On concentration polarization in fluidized bed membranereactorsCitation for published version (APA):Helmi, A., Voncken, R. J. W., Raijmakers, A. J., Roghair, I., Gallucci, F., & van Sint Annaland, M. (2018). Onconcentration polarization in fluidized bed membrane reactors. Chemical Engineering Journal, 332, 464-478.https://doi.org/10.1016/j.cej.2017.09.045

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Chemical Engineering Journal

journal homepage: www.elsevier.com/locate/cej

On concentration polarization in fluidized bed membrane reactors

A. Helmi, R.J.W. Voncken, A.J. Raijmakers, I. Roghair, F. Gallucci, M. van Sint Annaland⁎

Chemical Process Intensification, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, P.O. Box 513, 5612 AZ Eindhoven, TheNetherlands

G R A P H I C A L A B S T R A C T

A R T I C L E I N F O

Keywords:Fluidized bedPd membraneConcentration polarizationTFM

A B S T R A C T

Palladium-based membrane-assisted fluidized bed reactors have been proposed for the production of ultra-purehydrogen at small scales. Due to the improved heat and mass transfer characteristics inside such reactors, it iscommonly believed that they can outperform packed bed membrane reactor configurations. It has been widelyshown that the performance of packed bed membrane reactors can suffer from serious mass transfer limitationsfrom the bulk of the catalyst bed to the surface of the membranes (concentration polarization) when usingmodern highly permeable membranes. The extent of concentration polarization in fluidized bed membranereactors has not yet been researched in detail. In this work, we have quantified the concentration polarizationeffect inside fluidized bed membrane reactors with immersed vertical membranes with high hydrogen fluxes. ATwo-Fluid Model (TFM) was used to quantify the extent of concentration polarization and to visualize theconcentration profiles near the membrane. The concentration profiles were simplified to a mass transferboundary layer (typically 1 cm in thickness), which was implemented in a 1D fluidized bed membrane reactormodel to account for the concentration polarization effects. Predictions by the TFM and the extended 1D modelshowed very good agreement with experimental hydrogen flux data. The experiments and models show thatconcentration polarization can reduce the hydrogen flux by a factor of 3 even at low H2 concentrations in thefeed (10%), which confirms that concentration polarization can also significantly affect the performance offluidized bed membrane reactors when integrating highly permeable membranes, but to a somewhat lesserextent than packed bed membrane reactors. The extraction of hydrogen also affects the gas velocity and solidshold-up profiles in the fluidized bed.

http://dx.doi.org/10.1016/j.cej.2017.09.045Received 3 January 2017; Received in revised form 11 August 2017; Accepted 7 September 2017

⁎ Corresponding author.E-mail address: M.v.SintAnnaland@tue.nl (M. van Sint Annaland).

Chemical Engineering Journal 332 (2018) 464–478

Available online 11 September 20171385-8947/ © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

MARK

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1. Introduction

Currently, hydrogen is mainly produced on large scale via steamreforming of methane (SMR) [1]. In this process, methane is first re-formed with steam (Eq. (1)) in high temperature multi-tubular packedbed reactors. In a second step the carbon monoxide is converted via thewater gas shift (WGS) reaction (Eq. (2)) in packed bed reactors. Typi-cally, a two stage WGS is used to take advantage of fast reaction rates athigh temperatures (450 °C) and higher equilibrium conversions atlower temperatures (200 °C). Finally, the hydrogen produced is furtherpurified using pressure swing adsorption (PSA).

Steam methane reforming reaction (SMR):

+ ↔ + =CH H O CO H H3 Δ 206 kJ/molr4 2 2 (1)

Water gas shift reaction (WGS):

+ ↔ + = −CO H O CO H HΔ 41 kJ/molr2 2 2 (2)

The equivalent hydrogen efficiency of the whole process is ap-proximately 80% thanks to steam/electricity export [2]. The heat in-tegration between the different stages becomes more complicated atsmaller scales, while heat export cannot be realized in distributed hy-drogen production applications. For this reason the system becomesinefficient and uneconomical at smaller scales. The cost of the hydrogenproduced at large scale is around 0.2 €/Nm3, while it increases up to0.4–0.5 €/Nm3 at smaller scales [2].

The efficiency of the hydrogen production via methane reformingcan be increased by integrating hydrogen production and separation ina single multifunctional reactor. This can be achieved by using perm-

Nomenclature

A area (m2)c c,1 2 constants in frictional stress model (–)B exchange of fluctuation energy (kg m−1 s−3)C concentration (mol m−3)Cd drag coefficient (–)D diffusion/dispersion coefficient (m2 s−1)d diameter (m)Ea activation energy (J mol−1)e coefficient of restitution (–)f fraction (–)Fr constant in frictional stress model (N m−2)g gravitational acceleration (m s−2)g0 radial distribution function (–)H height (m)I unit tensor (–)J membrane flux (mol m−2 s−1)K mass transfer coefficient (m s−1)kd mass transfer coefficient bulk to membrane (m s−1)Mw Molecular weight (kg mol−1)N flux (mol m−2 s−1)P partial pressure (Pa)Pm permeability (mol m−1 s−1 Pa−0.5)Pm,0 permeation constant (mol m−1 s−1 Pa−0.5)p pressure (Pa)QPd permeance (mol m−2 s−1 Pa−0.5)R universal gas constant (J mol−1 K−1)r radial position (m)Re Reynolds number (–)S strain rate (s−1)S source term (kg m−3 s−1)Sh Sherwood number (–)t time (s)tm membrane thickness (m)T temperature (K)u velocity (m s−1)V volume (m3)X molar fraction (–)Y mass fraction (–)z axial position (m)

Greek letters

α volume fraction (–)β interphase drag coefficient (kg m−3 s−1)γ dissipation of granular energy (kg m−1 s−3)δ film layer thickness (m)θ granular temperature (m2 s−2)

κ conductivity of granular energy (kg m−1 s−1)λ bulk viscosity (kg m−1 s−1)μ Shear viscosity (kg m−1 s−1)ρ density (kg m−3)τ shear stress tensor (N m−2)ϕ fric angle of internal friction (°)

Subscripts & superscripts

avg averageb bubblebc bubble to cloudbe bubble to emulsionbulk bulkce cloud to emulsioncell cell(s)e emulsionfric frictionalg gash hydraulicm membranemax maximummf minimum fluidizationmin fr. minimum frictionmol molecularn number of CSTRsp particlepp particle-particlepw particle-wallperm permeater radialreac reactorrise rises solidsim simulationT transposedtot total

Abbreviations

CFD computational fluid dynamicsCSTR continuous stirred tank reactorFBMR fluidized bed membrane reactorKTGF kinetic theory of granular flowPSA pressure swing adsorptionSMR steam methane reformingTFM two-fluid modelWGS water gas shift

A. Helmi et al. Chemical Engineering Journal 332 (2018) 464–478

465

selective palladium-based membranes in membrane reactors.Recovering the hydrogen during the reaction, results in a shift of theequilibrium towards the products, thus allowing achieving much higherconversions at lower temperatures. The equilibrium displacement (LeChatelier’s principle) allows to minimize the reactor volume up to 80%for WGS [3] and maximize the efficiencies, as total conversion can beachieved already at lower temperatures [4].

