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.
L21-16
Review: 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:
M  W  a  V
A
Ar
c
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.
Review: Basic Molar Balance at
Spherical Pellet Surface
Flux:
bulk to
external
surface
x
External
S.A.
Actual rxn
rate per
=
unit total
S.A.
x
L21-17
external +
internal S.A.
MA  WAr r R ac V  rA  ac V  Sa b V 
per
volume
ac: external surface area per reactor volume (m2/m3)
V: reactor volume (m3)
: porosity of bed (void fraction)
-r’’A: rate of reaction per unit surface area (mol/m2·s)
-r’A: mol/g cat∙s
-rA: mol/volume∙s
Sa: surface area of catalyst per unit mass of catalyst (m2/g cat)
b: bulk density, catalyst mass/ reactor volume b=c(1-
r ' A  r '' A Sa
 rA  r ' A c
 rA  r '' A Sa c
per mass cat→
k 'n  k ''n Sa
kn  k 'n c
kn  k ''n Sa c
per surface
area
Cancel out V & ac ≈0 since external surface area usually <<< internal surface
area (surface area of internal pores)
 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.
L21-18
Review: Overall Molar Rate of Reaction
MA  WAr r R ac  rA Sa b
For external mass transport:
WAr r R  k c  CAb  CAs 
Internal diffusion resistance is significant, so the reactant conc at the internal
surface is lower that the reactant conc at the external surface:
: internal effectiveness factor
r ''A
observed rxn rate


   r ''As   r ''A
r ''As rxn rate if no internal diff limit
For a 1st order rxn:
-r’’A=-k1CAs
Plug flux & 1st order rxn rate back into the mass balance, solve for CAs:
k c CAbac
MA  k c  CAb  CAs  ac  k1CAsSa b  CAs 
k ''1 Sa b  k c ac
Insert CAs into –r’’A=k1CAs:
rA 
k1k c ac CAb
k c ac  k1Sa b
Overall 1st order rxn rate with
internal & external diffusion
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-19
Review: Overall Effectiveness Factors
Remember, the internal effectiveness factor is based on CAs

actual overall rate of reaction
rate of rxn if entire interior surface were exposed to the external surface conditions
The overall effectiveness factor is based on CAb:
Omega

actual overall rate of reaction
rate of reaction if entire interior surface were exposed to the bulk conditions

rA

rAb
k1CAb
1  k1Sa b k c ac

k1CAb


1  k1Sa b k c ac
   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.
L21-20
Review: Observed Rxn Rate Variation
vs FT0, dp & T
External
r ' A  k c  C Ab  C As 
diffusion limited:
12
13
 Udp      
D AB 
 2  0.6 

kc 
 

dp 
     D AB  



ck1Sa
ck1Sa  
3
Internal diffusion r  k C S  
coth  R
 R
  1
A
r As a

k
S
D
D
limited:
e
e 


R2 c 1 a 
De
Surface reaction
Type of Limitation
-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
Whether the rate varies when FT0 or particle size changes indicates tells us whether
external diffusion, internal diffusion, or the surface rxn is limiting (slowing down) the
observedEngr
rate
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular
Dept, University of Illinois at Urbana-Champaign.
Review: Rxn Rate Variation vs Reactor
Conditions
L21-21
When the observed rate of a reaction is limited by external diffusion,
internal diffusion, or the surface rxn, the observed reaction kinetics are:
12
13
Rate for external
 Udp      
D AB 
 2  0.6 

r ' A  k c  C Ab  C As  k c 
 

diff limited rxn:
dp 
     D AB  


kc: mass transfer coefficient DAB: diffusivity (m2/s)
dp: diameter
U: free-stream velocity (m/s),  to flow rate (FT, FT0) for constant CA0
n: kinematic viscosity (m2/s); n/
: fluid density (kg/m3) : viscosity


