L20-1
Review: Heterogeneous Catalyst
• We have looked at cases where
1) Adsorption, surface reaction, or desorption is rate limiting
2) External diffusion is rate limiting
3) Internal diffusion is rate limiting- today
• Next time: Derive an overall rate law for heterogeneous catalyst where the
rate limiting step as any of the 7 reaction steps. This new overall reaction
rate would be inserted into the design equation to get W, XA, CA, etc
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-2
Review: Types of Boundary Conditions
1. Concentration at the boundary (i.e., catalyst particle surface) is specified:
• If a specific reactant concentration is maintained or measured at the
surface, use the specified concentration
• When an instantaneous reaction occurs at the boundary, then CAs≈0
2. Flux at the boundary (i.e., catalyst particle surface) is specified:
a) No mass transfer at surface (nonreacting surface)
WA surface  0
b) Reaction that occurs at the surface is at steady state: set the molar
flux on the surface equal to the rate of reaction at the surface
WA surface  rA ''
reaction rate per unit surface area (mol/m2·sec)
c) Convective transport across the boundary layer occurs
WA boundary  k c  CAb  CAs 
3. Planes of symmetry: concentration profile is symmetric about a plane
• Concentration gradient is zero at the plane of symmetry
Radial diffusion
in a tube:
r
dCA
 0 at r=0
dr
r
Radial diffusion
in a sphere
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-3
Review: Transport & Rxn Limited Rates
r ''As 
-rA’
kr k c CAb
kr  k c
reaction limited regime: r '' As  kr CAb
Used kc(CAb-CAs)=krCAS to
solve for CAs & plugged
back into –r”As= krCAS
transport limited regime
(Convective transport across boundary layer)
r '' As  k c CAb
D

Ud
k c  AB Sh Sh  2  0.6Re1 2 Sc1 3 Re= p Sc 
dp
DAB

12
13 

Ud




D

p

k c  AB  2  0.6 
 

dp 
    DAB  

(U/dp)1/2
(fluid velocity/particle diameter)1/2
When measuring rates in the lab, use high velocities or small particles
to ensure the reaction is not mass transfer limited
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-4
Review: Mass Transfer Limited Rxn in
PBR
b
c
d
A B C D
a
a
a
A steady state mole balance on reactant A between z and z + z :
6 1  f 
FAz z  FAz z z  r ''A ac (A c z)  0 where ac 
dp
ac: external surface area of catalyst per volume of catalytic bed (m2/m3)
f: porosity of bed, void fraction
dp: particle diameter (m)
r’’A: rate of generation of A per unit catalytic surface area (mol/s·m2)
Ac: cross-sectional area of tube containing catalyst (m2)
1.
2.
3.
4.
5.
Divide out Acz and take limit as z→0
Put Faz and –rA’’ in terms of CA
Assume that axial diffusion is negligible compared to bulk flow
Assume molar flux of A to surface = rate of consumption of A at surface
Rearrange, integrate, and solve for CA and r’’A
 k c ac
CA  CA0 exp  
 U

z

 k c ac
r ''A  k c CA0 exp  
 U

z

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-5
Shrinking Core Model
• Solid particles are being consumed either by dissolution or reaction
• The amount of the material being consumed is shrinking
• Drug delivery (pill in stomach)
• Catalyst regeneration
• Regeneration of catalyst by burning off carbon coke in the presence of O2
• Begins at the surface and proceeds to the core
• Because the amount of carbon that is consumed (burnt off) is
proportional to the surface area, and the amount of carbon that is
consumed decreases with time
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-6
Catalyst Regeneration
Coking-deactivated catalyst particles are reactivated by burning off the carbon
R0
C  O2  CO2
•Oxygen (A) diffuses from particle surface
(r = R0, CA = CA0) through the porous pellet
r+r
matrix to the unreacted core (r = R, CA = 0)
•Reaction of O2 with carbon at the surface of
r
the unreacted core is very fast
R
•CO2 generated at surface of core diffuses out
•Rate of oxygen diffusion from the surface of
CO2
the pellet to the core controls rate of carbon
removal
r : radius R0:outer radius of particle R: radius of unreacted core r = 0 at core
O2
What is the rate of time required for the core to shrink to a radius R?
Though the core of carbon (from r = 0 to r = R0) is shrinking with time (unsteady
state), we will assume the concentration profile at any time is the steady state
profile over distance (R0- R): quasi-steady state assumption (QSSA)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-7
Mole Balance on O2 From r to r+r
R0 C  O2  CO2
O2
Rate in - rate out
WAr 4 r 2  WAr 4 r 2
r+r
r
r r
0 0
Oxygen reacts at the surface, not in this region
r
R
Divide by -4r: 
WAr r 2
Take limit as r→0: 
CO2
Put WAr in terms of
conc of oxygen (CA)
WA  De
r r
dCA 2 
d