In literature, both packed bed and fluidized-bed membrane reactorconfigurations have been proposed for SMR and WGS reactions. Thelatest developments in the fabrication of ultra-thin membranes withhigh permeation rates [5], have once more sparked the debate on theinherent bed-to-membrane mass transfer limitations (concentrationpolarization) in packed bed membrane reactors [6,7].

From an experimental point of view, Hara et al. [8] studied thedecline of hydrogen permeation in a packed bed membrane reactor byinjecting the reactor with H2-Ar and H2-CO mixtures. It was found thatthe reduction in hydrogen permeation was caused by CO poisoning ofthe Pd based membrane and concentration polarization near themembrane wall. It was concluded that in order to fairly predict themembrane reactor performance, concentration polarization needs to betaken into account.

Mori et al. [9] investigated the influence of concentration polar-ization on hydrogen production via SMR in a packed bed membranereactor with a highly permeable membrane. They performed experi-ments and compared them with a simple model that did not take intoaccount the effect of concentration polarization. By increasing the re-actor pressure, they found that the experimental methane conversionwas lower than the simulated conversion. This implies that concentra-tion polarization is occurring in the reactor and affects the methaneconversion. The presence of concentration polarization was confirmedwith experiments with a binary mixture of hydrogen and nitrogen.

Caravella et al. [6] made a model predicting the permeance of hy-drogen in a hydrogen-nitrogen mixture including the effect of con-centration polarization in an empty annular tube. It was found that theeffect of polarization is relevant not only for the very thin membranes(1–5 µm) with high fluxes but also for the thicker ones (100 µm) atcertain operating conditions.

In a Computational Fluid Dynamics (CFD) study by Nekhamkinaet al. [10], the mass transfer processes in two configurations werestudied: an empty reactor with (i) the membrane at the wall, and (ii) anannular cylinder with the membrane as the inner tube. A model wasdeveloped to predict the membrane flux considering the effect of con-centration polarization. A parameter Γ was defined which representsthe ratio of the diffusion to the permeation flux. It was concluded thatonly when Γ > 6 the effect of concentration polarization can be ne-glected.

To circumvent the mass transfer limitations typical of empty orpacked bed membrane reactor configurations, fluidized bed membranereactors were suggested, because of their improved heat and masstransfer characteristics. Patil et al. [11] and Gallucci et al. [12] suc-cessfully demonstrated this membrane reactor concept for the SMRreaction with relatively low flux membranes. No concentration polar-ization effects were reported, but the flux of the membrane used was5–10 times lower than recently available highly permeable membranes.

More recently, Helmi et al. [13] successfully demonstrated the longterm (>900 h) performance of a fluidized bed membrane reactor uti-lizing very high flux membranes for ultra-pure hydrogen production viaWGS. Although the long term stability of this membrane reactor hasbeen confirmed (with CO content in the permeate side

⎜ ⎟= ⎛⎝

−−

⎞⎠

N k CX

Xln

11H d tot

H m

H bulk

,

,2

2

2 (4)

with the mass transfer coefficient from the bulk to the membrane walldefined as

=k Dδd (5)

which can be determined from a Sherwood correlation:

=Sh k dD

d H(6)

where dH is the hydraulic diameter of the reactor (dH = dreac − dm).Thus, for the thickness of the film layer:

=δ dSh

H(7)

In literature, no Sherwood correlation was found that can describethe gas mass transfer from the bulk of a fluidized bed to an immersedwall inside the bed. Moreover, also no generally applicable correlationfor the radial gas dispersion in fluidized beds is available. Most of theproposed correlations in literature are derived to predict the solidsdispersion inside a gas-solid fluidized bed. They are 1D type equationsthat can describe the axial/radial movement of the solids inside afluidized bed at the investigated operating conditions [19–25].

On the other hand, the equations found for gas dispersion in flui-dized beds were derived for risers, circulating systems or fast fluidizedbeds, and for operating conditions that were often orders of magnitudehigher than the superficial gas velocities used in our experiments[26–32]. Furthermore, these equations are not useful for CFD models,because they do not contain local and instantaneous gas and solidsproperties. Therefore, the radial dispersion in the densified zone of themass boundary layer close to the membranes is estimated using thecorrelation by Tsotsas and Schlünder [33] for the dispersion coefficient

in packed beds (see Table A.1 in Appendix A). It should be noted thatthe dispersion coefficient is likely to be somewhat under-predicted.

The mass transfer of hydrogen through the selective dense Pd/Aglayer of the supported membrane is described with the solution diffu-sion mechanism. Following Sieverts’ law [34], the flux through thedense layer is proportional to the difference between the square-root ofthe H2 partial pressure at the retentate side (reaction zone) and thepermeate side (inside the tubes) of the membrane. The diffusionthrough the selective dense layer is considered as the rate limiting stepfor H2 permeation and it is assumed that there is no concentrationgradient (nor pressure gradient) across the porous ceramic supportlayer of the membrane, and also mass transfer limitations at thepermeate side are assumed to be negligible (Fig. 2). These assumptionsare valid in this work because the selective Pd-Ag layer was applied onthe outer side of the asymmetrical porous tube. Thus, there is no con-centration gradient over the porous support, since on the permeate sideonly virtually pure H2 is present (the ideal perm-selectivity was in theorder of 5000), and there can only be a very small pressure gradientover the porous support following the Dusty Gas model. It will beshown that the 1D model can already well describe the experimentalflux when using the experimentally determined pressure at thepermeate side, so that the pressure drop over the porous support canindeed be neglected, which corresponds very well with the findings byCaravella et al. (2016) [35]. If necessary, the model could be extendedto account for these factors.

