L21-1
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
• Next goal: 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.
L21-2
Review: Internal Diffusion Effects in
Spherical Catalyst Particles
• Internal diffusion: diffusion of reactants or products from particle surface
(pore mouth) to pellet interior
• Concentration at the pore mouth will be higher than that inside the pore
Step 1) Mole balance over the shell thickness r is:
CAb
External
diffusion
IN
Internal
diffusion
CAs
R
-
OUT
+
GEN =

ACCUM

WAr 4r 2 r  WAr 4r 2 r r  rA 4rm2r c  0
Volume of shell
r
Divide by -4/r & take

limit as r →0
r’A: rxn rate per mass of catalyst
c: catalyst density
rm: mean radius between r and r - r

d WAr r 2
dr
  r r 2
A
Differential BMB in
c  0 spherical catalyst particle
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
CAs
Review: Diffusion & Rxn in Spherical
Catalyst

d WAr r 2
R
dr
r
r+r
  r  r 2
A
c
L21-3
 0 (step 1, BMB)
System at steady state, so EMCD: WB = -WA
(otherwise A or B would accumulate)
WA  cDe
dy A
dC
 De A
dr
dr
Rate law:
 mol 
 mol 
n
r '' A 

k
C

-r
'
 r '' A S A
n A
A


2
m s
 g cat  s 
SA 
catalyst surface area
mass of catalyst
dCA 2  2
Solve for CA(r) & get
Insert diffusion eq & d 
n

D
r

r

S
k
"C

0
C A n A

WAr(r) from diffusion eq
rate eq into BMB:
dr  e dr
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-4
Review: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

n2 
 n2  n a c

CAs
De  CAs  0  R 
De
R
d2 
2  d 
2 n




 n  0
2
  d 
d
Boundary Conditions:
 =1 at =1
 =finite at =0
Thiele modulus for rxn of nth order ≡ n
Subscript n = reaction order
n2 
n is small: surface reaction is rate limiting
n is large: internal diffusion is rate limiting
The solution for   CA  1  sinh 1 
CAs   sinh 1 
a 1st order rxn:
"a" surface rxn rate
"a" diffusion rate
CA
C As
small 1
medium 1
large 1
small 1: surface rxn control, significant amount of reactant
R
diffuses into pellet interior w/out reacting
r=0
large 1: 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.
L21-5
Review: Internal Effectiveness Factor, 

eta
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 
Effectiveness factor vs n
knR2Sac CAsn1
n 
De
2
1
0.8
Reaction limited
0.6
• As particle diameter ↓,
n ↓, →1, rxn is
surface rxn limited
 0.4
0.2
Internal diffusion limited
0.1
0.2
1
2
1
4
6
8
10
• As particle diameter ↑,
n ↑, →0, rxn is
diffusion limited
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-6
Review: Effectiveness Factor & Rxn Rate
3
  2  1 coth 1  1
1
1  R
ck1Sa
De
     k1CAs  Sa
rA    rAs
R  1    1 surface-reaction-limited
when 1 ,(  30)  can be simplified to:  
De
3
3

,  1
1
R k1c Sa
1 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)

rA
3
  1 coth 1  1

rAs
12
rA    k1CAs  Sa
When internal-diffusion-limited: 
De
3
 rA 
k1CAs  Sa
R k1c Sa
 -rA 
De
3
R k1c Sa
3 De Sak1
CAs
R
c
Overall rate for 1st-order rxn
To increase the overall rate of a rxn limited by internal diffusion
(1) decrease the radius R
(2) increase the temperature
(3) increase the concentration of A
(4) increase the internal surface area
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-7
L21: Simultaneous Internal
Diffusion & External Diffusion
Goal: Derive a new rate eq that accounts for internal & external diffusion
• -r’A is a function of reactant concentration
• Reactant conc is affected by internal & external diffusion
• Express reactant conc in terms of diffusion-related constants & variables
→Use mole balance
At steady-state: transport of reactants from bulk
fluid to external catalyst surface is equal to net rate
CAs
CAb of reactant consumption in/on the pellet
C(r)
Molar rate of mass transfer from bulk fluid to
external surface:
MA  WAr  ac  V
reactor volume
molar flux
external surface area per unit reactor volume
This molar rate of mass transfer to surface is equal to net rxn rate on & in pellet!
MA  rA  external area  internal area 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-8
Basic Molar Balance at Pellet Surface
Flux:
bulk to
external
surface
x
External
S.A.
Actual rxn
rate per
=
unit total
S.A.
x
external +
internal S.A.
  rA

 ac V  Sa bV 
ac: external surface area per reactor volume (m2/m3)
V: reactor volume (m3)
-r’’A: rate of reaction per unit surface area (mol/m2·s)
Sa: surface area of catalyst per unit mass of catalyst (m2/g cat)
b: bulk density, catalyst mass/ reactor volume b=c(1-
: porosity of bed (void fraction)
c: catalyst density
 WAr r R

 ac V 
MA  WAr r R ac V  rA  ac V  Sa b V 
 MA  WAr r R ac  rA  ac  Sa b 
Typically external surface area <<< internal surface area
 MA  WAr r R ac  rA Sa b
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-9
Overall Molar Rate of Reaction
Overall rxn rate = flux to surface = rxn rate on & in pellet
MA  WAr r R ac  rA Sa b
For external mass transport:
WAr r R  k c  CAb  CAs 
Since internal diffusion resistance is also significant, the reactant conc at the
internal surface is lower that the reactant conc at the external surface:
r ''A
   r ''As   r ''A For a 1st order rxn: -r’’A=-k1CAs