D
r 0
 e
dr 
dr

 WAr r 2
r

d WAr r 2
dr
dCA
 y A  WA  WB 
dr
For every mole of O2 that enters, a
mol of CO2 leaves → WO2=-WCO2
Plug WAr into
mole balance:
+ gen = accum
r
0
0
De: effective diffusivity
dCA
dr
d  dCA 2 
 
r 0
dr  dr

 WA  De
Divide out –De:
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-8
Mole Balance on O2 From r to r+r (2)
O2
dCA 2
d  dCA 2 

r  K1
r   0
 
R0 C  O2  CO2
dr
dr  dr

K
K
r+r
  dCA   21 dr  CA   1  K 2
r
r
Use boundary conditions to determine the concentration
r
profile (CA/CA0) in terms of the various radii (R, R0 & r)
R
CO2
For any r:
At r = R0, CA= CA0 and at r = R, CA= 0
K
K
First use CA=0 when
CA  0   1  K 2
 1  K2
r = R to determine K2
R
R
K K
 1 1
Next solve for when
 C A  K1   
CA   1  1
r = R0 & CA=CA0
r
R
R r 
K
K
C A0   1  1
R0 R

CA
CA0
1 1 
Take the ratio to
 CA0  K1  

determine CA/CA0
R
R

0
CA
1 R 1 r
K1 1 R  1 r 



CA0 1 R  1 R0
K1 1 R  1 R0 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-9
Oxygen Concentration Profile & Flux
CA
1 R 1 r

CA0 1 R  1 R0
R0 C  O2  CO2
O2
CA: oxygen concentration
CAb = CA0
Oxygen concentration Profile at time t
r+r
1
r
R
CO2
0.8
0.6
CA
0.4
C A0
0.2
Finally determine the
0
flux of oxygen to the
0
10
20
R
surface of the core:
(center)
(core)
r
R0
dC A
WA  De
dr
d  CA0 1 R  1 r  
DeCA0
CA0 1 R  1 r 
 WA  De 
CA 
  WA 
dr
1
R

1
R
1 R  1 R0

0 
1 R  1 R0  r 2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-10
Mass Balance on Carbon (C)
R0 C  O2  CO2
O2
r+r
r
R
In – out + gen = accumulation
4

d   R3 CfC 
3

0  0  r ''c 4 R2  
dt
Elemental C does not enter or leave the surface
Change in the mass of the carbon core
r’’C: rate of C gen. per unit surface area of core (mol/s·m2)
C: density of solid C
fC: fraction of the volume of the core that is C
4 3

d   R CfC   r ''c  dR
Simplify mass balance:
3

CfC dt
r ''c 4 R2  
dt
The rate of carbon disappearance (-dR/dt) is equal to the rate of oxygen flux
to the surface of the core, -WO2 = WCO2, and this occurs at a radius of R so:
DeCA0
DeCA0
 r''C =
r''C  - WA  WB 
R  R 2 R0
1 R  1 R0  r 2
CO2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-11
Time Required to Shrink Core to Radius R
R0
O2
r ''c
dR

dt CfC
C  O2  CO2
r+r
R
R

2

t
  R  R R0 dR  
R
R0
DeCA0
R
R 


t
 
2 3R0 
CfC
R0
2
3
R  R 2 R0
Substitute r’’c into -dR/dt, dR D C 

1
e
A0



get like terms together, 
dt
CfC  R  R2 R0 
integrate, & solve for t
r
CO2
-r''c =
DeC A0
0
DeCA0
CfC
dt Integrate over
0 to t & R0 to R
R2 R3 R02 R03 DeC A0





t
2 3R0
2
3R0
CfC
R2 R3 R02 R02 DeCA0





t
2 3R0
2
3
CfC
3R0R2 2R3 3R02  2R02 DeCA0




t
6R0
6R0
6
CfC
Get common
denominators
3R0R2 2R3 R02 DeCA0




t
6R0
6R0
6
CfC
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-12
Time Required to Shrink Core to Radius R
R0
O2
C  O2  CO2
r+r
r
R
CO2
3R0R2 2R3 R02 DeCA0