The membrane flux is thus described by Sieverts’ law [34]:

= −J Pt

P P( ) [mol/m s]H mm

H m H per,0.5

,0.5 2

2 2 2 (8)

= −P P exp E RT( / )m m a0 (9)

in which Pm is the membrane permeability, Pm0 is the permeation con-stant, Ea is the membrane activation energy and tm is the membraneselective layer thickness.

Furthermore, it is assumed that there is only mass transfer from thebubble phase to the emulsion phase, not directly to the membrane,because of the relatively small bubble hold-up in bubbling fluidizedbeds and especially near the vertically immersed membrane tubes.From the emulsion phase the hydrogen transfers to the film layer andfrom there it permeates through the membrane. Therefore the compo-nent mass balance for the bubble phase reads:

= − ⎡⎣⎢

+ + − ⎤⎦⎥

dCdz f u

u Cdfdz

f Cdu

dzK f C C1 ( )b

b b riseb rise b

bb b

b risebe b b e

,,

,

(10)

The rise velocity of bubbles in a swarm (ub rise, ), the bubble fraction

Fig. 1. Schematic representation of the 1D phenomenological fluidized bed membranereactor model. It is assumed that the gas flow rate in the bubble phase (ub) is equal theexcess gas flowrate above that is required to keep the emulsion phase at minimum flui-dization velocity (umf). There is only mass transfer between the bubble phase and theemulsion phases (no direct mass transfer from the bubble phase to the membrane surfaceis considered).

Fig. 2. H2 concentration profile across the membrane.

A. Helmi et al. Chemical Engineering Journal 332 (2018) 464–478

467

( fb) and the bubble to emulsion phase mass transfer coefficient Kbe aredetermined from correlations reported in [36]. The total superficialvelocity in CSTR number n (utot n, ), is calculated by subtracting the flowthrough the membrane from the axial flow in CSTR number n− 1:

= −−u uJ AA C

.tot n tot n

H m

reac tot, , 1

2

(11)

where Am is the surface area of the membrane and Areac is the crosssectional area of the reactor.

=A πD Lm m m

=AπD

4reacr2

The emulsion phase exchanges hydrogen with the bubble phase andtransports it via the film layer to the membrane wall. This can be de-scribed as:

⎜ ⎟= ⎡⎣⎢

− − ⎛⎝

−−

⎞⎠

⎤⎦⎥

dCdz u A

f K A C C k πd C XX

1 ( ) ln 11

e

mf reacb be reac b e d m tot

m

e (12)

For each CSTR, one value for Xm and Xe will be calculated re-presenting the average concentration of H2 in that CSTR. The flux en-tering the film layer should be equal to the flux through the membrane,thus:

= − = ⎡⎣⎢

−−

⎤⎦⎥

J Pt

P P k C XX

( ) ln 11H

m

mH m H per d tot

m

e2,

0.52,

0.52 (13)

An overview of all the hydrodynamic parameters is provided inTable A.1 (Appendix A). For a detailed discussion on the model equa-tions and assumptions the interested reader is referred to [37].

2.2. Two-fluid model

To supplement the one-dimensional phenomenological model withan estimate of the thickness of the mass transfer boundary layer, si-mulations using the Two-Fluid model (TFM) have been performed,using OpenFOAM twoPhaseEulerFoam version 2.3.1. This solver hasbeen extended with gas-phase species balance equations and realisticmembrane models to simulate the selective extraction of hydrogen.

The TFM considers the gas and solids phases as interpenetratingcontinua. The governing and constitutive equations are presented inTable B.1 (see Appendix B). The gas phase is described as an ideal gaswith Newtonian behavior, whereas the rheology of the solids phase ismodeled with the Kinetic Theory of Granular Flow (KTGF). Extractionof mass via the membrane is accounted for with a source term (Sm) inthe gas phase continuity equation.

The drag between the solids and the gas phase is calculated with theGidaspow drag model[38], which combines the drag model of Ergun[39] and Wen & Yu [40]. Ergun’s model is valid for high solids hold-ups(20% and higher) and Wen & Yu’s model is valid at lower solids hold-ups (below 20%). The drag coefficient Cd is determined based on theparticle Reynolds number.

To approximate the rheological properties of the particulate phasein a fluidized bed, the KTGF closure equations are used. The closureequations used in this work are presented in Table B.2 [41]. A numberof closure equations were not available in the original OpenFOAM TFM,so they were added to the model. Further details on the TFM and KTGFcan be found in literature [38,42–46]. Detailed information on theOpenFOAM TFM specifically has also been published by other authors[47,48].

To model mass transfer phenomena and extraction of hydrogen viamembranes, a hydrogen species balance was added to the TFM (Eq.(14)). The effect of the membranes on the system was taken into ac-count via the source term, Sm, which is applied to the computationalcells adjacent to a membrane boundary (illustrated by the red cells inFig. 3). The source term in Eq. (14) is the membrane flux calculated

with Sieverts’ law, multiplied by the boundary cell’s area Acell, dividedby the cell volume Vcell, see Eq. (15). This approach to simulate perm-selective membranes was also used by Coroneo et al.[49].

∂∂

+ ∇ = ∇ ∇ +α ρ Y

tα ρ Y α ρ D Y Su·( ) ·( )

g g Hg g g H g g H H m

22 2 2 (14)

= −S AV

Q M X p X p·[( ) ( ) ]m cellcell

Pd w Hm

tot Hperm

tot0.5 0.5

2 2 (15)

Previous research has shown that extraction of gas from a fluidizedbed can create densified zones which may affect the flow patterns of thesolids [16,50]. In the case of selective hydrogen extraction, the removalof momentum from the system is expected to have a limited effect dueto low molecular weight of hydrogen. However, when modelling ex-traction or addition of a component with a higher molecular weight, theextraction of momentum may become more significant. Therefore, aboundary condition for the momentum balances was modified in theTFM which accounts for the extraction of momentum due to themembrane permeation. The boundary condition effectively imposes avelocity um, whose magnitude is correlated to the extracted mass sourceterm Sm, and in the normal direction to the membrane boundary asgiven in Eq. (16). The velocity is always imposed normal to the mem-brane surface and ensures that momentum is extracted from the mo-mentum equations in Table B.1.

