Review: Steps in a Heterogeneous
Catalytic Reaction
7. Diffusion of B
from external
surface to the bulk
fluid (external
diffusion)
1. Mass transfer
of A to surface
2. Diffusion of A
from pore mouth to
internal catalytic
surface
L19-1
3. Adsorption of A
onto catalytic surface
6. Diffusion of B
from pellet interior
to pore mouth
5. Desorption of product B
from surface
4. Reaction on surface
Ch 10 assumes steps 1,2,6 & 7 are fast, so only steps 3, 4, and 5 need to be considered
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
Review: Guidelines for Deducing
Mechanisms
L19-2
• More than 70% of heterogeneous reaction mechanisms are surface
reaction limited
• When you need to propose a rate limiting step, start with a surface
reaction limited mechanism unless you are told otherwise
• If a species appears in the numerator of the rate law, it is probably a
reactant
• If a species appears in the denominator of the rate law, it is probably
adsorbed in the surface
i jk
Generic equation:  r ' A 
kPP
i j
1  KiPi  K jPj  KkPk
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-3
L19: External Diffusion Effects
• Up until now we have assumed adsorption, surface reaction, or desorption
was rate limiting, which means there are no diffusion limitations
• In actuality, for many industrial reactions, the overall reaction rate is limited
by the rate of mass transfer of products and reactants between the bulk
fluid and the catalyst surface
• External diffusion (today)
• Internal diffusion (L20, L21 & L21b)
• Goal: Overall rate law for heterogeneous catalyst with external diffusion
limitations. This new overall reaction rate would be inserted into the design
equation to get W, XA, CA, etc
External diffusion
Internal diffusion
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-4
Mass Transfer
• Diffusion: spontaneous intermingling or mixing of atoms or molecules
by random thermal motion
• External diffusion: diffusion of the reactants or products between bulk
fluid and external surface of the catalyst
• Molar flux (W)
• Molecules of a given species within a single phase will always
diffuse from regions of higher concentrations to regions of lower
concentrations
• This gradient results in a molar flux of the species, (e.g., A), WA
(moles/area•time), in the direction of the concentration gradient
• A vector:
WA  iWAx  jWAy  kWAz
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-5
Molar Flux W & Bulk Motion BA
Molar flux consists of two parts
• Bulk motion of the fluid, BA
• Molecular diffusion flux relative to the bulk motion of the fluid
produced by a concentration gradient, JA
• WA = BA + JA (total flux = bulk motion + diffusion)
Bulk flow term for species A, BA: total flux of all molecules relative to fixed
coordinates (SWi) times the mole fraction of A (yA):
BA  y A  Wi
Or, expressed in terms of concentration of A & the molar average velocity V:
BA  CA V  BA  CA  yiVi
mol
m2  s

mol m

3 s
m
The total molar flux of A in a binary system composed of A & B is then:
WA  JA  CA V ←In terms of concentration of A
WA  JA  CA  yi Vi
WA  JA  y A  WA  WB  ←In terms of mol fraction A
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-6
Diffusional Flux of A, JA & Molar Flux W
WA = JA + BA (total flux = diffusion + bulk motion)
WA  JA  CA V
WA  JA  CA  yi Vi
WA  JA  y A  WA  WB 
Diffusional flux of A resulting from a concentration difference, JA, is related to
the concentration gradient by Fick’s first law:
mol
JA  cDABy A
2
m s
c: total concentration
DAB: diffusivity of A in B yA: mole fraction of A



 i  j k
gradient in rectangular coordinates
x y
z
Putting it all together:
WA  cDABy A  y A  Wi General equation
i
WA  cDABy A  y A  WA  WB  molar flux of A in binary system of A & B
Effective diffusivity, DAe: diffusivity of A though multiple species
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-7
Simplifications for Molar Flux
WA = JA + BA (total flux = diffusion + bulk motion)
General equation: WA  cDABy A  y A  Wi
WA  cDABy A  y A  WA  WB 
i
Molar flux of A in binary system of A & B
• For constant total concentration: cDAB yA = DAB CA
• When there is no bulk flow:  Wi  0
i
• For dilute concentrations, yA is so small that: y A  Wi
0
i
For example, consider 1M of a solute diffusing in water,
where the concentration of water is 55.6 mol water/dm3
yA 
CA
1

