- Email: [email protected]

Introduction

The laboratory application and investigation of different types of membrane reactors as a promising unit operation started in the 1980s (Marcano and Tsotsis, 2002; Seidel-Morgenstern, 2010). It is well known that the selectivity in reaction networks toward a target compound can be increased by properly adjusting the local concentration of the reactants involved. A membrane separation unit can be applied especially for adjustment of the reactant’s and/or product’s concentration simply by coupling the membrane separation unit with a chemical/biochemical reactor. This coupling or integration often is made in the same unit. During the last three decades, this technical concept has attracted substantial worldwide research and process development efforts (Marcano and Tsotsis, 2002). There are several books and reviews published in the fields of both membrane reactors (Reij et al., 1998; Coronas and Santamarı´a, 1999; Saracco et al., 1999; Julbe et al., 2001; Marcano and Tsotsis, 2002; Paturzo et al., 2002; Dittmeyer et al., 2004; Charcosset, 2006; Judd, 2006; McLearly et al., 2006; Ozdemir et al., 2006; Seidel-Morgenstern, 2010) and membrane bioreactors (Marcano and Tsotsis, 2002; Rios et al., 2004; Fenu et al., 2010; Santos et al., 2010). It is not the aim of this work to discuss in detail the properties, applications, or devices of this process. We look at this unit operation as deeply as needed for its mathematical modeling or description. On the other hand, considering the essential differences in the operation and behaviors between the membrane reactors and the membrane bioreactors, mainly due to the other kinetic models, especially for the living organisms, these two reactor types will be discussed separately. Basically, two configurations of the membrane reactor system can be applied: in the first case, the reactor and the membrane separation equipment are simply connecting in series, while in the second case, the real membrane reactor concept combines these two different processing units (namely, reactor and a membrane separator) into a single unit (Figure 8.1). The membrane can serve as a distributor of one of the reactants or as an active catalyst and permselective layer. The subject of this chapter is to analyze briefly the mass transport in this latter membrane reactor configuration.

8.2

Membrane Reactor Configurations

Membrane-based reactive separation processes are mostly applying thin permselective, porous, or dense layers prepared by means of materials that are organic, inorganic, metal, and so on. The choice of a porous or a dense film and the type of material Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00008-9 © 2012 Elsevier Inc. All rights reserved.

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Basic Equations of the Mass Transport through a Membrane Layer

used for manufacturing depends on the desired separation process, operating temperature, and driving force used for separation; the choice of material depends on the desired permeance and selectivity, and on thermal and mechanical stability requirements (Marcano and Tsotsis, 2002). There are various membrane reactor configurations at laboratory scale that focus on the reactant/product distribution in order to improve selectivity-conversion performances. According to Saracco et al. (1999) and Seidel-Morgenstern (2010), the membrane reactor concept can be divided into six groups which are schematically illustrated in Figure 8.2. Products in sweep fluid

Sweep fluid Reactants

Reactants

Catalytic and permselective membrane

Figure 8.1 Integrated membrane reactor system. (I)

(II) Catalytic membrane layer (A ⫹ B C)

Reactants

Homogeneous catalyst Ji

A

C

B

C

(III)

A

(IV)

A

B⫹C

Sweep

B

C

A

B

D

(V) A Sweep

A D⫹B

B⫹C

E

C E

B

(VI) A⫹B D⫹B JD

D E

E

A

D

B

A⫹B D⫹B

D E

JB

Figure 8.2 Illustration of most often-applied membrane reactor concepts (IVI).

D

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According to Figure 8.2, the essentials of this membrane concept are: 1. The reaction product is separated continuously from the homogeneous catalyst, thus, this process can be operating in continuous mode. 2. The membrane serves as a catalyst layer; reactants can supply in regular manner which enables the avoidance of side reactions. 3. The selective removal of the product B enables the enhancement of the productivity to shift the reaction to production of compound B. 4. This is a realization of the selective transport of product B which participates in a second reaction. 5. This concept enables the removal of the product in order to avoid the undesirable consecutive reaction. 6. This is a controlled addition of a reactant through the membrane in order to achieve higher selectivities and yields.

All these reactor configurations serve for better selectivity and yield of the chemical reaction. Thus, it is particularly important to understand the relation between local concentrations, temperatures, and the selectivity-conversion behavior. In the following section, a few basic expressions of chemical reaction engineering that are important for understanding how membrane reactor should act to achieve higher reaction efficiency are discussed.