=u S RTpM

VAm

m

w

cell

cell (16)

The experimental setup is a cylindrical fluidized bed reactor with asingle submerged membrane in the center of the reactor. This systemwas approximated with a 2D simulation where the bottom and top ofthe membrane are the axial height of the inlet and outlet of the TFMsimulations. A sketch of the experimental set-up and how it has beenapproximated with the model is presented in Fig. 4. Hydrogen wasextracted via the left boundary, to which the membrane velocityboundary condition described by Eq. (16) was applied. On the rightboundary a no-slip condition was imposed. For the solids phase, aJohnson & Jackson partial slip boundary condition with a specularitycoefficient of 0.50 was applied on both the left and right walls (seeTable B.3 in Appendix B).

The settings for the vertical membrane simulations are presented inTables B.4 and B.5 (in Appendix B). The domain width is equal to theradius of the experimental reactor and the domain height is equal to themembrane length. The selected grid was 0.5625 by 0.5625 mm, whichis sufficiently fine to yield converged solutions. Temporal discretizationwas done with the second order Crank-Nicolson scheme. A combinationof two second order schemes, the Gauss linear scheme and the Van Leer

Fig. 3. Schematic representation showing where the membrane source term andboundary condition have been applied.

A. Helmi et al. Chemical Engineering Journal 332 (2018) 464–478

468

scheme, were used for spatial discretization. The TFM simulations wereperformed at three different hydrogen molar fractions at the inlet andfour different reactor pressures at the outlet.

The 2D Cartesian approximation of the cylindrical fluidized bedmembrane reactor was applied because of its simplicity and reasonablesimulation times required to obtain the results. It is an often used ap-proximation when simulating fluidized beds [51–53]. However, anumber of phenomena will not be simulated fully realistically with this2D approach. To take into account all the hydrodynamic effects thatoccur in the fluidized bed with an immersed membrane, such as bubblespassing by around all sides of the membrane, 3D cylindrical grids arerequired with approximately 0.5·106–1·106 computational cells, whichis not in the scope of this study. Nonetheless, fast X-ray analysis ofcylindrical fluidized beds with inserted permeating internals performedby Helmi et al. (2017) [54] showed that most of the bubbles are pushedaway from the internals by the solids, which is similar to what wasfound in the 2D simulations.

Furthermore, in the Cartesian 2D approach, the increase in radialarea when moving from the membrane surface towards the reactor wallis not taken into account. Therefore, the concentration difference be-tween the membrane and bulk will be overestimated in the 2DCartesian simulations compared to a 3D cylindrical system. To take intoaccount the dependency of the hydrogen flux on the radial position, themolar balance in cylindrical coordinates can be integrated from themembrane surface (rm) to the radial positions where the bulk con-centration (rbulk = rm + δ). The result can be found in Eq. (17):

⎜ ⎟= ⎛⎝

−−

⎞⎠ + +( )

N D CX

X r δln

11

1

( )ln 1H H tot

H m

H bulk mδ

r

,

,m

2 22

2 (17)

This means that the film layer thickness, δ, can be estimated withEq. (18), which relates the 2D Cartesian TFM film layer thickness to theactual film layer thickness with radial dependence. The TFM film layerthickness, δTFM, can be obtained by using Eq. (4).

⎜ ⎟= + ⎛⎝

+ ⎞⎠

δ r δ δr

( )ln 1TFM mm (18)

In this work, the 2D approach serves as learning model to qualita-tively understand how concentration polarization can manifest itself incylindrical fluidized bed membrane reactor systems. A full 3D approachis required to capture all the details of the phenomena occurring incylindrical fluidized beds with immersed membranes. A possible al-ternative approach to full 3D cylindrical fluidized bed reactor simula-tions, while still partly accounting for the geometrical shape of cy-lindrical beds, is the 2.5D approach proposed by Li et al. (2015) [55].This 2.5D approach would be computationally much more efficientthan the 3D approach, but it remains to be investigated whether the2.5D approach fully and quantitatively resolves all the required hy-drodynamic features that prevail in a 3D cylindrical fluidized bed withimmersed membranes. A second alternative approach to full 3D cy-lindrical simulations is simulating the full diameter of the fluidized bedin 2D and placing the membrane in the middle of the bed. This ap-proach has been tested before adopting the approach used in this work,and it resulted in unrealistic gas flow profiles, where the gas regularlyflows downwards around the membrane and drags bubbles down pastboth sides of the membrane. This gas and bubble down flow in turncauses the concentration profiles between the wall and the membraneto become more diffuse, so a stable bulk concentration far from themembrane is never reached.

3. Experimental

Pt/Al2O3 particles with an average particle size of 200 µm anddensity 1400 kg m−3 were used for the experiments (provided by JM®).Detailed information on particle size measurement, minimum fluidi-zation velocity and Geldart classification can be found in [13]. In aprevious study it was ensured that particles do not chemically interactwith the Pd membrane surface [56].

A 365 mm long cylindrical stainless steel tube with an inner dia-meter of 45 mm was used for the experiments. The gas distributor was aporous stainless steel plate of 40 µm pore size. Two thermocouples wereplaced inside and outside of the membrane (close by the surface of themembrane). In the fluidized bed experiments, 180 g of Pt/Al2O3 par-ticles were integrated inside the reactor to ensure full immersion of themembrane at minimum fluidization conditions. For more detailed in-formation on the experimental setup, see [13].

In the center of the reactor a 113 mm long Pd0.85-Ag0.15 based mem-brane supported on porous Al2O3 (100 nm pore size at the surface) wasplaced. The outer diameter of the membrane was 1 cm and was placed 3 cmabove the distributor plate. The supported membrane was fabricated withan electroless plating technique with an average selective layer thickness of4.5 μm along the membrane. The membrane was first integrated into thereactor module without catalyst particles to activate the membrane (see[56]). Subsequently, the membrane permeation properties (Pm0 andEa) werecharacterized at different retentate side pressures (1.2–1.6 bar) and at dif-ferent temperatures (350, 400, and 450 °C) under pure hydrogen flow(Pm0 =1.76·10−8 mol m–1 s−1 Pa−0.5, Ea =7.1 kJ mol–1). The obtainedpermeation properties from the experiments were used in the models(phenomenological model and TFM) to describe the H2 flux through themembrane. The permeate side was kept at 1 bar for the entire character-ization period. To ensure that the membrane was leak tight, the nitrogenleakage rate was monitored during the experimental work at identical op-erating pressures (measured average ideal H2/N2 selectivity was 5000).

After the characterization procedure, experiments were performedwith binary gas mixtures of N2 and H2 to quantify the concentrationpolarization effect. Experiments with pure hydrogen at the inlet wereused to monitor the stability of the membrane over time. All the ex-periments were performed at 400 °C and hydrogen mole fractions of0.1, 0.25 and 0.45 were used. Finally, the relative fluidization velocity(u/umf) was varied between 1.3 and 3.3 and the membrane performancewas measured at constant pressure of 1.3 bar at the retentate side.

