Transcript Molecular-, Knudsen- and surface diffusion in pores

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 5
Chapter 3: other steps
Molecular-, Knudsen- and surface diffusion in pores
(mainly encountered
in zeolite catalysts)
[Adapted from Weisz, 1973]
Molecular-, Knudsen- and surface diffusion in pores
Molecular diffusion:
• Driven by composition gradient
• Mixture n components: Stefan-Maxwell:
M
 p   RT
i

ji
y N
j
D ,i
 y N
i
D, j
D'
D , ij
• 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
'
D ij   s D ij
catalyst particle
with εs: void fraction of the catalyst particle (m3f / m3cat)
= 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:
N K ,i  
D K ,i  p i
RT
l
• Knudsen diffusivity:
D K ,i 
2r
8 RT
3
M
and:
D K ,i   b D K ,i
'
i
• function of the pore radius
• independent of the total pressure
• varies with T1/2
D K ,i
DK,j

M
j
M
i
(Graham’s law)
Molecular-, Knudsen- and surface diffusion in pores
Simultaneous Molecular and Knudsen diffusion
and flux from viscous or laminar flow:
Darcy’s
permeability
Dusty gas model equation (kinetic gas theory)
constant
pi
N
K ,i
  RT  '

 D K ,i

yjN

 yi N
D ,i
D
ji
D, j
'
D , ij
Example: Binary mixture of A and B:
1
D
'
A


N
1  y A  1 
N

D
B
A
'
D , AB





1
'

y B p
  i o t p
'
t
μ
D

K ,i

Viscous flow term:
• generally negligible
• except when
B o p t /  D K , i > 10 – 20
(micron-size pores)
D K ,A
For equimolar counterdiffusion:
1
'
DA

1
'
D D , AB

1
'
D K ,A
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:
1
D
'
jm
m


k 1

Nk
1 
1
y 

y
k
j
'
 D'
N j
D jk 
K,j

Molecular-, Knudsen- and surface diffusion in pores
Surface diffusion:
• Hopping of molecules from one adsorption site to another
• Random walk model
kλ2
Ds 
Vary with temperature according
to the van ‘t Hoff exponential law
τ'
where:
•  : the jump length
•  ' : the correlation time for the motion
• k : a numerical proportionality factor
• pre-exponential factor: D s,a  k 20 /  ' 0
• energy factor: 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: N A   D eA
Sphere:
NA
dC
A
dz
  2C A
2 C A 

  D eA 



r
²
r

r


Diffusion in a catalyst particle
A pseudo-continuum model:
D eA 
D

'
A

s

DA
in:
m
3
f
m ps
« 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
Tracer pulse
Fixed bed
Out
Pulse response
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.A
Experimental 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 m3f / m3r
• Catalyst density, ρcat: 1080 kg cat/m3cat
s ,  ?
Hg-porosimetry, N2-adsorption and -desorption
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 Hg
Nitrogen 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 Melkebeke
and 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 radius
Tracer 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  D and  K
• Measurements possible in the absence or presence
of reactions
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 model
2. 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-pores
2. Through the micro-pores
3. Through interconnected macro-micro pores

2
D e   M D M  1   M
2
2
  1  3 M
2

2
M
DM 
1   M 
1M

D
2
D   2 2  M 1   M
with:
1
DM
or 

1
D AB



2
1   M 
2
D
1
D KM or D K 
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: f ( r ,  )
• Pellet flux: integrating flux in single pore with orientation l
and accounting for the distribution function:
N
j

 
l
N
j,l
f ( r ,  ) dr d 
with:  l : unit vector or direction cosine between l direction and coordinate axes
Example: mean binary diffusivity: N
with:
j,l
1
D jm
N
j
  D jm
N


k 1
   D jm  l  l . C j f ( r ,  ) dr d 
with:  l  l the tortuosity tensor
dC
dl
j
  D jm  l .  C
j

Nk
1 
1
y 

yj 
k

 D
D jk 
Nj
Kj

Diffusion in a catalyst particle
Structure models: The parallel cross-linked pore model
Limiting cases:
1) Perfectly communicating pores
Cj(z; r, Ω) = Cj(z)
(closest to usual types of catalyst particles)
N
j
 
