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Chemical Reactor Analysis and Design. 3th Edition. G.F. Froment, K.B. Bischoff † , J. De Wilde. Chapter 3. Transport Processes with Reactions Catalyzed by Solids. Part two Intraparticle Gradient Effects. Introduction. - PowerPoint PPT Presentation
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Chemical Reactor Analysis and Design
3th Edition
G.F. Froment, K.B. Bischoff†, J. De Wilde
Chapter 3
Transport Processes with Reactions Catalyzed by Solids
Part two Intraparticle Gradient Effects
Introduction
1. Transport of reactants A, B, ... from the main stream to the catalyst pellet surface.
2. Transport of reactants in the catalyst pores.3. Adsorption of reactants on the catalytic site.4. Chemical reaction between adsorbed
atoms or molecules.5. Desorption of products R, S, ....6. Transport of the products in the catalyst
pores back to the particle surface.7. Transport of products from the particle
surface back to the main fluid stream.
Steps 1, 3, 4, 5, and 7: strictly consecutive processes Steps 2 and 6: cannot be entirely separated !
Chapter 2: considers steps 3, 4, and 5Chapter 3: other steps
Molecular-, Knudsen- and surface diffusion in pores
[Adapted from Weisz, 1973]
(mainly encountered in zeolite catalysts)
Molecular-, Knudsen- and surface diffusion in pores
Molecular diffusion:• Driven by composition gradient• Mixture n components: Stefan-Maxwell:
M
ij ijD
jDiiDj
i D
yyRTp
'
,
,,NN
• Molecular diffusivities (binary):• independent of the composition• inversely proportional to the total pressure (gas)• proportional to T3/2
• Momentum transfer: by collisions between atoms or molecules• Fluxes: expressed per unit external surface of the catalyst particle ijsij DD '
with εs: void fraction of the catalyst particle (m3f / m3
cat)= fraction particle surface taken by pore mouths (Dupuit)
Molecular-, Knudsen- and surface diffusion in pores
Knudsen diffusion:• Mean free path of the components >>> pore dimensions• Momentum transfer: mainly collisions with the pore walls• Encountered
• at pressures below 5 bar• with pore sizes between 3 and 200 nm
• Knudsen diffusion flux of i : independent of the fluxes of the other components:
l
p
RT
DiiK
iK
,, N
• Knudsen diffusivity:
• function of the pore radius• independent of the total pressure• varies with T1/2
iiK M
RTrD
8
3
2, iKbiK DD ,
', and:
i
j
jK
iK
M
M
D
D
,
,
(Graham’s law)
Molecular-, Knudsen- and surface diffusion in pores
Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow:
tiK
toi
ijijD
jDiiDj
iK
iK
ip
D
pBy
D
yy
DRTp
'
,
'
,
,,
'
,
,
μ
NNN
Darcy’s permeability constant
Dusty gas model equation (kinetic gas theory)
Viscous flow term:• generally negligible• except when > 10 – 20 (micron-size pores)
iKto DpB ,/
Example: Binary mixture of A and B:
',
',
'
111
1
AKABD
A
BA
A DD
y
D
N
N
For equimolar counterdiffusion:
',
',
'
111
AKABDA DDD Additive resistance relation
(Bosanquet formula)
Molecular-, Knudsen- and surface diffusion in pores
Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow:Diffusion in a multicomponent mixture:Sometimes Stefan-Maxwell replaced by less complicated equivalent binary mixture equation:
m
k jKj
j
kk
jkjm Dyy
DD 1'
,''
111
N
N
Molecular-, Knudsen- and surface diffusion in pores
Surface diffusion:• Hopping of molecules from one adsorption site to another• Random walk model
'
2
τ
λkD
s
where:• : the jump length• : the correlation time for the motion• k : a numerical proportionality factor
'
Vary with temperature according to the van ‘t Hoff exponential law
• pre-exponential factor:• energy factor:
020 '/ kD as,
ED = 2Eλ – Eτ ~ 1/number of available sites
known for structured surfaces like zeolites, but much less for amorphous surfaces
• Depends on the surface coverage• More important in micro- than in macroporous material• Driving force: not fluid phase concentration gradient (Fickian law can not be applied)
Diffusion in a catalyst particle
A pseudo-continuum model:
Effective diffusivities:
Catalyst particles: very complicated (3D) pore structure
Model:• Pseudo-continuum• 1D• « Effective » diffusivity
Fick type law:
Pellet surface:
Sphere:
dz
dCDN A
eAA
r
C
rr
CDN AA
eAA
2
²
2
Diffusion in a catalyst particle
A pseudo-continuum model:
AsA
eA DD
D
'
sm
m
p
f3
in:
« Tortuosity » factor:• Tortuous nature of the pores• Eventual pore constrictions• Typical value: 2 - 3
