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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
Document license:CC BY
DOI:10.1016/j.cej.2017.09.045
Document status and date:Published: 15/01/2018
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Contents lists available at ScienceDirect
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: [email protected]
(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
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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.
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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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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].
A. Helmi et al. Chemical Engineering Journal 332 (2018)
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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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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
A. Helmi et al. Chemical Engineering Journal 332 (2018)
464–478
476
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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