D
jm
(r )  (r ) d s (r )
 C j
with: κ(r) : a reciprocal tortuosity
(results from the integration over Ω)
Proper diffusivity: weighted with respect
to the measured pore size distribution
2) Noncommunicating pores:
Pure diffusion at steady state, dNj/dz = 0 or Nj = constant
+ no assumption on communication of pores
L
N
jz
 dz  
0
 C jL
dr d     D jm dC
 C j 0

  l l f ( r ,  )
j

Diffusion in a catalyst particle
Structure models: The parallel cross-linked pore model
Limiting cases:
3) Pore size and orientation effects are uncorrelated
f (r ,  )  f (r ) f  ( )
with:
• f(r) : the pore size distribution
•  f  (  )d   1
N
j
 
D
jm
(r ) d s (r )
 
f  d 
 C j
Completely random pore orientations
=> tortuosity depends only on the vector component cos 
1
2
 f   d    f  cos  d   3

=3
Diffusion in a catalyst particle
Network models: A Bethe tree model:
• 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
branching network of pores:
• coordination number of 3
• no closed loops
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
Random number generator
• Disorder to be included: certain
(Monte Carlo simulation)
fraction of pores blocked
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:
2
Species continuity equation A: D eA
d Cs
dy
with boundary conditions: C s ( L )  C s
s
dC s ( 0 )
Solution:
dy
cosh
Cs ( y)
C
s
s
k s
y
D eA

cosh
k s
D eA
L
2
 k s C s  0
at the surface
 0 at the center line
Diffusion and reaction in a catalyst particle. A continuum model
First-Order Reactions. The Concept of Effectiveness Factor:
with:  ' 
modulus  L k  s / D eA 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:
 
rate of reaction with pore diffusion
resistance
rate of reaction at surface conditions
1
 
Wc
r
A
( C s ) dW c
s
rA ( C s )
Observed reaction rate:
First-order reaction
 
 rA obs
  rA ( C s )
s
tanh  '
'
Extension to more practical pellet geometries: cylinders or spheres:
e.g., sphere: D eA
1 d  2 dC As 
r
  rA  s
2
dr 
r dr 
Aris [1957]:  
V
k s
S
D eA
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)
1/ 2
s

V rA ( C s )  s 
'
'
'
  D eA ( C s ) rA ( C s )  s dC s 
 
S
2
 C s , eq

Depends on Cs !
• Numerical solution
s
Cs
Coupled multiple reactions:
1 .0
Nonane
0 .8
P i/P s
Numerical solution:
• finite difference
• orthogonal collocation
0 .6
W ilk e
0 .4
S te fa n -M a x w e ll
0 .2
0 .0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
D im e n sio n le ss ra d ia l co o rd in a te , r
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].
Falsification of rate coefficients and activation energy by diffusion limitations
Consider nth order reaction:
r A  obs
 
 k C
~
1

s
s
n
 
k C
s
s
n
Introduce generalized modulus:
 r A  obs

S
2
V
n 1
 
D eA k C
s
s
 n 1  / 2
observed rate: order (n+1)/2
only correct for 1st order reaction
Also: k obs   k

S
V
E obs  
effective diffusion
2
n 1
A
d ln  k obs
d 1 / T 
D

e

 E D / RT
A0 e
E  ED
2
 E / RT


1/ 2
E
2
(strong pore diffusion limitations)
Falsification of rate coefficients and activation energy by diffusion limitations
Weisz and Prater [1954]:
E obs
1
E
or:
E obs  E 
1 d ln 
2 d ln 
1
2
E
 ED

d ln 
d ln 
Languasco, Cunningham, and Calvelo [1972]:
n obs  n 
EXAMPLE 3.7.A
n  1 d ln 
2
d ln 
EFFECTIVENESS FACTORS FOR SUCROSE
INVERSION IN ION EXCHANGE RESINS