Experimental determination of effective diffusivities of a component and of the tortuosity
Pulse response technique:• column packed with catalyst (fixed bed)• ideal plug flow pattern ( dt/dp)• tracer pulse injected in carrier gas flow• pulse response measured (reactor outlet)
In Out
Tracer pulse Pulse response
Fixed bed
Pulse widens:• Dispersion in the bed:
• Adsorption on the catalyst surface• Effective diffusion inside the catalyst particle
• Three parameters to be estimated: method of moments
Experimental determination of effective diffusivities of a component and of the tortuosity
Wicke-Kallenbach cell:• Steady state or transient operation• Single catalyst particle used as membrane• Above membrane: steady flow of carrier gas• Tracer pulse injected into the carrier gas:
• Diffuses through the catalyst membrane• Swept in the compartment underneath by a carrier gas => to detector
• Two parameters to be estimated
Determination tortuosity:• Specific catalyst characterization equipment (mercury porosimetry & nitrogen–sorption and –desorption)
Experimental determination of effective diffusivities of a component and of the tortuosity
EXAMPLE 3.5.1.2.AExperimental determination of the effective diffusivity of a component and of the catalyst tortuosity by means of the packed column technique
• Pt-Sn-y-alumina catalyst (catalytic reforming of naphtha)• Column internal diameter: 10-2 m• Column length: 0.805 m• Particle radius: 0.975 × 10-3 m• Void fraction of the packing: 0.429 m3
f / m3r
• Catalyst density, ρcat: 1080 kg cat/m3cat
Hg-porosimetry, N2-adsorption and -desorption?,s
Hg porosimetry:• Pore volume as a function of the amount of intruded mercury• Pore radius: calculated from Washburn eq. (cylindrical pores)• At 2000 bar: all pores > 3.3 nm filled with HgNitrogen sorption:• Steep increase of the amount adsorbed at pressure where the macropores are filled by nitrogen through capillary condensation• From Washburn eq.: total volume of adsorbed N2
• The volume of N2 adsorbed until the sharp rise is the meso pore volume
Experimental determination of effective diffusivities of a component and of the tortuosity
From Van Melkebekeand Froment [1995]
Experimental determination of effective diffusivities of a component and of the tortuosity
Cumulative pore volume distribution:• Derived from N2 adsorption curve: Broekhoff-De Boer eq. (cylindrical pores)• Inflection point => differential pore volume distribution by a peak at mean pore radiusTracer pulse injected into packed column:• Fitting data: Kubin and Kucera-model => De, KA, and Dax
=>
Remarks:• Performing experiments at various total pressures => possible to distinguish between and • Measurements possible in the absence or presence of reactions
D K
Diffusion in a catalyst particle
Structure and Network models: (in contrast to Pseudo-continuum model)
• More realistic representation• More accurate
Structure models:• The random pore model• The parallel cross-linked pore model
Network models:1. A Bethe tree model2. Network models for disordered pore media
• Monte Carlo simulation• Effective Medium Approximation (EMA)
Diffusion in a catalyst particle
Structure models: The random pore model:• Macro- micro pore model [Wakao and Smith, 1962 & 1964]• Application: pellets manufactured by compression small particles• Void fraction- and pore radius distributions: each replaced by two averaged values, for the macro for the micro distribution (often a pore radius of ~100 Å is used as the dividing point between macro and micro)• Micro-pores particles: randomly positioned in pellet space• Macro-pores of the pellet: interstices• Diffusion flux: three parallel contributions:
1. Through the macro-pores2. Through the micro-pores3. Through interconnected macro-micro pores
DDDD
M
MM
M
MMMe 2
2
2
222
1122
11
DDM
MMM
1
3122
KKMABorM DorDDD
111with:
Diffusion in a catalyst particle
Structure models: The random pore model:
Diffusion areas in random pore model. Adapted from Smith [1970].
Diffusion in a catalyst particle
Structure models: The parallel cross-linked pore model• Pore size and orientation distribution function:• Pellet flux: integrating flux in single pore with orientation l and accounting for the distribution function:
),( rf
ddrrfN ljlj ),(,N
lwith: : unit vector or direction cosine between l direction and coordinate axes
Example: mean binary diffusivity:dl
dCDN j
jmlj , jljm CD .