ck1Sa
ck1Sa  
3
Rate for internal r  k C S  
coth  R
 R
  1
A
r As a

k
S
D
D
diff limited rxn:
e
e 


R2 c 1 a 
De
R: radius at
actual  observed overall rate of rxn
particle surface

rate of rxn if entire interior surface were exposed to C As & Ts
De: effective diffusivity
Rate for surface reaction limited rxn:
-r’A = kCA
Whether the rate varies when FT0 (at constant CT0) or particle size
changes indicates tells us whether external diffusion, internal diffusion,
Slides courtesy
Profsurface
M L Kraft, Chemical
& Biomolecular
Engr Dept,
University
Illinois at Urbana-Champaign.
orofthe
rxn is limiting
(slowing
down)
the of
observed
rate
L21-22
Observed Rxn Rate vs FT0, dp & T
12
13
 Udp      
D AB 
 2  0.6 

kc 
 

dp 
     D AB  

U: free-stream velocity (m/s),  to flow rate (FT, FT0) for constant CA0
Rate for external
r ' A  k c  C Ab  C As 
diff limited rxn:
dp: diameter


ck1Sa
ck1Sa  
3
Rate for internal r  k C S  
coth  R
 R
  1
A
r As a

k
S
D
D
diff limited rxn:
e
e 


R2 c 1 a 
R: radius at particle surface
Rate for surface reaction limited rxn:
De
-r’A= kCA
According to these equations, if we increase the flow rate (FT0) without
increasing the concentration of reactants in the feed, the observed rxn rate
will increase if the rxn is limited (slowed down) by:
a. External diffusion
Free-stream velocity (U), which is  to flow rate for
constant CA0, is only in rate eq for a external
b. Internal diffusion
diffusion limited reaction
c. The surface reaction
d. Either external & internal diffusion
e. Any of these (external diffusion, internal diffusion, or surface reaction)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-23
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
FT0=1000 mol/h, T=400K
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
external diffusion?
External diffusion limits the observed rate when increasing FT0 increases –r’A
• Need to find the points that have the same T and dp. If the rate increases when Fto
increases, the trial at the LOWER flow rate is limited by external diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 1000 mol/h has a lower rate than the trial with
T = 400K, dp= 0.8 cm & FT0= 1500 mol/h
Thus, rate is limited by external diffusion when T = 400K, dp= 0.8 cm & FT0= 1000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-24
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
FT0=1000 mol/h, T=400K
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
4
ext diff lim
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
external diffusion?
External diffusion limits the observed rate when increasing FT0 increases –r’A
• Need to find the points that have the same T and dp. If the rate increases when Fto
increases, the trial at the LOWER flow rate is limited by external diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 1500 mol/h has a lower rate than the trial with
T = 400K, dp= 0.8 cm & FT0= 2000 mol/h
Thus, rate is limited by external diffusion when T = 400K, dp= 0.8 cm & FT0= 1500 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-25
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
FT0=1000 mol/h, T=400K
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
external diffusion?
External diffusion limits the observed rate when increasing FT0 increases –r’A
• Need to find the points that have the same T and dp. If the rate increases when Fto
increases, the trial at the LOWER flow rate is limited by external diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 2000 mol/h has a lower rate than the trial with
T = 400K, dp= 0.8 cm & FT0= 3500 mol/h
Thus, rate is limited by external diffusion when T = 400K, dp= 0.8 cm & FT0= 2000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-26
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
ext diff lim
8
6
FT0=1000 mol/h, T=400K
FT0=1500 mol/h, T=400K
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
external diffusion?
External diffusion limits the observed rate when increasing FT0 increases –r’A
• Need to find the points that have the same T and dp. If the rate increases when Fto
increases, the trial at the LOWER flow rate is limited by external diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 3500 mol/h has the same rate as the trial with
T = 400K, dp= 0.8 cm & FT0= 4000 mol/h
Rate is NOT limited by external diffusion when T = 400K, dp= 0.8 cm & FT0= 3500 mol/h
or T =courtesy
400K,ofdProf
FT0= 4000
mol/h Engr Dept, University of Illinois at Urbana-Champaign.
Slides
M L cm
Kraft,&Chemical
& Biomolecular
p= 0.8
-rA (mol/g cat*h)
L21-27
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
ext diff lim
8
6
FT0=1000 mol/h, T=400K
FT0=1500 mol/h, T=400K
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