t
6R0
6R0
6
CfC
Solve for t:
 3R0R2 2R3 R02   CfC 



  
t
 6R0
6R0
6   DeCA0 

Factor out
R02/6
 3R2 2R3
 CfCR02 
  2  3  1 
 t Factor out -1

R
 6DeCA0 
R0
 0


 3R2 2R3  CfCR02
 1  2  3 
t
 R

R0  6DeCA0

0
2
3

R 
 R   CfCR02
 1  3 
t
  2
 

 R0 
 R0   6DeCA0

2
3
At the core of the

2
 0 
 0   CfCR02

f
R
C
C
0
catalyst particle,  t   1  3 
t
  2
 
R
R
6D
C


6DeCA0
 0
 0 
e A0
R=0, then:

Complete regeneration
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-13
L20: Internal Diffusion Effects in
Spherical Catalyst Particles
Internal diffusion: diffusion of the reactants or products from the external
pellet surface (pore mouth) to the interior of the pellet. (Chapter 12)
When the reactants diffuse into the pores within the catalyst pellet, the
concentration at the pore mouth will be higher than that inside the pore and
the entire catalytic surface is not accessible to the same concentration.
CAs
CA(r)
CAb
External
diffusion
Internal
diffusion
Though A is diffusing
inwards, convention of
Porous catalyst shell balance is flux is
in direction of
particle
increasing r. (flux is
positive in direction of
External surface increasing r). In
actuality, flux of A will
have a negative sign
since it moves inwards.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Basic Molar Balance for Differential
Element
CAs
R
L20-14
An irreversible rxn A→B occurs on the surface of
pore walls within a spherical pellet of radius R:
2
Rate of A in at r = WAr · area = WAr  4r r
r
r+r
2
Rate of A out at r - r = WAr · area = WAr  4r r r
The mole balance over the shell thickness r is:
Spherical shell of
IN
OUT
inner radius r & outer
radius r+r
WAr 4r 2 r  WAr 4r 2
+
r r
GEN


=ACCUM
 rA 4rm2r c  0
Volume of shell
r’A: rate of reaction per mass of catalyst (mol/g•s)
c: mass of catalyst per unit volume of catalyst (catalyst density)
rm: mean radius between r and r - r
d WAr r 2
Divide by -4r &
Differential BMB in
2
 rA r c  0 spherical catalyst particle
take limit as r →0 
dr


Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-15
Diffusion Equation (Step 2)
CAs
IN
-
OUT
+
GEN


=ACCUM
WAr 4r 2 r  WAr 4r 2 r r  rA 4rm2r c  0
R
Steady state assumption implies equimolar counter
diffusion, WB = -WA (otherwise A or B would
r+r
accumulate)
dy A
dC A
WA  cDe
 De
dr
dr
Must use effective diffusivity, De, instead of DAB to account for:
1) Tortuosity of paths
2) Void spaces
3) Pores having varying cross-sectional areas
r
DA bulk diffusivify
De  DAB
fpc

fp

~
pellet porosity (Vvoid space/Vvoid & solid) (typical ~ 0.4)
constriction factor (typical ~ 0.8)
tortuosity (distance molecule travels between 2 pts/actual
distance between those 2 pts) (typical ~ 3.0)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-16
Diffusion & Rxn in a Spherical Catalyst
BMB : WAr 4r 2 r  WAr 4r 2 r r  rA  4rm2r  c  0
C
As

R

d WAr r 2
dr
  r  r 2
A
c
0
Diffusion: WA  cDe
r
dy A
dC
 De A
dr
dr
Write the rate law
r '' A  k "n C A n
based on surface area:
Relate r’A to r’’A by:
BMB :

d WAr r 2
dr
mol
-r ' A  r '' A Sa
g cat  s
  r  r 2
A
c
0
dCA 2  2
d
n

D
r

r

S
k
"
C
0
e
C
a
n
A


dr 
dr

Sa 
mol
m2  s
catalyst surface area
mass of catalyst
Insert the diffusion eq & the
rate eq into the BMB:
Boundary Conditions:
CA finite at r=0
CA = CAs at r =R
Solve to get CA(r) and use the diffusion equation to get WAr(r)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-17
Dimensionless Variables
dCA 2  2
d
n