Kjj
j
kk
N
k jkjm Dy
N
Ny
DD
111
1
with:
ddrrfCD jlljmj ),(.N
l lwith: the tortuosity tensor
Diffusion in a catalyst particle
Structure models: The parallel cross-linked pore modelLimiting cases:
1) Perfectly communicating pores Cj(z; r, Ω) = Cj(z)
2) Noncommunicating pores:Pure diffusion at steady state, dNj/dz = 0 or Nj = constant
+ no assumption on communication of pores
),(00
rfdCDddrdzN ll
C
C
jjm
L
jz
jL
j
(closest to usual types of catalyst particles)
jsjmj CrdrrD )()()( N
with: κ(r) : a reciprocal tortuosity (results from the integration over Ω)Proper diffusivity: weighted with respect to the measured pore size distribution
Diffusion in a catalyst particle
Structure models: The parallel cross-linked pore modelLimiting cases:
3) Pore size and orientation effects are uncorrelated
)()(),( frfrf
with:• f(r) : the pore size distribution
• 1)( df
jsjmj CdfrdrD )()(N
Completely random pore orientations=> tortuosity depends only on the vector component cos
3
1cos2 dfdf
= 3
Diffusion in a catalyst particle
Network models: A Bethe tree model:
branching network of pores:• coordination number of 3• no closed loops
• Higher coordination numbers possible• Pores can have a variable diameter• Main advantage: can yield analytical solutions for the fluxes• Disadvantage: absence of closed loops not entirely realistic
Diffusion in a catalyst particle
Network models: Disordered pore media:• Amorphous catalysts: no regular or structured morphology• Sometimes structure modified during its application (pore blockage)
• Pore medium description:• Network of channels (preferably 3D)• Size distribution• Disorder to be included: certain fraction of pores blocked
Random number generator(Monte Carlo simulation)
Calculations repeated for same over-all blockage probability & average pore size => calculated set of values of De is averaged
• Effective Medium Approximation (EMA): construct small size network => relation between diffusivity & blockage without considering complete network
Diffusion and reaction in a catalyst particle. A continuum model
First-Order Reactions. The Concept of Effectiveness Factor:
Reaction and diffusion occur simultaneous: Process not strictly consecutive Both phenomena must be considered together
Example: first-order reaction, equimolar counterdiffusion, isothermal conditions, and steady-state: slab of thickness L:
Species continuity equation A: 02
2
sss
eA Ckdy
CdD
with boundary conditions:sss CLC )( at the surface
0)0(
dy
dCs at the center line
LD
k
yD
k
C
yC
eA
s
eA
s
ss
s
cosh
cosh)(
Solution:
' modulus eAs DkL /
Diffusion and reaction in a catalyst particle. A continuum model
First-Order Reactions. The Concept of Effectiveness Factor:
with: for a slab of thickness L
Diffusion and reaction in a catalyst particle. A continuum model
First-Order Reactions. The Concept of Effectiveness Factor:
Effectiveness factor:
conditionssurfaceatreactionofrate
resistancediffusionporewithreactionofrate
)(
)(1
ssA
csAc
Cr
dWCrW
Observed reaction rate: )( ssAobsA Crr
First-order reaction'
'tanh
Extension to more practical pellet geometries: cylinders or spheres:
e.g., sphere: sAAs
eA rdr
dCr
dr
d
rD
2
2
1
eA
s
D
k
S
V Aris [1957]:
Diffusion and reaction in a catalyst particle. A continuum model
First-Order Reactions. The Concept of Effectiveness Factor:
Effectiveness factors for slab (P), cylinder (C), and sphere (S) as functions of the Thiele modulus. Dots represent calculations by Amundson and Luss [1967] and Gunn [1965]. From Aris [1965].
Diffusion and reaction in a catalyst particle. A continuum model
More General Rate Equations. Single rate equation:
Analytical solution not possible
• Generalized modulus (10 – 15% error)
• Numerical solution
2/1
'''
,
)()(2
)(
ss
eqs
C
C
sssAseAs
ssA dCCrCD
Cr
S
V
Coupled multiple reactions:
Numerical solution:• finite difference• orthogonal collocation
Nonane
Dimensionless radial coordinate, r
0.0 0.2 0.4 0.6 0.8 1.0
Pi/P
s
0.0
0.2
0.4
0.6
0.8
1.0
Wilke
Stefan-Maxwell
Catalytic reforming of naphtha on Pt.Sn/alumina. Dimensionless partial pressure profile inside the particle for the reactant nonane. Total pressure: 7 bar, T = 510 °C, molar ratio H2/Hydrocarbons = 5. From Sotelo-Boyas and Froment [2008].
Depends on Cs !