8.3

Reaction Rate

The reaction rates are the key information required to quantify chemical reactions and to describe the performance of chemical reactors. The specific rate of a single reaction in which N components are involved is defined; for details, see Levenspiel (1999) and Westerterp et al. (1984): Qi 5

1 dci ; ξ i dt

i 5 1; . . . ; N

ð8:1Þ

where Qi is the reaction rate of component i (kg/m3s, kmol/m3s); c is the concentration (kg/m3, kmol/m3); and ξ i is the stoichiometric coefficient (equal to zero for inerts or diluents). Equation (8.1) is applied for a homogeneous catalyst reaction. In heterogeneous catalysis, often the mass or surface area of the catalyst is used for relation, thus the reaction rate measures as kmol/kgs, kmol/m2s, and so forth. Obviously, the chosen scaling quantity should be used consistently for calculation of the Qi reaction rate. The reaction rates can depend on temperature and the molar concentration change of reactants. Conversion for constant volume can be defined as Xi 5

coi 2ci coi

ð8:2Þ

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where coi is the initial or inlet concentration and Xi is the conversion. The reaction rate can be given for reaction of A 1 B2E 1 F reversible reaction as cA cB 2cE cF ð8:3Þ Q 5 k2 Keq where k2 is the reaction rate constant (m3/kmols); Keq is the equilibrium constant; and Q is the reaction rate (kmol/m3s). Selectivity of component E, σE, is the ratio between the amount of desired product E obtained and the amount of a key reactant, A, converted: σE 5

cP 2coPξ A coA 2cAξ P

ð8:4Þ

where coi is the initial concentration of components (kmol/m3) and ξi is the stoichiometric coefficient. Thiele modulus (ϑ), which is the ratio of characteristic time for radial diffusion to the characteristic time for reaction in the membrane, is described by the following equation: sﬃﬃﬃﬃﬃﬃﬃﬃﬃ δ2 Qi ð8:5Þ ϑ5 D i ci where δ is the membrane thickness; c-i is the average concentration of i (kmol/3); Qi is the reaction rate (kmol/m3s); and Di is the effective diffusion coefficient in the membrane layer (m2/s). For first-order chemical reaction, Q 5 k1c (k1 is the reaction rate constant (1/s)), thus the value of ϑ is as sﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ k1 D k1 δ 2 5 ϑ5 D βo

ð8:6Þ

Equation (8.6) is the well-known Ha-number for fluid phase, Ha (Ha ϑ, with β o5D/δ). Let us express ϑ for cylindrical space. Taking into account Eq. (3.8), the Thiele modulus for cylindrical space can be expressed as follows: rﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ k1 D k1 ro2 ϑ5 5 o D β lnð11δ=ro Þ

ð8:7Þ

with βo 5

D ro lnð11δ=ro Þ

ð8:8Þ

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197

The cylindrical mass transfer coefficient, β o, tends to the plane mass transfer coefficient, β o 5 D/δ, in limiting cases, namely when ro-N.

8.4

Modeling of Membrane Reactors

As seen in Figure 8.1, different membrane’s configurations can be obtained depending on the placement of a catalyst (tube and/or shell side or in/on membrane), and motion or not of catalyst particles (packed beds, fluidized beds). The mass balance equations should be given according to real operating conditions. The basis of the continuum models are the differential balance equation given by Eqs (7.1)(7.8) for the fluid zones, completing them by source or sink terms. Because fluid phases and catalyst and/or permselective membrane layers must be treated simultaneously, a multiphase approach is necessary (Seidel-Morgenstern, 2010). Accordingly, the balance equations should be given every three sections of a capillary membrane reactor, namely for the lumen and shell fluid phases as well as for the catalytic (or noncatalytic) membrane. Here we give the general equations that should be applied for every section, taking into account operating conditions of every section. The complete equations for conservation of momentum, total mass, component mass, and energy must be considered in every real fluid and membrane matrix. Thus, one needs for description of such a system continuity equation, momentum transport energy balance, and (n 21) species balance equations (n is number of components of the system). Accordingly, the starting equation system to be used is as it is given in Chapter 7. The equation of continuity (conservation of mass) in cylindrical coordinate system and with variable μ, ε, and ρ (Bird et al., 1960): @ðρεÞ 1 @ðrερυr Þ 1 @ðερυθ Þ @ðερυz Þ 1 1 1 50 @t r @r r @θ @z