external diffusion?
External diffusion limits the observed rate when increasing FT0 increases –r’A
• Need to find the points that have the same T and dp. If the rate increases when Fto
increases, the trial at the LOWER flow rate is limited by external diffusion
For all remaining trials, increasing FT0 does not increase the reaction rate, so no other
trial conditions are external diffusion limited.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-28
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
ext diff lim
8
6
FT0=1000 mol/h, T=400K
FT0=1500 mol/h, T=400K
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 3500 mol/h has a lower rate than the trial with
T = 400K, dp= 0.6 cm & FT0= 3500 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.8 cm & FT0= 3500 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-29
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
FT0=3500 mol/h, T=300K
10
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.8 cm & FT0= 4000 mol/h has a lower rate than the trial with
T = 400K, dp= 0.6 cm & FT0= 4000 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.8 cm & FT0= 4000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-30
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
FT0=3500 mol/h, T=400K
10
int diff lim FT0=3500 mol/h, T=300K
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.6 cm & FT0= 3500 mol/h has a lower rate than the trial with
T = 400K, dp= 0.2 cm & FT0= 3500 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.6 cm & FT0= 3500 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-31
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
12
int diff
lim
10
8
FT0=3500 mol/h, T=400K
int diff lim FT0=3500 mol/h, T=300K
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.6 cm & FT0= 4000 mol/h has a lower rate than the trial with
T = 400K, dp= 0.2 cm & FT0= 4000 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.6 cm & FT0= 4000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-32
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
12
10
8
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.2 cm & FT0= 3500 mol/h has a lower rate than the trial with
T = 400K, dp= 0.1 cm & FT0= 3500 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.2 cm & FT0= 3500 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-rA (mol/g cat*h)
L21-33
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
16
FT0=4000 mol/h, T=400K
14
FT0=4000 mol/h, T=300K
int diff lim
12
10
8
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.2 cm & FT0= 4000 mol/h has a lower rate than the trial with
T = 400K, dp= 0.1 cm & FT0= 4000 mol/h
Thus, rate is limited by internal diffusion when T = 400K, dp= 0.2 cm & FT0= 4000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-34
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
FT0=4000 mol/h, T=400K
16
int diff lim
int diff lim
-rA (mol/g cat*h)
14
12
10
8
FT0=4000 mol/h, T=300K
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
Trial with T = 400K, dp= 0.1 cm & FT0= 3500 mol/h has the SAME rate as the trial with
T = 400K, dp= 0.05 cm & FT0= 3500 mol/h
Rate israte
Thus,
NOT
is limited by internal diffusion when T = 400K, dp= 0.1
0.2 cm & FT0= 3500
4000 mol/h
or T =courtesy
400K,ofdProf
cm Chemical
& FT0= 3500
mol/h Engr Dept, University of Illinois at Urbana-Champaign.
Slides
M L Kraft,
& Biomolecular
p= 0.05
L21-35
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
FT0=4000 mol/h, T=400K
16
int diff lim
int diff lim
-rA (mol/g cat*h)
14
12
10
8
FT0=4000 mol/h, T=300K
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
internal diffusion?
Internal diffusion limits the observed rate when decreasing dp increases –r’A
• Need to find the points that have the same T. If the rate increases when dp
decreases but does not change with FT0, the trial at the larger dp is limited by
internal diffusion
For all remaining trials, decreasing dp does not increase the reaction rate, so no other
trial conditions are internal diffusion limited.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-36
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
FT0=4000 mol/h, T=400K
16
int diff lim
int diff lim
-rA (mol/g cat*h)
14
12
10
8
FT0=4000 mol/h, T=300K
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
the surface reaction?
The surface reaction limits the reaction rate when the observed rxn rate increases
when we increase T, but it does not increase when we decrease dp or increase FT0
without increasing CT0
For all remaining trials, neither decreasing dp nor increasing FT0 increases the
reaction rate. Therefore, the surface reaction limits (slows down) the rates of the
remaining trial conditions.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-37