Fig. 4. Schematic representation of the 2D simulation grid.

A. Helmi et al. Chemical Engineering Journal 332 (2018) 464–478

469

4. Results and discussion

The experiments reported hereafter were performed for various H2mole fractions. For every mole fraction experiments were carried out atdifferent H2 partial pressure differences across the membrane. First theexperimental results will be compared with the results obtained withthe TFM to obtain a proper estimation of the radial dispersion coeffi-cient for the fluidized suspension. As described in Section 2, in thephenomenological model the concentration polarization is modelled byassuming a mass transfer film layer with thickness δ around the mem-brane with an external mass transfer coefficient of kd. The thickness ofthe film layer will be determined using the optimized dispersion coef-ficient in the TFM.

The effect of densified zones on concentration polarization is lookedinto and the effect of the hydrogen mole fraction and reactor pressureon the boundary film layer thickness will also be investigated.Subsequently, results from the phenomenological model without con-sidering concentration polarization (referred to as 1D) and with ac-counting for concentration polarization (indicated by 1D/kd) will becompared with experimental results for identical conditions. Finally, itwill be discussed whether the bubble-to-emulsion, emulsion-to-mem-brane or the mass transfer across the membrane is the rate limiting stepfor gas extraction in fluidized beds.

4.1. Film layer thickness

In order to use the 1D/kd model which is developed in this work, thethickness of the film layer needs to be estimated. To help estimating themagnitude of δ, and the corresponding radial dispersion coefficient Dr,the Two-Fluid Model (TFM) was used. To the authors’ knowledge, thereare currently no generally accepted relations that describe the localradial dispersion in fluidized beds with internals as a function of(amongst others) the local solids hold-up. To estimate a minimum valuefor the radial dispersion in the densified zones close to the membranes,we have used a correlation for packed beds. The equation of Tsotsas andSchlünder in Table A.1 was used to calculate an average radial dis-persion coefficient at solids hold-ups between 0.05 and 0.60 and at an(interstitial) gas velocity of 0.1 m/s, yielding for the considered con-ditions a dispersion coefficient in the order of 5·10−5 to 1·10−4 m2 s−1.Variation of the gas velocity and particle diameter within the experi-mental range had only a very limited effect on the estimated dispersioncoefficient. Thus, we have found that using a single constant dispersioncoefficient based on the binary gas diffusion coefficient estimated withFuller’s equation was sufficiently accurate for the simulations per-formed in this work. Fig. 5 shows the computed averaged hydrogenfluxes versus the difference in the square-root of the hydrogen partialpressures across the membrane at various combinations of inlet com-positions and reactor pressures for the experiments, the TFM with adispersion coefficient of 5·10−5 and 1·10−4 m2 s−1, and the 1D model.

According to Fig. 5, a very good match between TFM and experi-mental observations was obtained when using a radial dispersioncoefficient of 1·10−4 m2 s−1. Therefore, and due to the fact that novalidated correlation exists for the radial dispersion coefficient insidefluidized bed membrane reactors, this value was used. The differencebetween the 1D model predictions and the experiments/TFM showsthat the 1D model over predicts the hydrogen flux and does not takeconcentration polarization into account, because the 1D model does notsimulate the concentration drop near the membrane surface.

Each set of experimental data (separated by the dashed boxes) inFig. 5 was measured at different moments in time. On each day andbefore each experiment, the membrane performance (permeability andideal H2/N2 selectivity) was checked to ensure it was stable throughoutthe entire experimental work. The data obtained from the daily testsshows that the permeability of the membrane was slightly fluctuatingaround an average value. On the other hand, the TFM results wereobtained using one average experimentally obtained permeation rate.

Due to these minor fluctuations in membrane behavior throughout theexperimental program, a slight difference between model predictionsand the experimental observations can be observed as both over andunder predictions.

Investigating the concentration profiles in the vicinity of the mem-brane as computed by the TFM, the concentration of hydrogen sig-nificantly decreases from a bulk concentration to a minimum valueclose to the membrane for all the cases. This confirms the existence of amass transfer boundary layer near the membrane imposing a masstransfer resistance from the bulk of the fluidized suspension to thesurface of the membranes. Simulations were performed for differentinlet H2 mole fractions of 0.1, 0.2, 0.45 and 1 to investigate thethickness of this boundary layer for various operating conditions (seeFigs. 6 and 7). The computed results clearly show a thinner film layer atthe bottom of the membrane that increases significantly as a function ofthe axial position. This shows that the assumption of a film layer with aconstant thickness is obviously a simplification. The description couldbe extended using boundary layer theory to account for this, but theresults shown later will show that the assumption of a constant filmlayer thickness is sufficient for this system.

The TFM film layer thickness was calculated with Eq. (4). The fluxesfor the three inlet mole fractions at 1.5 bar pressure are presented inFig. 8 and were used for the calculations. The flux strongly reduceswithin the first 2 cm of the membrane, and then stabilizes, which showsthat using a single value for the film layer thickness is a good initialestimation. When using all the flux values to calculate the film layerthickness and averaging the δTFM values, the TFM film layer thickness is1.55 cm. When using the average of the stable flux values abovez = 4 cm, the average TFM film layer thickness was found to be1.94 cm. The corrected film layer thicknesses δ, calculated via Eq. (18),are then 0.96 cm and 1.14 cm respectively. In the 1D/kd model, a δ of1 cm was used. The results for all film layer thickness values are sum-marized in Table 1.

4.2. Densified zones

The two-dimensional TFM simulations were used to investigate theformation of densified zones near the membrane and their effect onconcentration polarization. Non-consecutive snapshots of the in-stantaneous hydrogen mole fractions and gas bubbles (defined as re-gions with a gas porosity above 0.85) show that the bubbles do not

50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4X=0.45X=0.25X=1 Exp

1D model TFM 5x10-5

TFM 1x10-4

Flux

(mol

m-2 s

-1)

PΔ 0.5H2 (Pa0.5)

X=0.1

Fig. 5. Comparison of the experimentally determined and TFM computed membrane fluxas a function of the hydrogen partial pressure (sampled at the inlet of the reactor), twodifferent gas phase radial dispersion coefficients have been used in the TFM(Dr = 5 · 10−5 and 1 · 10−4 m2 s−1), Preactor = 1.5–1.8 bar, u/umf = 3.3. The obtainedresults from the 1D model (considering no concentration polarization) significantly overpredicts the experimentally determined hydrogen flux.