ð8:9Þ

The equation of motion in cylindrical coordinates (in terms of Newtonian fluid with constant ρ and μ and porosity ε; these equations are assumed to be valid for membrane or packed-bed reactors as well; the fluid phase in packed membrane reactors are considered as quasihomogeneous flowing phase): r-component: @ðρευr Þ υr @ðrρευr Þ υθ @ðρευr Þ ρευ2θ @ðρευr Þ @ðεpÞ 1υz 1 1 2 1 r r @θ r @t @r @r @r 2 0 1 0 1 0 13 1 @ @ @ðευr ÞA 1 @ @ @ðευr ÞA 2 @ðευθ Þ @ @ @ðευr ÞA5 μr 1 2 μ 2 2μ 1 μ 24 r @r @r r @θ @θ r @θ @z @z 2ερgr 5 2εfr ð8:10Þ

198

Basic Equations of the Mass Transport through a Membrane Layer

θ-component: 0

1 @@ðρευθ Þ 1υr @ðρευθ Þ 1 υθ @ðρευθ Þ 2ρε υr υθ 1υz @ðρευθ ÞA1 1 @ðεpÞ r r @t @r @θ @r r @θ 2 0 0 1 0 13 1 @ 1 @ 1 @ @ðευθ ÞA 2 @ðευr Þ @ @ @ðευθ ÞA5 2 2μ 1 μ 24 @ μr ðrυθ ÞA1 2 @μ @r r @r r @θ @θ r @θ @z @z 2ερgθ 5 2εfθ ð8:11Þ z-component: 0

1 @ðρευ Þ υ @ðrρευ Þ υ @ðρευ Þ @ðρευ Þ z r z θ z z @ A1 @ðεpÞ 1 1 1 υz r r @θ @t @r @z @z 2 0 1 0 1 0 13 1 @ @ðευ Þ 1 @ @ðευ Þ @ @ðευ Þ z z z @μr A1 @μ A1 @μ A52 ερgz 52εfz 24 r @r @r r 2 @θ @θ @z @z ð8:12Þ where the source terms in the momentum conservation equations (SeidelMorgenstern, 2010): fj 5 f1 υj 1 f2 υj jυj with j 5 r; q; z

ð8:13Þ

The fr ; fz source terms in the momentum conservation equations can be neglected for lumen and shell without solid particles. This is not the case in porous membranes or in packed-bed lumens or shells. Equation (8.13) expresses that friction and inertial forces caused by flow through pores lead to an additional loss of momentum, accounted for by the source term f. The parameters f1 and f2, taking into account the pressure drop during transport through membrane layer, can be calculated using the coefficient determined by application of the dusty gas model (Wesselingh and Krishna, 2000) in the case of gasfluid phase: f1 5

μf ; 2 =τ32Þ1ðDμ =pÞ ðεdpore f

f2 5 0

ð8:14Þ

where μf is the dynamic viscosity of the gas phase (Pa s); dpore is the pore size; p is pressure (Pa), ε is porosity; τ denotes the tortuosity; and D is diffusivity of the key component (m2/s). The factor f1 takes into account the viscous slip at pore walls, the parameter f2 can be set to zero due to the laminar character of the flow (SeidelMorgenstern, 2010).

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The component balance equation will be as: 0

