concentration profiles in the bulk of the gas phase are altered because of the intraparticle transport limitations. Whether the positive or negative effects of intraparticle mass-transfer limitations prevail in the PBMR is investigated next. 3.1. One-Dimensional, Heterogeneous PBMR Model. To study the effect of intraparticle mass-transport limitations on the integral PBMR-performance, we developed a one-dimensional, heterogeneous model. Obviously, the performance of the PBMR also depends on the overall design, as, for example, the

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Table 1. Model Equations for the One-Dimensional, Heterogeneous PBMR Model boundary conditions VFωi0 ) vFωi|z)0

gas-phase mass balance ∂/∂t(Fωi) ) -∂/∂z(Fωiv) - ans,i + φm,distr,i catalyst phase mass balance nr ∂/∂t(Fωi) ) 1/r2 ∂/∂r(r2Deff,i F∂ωi /∂r) + Mi∑ j)1 νij rj transmembrane flux φm,distr,i ) Φm,distrωdistr,i /(π/4)dt2L

tube diameter, the radial flow pattern (mainly determined by the particle to tube ratio), or the pressure drop over the length of the packed bed. However, to clearly elucidate the effects of intraparticle diffusion limitations, we have ignored these effects here. To facilitate the discussion of the effects. we have also seclected a constant and uniform distribution of the added flow. Model assumptions: • Adiabatic, one-dimensional plug flow model: The equations of change in the gas phase are calculated on the basis of the assumption of a one-dimensional plug flow model neglecting the axial dispersion and the pressure drop over the packed bed. Source terms of the component continuity equations are due to mass transfer between the bulk of the gas phase and the catalyst surface, and the added mass flow. Homogeneous gas-phase reactions have not been considered here. • Macroporous, symmetrical catalyst particle: The catalyst particles are assumed to be of spherical shape of constant particle diameter dP, so that a one-dimensional model can be used to describe the profiles inside the catalyst particle. The catalyst structure is assumed to be macroporous, so that transport mechanisms such as viscous transport or Knudsen diffusion can be neglected. It is assumed that the component mass transport inside the particle is described by Fick’s law of diffusion (due to the relatively low concentrations of the relevant components). The resulting model equations with the constitutive equations for the transmembrane flux and interphase mass transport have been listed in Table 1. At the reactor inlet and outlet, the standard Danckwerts’ boundary conditions have been used for the gas-phase mass balance, whereas symmetry at the particle center and continuity of the mass flux at the particle outer surface has been applied for the catalyst phase mass balance. The molecular diffusion coefficients were calculated with the equation by Wilke10 using binary diffusion coefficients estimated with relation by Fuller et al.11 Methane, formaldehyde, and carbon dioxide were chosen for reactant A, product P, and waste product W. For the calculation of the effective diffusion coefficient, a value of /τ ) 0.15 was assumed. The gas mixture properties were calculated following Reid et al.12 using the data set by AIChE Design Institute for Physical Property Data, DIPPR, New York, for the pure component properties. To obtain the steady-state solutions of both the mass balances in the bulk of the gas phase and for the catalyst phase, we have solved the equations with accumulation terms using a standard finite difference technique. The partial differential equations representing the gas and catalyst phase mass balances have been solved sequentially, where the equations are coupled via the concentration and concentration gradient at the particle outer surface. Typically, 100 equidistant axial grid cells for the gasphase mass balance has been used, whereas typically 50 cells on the particle radius with grid refinement toward the catalyst surface were used to solve the catalyst phase mass balance (by varying the number of grid cells, grid independence of solutions was demonstrated). The model parameter values used in the calculations have been summarized in Table 2. If not mentioned otherwise, the

∂ωi/∂z|z)L ) 0

Deff,iF∂ωi/∂r|r)R ) ns,i ∂ωi/∂r|r)0 ) 0 interphase mass transport ns,i ) Fks,i(ωi - ωs,i)