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come close to the membrane (Fig. 9). The hydrogen molecules thereforehave to depend on diffusion to reach the membrane, the distance of thebubbles to the membrane is too large to convectively refresh the hy-drogen at the membrane.

Fig. 10 presents the spatial average in axial direction of all time-averaged solids hold-up profiles for a fluidized bed injected with binarygas mixtures with 10, 25 and 45 mol% hydrogen. The 25 mol% casewas also performed for the same bed without hydrogen extraction.When hydrogen is extracted, the solids shift more towards the mem-brane. At higher hydrogen molar fractions, the solids hold-up near themembrane increases slightly compared to lower hydrogen molar frac-tions, because the momentum flux of the hydrogen towards the mem-brane is higher.

The solids hold-up at the wall opposite to the membrane

simultaneously decreases by about 1 to 2%, indicating that the solidsshift slightly more towards the membrane at higher extraction fluxes.However, when no extraction takes place, the solids hold-up near themembrane and right wall is higher than for the case with extraction.The extraction of hydrogen thus did not significantly alter the extent ofthe densified zones, and these small changes in the solids hold-up near

0.0 0.5 1.00.0

0.1

0.2

0.3

z/L=0.04 z/L=0.24 z/L=0.73 z/L=0.98 Average

H2 m

ole

frac

tion

[-]

x/width [-]

Fig. 6. Time-averaged TFM predicted hydrogen concentration profiles at different axialpositions and the average profile over the displayed positions, z: axial distance from themembrane bottom, L: the membrane length, x: distance from the membrane in radialdirection, ΔP: 0.5 bar, X = 0.25.

Fig. 7. Time averaged concentration profiles of H2 computed with the TFM (X = 0.25),the dashed lines refer to the axial positions where the lateral concentration profiles areshown in Fig. 6.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00

0.02

0.04

0.06

0.08

0.10

X=0.10 X=0.25 X=0.45

Axi

al p

ositi

on (m

)

Flux (mol m-2 s-1)

Fig. 8. Hydrogen flux at various axial positions for three inlet hydrogen mole fractions at1.5 bar reactor pressure.

Table 1TFM and corrected film layer thicknesses when taking all fluxes into account or only thestable flux. All values are in cm.

All fluxes Stable flux (above z = 4 cm)

δTFM at X = 0.10 1.71 2.21δTFM at X = 0.25 1.60 1.93δTFM at X = 0.45 1.35 1.68Average δTFM 1.55 1.94Average δ 0.96 1.14

Fig. 9. Two instantaneous non-consecutive TFM snapshots of hydrogen molar fractionand bubble contours (white).

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471

the membrane cannot be the main cause of concentration polarization.To better elucidate the effects of extraction, in Fig. 11 the gas ve-

locity profiles in an empty membrane tube with and without hydrogenextraction are compared, where all the conditions were kept the sameas for the FBMR case with 25 mol% hydrogen at 1.5 bar pressure butwithout solids. A clear shift of the gas velocity profile towards themembrane was observed, which explains the slight reduction in thesolids hold-up near the membrane and the overall shift of the solidshold-up profile towards the membrane.

4.3. 1D/kd model verification

In an independent experimental observation by Patil et al. [57] for aFBMR with a Pd-based membrane with much lower permeation prop-erties (Pm = 1.35·10−12 mol m−1 s−1 Pa) in comparison with themembrane used in this work, no influence of concentration polarizationwas reported. Therefore, it was investigated whether the 1D/kd model isable to describe these experiments with negligibly small film layerthickness. Fig. 12 compares the reported experimental observation for acase at 400 °C, with a superficial gas velocity between u0 = 3 andu0 = 5 cm s−1, and the H2 partial pressure drop was between ΔP = 0.5and ΔP = 3 bar, with the predictions from the phenomenological model

using 100 CSTR’s in series for both emulsion phase and the bubblephases.

The model without considering concentration polarization (1D)predicts the experimental results very well. To validate the 1D/kdmodel, a film layer thickness of 1·10−6 m around the membrane wasassumed with a radial dispersion coefficient of 1·10−4 m2 s−1, and themodel reduces indeed to the results from the 1D model confirming theabsence of concentration polarization for the membrane used in thework by Patil et al. at the specified operating condition.

4.4. Model vs. experiments

In this section results from the one-dimensional models (1D and 1D/kd) for the fluidized bed will be compared with the experimental ob-servations at identical operating conditions. Experiments in the flui-dized bed were performed with an inlet superficial gas velocity ofu0 = 0.05 m s−1 (u/umf = 3.3). The membrane permeability was de-termined to be Pm = 1.76·10−8 mol m−1 s−1 Pa−0.5, the operatingtemperature was 400 °C, the reactor pressure was varied between 1.44and 1.8 bar and the H2 mole fraction was varied between 0.1 and 1.0,and the model parameters were set up accordingly. Fig. 13 summarizesthe experimental observations in comparison with the obtained resultsfrom simulations with the 1D and 1D/kd models for different H2 partialpressure differences. The figure clearly shows that the 1D model ig-noring concentration polarization effects largely overestimates themembrane flux for all transmembrane pressure differences and thediscrepancies further increase for smaller hydrogen concentrations,whereas the 1D/kd model that accounts for concentration polarizationeffects accurately predicts the membrane flux for all the consideredcases.

Furthermore, the simulation results of the 1D/kd model for the casewithout bubble-to-emulsion mass transfer resistance were virtuallyidentical to the results when bubble-to-emulsion mass transfer wastaken into account, from which it can be concluded that the bubble-to-emulsion mass transfer resistance is negligible compared to the externalmass transfer resistance to the membrane wall for the considered cases.

The 1D model in this work uses the standard correlations from Kuniiand Levenspiel for the bubble-to-emulsion phase mass transfer (Kbe).The Davidson and Harrison correlation for the bubble-to-emulsionphase mass transfer coefficient was derived for single rising bubbles, soit will under predict the bubble-to-emulsion mass transfer for freelybubbling flows. In Fig. 14 the axial H2 mole fraction profiles along themembrane length are plotted for the bubble phase, emulsion phase andat the surface of the membrane. This is shown for one selected

Fig. 10. Time-averaged TFM solids hold-up data for a fluidized bed with membrane(X = 0.10, 0.25 and 0.45) and one case with membrane without gas extraction(X = 0.25), Preactor = 1.5 bar.