1

@@ðερxi Þ 1υz @ðερxi Þ 1υr @ðερxi Þ 1 υθ @ðερxi ÞA r @θ @t @z @r 8 0 1 0 1 0 19 = <1 @ @rDi @ðερxi ÞA1 1 @ @Di @ðερxi ÞA1 @ @Di @ðερxi ÞA 5 Mi Q^ i 2 ; : r @r @r r 2 @θ @θ @z @z ð8:15Þ where Q^ i denotes the reaction rate of component i related to the total volume of membrane (kmol/m3s); xi is the molfraction of component i; Mi is the mole weight of i (kg/kmol); and ρ is the fluid density (kg/m3). The equations of energy in terms of the transport properties (with Newtonian fluids of variable parameters) are 0 1 ^ ^ ^ ^ @@ðρC p εTÞ 1υz @ðρCp εTÞ 1 υr @ðrρC p εTÞ 1 υθ @ðρC p εTÞA @t @z @r @θ r r 8 0 1 0 1 0 19 ð8:16Þ n = X < @ kε @ðrTÞ 1 @ @T @ @T @ A1 @kε A1 @kε A 5 2 hi Q^ i :@r r @r r 2 @θ @θ @z @z ; i51 where C^ p denotes the heat capacity at constant pressure, per unit mass (kJ/kgK); k is the thermal conductivity (kJ/ms); hi is the molar enthalpy of formation of species i (kJ/ kmol); and M is the molecular weight (kg/kmol). Note that k 5 λρC^ p ; where λ is the thermal diffusivity (m2/s). Applying Eq. (8.16) for the membrane, it should be noted that Eq. (8.16) does not involve the heat capacity of the membrane matrix; its value assumed to be negligible. If it is not the case, this should also be taken into account. The relation between the superficial and interstitial (real) velocity in a porous medium is υz;o 5 ευz

ð8:17Þ

The same is true for the transverse velocity as well (υo 5 ευ). If we assume that the porosity on the membrane interface is equal to that in the membrane matrix, then υz,o or υr,o gives the velocity related to the total interface. It should be noted that the axial velocity is generally negligible in the membrane matrix (υz 5 0); only the transverse velocity should be taken into account. The conservation equations for mass and energy should be used in the reaction zone only. Chemical reactions can take place outside the membrane (lumen and/or shell sides—cases IIIVI in Figure 8.2, these are so called packed-bed membrane reactors) or in the catalytic membrane layer (case II in Figure 8.2, called catalytic membrane reactors). Simplification of the equation system, from Eqs (8.9)(8.17), is necessary for all practical cases in order to get the model as simple as possible. Equations (8.9)(8.17) can easily be given by variable parameters as well (Geraldes et al., 2001; Marriott and Sorensen, 2003; Seidel-Morgenstern, 2010, pp. 3334).

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Basic Equations of the Mass Transport through a Membrane Layer

8.4.1

Modeling of a Membrane Reactor with a Catalytic Membrane Layer

Because of its importance, let us look at the model equations of a catalytic membrane layer (case II, in Figure 8.2), assuming steady-state, highly diluted gas reaction system, and no gravity influence. A chemical reaction takes place in the membrane layer only. Model equations for cylindrical catalytic membrane layer for Newtonian fluid (steady state, with constant ρ, μ, C^ p ; k values) are listed in Table 8.1a with boundary conditions in Table 8.1b. Note that υr5ro 50 and υr5rδ 50 when the transmembrane pressure is equal to zero in every axial position. If there is a large volume change during the reaction, then this can create pressure difference between the membrane layer and the lumen and/or shell side, which can cause transverse and axial convective velocities in the membrane. Accordingly, the values of υr5ro and υr5rδ will differ from zero. On the other hand, looking at case II, in Figure 8.2, there is mass transfer of reactants on both sides of membrane; this fact is not involved in the boundary conditions for ro and rδ. Thus, the boundary can be expressed for mass and energy conservation of component B as follows: for r5rδ

DB;f

@cB @φ 5DB B ; @r @r

kB;f

@Tf @T 5kB @r @r

ð8:18Þ

Table 8.1a Model Equations for Cylindrical Catalytic Membrane Layer for Newtonian Fluid (Steady State, with Constant ρ, μ, C^ p , k values) Conversion of total mass [email protected] @ ðρrυr Þ1 ðρυz Þ 5 0 r @r @z Momentum conservation equations r-coordinate: @υr @υr [email protected] @υr υr @2 υr 1υz 2μ 2 2 1 2 5 2fr r ρ υr @r @r @r r @z r @r z-coordinate: @υz @υz @p [email protected] @υz @2 υz ρ υr 1υz 1 2μ 1 2 52fz r @r @z @r @z @z r @r Transport equation for species i: @φi @φi [email protected] @φi @ 2 φi υz 1υr 2Di 1 5 Q^ i r @z @r @r @z r @r Energy conservation equation X n @T @T @ 1 @T @2 T hi Q^ i 1υr 1 2k 1 5 ρC^ p υz @z @r @r r @r @z 1