The graph below shows the reaction rates obtained when the irreversible, liquidphase, catalytic reaction A→B was carried out in a PBR using the indicated catalyst dp,
T, and FT0. CA0 was the same in each trial.
FT0=4000 mol/h, T=400K
16
int diff lim
int diff lim
-rA (mol/g cat*h)
14
12
SRL
10
8
FT0=4000 mol/h, T=300K
int diff lim
FT0=3500 mol/h, T=400K
int diff
int diff lim FT0=3500 mol/h, T=300K
lim
int diff lim FT0=1000 mol/h, T=400K
ext diff lim
FT0=1500 mol/h, T=400K
6
FT0=2000 mol/h, T=400K
ext diff lim
ext diff lim
4
2
SRL
0
0.0
0.1
0.2
0.3
0.4 0.5
dp (cm)
0.6
0.7
0.8
0.9
Which, if any, of the conditions shown (flow rates, T, and dp) is the reaction limited by
the surface reaction?
For all remaining trials, neither decreasing dp nor increasing FT0 increases the
reaction rate. Therefore, the surface reaction limits (slows down) the rates of the
remaining trial conditions. Surface reaction limited (SRL):
T = 400K, dp= 0.1 cm & FT0= 3500 mol/h, T = 400K, dp= 0.05 cm & FT0= 3500 mol/h,
T = 400K, dp= 0.1 cm & FT0= 4000 mol/h, T = 400K, dp= 0.05 cm & FT0= 4000 mol/h,
T = 300K, all dp tested, & FT0= 4000 mol/h & T = 300K, all dp tested, & FT0= 3500 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-38
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For which, if any, of the conditions
shown (flow rates and temps) is the
reaction limited by external diffusion?
External diffusion limited where –r’A↑
linearly when T↑
Variation of Reaction Rate with:
Type of
Limitation
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
For FT0 = 10 mol/h, the rate of rxn increases approximately linearly with T over
the entire temperature range- external diffusion limited at FT0 = 10 and all T
For FT0 = 100 mol/h, the rate of rxn increases ~linearly with T when T > 360K.
The reaction is external diffusion limited when FT0 = 100 & T> 360K
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For which, if any, of the conditions
shown (flow rates and temps) is the
reaction limited by surface reaction
rate?
Rxn rates ↑ exp
with T but not
FT0 for FT0 =
100, 1000 &
5000 mol/h at
T< 360K
L21-39
ext diff lim
Surface rxn limited when –r’Aincreases
exponentially with T↑ but independent of superficial velocity (flow!)
Variation of Reaction Rate with:
Type of
Limitation
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
For conditions FT0 = 100, 1000 & 5000 mol/h at T< 360K, rxn rate is
independent of FT0 but exponentially dependent on T→ surface reaction limited
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For which, if any, of the conditions
shown (flow rates and temps) is the
reaction limited by surface reaction
rate?
For FT0 = 1000
& 5000 mol/h
at T< 366K, rxn
rates ↑ exp with
T but not FT0
L21-40
ext diff lim
Surface rxn limited when –r’Aincreases
exponentially with T↑ but independent of velocity
Variation of Reaction Rate with:
Type of
Limitation
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
For FT0 = 100, 1000 & 5000 mol/h at T< 360K → surface reaction limited
For conditions FT0 = 1000 & 5000 mol/h at T< 366K, rxn rate is independent of
FT0 but exponentially dependent on T→ surface reaction limited
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For which, if any, of the conditions
shown (flow rates and temps) is the
reaction limited by internal diffusion?
Rxn rates ↑ exp
with T but not
FT0 for FT0=
1000 & 5000
mol/h at T>
367K
L21-41
ext diff lim
Internal diffusion limited when –r’Aincreases
exponentially with T↑ & is independent of velocity
Variation of Reaction Rate with:
Type of
Limitation
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
For FT0 = 1000 & 5000 mol/h at T> 367K, rxn rate is roughly independent of
FT0 but exponentially dependent on T. The reaction rate is internal diffusion
limited at T> 370K for FT0 = 1000 & 5000 mol/h
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-42
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For which, if any, of the conditions
shown (flow rates and temps) is the
reaction limited by internal diffusion?
ext diff lim
Internal diffusion limited when –r’Aincreases
exponentially with T↑ & is independent of velocity
Variation of Reaction Rate with:
Type of
Limitation
Superficial velocity
Particle size
Temperature
External
U1/2
dp-3/2
Linear
Internal
Independent
dp-1
Exponential
Surface reaction
Independent
Independent
Exponential
How do we know it’s not surface rxn limited at FT0=1000 & 5000 mol/h & T>367K?
As T↑, the specific rate constant k↑, the rate of the surface rxn & consumption of
reactant ↑. Thus the reactant is more likely to be consumed before it reaches the
core.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.