Fig. 11. Gas velocity profiles for a gas reactor with and without extraction at X = 0.25and Preactor = 1.5 bar.

Fig. 12. 1D and 1D/kd model predictions for the H2 flux compared with experiments byPatil et al. [57].

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472

experiment, but all the other simulations showed the same trend. Sincethe concentration differences between the bubble and the emulsionphase and along the membrane are small compared to the concentra-tion differences between the emulsion and the membrane wall, it can beconcluded that the external bed-to-membrane mass transfer is rate

limiting, thus the under prediction in the bubble-to-emulsion masstransfer rate will not have a significant effect on the simulation results.Medrano et al. (2017) [58] have developed a correlation for bubble-to-emulsion mass transfer in freely bubbling beds, however only valid forpseudo-2D beds. To more accurately calculate the bubble-to-emulsionmass transfer, improved correlations for Kbe need to be developedspecifically for fully 3D fluidized beds and fluidization in the presenceof (permeating) internals.

To figure out the effect of inlet flow velocity on concentration po-larization, experiments were performed with different inlet flow velo-cities for three different inlet compositions. Binary mixtures of H2 andN2 were chosen with a H2 content of 10, 25 and 45% at the inlet. Foreach inlet flow composition, the inlet velocity was varied to investigatethe performance of the model at higher inlet flow rates in the bubblingfluidization regime (Fig. 15).

According to the obtained results, in general the 1D/kd model canpredict accurately the flux through the membrane for different inletflow rates and inlet gas compositions. Considering the fact that thethickness of the film layer was considered with a constant value of0.01 m, it can be concluded that this constant can be a good estimate forthe average thickness of the film layer for a wide range of inlet flowvelocities. Investigating the obtained modeling results with and withoutconsidering concentration polarization, the effect of concentration po-larization becomes more pronounced for higher inlet gas velocities andlower hydrogen inlet concentrations, and the developed 1D/kd modelcan accurately capture this.

Fig. 13. Experimental data versus model predictions for different H2 concentrations; (1D): one-dimensional model without considering concentration polarization; (1D/kd): one-di-mensional model accounting for concentration polarization considering a film layer thickness of 1 cm and a radial gas dispersion coefficient Dr of 1 · 10−4 m2/s.

Fig. 14. Axial hydrogen mole fraction profiles at the membrane surface, in the emulsionand in the bubble phase. Experiment: X = 0.25, u = 0.05 m s−1, Preactor = 1.44 bar,Pm = 1.76 · 10−8 mol m−1 s−1 Pa−0.5.

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In general, for the phenomenological fluidized bed model a lot moreresearch needs to be done. The influence of the membrane immersionon the hydrodynamic properties of the bed needs to be further in-vestigated. The 1D/kd model gives a very good prediction of the ex-perimental observations when using a radial dispersion coefficient es-timated from TFM simulations. However, an accurate correlation forthe radial dispersion in fluidized beds (and preferably accounting forthe presence of immersed objects) would facilitate the modelling.Another complicating factor is that the film layer thickness is quitelarge compared to the reactor diameter (δ= 0.01 m and dr = 0.045 m).It should be noted that for this research a lab-scale reactor with a re-latively small diameter was used, in larger reactors the effect of the filmlayer may be less pronounced. On the other hand, often membranemodules are inserted into the bed, where the effect of the presence andpermeation through neighboring membranes might need to be ac-counted for in the estimation of the mass transfer boundary layerthickness.

4. Conclusions

A simple one-dimensional two-phase phenomenological model wasdeveloped that captures the effect of concentration polarization influidized bed membrane reactors (1D/kd). In this model the fluidizedbed was divided into a number of CSTR’s in series for both the emulsionand bubble phase while accounting for mass transfer limitations fromthe bed bulk to the surface of the membranes, assuming that this occursentirely in a thin stagnant film layer with constant thickness around themembrane. The H2 flux through the membranes was described bySieverts’ law.

A more detailed Euler-Euler CFD model, the Two-Fluid Model, was

developed in OpenFOAM where the solver was extended with speciesmass balance equations and membrane models to simulate the selectiveextraction of hydrogen. The model was used to quantify the extent ofconcentration polarization in a lab-scale experimental reactor and todetermine the mass transfer boundary layer thickness which is requiredby the one-dimensional phenomenological model.

Comparing the results obtained from experiments with the TFMmodel a very good agreement was found when an appropriate value forthe gas phase dispersion coefficient was selected. The computed con-centration profiles near by the membrane, confirmed the existence of aconcentration boundary layer in the vicinity of the membrane thatimposes a mass transfer resistance from the bulk of the fluidized bed tothe surface of the membranes. Although the thickness of the film layerincreases with the axial position, and decreases slightly for higher molefractions, an average film layer thickness was estimated at 0.01 m forall the different operating conditions and was assumed constant. Thisfilm layer thickness and gas dispersion coefficient was used in thephenomenological 1D/kd model.

The results of the 1D/kd model for the membrane flux were com-pared with experimental observations over a wide range of inlet con-centrations, operating pressures and inlet gas velocities, and a verygood agreement was found, despite the fact that the film layer thicknesswas assumed constant. It was also found that the bubble-to-emulsionphase mass transfer limitations are much less pronounced relative tothe emulsion-to-membrane wall mass transfer resistances for the in-vestigated cases. Comparison with the 1D model results that do notaccount for concentration polarization, clearly indicates the very pro-nounced effect of concentration polarization, also for fluidized bedmembrane reactors.

Fig. 15. Model predictions versus experiments at different inlet velocities at constant H2 partial pressure differences (a) =P PaΔ ( ) 86H20.5 0.5 , (b) =P PaΔ ( ) 150H20.5 0.5 , (c) =P PaΔ ( ) 205H20.5 0.5 .

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Appendix A

Table A.1.

Appendix B

Tables B.1–B.5.

