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.