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Table 8.1b Boundary Conditions for Balance Equations in Table 8.1a Boundary Conditions Geometric Position

Momentum

Mass

Energy

Shell side, z50, υz5υδ,o, υr50 ro#r#rδ

φi 5φδ;i

T5Tδ

Tube side, z50, υz5υz,o, υr50 ro#r#rδ

φi 5φoi

T 5T o

Tube side, z50, ro#r#rδ

p5po

Shell side, z50, ro#r#rδ

p5poδ @υz 50 @z

@φi 50 @z

z5L, ro#r#rδ

υr50,

r5ro, all z

ρυr5roro 5 ρδυδ(ro1δ) Equation (8.18) or (8.19)

@φi 50 @z rδ 5ro1δ; re is radius of shell side (see Figure 8.3)

r5re, all z

υr5υz50

@T 50 @z Equation (8.18) or (8.20) @T 50 @z

where DB,f and DB are the diffusion coefficients in the fluid and membrane phases, respectively (m2/s); cB,f and φB are the concentration in the fluid phase and membrane layers, respectively (kmol/m3); and k is the thermal conductivity [kmol m/s3K or J/(Kms)]. If the radial convective velocity is higher than zero, then this fact should also be taken into account in Eq. (8.18) as for r 5 ro for the component balance: υw cB 2DB;f

@cB @φ 5 υw φ2DB @r @r

ð8:19Þ

or for heat transport at r5ro: ρf C^ p;f υw T 2kf

@T @T 5 ρC^ p υw T 2k @r @r

ð8:20Þ

where υw is the radial convective wall velocity at r 5 ro (m/s) and subscript f denotes the fluid phase. For the complex description of the system, the transport equations also must be given for the lumen and the shell side of the membrane. The exact absolute values of the boundary conditions, such as φoi or Tio or the differential quotient at the wall, can only be obtained by the simultaneous solution of the equation system given in all three regions. In the case of constant transport parameters, the inlet and outlet

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Basic Equations of the Mass Transport through a Membrane Layer

mass transfer rates of the membrane, J and Jδ, respectively (for details of these transfer rates see Chapters 36) given in Eqs (8.18)(8.20) can be expressed separately, then the lumen and shell sides of the balance equations can be solved separately. This solution should incorporate the mass transfer rates in the boundary conditions of these equations given for the membrane interfaces.

8.4.2

Some Typical Reactor Configurations: Packed-Bed Membrane Reactor

Let us look at an often-applied membrane reactor configuration where the membrane is not catalytic, and it serves as a separation layer. Its task is to remove one of the product components in order to improve the separation efficiency by shifting the reaction to higher conversion (e.g., case III in Figure 8.2). Such a process is the hydrogen enrichment during the watergas-shift reaction, namely, H2O 1 CO2CO2 1 H2 (Brunetti et al., 2007, 2009; Gosiewski and Tanczyk, 2010; Gosiewski et al., 2010). The steam methane (Pieterse et al., 2010) or ethane reforming (Mendes et al., 2010) is also used for hydrogen production, CH4 1 H2O2CO 13H2 and CH4 1 2H2O2CO2 14H2. The membrane reactor configuration is illustrated in Figure 8.3. This is the so-called packed-bed reactor, where the reforming reaction takes place on the catalyst particles filled in either the tube side, as the case in Figure 8.3, or on the shell side (Brunetti et al., 2007, 2009). Typically, the feed pressure of the membrane reactor equals the pressure at which natural gas is available from high-pressure pipelines, i.e., 4045 bars (Pieterse et al., 2010). Permeate pressure typically is in the range of 510 bars. To the permeate side, an inert sweep gas can be introduced either in cocurrent or in countercurrent configuration with respect to the flow direction in the reaction side. The removal of one of the reaction products, namely H2, shifts the reaction in production’s direction. Inorganic membranes that Pd–Ag membrane

Sweep ⫹ H2

Sweep H2

re δ r0

Feed H2O ⫹ CO

CO2 ⫹ H2

H2

Figure 8.3 Schematic illustration of the packed-bed membrane reactor with hydrogen removal by a selective membrane.