r ''As
actual  observed  overall rate of rxn
where the internal

effectiveness factor:
rxn rate if entire interior surface were exposed to C As & Ts
Plug flux & 1st order rxn rate back into the mass balance:
MA  k c  CAb  CAs  ac  k1CAsSa b
 kcCAbac  kcCAsac  k1CAsSa b
 k c CAbac  CAs k1Sa b  k c ac 
Solve mass balance for CAs
 kcCAbac  k1CAsSa b  kcCAsac

k c CAbac
 CAs
k1Sa b  k c ac
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-10
Overall Effectiveness Factors
CAs 
k c ac CAb
k c ac  k1Sa b
rA  k1CAs  rA 
Finally insert CAs into –r’’A
k1k c acC Ab
k c ac  k1Sa b
Overall rxn rate with internal
& external diffusion
Is this the overall rxn rate that we ALWAYS use for a
surface reaction that has internal & external?
(a) Yes, we should always use this rate equation for a surface reaction
(b) No, we should only use this rate eq for processes that use spherical
catalyst pellets
(c) No, we should only use this rate eq for processes that that involve
catalyst particles that have a constant density & even catalyst loading
on the surface
(d) No, we should only use this rate eq for 1st order irreversible reactions
(e) b, c, & d
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-11
Overall Effectiveness Factors
CAs 
k c ac CAb
k c ac  k1Sa b
rA  k1CAs  rA 
Finally insert CAs into –r’’A
k1k c ac CAb
k c ac  k1Sa b
Overall rxn rate with internal
& external diffusion
Remember, the internal effectiveness factor (based on CAs) is:

actual overall rate of reaction
rate of rxn if entire interior surface were exposed to the external surface conditions
The overall effectiveness factor (based on CAb) is defined as:
Omega

actual overall rate of reaction
rate of reaction if entire interior surface were exposed to the bulk conditions
k1CAb
rA

1  k1Sa b k c ac


rAb


1  k1Sa b k c ac
k1CAb
   r ''A Put into design eq to account for internal & external diffusion
   rAb
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-12
Rxn Rate Variation vs Reactor
Conditions
External diffusion
Internal diffusion
Surface reaction
Type of Limitation
12
13 

Ud




D
D
 
p

r 'A  k c  AB Sh  k c  AB  2  0.6 
 

dp
dp 
     DAB  

 ck1Sa
 ck1Sa  
3
rA  kr CAsSa  
coth  R
 1
 R

De
De  
2 ck1Sa 

R
De
-r’A=kCA
Variation of Reaction Rate with:
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-13
Consider an isothermal catalytic reaction in a PBR where there is no pressure
drop and the catalyst pellets are uniformly packed & spherical. The kinetics
are 1st order, and k, all physical parameters, and the inlet conditions (pure A
in feed, A→ products) are given. Derive an equation for XA, taking into
account the diffusion to and within each catalyst particle, but ignore diffusion
down the length of the reactor.
Rate must account for diffusion &
dX A
F


r
'
PBR design eq:
A0
A
be in terms of catalyst surface area
dW
1. Put rate in terms of the unit surface area: -r 'A  r ''A Sa
   -r 'A  r ''Ab Sa
2. Account for diffusion limitations in rate eq: r ''A    rAb
3. Rate is 1st order: r ''Ab  kCAb  -r 'A  kCAbSa
4. Put into design eq: FA0
dX A
 kC Ab Sa
dW
5. Put Cab in terms of XA: CAb  CAb0 1  X A   FA0
6. Integrate:
dX A
 kSaCAb0 1  X A 
dW
XA
W kS C
dX A
dX A kSaCAb0 1  X A 
a Ab0 dW
 
 


FA0
dW
FA0
0 1  X A 
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-14
Consider an isothermal catalytic reaction in a PBR where there is no pressure
drop and the catalyst pellets are uniformly packed & spherical. The kinetics
are 1st order, and k, all physical parameters, and the inlet conditions (pure A
in feed, A→ products) are given. Derive an equation for XA, taking into
account the diffusion to and within each catalyst particle, but ignore diffusion
down the length of the reactor.
W kS C
dX A
a Ab0 dW   ln 1  X   kSaC Ab0 W
6. Integrate: 
 
A
FA0
1

X
F


A
A0
0
0
XA
kSaCAb0 W

ln
1

X


7. Solve for XA:
A
FA0
 XA  1  e
kSaCAb0 W
FA0
 1  XA  e
kSaCAb0 W
FA0
kSa W
 XA  1  e
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-15
1st
XA for
order rxn executed in an isothermal PBR packed with spherical
catalyst particles with internal & external diffusion limitations
kSa W
XA  1  e
0
For same conditions, eq derived in Fogler (12-71) for XA at end of reactor of
length L is:
XA
kSa bL
U
 1 e
catalyst mass kg
where : b 
= 3
reactor volume m
L= z U=superficial velocity=
0
Ac
Are these equations the same?
They differ in the exponent:
W bL
W b  L  A c
 ?
 ?
0
0
0
0
Ac
L  A c  V
kSa W
XA  1  e
0
kSa W
0
?
kSa bL
U
W W V V W W
W b V


 ?
?
0
0
kSa bL
U
 1 e
0
0
0
0
 XA
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.