First-order reaction: C 12 H 22 O 11  H 2 O   C 6 H 12 O 6  C 6 H 12 O 6
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
dp (mm)
0.04
0.27
0.55
0.77
a
k s ( s
EFFECTIVENESS FACTORS FOR SUCROSE
INVERSION IN ION EXCHANGE RESINS
1
)
a
0.0193
0.0110
0.00664
0.00487
k / k 0 .04

  R k  s / D eA
1.0
0.570
0.344
0.252
0.53
3.60
7.35
10.3
1.0
0.600
0.352
0.263
Calculated on the basis of approximate normality of acid resin = 3N.
Separately measured: DeA = 2.69 × 10-7 cm2/s.
dp (mm)
0.04
0.27
0.55
0.77
Homogeneous acid solution
From data at 60 and 70°C. ED = 34 kJ/mol
E (kJ/mol)
105
84
75
75
105
theor. E obs =
105  34
2
= 70 kJ/mol
Influence of diffusion limitations on the selectivities of coupled reactions
Parallel, independent reactions:
[Wheeler, 1951]
A
 R , with order a 1
1
B
 S , with order a 2
2
Absence of diffusion limitations:
 rR

r
 S
With pore diffusion limitations:
 rR

r
 S

k 1 C As  1

a2
 k


C
2

Bs
a

 1 k 1 C As  1


a2


k


C
2 2
 obs
Bs
Compare:
 Diffusional resistance
decreases selectivity !
a
Strong pore diffusion limitations:  i ~ 1 /  i
 a  1 k D C
 rR 
eA
2
1


~

r 
s
 a 1  1 k 2 D eB C Bs
 S  obs
s
As


a1  1


a 2 1

First-order reactions & D eA
1/ 2
 rR 



r 
 s  obs
s
k 1 C As
s
k 2 C Bs
 D eB
Influence of diffusion limitations on the selectivities of coupled reactions
Consecutive first-order reactions:
[Wheeler, 1951]
A
 R 
 S
1
Absence of diffusion limitations:
rR
2
1
rA
k 2 C Rs
k 1 C As
With pore diffusion limitations:
Species continuity equations A and R, for slab geometry:
2
D eA
d C As
dz
2
 k 1  s C As
2
D eR
Selectivity
d C Rs
dz
2
  k 1  s C As  k 2  s C Rs
Influence of diffusion limitations on the selectivities of coupled reactions
Consecutive first-order reactions:
[Wheeler, 1951]
With pore diffusion limitations:
Selectivity:
L
 rR 
  
r 
 A  obs
L
r
R
dz
r
1

0
L
A
 k C
0
L
k C
dz
1
0
Compare:
 Diffusional resistance
decreases selectivity !
with:  i 

tanh  i
i
 k 2 C Rs dz
As
As
dz
0
1   2 /  1 
1
2
 
 1
 k 2 C Rs

 k Cs
 1 As
 i  L k i  s / D ei
with i = 1, 2
s
 2
  
 1
2

k 2 D eA
 

k 1 D eR

Strong pore diffusion limitation and for DeA = DeR:
s
 rR 
1
1 k 2 C Rs




~
s
r 
1
1 
 k 1 C As
 A  obs
1
k 2 / k1
s

k 2 C Rs
s
k 1 C As
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
1
2

L1
L2
with: L 
V
and:
S
robs 1
robs  2

1
2
No intraparticle diffusion limitations:  1   2 = 1
Strong intraparticle diffusion limitations:  = 1/  :
robs 1
robs  2

2
1

L2
L1
2) Weisz-Prater criterion [1954]:
First-order
reaction:
 r A  s  obs L
D eA C
s
s
2
 
Extendable via generalized modulus
2
 
No pore diffusion limitation:
2
 << 1, η = 1 =>  << 1;
Strong pore diffusion limitation:
2
 >> 1, η = 1/  =>    >> 1.
Combination of external and internal diffusion limitations

kg C  C
s
s

 dC s 
 D eA 

 dz  s
Nonisothermal particles
Practical situations:
• Internal temperature gradients unlikely
• Internal gradients unlikely to cause particle instability