Table 2. Model Parameters rate constants: n ) 0.5/m ) 1.0 rate constants: n ) 1.0/m ) 2.0 stoichiometric coefficients catalyst mass catalyst density particle diameter interphase mass-transfer coefficient reactor pressure and temperature premixed feed volumetric flow mole fraction of reactant A mole fraction of oxygen distributed feed volumetric flow mole fraction of oxygen a

k′1 ) 0.015 mol/(g s bar1+n) k′2 ) 0.045 mol/(g s bar1+m) k′1 ) 0.150 mol/(g s bar1+n) k′2 ) 4.500 mol/(g s bar1+m) ν1 ) 1 ν2 ) 1 mcat ) 0.1-0.32 g Fcat ) 1000 kg/m3 dp ) 0.1-3 mm ks ) 1 m/s p, T ) 1.013 bar, 550 K ΦV,0 ) 100 mL(STP)/min yA,0 ) 0.1 yO2,0 ) 0.03 ΦV,distr ) 35 mL(STP)/mina yO2,distr ) 0.2

STP ) 1.013 bar and 298 K.

concentration ratio in the premixed feed is yA/yO2 ) 10/3, and 70% of the oxygen is added in distributive mode to the PBMR, according to the distribution for optimized product yield. In this section, the effect of intraparticle mass-transfer limitations on the integral selectivity of a PBMR is studied. Because the effects of intraparticle mass-transport limitations on the product selectivity and the activity of the catalyst interfere, the situation is first studied in which the oxygen concentration was kept constant along the axial coordinate of the PBMR, regardless of the size of the catalyst particles. Subsequently, the magnitude of the distributed oxygen flow was kept constant along the axial coordinate of the reactor, resulting in different axial oxygen concentration profiles depending on the catalyst particle diameter. 3.2. Effect of Particle Size at Axially Constant Oxygen Concentration. First a constant and uniform oxygen concentration has been chosen, which was 20 times smaller than the inlet concentration of the hydrocarbon A. The resulting axial concentration profiles of A and product P are shown in Figure 8. For larger catalyst particles, the effects of intraparticle masstransfer limitations become more pronounced, as expected from the larger values for the modified Thiele modulus (Figure 9b). In regions of the PBMR where the oxygen concentration is small compared to the hydrocarbon concentrations, intraparticle mass-transport limitations increase the product selectivity, because the rate of the consecutive reaction is more decreased by the reduced oxygen concentration inside the particle than the primary reaction. In the first 10% of the PBMR, the concentration of P is well below that of oxygen. Thus the effect of the intraparticle mass-transfer limitations on the product selectivity is therefore negative at the inlet of the PBMR, i.e., F1-σ > 1 (see Figure 9a). Figure 10 shows a comparison of the calculated local selectivity ratios and particle efficiencies with the values determined in Section 2 for c/cA ) c/cP ) 0.1 and pnm ) 0.01. The question now arises whether the positive effect of intraparticle diffusion limitations at the PBMR outlet outweighs

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Figure 9. Axial profile of (a) the selectivity ratio and (b) the modified Thiele moduli for different particle sizes ranging from 0.1 to 1.0 mm.

Figure 10. Local selectivity ratio and particle effectiveness for oxygen consumption as a function of the modified Thiele modulus for four different particle diameters ranging from 0.1 to 1.0 mm. The dashed lines correspond to c/cA ) c/cP ) 0.1 and pnm ) 0.01.

Figure 11. Effect of the particle diameter on the selectivity-conversion plot.

the negative effect at the PBMR inlet. In Figure 11, the integral effect of the particle size on the reactant conversion and product selectivity is illustrated. Figure 11 shows that for larger particles, higher selectivities can indeed be achieved at the same conversion. However, the overall conversion was decreased becauseof the intraparticle mass-transport limitations. To arrive at the same overall product yield, we require more catalyst mass when selecting a larger particle diameter, which will be studied later. In these calculations, it was assumed that the oxygen concentration was axially constant. However, in most applications, the membrane flux will be (approximately) constant and not the oxygen concentration. This is studied in the next paragraph. 3.3. Effect of Particle Size at Axially Constant Oxygen Transmembrane Flux. If the membrane flux is independent of the reaction within the PBMR, e.g., a fixed mass flow of air is equally distributed over the length of the membrane, the reduced particle effectiveness factor (see Figure 12a) causes an increase in the oxygen concentration in the gas-phase further downstream in the PBMR (see Figure 12b) with a corresponding negative effect on the product selectivity. The two opposing effects of increased downstream gas-phase oxygen concentrations and intraparticle oxygen concentration