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can be applied for hydrogen removal are reviewed by Lu et al. (2007). For the mathematical description of the process, every three sections, namely the tube and shell sides (reaction and permeation sides) and the membrane layer should be modeled. The mathematical analysis can be based on the differential conservation equations of total mass, momentum chemical species, and enthalpy in steady, twodimensional, cylindrical flow. Thus, the balance equations listed in Table 8.1a can be regarded as starting equations adapting for flowing fluid phase in the packed-bed column. The reaction rate can be based on the LangmuirHinshelwood kinetic model (Brunetti et al., 2007; Mendes et al., 2010; Pieterse et al., 2010): 2QCO 5

kKCO2 KH2 O ðpCO pH2 O 2½ pCO2 pH2 =Ke Þ ð11KCO pCO 1KH2 O pH2 O 1KCO2 pCO2 Þ2

ð8:21Þ

where QCO is the rate of carbon monoxide consumption (kmol/skgcat); k is the rate constant; Ke is the equilibrium constant; Ki is the adsorption constant for species i (i 5 CO, H2O, H2, and CO2); and pi is the partial pressure of the component i. The values of Ki, k, and Ke can be given in papers of Brunetti et al. (2007) and Mendes et al. (2010). The k reaction rate is a strong function of temperature. This function can be expressed as k 5 ko e 2Ecat =RT

ð8:22Þ

where k is preexponential factor (kmol/skg); Ecat is activation energy of catalyst reaction (kJ/kmol); and R is the gas constant (kJ/kmol K). The rate of hydrogen permeation generally is calculated by the Sieverts’ equation (Brunetti et al., 2007; Lu et al., 2007; Ziaka and Vasieiadis, 2011): JH2 5

Do H n 2E=RT n n e ðp 2pδ Þ δ

ð8:23Þ

where JH2 is the permeation rate of H2 (kmol/m2s); P^H2 is the membrane permeability of H2 (kmol/[msPan]); E is the activation energy for permeation (equal to the sum of the diffusion energy and the heat dissolution) (kJ/kmol); p and pδ are the partial pressures of H2 on the reaction and permeate side, respectively, n 5 0.51; R is the gas constant; Do is diffusivity of hydrogen (m2/s); Hn is the hydrogen solubility in metal film (kmol/m3Pan); and δ is the membrane thickness (m). According to Eq. (8.8), the mass transfer rate, JH2 ; can be rewritten for cylindrical space as JH2 5

Do H n e 2E=RT ðpn 2pnδ Þ ro lnð11δ=ro Þ

ð8:24Þ

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Basic Equations of the Mass Transport through a Membrane Layer

The f1 and f2 friction coefficient for momentum loss in the catalytic bed can be calculated according to Ergun equations (Koukou et al., 2001; Brunetti et al., 2007; Seidel-Morgenstern, 2010): f1 5 150

ð12εÞ2 μ ; ε3 dp2

f2 5 1:75

ð12εÞ2 ρ ε3 dp

ð8:25Þ

where ε is the bed porosity (2); μ is the viscosity (Pas); dp is the particle size (m); and ρ is the fluid density (kg/m3). The question arises, how can we simplify the relatively complex equation system given for the reaction side. This simplified model should be valid for practical applications. Let us take the following main assumption for it: plug flow in the catalyst bed, negligible radial temperature and concentration gradients (one-dimensional model with negligible concentration polarization), negligible pressure drop in the reactor in the both sides, thus the momentum balance equation can also be neglected. The model equations are listed in Tables 8.2a and 8.2b. Note that the axial dispersion term is also neglected because of the relatively large axial convective velocity. Some remarks are needed for the balance equations in Table 8.2a. The Qi reaction rate does not involve the mass transport inside of the catalyst particle. The mass balance equation of the reaction side considers a particle as a point source or sink term. If the Thiele modulus is low, then the internal mass transport should also be taken into account (see Chapter 4). The axial diffusion or dispersion term (this latter can be formed during turbulent flow) often can be neglected due to the relatively high axial convective velocity. Estimation of the axial Peclet number can help to this decision. Similar consideration should be done in the radial direction. Regarding the relatively high permeation rate of hydrogen, it is also recommended to take into account the change of the volumetric flow rate. This is done by the total mass balance equation. Obviously, when the volumetric flow rate of the sweep gas is large enough, then this effect could also be neglected. The permeation rate is given here by the Sieverts’ equation [Eq. (8.23) or (8.24)] (see also Chapters 1.5 and 12.2). The superficial axial convective velocity is denoted here by u. The parameter ρcat denotes the density of catalyst particles (kg/m3), thus (12ε)ρcat defines the amount of catalyst related to the reactor volume.