 y A  0.018
CA  CW 1  55.6
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-8
Evaluation of Molar Flux
Type 1: Equimolar counter diffusion (EMCD)
• For every mole of A that diffuses in a given direction, one mole of B
diffuses in the opposite direction
• Fluxes of A and B are equal in magnitude & flow counter to each other:
WA = - WB
WA  cDABy A  y A  WA  WB 
0 bulk motion ≈ 0
 WA  cDABy A or for constant total concentration: WA  DABCA
Type 2: Dilute concentration of A: y A  Wi
0
i
WA  cDABy A  y A  WA  WB   WA  cDABy A or constant Ctotal :
0
WA  DABCA
Type 3: Diffusion of A though stagnant B: WB=0
1
cDABy A
WA  cDABy A  y A  WA  WB   WA 
1 yA
0
Type 4: Forced convection drives the flux of A. Diffusion in the direction
of flow (JA) is tiny compared to the bulk flow of A in that direction (z):

WA  cDABy A  CA Vz  WA  CA Vz  WA  CA
Ac
0 diffusion ≈ 0
volumetric flow rate
cross-sectional area
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-9
Boundary Conditions
• Boundary layer
• Hydrodynamics boundary layer thickness: distance from a solid
object to where the fluid velocity is 99% of the bulk velocity U0
• Mass transfer layer thickness: distance  from a solid object to
where the concentration of the diffusing species is 99% of the
bulk concentration
• Typically diffusive transport is modelled by treating the fluid layer
next to a solid boundary as a stagnant film of thickness 
CAb

CAs
CAs: Concentration of A at surface
CAb: Concentration of A in bulk
In order to solve a design equation that accounts for external
diffusion limitations we need to set the boundary conditions
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-10
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
dC A
 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.
L19-11
Correlation for Convective Transport
Across the Boundary Layer
For convective transport across the boundary layer, the boundary condition is:
WA boundary  k c  CAb  CAs 
The mass transfer coefficient for a single spherical particle is calculated from
the Frössling correlation:
D
k c  AB Sh
dp
kc: mass transfer coefficient
dp: diameter of pellet (m)
DAB: diffusivity (m2/s)
Sh: Sherwood number (dimensionless)
Sh  2  0.6Re1 2 Sc1 3

Udp
Schmidt number: Sc 
Reynold's number Re=
DAB

: kinematic viscosity or momentum diffusivity (m2/s); m/r
r: fluid density (kg/m3)
m: viscosity (kg/m·s)
U: free-stream velocity (m/s)
dp: diameter of pellet (m)
DAB: diffusivity (m2/s)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-12
Rapid Rxn on Catalyst Surface
CAb= 1 mol/L
• Spherical catalyst particle in PBR
• Liquid velocity past particle U = 0.1 m/s
0.01m
• Catalyst diameter dp= 1 cm = 0.01 m
• Instantaneous rxn at catalyst surface CAs≈0
CAs=0
• Bulk concentration CAb= 1 mol/L
Determine the flux of A to
•  ≡ kinematic viscosity = 0.5 x 10-6 m2/s
the catalyst particle
• DAB = 1x10-10 m2/s
The velocity is non-zero, so we primarily have convective mass transfer to the
catalyst particle:
WA boundary  k c  CAb  CAs 
D

Ud
Compute kC from
k c  AB Sh Sh  2  0.6 Re1 2 Sc1 3 Re= p Sc 
dp
DAB
Frössling correlation:

Re 
0.1m s  0.01m 
0.5  10
6
2
m s
 Re  2000 Sc 
Sh  2  0.6  2000 
12
0.5  10 6 m2 s
1 10
 5000 1 3
10
2
 Sc  5000
m s
 Sh  461
1 1010 m2 s
6 m
kc 
461  k c  4.61 10
s
0.01m
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-13
Rapid Rxn on Catalyst Surface
CAb= 1 mol/L
• Spherical catalyst particle in PBR
• Liquid velocity past particle U = 0.1 m/s
0.01m
• Catalyst diameter dp= 1cm = 0.01m
• Instantaneous rxn at catalyst surface CAs≈0
CAs=0
• Bulk concentration CAb= 1 mol/L
Determine the flux of A to
•  ≡ kinematic viscosity = 0.5 x 10-6 m2/s
the catalyst particle
• DAB = 1x10-10 m2/s
The velocity is non-zero, so we primarily have convective mass transfer to the
catalyst particle:
WA boundary  k c  CAb  CAs 
Computed kC from
Frössling correlation:
WA boundary  4.61 106
kc 
D AB
Sh
dp
k c  4.61 10 6
m
s

m  mol  1000L 
3 mol

W

4.61

10
1

0
A boundary

s  L  m3 
m2  s

Because the reactant is consumed as soon as it reaches the surface
mol
WA boundary  rAs ''  4.61 10 3 2
m s
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
For the previous example, derive an equation for the flux if the reaction L19-14
were not instantaneous, and was instead at steady state (WA|surface =-rA”)
and followed the kinetics: -rAS’’=krCAs (Observed rate is not diffusion limited)
k c  CAb  CAs   WA boundary
rAs ''  kr CAs
Because the reaction at the surface is at the steady state & not instantaneous:
WA boundary  rAs ''  kr CAs
CAs  0
So if CAs were in terms of measurable species, we would know WA,boundary
Use the equality to put CAs in terms of measurable species (solve for CAs)
k c  CAb  CAs   k r CAs  k c C Ab  k c CAs  k r CAs  k c C Ab  k r CAs  k c CAs
k c CAb

 CAs Plug into -r’’As
 k c CAb  CAs  kr  k c 
kr  k c
kr k c CAb
WA boundary  r '' As  k r CAs  WA boundary  r '' As 
kr  k c
kk C
kk C
Rapid rxn, kr>>kc→ kc in
r ''As  r c Ab  r ''As  r c Ab  r ''As  kcCAb
denominator is negligible
kr  k c
kr
Diffusion limited
kk C
kk C
Slow rxn, kr<<kc→ kr in
r ''As  r c Ab  r '' As  r c Ab  r ''As  kr CAb
denominator is negligible
kr  k c
kc
Reaction limited
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-15
Mass Transfer & Rxn Limited Reactions
r ''As
-rA’
kr k c CAb

kr  k c
reaction limited regime
r ''As  kr CAb
transport limited regime
r ''As  kcCAb
kc 
D AB
Sh
dp
12
Sh  2  0.6Re
13
Sc
Re=
Udp

Sc 

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.
L19-16
Mass Transfer & Rxn Limited Reactions
r ''As
-rA’
kr k c CAb

kr  k c
reaction limited regime
r ''As  kr CAb
transport limited regime
r ''As  kcCAb
kc 
D AB
Sh
dp
12
Sh  2  0.6Re
13
Sc
Re=
Udp

Sc 

DAB
12
13 

Ud




D

p

k c  AB  2  0.6 
 

dp 
    DAB  

(U/dp)1/2 = (fluid velocity/particle diameter)1/2
12
12
23
16
2 3  12 


d
D
U  D
  
k
U
  c2   2   AB2   1   p1 
k c  AB 
k c1  U1   D AB1    2   dp2 
 1 6  dp1 2 
Proportionality is useful for assessing parameter sensitivity
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-17
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)
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.
L19-18
Mass Transfer Limited Rxn in PBR
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/ (s-1) : 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
dC A 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.
L19-19
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.
L19-20
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
dC A
 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.
L19-21
Review: Transport & Rxn Limited Rates
kk C
r ''As  r c Ab
kr  k c
-rA’
reaction limited regime: r '' As  k r C Ab
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 C Ab
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.
L19-22
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  C A0 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.
L19-23
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.
L19-24
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.
L19-25
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.
L19-26
Mole Balance on O2 From r to r+r (2)
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)
O2
R
CO2
For any r:
CA0
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
 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 