L21-43
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For a flow rate of 10 g mol/h,
determine the overall effectiveness
factor  at 360K
actual overall rxn rate
rxn rate if entire interior surface were exposed to the bulk conditions
Rxn w/out diffusion limitations

rA
   0.26    0.37

0.70
rAb
What do we use for the rate of reaction if the interior was exposed to bulk conditions?
Use the rxn rate obtained under surface reaction limited conditions
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.

L21-44
-r’A (mol/gcat·s)
The catalytic reaction A→B takes
place in a fixed bed reactor containing
spherical porous catalyst X22. The
overall rxn rates at a point in the
reactor are shown in the graph below.
For FT0= 5000 g mol/h, estimate the
internal effectiveness factor  at 367K
actual overall rate of reaction
rate of rxn if entire interior surface were exposed to the external surface conditions

r ''A
   1.2
r ''As
1.4
Rxn w/out internal diffusion limitations
   0.86
What do we use for the rate of reaction if the interior was exposed to the conditions at
the surface of the pellet?
Extrapolate the line for the surface reaction limited regime of the FT0 = 5000 mol/h plot
to estimate the rxn rate that would be obtained without internal diffusion
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-45
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
Type of Limitation
XA2=? for dp1/3, 1.5z1, and 40,1
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.
L21-46
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=? for dp1/3, 1.5z1, and 40,1
Need to relate XA to reactor length in the presence of an external diffusion limit
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-47
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   
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)
: porosity of bed, void fraction
dp: particle diameter (m)
r’’A: rate of generation of A per unit catalytic surface area (mol/s·m2)
Divide out FAz z  FAz z z
Take limit  1  dFAz   r '' a  0
 r ''A ac  0


A c
A
dz
Acz:
as
z→0:

A c z
c
Put Faz and –rA’’ in terms of CA: FAz  WAz Ac  (JAz  BAz )Ac
Axial diffusion is negligible compared to bulk flow (convection)
FAz  BAz Ac  UCA Ac Substitute into the mass balance
dCA
dU 
d UCA 
 dCA


U
 r ''A ac  0


U

C

r
''
a

0

 r ''A ac  0


A
A c
dz
dz 
dz
 dz
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-48
Review: Mass Transfer Limited Rxn in
PBR (continued)
b
c
d
A B C D
a
a
a
U
dCA
 r '' A ac  0
dz
At steady-state:
Molar flux of A to particle surface = rate of disappearance of A on the surface
r ''A  WAr  k c  CA  CAs  Substitute
mass transfer coefficient kc =DAB/d (s-1) d: boundary layer thickness
CAs: concentration of A at surface
CA: concentration of A in bulk
dCA
CAs ≈ 0 in most mass transfer-limited rxns
U
 k c ac  C A  C As   0
dz
dC
Rearrange & integrate to find how CA and the r’’A
 U A  k c ac C A  0
varies with distance down reactor
dz
CA
k c ac
CA
dCA
dCA z k c ac

ln


z
 U
 k c ac C A  
 
dz
CA0
U
dz
CA
U
C
0
A0

CA
 k a
 exp   c c
CA0
 U

 k a
z   CA  CA0 exp   c c

 U

 k a 
z  r ''A  k c CA0 exp   c c z 
 U 

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-49
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
For an external diffusion
limited rxn in a PBR, we
found (L19):
XA2=? for dp1/3, 1.5z1, and 40,1
CA
 k a
 exp   c c
CA0
 U

z

 k a 
In terms of XA: CA0 1  X A 
 exp   c c z 
CA0

U

ac: external surface
area of catalyst per a  6 1   
catalyst bed volume c
dp
: porosity of bed
k a
 ln 1  X A    c c z
U
ln 1  X A1 
k c1ac1zU2
Express XA at 2
Relate U to 0 & ac to dp

reaction conditions ln 1  X A2  k c2ac2 1.5z  U1
as a ratio:
6 1   
U=0 Ac where Ac  cross-sectional area of PBR
dp1
ac1
U1 0,1 A c
U1 0,1
ac1 dp2




ac2 6 1     a  d
U2 0,2 A c
U2 0,2
c2
p1
dp2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-50
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=? for dp1/3, 1.5z1, and 40,1
ln 1  X A1 
k c1ac1zU2