Falsification of rate coefficients and activation energy by diffusion limitations
Consider nth order reaction:
nssobsA Ckr
nssCk
1
~
2/1
1
2
nsseAobsA CkD
nV
Sr
Introduce generalized modulus:
observed rate: order (n+1)/2
only correct for 1st order reaction
Also: kkobs
2/1/0
/
1
2 RTERTED eAeA
nV
SD
22/1
ln EEE
Td
kdE Dobs
obs
effective diffusion
(strong pore diffusion limitations)
Falsification of rate coefficients and activation energy by diffusion limitations
ln
ln
2
11
d
d
E
Eobs
ln
ln
2
1
d
dEEEE Dobs
Weisz and Prater [1954]:
or:
Languasco, Cunningham, and Calvelo [1972]:
lnd
lnd
2
1obs
nnn
EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS
First-order reaction: 612661262112212 OHCOHCOHOHC H
(sucrose) (glucose) (fructose)
Studied in particles with different size [Gilliland, Bixler, and O’Connell, 1971]
Falsification of rate coefficients and activation energy by diffusion limitations
EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS
dp (mm) as sk )( 1 04.0/ kk
eAs DkR /
0.04 0.0193 1.0 0.53 1.0 0.27 0.0110 0.570 3.60 0.600 0.55 0.00664 0.344 7.35 0.352 0.77 0.00487 0.252 10.3 0.263
a Calculated on the basis of approximate normality of acid resin = 3N.DeA = 2.69 × 10-7 cm2/s.Separately measured:
dp (mm) E (kJ/mol) 0.04 105 0.27 84 0.55 75 0.77 75
Homogeneous acid solution 105 From data at 60 and 70°C. ED = 34 kJ/mol obsE =
2
34105 = 70 kJ/mol theor.
Compare: Diffusional resistance decreases selectivity !
Influence of diffusion limitations on the selectivities of coupled reactions
[Wheeler, 1951]Parallel, independent reactions:
RA 1 , with order 1a
SB 2 , with order 2a
Absence of diffusion limitations:
2
1
2
1a
Bs
aAs
S
R
C
C
k
k
r
r
With pore diffusion limitations:
2
1
22
11a
Bs
aAs
obsS
R
C
C
k
k
r
r
Strong pore diffusion limitations: i ~ i/1
2/1
1
1
2
1
1
2
2
1
1
1a~
asBs
asAs
eB
eA
obsS
R
C
C
D
D
k
k
ar
r
First-order reactions & eBeA DD
sBs
sAs
obss
R
C
C
k
k
r
r
2
1
Influence of diffusion limitations on the selectivities of coupled reactions
[Wheeler, 1951]Consecutive first-order reactions:
SRA 21
Absence of diffusion limitations:
As
Rs
A
R
C
C
k
k
r
r
1
21
With pore diffusion limitations:
Species continuity equations A and R, for slab geometry:
AssAs
eA Ckdz
CdD 12
2
RssAssRs
eR CkCkdz
CdD 212
2
Selectivity
L
As
L
RsAs
L
A
L
R
obsA
R
dzCk
dzCkCk
dzr
dzr
r
r
0
1
0
21
0
0
Selectivity:
Influence of diffusion limitations on the selectivities of coupled reactions
[Wheeler, 1951]Consecutive first-order reactions:
With pore diffusion limitations:
sAs
sRs
C
C
k
k
1
2
1
212
1
/1
with:i
ii
tanh eisii DkL /
eR
eA
Dk
Dk
1
2
2
1
2
with i = 1, 2
Strong pore diffusion limitation and for DeA = DeR:
obsA
R
r
r
~ s
As
sRs
C
C
k
k
1
21
1
1
sAs
sRs
C
C
k
k
kk 1
2
12 /1
1
Compare: Diffusional resistance decreases selectivity !
Criteria for the importance of intraparticle diffusion limitations
Determining kinetic parameters from experimental data:• kρs not available yet• Criteria importance pore diffusion not explicitly containing kρs also needed !1) Experiments with two different sizes of catalyst:
Assume kρs and DeA same for both sizes
2
1
2
1
L
L
S
VL with:
2
1
2
1
obs
obs
r
rand:
No intraparticle diffusion limitations: 21 = 1
Strong intraparticle diffusion limitations: = 1/
1
2
1
2
2
1
L
L
r
r
obs
obs
:
2) Weisz-Prater criterion [1954]:
2
2obs
sseA
sA
CD
LrFirst-orderreaction:
No pore diffusion limitation: << 1, η = 1 => 2 << 1; Strong pore diffusion limitation: >> 1, η = 1/ => 2 >> 1.
Extendable via generalized modulus
Combination of external and internal diffusion limitations
s
seA
ssg dz
dCDCCk
Nonisothermal particles
Practical situations:• Internal temperature gradients unlikely• Internal gradients unlikely to cause particle instability