profiles due to mass-transfer limitations can cancel each other for smaller particles, as shown by the comparison of the conversion versus selectivity plot in Figure 13. For large values of the modified Thiele moduli, the reduced particle effectiveness factor results in an overall selectivity loss due to the strongly increased oxygen concentrations at the reactor outlet. This means that the integral effect of intraparticle mass-transport limitation is opposite to the intrinsic effect outlined in the previous section because of its effect on the axial oxygen concentration profile. Figure 14a shows that the particle effectiveness factor can still be described by eq 13, despite the fact that the oxygen concentration is not small in all regions of the PBMR compared to those of the hydrocarbon reactant and intermediate product. The reactant A is fed to the packed bed in a ratio of yA/yO2 ) 10. For the case of 0.1 mm diameter particles, this ratio initially increases to a maximum value of 26 located close to the minimum oxygen concentration and subsequently slowly decreases to 1.9. The ratio of the intermediate product yP/yO2 is at the inlet zero and reaches 8.4 at maximum (at about 30% of the reactor length) and 4.3 at the reactor outlet. Near the reactor inlet, the intraparticle concentration gradients of the intermediate product result in a selectivity loss (yp/yO2 < 2 for the case of dP ) 0.1 mm) or a smaller selectivity improvement (see values for dP)0.3 mm in Figure 14b; the dashed line shows results from Section 2 for yA/yO2 ) yP/yO2 ) 10 and pnm ) 0.01). For a larger particle with a diameter of dP ) 1 mm, the increased oxygen concentrations in the gas phase (i.e., decreased ratios of hydrocarbons to oxygen concentrations) result over the full length of the reactor in a negative effect of the intraparticle hydrocarbon concentration profiles on the intrinsic selectivity. The effect of intraparticle mass-transport limitations on the performance of PBMR was also studied for other series of reaction kinetics with reaction orders n ) 1, m ) 2, and for both sets of reaction orders for a higher catalyst mass (0.32 g).6 A summary of the most important observations:

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Figure 12. Influence of the particle size on the axial profiles of (a) the particle effectiveness for oxygen consumption and (b) the oxygen concentration for a kinetic system with n ) 0.5 and m ) 1.

Figure 13. Influence of the particle size on the selectivity-conversion plot for a reaction system with oxygen reaction orders n ) 0.5 and m ) 1.

• For φ′ < 0.5, the effect on the effectiveness factor of oxygen is small and the effect on product selectivity and reactant conversion is negligible. • Further increase in the modified Thiele modulus (φ′ < 1.1) results in an increase of the oxygen concentrations downstream in the PBMR, which results in a selectivity loss. The integral loss of product yield for a particle diameter of 1 mm is about 5%. • For the cases with increased catalyst mass (corresponding to lower axial oxygen concentrations), the intrinsic selectivity improvements due to intraparticle mass-transfer limitations are higher. Therefore, the integral loss of product yields (dp ) 1 mm) decreases below 2%. Thereby, the modified Thiele modulus even increases for the reaction orders n ) 0.5, m ) 1 to φ′ < 1.6. 3.4. Effect of Increased Catalyst Mass to Account for the Reduced Activity at Axially Constant Membrane Flux. If the reduced particle effectiveness is accounted for by distributing the same total flow of oxygen over a larger amount of catalyst, the selectivity in a PBMR can be slightly improved because of the intraparticle mass-transport limitations; however, it is at the expense of increased investment costs. This is illustrated in Figure 15. The increase in the particle diameter by a factor 30 is compensated for by an increase in the catalyst mass by a factor of 3.2, such that the oxygen concentration at the reactor outlet is about the same (see Figure 15b). However, the resulting slight improvement in the product selectivity is small compared to the improvement that could have been obtained if the catalyst mass was increased still using the smaller particle diameter. With the smaller particle diameter, the oxygen concentrations downstream in the gas phase in the PBMR are much smaller, with a corresponding strong increase in product selectivity. It can be concluded that increasing the particle diameter is not an efficient way to improve the product selectivity in a PBMR. On the other hand, if the design of the PBMR requires

the application of larger catalyst particles (e.g., because of pressure drop considerations), the effect of possible intraparticle mass-transfer limitations on the product selectivity is not disadvantageous, provided that the oxygen concentration is small compared to those of the hydrocarbons. Because of intraparticle diffusion limitations, however, a larger amount of catalyst is required to achieve the same conversion. 3.5. Influence of the Stoichiometric Coefficients. A case with ν1 ) 0.5 and ν2 ) 10 was selected to demonstrate the influence of the stoichiometric coefficients. Ethylbenzene, styrene, and carbon dioxide were selected as reactant A, product P, and waste product W. In the preceding sections, the same effective diffusion coefficients were assumed for the hydrocarbons and oxygen. In the following model calculations, the effect of the size of the hydrocarbon molecule on the diffusion coefficient is included. The ratio of the effective molecular diffusion coefficients of reactant A and oxygen decreases with the size and the molecular weight of reactant A. To estimate this ratio, we considered a ternary mixture of reactant A (yA ) 0.1), oxygen (yO2 ) 0.01), and nitrogen. Assuming that the intraparticle transport is dominated by molecular diffusion and applying the equation by Wilke,10 the ratio is given by