ln 1  X A2  k c2ac2 1.5z  U1
ac1 dp2

ac2 dp1
U1 0,1

U2 0,2
How are kc1 and kc2
related?
12
13

Ud
 p  n  
D AB
 2  0.6 
 Typically the 2 is negligible so
kc 
 

dp 
 n   DAB  

12
 Udp 
DAB
 kc 
 0.6  

dp
n


DAB
23
13
 n 
D 
 AB 
 k c  0.6
DAB2 3 U1 2
n 1 6 dp1 2
U11 2
12
k c1 0.6 n 1 6 dp,1

k c2 0.6 DAB2 3 U21 2
n 1 6 dp,21 2
12
k c1  U11 2  dp,2 




1
2
1
2
k c2  dp,1  U2 



Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-51
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=? for dp1/3, 1.5z1, and 40,1
ln 1  X A1 
k c1ac1zU2

ln 1  X A2  k c2ac2 1.5z  U1
ac1 dp2

ac2 dp1

ln 1  X A1 
ln 1  X A 2 

U1 0,1

U2 0,2

ln 1  X A1  k c1  ac1   z  U2




ln 1  X A2  k c2  ac2   1.5z  U1
12
k c1  U11 2  dp,2 



1
2
1
2
k c2  dp,1  U2 



12
ln 1  X A1   U11 2  dp,2   dp2   z  0,2






1
2
1
2
ln 1  X A2   dp,1  U2   dp1   1.5z  0,1



 U 1 2 dp,23 2   1  
  1 2 dp,23 2   1  
ln
1

X


0,2
0,2
A1

 0,1

  11 2




 U2 dp,13 2   1.5  0,1
ln 1  X A2   0,21 2 dp,13 2   1.5  0,1




ln 1  X A1 
ln 1  X A2 
 4 1 2 d 3 3 2 
 0,21 2dp,23 2   1 
ln 1  0.632   0,1  p,1    1 







12
32


1
2
3
2
ln
1

X
1.
5




 0,1 dp,1   1.5 

d


A2
0,1
p,1




Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-52
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=? for dp1/3, 1.5z1, and 40,1


 4 1 2 d 3 3 2 
ln 1  0.632   0,1
p,1
 1   ln 1  0.632   2 1 3 3 2 0.667



  1.5 
ln 1  X A2 
ln 1  X A2   0,11 2dp,13 2



ln  0.368 
 0.257
ln 1  X A2 
 3.8898  ln 1  X A2 

0.99967
 0.257
ln 1  X A2 
 e3.8898  1  XA2
 0.0204  1  XA2
 XA2  0.98
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-53
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=0.98 for dp1/3, 1.5z1, and 40,1
Hint for T: The conversion of 0.98 is dependent on the reaction still being external
diffusion-limited. How can we adjust the T, CA, and  to make sure that the process
is not instead slowed down by the surface reaction, but without slowing down
external diffusion?
Type of Limitation
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
• To keep the rate of Cl2 consumption (surface reaction) faster than external diffusion
(still in external diffusion limited regime), use high T
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L21-54
Cl2 is removed from a waste stream by passing the effluent gas over a solid granular
absorbent in a tubular PBR. Presently 63.2% is removed and the reaction is external
diffusion limited. If the flow rate were increases by a factor of 4, the particle diameter
were decreased by a factor of 3, and the tube length (z) were increased by 1.5x, what
percentage of Cl2 would be removed (assume still external diffusion limited)? What
guidelines (T, CA, ) do you propose for efficient operation of this bed?
XA1=0.632 for dp, z, & 0
XA2=0.98 for dp1/3, 1.5z1, and 40,1
Hint: How does changing CA and  influence the rate of external diffusion and the
surface reaction?
• The mass transfer rate can be increased by increasing the concentration gradient,
which is achieved by increasing the bulk concentration of A
k c  0.6
DAB
23
n1 6
0,1
Ac 
12
dp1 2
• Increasing the volumetric flow rate 0 increases the mass transfer coefficient but
reduces the spacetime, and therefore XA. The process also becomes reaction
limited instead of external diffusion limited. XA,mass x-fer a kc a 01/2 but XA,reaction a t a
01 so the increase in 0 may be offset by a reaction-limited decrease in
conversion, assuming constant packed-bed properties. We would need the
parameters for the reaction to evaluate whether increasing 0 is a good idea.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.