D
r

r

S
k
"
C
e
C A n A  0 Put into dimensionless form


dr 
dr

n
2
n1
CA
k
"
S

R
C
k
"
S

R
C
r
n a c
As
As
(Psi)  
fn2 
 fn2  n a c

C As
De  CAs  0  R 
De
R
d2 
2  d 
2 n


f

 n  0
2
  d 
d
Boundary Conditions:
 =1 at =1
 =finite at =0
Thiele modulus for rxn of nth order ≡ fn
Subscript n = reaction order
fn2 
fn is small: surface reaction is rate limiting
fn is large: internal diffusion is rate limiting
The solution for   CA  1  sinh f1 
CAs   sinh f1 
a 1st order rxn:
"a" surface rxn rate
"a" diffusion rate
CA
C As
small f1
medium f1
large f1
small f1: surface rxn control, significant amount of reactant
R
diffuses into pellet interior w/out reacting
r=0
large f1: surface rxn is rapid, reactant is consumed very closed to the external surface of
pellet (A waste of precious metal inside of pellet)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Internal Effectiveness Factor, 
L20-18
Internal effectiveness factor:
(1) the relative importance of diffusion and reaction limitations
(2) a measurement of how far the reactant diffuses into the pellet before reacting
actual  observed overall rate of rxn

rate of reaction if entire interior surface were exposed to CAs & Ts
r 'A  mass of catalyst 
rA
r ''A



rAs r ''As r 'As mass of catalyst 
For example, when n=1 (1st order kinetics, -r’’As )


4R2  WAs r R
4 3

r
 As  R
3



 dC A 
 df 
2
4

R
D
C
4R De 
e As 
 d  1
 dr r R


4 3
4 3
R c Sak "1 CAs
c Sak "1 CAs R
3
3
2


x
x
e

e
 3 
cosh x
    2   f1 coth f1  1 where coth x=

f 
sinh x
e x  e x
 1 
 2  e x  e x
 2 e x  e x
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-19
Internal Diffusion & Overall Rxn Rate
r ''A    r '' As   quantifies how internal diffusion affects the overall rxn rate
Effectiveness factor vs fn
1
0.8
Reaction limited
0.6
 0.4
0.2
Internal diffusion limited
0.1
0.2
1
2
f1
4
6
8
10
As particle diameter ↓, fn ↓, →1, rxn is surface rxn limited
As particle diameter ↑, fn ↑, →0, rxn is diffusion limited
This analysis was for spherical particles. A similar approach can be used to
evaluate other geometries, non-isothermal rxn, & more complex rxn kinetics
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-20
Effectiveness Factor & Rxn Rate
3
  2  f1 coth f1  1
f1
f1  R
ck1Sa
De
     k1CAs  Sa
rA    rAs
R  f1    1 surface-reaction-limited
when f1 ,(  30)  can be simplified to:  
De
3 3

,  1
f1 R k1c Sa
f1 is large, diffusion-limited reaction inside the pellet (external diffusion will have a
negligible effect on the overall rxn rate because internal diffusion limits the rxn rate)
r 
3
  A  2  f1 coth f1  1

rAs
f1
internal-diffusion-limited:
De
3



r

k1CAs  Sa
A
rA    k1CAs  Sa
R k1c Sa

 -rA 
De
3
R k1c Sa
3 DeSak1
CAs
R
c
Overall rate for 1st-order rxn
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-21
Clicker Question
rA    k1CAs  Sa
Overall rate for
1st-order rxn
De
3

R k1c Sa
 rA 
De
3
k1CAs  Sa
R k1c Sa
When the overall rate of rxn when the reaction is limited by
internal diffusion, which of the following would decrease
the internal diffusion limitation?
(a) decreasing the radius R of the particle
(b) increasing the concentration of the reactant
(c) increasing the temperature
(d) increasing the internal surface area
(e) Both a and b
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L20-22
Total Rate of Consumption of A
in Pellet, MA (mol/s)
• At steady state, net flow of A into pellet at the external surface completely
reacts within the pellet
• Overall molar rxn rate = total molar flow of A into catalyst pellet
• MA = (external surface area of pellet) x (molar flux of A into pellet at
external surface)
• MA =the net rate of reaction on and within the catalyst pellet
 CA 
d

C
C
dC


 As 
As
A 
dCA 
2
2
 R 
MA  4 R WAr r R  MA  4 R  De

dr

 d r 
dr  r R

 
R
 MA  4 R
2 De C As
R
 CA 
d

C
 As 
r
d 
R
r R
 MA  4 RDe CAs
d
d  1
r R
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.