y N2 yA + Dm,A 1 - yA DO2A DO2N2 Deff,A ) ) ) yN2 Deff,O2 Dm,O2 1 - yO2 yO2 + DAO2 DAN2 1 - yA 1 - y O2

y A + y N2

DAO2 DO2N2

y O2 + y N2

DAO2

(17)

DAN2

which after substitution of the binary diffusion coefficients approximated by Fuller et al.11 results in

1 - yA yA + 0.357yN2 Deff,A ) ) 0.44 Deff,O2 1 - yO2 yO2 + 0.965yN2

(18)

As a result, the hydrocarbons will form even stronger intraparticle concentration profiles, with a negative consequence for the intrinsic product selectivity. Because of the higher consumption rate of oxygen in the consecutive reaction (despite the lower consumption rate in the primary reaction), the distributive flow had to be increased to ΦV,distr ) 70 mL/min. The other model parameters were as listed in Table 2. The effect of the decreased hydrocarbon diffusivity is illustrated in Figure 16. The particle efficiency, ηO2, and the

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Figure 14. Influence of the particle size on (a) the particle effectiveness for oxygen consumption and (b) the selectivity ratio as a function of the modified Thiele modulus φ′ for reaction kinetics with oxygen reaction orders n ) 0.5 and m ) 1.

Figure 15. Effect of an increased particle size or a reduced oxygen concentration on (a) the axial oxygen concentration profile and (b) selectivityconversion plot for arbitrary kinetics with oxygen reaction orders n ) 0.5 and m ) 1.

Figure 16. Influence of the particle size on (a) the particle effectiveness for oxygen consumption and (b) the selectivity ratio as a function of the modified Thiele-modulus φ′ for reaction kinetics with n ) 0.5/m ) 1 and ν1 ) 0.5/ν2 ) 10.

selectivity ratio, F1-σ, show considerable differences from the values computed in section 2, where equal effective diffusion coefficients were assumed. The particle efficiency for the oxygen consumption even takes values above unity near the reactor inlet, which shows the strong contribution of the consecutive reaction in the oxygen consumption. For a particle diameter of 1 mm, the product yield reduced by 12% (relative value). For a higher catalyst mass and reduced inlet oxygen concentration, the ratio of hydrocarbon to oxygen concentration is increased so far that the particle properties are dominated by the intraparticle concentration profiles of oxygen. After the first 20% of the reactor length, the particle efficiency and selectivity ratio agree well with the values from section 2, and the loss of product yield reduces to only 5%.6 3.6. External Transport Limitations and Temperature Effects. For the model calculations presented in this paper, an external mass-transfer coefficient for the transport from the bulk gas phase to the catalyst surface of ks ) 1 m/s was assumed. This value was chosen such that the external mass transfer had a negligible effect on the performance of the catalyst particle. Transport limitations from the gas bulk to the catalyst surface

result in a decrease in the concentrations of oxygen and reactant A and an increase in the concentration of product P. Whether the negative effect of the change in the hydrocarbon concentrations or the positive effect of the decrease in the oxygen concentration on the intrinsic particle selectivity prevails depends on the ratio of the gas-phase concentrations. From Figure 18a, it can be observed in the region of the PBMR where the intraparticle mass transfer coefficients have a negative effect on the intrinsic product selectivity (F1-σ > 1) that this effect is further increased by the additional external mass-transfer limitation (ks ) 0.1 m/s). Likewise, the selectivity improvement is intensified in the region in which the positive effect of the intraparticle transport limitations prevails. Obviously, external mas-transfer limitations result in an activity loss, which is shown in Figure 18b. In summary, it can be concluded that external mass-transfer limitation intensifies the effect of intraparticle mass-transfer limitations. Finally, heat-transport limitations from the bulk gas phase to catalyst surface are more significant than intraparticle heattransport limitations, especially in laboratory reactors,13 so the temperature inside the particle is relatively uniform. The effect

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Figure 17. Influence of the particle size on (a) the axial oxygen concentration profile and (b) the selectivity-conversion plot for arbitrary kinetics with ν1 ) 0.5/ν2 ) 10.