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Table 8.2a Basic Balance Equations of Modeling the Packed-Bed Membrane Reactor Investigated Reaction (lumen) side—: mass balances of reactive components (ξi is the stoichiometric factor): @ðuci Þ [email protected] @ci 5 6ð12εÞρcat ξ i Qi 1ε rDf;i ε @r @z r @r Total mass balance: @ðρuÞ 2 5 2 Ji ; @z ro

i 5 H2

Boundary conditions: z50 ρu5ρouo

for all r

@ci 50 for all z for all reactants @r @ci 5Ji for all z with i5H2 2 Df;i @r @ci 50 for all z; i5CO, CO2, H2O Df;i @r

r50 Df;i r5ro r5ro

Permeate (shell) side: @ðue ce;i Þ 1 @ @ce;i 5 0; 1 rDf;i @r @z r @r

i5H2

Total mass balance: @ðρe ue Þ 2 5 2 Ji @z ro Boundary conditions: z50 ρeue 5ρe,oue,o for all r @φi @ce;i 5 Def;i for all z for i5H2 @r @r @ce;i Def;i 5 0 for all z for i5H2 @r

r5ro1δ r5re

Di

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Basic Equations of the Mass Transport through a Membrane Layer

Table 8.2b Heat Balance of the Packed-Bed Reactor Reaction (lumen) side: @T @ 1 @T @2 T 1 5 2ΔHð12εÞρcat QCO ρC^ p υz ε 2kf @z @r r @r @z Boundary conditions: z50 T5To

for all r

@T 5 0 for all z @r @T @T kf 5k for all z @r @r

r50 kf r5ro

For the membrane layer: 2 @ 1 @T @ T 1 50 k @r r @r @z Boundary conditions: z50 T5To

for all r

@T 5 0 for all r @z @T @T r5ro1δ k 5 ke for all z @r @r

z5L

k

Shell (permeate) side: @T @ 1 @T 2ke 50 ðρC^ p uεÞe @z @r r @r Boundary condition: z50 T5To z5L r5re

for all r

@T 5 0 for all r @z @T ke 5 αe ðTe 2Tsurr Þ @r

ke

for all z

ΔH denotes the heat of reaction (kJ/kmol); Q is the reaction rate (kmol/skgcat); ρcat 5 mcat/Vcat (kg/m3); Te and Tsurr are the wall temperature at r5re and the outside temperature, respectively; αe is the heat transport coefficient through the wall at r5ro (kJ/m2Ks); and L is the length of the membrane tube (m). The k thermal conductivity for the catalytic membrane can be estimated as k 5 εkf 1 ð12εÞkp

ð8:26Þ

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207

r0 δ1

CH4,

Butane

Ji Catalytic layer

δ

Distributor

re O2

Je , O2

Figure 8.4 Membrane reactor for oxidative dehydrogenation.

where kf is the fluid phase conductivity (kJ/ms) and kp is the conductivity of the catalyst particles (kJ/ms). This means further simplifications when the radial component diffusivity and the thermal conductivity as well as the axial thermal conductivity are neglected, as done by Brunetti et al. (2007) and Gosiewski et al. (2010).

8.4.3

Catalytic Membrane Reactor

As an example, the oxidative dehydrogenation of paraffin hydrocarbons, from methane to butane, was chosen (Te´llez et al., 1999; Bottino et al., 2002; Pedernera et al., 2002; Rodriguez et al., 2010). This reaction can also be carried out in a packed-bed reactor (van Dyk et al., 2003; Rodriguez et al., 2010). The mass transfer equations given in Section 8.4.2 can also be used here. The oxygen transport through the membrane can be estimated by Eqs (1.48)(1.56). In this section, the reactor configuration is discussed where the chemical reaction takes place in the catalytic membrane layer as illustrated in Figure 8.4. The membrane layer is asymmetric: it contains a diffuser layer for distribution of the oxygen (diffusion zone) and a thin active layer where the dehydrogenation reaction takes place (catalytic zone; Julbe et al., 2001; Bottino et al., 2002; Pedernera et al., 2002). For the mathematical model, let us make the following simplification (Table 8.3): steady-state process; one-dimensional diffusion in the membrane; pressure gradient, along the tube and the shell side, is neglected; the radial velocity (plug flow), mass, and temperature gradient are neglected on the tube and shell sides; there is no reactant transport from the tube side to the shell side, pe2pt . 0 (pe and pt are the shell and the tube side pressures, respectively); the kinetics of dehydrogenation reactions are not discussed here; see Te´llez et al. (1999) and Heracleous and Lemonidou (2006).