Table A.1Summary of the hydrodynamic parameters used in the 1D phenomenological model [59]

Parameter Equation

Archimedes number=

−Ar

dpρg ρp ρg g

μg

3 ( )2

Minimum fluidization velocity⎜ ⎟= ⎛⎝

⎞⎠

+ −u Ar( (27.2) 0.0408 27.2)mfμg

ρg dp2

Bed voidage at minimum fluidization velocity⎜ ⎟= ⎛⎝

⎞⎠

−ε Ar0.586mfρgρp

0.0290.021

Initial bubble diameter (porous plate Distributor) = −d u u0.376( )b mf0 0 2

Maximum bubble diameter= ⎛

⎝− ⎞

⎠( )d d d u umin ,0.65 ( )b Hπ

H mf,max 42

00.4

Average bubble diameter= − −

⎡⎣⎢

− ⎤⎦⎥d d d d e( )b avg b b b

Hdh, ,max ,max ,0

0.15

Bubble diameter= − −

−d d d d e( )b b b b

Hdh,max ,max ,0

[ 0.3 ]

Velocity of rise of swarm of bubbles = − +u u u gd0.711( )b avg mf b avg, 0 , 1/2

Bubble phase fraction =−

fbu umf

ub avg

0

,

Emulsion phase fraction = −f f1e bGas exchange coefficient

⎜ ⎟= ⎛⎝

⎞⎠

+ ⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ = +K K4.5 5.85 6.77bc

umfdb avg

D g

db avgce

αmf Dub avgdb avg Kbe Kbc Kce,

1/4

,5/4

,

,3

1/21 1 1

Fuller equation

=+

+D 0.001mol

TMw A Mw B

p VA VB

1.75 1,

1,

( 1/3 1/3)2

Totsas and Schlünder = − − +D α D(1 1 )r mf molu dp0

8

Table B.1Summary of all governing and constitutive equations used in the TFM.

Continuity equation of the gas phase

+ ∇ =∂

∂α ρ Su·( )

αg ρgt g g g m

Continuity equation of the solids phase

+ ∇ =∂∂

α ρ u·( ) 0αsρst s s s

Momentum equation gas phase

+ ∇ = − ∇ − ∇ − − +∂

∂τ gα ρ α α p β α ρu u u u·( ) ( · ) ( )

αg ρg gt g g g g g g g g s g g

u

Momentum equation solids phase

+ ∇ = − ∇ − ∇ −∇ + − +∂∂

τ gα ρ α α p p β α ρu u u u·( ) ( · ) ( )αsρs st s s s s s s s s g s s s

u

Granular temperature equation (non-equilibrium)

+ ∇ = − + ∇ + ∇ ∇ − −∂∂( ) τα ρ θ p I α κ θ γ Bu u·( ) ( ): ·( )αsρsθt s s s s s s s s s s32 ( )

Viscous stress tensor gas phase

= −⎡⎣ ∇ + ∇ + ∇ ⎤⎦τ μ μ Iu u u( ( ) ) ( · )g g g gT

g g23

Viscous stress tensor solids phase

= −⎡⎣

∇ + ∇ + − ∇ ⎤⎦( )τ μ λ μ Iu u u( ( ) ) ( · )s s s s T s s s23

Inter-phase drag coefficient

= +−

β 150 1.75αs μgαg dp

αsρg g sdp

u u2

2| |

for ⩾α 0.20s

= − −β C αu u| |dαg αsρg

dp g s g34

2.65 for Re 1000p

=−

αRep gρg dp g s

μg

u u| |

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475

Table B.3Boundary conditions of the TFM simulations.

ug us p α α,g s YH2 θ

Inlet Dirichlet (interstitial inlet velocity) Dirichlet (zero) Neumann Neumann Dirichlet (Table B.4) Neumann (initial value set at t = 0)Outlet Neumann Dirichlet (zero) Dirichlet (Table B.4) NeumannMembrane =um

SmRTpMw

VcellAcell

Partial slip (spec. coef.= 0.50) Neumann Partial slip (spec. coef.= 0.50)

Right wall Dirichlet (no-slip)

Table B.4Simulation settings for the 2D systems.

Quantity Setting

Width (x) 0.0225 mHeight (z) 0.113 mNcells width 40Ncells height 200dp 200 µm

ρp 1400 kg/m3epp, epw 0.90u/umf 3.33DH2 1 · 10−4 m2/sQPd 4.3 · 10−3 mol/m2/s/Pa0.5

Am 1.836 · 10−3 m2

T 405 °Ctsim 15 sΔt 2 · 10−5 s

Table B.2Summary of all KTGF closure equations used in the TFM [41].

Solids shear viscosity

= + +⎛⎝

+ + ⎞⎠

⎛⎝

+ ⎞⎠μ πρ d α ρ d g e1.01600 (1 )s s p

θπ

e αsg αsg

αsg s s pθπ

596

1 85

(1 )2 0

1 85 0

0

45 0

Solids bulk viscosity

= +λ α ρ d g e(1 )s s s pθπ

43 0

Solids pressure= + +p α ρ θ e α g(1 2(1 ) )s s s s 0

Frictional pressure

= = = = =−− −

p Fr α Fr c c· with: 0.50, 0.05, 2, 3sfric αs αs

fric c

αs αs cs

fric[max((min. ),0)] 1

[max(( max ),5·10 2)] 2min.

1 2

Frictional shear viscosity

= = ∇ + ∇ − ∇ = °+

Sμ I ϕu u uwith: (( ) ( ) ) · with: 28S S

sfric ps

fric ϕ fric

αsθ

dp

s s T s fric2 sin

2 : 2

12

13

Radial distribution function

= =

⎜ ⎟

+ + +

⎡

⎣⎢⎢

− ⎛⎝

⎞⎠

⎤

⎦⎥⎥

g αwith: 0.62αs αs αsαs

αs

s01 2.5 4.5904 2 4.515439 3

1 max

3 0.67802max

Conductivity of fluctuation energy

= + +⎛⎝

+ + ⎞⎠

⎛⎝

+ ⎞⎠κ πρ d α ρ d g e1.02513 2 (1 )s s p

θπ

e αsg αsg

αsg s s pθπ

75384

1 125

(1 )2 0

1 125 0

0 0

Dissipation of granular energy

= − ⎡⎣

− ∇ ⎤⎦

γ e α ρ g θ u3(1 ) ( · )s s s dpθπ s

2 20

4

Fluctuating velocity/force correlation=B βθ3s

Table B.5Simulated hydrogen mole fraction and outlet pressures for the 2D systems.

XH2 Poutlet Pperm[–] [Pa] [Pa]

0.10 1.5 · 105 0.01 · 105

0.25 1.6 · 105

0.45 1.7 · 105

1.8 · 105

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On concentration polarization in fluidized bed membrane reactorsIntroductionModeling1D phenomenological modelTwo-fluid model

ExperimentalResults and discussionFilm layer thicknessDensified zones1D/kd model verificationModel vs. experiments

ConclusionsAppendix AAppendix BReferences