Figure 18. Axial profiles of (a) the selectivity ratio and (b) particle effectiveness for oxygen consumption for different external mass-transfer coefficients.

of increased catalyst temperature on intrinsic selectivity of the catalyst particle depends on the difference ∆EA ) EA,2 - EA,1 between the activation energies of consecutive (EA,2) and primary reaction (EA,2). For ∆EA > 0, the reaction rate of the consecutive reaction increases more strongly with temperature, and energy-transport limitations results in additional selectivity losses. However, the increased reactivity also results in an increase in intraparticle mass-transfer limitations with a positive influence on the product selectivity. A more detailed discussion of temperature effects is beyond the scope of this article. 4. Summary and Conclusions A PBMR can be applied to improve the selectivity to the intermediate product in a partial oxidation reaction system. A low oxygen concentration throughout the entire reactor reduces product losses by the consecutive combustion reaction, provided that the oxygen dependency of the target product formation reaction rate is smaller than that of the consecutive reaction (n < m). If the oxygen concentration is small compared to the hydrocarbon concentrations, the particle effectiveness for oxygen consumption and the intrinsic product selectivity are determined by the oxygen concentration gradients inside the particle. Intraparticle mass-transfer limitations then result in an improvement of the product selectivity to the intermediate product. For this case, a modified Thiele modulus can be defined using the concentrations in the gas bulk

R φ′ ) 3

x

n+1 m+1 ν k c cn-1 + ν2k2cPbcm-1 b 2 1 1 Ab b 2 (19) D

and the particle effectiveness factor for oxygen consumption is well-approximated with the standard correlation for a single, first-order reaction for a wide range of different reaction orders

Figure 19. Influence of intraparticle-transport limitations on the average product selectivity as a function of the modified Thiele modulus for different values of pnm (cb/cAb ) cb/cPb ) 0.1, ν1,2 ) 1) and orders n,m as indicated in the figure.

(n and m) and ratios of the reaction rates of the undesired and desired reactions (pnm). However, the product selectivity is not a simple function of the above-described modified Thiele modulus. It also depends on the ratio of the secondary to the primary reaction rates (pnm) and on the reaction orders n and m, as summarized in Figure 19. However, the intraparticle mass-transfer limitations will have their effect on the axial oxygen concentration profile, which needs to be taken into account to quantify the overall effect of intraparticle diffusion limitations on the integral product selectivity. When studying the integral effect on the performance of the PBMR, two aspects have to be considered that both have a negative influence on the final product selectivity. First, in the inlet zone of the PBMR (until the concentration of the produced product P exceeds that of oxygen), the intrinsic product selectivity is decreased for larger catalyst particles. Second, an even stronger loss in the integral product selectivity is caused by the reduced consumption of oxygen in large particles because of the intraparticle diffusion limitations. The increased downstream gas-phase concentration counteracts the improvement of

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7523

the intrinsic selectivity for larger particles. For small modified Thiele moduli (φ′ < 0.5), the effect of intraparticle mass-transfer limitations can be neglected. If the modified Thiele modulus in the PBMR is above φ′ > 1, the overall effect of the intraparticle mass-transfer limitations is negative. However, intraparticle mass transfer can still result in improved integral product selectivity, if the reduced catalyst activity is counteracted, e.g., by increasing the catalyst mass. However, with the same increase in the catalyst mass while still using the small catalyst particles, the product selectivity can be enhanced much more because of a stronger decrease in the oxygen concentration level in the PBMR. Nonetheless, if the design of the PBMR requires the application of large particles, for example, because of pressure drop limitations, negative consequences on the product selectivity can be prevented if the catalyst mass is increased to account for the reduced particle effectiveness. To avoid negative effects of the intraparticle diffusion limitations on the integral product selectivity, we should choose the particle size such that the modified Thiele modulus φ′ < 1. Notation A,P,W ) hydrocarbon reactant, target, and waste product a ) specific surface area per volume of packed bed (m2/m3) c ) concentration (mol/m3) dP ) diameter of catalyst particle (m) dt ) tube diameter (m) D ) diffusion coefficient (m2/s) F ) selectivity ratio k ) reaction rate constant (mol m-3 s-1 bar1+n, mol m-3 s-1 bar1+m) k′ ) reaction rate constant (mol g-1 s-1 bar1+n, mol g-1 s-1 bar1+m) ks ) interphase mass transfer coefficient (m/s) L ) length of the packed bed (m) m ) reaction order in oxygen of secondary reaction forming the waste product; mcat ) catalyst mass (kg) M ) molar mass (kg/mol) n ) reaction order in oxygen of primary reaction forming the target product; nr ) number of reactions ns ) interphase mass flux (kg m-2 s-1) p ) pressure (bar) pnm ) ratio of reaction rates in the secondary and primary ) reaction () (k2cPb/k1cAb)cm-n b PBMR ) packed bed membrane reactor r ) radial direction (m) rj ) rate of reaction j (mol m-3 s-1) R ) radius of catalyst particle (m) t ) time (s) T ) temperature (K) V ) velocity (m/s) y ) molar fraction z ) axial coordinate of the packed bed (m) Greek Symbols γ ) dimensionless concentration () c/cR)  ) porosity η ) yield, particle effectiveness factor ν ) stoichiometric constant