208

Basic Equations of the Mass Transport through a Membrane Layer

Table 8.3 Model Equation of Oxidative Dehydrogenation Applying Catalytic Membrane Layer Lumen side mass balance Total mass balance :

n dðut ρÞ 2X Jj ; 5 dz ro j51

Component balance :

dðut ci Þ 2 5 2 Ji dz ro

j 6¼ O2 with i 5 1; . . . ; n; i 6¼ O2

dT 2 ðρC^ p uÞt 5 αðTt 2Tw;ro Þ dz ro

Energy balance :

Shell side mass balance due 1 2 5 Je;i ; dz ρ ro 1δ

Total mass balance :

dðue ce;i Þ 2 Je;i ; 52 dz ro 1δ

Species mass transport : ðρC^ p uÞe

Energy balance :

i 5 O2 i 5 O2

dT 2 5 αðTw;r1δ 2Te Þ dz ro

(Note: t denotes tube (lumen), e denotes shell) Membrane catalyst layer Component balance :

Di

ρC^ p u

Energy balance :

n X dφi ξ i Qi 5 dy i51

nr X @T d2 T Qj ð2ΔHj Þ 1 k 2 5 ρcat @z dy j51

Membrane porous layer Component balance ði 5 O2 Þ : ^ 5 ci Þ (Note: pi =RT Energy balance :

kpor

with i 5 1; . . . ; n

Ji 5

ε dp2 DK;i pe;i ln ðpe;i 2pt;i Þ1 ^ pt;i δτ 32 δRT

1 @ @ðrT Þ 50 r @r @r

Boundary conditions, e.g., concurrent mode ´ ;o ; z50 then ut 5 ut r5ro

then c5ct;

ue 5ue;o ;

φ5Hct,;

T 5To ;

for all r

Ji 5 2Di

dφi dy

T5Tt

r5r1δ1 then H1φ 5 Hporcpor; T 5 Tpor; r5r 1 δ1 then k

ci 5coi ;

dT dT 5 kpor dy dr

(Note: porous layer is cylindrical, catalyst layer is plane)

i 5 O2

Membrane Reactor

209

Where n is the number of reactant (e.g., in case of ethane dehydrogenation the components are: C2H6, C2H4, O2, H2O, CO2; Rodriguez et al., 2010); Di is the effective diffusion coefficient for i (m2/s); δ is the thickness of the catalyst layer (m); α is the heat transfer coefficient (kJ/m2sK); subscript “e” denotes the shell side, subscript “por” is the porous distributor layer; Jj is the component transport rate (kmol/m2 s) (the oxygen transfer rate depends on the membrane properties, thus the value of Ji, i 5 O2, in the porous layer is valid for the Knudsen flow regime; see Chapter 12), that can enter or leave the membrane or the lumen fluid; δ is the thickness of the porous layer of the membrane (m); δ1 is the thickness of the catalyst layer (m); kpor is the thermal conductivity in the porous membrane layer for oxygen distribution (kJ/K m s); and H is the solubility coefficient (2). Practically, the overall membrane thickness is the same as the thickness of the membrane because the catalyst layer is generally very thin. Assuming that the catalyst membrane layer is very thin (δ/ro{1), the cylindrical effect is negligible. On the other hand, if the thickness of the membrane distributor is relatively large, the cylindrical effect can have significant effect of the transport process. In the energy balance of the porous layer, the effect of the countercurrent viscous flow of oxygen is neglected. The component balance equation of the porous membrane layer does not involve the cylindrical effect. Its thickness can be several hundred μm. In the case of a tubular inorganic membrane, where the inner diameter of the membrane is about 0.0080.01 m, this effect is negligible. When one applies a capillary membrane with an inner radius of 100300 μm (Bottino et al., 2002; van Dyk et al., 2003), this effect can be significant, thus Eq. (1.56) can be applied for description of the membrane mass transport.

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