C
K1 1 R  1 R0 
A0 1 R  1 R0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-27
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.
L19-28
Mass Balance on Carbon (C)
R0 C  O2  CO2
O2
r+r
r
R
In – out + gen = accumulation
4

d   R3 rCfC 
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)
rC: density of solid C
fC: fraction of the volume of the core that is C
4 3

d   R rCfC   r ''c  dR
Simplify mass balance:
3

rCfC 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.
L19-29
Time Required to Shrink Core to Radius R
R0
O2
r ''c
dR

dt rCfC
C  O2  CO2
r+r
R
R

2

t
R
R0
DeCA0
R
R 


t
 
2 3R0 
rCfC
R0
3
DeCA0
dt Integrate over
0 to t & R0 to R
0 rCfC
  R  R R0 dR  
2
R  R 2 R0
Substitute r’’c into -dR/dt, dR D C 

1
e
A0



get like terms together, 
dt
rCfC  R  R2 R0 
integrate, & solve for t
r
CO2
-r''c =
DeCA0
R2 R3 R02 R03 DeCA0





t
2 3R0
2
3R0
rCfC
R2 R3 R02 R02 DeC A0





t
2 3R0
2
3
rCfC
3R0R2 2R3 3R02  2R02 DeC A0




t
6R0
6R0
6
rCfC
Get common
denominators
3R0R2 2R3 R02 DeC A0




t
6R0
6R0
6
rCfC
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-30
Time Required to Shrink Core to Radius R
R0
O2
C  O2  CO2
r+r
r
R
CO2
3R0R2 2R3 R02 DeC A0




t
6R0
6R0
6
rCfC
Solve for t:
 3R0R2 2R3 R02   rCfC 



  
t
 6R0
6R0
6   DeCA0 

Factor out
R02/6
 3R2 2R3
 rCfCR02 
  2  3  1 
 t Factor out -1

R
 6DeCA0 
R0
 0


 3R2 2R3  rCfCR02
 1  2  3 
t
 R

R0  6DeCA0

0
2
3

R 
 R   rCfCR02
 1  3 
t
  2
 

 R0 
 R0   6DeCA0

2
3
At the core of the

2
 0 
 0   rCfCR02
r
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.
L19-31
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
L19-32
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 rc  0
Volume of shell
r’A: rate of reaction per mass of catalyst (mol/g•s)
rc: 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 rc  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.
L19-33
Diffusion Equation (Step 2)
CAs
IN
WAr 4r 2
R
r
OUT
 WAr 4r 2
+
r r
GEN


=ACCUM
 rA 4rm2r rc  0
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.
L19-34
Diffusion & Rxn in a Spherical Catalyst
BMB : WAr 4r 2 r  WAr 4r 2 r r  rA  4rm2r  rc  0
C
As

R

d WAr r 2
dr
  r  r 2r
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 2r
A
c
0
dCA 2  2
d
n

D
r

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.
L19-35
Dimensionless Variables
dCA 2  2
d
n

D
r

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
R
C
k
"
S
r
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, 
L19-36
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 rc Sak "1 CAs
rc 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.
L19-37
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.
L19-38
Effectiveness Factor & Rxn Rate
3
  2  f1 coth f1  1
f1
f1  R
rck1Sa
De
     k1CAs  Sa
rA    rAs
R  f1    1 surface-reaction-limited
when f1 ,(  30)  can be simplified to:  
De
3 3

,  1
f1 R k1rc 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 k1rc Sa

 -rA 
De
3
R k1rc Sa
3 DeSak1
CAs
R
rc
Overall rate for 1st-order rxn
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L19-39
Clicker Question
rA    k1CAs  Sa
Overall rate for
1st-order rxn
De
3

R k1rc Sa
 rA 
De
3
k1CAs  Sa
R k1rc 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.
L19-40
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.