F ) density (kg/m3), dimensionless coordinate in radial direction () r/R) σ ) selectivity τ ) tortuosity φ′ ) modified Thiele modulus )

+ ((m+1)/2)ν2k2cPbcm-1 R/3x((n+1)/2)ν1k1cAbcn-1 b b

D φm,distr ) mass flow added distributively per volume of packed bed (kg m-3 s-1) ΦV ) volumetric flow rate (m3/s or mL/min) Φm ) mass flow rate (kg/s) ω ) mass fraction Subscripts 0 ) at reactor inlet conditions A,P,W ) hydrocarbon reactant, target, and waste product b ) at bulk conditions cat ) catalyst distr ) distributed eff ) effective max ) maximum value O2 ) oxygen p ) particle R ) at the outer radius of a catalyst particle s ) at catalyst surface Literature Cited (1) Lafarga, D.; Al-Juaied, M. A.; Bondy, C. M.; Varma, A. Ethylene epoxidation on Ag-Cs/R-Al2O3 catalyst: experimental results and strategy for kinetic parameter determination. Ind. Eng. Chem. Res. 2000, 39, 2148. (2) Pedernera, M.; Mallada, R.; Mene´ndez, M.; Santamarı´a, J. Simulation of an inert membrane reactor for the synthesis of maleic anhydride. AIChE J. 2000, 46, 2489. (3) Shakhnovich, G. V.; Belomestnykh, I. P.; Nekrasov, N. V.; Kostyukovsky, M. M.; Kiperman S. L. Kinetics of ethylbenzene oxidative dehydrogenation to styrene over vanadia/magnesia catalyst. Appl. Catal. 1984, 12, 23. (4) Diakov, V.; Blackwell, B.; Varma, A. Methanol oxidative dehydrogenation in a catalytic packed-bed membrane reactor: experiments and model. Chem. Eng. Sci. 2002, 57, 1563. (5) Deshmukh, S. A. R. K.; van Sint Annaland, M.; Kuipers, J. A. M. Kinetics of the partial oxidation of methanol over a Fe-Mo catalyst. Appl. Catal., A 2005, 289, 240. (6) Ku¨rten, U. Ph.D. Thesis, University of Twente, Enschede, The Netherlands, 2003. (7) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (8) Reyes, S. C.; Kelkar, C. P.; Iglesia, E. Kinetic-transport models and design of catalysts and reactors for the oxidative coupling of methane. Catal. Lett. 1993, 19, 167. (9) Follmer, G.; Lehmann, L.; Baerns, M. Effect of transport limitations on C2+ selectivity in the oxidative methane coupling reaction using a NaOH/ CaO catalyst. Catal. Today 1989, 4, 323. (10) Wilke, C. R. Diffusional properties of multicomponents gases. Chem. Eng. Prog. 1950, 92, 1. (11) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. A new method for prediction of binary gas-phase diffusion coefficients. Ind. Eng. Chem. 1966, 58, 19. (12) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill Book Company: New York, 1988. (13) Mears, D. E. Tests for transport limitations in experimental catalytic reactors. Ind. Eng. Chem. Proc. Des. DeV. 1971, 10, 541.

ReceiVed for reView May 7, 2007 ReVised manuscript receiVed June 13, 2007 Accepted June 19, 2007 IE070638K